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The stationary points of the potential energy function of the φ⁴ model on a two-dimensional square lattice with nearest-neighbor interactions are studied by means of two numerical methods: a numerical homotopy continuation method and a globally convergent Newton-Raphson method. We analyze the properties of the stationary points, in particular with respect… (More)

One of the most challenging and frequently arising problems in many areas of science is to find solutions of a system of multivariate nonlinear equations. There are several numerical methods that can find many (or all if the system is small enough) solutions but they all exhibit characteristic problems. Moreover, traditional methods can break down if the… (More)

—The manuscript addresses the problem of finding all solutions of power flow equations or other similar nonlinear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly in the context of direct methods for transient stability analysis and voltage stability assessment. We introduce a novel form of… (More)

Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. We translate this into an algebraic geometry problem and use numerical methods to find all of the equilibria for various choices of coupling constants K, natural frequencies, and on different graphs. We note… (More)

— In this paper we investigate how the equilibrium characteristics of conventional power systems may change with an increase in wind penetration. We first derive a differential-algebraic model of a power system network consisting of synchronous generators, loads and a wind power plant modeled by a wind turbine and a doubly-fed induction generator (DFIG).… (More)

Typically, there is no guarantee that a numerical approximation obtained using standard nonlinear equation solvers is indeed an actual solution, meaning that it lies in the quadratic convergence basin. Instead, it may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the corresponding stationary point when… (More)

—The power flow equations, which relate power injections and voltage phasors, are at the heart of many electric power system computations. While Newton-based methods typically find the " high-voltage " solution to the power flow equations, which is of primary interest, there are potentially many " low-voltage " solutions that are useful for certain… (More)

— The power flow equations are at the core of most of the computations for designing and operating electric power systems. The power flow equations are a system of multivariate nonlinear equations which relate the power injections and voltages in a power system. A plethora of methods have been devised to solve these equations, starting from Newton-based… (More)

- Joseph Cleveland, Jeffrey Dzugan, Jonathan D. Hauenstein, Ian Haywood, Dhagash Mehta, Anthony Morse +2 others
- ArXiv
- 2014

A challenging problem in computational mathematics is to compute roots of a high-degree univariate random polynomial. We combine an efficient multiprecision implementation for solving high-degree random polyno-mials with two certification methods, namely Smale's α-theory and one based on Gerschgorin's theorem, for showing that a given numerical… (More)

A large amount of research activity in power systems areas has focused on developing computational methods to solve load flow equations where a key question is the maximum number of isolated solutions. Though several concrete upper bounds exist, recent studies have hinted that much sharper upper bounds that depend the topology of underlying power networks… (More)