Theoretical convergence guarantees versus numerical convergence behavior of the holomorphically embedded power flow method

Abstract The holomorphic embedding load flow method (HELM) is an application for solving the power-flow problem based on a novel method developed by Dr. Trias. The advantage of the method is that it comes with a theoretical guarantee of convergence to the high-voltage (operable) solution, if it exists, provided the equations are suitably framed. While theoretical convergence is guaranteed by Stahl’s theorem, numerical convergence is not; it depends on the analytic continuation algorithm chosen. Since the holomorphic embedding method (HEM) has begun to find a broader range of applications (it has been applied to nonlinear structure-preserving network reduction, weak node identification and saddle-node bifurcation point determination), examining which algorithms provide the best numerical convergence properties, which do not, why some work and not others, and what can be done to improve these methods, has become important. The numerical Achilles heel of HEM is the calculation of the Pade approximant, which is needed to provide both the theoretical convergence guarantee and accelerated numerical convergence. In the past, only two ways of obtaining Pade approximants applied to the power series resulting from power-system-type problems have been discussed in detail: the matrix method and the Viskovatov method. This paper explores several methods of accelerating the convergence of these power series and/or providing analytic continuation and distinguishes between those that are backed by the theoretical convergence guarantee of Stahl’s theorem (i.e., those computing Pade approximants), and those that are not. For methods that are consistent with Stahl’s theoretical convergence guarantee, we identify which methods are computationally less expensive, which have better numerical performance and what remedies exist when these methods fail to converge numerically.

[1]  Yang Feng,et al.  The Holomorphic Embedding Method Applied to the Power-Flow Problem , 2016, IEEE Transactions on Power Systems.

[2]  R D Zimmerman,et al.  MATPOWER: Steady-State Operations, Planning, and Analysis Tools for Power Systems Research and Education , 2011, IEEE Transactions on Power Systems.

[3]  Stefan Paszkowski,et al.  Recurrence relations in Padé-Hermite approximation , 1987 .

[4]  O. Alsac,et al.  Fast Decoupled Load Flow , 1974 .

[5]  P. Wynn,et al.  On a Device for Computing the e m (S n ) Transformation , 1956 .

[6]  Claude Brezinski,et al.  Extrapolation algorithms and Pade´ approximations: a historical survey , 1996 .

[7]  Daniel Tylavsky,et al.  Estimating the saddle-node bifurcation point of static power systems using the holomorphic embedding method , 2017 .

[8]  J. W. Walker,et al.  Direct solutions of sparse network equations by optimally ordered triangular factorization , 1967 .

[9]  Daniel Tylavsky,et al.  Nonlinear Structure-Preserving Network Reduction Using Holomorphic Embedding , 2018, IEEE Transactions on Power Systems.

[10]  Daniel Tylavsky,et al.  Two-Bus Holomorphic Embedding Method-based Equivalents and Weak-Bus Determination , 2017 .

[11]  Daniel Tylavsky,et al.  Nonlinear network reduction for distribution networks using the holomorphic embedding method , 2016, 2016 North American Power Symposium (NAPS).

[12]  T. Gamelin.  Complex Analysis , 2001 .

[13]  Herbert Stahl,et al.  The convergence of Padé approximants to functions with branch points , 1997 .

[14]  Herbert Stahl,et al.  On the convergence of generalized Padé approximants , 1989 .

[15]  P. Wynn,et al.  The epsilon algorithm and operational formulas of numerical analysis : (mathematics of computation, _1_5(1961), p 151-158) , 1961 .

[16]  Walter Van Assche,et al.  Pade and Hermite-Pade approximation and orthogonality , 2006, math/0609094.

[17]  B. Burrows,et al.  Lower bounds for quartic anharmonic and double‐well potentials , 1993 .

[18]  Brian Stott,et al.  Decoupled Newton Load Flow , 1972 .

[19]  Ana C. Matos,et al.  A derivation of extrapolation algorithms based on error estimates , 1996 .

[20]  Antonio Trias,et al.  The Holomorphic Embedding Loadflow Method for DC Power Systems and Nonlinear DC Circuits , 2016, IEEE Transactions on Circuits and Systems I: Regular Papers.

[21]  A note on evaluation of coefficients in the polynomials of Padé approximants by solving systems of linear equations , 1982 .

[22]  Claude Brezinski,et al.  Convergence acceleration methods: The past decade☆ , 1985 .

[23]  D. Shanks.  Non‐linear Transformations of Divergent and Slowly Convergent Sequences , 1955 .

[24]  R. G. Brookes.  The quadratic hermite-padé approximation , 1989, Bulletin of the Australian Mathematical Society.

[25]  D. E. Roberts,et al.  The epsilon algorithm and related topics , 2000 .

[26]  P. Wynn,et al.  Sequence Transformations and their Applications. , 1982 .

[27]  Ernst Joachim Weniger.  Prediction properties of Aitken's iterated D 2 process, of Wynn's epsilon algorithm, and of Brezinski's iterated theta algorithm , 2000 .

[28]  William F. Tinney,et al.  Power Flow Solution by Newton's Method , 1967 .

[29]  Stefan Paszkowski.  Hermite-Pade´ approximation (basic notions and theorems) , 1990 .

[30]  J. B. Ward,et al.  Digital Computer Solution of Power-Flow Problems [includes discussion] , 1956, Transactions of the American Institute of Electrical Engineers. Part III: Power Apparatus and Systems.

[31]  Yang Feng,et al.  Solving for the Low-Voltage/Large-Angle Power-Flow Solutions by Using the , 2015 .

[32]  Daniel Tylavsky,et al.  A novel method to converge to the unstable equilibrium point for a two-bus system , 2013, 2013 North American Power Symposium (NAPS).

[33]  Ernst Joachim Weniger,et al.  Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series , 1989 .

[34]  A. Trias,et al.  The Holomorphic Embedding Load Flow method , 2012, 2012 IEEE Power and Energy Society General Meeting.