VARIABLE speed drive systems are essential in

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1 1 Senorle Field Orientation Control of Induction Motor Uing Reduced Order Oberver R. Mehram, S. Bahadure, S. Matani, G. Datkhile, T. Kumar, S. Wagh Electrical Engineering Department Veermata Jijabai Technological Intitute, Mumbai Abtract The compoite method of the flux oberver and reduced order peed oberver and it deign procedure for a nonlinear dynamic ytem i preented. The traditional approach with flux and peed oberver and derive ix order nonlinear model that decribe motor model in field oriented coordinate. The model take into the error in flux etimation. For flux regulation traditional PI controller are ued. For peed regulation, model i implified by auming that the flux regulation i relatively fat and high gain PI controller to regulate q- axi current to it command. Thi outcome i in a third order nonlinear model in which the peed and two flux etimation error are tate variable, the q-axi current i a control input and peed etimate provided by oberver i a meaured output. Thi nonlinear model i a main contribution to paper. The deign of controller i take place via linearization. The analyi concede an important role by the teady tate product of flux frequency and q-axi current in determining control property of ytem. Index Term Field Oriented Control, Induction Motor drive (IM), Nonlinear ytem, Oberver, Reduced order ytem, Singular perturbation, Speed etimation I. INTRODUCTION VARIABLE peed drive ytem are eential in many indutrial application. In the pat, DC motor were ued extenively in area were veritable peed operation wa required, ince there flux and torque could be controlled eaily by the field and armature current. Due to exitence of the bruhe and commutator DC motor have certain diadvantage becaue of that they require periodic maintenance. Thee problem can be overcome by the application of alternating current motor, which can have imple and rugged tructure, high maintainability, and economy; they are alo robut and immune to heavy overloading. Thee advantage have recently made induction machine widely ued in indutrial application. There are numerou control technique for Induction motor control uch a open loop V/f control,v/f control with lip compenation, but thee technique are leat preferred becaue at low peed motor will conume exceive current. Thi can be eaily achieved by mean of vector control methodology. Field orientation control of an Induction Motor i motly preferred control trategy in the indutry due to it high reliability. The implementation of vector control (tator flux R. Mehram and S. Bahadure are reearch cholar in Electrical Engineering Department V.J.T.I. Matunga, Mumbai, India (raginimehram18@gmail.com). S. Matani, G. Datkhile, T. Kumar are MTech cholar in Electrical Engineering Department V.J.T.I. Matunga, Mumbai. S. Wagh i At Profeor at EED, V.J.T.I. Matunga, Mumbai and Viiting Reearch Scholar, Tuft Univerity, USA. oriented control, rotor flux oriented control or magnetizing flux oriented control) require information of magnitude and pace angle of the tator flux, rotor flux or magnetizing flux pace phaor repectively tated in [1]-[3]. The control can be performed in a reference frame fixed to the tator flux, rotor flux or magnetizing flux pace phaor repectively, direct and quadrature axi tator current are obtained in the correponding reference frame. Thee tator current ued independently to control the flux and torque. A explained in [4][5] the vector control method of the induction machine can be claified a the indirect method and the direct method. In the indirect method, the lip angular frequency of the induction machine i controlled and the flux and the torque component current are indirectly adjuted through the lip angular frequency. In the direct method, the flux and the torque component current are directly adjuted through the rotor peed. Indirect control technique i motly preferred. A nonlinear deign method i ued in IM to deign etimator. Parameter and tate etimation or adaptation technique i ued in oberver deign. Etimation method can be claified baed on full order and reduced order oberver. Some of full-order oberver for the IM are preented in [6][7] where they ue the entire et of tate pace equation. Reduced-order oberver in [8][9], offer ome parameter of tate vector to reduce computational effort where a other part i computed algebraically. Adaptive peed oberver provide trong robutne along with high accuracy over a wide peed range with probably imple implementation. Thi paper focue on the cloed-looped oberver baed enorle technique. The principle of the peed oberver give improved dynamic performance of peed etimation with the reduced order oberver. Thi paper aim to give a comprehenive evaluation of the oberver from everal apect, including teady tate for high and low peed operation etc. In many of reference adaptive ytem analyi i limited to local linear model, model uncertainty i uually ignored, no analyi of cloed loop ytem. The goal of thi paper to tackle the preceding drawback. In [10][11] tudy a traditional field orientation control where flux oberver i ued to etimate rotor flux. Here main focu i on peed control problem, where motor peed required to track the given peed command for any unknown load. The key element of thi approach are: To keep track the error in etimating rotor flux, the field

2 2 orientation i performed uing the etimated flux angle and two additional tate variable are added a the flux etimation error into the field oriented ytem To etimate peed from current meaurement a high gain oberver i deign Hence the motor in field orientated coordinate i decribed by ixth order model and put together the flux and peed regulation problem. For flux regulation problem a traditional approach of PI controller i ued and for peed regulation problem the model i implified by auming flux regulation take place relatively fat and by uing high gain PI controller the q-axi current to it command. Thi lead to a third order nonlinear model in which peed and two fluxe error are the tate variable. The q-axi current i a control input and the peed etimate from high gain oberver i meaured output. The nonlinear model i a main contribution to paper becaue it ineffectual to analye cloed loop ytem under different controller. PI controller are deigned via linearization and it can be deign to tabilize the nonlinear third order model at deired equilibrium point. The analyi let out the important role played by flux frequency and q-axi current to determine control propertie of ytem. When thi ytem i zero it i impoible to tabilize ytem by PI controller. In actual fact, it i impoible to robutly tabilize ytem by any controller which ue integral action. When it i poitive the ytem i in minimum phae and PI controller can be deigned to achieve good performance and robutne. In the end when it i negative, the ytem i non minimum phae and PI controller can not tabilize the ytem. Simulation reult are preented uing MATLAB/Simulink to confirm analyi finding. II. IM MODEL ALIGNED WITH ROTOR FLUX DERIVATIVE Fig. 1 how ytem coniting of an induction motor connected to the three phae power upply through a three phae diode rectifier, dc-link and VSI. C i the dc ide capacitor while L and R L repreent the mall inductance and reitance that appear in dc-link. Uing the ynchronouly rotating d-q reference frame all inuoidal quantitie are tranformed into dc quantitie in teady tate. The dynamic equation of the quirrel cage induction motor are given a follow: Fig. 1: Schematic diagram of VSI-fed induction motor V d = R i d + λ d ω λ q V q = R i q + λ q + ω λ d 0 = R r i dr + λ dr (ω pω r )λ qr 0 = R r i qr + λ qr + (ω pω r )λ dr (1) The fluxe are combined with the current according to the following expreion: λ d = L i d + L m i dr λ q = L i q + L m i qr λ dr = L r i dr + L m i d λ qr = L r i qr + L m i q (2) Along with, the torque equation that decribe the relationhip between the motor torque and the rotor peed i a follow: T e = T L + J ω r + f v ω r (3) Wherea the electromagnetic torque of motor i given a follow: T e = 3 2 pl m L r (λ dr i q λ qr i d ) (4) Indirect rotor field orientation control technique relie on operating the induction motor a a eparately excited dc motor. In mot application, the indirect field orientation i applied where the total rotor flux λ r i aligned on the ynchronou rotating d-axi λ r = λ dr, λ qr = 0 (5) Thi operation decouple the rotor flux dynamic equation leading to implified control deign. Applying (5) to the nonlinear dynamic model (1) and auming teady-tate operation we can determine the deired lip frequency from the 4th equation of (1) a follow: ω pω r = ω l = L m τ r λ r i q (6) Becaue the rotor flux cannot be meaured, the amplitude of the flux i uually etimated from the 3rd equation of ytem (1) a follow: ˆ λ r + ˆλ r = L m i d (7) where ˆλ r i the etimation of the total rotor flux. Auming the induction motor i operated below the rated peed, the rotor flux hould be maintained contant and therefore the reference ignal are the deired motor peed ω rref and the rotor flux ˆλ r. Auming teady tate operation, ˆλ r = λ r we can define from (7) the contant d-axi current i d a follow: λ r = L m i d (8) Thu, conventional IR-FOC technique require the ue of three PI controller wherein: 1) a PI controller i ued at the tator input voltage V d to maintain the d-axi current at i d and 2) two cacaded PI controller are ued to control the rotor peed through the q-axi current of the tator. In particular, the firt inner-loop PI controller, ued at the tator voltage

3 3 V q, regulate current i q at it reference i q, wherea the econd outer loop PI controller provide the i q value by comparing the meaured motor peed ω r with the deired reference ω reef. Inner loop PI controller deign i baed on the linear ytem theory and therefore decoupling term are uually added to provide tability. Thi decoupling i alo needed to enure precie field orientation during tranient, i.e., to enure the rotor flux to be kept contant at all time. Although the ue of compenating term reult in a linear ytem that permit a direct gain election for the inner loop PI controller, the dependence of the control deign from the ytem parameter increae, thu increaing the controller enitivity in parameter uncertaintie. Furthermore, becaue the final control i implemented on the duty-ratio ignal input of the VSI, aturator that can limit the controller output in the permitted range are needed. III. DESIGN OF FLUX AND REDUCED ORDER SPEED OBSERVERS FOR SIXTH ORDER IM MODEL The induction motor model in the complex vector form in tationary frame of reference by equation [4] di dt = { R (1 σ) + }i + L m σl στ r dλ r dt 1 λ r σl L r τ r L m jω r λ r + 1 V (9) σl L r σl = L m τ r i 1 τ r λ r + jω r λ r (10) Where i R 2 i the tator current, λ r R 2 i the rotor flux, V R 2 i the tator voltage and ω r i the rotor peed. The R and R r are tator and rotor reitance, L, L r and L m denote the tator, rotor and mutual inductance, σ = 1 L 2 m/l L r i the leakage coefficient, τ r and τ are rotor and tator time contant, J i a rotor moment of inertia, f v i a friction coefficient and p i number of pole pair. The kew ymmetric matrice I and J are defined by [ ] [ ] I =, J = Flux Oberver: In rotor field orientation, the controller i deigned by tranforming all variable into d-q coordinate where the d-axi of the rotating frame of reference[13] i oriented along the rotor flux λ r [10]. We ue the open loop oberver ince λ r i not meaured. ˆλ r = ( 1 τ r I + pω 1 J ) ˆλ r + L m τ r i (11) The ω 1 i available and would have taken a ω reef in (11), would have been an aymptotic oberver that taking lim t [λ r (t) ˆλ r (t)] = 0. A vector λ r = [ ] T λ αr λ βr in the tator frame of reference i tranformed into the rotating frame of reference by change of variable [ ] [ ] [ ] λdr coρ inρ λαr = (12) inρ coρ λ qr λ βr where ρ = ˆλ r and error e = λ r ˆλ r. Orienting the vector ˆλ r denote d-axi component by λ dr = ˆλ r and q-axi component by λ qr = 0. The machine model i repreented by equation a λ d = αλ dr + αl m i d (13) i d = αβλ dr ηi d + pω 1 i q + αl m i 2 q/λ dr + V d /σl + αβe d + βpωe q (14) i q = βpωλ dr pω 1 i d ηi q αl m i d i q /λ dr + V q /σl βpωe d + αβe q (15) ω = (3pL m )/(2JL r )[i q (λ dr + e d ) i q e q ] f v ω T L /J (16) ė d = αe d + (pω 1 pω + αl m i q /λ dr )e q (17) ė q = (pω 1 pω + αl m i q /λ dr )e d αe q + p(ω ω 1 )λ dr (18) where α = 1 τ r,β = 1 σ σl m and η = 1 σ. The variable λ dr, i d and i q are available ince they can be calculated from V and i, while ω, e d and e q are not available. Speed Oberver: Since peed i not available for meaurement, it need to ue oberver for etimation. The information about ω i available in (15), rewrite(15) and (16) a where i q = βpλ dr ω f 1 (λ dr, i d, i q, V q, ω 1 ) + δ 1 (19) ω = (3pL m )/(2JL r )i q λ dr f v ω T L /J (20) f 1 (λ dr, i d, i q, V q, ω 1 ) = pω 1 i d + ηi q + αl m i d i q /λ dr V q /σl i available on line and δ 1 i uncertain term given by δ 1 = ηi q βpωe d + αβe q I q i a meaured output and ued with model (19) and (20) to etimate ω in oberver. The robut high gain oberver can deign if uncertain term are appear only in (20) but preence of δ 1 in (19) violate thi condition hence the change of variable take place Ω = ω δ ( ) 1 λdr + e d = ω + 1 [ηi q αβe q ] βpλ dr βpλ dr λ dr (19) and (20) can be write in form a (21) i q = βpλ dr Ω f 1 (λ dr, i d, i q, V q, ω 1 ) (22) Ω = (3pL m )/(2JL r )i q λ dr f v Ω + δ 2 (23) which i acceptable for high gain oberver deign, where δ 2 = T L /J f vδ 1 d ( ) δ1 βpλ dr dt βpλ dr def = f 2 (λ dr, i d, i q, ω, V q, e d, e q, T L, ω 1 ) and f 2 i a continuou function. The change of variable (21) i invertible, provided λ dr + e d 0. The high gain oberver

4 4 i modelled a î q = βpλ dr ˆΩ f1 (λ dr, i d, i q, V q, ω 1 ) + ( α1 ɛ ˆΩ = (3pL m )/(2JL r )i q λ dr f v ˆΩ ( α2 ɛ 2 pβλ dr ) (i q î q ) (24) ) (i q î q ) (25) where ɛ i mall poitive parameter and α 1 and α 2 are poitive contant that aign the root of 2 +α 1 +α 2 = 0 at deired location in left half plane. The etimation error caled a η 1 = i q î q, η 2 = Ω ɛ ˆΩ which aure the equation ɛ η 1 = α 1 η 1 βpλ dr η 2 (26) ( ) α2 ɛ η 2 = η 1 ɛf v η 2 + ɛδ 2 (27) βpλ dr For mall ɛ, the cloed loop ytem i ingularly perturbed one with η 1 and η 2 a the fat variable. A pecified by ingular perturbation theory from [14], the tability i determined by the matrix [ ] α1 βpλ dr α 2 βpλ dr 0 where λ dr > 0 i referred a contant. The characteritic equation 2 + α 1 + α 2 = 0 i Hurwitz. From the high gain oberver theory tated in [12], if the control imput V i bounded in ɛ then the etimation error Ω Ω 1 will be zero after a hort tranient period [0, T (ɛ)], where lim ɛ 0 T (ɛ) = 0. Furthermore, the cloed loop ytem with feedback from ˆΩ regain the performance of the cloed loop ytem with feedback from Ω a ɛ tend to zero. Hence, deign of peed controller a if Ω i available for feedback. By limiting V to rated voltage, the boundedne of V uniformly in ɛ i enured. IV. CONTROLLER STRATEGY FOR SENSORLESS IM MODEL Flux Controller: The field orientation i guaranteed when flux λ dr i regulated to reference flux λ ref > 0, which i choen a contant. The traditional approach of uing PI controller i ued. In (13) i d i a control input and deign a controller a i d = (K fp + K fi ) [λ ref λ dr ] The econd controller i to derive reference voltage command and i deigned a V d = (K dp + K di ) [i d i d ] With tight feedback loop, the regulation of λ dr to λ ref can be guarantee for a wide range of variation in term (pω 1 i q + αl m i 2 q/λ dr + αβe d + βpωe q ), which act a an input to (14). The deign hould enure that λ dr tart at poitive value and approache λ ref monotonically in order that it i alway poitive. The initial condition of λ dr i determined by the initial condition of the flux oberver. Speed Controller: The firt aumption i that flux controller i act fat relative to peed to regulate λ dr to λ ref. Aume that λ dr = λ ref and i d = λ ref /L m and eliminate (13) and (14). From (15) for any current command i q the deign of V q a the PI controller V q = (K qp + K qi ) [i q i q ] with ufficiently large gain to regulate i q to i q. Thi allow to view i q a a control input. Thu, the peed controller can be deigned a i q = (K ωp + K ωi ) [ω ref Ω] Here the important role play by ωi q in the control deign. When ωi q = 0, the plant ha zero at origin. Hence it i impoible to deign any controller with integral action. When ωi q < 0, the plant ha a real zero in the right half plane, hence it i non minimal phae. When ωi q > 0 in thi cae plant i a minimal phae and deign of PI controller with high gain feedback to tabilie the cloed loop ytem i poible to achieve good tracking propertie. The condition ωi q > 0 i atified when motor i operating in motoring or braking mode, but not in a generating mode. The condition ωi q = 0 i atified if i q = 0 or ω = 0.The cae ω = 0 indicate operation at zero frequency that i in braking mode correponding to certain peed and torque. Which i well known in induction motor on enorle control that zero frequency operation i challenging. The cae i q = 0 inconiderate of peed indicate that the power into the machine i negative. V. PERFORMANCE VALIDATION OF SENSORLESS IM USING OBSERVER TECHNIQUE The induction motor parameter ued for imulation i hown in Table I. Fig. 2 how the overall block diagram of enorle vector controlled induction motor drive uing oberver. Fig. 2: Cloed loop control block diagram with oberver In thi imulation reference d-q tator current are obtained according to reference load torque and peed waveform. They are compared with the actual motor current and the error are input to the PI controller to obtained reference voltage. Thee voltage are given to motor through VSI. Since the

5 5 oberver i in feedback path of the peed and rotor flux loop, oberver behaviour will effect on a ytem performance. The etimator generate accurate etimation in teady tate to achieve required accuracy. The dynamic of etimator hould ufficiently fat to meet tranient performance of vector control drive. The deign of parameter for nominal cae with bet choice of ω 1 a ω ref, the contant of PI controller of V d, i d and V q choen uing linearied model are K dp = 50, K di = 90,K fp = 40, K fi = 50, K qp [ = 34, ] K qi = 50. The flux oberver (11) i initiated a ˆλ 0.1 r (0) = 0 o that λ dr (0) = 0.1. The peed oberver (22) and (23) i implimated with α 1 = 55, α 2 = 750, ɛ = For all the reported reult the controller i not aturated, except for a hort period during the peed oberver tranient. The peed reference i taken a tep input. Fig. 3 how performance of flux and peed oberver. In the cae, peed i commanded at 140 rad/ec applied at zero time with the rated load of 12 Nm. The etimation and control of flux and peed how atifactory performance. It i noted from Fig. 3 that the etimateion error of both flux and peed i around the zero croing of the peed i taking large. The performance for low peed applied i hown in Fig. 4. At low peed with load of 12 Nm i hown. In thi cae motor peed i controlled at 4% rated peed. In the cae, the d-axi rotor flux i converge to reference value and q-axi rotor flux i maintained at zero hence field orientation occur. The q-axi tator current component i determined accordingly to the reference torque. From Fig. 4 the error of both flux i around zero. It can be aid from the imulation reult provide proper etimation of tate for eneorle vector control of induction motor drive. TABLE I: Induction Motor Parameter 3-phae, 4kW, 400V, 4 pole, 50 Hz Rated flux 0.65 Wb Stator reitance Ω Rotor reitance Ω Stator leakage inductance mh Rotor leakage inductance mh Magnetizing inductance H Moment of Inertia kg m 2 Friction coefficient kg m 2 /ec Fig. 3: Performance tudy of propoed model with oberver at 140 rad/ec with 12 Nm load VI. CONCLUSION The main contribution i to introduce a high gain peed oberver and ued it together with flux oberver for development of ixth order nonlinear model. It alo provide cloed loop analyi of traditional field oriented control where flux i directed to a reference. Reduced oberver model of third order where known parameter are eliminated i jutified uing ingular perturbation theory. To maintain the tability at deired equilibrium point the condition of ωi q > 0 hould atify at teady tate. Thi condition fail for machine in generating mode. To clarify the benefit of thi model, the main focu i on analyi of PI controller and it condition which could have not been obtained by imple linear model. For future work, a formidable problem would be to deign nonlinear controller for nonlinear model and etimate region of attraction.

6 6 VII. ACKNOWLEDGMENT The author would like to acknowledge the upport of Centre of Excellence in Complex and Nonlinear Dynamic Sytem (CoE-CNDS), VJTI, Matunga, Mumbai, India under TEQIP-II (ubcomponent 1.2.1). The author would like to Thank Mr. R. Rane, Head Power Electronic & Automation Technology Centre, Electrical & Automation, Laren & Toubro Limited for hi regiourou technical dicuion and input. REFERENCES [1] R. D. Lorenz, T. A. Lipo, and D. W. Novotny. Motion control with induction motor.proceedign of IEEE: epecial iue on power electronic and motion control,82(8): , Aug [2] R. W. de Doncker and D. W. Novotny, The Univeral Field Oriented Controller.IEEE Tran. on Indutry Appliaction,30(1):92-100,Jan./Feb.1994 [3] H. Kubota and K. Matue, Speed enorle field oriented control of induction motor with rotor reitance adaption.ieee Tran.on Indutry Application, 30(5): ,Sep./Oct,1994. [4] P. Va, The control of AC machine, Oxford Univ., [5] S. H. Kim and S. K. Sul, Voltage control trategy for maximum torque operation of an induction machine in the field weakening region. IEEE Tran. Ind. Electron., vol. 44, no. 4, pp , Aug [6] M. Hinkkanen, Analyi and Deign of Full Order Flux Oberver for Senorle Induction Motor. IEEE Tranaction on Indutrial Electronic, vol. 51, no. 5, Oct. 2004, pp [7] P. Vaclavek, P. Blaha, Lyapunov function baed flux and peed oberver for AC induction motor enorle control and parameter etimation. IEEE Tranaction on Indutrial Electronic, vol. 53, no. 1, Feb. 2006, pp [8] L. Harnefor, M. Hinkkanen, Complete tability of reduced order and full order oberver for IM drive. IEEE Tranaction on Indutrial Electronic, vol. 55, 2008, pp [9] M. Hinkkanen, L. Harnefor, J. Luomi, Reduced-order flux oberver with tator reitance adaptation for peed enorle induction motor drive, IEEE Energy Converion Congre and Expoition, pp [10] W. Leonhard, Control of Electrical Drive, 2nd ed. New York: Springer, 1996 [11] R. Krihnan Electrical Motor Drive Prentice Hall, 2001 [12] Gilda Beancon (Ed.), Nonlinear Oberver and Application Springer- Verlag Berlin Heidelberg New York 2007 [13] G. C. Verghee and S. R. Sander, Oberver for flux etimation in induction machine, IEEE Tran. Ind. Electron., vol. 35, no. 1, pp. 85?94, Feb [14] P. V. Kokotovic, H. K. Khalil and J. O Reilly, Singular Perturbation Method in Control: Analyi and Deign. New York: Academic Pre, 1986 Fig. 4: Performance of propoed model with oberver at low peed (4 % of rated peed) with 12 Nm load