Julia: How to introduce binary integers for mixed integers optimization problems with JuMP?

Question

Problem description

I am trying to do a mixed-integer optimization for a "Unit Commitment" problem in Julia with Jump. But JuMP expects my introduction of the unit activation variable, x[1:N], to be a number and not a variable. However, the unit activation is a binary integer decision variable for the optimization problem so I have trouble including the variable into the optimization problem.

What am I doing wrong?

My approaches have been:

Approach 1: Include x[1:N] as part of @variable macro for P_G.

m = Model(Cbc.Optimizer)                                        # Model
@variable(m, x[1:N], Bin)                                       # Unit activation
@variable(m, P_C[i,1]*x[i] <= P_G[i=1:N,1:T] <= P_C[i,2]*x[i])  # Unit generation limit
for i in 1:T                                                    # Load balance
    @constraint(m, sum(P_G[:,i]) == P_D[i])
end
@objective(m,Min,sum(P_G[:,1:T].*F[1:N]*x[1:N]))                # Objective function
optimize!(m)                                                    # Solve

This leads to the following error: LoadError: InexactError: convert(Float64, 50 x[1]).

Approach 2: Define the feasible region for P_G as a constraint including x[1:N]:

m = Model(Cbc.Optimizer)                                        # Model
@variable(m, x[1:N])                                            # Unit activation
@variable(m, P_G[i=1:N,1:T] )                                   # Unit generation limit
for i in 1:T                                                    # Load balance
        @constraint(m, sum(P_G[:,i]) == P_D[i])
end
for i in 1:N                                                    # Unit generation limit
        for j in 1:T
                @constraint(m, P_C[i,1]*x[i] <= P_G[i,j] <= P_C[i,2]*x[i])
        end
end
@objective(m,Min,sum(P_G[:,1:T].*F[1:N]*x[1:N]))                # Objective function
optimize!(m)                                                    # Solve

This leads to: LoadError: [..] '@constraint(m, $(Expr(:escape, :(P_C[i, 1]))) * $(Expr(:escape, :(x[i]))) <= P_G[i, j] <= $(Expr(:escape, :(P_C[i, 2]))) * $(Expr(:escape, :(x[i]))))': Expected 50 x[1] to be a number.

NB: there might be more proper iteration methods but this should be idiot proof to Julia and JuMP newbies like me.

Working code without mixed-integer optimization

using JuMP, Cbc                 # Optimization and modelling
using Plots, LaTeXStrings       # Plotting

# DATA
P_C  = [50 200;                                         # Power capacity [:, (min, max)]
        25 200;
        100 200;
        120 500;
        10 500;
        20 500;
        200 800;
        200 800;
        100 800;
        200 1000;]
P_D = LinRange(sum(P_C[:,1]), sum(P_C[:,2]), 100)       # Power demand
F = rand(100:500,10)                                    # Random prod. prices
T = length(P_D)                                         # Number of time steps
N = length(P_C[:,1])                                    # Number of generators

# MODEL
m = Model(Cbc.Optimizer)                                # Model

@variable(m, x[1:N], Bin)                               # Unit activation
@variable(m, P_C[i,1] <= P_G[i=1:N,1:T] <= P_C[i,2])    # Unit generation limit
for i in 1:T                                            # Load balance
    @constraint(m, sum(P_G[:,i]) == P_D[i])
end
@objective(m,Min,sum(P_G[:,1:T].*F[1:N]))               # Objective function
optimize!(m)                                            # Solve

# PLOT
plt = plot(P_D[:],value.(P_G[:,1:T])', xlab = L"P_{load} [MW]", ylab = L"P_{unit} [MW]")
@show plt

Which should produce something similar:

The expected outcome of introducing the unit activation variable would be that each unit is not required to generate power in the the lower region of the P_load.

Preliminary

I have introduced the basics of the problem with success:

• Objective function: Minimize the cost of power generation
• Variable, P_G: for power generation (feasible region defined by min and max capacity, P_C)
• Production cost, F (as constant only!)
• Power demand, P_D, is set to be a linear space from the min power cap. to the max cap.

Mathematically expressed:

Answer 1

Variable bounds cannot include other variables. Do:

m = Model(Cbc.Optimizer)
@variable(m, x[1:N], Bin)
@variable(m, P_G[i=1:N,1:T])
@constraint(m, [i=1:N, t=1:T], P_C[i, 1] * x[i] <= P_G[i, t])
@constraint(m, [i=1:N, t=1:T], P_G[i, t] <= P_C[I, 2] * x[i])