On the Holomorphic Embedding Power Flow Method: Theoretical Foundation, Limitations, Extensions and Implementation

潮流研究在电力系统运行和规划中有着广泛的应用。它涉及求解一组非线性潮流（代数）方程。潮流发散问题是潮流研究中的一大难题。这是一个长期存在的问题。新型电力系统设备与可再生能源的融合加剧了潮流发散问题。由于可再生能源的发展，未来的电网必须支持各种各样的潮流模式。目前可用来克服潮流发散的传统选择相当有限。此外，当潮流发散发生时，需要解决以下问题：-Issue EN（存在）：潮流解确实存在，但通常使用的牛顿法等数值方法无法根据给定的初始猜测计算出潮流解；-Issue NE（不存在）：在给定的网络拓扑和负载条件下，潮流解不存在。最近，提出了一种全纯嵌入潮流（HEPF）方法，并获得了美国专利，宣称该方法具有智能和鲁棒性。具体来说，提出了以下三个主要但重要的主张：(i)它是确定性的和非迭代的，（ii）如果它的存在得到保证，它始终如一地计算正确的潮流解决方案，（iii）如果它不能找到正确的潮流解决方案，它明确地表明潮流解决方案不存在。然而，几乎没有理论依据和技术分析来支持上述三大主张。HEPF如所宣称的那样，可以解决长期存在的潮流发散问题。然而，当前版本的HEPF方法存在一些局限性、不足和七个未解决的问题，上述权利要求不合理。本课题旨在为HEPF方法建立理论基础，考察HEPF方法的局限性，研究其适用范围，并将HEPF方法扩展到大型可再生能源电网。建议的工作将涵盖一系列的理论基础发展，解决方法设计和实际数值实现。开发智能鲁棒潮流求解器及其理论基础，并将其应用于大规模可再生能源电网。在工程和科学的许多实际应用中，一组非线性方程解的鲁棒和高效计算是一项重要的任务。因此，该研究的发展对许多实际应用具有广泛的影响，例如超大规模集成电路（VLSI）直流工作点的计算。这是VLSI电路模拟器中最重要也是最困难的任务之一。克服潮流发散问题的困难将具有深远的意义，因为该问题是输电能力计算、静态安全评估、最优潮流、动态安全评估等电力系统研究的核心问题。

Power flow study is used extensively in power system operations and planning. It involves solving a set of nonlinear power flow (algebraic) equations. A great challenge in power flow study is the problem of power flow divergence. This is a long standing problem. The integration of new power system devices and renewable energy aggravates the power flow divergence problem. Future power grids have to support a wide variety of power flow patterns due to renewable energies. The conventional options currently available to overcome power flow divergence are quite limited. In addition, the following issues need to be addressed when the power flow divergence occurs: -Issue EN (existence): a power flow solution does exist, but the numerical method, such as the commonly used Newton method has failed to compute it from a given initial guess, or -Issue NE (non-existence): no power flow solution exists with the specified network topology and loading conditions. Recently, a Holomorphic Embedding Power Flow (HEPF) method was proposed, awarded a U.S. patent, and declared to be smart and robust. Specifically, the following three major and yet important claims were made (i) it is deterministic and non-iterative, (ii) it consistently computes the correct power flow solution if its existence is ensured, (iii) it unambiguously signals the nonexistence of a power flow solution if it cannot find the correct power flow solution. However, there is little theoretical basis and technical analysis to support the above three major claims. HEPF, if works as claimed, can resolve the long-standing power flow divergence problem. However, the current version of HEPF method has several limitations, deficiencies and seven unresolved issues and the above claims are unjustified. This proposal aims to establish a theoretical foundation for the HEPF method, examine the limitations of HEPF, study the scope of its applicability and extend the HEPF method to deal with large-scale power networks with renewable energy.The proposed work will cover a range of theoretical foundation development, solution methodology design and practical numerical implementation. A smart and robust power flow solver and its theoretical foundation will be developed and applied to large-scale power grids with renewables. Robust and efficient computation of the solutions of a set of nonlinear equations is an important task in many practical applications in engineering and sciences. Hence, the development of the proposed study has broad impacts on many practical applications such as computation of direct current (DC) operating points of very large scale integration (VLSI) circuits. This is one of the most important and yet difficult tasks in VLSI circuit simulators. Overcoming the difficulties of power flow divergence problems will have a far-reaching benefit since the problem is at the heart of several power system studies such as power transfer capability calculations, static security assessment, optimal power flow, dynamic security assessment, etc.