POWER systems have been traditionally analyzed using

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1 Comparison Between Classic and Space Vector Models of the Induction Motor during Start Up at Industrial Frequency Aurimar Angulo, José M. Aller, José J. Vásquez, and Alexander Bueno Abstract A comp arison between space vector and classic steady state models of the induction motor is performed during the start up condition at industr ial full voltage and frequency. This work shows several imprecisions obtained when the impact over the system is analyzed using classic mod els, specially during the start up. Pulsating torque effects and magnitude of maximum torque are considered. A deep bars model of the squirrel cage motor in space vector coordinates was developed, and its parametri c estimati on is performed to match the motor data sheet and experimental results with the machine performan ce in tran sient behavior. The wound rotor machine is modeled in space vector coordinates, compared with the classic model and its parameters obtained by an estimati on algorithm. Electri c transient behavior, harmonic analysis, dynamics and source unbalances can be better analyzed using space vector tran sformati on, because a reduce ord er syst em is obtained and power electronic converters could be included. Index Terms-- Deep bar rotor, induction motor, parametric estimati on, space vectors, squirrel cage motor, start up simulation, wound rotor. I. INTRODUCTION POWER systems have been traditionally analyzed using matrix transformations (Clark, Fortescue, Park, etc.), 2,, 4, 5, 6. The modal representation method decouples symmetric power system equations. This technique has been widely used to model electric systems in steady-state, under balanced and unbalanced operation, to analyze transients, dynamics and harmonics. However, modal decoupling of the system equations can be difficult in complex poly-phase networks. Hence, some authors have used non-transformed coordinates for power system analysis 7. During the last decade, the control of induction machines using space vector methods and field orientation has been thoroughly developed 8, 9, 0,, 2,. Space vector transformations are derived using the symmetrical components transformation, 5,, and lead to a concise representation. One advantage of the space vector transformation for induction machine analysis is the reduction from six real equations (three for the stator and three for the rotor) to a pair of complex equations (one for the stator and one for the rotor). The other advantage Aurimar Angulo & José J. Vásquez are with the Departamento de Ingeniería Eléctrica, Inelectra, Caracas - Venezuela ( aurimar.angulo@inelectra.com). José M. Aller is with the Departamento de Conversión y Transporte de Energía, Universidad Simón Bolívar Caracas - Venezuela ( jaller@usb.ve). Alexander Bueno is with the Departamento de Tecnología Industrial, Universidad Simón Bolívar Caracas - Venezuela ( bueno@usb.ve) /06/$ IEEE is the representation of three time-varying quantities with one space-time complex variable. Transient analysis during induction machine start up is required in order to specify transformers, feeders, soft starters and protections. Steady state model of the squirrel cage machine are widely used to perform these studies, by the second Newton s Law applied to rotational systems using numeric methods. Nevertheless, this model is not accurate for transient analysis because it neglects the time variation of the fluxes. The classical model use frequently in transient studies overestimates electric torque and the sub-estimate starting up time. Harmonic analysis, required for filter designs and power factor correction, can be performed using the classical model of the induction motor when superposition principle is applied, but a model for each harmonic need to be considered. Space vectors can be used to obtain a better analysis of the transient behavior and simpler power system models. Mathematical models of the power system elements can be represented by symmetric or cyclic matrices. Direct application of the space vector transformation simplify the dynamic equations representing three phase quantities by the use of complex variables and can decouple the modal representation. Only one sequence has to be analyzed using this technique, because the zero sequence is not excited when the neutral wire is not connected and the negative sequence is equal to the complex conjugated positive sequence. Transient analysis using space vector variables use the second Newton s Law solve the dynamic performance applying a precise electromagnetic model which accurately reproduce the instantaneous electric torque, fluxes and currents. The space vector model can consider harmonic excitation using all time a single model defined by lumped electric parameters like resistances, inductances, capacitors and dependent or independent sources. Space vector models of three phase sources, transformers, feeders, reactive compensators, passive loads and induction motors have been considered for the start up analysis of the electromechanical converter. Induction machine models developed in this paper consider deep bar, simple squirrel cage and wound rotors to obtain the starting up time, the voltage drop and the transient currents. The power transformer have been modeled using the space vector approach, considering the transformer connectivity and the harmonic flow between the source and the load. To obtain accurate results in the transient analysis studies, a non-linear parameter estimator was developed. This algorithm can estimate the induction motor parameters, considering deep

2 2 bar, squirrel cage and wound rotors. The estimator can use either experimental data or the data sheet supplied by the manufacturer. II. METHODOLOGY A. Space Vector Definition Traditionally power systems have been analyzed using symmetrical components, Clark transformation or any other modal transformation,, 4, 5, 6. These poly-phase transformations decouple symmetric and cyclic poly phase systems. In balanced power systems or in systems without a neutral return, the zero sequence component is zero. Positive- and negativesequences have similar behavior, especially in symmetric poly phase systems. The space vector transformation is defined as 2 x e j 2π e j 4π x a(t) x b (t) x α. (t)+jx β (t). x c (t) () 2 The coefficient is needed to conserve the instantaneous power between the coordinate systems. B. Power System Model Fig. shows the complete circuit for starting an induction motor. This electric power system is composed by the source, feeder, delta -Wye power transformer, passive load, reactive compensator and induction machine. Mechanical system dynamics are represented by the inertia and a polynomial function of the angular speed as load torque. Fig.. Power system diagram C. Source, load and reactive compensation model Fig. 2(a) shows a three phase model valid for delta connected sources, passive loads and reactive compensators. Fig. 2(b) shows the equivalent models for the Wye connection. The space vector model for delta or Wye connection can be expressed as 4 v s ke s +Z(p) M(p) i s (2) 2 e s e j 2π e j 4π p d dt. v (t) v 2 (t) v (t) t, For resistive, inductive or capacitive elements, operational impedances Z(p) and M(p) in (2) are shown in Table I. TABLE I IMPEDANCES IN WYE AND DELTA CONNECTION Element k Y Z Y (p) M Y (p) k Δ Z Δ (p) M Δ (p) Resistive R 0 Inductive Lp Mp Capacitive Cp 0 e j π 6 e j π 6 R 0 L p M p e j π 6 Cp 0 Fig. 2. Delta and Wye model of source, passive load and reactive compensator D. Feeder model The feeder model can be obtained from the transmission line model, including line-line, and line-ground capacitances, self- and mutual inductances and resistances. The transmission line model in space vector coordinates can be obtained using generalized circuit parameters (A, B, C, D) as vin A(p) B(p) vout () i in C(p) D(p) i out A(p) +R L (C M + C G ) p+(l L M L )(C M + C G ) p 2 C(p) B(p) R L +(L L M L ) p C M + C G + R L (C M + C G ) 2 p+ +(L L M L )(C M + C G ) 2 p 2 D(p) A(p) In () R L is the feeder resistance, L L is the feeder self inductance, M L represents phase to phase mutual inductance, C M is the mutual capacitance between conductors and C G is the ground capacitance. E. Transformer model An ideal Wye-delta transformer is shown in Fig.. Wyedelta connections can be analyzed using space vectors, as described by the following equation : v a V e jζ v 2 (4) a V and ζ are defined in Table II for several useful transformer connections.

3 R r Rr U 0 0 R r2 U Fig.. Ideal three phase Wye-delta transformer L ss L σs U + L ms S L σs U + L ms Lσr U+L L rr mr S L r2 S L r2 S L σr2 U+L mr2 S L sr L sr C L sr2 C TABLE II PARAMETERS ζ AND a V FOR MAIN TRANSFORMER CONNECTIONS n a V Yyn π - N N 2 Ydn π 5π 5π π N N 2 Dyn - π 5π 5π π N N 2 Ddn π - - N N 2 Equation (4) includes the phase change when a Wye-delta or a delta-wye connection is considered. F. Induction Machine Models Neglecting the effect of magnetic saturation, eddy currents, non sinusoidal magneto-motive force distribution, eccentricity and slotting, the induction machine model for squirrel cage and wound rotors can be expressed in space vector coordinates referred to stator frame as vs R s + L s p L sr p v r L sr (p j θ ) R r + L r (p j θ ) is i r T e L sr Im {i s i r } (5) Deep bar rotor can be modeled in original coordinates frame as vs Rs 0 is + (6) v r 0 R r i r Lss L sr is + p L sr t L rr i r T e 2 x r is i r t Lss L sr θ L sr t L rr x s x sa x sb x sc t xr x r2 is i r t xra x rb x rc t xr2a x r2b x r2c R s R s U R s cos θ cos ( ) θ 2π cos ( ) θ 4π C cos ( ) θ 4π cos θ cos ( ) θ 2π cos ( ) θ 2π cos ( ) θ 4π cos θ Applying space vector transformation, referred to the stator frame, to (6), the following deep bar model of the induction motor is obtained v s v r v r2 L s R s R r R r2 2 L sr + 2 L sr L r 2 L sr2 j θ 2 L sr2 2 L r2 2 L r2 L r L sr L r 2 L sr2 2 L r2 L r2 i s i r i r2 + p i s i r + (7) i r2 2 L r2 i s i r i r2 T e 2 Im { i s L sr i r + L sr2 i r2 } L x L σx + 2 L mx G. Mechanical model Mechanical load is modeled using a second order polynomial function of the rotor speed T m (ω m )k + k 2 ω m + k ω 2 m (8) Equation (8) can represent static, rotating friction and pumping torques. Choosing appropriate values for the parameters k, k 2 and k, a wide spectrum of mechanical loads can be modeled. The dynamic equation is obtained by applying the second Newton s Law J dω m dt T e T m (ω m ) (9)

4 4 H. Parameters Estimation of the Induction Machine Parameter estimation is performed using a steady state model of the induction machine, excited by a sinusoidal and three phase balanced source. Referring rotor circuits to the stator equation by Krön s reduction, an the impedance machine model in the stator terminals is obtained. Quadratic errors between the stator impedance and impedance calculated with the steady state model of the induction machine at several test condition give a cost function. The induction machine s parameters are obtained with a cost function optimization. The model of the induction machine in steady state are derived from (7) considering constant rotor speed θ ω r, derivative operator p jω s and rotor bars in short circuit. The resultant steady state model, also known as classic induction machine model, is vs A B is 0 B t (0) C i r A R s + jx s B jx sr jx sr2 Rr C s + jx r jx r2 jx r2 + jx r2 s ω s ω r ω s Applying Krön s reduction, the following stator impedance can be obtained Z s (P,s) v s A BC B t () i s P R X R R s R r R r2 X X s X sr X sr2 X r X r2 X r2 Parameter estimation is achieved by minimizing the error function n Ψ i R r2 s ( Z s (P,s i ) v )( s(s i ) Z s (P,s i ) v ) s(s i ) i s (s i ) i s (s i ) (2) Equation (2) is a non linear function and can be minimized using the Gauss-Newton algorithm. For wound or squirrel cage rotors, the second rotor equation is excluded and only five parameters are considered. III. RESULTS A. Power system modeling Fig. is an example of a power system used to start up an induction motor. Table III shows the main characteristics of the system elements. To simulate the power system in this example, the following per unit bases were chosen: S B MVA V BH 4.5kV V BL 4.6kV By using a per unit base a set of parameters are obtained for a deep bar induction motor, feeders, power transformer, P + jq load and reactive compensation of the motor bar. TABLE III POWER SYSTEM SPECIFICATIONS Source: SCC, 92 MVA 4.5kV Q.22% Feeder : 500MCM 540m 4.5kV Transformer: 0MVA 4.5/4.6kV Dy Z 5.75% Feeder 2: 4 500MCM 0m 4.6kV Feeder : 250MCM 572m 4.6kV Ind. Machine: 825kW 4kV 45A fp 85% 95rpm P-Q Load:, 7MV A fp 84.2 React. Compens.: 2000 kv AR 4.6 kv Mech. Load: T m ω 2 J (55+22)kg/m 2 ) Induction machine parameters: Calculation of the induction motor parameters was performed minimizing the cost function Ψ (2), using the data sheet supplied by the machine manufacturer. Resistances: R s.0078 R r.0 R r2.025 Inductances: L σs 0.04 L σr 0.04 L σr L ms.04 L msr.04 L msr L mr.2 L mr2.4 L r2. 2) Feeders: F eeder R L.69 0 F eeder 2 R L F eeder R L ) Transformer: R sc L sc a V ζ 5π 6 4) Load: R L.280 L L.796 5) Mechanical Load: T m ω 2 H IM.4287 H L.75 B. Steady state and dynamic models A comparison between the classic model of the induction motor and the space vector based model was performed. The results obtained for a conventional squirrel cage induction motor are shown in Fig. 4. The flux grow during start up produces lower torque than predicted by classic induction model which uses a constant flux hypothesis, valid only in steady state conditions. C. Parameter estimation Manufacturer s data is normally obtained in steady state at industrial frequency. Test of the induction motor at several non industrial frequencies is a powerful tool to obtain better parameter estimation for its dynamic behavior. These tests are particularly useful in the case of deep bar rotors. Fig. 5 shows a comparison between the toque-slip characteristics given in the factory data-sheet and the classic model obtained from the parameter estimator routine using an induction motor with a deep bar rotor. Matching the calculated parameters with the

5 5 Torque & Current (pu) Start up Time (s) Space Vector Model Classic Model 2 Space Vector Model Classic Model Magnetization Current time (pu) Fig. 4. models 2.5 Comparison between dynamic and steady state induction machine Estimator Data Sheet Feeder Section (MCM) Fig. 6. Comparison between start up time for classic and dynamic induction motor models Fundamental 5th Harmonics Electric Torque (pu) 2.5 Voltage Drop (%) Fig. 5. model Slip (pu) Torque-slip characteristics of the induction motor using the classic manufacturer data sheet shows a good approximation, since both data set are obtained maintaining constant flux in the air gap. D. Start up time Classical and dynamics models of the induction motor produce different start up times. This parameter is essential in the design of feeders and to adjust the current protection of motors, feeders and transformers. Over current and instantaneous protection must be adjusted using dynamic performance, because the real instantaneous torque is lower than the value predicted by classic or steady state models. Fig. 6 shows the start up time using classic and dynamic models for different feeder sections Feeder Section (MCM) Fig. 7. Voltage drop using the space vector model varying feeder section for fundamental and 5 th harmonics frequency E. Feeder voltage drop Fig. 7 shows a sensitivity analysis of the voltage droop at the induction motor bar, changing the feeder section with and without harmonics presence. Space vector model of the complete power system is a very useful tool for harmonic consideration because the same model is valid for any applied voltage if the power system is balanced. IV. CONCLUSIONS This paper shows the advantage of the space vector model over the use of the classical techniques. Transient behavior and harmonic analysis are fields space vector approach are a simple method to design and analyzed power systems. Transformers, feeders and induction motors can be modeled in a more precise and simple way using the space vector approach. Classical induction motor models introduce errors in electric torque and currents when they are used in transient analysis

6 6 because flux is not constant in these conditions. Only in steady state operation, the classic model predictions match well with the real machine behavior. Harmonics can be modeled using classic approach but the superposition principle has to be applied and several harmonic models need to be considered. Space vector models require only one model to perform a very precise harmonic analysis. Induction motors are build often with deep bar rotors in order to increase starting torques and reduce the start up time. The estimation parameters method developed in this work is a robust technique and can be applied using the data sheet supplied from manufactures. Use of some additional test at non industrial frequencies can improve the accurate of parameter estimation of the induction motor with deep bar rotors. REFERENCES E. Clarke, Circuit Analysis of AC Power Systems. New York: Jhon Wiley, R. H. Park, Two-reaction theory of synchronous machines, AIEE Transactions, vol. 48, p. 76, 929. C. L. Fortescue, Method of the symmetrical coordinates applied to the solution of polyphase networks, AIEE Transaction, vol. 7, H. Karrenbauer, Propagation of traveling waves on overhead lines for variation tower configuration and its effects on the shape of the recovery voltage for short line faults. PhD thesis, Munich, Germany, H. Edelmann, Algebraic eigenvalue theory of components systems utilizing network symmetries, Siemens Forschungs- und Entwicklungsberichte Siemens Reseach and Development Reports, vol. 0-6, H. W. Dommel and W. S. Meyer, Computation of electromagnetic transients, Proceedings of the IEEE, vol. 62-7, J. Arrillaga, C. P. Arnild, and B. J. Harker, Computer modelling of electrical power system. New York: Jhon Wiley, F. Blaschke, The principle of field orientation as applied to the new transkvector close-loop control system for rotating-field machines, Siemens Review, vol. 4, W. Leonhard, Control of Electrical Drives. Berlin: Springer-Verlag, S. Yamamura, Spiral Vector Theory of AC Circuits and Machines. New York: Oxford University Press, 992. D. Novotny and T. Lipo, Vector control and dynamics of AC drives. New York: Oxford University Press, A. Tryzanadlowski, The field orientation principle in control of induction motors, Boston: Kluwer Academic, 994. J. M. Aller, Simple matrix method to introduce space vector transformation and oriented field equations in electric machine courses, in Proc. ICEM International Conference on Electrical Machines (ICEM 96), (Vigo, Spain), pp , Sept J. M. Aller, A. Bueno, and T. Pagá, Power system analysis using space vector transformation, IEEE Transaction on Power System, vol. 7, no. 4, pp , BIBLIOGRAPHIES Aurimar Angulo was born in Barquisimeto, Venezuela in 977. She received the Eng. Degree in Electrical Engineering from Simón Bolívar University, Caracas,Venezuela, in At this moment, she works in the Department of Electrical Engineering of INELECTRA designing electrical substations; more specifically, working with One line diagrams, control schematics,layouts, etc. Her research interests include space vector application, electrical machine control, electrical substation and hydroelectric power plants design. José M. Aller was born in Caracas, Venezuela in 958. He received the Ing. Degree in Electrical Engineering from Simón Bolívar University, Caracas, Venezuela, in 980, the M.S. degree in Electrical Engineering from Universidad Central de Venezuela, Caracas, Venezuela in 982, and the Ph. D. degree in Industrial Engineering from Universidad Politécnica de Madrid, Madrid, Spain in 99. He has been a Lecturer at the Simón Bolívar University for 26 years, he is currently a full-time professor at the Energy Conversion Department. Was the General Secretary of the Simón Bolívar University between 200 and His research interests include space vector application, electrical machine control and power electronics. José J. Vásquez was born in Caracas, Venezuela in 957. He received the Electrical Engineering Degree from Simón Bolívar University, Caracas, Venezuela, in 980. At this moment, he works at the electrical engineering department of INELECTRA as Electrical Discipline Leader. His research interests include development of Consulting Engineering Project using new team works managment technics. Alexander Bueno was born in Caracas, Venezuela in 97. He received the Ing. and M.S. degree in Electrical Engineering from Universidad Simón Bolívar, Caracas, Venezuela, in 99 and 997, respectively. He has been a Lecturer of Electrical Power Engineering at the Industrial Technology Department of Simón Bolívar University for years. Currently, his research interests include space vector control applications, electrical machines, neural network estimators and power electronics.