Abstract:
Nonlinear systems of equations in complex domain are frequently encountered in applied mathematics, e.g., power systems, signal processing, control theory, neural networks and biomedicine, to name a few. The solution of these problems often requires a first- or second-order approximation of these nonlinear functions to generate a new step or descent direction to meet the solution iteratively. However, such methods cannot be applied to real functions of complex variables because they are necessarily non-analytic in their argument, i.e., the Taylor series expansion in their argument alone does not exist. To overcome this problem, the nonlinear function is usually redefined as a function of the real and imaginary parts of its complex argument so that standard methods can be applied. Although not widely known, it is possible to build an expansion of these nonlinear functions in its original complex variables by noting that functions of complex variables can be analytic in their argument and its complex conjugate as a whole. This property lies in the fact that if a function is analytic in the space spanned by ℜ{𝑥} and ℑ{𝑥}in ℝ, it is also analytic in the space spanned by 𝑥 and 𝑥* in ℂ. The main contribution of this work is the application of this methodology to a complex Taylor series expansions aiming algorithms commonly used for solving complex-valued nonlinear systems of equations emerged from power systems problems. In our proposal, a complex-valued power ow analysis (CV PFA) model solved by Newton-Raphson method is revisited and enhanced. Nonetheless, especially emphasis is addressed to Gauss-Newton method when derived in complex domain for solving power system state estimation (CV PSSE) problems, whichever they are applied in transmission or distribution systems. The factorization method of the complex Jacobian matrices emerged from CV PFA and CV PSSE approaches is the Three Angle Complex Rotation (TACR) algorithm that comes from the Givens Rotations algorithm in real domain. In this research one demonstrates that Wirtinger derivatives can lead to greater insights in the structure of both problems, i.e., CV PFA & CV PSSE. Moreover, it can often be exploited to mitigate computational overhead, storage cost and enhance the network's component modeling as FACTS devices, e.g., STATCOM, VSC-HVDC, besides easily handle PMU measurements and embedding new technologies towards smart grids. Finally, in order to add numerical robustness, a fourth-order Levenberg-Marquardt algorithm is employed to the CV PFA & CV PSSE approaches because of its nice bi-quadratic convergence property, instead of the well-known quadratic convergence property of the classical Newton-Raphson and Gauss-Newton algorithms. Recall that these latter algorithms are prone to collapse when the power system network is ill-conditioned, i.e., it is heavily loaded or presents branches with high R/X ratio. These results are partially presented in this thesis because they are still under study and development. But most of them will appear in forthcoming papers submitted to IEEE-PES Transactions on Power Systems and coming up Top Conferences.