This new edition addresses the needs of dynamic modeling and simulation relevant to power system planning, design, and operation, including a systematic derivation of synchronous machine dynamic models together with speed and voltage control subsystems. Reduced-order modeling based on integral manifolds is used as a firm basis for understanding the derivations and limitations of lower-order dynamic models. Following these developments, multi-machine model interconnected through the transmission network is formulated and simulated using numerical simulation methods. Energy function methods are discussed for direct evaluation of stability. Small-signal analysis is used for determining the electromechanical modes and mode-shapes, and for power system stabilizer design.
Power System Dynamics And Stability P W Sauer M A Pai.pdfl
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Time-synchronized high-sampling-rate phasor measurement units (PMUs) to monitor power system disturbances have been implemented throughout North America and many other countries. In this second edition, new chapters on synchrophasor measurement and using the Power System Toolbox for dynamic simulation have been added. These new materials will reinforce power system dynamic aspects treated more analytically in the earlier chapters.
Peter W. Sauer obtained his BS in Electrical Engineering from the University of Missouri at Rolla in 1969, and the MS and PhD degrees in Electrical Engineering from Purdue University in 1974 and 1977 respectively. He served as a facilities design engineer in the U.S. Air Force from 1969 to 1973. He is currently the Grainger Professor of Electrical Engineering at the University of Illinois, Urbana-Champaign where he has been since 1977. His main work is in modeling and simulation of power systems with applications to steady-state and transient stability analysis. He served as the program director for power systems at the National Science Foundation from 1990 to 1991. He was a cofounder of PowerWorld Corporation and the Power Systems Engineering Research Center (PSERC). He is a registered Professional Engineer in Virginia and Illinois, a Fellow of the IEEE, and a member of the U.S. National Academy of Engineering. M. A. Pai is Professor Emeritus in Electrical and Computer Engineering at the University of Illinois, Urbana-Champaign. He received his BE degree from Univ. of Madras, India in 1953, MS and PhD degrees from University of California, Berkeley in 1957 and 1961 respectively. He was with the Indian Institute of Technology, Kanpur, India from 1963 to 1981 and at the University of Illinois, Urbana-Champaign, from 1981 to 2003. His research interests are in dynamics and stability of power systems, smart grid, renewable resources and power system computation. He is the author of several text books and research monographs in these areas. He is a Fellow of IEEE, I.E. (India) and the Indian National Science Academy. Joe H. Chow is Professor of Electrical, Computer, and Systems Engineering at Rensselaer. He received his BS degrees in Electrical Engineering and Mathematics from the University of Minnesota, Minneapolis, in 1974, and his MS and PhD degrees from the University of Illinois, Urbana-Champaign, in 1975 and 1977. He worked in the power systems business at General Electric Company in 1978 and joined Rensselaer in 1987. His research interests include power system dynamics and control, voltage stability analysis, FACTS controllers, synchronized phasor measurements and applications, and integration of renewable resources. He is a fellow of IEEE, and past recipient of the Donald Eckman Award from the American Automatic Control Council, the Control Systems Technology Award from the IEEE Control Systems Society, and the Charles Concordia Power Systems Engineering Award from the IEEE Power and Energy Systems Society. Permissions Request permission to reuse content from this site
Generation rescheduling is a major action against abnormal system conditions such as small signal instability. The operating point of the power system impacts the small signal stability; therefore, this article proposes a novel sequential generation rescheduling model as a preventive control against low-frequency fluctuations. An optimal power flow (OPF) problem is utilized to determine the base operating point. Then, in the process of improving the small signal stability, the modal analysis specifies the system stability status in each step, and the proposed second-order convex redispatch model defines the optimal direction of generation rescheduling. Small signal stability is considered a constraint in this model. Using a sensitivity analysis, system generators and system buses are divided into two increasing and decreasing groups, which are included in the model according to their contribution coefficients in damping improvement. Finally, the alternating current (AC) power flow analysis provides the subsequent operating points. The redispatch model is solved using the quadratic programming algorithm, and the Newton algorithm is used to manage the nonlinear and nonconvex characteristics of the power flow model. The proposed method is simulated over the IEEE 9, 39, and 118 bus test systems. It is shown that the proposed method increases the damping of the power system from the unstable state to the desired condition by controlling the operating point.
In this paper, in order to deal with the discontinuity nature of the small signal stability index, a sequential solution approach is adopted, and also to reduce the computational burden and complexity, a second-order convex power generation rescheduling model is proposed. First, the OPF model is solved with the aim of minimizing the production cost of the generators. Power balance equations, branch flow limits as well as upper and lower limits of generation levels and voltage limits are considered as constraints. The results are used to extract a state-space model of the power system and calculate system eigenvalues. In case of instability or undesirable damping ratio, the generators are dispatched to provide the specified conditions. Critical eigenvalues are identified at each step to manage their cyclical behaviour. Also, to ensure the effectiveness of the small signal stability index linearization, the amount of change in the active power of the generators and the bus voltage magnitudes is limited.
The rest of this paper is organized as follows: Section 2 is dedicated to the problem formulation. Small signal stability and OPF problem are briefly introduced and modelled in Sections 2.1 and 2.2, respectively. In Section 3, the modelling of the proposed method is described. Section 4 evaluates the effectiveness of the proposed method by applying it to three different test systems. Section 5 concludes the article.
Change of system operating conditions affects the stability and economic efficiency of the power system, so to achieve various objectives of generation scheduling, security and economic indices need to be taken into account in dispatch strategies. The purpose of this section is to consider the small signal stability in economic generation dispatch or OPF problem.
Small signal stability refers to the ability of the power system to maintain synchronism under small disturbances such as load perturbation [26]. To study the small signal stability, the dynamic behaviour of power systems is described by a set of differential and algebraic equations (DAE) [27]. Generator dynamics as differential equations and the network model as the algebraic equations can be written in the following compact form:where and are vectors of system state and algebraic variables, respectively, and is the input to the system. The power system model is nonlinear, but it can be linearized at a specific operating point to analyse the small signal stability. The first-order approximation of the Taylor series of the power system equations around the equilibrium point leads to the following equation:
Assuming as a nonsingular matrix, can be removed from the set of equations, and we can achieve the following:where is known as the system state matrix. Once the system state matrix is specified, the stability condition of the linear system can be determined from the eigenvalue problem:where and are matrices of the left and right eigenvectors, respectively, and is a diagonal matrix of eigenvalues. According to Lyapunov theory, the system is stable in terms of small signal stability if the real part of all eigenvalues of the state matrix is nonpositive. In this article, the minimum damping ratio of the system is considered as a small signal stability index:
Since the small signal stability is evaluated after the linearization of the power system equations around a particular operating point, the direct solution of the optimization problem that creates a different operating point does not preserve the small signal stability analysis. As shown in Figure 1, we deal with this issue using a sequential approach. Using the standard OPF that takes into account the AC power flow, the initial operating point is determined and the subsequent operating points are determined based on the power flow. In solving the power flow problem, two known quantities associated with each bus should be determined. Since the amount of network load does not change during the stability improvement process, the load demand (P-Q) is known. In the case of regulated or voltage-controlled buses (PV buses), the active power and voltage magnitude are obtained by solving the redispatch model.
In each step, alternating current (AC) power flow is used to determine the operating point, which takes losses into account. However, for moving from one operating point to another point and determining the direction of redispatch, because the losses of the transmission system are insignificant and on the contrary, redispatch does not create a noticeable change in the losses, so it is ignored.
In order to move towards the stability region or improve the damping ratio of the system, the damping should be increased in each step. If is an optimization variable, using the Taylor expansion of the damping ratio, we have the following:where is the first-order sensitivity of the damping ratio. To determine the status of the voltage-controlled buses at the new operating point, the active power of the generators and the voltage magnitude of the buses are considered as optimization variables. Therefore, the sensitivity of the damping ratio to these two variables is determined, and the small signal stability constraint is considered according to (14) in the optimization problem as follows: 2ff7e9595c
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