Genetic Algorithm (GA) is an Artificial Intelligence (AI) based methodology. Other literature called it as a variation of an evolutionary algorithm. It is a non-mathematical, non-deterministic, instead, a stochastic process or algorithm for solving optimization problems, preferably, involving discrete variables and non-constraint objective function.

Optimization using genetic algorithm can be applied in various field like power engineering. I use genetic algorithm in my undergraduate thesis about the optimal allocation of fixed and switched capacitors for unbalanced radial distribution feeders .

Note: The following discussions are just ideas and formulations and haven’t proven yet. Everyone is invited to explore and, much better, implement these thought

The system of equations
System of equations are functions of atleast two variables. It can either be linear or non-linear, and to obtain a solution, the system should be non-singular and have a point in space where it coincides.

Generally, the systems of equations have the following form;

F1(x1,x2,x3,….,xn) = a1
F2(x1,x2,x3,….,xn) = a2
….
Fn(x1,x2,x3,….,xn) = an

where x are unknown variables, n is the number of unknown variables, and a are constants.

Formulation
The only information that are needed in genetic algorithm are objective function, and the representation scheme in a form of a coded string, containing information about the possible solutions. Evaluation of a possible solution was done after every set of genetic operations.

Therefore, the simultaneous systems of equations can be solve using genetic algorithm if we can expressed it in a form of an objective function And a solution is said to be correct, if it can satisfy all of the equations involving those variables. That’s it;

F1(x1,x2,x3,….,xn) - a1 = 0
F2(x1,x2,x3,….,xn) - a2 = 0
….
Fn(x1,x2,x3,….,xn) - an = 0

or approximately,

the
absolute value of (F1(x1,x2,x3,….,xn) - a1), AND
absolute value of (F2(x1,x2,x3,….,xn) - a2), AND
…
absolute value of (Fn(x1,x2,x3,….,xn) - an)

must be all at minimum.

The last set of equations (bold and italicized) are actually the objective function that is needed by Genetic Algorithm. Note, that genetic algorithm does not output the exact solution but very near to the actual solution.

Possible Problem that might be encounter
Genetic Algorithm specializes on the optimization of discrete solutions and non-constraint objective function. Obviously, the objective function that we had formed is not constrained, so, we have no problem with that. However, the unknown variables, which will contain the possible solution is not discrete. In fact, it is continuous from positive to negative infinity. This dilemma will cause a problem in representation scheme due to infinitely large sample space.