Conventionally, algorithms for solving various power system applications are developed in the real domain. Evidently, real-valued models are not natural representations of complex-valued voltage and current phasors; they lead to solution methods that may suffer from large computing times and ill-conditioned problems. Therefore, to circumvent these weaknesses, iterative and non-iterative algorithms carried out in the complex plane have recently proposed in published literature. Besides in power flow analysis, CR calculus has also been extended to power system state estimation. As of now, a number of methods have been proposed to solve ill-conditioned nonlinear system of equations. In a recent publication, the Levenberg Marquardt algorithm was demonstrated. This intricate algorithm was reported to have the potential to solve the problem posed in complex-valued power flow analysis. However, further research is necessitated to enhance the algorithm.
On this account, Brazilian researchers from the Institute of Electric Systems and Energy at Federal University of Itajubá: Professor Robson Pires and Dr. Guilherme Chagas in collaboration with Dr. Lamine Mili at the Virginia Polytechnic Institute and State University proposed to improve the numerical robustness of the Newton-Raphson power flow algorithm. They opted to use Barel’s format equation since it was based on the Jacobian instead of the gain matrix; an approach that helped them speed up the search of a solution. Their work is currently published in the research journal, Electrical Power and Energy Systems.
In their approach, the complex-valued Newton-Raphson and Levenberg Marquardt power flow algorithms, were developed by using Wirtinger calculus. The researchers then derived the bus models as applied to power flow analysis. The team then demonstrated the derivation of the robust complex-valued Levenberg Marquardt algorithm which emerged in unified complex conjugate coordinates system, including its Jacobian matrix. Lastly, a small example was presented followed by simulations carried out on well- and ill-conditioned IEEE-test systems.
The research team reported that few changes in the codes were required to transform the complex-valued Newton-Raphson power flow algorithm into the complex-valued Levenberg Marquardt power flow one. Consequently, using their approach, the team showed that the latter approach blended itself well to modeling new smart grid technologies while exhibiting a bi-quadratic convergence rate and superior performance as compared to the former procedure.
In summary, the study presented an advancement of the complex-valued Newton-Raphson and a robust complex-valued Levenberg Marquardt algorithm and referred versions, aimed at solving well- and ill-conditioned power flow problems, respectively. The study showed that the implementation of the algorithms was straightforward and was much easier to encode them in the complex plane than in the real domain. In fact, all the computations in the complex plane could be carried out in a very similar manner as those in the real domain, making many tools and methods developed for the latter readily available for the former domain.
Robson Pires, Lamine Mili, Guilherme Chagas. Robust complex-valued Levenberg-Marquardt algorithm as applied to power flow analysis. Electrical Power and Energy Systems, volume 113 (2019) page 383–392.