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
I VER
S
E
R
G
O F
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IT
E
UN
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N B U
Andreas Grothey
OPF contingency generation
USA August 2003
before 4.05pm
after 4.10pm
50 million people
disconnected within 5 min
Andreas Grothey
OPF contingency generation
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
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
Challenges for the Energy Industry
Power Systems operation faces new challenges
Competition/increasingly guided by market principles
Increasing integration of renewables (uncertainty)
Expansion of distributed
generation/smart grid
Transmission capacity does not keep
up, expansion is politically
contentious.
⇒ Secure operation of networks becomes increasingly important!
Andreas Grothey
OPF contingency generation
Challenges for the Energy Industry
Power Systems operation faces new challenges
Competition/increasingly guided by market principles
Increasing integration of renewables (uncertainty)
Expansion of distributed
generation/smart grid
Transmission capacity does not keep
up, expansion is politically
contentious.
⇒ 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 account
explicitly
⇒ Security constrained Optimal Power Flow
Andreas Grothey
OPF contingency generation
Challenges for the Energy Industry
Power Systems operation faces new challenges
Competition/increasingly guided by market principles
Increasing integration of renewables (uncertainty)
Expansion of distributed
generation/smart grid
Transmission capacity does not keep
up, expansion is politically
contentious.
⇒ 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
etc
Andreas Grothey
OPF contingency generation
Overview
The Problem
Power systems 101
Optimal Power Flow & Security constraints
AC OPF vs DC OPF
Linear Algebra Goodies
Interior Point Methods
Compact factorization
Iterative methods & preconditioners
Contingency Generation
Motivation
IPM, warmstarting & theory
Results
Andreas Grothey
OPF contingency generation
Optimal Power Flow
Andreas Grothey
OPF contingency generation
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 without
overloading the transmission network
Solved at a given time for known demand
Andreas Grothey
OPF contingency generation
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 without
overloading the transmission network
Solved at a given time for known demand
Need to model all lines in the network
Power flows are AC: represented by complex flows (or real and
reactive components)
⇒ nonlinear network constraints
Andreas Grothey
OPF contingency generation
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 without
overloading the transmission network
Solved at a given time for known demand
Need to model all lines in the network
Power flows are AC: represented by complex flows (or real and
reactive components)
⇒ nonlinear network constraints
Unlike other transportation
networks, operator has no
control over routing. Routing
is determined by physics!
⇒ Kirchhoffs Laws.
Andreas Grothey
OPF contingency generation
Power Systems Operation 101
Andreas Grothey
OPF contingency generation
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
δb
pbb′ , qbb′
pgG , qgG
Voltage level at bus b
Phase angle at bus b
Real and reactive power flow on line l = (bb ′ )
Real and reactive power output at generator g
Andreas Grothey
OPF contingency generation
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/reactive
demands, limits on power flow and voltages
Parameters
Gbb′ , Bbb′
conductance and susceptance of line l
Cb
PbD , QbD
bus susceptance
real and reactive power demand at bus b
(reciprocals of resistance and reactance)
Andreas Grothey
OPF contingency generation
Power System Operation
Operator decides on real generation level and voltages at
generators.
Power flows and reactive generation arrange themselves so as
to satisfy power flow equations (Kirchhoff Laws)
Constraints
Kirchhoff Voltage Law (KVL)
L
pbb
′ = vb [Gbb ′ (vb − vb ′ cos(δi − δj )) + Bbb ′ vb ′ sin(δb − δb ′ )]
L
qbb
′ = vb [−Gbb ′ vb ′ sin(δi − δj ) + Bbb ′ (vb ′ cos(δb − δb ′ ) − vb )]
Kirchhoff
X Current Law (KCL)X
pgG − PbD =
pbb′ ,
(b,b′ )∈L
g |og =b
X
g |og =b
qgG − QbD
=
X
∀b ∈ B,
qbb′ + Cb vb 2 ,
(b,b′ )∈L
Andreas Grothey
OPF contingency generation
∀b ∈ B
Power System Operation
Operator decides on real generation level and voltages at
generators.
Power flows and reactive generation arrange themselves so as
to satisfy power flow equations (Kirchhoff Laws)
Constraints
Kirchhoff Voltage Law (KVL)
L
pbb
′ = vb [Gbb ′ (vb − vb ′ cos(δi − δj )) + Bbb ′ vb ′ sin(δb − δb ′ )]
L
qbb
′ = vb [−Gbb ′ vb ′ sin(δi − δj ) + Bbb ′ (vb ′ cos(δb − δb ′ ) − vb )]
Kirchhoff
X Current Law (KCL)X
pgG − PbD =
pbb′ ,
(b,b′ )∈L
g |og =b
X
g |og =b
qgG − QbD
=
X
∀b ∈ B,
qbb′ + Cb vb 2 ,
(b,b′ )∈L
∀b ∈ B
L 6= p L , q L 6= q L due to line losses!
Note: pbb
′
b′ b
b′ b bb′
Andreas Grothey
OPF contingency generation
Power System Operation
Line flows and generator reactive output should satisfy
Operational Limits
Line Flow Limits at both ends of each line
2
2
(pb′ b )2 + (qb′ b )2 ≤ fl
(pbb′ )2 + (qbb′ )2 ≤ fl ,
Voltage limits at buses
V min ≤ vb ≤ V max
reactive generation limits at generators
Qgmin ≤ qgG ≤ Qgmax
Andreas Grothey
OPF contingency generation
Power System Operation
The variables of the OPF model can be divided into control and
state variables
Control u (set by the system operator)
pgG real power output at generators
vg voltage levels at generation buses
State x (determined by Kirchhoff’s laws)
vb
voltage levels at non-generating buses
G
qg
reactive power output at generators
δb
phase angles
′
′
pbb , qbb real and reactive power flows over lines
OPF model
minu∈U
s.t.
f (u)
KL(u, x) = 0
g (x) ≤ 0
Andreas Grothey
(Kirchhoff’s laws)
(Operational limits)
OPF contingency generation
Power System Operation
The variables of the OPF model can be divided into control and
state variables
Control u (set by the system operator)
pgG real power output at generators
vg voltage levels at generation buses
State x (determined by Kirchhoff’s laws)
vb
voltage levels at non-generating buses
G
qg
reactive power output at generators
δb
phase angles
′
′
pbb , qbb real and reactive power flows over lines
OPF model
minu∈U
s.t.
f (u)
KL(u, x) = 0
g (x) ≤ 0
(Kirchhoff’s laws)
(Operational limits)
Nonlinear! nonconvex! May have local solutions!
Andreas Grothey
OPF contingency generation
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 with
respect to network limits (line flows, bus voltages) even if any
one line should fail (while meeting given demand)
Andreas Grothey
OPF contingency generation
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 with
respect to network limits (line flows, bus voltages) even if any
one 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 computationally
very expensive!
Andreas Grothey
OPF contingency generation
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 with
respect to network limits (line flows, bus voltages) even if any
one 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
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 with
respect to network limits (line flows, bus voltages) even if any
one 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. No
objective contribution! No recourse action!
Andreas Grothey
OPF contingency generation
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 with
respect to network limits (line flows, bus voltages) even if any
one 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. No
objective contribution! No recourse action!
Observation
Only a few contingency scenarios are needed/active to determine
the solution. How to find them?
Andreas Grothey
OPF contingency generation
(n-1) secure OPF
In case of equipment (line/bus) failure, power flows will rearrange
themselves according to the power flow equations
Control variables u = (pgG , vg ) same for all contingencies,
Each contingency has its own set of state variables
G
, δb,c , pbb′ ,c , qbb′ ,c )
xc = (vb,c , qb,c
determined by Kirchhoff’s laws for the reduced network (KLc )
⇒ Seek a setting of control variables that does not lead to
violation of operational limits in any contingency
SCOPF model
minu∈U
s.t.
f (u)
KLc (u, xc ) = 0
gc (xc ) ≤ 0
Andreas Grothey
∀c ∈ C
∀c ∈ C
OPF contingency generation
Structure of SCOPF problem
SCOPF model
minu∈U
s.t.
f (u)
KLc (u, xc ) = 0
gc (xc ) ≤ 0
Structure of Jacobian
 ∂KL1








∂x1
∂g1
∂x1
∀c ∈ C
∀c ∈ C
∂KLc
∂u
..
0
..
.
.
∂KL|C|
∂x|C|
∂g|C|
∂x|C|
Andreas Grothey
∂KL|C|
∂u
0









OPF contingency generation
Structure of SCOPF problem
SCOPF model
minu∈U
s.t.
f (u)
KLc (u, xc ) = 0
gc (xc ) ≤ 0
Structure of Jacobian
 ∂KL1








∂x1
∂g1
∂x1
∀c ∈ C
∀c ∈ C
∂KLc
∂u
W1
T01
..
..
.
.
∂KL|C|
∂x|C|
∂g|C|
∂x|C|
∂KL|C|
∂u
W|C| T0|C|









Bordered block-diagonal matrix.
Andreas Grothey
OPF contingency generation
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
L = 0.
⇒ sin(δ1 − δ2 ) ≈ δ1 − δ2 , cos(δ1 − δ2 ) ≈ 1, ⇒ qbb
′
“DC”-OPF model
Kirchhoff Voltage Law
L
pbb
′ = −Bbb ′ (δb − δb ′ ),
∀(bb ′ ) ∈ L
Kirchhoff Current
Law X
X
G
L
D
pg =
p(b,b
′ ) + Pb ,
g |og =b
(b,b′ )∈L
Line Flow Limits: −fl ≤ plL ≤ fl ,
Andreas Grothey
∀b ∈ B
∀l ∈ L
OPF contingency generation
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
L = 0.
⇒ sin(δ1 − δ2 ) ≈ δ1 − δ2 , cos(δ1 − δ2 ) ≈ 1, ⇒ qbb
′
“DC”-OPF model
Kirchhoff Voltage Law
L
pbb
′ = −Bbb ′ (δb − δb ′ ),
∀(bb ′ ) ∈ L
Kirchhoff Current
Law X
X
G
L
D
pg =
p(b,b
′ ) + Pb ,
g |og =b
(b,b′ )∈L
Line Flow Limits: −fl ≤ plL ≤ fl ,
∀b ∈ B
∀l ∈ L
⇒ DC OPF is a linear programming problem
Andreas Grothey
OPF contingency generation
Structure of DC OPF problem
the DC-OPF problem can be written as
DC-OPF
min c ⊤ p G
s.t.
R · p L +A⊤ · δ
−J · p G
A · pL
=0
= −P D
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
Structure of DC OPF problem
the DC-OPF problem can be written as
DC-OPF
min c ⊤ p G
s.t.
R · p L +A⊤ · δ
−J · p G
A · pL
=0
= −P D
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
Structure of (n-1) secure DC OPF
SCOPF (DC version)
c ⊤pG
min
p G ,p L ,δ
s.t.
Rp1L +A⊤
1 δ1
L
A1 p1
..
−Jp G
=0
= PD
.
= ..
G
Rp|C|
+A⊤
|C| δ|C|
G
A|C| p|C|
−Jp G
=0
= PD
..
.
.
Andreas Grothey
OPF contingency generation
Structure of (n-1) secure DC OPF
SCOPF (DC version)
c ⊤pG
min
p G ,p L ,δ
s.t.
Rp1L +A⊤
1 δ1
L
1
A1 p1
W
T
−Jp1
G
..
..
.
.
=0
= PD
.
= ..
G
Rp|C|
+A⊤
=0
|C| δ|C|
G
G
|C|
|C|
A|C| p|C|
−Jp
= PD
W
Bordered block-diagonal matrix.
Andreas Grothey
OPF contingency generation
T
whistle stop tour of
Interior Point Methods
Andreas Grothey
OPF contingency generation
Interior Point Methods (for LP)
Linear Program
min c ⊤ x
s.t. Ax
x
= b
≥ 0
(LP)
KKT Conditions
c − A⊤ λ − s
Ax
XSe
x, s
X = diag(x), S = diag(s)
Andreas Grothey
=
=
=
≥
0
b
0
0
OPF contingency generation
(KKT)
Interior Point Methods (for LP)
Barrier Problem
min c ⊤ x − µ
X
ln xi
s.t. Ax
x
= b
≥ 0
(LPµ )
KKT Conditions
c − A⊤ λ − s
Ax
XSe
x, s
=
=
=
≥
0
b
µe
0
Introduce logarithmic barriers for x ≥ 0
Andreas Grothey
OPF contingency generation
(KKTµ )
Interior Point Methods (for LP)
Barrier Problem
min c ⊤ x − µ
X
ln xi
s.t. Ax
x
= b
≥ 0
(LPµ )
KKT Conditions
c − A⊤ λ − s
Ax
XSe
x, s
=
=
=
≥
0
b
µe
0
Introduce logarithmic barriers for x ≥ 0
(LPµ ) is strictly convex
System (KKTµ ) can be solved per Newton-Method
Andreas Grothey
OPF contingency generation
(KKTµ )
Interior Point Methods (for LP)
Barrier Problem
min c ⊤ x − µ
X
ln xi
s.t. Ax
x
= b
≥ 0
(LPµ )
KKT Conditions
c − A⊤ λ − s
Ax
XSe
x, s
=
=
=
≥
0
b
µe
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)
Andreas Grothey
OPF contingency generation
Interior Point Methods (for LP)
KKT conditions
c − A⊤ λ − s
Ax
XSe
x, s
Andreas Grothey
=
=
=
≥
0
b
µe
0
OPF contingency generation
(KKTµ )
Interior Point Methods (for LP)
KKT conditions
c − A⊤ λ − s
Ax
XSe
x, s
=
=
=
≥
0
b
µe
0
(KKTµ )
Newton-Step


 



0 A⊤ I
∆x
ξc
c − A⊤ λ − s
 A 0 0   ∆λ  =  ξb  :=  b − Ax

S 0 X
∆s
rxs
µ+ e − XSe
Andreas Grothey
OPF contingency generation
Interior Point Methods (for LP)
KKT conditions
c − A⊤ λ − s
Ax
XSe
x, s
=
=
=
≥
0
b
µe
0
(KKTµ )
Newton-Step


 



0 A⊤ I
∆x
ξc
c − A⊤ λ − s
 A 0 0   ∆λ  =  ξb  :=  b − Ax

S 0 X
∆s
rxs
µ+ e − XSe
Newton Step (reduced)
∆x
ξc − X −1 rxs
−Θ A⊤
=
A
0
∆y
ξb
where Θ = X −1 S,
X = diag(x),
Andreas Grothey
S = diag(s)
OPF contingency generation
Solving NLP by Interior Point Method
NLP
min f (x)
s.t. g (x)
Andreas Grothey
≤ 0
OPF contingency generation
(NLP )
Solving NLP by Interior Point Method
NLP
min f (x) − µ
X
ln zi
s.t. g (x) + z
z
Andreas Grothey
= 0
≥ 0
OPF contingency generation
(NLPµ )
Solving NLP by Interior Point Method
NLP
min f (x) − µ
X
ln zi
s.t. g (x) + z
z
= 0
≥ 0
(NLP )
Optimality conditions
∇f (x) − A(x)⊤ y
g (x) + z
XZe
x, z
=
=
=
≥
0
0
µe
0
Newton Step
Q(x, y ) A(x)⊤ ∆x
∇ − f (x) − A(x)⊤ y
=
A(x)
−Θ
∆y
−g (x) − µY −1 e
where
Q(x, y ) = ∇2xx (f (x) + y ⊤ g (x)),
Θ=X
−1
Z,
X = diag(x),
Andreas Grothey
A(x) = ∇g (x)
Z = diag(z)
OPF contingency generation
Linear Algebra of IPMs
Main work: solve
r
−Q − Θ A⊤ ∆x
=
A
0
∆y
h
|
{z
}
Φ (QP)
or
r
−Q(x, y ) A(x)⊤ ∆x
=
A(x)
Θ
∆y
h
{z
}
|
Φ (NLP)
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 in
these two steps
Andreas Grothey
OPF contingency generation
Structure of matrices A and Q for SCOPF:
Matrix A

 W1



W2






T1
T2
W|C| T|C|











Andreas Grothey

 Q1













Matrix Q

Q2
OPF contingency generation
Q|C|
Q0














Structures of A and Q imply structure of Φ:

W1⊤
Q1
 Q2







W
 1
 W
2



W2⊤
Q|C|
Q0 T1⊤ T2⊤
T1
T2
W|C|T|C|
Q A⊤
A 0





⊤
W|C|

⊤ 
T|C|







Andreas Grothey

Q1W1⊤
 W1


















T2 





Q|C|W ⊤

|C|

T|C| 
W|C|
T1
Q2W2⊤
W2
T1⊤
P
T2⊤
Q A⊤
A 0
OPF contingency generation

⊤
T|C|
P −1
Q0
Structures of A and Q imply structure of Φ:

W1⊤
Q1
 Q2







W
 1
 W
2



W2⊤
Q|C|
Q0 T1⊤ T2⊤
T1
T2
W|C|T|C|
Q A⊤
A 0





⊤
W|C|

⊤ 
T|C|








 Φ1












B1
P
B1⊤
Φ2
..
B2
.
···
Q A⊤
A 0



⊤
B2 


.. 
. 



⊤ 
Φ|C| B|C|

B|C|
Φ0
P −1
Bordered block-diagonal structure in Augmented System!
Andreas Grothey
OPF contingency generation

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 from
portfolio management with over 109 variables
(≈ 2h on 1280 processors)
Andreas Grothey
D11
D12
A
C31
C32
D1
D10 B11
B12
Primal Block Angular Structure










Dual Block Angular Structure











Primal Block Angular Structure

D30
D2
D21 D22 D23
OPF contingency generation
D20
B21 B22 B23
Exploiting Structure in IPM
Block-Factorization of Augmented System Matrix
10 1 0 1
0
Φ1
B1⊤
x1
b1
B ..
.. C B .. C B .. C
B
B C B C
.
. C
CB . C = B . C
B
⊤A @ A
@
@bn A
xn
Φn Bn
x
b0
B1 · · · Bn Φ0
0
{z
} | {z } | {z }
|
x
b
Φ
Solution of Block-system by Schur-complement
The solution to Φx = b is
P
x0 = C −1 b̃0 , b̃0 = b0 − i Bi Φ−1
i bi
−1
⊤
xi = Φi (bi − Bi x0 ),
i = 1, . . . , n
where C is the Schur-complement
n
X
⊤
C = Φ0 −
Bi Φ−1
i Bi
i =1
⇒ only need to factor Φi , not Φ
Andreas Grothey
OPF contingency generation
Exploiting Structure in IPM
Solution of Block-system by Schur-complement
The solution to Φx = b is
P
x0 = C −1 b̃0 , b̃0 = b0 − i Bi Φ−1
i bi
−1
⊤
xi = Φi (bi − Bi x0 ),
i = 1, . . . , n
where C is the Schur-complement
n
X
⊤
C = Φ0 −
Bi Φ−1
i Bi
i =1
Bottlenecks in this process are
Factorization of the Φi
P
⊤
Assembling ni=1 Bi Φ−1
i Bi
Andreas Grothey
OPF contingency generation
Structure of Augmented System Matrix
Bottlenecks in this process are
Factorization of the Φi ,




Φ=


B1⊤
B2⊤
..
.
Φ1
Φ2
..
B1
.
B2 · · ·
For DC-OPF
Wi =
Assembling
Φn
Bn
Bn⊤
R
Ai
AT
i
0
Φ0




,


,
Pn
−1 ⊤
i =1 Bi Φi Bi
Φi =
Xi−1 Si
Wi
0
J
Bi⊤
=
WiT
0
,
where
J ∈ IR |Bi |×|Gi | : bus-generator incidence matrix
Ai ∈ IR |Bi |×|Li | : node-arc incidence matrix for contingency i
R = diag(−V 2 /B1 , . . . , V 2 /B|L| ): resistances > 0
Andreas Grothey
OPF contingency generation
Block-Factorization for DC-OPF
Structure of Φi for DC-OPF
−1
R
Xi Si WiT
, Wi =
Φi =
Ai
Wi
0
AT
i
0
,
Bi⊤
=
0
J
Solve Φi x = b:
Wi is invertible and constant throughout IPM iterations
To solve Φi x = b only need to factorize Wi :
(0) (0) −1
x
b
Xi Si Wi⊤
=
(1)
Wi
0
x
b (1)
⇒ x (1) = Wi−1 b (1) ,
x (0) = Xi Si−1 (b (0) − Wi⊤ x (1) )
⊤
To build Bi Φ−1
i Bi
⊤
= −J ⊤ Wi−⊤ Xi Si−1 Wi−1 J
Bi Φ−1
i Bi
= −Vi Xi−1 Si Vi⊤ ,
Andreas Grothey
Vi⊤ = Wi−1 J
OPF contingency generation
,
Exploiting Structure in IPM
Solution of block-system by Schur-complement
The solution to Φx = b is
P
x0 = C −1 b̃0 ,
b̃0 = b0 − i Bi Φ−1
i bi
n
X
Vi⊤ = Wi−1 J
Vi Xi−1 Si Vi⊤ ,
C = Φ0 +
i =1
Forming Vi Xi−1 Si ViT and factorizing C is (still) expensive
Andreas Grothey
OPF contingency generation
Exploiting Structure in IPM
Solution of block-system by Schur-complement
The solution to Φx = b is
P
x0 = C −1 b̃0 ,
b̃0 = b0 − i Bi Φ−1
i bi
n
X
Vi⊤ = Wi−1 J
Vi Xi−1 Si Vi⊤ ,
C = Φ0 +
i =1
Forming Vi Xi−1 Si ViT and factorizing C is (still) expensive
⇒ Solve Cx0 = b̃0 by iterative method
Use (preconditioned) iterative method (e.g. PCG/GMRES)
⇒ Evaluating matrix-vector products C · x is cheap:
X
C · x = Φ0 x +
J ⊤ Wi−⊤ Xi−1 Si Wi−1 Jx
Multiplication with J is trivial: J is 0-1 matrix.
Wi has been factorized
Andreas Grothey
OPF contingency generation
Possible preconditioners
Single (base) contingency
(Qiu, Flueck ’05)
M = Φ0 + nV0 X0−1 S0 V0⊤
Sample Average Approximation
M = Φ0 +
(Anitescu, Petra ’12)
n X
Vi Xi−1 Si Vi⊤
|S|
i ∈S
where S = random selection of contingency scenarios
Active Contingencies
M = Φ0 +
(Chiang, G. ’13)
X
Vi Xi−1 Si Vi⊤
i ∈A
where A = set of contingencies considered active
Andreas Grothey
OPF contingency generation
Active Contingency Preconditioner
Active Contingencies
(Chiang, G. ’13)
M = Φ0 +
X
Vi Xi−1 Si Vi⊤
i ∈A
where A = set of contingencies considered active.
Based on the observation
Vi Xi−1 Si Vi⊤
Vi = J ⊤ Wi−⊤ constant
X = diag(x), x = vector of slacks
⇒ Large contribution Xi−1 Si ≡ small slack xi ≡ active
contingency
⇒ Concentrate on active contingencies.
Andreas Grothey
OPF contingency generation
Active Contingency Preconditioner
Active Contingencies
(Chiang, G. ’13)
M = Φ0 +
X
Vi Xi−1 Si Vi⊤
i ∈A
where A = set of contingencies considered active.
Based on the observation
Vi Xi−1 Si Vi⊤
Vi = J ⊤ Wi−⊤ constant
X = diag(x), x = vector of slacks
⇒ Large contribution Xi−1 Si ≡ small slack xi ≡ active
contingency
⇒ Concentrate on active contingencies.
Which contingencies should be considered active?
Andreas Grothey
OPF contingency generation
Active Contingency Preconditioner: Theoretical Result
Convergence speed of an iterative scheme with
system matrix C and preconditioner M,
is determined by the distribution of eigenvalues of
M −1 C
Aim:
All close to unity
Small numbers of clusters
Theorem
For a given δ > 0 let A consist of all contingencies i for which any
xi ,j < 1/δ. Then all eigenvalues λi of M −1 C satisfy
|λi (M −1 C ) − 1| = O(δ)
Related to Active Set Preconditioners (Forsgren, Gill, Griffin ’07)
Andreas Grothey
OPF contingency generation
Test Problem
Buses
3
26
56
118
300
iceland
100
200
300
400
500
Gen
4
5
7
54
69
35
25
50
75
100
125
Cont
2
40
79
177
322
72
180
370
565
760
955
Variables
17
2,630
10,648
53,811
229,077
28,725
50,344
210,779
488,534
881,339
1,389,194
Andreas Grothey
Constraints
14
2,626
10,642
53,758
229,009
28,691
50,320
210,730
488,460
881,240
1,389,070
OPF contingency generation
Nonzeros
35
7,929
31,060
172,019
683,542
77,059
165,649
701,249
1,635,824
2,960,399
4,674,974
Results
Results for Active Contingency preconditioner:
#Bus
3
26
56
118
300
iceland
100
200
300
400
500
#Scenarios
3
41
80
178
323
73
181
371
566
761
956
Time(s)
<0.1
0.27
1.26
6.68
24.83
3.12
9.09
50.63
205.79
523.65
823.95
Iters
8
13
15
13
13
13
20
28
39
55
46
|A|
2
2
6
7
12
5
7
9
20
20
25
Base case preconditioner only works for smallest two problems
Andreas Grothey
OPF contingency generation
Results
buses
3
26
56
118
300
iceland
100
200
300
400
500
Default
time(s) memory
<0.1
5.2MB
0.21
7.6MB
1.00 14.3MB
6.88
49MB
31.79
198MB
2.28
25MB
9.02 54.6MB
65.97
220MB
251.70
531MB
955.32
985MB
1552.80 1593MB
Direct
time(s) memory
<0.1
5.2MB
0.17
7.5MB
0.77 14.1MB
6.69
64MB
30.42
282MB
2.36
30MB
6.16 53.1MB
45.23
244MB
177.39
667MB
655.50 1380MB
1195.77 2467MB
Andreas Grothey
Iterative
time(s) memory
<0.01
5.2MB
0.27
7.4MB
1.26 13.5MB
6.88
24.83
3.12
9.09 43.6MB
50.63
163MB
205.79
387MB
523.65
715MB
823.95 1163MB
OPF contingency generation
Contingency Generation
Andreas Grothey
OPF contingency generation
Contingency Generation
Andreas Grothey
OPF contingency generation
Contingency Generation
“n-1”- (or even “n-2”-security) requires the inclusion of many
contingency 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 the
system (but which ones)?
Contingency Generation
Generate contingency scenarios dynamically when needed
Andreas Grothey
OPF contingency generation
Contingency Generation
Prototype Algorithm:
Set up the model with a few base contingency scenarios
Solve model to obtain optimal controls uk∗
repeat
Check for violated contingency scenarios.
Add violated scenarios to the model
∗ .
Re-solve model to obtain new controls uk+1
until no more violated contingencies
Andreas Grothey
OPF contingency generation
Contingency Generation
Prototype Algorithm:
Set up the model with a few base contingency scenarios
Solve model to obtain optimal controls uk∗
repeat
Check for violated contingency scenarios.
Add violated scenarios to the model
∗ .
Re-solve model to obtain new controls uk+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 partially
solved OPF.
Andreas Grothey
OPF contingency generation
Warmstarting Interior Point Methods
Aim: Use information from solution process of
min c ⊤ x s.t. Ax
x
= b
≥ 0
(LP)
to construct a starting point for (nearby problem)
min c̃ ⊤ x s.t. Ãx
x
= b̃
≥ 0
f
(LP)
where à ≈ A, b̃ ≈ b, c̃ ≈ c
f
It is not a good idea to use the solution of (LP) to start (LP).
Unlike for the Simplex/Active Set Method!
Andreas Grothey
OPF contingency generation
Interior Point Methods (for NLP)
Nonlinear Problem
min f (x)
s.t. g (x) = 0
x ≥ 0
(NLP)
KKT Conditions
∇c(x) − ∇g (x)⊤ λ − s
g (x)
XSe
x, s
=
=
=
≥
0
0
0
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)
Andreas Grothey
OPF contingency generation
Interior Point Methods (for NLP)
Barrier Problem
min f (x) − µ
X
ln xi
s.t. g (x) = 0
x ≥ 0
(NLPµ )
KKT Conditions
∇c(x) − ∇g (x)⊤ λ − s
g (x)
XSe
x, s
=
=
=
≥
0
0
µe
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)
Andreas Grothey
OPF contingency generation
Interior Point Methods (for NLP)
KKT conditions
∇c(x) − ∇g (x)⊤ λ − s
g (x)
XSe
x, s
=
=
=
≥
0
0
µe
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)
where
N (κ) := {(x, λ, s) : Eµ (x, λ, s) ≤ κµ}
Eµ (x, λ, s) := max
k∇f (x) − ∇g (x)λ − sk
kXS − µek
, kg (x)k,
sd
sc
Andreas Grothey
OPF contingency generation
Path Following Methods (for NLP)
choose x0 , λ0 , s0 > 0, µ0 = x0⊤ s0 /n, k = 0
Outer iteration
update µ:
µ+ = σ
⊤
x+
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 compute
stepsizes and take step in Newton direction
until
Eµk (xk , λk , sk ) ≤ κµk
Andreas Grothey
OPF contingency generation
Why?
Hippolito (1993): Search direction is parallel to nearby constraints
Modified Problem
Original Problem
Andreas Grothey
OPF contingency generation
Why?
Hippolito (1993): Search direction is parallel to nearby constraints
Modified Problem
Original Problem
⇒ only small step in search direction can be taken
Andreas Grothey
OPF contingency generation
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)
Andreas Grothey
OPF contingency generation
Interior Point Warmstarts: Theoretical Results
A typical warmstart results is (Assume à = A):
Lemma (based on Gondzio/G. ’03)
Let (x, λ, s) ∈ N−∞ (γ0 ) for problem (LP) then a full modification
f is feasible and
step (∆x, ∆λ, ∆s) in the perturbed problem (LP)
(x + ∆x, λ + ∆λ, s + ∆s) ∈ Ñ−∞ (γ)
provided that
δbc ≤
√
γ0 − γ
γ0 µ3/2
2
2B∞
⇒“small δbc , large µ”
where
δbc
:= kc̃ − ÃT y − sk2 + kÃT (ÃÃT )−1 (b̃ − Ãx)k2
:= Pc̃−ÃT y−s (s) + Pb̃−Ãx (x)
is the orthogonal distance of the warmstart point from primal-dual
feasibility in the perturbed problem
Andreas Grothey
OPF contingency generation
Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios
Solve model to obtain optimal controls u0∗ .
repeat
Check for violated contingency scenarios.
Add violated scenarios to the model
∗ .
Re-solve model to obtain new controls uk+1
until no more violated contingencies
Andreas Grothey
OPF contingency generation
Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios.
Solve model to obtain controls u0∗ .
repeat
Check for violated contingency scenarios.
Add violated scenarios to the model.
∗
Re-solve model to obtain new controls uk+1
.
until no more violated contingencies
Andreas Grothey
OPF contingency generation
Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios. µ0 = µ̄, 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 + 1
Warmstart and iterate to find µk -center. ⇒ controls uµk .
until no more violated contingencies
Andreas Grothey
OPF contingency generation
Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios. µ0 = µ̄, 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 = uµk and solve for
µk -center
end for
Add violated scenarios to the model.
Patch together warmstart point for expanded problem.
Choose µk+1 : 0 < µk+1 < µk , k ← k + 1
Warmstart and iterate to find µk -center. ⇒ controls uµk .
until no more violated contingencies
Andreas Grothey
OPF contingency generation
Contingency Generation
Prototype Algorithm:
Set up the model with a few base scenarios. µ0 = µ̄, 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 = uµk and solve for
µk -center
end for
Add violated scenarios to the model.
Patch together warmstart point for expanded problem.
Choose µk+1 : 0 < µk+1 < µk , k ← k + 1
Warmstart and iterate to find µk -center. ⇒ controls uµk .
until no more violated contingencies
Claim
We can ensure that the assembled warmstart point is in the
neighbourhood N (κ)
⇒ IPM convergence not negatively affected (warmstart successful)
Andreas Grothey
OPF contingency generation
Derivation of Algorithm
IPM applied to SCOPF model
minu,xc ,sc
s.t.
P
P
f (u) − µ c∈C i ln sc,i
KLc (u, xc ) = 0
gc (xc ) + sc = 0
Partially decompose: C = C0 ∪ C0
min
u,xc ,sc
s.t.
f (u) − µ
P P
i
c∈C0
ln sc,i +
P
∀c ∈ C
∀c ∈ C
Vc,µ (u)
c∈C0
KLc (u, xc ) = 0
gc (xc ) + sc = 0
∀c ∈ C0
∀c ∈ C0
where
Vc,µ (u) = min −µ
xc ,sc
s.t.
P
i
ln sc,i



KLc (u, xc ) = 0  ∀c ∈ C0

gc (xc ) + sc = 0
Andreas Grothey
OPF contingency generation
Active/Inactive contingencies
KKT conditions for contingency problem (Vc,µ (u))
primal feasibility
dual feasibility
KLc (u, xc )=0 ∇x KLc (u, xc )⊤ λc +∇gc (xc )⊤ νc =0
gc (xc ) + sc =0
Andreas Grothey
OPF contingency generation
centrality
sc,i νc,i =µe
s c , νc ≥ 0
Active/Inactive contingencies
KKT conditions for contingency problem (Vc,µ (u))
primal feasibility
dual feasibility
KLc (u, xc )=0 ∇x KLc (u, xc )⊤ λc +∇gc (xc )⊤ νc =0
gc (xc ) + sc =0
sc,i νc,i = µ ⇒ νc,i = µ/sc,i
centrality
sc,i νc,i =µe
s c , νc ≥ 0
∗ > 0 and thus
if contingency c is never active then sc,i → sc,i
kνc k = O(µ) → 0 ⇒ kλc k = O(µ) → 0
Andreas Grothey
OPF contingency generation
Active/Inactive contingencies
KKT conditions for contingency problem (Vc,µ (u))
primal feasibility
dual feasibility
KLc (u, xc )=0 ∇x KLc (u, xc )⊤ λc +∇gc (xc )⊤ νc =0
gc (xc ) + sc =0
sc,i νc,i = µ ⇒ νc,i = µ/sc,i
centrality
sc,i νc,i =µe
s c , νc ≥ 0
∗ > 0 and thus
if contingency c is never active then sc,i → sc,i
kνc k = O(µ) → 0 ⇒ kλc k = O(µ) → 0
if contingency is active at solution then
kνc k → kνc∗ k > 0 ⇒ kλc k → kλ∗c k > 0
Andreas Grothey
OPF contingency generation
Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
where
N (κ) := {(x, λ, s) : Eµ (x, λ, s) ≤ κµ}
Eµ (x, λ, s) := max






k∇f (x) − ∇g (x)λ − sk∞ , kg (x)k∞ , kXS − µek∞

{z
} | {z } |
{z
}

|
dual feasibility
Andreas Grothey
primal feasibility
OPF contingency generation
centrality
Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
where
N (κ) := {(x, λ, s) : Eµ (x, λ, s) ≤ κµ}
Eµ (x, λ, s) := max






k∇f (x) − ∇g (x)λ − sk∞ , kg (x)k∞ , kXS − µek∞

{z
} | {z } |
{z
}

|
dual feasibility
primal feasibility
centrality
Central Path conditions for SCOPF problem
primal feasibility
dual
centrality
P feasibility ⊤
KLc (u, xc ) =0
∇f (u)− c ∇u KLc (u, xc ) λc =0 sc,i νc,i =µe
gc (xc ) + sc =0 ∇x KLc (u, xc )⊤ λc +∇gc (xc )⊤ νc =0 sc , νc ≥0
Andreas Grothey
OPF contingency generation
Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
where
N (κ) := {(x, λ, s) : Eµ (x, λ, s) ≤ κµ}
Eµ (x, λ, s) := max






k∇f (x) − ∇g (x)λ − sk∞ , kg (x)k∞ , kXS − µek∞

{z
} | {z } |
{z
}

|
dual feasibility
primal feasibility
centrality
Central Path conditions for SCOPF problem
primal feasibility
dual
centrality
P feasibility ⊤
KLc (u, xc ) =0
∇f (u)− c ∇u KLc (u, xc ) λc =0 sc,i νc,i =µe
gc (xc ) + sc =0 ∇x KLc (u, xc )⊤ λc +∇gc (xc )⊤ νc =0 sc , νc ≥0
Satisfied within κµ by construction
Andreas Grothey
OPF contingency generation
Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
where
N (κ) := {(x, λ, s) : Eµ (x, λ, s) ≤ κµ}
Eµ (x, λ, s) := max






k∇f (x) − ∇g (x)λ − sk∞ , kg (x)k∞ , kXS − µek∞

{z
} | {z } |
{z
}

|
dual feasibility
primal feasibility
centrality
Central Path conditions for SCOPF problem
centrality
primal feasibility
dual
P feasibility ⊤
∇f (u)− c ∇u KLc (u, xc ) λc =0 sc,i νc,i =µe
KLc (u, xc ) =0
gc (xc ) + sc =0 ∇x KLc (u, xc )⊤ λc +∇gc (xc )⊤ νc =0 sc , νc ≥0
Residual is ∇u KLc (u, xc )⊤ λc for added contingency
Andreas Grothey
OPF contingency generation
Combined point is in neighbourhood N (κ)
Neighbourhood of the Central Path
where
N (κ) := {(x, λ, s) : Eµ (x, λ, s) ≤ κµ}
Eµ (x, λ, s) := max






k∇f (x) − ∇g (x)λ − sk∞ , kg (x)k∞ , kXS − µek∞

{z
} | {z } |
{z
}

|
dual feasibility
primal feasibility
centrality
Central Path conditions for SCOPF problem
centrality
primal feasibility
dual
P feasibility ⊤
∇f (u)− c ∇u KLc (u, xc ) λc =0 sc,i νc,i =µe
KLc (u, xc ) =0
gc (xc ) + sc =0 ∇x KLc (u, xc )⊤ λc +∇gc (xc )⊤ νc =0 sc , νc ≥0
Residual is ∇u KLc (u, xc )⊤ λc for added contingency
⇒ ok, if added while λc is small (compared to µk )
Andreas Grothey
OPF contingency generation
Statement of Algorithm
Choose initial contingency set C0 ⊂ C. Choose µ0
repeat
Do IPM iteration in the master problem to get approximate
solution u k to (PC0 ,µk ).
For all c ∈ C0 solve Vc,µk (u k ) approximately to get sck , λkc
if kλkc k > 12 κµk for any c then
set C0 ← C0 ∪ {c}.
take combined solution of (PC0 ,µk ) and (Vc,µk (u k )) as
starting point for next IPM iteration
end if
µk+1 = σµk
until convergence in master problem
Andreas Grothey
OPF contingency generation
Statement of Algorithm
Choose initial contingency set C0 ⊂ C. Choose µ0
repeat
Do IPM iteration in the master problem to get approximate
solution u k to (PC0 ,µk ).
For all c ∈ C0 solve Vc,µk (u k ) approximately to get sck , λkc
if kλkc k > 12 κµk for any c then
set C0 ← C0 ∪ {c}.
take combined solution of (PC0 ,µk ) and (Vc,µk (u k )) as
starting point for next IPM iteration
end if
µk+1 = σµk
until convergence in master problem
Approximate means a point in the neighbourhood N (κ)
Andreas Grothey
OPF contingency generation
Statement of Algorithm
Choose initial contingency set C0 ⊂ C. Choose µ0
repeat
Do IPM iteration in the master problem to get approximate
solution u k to (PC0 ,µk ).
For all c ∈ C0 solve Vc,µk (u k ) approximately to get sck , λkc
if kλkc k > 12 κµk for any c then
set C0 ← C0 ∪ {c}.
take combined solution of (PC0 ,µk ) and (Vc,µk (u k )) as
starting 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 a
single iteration in (Vc,µk (u k )) starting from the previous iterate.
Andreas Grothey
OPF contingency generation
Statement of Algorithm
Choose initial contingency set C0 ⊂ C. Choose µ0
repeat
Do IPM iteration in the master problem to get approximate
solution u k to (PC0 ,µk ).
For all c ∈ C0 solve Vc,µk (u k ) approximately to get sck , λkc
if kλkc k > 12 κµk for any c then
set C0 ← C0 ∪ {c}.
take combined solution of (PC0 ,µk ) and (Vc,µk (u k )) as
starting 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.
Andreas Grothey
OPF contingency generation
Regarding Convergence & Efficiency of Algorithm
Issues
warmstart is always successful
(i.e. the patched point is in the N (κ) neighbourhood
Andreas Grothey
OPF contingency generation
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
1
since for active contingencies λc → λ∗c > κµ → 0
2
Andreas Grothey
OPF contingency generation
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
1
since for active contingencies λc → λ∗c > κµ → 0
2
Contingency subproblems can be solved efficiently
Contingency subproblem is just equation solving ⇒ only a
single (usually) Newton step needed. ⇒ FDLF
Andreas Grothey
OPF contingency generation
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
1
since for active contingencies λc → λ∗c > κµ → 0
2
Contingency subproblems can be solved efficiently
Contingency subproblem is just equation solving ⇒ only a
single (usually) Newton step needed. ⇒ FDLF
Only a very small fraction of contingencies needed
only check for contingencies periodically.
pre-screen and discard contingencies which are likely to never
be active:
Contingency Screening:
Ejebe, Irisarri, Mokhtari (1995), Vaahedi, Fuchs, Xu, Mansour
(1999), Capitanescu, Wehenkel (2007, 2008), Dent, Ochoa,
Harrison, Bialek (2010)
Andreas Grothey
OPF contingency generation
Contingency Generation: Results
Prob
6bus
IEEE 24
IEEE 48
IEEE 73
IEEE 96
IEEE 118
IEEE 192
Sce
2
38
78
117
158
178
318
Default
time(s)
iters
<0.1
13
5.7
41
52.3
71
204.1
97
351.5
106
- >200
2393
132
Contingency Generation
time(s) iters
ActS
<0.1
13
2
3.9
30
6
32.3
52
15
156.7
92
25
252.9
76
27
1225
75
46
1586
92
40
This is for the (nonlinear) (n − 1) AC-SCOPF problem
Work in progress: currently scans all (inactive) scenarios at
every iteration.
Andreas Grothey
OPF contingency generation
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 iterations
without leaving ’safe’ neighbourhood.
Andreas Grothey
OPF contingency generation
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 iterations
without leaving ’safe’ neighbourhood.
Extension:
Much further scope for efficiency gains (contingency
screening, efficient load flow solver for subproblems)
Transmission constraints and security planning will become
increasingly important in future with decentralised and
intermittent generation.
Many operational and planning models in power systems (unit
commitment, transmission switching/islanding) should take
contingency constraints into account but currently do not
(due to algorithmic complexity).
Andreas Grothey
OPF contingency generation
Thank You!
Andreas Grothey
OPF contingency generation
Bibliography
M. Colombo, A. Grothey: A decomposition-based warm-start
method for stochastic programming, Computational
Optimization and Applications, Volume 55, Issue 2 (2013),
Page 311-340.
N.-Y. Chiang, A. Grothey: Solving Security Constrained
Optimal Power Flow Problems by a Structure Exploiting
Interior Point Method. Optimization and Engineering,
published online February 2014.
N.-Y. Chiang: Structure-Exploiting Interior Point Methods for
Security Constrained Optimal Power Flow Problems, PhD
Thesis, University of Edinburgh (2013).
http://www.maths.ed.ac.uk/ERGO/preprints.html
Andreas Grothey
OPF contingency generation