AMPS: Algebraic Geometry under Uncertainty for Power Flow Systems

美国国家工程院将大范围的电气化列为20世纪最伟大的工程成就之一，但2006年的一项研究估计，全国每年因停电造成的损失约为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这个项目中研究这如何影响他们的解。这些方程模拟电力系统中电压和有功和无功注入之间的非线性关系，可以用多变量二次曲线来表示。现在的目标是在多项式的系数(导数)中引入不确定性，并研究它们的实解的数量是如何变化的，限定它，并获得在这种不确定性下寻找所有这些解的实用算法。对于一般的多项式系统，在系数为iid高斯的假设下，实解的个数分布是已知的，但潮流系统的实解似乎比这些结果所预测的要少。该小组打算将这项工作推广到我们的特殊系统，了解这些现象，然后将它们应用到实际的电力系统中。

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.