This report presents the development of a new family of difference equations for the numerical solution of systems of differential equations arising in electric power circuits. The main advantage of these difference equations is their adjustability, which allows for the tuning of the resulting formulas while using large time steps. The tuned formulas have been used successfully in the time-domain simulation of 50/60-Hz solutions of electric power circuits for linear and non-linear problems. The linear steady state and the non-linear power flow problems can be solved, with no error either in magnitude or in phase, at a comparable computational cost to traditional techniques such as the well-known phasor analysis and Newton-Raphson iterations. Additionally, the techniques developed do not use complex numbers, resulting in a faster matrix factorization for linear systems, or a smaller Jacobian matrix for non-linear systems. The main advantage of the proposed techniques is that due to the time-domain nature of the simulation, where the transition from one time solution to the next represents a small transition between states, the technique is able to closely follow the trajectory of the system even for ill-conditioned power flow problems where traditional Newton-Raphson iterations fail to converge, such as when the system buses are beyond or near the point of voltage collapse. With the proposed novel formulas, the computational cost of generating PV curves across all busses of the power system network is greatly reduced, and the resulting curves clearly show the point of voltage collapse as well as any unstable voltage/power combination thereafter. The ability to quickly obtain PV curves makes the proposed techniques particularly well suited for online power system monitoring of operating states. Also, since these new difference equations introduce very little error for frequencies lower than the tuned frequency, they are also well suited for the solution of dynamic power flow problems such as those resulting from loads variation, transformers tap change, and outages.