Modeling of saturation in induction machines using EMTP, PSpice and a dedicated computer program 1:;

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1 '-l LSVR lectric Power Systems Research 3 (1994) LCTAC QWA SiSTmS ASr:1ACH Modeling of saturation in induction machines using MTP PSpice and a dedicated computer program 1:; J.F. Reynaud P. Pillay Department of lectrical ngineering University of New Orleans New Orleans LA 7148 USA Abstract The effects of saturation in an induction motor's performance during a direct on-line start are studied using three different techniques. n the first method the motor is modeled using the well-known lectromagnet Transients Program (MTP) with saturation included in the magnetizing branch only. Secondly the motor is modeled using PSpice a popular circuit simulation software package where magnetic-core saturation is modeled more accurately than in MTP. n the final method the motor is modeled by taking into account cross- as well as magnetic-core saturation and implemented in a dedicated computer program. A comparative evaluation of the results is presented. Keywords: Saturation modeling; nduction motors 1. ntroduction n an industrial environment it is important to be able to predict as accurately as possible the mechanical and electrical response of electrical machinery under different power supply disturbances. The effects of unbalanced voltages have been well documented in the technical literature [ 2]. Supply short-circuit faults have been recognized as the cause of severe transients on electrical machines. Upon reapplication of the supply even more severe transients can occur depending on the magnitude and phase of the motor's residual back emf. The current transient could easily exceed even that produced at starting which itself is typically five or six times the rated value. t is during these conditions of starting or reconnection of the supply that machines encounter deep levels of saturation. t then becomes mandatory to accurately model saturation effects for the correct determination of motor transients. n this paper several different methods of modeling saturation are studied and evaluated including the use of MTP PSpice and a dedicated computer program... Presented at the Third Biennial Symposium on ndustrial lectric Power Applications New Orleans LA USA November ] nduction motor model 2.1. MTP implementation The lectromagnetic Transients Program (MTP) is a powerful software package for the simulation of power systems. ncluded in the package is a Universal Machine (UM) model. The model used in MTP is as follows: V = Ri + L :t i + WrGi (1) where V= VdS ds Vdr ldr i= Vqs lqs [ Vqr ] [ iqr] R = Rr Rs (2) [R' Rr ] Lmo L= Lmo Lro LsQ (3) [L'D LmQ LrQ L] lsevier Science S.A. SSD (94 )86 7-4

2 28 i.f. Reynaud P. Pil/ay / lectric Power Systems Research 3 (/994) G= r -LmQ LsD = Ls + LmD LSQ = Ls + LmQ LrD = Lr + LmD -LrQ LmD LrD ] (4) (5) (6) LrQ = Lr + LmQ (8) R is the resistance matrix (constant) LsD and LsQ are the D and Q axis stator self inductances LrD and LrQ are the D and Q axis rotor self inductances and Wr is the rotor speed. JOe electromagnetic torque is defined for both saturated and unsaturated conditions as (9) Te = -2"2 Lm(isQirD dwr Te = J dt + flwr + 1; - isdirq) (7) (1) n MTP saturation is approximated by a two-segment piecewise curve of the magnetizing characteristics of the motor (Fig. 1). During saturation the inductance matrix L can only take two values for Lm. Fig. 2 shows the approximation of the magnetizing curve of the motor by a two-segment piecewise curve. The user must select a point in the magnetizing curve as the border between saturated and unsaturated operation. By inspecting this magnetizing curve it was determined that the motor operates in the saturated region when 1m> 1.73 A. This point is labeled as P in Fig JOO "... " " > uc '" " :: Fig. 2. Approximated l.a9nolizing Curron! (A) magnetizing curve. Fig. 3 shows the circuit representation MTP PSpice implementation as used in n PSpice the induction motor is modeled using the same D Q axis equations given in Section 2.1. However saturation is accounted for in the magnetic core more accurately than in MTP. A slightly different form of the torque equation is used in PSpice for ease of implementation: Te = -2"2 (Aqridr - Adriqr) () Aqr= Llriqr + LmQ(iqs+ iqr) (12) c:... " :: Fig. 1. Magnetizing curve. l.agnetizing Currenl (A) Adr = Llr idr + LmD (idr + id.) dwr Te = J dt + flwr + T (13) (14) n (12) and (13) LmD and LmQ are considered saturable and can be modeled as nonlinear functions of their currents. The inductance curve for the motor under study is approximated by a nonlinear function of the magnetizing currents (Fig. 4):.38 ( 15) LmD = iml L mq--.38 (16) imq Since PSpice is a circuit simulation package an equivalent circuit must be generated to represent the D

3 J.F. Reynaud P. Pillay / lectric Power Systems Research 3 (1994) R. A R).f 281 Rb Rc A T UR T v. Vb Vc RCnd Rom.1 Fig. 3. MTP equivalent circuit. - Lm (True & Approx.) O.!>.4 Vl :1 Rw Lt.. u.. R- Urd.. 'C V3 2 Rw L.. L.. Fr.1 Unq a m Fig. 4. nductance curves. (A-rrns) Fig. 5. PSpice D Q axis equivalent circuits. Q axis equations. These are shown in Fig. 5. The circuit representation of q. (1) is given in Fig. 6 where (17) Tel = "2'2 Adriqr re2 ="2 '2 Aqridr (18) V through Vs in Figs. 5 and 6 are dummy (zero) voltage sources to measure currents. and 2 are current-controlled voltage sources related to the G matrix in q. (4). Tel and Te2 are current-controlled voltage sources representing the electromagnetic torque of the motor. Rte and LJ are respectively the friction coefficient and the inertia of the motor. n order to implement saturation in the magnetizing branch of the circuit a nonlinear inductor was modeled as shown in V5 Fig. 6. Torque equivalent circuit. - Ri:e the Appendix. The nonlinear inductor is defined as a unit inductance in series with a current-controlled voltage source whose instantaneous value represents the inductance curve for the motor. LJ

4 282 J.F. Reynaud P. Pillay / lectric Power Systems Research 3 (/994) Modeling and implementation using a dedicated program LmQ imq = Lm + -:-- Ldq md (25) The most'accurate method [3-51 to account for core saturation is to include cross saturation between the D and Q axes which is absent under the assumption of linearity of the magnetic circuit. q. (1) characterizes the induction motor in the saturated n this case the Land G matrices change to account for the cross-saturation terms: [ LsD LmD Ldq L = LmD Lrd Ldq G= [ Ldq Ldq LSQ Ldq Ldq LmQ -Lm Ldq Ldq LmQ LrQ Lm Lr -Lr The components of Land ] ] (19) (2) G are defined as follows: Lm = j (21) imd imq dlm Ldq = 1'm dl'm Lr = Lrl+ Lm LmD imd = Lm + -:-- Ldq mq (22) (23) (24) LsD = LS + LmD (26) LsQ = LS + LmQ LrD = Lrl + LmD LrQ = Lrl + LmQ (27) (28) (29) where Lm is the magnetizing inductance fm' imd imq and 'm are the magnetizing flux linkage D and Q axis magnetizing currents and the resultant magnetizing current vector respectively and Ldq is the cross-saturation inductance. The flux of the machine can then be expressed in terms of the resultant magnetizing current vector as follows: t/1m =.822 tan-l(.652iml) +.54jiml (3) The electromagnetic torque is defined for both saturated and unsaturated conditions as Te = Lm(isQird - isdirq) (31) dwr f3 Te=Jd(+ wr+1] (32) The introduction of the Ldq term in the inductance matrix due to the effects of cross saturation between the D and Q axes induces a much greater computational burden on the model a z '-' Qj =' " h 2iJ -1 GO GO Time (msec) Fig. 7. Torque variations for the MTP unsaturated

5 J.F. Reynaud P. Pillay / lectric Power Systems Research 3 (1994) '" Z '" W J [j L : : "''-..j" ' Oms 2ms 4ms 6ms 8ms looms a v(34) Tim Fig. 8. Torque variations for the PSpice unsaturated n order to solve q. (1) a fourth-order Runge- Kutta method is used; hence q. (1) is arranged in state space form: i i = L- ( V - Ri - w dt Gi) r dwr = Te - 13wr- r. dt J 3. Results (33) (34) ::> C" /"' "'. 36 J / '" ) '- "' The motor used in the simulation is a four-pole 5 hp induction motor with the following parameters: Rs = 1.86 Q Rr=2.12Q Ls =.11 H Lr =.6 H Lm(unsaturated) =.4699 J =.625 kg m2fs2 The simulation was carried out at 36 V to emphasize the effects of saturation on the machine performance. A direct on-line start under this input voltage was simulated. Figs. 7 8 and 9 show the results for MTP PSpice and the dedicated program respectively under linear conditions. t can be seen that all results agree as expected. The electric torque reaches about Time (s) Fig. 9. Torque variations for the dedicated program unsaturated case 28 N m at startup. When saturation is included though the results change. Fig. 1 shows the results for MTP. t can be seen that torque peaks are reduced due to the inclusion of saturation in the magnetic core. However higher torque peaks than the other two cases are observed. This is due to the fact that the magnetizing curve of the machine is approximated in a rather inaccurate way; thus the motor operates longer in the unsaturated region producing higher torque values at startup.. The PSpice results (Fig. 11) show the same trend. Torque peaks have decreased even more than in MTP which can be attributed to the more accurate representation of the magnetizing curve of the motor.

6 284 J.F. Reynaud P. Pil/ay / lectric Power Systems Research 3 (/994) ;".; a '-' z Q) C" -< -t 1 5 '-"....:) no "" no..no'.. Fig. 1. Torque variations for the MTP saturated 2 1 ro so Time (msec) 3; t... 2C+ + Z 'V U L f ' :. :. " Orns 2ms "Oms 6ms 8!)ms looms D v(34) Time Fig. 11. Torque variations for the PSpice saturated With cross and magnetic-core nonlinearities included in the machine dynamics the motor shows less developed torque agreeing with the previous results; however a more accurate response is obtained due to the cross-saturation terms in the L matrix. Fig. 12 shows the results for the dedicated program. 4. Conclusions This paper has examined how saturation affects the machine performance during startup conditions. t is shown that the inclusion of cross- and magnetic-saturation factors in the model gives the most accurate re-

7 J.F. Reynaud P. Pillay / lectric Power Systems Research 3 (1994) ( "'t ( 12! -;... j '...!. 36 / / Time (5) L 1 H HL (v(k)-1)*v( i j) Fig. 12. Torque variations for the dedicated program saturated suts. Although saturation was only included in the main flux using a continuous magnetizing curve the PSpice results appear to be more accurate than the results obtained using MTP in which the magnetizing curve of the motor is approximated with a two-segment curve. Simulation time is always a constraint that must be satisfied in all computer models. n this respect MTP shows the best performance followed by the dedicated program. Although it is much simpler to simulate the interaction of the electrical network with the electrical machinery and its loads using an off-the-shelf package such as PSpice or MTP it is shown in this paper that a dedicated computer program provides the most accurate results. Furthermore all models used in this paper are suitable for the calculation of machine performance in a variety of situations such as bus transfer and power disturbance studies. Acknowledgements The work reported in this paper was supported by the lectric Power Research nstitute the ntergy Corporation Louisiana Power and Light Co. and the Mobil Oil Chalmette Refinery. Fig. A. Nonlinear inductor. Appendix The model of the nonlinear inductor used to implement saturation in the magnetizing branch is shown in Fig. A where k is the voltage source node defining LmD or LmQ' References [] J.e. Das ffects of momentary voltage dips on the operation of induction and synchronous motors Trans. nd. Appl. 26 (199) [2] K.J. Sanghani Protection of large induction motors against restarting on out-of-phase power 2nd Biennial Symp. lectric ndustry Applications New Orleans LA USA 199. [3] J.. Brown K.P. Kovacs and P. Vas A method of including the effects of main flux path saturation in the generalized equations of A.C. machines Trans. Power Appar. 3yst. PAS-12 (1983) [4] K.-. Hallenius P. Vas and J.. Brown The analysis of a saturated self-excited asynchronous generator Trans. nergy Conv. 6 (1991) [5] J.O. Ojo and A. Lipo An improved model of saturated induction machines Trans. nd. Appl. 26 (199)