Model of Induction Machine to Transient Stability Programs

Model of Induction Machine to Transient Stability Programs Pascal Garcia Esteves Instituto Superior Técnico Lisbon, Portugal Abstract- this paper reports the work performed on the MSc dissertation Model of Induction Machine to Transient Stability Programs. This work is divided into two parts, the first is a numerical method for the estimation of induction motor doublecage model parameter from standard manufacturer data. The second part is a continuation in the development of a studentgrade program for transient stability analysis, implemented in MATLAB environment. The goal was to make the program capable of dealing with pumping systems in the networks. The dynamic models that already existed in the software were revisited and two new dynamic models were also implemented. The validation of the software s is performed with a side by side comparison with a program developed in [2] and PSS/E TM. N I. INTRODUCTION owadays induction machines are extensively applied in all sectors of industry due to their ruggedness, flexibility and low price. The presence of large induction machine in power systems plays a significant role in transient stability. Accurate machine parameters are crucial for analysis of the systems behaviour. Squirrel-cage-type induction machines are modelled with double-cage model, because it s only possible predict starting current and torque from manufacturer data with doublecage rotor. The usual information about the motor is given by the catalog data, such as full load active and reactive power, full load efficiency, breakdown torque, and starting current and torque. The conversion of these data into the equivalent circuit is a difficult task. A computer program for solving the non-linear equations was implemented and described in section II. After known the equivalent circuit parameters was developed the two models of the induction machine. This models were the main goal of this work, with the aimed of further expand a dynamic simulation program which has been under development in previous works performed by former I.S.T master students. This simulation package is intended for educational use, while accomplishing an approximate or even similar level of precision when compared to existing commercial packages. The nomenclature given on the dynamic models is the same as the one used by PSS/E TM, since this is the simulation software used as a reference. The software developed in [2] and PSS/E TM are used to recognize the developed software s through a side by side comparison between results obtained by the programs. The paper is organized in five section. In section II are present the models and the computer program for solve the nonlinear equation. In section III the differential representation of the dynamics models is given. Section IV reports the simulation results and discuss the obtained results. Section V gives the conclusions. A. Single-Cage Model II. Induction Machine Model Fig. (a) shows the equivalent circuit of a single-cage induction machine. This circuit has five different parameters. In TABLE I shows the manufacturer data of the induction motor and is well established that there only five independent parameters [2]. The single-case model cannot approximate the starting current and torque values. This is only possible with a double-cage model. Thus, there are only three pieces of useful manufacturer data (P mfl, Q FL and T M) for the correct estimation of the four independent parameters. Fig. (b) shows the equivalent circuit of a double-cage induction machine. This circuit has seven different parameters. Fig.. Steady-state star equivalent circuit for the three-phase induction machine: (a) single-cage model (five-parameter circuit), and (b) double-cage model (seven-parameter circuit) [] Therefore, relations between the parameters must be imposed. Generally, it s supposed that the stator and rotor leakage reactance are related. It s necessary another relation and following the sensitivity analysis done in [], it can be observed that the stator resistence influence on magnitudes P m(s FL), Q(s FL) and T(s M) has very little importance. Then, it s supposed that the stator resistence is linked with the rotor resistence X rd = K x X sd R s = K r R r ()

2 The determination of the single-cage parameters is essential because the calculated values are a good starting guess for the double-cage estimation. The problem formulation uses the three pieces of the manufacturer data P mfl, Q FL, T M to find the five single-cage parameters Rs, R r, X m, X sd by solving the nonlinear system of the form F(x) = f (x) = P mfl P m (s FL ) = f 2 (x) = Q FL Q(s FL ) = f 3 (x) = T M T(s M ) = Where F = (f,f 2,f 3) and x = (R r, X m, X sd), and the restrictions in () are take into account. Furthermore, it is important to impose that all of the variables are always positive, so the algorithm have imposed the restrictions: R k > X k > TABLE I MANUFACTURER DATA [] (2) (3) f (x) = P mfl P m (s FL ) = f 2 (x) = Q FL Q(s FL ) = f 3 (x) = T M T(s M ) = f 4 (x) = I st I() = f 5 (x) = T st T() = f 6 (x) = dt(s M) = ds Where F = (f, f 2, f 3, f 4, f 5, f 6) and x = (R, R 2, X m, X sd, X d, s M). Appendix contains the detailed expression of functions P m(s), Q(s), I(s) and T(s) used in formulation of the both models. C. Numerical Method The modified Newton-Raphson method was chosen for solving the nonlinear system of equations of the form (7) f(x) = (8) In the algorithm, the restrictions present in (4), (5) and (6) are easily included with a simple change of variables x = R R = x x 2 = R R 2 R 2 = x + x 2 x 3 = X m X m = x 3 x 4 = X sd X sd = x 4 x 5 = X d K x X sd X d = K x x 4 + x 5 (9) B. Double-Cage Model As mentioned before, with double-cage model is possible obtain the starting torque and current. Furthermore, with the sensitivity analysis done in [], new relations between parameters need be imposed X 2d = K x X sd R s = K r R Then, with (4) are five independent parameters. From the manufacturer data retires five useful pieces P mfl, Q FL, T M, I st and T st for the correct estimation of the parameters. In the double-cage problem, it s necessary force restrictions between the variables. In the rotor, the inner cage has always a greater leakages flux than the outer-cage, then (5) must be respected. When the motor starts, the outer cage impedance R 2 and X 2d prevails over the inner cage impedance R and X d, therefore (6) must be respected. (4) X d > X 2d (5) R 2 > R (6) The conditions in (3) are applied by using the absolute value of the variables as function input F( x, x 2, x 3, x 4, x 5 ) = () The iterative algorithm used is x (k+) = x (k) h n DF(x k, s k FL ) F(x k, s FL ) = () Where the Newton-Raphson method is modified by including a coefficient h n. This coefficient is calculated in each step to verify that F(x (k+), s M k ) < F(x k, s M k ) (2) The converge of the algorithm is controlled by calculating the norm of the error, F(x k ), and ensuring that the error in each iteration decreases. In the double-cage model, the expression for the slip at the point of maximum torque isn t simple. Then, the slip of maximum torque is calculated separately. Once the algorithm is iterative, there is always a value for slip from which the point of maximum torque can be calculated The problem formulation uses the five pieces of manufacturer data to find the seven double-cage parameters R s, R, R 2, X m, X sd, X d, X 2d by solving a nonlinear system of the form F(x)= T k m = (s k, x) T k m > T M (3)

3 IV. DIFFERENTIAL-ALGEBRAIC MODEL In this section presents the dynamic models in their differential and algebraic forms. In this paper only the description of the single-cage and double-cage dynamic models is given. The other models present in the simulations is not given due the large number of dynamic models presented. These models and the implemented algorithm are described in [5]. A. Single-Cage Model In order to simplify the forms, the models were formulated using the Park s approach. Equations are formulated in terms of the real(d-) and imaginary (q-) axe, with respect to the network reference angle [4]. The simplified electrical circuit for the single-cage induction motor is given in Figure 2. Where X, X and T are determined from the circuit resistances and reactance s estimated in a previous method: X = X s + X m X = X s + X rx m X r + X m T = X r + X m w s R r The mechanical equation is as follows: s = (T m(s) T e ) 2H (9) (2) Where the electrical and mechanical torque are given as by: T e = E d i d + E q i q T m = a( + s) 2 (2) B. Double-Cage Model Fig. 2 Electrical circuit of the first-order induction machine model [4] In a synchronously rotating reference frame, the link between the network and the stator machine voltage is as follows [4] v d = v h sin θ v q = v h cos θ Using the notation of Fig.2, the power absorptions are: p h = (v d i d + v q i q ) q h = (v q i d v d i q ) (5) (6) And the link between voltages, currents and state variables is as follows: v d = E d + r s i d X i q v q = E q + r s i q + X i d (7) The differential equations in terms of the voltage behind the stator resistance are: de d de q = w s se q E d + (X X )I qs T = w s se d E q (X X )I ds T ds = (T m(s) T e ) 2H (8) (8) The electrical circuit for the double-cage induction machine model is shown in Fig.3. As for the single-cage model, real and imaginary axes are defined with respect to the network reference angle, and (5) and (6) apply, [4]. Using the notation of Fig. 3, the link between voltages, currents and state variables is as follows v d = E d + r s i d X i q (22) v q = E q + r s i q + X i d And the two voltages behind the stator resistance model the cage dynamics, as follows [5] de d de q de d de q = w s se q E d + (X X )I qs T = w sse d E q (X X )I ds T = w s s(e q E q) + de d E d E q (X X )I qs T = w s s(e d E d ) + de q ds = (T m (s) T e ) 2H E q E d + (X X )I ds T (23) Where X and T, as in single-cage models, can be obtained from the motor parameters estimated in the previous method: X X r X r2 X m = X s + X r X r2 + X m X r + X m X r2 (24) X r2 + X rx m T X = r + X m w s X r2

4 The differential equations for the slip and mechanical torque are given by (2) and (2). Although, to the double-cage model the electrical toques is defining as follows T e = E d i ds + E q i qs (25) C = [ ] u = [ ] And for the double-cage dynamic model Fig. 3 Electrical circuit of the fifth-order induction machine model [4] C. Models in Algebraic Forms The differential expressions presented in the last section need to be converted into an algebraic state equation so that the implemented integration algorithm in [5] can be applied in the simulation software. The state function as the form of the following algebraic expression f = [x ] = [A][x] + [R] + [C][u] (26) This outline takes into account that the time constants and the parameters associated with the dynamic models remain constant throughout the simulation process and, therefore, do not need to be computed in every time step [5]. Matrix [A] contains the dependent associated time constants and model parameters, constants in all the computation. Matrix [R] includes the non-constant terms, these matrix is calculated in every step of the computation. Matrix [C] contains the independent terms, related with matrix [u], which includes the fixed inputs. R = [ A = E d E q x = E d E q [ s ] [ ] w s se q (X X )I qs w s se d (X X )I ds w s s(e q E q) + R() + (X X )I qs w s s(e d E d) + R(2) (X X )I ds (T m (s) T e ) 2H ] C = [ ] u = [ ] (28) The single-cage dynamic model represented by the differential equations (8) and (2), in its algebraic form, is given by R = E d x = [ E q ] s A = [ ] w s se q (X X )I qs w s se d (X X )I ds (T m (s) T e ) [ 2H ] (27) IV. NUMERICAL RESULTS, SIMULATION AND DISCUSSION After implemented the different models above some simulations and numerical analysis were carried out to prove the veracity. A. Estimation of induction motor parameters In this simulation, the manufacturer data represented in Table I has Kw as system base, nominal voltage 38 V and the nominal frequency is 5 Hz. The table II have the estimated parameters side by side with the software developed in [] and [2].

5 Table II Estimated Parameters of induction motor from Manufacturer data in table I Estimated Parameters Program [] Error (%) Magnitudes Program Rs=.2 Xsd=.638 Xm=2.7439 R=.8 Xd=.52 R2=.5 Rs=.2 Xsd=.638 Xm=2.725 R=.8 Xd=.5 R2=.5.69.9 PmFL=9.92 QFL=52.788 Ist/IFL= 8.293 Tst/TFL=2.399 TM/TFL=3. ηfl=.958 ---- X2d=.39 X2d=.39 --- Error (%) -.86 -.88 -.67.29 In the figures below the result of the developed program are represented by a black continuous line, while the PSS/E TM results are represented by a red continuous line. ) Single-Cage Validation Fig. 4 to Fig. 8 shows the response of the active power (P G6), reactive power (Q G6), voltage (V 6) and the speed deviation ( w) of the single-cage motor. The Table II shows the numerical results obtained by the two different programs. The results above for the estimation of the parameters from the manufacturer data represented in Table I are close and the error obtained to the magnitudes is small. B. Validation of the Dynamic Models To have the validation of the dynamic models, in Table III is given the results obtained in the load flow computation. In Fig.6 to Fig. 3 is given the simulation of the symmetric short circuit in Bus 4. This short circuit is cleared by removing a brunch connected to the faulted bus. Fig. 6 Single-Cage Active Power Fig. 5-Bus network Table III- Power Flow results for the 5-bus simulation Power Flow Results Bus Voltage [Pu] Angle [º] PG [MW] QG [Mvar] PL [MW] QL [Mvar]. Swing.6. 9.97 94.8 --- --- 2. PV. -.59 4. -7.4 2.. 3. PQ.9933-3.6 --- --- 2.. 4. PQ.989-4.3 --- --- 4. 5. 5. PQ.9735-5.3 --- --- 6.. 6. PV.999-3.72-4.9-3. --- --- Fig. 7 Single-Cage Reactive Power

6 Fig. 8 Single-Cage Voltage Fig. Double-Cage Active Power Fig. 9 Single-Cage speed deviation Fig. Double-Cage Reactive Power Before the fault until t= s, the system is stable how the Fig.4 to Fig. 9 shows. In the moment of the fault t= s, P G and Q G decreases and because of it the demand cannot be supplied. Then a mismatch between the mechanical and electrical torque occurs, which result in an increase of the speed machine. When the disturbance occurs, the voltage droops too to values near zero because of the low impedance caused by the sortcircuit. After clearance of the fault, all variables start to rise and oscillate close to the initial values and in t=2 s they reach its stationary and initial values. 2) Double-Cage Validation Fig. to Fig. 3 shows the response of the active power (P G6), reactive power (Q G6), voltage (V 6) and the speed deviation ( w) of the double-cage motor. Fig. 2 Double-Cage Voltage

7 APPENDIX Single-Cage Model Equations To the circuit of the Fig. the star equivalent circuit is considered and Thévenin was apply [], then Z th = + = R th + jx th R s + jx sd jx m (29) Fig. 3 Double-Cage speed deviation Also as in the single-cage case, the system until t= s is stable. In t= s, occurs the fault and the variables as shown in the figures above goes down. The P G6 and Q G6 decreases and with this the demand cannot be supplied. Then a mismatch between the mechanical and electrical torques occurs, which results in an increase of the speed machine. The voltage droops and goes to values near zero because of the low value of the impedance caused by the short-circuit. After clearance of the fault, all variables start to rise and oscillate close to the initial values and in t=2 s they reach its stationary and initial values. Where And { R th = real(z th ) X th = imag(z th ) V s = U 3 Z p (s) = + { R r /s + jx rd jx m Stator and rotor currents I s and I r in terms of slip s are (3) (3) V. CONCLUSION Through the observation of the results provided in section IV, it s possible to conclude that the estimation parameters for induction motors and the models implemented in the software that had been developed by students from Instituto Superior Técnico are correct and in concordant with PSS/E TM and the software in [2]. The results obtained to the estimation of induction motor parameters from the program developed and the program in [2] are similar and with low error. However, in some manufacturer data exists greater errors due do not be used the same relations K x and K r. The results obtained from the software and from the PSS/E TM are very close. Some errors are visible in the figures above. These errors are due mainly to the fact that variables have small values and shorts variations. Despite all of these errors and differences, the implementation of the software s presents precision very close to other simulation package with a commercial nature. V s I s (s) = R s + jx sd + Z p (s) I r (s) = Z p(s)i s (s) { R r /s + jx rd The torque in terms of slip is given by [] T(s) = 3p (I w r (s)) R 2 r s s (32) (33) Deriving (33) in order to s is obtained the value of slip which the breakdown torque is obtained s M = R th R r 2 + (X th + X rd ) 2 (34) Then, the expression for mechanical power, reactive power and breakdown torque required to estimate the error functions in (2) are P m (s FL ) = T(s FL ) w p p ( s FL) Q(s FL ) = 3 Imag{V s [I s (s FL )] } T(s { M ) = 3p R r (I w s s r (s M )) 2 M (35)

8 In the case of the double-cage model, Fig.2, like the singlecage model the star equivalent circuit is considered, then V s = U 3 Z p (s) = + + { R /s + jx d jx m R 2 /s + jx 2d (36) [4] Dr. Federico Milano, Power System Modelling and Scripting, Spain : ETSII, University os Castilla La Mancha [5] Pedro Araújo, Dynamic Simulations in Realistic-Size Networks. Lisboa: Instituto Superior Técnico, 2. [6] PSS/E, Program Application Guide 34., vol. II. [7] PSS/E, Model Library 34. Stator and rotor currents I s, I and I 2 in terms of slip is given by V s I s (s) = R s + jx sd + Z p (s) I (s) = Z p(s)i s (s) R /s + jx d I { 2 (s) = Z p(s)i s (s) R 2 /s + jx 2d (37) And the torque in terms of slip is T s (s) = 3p w s ((I (s)) 2 R s + (I 2(s)) 2 R 2 s ) (38) Then, the expression for mechanical power, reactive power, breakdown torque, starting torque and starting current required to estimate the error functions in (7) are P m (s FL ) = T(s FL ) w s p ( s FL) Q(s FL ) = 3 Imag{V s [I s (s FL )] } T(s M ) = 3p ((I w (s M )) 2 R + (I s s 2 (s M )) 2 R 2 ) M s M V s I s () = R s + jx sd + Z p () T() = 3p ((I { w ()) 2 R s + (I 2()) 2 R 2 ) (39) References [] Joaquín Pedra, On the Determination of Induction Motor Parameters from Manufacturer Data for Electromagnetic Trasient Programs, IEEE Transactions on Power Systems, Vol.23, No.4 November 28. [2] Joaquín Pedra and Felipe Corcoles, Estimation of Induction Motor Double-Cage Model Parameters From Manufacturer Data, IEEE Transactions on Energy Conversion, Vol.9, No.2 June 24. [3] José P. Sucena Paiva, Redes de Energia Eléctrica: uma Análise Sistémica. Lisboa: IST Press, 25.