A GENERALIZED MATHEMATICAL MODEL OF THE TWO-LEVEL CASCADED INVERTERS FEEDING THE OPEN-END STATOR WINDING INDUCTION MACHINE

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1 Journal of lectrical ngineering GNRLIZD MTHMTICL MDL F TH TWLL CCDD INRTR FDING TH NND TTR WINDING INDUCTIN MCHIN. NYLI. GUIZNI University of tunis, TT, Tunisia. University of l Manar, IIM, Tunisia. mail : n.ayli@hotmail.fr, Guizani_sami@yahoo.fr, F. BN MMR University of Carthage, MM Laboratory, INT, Tunisia mail : Faouzi.Benamar@insat.rnu.tn bstract: In this paper the authors propose the supply of the openend stator winding asynchronous machine by voltage converter to each input, these converters constituted by two or several level cascaded inverters. generalized mathematical model of the «n» levels inverters feeding the openend stator winding induction machine is presented. n extended generalized WM strategy is carried out to control the converter. Key words: penend winding induction machine, twolevel cascaded inverter, generalized mathematical model of cascaded inverters.. Introduction The power segmentation, the improvement of reliability and availability become priority in the industrial drives at variable speeds such as railways applications, aeronautics, electrical propulsion of ships and electrical vehicles []. The drive system is primarily based on the threephase synchronous or asynchronous machines associated with threephase level inverters but this association is not without disadvantage. To improve reliability and consequently the availability of this association inverter machine, everal researches have also been developed in inverter structures : HBridge inverters [4]. Twolevel cascaded inverters [5]. Different cascaded inverters [3]. Multilevel inverters with flyingcapacitor [4]. r in machine structures: Multiphase asynchronous or synchronous machine, where each phase is fed by its own voltage inverter. [57]. Multistar asynchronous or synchronous machine, where each star is fed by threephase voltage inverter. [89]. The openend stator winding asynchronous or synchronous machine, where each windings extremity is fed by threephase voltage inverter. [34] and [0]. This last type of machine, associated with the threephase level cascaded inverters will be the subject of this paper. In the first part, the modelling of the openend winding induction machine is successively presented in Clark (α,β) and ark (d,q) reference frames. The second part is devoted to the implementation of a simulation model of the machine. The model will be validated in the environment of «Matlab imulink». In the third part, the authors devote this paper to the supply of the openend stator winding asynchronous machine by multilevel structures with level cascaded inverters. The type WM strategy is used phase disposition WM. THD analysis is presented for the feeding the machine by level cascaded inverters. In the fourth part, this article gives the mathematical model for the supply of the openend stator winding asynchronous machine by level cascaded inverters and an extended generalized WM strategy is carried out to control the converter. Finally, the authors present the advantages of the association openend winding induction machine level cascaded inverters.

2 Journal of lectrical ngineering. imulation model of openend winding induction machine for voltage supply The supply of the openend stator winding induction machine is represented in the figure. The voltages of the machine are defined as: T s s s s3 T oltage vector inverter. s s s3 s oltage vector inverter. T oltage s s s s s s3 s3 vector of a stator winding of the machine. T Inverter Inverter T T3 T T T 3 Is 3 Is Is s 3 s s IM s s s 3 Fig.. penend stator winding asynchronous machine supplied by voltage source inverters. The mathematical flux model is written in (d,q) reference frame, and described by the following state equations representation: dx(t) (ω(ω dq X(t) [B].U(t) dt () Y(t) CX(t) Where: T X(t) sd sq rd rq : The tate vector U(t) U (t) U (t) T sd sd sq sq : The Control vector I I I I I T Y(t) sd sq rd rq : The output vector The functional diagram is given by figure : U U B T3 T 3 T T X Y C Fig.. Functional diagram of the openend winding machine. we consider the following parameters: L τ s s R Constant of time for the stator s T T L r r R Constant of time for the rotor r M ² sr : Coefficient of dispersion of Blondel L s L r Rr: Rotor resistance Rs: tator resistance Ls: tator cyclic Inductance Lr: Rotor cyclic Inductance Msr: Mutual Maximal cyclic inductance between stator and rotor. The equation of current vector is: I L () we obtained for the matrices,b et [C] are: Msr 0 s dq slr Msr dq 0 s slr (3) Msr 0 rls r dq Msr 0 ( ) rls dq r B (4) Msr 0 0 σl s σlslr Msr 0 0 σls σlslr C L (5) Msr 0 0 σlrls σlr Msr 0 0 σlrls σlr The drive mechanical equation is given as follows: T em d T r = j f (6) dt 3 T em = p(φ I Φ I ) (7) sα sβ sβ sα T em : lectromagnetic torque T r : Load torque 3. Feeding the openend stator winding induction machine by three phase inverters in cascade. To circumvent the balancing problems of floating sources and congestion encountered in the inverters

3 Torque (Nm) peed (rpm) Current () Current () oltage () oltage () Journal of lectrical ngineering with flyingcapacitor, then we will use the three phase cascaded inverters because they are modular inverters which therefore allows to reduce congestion. 3.. Feeding the openend stator winding machine by two level cascaded inverters The openend stator winding induction machine is fed by two levels inverters in cascade based on /f law, is shown in figure 3. T T Converter T3 T T T 3 Inv Figure 4 shows two triangular carriers and a signal reference. Fig. 5. ignal reference and carriers vertically shifted. Network 3 ~ T T T3 T T T 3 Inv The simulation results of the openend winding induction machine where each entry is supplied by two level cascaded inverters as shown in the figure 6. The load torque is the type kn². Induction T T T3 Inv T T T 3 Network 3 ~ T T T3 Inv T T T 3 Converter Fig. 3. penend stator winding asynchronous machine supplied with two level cascaded inverters. To control the two three phase cascaded inverters, the three reference signals of frequency fm and amplitude m are compared with two triangular carriers of frequency fp and amplitude shown by figure 4. p, as Fig. 6. The pahsetophase machine voltage, stator currents, speed and torque for Tr = kn². Fig. 4. rinciple of the phasedisposition WM for control two threephase cascaded inverters. Figure 7 shows the voltage (s s ), (s s ) and phasetophase machine voltage U which is 5 levels to supply with two threelevel inverters. With: s, s simple voltage of inverter s s pole voltage of inverter s, s simple voltage of inverter s s pole voltage of inverter. U = (s s ) (s s ) pole voltage of the machine. 3

4 β xis β xis Flux of rotor (Wb) Flux b of stator (Wb) Flux a of stator (Wb) oltage () oltage () oltage () Journal of lectrical ngineering Figure 0 shows the waveform of the phasetophase machine voltage and the harmonic content of the voltage. Fig. 7. ole voltage inverter and phasetophase machine oltage. Figure 8 shows the waveform of the stator and rotor flux. Fig. 0. Waveform and harmonic ration of machine voltage. Figure shows the waveform of the stator current and the harmonic content of the current. Fig.. Waveform and harmonic ration of current Fig. 8. Waveform of flux. Figure 9 shows space vector voltage of the two level cascaded inverters and machine. lot(α,β) lot(α,β) It is clear that the use of the level cascaded inverters improves THD voltage, THD current, torque undulations, and especially increases the degrees of liberty in degraded mode, while keeping the advantage of the power segmentation offered by the association of these structure inverters with the openend winding induction machine, indeed any number of the inverters put in cascaded, they will be of the dimension inferior or equal to the power half of the machine. t the end of this article, we show of the tables that summarize the dimensioning of the inverters and their advantages for the different number of the inverters put in cascaded. 3.. Feeding the openend stator winding machine by 6 level cascaded inverters α xis α xis Fig. 9. pace vector converter (Left) and machine (Right) The structure of supply of the openend stator winding induction machine by six level cascaded inverters is shown figure. 4

5 oltage () oltage () oltage () oltage () Journal of lectrical ngineering Fig.. penend stator winding asynchronous machine supplied by 6 threephase cascaded inverters Figure 3 shows three triangular carriers and signals references for control the six level cascaded inverters. Figure 5 shows the waveform of the phasetophase machine voltage and the harmonic content of the voltage. t(s) Fig. 3. ignal reference with three triangular carriers. The simulation results of the voltage (s s ), (s s ) and phasetophase machine voltage U which is 3 levels to supply with six level cascaded inverters of voltage is shown by figure 4. Fig. 5. Waveform and harmonic ration of voltage Figure 6 shows the waveform of the current machine and the harmonic content of the current. Fig. 6. Waveform and harmonic ration of current. Fig. 4. The phasetophase voltage inverter and phasetophase machine oltage. Table I summarizes the results for different levels of the phase to phase converter, and phase to phase machine voltage and harmonic ration of machine voltage for different structure to supply by the level cascaded inverters with phase disposition WM strategy. It is possible to observe the increase level of machine voltage and lower THD. 5

6 Journal of lectrical ngineering Inverter n Levels of voltage Inverter n N THD Current ( %) THD oltage ( %) In the phase disposition WM strategy, the reference signals modulating of frequency fm and amplitude m are compared with «p» triangular carriers, with the same frequency fp and amplitude shown in the figure 8. p p p as p Table I. Different levels of voltage and THD voltage. Carrier p Inverter p 4.Generalized phasedisposition WM strategy for control «p» inverters in cascade In figure 7, we represent the feeding of the openend winding induction machine by p inverters in cascade at entry and p inverters in cascade at entry. For the DC bus, the entry of two converters becomes = =, with the input of the «p» p Carrier Inverter cascaded inverters are p. T p T p Converter T 3p T p T p T 3p Inv p ref p Carrier ref ref 3 Inverter T T T 3 Inv Fig. 8. General principle of the phasedisposition WM strategy for inverter by «p» cascaded inverters. Network 3 ~ T T T 3 T Induction T T 3 T T T 3 Inv 5. Generalized mathematical model of level cascaded inverters We presented the generalized mathematical model of level cascaded inverters. That the DC bus for each inverter is. Figure 9 shows the structure of the two level cascaded inverters. T T T 3 T T T3 T T T 3 Inv B C T T T 3 Inv Network 3 ~ T T T 3 T T T 3 Inv T T T3 B C T T T 3 Inv T T T 3 T T T 3 Inv p Induction n Bn Cn Converter Fig. 7. penend stator winding induction machine supplied by «p» level cascaded inverters. Fig. 9. synchronous machine supplied by two level cascaded inverters. n 6

7 Journal of lectrical ngineering Notations: n : fictitious neutral. i : index indicating the number of the phase i, i = {,,3} j : index indicating the number of the switch j, j = [ p] T (i,j), T (i,j) : power switches. (i,j) : the corresponding switching signal of the T(i,j) (i,j) : the corresponding switching signal of the T (i,j). The (i, j) and (i, j) are complementary signals. From the scheme of Figure 9, we determinate the following equations: ' ( ) ( ) ( ) (8) We assume the DC bus for each inverter is and it is written by the following relation: For inverter : T' For inverter : 3 T' 3 ( ) (9) (0) In this case, the openend winding induction machine of power supplied to each entry by three phase cascaded inverters, there will be two inverters Inv and Inv dimensioned for a power also the two other inverters Inv and Inv dimensioned for a power 4. To put three cascaded inverters is the first power, the second is 3 power, the third is 6 power and so on. Thereafter figure 0 shows the structure of the three level cascaded inverters, we obtained the following equations: ' ( ) () ' ( ) 3 3 () Fig. 0. synchronous machine supplied by three cascaded inverters. That is to say: 3 (3) From the general case of «p» cascaded inverters, we obtained the general equation: p p j (4) j j The three voltages are obtained directly by the following expressions: n n B n C n n B (5) n C n B C 0 (6) n n ( B C ) (7) n 3 That is to say: ( B C ) (8) n The three voltages are described by the following matrix: n B n C n 3 3 T3 T Induction T T3 T T T 3 T T3 Converter T T33 3 B3 C3 T 3 T 3 T 33 B C B T3 T T T 3 p L j L j p L j L j p L j L j 3 n C n Bn Cn Inv 3 Inv Inv (9) 7

8 oltage () oltage () oltage () Torque (Nm) peed (rpm) Current () Current () oltage () Journal of lectrical ngineering For feeding the openend winding induction machine by cascaded inverters, the three voltages are written by the following matrix. For converter : n B n C n 3 For converter : p L j L j p L (0) j L j p L 3j L j p L j L j p L n () B 3 j L j n C n p L 3j L j The three voltages of phase machine are described by the following matrix: B B C C 3 p L p L j j L j L j p L p L j j L j L j p L p L 3j 3j L j L j () Figure shows the validation of the mathematical model correspond to the voltage (s s ), (s s ) and phasetophase machine voltage U which is 3 levels to supply with six level cascaded inverters. Fig.. The phasetophase inverter voltage and phasetophase machine oltage from the mathematical model. We used the mathematical model of three cascaded inverters to supply the induction motor. The simulation results are shown by figure. Fig.. The phase to phase voltage of the machine, stator currents, speed, and torque, for a load torque Tr = kn². The characteristics of the machine used: Nominal power = 45 KW. peed n = 450 rpm. Resistance of stator Rs = 0.5 Ω. Resistance of rotor Rr = Ω. Inductance of stator Ls = 7.9 mh. Inductance of rotor Lr = 8.6 mh. Mutual inductance Msr = 7. mh. 6. dvantage of the association openend winding induction machine level cascaded inverters If we consider that the induction machine (IM) and the openend winding induction machine (WIM) have a power ( = 45 kw (55 k) for our application), the table II summarizes the power of each inverter, the current and the voltage of the switches ( T ) for the three phase inverter. tructure Inverter owers Current I () switches oltage () IM WIM / Table II. Dimensioning of the inverter. 8

9 Journal of lectrical ngineering The table II show the openend winding induction machine is a good solution for the great power; indeed its two inverters are dimensioned to a half power of the machine. nd more, this structure allows the operation in degraded mode but with only one degree of liberty, since a single failure in one of the two inverters is tolerated. The table III summarizes the power of each inverter, the current and the voltage of the switches for the two three phase cascaded inverters. Inverters Inverter Current I witches owers () voltage () Inver and Inver / 80 T = 70 Inver and Inver /4 80 T =35 Table III. Dimensioning of the two three phase cascaded inverters The table III shows the use of the two cascaded inverters supply for the openend winding induction is not a constraint for the power segmentation since the cascaded inverters (Inv ) and lower power ( / 4 ) that principal inverter (Inv ), and more this structure improves the THD voltage and THD current, reduces torque undulations and especially increases the degrees of the liberty in degraded mode [5]. The table I summarizes the power of each inverter, the current and the switches voltage for the six three phase cascaded inverters. Inverters Inverter Current witches owers I () voltage () Inver and Inver / 80 T = 70 Inver and Inver 5/ 80 T = 5 Inver 3 and Inver 3 /3 80 T3 = 80 Inver 4 and Inver 4 /4 80 T4 =35 Inver and Inver /6 80 T5 = 90 Inver and Inver / 80 T6 = 45 Table I. Dimensioning of the six three phase cascaded inverters Table I shows that when increasing the number of cascaded inverters, the latters are of the power increasingly reduced, which does not render the very high cost of this structure type, on the other hand we increase of the degrees of system liberty of the machine supply in a degraded mode. Notes: We do not treat in this paper the cost of the inverters although it is evident that cascaded inverters especially for the number superior than associated with the WIM become costliest, this is the price to pay to increase the degrees of liberty and consequently improve the continuity of service of the variable speed drive which is our principal objective. 7. Conclusion We implemented the simulation model of the openend winding asynchronous machine supplied by two voltage sources cascaded inverters in the Matlab imulink environment. The simulation results show that the openend stator winding asynchronous machine has the advantage of increasing the level of phasetophase machine voltage. Indeed if we supply the machine by two «n» levels inverters, the level of phase to phase machine voltage is equal to N = n. The supply of the openend winding induction machine by level cascaded inverters clearly improves the THD voltage and current of stator. If we want to increase the level voltage just add lower power inverters in cascade. We developed a mathematical model generalized for the supply with level cascaded inverters of this openend stator winding machine. This model was validated in the «Matlab imulink» environment, it reduced the simulation time and simplified the schemes realised with ower ystem Blocks. The association openend winding induction machine level cascaded inverters ensure the power segmentation and modular inverters. It is also very interesting to improve the reliability of a variable speed drive system and without forgetting the redundant degrees it can offer. References. Blanke, M., andberg, T.J.: lectrical teering of vehicles faulttolerant analysis and design. lsevier Microelectronics Reliability (006), ol. 46, Issue 9, eptember 006, pp zli, N.., Nourdine, N.M., Idris N.R.: Direct torque control of multilevel inverter fed induction machine. Journal of Theoretical and pplied Information Technology JTIT (0), ol. 4 N., July 0, pp McGrath, B.., Holmes, D.G., Kong, W.Y.: Decentralized Controller rchitecture for a Cascaded HBridge Multilevel Converter. I Transactions on Industrial lectronics (04), ol. 6, Issue 3, pp

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