Stochastic Optimization for Unit Commitment A Review Qipeng P. Zheng +, Jianhui Wang and Andrew L. Liu + Department of Industrial Engineering & Management Systems University of Central Florida Argonne National Laboratory School of Industrial Engineering, Purdue University INFORMS Annual Meeting 2014, San Francisco, CA Based on the paper Stochastic Optimization for Unit Commitment A Review to appear in IEEE Trans on Power Systems. DOI: 10.1109/TPWRS.2014.2355204 Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 1 / 20
Outline Outline 1 Introduction 2 Uncertainty Modeling for UC 3 UC models under uncertainty and solution algorithms 4 Market Operations 5 Conclusions and Future Research Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 2 / 20
Introduction Power Generation Unit Commitment One the key applications in power system operations. (ISO and GENCO). Optimal commitment status of the generating units. It is a NP-hard problem. Two waves of revolutions of UC research: Mixed integer programming solution algorithm. Deterministic to stochastic optimization. Increasing demands to deal with uncertainties in the new age of power generation. High renewable energy penetration (e.g., wind, solar, etc.). Electricity market deregulation. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 3 / 20
Introduction Our focus of this review Because of a large number of papers related to this subject, we focus on: Short-term unit commitment (hours-ahead to day-ahead) rather than longer-term unit commitment (weekly, seasonal and yearly); SO techniques in the formulation and solution of UC, instead of deterministic UC problems with additional constraints incorporating inputs derived from statistical methods (e.g., reserve requirements calculated based on probabilistic forecasts); Optimization algorithms that have explicit formulations and can lead to exact solutions rather than metaheuristic methods such as genetic algorithms, simulated annealing, or swarm-based approaches, etc. Disclaimer: Due to the vastness and complexity of the literature, any omissions or inaccurate characterization of the works cited is strictly due to the limitation of the authors knowledge. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 4 / 20
Uncertainty Uncertainty Modeling Scenarios A scenario is a realization of the all uncertainties. Usually need a large number of them for stochastic programming UC models. Monte Carlo simulation is usually used to generate them based on a given distribution or stochastic process. Trade off between desired accuracy and computational performance. Probabilistic forecasting Upper and lower quantiles (can be used as the inputs for uncertainty sets). Quantile regression, Kernel density estimators, qantile-copula estimator, etc. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 5 / 20
Uncertainty Modeling Uncertainty Uncertainty Sets Box intervals [max{0, d + z α σ}, d + z β σ]. Polyhedral sets (reduces conservativeness). Ellipsoidal sets using expectations and covariance matrix. Discrete sets, e.g., contingencies, wind power outputs, etc. Generate uncertainty sets by risk measures (e.g., VaR and CVaR). Particularly, constraints on CVaR can be transformed to polyhedral sets for certain distributions. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 6 / 20
Models and Algorithms UC models under uncertainty and solution algorithms Stochastic Programming Two Stage Stochastic Programming Multi Stage Stochastic Programming Risk Consideration Robust Optimization Stochastic Dynamic Programming Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 7 / 20
Models and Algorithms Two Stage Stochastic Programming Models for UC Day-ahead unit commitment schedules (first stage) Commitment technical constraints U. ξ is the random vector/variables. Expected real time cost (e.g., fuel cost). Real-time dispatch decisions (second stage) min u U ct u + E ξ [F (u, ξ)]. (1) F (u, s) = min p s,f s f (p s ) (2a) s.t. A s u + B s p s + H s f s d s, (2b) s denotes a specific scenario/realization of the r.v. Technical constraints (e.g., load balancing, ramping, etc.) (A s, B s, H s ) model contingencies and/or rescheduling. (d s ) model demands and/or renewable energy outputs. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 8 / 20
Models and Algorithms Solving the two stage models Use discrete scenarios and the models reduce to a large-scale (a great amount of scenarios) deterministic optimization problem. Decomposition algorithms. Benders decomposition or L-shaped method use cutting planes approximate the expected real-time cost function, E ξ [F (u, ξ)]. (acceleration techniques) Lagrangian relaxation. Relax the unit commitment constraints and the scenarios are decoupled. Relax the demand and reserve constraints and the units are decoupled. Bundle or regularization method to accelerate computation. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 9 / 20
Models and Algorithms Multi Stage Stochastic Programming Models n4 n3 n5 n2 n7 n1 n6 n10 n8 n11 n9 n12 n13 n14 n15 t 1 t 2 t 3 t 4 Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Scenario 6 Scenario 7 Scenario 8 Capture dynamics of the uncertainty and decisions are made along the process of unfolding uncertainties. Scenario tree formulations min s S Prob s f (u s, p s, r s, x s ) s.t. (u s, p s, r s, x s ) U s, s S (u s,t, p s,t, r s,t, x s,t ) = (u n, p n, r n, x n ), (s, t) S n, n N, Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 10 / 20
Models and Algorithms Solving the multi stage models Computationally very challenging because of the number of scenarios grow exponentially (e.g., two outcomes at each node/hour, then 16 million scenarios for 24 hours). Scenarios aggregations/reductions. But still need to use decomposition methods. Column Generation and Lagrangian relaxation (Progressive Hedging). Decompose by scenarios through relaxing the non-anticipativity constraints. Decompose by units through relaxing the demand/reserve constraints. Acceleration methods (stabilization, bundle, regularization, etc.) Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 11 / 20
Models and Algorithms Risk Consideration in Stochastic Programming Models Risk averse models in addition to expectation minimization. Various risk measures and challenges: Expected Load Not Served (ELNS). Variance of total profit. Loss of Load Probability (LOLP). Value at Risk (VaR) as same as LOLP or chance-constrained programs. Computationally challenging and can use Sampling Average Approximation (SAA). Conditional Value at Risk (CVaR) (convex constraints and only continuous variables). Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 12 / 20
Models and Algorithms Robust Optimization UC models Minimize the worst-case minimal cost (v.s. expected cost in stochastic programming). { } min c T u + max [F (u, v)], (3) u U v V where F (u, v) = min p,f q T p (4a) s.t. A v u + B v p + H v f d v (4b) Conservative solutions but no scenarios enumeration. Various types of uncertain sets for demands, renewable energy outputs, contingencies, demand side management, etc. Budget of uncertainty constraints to adjust the robustness/conservativeness. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 13 / 20
Models and Algorithms Solving Robust Optimization UC Models Reformulating the minimal cost problem by taking its dual. max v,π (d v A v u) T π (5a) s.t. H v π 0 (5b) B v π q v V, (5c) (5d) A bilinear programming problem with bilinear terms only appearing in objective function. Guarantee extreme-point optimal solutions for both π and v. Benders decomposition type of approach. z (d v A v u) T π, Column-and-constraint generation or constraints generation approach. z q T p i A v u + B v p i + H v f i d v. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 14 / 20
Models and Algorithms Stochastic Dynamic Programming UC models The stochastic dynamic programming framework [ T 1 ] inf V π(s 0 ) := E C t (s t, µ t (s t ), ξ t ) + C T (u T ), (6) π Π t=0 Bellman backward reduction approach for deterministic approach. It is hard for stochastic UC to take policy enumeration approach. Approximate dynamic programming. Value Function Approximation. Policy Function Approximation (e.g., Model Predictive Control). State-space Approximation. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 15 / 20
Market Operations Market Operations UC can be used to aid day-ahead market clearing, and to improve post-day- ahead and intraday reliability. Stochastic market clearing needs to ensure revenue adequacy. Cooptimiztion of energy and ancillary service needs further investigation. Revenue adequacy and associated issues such as pricing, settlement, market power, and uplift charges all need to be addressed before any model can be put into use in practice. Models should be fair, transparent, and comprehensible to all participants. Definitions of scenarios, uncertainty sets all will be highly contentious among participants and be barrier to implement SO UC approaches. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 16 / 20
Conclusions and Future Research Conclusions and Future Research 1. Uncertain Modeling Better renewable energy output forecasting (e.g., wind and solar). A delicate balance needs to be achieved between economics and reliability. Improvements of existing approaches, such as scenario selection, reduction, and evaluation. New uncertainty modeling concepts (e.g., data-driven optimization, distributionally robust optimization etc.). Muliscale modeling (e.g., detailed modeling of different time-scale decisions will improve model fidelity). Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 17 / 20
Conclusions and Future Research Conclusions and Future Research 2. Computational Challenges Efficient convexification method for nonconvex second stage problems. Acceleration techniques for different types of decomposition algorithms. Advanced techniques based on reformulation and approximation for robust models with multiple stages and distributionally robust models. Value function approximation using post-state decision variables for SDP. Model Predictive Control approach to SDP UC need further study. 3. Market Design Fair and Transparent settlement rules and pricing. Link forecasting errors with prices, requirements, compensation for reserves. Incorporating sustainability and environmental concerns. Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 18 / 20
Conclusions and Future Research Acknowledgements We would like to thank the discussion with Dr. Feng Qiu. We would like to thank the editors and the reviewers for suggestions. Dr. Liu would like to thank NSF for support (CMMI-1234057). Dr. Wang would like to thank DOE Office of Electricity Delivery and Energy Reliability for support. Dr. Zheng would like to thank NSF for support (CMMI-1355939). Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 19 / 20
Thank you! Conclusions and Future Research Thank you! Questions? Zheng(UCF)&Wang(Argonne)&Liu(Purdue) () Stochastic Optimization for UC November 11 th, 2014 20 / 20