Ijesp Hopf Bifurcations in the IEEE Second Benchmark  Model for SSR studies
7 Pages
English
Downloading requires you to have access to the YouScribe library
Learn all about the services we offer

Ijesp Hopf Bifurcations in the IEEE Second Benchmark Model for SSR studies

-

Downloading requires you to have access to the YouScribe library
Learn all about the services we offer
7 Pages
English

Description

;:áàÙ HOPF BIFURCATIONS IN THE IEEE SECOND BENCHMARK MODEL FOR SSR STUDIES Yasar Kucukefe Adnan Kaypmaz ITU ITU Istanbul,Turkey Istanbul,urkey yasar.kucukefe@ieee.org kaypmaz@elk.itu.edu.tr Abstract – In this paper, the first system of the IEEE Hopf bifurcation is defined as the birth of a limit second benchmark model (SBM) for subsynchronous cycle from equilibrium in a dynamical system governed resonance (SSR) studies is analyzed using the bifurcation by autonomous ODEs (Ordinary Differential Equations) theory. A nonlinear model for the SBM which consists of a under variation of one or more parameters on which the synchronous generator connected to an infinite busbar dynamic system is dependent. Supercritical Hopf bifur-through two parallel transmission lines, one of which is cations result in birth of stable limit cycles. On the con-equipped with a series capacitor, has been developed in trary, unstable limit cycles are born in the case of sub-MATLAB. Generator damper windings are also included critical Hopf bifurcation. Floquet theory is widely used in the model. The existence of Hopf bifurcations in the model is verified. The stability of limit cycles (i.e. subcriti- to study the stability of limit cycles. The procedure cal or supercritical Hopf bifurcation) is investigated by involves the calculation of steady-state solutions, Hopf computing the first Lyapunov coefficient. The impact of bifurcation points, the ...

Subjects

Informations

Published by
Reads 28
Language English

Exrait

HOPF BIFURCATIONS IN THE IEEE SECOND BENCHMARK MODEL FOR SSR STUDIES Yasar KucukefeAdnan Kaypmaz ITU ITU Istanbul, TurkeyIstanbul, Turkey yasar.kucukefe@ieee.org kaypmaz@elk.itu.edu.tr Abstract – In this paper, the first system of the IEEEHopf bifurcation is defined as the birth of a limit second benchmark model (SBM) for subsynchronous cycle from equilibrium in a dynamical system governed resonance (SSR) studies is analyzed using the bifurcation by autonomous ODEs (Ordinary Differential Equations) theory. A nonlinear model for the SBM which consists of a under variation of one or more parameters on which the synchronous generator connected to an infinite busbar dynamic system is dependent. Supercritical Hopf bifur-through two parallel transmission lines, one of which is cations result in birth of stable limit cycles. On the con-equipped with a series capacitor, has been developed in trary, unstable limit cycles are born in the case of sub-MATLAB. Generator damper windings are also included critical Hopf bifurcation. Floquet theory is widely used in the model. The existence of Hopf bifurcations in the model is verified. The stability of limit cycles (i.e. subcriti-to study the stability of limit cycles. The procedure cal or supercritical Hopf bifurcation) is investigated byinvolves the calculation of steady-state solutions, Hopf computing the first Lyapunov coefficient. The impact of bifurcation points, the branches of periodic orbits which the field voltage, the mechanical torque input to the gene-emanate from the Hopf bifurcation points and the com-rator and the infinite-bus voltage on the first Lyapunov putation of themonodromy matrixThe [7].Floquet coefficient is also explored. In the model with Automatic multipliers arethe eigenvalues of the monodromy ma-Voltage Regulator (AVR) and generic Power System Stabi-trix. Then by tracing the evolution of the Floquet mul-lizer (PSS), the first Lyapunov coefficient becomes nega-tipliers, one can observe the stability of these solutions. tive but remains near zero indicating the existence of a One of the multipliers is always unity for an autonom-generalized Hopf bifurcation. Time domain simulations verify the analytical findings.ous system. If all the other multipliers are inside the unit circle on the complex plane, then the limit cycle is orbi-Keywords: Hopf bifurcations, subsynchronous re-tally stable. sonance (SSR), first Lyapunov coefficient, torsionalIn this paper, we employ an alternative method to in-oscillation, eigenvalues, nonlinear systemsvestigate the stability of limit cycles. The method in-volves the computation of the first Lyapunov coefficient (l1(0)). Positive / negative sign ofl1(0) corresponds to subcritical / supercritical Hopf bifurcation. Ifl1(0) va-1INTRODUCTION nishes then the generalized Hopf bifurcation occurs. Series capacitor compensation of AC transmission This paper is organized as follows. Section 2 gives a systems is an effective way of increasing load carrying brief review of the bifurcation theory. The SMIB power capacity and enhancing transient stability. However, system under study is described and its mathematical capacitors in series with transmission lines can cause model is obtained in Section 3. Hopf bifurcation analy-subsynchronous resonance (SSR) that can lead to tur-sis of the system is presented in Section 4. Then in Sec-bine-generator shaft failures as occurred at the Mohave tion 5, the bifurcation analysis is carried out by includ-Generating Station in Southern Nevada in the USA in ing the AVR and PSS in the model. The results show 1971 [1]. Since then, considerable effort has been de-that the generalized Hopf bifurcation occurs in the mod-voted to the understanding and analysis of the SSR el. Time domain simulations using MATLAB-Simulink phenomenon by researchers and utility professionals. are carried out to verify the analytical findings. IEEE SSR Working Group has constructed three benchmark models for computer simulation of the SSR 2REVIEW OF BIFURCATION THEORY [2,3]. Analytical tools for studying the SSR involve frequency scanning, eigenvalue technique, time domain 2.1Nonlinear Systems and Parameter Dependence simulation programs [4] and the complex torque coeffi-By definition, a nonlinear system is a system which cient method [5]. does not satisfy the superposition principle. The most A single-machine-infinite-busbar (SMIB) power sys-common way to define a continuous-time nonlinear tem with series capacitor compensation is inherently dynamical system is to represent the system in the form nonlinear and can be modeled by sets of ordinary diffe-of autonomous ordinary differential equations (ODEs). rential equations. The nonlinear model including the Consider a continuous-time nonlinear system depending dynamics of the turbine-generator shaft system can be on a parameter vector. analyzed using the bifurcation theory. Zhu et al. [6]௡ ௠ ݔԹא,ݔߙ,ݔ݂,אԹߙ (1) demonstrated the existence of Hopf bifurcation in a where݂smooth with respect to isݔ andߙ. If varying SMIB experiencing SSR. the parameter vectorߙ resultsin qualitative changes in
the system dynamic behavior in a way that different behaviors (aperiodic, periodic, chaotic, etc.) and stabili-ty conditions are introduced, these changes are called bifurcationsthe parameter vector values at which and the changes occur are calledbifurcation (critical) val-ues.
2.2Stability of Equilibrium Solutions Suppose that dynamical system (1) has an equili-଴ ଴brium atݔ (i.e.ൌ0ߙ ሻ݂ሺݔ ,), and denote byA the Jacobian matrix of݂ሺݔሻ evaluatedat the equilibrium, ଴ ଴ A=݂. If ݔ ,ߙall the eigenvaluesλ1,λ2,…,λn ofA଴ ଴ satisfy Re(λi)<0 for i=1,2,..,n, then the systemߙ ሻ݂ሺݔ ,is asymptotically stable.
2.3Bifurcation Mechanisms There are different types of bifurcations. The most important ones are fold bifurcation, pitchfork bifurca-tion, transcritical bifurcation and Hopf bifurcation [8]. Fold bifurcations are associated with dynamic systems which have Jacobian matrix with a single zero eigenva-lue while all the other eigenvalues remain on the left half plane. This type of bifurcation has also other names such as saddle-node bifurcation and turning point. Tran-scritical bifurcation is characterized by the intersection of two bifurcation curves. Pitchfork bifurcations often occur in systems with some symmetry, as a manifesta-tion of symmetry braking. The bifurcation correspond-ing to the presence of distinct pair of purely imaginary eigenvaluesλ1,2 =±i߱0,߱0>0, of the Jacobian matrix ଴ ଴ ݂ ,ߙ ሺݔ ሻis called a Hopf (or Andronov-Hopf) bifurca-tion [9].
2.4Supercritical and Subcritical Hopf Bifurcation Limit cycles are periodic orbits that represent regular motions in a dynamical system. Hopf bifurcations gen-erate limit cycles from equilibrium. Supercritical Hopf bifurcation results in a stable limit cycle. On the other hand, unstable limit cycle is born in case of subcritical Hopf bifurcation. In both cases, loss of equilibrium condition occurs. Whether a Hopf bifurcation is supercritical or sub-critical can be determined by the sign of the first Lya-punov coefficient (l1(0)) of the dynamical system near the equilibrium. This coefficient can be computed as follows [10]: ଴ ଴ Supposeߙ ሻሺݔ ,is an equilibrium point of (1) where the Jacobian matrixAa distinct pair of complex has eigenvalues on the imaginary axis,λ1,2= ±i߱0,߱0>0, and these eigenvalues are the only eigenvalues ofAwith zero real parts. Letݍ א ܥ bea complex eigenvector corresponding toλ1: ݅߱ݍ,ܣݍ݅߱ݍ ܣݍ ൌ଴ ଴ (2) Introduce also the adjoint eigenvector݌ א ܥ having the properties: ் ் ݅߱݌,ܣ݌ҧ݅߱݌ҧ ܣ ݌଴ ଴ (3) and satisfying the normalization ۃ݌, ݍۄ1 (4) ۃ݌, ݍۄൌ∑ ݌ҧݍ where௜ ௜ isthe standard scalar product ௜ୀଵ inܥ.
Then the following invariant expression gives the first Lyapunov coefficient,l1(0): ିଵ  Re[ݍݍ,݌ۃܥ,ݍ,ݍܤۄ2ۃۄ,ݍܣݍ,ܤ݌,ଶఠ ିଵ  +ܫ ‐ܣሻሻሻۄۃ݌, ܤሺݍത, ሺ2] (5) ݅߱଴ ௡ܤሺݍ, ݍ whereܤ andܥsymmetric aremultilinear vectorfunc-tions ofݔ, ݕ, ݖ ߳ ܴ. With a shift of coordinates in (1), 0 ߦ=x-x: ଶ ଴ ߲ ݂ሺߦ, ߙܤ ݔݕ ሺݔ, ݕሻൌ෍ |కୀ଴ ௝ ሺ6ሻ߲ߦ ߲ߦ ௝ ௞ ௝,௞ୀଵ ଷ ଴ ߲ ݂ሺߦ, ߙܥ ሺݔ, ݕ, ݖሻ ൌ ෍| ݔݕ ݖሺ7ሻ௜ కୀ଴௝ ௞ ௟ ߲ߦ ߲ߦ ߲ߦ ௝ ௞ ௟ ௝,௞,௟ୀଵ fori=1,2,…,n. The software routines we have developed in MATLAB for the analytic calculation ofl1(0) in the model make extensive use of the algorithms available in the continuation software MatCont [11]. Positive / nega-tive sign ofl1(0) corresponds to the occurrence of sub-critical / supercritical Hopf bifurcations. Ifl1(0) vanish-es with nonzero second Lyapunov coefficient (l2(0)), then the generalized Hopf bifurcation occurs [12].
3SMIB POWER SYSTEM MODEL 3.1Model Description The first system of the IEEE second benchmark model for SSR studies is analyzed. Fig.1 shows the configuration of the SMIB power system. Series capaci-tor compensation is applied in one of the parallel trans-mission lines.
Figure 1:SMIB power system (IEEE second benchmark model for SSR studies – System-1)
The single shaft turbine generator mechanical system consists of a high-pressure (HP) turbine, a low-pressure turbine (LP), a generator and an exciter (Exc.), as shown in Fig. 2. The coefficients K and D represent stiffness and damping. Rotational inertia is represented by M. The numerical parameters are given in the Appendix.
Figure 2:Schematic diagram of the mechanical system con-sisting of a high-pressure (HP) turbine, a low-pressure turbine (LP), a generator and an exciter (Exc.)
3.2Mathematical Model of the System We include the dynamics of the generator damper windings on the q- and d-axes but neglect the effect of saturation and dynamics of the turbine governor. Using
direct and quadrature (d-q) axes and Park’s transforma-tion, the complete mathematical model describing the dynamics of the electrical and mechanical systems can be written as follows [13]: (a) Electrical System Define the state variables: ் ହ ࢏ ൌሾ݅݅ ݅ ݅݅ ሿ࢏ אԹ ௤ ௙ ௞௤ ௞ௗࢍ ௗ,் ଶ ࢂ ൌሾܸܸ ሿ ࢉ ௖ௗ௖௤,א ԹThe equations in state space form: ݀࢏ -1 ߱ ሺ=BC+D) (8) ݀ݐ ݀ࢂ ߱ ࢂ =E+F) (9) ݀ݐ where: ‐ሺXX0 ܺሻ 0 ܺ d Eafd akd 0 ‐ሺܺ൅ܺ ሻ0 ܺ0 q Eୟ୩୯ 0 ܺBൌ ‐ܺ0 ܺ (10) afd ffdkd 0 ‐ܺ0 ܺ0 akq kkq ۏ ‐ܺ0 ܺ 0ܺ ے akd kdkkd r‐ሺ൅ܴ ሻX൅߱ ܺ ሻܺ 00 ߱ a EE ௥q ௥akq X൅߱ ܺ ሻr0 ‐߱‐߱ ܺܺ൅ܴ ሻ E ௥d aE ௥afd ௥akd Cൌ 00 ‐r0 0fd 0 00 ‐r0 kq ۏ 00 00 ‐rے kd  (11) ܸsinߜܸ ଴ ௥௖ௗ ܸcosߜܸ ଴ ௥௖௤ Dൌrܧ /ܺ (12) fd ௙ௗafd 0 ۏ ے 0 ൌ ൅kXX XEXbt L1 (13) RRkRR (14) E tL1 b ߤ=ܺ /ܺ (15) ௖ ௅ଵ ଶ ଶ ඥܴ ൅ܺ 2 L2 ݇ ൌ(16) ଶ ଶ ඥሺܴ ൅ܴ ሻ൅ ሺܺ൅ ܺെ ߤܺ1 2L1 L2L1 0 0 μkܺ10 0 L Eൌ൤ ൨ (17) ܺ 0μkL10 0 0 0 1 Fൌ൤ ൨ (18)‐1 0 (c) Mechanical System ் ଼ ࡾ ሾ߱ߠ ߱Define=ଵ ଵߠ ߱ ߜ ߱ ߠ,א Թ.In ଶ ௥௥ ସstate space form: ݀ࡾ =G+H(19) ݀ݐ where
-D -KK 1 1212 0 00 00 M MM 1 11 ω0 0 0 00 00 b K -D-(K +KK 12 212 2323 0 00 0 M MM M 2 22 2 0 0ω0 0 00 0 b G= K -D-(K +KK 23 323 3434 0 0 00 M M MM 3 3 33 0 0 00ω00 0 b K -D-K 34 434 0 0 00 0 M MM 4 44 ۏ0 0 00 0 0ω0ے b (20) D D ሺܶ‐ܶ ൅Dሻ D 1 2௠ ௘3 4 Hൌ൤ ‐ω‐ω ‐ω‐ω ൨b bb b M MM M 1 23 4 (21) In (21),ܶ representsthe electromechanical torque and it is expressed as follows: ሻ ݅݅ ܺ݅ ݅ܺ ݅ ݅ ܶ=(ܺ-ܺafd݅d+ܺ ݅-akq௞௤ ௗ+akd௞ௗ ௤ (22) q Finally, in order to describe the complete model in state space representation form (1), we define the state ் ் ்் ଵହ ࢞ ൌሾ࢏ࢂ ࡾ vectorࢍ ࢉሿ ,࢞ א Թand combine state equations (8), (9) and (19) as: ‐1 ሺ ࢏൅ ሻ B ߱CD ࢞ሶ ൌ቎߱ ሺE࢏ ൅Fࢂ ሻ (23) ௕ ࢍGࡾ ൅H The control parameter vector consists of the series ߤ ܺcompensation factor (=௖ ௅ଵ), the mechanical torque ܶ ሻ,( thefield voltage (ܧ௙ௗthe network voltage and level (ܸ.
4ANALYSIS OF HOPF BIFURCATIONS IN THE MODEL 4.1Bifurcation Analysis ߤ ܺ/ We use the series compensation factor (=ܺ௅ଵ) as the bifurcation parameter and carry out bifurcation analysis. The other control parameters are kept constant ܧ ܸ (ܶ=0.91,௙ௗ=2.2 and=1.0). Fig. 3 shows the oscillatory modes of the model. Su-persynchronous and subsynchronous electrical modes have frequencies dependent on the series compensation factor. There are three torsional modes with frequencies of 24.7, 32.4 and 51.1 Hz.The local swing mode has the frequency of 1.53 Hz. In local swing mode, the turbine-generator shaft sections oscillate as a rigid body. In case the torsional modes are excited, on the other hand, some of the shaft masses oscillate against the others causing loss of fatigue life and eventually the shaft damage [14]. Asߤthe subsynchronous electrical mode increases, frequency decreases and interacts with all three torsion-al modes. The interaction results in movement of the real part of the corresponding eigenvalues towards to the zero-axis, as shown in Fig. 4.
Figure 3:Oscillatory modes of the model
Figure 4:Real parts of the torsional mode eigenvalues
The interaction with the third torsional mode occurs atߤൌ0.07, without causing instability. The real part of the second torsional mode eigenvalue crosses the zero-axis atߤൌ0.5184, as a result of interaction with the subsynchronous electrical mode, and the system stabili-ty is lost through a Hopf bifurcation. Though the second torsional mode regains stability atߤൌ0.8110, the overall system stability is not regained because of the Hopf bifurcation occurring atߤൌ0.7283 in the first torsional mode which interacts with the subsynchronous electrical mode. Using (5), the first Lyapunov coefficient for the Hopf bifurcation occuring atߤ=0.5184 is computed as -5 1.44x10 .From the positive sign ofl1(0), we conclude that the type of Hopf bifurcation is subcritical. Time domain simulations using MATLAB-Simulink are carried out to verify the analysis results. Fig. 5 shows the generator rotor speed response to a distur-bance of 0.46 p.u. negative pulse torque on the genera-tor shaft at t=5s for a duration of 0.5s at the Hopf bifur-ߤ cation point (=0.5184). Following the disturbance, the generator rotor speed oscillates at decaying magni-tudes until the born of limit cycles of small magnitude. The simulation is repeated forߤ=0.55. Fig. 6 shows that the limit cycles do not reach to a stable orbit. It should be noted that in a real system the generator would lose synchronism following the disturbance.
߱ Figure 5:Generator rotor speed () response to the disturbance at the Hopf bifurcation point (ߤ=0.5184)
Figure 6:Generator rotor speed (߱) response to the disturbance atߤ=0.55 (ߤ=0.5184) 4.2Parameter Dependency of the Bifurcation Point and the First Lyapunov Coefficient ߤ In addition to, the other control parametersܶ,ܸܧ and௙ௗ canaffect the dynamic response of the system under study. In this section, we analyze the impact of these parameters on the Hopf bifurcation points and the first Lyapunov coefficient, thereby on the stability of limit cycles arising from the Hopf bifurcation points. Table 1 shows the Hopf bifurcation points and the first Lyapunov coefficients for the varying values ofܶfrom 0.50 to 1.0 p.u. The other control parametersܧ௙ௗܸ andare kept constant. It is evident from Table 1 that ߤandl1(0) increase asTmis raised. Torsional Mode-1Torsional Mode-2 μl1(0)μl1(0) -5 -5 0.50 0.6760-0.69x10 0.49590.97x10 -5 -5 0.60 0.6938-0.21x10 0.50261.08x10 -5 -5 0.70 0.70750.65x10 0.50831.19x10 -5 -5 0.80 0.71841.91x10 0.51331.30x10 -5 -5 0.90 0.72743.71x10 0.51801.42x10 -5 -5 1.00 0.73567.02x10 0.52261.65x10 Table 1:Hopf bifurcation points and the first Lyapunov ܶ ܧ coefficients at differentvalues (௙ௗ=2.2,ܸ=1.0)
In Table 2, the Hopf bifurcation points and the first Lyapunov coefficients are given for the values of the ܧ ሻܶ ܸ field voltage (௙ௗ2.2 to 2.8 p.u. from and are ߤ kept constant. The results show thatl1(0) andof the ܧ torsional mode-1 decrease as௙ௗis raised. Torsional Mode-1Torsional Mode-2 ࢌࢊμl1(0)μl1(0) -5 -5 2.0 0.73247.17x10 0.51961.63x10 -5 -5 2.2 0.72833.95x10 0.51841.44x10 -5 -5 2.4 0.72522.14x10 0.51811.38x10 -5 -5 2.6 0.72260.72x10 0.51841.36x10 -5 -5 2.8 0.7203-0.55x10 0.51901.34x10 Table 2:Hopf bifurcation points and the first Lyapunov ܸ coefficients at different field voltages (ܶ=0.91,=1.0)
The bifurcation analysis results at various levels of the network voltage are shown in Table 3 for constant ܶ ܸ 0) decrease as the values ofand. Bothߤ andl1( network voltage raises to 1.02 from 0.98 p.u. Torsional Mode-1Torsional Mode-2 μl1(0)μl1(0) -5 -5 0.98 0.73124.49x10 0.52031.47x10 -5 -5 0.99 0.72974.21x10 0.51941.45x10 -5 -5 1.00 0.72833.95x10 0.51841.44x10 -5 -5 1.01 0.72693.71x10 0.51741.42x10 -5 -5 1.02 0.72543.48x10 0.51651.41x10 Table 3:Hopf bifurcation points and the first Lyapunov ܧ ܶ coefficients for various network voltages (௙ௗ=2.2,=0.91)
It is interesting to see that the impact of the control ܧ ܸ parametersTm,௙ௗandon the first Lyapunov coeffi-cient is evident. Though not very significant, the changes in the computed first Lyapunov coefficients as a result of an increase or decrease in one of the control parameters exhibit a regular pattern. Hence, the accura-cy of the computed first Lyapunov coefficient is found to be adequate for determining the type of Hopf bifurca-tion. The analysis results show that the first Lyapunov coefficients remain positive and/or near zero. Therefore, the Hopf bifurcations in the model are found not super-critical. The emphasis is given to identifying any occur-rence of supercritical Hopf bifurcation condition from which stable limit cycle emanates.
5THE MODEL WITH AVR AND PSS Automatic Voltage Regulator (AVR) of type DC1A and power system stabilizer (PSS) described in [15] are added to the model with minor modifications. The exci-ter saturation effect is neglected and the limiters are omitted. Fig. 7 shows the block diagram of the excita-tion system with AVR and PSS. The numerical values for the AVR and the PSS parameters are provided in the Appendix.
Damping ൅ ݏܭ ி ܸݎ݂݁Σ1 ൅ ݏܶ ி ܸܨ െ ܭ 1ܣ ܸܧ ݐΣ݂݀ 1 ൅ ݏܶ 1 ൅ ݏܶܣ ܴܸ ܥ Regulator Transducer ܸ ܺ 1 ൅ ݏܶ ݏܶ1 ܹ ߱ ܭ ܵݎ 1 ൅ ݏܶ 1 ൅ ݏܶ2 ܹ PSS Gain Phase Washout Compensation Figure 7:the excitation system with AVRBlock diagram of and PSS We include the AVR and PSS dynamics into the ma-thematical model (23) as follows. First define the state variables vector for the excitation system as ் ૞ ࢂ ൌሾܸܺ ܸܸ ܧ,Թࢂ א.The state equa-ࢋ࢞ࢉ ஼ௐ ௦ி ௙ௗࢋ࢞ࢉ tion describing the dynamics of the excitation system with AVR and PSS can be written as follows: ݀ࢂ ࢋ࢞ࢉ =Pࢋ࢞ࢉ+Q (24)݀ݐ where ‐1 0 00 0 ܶ ‐1 0 00 0 ܶ ܭ ܶ‐1 ௌ ଵ Pൌ 0ሺ1‐ ሻ0 0(25) ܶ ܶ ܶ ଶ ௐ‐1 0 00 0 ܶ ி ‐ܭ ‐ܭ‐1 ஺ ஺ 0 0 ۏ ܶܶ ܶے ஺ ஺ܸ ܭܶ ܭܭ ௧ ௌଵ ிሺ ߱ሻ ሺܧ ሻሺ ܸQ=߱௥ ௥௙ௗ ௥௘௙(26) ܶ ܶ ܶ ܶ ோ ଶி ஺ ܸ(26) is the generator terminal voltage. Neglecting in the transients, it can be expressed as: ଶ ଶ ܸݎ݅ܺ݅ݎ௔ ௤௔௙ௗ ௙ௗௗ ௗ(27) ݅ ܺ݅ ൅ܺ ݅ ௧ ௔ௗ ௤Combining (24) with (23), the complete mathemati-cal model for the SMIB power system with AVR and PSS is obtained as ‐1 B ߱C࢏ ൅D௕ ࢍ ߱ ሺE࢏ ൅Fࢂ ሻ ௕ ࢍ࢞ሶ ൌ (28) GH ۏPࢋ࢞ࢉ+Qے The final model has 20 state variables and these are ݅ ݅݅ ܸܸ ߱ߠ ߱ ߠ ߱ ߜ ߱ ݅,݅,,௞௤,௞ௗ,௖ௗ,௖௤,,,,,,,, ߠ ܸܺ ܸܧ ,ܸ,,,ிand௙ௗ. The control parameter vector
K K consists of the AVR gain (Ā), the PSS gain (), the reference voltage (V୰ୣ୤), the series compensation factor µ ܶ ( ),the mechanical torque input (the network and ܸ ሻ voltage level (.
5.1Bifurcation Analysis ܸ ܸ Withܶ=0.91,௥௘௙=1.0982 and=1.0, the same operating conditions as in section 4.1, the bifurcation analysis is carried out. Fig. 8 and Fig. 9 show the oscil-latory modes and the real parts of the torsional mode eigenvalues of the model with AVR and PSS, respec-tively.
Figure 8:Oscillatory modes of the model with AVR and PSS
Figure 9:Real parts of the torsional mode eigenvalues of the model with AVR and PSS
Two more oscillatory modes with frequencies of 8.8 Hz and 0.22 Hz appear in the model because of the AVR and PSS.These oscillatory modes have sufficient damping and do not interact with the subsynchronous electrical mode. Tracking the movement of the eigenvalue real parts reveals that the real part of the second torsional mode crosses the zero axis atߤൌ0.5219. The first Lyapunov -5 coefficient is computed as -3.59x10. Similarly, the first torsional mode undergoes a Hopf bifurcation at -7 ߤൌ0.7403.l1. The(0) value is calculated as -1.82x10 Bifurcation of this type in which the first Lyapunov coefficient vanishes is called the generalized Hopf bi-furcation. In conclusion, the Hopf bifurcations in the model with AVR and PSS are not supercritical.
5.2Time Domain Simulations in MATLAB-Simulink The SMIB power system model with AVR and PSS under study is modeled using the software MATLAB-Simulink. The simulation is started in steady state with ܶ ܸܸ =0.91,௥௘௙=1.0982 and=1.0. The simulated dis-turbance is identical to the one applied in Section 4. Fig. 10 shows the generator rotor speed response to the dis-ߤ turbance at the Hopf bifurcation point (=0.5219). Similar to the model without AVR and PSS, the limit cycles reach to a stable orbit following the initial oscil-lations.
Figure 10:Generator rotor speed response to the disturbance ߤ at the Hopf point (=0.5219)
 ܸܸ With the same values ofܶ,௥௘௙and, the simula-tion is repeated atߤ=0.55. Fig. 11 shows the response of the generator rotor speed to the disturbance. As a result of the generalized Hopf bifurcation, the model dynamic response bifurcates into a torus following high magni-tude oscillations. This phenomenon is also called the secondary Hopf bifurcation. It is evident from Fig. 10 and Fig. 11 that the value of the series compensation factor affects dynamic response of the model to the identical disturbances.
Figure 11:Generator rotor speed response to the disturbance ߤ at =0.55(ߤ=0.5219)
In the future part of this study, control techniques for stabilizing the unstable torsional modes in power sys-tems susceptible to the SSR will be investigated.
6CONCLUSIONS In this paper, we have investigated the stability of limit cycles emanating from the Hopf bifurcations in the IEEE Second Benchmark Model for SSR studies by computing the first Lyapunov coefficients. Generator damper windings are included in the nonlinear model. Taking the series compensation factor as bifurcation parameter, we have verified the existence of Hopf bifur-cations in model. The first Lyapunov coefficients are computed and the results reveal that the Hopf bifurca-tions existing in the model are not supercritical. The impact of the mechanical torque input, the network voltage and the excitation field voltage on the bifurca-tion point and the first Lyapunov coefficient is also investigated. Moreover, the AVR and PSS are included in the model and bifurcation analysis results are presented. Two new oscillatory modes appear because of the AVR and PSS. The first Lyapunov coefficients are computed and the results indicate that the generalized Hopf bifur-cation occurs in the model with AVR and PSS. The contribution we have made with this study is to use the first Lyapunov coefficient to determine the type of Hopf bifurcations existing in a SMIB power system susceptible to SSR. The applied method provides ana-lytic results for studying the limit cycles. Time domain simulations have been carried out to verify that the method gives accurate results. REFERENCES [1]IEEE Committee Report, “Reader’s Guide to Sub-synchronous Resonance”, IEEE Trans. on Power Systems, Vol. 7, No. 1, pp. 150-157, February 1992 [2]IEEE SSR Working Group, “First Benchmark Mod-el for Computer Simulation of Subsynchronous Re-sonance”, IEEE Trans. Power Apparatus and Sys-tems, vol. 96, pp. 1565-1572, September 1977 [3]IEEE SSR Working Group, “Second Benchmark Model for Computer Simulation of Subsynchronous Resonance”, IEEE Trans. Power Apparatus and Sys-tems, vol. 104, pp. 1057-1066, May 1985 [4]P. M. Anderson, B. L. Agrawal and J. E. Van Ness, “Subsynchronous Resonance in Power Systems”, New York, IEEE Press, ISBN 0-87942-258-0, pp. 11-14, 1990 [5]I. M. Canay, “A novel approach to the torsional interaction and electrical damping of synchronous machine, part I: Theory”, IEEE Trans. Power Appa-ratus and Systems, vol. PAS-101, No. 10, pp. 3630-3638, October 1982. [6]W. Zhu, R. R. Mohler, R. Spee, E. A. Mittelstadt and D. Maratukulam, “Hopf Bifurcations in a SMIB Power System with SSR”, IEEE Proceeding of the 1995 Summer Meeting, pp. 531-534, PWRS [7]V. Ajjarapu and B. Lee, “Bifurcation theory and its application to nonlinear dynamical phenomena in an Electrical Power System”, IEEE Transactions on
Power Systems, vol. 7, No. 1, pp. 424-431, February 1992 [8]P. G. Drazin, “Nonlinear Systems”, Cambridge, Cambridge University Press, ISBN 0-521-40489-4, pp. 6-22, 1992 [9]R. Seydel, “Practical Bifurcation and Stability Analysis, From Equilibrium to Chaos”, New York, Springer-Verlag, ISBN 0-387-94316-1, p.72, 1994. [10] Y. A. Kuznetsov, “Elements of Applied Bifurcation Theory”, New York, Springer-Verlag, ISBN 0-387-21906-4, pp. 177-180, 2004 [11] W.Govaerts, Yu.A. Kuznetsov, “Continuation Software in Matlab: MatCont” [Online]. Available: www.matcont.ugent.be [12] J. Guckenheimer, Yu. A. Kuznetsov, “Bautin bifur-cation” [Online]. Available: www.scholarpedia. org/article/Bautin_bifurcation [13] A.M. Harb, M.S. Widyan, “Modern Nonlinear Theory As applied to SSR of the IEEE Second Benchmark Model”, Bologna PowerTech 2003 Con-ference, Bologna-Italy, 2003 [14] P. Kundur,“Power System Stability and Control”, Electric Power Research Institute, McGraw-Hill, New York, pp. 1061-1065, ISBN 0-07-035958-X, 1994 [15] "RecommendedPractice for Excitation System Models for Power System Stability Studies", IEEE Standard 421.5-1992, August 1992 APPENDIX The numerical parameters of the system are as follows (All units except AVR and PSS parameters in p.u. on the base of the generator ratings): 1) Synchronous generator Xd1.65Xq=1.59 X1.51X=1.45 akd akq X X kkd1.642kkq=1.5238 X1.6286X=1.51X=1.51 ffd afdkd r r=0.0045fd=0.00096 a r=0.016r=0.0116 kd kq 2) Network X0.12RT=0.0012 T X0.48R=0.0444 L1 1 XL20.4434R2=0.0402 X R b0.18b=0.0084 3) Mechanical System (Damping, Inertia and Stiffness constants) D ൌM K 10.04981=0.49812=42.6572 D ൌM 20.0312=3.1004K23=83.3823 ൌ M D30.17583=1.7581K34=3.7363 M D ൌ0.00144=0.0138 4 4) AVR and PSS KT F0.001F=0.1KA187 T KTR0.020A=0.001S10 TT T 10.0502=0.020W=10