A new methodology for the stability analysis of large-scale power electronics systems

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A new methodology for the stability analysis of large-scale power electronics systems
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  IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 4, APRIL 1998 377 A New Methodology for the Stability Analysis of Large-Scale Power Electronics Systems Phuong Huynh and Bo H. Cho,  Senior Member, IEEE   Abstract— A new methodology is proposed to investigate thelarge-signal stability of interconnected power electronics systems.The approach consists of decoupling the system into a sourcesubsystem and a load subsystem, and stability of the entiresystem can be analyzed based on investigating the feedback loopformed by the interconnected source/load system. The proposedmethodology requires two stages: 1) since the source and the loadare unknown nonlinear subsystems, system identification, whichconsists of isolating each subsystem into a series combinationof a linear part and a nonlinear part, must be performed; and2) stability analysis of the interconnected system is conductedthereafter based on a developed stability criterion suitable forthe nonlinear interconnected source-load model. Applicability of the methodology is verified through the stability analysis of atypical power electronics system.  Index Terms—  DC–DC converters, nonlinear systems, powerelectronics, stability, system indentifiction. I. I NTRODUCTION P RESENTLY, existing methods for the stability analysisof large-scale power electronics systems are based onlinearization around the system steady-state operating point,from which linear analysisthe so-called  impedance compar-ison technique can be performed [1]–[3]. Even though thelinearization technique is a powerful tool, it can only predictthe system local stability due to many nonlinearities presentin power electronics systems. On the other hand, [4] and [5]have approached the large-signal stability issue of switchingregulators using the phase-plane trajectories technique, whichlends itself to the analysis of simple systems. Thus, new tech-niques are needed to investigate the large-signal behaviors andthe global stability of large-scale power electronics systems.In general, design and analysis of power electronics systemsadopt a modular approach. A large-scale power electronicssystem consists of several interconnected subsystems or mod-ules. First, each module is designed and optimized for stabilityat the subsystem level; then, the next step is to integratethese modules. However, integration of these modules doesnot guarantee that the interconnected system is stable due tointeraction among modules. Therefore, interaction analysis ateach system interface is needed to ensure the overall system Manuscript received September 19, 1995; revised December 12, 1996. Thispaper was recommended by Associate Editor A. Ioinovici.P. Huynh was with Philips Research, Briarcliff Manor, NY 10510 USA.He is now with Cadence Design Systems, Columbia, MD 21046 USA.B. H. Cho is with the Department of Electrical Engineering, Seoul NationalUniversity, Seoul, Korea.Publisher Item Identifier S 1057-7122(98)02110-2.Fig. 1. Source and load decomposition.Fig. 2. Source and load interconnection.Fig. 3. Feedback connection of source and load. stability. This interaction analysis consists of decoupling thesystem at an interface of interest into a source that comprisesall the upstream modules and a load that consists of all thedownstream modules, as illustrated in Fig. 1, and investigat-ing any possible interaction between both subsystems. Thisprocess is repeated at all the interfaces of the interconnectedsystem to ensure the overall system stability.When interconnecting a source and a load, instability canarise due to existence of a feedback loop. Indeed, if oneconsiders both source and load as two unterminated one-portnetworks (Fig. 2), where and denote perturbation at theinterface, and the operators and represent the sourceimpedance and the load admittance, respectively, a feedback loop system, as shown in Fig. 3, is obtained thereby. Sinceand are nonlinear, nonlinear analysis is needed to investigatethe stability property of the interconnected system.This paper will address two issues associated with thenonlinear feedback loop system shown in Fig. 3: 1) character- 1057–7122/98$10.00  󰂩  1998 IEEE  378 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 4, APRIL 1998 Fig. 4. Feedback representation in factorization form. ization of and in such a way that they are suitable for thestability analysis; and 2) development of a nonlinear stabilitycriterion for the stability analysis of the interconnected system.Section II will address the theoretical development of theproposed methodology, and Section III will discuss applicationof the methodology to a particular power electronics system.Concluding remarks are shown in Section IV.II. T HEORETICAL  D EVELOPMENT OF THE  M ETHODOLOGY First, characterization of and is necessary for the sta-bility analysis. Assuming that and are stable subsystems,one can factorize and into series combinations of stablelinear parts, and stable nonlinear parts, as illustratedin Fig. 4, where represents the linearized system aroundthe steady-state operating point, and represents the large-signal nonlinear dynamic. This implies that under small-signalinput perturbation, has little contribution to the outputresponse and becomes an identity operator, The reasonfor the factorization is to improve the conservativeness of theto-be-developed nonlinear stability criterion and to establishstability conditions that are in similar form to the Nyquistcriterion.  A. Linear Part Identification Identification of the linearized model, can be doneby injecting small-signal perturbation around the steady-stateoperating point and collecting the system response for modelidentification. Many existing algorithms in the literature can beapplied to obtain a small-signal linear model for the unknownsystem, either in the form of a transfer function or state-spacerepresentation; one approach is based on the least-squaresfit [8].  B. Nonlinear Sector Gain Identification Detailed characterization of a nonlinear system is muchmore difficult. Nevertheless, for this proposed methodology,one needs only to estimate the gain of the nonlinear system.From Appendix I, a nonlinear system with an input andan output which belongs to the conic sector mustsatisfy Therefore, the conic radius, as afunction of the conic center, is given by(1) Fig. 5. System identification. which, according to the definition of the 2-norm in (20), canbe expanded to(2)or(3)where denotes the average power of the signal asdefined in (2). Furthermore, since can also be computedfrom the power spectrum density function [12]; namely(4)the conic radius can be approximated by(5)where can be computed from the FFT algorithm.Thus, once is identified, the sector gain of canbe approximated. Under large-signal perturbation, the non-linear effects in cannot be neglected. Consequently, if represents the large-signal response of thesystem around a steady-state operating point, the output of the linear part, (Fig. 5), can be computed fromand the sector gain of can be computed by usingin (5). The outcome is a set of conic radii fordifferent conic centers. C. A Large-Signal Stability Criterion for the Interconnected Source-Load System Let us assume that the nonlinear parts and in Fig. 4have conic sectors and respectively; thenexpanding and as mentionedin Appendix I yields the block structure representation shownin Fig. 6, where and represent the nonlinear operatorswhich belongs to the conic sector and  HUYNH AND CHO: STABILITY ANALYSIS OF LARGE-SCALE POWER ELECTRONICS SYSTEMS 379 respectively. Consequently, the system has the following in-put–output representation:(6)(7)Rearranging (6) and (7) yields(8)where represents the small-signal loop gain and is givenby Applying the norm elementwise, the triangularinequality, and regrouping leads to(9)where and are the gains of the nonlinear operators,and respectively. Equation (9) has the followingrepresentation:(10)Therefore, if is invertible and all elements of arepositive and finite, then(11)It follows from (9) that the conditions required for input-outputstability are(12)(13)(14) Fig. 6. Conicity decomposition of the nonlinear parts.Fig. 7. Multiplier insertion. The above three sufficient conditions represent a stabilitycriterion for the nonlinear feedback loop shown in Fig. 4,and (12) and (13) have graphical interpretations similar tothat from Theorem 1 in Appendix I. How far the outcomeof this criterion is from the necessary condition dependsupon systems. Yet, reduction of the conservativeness of thisnew stability criterion can be achieved further by inserting a multiplier  , a linear operatoras shown in Fig. 7. Properselection of can shape the new linear part at thecost of enlarging or reducing the gain of the new nonlinearpart in such a way as to enhance the inequality in(12) and (13). Nevertheless, selection of must result instable operators and which implies thatmust contain only poles and zeros in the left-half plane.III. S TABILITY  A NALYSIS OF A  P OWER  E LECTRONICS  S YSTEM In this section, interaction between a dc–dc converter unit(DDCU) and a particular load subsystem (load) is analyzedbased on the proposed methodology. A simulation model of the entire system is built for the analysis purpose, and Fig. 8illustrates the key points of the interconnected system. Thesource input voltage is set at 115 V and the source output isregulated at 124 V. The load converter input filter is selectedso that instability occurs under large-signal perturbation for acertain value of the damping resistor, even though theinterconnected system is designed to be stable around thesteady-state operating point.To analyze the system stability, small-signal models for bothsource and load are needed first. By injecting small-signalperturbation around the steady-state operating pointV, A) and applying the small-signal identificationtechnique, the source impedance, is found as shown in(15). Likewise, the load impedance at(15)  380 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 45, NO. 4, APRIL 1998 (a)(b)Fig. 8. An interconnected system. (a) Switching regulator and load converter. (b) a power electronics system. is found as(16)Fig. 9(a) illustrates the impedance comparison of the sourceand the load; both curves do not overlap. According to theimpedance comparison technique [3], the Nyquist plot of thesmall-signal loop gain, [Fig. 9(b)] notencircling the point indicates that the system isstable in the small-signal sense. However, injection of a binarypseudorandom signal, with an amplitude of 50 A can causethe source output voltage to oscillate, as illustrated in Fig. 9(c).The reason is due to the duty cycle saturation in the sourceconverter, which is a nonlinearity that has not been takeninto account in the small-signal analysis. In fact, the dampingresistor of the input filter, is selected basedon trials and errors such that the system is almost exactly onthe borderline between large-signal instability and stability.Thus, because of the nonlinear interaction between the sourceand the load, small-signal linear analysis fails to predict thesystem stability.On the other hand, investigation of the system large-signalstability can be performed based on the proposed methodology, (a) (b)(c)Fig. 9. Instability for             : (a) impedance comparison, (b) Nyquistplot of loop gain, and (c) simulation results.  HUYNH AND CHO: STABILITY ANALYSIS OF LARGE-SCALE POWER ELECTRONICS SYSTEMS 381 (a)(b)Fig. 10. Nonlinear part identification results. (a) Sector gains. (b) Effect of inclusion of         which requires identification of the nonlinear parts, in additionto the already-identified linear parts, and in (15)and (16). To identify the nonlinear parts, large-signal randomperturbations with uniform disturbution are introduced intothe systems and the input–output data are recorded. Then,the sector gains of both source and load can be computedautomatically from a MATLAB program (Appendix II) andare shown in Fig. 10(a). Given these data, the stability testcan be performed. During the process, it is realized that byinserting a two-pole two-zero multiplier, as in Fig. 7,where(17)conservativeness of the stability test can be reduced. As illus-trated in Fig. 10(b), insertion of helps reducing the mag-nitude of the small-signal loop gain, atthe cost of expanding the sector gain of the source [Fig. 10(a)].However, efficiency of the stability test is enhanced by appli-cation of Finally, selections of the pairs andfor optimum stability results are given in Table I.Two cases are selected for the large-signal stability test: 1)and 2) For case 2, the small-signalload impedance is found as(18)The graphical tests for all three stability conditions [(12)–(14)]are illustrated in Fig. 11, where the subscripts and denotethe 0.09 case and the 0.19 case, respectively; and theshaded regions are the forbidden regions, meaning the testwill fail if the curve crosses those regions. It is observedthat all three stability conditions for are notmet, whereas the 0.19 case meets the large-signal stabilitytest. This value is approximately double of 0.09 which, asmentioned earlier, indicates the actual instability margin. Theabove results indicate that the methdology is applicable andits outcome is not too conservative.Furthermore, the development of this new methodologyprovides a significant understanding in the know-how of designing the small-signal loop gain to achieve large-signalstability for the interconnected system. Indeed, one can selectand a small sector gain translates into a smallcircle that contains the point in the Nyquist plane,as illustrated in Fig. 12(a). However, when the sector gainbecomes larger, the forbidden region expands and occupiesthe entire left half plane, when Furthermore, when
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