BCU Method Research Articles

A Homotopy-Based Method for Robust Computation of Controlling Unstable Equilibrium Points

This paper proposes a homotopy-based method for efficiently computing and precisely determining controlling unstable equilibrium points (CUEPs) in transient stability analysis and screening using direct methods. The proposed method is intended to overcome potential numerical convergence problems associated with computing CUEPs. These convergence problems are mostly due to irregularities in the region of convergence of CUEPs which make it difficult to determine suitable initial points. The most commonly used initial points in computing CUEPs are minimum gradient points (MGPs) on the stability boundary which in turn rely on the determination of exit points (EPs-points at which sustained-fault trajectories exit system stability boundaries). However, determination of MGPs and EPs are computationally involved and sophisticated numerical methods have been introduced such as stability-boundary-following procedure and shadowing method to correct the solution trajectory. The proposed method does not depend on MGPs and also does not require accurate EPs. The proposed method maps the solution from EPs to the CUEPs. The proposed method is applied on several known systems including the NE 39 bus system and the reduced WECC system and the results are provided. The results are compared with the traditional methods such as Boundary of stability region based CUEP method (BCU method).

Design and Implementation of on‐line dynamic security assessment

AbstractDespite possible changes in energy sources, the importance of maintaining the stability of interconnected power systems will not change. Technologies for maintaining power system stability must evolve with renewable energy technology. As renewable energy increases, analysis and control methods for transient stability assessment must be reexamined and improved. This paper presents an overview of on‐line dynamic security assessment (DSA) systems and related direct methods. The controlling unstable equilibrium point (UEP) method and the boundary of the stability‐region‐based controlling UEP (BCU) method are briefly described in order to provide a framework of understanding the concept of a transient stability‐screening program for DSA systems. The integrated package TEPCO‐BCU, a very advanced practical program based on the theory of the controlling UEP method, BCU method, energy function method, and BCU‐guided time‐domain method, is described from the viewpoint of on‐line performance. Finally, the importance of validating the assessment results of the BCU method is discussed. Copyright © 2009 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

A new method for transient stability analysis

AbstractThis paper proposes a brand‐new method for transient stability analysis in power systems. The proposed method directly computes the critical trajectory for a given contingency to obtain the critical condition of the studied system. Although the method may be useful for general nonlinear dynamic systems, it is applied to the problem of obtaining a controlling UEP, unstable equilibrium point, which provides inevitable information for the energy function methods to assess transient stability. Namely, the proposed method effectively yields a critical trajectory on PEBS together with the controlling UEP, thus improving the conventional BCU method. The effectiveness of the proposed method is demonstrated in 3‐machine 9‐bus and 6‐machine 30‐bus systems. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn, 159(3): 26–33, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20245

過渡安定性評価のための新しい解析手法の提案

This paper proposes a brand-new method for transient stability analysis in power systems. The proposed method directly computes the critical trajectory for a given contingency to obtain the critical condition of the studied system. Although the method may be useful for general nonlinear dynamic systems, it is applied to the problem of obtaining a controlling UEP, unstable equilibrium point, which provides inevitable information for the energy function methods to assess transient stability. Namely, the proposed method effectively yields a critical trajectory on PEBS together with the controlling UEP, thus improving the conventional BCU method. The effectiveness of the proposed method is demonstrated in 3-machine 9-bus and 6-machine 30-bus systems. © 2007 Wiley Periodicals, Inc. Electr Eng Jpn, 159(3): 26–33, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20245

AN EXTENDED BCU METHOD TO POWER SYSTEM TRANSIENT STABILITY ASSESSMENT

This paper shows that under some circumstances the so called BCU method may not be able to find the true CUEP for a specific fault-on trajectory of a power system. The reasons to that malfunction are analysed, and a methodology than can improve the capability of the BCU method to successfully find the CUEP is presented. Finally, several applications of the proposed method to power electrical systems, are shown.

Application of Melnikov's method for computing heteroclinic orbits in a classical SMIB power system model

The main concern of this paper is the application of the Melnikov method for computing heteroclinic orbits in a classical SMIB power system model and testing the one-parameter transversality condition required in the BCU method for transient stability analysis. First, the Melnikov method is used to give a first approximation for the minimum damping coefficient or, alternatively, for the maximum input mechanical power that should be associated with the system so that the transversality condition is satisfied. This first approximation is obtained without solving any differential equations. After that, this approximation is used as an initial condition for a monotonically convergent process which is very easy to implement and does not require knowledge of fault trajectory at all.

Generic properties, one-parameter deformations, and the BCU method [power system stability

This paper discusses the theoretical justification of the BCU method for the evaluation of power system stability. Recent literature has suggested that the method can be mathematically justified in terms of a one-parameter deformation between two systems. This paper shows that the assumptions of this method do not hold generically for power system models. In particular, the one-parameter transversality conditions inherent in the deformation argument do not apply for a large region in parameter space, thus, the validity of the BCU method cannot be theoretically justified by this procedure.

An improved technique to determine the controlling unstable equilibrium point in a power system

The accuracy of stability assessment provided by the transient energy function (TEF) method depends on the determination of the controlling unstable equilibrium point (UEP). The technique that currently determines the controlling UEP in the TEF method is based on the so-called exit point method and has also been recently labeled the BCU method. The exit point method consists of two basic steps. First, the exit point is approximated by the point /spl theta//sup egsa/, where the first maximum of the potential energy along the fault-on trajectory is encountered. Second, the minimum gradient point /spl theta//sup mgp/ along the trajectory from /spl theta//sup egsa/ is computed. The controlling UEP is then obtained by solving a system of nonlinear algebraic equations with /spl theta//sup mgp/ as an initial guess. It has been observed that this method lacks robustness in the sense that the following two problems may occur: (1) there may be no detection of the minimum gradient point /spl theta//sup mgp/ and hence, no determination of the controlling UEP, (2) if /spl theta//sup mgp/ is found, then based on the definition of the controlling UEP, it may not be in the domain of convergence of the controlling UEP for the particular solving algorithm used. Hence, another equilibrium point, possibly a stable equilibrium point, not the controlling UEP will be located. This results in a flawed transient stability assessment. The result of this research has been the development of a new numerical technique for determining the controlling UEP. With an initial starting point that is close to the exit point this technique efficiently produces a sequence of points. An analytical foundation for this method is given which shows that under certain assumptions this sequence will converge to the controlling UEP. Hence this new technique exhibits a substantial improvement over the exit point method because of the following reasons: (1) the technique does not attempt to detect the point /spl theta//sup mgp/, (2) the technique can produce a point that is close to the controlling UEP thus avoiding a domain of convergence problem. The analytical foundation is provided for the unloaded gradient system, but an application of the technique to the IEEE 50-generator system shows that satisfactory stability assessment is also obtained for more general systems, for which the exit point method fails.

Some clarifications in the transient energy function method

The purpose of this paper is to clarify the evaluation of path dependent integrals in the energy function method for stability analysis in power systems. In the literature these are handled in an approximate manner through straight line approximation leading to closed form analytic expressions of the energy functions. This may not always be accurate. Here it is compared with the trapezoidal method of integration along the faulted trajectory as originally proposed by Athay et al. The paper also emphasizes some tutorial aspects for explaining the PEBS and the BCU method.

Theoretical foundation of the BCU method for direct stability analysis of network-reduction power system. Models with small transfer conductances

The controlling unstable equilibrium point (UEP) method is considered to be the most viable for direct stability analysis of practical power systems. The success of the controlling UEP method, however, hinges upon its ability to find the (correct) controlling UEP. Recently, a systematic method, called the boundary of stability region based controlling unstable equilibrium point method (BCU method), to find the controlling UEP was developed. The BCU method has been evaluated in a large-scale power system and it compared favorably with other methods in terms of its reliability and the required computational efforts. In this paper, a theoretical foundation of the BCU method for the network-reduction classical machine stability model is developed. Sufficient conditions for the BCU method to find the correct controlling UEP relative to a given fault are derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Direct stability analysis of electric power systems using energy functions: theory, applications, and perspective

Stability analysis programs are a primary tool used by power system planning and operating engineers to predict the response of the system to various disturbances. Important conclusions and decisions are made based on the results of stability studies. This paper presents a theoretical foundation of direct methods for both network-reduction and network-preserving power system models. In addition to an overview, new results are offered. A systematic procedure of constructing energy functions for both network-reduction and network-preserving power system models is proposed. An advanced method, called the BCU method, of computing the controlling unstable equilibrium point is presented along with its theoretical foundation. Numerical solution algorithms capable of supporting online applications of direct methods are provided. Practical demonstrations of using direct methods and the BCU method for online transient stability assessments on two power systems are described. Further possible improvements, enhancements and other applications of direct methods are outlined.

A BCU method for direct analysis of power system transient stability

A boundary of stability region based controlling unstable equilibrium point method (BCU method) for direct analysis of power system transient stability is presented. Features distinguishing the BCU method from the existing direct methods are that it consistently finds the exact controlling unstable equilibrium point (UEP) relative to a fault-on trajectory and that it has a sound theoretical basis. Moreover, the BCU method appears to be fast. This method is based on the relationship between the boundary of stability region of a power system and that of a reduced system. The BCU method finds the controlling UEP of the original system via a reduced system whose controlling UEP is easier and cheaper to compute. Effective numerical schemes to speed up the presented method are also proposed. This method has been tested on several power systems with very promising results. Simulation results on a 50 generator, 145 bus system are presented along with a comparison between the results obtained using the BCU method and another existing method. >

On the nature of unstable equilibrium points in power systems

The application of direct methods for transient stability analysis of stressed power systems is considered. One of the most important issues in this area is to find the controlling unstable equilibrium point for the disturbance under consideration. A conceptual framework for discussing stable and unstable equilibrium points in power systems based on simple topological arguments is provided. It is shown that every conceivable case of system separation can be related to a specific unstable equilibrium point (UEP) in an unloaded system. This result can be seen as a verification of the soundness of the often-used corrected corner point approximation and ray point approximation, since these two approximations would give the same unstable equilibrium points in an unloaded system. Moreover, an example is given to show that some of the unstable equilibrium points can disappear when the loading of the system increases. The implications of the findings are discussed for the so-called MOD method of finding the controlling UEP, which is used in some software packages. A method based on a combination of the MOD method and the BCU method is outlined and proposed as the subject of future investigation.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A sensitivity-based BCU method for fast derivation of stability limits in electric power systems

For a given power system with a given contingency, called the base case, if parameters (such as real power generation) in the power system change (due to some emergency, perhaps), to form what is referred to as the new case, one would like to know how these changes will affect the system stability as quickly as possible. Also, there is a growing need of a fast approach to determine the transient stability loading limits of a system and to decide how to make generation rescheduling and load shedding in order to improve the system's transient stability. In this paper, a sensitivity-based BCU method is proposed to resolve these problems. One feature that distinguishes the proposed method from the existing methods is that it does not rely on the invalid assumption that the mode of the controlling u.e.p. does not change for any new case. A counter-example to this assumption is also provided. The proposed method has been tested on a 50-generator system. A good, fast assessment of the system stability and the stability loading limits has been obtained. >