heuristic)minlp)for)solving) optimal)power)flow)problems
TRANSCRIPT
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Heuristic)MINLP)for)Solving)
Optimal)Power)Flow)Problems
Carleton)Coffrin)&)Hassan)Hijazi
1
OpBmisaBon)Research)Group
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NICTA)H)Optimisation)Research)Group
2
We#Love#Optimization#Solvers
Mostly#Computer#Scientists
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Optimization)Solver)Idea
2
SOLVER
Input)Data1
!Solution)1
"S1Sij =
Y ⇤ij � i
blij2
!ViV ⇤
i
TijT ⇤ij
� Y ⇤ij
ViV ⇤j
T ⇤ij
(i, j) 2 E
Sji =
Y ⇤ij � i
blij2
!VjV
⇤j � Y ⇤
ijVjV ⇤
i
Tij(i, j) 2 E
|Sij | |Sij | 8(i, j) 2 E [ ER
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Optimization)Solver)Idea
2
SOLVERBranch#and#Bound#
Algorithms
Highly#OpAmized
Structure#Structure#StructureInput)Data1
!Solution)1
"S1Sij =
Y ⇤ij � i
blij2
!ViV ⇤
i
TijT ⇤ij
� Y ⇤ij
ViV ⇤j
T ⇤ij
(i, j) 2 E
Sji =
Y ⇤ij � i
blij2
!VjV
⇤j � Y ⇤
ijVjV ⇤
i
Tij(i, j) 2 E
|Sij | |Sij | 8(i, j) 2 E [ ER
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Why)Optimization)Solvers?
2
http://wwwH03.ibm.com/software/products/nl/ibmilogcpleoptistud
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Call)for)Competition)on)
“Application)of)Modern)Heuristic)
Optimization)Algorithms)for)Solving)
Optimal)Power)Flow)Problems”
2
SOLVER?Genetic#Alg
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Applying)BlackHBox)Solvers)to)the)OPF
• Need)to)be)able)to)write)down)the)system)of)
equations)and)constraints)• 3)Problems)H)ORPD,)OARPD,)WPP)
• Simulation)Based)on)Matpower)
!
• Still)unclear)if)the)solver)would)work)well…)• These)problems)are)on)the)frontier)of)solver)
technology
2
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Modeling)the)Competition)OPF
2
��������
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1
An)Optimization)Model
Capital)Le[er)H)Complex)
Script'('Variable'Bold%Script%,%Data
Variables:
Sgi 8i 2 N
Vi 8i 2 N
Minimize:
X
i2N
c2i (<(Sgi ))
2+ c1i<(S
gi )
Subject to:
Sgi Sg
i Sgi 8i 2 N
|Vi| |Vi| |Vi| 8i 2 N
Sdi � Sg
i =
X
(i,j)2E[ER
�Sij 8i 2 N
Sij = Y ⇤ijViV
⇤i � Y ⇤
ijViV⇤j (i, j) 2 E
Sji = Y ⇤ijVjV
⇤j � Y ⇤
ijVjV⇤i (i, j) 2 E
|Sij | |Sij | 8(i, j) 2 E [ ER
Choices)
SimulaBon
Goal
Constraints
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1
Publication)OPF
Variables:
Sgi 8i 2 N
Vi 8i 2 N
Minimize:
X
i2N
c2i (<(Sgi ))
2+ c1i<(S
gi )
Subject to:
Sgi Sg
i Sgi 8i 2 N
|Vi| |Vi| |Vi| 8i 2 N
Sdi � Sg
i =
X
(i,j)2E[ER
�Sij 8i 2 N
Sij = Y ⇤ijViV
⇤i � Y ⇤
ijViV⇤j (i, j) 2 E
Sji = Y ⇤ijVjV
⇤j � Y ⇤
ijVjV⇤i (i, j) 2 E
|Sij | |Sij | 8(i, j) 2 E [ ER
GeneraBon
Voltage
GeneraBon)Bounds
Cost
Voltage)Bounds
KCL
Ohm’s)Law
Thermal))Limit
Capital)Le[er)H)Complex)
Script'('Variable'Bold%Script%,%Data
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1
Real)OPF
Variables:
Sgi 8i 2 N
Vi 8i 2 N
Minimize:
X
i2N
c2i (<(Sgi ))
2+ c1i<(S
gi )
Subject to:
Sgi Sg
i Sgi 8i 2 N
|Vi| |Vi| |Vi| 8i 2 N
Sdi + Y s
i ViV⇤i � Sg
i =
X
(i,j)2E[ER
�Sij 8i 2 N
Sij =
Y ⇤ij � i
blij2
!ViV ⇤
i
TijT ⇤ij
� Y ⇤ij
ViV ⇤j
T ⇤ij
(i, j) 2 E
Sji =
Y ⇤ij � i
blij2
!VjV
⇤j � Y ⇤
ijVjV ⇤
i
Tij(i, j) 2 E
|Sij | |Sij | 8(i, j) 2 E [ ER
Bus)Shunts
Transformers
Line)Charging
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Competition)OPF
Bus)Shunts
Bus)Shunt)On/Off
Tap)RaBo
Transformers
Variables:
Sgi 8i 2 N
Vi 8i 2 N
zi 2 {0, 1} 8i 2 N
Tij = {0.90, 0.91, . . . , 1.09, 1.1} (i, j) 2 E
Minimize:
X
i2N
c2i (<(Sgi ))
2+ c1i<(S
gi )
Subject to:
Sgi Sg
i Sgi 8i 2 N
|Vi| |Vi| |Vi| 8i 2 N
Sdi + zi (Y
si ViV
⇤i )� Sg
i =
X
(i,j)2E[ER
�Sij 8i 2 N
Sij =
Y ⇤ij � i
blij2
!ViV ⇤
i
T 2ij
� Y ⇤ij
ViV ⇤j
Tij(i, j) 2 E
Sji =
Y ⇤ij � i
blij2
!VjV
⇤j � Y ⇤
ijVjV ⇤
i
Tij(i, j) 2 E
|Sij | |Sij | 8(i, j) 2 E [ ER
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Competition)OPF)+)NH1Variables:
Sgi 8i 2 N
Vis 8i 2 N, 8s 2 S
zi 2 {0, 1} 8i 2 N
Tij = {0.90, 0.91, . . . , 1.09, 1.1} (i, j) 2 E
Minimize:
X
i2N
c2i (<(Sgi ))
2+ c1i<(S
gi )
Subject to:
Sgi Sg
i Sgi 8i 2 N
|Vis| = |Vit| 8i 2 G, 8s 2 S, 8t 2 S
8s 2 S
|Vi| |Vis| |Vi| 8i 2 N
Sdi + zi (Y
si VisV
⇤is)� Sg
i =
X
(i,j)2Es[ERs
�Sijs 8i 2 N
Sijs =
Y ⇤ij � i
blij2
!VisV ⇤
is
T 2ij
� Y ⇤ij
VisV ⇤js
Tij(i, j) 2 Es
Sjis =
Y ⇤ij � i
blij2
!VjsV
⇤js � Y ⇤
ijVjsV ⇤
is
Tij(i, j) 2 Es
|Sijs| |Sijs| 8(i, j) 2 E [ ER
Scenario))
Replicates
Generator)Voltages
Scenario)Voltages
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Solving)the)Competition)OPF
2
Bonmin#(IPOPT)
Input)Matpower
!Solution
"S
Variables:
Sgi 8i 2 N
Vi 8i 2 N
zi 2 {0, 1} 8i 2 N
Tij = {0.90, 0.91, . . . , 1.09, 1.1} (i, j) 2 E
Minimize:
X
i2N
c2i (<(Sgi ))
2+ c1i<(S
gi )
Subject to:
Sgi Sg
i Sgi 8i 2 N
|Vi| |Vi| |Vi| 8i 2 N
Sdi + zi (Y
si ViV
⇤i )� Sg
i =
X
(i,j)2E[ER
�Sij 8i 2 N
Sij =
Y ⇤ij � i
blij2
!ViV ⇤
i
T 2ij
� Y ⇤ij
ViV ⇤j
Tij(i, j) 2 E
Sji =
Y ⇤ij � i
blij2
!VjV
⇤j � Y ⇤
ijVjV ⇤
i
Tij(i, j) 2 E
|Sij | |Sij | 8(i, j) 2 E [ ER
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NO!
Global)Optimality?
Why?
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Continuous)nonHlinear)relaxation)(IPOPT))is)not)a)Lower)Bound)(nonHconvex)!)
!Bonmin’s)Branch)and)Bound))))))))))))))))))))))))))))not)guaranteed)to)find)OPT)
!
Still)a)Heuristic,)))))))))))))))))))))))))))))))))))))))
Heuristic%MINLP)!
Good)Enough?
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Results
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Wind)Power)Plant)Case)(96)scenarios)
2
))))))))))))))))))))))))))))Obj.))H)))Time)))H))ViolaBons
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IEEE)118)H)ORPD
2
))))))))))))))))))))))))))))Node)))))))))))))Obj.)))))Node)Iter.)H)Bme
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IEEE)118)Case)H)ORPD)H)Comparisons
2
Model Cont. ObjecBve Time
NLP Basecase 114.68 0.085s
MINLP Basecase 114.72 2.4s
NLP NH1 115.48 0.6s
MINLP NH1 115.54 15s
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Conclusions
• Shocked)by)how)effective)Bonmin/IPOPT)are)at)solving)these)problems.)
!
• Would)love)to)see)phase#shifters)added)to)the)decision)variables.)
!
• Great)exercise,)BIG)thanks)the)organizers!)• (hope)this)becomes)an)annual)event)
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