Comparative Evaluation of Direct Torque Control Strategies for Permanent Magnet Synchronous Machines Feng Niu and Kui Li
Bingsen Wang and Elias G. Strangas
Province-Ministry Joint Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability Hebei University of Technology Tianjin 300130, China Email:
[email protected] [email protected]
College of Engineering Michigan State University East Lansing, MI 48824, USA Email:
[email protected] [email protected]
Abstract—This paper presents a detailed comparative evaluation of several direct torque control (DTC) strategies for permanent magnet synchronous machines (PMSMs), namely basic DTC, model predictive DTC (MPDTC) and DTC with duty ratio modulation (DTC-duty). Moreover, field orient control (FOC) is also included in this study. The aforementioned control strategies are reviewed and their performances are analyzed and compared. The comparison is carried out through simulation of a 60 kW PMSM fed by a two-level voltage source inverter (VSI). With the intent to fully reveal advantages and disadvantages of each control strategy, critical evaluation has been conducted on the basis of several criteria: torque and stator flux ripple, inverter switching frequency, machine parameter sensitivity, computational complexity and stator current total harmonic distortion (THD). The choice of control scheme can be determined based on specific requirements of particular application under consideration.
I. I NTRODUCTION Since direct torque control (DTC) was firstly proposed for induction machine in the 1980s by Takahashi and Noguchi [1] and Depenbrock [2], it has become a powerful and widely adopted control strategy. In DTC, the decoupling of field oriented control (FOC) is replaced with the bang-bang control, which meets very well with switched-mode operation of power inverters. Numerous merits of DTC have attracted extensive research attention [3]. In the late 1990s, application of DTC to permanent magnet synchronous machine (PMSM) was presented [4]. In comparison to field oriented control (FOC) strategy, DTC does not require any explicit current regulator, coordinate transformation and space vector modulation. Furthermore, the rotor position sensing that is essential for FOC is not required for DTC to operate properly, which greatly simplifies implementation of the sensorless control in DTC. In spite of the simplicity, DTC achieves adequate torque control performance under both steady-state and dynamic conditions. In addition, DTC features low sensitivity to accuracy of machines parameter estimation. On the other hand, it is widely known that the basic DTC has several disadvantages. The prominent one is that the performance of basic DTC
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deteriorates at low speed due to reduced controllability, high torque and current ripple, and variable switching frequency. Furthermore, undesirable high sampling frequency (HSF) is necessary for digital implementation of the controller. The implementation challenge is elevated when HSF is coupled with increased computation burden in each sampling period. During the past few decades, many researchers have been working on solving the challenges associated with basic DTC [5]-[14]. In [6], a DTC scheme combined with space vector modulation (SVM), namely DTC-SVM, is proposed to achieve constant switching frequency while obtain the desired torque and stator flux values in one sampling period by synthesizing a suitable voltage vector through SVM. Unlike the conventional switching-table-based DTC, which is composed of limited number of voltage vectors with fixed magnitudes and positions, DTC-SVM can generate any arbitrary voltage vector within its linear range. In relatively recent development, model predictive control (MPC) has been attracting research attention due to improved control performance as it is integrated with DTC [7]-[11]. The fundamental principle of MPC is that it predicts a constant number of the future machine states, which is called prediction horizon N, using a discrete system model. In [7], an MPDTC scheme is proposed for three-level-inverter-fed PMSM, which utilizes a prediction horizon greater than one to achieve reduced switching frequency while the torque and stator flux are kept within their respective hysteresis bounds. Another approach presented by Zhang et al. applies duty ratio modulation to basic DTC [12]. The torque ripple has been significantly reduced by adjusting the duty ratio of the active voltage vectors. The key point of this strategy is to determine the duty ratio. Several different methods have proposed to obtain the duty ratio based on different optimization objectives and voltage vector numbers [13]-[14]. These improved DTC strategies result in varied performance in terms of torque ripple, and switching frequency. In practical applications, a particular set of performance characteristics of
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the controllers may be preferred over others. For instance, in high power application, reducing the switching frequency of power devices is very likely of top-priority significance. The majority of published papers only focus attention on the comparison between basic DTC and modified DTC, or among modified DTC schemes of the same type [15]-[17]. But very few have conducted the comparative evaluation of different types of improved DTC schemes, especially relatively new methods such as MPDTC. In this paper, these improved DTC strategies and FOC will be comparatively investigated through simulations using various criteria, and a comprehensive conclusion is drawn to summarize the results. The results will provide valuable guidance for the users to determine which scheme shall be employed in order to achieve the desired objectives. The rest of the paper is organized as the follows: Section II presents the PMSM model. In Section III, the control strategies, i.e., basic DTC, MPDTC, DTC-duty and FOC are reviewed. The overall control performance of each control scheme is also included in this section. The comparative evaluation with various criteria is presented and analyzed in Section IV. The paper is concluded with a summary in Section V. II. M ODEL OF PMSM S The state equations of a PMSM in d-q reference frame are expressed as follows us,dq = Rs is,dq +
d ψs,dq + Fψs,dq dt
(1)
where Rs is the stator resistance; ψs,dq is the stator flux; us,dq = [usd usq ]T and is,dq = [isd isq ]T are stator voltage and current, respectively. The stator flux is T
ψs,dq = [ψsd ψsq ] = Gis,dq + ψr,dq F=
(2)
A. Basic DTC For AC machines, the electromagnetic torque is proportional to the cross product of ψs and ψr . The basic DTC controls the magnitude and angle of stator flux by applying appropriate voltage vector to obtain desired electromagnetic torque. The block diagram of basic DTC is shown in Fig. 1.
ωr* + _
ωr
Te* +
PI
Tee
_
ψ s* +
LUT
ψ se
_ Te
|ψ s |
sector uabc Torque&Flux iabc Observer
PM SM Fig. 1. Block diagram of basic DTC.
The desired stator flux and torque are compared with the actual values in separate hysteresis controllers. The flux controller is a two-level comparator while the torque controller is a three-level comparator. According to the stator flux position and the output signals of hysteresis controllers, there exists an optimal voltage vector to be applied to the stator winding, which will minimize the error of torque and stator flux in every control period. The selection of optimal voltage vector is implemented with reference to Table I, which requires three inputs and provides one output. The three inputs are: desired stator flux variation; desired torque variation; and the sector that stator flux stays within. The single output is the optimal voltage vector to be applied to the machine.
i Ld 0 0 −ωr ,G = 0 Lq ωr 0
h
TABLE I VOLTAGE V ECTOR L OOKUP TABLE OF BASIC DTC
with ωr being electrical rotor angular speed, Ld and Lq being the direct-axis and quadrature-axis stator inductance, T respectively; ψr,dq = [ψf 0] with ψf being the permanent magnet flux. The electromagnetic torque produced by the machine is Te =
3 p (ψd iq − ψq id ) 2
(3)
where p is the number of pole pairs. III. C ONTROL S TRATEGIES In this section, the DTC based control strategies will be briefly reviewed to lay foundation for more detailed comparative study in Section IV. The simulation results of overall control performance for each control strategy will also be presented.
ψse
Tee
Sector ® ¯
¬
°
±
1
1 0 −1
us2 us0 us6
us3 us7 us1
us4 us0 us2
us5 us7 us3
us6 us0 us4
us1 us7 us5
−1
1 0 −1
us3 us7 us5
us4 us0 us6
us5 us7 us1
us6 us0 us2
us1 us7 us3
us2 us0 us4
In Fig. 2, the performance of basic DTC are shown. The machine parameters are listed in Table II. It can be observed that the torque and stator flux are limited within the hysteresis bounds. The switching frequency is not constant at the beginning, but it will become stabilized gradually. In addition, the switching frequency varies according to the machine speed, load and the hysteresis bands of torque and stator flux.
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the following discrete-time model of PMSM Rs Ts ψr,dq Ld (5) where I denotes the identity matrix; and Ts is the sampling interval. The prediction of stator flux can be achieved by (5). Furthermore, the prediction models of stator current and electromagnetic torque are shown as (6) and (7). ψs,dq (k + 1) = (I − DTs ) ψs,dq (k)+Ts us,dq (k)+
is,dq (k + 1) = E (ψs,dq (k + 1) − ψr,dq ) R
s
D = Ld ωr
(6)
1 0 Ld 1 Rs , E = 0 Lq Lq
−ωr
3 pψs,dq (k + 1) ⊗ is,dq (k + 1) (7) 2 2) Prediction Algorithm: The prediction algorithm of MPDTC is shown in Fig. 4. y ∗ = [Te∗ ψs∗ ]T is the vector of desired torque and desired stator flux; y = [Te ψs ]T is the vector of actual torque and actual stator flux; ∆y = [∆T ∆ψs ]T is the vector of hysteresis bounds of torque and stator flux; ye = [|Te∗ − Te | |ψs∗ − ψs |]T is the vector of absolute values of torque error and stator flux error; N is prediction horizon; usm is the non-zero voltage vector of two-level VSI. Te (k + 1) =
Fig. 2. Starting and sudden load change response of basic DTC. Mechanical speed: 600 rpm; load: 0.4 p.u. (72 Nm), applied at 0.03s. The graphs are (from top): electromagnetic torque (ripple: 7.2 Nm); stator flux (ripple: 0.0014 Wb); current of phase A; switching frequency (2.8 kHz).
B. MPDTC The block diagram of MPDTC is shown in Fig. 3. It is almost the same as the diagram of basic DTC, and the hardware of basic DTC does not need any change to implement MPDTC. The only difference is that the MPDTC model is employed to replace the hysteresis controllers and voltage vector lookup table (LUT). The MPDTC model mainly includes prediction algorithm and cost function, which will be described in the following.
ωr* + _
ωr
PI
Begin (k-1) state measurement k state prediction prediction states (k+n) when voltage vector usm is applied (n=1,2...N;m=1...6)
Te*
ψ s*
| y* − y ( k + n) |< Δy or
MPDTC Te
ye (k + n) < ye (k + n − 1)
|ψ s |
n=n+1 n>N
Torque&Flux uabc iabc Observer
N
Y
N m=m+1 n=1
PM SM
N m>6 Y cost function
Fig. 3. Block diagram of MPDTC.
1) Prediction Model: From (1) and (2), the state-space equation of PMSM can be obtained as d Rs ψs,dq = us,dq − Dψs,dq + ψr,dq dt Ld
Y
End Fig. 4. Flow chart of the prediction algorithm of MPDTC.
(4)
Application of forward Euler approximation approach yields
The prediction algorithm is carried out one sampling interval ahead so that real-time control of whole system can
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be implemented given the computation burden. The output voltage vector is determined at kth sampling instant, but it is not applied until (k + 1)th sampling instant. The execution of MPDTC algorithm works as follows: The necessary machine variables are measured in period (k − 1), and the control algorithm is executed in (k − 1). The first step is to compute the system states of period k according to the current voltage vector applied to the machine. Then the possible future system states within prediction horizon are computed for the periods (k + 1) , . . . , (k + N ) while all admissible voltage vectors are considered. Finally, the prediction results are evaluated with a cost function and the voltage vector with lowest cost will be applied in period k. At the next sampling instant, this procedure is repeated with updated measurements.
simplified when (8) is used as the cost function. The detailed prediction algorithm is explained in [7] and the strategies to reduce the computational burden are introduced in [10]-[11]. Fig. 5 shows the overall control performance of MPDTC. The torque and stator flux are controlled within the pre-designed boundaries. Like basic DTC, the inverter switching frequency varies according to the machine speed, load and tolerance of hysteresis bounds. C. DTC with duty ratio modulation For basic DTC and MPDTC, non-zero voltage vector is applied to the machine during the whole sampling period, which is the main cause of high torque ripple. Therefore, duty ratio modulation is introduced to basic DTC and leads to an improved DTC algorithm, namely DTC-duty. For DTC-duty, the time duration of non-zero voltage vector varies from zero to the whole period, which is equivalent to change the voltage vector amplitude. Zero vector is applied during the remaining time to maintain torque and stator flux.
Fig. 5. Starting and sudden load change response of MPDTC. Mechanical speed: 600 rpm; load: 0.4 p.u. (72 Nm), applied at 0.03s. The graphs are (form top): electromagnetic torque (ripple: 4.8 Nm); stator flux (ripple: 0.0014 Wb); current of phase A; switching frequency (4.4 kHz).
3) Cost Function: There are different forms of cost function [7]-[8]. Authors of [7] proposed a cost function whose main objective is to reduce the switching frequency while torque and stator flux are limited within their hysteresis bounds. The cost function expressed by (8) combines multiple control criteria and corresponding weight coefficients to achieve different objectives by adjusting the weight coefficients [8]. k+n P
cost (m) =
2
2
λT (Te (i) − Te∗ ) + λF (ψs (i) − ψs∗ )
Fig. 6. Starting and sudden load change response of DTC-duty. Mechanical speed: 600 rpm; load: 0.4 p.u. (72 Nm), applied at 0.03s. The graphs are (form top): electromagnetic torque (ripple: 2.7 Nm); stator flux (ripple: 0.002 Wb); current of phase A; switching frequency (8.1 kHz); duty.
The duty ratio d is determined by (9)
i=k+1
n
(8) where λT and λF are weighting coefficients. The objectives of (8) are the same as that of the prediction algorithm shown in Fig. 4, which means that the prediction algorithm could be
2
2
cost (m) = λT (Te (k + 1) − Te∗ ) + λF (ψs (k + 1) − ψs∗ ) (9) where λT and λF are weighting coefficients. The cost will will change as the voltage vector and duty ratio d vary. The
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optimal voltage vector and duty ratio is determined to obtain the minimum cost. The details about different methods of calculation for optimal duty ratio to achieve various objectives can be found in [12]-[14]. Fig. 6 shows the overall performance of the system based on DTC-duty strategy. It is worth noting that the switching frequency is constant in both dynamic and steadystate conditions. The duty ratio varies during transient period as the machine speed or load change and settles to constant value during steady-state.
TABLE II M AIN PARAMETERS OF S YSTEM Electrical Machine Type Rated power P [kW] Rated current I [A] Rated torque Te [N·m] Max speed ωm [rpm] Stator resistance R [Ω] Inductance (d axis) Ld [mH] Inductance (q axis) Lq [mH] PM rotor flux ψf [Wb] Pole pairs p
D. FOC In order to clearly reveal the advantages and disadvantages of DTC, the most commonly used FOC strategy is also included to serve as the benchmark. The control scheme is implemented in the rotor flux reference frame using classical PI regulators. Fig. 7 shows the performance of FOC. The constant switching frequency is observed.
IPMSM 60 140 180 9000 0.033 0.2 0.35 0.07 6
Inverter Type DC-link voltage Udc [V] Max current I [A]
2-level VSI 300 200
A. Torque Ripple First, torque ripple difference under same switching frequency and stator flux ripple is presented in Fig. 8. Table III summarizes the quantitative index of various strategies.
Fig. 7. Starting and sudden load change response of FOC. Mechanical speed: 600 rpm; load: 0.4 p.u. (72 Nm), applied at 0.03s. The graphs are (form top): electromagnetic torque (ripple: 2.3 Nm); current of phase A; switching frequency (10 kHz).
Fig. 8. Torque ripple difference between the control strategies when switching frequency and stator flux ripple stay consistent. Mechanical speed: 1000 rpm; load: 0.3 p.u. (54 Nm); stator flux ripple: 0.002 Wb.
IV. P ERFORMANCE E VALUATION
TABLE III T ORQUE R IPPLE D IFFERENCE B ETWEEN T HE C ONTROL S TRATEGIES U NDER S AME S WITCHING F REQUENCY
The four control strategies explained in Section III will be comparatively investigated through MATLAB/Simulink model. The parameters of the machine and inverter in the simulation model are listed in Table II. For basic DTC and MPDTC, the magnitudes of the torque hysteresis bounds are adjusted to achieve the same effective switching frequency. For DTC-duty and FOC, the sampling interval is adjusted to obtain the switching frequency equal to basic DTC and MPDTC. It is widely known that the magnitude variation of stator flux ripple will also affect the switching frequency. In order to make a fair comparison, the magnitudes of stator flux ripple of these methods are kept as consistent as possible.
Methods
5 kHz
Torque ripple [N·m] 7.5 kHz 10 kHz
15 kHz
Basic DTC
6.5
3.9
2.6
1.5
MPDTC
6.5
3.7
2.3
1.2
DTC-duty
6
4
3
2.5
FOC
7.2
5
3.9
2.9
It can be observed that the torque ripple decreases as the switching frequency increases. At low switching frequency of 5 kHz, DTC-duty results in the lowest torque ripple because of its large sample interval. The torque ripple levels of basic DTC and MPDTC are comparable while FOC features the
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most pronounced torque ripple. The torque ripple of DTCduty, MPDTC and basic DTC are, respectively, 83%, 90% and 90% of the torque ripple of FOC. With increased switching frequency, the torque ripple of MPDTC decreases relatively faster than the other three methods. In contrast, DTC-duty has the slowest rate of decline. At higher switching frequency of 15 kHz, the lowest torque ripple is achieved by MPDTC followed by basic DTC and DTC-duty. FOC still features the highest torque ripple. The torque ripple of MPDTC, basic DTC and DTC-duty are 42%, 52% and 86% of the torque ripple of FOC. One conclusion to be drawn from the preceding discussion is that MPDTC consistently outperforms basic DTC, which is particularly true at high switching frequency. On the other hand, FOC consistently underperforms the DTC methods in terms of torque ripple level. However, the sampling period in FOC could be allowed much longer than basic DTC and MPDTC. The allowed lower sampling frequency in FOC is advantageous for experimental implementation. The performance of DTC-duty deteriorates as switching frequency increases.
(a)
B. Switching Frequency (b)
The impact of DTC methods on switching frequency is investigated while torque ripple and stator flux ripple stay consistent. The results are illustrated in Fig. 9 and tabulated in Table IV.
(c)
Fig. 9. Switching frequency difference between the control strategies when torque ripple and stator flux ripple stay consistent. Mechanical speed: 1000 rpm; load: 0.3 p.u. (54 Nm); stator flux ripple: 0.002 Wb. (d) Fig. 10. Steady response of (a) basic DTC; (b) MPDTC; (c) DTC-duty and (d) FOC with machine parameters exceeding the actual value by 10%. Mechanical speed: 1000 rpm; load: 0.3 p.u. (50 Nm); switching frequency: 5 kHz.
TABLE IV S WITCHING F REQUENCY D IFFERENCE B ETWEEN T HE C ONTROL S TRATEGIES U NDER S AME T ORQUE R IPPLE Methods
Switching frequency [kHz] 2 Nm 5 Nm 7 Nm 10 Nm
Basic DTC
12.1
6
4.8
3.8
MPDTC
11.2
5.8
4.7
3.8
DTC-duty
13.5
5.8
4.6
3.4
FOC
20
7.5
5.2
3.7
Fig. 9 shows significant switching frequency difference among the DTC methods when the torque ripple is low (2
Nm). FOC exhibits the highest switching frequency followed by DTC-duty, basic DTC and MPDTC, and the reductions in the switching frequency are 32.5%, 39.5% and 44%, respectively. With the increasing of torque ripple, the switching frequency monotonically decreases. FOC achieves the fastest rate of decline followed by DTC-duty, basic DTC and MPDTC. When the torque ripple is 10 Nm, the switching frequency
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of basic DTC, MPDTC and FOC are almost the same. And DTC-duty presents a 10% reduction of switching frequency in comparison to the other three. FOC still underperforms all DTC methods while MPDTC outperforms the rest in terms of switching frequency. It is worth mentioning that higher switching frequency implies higher switching losses. The performance of DTC-duty improves with increased torque ripple level. C. Parameter Sensitivity (a)
The parameter sensitivity is of substantial significance in practical implementation the control strategies under consideration. As a result, the performance of the control strategies with intended estimation error in the machine parameters is evaluated through simulations. Fig. 10 shows the stator flux ripple and torque ripple of different strategies with the values of stator resistance and inductance exceeding the actual value by 10%. The simulation results reveal that, for all methods, the performance deterioration of stator flux is insignificant when the machine parameters vary. Similarly, the torque ripple of basic DTC almost stays independent from the machine parameter variations. In contrast, the torque ripple of MPDTC and DTC-duty shows higher sensitivity to mainly inductance estimation error. It should be noticed that the torque ripple of FOC is even a little smaller with the mismatched parameters. Such observation is due to the fact that the determination of proportional gain and integral gain of the current regulator is related to the values of stator resistance and inductance. Therefore, the machine parameter variation will indirectly affect the performance of FOC even though the strategy directly employs neither of them.
(b)
D. Computational Complexity
(c)
Among the four methods, basic DTC demands minimum amount of computation. The main part of its computation is the integration for estimating the stator flux. On the other end, MPDTC is the most complicated because of its iterative algorithm to achieve the prediction. The computational complexity of MPDTC is proportional to the number of admissible voltage vectors and the prediction horizon. The number of admissible voltage vector is determined by the inverter topology, i.e., multi-level inverter has more admissible voltage vector than two-level inverter. The prediction horizon could be adjusted according to the required performance in certain application. Long prediction horizon will greatly boost the performance of MPDTC, such as reduced switching frequency and torque ripple. But at the same time, large prediction horizon will also lead to substantial increase of calculation burden. Several approaches to reducing the amount of computation for MPDTC have been proposed such as branch and bound algorithm for MPDTC [10]. Further simplification is necessary in order to allow MPDTC to be implemented in broader range of applications. The computation complexity of DTC-duty and FOC are essentially equivalent. Their main tasks are almost the same,
(d) Fig. 11. THD of stator current for (a) basic DTC; (b) MPDTC; (c) DTC-duty and (d) FOC. Mechanical speed: 1000 rpm; load: 0.3 p.u. (54 Nm); switching frequency: 5 kHz.
which involve calculating the duty ratio of different voltage vectors. E. Stator Current THD The spectrum of stator currents affects the iron losses of the machine and the total harmonic distortion (THD) is the
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widely adopted merit figure of spectrum performance. The stator current and its THD at steady state are shown as Fig. 11. The simulation condition is set to the rotor speed of 1000 rpm and the loading torque of 0.3 p.u. (54 Nm) for different strategies. The THD is calculated up to 10 kHz. It can be observed that the four control strategies feature different stator current and torque ripple under same switching frequency. The DTC-duty results in highest THD while its torque ripple is lowest. The higher stator current THD of DTCduty is mainly caused by the large stator flux ripple, which could be regulated by adjusting the coefficients λT and λF . It is a tradeoff between the stator flux ripple and torque ripple that needs to be balanced according to application requirement. The stator current THDs of DTC-duty and FOC are higher than basic DTC and MPDTC.
[5]
[6]
[7]
[8]
V. C ONCLUSION Four control strategies that include three DTC-based strategies and FOC for PMSM have been compared against several performance metrics. The comparative study clearly reveals advantages and disadvantages associated with each control scheme and will provide valuable guidance to decide the most suitable control scheme for a specific application. Generally speaking, basic DTC features the simplest structure among the control strategies, and its switching frequency is low, which collectively indicates that it is suitable for high power applications. But high sampling frequency is required to obtain adequately low torque ripple. MPDTC achieves lower switching frequency and torque ripple than basic DTC, but its computation complexity is substantially elevated while it is also relatively sensitive to parameter changes. Hence, it could be employed for high accuracy control in high power occasions. FOC has simple structure which makes it easy to be implemented. Moreover, the switching frequency of FOC is inherently constant. The switching frequency that is higher than the other three methods is necessary to achieve comparable torque ripple, which means suitability for low power applications. DTC-duty plays the role as a compromised approach between FOC and basic DTC.
[9]
[10]
[11]
[12]
[13]
[14]
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