MEDIUM-TERM HYDROPOWER SCHEDULING BY STOCHASTIC DUAL DYNAMIC INTEGER PROGRAMMING: A CASE STUDY

Transcription

1 MEDIUM-TERM HYDROPOWER SCHEDULING BY STOCHASTIC DUAL DYNAMIC INTEGER PROGRAMMING: A CASE STUDY Martin Hjelmeland NTNU Norwegian University of Science and Technology, Trondheim, Norway. martin.hjelmeland@ntnu.no Hydro Power Scheduling Workshop 2018, Stavanger, Norway

2 Background

3 Lysebotn-2 Research project initiated from the Hydropower Scheduling Workshop in Medium-term hydropower scheduling in multiple markets Affects on Water Values 3

4 The Medium-Term Hydropower Scheduling (MTHS) Problem

5 Medium-Term Hydropower Scheduling Challenges Uncertainty of inflow and future energy/capacity prices Level of details? Individual generators Unit commitment Water head Other system constraints 5

6 Medium-Term Hydropower Scheduling Objective: Maximize income from energy and capacity reserves Main scope of my PhD thesis Capacity reserves/spinning reserves Primary Secondary 6

7 Nonconvex Generation Function Generation function for each hydropower station Head dependencies? P t q t q Min q Max 7

8 Stochastic Dual Dynamic integer Programming (SDDiP) J. Zou, X. A. Sun and S. Ahmed, «Stochastic dual dynamic integer programming», Mathematical Programming 2018.

9 One-Stage Dispatch Problem (Weekly Decision Problem) «Generic» Stochastic Dual Dynamic Programming (SDDP) φ t (x t ) Stochastic Dual Dynamic Integer Programming (SDDiP) ψ t (x t ) x x 9

10 Cut Families 1 Benders Cuts Traditional Used in generic SDDP 2 Integer Optimality Cuts 3 Lagrangian Cuts 3 Strengthened Benders Cuts 10

11 Cut Families Benders Cuts Integer Optimality Cuts Lagrangian Cuts Evaluates that exact solution Large coefficients Increased CPU time Not applicable in practice EFP Integer optimality cut 3 Strengthened Benders Cuts True EFP x t 11

12 Cut Families Benders Cuts Integer Optimality Cuts Lagrangian Cuts Lagrangian relaxation is normally used for simplifying difficult constraints Here the problem is to find the Lagrangian multipliers that are used to generate tight cuts Relaxing time-linking constraint Computational demanding Approx 30 MIP problems for subproblem 3 Strengthened Benders Cuts 12

13 Cut Families 1 Benders Cuts L π + πx t 1 2 Integer Optimality Cuts 3 Lagrangian Cuts 3 Strengthened Benders Cuts Q i ( ) π LP π Opt π 13

14 Cut Families 1 Benders Cuts Parallel to Benders Cuts Requires to solve one additional MIP problem 2 Integer Optimality Cuts EFP C1 φ t x t 3 Lagrangian Cuts A C2 φ t x t 3 Strengthened Benders Cuts x t 14

15 Cut Families 1 Benders Cuts Can be computed without having binary state variables Does not guarantee convergence but might provide a satisfying solution. 2 Integer Optimality Cuts 3 Lagrangian Cuts 3 Strengthened Benders Cuts 15

16 16 Implications of SB on the Water Values

17 Case Study M. N. Hjelmeland, J. Zou, A. Helseth and S. Ahmed, "Nonconvex Medium-Term Hydropower Scheduling by Stochastic Dual Dynamic Integer Programming," IEEE Transactions on Sustainable Energy, 2018.

18 Case Studies Sales of both energy and reserve capacity 2 years Weekly decision stages Uncertainty of inflow and energy price VAR-1 model McCormick Envelopes for bilinear terms Nonconvex generation function Only dependent on discharge Minimum generation limits Spillage Reservoir operation 7% 15 MW 24% 91% 400 MW 51% Reservoir 1 Power Station 1 Reservoir 2 Reservoir 3 25% 2% Power Station 2 18

19 Results

20 20 Validation of SDDiP with a Test Case

21 21 Lagrangian cuts too comprehensible.

22 Adding Head Dependencies

23 Head Function Head(Volume) Head(Cross section) 23

24 Head Dependencies Multiple turbines P t Adding to the nonconvexities Splitting the generation function into convex segments Efficient way of doing this? q Min q Max q t h t net h net,max h net,min q t q Min q Max 24

25 Visualization of the EFP Function

26 Visualize the EFP Function Simple Idea Solve the extensive form of the problem Iterate over different starting reservoirs Add different nonlinearities and observe Inflow Energy Price 26

27 Visualize the EFP Function Energy & Capacity Reserves Energy Head Function 27

28 Case Study 2.0

29 Case Studies Sales of both energy and reserve capacity A. Downward, years Weekly decision stages Uncertainty of inflow, energy & capacity price VAR-1 model McCormick Envelopes for bilinear terms Nonconvex generation function Dependent on discharge and head Minimum generation limits Spillage Reservoir operation 7% 15 MW 24% 91% 400 MW 51% Reservoir 1 Power Station 1 Reservoir 2 Reservoir 3 25% 2% Power Station 2 29

30 Convergence B vs SB Significantly improved Gap Slightly improvement of policy 30

31 Water Values Assuming all but one reservoir fixed Water values for a given week using either B or SB Generally lower water values by SB Does not imply lower operating states Need more testing and validation 31

32 Thank you for your attention!

33 References J. Zou, A. Sun and S. Ahmed, "Stochastic dual dynamic integer programming," Mathematical Programming, Mar ICSP 2016 Presentation on SDDiP by Shabbir Ahmed: R. Baucke, A. Downward, G. Zakeri, A deterministic algorithm for solving stochastic minimax dynamic programmes, Optimization-online. URL: A. Downward, O. Dowson, R. Baucke, Stochastic dual dynamic programming with stagewise dependent objective uncertainty, Optimization-online. URL: M. N. Hjelmeland, A. Helseth, M. Korpås, Impact of modelling details on the generation function for a norwegian hydropower producer, Journal of Physics: Conference Series 1042 (1) (2018) doi: / /1042/1/