A solution to the optimal power flow (OPF) problem for electrical generation and distribution systems utilizes a re-configuration of the OPF problem that allows for a simplified analysis and resolution of a network-based OPF problem in a minimal number of iterations. The standard mixed integer quadratic problem (MIQP) definition is be reconfigured, using the chain rule, to a relatively compact linear system of six equations with six unknowns (the smallest reducible (atomic) problem). Advantageously, the reduction in the complexity of the problem does not require any assumptions and yields a solution equivalent to the original problem.