Bilevel Programming-Based Unit Commitment for Locational Marginal Price Computation

Transcription

1 Bilevel Programming-Based Unit Commitment for Locational Marginal Price Computation Presentation at 50 th North American Power Symposium Abdullah Alassaf Department of Electrical Engineering University of South Florida University of Hail Dr. Lingling Fan Department of Electrical Engineering University of South Florida Septemper 10, 2018

2 Outline Introduction Model Formulation The Conventional Unit Commitment Model The Bilevel Unit Commitment Model The Upper-Level Problem The Lower-Level Problem The Single-Level Equivalent Problem Case Studies 5-Bus PJM System Case Study IEEE 30-Bus System Case Study Conclusion

3 Outline Introduction Model Formulation The Conventional Unit Commitment Model The Bilevel Unit Commitment Model The Upper-Level Problem The Lower-Level Problem The Single-Level Equivalent Problem Case Studies 5-Bus PJM System Case Study IEEE 30-Bus System Case Study Conclusion

4 Introduction In the US, locational marginal pricing system is applied in all ISOs, independent system operators [B. Eldridge, 2017]. Locational marginal price (LMP) is the cost of the next increased demand unit at a bus. Generation company bidding is based on the LMP. The LMP is the dual variable of the power balance equation. In the day-ahead market, generation companies determine on/off status of each unit for the next 24 hours. Unit status causes nonconvexity in formulating unit commitment participating in the day-ahead market.

5 Introduction Issues Commercial solvers cannot reach the problem dual variables. Traditionally, the LMP is obtained by solving two problems. The first one is unit commitment. The second problem is DCOPF, where units statuses are known. Solution We propose a bilevel model that solves the unit commitment problem and attains the LMP and the other dual variables. We present solving the bilevel problem using primal-dual relationship. The solution of the bilevel model and the conventional model exactly match.

6 Outline Introduction Model Formulation The Conventional Unit Commitment Model The Bilevel Unit Commitment Model The Upper-Level Problem The Lower-Level Problem The Single-Level Equivalent Problem Case Studies 5-Bus PJM System Case Study IEEE 30-Bus System Case Study Conclusion

7 Model Formulation The comparison diagram of the two models The Conventional Method The Bilevel Method The Mixed-Integer Unit Commitment Problem The Bilevel Unit Commitment Problem ON/OFF Unit Variables DCOPF Problem The Locational Marginal Price The Locational Marginal Price

8 The Conventional Unit Commitment Model min p f l,pg j,δn j J K g j pg j + j J c u j (1a) subject to c u j Kj u (v j v j0 ); j J (1b) c u j 0; j J (1c) v j {0, 1}; j J (1d) p g j + p f l p f l = Pn; d n N (1e) j J l d(l)=n l o(l)=n p f l = 1 x l (δ o(l) δ d(l) ); l L P f l p f l P f l ; l L P g j v j p g j P g j v j ; j J (1f) (1g) (1h)

9 The Bilevel Unit Commitment Model

10 The Bilevel Unit Commitment Model The Upper-Level Problem max v j n N P d nλ n j J c u j (2a) subject to c u j K u j (v j v j0 ); j J (2b) c u j 0; j J v j {0, 1}; j J (2c) (2d)

11 The Bilevel Unit Commitment Model The Lower-Level Problem min p f l,pg j,δn K g j pg j j J (3a) subject to p f l = 1 x l (δ o(l) δ d(l) ); (ν l ); l L (3b) p g j + j J P f l p f l P f l ; (φ min l, φ max l ); l L (3c) P g j v j p g j P g j vj ; (θj min, θj max ); j J (3d) l d(l)=n p f l l o(l)=n p f l = P d n; (λ n ); n N (3e)

12 The Bilevel Unit Commitment Model The structure of the bilevel model Upper-Level Problem (2) Maximize The Production Profit. Determine: The Optimal Unit Status. v j λ n Lower-Level Problem (3) Minimize The Operation Cost. Determine: The Optimal Dispatch.

13 The Bilevel Unit Commitment Model The lower-level dual problem max λ,ν,φ,θ Z D = n N P nλ d n l L P f l (φ max l subject to + φ min l K g j λ n + θj max λ o(l)=n λ d(l)=n ν l + φ max l o(l)=n 1 x l ν l l ) + j J v j (θmin P g j θmax l d(l)=n j j P g j ) (4) θ min j = 0; j J (5a) φ min l = 0; l L (5b) 1 ν l = 0; l L x l (5c) θj max, θj min 0; j J (5d) φ max l, φ min l 0; l L (5e)

14 The Bilevel Unit Commitment Model The single-level equivalent j J P d nλ n + j J max p f l,pg j,δn,λn,ν l,φ l,θ j n N P d nλ n j J c u j (6a) subject to Constraints (2b) (2d) (6b) (P g j θmin j v j P g j θj max v j ) l L Constraints (3b) (3e) (6c) Constraints (5a) (5e) (6d) P f l (φ max l + φ min l ) = K g j pg j (6e) n N

15 The Bilevel Unit Commitment Model Constraint (6e) is linearized j J P d nλ n + j J(P g j b j P g j a j ) l L P f l (φ max l + φ min l ) = n N K g j pg j (7a) 0 a j θ max j v j (7b) 0 θ max j a j θ max j (1 v j ) (7c) 0 b j θ min j v j (7d) 0 θ min j b j θ min j (1 v j ) (7e)

16 The Bilevel Unit Commitment Model The linearized single-level equivalent max p f l,pg j,δn,λn,ν l,φ l,θ j n N P d nλ n j J c u j (8a) subject to Constraints (2b) (2d) (8b) Constraints (3b) (3e) (8c) Constraints (5a) (5e) (8d) Constraints (7a) (7e) (8e)

17 Outline Introduction Model Formulation The Conventional Unit Commitment Model The Bilevel Unit Commitment Model The Upper-Level Problem The Lower-Level Problem The Single-Level Equivalent Problem Case Studies 5-Bus PJM System Case Study IEEE 30-Bus System Case Study Conclusion

18 Case Studies The model (8) has been tested on different scale systems. The tested cases are nesta_case5_pjm and nesta_case30_ieee, taken from [C. Coffrin, 2014]. The model has been implemented in Matlab with CVX using solver Gurobi 7.51.

19 5-Bus PJM System Case Study 10 $/MW 0 p g MW 5 Limit = 240 MW 40 $/MW 0 p 4 g 200 MW p 4 d = 400 MW 14 $/MW 0 p g MW 1 15 $/MW 0 p g MW Limit = 400 MW 2 p 2 d = 300 MW 3 30 $/MW 0 p 3 g 520 MW p 3 d = 300 MW

20 The System Normal Operation Bus LMP ($/MWh) Total Revenue = $32,892 Table: 5-Bus System LMPs of The Buses and The Revenue ON/OFF Output (MW) θ max θ min G G G G G Total Generation Cost = $17,480 Table: 5-Bus System Generator Primal and Dual Components

21 The System Normal Operation p f 12 p f 14 p f 15 p f 23 p f 34 p f 45 Flow (MW) φ max ($/MWh) φ min ($/MWh) Table: 5-Bus System Primal and Dual Components of The Lines

22 IEEE 30-Bus System Case Study The system contains 6 generator units and 41 transmission lines. The system is tested under normal operation. The system is examined under congestion operation. The maximum capacity limit of line 1-3 is set 72.5 MW.

23 IEEE 30-Bus System Case Study The LMPs of the system buses in both normal and congested operations.

24 IEEE 30-Bus System Case Study ON/OFF Output (MW) θ max θ min G G Total Generation Cost = $ Table: Generator Primal and Dual Components ON/OFF Output (MW) θ max θ min G G Total Generation Cost = $ Table: Generator Primal and Dual Components (Congested)

25 Outline Introduction Model Formulation The Conventional Unit Commitment Model The Bilevel Unit Commitment Model The Upper-Level Problem The Lower-Level Problem The Single-Level Equivalent Problem Case Studies 5-Bus PJM System Case Study IEEE 30-Bus System Case Study Conclusion

26 Conclusion The nonconvexity of unit commitment problem causes difficulty of achieving the problem dual variables that play an important rule in market participation. We develop a bilevel model that consists of two level problems, namely, the upper-level and the lower-level. The upper-level has binary decision variables. The lower-level has only continuous decision variables. The bilevel problems are transformed to a single-level problem and solved efficiently using Gurobi.