Gekko Variable Definition - Primary vrs. Utility Decision Variable

Question

I am trying to formulate and solve an optimization problem based on an article. The authors introduced 2 decision variables. Power of station i at time t, P_i,t, and a binary variable X_i,n which is 1 if vehicle n is assigned to station i. They introduced some other variables, called utility variables. For instance, energy delivered from station i up to time t for vehicle n, E_i,t,n which is calculated based on primary decision variables and a few fix parameters.

My question is should I define the utility variables as Gekko variables? If yes, which type is more appropriate?

I = 4 # number of stations
T = 24 # hours of simulation 
N = 5 # number of vehicles

p = m.Array(m.Var,(I,T),lb=0,ub= params.ev.max_power)
x = m.Array(m.Var,(I,N),lb=0,ub=1, integer = True)

Should I define E as follow to solve these equations as an example? This introduces extra variables that are not primary decision variables and are calculated based on other terms that depend on the primary decision variable.

E = m.Array(m.Var,(I,T,N),lb=0)

for i in range(I):
    for n in range(N):
        for t in range(T):
                    m.Equation(E[i][t][n] >= np.sum(0.25 * availability[n, :t] * p[i,:t]) - (M * (1 - x[i][n])))
                    m.Equation(E[i][t][n] <= np.sum(0.25 * availability[n, :t] * p[i,:t]) + (M * (1 - x[i][n])))
                    m.Equation(E[i][t][n] <= M * x[i][n])
                    m.Equation(E[i][t][n] >= -M * x[i][n])

Answer 1

All of those variable definitions and equations look correct. Here are a few suggestions:

• There is no availability[] variable defined yet. If availability is a function of other decision variables, then it is generally more efficient to use an m.Intermediate() definition to define it.
• As the total number of total decision variables increase, there is often a large increase in computational time. I recommend starting with a small problem initially and then scale-up to the larger sized problem.
• Try the gekko m.sum() instead of sum or np.sum() for potentially more efficient calculations. Using m.sum() does increase the model compile time but generally decreases the optimization solve time, so it is a trade-off.