The power-flow problem has been solved using the Newton-Raphson and Gauss-Seidel methods. Recently the holomorphic embedding method (HEM), a recursive (non-iterative) method applied to solving nonlinear algebraic systems, was applied to the power-flow problem. HEM has been claimed to have superior properties when compared to the Newton-Raphson and other iterative methods in the sense that if the power-flow solution exists, it is guaranteed that a properly configured HEM can find the high voltage solution and, if no solution exists, HEM will signal that unequivocally. Provided a solution exists, convergence of HEM in the extremal domain is claimed to be theoretically guaranteed by Stahl’s convergence-in-capacity theorem, another advantage over other iterative nonlinear solver.In this work it is shown that the poles and zeros of the rational function from fitting the local PMU measurements can be used theoretically to predict the voltage-collapse point. Different numerical methods were applied to improve prediction accuracy when measurement noise is present. It is also shown in this work that the dc optimal power flow (DCOPF) problem can be formulated as a properly embedded set of algebraic equations. Consequently, HEM may also be used to advantage on the DCOPF problem. For the systems examined, the HEM-based interior-point approach can be used to solve the DCOPF problem. While the ultimate goal of this line of research is to solve the ac OPF; tackled in this work, is a precursor and well-known problem with Padé approximants: spurious poles that are generated when calculating the Padé approximant may, at times, prevent convergence within the functions domain. A new method for calculating the Padé approximant, called the Padé Matrix Pencil Method was developed to solve the spurious pole problem. The Padé Matrix Pencil Method can achieve accuracy equal to that of the so-called direct method for calculating Padé approximants of the voltage-functions tested while both using a reduced order approximant and eliminating any spurious poles within the portion of the function’s domain of interest: the real axis of the complex plane up to the saddle-node bifurcation point.

The grounding system in a substation is used to protect personnel and equipment. When there is fault current injected into the ground, a well-designed grounding system should disperse the fault current into the ground in order to limit the touch potential and the step potential to an acceptable level defined by the IEEE Std 80. On the other hand, from the point of view of economy, it is desirable to design a ground grid that minimizes the cost of labor and material. To design such an optimal ground grid that meets the safety metrics and has the minimum cost, an optimal ground grid application was developed in MATLAB, the OptimaL Ground Grid Application (OLGGA).

In the process of ground grid optimization, the touch potential and the step potential are introduced as nonlinear constraints in a two layer soil model whose parameters are set by the user. To obtain an accurate expression for these nonlinear constraints, the ground grid is discretized by using a ground-conductor (and ground-rod) segmentation method that breaks each conductor into reasonable-size segments. The leakage current on each segment and the ground potential rise (GPR) are calculated by solving a matrix equation involving the mutual resistance matrix. After the leakage current on each segment is obtained, the touch potential and the step potential can be calculated using the superposition principle.

A genetic algorithm is used in the optimization of the ground grid and a pattern search algorithm is used to accelerate the convergence. To verify the accuracy of the application, the touch potential and the step potential calculated by the MATLAB application are compared with those calculated by the commercialized grounding system analysis software, WinIGS.

The user's manual of the optimal ground grid application is also presented in this work.