Direct Stability Analysis of Electric Power Systems Using Energy Functions: Theory, Applications, and Perspective HSIAO-DONG CHIANG,

CHIA-CHI CHUAND GERRY CAULEY

ABSTRACT Problem Stability analysis for system planning and operation Solution Primary power tools are programs for predicting system response to various disturbances. Background/preamble The methods of stability analysis for a given disturbance: 1. the conventional approach calculates the transient behavior of generators to determine whether stability has been maintained or lost. 2. Online direct stability analysis method compares present system state to a threshold to check if the system will remain stable once disturbance is removed. 3. Direct methods also provide a quantitative measure of the degree of system stability but are not as time consuming as conventional methods The provision of additional information makes direct methods very attractive when the relative stability of different plans must be compared or when stability limits must be calculated quickly. Important conclusions and decisions are made based on the results of stability studies.
Table 1 Time-Domain Approach applicable to general power system stability models provide time responses of all state variables slow in computation no measure of degree of stability no useful information regarding how to derive preventive control Direct Methods (Based on Energy Functions) fast in computation measure the degree of stability or instability provide useful information regarding how to derive preventive control only applicable to power system stability models having energy functions provide no time response of any state variables of the post-fault system

Paper Outline This paper starts with an intuitive understanding of direct methods and then develops energy function theory, followed by a theoretical foundation for direct methods for both network-reduction and network-preserving power system models. An advanced method, called the BCU method, of computing the controlling unstable equilibrium point (u.e.p.), which is essential in determining the critical energy value, is presented along with its theoretical foundation. In addition to an overview, new material is offered. Numerical solution algorithms capable of supporting online applications of direct methods are provided. These algorithms provide practical solutions to problems that have in the past made direct methods infeasible for on-line applications. A major limitation of direct methods in the past has been the simplicity of the models used in various direct method implementations in software programs. Much of this limitation has been overcome and is presented in this paper. Another limitation has been that direct methods apply to first swing instability only. However, one of the most viable direct methods, the controlling u.e.p. method, is now shown in this paper to provide useful information for identifying multiswing unstable cases. In practical applications, the controlling u.e.p. method in conjunction with the BCU method has shown promise as a tool for fast approximate contingency screening (thereby improving performance) and efficiently computing operating limits. *This paper describes emerging applications of direct methods for on-line applications. Two examples include online transient stability assessments at the control center of Northern States Power Company and on-line transfer limit calculations at Ontario Hydro.

A major activity in utility system planning and operations is to test system transient stability relative to disturbances. Transient stability analysis programs are used by power system planners and operators to predict system response to various disturbances. The behavior of a power system via program simulations evaluation determines stability or operating limits or perhaps the need for additional facilities. As important conclusions and decisions are based on the results of stability studies it is important that the stability studies results are timely and accurate as possible. Until recently power companies perform transient stability analysis exclusively via time-domain approach, numerical integrations that calculate generator behaviors for any given disturbance. By examining the behavior of the generators, system stability is determined as lost or maintained. Several advantages are inherent in this approach: 1) it is directly applicable to any level of detail of power system models, 2) all the information of state variables during transient and steady-state is available, and 3) simulation results can be readily interpreted by system operators. The chief disadvantages of this practice include: 1) intensive time-consuming numerical integration, that is unsuitable for on-line applications, 2) does not provide degree of stability information (when the system is stable or unstable) or 3) information as regards preventive control ( system deemed unstable)

Each council establishes the types of disturbances which the system must withstand without cascading outages. The following conditions, although conservative in nature, can ensure cascading outages will not occur: 1) when any of a specified set of disturbances occurs, the system will survive the ensuing transient and move into a steadystate condition, 2) no bus voltage magnitudes during transients are outside their permissible ranges, 3) in this new steady-state condition, no control devices, equipment or transmission lines are overloaded and no bus voltage magnitudes are outside their permissible ranges (say 5% of nominal). The first condition is related to the transient stability problem while the second one is related to the voltage-dip problem. The conditions are referred to as dynamic security. The third condition is referred to as static security. Power system security analysis deals with the power system dynamic response to disturbances. In dynamic security analysis, the transition from the present operating condition to a new operating condition and the fact that during the transient cascading outages will not be triggered are of concern. In static security analysis, the transition to a new operating condition is assumed, and the analysis is focused on the satisfaction of both operating and engineering constraints (overloading, voltage, etc.) The present-day power system operating environment has contributed to the increasing importance of the problems associated with dynamic security assessment of power systems. To a large extent, this is due to the fact that most of the major power system breakdowns are caused by problems relating to system dynamic responses [32] which mainly come from disturbances in power systems. In practice it is expedient to do many power system stability studies to reveal the effects of different faults based on locations and types, various operating conditions, different network topologies and control device characteristics. For a large power system, many nonlinear differential and algebraic equations must be solved recursively taking valuable computer time. This computational intensity/expense imposes severe limits on the number of cases that can be studied. Moreover, the current power system operating environment motivates moving transient stability assessment from the off-line planning area into the online operating environment. Many significant benefits are realized with online application. 1. operation with margins reduced by a factor of at least 10 are possible if the security assessment is based on actual system configuration and operating conditions, instead of assumed worst case conditions, as is done in offline studies. 2. As a simple example, a power transfer corridor whose actual stability boundary on a given hour can be computed to be 2500 MW could be forced to operate at a limit of 2000 MW based on the conservative assumptions inherent to off-line analysis. An on-line stability assessment using the actual system topology and real-time data could

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compute a limit of 2450 MW with enough certainty to allow the operator to load the system to that point. The savings realized in increased MW transaction capability can be $10,000 per hour. Another benefit of on-line analysis is that the amount of analysis can be reduced to those cases relevant to actual operating conditions, thereby freeing engineering resources for other critical activities.

Thus there is always considerable incentive to find superior calculation methods for stability analysis. An alternative approach to transient stability analysis employing energy functions, called direct methods, was originally proposed by Magnusson [49] in the late 1940's, and pursued in the 1950's by Aylett [9] and in the 1960's by Gless [38], and ElAbiad and Nagappan [30]. In contrast to the time-domain approach, direct methods determine system stability directly based on energy functions. This method determines whether or not a system remains stable once the fault is cleared by comparing the system energy (when the fault is cleared) to a critical energy value. Direct methods not only avoid the time-consuming solutions of step-by-step time-domain stability analysis of the post-fault system, but also provide a quantitative measure of the degree of system stability. This additional information makes direct methods very attractive when the relative stability of different plans must be compared or when stability limits must be calculated quickly. Obviously a full step-by-step time-domain simulation cannot be run for each possible contingency. The operator needs to know if the system is safe in the present state and to what limits the system can be safely loaded to take advantage of, for example, economic opportunities. Direct methods can meet this need. A merit comparison between direct methods and the time-domain approach is summarized in Table 1. Direct methods have a long development history spanning four decades, but until recently were thought by many to be impractical for large-scale power systems analysis with detailed models. However, recent developments have made direct methods a more practical means of solving large-scale power systems with network-preserving models. As seen in these early applications, direct methods provide several key advantages in performing on-line stability assessment using the actual power system configuration and on-line state estimated data. One key advantage is their ability to assess the degree of stability (or instability). The second advantage is the ability to calculate sensitivities of the stability margin to power system parameters, allowing for efficient computation of operating limits.