interior point methods for optimal power flow: linear

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

E

U N I V E R S

I TY

OF

ED I N B U

RG

H

Andreas Grothey OPF contingency generation

USA August 2003

before 4.05pm

after 4.10pm

50 million peopledisconnected within 5 min


US-Canada Power System Outage Task Force: “Interim Report: Causes of the August 14th Blackout in the

United States and Canada”, November 2003


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








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

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


Optimal Power Flow


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






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.


Power Systems Operation 101


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



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



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



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!



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



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)



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!



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!







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!










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?


(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


Structure of SCOPF problem

SCOPF model


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


Structure of SCOPF problem

SCOPF model


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.


W1

W|C|

T1

T|C|

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


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


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|)


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!


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


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.


W1

W|C|

T1

T|C|

whistle stop tour of

Interior Point Methods


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)



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



Barrier Problem

min c⊤x − µ∑


(LPµ)

KKT Conditions

c − A⊤λ− s = 0Ax = b


(KKTµ)


(LPµ) is strictly convex

System (KKTµ) can be solved per Newton-Method



Barrier Problem

min c⊤x − µ∑


(LPµ)

KKT Conditions

c − A⊤λ− s = 0Ax = b


(KKTµ)


(LPµ) is strictly convex


For µ→ 0 solution of (LPµ) converges to solution of (LP)



KKT conditions

c − A⊤λ− s = 0Ax = b


(KKTµ)



KKT conditions

c − A⊤λ− s = 0Ax = b


(KKTµ)

Newton-Step

0 A⊤ IA 0 0S 0 X

∆x∆λ∆s

=

ξc

ξb

rxs

:=

c − A⊤λ− sb − Axµ+e − XSe



KKT conditions

c − A⊤λ− s = 0Ax = b


(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)


Solving NLP by Interior Point Method

NLP

min f (x) s.t. g(x) ≤ 0 (NLP )



NLP

min f (x)− µ∑

ln zi s.t. g(x) + z = 0z ≥ 0

(NLPµ)



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)


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


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


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


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!


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


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



Solution of Block-system by Schur-complement


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


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

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



Solution of block-system by Schur-complement


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



Solution of block-system by Schur-complement


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


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


Active Contingency Preconditioner


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.


Active Contingency Preconditioner


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?


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)


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


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


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


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)?


Generate contingency scenarios dynamically when needed



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




Set up the model with a few base contingency scenariosSolve model to obtain optimal controls u∗

k

repeat



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.


Warmstarting Interior Point Methods

Aim: Use information from solution process of


(LP)

to construct a starting point for (nearby problem)


(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!


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)



For µ→ 0 solution of (NLPµ) converges to solution of (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)



For µ→ 0 solution of (NLPµ) converges to solution of (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

}


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


Why?

Hippolito (1993): Search direction is parallel to nearby constraints

Original Problem

Modified Problem


Why?

Hippolito (1993): Search direction is parallel to nearby constraints

Modified Problem

Original Problem

⇒ only small step in search direction can be taken


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)


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




Set up the model with a few base scenariosSolve model to obtain optimal controls u∗

0 .repeat






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∗





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





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






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


Claim

We can ensure that the assembled warmstart point is in theneighbourhood N (κ)⇒ IPM convergence not negatively affected (warmstart successful)


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


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








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


Combined point is in neighbourhood N (κ)

Neighbourhood of the Central Path



‖∇f (x)−∇g(x)λ− s‖∞︸︷︷︸

dual feasibility

, ‖g(x)‖∞︸︷︷︸primal feasibility

, ‖XS − µe‖∞︸︷︷︸centrality






‖∇f (x)−∇g(x)λ− s‖∞︸︷︷︸

dual feasibility



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







‖∇f (x)−∇g(x)λ− s‖∞︸︷︷︸

dual feasibility








⊤νc =0


Satisfied within κµ by construction






‖∇f (x)−∇g(x)λ− s‖∞︸︷︷︸

dual feasibility








⊤νc =0


Residual is ∇uKLc (u, xc)⊤λc for added contingency






‖∇f (x)−∇g(x)λ− s‖∞︸︷︷︸

dual feasibility








⊤νc =0


Residual is ∇uKLc (u, xc)⊤λc for added contingency

⇒ ok, if added while λc is small (compared to µk)


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




repeat



c

if ‖λkc ‖ > 1



end if

µk+1 = σµk


Approximate means a point in the neighbourhood N (κ)




repeat



c

if ‖λkc ‖ > 1



end if

µk+1 = σµk


Again a point in the neighbourhood N (κ). Usually only needs asingle iteration in (Vc,µk (uk)) starting from the previous iterate.




repeat



c

if ‖λkc ‖ > 1



end if

µk+1 = σµk


The combined point is still in the N (κ) neighbourhood.⇒ successful warmstart of enlarged problem.


Regarding Convergence & Efficiency of Algorithm

Issues

warmstart is always successful(i.e. the patched point is in the N (κ) neighbourhood



Issues


All active contingencies will be included eventually

since for active contingencies λc → λ∗c >

1

2κµ→ 0



Issues




1

2κµ→ 0

Contingency subproblems can be solved efficientlyContingency subproblem is just equation solving ⇒ only asingle (usually) Newton step needed. ⇒ FDLF



Issues




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)


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.


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.


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).


Thank You!


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


interior point methods for optimal power flow: linear

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