An efficient method to locate all the load flow solutions - revisited

Lesieutre, Bernard C.; Wu, Dan

doi:10.1109/allerton.2015.7447029

Cited by 29 publications

(30 citation statements)

References 16 publications

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“…However, the robustness proof showing that it locates all equilibria for this more general class of problems was shown to be flawed [9] with a counterexample presented in [30]. In [24], a modification of the method based on an elliptical reformulation of equations was shown to have improved robustness. There currently does not exist a robustness proof for this modification or a known counterexample, so the capabilities of this method remain to be fully characterized.…”

mentioning

confidence: 99%

Locating and Counting Equilibria of the Kuramoto Model with Rank-One Coupling

Coss¹,

Hauenstein²,

Hong³

et al. 2018

SIAM J. Appl. Algebra Geometry

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Abstract. The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phase-coherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, n, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of n. For these, the equilibria can be located by finding the real solutions of a system of polynomial equations utilizing techniques from algebraic geometry. However, typical methods for solving such systems locate all complex solutions even though only the real solutions give equilibria. In this paper, we present an algorithm to locate only the real solutions of the model, thereby shortening computation time by several orders of magnitude in certain situations. This is accomplished by choosing specific equilibria representatives and the consequent algebraic decoupling of the system. The correctness of the algorithm (that it finds only and all the equilibria) is proved rigorously. Additionally, the algorithm can be implemented using interval methods, so that the equilibria can be approximated up to any given precision without significantly more computational effort. We also compare this solution approach to other computational algebraic geometric methods. Furthermore, analyzing this approach allows us to prove, asymptotically, that the maximum number of equilibria grows at the same rate as the number of complex solutions of a corresponding polynomial system. Finally, we conjecture an upper bound on the maximum number of equilibria for any number of oscillators, which generalizes the known cases and is obtained on a range of explicitly provided natural frequencies.

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mentioning

confidence: 99%

Locating and Counting Equilibria of the Kuramoto Model with Rank-One Coupling

Coss¹,

Hauenstein²,

Hong³

et al. 2018

SIAM J. Appl. Algebra Geometry

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show abstract

“…To make this article self-sustained, we briefly explain the search strategies in the outer loop and summarize it in Algorithm 1. Interested readers can refer to [20]. x 0 ← x k ⊲ Update starting solution 9: for l = 1, 2, · · · , N eqn do 10:…”

Section: A Outer Loop For Solution Searchmentioning

confidence: 99%

“…In this part, we testify the traditional full predictor-corrector method from [20] and the proposed HEBC method on the same set of test cases, and compare their numerical performances. 12 Tap ratios are removed in this case to reduce the number of solutions.…”

Section: B Comparison To Full Predictor-corrector Algorithmmentioning

confidence: 99%

“…Though efficient, [19] provided a counter-example for [17]. To rectify their method, an elliptical formulation of the power flow problem is used in [20] to restrict the curve design on high dimensional ellipses. It helps solve all the standard IEEE test cases which can be verified by the homotopy method in a reasonable time 3 , including the counter-example in [19].…”

Section: Introductionmentioning

confidence: 99%

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Holomorphic Embedding Based Continuation Method for Identifying Multiple Power Flow Solutions

Wang

2019

IEEE Access

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In this paper, we propose an efficient continuation method for locating multiple power flow solutions. We adopt the holomorphic embedding technique to represent solution curves as holomorphic functions in the complex plane. The holomorphicity, which provides global information of the curve at any regular point, enables large step sizes in the path-following procedure such that non-singular curve segments can be traversed with very few steps. When approaching singular points, we switch to the traditional predictor-corrector routine to pass through them and switch back afterward to the holomorphic embedding routine. We also propose a warm starter when switching to the predictor-corrector routine, i.e. a large initial step size based on the poles of the Padé approximation of the derived holomorphic function, since these poles reveal the locations of singularities on the curve. Numerical analysis and experiments on many standard IEEE test cases are presented, along with the comparison to the full predictor-corrector routine, confirming the efficiency of the method.

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“…This section applies an efficient method [21], [22] to solve (12) for multiple equilibra of the DAE model (7). We gradually change the load model from constant power to constant impedance and observe the change of equilibra.…”

Section: Equilibria With Different Load Modelsmentioning

confidence: 99%

Influence of Load Models on Equilibria, Stability and Algebraic Manifolds of Power System Differential-Algebraic System

Wang

2019

2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Load models have a great impact on voltage behaviors as well as power system transient dynamics. Extensive work has been done on this topic, proposing appropriate load models and capturing better load behaviors during transient. This paper presents a comprehensive study to investigate the geometric and topological changes induced by different load models for the traditional power system differential-algebraic equations. Specifically, we attempt to reveal the deformation of equilibria, stability regions, and algebraic manifolds during a continuous evolution of load model. Several findings are presented in the paper, some of which countering traditional recognitions and intuitions. A major discovery is that the load model with a large proportion of constant impedance and a small proportion of constant power exhibits much more complex features than the load model with the reversed proportions of impedance and power. The increase of complexity is thoroughly recorded and investigated by the changes of geometric properties and mutations of topological invariants in the sense of equilibria, stability regions, and algebraic manifolds for the DAE system. However, most of the changes seem to occur on unstable components of algebraic manifolds or near the singular boundary surfaces, suggesting a limited variation of dynamical behaviors on the stable component.

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An efficient method to locate all the load flow solutions - revisited

Cited by 29 publications

References 16 publications

Locating and Counting Equilibria of the Kuramoto Model with Rank-One Coupling

Locating and Counting Equilibria of the Kuramoto Model with Rank-One Coupling

Holomorphic Embedding Based Continuation Method for Identifying Multiple Power Flow Solutions

Influence of Load Models on Equilibria, Stability and Algebraic Manifolds of Power System Differential-Algebraic System

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