Indices to Detect Hopf Bifurcations in Power Systems. N. Mithulananthan Claudio A. Ca~nizares John Reeve. University ofwaterloo

Transcription

1 NAPS-, Waterloo, ON, October Indices to Detect Hopf Bifurcations in Power Systems N. Mithulananthan laudio A. a~nizares John Reeve University ofwaterloo Department of Electrical & omputer Engineering Waterloo, ON, anada NL G c.canizares@ece.uwaterloo.ca Abstract Simple performance indices with predictable shapes, for detecting and predicting stability problems such as voltage collapse or system oscillations, have been a concern of planning and operating engineers in power utilities. In this paper, two simple indices to detect oscillatory problems (Hopf Bifurcations) in power systems are proposed, based on the system state matrix as well as an augmented system matrix. Oscillatory problems associated with Hopf bifurcations for various contingencies and dierent static load models are studied using the proposed indices in three dierent test systems. Keywords: On-line performance indices, system oscillations, Hopf bifurcations, singular value decomposition. I. Introduction Nonlinear phenomena such as bifurcation and chaos may occur in power system models, as has been reported in several studies [,,, ]. Among the various types of bifurcations, saddle-node, limit induced, and Hopf bifurcations have been identied as some of the main reasons behind instability in power systems []. Saddle-node and limit-induced bifurcations basically consist of loss of system equilibria, which is typically correlated with the lack of power ow solutions []. In the case of saddle-node bifurcations, a singularity of a system Jacobian and/or state matrix results in the disappearance of such \static" solutions, whereas in the case of limitinduced bifurcations, the lack of steady state solutions are due to system controls reaching limits (e.g., generator reactive power limits). These bifurcations lead to a monotonic collapse of the system variables, and hence are typically associated with voltage collapse problems. Hopf bifurcations, on the other hand, produce \limit cycles" (periodic orbits), leading to oscillatory problems and possible instabilities. These types of bifurcations have been detected in a variety ofpower system models [,, ], as well as in actual power systems [9, ], and could arise due to variable net damping, lossy transmission lines, frequency dependence of the electrical torque, and voltage control issues (e.g., fast acting voltage regulators) []. The rst and last reasons are the most common in power systems the rst one is associated with the electromechanical modes of generators, whereas the last one is associated with exciter modes in general. Load models greatly inuence bifurcations, as these have a large eect on the system eigenvalues [, ]. All these bifurcations are usually triggered by contingencies in the system. In general, bifurcations occur on very stressed systems, i.e., heavily loaded systems operating close to the tip of the nose cure or PV curve. At these operating conditions, the region of attraction of the operating point becomes very small [], hence the system is unable to withstand perturbations. This is very relevant for present power systems, as many of today's networks operate near their stability limits due to economical and environmental reasons. Some recent system collapses (e.g., [9, ]) can be explained using bifurcation theory these collapses have resulted in major outages. With a way of predicting and controlling bifurcations, these incidents could have been minimized or avoided all together. Many indices to detect proximity to saddle-node and limits induced bifurcations have been proposed, as described in detailed in []. These indices are currently being used in actual operating environments to determine how close a system is to these types of bifurcations, so that preventive and/or corrective actions can be taken to avoid consequential system collapses. Great attention has been given to the issue of controlling the oscillatory problems associated with Hopf bifurcations [,,,, ]. However, very scant attention has been paid to predicting proximity to such bifurcations. Some work has been done in directly detecting these bifurcations. For example, in [], generic techniques to directly detect Hopf bifurcations are discussed however, the authors demonstrate the diculty of applying these techniques to practical power systems. In [], an optimization technique is used to predict the closest Hopf bifurcation in power systems. No particular indices have been proposed for detecting proximity to Hopf bifurcations. A simple test function that can be used to detect Hopf bifurcations is proposed in [9] for generic nonlinear systems it is simply based on the maximum of the real parts of the eigenvalues. The problem with applying such a test function to power systems is that the computation of the eigenvalues is extensive and, the system eigenvalues are highly nonlinear []. Thus, a smooth and predictable index for fast Hopf bifurcation detection at low computational eort is needed in power systems, so that system operators can determine, on-line, proximity to oscillatory problems to apply preventive and/or corrective actions. Hence, in this paper, two such indices are proposed, both based on singular values of \extended" system state matrices. Both of them show a smooth and predictable shape, and can be readily computed so that predictions can be made on-line. This paper is organized as follows: Section II explains the basic theory associated with nonlinear power system models used in this paper Hopf bifurcations are explained in detail in this section. The proposed indices are presented in Section III. In Section IV, the results obtained for various test systems are presented and discussed, together with a brief description of the analysis tools used to obtain these results. Finally, the major contributions made in this paper are highlighted in Section V some future research directions are also discussed in this last section.

2 II. Basic Background Imaginary A. Power System Modeling and Eigenvalues In general, power systems are modeled by the following set of dierential and algebraic equations (DAE): _x = f(x y p) () = g(x y p) where x < n is a vector of state variables associated with dynamic states of generators, loads, and other system controllers y < m is a vector of algebraic variables associated with steady-state variables by neglecting fast dynamics, e.g., most load voltage phasor magnitudes and angles, HVD links, etc. <` is a set of uncontrollable parameters, such as variations in active and reactive power of loads and p < k is a set of controllable parameters such as tap and Automatic Voltage Regulator (AVR) settings, SV reference voltages, etc. Bifurcation analysis is based on an eigenvalue analysis [9], also referred to as small signal stability or steady state stability analysis in power systems [], as some system parameters and/or p change in () []. Hence, linearization of these equations is needed at an equilibrium point(x o y o ) for given values of the parameters ( p), i.e., at an operating point such that [f(x o y o p) g(x o y o p)] T =. Thus, on linearizing () at (x o y o p) it follows that _x = J J J J {z } J x y where J is the system Jacobian, J J J is nonsingular, the system eigenvalues can be readily computed by eliminating the vector of algebraic variable y in (), i.e., _x =(J ; J J ; J )x = A x Thus, the DAE system can then be reduced to a set of ODE equations []. Hence, bifurcations on power system models are typically detected by monitoring the eigenvalues of matrix A as the system parameters ( p) change. Once the reduced system state matrix A is established, the following equation can be used to compute the eigenvalues of the system and thus analyze the steady state stability of the equilibrium point: () Av= v () where is the eigenvalue, and v is the eigenvector associated with. The main drawback of this method is the need for obtaining the inverse of J, and hence the associated loss of sparsity. However, the augmented system equations can be used directly, sothatecient sparsity-based algorithms may speed up the eigenvalue computation [] thus, the eigenvalue problem can be restated as J J J J v v = v where is the eigenvalue and [v v ] T are the augmented eigenvectors of. By eliminating v from (), () can be readily obtained. () λ λ Hopf Bifurcation Fig.. Locus of the critical eigenvalue on a Hopf bifurcation. Real B. Hopf Bifurcations Hopf bifurcations are also known as oscillatory bifurcations. Such bifurcations are characterized by periodic orbits emerging around an equilibrium point, and can be studied with the help of linearized analyses, with () having a pair of purely imaginary eigenvalues of the state matrix A [9]. onsider the dynamical power system modeled by (). When the parameters and/or p vary, the equilibrium points (x o y o )change, as do the eigenvalues of the corresponding system state matrix. These equilibrium points are asymptotically stable if all the eigenvalues of the system state matrix have negative real parts. As the parameters change, the eigenvalues associated with the corresponding equilibrium point change as well. The point where a complex conjugate pair of eigenvalues reach the imaginary axis with respect to the changes in ( p), say (x o y o o p o ), is known as a Hopf bifurcation point. This phenomenon is illustrated in Fig. using the locus of the \critical" eigenvalues in the complex plane, i.e., the \bifurcating" complex conjugate pair of eigenvalues. At a Hopf bifurcation point (x o y o o p o ), the following transversality conditions should be satised [9]:. [f(x o y o o p o ) g(x o y o o p o )] T =.. The Jacobian matrix evaluated at (x o y o o p o ) should only have a simple pair of purely imaginary eigenvalues = j.. The rate of change of the real part of the purely imaginary eigenvalues with respect to a varying system parameter, say i, should be nonzero at this point, i.e., drefg = d i o If this is the case, there is a birth of limit cycles at (x o y o o p o ) with an initial period of T o =

3 These conditions basically state that a Hopf bifurcation corresponds to a system equilibrium with a pair of purely imaginary eigenvalues with all other eigenvalues having non-zero real parts, and that the pair of bifurcating or critical eigenvalues cross the imaginary axis with nonzero speed. ~ BUS_ R + j X BUS_ III. Hopf Bifurcation Indices Since at a Hopf bifurcation point the system Jacobian has a simple pair of purely imaginary eigenvalues, the problem can be restated as follows: For the system state matrix A, a complex pair of eigenvalue for can be rewritten in the following form: A [v R jv I ]= j [v R jv I ] where and are the real and imaginary parts of the eigenvalue, respectively, and v R jv I are the associated eigenvectors. If real and imaginary parts are separated, it follows that ) A ; I n ;I n (A ; I n )v R + v I = (A ; I n )v I ; v R = I n A ; I n A +I n ;I ;I n A n {z } Am v R vi v R v I = = Since [v R v I ] T =, at a Hopf bifurcation where =, detf A m ; I n g = i.e., the modied matrix A m becomes singular at this point. Observe that this matrix is also singular at a saddle-node bifurcation, as = = in this case. Following the same criteria previously proposed to de- ne indices for saddle-node bifurcations [], the singular value of the modied state matrix is used here as an index for detecting Hopf bifurcations. Hence, the rst Hopf bifurcation index (HBI) is dened as follows: HBI = min (A m ) () where min is the minimum singular value of the modied state matrix A m,which becomes zero at a Hopf or saddlenode bifurcation point. Results of applying this index to three dierent test systems are presented in the next section. The HBI index proposed here has the problem that it requires the state matrix A, which is computationally expensive, as previously discussed. This problem can be avoided if the full matrix is used as dened in (). In this case, for a complex pair of eigenvalues: J J J J vr jv I v R = j jv I vr jv I By separating the real and imaginary parts and rearranging these equations: J ; I J I J J ;I J ; I J J J v R v R v I v I = Fig.. Two-bus test system. Since at a Hopf bifurcation =, the matrix J m = J J I J J ;I J J J J becomes singular notice that this also holds at a saddlenode bifurcation point. Therefore, the minimum singular value of the modied full Jacobian matrix J m can be used as another index to indicate proximity to a Hopf or a saddle-node bifurcation. Thus, the following HBI is proposed here: HBI = min (J m ) () This index is computationally less involved than HBI in (), and full advantage can be taken of the sparsity ofj m. A. Test Systems IV. Results In order to illustrate the use of the proposed indices, three dierent test systems were used:. The -bus test system of Fig. represents a simple generator and load system with shunt capacitance at the load side. The shunt capacitor was varid to force the appearance of a Hopf bifurcation. Typical dynamic data for the generator model as well as its exciter and AVR were extracted from [] the load was represented as a constant PQ load.. A one line diagram of the IEEE bus test system used here is given in Fig.. It consists of generators with exciters and AVRs, transformers, buses and lines. Static data for this system can be found in []. The data for the generator dynamic model, including exciters and AVRs, was chosen based on typical data given in [].. The one-line diagram of a two-area system proposed in [] for oscillation studies is depicted in Fig.. It consists of generators, transformers, buses, including two load buses with static capacitors, and lines. Static and generator dynamic data for this system were taken from []. Simple exciter and AVR models were used for each machine, except G, and the loads were modeled as PQ loads. All these test systems were chosen and/or set up so that Hopf bifurcations, i.e., oscillatory problems, could appear under certain system conditions, as discussed below.

4 G GENERATORS SYNHRONOUS ONDENSERS G 9 Saddle node Bifurcation G THREE WINDING TRANSFORMER EQUIVALENT Fig.. PV curves for the -bus test system. Fig.. IEEE -bus test system. G ~ 9 ~ ~ G L Area Area Fig.. Two area system from [] B. Power System Analysis Tools Steady state equilibrium points at each loading level were calculated with the help of UWPFLOW [], and the corresponding eigenvalues for each test system were calculated using the Multi-Area Small Signal Stability (MASS) program [], and the Power System Toolbox (PST) []. In addition to this, a MATLAB function was written to construct the matrices A m and J m, and to compute the corresponding singular values to obtain the desired indices HBI and HBI as dened in () and (), respectively. UWPFLOW is a research tool that has been designed to calculate local fold bifurcations such as saddle-node and limit-induced bifurcations in power systems. This program was developed based on continuation and direct computational methods. The program generates a variety of outputs that allow for further analyses, such as tangent vectors, left and right eigenvectors associated with the smallest real eigenvalue of the static Jacobian, power ow solutions at dierent loading levels, a variety of voltage stability indices, bus voltage proles, etc. The program has detailed static models of various power system elements such as generators, loads, HVD links, and various FATS, including their \true" control limits. MASS is part of EPRI's Small Signal Stability Program (SSSP), from the Power System Analysis Package (PSAPA). It can be used to analyze the steady state 9 L9 G G ~ stability ofapower system by calculating all its eigenvalues. It also yields several outputs such as the system state matrix A, the participation matrix, left and right eigenvectors, etc. However, itdoesnotprovide information regarding the full DAE Jacobian matrix J for this, the PST package was used. PSTisaMATLAB based power system analysis toolbox initially developed to perform power system analyses based on user dened models. It has several graphical tools, namely, avoltage stability, a transient stability and a small signal stability tool. The small signal stability program has two versions the rst one can be used to calculate the eigenvalues of the system numerically, whereas the second one calculates the eigenvalues analytically by forming all Jacobians for the chosen DAE model.. Simulation Results. Two-bus System The HBI index was rst calculated for the -bus test system. A Hopf bifurcation was created at a loading factor =: p.u. by varying the shunt capacitor compensation level and the voltage regulator gains. The modied matrix A m was then constructed using MATLAB, based on the state matrix A produced by MASS. The minimum singular value of A m was calculated at dierent loading levels by varying the active and reactive loadpowers at the same initial ratio until a Hopf bifurcation point in the system was reached. All load power changes were picked up by the only generator in the system. Figure shows the voltage prole for the test system, as produced by UWPFLOW. The corresponding HBI index is depicted in Fig. observe the almost linear relationship with respect to the loading factor. In these gures, = corresponds to a base load of about MW. The load was modeled as constant PQ.. -bus IEEE System Based on the dynamic data used, the system presents a Hopf bifurcation at =: p.u., as shown on the PV curve andhbi index plot in Figs. and, respectively. The loads in the system were modeled as constant PQ loads, and any changes in the load power were picked up by the swing bus, i.e., one of the generators in the system (BUS-) the loads were increased from initial active and reactive power values. In this case,

5 Normalized HBI_... Normalized HBI_ Fig.. HBI index for a the -bus test system Fig.. HBI index for the IEEE -bus test system. Base case. Saddle node bifurcation point is shown in gures as limits become active, the slope of the index signicantly changes.. Two-area System The two-area system was used to illustrate the use of the HBI index, and compare its performance with respect to the HBI index. Thus, Figs. and show the static loading margin of the system and corresponding Hopf bifurcation indices HBI and HBI. As can be seen in Fig., both indices present an almost linear behavior with respect to changes in the loading parameter. In this case, = corresponds to a loading level of about, MW both system loads were modeled as constant PQ loads and then increased based on their initial values. V. onclusions Fig.. Voltage prole at the weakest bus for the IEEE -bus test system. Base case. = p.u. corresponds to a total initial load of about 9 MW. The proposed HBI index was calculated for various system contingencies. Figures 9 and illustrate the PV curves and corresponding HBI plots for the dierent contingencies, respectively. It can be seen that Hopf bifurcations occur at lower loading levels depending on the severity of the line contingencies in the system. From these plots, it is clear that the proposed index presents an almost linear relationship with respect to the loading factor. Figure shows the HBI index for dierent static load models in the system. Three dierent static load models were considered, namely, constant power (P), constant current (I), and constant impedance (Z). Observe thatdif- ferent static load models yield dierent dynamic stability margins, i.e., Hopf bifurcation points. The onstant PQ load model has the smallest margin, followed by the constant impedance and then the constant current model, would be expected, since PQ loads lack sensitivity tobus voltage changes. The eect of system limits on the index This paper proposes two indices to detect proximity to Hopf bifurcations in dynamic power system models, with one of them (HBI ) presenting signicant computational advantages that make it suitable for on-line applications. The proposed indices can also be used to detect proximity to saddle-node bifurcations. The indices show a smooth and predictable (linear) behavior with respect to loading changes, making them very adequate for predicting proximity to oscillatory problems in power systems. However, the eect of limits on the proposed indices needs to be studied further, as these have a signicant eect on the shape of the index possible improvements to the proposed indices should also be considered at this point. Furthermore, studies on larger system models have to be performed to better understand the behavior of the index as well as related computational issues in practical applications. References [] H. G. Kwatny, A. K. Pasrija and L. Y. Bahar, \Static bifurcation in electric power networks: Loss of steady-state stability and voltage collapse," IEEE Trans. Power Systems, Vol., pp. 9{99, 9. [] H-D. hiang, I. Dobson, R. J. Thomas, J. S. Thorp and L. Fekih-Ahmed, \On voltage collapse in electric power system," IEEE Trans. Power Systems, Vol., pp. {, 99. [] H-D. hiang, -W. Liu, P. P. Varaiya, Felix F. Wu and M. G.

6 s Base ase Line ( ) Outage Line ( ) Outage Saddle node bifurcation Limit induced bifurcation Fig. 9. Voltage proles at the weakest bus for various contingencies for the IEEE -bus test system. 9 x Base ase Line ( ) Outage Line ( ) outage Fig.. Voltage proles at bus for the two-area system. HBI_ s HBI_ HBI_ Fig.. HBI index for various contingencies for the IEEE -bus test system onstant PQ onstant I onstant Z Hopf Bifurcation Index.... HBI_.. s Fig.. HBI and HBI indices for the two-area system Fig.. HBI index for dierent load models for the IEEE -bus test system.

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Kundur, Power System Stability and ontrol, McGraw Hill, New York, 99. [] D. J. Hill and I. M. Y. Mareels, \Stability Theory for Dierential/Algebraic Systems with Application to Power Systems," IEEE Trans. ircuits and Systems, Vol., No., pp. {, Nov. 99. [] T. Yong and R. Lasseter, \Optimal power ow formulation in market of retail wheeling," Power Engineering Society Winter Meeting, IEEE, vol., pp. 9-9, 999. [] P. M. Anderson and A. A. Fouad, Power System ontrol and Stability, IEEE Press, 99. []. A. a~nizares, et. al, \PFLOW: ontinuation and Direct Methods to Locate Fold Bifurcations in A/D/FATS Power Systems," University of Waterloo, August 99. Available at [] \Small Signal Stability Analysis Program Ver..: User's Manual," EPRI, TR--VR, May 99. [] \Power System Toolbox Ver..: Dynamic Tutorial and Functions," herry Tree Scientic Software, olborne, Ontario, 999. Nadarajah Mithulananthan was born in Sri Lanka. He received his B.Sc. (Eng.) and M.Eng. degrees from the University ofper- adeniya, Sri Lanka, and the Asian Institute of Technology, Thailand, in May 99 and August 99, respectively. Mr. Mithulananthan has worked as an Electrical Engineer at the Generation Planning Branch of the eylon Electricity Board, and as a Researcher at hulalongkorn University, Thailand. He is currently a full time Ph.D. student attheuniversity ofwaterloo working on applications and control design of FATS controllers. laudio A. a~nizares received in April 9 the Electrical Engineer diploma from the Escuela Politecnica Nacional (EPN), Quito- Ecuador, where he held dierent teaching and administrative positions from 9 to 99. His M.Sc. (9) and Ph.D. (99) degrees in Electrical Engineering are from the University of Wisconsin{ Madison. Dr. a~nizares is currently an Associate Professor at the University of Waterloo, E&E Department, and his research activities are mostly concentrated in studying stability, modeling and computational issues in ac/dc/fats systems. John Reeve received the B.Sc., M.Sc., Ph.D. and D.Sc. degrees from the University of Manchester (UMIST). After employment in the development of protective relays for English Electric, Staord, between 9 and 9, he was a lecturer at UMIST until joining the University ofwaterloo in 9, where he is currently an Adjunct Professor in the Department of Electrical & omputer Engineering. He was a project manager at EPRI, 9-, and was with IREQ, His researchinterests since 9 have been HVD transmission and high power electronics. He is the President of John Reeve onsultants Limited. Dr. Reeve was chair of the IEEE D Transmission Subcommittee for years, and is a member of several IEEE and IGRE ommittees on dc transmission and FATS. He was awarded the IEEE Uno Lamm High Voltage Direct urrent Award in 99.