AMPS: Algebraic Geometry under Uncertainty for Power Flow Systems

美国国家工程院将广泛的电气化列为世纪最伟大的工程成就之一，但2006年的一项研究估计，全国每年因电力中断而造成的损失约为790亿美元。潮流方程是电力系统工程师用来维护可靠、经济运行的电力系统的所有工具的核心。这些方程具有可变性，我们的工程师和数学家团队将在本项目中讨论这如何影响他们的解决方案。这些方程模拟了电力系统中电压与有功和无功功率注入之间的非线性关系，可以用多元二次型表示。现在的目标是将不确定性引入到多项式的系数中（阻抗），并研究它们的真实的解的数量如何变化，限制它，并获得在不确定性下找到所有这些解的实用算法。一般多项式系统的真实的解的数量分布是已知的，假设下，如系数是iid高斯，但电力流系统似乎有更少的真实的解决方案比这些结果将预测。该小组打算将这项工作推广到我们的特殊系统，理解这些现象，然后将其应用于实际的电力系统。

The National Academy of Engineering has classified widespread electrification as one of the greatest engineering achievements of the 20th century, but a 2006 study estimated that the national annual cost of power interruptions was about $79 billion dollars. The power flow equations are at the heart of all tools used by power system engineers to maintain reliable, economically operated power systems. These equations have variability, and our team of engineers and mathematicians will address in this project how this affects their solutions.These equations model the nonlinear relationship between voltages and active and reactive power injections in a power system and can be represented by multivariate quadratics. The goal now is to introduce uncertainty into the coefficients of the polynomials (the susceptances) and to investigate how their number of real solutions varies, bound it, and obtain practical algorithms for finding all these solutions under that uncertainty. The distribution of numbers of real solutions is known for general polynomial systems under assumptions such as that the coefficients are iid Gaussian but the power flow systems appear to have fewer real solutions than those results would predict. The group intends to generalize this work to our special systems, understand the phenomena, and then apply them to actual power systems.