Interior Point Methods for Optimal Power Flow: Linear Algebra Goodies. OPF contingency generation

Transcription

1 Interior Point Methods for Optimal Power Flow: Linear Algebra Goodies and Contingency Generation (joint work with Nai Yuan Chiang, Argonne National Lab) School of Mathematics, University of Edinburgh IMA Numerical Analysis 2014, 3-5 September, Birmingham T H E U N I V E R S I T Y O H F E D I N B U R G

2 USA August 2003 before 4.05pm after 4.10pm 50 million people disconnected within 5 min

3 US-Canada Power System Outage Task Force: Interim Report: Causes of the August 14 th Blackout in the United States and Canada, November 2003

4 US-Canada Power System Outage Task Force: Interim Report: Causes of the August 14 th Blackout in the United States and Canada, November 2003

5 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!

6 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

7 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

8 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

9 Optimal Power Flow

10 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

11 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

12 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.

13 Power Systems Operation 101

14 Power System Operation Network b B Buses l = (bb ) L Lines g G Generators (at bus o g ) Power flows are AC. Described by real and reactive flows over lines, voltage and phase angle at buses Variables v b Voltage level at bus b δ b Phase angle at bus b p bb,q bb Real and reactive power flow on line l = (bb ) pg G, qg G Real and reactive power output at generator g

15 Power System Operation Network b B Buses l = (bb ) L Lines g G Generators (at bus o g ) Parameters of the model are: line/bus characteristics, real/reactive demands, limits on power flow and voltages Parameters G bb,b bb C b P D b,qd b conductance and susceptance of line l (reciprocals of resistance and reactance) bus susceptance real and reactive power demand at bus b

16 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) p L bb = v b[g bb (v b v b cos(δ i δ j )) + B bb v b sin(δ b δ b )] q L bb = v b[ G bb v b sin(δ i δ j ) + B bb (v b cos(δ b δ b ) v b )] Kirchhoff Current Law (KCL) pg G PD b = g o g=b g o g=b q G g Q D b = (b,b ) L (b,b ) L p bb, b B, q bb + C b v b 2, b B

17 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) p L bb = v b[g bb (v b v b cos(δ i δ j )) + B bb v b sin(δ b δ b )] q L bb = v b[ G bb v b sin(δ i δ j ) + B bb (v b cos(δ b δ b ) v b )] Kirchhoff Current Law (KCL) pg G PD b = g o g=b g o g=b q G g Q D b = (b,b ) L (b,b ) L p bb, b B, q bb + C b v b 2, b B Note: p L bb p L b b,ql bb q L b b due to line losses!

18 Power System Operation Line flows and generator reactive output should satisfy Operational Limits Line Flow Limits at both ends of each line (p bb ) 2 + (q bb ) 2 f l 2, (pb b) 2 + (q b b) 2 f l 2 Voltage limits at buses V min v b V max reactive generation limits at generators Q min g q G g Q max g

19 Power System Operation The variables of the OPF model can be divided into control and state variables Control u (set by the system operator) pg G real power output at generators v g voltage levels at generation buses State x (determined by Kirchhoff s laws) v b voltage levels at non-generating buses qg G reactive power output at generators δ b phase angles p bb,q bb real and reactive power flows over lines OPF model min u U f (u) s.t. KL(u,x) = 0 (Kirchhoff s laws) g(x) 0 (Operational limits)

20 Power System Operation The variables of the OPF model can be divided into control and state variables Control u (set by the system operator) pg G real power output at generators v g voltage levels at generation buses State x (determined by Kirchhoff s laws) v b voltage levels at non-generating buses qg G reactive power output at generators δ b phase angles p bb,q bb real and reactive power flows over lines OPF model min u U f (u) s.t. KL(u,x) = 0 (Kirchhoff s laws) g(x) 0 (Operational limits) Nonlinear! nonconvex! May have local solutions!

21 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)

22 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!

23 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)

24 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!

25 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?

26 (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 = (p G g,v g) same for all contingencies, Each contingency has its own set of state variables x c = (v b,c,q G b,c,δ b,c,p bb,c,q bb,c) determined by Kirchhoff s laws for the reduced network (KL c ) Seek a setting of control variables that does not lead to violation of operational limits in any contingency SCOPF model min u U f (u) s.t. KL c (u,x c ) = 0 c C g c (x c ) 0 c C

27 Structure of SCOPF problem SCOPF model min u U f (u) s.t. KL c (u,x c ) = 0 c C g c (x c ) 0 c C Structure of Jacobian KL 1 x 1 KL c u g 1 x KL C KL C x C u g C x C 0

28 Structure of SCOPF problem SCOPF model min u U Structure of Jacobian f (u) s.t. KL c (u,x c ) = 0 c C g c (x c ) 0 c C KL 1 KL c x 1 u g 1 x 1 T0 1 W KL C KL C x C u g C x C 0 W C T C Bordered block-diagonal matrix.

29 The DC OPF problem The OPF model can simplified under the following assumptions: Voltage level at all buses is the same: v b = 1, b B. The resistance of each line is small compared to reactance: B bb G bb assume G bb = 0. Phase angle difference across each line is small sin(δ 1 δ 2 ) δ 1 δ 2,cos(δ 1 δ 2 ) 1, q L bb = 0. DC -OPF model Kirchhoff Voltage Law p L bb = B bb (δ b δ b ), Kirchhoff Current Law pg G = g o g=b (b,b ) L Line Flow Limits: f l p L l f l, p L (b,b ) + PD b, l L (bb ) L b B

30 The DC OPF problem The OPF model can simplified under the following assumptions: Voltage level at all buses is the same: v b = 1, b B. The resistance of each line is small compared to reactance: B bb G bb assume G bb = 0. Phase angle difference across each line is small sin(δ 1 δ 2 ) δ 1 δ 2,cos(δ 1 δ 2 ) 1, q L bb = 0. DC -OPF model Kirchhoff Voltage Law p L bb = B bb (δ b δ b ), Kirchhoff Current Law pg G = g o g=b (b,b ) L Line Flow Limits: f l p L l f l, p L (b,b ) + PD b, l L (bb ) L b B DC OPF is a linear programming problem

31 Structure of DC OPF problem the DC-OPF problem can be written as DC-OPF where min c p G s.t. R p L +A δ = 0 A p L J p G = P D bus/generator incidence matrix J IR B G node/arc incidence matrix A IR B L R = diag(1/b 1,...,1/B L )

32 Structure of DC OPF problem the DC-OPF problem can be written as DC-OPF where min c p G s.t. R p L +A δ = 0 A p L J p G = P D bus/generator incidence matrix J IR B G node/arc incidence matrix A IR B L R = diag(1/b 1,...,1/B L ) Augmented system like structure!

33 Structure of (n-1) secure DC OPF SCOPF (DC version) min c p G p G,p L,δ s.t. Rp1 L +A 1 δ 1 = 0 A 1 p1 L Jp G = P D.... =. Rp C G +A C δ C = 0 A C p C G Jp G = P D

34 Structure of (n-1) secure DC OPF SCOPF (DC version) min c p G p G,p L,δ s.t. Rp1 L +A 1 δ 1 = 0 A 1 W p1 L 1 T Jp 1 G = P D.... =. Rp C G +A C δ C = 0 A C W p C G C T Jp C G = P D Bordered block-diagonal matrix.

35 whistle stop tour of Interior Point Methods

36 Interior Point Methods (for LP) Linear Program KKT Conditions minc x s.t. Ax = b x 0 c A λ s = 0 Ax = b XSe = 0 x,s X = diag(x), S = diag(s) 0 (LP) (KKT)

37 Interior Point Methods (for LP) Barrier Problem minc x µ lnx i s.t. Ax = b x 0 (LP µ ) KKT Conditions c A λ s = 0 Ax = b XSe = µe x,s 0 (KKT µ ) Introduce logarithmic barriers for x 0

38 Interior Point Methods (for LP) Barrier Problem minc x µ lnx i s.t. Ax = b x 0 (LP µ ) KKT Conditions c A λ s = 0 Ax = b XSe = µe x,s 0 (KKT µ ) Introduce logarithmic barriers for x 0 (LP µ ) is strictly convex System (KKT µ ) can be solved per Newton-Method

39 Interior Point Methods (for LP) Barrier Problem minc x µ lnx i s.t. Ax = b x 0 (LP µ ) KKT Conditions c A λ s = 0 Ax = b XSe = µe x,s 0 (KKT µ ) Introduce logarithmic barriers for x 0 (LP µ ) is strictly convex System (KKT µ ) can be solved per Newton-Method For µ 0 solution of (LP µ ) converges to solution of (LP)

40 Interior Point Methods (for LP) KKT conditions c A λ s = 0 Ax = b XSe = µe x,s 0 (KKT µ )

41 Interior Point Methods (for LP) KKT conditions c A λ s = 0 Ax = b XSe = µe x,s 0 (KKT µ ) Newton-Step 0 A I A 0 0 S 0 X x λ s = ξ c ξ b r xs := c A λ s b Ax µ + e XSe

42 Interior Point Methods (for LP) KKT conditions c A λ s = 0 Ax = b XSe = µe x,s 0 (KKT µ ) Newton-Step 0 A I A 0 0 S 0 X x λ s = ξ c ξ b r xs := c A λ s b Ax µ + e XSe Newton Step (reduced) [ Θ A A 0 ][ x y ] [ ξc X = 1 r xs where Θ = X 1 S, X = diag(x), S = diag(s) ξ b ]

43 Solving NLP by Interior Point Method NLP min f (x) s.t. g(x) 0 (NLP )

44 Solving NLP by Interior Point Method NLP minf (x) µ lnz i s.t. g(x) + z = 0 z 0 (NLP µ )

45 Solving NLP by Interior Point Method NLP minf (x) µ lnz i s.t. g(x) + z = 0 z 0 (NLP ) Optimality conditions Newton Step [ ][ ] Q(x,y) A(x) x A(x) Θ y where f (x) A(x) y = 0 g(x) + z = 0 XZe = µe x,z 0 = Q(x,y) = 2 xx (f (x) + y g(x)), [ f (x) A(x) y g(x) µy 1 e A(x) = g(x) Θ = X 1 Z, X = diag(x), Z = diag(z) ]

46 Linear Algebra of IPMs Main work: solve [ ] Q Θ A A 0 }{{} Φ (QP) [ ] x = y [ ] r h or [ ] Q(x,y) A(x) A(x) Θ }{{} Φ (NLP) [ ] x = y [ ] r h 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

47 Structure of matrices A and Q for SCOPF: Matrix A W 1 W 2 W C T 1 T 2 T C Matrix Q Q 1 Q 2 Q C Q 0

48 Structures of A and Q imply structure of Φ: Q 1 Q 2 W 1 W 2 Q C Q0 T 1 T 2 W C T C ( Q A A 0 W 1 W 2 T 1 T 2 ) W C T C Q 1 W1 W 1 Q 2 W2 W 2 T 1 T 2 ( Q A P A 0 Q C W C W C T 1 T 2 T C T C Q 0 ) P 1

49 Structures of A and Q imply structure of Φ: Q 1 Q 2 W 1 W 2 Q C Q0 T 1 T 2 W C T C ( Q A A 0 W 1 W 2 T 1 T 2 ) W C T C Φ 1 Φ 2... B 1 B 2. Φ C B C B 1 B 2 B C ( ) Q A P P A 0 1 Φ 0 Bordered block-diagonal structure in Augmented System!

50 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 10 9 variables ( 2h on 1280 processors) Primal Block Angular Structure Dual Block Angular Structure D1 Primal Block Angular Structure D11 D12 D10 B11 B12 D21 D22 D23 D20 B21 B22 B23 D2 A C31 C32 D 30

51 Exploiting Structure in IPM Block-Factorization of Augmented System Matrix 0 Φ 1 B x 1 b B C B. C = B. Φ n Bn x n b n A B 1 B n Φ 0 x 0 b 0 {z } {z } {z } Φ x b Solution of Block-system by Schur-complement The solution to Φx = b is x 0 = C 1 b 0, b 0 = b 0 i B iφ 1 i b i x i = Φ 1 i (b i Bi x 0 ), i = 1,...,n where C is the Schur-complement n C = Φ 0 B i Φ 1 i Bi i=1 only need to factor Φ i, not Φ

52 Exploiting Structure in IPM Solution of Block-system by Schur-complement The solution to Φx = b is x 0 = C 1 b 0, b 0 = b 0 i B iφ 1 i b i x i = Φ 1 i (b i Bi x 0 ), i = 1,...,n where C is the Schur-complement n C = Φ 0 B i Φ 1 i Bi Bottlenecks in this process are Factorization of the Φ i Assembling n i=1 B iφ 1 i B i i=1

53 Structure of Augmented System Matrix Bottlenecks in this process are Factorization of the Φ i, Φ 1 B1 Φ 2 B2 Φ =.... Φ n Bn B 1 B 2 B n Φ 0 Assembling n i=1 B iφ 1 i B i [ X 1, Φ i = i S i Wi T W i 0 ] For DC-OPF W i = [ R A T i A i 0 ] [, Bi 0 = J ], where J IR B i G i : bus-generator incidence matrix A i IR B i L i : node-arc incidence matrix for contingency i R = diag( V 2 /B 1,...,V 2 /B L ): resistances > 0

54 Block-Factorization for DC-OPF Structure of Φ i for DC-OPF [ X 1 Φ i = i S i Wi T W i 0 ], W i = [ R A T i A i 0 ] [, Bi 0 = J ], Solve Φ i x = b: W i is invertible and constant throughout IPM iterations To solve Φ i x = b only need to factorize W i : [ X 1 i S i Wi ][ ] [ ] x (0) b (0) = W i 0 x (1) b (1) x (1) = W 1 i b (1), x (0) = X i S 1 i (b (0) Wi x (1) ) To build B i Φ 1 i Bi B i Φ 1 i Bi = J W i X i S 1 i = V i X 1 i S i Vi, W 1 i J V i = W 1 i J

55 Exploiting Structure in IPM Solution of block-system by Schur-complement The solution to Φx = b is x 0 = C 1 b0, b0 = b 0 i B iφ 1 i b i n C = Φ 0 + V i X 1 i S i Vi, Vi = W 1 i J i=1 Forming V i X 1 i S i Vi T and factorizing C is (still) expensive

56 Exploiting Structure in IPM Solution of block-system by Schur-complement The solution to Φx = b is x 0 = C 1 b0, b0 = b 0 i B iφ 1 i b i n C = Φ 0 + V i X 1 i S i Vi, Vi = W 1 i J i=1 Forming V i X 1 i S i Vi T Solve Cx 0 = b 0 by iterative method and factorizing C is (still) expensive Use (preconditioned) iterative method (e.g. PCG/GMRES) Evaluating matrix-vector products C x is cheap: C x = Φ 0 x + J W i X 1 i S i W 1 Jx Multiplication with J is trivial: J is 0-1 matrix. W i has been factorized i

57 Possible preconditioners Single (base) contingency (Qiu, Flueck 05) M = Φ 0 + nv 0 X 1 0 S 0 V 0 Sample Average Approximation (Anitescu, Petra 12) M = Φ 0 + n S i S V i X 1 i S i Vi where S = random selection of contingency scenarios Active Contingencies (Chiang, G. 13) M = Φ 0 + i A V i X 1 i S i Vi where A = set of contingencies considered active

58 Active Contingency Preconditioner Active Contingencies (Chiang, G. 13) M = Φ 0 + i A V i X 1 i S i Vi where A = set of contingencies considered active. Based on the observation V i X 1 i S i Vi V i = J W i constant X = diag(x), x = vector of slacks Large contribution X 1 i S i small slack x i active contingency Concentrate on active contingencies.

59 Active Contingency Preconditioner Active Contingencies (Chiang, G. 13) M = Φ 0 + i A V i X 1 i S i Vi where A = set of contingencies considered active. Based on the observation V i X 1 i S i Vi V i = J W i constant X = diag(x), x = vector of slacks Large contribution X 1 i S i small slack x i active contingency Concentrate on active contingencies. Which contingencies should be considered active?

60 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 x i,j < 1/δ. Then all eigenvalues λ i of M 1 C satisfy λ i (M 1 C) 1 = O(δ) Related to Active Set Preconditioners (Forsgren, Gill, Griffin 07)

61 Test Problem Buses Gen Cont Variables Constraints Nonzeros ,630 2,626 7, ,648 10,642 31, ,811 53, , , , ,542 iceland ,725 28,691 77, ,344 50, , , , , , ,460 1,635, , ,240 2,960, ,389,194 1,389,070 4,674,974

62 Results Results for Active Contingency preconditioner: #Bus #Scenarios Time(s) Iters A 3 3 < iceland Base case preconditioner only works for smallest two problems

63 Results Default Direct Iterative buses time(s) memory time(s) memory time(s) memory 3 < MB < MB < MB MB MB MB MB MB MB MB MB MB MB iceland MB MB MB MB MB MB MB MB MB MB MB MB MB MB MB MB MB

64 Contingency Generation

65 Contingency Generation

66 Contingency Generation n-1 - (or even n-2 -security) requires the inclusion of many contingency scenarios. Pan-European system has nodes and lines Resulting SCOPF model would have variables. Only a few contingencies are critical for operation of the system (but which ones)? Contingency Generation Generate contingency scenarios dynamically when needed

67 Contingency Generation Prototype Algorithm: Set up the model with a few base contingency scenarios Solve model to obtain optimal controls u k 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

68 Contingency Generation Prototype Algorithm: Set up the model with a few base contingency scenarios Solve model to obtain optimal controls u k 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 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.

69 Warmstarting Interior Point Methods Aim: Use information from solution process of minc x s.t. Ax = b x 0 (LP) to construct a starting point for (nearby problem) min c x s.t. Ãx = b x 0 ( LP) where à 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!

70 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 = 0 g(x) = 0 XSe = 0 x,s 0 (KKT µ ) Introduce logarithmic barriers for x 0 System (KKT µ ) can be solved per Newton-Method For µ 0 solution of (NLP µ ) converges to solution of (NLP)

71 Interior Point Methods (for NLP) Barrier Problem minf (x) µ lnx i s.t. g(x) = 0 x 0 (NLP µ ) KKT Conditions c(x) g(x) λ s = 0 g(x) = 0 XSe = µe x,s 0 (KKT µ ) Introduce logarithmic barriers for x 0 System (KKT µ ) can be solved per Newton-Method For µ 0 solution of (NLP µ ) converges to solution of (NLP)

72 Interior Point Methods (for NLP) KKT conditions c(x) g(x) λ s = 0 g(x) = 0 XSe = µe x,s 0 (KKT µ ) Central Path The set of all solutions to (KKT µ ) 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) : E µ (x,λ,s) κµ} where { f (x) g(x)λ s E µ (x,λ,s) := max, g(x), s d } XS µe s c

73 Path Following Methods (for NLP) choose x 0,λ 0,s 0 > 0,µ 0 = x 0 s 0/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 (KKT µ) and given µ k. Do line search with merit function or filter to compute stepsizes and take step in Newton direction until E µk (x k,λ k,s k ) κµ k

74 Why? Hippolito (1993): Search direction is parallel to nearby constraints Modified Problem Original Problem

75 Why? Hippolito (1993): Search direction is parallel to nearby constraints Modified Problem Original Problem only small step in search direction can be taken

76 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)

77 Interior Point Warmstarts: Theoretical Results A typical warmstart results is (Assume à = A): Lemma (based on Gondzio/G. 03) Let (x,λ,s) N (γ 0 ) 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) Ñ (γ) provided that γ0 γ δ bc γ 0 µ 3/2 2B 2 small δ bc, large µ where δ bc := c à T y s 2 + à T (Ãà T ) 1 ( b Ãx) 2 := P c à T y s (s) + P b Ãx (x) is the orthogonal distance of the warmstart point from primal-dual feasibility in the perturbed problem

78 Contingency Generation Prototype Algorithm: Set up the model with a few base scenarios Solve model to obtain optimal 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

79 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

80 Contingency Generation Prototype Algorithm: Set up the model with a few base scenarios. µ 0 = µ, k = 0. Solve model to obtain µ 0 -center. controls u µ0. repeat Check for violated contingency scenarios. Add violated scenarios to the model. Choose µ k+1 : 0 < µ k+1 < µ k, k k + 1 Warmstart and iterate to find µ k -center. controls u µk. until no more violated contingencies

81 Contingency Generation Prototype Algorithm: Set up the model with a few base scenarios. µ 0 = µ, k = 0. Solve model to obtain µ 0 -center. controls u µ0. repeat Check for violated contingency scenarios. for all violated scenarios do Set up single scenario problem with u = u µk and solve for µ k -center end for Add violated scenarios to the model. Patch together warmstart point for expanded problem. Choose µ k+1 : 0 < µ k+1 < µ k, k k + 1 Warmstart and iterate to find µ k -center. controls u µk. until no more violated contingencies

82 Contingency Generation Prototype Algorithm: Set up the model with a few base scenarios. µ 0 = µ, k = 0. Solve model to obtain µ 0 -center. controls u µ0. repeat Check for violated contingency scenarios. for all violated scenarios do Set up single scenario problem with u = u µk and solve for µ k -center end for Add violated scenarios to the model. Patch together warmstart point for expanded problem. Choose µ k+1 : 0 < µ k+1 < µ k, k k + 1 Warmstart and iterate to find µ k -center. controls u µk. 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)

83 Derivation of Algorithm IPM applied to SCOPF model min u,xc,s c f (u) µ c C i lns c,i s.t. KL c (u,x c ) = 0 c C g c (x c ) + s c = 0 c C Partially decompose: C = C 0 C 0 where min u,x c,s c f (u) µ c C 0 i lns c,i + c C 0 V c,µ (u) s.t. KL c (u,x c ) = 0 c C 0 g c (x c ) + s c = 0 c C 0 V c,µ (u) = min µ x i lns c,i c,s c s.t. KL c (u,x c ) = 0 g c (x c ) + s c = 0 c C 0

84 Active/Inactive contingencies KKT conditions for contingency problem (V c,µ (u)) primal feasibility KL c (u,x c )=0 g c (x c ) + s c =0 dual feasibility x KL c (u,x c ) λ c + g c (x c ) ν c =0 centrality s c,i ν c,i =µe s c,ν c 0

85 Active/Inactive contingencies KKT conditions for contingency problem (V c,µ (u)) primal feasibility KL c (u,x c )=0 g c (x c ) + s c =0 dual feasibility x KL c (u,x c ) λ c + g c (x c ) ν c =0 centrality s c,i ν c,i =µe s c,ν c 0 s c,i ν c,i = µ ν c,i = µ/s c,i if contingency c is never active then s c,i s c,i > 0 and thus ν c = O(µ) 0 λ c = O(µ) 0

86 Active/Inactive contingencies KKT conditions for contingency problem (V c,µ (u)) primal feasibility KL c (u,x c )=0 g c (x c ) + s c =0 dual feasibility x KL c (u,x c ) λ c + g c (x c ) ν c =0 centrality s c,i ν c,i =µe s c,ν c 0 s c,i ν c,i = µ ν c,i = µ/s c,i if contingency c is never active then s c,i sc,i > 0 and thus ν c = O(µ) 0 λ c = O(µ) 0 if contingency is active at solution then ν c νc > 0 λ c λ c > 0

87 Combined point is in neighbourhood N(κ) Neighbourhood of the Central Path N(κ) := {(x,λ,s) : E µ (x,λ,s) κµ} where E µ (x, λ, s) := max f (x) g(x)λ s, g(x), XS µe }{{}}{{}}{{} dual feasibility primal feasibility centrality

88 Combined point is in neighbourhood N(κ) Neighbourhood of the Central Path N(κ) := {(x,λ,s) : E µ (x,λ,s) κµ} where E µ (x, λ, s) := max f (x) g(x)λ s, g(x), XS µe }{{}}{{}}{{} dual feasibility primal feasibility centrality Central Path conditions for SCOPF problem primal feasibility KL c (u,x c ) =0 g c (x c ) + s c =0 dual feasibility f (u) c ukl c (u,x c ) λ c =0 x KL c (u,x c ) λ c + g c (x c ) ν c =0 centrality s c,i ν c,i =µe s c,ν c 0

89 Combined point is in neighbourhood N(κ) Neighbourhood of the Central Path N(κ) := {(x,λ,s) : E µ (x,λ,s) κµ} where E µ (x, λ, s) := max f (x) g(x)λ s, g(x), XS µe }{{}}{{}}{{} dual feasibility primal feasibility centrality Central Path conditions for SCOPF problem primal feasibility KL c (u,x c ) =0 g c (x c ) + s c =0 dual feasibility f (u) c ukl c (u,x c ) λ c =0 x KL c (u,x c ) λ c + g c (x c ) ν c =0 centrality s c,i ν c,i =µe s c,ν c 0 Satisfied within κµ by construction

90 Combined point is in neighbourhood N(κ) Neighbourhood of the Central Path N(κ) := {(x,λ,s) : E µ (x,λ,s) κµ} where E µ (x, λ, s) := max f (x) g(x)λ s, g(x), XS µe }{{}}{{}}{{} dual feasibility primal feasibility centrality Central Path conditions for SCOPF problem primal feasibility KL c (u,x c ) =0 g c (x c ) + s c =0 dual feasibility f (u) c ukl c (u,x c ) λ c =0 x KL c (u,x c ) λ c + g c (x c ) ν c =0 centrality s c,i ν c,i =µe s c,ν c 0 Residual is u KL c (u,x c ) λ c for added contingency

91 Combined point is in neighbourhood N(κ) Neighbourhood of the Central Path N(κ) := {(x,λ,s) : E µ (x,λ,s) κµ} where E µ (x, λ, s) := max f (x) g(x)λ s, g(x), XS µe }{{}}{{}}{{} dual feasibility primal feasibility centrality Central Path conditions for SCOPF problem primal feasibility KL c (u,x c ) =0 g c (x c ) + s c =0 dual feasibility f (u) c ukl c (u,x c ) λ c =0 x KL c (u,x c ) λ c + g c (x c ) ν c =0 centrality s c,i ν c,i =µe s c,ν c 0 Residual is u KL c (u,x c ) λ c for added contingency ok, if added while λ c is small (compared to µ k )

92 Statement of Algorithm Choose initial contingency set C 0 C. Choose µ 0 repeat Do IPM iteration in the master problem to get approximate solution u k to (P C0,µ k). For all c C 0 solve V c,µ k(u k ) approximately to get s k c,λ k c if λ k c > 1 2 κµk for any c then set C 0 C 0 {c}. take combined solution of (P C0,µ k) and (V c,µ k(uk )) as starting point for next IPM iteration end if µ k+1 = σµ k until convergence in master problem

93 Statement of Algorithm Choose initial contingency set C 0 C. Choose µ 0 repeat Do IPM iteration in the master problem to get approximate solution u k to (P C0,µ k). For all c C 0 solve V c,µ k(u k ) approximately to get s k c,λ k c if λ k c > 1 2 κµk for any c then set C 0 C 0 {c}. take combined solution of (P C0,µ k) and (V c,µ k(uk )) as starting point for next IPM iteration end if µ k+1 = σµ k until convergence in master problem Approximate means a point in the neighbourhood N(κ)

94 Statement of Algorithm Choose initial contingency set C 0 C. Choose µ 0 repeat Do IPM iteration in the master problem to get approximate solution u k to (P C0,µ k). For all c C 0 solve V c,µ k(u k ) approximately to get s k c,λ k c if λ k c > 1 2 κµk for any c then set C 0 C 0 {c}. take combined solution of (P C0,µ k) and (V c,µ k(uk )) as starting point for next IPM iteration end if µ k+1 = σµ k until convergence in master problem Again a point in the neighbourhood N(κ). Usually only needs a single iteration in (V c,µ k(u k )) starting from the previous iterate.

95 Statement of Algorithm Choose initial contingency set C 0 C. Choose µ 0 repeat Do IPM iteration in the master problem to get approximate solution u k to (P C0,µ k). For all c C 0 solve V c,µ k(u k ) approximately to get s k c,λ k c if λ k c > 1 2 κµk for any c then set C 0 C 0 {c}. take combined solution of (P C0,µ k) and (V c,µ k(uk )) as starting point for next IPM iteration end if µ k+1 = σµ k until convergence in master problem The combined point is still in the N(κ) neighbourhood. successful warmstart of enlarged problem.

96 Regarding Convergence & Efficiency of Algorithm Issues warmstart is always successful (i.e. the patched point is in the N(κ) neighbourhood

97 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 λ c λ c > 1 2 κµ 0

98 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 λ c λ c > 1 2 κµ 0 Contingency subproblems can be solved efficiently Contingency subproblem is just equation solving only a single (usually) Newton step needed. FDLF

99 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 λ c λ c > 1 2 κµ 0 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)

100 Contingency Generation: Results Default Contingency Generation Prob Sce time(s) iters time(s) iters ActS 6bus 2 < < IEEE IEEE IEEE IEEE IEEE > IEEE This is for the (nonlinear) (n 1) AC-SCOPF problem Work in progress: currently scans all (inactive) scenarios at every iteration.

101 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.

102 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).

103 Thank You!

104 Bibliography M. Colombo, A. Grothey: A decomposition-based warm-start method for stochastic programming, Computational Optimization and Applications, Volume 55, Issue 2 (2013), Page 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 N.-Y. Chiang: Structure-Exploiting Interior Point Methods for Security Constrained Optimal Power Flow Problems, PhD Thesis, University of Edinburgh (2013).