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# 9291_c011 - 11 Direct Stability Methods 11.1 11.2 11.3 11.4...

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11 Direct Stability Methods Vijay Vittal Arizona State University 11.1 Rev iew of Literature on Direct Methods ....................... 11 -2 11.2 The Power System Model ............................................... 11 -4 Review of Stability Theory 11.3 The Transient Energ y Function ...................................... 11 -8 11.4 Transient Stabilit y Assessment ....................................... 11 -9 11.5 Determination of the Controlling UEP ......................... 11 -9 11.6 The BCU (Boundar y Controlling UEP) Method ....... 11 -10 11.7 Applications of the TEF Method and Modeling Enhancements .............................................. 11 -11 Direct methods of stability analysis determine the transient stability (as defined in Chapter 7 and described in Chapter 8 ) of power systems wi thout explicitly obtaining the solutions of the differential equations governing the dynamic behav ior of the system. The basis for the method is Lyapunov’s second method, also known as Lyapunov’s direct method, to determine stability of systems governed by differential equations. The fundamental work of A.M. Lyapunov (1857–1918) on stability of motion was published in Russian in 1893, and was translated into French in 1907 (Lyapunov, 1907). This work received little attention and for a long time was forgotten. In the 1930s, Soviet mathematicians revived these investigations and showed that Lyapunov’s method was applicable to several problems in physics and engineering. This revival of the subject matter has spaw ned several contributions that have led to the fur ther development of the theor y and application of the method to physical systems. The follow ing example motivates the direct methods and also provides a comparison wit h the conventional technique of simulating the differential equations governing the dynamics of the system. Figure 11.1 shows an illustration of the basic idea behind the use of the direct methods. A vehicle, initially at the bottom of a hill, is given a sudden push up the hill. Depending on the magnitude of the push, the vehicle w ill either go over the hill and tumble, in which case it is unstable, or the vehicle will climb only part of the way up the hill and return to a rest position (assuming that the vehicle’s motion w ill be damped), i.e., it w ill be stable. In order to determine the outcome of distur bing the vehicle’s equilibrium for a given set of conditions (mass of the vehicle, magnitude of the push, heig ht of the hill, etc.), two different methods can be used: 1. Know ing the initial conditions, obtain a time solution of the equations describing the dynamics of the vehicle and track the position of the vehicle to determine how far up the hill the vehicle will travel. This approach is analogous to the traditional time domain approach of determining stability in dynamic systems. 2. The approach based on Lyapunov’s direct method would consist of characterizing the motion of the dynamic system using a suitable Lyapunov function. The Lyapunov function should satisfy certain sign definiteness properties. These properties will be addressed later in this subsection. A natural choice for the Lyapunov function is the system energy. One would then compute the ß 2006 by Taylor & Francis Group, LLC.

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9291_c011 - 11 Direct Stability Methods 11.1 11.2 11.3 11.4...

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