Multi-Area Stochastic Unit Commitment for High Wind Penetration

Transcription

1 Multi-Area Stochastic Unit Commitment for High Wind Penetration Workshop on Optimization in an Uncertain Environment Anthony Papavasiliou, UC Berkeley Shmuel S. Oren, UC Berkeley March 25th, 2011

2 Outline 1 2 Unit Commitment Multi-Area Wind Scenario Selection Algorithm 3 4

3 Unit Commitment and Economic Dispatch Unit commitment (day-ahead market, first stage) 1 Electricity suppliers and consumers place supply bids and ask bids 2 System operator determines commitment and production schedule of generators in order to maximize social welfare 3 Resulting problem is a mixed integer linear program 4 Problem is solved one day in advance of operating day Economic dispatch (real-time market, recourse) 1 Fast units are committed 2 Production levels of all units are adjusted 3 Solved 75 minutes in advance of operating interval

4 Motivation for Stochastic Unit Commitment Drawbacks of renewable power integration Unpredictability Non-controllability Merits of stochastic unit commitment Captures operating costs of renewable power integration Captures capital costs for renewable power integration under uncertainty

5 Renewable Supply Uncertainty

6 Renewable Supply Variability

7 Contingencies C12 C5 C6 C13 C7 C2 C11 C14 C8 C9 C3 C4 C10

8 Components Outcomes Scenario selection Representative outcomes Stochastic UC Wind model Stoch < Det? Outcomes Deterministic UC Slow gen UC schedule Slow gen UC schedule Economic dispatch Min load, startup, fuel cost

9 Deterministic Versus Stochastic Unit Commitment Deterministic model 1 Reserve requirements s gt + f gt T req t, f gt F req t, t T g G s g G f 2 Import constraints g G Two-stage stochastic model l IG j γ jl e lt IC j, j IG, t T 1 Day-ahead commitment w gt for slow generators, g G s 2 Uncertainty reveals (net demand D nst, line availability B ls, generator availability P + gs, P gs) 3 Real-time adjustment of fast generator commitment u gst, g G f, and production levels p gst, g G

10 Unit Commitment Multi-Area Wind Scenario Selection Algorithm Stochastic Unit Commitment Formulation min g G s.t. π s (K g u gst + S g v gst + C g p gst ) s S t T e lst + p gst = D nst + e lst, n N, s S, t T (2) l LI n g G n l LO n e lst = B ls (θ nst θ mst ), l = (m, n) L, s S, t T (3) e lst TC l, l L, s S, t T (4) TC l e lst, l L, s S, t T (5) p gst P gsu + gst, g G, s S, t T (6) Pgsu gst p gst, g G, s S, t T (7) (1)

11 Unit Commitment Multi-Area Wind Scenario Selection Algorithm Stochastic Unit Commitment Formulation (II) p gst p gs,t 1 R + g, g G, s S, t T (8) p gs,t 1 p gst Rg, g G, s S, t T (9) t v gsq u gst, g G f, s S, t UT g (10) q=t UT g+1 t+dt g q=t+1 v gsq 1 u gst, g G f, s S, t N DT g (11) t q=t UT g+1 z gq w gt, g G s, t UT g (12)

12 Unit Commitment Multi-Area Wind Scenario Selection Algorithm Stochastic Unit Commitment Formulation (III) t+dt g q=t+1 z gq 1 w gt, g G s, t N DT g (13) z gt 1, g G s, t T (14) v gst 1, g G, s S, t T (15) z gt w gt w g,t 1, g G s, t T (16) v gst u gst u gs,t 1, g G f, s S, t T (17) π s u gst = π s w gt, g G s, s S, t T (18) π s v gst = π s z gt, g G s, s S, t T (19) p gst, v gst 0, u gst {0, 1}, g G, s S, t T (20) z gt 0, w gt {0, 1}, g G s, t T (21)

13 Wind Data Source Unit Commitment Multi-Area Wind Scenario Selection Algorithm 2 wind integration cases: 7.1% energy integration (2012), 14% energy integration (2020) California ISO interconnection queue lists locations of planned wind power installations NREL Western Wind and Solar Interconnection Study archives wind speed - wind power for Western US County Existing 7.1% wind 14% wind Altamont ,086 Clark - - 1,500 Imperial - - 2,075 Solano ,149 Tehachapi 1,346 4,886 8,333 Total 2,766 6,688 14,143

14 Wind Sites Unit Commitment Multi-Area Wind Scenario Selection Algorithm

15 Calibration Steps Unit Commitment Multi-Area Wind Scenario Selection Algorithm wind speed instead of wind power due to nonlinearity of wind speed - wind power relation 1 Fit nonparametric distribution F k ( ) to wind speed data v kt 2 Transform wind speed data to Gaussian: v kt = N 1 (F k (v kt )) 3 Remove seasonal and diurnal effects: ˆv kt = v kt µ kmt σ kmt 4 Obtain autoregressive parameters from Yule Walker 2 equations: ˆv k,t+1 = φ kj ˆv k,t j + ω kt j=0 5 Estimate autocorrelation Σ of residual noise

16 Data Fit Unit Commitment Multi-Area Wind Scenario Selection Algorithm Data Data Data Output (MW) Output (MW) Output (MW) Data 0.8 Data Output (MW) Output (MW) Load duration curves for Altamont, Clark County, Imperial, Solano and Tehachapi, and power curve at the Tehachapi area

17 Scenario Selection Literature Unit Commitment Multi-Area Wind Scenario Selection Algorithm J. Dupacova, N. Growe-Kuska, and W. Romisch, Scenario Reduction in Stochastic Programming: An Approach Using Probability Metrics N. Growe-Kuska, K. C. Kiwiel, M. P. Nowak, W. Romisch and I. Wegner, Power Management in a Hydro-Thermal System Under Uncertainty by Lagrangian Relaxation H. Heitsch and W. Romisch, Scenario Reduction Algorithms in Stochastic Programming J. M. Morales, S. Pineda, A. J. Conejo and M. Carrion, Scenario reduction for futures trading in electricity markets

18 Scenario Selection Motivation Unit Commitment Multi-Area Wind Scenario Selection Algorithm Traditional scenario selection methods do not capture possibility of highly adverse wind outcomes, motivated us to have modeler specify which wind production characteristics are relevant Traditional scenario selection methods do not guarantee match of moments, which is crucial due to predominant role of fuel costs, which motivates choosing weights to match moment as closely as possible

19 Scenario Selection Unit Commitment Multi-Area Wind Scenario Selection Algorithm 1 Select most prominent contingencies and assign probability 10 4 for their occurrence 2 Generate 1,000 samples of wind production and select representative wind outcomes 3 Weigh wind outcomes by solving min ( π s a π s R S t s µ t ) 2 t T s S s.t. π s = 1 s S π s 0.01, 4 If scenario s corresponds to a contingency, its weight is 10 4 π s, otherwise ( )π s

20 Key idea: relax non-anticipativity constraints on both unit commitment and startup variables 1 Balance size of subproblems 2 Obtain lower and upper bounds at each iteration Lagrangian: L = π s (K g u gst + S g v gst + C g p gst ) g G s S t T + π s (µ gst (u gst w gt ) + ν gst (v gst z gt )) g G s s S t T

21 Second-Stage Decision Subproblem (Per Scenario) (P1 s ) : min π s (K g u gst + S g v gst + C g p gst ) g G t T + π s (µ gst u gst + ν gst v gst ) s.t. g G s t T (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (15), (17) p gst 0, v gst 0, u gst {0, 1}, g G, t T

22 First Stage Decision Subproblem and Multiplier Update First-stage decision subproblem (P2) : min π s (µ gst w gt + ν gst z gt ) s.t. Multiplier update g G s s S t T (12), (13), (14), (16) w gt {0, 1}, z gt 0, g G s, t T µ k+1 gst = µ k gst + α k π s (w k gt u k gst), g G s, s S, t T ν k+1 gst = ν k gst + α k π s (z k gt v k gst), g G s, s S, t T.

23 Parallelization Second-stage subproblems, second-stage feasibility runs and economic dispatch simulations can be parallelized Implemented in PVM, CPLEX

24 California ISO Characteristics System summary 1 55,027 MW power plant capacity 2 50,270 MW record peak demand 3 30,000 market transactions per day 4 30 million people served Reduced model units (82 fast, 40 slow) 2 53,665 MW power plant capacity buses transmission lines 5 42 scenarios (3 wind scenarios, 14 contingencies) 6 11 wind scenarios in previous work without transmission network

25 Competing Reserve Rules Perfect foresight: anticipates outcomes in advance Percent-Of-Peak-Load rule: commit total reserve T req at least x% of peak load, F req = 0.5T req 3+5 rule: commit fast reserve F req at least 3% of hourly forecast load plus 5% of hourly forecast wind, T req = 2F req

26 Day Types 8 day types considered, one for each season, one for weekdays/weekends Day types weighted according to frequency of occurrence Net Load (MW) x 10 4 FallWD FallWE WinterWD WinterWE SummerWD SummerWE SpringWD SpringWE Hour

27 Policy Comparison - 7.1% Wind Cost ($) Cost ($) Cost ($) Cost ($) Stoch Foresight 20% peak 3+5 WinterWD 7,716, ,003 36,103-35,381 SpringWD 7,655, ,622 73,382 74,532 SummerWD 13,008, , , ,209 FallWD 9,582, , , ,697 WinterWE 4,829,516-65,749 9,024 12,671 SpringWE 4,579,470-69,794 26,303 25,140 SummerWE 9,950, ,428 97, ,041 FallWE 6,618,946-97,468 35,771 53,293 Total 8,634, , , ,297 improv. (%)

28 Policy Comparison - 14% Wind Cost ($) Cost ($) Cost ($) Cost ($) Stoch Foresight 30% peak 3+5 WinterWD 5,480, ,043 94,612 88,452 SpringWD 5,369, ,762 8,289 50,978 SummerWD 11,267, , , ,904 FallWD 7,988, , , ,851 WinterWE 3,124, ,177 17,702 14,086 SpringWE 2,815, ,497 11,803 19,254 SummerWE 8,280, , , ,777 FallWE 5,192,561-92,915 62,197 61,849 Total 6,762, , , ,281 improv. (%)

29 Jensens-Inequality Effect Deterministic UC under-commits slow reserves Stochastic unit commitment wastes more wind

30 Running Times CPLEX DELL Poweredge 1850 servers (Intel Xeon 3.4 GHz, 1GB RAM) Average running time of 5,685 seconds on single machine Average MIP gap of 0.8%

31 Conclusions Scenario selection is validated by demonstrating that it outperforms two common deterministic reserve rules Compared to best determinstic policy, stochastic unit commitment achieves 55.9% of perfect foresight savings for 7.1% wind integration, 1.94% cost reduction 54.5% of perfect foresight savings for 14% wind integration, 2.61% cost reduction Proposed decomposition algorithm can be parallelized, speeding up computation

32 No Transmission Constraints - 7.1% Wind Cost ($) Cost ($) Cost ($) Cost ($) Stoch Clair 20% peak 3+5 WinterWD 5,970,040-21,816 20,484 39,747 SpringWD 6,003,520-37,218-2,147 14,870 SummerWD 11,272,575-62, , ,793 FallWD 8,081,245-39,921 15,751 21,618 WinterWE 3,166,890-32,214 1,346-3,587 SpringWE 2,642,864-28,857-12,622 14,463 SummerWE 7,595,842-42,179 46,661 46,892 FallWE 5,106,143-25,350-1,806 1,002 Total 6,916,442-38,041 26,733 37,596 improv. (%)

33 No Transmission Constraints - 14% Wind Cost ($) Cost ($) Cost ($) Cost ($) Stoch Clair 30% peak 3+5 WinterWD 4,121, ,553 27,642 55,358 SpringWD 3,906, ,016 71, ,306 SummerWD 9,773, , ,861 67,553 FallWD 6,125,650-89,470 27,721 34,900 WinterWE 1,967,672-75,346 92,732 3,619 SpringWE 1,482,317-57, ,434 96,514 SummerWE 6,309,549-78,993 79,931 39,757 FallWE 3,524,599-78,288-2,508 2,443 Total 5,551, ,389 69,309 56,795 improv. (%)