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Generalized Two-Vector-Based Model-Predictive Torque Control of Induction Motor Drives Yongchang Zhang, Member, IEEE, and Haitao Yang, Student Member, IEEE
Abstract—Conventional single-vector-based model-predictive torque control (MPTC) has been widely studied owing to its intuitive concept and quick response. To improve the steady-state performance, recently, the concept of duty cycle control was introduced in MPTC by inserting a null vector along with an active vector during one control period. However, this still fails to reduce the torque error to a minimal value due to the imposed restriction on vector combination and the cascaded processing of vector selection and vector duration. This paper proposes a generalized two-vectors-based MPTC (GTV-MPTC) by relaxing the vector combination to two arbitrary voltage vectors. By evaluating the vector combination and their durations simultaneously in the predefined cost function, global minimization of torque error can be obtained in theory. However, the computational burden is also significantly increased. By choosing a proper method to determine the vector durations, the redundant vector combinations can be eliminated, which makes the proposed GTV-MPTC suitable for real-time implementation. Both simulation and experimental results were carried out to verify the effectiveness of the proposed method. The presented results show that, compared to prior MPTC with or without duty cycle control, the proposed GTV-MPTC achieves much better performance with lower sampling frequency over a wide speed range. Furthermore, the average switching frequency is even lower than that of conventional MPTC in the medium speed range. Index Terms—Duty cycle control, induction motor drives, model-predictive torque control, ripple reduction, two vectors.
I. INTRODUCTION IRECT torque control (DTC) is featured with fast dynamic response and simple structure. It selects voltage vector by using two hysteresis comparators and a predefined switching table. However, the vector selected in this heuristic way is not very effective to satisfy the demand of torque, flux, and other control aims [1], [2]. Recently, MPTC was introduced to cope with these problems, and it has gained increasing attention throughout the world, owing to its intuitive concept, fast dynamic response, and flexibility to incorporate various constrains [3]–[7]. Based on an internal system model, MPTC can predict future behavior of the controlled variables, such as torque, stator flux,
D
Manuscript received June 12, 2014; revised July 17, 2014; accepted August 13, 2014. Date of publication August 20, 2014; date of current version February 13, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 51207003 and Grant 51347004, and in part by Beijing Nova Program under Grant xx2013001. The part of this paper was presented at the IEEE Energy Conversion Congress and Exposition, Pittsburgh, PA, USA, 2014. Recommended for publication by Associate Editor Y. W. Li. The authors are with the Power Electronics and Motor Drives Engineering Research Center of Beijing, North China University of Technology, Beijing 100144, China (e-mail:
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2014.2349508
switching transitions, etc. A cost function related to the control variables is evaluated for each possible voltage vector, and the one minimizing the cost function is selected as the best voltage vector [3]. Compared to DTC, the voltage vector selected in MPTC is more accurate and effective owing to the online optimization of a cost function. Additionally, the use of cost function offers a flexible and simple way to incorporate various constraints and other control objectives, such as decreasing switching frequency [8], reducing common-mode voltage [9], and restricting the peak current [3]. Despite that more accurate vector is selected by MPTC, it still suffers from relatively high torque ripple, variable switching frequency, and high sampling frequency requirement [10] because only one voltage vector is applied during the whole control period. In DTC, it has been well known that the steady-state performance of torque can be improved by introducing duty cycle control [2], [11], [12]. In other words, one active vector and one null vector are applied in each control period, and the vector durations can be determined based on various principles [2]. Recently, the concept of duty cycle control has also been introduced in MPTC to achieve torque ripple reduction [4], [13], [14]. The best active voltage vector is first selected based on cost function minimization, and then, its duty ratio is subsequently optimized according to some principles. It is shown that steady-state performance is much improved with duty ratio optimization. However, this cascaded processing of vector selection and vector duration cannot ensure global minimization of torque error because the selected best voltage vector may no longer remain optimal if it is not applied for the whole control period. To address this problem, it is suggested in [10] to consider both vector duration and vector selection in the cost function. Much better steady-state performance, especially at low speed, can be obtained even if the sampling frequency is reduced by half [10]. However, the second voltage vector is fixed to a null vector in prior methods [4], [10], [13]. This poses some limitations on the performance improvement because the best second voltage vector is not necessarily a null vector. In fact, the combination of an active vector and a null vector is only a subspace of possible voltage vector combinations, which cannot achieve global minimization of cost function in true sense. This paper proposes a generalized two-vectors-based MPTC (GTV-MPTC) for induction motor (IM) drives, which is a generalization of single-vector-based finite control set modelpredictive control (MPC) [15]. The restrictions on vector combinations in prior methods [4], [10], [13] are relaxed to two arbitrary voltage vectors. Hence, it is possible to achieve better performance than prior MPTC with [4], [10], [13] or without [3] duty cycle control. For a two-level inverter, two arbitrary
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ZHANG AND YANG: GENERALIZED TWO-VECTOR-BASED MODEL-PREDICTIVE TORQUE CONTROL OF INDUCTION MOTOR DRIVES
voltage vectors lead to 7×7 = 49 vector combinations (two null vectors are considered as the same vector), which is much more than the vector combinations in [13]. Furthermore, both the vector selection and vector durations are simultaneously evaluated in the cost function, which produces the global optimization of torque error in theory. To cope with the computational burden caused by increased vector combinations in the proposed GTV-MPTC, the vector duration is optimized to make the instantaneous torque reach its reference value at the end of the next control period, acting in a deadbeat fashion rather than torque ripple minimization in [13]. The resulting vector duration is independent of sequence order of vectors. This brings the benefits of elimination of the redundant vector combinations and the possibility of switching frequency reduction. The presented GTV-MPTC is compared with conventional MPTC [3] and prior MPTC methods with duty ratio optimization [10], [13]. The presented simulation and experimental results confirm the effectiveness of the proposed method. II. DYNAMIC EQUATIONS OF IM
Fig. 1.
Control diagram of the proposed MPTC.
where tsc is control period, xkp +1 is the predictor–corrector T of state vector, and xk +1 = iks +1 ψ ks +1 is predicted state vector for stator current and stator flux. The rotor flux at (k+1)th instant can be estimated from stator flux ψ ks +1 and current iks +1 as ψrk +1 =
Take stator flux and stator current as state variables, the dynamic equations of IM can be expressed in stationary frame as [16]
Lr k +1 1 k +1 ψ − i Lm s λLm s
(1)
T where x = is ψ s are state variables of stator current and stator flux, u = us is stator voltage vector and
−λ(Rs Lr + Rr Ls ) + jωr A= −Rs λLr B= 1
λ(Rr − jLr ωr ) (2) 0 (3)
R s , Rr Ls , Lr , Lm
stator resistance, rotor resistance; stator inductance, rotor inductance, and mutual inductance; electrical ωr rotor speed; λ = 1/ Ls Lr − L2m . To predict torque and flux at (k + 1)th instant, (1) must be discretized in digital implementation. A simple way to discretize (1) is first-order Euler method. However, its accuracy is relatively limited especially during high-speed operation [17]. In [3], an improved discrete model of IM is obtained by using the Cayley–Hamilton theorem to compute the matrix exponential. However, it is quite complicated for real-time implementation. To achieve good accuracy with relatively low computation burden, the so-called Heun’s method [18], [19] is employed in this paper, which is an improvement of simple Euler’s method, and it is expressed as [10], [13] ⎧ k +1 ⎨xp
= xk + tsc (Axk + Buks )
⎩ k +1 x
= xkp +1 +
(4) ts c 2
A(xkp +1 − xk )
(5)
and the electromagnetic torque can be predicted as [7] Tek +1 =
x˙ = Ax + Bu
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3 Np ψ ks +1 ⊗ iks +1 . 2
(6)
III. PRINCIPLE OF THE PROPOSED MPTC The control diagram of the proposed MPTC is shown in Fig. 1, which is mainly composed of five parts: estimation of torque and flux, optimization of duty cycle, prediction of torque and flux, cost function minimization, and pulse generation. Torque reference is generated by an outer speed control loop using a PI controller. The stator flux reference is constant because the fieldweaken operation is not considered in this paper. The detailed introduction of the control diagram will be elaborated in the following text. A. Estimation and Prediction of Torque and Flux Accurate estimation of torque and flux is essential to achieve good performance of MPTC. In this paper, a full-order observer for the flux and torque estimation is adopted owing to its good accuracy and parameter robustness over a wide speed range. The mathematical model of the observer is expressed as [20] dˆ x = Aˆ x + Bus + G(is − ˆis ). (7) dt ˆs T are state variables representing the esˆ = ˆis ψ where x timated stator current and stator flux. A constant gain matrix G is employed to improve stability of the observer, which is expressed as 2b (8) G=− b/(λLr ) where b is a negative constant gain [20]. This gain matrix is very effective and simple to implement.
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After obtaining the estimated stator current and stator flux at kth instant from (7), the next step is to predict the value of toque and stator flux at (k+1)th instant. As stator current is not directly controlled in MPTC, it is preferable to eliminate the complicated prediction of stator current for the aim of reducing computation burden. Hence, the torque at (k+1)th instant can be calculated as [21] Tek +1 = 1.5Np ψ ks +1 ⊗ iksλ
(9)
where Np is pole pairs; iksλ is calculated based on the variables at kth instant, and it is expressed as iksλ = (1 − λRr Ls tsc + jωr tsc ) iks +λ (Rr tsc − Lr − jωr Lr tsc ) ψ ks .
(10)
The stator flux at (k + 1)th instant can be predicted from the stator voltage equation as ψ ks +1 = ψ ks + tsc (uks − Rs iks ).
(11)
where tsc is the control period. As there are more vector combinations than conventional MPTC to be evaluated, the simple Euler integration is used in (11) for the sake of efficiency, and it does not cause much influence on performance, as shown in the simulation and experimental results. It should be noted that by using (9) and (11), the prediction process is greatly simplified because only the stator flux is necessary to be predicted. Otherwise, the computation burden will be huge as there are much more voltage vector combinations in the proposed MPTC than that in conventional MPTC. B. Vector Duration For two-vectors-based MPTC, the key aspects include selecting appropriate vectors and determining their respective duration during one control period based on some principles. In this paper, the vector durations are calculated based on the principle of nullifying the tracking error of torque at the end of the next control period, namely deadbeat torque control. The vector selection is introduced in Section III-C. Under the assumption that the vector combination is composed of u1 , u2 and their respective durations are t1 and tsc − t1 , the torque at (k + 1)th instant can be obtained from (9) to (11) as Tek +1
=
1.5Np ψ ks +1
⊗
iksλ
= 1.5Np (ψ ks − Rs iks tsc + t1 u1 + (tsc − t1 )u2 ) ⊗ iksλ . (12) As the principle of deadbeat torque control is employed, solving Tek +1 = Teref , the optimal duration t1 of u1 can be obtained as t1 =
Teref /(1.5Np ) − T0 − tsc (u2 ⊗ iksλ ) (u1 − u2 ) ⊗ iksλ
(13)
where T0 = (ψ ks − Rs iks tsc ) ⊗ iksλ . The duration of u2 is subsequently obtained as t2 = tsc − t1 =
Teref /(1.5Np ) − T0 − tsc (u1 ⊗ iksλ ) . (14) (u2 − u1 ) ⊗ iksλ
Fig. 2. Voltage vectors and corresponding switching states of the two-level inverter.
Apart from simplicity, one of the main advantages of deadbeat torque control is that exchanging the sequence order of vector combinations does not affect their respective durations. For example, if the vector combination is changed from u1 , u2 to u2 , u1 , now the equation to solve the duration of the first vector is rewritten as
Tek +1 = 1.5Np (ψ ks − Rs iks tsc + t2 u2 + (tsc − t2 )u1 ) ⊗ iksλ . (15) Omitting the tedious deduction process, it can be found that the deduced t2 is the same as t2 expressed in (14). On the contrary, if other principle such as torque ripple minimization [11], [13] is used, the duration of each vector will be different. As t2 = t2 = tsc − t1 , Tek +1 is also not affected by the order of two vectors, which can be seen from (12) and (15). This means that the combinations of (u1 , u2 ) and (u2 , u1 ) have the same effect from the view of flux and torque control. This is very helpful to eliminate the redundant vector combinations, as shown in Section III-C. C. Vector Selection In this paper, the cost function in (16) is used as the criterion to select the best voltage vectors among all the possible vector combinations k +1 || J = |Teref − Tek +1 | + kψ ||ψ ref s | − |ψ s
(16)
where kψ is weighting factor for stator flux. The main difference between the proposed GTV-MPTC and prior MPTC with duty cycle control [4], [10], [13] is that the vector selection is relaxed to two arbitrary voltage vectors. For two-level inverter-fed IM drives, there are eight switching states but only seven different voltage vectors, as shown in Fig. 2. As a result, there are 7 × 7 = 49 vector combinations. If the two vectors are the same, the proposed MPTC is exactly the same as conventional single-vector-based MPTC. In other words, conventional MPTC can be considered as a special case of the proposed GTV-MPTC. As indicated in Section III-B, the order of two vectors does not affect the final result, which greatly reduces the number of vector combinations to be evaluated. The remaining vector combinations can be summarized as (ui , uj )
ZHANG AND YANG: GENERALIZED TWO-VECTOR-BASED MODEL-PREDICTIVE TORQUE CONTROL OF INDUCTION MOTOR DRIVES
(i = 1, 2, ..., 7, j = i, i + 1, i + 2, ..., 7), and the total number is 7 + 6 + · · · + 1=28. Among the 28 combinations, the combinations of (ui , ui+3 ) (i = 1, 2, 3) would lead to large du/dt, and all three arm of the inverter would switch once during one control period, which is unfavorable in practical application. In fact, after in-depth analysis, we can find that the combinations of (ui , ui+3 ) (i = 1, 2, 3) can be equivalently replaced by the combinations of (ui , u0 ) (i = 1, 2, ..., 6) because ui and ui+3 are opposite to each other. Take (u1 , u4 ) and (u1 , u0 ) as an example, assuming that the duration of u1 in the two combinations are t14 and t10 , respectively. Based on the principle of deadbeat torque control and considering that u1 = −u4 , the durations of t14 and t10 obtained from (13) have the following relationship: t14 = 0.5(t10 + tsc ).
= ψ ks − Rs iks tsc + t10 u1 .
TABLE I MACHINE AND CONTROL PARAMETERS Ud c PN UN fN TN Np Rs Rr Lm Ls Lr |ψ s∗ | kψ
DC-bus voltage Rated power Rated voltage Rated frequency Rated torque Number of pole pairs Stator resistance Rotor resistance Mutual inductance Stator inductance Rotor inductance Flux amplitude reference Weighting factor for flux
540 V 2.2 kW 380 V 50 Hz 14 N · m 2 3.126 Ω 1.879 Ω 0.221 H 0.230 H 0.230 H 0.91 Wb 100
(17)
Substitute (17) into the stator flux equation in (11), we can obtain that ψ ks +1 = ψ ks − Rs iks tsc + t14 u1 + (tsc − t14 )u4
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(18) (19)
This means that using the duty ratio optimization method in (13) and (14), the vector combinations of (u1 , u4 ) and (u1 , u0 ) would produce the same value of ψ ks +1 and Tek +1 . As the combination of (u1 , u0 ) has lower du/dt and less switching transitions, it is preferable in practical application than (u1 , u4 ). It should be noted that if t10 < 0, (u1 , u4 ) can be equivalently replaced by (u4 , u0 ); the derivation process is similar. By further eliminating the three redundant combinations (ui , ui+3 ) (i = 1, 2, 3), the final number of vector combinations is reduced to 25, which is nearly half of initial 49 combinations. To elaborate the process of vector selection clearly, the procedure of the proposed GTV-MPTC is summarized in the following steps: 1) for a candidate vector combination, the duty cycles of two vectors are optimized based on (13) and (14); 2) once the duty cycle is obtained, the value of Tek +1 and |ψ ks +1 | can be predicted from (9) to (11); 3) subsequently, the predicted Tek +1 and |ψ ks +1 | are sent to the cost function in (16) to evaluate the candidate vector combination as well as its duty cycle; 4) the steps 1–3 are repeated for all of the 25 vector combinations, and the vector combination along with its vector durations producing the minimal value of cost function is selected as the optimal solution. It can be seen that the vector combinations and their durations are evaluated as a whole in the cost function in the proposed GTV-MPTC. On the contrary, in the prior method [13], the cost function is only associated with the vector selection, and the vector duration is subsequently calculated. This separate and cascaded processing of vector selection and duty ratio optimization fails to consider the influence of duty ratio optimization on the final cost function; thus, the obtained solution is not “global” optimal. It is possible to use three vectors during one control period, which should provide more freedom to optimize torque and stator flux performance. However, the vector combinations
are increased significantly and high requirements on hardware are posed. The current study on three-vector-based method is focused on the SVM-based method [22] and switching-tablebased DTC [23], [24]. Developing generalized three-vectorsbased MPTC is a very challenging work and will not be further discussed in this paper. D. Pulse Generation After the optimal combination of two vectors and their durations are determined, switching pulses can be generated according to the vector sequences. If one of the selected voltage vectors is a null vector, the one (u0 (000) or u7 (111)) producing less switching jumps should be applied. For example, if the selected two vectors are u2 (110) and u0 (000), then u0 (000) should be changed to u7 (111). Furthermore, as shown in Section III-B, the final results are not affected by the sequence order of the two vectors, which provides an extra freedom to obtain switching frequency reduction without performance degradation. For example, if the last vector applied during the previous period is u0 (000), and the vector combinations to be applied in the next control period are u2 (110) and u3 (010), in that case, u3 (010) instead of u2 (110) will be applied first to decrease the switching jumps. The vector durations should be exchanged accordingly if there is a vector sequence exchange. IV. SIMULATION STUDY In this section, the proposed MPTC is simulated in the environment of MATLAB/Simulink. The performance of conventional MPTC [3], MPTC with separate duty ratio optimization [13], MPTC with simultaneous optimization of vector selection and vector duration [10], and the proposed GTV-MPTC will be compared in detail. For simplicity, the prior methods in [13] and [10], which use an active voltage vector and a zero vector, are named as Duty-MPTC I and Duty-MPTC II, respectively, in the following text. The control diagram of proposed GTV-MPTC is shown in Fig. 1, and the parameters of motor and control system are listed in Table I. The sampling frequency is 20 kHz for both conventional MPTC and Duty-MPTC I, while for Duty-MPTC II and the proposed GTV-MPTC, the sampling frequency is set to 10 kHz to show their superiority.
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Fig. 4. Average switching frequencies of conventional MPTC, Duty-MPTC I, and GTV-MPTC at steady state of 1500 r/min with rated load.
Fig. 3. Simulated responses of torque, stator flux, and stator current at 1500 r/min with sudden load change for (a) conventional MPTC at 20-kHz sampling frequency, (b) duty-MPTC I at 20-kHz sampling frequency, and (c) GTV-MPTC at 10-kHz sampling frequency.
First, a comparative study of conventional MPTC, DutyMPTC I, and GTV-MPTC is carried out. The machine runs at 1500 r/min, and then, an external load of 14 N · m (100% rated torque) is suddenly applied at t = 0.1 s. From Fig. 3, it is clearly seen that Duty-MPTC I has much lower torque ripple than that in conventional MPTC, while the stator flux ripples of both methods are similar. The proposed GTV-MPTC presents the best steady-state performance in terms of lower torque/flux ripples and less current harmonics, especially in the stator flux ripple. Considering that the sampling frequency of GTV-MPTC is reduced by half with respect to MPTC and Duty-MPTC I, it can be concluded that the proposed GTV-MPTC performs best among the three MPTC schemes. It should be noted that at low speed, Duty-MPTC II are completely equivalent to GTV-MPTC, so the results obtained from Duty-MPTC II is not presented due to page limitations. The average switching frequencies fav of three methods at steady state of 1500 r/min with rated load (14 N · m) are presented and compared in Fig. 4. fav is obtained by counting the total switching jumps N of six legs of two-level inverter over a fixed period of 0.05 s, namely that fav = N/6/0.05. The method to calculate the switching frequency has been verified in our prior study on DTC for PMSM drives [2], [24]. The average switching frequencies during the recorded time range are 4.97, 2.63, and 2.41 kHz for Duty-MPTC I, GTV-MPTC, and conventional MPTC, respectively. It is seen that, compared to conventional MPTC, much better performance is obtained in the proposed GTV-MPTC while the switching frequency is only slightly increased up to 9.1%. On the contrary, the steady-state performance improvement in Duty-MPTC I is obtained at the cost of more than double of the average switching frequency of conventional MPTC. The responses of startup for the proposed GTV-MPTC are shown in Fig. 5. From top to bottom, the curves shown in Fig. 5 are rotor speed, electromagnetic torque, stator flux, and onephase current. To prevent overcurrent and provide sufficient
ZHANG AND YANG: GENERALIZED TWO-VECTOR-BASED MODEL-PREDICTIVE TORQUE CONTROL OF INDUCTION MOTOR DRIVES
Fig. 5. Simulated starting responses from standstill to 1500 r/min without load for the proposed GTV-MPTC.
Fig. 6.
Selected two vectors for GTV-MPTC during startup.
torque during startup, the scheme of preexcitation is employed [20]. During the process of preexcitation, a fixed voltage vector u1 (100) is applied to establish the stator flux. However, if the stator current is larger than the limitation setting, a zero voltage vector u0 (000) will be applied to decrease the current, which acts in a bang-bang fashion. When the stator flux reaches the predetermined value, which is set as 95% of the reference in the paper, the preexcitation terminates and the machine can start with sufficient torque. It is seen in Fig. 5 that the motor accelerates quickly to 1500 r/min without large starting current after preexcitation. To validate that the best voltage vector combinations may be not the combination of an active vector and a null vector, Fig. 6 illustrates the number of selected two vectors during the startup response. The number of two null vectors are both indicated by 0 in Fig. 6. It is seen that the first voltage vector is always selected as an active voltage vector, and the number of second voltage vector varies from 0 to 6. This indicates that the op-
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Fig. 7. Selected vectors and pulse signal for GTV-MPTC during stepped load torque change.
timal second voltage vector is not always zero voltage vector, especially during medium and high-speed operation. However, it can be observed that the optimal second voltage vector is always a zero vector in the region around t = 0.1s, where the motor runs at low speed (less than 400 r/min). The reason is that the back electromotive force is small during low-speed operation; thus, a zero vector providing lower stator voltage is better to satisfy the control of torque and flux. This is in accordance to the results of Duty-MPTC II shown in [10], where the performance improvement is very significant at low speed. On the contrary, the performance improvement is limited at high speed for Duty-MPTC II because the second optimal voltage vector is not necessarily a null vector at medium and high speeds, as shown in Fig. 6. Additionally, when the machine runs at 1500 r/min, the number of selected two vectors and the corresponding pulse signal for one switch of inverter during stepped torque change is illustrated in Fig. 7. It is clearly seen that the second vector may be a nonzero vector during both steady and dynamic process, confirming that relaxing the vector combinations to two arbitrary vectors in the proposed GTV-MPTC leads to different selection of two vectors (generally more effective) compared to Duty-MPTC II. In fact, if the second optimal voltage vector is fixed to a zero vector, the proposed GTV-MPTC would be exactly the same as Duty-MPTC II. Hence, the latter can be considered as a special case of the proposed method. The responses of torque, stator flux, and current for Duty-MPTC II and GTV-MPTC at high speed of 1500 r/min with rated load are compared and presented in Fig. 8. It is clearly seen that the proposed GTVMPTC presents much lower stator flux ripple and the torque spikes in Duty-MPTC II are eliminated. A quantitative comparison between the two methods in terms of torque ripple, stator flux ripple, and total harmonic distortion (THD) of stator current is summarized in Table II. It is seen from Table II that, by relaxing the vector combination to two arbitrary voltage vectors, GTV-MPTC has overall better performance than
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Fig. 8. Simulated high-speed operation at 1500 r/min with rated load for (a) Duty-MPTC II and (b) GTV-MPTC.
TABLE II QUANTITATIVE COMPARISON OF TWO MPTC METHODS method Duty-MPTC II GTV-MPTC
T r i p (N · m)
ψ r i p (Wb)
THD of i s
0.289 0.144
0.0104 0.0066
6.88% 4.22%
Duty-MPTC II, confirming its effectiveness in steady-state performance improvement. V. EXPERIMENTAL RESULTS Apart from simulation studies, the proposed GTV-MPTC was also experimentally tested on a two-level inverter-fed IM drive platform. A 32-bit floating point DSP TMS320F28335 is employed to accomplish the developed control algorithm. The control and system parameters are the same as those listed in Table I. The sampling frequencies of conventional MPTC,
Duty-MPTC I, Duty-MPTC II, and the proposed GTV-MPTC are 20, 20, 10, and 10 kHz, respectively, which are the same as those used in simulations. In the following experimental tests, all variables are displayed on digital oscilloscope via on-board DA converter except the stator current, which is directly measured by a current probe. Fig. 9 presents the responses of torque, stator flux, and stator current at 150 r/min with rated torque (14 N · m). Conventional MPTC presents relatively large torque ripple, and there are much harmonics in the stator current. The torque ripple of Duty-MPTC I is lower than that of conventional MPTC, but its stator current is somewhat distorted, which is mainly caused by the cascaded processing of vector selection and vector durations. The proposed GTV-MPTC exhibits the best steady-state performance in terms of torque and stator flux ripples as well as current harmonics among the three MPTC methods, even if its sampling frequency is only half of the other two methods. A numerical comparisons of torque ripple and THD of stator current are shown in Fig. 10, which are obtained at various speeds with rated torque. It is seen that the conventional MPTC presents the highest torque ripple and current THD, followed by Duty-MPTC I. The proposed GTV-MPTC has the lowest torque ripple and current THD at different speeds. The average switching frequency of the three methods are illustrated in Fig. 11. It is clearly seen that the average switching frequency of conventional MPTC varies with speed significantly. The Duty-MPTC I presents relatively constant switching frequency, but still variable with speed, especially at high speed range. Although the torque ripple of Duty-MPTC I is improved compared to conventional MPTC, its switching frequency is considerably increased by more than double. On the contrary, the average switching frequency of GTV-MPTC is almost constant over a wide speed range and is even lower than that of conventional MPTC in the medium speed range. Hence, it is easy to conclude that GTVMPTC performs best by providing the lowest torque/flux ripples and current harmonics with almost constant (and even lower) switching frequency. One of the main benefits of using two voltage vectors during one control period is that better current harmonic spectrum can be obtained. It is seen from Fig. 12 that the current harmonics of conventional MPTC are distributed over a wide frequency range, and the amplitude of low-order harmonics is relatively large. The lower order harmonics are reduced, and the current harmonics now concentrates on the sampling frequency of 20 and 10 kHz for Duty-MPTC I and the proposed GTV-MPTC, respectively, which is similar to SVM-based methods. This fact has been reported in our prior study for Duty-MPTC I [4], [10] and DTC with duty cycle control [24]. Similar to the simulation study shown in Fig. 8, to validate the effectiveness of relaxing vector combinations to two arbitrary voltage vectors, the performance of the proposed GTV-MPTC is further compared to Duty-MPTC II. Fig. 13 presents the responses of high-speed operation at 1500 r/min with rated torque for conventional MPTC, Duty-MPTC II, and the proposed GTVMPTC. The torque performance improvement in Duty-MPTC II is limited compared to conventional MPTC. On the contrary, the torque ripple reduction in the proposed GTV-MPTC is very
ZHANG AND YANG: GENERALIZED TWO-VECTOR-BASED MODEL-PREDICTIVE TORQUE CONTROL OF INDUCTION MOTOR DRIVES
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Fig. 10. Comparisons of torque ripple and current THD for conventional MPTC, Duty-MPTC I, and GTV-MPTC operating at different speeds with rate load.
Fig. 11. Average switching frequencies for conventional MPTC (marked with “O”), Duty-MPTC I (marked with “*”), and GTV-MPTC (marked with “Δ”) at different speeds with rated load.
Fig. 9. Low-speed operation at 150 r/min with rated torque for (a) conventional MPTC, (b) Duty-MPTC I, and (c) GTV-MPTC.
significant, and there are much less current harmonics, which is in accordance to the simulation results in Fig. 8. A quantitative comparison between GTV-MPTC and DutyMPTC at various speeds is shown in Fig. 14. It is seen that, the performance difference between the two methods at low speeds is minor, which is in accordance to the analysis presented in Section IV. However, the performance of the proposed GTV-MPTC becomes much better than Duty-MPTC II when the speed increases. This confirms that the combination of two nonzero voltage vectors are more suitable at medium and high speeds than the combination of an active vector and a null vector. Furthermore, it should be noted that the performance of the proposed GTV-MPTC does not change much over a wide speed range, validating its effectiveness at various speeds.
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Fig. 12. Harmonic spectrum of one-phase stator current for conventional MPTC, Duty-MPTC I, and the proposed GTV-MPTC at 750 r/min without load.
Apart from the steady-state performance comparison, the dynamic tests of the proposed GTV-MPTC were also carried out and compared to conventional MPTC. The responses during startup are shown in Fig. 15. From top to bottom, the curves shown in Fig. 15 are rotor speed, electromagnetic torque, stator flux, and stator current. As introduced in Section IV, the stator flux is first established before starting the machine. It is seen that the motor accelerates quickly to 1500 r/min without large starting current and decoupled control of torque, and stator flux is achieved in both methods. The proposed GTV-MPTC has very similar quick torque response to that in conventional MPTC, while its torque ripple is much lower. This can be more clearly seen in Fig. 16, where the torque reference is changed from zero to 120% rated value suddenly at t = 2 ms. The settling time (from zero to 120% rated torque) is around 0.25 ms for each MPTC method, and the difference among various MPTC methods is very insignificant. This confirms that the proposed MPTC can achieve significant steady-state performance improvement without degrading the dynamic response. The responses of speed reversal are presented in Fig. 17, where the motor runs at −1500 r/min without load first and then changes to 1500 r/min quickly. It is clear that proposed GTVMPTC exhibits much lower torque and flux ripples as well as current harmonics, even in the dynamic process with much lower sampling frequency. Finally, very low-speed operation at 6 r/min is demonstrated for the proposed GTV-MPTC, and the recorded waveforms are shown in Fig. 18. It is seen that the proposed GTV-MPTC is stable and performs very well. The magnitude of stator flux is kept at its reference value, and the current is sinusoidal in shape. It should be noted that in Fig. 18(a), there are some glitches in the measured speed, which are mainly caused by the relatively poor accuracy of speed measurement, limited resolution of onboard DA converter and the measurement noise. Actually, the motor runs at a steady-state speed in the experimental tests,
Fig. 13. High-speed operation at 1500 r/min with rated torque for (a) conventional MPTC, (b) Duty-MPTC II, and (c) GTV-MPTC.
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Fig. 16. Short time response of torque for conventional MPTC, Duty-MPTC I, Duty-MPTC II, and the proposed GTV-MPTC.
Fig. 14. Comparisons of torque ripple and current THD for Duty-MPTC II and GTV-MPTC, operating at different speeds with rated load.
Fig. 17. Responses during speed reversal at 1500 r/min for (a) conventional MPTC and (b) GTV-MPTC.
Fig. 15. Starting from standstill to 1500 r/min for (a) conventional MPTC and (b) GTV-MPTC.
which can be confirmed by the constant estimated stator flux and sinusoidal current measured by a current probe. The motor can operate at 6 r/min with rated load, as shown in Fig. 18(b). The external load is applied by a magnetic powder brake, which is not accurately constant, so there are some oscillations in the
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(2) Both vector combination and vector durations are considered as a whole in the cost function, hence improving the steady-state performance of MPTC with duty cycle control significantly, especially in the low speed range. (3) As there are much more vector combinations in the proposed GTV-MPTC, the computational burden is increased significantly. This problem is solved by using deadbeat torque control to obtain the vector durations; thus, the obtained vector durations are independent of vector sequence. This brings the benefits of eliminating the redundant vector combinations and makes the proposed MPTC suitable for practical applications. (4) The proposed GTV-MPTC is compared with prior MPTC methods with or without duty cycle control, and its effectiveness is confirmed by the presented simulation and experimental results.
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Fig. 18. Stead-state operation at 6 r/min for the proposed GTV-MPTC, (a) without load and (b) with full load.
measured speed. However, the magnitude of stator flux is still constant, and the current is very sinusoidal. The results presented in Fig. 18 confirm that the proposed GTV-MPTC exhibits good performance even at very low speed. VI. CONCLUSION This paper proposes a generalized two-vectors-based MPTC, which uses two arbitrary vectors as a combination during one control period. Compared to prior MPTC with or without duty cycle control, the proposed GTV-MPTC has the following merits: (1) By relaxing the vector combinations to two arbitrary voltage vectors, much better steady-state performance in terms of torque and flux ripples as well as current harmonics can be obtained at various speeds. The average switching frequency is almost constant and even lower than conventional single-vector-based MPTC at medium speeds.
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Yongchang Zhang (M’ 10) received the B.S. degree from Chongqing University, Chongqing, China, in 2004, and the Ph.D. degree from Tsinghua University, Beijing, China, in 2009, both in electrical engineering. From August 2009 to August 2011, he was a Postdoctoral Fellow at the University of Technology, Sydney, Australia. He joined the North China University of Technology, Beijing, in August 2011 and became an Associate Professor in power electronics and motor drives. He has published more than 70 technical papers in the area of motor drives, pulsewidth modulation, and ac/dc converters. His current research interests include model-predictive control for power converters and motor drives.
Haitao Yang (S’14) was born in 1987. He received the B.S. degree from the HeFei University of Technology, Hefei, China, in 2009. He is currently working toward the master’s degree in electrical engineering at the North China University of Technology, Beijing, China. His research interest includes model-predictive control of motor drives.
