I am modelling a battery and interested in finding the best current that maximizes the energy output, using fmincon. I will summarize the idea:
- The battery is a system of differential-algebraic equations, so we employ ode15s.
- The cell potential will be obtained upon solving the DAE's, and this value is a subtraction of the integrated variables. Say it is y(20) - y(19) (the cathode minus the anode).
- We integrate the cell potential across the capacity, the capacity being "time multiplied by the current", so that the energy is trapz(t.*I, y(:, 20) - y(:, 19)).
- One of the restrictions in fmincon is that the time for the discharge doesn't exceed some limit. Thus, we would have something like t(end) = 10 in the ceq vector in nonlcon within fmincon.
The system can be solved only if we give a value to the electric current I. The most direct way is a constant discharge, so we just give ode15s some constant value of I. Another way can be to set up a function of the current, say I(t) = a*cos(t) + b*t. This function doesn't make sense, but anyways, fmincon will vary the parameters a and b in order to maximize the energy given by the said integral.
However, my question is the following. Lets say I don't want to set any time-dependent function for the current I(t). I would like to maximize the energy with a current that is an array of numbers. So, at the times given at tspan, we would have a different value I. Thus, we would have obtained the "optimal values" for the current at every instant of time, that maximizes the integral. In other words, instead of obtaining a set of parameters with fmincon, its output would be a vector of I = [I(t = 0), I(t = t1), ..., I(t = end)].
Is it possible to implement this? Maybe fmincon is not the correct way to do this, and there is other path, which of course I am interested in knowing. Thanks in advance
