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

电力网络是产生、传输和消耗电能的关键基础设施，对现代生活的各个方面都至关重要。因此，提高电力网络的效率、稳定性和弹性对我们的社会具有重大意义。一个单一的电力网络可以运行在许多不同的模式-一些导致有效的操作，而其他导致灾难性的故障。理解所有这些操作模式以及大型复杂电网的可能过渡仍然是一个困难的数学问题，可能会产生重要的现实后果。该项目旨在利用数学中的最新发现开发新的方法和计算工具来解决这个问题。该项目将为研究生和本科生提供培训，并向更大的社区提供开源软件。电力网络分析的核心是求解潮流方程的数学问题：描述电力网络中复杂的电力平衡条件的非线性方程组。潮流方程的解描述了电力网络理论上可能的运行模式，这在严格的电力网络分析中，特别是在评估网络稳定性问题中至关重要。尽管几十年的积极研究，完整的潮流解决方案的分析仍然是一个困难的，往往计算不切实际的任务。通过利用凸几何，热带几何和同伦方法开发的新工具，该项目旨在开发一种灵活的分治方法，用于完全求解潮流方程，并创建实用的软件实现。由此产生的理论框架和软件包将使研究人员能够对全套潮流解决方案进行全面分析，从而了解电力网络的稳定性和弹性。此外，该项目为我们提供了机会，拓宽了我们对凸多面体在复杂网络中涌现现象研究中的作用的理解，这可能在更广泛的背景下具有价值。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.