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5th Southeast Symposium on Contemporary Engineering Topics (SSCET), 2014 A Parallel Solution to Stochastic Power System Operation with Renewable Energy Yong Fu, Ph.D. Associate Professor Electrical and Computer Engineering Mississippi State University New Orleans, LA September 19th, 2014 Parallel Computing o With development of high performance computing technique, parallel computing technique can significantly improve computational efficiency of optimization problem with utilization of multi-processors and multi-threads. o These improvements cannot be achieved by the architectures of the machines alone, it is equally important to develop suitable mathematical algorithms and proper decomposition & coordination technique in order to effectively utilize parallel architectures A Typical Power System Operation Problem – Security Constrained Unit Commitment Objective Function – Minimize Generation and startup/shutdown costs Generating Unit Constraints Load (MW) 300 Unit 3 250 Generation capacity Minimum ON/OFF time limits Ramping UP/DOWN limits Must-on and area protection constraints Forbidden operating region of generating units Unit 2 200 Unit 1 150 100 1 4 7 10 13 16 19 22 Hours System Operation Constraints Power balance System reserve requirements Power flow equations Transmission flow and bus voltage limits Limits on control variables Limits on corrective controls for contingencies Large Scale, Non-Convex, Mixed Integer Nonlinear Problem Who Use SCUC and How? ISOs: PJM, MISO, ISO New England, California ISO, New York ISO and ERCOT GENCOs TRANSCOs DISTCOs ISO Security-Constrained Unit Commitment ISO (SCUC) and Market Participants Day Ahead Market (DAM) determines the 24-hourly status of the generating units for the following day based on financial bidding information such as generation offers and demand bids. Day Ahead UC for Reliability (RUC), which focuses on physical system security based on forecasted system load, is implemented daily to ensure sufficient hourly generation capacity at the proper locations. Look-Ahead UC (LAUC), as a bridge between dayahead and real-time scheduling, constantly adjusts the hourly status of fast start generating units to be ready to meet the system changes usually within the coming 3-6 hours. Real-Time Market (RTM) further recommits the very fast start generating units based on actual system operating conditions usually within the coming two hours in 15-minute intervals. Stochastic SCUC In stochastic programming, the decision on certain variables has to be made before the stochastic solution is disclosed, whereas others could be made after. The set of decisions is then divided into two groups: A number of decisions are made before performing experiments. Such decisions are called firststage decisions and the period when these decisions are made is called the first stage. A number of second-stage decisions are made after the experiments in the second stage. Stochastic models containing above two groups of variables, first-stage and second-stage decision variables, are called two-stage stochastic programming. Min x , y1 ,, yk K c x Min p k d T y k T k 1 s.t. Ax b, x is binary, Hx Fy k ek , k 1, , K Stochastic SCUC --- Example G1 13 $/MWh 40MW~80MW G2 42 $/MWh 15MW~ 40MW 1 2 L1 75MW L2 75MW G3 16 $/MWh 10MW~40MW Load W System Base Case 50 MW Wind (WM) Load (MW) Base case - 20 100 Scenario 1 G3 15 105 Scenario 2 L2 23 95 G3 can adjust dispatches by 5 MW G2 is quick-start unit with 30 MW QSC Scenario 1 Scenario 2 0 MW ? MW 52.5 MW 0 MW 15 MW 0 MW 50 MW 52.5 MW Base Case 105 MW 95 MW 23 MW Scenario 1 Scenario 2 60 MW 42.5 MW 15 MW 52 MW 52.5 MW 0 MW 30 MW 0 MW 42.5 MW 20 MW 75 MW 15 MW 65 MW 0 MW ? MW 100 MW 20 MW Solution 2 Equipment Outage 75 MW 80 MW Solution 1 Cases 20 MW 75 MW 0 MW 52.5 MW 100 MW 15 MW 105 MW 95 MW 23 MW Current Work Start Unit=1:NG Unit Commitment calculation Unit Commitment calculation Unit Commitment calculation Unit Commitment calculation Optimal Power Flow Calculation Optimal Power Flow Calculation Time=1:NG Optimal Power Flow Calculation Optimal Power Flow Calculation End o Amdahl’s law: an upper bound on the relative speedup achieved on a system with multi-processors is decided by the execution time of the application operating sequentially. Proposed Approach Start Time=1:NG Unit=1:NG Unit Commitment calculation Unit Commitment calculation Unit Commitment calculation Unit Commitment calculation Optimal Power Flow Calculation Optimal Power Flow Calculation Optimal Power Flow Calculation Optimal Power Flow Calculation End o o Structure of Algorithm: Scenario-based stochastic model is adopted to analyze the uncertainties of load and wind energy in this paper. Instead of master-and-slave structure, UC and OPF subproblems are solved simultaneously in the proposed parallel calculation method. Convergence performance: In an iterative solution process, the number of iterations affects the overall computational time. Several convergence acceleration options, including initialization and update of penalty multipliers, truncated auxiliary problem principle and trust region technique, are used to improve the convergence performance and efficiency in a scenario-based study. Decomposition Strategy Mathematically, the stochastic SCUC can be formulated as a mixed integer programming (MIP) problem as shown in NS s Fs ( x, y s ) Min s 0 s.t. Ax bys d , Eys h Variable Duplication Technique NS s Fs ( x, y s ) Min s.t. s 0 Ax bys d , Eyˆ s h, y s yˆ s Augmented Lagrangian Method NS Min s.t. T s Fs ( x, y s ) ( yˆ s y s ) s 0 Ax bys d , Eyˆ s h c yˆ s y s 2 2 Algorithms for Parallel Solutions Auxiliary Problem Principle (APP) Method Diagonal Quadratic Approximation (DQA) Method Alternating Direction Method of Multipliers (ADMM) Analytical Target Cascading (ATC) Method Iterative Solution Procedure Decomposition structure: Original Augmented Lagrangian Problem Optimal Power Flow Subproblem Unit Commitment Subproblem UC Subproblem Unit 2 UC Subproblem Unit 1 UC Subproblem Unit NG OPF Subproblem Period 1 OPF Subproblem Period NT-1, Scenario 0 OPF Subproblem Period NT-1 OPF Subproblem Period NT-1, Scenario 1 Two separated auxiliary problem: Decision variables for the current iteration NS NT NG s 0 t 1 i 1 OPF Subproblem Period NT OPF Subproblem Period NT-1, Scenario NS Given values from the previous iteration 1) 1) Luc Min s [ F ( Pg ,its , I it ) SUDit c g ,its ( Pg ,its ) 2 (its c g ,its ( Pˆg( k,its Pg( k,its )) Pg ,its ] NS NT NG s 0 t 1 i 1 1) 1) ˆ Lopf Min s [c g ,its ( Pˆg ,its ) 2 (its c g ,its ( Pˆg( k,its Pg( k,its )) Pg ,its ] Case Study – IEEE 118-bus Testing System o Case 1: Deterministic case o Case 2: Stochastic case with 3 scenarios Zone 1 Zone 2 7 2 13 1 33 55 11 56 34 53 14 46 17 4 54 45 12 6 54 thermal units 3 wind farms 118 buses 186 branches 44 117 15 3 43 36 35 37 18 57 52 47 42 39 16 58 51 19 41 5 59 48 40 49 50 60 38 8 9 30 10 20 113 31 29 73 32 66 21 62 69 67 64 65 28 114 71 26 22 75 118 76 77 115 25 27 68 23 80 116 24 98 70 78 79 97 86 85 90 63 72 74 87 61 81 88 89 96 84 83 82 95 93 91 112 94 107 92 106 100 102 101 106 109 105 103 108 Zone 3 111 104 110 99 Deterministic Case Study The converged result is obtained after 39 iterations. 150 Unit 36 at Hour 5 150 Power(MW) Power(MW) 200 100 P op f,36,5 50 Unit 45 at Hour 5 100 50 P op f,45,5 P uc,36,5 0 10 20 Iterations 30 0 40 300 300 250 250 200 Unit 36 at Hour 21 150 100 P op f,36,21 50 0 P uc,36,21 0 10 20 Iterations 30 40 Power(MW) Power(MW) 0 P uc,45,5 0 10 200 20 Iterations 40 Unit 45 at Hour 21 150 100 P op f,45,21 50 0 30 P uc,45,21 0 10 20 Iterations 30 40 Deterministic Case Study Items Centralized SCUC Parallel SCUC Changes Total Cost ($) 1,583,700 1,584,997 +0.08% Time (Seconds) 19 8 -58% Stochastic Case Study (3 scenarios) Items Centralized SCUC Parallel SCUC Changes Cost ($) 1,582,840 1,583,565 +0.046% Time (Seconds) 1,083 20 -96% Case Study – A 1168-bus Power System 1200 1000 Power Output (MW) o A practical 1168-bus power system with 169 thermal units, 10 wind farms, 1474 branches, and 568 demand sides. o It could be nearly impossible to get a near-optimal stochastic SCUC solution for this system by applying a traditional centralized SCUC algorithm. o However, the proposed parallel stochastic SCUC algorithm provides solutions. 800 600 400 Popf,8,1,1 200 Puc,8,1,1 0 0 50 100 150 Iterations Unit 8 at Hour 1 200 250 Case Study – A 1168-bus Power System # of Scenarios # of Iteration Total Time (sec.) 0 1 2 3 4 5 6 7 8 9 10 315 330 299 327 277 278 248 243 242 237 231 109.55 146.06 139.31 163.47 142.28 140.28 139.52 130.94 133.25 148.33 131.95 Conclusions o The proposed stochastic SCUC approach minimizes the operation cost of system by possibility expectation of each scenarios, which can adaptively and robustly adjust generation dispatch in response to constraints in different scenarios. o In comparison with traditional stochastic SCUC, optimal power flow problem does not have to wait for unit commitment decision, both problems can be solved simultaneously, which is more computational efficient in both day-head and real-time power markets. o The ideas can be applied to various power system applications: state estimation, economic dispatch, and planning. 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