Baljinnyam Sereeter

During the normal operation, control and planning of the power system, grid operators employ numerous tools including the Power Flow (PF) and the Optimal Power Flow (OPF) computations to keep the balance in the power system. The solution of the PF computation is used to assess whether the power system can function properly for the given generation and consumption, whereas the OPF problem provides the optimal operational state of the electrical power system, while satisfying system constraints and control limits.

In this thesis, we study advanced models of the power system that transform the physical properties of the network into mathematical equations. Furthermore, we develop new mathematical formulations and algorithms for fast and robust power system simulations, such as PF and OPF computations, that can be applied to any balanced single-phase or unbalanced three-phase network.

The power flow, or load flow, problem is the problem of computing the voltages in each bus of a power system where the power generation and consumption are given. Mathematically, the power flow problem comes down to solving a nonlinear system of equations where all variables are given in complex numbers. In practice, the Newton power flow method using the power balance equations in polar coordinates is preferred in terms of quadratic convergence. In order to obtain the required fast and robust PF solution method for an changing electrical power system, we examine all six formulations of the PF problem using two different mismatch formulations: the current and power balance equations, and three different coordinate systems: Cartesian, polar, and complex form. Moreover, we develop new versions of the Newton power flow method based on all six formulations of the PF problem. Our newly developed versions are compared with the existing variants of the Newton power flow method for both balanced single-phase and unbalanced three-phase networks in terms of the computational speed and robustness. Two Newton power flow variants developed in this thesis are proven to be faster and more robust than the existing Newton power flow methods.

We introduce the new approach to linearize the original nonlinear PF problem using the connection between actual buses in the network to artificial ground buses. Direct and iterative methods are developed in this research work to solve the resulting Linear Power Flow (LPF) problem. Accuracy and efficiency of both direct and iterative linear approaches are validated by comparing them with the conventional Newton power flow algorithm on various transmission and distribution networks. The direct LPF method is further improved with Numerical Analysis (NA) techniques to solve very large LPF problems with 27 million buses simulating both the entire LV and MV networks in a single simulation. Reordering technique (RCM), a couple of direct solvers (Cholesky, IC, LU, and ILU), and various Krylov subspace methods (CG, PCG, GMRES, and BiCGSTAB) are used to improve the computational time of the direct LPF method. We confirm that our LPF algorithms are very fast and user friendly for power flow computations on a large distribution network.

The OPF problem is an optimization problem that has an objective function, equality, and inequality constraints. There is no method that is the best for all OPF problems, because each OPF problem results in an optimization problem with different properties depending on the choice of objective functions, control variables and system constraints. In this thesis, we consider the OPF problem with minimization of active power generation costs as an objective function, nonlinear power flow equations as equality constraints, and squared apparent power limits as inequality constraints. Furthermore, we study four equivalent mathematical formulations of the OPF problem and their computational impacts on the performance of the OPF solution methods. In order to identify the formulation that results in the best convergence characteristics for the solution method, we apply MIPS (Matpower’s Interior Point Method), KNITRO (Commercial software package for solving large scale nonlinear optimization problems), and FMINCON (Matlab’s optimization solver) on various test cases. We compare all four formulations in terms of impact factors on the solution method such as a number of nonzero elements in the Jacobian and Hessian matrices, the number of iterations and computational time on each iteration. Our numerical results verify that the performance of any OPF solution method can be improved by changing the mathematical formulation used to specify the OPF problem while keeping the same algorithm.

Mathematical formulations and computational methods based on this thesis are implemented in Matpower 7.0 for future research and practical use.