The MIDAS touch: EPOC Winter Workshop 9th September solving hydro bidding problems using mixed integer programming

Transcription

1 The MIDAS touch: solving hydro bidding problems using mixed integer programming EPOC Winter Workshop 9th September 2015 Faisal Wahid, Cédric Gouvernet, Prof. Andy Philpott, Prof. Frédéric Bonnans 0 / 30

2 Acknowledgements Anes Dallagi, EDF R&D France Andrew Kerr, Meridian Energy New Zealand PGMO OSIRIS EDF R&D FMJH 0 / 30

3 1 Motivation 2 Hydro-bidding problem 3 HERBS 4 MIDAS 5 Results 6 Conclusions 7 Questions 0 / 30

4 Motivation Problem Description 1 / 30

5 Motivation Problem Description How do you construct oers for hydro producing agents? 1 / 30

6 Motivation Problem Description How do you construct oers for hydro producing agents? Given that: 1 / 30

7 Motivation Problem Description How do you construct oers for hydro producing agents? Given that: 1 / 30

8 Motivation Problem Description How do you construct oers for hydro producing agents? Given that: Price-taking agents 1 / 30

9 Motivation Problem Description How do you construct oers for hydro producing agents? Given that: Price-taking agents Discrete & nite oer stacks 1 / 30

10 Motivation Problem Description How do you construct oers for hydro producing agents? Given that: Price-taking agents Discrete & nite oer stacks Cleared oers aect exibility on future operations of the hydro-scheme 1 / 30

11 Motivation Problem Description How do you construct oers for hydro producing agents? Given that: Price-taking agents Discrete & nite oer stacks Cleared oers aect exibility on future operations of the hydro-scheme EDF & Meridian: 1 / 30

12 Motivation Problem Description How do you construct oers for hydro producing agents? Given that: Price-taking agents Discrete & nite oer stacks Cleared oers aect exibility on future operations of the hydro-scheme EDF & Meridian: EDF participate in the French Balancing market using their hydro-schemes 1 / 30

13 Motivation Problem Description How do you construct oers for hydro producing agents? Given that: Price-taking agents Discrete & nite oer stacks Cleared oers aect exibility on future operations of the hydro-scheme EDF & Meridian: EDF participate in the French Balancing market using their hydro-schemes Meridian participate in the NZ Spot market using the Waitaki hydro-scheme 1 / 30

14 Motivation Problem description Produce optimal oers: 2 / 30

15 Motivation Problem description Produce optimal oers: Each oer needs to be feasible across the hydro-scheme Treat oer prices as exogenous random variable 2 / 30

16 Motivation Problem description Produce optimal oers: Each oer needs to be feasible across the hydro-scheme Treat oer prices as exogenous random variable Hydro-bidding problem: Combines oer optimization problem with hydro-scheduling problem 2 / 30

17 Motivation Problem description Produce optimal oers: Each oer needs to be feasible across the hydro-scheme Treat oer prices as exogenous random variable Hydro-bidding problem: Combines oer optimization problem with hydro-scheduling problem 2 / 30

18 Motivation Problem description Produce optimal oers: Each oer needs to be feasible across the hydro-scheme Treat oer prices as exogenous random variable Hydro-bidding problem: Combines oer optimization problem with hydro-scheduling problem 1. Each period solve single stage hydro-bidding model; 2 / 30

19 Motivation Problem description Produce optimal oers: Each oer needs to be feasible across the hydro-scheme Treat oer prices as exogenous random variable Hydro-bidding problem: Combines oer optimization problem with hydro-scheduling problem 1. Each period solve single stage hydro-bidding model; 2. Then observe market clearing prices & dispatch; 2 / 30

20 Motivation Problem description Produce optimal oers: Each oer needs to be feasible across the hydro-scheme Treat oer prices as exogenous random variable Hydro-bidding problem: Combines oer optimization problem with hydro-scheduling problem 1. Each period solve single stage hydro-bidding model; 2. Then observe market clearing prices & dispatch; 3. Solve for next period based on observed prices & dispatch 2 / 30

21 Hydro-bidding problem Hydro-bidding model 3 / 30

22 Hydro-bidding problem Hydro-bidding model Objective Maximize expected prot from oerstack, & expected future value of the remaining water, conditioned on observed prices. 3 / 30

23 Hydro-bidding problem Hydro-bidding model Objective Maximize expected prot from oerstack, & expected future value of the remaining water, conditioned on observed prices. Constraints 3 / 30

24 Hydro-bidding problem Hydro-bidding model Objective Maximize expected prot from oerstack, & expected future value of the remaining water, conditioned on observed prices. Constraints Respect physical constraints of the hydro-scheme (i.e. storage bounds and water ow balance) 3 / 30

25 Hydro-bidding problem Hydro-bidding model Objective Maximize expected prot from oerstack, & expected future value of the remaining water, conditioned on observed prices. Constraints Respect physical constraints of the hydro-scheme (i.e. storage bounds and water ow balance) Respect the operational constraints of the turbines (i.e. plant capacity) 3 / 30

26 Hydro-bidding problem Hydro-bidding model Objective Maximize expected prot from oerstack, & expected future value of the remaining water, conditioned on observed prices. Constraints Respect physical constraints of the hydro-scheme (i.e. storage bounds and water ow balance) Respect the operational constraints of the turbines (i.e. plant capacity) Respect the physical constraints of the hydro-scheme 3 / 30

27 Hydro-bidding problem Hydro-bidding model Objective Maximize expected prot from oerstack, & expected future value of the remaining water, conditioned on observed prices. Constraints Respect physical constraints of the hydro-scheme (i.e. storage bounds and water ow balance) Respect the operational constraints of the turbines (i.e. plant capacity) Respect the physical constraints of the hydro-scheme Meet ancillary requirements: Frequency containment reserve Frequency restoration reserve 3 / 30

28 HERBS Description of HERBS 4 / 30

29 HERBS Description of HERBS Stands for Hydro Electric Reservoir Bidding System 4 / 30

30 HERBS Description of HERBS Stands for Hydro Electric Reservoir Bidding System Stochastic dynamic programming hydro-bidding model 4 / 30

31 HERBS Description of HERBS Stands for Hydro Electric Reservoir Bidding System Stochastic dynamic programming hydro-bidding model Uncertainty is the market clearing price 4 / 30

32 HERBS Description of HERBS Stands for Hydro Electric Reservoir Bidding System Stochastic dynamic programming hydro-bidding model Uncertainty is the market clearing price Market clearing price modelled as a Markov chain of price distributions 4 / 30

33 HERBS Description of HERBS Stands for Hydro Electric Reservoir Bidding System Stochastic dynamic programming hydro-bidding model Uncertainty is the market clearing price Market clearing price modelled as a Markov chain of price distributions Partition overall price distribution into segments 4 / 30

34 HERBS Description of HERBS Stands for Hydro Electric Reservoir Bidding System Stochastic dynamic programming hydro-bidding model Uncertainty is the market clearing price Market clearing price modelled as a Markov chain of price distributions Partition overall price distribution into segments Price segment represents conditional distribution, based on the observed price of previous period 4 / 30

35 HERBS Description of HERBS Stands for Hydro Electric Reservoir Bidding System Stochastic dynamic programming hydro-bidding model Uncertainty is the market clearing price Market clearing price modelled as a Markov chain of price distributions Partition overall price distribution into segments Price segment represents conditional distribution, based on the observed price of previous period Assumptions: Price taking hydro power producers Deterministic inow Discrete nite oers (i.e. an oer stack) 4 / 30

36 HERBS Formulation I: Modelling price Market clearing price p(t) is random variable p(t) 5 / 30

37 HERBS Formulation I: Modelling price Market clearing price p(t) is random variable Partition price p(t) into M number of segments p i p i+1 p(t) 5 / 30

38 HERBS Formulation I: Modelling price Price segments: {[p 1, p 2 ), [p 2, p 3 ),..., [p M 1, p M ), [p M, p M+1 )} 6 / 30

39 HERBS Formulation I: Modelling price Price segments: {[p 1, p 2 ), [p 2, p 3 ),..., [p M 1, p M ), [p M, p M+1 )} Each segment represents the price of each oer 6 / 30

40 HERBS Formulation I: Modelling price Price segments: {[p 1, p 2 ), [p 2, p 3 ),..., [p M 1, p M ), [p M, p M+1 )} Each segment represents the price of each oer Transition probability of moving from segment i to segment j: P i,j (t) = P[p(t) [p j, p j+1 ) p(t 1) [p i, p i+1 )]. 6 / 30

41 HERBS Formulation I: Modelling price Price segments: {[p 1, p 2 ), [p 2, p 3 ),..., [p M 1, p M ), [p M, p M+1 )} Each segment represents the price of each oer Transition probability of moving from segment i to segment j: P i,j (t) = P[p(t) [p j, p j+1 ) p(t 1) [p i, p i+1 )]. Conditionally expected price of segment j: π t,j = E[p(t) p(t) [p j, p j+1 )]. 6 / 30

42 Formulation II: Modelling oerstack HERBS 7 / 30

43 HERBS Formulation II: Modelling oerstack Decision variable: Discrete oer quantities (i.e. oerstack) q t = (q t,1,..., q t,m ) 7 / 30

44 HERBS Formulation II: Modelling oerstack Decision variable: Discrete oer quantities (i.e. oerstack) q t = (q t,1,..., q t,m ) Expected prot from oer stack: M P i,j (t)π t,j q t,j. j=1 7 / 30

45 HERBS Formulation II: Modelling oerstack Decision variable: Discrete oer quantities (i.e. oerstack) q t = (q t,1,..., q t,m ) Expected prot from oer stack: M P i,j (t)π t,j q t,j. Price j=1 p 5 p 4 p 3 p 2 p 1 q t,1 q t,2 q t,3 q t,4 π t,4 = E[p(t) p(t 1) [p 4, p 5 )] π t,3 π t,2 π t,1 Quantity 7 / 30

46 HERBS Formulation III: HERBS Objective function 8 / 30

47 HERBS Formulation III: HERBS Objective function State variable: Storage of reservoir r starting period t + 1 for each oer j 8 / 30

48 HERBS Formulation III: HERBS Objective function State variable: Storage of reservoir r starting period t + 1 for each oer j x t+1,j,r [x r, x r ] 8 / 30

49 HERBS Formulation III: HERBS Objective function State variable: Storage of reservoir r starting period t + 1 for each oer j x t+1,j,r [x r, x r ] x t,i : Observed storage level from cleared oer q t 1,i in previous period t 1 8 / 30

50 HERBS Formulation III: HERBS Objective function State variable: Storage of reservoir r starting period t + 1 for each oer j x t+1,j,r [x r, x r ] x t,i : Observed storage level from cleared oer q t 1,i in previous period t 1 Prot objective for period t (i.e. objective function): V t,i (x t,i ) = 8 / 30

51 HERBS Formulation III: HERBS Objective function State variable: Storage of reservoir r starting period t + 1 for each oer j x t+1,j,r [x r, x r ] x t,i : Observed storage level from cleared oer q t 1,i in previous period t 1 Prot objective for period t (i.e. objective function): V t,i (x t,i ) = max E[Prot(q t )] M P i,j (t)π t,j q t,j j=1 8 / 30

52 HERBS Formulation III: HERBS Objective function State variable: Storage of reservoir r starting period t + 1 for each oer j x t+1,j,r [x r, x r ] x t,i : Observed storage level from cleared oer q t 1,i in previous period t 1 Prot objective for period t (i.e. objective function): V t,i (x t,i ) = max E[Prot(q t )] +E[FutureProt(x t+1,j )] M M P i,j (t)π t,j q t,j + P i,j (t)v t+1,j (x t+1,j ) j=1 j=1 8 / 30

53 HERBS Formulation IV: Water-balance constraint 9 / 30

54 HERBS Formulation IV: Water-balance constraint Enforce the water-balance constraint: x t,i,r x t,i,r 9 / 30

55 HERBS Formulation IV: Water-balance constraint Enforce the water-balance constraint: x t,i,r +upstream ow u t,j,s1 x t,i,r 9 / 30

56 HERBS Formulation IV: Water-balance constraint Enforce the water-balance constraint: x t,i,r +upstream ow station discharge u t,j,s1 x t,i,r u t,j,sr 9 / 30

57 HERBS Formulation IV: Water-balance constraint Enforce the water-balance constraint: x t,i,r +upstream ow station discharge +inow u t,j,s1 ω t,r x t,i,r u t,j,sr 9 / 30

58 HERBS Formulation IV: Water-balance constraint Enforce the water-balance constraint: x t+1,j,r = x t,i,r +upstream ow station discharge +inow u t,j,s1 ω t,r x t,i,r x t+1,j,r u t,j,sr 9 / 30

59 HERBS Formulation V: Discrete power production 10 / 30

60 HERBS Formulation V: Discrete power production (θ t,s,l, η t,s,l ): Discrete pairs of waterow (cubic meters) and power generation quantities (MWhs) 10 / 30

61 HERBS Formulation V: Discrete power production (θ t,s,l, η t,s,l ): Discrete pairs of waterow (cubic meters) and power generation quantities (MWhs) z j,s,l : Binary variable used to select a single pair of discrete production 10 / 30

62 HERBS Formulation V: Discrete power production (θ t,s,l, η t,s,l ): Discrete pairs of waterow (cubic meters) and power generation quantities (MWhs) z j,s,l : Binary variable used to select a single pair of discrete production Production levels Electricity Production (ηt,s,l) Water discharge (θt,s,l) 10 / 30

63 HERBS Formulation V: Discrete power production (θ t,s,l, η t,s,l ): Discrete pairs of waterow (cubic meters) and power generation quantities (MWhs) z j,s,l : Binary variable used to select a single pair of discrete production Production levels Electricity Production (ηt,s,l) Water discharge (θt,s,l) 10 / 30

64 HERBS Formulation V: Discrete power production (θ t,s,l, η t,s,l ): Discrete pairs of waterow (cubic meters) and power generation quantities (MWhs) z j,s,l : Binary variable used to select a single pair of discrete production Production levels Electricity Production (ηt,s,l) Water discharge (θt,s,l) 10 / 30

65 HERBS Formulation V: Discrete power production (θ t,s,l, η t,s,l ): Discrete pairs of waterow (cubic meters) and power generation quantities (MWhs) z j,s,l : Binary variable used to select a single pair of discrete production Production levels Electricity Production (ηt,s,l) Water discharge (θt,s,l) 10 / 30

66 HERBS Formulation V: Discrete power production 11 / 30

67 HERBS Formulation V: Discrete power production Enforce discrete generation: 11 / 30

68 HERBS Formulation V: Discrete power production Enforce discrete generation: u t,j,s = 11 / 30

69 HERBS Formulation V: Discrete power production Enforce discrete generation: L t,s u t,j,s = θ t,s,l z j,s,l l=1 11 / 30

70 HERBS Formulation V: Discrete power production Enforce discrete generation: L t,s u t,j,s = θ t,s,l z j,s,l l=1 Enforce the j'th oer quantity as total generation of the hydro-scheme: 11 / 30

71 HERBS Formulation V: Discrete power production Enforce discrete generation: L t,s u t,j,s = θ t,s,l z j,s,l l=1 Enforce the j'th oer quantity as total generation of the hydro-scheme: q t,j = 11 / 30

72 HERBS Formulation V: Discrete power production Enforce discrete generation: L t,s u t,j,s = θ t,s,l z j,s,l l=1 Enforce the j'th oer quantity as total generation of the hydro-scheme: q t,j = L S t,s η t,s,l z j,s,l s=1 l=1 11 / 30

73 HERBS Formulation V: Discrete power production Enforce discrete generation: L t,s u t,j,s = θ t,s,l z j,s,l l=1 Enforce the j'th oer quantity as total generation of the hydro-scheme: q t,j = Monotonicity of oers: L S t,s η t,s,l z j,s,l s=1 l=1 11 / 30

74 HERBS Formulation V: Discrete power production Enforce discrete generation: L t,s u t,j,s = θ t,s,l z j,s,l l=1 Enforce the j'th oer quantity as total generation of the hydro-scheme: q t,j = Monotonicity of oers: L S t,s η t,s,l z j,s,l s=1 l=1 q t,j 11 / 30

75 HERBS Formulation V: Discrete power production Enforce discrete generation: L t,s u t,j,s = θ t,s,l z j,s,l l=1 Enforce the j'th oer quantity as total generation of the hydro-scheme: q t,j = Monotonicity of oers: L S t,s η t,s,l z j,s,l s=1 l=1 q t,j q t,j+1 11 / 30

76 HERBS Value function 12 / 30

77 HERBS Value function V t+1,j (x t+1,j ) is monotonic & non-concave 12 / 30

78 HERBS Value function V t+1,j (x t+1,j ) is monotonic & non-concave Due to discrete production & oers = total block dispatch 12 / 30

79 HERBS Value function V t+1,j (x t+1,j ) is monotonic & non-concave Due to discrete production & oers = total block dispatch Creates plateaus of similar value function values 12 / 30

80 HERBS Value function V t+1,j (x t+1,j ) is monotonic & non-concave Due to discrete production & oers = total block dispatch Creates plateaus of similar value function values Makes HERBS hard to solve 12 / 30

81 MIDAS Solution approaches 13 / 30

82 MIDAS Solution approaches Solve the HERBS via existing algorithms: 13 / 30

83 MIDAS Solution approaches Solve the HERBS via existing algorithms: Stochastic dynamic program (SDP) (Pritchard & Zakeri [1]) Limited by the `curse of dimensionality' 13 / 30

84 MIDAS Solution approaches Solve the HERBS via existing algorithms: Stochastic dynamic program (SDP) (Pritchard & Zakeri [1]) Limited by the `curse of dimensionality' Stochastic approximate dynamic program (SADP) (Löhndorf et al. [2]) Dicult to nd appropriate basis functions to represent the value function 13 / 30

85 MIDAS Solution approaches Solve the HERBS via existing algorithms: Stochastic dynamic program (SDP) (Pritchard & Zakeri [1]) Limited by the `curse of dimensionality' Stochastic approximate dynamic program (SADP) (Löhndorf et al. [2]) Dicult to nd appropriate basis functions to represent the value function Stochastic dual dynamic program (SDDP) (Pereira & Pinto [3]) Assumes concave value functions 13 / 30

86 MIDAS Solution approaches Solve the HERBS via existing algorithms: Stochastic dynamic program (SDP) (Pritchard & Zakeri [1]) Limited by the `curse of dimensionality' Stochastic approximate dynamic program (SADP) (Löhndorf et al. [2]) Dicult to nd appropriate basis functions to represent the value function Stochastic dual dynamic program (SDDP) (Pereira & Pinto [3]) Assumes concave value functions Linear decision rules (Braathen & Eriksrud [2]) Optimal policy not guaranteed, due to restriction of the decision space 13 / 30

87 MIDAS Description of MIDAS 14 / 30

88 MIDAS Description of MIDAS Mixed-integer Dynamic Approximation Scheme 14 / 30

89 MIDAS Description of MIDAS Mixed-integer Dynamic Approximation Scheme Stochastic optimization algorithm designed to solve sequential decision problems under uncertainty 14 / 30

90 MIDAS Description of MIDAS Mixed-integer Dynamic Approximation Scheme Stochastic optimization algorithm designed to solve sequential decision problems under uncertainty Creates an outer approximation of V t+1,j (x t+1,j ) 14 / 30

91 MIDAS Description of MIDAS Mixed-integer Dynamic Approximation Scheme Stochastic optimization algorithm designed to solve sequential decision problems under uncertainty Creates an outer approximation of V t+1,j (x t+1,j ) Uses piece-wise constant functions to represent the plateaus of V t+1,j (x t+1,j ) 14 / 30

92 MIDAS Formulation I 15 / 30

93 MIDAS Formulation I ˆv t+1,j : Variable that approximates the value function V t+1,j (x t+1,j ) 15 / 30

94 MIDAS Formulation I ˆv t+1,j : Variable that approximates the value function V t+1,j (x t+1,j ) H t+1 : Set of piece-wise constant functions 15 / 30

95 MIDAS Formulation I ˆv t+1,j : Variable that approximates the value function V t+1,j (x t+1,j ) H t+1 : Set of piece-wise constant functions a j,h,r : Points in the state space where piece-wise function h starts in dimension r 15 / 30

96 MIDAS Formulation I ˆv t+1,j : Variable that approximates the value function V t+1,j (x t+1,j ) H t+1 : Set of piece-wise constant functions a j,h,r : Points in the state space where piece-wise function h starts in dimension r b j,h : Upper bounded estimate of ˆv t+1,j 15 / 30

97 MIDAS Formulation I ˆv t+1,j : Variable that approximates the value function V t+1,j (x t+1,j ) H t+1 : Set of piece-wise constant functions a j,h,r : Points in the state space where piece-wise function h starts in dimension r b j,h : Upper bounded estimate of ˆv t+1,j Use series of binary variables & Big M constraints to couple x t+1,j to ˆv t+1,j using a j,h,r & b j,h 15 / 30

98 MIDAS Formulation II For period t + 1 & price segment j: 16 / 30

99 MIDAS Formulation II For period t + 1 & price segment j: x 2 x 1 16 / 30

100 MIDAS Formulation II For period t + 1 & price segment j: x 2 b 4 = 30 b 2 = 20 b 3 = 25 b 1 = 10 x 1 16 / 30

101 MIDAS Formulation II For period t + 1 & price segment j: x 2 a 4 = (8, 8) 8 b 4 = 30 b 2 = 20 b 3 = 25 b 1 = 10 8 x 1 16 / 30

102 MIDAS Formulation II For period t + 1 & price segment j: x 2 a 4 = (8, 8) a 3 = (6, 4) 8 b 4 = 30 b 2 = 20 4 b 3 = 25 b 1 = x 1 16 / 30

103 MIDAS Formulation II For period t + 1 & price segment j: x b 4 = 30 b 2 = 20 b 3 = 25 b 1 = 10 a 4 = (8, 8) a 3 = (6, 4) a 2 = (4, 6) a 1 = (2, 2) x 1 16 / 30

104 MIDAS Formulation II For period t + 1 & price segment j: x ˆv x 1 16 / 30

105 MIDAS Formulation II For period t + 1 & price segment j: x ˆv 30 ˆv x 1 16 / 30

106 MIDAS Formulation II For period t + 1 & price segment j: x ˆv 10 ˆv 20 ˆv 30 ˆv x 1 16 / 30

107 MIDAS Formulation II For period t + 1 & price segment j: ˆv 30 x ˆv 20 2 ˆv 10 ˆv 30 ˆv 25 x x 1 > 6; Or x 1 > 4 Or x 1 > 0 x 2 > 6; Or x 2 > 4 Or x 2 > x 1 16 / 30

108 MIDAS Formulation II For period t + 1 & price segment j: x 2 ˆv 30; And ˆv ˆv 30 x 1 > 4 x 2 > ˆv 10 ˆv 20 ˆv 25 x x 1 16 / 30

109 Description of the MIDAS algorithm MIDAS 17 / 30

110 Description of the MIDAS algorithm 1. Forward simulation: for t = 1,..., T MIDAS 17 / 30

111 Description of the MIDAS algorithm 1. Forward simulation: for t = 1,..., T 1.1 Generates a scenario of market clearing prices MIDAS 17 / 30

112 Description of the MIDAS algorithm MIDAS 1. Forward simulation: for t = 1,..., T 1.1 Generates a scenario of market clearing prices 1.2 Solves HERBS sub-problem & computes series of storage levels (state trajectory) 17 / 30

113 Description of the MIDAS algorithm MIDAS 1. Forward simulation: for t = 1,..., T 1.1 Generates a scenario of market clearing prices 1.2 Solves HERBS sub-problem & computes series of storage levels (state trajectory) x t t 17 / 30

114 Description of the MIDAS algorithm 1. Backward recursion: for t = T,..., t & m = 1,..., M MIDAS x t t 18 / 30

115 Description of the MIDAS algorithm 1. Backward recursion: for t = T,..., t & m = 1,..., M 1.1 Explores around the neighbourhood of state trajectory MIDAS x t t 18 / 30

116 Description of the MIDAS algorithm 1. Backward recursion: for t = T,..., t & m = 1,..., M 1.1 Explores around the neighbourhood of state trajectory 1.2 Computes b j,h & a j,h parameters MIDAS x t t 18 / 30

117 Description of the MIDAS algorithm 1. Backward recursion: for t = T,..., t & m = 1,..., M 1.1 Explores around the neighbourhood of state trajectory 1.2 Computes b j,h & a j,h parameters 1.3 Adds (b j,h, a j,h ) to H t to period t 1 sub-problem MIDAS x t t 18 / 30

118 Description of the MIDAS algorithm 1. Convergence testing: After a certain number forward simulation & backward recursion MIDAS Policy Value Upper bound Optimal value Lower bound iteration 19 / 30

119 Description of the MIDAS algorithm 1. Convergence testing: After a certain number forward simulation & backward recursion 1.1 Simulates policy over sample of price scenarios MIDAS Policy Value Upper bound Optimal value Lower bound iteration 19 / 30

120 Description of the MIDAS algorithm 1. Convergence testing: After a certain number forward simulation & backward recursion 1.1 Simulates policy over sample of price scenarios 1.2 Compute condence interval (i.e. Lower bound) MIDAS Policy Value Upper bound Optimal value Lower bound iteration 19 / 30

121 Description of the MIDAS algorithm 1. Convergence testing: After a certain number forward simulation & backward recursion 1.1 Simulates policy over sample of price scenarios 1.2 Compute condence interval (i.e. Lower bound) 1.3 Checks if rst stage objective (i.e. Upper bound) is within condence interval Policy Value MIDAS Upper bound Optimal value Lower bound iteration 19 / 30

122 Results Hydro-scheme data 20 / 30

123 Results Hydro-scheme data Two identical, cascaded reservoirs r 1 s 1 r 2 s 2 20 / 30

124 Results Hydro-scheme data Two identical, cascaded reservoirs T = 4 stages (boundary value function zero) M = 3 price segments OR 3 tranche oer stacks r 1 s 1 r 2 s 2 20 / 30

125 Results Hydro-scheme data Two identical, cascaded reservoirs T = 4 stages (boundary value function zero) M = 3 price segments OR 3 tranche oer stacks r 1 Production levels s 1 r 2 Electricity Production (MWhs) s Water discharge (Cubic Metres) 20 / 30

126 Results Comparison of policy value & approximation 21 / 30

127 Results Comparison of policy value & approximation Solved & simulated oer policy of MIDAS & SDDP for range of initial storage levels 21 / 30

128 Results Comparison of policy value & approximation Solved & simulated oer policy of MIDAS & SDDP for range of initial storage levels MIDAS policy: On average 98% of optimal value SDDP policy: On average 93% of optimal value Mean Median Upper quartile Lower quartile SDDP MIDAS / 30

129 Results Approximation of value function 22 / 30

130 Results Approximation of value function 22 / 30

131 Results Approximation of value function 22 / 30

132 Results Comparison of an oer 23 / 30

133 Results Comparison of an oer Price scenario: $20 $20 $25 $20 Storage: x t,i = (50, 60) 23 / 30

134 Results Comparison of an oer Price scenario: $20 $20 $25 $20 Storage: x t,i = (50, 60) Oer MIDAS $20 $25 $30 Oer quantity Trading reriod 23 / 30

135 Results Comparison of an oer Price scenario: $20 $20 $25 $20 Storage: x t,i = (50, 60) Oer MIDAS Oer SDDP $20 $25 $30 $20 $25 $30 Oer quantity Oer quantity Trading reriod Trading reriod 23 / 30

136 Results Comparison of an oer Price scenario: $20 $20 $25 $20 Storage: x t,i = (50, 60) Oer MIDAS Oer SDDP Oer SDP $20 $25 $30 $20 $25 $30 $20 $25 $30 Oer quantity Oer quantity Oer quantity Trading reriod Trading reriod Trading reriod 23 / 30

137 Results Comparison of computation I Extend the number of periods to / 30

138 Results Comparison of computation I Extend the number of periods to 10 High storage: x t,i = (200, 90) 24 / 30

139 Results Comparison of computation I Extend the number of periods to 10 High storage: x t,i = (200, 90) MIDAS takes more iterations to get close to optimal value Upper bound value Convergence comparison SDDP MIDAS Optimal Upper bound value Convergence comparison SDDP MIDAS Optimal Iteration Iteration 24 / 30

140 Results Comparison of computation II Low storage: x t,i = (100, 50) 25 / 30

141 Results Comparison of computation II Low storage: x t,i = (100, 50) SDDP takes more iterations to get close to optimal value 25 / 30

142 Results Comparison of computation II Low storage: x t,i = (100, 50) SDDP takes more iterations to get close to optimal value 10 4 Convergence comparison Convergence comparison Upper bound value SDDP MIDAS Optimal Upper bound value 7,900 7,800 7,700 7,600 SDDP MIDAS Optimal 1 7, Iteration Iteration 25 / 30

143 Results Comparison of computation III 26 / 30

144 Results Comparison of computation III Value function at high storage levels has convexity-like structure 26 / 30

145 Results Comparison of computation III Value function at high storage levels has convexity-like structure MIDAS not able to exploit the gradient of the value function (unlike SDDP) 26 / 30

146 Results Comparison of computation III Value function at high storage levels has convexity-like structure MIDAS not able to exploit the gradient of the value function (unlike SDDP) 26 / 30

147 Results Comparison of computation III Value function at high storage levels has convexity-like structure MIDAS not able to exploit the gradient of the value function (unlike SDDP) 26 / 30

148 Conclusions Conclusion 27 / 30

149 Conclusions Conclusion MIDAS has better representation of the structure of future value function compared to SDDP 27 / 30

150 Conclusions Conclusion MIDAS has better representation of the structure of future value function compared to SDDP MIDAS produces good policy and accurate value function approximation 98% of optimal value 27 / 30

151 Conclusions Conclusion MIDAS has better representation of the structure of future value function compared to SDDP MIDAS produces good policy and accurate value function approximation 98% of optimal value For high storage MIDAS takes many iterations to get close to optimal value : 27 / 30

152 Conclusions Conclusion MIDAS has better representation of the structure of future value function compared to SDDP MIDAS produces good policy and accurate value function approximation 98% of optimal value For high storage MIDAS takes many iterations to get close to optimal value : Not able to exploit the gradient of the value function (unlike SDDP) 27 / 30

153 Conclusions Conclusion MIDAS has better representation of the structure of future value function compared to SDDP MIDAS produces good policy and accurate value function approximation 98% of optimal value For high storage MIDAS takes many iterations to get close to optimal value : Not able to exploit the gradient of the value function (unlike SDDP) Solving increasingly large MIPs with iterations 27 / 30

154 Conclusions Conclusion MIDAS has better representation of the structure of future value function compared to SDDP MIDAS produces good policy and accurate value function approximation 98% of optimal value For high storage MIDAS takes many iterations to get close to optimal value : Not able to exploit the gradient of the value function (unlike SDDP) Solving increasingly large MIPs with iterations Solving MIPs with many Big M constraints & tight integer feasibility tolerance 27 / 30

155 Questions Questions Thank you for listening! Questions? 28 / 30

156 References References I [1] Pritchard, G. and Zakeri, G. Market oering strategies for hydroelectric generators. Operations Research, 51(4):602612, [2] Löhndorf, N., Wozabal, D. and Minner, S. Optimizing Trading Decisions for Hydro Storage Systems Using Approximate Dual Dynamic Programming. INFORMS, 61(4): , [3] Pereira M.V.F. and Pinto L.M.V.G. Multi-stage stochastic optimization applied to energy planning. Mathematical Programming, 52(1):359375, [4] Boomsma, T.K., Juul N. and Fleten S.E. Bidding in sequential electricity markets: The Nordic cas European Journal of of Operation Research, 238(3):797809, / 30

157 References References II [1] Fleten, S.E. and Kristoersen, T.K. Stochastic programming for optimizing bidding strategies of a Nordic hydropower producer. European Journal of Operational Research, 181(2):916928, [2] Braathen, J and Eriksrud, A.L. Hydropower Bidding Using Linear Decision Rules. Institutt for industriell økonomi og teknologiledelse, / 30