Equilibria analysis of power systems using a numerical homotopy method

Chandra, Souvik; Mehta, Dhagash; Chakrabortty, Aranya

doi:10.1109/pesgm.2015.7285823

Cited by 13 publications

(6 citation statements)

References 7 publications

(10 reference statements)

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“…Solving the system at many parameter points can then help to visualise how the system varies with the parameters. We will exploit this feature of the NPHC method, called the parameter homotopy in separate papers [28–30].…”

Section: Polynomial Homotopy Continuation Methodsmentioning

confidence: 99%

“…It should be pointed out that the previous works demonstrating similar methods to the PF equations either scaled badly with the system size [10] or were proved not to find all the solutions in practice [2, 14, 27]. Moreover, the NPHC method can store the general form of solutions that only depend on the system parameters, the feature which we will exploit in the PF problems elsewhere [28–30].…”

Section: Introductionmentioning

confidence: 99%

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Numerical polynomial homotopy continuation method to locate all the power flow solutions

Mehta

Nguyen

Turitsyn

2016

IET gener. transm. distrib.

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Abstract-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 homotopy continuation method called the numerical polynomial homotopy continuation (NPHC) method that is mathematically guaranteed to find all the solutions without ever encountering a bifurcation. The method is based on embedding the real form of power flow equation in complex space, and tracking the generally unphysical solutions with complex values of real and imaginary parts of the voltage. The solutions converge to physical real form in the end of the homotopy. The so-called γ-trick mathematically rigorously ensures that all the paths are wellbehaved along the paths, so unlike other continuation approaches, no special handling of bifurcations is necessary. The method is embarrassingly parallelizable and can be applied to reasonably large sized systems. We demonstrate the technique by analysis of several standard test cases up to the 14-bus system size. Finally, we discuss possible strategies for scaling the method to large size systems, and propose several applications for transient stability analysis and voltage stability assessment.

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Section: Polynomial Homotopy Continuation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical polynomial homotopy continuation method to locate all the power flow solutions

Mehta

Nguyen

Turitsyn

2016

IET gener. transm. distrib.

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“…, N }. The equilibrium for each of the different buses are obtained by considering power balance with the neighboring buses [22]. The superscript e for any variable is used to indicate its equilibrium value(s).…”

Section: Power Flow Solution Boundaries In Power Systems With Renewab...mentioning

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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“…As we explain in Section 8, ours is not the first algebraisation of the system (1), cf. [37,4,3,2], and there is a long history [34,14,26,23,31,29,9,39,41] of the use of homotopycontinuation methods.…”

Section: Introductionmentioning

confidence: 99%

Power Flow as an Algebraic System

Marecek,

McCoy,

Mevissen

2014

Preprint

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Steady states of alternating-current (AC) circuits have been studied in considerable detail. In 1982, Baillieul and Byrnes derived an upper bound on the number of steady states in a loss-less AC circuit [IEEE TCAS, 29(11): 724-737] and conjectured that this bound holds for AC circuits in general. We prove this is indeed the case, among other results, by studying a certain multi-homogeneous structure in an algebraisation.

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Equilibria analysis of power systems using a numerical homotopy method

Cited by 13 publications

References 7 publications

Numerical polynomial homotopy continuation method to locate all the power flow solutions

Numerical polynomial homotopy continuation method to locate all the power flow solutions

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

Power Flow as an Algebraic System

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