interior point methods for optimal power flow: linear
TRANSCRIPT
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Interior Point Methods for Optimal Power Flow:Linear Algebra Goodies
and Contingency Generation
Andreas Grothey(joint work with Nai Yuan Chiang, Argonne National Lab)
School of Mathematics, University of Edinburgh
IMA Numerical Analysis 2014, 3-5 September, Birmingham
TH
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U N I V E R S
I TY
OF
ED I N B U
RG
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Andreas Grothey OPF contingency generation
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USA August 2003
before 4.05pm
after 4.10pm
50 million peopledisconnected within 5 min
Andreas Grothey OPF contingency generation
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US-Canada Power System Outage Task Force: “Interim Report: Causes of the August 14th Blackout in the
United States and Canada”, November 2003
Andreas Grothey OPF contingency generation
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US-Canada Power System Outage Task Force: “Interim Report: Causes of the August 14th Blackout in the
United States and Canada”, November 2003
Andreas Grothey OPF contingency generation
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Challenges for the Energy Industry
Power Systems operation faces new challenges
Competition/increasingly guided by market principles
Increasing integration of renewables (uncertainty)
Expansion of distributedgeneration/smart grid
Transmission capacity does not keepup, expansion is politicallycontentious.
⇒ Secure operation of networks becomes increasingly important!
Andreas Grothey OPF contingency generation
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Challenges for the Energy Industry
Power Systems operation faces new challenges
Competition/increasingly guided by market principles
Increasing integration of renewables (uncertainty)
Expansion of distributedgeneration/smart grid
Transmission capacity does not keepup, expansion is politicallycontentious.
⇒ Secure operation of networks becomes increasingly important!
Power System operation needs to be designed robustly
Traditionally achieved by use of safety margins
Increasing need to take network contingencies into accountexplicitly
⇒ Security constrained Optimal Power FlowAndreas Grothey OPF contingency generation
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Challenges for the Energy Industry
Power Systems operation faces new challenges
Competition/increasingly guided by market principles
Increasing integration of renewables (uncertainty)
Expansion of distributedgeneration/smart grid
Transmission capacity does not keepup, expansion is politicallycontentious.
⇒ Secure operation of networks becomes increasingly important!
Security Constrained Optimal Power Flow (SCOPF)
Important both in its own right and as a subproblem for
Unit Commitment
Transmission switching
etcAndreas Grothey OPF contingency generation
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Overview
The ProblemPower systems 101Optimal Power Flow & Security constraintsAC OPF vs DC OPF
Linear Algebra GoodiesInterior Point MethodsCompact factorizationIterative methods & preconditioners
Contingency GenerationMotivationIPM, warmstarting & theoryResults
Andreas Grothey OPF contingency generation
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Optimal Power Flow
Andreas Grothey OPF contingency generation
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Security Constrained Optimal Power Flow
Optimal Power Flow (OPF)
Optimal Power Flow is the problem of deciding on (cost)optimal electricity generator outputs to meet demand withoutoverloading the transmission network
Solved at a given time for known demand
Andreas Grothey OPF contingency generation
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Security Constrained Optimal Power Flow
Optimal Power Flow (OPF)
Optimal Power Flow is the problem of deciding on (cost)optimal electricity generator outputs to meet demand withoutoverloading the transmission network
Solved at a given time for known demand
Need to model all lines in the networkPower flows are AC: represented by complex flows (or real andreactive components)
⇒ nonlinear network constraints
Andreas Grothey OPF contingency generation
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Security Constrained Optimal Power Flow
Optimal Power Flow (OPF)
Optimal Power Flow is the problem of deciding on (cost)optimal electricity generator outputs to meet demand withoutoverloading the transmission network
Solved at a given time for known demand
Need to model all lines in the networkPower flows are AC: represented by complex flows (or real andreactive components)
⇒ nonlinear network constraintsUnlike other transportationnetworks, operator has nocontrol over routing. Routingis determined by physics!
⇒ Kirchhoffs Laws.
Andreas Grothey OPF contingency generation
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Power Systems Operation 101
Andreas Grothey OPF contingency generation
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Power System Operation
Network
b ∈ B Buses
l = (bb′) ∈ L Lines
g ∈ G Generators (at bus og )
Power flows are AC. Described by real and reactive flows over lines,voltage and phase angle at buses
Variables
vb Voltage level at bus bδb Phase angle at bus bpbb′ , qbb′ Real and reactive power flow on line l = (bb′)pGg , qG
g Real and reactive power output at generator g
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Power System Operation
Network
b ∈ B Buses
l = (bb′) ∈ L Lines
g ∈ G Generators (at bus og )
Parameters of the model are: line/bus characteristics, real/reactivedemands, limits on power flow and voltages
Parameters
Gbb′ ,Bbb′ conductance and susceptance of line l(reciprocals of resistance and reactance)
Cb bus susceptancePD
b ,QDb real and reactive power demand at bus b
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Power System Operation
Operator decides on real generation level and voltages atgenerators.Power flows and reactive generation arrange themselves so asto satisfy power flow equations (Kirchhoff Laws)
Constraints
Kirchhoff Voltage Law (KVL)
pLbb′ = vb[Gbb′(vb − vb′ cos(δi − δj)) + Bbb′vb′ sin(δb − δb′)]
qLbb′ = vb[−Gbb′vb′ sin(δi − δj ) + Bbb′(vb′ cos(δb − δb′)− vb)]
Kirchhoff Current Law (KCL)∑
g |og =b
pGg − PD
b =∑
(b,b′)∈L
pbb′ , ∀b ∈ B,
∑
g |og=b
qGg − QD
b =∑
(b,b′)∈L
qbb′ + Cbvb2, ∀b ∈ B
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Power System Operation
Operator decides on real generation level and voltages atgenerators.Power flows and reactive generation arrange themselves so asto satisfy power flow equations (Kirchhoff Laws)
Constraints
Kirchhoff Voltage Law (KVL)
pLbb′ = vb[Gbb′(vb − vb′ cos(δi − δj)) + Bbb′vb′ sin(δb − δb′)]
qLbb′ = vb[−Gbb′vb′ sin(δi − δj ) + Bbb′(vb′ cos(δb − δb′)− vb)]
Kirchhoff Current Law (KCL)∑
g |og =b
pGg − PD
b =∑
(b,b′)∈L
pbb′ , ∀b ∈ B,
∑
g |og=b
qGg − QD
b =∑
(b,b′)∈L
qbb′ + Cbvb2, ∀b ∈ B
Note: pLbb′ 6= pL
b′b, qLbb′ 6= qL
b′b due to line losses!
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Power System Operation
Line flows and generator reactive output should satisfy
Operational Limits
Line Flow Limits at both ends of each line
(pbb′)2 + (qbb′)2 ≤ fl2, (pb′b)
2 + (qb′b)2 ≤ fl
2
Voltage limits at buses
V min ≤ vb ≤ V max
reactive generation limits at generators
Qming ≤ qG
g ≤ Qmaxg
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Power System Operation
The variables of the OPF model can be divided into control andstate variables
Control u (set by the system operator)pGg real power output at generators
vg voltage levels at generation busesState x (determined by Kirchhoff’s laws)vb voltage levels at non-generating busesqGg reactive power output at generators
δb phase anglespbb′ , qbb′ real and reactive power flows over lines
OPF model
minu∈U f (u)s.t. KL(u, x) = 0 (Kirchhoff’s laws)
g(x) ≤ 0 (Operational limits)
Andreas Grothey OPF contingency generation
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Power System Operation
The variables of the OPF model can be divided into control andstate variables
Control u (set by the system operator)pGg real power output at generators
vg voltage levels at generation busesState x (determined by Kirchhoff’s laws)vb voltage levels at non-generating busesqGg reactive power output at generators
δb phase anglespbb′ , qbb′ real and reactive power flows over lines
OPF model
minu∈U f (u)s.t. KL(u, x) = 0 (Kirchhoff’s laws)
g(x) ≤ 0 (Operational limits)
Nonlinear! nonconvex! May have local solutions!
Andreas Grothey OPF contingency generation
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Security Constrained Optimal Power Flow
Robustification of OPF
Things go wrong: Lines do fail
⇒ Security-constrained OPF (SCOPF)
Find a (cost optimal) generation schedule that is feasible withrespect to network limits (line flows, bus voltages) even if anyone line should fail (while meeting given demand)
Andreas Grothey OPF contingency generation
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Security Constrained Optimal Power Flow
Robustification of OPF
Things go wrong: Lines do fail
⇒ Security-constrained OPF (SCOPF)
Find a (cost optimal) generation schedule that is feasible withrespect to network limits (line flows, bus voltages) even if anyone line should fail (while meeting given demand)
Contingency Scenarios
Typically consider all (single) line and bus (node) failures⇒ (n − 1) secure operation
Many operators require (n − 2). Becomes computationallyvery expensive!
Andreas Grothey OPF contingency generation
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Security Constrained Optimal Power Flow
Robustification of OPF
Things go wrong: Lines do fail
⇒ Security-constrained OPF (SCOPF)
Find a (cost optimal) generation schedule that is feasible withrespect to network limits (line flows, bus voltages) even if anyone line should fail (while meeting given demand)
Two-Stage Setup (like Stochastic Programming)
First stage decides on generation levels for all generators
Second stage corresponds to contingency scenarios(evaluate consequences of line/bus failures)
Andreas Grothey OPF contingency generation
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Security Constrained Optimal Power Flow
Robustification of OPF
Things go wrong: Lines do fail
⇒ Security-constrained OPF (SCOPF)
Find a (cost optimal) generation schedule that is feasible withrespect to network limits (line flows, bus voltages) even if anyone line should fail (while meeting given demand)
Two-Stage Setup (like Stochastic Programming)
First stage decides on generation levels for all generators
Second stage corresponds to contingency scenarios(evaluate consequences of line/bus failures)
Scenarios only evaluate feasibility of first stage decisions. Noobjective contribution! No recourse action!
Andreas Grothey OPF contingency generation
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Security Constrained Optimal Power Flow
Robustification of OPF
Things go wrong: Lines do fail
⇒ Security-constrained OPF (SCOPF)
Find a (cost optimal) generation schedule that is feasible withrespect to network limits (line flows, bus voltages) even if anyone line should fail (while meeting given demand)
Two-Stage Setup (like Stochastic Programming)
First stage decides on generation levels for all generators
Second stage corresponds to contingency scenarios(evaluate consequences of line/bus failures)
Scenarios only evaluate feasibility of first stage decisions. Noobjective contribution! No recourse action!
Observation
Only a few contingency scenarios are needed/active to determinethe solution. How to find them?
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(n-1) secure OPF
In case of equipment (line/bus) failure, power flows will rearrangethemselves according to the power flow equations
Control variables u = (pGg , vg ) same for all contingencies,
Each contingency has its own set of state variables
xc = (vb,c , qGb,c , δb,c , pbb′,c , qbb′,c)
determined by Kirchhoff’s laws for the reduced network (KLc )
⇒ Seek a setting of control variables that does not lead toviolation of operational limits in any contingency
SCOPF model
minu∈U f (u)s.t. KLc(u, xc ) = 0 ∀c ∈ C
gc(xc ) ≤ 0 ∀c ∈ C
Andreas Grothey OPF contingency generation
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Structure of SCOPF problem
SCOPF model
minu∈U f (u)s.t. KLc(u, xc ) = 0 ∀c ∈ C
gc(xc ) ≤ 0 ∀c ∈ C
Structure of Jacobian
∂KL1∂x1
∂KLc
∂u∂g1∂x1
0. . .
...∂KL|C|
∂x|C|
∂KL|C|
∂u∂g|C|∂x|C|
0
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Structure of SCOPF problem
SCOPF model
minu∈U f (u)s.t. KLc(u, xc ) = 0 ∀c ∈ C
gc(xc ) ≤ 0 ∀c ∈ C
Structure of Jacobian
∂KL1∂x1
∂KLc
∂u∂g1∂x1
0. . .
...∂KL|C|
∂x|C|
∂KL|C|
∂u∂g|C|∂x|C|
0
Bordered block-diagonal matrix.
Andreas Grothey OPF contingency generation
W1
W|C|
T1
T|C|
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The DC OPF problem
The OPF model can simplified under the following assumptions:
Voltage level at all buses is the same: vb = 1, ∀b ∈ B.
The resistance of each line is small compared to reactance:⇒ |Bbb′ | ≫ |Gbb′ | ⇒ assume Gbb′ = 0.
Phase angle difference across each line is small⇒ sin(δ1 − δ2) ≈ δ1 − δ2, cos(δ1 − δ2) ≈ 1, ⇒ qL
bb′ = 0.
“DC”-OPF model
Kirchhoff Voltage Law
pLbb′ = −Bbb′(δb − δb′), ∀(bb′) ∈ L
Kirchhoff Current Law∑
g |og=b
pGg =
∑
(b,b′)∈L
pL(b,b′) + PD
b , ∀b ∈ B
Line Flow Limits: −fl ≤ pLl ≤ fl , ∀l ∈ L
Andreas Grothey OPF contingency generation
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The DC OPF problem
The OPF model can simplified under the following assumptions:
Voltage level at all buses is the same: vb = 1, ∀b ∈ B.
The resistance of each line is small compared to reactance:⇒ |Bbb′ | ≫ |Gbb′ | ⇒ assume Gbb′ = 0.
Phase angle difference across each line is small⇒ sin(δ1 − δ2) ≈ δ1 − δ2, cos(δ1 − δ2) ≈ 1, ⇒ qL
bb′ = 0.
“DC”-OPF model
Kirchhoff Voltage Law
pLbb′ = −Bbb′(δb − δb′), ∀(bb′) ∈ L
Kirchhoff Current Law∑
g |og=b
pGg =
∑
(b,b′)∈L
pL(b,b′) + PD
b , ∀b ∈ B
Line Flow Limits: −fl ≤ pLl ≤ fl , ∀l ∈ L
⇒ DC OPF is a linear programming problem
Andreas Grothey OPF contingency generation
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Structure of DC OPF problem
the DC-OPF problem can be written as
DC-OPF
min c⊤pG
s.t. R · pL +A⊤ · δ = 0A · pL −J · pG = −PD
where
bus/generator incidence matrix J ∈ IR |B|×|G|
node/arc incidence matrix A ∈ IR |B|×|L|
R = diag(1/B1, . . . , 1/B|L|)
Andreas Grothey OPF contingency generation
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Structure of DC OPF problem
the DC-OPF problem can be written as
DC-OPF
min c⊤pG
s.t. R · pL +A⊤ · δ = 0A · pL −J · pG = −PD
where
bus/generator incidence matrix J ∈ IR |B|×|G|
node/arc incidence matrix A ∈ IR |B|×|L|
R = diag(1/B1, . . . , 1/B|L|)
Augmented system like structure!
Andreas Grothey OPF contingency generation
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Structure of (n-1) secure DC OPF
SCOPF (DC version)
minpG ,pL,δ
c⊤pG
s.t. RpL1 +A⊤
1 δ1 = 0A1p
L1 −JpG = PD
. . .... =
...RpG
|C| +A⊤|C|δ|C| = 0
A|C|pG|C| −JpG = PD
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Structure of (n-1) secure DC OPF
SCOPF (DC version)
minpG ,pL,δ
c⊤pG
s.t. RpL1 +A⊤
1 δ1 = 0A1p
L1 −JpG = PD
. . .... =
...RpG
|C| +A⊤|C|δ|C| = 0
A|C|pG|C| −JpG = PD
Bordered block-diagonal matrix.
Andreas Grothey OPF contingency generation
W1
W|C|
T1
T|C|
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whistle stop tour of
Interior Point Methods
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Interior Point Methods (for LP)
Linear Program
min c⊤x s.t. Ax = bx ≥ 0
(LP)
KKT Conditions
c − A⊤λ− s = 0Ax = b
XSe = 0x , s ≥ 0
(KKT)
X = diag(x), S = diag(s)
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Interior Point Methods (for LP)
Barrier Problem
min c⊤x − µ∑
ln xi s.t. Ax = bx ≥ 0
(LPµ)
KKT Conditions
c − A⊤λ− s = 0Ax = b
XSe = µex , s ≥ 0
(KKTµ)
Introduce logarithmic barriers for x ≥ 0
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Interior Point Methods (for LP)
Barrier Problem
min c⊤x − µ∑
ln xi s.t. Ax = bx ≥ 0
(LPµ)
KKT Conditions
c − A⊤λ− s = 0Ax = b
XSe = µex , s ≥ 0
(KKTµ)
Introduce logarithmic barriers for x ≥ 0
(LPµ) is strictly convex
System (KKTµ) can be solved per Newton-Method
Andreas Grothey OPF contingency generation
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Interior Point Methods (for LP)
Barrier Problem
min c⊤x − µ∑
ln xi s.t. Ax = bx ≥ 0
(LPµ)
KKT Conditions
c − A⊤λ− s = 0Ax = b
XSe = µex , s ≥ 0
(KKTµ)
Introduce logarithmic barriers for x ≥ 0
(LPµ) is strictly convex
System (KKTµ) can be solved per Newton-Method
For µ→ 0 solution of (LPµ) converges to solution of (LP)
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Interior Point Methods (for LP)
KKT conditions
c − A⊤λ− s = 0Ax = b
XSe = µex , s ≥ 0
(KKTµ)
Andreas Grothey OPF contingency generation
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Interior Point Methods (for LP)
KKT conditions
c − A⊤λ− s = 0Ax = b
XSe = µex , s ≥ 0
(KKTµ)
Newton-Step
0 A⊤ IA 0 0S 0 X
∆x∆λ∆s
=
ξc
ξb
rxs
:=
c − A⊤λ− sb − Axµ+e − XSe
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Interior Point Methods (for LP)
KKT conditions
c − A⊤λ− s = 0Ax = b
XSe = µex , s ≥ 0
(KKTµ)
Newton-Step
0 A⊤ IA 0 0S 0 X
∆x∆λ∆s
=
ξc
ξb
rxs
:=
c − A⊤λ− sb − Axµ+e − XSe
Newton Step (reduced)[−Θ A⊤
A 0
] [∆x∆y
]=
[ξc − X−1rxsξb
]
where Θ = X−1S , X = diag(x), S = diag(s)
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Solving NLP by Interior Point Method
NLP
min f (x) s.t. g(x) ≤ 0 (NLP )
Andreas Grothey OPF contingency generation
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Solving NLP by Interior Point Method
NLP
min f (x)− µ∑
ln zi s.t. g(x) + z = 0z ≥ 0
(NLPµ)
Andreas Grothey OPF contingency generation
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Solving NLP by Interior Point Method
NLP
min f (x)− µ∑
ln zi s.t. g(x) + z = 0z ≥ 0
(NLP )
Optimality conditions
∇f (x)− A(x)⊤y = 0g(x) + z = 0
XZe = µex , z ≥ 0
Newton Step[Q(x , y) A(x)⊤
A(x) −Θ
] [∆x∆y
]=
[∇− f (x)− A(x)⊤y−g(x)− µY−1e
]
whereQ(x , y) = ∇2
xx(f (x) + y⊤g(x)), A(x) = ∇g(x)
Θ = X−1Z , X = diag(x), Z = diag(z)
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Linear Algebra of IPMs
Main work: solve[−Q −Θ A⊤
A 0
]
︸ ︷︷ ︸
Φ (QP)
[∆x∆y
]=
[rh
]or
[−Q(x , y) A(x)⊤
A(x) Θ
]
︸ ︷︷ ︸
Φ (NLP)
[∆x∆y
]=
[rh
]
for several right-hand-sides at each iteration
Two stage solution procedure
factorize Φ = LDL⊤
backsolve(s) to compute direction (∆x ,∆y) + corrections
⇒ Φ changes numerically but not structurally at each iteration
Key to efficient implementation is exploiting structure of Φ inthese two steps
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Structure of matrices A and Q for SCOPF:
Matrix A Matrix Q
W1
W2
W|C|
T1
T2
T|C|
Q1
Q2
Q|C|
Q0
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Structures of A and Q imply structure of Φ:
Q1
Q2
Q|C|
Q0
W⊤1
W⊤2
W⊤|C|
T⊤1 T⊤
2 T⊤|C|
W1
W2
W|C|
T1
T2
T|C|
Q1
Q2
Q|C|
Q0
W1
W2
W|C|
T1
T2
T|C|
W⊤1
W⊤2
W⊤|C|
T⊤1 T⊤
2T⊤|C|
(Q A⊤
A 0
)P
(Q A⊤
A 0
)P−1
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Structures of A and Q imply structure of Φ:
Q1
Q2
Q|C|
Q0
W⊤1
W⊤2
W⊤|C|
T⊤1 T⊤
2 T⊤|C|
W1
W2
W|C|
T1
T2
T|C|
Φ0
Φ1
Φ2
. . .
Φ|C|
B⊤1
B⊤2
...
B⊤|C|
B1 B2 · · · B|C|
(Q A⊤
A 0
)P
(Q A⊤
A 0
)P−1
Bordered block-diagonal structure in Augmented System!
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OOPS: Object Oriented Parallel Solver
OOPS
OOPS is an IPM implementation, that can exploit (nested)block structures through object oriented linear algebra
Solved (multistage) stochastic programming problems fromportfolio management with over 109 variables(≈ 2h on 1280 processors)
Dua
l Blo
ck A
ngul
ar S
truc
ture
Prim
al B
lock
Ang
ular
Str
uctu
re
Prim
al B
lock
Ang
ular
Str
uctu
re
D30
C31A
D1 D2
D D1211 D10 B B11 12 D D D D B B B21 22 23 20 21 22 23
C32
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Exploiting Structure in IPM
Block-Factorization of Augmented System Matrix0
B
B
B
@
Φ1 B⊤1
. . ....
Φn B⊤n
B1 · · · Bn Φ0
1
C
C
C
A
| {z }
Φ
0
B
B
B
@
x1
...xn
x0
1
C
C
C
A
| {z }
x
=
0
B
B
B
@
b1
...bn
b0
1
C
C
C
A
| {z }
b
Solution of Block-system by Schur-complement
The solution to Φx = b is
x0 = C−1b0, b0 = b0 −∑
i BiΦ−1i bi
xi = Φ−1i (bi − B⊤
i x0), i = 1, . . . , n
where C is the Schur-complement
C = Φ0 −n∑
i=1
BiΦ−1i B⊤
i
⇒ only need to factor Φi , not Φ
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Exploiting Structure in IPM
Solution of Block-system by Schur-complement
The solution to Φx = b is
x0 = C−1b0, b0 = b0 −∑
i BiΦ−1i bi
xi = Φ−1i (bi − B⊤
i x0), i = 1, . . . , n
where C is the Schur-complement
C = Φ0 −n∑
i=1
BiΦ−1i B⊤
i
Bottlenecks in this process are
Factorization of the Φi
Assembling∑n
i=1 BiΦ−1i B⊤
i
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Structure of Augmented System Matrix
Bottlenecks in this process are
Factorization of the Φi , Assembling∑n
i=1 BiΦ−1i B⊤
i
Φ =
Φ1 B⊤1
Φ2 B⊤2
. . ....
Φn B⊤n
B1 B2 · · · Bn Φ0
, Φi =
[X−1
i Si W Ti
Wi 0
]
For DC-OPF
Wi =
[R AT
i
Ai 0
], B⊤
i =
[0J
],
whereJ ∈ IR |Bi |×|Gi |: bus-generator incidence matrixAi ∈ IR |Bi |×|Li |: node-arc incidence matrix for contingency iR = diag(−V 2/B1, . . . ,V
2/B|L|): resistances > 0Andreas Grothey OPF contingency generation
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Block-Factorization for DC-OPF
Structure of Φi for DC-OPF
Φi =
[X−1
i Si W Ti
Wi 0
], Wi =
[R AT
i
Ai 0
], B⊤
i =
[0J
],
Solve Φix = b:
Wi is invertible and constant throughout IPM iterations
To solve Φix = b only need to factorize Wi :[
X−1i Si W⊤
i
Wi 0
] [x(0)
x(1)
]=
[b(0)
b(1)
]
⇒ x(1) = W−1i b(1), x(0) = XiS
−1i (b(0) −W⊤
i x(1))
To build BiΦ−1i B⊤
i
BiΦ−1i B⊤
i = −J⊤W−⊤i XiS
−1i W−1
i J
= −ViX−1i SiV
⊤i , V⊤
i = W−1i J
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Exploiting Structure in IPM
Solution of block-system by Schur-complement
The solution to Φx = b is
x0 = C−1b0, b0 = b0 −∑
i BiΦ−1i bi
C = Φ0 +n∑
i=1
ViX−1i SiV
⊤i , V⊤
i = W−1i J
Forming ViX−1i SiV
Ti and factorizing C is (still) expensive
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Exploiting Structure in IPM
Solution of block-system by Schur-complement
The solution to Φx = b is
x0 = C−1b0, b0 = b0 −∑
i BiΦ−1i bi
C = Φ0 +n∑
i=1
ViX−1i SiV
⊤i , V⊤
i = W−1i J
Forming ViX−1i SiV
Ti and factorizing C is (still) expensive
⇒ Solve Cx0 = b0 by iterative method
Use (preconditioned) iterative method (e.g. PCG/GMRES)
⇒ Evaluating matrix-vector products C · x is cheap:
C · x = Φ0x +∑
J⊤W−⊤i X−1
i SiW−1i Jx
Multiplication with J is trivial: J is 0-1 matrix.Wi has been factorized
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Possible preconditioners
Single (base) contingency (Qiu, Flueck ’05)
M = Φ0 + nV0X−10 S0V
⊤0
Sample Average Approximation (Anitescu, Petra ’12)
M = Φ0 +n
|S|∑
i∈S
ViX−1i SiV
⊤i
where S = random selection of contingency scenarios
Active Contingencies (Chiang, G. ’13)
M = Φ0 +∑
i∈A
ViX−1i SiV
⊤i
where A = set of contingencies considered active
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Active Contingency Preconditioner
Active Contingencies (Chiang, G. ’13)
M = Φ0 +∑
i∈A
ViX−1i SiV
⊤i
where A = set of contingencies considered active.
Based on the observation
ViX−1i SiV
⊤i
Vi = J⊤W−⊤i constant
X = diag(x), x = vector of slacks
⇒ Large contribution X−1i Si ≡ small slack xi ≡ active
contingency
⇒ Concentrate on active contingencies.
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Active Contingency Preconditioner
Active Contingencies (Chiang, G. ’13)
M = Φ0 +∑
i∈A
ViX−1i SiV
⊤i
where A = set of contingencies considered active.
Based on the observation
ViX−1i SiV
⊤i
Vi = J⊤W−⊤i constant
X = diag(x), x = vector of slacks
⇒ Large contribution X−1i Si ≡ small slack xi ≡ active
contingency
⇒ Concentrate on active contingencies.
Which contingencies should be considered active?
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Active Contingency Preconditioner: Theoretical Result
Convergence speed of an iterative scheme withsystem matrix C and preconditioner M,
is determined by the distribution of eigenvalues of
M−1C
Aim: All close to unity
Small numbers of clusters
Theorem
For a given δ > 0 let A consist of all contingencies i for which anyxi ,j < 1/δ. Then all eigenvalues λi of M−1C satisfy
|λi (M−1C )− 1| = O(δ)
Related to Active Set Preconditioners (Forsgren, Gill, Griffin ’07)
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Test Problem
Buses Gen Cont Variables Constraints Nonzeros3 4 2 17 14 35
26 5 40 2,630 2,626 7,92956 7 79 10,648 10,642 31,060
118 54 177 53,811 53,758 172,019300 69 322 229,077 229,009 683,542
iceland 35 72 28,725 28,691 77,059100 25 180 50,344 50,320 165,649200 50 370 210,779 210,730 701,249300 75 565 488,534 488,460 1,635,824400 100 760 881,339 881,240 2,960,399500 125 955 1,389,194 1,389,070 4,674,974
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Results
Results for Active Contingency preconditioner:
#Bus #Scenarios Time(s) Iters |A|3 3 <0.1 8 2
26 41 0.27 13 256 80 1.26 15 6
118 178 6.68 13 7300 323 24.83 13 12
iceland 73 3.12 13 5100 181 9.09 20 7200 371 50.63 28 9300 566 205.79 39 20400 761 523.65 55 20500 956 823.95 46 25
Base case preconditioner only works for smallest two problems
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Results
Default Direct Iterativebuses time(s) memory time(s) memory time(s) memory
3 <0.1 5.2MB <0.1 5.2MB <0.01 5.2MB26 0.21 7.6MB 0.17 7.5MB 0.27 7.4MB56 1.00 14.3MB 0.77 14.1MB 1.26 13.5MB
118 6.88 49MB 6.69 64MB 6.88300 31.79 198MB 30.42 282MB 24.83
iceland 2.28 25MB 2.36 30MB 3.12100 9.02 54.6MB 6.16 53.1MB 9.09 43.6MB200 65.97 220MB 45.23 244MB 50.63 163MB300 251.70 531MB 177.39 667MB 205.79 387MB400 955.32 985MB 655.50 1380MB 523.65 715MB500 1552.80 1593MB 1195.77 2467MB 823.95 1163MB
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Contingency Generation
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Contingency Generation
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Contingency Generation
“n-1”- (or even “n-2”-security) requires the inclusion of manycontingency scenarios.
Pan-European system has 13000 nodes and 20000 lines
⇒ Resulting SCOPF model would have ≈ 1010 variables.
Only a few contingencies are critical for operation of thesystem (but which ones)?
Contingency Generation
Generate contingency scenarios dynamically when needed
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Contingency Generation
Prototype Algorithm:
Set up the model with a few base contingency scenariosSolve model to obtain optimal controls u∗
k
repeat
Check for violated contingency scenarios.Add violated scenarios to the modelRe-solve model to obtain new controls u∗
k+1.until no more violated contingencies
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Contingency Generation
Prototype Algorithm:
Set up the model with a few base contingency scenariosSolve model to obtain optimal controls u∗
k
repeat
Check for violated contingency scenarios.Add violated scenarios to the modelRe-solve model to obtain new controls u∗
k+1.until no more violated contingencies
Can we do this with Interior Point Methods?
Interior Point Methods are efficient for large scale (AC)OPF.
IPMs are bad at resolving a modified problem instance(warmstarting)
⇒ attempt to dynamically add contingencies into the partiallysolved OPF.
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Warmstarting Interior Point Methods
Aim: Use information from solution process of
min c⊤x s.t. Ax = bx ≥ 0
(LP)
to construct a starting point for (nearby problem)
min c⊤x s.t. Ax = bx ≥ 0
(LP)
where A ≈ A, b ≈ b, c ≈ c
It is not a good idea to use the solution of (LP) to start (LP).
Unlike for the Simplex/Active Set Method!
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Interior Point Methods (for NLP)
Nonlinear Problem
min f (x) s.t. g(x) = 0x ≥ 0
(NLP)
KKT Conditions
∇c(x)−∇g(x)⊤λ− s = 0g(x) = 0XSe = 0x , s ≥ 0
(KKTµ)
Introduce logarithmic barriers for x ≥ 0
System (KKTµ) can be solved per Newton-Method
For µ→ 0 solution of (NLPµ) converges to solution of (NLP)
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Interior Point Methods (for NLP)
Barrier Problem
min f (x)− µ∑
ln xi s.t. g(x) = 0x ≥ 0
(NLPµ)
KKT Conditions
∇c(x)−∇g(x)⊤λ− s = 0g(x) = 0XSe = µex , s ≥ 0
(KKTµ)
Introduce logarithmic barriers for x ≥ 0
System (KKTµ) can be solved per Newton-Method
For µ→ 0 solution of (NLPµ) converges to solution of (NLP)
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Interior Point Methods (for NLP)
KKT conditions
∇c(x)−∇g(x)⊤λ− s = 0g(x) = 0XSe = µex , s ≥ 0
(KKTµ)
Central Path
The set of all solutions to (KKTµ) for all µ > 0.Central Path joins the analytical center (for µ=∞)with the NLP solution (for µ = 0).
Neighbourhood of the central path (as used by IPOPT)
N (κ) := {(x , λ, s) : Eµ(x , λ, s) ≤ κµ}where
Eµ(x , λ, s) := max
{‖∇f (x)−∇g(x)λ− s‖sd
, ‖g(x)‖, ‖XS − µe‖sc
}
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Path Following Methods (for NLP)
choose x0, λ0, s0 > 0, µ0 = x⊤0 s0/n, k = 0
Outer iteration
update µ:
µ+ = σx⊤+ s+
n, 0 < σ < 1, k ← k + 1
Inner iteration
compute Newton step (∆x , ∆s, ∆λ) for (KKTµ) and given µk .
Do line search with merit function or filter to computestepsizes and take step in Newton direction
until
Eµk(xk , λk , sk) ≤ κµk
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Why?
Hippolito (1993): Search direction is parallel to nearby constraints
Original Problem
Modified Problem
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Why?
Hippolito (1993): Search direction is parallel to nearby constraints
Modified Problem
Original Problem
⇒ only small step in search direction can be taken
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Warmstarting Heuristics
Idea: Start close to the (new) central path, not close to the (old) solution
Modified Problem
Original Problem
⇒ Start from a previous iterate and do additional modification step.
Ok, as long as change to problem is small (compared to µ)
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Interior Point Warmstarts: Theoretical Results
A typical warmstart results is (Assume A = A):
Lemma (based on Gondzio/G. ’03)
Let (x , λ, s) ∈ N−∞(γ0) for problem (LP) then a full modification
step (∆x ,∆λ,∆s) in the perturbed problem (LP) is feasible and
(x + ∆x , λ + ∆λ, s + ∆s) ∈ N−∞(γ)
provided that
δbc ≤√
γ0 − γ
2B2∞
γ0µ3/2
⇒“small δbc , large µ”
where δbc := ‖c − AT y − s‖2 + ‖AT (AAT )−1(b − Ax)‖2:= Pc−AT y−s(s) + Pb−Ax(x)
is the orthogonal distance of the warmstart point from primal-dual
feasibility in the perturbed problem
Andreas Grothey OPF contingency generation
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Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenariosSolve model to obtain optimal controls u∗
0 .repeat
Check for violated contingency scenarios.Add violated scenarios to the modelRe-solve model to obtain new controls u∗
k+1.until no more violated contingencies
Andreas Grothey OPF contingency generation
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Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios.Solve model to obtain controls u∗
0 .repeat
Check for violated contingency scenarios.
Add violated scenarios to the model.
Re-solve model to obtain new controls u∗
k+1.until no more violated contingencies
Andreas Grothey OPF contingency generation
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Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios. µ0 = µ, k = 0.Solve model to obtain µ0-center. ⇒ controls uµ0 .repeat
Check for violated contingency scenarios.
Add violated scenarios to the model.
Choose µk+1 : 0 < µk+1 < µk , k ← k + 1Warmstart and iterate to find µk -center. ⇒ controls uµk
.until no more violated contingencies
Andreas Grothey OPF contingency generation
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Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios. µ0 = µ, k = 0.Solve model to obtain µ0-center. ⇒ controls uµ0 .repeat
Check for violated contingency scenarios.for all violated scenarios do
Set up single scenario problem with u = uµkand solve for
µk -centerend for
Add violated scenarios to the model.Patch together warmstart point for expanded problem.Choose µk+1 : 0 < µk+1 < µk , k ← k + 1Warmstart and iterate to find µk -center. ⇒ controls uµk
.until no more violated contingencies
Andreas Grothey OPF contingency generation
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Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios. µ0 = µ, k = 0.Solve model to obtain µ0-center. ⇒ controls uµ0 .repeat
Check for violated contingency scenarios.for all violated scenarios do
Set up single scenario problem with u = uµkand solve for
µk -centerend for
Add violated scenarios to the model.Patch together warmstart point for expanded problem.Choose µk+1 : 0 < µk+1 < µk , k ← k + 1Warmstart and iterate to find µk -center. ⇒ controls uµk
.until no more violated contingencies
Claim
We can ensure that the assembled warmstart point is in theneighbourhood N (κ)⇒ IPM convergence not negatively affected (warmstart successful)
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Derivation of Algorithm
IPM applied to SCOPF model
minu,xc ,sc f (u)− µ∑
c∈C
∑i ln sc,i
s.t. KLc(u, xc ) = 0 ∀c ∈ Cgc (xc) + sc = 0 ∀c ∈ C
Partially decompose: C = C0 ∪ C0
minu,xc ,sc
f (u)− µ∑
c∈C0
∑i ln sc,i +
∑c∈C0
Vc,µ(u)
s.t. KLc(u, xc) = 0 ∀c ∈ C0gc(xc) + sc = 0 ∀c ∈ C0
where
Vc,µ(u) = minxc ,sc
−µ∑
i ln sc,i
s.t. KLc(u, xc) = 0gc(xc ) + sc = 0
∀c ∈ C0
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Active/Inactive contingencies
KKT conditions for contingency problem (Vc,µ(u))
primal feasibilityKLc(u, xc)=0gc(xc) + sc=0
dual feasibility∇xKLc (u, xc)
⊤λc +∇gc(xc)⊤νc=0
centralitysc,iνc,i =µesc , νc ≥0
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Active/Inactive contingencies
KKT conditions for contingency problem (Vc,µ(u))
primal feasibilityKLc(u, xc)=0gc(xc) + sc=0
dual feasibility∇xKLc (u, xc)
⊤λc +∇gc(xc)⊤νc=0
centralitysc,iνc,i =µesc , νc ≥0
sc,iνc,i = µ⇒ νc,i = µ/sc,i
if contingency c is never active then sc,i → s∗c,i > 0 and thus
‖νc‖ = O(µ)→ 0⇒ ‖λc‖ = O(µ)→ 0
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Active/Inactive contingencies
KKT conditions for contingency problem (Vc,µ(u))
primal feasibilityKLc(u, xc)=0gc(xc) + sc=0
dual feasibility∇xKLc (u, xc)
⊤λc +∇gc(xc)⊤νc=0
centralitysc,iνc,i =µesc , νc ≥0
sc,iνc,i = µ⇒ νc,i = µ/sc,i
if contingency c is never active then sc,i → s∗c,i > 0 and thus
‖νc‖ = O(µ)→ 0⇒ ‖λc‖ = O(µ)→ 0
if contingency is active at solution then
‖νc‖ → ‖ν∗c ‖ > 0⇒ ‖λc‖ → ‖λ∗
c‖ > 0
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Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
N (κ) := {(x , λ, s) : Eµ(x , λ, s) ≤ κµ}where
Eµ(x , λ, s) := max
‖∇f (x)−∇g(x)λ− s‖∞︸ ︷︷ ︸
dual feasibility
, ‖g(x)‖∞︸ ︷︷ ︸primal feasibility
, ‖XS − µe‖∞︸ ︷︷ ︸centrality
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Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
N (κ) := {(x , λ, s) : Eµ(x , λ, s) ≤ κµ}where
Eµ(x , λ, s) := max
‖∇f (x)−∇g(x)λ− s‖∞︸ ︷︷ ︸
dual feasibility
, ‖g(x)‖∞︸ ︷︷ ︸primal feasibility
, ‖XS − µe‖∞︸ ︷︷ ︸centrality
Central Path conditions for SCOPF problem
primal feasibilityKLc(u, xc) =0gc(xc) + sc =0
dual feasibility∇f (u)−∑
c∇uKLc(u, xc )⊤λc =0
∇xKLc (u, xc)⊤λc +∇gc(xc)
⊤νc =0
centralitysc,iνc,i =µesc , νc ≥0
Andreas Grothey OPF contingency generation
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Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
N (κ) := {(x , λ, s) : Eµ(x , λ, s) ≤ κµ}where
Eµ(x , λ, s) := max
‖∇f (x)−∇g(x)λ− s‖∞︸ ︷︷ ︸
dual feasibility
, ‖g(x)‖∞︸ ︷︷ ︸primal feasibility
, ‖XS − µe‖∞︸ ︷︷ ︸centrality
Central Path conditions for SCOPF problem
primal feasibilityKLc(u, xc) =0gc(xc) + sc =0
dual feasibility∇f (u)−∑
c∇uKLc(u, xc )⊤λc =0
∇xKLc (u, xc)⊤λc +∇gc(xc)
⊤νc =0
centralitysc,iνc,i =µesc , νc ≥0
Satisfied within κµ by construction
Andreas Grothey OPF contingency generation
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Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
N (κ) := {(x , λ, s) : Eµ(x , λ, s) ≤ κµ}where
Eµ(x , λ, s) := max
‖∇f (x)−∇g(x)λ− s‖∞︸ ︷︷ ︸
dual feasibility
, ‖g(x)‖∞︸ ︷︷ ︸primal feasibility
, ‖XS − µe‖∞︸ ︷︷ ︸centrality
Central Path conditions for SCOPF problem
primal feasibilityKLc(u, xc) =0gc(xc) + sc =0
dual feasibility∇f (u)−∑
c∇uKLc(u, xc )⊤λc =0
∇xKLc (u, xc)⊤λc +∇gc(xc)
⊤νc =0
centralitysc,iνc,i =µesc , νc ≥0
Residual is ∇uKLc (u, xc)⊤λc for added contingency
Andreas Grothey OPF contingency generation
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Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
N (κ) := {(x , λ, s) : Eµ(x , λ, s) ≤ κµ}where
Eµ(x , λ, s) := max
‖∇f (x)−∇g(x)λ− s‖∞︸ ︷︷ ︸
dual feasibility
, ‖g(x)‖∞︸ ︷︷ ︸primal feasibility
, ‖XS − µe‖∞︸ ︷︷ ︸centrality
Central Path conditions for SCOPF problem
primal feasibilityKLc(u, xc) =0gc(xc) + sc =0
dual feasibility∇f (u)−∑
c∇uKLc(u, xc )⊤λc =0
∇xKLc (u, xc)⊤λc +∇gc(xc)
⊤νc =0
centralitysc,iνc,i =µesc , νc ≥0
Residual is ∇uKLc (u, xc)⊤λc for added contingency
⇒ ok, if added while λc is small (compared to µk)
Andreas Grothey OPF contingency generation
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Statement of Algorithm
Choose initial contingency set C0 ⊂ C. Choose µ0
repeat
Do IPM iteration in the master problem to get approximatesolution uk to (PC0,µk ).
For all c ∈ C0 solve Vc,µk (uk) approximately to get skc , λk
c
if ‖λkc ‖ > 1
2κµk for any c then
set C0 ← C0 ∪ {c}.take combined solution of (PC0,µk ) and (Vc,µk (uk)) asstarting point for next IPM iteration
end if
µk+1 = σµk
until convergence in master problem
Andreas Grothey OPF contingency generation
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Statement of Algorithm
Choose initial contingency set C0 ⊂ C. Choose µ0
repeat
Do IPM iteration in the master problem to get approximatesolution uk to (PC0,µk ).
For all c ∈ C0 solve Vc,µk (uk) approximately to get skc , λk
c
if ‖λkc ‖ > 1
2κµk for any c then
set C0 ← C0 ∪ {c}.take combined solution of (PC0,µk ) and (Vc,µk (uk)) asstarting point for next IPM iteration
end if
µk+1 = σµk
until convergence in master problem
Approximate means a point in the neighbourhood N (κ)
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Statement of Algorithm
Choose initial contingency set C0 ⊂ C. Choose µ0
repeat
Do IPM iteration in the master problem to get approximatesolution uk to (PC0,µk ).
For all c ∈ C0 solve Vc,µk (uk) approximately to get skc , λk
c
if ‖λkc ‖ > 1
2κµk for any c then
set C0 ← C0 ∪ {c}.take combined solution of (PC0,µk ) and (Vc,µk (uk)) asstarting point for next IPM iteration
end if
µk+1 = σµk
until convergence in master problem
Again a point in the neighbourhood N (κ). Usually only needs asingle iteration in (Vc,µk (uk)) starting from the previous iterate.
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Statement of Algorithm
Choose initial contingency set C0 ⊂ C. Choose µ0
repeat
Do IPM iteration in the master problem to get approximatesolution uk to (PC0,µk ).
For all c ∈ C0 solve Vc,µk (uk) approximately to get skc , λk
c
if ‖λkc ‖ > 1
2κµk for any c then
set C0 ← C0 ∪ {c}.take combined solution of (PC0,µk ) and (Vc,µk (uk)) asstarting point for next IPM iteration
end if
µk+1 = σµk
until convergence in master problem
The combined point is still in the N (κ) neighbourhood.⇒ successful warmstart of enlarged problem.
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Regarding Convergence & Efficiency of Algorithm
Issues
warmstart is always successful(i.e. the patched point is in the N (κ) neighbourhood
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Regarding Convergence & Efficiency of Algorithm
Issues
warmstart is always successful(i.e. the patched point is in the N (κ) neighbourhood
All active contingencies will be included eventually
since for active contingencies λc → λ∗c >
1
2κµ→ 0
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Regarding Convergence & Efficiency of Algorithm
Issues
warmstart is always successful(i.e. the patched point is in the N (κ) neighbourhood
All active contingencies will be included eventually
since for active contingencies λc → λ∗c >
1
2κµ→ 0
Contingency subproblems can be solved efficientlyContingency subproblem is just equation solving ⇒ only asingle (usually) Newton step needed. ⇒ FDLF
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Regarding Convergence & Efficiency of Algorithm
Issues
warmstart is always successful(i.e. the patched point is in the N (κ) neighbourhood
All active contingencies will be included eventually
since for active contingencies λc → λ∗c >
1
2κµ→ 0
Contingency subproblems can be solved efficientlyContingency subproblem is just equation solving ⇒ only asingle (usually) Newton step needed. ⇒ FDLFOnly a very small fraction of contingencies needed
only check for contingencies periodically.pre-screen and discard contingencies which are likely to neverbe active:Contingency Screening:Ejebe, Irisarri, Mokhtari (1995), Vaahedi, Fuchs, Xu, Mansour
(1999), Capitanescu, Wehenkel (2007, 2008), Dent, Ochoa,
Harrison, Bialek (2010)
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Contingency Generation: Results
Default Contingency Generation
Prob Sce time(s) iters time(s) iters ActS
6bus 2 <0.1 13 <0.1 13 2IEEE 24 38 5.7 41 3.9 30 6IEEE 48 78 52.3 71 32.3 52 15IEEE 73 117 204.1 97 156.7 92 25IEEE 96 158 351.5 106 252.9 76 27IEEE 118 178 - >200 1225 75 46IEEE 192 318 2393 132 1586 92 40
This is for the (nonlinear) (n − 1) AC-SCOPF problem
Work in progress: currently scans all (inactive) scenarios atevery iteration.
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Conclusions & Outlook
Conclusions:“Active Contingencies” preconditioner is effective
Only a few contingency scenarios are active at the solution⇒ Contingency generation
Contingencies can be added throughout IPM iterationswithout leaving ’safe’ neighbourhood.
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Conclusions & Outlook
Conclusions:“Active Contingencies” preconditioner is effective
Only a few contingency scenarios are active at the solution⇒ Contingency generation
Contingencies can be added throughout IPM iterationswithout leaving ’safe’ neighbourhood.
Extension:Much further scope for efficiency gains (contingencyscreening, efficient load flow solver for subproblems)
Transmission constraints and security planning will becomeincreasingly important in future with decentralised andintermittent generation.
Many operational and planning models in power systems (unitcommitment, transmission switching/islanding) should takecontingency constraints into account but currently do not(due to algorithmic complexity).
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Thank You!
Andreas Grothey OPF contingency generation
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Bibliography
M. Colombo, A. Grothey: A decomposition-based warm-startmethod for stochastic programming, ComputationalOptimization and Applications, Volume 55, Issue 2 (2013),Page 311-340.
N.-Y. Chiang, A. Grothey: Solving Security ConstrainedOptimal Power Flow Problems by a Structure ExploitingInterior Point Method. Optimization and Engineering,published online February 2014.
N.-Y. Chiang: Structure-Exploiting Interior Point Methods forSecurity Constrained Optimal Power Flow Problems, PhDThesis, University of Edinburgh (2013).
http://www.maths.ed.ac.uk/ERGO/preprints.html
Andreas Grothey OPF contingency generation