A Multi-Stage Stochastic Programming Model for Managing Risk-Optimal Electricity Portfolios. Stochastic Programming and Electricity Risk Management

Transcription

1 A Multi-Stage Stochastic Programming Model for Managing Risk-Optimal Electricity Portfolios SLIDE 1

2 Outline Multi-stage stochastic programming modeling Setting - Electricity portfolio management Electricity Spot & Demand Process Specifying underlying uncertainty - Scenario tree generation Model Description & Application Scenarios Numerical results Supply contracts (Swing option) Real option (Power Plant) and Risk Management Conclusions SLIDE 2

3 Multistage modeling - Postmodernism Reject objective truth and global cultural narrative: complete decoupling of the multi-stage modeling process and the solution approach! 1 Optimization under uncertainty modeling approach: expectation-based multi-stage stochastic programming, worst-case robust optimization,... 2 Underlying solution technique: scenario tree-based deterministic equivalent formulation, primal/dual linear decision rules,... SLIDE 3

4 Multistage modeling - Old School Complete decoupling of scenario tree modeling and handling from the decision problem modeling process: Decision problem layer. Decision problem modeler only concerned with actions/decisions at stages. In case of trees and deterministic equivalent formulations: explicit decoupling of modeling and (scenario) tree handling. Scenario tree layer. Creating a scenario tree which optimally represents the subjective beliefs of the decision taker at each node. Data layer. Data structures, how to (memory-)optimally store large trees, and access ancestor tree nodes fast,... SLIDE 4

5 Multistage modeling - Old School (1) Modeler View (Stage) (2) Stochastic View (Tree) (3) Data View (Node) Root Stage Recourse Stage Terminal Stage SLIDE 5

6 Multistage modeling - New Age Complete decoupling of scenario tree modeling and handling from the decision problem modeling process: Decision problem layer. Decision problem modeler only concerned with actions/decisions at stages. In case of linear decision rules: 1 A. Georghiou, W. Wiesemann, and D. Kuhn. Generalized Decision Rule Approximations for Stochastic Programming via Liftings. Optimization Online D. Kuhn, W. Wiesemann, and A. Georghiou. Primal and Dual Linear Decision Rules in Stochastic and Robust Optimization. Mathematical Programming, online first SLIDE 6

7 Multistage modeling - New Age (1) Modeler View (Stage) (2) Stochastic View (Upper/Lower Approximation) Root Stage Recourse Stage Terminal Stage SLIDE 7

8 Multi-stage modeling Complete decoupling of scenario tree modeling and handling from the decision problem modeling process: Decision problem layer. Decision problem modeler only concerned with actions/decisions at stages. In this talk, we will conduct an old-school approach using multi-stage scenario trees and use deterministic equivalent formulations to solve the problem, with a focus on communicability of models (electrical engineers and optimization wizards), and modular modeling structure (using an abstract model generator). SLIDE 8

9 Electricity Portfolio Optimization Classical bi-critertia risk-return portfolio optimization problems (minimize risk, maximize return) in a multi-stage stochastic setting. Example: Large consumer satisfies demand by: 1 buying at the uncertain spot market on day-ahead basis, 2 buying energy futures (base or peak), 3 buying contracts for delivery of energy in advance, 4 (optional) own production with some power plant. Stochastic optimization: 1 Uncertainty: spot price and demand. 2 Decision: composition of the electricity portfolio. SLIDE 9

10 Electricity Portfolio Optimization Daily (day-ahead hourly) electricity portfolio changes massively multi-stage! Trade-off between level of model detail (variables and constraints) and underlying uncertainty (dimension of underlying scenario tree), and solvability of the problem. Design of uncertainty: Instead of modeling price and demand as uni-variate time series (i.e stages per year), we define one stage per day (24 dimensional time series with 365 stages per year). Hour-Blocks: To simplify the model we group the uncertainty in 6 4-hour blocks, i.e. we are dealing with a 6-dimensional instead of a 24-dimensional process. SLIDE 10

11 Electricity Portfolio Optimization 1 Model the uncertain spot price process. 2 Generate possible scenarios for the spot price movement and the demand by simulating disturbance in the respective models. 3 Construct a scenario tree from the fan of simulated trajectories. 4 Choose an appropriate risk measure. 5 Model decision problem as multi-stage stochastic optimization program. 6 Solve the optimization problem. Goals: Semi-automatization (workflow orchestration), and (multi-stage) modeling simplification/modularization. SLIDE 11

12 Spot & Demand Process The hourly spot prices at the EEX are modeled via linear regression. The main explainationary factors are 1 weekday, hour of the day, season and all combinations thereof 2 temperature 3 future prices The above factors model the average behavior of the price process. The peaks are captured by fitting a stable distribution to the residuals. The model is calibrated with data ranging from Demand is modeled in a similar setting (fitted with one year of data). SLIDE 12

13 Simulations Both models may now be used to simulate possible future price and demand scenarios. Since we assume independence in the error terms of the two models we can pairwise merge the scenarios for demand and price developement. The resulting scenarios are merged into a tree structure. SLIDE 13

14 Model: Portfolio The energy portfolio in every hour block h and in every stage t consists of 1 contracted volume c t,h, 2 energy out of future contracts f t,h, 3 energy bought on the spot market s t,h and 4 energy produced with a power plant p t,h (optional). Note: We do not use (scenario tree) node-based formulations, which greatly enhances the readability. The conversion to a node-based formulation is done semi-automatically. SLIDE 14

15 Model: Objective Function The expenditure (over the whole time horizon) is given by e T = t T,h H e t,h where e t,s is the expenditure at stage t in hour block h given by e t,h = c t,h C + f t,h F t,h + s t,h S t,h + p t,h P with no recourse, where C, F h, S t,h, P are the prices of one MWh of energy from the supply contract, the future contract, the spot market and the power plant respectively. Our aim is to minimize e T + κavar α (e T ) i.e. to perform a bi-criteria optimization whichs keeps expected expenditure and its AV@R (risk) low. SLIDE 15

16 Model: Constraints Demand d t,h (stochastic variable) has to be met at every stage t and every hour h c t,h + f t,h + s t,h + p t,h d t,h, t T, h H Supply contract. Amount bought every hour is restrained by the constant γ u and γ h and the overall contracted volume C h for that hour block h (over the whole planning horizon) in the following way γ l C h c t,h γ u C h, t T, h H Supply contract. amount of energy bought in every scenario has to be within a certain range around the contracted value, i.e. δ l C h c t,h δ u C h t T,h H SLIDE 16

17 Model: Stylized Power Plant production p t,h is limited by a maximum amout π, i.e. p t,h π, t T, h H production of the power plant cannot change more than β MWh per hour block, i.e. p t,h p t,h 1 β, h = 2,..., 5, t T and p t,1 p t 1,6 β, t = 2,..., T. SLIDE 17

18 Model: Putting it all together minimize e T + AVaR α (e T ) subject to d t,h c t,h + f t,h + s t,h + p t,h c t,h C + f t,h F t,h + s t,h S t,h + p t,h P = e t,h γ l C h c t,h γ u C h δ l C h T,H c t,h δ u C h p t,h π p t,h p t,h 1 β p t,1 p t 1,6 β T,H e t,h = e T In the above program all variables are non-negative. SLIDE 18

19 Model: Putting it all together minimize s S P sc s + κ(q α + P s(n) z n n N (T) 1 α ) subject to d n,h c n,h + f n,h + s n,h + p n,h c n,h C + f n,h F n,h + s n,h S n,h + p n,h P = e n,h γ l C h c n,h γ u C h δ l C h n N (s),h=1,...,6 c n,h δ u C h p n,h π p n,h p n,h 1 β p n,1 p pred(n),6 β n N (s),h=1,...,6 e n,h = e s c s q α z n In the above program all variables are non-negative. SLIDE 19

20 Numerical Results The basic model defined above will be adapted to two application examples: pricing supply contracts (swing option), real options (power plant) and risk management. Even in this basic model, the number of possible applications and parameter studies is huge - disregarding different approaches to modeling underlying uncertainty (different scenario trees), i.e. only one specific tree was used: 184 stages, nodes, 426 scenarios, 12 dimensions (6 spot, 6 demand). SLIDE 20

21 Numerical Results - Implementation Workflow, Simulation/Estimation: MatLab R2007a (7.4) Scenario Generator: MatLab Model Generator: Python (Multi-stage) Modeling: AMPL Solver: MOSEK 4 (LP IPM), parameter tweaked Intel(R) Pentium(R) 4 CPU 3.00GHz 4 GB RAM Debian Stable (Etch) SLIDE 21

22 Numerical Results - Main parameters Risk Parameters: α = 0.85, κ = 1 EEX Future Prices 07/ / / / Supply Contract: Contract Price: 70 EUR/MWh (Total) Factor Upper/Lower: δ = (0.9, 1.1) (Daily) Gamma Upper/Lower: γ = (0, 0.025) SLIDE 22

23 Results - Supply contract (without γ) SLIDE 23

24 Results - Supply contract (without γ) SLIDE 24

25 Results - Supply contract (with γ) SLIDE 25

26 Results - Supply contract (with γ) SLIDE 26

27 Results - Supply contract (with γ) SLIDE 27

28 Results - Supply contract (with γ) SLIDE 28

29 Results - Supply contract (with γ) SLIDE 29

30 Results - Supply contract (with γ) SLIDE 30

31 Results - Supply contract (with γ) SLIDE 31

32 Pricing Supply Contracts The aim is to price a energy supply contract for a large energy consumer. The contract is issued by a energy broker, that obtains energy from the EEX (spot and future markets). The price is chosen such that the AV@R α of the expected costs is lower than µ. The supply contract has to cover all the (stochastic) energy demand d t,h. Nominal consumption is limited by upper and lower bounds for every hour block. Consumption that exceeds these limits is priced at a higher price X. The consumption of energy has to be matched by the energy bought on the spot and the future markets. SLIDE 32

33 Pricing Supply Contracts - Model The aim is to minimize the price of the contract C. The overshooting over the maximum allowed consumption ν h is measured with the variable x n,h [d n,h ν h ] + = x n,h The expenditure in every node is given by e n,h = f n,h F n,h + s n,h S n,h d n,h C x n,h X The AV@R α constraint reads qα + n N (T) P s(n) z n 1 α µ SLIDE 33

34 Pricing Supply Contracts - Model minimize C + n N,h=1,...,6 x n,hx subject to d n,h f n,h + s n,h f n,h F n,h + s n,h S n,h d n,h C x n,h X = e n,h d n,h ν h x n,h n N (s),h=1,...,6 e n,h = e s µ c ( s q α z n q α + ) P s(n) z n n N (T) 1 α Besides the minimal price the model yields an optimal hedge in terms of EEX futures. SLIDE 34

35 Pricing Supply Contracts - Results SLIDE 35

36 Results - Power Plant (real option) Stylized Power Plant: Cost of production: 50 EUR/MWh Minimum production per hour block: 0 Maximum production per hour block π: Maximum change of production from hour block to the next: π/5 SLIDE 36

37 Results - Power Plant (real option) SLIDE 37

38 Results - Solver Run-Time Size of LPs and run-time with our specific scenario tree: Power Plant No No Yes Contract Gamma No Yes Yes Constraints Variables Nonzeros Solution Time [sec] Pricing Supply Contracts: variables, constraints, non-zeros, seconds SLIDE 38

39 Thank you for your attention Content of this talk appeared as: R. Hochreiter and D. Wozabal. A multi-stage stochastic programming model for managing risk-optimal electricity portfolios. Handbook of Power Systems II. Volume 4 of Energy Systems: Springer, Download at: SLIDE 39

40 Thank you for your attention Ronald Hochreiter Department of Finance, Accounting and Statistics Institute for Statistics and Mathematics URL: WU Wirtschaftsuniversität Wien Augasse 2 6, A-1090 Wien SLIDE 40