Energy and Power Engineering, 2011, 3, 18 doi:10.4236/epe.2011.31001 Published Online February 2011 (http://www.SciRP.org/journal/epe) Copyright © 2011 SciRes. EPE Singular Hopf Bifurcations in DAE Models of Power Systems Wieslaw Marszalek1, Zdzislaw Trzaska2 1College of Engineering & Information Sciences, DeVry University, North Brunswick, USA 2Department of Management and Production Engineering, Warsaw University of Ecology and Management, Warsaw, Poland Email: wmarszalek@devry.edu, zdzislaw.trzaska@netlandia.pl Received August 25, 2010; revised October 10, 2010; accepted October 15, 2010 Abstract We investigate an important relationship that exists between the Hopf bifurcation in the singularly perturbed nonlinear power systems and the singularity induced bifurcations (SIBs) in the corresponding different tialalgebraic equations (DAEs). In a generic case, the SIB phenomenon in a system of DAEs signals Hopf bifurcation in the singularly perturbed systems of ODEs. The analysis is based on the linear matrix pencil theory and polynomials with parameter dependent coefficients. A few numerical examples are included. Keywords: Power Systems, Singularly Perturbed Systems, DAEs, Bifurcations, Matrix Pencils 1. Introduction In an effort to better understand dynamical properties of power systems, their stability features and the impact of various parameters, the DAEs approach seems to be very important as was shown in a number of recent papers (see for example [110]). Some of these papers deal with the existence of Hopf bifurcations in the singularly perturbed systems of non linear ODEs. DAEs are closely related to singularly per turbed ODEs, therefore it is natural to expect similar types of behavior. However, there are also obvious dif ferences between the qualitative properties of DAEs and singularly perturbed ODEs. One can mention, for exam ple, the SIB phenomenon, which is present in DAEs and not in ODEs. Also, singularities in DAEs add another layer of difficulty in the bifurcation analysis, in particular in the Hopf bifurcation. Because of a local nature of these phenomena, it is feasible to consider the singular Hopf bifurcation and the SIBs by analyzing certain properties of linear matrix pencils and their characteristic polynomials [9,1114]. The matrix pencil and characteristic polynomial ap proach used in the analysis of the singularly perturbed systems and DAEs models has one obvious drawback: the available results give sufficient conditions only and, as of today, no significant results are available for multipa rameter bifurcations of the underlying DAEs. The fun damental SIB theorem for DAEs reported in [9] has been slightly improved in [12,13,15] to weaken some of the sufficient conditions and to include the case of second order slow subsystems (typical in power systems). In [16] the sufficient conditions of the SIB phenomenon were given for quasilinear DAEs and other interesting appli cations of DAEs have been reported in [17,18]. In this paper we extend some of the results presented in [1]. In particular, we present further studies of the singularly perturbed power systems, their SIB, singular Hopf bifurcations and interactions between them. The topic of bifurcations in power systems (in particular the Hopf bifurcation) is one of the most important and inter esting ones, and, at the same time quite difficult, as any realistic power grids have thousands of buses, generators and other devices. Even small failures of various devices of secondary importance may yield catastrophic conse quences, power outages, or unstable systems’ behavior [7,8]. Section 2 provides a short introduction into the DAEs and singularly perturbed ODEs as models of power sys tems. Section 3 constitutes the main part of this paper. First, we discuss the SIBs and Hopf bifurcations in the context of singularities of DAEs. Next, several results related to these bifurcations in power systems are given and illustrated by several numerical examples. Section 4 provides a summary of the results and a few suggestions for possible extensions. The three part appendix is an
W. MARSZALEK ET AL. 2 important addition as it provides several properties of linear matrix pencils and polynomials with parameter dependent coefficients that are frequently used in our analysis in Section 3. 2. DAEs as Models of Power Systems The DAE model of a power system is as follows [1,2, 19,20] 2 2,1,, ,1, , ,,1, , i iiimi G idiGG PV idii diGPVG PVPQ dd i MDfP i N dt dt fP iNNN P gQ iNNNNN (1) where fi and gi form the load flow equations and are de fined as (n ≡ NG + NPV + NPQ) 1 1 sin cos 1,, sin cos 1,, n iijijijijiji j n iijijijijiji j GV fVVBVVG i n gVVG VVB iNN n j j (2) with the following notation: Mi is the rotor inertia of the ith generator, Di is the damping coefficient of the ith generator, Vi is the voltage magnitude of the ith bus, αi is the angle of the ith bus, Bij, Gij are the transfer susceptances and conductances, Pmi is the turbine mechanical power injection of the ith generator, and Pdi, Qdi are the real and reactive power loads at the ith bus. The above system can easily be written in the following DAE form 2 2,, ,0,, dx dx MDfxy gxy dt dt (3) with x ≡ [α1,, αNG]T , y ≡ [αNG+1,, αn, V1,, Vn]T , λ is the vector of k parameters, and the matrices M, D are positive and semipositive definite, respectively. One of the generator buses is usually considered as a swing bus. This allows us to introduce a set of new vari ables (relative angles) and reduce the number of differ ential equations in the first subsystem in (3) by one [3]. The DAE model (3) and its bifurcations are closely related to the behavior of the singularly perturbed ODE model of the form 2 2,, ,,, dx dxdy MDfxy gxy dt dt dt when ε ≠ 0. We shall examine the singular Hopf bifurca tion of (4) for small 0 < ε << 1. The singularly perturbed ODEs (4), when linearized, yield linear matrix pencils. Either first or second order matrix pencils can be used depending on an individual preference (see Appendices 1.1 and 1.2 for more details). A second order matrix pencil would normally be preferred for power systems as it may directly lead to double SIB points [1,21]. 3. Bifurcations of Power Systems at Singularities 3.1. The Singularity Induced Bifurcation The SIB is a phenomenon attributed exclusively to DAEs. Since DAEs are used quite often as models of electrical power systems, it is important to study the changes of stability of those systems to better understand their dy namical properties. This may help to prevent future fail ures and catastrophic events such as, for example, mas sive power outages. It is also known that DAEs may ex hibit other types of bifurcations that are widely found in nonlinear ODEs. One of such bifurcations is the well known Hopf bifurcation when a linearized system has a pair of complex conjugate eigenvalues on the imaginary axis with all other eigenvalues lying off the axis. The bifurcating parameter may cause the subcritical or super critical Hopf bifurcations. The bifurcation is called sub critical if the periodic solutions are unstable and super critical if the periodic solutions are stable. The Hopf bifur cation in power systems has been analyzed in [2,22]. The singularity of nonlinear semiexplicit parameter dependent DAEs = f(x, y, λ), 0 = g(x, y, λ) occurs when det(gy) = 0 for some (x∗, y∗, λ0) Rn+m+1, x(t) x Rn, y(t) Rm, λ R, dtdxx and dydgg y. The λ is a slowly varying parameter. Equivalently, the linear matrix pencil 0, 00 AB CD (5) where A(λ) = fx, B(λ) = fy, C(λ) = gx, D(λ) = gy, has index greater than 1 at (x∗, y∗, λ0) (see Appendices 1.1 and 1.2). For the singularly perturbed ODEs we have= f(x, y, λ), x y = g(x, y, λ) where 0 < ε << 1 is small and constant. A natural extension to secondorder DAEs and singularly perturbed ODEs is to have),,,,( yxxfx y = g(x, y, λ). (4) It turns out that under the assumption of the SIB theo rem (with an algebraically simple zero of gy(x ∗ , y ∗ , λ0) [9]), the SIB phenomenon for semiexplicit parameter Copyright © 2011 SciRes. EPE
W. MARSZALEK ET AL.3 dependent DAEs may be equivalent to Hopf bifurcation of the singularly perturbed ODEs. The following example illustrates such a case. Example 1: Consider the following singularly per turbed parameter dependent DAEs 2 1 , 2 yyx y (6) Two solutions of (6) for λ = 0 are shown in Figures 12. If ε = 0, then the DAEs (6) undergoes the SIB as the pa rameter λ → 0. The equilibrium locus is (λ2/2, λ), i.e. x = λ2/2, y = λ, and an eigenvalue −1/λ diverges through in finity as λ → λ0 = 0. The stability of the system is changed at λ0. The point λ0 is called a single SIB point (see [11]). The system is asymptotically stable for λ > λ0 and unstable for λ < λ0. This relatively simple case with scalar f(x, y, λ) and g(x, y, λ) is covered by Lemma 1 in [15] which, under the conditions gy(λ0) = 0, g′y(λ0) ≠ 0 along the equilibrium locus, and gx(λ0) fy(λ0) = −ω2 < 0, predicts a pair of com plex eigenvalues 2 0 2 ,1 2 x f i  1 O (7) as ε → 0. The ′ = d/dλ and 1i. For (6) we have fx(λ0) = 0, gy(λ) = −λ, g′y (λ) = −1 and ω2= 1. Thus, the eigenvalues (7) are: α(ε, λ(ε)) ~ ±ω √ ε + Figure 1. Trajectories for ε = 0.01. Figure 2. Trajectories for ε = 0.09. O(1). The linearization of these DAEs along the equi librium locus (λ2/2, λ) yields the matrix pencil (see Ap pendix A) 100 1 , 001 (8) which has index 1 for λ ≠ 0 and index 2 for λ = 0 (Ap pendix A). The CB = −1, v = 1, and CBv Im(D) at λ = 0. Therefore, at λ = 0, the index jumps by one, the well established fact for DAEs undergoing the SIB. Also, the characteristic equation of the pencil (8) is (see Appendix 1.1, case k = 1) −λs + 1 = 0 with the root s = 1/λ diverging through ±∞ as λ → 0. Example 2: Now, suppose that d2x/dt2 +γ dx/dt =λ−y, εdy/dt = x −12y2, with γ > 0. The eigenvalues are given by 2 221 , and the system is asymp totically stable for λ > 0 and unstable for λ < 0. It can be checked that the linear system has index 1 if λ ≠ 0, and the index jumps to 3 at λ = 0. If λ ≠ 0, then along the equilib rium locus (λ2/2, λ, λ) we have (see Appendix 1.2): det(E(0)s − L(λ)) = s2λ + sγλ + 1, while at the SIB point λ = 0 we have det(E(0)s − L(0)) = 1. This drop by 2 in the degree of det(E(0)s − L(λ)) is equivalent to the index increase from 1 (at λ ≠ 0) to 3 (at λ = 0). If λ < 0 one of the eigenvalues is unstable while the other remains stable. At λ = 0 the eigenvalues diverge through ±∞. The point λ = 0 is called the double SIB point, the concept introduced by Beardmore in [11]. For λ ≠ 0 the Kronecker Normal Form of the pencil (E(0), L(λ)) is 10001 0 0010,1 0 00000 1 E L (9) and for λ = 0 the structure of the pencil changes to 010 100 001,0010 000 001 E L (10) 3.2. Properties of DAEs with SIB Phenomenon The main theorem indicating Hopf bifurcation in the singularly perturbed ODEs is due to Beardmore [15], as follows. THEOREM: Suppose that the singularly perturbed ODEs x f(x, y, λ), y =g(x, y, λ) have a trivial equi librium for λ = λ0 and 1) 0y ker gv with 0 Im v vg , 2) 00 Im yy gvg , Copyright © 2011 SciRes. EPE
W. MARSZALEK ET AL. 4 3) det 0 xy xy ff gg 4) 0 T y ker gu and 00 20 T xy T ugf v uv . Then, there exists ε1 > 0 such that ε1 > ε > 0 there is a λ0(ε) such that the singularly perturbed linearized ODEs have a pair of purely imaginary eigenvalues at λ = λ0(ε) and λ0(ε) is continuous in ε at 0 with λ0(0) = λ0. In the above, the ker, Im and det denote the kernel, image and determinant of a matrix, respectively, the < > is a column vector and T denotes the transposition. It is easy to check that the above sufficient conditions are satisfied for (6) with v = u = 1, g′y (0) = −1, fxgy − fygx = 1, gx(0)fy(0) = −1, yielding ω = 1. LEMMA 1: Suppose that in (18) in Appendices 1.2 we have for ε = 0: ind(E(0), L(λ)) = 1 for λ ≠ λ0, detL(λ0) ≠ 0, kerD(λ0) ≡ vRm, D′(λ0)vImD(λ0) and C(λ0)B(λ0)v ImD(λ0). Then, 10 det 0, nm nm EsLasa sa (11) 2 010 det 0, nm EsLsd sd 00 (12) and the two eigenvalues of the matrix pencil (E(0), L(λ)) diverging through ±∞ are the zeros of the seconddegree polynomial 0 2 2 0 nm nm nm as a asbsc sd (13) with a(λ) = an−m(λ), a(λ0) = b(λ0) = 0, and c(λ0) ≠ 0. The index of (E(0), L(λ0)) is 3. Lemma 1 is a consequence of the above theorem. In particular, from the Kronecker Normal Form of (E(0), L(λ)) and the assumption that ind(E(0), L(λ)) = 1 for λ ≠ λ0, we obtain (11). The other assumptions are needed to have the two eigenvalues divergent through ±∞ [11], and as a consequence, for λ = λ0 we have (12). Notice that in example 2 det(E(0), L(λ)) = s2λ + sγλ + 1 and det(E(0), L(0)) = −1, therefore a(λ) = λ, b(λ) = γλ, c(λ) = 1 and the two diverging eigenvalues are the roots of s2λ + sγλ + 1 = 0. As a direct consequence of the above lemma we have the following result. LEMMA 2: If the assumptions of lemma 1 are satis fied and if Γ(λ) ≡ 0 in (18) (see Appendix 1.2), then b(λ) ≡ 0 and the two diverging eigenvalues are purely imaginary. If c(λ)/a(λ) changes its sign from positive to negative at λ = λ0, then the system undergoes transition from a center to a saddle, or viceversa, from a saddle to a center if c(λ)/a(λ) changes sign from negative to positive. The proof of the above lemma follows from the fact that the two diverging eigenvalues are 1,2 sca . No damping term exists in the system. A similar behavior in active RLC circuits has been reported in [23]. 3.3. An Illustrative Example The following example illustrates the above described features of a power system undergoing the singularity induced bifurcation. Example 3: Consider the 4bus, 3 generator power system shown in Figure 3. If we assume that the bifurca tion parameter λ defines the real and reactive powers at bus 4 with P = P0 (1 + λ), Q = Q0 (1 + λ), (P0 = const, Q0 = const), then the DAE model (3) of this power system undergoes the SIB at λ0 = 0.537305. Assuming that the rotor inertia of all generators equals 1, the line transfer conductances Gij are all zero, the damping generator coefficients to be γ, and considering bus 1 as a swing bus, the DAE system is as follows 2 11 21 21121 2 1313 22424 31 2 22 31 13132 2 12 1 2134 3432 4224 314334 32 4242 3143343 2sin sin sin 2sin sin sin 0sin sin 0sin sin mm mm dd PP BVV dt dt BVV BVV dd PP BVV dt dt BVV BVV BVV BVVP BVV BVV 2 2444 BV Q (14) with the relative angles θi ≡ αi+1 − α1 for i = 1, 2, 3. Suppose that the parameters are: B12 = B13 = 1, B24 = B34 = 2, B44 = −4, Bij = Bji, Pm2 − Pm1 = 1, Pm3 − Pm1 = 2, P0 = 1, Q0 = 0.3, γ = 1.5 and Vi = 1 for i = 1, 2, 3. Then, the DAEs (14) have an equilibrium placed at the singularity for Figure 3. A simple 4bus, 3generator power system. Copyright © 2011 SciRes. EPE
W. MARSZALEK ET AL. Copyright © 2011 SciRes. EPE 5 1234 ,,,,0.8496, 2.3761, 2.4940, 0.6334, 0.5373V For the singularly perturbed DAEs with P = P0 (1 + λ) and Q = Q0 (1 + λ) + εdθ4/dt we get the same equilibrium. Solutions θi(t), i = 1, 2, 3, for ε = 0.002 and various values of γ are shown in Figure 4. Notice that V4(t) can be eliminated from (14) and the system reduces to two sec ondorder differential equations and one firstorder dif ferential equation containing εdθ4/dt. It turns out that the above equilibrium is unstable and at λ = 0.537305, ε = 0.002, γ = 0 the pencil (18) has the following eigenvalues −5.6245 ± 9.7933i, ±0.1342i, 11.2487. The solutions in Figure 4 escape the above equilibrium and oscillate around another equilibrium (not on the sin gularity) 1234 ,,,,0.6812, 1.0816, 1.2770, 1.1018, 0.5373V with the eigenvalues ±1.3238i, ±1.5720i, −4742.9260 for γ = 0 and ε = 0.002. For ε = 0 and γ ≠ 0 the characteristic equation of the (a) (b) (c) (d) (e) (f) Figure 4. (a, c, e) The θi(t), i = 1, 2, 3, for γ = 0; (b, d, f) the θi(t), i = 1, 2, 3, for γ = 0.05.
W. MARSZALEK ET AL. Copyright © 2011 SciRes. EPE 6 linear matrix pencil (18) is (see (19)(21) in Appendix 1.3): a4(λ)s4 + a3(λ)s3 + a2(λ)s2 + a1(λ)s + a0(λ) = 0 and for λ = λ0 = 0.537305 the equation reduces to 2.869398s2 + 4.304097s + 0.051705. Clearly, we have the case k = 2 (see Appendix 1.3) and in (21) we have c2(λ0) = c1(λ0) = 0, c0(λ0) = 2.869398, b1(λ0) = γ = 1.5000, and b0(λ0) = 0.018019. For λ → λ0 the DAEs have two eigenvalues (divergent through in finity) 2 110 1,22 22 2 24 ccc scc c (15) whose real parts are such that lim 2cc 012 and the imaginary parts diverge to infinity as λ → λ0. The point λ0 = 0.537305 is called a double SIB point and the linear ma trix pencil has index 3 at λ = λ0 and 1 if λ ≠ λ0. One can also examine matrix R(λ) in Appendix 1.4 to check that detR(λ) = 0 if the characteristic polynomial of the matrix pencil (18) admits a pair of purely imaginary roots. Figure 5 shows detR(λ) for system (14). Clearly, this determinant is 0 for λ = 0.537305 because of the two complex eigenvalues ±0.1342i of the matrix pencil. 4. Conclusions In the paper we have examined sufficient conditions un der which the single and double SIB points in systems of DAEs are responsible for the Hopf bifurcation in singu larly perturbed systems of ODEs. Both, the DAEs and singularly perturbed ODEs are widely used as dynamical models of power systems making their analysis very relevant in the studies of stability, reliability and sensi tivity to various parameters. The analysis of power system models around their singularities is important from both the qualitative and numerical points of view. The quali tative analysis presented in this paper should be supple mented by a numerical one [17], as a typical power system Figure 5. The det R(λ) as a function of λ (see Appendices 1.4). model includes hundreds or even thousands of equations [20]. It is also our opinion that the future work in the quali tative analysis of the SIB and singular Hopf phenomena via matrix pencils and parameter dependent polynomials should focus on the formulation of the necessary and sufficient conditions and the analysis of multiparameter bifurcations at singularities. To the authors’ knowledge no formulation of the necessary conditions of the SIB phenomenon exists at the present time. 5. References [1] W. Marszalek and Z.W. Trzaska, “SingularityInduced Bifurcations in Electrical Power Systems,” IEEE Trans actions on Power Systems, Vol. 20, No. 1, 2005, pp. 312 320. doi:10.1109/TPWRS.2004.841244 [2] H. G. Kwatny, R. F. Fischl and C. O. Nwankpa, “Local Bifurcation in Power Systems: Theory, Computation, and Applications,” Proceeding of IEEE, Vol. 83, No. 11, 1995, pp. 14561483. doi:10.1109/5.481630 [3] H. G. Kwatny, A. K. Pasrija and L. Y. Bahar, “Static Bifurcations in Electric Power Networks: Loss of Steady state Stability and Voltage Collapse,” IEEE Transactions on Circuits and Systems, Vol. CAS33, No. 10, 1986, pp. 981991. [4] H. G. Kwatny and X.M. Yu, “Energy Analysis of LoadInduced Flutter Instability in Classical Models of Electric Power Networks,” IEEE Transactions on Cir cuits and Systems, Vol.36, No.12, 1989, pp. 15441557. [5] S. Ayasun, C. O. Nwankpa and G. G. Kwatny, “Compu tation of Singular and Singularly Induced Bifurcation Points of DifferentialAlgebraic Power System Model,” IEEE Transactions on Circuits and Systems I, Vol. 51, No. 8, 2004, pp. 15251538. [6] D. J. Hill and I. M. Y. Mareels, “Stability Theory for Dif ferential/Algebraic Systems with Application to Power System,” IEEE Transactions on Circuits and Systems, Vol. CAS37, No. 11, 1990, pp. 14161423. doi:10.1109/ 31.62415 [7] C. A. Canizares, N. Mithulananthan, F. Milano and J. Reeve, “Linear Performance Indices to Predict Oscilla tory Stability Problems in Power Systems,” IEEE Trans actions on Power System, Vol. 19, No. 2, 2004, pp. 1104 1114. doi:10.1109/TPWRS.2003.821460 [8] I. Dobson, J. Zhang, S. Greene, H. Engdahl and P. W. Sauer, “Is Strong Modal Resonance a Precursor to Power System Oscillations?” IEEE Transactions on Circuits and Systems, Vol. 48, No. 3, 2001, pp. 340349. [9] V. Vekatasubrumanian, H. Schattler and J. Zaborszky, “A Stability Theory of Large Differential Algebraic Systems: A Taxonomy,” Report SSM 9201 — Part I, Washington University, St. Louis, 1992. [10] V. Vekatasubrumanian, H. Schattler and J. Zaborszky, “Analysis of Local Bifurcation Mechanisms in Large Dif ferentialAlgebraic Systems such as the Power System,”
W. MARSZALEK ET AL.7 0 I Proceedings of 32nd Conference on Decision and Con trol, San Antonio, December 1993, pp. 37273733. [11] R. E. Beardmore and R. Laister, “The Flow of a Differen tialAlgebraic Equation near a Singular Equilibrium,” SIAM Journal on Matrix Analysis, Vol. 24, No. 1, 2002, pp. 106120. doi:10.1137/S0895479800378660 [12] R. E. Beardmore, “The SingularityInduced Bifurcation and Its Kronecker Normal Form,” SIAM Journal on Ma trix Analysis, Vol. 23, No. 1, 2001, pp. 112. [13] L. J. Yang and Y. Tang, “An Improved Version of the Singularity Induced Bifurcation Theorem,” IEEE Trans actions on Automatic Control, Vol. AC46, No.9, 2001, pp. 14831486. doi:10.1109/9.948482 [14] W. Marszalek and S. L. Campbell, “DAEs Arising from Traveling Wave Solutions of PDEs II,” Computers and Mathematics with Applications, Vol. 37, 1999, pp. 1534. doi:10.1016/S08981221 (98)002387 [15] R. E. Beardmore, “Double SingularityInduced Bifurca tion Points and Singular Hopf Bifurcations,” Dynamics and Stability of Systems, Vol. 15, No. 4, 2000, pp. 319342. [16] R. Riaza, “On the SingularityInduced Bifurcation Theo rem,” IEEE Transactions on Automatic Control, Vol. AC47, No. 9, 2002, pp. 15201523. doi:10.1109/TAC. 2002.802757 [17] S. L. Campbell and W. Marszalek, “Mixed Symbolic numerical Computations with General DAEs: An Appli cations Case Study,” Numerical Algorithms, Vol. 19, No. 14, 1998, pp. 8594. doi:10.1023/A:1019106507166 [18] S. L. Campbell and W. Marszalek, “DAEs Arising from Traveling Wave Solutions of PDEs I,” Journal of Com putational Applied Mathematics, Vol. 82, No. 12, 1997, pp. 4158. doi:10.1016/S03770427(97)000848 [19] K. L. Praprost and K. A. Loparo, “An Energy Function Method for Determiningvoltage Collapse during a Power System Transient,” IEEE Transactions on Circuits Sys tem, Vol. CAS41, No. 11, 1994, pp. 635651. [20] M. M. Begovic and A. G. Phadke, “Voltage Stability as Sessment of a Reduced State Vector,” IEEE Transactions on Circuits System, Vol. 5, No. 1, 1990, pp. 198203. doi: 10.1109/59.49106 [21] R. Riaza, “Double SIB Points in DifferentialAlgebraic Systems,” IEEE Transactions on Automatic Control, Vol. AC48, No. 9, 2003, pp. 16251629. doi:10.1109/TAC. 2003.817002 [22] Yang Lijun and Zeng Xianwu, “Stability of Singular Hopf Bifurcations,” Journal of Differential Equations, Vol. 206, No. 1, 2004, pp. 3054. doi:10.1016/j.jde.2004. 08.002 [23] R. Riaza, S. L. Campbell and W. Marszalek, “On Singu lar Equilibria of Index1 DAEs,” Circuits, Systems, and Signal Processing, Vol. 19, No. 2, 2000, pp. 131157. doi: 10.1007/BF01212467 [24] P. Lancaster and M. Tismenetsky, “The Theory of Matri ces,” London Academic Press, London, 1985, pp. 454 474 [25] J. Guckenheimer, M. Myers and B. Strumfels, “Comput ing Hopf Bifurcations,” SIAM Journal on Numerical Analysis, Vol. 34, No. 1, 1997, pp. 12. doi:10.1137/S00 36142993253461 1. Appendix 1.1. Matrix Pencils for First Order Systems For a pair of constant square n × n matrices, say E and L, with detE = 0, if det(sE − L) ≠ 0, then there exist non singular matrices U and V, such that  0 , 00 mm nm nm IJ UEV ULV N (16) where N n−m is a nilpotent matrix of size (n − m) × (n − m) and index ν ≤ n − m. That is, ν is a positive integer such that Nν = 0, Nν−1 ≠ 0. The Ik is the unit matrix of size k × k. The pencil (UEV, ULV) is said to be in Kronecker Normal (Canonical) Form and σ(E, L) = σ(Jm), i.e. the finite spectrum of the pencil (E, L) is the same as the spectrum of matrix Jm. It is also wellknown that for DAEs with the matrix pencil 0, 00 AB UEV ULVCD (17) and detL ≠ 0, the index(UEV, ULV)=1 if and only if detD ≠ 0. If ker(D) = <v > ≠ 0 and CBv Im(D), then the index (UEV, ULV) = 2. 1.2. Matrix Pencils for Second Order Systems The linearization of ),,,,( yxxfx y = g(x, y, λ) yields the matrix pencil (E(ε), L(λ)) with Copyright © 2011 SciRes. EPE
W. MARSZALEK ET AL. 8 000 0 () 00,()()()() 00 ()0D() n n m II n EI LAΓB IC (18) and A(λ) = fx, B(λ) = fy, C(λ) = gx, D(λ) = gy, Γ(λ) =xand E(ε), L(λ)R2n+m and det(E(ε)s − L(λ)) = det(E(ε)s − L(λ)) = an+m sn+m + an+m−1sn+m−1 + …+ a1s + a0 with ai = ai(ε, λ). f 1.3. Parameter Dependent Polynomials The change of the index of DAEs is related to certain properties of the characteristic polynomial of the pencil (UEV, ULV). Let the polynomial be 1 11 ,rr rr psasasa sa 0 , (19) with ar(λ0) = ar−1(λ0) = = ar−k+1(λ0) = 0, ar−k(λ0) ≠ 0. Polynomial (19) admits a real decomposition [13] 12 ,,psp sps (20) with 1 11 1 21 , , kk kk rk rk rk psc scsc pss bsb 0 0 (21) where ck, , c0 and br−k−1, , b0 are real and smooth, and ck(λ) = ar(λ), ci(λ0) = 0 for 1 ≤ i ≤ k, c0(λ0) = ar−k(λ0) ≠ 0. If k = 1, then one root of p1(λ, s) diverges through ∞ as λ → λ0. This is easily seen by observing that with k = 1 we have c1(λ) = ar(λ) = (−1)detD, and the root −c0(λ)/c1(λ) → ∞ as λ → λ0. This divergence is equivalent to the index jump of the matrix pencil from 1 at λ ≠ λ0 to 2 at λ = λ0. If k = 2, then two roots of p1(λ, s) diverge through ∞ and the index of the corresponding matrix pencil jumps from 1 at λ ≠ λ0 to 3 at λ = λ0. The two diverging eigenvalues, say s1,2, are such that limλ→λ0s2 1,2(λ)(λ − λ0) = μ with μ = −(uTC(λ0)B(λ0)v)/(uTD′(λ0)v), u ≡ kerD(λ0)T , v ≡ kerD(λ0), and D′ ≡ dD(λ)/dλ (see [1] for more details). 1.4. Crossing the Imaginary Axis: Loss of Stability Applying the LiénardChipart splitting [24], polynomial (19) with continuous real coefficients ai(λ) can be written as p(λ, s) = peven(λ, s2) + spodd(λ, s2), where 224 02 4 224 13 5 , , evens j odds j paasas paasas (22) Polynomial p(λ, s) has purely imaginary roots ±jω if and only if peven(λ,−ω2) = 0 and podd(λ,−ω2) = 0. These conditions imply that the resultant matrix is singular [25]. Determinant detR(λ) = ±detHr(λ), where Hr is the Hurwitz 135 1 13 31 135 31 24 20 2420 24 420 000 0 00 00 000 00 00 rr r rr rr r rr r rr rr r aaa a aa aa aaa aa Raa aa a aaa aa a aaaaa 00 0 0 matrix of (19). Note that detHr(λ) = a0detHr−1 and 12 1 1,1 det 1, rr r rr iki k r ik as s (23) This implies that detHr−1 is zero not only for the purely imaginary roots ±jω, but also for a pair of real symmetric roots s1,2 = ±σ and for two complex and symmetric pairs of roots s1,2 = σ ± jω and s3,4 = −σ ± jω. If, during the bifurcation process with λ → λ0, a power system oper ates initially in a stable mode (all eigenvalues are in the open left half plane), then, providing that none of the eigenvalues diverges through ∞, the eigenvalues have to cross the imaginary axis first (at λ0) before reaching unstable locations in either of the symmetric patterns with respect to the origin. Thus, in a typical situation, the increased illconditioning of the matrix Hr(λ) with the changing parameter λ will indicate an imminent possi bility of the Hopf bifurcation as some of the eigenvalues approach the imaginary axis and become ±jω at λ = λ0 for finite ω. Copyright © 2011 SciRes. EPE
