E cient Management of HVAC Systems - Automatica - Università ...

1

1

1

1

1

1

1

1

1

1

1

m 2

m 2

1 ≈ 1005 J

1 ≈ 1005 J

1 ≈ 1005 J

2

2

2

2

2

CO2

CO2

CO2

≥ ≥
≤ ≤ ≤
≤ ≤ ≤
≤ ≤ ≤
≤ ≤ ≤
≤ ≤ ≤

3

3

3

3

3

3

3

3

3

3

∆T
∆T

∆T
∆T

∆T
∆T

∆T
∆T

∆T
∆T

∆T
∆T

4

cp
e
ec
ep
f
L
˙m
Q
s
tc
T o
V m 3
ρ m 3
τ
f
H
L
C
i
o
k

˙mk,i − ˙mk,o = 0 .
˙mk,i = ˙mk,o = ˙mk .
t
t
dQk
dτ
− Lk
dτ = − ˙mk,i (cpTk,i + ep,k,i + ec,k,i)
+ ˙mk,o (cpTk,o + ep,k,o + ec,k,o)
+ ∂
∂τ
ˆ Vk
0
eρdv .

τ
dTk,f
− ˙mk,icpTk,i + ˙mk,ocpTk,f + fkρVkcp
dτ
= 0 ,
fk
dv t
τ
− ˙mk,icpTk,f(τ) + ˙mk,ocpTk,o(τ) + ∂
∂τ
ˆ Vk
fkV
ρcpTk,i(t)dv = 0 .
v
t
v
τ

(1 − fk)Vk
v = fkVk + (τ − t) ,
tc,k
tc
tc,k = (1 − fk) ρVk
˙mk,i
.
− ˙mk,icpTk,f(τ) + ˙mk,ocpTk,o(τ) + ∂
∂τ
ˆ τ−tc
τ
(1 − fk)Vk
−ρcp Tk,f(t)dt = 0 ,
˙mk,ocpTk,o(τ) = ˙mk,icpTk,f(τ − tc) .
Wk(s) = Tk,o(s)
Tk,i(s) =
e −stc
1 + s fkρVk
˙mk
tc,k
.
k − τ
− ˙mk,icpTk,i + ˙mk,ocpTk,o = dQk
dτ = Pk ,

˙mP = ˙mS ˙mP < ˙mS
˙mCh1, ..., ˙mChn n
˙mS = ˙mCh1 + ... + ˙mChn
˙mP ∈ 0, ˙mCh1, ˙mCh1 + ˙mCh2, ...,
i∈ON ˙mChi
˙mP ≤ ˙mS
˙mP = ˙mS

PP = ˙mP cp(TP,o − TP,i) ,
PS = ˙mScS(TS,o − TS,i) ,
PP = PS .
TP,o = TS,i ,
TP,i = TS,o .
˙mP < ˙mS
(TP,i − TP,o) ˙mP = (TS,o − TS,i) ˙mS ,
TP,i = TS,o .
TS,i = TS,o − ˙mP
˙mS
(TS,o − TP,o) ≡ TS,o − ˙mP
˙mS
(TP,i − TP,o) .

˙mCh1cpTchw1,o + ... + ˙mChncpTchwn,o = ˙mP cpTP,o ,
TP,o = ˙mCh1cpTchw1,o + ... + ˙mncpTchwn,o
˙mP
.

ɛ

ɛ

ɛ

ɛ

5

EER = f(P LR) .
Qi
QCL
arg max
P LRi
EERi ,
i
Qi = QCL ,
i
i ∈ {1, ..., nch} nch

15
2 15
arg min
statusi
InputP oweri , si ∈ {}
i
Qi = QCL .
i
P LRmin,i ≤ P LRi ≤ P LRMax,i ,
P LRmin,i P LRMax,i P LRi
P LRMax,i

Pefull Pcfull
Peful (t) = ae + beTchwr(t) + ceTair(t) + de ˙mw(t) + eeTchwr(t) ˙mw(t) .
Pcfull (t) = ac + bcTchwr(t) + ccTair(t) + dc ˙mw(t) + ecTchwr(t) ˙mw(t) .
ae, be, ce, de, ee, ac, bc, cc, dc, ec
EERfull

∆τ
EERfull = Pc,full
Pe,full
,
P LR
P LR = Pc
Pc,full
,
EERfull
P LF P LR
P LF
P LF = 1 − cd(1 − P LR)
cd
0.25
∆τ
Pc,P LF = P LR · Pc,full ,
Pe,P LF =
P LF
P LR · Pefull i .
P LF P LR

EER P LR Tair
Tchwr

Y Z P LR Tair
Y = EERcyc
EERfull
EERcyc
EERfull EER
P LRcyc = Pc,cyc
Pc,full
Pc,cyc Pc,full
Z
Pe,cyc

1.4
1.2
1
0.8
0.6
0.4
0.2
T air = 35 °C
T air = 20 °C
T air = 20 °C
T air = 35 °C
0
0 0.2 0.4 0.6 0.8 1
PLR
Y Z P LR Tair
Pe,full
Z = Pe,cyc
Pe,full
Y Z
P LRcyc
Y
Z
Y =
Z
Y Z
Ycurve Zcurve P LR
Tair
Y Z
Ycurve Zcurve P LR
Tair
Zcurve
Pc,cyc = P LRcyc · Pc,full ,

Pe,cyc = Z · Pe,full .
EER ∆τ
EER = Pc,cyc∆τ
Pe,cyc∆τ
.

6
n m
Ch1,j, ...
, Chi,j, ..., Chn,j 1 ≤ j ≤ m t, Ch1,m, ...
, Chi−1,m, Chi,s
Chi,s+1
t, Ch1,s, ..., Chi,s, Chi+1,s−1,
..., Chn,s−1 Chi+1,s

Capacity steps [−]
6
5
4
3
2
1
0
Sequential Strategy − MS
6
5
4
3
2
1 7
1 2
Chiller [−]
Capacity steps [−]
4
3
2
1
0
Symmetric Strategy − SS
7
5
3
1
1 2
Chiller [−]
DM DR
∆T
6
4
2

Capacity steps [−]
6
5
4
3
2
1
0
Sequential Strategy − MS
6
5
4
3
2
1 7
1 2
Chiller [−]
Capacity steps [−]
4
3
2
1
0
Symmetric Strategy − SS
7
5
3
1
1 2
Chiller [−]
DM DR
∆T
6
4
2

T
T
T

T
Wvt(s) = e−sτvt
1 + sTvt
.
Hvt(z) =
k1
1 + z −1 k2
z −τvt
Ts .

1
TCh_out_set−point
DM
Relay Logic
Virtual Tank
vt Tvt k1 k2 ÷
τvt Tvt k1 k2
7.69 3.511 · 10 −2
1.745 · 10 −2
1.161 · 10 −2
8, 83 · 10 −3
7.04 · 10 −3
5.855 · 10 −3
5.001 · 10 −3
4.375 · 10 −3
3.885 · 10 −3
3.493 · 10 −3
vt Tvt k1 k2
Chiller
1
TCh_out

∆τ
arg min
(P LRi, statusi)
i
Pe,i ,
Pc,i = ˆ PL ,
i
P LRi − P LRiprev
≦ κi , i = 1, ..., n .
Pe,i Pc,i i−
ˆ PL ∆τ
P LRi i
P LRi P LRiprev
i−

P LRi = Pc,cyc
Pc,full i
Zi = Pe,cyc
Pe,full i
Pc,i = P LRi · Pc,full| i ,
Pe,i = Zi · Pe,full| i .
EERi ∆τ
EERi = Pc,i∆τ
Pe,i∆τ

P MV = 0.303 · e −0.036M + 0.028 {(M − W ) − 3.05 · 10 −3 [5733+
−6.99 (M − W ) − pa] − 0.42 [(M − W ) − 58.15] +
−1.7 · 10 −5 M (5867 − pa) − 1.4 · 10 −3 M (34 − Ta) +
−3.96 · 10 −8 fcl
−fclhc (Tcl − Ta) } ;
(Tcl + 273) 4 − (Tmrt + 273) 4 +
Tcl = 35.7 − 0.028(M − W ) − 0.155Icl{3.96 · 10 −3 fcl[(Tcl + 273) 4 −
hc =
−(Tmrt + 273) 4 ] − fclhc(Tcl − Ta)] ,
⎧
⎪⎨
⎪⎩
2.38(Tcl − Ta) 0.25 2.38(Tcl + Ta) 0.25 ≥ 12.1 √ Vair
12.1 √ Vair
2.38(Tcl + Ta) 0.25 ≤ 12.1 √ Vair
M 1 met = 58.2 W/m 2
W W/m 2
,

pa Pa)
ta C
fcl
hc W/m 2 /C
Tcl C
Icl 1 clo = 0.155Km 2 /W
Tmrt C
Vair m/s
M W tcl hc tr
∆τ
arg min
(P LRi, statusi) Fobj ,

Fobj hobj
i = 1, ..., n ,
i
Pe,i∆τ
νobj
+ herr Pc,i −
ˆ
PL ∆τ
+ hreg max 0,
P LRi − P LRiprev
− κi
νreg i
i
νerr
+
hobj νobj
herr νerr
hreg νreg
TP,i TP,o
TL,i
dTL,i(t)
˙mP cp(TP,i(t) − TP,o(t)) + ρcpVtank
dt − PL = 0

PL
⎧
⎪⎨
⎪⎩
˙
PL = 0
˙
TL,i =
1
ρcpVtank
PL + ˙ mP
TP,o −
ρVtank
˙ mP
TP,i
ρVtank
PL(n)Ts = ˙mP cp(TP,i(n) − TP,o(n))Ts + ρcpVtank(TL,i(n + 1) − TL,i(n)) ,
Ts n ∈ Z(Ts)
Σ
ˆ PL
⎧
⎪⎨
PL(n + 1) = PL(n)
Ts
⎪⎩ TL,i(n + 1) = PL(n) + TL,i(n) +
ρcpVtank
˙mP Ts
TP,o −
ρVtank
˙mP Ts
TP,i
ρVtank
′
TP,o TP,i Σ TL,i
Σ Σ(A, B, C, D)
A =
1 0
Ts
ρcpVT ank
1
, B =
0 0
˙mP Ts
ρVT ank
C = 0 1 , D = 0 .
− ˙mP Ts
ρVT ank
dx(t)
dt ≈ xn+1 − xn
Ts
,

QL [KW]
200
150
100
50
0
Load Estimation
−50
0 2 4 6 8 10 12
t [h]
14 16 18 20 22 24
(F, G, H, J) u Σ =
QL
ˆQL
TP,o TP,i TL,i
Σ
′
yˆ Σ = ˆPL
ˆTL,i
F = [A − LC] , G = B L , H = I2×2 , J = 02×3 .
L
L
′

nph
ng/ph

nph,Max
ng/ph = nph,Max
nph
jph = 1, ..., nph−1,
L
L
(1 − L) 0 L 1 L
n
m
nph L1 L2,
P LRi statusi
P LRi statusi TspiGA i

jph = 1...nph − 1
⎧
⎨ L
⎩
(1 − L)
jph = nph
⎧
L
⎪⎨
⎪⎩
(1 − L)
⎧
⎪⎨
⎪⎩
(1 − L)L1
⎧
⎨
⎩
(1 − L)(1 − L1)
(1 − L)L1L2
(1 − L)L1(1 − L2)
(10 + 1) · n
P LR (1)
1 ... P LR (10)
1 status1 ... ... P LR (1)
n ... P LR (10)
n
statusn

TspiGA = ˆ TP,i − P LRi · ∆T ,
ˆTP,i = TP,o + ˆ
P LRtot · ∆T ,
P LRtot
ˆ = ˆ QL
,
QMax
∆T ˆ QL
QMax
Ycurve − Zcurve
TL,i
eTsp ¯ TL,isp
Tspi = TspiGA
deTsp
+ KpeTsp + Kd
dt
+ Ki
ˆ
eTspdt
eTsp = ¯ TL,isp −TL,i

kW )
kW i
P LRi
kW i = ai + biP LRi + ciP LR 2 i ,
ai, bi, ci kW P LR
J =
I
I
i=1
I
KWi ,
i=1
P LRi · CCi = CL ,
CCi CL
kW

λ
L =
⎡
⎤
I
I
KWi + λ ⎣CL − P LRi · CCi⎦
.
i=1
L
P LRi P LRi
P LRi = λCCi − bi
.
2ci
I
i=1
I
P LRi · CCi = λ
i=1
i=1
CC 2
i
2ci
−
I bi
CCi
2ci
i=1
λ =
2CL +
I
i=1
I
i=1
bi
ci
2
i
ci
CCi
.
CL λ
CCi CL ai , bi ci
P LRi
P LRi
P LRi 1
kW P LR
kW − P LR

ai bi ci CCi[kW ]

CCi kW CCi kW

7
Tsam
Tsup Tsup > Tsam
Tsam = 60 Tsup = 600

1
Tair
2
TChwr
3
m_dot
4
step
Tair
TChwr
m_dot
step
Pc
Pe
Chiller : cooling and electric powers
CoolingPower
ElectricPower
−K−
1/cp*m_dot
DeltaT _Ch
TChws = TChwr − Pc
1
cp ˙mch,o
1
TChws

Tair
hour
TChw 1,o
NoOp
Carico edificio
From1
T_set
hour
Tair
T_P_i
Tp,o
T_P_o
TChw 1,n
...
2
NoOp
status
m_dot_P_i
m_dot_P_o
m_dot_P_o
m_dot_Ch1n
Tchw n,o
1
Collector
Twt_o
T_S_i
T_S_i
Ts,o
m_dot_Ch 1
Tchwr
m_dot_wt
m_dot_S_i
m_dot_S_i
m_dot_S_o
...
In1Out1
Water Tank
ByPass
m_dot_Ch n
m_dot_P,i
PID
Relay Logic + Parallel chillers
Tset1_n
Tair
T_L_i
Tchwr
From4
[Tair]
m_dot_L_i
T_set1_n
Status _1_n
PL_hat
MPGA
Coolig Load
T_L_o
m_dot_L_o
staust 1_n
m_dot_P_i
T_P_o
PL_hat
T_L_ot
T_L_i
Stimatore

Tair
hour
TChw 1,o
NoOp
Carico edificio
From1
T_set
hour
Tair
T_P_i
Tp,o
T_P_o
TChw 1,n
...
2
NoOp
status
m_dot_P_i
m_dot_P_o
m_dot_P_o
m_dot_Ch1n
Tchw n,o
1
Collector
Twt_o
T_S_i
T_S_i
Ts,o
m_dot_Ch 1
Tchwr
m_dot_wt
m_dot_S_i
m_dot_S_i
m_dot_S_o
...
In1Out1
Water Tank
ByPass
m_dot_Ch n
m_dot_P,i
PID
Relay Logic + Parallel chillers
Tset1_n
Tair
T_L_i
Tchwr
From4
[Tair]
m_dot_L_i
T_set1_n
Status _1_n
PL_hat
MPGA
Coolig Load
T_L_o
m_dot_L_o
staust 1_n
m_dot_P_i
T_P_o
PL_hat
T_L_ot
T_L_i
Stimatore

Tair = 35 C Twater =
12 C
ac
ae
bc
be
cc
ce
dc
de

P LRi

1.4
1.2
1
0.8
0.6
0.4
0.2
T air = 35 °C
T air = 20 °C
T air = 20 °C
T air = 35 °C
0
0 0.2 0.4 0.6 0.8 1
PLR
T air [°C]
28
26
24
22
20
18
16
1
0.8
PLR [−]
0.6
0.4
0.2
0
Y
Z
[−]
1.4
1.2
1
0.8
0.6
0.4
0.2
T air = 20 °C
T air = 35 °C
T air = 35 °C
T air = 20 °C
0
0 0.2 0.4 0.6 0.8 1
PLR
Y
Z
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Z [−]
27
26
25
24
23
22
21
20
19
18
17

L1
L2
hobj ÷
nph νobj
ng/ph herr ÷
νerr
hreg ÷
νreg
L ki ÷
÷
EER
PCL − ˆ PCL

[KW]
EER [−]
1800
1600
1400
1200
1000
800
600
400
200
Cooling Power
MCM
SS
Q L
0
0 2 4 6 8 10 12
[hours]
14 16 18 20 22 24
5.2
5
4.8
4.6
4.4
4.2
4
3.8
3.6
3.4
EER
MCM
SS
0 2 4 6 8 10 12
[hours]
14 16 18 20 22 24

∆
E PCL − ˆ
PCL
V ar PCL − ˆ
PCL
σ PCL − ˆ
PCL
SS kW kW 2 kW
MCM kW kW 2 kW
[°C]
8
7.8
7.6
7.4
7.2
7
6.8
6.6
6.4
6.2
Temperature
Set−Point
MCM
SS
6
0 2 4 6 8 10 12
[hours]
14 16 18 20 22 24

Step [−]
Step [−]
Step [−]
Step [−]
Step [−]
Step [−]
4
3
2
1
Chiller switching
0
6 8 10 12 14 16 18 20
4
3
2
1
0
6 8 10 12 14 16 18 20
4
3
2
1
0
6 8 10 12 14 16 18 20
4
3
2
1
0
6 8 10 12 14 16 18 20
4
3
2
1
0
6 8 10 12 14 16 18 20
4
3
2
1
MCM
SS
0
6 8 10 12 14 16 18 20
[hours]

∆
E PCL − ˆ
PCL
V ar PCL − ˆ
PCL
σ PCL − ˆ
PCL
MS kW kW 2 kW
MCM kW kW 2 kW
EER
PCL − ˆ PCL

[KW]
[KW]
2000
1800
1600
1400
1200
1000
800
600
400
200
Cooling Power
MCM
MS
Q L
0
0 2 4 6 8 10 12
[hours]
14 16 18 20 22 24
600
500
400
300
200
100
Electric Power
MCM
MS
0
0 2 4 6 8 10 12
[hours]
14 16 18 20 22 24

[°C]
8.5
8
7.5
7
6.5
Temperature
MCM
MS
6
0 2 4 6 8 10 12
[hours]
14 16 18 20 22 24
EER
EER
PCL − ˆ PCL

step [−]
step [−]
step [−]
6
5
4
3
2
1
0
6 8 10 12 14 16 18 20
6
5
4
3
2
1
0
6 8 10 12 14 16 18 20
6
5
4
3
2
1
MCM
MS
0
6 8 10 12 14 16 18 20
[hours]
∆
∆

E PCL − ˆ
PCL
V ar PCL − ˆ
PCL
σ PCL − ˆ
PCL
MS kW kW 2 kW
SS kW kW 2 kW
MCM kW kW 2 kW
[KW]
EER [−]
2000
1800
1600
1400
1200
1000
800
600
400
200
Cooling Power
MCM
MS
SS
PL
0
0 2 4 6 8 10 12
[hours]
14 16 18 20 22 24
4.6
4.4
4.2
4
3.8
3.6
3.4
EER
MCM
MS
SS
8 10 12 14
[hours]
16 18 20

[°C]
8
7.5
7
6.5
6
5.5
MCM
MS
SS
Temperature
5
0 2 4 6 8 10 12
[hours]
14 16 18 20 22 24
F lt
F lt

Step [−]
Step [−]
Step [−]
Step [−]
6
5
Screw 1
4
3
2
1
0
6 8 10 12 14 16 18 20
Screw 2
6
5
4
3
2
1
0
6
4
8 10 12 14
Scroll 1
16 18 20
3
2
1
0
6
4
8 10 12 14
Scroll 2
16 18 20
3
2
1
MCM
MS
SS
0
6 8 10 12
[hours]
14 16 18 20
MCMF lt
MCMF lt
MCMF lt
∆ (MCM − MS)
∆ (MCM − SS)
∆ (MCMF lt − MCM)

T Set−Point [°C]
T L,i [°C]
16
15
14
13
12
11
10
9
8
7
6
Float Set−Point
5
0 0.2 0.4 0.6
PLR [−]
0.8 1
16
15
14
13
12
11
10
9
8
T L,i GA
T Set−Point
T Set−Point GA
T MS
T MS
200 400 600 800
Time [min]
1000 1200 1400

n

Time [s]
45
40
35
30
25
20
15
MPGA Average Elapsed Time
Population Size= 200
10
2 4 6 8 10 12 14 16 18 20
Number of chillers [−]
Time [s]
350
300
250
200
150
100
50
0
500
400
300
Population Size
200
MPGA Average Elapsed Time
100
2
4
6
8
10
12
14
16
Number of Chillers [−]
18
20
300
250
200
150
100
50
[s]

Memory [bytes]
4.5
4
3.5
3
2.5
2
1.5
1
0.5
500
x 10 6
450
400
Population Size
350
300
250
Average Memory Request
200
2
4
6
8
10
12
14
16
Number of Chillers [−]
x 10 6
[bytes]
18
4
3.5
3
2.5
2
1.5
20
1

Memory [bytes]
4.5
4
3.5
3
2.5
2
1.5
1
0.5
500
x 10 6
450
400
Population Size
350
300
250
Average Memory Request
200
2
4
6
8
10
12
14
16
Number of Chillers [−]
x 10 6
[bytes]
18
4
3.5
3
2.5
2
1.5
20
1

A

0 ⇒ 1
1 ⇒ 0

Nind
Nind × Lind
0, 1

F (x) = g(f(x)) ,
f(·) g(·)
F (·)
F (xi)
f(xi)
F (xi) =
f(xi)
Nind i=1
Nind xi
f(xi) ,
i
F (x) = af(x) + b ,
a
b

MAX
MIN = 2.0 − MAX
INC = 2.0 × (MAX − 1.0)/Nind
LOW = INC/2.0
MIN INC
LOW
MAX [1.1, 2.0]
Nind = 40 MAX = 1.1 MIN = 0.9
INC = 0.05 LOW = 0.025
F (xi) = 2MAX + 2(MAX1) xi − 1
,
Nind − 1
xi i

Sum
[0, Sum]
[0, Sum]
N N

[0 Sum/N] ptr
N 1 [ptr, ptr + 1, ..., ptr + N − 1]
O(N log N)
O(N)
m ki ∈
{1, 2, ..., l − 1} ki l
0.001 0.01

[0 Sum/N] ptr
N 1 [ptr, ptr + 1, ..., ptr + N − 1]
O(N log N)
O(N)
m ki ∈
{1, 2, ..., l − 1} ki l
0.001 0.01

[0 Sum/N] ptr
N 1 [ptr, ptr + 1, ..., ptr + N − 1]
O(N log N)
O(N)
m ki ∈
{1, 2, ..., l − 1} ki l
0.001 0.01

[0 Sum/N] ptr
N 1 [ptr, ptr + 1, ..., ptr + N − 1]
O(N log N)
O(N)
m ki ∈
{1, 2, ..., l − 1} ki l
0.001 0.01

function genot = rv2bs(var,fieldD)
%
% This function decodes vectors of reals (phenotype) into genotype. The
% chromosomes are made of binary strings of given
% length using ONLY standard binary (NO Gray decoding).
% The real numbers must given in a specified interval.
%
% INPUT:
%
% var: matrix containing in each row the vector of reals of the current
% population.
% fieldD: matrix describing the length and how to decode each substring
% in the chromosome. It has the following structure:
%
% [len; (num)
% lb; (num)
% ub; (num)
% code; (0=binary | 1=gray)
% scale; (0=arithmetic | 1=logarithmic)
% lbin; (0=excluded | 1=included)
% ubin]; (0=excluded | 1=included)
%
% where
% len: row vector containing the length of each substring in Chrom.
% sum(len) should equal the individual length.
% lb, ub: lower and upper bounds for each coded variable.
% code: row vector indicating how each substring is to be decoded.
% ONLY BINARY is allowed
% scale: binary row vector indicating where to use arithmetic
% and/or logarithmic scaling. ONLY ARITHMETIC is allowed
% lbin, ubin: binary row vectors indicating whether or not to include
% each bound in the representation range
%
% OUTPUTS:
%
% genot: matrix containing in each row the individual's concatenated
% binary string representation.
% Leftmost bits are MSb and rightmost are LSb.
%
% Author: Marco Bertinato and Mirco Rampazzo
% Date: 20/09/09
% Identify the population size (Nind)
% and the number of variable (Nvar)
[Nind,Nvar] = size(var);
% Identify the number of decision variables (Nvar)
[seven,NvarF] = size(fieldD);
if Nvar = NvarF
error('var must have the number of variables described in fieldD.');
end
if seven = 7
error('fieldD must have 7 rows.');
end
% Get substring properties
len = fieldD(1,:);
lb = fieldD(2,:);
ub = fieldD(3,:);
code = ¬(¬fieldD(4,:));
scale = ¬(¬fieldD(5,:));
lin = ¬(¬fieldD(6,:));
uin = ¬(¬fieldD(7,:));
% number of bit each genotypic representation
Lind = sum(len);

% preallocating for speed
genot = zeros(Nind,Lind);
% vector with the index of last bit for each variables
lf = cumsum(len);
% vector with the index of first bit for each variables
li = cumsum([1 len]);
% for logarithmic scaling
logsgn = sign(lb(scale));
lb(scale) = log( abs(lb(scale)) );
ub(scale) = log( abs(ub(scale)) );
∆ = ub − lb;
% vector with the quantum for the representation of each variables
Prec = .5 .^ len;
% = quantum if lb is not included
% num
% = 0 if lb is included
num = (¬lin) .* Prec;
% = quantum if lb and ub are included
% den = −quantum if neither lb nor up are included
% = 0 if lb is icluded && ub not or viceversa
den = (lin + uin − 1) .* Prec;
% initializing at zero evry bit of genotype
genDec = zeros(Nind,Nvar);
for i = 1:Nvar,
% scaling of the real values of each variables into [0 1]
% for all the row of var matrix
genDec(:,i) = (var(:,i)−lb(i))./∆(i) − num(i) ./ (1−den(i));
for n = 1:Nind
for b = 1:len(i)
% performing consecutive divisions to convert the rv
% into binary representation.
if ( genDec(n,i)/(.5^b) ) ≥ 1
% put 1 in correct position in genot: LSB @ li(i)
% MSB at lf(i)
genot(n,b+li(i)−1) = 1;
genDec(n,i) = genDec(n,i) − .5^b;
end
end
end
end

B
arg (x∈F ⊆S⊆R n ) min f(x) ,
gi(x) ≤ 0 i = 1, ..., q ;
hj(x) j = q + 1, ..., m ;
x F S
q m − q f(x)
¯x
gi(x) = 0
x
x ∗ ∈ F f(x ∗ ) ≤ f(x) x ∈ F

eval(x) =
f(x) , x ∈ F
f(x) + p(x) ,
p(x) p(x)
eval(x)
eval(x) =
f(x) , x ∈ F
f(x) · p(x) , .
p(x)
NP
ϕ(x) = f(x) +
q
i=1
riGi +
m
j=q+1
cjLj
,
(x) Gi Lj
gi(x) hj(x) ri cj
Gi Lj
Gi = max [0, gi(x)] β ,
Lj = |hj(x)| γ ,
g(x) ≤ 0
max [0, gi(x)] (x)
gi(x) > 0 hj(x) = 0
(x)

i cj

C
Nch QL Tair
P LRs
ON/OF F
{status, PLR, Q, Pe, E} = QL, Tair
CumPf := 0 ⊲
{} = (CumPf, 1) ⊲

i
{} = cumPf, i
cumPf + P (1C)
f,i > QL ⊲ i
statusi = 1
Qi = QL − cumPf
⊲
P LRi = Qi/P (2C)
f,i
P LRi = P LRi/P LR∗
i
k∗ F,i Pe,i
Ei
statusi+1 = . . . = statusNch = 0 ⊲
Qi+1 = . . . = QNch = Pe,i+1 = . . . = Pe,Nch = Ei+1
= . . . = ENch = 0
statusi = 1
P LRi = P LR∗ i ⊲
Qi = P (1C)
f,i
⊲
Pe,i Ei
i < Nch
{} = (CumPf + Qi, i + 1) ⊲
{} = (CumPf + Qi, 1) ⊲

i
{} = cumPf, i
cumPf = cumPf − P (1C)
f,i
⊲
cumPf + P (2C)
f,i > QL ⊲ i
Qi = QL − cumPf ⊲
P LRi = Qi/P (2C)
f,i
kF,i Pe,i Ei
P LRi = 1 ⊲
Qi = P (2C)
f,i
⊲
Pe,i Ei
i < Nch
{} = (CumPf + Qi, i + 1) ⊲
⊲

i
{} = cumPf, i
cumPf + P (2C)
f,i > QL ⊲ i
statusi = 1
Qi = QL − cumPf
⊲
P LRi = Qi/P (2C)
f,i
P LRi ≤ P LR∗
i
P LRi = P LRi/P LR∗
i
k∗ F,i Pe,i
Ei
⊲
kF,i Pe,i Ei
⊲
statusi+1 = . . . = statusNch = 0 ⊲
Qi+1 = . . . = QNch = Pe,i+1 = . . . = Pe,Nch = Ei+1
= . . . = ENch = 0
statusi = 1 ⊲
P LRi = 1
Qi = P (2C)
f,i
⊲
Pe,i Ei
i < Nch
{} = (CumPf + Qi, i + 1) ⊲
⊲

D

D

Specification
Controller
design
Model
identification
Reference
+
Controller
Controller
parameters
Process
Output
-
Process
parameters
Auto-tuning regulator

Specification
Controller
design
Model
identification
Reference
+
Controller
Controller
parameters
Process
Output
-
Process
parameters
Auto-tuning regulator

(t) y(t)
+
Gc Gp
-

y ss
System model
SOPDT
0
0 50
Normalized time [−]
100
Gc(s)
Gp(s)
Gc(s) = Kc
Gp(s) =
1 + 1
,
Tis
Kpe −dps
1 + τps
.
Kc Ti
Gcl(s) =
Y (s)
R(s) =
Ke−ds τ 2s2 .
+ 2ζτs + 1
Kp dp τp

0.1 < dp/τp < 1
Kc
Ti
Td
1.048
Kp
−0.897 dp
τp
τp
1.195 − 0.368 · dp/τp
0.489τp
0.888 dp
τp
1.468
−0.970 dp
Kp τp
τp
−0.725 τp
0.942 dp
0.939 dp
0.443τp
ˆ ∞ n 2
Jn(θ) = t e(θ, t)] dt ,
0
e(θ, t) θ
Jn(θ)
RM
τp

Temperature [°C]
11.5
11
10.5
10
RM id ZAS
Kp 1.27 Kc 3.5 2.2 2.5
dp 29 Ti 97 110 92
τp 100 Td 12 16
PID RM
On−line identification
under PI id
Setpoint
Superheat
PID ZAS
ISE=17.66 ISE=16.36
9.5
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Normalized time [−]
id
ZAS

RM 4.4 · 10 7 2.3 · 10 9 1.9 · 10 7
ZAL 2.1 · 10 7 4.6 · 10 8 8.1 · 10 6
ZAL Kc = 3.8 Ti = 43
Td = 14
RM ZAL
ZAL
RM

Temperature [°C]
Valve position [%]
85
80
75
disturbance
RM
ZA
70
0 100 200 300 400 500 600 700
Normalized time [−]
13.5
13
12.5
12
11.5
11
10.5
10
9.5
RM
ZA
9
0 100 200 300 400 500 600 700
Normalized time [−]
RM Kc = 3.5 Ti = 97
Td = 12 ZAL Kc = 3.8 Ti = 43 Td = 14

Temperature [°C]
18
16
14
12
10
8
6
RM
ZA
T airin
4
0 500 1000 1500
Normalized time [−]
Temperature [°C]
13
12
11
10
9
8
7
RM
ZA
T airin
6
0 100 200 300
Normalized time [−]
400 500 600

Gcl(s) =
Y (s)
R(s) =
Ke−ds τ 2s2 ,
+ 2ζτs + 1
K = yss
,
rr
ρ = − 1
2π ln
yp2 − yss
,
yp1 − yss
ζ =
ρ2 ,
1 + ρ2
τ = (tp2 − tp1) 1 − ζ 2
2π
,
d = Sc
− 2ζτ ,
yss
yss yp1 yp2 tp1 tp2
rr Sc
ˆ +∞
Sc =
0
[yss − y(t)]dt ,
K ζ τ d
Gcl(jω)
Gcl(jω) Gc(jω)
Gp(jω)
ωc
−dωc − arctan 2ζτωc
1 − τ 2ω2 c
= −π ,
|Gcl(jωc)|
ωc

Process output
y ss
0
0
t p1
y p1
y p2
t p2
Time
|Gc(jωc)Gp(jωc)| = |Gcl(jωc)|
.
1 + |Gcl(jωc)|
|Gcl(jωc)| = M =
K
(1 − τ 2ω2 c ) 2 + (2τζωc) 2
,
Kp = Ti
KcSc
τp =
(KcKp) 2 (1 + T 2
i ω2 c ) (1 + M) 2 − (MTiωc) 2
Mω 2 c Ti
dp = 1
ωc
1
arctan (Tiωc) + arctan
yss ,
τpωc
,
,

Process output
y ss
0
0
t p1
y p1
y p2
t p2
Time
|Gc(jωc)Gp(jωc)| = |Gcl(jωc)|
.
1 + |Gcl(jωc)|
|Gcl(jωc)| = M =
K
(1 − τ 2ω2 c ) 2 + (2τζωc) 2
,
Kp = Ti
KcSc
τp =
(KcKp) 2 (1 + T 2
i ω2 c ) (1 + M) 2 − (MTiωc) 2
Mω 2 c Ti
dp = 1
ωc
1
arctan (Tiωc) + arctan
yss ,
τpωc
,
,

Process output
y ss
0
0
t p1
y p1
y p2
t p2
Time
|Gc(jωc)Gp(jωc)| = |Gcl(jωc)|
.
1 + |Gcl(jωc)|
|Gcl(jωc)| = M =
K
(1 − τ 2ω2 c ) 2 + (2τζωc) 2
,
Kp = Ti
KcSc
τp =
(KcKp) 2 (1 + T 2
i ω2 c ) (1 + M) 2 − (MTiωc) 2
Mω 2 c Ti
dp = 1
ωc
1
arctan (Tiωc) + arctan
yss ,
τpωc
,
,

Process output
y ss
0
0
t p1
y p1
y p2
t p2
Time
|Gc(jωc)Gp(jωc)| = |Gcl(jωc)|
.
1 + |Gcl(jωc)|
|Gcl(jωc)| = M =
K
(1 − τ 2ω2 c ) 2 + (2τζωc) 2
,
Kp = Ti
KcSc
τp =
(KcKp) 2 (1 + T 2
i ω2 c ) (1 + M) 2 − (MTiωc) 2
Mω 2 c Ti
dp = 1
ωc
1
arctan (Tiωc) + arctan
yss ,
τpωc
,
,

Process output
y ss
0
0
t p1
y p1
y p2
t p2
Time
|Gc(jωc)Gp(jωc)| = |Gcl(jωc)|
.
1 + |Gcl(jωc)|
|Gcl(jωc)| = M =
K
(1 − τ 2ω2 c ) 2 + (2τζωc) 2
,
Kp = Ti
KcSc
τp =
(KcKp) 2 (1 + T 2
i ω2 c ) (1 + M) 2 − (MTiωc) 2
Mω 2 c Ti
dp = 1
ωc
1
arctan (Tiωc) + arctan
yss ,
τpωc
,
,

Process output
y ss
0
0
t p1
y p1
y p2
t p2
Time
|Gc(jωc)Gp(jωc)| = |Gcl(jωc)|
.
1 + |Gcl(jωc)|
|Gcl(jωc)| = M =
K
(1 − τ 2ω2 c ) 2 + (2τζωc) 2
,
Kp = Ti
KcSc
τp =
(KcKp) 2 (1 + T 2
i ω2 c ) (1 + M) 2 − (MTiωc) 2
Mω 2 c Ti
dp = 1
ωc
1
arctan (Tiωc) + arctan
yss ,
τpωc
,
,