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PDF | Power System Dynamics: Stability and Control, Second Edition, John Wiley & Sons Ltd, , pages Jan Machowski, Warsaw University of. Power System Dynamics and Stability Peter W. Sauer M. a. Pai - Ebook download as PDF File .pdf) or read book online. stability and power oscillation damping, are all operated in a decentralized DERs on power system dynamics through numerical simulations.

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POWER SYSTEM DYNAMICS. AND STABILITY. Peter W. Sauer and M. A. Pai. Department of Electrical and Computer Engineering. The University of Illinois at. POWER SYSTEM DYNAMICS Stability and Control Second Edition Jan Machowski Warsaw University of Technology, Poland Janusz W. Bialek The University. Power System Dynamics and StabilityPETER W. SAUER M. A. PAl Department of Electrical and Computer Engineering Univers.

Classic power system dynamics text now with phasor measurement and simulation toolbox. 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.

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. India and the Indian National Science Academy. Joe H. He worked in the power systems business at General Electric Company in and joined Rensselaer in His research interests include power system dynamics and control, voltage stability analysis, FACTS controllers, synchronized phasor measurements and applications, and integration of renewable resources.

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If the address matches an existing account you will receive an email with instructions to retrieve your username. Skip to Main Content. Power System Dynamics and Stability: Sauer M.

Pai Joe H. First published: Print ISBN: All rights reserved. The limit constraints on VRi are also deleted, since we wish to concentrate on modeling and simulation. The resulting differential-algebraic equations follow from 6. When the stator transients were neglected, the electrical torque became equal to the per-unit power associated with the internal voltage source.

Algebraic Equations The algebra. The stator algebra. The network equations written at the n buses are in complex form.

From 6.

Thus, 7. Equation 7. Seven differential equations d. One complex stator algebraic equation 7. One complex network equation 7. V nl ' Funct ionally, the refore, the differential equations 7. We now formally put 7. The idea is to express Idi Iqi in terms of the state and network variables. Both the polar form and the rectangular form will be explained. If we multiply 7. Thus i Alternatively, the right-hand side of 7.

Symbolically, 7. Graphical representation 7. The machine-network transformation is given by 7. Figure 7. Using 7. Note that, in this formulation , algebraic equation 7. The latter form is more popular with the industry software packages. We discuss both of them now. The network equations 7. The differential equations 7. The stator algebraic equations of the form 7. This is t he differelltialalgebraic equation DAE analytical model with the network algebraic variables in the polar form.

We prefer this form, since in load-flow equations the voltages are geIlera. Equations 7. This is also the system app earing in [74] and widely used in the literature. The base MVA is , and system frequency is 60 Hz. The Ybus for the network also denoted as Y N can be written by inspection from Figure 7. The machine data and the exciter data are given in Table 7.

Assume w. OKV Gen2 1. Electric Power Research Institute. Power System Dynamic Analysis, Phase 1. Reprinted with Permission. Table 7. Y N for the Network in Figure 7. IIU The constant power loads are treated as injected into the buses.

V7 cosB. V7 sinO. Vg cos 8. V5 sin 1i. Vs cosli. Vs cos 1i. T his is left as an exercise for the reader. The network equations are 7. The principal difference is in terms of suitably rearranging the equations from a programming point of view. This will be referred to as the industry model. From 7. For ease in programming, the electric power output of machine i in 7.

The terminal voltage Vi is 7. With these definitions, we can express the differential equations 7. Thus, assembling 7. A x , B , and C are matrices having a block structure with each block belonging to a machine. G contains the stator algebraic equations and intermediate equations for and Vi in terms of E, V. This formulation is easy to program. Alternatively, we can substitute 7. The DAE model with the network equations in the current-balance form is the preferred industry model, since the network-admittance matrix has to be refactored only if there is a network change.

Otherwise , the initial factorization will remain. The DAE model with network equati ons in powerbalance form discussed in Section 7. EN' Mac l1ne Mt,ci ,ille. Block diagram conceptualization of 7.

Thus 7. A similar substitution for iJ;, Iq , in 7. They are also amenable to a network theoretic approach. Using these simplifications, we develop later 1 the structure-preserving flux-decay model, 2 the structure-preserving classical model, and 3 the internal node model using the classical machine model and constant impedance loads.

Both 1 and 2 can have nonlinear load representations. These can be done independently or together, resulting in simplified models.

Simplification 1 neglecting transient saliency in the synchronous machine Transient saliency corresponds to different values of X di and X di , then the stator algebraic equation 7. Writing the KVL equation for Figure 7. In the differential equations, the expression for electric power P" given by 7. Equivalent circuit all quantities in network reference frame It can be verlfied that the right-hand side of 7. Simplification 2 constant impedance load in the transmission system Here, the loads are assumed to be of the constant imp edance type, i.

Load admittance representation Because of the orientation of Vi and I Li, there is a negative sign in front of the load admittance y" in 7.

From Figure 7. Yii is the complex admittance due to the loads. If we make only the constant impedance approximation, then we can obtain a passive reduced network, as in Figure 7.

If we make both assumptions, then we obtain a passive reduced network, as in Figure 7. Solution Note that the load specined in Table 7. Hl -j The resulting Y red is given by 1. The loadflow equations are part of t he network equations, as shown below. Load-Flow Formulation We now revert to the formulation of the network equ ations that were written as real and r eactive power-balance equation s at the nodes, i. We reproduce them below by writing real power equations first, followed by the reactive power equations.

It can be shown from 7. Vj q' Figure 7. Synchronous machine dynamic circuit We further define net injected power at a bus as 7. We now define standard load flow as follows, using 7. Loads are of the constant p ower t yp e. Specify bus voltage magnit udes numbered 1 to m. Specify bus voltage angle at bus number 1 slack bus. Specify net injected real power Pi a t buses numbered 2 to m.

The following equations result from 7. These are known as the load-flow equations. The standard load flow solves 7. It can also include inequality constraints on quantities such as QG.

For details, refer to [15, 16, 18]. One important point about load flow should be emphasized. Load flow is normally used to evaluate operation at a specific load level specified by a given set of powers. For a specified load and generation schedule, the solution is independent of the actual load model.

That is, it is certainly possible to evaluate the voltage at a constant impedance load for a specific case where that impedance load consumes a specific amount of power. Thus, the use of constant power" in load-flow analysis does not require or even imply that the load is truly a constant power device. It merely gives the voltage at the buses when the loads any type consume a specific amount of power. The load characteristic is important when the analyst wants to study the system in response to a change, such as contingency analysis o.

For these purposes, standar d load flow is computed on the basis of constant PQ loads and usually provides the "initial conditions" for t he dynamic system.

In power system dynamic analysis, t he fixed inputs and initial conditions are normally found from a base case load-flow solution. That is , the values of Vrefi are computed such that the m generator voltages are as specified in the load flow. The values of TMi are computed such that the m generator real power outputs PGi are as specified and computed in the load flow for rated speed w,. To see how this is done, we assume that a load-flow solution as defined in previous section has been found, i.

The first step in computing the initial conditions is normally the calculation of generator currents from 7. In steady state, aU the derivatives are zero in the differential equations 7.

The first step is to calculate t he rotor angles 6i at all the machines. We use the complex stator algebraic equation 7.

Power system dynamics: stability and control

The complex number representation fOT computing ai from 7. Thls representation is generally known as "phasor diagram" in the literature and "locating the q axis" of the machine. Representation of stator algebraic equations in steady state Step 2 Oi is computed as O.

Step 3 Compute I d;, 1. The mechanical states Wi and TMi are found from 7.

Thus, we have computed x o '-1! For a given disturbance, the inputs remain fixed throughout the simulation. If the disturbance occurs due to a fault or a network change, the algebraic states must change instantaneously.

The dynamic states cannot change instantaneously. Thus, it will be necessary to solve all the algebraic equations inclusive of the stator equations with the dynamic states specified at their values just prior to the disturbance as initial conditions to determine the new initial values of the algebraic states.

From the above description, it is clear that once a standard load-flow solution is found , the remaining dynamic states and inputs can be found in a systematic manner. The machine rotor angles O. In this case, the state that is limited would have to be fixed at its limiting value and a corresponding new steady-state solution would have to be found. This would require a new load flow by specifying either different values of generator voltages , different generator real powers , or possibly specifying generator-reactive power injections , thus allowing generator voltage to be a part of the load-ftow solution.

In fact , the use of reactive power limits in load flow can usually be traced to an attempt to consider excit ation system limits or generator capability limits. Example 7. The solved load-flow data are given in Table 7. Furthermore, if the inertia constant on this reference angle machine is infinity, the order of the system can be reduced to 7m This is also possible if the machines have zero or uILiform damping.

Finally, we can also use center-of-angle formulation instead of relative rotor angle formulation. T he center-of-angle formulation is discussed in Chapter 9. Power-Balance Form The number of algorithms that have been proposed for the numerical solution of t he DAB syst em of equations is very large. There are basically two approaches used in p ower system simulation packages. Simultaneous-implicit 51 method 2.

Partitioned-explicit PE method 7. We illustrate the S1 method on the WSCC system with a two-axis model for the machine and with the exciter on all the three machines. In the S1 method, the differential equations in 7. These resulting algebraic equations are then solved simultaneously with the remaining algebraic equations 7.

Review of Newton's Method Let! III a Taylor 7. In an expanded version: Then from 7. IT this is 7. Rearranging 7. The Newton iterates are 7m 2m 2n 7m 1, 2m J. The iterations are continued until the norm of the mismatch vector [F F 2 , F3 ]' is close to zero. But if there is a disturbance in the network , a different procedure has to be adopted, as explained below. The algebraic variables can change instantaneously, whereas the state variables do not.

If there is a short circuit at bus i, then set Vi - 0 and delete the P and Q equations at bus i from 90 X, Id-q , V to obtain 7. In the second case, a similiar procedure is followed. The PE scheme , although conceptually simple, has numerical convergence problems such as interface errors , etc.

Current-Balance Form In this section, we explain the industrial approach to implement the SI method forming the structured approach to the problem. The current- balance approach is favored, since the bus admittance matrix is easily formed and factorized. The DAE model from 7. Associate Id-q,; with the respective x; so that X; [xl I,L q;]'.

With this, the application of Newton 's method yields the following equations. The right-hand sides of 7. We recognize that the nonzero columns in BGVi correspond to t. V , and that those in CGVi correspond to t. Thus, from 7. Xi from 7. The same computations are then repeated at the next time instant and so on. Let J;: Solving 7. This reduces the overall cost of comp utation. The choice of when to r eevaluate the Jacobi an is based on exper ience.

The Jacobian must be reevaluated at any major system change. Between time st ep s, it is reevaluated if the pr evious time-step iteration was considered too slow took t hree or more iterations. Within a time step , a m aximum of five iterations are taken using the same Jacobian.

Prediction [70J Whenever a Jacobian is evaluated at of taking the converged values of the ear prediction for generator variables voltages. Thus, at time instant tnH, of the Jacobian would be t he beginning of a time step, instead previous ti me step , we can use a linand geometric prediction for network the initial estimate in the evaluation 7.

Flux-decay model with a fast exciter. T he network st ructure is preserved. Structure-preserving model with a classical m achine m odel. Classical m odel with network nodes eliminated. If the damper-winding constants are very small, then we can set them to zero i. The synchronous machine dynamic circuit is modified as shown in Figure 7. It is also common, while using the flux-decay model, to have a simplified exciter with one gain and one time constant, as shown in Figure 7.

Synchronous machine flux-decay model dynamic circuit Vi Figure 7.

Power System Dynamics and Stability Peter W. Sauer M. a. Pai

Static e: Initial Conditions The steps to compute the initial conditions of the multimachine flux-decay model DAE system are given below. Step 1 From the load flow, computeiGieh, as in 7.

Ignoring 7. The stator algebraic equations 7. Constant voltage behind transient reactance This forms the basis of the structure-preserving model which, is discussed next. It consists of the swing equations 7. In Figure 7. The complex power output at the ith internal node in Figure 7. Structure-preserving model with constant voltage behind reactance 7. At the generator buses 1, This is shown as follows. The model given by 7. An interesting observation from 7. ViV;Bij COs 8. Note that, in this model , we are allowed to have nonlinear load representation.

This model is used in voltage stability studies by means of the energy function method [11 7]. Adding these to the diagonal elements of the Y NI matrix in 7. The n network buses can be eliminated, since there is no current injection at these buses. J m for ease of notation. Step 2 Using 7. Step 3 From 7.

Thus, computation of initial conditions is simple when a classical model is used. The computation of in itial conditions and solution methodology for the two-axis model using t he simultaneous-implicit method has been discussed.

The reduced-order flux-decay model and the classical model, as well as the computation of initial conditions, are discussed. We also discussed the structure-preserving classical model. Plot the voltage magnitude at bus 5 as a fun ction of the load.

This is called a PV curve. In practice, the AGe system allocates it t hrough the area control error, etc, As an alternative, consider allocating the increased load in proportion to the inertias of the machine, i. Again draw t he PV curve. Draw the PV curves for the pre-contingency state and the contingent case on the same graph. As explained at the end of Example 7.

Express the resulting equations in the form: See Section 7. With the new load -flow compute the initial conditions for the variables, Id. You may use the model obtained in Problems 7. Do the simulation as in Problem 7. Instability occurs when relative rotor angles diverge. Also find the critical-clearing time tcr. Write the equations in the form 7. This is very much a function of the operating cond. Oscillations, even if undamped at low frequencies, are undesirable because they limit power transfers on transmission lines and, in some cases induce stress the mechanical shaft.

The source of inter-area oscillations is difficult to diagnose. Extensive research has been done in both of these areas. In recent years, there has been considerable interest in dynamic voltage collapse.

As regional transfers vary over a wide range due to restructuring and open transmission access , certain parts of the system may face increased loading conditions.

Earlier, this phenomenon was analyzed purely on the basis of static considerations , i. In this chapter, we develop a comprehensive dynamic model to study both low-frequency oscillations and voltage stability using a two-axis model with IEEE-Type I exciter, as well as the flux-decay model with a high-gain fast exciter. Both the electromechanical oscillations and t heir damping , as well as dynamic voltage stability, are discussed.

The electromechanical oscillation is of two types: Local m ode, typically in the 1 to 3-Hz range between a remotely located power station and the rest of the system. Inter-area oscillations in the range of less than 1 Hz.

Two kinds of analysis are possible: Dynamic voltage stability is analyzed by monit oring the eigenvalues of the linearized system as a power system is progressively loaded.

Instability occurs when a pair of complex eigenvalues cross to the right-half plane. This is referred to as dynamic voltage instability. Mathematically, it is called Hopf bifurcation. Also discussed in this chapter is the role of a power system stabilizer that stabilizes a m achine with respect to the local mode of oscillation.

A brief review of the approaches to the design of t he stabilizers is given. For detailed design procedures, it is necessary to r efer to the literature. References [77J and [78J are the basic works in this area. The nonlinear mo del derived in Chapter 7, with a two-axis model with IEEE-Type I exciter or flux-decay model wi t h a static exciter , is of t he form: Thus, 8.

Equation 8. To show explicitly the traditional load-flow equat ions and t he other algebraic equations, we partition y as y - [IL. OJ V,. VnJ' 83 8. Bus 1 is taken as the slack bus and buses 2, The dimension of x is 7m. Linearizing 8. The model represented by 8. We use the above formula tion for a multimachine system with a twoaxis machine model and the IEEE-Type I exciter model A and indicate a similiar extension for the flux-decay model with a fast exciter model B.

The methodology is based on [79J and [80J- 8. The differential and algebraic equations follow. The linearization of the differential equations 8. Ed; dt db. KAi Ai 6. Vrefi 8. Writing 8. VRi f. Rp, TFi E;. D enotmg [6h 1. RF' [ Md; ]! Linearizing the network equations " 8. Vi - ldio Via sin 0';0 - 8io. Bi - Via sin 6io - Bio.

Vi - IqioViocos 6io - 8io 1: In matrix notation, 8. N ote that C 2 , D3 are block diagonal, whereas D 4 , Ds are full matrices. Linearizing network equations 8. Rewriting 8.

VI L':. Equations 8. This model is quite general and can easily be expanded to include frequency or V dependence at the load buses. The power system stabilizer PSS dynamics can also be included easily. In the above model, 6 I g is not of interest and, hence, is eliminated from 8.

Thus, from 8. Note that we have eliminated the stator algebraic variables. In any rotational system, the reference for angles is arbitrary. The order of the dynamical system in 8. The differential equations 66, Wi i 2, This process is analytically neat, but is not carried out in linear analysis packages.

Since angles appear as differences, they are computed to within a constant. Hence, a zero eigenvalue is always present. Recognizing this fact, we retain the formulation corresponding to 8. We rewri te the differential-algehraic DAE system 8. Note that, compared to the algebraic Jacobian JAE of 8.

The system Asys matrix is obtained as 6. E and J LF for 1 constant power case, 2 constant current case, and 3 constant impedance case for the models A and B. We now show that the model in 8. These are 7.

Power system dynamics: stability and control - PDF Free Download

Vm I O J v Figure 8. Static exciter model Differential Equations do. The linearization of this model is done in the same manner as in model A and, hence, is not discussed.

The appropriate definition and determination as to which state variables significantly participate in the selected modes become very important. This requires a tool for identifying the state variables that have significant participation in a selected mode. It is natural to suggest that the significant state variables for an eigenvalue Ai are those that correspond to large entries in the corresponding eigenvector v;. But the entries in the eigenvector are dependent on the dimensions of the state variables which are, in general, incommensurable for example, angle, velocities, and flux.

Verghese et al. Participation factor analysis aids in the identification of how each dynamic variable affects a given mode or eigenvalue. The participation factor may also be defined by 8. We establish the equivalence as follows.

It is our goal to examine the sensitivity of the eigenvalue to a diagonal element of the A matrix. From 8. Aivi 8. An eigenvector may be scaled by any value resulting in a new vector which is also an eigenvector. In any case since 2: To handle participation factors corresponding to complex eigenvalues, we introduce some modifications as follows.

The eigenvectors corresponding to a complex eigenvalue will have complex elements. Hence, Pki is defined as 8. Example 8. The right and left eigenvectors corresponding to the complex eigenvalue. We can normalize with respect to Pn by making it unity, in wh. Compute the eigenvalues, as well as the participation factors, for the eigenvalues for the nominal loading of Example 7.

The damping Di '" a i 1,2,3. The machine and exciter data are given in Table 7. Loads are assumed as constant power type. The eigenvalues are shown in Table 8. The participation factors associated with the eigenvalues are given in Table 8.

Only the participation factors greater than 0. Also shown are the state variables and the machines associated with these state variables. From a practical point of view, this information is very useful. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access.

Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract This introduction presents an overview of the key concepts discussed in the subsequent chapters of this book. Related Information. Email or Customer ID. Forgot password? Old Password. New Password. Your password has been changed.

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