Congestion and Price Prediction in Locational Marginal Pricing Markets Considering Load Variation and Uncertainty

Transcription

1 The Department of Electrical Engineering and Computer Science Congestion and Price Prediction in Locational Marginal Pricing Markets Considering Load Variation and Uncertainty Dissertation Defense Rui (Ray) Bo EECS Department University of Tennessee, Knoxville August 3,

2 Outline Background Motivation Problem Statement Research Outline: 3 Research Stages Conclusions and Future Work 2

3 Background Deregulation Generation Side Transmission Network Customer Side 3

4 Background Energy Market Day-ahead market Real-time market Security Constrained Unit Commitment (SCUC) Security Constrained Economic Dispatch (SCED) Locational Marginal Price (LMP) Day-ahead market Real-time market SCUC/SCED -- based on OPF Optimal Power Flow (OPF) Alternating Current OPF (ACOPF) Direct Current OPF (DCOPF) 4

5 Motivation Congestion More vulnerable Economic inefficiency: $1.06 billion, or, 8% total PJM billing (2006) Price volatility; price spike LMP is calculated from solving OPF OPF parameters Physical limits, offers/bids, availability, load Load Forecasting and Uncertainty 5

6 Problem Statement Investigate the approaches for predicting power system steady state statuses (especially, congestions and LMPs) under load variation, by exploring the OPF models which determines the generation dispatch, system statuses, and price as well. In addition, examine the risk associated with the prediction w.r.t. load forecasting uncertainty. 6

7 A Look-ahead What do you expect the LMP vs. Load curve to be? 7

8 A Look-ahead 40 LMP at Five Buses 35 LMP ($/MWh) Load (MW) A B C D E 8

9 A Look-ahead LMP at Five Buses Price may decrease LMP ($/MWh) Load (MW) A B C D E Change of Bidding Blocks What happened here (Critical Load Levels)? A B Congestion 9 Shift of Marginal Unit

10 A Look-ahead How to find CLLs? Efficient? Effective? Caught price spike One option: Repetitive OPF runs Brute-force Depend on simulation resolution Missing price spike Effectiveness Efficiency 10

11 Research Outline Three Stages Stage 1: Optimal Power Flow (OPF) Models and LMP Calculation Stage 2: Congestion and Price Prediction under Load Variation Stage 3: Probabilistic LMP forecast under Load Uncertainty 11

12 Dissertation Structure Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 12

13 Stage 1: OPF Models and LMP Calculation Traditional OPF Models Lossless DCOPF ACOPF DCOPF with Loss A Proposed FND-based DCOPF Model Summary 13

14 We are getting here Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 14

15 Recall: Power Flow Calculation AC power flow Nonlinear VV i j,sinθ, cosθ ij ij P = P i Q V i = Q = V θ = 0 i i Gi Gi P given i known Di Q = V Di = V n i j= 1 V ( G V ( G ( i PVbusor referencebus ) ( i referencebus ) n i j= 1 j j unknowns ij cosθ + B ij ij sinθ B ij ij sinθ ) ij ij cosθ ) ij ( i PQbusor PVbus) ( i PQbus) DC power flow Linear Assumptions: V=1, θis small, ignore line resistance and shunt capacitor F ij = θ θ i x P = B θ known ij unknowns j F = GSF P NOTE: GSF denotes one matrix, not multiplication of matrices 15

16 Traditional OPF Models: (1) Lossless DCOPF 16

17 Traditional OPF Models: (2) ACOPF 17

18 Traditional OPF Models: (3) DCOPF with Loss Solving technique: Iterative LP algorithm 18

19 LMP Calculation ACOPF DCOPF with Loss Lossless DCOPF 19

20 Issues with DCOPF with Loss Model Mismatch at the reference bus Loss is not considered in power flow equations 20

21 We are getting here Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 21

22 New DCOPF with Loss Model: FND-Based DCOPF Fictitious Nodal Demand (FND) 22

23 FND-Based DCOPF Model Solving technique: Iterative LP algorithm 23

24 FND-Based DCOPF Model Total loss is distributed into each individual line And represented by FND at each bus 24

25 Benchmarking FND-DCOPF and Lossless DCOPF with ACOPF 25

26 LMP Sensitivity Study using FND-DCOPF Model LMP Sensitivity ($/MWh 2 ) w.r.t. Load at Bus B (MWh) Bus A Bus B Bus C Bus D Bus E No loss case With change of marginal units 26 With loss case

27 Some Real-world Data From NYISO Thanks to Zhiqiang Jin Real-time Zonal Price at zone CAPITL 10:00am~11:00am D1 D2 D3 07/22/ /24/

28 Summary of Research Stage 1 Prove doubled losses incurred by marginal loss factor Propose FND-based DCOPF model Eliminate the nodal mismatch at reference bus Distribute total losses into each individual line Fit into the LMP decomposition framework Benchmark FND-based DCOPF and Lossless DCOPF with ACOPF FND-based DCOPF outperforms Lossless DCOPF Rule of thumb: difference in marginal unit set LMP sensitivity Study Conclusions LMP step change phenomenon (at CLLs) Related to change of marginal unit set 28

29 Stage 2: Congestion and Price Prediction under Load Variation Problem 40 LMP at Five Buses 35 LMP ($/MWh) Load (MW) A B C D E How to predict the Critical Load Levels (CLLs)? Repetitive OPF-run Approach: Brute-force Desire: more efficient and effective approach 29

30 Outline of Research Stage 2 Solution for Lossless DCOPF framework Simplex-like Method Solution for ACOPF framework Quadratic Interpolation Method Solution for FND-DCOPF framework Variable Substitution Method Summary 30

31 We are getting here Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 31

32 Research Stage 2: Congestion and Price Prediction For lossless DCOPF framework Study For Lossless DCOPF Model Observation from Previous LMP simulation results At a CLL: change of marginal units (and congestion) Between two adjacent CLLs: LMP constants Conduct the study in 3 steps Study the phenomenon around current operating point Locate the next and previous CLL Find what happens beyond the CLL 32

33 Solution Step 1 Revisit Lossless DCOPF Model Changing 33

34 Solution Step 1 Mathematical Interpretation Case 1 O O O Case 2 O I Simplex Method Proposed Simplex-like Method 34

35 Solution Step 1 Classification Problem Interpretation from Math perspective How will optimal solution and objective function change with parameter D Parametric Analysis Classification of Generators Marginal Unit (MU): Variable generation Non-marginal Unit (NU) : Fixed generation Classification of Transmission Lines Congested Line (CL) Un-congested Line (UL) Sets of MUs, NUs, CLs, ULs may change at CLLs 35

36 Solution Step 1 Reformulate the Model Energy balance constraints Basic Variables Transmission line constraints Add slack variables Non-basic Variables For Congested Lines For Un-congested Lines Non-basic Variables Basic Variables 36

37 Solution Step 1 Matrix formulation Non-basic Variables Basic Variables Changing Parameter Where N MG N MG matrix since 37

38 Solution Step 1 Matrix formulation (cont ) Objective function Function of D, NG, CL. No MG is present! 38

39 Solution Step 1 Define Load Variation Pattern Load Variation Participation Factor: Example: f i defined as constants Example: f i =1, f j =0,j i N Independent Parameters D One Parameter D 39

40 Rewrite the equations in difference forms in terms of D Σ Sensitivities for basic variables: Constant vectors! Not only for current operating point, but also for the load range within two adjacent CLLs 40

41 Calculate LMP Sensitivities w.r.t. non-basic variables LMP is a constant value! 41

42 Solution Step 2: Locate Next and Previous Critical Load Level What happens at Critical load levels? New binding constraint Either: Marginal unit becomes non-marginal Or: Un-congested line becomes congested D 2 D 3 42 D 1

43 Besides new binding constraint New unbinding constraint Either: Non-marginal unit becomes marginal Or: Congested line becomes un-congested Could NOT be identified using sensitivity information as was used in identifying new binding constraint! Paired with the new binding constraint Case 1: Marginal unit becomes non-marginal Case 2: Un-congested line becomes congested 43

44 Case 1: Marginal Unit Becomes Non-marginal This basic variable becomes non-basic variable l th marginal unit becomes non-marginal MG l =0 One of the non-basic variables has to become basic variable If the j th non-marginal unit will become marginal If the k th congested line will become un-congested Sensitivity of nonbasic variables w.r.t. load 44

45 Case 1: Marginal Unit Becomes Non-marginal (cont ) Expected incremental cost vector Identify the unbinding constraint choose the smallest positive one in load growth case (or, largest negative one in load drop case), and the corresponding j (or k) implies the new marginal unit (or new un-congested line) 45

46 Case 2: Un-congested Line Becomes Congested This basic variable becomes non-basic variable r th un-congested line becomes congested UL r =0 One of the non-basic variables has to become basic variable If the j th non-marginal unit will become marginal If the k th congested line will become un-congested Sensitivity of nonbasic variables w.r.t. load 46

47 Case 2: Un-congested Line Becomes Congested (cont ) Expected incremental cost vector Identify the unbinding constraint choose the smallest positive one in load growth case (or, largest negative one in load drop case), and the corresponding j (or k) implies the new marginal unit (or new un-congested line) 47

48 Solution Step 3: Find what happens beyond CLL Update Marginal Unit Set and Congestion Line Set Set the next critical load level as the current operating point Repeat Step 1 and Step 2 48

49 We could also do Update MG sensitivity directly Update LMP directly 49

50 Case Study PJM 5 bus system 50

51 Case Study (Cont ) LMP 51

52 Case Study (Cont ) Marginal Units and Congestions versus Load 52

53 Performance Speedup Binary Search Average-case performance: ( ( )) O log 2 n 53

54 We are getting here Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 54

55 Research Stage 2: Congestion and Price Prediction For ACOPF framework Study for ACOPF Model Overview of the Interpolation Method 55

56 Polynomial Curve-fitting Marginal Unit Generation (MW) Need OPF solutions at m load levels Line Flow (MVA) 56

57 Quadratic Curve-fitting Results For a Modified IEEE 30-bus System Marginal unit at Bus 22 Line flow through line Observations: Quadratic curvefitting is sufficiently accurate. 57

58 Prediction of CLLs Generator output bounds Line thermal limit 58

59 Quadratic Interpolation Method Seek three load levels Need OPF solutions at THREE load levels Three-point Quadratic Interpolation 3 by 3 linear system 59

60 Case Study Results For PJM 5-bus System Very close in values 60

61 We are getting here Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 61

62 62 Study for FND-based DCOPF Model Overview of the Variable Substitution Method c b a G + + D D numerically 2 0 ), ( 0 ), ( 0 ), ( 2 1 = = = D f D f D f n G G G M c b a G + + = D D assume 2 0 ),,, ( 0 ),,, ( 0 ),,, ( 2 1 = = = D g D g D g m c b a c b a c b a M Research Stage 2: Congestion and Price Prediction For FND-based DCOPF framework

63 Characteristic Constraints Equalities of Binding Constraints (, ) = 0 f e MG D max (, D ) = Fk k Bc f, k MG l, 63

64 Substitution of Variables Make assumptions on MG j (, ) = 0 f e MG D f (, D ) l, k MG = F max k, k B c h l, k ( { a, a, a, j MG}, = 2, j F 1, j max k, k 0, j Bc D Σ ) 64

65 New Set of Nonlinear Equations (M+1) eqns (M+1) eqns (M+1) eqns Solve above equations numerically for Use the coefficients to predict CLLs Need OPF solutions at only ONE load level (3M+3) unknowns 65

66 Case Study Results For PJM 5-bus System Marginal unit Sundance Marginal unit Brighton 66

67 Case Study Results For PJM 5-bus System (Cont ) close in values, yet not as close as those by the interpolation method 67

68 Case Study Results For PJM 5-bus System (Cont ) 68

69 Summary of Research Stage 2 For Lossless DCOPF Model: Simplex-like Method Systematic and analytical approach Very efficient in computation. Only the initial DCOPF solution is needed, no more DCOPF run! Within two adjacent CLLs: MG sensitivity are shown to be constants; LMPs are constants For ACOPF Model: Interpolation Method Numeric approach; accurate results Computational efficient. Need OPF solutions at three load levels. Within two adjacent CLLs: MG exhibits quadratic pattern. For FND-based DCOPF Model: Variable Substitution Method Analytical approach; numeric solution; less accurate Computational efficient. Need only the initial OPF solution. 69

70 Stage 3: Load Uncertainty Impact Study Problem 40 LMP at Five Buses 35 LMP ($/MWh) Load (MW) A B C D E How to quantify the risk associated with a forecasted LMP? 70

71 Outline of Research Stage 3 Study for Lossless DCOPF framework Study for ACOPF framework Study for FND-DCOPF framework Summary 71

72 We are getting here Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 72

73 Research Stage 3: Load Uncertainty Impact Study For lossless DCOPF model Probabilistic Model of Load Load at hour t: D t Assume: Normal distribution 73

74 Model for LMP versus Load Curve 74

75 Look at two curves at a time 75

76 Probability Mass Function of Probabilistic LMP LMP t is a discrete random variable Named Probabilistic LMP 76

77 Alignment Probability (AP) Probability that the deterministically forecasted LMP and the actual LMP are the same Deterministically forecasted LMP Alignment Probability with α% price tolerance 77

78 Expected Value of Probabilistic LMP Finite bounds 78

79 Cast Study Results PMF of LMP t 40 LMP at Five Buses 35 LMP ($/MWh) Load (MW) A B C D E 730MW 900MW 79

80 Alignment Probability Curve 80

81 Expected Value of Probabilistic LMP versus Forecasted Load 40 LMP at Five Buses 35 LMP ($/MWh) Load (MW) A B C D E Deterministic LMP Curve 81

82 Impact of Load Forecasting Accuracy Forecasted Load=730MW Bus B 82

83 We are getting here Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 83

84 Research Stage 3: Load Uncertainty Impact Study Study for ACOPF framework Different from the study for lossless DCOPF framework Effect of losses For ACOPF framework LMP versus Load curve is piece-wise linear Not constant value 84

85 Probabilistic LMP LMP t is a piece-wise continuous random variable 85

86 CDF and PDF of LMP t Cumulative Density Function (CDF) Probability Density Function (PDF) differentiable almost everywhere 86

87 Alignment Probability (AP) Probability that the deterministically forecasted LMP and the actual LMP are the same Deterministically forecasted LMP Alignment Probability with α% price tolerance 87

88 Expected Value of Probabilistic LMP Finite lower-bound and upper-bound 88

89 Cast Study Results CDF of LMP t 89

90 Cast Study Results PDF of LMP t 90

91 Cast Study Results PMF of LMP t 900MW 730MW 91

92 Alignment Probability Curve 92

93 Expected Value of Probabilistic LMP versus Forecasted Load Deterministic LMP Curve 93

94 Impact of Load Forecasting Accuracy Forecasted Load=730MW Bus B 94

95 We are getting here Work done after the proposal Research Stage 1 Research Stage 2 Research Stage 3 95

96 Research Stage 3: Load Uncertainty Impact Study Study for FND-based DCOPF framework Overview LMP versus Load curve is piece-wise linear The entire methodology for ACOPF framework fits into FND-based DCOPF framework Case Study For FND-based DCOPF framework Similar observations as those for ACOPF framework 96

97 LMP Curve and its Piece-wise Linear Approximation Bus B 97

98 Alignment Probability Curve 98

99 Impact of Load Forecasting Accuracy Forecasted Load=730MW Bus B 99

100 Summary of Research Stage 3 For Lossless DCOPF Framework Probabilistic LMP concept is introduced A systematic approach is proposed to quantify the risk in deterministically forecasted LMP due to load forecasting error Sensitivity of expected value of probabilistic LMP is proved to be bounded step change does not appear For ACOPF Framework Effect of losses brings in difficulty Probabilistic LMP is a piece-wise continuous random variable CDF and PDF are formulated Expected value of probabilistic LMP is derived. Its sensitivity is proved to be bounded For FND-based DCOPF Framework The same methodology for ACOPF framework applies 100

101 Final Conclusions 101

102 Publications During Ph.D. Program---Journal Papers Published [1]. Rui Bo and Fangxing Li, "Probabilistic LMP Forecasting Considering Load Uncertainty," IEEE Transactions on Power Systems, vol. 24, no. 3, pp , August [2]. Fangxing Li and Rui Bo, "Congestion and Price Prediction under Load Variation," IEEE Transactions on Power Systems, vol. 24, no. 2, pp , May [3]. Fangxing Li and Rui Bo, "DCOPF-based LMP Calculation: Algorithms, Comparison with ACOPF, and Sensitivity," IEEE Transactions on Power Systems, vol. 22, no. 4, pp , November Under Review [1]. Rui Bo, Fangxing Li and Kevin Tomsovic, "Prediction of Critical Load Levels Using Quadratic Interpolation" submitted to IEEE Transactions on Power Systems. 102

103 Publications During Ph.D. Program---Journal Papers (cont ) Under Preparation [1]. Rui Bo and Fangxing Li, An Open Source Educational Tool for Market Simulation," to be submitted to IEEE Transactions on Power Systems. [2]. Rui Bo and Fangxing Li, Critical Load Levels for DCOPF with losses modeled," to be submitted to IEEE Transactions on Power Systems. 103

104 Publications During Ph.D. Program---Conference Papers Conference Papers [1]. Rui Bo, Fangxing Li, "Impact of Load Forecast Uncertainty on LMP," Proceedings of 2009 IEEE PES Power Systems Conference & Exposition, Seattle, Washington, USA, [2]. Rui Bo, Fangxing Li, "Power Flow Studies Using Principle Component Analysis," Proceedings of the North American Power Symposium 2008, Calgary, Canada, [3]. Rui Bo, Fangxing Li and Chaoming Wang, "Congestion Prediction for ACOPF Framework Using Quadratic Interpolation," Proceedings of the IEEE Power Engineering Society General Meeting 2008, Pittsburgh, USA, [4]. Rui Bo and Fangxing Li, "Sensitivity of LMP Using an Iterative DCOPF Model," Proceeding of the 3rd IEEE International Conference on Deregulation, Restructuring, and Power Technology (DRPT2008), Nanjing, China, [5]. Rui Bo and Fangxing Li, "Comparison of LMP Simulation Using Two DCOPF Algorithms and the ACOPF Algorithm," Proceeding of the 3rd IEEE International Conference on Deregulation, Restructuring, and Power Technology (DRPT2008), Nanjing, China, [6]. Fangxing Li, Rui Bo, Wenjuan Zhang, "Comparison of Different LMP Calculations in Power Market Simulation," Proceeding of 2006 International 104 Conference on Power System Technology, Chongqing, China, 2006.

105 Awards Received During Ph.D. Program Awards 2nd Prize Award at Student Poster Contest at PSCE09, Seattle, WA, March UT Citation Award in Extraordinary Professional Promise, April

106 Some Relevant Works During Ph.D. Program Developed a software package --- PMSP A MATLAB TM -based Power Market Simulation Package (PMSP) Design and implementation from 01/2006 ~ present Version is to be published Open Source PMSP ~170 source-code files written in MATLAB TM language and organized in 7 directories ~1.7 megabytes in size 6 modules Generic program: no hard-coding Highly customizable 106

107 Structure of PMSP 107

108 GUI of PMSP 108

109 Thank you! Questions? 109

110 Q&A Hourly load w.r.t. the peak load (IEEE-RTS-1996 spring/fall weekday) Price 100 Forecasted Load= D 90 Load in Percentage Forecasted Load= D Hour At Time t 110

111 Q&A (Prob LMP for IEEE 30-bus System) 111

112 Q&A (Prob LMP for IEEE 118- bus System) 112