Untitled - HEP Grupa

06/07ENERGIJAIZDAVA»Hrvatska elektroprivreda d.d., ZagrebZA IZDAVA»AMr. sc. Ivan MravakSUIZDAVA»ISveuËilište u Zagrebu, Fakultet elektrotehnike i raËunarstvaMinistarstvo gospodarstva, rada i poduzetništvaPOMOΔ U IZDAVANJUMinistarstvo znanosti, obrazovanja i πportaURE–IVA»KI SAVJETMr. sc. Kaæimir VrankiÊ (predsjednik) - doc. dr. sc. AnteΔurkoviÊ - prof. dr. sc. Igor DekaniÊ Zagreb - prof. dr. sc. DaniloFeretiÊ, Zagreb - mr. sc. Nikica GrubiπiÊ, Zagreb - prof. dr. sc.Slavko Krajcar, Zagreb - doc. dr. sc. Æeljko TomπiÊ, Zagreb - dr.sc. Mladen Zeljko, ZagrebURE–IVA»KI ODBORGlavni urednik ∑ mr. sc. Goran Slipac, ZagrebGlavni tajnik ∑ mr. sc. Slavica Barta-Koπtrun, ZagrebLektor ∑ ©imun »agalj, prof., ZagrebMetroloπka recenzija ∑ Dragan BorojeviÊ, dipl. ing., ZagrebPrijevod ∑ Hrvatsko druπtvo znanstvenih i tehniËkih prevoditelja∑ Prevoditeljski centar, ZagrebUREDNI©TVO I UPRAVAHEP d.d. - Energija, UreappleivaËki odborUlica grada Vukovara 37, 10000 Zagreb, HrvatskaTelefoni: +385 (1) 6171291 i 6322641Telefaks: +385 (1) 6322143e-mail: goran.slipac@hep.hr; slavica.barta@hep.hr;www.hep.hrGodiπnje izlazi 6 brojeva.Godiπnja pretplata bez PDV-a (22 %) iznosi:∑ za pojedince 250 kn∑ za poduzeÊa 400 kn∑ za studente 60 knÆiro raËun kod ZagrebaËke banke broj:2360000-1400129978Godiπnja pretplata za inozemstvo iznosi USD 95.Devizni raËun:ZagrebaËka banka broj: 2000006299GrafiËko ureappleenje omota ∑ mr. sc. Kaæimir VrankiÊ, ZagrebGrafiËko ureappleivanje ∑ Bestias dizajn d.o.o., ZagrebTisak ∑ Intergrafika d.o.o, ZagrebNaklada ∑ 1 500 primjeraka Godiπte 56(2007)Zagreb, 2007Broj 6., str. 637∑758Oglasi su veliËine jedne stranice. Cijena oglasa je 3 000 knbez PDV (22%).ENERGIJAPUBLISHED BYHrvatska elektroprivreda d.d., ZagrebPUBLISHER’S REPRESENTATIVEIvan Mravak, MScCO-PUBLISHED BYUnivesity of Zagreb, Fakulty of Electrical Engineering and ComputingMinistry of Economy, Labour and EntrepreneurshipSUPPORTED BYMinistry of Science, Education and SportEDITORIAL COUNCILKaæimir VrankiÊ, MSc (Chairman), HEP d.d., Zagreb - AssistantProf Ante ΔurkoviÊ, PhD, Zagreb - Prof Igor DekaniÊ, PhD, Zagreb- Prof Danilo FeretiÊ, PhD, Zagreb - Nikica GrubiπiÊ, MSc, Zagreb- Prof Slavko Krajcar, PhD, Zagreb - Assistant Prof Æeljko TomπiÊ,PhD, Zagreb - Mladen Zeljko, PhD, ZagrebEDITORIAL BOARDEditor-in-Chief ∑ Goran Slipac, MSc, ZagrebSecretary ∑ Slavica Barta-Koπtrun, MSc, ZagrebLanguage Editor ∑ ©imun »agalj, prof., ZagrebMetrology ∑ Dragan BorojeviÊ, dipl. ing., ZagrebTranslation ∑ Croatian Association of Scientific and TechnicalTranslators ∑ Croatian Translation Agency, ZagrebHEAD OFFICE AND MANAGEMENTHEP d.d. - Energija, Editorial BoardUlica grada Vukovara 37, 10000 Zagreb, CroatiaTelephone: +385 (1) 6171291 and 6322641Fax: +385 (1) 6322143e-mail: goran.slipac@hep.hr; slavica.barta@hep.hr;www.hep.hrAppears 6 times a year.Annual subscription fee excl. VAT (22 %):∑ for individual subscribers HRK 250∑ for companies HRK 400∑ for students HRK 60Number of gyro account with ZagrebaËka Banka:2360000-1400129978Annual subscription fee for the overseas: USD 95.Number of foreign currency account with ZagrebaËka Banka:2000006299Cover design ∑ Kaæimir VrankiÊ, MSc, ZagrebGraphic layout ∑ Bestias Dizajn d.o.o., ZagrebPrinted by ∑ Intergrafika d.o.o., ZagrebCirculation ∑ 1,500 copies Volume 56(2007)Zagreb, 2007No 6., pp. 637∑758Ads are the size of page. The price of an ad is HRK 3 000 excl.VAT (22%).SADRÆAJCONTENTS642-675676-699700-711712-729730-753Rajπl, I., Krajcar, S., Krpan, K.PRIMJENA VI©EAGENTSKIH SUSTAVA U SIMULATORIMATRÆI©TA ELEKTRI»NOM ENERGIJOM(pregledni Ëlanak)GoiÊ, R., Jakus, D., MudniÊ, E.PRORA»UN GODI©NJIH GUBITAKA RADNE ENERGIJEU DISTRIBUCIJSKOJ MREÆI S PRIKLJU»ENOMVJETROELEKTRANOM(izvorni znanstveni Ëlanak)VujeviÊ, D.PRIMJENA MÖBIUSOVE VRPCE U ELEKTROTEHNICI(prethodno priopÊenje)ΔuÊiÊ, B.3D PRORA»UN KVAZISTATI»KOG MAGNETSKOG POLJAOKO VODI»A I FEROMAGNETSKE PLO»E INTEGRALNIMJEDNADÆBAMA(izvorni znanstveni Ëlanak)BariÊ, T., ©ljivac, D., Stojkov, M.GRANICE VALJANOSTI IZRAZA ZA MJERENJA SPECIFI»NOGOTPORA TLA WENNEROVOM METODOM PREMA IEEE NORMIStd. 81-1983(izvorni znanstveni Ëlanak)Rajπl, I., Krajcar, S., Krpan, K.APPLICATION OF MULTI-AGENT SYSTEMS IN ELECTRICITYMARKET SIMULATORS(review article)GoiÊ, R., Jakus, D., MudniÊ, E.CALCULATION OF ANNUAL ACTIVE ENERGY LOSSES IN ADISTRIBUTION NETWORK WITH A CONNECTED WIND POWERPLANT(original scientific article)VujeviÊ, D.APPLICATION OF THE MÖBIUS STRIP IN ELECTRICAL ENGINEERING(preliminary information)ΔuÊiÊ, B.THE 3D CALCULATION OF THE QUASISTATIC MAGNETICFIELD AROUND A CURRENT CARRYING CONDUCTOR ANDFERROMAGNETIC PLATE BY MEANS OF INTEGRAL EQUATIONS(original scientific article)BariÊ, T., ©ljivac, D., Stojkov, M.VALIDITY LIMITS OF THE EXPRESSION FOR MEASURING SOILRESISTIVITY BY THE WENNER METHOD ACCORDING TO IEEESTANDARD 81-1983(original scientific article)638»asopis je ubiljeæen u Ministarstvu znanosti, obrazovanja i sporta pod The magazine is registered with the Ministry of Science, Education andbrojem 161 od 12.11.1992.Sport under No. 161 since 12.11.1992.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, »asopis I., Krajcar, je indeksiran S., Krpan, u sekundarnom K., Application bibliografskom of Multi-Agent izvoru System INSPEC in …, Energija, The vol. 56(2007), magazine No. is indexed 6, pp. 642-675 with the secondary reference source of INSPEC∑ The Institution of Electrical Engineering, England.∑ The Institution of Electrical Engineering, England.

URE–IVA»KA POLITIKA»asopis Energija znanstveni je i struËni Ëasopiss dugom tradicijom viπe od 50 godina. PokrivapodruËje elektroprivredne djelatnosti i energetike.»asopis Energija objavljuje izvorne znanstvene istruËne Ëlanke πirokoga podruËja interesa, od specifiËnihtehniËkih problema do globalnih analizaprocesa u podruËju energetike.U vrlo πirokom spektru tema vezanih za funkcioniranjeelektroprivredne djelatnosti i opÊenito energetikeu træiπnim uvjetima i opÊoj globalizaciji, Ëasopisima poseban interes za specifiËne okolnostiostvarivanja tih procesa u Hrvatskoj i njezinu regionalnomokruæenju. Funkcioniranje i razvoj elektroenergetskihsustava u srediπnjoj i jugoistoËnojEuropi, a posljediËno i u Hrvatskoj, optereÊenoje mnogobrojnim tehniËko-tehnoloπkim, ekonomskim,pravnim i organizacijskim problemima. Namjeraje Ëasopisa da postane znanstvena i struËnatribina na kojoj Êe se kritiËki i konstruktivno elaboriratinavedena problematika i ponuditi rjeπenja.»asopis je posebno zainteresiran za sljedeÊu tematiku:opÊa energetika, tehnologije za proizvodnjuelektriËne energije, obnovljivi izvori i zaπtitaokoliπa; koriπtenje i razvoj energetske opreme i sustava;funkcioniranje elektroenergetskoga sustavau træiπnim uvjetima poslovanja; izgradnja elektroenergetskihobjekata i postrojenja; informacijskisustavi i telekomunikacije; restrukturiranje i privatizacija,reinæenjering poslovnih procesa; trgovanjei opskrba elektriËnom energijom, odnosi s kupcima;upravljanje znanjem i obrazovanje; europska iregionalna regulativa, inicijative i suradnja.Stranice Ëasopisa podjednako su otvorene iskusnimi mladim autorima, te autorima iz Hrvatske iinozemstva. Takva zastupljenost autora osiguravaznanje i mudrost, inventivnost i hrabrost, te pluralizamideja koje Êe Ëitatelji Ëasopisa, vjerujemo,cijeniti i znati dobro iskoristiti u svojem profesionalnomradu.EDITORIAL POLICYThe journal Energija is a scientific and professionaljournal with more than a 50-year tradition. Coveringthe areas of the electricity industry and energy sector,the journal Energija publishes original scientificand professional articles with a wide area ofinterests, from specific technical problems to globalanalyses of processes in the energy sector.Among the very broad range of topics relating to thefunctioning of the electricity industry and the energysector in general in a competitive and globalizingenvironment, the Journal has special interest in thespecific circumstances in which these processesunfold in Croatia and the region. The functioningand development of electricity systems in Centraland South Eastern Europe, consequently in Croatiatoo, is burdened with numerous engineering, economic,legal and organizational problems. The intentionof the Journal is to become a scientific andprofessional forum where these problems will becritically and constructively elaborated and wheresolutions will be offered.The Journal is especially interested in the followingtopics: energy sector in general, electricityproduction technologies, renewable sources andenvironmental protection; use and development ofenergy equipment and systems; functioning of theelectricity system in competitive market conditions;construction of electric power facilities and plants;information systems and telecommunications;restructuring and privatization, re-engineering ofbusiness processes; electricity trade and supply,customer relations; knowledge management andtraining; European and regional legislation, initiativesand cooperation.The pages of the Journal are equally open to experiencedand young authors, from Croatia and abroad.Such representation of authors provides knowledgeand wisdom, inventiveness and courage as well aspluralism of ideas which we believe the readers ofthe Journal will appreciate and know how to put togood use in their professional work.639

UVODINTRODUCTIONDragi Ëitatelji,u rukama Vam je zadnji, πesti po redu, ËasopisEnergija kojim zakljuËujemo izdavanje u 2007.godini. Objavljeno je 29 Ëlanaka koje potpisuje71 autor iz Hrvatske i 11 autora iz inozemstva.Posebno je zadovoljstvo istaÊi kako je od ukupnogbroja objavljenih Ëlanaka 9 Ëlanaka u kategorijiizvornih znanastveih radova. Objavljeni Ëlanciodraæavaju ureappleivaËku politiku Ëasopisa Energija,a uvjereni smo kako je moguÊe proπiriti podruËjainteresa i obuhvatiti razliËite aspekte regionalnogtræiπta energije, ali i træiπta energije u Europskojuniji i ostalim globalnim træiπtima energije.Tijekom 2008. godine oËekuju nas znaËajnidogaappleaji na podruËju restrukturiranja energetskogsektora u Republici Hrvatskoj, jugoistoËnoj Europi,a oËekuju se i znaËajni pomaci u definiranjuenergetske strategije u jugoistoËnoj Europi i naravnou Europskoj uniji, πto, nadamo se, predstavljadovoljan motiv za nova istraæivanja.U ovom broju Ëasopisa Energija, objavljujemo triizvorna znanstvena Ëlanka, jedan pregledni i jedanËlanak kao prethodno priopÊenje:− Primjena viπeagentskih sustava u simulatorimatræiπta elektriËnom energijom,− ProraËun godiπnjih gubitaka radne energijeu distribucijskoj mreæi s prikljuËenomvjetroelektranom,− Primjena Möbiusove vrpce u elektrotehnici,− 3D proraËun kvazistatiËkog magnetskog poljaoko vodiËa i feromagnetske ploËe integralnimjednadæbama,− Granice valjanosti izraza za mjerenja specifiËnogotpora tla Wennerovom metodom premaIeee normi Std. 81-1983.U prvom Ëlanku opisuje se jedan noviji pristupsimulaciji i analizi træiπta energije koji se razvioiz teorije igara. U uvodnom dijelu se postupnoopisuju temeljna polaziπta kao i razlike prema optimizacijskimmodelima koji su joπ uvijek u vrloπirokoj uporabi u analizama træiπta. Viπeagentskisustavi se temelje na pretpostavci da svaki sudionikutjeËe na ciljeve i ponaπanje ostalih sudionikau igri, odnosno na træiπtu za koje se takoappleerDear Readers,You are holding the sixth and final issue of the journalEnergija for the year 2007. During the past year,we have published twenty-nine articles signed byseventy-one authors from Croatia and eleven authorsfrom other countries. It is with particular satisfactionthat we point out that nine of these articles wereoriginal scientific papers. The articles publishedexpress the editorial policy of the journal Energija.We are convinced that it is possible to expand thearea of interest and encompass various aspects ofthe regional energy markets as well as those in theEuropean Union and elsewhere.During the year 2008, significant events are anticipatedin the restructuring of the energy sector in the Republicof Croatia and South East Europe, as well as significantshifts in the definition of the energy strategy in SouthEast Europe and the European Union, which, we hope,will provide sufficient occasion for new research.In this issue of Energija, we are publishing threeoriginal scientific articles, one overview article andone article as preliminary information:−−−−−Application of Multi-Agent Systems in ElectricityMarket Simulators,Calculation of Annual Active Energy Losses ina Distribution Network with a Connected WindPower Plant,Application of the Möbius Strip in ElectricalEngineering,A 3D Calculation of the Quasistatic Magnetic Fieldaround a Current-Carrying Conductor and FerromagneticPlate by Means of Integral Equations,Validity Limits of the Expression for MeasuringSoil Resistivity by the Wenner Method Accordingto IEEE Standard 81-1983.The first article presents a new approach to the simulationand analysis of electricity markets developedfrom the game theory. In the introductory section,there is a description of the current situation as wellas the differences between this approach and the optimizationmodels still widely used in market analyses.Multi-agent systems are based upon the assumptionthat each participant has an impact on the goals andbehavior of the other participants in the game, i.e. in640

moæe definirati odreappleeni okvir. U radu se isto takoopisuje i jedan od najznaËajnijih raËunarskih modelatemeljenih na viπeagentskim sustavima tzv.EMCAS Ëija je implementacija u hrvatske znastvenoistraæivaËke organizacije kao i Hrvatsku elektroprivreduzapoËela krajem 2007. godine.Sasvim je jasno da prikljuËak Ëak i malih izvoraenergije na distribucijsku mreæu izaziva promjeneu tokovima snaga u mreæi, a πto se u konaËnici oËitujei kroz promjenu gubitaka radne energije. ProraËungubitaka radne energije, i to kada je na distribucijskumreæu prikljuËena vjetrolektrana, opisanje u drugom Ëlanku. U radu je isto tako razraappleenametodologija proraËuna godiπnjih gubitaka radneenergije, a kao model je analizirana realna mreæauz simuliranu vjetrolektranu. Analizirane su mnogeutjecajne veliËine koje zasigurno imaju utjecajna godiπnji iznos gubitaka radne energije u mreæi.TreÊi Ëlanak opisuje pomalo neobiËnu temu vezanuuz primjenu Möbiusove vrpce koja je napoËetku bila zanimljiva samo matematiËarimai koju je osim u umjetnosti moguÊe primijeniti iu elektrotehnici. Uglavnom se njena primjenaodnosi na izradu otpornika malih otpora i veÊihsnaga, a posebna prednost pri izradi tih otpornikaje moguÊnost razliËitih oblika i to bez utjecaja pojedinihnjihovih dijelova ili okolnih predmeta.Model 3D magnetskog polja za sluËaj vodiËa iferomagnetske ploËe opisan pomoÊu integralnihjednadæbi za kvazistatiËki sluËaj prikazan je uËetvrtom Ëlanku. Posebna vrijednost proraËuna jeËinjenica kako se rezultati proraËuna slaæu s rezultatimamjerenja.Posljednji Ëlanak bavi se analizom matematiËkogizraza za prividni specifiËni otpor tla naveden umeappleunarodnoj normi IEEE Std. 81-1983. Naime,za analizu rezultata i obavljanje mjerenja specifiËnogotpora tla inæenjerima su dane na raspolaganjesmjernice i naputci razliËitih meappleunarodnihnormi. Prilikom koriπtenja matematiËkih izraza iznavedenih normi Ëesto nisu jasne okolnosti podkojima su dobiveni navedeni izrazi. Jedan takavsluËaj opisan je u ovom Ëlanku, a odnosi se na teorijskimodel i predviappleanje mjernih rezultata dobivenihWennerovim mjernim rasporedom elektroda. Iztog razloga u Ëlanku je detaljno prikazan izvod izrazaza prividni specifiËni otpor tla u sluËaju kada semjerenje obavlja Wennerovom mjernom tehnikom.Glavni urednikMr. sc. Goran Slipacthe market, for which a specific framework can alsobe defined. This article also provides a description ofone of the most significant computer models, basedupon the Electricity Market Complex Adaptive System(EMCAS), which began to be used in Croatian scientificresearch organizations as well as HEP in late 2007.The connection of even small power sources to adistribution network causes redistribution in the loadflows, ultimately evident in changes in the active energylosses. The calculation of active energy losseswhen a wind power plant is connected to a distributionnetwork is described in the second article. Amethodology has also been devised for calculatingannual active energy losses, using an existing networkwith a simulated wind power plant as a model.Various values are analyzed that affect annual activeenergy losses in a distribution network.The third article concerns a somewhat unusual topicin connection with the application of the Möbiusstrip, which initially was only of interest to mathematiciansand which, in addition to the arts, is alsoapplied in electrical engineering. The Möbius stripis generally used in low-ohm non-inductive resistors,particularly in high frequency devices. The specificadvantage in the construction of such resistors isthat they can be fashioned into various forms andshapes, without coupling electromagnetically tothemselves or surrounding metallic objects.The fourth article presents a 3D model of a magneticfield in the case of a conductor and ferromagnetic plateby means of integral equations in the quasistatic case.The particular merit of the calculation is the fact thatits results are in agreement with measurement results.The last article concerns the analysis of the mathematicalexpression for apparent soil resistivity providedin the international IEEE Standard 81-1983.Engineers have guidelines and instructions from variousinternational standards at their disposal for theanalysis of soil resistivity results and measurement.When using mathematical expressions from thesestandards, the circumstances under which they wereobtained are not always clear. One such instance isdescribed in this article and concerns a theoreticalmodel and the prediction of the measurement resultsobtained by using the Wenner method of electrodearrangement. The derivation of the expression for apparentspecific soil resistivity when measurement isperformed using the Wenner method is presented indetail. An alternative expression is presented by theauthors for the determination of the apparent resistivityof two-layer soil measured by the Wenner method,which takes the actual geometry of electrodes andother measurement procedures into account.Editor-in-chiefGoran Slipac, MSc641

PRIMJENA VIŠEAGENTSKIHSUSTAVA U SIMULATORIMATRŽIŠTA ELEKTRI»NOM ENERGIJOMAPPLICATION OF MULTI-AGENTSYSTEMS IN ELECTRICITYMARKET SIMULATORSIvan Rajπl, dipl. ing., SveuËiliπte u Zagrebu,Fakultet elektrotehnike i raËunarstva, Unska 3, 10000 Zagreb, HrvatskaProf. dr. sc. Slavko Krajcar, SveuËiliπte u Zagrebu,Fakultet elektrotehnike i raËunarstva,Unska 3, 10000 Zagreb, HrvatskaKreπimir Krpan, dipl. ing., Emerson Network Power,Selska cesta 93, 10000 Zagreb, HrvatskaKako procesi liberalizacije i deregulacije træiπta elektriËne energije diljem svijeta uzimaju sve viπemaha, tako se broj sudionika na træiπtu, a ujedno i njihova raznolikost, naglo poveÊava. U tradicionalnommonopolistiËkom modelu træiπta svi su procesi, koji se odvijaju u elektroenergetskomsustavu (EES), bili pod nadzorom vertikalno integriranih kompanija. No, kako se træiπna logikamijenja od one usmjerene na πto je niæe moguÊe troπkove proizvodnje elektriËne energije na teænjuk πto veÊem profitu, tako se i rizik raspodjeljuje meappleu svim træiπnim sudionicima podjednako i nijeviπe u potpunosti na leappleima krajnjih potroπaËa. U takvim nesigurnim uvjetima, u kojima se cijenaelektriËne energije mijenja iz sata u sat, svaki træiπni sudionik æeli se osloniti na pomoÊ pouzdanogalata koji bi mu pomogao pri donoπenju optimalnih i kvalitetnih odluka i strateπkih nastupa.Since the liberalization and deregulation of the electricity markets throughout the world are infull swing, the number of market participants, as well as their diversity, is sharply increasing. Inthe traditional monopolistic market model, all the processes that occur in a power system havebeen supervised by vertically integrated companies. Since market logic is changing from thatoriented toward minimizing generation costs for electrical energy to the trend of maximizingprofit, the risk is being distributed among all the market participants uniformly and is no longerentirely borne by the final consumers. Under such uncertain conditions, in which the pricesof electrical energy change from hour to hour, each market participant wants a reliable tool tofacilitate optimal and quality decisions and strategic performance.KljuËne rijeËi: agenti, EMCAS simulacijski model, simulatori træiπta elektriËne energije,træiπte elektriËne energijeKey words: agents, Electricity Market Complex Adaptive System (EMCAS), electricity marketsimulators, electricity market.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br.6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675642

1 UVODKako bi bilo moguÊe modelirati bilo koji ekonomskisustav nuæno je poznavati mikro ponaπanjasudionika u njemu, naËine komunikacije i interakcijemeappleu sudionicima te globalne zakonitosti ipravilnosti promatranog ekonomskog sustava. EkonomistiveÊ stoljeÊima pokuπavaju πto je moguÊebolje modelirati ekonomske procese. Takvi modelitrebali bi biti sposobni nositi se sa stvarnim problemimaprisutnim u ekonomskim sustavima kaoπto su nesimetriËnost informacija, nesavrπenakompetitivnost, strateπka interakcija, kolektivnouËenje, moguÊnost postojanja viπestruke ravnoteæesustava i drugima. Naime, ekonomski æivot sastojise od mnogo strateπkih interakcija meappleu tvrtkama,kuÊanstvima, upravnim tijelima i drugima.PodruËja analize ponaπanja træiπnih sudionika uekonomiji naziva se teorijom igara (engl. GameTheory). Teorija igara analizira naËine na koje dvaili viπe sudionika, odnosno dvije ili viπe strana,koje sudjeluju u takvoj strukturi kao πto je træiπte,odabiru poteze ili strategije koje imaju utjecaja nasvakog sudionika. Ovu teoriju je razvio John vonNeumann (1903.∑1957.), maapplearski matematiËkigenij.Za analizu i simulaciju ovakvih træiπnih stanja zasigurnoje potrebno koristiti simulacijske alate kojiprelaze okvire optimizacijskih modela koji su se koristiliu veÊoj mjeri u monopolistiËkom okruæenju,ali se i danas joπ uvijek koriste. Dosadaπnji napredaku razvoju analitiËkih i proraËunskih alataomoguÊuje nove i naprednije pristupe kvantitativnimstudijama ekonomskih sustava. Jedan odtakvih pristupa je raËunski model ekonomskogsustava zasnovan na agentima (engl. Agent-basedComputational Economics, ACE).1 INTRODUCTIONIn order to model any economic system whatsoever,it is necessary to be acquainted with themicro behavior of the participants, the manner ofcommunication and interaction among the participantsand the global regularity of the economicsystem being studied. Economists have beentrying to model economic processes accuratelyfor centuries. Such models should be capable ofconfronting the actual problems present in economicsystems such as information asymmetry,imperfect competition, strategic interaction, collectivelearning, the possibility of the existenceof multiple-equilibrium systems etc. Economiclife consists of many strategic interactions amongcompanies, households, administrative bodies andothers. The area of the analysis of the behaviorof market participants in the economy is knownas the game theory. The game theory analyzes themanner in which two or more participants, i.e. twoor more sides who participate in a structure suchas a market, choose moves or strategies that havean impact upon each participant. This theory wasdeveloped by the Hungarian mathematical geniusJohn von Neumann (1903∑1957).For the analysis and simulation of such marketstates, it is certainly necessary to use simulationtools that exceed the framework of the optimizationmodels that were used in monopolistic environmentsto a great extent but are also still inuse today. Advances to date in the development ofanalytical and computational tools permit new andmore advanced approaches to quantitative studiesof economic systems. One such approach isagent-based computational economics, ACE.2 OPΔENITO O TRÆI©TUTræiπte je ustrojstvo kojim kupci i prodavaËimeappleusobno djeluju da bi odredili ravnoteænu cijenui koliËinu dobara ili usluga [1]. U træiπnojekonomiji niti jedan pojedinac ili organizacija nijeodgovoran za proizvodnju, potroπnju, razdiobu iodreappleivanje cijena. U træiπnom sustavu sve imacijenu koja predstavlja vrijednost nekog dobra.Upravo te cijene usklaappleuju odluke proizvoappleaËai potroπaËa na træiπtu. Viπe cijene ohrabrujuproizvodnju, ali dovode do smanjenja kupovinepotroπaËa. Istodobno niæe cijene ohrabrujupotroπnju, ali obeshrabruju proizvodnju. Cijene sudakle kotaË ravnoteæe u træiπnom ustrojstvu.2 GENERAL MARKETINFORMATIONA market is a structure according to which buyersand sellers mutually act in order to establishprice equilibrium and the quantity of goods orservices [1]. In a market economy, no individualor organization is responsible for the production,consumption, distribution and determination ofprices. In a market system, everything has a pricethat represents the value of the commodity. Theseprices actually coordinate the decisions of theproducers and consumers on the market. Higherprices encourage production but lead to a reductionin consumer purchases. At the same time,lower prices encourage consumption but discourageproduction. Prices are thus a balance wheel inthe market structure.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675644

Odlike træiπnog mehanizma potpuno se ostvarujusamo onda kad su prisutni nadzori i uravnoteæenjasavrπene konkurencije, πto znaËi da niti jednopoduzeÊe ili potroπaË nije sposobno utjecati natræiπnu cijenu. Suprotno tome, nesavrπena konkurencijadovodi do cijena koje podiæu troπkovei smanjuju kupovinu potroπaËa ispod uËinkovitihrazina. Model sa suviπe visokom cijenom i suviπeniskom proizvodnjom obiljeæje je prisutnostinesavrπene konkurencije.Kroz godine i godine træiπte je evoluiralo od jednostavnihmjesta na kojima su se razmjenjivala dobrado virtualnih domena u kojima informacije kruæe uelektroniËkom obliku, tako da se danas ugovori okupnji ili prodaji mogu sklopiti tek jednostavnimklikom miπa na burzama prisutnim na internetu.No, unatoË svim tim, prvenstveno tehniËkimpromjenama, osnovni princip i namjena træiπta jeostala ista: træiπte je mjesto sastajanja kupaca iprodavaËa koji pokuπavaju ostvariti trgovinu pocijeni koja zadovoljava obadvije strane.The characteristics of the market mechanism areonly completely fulfilled when supervision andperfect competitive equilibrium are present, whichmeans that no enterprise or consumer is capableof affecting the market price. On the contrary, imperfectcompetition leads to prices that raise costsand reduce consumer buying to below the effectivelevels. A model with an excessively high price andexcessively low production is characteristic of thepresence of imperfect competition.Through the years, the market has evolved from asimple place where goods are exchanged to virtualdomains in which information circulates in electronicform, so that contracts on buying or sellingcan be entered today with a simple mouse click onInternet markets. However, despite all of this, especiallytechnical changes, the basic principle andpurpose of the market have remained the same: themarket is a meeting place for buyers and sellerswho are attempting to conduct trade at a price thatsatisfies both sides.3 ELEKTRI»NA ENERGIJA KAOSPECIFI»AN PROIZVODKao i svaki drugi proizvod i elektriËna energija jeobjekt trgovinske razmjene, no s nekim bitnim posebnostima.Razvoj træiπta elektriËne energije zasnivase na pretpostavci da se elektriËna energijamoæe smatrati obiËnim proizvodom. Ipak postojebitne razlike izmeappleu nje i ostalih proizvoda kao πtosu na primjer energenti (nafta, plin i dr.). Razlikase prvenstveno oËituje u brzini kojom se odvijarazmjena elektriËne energije i drugih proizvoda.ElektriËna energija se prenosi brzinom svjetlosti,dakle gotovo trenutaËno, dok je za razmjenu svihdrugih dobara potrebno puno viπe vremena. TakodinamiËan sustav kao πto je elektroenergetskizahtijeva jednakost proizvodnje i potroπnje u realnomvremenu.Ukoliko takva jednakost iz bilo kojeg razloga nijeprisutna moæe doÊi do sloma sustava koji za sobompovlaËi katastrofalne posljedice. Takvi poremeÊajisu apsolutno neprihvatljivi, ne samo zato πtotræiπte elektriËne energije prestaje s radom, negoi Ëitava zahvaÊena regija ili dræava ostaje bez napajanjaelektriËnom energijom kroz nekoliko satiili duæe.Vrijednost neisporuËene elektriËne energije (engl.Value Of Lost Load - VOLL) procjenjuje sa naiznose i do stotinu puta veÊe od vrijednosti isporuËeneelektriËne energije. Zbog svih ovih razlogajasno je da upravljanje neravnoteæom izmeappleupotroπnje i proizvodnje elektriËne energije mora3 ELECTRICITY AS A SPECIFICPRODUCTLike every other product, electricity is an objectof market exchange but also has some importantparticularities. The development of the electricitymarket is based upon the assumption that electricitycan be considered to be an ordinary product.Nonetheless, there are significant differences betweenit and other products such as, for example,energy-generating products (oil, gas etc.). The differenceis primarily evident in the speed in whichthe exchange of electricity occurs, as opposed toother products. Electricity is transmitted at thespeed of light, i.e. nearly instantaneously, whilethe exchange of all other goods requires muchmore time. A dynamic system such as an electricitysystem requires equal production (generation) andconsumption in real time.Insofar as such equality is not present for any reasonwhatsoever, a breakdown in the system mayoccur, leading to catastrophic consequences. Suchdisruptions are absolutely unacceptable, not merelybecause the electricity market ceases operationsbut because the entire affected region or countryremains without its electricity supply for severalhours or longer.The value of lost load (VOLL) is estimated atamounts of up to a hundred times greater than thevalue of delivered electricity. For all these reasons,it is clear that the management of the imbalancebetween the consumption and generation of elec-645Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

iti povjereno nezavisnom mehanizmu koji se neoslanja na træiπte i mora se provesti bez obzira natroπkove koje iziskuje. No, mehanizam stvaranjacijene ne moæe djelovati dovoljno brzo da bi uspiouravnoteæiti proizvodnju i potroπnju u realnomvremenu. Cijena elektriËne energije se zbog togakreira ili prije realnog vremena, ex ante, ili nakonrealnog vremena, ex post [2].Pri kupnji bilo kojeg drugog proizvoda kupacmoæe uÊi u trag proizvoappleaËu tog proizvoda. SelektriËnom energijom nije takav sluËaj. TakopotroπaË nije u stanju koristiti energiju iskljuËivojednog proizvoappleaËa elektriËne energije. Sva proizvedenaenergija se ujedinjuje, a to je moguÊe,jer ne postoji razlika izmeappleu jedinica elektriËneenergije proizvedenih od razliËitih generatora. Ovoujedinjenje je poæeljno, jer dovodi do ekonomijerazmjera: maksimalni ukupni proizvodni kapacitetimoraju biti razmjerni maksimalnoj agregatnoj(zbirnoj) potraænji. Ipak, mana je u tome πtoruπenje sustava u kojem se ujedinjuje elektriËnaenergija utjeËe na sve sudionike tog sustava (ukolikone bi bilo ujedinjavanja tada bi ispad jednoggeneratora utjecao samo na potroπaËe koji troπeelektriËnu energiju koju on proizvodi).Potroπnja elektriËne energije pokazuje odreappleenepravilnosti u dnevnim i tjednim cikliËkim varijacijama.Na slikama 1 i 2 se nalaze prikazi tjednih imjeseËnih dijagrama optereÊenja [3].tricity must be entrusted to an independent mechanismthat does not rely upon the market and mustbe conducted regardless of the costs that ensue.Nonetheless, the mechanism for creating pricescannot act quickly enough in order to succeed inbalancing generation and consumption in real time.Therefore, electricity prices are created either priorto real time, ex ante, or after real time, ex post [2].In purchasing any other product, a customer is ableto trace the producer of that product. This is not thecase with electrical energy. Thus, the consumer isnot in a situation to use energy exclusively from oneelectricity generator enterprise. All the generatedenergy is pooled, which is possible because thereis no difference between units of electrical energyproduced by various generators. This pooling is desirablebecause it leads to an economy of scale themaximum total generation by the facilities must beproportionate to the maximum aggregate demand.Nonetheless, an inherent shortcoming is that abreakdown in a system in which there is pooling ofelectricity has an impact upon all the participants inthat system (if there were no pooling, the breakdownof one generator would only affect the consumerswho consume the electricity that it produces).The consumption of electricity shows certain regularitiesin the daily and weekly cyclical variations.Figures 1 and 2 present weekly and monthly loaddiagrams [3].Slika 13D dijagram tjednogoptereÊenjaFigure 13D weekly load diagramSnaga / Power Rating (MW)P 0Sati / Hours012543Dani / Days67Slika 23D dijagrammjeseËnogoptereÊenjaFigure 23D monthly loaddiagramSnaga / Power Rating (MW)6P 02418Sati / Hours126 0N7NN21N14Dani / Days2831Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675646

Za razliku od drugih proizvoda koji se moguskladiπtiti, elektriËna energija se troπi u trenutkukada se i proizvodi. Kako se potroπnja stalno mijenjapotrebni su proizvodni kapaciteti koji mogupratiti te promjene. LogiËno je dakle da neÊe sviproizvodni kapaciteti biti u pogonu tijekom Ëitavogdana. Za vrijeme niske potraænje u pogonu susamo najuËinkovitije proizvodne jedinice, a manjeuËinkovite jedinice koriste se za pokrivanje vrπnihoptereÊenja.Karakteristike elektroenergetskog sustava moguÊeje opisati matematiËkim funkcijama, zbog Ëinjeniceda je to interkonekcijski fiziËki sustav. ToomoguÊava detaljnije ekonomsko modeliranjeelektroenergetskog sustava nego πto je to moguÊeu drugim podruËjima gospodarstva.Unlike other products that can be stored, electricalenergy is consumed at the moment that it isproduced. Since consumption changes constantly,it is necessary to have generation facilities that canmonitor these changes. It is logical that all the generationfacilities will not be in operation throughoutthe entire day. For times of low demand, only themost efficient generating units are in operation, andthe less efficient units are only used to cover peakloads.The characteristics of a power system can be describedby mathematical functions because it isan interconnected physical system. This facilitatesmore detailed economic modeling of the power systemthan is possible in other areas of the economy.4 PROFITNA ORIJENTACIJAI PLANIRANJE NA TRÆI©TUELEKTRI»NE ENERGIJEElektroenergetski sektor proæivljava burno razdobljemijena i prestrukturiranja. Premda je velikbroj vertikalno organiziranih monopolistiËkihenergetskih kompanija i dalje prisutan u svijetu,træiπne prilike i procesi liberalizacije i deregulacijevode k formiranju kakvog takvog træiπnog natjecanja.Naime, zbog posebnosti elektriËne energijegotovo je nemoguÊe uspostaviti apsolutno kompetitivnotræiπte u kojem niti jedan træiπni sudionikne bi mogao utjecati na iznos træiπne cijene elektriËneenergije. Pri tome se prijenos i distribucijaelektriËne energije podrazumijevaju kao prirodnimonopoli u kojima uvoappleenje konkurencije ne bibilo ekonomski opravdano.Izmeappleu ostalih promjena koje uzrokuje liberalizacijai deregulacija dolazi i do promjena u pristupuplaniranja EES-a. Uvoappleenjem potpuno otvorenogtræiπta elektriËne energije nestalo je nekadaπnjestabilno monopolistiËko stanje u kojem se cijenaelektriËne energije odreappleuje uglavnom vladinimodlukama. Transformacijom ovih vertikalno integriranih,monopolistiËkih i uglavnom nacionalnihenergetskih kompanija uvodi se træiπni mehanizami natjecanje i u elektroenergetski sektor. Unovonastalim prilikama rizik viπe nije u potpunostina leappleima samih potroπaËa, nego se raspodjeljujemeappleu svim træiπnim sudionicima. Razlog tomu jeprvenstveno nova træiπna filozofija. Naime, umjestominimiziranja troπkova proizvodnje kao glavnogcilja kompanija u nacionalnom vlasniπtvu, tvrtkesudionici u elektroenergetskom sektoru poËinjurazmiπljati potpuno træiπno u teænji da maksimizirajusvoj profit. Isto tako pojavila se potreba zakoriπtenjem strateπkih alata u svrhu smanjenja4 PROFIT ORIENTATIONAND PLANNING ON THEELECTRICITY MARKETThe electrical energy sector is experiencing a turbulentperiod of change and restructuring. Althougha large number of vertically organized monopolisticenergy companies still exist in the world, marketconditions and the processes of liberalization andderegulation are leading to the formation of somemarket competition. Due to the specific nature ofelectrical energy, it is nearly impossible to establishan absolutely competitive market in which a singlemarket participant could not affect the marketprice of electricity. Moreover, the transmission anddistribution of electrical energy are understood asnatural monopolies in which the introduction ofcompetition would not be economically justified.Among the other changes being brought about byliberalization and deregulation are changes in theapproach to the planning of the power system.Through the introduction of a completely open electricitymarket, the formerly stable monopolistic state,in which the price of electricity was determinedchiefly by government decisions, has disappeared.Through the transformation of these vertically integrated,monopolistic and generally governmentownedenergy companies, market mechanisms andcompetition are being introduced into the electricitysector. Under the new conditions, risk is no longerborne entirely by the consumers but is distributedamong all the market participants. The reason forthis is primarily the new market philosophy. Insteadof minimizing generation costs as the main goal ofcompanies under government ownership, companyparticipants in the electrical energy sector are beginningto think in pure market terms in the pursuitof maximizing their profits. Similarly, the need hasarisen for the use of strategic tools for the purpose647Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

izika nastupanja na træiπtu. Cijena elektriËneenergije poËinje varirati ne samo tijekom dana,nego je ovisna i o godiπnjem dobu i atmosferskimprilikama.Sposobnost πto boljeg predviappleanja buduÊe situacije,odnosno stanja na træiπtu, od presudne jevaænosti za træiπne sudionike u novonastalim prilikama.Uz energetsku i tehniËku vrlo je vaæna i ekonomskauËinkovitost. Veoma je bitna i strategija zakoju se pojedini sudionici odluËuju, jer u ovakvomokruæenju postoji rizik raspodijeljen meappleu svimtræiπnim sudionicima, pa je naglaπena potrebaza pomoÊnim alatom pri uËinkovitom donoπenjuodluka i upravljanju rizicima. Upravo radi πto boljegpredviappleanja kretanja na træiπtu te odreappleivanjaodgovarajuÊeg nastupa i poteza træiπnih sudionikavaænu ulogu imaju simulatori træiπta, a u novijevrijeme i viπeagentski sustavi koji se razlikuju odklasiËnih simulatora træiπta u samom pogledu natræiπte. Dok klasiËni simulatori gledaju na træiπtekao na jednu cjelinu, viπeagentski sustavi se koncentrirajuna svakog sudionika posebno, a træiπtese simulira kao posljedica meappleusobne interakcijenezavinih programskih entiteta ∑ agenata. Ipak,ovi agenti nisu potpuno nezavisni, jer njihove reakcijeu nekoj mjeri ovise o reakcijama drugih agenata,Ëime se simulira stvarno stanje na træiπtu,πto upravo i jest njihov cilj.Tek bi se nadljudskim sposobnostima mogla toËnoodrediti neka buduÊa situacija, no kvalitetnopodeπeni agenti uvijek Êe optimalno reagirati nanovonastalu situaciju i sasvim je sigurno, da je uveÊoj prednosti onaj træiπni sudionik koji posjedujekvalitetniji i inteligentniji programski alat ∑viπeagentski sustav.of reducing the risk of appearing on the market. Theprice of electrical energy has begun to vary not onlywithin a week but is also according to the seasonand atmospheric conditions.The ability to forecast future situations as accuratelyas possible, i.e. the market situation, is of crucialimportance for market participants under thenew conditions. In addition to energy and technicalefficiency, economic efficiency is also very important.The strategy that an individual actor decidesupon is very important because the risk is distributedamong all the market participants under suchcircumstances. Therefore, there is a marked needfor an auxiliary tool for effective decision makingand risk management. In order to forecast trendson the market and determine the corresponding appearanceand moves by market participants, marketsimulators have an important role. Recently, thesehave included multi-agent systems that differ fromclassical market simulators in their perception ofmarkets. While classical simulators view a market asa whole, multi-agent systems concentrate on eachparticipant separately, and a market is simulatedas the consequence of the mutual interaction of independentprogram entities ∑ agents. Nonetheless,these agents are not completely independent becausetheir reactions to some degree are dependentupon the reactions of other agents, according towhich the actual situation on the market is simulated,which is precisely the goal.Some future situations could only be precisely predictedby superhuman abilities but properly attunedagents will always react in an optimal manner to anewly arisen situation. The market participant whopossesses a quality and intelligent program tool ∑ amulti-agent system ∑ is certainly at an advantage.5 AGENTSKI MODEL I AGENTIAgentski model je specifiËni, individualno baziraniproraËunski model koji je u Ëvrstoj vezi sviπeagentskim sustavima. Model se poËeo razvijatiod jednostavnog koncepta u 1940. godini. Glavnaideja vodilja jest stvoriti proraËunski ureappleaj(najËeπÊe zvan agent) koji bi oponaπao stvarnepojave. Povijest agentskog modela seæe joπ od VonNeumann-ovog stroja koji je trebao biti sposobanza reprodukciju.Agentski model se zasniva na postojanju dinamiËkeinterakcije izmeappleu viπe agenata koji imajusvoje zadatake. Takav sustav unutar kojeg se vrπiinterakcija agenata sposoban je kreirati i oponaπatikompleksne procese iz stvarnog æivota.5 AGENT-BASED MODELS ANDAGENTSAn agent-based model is a specific, individuallybased computational model that is closely connectedto multi-agent systems. This type of model developedfrom a simple concept discovered in 1940. The leadingconcept is to create a computational device, mostfrequently called an agent, which emulates actualphenomena. The history of the agent-based modelcan be traced back to the Von Neumann machine,which was supposed to be capable of reproduction.An agent-based model is based upon the existenceof dynamic interactions among several agents thathave their own tasks. Such a system, in which theagents interact, is capable of creating and emulatingcomplex processes from real life.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675648

Agenti su prisutni posvuda i svaki dan ljudi sesusreÊu s velikim brojem agenata (inteligentniagenti, informatiËki agenti, mobilni agenti i dr.).No, postavljaju se brojna pitanja: ©to su to agenti?∑ Je li zbilja moguÊe na neki naËin uvesti red uoËevidni nered koji nas okruæuje? ∑ Ima li neπtoπto je zajedniËko svim agentima? — Kako organiziratiagente da budu sposobni obavljati odreappleenezadatke?Izraz agent je priliËno teπko egzaktno definirati.Prema [4] agente se Ëesto opisuje kao entitetesa osobinama koje se smatraju korisnima uposebnim podruËjima. Tako inteligentni agentimogu oponaπati mentalne procese ili simuliratiracionalno ponaπanja; mobilni agenti su sposobnisamostalno putovati kroz meappleusobno povezanaokruæenja kako bi ostvarili postavljene ciljeve; informacijskiagenti mogu filtrirati i skladno organiziratirasprπene podatke i podatke koji nisu srodni;autonomni agenti mogu bez nadzora obaviti nekefunkcije.Prema Russell-u i Norvig-u (2003.), agent je sveono πto senzorima opaæa svoje okruæenje i utjeËena to okruæenje pomoÊu izvrπnih ureappleaja (aktuatora)[5].Agenti koji se stalno trude optimizirati svoje postupkenazivaju se racionalnim agentima. Tako semeappleu racionalne agente mogu svrstati ljudi (nekasu recimo oËi senzori, a ruke izvrπni ureappleaji), roboti(imaju kamere kao senzore, a kotaËe kao izvrπneureappleaje) ili programski agenti kojima je grafiËkosuËelje i senzor i izvrπni ureappleaj.5.1 Svojstva agenataCilj brojnih istraæivanja bio je upravo pronalazaknekog suvislog skupa osobina ili svojstava koje biagenti mogli i trebali imati.UobiËajena svojstva agenata su [4]:∑ adaptivnost: sposobnost uËenja i unapreappleivanjana temelju iskustva,∑ samostalnost: orijentiranost k cilju, proaktivnoi samoinicijativno ponaπanje,∑ suradniËko ponaπanje: sposobnost suradnje sdrugim agentima sa svrhom ostvarenja zajedniËkogcilja,∑ sposobnost zakljuËivanja: sposobnost reagiranjana apstraktne zahtjeve zadatka.∑ sposobnost komunikacije: sposobnost komunikacijes drugim agentima na naËin koji jesliËniji ljudskoj komunikaciji nego standardnomsimboliËkom program-program protokolu,∑ mobilnost: sposobnost prelaska u samostalnoodreappleenom smjeru s jedne na drugu domaÊinskuplatformu,Agents are present everywhere. Every day, peopleencounter a large number of agents (intelligentagents, information agents, mobile agents etc.)However, numerous questions arise: What are theseagents? Is it really possible to introduce some mannerof order into the evident disorder that surroundsus? Do all agents share something in common? Howcan agents be organized in order to become capableof performing certain tasks?The expression agent is fairly difficult to defineexactly. According to [4], agents are often definedas entities with attributes considered useful in aparticular domain . Such intelligent agents canemulate mental processes or simulate rational behavior.Mobile agents are capable of roaming networkingenvironments in order to fulfill their goals,information agents can filter and coherently organizescattered and unrelated data, and autonomousagents can accomplish some functions withoutsupervision.According to Russell and Norvig (2003), an agentis anything that can be viewed as perceiving itsenvironment through sensors and acting upon thatenvironment through actuators [5].Agents that constantly strive to optimize their behaviorare called rational agents. Therefore, it ispossible to include people (for some the eyes aresensors and the hands are actuators), robots (theyhave cameras as sensors and wheels as actuators)or program agents for whom the graphic interface isthe sensor and actuator.5.1 Agent attributesThe goal of numerous investigations has been preciselyto find some meaningful set of properties orcharacteristics that agents could and should have.The customary properties of agents are as follows[4]:∑ adaptivity: the ability to learn and improve withexperience,∑ autonomy: goal-directedness, proactive andself-starting behavior,∑ collaborative behavior: the ability to work withother agents to achieve a common goal,∑ inferential capability: the ability to act onabstract task specifications,∑ knowledge-level' communication ability: theability to communicate with other agents withlanguage more resembling human-like speechacts than typical symbol-level program-to-programprotocols,∑ mobility: the ability to migrate in a self-directedway from one host platform to another,649Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

∑∑∑osobnost: sposobnost oponaπanja manira ljudskogponaπanja,reaktivnost: sposobnost selektivnog percipiranjai djelovanja,vremenska trajnost: prisutnost agentova identitetai stanja kroz dugi vremenski period.∑ personality: the ability to manifest attributes ofa believable human character,∑ reactivity: the ability to selectively sense andact,∑ temporal continuity: persistence of identity andstate over long periods of time.5.2 Racionalni i reaktivni agentiAgent se smatra racionalnim ako uvijek odabirereakcije koje optimiziraju mjeru uËinkovitostiodreappleenu od strane korisnika (mjera uËinkovitostije stupanj uspjeπnosti rjeπavanja zadatka koji jekorisnik postavio). Te reakcije temeljene su natrenutaËnom znanju i iskustvu agenta. Racionalniagenti nazivaju se joπ i inteligentnim agentima.Neka je dan diskretni skup vremenskih intervalat =1, 2, ..., u kojima agent mora izabrati optimalnodjelovanje a t iz konaËnog skupa svih djelovanja Akoja ima na raspolaganju. Kako bi mogao reagiratiracionalno, agent mora uzeti u obzir kako proπlosttako i buduÊnost pri odabiru prikladnog djelovanja.Pri tome proπlost podrazumijeva dosadaπnja agentovaopaæanja i sva djelovanja koja je primjenjivaodo promatranog vremenskog intervala t. BuduÊnostse odnosi na agentova oËekivana opaæanja i oËekivanadjelovanja.Ukoliko se sa o T oznaËi agentovo opaæanje u vremenuT tada se proπlost uzima u obzira na naËinda za optimalan odabir djelovanja agent mora koristitisva proπla opaæanja o T i sva proπla djelovanjaa T pri Ëemu je T t. Funkcija:5.2 Rational and reactive agentsAn agent is considered to be rational if it alwayschooses reactions that optimize the measure of efficiencydetermined by the user (a measure of efficiencyis the degree of success in solving a taskposed by the user). These reactions are based uponthe current knowledge and experience of the agent.Rational agents are also called intelligent agents.Let us assume a discrete set of time steps t =1, 2, ...,in each of which the agent must choose the optimalaction a t from a finite set of all actions A that it hasavailable. In order to react rationally, an agent musttake into account both the past and future whenchoosing suitable action. The past is understood tomean the agent's perceptions up to the present andall the actions that it has applied up to time t. Thefuture refers to the agent's expected perceptionsand expected activities. If o T represents the agent'sperceptions at time T then the past is taken intoaccount in such a manner that all past perceptionso T and all past a T actions must be used for the optimalchoice of an action by the agent, where T t.The function,(1)zahtijeva pohranjivanje svih proπlih opaæanja idjelovanja, pridruæuje vremenskom intervalu t optimalnodjelovanje a t i naziva se agentovom smjernicom(engl. policy). Ovom funkcijom je rijeπendio optimalnog donoπenja odluka koji se odnosi naproπlost. No, definiranje ovakve funkcije i njezinapraktiËna primjena nisu tako jednostavni kao πtonaoko izgleda. Parovi proπlih opaæanja i djelovanjamogu biti veoma brojni i razlikuju se od zadatkado zadatka. Dakle, sama memorija stvara problemejer mora biti golema a dodatni veliki problemleæi u raËunskoj sloæenosti odreappleivanja funkcije π.Navedene Ëinjenice zahtijevaju jednostavnijesmjernice, odnosno jednostavniji princip pokojem agent odabire svoje postupke. Jedna odmoguÊnosti pojednostavljenja jest ignoriranje svihproπlih opaæanja osim zadnjeg o t pa djelovanje a tovisi samo o tom zadnjem opaæanju i odreappleuje senovom funkcijom:requires the storage of all past perceptions andactions up to time t and an optimal action a t andis called the agent's policy. According to this function,part of the optimal decision making is solvedregarding the past. Nonetheless, the definition ofsuch a function and its practical application are notas simple as they seem. Pairs of past perceptionsand actions can be very numerous and differ fromtask to task. Thus, memory itself creates problemsbecause it must be vast and additional major problemsare inherent in the computational complexityof the determination of function π.These facts require simpler policies, i.e. a simplerprinciple according to which the agent chooses itsprocedures. One of the possibilities for simplificationis ignoring all the past perceptions except forthe last one, o t so that action a t depends solely uponthe last perception and is determined by a newfunction:Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675650

.(2)Agent koji koristi ovakvu funkciju ovisnu samo ozadnjem opaæanju naziva se reaktivnim agentom,a njegova smjernica naziva se reaktivnom ili bezmemorijskom.Sasvim prirodno se nameÊe pitanjekoliko ovakav agent moæe biti uËinkovit? NjegovaneuËinkovitost moæe biti zadovoljavajuÊa u posebnimvrstama okruæenja.5.3 Agentovo okruæenjeIz dosada navedenog je oËigledno da su pojmoviagent i njegovo okruæenje meappleusobno Ëvrsto povezanii niti jedan od njih ne moæe biti definiran bez onogdrugog. Ponekad je Ëak teπko odrediti razliku izmeappleuagenta i okoline i promatrati ih odvojeno (slika 3).An agent that uses such a function dependent solelyupon the last perception is known as a reactiveagent, and its policy is called reactive or memoryless.It is quite normal to ask how such an agentcan be efficient. Its efficiency can be satisfactoryunder certain types of environments.5.3 Agent environmentsIn light of the above, it is evident that the conceptsof an agent and its environment are firmly connectedand neither of them can be defined without theother. Sometimes it is even difficult to determinethe difference between the agent and the environmentor to consider them separately (Figure 3).Okruženje / EnvironmentSlika 3Agent i njegovookruæenjeFigure 3An agent and itsenvironmentAgent /Agent• ciljevi / goals• postupci / actions• znanja iz domene/ domainknowledgeMoæe se promotriti okruæenje u kojem se nalaziviπe agenata i unutar kojeg oni opaæaju,razmiπljaju i djeluju. Kolektivna informacija ookruæenju u nekom vremenskom intervalu t a kojase tiËe trenutaËnog zadatka naziva se stanjemokruæenja i oznaËava sa s t . Skup svih moguÊih stanjau okruæenju tada se oznaËava sa S. Za primjermoæe posluæiti simulacija nogometne utakmiceputem agenata (robota). Tako stanja okruæenjamogu biti plan nogometnog igraliπta, trenutaËnipoloæaj, smjer i brzine agenata ili lopte, podacikoje pojedini agenti znaju jedni o drugima i drugiparametri bitni za donoπenje odluke kao πto je naprimjer ukupno vrijeme koje je proπlo od poËetkautakmice.It is possible to study an environment in which thereare several agents and in which they perceive, considerand act. Collective information on the environmentduring any time step t regarding the currenttask is called the state of the environment and isindicated as s t . The set of all the possible states inthe environment is indicated as S. For example, it ispossible to use a simulation of a soccer match viaagents (robots). Thus, the state of the environmentcan be the plan of the soccer field, the current position,direction and velocity of the agents or the ball,data which individual agents know about the othersand other parameters essential for decision makingsuch as, for example, the total time that haselapsed since the beginning of the match.Depending upon the nature of the problem, the environmentcan be discrete (there is a finite number651Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

Ovisno o prirodi problema okruæenja mogu biti diskretna(postoji konaËan broj stanja) ili kontinuirana(neprekidna s beskonaËno mnogo stanja).Osnovni naËin kojim se karakterizira okruæenjeiz agentove perspektive jest njegova percepcija.Okruæenje je potpuno mjerljivo pojedinom agentuako trenutaËno opaæanje tog agenta (o t ) potpuno otkrivatrenutaËno stanje okruæenja (s t ), tj. ukoliko je:of states) or continuous (there is an infinite numberof states).The basic manner in which an environment is characterizedfrom the agent's perspective is the agent'sperception. The environment is completely measurableby an individual agent if the current perceptionof that agent (o t ) fully reveals the current stateof the environment (s t ), i.e. insofar as.(3)S druge strane, u djelomiËno mjerljivom okruæenjutrenutaËno opaæanje agenta pruæa samo djelomiËnusliku trenutaËnog stanja okruæenja i to uformi uvjetne distribucije vjerojatnosti Ps t o t . ToznaËi da trenutaËno opaæanje agenta ne otkrivapotpuno stanje okruæenja, ali istodobno agent svakomstanju s t pridruæuje vjerojatnost Ps t o t kojakazuje koliko je vjerojatno da je baπ s t pravo stanjeokruæenja. Pri tome vrijedi:From the other side, in a partially measurable environment,the current perception of an agent only providesa partial picture of the current state of the environmentand this in a form of a conditional probabilitydistribution Ps t o t . This means that the current perceptionof an agent does not reveal the complete stateof the environment but at the same time the agentsimultaneously assigns probability Ps t o t to eachstate s t which indicates the probability that preciselys t is the true state of the environment. The followingexpressions apply:(4)iand.(5)Ovdje je dakle varijabla s t sluËajna varijabla kojamoæe poprimiti sve vrijednosti iz skupa S saodreappleenom vjerojatnoπÊu Ps t o t .Postoje dva glavna razloga koja mogu dovesti dodjelomiËne mjerljivosti okruæenja. Prvi razlog jeπum ili smetnje u senzorima putem kojih agentopaæa i mjeri okolinu, a drugi je percepcijsko preklapanje,tj. dva razliËita stanja agentu mogu bitijednaka.U potpuno mjerljivom okruæenju funkcija po kojojagent odluËuje o naËinu djelovanja je pridruæivanjepojedinih djelovanja pojedinim stanjima okruæenja.Veoma je korisna Ëinjenica da u veÊini sluËajevastanje okruæenja u nekom vremenskom intervaluomoguÊuje potpuni opis proπlosti. Takvo okruæenjekoje ima sposobnost sakupljanja svih proπlih informacijau svojem stanju naziva se MarkovimHere variable s t is a random variable that can assumeall the values from set S with a specifiedprobability Ps t o t .There are two main reasons that can lead to the partialmeasurability of an environment. The first reasonis the noise or interference in the sensors via whichthe agent perceives and measures the environment,and the second is perceptual aliasing, i.e. two differentstates may seem to be the same to the agent.In a fully measurable environment, the function accordingto which the agent decides upon the mannerof action is the mapping of the individual parts ofan environmental state. It is a very useful fact thatin the majority of cases the state of the environmentat some time step permits the complete descriptionof the past. Such an environment that has the abilityof gathering all the past information in its stateRajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675652

okruæenjem ili se kaæe da ima Markovo svojstvo(Andrei Andreyevich Markov (1856. ∑ 1922.) ruskimatematiËar). Tako u Markovom okruæenju agentmoæe koristiti bezmemorijske postupke donoπenjaodluka o djelovanju.is called a Markov environment or it is said to havethe Markov property (Andrei Andreyevich Markov,1856∑1922, a Russian mathematician). Thus, anagent can use memoryless procedures in decidingupon an action in a Markov environment.5.4 Prijelazni modelDosada je opisan utjecaj proπlih iskustava agentai njegovog okruæenja na njegovo odluËivanje. No,kao πto je veÊ reËeno za donoπenje optimalnih odlukapotrebno je uzeti u obzir i buduÊnost.U svakom vremenskom intervalu t agenta dakleodabire djelovanje a t iz konaËnog skupa djelovanjaA. Naravno da se pod utjecajem tog agentovogdjelovanja mijenja i okruæenje agenta. Prijelaznimodel precizira naËin na koji se okruæenje mijenjapod utjecajem djelovanja agenta.Ako je trenutaËno stanje okruæenja s t , a agent seodluËi za djelovanje a t tada se mogu razlikovatisljedeÊa dva sluËaja okruæenja:∑ deterministiËko okruæenje u kojem prijelaznimodel pridruæuje paru (s t , a t ) jedinstveno novostanje s t +1 . Tako u πahu na primjer svaki potezmijenja postavu na polju na deterministiËkinaËin. PoznavajuÊi dakle stanje okruæenjai buduÊi potez agenta u deterministiËkomokruæenju jednoznaËno je odreappleeno buduÊestanje okruæenja.∑ stohastiËko okruæenje u kojem prijelazni modelpridruæuje paru (s t , a t ) odreappleenu distribucijuvjerojatnosti Ps t +1 s t ,a t . Pri tome je s t +1sluËajna varijabla koja moæe poprimiti svevrijednosti iz skupa odreappleenih stanja S i to sapripadajuÊom vjerojatnoπÊu Ps t +1 s t ,a t . Uzpoznato stanje okruæenja i buduÊi potez agentau stohastiËkom okruæenju nije jednoznaËnoodreappleeno buduÊe stanje okruæenja, nego supoznate vjerojatnosti pojavljivanja odreappleenihstanja okruæenja.VeÊina aplikacija koje oponaπaju stvarni svijetzahtijevaju stohastiËki prijelazni model koji unosidodatne poteπkoÊe i joπ viπe kompliciraju procesoptimalnog odluËivanja agenta.VeÊinom je krajnji cilj kojem agent teæi nekoæeljeno stanje okruæenja. Tako je planiranje zapravodefiniranje optimalnog puta kroz sva moguÊastanja prema æeljenom stanju. U deterministiËkomse okruæenju podrazumijeva da agent daje prednoststanjima koja su bliæa æeljenim. MnogoopÊenitije, agent moæe imati razliËite sklonostiprema razliËitim stanjima.5.4 Transition modelThus far, the impact of the agent’s past experiencesand the environment upon the agent's decisionmaking has been described. However, as previouslymentioned, it is also necessary to take the futureinto account in order to make optimal decisions.In every time step t the agent chooses an actiona t from a finite set of activities A. Naturally, underthe influence of that agent's action, the agent environmentchanges. A transition model specifies themanner in which the environment changes underthe impact of the action of an agent.If the current state of the environment is s t and theagent decides upon action a t it is then possible todistinguish the following two environmental cases:∑∑a deterministic environment in which the transitionmodel maps a state-action pair (s t , a t )to a single new state s t +1 . Thus in chess, forexample, every move changes the configurationon the board in a deterministic manner. Beingacquainted with the state of the environmentand the future move of an agent in a deterministicenvironment unambiguously determinesthe future state of the environment,a stochastic environment in which the transitionmodel maps a state-action pair (s t , a t ) to a specificprobability distribution Ps t +1 s t ,a t . Heres t +1 is a random variable that can assume all thevalues from the set specific states S and thiswith the corresponding probability Ps t +1 s t ,a t .In addition to the known state of the environmentand the future move of the agent in astochastic environment, the future state of theenvironment is not unambiguously determinedbut the probabilities of the occurrence of certainstates of the environment are known.The majority of the applications that emulate the realworld require a stochastic transition model that introducesadditional difficulties and further complicatesthe process of optimal decision making by the agent.For the majority, the final goal to which the agent aspiresis some desired state of the environment. Thus,planning is actually defining the optimal path throughall the possible states toward the desired state. In adeterministic environment, it is understood that anagent affords preference to states that are close to thedesired one. Much more generally, an agent may havevarious preferences toward various states.653Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

5.5 Sklonosti agenataKako bi se formulirao ovaj pojam sklonosti agenatapojedinim stanjima, svakom stanju s se pridodajerealan broj U(s) koji predstavlja sklonostagenta tom specifiËnom stanju okruæenja. Tako jeU(s)>U(s') ako i samo ako je agent skloniji stanjus od stanja s' i U(s)=U(s') ako i samo ako je agentjednako sklon prema oba stanja.Ako se pretpostavi da agent ima ugraappleene sklonostiprema pojedinim stanjima, postavlja se pitanjekako agent moæe te sklonosti uËinkovito iskoristitipri donoπenju odluka. Dakle, pretpostavljenookruæenje je stohastiËko kao i prijelazni model,Ps t +1 s t ,a t , a agent se nalazi u stanju s t i pokuπavaodrediti djelovanje a t . Neka je U(s) sklonost ovogagenta (koji je jedini u promatranom okruæenju)prema stanju s.Donoπenje odluka bazirano na sklonostima temeljise na pretpostavci da optimalno djelovanje agentaa t * treba maksimizirati oËekivanu sklonost:5.5 Agent preferenceIn order to formulate this concept of agent preferencetoward individual states, to each state s a realnumber U(s) is added that represents the preferenceof the agent toward a specific environmentalstate. Thus, U(s)>U(s') if and only if the agent hasa greater preference toward state s than toward states' and U(s)=U(s') if and only if the agent has anequal preference toward both states.If it is assumed that the agent has intrinsic preferencestoward individual states, the question isposed how the agent can use these preferenceseffectively in making decisions. The assumed environmentis stochastic, as is the transition modelPs t +1 s t ,a t . The agent is situated in state s t and attemptsto determine action a t . Let U(s) representthe preference of this agent (who is the only one inthe environment being studied) toward state s.Decision making is based upon preferences based uponthe assumption that the optimal action of the agent a t *should maximize the anticipated preference:.(6)Kako bi se ocijenilo koliko je pojedino djelovanjedobro, potrebno je pomnoæiti sklonost ka postizanjunekog stanja sa vjerojatnoπÊu postizanja togstanja i sumirati sve dobivene vrijednosti za to djelovanje.Postupak se obavlja za sva moguÊa djelovanjai potom se optimalno djelovanje a t * odreappleujena temelju najveÊe dobivene sume.Ako svako stanje ima pridodanu vrijednost sklonostiagenta za postizanje tog stanja, agent moæena naveden naËin odrediti optimalna djelovanja zasva moguÊa stanja. Tako agent posjeduje funkcijupo kojoj svakom stanju pridruæuje odgovarajuÊedjelovanje na optimalan naËin. Optimalna smjernica(funkcija) djelovanja agenta moæe se zapisatina sljedeÊi naËin:In order to assess the goodness of an individualaction, it is necessary to multiply the preferencetoward the achievement of some state by the probabilityof achieving this state and add together allthe values obtained for this action. This procedureis performed for all possible actions and optimalaction a t * is determined on the basis of the highestsum obtained.If every state has the added value of the preferenceof the agent for achieving this state, the agent candetermine the optimal action for all possible statesin the previously described manner. Thus, an agentpossesses a function according to which the correspondingaction is added to each state in the optimalmanner. The optimal policy (function) of theagent’s action can be expressed as follows:,(7)gdje je U*(s) skup najviπih ostvarivih sklonosti(optimalnih sklonosti).where U*(s) is the set of the greatest feasible preferences(optimal preferences).Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675654

5.6 Neke moguÊe podjele agenataAgenti se mogu podijeliti na jake i slabe agente[6]. Slabi agenti koriste se pri rjeπavanju nekihinæenjerskih tipova problema gdje je potrebnoneπto malo inteligencije (niæa razina sloæenostiproblema). Jaki agenti su sposobni oponaπati nekeili sve osobine ljudskog uma (vrlo visoka razinasloæenosti problema).Slabi agenti veÊinom posjeduju autonomnost (nemapotrebe za ljudskim nadzorom), socijalne sposobnosti(moguÊnost komunikacije s drugim agentima),reaktivnost (reagiraju na promjene u okruæenju) iproaktivnost (sposobnost poduzimanja inicijative).Jaki agenti posjeduju viπe svojstava, kao πto suznanje, usaappleena uvjerenja, namjere ili æelje.SljedeÊa razlika moæe se podvuÊi izmeappleu agenataπireg djelokruga (engl. broad) i onih fokusiranihili specijaliziranih agenata. Specijalizirani agentisu posveÊeni posebnom zadatku ili okruæenju izahtijevaju razvijanje algoritma prilagoappleenog tomspecijalnom zadatku. Oni nemaju sposobnostprilagoappleavanja i uËinkovito rjeπavaju samo oneprobleme za koje su predviappleeni.Agenti πireg djelokruga su opÊenitiji i posjedujuπirok raspon razliËitih sposobnosti. Njihovo djelovanjeje priliËno plitko jer nisu posebno specijaliziraniniti za jedan zadatak.Neki agenti su stvarno fiziËki prisutni, tj. utjelovljenisu (to su uglavnom roboti koji imaju kamere idruge vrste senzora i motore kao aktuatore). Drugipak agenti nisu utjelovljeni i nisu prisutni u stvarnomsvijetu. To su programski agenti koji djelujuunutar svijeta simulacije i nemaju fiziËkih dijelova.Programski se agenti susreÊu s manje problema,jer se njihovi senzori i aktuatori mogu podesiti povolji uËinkovitima.5.7 Arhitektura agenataArhitektura agenta je zapravo struktura njegovaprograma (slika 4).Model samostalnog (u smislu ostvarenja cilja)agenta sastoji se iz osam jedinica i ukljuËuje [7]:5.6 Some possible agent classificationsAgents can be classified as strong or weak [6]. Weakagents are used in solving some engineering typesof problems where some intelligence is required (alower level of problem complexity). Strong agentsare capable of emulating some or all of the characteristicsof the human mind (a very high level ofproblem complexity).The majority of weak agents possess autonomy (donot require human supervision), social abilities (theability to communicate with other agents), reactivity(react to changes in the environment) and proactivity(the ability to take initiatives).Strong agents possess more attributes, such asknowledge, belief, intention or desires.The following distinction can also be made betweenbroad agents and focused or specialized agents.Specialized agents are devoted to a particular taskor environment and require the development of analgorithm adapted to the special task. They lack theability to adapt and can only solve those problemsefficiently for which they are intended.Broad agents are more generalized and have a broadrange of various abilities. Their action is fairly shallowbecause they are not specialized for a specifictask.Some agents are actually physically present, i.e.they are embodied (mainly robots that have camerasand other types of sensors and motors as actuators).Other agents are not embodied and are notpresent in the real world. These are program agentsthat act within the world of simulation and lackphysical parts. Program agents encounter fewerproblems because their sensors and actuators canbe adjusted to the desired efficiencies.5.7 Agent ArchitectureAgent architecture is actually the structure of itsagent program (Figure 4).A model of an independent (in the sense of achievinggoals) agent consists of eight units, as follows[7]:∑ jedinicu za opaæanje (engl. perception unit),∑ jedinicu za obradu (engl. process unit),∑ kontrolnu jedinicu (engl. control unit),∑ djelatnu jedinicu (engl. action unit),∑ jedinicu za komunikaciju (engl. communicationunit),∑ jedinicu za razumijevanje (engl. knowledgeunit),∑ raËunsku jedinicu (engl. compute unit),∑ jedinicu s podacima (engl. data unit).∑∑∑∑∑∑∑∑perception unit,process unit,control unit,action unit,communication unit,knowledge unit,compute unit,data unit.655Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

Slika 4Arhitekturasamostalnog agentaFigure 4Goal autonomousagent architecturebaza podataka /databasebaza znanja /knowledge basepodatak /dataproces /processznanje /knowledgeokruženje / environmentopažanje / perceptionproraËun /computekontrolni proces /controldjelovanje / actionokruženje / environmentkomunikacija /communicationJedinica za opaæanje oËitava podatke iz agentovogokruæenja. Ona sadræi listu pojedinaËnih stanjakoja definiraju ukupno stanje okruæenja. Ukolikose neko od pojedinih stanja promijeni, lista podatakase aæurira te se obavjeπtava kontrolnu jedinicuo promjenama kako bi poduzela odgovarajuÊemjere.Jedinica za obradu sadræi cilj ili viπe ciljeva te njihovemeappleusobne odnose.Djelatna jedinica sadræi sve djelatnosti koje jeagent sposoban obaviti.Kontrolna jedinica odluËuje koje Êe se djelatnostiobaviti radi ostvarenja cilja kojeg takoappleer ona odabire.Ukoliko jedinica za opaæanje uoËi promjene uokruæenju, tada kontrolna jedinica aktivira reakcijskadjelovanja na te promjene.U raËunskoj jedinici su definirane funkcije kojimase odabiru ciljevi i optimalna djelovanja. Takoappleersadræi algoritme i mehanizme odabiranja optimalnihdjelovanja.Kontrolna jedinica poziva funkcije iz raËunske jedinicekako bi se odredio slijedeÊi cilj i prikladnadjelovanja.Jedinica za razumijevanje nadgleda dio odgovoranza znanje agenta koje se primjenjuje pri rjeπavanjustvarnih problema.Jedinica s podacima omoguÊava pristup svim mehanizmimau bazu podataka.Komunikacijska jedinica definira naËine komunikacijeizmeappleu agenata.The perception unit reads data from the agent's environment.It contains a list of individual states thatdefine the overall state of the environment. If someof the individual states change, the data list is updatedand the control unit is informed of the changesin order to undertake the suitable measures.The process unit consists of a goal or several goalsand their mutual relationships.The action unit contains all the actions that theagent is capable of performing.The control unit decides which actions will beperformed in order to achieve the goal that it alsochooses. If the perception unit detects changes inthe environment, the control unit activates a reactiveaction against the changes.In the compute unit, functions are defined by whichgoals and optimal action are selected. It also containsalgorithms and mechanisms for selecting optimalactions.The control unit will call the functions in the computeunit according to a signal from the processunit.The knowledge unit monitors the part responsiblefor the knowledge of the agent which is applied tosolving actual problems.The data unit facilitates access to all the mechanismsin the data base.The communication unit defines the manner ofcommunication among agents.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675656

gleda /perceivesagent / agentakcija /actionSlika 5Arhitektura samostalnogagenta orijentiranog kciljuFigure 5Goal autonomous agentarchitecturesljedeÊe / nextstanje /statesenzor / sensorulazni podatak / input dataokolina /environmentakcija / actionizlazni podatak / output dataKao πto slika 5 [8] joπ jedanput naglaπava, osnovnidijelovi agenta prema funkcijama koje obavljajusu:As Figure 5 [8] reemphasizes, the basic parts ofthe agent according to the functions they performare as follows:∑∑dio za opaæanje (senzor),akcijski dio.∑∑sensor,action part.Prema arhitekturi agenti se mogu podijeliti nasljedeÊe vrste [9]:∑∑∑∑promiπljajuÊi agenti,reaktivni agenti,uvjerenje-æelja-namjera arhitekture (BDI, engl.Beliefs, Desires, Intentions),slojeviti agenti.PromiπljajuÊi agenti koriste, kao πto im samo imei govori, svjesno odluËivanje, odnosno logiËku dedukcijui matematiËku logiku pa se mogu koristitipri dokazivanju teorema. Kako ne mogu prouËavatipromjene okruæenja koje nastaju njihovim djelovanjem,nisu sposobni za simulaciju dogaappleaja ustvarnom vremenu.Reaktivni agenti reagiraju na poticaje iz okoline ukoju su smjeπteni. Jednostavni su i moguÊe je pratitiraËunanje, ali nemaju sposobnost uËenja takoda se pri odluËivanju o djelovanju koriste samopodaci dobiveni opaæanjem agentova okruæenja.According to the architecture, agents can be classifiedinto the following types [9]:∑∑∑∑deliberative agents,reactive agents,beliefs, desires and intentions (BDI),layered agents.Deliberative agents, as their name implies, use deliberativedecision making, i.e. logical deductionand mathematical logic, and can be used in provingtheories. Since they cannot take into accountchanges in the environment that occur due to theiraction, they are incapable of simulating events inreal time.Reactive agents react to stimuli from the environmentin which they are located. They are simple andthe computation process can be monitored but theydo not have the ability to learn. In decision makingregarding action, they only use data obtained fromthe perception of the agent's environment.657Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

Slika 6Agent temeljen naBDI arhitekturiFigure 6Agent based uponBDI architectureuvjerenja /beliefplan /planulaz / entranceprevoditelj /translatorizlaz / exitakcija / actionciljevi (želje) /goals(desires)namjere /intentionsNa slici 6 se nalazi forma agenta temeljenog naBDI arhitekturi. BDI arhitektura agenta zasniva sena praktiËnom zakljuËivanju. OdluËuje se o ciljevimakoje treba postiÊi i na koji naËin Êe biti postignuti.Agenti su ustrajni u svojim namjerama ione ograniËavaju buduÊe promiπljanje. Iz namjeratakoappleer proizlazi svrhovito promiπljanje.Slojevite arhitekture koriste se dekompozicijomproblema na dijelove, a svakom od njih se dodjeljujeodreappleeno ponaπanje. Topologija slojeva podijeljenaje s obzirom na razmjenu kontrolnih podataka.Postoji horizontalna i vertikalna topologija.Figure 6 presents the form of an agent based uponBDI architecture. BDI agent architecture is basedupon practical conclusion making. Decisions aremade on goals that must be achieved and in whatmanner they will be achieved. Agents are consistentin their intentions and they limit future deliberations.Goal based deliberation also issues from theintentions.Layered architecture is used for the decompositionof problems into parts, and each of them isassigned certain behavior. The topology of the layersis divided according to the exchange of controldata. There are horizontal and vertical topologies.6 VIŠEAGENTSKI SUSTAVIViπeagentski sustavi, MAS (engl. Multiagent Systems)spadaju u podruËje prouËavanja umjetneinteligencije, AI (engl. Artificial Intelligence) kojese bavi naËinima konstrukcije sloæenih sustavakoji koriste veÊi broj agenata te usklaappleivanjemponaπanja tih agenata [9]. To je podruËje prouËavanjaraspodijeljene umjetne inteligencije (engl.Distributed Artificial Intelligence DAI).Postoje brojna podruËja koja zahtijevaju uporabuviπeagentskih sustava. Tako se ovakvi sustavi koristeu zahtjevnim raËunalnim igrama, problemimatransporta, grafiËkim problemima, GIS (Geografskiinformacijski sustav ∑ Geographic InformationSystem) sustavima te u mnogim drugim podruËjima.Ukoliko se problem svodi na veÊi broj ljudi iliorganizacija s razliËitim (moguÊe i proturjeËnim)ciljevima i zasebnim informacijama, tada je nuænauporaba viπeagentskih sustava kako bi se upravljalonjihovom meappleudjelovanjem. Takvo podruËje jezasigurno i træiπte elektriËne energije.6 MULTI-AGENT SYSTEMSMulti-agent systems (MAS) are part of the field ofcomputational artificial intelligence, AI, which isengaged in the manner of the construction of complexsystems that use a large number of agents andcoordinate the behavior of these agents [9]. This isthe area of the study of distributed artificial intelligence,DAI.There are numerous areas that require the use ofmulti-agent systems. These systems are used indemanding computer games, transport problems,graphic problems, GIS (Geographic InformationSystem) systems and many other areas. If a probleminvolves a large number of persons or organizationswith various (possibly also contradictory) goalsand separate information, it is then necessary touse multi-agent systems in order to manage theirinteractions. The electricity market is certainly suchan area.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675658

Viπeagentski sustav se razlikuje od onog u kojemje samo jedan agent u postojanju veÊeg broja agenataod kojih svaki utjeËe na ciljeve i djelovanjasvih drugih (promatrano iz perspektive jednogagenta ostali su njegovo okruæenje). Takoappleer postojimoguÊnost direktne interakcije (komunikacije)izmeappleu dva ili viπe agenata. Ta interakcijaagenata moæe biti promatrana kao dio okruæenja ilipotpuno odvojeno od okruæenja agenata.Iz perspektive jednog agenta stvari se dramatiËnokompliciraju ulaskom u viπeagentski sustav s obziromna situaciju kada je potpuno sam u svomokruæenju. Naime ovdje svi ostali agenti svojimdjelovanjima utjeËu na promjenu okruæenja. Naslici 7 nalazi se prikaz dva agenta koji se nalaze uistom okruæenju i pri Ëemu svaki ima svoje ciljeve(meappleusobno razliËite), postupke djelovanja i znanjaiz odreappleene domene. Na istoj slici prikazanaje i moguÊnost direktne komunikacije izmeappleu ovadva agenta.A multi-agent system differs from a system in whichthere is only one agent due to the presence largenumber of agents, each of which influences the goalsand actions of all the others (from the perspective ofone agent the other constitute its environment). Thereis also the possibility of direct interaction (communication)among two or more agents. This agent interactioncan be studied as a part of the environment orcompletely separated from the agent environment.From the perspective of an agent, matters becomedramatically more complicated by entering a multiagentsystem due to the situation when it is completelyalone in its environment. All the other agentshere influence change in the environment with theiractions. In Figure 7, two agents are presented thatare in the same environment and each has its owngoals (mutually different), action procedure andknowledge of a certain domain. In the same figure,possibilities are presented for direct communicationbetween these two agents.Okruženje / EnvironmentSlika 7Dva agenta u istomokruæenjuFigure 7Two agents in the sameenvironmentAgent /AgentAgent /Agent• ciljevi / goals• postupci / actions• znanja iz domene/ domainknowledge• ciljevi / goals• postupci / actions• znanja iz domene/ domainknowledgeSasvim opÊenito, moguÊe je postojanje po voljivelikog broja agenata koji se meappleusobno viπeili manje razlikuju i mogu, ali i ne moraju, imatimoguÊnost komunikacije s ostalim agentima.»est je sluËaj da se agenti, koji saËinjavaju istiviπeagentski sustav, bitno razlikuju jedni od drugih.TipiËan primjer su programski agenti (engl.softbot) koji se razlikuju u realizaciji od strane razliËitihdizajnera. OpÊenito, razlike mogu biti bilohardverske bilo softverske. Ovakvi agenti nazivajuse heterogenima za razliku od homogenih koji sudizajnirani na identiËan naËin.U viπeagentskom sustavu prisustvo veÊeg brojaagenata uzrokuje veÊu dinamiËnost okoline, a timei sloæenost promatranog problema.Quite generally, it is possible for a large number ofagents to exist, as desired, who mutually more orless differ and can, but not necessarily, have thepossibility of communicating with the other agents.It is frequently the case that agents comprising thesame multi-agent system differ from each othersignificantly. Typical examples are program agents(software robots or softbots), which differ accordingto the designers. Generally, differences can existin hardware or software. Such agents are calledheterogeneous, in contrast to homogeneous agentsthat are designed in the identical manner.In a multi-agent system, the presence of a largenumber of agents makes the environment more dynamicand thereby increases the complexity of theproblem studied.659Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

Ukupna informacija koju primaju svi agentiviπeagentskog sustava je raspodijeljena kakoagenti opaæaju stanja koja se razlikuju prostorno,vremenski ili Ëak u naËinu interpretacije. Zbogsvega toga stanja okruæenja automatski postajudjelomiËno mjerljiva iz perspektive svakog pojedinaËnogagenta πto na razliËite naËine utjeËe naodluËivanje pojedinih agenata.Dodatni problem predstavlja mijeπanje podatakaprimljenih putem senzora te se nameÊe pitanjekako optimalno kombinirati percepcije pojedinihagenata s ciljem postizanja, πto potpunije kolektivneslike okruæenja.Joπ jedna razlika viπeagentskog sustava s obziromna sluËaj sustava sa samo jednim agentom jestdecentralizacija upravljanja. To znaËi da ne postojicentralni proces koji skuplja podatke od svihagenata i na temelju tih podataka odluËuje πto Êesvaki agent raditi. Tako odluËivanje svakog agentau najveÊoj mjeri ovisi o njemu samome.U viπeagentskim sustavima svaki pojedinaËniagent treba uzeti u obzir znanja koja posjedujudrugi agenti pri donoπenju vlastitih odluka. Zbogtoga je posebno vaæan koncept javnog znanjaprema kojem svaki agent zna neku Ëinjenicu, istotako svaki agent zna da svaki drugi agent zna tuËinjenicu, i tako dalje.Komunikacija u viπeagentskim sustavima podrazumijevadvosmjerni proces gdje su svi agenti potencijalnipoπiljatelji ili primatelji neke informacije.No, postavljaju se pitanja kojim se protokolimasluæiti kako bi razmijenjene informacije sigurno ina vrijeme stigle na odrediπte. Bitan je i jezik putemkojeg se agenti sporazumijevaju i razmjenjujuinformacije, pogotovo ako je rijeË o heterogenimagentima.U viπeagentskom sustavu gdje velik broj agenatadonosi odluke u isto vrijeme, veliku nepoznanicusvakom od agenata predstavlja pretpostavljanjeodluka drugih agenata. Tako djelovanje jednogagenta ovisi o akcijama svih drugih agenata.Donoπenje odluke agenata u viπeagentskom sustavutakoappleer je predmet veÊ spomenute teorije igara.Teorija pokuπava razumjeti i objasniti ponaπanjeagenata koji su u meappleusobnoj interakciji i suoËenisu s brojnim nesigurnostima. Teorija se zasnivana dvjema pretpostavkama, a to su da su agentiracionalni i da se ponaπaju strateπki πto znaËi dauzimaju u obzir moguÊe odluke drugih agenata privlastitom odluËivanju.S obzirom na naËine na koje agenti odluËuju o svojimakcijama mogu se razluËiti dvije vrste igara:The total information received by all the agents of amulti-agent system is distributed according to how theagents perceive the states which differ spatially, temporallyor even in the manner of interpretation. Therefore,the state of the environment automatically becomesincompletely measurable from the perspective of eachindividual agent, which in various ways influences thedecision making of the individual agents.An additional problem is represented by the mixingof data received via sensors. The question ariseshow to combine the perceptions of individual agentsin an optimal manner with the goal of achieving themost complete possible collective perception of theenvironment.One more difference between a multi-agent systemand a system with only one agent is the decentralizationof control. This means that there is no centralprocess that collects data from all the agentsand on the basis of these data decides what eachagent will do. Such decision making by each agentprimarily depends upon the agent alone.In multi-agent systems, each individual agentshould take into consideration the knowledge possessedby other agents when making its own decisions.Therefore, the concept of public knowledgeis particularly important, according to which eachagent knows some fact and, similarly, each agentknows that every other agent knows that fact etc.Communication in multi-agent systems is understoodto mean a two-directional process where allthe agents are potential senders or receivers ofsome information. However, the question is posedwhich protocol will be used so that the exchangedinformation reaches its destination securely and ontime. The language with which the agents communicateand exchange information is of importance,particularly concerning heterogeneous agents.In a multi-agent system where a large number of agentsmake decisions at the same time, a great unknown foreach agent is represented by the assumed decisionsmade by other agents. Such action by one agent dependsupon the actions of all the other agents.Decision making by an agent in a multi-agent systemis also a subject of the previously mentioned gametheory. The theory attempts to understand and explainthe behavior of agents who are confronted with numerousuncertainties in mutual interaction. The theory isfounded upon two assumptions, that the agents are rationaland that they behave strategically, which meansthat they take possible decisions by other agents intoaccount when making their own decisions.Regarding the ways in which agents decide on theiractions, two types of games can be differentiated:Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675660

U strateπkim igrama svaki se agent odluËuje zasvoju strategiju na poËetku igre i potom svi agentipoduzimaju akcije simultano. Ako je pak rijeË oekstenzivnim igrama agenti su u moguÊnosti ponovnorazmisliti o svojim planovima tijekom sameigre. Bitno je takoappleer kakve su spoznaje pojedinogagenta o aspektima koji ukljuËuju druge agente.Takve spoznaje mogu biti savrπene ili nesavrπene.Agenti se odluËuju za pojedine akcije koje formirajujednu zajedniËku akciju na temelju kojesvaki agent biva obavijeπten o svojoj uspjeπnosti(ishodu). Ta zajedniËka akcija se naziva ishodomigre. Iako su funkcije uspjeπnosti agenata javnopoznate, svaki agent nije upoznat s odabirom akcijaostalih agenata. Najviπe πto moæe uËiniti jepokuπati predvidjeti te akcije. Najbolje rjeπenje zaigru je predviappleanje ishoda igre temeljeno na pretpostavcida su svi agenti racionalni i razmiπljajustrateπki.In strategic games, each agent decides upon itsstrategy at the beginning of the game and then allthe agents undertake actions simultaneously. Inthe case of extensive games, agents are able to rethinktheir plans during the game itself. The kind ofknowledge of an individual agent regarding aspectsthat include other agents is also important. Suchknowledge can be perfect or imperfect.Agents decide on individual actions that forma cooperative action, on the basis of which eachagent is informed of its success (result). This cooperativeaction is called the game result. Althoughthe functions of the success of the agents are publiclyknown, each agent is not acquainted with thechoices of the actions of the other agents. The mostit can do is to attempt to predict these actions. Thebest game solution is to predict the game resultbased upon the assumption that all the agents arerational and think strategically.7 SIMULATORI TRÆIŠTAEkonomisti veÊ jako dugo prouËavaju strukturu,uËinkovitost i razvoj træiπta opÊenito. Tradicionalnipristupi koriste se matematiËkim modelima kaoπto je teorija igara kako bi odredili ravnoteænostanje dinamiËkih ekonomskih sustava. Pri tomese koriste razliËite vrste simulatora. No, ovakvipristupi nisu u moguÊnosti analizirati mikro interakcijemeappleu sudionicima træiπta pogotovo unovonastalim uvjetima kada se mijenja i pristupplaniranju EES-a.Simulatori træiπta, na temelju velikog broja podatakao elektroenergetskom sustavu, troπkovimaproizvodnje, prijenosnim ograniËenjima, træiπnommodelu, stupnju koncentracije i drugim karakteristikamapredviappleaju stanje na træiπtu u odreappleenomvremenskom razdoblju. Ipak simulator nije alat kojije sposoban u potpunosti zamijeniti planera i nemoæe bez Ëovjekove intervencije proizvesti træiπnustrategiju. Simulacijom se prvenstveno predviapplearavnoteæna cijena na træiπtu, no koriπtenjem prikladnogsimulatora mogu se razmatrati i sloæenijapitanja, primjerice utjecaj restrukturiranja iliπirenja træiπta, strateπki ciljevi u srednjem i duæemvremenskom razdoblju, ulaz novih proizvodnihkapaciteta na træiπte, ograniËenja u resursima,problemi vezani uz pouzdanost proizvodnih i prijenosnihkapaciteta, utjecaj bilateralnih ugovora natræiπte i sliËno. Takvi postupci zahtijevaju intenzivnuinterakciju Ëovjeka-planera i simulatora.VeÊina simulatora træiπta u osnovi primjenjujeneki od scenarijskih modela u simulaciji træiπtai klasiËne ekonomske teorije ravnoteæe. Pristupi7 MARKET SIMULATORSEconomists have long studied market structure, efficiencyand development in general. Traditional approachesuse mathematical models such as gametheory in order to determine the equilibrium of thedynamics of economic systems. Various types ofsimulators are used. Nonetheless, such approachescannot be used to analyze micro interactions amongmarket participants, particularly under the new circumstanceswhen the approach to the planning ofpower systems is changing.Market simulators, based upon extensive data onthe power system, production costs, transmissionlimitations, market model, degree of concentrationand other characteristics predict the state of themarket at a defined time period. Nevertheless, asimulator is not a tool that can replace a plannercompletely. Without human intervention, it cannotimplement market strategy. Simulation primarilypermits the prediction of the price equilibrium onthe market. However, by using a suitable simulatorit is also possible to address more complex questions,such as the influence of market restructuringor expansion, strategic goals in a medium-range andlong-range time period, the entry of new productionfacilities on the market, limitations in resources,problems in connection with the reliability of thecapacities of generation and transmission facilities,the impact of bilateral contracts on the market etc.Such approaches require intensive interaction betweenthe human planner and the simulator.The majority of market simulators basically applysome of the scenario models in the simulation of mar-661Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

su pritom razliËiti, no prevladava probabilistiËkimodel i Monte Carlo simulacija. Takvi ekonomskimodeli ipak nisu sposobni stvoriti uvid u strateπkoponaπanje træiπnih sudionika. Iako jedinstveno poprirodi dobara kojim se trguje, træiπte elektriËneenergije tu ne odskaËe od ostalih træiπta. Konvencionalnemetode imaju ograniËene sposobnostiuvida u strateπko ponaπanje tvrtki i imaju tendencijuprevelikog pojednostavljivanja prilika natræiπtu i tehniËkih karakteristika elektroenergetskogsustava.Meappleu najuspjeπnijim simulacijskim metodamadanas je koriπtenje programskih agenata. U simulacijizasnovanoj na agentima elementi stvarnogtræiπta modeliraju se agentima. Dvije su osnovneznaËajke ovakvih modela ∑ s jedne strane agentise ponaπaju autonomno i pokuπavaju ostvariti cilj,a s druge strane agenti mogu meappleusobno komunicirati.Svaki agent pritom ima vlastite, specifiËne,strateπke i operativne ciljeve.Kako bi viπeagentski simulacijski sustav prikladanza ekonomsko modeliranje bio uspjeπan, morapodræavati sljedeÊe [10]:∑ komunikaciju meappleu agentima, na naËin daagenti zadræe svoju autonomiju, informacijemogu biti dostupne samo ciljanim skupinamaagenata, u skladu s træiπnim pravilima,∑ upravljanje konfiguracijom i izvoappleenjemagenata,∑ praÊenje aktivnosti agenta,∑ upravljanje modelom i konzistentnostpodataka,∑ podrπku za razliËite vremenske korake usimulacijama,∑ zajedniËke servise za agente (pojedini aspektisimuliranog svijeta pojavljuju se u viπe agenataistodobno i na neki su naËin dijeljeni, pazato mogu biti dostupni kao usluge sustava.Ovo nije nuæan uvjet, no doprinosi softverskojuËinkovitosti ovakvih modela).Kod primjene na simulaciju træiπta elektriËneenergije redovito se koriste okruæenja za razvojviπeagentskih sustava, ali agenti i njihovaunutraπnja arhitektura razvijaju se posebno za tuprimjenu. Posebno su zanimljivi adaptivni agenti.Adaptivni agenti sposobni su za inovaciju, razvojuzoraka ponaπanja koji nisu unaprijed programirani,nasuprot uËenju samo na iskustvenoj bazigdje se iskljuËivo na osnovi doæivljenih iskustavaodabire postupak koji daje optimalan rezultat.Adaptivni agenti svojim postupcima mogu utjecatina okolinu, i time praktiËki provoditi eksperimentei zakljuËivati iz njih. S obzirom na priliËno zahtjevnezadane uloge modeliranja ljudskog ponaπanja,kets and the classical theory of economic equilibrium.The approaches differ but the probability model andMonte Carlo simulation predominate. Such economicmodels are nevertheless incapable of providing insightinto the strategic behavior of market participants.Although unique in terms of the nature of the goodsthat are traded, electrical energy markets do not differfrom other markets in this respect. Conventionalmethods have limited capabilities of insight into thestrategic behavior of companies and have a tendencyto oversimplify the conditions on the market and thetechnical characteristics of a power system.Among the most successful simulation methodsused today are program agents. In agent-based simulation,elements of the real market are modeledwith agents. There are two fundamental characteristicsof such models: from the one side, the agentsact autonomously and attempt to achieve the goaland from the other side, the agents can communicateamong themselves. Each agent has its ownspecific, strategic and operative goals.In order for a multi-agent simulation system suitablefor economic modeling to be successful, itmust support the following [10]:∑ communication among the agents, in such amanner that the agents retain their autonomy;information is only available to target groups ofagents, according to market rules,∑ configuration control and agent performance,∑ monitoring agent actions,∑ model control and data consistency,∑ support for various time steps in simulations,∑ ancillary services for agents (individual aspects ofthe simulated world occur in several agents simultaneouslyand are distributed in some manner,and therefore can be available as system services.This is not an essential condition but contributesto the software performance of such models).In an application for the simulation of electricalenergy markets, environments for the developmentof multi-agent systems are regularly used but theagents and their internal architecture are developedespecially for this application. Adaptive agents areparticularly interesting.Adaptive agents are capable of innovation, thedevelopment of behavior models that are not programmedin advance, as opposed to learning solelyon the basis of experience where an approach ischosen exclusively based upon experiences thatprovide an optimal result. Adaptive agents caninfluence the environment with their behavior andthereby practically conduct experiments and drawconclusions from them. Taking into account thefairly demanding given roles for the modeling of hu-Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675662

oËito je da agenti ukljuËuju algoritamske tehnikepoznate iz strojnog uËenja, robotike i pretraæivanjainformacija.Kako se jasnije vidi sa slika 8 i 9, bitna je razlikaizmeappleu klasiËnih simulatorskih modela i simulacijskihmodela temeljenih na viπeagentskim sustavimau njihovom pogledu na træiπte, odnosno unjihovom shvaÊanju pojma træiπta.Tržište / Marketman behavior, it is evident that the agents includealgorithm techniques known from machine learning,robotics and information searches.As seen more clearly from Figures 8 and 9, there isa significant difference between classical simulatormodels and simulator models based upon multiagentsystems in their view of the markets, i.e. intheir understanding of the concept of a market.Slika 8Træiπte elektriËneenergije iz perspektiveklasiËnih simulatorskihsustava baziranih namatematiËkim modelimatræiπtaFigure 8Electrical energy marketfrom the perspectiveof classical simulatorsystems based onmathematical marketmodelsTržište / MarketAgent / AgentSlika 9Træiπte elektriËneenergije iz perspektiveviπeagentskog sustavaFigure 9Electrical energy marketfrom the perspective ofa multi-agent systemKlasiËni simulatorski modeli zasnovani na matematiËkommodelu promatraju træiπte kao cjelinuunutar koje se svi procesi odvijaju po nekomodreappleenom zakonu. No, koriπtenje viπeagentskihsustava omoguÊilo je usmjeravanje paænje i fokusiranjena same sudionike træiπta. Tako je svaki sudionikpredstavljen jednim agentom, a interakcijommeappleu agentima posredno se simuliraju træiπneaktivnosti (kao u stvarnom æivotu).Classical simulator models based on a mathematicalmodel perceive the market as a whole, withinwhich all the processes take place according to adetermined law. However, the use of multi-agentsystems makes it possible to focus attention on themarket participants themselves. Thus, each participantis represented by an agent and the interactionamong the agents indirectly simulates market activities(as in real life).663Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

8 PRIMJENA VIŠEAGENTSKIHSUSTAVA ZA POTREBESIMULIRANJA TRÆIŠTAELEKTRI»NE ENERGIJEZbog trenutaËno intenzivnih procesa liberalizacijei deregulacije, sasvim je neizvjesna buduÊa poslovnastruktura industrije elektriËne energije i træiπtaelektriËnom energijom. Stoga se na brojne naËinepokuπava πto toËnije predvidjeti buduÊe stanje naelektroenergtskom sektoru kako bi bilo moguÊe navrijeme donijeti neke vaæne strateπke odluke.TrenutaËna predviappleanja moguÊih novih poslovnihstruktura uglavnom se zasnivaju na predviappleanjuodreappleenih buduÊih scenarija. ProuËavanje takvihscenarija, odnosno razliËitih ustrojstava træiπtaelektriËne energije, pruæa zauzvrat moguÊnostutvrappleivanja ispravnih strateπkih kapitalnih investicijau odreappleena træiπne sektore.Velika mana ovakvog pristupa kojim se prouËavajurazliËiti scenariji pri donoπenju optimalnih odlukajeste potpuna ovisnost o stvarnoj vjerojatnostipojavljivanja baπ tog odreappleenog promatranog scenarijaili neke promatrane familije pretpostavljenihscenarija.Dakle, donoπenje ispravnih strateπkih odluka temeljise na toËnosti predviappleanja buduÊih poslovnihstruktura na kojima Êe se zasnivati model træiπtaelektriËne energije. Upravo zbog takvog naËinadonoπenja odluka ovaj se pristup moæe nazvatipristupom od gore prema dolje (engl. top-downapproach).Za razliku od pristupa od gore prema dolje, postojii razvija se alternativni pristup od dolje premagore (engl. bottom-up approach). Ovakav pristupprouËavanja træiπnih strategija nije utemeljen napredviappleanjima raznih scenarija. Naprotiv, glavnitemelji bottom-up pristupa su fizikalni zakoni kojivrijede za tok elektriËne energije te cijena i dostupnostraznih moguÊih tehnoloπkih ili ekonomskihrjeπenja.Upravo u ovakvom naËinu odreappleivanja ispravnihstrateπkih poteza veliku pomoÊ predstavljajuagenti i viπeagentski sustavi. Naime, autonomnii adaptivni agenti mogu na vrlo uËinkovit naËinpredstaviti razliËite sudionike træiπta elektriËneenergije, ali isto tako i naËine na koje ti sudionicimeappleusobno djeluju (kooperacija, konkurencija,nadreappleenost, podreappleenost, nezavisnost itd.).Evolucijom agenata kroz izvrπavanje brojnih eksperimenatau simulacijama trebale bi se pokazatimoguÊe konfiguracije træiπta i industrije elektriËne8 THE APPLICATION OF MULTI-AGENT SYSTEMS FOR THEPURPOSES OF SIMULATING THEELECTRICAL ENERGY MARKETDue to the current intensive processes of liberalizationand deregulation, the future business structureof the electricity industry and the electricity marketis completely uncertain. Therefore, numerous variedattempts are made to forecast the future stateof the electricity sector as precisely as possible inorder to make some important strategic decisionsin time.The currently anticipated possible new businessstructures are generally based upon the forecast ofcertain future scenarios. Studying these scenarios,i.e. various structures of the electricity markets,makes it possible to determine correct strategiccapital investments within specific market sectors.A major shortcoming of such an approach in whichvarious scenarios are studied in making optimal decisionsis the complete dependence upon the actualprobability of the occurrence of precisely that specificscenario studied or a family of proposed scenarios.Therefore, making correct strategic decisions isbased upon the precision of the predicted futurebusiness structures upon which the model of theelectricity market will be established. Due to thismanner of decision making, this approach can becalled the top-down approach.Unlike the top-down approach, an alternative approachexists and is being developed, the bottomupapproach. Such an approach to studying marketstrategies is not based upon various predictedscenarios. On the contrary, the main foundationsof the bottom-up approach are physical laws thatapply electrical energy flow as well as the cost andavailability of various possible technological or economicsolutions.It is precisely in such a manner for determiningthe correct strategic moves that great assistanceis provided by agents and multi-agent systems.Autonomous and adaptive agents can provide highlyeffective representation for various participantsin the electricity market and can also represent theways in which these participants act together (cooperation,competition, dominance, subordination,independence etc.).Through the evolution of agents by conducting numerousexperiments in simulations, it should befeasible to demonstrate the possible configurationsof the electricity market and industry, dependingRajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675664

energije i to u ovisnosti o stupnju kooperacije,kompetitivnosti i regulacije.Osnovni cilj razvoja ovakvog pristupa i modela jestpomoÊi pojedinaËnim træiπnim sudionicima priodluËivanju o uvoappleenju novih poslovnih strategija,a koje se tiËu internih reorganizacija samih sudionikakao i njihova moguÊeg ulaska u nova partnerstvaili moguÊeg prodora u neke nove dijelovetræiπta. Isto tako je moguÊe kroz raËunalne simulacijeovog modela dobiti uvid u razvoj i evolucijuenergetskog sektora kao cjeline.8.1 Uloge agenataAgenti dakle trebaju igrati uloge stvarnih træiπnihsudionika pa tako svaki od njih mora imati posebnespecifikacije i vlastite ciljeve. ©to je preciznijedefinirana zadaÊa i metode razmiπljanja pojedinogagenta, on je bliæi stvarnom sudioniku, ali zato imnogo kompleksniji. Za potrebe simulacije træiπtaelektriËne energije u potpuno liberaliziranim uvjetimasasvim sigurno su potrebni sljedeÊi agenti[11]:∑ ekonomski agent (fiktivni agent),∑ agent proizvoappleaË,∑ agent potroπaË,∑ agent za informacije iz okruæenja (fiktivniagent),∑ agent trgovac,∑ agent opskrbljivaË,∑ agent regulator,∑ agent operator træiπta.upon the degree of cooperation, competition andregulation.The basic goal of the development of such an approachand model is to help individual marketplayers decide on the introduction of new businessstrategies, regarding internal reorganization of theparticipants themselves as well as their eventualentry into new partnerships or penetration intosome new parts of the market. Similarly, throughthe computer simulation of this model, it is possibleto obtain insight into the development andevolution of the energy sector as a whole.8.1 The roles of agentsThus, agents must play the roles of actual marketparticipants and each of them must have particularspecifications and its own goals. The more preciselydefined the task and methods of thinking of an individualagent, the closer the agent is to an actualparticipant but also much more complex. For thepurposes of simulating the electrical energy marketunder completely liberalized conditions, the followingagents are certainly necessary [11]:∑∑∑∑∑∑∑∑economy agent (fictive agent),generator agent,consumer agent,information environment agent (fictive agent),retailer agent,delivery agent,regulator agent,market operator agent.8.1.1.Ekonomski agentEkonomski agent prevodi razliËite podatke, poputekonomskih faktora, godiπnjeg doba, vremenskihuvjeta u varijable energetske potraænje. Te se vrijednostipotraænje potom πalju agentu za informacijeiz okruæenja.8.1.1.Economy agentThe economy agent translates various data, such aseconomic factors, season of the year and weatherconditions into energy demand variables. Thesedemand values are then sent to the information environmentagent.8.1.2 Agent proizvoappleaËAgenti koji predstavljaju proizvoappleaËe elektriËneenergije brinu se za proizvodnju dovoljne koliËineelektriËne energije za potrebe krajnjih potroπaËa.No, primarni cilj ovakvih agenata je ostvarivanjeπto veÊeg profita (træiπno su orijentirani). Ovisno okojoj vrsti proizvoappleaËa je rijeË, (nuklearna, hidro,termo ili neka druga elektrana), tj. ovisno o marginalnimtroπkovima proizvodnje i samim uvjetimavezanim uz proizvodnju elektriËne energije, agentiproizvoappleaËi pokuπavaju na optimalan naËin plasiratisvoju proizvodnju na træiπte. Pri tome svakiagent mora dobro predvidjeti poteze drugog agentaproizvoappleaËa. Ovim naËinom je ostvarena konkurencijai kompetitivnost u proizvodnom sektoru.8.1.2 Generator agentAgents that represent generators (producers) ofelectrical energy are concerned that sufficient energyis produced for the needs of the final consumers.However, the primary goal of such agents is toachieve maximum profits, since they are market oriented.Depending on the type of generator (nuclear,hydro, thermo or some other type of power plant),i.e. depending on the marginal production costsand the conditions connected with the generationof electrical energy, generator agents attempt toplace their production on the market in an optimalmanner. Each agent must anticipate the moves ofthe other generator agents well. In this manner,competition and competitiveness are achieved inthe generation sector.665Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

8.1.3 Agent potroπaËAgenti potroπaËi ne predstavljaju svakog pojedinaËnogpotroπaËa veÊ odreappleene grupe ili klasepotroπaËa poput kuÊanstava, industrije ili privatnogsektora. Svaki agent kupuje odreappleenu koliËinuenergije i mijenja udio energije na træiπtu ovisno ocijeni, potrebi, elastiËnosti potraænje i osjetljivosticijena. Energetska uËinkovitost takoappleer se uzimau obzir.8.1.3 Consumer agentConsumer agents do not represent each individualconsumer but rather groups or classes of consumerssuch as residential, commercial or the privatesector. Each agent purchases a specific quantity ofenergy, which changes the market share of energyaccording to prices, requirements, elasticity of demandand price sensitivity. Energy efficiency is alsotaken into account.8.1.4 Agent za informacije iz okruæenjaOvakav agent predstavlja neku vrstu oglasne ploËe,jer sadræi sve dostupne informacije o svim sudionicimatræiπta. Moæe se predoËiti kao posrednikkojem træiπni sudionici daju informacije u vezi sasvojim trenutaËnim radnjama i traæe informacijekako bi procijenili svoje buduÊe poteze. Osim komunikacijskevaænosti vrπi i sakupljanje podatakaod træiπnih sudionika pruæajuÊi tako jasne i transparentnepodatke.8.1.4 Information environment agentThis agent represents a type of bulletin board becauseit contains all the available information on all themarket participants. It can be seen as an intermediaryvia which the market participants provide informationin connection with their current actions and requestinformation in order for them to evaluate their futuremoves. In addition to communication importance, italso compiles data from market participants and thusprovides clear and transparent information.8.1.5 Agent trgovacGlavne unutarnje funkcije agenata trgovaca su:∑∑∑nadziranje vlastitog djelovanja u smislu profitabilnosti,kao i kretanja udjela na træiπtu,traæenje optimalne kombinacije odluka kako bipostigao bolje rezultate,unaprjeappleenje uËinkovitosti voappleenja posla.Glavni cilj ovakvog agenta jest postizanje maksimalnogprofita ne ugroæavajuÊi pri tome sigurnostopskrbe potroπaËa. Izrazito vaæna funkcija kojuposjeduje agent trgovac je strateπko planiranje.Kako bi to ostvarili, ovi agenti posjeduju unutarnjefunkcije za simuliranje træiπta i predviappleanje potezadrugih sudionika na træiπtu. To je postupak simulacijeunutar simulacije, te se temelji na svomznanju o ponaπanju ostalih sudionika koje agentposjeduje. Kako bi predvidio buduÊu potroπnju,a samim time i cijenu (πto je joπ vaænije) agentovog tipa ponekad koristi neuronske mreæe. Istotako koristi evolucijsku raËunsku simulaciju kakobi planirao buduÊnost i izveo optimalnu strategijuza πirenje posla i odreappleivanje cijena.8.1.6 Agent opskrbljivaËNjihove su zadaÊe πirenje mreæe kroz podruËjekako bi omoguÊili opskrbu novih potroπaËa, nazahtjev agenata trgovaca. ©irenje mreæe vrπi sepomoÊu funkcija kojima se optimizira put i profit,πto je takoappleer moguÊe u GIS platformama. Agentovog tipa ima vlastitu logiku i poput agenta trgovca,teæi k maksimizaciji profita (sigurnost opskrbepotroπaËa elektriËne energije nije pri tome upitna).Svojim akcijama agenti opskrbljivaËi potiËu8.1.5 Retailer agentThe chief internal functions of a retailer agent areas follows:∑∑∑monitoring its own performance in the sense ofprofitability, as well as market share movement,finding optimal decision combinations in orderto improve performance,improving management efficiency.The main goal of such an agent is to achieve maximumprofit without endangering the reliability of theconsumer supply. A particularly important functionof the retailer agent is strategic planning. In order toachieve this, these agents possess an internal functionfor market simulation and predicting the movesof other market participants. This is a simulation insidethe simulation process, based upon the agent'sknowledge of the behavior of the other participants.In order to predict future consumption, and therebythe price (which is more important), this type of agentsometimes uses neural networks. Similarly, the agentalso uses evolutionary computer simulation in order toplan the future and derive an optimal strategy for theexpansion of business and determination of prices.8.1.6 Delivery agentThe task of a delivery agent is to extend the networksover the territory in order to facilitate supply to newconsumers, at the request of retail agents. Expansionof the network is performed using functions that optimizepaths and profits, which is also possible inGIS platforms. An agent of this type has a logic of itsown and like a retailer agent seeks to maximize profits(the reliability of the supply to electricity consum-Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675666

natjecateljsko ponaπanje na træiπtu buduÊi daomoguÊavaju πirok izbor razliËitih ponuda krajnjimpotroπaËima za Ëiju se naklonost bore.8.1.7 Agent regulatorAgent regulator postavlja razliËita ograniËenja zavrijeme odvijanja simulacije poput ograniËavanjavremena izmeappleu uzastopnog mijenjanja cijenaenergije ili postavljanja vrπne cijene za energijuËime πtiti krajnje potroπaËe od neopravdano visokihcijena.ers is not in question). The action of these agentsstimulates competitive behavior on the market sinceit makes a wide choice of various offers available tothe final customers, for whose favor they compete.8.1.7 Regulatory agentThe regulatory agent imposes various restrictions duringthe simulation such as restrictions on the durationbetween successive movements in energy pricesor imposing price-caps for energy, thereby protectingthe final consumers from unjustifiably high prices.8.1.8 Agent operator træiπtaAgent operator træiπta odreappleuje cijene na træiπtu,ovisno o kojoj je organizaciji træiπta rijeË (træiπteza dan unaprijed, spot træiπte i dr.). On zapravopreuzima ulogu burze elektriËne energije jerpokuπava predvidjeti potroπnju, traæi ponude i imamoguÊnost prihvatiti te ponude.8.1.8 Market operator agentThe market operator agent determines the prices onthe market, depending on the type of market organization(day-ahead market, spot market etc.). It actuallyassumes the role of a wholesale electricity market becauseit attempts to predict consumption, find offersand has the capability of accepting these offers.8.2 Osnovni i sintetski agentiOvisno o podruËju djelovanja mogu se razlikovatidvije vrste agenata potrebne za opis elektroenergetskogsektora a to su [12]:8.2 Basic and synthetic agentsDepending on the area of action, it is possible todistinguish two types of agents needed for the descriptionof the energy sector, as follows [12]:∑∑osnovni agenti,sintetski agenti.∑∑basic agents,synthetic agents.Osnovni agenti su elementarni agenti opisaniskupom statiËkih i dinamiËkih parametara kao i snekim posebnim sposobnostima vezanim uz proraËune,komunikaciju i razumijevanje na temeljusvog znanja.Sintetski agent je agent nastao kombinacijom nekihosnovnih agenata koji djeluju pod njegovomkontrolom i to prema specificiranim strategijama.Skup osnovnih agenata ukljuËuje:∑ potroπaËa, C,∑ proizvoappleaËa, G,∑ operatora prijenosnog sustava, O,∑ operatora distribucijskog sustava, D,∑ operatora træiπta, M,∑ veletrgovca elektriËne energije, W,∑ trgovca, R,∑ regulatorno tijelo, T.Kako bi se mogle opisati i uzeti u obzir situacijekada jedan te isti træiπni sudionik ima viπe funkcija(npr. ako je jedan træiπni subjekt istodobno i operatordistribucijskog sustava i trgovac elektriËnomenergijom) koriste se sintetski agenti. Sintetskiagenti se sastoje od osnovnih agenata i koordinirajuakcijama svojih komponenti koje su usmjereneBasic agents are elementary agents described by aset of static and dynamic parameters, with somespecial abilities connected with computation, communicationand reasoning on the basis of theirknowledge.A synthetic agent is formed from a combination ofseveral basic agents that act under its control accordingto specified strategies.The set of basic agents includes the following:∑ consumer, C,∑ generator, G,∑ transmission system operator, O,∑ distribution system operator, D,∑ market operator, M,∑ trader/broker or wholesaler, W,∑ retailer, R,∑ regulatory body, T.In order to be able to describe and take into considerationa situation when one of these marketsubjects has several functions (e.g., if one marketsubject is simultaneously the distribution systemoperator and the electricity retailer), syntheticagents are used. Synthetic agents consist of basicagents and coordinate the actions of all their com-667Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

k nekom specifiËnom cilju, a odvijaju se u skladus nekom strategijom. Ipak interni, osnovni agentiostaju samostalni u pogledu vlastitih posebnosti isposobnosti.Sintetski agenti samo vrπe interakciju sa svojimsastavnim osnovnim agentima ili s drugim kombiniranimagentima, ali ne i sa svim drugim osnovnimagentima. Ukoliko se udruæuje viπe agenataiste klase nastaje posebna vrsta sintetskog agentakoja se predoËava kao jedan Ëlan.Postoji, dakle, skup od osam osnovnih agenata:ponents that are oriented toward a specified goaland according to specific strategies. Nonetheless,internal basic agents remain independent in respectto their own attributes and abilities.Synthetic agents only interact with their componentbasic agents or other synthetic agents but not withall other basic agents. If several agents of the sameclass become associated, a special type of syntheticagent occurs that is represented by a singleton.Therefore, the set of eight basic agents is asfollows:.(8)UzimajuÊi u obzir da je uloga regulatora iskljuËiva,tj. da jedan agent moæe predstavljati regulatornotijelo i niπta drugo, preostaje sedam ostalih osnovnihagenata pomoÊu kojih se mogu tvoriti sintetskiagenti. Taj skup je sljedeÊi:Taking into account that the role of the regulator isexclusive, i.e. that an agent can represent a regulatorybody and nothing else, seven basic agentsremain, with which it is possible to create syntheticagents. This set is as follows:.(9)Tako je ukupan broj razliËitih sintetskih agenatajednak:Thus, the total number of various synthetic agentsequals the following:.(10)Takvi sintetski agenti su sintetski agenti prvogstupnja. Meappleusobnom sintezom sintetskih agenataprvog stupnja nastaju sintetski agenti drugogstupnja. Hibridni sintetski agenti nastaju sintezomsintetskih agenata prvog i drugog stupnja.Tradicionalno, struktura EES-a bila je vertikalnointegrirana tako da su se svi procesi od proizvodnje,prijenosa i distribucije elektriËne energijeodvijali unutar iste kompanije. Mana ovakvogsustava je moguÊnost prebacivanja Ëitavog rizika(npr. rizik od preinvestiranosti u kapacitete EES-a)na potroπaËe.Na slici 10 nalazi se osnovni model EES-a koji sesastoji od agenata i njihove meappleusobne interakcijeprikazane njihovim shemama i komunikacijom.Such combined agents are degree 1 syntheticagents. Through mutual synthesis of degree 1 syntheticagents, degree 2 agents are formed. Hybridsynthetic agents are formed through the synthesisof degree 1 and degree 2 synthetic agents.Traditionally, the power system structure was verticallyintegrated so that all the processes of the generation,transmission and distribution of electricityoccurred within the same company. A shortcomingof this system is the possibility of passing on theentire risk (e.g., the risk of overinvestment in powersystem capacities) to the consumers.Figure 10 is a basic model of an power system thatconsists of agents and their mutual interactions,represented by their schemes and communication.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675668

Osnovni model postavlja okvire za izgradnju specifiËnihmodela elektroenergetskog sustava. Træiπnimodel je podmodel elektroenergetskog modela iodnosi se na transakcije vezane uz elektriËnu energijukao i na komercijalne aktivnosti na træiπtu.Model ukljuËuje agente i njihovu meappleusobnuinterakciju. Glavni elementi træiπta su sudionicitræiπta, teritorij, træiπna pravila, transakcije, proizvodi/uslugete fiziËko ili virtualno okruæenje.The basic model establishes frameworks for theconstruction of specific energy system models. Themarket model is a sub-model of the electricity modeland refers to transactions in connection with electricitysuch as commercial activities on the market.The model includes agents and their mutual interaction.The main market elements are market participants,territory, market rules, transactions, products/services, and physical or virtual environment.TMOSlika 10Osnovni model EES-aprikazan agentima injihovom meappleusobnominterakcijomFigure 10Basic model of a powersystem presented viaagents and their mutualinteractioni = 1 …. nj = 1 …. mq = 1 …. q rk = 1 …. pz = 1 …. z wG jC iD kR qW zU okviru modela, operator sustava, operatortræiπta, proizvoappleaËi i veletrgovci meappleusobno vrπeinterakciju kao osnovni agenti na razini trgovineelektriËnom energijom na veliko (od srednjeg kviπem naponu). Istodobno su operatori distribucijskihsustava, trgovci i potroπaËi u sliËnoj interakciji,ali na razinama napona od niæeg premasrednjem. Trgovci kupuju elektriËnu energiju odveletrgovaca, a krajnji potroπaËi kupuju pak energijuod trgovaca.Sintetski agent se stvara recimo u trenutku kadakrajnji potroπaË æeli kupiti elektriËnu energiju natræiπtu, tj. od veletrgovca i rijeË je o {C,R} sintetskomagentu (potroπaË i trgovac).MoguÊa je i situacija da potroπaË (ukoliko muje dovoljno velika potroπnja) kupuje elektriËnuenergiju direktno od proizvoappleaËa. Tada je rijeË o{C,R,W} sintetskom agentu. Na analogan naËinse modeliraju svi drugi multi-funkcionalni agentiunutar zadane arhitekture.Within the framework of the model, the system operator,market operator, generators and wholesalersengage in mutual interactions as basic agents onthe wholesale level of electricity commerce (mediumto high voltage). At the same time, the distributionsystem operators, retailers and consumers areengaged in similar interaction but at the level of lowto medium voltage. The retailers purchase electricityfrom the wholesalers, and the final consumerspurchase energy from the retailers.A synthetic agent is created, let us say, at the momentwhen the final consumer wants to buy electricityon the market, i.e. directly from the wholesaler,a {C,R} synthetic agent (consumer and retailer).A possible situation is that the consumer (if theconsumer has sufficiently high consumption) purchaseselectricity directly from the generator. Thisis a {C,R,W} synthetic agent. All the other multifunctionalagents within a given architecture aremodeled in an analogous manner.669Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

9 TRENUTA»NO DOSTUPNIALATIBuduÊi da prouËavanje agenata spada u relativnomladu znanstvenu disciplinu, na træiπtu nijeprisutan velik broj agentski zasnovanih aplikacijakojima bi se moglo simulirati træiπte elektriËneenergije. Ali to podruËje je interesna zona brojnihistraæivanja i uËinjeni su znaËajni pomaci. Cilj ovakvihalata jest opremiti sudionike træiπta moÊnimpomoÊnim sredstvom pri donoπenju vaænih,strateπkih odluka.Izmeappleu svih simulatora træiπta elektriËne energijezasnovanih na viπeagentskom sustavu, daleko jenajnapredniji i najrazvijeniji EMCAS.9 CURRENTLY AVAILABLETOOLSSince the study of agents belongs to a relativelyyoung scientific discipline, there are not manyagent-based applications on the market that can beused to simulate electricity markets. However, thisarea is an interesting zone for numerous investigationsand significant strides have been made. Thegoal of these tools is to equip market participantswith powerful means to help them make importantstrategic decisions.Among all the electricity market simulators basedupon the multi-agent system, the most advancedand developed is the EMCAS.9.1 EMCAS simulacijski modelZa razliku od konvencionalnih metoda i alata zaanalizu elektroenergetskog sektora, EMCAS (engl.Electricity Market Complex Adaptive System) sustavne zasniva se na modelu unutar kojeg samojedan subjekt donosi odluke i u kojem postojejedinstveni ciljevi za Ëitav sustav. Naprotiv, agenti,koje ovaj sustav podræava, imaju svoje vlastiteciljeve i vlastite naËine donoπenja odluka (decentraliziranorazmiπljanje) [13]. Koristi se dakle modeliranjei simulacija na razini agenta (sudionika)(ABMS, Agent Based Modeling and Simulation).EMCAS je razvijen od strane CEEESA (engl.Argonne’s Center for Energy, Environmental andEconomic, & Systems Analysis) [14]. Unutarsloæenog adaptivnog sustava (CAS, ComplexAdaptive System), simuliraju se agenti koji uËena temelju njihovih proπlih iskustava (znanje) isposobni su promijeniti vlastito ponaπanje ovisnoo novonastalim prilikama (inteligencija).Tako agenti mogu stalno prilagoappleavati svoje strategijeovisno o uspjehu proπlih napora i pokuπaja. Inteligencija,odnosno sposobnost uËenja dostupnaje svakom agentu putem metode genetskih algoritama(engl. Genetic Algorithms, GA). Svi agentiimaju vlastite skupove zadataka, naËine donoπenjaodluka i obiljeæja ponaπanja. U samom procesuodluËivanja agenti se mogu osloniti na niz proπlihinformacija (proπle cijene elektriËne energije npr.)ali i na neke predviappleene podatke (potroπnja za danunaprijed npr.) (slika 11).9.1 EMCAS simulation modelUnlike conventional methods and tools for analyzingthe electricity sector, the Electricity MarketComplex Adaptive System (EMCAS) is not basedupon a model in which only one subject makesdecisions and in which there is a single goal forthe entire system. On the contrary, the agents thatthis system supports have their own goals and theirown ways of making decisions (decentralized decisionstructures) [13]. Agent-Based Modeling andSimulation (ABMS) are used.EMCAS was developed by Argonne’s Center forEnergy, Environmental and Economic SystemsAnalysis (CEEESA) [14]. Within the ComplexAdaptive System (CAS), agents are simulated thatlearn on the basis of their past experience (knowledge)and are capable of altering their own behavior,depending upon new situations (intelligence).Thus, agents can constantly adapt their strategies,depending on the success of past efforts and attempts.Intelligence, i.e. the ability to learn, isavailable to every agent via the method of GeneticAlgorithms (GA). All agents have their own sets oftasks, manner of decision making and characteristicbehavior. In the decision making process, agentscan rely on a series of past data (e.g., past electricityprices) but also on some predicted data, e.g.,next-day consumption (Figure 11).Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675670

pogled u buduÊnost (predviappleanje cijena i potražnje) /perception of future (forecast of prices and demand)Slika 11NaËin donoπenja odlukaagenta u EMCAS sustavuFigure 11Decision making by anagent in the EMCASVrijeme / Timepogled prema drugimagentima i suparnicima /Perception of otheragents and rivalsagent /agentnaËini odluËivanja /types of decisionmakingdonešene odluke /decision makinguvid u prošlost (memorija) /insight into the past (memory)S ovakvim pristupom orijentiranim na agente,EMCAS je posebno dizajniran za simulacijuviπeagentskih træiπta te omoguÊuje koriπtenje ianalizu regulatornih struktura prije no πto se krenukoristiti u stvarnom okruæenju.EMCAS provodi simulacije kroz πest slojeva odluËivanjaprema vremenskim razdobljima koja idu odjednog sata do dugoroËnog planiranja. Na svakomsloju planiranja agenti donose skup odluka kaonpr. odreappleivanje potroπnje (agenti koji predstavljajupotroπaËe), vozni red elektrana (proizvoappleaËi),bilateralni ugovori (proizvoappleaËi i opskrbljivaËi) iraspored rada (vozni red) elektrana (operator sustava).Agenti zatim primjenjuju vlastita pravilaodluËivanja te ocjenjuju koliko dobro ta pravilaispunjavaju njihove ciljeve.Model ukljuËuje velik broj razliËitih agenata srazliËitim zadaÊama (razliËitih sudionika træiπtaelektriËne energije) kako bi se πto bolje prikazalai doËarala kompleksnost i heterogenost sudionikatræiπta. Dostupni su agenti koji predstavljaju proizvodnetvrtke (GenCos), prijenosne tvrtke (TransCos),distribucijske tvrtke (DistCos), nezavisnioperatori sustava (ISOs) ili organizatori regionalnogprijenosa (RTOs), tvrtke opskrbljivaËi (Dem-Cos), potroπaËi i regulatori. Ovi agenti su visokospecijalizirani kako bi obavljali razliËite zadatke,a kako bi se omoguÊila takva specijalizacija agentiposjeduju detaljizirana pravila. Isto tako korisnik(analizator) moæe definirati nova pravila i strategijete promotriti posljedice tih novih strategija natræiπne prilike.Svaki agent ima svoju djelatnu strategiju putemkoje odluËuje o tome kada i kako djelovati te nakoje naËine odrediti iznos cijene. Ove strategijese mogu mijenjati, dakle nisu statiËne. Naprotiv,With such an agent-oriented approach, EMCAS isespecially designed for the simulation of multiagentmarkets and facilitates the use and analysisof regulatory structures before they are applied inthe real environment.EMCAS performs simulation through six layers ofdecision making according to time periods thatrange from one hour to long-term planning. In everyplanning layer, agents make a set of decisions suchas, for example, determining consumption (agentswho represent consumers), power plant schedules(generators), bilateral contracts (generators andretailers) and the power plant schedule (systemoperator). Agents apply their own decision-makingrules and assess how well these rules meet theirgoals.The model includes a large number of various agentswith various tasks (electricity market participants)in order to present and conjure the complexity andheterogeneity of the market participants as well aspossible. Agents are available that represent generationcompanies (GenCos), transmission companies(TransCos), distribution companies (DistCos),independent system operators (ISOs) or regionaltransmission operators (RTOs), demand companies(DemCos), consumers and regulators. These agentsare highly specialized in order to perform varioustasks and have detailed rules in order to facilitatesuch specialization. Similarly the user (analyst) candefine new rules and strategies and observe theconsequences of these new strategies under marketconditions.Each agent has its own strategy, according to whichit decides when and how to act and how to set prices.These strategies can change and are not static.On the contrary, it is desirable for agents to change671Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

æelja i jest da agenti mijenjaju svoje strategije naosnovi steËenog znanja i adaptivnih sposobnosti.U EMCAS simulacijskom modelu, agenti uËe ovlastitom træiπnom ponaπanju kao i o ponaπanjimai akcijama drugih agenata putem dva zasebnanaËina uËenja:their strategies on the basis of knowledge acquiredand adaptive abilities.In the EMCAS simulation model, the agents learnabout their own market behavior as well as the behaviorand actions of other agents in two separatetypes of learning:∑∑uËenje zasnovano na promatranju,uËenje zasnovano na istraæivanju.∑∑observation-based learning,exploration-based learning.Pod uËenjem zasnovanim na promatranju, podrazumijevase da agenti razmatraju svoja proπla djelovanjai na osnovi uspjeπnosti tih djelovanja odluËujuo promjeni strategije, njenom zadræavanju ili samonekim preinakama u strategiji. UËenje temeljeno naistraæivanju omoguÊuje agentima da istraæuju i analizirajumoguÊe træiπne strategije te strategije ponude.Ukoliko agent pronaapplee zadovoljavajuÊu strategijuon ju poËinje primjenjivati te fino podeπavatiovisno o novim træiπnim prilikama.Under observation-based learning, it is understoodthat the agents consider their past actionsand on the basis of the success of these actionsdecide upon changing the strategy, retaining it oronly making some adjustments. Exploration-basedlearning allows agents to explore and analyze possiblemarket strategies and supply strategies. Ifan agent finds a satisfactory strategy, it begins toapply and refine it, depending on the new marketconditions.9.2 Interakcijske razine EMCAS modelaInterakcijske razine predstavljaju zapravo multidimenzionalnookruæenje unutar kojeg agentiobavljaju svoje djelatnosti. Agenti djeluju unutarnekoliko meappleusobno povezanih razina, a to su fizikalnarazina, nekoliko poslovnih razina te najviπa,regulatorna razina (slika 12).9.2 Interaction layers of the EMCAS modelInteraction layers actually represent a multidimensionalenvironment within which agents operate.Agents operate within several interconnected layersand these are the physical layer, several businesslayers and the highest layer, the regulatory layer(Figure 12).Slika 12Interakcijske razine uEMCAS sustavuFigure 12Interaction layers inthe EMCASRegulator / RegulatorRegulatorni sloj / Regulatory layerInformacije / InformationPoslovni sloj 1 − burza / Business layer 1 − pool marketsPoslovni sloj 2 − bilateralni ugovori / Business layer 2 − bilateral contract marketInformacije / InformationPoslovni sloj 3 − T&D tvrtke / Business layer 3 − T&D companiesFizikalni sloj − stvarni tok elektriËne energije / Physical Layer − actual load flowFizikalna razina sastoji se od agenata kojiobuhvaÊaju fiziËku proizvodnju, prijenos, distribucijui potroπnju elektriËne energije. PotroπaËi iproizvoappleaËi meappleusobno povezani prijenosnim vodovimaËine fizikalni dio træiπta elektriËnom energijom.Agent koji predstavlja ISO, odnosno RTO ufizikalnoj razini odgovoran je za povezivanje proizvodnjei potroπnje te za prilagoappleavanje promjenau potroπnji ili proizvodnji.Physical layers consist of agents, which include thephysical generation, transmission, distribution andconsumption of electrical energy. Consumers andgenerators interconnected by transmission linescomprise the physical part of the electricity market.An agent that represents an ISO or RTO in aphysical layer is responsible for linking generationand consumption and for adjusting to changes inconsumption or generation.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675672

Na slici 12 mogu se vidjeti i tri poslovne razinekoji zajedno predstavljaju poslovnu stranu træiπta.Na ovoj razini agenti proizvoappleaËi promiπljaju ouporabi svojih resursa, agenti opskrbljivaËi kupujuelektriËnu energiju od agenata proizvoappleaËa i prodajuje krajnjim potroπaËima.Agenti opskrbljivaËi mogu kupovati energiju odproizvoappleaËa bilo na burzi, kojom upravlja ISO/RTO,bilo putem bilateralnih ugovora koji se dogovarajuprivatno izmeappleu dva agenta. Razina prijenosnih idistribucijskih tvrtki je dizajnirana kako bi se uzelo uobzir vlasniπtvo nad tim sustavima kao i potraæivanjaovih tvrtki za koriπtenje njihovih postrojenja.Regulator je agent koji se nalazi u regulatornoj razini.On postavlja træiπna pravila i nadgleda træiπnazbivanja.9.3 MoguÊnosti simulacijskog modela EMCASSustav je u stanju simulirati tri vrste træiπta, a tuspadaju bilateralni ugovori, burza elektriËne energijei træiπte pomoÊnih usluga. U globalu bilateralniugovori se sklapaju izmeappleu jednog agentaproizvoappleaËa i jednog agenta opskrbljivaËa i mogubiti na bazi jednog sata ili nekoliko godina.U sluËaju trgovine elektriËne energije na burzi,agenti prilaæu svoje ponude i potraænje ISO agentu.Postoje dvije vrste formiranja cijene na burzi.U prvom naËinu svim se agentima plaÊa jednakacijena koja se dobiva iz ukupne potraænje i ponudeelektriËne energije. U drugom naËinu agentimaproizvoappleaËima isporuËena energija biva plaÊenapo ponuappleenoj cijeni (engl. pay as bid).U modelu se simuliraju tri tipa pomoÊnih træiπta:regulacija, rotirajuÊa rezerva i dodatna rezerva.EMCAS omoguÊava predviappleanje cijena elektriËneenergije i to poËevπi od satne pa sve do vremenskerazine od nekoliko godina [15]. Na taj naËin moæese analizirati bilo kratkoroËna bilo dugoroËna pozicijasudionika na træiπtu. Tako je EMCAS alat kojipomaæe pri upravljanju rizicima i odreappleivanju optimalnestrategije træiπnog nastupa. Pri tome je posebnapaænja posveÊena proizvoappleaËima elektriËneenergije kao sudionicima koji svojim strateπkimodlukama i potezima mogu ispitivati vlastitutræiπnu moÊ (fiziËko i ekonomsko suzdræavanje).Dodatna moguÊnost EMCAS-a jest dugoroËnoplaniranje za nekoliko godina. Decentraliziranimodlukama pojedinaËni agenti na temelju svojihinternih ciljeva planiraju izgradnju novih proizvodnihkapaciteta. ProizvoappleaË elektriËne energije natemelju svoje trenutne træiπne pozicije, sklonostiriziku i poslovnih ciljeva moæe ispitati i rangiratiFigure 12 shows three business layers that togetherrepresent the business side of the market. At thislevel, generator company agents decide upon theuse of their resources and demand company agentspurchase electricity from generator company agentsand sell it to the final consumers.Demand company agents can purchase energy fromgenerators either on a market administered by anISO/RTO or via bilateral contracts that are enteredprivately between two agents. The layer of the transmissionand distribution companies is designed totake the ownership of these systems into account aswell as the demand for the use of their facilities.The regulator is the agent in the regulatory layer. Itsets the market rules and monitors market events.9.3 EMCAS Model SimulationThe system can simulate three types of markets, includingbilateral contract, pool energy and ancillaryservices. Generally, bilateral contracts are enteredbetween a single generator company agent and asingle demand company agent, and can be basedupon one hour or several years.In the case of the pool energy market, agents submittheir buy and sell bids to the ISO agent. Thereare two types of price formation on the pool energymarket. In the first type, all the agents are paid thesame price, which is obtained from the total electricitysupply and demand. In the second type, generatorcompany agents delivers energy at the pricethat it bids, known as Pay-as-Bid.In the model, three ancillary services markets aresimulated: regulation, spinning reserve and replacementreserve markets.EMCAS facilitates the forecasting of electricityprices, ranging from hours to several years [15]. Inthis manner, it is possible to analyze both shorttermand long-term participant positions on themarket. Thus, EMCAS is a useful tool in risk managementand determining optimal strategies formarket performance. Particular attention is devotedto generators of electricity as participants who withtheir strategic decisions and moves can exploretheir own market power (physical and economicwithholding).An additional possibility of the EMCAS is long-termplanning for several years. Through decentralizeddecisions, individual agents plan the constructionof new production facilities based upon their internalgoals. An electricity generator company canreview and rank the available production optionsbased upon its current market position, risk profile673Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

dostupne opcije proizvodnje kojim bi eventualnoproπirio svoje proizvodne kapacitete.and business goals, according to which it couldeventually expand its generating capacities.10 ZAKLJU»AKNakon svega navedenog, moæe se gotovo sasigurnoπÊu ustvrditi da su klasiËni simulacijskimodeli træiπta elektriËne energije, temeljeni namatematiËkim modelima, naπli i viπe nego dostojnuzamjenu i da moraju svoje mjesto prepustitinovoj snaænoj i obeÊavajuÊoj metodi temeljenoj naviπeagentskom modelu træiπta. Takvim pristupommoguÊe je u simulaciji predstavljati i promatratisvakog træiπnog pojedinca i za svakog od njihpronaÊi optimalnu træiπnu strategiju.Kao i svaki drugi simulacijski model i ovaj zasnovanna agentima donosit Êe to bolje rezultate πtoje detaljnije i preciznije prikazano stanje realnogpromatranog træiπta. Naime svako træiπte ima svojespecifiËnosti po kojima se razlikuje od drugih,bilo da je rijeË o veliËini træiπta, udjelu elektriËneenergije dobivene iz razliËitih izvora ili pak zakonskimregulativma. Onaj træiπni sudionik koji budeposjedovao kvalitetniji simulacijski model koji jeprimjereno prilagoappleen spomenutim specifiËnostima,sasvim sigurno Êe biti u velikoj prednosti predostalom træiπnom konkurencijom.10 CONCLUSIONIn light of that which has been presented, it can besaid that classical simulation models for electricitymarkets, based upon mathematical models, nowhave a more than suitable replacement and theymust relinquish their place to the new, powerfuland promising methods based upon the multi-agentmarket model. With this approach, it is possibleto present and review every individual market participantin simulation and find the optimal marketstrategy for each one.As with every other simulation model, the quality ofthe results provided by this agent-based model willdepend upon the degree of detail and precision ofthe presentation of the real state of the investigatedmarket. Each market has its own specific characteristicsthat differentiate it from others, whether concerningthe market size, percentage of electricityobtained from various sources or legal regulations.The market player in possession of a quality simulationmodel adapted to these specificities will certainlyhave an advantage over market competition.Rajšl, I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675674

LITERATURA / REFERENCES[1] SAMUELSON, P. A., NORDHAUS, W.D., Economics, 15. izdanje, McGraw-Hill, Inc., hrvatski prijevod,izdavaË MATE, Zagreb, 2000.[2] WANGESTEEN, I., Power Markets, Predavanja iz kolegija Uvod u træiπte energentima, Fakultet elektrotehnikei raËunarstva, Zagreb, 2006.[3] STEFANOV, P. »., Eksploatacija elektroenergetskih sistema, (Power point prezentacija), Beograd,2004.[4] FLORES-MENDEZ, R. A., Towards a Standardization of Multi-Agent System Frameworks, 1999.,http://www.acm.org[5] VLASSIS, V., A Concise Introduction to Multiagent Systems and Distributed AI, University of Amsterdam,Amsterdam, 2003[6] BOURGNE, G., Affect-based Multi-Agent Architecture, the University of Hull, Hull, September2003[7] SHEN, Z. ET AL, Goal Autonomous Agent Architecture, Nanyang Technological University, Singapore,2004[8] KU©EK, M., Agentski orijentirano programsko inæenjerstvo, (Power point prezentacija), Fakultet elektrotehnikei raËunarstva, Zavod za telekomunikacije, Zagreb, oæujak 2006.[9] STONE, P., VELOSO, M., Multiagent Systems: A Survey from a Machine Learning Perspective, AutonomousRobotics, vol. 8, No. 3, July 2000[10] KEKO, H., Inteligentni viπeagentski sustavi u simulaciji træiπta elektriËne energije, kvalifikacijski doktorskirad, Fakultet elektrotehnike i raËunarstva, Zavod za visoki napon i energetiku (ZVNE), Zagreb,2006.[11] NAING WIN OO, N., MIRANDA, V., Multi-energy Retail Market Simulation with Intelligent Agents,Faculty of Engineering of the University of Porto, Portugal, 2005[12] GNANSOUNOU, E. et al, A multi-agent approach for planning activities in decentralized electricitymarkets, Knowledge ∑ Based Systems, 2006.[13] GUENTER, E. et al, Multi-Agent Power Market Simulation using EMCAS, Argonne National Laboratory,Argonne, 2004[14] Argonne National Laboratory, Center for Energy, Environmental, and Economic Systems Analysis,http://www.dis.anl.gov[15] ANDRO»EC, I., PUZAK, D., KRAJCAR, S., Modeliranje elektroenergetskog sustava bazirano na multiagentskimalgoritmima, 8. savjetovanje HRO CIGRE, Cavtat, 2007.Uredniπtvo primilo rukopis:2007-11-29PrihvaÊeno:2007-12-07Manuscript received:2007-11-29Accepted:2007-12-07675Rajšl,I., Krajcar, S., Krpan, K., Primjena višeagentskih sustava u …, Energija, god. 56(2007), br. 6., str. 642-675Rajšl, I., Krajcar, S., Krpan, K., Application of Multi-Agent System in …, Energija, vol. 56(2007), No. 6, pp. 642-675

PRORA»UN GODIŠNJIHGUBITAKA RADNE ENERGIJEU DISTRI BUCIJSKOJMREŽI S PRIKLJU»ENOMVJETROELEKTRANOMCALCULATION OF ANNUAL ACTIVEENERGY LOSSES IN A DISTRIBUTIONNETWORK WITH A CONNECTEDWIND POWER PLANTDr. sc. Ranko GoiÊ, Damir Jakus, dipl. ing., Dr. sc. Eugen MudniÊ,SveuËiliπte u Splitu, Fakultet elektrotehnike, strojarstva i brodogradnje,R. BoπkoviÊa b.b, 21000 Split, HrvatskaPrikljuËak distribuiranih izvora energije na distribucijsku mreæu, izmeappleu ostalih utjecaja, ima zaposljedicu preraspodjelu tokova snaga u mreæi, a time i promjenu iznosa gubitaka radne energije.U odreappleenom trenutku, ovisno o potroπnji u mreæi i snazi koju u mreæu injektira vjetroelektrana,gubici radne snage mogu biti veÊi ili manji u odnosu na stanje bez prikljuËene elektrane. Meappleutim,trenutaËna vrijednost gubitaka radne snage nije bitna, veÊ prvenstveno ukupni gubici radne energijena godiπnjoj razini. Zbog toga je u radu razraappleena metodologija proraËuna godiπnjih gubitakaradne energije u distribucijskoj mreæi u uvjetima prikljuËenog distribuiranog izvora, pri Ëemu se polaziod poznatih kronoloπkih krivulja optereÊenja i proizvodnje. OdgovarajuÊi proraËuni su izvrπenina primjeru realne distribucijske mreæe i vjetroelektrane koja se planira prikljuËiti na nju.Connecting distributed generation to a distribution network affects the load flows in the network,resulting in the redistribution of load flows in the network and thereby changing the amountof active energy losses. At a given moment, depending upon the network consumption and thepower injected from a wind power plant into the network, the active power losses can be higheror lower than they were before the wind power plant was connected. However, the aggregateannual energy losses, rather than instantaneous active power losses, are of significance.Therefore, a methodology has been developed for calculating the annual active energy lossesin a distribution network in the case of a distributed generation connection, assuming theknown chronological consumption and production curves. The corresponding calculations havebeen performed for an existing distribution network and a wind power plant scheduled to beconnected to it.KljuËne rijeËi: distribuirani izvor, gubici radne energije, vjetroelektranaKey words: Distributed generation, wind power plant, active energy lossesGoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br.6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699676

1 UVODU proteklih 10-tak godina svjedoci smo relativnovelikog rasta udjela distribuiranih izvora elektriËneenergije, tj. elektrana prikljuËenih na distribucijskumreæu, πto je primarno rezultat:∑ otvaranja træiπta elektriËne energije, praÊenogodgovarajuÊom zakonskom regulativom zaslobodan pristup distribucijskoj mreæi, πtoomoguÊava privatnim (malim) investitorimaulaganje i izgradnju manjih elektrana,∑ poticajnih cijena elektriËne energije (najËeπÊeu obliku fiksnih i garantiranih otkupnih cijena)koje se takoappleer odgovarajuÊom zakonskomregulativom redovito propisuju za obnovljiveizvore energije i kogeneracijsku proizvodnju,∑ tehnoloπkog napretka, odnosno razvoja efikasnijihi jeftinijih elektrana, posebice onihmanjih nazivnih snaga za koje prije, bez prethodnedvije stavke, nije postojala dovoljnamotivacija za razvoj.U zadnjih nekoliko godina, posebno je intenzivanrast novih distribuiranih izvora ostvaren u vjetroelektranama.Razlog tome je bio prvenstvenoekonomske naravi, tj. najkraÊe vrijeme povratainvesticije u odnosu na ostale obnovljive izvore, alii ostali faktori, kao πto su zanemariv utjecaj naokoliπ, kratko vrijeme izgradnje (tipski proizvod),veliki potencijalni resursi energije vjetra itd. Iakotakvo stanje vrijedi i danas, tendencije razvoja vjetroagregataidu prema veÊim jedinicama i veÊimvjetroelektranama koje se u tom sluËaju prikljuËujuna prijenosnu mreæu.PrikljuËak elektrane na distribucijsku mreæu [1] i[2], bez obzira na snagu elektrane i naponsku razinu,odnosno mjesto prikljuËka, dovode do znaËajnihpromjena pogonskih znaËajki distribucijskemreæe, koje se mogu grupirati u stacionarne i dinamiËkeznaËajke. PromatrajuÊi samo stacionarneznaËajke, valja istaknuti dva osnovna aspekta:∑∑naponske prilike duæ radijalnog kraka distribucijskemreæe na koju je prikljuËen jedan ili viπedistribuiranih izvora,utjecaj na gubitke snage i energije u distribucijskojmreæi.Oba navedena aspekta mogu imati pozitivne inegativne utjecaje, ovisno o tipu i reæimu radadistribuiranog izvora s jedne strane, te reæimuoptereÊenja i tehniËkim karakteristikama distribucijskemreæe s druge strane. U ovom radu ukratkosu opisane teoretske osnove utjecaja distribuiranihizvora na gubitke snage i energije u distribucijskojmreæi, te moguÊnosti njihova izraËuna. Pri tome jeod primarnog znaËenja πto toËniji izraËun promje-1 INTRODUCTIONDuring the past decade, we have witnessed a relativelyhigh increase in the share of distributed generation,i.e. power plants connected to a distributionnetwork, primarily due to the following:∑ electricity market liberalization, accompaniedby the corresponding legislative framework foropen access to the distribution network, whichhas made it possible for private (small) investorsto invest in and build small power plants,∑ incentive electricity prices, most commonly inthe form of fixed and guaranteed feed-in tariffs,which are also regularly stipulated by the legislationgoverning renewable energy sources andcogeneration,∑ technological achievements, i.e. the developmentof more efficient and less expensivepower plants, especially those with lower nominalpower, for which there had been little inducementprior to the above-mentioned marketliberalization and price incentives.Recently the growth of new distributed power sourcesfrom wind power plants has been particularlyintensive. The reason for this is primarily of an economicnature, i.e. the shortest period of investmentreturn among the other renewables as well other factors,such as negligible environmental impact, shortconstruction time (standardized product), the greatpotential of wind energy resources etc. Although allthis still applies today, the trend in wind turbinedevelopment is toward large units and wind powerplants that are connected to a transmission network.The connection of a power plant to a distributionnetwork [1] and [2], irrespective of the rated power,voltage level and connection point of the powerplant, results in significant changes in the operatingfeatures of the distribution network, which canbe classified as either stationary or transient features.Regarding the stationary features, two basicaspects should be emphasized:∑∑the voltage profile along the distribution networkradial feeder, to which one or more distributedsources are connected,the impact on power and energy losses in thedistribution network.Both of these aspects can have positive or negativerepercussions, depending upon the type of distributedgeneration and operating conditions, on theone side, and the technical characteristics of thedistribution network and load conditions, on theother side. In this work, the theoretical foundationsof the basic impact of distributed generation uponpower and energy losses in a distribution networkGoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699678

ne (poveÊanja/smanjenja) gubitaka radne energijena godiπnjoj razini nakon prikljuËka distribuiranogizvora na distribucijsku mreæu. Takoappleer, vrloje znaËajno iznalaæenje prave razine kompromisaizmeappleu detaljnog toËnog proraËuna koji zahtijevaveliku koliËinu ulaznih parametara i aproksimativnihizraËuna koji traæe manji opseg ulaznih parametarauz bræi i jednostavniji proraËun, ali uzneπto veÊu greπku u proraËunu.Primjer takvog proraËuna dan je za realnu distribucijskumreæu i vjetroelektranu koja se planiraprikljuËiti na nju, na naËin da je napravljen detaljnii aproksimativni izraËun.2 GUBICI SNAGE I ENERGIJE UDISTRIBUCIJSKOJ MREŽI2.1 OpÊenito o gubicima snage i energije u distribucijskojmreæiPrema naËinu nastanka, gubici elektriËne snage ienergije [3] dijele se na:∑ tehniËke gubitke koji su direktna su posljedicastavljanja postrojenja pod napon i protjecanjaelektriËne struje kroz elemente elektriËnemreæe. VeliËina ovih gubitaka ovisi o tehniËkimkarakteristikama elemenata elektriËne mreæe.TehniËki gubici se mogu vrlo toËno ustanovitimjerenjem ili proraËunom na odgovarajuÊemraËunalnom modelu, iako obje metode mogubiti vrlo zahtjevne. Mjerenje zahtijeva instaliranjemjernih ureappleaja koje se uobiËajeno nekoriste, tako da implicira dodatne troπkove.Mjerenjem se ne mogu ustanoviti gubici radneenergije u fazi planiranja prikljuËka distribuiranogizvora, dok je nakon prikljuËka tomoguÊe, ali samo uz dodatne raËunske tehnikekojima je moguÊe sa zadovoljavajuÊom razinomtoËnosti ustanoviti promjenu u odnosuna stanje bez distribuiranog izvora. Meappleutim,opÊenito mjerenjem nije moguÊe ustanovititehniËke gubitke u mreæi, prvenstveno zbogutjecaja komercijalnih gubitaka,∑ komercijalne (netehniËke) gubitke koji nastajuzbog neispravnosti mjernih ureappleaja ineovlaπtene potroπnje elektriËne energije.Redovito ih nije moguÊe direktno prepoznati ialocirati, veÊ samo izraËunati kao razliku ukupnoregistriranih gubitaka (razlika obraËunateelektriËne energije na mjestima preuzimanja iisporuke elektriËne energije) i izraËunatih tehniËkihgubitaka pomoÊu raËunalnog modela.TehniËki gubici, s obzirom na naËin nastankamogu se podijeliti na [4]:are briefly described, as well as possibilities fortheir calculation. Following the connection of a distributedpower plant to a distribution network, it isof primary importance to calculate the changes (increases/decreases)in energy losses on the annuallevel as precisely as possible. Furthermore, it is veryimportant to find the right level of compromise betweendetailed and precise calculation, requiring alarge number of input parameters, and approximatecalculation, requiring fewer input parameters andfacilitating faster and simpler calculation but with asomewhat higher margin of error.An example is provided for an existing distributionnetwork and wind power plant scheduled to be connectedto it, for which both detailed and approximatecalculations have been performed.2 POWER AND ENERGY LOSSESIN DISTRIBUTION NETWORKS2.1 Power and energy losses in distribution networksPower and energy losses are classified according tothe manner in which they occur [3], as follows:∑ technical losses are a direct consequence ofenergizing electrical facilities and current flowsthrough the electrical network elements. Themajority of these losses depend upon the technicalcharacteristics of the network elements. Technicallosses can be determined very preciselyby measurement or calculation using a suitablecomputer model, although both methods canbe highly demanding. Measurement requiresthe installation of measuring equipment, whichis generally not used and therefore impliesadditional costs. Measurement cannot determineactive energy losses during the preparatoryphase of a distributed generation connection.Measurement can be used after connection buta satisfactory level of accuracy in determiningchange in comparison to the situation withoutthe distributed source requires the use of additionalcalculation techniques. However, it isgenerally not possible to determine technical lossesin a network through measurement, primarilydue to the influence of commercial losses,∑ commercial (non-technical) losses occur dueto the malfunctioning of the measuring equipmentand illegal electricity consumption. Theycannot be recognized and allocated by regularmeans but only as the difference between thetotal registered losses (the difference betweenthe accounted energy at delivery points and theenergy delivered to final customers) and the679GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

∑naponski ovisni gubici (tzv. gubici neovisni ooptereÊenju) nastaju na popreËnim elementimaekvivalentne sheme promatranog elementaelektriËne mreæe. TipiËan i najznaËajniji primjersu gubici praznog hoda transformatorakoji su u ekvivalentnoj shemi predoËeni popreËnomgranom. OpÊeniti izraz za naponskiovisne gubitke snage dan je izrazom:technical losses calculated using a computermodel.Technical losses are classified according to themanner in which they occur [4], as follows:∑voltage-dependent losses (so-called load-independentlosses) occur in parallel elements ofthe network under consideration. The typicaland most significant examples are transformeriron losses, shown by a cross branch in equivalentschemes. The general expression for voltage-dependentlosses is as follows:(1)gdje je:S pop ∑ gubici prividne snage u popreËnojgrani elementa mreæe,P pop ∑ gubici radne snage u popreËnojgrani elementa mreæe,Q pop ∑ gubici jalove snage u popreËnojgrani elementa mreæe,V ∑ fazni napon elementa mreæe,Y ∑ popreËna admitancija elementamreæe.Ukupni naponski ovisni gubici snage u promatranomdijelu distribucijske mreæe danisu sumiranjem izraza (1) po svim Ëvorovimamreæe (N):where:S pop ∑ apparent power losses in a parallelnetwork element branch,P pop ∑ active and reactive power losses in aparallel network element branch,Q pop ∑ reactive power losses in a parallelnetwork element branch,V ∑ phase voltage of the networkelements,Y ∑ parallel admittance of the networkelements.The total voltage-dependent power losses in thepart of the distribution network under considerationis obtained as the sum of expression (1)for all the network nodes (N):(2)∑ Strujno ovisni gubici (tzv. gubici ovisni ooptereÊenju) nastaju na uzduænim elementimaekvivalentne sheme promatranog elementamreæe. TipiËni i najznaËajniji primjeri su gubiciu vodiËima zraËnih, odnosno kabelskihvodova i namotima transformatora. U ekvivalentnojshemi predoËeni su uzduænom granomelementa mreæe. OpÊeniti izraz za strujno ovisnegubitke snage dan je izrazom:∑ Current-dependent losses (so-called load-dependentlosses) occur in series elements of thenetwork equivalent scheme considered. The typicaland most significant examples are lossesin the conductors of overhead and cable powerlines and transformer windings. In an equivalentscheme, they are represented by a series branchof the network element. The general expressionfor current-dependent losses is as follows:(3)GoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699680

gdje je:S uzd ∑ gubici prividne snage u uzduænojgrani elementa mreæe,P uzd ∑ gubici radne snage u uzduænoj granielementa mreæe,Q uzd ∑ gubici jalove snage u uzduænoj granielementa mreæe,I ∑ struja koja protjeËe kroz elementmreæe,Z ∑ uzduæna impedancija elementa mreæe.Ukupni strujno ovisni gubici snage u promatranomdijelu distribucijske mreæe dani su sumiranjemizraza (3) po svim granama mreæe (M):where:S uzd ∑ apparent power losses in a seriesbranch of the network element,P uzd ∑ active power losses in a series branchof the network element,Q uzd ∑ reactive power losses in a seriesbranch of the network element,I ∑ network element current,Z ∑ network element series impedance.The total current-dependent power losses in thepart of the distribution network under considerationis obtained as the sum of expression (3)for all the network branches (M):(4)U razmatranju utjecaja distribuiranog izvora na gubitkesnage u distribucijskoj mreæi, popreËni gubicise mogu zanemariti, bez obzira πto se naponskeprilike u mreæi ipak mijenjaju. Naime, naponi umreæi u realnom sluËaju trebali bi biti u granicama±5 %, πto prema (1) znaËi moguÊe varijacijenaponski ovisnih gubitaka u granicama ±10 %. Toje daleko manje od moguÊih varijacija gubitakaovisnih o strujnom optereÊenju mreæe koje moæevarirati u rasponu 0 do 100 %. Zbog toga se daljeu tekstu promatraju samo strujno ovisni gubici, paje iz oznaka izostavljen indeks uzd koji je oznaËavaokomponentu uzduænih gubitaka, πto se daljepodrazumijeva.Izrazi (3) i (4) mogu se izraziti i preko snage kojaprolazi kroz element mreæe, tj.:Cconsidering the impact of the distributed sourceon the power losses in the distribution network,voltage-dependent losses can be ignored, althoughthe voltage conditions in the network nonethelesschange. Voltages in the existing situation should bewithin the limits of ±5 %, which according to (1)signify possible voltage-dependent power loss variationwithin the limits of ±10 %. This is far less thanthe possible variations in loss, depending upon thenetwork current loads that can vary within a rangeof 0 to 100 %. Therefore, only the current-dependentlosses will be considered henceforth. The superscriptuzd , which indicates the parallel power losscomponent, will not be written in the expressionsbut should be assumed.The expressions (3) and (4) can also be expressedas power through the network element, i.e.:(5)(6)gdje je:P ∑ radna snaga koja protjeËe kroz elementmreæe,Q ∑ jalova snaga koja protjeËe kroz elementmreæe,R ∑ radni otpor elementa mreæe,X ∑ induktivni otpor elementa mreæe,U ∑ linijski napon elementa mreæe.where:P ∑ active power through the network element,Q ∑ reactive power through the networkelement,R ∑ the resistance of the network element,X ∑ the reactance of the network element,U ∑ the phase-to-phase voltage of the networkelement.681GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

Tokovi radnih i jalovih snaga kroz elemente elektroenergetskemreæe, te naponi svih Ëvorova, dobijuse standardnim proraËunom tokova snaga i naponskihprilika, pod pretpostavkom da su poznatitehniËki parametri mreæe, topologija i uklopnostanje, te optereÊenje mreæe u svakom Ëvoru. Redovitose u programima za proraËun tokova snagaimplementira i proraËun za automatsko izraËunavanjegubitaka radne i jalove snage za svaki elementmreæe, te eventualno zbirno po naponskimnivoima, istovrsnim elementima mreæe, distribucijskim(napojnim) podruËjima, izvodima itd. [5]i [6]. Zbog toga je proraËun gubitaka snage u distribucijskojmreæi zapravo trivijalan zadatak, kojegje za manju distribucijsku mreæu moguÊe izvestiËak i ruËno. Meappleutim, poznavanje ∑ izraËun gubitakasnage u distribucijskoj mreæi za promatranooptereÊenje, opÊenito nije dovoljan pokazatelj, bezobzira πto se moæe izvesti primjerice za dva krajnjakarakteristiËna stanja maksimalnog i minimalnogoptereÊenja. Pravi znaËajni pokazatelj su gubicienergije u odreappleenom vremenskom razdoblju, primjericeu jednoj kalendarskoj godini.ToËan izraËun gubitaka elektriËne energije znatnousloænjava proraËun i to najmanje iz dva(meappleusobno povezana) razloga:∑∑potrebno je poznavati karakteristiku potroπnjeu svim Ëvorovima promatrane distribucijskemreæe, tj. kronoloπke krivulje optereÊenja linearnodiskretizirane na veÊe ili manje vremenskepodintervale (primjerice sate) ili odgovarajuÊekrivulje trajanja optereÊenja. Prva aproksimacija,bez veÊih utjecaja na greπku proraËuna,moæe se napraviti i koriπtenjem navedenihkrivulja promatrane distribucijske mreæe ucjelini, uz odgovarajuÊu raspodjelu potroπnjepo Ëvorovima koja odgovara potroπnji u trenutkuvrπnog optereÊenja ili u trenutku za kojeraspolaæemo izmjerenim podacima optereÊenjasvih Ëvoriπta,s obzirom na promjenljivost potroπnje, proraËuntokova snaga potrebno je napraviti za svakipromatrani vremenski podinterval ili eventualnoreducirati potreban broj proraËuna nanaËin da se koristi krivulja trajanja optereÊenjai njena linearizacija na veÊe podsegmente (uodnosu na bazni vremenski podinterval), priËemu je onda za oËekivati odreappleenu pogreπkuu rezultatu zbog takve aproksimacije.Dakle, ukupni gubici energije u promatranom vremenskomrazdoblju T mogu se dobiti iz izraza (6),sumiranjem gubitaka energije po svim vremenskimpodintervalima unutar ukupno promatranog razdoblja,pod pretpostavkom konstantnog optereÊenjau svakom podintervalu i istog trajanja svakog podintervala(t):The flows of active and reactive power through thepower network elements and all the node voltagesare obtained using the standard calculations ofthe load flow and voltage drop, assuming that thetechnical parameters of the network, topology, networkconfiguration and network loads in each nodeare known. In power flow software, automatic activeand reactive calculations of power losses areregularly performed for each network element, andeventually totaled according to voltage levels, thesame types of network elements, distribution areas,feeders etc. [5] and [6]. Therefore, the calculationof the power losses in a distribution network is actuallya trivial task, which can even be performedmanually for a small distribution network. However,the power loss in a distribution network for a particularload is generally not a sufficient indicator,although calculation can be performed, e.g. for twocharacteristic states of maximum and minimumloads. The truly significant indices are energy lossesduring a specific time period, e.g., during one calendaryear.The precise calculation of electrical energy lossessignificantly complicates analysis, for at least tworelated reasons:∑ it is necessary to know the load characteristicsfor all the nodes of the distribution networkconsidered, i.e. the chronological load curveslinearly discretized into larger or smaller timesubintervals (e.g., hours) or corresponding loadduration curves. The first approximation, withouta major impact on the calculation error,can be performed by using the aforementionedcurves of the considered distribution network inits entirety, with the corresponding load allocationon the nodes, which corresponds to consumptionat the moment of peak load or at themoment for which we have measured load dataavailable for all the nodes,∑ due to load variability, load flow calculationshould be performed for each time subintervalconsidered or eventually the required numberof calculations should be reduced in such amanner that the load duration curve and its linearizationare used on large subsegments (inrelation to the basic time subinterval), althougha certain degree of error should be expected inthe result due to such approximations.Therefore, total energy losses during the time periodT under consideration can be obtained usingexpression (6), by totaling the energy losses accordingto all the time subintervals within the total periodunder consideration, assuming a constant loadat each subinterval and the same duration of eachsubinterval (Δt):GoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699682

(7)U gornjem izrazu, napon se moæe pretpostavitijednak nazivnom naponu bez veÊeg utjecaja natoËnost izraËuna. U sluËaju proraËuna gubitakaenergije pomoÊu programa za tokove snaga,nema potrebe za takvom aproksimacijom. Akose proraËun radi temeljem aproksimirane krivuljetrajanja optereÊenja, vremenski podintervali topÊenito ne moraju biti jednaki, tako da je u izrazu(7) faktor t potrebno uvesti u sumu.In the above expression, the voltage can be assumedto be equal to the rated voltage, withoutmajor impact upon the precision of the calculation.In the calculation of energy losses using load flowsoftware, there is no need for such approximation.If the calculation is based upon the approximatedload duration curve, the time subintervals Δt generallydo not have to be equal, so that in expression(7), factor Δt must be entered into the sum.2.2 Utjecaj prikljuËka distribuiranog izvora na gubitkesnage i energije u distribucijskoj mreæiU tzv. pasivnoj distribucijskoj mreæi radne i jalovesnage u granama P i (t), Q i (t) rezultat su iskljuËivopotroπnje, odnosno potroπaËa prikljuËenih iza (uodnosu na pojnu toËku) promatrane grane mreæe,te gubitaka snage u granama (koji su obiËno redaveliËine do nekoliko postotaka). Zbog toga se naradijalnom izvodu distribucijske mreæe (npr. izvodu10 kV ili 20 kV) snage u granama smanjujukako se kreÊe od pojne toËke prema kraju izvoda,pa se i gubici koncentriraju na poËetnim granama,te opadaju prema kraju izvoda. Na slici 1 je prikazanprimjer 10(20) kV-nog izvoda napajanog izTS x/10(20) kV sa 5 Ëvorova (mjesta prikljuËka TSx/0,4 kV) i 5 grana (vodova). Grafikonom su prikazanitokovi radnih i jalovih snaga duæ izvoda, tj.po svim granama. Potroπnja radne i jalove snage ui-tom Ëvoriπtu oznaËena je sa P p-i , odnosno Q p-i .Ako se npr. u Ëvoru 4 prikljuËi elektrana, ona Êeu trenutku t proizvoditi radnu snagu P G (t) i jalovusnagu Q G (t), pa se takva mreæa, odnosno diomreæe od pojne toËke do Ëvora s prikljuËenomelektranom, naziva aktivna distribucijska mreæa.U sluËaju da nema potroπnje na izvodu, radna ijalova snaga koju proizvede elektrana ima tijekprema pojnoj toËki i to u jednakom iznosu prekosvih grana 1 do 4 (takoappleer zanemareni gubici).Superpozicijom tokova snaga u granama koje surezultat potroπnje i tokova snaga koje su rezultatproizvodnje elektrane, dobije se situacija prikazanana slici 2. Odabran je sluËaj kad je radna snagaelektrane upravo jednaka potroπnji na izvodu, dokje jalova snaga elektrane jednaka nuli, buduÊida je najËeπÊi sluËaj da distribuirani izvori rades faktorom snage cos = 1. Meappleutim, u nekimsluËajevima radi se o asinkronim generatorima bezkompenzacije jalove snage, tako da elektrana utom sluËaju predstavlja dodatnog potroπaËa jalovesnage, poveÊavajuÊi na taj naËin optereÊenje granajalovom snagom.2.2 The impact of the connection of a distributedsource on power and energy losses in a distributionnetworkIn a so-called passive distribution network, the activeand reactive power in branches P i (t) and Q i (t) areexclusively the result of the loads connected behindthe feeder supply point of the network branch considered,and the power losses in the branches (whichare usually of an order of magnitude of up to severalpercentage points). Therefore, on the radial distributionfeeder, e.g., a 10 kV or 20 kV feeder, the powerin the branches decreases as we move away from thefeeder supply point toward the feeder end, so that thelosses are concentrated at the beginning branchesand decrease toward the end of the feeder. Figure 1presents an example of a 10(20) kV feeder suppliedfrom an x/10(20) kV substation with 5 nodes (theconnection points of the x/10(20) kV substation) and5 branches (lines). The graphs present the active andreactive power flows along the feeder, i.e. along allthe branches. The active and reactive consumptionin the i th node is indicated by P p-i or Q p-i .If, for example, a power plant is connected to Node4, it will generate active power P G (t) and reactivepower Q G (t) at the moment of time t. Such a network,i.e. the part of the network from the feedersupply point to the node with the power plantconnection, is called the active distribution network.In the event that there are no loads on thefeeder, the active and reactive power generated bythe power plant will flow toward the feeder supplypoint in equal amounts along all the branches 1to 4 (also with negligible losses). By superpositionof the branch power flows resulting from consumptionand the power flows resulting from power plantgeneration, the situation presented in Figure 2 isobtained. A case has been selected in which theactive power of the power plant is equal to the loadson the feeder, while the reactive power of the powerplant is equal to zero, since the distributed sourcesusually operate with a power factor of cos = 1.683GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

However, some cases involve induction generatorslacking reactive power compensation, so that apower plant represents an additional reactive powerload, thereby increasing the reactive power load ofthe branches.Slika 1Tokovi snaga ugranama pasivnedistribucijske mreæeFigure 1Power flows in apassive distributionnetwork110(35) kV 10(20) kVMreža 110 kV /110 kV network P p-1 (t)Q p-1 (t)1 2 3 4 51 2 3 4 5P p-2 (t)Q p-2 (t)P p-3 (t)Q p-3 (t)P p-4 (t)Q p-4 (t)P p-5 (t)Q p-5 (t)Tokovi radnih snaga /Active power flowP 1 (t)P 2 (t)Tokovi jalovih snaga /Reactive power flowQ 1 (t)Q 2 (t)P 3 (t)Q 3 (t)P 4 (t)Q 4 (t)P 5 (t)Q 5 (t)1 2 3 4 5Slika 2Tokovi snaga ugranama aktivnedistribucijske mreæeFigure 2Power flows in anactive distributionnetworkMreža 110 kV /110 kV network110(35) kV 10(20) kVTokovi radnih snaga(bez elektrane) /Active power flow(without power plant)1 2 3 4 5P p-1 (t)Q p-1 (t)1 2 3 4 5P p-2 (t)Q p-2 (t)P p-3 (t)Q p-3 (t)P p-4 (t)Q p-4 (t)P G (t)Q G (t)P p-5 (t)Q p-5 (t)Tokovi radnih snaga(s elektranom) /Active power flow(with power plant)Tokovi jalovih snaga /Reactive power flowQ 1 (t)P 1 (t)=0Q 2 (t)P 2 (t)P 3 (t)Q 3 (t)P 4 (t)Q 4 (t)P 5 (t)Q 5 (t)1 2 3 4 5Prikazani primjer ilustrira tipiËnu situaciju u aktivnojdistribucijskoj mreæi, u kojoj se mijenja iznos ismjer tokova radnih snaga u svim granama od pojnetoËke do mjesta prikljuËka elektrane u mreæi.Sa stanoviπta gubitaka radne snage, elektrana uovakvom sluËaju ima pozitivni uËinak, buduÊi dau cjelini smanjuje tokove radnih snaga u mreæi, patako i gubitke radne snage. Meappleutim, u sluËaju daje proizvodnja radne snage elektrane znatno veÊaod ukupne potroπnje radne snage na izvodu, dobitÊe se obrnuti efekt, tj. poveÊanje ukupnih tokovaradne snage po granama, tako da Êe ukupni gubiciradne snage biti veÊi.Doprinos rada prikljuËene elektrane na smanjenje/poveÊanje gubitaka radne snage na promatranomThe example shown illustrates a typical situation inan active distribution network, in which the amountand direction of the active power flows change inall the branches from the feeder supply point tothe point at which the power plant is connected tothe network. From the standpoint of active powerlosses, in such a case the power plant has a positiveimpact, since it decreases the active powerflows in the network as a whole, and thereby alsothe active power losses. However, in the event thatthe generation of active power by the power plantis significantly greater than the total active powerload at the feeder, the opposite effect occurs, i.e.the increase of the total active power flows of thebranches, so that the total active power losses willbe greater.GoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699684

izvodu distribucijske mreæe za promatrani trenutaklako se izraËunava na naËin da se gubici izraËunajuu obje varijante (bez elektrane i sa elektranom).Meappleutim, za izraËun gubitaka radne energije uduæem vremenskom razdoblju, takav proraËun jepotrebno ponoviti za sve kombinacije potroπnjena izvodu i proizvodnje elektrane, buduÊi da supotroπnja i proizvodnja promjenljive u vremenu i unaËelu ne postoji nikakva meappleusobna ovisnost (korelacija).Dakle, prilikom raËunanja gubitaka radneenergije bitno je uvaæiti kronoloπku podudaranostpotroπnje s proizvodnjom distribuiranog izvora[7] i [8], pri Ëemu smanjenju gubitaka pridonosinjihova vremenska podudaranost (maksimalnooptereÊenje ∑ maksimalna proizvodnja, minimalnooptereÊenje ∑ minimalna proizvodnja).The contribution of the operation of the connectedpower plant to the decrease/increase in the activepower losses at a distribution network feeder at agiven moment can be easily determined by calculatingthe losses for both variants, with and withoutthe power plant. However, for the calculation of theactive power losses over a longer period of time,it is necessary to repeat such calculations for allthe combinations of feeder loads and power plantgeneration, since consumption and generation arevariable in time and in principle are not correlated.Therefore, when calculating active energy losses, itis essential to take the chronological correlation ofconsumption and generation into account [7] and [8].Losses are decreased when consumption and generationare synchronous (maximum load ∑ maximumgeneration, minimum load ∑ minimum generation).3 PRORA»UN GUBITAKARADNE SNAGE UDISTRIBUCIJSKOJ MREŽIS PRIKLJU»ENOMVJETROELEKTRANOM3.1 Ulazni parametriPrimjer proraËuna utjecaja prikljuËka elektrane nagubitke radne snage u distribucijskoj mreæi izraappleenje na realnoj mreæi 30 kV, tj. zraËnom vodu TS Bilice∑ Primoπten ∑ Rogoznica. Na taj vod se planira,kao jedna od opcija, prikljuËak vjetroelektraneOrlice snage 11 x 900 kW (9,9 MW), koji bi seizveo kabelom presjeka 3x185 mm 2 Al od lokacijevjetroelektrane do prikljuËnog mjesta na jednomstupu navedenog nadzemnog voda 30 kV koji sekoristi se za napajanje TS 30/10 kV Primoπten iTS 30/10 kV Rogoznica.Jednopolna shema promatrane distribucijskemreæe i VE Orlice prikazana je na slici 3. Plavombojom prikazana je transformacija 110/30 kV uTS 220/110/30 kV Bilice odakle se napaja DV 30kV Bilice ∑ Primoπten ∑ Rogoznica koji je prikazanzelenom bojom. Crvenom bojom prikazana je internakabelska mreæa 30 kV VE Orlice zajedno sprikljuËnim kabelom. Topologija interne kabelskemreæe VE Orlice rezultat je posebne analize kojomsu definirane trase, presjeci i naËin povezivanjavjetrogeneratora s obzirom na pozicije vjetrogeneratorai pristupne ceste. Oznake vjetrogeneratoraVE I-x i VE II-x odnose se na prvu fazu (6 VG)i drugu fazu (5 VG) izgradnje VE Orlice, uz izostavljenevjetrogeneratore druge faze VE II-3 i VEII-4.3 CALCULATION OFACTIVE POWER LOSS IN ADISTRIBUTION NETWORK WITHA CONNECTED WIND POWERPLANT3.1 Input parametersAn example of the calculation of the impact of powerplant generation on the active power losses in a distributionnetwork has been performed for an existing30 kV network, i.e. the overhead line supplyingthe Primoπten and Rogoznica 30/10 kV substationsfrom the Bilice 110/30 kV substation. Among theoptions is the connection of the Orlice Wind PowerPlant (WPP) with a rated power of 11 x 900 kW(9,9 MW) using a 3x185 mm 2 aluminum cable fromthe wind power plant site to the connection point atone of the poles of the 30 kV overhead line used tosupply the Primoπten and Rogoznica substations.A single-line diagram of this distribution networkat the Orlice WPP is presented in Figure 3.Transformation of 110/30 kV is shown in blue atthe Bilice 220/110/30 kV substation, which suppliesthe Bilice ∑ Primoπten ∑ Rogoznica 30 kV line,shown in green. The 30 kV internal cable network ofthe Orlice WPP together with the connection cableis shown in red. The internal cable network topologyof the Orlice WPP is the result of a separate analysisaccording to which the routes, cross-sections andwind turbine interconnection method are defined,regarding the positions of the wind turbines and accessroads. The designations VE I-x and VE II-x ofthe wind turbines refer to the first phase (6 windturbines) and the second phase (5 wind turbines)of the construction of the Orlice Wind Power Plant,685GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

Mjesto prikljuËka je oko 15 km od poËetka voda.TehniËki parametri srednjonaponske (SN) mreæe iinterne kabelske mreæe vjetroelektrane dani su utablicama 1 i 2.omitting the VE II-3 and VE II-4 wind turbines ofthe third phase.The connection point is approximately 15 km fromthe beginning of the overhead line ∑ feeder. Thetechnical parameters of the medium-voltage networkand internal cable network of the wind powerplant are presented in Tables 1 and 2.Slika 3Jednopolna shemadistribucijske mreæe30 kV Bilice-Primoπten-Rogoznicai VE OrliceFigure 3Single-line diagramof the Bilice-Primoπten-Rogoznica30 kV network andthe Orlice WPPPOSTOJEΔA MREŽA /EXISTING NETWORKVE ORLICE /ORLICE WPP(11 x 0,9 MW)Tablica 1∑ Parametri vodovaTable 1∑ Power line dataTip voda /Line typeI max(A)R d / R 0(Ω/km)X d / X 0(Ω/km)B d / B 0(μS/km)Al/» 3x185 535 0,157 / 0,471 0,38 / 1,14 3,25 / 3,25Al/» 3x120 345 0,253 / 0,403 0,35 / 1,47 3,3 / 1,98Cu 3x120 440 0,155 / 0,465 0,38 / 1,14 3,08 / 3,08XHE 49 3x185 361 0,164/1,07 0,11/0,36 88/88GoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699686

Tablica 2 ∑ Parametri transformatoraTable 2 ∑ Transformer danaTransformator /TransformerKomada /NumberU n1 /U n2 /U n3(kV)S n1 /S n2 /S n3(MVA)P ksP-S /P ksP-T /P ksS-T(KW)u kP-S /u kP-T /u kS-T(%)P 0(kW)I 0(%)Grupa spoja /ConnectionBilice 2 110 / 30 / 10 63 / 63 / 21 310 / 80 / 67 11 / 12 / 7,7 52,5 0,5 Ynyn0d11Primoπten 2 30 / 10 8 54 7 9,4 4 Yd5Rogoznica 1 30 / 10 8 54 7 9,4 4 Yd5VE Orlice 11 0.4 / 30 1 10 6 1,1 2 Yd5Vrπna snaga TS 30/10 kV Primoπten iznosi 6 + j 2MVA, a TS 30/10 kV Rogoznica 2,5 + j 0,8 MVA,tako da je ukupno neistodobno vrπno optereÊenjeizvoda 8,5 + j 2,8 MVA, a faktor istodobnostije blizu 1, buduÊi da se radi o ljetnom vrπnomoptereÊenju konzuma praktiËki istih karakteristika.U odnosu na maksimalnu planiranu snagu VE Orlice,vrπno radno optereÊenje je oko 15 % manje.BuduÊi da se radi o ljetnom vrπnom optereÊenjuu kojem se vrlo rijetko moæe oËekivati maksimalniangaæman VE Orlice, za oËekivati je relativno Ëestesituacije:∑ maksimalne distribucijske potroπnje i minimalneproizvodnje VE Orlice (ljeti),∑ minimalne distribucijske potroπnje i maksimalneproizvodnje VE Orlice (proljeÊe, jesen).Svi proraËuni gubitaka radne snage i energije udistribucijskoj mreæi ograniËeni su na DV 30 kVBilice ∑ Primoπten ∑ Rogoznica i transformatore30/10 kV Primoπten i Rogoznica, dakle bez transformacije110/30 kV i prijenosne mreæe.3.2 Rezultati proraËunaProraËun gubitaka radne snage u prethodno opisanojmreæi 30 kV napravljen je uz pretpostavkuprikljuËene VE Orlice, pri Ëemu je pretpostavljenreæim potpune kompenzacije jalove snage, tj. svakivjetroagregat radi s faktorom snage cos = 1.ProraËuni su napravljeni na modelu u programskompaketu PowerCAD za razliËite kombinacijepotroπnje u mreæi i proizvodnje VE Orlice.Tablica 3 prikazuje izraËunate gubitke radne snageu ovisnosti o optereÊenju (potroπnja TS Primoπteni TS Rogoznica), πto je osnovna referenca za daljnjeproraËune gubitaka radne snage i energije.The peak power of the Primoπten 30/10 kV substationamounts to 6 + j 2 MVA, and that of theRogoznica 30/10 kV substation is 2,5 + j 0,8 MVA,so that the total non-simultaneous feeder peak loadis 8,5 + j 2,8 MVA, and the coincidence load factoris close to 1, since this is a case of a summer peakload of two substations supplying similar loads. Inrelation to the maximum planed power rating of theOrlice WPP, the active power peak load is approximately15 % lower. Since this is a case of a summerpeak load in which the maximum generation of theOrlice WPP will occur very rarely, the following situationscan be expected to occur relatively often:∑∑maximum distribution consumption and minimumgeneration of the Orlice WPP (summer),minimum distribution consumption and maximumgeneration of the Orlice WPP (spring andautumn).All the calculations of the active power and energylosses in the distribution network are limited to theBilice ∑ Primoπten ∑ Rogoznica 30 kV overhead lineand the Primoπten and Rogoznica 30/10 kV transformers,i.e. without 110/30 kV transformation andthe transmission network.3.2 Calculation resultsCalculations of active power losses in the previouslydescribed 30kV network have been performed underthe assumption that the Orlice WPP is connected,and it was also assumed that the reactive powerwas fully compensated, i.e. each wind generatoroperates at a power factor of cos = 1. The calculationswere performed on a model using PowerCADsoftware for various load combinations in the networkand generation of the Orlice WPP.Table 3 presents the calculated load-dependentactive power losses (consumption of the Primoπtenand Rogoznica substations), which is the basicreference for the further calculation of the reactivepower and energy losses.687GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

Tablica 3 ∑ Gubici snage u mreæi 30 kV bez VE OrliceTable 3 ∑ Power losses in the 30 kV network without the Orlice WPPPotroπnja / Load (%) 30 40 50 60 70 80 90 100Gubici snage / Active power loss (kW) 87 127 176 240 318 412 524 656U tablici 4, prikazani su rezultati proraËuna gubitakaradne snage u mreæi 30 kV na naËin da jevarirana proizvodnja VE Orlice od 10 % P n do P n ,te potroπnja od 30 % P peak do P peak . Gubici radnesnage raËunati su za svaku kombinaciju potroπnjei proizvodnje, tako da su rezultati prikazani u matriËnomobliku. Prvi redak pribliæno odgovara stanjubez VE Orlice (proizvodnja VE Orlice = 0), alisu gubici radne snage neznatno manji u odnosu navrijednosti iz tablice 3, buduÊi da kabelska mreæa30 kV VE Orlice u praznom hodu proizvodi jalovusnagu i injektira je u zraËni vod 30 kV, smanjujuÊina taj naËin tokove jalovih snaga.In Table 4, the results of the calculation of the activepower losses in the 30 kV network are presentedin such a manner that the Orlice WPP generationvaried from 10 % P n to P n , and the load from 30 %P peak to P peak . Active power losses were calculated foreach combination of load and generation and theresults were presented in matrix form. The first rowapproximately corresponds to the situation withoutthe Orlice WPP (Orlice WPP generation = 0). Theactive power losses are insignificantly lower thanthe values from Table 3, since the unloaded 30 kVcable network of the Orlice WPP generates reactivepower and injects it into the 30 kV overhead line,thereby reducing the reactive power flows.Tablica 4 ∑ Gubici radne snage u mreæi 30 kV ovisno o angaæiranju VE Orlice (kW)Table 4 ∑ Active power losses in the 30 kV network in relation to generation of the Orlice WPPPotroπnja / Load (%)30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 %0 82 121 169 231 308 400 510 640Snaga VE Orlice / Orlice WPP power (MW)0,99 67 99 140 195 264 348 449 5681,98 59 84 119 167 229 304 397 5072,97 58 77 106 147 201 270 353 4553,96 65 77 100 135 182 243 319 4124,95 78 85 101 130 170 224 293 3785,94 97 99 109 131 166 213 274 3526,93 124 120 124 140 168 209 264 3347,92 156 147 145 156 178 213 261 3248,91 195 181 172 178 195 223 265 3229,90 240 221 206 206 218 241 277 327Rezultati iz tablice 4 prikazani su grafiËki na slikama4, 5 i 6. Na slici 4 prikazana je krivulja gubitakaradne snage u mreæi 30 kV ovisno o optereÊenjumreæe i proizvodnji VE Orlice. Tamno plava linijaprikazuje gubitke radne snage kad VE Orlice nijeprikljuËena na distribucijsku mreæu. PodruËjeosjenËano svjetlo plavom bojom prikazuje granicuu kojoj se kreÊu gubici radne snage u mreæi 30 kVu sluËaju da je prikljuËena VE Orlice, a raspon seodnosi na angaæiranje VE Orlice 0 do P n .The results from Table 4 are presented graphicallyin Figures 4, 5 and 6. Figure 4 presents the activepower loss curve in the 30 kV network as a functionof the network load and generation of the OrliceWPP. The dark blue line shows the active powerlosses when the Orlice WPP is not connected to thedistribution network. The light blue shaded areashows the range of the active power losses in the30 kV network when the Orlice WPP is connected,as a function of its generation, 0 to P n .GoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699688

Na slikama 5 i 6 prikazane su apsolutne, odnosnopostotne promjene ukupnih gubitaka snagena DV Bilice ∑ Primoπten ∑ Rogoznica, ovisno ooptereÊenju mreæe i veliËini angaæiranja VE Orlice.Kao referenca za promjenu gubitaka uzeti su podaciiz tablice 3, tj. gubici bez VE Orlice, premapostojeÊem stanju mreæe.Iz slika 4 do 6 jasno se uoËava uglavnom pozitivanutjecaj prikljuËka VE Orlice u smislu smanjenjagubitaka radne snage u distribucijskoj mreæi nakoju se prikljuËuje. PrikljuËak VE Orlice poveÊavagubitke radne snage jedino u kombinacijama niskepotroπnje i visoke proizvodnje.Figures 5 and 6 present the absolute or percentagechanges in the total power losses on the Bilice ∑Primoπten ∑ Rogoznica overhead line, as a functionof network load and generation of the Orlice WPP.Data from Table 3, i.e. energy losses prior to the connectionof the Orlice WPP, were used as referencevalues indicating the existing status of the network.From Figures 4 to 6, the generally positive impactof the connection of the Orlice WPP is evident inreducing active power losses in the distribution networkto which it is connected. Connection of theOrlice WPP only increases active power losses incombinations of low loads and high generation.Gubici snage / Power loss(kW)700600500400300200bez VE / without WPPsa VE / with WPPSlika 4Raspon gubitakaradne snage u mreæi30 kV u sluËajuprikljuËka VE OrliceFigure 4Active power lossesin the 30 kV networkwith the Orlice WPPconnected100030405060708090100Potrošnja / Load (%)Razlika gubitaka snage / Difference in active power loss(kW)2001000-100-200-300-40000,991,982,973,964,955,946,93 7,92 8,91 9,9Potrošnja / Load (%)30 %40 %50 %60 %70 %80 %90 %100 %Slika 5Apsolutne promjenegubitaka radne snageu mreæi 30 kV usluËaju prikljuËka VEOrliceFigure 5Absolute activepower loss changesin the 30 kV networkwith the Orlice WPPconnectedSnaga vjetroelektrane / WPP power (MW)689GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

Slika 6Postotne promjenegubitaka radne snageu mreæi 30 kV usluËaju prikljuËka VEOrliceFigure 6Active power lossesin the 30 kV networkwith the Orlice WPPconnected (%)Razlika gubitaka snage / Difference in active power loss(%)1901409040-10-6000,99 1,98 2,97 3,96 4,95 5,94 6,93 7,92 8,91 9,9Potrošnja / Load (%)30 %40 %50 %60 %70 %80 %90 %100 %Snaga vjetroelektrane / WPP power (MW)4 PRORA»UN GODIŠNJIHGUBITAKA RADNE ENERGIJEU DISTRIBUCIJSKOJMREŽI S PRIKLJU»ENOMVJETROELEKTRANOM4 CALCULATION OF ANNUALACTIVE ENERGY LOSS IN ADISTRIBUTION NETWORKWITH A WIND POWER PLANTCONNECTED4.1 Ulazni parametriZa toËan proraËun godiπnjih gubitaka radne energijeu istom primjeru iz prethodnog poglavlja potrebnoje poznavati:4.1 Input parametersFor the precise calculation of annual active energylosses in the same example from the previous chapter,it is necessary to know the following:∑∑kronoloπku krivulju optereÊenja TS 30/10 kVPrimoπten i TS 30/10 kV Rogoznica, tj. satnepotroπnje u razdoblju od 1 godine,kronoloπku krivulju proizvodnje VE Orlice, tj.satne proizvodnje u razdoblju od 1 godine.∑ the chronological load curve of the 30/10 kVPrimoπten and 30/10 kV Rogoznica subtations,i.e. hourly loads during a 1-year period,∑ the chronological generation curve of the OrliceWPP, i.e. hourly generation during a 1-year period.S obzirom da za promatranu mreæu nije bila dostupnakronoloπka krivulja potroπnje, koriπteni suizmjereni podaci iz druge TS 35/10 kV (prosjeËnasatna optereÊenja), normirani po vrπnoj snazina poznato vrπno optereÊenje DV 30 kV Bilice ∑Primoπten ∑ Rogoznica. Jalova snaga potroπaËapretpostavljena je proporcionalna radnoj, uz konstantanfaktor snage cos = 0,95.Za simulaciju proizvodnje VE Orlice nisu koriπtenipodaci o brzinama vjetra s lokacije Orlice, veÊ sukoriπteni izmjereni jednogodiπnji podaci s drugelokacije (brzina vjetra usrednjena na satneintervale). Zbog toga se izraËunati parametri nemogu direktno primijeniti na promatranu mreæu ivjetroelektranu, veÊ samo kao ilustracija opisanemetodologije.Since a chronological load curve was not availablefor the network considered, measured data (averagehourly loads) from another 35/10 kV substationwere used, normalized according to peak power forthe known peak load of the 30 kV Bilice-Primoπten-Rogoznica line. The reactive power loads were assumedto be proportional to the active power loads,with a constant power factor of cos = 0,95.For the simulation of the Orlice WPP generation,wind speed data from the Orlice location were notused. Instead, data measured for one year fromanother location were employed (wind speed averagedat hourly intervals). Therefore, the calculationparameters cannot be applied directly to the considerednetwork and wind power plant but only asan illustration of the described methodology.GoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699690

Kronoloπka krivulja proizvodnje VE Orlice slijedi izkronoloπke krivulje brzina vjetra i P-v karakteristikevjetroagregata Enercon E-44 koji je pretpostavljenza VE Orlice. Radi se o sinkronom generatoru nazivnesnage 900 kW, s direktnim pogonom i indirektnimprikljuËkom na mreæu (preko pretvaraËa).U proraËunu je pretpostavljena jednaka brzinavjetra na svih 11 lokacija vjetroagregata, te reæimpotpune kompenzacije jalove snage vjetrogeneratoraVE Orlice (cos = 1).Da bi se reducirala potreba za izvoappleenjem8 760 proraËuna tokova snaga na razumnu mjeru,proraËun je izveden pomoÊu krivulje trajanjapotroπnje i krivulje trajanja proizvodnje VE Orlice,aproksimacijom krivulja na manji broj segmenatakonstantne snage. Meappleutim, istodobnost distribucijskepotroπnje i proizvodnje VE Orlice nije na tajnaËin izgubljena, veÊ je modelirana izdvojeno zasebnimproraËunom koji Êe naknadno biti opisan.Godiπnja krivulja trajanja potroπnje formirana je naosnovi kronoloπke krivulje potroπnje, te je segmentiranana 8 diskretnih iznosa snaga u rasponu od30 % do 100 % vrπnog optereÊenja s korakom10 % (slika 7). Pri tome svaki segment koji jepredstavljen snagom P i reprezentira (aproksimira)snage u rasponu P i - 5%P peak P i P i + 5%P peak . SaP peak oznaËena je snaga koja odgovara prvom segmentui neπto je manja od stvarnog maksimalnogoptereÊenja.The chronological generation curve of the OrliceWPP is derived from the chronological wind speedcurve and P-v curve of the Enercon E-44 WindGenerator, which is assumed for the Orlice WPP.This is a synchronous generator with a rated powerof 900 kW, direct drive and an indirect networkconnection using a converter. In the calculation, auniform wind speed was assumed for all 11 windgenerator locations, and the reactive power of theOrlice WPP was fully compensated (cos = 1).In order to reduce the necessary 8 760 load-flowcalculations to a reasonable number, calculationwas performed using the load duration curve andthe generation duration curve of the Orlice WPP,with an approximated curve for a smaller number ofsegments with constant power. However, the coincidenceof the distribution consumption and generationof the Orlice WPP was not lost in this mannerbut modeled separately by a calculation that will bedescribed subsequently.The annual load duration curve is formed on the basisof the chronological load curve and segmentedinto 8 discrete parts with constant power, rangingfrom 30 % to 100 % of the peak load, with a 10 %increment (Figure 7). Moreover, each segment designatedby power P i represents (approximates)power in the range of P i - 5%P peak P i P i + 5%P peak .P peak stands for power that corresponds to the firstsegment and is somewhat lower than the existingmaximum load.Potrošnja / Load(%)100806040Slika 7Aproksimacijakrivulje trajanjapotroπnjeFigure 7Approximated loadduration curve2000 8 760Vrijeme / Time (h)Godiπnja krivulja trajanja proizvodnje VE Orlicetakoappleer je formirana na osnovi kronoloπke krivuljetrajanja proizvodnje izraËunate preko kronoloπkekrivulje brzine vjetra i P-v krivulje odabranog vjetroagregata,a segmentirana je na 11 diskretnihiznosa snaga pojedinaËnog vjetroagregata u rasponuod 0 do 900 x 11 kW uz korak 90 x 11The annual generation duration curve of the OrliceWPP is also formed on the basis of the chronologicalgeneration curve calculated using the chronologicalwind speed curve and the P-v curves of theselected wind turbine, and segmented into 11 discretepower units of an individual wind power plantranging from 0 to 900 x 11 kW, with an increment691GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

kW (slika 8), tj. u rasponu od 0% P n do 100% P ns korakom 10% P n . Svaki segment predstavljensnagom P j reprezentira (aproksimira) proizvodnjuVE Orlice u rasponu P j - 5%P n P j P i + 5%P n ,osim prvog i zadnjeg segmenta za koje je P 1 = P n ,odnosno P 11 = 0.of 90 x 11 kW (Figure 8), i.e. ranging from 0 %P n to 100 % P n , with an increment of 10 % P n .Each segment designated by power P j represents(approximates) the generation of the Orlice WPPwith a range of P j - 5%P n P j P i + 5%P n , with theexception of the first and last segments, for whichP 1 = P n and respectively P 11 = 0.Slika 8Aproksimacijakrivulje trajanjaproizvodnje VE OrliceFigure 8Approximated OrliceWPP generationduration curveSnaga VE Orlice / Orlice WPP power(%)1008060402000 8 760Vrijeme / Time (h)4.2 Modeliranje istodobnosti distribucijskepotroπnje i proizvodnje vjetroelektraneIstodobnost distribucijske potroπnje i proizvodnjeVE Orlice uvaæena je formiranjem matrice uËestalostiistodobne pojave optereÊenja koji odgovarai-tom segmentu krivulje sa slike 7 i proizvodnjeVE Orlice koja odgovara j-tom segmentu krivuljesa slike 8. Dakle, na temelju kronoloπkih krivuljaoptereÊenja distribucijske mreæe i proizvodnje VEOrlice odreappleuje se godiπnji postotni iznos vremenaza svaku kombinaciju optereÊenja distribucijskemreæe i proizvodnje VE Orlice kako slijedi:∑∑formiraju se vektori prosjeËne satne snage distribucijskemreæe i satne snage VE Orlice: [P distr ] (8760 x 1) , gdje je P distr (k) prosjeËna satnasnaga distribucijske potroπnje u k-tom satu [P VE ] (8760 x 1) , gdje je P VE (k) prosjeËna satnasnage VE Orlice u k-tom satu,raËunaju se dvije matrice:[p distr ] (8 760 x 8) ,gdje je element matrice definiran sa:4.2 Modeling of the coincidence of the distributionconsumption and wind power plant generationThe coincidence of distribution consumption andOrlice WPP generation has been taken into accountin the formation of a frequency matrix of the coincidenceof consumption, corresponding to the i-th curvesegment in Figure 7, and Orlice WPP generation,corresponding to the j-th curve segment in Figure 8.Therefore, based upon the chronological load curvesof the distribution network and Orlice WPP generation,the annual percentages of the duration of eachcombination of loads on the distribution network andOrlice WPP generation are determined, as follows:∑ the vectors of the average hourly load of thedistribution network and wind power plant generationare formed as follows:∑ [P distr ] (8760x1) , where P distr (k) is the average hourlyload of the distribution network at hour k [P VE ] (8760x1) , where P VE (k) is the average OrliceWPP generation at hour ktwo matrices are calculated:[p distr ] (8760x8) ,where the matrix element is defined as:ako je / ifinaËe / otherwiseGoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699692

[p VE ] (8760 x 11) ,gdje je element matrice definiran sa:[p VE ] (8760x11) ,where the matrix element is defined as:ako je / ifinaËe / otherwiseako je / ifinaËe / otherwise, za j = 2, ….10.ako je / ifinaËe / otherwiseJedinica u matricama [p distr ] i [p VE ] zapravo predstavljapojavljivanje odgovarajuÊe potroπnje/proizvodnje u k-tom satu unutar odgovarajuÊegsegmenta aproksimiranih krivulja potroπnje i proizvodnje.Dakle, u svakom retku navedenih matricapojavljuje se toËno jedna jedinica, i to u onomstupcu koji predstavlja i-ti odnosno j-ti segmentaproksimirane krivulje distribucijske potroπnje,odnosno proizvodnje VE Orlice. Mnoæenjem matrica[p VE ] T i [p distr ] dobije se matrica uËestalostiistodobne pojave optereÊenja distribucijske mreæei proizvodnje VE Orlice:[p VE ] T [p distr ][p distr_VE ] (11x 8) = ,8760za koju vrijedi:∑p distr_VE (j,i) predstavlja vjerojatnost istodobnogpojavljivanja proizvodnje VE Orlice u j-tomintervalu aproksimirane krivulje trajanja proizvodnjes distribucijskom potroπnjom u i-tom intervalu aproksimirane krivulje trajanjapotroπnje,∑ .In the matrices [p distr ] and [p VE ], the integer 1 actuallyrepresents load/generation at hour k within thecorresponding segments of the approximated load/generation curves. Thus, in each row of these matrices,the integer 1 occurs only once, in the columnrepresenting the i-th or j-th segment of the approximatedload distribution curve or the Orlice WPPgeneration curve. By multiplying matrices [p VE ] Tand [p distr ], the frequency matrix of the simultaneousdistribution network power load and Orlice WPPgeneration is obtained:[p VE ] T [p distr ][p distr_VE ] (11x 8) = ,8760for which the following apply:∑p distr_VE (j,i) represents the probability of thesimultaneous occurrence of Orlice WPP generationin the j-th interval of the approximatedgeneration duration curve and distribution consumptionin the j-th interval of the approximatedload duration curve.∑ .4.3 Rezultati proraËunaIzraËunata matrica [p distr_VE ] prikazana je u tablici5, na naËin da su njeni elementi (odgovarajuÊevjerojatnosti) dani u postotnim iznosima, a takoappleerje izraËunata i suma vjerojatnosti po redcima istupcima, πto odgovara udjelima pojedinih segmenataaproksimirane krivulje trajanja distribucijskepotroπnje i proizvodnje VE Orlice (slike 9 i 10).4.3 Calculation resultsThe calculated matrix [p distr_VE ] is presented in Table5 in such a manner that its elements (correspondingvalues) are given in percentages and the sumsof the probabilities are calculated for the rows andcolumns, corresponding to the percentages of theindividual segments of the approximated distributionconsumption duration curve and the OrliceWPP generation curve (Figures 9 and 10).693GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

Tablica 5 ∑ Matrica uËestalosti istovremenog pojavljivanja distribucijske potroπnje i proizvodnje VE OrliceTable 5 ∑ Frequency matrix of the simultaneous distribution network power load and Orlice WPP generationPotroπnja / Load (%)30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 % ∑Snaga VE Orlice / Orlice WPP power (MW)0 0,93 % 1,83 % 2,11 % 2,36 % 3,50 % 3,47 % 1,57 % 0,14 % 15,9 %0,99 2,75 % 4,95 % 5,16 % 6,48 % 10,53 % 9,42 % 4,07 % 0,53 % 43,9 %1,98 0,51 % 0,90 % 0,65 % 1,11 % 1,60 % 2,25 % 0,76 % 0,19 % 8,0 %2,97 0,39 % 0,39 % 0,72 % 0,83 % 1,25 % 1,25 % 0,60 % 0,00 % 5,4 %3,96 0,42 % 0,63 % 0,46 % 0,81 % 0,88 % 1,16 % 0,56 % 0,07 % 5,0 %4,95 0,23 % 0,37 % 0,39 % 0,58 % 0,74 % 1,02 % 0,46 % 0,07 % 3,9 %5,94 0,28 % 0,46 % 0,42 % 0,65 % 0,86 % 0,83 % 0,32 % 0,05 % 3,9 %6,93 0,14 % 0,44 % 0,35 % 0,35 % 0,60 % 0,53 % 0,32 % 0,00 % 2,7 %7,92 0,14 % 0,32 % 0,35 % 0,37 % 0,76 % 0,65 % 0,16 % 0,02 % 2,8 %8,91 0,21 % 0,44 % 0,76 % 0,56 % 1,18 % 1,09 % 0,53 % 0,07 % 4,8 %9,90 0,12 % 0,49 % 0,49 % 0,60 % 0,76 % 0,76 % 0,44 % 0,07 % 3,7 %∑ 6,11 % 11,23 % 11,85 % 14,70 % 22,66 % 22,43 % 9,81 % 1,20 % 100 %Slika 9Postotni udjeli trajanjasegmenta krivuljetrajanja optereÊenjaFigure 9Frequency of loadduration curvesegmentsGodišnji udio vremena / Duration in one year(%)2520151056,1%11,2%11,9%14,7%22,7% 22,4%9,8%1,2%03040 50 60 70 80 90 100Potrošnja / Load (%)Slika 10Postotni udjeli trajanjasegmenta krivuljetrajanja proizvodnjeFigure 10Frequency ofgeneration durationcurve segmentsGodišnji udio vremena / Duration in one year(%)504540353025201510515,9%43,9%8,0%5,4% 5,0% 3,9% 3,9% 2,7% 2,8%4,8% 3,7%00 0,99 1,98 2,97 3,96 4,95 5,94 6,93 7,92 8,91 9,9Snaga VE Orlice / Orlice WPP power (MW)GoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699694

ZnajuÊi matricu uËestalosti [p distr_VE ], izraËungodiπnjih gubitaka radne energije u promatranomdijelu distribucijske mreæe svodi se na proraËun11 x 8 = 88 varijanti tokova snaga, za svaku kombinacijudistribucijske potroπnje P i i snage VE OrliceP j . Mnoæenjem u svakoj varijanti izraËunatihgubitaka radne snage s odgovarajuÊim vjerojatnostimadefiniranim preko matrice [p distr_VE ] i ukupnimbrojem sati u godini, dobiju se gubici radneenergije u podrazdobljima. Konkretne vrijednostigubitaka radne snage za svaku kombinaciju veÊsu izraËunate u poglavlju 3 i prikazane u tablici 4.Sumiranjem gubitaka energije za sve kombinacijedobiju se ukupni godiπnji gubici radne energije upromatranom dijelu distribucijske mreæe:If the frequency matrix [p distr_VE ] is known, the calculationof annual active energy losses in part ofthe distribution network considered is reduced tothe calculation of 11 x 8 = 88 variants of the powerflows, for each combination of load P i and thepower rating of the Orlice WPP P j . By multiplyingthe calculated active power losses in each variantby the corresponding probabilities defined by thematrix [p distr_VE ] and the total number of hours in ayear, the active energy losses are obtained for thesubintervals. The values of the active power lossesfor each combination have already been calculatedin Chapter 3 and presented in Table 4. By addingtogether the energy losses for each combination,the total annual active energy losses for the part ofthe distribution network considered are obtained:(8)gdje je:P(j,i) oznaka za gubitke radne snage u mreæi zakombinaciju proizvodnje P j i potroπnje P i .U tablici 6 prikazani su godiπnji gubici radne energijeu promatranom dijelu distribucijske mreæe uvarijanti kad nije prikljuËena VE Orlice, dok suu tablici 7 prikazani odgovarajuÊi gubici radneenergije u sluËaju prikljuËka VE Orlice izraËunatiprethodno opisanom metodom.where:P(j,i) designates the active power losses in thenetwork for the combination of generation P j andload P i .Table 6 presents the annual active energy lossesin the considered part of the distribution networkwhen the Orlice WPP is not connected, while Table7 presents the corresponding active energy losseswhen the Orlice WPP is connected, as calculated bythe previously described method.Tablica 6 ∑ Ukupni godiπnji gubici energije u mreæi 30 kV bez VE OrliceTable 6 ∑ Annual active energy losses in the 30 kV network without the Orlice WPP (MWh)Potroπnja / Load (%) 30 40 50 60 70 80 90 100 ∑Gubici energije /Energy losses (MWh)46,66 124,90 182,72 309,03 631,29 809,54 450,52 69,17 2 624695GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

Tablica 7 ∑ Ukupni godiπnji gubici energije u mreæi 30 kV sa VE Orlice (MWh)Table 7 ∑ Annual active energy losses in the 30 kV network with the Orlice WPP (MWh)Potroπnja / Load (%)30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 % ∑0 6,68 19,38 31,19 47,78 94,31 121,67 70,32 7,79 399,10,99 16,14 42,95 63,31 110,72 243,58 287,21 160,24 26,49 950,6Snaga VE Orlice / Orlice WPP power (MW)1,98 2,63 6,67 6,76 16,25 32,04 59,80 26,57 8,22 158,92,97 2,01 2,66 6,66 10,73 22,01 29,57 18,61 0,00 92,33,96 2,35 4,24 4,06 9,58 14,02 24,64 15,52 2,51 76,94,95 1,57 2,75 3,48 6,59 11,03 19,99 11,88 2,30 59,65,94 2,37 4,01 3,98 7,44 12,45 15,55 7,78 1,43 55,06,93 1,51 4,62 3,77 4,26 8,86 9,75 7,49 0,00 40,37,92 1,90 4,17 4,41 5,06 11,91 12,09 3,70 0,66 43,98,91 3,56 6,97 11,51 8,66 20,17 21,25 12,36 1,96 86,49,90 2,43 9,41 8,77 10,86 14,59 16,13 10,67 1,99 74,9∑ 43,2 107,8 147,9 237,9 485,0 617,6 345,2 53,3 2037,9Dakle, izraËunati godiπnji gubici radne energijeu promatranom dijelu mreæe 30 kV bez VE Orliceiznose 2 624 MWh (πto je u odnosu na ukupnugodiπnju distribucijsku potroπnju oko 5,3 %), dokse u sluËaju prikljuËka VE Orlice smanjuju na 2038 MWh, tj. za 22 %.S obzirom da ukupna godiπnja proizvodnja VEOrlice iznosi 41 % distribucijske potroπnje, smanjenjegodiπnjih gubitaka radne energije je bilo iintuitivno za oËekivati zbog smanjenja optereÊenjadistribucijske mreæe.Neproporcionalno smanjenje gubitaka u odnosu nasmanjenje ukupno preuzete energije iz transformacije110/30 kV rezultat je poveÊanja gubitaka urazdobljima visokog angaæmana VE Orlice i niskogoptereÊenja u mreæi, te prvenstveno Ëinjenice dasu gubici na DV 30 kV i transformatorima 30/0,4kV dalje od mjesta prikljuËka VE Orlice (oko 40 %od ukupnih gubitaka radne energije) ostali nepromijenjeni.Ukoliko bi promatrali samo dio mreæe30 kV od pojne toËke do mjesta prikljuËka VE Orlice,godiπnji gubici radne energije se smanjuju za37 %.ZnaËajan doprinos smanjenju godiπnjih gubitakaradne energije zbog rada VE Orlice moæe setakoappleer oËekivati i u transformatoru 110/30 kV, πtonije ukljuËeno u prethodni proraËun. Naime, kodtransformatora 110/30 kV gubici radne snage usvakom trenutku se smanjuju injektiranjem snageiz VE Orlice, buduÊi da je minimalno optereÊenjetransformacije 110/30 kV veÊe od maksimalnesnage VE Orlice.Thus, the calculated annual active energy losses inthe considered part of the 30 kV network withoutthe Orlice WPP amount to 2 624 MWh , which isapproximately 5,3 % of the total annual distributionnetwork consumption and is reduced to 2 038MWh, i.e. 22 %, when the Orlice WPP is connected.Since the total annual generation of the Orlice WPPamounts to 41 % of the distribution consumption,reduction of the annual active energy losses couldbe anticipated intuitively due to the reduced distributionnetwork load.Disproportional reduction in losses in comparisonto the reduction in the total energy input from the110/30 kV substation is due to increased lossesduring periods of high generation by the Orlice WPPand low network consumption, and especially thefact that the losses on the 30 kV overhead lines andthe 30/0,4 kV transformers after the connectionpoint of the Orlice WPP (approximately 40 % of thetotal active energy losses) remained unchanged.If we consider only the part of the 30 kV networkfrom the supply point to the connection point ofthe Orlice WPP, annual active energy losses are reducedby 37 %.A significant contribution to the reduction in theannual active energy losses due to the operationof the Orlice WPP can also be anticipated in the110/30 kV transformer, which was not included inthe previous calculation. The active power losses ofthe 110/30 kV substation are continuously beingreduced by the injection of power from the OrliceWPP, since the minimum substation loading isgreater than the maximum power of the Orlice WPP.GoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699696

4.4 MoguÊnosti daljnjeg pojednostavljenjaproraËunaPrikazana metoda pruæa moguÊnost toËnog izraËunagodiπnjih gubitaka radne energije u distribucijskojmreæi na koju je prikljuËena vjetroelektrana,na naËin da se znatno reducira potreban broj proraËunatokova snaga (u primjeru sa 8 760 na 88proraËuna). Meappleutim, joπ uvijek je u velikoj mjerivremenski zahtjevna i nepraktiËna, jer je ipakpotrebno poznavati godiπnju kronoloπku krivuljutrajanja optereÊenja i godiπnju kronoloπku krivuljutrajanja brzina vjetra (snaga vjetroelektrane), πtoËesto nije raspoloæivo. Zbog toga se moæe postavitipitanje koliku je pogreπku za oËekivati u sluËajupotpunog zanemarenja istodobnosti distributivnepotroπnje i proizvodnje vjetroelektrane, tj. ukolikose proraËun vrπi samo na osnovi aproksimiranihkrivulja trajanja distribucijske potroπnje i proizvodnjeVE. U tom sluËaju izraËun je moguÊ na naËinda se formira fiktivna matrica [p distr_VE ] f direktno iziznosa udjela pojedinog segmenta aproksimiranekrivulje trajanja distribucijske potroπnje, odnosnoproizvodnje vjetroelektrane:4.4 Possibilities for further simplifying calculationThe method presented facilities the precise calculationof the annual active energy losses in a distributionnetwork to which a wind power plant is connected,in a manner that significantly reduces therequired number of power flow calculations (from8 760 to 88 in the example provided). However,it still takes considerable time and is impracticalbecause it requires the annual chronological loadduration curves and the annual chronological windspeed curves, i.e. annual chronological WPP generationcurves, which are frequently not available.Therefore, the question may be asked how great amargin of error can be expected if the coincidenceof the distribution consumption and the windpower plant generation are completely ignored, i.e.if the calculation is performed solely on the basisof the approximated distribution consumption durationand wind power plant generation curve. Inthis case, calculation is possible if a fictive matrix[p distr_VE ] f is directly formed from the percentage ofan individual segment of an approximated curve ofdistribution consumption duration or wind powerplant generation:gdje su:∑ vektor koji sadræi jediniËnu vrijednostvremenskog trajanja segmenataaproksimirane krivulje trajanja proizvodnjevjetroelektrane (t j /8 760),∑ vektor koji sadræi jediniËnu vrijednostvremenskog trajanja segmenataaproksimirane krivulje trajanjadistribucijske potroπnje (t i /8 760).Na takav naËin fiktivna matrica [p distr_VE ] f ima istoznaËenje kao i stvarna matrica [p distr_VE ] i vrijedi:where:∑ is a vector containing the p.u. valuesof the segment duration of an approximatedgeneration duration curve(t j /8 760) for a wind power plant, and∑ is a vector containing the p.u. valuesof the segment duration of anapproximated distribution consumptionduration curve (t i /8 760).In this manner, the fictive matrix [p distr_VE ] f has thesame significance as the actual matrix [p distr_VE ], andthereforesamo πto vjerojatnosti pojave distribucijskepotroπnje i proizvodnje vjetroelektrane ne odgovarajustvarnim vrijednostima buduÊi da nisu dobiveneiz kronoloπkih krivulja.Godiπnji gubici radne energije izraËunati prekomatrice [p distr_VE ] f , na isti naËin kao i preko matrice[p distr_VE ], za promatranu distribucijsku mreæu iznose2 050 MWh, πto je u odnosu na toËnu vrijednostod 2 038 MWh zanemariva razlika, neusporedivomanja od reda veliËine nepouzdanosti ulaznih podataka.Takoappleer, Ëak i u toËnom proraËunu dobivenirezultat je primjenjiv samo na godinu za kojuraspolaæemo s kronoloπkom krivuljom brzina vjetraexcept that the probabilities of the distribution consumptionand wind power plant generation do notcorrespond to the actual values, since they are notobtained from chronological curves.Annual active power losses calculated using the matrix[p distr_VE ] f , in the same manner as if using matrix[p distr_VE ], amount to 2 050 MWh for the distributionnetwork considered. The difference between thisvalue and the more precise value of 2 038 MWhis negligible. Furthermore, even with more precisecalculation, the result obtained is only applicablefor the year that a chronological wind speed curve697GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

na lokaciji promatrane vjetroelektrane, πto moæeznaËajno varirati iz godine u godinu.Zbog toga se moæe zakljuËiti da je za praktiËne (ibrze) izraËune, sasvim dovoljna (Ëak i vrlo toËna)metoda s fiktivnom matricom [p distr_VE ] f , bez obziraπto je teoretski netoËna. Jasno, ovisno o karakteristikamakonkretne distribucijske mreæe i distribuiranogizvora (mjesto prikljuËenja, omjer nazivnesnage distribuiranog izvora i vrπnog optereÊenjaizvoda, omjer godiπnje proizvodnje distribuiranogizvora i godiπnje potroπnje na izvodu, reæim radadistribuiranog izvora itd.), ipak je moguÊe oËekivatii veÊe pogreπke pri takvoj zanemarenosti.is available for the location of the WPP site, whichcan significantly vary from year to year.Therefore, it can be concluded that for practicaland rapid calculations, the method with the fictivematrix [p distr_VE ] f is quite sufficient and even very precise,albeit theoretically imprecise. Depending onthe characteristics of the specific distribution networkand the distributed source (connection point,the ratio of the rated power of the power plant andthe feeder peak load, the ratio of annual powerplant generation and feeder consumption, the powerplant operating regime etc.), a greater degree oferror can clearly be anticipated from such simplifiedcalculations than in the example presented.5 ZAKLJU»AKU ovom radu opisan je utjecaj prikljuËka distribuiranogizvora elektriËne energije na poveÊanje/smanjenje godiπnjih gubitaka radne energije udistributivnoj mreæi. Utjecajni faktori su brojni,a prvenstveno konfiguracija (topologija) distributivnemreæe, njene tehniËke karakteristike, vrstadistribuiranog izvora, odnosno reæim rada, mjestoprikljuËka na mreæu itd.Uz teoretske osnove nuæne za razumijevanjeproblema, razraappleena je metodologija izraËunagodiπnjih gubitaka radne energije u distribucijskojmreæi na koju je prikljuËena vjetroelektrana, πtoje ilustrirano na primjeru realne distribucijskemreæe 30 kV i vjetroelektrane nazivne snage 9,9MW. Metodologija zahtijeva poznavanje kronoloπkekrivulje distribucijske potroπnje i kronoloπke krivuljebrzina vjetra na lokaciji vjetroelektrane, πtoje nuæan preduvjet za toËan izraËun gubitaka kojiuvaæava istodobnost potroπnje i proizvodnje. Iakoje potreban broj proraËuna tokova snaga reduciranna prihvatljiv broj, opisana metoda je ipak vremenskizahtjevna i nepraktiËna ako nisu poznatenavedene kronoloπke krivulje. Meappleutim, pokazanoje da u sluËaju potpunog zanemarivanja istodobnostipotroπnje i proizvodnje, pogreπka u rezultatumoæe biti praktiËki zanemariva, tako da se proraËunmoæe znatno pojednostavniti koriπtenjemsamo aproksimiranih krivulja trajanja potroπnje iproizvodnje.Iako u promatranom primjeru prikljuËak distribuiranogizvora smanjuje gubitke radne energije uSN mreæi, to nikako ne moæe biti i generalni zakljuËak.Obrnuti efekt se moæe oËekivati u sluËajuveÊeg iznosa ukupno injektirane energije od stranedistribuiranog izvora (πto je u primjeru iznosilo oko41 % ukupne distribucijske potroπnje).5 CONCLUSIONIn this article, the impact of a connected distributedenergy source on increasing/decreasing annualactive power losses in a distribution networkis described. There are numerous influential factors,primarily the configuration (topology) of thedistribution network, the technical characteristicsthereof, types of the distributed source or powerplant operation regime, connection point to thenetwork etc.In addition to the theoretical foundations essentialfor understanding the problem, a methodology hasbeen developed for the calculation of annual activeenergy losses in a distribution network to which awind power plant is connected, illustrated using theexample of an existing 30 kV distribution networkand a wind power plant with a rated power of 9,9 MW.The methodology requires the chronological loaddistribution curve and the chronological wind speedcurve at the site of the wind power plant, which areessential prerequisites for the precise calculationsof losses that take the coincidence of consumptionand generation into account. Although the necessarypower flow calculations have been reducedto an acceptable number, the method describedstill requires considerable time and is impracticalif the aforementioned chronological curves are notavailable. However, it has been shown that whenthe coincidence of consumption and generation arecompletely ignored, the error in the result can bepractically negligible. Therefore, calculation can besignificantly simplified by using only the approximatedload and generation duration curves, withouttaking account of load and generation coincidence.Although the connected distributed source reducedactive energy losses in the medium-voltage networkin the example presented, this result can in no wayGoiÊ, R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699698

Takoappleer, ne moæe se ni pribliæno ocijeniti gdje bibila granica pri kojoj je utjecaj prikljuËka distribuiranogizvora neutralan s obzirom na godiπnjegubitke radne energije u mreæi, i to zbog brojnihutjecajnih parametara: topologija mreæe, tehniËkekarakteristike mreæe, mjesto prikljuËka distribuiranogizvora, reæim rada distribuiranog izvora uodnosu na distributivnu potroπnju itd.be taken as a general conclusion. The opposite effectcan be expected when a large amount of totalenergy is injected from a distributed source (whichamounted to approximately 41 % of the total distributionconsumption in the example presented).Furthermore, it is not possible to make even an approximatedetermination of the point at which theimpact of a connected distributed source would beneutral in relation to the annual active power lossesin a network, due to a number of parameters: thetopology of the network, the technical characteristicsof the network, the site of the connection ofthe distributed source, the operation regime of thedistributed source in relation to the distributionconsumption etc.LITERATURA / REFERENCES[1] Solutions for the connection and operation of distributed generation, Technical report, EA TechnologyLtd. & DTI , London, 2003[2] BEGOVIΔ, M. at al, Impact of Renewable Distributed Generation on Power Systems, Proc. of the 34thHawaii International Conference on System Sciences, 2001[3] PAVIΔ, A., TRUPINIΔ, K., Gubici elektriËne energije u distribucijskoj mreæi, Energija, god. 56(2007),br. 2., 2007.[4] Network losses and distribution generation, Technical report, EA Technology Ltd. & DTI, London,2006[5] GOIΔ, R., MUDNIΔ, E., JADRIJEV, Z., Primjena programskog paketa PowerCAD za analizu gubitakasnage i energije u distributivnim mreæama , V. savjetovanje HK CIGRE, Cavtat, 2001.[6] GOIΔ, R., MUDNIΔ, E., MILUN, D., Struktura gubitaka snage i energije u srednjenaponskoj distributivnojmreæi, IV. simpozij o elektrodistribucijskoj djelatnosti HK CIGRE, Pula, 2002.[7] GOIΔ, R., Utjecaj vjetrogeneratora na varijacije napona i gubitke snage u razdjelnoj mreæi, V. simpozijo elektrodistribucijskoj djelatnosti HO CIGRE, Zadar, 2004.[8] RAKI–IJA, T., Utjecaj rada vjetroelektrana na statiËke varijacije napona i gubitke u distribucijskojmreæi, diplomski rad, FESB, 2005.Uredniπtvo primilo rukopis:2007-11-26PrihvaÊeno:2008-01-15Manuscript received on:2007-11-26Accepted on:2008-01-15699GoiÊ,R., Jakus, D., MudniÊ, E., ProraËun godišnjih gubitaka radne energije …, Energija, god. 56(2007), br. 6., str. 676 ∑ 699GoiÊ, R., Jakus, D., MudniÊ, E., Calculation of annual active energy losses …, Energija, vol. 56(2007), No. 6, pp. 676 ∑ 699

PRIMJENA MÖBIUSOVEVRPCE U ELEKTROTEHNICIAPPLICATION OF THEMÖBIUS STRIP INELECTRICAL ENGINEERINGDr. sc. Duπan VujeviÊ,Cankarova 2 a,10000 Zagreb, HrvatskaJednostrana dvodimenzionalna povrπina nazvana Möbiusovom vrpcom osim u umjetnostii ostalim granama tehnike rabi se i u elektrotehnici. NajËeπÊe za izradu nisko omskihneinduktivnih otpornika, posebice u visokofrekvencijskim i impulsnim ureappleajima, zatimsustava kondenzatora u visokofrekvencijskim sklopovima, mikrovalnih rezonatora i filtara.O tomu postoje Ëetiri patenta prijavljena u SAD-u.The one-sided two-dimensional surface known as the Möbius strip, in addition to applicationsin the arts and various branches of technology, is also used in electrical engineering,most frequently in the construction of low-ohm non-inductive resistors, particularlyin high frequency and pulse devices, as well as capacitor systems in high frequency unitsand microwave resonators and filters, for which there are four registered patents in theUnited States.KljuËne rijeËi: kondenzator, neinduktivni otpornik, rezonatorKey words: capacitor, noninductive resistor, resonatorVujeviÊ, D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711700

1 UVODNjemaËki astronom i matematiËar August FerdinandMöbius (1790. ∑ 1868.) uz Georga FriedrichaBernharda Riemanna (1826. ∑ 1866.) smatrase jednim od njemaËkih pionira neeuklidskegeometrije i topologije. Topologija (grËki: topos ∑mjesto, logos ∑ prouËavanje) je, s viπe grana, dioËiste matematike, zapravo moderna geometrija,koja se bavi svojstvima objekata koja su saËuvanapri njihovoj deformaciji, uvrtanju i rastezanju, dokrezanje ili deranje nije dopuπteno.Tako je npr. kruænica topoloπki jednaka elipsi, akugla elipsoidu, jer su potonji nastali rastezanjemkruænice odnosno kugle. Möbius je 1858. godineotkrio, a 1865. godine obznanio znaËajke jednostranedvodimenzionalne povrπine nazvane ponjemu Möbiusovom vrpcom (Möbius strip, Möbiusband).Te znaËajke je istovremeno, a po nekim autorima inekoliko godina ranije, neovisno o Möbiusu, otkrioi njemaËki svestrani znanstvenik Johann BenedictListing (1808. ∑ 1882.), ali ga se u literaturi, stim u svezi, rijeapplee spominje. Listing je, meappleu ostalim,prvi uporabio izriËaje topologija i mikron.Sva trojica spomenutih znanstvenika bili su uËenicii/ili suradnici velikog njemaËkog matematiËara,astronoma, geodeta, fiziËara, topografa itd. JohannaFriedricha Carla Gaussa (1777. ∑ 1855.). Natog velikana podsjeÊa nas viπe desetaka nazivljas njegovim prezimenom iz podruËja matematike,fizike, astronomije itd. Spomenimo neka: zakon uelektrotehnici, teorem o divergenciji u vektorskojanalizi, stara jedinica za magnetsku indukciju,razdioba odnosno krivulja vjerojatnosti i krater naMjesecu. Njegov se lik viπekratno pojavljivao napoπtanskim markama, nalazio se na novËanici od10 DM itd.Nije rijetkost da mnogi pronalasci na jednom podruËjuljudske djelatnosti nakon viπe desetljeÊa iliËak stoljeÊa daju ideje za druge, na njima temeljenim,pronalascima iz sasvim drugih podruËja. Toje sluËaj i s Möbiusovom vrpcom, koja je prvotnobila zanimljiva samo matematiËarima, a ovdje jeopisana njezina primjena u elektrotehnici.1 INTRODUCTIONThe German astronomer and mathematician AugustFerdinand Möbius (1790∑1868) is consideredone of the German pioneers of non-Euclidean geometryand topology, together with Georg FriedrichBernhard Riemann (1826∑1866). Topology(Greek: topos ∑ place, logos ∑ study) is, with severalbranches, a part of pure mathematics, actuallymodern geometry, engaged in the properties ofobjects that are retained during their deformation,twisting and stretching, while cutting or tearing arenot permitted.Thus, for example, a circle is topologically equalto an ellipse and a sphere is equal to an ellipsoid,since an ellipse occurs with the stretching ofa circle and an ellipsoid with the stretching of asphere. In 1858, Möbius discovered and in 1865published the characteristics of a one-sided twodimensionalsurface named after him, the Möbiusstrip or Möbius band.According to some authors, at the same time or severalyears earlier, independently of Möbius, thesecharacteristics had been discovered by the Germanall-around scientist Johann Benedict Listing(1808∑1882), who is, however, less frequentlymentioned in this context in the literature. Listing,among other things, was the first to use the termstopology and micron.All three of the scientists mentioned were eitherstudents or associates of the great Germanmathematician, astronomer, geodesist, physicistand topographist Johann Friedrich Carl Gauss(1777∑1855). We are reminded of this great manby more than ten terms using his surname from thefields of mathematics, physics, astronomy etc. Wemention Gauss' law in electrical engineering, Gauss'theorem of divergence in vector analysis, the oldGauss unit for magnetic induction, Gauss distribution,i.e. the Gaussian probability distribution curveand a crater on the moon. His portrait has appearedmany times on postage stamps and could be foundon 10 DM bills etc.It is not uncommon for many discoveries in onearea of human activity to provide ideas for discoveriesin other areas after many decades or even centuries.This is the case with the MÐbius strip, whichwas initially only of interest to mathematicians butalso has applications in electrical engineering, asdescribed in this article.VujeviÊ, D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711702

2 MÖBIUSOVA VRPCAAko se jedan kraj, npr. pravokutne papirnatevrpce, zakrene uzduæ dulje osi za pola okretaja, tj.za 180º i spoji s drugim krajem dobije se posebanoblik, s jednom stranom i rubom, tzv. Möbiusovavrpca ili petlja (slika 1). Da ima samo jednu stranulako je dokazati bilo kojim pisalom. Ako se iz jednetoËke, sredinom vrpce, pisalom zapoËne crtatilinija vratit Êe se u istu toËku bez prijelaza prekoruba vrpce. SliËnim postupkom, npr. oznaËavanjemmarkerom, moæe se dokazati da taj oblik imasamo jedan rub. Zanimljivo je, πto Êe se dobiti akose πkarama reæe uzduæ srediπnje linije tog oblika,ili linijom povuËenom treÊinom πirine vrpce.Möbiusova vrpca ima πiroku primjenu. U starimindustrijskim objektima snaga jednog pogonskogstroja prenosila se na viπe radnih strojeva koænatimili gumenim remenima. Ako je remen bio u oblikuMöbiusove vrpce jednoliko su se troπile obje strane,dok se s obiËnim remenom troπila samo jednanjegova strana. Isto se tako magnetofonska vrpcau obliku Möbiusove vrpce rabila u ureappleajima zaneprekinuto snimanje, jer se time postizalo dvostrukovrijeme registracije, odnosno reprodukcije.2 THE MÖBIUS STRIPIf one end of a strip of paper is twisted along itslength by a half turn, i.e. by 180 degrees, andconnected to the other end, a special shape is obtained,with one side and one edge, the so-calledMöbius strip or band (Figure 1). It is easy to demonstratethat it has only one side with any writingimplement. If from one point in the center of thestrip one starts to draw a line, it will return to thesame point without crossing the edge of the strip.Through a similar approach, for example markingwith a marker, it can be demonstrated that thisform has only one edge. It is interesting what willbe obtained if one cuts along the center of the lineof this form, or along the line at a third of the widthof the strip.The Möbius strip has wide applications. In old industrialfacilities, the power from a drive machinewas transmitted to several other machines withleather or rubber belts. If the belt was in the shapeof a Möbius strip, it would wear out evenly, whilean ordinary belt would wear out on only one of itssides. Similarly, recording tapes in the shape of aMöbius strip are used in devices for uninterruptedrecording because they provide double recordingand playing times.Slika 1Möbiusova vrpca(petlja)Figure 1Möbius strip(band)Zakrene li se jedan kraj vrpce za dva poluokreta,tj. umjesto 180º za 360º, dobije se dvostranadvorubna vrpca. Dakle, Möbiusova vrpca postiæese samo s neparnim brojem poluzakretaja jednogakraja.Veliki je broj primjena Möbiusove vrpce u kiparstvu,grafici i ostalim granama umjetnosti, tehniciitd., a nalazi se i na brazilskoj poπtanskojmarki. Zanimljivo je, da se u Ëasopisu Nature od23.5.2002. godine spominje, da su pronaappleeni kri-If one end of a strip is twisted 360º by two turns, insteadof 180º, a double two-sided strip is obtained.Thus, a Möbius strip is only obtained with an oddnumber of half twists of one end.There are a large number of applications of theMöbius strip in sculpture, graphics and otherbranches of art, technology etc., and one is picturedon a Brazilian postage stamp. It is interestingthat the May 23, 2002 issue of the journalNature mentions that crystals have been found of703VujeviÊ,D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711

stali nekih kemijskih spojeva u obliku te vrpce, πtosvakako nema veze sa samim Möbiusom.Najpoznatija primjena Möbiusove vrpce jemeappleunarodni znak za reciklaæu, u obliku trokuta stri strjelice (slika 2), koji se nalazi na odgovarajuÊojambalaæi. Veliki broj tekstova o Möbiusovoj vrpcimogu se naÊi na internetu [1] i [2].Valja napomenuti da se prezime Möbius u tekstovimana engleskom jeziku piπe dvojako, kao Mobiusi Moebius.some chemical compounds in the form of this strip,which certainly has no connection whatsoever withMöbius.The best known application of the Möbius strip isthe international symbol for recycling, in the formof a triangle with three arrows (Figure 2), used onpackaging materials. A large number of texts on theMöbius strip can be found on the Internet, for example[1] and [2].It should be mentioned that the surname of Möbiuscan also be written as Mobius and Moebius inEnglish texts.Slika 2Meappleunarodni znakza reciklaæuFigure 2Internationalrecycling symbol3 PRIMJENA MÖBIUSOVEVRPCE U ELEKTROTEHNICIKoliko je poznato, do danas je u SAD-u patentiranoviπe naprava, ureappleaja i elemenata koji se temeljena naËelu Möbiusove vrpce, meappleu kojima su itri elektriËna elementa. I Teslin patent za namotelektromagneta iz 1894. godine takoappleer se pripisujenaËelu Möbiusove vrpce.3 APPLICATION OF THEMÖBIUS STRIP IN ELECTRICALENGINEERINGSeveral devices and elements based upon theMöbius strip have been patented in the UnitedStates, including three electrical elements. Tesla's1894 patent for a coil for electromagnets is alsoattributed to the principle of the Möbius strip.3.1 Möbiusov neinduktivni otpornikElementi elektriËnih ureappleaja otpornici, kondenzatorii svitci, posebno oni precizni, trebaju imati πtomanje parazitskih sastavnica (komponenata). Zbogstruje kroz otpornik nastaje unutar i izvan njegamagnetsko polje, pa stoga otpornik ima odreappleenisamoinduktivitet L, kojeg se moæe zamisliti u serijis otporom R. Taj induktivitet dolazi do izraæajaveÊ pri niskim frekvencijama. Izmeappleu zavoja, kaoi izmeappleu razliËitih dijelova otpornika i susjednihmetalnih predmeta postoji mnoπtvo kapaciteta malihvrijednosti, koje se moæe nadomjestiti jednim3.1 The Möbius noninductive resistorThe elements of electrical devices such as resistors,capacitors and coils, particularly those which are moreprecise, should have as few parasitic components aspossible. Due to current passing through a resistor, amagnetic field is formed on the inside and outside,and thus the resistor has a certain self-inductance L,which we can imagine in a series with resistance R.This inductance is already evident at low frequencies.Between the turns, as well as between various partsof a resistor and neighboring metal objects, there aremany low value capacitances, which can be substi-VujeviÊ, D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711704

kapacitetom C izmeappleu poËetka i kraja otpornika.Vlastiti induktivitet i kapacitet otpornika izazvatiÊe u strujnom krugu fazni pomak:tuted by the equivalent capacitance C between thebeginning and the end of the resistor. The self inductanceand capacitance of a resistor causes a phaseshift between the current and the voltage in a circuit:(1)izmeappleu struje i napona, pa Êe otpornik, posebicekod viπih frekvencija, djelovati kao impedancija.Stoga se posebnim naËinima izrade preciznihotpornika nastoji smanjiti vlastiti induktivitet ikapacitet, jer se oni ne mogu potpuno otkloniti.Meappleutim, meappleusobno se mogu tako uskladiti dacijeli otpornik djeluje u strujnom krugu kao djelatniotpor. KakvoÊa otpornika za izmjeniËnu struju,pri kruænoj frekvenciji ω=2πf s obzirom na parazitskesastavnice, iskazuje se vremenskom stalnicom(konstantom) τ [3]:Thus, the resistor behaves as an impedance, especiallyat higher frequencies. Therefore, high-precisionresistors are constructed in a specific mannerin order to reduce self-inductance and capacitance,since they cannot be completely eliminated.However, they can be matched so that the wholeresistor acts as an active resistor in a circuit. Thequality of a resistor in an alternating current circuitin respect to its parasitic components at theangular frequency ω=2πf is expressed by the timeconstant τ [1]:(2)Za frekvencije do 20 kHz je ω 2 LC

elektroniËkim sklopovima, npr. u tadaπnjim radarima,koji su radili na frekvencijama do nekolikogigaherca. Otpornik se sastoji od dvije vrpce prikladnogotpornog materijala, iste duljine i πirine,uËvrπÊene na suprotnim stranama jedne izolacijskevrpce. Jedni krajevi te kombinacije zakrenu seza 180º i spoje s drugim krajevima, tj. oblikujese Möbiusova vrpca (slika 3). Krajevi otpornihvrpca spoje se lemljenjem. Umjesto vrpce moæese rabiti izolirana otporna æica, npr. manganinska.PrikljuËci na otporniËke vrpce, πto je od posebnevaænosti, moraju biti toËno jedan nasuprotdrugome. U suprotnome otpornik ima induktivnusastavnicu koja je najveÊa kada su prikljuËci razmaknutiza polovicu duljine petlje. Struje, odnosnoimpulsi u sklopovima, od prikljuËka teku otpornimvrpcama u suprotnim pravcima, tako da senjihova elektromagnetska polja poniπtavaju, pa sedobije neinduktivni otpornik vrlo male vremenskestalnice. Jedna i druga vrpca zapravo su paralelnospojene. PopreËni presjek otpornika pokazuje daje to kondenzator, pa postoji odreappleena kapacitivnasastavnica.pulse electronic circuits, for example in the radarinstallations of the time, which operated at frequenciesof up to several GHz. The resistor consistsof two ribbons of a suitable resistive material of thesame length and width, affixed on opposite endsof a strip of dielectric. One end of this assemblyis twisted 180º and is joined to the other end, i.e.a Möbius strip is formed (Figure 3). The ends ofthe resistive ribbons are soldered together. Insteadof ribbons, it is possible to use resistive wire, e.g.,Manganin. The connection points on resistive ribbonsmust be precisely one opposite the other,which is of particular importance. Otherwise, theresistor would have an inductive component whichis greatest when the distance between the connectionpoints is half the loop length. Currents orpulses flow through the resistive ribbons in oppositedirections, so that their electromagnetic fields canceleach other. Thus, a non-inductive resistor witha very small time constant is obtained. Actually, theribbons are connected in parallel. A cross-sectionalview reveals that it is a capacitor, and there is acertain capacitive component.Slika 3Möbiusov otpornikFigure 3Möbius resistor32121 − izolacijska vrpca /insulating ribbon2 − otporna vrpca /resistive ribbon3 − prikljuËci /connecting pointsEksperimentalni primjerci takvih otpornika, otporareda veliËine 10 Ω, imali su induktivitete redaveliËine 10 nH i kapacitete reda 0,1 pF, daklevremenske stalnice reda veliËine nanosekunde.Vrijeme porasta impulsa, tj. vrijeme potrebno daimpuls od 10 % dostigne razinu od 90 % svojekonaËne vrijednosti, kod 1 kV, bilo je oko 0,1 μs.Na jednoj izolacijskoj vrpci mogu se naËiniti dvaili viπe takvih otpornih elemenata, s meappleusobimrazmacima od oko 2 mm, koji se mogu, po æelji,spajati serijiski ili paralelno (slika 4). Sastavnicetakvog sustava nemaju meappleusobnog utjecaja, kaoniti utjecaja bliskih metalnih objekata i vanjskihpolja. ZnaËajke Möbiusovog otpornika ne mijenjajuse njegovom duljinom ili oblikom. To znaËi da seExperimental samples of such resistors, of an orderof magnitude of 10 ohms, had inductances of anorder of magnitude of 10 nH and capacitances ofan order of magnitude of 0,1 pF, and thus a timeconstant of an order of magnitude of a nanosecond.The pulse rise time, i.e., the time necessary for apulse to increase from 10 % to 90 % of its peakvalue, was approximately 0,1 μs at 1 kV.One or more such resistive elements can be applied toan insulating ribbon with approximately 2 mm spacings.Resistive elements can be connected in seriesor in parallel (Figure 4). The components of such asystem do not affect each other and do not coupleelectromagnetically to other metallic objects. Thecharacteristics of a Möbius resistor do not change withVujeviÊ, D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711706

Möbiusov otpornik moæe omotati oko valjkastog tijelaili tanke kartice, pa Ëak ga oblikovati u kuglu.its length or form. This means that a Möbius resistorcan be wound around a cylindrical core or a thin card,and can even be formed into a ball.12Slika 4Möbiusovviπestruki otpornikFigure 4Möbius combinedresistor21 − izolacijska vrpca /insulating ribbon2 − otporna vrpca /resistive ribbon3.2 Möbiusov kondenzatorDvadeset godina nakon patenta za neinduktivniMöbiusov otpornik, na njegovoj je osnovi ThomasJ. Brown patentirao 8.7.1986. godine Möbiusovkondenzator [5]. Jednostavno je na, prije spomenuteotporne vrpce, ili opÊenito vrpce od vodljivamaterijala Möbiusova otpornika, stavio izolacijskeslojeve i na njima vodljive vrpce, tako da sute kombinacije kraÊe od vrpci samog otpornika inalaze se jedna nasuprot druge (slika 5). VodiËimase na vanjskim vodljivim vrpcama (A,B,C i D)prikljuËuju u elektriËni krug. Mogu biti jedan ilidva para takvih kondenzatora uzduæ opsega ovogsloæenog sustava. Oni imaju ukupno sedam vrpci,izolacijskih i vodljivih. Duljinom i πirinom vrpci,te debljinom izolacije, mogu se ugaappleati vrijednostikapaciteta.A3.2 Möbius capacitorTwenty years after the patent was issued for the noninductiveMöbius resistor, Thomas J. Brown patentedthe Möbius capacitor on July 8, 1986 [5]. On top ofthe previously mentioned resistive ribbons or, generally,the ribbon conductors of a Möbius resistor, hesimply layered a dielectric material and then layeredthis with ribbon conductors, so that these combinationsare shorter than the ribbons of the resistor andlocated one opposite the other (Figure 5). These combinationsare connected to an electric circuit via leadsattached to the outside ribbon conductors (A, B, Cand D). There can be one or two capacitive enclosuresalong the circumference of this complex system. Theyhave a total of seven ribbons, dielectric and conductive.The capacitance can be adjusted by changingthe length and width of the ribbons and the dielectricthickness.Slika 5MöbiusovkondenzatorFigure 5Möbius capacitorBCDA, B, C, D − vanjski oblozi /outer plates707VujeviÊ,D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711

BuduÊi da je Möbiusovom kondenzatoru osnovicaMöbiusov otpornik, za kojeg je spomenuto daima kapacitivnu sastavnicu, razmotrit Êemo πto sedogaapplea kad se na prikljuËnice 3 otpornika (slika3) prikljuËi izmjeniËni napon u tjemene vrijednostiU m . Neka je l duljina Möbiusova otpornika (kadase petlja prereæe), c brzina πirenja vala elektriËnogpolja izmeappleu prikljuËnica 3 1 i 3 2 otpornika i vrijemeT=l/c putovanja vala izmeappleu 3 1 i 3 2 . Ako je frekvencijaprikljuËenog napona f bit Êe, u vremenu t,potencijal prikljuËnice 3 1 :Since the Möbius capacitor employs the principle ofthe Möbius resistor, which as previously mentionedhas a capacitive component, we shall discuss whatwill happen when an alternating voltage u with apeak value U m is applied between the connectionpoints of the resistor according to Figure 3. Let lrepresent the length of the Möbius resistor (whenthe loop is cut), the propagation speed of the electricfield wave between connection points 3 1 and3 2 of the resistor, and T=l/c the wave propagationtime between 3 1 and 3 2 . If f is the frequency of theapplied voltage, than in time t, the potential of theconnection point 3 1 is as follows:(4)a prikljuËnice 3 2 :and of the connection point 3 2 :(5)Napon na prikljuËnicama 3 je jednak razlici potencijalaprikljuËnica 3 1 i 3 2 :The voltage at 3 is equal to the potential differencebetween 3 1 and 3 2 :(6)Otpornik Êe djelovati kao kondenzator kapacitetaC, pa je izmeappleu prikljuËnica 3 kapacitivna strujai C =C(du/dt). Derivacijom po vremenu t jednadæbe(6) i uvrπtenjem f=c/l, dobiva se i C =0. Stoga ÊeprikljuËivanjem Möbiusova otpornika u elektriËnisklop, pri visokim frekvencijama, kada su valneduljine jednake ili blizu duljini petlje, ili njihovimcjelobrojnim viπekratnicima, on propuπtati rezonantnufrekvenciju i njezine harmonike, a priguπitiostale frekvencije. Möbiusovi sustavi kondenzatoramogu, prema tvrdnji autora patenta, sluæiti za viπesvrha, meappleu ostalim za filtriranje pravokutnih ipilastih valnih oblika, ispitivanje jednakosti i istofaznostidvaju signala itd.3.3 Möbiusov rezonator i filtarPri vrlo visokim frekvencijama, kada su duljinevala reda veliËine desetak centimetara ili manje,zbog velikih gubitaka, umjesto vodiËa u oblikuæica elektromagnetski valovi prostiru se valovodima.To su metalne ili dielektriËne, ali i mjeπovite,The resistor will act as a capacitor of the capacitanceC. Thus, current i C =C(du/dt) flows betweenconnection points 31 and 32. The derivative ofequation (6) with respect to time t and for f=c/l, isi C =0. Therefore, when a Möbius resistor is placed ina high frequency electrical circuit, when the wavelengthsare equal to or approximately the length ofthe loop or their integer multiples, it passes the resonantfrequency and its harmonics and attenuatesother frequencies. Möbius capacitors can, accordingto the inventor, serve many purposes, including thefiltering of square and sawtooth waveforms, testingwhether two signals are equal and in-phase etc.3.3 Möbius resonator and filterAt very high frequencies, when the wavelength is ofan order of magnitude of ten centimeters or less,due to high losses, instead of a conductor in theform of a wire, electromagnetic waves propagate inwaveguides. These are metallic or dielectric, butalso tubes of various cross-sectional dimensions [6]VujeviÊ, D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711708

cijevi razliËitih izmjera presjeka [6] i [7]. Presjecivalovoda, iz praktiËnih su razloga pravokutni ilikruæni. Izmjere valovoda ovise o frekvenciji. Ëim jefrekvencija niæa, izmjere su valovoda veÊe. Energijase πiri medijem unutar valovoda, refleksijamaod zida do zida, a samo njezin manji dio ulazi uzidove i gubi se u obliku topline. Podjela valovodatemelji se na longitudinalnoj sastavnici poljausmjerenoj duæoj osi Z. Ako nema elektriËnog poljau smjeru propagacije vala tada on nosi oznakuTE (transverzalni elektriËni val), a onaj koji nemamagnetskog polja u smjeru propagacije naziva seTM (transverzalni magnetski val). Jedan od poznatijihdielektriËkih valovoda je svjetlovod, koji jedanas u opseænoj uporabi.Valovod potpuno zatvoren sa svih strana, ispunjendielektrikom sa zanemarivim gubicima i savrπenovodljivih zidova, ima svojstva elektromagnetskogrezonatora.Jeffrey M. Pond patentirao je 3.9.2002. godineMöbiusov rezonator i filtar [8]. Rezonator Ëini pravokutnivalovod Ëiji je jedan kraj zakrenut uzduæosi za 180º i spojen s drugim krajem. To zakretanjedovodi do dodatnog faznog pomaka elektromagnetskogvala u valovodu πto olakπava uvjeterezonancije u malom obujmu. Filtri mogu bitiniskopropusni, visokopropusni i pojasnopropusni.Niskopropusni filtri propuπtaju sve frekvencije odnulte do odreappleene gornje graniËne frekvencije, adruge priguπuju. Visokopropusni filtri propuπtajusve frekvencije viπe od donje graniËne frekvencije,a pojasnopropusni propuπtaju sve frekvencijeizmeappleu donje i gornje graniËne frekvencije. Filtrisu graappleeni od kombinacija induktiviteta i kapaciteta.Mikrovalni filtri, osim onih naËinjenih odprijenosnih linija, ukljuËuju jedan ili viπe spojenihrezonatora s nizom dijafragmi (prozora) u valovodukoji djeluju kao induktivni ili kapacitivni elementi,kako bi se ostvarilo æeljeno frekvencijskorazdvajanje.3.4 Namot elektromagnetaNikola Tesla je 9.1.1894., kao svoj 56 patent u18 godina, patentirao namot za elektromagnet [9].U ovom patentu navodi da takvi svitci imaju zbogsamoinduktiviteta znaËajnu jalovu sastavnicu,koja se moæe kompenzirati prikladnim kondenzatorima.Kako bi se izbjegla uporaba, u ono dobaskupih i glomaznih kondenzatora, Tesla predlaæesvoj svitak koji ne bi imao induktivnu sastavnicu,jer bi bila kompenzirana kapacitetom samogasvitka za odreappleenu frekvenciju i napon. Inovacijase, prikazana naËelnim primjerom, sastoji u tomeda je zavojnica bifilarna, tj. paraleno se namatajudva izolirana vodiËa A i B (slika 6). Kraj vodiËa Bspaja se na poËetak vodiËa A. Razlika potencijalaand [7]. Waveguide cross sections are rectangularor circular for a practical reason. Waveguide dimensionsdepend on frequency. When the frequencyis lower, waveguide dimensions are higher. Energypropagates through a medium in waveguides andis reflected from wall to wall. Only a small amountenters the walls and is lost in the form of heat. Theclassification of waveguides is based upon the longitudinalfield component along axis Z. If there isno electric field in the direction of the propagationof the wave, it is designated as the transverse electricwave (TE), and if there is no magnetic field inthe direction of the propagation of the wave, it isdesignated as the transverse magnetic wave (TM).One of the better known dielectric waveguides is anoptical waveguide, which is widely used today.A waveguide that is completely closed on all sides,filled with dielectric, with negligible losses andideally conducting walls, has the properties of anelectromagnetic resonator.On September 3, 2002, Jeffrey M. Pond patentedthe Möbius resonator and filter [8]. The resonatorconsists of a rectangular waveguide, one end ofwhich is twisted along its axis and connected tothe other end. This twist provides additional phaseshift of the electromagnetic wave in the waveguide,which facilitates a resonant condition in a smallervolume. Filters can be low-pass, band-pass or highpass.Low-pass filters allow all frequencies fromzero up to a cutoff frequency to pass through, andattenuate others. High-pass filters let through allfrequencies higher than a low cutoff frequency,and band-pass filters let through all frequenciesbetween the lower and upper cutoff frequencies.The filters consist of a combination of inductive andcapacitive components. Microwave filters, with theexception of those consisting of transmission lines,include one or more connected resonators with lowdiaphragms (windows) in the waveguide that serveas inductive or capacitive elements, in order toachieve the desired frequency separation.3.4 Coil for electromagnetsOn January 9, 1894, Nikola Tesla obtained his 56 thpatent in 18 years, a coil for electromagnets [9]. Inthis patent, it is stated that such coils have a significantreactive component due self-inductance, whichcan be compensated by suitable capacitors. In orderto avoid the use of capacitors, which at the time wereexpensive and cumbersome, Tesla proposed a coilthat would not have an inductive component, sinceit would be compensated for specific frequency andvoltage by the capacity of the coil. The innovation,presented with a general example, consists of thefact that the coil is bifilar, i.e. with two insulatedconductors A and B wound in parallel (Figure 6). The709VujeviÊ,D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711

izmeappleu bilo kojih susjednih toËaka tih dvaju vodiËau zavojnici jednaka je polovici prikljuËenognapon na svitke. Kod obiËne zavojnice razlikapotencijala izmeappleu dviju susjedniih toËaka dvajuzavoja jednaka je naponu prikljuËenom na zavojnicupodijeljenom s ukupnim brojem zavoja. Zbogtijesno namotanih vodiËa odijeljenih relativnotankom izolacijom, kapaciteti su veliki. Energijapohranjena u takvom kondenzatoru razmjerna jekvadratu razlike potencijala izmeappleu obloga.end of conductor B is connected to the starting pointof conductor A. The potential difference between anyneighboring points of these two conductors in thecoil is equal to half the applied voltage to the coil.With ordinary coils, the potential difference betweentwo contiguous points is equal to the applied voltageto the coil divided by the total number of turns(convolutions). Due to the tightly wound conductors,separated by relatively thin insulation, the capacitiesare high. The energy stored in such a capacitor isproportional to the square of the potential differencebetween adjacent turns.Slika 6NaËelo izrade namotaelektromagnetaFigure 6Coil forelectromagnetsAABBKako je razlika potencijala viπestruko veÊa negou obiËnoj zavojnici pa je i kompenzacija samoinduktivitetarazmjerno veÊa. Kompenzacija naovaj naËin je pogodnija, jer su kapaciteti ravnomjernorasporeappleeni. Ovisno o namjeni, svitci semogu razliËito namatati i pritom postiÊi æeljenukompenzaciju.Since the difference in potential is many timesgreater than in an ordinary coil, the compensationof the self-inductance is proportionally greater.Compensation is, thereby, improved because the capacitiesare evenly distributed. Depending upon theintended purpose, the coils can be wound in variousways in order to obtain the desired compensation.VujeviÊ, D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711710

4 ZAKLJU»AKNizu postupaka u izvedbi pasivnih dijelova elektriËnihsklopova sa smanjenim parazitskim sastavnicama,od kojih su neki znani s kraja 19. stoljeÊa,pridruæili su se novi. Suvremene izvedbe otpornikaza visoke frekvencije, kao πto su one u tehnici tankogfilma, sendviË itd., veÊinom su prikladne zaotpore veÊih od 10 Ω i relativno malih snaga.Opisani Möbiusovi otpornici, prema svojoj izvedbi,Ëini se da su prikladni za male otpore i veÊe snagepri visokim frekvencijama. Jedna je od prednostiovih otpornika i kondenzatora πto se mogu razliËitooblikovati. Gotove vrpce otpornika i sustava kondenzatoramogu se omotani oko tijela razliËitihoblika, ili Ëak ih oblikovati u kugle, a da pritomnema meappleusobnih utjecaja pojedinih njihovih djelova,ili utjecaja okolnih predmeta.4 CONCLUSIONProcedures for devising electric circuit componentswith reduced parasitic components, some of whichhave been known since the end of the 19 th century,are being joined by new ones. The majority ofmodern high frequency resistors, such as thin-film,sandwich etc., are suitable for resistance of greaterthan 10 Ω and relatively low power.The Möbius resistors described appear to be moresuitable for low resistance and high power at highfrequencies, due to their construction. One of theadvantages of these resistors and capacitors is thatthey can be shaped in various ways. Finished resistorstrips and capacitor systems can be woundaround objects of a variety shapes or even formedinto a sphere, without the components being affectedby each other or coupled to surroundingobjects.LITERATURA / REFERENCES[1] http://www.math.unh.edu[2] http://scdiv.bcc.ctc.edu[3] BEGO, V.: Mjerenja u elektrotehnici, Graphis, Zagreb, 2003.[4] DAVIS, R.D., Non-inductive electric resistor, US Patent 3 267 406[5] BROWN, T.J., Mobius capacitor, US Patent 4 599 586[6] BOSANAC, T.,Teoretska elektrotehnika 1, TehniËka knjiga, Zagreb, 1970.[7] SMRKIΔ, Z., Mikrovalna elektronika, ©kolska knjiga, Zagreb, 1986.[8] POND, J.M., Mobius resonator and filter, US Patent 6 445 264[9] TESLA, N., Coil for electro-magnets, US Patent 512 340Uredniπtvo primilo rukopis:2007-10-22PrihvaÊeno:2007-11-28Manuscript received:2007-10-22Accepted:2007-11-28711VujeviÊ,D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711

3D PRORA»UN KVAZISTATI»KOGMAGNETSKOG POLJA OKO VODI»AI FEROMAGNETSKE PLO»EINTEGRALNIM JEDNADÆBAMATHE 3D CALCULATION OF THEQUASISTATIC MAGNETIC FIELDAROUND A CURRENT CARRYINGCONDUCTOR AND FERROMAGNETICPLATE BY MEANS OF INTEGRALEQUATIONSMr. sc. Branimir ΔuÊiÊ, KonËar ∑ Distributivni i specijalni transformatori,MokroviÊeva 8, 10090 Zagreb, HrvatskaZa model vodiËa i feromagnetske ploËe napravljen je 3D proraËun magnetskog polja pomoÊu integralnihjednadæbi za kvazistatiËki sluËaj. Usporeappleeni rezultati proraËuna i mjerenja dobro se slaæu.Pristup preko integralnih jednadæbi pokazuje se vrlo efikasnim u prostoru u kojem se traæi poljedaleko od izvora i u kojem nema puno razliËitih feromagnetskih materijala. Pri tome se pretpostavljada su vodljivost i permeabilnost feromagnetskog materijala konstantni.Takoappleer je prikazan utjecaj feromagnetskog materijala na magnetsko polja vodiËa.The 3D calculation of the magnetic field around a model of a current carrying conductor andferromagnetic plate has been performed by means of integral equations in the quasistatic case.There is good agreement between the calculated and measured results. The integral equation approachis very effective in a space in which a field distant from the source is calculated and thereare not many different types of ferromagnetic materials. The conductivity and permeability of theferromagnetic materials are assumed to be constant. Furthermore, the influence of the ferromagneticmaterials on the magnetic field of a conductor is shown.KljuËne rijeËi: 3D proraËun, feromagnetski materijal, integralne jednadæbe,kvazistatiËko magnetsko poljeKey words: 3D calculation, ferromagnetic material, integral equations, quasistaticmagnetic fieldΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729712

1 UVODPri numeriËkom proraËunu magnetskog polja metodomkonaËnih elemenata u prostoru u kojemse traæi polje daleko od izvora, moæe se pojavitiproblem vrlo velikog broja potrebnih elemenata zamodeliranje kao i problem definiranja rubnih uvjetana granici domene proraËuna. Nasuprot tomeu pristupu preko integralnih jednadæbi potrebnoje modelirati samo feromagnetske materijale i netreba postavljati rubne uvjete, jer je domena proraËunabeskonaËna. Ëitav se prostor promatra kaoslobodni (μ=μ 0 ) u kojem postoje primarni izvori(stvarni izvori) i sekundarni izvori (izvori koji modelirajumaterijal).Prikazat Êe se 3D proraËun magnetskog polja prekointegralnih jednadæbi na najjednostavnijem modelu;tanki vodiË kroz koji teËe struja, a u Ëijoj se blizininalazi feromagnetska ploËa u obliku kvadra.1 INTRODUCTIONIn the numerical calculation of a magnetic field bymeans of the commonly used finite element methodin a space in which a field distant from the sourceis being sought, problems in connection with thevery large number of elements required for modelingand the definition of boundary conditions mayarise. However, when using integral equations, onlythe ferromagnetic materials need to be modeledand it is not necessary to set boundary conditions,since the domain is infinite. All the space in whichthe primary (real sources) and secondary sourcesexist is regarded as free (μ=μ 0 ).The 3D calculation of a magnetic field by meansof integral equations will be shown for the simplestmodel; a current carrying conductor and nearby ferromagneticcuboid plate.2 MODEL ZA 3D PRORA»UNNa slici 1 prikazan je ispitivani model, a na slici2 pojednostavljeni model za proraËun u kojem suzanemareni progib i promjer vodiËa.Struja I 0 frekvencije f teËe kroz vodiË duljine l usmjeru +x osi. Ispod vodiËa simetriËno je postavljenaferomagnetska ploËa stalne specifiËne vodljivostiγ i stalne permeabilnosti μ. Ishodiπte koordinatnogsustava se postavlja u srediπte vodiËaprema slikama 1 i 2.Zanemaruje se utjecaj svih ostalih izvora i materijalana polje promatranog vodiËa osim sameferomagnetske ploËe. Polje Êe se raËunati samou smjeru osi z .2 MODEL FOR 3D CALCULATIONThe model studied is shown in Figure 1, while asimplified model for calculation is shown in Figure2, in which the deflection and diameter of the conductorare ignored.Current I 0 of frequency f flows through the conductorof length l in the direction of the +x axis. A ferromagneticplate is placed symmetrically beneaththe conductor. The plate has constant conductivityγ and constant permeability μ. The origin of the coordinatesystem is in the center of the conductor,according to Figures 1 and 2.The influence of other sources and materials (exceptthe ferromagnetic plate) on the magnetic fieldof the conductor is ignored. The field will only becalculated along the z axis.ΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729714

Slika 1Ispitivani model vodiËa iferomagnetske ploËeFigure 1Studied model ofa conductor andferromagnetic plateSlika 2Pojednostavljeni modelza proraËun magnetskogpoljaFigure 2Simplified modelfor magnetic fieldcalculationI 0 ∑ efektivna vrijednost struje koja teËe kroz vodiË,f ∑ frekvencija,l ∑ duljina vodiËa,p ∑ progib vodiËa,d ∑ promjer vodiËa,a ∑ duljina feromagnetske ploËe u smjeru x-osi,b ∑ duljina feromagnetske ploËe u smjeru y-osi,c ∑ duljina feromagnetske ploËe u smjeru z-osi,z p ∑ z koordinata donje plohe feromagnetske ploËe,γ ∑ vodljivost feromagnetske ploËe,μ ∑ permeabilnost feromagnetske ploËe (μ=μ 0·μ r ),μ 0 ∑ permeabilnost zraka (μ 0 = 1,256·10 -6 H/m).Zbog kvazistatiËnosti polja ( f= 50 Hz), proraËunÊe se provesti u kompleksnom podruËju pa Êe svevremenski ovisne veliËine imati toËku iznad naziva(fazor). Ako veliËina ima i vektorski karakter, ondaÊe naziv biti podebljan (bold) (npr. H ).I 0 ∑ effective value of current flowing throughconductor,f ∑ frequency,l ∑ length of conductor,p ∑ deflection of conductor,d ∑ diameter of conductor,a ∑ length of the ferromagnetic plate in the x-axisdirection,b ∑ length of the ferromagnetic plate in the y-axisdirection,c ∑ length of the ferromagnetic plate in the z-axisdirection,z p ∑ z-coordinate of the bottom surface of theferromagnetic plate,γ ∑ conductivity of the ferromagnetic plate,μ ∑ permeability of the ferromagnetic plate(μ=μ 0·μ r )μ 0 ∑ permeability of the air (μ 0 = 1,256·10 ∑6 H/m)Because the field is quasistatic ( f = 50 Hz), thecalculation will be performed in a complex plane sothat all time-dependent variables will have a pointabove the symbol (phasor). If the variable is a vector,it will have a bold symbol (e.g. H ).715ΔuÊiÊ,B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729

3 PRORA»UN MAGNETSKOGPOLJA VODI»A (BEZFEROMAGNETSKE PLO»E)Dogovorno Êe sve veliËine koje uzrokuje primarniizvor (vodiË) I 0 imati indeks 0. Vektorski magnetskipotencijal A 0 i jakost magnetskog polja H 0 proraËunatÊe se prema Biot-Savartovu zakonu. BuduÊida struja u vodiËu teËe samo u smjeru +x osi, upromatranoj toËki prostora vektorski magnetskipotencijal imat Êe samo x komponentu, a jakostmagnetskog polja y i z komponentu.Vektorski magnetski potencijal i jakost magnetskogpolja u bilo kojoj toËki (x,y,z) dani su sljedeÊimizrazima:3 CALCULATION OF A 0 AND H 0FOR A CONDUCTOR (WITHOUTA FERROMAGNETIC PLATE)All the variables which are caused by primarysource I 0 will have an index of 0. (vector magneticpotential) and (magnetic field intensity) will be calculatedaccording to the Biot-Savart law. Since thecurrent in a conductor only flows in the directionof the +x axis, the vector magnetic potential at theobservation point on the z-axis will only have an xcomponent, while the magnetic field intensity willonly have a y and z component.Vector magnetic potential A0 and magnetic field intensityat any point (x,y,z) are calculated as follows:(1)(2)ΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729716

4 POSTAVLJANJE INTEGRALNIHJEDNADÆBI ZA MODELNeka se prema slikama 1 i 2 promatra navedenimodel vodiËa i feromagnetske ploËe za kvazistatiËkisluËaj.Pretpostavlja se da su sve vremenske promjenesinusne s kruænom frekvencijom ω.4 SETTING THE INTEGRALEQUATIONS FOR THE MODELLet us consider the model of the current carryingconductor and ferromagnetic plate for the quasistaticcase according to Figures 1 and 2.It is assumed that all the time variations are sinusoidalwith angular frequency ω:(3)Vanjski vremensko promjenjivi izvor I 0 u feromagnetskojploËi uzrokuje magnetizaciju i vrtloænestruje πto se modelira sljedeÊim sekundarnimizvorima [1] i [2]:J v ∑ gustoÊa vrtloænih struja (A/m 2 ) koja se razvijaunutar volumena V feromagnetske ploËe,K m ∑ linijska gustoÊa magnetizacijske struje (A/m)koja se javlja na povrπini S feromagnetskeploËe (magnetizacijski strujni oblog),σ v ∑ ploπna gustoÊa elektriËnog naboja (C/m 2 )koja se pojavljuje na povrπini S feromagnetskeploËe,Prema [1] i [2], ukupno magnetsko polje kojegstvaraju navedeni primarni i sekundarni izvori ubilo kojoj toËki prostora je:The outer time varying source I 0 causes magneticpolarization and eddy currents in the ferromagneticplate, which can be modeled by the following secondarysources [1] and [2]:J v ∑ surface eddy current density (A/m 2 ), whichappears inside the volume V of the ferromagneticplate,K m ∑ line magnetizing current density (A/m), whichappears on the surface S of the ferromagneticplate,σ v ∑ surface electric charge density (C/m 2 ), whichappears on the surface S of the ferromagnetic plate.According to [1] and [2], the total magnetic fieldcaused by the previously mentioned primary and secondarysources at any point of the space is as follows:(4)(5)gdje je:H 0(r) ∑ magnetsko polje koje u toËki r stvara primarniizvor I 0 ,H v(r) ∑ magnetsko polje koje u toËki r stvara sekundarniizvor J v ,H s(r) ∑ magnetsko polje koje u toËki r stvara sekundarniizvor K m,r ∑ radij vektor promatrane toËke (slika 3) ukojoj se raËuna polje (moæe biti bilo gdje uprostoru; unutar volumena V na povrπini Sili izvan volumena V ),where:H 0(r) ∑ the magnetic field at point r caused by theprimary source I 0 ,H v(r) ∑ the magnetic field at point r caused by thesecondary source J v ,H s(r) ∑ the magnetic field at point caused by thesecondary source K m,r ∑ the radius vector of the observation point(Figure 3), in which the field is calculated(can be anywhere in the space; inside thevolume V, on the surface S or outside volumeV ),717ΔuÊiÊ,B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729

’ ∑ radij vektor tekuÊe toËke integracije (slika3) (moæe biti samo unutar volumena V ilina povrπini S),R ∑ vektor udaljenosti (slika 3).r’ ∑ radius vector of the integration point (Figure3) (can be either inside the volume V or onthe surface S),R ∑ connecting vector (Figure 3).(6)(7)(8)(9)Slika 3Radij vektoriFigure 3Radius vectorsVektorski magnetski potencijal u bilo kojoj toËkiprostora dan je sljedeÊim izrazom [1] i [2]:The vector magnetic potential at any point of thespace can be calculated as follows [1] and [2]:(10)(11)ΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729718

gdje je:A 0(r) - vektorski magnetski potencijal kojeg utoËki r stvara primarni izvor I 0 ,A v(r) - vektorski magnetski potencijal kojeg utoËki r stvara sekundarni izvor J v ,A s(r) - vektorski magnetski potencijal kojeg utoËki r stvara sekundarni izvor K m.Ploπni naboj σ v stvara skalarni elektriËni potencijal koji je u bilo kojoj toËki prostora dan izrazom[1] i [2]:where:A 0(r) ∑ the vector magnetic potential at point r causedby the primary source I 0 ,A v(r) ∑ the vector magnetic potential at point r causedby the secondary source J v .A s(r) ∑ the vector magnetic potential at point r causedby the secondary source K m.Surface charge σ v creates electric scalar potential. At any point of the space, this potential can becalculated, as follows [1] and [2]:(12)gdje je ε 0 relativna dielektriËna konstanta u zraku:ε 0 = 8,854·10 -12 F/m.Djelovanjem operatora grad na potencijal , zatoËku koja ne leæi na S dobiva se:where ε 0 is the relative dielectric constant in the airε 0 = 8,854·10 -12 F/m.If the operator grad is applied to the potential , Scan be written as follows for any point that does notlie on the surface:(13)Za toËku na graniËnoj plohi S vrijedi (s unutraπnjestrane povrπine S feromagnetskog materijala) [1]:For a point on the surface S (from the inner sideof the surface S of the ferromagnetic material), thefollowing can be written [1]:(14)U bilo kojoj toËki r unutar volumena V vrijedi [3]:At any point r inside the volume V, the followingcan be written:(15)719ΔuÊiÊ,B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729

Uvrπtavanjem izraza (11) i (13) u (15) dobiva se:Inserting expressions (11) and (13) into (15), itfollows:(16)U bilo kojoj toËki r na povrπini S vrijedi [1]:At any point r on the surface S, the following canbe written [1]:(17)gdje je:n(r) - okomica na povrπinu u toËki r.Usmjerena je iz feromagnetskog materijala premavan.where:n(r) ∑ Normal at the point r.The direction is from the ferromagnetic materialtowards the outside:(18)Uvrπtavanjem izraza (5) u (17), dobiva se:Inserting expression (5) into (17), it follows:(19)Takoappleer se za bilo koju toËku r na povrπini S moæenapisati [3]:At any point r on the surface S, the following canalso be written [3]:(20)ΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729720

Iz uvjeta na granici feromagnetski materijal-zrakdobiva se [1]:From the conditions at the boundary between theferromagnetic material and the air, it follows [1]:(21)Mnoæenjem (20) sa n(r) i uvrπtavanjem (11), (14)i (21) u (20), slijedi:Multiplying (20) by n(r) and inserting (11), (14)and (21) into (20), it follows:(22)Izrazi (16), (19) i (22) su integralne jednadæbe izkojih se dobivaju traæeni sekundarni izvori.Expressions (16), (19) and (22) are integral equationsfrom which secondary sources are calculated.5 DISKRETIZACIJAINTEGRALNIH JEDNADÆBIDa bi se odredili J v (unutar volumena V), K m i σ v(na povrπini S feromagnetske ploËe), volumen Vse dijeli N V na volumnih elemenata, a povrπina S(oploπje kvadra) na N S povrπinskih elemenata (slika4). N X je broj podjela u smjeru x osi, N Y broj podjelau smjeru y osi i N Z broj podjela u smjeru z osi.Kao elementarni volumen V uzet Êe se kvadarzbog konstantnih granica integracije. Elementarnapovrπina S je pravokutnik.5 DISCRETIZATION OFINTEGRAL EQUATIONSIn order to find J v (inside the volume V), K m and σ v (onthe surface S of the ferromagnetic plate), volume V isdivided into N V volume elements, while the surfaceS (surface of the cuboid) is divided into N S surfaceelements (Figure 4). N X is the number of divisions inthe direction of the x-axis, N Y is number of divisionsin the direction of the y-axis and N Z is the number ofdivisions in the direction of the z-axis. Elementaryvolume V is cuboid because of the constant bordersof integration. Elementary surface S is rectangular.Slika 4Podjela volumena V ipovrπine S feromagnetskeploËe na elementarnevolumene V ielementarne povrπine SFigure 4Division of the volumeV and surface S of theferromagnetic plate intoelementary volumes Vand elementary surfacesS.721ΔuÊiÊ,B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729

Prema slici 4 vrijedi:According to Figure 4, it follows:(23)(24)Pretpostavit Êe se da je J v konstantan unutar svakogelementarnog volumena V te da su K m i σ vkonstantni na svakoj elementarnoj povrπini S.Integralne jednadæbe (16), (19) i (22) napisat Êese u obliku skalarnih jednadæbi po komponentama.Jednadæbe Êe se postavljati u teæiπtima volumnih iploπnih elemenata.Za bilo koju toËku r i teæiπta elementarnog volumenaV i vrijedi (r i ' je teæiπte tekuÊeg volumnog ilipovrπinskog elementa po kojem se integrira):It will be assumed that J v is constant inside everyelementary volume V and that K m and σ v are constanton every elementary surface S.Integral equations (16), (19) and (22) will be writtenin the form of scalar equations for the components.The equations will be set in the center pointsof the volume and surface elements.For any center point r i of elementary volume V i thefollowing can be written (r i ' is the center point ofeither volume or surface element being integrated):(25)(26)(27)ΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729722

Za bilo koju toËku r i teæiπta elementarne povrπineS i vrijedi:For any center point r i of elementary surface S i thefollowing can be written:(28)(29)723ΔuÊiÊ,B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729

(30)(31)ΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729724

BuduÊi da ukupno postoji 3N V + 4N S nepoznatihsekundarnih izvora u teæiπtima volumnih ipovrπinskih elemenata, potrebno je napisati istotoliko skalarnih integralnih jednadæbi. U tu sesvrhu svaka od jednadæbi (25), (26) i (27) postavljau svim teæiπnim toËkama volumnih elemenata(ukupno 3N V skalarnih jednadæbi), dok se svaka odjednadæbi (28), (29), (30) i (31) postavlja u svimteæiπnim toËkama povrπinskih elemenata (ukupno4N S skalarnih jednadæbi).Rjeπavanjem linearnog sustava od 3N V + 4N Sjednadæbi sa 3N V + 4N S nepoznanica, dobivaju setraæeni sekundarni izvori u teæiπtima volumnih ipovrπinskih elemenata.Magnetsko polje u bilo kojoj toËki r prostora je:Since there are 3N V + 4N S of unknown secondarysources in the center points of the volume andsurface elements, it is necessary to write the samenumber of scalar integral equations. Therefore,equations (25), (26) and (27) are set in all thecenter points of the volume elements (a total of 3N Vscalar equations), while equations (28), (29), (30)and (31) are set in all the center points of the surfaceelements (a total of 4N S scalar equations).By solving the linear system of 3N V + 4N S equationswith 3N V + 4N S unknowns, the secondary sources atthe center points of volume and surface elementsare obtained.The magnetic field at any point r of the space is asfollows:(32)(33)(34)Za navedeni model u kojem se polje raËuna usmjeru osi z je H 0X = H 0Z = 0.For the defined model, H 0X = H 0Z = 0.725ΔuÊiÊ,B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729

Magnetska indukcija B se dobiva mnoæenjem magnetskogpolja sa μ 0 :Magnetic flux density B is obtained by multiplyingthe magnetic field intensity by μ 0 :(35)(36)(37)Efektivna vrijednost magnetske indukcije u bilokojoj toËki prostora je:The effective value of the magnetic flux density atany point of the space is as follows:(38)6 PRIKAZ REZULTATAMJERENJAMjerenje magnetskog polja obavljeno je na petmodela prema slici 1. Podaci o modelima nalazese u tablici 1.6 MEASUREMENT RESULTSMagnetic field measurement was performed on fivemodels according to Figure 1. Model data are givenin Table 1.Tablica 1 ∑ Podaci modelaTable 1 ∑ Model dataMODEL /ModelI 0(A)f(Hz)l(m)p(m)1 5,26 50 7,3 0,175 0,005 - - - -2 5,26 50 7,3 0,175 0,005 0,51 0,80 0,00125 -0,1463 5,26 50 7,3 0,175 0,005 0,51 0,80 0,00250 -0,1464 5,26 50 7,3 0,175 0,005 0,80 0,51 0,00125 -0,1465 5,26 50 7,3 0,175 0,005 0,80 0,51 0,00250 -0,146d(m)a(m)b(m)c(m)z p(m)Krivulje izmjerenih vrijednosti magnetske indukcijepo z osi na modelima prikazane su na slici5. Vidljivo je da se umetanjem feromagnetskeploËe ispod vodiËa smanjuje magnetsko polje ispodploËe, dok se magnetsko polje iznad vodiËaneznatno poveÊava. ©to je ploËa deblja, priguπenjepolja ispod ploËe je veÊe.The curves of the measured values of the magneticflux density along the z-axis are given in Figure 5.It can be seen that the inserted ferromagnetic platebeneath the conductor decreases the magnetic fieldunder the plate and slightly increases the magneticfield above the conductor. The thicker the plate, thegreater the attenuation of the field under the plate.ΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729726

10,5z(m)Slika 5Krivulje izmjerenihvrijednosti B po z osi namodelimaFigure 5Curves of the measuredvalues of B along thez-axis for the models01,52 2,5 3 3,544,5B(μT)model 5 / model 5-0,5model 4 / model 4model 3 / model 3model 2 / model 2model 2 / model 2-17 USPOREDBA REZULTATAPRORA»UNA I MJERENJAKao πto je veÊ reËeno, proraËun magnetskog poljana modelu napravljen uz zanemarenje progiba ipromjera vodiËa te zanemarenje utjecaja svih ostalihizvora i materijala na polje vodiËa (osim navedeneferomagnetske ploËe). Krivulje usporedbemjerenih i raËunskih vrijednosti B za modele 1 do5 prikazane su na slikama 6 do 10. Usporeappleenekrivulje dobro se slaæu.Odstupanja od oko 15 % postoje u najviπim mjernimtoËkama iznad vodiËa (cca na 1 m iznad vodiËa)zbog utjecaja polja ostalih izvora.10,5z(m)7 COMPARISON OF THECALCULATED AND MEASUREDDATAAs previously stated, the deflection and diameter ofthe conductor are ignored in the calculation of themagnetic field intensity of the model. Furthermore,the influence of other sources and materials (exceptthe ferromagnetic plate) on the magnetic field of theconductor is ignored. The measured and calculatedcurves of B for models 1 to 5 are shown in Figures 6to 10. The compared curves have good matching.Deviation of approximately 15 % exists only in thehighest measured points above the conductor (approximately1 m above the conductor) due to theinfluence of the other conductors.Slika 6Usporedba krivuljaizmjerenih i raËunskihvrijednosti B za model 1Figure 6Comparison of themeasured and calculatedcurves of B for Model 101,52 2,5 3 3,544,5B(μT)-0,5-1Model 1 / Model 1Samo vod bez feromagnetske ploËe /Only conductor without ferromagnetic plateLegenda / Legendmjerenje / measurementproraËun / calculation727ΔuÊiÊ,B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729

Slika 7Usporedba krivuljaizmjerenih i raËunskihvrijednosti B zamodel 2Figure 7Comparison ofthe measured andcalculated curves of Bfor Model 2z(m)10,501,52 2,5 3 3,544,5B(μT)-0,5-1Model 2 / Model 2a=0,51 m, b=0,8 m, c=1,25 mmN x =6, N y =7, N z =3γ=6,4 M S/m, μ r =400Legenda / Legendmjerenje / measurementproraËun / calculationSlika 8Usporedba krivuljaizmjerenih i raËunskihvrijednosti B zamodel 3Figure 8Comparison ofthe measured andcalculated curves of Bfor Model 3z(m)10,501,52 2,5 3 3,544,5B(μT)-0,5-1Model 3 / Model 3a=0,51 m, b=0,8 m, c=2,5 mmN x =6, N y =7, N z =3γ=6,4 M S/m, μ r =400Legenda / Legendmjerenje / measurementproraËun / calculationSlika 9Usporedba krivuljaizmjerenih i raËunskihvrijednosti B zamodel 4Figure 9Comparison ofthe measured andcalculated curves of Bfor Model 4z(m)10,501,52 2,5 3 3,544,5B(μT)-0,5-1Model 4 / Model 4a=0,8 m, b=0,51 m, c=1,25 mmN x =7, N y =6, N z =3γ=6,4 M S/m, μ r =400Legenda / Legendmjerenje / measurementproraËun / calculationΔuÊiÊ, B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729728

z(m)10,501,52 2,5 3 3,544,5B(μT)Slika 10Usporedba krivuljaizmjerenih i raËunskihvrijednosti B zamodel 5Figure 10Comparison ofthe measured andcalculated curves of Bfor Model 5-0,5-1Model 5 / Model 5a=0,8 m, b=0,51 m, c=2,5 mmN x =7, N y =6, N z =3γ=6,4 M S/m, μ r =400Legenda / Legendmjerenje / measurementproraËun / calculation8 ZAKLJU»AK3D proraËun kvazistatiËkog magnetskog poljapreko integralnih jednadæbi pokazuje se vrlo efikasnimu prostoru u kojem se traæi polje daleko odizvora i u kojem nema puno razliËitih feromagnetskihmaterijala. Pri tome se pretpostavlja da suvodljivost i permeabilnost feromagnetskog materijalakonstantni.Za navedeni model vodiËa i feromagnetske ploËepostoji vrlo dobro slaganje rezultata mjerenja i proraËuna.Feromagnetska ploËa, umetnuta ispod vodiËa,smanjuje magnetsko polje vodiËa u podruËjuispod feromagnetske ploËe, a neznatno poveÊavamagnetsko polje iznad vodiËa.8 CONCLUSIONThe 3D calculation of a quasistatic magnetic fieldby means of integral equations is very effective fora space in which the field to be calculated is farfrom the source and in which there are not manydifferent ferromagnetic materials. The conductivityand permeability of the ferromagnetic material areassumed to be constant.For the defined model of the current carrying conductorand ferromagnetic plate, the results of themeasurements and calculations are in good agreement.The insertion of a ferromagnetic plate beneathunder the conductor decreases the magneticfield under the plate and slightly increases the magneticfield above the conductor.LITERATURA / REFERENCES[1] MATJAN, J., Rjeπavanje zadaÊa elektromagnetskog polja integralnim jednadæbama, doktorska disertacija,ETF, Zagreb, 1977.[2] TOZONI, O. V., I. D. MAYERGOIZ, Calculation of three-dimensional Electromagnetic Fields, Tekhnika,Kiev, 1974[3] HAZNADAR, Z. i ©TIH, Æ., Elektromagnetizam 1, ©kolska knjiga, Zagreb, 1997.[4] HAZNADAR, Z. i ©TIH, Æ., Elektromagnetizam 2, ©kolska knjiga, Zagreb, 1997.Uredniπtvo primilo rukopis:2007-10-23PrihvaÊeno:2007-12-04Manuscript received on:2007-10-23Accepted on:2007-12-04729ΔuÊiÊ,B., 3D proraËun kvazistatiËkog magnetskog polja …, Energija, god. 56(2007), br. 6., str. 712-729ΔuÊiÊ, B., The 3D Calculation of the Quasistatic Magnetic Field …, Energija, vol. 56(2007), No. 6, pp. 712-729

GRANICE VALJANOSTI IZRAZA ZAMJERENJA SPECIFI»NOG OTPORATLA WENNEROVOM METODOMPREMA IEEE NORMI Std. 81-1983VALIDITY LIMITS OF THE EXPRESSIONFOR MEASURING SOIL RESISTIVITY BYTHE WENNER METHOD ACCORDINGTO IEEE STANDARD 81-1983Mr. sc. Tomislav BariÊ, dr. sc. Damir ©ljivac,SveuËiliπte J. J. Strossmayer, ElektrotehniËki Fakultet,Kneza Trpimira 2b, 31000 Osijek, HrvatskaDr. sc. Marinko Stojkov, HEP Operator distribucijskog sustava d.o.o.,P. Kreπimira IV 11, 35000 Slavonski Brod, HrvatskaZa analizu rezultata i obavljanje mjerenja specifiËnog otpora tla inæenjerima su dane na raspolaganjesmjernice i naputci razliËitih meappleunarodnih normi. Prilikom koriπtenja matematiËkih izrazaiz navedenih normi Ëesto nisu jasne okolnosti pod kojima su dobiveni navedeni izrazi. Jedan takavsluËaj opisan je u ovom Ëlanku, a odnosi se na meappleunarodnu normu IEEE Std. 81-1983, izrazza prividni specifiËni otpor tla, a koji se odnosi na teorijski model i predviappleanje mjernih rezultatadobivenih Wennerovim mjernim rasporedom elektroda. U IEEE Std. 81-1983 πtapne elektrode sumodelirane kao kuglaste, πto u konaËnici rezultira jednostavnim matematiËkim izrazom za prividnispecifiËni otpor tla. Zbog navedenog pojednostavljenja inæenjeri nemaju uvid u granice valjanostiprema normi dobivenog izraza, odnosno njegovu toËnost.For the analysis of soil resistivity results and measurement, engineers have guidelines and instructionsfrom various international standards at their disposal. The circumstances under which themathematical expressions in these standards were obtained are not always clear. One such case isdescribed in this article and refers to the international IEEE Standard81-1983, an expression for apparent soil resistivity, and refers to a theoretical model and the predictionand interpretation of measurement results obtained by using the Wenner method of electrodearrangement. In IEEE Std. 81-1983, rod electrodes are modeled as spherical, which ultimatelyresults in a simplified mathematical expression for apparent soil resistivity. Due to this simplification,the expression in this standard does not provide engineers with insight into its validity limits,i.e. accuracy.KljuËne rijeËi: IEEE std. 81-1983, prividni otpor tla, specifiËni otpor tla, Wennerov raspored elektrodaKey words: apparent soil resistivity, IEEE Std. 81-1983, soil resistivity, Wenner electrode arrangementBariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753730

1 UVODReforma πkolstva fakultetskog i srednjeg obrazovanjakontinuiran je proces koji se odvija u cijelomesvijetu. Reforme najËeπÊe imaju trend smanjivanjaopsega nastavnog gradiva, koji se provodismanjenjem udjela satnice teorijskih predmeta, apoveÊanjem udjela nastavnih sati praktiËne nastave,odnosno znanstveno usmjeravajuÊi predmeti.Inæenjeri πkolovani u takvom okruæju suoËeni sus poteπkoÊama u tumaËenju i koriπtenju matematiËkoginstrumentarija pri sloæenijim zadaÊama, akoje im se nameÊu u inæenjerskoj praksi. Iz tograzloga razliËita normativna meappleunarodna tijelaprilikom izdavanja normi, Ëije smjernice i naputkei mi slijedimo, koriste vrlo jednostavne matematiËkeizraze u svojim izdanjima. Takvi izraziizvedeni su pod brojnim pretpostavkama, kakofizikalnim, tako i matematiËkim. Navedena pojednostavljenjafizikalne slike opravdana su pododreappleenim uvjetima koji se u normama Ëesto nenavode. Naime, mnoge norme pisane su u vrijemekada je nastavni program u veÊini zemalja svijetasadræajno bio bogatiji teorijskim predmetima teveÊom satnicom matematike. Inæenjeri kao korisnicinormi u takvim uvjetima Ëesto nemaju potpunupredodæbu o moguÊoj pogreπci pri koriπtenjunormom preporuËenih izraza za proraËune ukolikose ista koristi u graniËnim uvjetima, ili izvan njih.Osim navedenog, znatan udio meappleunarodnih normipisan je unazad nekoliko desetljeÊa, u vrijemekada su osobna raËunala bila dostupna uæem kruguinæenjerske populacije. Naime, tek od kolovoza1981. godine kada je IBM na træiπte lansiraoosobno raËunalo (PC) zapoËela je era osobnihraËunala. Danas, 25 godina poslije, kada gotovosvaki inæenjer u razvijenom svijetu posjeduje osobnoraËunalo nije potrebno da se za inæenjerskeproraËune koriste matematiËki pojednostavljeniizrazi, koji vrijede u vrlo uskim granicama fizikalnihvarijabli. Ovaj Ëlanak daje osvrt na jedan takavizraz opisan meappleunarodnom normom IEEE Std.81-1983 [1] za interpretaciju mjernih rezultatamjerenjem specifiËnog otpora tla Wennerovommetodom. Da bi se odredila toËnost izraza premameappleunarodnoj normi IEEE Std. 81-1983 odreappleenje toËan izraz za prividni specifiËni elektriËni otpordvoslojnog tla mjeren Wennerovom tehnikom [2].Na nekoliko primjera koji se mogu javiti u praksiodreappleena je pogreπka predviappleanja prividnog specifiËnogotpora tla prema normi IEEE Std. 81-1983u odnosu na toËniji pristup prikazan u ovom Ëlanku.Rezultati oba pristupa prikazani su analitiËki,grafiËki te diskutirani.1 INTRODUCTIONReform of higher and secondary education is an ongoingprocess throughout the world. Reforms mostoften tend to reduce the range of material taughtand are implemented by reducing the percentage ofthe class hours devoted to theoretical subjects andincreasing the percentage of class hours devotedto practical subjects, i.e. scientifically orientedsubjects. Engineers schooled in such an environmentare confronted by difficulties in interpretingand using mathematical instrumentation in thecomplex tasks they encounter in practice. For thisreason, various international institutions use highlysimplified mathematical expressions when issuingthe standards whose guidelines and instructions wefollow. Such expressions have been derived undernumerous assumptions, both physical and mathematical.This simplification of the physical pictureis justified under certain circumstances, which arenot frequently mentioned in the standards. Manystandards were written at a time when the curriculain the majority of the countries in the world includedmore theoretical subjects and hours of mathematics.Engineers, as the users of these standardsunder such circumstances, are frequently not fullyaware of the errors that are possible when usingthe expressions recommended by the standards forcalculations under or outside boundary conditions.Moreover, a significant percentage of the internationalstandards were written several decades ago,at a time when personal computers (PCs) were onlyavailable to small group of the engineering population.It has only been since August 1981 that IBMlaunched the personal computer on the marketand launched the PC era. Today, twenty-five yearslater, when nearly every engineer in the developedworld posses a personal computer, it is not necessaryto use mathematically simplified expressionsfor engineering calculations, which are only validunder very narrow limits of physical variables. Thisarticle provides a review of such an expression describedby an international standard, IEEE Std. 81-1983 [1] for the interpretation of the results of themeasurement of soil resistivity using the Wennermethod. In order to determine the precision of theexpression according to IEEE Std. 81-1983, a preciseexpression has been determined for the apparentresistivity of a two-layer soil model using theWenner technique [2]. The error in predicting andinterpreting apparent soil resistivity in several examplesthat can occur in practice was determinedaccording to IEEE Std. 81-1983 and compared tothe precise approach presented in this article. Theresults of both approaches are presented graphicallyand analytically, and are discussed.BariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753732

2 MJERENJE SPECIFI»NOGOTPORA TLANajËeπÊe preporuËivana i koriπtena mjerna tehnikaelektriËnog mjerenja specifiËnog otpora tla jeWennerova metoda [2] (slika 1). Razlog rasprostranjenostiuporabe navedene mjerne tehnike jeu jednostavnosti interpretacije mjernih rezultata[3] i [4].2 MEASUREMENT OF SOILRESISTIVITYThe most frequently recommended and used measurementtechnique for the electrical measurementof soil resistivity is the Wenner method [2] (Figure1). The reason for the widespread use of this measurementtechnique is the simplicity of the interpretationof the measured results [3] and [4].gornji sloj tla /upper soil layerdonji sloj tla /lower soil layerImedij a /A B V C Dmedium aa a a gh dmedij b /medium bmedij c /medium cSlika 1Dvoslojno tlo i Wennerovraspored elektrodaFigure 1Two-layer soil andthe Wenner electrodearrangementNavedena mjerna tehnika veoma je prikladna zaodreappleivanje parametara dvoslojnog tla. Postupakmjerenja obavlja se za tu namjenu specijaliziranimbaterijski napajanim mjernim ureappleajem.Mjerni ureappleaj u sebi sadræi strujni izvor sinusnopromjenjive struje kojemu je frekvencija razliËitaod mreæne, tj. od 50/60 Hz, koja se ne podudaras moguÊim harmonicima mreæe, filtar kojim se izdvajaizmjeniËni mjerni napon frekvencije strujnogizvora mjeren voltmetrom, a koji potiskuju ostalefrekvencije te ampermetar.This measurement technique is highly suitablefor determining the parameters of two-layer soil.The measurement procedure is performed for thispurpose using a specialized battery-operated measuringdevice. The measuring device consists of asinusoidal current source of variable amplitude, thefrequency of which differs from that of the mainsfrequency, i.e. 50/60 Hz and does not coincide withpossible mains harmonics, a filter that separates alternatingmeasuring voltage of the frequency of thecurrent source measured by a voltmeter and attenuatesother frequencies, and an amp meter.733BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

3 STATI»KI MODEL STRUJNOGPOLJATumaËenje dobivenih mjernih rezultata temelji sena statiËkom modelu strujnog polja u tlu [5]. U tusvrhu koristi se tehnika odslikavanja izvora strujnihpolja na granicama diskontinuiteta specifiËnogelektriËnog otpora tla (slika 2a). Utiskivanjemstruje I kroz strujne elektrode u tlo, nastalo strujnopolje u dvoslojnom tlu mora zadovoljiti rubneuvjete na granicama diskontinuiteta specifiËnogelektriËnog otpora: tlo-zrak, te gornjeg i donjegsloja tla, a koji glase [5] i [6]:3 STATIC MODEL OF ACURRENT FIELDInterpretation of the measurement results is basedupon a static model of a current field in the soil [5].For this purpose, the method of images is used todescribe the discontinuity boundaries of soil resistivity(Figure 2a). The current field resulting fromthe injection of current I through the current electrodesin the two-layer soil must satisfy the boundaryconditions at the discontinuity boundaries of resistivity:soil-air, the upper soil layer and the lowersoil layer, as stated in [5] and [6]:(1)(2)gdje je:Where:nJ aJ bJ c∑ vektor normale na granice diskontinuitetaspecifiËnog elektriËnog otpora,∑ gustoÊa struje u mediju a (zrak) na granicidiskontinuiteta specifiËnog elektriËnog otpora,∑ gustoÊa struje u mediju b (gornji sloj tla) nagranici diskontinuiteta specifiËnog elektriËnogotpora i∑ gustoÊa struje u mediju c (donji sloj tla) nagranici diskontinuiteta specifiËnog elektriËnogotpora.nJ aJ bJ c∑ is the normal vector at the discontinuity boundaryof resistivity,∑ is the current density in medium a (air) atthe discontinuity boundary of resistivity,∑ is the current density in medium b (uppersoil layer) at the discontinuity boundary ofresistivity and∑ is the current density in medium c (lowersoil layer) at the discontinuity boundary ofresistivity.Zrak (medij a) ima visok specifiËni elektriËni otporza koji se misli da moæe biti 10 18 Ωm. Iz tog razlogamoæe ga se gotovo bez ikakve pogreπke smatratiizolatorom [7] i [8]. Poznavanje rubnih uvjeta (1) i(2) na granicama diskontinuiteta specifiËnog elektriËnogotpora omoguÊava matematiËko rjeπavanjezadaÊe i odreappleivanje potencijala na naponskimmjernim elektrodama (slika 1).Air (medium a) has high specific resistivity, and isthought to be 10 18 Ωm. For this reason, it can beconsidered as an insulator, with practically no error[7] and [8]. Knowing the boundary conditions (1)and (2) at the discontinuity boundary of specific resistivityfacilitates the mathematical solution of thetask and the determination of the potential on thevoltage measurement electrodes (Figure 1).BariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753734

a) b)PodruËje a / Area a2hha a aA B C D ldl gPodruËje b / Area bAT 1 BT 2PodruËje a / Area a 1 2CPodruËje b / Area b gDSlika 2Wennerov rasporedelektroda, dvoslojno tloi odslikavanje strujnihelektrodaFigure 2Wenner electrodearrangement, two-layersoil images of currentelectrodeshPodruËje c / Area c dPodruËje c / Area c dPrema teoriji odslikavanja [5] izvori strujnog polja,tj. strujne elektrode A i D odslikavaju se od granicadiskontinuiteta specifiËnog elektriËnog otpora.Prvo odslikavanje odvija se na granici tlo-zrak.Ukopani dio strujnih elektroda duljine l odslikavase od granice tlo-zrak, u elektrodu duljine l, a izkoje istjeËe jednaka struja kao iz originalne elektrode.S obzirom da jednaka struja istjeËe iz originalneelektrode i njezine slike u podruËju a, tekako se one dodiruju na granici tlo-zrak, moæemoih za proraËune u podruËju b promatrati kao jednucjelinu, tj. kao elektrodu dvostruke duljine2l, a iz koje istjeËe dvostruka struja. Tako nastalikonglomerati originalnih strujnih elektroda A i Ds njihovim slikama koje se nalaze u podruËju aodslikavaju se od granice izmeappleu gornjeg i donjegsloja tla, tj. od granice izmeappleu podruËja b i c u podruËjec. Novonastale slike nalaze se u podruËju c(donji sloj tla), a srediπte slika πtapnih elektrodaudaljeno je od granice tlo-zrak za dvostruku debljinugornjeg sloja tla, te iznosi 2h. Iz ove slike elektrodeA viπe ne istjeËe struja 2I, nego β·2I, gdje jeβ koeficijent odslikavanja izmeappleu podruËja b i c(β = ( d - g )/( d + g )).U iduÊem koraku potrebno je odslikati zamiπljeneslike iz podruËja c u podruËje a. Odslikavanje izpodruËja c u podruËje a odvija se na granici tlozrak,a srediπta novonastalih zamiπljenih slika upodruËju a nalaze se na jednakoj udaljenosti odgranice tlo-zrak, kao i zamiπljene slike iz prethodnogkoraka u podruËju c, a koja odgovara dvostrukojdebljini gornjeg sloja tla te iznosi 2h. U ovomsluËaju odslikavanje iz podruËja c u podruËje ane mijenja iznos struje koja istjeËe iz zamiπljeneelektrode u podruËju a, jer je koeficijent odslikavanjajednak jedinici s obzirom da se podruËje a(zrak) moæe smatrati izolatorom. Tako dobivenezamiπljene slike u podruËju a (zrak) potrebno jeponovno odslikati od granice izmeappleu gornjeg i do-According to the method of images theory [5], thesources of the current field, i.e. current electrodesA and D are mirrored at the discontinuity boundaryof resistivity. The first application of the method ofimages occurs at the soil-air boundary. The parts ofthe current electrodes buried in the soil of lengthl are mirrored at the soil-air boundary, in an electrodeof length l from which equal current flows outas from the original electrode. Since equal currentflows from the original electrode and its image inArea a, and since they touch the soil-air boundary,for calculations in Area b we can consider them tobe a single entity, i.e. an electrode of double length2l from which double current flows. Thus, suchconglomerates of original current electrodes A andD, with their images in Area a are by the methodof images mirrored at the boundary between theupper and lower soil layers, i.e. from the boundarybetween Area b and Area c in Area c. The newimages are located in Area c (lower soil layer) andthe center of the image of the rod electrodes is ata distance from the soil-air boundary of two timesthe thickness of the upper soil layer, amounting to2h. Current β·2I instead of 2I flows from this imageof electrode A, where β is the reflection coefficient[1] (reflection factor [3]) between Areas b and c(β = ( d - g )/( d + g )).In the next step, it is necessary to apply the methodof images on the imaginary image in Area c to mirrorthem in Area a. Mirroring from Area c to Area aoccurs at the soil-air boundary, and the centers ofthe new imaginary images in Area a are at the samedistance from the soil-air boundary as the imaginaryimages from the previous step in Area c, which correspondsto a thickness twice that of the upper soillayer and equals 2h. In this case, the mirroring fromArea c to Area a does not alter the amount of currentthat flows from the imaginary electrode in Areaa, because the reflection coefficient is equal to 1,735BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

njeg sloja tla, tj. podruËja b i c u podruËje c, skoeficijentom odslikavanja β, te se takav postupaknastavlja unedogled. Prilikom svakog odslikavanjaod granica diskontinuiteta elektriËnih znaËajki(specifiËnog elektriËnog otpora tla) poveÊava seudaljenost novonastalih zamiπljenih izvora poljaod granice tlo-zrak te se njihova srediπta nalazena udaljenosti od granice tlo-zrak h, 2h, 4h, ..., 2nh,gdje je n redni broj zamiπljene slike.Za razliku od prikazanog odslikavanja u kojemu jeoËuvana stvarna geometrija strujnih izvora polja(elektrode A i D) prema slici 2a, u meappleunarodnojnormi IEEE Std. 81-1983 [1] navedeno odslikavanjeje znatno pojednostavljeno. Naime, prvapretpostavka koja je uËinjena u fizikalnom modelunorme IEEE Std. 81-1983, je zamjena πtapnihelektroda s polukuglama, koje su se stopile sa svojomslikom iz podruËja a, koja je takoappleer polukugla,pa se daljnje odslikavanje provodi s kuglama(slika 3).since Area a (air) can be considered as an insulator.Such obtained imaginary images in Area a (air)must be mirrored again from the boundary betweenthe upper and lower soil layers, i.e. Areas b and c inArea c, with the reflection coefficient β, and such aprocedure is continued indefinitely. On the occasionof every mirroring from the discontinuity boundaryof soil resistivity, the distance increases between thenew imaginary field sources and the soil-air boundary,and their centers are located at distances fromthe soil-air boundary of h, 2h, 4h, ..., 2nh where n isthe ordinal number of the imaginary image.Unlike the mirroring presented in which the actualgeometry of the current source fields (electrodesA and D) is according to Figure 2a, this mirroringis significantly simplified in the internationalIEEE Std. 81-1983 [1]. The first assumption inthe physical model of IEEE Std. 81-1983 is thereplacement of rod electrodes with hemisphericalones, which merge with their image from Area a,which is also a hemisphere, and further mirroring isconducted with spheres (Figure 3).Slika 3Odslikavanje udvoslojnom tluekvivalentnimkuglama i Wennerovraspored elektrodaFigure 3Mirroring in two-layersoil with equivalentspheres and theWenner electrodearrangement(2I)β 22h(2I)βaa a2hA B C D(2I)hh(2I)β2h(2I)β 2 zraka/air g dPodruËje a / Area aPodruËje b / Area bPodruËje c / Area cPredstavljanjem πtapnih (cilindriËnih) elektrodastrujnih mjernih sondi s kuglama omoguÊeno jekoriπtenje izraza za potencijal u okoliπu kugle,odnosno toËkaste izvore polja, jer se potencijalodreappleuje izvan ekvivalentne kugle. Izrazi za elektriËnipotencijal u prostoru oko toËkastih izvorapolja su matematiËki vrlo jednostavni. Zbog navedenogpojednostavljenja izraz za prividni specifiËniotpor tla prema IEEE normi Std. 81-1983 [1]je valjan ukoliko je dovoljno velik razmak izmeappleususjednih elektroda a. Naime, ukoliko je dovoljnovelik razmak izmeappleu elektroda a, tada su ekvipotencijalnekrivulje koje su uzrokovane strujnimelektrodama cilindriËnog oblika sliËne ekvipotencijalnimkrivuljama toËkastih izvora polja.Presenting the rod (cylindrical) electrodes of currentmeasuring probes with spheres permits theuse of the expression for potential in the spherevicinity, i.e. field point sources, because the potentialis determined outside the equivalent sphere.Expressions for electric potential around the pointsources are mathematically very simple. Therefore,the cited simplified expression for apparent soilresistivity according to IEEE Std. 81-1983 [1] isvalid if there are is a sufficiently large spaces betweenthe neighboring electrodes a. If there is asufficiently large space between electrodes a, theequipotential curves that are caused by cylindricalcurrent electrodes are similar to the equipotentialcurves of the point sources.BariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753736

Ekvipotencijalne krivulje kojeg uzrokuju cilindriËneelektrode su oblika elipse, a ukoliko je toËka promatranjadaleko od elektrode ekvipotencijalnakrivulja koja prolazi kroz nju je pribliæno kruænogoblika, πto odgovara toËkastom izvoru polja (slika2b). Iz navedenog je jasan fizikalni smisao pojednostavljenja,ali ne i utjecaj navedenog pojednostavljenjana pogreπku pri odreappleivanju specifiËnogotpora tla, pogotovo u sluËaju kada su mjerne elektroderelativno blizu jedna drugoj. Da bi se dobioodgovor na ovo pitanje potrebno je odrediti izrazza prividni specifiËni otpor tla mjeren Wennerovimspojem u kojemu su mjerne elektrode modeliranekao πtapovi (tanki cilindri). Zatim je na nekolikoprimjera iz prakse odreappleena pogreπka koja nastajezbog modeliranja πtapnih elektroda s kuglama teutjecaj blizine elektroda na mjerenje.Equipotential curves caused by cylindrical electrodesare in the form of an ellipse, and if the observationpoint is distant from the electrode, the equipotentialcurve that passes through it is approximately round inshape, which corresponds to the point source (Figure2b). Therefore, the physical purpose of simplificationis clear but not the impact of this simplificationon the error in determining soil resistivity, especiallywhen measurement electrodes are relatively close toeach other. In order to obtain an answer to this question,it is necessary to define an expression for apparentsoil resistivity measured by the Wenner methodin which the measurement electrodes are modeledas rods (thin cylinders). In several actual examples,errors were determined that occurred due to modeledrod electrodes with spheres and the influence of thevicinity of the electrodes upon measurement.4 ODRE–IVANJE TO»NOGIZRAZA ZA PRIVIDNISPECIFI»NI OTPOR TLAPostupak odreappleivanja toËnog izraza za prividnispecifiËni elektriËni otpor tla sastoji se od nekolikokoraka. U prvom koraku potrebno je pronaÊineki jednostavniji izraz uz pomoÊ kojega se lakoodreappleuje potencijal u okolini πtapne elektrodeiz koje istjeËe struja I, kada se ona nalazi u neograniËenommediju. U drugom koraku potrebnoje uvaæiti utjecaj granice tlo-zrak koriπtenjemtehnike odslikavanja, te odreappleivanje izraza zaskalarni elektriËni potencijal na mjestu jedne odnaponskih mjernih elektroda. TreÊi korak sastojise od uvaæavanja joπ jedne granice diskontinuitetaspecifiËnog elektriËnog otpora tla, a koja se nalaziispod gornjeg sloja tla. Kao i u prethodnomkoraku primjenom tehnike odslikavanja, potrebnoje odrediti potencijal u gornjem sloju tla (podruËjeb) na mjestu jedne od naponskih mjernih elektroda.U posljednjem koraku na temelju prethodnoodreappleenih izraza potrebno je odrediti izraz za prividnispecifiËni elektriËni otpor dvoslojnog tla mjereneWennerovim rasporedom elektroda (slika 1).Skalarni elektriËni potencijal u okolini πtapne elektrodekonaËne duljine l iz koje istjeËe struja I, akoji se nalazi u neograniËenom mediju specifiËnogotpora g prikazan slikom 4a prema [5] glasi:4 DEFINITION OF A PRECISEEXPRESSION FOR APPARENTSOIL RESISTIVITYThe procedure for defining a precise expression forapparent soil resistivity consists of several steps. Inthe first step, it is necessary to find a simpler expressionwith the help of which potential can be easilydefined in the vicinity a rod electrode from which currentI flows, when it is in an infinite medium. In thesecond step, it is necessary to take into account theinfluence of the soil-air boundary by using the methodof images and the determination of an expressionfor electrical scalar potential on the site of one of thevoltage measurement electrodes. The third step consistsof taking into account one more discontinuityboundary of soil resistivity, which is found below theupper soil layer. As in the previous step in the appliedmethod of images, it is necessary to define the potentialin the upper soil layer (Area b) at the site of oneof the voltage measurement electrodes. In the finalstep, based on the previously defined expressions, itis necessary to define an expression for the apparentresistivity of two-layer soil measured using theWenner electrode arrangement (Figure 1).The electrical scalar potential in the vicinity of a rodelectrode of the final length l, from which current Iflows, located in an infinite medium of resistivity gshown in Figure 4a according to [5], is as follows:(3)737BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

gdje je:where:I glc∑ struja koja istjeËe iz povrπine vodiËa (A),∑ specifiËni elektriËni otpor gornjeg sloja tla(Ωm),∑ duljina vodiËa elektrode (m),∑ pola duljine vodiËa strujne elektrode (l=2c).I glc∑ the current that flows from the conductorsurface (A),∑ the resistivity of the upper soil layer (Ωm),∑ the length of the electrode (m),∑ half the length of the current electrode(l=2c) (m).Slika 4Ravni πtap irelevantna geometrijaFigure 4Straight rod and therelevant geometrya) b)yT(x,y)lxl'yl=2clT(x,y)xStapanjem strujne elektrode duljine l s njezinomslikom novonastala elektroda ima duljinu l'=2l(slika 4b), a ukupna struja koja istjeËe iz nastalogkonglomerata iznosi I=2I, te je potencijal uokolini takve elektrode prema slici 4b za x>0 danizrazom:With the merging of a current electrode of the length l ofwith its image, the new electrode has the length of l'=2l(Figure 4b), and the total current that flows from thenew conglomerate amounts to I=2I, and the potential inthe vicinity of such an electrode according to Figure 4bfor x>0 is given with the following expression:(4)Naravno, pod duljinom elektrode koja se odslikalaod granice tlo-zrak razmatra se samo ukopanidio strujne elektrode u tlu (ono πto je izvor polja).Potencijal na mjestu naponske elektrode B zbogistjecanja struje iz strujne elektrode A stopljenesa svojom slikom koja se nalazi u podruËju a (slika4b), i utjecanja struje u strujnu elektrodu D stopljenesa svojom slikom koja se nalazi u podruËjua (slika 4b), u toËki T 1 (slika 2b) u jednoslojnomtlu iskazan geometrijom mjernog spoja glasi:Naturally, along the length of the electrode that ismirrored at the soil-air boundary, only the part ofthe electrode buried in the soil is considered (thatwhich is the source of the field). The potential atthe site of voltage electrode B due to the flow ofcurrent from current electrode A merged with itsimage that is located in Area a (Figure 4b), andthe inflow of current in electrode D merged with itsimage that is located in Area a (Figure 4b), at pointT 1 (Figure 2b) in one layer soil is expressed by thegeometry of the measurement circuit, as follows:(5)BariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753738

Zbog simetrije potencijal elektrode C suprotnogje predznaka od potencijala elektrode B, te vrijedi: C =– B . Potencijal B , odnosno potencijal Codreappleeni su na uzduænim osima naponskih mjernihelektroda B i C u toËki koja dodiruje granicutlo-zrak (slika 2b). Meappleutim, treba imati na umuda potencijal treba odrediti metodom srednjegpotencijala, jer se prema izrazu (4) potencijalmijenja duæ uzduæne osi naponske elektrode.NaoËigled je ova konstatacija zbunjujuÊa, jer sesve toËke na idealnom vodiËu pri elektrostatskimuvjetima nalaze na jednakom potencijalu. Da bisve toËke vodiËa naponskih elektroda bile na jednakompotencijalu, bilo bi potrebno vrπiti odslikavanjeod cilindriËne granice, tj. strujnih elektrodaod povrπine naponskih elektroda. Meappleutim, kakoje duljina naponskih mjernih elektroda konaËna,takvo odslikavanje je teπko provedivo. Zanemarivanjemodslikavanja strujnih elektroda od naponskihmjernih elektroda, vodiËi naponskih mjernih elektrodapresijecaju razliËite ekvipotencijalne krivuljeuzrokovanih strujnim elektrodama (slika 2b). Iakosu sve toËke na idealnom vodiËu u statiËkom poljuna jednakom potencijalu, matematiËki nije jednostavnoodrediti iznos tog potencijala. U takvimsluËajevima ukoliko je moguÊe koristi se metodasrednjeg potencijala. Srednja vrijednost potencijalapredstavlja nepristranu procjenu potencijalana kojemu se nalazi vodiË naponske mjerneelektrode.Uzmimo da je duljina strujnih mjernih elektroda l,a naponskih mjernih elektroda d (slika 2a).Tada srednji potencijal na mjestu naponske elektrodeB zbog struje konglomerata elektrode A injezine slike u podruËju a, te konglomerata strujneelektrode D i njezine slike u podruËju a glasi:Due to the symmetry, the potential of electrode C isopposite to that of the of the potential of electrodeB, as follows: C =– B . Potential B , or potential Care defined along the axes of voltage measurementelectrodes B and C at the points that touch the soilairboundary (Figure 2b). However, it is necessary tobear in mind that the potential should be determinedby the mean potential method, because according toExpression (4), the potential changes along the lengthof the axis of the voltage electrode. Apparently thisstatement is confusing because all the points on theideal conductor under electrostatic conditions are locatedat the same potential. In order for all the pointson voltage electrodes to be at the same potential, itwould be necessary to perform the method of imagesfrom the cylindrical boundary, i.e. current electrodesfrom the surface of the voltage electrodes. However,since the length of the voltage measurement electrodesis finite, in such case it is difficult to performthe method of images. By ignoring the image of thecurrent electrodes from the voltage measurementelectrodes, the buried part (conductor) of the voltagemeasurement electrodes intersect with various equipotentialcurves caused by the current electrodes (Figure2b). Although all the points on an ideal conductor in astatic field are of an equal potential, mathematically itis not simple to determine the value of this potential.In such cases, insofar as possible, the mean potentialmethod is used. The mean potential value representsan unbiased assessment of the potential of the buriedpart of the voltage measurement electrode.Let us say that the length of the current measurementelectrodes is l and the voltage measurementelectrodes is d (Figure 2a).Then the mean potential at the site of the voltageelectrode B due to the current conglomerate ofelectrode A and its image in Area a, and the conglomerateof current electrode D and its image inArea a is as follows:(6)Srednji potencijal elektrode B zbog struja konglomerataoriginalne elektrode A, koja se nalazi upodruËju b i njezine zamiπljene slike koja se nalaziu podruËju a, te konglomerata originalne elektrodeD, koja se nalazi u podruËju b i njezine zamiπljeneslike koja se nalazi u podruËju a glasi:The mean potential of electrode B, due to the currentsfrom conglomerates of the original electrode A, whichis located in Area b and its imaginary image that is locatedin Area a, and the conglomerates of the originalelectrode D, which is located in Area b and its imaginaryimage that is located in Area a, is as follows:(7)739BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

Umjesto rjeπavanja integrala u izrazu (7), za kojegpostoji analitiËko rjeπenje, prikladnije je ishoditinumeriËko rjeπenje, tj. provesti numeriËku integraciju.Kako je podintegralna funkcija glatkafunkcija, umjesto sloæenijih postupaka numeriËkihintegracija (Trapezno pravilo, Simpsonovo pravilo,Romberg ) prikladnije je provesti jednostavnu integracijuprema pravokutnom pravilu [9]:Instead of solving the integral in Expression (7),for which there is an analytical solution, it is moreappropriate to obtain a numerical solution, i.e. toperform numerical integration. Since the subintegralfunction is a smooth function, instead of morecomplex procedures of numerical integration (thetrapezoidal, Simpson and Romberg rules), it ismore appropriate to perform a simple integrationaccording to the rectangular rule [9]:,(8)gdje je: ∑ korak tj., prostorna udaljenost izmeappleu dvijesusjedne toËke u kojima se uzima uzorakfunkcije f(x), a koji se nalazi na krivuljiintegriranja, x 1 i x 2 poËetna i krajnja toËkaduæ krivulje na kojoj se integrira funkcijaf(x),O() ∑ numeriËka pogreπka integracije, tj. odstupanjeod toËne vrijednosti integrala,M ∑ broj uzoraka funkcije na intervalu izmeappleutoËaka x 1 i x 2 .Naravno, zbog jednostavnije sheme numeriËkogintegriranja potreban je neπto veÊi broj uzoraka Mfunkcije f(x), ali je iskazivanje rjeπenja analitiËkimizrazom olakπano.Primjenom izraza (8) na izraz (7) srednji potencijalelektrode B zbog konglomerata originalnih strujnihelektroda A i D, koje se nalaze u podruËju b i njihovihzamiπljenih slika koje se nalaze u podruËjua glasi:where: ∑ step, i.e. the spatial distance between twoneighboring points in which a sample of functionf(x) is taken, which is on the integrationcurve, x 1 and x 2 the initial and final pointsalong the curve on which the function f(x) isintegrated,O() ∑ the numerical integration error, i.e. the deviationfrom the precise value of the integral,M ∑ the number of the samples of the functionbetween points x 1 and x 2 .Naturally, due to the simpler scheme of numerical integration,a somewhat larger number of samples of theM function f(x) are required, but the presentation ofthe solution is easier with the analytical expression.By applying Expression (8) to Expression (7), themean potential of electrode B, due to the conglomerateof the original current electrodes A and D,which are located in Area b and their imaginary imagesthat are located in Area a, is as follows:(9)gdje je korak duæ osi naponske mjerne elektrodeB dan izrazom:where step along the axis of the voltage measurementelectrode B is given with the following expression:(10)BariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753740

u kojemu je:dM∑ duljina ukopanog dijela naponske mjerneelektrode B,∑ broj uzoraka, tj. toËaka promatranja potencijala,a koje se nalaze na uzduænojosi ukopanog dijela naponske elektrode B.U dvoslojnom tlu situacija je neπto sloæenija, jerna potencijal naponske mjerne elektrode utjeËuzamiπljene slike zbog viπestrukih odslikavanja, akoje se nalaze u podruËjima a i c. Utjecaj slikastrujnih elektroda A i D u podruËjima a i c moæese uvaæiti neovisno te se u konaËnom izrazu za potencijalalgebarski zbrajaju zajedno s potencijalomodreappleenim izrazom (9). Srednji potencijal zbogslika iz podruËja a prema izrazu (8) iznosi:in which:dM∑ the length of the buried part of voltage measurementelectrode B,∑ the number of samples, i.e. the points inwhich potential is observed, which are locatedon the longitudinal axis of the buriedpart of charge electrode B.In two-layer soil, the situation is somewhat morecomplex because the potential of the voltage measurementelectrode is influenced by the imaginary imagedue to multiple image sources, which are locatedin Areas a and c. The influence of the images of thecurrent electrodes A and D in Areas a and c can beconsidered independently and in the final expressionfor the potential, they are summed algebraically togetherwith the potential defined by Expression (9).The mean potential due to the image from Area aaccording to Expression (8) is as follows:(11)Srednji potencijal zbog zamiπljenih slika iz podruËjac prema izrazu (8) iznosi:The mean potential due to the imaginary images fromArea c according to Expression (8) is as follows:(12)ili prema:or according to:(13)Da bi se mogao odrediti izraz za srednji potencijalnaponskih mjernih elektroda prema posljednjimizrazima potrebno je odrediti analitiËki oblik izrazaza potencijal (x,y) u dvoslojnom tlu za podruËjeb, u kojemu se nalaze naponske mjerne elektrode,a zbog odslikanih strujnih elektroda A i D, koje senalaze u podruËjima a i c. Potencijal u podruËjub, na granici tlo-zrak na mjestu naponske mjerneelektrode B u toËki T 1 prema slici 2b, a zboguvaæenih N slika strujnih elektroda A i D koje senalaze u podruËju a glasi:In order to define the expression for the mean potentialof the voltage measurement electrodes accordingto the previous expressions, it is necessaryto define the analytical form of the expression forpotential (x,y) in two-layer soil for Area b, in whichthe voltage measurement electrodes are located,and due to the images of current electrodes A andD, which are located in Areas a and c. The potentialin Area b, at the soil-air boundary at the site ofthe voltage measurement electrode B in point T 1according to Figure 2b, and due to N images of thecurrent electrodes A and D that are located in Areaa, which are taken into consideration, is as follows:741BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

.(14)Potencijal u podruËju b, na granici tlo-zrak namjestu naponske mjerne elektroda B zbog slikastrujnih elektroda A i D, koje se nalaze u podruËjuc glasi:The potential in Area b, at the soil-air boundary atthe site of voltage measurement electrode B due tothe images of the current electrodes A and D, whichare located in Area c, is as follows:(15)Kao πto se to moæe zapaziti izrazi (14) i (15) suidentiËni, jer je potencijal odreappleen u toËki nagranici tlo-zrak. Meappleutim, kada se potencijalodreappleuje u bilo kojoj drugoj toËki unutar podruËjab izrazi (14) i (15) Êe se razlikovati. Prema slici2b izrazi (14) i (15) odnose se na potencijal utoËki T 1 . Ukoliko se prema navedenim izrazimapotencijal odreappleuje u toËki T 2 , tada je izraze (14)i (15) potrebno modificirati. U izrazu (14) umjesto2nh+l treba pisati 2nh+l+d, a umjesto 2nh-l trebapisati 2nh-l+d. U izrazu (15) umjesto 2nh+l trebapisati 2nh+l-d, a umjesto 2nh-l treba pisati 2nh-l-d.Na temelju iznesene logike razmiπljanja dolazi sedo izraza za srednje potencijale elektrode B zbogslika strujnih elektroda A i D, koje se nalaze u podruËjimaa i c, koji glase:As can be observed, Expressions (14) and (15) areidentical because the potential is defined at the pointat the soil-air boundary. However, when the potential isdefined at any other point within Area b, Expressions(14) and (15) will differ. According to Figure 2b,Expressions (14) and (15) refer to the potential atpoint T 1 . If according to the cited expressions the potentialis defined at point T 2 , then Expressions (14)and (15) must be modified. In Expression (14), insteadof 2nh+l it is necessary to write 2nh+l+d, andinstead of 2nh-l it is necessary to write 2nh-l+d. InExpression (15), instead of 2nh+l it is necessary towrite 2nh+l-d, and instead of 2nh-l it is necessary towrite 2nh-l-d. Based upon the logic presented, theexpression is obtained for the mean potential of electrodeB due to the images of the current electrodes Aand D, which are located in Areas a and c, as follows:(16)(17)u kojima je K=I g /4πl, a korak odreappleen izrazom(10).in which K=I g /4πl, and step is defined byExpression (10).BariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753742

Ukupni srednji potencijal zbog zamiπljenih slikaelektroda A i D koje se nalaze u podruËjima a i c tekonglomerata strujnih elektroda A i D s njihovomslikama koje se nalaze podruËju a, jednak je zbrojusrednjih potencijala —a B i —c B sa —b B. Takav bi izrazzbog svoje duljine postao nepregledan, a ujednone bi bio prilagoappleen za pisanje programa za raËunalokoriπtenjem programskih petlji. Naime, trebaimati na umu da izraz treba svesti na takvu formukoja je prikladna za pisanje programa na raËunalu,pogotovo za koriπtenje programskih petlji.SreÊom, moguÊe je napisati jezgrovitiji izraz zasrednji potencijal zbog slika iz podruËja a i c, akoji je istodobno prikladan za pisanje programa zaosobno raËunalo (PC) te on glasi:The total mean potential due to the imaginary imagesof electrodes A and D in Areas a and c and theconglomerate of current electrodes A and D with theirimages located in Area a, is equal to the sum of themean potentials —a B and —c B with —b B . Due to the lengthof such an expression, it is not easy to comprehendand would not be suitable for writing programs for acomputer using program loops. It is necessary to bearin mind that an expression must be reduced to a formthat is suitable for writing programs on a computer,especially for using program loops.Fortunately, it is possible to write a more concise expressionfor mean potential due to the images fromAreas a and c, which at the same time is suitable forwriting programs for a personal computer, as follows:(18)Kratkim osvrtom na prethodni izraz uoËavaju sesliËni Ëlanovi u razlomku pod logaritmom, πtootvara moguÊnost daljnjeg saæimanja dobivenogizraza na joπ jezgrovitiji oblik:Briefly reviewing the previous expression, it is evidentthat there are similar articles in the fractionunder the logarithm, thereby opening the possibilityfor further reduction of the expression obtained intoan even more concise form:(19)Ukoliko se u konaËnom izrazu za potencijal namjeravakoristiti jednadæba za srednji potencijalu obliku prikazanom jednadæbom (19), tada jejednadæbu (9) potrebno svesti na takav oblik, tj.:If it is intended to use an equation for the meanpotential in the form presented by equation (19) inthe final expression for potential, then Expression(9) must be reduced to such a form, i.e.:(20)Ukupni srednji potencijal naponske mjerne elektrodeB iznosi:The total mean potential of the voltage measurementelectrode B amounts to:(21)743BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

Kako je zbog geometrijske i elektriËne simetrije(slika 2a) potencijal elektrode C suprotnog predznakaod potencijala elektrode B, a time i srednjipotencijal, tj. — C = – — B, napon U — BC izmeappleu naponskihelektroda B i C, mjeren voltmetrom (slika 1)iznosi U — V = U — BC = 2 — B, a iskazan geometrijom mjernogspoja glasi:Since due to geometric and electrical symmetry(Figure 2a), the potential of electrode C is of theopposite sign of the potential of electrode B, andthereby the mean potential, i.e. — C = – — B, then voltageU — BC between voltage electrodes B and C measuredwith a voltmeter (Figure 1) amounts to U — V =U — BC = 2 — B, and expressed by the geometry of themeasurement circuit is as follows:(22)Ukoliko je iz nekog razloga prikladnija proπirenijaforma, napon U — BC glasi:If for some reason an expanded form is more suitable,U — BC is as follows:(23)Jednom odreappleen napon U BC = U — BC iskazan geometrijommjernog spoja omoguÊava odreappleivanjeizraza za prividni specifiËni elektriËni otpor (a)za svaku pojedinu mjernu tehniku, tj. rasporedelektroda. Prividni specifiËni otpor tla (a) iskazangeometrijskim faktorom F po definiciji glasi:Once voltage U BC = U — BC is determined, expressedby the geometry of the measurement circuit, it ispossible to determine an expression for apparentsoil resistivity (a) for each individual measurementtechnique, i.e. electrode arrangement. Apparentsoil resistivity (a) expressed by the geometric factorF is by definition as follows:(24)Kada je tlo homogeno, tada prividni specifiËni otportla (a) ne ovisi o razmaku elektroda, te je onjednak specifiËnom otporu tla gornjeg sloja tla, tj.(a) = g , odnosno vrijedi:When the soil is homogeneous, the apparent soilresistivity (a) does not depend upon the distancebetween the electrodes and is equal to the soil resistivityof the upper soil layer, i.e. (a) = g , or:(25)Odatle je:Thus:BariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753744

(26)Kada je specifiËni elektriËni otpor gornjeg slojatla g jednak specifiËnom elektriËnom otporu donjegsloja tla d tada je koeficijent odslikavanja βjednak nuli. Tada u jednadæbi (22) za napon U BCotpada Ëlan uz β, te se dobiva izraz za napon U — BCkada je tlo jednoslojno te on glasi:When the specific resistivity of the upper soil layer g equals the resistivity of the lower soil layer d ,the reflection coefficient (reflection factor) β equalszero. Thus, in equation (22) for voltage U BC the articlenext to β is eliminated and the expression U — BC isobtained for a one-layer soil, as follows:(27)Uvrπtavanjem dobivenog izraza (27) u jednadæbu(26) dobiva se:By including the obtained Expression (27) inExpression (26), the following is obtained:(28)ili u obliku:or in the following form:(29)gdje je novi geometrijski faktor C dan izrazom:where the new geometric factor C is given by thefollowing expression:(30)Sada je moguÊe koriπtenjem izraza (22), (24) i(29) odrediti prividni specifiËni elektriËni otpordvoslojnog tla u sluËaju kada je uvaæena duljinastrujnih i naponskih mjernih elektroda:Using Expressions (22), (24) and (29), it is nowpossible to determine the apparent resistivity of twolayersoil when the lengths of the current and voltagemeasurement electrodes are taken into account:(31)745BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

gdje je koeficijent odslikavanja β dan izrazom:where the imaging coefficient β is given by the followingexpression:(32)Dobiveni izraz je najtoËniji izraz kojim se odreappleujeprividni specifiËni otpor tla mjeren Wennerovimspojem. Vrijedi kako za male razlike izmeappleu elektrodaaa, odnosno, vrijediza sve razmake mjernih elektroda, jer je uvaæenutjecaj stvarne geometrije strujnih elektroda zaodreappleivanje napona U BC . Zbog svoje toËnosti dobiveniizraz Êe se smatrati referentnim izrazom,uz pomoÊ kojeg Êe se provjeravati toËnost izrazaza prividni specifiËni otpor tla prema IEEE normi.Jedina pretpostavka, a time i ograniËenje prikoriπtenju dobivenog izraza je da mjerne elektrodene prodiru u donji sloj tla.Meappleunarodnom normom IEEE Std. 81-1983 [1],prividni otpor tla odreappleuje se izrazom:The expression obtained is the most precise expressionwith which apparent soil resistivity is determinedwhen measured by the Wenner method. Itapplies for small differences between electrodesaa, i.e. it is valid for allspacing between measurement electrodes, becausethe influence of the actual geometry of the currentelectrodes for determining voltage U BC is takeninto account. Due to the precision of the expressionobtained, it will be considered as a referenceexpression, according to which the precision of theexpression for apparent soil resistivity according tothe IEEE standard will be checked. The only assumption,and thereby limitation, in using theobtained expression is that the measurement electrodesdo not penetrate into the lower soil layer.With the international IEEE Std. 81-1983 [1], apparentsoil resistivity is determined with the followingexpression:(33)koji je izveden pod pretpostavkom da se πtapneelektrode mogu zamijeniti s kuglastim. Izraz (33)ima upitnu toËnost za male razmake izmeappleuelektroda, tj. kada je a

5 NUMERI»KI PRIMJERIKoriπtenjem izraza (31) za prividni specifiËnielektriËni otpor dvoslojnog tla moguÊe je na numeriËkomprimjeru lako utvrditi granice valjanostii pogreπku koja nastaje koriπtenjem pojednostavljenogizraza prema meappleunarodnoj normi (33) [1]za razliËite sluËajeve koji mogu nastupiti u praksi.Kao πto je to i uobiËajeno granice valjanosti nekogizraza i/ili modela umjesto matematiËkog dokazanajlakπe je utvrditi usporeappleujuÊi dobivene rezultates toËnim modelom. Neka kao primjer posluæisluËaj kada je tlo dvoslojno, debljina gornjeg slojaiznosi 1 m, 2 m i 5 m, njegov specifiËni elektriËniotpor 200 Ωm ( g = 200 Ωm). Donji sloj tla nekaima specifiËni elektriËni otpor 100 Ωm ( d = 100Ωm), te se proteæe u beskonaËnost. Neka su sveËetiri elektrode mjernog spoja (sonde) jednake kaoi njihov ukopani dio. Duljina ukopanog dijela mjernihelektroda iznosi 0,4 m (l = d = 0,4 m).Za oËekivati je da Êe najveÊa pogreπka priodreappleivanju specifiËnog elektriËnog otpora tla teorijskimmodelom koji nadomjeπta πtapne elektrodes kuglastim nastupiti pri malim razmacima izmeappleustrujnih elektroda, tj. kada je udaljenost a izmeappleususjednih elektroda mjerljiva s duljinom strujnihi/ili naponskih mjernih elektroda. Iz tog razlogaanaliza je ograniËena na promatranje krivuljeprividnog specifiËnog elektriËnog otpora za malerazmake izmeappleu susjednih elektroda koje iznosiproizvoljno 10 m. Promatra se sluËaj kada je duljinacijelog oæiËenja mjernog spoja 30 m (slika 1).Pogreπka Êe ovisiti o nekoliko parametara: debljinigornjeg sloja tla, specifiËnom elektriËnom otporugornjeg sloja tla, specifiËnom elektriËnom otporudonjeg sloja tla, duljini naponskih mjernih elektrodate duljini strujnih mjernih elektroda i naravno oudaljenosti izmeappleu susjednih elektroda.Slika 5 prikazuje krivulje prividnih specifiËnih otporatla u funkciji udaljenosti izmeappleu susjednihelektroda. Krivulja prividnog specifiËnog otporatla dobivena prema izrazu (31) prikazana je crvenombojom, a ona dobivena prema IEEE normitj. prema izrazu (33) prikazana je plavom bojom.SpecifiËni otpor gornjeg sloja tla iznosi g = 200Ωm, njegova debljina h = 1 m, a specifiËni otpordonjeg sloja tla d = 100 Ωm. Brojevi iteracija suM = 5 i N = 20.5 NUMERICAL EXAMPLESBy using Expression (31) for the apparent resistivityof two-layer soil, it is easily possible to determinethe validity limits and error that occur on the numericalexample when using the simplified expressionaccording to the international standard (33)[1] for various instances that occur in practice.As is usually the case, the validity limits of an expressionand/or model instead of a mathematicalproof can most easily be determined by comparingthe results obtained with a precise model. A case inwhich the soil is two-layer, the thickness of the soilis 1 m, 2 m and 5 m; and its resistivity is 200 Ωm( g = 200 Ωm) is used as an example. Let the lowersoil layer have a resistivity of 100 Ωm ( d = 100Ωm) and extend to infinity. Let all four electrodes(probes) of the measurement circuit be equal, aswell as their buried part. The length of the buriedpart of the measurement electrodes amounts to 0,4m (l = d = 0,4 m).It is to be expected that the greatest error in determiningthe soil resistivity with a theoretical modelthat replaces rod electrodes with spherical oneswill occur when the spacing between the currentelectrodes is small, i.e. when distance a betweenneighboring electrodes is approximately the samelength of the current and/or voltage measurementelectrodes. For this reason, analysis is limited tostudying the apparent resistivity curve for smallspaces between neighboring electrodes that arbitrarilyamount to 10 m. A case is studied whenthe length of the entire wiring of the measurementcircuit is 30 m (Figure 1). The error will dependupon several parameters: the thickness of the uppersoil layer, the resistivity of the upper soil layer, theresistivity of the lower soil layer, the length of thevoltage measurement electrodes and the length ofthe current measurement electrodes and, naturally,the distance between neighboring electrodes.Figure 5 presents apparent soil resistivity curvesas functions of the distance between the neighboringelectrodes. The apparent soil resistivity curveobtained according to Expression (31) is shown inred, and that obtained according to the IEEE standard,i.e. according to Expression (33), is shown inblue. The resistivity of the upper soil layer amountsto g = 200 Ωm, its thickness is h = 1 m, and theresistivity of the lower soil layer is d = 100 Ωm.The number of iterations are M = 5 and N = 20.747BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

Slika 5Prividni specifiËniotpor tla premaizrazima (31) i (33)Figure 5Apparent soilresistivity accordingto expressions (31)and (33)Prividni specifiËni otpor /Apparent soil resistivity(Ωm)200180160140120IVh=1 m a a a g =200 Ωm d =100 Ωmd=l=0,4 mM=5, N=20prema izrazu (31)according toexpression (31)prema izrazu (33)according toexpression (33)1000 1 2 3 4 5 6 7 8 9 10(m)Udaljenost izmeappleu susjednih elektroda a / Spacing betweeen the neighboring electrodes aNa slici 6 prikazana je krivulja postotne pogreπkeu odreappleivanju prividnog specifiËnog otpora tla premaIEEE normi, tj. prema izrazu (33) u odnosu naizraz (31), u funkciji razmaka izmeappleu susjednihelektroda za parametre tla prema slici 5.In Figure 6, the percentage error curve is shown whenapparent soil resistivity is determined according tothe IEEE standard, i.e. according to Expression (33),in comparison to Expression (31), in the function ofthe distance between the neighboring electrodes forsoil parameters according to Figure 5.Slika 6Postotna pogreπkaIEEE izraza uodreappleivanju prividnogspecifiËnog otporaFigure 6Percentage error ofthe IEEE expressionin determiningapparent resistivityPostotna pogreπka /Percentage error(%)10-1-2 prema izrazu (33)/according to expression (33) (a) - prema izrazu (31)/according to expression (31) (a)p % (a)= ·100% prema izrazu (31)/according to expression (31) (a)-3-40 1 2 3 4 5 6 7 8 9 10(m)Udaljenost izmeappleu susjednih elektroda a / Spacing betweeen the neighboring electrodes aSlika 7 prikazuje krivulje prividnih specifiËnih otporatla u funkciji udaljenosti izmeappleu susjednihelektroda. Krivulja prividnog specifiËnog otporatla dobivena prema izrazu (31) prikazana je crvenombojom, a ona dobivena prema IEEE normi,tj. prema izrazu (33) prikazana je plavom bojom.SpecifiËni otpor gornjeg sloja tla iznosi g = 200Ωm, njegova debljina h = 2 m, a specifiËni otpordonjeg sloja tla d = 100 Ωm.Figure 7 shows the apparent soil resistivity curvesthe functions of the distance between neighboringelectrodes. The apparent soil resistivity curve obtainedaccording to Expression (31) is shown in redand that obtained according to the IEEE standard,i.e. according to Expression (33), is shown in blue.The resistivity of the upper soil layer amounts to g= 200 Ωm, its thickness is h = 2 m, and the resistivityof the lower soil layer is d = 100 Ωm.BariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753748

Prividni specifiËni otpor /Apparent soil resistivity(Ωm)200180160140120IVa a ah=2 md=l=0,4 mM=5, N=20Slika 7Prividni specifiËni otportla prema izrazima (31)i (33)Figure 7Apparent soil resistivityaccording to Expressions(31) and (33)prema izrazu (31)according toexpression (31)prema izrazu (33)according toexpression (33)1000 1 2 3 4 5 6 7 8 9 10(m)Udaljenost izmeappleu susjednih elektroda a / Spacing betweeen the neighboring electrodes aNa slici 8 prikazana je krivulja postotne pogreπkeu odreappleivanju prividnog specifiËnog otpora tla premaIEEE normi, tj. prema izrazu (33) u odnosu naizraz (31), u funkciji razmaka izmeappleu susjednihelektroda za parametre tla prema slici 7.In Figure 8, the percentage error curve is shown inthe determination of apparent soil resistivity accordingto the IEEE standard, i.e. according to Expression(33) in relation to Expression (31), as the function ofthe distance between the neighboring electrodes forthe soil parameters according to Figure 7.Postotna pogreπka /Percentage error(%)10-1-2-3-4p % (a)= ·100% prema izrazu (33)/according to expression (33) (a) - prema izrazu (31)/according to expression (31) (a) prema izrazu (31)/according to expression (31) (a)Slika 8Postotna pogreπka IEEEizraza u odreappleivanjuprividnog specifiËnogotporaFigure 8Percentage error ofthe IEEE expression incomparison to apparentresistivity-5-60 1 2 3 4 5 6 7 8 9 10(m)Udaljenost izmeappleu susjednih elektroda a / Spacing betweeen the neighboring electrodes aSlika 9 prikazuje krivulje prividnih specifiËnih otporatla u funkciji udaljenosti izmeappleu susjednihelektroda. Krivulja prividnog specifiËnog otporatla dobivena prema izrazu (31) prikazana je crvenombojom, a ona dobivena prema IEEE normi,tj. prema izrazu (33) prikazana je plavom bojom.SpecifiËni otpor gornjeg sloja tla iznosi g = 200Ωm, njegova debljina h = 5 m, a specifiËni otpordonjeg sloja tla d = 100 Ωm.Figure 9 presents apparent soil resistivity curves asfunctions of the distance between neighboring electrodes.The apparent soil resistivity curve obtainedaccording to Expression (31) is shown in red, andthe one obtained according to IEEE standard, i.e.according to Expression (33), is shown in blue. Theresistivity of the upper soil layer amounts to g =200 Ωm, its thickness is h = 5 m, and the resistivityof the lower soil layer is d = 100 Ωm.749BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

Slika 9Prividni specifiËniotpor tla premaizrazima (31) i (33)Figure 9Apparent soilresistivity accordingto Expressions (31)and (33)Prividni specifiËni otpor /Apparent soil resistivity(Ωm)200180160140120IVa a ah=5 md=l=0,4 mM=5, N=201000 1 2 3 4 5 6 7 8 9 10prema izrazu (31)according toexpression (31)prema izrazu (33)according toexpression (33)(m)Udaljenost izmeappleu susjednih elektroda a / Spacing betweeen the neighboring electrodes aNa slici 10 prikazana je krivulja postotne pogreπkeu odreappleivanju prividnog specifiËnog otpora tla premaIEEE normi, tj. prema izrazu (33) u odnosu naizraz (31), u funkciji razmaka izmeappleu susjednihelektroda za parametre tla prema slici 9.In Figure 10, the percentage error curve in the determinationof apparent soil resistivity accordingto the IEEE standard, i.e. according to Expression(33), is shown in comparison to Expression (31), asthe function of the distance between the neighboringelectrodes for the soil parameters according toFigure 9.Slika 10Postotna pogreπkaIEEE izraza uodreappleivanju prividnogspecifiËnog otporaFigure 10Percentage error ofthe IEEE expressionin the determinationof apparent resistivityPostotna pogreπka /Percentage error(%)10-1-2-3-4 prema izrazu (33)/according to expression (33) (a) - prema izrazu (31)/according to expression (31) (a)p % (a)= ·100% prema izrazu (31)/according to expression (31) (a)-5-60 1 2 3 4 5 6 7 8 9 10(m)Udaljenost izmeappleu susjednih elektroda a / Spacing betweeen the neighboring electrodes aBariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753750

6 ANALIZA REZULTATASlike 6, 8 i 10 ukazuju da pogreπka u odreappleivanjuprividnog specifiËnog otpora tla prema izrazu (33)u odnosu na izraz (31) moæe biti pozitivna, ili negativna,ovisno od sluËaja do sluËaja.Na slici 6 navedena pogreπka pri vrlo malim razmacimaizmeappleu dvije susjedne elektrode je pozitivna,zatim naglo postaje negativna te maksimumpogreπke nastupa kada je razmak izmeappleu susjednihelektroda pribliæno 2 m. Daljnjim poveÊanjemrazmaka izmeappleu susjednih elektroda pogreπka sesmanjuje te teæi ka nuli. Ovako vladanje pogreπkeu odreappleivanju prividnog specifiËnog otpora tla premaizrazu (33) u odnosu na izraz (31) tipiËno je iza ostale analizirane sluËajeve.Kada su mjerne elektrode jako blizu (a

Osim toga, zbog velike geometrijske udaljenostiodslikanih zamiπljenih slika strujnih elektroda odnaponskih elektroda, neznatna je promjena potencijaladuæ uzduænih osi naponskih elektroda. Iztog razloga nije velika razlika izmeappleu potencijalaodreappleenog u jednoj toËki na granici tlo-zrak, ilimetodom srednjeg potencijala kada je velik razmakizmeappleu mjernih elektroda. Zanimljivo je zaprimijetiti da maksimum pogreπke nastupa kadaje razmak izmeappleu susjednih elektroda pribliænojednak dvostrukoj debljini gornjeg sloja tla.Prve slike izvora polja elektroda A i D u podruËjimaa i c imaju najveÊi doprinos korekciji potencijala,u prvom redu zbog njihove relativne blizinespram naponskih mjernih elektroda u odnosu naostale slike, te neznatno oslabljene struje βI, kojase uvaæava u raËunu. Ekvipotencijalne krivulje prvihslika elektroda A i D u podruËjima a i c namjestu naponskih elektroda presijecaju uzduænuos naponskih mjernih elektroda u viπe toËaka (slika2b). Navedeni utjecaj uvaæen je koriπtenjemmetode srednjeg potencijala. Kako su prve slikestrujnih elektroda A i D udaljene od granice tlozrakupravo za dvostruku debljinu gornjeg sloja tlarazumljiva je pojava maksimuma pogreπke kada jerazmak izmeappleu susjednih elektroda jednak dvostrukojdebljini gornjeg sloja tla. Upravo pri tomrazmaku najizraæeniji je utjecaj prvih slika strujnihelektroda.Zanimljivo je za primijetiti da je pogreπka IEEEizraza prevladavajuÊe negativna. Razlog tome jeπto potencijal odreappleen izrazom za πtap logaritamskiopada pri poveÊanju udaljenosti toËaka promatranjaod njegove uzduæne osi, a za kuglu, tj.toËkaste izvore s kvadratom udaljenosti, odnosnopri malim udaljenostima, opada bræe nego li zaπtap. Na taj naËin je potencijal izraËunat izrazimaza toËkaste izvore polja pri istoj struji manji, te jeiz tog razloga i specifiËni elektriËni otpor niæi.7 ZAKLJU»AKIznenaappleujuÊe je da pogreπka koja se dobivakoriπtenjem izraza iz IEEE norme nije prelazila10 %, za brojne analizirane sluËajeve, odnosno6 % u prikazanim primjerima. Na temelju ovogËlanka i njegovih rezultata neosporna i razvidnaje svrhovitost uporabe IEEE izraza. Autori su dalisvoj doprinos u odgonetanju desetljeÊima starezagonetke o toËnosti IEEE izraza za prividni specifiËnielektriËni otpor dvoslojnog tla. Ujedno, danje alternativni izraz za odreappleivanje prividnog specifiËnogotpora dvoslojnog tla mjeren Wennerovimspojem, koji je prema trenutnim spoznajama autorajedinstven.Moreover, due to the great geometric distance of theimaginary images of the current electrodes from thevoltage electrodes, the change in potential is insignificantalong the longitudinal axis of the voltageelectrodes. For this reason, there is not a large differencebetween the potential determined at one pointon the soil-air boundary or by the mean potentialmethod when there is a great distance between themeasurement electrodes. It is interesting that themaximum error occurs when the distance betweenneighboring electrodes is approximately twice thethickness of the upper soil layer.The first images of the field sources of electrodes A andD in Areas a and c make the greatest contribution tothe correction of the potential, in the first place dueto their relative vicinity to the voltage measurementelectrodes in comparison to the other images, and theinsignificant reduction in current βI, which is taken intoaccount in the calculation. The equipotential curves ofthe first images of electrodes A and D in Areas a andc at the site of the voltage electrodes cross the longitudinalaxis of the voltage measurement electrodes atseveral points (Figure 2b). This influence is taken intoaccount by using the mean potential method. Since thefirst images of current electrodes A and D are at a distancefrom the soil-air boundary of twice the thicknessof the upper soil layer, the occurrence of the maximumerror is understandable when the distance between theneighboring electrodes is equal to twice the thicknessof the upper soil layer. It is precisely at this distancethat the influence of the first images of the currentelectrodes is the most marked.Interestingly, the error of the IEEE expression ispredominantly negative. The reason for this is thatthe potential determined by the expression for therod is logarithmically decreased with an increase inthe distance of the observation point from its longitudinalaxis, and for a sphere, i.e. point sourceswith the square of the distance or at small distancesdecreases more rapidly than for a rod. In this manner,the potential calculated with the expressionsfor point sources at the same current is lower, andfor this reason the resistivity is lower.7 CONCLUSIONIt is surprising that the error obtained by using theexpression from the IEEE standard did not exceed10% in numerous cases analyzed, i.e. 6% in the examplespresented. Based upon this article and theirresults, the purpose of using the IEEE expression isindisputable and clear. The authors have contributedto deciphering the decades-old enigma regarding theprecision of the IEEE expression for the apparentresistivity of two-layer soil. An alternative expressionBariÊ, T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753752

Iako je ovim Ëlankom potvrappleena primjenjivostIEEE izraza, to ne znaËi da svaki izraz predloæennormom treba prihvatiti bez kritiËkog osvrta. Naime,pri izdavanju normi i preporuËenih izraza zauporabu treba imati na umu fizikalne pretpostavkepod kojim one vrijede, te ponekad i ograniËenjamatematiËkog instrumentarija koji su ukljuËeni unjihovu uporabu.Iako se dobiveni izraz (31) za prividni specifiËnielektriËni otpor, moæe Ëiniti znatno sloæenijim odizraza (33) prema IEEE normi, treba imati na umuda se danas proraËuni ove vrste odvijaju iskljuËivona osobnom raËunalu (PC). Iz tog razloga neπtosloæeniji izraz nije zapreka njegovoj uporabi, pogotovoukoliko se ima na umu da je on svojomformom prilagoappleen pisanju programskih petlji, teje time programiranje elegantno.Autori ovog Ëlanka se nadaju da Êe dobiveni alternativniizraz kao i prikazana metodologijauvaæavanja stvarne geometrije elektroda i pri nekimdrugim mjernim postupcima omoguÊiti zainteresiranimËitateljima da na beskompromisannaËin pristupaju svojim zadaÊama.has also been provided for the determination of theapparent resistivity of two-layer soil measured by theWenner method, which is unique to the best knowledgeof the authors.Although this article affirms the applicability ofthe IEEE expression, this does not mean that everyexpression proposed by a standard should be acceptedwithout critical examination. When assessingstandards and recommended expressions foruse, it is necessary to bear in mind the physicalassumptions according to which they are valid, andsometimes the limitations of the mathematical instrumentationthey use.Although the obtained Expression (31) for apparentresistivity may seem significantly more complex thanExpression (33) according to the IEEE standard, itshould remembered that calculations of this type areperformed today exclusively on personal computers.For this reason, the somewhat greater complexity ofthe expression should not be an obstacle to its use,especially because its form has been adapted to writingprogram loops and elegant programming.The authors of this article hope that the alternativeexpression and methodology presented, whichtake the actual geometry of electrodes and othermeasurement procedures into account, will helpinterested readers approach their tasks in an uncompromisingmanner.LITERATURA / REFERENCES[1] IEEE Std 81-1983, IEEE Guide for Measuring Earth Resistivity, Ground Impedance and Earth SurfacePotentials of a Ground System, (Revision of IEEE Std. 81 -1962), The Institute of Electrical andElectronics Engineers, Inc., New York, 1983[2] WENNER, F., A Method for Measuring Earth Resistivity, Bureau of Standards scientific paper, 1915,No. 258, Washington, D.C., 1915[3] LAGACÉ, P. J., FORTIN, J., CRAINIC, E. D., Interpretation of Resistivity Sounding Measurements in N-Layer Soil using Electrostatic Images, IEEE Transactions on Power Delivery, Vol.11, No. 3, July 1996[4] VAN NOSTRAND, R. G.,COOK, K. L., Interpretation of Resistivity Data, Geological Survey professionalpaper 499, US Department of the Interior, Washington, D.C., 1966[5] HAZNADAR, Z., ©TIH, Æ., Elektromagnetizam 1, ©kolska knjiga, Zagreb, 1997.[6] SENIOR, T. B. A., VOLAKIS, J. L., Approximate boundary conditions in electromagnetics, The Institutionof electrical Engneers, London, 1995[7] BARIΔ, T., JOZSA, L., GLAVA©, H., Kapacitivni utjecaj visokonaponskih nadzemnih vodova na mjerenjespecifiËnog otpora tla, Energija, god. 56(2007), broj 2, Zagreb, 2007.[8] BARIΔ, T., BORAS, V., GALIΔ, R., Nadomjesni model tla zasnovan na umjetnim neuronskim mreæama,Energija, god. 56(2007), broj 1, Zagreb, 2007.[9] HARRIS, J. W., STOCKER, H., Handbook of Mathematics and Computational Science, 1998 Springer∑ Verlag, New York, Inc.Uredniπtvo primilo rukopis:2007-10-09PrihvaÊeno:2007-11-28Manuscript received on:2007-10-09Accepted on:2007-11-28753BariÊ,T., Šljivac, D., Stojkov, M., Granice valjanosti izraza za mjerenja specifiËnog otpora …, Energija, god. 56(2007), br. 6., str. 730-753BariÊ, T., Šljivac, D., Stojkov, M., Validity Limits of the Expression for Measuring Soil Resistivity …, Energija, vol. 56(2007), No. 6, pp. 730-753

UPUTSTVO ZA RUKOPISUPUTEAUTORIMA1. »asopis Energija objavljuje Ëlanke koji do sada nisu objavljeni u nekom drugom Ëasopisu.2. Radovi se piπu na hrvatskom ili engleskom jeziku, u treÊem licu, na jednoj stranici papira, poËinju suvodom i zavrπavaju sa zakljuËkom, u dvostrukom proredu i s dostatnim marginama. Stranicese oznaËavaju uzastopnim brojevima.3. Radovi u pravilu ne mogu biti dulji od 14 stranica Ëasopisa Energija (oko 9000 rijeËi).4. Ime i prezime autora, znanstvena ili struËna titula, naziv i adresa tvrtke u kojoj autor radi i e-mailadresa navode se odvojeno.5. Iznad teksta samoga rada treba biti saæetak od najviπe 250 rijeËi. Saæetak treba biti zaokruæena cjelinarazumljiva prosjeËnom Ëitatelju izvan konteksta samoga rada. Nakon saæetka navode se kljuËne rijeËi.6. »lanci se piπu u Word-u sa slikama u tekstu ili u posebnim file-ovima u tiff formatu, 1:1, rezolucijenamanje 300 dpi.7. »lanci se piπu bez biljeπki na dnu stranice.8. MatematiËki izrazi, grËka slova i drugi znakovi trebaju biti jasno napisani s dostatnim razmacima.9. Literatura koja se koristi u tekstu navodi se u uglatoj zagradi pod brojem pod kojim je navedena nakraju Ëlanka. Koriπtena literatura navodi se na kraju Ëlanka redom kojim je spomenuta u Ëlanku. Akorad na koji se upuÊuje ima tri ili viπe autora, navodi se prvi autor i potom et al. Nazivi Ëasopisa senavode u neskraÊenom obliku.»asopis[1] FRAZIER, L., FODOR, J. D., The sausage machine: A new two-stage parsing model. Cognition,6 (1978), 291∑ 325Knjiga[5] NAGAO, M., Knowledge and Inference. Academic Press, Boston, 1988Referat[7] WATROUS, R. L., SHASTRI, L., Learning phonetic features using connectionist networks:An experiment in speech recognition. Presented at the Proceedings of the IEEE InternationalConference on Neural Networks, (1987) San Diego, CANeobjavljeno izvjeπÊe/teze[10] ROZENBLIT, J. W., A conceptual basis for model-based system design. PhD Thesis, WayneState University, Detroit, Michigan, 198510. »lanak je prihvaÊen za objavljivanje ako ga pozitivno ocijene dva struËna recenzenta. U postupkurecenzije Ëlanci se kategoriziraju na sljedeÊi naËin:∑ izvorni znanstveni Ëlanci ∑ radovi koji sadræe do sada joπ neobjavljene rezultate izvornih istraæivanja upotpunom obliku,∑ prethodna priopÊenja ∑ radovi koji sadræe do sada joπ neobjavljene rezultate izvornih istraæivanja upreliminarnom obliku,∑ pregledni Ëlanci ∑ radovi koji sadræe izvoran, saæet i kritiËki prikaz jednog podruËja ili njegova dijelau kojem autor i sam aktivno sudjeluje ∑ mora biti naglaπena uloga autorovog izvornog doprinosa u tompodruËju u odnosu na veÊ objavljene radove, kao i pregled tih radova,∑ struËni Ëlanci ∑ radovi koji sadræe korisne priloge iz struke i za struku, a ne moraju predstavljatiizvorna istraæivanja.11. »lanci se lektoriraju i provodi se metroloπka recenzija.12. »lanci se dostavljaju u elektroniËkom obliku i 1 primjerak u tiskanom obliku na adresu:HEP d.d. - EnergijaN/r tajnika UreappleivaËkog odbora ∑ mr. sc. Slavica Barta-KoπtrunUlica grada Vukovara 37, 10000 Zagreb, HrvatskaTel.: +385 (1) 632 2641Faks: +385 (1) 617 0438e-mail: slavica.barta@hep.hrKOREKTURA I AUTORSKI PRIMJERCI1. Autori su duæni izvrπiti korekturu svoga rada prije objavljivanja. VeÊe promjene teksta u toj fazi neÊese prihvatiti.2. Autori dobivaju besplatno 5 primjeraka Ëasopisa u kojemu je obljavljen njihov Ëlanak. Naknada zaobjavljeni Ëlanak obraËunava se prema Odluci o visini autorskih honorara Ëasopisa Energija.AUTORSKO PRAVO1. Autorsko pravo na sve obljavljene materijale ima Ëasopis Energija.2. Autori moraju telefaksom dostaviti popunjeni obrazac o autorskom pravu nakon prihvaÊanja Ëlanka.3. Autori koji æele koristiti materijale koji su prethodno objavljeni u Ëasopisu Energija trebaju seobratiti izdavaËu.VujeviÊ, D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711754

MANUSCRIPTS1. Energija journal publishes articles never before published in another periodical.2. Articles are written in Croatian or English, in the third person, on one paper side, beginning withan introduction and ending with a conclusion, with double line spacing and adequate margins. Pagesare numbered consecutively.3. As a rule articles cannot exceed 14 pages of the Energija journal (about 9000 words).4. The name of the author and his/her academic title, the name and address of thecompany of the author’s employment, and e-mail address, are noted separately.5. The text of the article is preceded by a summary of max. 250 words. The summary is a rounded offwhole comprehensible to an average reader apart from the context of the article. The summary isfollowed by the listing of the key words.6. Articles are written in MS Word with pictures embedded or as separate TIFF fi les, 1:1, min. 300 dpi.7. Articles are written without bottom-of-page footnotes.8. Mathematical expressions, Greek letters and other symbols must be clearly written with suffi cientspacing.9. The sources mentioned in the text of the article are only to be referenced by the number under whichit is listed at the end of the article. References are listed at the end of the article in the order in whichthey are mentioned in the text of the article. If a work referenced has three or more authors, the fi rstauthor is mentioned followed by the indication et al. Names of journals are given in full.INSTRU-CTIONS TOAUTHORSJournal[1] FRAZIER, L., FODOR, J. D., The sausage machine: A new two-stage parsing model.Cognition, 6 (1978), 291∑325Book[5] NAGAO, M., Knowledge and Inference. Academic Press, Boston, 1988Conference paper[7] WATROUS, R. L., SHASTRI, L., Learning phonetic features using connectionist networks:An experiment in speech recognition. Presented at the Proceedings of the IEEE International Conference on Neural Networks, (1987) San Diego, CAUnpublished report/theses[10] ROZENBLIT, J. W., A conceptual basis for model-based system design. PhD Thesis, WayneState University, Detroit, Michigan, 198510. An article will be accepted for publication if it is positively evaluated by two professional reviewers. Inthe review, articles are categorised as follows:∑ original scientifi c articles ∑ works containing hitherto unpublished full results of original research,∑ preliminary information ∑ works containing hitherto unpublished preliminary results of originalresearch,∑ review articles ∑ works containing the original, summarized and critical review from the fi eld orfrom a part of the fi eld in which the author of the article is himself/herself involved ∑ the role of theauthor’s original contribution to the fi eld must be noted with regard to already published works,and an overview of such works provided,∑ professional articles ∑ works containing useful contributions from the profession and for theprofession, not necessarily derived from original research.11. Articles will undergo language editing and metrological reviews.12. Articles are to be submitted in a machine-readable form plus one printout to the following address:HEP d.d. - EnergijaN/r tajnika UreappleivaËkog odbora ∑ mr. sc. Slavica Barta-KoπtrunUlica grada Vukovara 37, 10000 Zagreb, CroatiaTel.: +385 (1) 632 2641Fax: +385 (1) 617 0438e-mail: slavica.barta@hep.hrCORRECTIONS AND FREE COPIES FOR AUTHORS1. Authors are required to make the corrections in their works prior to publication. Major alterationsof the text at the stage of publication will not be accepted.2. Authors will receive free of charge 5 copies of the Journal in which their respective articlesappear. The fee for an article published will be calculated in accordance with the Decision on theFees for the Authors of the Energija journal.COPYRIGHT1. The copyright on all the materials published belongs to the Energija journal.2. Authors must fax in a fi lled out copyright form when their articles have been accepted.3. Authors wishing to use the materials published in the Energija journal need to contact thepublisher.VujeviÊ, 755D., Primjena Möbiusove vrpce …, Energija, god. 56(2007), br. 6., str. 700-711VujeviÊ, D., Application of the Möbius Strip …, Energija, vol. 56(2007), No. 6, pp. 700-711

ENERGIJA, godiπte 56(2007)SADRÆAJENERGIJA, volume 56(2007)CONTENTSENERGIJA 0101-3940-6364-9596-113114-133©tritof, I., KleËina, F.MODEL POTICAJNE REGULACIJE U PRIJENOSU ELEKTRI»NEENERGIJE U REPUBLICI HRVATSKOJ(pregledni Ëlanak)Krajcar, S., Skok, M., AndroËec, I., Blagajac, S., Solem, G.,Livik, K., Morland, B.TRENING CENTAR ZA TRGOVANJE ENERGIJOM(pregledni Ëlanak)MileusniÊ, E.PRIJEDLOG UNAPRE–ENJA PRAVILNIKA O ZA©TITI ODELEKTROMAGNETSKIH POLJA(pregledni Ëlanak)BariÊ, T., Boras, V., GaliÊ, R.NADOMJESNI MODEL TLA ZASNOVAN NA UMJETNIMNEURONSKIM MREÆAMA(prethodno priopÊenje)Uran, V.PRINCIP IZVO–ENJA OPCIJA NA TRÆI©TU ELEKTRI»NEENERGIJE(pregledni Ëlanak)©tritof, I., KleËina, F.MODEL FOR INCENTIVE REGULATION IN THE TRANSMISSION OFELECTRICAL ENERGY IN THE REPUBLIC OF CROATIA(review article)Krajcar, S., Skok, M., AndroËec, I., Blagajac, S., Solem, G., Livik,K., Morland, B.TRAINING CENTRE FOR ENERGY TRADING(review article)MileusniÊ, E.PROPOSAL FOR REVISING THE CROATIAN REGULATIONS ONPROTECTION FROM ELECTROMAGNETIC FIELDS(review article)BariÊ, T., Boras, V., GaliÊ, R.SUBSTITUTIONAL MODEL OF THE SOIL BASED ON ARTIFICAL NEURALNETWORKS(preliminary information)Uran, V.THE PRINCIPLE OF EXERCISING OPTIONS ON THE ELECTRICITYMARKET(review article)ENERGIJA 02144-181182-215216-231232-249250-259IliÊ, I., MaljkoviÊ, Z., Gaπparac, I., Pavlica, M., IliÊ-ZuboviÊ,D., JariÊ, V., ViπkoviÊ, A., BelobrajdiÊ, R.METODOLOGIJA ODRE–IVANJA KORISNI»KE POGONSKE KARTEHIDROAGREGATA(pregledni Ëlanak)PaviÊ, A., TrupiniÊ, K.GUBICI ELEKTRI»NE ENERGIJE U DISTRIBUCIJSKOJ MREÆI(pregledni Ëlanak)JerbiÊ, G.PRIMJENA TRANSFORMATORA S POPRE»NOM REGULACIJOM UHRVATSKOM ELEKTROENERGETSKOM SUSTAVU(pregledni Ëlanak)BariÊ, T., Glavaπ, H., Jozsa, L.KAPACITIVNI UTJECAJ VISOKONAPONSKIH NADZEMNIHVODOVA NA MJERENJE SPECIFI»NOG OTPORA TLA(izvorni znanstveni Ëlanak)KuzmanoviÊ, B., FerkoviÊ, L., Baus, Z.DEFINICIJA AMPERA JE ZBUNJUJUΔA(izvorni znanstveni Ëlanak)IliÊ, I., MaljkoviÊ, Z., Gaπparac, I., Pavlica, M., IliÊ-ZuboviÊ, D.,JariÊ, V., ViπkoviÊ, A., BelobrajdiÊ, R.METHODOLOGY FOR DETERMINING THE ACTUAL PQ DIAGRAM OF AHYDROGENERATOR(review article)PaviÊ, A., TrupiniÊ, K.ELECTRICAL ENERGY LOSSES IN THE DISTRIBUTION NETWORK(review article)JerbiÊ, G.APPLICATION OF PHASE SHIFTING TRANSFORMERS IN THECROATIAN POWER SUPPLY SYSTEMS(review article)BariÊ, T., Glavaπ, H., Jozsa, L.CAPACITIVE INFLUENCE OF HIGH VOLTAGE OVERHEADTRANSMISSION LINES ON THE MEASUREMENT OF SOIL RESISTIVITY(original scientifical article)KuzmanoviÊ, B., FerkoviÊ, L., Baus, Z.THE DEFINITION OF AMPER IS CONFUSING(original scientifical article)756

ENERGIJA 03268-291292-327328-345346-373374-387Kennedy M. W., StaniÊ, Z.ENERGETSKA POLITIKA U EUROPI I NJEN UTJECAJ NA OPSKRBUELEKTRI»NOM ENERGIJOM(pregledni Ëlanak)Zeljko, M., »anoviÊ, M.ENERGETSKI SEKTOR CRNE GORE ∑ STANJE I PERSPEKTIVE(pregledni Ëlanak)De Paoli, L., ViπkoviÊ A.JAVNA POTPORA RAZVITKU OBNOVLJIVIH IZVORA ENERGIJE(pregledni Ëlanak)LugariÊ, L., Krajcar, S., ΔurkoviÊ, A.ANALIZA FINANCIJSKOG RIZIKA U VREDNOVANJU PROJEKATAIZGARDNJE VJETROELEKTRANA(izvorni znanstveni Ëlanak)KuzmanoviÊ, B., Baus, Z., FerkoviÊ, L.RA»UNANJE SREDNJE I EFEKTIVNE VRIJEDNOSTI STRUJE(NAPONA) SLOÆENIH VALNIH OBLIKA(struËni Ëlanak)Kennedy M. W., StaniÊ, Z.ENERGY POLICY IN EUROPE AND ITS IMPACT ON ELECTRICITYSUPPLY(review article)Zeljko, M., »anoviÊ, M.THE CURRENT SITUATION AND FUTURE POTENTIAL OF THEMONTENEGRIN ENERGY SECTOR(review article)De Paoli, L., ViπkoviÊ A.PUBLIC SUPPORT FOR THE DEVELOPMENT OF RENEWABLE ENERGYSOURCES(review article)LugariÊ, L., Krajcar, S., ΔurkoviÊ, A.RISK ANALYSIS METHODOLOGIES FOR THE FINANCIAL EVALUATIONOF WIND ENERGY POWER GENERATION PROJECTS(original scientifical article)KuzmanoviÊ, B., Baus, Z., FerkoviÊ, L.CALCULATION OF THE MEAN AND EFFECTIVE CURRENT(VOLTAGE9VALUES OF COMPLEX WAVEFORMS(professional article)ENERGIJA 04398-431432-455456-489490-517Bogdan, Æ., ÆivkoviÊ, S. A., DokmanoviÊ, V., MeriÊ, J.TEHNOLOGIJE »ISTOG UGLJENA U STRATEGIJI RAZVOJAELEKTROENERGETSKOG SUSTAVA(pregledni Ëlanak)SprËiÊ, P., Krajcar, S.PRIMJENA IZVEDENICA U UPRAVLJANJU CJENOVNIM RIZIKOM UENERGETSKIM KOMPANIJAMA(pregledni Ëlanak)IliÊ, I., MaljkoviÊ, Z., Gaπparac, I., Pavlica, M., IliÊ-ZuboviÊ, D.,JariÊ, V., ViπkoviÊ, A., BelobrajiÊ, R.PRIMJER PROVEDBE ALGORITMA IZRADE KORISNI»KEPOGONSKE KARTE HIDROAGREGATA(izvorni znanstveni Ëlanak)MuæiniÊ, F., ©krlec, D.MODELIRANJE PROJEKTNIH RIZIKA U RAZVOJU PROJEKTAVJETROELEKTRANE(izvorni znanstveni Ëlanak)Bogdan, Æ., ÆivkoviÊ, S. A., DokmanoviÊ, V., MeriÊ, JCLEAN COAL TECHNOLOGIES IN THE STRATEGIC DEVELOPMENT OFTHE ELECTRICAL ENERGY SYSTEM(review article)SprËiÊ, P., Krajcar, S.THE APPLICATION OF DERIVATIVES BY ENERGY COMPANIES INPRICE RISK MANAGEMENT(review article)IliÊ, I., MaljkoviÊ, Z., Gaπparac, I., Pavlica, M., IliÊ-ZuboviÊ, D.,JariÊ, V., ViπkoviÊ, A., BelobrajiÊ, R.AN EXAMPLE OF APPLYING ALGORITHM TO CREATE THE ACTUAL PQDIAGRAM OF A HYDRO-GENERATOR(original scientifical article)MuæiniÊ, F., ©krlec, D.MODELING PROJECT RISKS IN THE DEVELOPMENT OF A WINDPOWER PLANT PROJECT(original scientifical article)757

526-553554-583584-607608-617618-633ENERGIJA 05Klepo, M.ULOGA REGULATORNOG TIJELA U DONO©ENJU TARIFNIHSTAVOVA: OKRUGLI STOL ∑ PRIKAZ I ZAKLJU»CI(pregledni Ëlanak©tritof, I., KleËina, F.REGULATORNA POLITIKA I NJEN UTJECAJ NA PLANOVERAZVOJA I IZGRADNJE REGULATORNIH ENERGETSKIHSUBJEKATA(pregledni Ëlanak)TokiÊ, A., UgleπiÊ, I.NUMERI»KI PRORA»UN NISKO FREKVENTNIHELEKTROMAGNETSKIH PRIJELAZNIH POJAVA U ENERGETSKIMTRANSFORMATORIMA(izvorni znanstveni Ëlanak)Papazoglou, Th. M., Papazoglou, M. T., Thalassinakis, E. J.,Tsichlakis, Ch., Hatziargyriou, N. D.DIJAGNOSTI»KI PREGLED JEDNOG ISPADAELEKTROENERGETSKOG SUSTAVA NA RODOSU(struËni Ëlanak)SutloviÊ, E., Petric, K.BAZA PODATAKA RELEJNE ZA©TITE(struËni Ëlanak)Klepo, M.THE ROLE OF THE REGULATORY AGENCY IN THE ADOPTION OF TARIFFSYSTEMS: ROUND TABLE DISCUSSION ∑ REPORT AND CONCLUSION(review article)©tritof, I., KleËina, F.REGULATORY POLICY AND ITS IMPACT ON THE DEVELOPMENT ANDCONSTRUCTION PLANS OF REGULATED ENERGY ENTITIES(original scientifical article)TokiÊ, A., UgleπiÊ, I.THE NUMERICAL CALCULATION OF LOW FREQUENCYELECTROMAGNETIC TRANSIENT PHENOMENA IN POWERTRANSFORMERS(original scientifical article)Papazoglou, Th. M., Papazoglou, M. T., Thalassinakis, E. J.,Tsichlakis, Ch., Hatziargyriou, N. D.DIAGNOSTIC REVIEW OF A BLACKOUT IN RHODES(professional article)SutloviÊ, E., Petric, K.RELAY PROTECTION DATABASE(profesional article)642-675676-699700-711712-729730-753ENERGIJA 06Rajπl, I., Krajcar, S., Krpan, K.PRIMJENA VI©EAGENTSKIH SUSTAVA U SIMULATORIMATRÆI©TA ELEKTRI»NOM ENERGIJOM(pregledni Ëlanak)GoiÊ, R., Jakus, D., MudniÊ, E.PRORA»UN GODI©NJIH GUBITAKA RADNE ENERGIJEU DISTRIBUCIJSKOJ MREÆI S PRIKLJU»ENOMVJETROELEKTRANOM(izvorni znanstveni Ëlanak)VujeviÊ, D.PRIMJENA MÖBIUSOVE VRPCE U ELEKTROTEHNICI(prethodno priopÊenje)ΔuÊiÊ, B.3D PRORA»UN KVAZISTATI»KOG MAGNETSKOG POLJAOKO VODI»A I FEROMAGNETSKE PLO»E INTEGRALNIMJEDNADÆBAMA(izvorni znanstveni Ëlanak)BariÊ, T., ©ljivac, D., Stojkov, M.GRANICE VALJANOSTI IZRAZA ZA MJERENJA SPECIFI»NOGOTPORA TLA WENNEROVOM METODOM PREMA IEEE NORMIStd. 81-1983(izvorni znanstveni Ëlanak)Rajπl, I., Krajcar, S., Krpan, K.APPLICATION OF MULTI-AGENT SYSTEMS IN ELECTRICITY MARKETSIMULATORS(review article)GoiÊ, R., Jakus, D., MudniÊ, E.CALCULATION OF ANNUAL ACTIVE ENERGY LOSSES IN ADISTRIBUTION NETWORK WITH A CONNECTED WIND POWER PLANT(original scientific article)VujeviÊ, D.APPLICATION OF THE MÖBIUS STRIP IN ELECTRICAL ENGINEERING(preliminary information)ΔuÊiÊ, B.THE 3D CALCULATION OF THE QUASISTATIC MAGNETIC FIELDAROUND A CURRENT CARRYING CONDUCTOR AND FERROMAGNETICPLATE BY MEANS OF INTEGRAL EQUATIONS(original scientific article)BariÊ, T., ©ljivac, D., Stojkov, M.VALIDITY LIMITS OF THE EXPRESSION FOR MEASURING SOILRESISTIVITY BY THE WENNER METHOD ACCORDING TO IEEESTANDARD 81-1983(original scientific article)758