AMPS: Collaborative Research: A Convex Geometry and Homotopy Approach for Power-Flow Equations

电网是电能产生、传输和消耗的关键基础设施，对现代生活的各个方面都至关重要。因此，提高电网的效率、稳定性和弹性是我们社会非常感兴趣的问题。一个单一的电力网络可以在许多不同的模式下运行——一些模式导致高效运行，而另一些模式导致灾难性故障。对于大型复杂的电网来说，理解所有这些运行模式以及它们之间可能的转换仍然是一个困难的数学问题，可能会对现实世界产生重要影响。本项目旨在利用数学上的最新发现，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，并为更大的社区提供开源软件。电网分析的核心在于求解潮流方程的数学问题：描述电网中电力复杂平衡条件的非线性方程组。潮流方程的解描述了一组理论上可能的电网运行模式，对于电网的严格分析，特别是对电网稳定性的评估问题具有至关重要的意义。尽管经过了几十年的积极研究，但对潮流解的完整分析仍然是一项困难且通常在计算上不切实际的任务。通过利用凸几何、热带几何和同伦方法中开发的新工具，该项目旨在开发一种灵活的分治方法，以完全解决功率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电力流解决方案进行完整的分析，从而了解电网的稳定性和弹性。此外，该项目提供了扩大我们对凸多面体在复杂网络中突发现象研究中的作用的理解的机会，这可能在更广泛的背景下具有价值。该奖项反映了美国国家科学基金会的法定使命，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.