AMPS: Collaborative Research: A convex geometry and homotopy approach for power-flow equations

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Power networks are critical infrastructures for generating, transferring, and consuming electric energy, and they are of fundamental importance to every aspect of modern life. Improving the efficiency, stability, and resilience of power networks is therefore of great interest to our society. A single power network can operate on many different modes --- some lead to efficient operations while others lead to catastrophic failures. Understanding all such operation modes and the possible transition among them for large and complex power networks remains a difficult mathematical question that could have important real-world consequences. This project aims to develop new methodology and computational tools for solving this problem by utilizing recent discoveries in mathematics. The project will provide training to graduate and undergraduate students, and open source software to the larger community. At the heart of power network analysis lies the mathematical problem of solving power-flow equations: systems of nonlinear equations that describe the intricate balancing conditions of electric power in a power network. The solutions to power-flow equations describe the set of theoretically possible operating modes for a power network, which are of crucial importance in the rigorous analysis of power networks, especially in the problem of assessing network stability. Despite many decades of active research, the complete analysis of power-flow solutions is still a difficult and often computationally impractical task. By leveraging new tools developed in convex geometry, tropical geometry, and homotopy methods, this project aims to develop a flexible divide-and-conquer approach for completely solving power-flow equations and to create practical software implementations. The resulting theoretical framework and software packages would allow researchers to conduct a complete analysis of the full set of power-flow solutions and thus understand the stability and resilience of power networks. Moreover, this project provides opportunities for broadening our understanding of the role of convex polytopes in the study of emergent phenomena in complex networks that may be of value in a much broader context.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.