Biobjective hydro-thermal
This tutorial was generated using Literate.jl. Download the source as a .jl file. Download the source as a .ipynb file.
using SDDP, HiGHS, Statistics, Test
function biobjective_example()
model = SDDP.LinearPolicyGraph(;
stages = 3,
lower_bound = 0.0,
optimizer = HiGHS.Optimizer,
) do subproblem, _
@variable(subproblem, 0 <= v <= 200, SDDP.State, initial_value = 50)
@variables(subproblem, begin
0 <= g[i = 1:2] <= 100
0 <= u <= 150
s >= 0
shortage_cost >= 0
end)
@expressions(subproblem, begin
objective_1, g[1] + 10 * g[2]
objective_2, shortage_cost
end)
@constraints(subproblem, begin
inflow_constraint, v.out == v.in - u - s
g[1] + g[2] + u == 150
shortage_cost >= 40 - v.out
shortage_cost >= 60 - 2 * v.out
shortage_cost >= 80 - 4 * v.out
end)
# You must call this for a biobjective problem!
SDDP.initialize_biobjective_subproblem(subproblem)
SDDP.parameterize(subproblem, 0.0:5:50.0) do ω
JuMP.set_normalized_rhs(inflow_constraint, ω)
# You must call `set_biobjective_functions` from within
# `SDDP.parameterize`.
return SDDP.set_biobjective_functions(
subproblem,
objective_1,
objective_2,
)
end
end
pareto_weights =
SDDP.train_biobjective(model; solution_limit = 10, iteration_limit = 10)
solutions = [(k, v) for (k, v) in pareto_weights]
sort!(solutions; by = x -> x[1])
@test length(solutions) == 10
# Test for convexity! The gradient must be decreasing as we move from left
# to right.
gradient(a, b) = (b[2] - a[2]) / (b[1] - a[1])
grad = Inf
for i in 1:9
new_grad = gradient(solutions[i], solutions[i+1])
@test new_grad < grad
grad = new_grad
end
return
end
biobjective_example()-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : false
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 5]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 0.000000e+00 0.000000e+00 8.987904e-03 36 1
10 0.000000e+00 0.000000e+00 2.858281e-02 360 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 2.858281e-02
total solves : 360
best bound : 0.000000e+00
simulation ci : 0.000000e+00 ± 0.000000e+00
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 7]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 6.750000e+02 5.500000e+02 2.787828e-03 407 1
10 4.500000e+02 5.733959e+02 6.219983e-02 731 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 6.219983e-02
total solves : 731
best bound : 5.733959e+02
simulation ci : 5.000000e+02 ± 1.079583e+02
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 14]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 4.850000e+02 3.349793e+02 3.133059e-03 778 1
10 3.550000e+02 3.468286e+02 3.127694e-02 1102 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 3.127694e-02
total solves : 1102
best bound : 3.468286e+02
simulation ci : 3.948309e+02 ± 7.954180e+01
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 19]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 1.887500e+02 1.995243e+02 2.788067e-03 1149 1
10 2.962500e+02 2.052855e+02 2.994609e-02 1473 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 2.994609e-02
total solves : 1473
best bound : 2.052855e+02
simulation ci : 2.040201e+02 ± 3.876873e+01
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 25]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 3.737500e+02 4.626061e+02 3.342152e-03 1520 1
10 2.450000e+02 4.658509e+02 3.395200e-02 1844 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 3.395200e-02
total solves : 1844
best bound : 4.658509e+02
simulation ci : 3.907376e+02 ± 9.045105e+01
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 33]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 1.675000e+02 1.129545e+02 3.429174e-03 1891 1
10 1.362500e+02 1.129771e+02 3.066206e-02 2215 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 3.066206e-02
total solves : 2215
best bound : 1.129771e+02
simulation ci : 1.176375e+02 ± 1.334615e+01
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 36]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 2.562500e+02 2.788373e+02 3.242016e-03 2262 1
10 2.375000e+02 2.795671e+02 3.343606e-02 2586 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 3.343606e-02
total solves : 2586
best bound : 2.795671e+02
simulation ci : 2.375000e+02 ± 3.099032e+01
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 41]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 3.812500e+02 4.072952e+02 3.648043e-03 2633 1
10 5.818750e+02 4.080500e+02 3.555894e-02 2957 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 3.555894e-02
total solves : 2957
best bound : 4.080500e+02
simulation ci : 4.235323e+02 ± 1.029245e+02
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 47]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 8.525000e+02 5.197742e+02 3.473997e-03 3004 1
10 4.493750e+02 5.211793e+02 3.642893e-02 3328 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 3.642893e-02
total solves : 3328
best bound : 5.211793e+02
simulation ci : 5.268125e+02 ± 1.227709e+02
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 1.33100e+03
existing cuts : true
options
solver : serial mode
risk measure : SDDP.Expectation()
sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
VariableRef : [9, 9]
AffExpr in MOI.EqualTo{Float64} : [2, 4]
AffExpr in MOI.GreaterThan{Float64} : [3, 53]
VariableRef in MOI.EqualTo{Float64} : [1, 1]
VariableRef in MOI.GreaterThan{Float64} : [8, 8]
VariableRef in MOI.LessThan{Float64} : [5, 6]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
1 3.437500e+01 5.937500e+01 3.152847e-03 3375 1
10 3.750000e+01 5.938557e+01 3.139281e-02 3699 1
-------------------------------------------------------------------
status : iteration_limit
total time (s) : 3.139281e-02
total solves : 3699
best bound : 5.938557e+01
simulation ci : 5.906250e+01 ± 1.352595e+01
numeric issues : 0
-------------------------------------------------------------------