RUI: Robust Feasibility and Robust Optimization using Algebraic Topology and Convex Analysis

RUI：使用代数拓扑和凸分析的鲁棒可行性和鲁棒优化

• 批准号: 1819229
• 负责人: Bala Krishnamoorthy
• 金额: $ 20万
• 依托单位: Washington State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-15 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819229&HistoricalAwards=false
• 关键词: RUI; Robust; Feasibility; Optimization; using; RUI; Robust; Feasibility; Optimization; using

Solving systems of equations and optimizing a function over such systems are ubiquitous in computational mathematics. The functions or equations being nonlinear and/or nonconvex often make these tasks challenging. Further, uncertainty in problem parameters adds to the problem complexity. A crucial part of modern society where such problems are prevalent is power systems. Two central computations in power systems operations are power flow (PF) studies and optimal power flow (OPF). PF studies ensure the power grid state (i.e., voltages and flows across the network) will remain within acceptable limits in spite of contingencies (e.g., loss of a generator or a transmission line) and other uncertainties (e.g., shifting demand or renewable sources of power such as wind and solar). OPF seeks further to choose values for controllable assets in the system (e.g., generators whose rate of power production could be controlled) so as to meet demand at minimum cost. These problems have inherent nonlinearities and nonconvexities, making them hard to solve in their natural form. This project uses ideas from algebraic topology and nonlinear analysis to develop efficient algorithms for robust feasibility and robust optimization. In particular, the investigator will develop a framework to derive mathematically rigorous guarantees for robust feasibility and optimization in nonlinear systems using scalable algorithms. The investigator will employ these algorithms to characterize the effects of uncertainties in nonlinear models of power systems. The investigator will also demonstrate the efficacy of the framework by testing it on large scale OPF problems.The rapid adoption of renewable energy sources such as wind and solar energy is creating increased uncertainty in modern power systems. In this project, the investigator will take a robust viewpoint of uncertainty: the worst-case impact of the uncertainty on feasibility and optimization problems will be quantified. To this end, the investigator will use ideas from algebraic topology and nonlinear analysis -- specifically Borsuk's theorem (a generalization of the intermediate value theorem) and topological degree theory -- to develop efficient algorithms for robust versions of the PF and OPF problems. On the computational side, the investigator will develop efficient implementations of these algorithms capable of scalably solving large instances of PF and OPF problems. The novel framework will combine rigorous guarantees, efficient algorithms, and the ability to handle nonlinearities. Such a framework is critical for operating modern power systems with significant uncertainty. While power systems are used as the main application area, the methods to be develop are fairly general, and could be applied to problems in other domains as well, e.g., gas distribution networks. More broadly, this project could have a direct impact on how complex and large scale infrastructure systems are handled, especially under increasing uncertainties created by the environment.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.

在计算数学中，求解方程式和优化功能在此类系统上的功能无处不在。函数或方程是非线性和/或非凸的，通常会使这些任务具有挑战性。此外，问题参数的不确定性增加了问题的复杂性。这种问题普遍存在的现代社会的关键部分是电力系统。电力系统操作中的两个中央计算是功率流（PF）研究和最佳功率流（OPF）。 PF研究确保电网状态（即，网络上的电压和流动）仍将在可接受的限制范围内保持在可接受的限制之内，尽管发生了突发事件（例如，发电机或传输线的损失）和其他不确定性（例如，需求转移或可再生能力来源，例如风和Solar）。 OPF寻求进一步选择系统中可控资产的值（例如，可以控制发电速率的发电机），以便以最低成本满足需求。这些问题具有固有的非线性和非洞穴性，因此很难以自然形式解决。该项目使用代数拓扑和非线性分析中的想法来开发有效的算法，以促进可行性和鲁棒优化。 特别是，研究人员将开发一个框架，以使用可伸缩算法在非线性系统中进行数学上严格的保证，以确保可行性和优化。 研究人员将采用这些算法来表征在功率系统非线性模型中不确定性的影响。 研究人员还将通过大规模OPF问题进行测试来证明该框架的功效。快速采用可再生能源（例如风能和太阳能）正在引起现代电力系统的不确定性。在这个项目中，研究人员将采取不确定性的强大视点：不确定性对可行性和优化问题的最坏情况将被量化。为此，研究者将使用代数拓扑和非线性分析中的思想 - 特别是Borsuk的定理（中间值定理的概括）和拓扑学位理论 - 用于开发PF和OPF问题的强大版本的有效算法。在计算方面，研究人员将开发这些算法的有效实现，能够可稳定地解决PF和OPF问题的大量实例。新颖的框架将结合严格的保证，有效的算法和处理非线性的能力。这样的框架对于操作具有重大不确定性的现代电力系统至关重要。尽管电源系统被用作主要的应用领域，但要开发的方法相当一般，也可以应用于其他域中的问题，例如气体分配网络。从更广泛的角度来看，该项目可能会直接影响复杂和大规模的基础架构系统的处理方式，尤其是在环境造成的不确定性越来越大的情况下。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准通过评估来进行评估的。