INTERACTIONS among power converters, input/output filters,

Transcription

1 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL Reaching Criterion of a Three-Phase Voltage-Source Inverter Oerating With Passive and Nonlinear Loads and Its Imact on Global Stability Kaustuva Acharya, Student Member, IEEE, Sudi K. Mazumder, Senior Member, IEEE, and Ishita Basu, Student Member, IEEE Abstract We develo and demonstrate a technique based on comosite Lyaunov functions (CLFs) to analyze the imacts of assive (RL and RC) and nonlinear (diode rectifier) loads on the reaching dynamics of a three-hase voltage-source inverter (VSI). The reaching criterion (which ensures convergences of state trajectories to an orbit) is synthesized using iecewise linear models of the VSI and the loads and conditions for switching among the various models (corresonding to the different switching states). Once orbital existence is ensured using the reaching criterion, we extend the CLF-based aroach to redict the stability of the nominal (eriod-) orbit of the system (comrising the three-hase VSI and the load) and comare these redictions with those obtained using a conventional imedance-criterion technique that is develoed based on a linearized averaged model. Overall, we demonstrate the significance of analyzing the reaching condition from the standoint of orbital existence and why such a criterion is necessary for analyzing global stability. On a broader note, the methodology outlined in this aer is useful for analyzing the global stability of multihase inverters, otentially leading to advanced control design of VSI for alications including uninterruted ower sulies, telecommunication ower sulies, grid-connected inverters, motor drives, and active filters. Index Terms Converter load interactions, imedance criterion, linear matrix inequality (LMI), Lyaunov s stability, iecewise linear (PWL) systems, reaching conditions, three-hase inverters. I. INTRODUCTION INTERACTIONS among ower converters, inut/outut filters, and their loads have been investigated in great detail over the years for dc/dc converters [ [6 and, recently, for multivariable systems including three-hase inverters and rectifiers in the synchronous reference frame [7. These techniques redict the regions of stable oeration with varying loads and inut conditions and are used to determine control strategies and controller bandwihs/gains to ensure oeration within stability bounds. However, these techniques are based on the Manuscrit received March 3, 7; revised December 8, 7. This work was suorted by the National Science Foundation (NSF) under CAREER Award 393 and in art by the Office of Naval Research (ONR) under Young Investigator Award N The authors are with the Laboratory for Energy and Switching-Electronics Systems, Deartment of Electrical and Comuter Engineering, University of Illinois, Chicago, IL 667 USA ( mazumder@ece.uic.edu). Color versions of one or more of the figures in this aer are available online at htt://ieeexlore.ieee.org. Digital Object Identifier.9/TIE.8.97 linear systems theory, and their redictions may be inadequate for certain oerating conditions of switching ower converters, as demonstrated, for instance, in [8 for dc/dc converters. Moreover, these techniques cannot account for cases where nonlinear loads (e.g., diode rectifier loads) are connected to the converter. To address the issues of load nonlinearities, a technique is develoed in [9 where the nonlinear load is decomosed into several linear (harmonic) loads and the imedance criterion is extended for each of the decomosed harmonics. Although the techniques in [8 and [9 can handle nonlinearities due to switching and nonlinear loads, resectively, they (as well as the methodologies outlined in [ [7) assume orbital existence. Therefore, these techniques cannot be used to redict the global stability of converters. For global stability redictions, the reaching criterion for orbital existence has to be established first. Traditionally, the reaching dynamics of the system are investigated using time-domain simulations of the system model [ [. Over the years, significant research effort has been devoted to develoing techniques for faster simulations; for instance, by using s-domain techniques to solve the differential equation that governs the system dynamics in each switching state [3 or by tracking the enveloe of the state trajectories during large-signal erturbations [4. However, these simulation-based aroaches sacrifice the accuracy of the system models (as in [4) either by ignoring the switching discontinuities or because of the limitations of the simulation algorithm (samling time and truncation errors as in [3). Furthermore, deending on how long the model has to be simulated and due to the ossibility of a large number of initial conditions, simulation-based techniques for large-signal stability analyses can have a significant comutational overhead, articularly as the dimension and number of switching states of the model increase. Therefore, there exists a need to develo analytical techniques to analyze the large-signal stability of ower converters. Some such analytical aroaches have been develoed in [5 [8. However, all of these techniques are either comutationally cumbersome [5 for large-dimension systems or are limited to certain secific convergence mechanisms [6 [8. The limitations of some of the existing analytical techniques for reaching condition analyses have been summarized in detail in [9, where the authors describe a technique based on comosite Lyaunov functions (CLFs) to determine the reaching condition for orbital existence of switching ower converters. In conjunction with existing linear and nonlinear techniques 78-46/$5. 8 IEEE

2 796 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 Fig.. Schematic of the overall VSI-load system, comrising the ower stage and feedback controller connected to two assive load tyes and a three-hase diode rectifier load. The conventional symmetrical sace-vector modulation scheme is used [4. For the assive loads, the arasitic resistances of the inductor and the caacitor C RC are assumed to be lumed with the series resistors (R RL and C RC, resectively). for steady-state stability, the reaching criterion in [9 can be alied to analytically determine the global stability of certain classes of switching ower converters [. In this aer, we investigate the imacts of different load tyes (constant and ulsating assive RL and RC loads, and a nonlinear diode rectifier load) on the reaching conditions of a three-hase voltage-source inverter (VSI) by modifying the techniques develoed in [9. Section II describes the iecewise linear (PWL) models of the three-hase VSI with different load tyes. In Section III, we develo the reaching criterion for the VSI-load system using a CLF-based aroach. In Section IV, we describe how this technique can be extended to redict the stability of the nominal (eriod-) orbit of the VSI-load system. Finally, in Section V, we resent the results of the analyses and determine the reachability bounds of the VSI-load system. Furthermore, we demonstrate how the lack of knowledge of reachability can yield inaccurate global stability results. From the oint of view of multihase inverter design, global-stability redictions using the CLF-based methodology is a owerful tool for robust stability analysis and control design for a wide range of alications including uninterruted ower sulies, telecommunication ower sulies, grid-connected inverters, motor drives, and active filters [ [3. II. PWL MODELS OF VSI WITH PASSIVE AND DIODE RECTIFIER LOADS In this section, we derive PWL state-sace models of the three-hase VSI and the different loads and define the conditions that govern switching among these PWL models. The reaching criterion develoed in Section III and the CLF-based orbital stability analyses technique in Section IV are derived using these models. Note that, in this aer, we ignore the notation of time, i.e., we reresent any state y(t) as y. A. Power-Stage Model Fig. illustrates the schematic of a three-hase VSI with two tyes of loads: ) linear assive loads (RC and RL) and ) a nonlinear load (three-hase diode rectifier). The voltage at oints a, b, and c of the VSI are given by v a = 3 (S a S b S c )V in v b = 3 ( S a +S b S c )V in v c = 3 ( S a S b +S c )V in (a) (b) (c) where S a, S b, and S c are the switching functions of the ower devices of the three-hase VSI and are determined based on the oututs of the feedback controller, and S a + S a =, S b + S b =, and S c + S c =(neglecting dead times between the turn-on and turn-off conditions of the ower devices in the same leg). The dynamics of the ower stage of the threehase VSI (shown in Fig. ) can be described by the following vector differential equations: di abc dv abc = K i abc + K i abc + K 3 v abc + K 4 v abc +K 5 V in (a) = K 6 i abc + K 7 i abc + K 8 v abc + K 9 v abc +K V in (b) where any vector y abc r can be reresented as y abc r = [ y a_r y b_r y c_r T. Matrices K K are described in the

3 ACHARYA et al.: REACHING CRITERION OF THREE-PHASE VSI WITH PASSIVE AND NONLINEAR LOADS 797 Aendix. We note that, in (a), matrix K 5 contains switching functions S a, S b, and S c. The following differential equations describe the dynamics of three commonly used loads: Inductive Load : Caacitive Load : Diode rectifier Load : di abc dv abc di abc = L i abc + L v abc + L 3 i abc (3a) = L 4 i abc + L 5 v abc + L 6 v abc = L 7 v abc + L 8 i abc + L 9 v D (3b) dv D = L i abc_ + L v D (3c) where v D R is the voltage across the outut caacitor of the diode rectifier, and matrices L L are defined in the Aendix. For the diode rectifier load, the terms L 9 and L contain the switching functions of the diodes S a, S b, and S c, due to which the model in (3c) is nonlinear. However, unlike the switching states of the VSI, which are determined based on the oututs of the feedback controller, the switching states of the diode rectifier deend on the states of the VSI and load ower stages (v abc, i abc, and v D ) and are described in the Aendix. We note that, for the models of the VSI in (a) and (b), as well as for the diode rectifier model in (3c), we assume that the switches are ideal (i.e., they have no ON-state dros and have negligible rise and fall times, as well as no reverse recovery effects in case of the diodes). Combining (a) and (b) with (3a) (3c), the state-sace equations of the overall ower stage can be exressed as where x abc dx abc = A abc _jx abc + B_j abc (4) =[(i abc ) T (v abc ) T (i abc ) T (v abc ) T v D T. The value of subscrit j deends on the switching functions of the VSI. Using Park s transformation [4, we convert the system of equations in the stationary reference frame (4) to the synchronous reference frame. The resulting state-sace equation can be exressed as dx dq = A dq _i xdq + B dq _i (5) where i reresents the switching states of the synchronous reference frame model of the VSI-load system, x dq = T x abc reresent the states of the ower stage in the synchronous reference frame, and T is the Park s transformation matrix. Matrices A dq _i and Bdq _i are defined in the Aendix. B. Power-Stage Model for Periodically Pulsating Loads The state-sace equation (5) describes the dynamics of the VSI when the load does not change dynamically. For timevarying loads (for instance, a eriodically ulsating load, as illustrated in Fig. ), the state-sace (5) has to be modified. Fig.. Schematic illustrating a ulsating load d and its reresentation as a sum of harmonic comonents. For the latter, we ignore harmonic comonents n>. As illustrated, d and d are the two load levels, d is the magnitude of the load ulse erturbation ( d = d d ), T and D are the time eriod and duty ratio of the ulse, resectively. The dynamics of the eriodically-ulsating load (d) can be reresented by a Fourier series as d = a + {a n cos ω n t + b n sin ω n t} (6) n= where the Fourier coefficients are given by a n = (/T ) T d cos(ω n t) and b n =(/T ) T d sin(ω n t), and ω n =(nπ/t ) reresents the frequency of the nth harmonic comonent. The dimension of a n and b n is the same as that of x dq. For the ulsating load in Fig., the exressions for the Fourier coefficients are given as follows [5: a = d a n = sin(nπd ) (d d ) nπ b n = nπ (d d ) { cos(nπd )} where d is the average value of the load, D is the duty ratio of the load ulse, and d and d are the two levels of the ulsating load. Using the exression for d in (6), the state-sace equations of the VSI with eriodic loads can be exressed as dx dq = A dq _i xdq + + B dq _i ( a + ) a n cos ω n t + b n sin ω n t. (7) n= Next, for each frequency comonent in (7), we define an additional state y n = a n cos ω n t + b n sin ω n t. The dynamics of the states of the nth harmonic comonent can be described as [6 y n = a n cos ω n t + b n sin ω n t dy n = a n ω n sin ω n t + b n ω n cos ω n t = y n dy n = a n ωn cos ω n t b n ωn sin ω n t = ω ny n. (8)

4 798 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 Using (8), the state-sace (7) can be modified as dx dq ( ) = A dq _i xdq + B dq _i + a + y n n= with the introduction of additional state-sace equations to describe the dynamics of each harmonic comonent. Using (8) and (9), the augmented state-sace equation of the overall VSIload system can be exressed as where ˆx dq dˆx dq = Âdq _iˆxdq + (9) ˆB dq _i () =[x dq y y y y T [( ˆB dq _i = B dq _i + a A dq  dq _i = ) T _i ω ω The dimensions of the matrices and vectors in () are infinite. However, for ractical uroses, we use only the dominant harmonic comonents for our analyses to reduce comutational overhead. C. Controller Model The closed-loo controller for the VSI-load system is imlemented in the synchronous reference frame, as illustrated in Fig.. The d-axis controller consists of a voltage loo, which generates the reference for the current loo. The reference for the q-axis current loo is internally generated, deending on the tye of load that is connected to the VSI. For all the control loos, conventional linear comensators are used. The oututs of the d-axis and q-axis current loos are modulated using a symmetrical sace-vector modulation scheme [4. Note that the analysis techniques described in this aer can be alied to other modulation schemes as well. As illustrated in Fig., the comensators in the d-axis and q-axis control loos introduce additional states. The state-sace equation of the controller is given by dx c = A cx c + H x dq + B c () where x c =[ξ ξ ξ 3 ξ 4 ξ 5 ξ 6 T are the states of the controller, as shown in Fig. ; B c = [ v dref i qref T ; H is the matrix of the sensor gains (given in the Aendix for the different load tyes); and ω dv K A c = dv K dv ω zdv ω di. ω qi D. Overall Closed-Loo System Model Combining (5) and () for the three loads described in Section II-A and combining () and () for the eriodically ulsating load described in Section II-B, the overall state-sace equation of the system is described as dx = A ix + B i () [ A dq _i [ B dq _i H A c B c [ x dq where x =, A x i = r s, and B i = c H A c [ B c ˆx dq for the loads described in Section II-A, and x =, A x i = [ [ c Âdq _i r s, and B i = ˆBdq _i for the ulsating load described in Section II-B. Here, all of the elements of matrix r s are zeros. For the VSI, the switching conditions of the ower devices in the synchronous reference frame deend on the oututs of the feedback controller states and are given by {, (Kdi ξ S d = 3 + K di ω zdi ξ 4 ) V mod, (K di ξ 3 + K di ω zdi ξ 4 ) V mod > {, (Kqi ξ S q = 5 + K qi ω zqi ξ 6 ) V mod, (K qi ξ 5 + K qi ω zqi ξ 6 ) V mod >. (3) The diode rectifier load introduces additional switching states in the state-sace equations of the overall system, aart from those of the original VSI. These switching states are due to the uncontrolled switching of the diodes and deend on the voltage across the diode. The switching states of the diode rectifier load can be exressed as (4), shown at the bottom of the next age. Variables a 35 a 47 are given in the Aendix. III. SYNTHESIS OF REACHING CRITERION As illustrated in Fig. 3, the dynamics of the three-hase VSI consists of ) the reaching hase, which describes the dynamics of the state trajectories from a given initial condition to the orbit, and ) the steady-state hase, where the error trajectories of the system corresond to a eriodic orbit. To derive the reaching criterion of the VSI-load system, we first translate the state-sace equation of the overall system given by () to the error coordinates using e = x x, where e reresents the error vector and x reresents the steady-state

5 ACHARYA et al.: REACHING CRITERION OF THREE-PHASE VSI WITH PASSIVE AND NONLINEAR LOADS 799 Fig. 3. Schematic illustrating the dynamics of a three-hase VSI (or any switching ower converter, in general) in the reaching and steady-state (corresonding to a eriodic orbit) hases of oeration. Symbol e sw =describes a switching surface, whereas e reresents the dynamics of one of the states of the VSI in error coordinates. values (corresonding to infinite switching frequency) of the states. The modified state-sace equation is given by TABLE I SWITCHING STATES OF THE VSI-LOAD SYSTEM de = A ie + B i (5) where B i = (B i + A i x ). The reaching criterion of the VSI-load system deends on the number of nonreetitive and nonredundant switching sequences generated by the noncomlementary switching functions of the VSI (S d and S q ) and the diode rectifier load (S d and S q ). The ossible switching states of the VSI-load system for the assive and diode rectifier loads are listed in Table I. For the case of the assive load, N =4, whereas, for the diode rectifier load, N =6. The total number of nonreetitive and nonredundant switching sequences is given by [9, [ M = N l= ( ) N C l = N l= ( ( N ) )! l!( N l) (6) where N is the total number of switching states of the overall system. To determine the reaching criterion of the VSI-load system, we define a CLF V k (e) > (for the kth switching sequence, k =,,...,M), which is a weighted sum of the Lyaunov functions (e T P ki e) in each switching state of a given sequence k and is given by [7 V k (e) = h α ki e T P ki e, k =,,...,M (7) i= where h is the number of switching states in a given switching sequence k, α ki are the weights of the Lyaunov functions in each switching state such that α ki and h i= α ki =, and P ki = Pki T is a ositive-definite matrix. Now, taking the derivative of V k (e) in (7) and using (5), we obtain the following: dv k (e) = = h i= ( ) de T α ki P kie + e T de P ki ( h [ e α ki i= T [ A T i P ki + P ki A i P ki B i B T i P ki [ ) e. (8) {, {( + a35 L S d = )v d (r L + a 3 L )i d +(L a 37 )v D L (a 3 i q + a 33 i d + a 34 i q + a 36 v q )}, {( + a 35 L )v d (r L + a 3 L )i d +(L a 37 )v D L (a 3 i q + a 33 i d + a 34 i q + a 36 v q )} > {, {( + a46 L S q = )v q (r L + a 4 L )i q +(L a 47 )v D L (a 4 i d + a 43 i d + a 44 i q + a 45 v d )}, {( + a 46 L )v q (r L + a 4 L )i q +(L a 47 )v D L (a 4 i d + a 43 i d + a 44 i q + a 45 v d )} > (4)

6 8 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 As er Lyaunov s method, the error trajectories of the system converge toward the orbit for finite switching frequency (or the equilibrium oint for infinite switching frequency), rovided that (dv k (e)/) <. This condition is satisfied by (8), rovided that the following matrix inequality is satisfied: TABLE II PARAMETERS OF THE THREE-PHASE VSI USED FOR SIMULATIONS h [ A T α i P ki + P ki A i P ki B i ki B T <. (9) i P ki i= If there are no solutions of P ki for (9), we investigate the dual of V k (e) to confirm that the error trajectories of the VSI do not converge to the orbit [8, [9. We define the dual of V k (e) as V Dk (e) = h λ ki e T Q ki e, k =,,...,M () i= where λ ki, h i= λ ki =, and Q ki = Q T ki is a ositive-definite matrix. For the kth switching sequence, the error trajectories of the VSI do not converge to the orbit, rovided that V Dk (e) satisfies the following criteria: or V Dk (e) = V Dk (e) > h i= and dv Dk (e) > (a) λ ki e T ( Q ki )e< and dv Dk(e) Following (8) and (9), (b) is satisfied, rovided that <. (b) h [ A T λ i Q ki Q ki A i Q ki B i ki B T <. () i Q ki i= If there are no solutions of P ki for (9) but there exist solutions of Q ki for (), we conclude that the system does not satisfy the reaching criterion for orbital existence. IV. STEADY-STATE STABILITY CRITERIA The reaching criterion (9) derived in Section III redicts orbital existence. Once this criterion is satisfied, the next goal is to analyze the stability of the nominal (eriod-) orbit of the VSI-load system. Because the dynamics of this system is fundamentally governed by a line-frequency (slow) and a switching-frequency (fast) [8 comonent, the frequency of the nominal (eriod-) orbit is determined by the least common multile of the two frequency comonents [3, [3. Stability of the nominal orbit can be investigated using the CLF-based aroach (which is exlained here) by investigating the change of (7) over one nominal time eriod. The nominal orbit is stable, rovided that dv k (e)/ =. Such an aroach can account for both the fast-scale and slow-scale dynamics of the system [8. Because the two frequency comonents in our case (and as shown in Table II) are several orders aart, for ractical uroses, we assume that the frequency of the nominal orbit is the same as the slow-frequency comonent and subsequently investigate the change in CLF (7) over this eriod. By equating the left-hand side of (9) to zero, the condition for the stability of the nominal orbit can be exressed as h [ A T α i P ki + P ki A i P ki B i ki B T = (3) i P ki i= where P ki are ositive-definite matrices, and α ki ( α ki and h i= α ki =) deends on the time sent in each switching state and can be obtained using the numerical search algorithm. The switching states of the overall system are given in Table I. If there are no ositive-definite matrices P ki that satisfy (3), we conclude that the nominal orbit is unstable. We note that, unlike the conventional ma-based aroach [8, [3, [33, which ascertain the stability of (5) using a ma that is derived by sequential atching of solutions corresonding to the ith switching state and hence requires knowledge of the modulation scheme, the CLF-based aroach deends only on the switching states of the nominal sequence. Even if the nominal sequence changes (e.g., due to a change in the modulation scheme), (3) is true as long as the switching states in the sequence are the same. In contrast to the CLF-based method, the conventional technique to analyze the orbital (or steady-state) stability of (5) is based on the imedance criterion [ that is derived using a linearized averaged model of the system. According to this criterion, the stability of the VSI-load system is guaranteed, if the inut imedance of the load Z i_load and the closedloo outut imedance of the VSI Z o_inv satisfy the following inequality: Z i_load > Z o_inv. (4) If (4) is violated, then, for certain cases, an extended analysis of the loo gain using the Nyquist criterion needs to be carried

7 ACHARYA et al.: REACHING CRITERION OF THREE-PHASE VSI WITH PASSIVE AND NONLINEAR LOADS 8 Fig. 4. (a) Plot showing the variation of the minimum eigenvalues of the augmented P and Q matrices [obtained by solving LMIs (9) and (), resectively with the load hase angle. The lot shows the reachable and unreachable regions of oeration for the inductive and caacitive loads as well as the conditions for the sliding mode and asymtotic mode of convergence. The oints marked (i) (iv) will be used later to comare the redictions of the reaching criteria with the steady-state stability results. Simulation results illustrating the (b) reachable and (b) unreachable dynamics of the three-hase VSI for four oerating conditions. out to determine that the system is unstable [4, [5. Note that this criterion is based on the averaged model and hence cannot account for the effect of fast-scale (switching-frequency) dynamics on the stability of the overall system. V. R ESULT In this section, we investigate the global stability of the threehase VSI with assive (constant and ulsating) and nonlinear loads using the reaching criterion discussed in Section III and the steady-state stability analysis techniques discussed in Section IV using the CLF-based criterion and the small-signal imedance criterion. Matrices P P h are obtained by solving linear matrix inequality (LMI) (9) (for the reachable case) and matrix equality (3) (when the nominal orbit is stable). Similarly, matrices Q Q h are obtained by solving LMI () for the unreachable case. Because the minimum eigenvalue of a ositive-definite matrix is ositive, we lot the variations of the minimum P eigenvalues of the augmented P =... and P h

8 8 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 Fig. 5. (a) Plot showing the variation of the minimum eigenvalue of the augmented P matrix [obtained by solving matrix equality (3) of a three-hase VSI with load hase angle for oerating conditions where eriod- stability is satisfied. Variations of the small-signal outut imedance of the VSI illustrating (b) stable and (b) unstable characteristics for inductive and caacitive loads, resectively, for the oerating conditions identified in the figure.

9 ACHARYA et al.: REACHING CRITERION OF THREE-PHASE VSI WITH PASSIVE AND NONLINEAR LOADS 83 Fig. 6. (a) Variation of the small-signal outut imedance of the VSI and the inut imedance of the inductive load for a hase angle of φ =35. (b) Simulation results illustrating the d-axis voltage-error of a three-hase VSI with time. Q Q =... matrices to see if the matrices obtained Q h by solving LMIs (9) and (), and matrix equality (3) are ositive definite. The redictions of the reaching criterion and the steady-state stability criteria are verified by timedomain simulations in Simulink. The nominal values of the ower stage and controller arameters are given in Table II. A. Passive Loads First, we investigate the imacts of the load hase angle on the reaching conditions of the three-hase VSI with a assive load. The variations of the minimum eigenvalues of the augmented P and Q matrices [obtained by solving LMIs (9) and (), resectively with the load hase angle are illustrated in Fig. 4(a). Distinction between the sliding and asymtotic modes of convergences can be obtained by using the criteria in [9 and [. Because sliding-mode convergence is tyically faster than asymtotic convergence, this distinction can be used to design controllers with suerior dynamic erformance. The simulation results in Fig. 4(b) and (b) show the evolution of the d-axis voltage-error trajectories for the unreachable and reachable cases, resectively, for the RC and RL loads. In the simulation results, the marker (x) indicates the desired equilibrium (corresonding to zero error at infinite frequency), whereas the arrows indicate how the state-error trajectories evolve with time. For these simulations, any arbitrary initial condition is chosen. These results validate the redictions of the reaching criterion, as illustrated in Fig. 4(a). The result in Fig. 4(a) only describes the reaching conditions for orbital existence of the VSI-load system. To comment on the global stability, we need to further analyze the stability of the nominal (eriod-) orbit. Fig. 5(a) shows the variation of the minimum eigenvalues obtained by solving matrix equality (3). To obtain these results, we examine the energy balance The simulation models of the three-hase inverter have been verified by exerimental results, which have been resented in our revious aers [9, [. of the system over one line cycle. However, because, in this case, the switching frequency is not an integer multile of the line frequency, there is a small discreancy in the energy balance over one line cycle. To avoid the large comutational overhead associated with comuting the energy balance over a long eriod of time, we consider that the nominal orbit of the system is stable if the change in V k (e) over one line cycle is bounded by a small value ε [35. For the cases considered in this aer, these results agree with the results of the small-signal imedance criterion, given by (4), as illustrated in Fig. 5(b) and (b), for the inductive and caacitive loads. Comarison of the redictions of orbital stability (Fig. 5) with those of the reaching criterion [Fig. 4(a) indicates a discreancy. For the inductive load, there are three arameterdeendent regions, i.e., the reachable and stable (φ <3 ), unreachable but stable (φ =35 ), and unreachable and unstable (φ >35 ) regions. For the oerating condition corresonding to φ =35 [marked (iii) in Figs. 4(a) and 5(a), the steadystate analysis techniques redict that the system is stable, as illustrated in Figs. 5(a) and 6(a). However, the reaching criterion is not satisfied for this case. Using time-domain simulations, Fig. 6(b) illustrates that the error trajectories reach the orbit, rovided that the initial condition lies within its vicinity. However, for any other arbitrary initial condition, which is not near the orbit, the error trajectories do not converge, as illustrated in Fig. 4(b) for the corresonding case. These results demonstrate the need for reaching condition analyses, in addition to the steady-state orbital stability to redict the global stability of the VSI-load system. Next, we investigate the imacts of load variations on the reaching conditions of the overall system for two different values of the d-axis voltage-loo comensator gain K dv.fig.7 illustrates that the region of reachable oeration shrinks with decrease in the load ower. In addition, the region of reachable oeration further reduces as K dv is increased. By investigating the effects of other controller arameters on the reaching conditions, one can determine aroriate controller arameters for the VSI-load system for different alications. Finally, we investigate whether the rate of change of the load has any imact on the reaching conditions of the system. Here, we note

10 84 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 Fig. 7. Variation of the reaching conditions of the VSI-load system with changes in the load ower and the d-axis voltage-loo comensator gain for (a) RL and (b) RC loads. Fig. 8. Simulation results illustrating the dynamics of the three-hase VSI for a 5% % (of the full load) load transient for two different slew rates: (a) and (b) 4 A/s. that both load levels satisfy the reaching criterion for orbital existence. Fig. 8 illustrates two cases: The rates of change of the load are ) A/s and ) 4 A/s. For both cases, the initial and final load conditions satisfy LMI (9), i.e., the dynamics are reachable. From Fig. 8, we observe that the reaching condition does not change with ste variation of the rate of change of load. This result is true as long as the rate of change of the load is sufficiently slower than the switching frequency of the VSIload system. B. Pulsating Load In the revious section, we investigated the imacts of load variations on the reaching conditions of a VSI with assive loads. In this section, we investigate the imacts of a ulsating RL load (as shown in Fig. ) on the reaching conditions of such a system. The ulsating load is reresented by a sum of harmonic comonents, as described in Section II-B. For the case considered in this aer, we observe that the ulsating load is adequately described by harmonic comonents from n = to n =. Therefore, to avoid a higher comutation We note that similar analyses can be erformed for ulsating RC and diode rectifier loads as well. Fig. 9. Variation of the maximum allowable duty ratio (corresonding to the reachable-region boundary) with variations of the load erturbation magnitude and frequency. The duty ratio of the ulsating load is defined as the ratio of the time sent at load level d with resect to load level d. overhead, we ignore the higher order harmonic comonents. Fig. 9 illustrates the variation of the maximum allowable duty ratio, which corresonds to the reachable-region boundary, with variations of the magnitude and frequency of the ulsating load. The region below this surface corresonds to the reachable region. From the figure, we observe that the reachable region

11 ACHARYA et al.: REACHING CRITERION OF THREE-PHASE VSI WITH PASSIVE AND NONLINEAR LOADS 85 Fig.. (a) Variations of the maximum allowable duty ratio with erturbation magnitude for f =Hz and f = Hz. (b) and (c) Simulation results illustrating the dynamics of the three-hase VSI with a ulsating load for (b) duty ratio = 5%and (c) duty ratio = 5%. For the simulation results in (b) and (c), d =A, and f =Hz. can vary with the magnitude and frequency of the load ulse. In other words, the reachable region significantly deends on the amount of time sent at each load level of the ulsating load. Fig. (a) illustrates the variation of the reachable-region boundary with the magnitude of the ulsating load for two oerating frequencies (f =Hz and f = Hz). Here, one of the load levels d satisfies the reaching conditions (from the results in Section V-A), whereas the other load level d does Fig.. (a) Variations of the maximum allowable duty ratio with erturbation frequency for ulsating load magnitudes of d =A, d =8A, and d =6A. (b) and (c) Simulation results illustrating the dynamics of the threehase VSI with a ulsating load for (b) f =Hz and (c) f = Hz. For the simulation results in (b) and (c), d =A, and D = 5%. not satisfy the reaching condition. We observe that, for low duty ratios, the system satisfies the reaching condition for orbital existence. In other words, if the time sent at load level d is small, the state-error trajectories reach the orbit from an arbitrary initial condition. The simulation results in Fig. (b) and (c) validate these redictions. Fig. also indicates that the reachable region can vary with the frequency of the load ulse. We lot the variation of the reachable region with the load ulse frequency in Fig..

12 86 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 Fig.. Variations of the maximum allowable duty ratio with erturbation magnitude for K dv =5, K dv =5,andK dv =. From Fig., we observe that, for high magnitudes of the ulsating load amlitude, the region of reachable oeration significantly reduces for low ulsating load frequencies. For the case considered in this aer, the region of reachable oeration does not change with frequency for ulsating load magnitudes lower than d =6A. The simulation results in Fig. (b) and (c) validate these redictions. We note that the results resented so far in this section are for the nominal arameters of the closed-loo controller given in Table II. The region of reachable oeration can also be altered by variations of the controller arameters. For instance, Fig. illustrates the variation of the maximum allowable duty ratio of the ulsating load with variations of its magnitude for three different values of the d-axis voltage-loo gain K dv.we observe that the region of reachable oeration increases as K dv decreases. Similar analyses can be erformed for other controller arameters as well, and aroriate controller arameters can be chosen, deending on the alication. C. Diode Rectifier Load Next, we investigate how the reaching criterion can be extended to the case of a nonlinear diode rectifier load. The PWL model of the overall system (comrising the three-hase VSI and the diode rectifier) is described in Section II. For the diode rectifier load, there are additional switching states due to the voltage-deendent switching of the diodes. Fig. 3 illustrates two simulation results that are obtained by imlementing the PWL models of the overall system. Clearly, for one case, the d-axis voltage-error trajectories converge to the orbit, whereas, for the other case, they do not. As for assive loads, the simulation results agree with the redictions of the reaching criterion derived in Section III. Fig. 4 illustrates the variations of the minimum eigenvalues of the augmented P and Q matrices [obtained by solving LMIs (9) and (), resectively with variation of the VSI outut voltage (v a_, v b_, v c_) total harmonic distortion (THD) for the diode rectifier load. As in the case of assive loads, distinction among the sliding-mode and asymtotic mode of convergence is obtained using the criteria described in [. The THD is varied by changing the values of the load inductors (L a, L b, and L c ) and the outut caacitor (C D ) of the diode rectifier load. In Fig. 4, Fig. 3. Simulation results illustrating the d-axis voltage-error trajectories for a three-hase VSI connected to a diode rectifier with a voltage-loo gain of K dv = 5 for the following load THDs: (a) 3.5% and (b) 9.8%. Fig. 4. Plot showing the variation of the minimum eigenvalues of the augmented P and Q matrices [obtained by solving LMIs (9) and (), resectively with the outut-voltage THD for a three-hase VSI with diode rectifier load. The solid and dashed lines corresond to the reachable and unreachable regions, resectively. the range of THD is thus chosen, because it accounts for the reachable and unreachable regions of oeration. Time-domain results are only illustrated for one reachable case and one unreachable case. Above a THD of 9%, the dynamics are exected to be unreachable for the VSI-load system considered in this aer. Comarison of these results with those in Fig. 4(a) illustrates that, for a given value of voltage-loo gain K dv (for a load-hase angle φ =3 and an aarent ower of.5 kva),

13 ACHARYA et al.: REACHING CRITERION OF THREE-PHASE VSI WITH PASSIVE AND NONLINEAR LOADS 87 Fig. 5. Variation of the reaching conditions of the VSI-load system with changes in the load ower and the d-axis voltage-loo comensator gain for a diode rectifier load. the reachable region for the assive load is higher than that for the nonlinear loads. Furthermore, variation of the reaching conditions of the VSI with load ower and the gain of the d-axis voltage-loo comensator is illustrated in Fig. 5. From Fig. 5, we observe that the region of reachable oeration shrinks with decrease in the load ower and is also deendent on K dv. Similar analyses of the reaching conditions with other controller arameters can hel in designing controllers that ensure the global stability of the VSI-load system for different oerating conditions. Finally, the steady-state stability of the system can be redicted by using the CLF-based aroach [given by (3), as illustrated in Fig. 6(a), and the imedance criterion [given by (4), as illustrated in Fig. 6(b). The small-signal imedance of the diode rectifier is obtained using the imedance matching rocedure described in [36. In Fig. 6(b), we illustrate only the ositive-sequence small-signal imedance. The negativesequence and zero-sequence small-signal imedances also yield similar results. For this case, the region of steady-state stable oeration and the region of reachable oeration are the same. VI. SUMMARY AND CONCLUSION We develo a methodology to investigate the imacts of assive (RL and RC) and nonlinear (diode rectifier) loads on the reaching conditions of a three-hase VSI. Using CLFs, the reaching criterion is develoed for PWL models of the VSI and the loads. The redictions of the reaching criterion are validated by time-domain simulation results. Furthermore, the CLF-based aroach is extended to investigate the stability of the nominal (eriod-) orbit. Such a technique can be used to redict the fast-scale as well as slow-scale dynamics of the system without the comutational overhead of solving comlex nonlinear mas. Comarisons of the redictions of the reaching criterion with those of the steady-state stability analyses (based on CLF as well as the small-signal imedance criterion) demonstrate the need for the reaching criterion to redict the global stability of the VSI-load system. Using some secific examles, we demonstrate how, under certain oerating conditions, although the system satisfies the steady-state stability criteria, its reaching conditions are not guaranteed. The simulation results show that the state-error trajectories will aroach the equilibrium orbit, rovided that the initial condition lies within Fig. 6. (a) Variation of the minimum eigenvalue of the augmented P matrix [obtained by solving (9) of a three-hase VSI (with a voltage-loo gain of K dv = 5) with the THD for oerating conditions where the system exhibits eriod- stability. (b) Variation of the small-signal outut imedance of the VSI and the inut imedance of the diode rectifier load for the cases marked (v) and (vi) in Figs. 4 and 6(a). its vicinity; however, its reachability from any arbitrary initial condition is not guaranteed. This discreancy is because the steady-state stability analysis techniques assume the existence of an orbit. However, for global stability of the system, it is necessary to first ascertain orbital existence (which can be redicted by the reaching criterion described in this aer).

14 88 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 Furthermore, the redictions of the reaching criterion for the VSI-load system indicate that the range of reachable oerating conditions can significantly vary for different tyes of loads. For instance, for the same aarent ower and load hase angles, higher values of voltage-loo gains are required to ensure the reachability for the case of the diode rectifier load, as comared to the assive loads. We observe that variations of the nature of the load and the ower drawn by it can also have an imact on the reaching conditions of the overall system. Furthermore, we demonstrate that, for eriodically ulsating loads, the reaching conditions can vary (comared to reaching condition redictions with constant loads) with changes in the duty ratio of the load ulse. The analysis in this aer demonstrates that the reaching conditions of the system vary with changes in the voltage-loo comensator gains. Similar results can be obtained for the variation of other controller arameters as well. Thus, the redictions of the reaching criterion can be used to select controller gains and bandwihs that increase the region of globally stable oeration of the VSI for different load/oerating conditions. However, the analysis rocedure resented in this aer has to be further modified to account for smooth nonlinear loads (for instance, an inverter sulying a motor load), where the system equations are described by iecewise nonlinear models. The analysis rocedure can also be extended to other ower converter tyes as well as interconnected networks of converters, such as arallel or cascaded connection, with aroriate modifications to account for the interconnection effects. These issues are art of our current research focus. APPENDIX DEFINITIONS OF MATRICES FOR THE PWL MODELS OF THE THREE-PHASE VSI WITH DIFFERENT LOAD TYPES A. Inductive Load For the inductive load, the definitions of the various vectors and matrices that are described in Section II are given as follows: i abc = i abc = i a_ i b_ i c_ i a_ i b_ i c_ v abc = v abc = v a_ i b_ i c_ v a_ i b_ i c_ (r L+r C ) L (r L+r C ) L K = (r L+r C ) L K = r C L r C L r C L L K 3 = L K 4 = 3 3 L K 5 = 3L (S a S b S c ) 3L (S b S c S a ) 3L (S c S a S b ) C K 6 = C C K 7 = C C C K = 3. r C L = r C L = r C K 8 = 3 3 K 9 = 3 3 (R RL+r C ) L (R RL+r C ) L (R RL+r C ) L 3 = and L 5 L are 3 3. The overall state-sace equations of the system in the synchronous reference frame for the inductive load are described by (). The definitions of the vectors and matrices in () are shown at the bottom of the next age. The sensor gain matrix is given by H dv H H = di. H qi B. Caacitive Load For the caacitive load, the definitions of the various vectors and matrices that are described in Section II are given as follows: i abc = i a_ i b_ v abc = v a_ i b_ i abc = i c_ i a_ i b_ i c_ v abc = i c_ v a_ i b_ i c_ L (r L+r C ) L (r L+r C ) L K = (r L+r C ) L K = r C L r C L r C L

15 ACHARYA et al.: REACHING CRITERION OF THREE-PHASE VSI WITH PASSIVE AND NONLINEAR LOADS 89 L K 3 = L K 4 = 3 3 L K 5 = 3L (S a S b S c ) 3L (S b S c S a ) 3L (S c S a S b ) C K 6 = C C C K 7 = C C K = 3. C RC L 4 = C RC C RC and L L 3 and L 5 L are 3 3. K 8 = 3 3 K 9 = 3 3 The overall state-sace equations of the system in the synchronous reference frame for the caacitive load are described by (). The definitions of the vectors and matrices in () are shown at the bottom of the age. The sensor gain matrix is given by H dv H = H di. H qi C. Diode Rectifier Load For the diode rectifier load, the definitions of the various vectors and matrices that are described in Section II are given as follows: i abc = i abc = [ ia_ i b_ i [ c_ ia_ i b_ i c_ v abc = v abc = [ va_ i b_ i [ c_ va_ i b_ i c_ Inductive Load: x dq (t) =[i d i q i d i q v d v q T [ ( 4 B dq_i = S d ) ( 3L S 4 q V in S q ) T 3L S d V in r L (r L + r C ) ω C L L ω r L (r L + r C ) C L L r C L A dq_i = RL (R RL + r C ) ω r C ω (R RL + r C ) C C ω C C ω Caacitive Load: x dq (t) =[i d i q v d v q v d v q T ( 4 B dq_i =[ 3L Sd S ) ( 4 q Vin 3L Sq S ) d Vin T A dq_i L (r L + R RCr c ( ) RRC C = R RC +r c ( ) RRC C RC R RC +r c ω ) ω L (r L + R RCr c R+r c R RC +r c ) ( ) RRC C R RC +r c ( ) C RC R RC +r c ( ) R RC L (R RC +r c ) r c L R RC +r c ( R RC L (R RC +r c ) L C (R RC +r c ) ω C (R RC +r c ) ) r c R RC +r c ω C (R RC +r c ) C (R RC +r c ) C RC (R RC +r c ) C RC (R RC +r c ) ω RRC C RC (R RC +r c ) ω C RC (R RC +r c )

16 8 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 (r L+r C ) L (r L+r C ) L K = (r L+r C ) L K = r C L r C L r C L L K 3 = L K 4 = 3 3 K 5 = L 3L (S a S b S c ) 3L (S b S c S a ) 3L (S c S a S b ) C K 6 = C C K 7 = C C C K = 3. r C L L 7 = r C L r C L r L L r L 8 = L L r L L 3L (S a S b S c ) L 9 = 3L (S b S c S a ) 3L (S c S a S b ) L =[ S a S b S c C D C D C D K 8 = 3 3 K 9 = 3 3 L = /R D C D, and L L 6 are 3 3. The switching functions of the diode rectifier load can be described as follows: {, va3 v D > S a =, v a3 v D <, v a3 v D = {, vb3 v D > S b =, v b3 v D <, v b3 v D = {, vc3 v D > S c =, v c3 v D <, v c3 v D = where v a3 = v a L di a_ v b3 = v b L di b_ i a_r L i b_r L di c_ v c3 = v c L i c_r L. To test the efficacy of the PWL model, we comare the states of this model with that redicted by Saber, which is a standard Fig. 7. Comarison of the outut voltage and inut current of the diode rectifier redicted by the PWL model with those redicted by a Saber [37 simulation of the diode rectifier load. simulation ackage for ower electronics simulations. We note that, in the Saber model, we use ideal diodes (i.e., that they have zero ON-state dro, negligible rise and fall times, and no reverse recovery). Fig. 7 illustrates that there is a one-to-one match between the two models. We note here that, for both the models, we choose a fixed samling ste (i.e., µs). The overall state-sace equations of the system in the synchronous reference frame for the diode rectifier load are described by (). The definitions of the vectors and matrices in () are given as follows: x dq (t) =[i d i q i d i q v d v q v D T B dq_i = [ S d S L V q in L V in T a a a 3 a 4 a 5 a 6 a 7 a a a 3 a 4 a 5 a 6 a 7 a 3 a 3 a 33 a 34 a 35 a 36 a 37 A dq_i = a 4 a 4 a 43 a 44 a 45 a 46 a 47 a 5 a 5 a 53 a 54 a 55 a 56 a 57 a 6 a 6 a 63 a 64 a 65 a 66 a 67 a 7 a 7 a 73 a 74 a 75 a 76 a 77 where the elements of A dq_i are given as follows: a = ) r C (r L + L +ω rc C ( C r ) C a = ω L ( + ω rc C )

17 ACHARYA et al.: REACHING CRITERION OF THREE-PHASE VSI WITH PASSIVE AND NONLINEAR LOADS 8 r C a 3 = L +ω rc C a 5 = L +ω rc C ( a 7 = a = ω a 4 = L ωr C C ( + ω r C C ) ωr C C a 6 = L ( + ω rc C ) C rc ) L ( + ω rc C ) a = ) r C (r L + L +ω rc C a 3 = L ωr C C ( + ω r C C ) a 4 = L r C ( + ω r C C ) a 5 = ωr C C L ( + ω r C C ) a 6 = a 7 = a 3 = r C L ( + ω rc C ) a 3 = ωrc C L ( + ω rc C ) a 33 = ( r C L +ω rc + r L + 3 ) R L r C S C d R L + r C a 34 = ( ωl 3 R L r C ωrc S d S q C ) L R L + r C ( + ω rc C ) a 35 = L ( + ω r C C ) a 36 = L ωr C C ( + ω r C C ) a 37 = R L S d a 4 = ωrc C L R L + r C L ( + ω rc C ) a 4 = r C L ( + ω rc C ) a 43 = ( ωl + 3 R D r C ωrc S d S q C ) L R D +r C ( + ω rc C ) a 44 = ( ) r C L ( + ω rc C ) + r L a 45 = ωr C C L ( + ω rc C ) a 46 = L ( + ω rc C ) ( ) a 47 = R D L S q a 5 = C ( + ω r C C ) ωr C a 5 = ( + ω rc C ) a 53 = C ( + ω rc C ) ωr C a 54 = ( + ω rc C ) a 55 = ω r C C ( + ω rc C ) ω ωr C a 56 = ( + ω rc C ) a 57 = a 6 = ( + ω rc C ) a 6 = ( ) ωr C C ( + ω rc C ) a 63 = ( + ω rc C ) ω a 64 = C ( + ω rc C ) a 65 = ( + ω rc C ) a 66 = ω r C C ( + ω r C C ) a 67 = R D a 7 = a 7 = a 73 = 3 S d R D C D + r C C D R D a 74 = 3 S q a 75 = a 76 = R D C D + r C C D a 77 =. R D C D + r C C D The sensor gain matrix is given by H dv H H = di. H qi ACKNOWLEDGMENT Any oinions, findings, conclusions, or recommendations exressed herein are those of the authors and do not necessarily reflect the views of NSF and ONR. REFERENCES [ R. D. Middlebrook, Inut filter considerations in design and alication of switching regulators, in Advances in Switch-Mode Power Conversion, nd ed. Pasadena, CA: TESLAco, 983, [ S. S. Kelkar and F. C. Lee, A novel inut filter comensation scheme for switching regulators, in Proc. IEEE Power Electron. Sec. Conf., Jun. 98, [3 Y. E. Sandra and W. A. Polivka, Inut filter design criteria for currentrogrammed regulators, IEEE Trans. Power Electron., vol. 7, no.,. 43 5, Jan. 99. [4 L. R. Lewis, B. H. Cho, F. C. Lee, and B. A. Carenter, Modeling, analysis and design of distributed ower systems, in Proc. IEEE Power Electron. Sec. Conf., Jun. 989, vol., [5 C. M. Wildrick, F. C. Lee, B. H. Cho, and B. Choi, A method of defining the load imedance secification for a stable distributed ower system, IEEE Trans. Power Electron., vol., no. 3,. 8 85, May 995. [6 X. Wang, R. Yao, and F. Rao, Three-ste imedance criterion for smallsignal stability analysis in two-stage DC distributed ower systems, IEEE Power Electron. Lett., vol., no. 3, , Se. 3. [7 S. Chandrasekaran, D. Borojevic, and D. K. Lindner, Inut filter interaction in three-hase AC-DC converters, in Proc. IEEE Power Electron. Sec. Conf., Jun. 999, vol., [8 S. K. Mazumder, A. H. Nayfeh, and D. Boroyevich, Theoretical and exerimental investigation of the fast- and slow-scale instabilities of a DC- DC converter, IEEE Trans. Power Electron., vol. 6, no.,. 6, Mar.. [9 R. Zhang, High erformance ower converter systems for nonlinear and unbalanced load/source, Ph.D. dissertation, Det. Elect. and Comut. Eng. Virginia Polytechnic Inst. and State Univ., Blacksburg, VA, Nov [ H. R. Visser and P. P. J. van den Bosch, Modelling of eriodically switching networks, in Proc. IEEE Power Electron. Sec. Conf., Jun. 99, [ R. Leyva, I. Queinnec, S. Tarbouriech, C. Alonso, and L. Martinez- Salamero, Large-signal stability in high-order switching converters, in Proc. IEEE Amer. Control Conf., Jun. 4,

18 8 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 4, APRIL 8 [ D. Casadei, J. Clare, L. Emringham, G. Serra, A. Tani, A. Trentin, P. Wheeler, and L. Zarri, Large-signal model for the stability analysis of matrix converters, IEEE Trans. Ind. Electron., vol. 54, no., , Ar. 7. [3 C. C. Chan and K.-T. Chau, A fast and exact time-domain simulation of switched-mode ower regulators, IEEE Trans. Ind. Electron., vol. 39, no. 4, , Aug. 99. [4 S. Ben-Yaakov, S. Glozman, and R. Rabinovici, Enveloe simulation by SPICE-comatible models of electric circuits driven by modulated signals, IEEE Trans. Ind. Electron., vol. 47, no.,. 5, Feb.. [5 R. W. Erickson, S. Cuk, and R. D. Middlebrook, Large-signal modelling and analysis of switching regulators, in Proc. IEEE Power Electron. Sec. Conf., Jun. 98, [6 V. Utkin, Sliding Modes in Control Otimization. New York: Sringer- Verlag, 99. [7 A. F. Filiov, Differential Equations With Discontinuous Right Hand Sides. Dordrecht, The Netherlands: Kluwer, 988. [8 D. Milosavljevic, General conditions for the existence of a quasi-sliding mode on the switching hyerlane in discrete variable structure systems, Autom. Remote Control, vol. 46, no. 3, , 985. [9 S. K. Mazumder and K. Acharya, Multile Lyaunov function based reaching condition analyses of switching ower converters, in Proc. IEEE Power Electron. Sec. Conf., Jun. 6, [ S. K. Mazumder and K. Acharya, Multile Lyaunov function based reaching condition for orbital existence of switching ower converters, IEEE Trans. Power Electron., May 8, to be ublished. [ G. Escobar, P. Mattavelli, A. M. Stankovic, A. A. Valdez, and J. Leyva- Ramos, An adative control for UPS to comensate unbalance and harmonic distortion using a combined caacitor/load current sensing, IEEE Trans. Ind. Electron., vol. 54, no., , Ar. 7. [ G. Willmann, D. F. Coutinho, L. F. A. Pereira, and F. B. Libano, Multile-loo H-infinity control design for uninterrutible ower sulies, IEEE Trans. Ind. Electron., vol. 54, no. 3,. 59 6, Jun. 7. [3 M. Dai, M. N. Marwali, J. W. Jung, and A. Keyhani, Power flow control of a single distributed generation unit with nonlinear local load, in Proc. IEEE Power Syst. Conf. Exo., Oct. 4, vol., [4 S. K. Mazumder, A novel discrete control strategy for indeendent stabilization of arallel three-hase boost converters by combining sacevector modulation with variable-structure control, IEEE Trans. Power Electron., vol. 8, no. 4,. 7 83, Jul. 3. [5 [Online. Available: htt://mathworld.wolfram.com/fourierseries.html [6 R. Marino, G. L. Santosuosso, and P. Tomei, Robust adative comensation of biased sinusoidal disturbances with unknown frequency, Automatica, vol. 39, no., , Oct. 3. [7 M. Johansson, Piecewise Linear Control Systems. New York: Sringer- Verlag,. [8 G. Feng, Stability analysis of iecewise discrete-time linear systems, IEEE Trans. Autom. Control, vol. 47, no. 7,. 8, Jul.. [9 R. Goebel, T. Hu, and A. R. Teel, Dual matrix inequalities in stability and erformance analysis of linear differential/difference inclusions, in Current Trends in Nonlinear Systems and Control. Boston, MA: Birkhauser, 5. [3 S. K. Mazumder, A. H. Nayfeh, and D. Boroyevich, A new aroach to the stability analysis of boost ower-factor-correction circuits, J. Vib. Control Secial Issue Nonlinear Dynamics Control, vol. 9, no. 7, , 3. [3 A. H. Nayfeh and B. Balachandran, Alied Nonlinear Dynamics. New York: Wiley, 995. [3 Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, S. Banerjee and G. C. Verghese, Eds. New York: IEEE Press,. [33 C. K. Tse, Comlex Behavior of Switching Power Converters. New York: CRC Press, 4. [34 K. Acharya, S. K. Mazumder, and M. Tahir, Communication faulttolerant wireless network control of a load-sharing multihase interactive ower network, in Proc. IEEE Power Electron. Sec. Conf., Jun. 6, [35 V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Practical Stability of Nonlinear Systems. Hackensack, NJ: World Scientific, 99. [36 Z. Bing, K. Karimi, and J. Sun, Inut imedance modeling and analysis of line-commutated rectifiers, in Proc. IEEE Power Electron. Sec. Conf., Jun. 7, [37 [Online. Available: htt:// saber/saber.html Kaustuva Acharya (S ) received the B.Eng. degree in electronics and communication engineering from the Regional Engineering College (now the National Institute of Technology), Bhoal, India, in and the M.Sc. degree in electrical engineering from the University of Illinois, Chicago, in 3. He is currently working toward the Ph.D. degree in electrical engineering at the University of Illinois. He is a Research Assistant in the Laboratory for Energy and Switching-Electronics Systems, Deartment of Electrical and Comuter Engineering, University of Illinois. He has ublished more than 5 refereed international journal and conference aers. His research interests include ower electronics for renewable and alternate energy sources, and modeling, analysis, and control of interactive ower networks for distributed ower systems. Mr. Acharya is a Reviewer for the IEEE TRANSACTIONS ON POWER ELECTRONICS and the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, and several international conferences. He coresented a tutorial entitled Global stability methodologies for switching ower converters at the IEEE Power Electronics Secialists Conference 7. Sudi K. Mazumder (SM ) received the M.S. degree in electrical engineering from Rensselaer Polytechnic Institute (RPI), Troy, NY, in 993, and the Ph.D. degree in electrical and comuter engineering from the Virginia Polytechnic Institute and State University, Blacksburg, in. He is currently with the Deartment of Electrical and Comuter Engineering, University of Illinois, Chicago, as an Associate Professor and the Director of Laboratory for Energy and Switching-Electronics Systems. He has more than ten years of rofessional exerience and has held R&D and design ositions in leading industrial organizations. He has been the Editor-in-Chief for the International Journal of Power Management Electronics since 6. He has ublished more than 6 refereed and invited journal and conference aers and is a Reviewer for six international journals. His current research interests are interactive owerelectronics/ower networks, renewable and alternate energy systems, hotonically triggered ower semiconductor devices, and systems-on-chi/module (i.e., SoC/SoM). Dr. Mazumder has also been an Associate Editor for the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS since 3 and was the Associate Editor for the IEEE POWER ELECTRONICS LETTERS until 5. He has been invited by both the IEEE and the American Society of Mechanical Engineers for several keynote and lenary lectures. He resented a tutorial entitled Global stability methodologies for switching ower converters at the IEEE Power Electronics Secialists Conference 7. He was the reciient of the Faculty Research Award (8) and Diamond Award (6) from the University of Illinois for his outstanding research erformance, the ONR Young Investigator Award (5), NSF CAREER Award (3), the DOE SECA Award (), and the Prize Paer Award () from the IEEE TRANSACTIONS ON POWER ELECTRONICS and the IEEE Power Electronics Society. He was a coreciient (with his Ph.D. student M. Tahir) of the 7 IEEE Outstanding Student Paer Award at the IEEE International Conference on Advanced Information Networking and Alications. Ishita Basu (S 6) received the B.Sc. degree in instrumentation and electronics engineering from Jadavur University, Kolkata, India, in 5. She is currently working toward the Ph.D. degree in electrical engineering from the Laboratory for Energy and Switching-Electronics Systems, Deartment of Electrical and Comuter Engineering, University of Illinois, Chicago.