Exploring the Impact of Wind Penetration on Power System Equilibrium Using a Numerical Continuation Approach

IEEE Trans. Power Syst.

2018

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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 may exist. This paper establishes such a topology dependent solution bound which is actually the best possible bound in the sense that it is always attainable. We also develop a geometric construction called adjacency polytope which accurately captures the topology of the underlying power network and is immensely useful in the computation of the solution bound. Finally we highlight the significant implications of the development of such solution bound in solving load flow equations.2. Algebraic formulation. In this paper, we focus on the mathematical abstraction of a power network which is captured by a graph G = (B, E) together with a complex matrix Y = (Y ij ). Here B is the finite set of nodes representing the "buses", E is the set of edges (also called lines or branches) representing the connections between buses, and the matrix Y is the nodal admittance matrix 1 which assigns a nonzero †

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

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On the Network Topology Dependent Solution Count of the Algebraic Load Flow Equations

Chen

IEEE Trans. Power Syst.

2018

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“…The Numerical Polynomial Homotopy Continuation (NPHC) method [43]- [45] has recently gained attention as it successfully found all the solutions of the power flow equations (2) for the IEEE test systems with up to 14 buses [35], [36]. The method has also been applied [37] to find all equilibria of the Kuramoto model [46], a prototypical model for the power flow equations [47], for up to an 18bus case with different network topologies.…”

Section: The Numerical Polynomial Homotopy Continuation Methodsmentioning

confidence: 99%

“…As described in Section III, the most computationally tractable of these methods are based on numerical polynomial homotopy continuation (NPHC). Existing techniques are tractable for systems with up to 14 buses [35], [36] (and the equivalent of 18 buses for the related Kuramoto model [37]). The NPHC methods use continuation to trace all complex solutions from a selected "simple" polynomial system for which all solutions can be easily calculated to the solutions of the specified target system.…”

Section: Introductionmentioning

confidence: 99%

Recent advances in computational methods for the power flow equations

2016 American Control Conference (ACC)

Molzahn

Turitsyn

2016

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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 methods to homotopy continuation and other optimization-based methods. While many of these methods often efficiently find a high-voltage, stable solution due to its large basin of attraction, most of the methods struggle to find low-voltage solutions which play significant role in certain stability-related computations. While we do not claim to have exhausted the existing literature on all related methods, this tutorial paper introduces some of the recent advances in methods for solving power flow equations to the wider power systems community as well as bringing attention from the computational mathematics and optimization communities to the power systems problems. After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method, Gröbner basis techniques, and moment/sum-of-squares relaxations using semidefinite programming. In passing, we also emphasize the importance of an upper bound on the number of solutions of the power flow equations and review the current status of research in this direction.

Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis

Chaos: An Interdisciplinary Journal of Nonlinear Science

Daleo

Dörfler

et al. 2015

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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 that for even modest sizes (N ∼ 10–20), the number of equilibria is already more than 100 000. We analyze the stability of each computed equilibrium as well as the configuration of angles. Our exploration of the equilibrium landscape leads to unexpected and possibly surprising results including non-monotonicity in the number of equilibria, a predictable pattern in the indices of equilibria, counter-examples to conjectures, multi-stable equilibrium landscapes, scenarios with only unstable equilibria, and multiple distinct extrema in the stable equilibrium distribution as a function of the number of cycles in the graph.