Stochastic Analysis of Water Supply Systems Using ... - eWISA

STOCHASTIC ANALYSIS OF WATER SUPPLY SYSTEMS
USING MOCASIM II
H.A. Kretzmann, J.E. van Zyl and J. Haarhoff
RAU Water Research Group, Department of Civil and Urban Engineering, Rand Afrikaans University,
PO Box 524, Auckland Park 2006. Tel: (011) 489-2345. Fax: (011) 489-2148.
E-mail: jevz@ing.rau.ac.za
ABSTRACT
A software package called Mocasim II has been developed to perform stochastic analysis on water
supply systems. This allows the relationship between the reliability of the supply system and the
capacity of its service reservoir(s) to be quantified using Monte Carlo analysis. In a Monte Carlo
analysis the factors which influence the reliability of the system such as water demand, fires, and
pipe failures are simulated stochastically over a long period of time. A reliability/capacity
relationship is quantified by analyzing the failure behavior of different service reservoir sizes. A
previous version of Mocasim used a simple mass balance model for calculating the flows in simple,
linear distribution networks. Mocasim II extends the functionality of its predecessor by enabling the
probabilistic modeling of more complex water distribution models. This was achieved by
integrating the stochastic modelling technique into the Epanet hydraulic analysis software package.
Mocasim II was designed using an object-oriented model which has various advantages such as
ease of testing, upgrading and maintaining as well as minimum repetitive code and a logical
structure. Additional capabilities of Mocasim II include the determination of probability distributions
for network properties such as flow rate, pressure and water quality at any node in the network.
This will assist in estimating the levels of service of a water supply system.
INTRODUCTION
Water planners and managers in South Africa are faced with new challenges due to the country’s
water-stressed resources and the large percentage of the South African population living a
considerable distance away from the closest water source. Large amounts of capital are being
invested into providing all South Africans with a basic water supply. It is therefore important to
optimize the design of water supply systems while still providing acceptable and affordable service.
Service reservoirs are an important component of a water distribution system. They are storage
containers for water, allowing the source to produce water at a constant rate while the consumer
demand varies over time. The importance of a service reservoir is highlighted by the fact that when
the supply from the source is interrupted, the reservoir is still able to supply consumers for a
certain period of time.
Table 1 lists some guidelines used in South Africa when designing a water distribution system. As
can be seen from this table, the size of a reservoir is usually a function of the annual average daily
demand (AADD).
Although these guidelines have served the industry and public well, they do not take the supply
area’s unique properties, such as size and land use into account. Another limitation is that the
relationship between the feeder pipe and the service reservoir is fixed, preventing the designer
from exploring the most economic combination of feeder pipe and service reservoir. The service
reservoirs probability of failure is also not considered when making use of such rigid guidelines.
Proceedings of the 2004 Water Institute of Southern Africa (WISA) Biennial Conference 2 –6 May 2004
ISBN: 1-920-01728-3 Cape Town, South Africa
Produced by: Document Transformation Technologies Organised by Event Dynamics

Table 1. Typical South African guidelines for sizing reservoirs and feeder pipes.
Authority
Nature of supply
Feeder
capacity
Service reservoir
capacity
gravity feed 1.5 x AADD 24h of AADD
Department of Water
Affairs pumped main 1.5 x AADD 48h of AADD
Co-operation & gravity feed 1.5 x AADD 24h of AADD
Development pumped main 1.5 x AADD 48h of AADD
National Building one source 1.5 x AADD 48h of AADD
Institute two sources 1.5 x AADD 36h of AADD
National Housing
Supply
pipe size
gravity feed 1.5 x AADD 20h of AADD
pumped main 1.5 x AADD 30h of AADD
two sources 1.5 x AADD 66% of capacity with
one source
Reliability
Service
reservoir size
Figure 1. Interrelationship amongst bulk water supply variables.
Figure 2. Typical water demand pattern.

A triangular relationship is shown in Figure 1 between the reliability of a bulk water supply system,
the capacity of the service reservoir(s) and the size of the supply pipeline. The size of the service
reservoir depends directly on the capacity of the bulk water supply pipeline. Figure 2 is a typical
water demand pattern. If the supply rate is Qmax, no reservoir is needed, because the peak demand
can be supplied directly from the pipeline. If the supply rate is less than Qave, an infinitely large
reservoir is required, because the long-term demand exceeds the supply capacity. Naturally, it
follows that larger reservoirs would require smaller pipelines, and vice versa.
A water distribution system has a great effect on the quality of life of its consumers. It is important
that consumers are supplied with a sufficient amount of water that is of an acceptable quality. As
seen in Figure 1, the reliability of supply plays a major role. Lewis (1) described reliability as
follows: “In the broadest sense, reliability is associated with dependability, with successful
operation, and with the absence of breakdowns or failures. It is necessary for engineering analysis
however, to define reliability quantitatively as a probability. Thus reliability is defined as the
probability that a system will perform its intended function for a specified period of time under a
given set of conditions. A product or system is said to fail when it ceases to perform its intended
function.”
Since the main function of the bulk supply systems is to supply reservoirs, and not consumers, the
reliability of these systems can be defined in terms of their ability to maintain water in the reservoir.
A reservoir that runs dry would be a failure event in the bulk water supply system. The reliability of
a bulk supply system can therefore be described in terms of the failure behaviour of its reservoir,
for instance in terms of the annual number of failure events, the total annual fail time, or the
average failure duration.
Larger reservoirs would fail less often and would thus provide a higher level of reliability. However,
the higher reliability comes at a cost of increased capital expenditure and the potential for water
quality problems due to longer retention times in the reservoir. In some cases it might be better to
improve the reliability of the system by increasing the capacity of the supply pipelines, changing
the pipe configuration, or reducing the time for finding and repairing pipe bursts.
MOCASIM II
Mocasim II is a software package that combines probabilistic analysis with hydraulic analysis. It
allows various types of analysis to be performed such as capacity versus reliability analysis for
reservoirs. A previous version of Mocasim used a simple mass balance model for calculating the
flows in simple, linear distribution networks. Mocasim II extends the functionality of its predecessor
by enabling the probabilistic modelling of more complex water distribution models. This was
achieved by integrating a stochastic modelling technique into the Epanet hydraulic analysis
software package. Mocasim II was designed using an object-oriented model which has various
advantages such as ease of testing, upgrading and maintaining as well as minimum repetitive code
and a logical structure.
The methodology used in Mocasim II was developed as part of a research initiative by the Water
Research Group at the Rand Afrikaans University. The motivation behind the methodology was to
enable the analysis of service reservoir reliability using stochastic analysis. Stochastic analysis is
useful for evaluating system reliability in a wide range of fields, including water supply distribution,
electrical power distribution, and communications networks (2). It is frequently used in the analysis
of complex systems where risk and uncertainty play important roles.
A stochastic model is one whose outputs are predictable only in a statistical sense. With a
stochastic model, repeated use of a given set of model inputs produces outputs that are not the
same but follow certain statistical patterns (3). The stochastic method used in this methodology is
the Monte Carlo simulation method. Monte Carlo methods have been used for centuries, but only
in the past several decades has the technique gained the status of a recognised numerical method
capable of addressing the most complex applications.

Monte Carlo simulation entails the repeated calculation of the system performance, each time with
a different combination of input parameters. The input parameters are randomly selected from
probability distribution functions. For example, we may know that the minimum time required to
repair a burst on a particular pipe is 3 hours, the maximum 36 hours and the median 12 hours. By
fitting a suitable function to these values, a probability distribution function may be defined which
can then be used in the stochastic analysis to simulate the pipe’s failure duration behaviour.
The input parameters are those stochastic variables, which affect the system: consumer demand,
fire-fighting demand and emergency storage requirements. Once estimates of the probability
distributions of these components are available, Monte Carlo simulation is used to generate
stochastic combinations of these variables for any required number of iterations, each iteration
covering a time step of arbitrary length, typically one hour time steps over 10 000 years. The
performance of the system is then reported statistically.
The performance criterion used for bulk water supply systems is whether the amount of water in
storage at the beginning of a time step, plus the inflow during the step, is more than the water
leaving the tank. If so, the supply to the customers is not interrupted and the system is successful.
If not, it implies that the tank will be empty for at least a part of the time step and a failure is
recorded. The process is repeated for each successive time step, with the tank level adjusted for
the net in/outflow after each step.
INPUT PARAMETERS
Very few reservoirs perform only one function; they have to provide more than one or all of the
functions listed below:
Emergency Storage
Volume must be provided to guarantee the supply to the consumers, even when the supply to the
service reservoir is partially or completely discontinued. Such events may be scheduled for
maintenance purposes, which is not a stochastic variable. More important for probabilistic analysis
are the unscheduled events due to pipe, power or source failures. The volume required for these
unscheduled events is stochastic, as neither the time of occurrence nor the duration of the
interruption can be predicted.
Fire Storage
It is essential to have an adequate water supply available for fire-fighting. This is mostly supplied
through the water reticulation system, and is thus also drawn from the service reservoir. Most
guidelines will specify an additional volume of water for which allowance must be made in the
storage tank. This approach is based on the conservative assumption that the fire demand will
coincide with a period of maximum consumer demand. The required fire storage is stochastic, as
neither the time of occurrence nor the actual volume required can be predicted.
Demand Storage
Consumers draw water from a service reservoir at a variable rate, whereas the supply line usually
delivers water at a constant rate. The reservoir has to absorb the difference between inflow and
outflow rates. The flow from the supply line into the tank is easily determined and usually well
controlled. The consumer demand, on the other hand, is determined by the cumulative effect of a
multitude of stochastic variables and is therefore itself a stochastic variable.
Operational Requirements
There could be additional requirements for service reservoir volume, such as freeboard (dependent
on the sophistication of level sensing and control equipment), bottom storage (dependent on the
potential of air or sediment entrainment at the outlet), or a pump control band (required for
automatic switching of pumps if water is being pumped to or from the reservoir). These
components are all deterministic, i.e. they can be calculated once and simply added to the volume
required for the stochastic components described above.

MODEL FOR EMERGENCY STORAGE
The volume required for emergency storage is equivalent to the total water demand for the
duration of the supply interruption. To determine the emergency storage volume, we need to
estimate two factors; the first is the frequency of supply interruptions and the second is the duration
of the interruption. The supply to a service reservoir is interrupted when the raw water source or
water treatment plant fails, pumping installations fail (e.g. due to power cuts), or the feeder pipe
system fails (e.g. pipe breaks). These reasons for supply interruptions are the most common,
however only pipe failures are considered here, as they are the most common cause of prolonged
supply interruptions in practice.
Frequency of Supply Interruptions
To model the frequency of pipe breakages, it is necessary to know the length of the pipe and the
average pipe breakage rate. From these two values the average number of breaks per year can be
calculated. On any given day, the probability of a break is then the number of breaks per year
divided by the number of days per year. Now, a random number between 0 and 1 can be
generated – if the number is less than the probability, a break is assumed. If a break is assumed,
the duration of the interruption must then be estimated.
Duration of Supply Interruptions
The total interruption time associated with a pipe break is assumed to be the sum of:
• Reporting time from the instant of break until the maintenance centre is notified
• Mobilization time to assemble crew, spares, equipment and vehicles
• Travel time from maintenance centre to point of break
• Preparation time, i.e. shutting valves, draining pipeline, and excavation to the level of the pipe
break
• Repair time, and
• Commissioning time, i.e. disinfection, air bleed, and controlled restoration of full pipeline
pressure
The total interruption time for rural and urban areas will be dominated by completely different
factors. In urban situations, the reporting and travel times will be minimal, as any break will be
immediately visible and the maintenance centre is usually situated close by within the urban area.
It can be expected that the excavation and repair time in a congested area, shared by numerous
other services, will have to be slow and careful. For rural areas, the total interruption time is
dominated by reporting and travel times due to remote location, absence of the people, and poor
roads and communications. Excavation and repair time, however, will be shorter due to
unrestricted room and mostly smaller pipeline diameters.
To model the duration of these interruptions in a probabilistic way, it is necessary to have: a
statistical distribution that adequately fits the observed or anticipated interruption, the mean
interruption time, and the coefficient of variation describing the variability of the interruption time
about the mean.
MODEL FOR FIRE STORAGE
The total volume of water used for each fire is required to determine the effect of fire-fighting on the
reliability of bulk water supply systems. The fire volume can be calculated as the fire duration
multiplied by the fire flow rate.
Fire Duration and Flow Rate
The mean, statistical distribution and standard deviation of the fire duration and fire flow rate must
be obtained to get a probabilistic estimate of the volume required for fire-fighting.
To get a true probabilistic estimate, the fire duration as well as the fire flow rate has to be described
in statistical terms. The mean, the appropriate statistical distribution and the standard deviation
must be obtained from municipal records, if available.

Such data is rarely available in South Africa, and their retrieval and analysis are extremely tedious.
Furthermore, as large fires are rather infrequent, most of the sources will yield too few data points
to allow meaningful statistical analysis. A much simpler, more direct approach is to get a
deterministic estimate from the relevant fire codes. These codes usually specify both the maximum
fire flow rate and the fire duration as a function of the land use category, population density, or
other parameters. By multiplying these values, an estimate can be obtained.
Probability of a Fire Occurring
It is difficult to estimate the probability of a fire occurring in the supply area of a water supply
system, even if historical records would be available. Most fires require very little or no fire water
from the water supply system, and these smaller fires are obviously of no consequence for the
design of the supply system. What is really needed, is the probability of a large fire occurring; a fire
that will exert a substantial demand from the supply system. This is especially important if the fire
water volume is estimated from the local fire code, as this value indicates an extreme fire event.
MODEL FOR WATER DEMAND
An actual water demand pattern is the sum of two sets of factors: (1) a number of cyclic factors,
introduced by the seasons of the year or the days of the week, and (2) a random factor, which is
introduced by the cumulative effects of unaccounted-for variables such as temperature, rainfall,
holidays, special events within the supply area, etc.
The first step in estimating the daily volume of water used is to select a random value from an
appropriate distribution function with a mean and standard deviation.
An important factor to include is that of serial correlation. Serial correlation (also called
autocorrelation) occurs when residuals from adjacent measurements in a time series are not
independent of one another (that is, if the i th residual is high, it is likely that the (i+1) th residual will
also be high, and likewise low residuals tend to follow other low residuals).
Daily water demand data normally exhibit a strong degree of serial correlation. For example, if
yesterday had a higher-than-average demand, today is also likely to have a higher-than-average
demand. Behavior of this kind is statistically described with a serial correlation coefficient Φ. The
correlation coefficient measures the similarity of variation of two residuals. Correlation coefficients
range from +1 (which indicates a perfect linear relationship between two residuals), to 0 (which
indicates absolutely no relationship), to –1 (for a perfect inverse linear relationship). A correlation
coefficient between 0 and +1 means that the variables increase or decrease together, but not
identically.
The analysis of numerous South African data sets have shown that it is only necessary to take one
preceding day into account as further serial correlation coefficients are not significant.
The cyclic factors must be incorporated. These factors are the following:
• systematic day-to-day variations found in a typical week. The average daily volume for a
particular day is calculated by multiplying by a peak factor PFMon…Sun.
• systematic annual seasonal variations which can be described in terms of 12 months of 52
weeks. The average daily volume for each of these time units, is calculated by multiplying by
an appropriate peak factor PF1…13.
Finally, the daily volume must be dimensionalized by multiplying it with the Annual Average Daily
Demand AADD.
If the simulation is done with a time step of one day, the above model is complete. If a time step of
one hour is used the approach is to generate a daily volume as indicated above, and then to
disassemble the daily volume into 24 hourly volumes according to a fixed hourly pattern. The fixed
hourly pattern is defined by PF1…24.

APPLICATION EXAMPLE
A network similar to that in an EPANET tutorial has been chosen to demonstrate the capabilities of
MOCASIM II. The network consists of a lake and a river as two sources, three service reservoirs,
92 nodes and pipes with a total length of approximately 80km. The network is shown in Figure 3.
The average annual daily demand for the entire network is 685 l/s.
Figure 3. Example network.
The same failure model was applied to all the pipes. This model has an average failure rate of 0.23
failures per km per year. One demand model was used which consisted of an hourly demand
pattern and a random component. The demand model was applied to four nodes at peak demand.
The hourly demand pattern is depicted graphically in Figure 4 and Table 2 lists the input
parameters used in the reliability analysis of the system.
2.5
2
1.5
1
0.5
0
1
3
5
Ratio of Hourly Average
7
9
11
13
15
Hour of the day
17
19
Figure 4. Hourly demand pattern.
21
23

Demand Model
Fail Model
Service Reservoirs
Table 2. MOCASIM II parameters.
Units Parameter
ID FailM1
Patterns
Seasonal Pattern none
Day of Week Pattern none
Hourly Pattern 1
Correlation Coefficient 0.4
Random Component
Distribution normal
Average 1
Standard Deviation 0.2
ID
Fail Frequency
Distribution lognormal
Average #/km/year 0.23
Standard Deviation 0.0322
Fail Duration
Distribution lognormal
Average h 6
Standard Deviation 0.84
ID 1
Elevation m 40
Min storage size kl 14778.8
Max storage size kl 500000
Storage size factor 1
Storage size additional 14778.8
ID 2
Elevation m 36
Min storage size kl 14778.8
Max storage size kl 500000
Storage size factor 1
Storage size additional 14778.8
ID 3
Elevation m 39
Min storage size kl 14778.8
Max storage size kl 5000000
Storage size factor 1
Storage size additional 14778.8
The simulation was run for a duration of 1000 years for each service reservoir. This duration was
adequate to get accurate enough results as the simulation was applied to the nodes at peak
demand. The results of the reliability for each service reservoir can be seen in Figure 5. It must be
noted that the AADD used for each reservoir is that of the entire system.
It can be seen from the graph that the reliability of service reservoir 3 (0.02 failures per year at 72
hours AADD) is much better than that of service reservoir 1 which has a failure rate of 244 per year
at a capacity of 72 hours AADD, and service reservoir 2 which has a failure rate of 0.4 at the same
capacity. The reason for this is due mainly to two factors. Firstly the distance of the reservoir from
the source plays an important role as the closer the reservoir is to the source, the fewer failures will

occur resulting in an improved reliability. The second factor is that of the size of the area which is
supplied by the reservoir, because the AADD for the entire system was used to size the reservoirs.
Service reservoir 1 supplies a larger area than that of service reservoir 2, thus resulting in a lesser
reliability.
No. failures (p.a)
1000
100
10
1
0.1
0.01
6 12 18 24 30 36 42 48 54 60 66 72 80 84
Reservoir Capacity (Hours AADD of System)
Figure 5. Reliability analysis of three service reservoirs.
Reservoir 1
Reservoir 2
Reservoir 3
Another capability of MOCASIM is demonstrated in Figure 6 and Figure 7, which show the demand
distributions of two nodes. Figure 8 and Figure 9 show the pressure distribution of the same two
nodes. These distributions quantify the service provision of the system. It could be seen from the
results obtained that the pressure at node 15 only dropped below 24 m (the minimum allowable
pressure for a distribution system) for 0.2 % of the time, but never dropped below 20 m. Similar
deductions can be made from the distribution of node 121. The statistical properties of these
distributions appear in Table 3.
Probability
Probability
1.2
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 25 30 35 40 45
Demand (l\s)
Node 15
Figure 6. Demand distribution of node 15
1.2
1
0.8
0.6
0.4
0.2
0
0 1 2 3 4 5 6 7 8 9 10
Demand (l/s)
Node121
Figure 7. Demand distribution of node 121.

Probability
Probabilty
1.2
1
0.8
0.6
0.4
0.2
0
0 202326293235384144475053565962
Pressure (m)
Node 15
Figure 8. Pressure distribution of node 15.
1.2
1
0.8
0.6
0.4
0.2
0
35 38 41 44 47 50 53 56 59 62 65 68 71 74
Pressure (m)
Node 121
Figure 9. Pressure distribution of node 121.
Table 3. Statistical properties of distributions.
Property Node Average (l/s) Std Deviation Minimum (l/s) Maximum (l/s)
Demand 15 17.51 5.65 6.04 44.34
Demand 121 2.99 0.89 1.79 5.43
Pressure 15 38.70 5.28 20.00 60.19
Pressure 121 53.67 4.18 39.18 72.35
Table 4 also contains useful results for analysing distribution systems. It lists 6 pipes in the
network, the number of failures that occurred per year and the failure duration statistics for each
pipe. We can see from the table that longer pipes fail more often than shorter pipes.
Table 4. Failure behaviour of pipes.
Duration Statistics
Pipe Length (m) No of failures Average (h) Std Deviation Minimum (h) Maximum (h)
20 302 0.08 5.93 0.52 5.19 6.55
103 411.5 0.09 5.49 0.78 4.05 6.45
114 2000 0.45 5.91 0.66 4.63 7.21
131 987.5 0.22 6.24 0.68 4.94 7.93
204 1380.7 0.32 5.94 0.74 4.21 7.35
247 1306 0.3 5.91 0.77 4.61 8.09
It is important to note that in a stochastic model such as this, the values obtained will not be
precisely the same for different simulations of a particular set of input data, but the results from
each simulation will lead to the same conclusions.

CONCLUSION
A software package, MOCASIM II was introduced which combines probabilistic analysis and
hydraulic analysis of water distribution systems. It allows various types of analysis to be performed
such as capacity versus reliability analysis for service reservoirs. From the application example it is
clear that the software is a powerful tool to investigate different scenarios for upgrading or
enlarging an existing water supply system, while maintaining an acceptable and predetermined
probability of failure.
REFERENCES
1. E. Lewis, Introduction to Reliability Engineering, John Wiley & Sons, Inc. (1996).
2. S. Yang, N.-S. Hsu, P. Louie and W.-G. Yeh, Water distribution network reliability: Stochastic
simulation, Journal of Infrastructure Systems, 2 (2). (1996).
3. C. T. Haan, Statistical Methods in Hydrology, The Iowa State University Press/ Ames (1977).