solving a circuit with multiple-objective optimization algorithm to find resistors and voltages values

I have been tasked with optimizing the circuit parameters (R1, R2, R3, V1, and V2) to maximize power across a 2Ω resistor (Pmax) using both a single-objective optimization algorithm and a multiple-objective algorithm in MATLAB. The permissible values for resistors are between 0Ω and 5Ω, and for voltages between 0V and 10V. The optimization process should involve a population of 20 with 50 iterations.

For the single-objective optimization, I successfully employed the Genetic Algorithm to find R1, R2, R3, V1 and V2 when the power (P) is maximum over the 2Ω resistor. the power function over 2Ω resistor as a function of R1, R2, R3, V1 and V2 has been found

P2Ω = 2*((280*R1 + 200*R2 + 270*R3 + 100*V1 + 20*R1*R2 + 78*R1*R3 + 60*R2*R3 + 60*R1*V1 + 10*R1*V2 + 50*R2*V1 + 30*R3*V1 + 4*R1*R2*R3 + 5*R1*R2*V1 + 17*R1*R3*V1 + 2*R1*R3*V2 + 15*R2*R3*V1 + R1*R2*R3*V1 + 900)/(520*R1 + 450*R2 + 210*R3 + 45*R1*R2 + 149*R1*R3 + 135*R2*R3 + 9*R1*R2*R3 + 700) - (220*R1 + 795*R3 + 100*V1 + 53*R1*R3 + 10*R1*V1 + 45*R1*V2 + 30*R3*V1 + 2*R1*R3*V1 + 9*R1*R3*V2 + 2650)/(520*R1 + 450*R2 + 210*R3 + 45*R1*R2 + 149*R1*R3 + 135*R2*R3 + 9*R1*R2*R3 + 700))^2

After finding P2Ω I used Genetic algorithm to find R1, R2, R3, V1 and V2.

Now, I am seeking guidance on how to solve this problem with multiple-objective algorithm. Any help would be greatly appreciated.

[Torsten]

And what are the "multiple objectives" in your case ?

[Moussa]

I only know that to increase power, I can use one of these equations: P = V * I, P = R * I^2, or P = V^2/R.

So I dont know how can I find other objective functions

[Torsten]

Usually when you do multiobjective optimization, you have several objectives you have to consider which in general are in conflict with each other. This means an improvement in one objective means a deterioration in another objective. Are there goals in your application that compete with each other ? E.g. more use of energy means more damage to environment where you have to find a compromise ?

https://en.wikipedia.org/wiki/Multi-objective_optimization

Or is it something that is described here that you are aiming at ?

https://www.sciencedirect.com/science/article/pii/S0950705122010802

[Sam Chak]

I only know that to increase power,...

Since the power formula is given by , if both V and I have the same sign, then the maximum power is obtained by . Would you like to validate your calculation with the result from the optimization algorithm?"

Answer 1

[Sam Chak]

Hi @Moussa

Update: This is not a complete solution, but it could serve as a starting point by applying Kirchhoff's Voltage Law, .

Additionally, I suggest calculating the power across the 2-ohm resistor using the formula , where represents the net current flowing through the resistor. Therefore, , and the goal is to maximize the cost .

Ideally, you should maximize and minimize , or vice versa. This is likely why your professor instructed you to perform multi-objective optimization. You need to formulate the Multi-objective Optimization Problem and solve it using the 'gamultiobj()' command, which is the multiobjective genetic algorithm function.

Loop 1

Loop 2

Loop 3

Loop 4

syms R1 R2 R3 V1 V2 I1 I2 I3 I4
eqns    = [ V1 -  7*I1 + 4 - 2*(I1 - I3)             == 0;  % Loop 1
           - 4 - R3*I2 + 3 - 5*(I2 - I4)             == 0;  % Loop 2
           - R2*I3 - 2*(I3 - I1) - R1*(I3 - I4) + 5  == 0;  % Loop 3
           - R1*(I4 - I3) - 5*(I4 - I2) - 10*I4 + V2 == 0]; % Loop 4
vars    = [I1; I2; I3; I4]                  % vector x

vars = 

[A, b]  = equationsToMatrix(eqns, vars)     % A·x = b

A = 

b = 

I'm unsure where to go from here. It is possible to set up the for-loop method to brute-force search for the best values in the specified search regions for , , , , . Perhaps, @Torsten can provide some advice.

Update 2: I didn't check your P2ohm equation, but this is likely how you derived the power equation from and .

x       = linsolve(A, b);
x       = simplify(x, 'steps', 100)

x = 

22/Dec/2023: @Moussa, Can you validate the GA result by making direct substitution of the values found for R1, R2, R3, V1, V2, and measure the power across the 2ohm resistor?

objfun    = @costfun;
PopSize   = 20;
MaxGens   = 50;

nvars     = 5;
A         = - eye(nvars);
b         = zeros(nvars, 1);  % A*x <= b
Aeq       = [];               
beq       = [];
lb        = [0 0 0  0  0];
ub        = [5 5 5 10 10];
nonlcon   = [];

options   = optimoptions(@ga,'PopulationSize', PopSize, 'MaxGenerations', MaxGens);
[x, fval] = ga(objfun, nvars, A, b, Aeq, beq, lb, ub, nonlcon, options)

ga stopped because it exceeded options.MaxGenerations.

x = 1×5

0 0 0.4999 10.0000 0

fval = -12.5000

%% R1 = 0, R2 = 0, R3 = 0.8789, V1 = 0.0585, V2 = 2.6896

function P2ohm = costfun(x)
    R1    = x(1);
    R2    = x(2);
    R3    = x(3);
    V1    = x(4);
    V2    = x(5);
    P2ohm = - (2*((280*R1 + 200*R2 + 270*R3 + 100*V1 + 20*R1*R2 + 78*R1*R3 + 60*R2*R3 + 60*R1*V1 + 10*R1*V2 + 50*R2*V1 + 30*R3*V1 + 4*R1*R2*R3 + 5*R1*R2*V1 + 17*R1*R3*V1 + 2*R1*R3*V2 + 15*R2*R3*V1 + R1*R2*R3*V1 + 900)/(520*R1 + 450*R2 + 210*R3 + 45*R1*R2 + 149*R1*R3 + 135*R2*R3 + 9*R1*R2*R3 + 700) - (220*R1 + 795*R3 + 100*V1 + 53*R1*R3 + 10*R1*V1 + 45*R1*V2 + 30*R3*V1 + 2*R1*R3*V1 + 9*R1*R3*V2 + 2650)/(520*R1 + 450*R2 + 210*R3 + 45*R1*R2 + 149*R1*R3 + 135*R2*R3 + 9*R1*R2*R3 + 700))^2);
end

[Moussa]

but, I think this code is for single-objective algorithm. Isn't it?

[Torsten]

As long as you don't know what the "multi-objectives" in your optimization are, we cannot help you.

[Sam Chak]

Hi @Moussa

I've given some thought to the problem and updated my answer above.

[Sam Chak]

Hi @Moussa

By the way, is the issue resolved?