IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 3, SEPTEMBER

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1 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 3, SEPTEMBER Advanced Integated Modeling and Analysis fo Adjustable Speed Dives of Induction Motos Opeating With Minimum Losses Antonio T. Alexandidis, Membe, IEEE, Geoge C. Konstantopoulos, Membe, IEEE, and Qing-Chang Zhong, Senio Membe, IEEE Abstact The nonlinea induction moto model is appopiately integated by incopoating the dynamics of the powe electonic convete in a manne that pemits the design of stable field-oiented contol (FOC) opeating with minimum losses. As aleady poven, the challenging issue of opeating the induction machine with minimum coppe losses equies a vaying oto flux opposed to the standad FOC technique, which keeps the oto field magnitude constant and tacks the electic toque to the desied level. To this end, exploiting the Hamiltonian stuctue of the developed moto/convete model, an innovated nonlinea contolle is poposed that guaantees the technical limits of the convete (linea modulation) and simultaneously opeates unde FOC at steady state to achieve accuate speed egulation with vaying oto flux accoding to the minimal losses equiements. Unde these cicumstances, the conventional FOC stability analysis does not hold anymoe, and theefoe fo the fist time, a new igoous analysis is povided that poves stability and convegence to the desied equilibium fo the complete closed-loop moto convete system. Finally, the theoetical contibution is examined in compaison to the taditional FOC opeation by simulations obtained fo an industial size induction moto, while it is futhe evaluated by eal-time esults of a moto with simila paametes. Index Tems Convete-machine modeling, induction moto, minimum losses, nonlinea contol, stability. I. INTRODUCTION THE THREE-PHASE induction moto epesents one of the most commonly used electic machines in industial applications. The integation of suitable powe electonic devices, paticulaly ac/dc voltage souce convetes (VSC), have decisively inceased the aea of applications fo the induction machine and has opened a new field in design and analysis. Vecto o field-oiented contol (FOC), though complex and demanding Manuscipt eceived May 6, 04; evised Novembe 4, 04; accepted Decembe 7, 04. Date of publication Febuay 5, 05; date of cuent vesion August 8, 05. This wok was patially suppoted by the Engineeing and Physical Sciences Reseach Council, U.K., unde Gant EP/J0558X/. Pape TEC A. T. Alexandidis is with the Depatment of Electical and Compute Engineeing, Univesity of Patas, Patas 6500, Geece ( a.t.alexandidis@ ece.upatas.g). G. C. Konstantopoulos is with the Depatment of Automatic Contol and Systems Engineeing, Univesity of Sheffield, Sheffield, S 3JD, U.K. ( g.konstantopoulos@sheffield.ac.uk). Q.-C. Zhong is with the Depatment of Automatic Contol and Systems Engineeing, Univesity of Sheffield, Sheffield, S 3JD, U.K., and also with the Depatment of Electical and Compute Engineeing, Illinois Institute of Technology, Chicago, IL 6066 USA ( zhongqc@ieee.og). Colo vesions of one o moe of the figues in this pape ae available online at Digital Object Identifie 0.09/TEC technique, emains a poweful tool fo adjustable induction moto speed dives [], [], since it esults in constant oto flux magnitude that simplifies the electomagnetic toque expession as in a sepaately excited dc moto. On the othe hand, the moto efficiency enhancement, esulting fom the eduction of coppe losses, has become a cucial opeating task caused by envionmental o special application easons such as wind tubine o electic vehicle efficient opeation [3] [9]. As it has been shown [0] [7], powe losses minimization cetainly equies a vaying oto flux magnitude in accodance to the paticula opeating point. Additionally, in all induction moto applications, stability plays a cucial ole in system opeation and should be always guaanteed. Seveal eseaches have poposed FOC methods to guaantee stability of the induction moto using the educedode cuent-fed model of the moto, i.e., by consideing a thid-ode system with states the oto fluxes and the moto speed, while the stato cuents ae the contol inputs [8] []. Adding to the model analysis the stato cuent dynamics, asymptotic stability has been poven only unde the use of paamete o load toque estimatos o adaptation mechanisms [] [9]. This means that additional dynamic designs ae needed, while when the induction moto should opeate with minimum losses, it becomes a cumbesome task to conduct a simila stability analysis, since the contolle opeation is fa fom the conventional FOC design. Futhemoe, in all the existing liteatue, the convete dynamics, though cucial fo the system stability, ae fully omitted, often because of thei nonlinea stuctue which inceases the difficulty [30] [3]. It is theefoe obvious that a complete system modeling is equied that takes into account the nonlinea stuctue of both the VSC and the induction moto. Moeove, on this complete model, simple contol designs without additional dynamic pats that cannot be easily implemented have to be developed, while thei dynamic pefomance should cetainly ensue system stability unde opeation with constant o vaying flux magnitude. In this fame, some ealy esults have been poposed by the authos in [33] fo a standad FOC appoach. In this study, a complete VSC-fed induction moto dive is consideed. Using advanced nonlinea Hamiltonian modeling and aveage analysis [7], [3], [33], the complete nonlinea dynamic model of the system is obtained in a seventh-ode state-space nonlinea fom that includes both the ac moto and the VSC dynamics in the synchonously otating d-q efeence fame [3], [33]. The contolled inputs ae diectly the VSC IEEE. Pesonal use is pemitted, but epublication/edistibution equies IEEE pemission. See standads/publications/ights/index.html fo moe infomation.

2 38 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 3, SEPTEMBER 05 duty-atio d-q components while the supplied voltage and the load toque ae consideed as extenal inputs with the second one consideed to be completely unknown with step-vaying magnitude. Adopting the analysis pesented in [7, Sec ], it is fist concluded that in ode to achieve minimum coppe losses, the squae of the oto flux magnitude should be linealy dependent fom the electic toque. Extending the method unde FOC steady-state opeation, it is poven that the d-q duty-atio input components can be used fo the speed egulation though the q cuent tacking, while the moto efficiency takes its maximum value by egulating the d cuent component in a constant atio with espect to the q cuent component. Howeve, since in this case, the classical vecto contol analysis cannot guaantee stability with the application of simple Popotional-Integal (PI) contolles [], [], [], [3], the main contibution of the pesent wok is that the poposed system modeling pemits the design of new simple nonlinea dynamic contolles, capable of ovecoming these dawbacks. As poven in the pesent pape, the closed-loop system is stable and conveges to the desied equilibium with the oto flux magnitude following the load vaiations in a manne that ensues minimum losses and field oientation at steady state. It is emakable that an intemediate esult fom the adopted minimum losses analysis is that of constant slip fequency equiement unde vaious load conditions. It is also significant to note that in this case, as it is shown in the pape, stability is achieved without needing any flux magnitude and angle estimation while the contolle paametes ae fully independent fom the system paametes and all the othe state vaiables except than the contolled ones; this substantially leads to simple contol designs, easily and diectly implemented on the d-q duty-atio components although thei dynamics ae nonlinea. The oveall design can also guaantee that the contol signals, namely the duty-atio signals, ae inheently bounded in the pedefined ange whee the convete opeates with linea modulation. An industial size.4-kw induction moto fed by a VSC is used to illustate the poposed appoach while compaisons with the taditional FOC technique veify the poposed minimum losses opeation. Simulation esults obtained with MATLAB/Simulink o a lab eal-time OPAL RT system development, ae povided to veify the poposed appoach. The pape is oganized as follows. In Section II, the complete system model consisting of the nonlinea convete dynamics and the d-q dynamics of the induction moto is obtained, while the existing FOC is biefly pesented. In Section III, an investigation of the system opeating with minimum losses is povided, and the nonlinea contolle is poposed to achieve this goal. In the sequel, closed-loop system stability analysis is poven fo the complete system fo the fist time in the liteatue. Seveal esults compaing the poposed contol scheme with minimum losses ove the same contolle with standad FOC ae pesented in Section IV, while finally in Section V, some conclusions ae dawn. II. VSC CONVERTER AND INDUCTION MOTOR ANALYSIS A. Complete Dynamic Model The system unde consideation consists of a diode ectifie, a dc-link, and a thee-phase VSC feeding a thee-phase induction Fig.. Schematic diagam of the system unde consideation. moto, as shown in Fig.. A dc-link capacito C and a smoothing inducto L ae used in the dc-link along with thei paasitic esistances R C and R L, espectively. Using the synchonously otating d-q efeence fame [] and assuming as state vaiables of the induction moto the stato cuents i ds, i qs, the oto fluxes λ d, λ q, and the moto speed ω, the widely used dynamic model of the induction moto can be obtained as ( R L ) m σ i ds = L + R s i ds + σω s i qs + R L m L λ d + L m pλ q ω + V ds L ( R L ) m σ i qs = L + R s i qs σω s i qs + R L m L λ q λ d = R L m L λ q = R L m L L m L pλ d ω + V qs i ds R L λ d +(ω s pω ) λ q () i qs R L λ q (ω s pω ) λ d J m ω = 3L m L pλ q i ds + 3L m L pλ d i qs bω T L whee R s and R ae the stato and oto esistances, espectively, L s and L ae the stato and oto inductances espectively, L m is the mutual inductance, ω s is the synchonous speed (efeence fame), p is the numbe of pole pais, J m is the total moto and load inetia, b is the fiction coefficient, T L is the load toque, and σ = L s L m L.Thed- and q-axis components of the stato voltages ae denoted as V ds and V qs, espectively. Using again the d-q tansfomation, the dynamic equations of the VSC with the dc-link can be witten in the fom L i = R L i V dc + V ec C V dc = 3 4 m dsi ds 3 4 m qsi qs V dc + i () R C whee i and V dc ae the dc-link cuent and voltage, espectively, and V ec is the output voltage of the diode ectifie (almost constant). m ds and m qs ae the d- and q-axis components of the duty-atio signals of the VSC and ae given as m ds = V ds V dc and

3 ALEXANDRIDIS et al.: ADVANCED INTEGRATED MODELING AND ANALYSIS FOR ADJUSTABLE SPEED DRIVES OF INDUCTION MOTORS 39 m qs = V qs V dc fo which it holds tue that [] m a = m ds + m qs (3) with m a being the switching duty atio of phase-a of the induction moto voltage in a peiod unde pulse width modulation (PWM) egulation (modulation index) and Δφ is the initial phase of the induction moto phase-a voltage mqs Δφ = actan. (4) m ds As a esult, by combining the moto dynamics () with the convete dynamics () and taking into account the duty atio m ds and m qs definitions, the complete system takes the following nonlinea matix fom: Mẋ =(J (x, m ds,m qs ) R) x + Gɛ (5) with the system matices given in (6) at the bottom of the page whee the state vecto is x = [ ] T i ds i qs λ d λ q ω iv dc and the extenal uncontolled input vecto is ɛ = [ 3 T L 3 V ] T ec which consists of bounded constant o piecewise constant signals. The only contol input signals ae the duty-atio components of the VSC, i.e., m ds and m qs, which appea in nonlinea tems of J. In the majoity of contol applications, it is pefeed that the invete opeates in the linea modulation aea [34] in ode to avoid the existence of highe hamonics. This means that m a. Theefoe, taking into account (3), it should be [] m ds + m qs. (7) Futhemoe, since the synchonous speed ω s is poduced fom the invete, it can be also specified by the contol opeato [] accodingly to the desied slip fequency ω sl ω s = pω + ω sl (8) It should be noted that in (5), matix M is positive definite, J is skew-symmetic, and R is positive definite. Using the stoage function H (x) = xt Mx, it can be easily poven that system (5) is equivalent to the genealized Hamiltonian-passive fom as detemined in [3]. B. Taditional FOC Stategy FOC technique elies on opeating the induction moto as a sepaately excited dc moto. In most applications, the indiect oto field oientation is applied whee the total oto flux λ is { M = diag σ, σ, L, L, J m 3, L 3, C 3 L m 0 σω s 0 0 pλ q 0 L σω s L m pλ d 0 L ω s pω J = L 0 0 ω s pω L L m L m pλ q pλ d L L m ds m qs [ ] T G = and R L m L + R s 0 R L m L R L m 0 L + R s 0 R L m L R L m R R = L 0 L R L m R L 0 L b R L 0 } R C m ds m qs (6)

4 40 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 3, SEPTEMBER 05 aligned on the synchonous otating d-axis, i.e. λ = λ d and λ q =0. (9) In this fame, the oto flux dynamics can be decoupled and lead to a simplified contol scheme. Assuming steady-state opeation, one can detemine the desied slip fequency fom the fouth equation of (5) as ω s pω ω sl = L m i qs (0) τ λ whee τ = L R is the oto time constant. Since the oto flux cannot be measued, the amplitude of the flux is usually estimated fom the thid equation of system (5) as τ ˆλ + ˆλ = L m i ds () whee ˆλ is the estimation of the total oto flux []. In ode to achieve fast dynamics below the ated speed, the oto flux is usually maintained constant, and theefoe, and the oto flux ˆλ. Assuming steady-state opeation, ˆλ = λ, one can define fom () the d-axis cuent ī ds as follows: the efeence signals ae the desied moto speed ω ef λ = L m ī ds. () Theefoe, taditional FOC techniques povide a constant efeence value fo the flux which can be tansfomed into a constant efeence fo the cuent ī ds = i ef ds. Taditional PI and cascaded PI contolles as well as nonlinea contolles, as those poposed by the authos in [33], can be used to achieve the desied egulation. Howeve, in many cases whee powe losses play a key ole in the system opeation, a modified contol stategy that also achieves this task is equied [7]. In the following section, using a suitable nonlinea contolle, the wok of [33] is extended to achieve minimum losses and field oientation at steady state with guaanteed nonlinea closed-loop system stability and convegence to the desied equilibium. III. NONLINEAR CONTROLLER FOR ACHIEVING MINIMUM LOSSES A. Opeation with Minimum Losses A lot of eseach has been conducted in the liteatue fo opeating the induction machine with minimum losses [0] [6]. Induction moto losses ae given fom the following expession: P loss = P supplied P mech 3 (V dsi ds + V qs i qs ) T e ω (3) whee T e is the electomagnetic toque. Taking into account the induction moto model (), it can be easily shown that P loss minimization equies an optimal oto flux λ opt as given by the following expession [7, Sec ]: λ opt L = p + R L m R s p T d (4) whee hee it is consideed at steady state that T d = 3 T e, which implies that the squae of the oto flux magnitude should be linealy dependent fom the electomagnetic toque. In this pape, the main tasks ae to achieve accuate speed egulation with both field oientation and losses minimization at steady state. Assuming FOC opeation at steady state whee the stato cuents ae denoted by ī ds and ī qs, the electomagnetic toque is given as T e = 3 pl m ī ds ī qs (5) L whee now the optimal flux as given by () is not locked on a constant value. Theefoe, consideing a vaying flux magnitude λ opt = L m ī ds (6) with ī ds appopiately vaying to satisfy (4) and substituting (5) and (6) into (4) one aives at L L m ī ds = p + R L m R s p pl m L ī ds ī qs. Defining the stato d-q cuents atio as l opt = īqs ī ds, one can easily obtain l opt = īqs ī ds = + R L m R s L. (7) At this point, it is noted that ecalling FOC theoy of Section II-B, by substituting () into (0), the steady-state slip fequency that leads to optimal opeation with minimum losses has to be constant as the following expession indicates: ω opt sl = τ ī qs ī ds = lopt τ = const. (8) As a esult, if the stato cuents ae contolled in accodance to (7), then opeation with globally minimum losses is achieved as long as the esulting flux magnitude is below its uppe allowable bound. In pactical applications, howeve, the calculated l opt value is too small, esulting in unacceptable high flux magnitudes; then, the meaning of (7) is to select the smallest possible value fo the l opt (bigge than the one calculated fom (7)) which can ensue opeation with locally minimum losses and flux magnitude stengthened close to its uppe bound. Unde these cicumstances, stability analysis should be always guaanteed not only at steady state but duing the tansient pefomance as well. Such an analysis based on a contolle that achieves all the pevious tasks and which povides accuate speed egulation is given in the following section. B. Nonlinea Contolle Design and Analysis Since the complete dynamic system (5) is nonlinea, in ode to obtain closed-loop stability, a nonlinea contolle is equied. The poposed contolle pesented in this section, which is applied at the contol inputs m ds and m qs of the convete, is poven to achieve pecise moto speed egulation at ω ef and minimum losses in a field-oiented opeation, as discussed in

5 ALEXANDRIDIS et al.: ADVANCED INTEGRATED MODELING AND ANALYSIS FOR ADJUSTABLE SPEED DRIVES OF INDUCTION MOTORS 4 the pevious section. The contol stuctue is given as whee m ds = z (9) m qs = z (0) ż = A cont (z,ω,i ds,i qs ) z () with A cont ( 0 0 k i ds i ) qs l opt = 0 ( 0 k ω ω ef k i ds i ) qs ( l opt k ω ω ef c z + z + z3 ). States z = [ ] T z z z 3 epesent the contolle dynamics, k, k ae two nonzeo constant gains, and c is positive constant. It becomes clea fom the contolle dynamics () that if ω is egulated at ω ef and the atio of i qs and i ds is maintained constant and equal to l opt at steady state, then the contolle states will convege to thee equilibium constant values z, z, z3. It can be easily seen that the poposed contolle () implies that the contol law acts as an attactive limit cycle fo the contolle states z, z, z 3 on the suface of a sphee C with cente the oigin and adius equal to, i.e. C = { z,z,z 3 : z + z + z 3 = }. As it is obvious, by intoducing the tem c ( z + z + z3 ) in (), the contolle states ae attacted and emain all time theeafte on sphee C.This means that the z, z, z 3 tajectoies ove ove till each an equilibium z, z, z3 as ω and i ds appoach thei efeence values, while the obustness of z, z, z 3 cucially inceases in the sense that, if fo any eason z, z, z3 ae distubed, they cannot leave C. Since the contolle states ae esticted on the suface of sphee C, then obviously each state is bounded in the set [, ] and as a esult (z (t)) +(z (t)) = (z 3 (t)) o (z (t)) +(z (t)) which implies that (7) is obviously satisfied. Theefoe, the poposed contolle can guaantee that the VSC will always opeate in the linea modulation aea whee 0 m a, i.e., the technical limits of the duty-atio input ae fulfilled. Fo futhe details about the geneal concept of the contolle opeation, the eade is efeed to [33], though the contolle dynamics in the pesent pape ae quite diffeent fom those pesented in [33], as imposed by the diffeent contolle tasks. C. Closed-Loop System Modeling and Stability Analysis Afte incopoating the contolle dynamics () to the complete VSC-moto system (5), the esulting closed-loop system is still given in the genealized nonlinea Hamiltonian-passive fom M x = J ( x) R x + Gɛ () Fig.. losses. Fig. 3. Nonlinea contolle fo achieving speed egulation with minimum Poposed nonlinea contolle implementation. whee the closed-loop system is of tenth ode with matices M = [ ] M 07 3 [ ], G = G T T I 3 [ ] [ ] R R =, with R cl = R cl 0 c ( z + z + z3 ) and J as given in Appendix. The closed-loop state vecto is x = [ x T z ] T T [ ] T = ids i qs λ d λ q ω iv dc z z z 3. One can easily see that matix M emains symmetic and positive definite, J is still skew-symmetic, and R matix is symmetic. Also, the system dynamics emain in its full fom without any simplification caused by the field-oiented demand. The schematic diagam of the poposed nonlinea contolle applied on the complete system is given in Fig., while the contolle implementation is depicted in Fig. 3.

6 4 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 3, SEPTEMBER 05 To begin with the stability analysis, conside fist the unfoced closed-loop system (ɛ =0) M x = J ( x) R x (3) with stoage function candidate V ( x) = xt Mx+ ( z 4 + z + z3 ). (4) Then, the time deivative of V is calculated as V = x T Mẋ + ( z + z + z3 ) (z ż +z ż +z 3 ż 3 ) = x T Rx c ( z + z + z 3 ) z 3 0 (5) which poves the stability of system (3) as follows. System (3) is a nonlinea autonomous system and since the stoage function (4) is adially unbounded with nonpositive deivative ove the whole state space, then accoding to the Global Invaiant Set Theoem 3.5 descibed in [35], all solutions unifomly globally asymptotically convege to the lagest invaiant set N in E. Set E is defined as the set whee V =0and since R>0, then it consists of the union of x =0and z, z, z 3 constained on sphee C. Futhemoe, fom the matix-diagonal fom of matices M, J, and R, it becomes clea that the induction moto-convete system (plant) and the contolle system can be handled sepaately in both the closed-loop unfoced system (3) and the oiginal one, as given by (). Paticulaly, as shown in the pevious analysis, the contolle system opeates as an attactive sphee fo the states z, z, and z 3, independently fom the contolled plant states i ds, i qs, ω, and the extenal input ɛ. Thus, exploiting this significant contolle popety to the closed-loop system analysis, the contolle state vecto z is consideed as a bounded time-vaying vecto fo the plant system Mẋ =(J (x, z(t)) R) x + Gɛ (6) which is handled independently as a nonautonomous nonlinea system with extenal input vecto ɛ. Fo system (6) conside the Lyapunov function candidate Taking the time deivative of W, it yields W = xt Mx. (7) Ẇ = x T Rx + x T Gɛ. (8) Then, it can be easily shown [36] that thee exist 0 <θ< such that Ẇ ( θ)λ min (R) x ɛ x (9) θλ min (R) whee λ min (R) is the smallest eigenvalue of the constant positive definite matix R and, in this case, Gɛ = ɛ. Inequality (9) poves that system (6) is input-to-state stable with espect to the extenal input ɛ [37], [36]. Theefoe, since the extenal input vecto ɛ is a bounded signal consisting of the diode ectifie voltage and the load toque, then the convete-moto system is TABLE I SYSTEM PARAMETERS moto ated powe P n.4 kw ated stato voltage V s 30 V ated stato cuent I s 39.5 A ated speed ω m 68 /min diode ectifie output V ec 670 V dc-link capacitance C.8 mf dc-link inductance L 3 mh dc-link esistance R L 0.05 Ω stato inductance L s 44. mh oto inductance L 4.7 mh mutual inductance L m 4 mh stato esistance R s 0.94 Ω oto esistance R 0.56 Ω pole pais p 3 moto-load inetia J m 0.4 kg m fiction coefficient b N m s/ad stable in the sense of boundedness, i.e., x is bounded. Futhemoe, since z, z, and z 3 ae attacted on a closed set descibed by sphee C, then it is concluded that the whole closed-loop state vecto x is bounded. Assuming that the efeence signals and the extenal input vecto ɛ ae povided in a manne that technically esult in an equilibium point (cetainly within the convete-moto system equiements), then fo the closed-loop system which is given in the Hamiltonian-passive fom [3] and unde some common assumptions mentioned in [3], it is poven that the bounded states x will eventually convege to the desied equilibium. IV. RESULTS A. Validation with MATLAB/Simulink Results In ode to evaluate the poposed contolle pefomance, an industial size.4-kw induction moto fed by a VSC is consideed. The poposed contolle is used to achieve minimum losses and is compaed to the contolle with FOC as descibed in [33] fo the same speed egulation scenaio. The system paametes ae analytically pesented in Table I. Stating fom a moto speed equal to 80 ad/s, at the time instant t =3s the efeence speed is set to ω ef = 0 ad/s, at t =6s it dops to 90 ad/s and at t =9s it changes again to 00 ad/s. At time instant t =sand while the efeence speed is maintained at ω ef = 00 ad/s, the load toque T L dops by 7% and 3 s late, it inceases apidly by 5%. These scenaio is pefomed to check the pefomance unde seveal efeence and load changes. In all cases, the paametes of both nonlinea contolles ae k = A, k =0.05 s/ad, and c = 000. Fom (7), the calculated value l opt =0.83 is too small, esulting in flux magnitude bigge than its uppe allowable bound of Wb and, theefoe, following the analysis descibed in Section III-A, l opt =.85 is used. Fig. 4 shows the time esponse of the complete convetemoto system states and input. The esponses of the stato cuents ae povided in Fig. 4(a) and (b), whee it is clea that the contolle with FOC egulates the d-axis cuent to the desied value i ef ds =9A, as descibed in Section II-B. On the othe

7 ALEXANDRIDIS et al.: ADVANCED INTEGRATED MODELING AND ANALYSIS FOR ADJUSTABLE SPEED DRIVES OF INDUCTION MOTORS 43 Fig. 5. Powe losses using the nonlinea contolle with minimum losses and the nonlinea contolle with standad FOC. Fig. 6. Steady-state powe losses fo diffeent load toque values and fo constant moto speed ω = 00 ad/s using the nonlinea contolle with minimum losses and the nonlinea contolle with standad FOC. Fig. 4. Complete system esponse fo the poposed nonlinea contolle with minimum losses and the nonlinea contolle with standad FOC. (a) d-axis stato cuent. (b) q-axis stato cuent. (c) d-axis oto flux. (d) q-axis oto flux. (e) Moto speed. (f) DC-link capacito voltage. (g) Modulation index. hand, the nonlinea contolle with minimum losses egulates i ds and i qs in such values in ode to have a constant atio equal to l opt at steady state. Fom the oto flux esponses in Fig. 4(c) and (d), it is clea that both contolles achieve field oientation at steady state since λ q =0. Additionally, it is obseved that the poposed contolle povides opeation with locally minimum losses and flux magnitude stengthened below its uppe bound (Fig. 4(c)). Both contolles achieve pecise moto speed egulation fo evey efeence change and load distubances as shown in Fig. 4(e). Howeve, the contolle with minimum losses povides a slightly slowe esponse but this is the cost fo minimizing the losses at steady state. The dc-link voltage V dc, shown in Fig. 4(f), is egulated accoding to the system equilibium whee the fluctuations ae caused fom the fact that the output voltage V ec of the diode ectifie is not stictly constant. Finally, in Fig. 4(g), it is veified that the poposed contolle foces the duty-atio signals to emain bounded and opeate the VSC in the linea PWM mode, as analytically pesented in this pape. In ode to check the main task of the poposed contolle, the system powe losses ae calculated and pesented in Fig. 5 fo the whole scenaio. It can be easily veified that the poposed contolle significantly educes the powe losses with espect to the taditional FOC techniques. Especially afte the final load toque incease, the powe losses ae educed by 5%. The steady-state losses fo a given moto speed ω = 00 ad/s and fo seveal load toque values T L (in pecentage ove the ated toque T N ) ae summaized in Fig. 6, whee it is obseved the significant eduction of moto losses using the poposed appoach compaed to the taditional FOC technique. Theefoe, the same speed egulation can be achieved with slightly sacificing the tansient pefomance, but the steady-state losses will be significantly educed by the technique poposed in the pesent pape. B. Validation with Real-Time Results To futhe validate the poposed contol appoach, ealtime esults have been obtained using the eal-time system of OPAL-RT fo a convete-induction moto system with simila

8 44 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 3, SEPTEMBER 05 Fig. 7. Real-time esults fo the nonlinea contolle with standad FOC. (a) d, q-axis stato cuents and oto fluxes. (b) Moto speed, powe losses, DC-link capacito voltage and modulation index. Fig. 8. Real-time esults fo the nonlinea contolle with minimum losses. (a) d, q-axis stato cuents and oto fluxes. (b) Moto speed, powe losses, DC-link capacito voltage and modulation index. paametes (in pu). Once again, both the poposed contolle fo minimum losses and the contolle with FOC ae compaed. In this scenaio, the efeence moto speed is initially set to ω ef =p.u., at time instant t =s it dops to ω ef = 0.8 p.u. and at t =8sitisset back to p.u. Finally, at t =4s, a 5% incease is applied at the load toque. Note that fo the contolle with FOC, the efeence d-axis cuent is set at i ef ds =p.u. duing the whole opeation. The esults fo the nonlinea contolle with standad FOC ae shown in Fig. 7, while the same scenaio with the poposed contolle opeating with minimum losses is shown in Fig. 8. As it is shown in Fig. 7(a), the contolle with FOC egulates the d-axis cuent at the desied value and achieves field oientation at all times since λ q is egulated at zeo. The est of the contolle states ae egulated at thei steady-state values with vey small tansients. Additionally, the contolle suitably egulates the moto speed at its desied value unde the efeence change and unde the sudden load change, as shown in Fig. 7(b). The dc-bus voltage stays always constant at p.u., the modulation index emains below at all times, and the esponse of the losses (P loss ) is obseved in the same figue. On the othe hand, the esults fo the poposed contolle opeating with minimum losses ae shown in Fig. 8. Field oientation is also achieved at steady state since the q-axis oto flux is egulated at zeo, i.e., λ q =0 at steady state and the stato cuents ae contolled to have a constant atio l opt = i qs i ds at steady state, as shown in Fig. 8(a). Howeve, a slightly lage tansient is obseved compaed to the contolle with the standad FOC. As it is veified in Fig. 8(b), the moto speed is egulated at its efeence value afte the efeence and the load changes, the dc-bus voltage is constant at p.u., and once again the modulation index stays below, as imposed by the contolle opeation. Fom Fig. 8(b), it is fully veified in ageement with the theoetical analysis, that the poposed method achieves significant eduction of the machine losses P loss, duing the whole opeation (steady state and tansient). Nevetheless, it becomes clea fom the compaison of the moto speed esponses, shown in Figs. 7(b) and 8(b) between the case of using the contolle with the standad FOC and the case of using the contolle which constains the system to opeate with minimum losses, espectively, that the cost to pay, is only a slightly lage tansient; howeve, the simple design and the oveall system pefomance, seems to be in favou of applying the poposed appoach. V. CONCLUSION In this pape, a complete nonlinea system modeling of a VSC-fed induction moto was pesented, and a nonlinea

9 ALEXANDRIDIS et al.: ADVANCED INTEGRATED MODELING AND ANALYSIS FOR ADJUSTABLE SPEED DRIVES OF INDUCTION MOTORS 45 L m 0 σω s 0 0 pλ q 0 L z 0 3 σω s L m pλ d 0 L z 0 3 l opt τ L J = 0 0 lopt τ L L m L m pλ q pλ d L L z z J c dynamic contolle was poposed to achieve pecise moto speed egulation with minimum losses. Taking into account the vaying oto flux theoy, the elationship between the stato cuents is obtained to minimize the powe losses and achieve field oientation at steady state. Both the cuent and speed egulation is implemented by a nonlinea contolle that is fully independent fom the system paametes and poduces bounded duty-atio signals always within the pedefined limits as they ae set by the linea modulation aea of the convete. Based on the complete system dynamics and the contolle stuctue, nonlinea closed-loop system stability is poven whee the system states ae guaanteed to emain bounded and convege to the desied equilibium. Adding to the igoous stability analysis the fact that the contolle can be easily implemented with no flux measuements o estimation needed, this contol scheme establishes a clealy innovated step in ac moto dives that decisively enhances the existing techniques. APPENDIX Appendix equation shown at the top of the page whee ω s = pω + ω opt sl and ( 0 0 k i ds i ) qs ( l opt ) J c = 0 0 k ω ω ef ( k i ds i ) qs. l opt k ω ω ef 0 REFERENCES [] B. K. Bose, Moden Powe Electonics and AC Dives. Uppe Saddle Rive, NJ, USA: Pentice-Hall, 00. [] N. P. Quang and J.-A. Dittich, Vecto Contol of Thee-Phase AC Machines: System Development in the Pactice. New Yok, NY, USA: Spinge, 008. [3] G. D. Maques and D. M. Sousa, Stato flux active damping methods fo field-oiented doubly fed induction geneato, IEEE Tans. Enegy Conves., vol. 7, no. 3, pp , Sep. 0. [4] Y. Zhou, P. Baue, J. A. Feeia, and J. Pieik, Opeation of gidconnected DFIG unde unbalanced gid voltage condition, IEEE Tans. 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Matins, D. M. Sousa, V. F. Pies, and R. A.,, Reducing the powe losses of a commecial electic vehicle: Analysis based on an asynchonous moto contol, in Poc. 4th Int. Conf. Powe Eng. Enegy Elect. Dives, 03, pp [] B.-A. Chen, T.-K. Lu, C. W.-L., and Z.-C. Lee, An analytical appoach to maximum powe tacking and loss minimization of a doubly fed induction geneato consideing coe loss, IEEE Tans. Enegy Conves., vol. 7, no., pp , Jun. 0. [] R. Leidhold, G. Gacia, and M. I. Valla, Field-oiented contolled induction geneato with loss minimization, IEEE Tans. Ind. Electon., vol. 49, no., pp , Feb. 00. [3] Y. Lei, A. Mullane, G. Lightbody, and R. Yacamini, Modeling of the wind tubine with a doubly fed induction geneato fo gid integation studies, IEEE Tans. Enegy Conves., vol., no., pp , Ma [4] Z. Qu, M. Ranta, M. Hinkkanen, and J. Luomi, Loss-minimizing flux level contol of induction moto dives, IEEE Tans. Ind. Appl., vol. 48, no. 3, pp , May/Jun. 0. [5] C. Canudas de Wit, and J. 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10 46 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 30, NO. 3, SEPTEMBER 05 [] M. N. Mawali, A. Keyhani, and W. Tjanaka, Implementation of indiect vecto contol on an integated digital signal pocesso-based system, IEEE Tans. Enegy Conves., vol. 4, no., pp , Jun [3] H. A. Toliyat, E. Levi, and M. Raina, A eview of RFO induction moto paamete estimation techniques, IEEE Tans. Enegy Conves., vol. 8, no., pp. 7 83, Jun [4] R. Otega, P. J. Nicklasson, and G. Espinosa-Péez, On speed contol of induction motos, Automatica, vol. 3, no. 3, pp , 996. [5] S. Peesada, A. Tonielli, and R. Moici, High-pefomance indiect field-oiented output-feedback contol of induction motos, Automatica, vol. 35, no. 6, pp , 999. [6] D. Kaagiannis, A. Astolfi, R. Otega, and M. Hilaiet, A nonlinea tacking contolle fo voltage-fed induction motos with uncetain load toque, IEEE Tans. Contol Syst. Technol., vol. 7, no. 3, pp , May 009. [7] P. J. Nicklasson, R. Otega, G. Espinosa-Peez, and C. Jacobi, Passivitybased contol of a class of Blondel-Pak tansfomable electic machines, IEEE Tans. Autom. Contol, vol. 4, no. 5, pp , May 997. [8] S. Peesada and A. Tonielli, High-pefomance obust speed-flux tacking contolle fo induction moto, Int. J. Adaptive Contol Signal Pocess., vol. 4, nos. /3, pp , 000. [9] A. Behal, M. Feemste, and D. Dawson, An impoved indiect fieldoiented contolle fo the induction moto, IEEE Tans. Contol Syst. Technol., vol., no., pp. 48 5, Ma [30] T.-S. Lee, Lagangian modeling and passivity-based contol of theephase AC/DC voltage-souce convetes, IEEE Tans. Ind. Electon., vol. 5, no. 4, pp , Aug [3] G. C. Konstantopoulos and A. T. Alexandidis, Genealized nonlinea stabilizing contolles fo Hamiltonian-passive systems with switching devices, IEEE Tans. Contol Syst. Technol., vol., no. 4, pp , Jul. 03. [3] G. C. Konstantopoulos and A. T. Alexandidis, Design and analysis of a novel bounded nonlinea contolle fo thee-phase ac/dc convetes, in Poc. IEEE 5nd Annu. Conf. Decision Contol, 03, pp [33] G. C. Konstantopoulos, A. T. Alexandidis, and E. D. Mitonikas, Bounded nonlinea stabilizing speed egulatos fo VSI-Fed induction motos in field-oiented opeation, IEEE Tans. Contol Syst. Technol., vol., no. 3, pp., May 04. [34] N. Mohan, T. M. Undeland, and W. P. Robbins, Powe Electonics. Hoboken, NJ, USA: Wiley, 003. [35] J.-J. E. Slotine and W. LI, Applied Nonlinea Contol. Englewood Cliffs, NJ, USA: Pentice-Hall, 99. [36] H. K. Khalil, Nonlinea Systems. Englewood Cliffs, NJ, USA: Pentice- Hall, 00. [37] E. D. Sontag, Smooth stabilization implies copime factoization, IEEE Tans. Autom. Contol, vol. 34, no. 4, pp , Ap Antonio T. Alexandidis (M 88) eceived the Diploma degee in electical engineeing fom the Depatment of Electical Engineeing, Univesity of Patas, Rion, Geece, in 98, and the Ph.D. degee in electical engineeing fom the Depatment of Electical and Compute Engineeing, West Viginia Univesity, Mogantown, WV, USA, in 987. He joined the Depatment of Electical and Compute Engineeing, Univesity of Patas, in 988, whee he is cuently a Pofesso and the Head of the Powe Systems Division. Duing 998, he joined the Contol Engineeing Reseach Cente, City Univesity London, London, U.K., as a Visiting Reseache. He has authoed moe than 30 intenational jounal and confeence papes. His eseach inteests include contol theoy, nonlinea dynamics, optimal contol, eigenstuctue assignment, passivity, and advanced contol applications on powe systems and dive systems. His cuent eseach inteests include enewable powe geneation contol and stability (wind geneatos, PV systems, and micogids). Pof. Alexandidis is a Membe of the National Technical Chambe of Geece. Geoge C. Konstantopoulos (S 07 M 3) eceived the Diploma and Ph.D. degees in electical and compute engineeing fom the Depatment of Electical and Compute Engineeing, Univesity of Patas, Rion, Geece, in 008 and 0, espectively. He has been with the Depatment of Automatic Contol and Systems Engineeing, Univesity of Sheffield, Sheffield, U.K., since 03, whee he is cuently a Reseach Fellow. His cuent eseach inteests include nonlinea modeling, contol, and stability analysis of powe convete and electic machine systems with an emphasis in micogid opeation, enewable enegy systems, and moto dives. D. Konstantopoulos is a Membe of the National Technical Chambe of Geece. Qing-Chang Zhong (M 03 SM 04) eceived the Ph.D. degee in contol theoy and engineeing fom Shanghai Jiao Tong Univesity, Shanghai, China, in 000, and the Ph.D. degee in contol and powe engineeing fom Impeial College London, London, U.K., in 004. He holds the Max McGaw Endowed Chai Pofesso in Enegy and Powe Engineeing at the Illinois Institute of Technology, Chicago, IL, USA, and a Reseach Chai at the Univesity of Sheffield, Sheffield, U.K. He (co-)authoed thee eseach monogaphs, including Contol of Powe Invetes in Renewable Enegy and Smat Gid Integation (Wiley, 03) and Robust Contol of Time-delay Systems (Spinge, 006), and poposed the achitectue fo next-geneation smat gids to unify the inteface of all playes with the gid though the synchonisation mechanism of conventional synchonous machines. His eseach inteests include advanced contol theoy and its applications to vaious sectos including powe and enegy engineeing, chemical engineeing, and mechatonics. Pof. Zhong is a Distinguished Lectue of the IEEE Powe Electonics Society, and is the U.K. Repesentative to the Euopean Contol Association. He seves as an Associate Edito fo IEEE TRANSACTIONS ON AU- TOMATIC CONTROL/POWER ELECTRONICS/INDUSTRIAL ELECTRONICS/CONTROL SYSTEMS TECHNOLOGY, IEEE ACCESS, IEEE JOURNAL OF EMERGING AND SE- LECTEDTOPICS IN POWER ELECTRONICS,andthe Euopean Jounal of Contol.