MATLAB: Energy storage optimisation problem – separate charge and discharge constraint

chargingdischargingenergy storageMATLABoptimization problem

Hi all!
I am currently working on an optimization problem to maximize the revenue from a combined wind turbine and energy storage system. With the code below, the system charges and discharges simultaneously at certain times. I am able to change the charge and discharge variables to a single variable with the lowerbound as a negative and positive number corresponding to discharging and charging, respectively. But in doing this, I cannot accurately apply the charge and discharge efficiencies of the storage device.
prob = optimproblem;
% Decision variables
% Energy storage system decision variables
ESS_ch = optimvar('ESS_ch',T,'LowerBound',0,'UpperBound',ESS_Pmax);
ESS_disch = optimvar('ESS_disch',T,'LowerBound',0,'UpperBound',ESS_Pmax);
ESS_SOC = optimvar('ESS_SOC',T,'LowerBound',0,'UpperBound',ESS_Cmax);
% Output power to the grid variables
Grid_E = optimvar('Grid_E',T,'LowerBound',0);
% Energy storage operational constraints
prob.Constraints.energyStorage  = optimconstr(T);
prob.Constraints.energyStorage(1) = ESS_SOC(1) == 10;
prob.Constraints.energyStorage(2:T) = ESS_SOC(2:T) == ESS_SOC(1:T-1)*(1-d) - ESS_disch(1:T-1)/eff + ESS_ch(1:T-1)*eff;
% Energy balance constraint (WT_E = wind turbine power output as vector of
% length T)
prob.Constraints.EnergyBalance = Grid_E == WT_E + ESS_disch - ESS_ch;
prob.ObjectiveSense = 'maximize';
prob.Objective = sum(Grid_E.*Price);
I have also attempted to apply a constraint that dictates the charge power multilied by the discharge power of the storage device is equal to zero at any single time step, see below.
prob.Constraints.chargeonoff  = optimconstr(T);
for i = 1:T
    prob.Constraints.chargeonoff(i) = ESS_disch(i)*ESS_ch(i) == 0;
end
However, when I add this the following error occurs.
Error using optim.problemdef.OptimizationProblem/solve
SOLVE requires a non-empty initial point structure to solve a nonlinear problem.
Any assistance or insight on how to force the program to either charge or discharge at each time step would be greatly appreciated.

Best Answer

The error message is spot on: when you multiply variables, the problem becomes nonlinear. However, I am sure that you would rather keep the problem linear if you can.
Choose a value M that upper bounds a nonnegative decision variable x. I mean x is ESS_disch or ESS_ch. Create a binary variable z_x that will indicate when x is positive. Set a linear constraint so that z tracks whether x is positive or not.
x <= M*z_x; % if x is positive, z_x has to be positive
Include z in the objective function, so the objective is minimized when z_x = 0.
Now to make the constraint that x and y cannot both be positive, create the linear constraint
z_x + z_y <= 1;
I may have some detail wrong, but I think that you see the idea. You are creating extra integer variables and linear constraints, and turning the problem from a linear programming problem to a mixed-integer linear programming problem. That keeps the problem out of the general nonlinear framework, which is a good thing.
Alan Weiss
MATLAB mathematical toolbox documentation
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