Managing Uncertainty and Security in Power System Operations: Chance-Constrained Optimal Power Flow

Managing Uncertainty and Security in Power System Operations: Chance-Constrained Optimal Power Flow Line Roald, November 4 th 2016 Line Roald 09.11.2016 1

Outline Introduction Chance-Constrained Optimal Power Flow Analytical reformulation Corrective control for forecast uncertainties Extension towards AC optimal power flow Modelling Risk of Constraint Violations Conclusions Line Roald 09.11.2016 2

Power Systems Laboratory, ETH Zürich dynamic simulation distribution system planning privacy hydropower optimization FACTS HVDC transmission system security large scale stochastic optimization battery modelling Focus on AC OPF transmission system flexibility Models Methods Analysis Tools protection demand response Prof. Göran Andersson storage Prof. Gabriela Hug state estimation Line Roald 09.11.2016 3

Introduction UMBRELLA Project Provide reliable tools for future operation of the pan-european grid TSC partners ENTSO-E 4 year project funded by European Commission 5 Universities 9 Transmission System Operators Line Roald 09.11.2016 4

Expected loss Introduction UMBRELLA Project: Risk-Based Security Assessment Goal: Maintaining power system security while facilitating integration of renewable energy and market operations Method development: Improve security assessment and planning tools Utilize information about - forecast uncertainties - outage probabilities - availability of corrective measures Develop risk-based measures of power system security Probability Figure 1: Risk zones in operation based on different risk levels. [UCTE OH] Line Roald 09.11.2016 5

European Power System Three main drivers for change: 1. Renewables 2. Declining nuclear 3. Market liberalization New and changing power flow patterns Higher need for transmission capacity Larger and more frequent deviations from schedules 4 GW Line Roald 09.11.2016 6

Transmission System Operational Planning Operational Planning: ~1 day to 15 min ahead of real time Two main tasks: 1. Balancing consumed and produced power 2. Managing transmission constraints Line Roald 09.11.2016 7

Transmission System Operational Planning Forecast errors and intra-day trading cause fluctuations in the power flows across the system. How to handle congestion? What do we mean by security? Overloads? [MW] [MW] Line Roald 09.11.2016 9

DC Optimal Power Flow (OPF) Goal: Low cost operation, while enforcing technical limits min s.t. C G T P G N B i=1 P G(i) + P W(i) P D(i) = 0 P G(g) P max G g, min, P G(g) P G g g = 1,, N G minimize generation cost balanced operation generator limits A (l, ) P G + P W P D A (l, ) P G + P W P D max, max, P L(l) P L l transmission line limits l = 1,, N L Line Roald 09.11.2016 11

Modelling wind power fluctuations Wind power generation: Forecasted power P Wi = P Wi + ω i Fluctuation Conventional generators: Scheduled generation P Gi = P Gi α i Ω Balancing wind power fluctuations where Ω = ω α = 1 total wind power deviation balanced system (AGC) Line Roald 09.11.2016 12

Formulation of Chance constraints Deterministic line constraint: A (l, ) P G + P W P D P max L(l), Changes in the wind in-feed influences the line flow: A (l, ) P G + P W P D + D l, ω P max L(l), Change due to fluctuations ω Confidence level ω is a random variable chance constraint: P A (l, ) P G + P W P D + D l, ω P max L(l) 1 ε Chance constraints limit the probability of constraint violation Line Roald 09.11.2016 14

Chance-Constrained Optimal Power Flow min s.t. C G T P G N B i=1 P G(i) + P W(i) P D(i) = 0 cost function power balance i P P G(g) i P P G(g) αω P G(g) 1 ε, αω P G(g) 1 ε, g = 1,, N G generation constraints all constraints that are impacted by uncertainty are formulated as chance constraints! P A l, P G + P W P D + D l, ω P max L(l) max P A l, P G + P W P D + D l, ω P L(l) l = 1,, N L 1 ε, 1 ε, line constraints Line Roald 09.11.2016 15

Chance-Constrained Optimal Power Flow min s.t. C G T P G + C R T R G N B i=1 P G(i) + P W(i) P D(i) = 0 cost function power balance P min G P G + R G P max G, P α (g) Ω R G(g) 1 ε, P α g Ω R G(g) 1 ε, g = 1,, N G generation and reserve constraints P A l, P G + P W P D + D l, ω P max L(l) max P A l, P G + P W P D + D l, ω P L(l) l = 1,, N L 1 ε, 1 ε, line constraints Line Roald 09.11.2016 16

Chance-Constrained Optimal Power Flow min s.t. C G T P G + C R T R G N B i=1 P G(i) + P W(i) P D(i) = 0 cost function power balance P min G P G + R G P max G, P α g Ω R G g, g = 1,, N G 1 ε, P α g Ω R G g, g = 1,, N G 1 ε, generation and reserve constraints jointly satisfied (all or none are violated) P A l, P G + P W P D + D l, ω P max L(l) max P A l, P G + P W P D + D l, ω P L(l) 1 ε, 1 ε, line constraints l = 1,, N L separate chance constraints Line Roald 09.11.2016 17

Chance-Constrained Optimal Power Flow min s.t. C G T P G + C R T R G N B i=1 P G(i) + P W(i) P D(i) = 0 cost function power balance P min G P G + R G P max G, P α g Ω R G g, g = 1,, N G 1 ε, P α g Ω R G g, g = 1,, N G 1 ε, generation and reserve constraints probability of reserve insufficiency P A l, P G + P W P D + D l, ω P max L(l) max P A l, P G + P W P D + D l, ω P L(l) l = 1,, N L 1 ε, 1 ε, line constraints probability of transmission line overload Line Roald 09.11.2016 18

Outline Introduction Chance Constrained Optimal Power Flow Analytic reformulation of chance constraints Corrective control for forecast uncertainties Extension towards AC optimal power flow Modelling Risk of Constraint Violations Conclusions Line Roald 09.11.2016 19

Chance constraint reformulation P A (l, ) P G + P W P D + D l, ω P max L(l) 1 ε Chance constraints must be reformulated to become computationally tractable: Sample-based, no assumption about distribution (M. Vrakopoulou et al, 2012) Analytic reformulation for normal distribution (L. Roald, F. Oldewurtel, T. Krause and G. Andersson, 2013) (D. Bienstock, M. Chertkov and S. Harnett, 2014) Analytic reformulation for (partially) unknown distributions (L. Roald, F. Oldewurtel, B. Van Parys, and G. Andersson, 2015) Adaptive (online) approaches, robust counterparts Line Roald 09.11.2016 20

Chance constraint reformulation Sample-based reformulation + Does not require any assumption about forecast error distribution Analytic reformulation - Requires some knowledge about forecast error distribution - For large systems, number of samples might be prohibitive + Scalable, even to large systems and many uncertainty sources - Solution tends to be conservative + Can leverage available information about the distributions to obtain a less conservative solution - Solution is stochastic (depends on selected samples) + Solution is deterministic (always the same, more transparent) Line Roald 09.11.2016 21

Analytical Reformulation with Gaussian ω P A (l, ) P G + P W P D max + D l, ω P L(l) 1 ε Scale to standard Gaussian variable Apply inverse cumulative distribution function Rearrange terms deterministic constraint uncertainty margin A (l, ) P G + P W P D P max L l Φ 1 1 ε D l, Σ 1 2 2 D (l, ) μ Line Roald 09.11.2016 22

Analytical Reformulation with Gaussian ω deterministic constraint uncertainty margin A (l, ) P G + P W P D P max L l Φ 1 1 ε D l, Σ 1 2 2 D (l, ) μ Uncertainty margin = security margin against uncertainty Uncertainty margin represents a decrease in available transmission capacity Lower transmission capacity = higher cost Line Roald 09.11.2016 23

Cumulative Probability Uncertainty margin for normally distributed ω Violation probability ε = 0.05 1 0.8 0.6 0.4 Exp. Value SCOPF Line Limit CDF SCOPF 0.2 0 380 400 420 440 460 480 500 520 540 Line Flow [MW] Line Roald 09.11.2016 24

Cumulative Probability Uncertainty margin for normally distributed ω Violation probability ε = 0.05 1 0.8 0.6 0.4 ɛ = 0.5 Exp. Value SCOPF Line Limit CDF SCOPF 0.2 0 380 400 420 440 460 480 500 520 540 Line Flow [MW] Line Roald 09.11.2016 25

Cumulative Probability Uncertainty margin for normally distributed ω Violation probability ε = 0.05 1 0.8 ɛ = 0.05 0.6 0.4 ɛ = 0.5 Exp. Value SCOPF Line Limit CDF SCOPF 0.2 0 380 400 420 440 460 480 500 520 540 Line Flow [MW] Line Roald 09.11.2016 26

Cumulative Probability Uncertainty margin for normally distributed ω Violation probability ε = 0.05 1 0.8 ɛ = 0.05 0.6 0.4 ɛ = 0.5 Uncertainty margin Exp. Value SCOPF Exp.Value pscopf Line Limit CDF pscopf CDF SCOPF 0.2 0 380 400 420 440 460 480 500 520 540 Line Flow [MW] Uncertainty margin leads to the desired violation probability! Line Roald 09.11.2016 27

Decrease [MW] Uncertainty margin for normally distributed ω Deterministic constraint: A (l, ) P G + P W P D max P L l Probabilistic constraint: A (l, ) P G + P W P D P max L l Φ 1 1 ε D l, Σ 1 2 2 D (l, ) μ Uncertainty margin 50 40 24 30 20 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 19 23 14 15 16 17 18 20 21 22 Line Number 25 26 27 28 29 30 31 32 33 34 35 36 37 38 Lines most influenced by wind in-feed deviations have largest decrease! Line Roald 09.11.2016 28

Reformulation with Non-Gaussian Uncertainty P A (l, ) P G + P W P D max + D l, ω P L(l) 1 ε Scale to random variable with zero mean, unit variance Apply probabilistic inequality (e.g., Chebyshev bound) Rearrange terms deterministic constraint uncertainty margin A (l, ) P G + P W P D P max L l f 1 1 ε D l, Σ 1 2 2 D (l, ) μ Line Roald 09.11.2016 29

Reformulation with Non-Gaussian Uncertainty deterministic constraint Uncertainty margin A (l, ) P G + P W P D P max L l f 1 1 ε D l, Σ 1 2 2 D (l, ) μ Different (unknown) distributions of ω lead to different expressions for f 1 (1 ε)! If multivariate normal (or elliptical): Exact reformulation based on inverse CDF If only partially known: Probabilistic inequalities Line Roald 09.11.2016 30

Value of f 1 1 ε A (l, ) P G + P W P D P max L l f 1 1 ε D l, Σ 1 2 2 D (l, ) μ Exact reformulation: Normal distribution t distribution Distributionally robust: Symmetric, unimodal with known μ & Σ Unimodal with known μ & Σ Chebyshev (known μ & Σ) Confidence level 1 ε Line Roald 09.11.2016 31

Value of f 1 1 ε A (l, ) P G + P W P D P max L l f 1 1 ε D l, Σ 1 2 2 D (l, ) μ More information about the distribution leads to smaller uncertainty margin! Exact reformulation: Normal distribution t distribution Confidence level 1 ε Distributionally robust: Symmetric, unimodal with known μ & Σ Unimodal with known μ & Σ Chebyshev (known μ & Σ) Line Roald 09.11.2016 32

Case study: IEEE 118 bus system 54 uncertain in-feeds μ, Σ based on samples of historical data from APG ε = 0.1 Constant D l, (LP) Different assumptions about ω Not normally distributed! Line Roald 09.11.2016 33

Case study: Cost and empirical violations Lower violation probability is related to higher cost Goal: Meet acceptable violation probability as close as possible Line Roald 09.11.2016 34

Case study: Testing distributional assumptions The distributions are unimodal. They are not normal. However, using normal assumption provides the closest guess? Distribution is close to normal! «Law of large numbers» Line Roald 09.11.2016 35

Chance Constrained Optimal Power Flow Computational complexity of the simplest, analytic chance constrained OPF is the same as for the deterministic problem Accounting for forecast uncertainty leads to a decrease in transfer capacity uncertainty margin Due to the uncertainty margin, OPF cost increases. Try to keep uncertainty margin as tight as possible. Line Roald 09.11.2016 36

Corrective Control to Forecast Uncertainty Inexpensive means of control: Phase-shifting transformer (PST) tap changing HVDC set-points adjustment Phase Shifting Transformers Planned HVDC Line Roald 09.11.2016 38

Corrective Actions with HVDC and PSTs Corrective control with HVDC and PSTs: reaction to contingencies reaction to forecast errors p DC = p DC + δ ij DC + α DC ω γ = γ + δ γ ij + α γ ω Affine Control Policy reacts separately to each deviation «local control» Influence on the uncertainty margin 0,ij P ij max Pij Φ 1 1/2 1 ε Σ W AL l, C G α g, C W + C DC α DC + B γ α γ + b γ α γ 2 controllable uncertainty margin: Reduce impact on congested lines! Line Roald 09.11.2016 39

Case Studies IEEE 118 bus system 3 HVDC and 3 PSTs 99 uncertain net loads N-1 security constraints + post-contingency corrective action Computational aspects: # constraints same as for DC SCOPF SOC instead of linear Sequential SOCP algorithm: - Solve problem without SOC constraints - Check SOC violations and add most violated (L.Roald, S.Misra, T.Krause and G.Andersson, IEEE TPWRS, in press) Line Roald 09.11.2016 40

Case Study Impact on Cost and Operation We are able to reduce cost while maintaining similar security level! without with without with empirical violation probability (Monte Carlo) Line Roald 09.11.2016 41

Corrective control for forecast uncertainty Corrective control (recourse) reduces the cost of integrating uncertain and variable generation Reduces the impact of uncertainty on important lines Increases nominal power transfer Reduces overall cost Line Roald 09.11.2016 43

Chance-Constrained AC Optimal Power Flow min P G i G c 2,i P 2 G,i + c 1,i P G,i + c 0,i AC power flow equations s.t. f Θ, V, P, Q = 0, ω U max P P G,g P G,g 1 ϵ, g G min P P G,g P G,g 1 ϵ, g G max P Q G,g Q G,g 1 ε, g G min P Q G,g Q G,g 1 ϵ, g G Chance constraints on active and reactive power generation current and voltage magnitudes max P I L,j I L,j max P V i V i min P V i V i 1 ε, j L 1 ε, i B 1 ε, i B Formulation is based on full, nonlinear AC power flow equations Power injections P, Q are uncertain Voltages Θ, V and currents I are uncertain! Chance constraints limit the probability of constraint violation (H. Qu, L. Roald, G. Andersson, 2015) & (J. Schmidli, L. Roald, S. Chatzivasileiadis, G. Andersson, 2016) & (L.Roald, 2016) Line Roald 09.11.2016 45

Approximate Chance-Constraint Reformulation Step A: Linearization around expected operating point - Formulate power flow for expected P, Q: f Θ, V, P, Q = 0 Full AC equations - Linearize with respect to ΔP, ΔQ: Full AC I L,j I L,j + Γ I(.,j) ΔP ΔQ - Approximate chance-constraint: Linear Γ I sensitivity factor P I L,j + Γ I(.,j) ΔP ΔQ I max L,j 1 ε ((Vorname Nachname)) 09.11.2016 46

Approximate Chance-Constraint Reformulation Step A: Linearization around expected operating point - Formulate power flow for expected P, Q: f Θ, V, P, Q = 0 - Linearize with respect to ΔP, ΔQ: Full AC I L,j I L,j + Γ I(.,j) ΔP ΔQ - Approximate chance-constraint: P I L,j + Γ I(.,j) ΔP ΔQ I max L,j Full AC equations Linear Γ I sensitivity factor 1 ε Step B: Analytic reformulation - Assume Gaussian forecast errors ΔP, ΔQ - Analytical reformulation: Nominal solution (full AC) I L,j I max L,j Φ 1 1 ε Uncertainty margin based on linearization Γ I.,j Σ W Γ I.,j - Formulate as constraint tightening: I L,j I max L,j δ ij δ ij = Φ 1 1 ε Uncertainty margin Γ I.,j Σ W Γ I.,j ((Vorname Nachname)) 09.11.2016 47

Implementation using Iterative Approach Iterative solution scheme = Outer loop on existing AC OPF Initialize uncertainty margins δ k = 0 Solve AC OPF with δ k Calculate δ k+1 Yes: k = k + 1 Is max δ k+1 δ k η? No: Solution found Line Roald 09.11.2016 48

Implementation using Iterative Approach Iterative solution scheme = Outer loop on existing AC OPF Use your favourite AC OPF solver! Initialize uncertainty margins δ k = 0 Solve AC OPF with δ k Calculate δ k+1 Yes: k = k + 1 Is max δ k+1 δ k η? No: Solution found Line Roald 09.11.2016 49

Implementation using Iterative Approach Iterative solution scheme = Outer loop on existing AC OPF Initialize uncertainty margins δ k = 0 Not restricted to analytical chance constraints Solve AC OPF with δ k Calculate δ k+1 Yes: k = k + 1 (scenario approach, monte carlo...) Is max δ k+1 δ k η? No: Solution found Line Roald 09.11.2016 50

Case Study Computation time RTS 96 118 Bus 300 Bus Polish Time 0.54s 1.15s 3.37s 31.89s Iterations 5 4 5 4 Cost 40 127 3575.3 17 143 802 238 Line Roald 09.11.2016 51

IEEE RTS 96 Accuracy and Performance Uncertainty margin based on linearization Reasonably accurate, except for lines with flow reversal Performance of AC CC-OPF Violation probability reduced from 50% with deterministic, to ~5% with chance-constraints Line Roald 09.11.2016 52

AC OPF with Approximate Chance Constraints Accounts for voltage and reactive power Reasonably accurate and computationally tractable Can be solved using your favorite AC OPF solver Possible to use different types of uncertainty representations Line Roald 09.11.2016 53

f(y x, ω ) Weighted Chance Constraints WCC y x, ω > 0 ε Overloads y x, ω : transmission line overload, reserve insufficiency... Weighted Chance Constraint (WCC) P y x, ω f y x, ω P y x, ω dω ε where f. is a risk function and P. the probability distribution. y x, ω [L. Roald, S. Misra, M.Chertkov, G. Andersson, 2015] Line Roald 09.11.2016 55

f(y x, ω ) Weighted Chance Constraints WCC y x, ω > 0 ε Overloads y x, ω : transmission line overload, reserve insufficiency... Weighted Chance Constraint (WCC) P y x, ω f y x, ω P y x, ω dω ε Expected risk! where f. is a risk function and P. the probability distribution. y x, ω [L. Roald, S. Misra, M.Chertkov, G. Andersson, 2015] Line Roald 09.11.2016 56

Example Risk Functions f y ω P ω dω ε Standard chance constraint - Probability of violation [-] - Does not consider size of violation - Non-convex Linear risk function - Expected risk of overload [MW] - Accounts for size of violation - Convex Quadratic risk function - Faster increase in risk for higher overloads - Accounts for size of violation - Convex Similar ideas in risk-based OPF! [G. Hug 2012], [F. Xiao and J. McCalley 2009] Line Roald 09.11.2016 57

Example Risk Functions f y ω P ω dω ε Standard chance constraint Linear risk function Quadratic risk function - Standard chance constraint meets violation probability [L. Roald, S. Misra, M.Chertkov, G. Andersson, 2015] Line Roald 09.11.2016 58

Example Risk Functions f y ω P ω dω ε Standard chance constraint Linear risk function Quadratic risk function - Standard chance constraint meets violation probability - Linear and quadratic: more small, fewer large overloads [L. Roald, S. Misra, M.Chertkov, G. Andersson, 2015] Line Roald 09.11.2016 59

Evaluation of Weighted Chance Constraints Convex risk function convex constraint! Regardless of distribution Very general control policies Opens up new modelling possibilities within the OPF: Manual activation of tertiary reserves Limiting wind power output Line Roald 09.11.2016 60

Evaluation of Weighted Chance Constraints Convex risk function convex constraint! Regardless of distribution Very general control policies Evaluation can be time consuming Assume normal distribution closed form General distributions: Monte Carlo sampling μ y 1 Φ μ y σ y Σ b 0,1 K 0 S 1,b 1 + σ y 1 μ y 2π e 2 S K,bK σ y 2 ε y P y, ω C dω C dy ε Cutting planes algorithm Very effective for OPF problems [Bienstock, Chertkov, Harnett 2014] Requires convexity Line Roald 09.11.2016 61

(AC) Optimal Power Flow with (Weighted) Chance Constraints Chance constrained OPF is one possible way of accounting for forecast uncertainty in operational planning Analytically reformulated chance constraints yields computationally tractable formulations Corrective control can be used to react to forecast errors, and to decrease operational cost Approximate chance constraints for AC optimal power flow can be captured through linearization Weighted chance constraints can account for the size of possible overloads through the definition of a risk function Line Roald 09.11.2016 63

Also applied to Coordination and scheduling of coupled gas-electric infrastructures under uncertainty A. Zlotnik, L. Roald, S. Backhaus, M. Chertkov and G. Andersson, Coordinated Scheduling for Interdependent Natural Gas and Electric Infrastructures, IEEE Transactions on Power Systems, in press L. Roald (presenter), Optimization of integrated gas-electric systems under uncertainty, EURO Conference, 2016 Chance-constrained Unit Commitment with consideration of N-1 security K. Sundar et al, Unit Commitment with N-1 Security and Wind Uncertainty, Power Systems Computation Conference (PSCC), 2016 Optimized risk limits to ensure efficient cost-security trade-off Andrew Morrison, Optimized Risk Limits for Stochastic Optimal Power Flow, ETH Master Thesis 2016 Integrated balancing and congestion management with uncertainty L. Roald, T. Krause and G. Andersson, Integrated balancing and congestion management under forecast uncertainty, IEEE EnergyCon, 2016 Line Roald 09.11.2016 64

References M. Vrakopoulou et al, Probabilistic guarantees for the N-1 security of systems with wind power generation, Proceedings of PMAPS 2012, Istanbul, Turkey, June 10-14, 2012 L. Roald, F. Oldewurtel, T. Krause and G. Andersson, Analytical Reformulation of Security Constrained Optimal Power Flow with Probabilistic Constraints, IEEE Powertech, Grenoble, France, 2013 D. Bienstock, M. Chertkov and S. Harnett, Chance-Constrained Optimal Power Flow: Risk-Aware Network Control under Uncertainty, SIAM Review, Vol. 56, No. 3, pp. 461-495, 2014 L. Roald, F. Oldewurtel, B. Van Parys, and G. Andersson, Security-Constrained Optimal Power Flow with Distributionally Robust Chance Constraints, arxiv: http://arxiv.org/abs/1508.06061 H. Qu, L. Roald and G. Andersson, Uncertainty Margins for Probabilistic AC Security Assessment, IEEE PowerTech Eindhoven, Netherlands, 2015 J. Schmidli, L. Roald, S. Chatzivasileiadis and G. Andersson, Stochstic AC Optimal Power Flow with Approximate Chance-Constraints, IEEE PES General Meeting, Boston, Massachusetts, 2016 L. Roald, T. Krause and G. Andersson, Integrated Balancing and Congestion Management under Forecast Uncertainty, IEEE EnergyCon, Leuven, Belgium, 2016 L. Roald, S. Misra, T. Krause and G. Andersson, Corrective Control to Handle Forecast Uncertainty: A Chance Constrained Optimal Power Flow, submitted to IEEE Transactions on Power Systems (2 nd round review) L. Roald, S. Misra, M.Chertkov and G. Andersson, Optimal Power Flow with Weighted Chance Constraints and General Policies for Generation Control, IEEE Conference on Decision and Control (CDC), 2015 L. Roald, S. Misra, M.Chertkov and G. Andersson, Optimal Power Flow with Wind Power Control and Limited Expected Risk of Overloads, Power Systems Computation Conference (PSCC), 2016 G. Hug, Generation Cost and System Risk Trade-Off with Corrective Power Flow Control, Allerton, Illinois, USA, 2012 F. Xiao and J. McCalley, Power system assessment and control in a multi-objective framework, IEEE Trans. Power Systems, vol. 24, pp.78 85, 2009 Line Roald 09.11.2016 65

Thank you! www.e-umbrella.eu Line Roald 09.11.2016 66