Modeling of a Twelve-phase Synchronous Machine Using Matlab/SimPowerSystems Liu Sheng Cheng Yinzhong College of Automation Harbin Engineering University Harbin Heilongjiang [email protected] College of Automation Harbin Engineering University Harbin Heilongjiang [email protected] Abstract—Electric propulsion is now gaining more and more interests in both civil and military ships. Multiphase machine, with many inherent advantages, is supposed to be a proper choice of the propulsion motor for electric propulsion systems. In this paper, mathematical model of twelve-phase synchronous machine with four Y-connected three-phase symmetrical windings displaced in turn by 15 degree is obtained. Then simulation model of the machine is established in Malabo using SimPowerSystems toolbox. In order to validate the simulation model, a simulation case is constructed and implemented, and the simulation results show that the simulation model of the twelvephase synchronous machine is usable. Keywords-twelve-phase synchronous eletric propulsion; model validation I. machine; modeling; INTRODUCTION Electric propulsion is now gaining more and more interests in both civil and military ships. Ships with Electric propulsion will be powered through a combined propulsion and electrical power distribution system. This revolutionary change will bring a lot of benefits for future ships [1]. 1) Unlocking of propulsion power, releasing the prime movers from ship propulsion; 2) Lower fuel cost, significant fuel savings can be realized through the use of electric propulsion; 3) Reducing manpower cost, reducing the amount of installed equipment reduces manpower requirements and therefore cost; 4) Design flexibility, reduced number of prime movers and auxiliary equipment brings an enormous amount of ship volume; 5) Improve the survivability and reliability. Therefore, significant effort is made to develop electric propulsion system all over the world. Electric propulsion presents important opportunities for overall ship improvement, but they are accompanied by unique challenges as well [2], such as increased system complexity, increased construction cost, increased requirement on components’ power density. One of the major challenges is the choice of the propulsion motor. Multiphase machine has many inherent advantages including ability to obtain high power in lower voltage level, fault tolerance, reducing the This paper is funded by the International Exchange Program of Harbin Engineering University for Innovation-oriented Talents Cultivation 978-1-4577-0321-8/11/$26.00 ©2011 IEEE 2131 torque ripple and so on, which make it be a proper choice for electric propulsion systems. The roots of multiphase variable speed drives can be traced back to the late 1960s. The pace of research started accelerating in the second half of the 1990s, predominantly due to the developments in the area of electric ship propulsion, which remains nowadays one of the main application areas for multiphase variable-speed drives. In view of the development of manufacturing twelve-phase synchronous machine is chosen to be the research object in this paper. In [3] Zhou presents the vector control of the twelve-phase synchronous machine, but the mathematical model given is just a part. Lin gives the whole mathematical model of the twelve-phase synchronous machine [4], but the model is not validated. In this paper, mathematical model of twelve-phase synchronous machine with four Y-connected three-phase symmetrical windings displaced in turn by 15 degree is obtained by deduction. Then simulation model of the twelvephase synchronous machine is established using SimPowerSystems toolbox in Matlab. A simulation case is implemented to validate the simulation model of the twelvephase synchronous machine. II. MATHEMATICAL MODEL The twelve-phase synchronous machine in this paper has four Y-connected three-phase symmetrical windings displaced in turn by 15 degree. The stator winding arrangement [5] of the twelve-phase synchronous machine is shown in Fig. 1. With some assumptions that the four three-phase windings are identical to each other and the structure of the rotor is symmetrical in d-q axis. Another assumption, also leading to a negligible error, is to set the permeability of the iron core to in¿nity and to ignore the parasitic interaction between the different winding systems. In order to model the machine, the first step is to transform the stator variables to a reference frame by Park’s transformation. In this paper, the rotor reference frame is chosen to be the reference frame [6]. Use the subscript “i” to represent four windings respectively. So we have: i=1, 2, 3, 4. The transformation equation can be written as: K S = [ K1 , K 2 , K 3 , K 4 ] (1) id 1 = id 2 = id 3 = id 4 ® ¯iq1 = iq 2 = iq 3 = iq 4 (5) λd 1 = λd 2 = λd 3 = λd 4 ® ¯λq1 = λq 2 = λq 3 = λq 4 (6) Where “i” stands for the current and Ȝ represents the flux linkage. All the variables are transformed to the rotor reference frame by using the transformation equation, We can obtain the voltage equations in d-q axis: udi = − rs idi − ωr λqi + pλdi °u = − r i + ω λ + pλ s qi r di qi °° qi ®u fd = rfd i fd + pλ fd °u = r i + pλ kd ° kd kd kd °̄ukq = rkq ikq + pλkq Figure 1. Stator winding arrangement of twelve-phase synchronous machine is: For the “i”th winding, the voltage in rotor reference frame Where ªu Ai º ªudi º « » = K i u Bi «uqi » «u » ¬ ¼ ¬ Bi ¼ π ª « cos[θ − ( i − 1) 12 ] « 2 π 2π K i = « cos[θ − − ( i − 1) ] « 3 12 3 « π 2π « cos[θ + − ( i − 1) ] 3 12 ¬ Where rs is the stator resistance, Ȧe is the rotor electrical angular velocity, the subscript “fd”, “kd” and “kq” denote the field winding and the d-q axis damper windings respectively. And the expressions for the flux linkages are: (2) º » 12 » π − ( i − 1) ] » 12 » » π − ( i − 1) ] » 12 ¼ − sin[θ − ( i − 1) − sin[θ − − sin[θ + 2π 3 2π 3 π T λdi = − Lls idi + Lmd ( −4idi + i fd + ikd ) ° ° λqi = − Lls iqi + Lmq ( −4iqi + ikq ) ° ® λfd = Llfd i fd + Lmd ( −4idi + i fd + ikd ) °λ = L i + L ( −4 i + i + i ) lkd kd md di fd kd ° kd °̄ λkq = − Llkq ikq + Lmq ( −4iqi + ikq ) ] The voltage of each stator winding can be written as: π °u Ai = 2U cos(ωt − (i − 1) 12 ) ° 2π π ° ®u Bi = 2U cos(ωt − − (i − 1) ) 3 12 ° °u = 2U cos(ωt + 2π − (i − 1) π ) °̄ Ci 3 12 (8) Where the Lls is the stator leakage inductance and Lmd and Lmq are the magnetizing inductances of the stator windings in d-q axis, respectively. Llfd and Llkd are the leakage inductance of the field winding and the damper winding in d axis respectively. Llkq is the leakage inductance of the damper winding in q axis. (3) Then we can obtain the expression of the electromagnetic torque: Applying (3) into (2) we have the relation of the voltage in d-q axis: ud 1 = ud 2 = ud 3 = ud 4 ® ¯uq1 = uq 2 = uq 3 = uq 4 (7) 3 2 p 2 4 Te = ( )( ) ¦ (λdi iqi − λqi idi ) i =1 (9) The torque and the rotor speed are relatived by: (4) 2 d ωr + Tl p dt Te = − J ( ) Similarly, we can obtain the currents and flux linkages relations of four windings in d-q axis: 2132 (10) Where P is the pole pairs of the machine, J is the inertia, and Tl is the load torque on the shaft. III. SIMULATION MODEL It is often convenient to express machine parameters and variables as per unit quantities. In order to obtain the simulation model of the twelve-phase synchronous machine, the equations of the machine are per unitized, and then the equations convenient for simulating the twelve-phase synchronous machine are established by solving the flux linkage per second equations for the current. The simulation model of the machine can be expressed as follows: rs ωb °ψ di = p [ u di + ω rψ qi + L (ψ md − ψ di )] ls ° rs ωb ° °ψ qi = p [ u qi − ω rψ di + L (ψ mq − ψ qi )] ls ° r ° ωb fd [ u fd + (ψ md − ψ fd )] ®ψ fd = p L lfd ° ° r ω °ψ kd = b [u kd + kd (ψ md − ψ kd )] p Llkd ° ° r ωb [ u kq + kd (ψ mq − ψ kq )] °ψ kq = p Llkd ¯ (11) 1 4 °°Te = 4 i¦=1 ( iqiψ di − idiψ qi ) ® °ω r = − ω b (Te − Tl ) 2 Hp °̄ (14) Where H is the inertia constant, Ȧb is the base electrical angular velocity, p is the differential operator. According to the equations convenient for simulating the machine, the simulation model of the twelve-phase synchronous machine is constructed in Matlab software using Simpowersystems toolbox. Fig. 2 shows the configuration of the simulation model of the twelve-phase synchronous machine, where the controlled-current source blocks and the voltage measurement blocks are used in the “interface” subsystem to interface the Simulink components with the Simpowersystems components of the simulation model [7]. Where p is the differential operator, ȥ denotes the flux linkage per second, and ȥmdand ȥmqare the magnetizing flux linkage per second of d-axis and q-axis respectively. The current can be expressed as: 1 (ψ di − ψ md ) °idi = − L ° ls ° 1 (ψ qi − ψ mq ) °iqi = − L ls ° ° 1 (ψ fd − ψ md ) ®i fd = − L lfd ° ° 1 (ψ kd − ψ md ) °ikd = − Llkd ° ° 1 (ψ kq − ψ mq ) °ikq = − L lkq ¯ (12) Figure 2. Configuration of the simulation model IV. When constructing and using a simulation model, it is important to consider its validity and how the validity can be evaluated. In this section, a simulation case is configured and implemented to validate the simulation model. In the above equations, we have: 4ψ di ψ fd ψ + + kd ) °ψ m d = L a d ( L L L lkd ls lfd ° ° 4ψ q i ψ kq ) + °ψ m q = L aq ( L ls L lkq ° ® ° L = ( 1 + 4 + 1 + 1 ) −1 ° ad Lm d L ls L lfd L lkd ° ° L = ( 1 + 4 + 1 ) −1 ° ad Lm q L ls L lkq ¯ MODEL VALIDATION This case investigates the starting of the synchronous machine. The configuration of the simulation case is shown in Fig. 3. The machine is powered by four three-phase programmable voltage source blocks. The rated power of the machine in the simulation is 45MW, other parameters we have p=6, UN=3300V, fN=18Hz, H=0.8. The simulation results of the rotor electrical angular velocity, the electromagnetic torque and the current in q-axis are shown in Fig. 4, Fig. 5 and Fig. 6 respectively. (13) The electromagnetic torque equation and the relation between the torque and rotor speed are: 2133 From these figures, it can be observed that the simulation results accord with the actual, which proves the validity of the simulation model. 6 4 q - a x is c u r re n t / p u 2 0 -2 -4 -6 -8 0 2 4 6 8 10 12 14 16 18 t/s 20 Figure 6. Simulation results of the current in q-axis Figure 3. Configuration of the simulation case V. In this paper, mathematical model of a twelve-phase synchronous machine with four Y-connected three-phase symmetrical windings displaced in turn by 15 degree is obtained. By using Simpowersystems toolbox of Matlab software, the simulation model of the twelve-phase machine is established. In order to validate the simulation model, a simulation case of the twelve-phase synchronous machine is constructed and implemented. The simulation results show that the simulation model of the twelve-phase synchronous machine is usable. 1.2 1 0.8 a n g u l a r v e lo c it y / p u CONCLUSION 0.6 0.4 0.2 ACKNOWLEDGMENT 0 -0.2 0 2 4 6 8 10 12 14 16 18 This paper is funded by the International Exchange Program of Harbin Engineering University for Innovationoriented Talents Cultivation. 20 t/s Figure 4. Simulation results of the rotor electrical angular velocity REFERENCES 3 [1] T o rq u e / p u 2 1 [2] 0 [3] -1 [4] -2 [5] -3 0 2 4 6 8 10 12 14 16 18 20 t/s [6] Figure 5. Simulation results of the electromagnetic torque [7] 2134 M. Benatmane, LCDR T. McCoy, T. Dalton and T.L. Cooper, “Electric power generation and propulsion motor development for U.S. Navy surface ships” Proceedings All Electric Ship: Developing Benefits for Maritime Applications, The Institute of Marine Engineers, (IMarE) Conference, Vol. 110, (2), pp 53-61, 29-30 September 1998. Kyle Schmitt, “Modeling and Simulation of an All-Electric Ship in Random Seas” Master thesis of MIT, 2010. Zhou Dejia, Zhao Zhengming, Wang Shanming and CHAI Jianyun, “Vector Control for a Twelve-Phase Synchronous Motor” Proceeding of International Conference on Electrical Machines and Systems 2007, Oct. 8~11, Seoul, Korea. Lin Hua and Zou Yunping, “Mathematical Models of 12-phase Synchronous Machine with Air Gap Flux-oriented Control” S&M Electrical Machines. 2004. Zhang Jingnan, Cong Wang and Dai Ke, “Research on a mathematical model of a multi_phase Y-coil synchronous motor” Journal of Harbin Engineering University, July 2007. Paul C. Krause, Oleg Wasynczuk and Scott D. Sudhoff, Analysis Of Electric Machinery And Drive Systems Second Edition. IEEE Press Power Engineering Series 2002. The Mathworks, Simulink User’s Guide. 2010, http://www.mathwor ks.com/help/toolbox/simulink/.