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Stochastic Programming for Hydropower Operations Modeling and Algorithms Martin Biel KTH - Royal Institute of Technology JUNE 28, 2018 Price forecasts Irregular power production: solar and wind Nuclear power phase-out Common:

  1. • No clearcut way to calculate stochastic measures: EVPI, VSS • The model creation is somewhat infmexible • Parallel L-shaped using the Distributed module in Julia… • …but StructJuMP relies on MPI • Creating a new hydromodel involves reimplementing a new type Initial Approach - Issues Martin Biel (KTH) Stochastic Programming for Hydropower 7 / 25 • A lot of code repetition

  2. • The model creation is somewhat infmexible • Parallel L-shaped using the Distributed module in Julia… • …but StructJuMP relies on MPI • Creating a new hydromodel involves reimplementing a new type Initial Approach - Issues Martin Biel (KTH) Stochastic Programming for Hydropower 7 / 25 • A lot of code repetition • No clearcut way to calculate stochastic measures: EVPI, VSS

  3. • Parallel L-shaped using the Distributed module in Julia… • …but StructJuMP relies on MPI • Creating a new hydromodel involves reimplementing a new type Initial Approach - Issues Martin Biel (KTH) Stochastic Programming for Hydropower 7 / 25 • A lot of code repetition • No clearcut way to calculate stochastic measures: EVPI, VSS • The model creation is somewhat infmexible

  4. • …but StructJuMP relies on MPI • Creating a new hydromodel involves reimplementing a new type Initial Approach - Issues Martin Biel (KTH) Stochastic Programming for Hydropower 7 / 25 • A lot of code repetition • No clearcut way to calculate stochastic measures: EVPI, VSS • The model creation is somewhat infmexible • Parallel L-shaped using the Distributed module in Julia…

  5. • Creating a new hydromodel involves reimplementing a new type Initial Approach - Issues Martin Biel (KTH) Stochastic Programming for Hydropower 7 / 25 • A lot of code repetition • No clearcut way to calculate stochastic measures: EVPI, VSS • The model creation is somewhat infmexible • Parallel L-shaped using the Distributed module in Julia… • …but StructJuMP relies on MPI

  6. Initial Approach - Issues Martin Biel (KTH) Stochastic Programming for Hydropower 7 / 25 • A lot of code repetition • No clearcut way to calculate stochastic measures: EVPI, VSS • The model creation is somewhat infmexible • Parallel L-shaped using the Distributed module in Julia… • …but StructJuMP relies on MPI • Creating a new hydromodel involves reimplementing a new type

  7. ∘ Flexible model creation ∘ Parallel capabilities based on the Distributed module ∘ Stochastic programming constructs ∘ Model creation focused on data and optimization formulation ∘ Effjcient model reinitialization ∘ Predefjned models • Short-term production planning • Optimal orders on the day-ahead market New Approach Martin Biel (KTH) Stochastic Programming for Hydropower 8 / 25 • StochasticPrograms.jl • HydroModels.jl

  8. ∘ Parallel capabilities based on the Distributed module ∘ Stochastic programming constructs ∘ Model creation focused on data and optimization formulation ∘ Effjcient model reinitialization ∘ Predefjned models • Short-term production planning • Optimal orders on the day-ahead market New Approach Martin Biel (KTH) Stochastic Programming for Hydropower 8 / 25 • StochasticPrograms.jl ∘ Flexible model creation • HydroModels.jl

  9. ∘ Stochastic programming constructs ∘ Model creation focused on data and optimization formulation ∘ Effjcient model reinitialization ∘ Predefjned models • Short-term production planning • Optimal orders on the day-ahead market New Approach Martin Biel (KTH) Stochastic Programming for Hydropower 8 / 25 • StochasticPrograms.jl ∘ Flexible model creation ∘ Parallel capabilities based on the Distributed module • HydroModels.jl

  10. ∘ Model creation focused on data and optimization formulation ∘ Effjcient model reinitialization ∘ Predefjned models • Short-term production planning • Optimal orders on the day-ahead market New Approach Martin Biel (KTH) Stochastic Programming for Hydropower 8 / 25 • StochasticPrograms.jl ∘ Flexible model creation ∘ Parallel capabilities based on the Distributed module ∘ Stochastic programming constructs • HydroModels.jl

  11. ∘ Effjcient model reinitialization ∘ Predefjned models • Short-term production planning • Optimal orders on the day-ahead market New Approach Martin Biel (KTH) Stochastic Programming for Hydropower 8 / 25 • StochasticPrograms.jl ∘ Flexible model creation ∘ Parallel capabilities based on the Distributed module ∘ Stochastic programming constructs • HydroModels.jl ∘ Model creation focused on data and optimization formulation

  12. ∘ Predefjned models • Short-term production planning • Optimal orders on the day-ahead market New Approach Martin Biel (KTH) Stochastic Programming for Hydropower 8 / 25 • StochasticPrograms.jl ∘ Flexible model creation ∘ Parallel capabilities based on the Distributed module ∘ Stochastic programming constructs • HydroModels.jl ∘ Model creation focused on data and optimization formulation ∘ Effjcient model reinitialization

  13. New Approach Martin Biel (KTH) Stochastic Programming for Hydropower 8 / 25 • StochasticPrograms.jl ∘ Flexible model creation ∘ Parallel capabilities based on the Distributed module ∘ Stochastic programming constructs • HydroModels.jl ∘ Model creation focused on data and optimization formulation ∘ Effjcient model reinitialization ∘ Predefjned models • Short-term production planning • Optimal orders on the day-ahead market

  14. StochasticPrograms.jl - Simple Example where Stochastic Programming for Hydropower Martin Biel (KTH) s.t. minimize Q ( x 1 , x 2 , 𝜊) = min y 1 , y 2 ∈ℝ x 1 , x 2 ∈ℝ s.t. 9 / 25 100 x 1 + 150 x 2 + 𝔽 𝜕 [ Q ( x 1 , x 2 , 𝜊)] x 1 + x 2 ≤ 120 x 1 ≥ 40 x 2 ≥ 20 q 1 (𝜊) y 1 + q 2 (𝜊) y 2 6 y 1 + 10 y 2 ≤ 60 x 1 8 y 1 + 5 y 2 ≤ 80 x 2 0 ≤ y 1 ≤ d 1 (𝜊) 0 ≤ y 2 ≤ d 2 (𝜊)

  15. StochasticPrograms.jl - Simple Example Martin Biel (KTH) Stochastic Programming for Hydropower 10 / 25 sp = StochasticProgram(solver=ClpSolver()) @first_stage sp = begin @variable(model, x₁ >= 40) @variable(model, x₂ >= 20) @objective(model, Min, 100*x₁ + 150*x₂) @constraint(model, x₁+x₂ <= 120) end @second_stage sp = begin @decision x₁ x₂ s = scenario @variable(model, 0 <= y₁ <= s.d[1]) @variable(model, 0 <= y₂ <= s.d[2]) @objective(model, Min, s.q[1]*y₁ + s.q[2]*y₂) @constraint(model, 6*y₁ + 10*y₂ <= 60*x₁) @constraint(model, 8*y₁ + 5*y₂ <= 80*x₂) end

  16. StochasticPrograms.jl - Simple Example Creates a generator function for the fjrst stage model Stochastic Programming for Hydropower Martin Biel (KTH) 10 / 25 sp = StochasticProgram(solver=ClpSolver()) @first_stage sp = begin @variable(model, x₁ >= 40) @variable(model, x₂ >= 20) @objective(model, Min, 100*x₁ + 150*x₂) @constraint(model, x₁+x₂ <= 120) end @second_stage sp = begin @decision x₁ x₂ s = scenario @variable(model, 0 <= y₁ <= s.d[1]) @variable(model, 0 <= y₂ <= s.d[2]) @objective(model, Min, s.q[1]*y₁ + s.q[2]*y₂) @constraint(model, 6*y₁ + 10*y₂ <= 60*x₁) @constraint(model, 8*y₁ + 5*y₂ <= 80*x₂) end

  17. StochasticPrograms.jl - Simple Example Creates a generator function for the second stage model Stochastic Programming for Hydropower Martin Biel (KTH) 10 / 25 sp = StochasticProgram(solver=ClpSolver()) @first_stage sp = begin @variable(model, x₁ >= 40) @variable(model, x₂ >= 20) @objective(model, Min, 100*x₁ + 150*x₂) @constraint(model, x₁+x₂ <= 120) end @second_stage sp = begin @decision x₁ x₂ s = scenario @variable(model, 0 <= y₁ <= s.d[1]) @variable(model, 0 <= y₂ <= s.d[2]) @objective(model, Min, s.q[1]*y₁ + s.q[2]*y₂) @constraint(model, 6*y₁ + 10*y₂ <= 60*x₁) @constraint(model, 8*y₁ + 5*y₂ <= 80*x₂) end

  18. StochasticPrograms.jl - Simple Example Explicitly denote that some variables originate from the fjrst stage Stochastic Programming for Hydropower Martin Biel (KTH) 10 / 25 sp = StochasticProgram(solver=ClpSolver()) @first_stage sp = begin @variable(model, x₁ >= 40) @variable(model, x₂ >= 20) @objective(model, Min, 100*x₁ + 150*x₂) @constraint(model, x₁+x₂ <= 120) end @second_stage sp = begin @decision x₁ x₂ s = scenario @variable(model, 0 <= y₁ <= s.d[1]) @variable(model, 0 <= y₂ <= s.d[2]) @objective(model, Min, s.q[1]*y₁ + s.q[2]*y₂) @constraint(model, 6*y₁ + 10*y₂ <= 60*x₁) @constraint(model, 8*y₁ + 5*y₂ <= 80*x₂) end

  19. StochasticPrograms.jl - Simple Example Injection point for scenario data Stochastic Programming for Hydropower Martin Biel (KTH) 10 / 25 sp = StochasticProgram(solver=ClpSolver()) @first_stage sp = begin @variable(model, x₁ >= 40) @variable(model, x₂ >= 20) @objective(model, Min, 100*x₁ + 150*x₂) @constraint(model, x₁+x₂ <= 120) end @second_stage sp = begin @decision x₁ x₂ s = scenario @variable(model, 0 <= y₁ <= s.d[1]) @variable(model, 0 <= y₂ <= s.d[2]) @objective(model, Min, s.q[1]*y₁ + s.q[2]*y₂) @constraint(model, 6*y₁ + 10*y₂ <= 60*x₁) @constraint(model, 8*y₁ + 5*y₂ <= 80*x₂) end

  20. StochasticPrograms.jl - Simple Example Martin Biel (KTH) Stochastic Programming for Hydropower 11 / 25 struct SimpleScenario <: AbstractScenarioData p:: Float64 d:: Vector { Float64 } q:: Vector { Float64 } end StochasticPrograms.probability(s::SimpleScenario) = s.p

  21. StochasticPrograms.jl - Simple Example Add two scenarios to the stochastic program Martin Biel (KTH) Stochastic Programming for Hydropower 11 / 25 struct SimpleScenario <: AbstractScenarioData p:: Float64 d:: Vector { Float64 } q:: Vector { Float64 } end StochasticPrograms.probability(s::SimpleScenario) = s.p s1 = SimpleScenario(0.4,[500.0,100],[-24.0,-28]) s2 = SimpleScenario(0.6,[300.0,300],[-28.0,-32]) append!(sp,[s1,s2])

  22. StochasticPrograms.jl - Simple Example Martin Biel (KTH) Stochastic Programming for Hydropower 11 / 25 print(sp)

  23. StochasticPrograms.jl - Simple Example Martin Biel (KTH) Stochastic Programming for Hydropower 11 / 25 print(sp) First-stage ============== Min 100 x₁ + 150 x₂ Subject to x₁ + x₂ ≤ 120 x₁ ≥ 40 x₂ ≥ 20 Second-stage ============== Subproblem 1: Min -24 y₁ - 28 y₂ Subject to 6 y₁ + 10 y₂ - 60 x₁ ≤ 0 8 y₁ + 5 y₂ - 80 x₂ ≤ 0 0 ≤ y₁ ≤ 500 0 ≤ y₂ ≤ 100 Subproblem 2: Min -28 y₁ - 32 y₂ Subject to 6 y₁ + 10 y₂ - 60 x₁ ≤ 0 8 y₁ + 5 y₂ - 80 x₂ ≤ 0 0 ≤ y₁ ≤ 300 0 ≤ y₂ ≤ 300

  24. • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25

  25. • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly

  26. • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions

  27. • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks

  28. • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks

  29. • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords

  30. • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds

  31. • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds

  32. • Effjcient parallel implementation • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation

  33. • Versatility Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation • Effjcient parallel implementation

  34. Implementation Details Deferred model creation Data injection Implications Martin Biel (KTH) Stochastic Programming for Hydropower 12 / 25 • JuMP models are not created instantly • Model defjnitions are stored in generating lambda functions • These model recipes are then used as building blocks • The generating functions contain certain placeholders keywords • Upon model creation, the keywords contain the required data fjelds • Flexible model creation and reformulation • Effjcient parallel implementation • Versatility

  35. dep = DEP(sp) Minimization problem with: * 5 linear constraints * 6 variables Solver is ClpMathProg • First stage generator • Second stage generator on all available scenarios • Connections possible due to the @decision annotation • DEP model is cached internally until new scenarios are added Stochastic Programming for Hydropower Martin Biel (KTH) StochasticPrograms.jl - Deterministically Equivalent Model minimize Ax = b s.t. c T x + 𝔽 𝜕 [ Q ( x , 𝜊(𝜕))] x ∈ℝ n 13 / 25

  36. • First stage generator • Second stage generator on all available scenarios • Connections possible due to the @decision annotation • DEP model is cached internally until new scenarios are added StochasticPrograms.jl - Deterministically Equivalent Model minimize Stochastic Programming for Hydropower Martin Biel (KTH) 13 / 25 Ax = b s.t. c T x + 𝔽 𝜕 [ Q ( x , 𝜊(𝜕))] x ∈ℝ n dep = DEP(sp) Minimization problem with: * 5 linear constraints * 6 variables Solver is ClpMathProg

  37. • Second stage generator on all available scenarios • Connections possible due to the @decision annotation • DEP model is cached internally until new scenarios are added StochasticPrograms.jl - Deterministically Equivalent Model minimize Stochastic Programming for Hydropower Martin Biel (KTH) 13 / 25 s.t. Ax = b c T x + 𝔽 𝜕 [ Q ( x , 𝜊(𝜕))] x ∈ℝ n dep = DEP(sp) Minimization problem with: * 5 linear constraints * 6 variables Solver is ClpMathProg • First stage generator

  38. • Connections possible due to the @decision annotation • DEP model is cached internally until new scenarios are added StochasticPrograms.jl - Deterministically Equivalent Model minimize Stochastic Programming for Hydropower Martin Biel (KTH) 13 / 25 Ax = b s.t. c T x + 𝔽 𝜕 [ Q ( x , 𝜊(𝜕))] x ∈ℝ n dep = DEP(sp) Minimization problem with: * 5 linear constraints * 6 variables Solver is ClpMathProg • First stage generator • Second stage generator on all available scenarios

  39. • DEP model is cached internally until new scenarios are added StochasticPrograms.jl - Deterministically Equivalent Model minimize Stochastic Programming for Hydropower Martin Biel (KTH) 13 / 25 s.t. Ax = b c T x + 𝔽 𝜕 [ Q ( x , 𝜊(𝜕))] x ∈ℝ n dep = DEP(sp) Minimization problem with: * 5 linear constraints * 6 variables Solver is ClpMathProg • First stage generator • Second stage generator on all available scenarios • Connections possible due to the @decision annotation

  40. StochasticPrograms.jl - Deterministically Equivalent Model minimize Stochastic Programming for Hydropower Martin Biel (KTH) 13 / 25 Ax = b s.t. c T x + 𝔽 𝜕 [ Q ( x , 𝜊(𝜕))] x ∈ℝ n dep = DEP(sp) Minimization problem with: * 5 linear constraints * 6 variables Solver is ClpMathProg • First stage generator • Second stage generator on all available scenarios • Connections possible due to the @decision annotation • DEP model is cached internally until new scenarios are added

  41. StochasticPrograms.jl - Deterministically Equivalent Model Martin Biel (KTH) Stochastic Programming for Hydropower 13 / 25 print(dep)

  42. StochasticPrograms.jl - Deterministically Equivalent Model Martin Biel (KTH) Stochastic Programming for Hydropower 13 / 25 print(dep) Min 100 x₁ + 150 x₂ - 9.6 y₁_1 - 11.2 y₂_1 - 16.8 y₁_2 - 19.2 y₂_2 Subject to x₁ + x₂ ≤ 120 6 y₁_1 + 10 y₂_1 - 60 x₁ ≤ 0 8 y₁_1 + 5 y₂_1 - 80 x₂ ≤ 0 6 y₁_2 + 10 y₂_2 - 60 x₁ ≤ 0 8 y₁_2 + 5 y₂_2 - 80 x₂ ≤ 0 x₁ ≥ 40 x₂ ≥ 20 0 ≤ y₁_1 ≤ 500 0 ≤ y₂_1 ≤ 100 0 ≤ y₁_2 ≤ 300 0 ≤ y₂_2 ≤ 300

  43. • Extended form solve(sp,solver=ClpSolver()) :Optimal getobjectivevalue(sp) -855.83 • L-shaped solve(sp,solver=LShapedSolver(:ls,ClpSolver())) L-Shaped Gap Time: 0:00:01 (4 iterations) Objective: -855.8333333333358 Gap: 2.1229209144670507e-15 Number of cuts: 5 :Optimal • Convenience function (Value of the recourse problem) VRP(sp,solver=ClpSolver()) -855.83 Stochastic Programming for Hydropower StochasticPrograms.jl - Solving Models Martin Biel (KTH) 14 / 25

  44. • L-shaped solve(sp,solver=LShapedSolver(:ls,ClpSolver())) L-Shaped Gap Time: 0:00:01 (4 iterations) Objective: -855.8333333333358 Gap: 2.1229209144670507e-15 Number of cuts: 5 :Optimal • Convenience function (Value of the recourse problem) VRP(sp,solver=ClpSolver()) -855.83 Stochastic Programming for Hydropower Martin Biel (KTH) StochasticPrograms.jl - Solving Models 14 / 25 • Extended form solve(sp,solver=ClpSolver()) :Optimal getobjectivevalue(sp) -855.83

  45. • Convenience function (Value of the recourse problem) VRP(sp,solver=ClpSolver()) -855.83 StochasticPrograms.jl - Solving Models Stochastic Programming for Hydropower Martin Biel (KTH) 14 / 25 • Extended form solve(sp,solver=ClpSolver()) :Optimal getobjectivevalue(sp) -855.83 • L-shaped solve(sp,solver=LShapedSolver(:ls,ClpSolver())) L-Shaped Gap Time: 0:00:01 (4 iterations) Objective: -855.8333333333358 Gap: 2.1229209144670507e-15 Number of cuts: 5 :Optimal

  46. StochasticPrograms.jl - Solving Models Martin Biel (KTH) Stochastic Programming for Hydropower 14 / 25 • Extended form solve(sp,solver=ClpSolver()) :Optimal getobjectivevalue(sp) -855.83 • L-shaped solve(sp,solver=LShapedSolver(:ls,ClpSolver())) L-Shaped Gap Time: 0:00:01 (4 iterations) Objective: -855.8333333333358 Gap: 2.1229209144670507e-15 Number of cuts: 5 :Optimal • Convenience function (Value of the recourse problem) VRP(sp,solver=ClpSolver()) -855.83

  47. ws = WS(sp,s1) Minimization problem with: * 3 linear constraints * 4 variables Solver is ClpMathProg • First stage generator • Second stage generator on the given scenario StochasticPrograms.jl - Wait-And-See Models Stochastic Programming for Hydropower Martin Biel (KTH) 𝜊 minimize ̃ for given x ≥ 0 Ax = b s.t. 𝜊) x ∈ℝ n 15 / 25 c T x + Q ( x , ̃

  48. • First stage generator • Second stage generator on the given scenario StochasticPrograms.jl - Wait-And-See Models ̃ Stochastic Programming for Hydropower Martin Biel (KTH) minimize 𝜊 for given x ≥ 0 Ax = b s.t. 𝜊) x ∈ℝ n 15 / 25 c T x + Q ( x , ̃ ws = WS(sp,s1) Minimization problem with: * 3 linear constraints * 4 variables Solver is ClpMathProg

  49. • Second stage generator on the given scenario StochasticPrograms.jl - Wait-And-See Models for given Stochastic Programming for Hydropower Martin Biel (KTH) minimize ̃ 𝜊 x ≥ 0 Ax = b s.t. 𝜊) x ∈ℝ n 15 / 25 c T x + Q ( x , ̃ ws = WS(sp,s1) Minimization problem with: * 3 linear constraints * 4 variables Solver is ClpMathProg • First stage generator

  50. StochasticPrograms.jl - Wait-And-See Models for given Stochastic Programming for Hydropower Martin Biel (KTH) minimize ̃ 𝜊 x ≥ 0 Ax = b s.t. 𝜊) x ∈ℝ n 15 / 25 c T x + Q ( x , ̃ ws = WS(sp,s1) Minimization problem with: * 3 linear constraints * 4 variables Solver is ClpMathProg • First stage generator • Second stage generator on the given scenario

  51. StochasticPrograms.jl - Wait-And-See Models Martin Biel (KTH) Stochastic Programming for Hydropower 15 / 25 print(ws)

  52. StochasticPrograms.jl - Wait-And-See Models Martin Biel (KTH) Stochastic Programming for Hydropower 15 / 25 print(ws) Min 100 x₁ + 150 x₂ - 24 y₁ - 28 y₂ Subject to x₁ + x₂ ≤ 120 6 y₁ + 10 y₂ - 60 x₁ ≤ 0 8 y₁ + 5 y₂ - 80 x₂ ≤ 0 x₁ ≥ 40 x₂ ≥ 20 0 ≤ y₁ ≤ 500 0 ≤ y₂ ≤ 100

  53. function expected(scenarios:: Vector {SimpleScenario}) return SimpleScenario(sum([s.p for s in scenarios]), sum([s.p*s.d for s in scenarios]), sum([s.p*s.q for s in scenarios])) end StochasticPrograms.jl - Expected Value Problems Must be possible to take expectation over scenarios Stochastic Programming for Hydropower Martin Biel (KTH) 𝜊 = 𝔽 𝜕 [𝜊(𝜕)] minimize ̄ where x ≥ 0 Ax = b s.t. 𝜊) x ∈ℝ n 16 / 25 c T x + Q ( x , ̄

  54. StochasticPrograms.jl - Expected Value Problems where Stochastic Programming for Hydropower Martin Biel (KTH) Must be possible to take expectation over scenarios minimize ̄ 𝜊 = 𝔽 𝜕 [𝜊(𝜕)] x ≥ 0 Ax = b s.t. 𝜊) x ∈ℝ n 16 / 25 c T x + Q ( x , ̄ function expected(scenarios:: Vector {SimpleScenario}) return SimpleScenario(sum([s.p for s in scenarios]), sum([s.p*s.d for s in scenarios]), sum([s.p*s.q for s in scenarios])) end

  55. StochasticPrograms.jl - Expected Value Problems Martin Biel (KTH) Stochastic Programming for Hydropower 16 / 25 evp = EVP(sp) Minimization problem with: * 3 linear constraints * 4 variables Solver is ClpMathProg print(evp)

  56. StochasticPrograms.jl - Expected Value Problems Martin Biel (KTH) Stochastic Programming for Hydropower 16 / 25 evp = EVP(sp) Minimization problem with: * 3 linear constraints * 4 variables Solver is ClpMathProg print(evp) Min 100 x₁ + 150 x₂ - 26.4 y₁ - 30.4 y₂ Subject to x₁ + x₂ ≤ 120 6 y₁ + 10 y₂ - 60 x₁ ≤ 0 8 y₁ + 5 y₂ - 80 x₂ ≤ 0 x₁ ≥ 40 x₂ ≥ 20 0 ≤ y₁ ≤ 380 0 ≤ y₂ ≤ 220

  57. x ̂ = [50,50]; eval_decision(sp,x ̂ ,solver=ClpSolver()) 356.0 • Create fjrst stage variables using generator • Fixate them to the given values • Generate the second stage problems • Again, linking handled through @decision • Solve resulting JuMP model Stochastic Programming for Hydropower Martin Biel (KTH) StochasticPrograms.jl - Decision Evaulation x + 𝔽 𝜕 [ Q ( ̂ x , 𝜊(𝜕))] 17 / 25 c T ̂

  58. • Fixate them to the given values • Generate the second stage problems • Again, linking handled through @decision • Solve resulting JuMP model StochasticPrograms.jl - Decision Evaulation Stochastic Programming for Hydropower Martin Biel (KTH) 17 / 25 x + 𝔽 𝜕 [ Q ( ̂ x , 𝜊(𝜕))] c T ̂ ̂ x = [50,50]; eval_decision(sp,x ̂ ,solver=ClpSolver()) 356.0 • Create fjrst stage variables using generator

  59. • Generate the second stage problems • Again, linking handled through @decision • Solve resulting JuMP model StochasticPrograms.jl - Decision Evaulation Stochastic Programming for Hydropower Martin Biel (KTH) 17 / 25 x , 𝜊(𝜕))] x + 𝔽 𝜕 [ Q ( ̂ c T ̂ ̂ x = [50,50]; eval_decision(sp,x ̂ ,solver=ClpSolver()) 356.0 • Create fjrst stage variables using generator • Fixate them to the given values

  60. • Again, linking handled through @decision • Solve resulting JuMP model StochasticPrograms.jl - Decision Evaulation Stochastic Programming for Hydropower Martin Biel (KTH) 17 / 25 x + 𝔽 𝜕 [ Q ( ̂ x , 𝜊(𝜕))] c T ̂ ̂ x = [50,50]; eval_decision(sp,x ̂ ,solver=ClpSolver()) 356.0 • Create fjrst stage variables using generator • Fixate them to the given values • Generate the second stage problems

  61. • Solve resulting JuMP model StochasticPrograms.jl - Decision Evaulation Stochastic Programming for Hydropower Martin Biel (KTH) 17 / 25 x , 𝜊(𝜕))] x + 𝔽 𝜕 [ Q ( ̂ c T ̂ ̂ x = [50,50]; eval_decision(sp,x ̂ ,solver=ClpSolver()) 356.0 • Create fjrst stage variables using generator • Fixate them to the given values • Generate the second stage problems • Again, linking handled through @decision

  62. StochasticPrograms.jl - Decision Evaulation Martin Biel (KTH) Stochastic Programming for Hydropower 17 / 25 x + 𝔽 𝜕 [ Q ( ̂ x , 𝜊(𝜕))] c T ̂ x ̂ = [50,50]; eval_decision(sp,x ̂ ,solver=ClpSolver()) 356.0 • Create fjrst stage variables using generator • Fixate them to the given values • Generate the second stage problems • Again, linking handled through @decision • Solve resulting JuMP model

  63. • Expected wait-and-see solution (EWS) EWS(sp,solver=ClpSolver()) -1518.75 • Expected value of perfect information (EVPI = VRP - EWS) EVPI(sp,solver=ClpSolver()) 662.92 • Value of the stochastic solution (VSS = EEV - VRP) VSS(sp,solver=ClpSolver()) 286.92 Stochastic Programming for Hydropower Martin Biel (KTH) Many of the required calculations are embarassingly parallel StochasticPrograms.jl - Stochastic Measures 18 / 25 • Expected value of using the expected solution (EEV) EEV(sp,solver=ClpSolver()) -568.92

  64. • Expected value of perfect information (EVPI = VRP - EWS) EVPI(sp,solver=ClpSolver()) 662.92 • Value of the stochastic solution (VSS = EEV - VRP) VSS(sp,solver=ClpSolver()) 286.92 Stochastic Programming for Hydropower Martin Biel (KTH) Many of the required calculations are embarassingly parallel StochasticPrograms.jl - Stochastic Measures 18 / 25 • Expected value of using the expected solution (EEV) EEV(sp,solver=ClpSolver()) -568.92 • Expected wait-and-see solution (EWS) EWS(sp,solver=ClpSolver()) -1518.75

  65. • Value of the stochastic solution (VSS = EEV - VRP) VSS(sp,solver=ClpSolver()) 286.92 StochasticPrograms.jl - Stochastic Measures Stochastic Programming for Hydropower Martin Biel (KTH) Many of the required calculations are embarassingly parallel 18 / 25 • Expected value of using the expected solution (EEV) EEV(sp,solver=ClpSolver()) -568.92 • Expected wait-and-see solution (EWS) EWS(sp,solver=ClpSolver()) -1518.75 • Expected value of perfect information (EVPI = VRP - EWS) EVPI(sp,solver=ClpSolver()) 662.92

  66. StochasticPrograms.jl - Stochastic Measures Many of the required calculations are embarassingly parallel Stochastic Programming for Hydropower Martin Biel (KTH) 18 / 25 • Expected value of using the expected solution (EEV) EEV(sp,solver=ClpSolver()) -568.92 • Expected wait-and-see solution (EWS) EWS(sp,solver=ClpSolver()) -1518.75 • Expected value of perfect information (EVPI = VRP - EWS) EVPI(sp,solver=ClpSolver()) 662.92 • Value of the stochastic solution (VSS = EEV - VRP) VSS(sp,solver=ClpSolver()) 286.92

  67. StochasticPrograms.jl - Stochastic Measures Many of the required calculations are embarassingly parallel Stochastic Programming for Hydropower Martin Biel (KTH) 18 / 25 • Expected value of using the expected solution (EEV) EEV(sp,solver=ClpSolver()) -568.92 • Expected wait-and-see solution (EWS) EWS(sp,solver=ClpSolver()) -1518.75 • Expected value of perfect information (EVPI = VRP - EWS) EVPI(sp,solver=ClpSolver()) 662.92 • Value of the stochastic solution (VSS = EEV - VRP) VSS(sp,solver=ClpSolver()) 286.92

  68. LShapedSolvers.jl L-shaped algorithm variants Martin Biel (KTH) Stochastic Programming for Hydropower 19 / 25 • L-shaped [Van Slyke,Wets] • Multicut L-shaped [Birge,Louveaux] • Regularized decomposition [Ruszczyński] • Trust-region L-shaped [Linderoth,Wright] • Level-set [Fábián,Szőke]

  69. • Distributed L-shaped variants 1. Distributed L-shaped with multiple cuts: LShapedSolver(:dls) 2. Distributed regularized L-shaped: LShapedSolver(:drd) 3. Distributed L-shaped with trust region: LShapedSolver(:dtr) 4. Distributed L-shaped with level sets: LShapedSolver(:dlv) • Trait based implementation. Every solver is a combination of a: ∘ Regularization trait ∘ Parallelization trait • Subproblems are solved using MathProgBase solvers Stochastic Programming for Hydropower Martin Biel (KTH) LShapedSolvers.jl 20 / 25 • L-shaped variants 1. L-shaped with multiple cuts (default): LShapedSolver(:ls) 2. L-shaped with regularized decomposition: LShapedSolver(:rd) 3. L-shaped with trust region: LShapedSolver(:tr) 4. L-shaped with level sets: LShapedSolver(:lv)

  70. • Trait based implementation. Every solver is a combination of a: ∘ Regularization trait ∘ Parallelization trait • Subproblems are solved using MathProgBase solvers LShapedSolvers.jl Stochastic Programming for Hydropower Martin Biel (KTH) 20 / 25 • L-shaped variants 1. L-shaped with multiple cuts (default): LShapedSolver(:ls) 2. L-shaped with regularized decomposition: LShapedSolver(:rd) 3. L-shaped with trust region: LShapedSolver(:tr) 4. L-shaped with level sets: LShapedSolver(:lv) • Distributed L-shaped variants 1. Distributed L-shaped with multiple cuts: LShapedSolver(:dls) 2. Distributed regularized L-shaped: LShapedSolver(:drd) 3. Distributed L-shaped with trust region: LShapedSolver(:dtr) 4. Distributed L-shaped with level sets: LShapedSolver(:dlv)

  71. • Subproblems are solved using MathProgBase solvers LShapedSolvers.jl Stochastic Programming for Hydropower Martin Biel (KTH) 20 / 25 • L-shaped variants 1. L-shaped with multiple cuts (default): LShapedSolver(:ls) 2. L-shaped with regularized decomposition: LShapedSolver(:rd) 3. L-shaped with trust region: LShapedSolver(:tr) 4. L-shaped with level sets: LShapedSolver(:lv) • Distributed L-shaped variants 1. Distributed L-shaped with multiple cuts: LShapedSolver(:dls) 2. Distributed regularized L-shaped: LShapedSolver(:drd) 3. Distributed L-shaped with trust region: LShapedSolver(:dtr) 4. Distributed L-shaped with level sets: LShapedSolver(:dlv) • Trait based implementation. Every solver is a combination of a: ∘ Regularization trait ∘ Parallelization trait

  72. LShapedSolvers.jl Martin Biel (KTH) Stochastic Programming for Hydropower 20 / 25 • L-shaped variants 1. L-shaped with multiple cuts (default): LShapedSolver(:ls) 2. L-shaped with regularized decomposition: LShapedSolver(:rd) 3. L-shaped with trust region: LShapedSolver(:tr) 4. L-shaped with level sets: LShapedSolver(:lv) • Distributed L-shaped variants 1. Distributed L-shaped with multiple cuts: LShapedSolver(:dls) 2. Distributed regularized L-shaped: LShapedSolver(:drd) 3. Distributed L-shaped with trust region: LShapedSolver(:dtr) 4. Distributed L-shaped with level sets: LShapedSolver(:dlv) • Trait based implementation. Every solver is a combination of a: ∘ Regularization trait ∘ Parallelization trait • Subproblems are solved using MathProgBase solvers

  73. • The user creates a model recipe using the @hydromodel macro • Defjne model indices • Defjne model data • Defjne modelindices(::AbstractHydroModelData, ::Horizon, args...) • Defjne optimization problem • horizon : the time horizon if the model • indices : structure with model indices • data : structure with model data HydroModels.jl Creating a Planning Problem Data injection keywords Martin Biel (KTH) Stochastic Programming for Hydropower 21 / 25 • Also based on deferred model creation and data injection

  74. • Defjne model indices • Defjne model data • Defjne modelindices(::AbstractHydroModelData, ::Horizon, args...) • Defjne optimization problem • horizon : the time horizon if the model • indices : structure with model indices • data : structure with model data HydroModels.jl Creating a Planning Problem Data injection keywords Martin Biel (KTH) Stochastic Programming for Hydropower 21 / 25 • Also based on deferred model creation and data injection • The user creates a model recipe using the @hydromodel macro

  75. • horizon : the time horizon if the model • indices : structure with model indices • data : structure with model data HydroModels.jl Creating a Planning Problem Data injection keywords Martin Biel (KTH) Stochastic Programming for Hydropower 21 / 25 • Also based on deferred model creation and data injection • The user creates a model recipe using the @hydromodel macro • Defjne model indices • Defjne model data • Defjne modelindices(::AbstractHydroModelData, ::Horizon, args...) • Defjne optimization problem

  76. HydroModels.jl Creating a Planning Problem Data injection keywords Martin Biel (KTH) Stochastic Programming for Hydropower 21 / 25 • Also based on deferred model creation and data injection • The user creates a model recipe using the @hydromodel macro • Defjne model indices • Defjne model data • Defjne modelindices(::AbstractHydroModelData, ::Horizon, args...) • Defjne optimization problem • horizon : the time horizon if the model • indices : structure with model indices • data : structure with model data

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