A Novel Model Predictive Current Control for Asymmetrical Six-Phase PMSM Drives With an Optimum Duty-Cycle Calculation Scheme

This paper presents a novel model predictive current control (MPCC) with an optimum duty-cycle calculation scheme for asymmetrical six-phase permanent magnet synchronous machine (ASPMSM) drives. The proposed method takes advantages of the steady-state performance improvement and the weighting factor elimination. Both merits are owing to the optimum duty-cycle calculation scheme. On the one hand, for the <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> - <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> subspace, the optimal voltage vector set (VVS) is chosen from twelve ones and the corresponding optimal duty cycles are calculated using the proposed scheme. On the other hand, for the x-y subspace, the VVS can be determined according to that of <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> - <inline-formula> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> subspace and the optimal duty cycles are deduced using the same scheme. The voltage vector references in two subspaces are then obtained, and they are transformed to six-phase voltages for controlling the six-phase inverter. The proposed method is verified by experimental results based on a 2 kW ASPMSM prototype.

The ASPMSMs have two sets of three-phase wingdings spatially shifted by 30 • , accordingly eliminating the torque pulsation related to sixth order harmonics [6]. Based on the vector space decomposition (VSD) technique [7], six-phase stator currents can be mapped into three subspaces, namely αβ subspace (12k±1th components, k = 1, 3, 5, . . . , relating to torque generation), x-y subspace (6k±1th components, The associate editor coordinating the review of this manuscript and approving it for publication was Pinjia Zhang . k = 1, 2, 3), and z1-z2 subspace (zero-sequence components, which can normally be omitted from consideration). The components in x-y subspace are advantageous for faulttolerant operation [8], [9], [10], while they merely result in extra heat loss in case of normal operation.
In recent years, traditional methods such as FOC and DTC for three-phase machines have been successfully extended to the ASPMSM drives [11], [12]. In order to suppress the harmonic currents, additionally integrated controllers are required, such as PI controller or improved switching table.
In comparison with the foresaid methods, model predictive control (MPC) is more powerful to realize multi-objective and nonlinearity control [13]. In machine drives, model predictive current control (MPCC) and model predictive torque control (MPTC) are the main classes [14], [15], by using cost functions constraining the stator currents and the torque/flux (this requires a weighting factor to make a trade-off between two variables), respectively. If employing MPC methods to ASPMSM drives, the components of x-y subspace should also be involved, with an additional weighting factor placed in front of them [16]. However, the tuning work of weighting factor is still complex and time-consuming because of the lack of theoretical support at present.
In order to eliminate the harmonic currents of x-y subspace without introducing weighting factors, many improved strategies have been proposed. For example, [17] provides a model predictive flux control (MPFC), of which, not only the torque and flux control is replaced by flux vector control but the harmonic currents are transformed to the harmonic flux vector as well. Alternatively, virtual vector construction is also a permissible way for harmonic currents elimination, on the basis of the fact that a switching state can produce two different voltage vectors in α-β and x-y subspaces [18], [19], [20]. That is, a set of virtual vectors can be properly constructed by synthesizing actual voltage vectors in the α-β subspace and all these correspond to the zero-voltage vector in the x-y subspace. Applying these virtual vectors to the inverter does not generate harmonic voltages and hence harmonic currents. By doing so, the tuning work of weighting factors and the computation burden increase are circumvented, but a reduced dc-link voltage utilization is inevitable.
On the other hand, it is well acknowledged that the traditional MPC methods suffer from the problem of bad steadystate performance because only one voltage vector is chosen and applied during each sampling period [13]. A huge body of publications focus on this issue, and the solutions are usually to replace the original single vector by a set of vectors. The duty cycles of vectors can be addressed based on numerus criterions. Additionally inserting one and two zero vectors during each sampling period are investigated in [14] and [18], respectively. The duty cycles of the active vector and the zero vector(s) are determined by solving the partial derivatives of the cost function. The results show an obvious enhancement in the steady-state performance; however, the cost is a larger calculation burden accompanied by solving the duty cycles.
Dissimilarly, a novel duty-cycle optimization scheme is proposed in [21] for a two-level three-phase grid-connected converter. The scheme is cascaded with the standard MPC and can be seen as a re-optimization process, which is exactly akin to the process of finding the minimized cost function in the standard MPC. The principle relies on a fact that there always exists an optimum set of duty cycles, for a certain set of voltage vectors, obtains the optimum control performance [22]. In [21], the optimum duty cycles are calculated for all sets of voltage vectors, and the so-called optimum control performance refers to the minimum root-mean-square (RMS) value associated with the cost function values of each set of voltage vectors. Then, the optimum set of voltage vectors can be readily determined by comparing the minimum RMS values. The drawback of this scheme is the non-negligibly additional computation burden due to the re-optimization process. For this reason, [23] proposes an improved duty-cycle optimization scheme for an ASPMSM drive. The scheme can avoid the additional computation burden to a large extent in a manner where the cost functions values directly determine the optimum set of voltage vectors. The extra computation burden is merely caused by the duty-cycles' calculation. Meanwhile, in [23], 12 active and 1 zero virtual vectors are constructed to tackle the issue of harmonic currents elimination, therefore a low dc-link voltage utilization occurs.
In this paper, a novel MPCC in conjunction with the optimum duty-cycle calculation scheme is proposed, taking advantages of the steady-state performance improvement and the weighting factor elimination. In the proposed method, the optimum duty-cycle calculation scheme is utilized for both currents in α-β and x-y subspaces, without introducing weighting factors. By doing so, the optimal duty-cycles are obtained and the voltage vector references in two subspaces are further calculated. The voltage vector references are then transformed to six-phase voltages that are outputted to the inverter using a pulse width modulation (PWM) module.
The rest of the paper is organized as follows. The mathematical modelling based on VSD technique of ASPMSM is described in Section II. In section III, the proposed MPCC method is elaborated. Consequently, the experimental results are presented to verify the method in Section IV. Finally, conclusion is made in Section V.

II. MODELLING OF ASPMSM DRIVE AND PRINCIPLE OF MPCC A. MATHEMATICAL MODEL OF ASPMSM DRIVE
The topology of an ASPMSM fed by a six-phase two-level power converter can be depicted in Fig. 1. Using the VSD technique, the machine variables can be mapped into two orthogonal subspaces, namely α-β and x-y subspaces. Considering the invariant amplitude criterion, the Clarke's transformation matrix based on VSD for the ASPMSM is Besides, the variables of α-β subspace are usually converted into d-q frame by the Park's transformation for the purpose of independent torque and flux control, as where θ is the angle of d-axis apart from α-axis.  On this basis, the mathematical model of the ASPMSM drive can be shown as follows.
where the variables v, i, and ψ are the stator voltage, current and flux, respectively; the subscripts d, q, x and y mean d-, q-, x-and y-axes components, respectively. Besides, L d , L q and L l are the d-axis, q-axis and leakage inductances, respectively; ψ f is the permanent magnet flux; R is the inner resistance of stator windings. Then, ω r , T e and P n are the rotor electrical angular speed, the electromagnetic torque and the number of pole pairs, respectively. In the equations, the stator voltages mapped onto different axes are controlled by the inverter, which contains six switch legs with two states, 1 and 0, corresponding to closing the upper and the lower switches, respectively. Customarily, the terminal voltages can be denoted using switching states. Here, the octal number of [S a S b S c ]-[S u S v S w ] is adopted, where S i =1 or 0 (i = a, b, c, u, v, w). Each switching state generates a voltage vector in both subspaces, as seen in Fig. 2. In each subspace, there are 64 voltage vectors divided into five groups according to their amplitudes, namely L 1 (0.644u dc ), L 2 (0.471u dc ), L 3 (0.333u dc ), L 4 (0.173u dc ) and zero voltage vectors.

B. TRADITIONAL MPCC FOR ASPMSM DRIVE
The MPC method is a model-based optimization method. When this method is applied to power converters, the so-called model is the dynamic model containing differential terms; and the optimization problem is always solved using enumeration, due to finite switching states can be generated by the switch-based converters. In order to do prediction for the future behavior of system, the discretized model should be deduced. In this paper, the forward Euler equation is applied, as where x denotes an arbitrary variable, such as current; ''k'' and ''k + 1'' mean the current and the next sampling periods, respectively; T s is the sampling period. Then, (3) and (4) can be discretized as 8098 VOLUME 11, 2023 The next step is to decide the optimum switching state from 64 ones, which can be realized by minimizing a cost function. With regard to the ASPMSM drive, two objectives must be considered, regulating the torque and eliminating the harmonic currents of x-y subspace. The torque control can be realized by adjusting the currents in α-β subspace. This leads to a cost function as where and the subscript '' * '' denotes the reference values. Similarly, the purpose of harmonic current elimination can be achieved by where Eventually, a multi-objective cost function can be obtained by algebraically summing (11) and (13) with a weighting factor λ placed in front of g 2 , which can be given as To determine which switching state is optimum, the problem can be readily solved in an enumerated manner. That is, predicting the currents is carried out using (9) and (10) for all possible voltage vectors and then evaluating their effects is done by calculating the cost function (15). The switching state corresponding to the minimum cost function value is the optimum one, which is then applied to the inverter. The enumeration is exactly straightforward in principle, while it causes a very large computation burden due to 64 possible choices; at the same time, the tuning work of the weighting factor λ is very demanding as well.

III. PROPOSED MPCC METHOD
This section elaborates the proposed MPCC method. At first, the optimum duty-cycle calculation scheme is discussed. Thereafter, the scheme is used for both α-β and x-y subspaces to calculate the voltage vector references in two subspaces. The references are then transformed to six-phase voltages, which are outputted to the inverter using PWM module.

A. PRINCIPLE OF THE OPTIMUM DUTY-CYCLE CALCULATION SCHEME
The optimum duty-cycle calculation scheme can be seen as the re-optimization process akin to that for finding the minimized cost function in the standard MPC. First of all, the voltage vector sets (VVSs) to be assessed should be clarified, and the number of which is relevant to the additional computation burden. In [21] and [22], three adjacent vectors forming a closed triangle in shape is included in an SVV. In this condition, the re-optimization process contains 6 and 84 repeated calculations for three-phase and six-phase twolevel converters, respectively. It is surely no longer affordable to apply this scheme to the ASPMSM drive, as a result of too large additional computation burden. In order to tackle this issue, this paper only concerns 12 voltage vectors with largest magnitude in α-β subspace (namely group L 1 ) and a zero vector. The vectors are called V 0 ∼ V 12 , as presented in Fig. 3(a). The VVSs are constructed using the same way as in [21] and [22], written by As a result, the re-optimization process has 12 repeated calculations, extremely reducing the additional computation burden. It should be mentioned that the group L 1 corresponding to the group L 4 in x-y subspace, where the voltage vectors are renumbered by V h1 ∼ V h12 , as shown in Fig. 3(b). The vectors V 1 ∼ V 12 corresponds to V h1 ∼ V h12 according to the switching states. The effects of V 1 ∼ V 12 on the dq-axes currents are predicted and evaluated using (9) and (11), respectively. The cost function value associated with V i can also be VOLUME 11, 2023 expressed by where e i is the current tracking error due to applying V i If an exceptionally small sampling period in the digital implementation is considered, the tracking error can be linearly approximated by the duty-cycle-weighted value d i e i [21].
In case of employing a VVS during one sampling period, their optimum duty cycles can be solved through minimizing the RMS value associated with the duty-cycle-weighted values, which can be governed by Minimizing (18) is equivalent to The solution of (19) can be readily deduced using the wellknown Lagrange multiplier method [23], obtaining the optimum duty cycles as In the following, the optimum VVS should be determined.
In conjunction with the foresaid optimum duty cycles, the minimized RMS value shown in (19) can be derived as Substituting (20) into (21), the minimized RMS value can be further acquired as It is evident that the lower the values of g 1 (V i ) and g 1 (V i+1 ) are, the lower the minimized RMS value is. Therefore, the optimal VVS can be readily determined according to the cost function values of the 12 voltage vectors shown in Fig. 3(a). With regard to a drive system, the applied voltage should change in a continuous manner in space. Therefore, the optimum VVS is chosen with two steps: 1) acquiring the optimum voltage vector corresponding to the minimized cost function value; 2) choosing the sub-optimum voltage vector from the two ones adjacent to the optimum one by comparing their cost function values.

B. CALCULATIONS OF VOLTAGE VECTOR REFERENCES IN TWO SUBSPACES
It should be noted that the aforementioned analyses are based on the cost function g 1 . This means that the deduced optimal duty cycles are only effective for regulating the components in α-β subspace. However, the components in x-y subspace are not under control. In this subsection, the obtained optimal VVS along with the optimal duty cycles are further exploited for the harmonic current elimination purpose.
Regarding the α-β subspace, the desired voltage vector can be readily calculated through vectorially summing the three vectors in the optimal VVS, weighted by the corresponding duty cycles, expressed as (23) where V * αβ is the voltage vector reference in α-β subspace. V * α is the component of the foresaid vector reference on the α-axis; V * β is the component of the foresaid vector reference on the β-axis.
In terms of the x-y subspace, the voltage vector reference is derived by a way similar to that of α-β subspace. Actually, the optimal VVS in α-β subspace refers to an optimal VVS in x-y subspace. For example, {V 0 , V 1 , V 2 } corresponds to {V h0 , V h1 , V h2 }, as highlighted in Fig. 3. The optimal VVS in x-y subspace is expressed by Note that two adjacent vectors in the αβ subspace correspond to two vectors 150 degrees apart in the xy subspace, such as V h1 &V h2 , V h3 &V h4 , and others. It is interesting that the synthesis of two vectors nearly in the opposite directions and the zero vector can suppress the xy axes voltages to a large extent. This feature is very similar to that of the virtual vector construction. Then, the optimal duty cycles of these three vectors can be deduced by the scheme proposed in Section III-A. Please note that the cost function for calculation is g 2 here. In this manner, the optimal duty cycles are (25), as shown at the bottom of the next page, Thus, the voltage vector reference in x-y subspace should be where V * xy is the voltage vector reference in x-y subspace. V * x is the component of the foresaid vector reference on the 8100 VOLUME 11, 2023 x-axis; V * y is the component of the foresaid vector reference on the y-axis.
The harmonic currents can be suppressed effectively by using this vector. The combination of V * αβ and V * xy can realize the control of ASPMSM drive. The two references can be transformed to six-phase stator voltages using the anti-Clarke's transformation matrix, as with and

C. STABILITY ANALYSIS
Stability analysis is undoubtedly a crucial issue for control strategies. However, publications involve the stability verification of MPC for power electronics are still rare [24], since most of the designed cost functions do not conform to the definition of the Lyapunov function. In order to ensure the stability, some works in this field design the cost function on the basis of the Lyapunov function stability theory [25], [26], [27], [28]. In [28], the control law based on continuous input is developed from the control-Lyapunov function of feedback control law and online adaptive law. The continuous control law is then transformed into the discreate constraints of MPC. However, transforming the control law from the continuous one to the discreate one may violate the conditions of Lyapunov stability theory. In [29], ''practical stability'' is mentioned to emphasize that only stability to a bounded neighborhood of the desired equilibrium state can be guaranteed. In short, the so-called practical stability implies that the state(s) of a system is controlled within a band, a circle or a ball. Here, according to the foresaid thought, the proposed MPCC method is proven to be practically asymptotically stable. Considering the dq-axes components at first, the dq-axes currents are regulated by the cost function (11). It should be noted that the vector corresponding to zero value of the cost function is the best one, which makes the currents equal to their references. The best vector can be calculated according to the deadbeat concept [30], and is marked by In essence, the traditional MPCC is actually to choose the vector, denoted by optimal vector V opt , closest to the best vector from a finite set. In our work, differently, the applied vector is determined by the optimal vector V opt , the suboptimal vector V sub , the zero vector V 0 and their durations, as expressed in (23). The diagram showing the relationship among the best vector V op , the optimal vector V opt and the applied vector V * αβ is presented in Fig. 4. In principle, the applied vector is more precise than the optimal vector. However, it is hard to ensure that the applied vector can overlap the best voltage vector. The applied vector can be expressed by the sum of the optimal vector and a quantization error, as where ε donates the quantization-error vector. Note that V * αβ is dependent on three vectors bounded within a finite set and their durations ranging from [0, 1], and V op , in relation to the dc bus voltage, is also bounded. Thus, the quantization-error vector is bounded as well [31], expressed with where ϕ > 0 is the upper bound. Then, substituting (30) into the prediction model of (9) yields  where ε d and ε q are the dq-axes components of the quantization-error vector.
Following this, substituting (32) into the cost function (11), considering the prediction model in (9) and that the studied PMSM is a salient-pole machine (this implies L d < L q ) can obtain From this inequality, it can be concluded that the cost function is ultimately bounded, which means that the desired equilibrium state of the cost function bounded by T s ϕ/L d can be reached in infinite time steps [30]. The xy-axes components can be analyzed by the same way. Hence, the proposed MPCC method is practically asymptotically stable.

D. SUMMARY OF THE PROPOSED METHOD
In terms of controlling dq-and xy-axes currents, the proposed MPCC method is a two-step cascaded method, rather than a weighted-optimization one (the traditional MPCC) or a separate-optimization one (the MPCC with the virtual vector construction like that proposed in [23]). Comparatively, the proposed method is an improved version of the two existing ones. The proposed method is able to not only eliminate the weighting coefficient in the weighted-optimization method but also improve the performance of xy currents by replacing the open-loop control due to the virtual vector construction with the online optimization. The overall control diagram of the proposed MPCC is shown in Fig. 5. The following steps are required in order to implement the proposed method.
Step 1. Measure six-phase stator currents, machine speed, rotor position and dc-link voltage at kth.
Step 2. Calculate the αβ-axes and xy-axes components of stator currents using (1), and further compute the dq-axes components using (2).
Step 3. Obtain the cost function values using (11) for the voltage vector in Fig. 3(a), and then choose the optimal VVS and calculate the optimum duty cycles using (20).
Step 4. Achieve the desired voltage vector in α-β subspace by (20); acquire the optimum duty cycles and the  voltage vector reference in x-y subspace using (25) and (26), respectively.
Step 5. Transform the voltage vectors in two sub-spaces into six-phase voltages using (27), and output the corresponding PWM signal by comparing the six-phase voltages with pre-loaded carriers.

IV. EXPERIMENTAL RESULTS
In order to demonstrate the feasibility and effectiveness of the proposed MPCC method, an experimental prototype is built, as shown in Fig. 6. The ASPMSM under test is with the rated power of 2 kW. Table 1 presents the main parameters of the machine. The six-phase inverter consists of six FF300R12ME4 (from Infineon) modules, and it is powered by an adjustable dc power supply, whose maximum voltage and rated power are 300 V and 3 kW, respectively. Furthermore, the dc-link voltage is obtained by a voltage sensor WHV05AS3S6 and the six-phase stator currents are sampled using six current sensors WHB25LSP3S1. In addition, a digital signal processor (DSP) TMS320F28335 from TI is chosen to implement the real-time control code that is developed using C language in the Code Composer Studio 6.0 software. In experiments, both the sampling frequency and the carrier frequency are set to 10 kHz.

A. TWO METHODS FOR COMPARISON
To verify the effectiveness of the proposed method, two existing methods are also implemented for comparison. One is the conventional MPCC, called ''method #1'' in what follows. It should be noted that the optimal vector is chosen from only 13 voltage vectors (twelve large ones and a zero one, as shown in Fig. 3(a)), in order to guarantee a control frequency of 10 kHz. The cost function is the same as (15).  Another one is that proposed in [23], denoted by ''method #2'' in the following. The method is also based on the optimum duty-cycle calculation scheme described in Section III, while the virtual vector construction is employed to address the harmonic current elimination. The control set is formed with the same way as in [23], as well as the voltage vector sets. What should be mentioned is that the flux control in that publication is replaced by the current control for fair comparison, therefore the cost function (11) is adopted in experiments.
Besides, for the sake of convenient description, the proposed method is termed as ''method #3'' in the following text.

B. STEADY-STATE PERFORMANCE
At first, the three methods are compared in terms of the steady-state performance. The speed command is set to 500 revolutions per minute (rpm) and the load is adjusted to 4 N·m. The captured waveforms of stator currents i a and i u , the outputted torque T e and the speed n are presented in Fig. 7. The currents on d-, q-, x-and y-axes are shown in Fig. 8. It can be seen that all methods can obtain sinusoidal stator currents, with the peak value of about 5 A. The torque and the speed can track their respective references, as well as the currents in different subspaces. The purpose of harmonic currents elimination is also attained. It can be easily observed that method #3 can regulate the harmonic current within ±0.4 A, showing the effectiveness of using the optimum duty-cycle calculation scheme for xy-axes currents control.
In order to further compare the methods in steady-state performance, two operation conditions are also conducted: 800 rpm with 2 N·m and 300 rpm with 8 N·m. The evaluation criterions, including the THD of i a , the torque ripple, as well as the peak-to-peak value of xy-axes currents, are concluded in Tables 2, 3 and 4, where condition #1, condition #2 and condition #3 corresponds to 500 rpm with 4 N·m, 800 rpm with 2 N·m and 300 rpm with 8 N·m, respectively. From the tables, it is verified that method #3 is commensurate with method #2, in terms of the behavior of torque control (namely the components in α-β subspace); meanwhile, method #3 shows a considerable improvement for the harmonic current elimination. At the same time, it indicates that the performance in terms of the harmonic current elimination is nearly irrelevant to the operating condition.

C. DYNAMIC PERFORMANCE
Then, two experiments are carried out to compare the dynamic performances of three methods. Fig. 9 corresponds  to the load variation operation in a manner where the load alters from 4 to 2 N·m. Each waveform consists of the stator current i a , the outputted torque T e and the speed n. As seen, the torque can track the load command promptly, and the speed shows a little rise. The response processes of three methods take nearly the same time about 400 ms.  The deceleration test is conducted by changing the reference speed from 500 rpm to 300 rpm. The obtained results are presented in Fig. 10, from which, it can be found that a smooth speed response accompanied by an acceptable speed overshoot can be achieved for all methods. The durations of response processes are still the same, about 80 ms.
The two experiments verify that the proposed MPCC maintains the satisfactory dynamic behavior of the traditional MPCC.

D. ROBUSTNESS ANALYSIS
The robustness is especially worth-noted in MPC methods for motor drives. All the parameters R, L d , L q and L l vary with the operating conditions. Of them, the resistance R changes due to temperature variation, while many publications have mentioned that the influence of the resistance change on the performance of MPC methods is very limited.
Differently, the inductances L d , L q and L l are mainly associated with the degree of magnetic saturation. Mismatches of the inductances are dominant to be considered when analyzing robustness. Therefore, the performance of the proposed MPCC considering inductance mismatches is tested. Note that the increase of the current leads to a higher degree of magnetic saturation and hence a smaller actual inductance, in which case, the value used in the program would be higher than the actual value. So, three experiments are taken into account, i.e. L d = 150%L * d , L q = 150% L * q and L l = 150% L * l . The speed reference and the load are the same as condition #1.
The results are presented in Fig. 11. It can be observed that the mismatches of inductances can cause larger current ripples. The ripples of i d , i q and i x /i y are increased from 0.6 A to 1.2 A, from 0.6 A to 1.3 A and from 0.4 A to 0.6 A when L d L q and L l are increased by 50%, respectively. Besides, mismatches of dq-axes inductances result in slight steady-state errors of dq axes currents. Nevertheless, the system is still stable in three cases and the performance is also acceptable. There are also some works dedicated to robustness enhancement of MPC. In [32] and [33], parameter identification strategies or disturbance observers are integrated into the MPC methods. Such ways are feasible theoretically but lead to additional computations unavoidably. Another way is to remove the model parameters from the controller, called model-free predictive control (MFPC) [34], [35]. The MFPC method is actually data-driven but not model-based, which implies that this method is hard to work if the sampling  frequency is low or the sampling accuracy is deficient. Overall, how to ensure and enhance the robustness of MPC methods is still left.

E. COMPUTATION BURDEN AND EFFICIENCY
The computation burdens of three method are tested and compared. In the digital implementation, the execution time of a code can be directly measured by the number of the used clock cycles. By doing so, the execution time of three methods is obtained as in Fig. 12.
As presented, method #2 needs the least execution time about 42.67 µs, because the virtual vector construction in practice leads to an open-loop control for xy-axes currents. The implementation of method #1 needs about 48.58 µs, although only 13 switching states are considered. Method #3 ranks second, with the execution time about 44.83 µs. It can be concluded that compared to method #2, the proposed method can highly improve the control performance of xy-axes currents, at a cost of a negligible additional computation burden.
Finally, the drive system efficiency under three operation conditions is estimated, by using η = P m P dc × 100% with P m = nT e × π 30 (35) where η is the drive system efficiency; P m is the outputted mechanical power, which can be estimated using (35); P dc is the input power, which can be read from the control panel of the dc power supply. The drive system efficiency comparison is shown in Fig. 13. It can be observed that due to the lowspeed operations, the efficiency is lower than 80%. Nevertheless, the proposed method ranks first in all three conditions, owing to lower harmonic contents.

V. CONCLUSION
In this work, a novel MPCC with an optimum duty-cycle calculation scheme is proposed for ASPMSM drives. Due to the use of the optimum duty-cycle calculation scheme for controlling both components in α-β and x-y subspaces, the proposed method is characterized with the steady-state performance improvement and the weighting factor elimination. The merits of the proposed method are verified by experiments based on a 2 kW ASPMSM drive. Two existing methods are also conducted for comparison. The results show that the proposed method takes a considerable improvement in terms of steady-state performance, in particular for the harmonic current elimination. Meanwhile, the dynamic performance of the proposed method is also satisfactory. The robustness considering inductance mismatches is also tested. It shows that when L d , L q and L l are increased by 50%, the ripples of i d , i q and i x /i y would be increased, while the system is still stable in three cases and the performance is also acceptable. Besides, regarding the computation burden, the proposed method needs about 44.83 µs, which ranks second among the tested three methods. The negligible additional computation burden is the cost for the significantly improved steady-state performance. Finally, the drive system efficiency comparison indicates that the proposed MPCC method ranks first due to lower harmonic contents.