Compact Complex-Valued Formulation of Power Flow Equations

Power flow analysis is a fundamental tool for the operation and planning of electrical power systems. The conventional approach relies heavily on polar coordinate representations of voltage and current, which are used to compute Jacobian matrices and solve the system of equations. However, this paper emphasizes the advantages of retaining the complex-valued representation of the equations. The complex formulation provides a more compact and efficient way to express the problem while facilitating a straightforward matrix representation for computational implementations. This compact notation enables faster and simpler derivations of solver algorithms, such as Newton-Raphson and its adaptations, like Fast Decoupled Power Flow (FDPF) and its variants for networks whose resistive part is not negligible. Furthermore, the decoupling step in FDPF can be avoided to exploit the efficiency of direct complex-domain formulations considering that decoupling is less important when Jacobian matrix is constant (as in FDPF). By implementing and comparing several algorithms with different formulations it is shown that the matrix-based complex formulations are computationally superior, offering significant reductions in execution times.