The holomorphic embedding method(HEM)stands as a mathematical technique renowned for its favorable convergence properties when resolving algebraic systems involving complex variables.The key idea behind the HEM is to ...The holomorphic embedding method(HEM)stands as a mathematical technique renowned for its favorable convergence properties when resolving algebraic systems involving complex variables.The key idea behind the HEM is to convert the task of solving complex algebraic equations into a series expansion involving one or multiple embedded complex variables.This transformation empowers the utilization of complex analysis tools to tackle the original problem effectively.Since the 2010s,the HEM has been applied to steady-state and dynamic problems in power systems and has shown superior convergence and robustness compared to traditional numerical methods.This paper provides a comprehensive review on the diverse applications of the HEM and its variants reported by the literature in the past decade.The paper discusses both the strengths and limitations of these HEMs and provides guidelines for practical applications.It also outlines the challenges and potential directions for future research in this field.展开更多
What is known as StahPs Theorem in power engineering circles is used to justify a convergence guarantee of the Holomorphic Embedding Method(HEM)as it applies to powerflow(PF)problem.In this two-part paper,we examine i...What is known as StahPs Theorem in power engineering circles is used to justify a convergence guarantee of the Holomorphic Embedding Method(HEM)as it applies to powerflow(PF)problem.In this two-part paper,we examine in more detail the implications of Stahl's theorems to both theoretical and numerical convergence for a wider range of problems to which these theorems are now being applied.We show that the difference between StahPs extremal domain and the function's domain is responsible for theoretical nonconvergence and that the fundamental cause of numerical nonconvergence is the magnitude of logarithmic capacity of the branch cut,a concept central to understanding nonconvergence.We introduce theorems using the necessary mathematical parlance and then translate the language to show its implications to convergence of nonlinear problems in general and specifically to the PF problem.We show that,among other possibilities,the existence of Chebotarev points,which are embedding specific,are a possible theoretical impediment to convergence・The theoretical foundation of Part I is necessary for understanding the numerical behavior of HEM discussed in Part II.展开更多
What has become known as Stahl's Theorem in power-engineering circles has been used to justify a convergence guarantee of the Holomorphic Embedding Method(HEM)as it applies to the power-flow problem.In this,the se...What has become known as Stahl's Theorem in power-engineering circles has been used to justify a convergence guarantee of the Holomorphic Embedding Method(HEM)as it applies to the power-flow problem.In this,the second part of a two-part paper,we examine implications to numerical convergence of the HEM and the numerical properties of a Pade approximant algorithm.We show that even if the point of interest is within the convergence domain,numerical convergence of the sequence of Pade approximants computed with finite precision is not guaranteed.We propose a convergence factor equation that can be used to both estimate the convergence rate and the capacity of the branch cut.We also show that the study of convergence properties of the Pade approximant is the study of the location of the branch-points of the function,which in turn dictate branch-cut topology and capacity and,therefore,convergence rate.展开更多
文摘The holomorphic embedding method(HEM)stands as a mathematical technique renowned for its favorable convergence properties when resolving algebraic systems involving complex variables.The key idea behind the HEM is to convert the task of solving complex algebraic equations into a series expansion involving one or multiple embedded complex variables.This transformation empowers the utilization of complex analysis tools to tackle the original problem effectively.Since the 2010s,the HEM has been applied to steady-state and dynamic problems in power systems and has shown superior convergence and robustness compared to traditional numerical methods.This paper provides a comprehensive review on the diverse applications of the HEM and its variants reported by the literature in the past decade.The paper discusses both the strengths and limitations of these HEMs and provides guidelines for practical applications.It also outlines the challenges and potential directions for future research in this field.
基金supported by the Science and Technology Project of SGCC(No.5455HJ160007).
文摘What is known as StahPs Theorem in power engineering circles is used to justify a convergence guarantee of the Holomorphic Embedding Method(HEM)as it applies to powerflow(PF)problem.In this two-part paper,we examine in more detail the implications of Stahl's theorems to both theoretical and numerical convergence for a wider range of problems to which these theorems are now being applied.We show that the difference between StahPs extremal domain and the function's domain is responsible for theoretical nonconvergence and that the fundamental cause of numerical nonconvergence is the magnitude of logarithmic capacity of the branch cut,a concept central to understanding nonconvergence.We introduce theorems using the necessary mathematical parlance and then translate the language to show its implications to convergence of nonlinear problems in general and specifically to the PF problem.We show that,among other possibilities,the existence of Chebotarev points,which are embedding specific,are a possible theoretical impediment to convergence・The theoretical foundation of Part I is necessary for understanding the numerical behavior of HEM discussed in Part II.
基金supported by the Science and Technology Project of SGCC(No.5455HJ160007).
文摘What has become known as Stahl's Theorem in power-engineering circles has been used to justify a convergence guarantee of the Holomorphic Embedding Method(HEM)as it applies to the power-flow problem.In this,the second part of a two-part paper,we examine implications to numerical convergence of the HEM and the numerical properties of a Pade approximant algorithm.We show that even if the point of interest is within the convergence domain,numerical convergence of the sequence of Pade approximants computed with finite precision is not guaranteed.We propose a convergence factor equation that can be used to both estimate the convergence rate and the capacity of the branch cut.We also show that the study of convergence properties of the Pade approximant is the study of the location of the branch-points of the function,which in turn dictate branch-cut topology and capacity and,therefore,convergence rate.
