Abstract

Nonlinear systems of equations in complex plane 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 nonlinear functions to generate a new step or descent direction to meet the solution iteratively. However, such methods cannot be applied to functions of complex and complex conjugate variables because they are necessarily nonanalytic. To overcome this problem, the Wirtinger calculus allows an expansion of nonlinear functions in its original complex and complex conjugate variables once they are analytic in their argument as a whole. Thus, the goal is to apply this methodology for solving nonlinear systems of equations emerged from applications in the industry. For instances, the complex-valued Jacobian matrix emerged from the power flow analysis model which is solved by Newton-Raphson method can be exactly determined. Similarly, overdetermined Jacobian matrices can be dealt, e.g., through the Gauss-Newton method in complex plane aimed to solve power system state estimation problems. Finally, the factorization method of the aforementioned Jacobian matrices is addressed through the fast Givens transformation algorithm which means the square root-free Givens rotations method in complex plane.

Highlights

  • IntroductionThis work is a tribute to Steinmetz’s contribution [1]. The reasons and motivations are stated throughout the whole document once the numerical solutions for solving power system applications are typically carried out in the real domain

  • The solution methods of these problems often require a first- or second-order approximation of the set of power flow equations; such methods cannot be applied to nonlinear functions of Advances in Complex Analysis and Applications complex variables because they are nonanalytic in their arguments

  • Taken into account the Wirtinger calculus, this chapter shows how the Jacobian matrix patterns emerge in complex plane corresponding to the steady-state models of power flow analysis and power system state estimation, respectively

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Summary

Introduction

This work is a tribute to Steinmetz’s contribution [1]. The reasons and motivations are stated throughout the whole document once the numerical solutions for solving power system applications are typically carried out in the real domain. The solution methods of these problems often require a first- or second-order approximation of the set of power flow equations; such methods cannot be applied to nonlinear functions of Advances in Complex Analysis and Applications complex variables because they are nonanalytic in their arguments. For these functions Taylor series expansions do not exist. In this chapter the classical Newton-Raphson and Gauss-Newton methods in complex plane aiming the numerical solution of the power flow analysis and power system state estimation are derived, respectively.

Theoretical foundation
CR-Calculus or Wirtinger calculus
Solution of the problem
Three-angle complex rotation algorithm
Complex-valued fast givens rotations
Complex-valued Newton-Raphson method
Jacobian matrix factorization
Nodal equation
Small example
Performance in larger systems
Conclusions and future developments
Full Text
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