Power Flow as an Algebraic System

Marecek, Jakub; McCoy, Timothy; Mevissen, Martin

doi:10.48550/arxiv.1412.8054

Cited by 4 publications

(7 citation statements)

References 37 publications

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“…[3] and then for the general case in Ref. [42] (see [46] for a recent alternative derivation of this bound). We shall refer to this bound as the Baillieul-Byrne-Li-Sauer-Yorke (BBLSY) bound.…”

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confidence: 96%

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

Chen

Mehta

2018

IEEE Trans. Power Syst.

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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 †

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confidence: 96%

“…Other polynomial formulations of the load flow equations have been also known (see, e.g., [2,3,8,9,46,49]).…”

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confidence: 99%

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

Chen

Mehta

2018

IEEE Trans. Power Syst.

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“…Recent reviews on the existing results on upper bounds are provided in [27] and [28]. An upper bound of 2N −2 N −1 was computed in [29]- [31] for a generic power flow problem with N buses, although it did not still exploit the network topologies. In [32], the number of complex solutions for networks with cliques with exactly one common node was shown to be equal to the product of number of complex solutions for the individual cliques as independent networks.…”

Section: Parameter Homotopy Continuation Algorithmmentioning

confidence: 99%

Locating Power Flow Solution Space Boundaries: A Numerical Polynomial Homotopy Approach

Chandra¹,

Mehta²,

Chakrabortty³

2017

Preprint

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The solution space of any set of power flow equations may contain different number of real-valued solutions. The boundaries that separate these regions are referred to as power flow solution space boundaries. Knowledge of these boundaries is important as they provide a measure for voltage stability. Traditionally, continuation based methods have been employed to compute these boundaries on the basis of initial guesses for the solution. However, with rapid growth of renewable energy sources these boundaries will be increasingly affected by variable parameters such as penetration levels, locations of the renewable sources, and voltage set-points, making it difficult to generate an initial guess that can guarantee all feasible solutions for the power flow problem. In this paper we solve this problem by applying a numerical polynomial homotopy based continuation method. The proposed method guarantees to find all solution boundaries within a given parameter space up to a chosen level of discretization, independent of any initial guess. Power system operators can use this computational tool conveniently to plan the penetration levels of renewable sources at different buses. We illustrate the proposed method through simulations on 3-bus and 10-bus power system examples with renewable generation.

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“…This bound was extended to (potentially lossy) systems of PQ buses in [40] and to general power systems in [41]. (See [42] for an alternative proof of this bound. )…”

Section: Introductionmentioning

confidence: 99%

“…Whereas the bounds in [39]- [42] are network agnostic, the monomial structure of the power flow equations is determined by the network topology. Topology-dependent upper bounds have the potential to be tighter than previous bounds for specific problems.…”

Section: Introductionmentioning

confidence: 99%

Investigating the Maximum Number of Real Solutions to the Power Flow Equations: Analysis of Lossless Four-Bus Systems

Molzahn,

Niemerg,

Mehta

et al. 2016

Preprint

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The power flow equations model the steady-state relationship between the power injections and voltage phasors in an electric power system. By separating the real and imaginary components of the voltage phasors, the power flow equations can be formulated as a system of quadratic polynomials. Only the real solutions to these polynomial equations are physically meaningful. This paper focuses on the maximum number of real solutions to the power flow equations. An upper bound on the number of real power flow solutions commonly used in the literature is the maximum number of complex solutions. There exist two-and three-bus systems for which all complex solutions are real. It is an open question whether this is also the case for larger systems. This paper investigates four-bus systems using techniques from numerical algebraic geometry and conjectures a negative answer to this question. In particular, this paper studies lossless, four-bus systems composed of PV buses connected by lines with arbitrary susceptances. Computing the Galois group, which is degenerate, enables conversion of the problem of counting the number of real solutions to the power flow equations into counting the number of positive roots of a univariate sextic polynomial. From this analysis, it is conjectured that the system has at most 16 real solutions, which is strictly less than the maximum number of complex solutions, namely 20. We also provide explicit parameter values where this system has 16 real solutions so that the conjectured upper bound is achievable.

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Power Flow as an Algebraic System

Cited by 4 publications

References 37 publications

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

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

Locating Power Flow Solution Space Boundaries: A Numerical Polynomial Homotopy Approach

Investigating the Maximum Number of Real Solutions to the Power Flow Equations: Analysis of Lossless Four-Bus Systems

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