Stochastic Dual Dynamic Programming with CVaR Risk Constraints Applied to Hydrothermal Scheduling. ICSP 2013 Bergamo, July 8-12, 2012

Transcription

1 Stochastic Dual Dynamic Programming with CVaR Risk Constraints Applied to Hydrothermal Scheduling Luiz Carlos da Costa Junior Mario V. F. Pereira Sérgio Granville Nora Campodónico Marcia Helena Costa Fampa ICSP 2013 Bergamo, July 8-12, 2012

2 Outline Introduction Stochastic Dual Dynamic Programming Risk aversion approaches Risk measures Proposed methodology Conclusions and future work 2

3 Introduction Generation scheduling Objective: optimize the use of existing resources (hydro, natural gas, renewables etc.) over a planning horizon Characteristics: Time-coupled: it is possible to store water in the reservoirs for future Strong stochastic components Trade-off: minimum cost x supply reliability Minimum cost operation least reliable Most reliable operation most expensive 3

4 Operation decision under uncertainty Problem Decision Future inflow Consequences Which plant to use? Do not use hydro storage Use hydro storage wet dry wet dry Spillage Ok Ok Expensive themal 4

5 Stochastic programming framework A real life generation scheduling problem has a ten-year horizon and monthly steps (120 stages). The main uncertainties are inflows to hydro plants, load growth and fuel prices The decision tree has ~ nodes Traditional solution: stochastic dynamic programming (SDP) Decomposition in time stages future cost function (FCF) State space formulation 5

6 Immediate and future cost functions Cost Future Cost Function Immediate Cost Function Final storage 6

7 Approximation of FCF: Stochastic DP Cost Future Cost Function Problem: curse of dimensionality Final storage 7

8 SDDP algorithm (Pereira & Pinto, 1991) 1. Forward simulation i. (Monte Carlo) simulation considering a set of scenarios ii. iii. New states are obtained and used as hot states in the backward recursion Feasible solution upper bound 2. Backward recursion i. FCF improvement: new cuts are generated around the hot states ii. First stage solution (IC + FC approximation) lower bound 3. Convergence check 8

9 Approximation of FCF: Stochastic DP Cost Future Cost Function SDDP avoids the curse of dimensionality by using a piecewise linear approximation to the FCF Similar to multi-stage stochastic Benders decomposition Final storage 9

10 SDDP forward and backward steps t = 0 t = 1 t = T-1 t = T 10

11 SDDP forward and backward steps $ V 2,T V 1,T V T t = 0 t = 1 t = T-1 t = T 11

12 SDDP forward and backward steps t = 0 t = 1 t = T-1 t = T 12

13 SDDP forward and backward steps All nodes of the forward and backward steps are solved independently t = 0 t = 1 t = T-1 t = T 13

14 Formulation: one stage problem Objective function α t v t = Min c j g tj J j=1 + α t+1 v t+1 Hydro constraints v t+1 = v t + a t u t s t v t+1 v u t u Thermal constraints e t = ρu t g tj g j j J Load supply e t + g tj = d t j=1 14

15 Energy shortages In some situations, the hydrothermal dispatch may be infeasible: the total generation (hydro + thermal) is not sufficient to supply the load In these cases, it is necessary to reduce the load (deficit): Load supply e t + J j=1 g tj + r t = d t Penalization in the objective function? Depending on the penalization, the operation policy becomes more conservative (more thermal dispatch) higher energy costs 15

16 Risk neutral x risk averse approaches Expected value risk neutral approach Gives the best decision in average From the economical point of view, it may be acceptable to face some energy deficit situations in order reduce the operation costs Problem Decision Future inflow Consequences Which plant to use? Do not use hydro storage Use hydro storage wet dry wet dry Spillage Ok Ok Expensive Shortage thermal 16

17 Risk neutral x risk averse approaches However, in practice, energy shortages can be very disastrous for society For this reason, system operators prefer to adopt risk averse policies Problem Decision Future inflow Consequences Which plant to use? Do not use hydro storage Use hydro storage wet dry wet dry Spillage Ok Ok Expensive Shortage thermal 17

18 How to avoid energy shortages? (Direct) penalization of the deficit variable Min E[ c g + μ r ] s.t.: Operation constraints What if the estimated deficit cost is not enough to ensure the desired reliability level? Deficit cost: estimation of the impacts of energy shortage for society Additional actions taken in Brazil: save water in the reservoirs in order to be protected against unpredicted or unlikely dry scenarios Minimum storage curve for reservoirs (Risk Aversion Curve) External procedure to increase thermal generation and save hydro generation, ensuring minimum reservoir levels in the short term 18

19 How to avoid energy shortages? (cont d) Minimization of a risk measure ρ[.] (explicit cost) Min ρ[ c g + μ r ] s.t.: Operation constraints Example: Combined mean-risk optimization Philpott & Matos, 2010, Shapiro, 2011, Guigues & Römisch, 2010, Guigues & Sagastizábal, 2012, Shapiro et al Nested CVaR: Min (1 λ) E[ c g + μ r ] + λ CVaR[c g + μ r ] 19

20 How to avoid energy shortages? (cont d) Proposed approach 1: Minimization of expected value of operating costs subject to a risk constraint (implicit cost) Min E[ c g + μ r ] s.t.: Operation constraints ρ[r] ρ max φ 20

21 Risk measures Different risk measures on energy shortage can considered, for example: Probability of occurrence ( Energy shortage risk ) Expected value over a given horizon Value at Risk Conditional Value at Risk 21

22 CVaR ( Conditional Value at Risk ) Expected value over the α% worst scenarios (focus on the tail of the function) E[Z] VaR α [Z] α CVaR α [Z] Z 22

23 Optimization of CVaR Linear optimization model (Rockafellar & Uryasev, 2000): CVaR α R = Min b + 1 p α s y s s S s. t. : y s r s b s S Coherent risk measure (Artzner et al., 1997): satisfies properties of monotonicity, sub-additivity, homogeneity and translational invariance CVaR is convex and can be applied in the SDDP framework 23

24 Formulation CVaR constraint (stage t) CVaR limit: b t + 1 α p s y t,s CVaR α s S y t,s r t,s b t y t,s 0 s S s S coupling between the scenarios

25 CVaR constraint Considering the deterministic equivalent problem, the CVaR limit can be easily incorporated as a set of linear constraints However, deterministic equivalent only can be solved in case of small scenario trees A constraint coupling all scenarios prevents the application of traditional SDDP algorithm SDDP decomposition method is based on the solution of independent sub-problems for one stage and one scenario Proposed approach 2: dualization of the CVaR limit constraint 25

26 Dualization of CVaR limit constraint: example p 31 p 21 p 32 p 33 p 22 p 34 26

27 Dualization of CVaR limit constraint: example Objective function Min c 1 g μr p 21 c 2 g μr 2 +p 22 c 2 g μr 2 +p 31 c 3 g μr 3 +p 32 c 3 g μr 3 +p 33 c 3 g μr 3 +p 34 c 3 g μr 3 1 st stage 2 nd stage 2 nd stage 3 rd stage 3 rd stage 3 rd stage 3 rd stage 27

28 Dualization of CVaR limit constraint: example CVaR constraint (stage 3) b α p 31y p 32 y p 33 y p 34 y 3 4 CVaR α y 1 3 r 1 3 b 3 y node 1 y 2 3 r 2 3 b 3 y node 2 y 3 3 r 3 3 b 3 y node 3 y 41 3 r 4 3 b 3 y node 4

29 Dualization of CVaR limit constraint: example Objective function Min c 1 g dr p 21 c 2 g μr 2 +p 22 c 2 g μr 2 +p 31 c 3 g μr 3 +p 32 c 3 g μr 3 +p 33 c 3 g μr 3 +p 34 c 3 g μr 3 φ 3 CVaR α b 3 1 α p 31y p 32 y p 33 y p 34 y

30 Dualization of CVaR limit constraint: example Objective function (rearranged) Min c 1 g μr 1 1 φ 3 CVaR α + φ 3 b 3 +p 21 c 2 g μr 2 1 +p 22 c 2 g μr 2 2 Decoupled problem +p 31 c 3 g μr φ 3 α y 3 1 +p 32 c 3 g μr φ 3 α y 3 2 +p 33 c 3 g μr φ 3 α y 3 3 +p 34 c 3 g μr φ 3 α y

31 SDDP with dualized CVaR constraint: remarks For a given Lagrangian multiplier φ, it is possible to apply standard SDDP algorithm: Iterative process to construct a FCF approximation Cuts are obtained for this given φ If φ φ, the solution may not meet the CVaR limit constraint Standard algorithm to find optimal multiplier φ (bisection) Proposed approach 3: adjust Benders cuts and reuse them at each λ iteration, avoiding restart SDDP from scratch 31

32 Economic interpretation There is a strong direct relationship between the optimal φ for the dualized CVaR constraint and the segment of the piecewise linear function for the deficit cost curve $ Slope = φ 3 α b y R 32

33 Economic interpretation Piecewise deficit cost curve Function associate to E[r] Slope = μ $ Piecewise linear function associate to CVaR α CVaR depth of the first segment of the deficit curve r 33

34 Conclusions Policy makers want to be protected against unlikely events with bad consequences ISO's main concern is the risk of energy shortages The risk level may not depend on other costs (for example, fuel costs) Expectation can be a naive (risk neutral) measure: For example, it cannot distinguish between two energy shortages of 100 MW or one of 200 MW CVaR is a good (risk averse) alternative: Sensitive to the tail of the distribution, representing a protection against extreme scenarios Coherent risk measure Convex: can be incorporated in decomposition schemes like SDDP 34

35 Conclusions - Risk aversion approaches Direct penalization of energy shortage Penalization impacts directly on the energy shortage variable However, the energy shortage cost is decided a priori and the risk level is checked afterwards (consequence) Expectation is used in the objective function Risk aversion curve / External procedure Constraints directly apply over reservoir levels to indirect control the risk of energy shortage The definition of minimum storage curve may be a challenging task Expectation is used in the objective function 35

36 Conclusions - Risk aversion approaches (cont d) Minimization of a risk measure (CVaR, for example) Requires the definition of a weight for the CVaR on the objective function (parameter definition is a challenging task) The risk level is a consequence Minimization of E[.] considering a CVaR constraint CVaR constraint is directly applied over the energy shortage to ensure the desired risk level The definition of the acceptable risk level is straightforward Implicit cost function for energy shortages is a result 36

37 Future work (in progress) Extend the CVaR constraint to limit the energy shortage in a sequence of periods The application of this method larger systems (Brazilian system, especially) are under development and will be reported soon In preparation: Stochastic Dual Dynamic Programming with CVaR Risk Constraints Applied to Hydrothermal Scheduling, Costa Jr. L. C., Pereira, M., Granville S., Compodónico N., Fampa, M. 37

38 Thank you! Comments, suggestions, questions? Luiz Carlos da Costa Junior