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2010 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2010), October 3-5, 2010, Penang, Malaysia Study into Effects of Stator Resistance Variation on Direct Torque Controlled Surface Permanent Magnet Synchronous Motor Drive Ali Jafarian Abianeh, Hew Wooi Ping Department of Electrical Engineering, University of Malaya, Kuala Lumpur, Malaysia E-mail: [email protected] Abstract— in the recent past decade, direct torque controlled (DTC) permanent magnet synchronous motor drives have been widely utilized in many relevant applications, as they offer numerous advantages compared to other control schemes. In fact, for the stator flux estimation of such scheme the motor parameter dependency is much less than the previous control schemes and it is just concerned with stator resistance. In practice, the stator resistance is prone to large variations caused by temperature which indeed are very likely to occur. The mismatch between actual and estimated resistance could deteriorate the performance of the DTC drive by introducing errors in the stator flux linkage estimation especially at lower speeds. This paper analytically investigates different effects of stator resistance variations on the performance of the direct torque controlled SPMSM drives while the simulation results are also provided. In order to reduce the level of torque and flux ripples, some researchers have proposed using multilevel inverters [5], and some others have focused on SVM based inverter switching technique [6]. In this regard, Intelligent based SVM-DTC schemes [7] and adaptive based SVM-DTCs [8], [9] have shown a superior performance as they have the capability of dealing with uncertainties and nonlinearities present in motor dynamics. The rest of DTC problems could be referred as estimation related issues. In fact, in a DTC scheme the stator flux linkage is estimated through integration of the difference between input voltage and resistive voltage drop as given by (1). Keywords- permanent magnet synchronous machine, stator resistance, torque control. Where φs0 is the initial stator flux linkage vector and Rs, vs and is are the stator resistance, applied stator voltage vector and stator current respectively. Due to presence of integrator in equation (1) any dc offset in the measured stator current and voltage could lead to drift of the estimated stator flux linkage. In this regard many researchers have proposed different techniques which each one has its own pros and cons [4]. Furthermore, according to (1) for a desirable DTC drive scheme, the initial rotor position must be known. In fact, unkown initial rotor position even could lead to wrong direction of motor rotation on the starting stage. Numerous estimation methods are proposed which managed to detect the initial rotor position in spite of their pertaining limitations [4]. From equation (1) it could be clearly seen that the only motor parameter employed for the flux estimation, is the stator resistance. Despite less motor parameter dependency of DTC drives compared to other schemes, the error in flux estimation could also occur due to stator resistance variation. Actually, such a resistance changes is mainly due to temperature variation which is caused by motor losses. In this regard, copper loss is more dominant and it is proportional to stator current value. Therefore, the larger the stator current is the higher resistance variation could occur. Basically, the stator resistance could change up to 1.5-1.7 times of its nominal value. As a result, a mismatch between estimated and actual stator resistance value happens, and therefore the estimated flux and torque provides wrong values for the DTC scheme, which deteriorates the performance of such motor drives. It should be noted that in the case of lower speed and higher ϕ s = ∫ (vs − Rs is )dt + ϕ s 0 I. INTRODUCTION In the past few decades, the field oriented control (FOC) scheme has been widely used in different applications of electric motor drives. However, such a control scheme involves algorithm complexity as well as sluggish response [1]. In the late 80’s Takahashi and Depenbreck proposed a new control scheme that was based on direct control of torque and stator flux linkage by employing the proper voltage vectors which was referred as direct torque control (DTC). The first proposed DTC schemes were designed for the Induction motors. However, in the late 90’s the PMSM drives based on DTC scheme appeared in the literature [2], [3], which combined the advantages of DTC and PMSMs so as to satisfy the requirements for high performance drive applications. In DTC scheme, the desired torque value could be obtained through controlling the amplitude and rotational speed of stator flux. As a result, fast torque response with a simpler control algorithm is attainable. However, there are a few shortcomings to DTC schemes which in order to provide a high performance drive, it has to be addressed properly. In this regard, excessive torque and flux ripples, drift in stator flux linkage estimation due to measurement errors, position sensor requirement for locating the initial rotor position as well as error in stator flux linkage estimation due to resistance variations, could be pointed out [4]. 978-1-4244-7647-3/10/$26.00 ©2010 IEEE 361 (1) load torque values, this effect could be more severe as even it can makes the drive to become unstable. In the recent decades, many researchers have studied different effects of stator resistance variations on DTC drives. Some has just focused on Induction motor DTC drives [10], [11] and some others evaluated such a behavior for PMSMs [12], [13]. However, in terms of PMSMs DTC drives, most of the reported papers are restricted to interior mounted permanent magnet synchronous motor (IPMSM) and lack of enough research for surface mounted permanent magnet synchronous motors (SPMSM) is noticeable. Although, the response of DTC SPMSM drive to stator resistance variations might be similar to the field weakening-based DTC IPMSM drives, still there are some considerable differences between them especially in terms stator current changes. In this paper, different effects of stator resistance variation on performance of DTC SPMSM drives are analytically investigated. Moreover, the expected responses to such a variation in terms of electromagnetic torque, stator flux and stator current are confirmed through simulation results. It is also demonstrated that the lower values of actual stator resistance could lead to unstable drive conditions in SPMSM DTC drives as previously was reported for IMs and IPMSMs DTC drives [11], [12]. In simulation stage, two different types of stator resistance variations are employed. First, it has been changed by a single step so as to provide a perfect comprehension of expected response changes. Then, a linear stator resistance variation, which represents more realistic resistance changes, is applied so as to display the problem caused in practical DTC SPMSMs drives. II. PRINCIPLES OF DIRECT TORQUE CONTROL The equations of surface mounted permanent magnet synchronous machine (SPMSM) in the rotor reference frame are as follows: ϕ sd = Lsd isd + ϕ m (2) ϕ sq = Lsq i sq (3) ϕ s = ϕ sd 2 + ϕ sq 2 (4) result, it is preferred for the flux and torque estimation to be carried out in the stationary reference frame which enables elimination of coordinate transformation and quicker dynamic response. The SPMSM equations in the α-β reference frame could be outlined as below: ϕ sα = ∫ (v sα − Rs i sα )dt + ϕ sα 0 (6) ϕ sβ = ∫ (v sβ − R s i sβ )dt + ϕ sβ 0 (7) ϕ s = ϕ sα 2 + ϕ sβ 2 T= 3 p(ϕ sα i sβ − ϕ sβ i sα ) 2 (8) (9) Where Rs is the stator resistance, φsα0 and φsβ0 are the initial components of stator flux, φsα and φsβ are the two components of stator flux, isα , isβ are the two components of stator current, vsα , vsβ are the two components of stator voltage in the stationary reference frame. Block diagram of a classical DTC SPMSM drive is shown in fig 1. It could be clearly seen that such a control scheme has some main parts including hysteresis controllers, switching table and Flux-torque estimation unit. Also, it should be noted that besides the initial flux vector detector (encoder), just two current sensors and one dc bus voltage sensor are employed for measurement purposes. Figure 1. Block diagram of classical DTC SPMSM drive T= 3 pϕ m i sq 2 (5) Where φsd and φsq are the direct and quadrature axis components of stator flux, Lsd , Lsq are the direct and quadrature axis components of stator inductance, isd , isq are the direct and quadrature axis components of stator current in the rotor flux reference frame. Also, φm is the magnetic rotor flux, p is the pair of motor poles and T is the produced electromagnetic torque. In the case of DTC scheme, the stator voltage could be obtained through DC bus voltage and inverter switching states. Thus, for measurement purposes just two current sensors and a dc bus voltage sensor are sufficient. As a 362 At first stage of such scheme, the command torque and stator flux values are compared with their corresponding estimated values. Then, the obtained torque and flux error values are applied to three-band and two-band hysteresis controllers respectively. The outputs of controllers along with stator flux position sector are employed to index a look up table of optimum voltage vectors. In fact, the selection of the voltage vectors are made so as to restrict the torque and flux error values in their allowable range of hysteresis bands. Consequently, a proper motor drive control with quick torque response could be attained. III. ϕ se = ∫ (v s − Rse is )dt ANALYSIS OF THE STATOR RESISTANCE VARIATION EFFECTS ON DTC SPMSM DRIVE PERFORMANCE As it is mentioned earlier, one of the main problems associated with DTC drive schemes is the stator winding resistance variation. In practice, the stator resistance changes as a result of temperature variation caused by machine losses. In this regard, stator current could substantially affect the losses amount as the produced copper loss is proportional to it. The resistance variation caused by machine losses could be represented by equation (10) as follows: Rs = Rs 0 [1 + α (T − T0 )] (10) Where Rs and Rs0 are the respective actual and initial stator resistance, α is the temperature coefficient of resistance and T and T0 are the actual and initial stator winding temperature respectively. As the resistance value varies by temperature, a mismatch between actual and estimated stator resistance occurs which can deteriorate the performance of DTC drives to some different extent. For instance, in the case of higher actual resistance values (caused by higher temperature), the produced effects might just be restricted to worsened stator flux response and higher copper losses resulted from higher stator current values. However, it should be noted that these effects could become more severe depend on torque limit assigned by speed controller, as even an inappropriate choice of these values could make the drive to become unstable for such a resistance variations. On the other hand, for the actual resistance smaller than the estimated one (cold environments), the unstable condition usually occurs as also reported for IMs and IPMSMs DTC drives [11], [12]. In fact, this instability is caused by large voltage vectors chosen based on wrong stator flux and torque error values. In this section, different effects of stator resistance variations on the performance of DTC SPMSM drives will be analytically explained in details by means of motor equations and stator current d-q plane. Basically, as the actual stator resistance of PMSMs ( ) increases, from (11) it is understood that due to constant applied voltage vector (vs) on that specific moment, the stator current (is) momentarily decreases. Thus, according to (12) since the estimated resistance ( e) is kept at its initial value, the resistive voltage drop on the estimator is also reduced. Therefore, the smaller voltage vectors (vs) are chosen by drive control scheme so as to compensate for the lower voltage drop caused and ensure providing estimated flux equal to desired value. Consequently, the smaller chosen voltage vectors results in smaller actual flux (φsa). The stator flux response to lower actual stator resistance values could also be similarly justified. In fact, for smaller actual resistance ( ) values, higher voltage vectors (vs) are chosen which results in higher actual stator flux linkage. In this case, the wrongly chosen large voltage vectors are known as the source of DTC drive instability for such a resistance variation, which later will be clearly demonstrated by means of simulation results. ϕ sa = ∫ (v s − Rsa is )dt (11) 363 (12) The effects of stator resistance variations on the stator flux linkage could be summarized as follows: ⎧ Rsa > Rse ⇒ ϕ a < ϕ e ⎨ ⎩ Rsa < Rse ⇒ ϕ a > ϕ e (13) In terms of the electromagnetic torque, the produced effects are supposed to be similar with stator flux linkage. However, since the motor drive is working as speed closed loop with constant load torque, the actual output torque would not be affected much. It is due to the fact that the applied load forces the actual torque to be constant and provides equilibrium with the applied load torque. As a result, just some negligible variation of actual torque might occur. On the other hand, the estimated torque will be pushed to the higher values for larger actual resistance values so as to enable the actual torque to be constant. Similarly, for smaller actual resistance, the estimated torque value will reduce. Thus: ⎧ Rsa > Rse ⇒ Ta < Te ⎨ ⎩ Rsa < Rse ⇒ Ta > Te (14) In order to investigate effects of stator resistance variation on the stator current, the constant torque and flux curves for the SPMSM has to be drawn on the stator current d-q plane. In fact these curves could be drawn based on the stator flux and torque equations of SPMSMs which is represented by equations (15) and (16) as follows: T= ϕ= ∫ (L i d d 3 pϕ m iq 2 (15) + ϕ m ) 2 + ( Lq i q ) 2 (16) The constant torque and flux curves in stator current d-q plane are drawn as shown in fig 2. Thus, by means of drawn constant flux and torque curves, and previously explained expected actual and estimated torque and flux responses, the expected changes on stator current could be easily understood. Figure 2. Constant torque and flux curves for SPMSMs As it is mentioned earlier, for any stator resistance variation in DTC drives, the actual flux changes while the actual torque is kept constant equal to load torque. Consequently, the new intersection point of constant flux and torque curves determines the new operating point of stator current components and the resultant stator current value respectively. In the case of SPMSMs, since the direct axis component of stator current does not need to build up flux, it could be set to about zero. Thus, the stator current is just including quadrature axis component (torque producing component) of stator current. It also has the advantage of providing maximum torque per ampere (MTPA) for the SPMSMs drive, since the quadrature components is the only torque producing current component in the case of SPMSMs. Thus, it could be concluded that by choosing reference stator flux equal to magnet flux, the intersection point of constant flux and torque curves is almost placed on q axis (MTPA path) so as to ensure maximum torque per ampere which is desirable for many relevant applications. If the stator resistance increases, the actual flux is lessened while the actual torque is unchanged. Thus, according to fig 2 the operating point of stator current in d-q plane move to the left hand side which implies that some negative direct axis current component is produced while the quadrature component is maintained at the initial value. Consequently, the stator current increases for the higher stator resistance values. On the other hand, for the reduced stator resistance values, the actual stator flux is increased while the actual torque is still not varied. Thus, the stator current operating point moves to the right hand side which produce positive direct axis current component with the same quadrature component value. Therefore, the stator current for the lower stator resistance is also supposed to be risen up. However, due to unstable conditions caused by large chosen voltage vectors, the operating point will not be stabilized and just fluctuate on different positive and negative values of direct axis component of stator current as it would be clearly illustrated by simulation results. The expected displacement of the stator current operating point for different stator resistance variations is shown in fig 3. It should be noted that the initial operating point is set to MTPA path which is a case for many applications of DTC SPMSM drives. Figure 3. Movements of stator current operating point in d-q plane for certain stator resistance variations 364 Therefore, the expected stator current changes for different variation of stator resistance could be summarized as below: ⎧ Rsa > Rse ⇒ I a ↑ ⎨ ⎩ Rsa < Rse ⇒ I a ↑ (17) According to the presented effects of stator resistance variation on DTC SPMSM drives performance, it is understood that stator resistance changes affect the stator current in some different way compared to DTC IPMSM drives, which in fact is due to unsimilar operating point of stator current components in d-q plane. Basically, in most of DTC IPMSM drives applications, the operating point is set to be on MTPA curves. However, for IPMSMs such a path is placed on the negative side of stator current d-q plane. Therefore, for the increased stator resistance values, the similar behavior with DTC SPMSM drives is expected (stator current rise up). However, for lower stator resistance the stator current decreases. In both DTC drives, lower actual resistance values lead to unstable drive conditions. It should be also mentioned that the stator resistance variation has same effects on IPMSMs as SPMSMs in terms of the actual torque and stator flux. Besides the considered effects on stator current, stator flux and produced torque values, a resistance mismatch would result in wrong voltage vector selection which can substantially affect the DTC drive performance. In fact, on the moment of region transition a wrong voltage vector would be chosen which does not satisfy the present stator flux and torque error. Therefore, each transition will be a source of further torque and flux ripples which is not desirable. In this section different effects of stator resistance variation on DTC SPMSM drives performance have been analytically investigated. In order to have more comprehensive impression of the produced changes by such a resistance mismatch, the resistance variation effects should be studied through simulation results. The next section is devoted to simulation studies of DTC SPMSM drive performance as the stator resistance is varied by step and linear function. IV. SIMULATION RESULTS This section deals with simulation studies of stator resistance variation effects on the performance of DTC SPMSM drive. In this regard, drive response to some different type of resistance mismatches (produced by step and linear variation of actual resistance) is evaluated in terms of stator flux linkage, electromagnetic torque and stator current. It could be clearly seen that the drive responses follows the earlier justified effects as it was expected. In order to change the stator resistance of the pertaining PMSM, the powersim model of this motor is manipulated in a manner so as to provide the access to its stator resistance by different desired function. The employed PMSM parameters are shown in table 1. The operating condition of the drive is set to 300 rpm (low speed) and 4 n.m. TABLE I. PMSM MOTOR PARAMETERS 5.25 mH 0.1827 wb Rotor flux linkage Stator Resistance (Ohms) Stator Resistance (Ohms) Times(s) Times(s) Times(s) Times(s) Times(s) Times(s) Times(s) Times(s) Stator Current (A) Stator Current (A) Times(s) Electromagnetic Torque (n.m) Times(s) Electromagnetic Torque (n.m) Times(s) Stator Flux (wb) Stator Flux (wb) Times(s) Times(s) Stator Current (A) The simulation results for the higher actual stator resistance values are shown in fig 4. At first, the stator resistance is increased by 50% with a step function which is applied on 0.5 sec. then, the stator resistance is raised linearly up to 50% of its nominal value and the produced effects are observed. It could be clearly seen that with such a resistance variation, as justified earlier, Actual stator flux reduces and the estimated torque increases while the actual torque and estimated flux are remained unaffected. Actually, there is some overshoot and oscillation for step responses of the varied signals which is mainly due to delay for selection of the appropriate voltage vectors after the applied stator resistance variation. Stator Resistance (Ohms) 5.25 mH q-axis stator inductance Lq Stator Flux (wb) 0.9585 Ω Electromagnetic Torque (n.m) Stator resistance Rs d-axis stator inductance Ld Stator Current (A) 4 Stator Resistance (Ohms) 300 v Magnetic pole pairs p Stator Flux (wb) 1.675 kw Electromagnetic Torque (n.m) Rated power PN Rated voltage UN Additionally, the stator resistance is decreased by 30% of its nominal value with a step and linear function and the produced effects are shown in fig 5. As it was mentioned earlier, for lower actual resistance values drive becomes unstable which in fact is due to large voltage vectors that is wrongly chosen by drive control scheme. Times(s) Stator Flux Trajectories Figure 5. Simulation results for DTC SPMSM drive responses to step and linear resistance reduction (30% of nominal value) Times(s) Blue-colored: actual value, red colored: estimated value The effects of stator resistance variations on the stator current could be more clarified through simulating the operating point of stator current on d-q plane. As it is shown in fig 6, with unchanged stator resistance the operating point is placed on q axis which implies almost zero direct axis current components. However, with any types of resistance changes it will be deviated to either left (for higher actual resistance) or right (for lower actual resistance) as it is clearly demonstrated in fig 6. In both cases, some direct axis component of stator current is produced which results in larger stator current values. In the case of reduced stator resistance, since the drive has become unstable, the stator current operating point starts to fluctuate for different values of direct axis component of stator current while maintains the same value of quadrature axis component. Times(s) Stator Flux Trajectories Figure 4. Simulation results for DTC SPMSM drive responses to step and linear resistance increase (50% of nominal value) Blue-colored: actual value, red colored: estimated value 365 REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] Figure 6. Simulation results for displacement of stator current operating point of DTC SPMSM drive for different types of stator resistance variations [10] V. CONCLUSION In this paper, different effects of stator resistance variation on the performance of SPMSM DTC drive have been studied in details. In this regard, the expected effects are analytically investigated at the first stage and then, they have been confirmed by simulation results obtained through applied step and linear resistance variations. It is understood that the performance of drive could be highly influenced on the low operating speeds with high load torque values. In order to address the problems caused by stator resistance mismatch, some mitigation methods (compensators, estimators) could be employed which demands a comprehensive understanding of the produced effects of stator resistance variation on the drive performance which is carried out in this paper. 366 [11] [12] [13] M. Meyer and J. Böcker, “Optimum Control for Interior Permanent Magnet Synchronous Motors (IPMSM) in Constant Torque and Flux Weakening Range”, 12th International Power Electronics and Motion Control Conference (EPE-PEMC), Portoroz, Slowenia, August 2006. L. Zhong, M. F. Rahman, W. Y. Hu, and K. W. 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