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8/14/2019 5.3 the Optimal Power Flow Problem
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OPF IntroductionThe idea of minimizing the total generation cost under full consideration of the
power flow equations goes back to the decade between 1960 and 1970. The
mathematical formulation and first solutions of this optimization problem have been
given by Carpentier (1962) and Tinney, Dommel (1967).
5.3.15.3 The optimal power flow problem
Looking back to the Lagrangian function which has been used in economic dispatch;
N1i)PPP()P(FLi
GilossLGii
i K=++=
We can realize, that the power flow in the network has been reduced to one simple
equality constraint: =+
iGilossL 0PPP
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The idea of Carpentier was to replace the simplifying constraint
load plus losses equals generation by the power flow equations for each node in
the network. This formulation of the minimization problem including the power flow
equations is called Optimal Power Flow (OPF).
Important applications of the OPF today:
Calculation of the optimum generation pattern to achieve the minimum total cost
of generation. Calculation of the optimum generation pattern to minimize air pollution by thermal
generating units.
Using the network losses as objective, the OPF algorithm can find the optimum
reactive power injections of generators, optimum settings of transformer taps and
switched capacitors.
5.3.25.3 The optimal power flow problem
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5.3.35.3 The optimal power flow problem
PG2PL2 =100 MW
V2 = 224 kV
PG1PL1 =100 MW
V1 = 224 kV; 1=0
PG3PL3 = 600 MW
V3 = 224 kV
1 2
3
~
~ ~L12
L23L13
OPF Example:Find the operating pattern with minimum total fuel costs if the total load of 800 MW is
distributed on the nodes PL1 = 100 MW, PL2 = 100 MW, PL3 = 600 MW.
( ) [ ]( ) [ ]
( ) [ ]h$2
3G3G3G3
h$2
2G2G2G2
h
$2
1G1G1G1
P012.0P162000PF
P01.0P141500PFP008.0P121000PF
++=
++=
++=
Line impedances:( )+=== 6164841412
2 31 31 2
.j.ZZZ
Line shunt admittances:
0=+ ikik jBSGS
Network voltage:
kV224VVV 321 ===
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5.3.45.3 The optimal power flow problem
Solution:
2
33
2
22
2
11
01201620000101415000080121000 GGGGGG P.PP.PP.PF ++++++++=
Objective function:
Equality constraints (power flow equations, active power):
( ) ( )[ ] 32102 ,,iPPsinBcosGVVGV LiGiiiiiiiii ==+++
( ) ( ) ( ) ( ) ( ) 01001311 3311 3211 2211 21 31 2
=++++ GPsinbcosasinbcosaaa
( ) ( ) ( ) ( ) ( ) 01002322 3322 3122 1122 12 32 1
=++++ GPsinbcosasinbcosaaa
( ) ( ) ( ) ( ) ( ) 06003233 2233 2133 1133 13 23 1
=++++ GPsinbcosasinbcosaaa
MW.aaaaaa 42473 22 33 11 32 11 2
======
MW.bbbbbb 99683 22 33 11 32 11 2
======
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5.3.55.3 The optimal power flow problem
Variables:
Lagrange function:
( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )[ ]600xxxsinbxxcosaxsinbxcosaaa
100xxxsinbxxcosaxsinbxcosaaa
100xxsinbxcosaxsinbxcosaaa
x012.0x162000x01.0x141500x008.0x121000L
34532453253153132313
25423542342142123212
151351341241213121
233
212
211
+++++
+++++
+++++
++++++++=
3
2
1
3
2
3
2
1
3
2
1
5
4
3
2
1
=
G
G
G
P
PP
x
x
x
xx
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5.3.65.3 The optimal power flow problem
Necessary conditions for extremum:
( ) ( )[ ]
( ) ( ) ( ) ( )[ ]
( ) ( )[ ] 0453 2453 23
542 3542 342 142 12
41 241 21
=+
++++
xxcosbxxsina
xxcosbxxsinaxcosbxsina
xcosbxsina:x
L0
4
=
:x
L0
5
=
:x
L0
1
=
0016012
11
=+ x. (1)
:xL 0
2
= 002014
22
=+ x. (2)
:x
L0
3
=
0024016
33
=+ x. (2)
(5)
(4)
( ) ( )[ ]
( ) ( )[ ]
( ) ( ) ( ) ( )[ ] 0xxcosbxxsinaxcosbxsina
xxcosbxxsina
xcosbxsina
453245325315313
542354232
5135131
=++++
+
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5.3.75.3 The optimal power flow problem
:L
0
1
=
( ) 0
5411
=x,x,xh
(6)( ) ( ) ( ) ( ) ( ) 0100151 351 341 241 21 31 2 =++++ xxsinbxcosaxsinbxcosaaa
:L
0
2
=
(7)
( ) 05422
=x,x,xh
( ) ( ) ( ) ( ) ( ) 01002542 3542 342 142 12 32 1
=++++ xxxsinbxxcosaxsinbxcosaaa
:
L0
3
=
(8)
( )0
5433
=x,x,xh
( ) ( ) ( ) ( ) ( ) 06003453 2453 253 153 13 23 1
=++++ xxxsinbxxcosaxsinbxcosaaa
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5.3.85.3 The optimal power flow problem
000
X
h
X
h100
000X
h
X
h010
000X
h
X
h001
X
h
X
h
X
h
XX
L
XX
L
000
X
h
X
h
X
h
XX
L
XX
L000
10000024.000
010000020.00
0010000016.0
0G
GL
5
3
4
3
5
2
4
2
5
1
4
1
5
3
5
2
5
1
55
2
54
24
3
4
2
4
1
45
2
44
2
T
2
=
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5.3.95.3 The optimal power flow problem
[ ] [ ] [ ]
[ ] [ ] [ ]( )
( )
( )5433
5422
5411
321
321
33
22
11
024016
020014
016012
x,x,xh
x,x,xh
x,x,xh
x.
x.
x.
h
L
++
++
+
+
+
=
Vector of right
hand side:
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5.3.105.3 The optimal power flow problem
63901
.x =
Result of
iterative
procedure:
62372
.x =
91953
.x =
052104
.x =
242405 .x =
250181
.=
752182 .=
701203
.=
MW.PG 63901 =
MW.PG 62372 =
MW.PG 91953 =
= 98722
.
= 889133 .
MWh$.25018
1=
MWh
$
.75218
2
=
MWh$.70120
3=
h
$
.F9617892=
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5.3.115.3 The optimal power flow problem
PG1=390.6 MW
V1 = 224 kV ; 1=0
PL1 =100 MW
1=18.250 $/MWh 1 2
3
~
~ ~
Ploss=24.06 MW
F = 17892.96 $/h
PG2=237.6 MWV2 = 224 kV ; 2=-2.987
PL2 =100 MW
2=18.752 $/MWh
PG3=195.9 MW
V3 = 224 kV ; 3=-13.889
PL3 =600 MW3=20.701 $/MWh
50.82 50.15
239.81
225.35
187.72
178
.79
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5.3.125.3 The optimal power flow problem
Locational Marginal Price (LMP)
Cost to supply the next unit of energy in the most economic way at a particular location in
the network
PG1 = 0.351; PG2 = 0.248; PG3 = 0.420PL3=1
PG1 = 0.320; PG2 = 0.414; PG3 = 0.249PL2=1
PG1 = 0.471; PG2 = 0.256; PG3 = 0.234PL1=1
LMPPGi (result of OPF calculation)PLiNode
1
1
25018 ==
MWh
$
L
.P
F
2
2
75218 ==
MWh
$
L
.P
F
3
3
70120 ==
MWh
$
L
.P
F
1
2
3