Modelagem e otimização de redes elétricas de transmissão visando o balanço de potência via teoria espectral de grafos

Abstract

Electricity is essential for human development, directly impacting key sectors such as education, healthcare, transportation, and industry. However, the increasing integration of renewable sources, such as wind and solar power, has made power system management increasingly complex, requiring robust infrastructure to ensure stability and prevent supply interruptions. In this work, we propose a new approach to the load balancing problem in transmission networks, using tools from spectral graph theory to explore the network topology and analyze the active power distribution in transmission lines. By linearizing the power flow equations and analyzing the network’s Laplacian matrix, it is possible to identify critical lines and optimize the power flow more efficiently. To achieve this, we apply a mathematical method based on the Karush-KuhnTucker (KKT) conditions, solving the original problem from the literature through elementary matrix operations. This strategy significantly reduces the computational time required to solve adapted IEEE test instances for the load balancing problem, with a computation time of less than 0.5 seconds for a 2383-node network. Furthermore, we develop an original linear mathematical model that promotes a more balanced load distribution by minimizing the maximum power in the lines, unlike the conventional approach of minimizing the squared norm of power flows. As a result, for the IEEE 2383 node network, we achieved an approximately 23% reduction in the maximum line power, along with significant gains in computational performance. Furthermore, the proposed model incorporates operational constraints such as power limits on the lines and power generation on the generator units, enabling a formulation that better aligns with real-world power system conditions. The new model also facilitates automated monitoring of critical lines operating above acceptable risk thresholds of their nominal capacities, helping identify network segments requiring investment. Consequently, this work enables new approaches to modernize the electrical infrastructure, improve operational efficiency, and support the imminent integration of renewable energy sources into the transmission system, with positive impacts on the reliability and performance of the grid.