A direct torque control for a PMSM

A Direct Torque Control for a PMSMG. Mino-Aguilar, A. Michelle Domínguez, R. Maya, R. Alvarez, L. Cortez, G. Muñoz, F. Guerrero,S. Maya, A. M. Rodriguez, F. Portillo, H. Azucena.Benemérita Universidad Autónoma de Puebla, Mexico.gerardo.mino@ece.buap.mxAbstractFor a long time, permanent magnet synchronous motors (PMSM) were only applied in some particular fields, e.g. for servo drives. However, in recent years PMSM gained increasing importance in novel domains like automotive hybrid drive trains [1]. For Mexico, this boom represents a valuable opportunity for universities and research institutions, so that together with Mexican industry of the Electric Vehicle (EV) forming partnerships that result in an infrastructure that will enable the country to develop its own technology in terms of EV. This paper presents a competent and not complex direct torque controller (DTC) for PMSM drives.1. Introduction.The study of the electric machine and control has been toughened by two major universal problems: pollution (water, ground, and the atmosphere) and overpopulation. This last one has created in part extremely high demands of power and personal means of travel, and this has inevitably resulted in the first [2].An increase of electric motors efficiency, which consumes more than half the electricity generated, can result in an important electrical energy savings [3]. In recent years the trend has changed from not only improving the design characteristics of the electric machine by developing better ways to control them, but also using digital systems, power electronics and advanced techniques of control [4].In particular, the three-phase induction motors (IM) are commonly used in industry, because not only do they have a simple design, but also they are reliable and inexpensive. The permanent magnet synchronous motors (PMSM) are increasing in popularity due to their high efficiency and power density. Many motor applications require a precise control of both speed and torque, which is possible by using frequency converters between the power lines and the motor. Another way to save electricity is by adjusting the motor speed [5]. Nowadays it can be said that the trend of use variable speed drives has increased significantly, especially where the torque and speed are to be varied to control the speed, position, flow and torque.Among the various methods used to control motors, the direct torque control (DTC) technique has an important place. This technique involves the respective control of the torque and flux linkage by selecting a proper voltage vector.The essential idea of DTC is to pick an appropriate voltage vector using a predefined switching table. This is achieved by calculating the instantaneous torque and flux from the stator machine variables. The torque and flux are controlled directly and independently through the selection of optimum inverter switching states and limiting the errors of the flux and torque by hysteresis controllers. DTC’s operation is illustrated in Fig. 1.The DTC has advantages for implementing a driver of low-cost and high performance, because the only parameter of the engine involved is the stator resistance. The well-known DTC is implemented in Matlab/Simulink to observe the performance of the PMSM at different speeds and loading conditions. Blocks were implemented for the models such as the MSIP, the DTC, the park transformation and the inverter.2. Direct Torque Control (DTC).Direct torque control, is a vector control method, Figure 1. Diagram of a typical DTC−PMSM drive.which incorporates the control voltage converter inside the speed control algorithm. The optimal switching sequence of the inverter transistors (VSI) is directly based on the state of flux and electromagnetic torque of the PMSM. The necessary calculations are performed using the equations from the model of the PMSM as well as currents and voltages in the stator terminals [6].The general DTC premises are [8]-The estimated instantaneous flux vector should remain close to its reference.- The estimated torque should remain close to its reference.-The dynamic engine behavior can be studied from sudden changes that may occur both in the reference speed of the rotor or from the torque reference value. These assumptions involve four major tasksa)Determine the appropriate values of the references.b)Calculate the actual values of the flux vector andelectromagnetic torque.c)Make an appropriate interpretation of the errorsbetween estimated values and their references.d)Manipulate the state of the inverter to reduce thespeed error.Fig. 1 shows the DTC for a PMSM block diagram. One can see, that once one has the estimated and reference instantaneous values of electromagnetic torque and stator flux, we proceed to calculate the error between them; these error are used as input for the hysteresis controllers. The output levels achieved in this stage of the control are input signals to the block that is responsible for finding the right vector to get rid of the error rate. This procedure is made for each sampling instant for driving the PMSM to the desired speed value.3. Flux and torque reference.The hysteresis controller output is generated after the estimated flux is compared with its reference. However, this reference value is obtained based on the electromagnetic torque reference [7]. The stator flux module can be expressed asІψsІ= ( ψsD 2 + ψsQ2 ) 1/2 (1) where ψs is the stator flux vector, ψsD and ψsQ are the sD and sQ axes stator fluxes given byψsD = L d i sD + ψm , (2)ψsQ = L q i sQ . (3)L d and L q are the direct and quadrature inductances, i sD and i sQ are the sD and sQ stator currents and ψm is the flux from the permanent magnets.Using (1) - (3) the magnitude of stator flux can be expressed asІψsІ= [ ( L d i sD + ψm) 2 + ( L sQ i sQ ) 2 ] 1/2 , (4) and considering that the electromagnetic torque is given by (5) where P is the number of pole pairst e = 3 / 2 P( ψm i sQ + ( L d – L q) i sD i sQ ) , (5)in which, the maximum torque per unit of current is obtained by considering i sD = 0, then (5) is ast e = 3 / 2 Pψm i sQ.(6) Thus the current produced by the torque isi sQ = 2t e / 3Pψm.(7)Substituting (7) in (4) the function required to obtain the reference flux isІψsref І = [ ( L d i sD + ψm) 2 + L q2( 2t eref / 3Pψm ) 2 ] 1/2. (8)The reference torque is generated through the PI controller for speed.4. Estimated flux.Using the position of the rotor and the stator currents, the stator flux estimation is given by the following equation [7] where θr is the rotor angle, Ls and is are inductance matrix and current vector of the statorψs =ІψsІe jθr = ψsD +jψsD = L s i s +ІψmІe jθr, (9) 5. Estimated torque.Using the estimated fluxes and the measured stator currents transformed into the stationary reference frame the electromagnetic torque can be calculated using [7]t e = ( 3/2 ) P( ψsD i sD – ψsQ i sQ ). (13)6. Estimated angle.It is possible to estimate the stator flux angle ρs through its stationary reference frame components so the angle is calculated usingρs = tan-1( ψsQ /ψsD). (14)7. Hysteresis controllers.When the estimated value of the variable is far from its reference, it will be necessary to use a suitable voltage vector to modify it so it stays within certain limits around the reference. Hysteresis controllers aim to maintain the torque and flux errors within upper and lower limits allowed, so that when evaluating within these limits an output level is obtained to know the status of the variable.8. Switching table.The main objective of the switching table is the get the fastest torque response. The switching table presented by Takahashi and Noguchi (Table 3) takes into account the sector (α(1),α(2),…,α(6)) in which the stator flux vector is located in stationary reference frame (sD, sQ) and the digital output levels of the flux and torque hysteresis controllers (d ψs, dTe).Fig. 2 shows that each sector covering 60 electrical degrees by the regions α(1),α(2),…,α(6), where the sector 1 covers from –30° up to 30°, sector 2 from 30° to 90° and so on until 330° is reached. The plane is defined by the axes sD and sQ where sD is at zero degrees [7].As shown in Fig. 3, the determination of the sector is made with res pect to the angle ρs at which the flux vector ψs is located.Once located the sector and having the state of the signals d ψs, dTe which indicate whether the variable should be increased or decreased, the voltage vector applied at each instant is to be determinate. Fig. 3 shows a situation in which the flux vector ψs is in the sector α(2) at the lower limit of hysteresis band. If a flux increment is required, it is necessary to apply a voltage vector to increase the radius of this circle, the vectors that achieve this goal are 1, 2 and 3 [9].Table 3. Commutation table by Takahashi andNoguchiFigure 2. Sectors distribution to locate fluxposition.Figure 3. Example of positioning the flux vectorin sector 2.To vary the torque Te an action is required on the load angle (δ), that is, a vector which causes a displacement on ψs must be chosen. In the case of Fig. 3, vectors 3 and 4 will increase the torque, 5 and 6 will decrease it. Accordingly, the application of different vectors will change the ψs behavior in different ways (size and position).9. Simulation.The dynamic model equations of the PMSM and all procedures involved in the DTC can be implemented thru simulation.The PMSM is described using SIMULINK blocks. The initial position and the load are considered as inputs to system and the engine’s mathematical relationships aredefined through subsystems.Changes between reference frames are performed through Park transformations.To control the PMSM, is made thru the implementation of the DTC. Some of the blocks include the estimators of flux, torque and position, the selection of the sector in which the stator flux vector is positioned, hysteresis controllers, the PI controller for the speed, as well as others.Fig. 4 shows the DTC implementation. First current and rotor angle are sampled. Once sampled an axis transformation of the currents takes place, those currents are used to calculate the stator flux components which together with the currents are used to obtain the estimated flux, torque and angle. These estimations of flux and torque are entered along with their corresponding references as input signals to the blocks called Subsystem and Subsystem1, which are the hysteresis controllers of the DTC. The output signals of these controllers together with the sector selection block output go through the switching table where the optimal vector shall be obtained, thereby achieving necessary changes in the motor flux and torque to reach the reference speed.To find the sector in which the stator flux is located, the block called “Look-Up Table” is used, which evaluates the input with respect to certain key points previously established. The “Look-Up Table” simply delivers the output data that indicates the matching sector.10. Results.Fig. 5 shows the actual motor speed compared with its reference previously set to a constant value of 1500 rpm. Fig. 6a shows the flux behavior and Fig. 6b illustrates the torque behavior, it can be seen how both remain close to their reference values.Figure 4. SIMULINK block diagram of the DTC.Figure 5. Constant reference speed at 1500 rpm.Figure 6. Comparison of estimated valuesagainst their references.With constant reference speed and changes in load, in Fig 7 can be seen that when load changes there is a minor repercussion in speed. At the moment when the load increases, the rotor speed tends to drop, this can clearly be seen at the 1.5 seconds, where the load was increased from 1 Nm to 3 Nm, then there is an opposite effect when load goes down, the speed increases for a few moments to re-establish its reference.Fig. 8 shows the estimated torque as it tries to follow its reference formed around the load.In Fig. 9a a variable speed is introduced, the motor speed follows its reference and there are only minor variations when load is increased or decreased. Fig 9b illustrates the load applied to the motor. In Fig. 9c can be seen how the electromagnetic torque has to compensate both speed and load increases, so that for a few seconds the reference torque has more noticeable variation, this can be seen at 0.5, 2 and 2.5 sec when changes in speed are introduced.Figure 7. (a) Constant reference speed at 1000rpm, (b) load variations.Figure 8. Behavior of the electromagnetic torquedue to changes in load.Figure 9. (a) Variable reference speed, (b) load variations, (c) electromagnetic torque behavior. 11. Conclusions.Although several techniques of control exist, the main objective of this paper was the implementation of DTC for a PMSM, which is based on the control of the motor’s torque and flux separately. Both torque and flux estimators were been implemented, stator flux angle was also calculated. From different reference speeds has been obtained necessary reference values for both electromagnetic components. In addition to this, also, through the blocks Look-up Table and Direct Look-up table, were derived both the area in which the flux vector is located and the optimal switching vector applied to the engine.The performance of this technique was tested by simulations driven in Matlab/Simulink. The simulation results show desirable behavior. It can be seen that the control meets the demands of the system quickly, what can be translated as a friendly speed control. It was illustrated how the stator flux and the motor torque can be held under control through voltage space vectors.It has been demonstrated that both instantaneous stator flux vector, and electromagnetic torque produced in a PMSM, can be estimated directly from the currents and voltages sensed in terminals of the motor.12. References.[1] Ouassaid, M., Cherkaoui, M., Nejmi, A. and Maarou, M. “Nonlinear Torque Control for PMSM: A Lyapunov Technique Approach,” Proceedings of World Academy of Science, Engineering and Technology, vol.6, pp. 118-121, June 2005. [2] Kosow, I., Control de máquinas eléctricas, Mexico: Reverte, 1977.[3] Viqueira Landa, J., “La generación y transmisión de energía eléctrica en México,” unpublished.[4] Ponce Cruz, P., and Sampé López, J., Máquinas eléctricas y técnicas modernas de control, D.F., Mexico: Alfaomega, 2008.[5] de Almeida, A. 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