Performance Prediction of Direct Torque-Controlled PMSM Drives Considering Different Pulse Selectors Using FEA-Based Model

Despite the robustness and fast dynamic response, direct torque control (DTC) exhibits relatively large ripples in torque and flux during steady-state operation. Mitigating these ripples depends on designing a pulse selector to identify the optimal voltage vector at every control cycle, which has been reported with different structures in the literature. Hence, it is becoming increasingly crucial to comprehensively understand the criteria used in pulse selection and analyze their impact on the overall DTC performance. These are challenging in permanent magnet synchronous motor (PMSM) drives due to the nonlinear torque/current characteristics and spatial harmonics of the magnetic flux. This article exploits the high fidelity and computational efficiency of the finite-element analysis (FEA)-based model to present a performance prediction strategy for DTC of PMSM under different pulse selectors. First, this model is employed to thoroughly analyze the influence of each switching pulse on the torque and flux deviations under changing operating conditions considering the machine's nonlinearities. Then, the pulse selection schemes used in switching-table-based DTC are comparatively evaluated, highlighting the advantages and limitations of each. Based on this evaluation, an advanced pulse selector with a variable-structure switching table (VSST) is developed, reducing torque/flux ripples and simplifying the control implementation while maintaining the fast dynamic response of DTC.

The basic principle of the pulse selector is to use a look-up table to set the switching states of the inverter depending on rough estimation of the stator-flux angular position and indication of the demanded deviation direction in torque and flux using two hysteresis regulators. The selected switching pulse is applied for the entire control cycle, ensuring almost the theoretical maximum dynamic performance. However, this leads to pronounced torque and flux ripples during steady-state operation, especially in the case of two-level inverters [6], [7]. Torque tracking with minimum steady-state ripples can avoid the acoustic noise and vibration that cause wear and tear of the mechanical components (such as the shaft, bearings, and loads) [8]. To minimize the torque and flux ripples, many researchers have attempted to improve the criteria used in pulse selection.
In [9], [10], [11], [12], [13], the look-up table structure is modified in terms of either the included switching positions (i.e., the admitted voltage vectors), the number of levels for the torque hysteresis regulator, or the sector definition needed to indicate the stator-flux angular position. Furthermore, dutyratio regulators have been employed in [14], [15], [16], [17], [18] to apply two or more voltage vectors at every control period during steady-state operation. Although these modifications can effectively reduce the torque ripple, they introduce other performance issues, such as increased flux ripple and current harmonics, and complicated control implementation (by introducing additional control parameters).
Other researchers attempted to ensure selecting the optimal switching pulse that reduces both the torque and flux ripple using the model predictive control (MPC) solution [19], [20]. MPC uses the machine model to predict the torque and flux trajectories under all admissible voltage vectors, and the optimal options are selected based on a cost-function evaluation. However, the need for the predictions and the weighting factor to tune the importance of each control objective increases the computational complexity and degrades the DTC robustness [21]. The need for a weighting factor can be eliminated using sequential and parallel optimization strategies, as presented in [22], [23], [24]. The cost function of each control objective are separately considered. From the point of view of each cost function, the admissible switching pulses are ranked in groups of a predefined number of candidates, from which the optimal option is selected. Moreover, the computation complexity of MPC can be reduced by artificially preselecting the candidate switching states before the prediction phase, as described in [25], [26], [27]. Despite the ability of these methods to optimize the switching pulses selection, they fails to identify the pulse patterns that minimize the flux ripple and current harmonics [28].
Alternatively, a space vector modulator (SVM) has been utilized in [29] to synthesize the exact voltage vector that accurately satisfies the required deviations in torque and flux. This voltage vector is identified from the demanded stator flux vector, the angular position of which is derived from the desired torque deviation via a proportional-integral (PI) controller. To avoid degradation of the DTC dynamic performance, a deadbeat (DB) controller replaces the PI regulator in [30], where the demanded stator-flux vector is determined using the machine model. In [31], the computation complexity of the DB controller has been reduced by directly deriving the reference voltage vector from the torque error. Instead, a sliding mode controller (SMC) is employed in [32], which can handle the unknown disturbances and modeling inaccuracies. Nevertheless, compared to the switching look-up table DTC, the use of a SVM complicates the control implementation and results in a higher switching frequency which is undesired in medium-and high-voltage drives [33].
With the wide variety of criteria used for pulse selection in DTC and the corresponding objectives, it is becoming increasingly crucial to comprehensively analyze their impact on control performance. A machine model with several simplifying assumptions has been employed in the aforementioned literature (such as [15], [16], [18], [26], [27]) to investigate the effect of each switching pulse on the torque and flux modulus, then to predict the control performance under the presented pulse selector. This model fails to accurately represent the dynamic behavior of PMSM because of the nonlinear torquecurrent characteristics and rotor position-dependent variation in the magnetic energy that results in flux harmonics and cogging effects [34]. These inaccuracies can seriously compromise the prediction quality of the PMSM-drive control performance. Therefore, accurate model-based analysis for the pulse selection criteria, can be useful during the control design to thoroughly evaluate the control behavior. Among the competing high-fidelity modeling techniques of PMSM [34], [35], [36], [37], [38], the finite-element analysis (FEA)-based model in synchronous rotating coordinates (dq), can provide the highest computational efficiency while considering various non-ideal characteristics, such as magnetic saturation, iron losses, and spatial harmonics. Therefore, this article proposes a detailed performance analysis for the pulse selection criteria in DTC of PMSM drives using the FEA-based model. It addresses different structures of the pulse selectors, including their basic principles, control objectives, and factors that cause control performance issues. The main contribution of the current work can be summarized as follows: r The FEA-based PMSM model is used to theoretically investigate the effects of each switching pulse admitted by the converter on the torque and flux responses under different operating conditions considering the magnetic saturation of the machine. From the perspective of this investigation, different pulse selectors are comparatively evaluated by highlighting the main features and limitations of each.
r To overcome these limitations, a variable-structureswitching table (VSST)-based strategy is proposed for pulse selection in DTC. This method not only ensures reduced torque and flux ripples but also simplifies the control implementation and reduces the average switching frequency while maintaining the fast dynamic response.
r By synthesizing the voltage vectors selected by the DTC schemes via the converter switching function, the FEAbased model is exploited to obtain a fast and high-fidelity prediction for the performance of the PMSM drive system. Simulations in Matlab/Simulink via the FEA-based PMSM model is compared with those obtained by the simplified model to verify the feasibility of the proposed performance prediction strategy. Then, experimental evaluations for different DTC pulse selectors are performed on a laboratory prototype of a PMSM drive system to verify the effectiveness of the proposed pulse selection technique with the VSST.
The rest of this article is organized as follows. Section II describes the basic principle of different pulse selectors used in DTC scheme. In Section III, the FEA-based PMSM model is used to analyze these pulse selectors, and the performance prediction strategy is demonstrated. Then, an advanced VSSTbased pulse selection method is proposed in Section IV. To validate the outcomes of the performance analysis, simulation results are discussed in Section V. After that, experimental verification of the proposed VSST-based method is presented in Section VI. Finally, conclusions are drawn in Section VII.

II. PULSE SELECTION IN DTC OF PMSM A. PRINCIPLE OF DTC
The dynamic behavior of the DTC's controlled variables (i.e., torque and flux) depends on the voltage vector applied to the machine terminals, which can be represented by different mathematical models for PMSM [37]. If the flux-linkagebased model in the stationary coordinates (αβ) is employed, the stator flux vector (ψ αβ = ψ α + jψ β ) and the electromagnetic torque (T e ) can be expressed as where v αβ and i αβ are the stator voltage and current vectors, respectively, R s represents the stator phase resistance, and p denotes the number of pole pairs. From (1), when DTC applies an arbitrary voltage vector v * αβ to the stator winding for a sampling period T s , it deviates the stator flux vector by ψ αβ = v * αβ T s (assuming that R s is negligible). This deviation adjusts the amplitude (ψ s = ψ αβ ) and angular position (θ s = arctan(ψ β /ψ α )) of the stator flux vector. By adjusting θ s , the (load) angle between the stator and rotor flux vectors can be directly manipulated due to the fast dynamics of the stator flux compared to that of the machine rotor [3].
To illustrate the effect of the direct manipulation of the load angle on the torque, (2) needs to be simplified by neglecting the magnetic saturation effects (i.e., the stator inductance L s is constant). As a result, the stator current vector can be written where ψ pm is the permanent magnet flux, and θ r is the rotor angular position. Substituting (3) into (2), the torque equation can be rewritten as This implies that increasing or decreasing the load angle (i.e., (θ s − θ r )) can increase ↑ or decrease ↓ the torque if the stator flux amplitude ψ s is maintained constant. Consequently, the direct control of torque and stator flux amplitude is accomplished by controlling the stator flux vector along its tangential and radial directions, as illustrated by Fig. 1. Thus, the proper selection of the voltage vector applied to the stator winding is the key to providing optimal control performance.

B. CRITERIA USED IN PULSE SELECTION
Aiming at robustness against parameter variation and fast dynamic response, the DTC employs a pulse selector to set a basic voltage vector among those admitted by the power converter, as shown in Fig. 2 where x ∈ {0, 1.., 7} is the voltage-vector index. Two of these voltage vectors (x ∈ {0, 7}) are zero voltage vectors (ZVVs), and the others are active voltage vectors (AVVs), which are defined in Fig. 3(a). By considering the influence of each vector on the deviation direction of the stator flux vector, the most feasible option can be properly selected. Because this effect relies on the angular position of the stator flux vector θ s , the αβ-plane is typically divided into six sectors (n ∈ {1, 2.., 6}) with two different boundary definitions n 1 and n 2 , as shown in Fig. 3(b) and (c). Therefore, a sector detector is employed to identify that position using one of the following expressions: Sector n 1 : (2n 1 − 3)π 6 < θ s ≤ (2n 1 − 1)π 6 (6) Sector n 2 : (2n 2 − 2) π 6 < θ s ≤ 2n 2 π 6 (7) For both sector definitions, Fig. 3(b) and (c) give an illustration of the influence of the voltage vectors on the torque T e and stator flux amplitude ψ s when the stator flux vector is located in Sector n = 1. The ZVV's (•) effects on torque and flux are considered negligible. The AVV indexed x = 2 ( ) and x = 5 ( ) respectively increases and decreases both T e and ψ s . With the sector definition n 1 , the vector x = 3 ( ) increases T e and decreases ψ s , and vice versa for x = 6 ( ). These two vectors are replaced in the sector definition n 2 with those indexed x = 4 ( ) and x = 1 (•) to satisfy a more pronounced impact on ψ s . As the phase shift between the voltage vectors matches the spatial range of the sectors (i.e., π/3), these impacts of the AVVs indexed as 2, 5, 3, 6, 4, and 1 will be respectively similar to those of 3, 6, 4, 1, 5, and 2 when the stator flux vector is located in Sector n = 2. Thus, the voltage vector index x can be expressed using the sector identifiers to generalize the description of their influence on the controlled variables as follows.
then T e ↓ ψ s ↑ n 1 + 4 or n 2 + 4, then T e ↓ ψ s ↓ As the DTC chooses the switching pulses of the voltage vector to force the torque and flux magnitude to track their reference signals T re f e and ψ re f s , two-or three-level hysteresis regulators are used to indicate the demanded deviation direction (ε ψ s and ε T e ), such that ↑, →, and ↓ represent the three possible directions. ε ψ s and ε T e are obtained based on the error between the estimated and reference values and the hysteresis bands (±h ψ s and ±h T e ), as depicted in Fig. 4. According to the output of the hysteresis regulators and the indicated sector n, the index of the most feasible voltage vector can be chosen from a look-up table among those shown in Table 1.
Four different look-up tables are presented in Table 1 [3], [9], [10], [11]. The basic switching table (BST) and the modified BST (MBST) use a three-level hysteresis torque regulator and employ the ZVVs when the torque needs to be maintained constant (ε T e =→). Unlike the other switching tables, MBST uses (7) instead of (6) to detect the sector n. Thus, it uses the AVVs indexed [n+3] and [n] rather than [n+2] and [n+5] to respectively increase and decrease the torque. Table 1(c) and (d) exclude one torque state (ε T e =→) by using two-level hysteresis regulator. The former employs the AVVs only, while the other utilizes the ZVVs when both the torque and the flux modulus has to be decreased (i.e., ε T e =↓ and ε ψ s =↓). For convenience, these switching tables are respectively specified in this work as AST and ZST.
Accordingly, the criteria used for pulse selection in DTC can change the control performance by considering a number of factors that can be summarized as follows: 1) The sector definition used to indicate the location of the stator voltage vector in the αβ plane.
2) The number of levels of the hysteresis regulators and the width of hysteresis bands.
3) The structure of the switching look-up table and the included voltage vectors. For setting these criteria, only the influence of the switching pulses on the stator flux vector has been considered in the conventional DTC by assuming that it depends only on the direction of the corresponding voltage vectors. However, by thoroughly analyzing these influences under different operating conditions (such as the rotor speed and the load torque), the criteria used for pulse selection in DTC can be better set, improving the overall control performance. To achieve that, a performance prediction strategy for the DTC is presented in the following section.

III. PERFORMANCE PREDICTION OF DTC
The impact of the switching pulses on the torque and flux responses can be principally predicted based on the PMSM model, which has been represented mathematically by simplified or FEA-based models. In this section, predictions for the torque and flux responses of PMSM using both models are presented first. Then, the pulse selectors presented in Section II are analyzed under different operating conditions to predict the DTC performance, highlighting the advantages and disadvantages of each.

A. SIMPLIFIED PREDICTION OF TORQUE AND FLUX RESPONSES
To simplify the prediction of the torque and flux responses, various assumptions have been considered for the PMSM model (3) and (4), [14], [15], [16], [17], [18], [27]. For instance, the rotor saliency is ignored, the hysteresis and saturation properties of the magnetic core are neglected, and the permanent magnet flux (ψ pm ) is assumed to be sinusoidally distributed in the air gap and aligned with the d-axis direction. As a result, the magnetic saturation and flux harmonics caused by the magnetic energy variation with the rotor position are ignored. The d-and q-axis components of the stator inductance (L d and L q ) have been dealt as equivalent and constant values (i.e., L d = L q = L s ).
Consequently, the variations of torque and stator flux magnitude over the sampling period T s due to applying the voltage vector v αβ [x] can be derived from (1)-(3) as [15]: where ψ r = ψ pm e jθ r . Usage of (9) and (10) to predict the impact of the switching pulses on the torque and flux amplitude responses provides acceptable accuracy for the surfacemounted PMSM (SPMSM) owing to its non-salient rotor and the relatively large air gap between the stator and rotor cores. However, with the above-mentioned assumptions, these expressions can seriously compromise the prediction quality of the control performance for the interior PMSM (IPMSM) because of the rotor saliency that not only introduces nonlinear torque-current characteristics but also causes a rotor positiondependent variation of the magnetic energy, leading to flux harmonics and cogging effects [39]. To overcome this issue, a fast and accurate PMSM model is needed for the performance prediction of DTC. Although the FEA can satisfy the required accuracy, it is computationally time-consuming. Reducing the calculation time significantly is feasible by pre-calculating and storing the FEA results for offline processing. Thus, the FEA-based PMSM model [34] can be a promising solution to predict the DTC performance.

B. PROPOSED PERFORMANCE PREDICTION STRATEGY USING FEA-BASED MODEL
The FEA-based model can capture the nonlinear characteristics of PMSM by extracting the d-and q-components of the stator flux (ψ d and ψ q ) and the torque from FEA at different values of d-and q-axis currents (i d and i q ) and rotor angular position θ r . To ensure the computational efficiency of this modeling scheme, the extracted data can be represented by either curve fitting or lookup tables as In this model, the rotating frame (dq) representation of the machine variables (i.e., stator flux, current, voltage) reduces the amount of data need to be extracted from FEA [36]. As these variables can be obtained from their αβ presentation using Park's transform (f dq = f αβ e −jθ r ), the stator flux equation (1) is rewritten as: where ω r = dθ r /dt denotes the rotor angular speed, and v dq is the applied voltage vector that can be obtained from (5) as v αβ e −jθ r . According to (13), the impact of the voltage vector on the stator flux can be predicted if the dq components of the stator-current vector i dq are obtained as functions of the corresponding flux linkages and the rotor position as A data set for these functions can be obtained by inverting (11) using, for example, MATLAB's gridfit function [40]. Based on (12)- (14), the schematic representation of the FEA-based PMSM model can be illustrated by Fig. 5. The implementation and verification steps for the FEA-based model of the 80-kW PMSM prototype under study, with the parameters given in the Appendix, are well described in [41] and not repeated here for the sake of space.
is applied at the kth control sample, the d-and q-axis flux linkages can be predicted from the discrete form of (13) as where ψ k d and ψ k q are determined from (11) according to the loading condition represented by the currents i k d and i k q . Based on (15), the d-and q-components of the current vector at the sampling instant k + 1 can be predicted using (14), and the deviations in torque and stator-flux modulus can be obtained for each admissible voltage vector x as where θ k+1 r = θ k r + ω k r T s . In this way, the impact of the voltage vector on the torque and flux-amplitude responses can be analyzed using the FEA data instead of the simplified model, resulting in more accurate predictions for the DTC control performance.
To illustrate, T e produced by v αβ [x = 1] throughout the whole spatial range of θ s under different operating conditions is calculated using the simplified and FEA-based models of the 80-kW PMSM prototype, as shown in Fig. 6. For these computations, the sampling interval T s is set to 50 μs, and only the magnetic saturation is considered (i.e., θ r is maintained zero in (11)-(14)). From Fig. 6, it can be seen that the simplified model exhibits lower absolute torque deviations than the FEA-based model over a wide range of θ s under all operating conditions [see the dotted areas in Fig. 6]. Consequently, if the former is employed to assess the torque control performance of DTC strategies, it will show lower torque ripple and slower dynamic response, as will be confirmed in Section V. Thus, the FEA-based model is employed to evaluate the DTC performance under different pulse selectors as follows.

C. COMPARATIVE ANALYSIS OF PULSE SELECTION CRITERIA USING FEA-BASED MODEL
The criteria used in pulse selection of DTC are theoretically analyzed using the FEA-based model considering only the magnetic saturation. The torque and flux variations generated by each voltage vector included in the switching tables [see Table 1] are calculated using (16) and (17) at a control sample of T s = 50 μs. These deviations are determined at different operating conditions over the entire spatial range of both sector definitions (6) and (7), as shown in Figs. 7 and 8.
As can be noticed from Figs. 7 and 8, the voltage vectors employed with the sector definition (7) produce lower torque deviations and higher flux variations than those used with (6), regardless the operating conditions. Consequently, employing MBST for DTC of PMSM can cause lower torque ripple, slower dynamic response, and higher flux ripple compared to the other switching tables (BST, AST, and ZST).
By comparing Fig. 7(b) with Fig. 7(a), it can be noted that, with the speed increase in the anticlockwise direction (+ve speed), the positive torque deviations produced by either the AVVs  strengthen. Thus, the torque decay will be faster than the torque rise for all pulse selectors. Also, both voltage vectors [n+3] and [n] fails to respectively increase and decrease the torque at high-speed (±3000 r/min) and loading conditions, which can result in failure of MBST to maintain stable operation, as will be confirmed in Sections V and VI.
From Fig. 7(b) and (c), the ZVV produces torque deviations that are always in a direction opposite to that of the rotation. These deviations are lower than those developed by [n+4] and [n+5] for positive speed and [n+1] and [n+2] for negative speed. Consequently, ZST can provide reduced torque ripple compared to AST, as the former utilizes the ZVVs at one of the table states. However, applying a ZVV at that state results in a slower torque response when a torque decay is demanded. Also, ZST cannot allow speed reversal as the torque deviation produced by ZVV becomes positive when the rotation direction changes, i.e., the ZVV fails to satisfy the condition ε T e =↓ in ZST. Therefore, a voltage-vector selection strategy based on a variable-structure switching table is presented in the following section.

IV. PROPOSED PULSE SELECTION STRATEGY WITH A VARIABLE-STRUCTURE SWITCHING TABLE (VSST)
As discussed in Section III-C, the torque and flux deviations depend not only on the angular position of the stator-flux vector but also the operating conditions, such as the direction of rotation. Hence, selecting the switching pulse from a look-up table with a changing structure based on these conditions can be a desired feature from the point of view of steady-state and dynamic performances of DTC.
The proposed strategy detects the sector n using (6) to avoid the performance issues associated with MBST at high speeds. When the motor speed is positive, applying a ZVV to decrease the torque (ε T =↓) instead of the AVVs [n+4] or [n+5] can improve the steady-state performance in terms of the torque and flux ripples because of two reasons. The ZVVs produce lower negative torque deviations [see Fig. 7

(a)]
and their impact on the flux is negligible [see Fig. 8]. The steady-state-performance improvement can be maintained in the backward direction of rotation (i.e., −ve speed) by employing ZVVs to increase the torque (when ε T =↑) instead of the AVVs [n+1] or [n+2]. The ZVV is decided (0 or 7) to minimize the number of switching jumps, such that v αβ [0] follows the voltage vectors indexed as 0, 1, 3, and 5, while the remaining vectors are followed by v αβ [7]. Because the ZVVs degrade the dynamic performance if a decay in the absolute torque is demanded, the AVVs are the best option at all the table states (as in the case of AST). Hence, the proposed strategy uses the variable-structure switching table (VSST) shown in Table 2, where the voltage vector index needs to be artificially selected considering the operating state and the rotation direction.
To achieve that, a simple strategy can be utilized for pulse selection, which is illustrated by the flowchart shown in Fig 9. The dynamic state of the system can be activated by detecting the demanded-torque transition ( T re f e ). To guarantee the speed reversal, the activation of the dynamic state has to be maintained using a " f lag" till the satisfaction of two conditions: (1) The estimated torque crosses the reference signal (i.e., sgn(T k err ) = sgn(T k−1 err )); (2) The direction of rotation is similar to that of the reference torque (T re f e · ω k r ≥ 0). According to the VSST [ , respectively, when θ s is in the range of the Subsector I ∈ [(2n − 3)π/6, (2n − 3)π/6 + π/12). Conversely, the AVVs [n+1] and [n+4] give higher absolute flux deviations when θ s is in the range of the Subsector II ∈ ((2n − 1)π/6 − π/12, (2n − 1)π/6]. As a result, the switching between the two possible AVVs during the transition from one sector to another leads to a considerable change in the stator flux amplitude, degrading the flux control performance. To avoid this issue, the vector index selected from VSST needs to be replaced according to the conditions shown in Fig. 9. The pulse replacement not only alleviates the occasional compromise of the flux regulation but also avoids the undesired torquedeviation directions produced by the replaced voltage vectors at high-speed conditions.
With this simple strategy, the switching-table structure is made variable depending on the operating state and direction of rotation, achieving improved steady-state torque performance while retaining the classical methods' fast dynamic response and robustness. Moreover, the control implementation is simplified by using the sign of the torque and flux error (T err = T re f e − T e and ψ err = ψ re f s − ψ s ) instead of the hysteresis regulators and the corresponding tuning of their band limits. As will be discussed in the following sections, the simulations and experimental results confirm the effectiveness of the proposed strategy.

A. SYSTEM DESCRIPTION AND SIMULATION PARAMETERS
The PMSM drive system shown in Fig. 10 is used to predict the DTC performance under different pulse selectors to verify the theoretical analysis discussed in Section III-C, and validate the effectiveness of the proposed pulse selection strategy presented in Section IV. The PMSM is driven in torque control mode where the operating speed (ω r ) is controlled via a virtual dynamometer, from which the rotor angular position θ r can be indicated. The machine states (i.e., flux and torque) are given to the pulse selector of DTC to decide the stator voltage that forces them to follow the predefined reference signals. By using (5), the stator voltage can be fed to the FEA-based model [ Fig. 5] to predict the torque and flux control performances in terms of steady-state ripples and average switching frequency ( f av ). The torque and flux ripples can be represented using the standard deviation (σ T e and σ ψ s ), as And f av can be determined by averaging the total number of the inverter legs' switching instants over a fixed period of time (10 ms in this work). These performance indices are also compared with those resulted using the simplified PMSM model to validate the analysis presented in Section III-B. The sampling frequency is maintained at 20 kHz for consistency with the theoretical analysis. The hysteresis band limits of the torque and flux regulators needed for the pulse selectors (BST, MBST, AST, and ZST) are arbitrarily set to 2% of the rated torque and permanent magnet flux.

B. COMPARISON OF STEADY-STATE PERFORMANCE
Figs. 11 and 12 present the steady-state waveforms of torque, stator-flux amplitude, and average switching frequency obtained using the FEA-based and simplified models at different speeds and loading conditions.
As can be noticed in these figures, the performance prediction for all DTC strategies using the simplified model showed a considerably lower torque ripple at lower average switching frequencies (in most cases) than that employing the FEA-based model. In other words, the simplified model gives optimistic predictions of the steady-state performance compared to the FEA-based model due to neglecting the magnetic saturation, as discussed in Section III-B. These confirm the  necessity of employing the latter for performance prediction of DTC.
According to the FEA-based-model predictions of DTC performance shown in Figs. 11 and 12, the proposed VSSTbased strategy outperforms the other methods regarding the torque and flux ripples, especially at high-speed (3000 r/min) or -loading conditions. Its average switching frequency is slightly higher than BST and ZST only at high-speed operation. The AST-based method produces the largest torque ripple with the highest average switching frequency because it uses the AVVs only, proving the importance of the ZVV inclusion in the look-up table. MBST can produce a lower torque ripple than BST and AST at the cost of a significant rise in the flux ripple, as shown in Figs. 11(a) and 12(a). Also, as can be noticed in Figs. 11(b) and 12(b), MBST causes unstable operation at 3000 r/min speed because the voltage vector indexed [n + 3] fails to increase the torque over a wide range of θ s at high speeds [see Fig. 7(b)], as discussed in Section III-C A detailed quantitative evaluations for σ T e , σ ψ s , and f av under various operating speeds are given in Fig. 13 for no load and 75% of full load. The simplified-model-based performance prediction showed that the proposed VSST produces a slightly higher torque ripple at no load than BST by average of 2% with almost the same average switching frequency. Also, at 0.75 of full load, it is detected that BST gives a lower torque ripple than MBST by around 0.5%. On the other hand, the FEA-model-based performance predictions indicated that the torque ripple produced by VSST is around 7% less than that of the BST with 25% lower average switching frequency at no load. At high load (0.75 of full load), MBST introduces lower torque ripple than BST by roughly 17% on average. For all DTC schemes under test, the simplified model predicts lower torque ripple and average switching frequency than the FEA-based model by averages of 22.5% and 17%, respectively. These results verify the analysis presented in Section III-B and the need for employing the FEA model for DTC performance prediction.
From the FEA-model-based predictions presented in Fig. 13(a), the proposed VSST gives lower torque ripple at no load than AST and ZST by averages of 48%, and 34%, respectively. Although the proposed method produces higher torque ripple than MBST by around 12%, the flux ripple and the average switching frequency are dramatically reduced by averages of 35% and 30%, respectively. From Fig. 13(b), the torque ripple, flux ripple, and average switching frequency of the proposed strategy are the lowest at 75% of full load, which are respectively 30%, 18%, and 29%, on average, less than those of the other DTC schemes. These results verify the efficacy of the proposed VSST-based pulse selection strategy to enhance the steady-state performance of DTC. Fig. 14 shows the FEA-model-based predictions of the torque dynamic performance when the demanded torque is stepped up from 0 to 300 N · m and down from 300 to 0 N · m under different rotor speeds, 100 and 3000 r/min.

C. COMPARISON OF THE DYNAMIC PERFORMANCE
Only MBST is tested at 2000 r/min instead of 3000 r/min to avoid the unstable operation mentioned in Section III-B.
It can be noticed in Fig. 14 that when the torque is increased, all the pulse selectors allow comparable torque responses for DTC, excluding MBST which causes slower torque dynamics. When the torque is stepped down, the dynamic response of ZST-DTC degrades significantly because of employing ZVV to decrease the torque, as discussed in Section III-C. This issue is avoided in the proposed strategy by applying AVVs during transient conditions.
By comparing Fig. 14(b) with Fig. 14(a), it can be noticed that the rising time increases at high speed (3000 r/min), while the falling time decays because the positive and negative torque deviations produced by AVVs weaken and strengthen, respectively, when the speed rises. Furthermore, it can be noticed in Fig. 14(b) that the falling time is shorter than the rising time as the negative deviations in torque is larger than the positive deviations when ω r is positive [see Fig. 7(b)]. These simulation results are consistent with the theoretical analysis presented in Section III-C and confirm that the proposed VSST-based pulse selection strategy can retain the fast dynamic response of DTC.

VI. EXPERIMENTAL VERIFICATION
In this section, the experimental results are presented to verify the outcomes of the comparative analysis presented in Section III-C and the effectiveness of the proposed method [Section IV] under different operating conditions. As the pulse selectors are the main focus of this article, a 0.75-kW SPMSM (with the parameters are listed in the Appendix) can be a feasible option for the experimental verification to avoid the implementation complexity and the stability issues associated with flux estimation of IPMSM [42], [43].
As shown in Fig. 15, the motor is fed from 220-Volt DC supply and 200-μF DC-link capacitors through a 2L-VSI. Two LEM sensors detects the stator-winding currents and the DC-link voltage, while an incremental encoder with a precision of 2500 pulse per revolution determines the rotor angular position. The motor shaft is mechanically coupled to a powder brake supplied by a 24-Volt variable DC source to manipulate the load torque.
Two software are used to realize the experimental implementation of the DTC strategies. Before the experiment, Matlab/Simulink is utilized to establish a model for the control algorithm under test and generate the corresponding control code. Then, dSPACE ControlDesk is utilized to download the code to the controller processing unit and directly change the reference signals and control parameters through the host PC. It is also employed to capture the experimental results for post processing, analysis, and drawing using Matlab. The sampling period is maintained at 50 μs. For the sake of accuracy, the one-step delay due to the digital implementation is remedied for all the DTC strategies such that the switching pulse is selected based on the predicted torque and flux at the (k + 1)th sampling instant [44].

A. STEADY-STATE PERFORMANCE TESTS
Figs. 16 and 17 represent the experimental waveforms of electromagnetic torque, stator-flux amplitude, phase current, and the selected-vector index for the DTC schemes at 750 r/min and 2250 r/min, respectively. As the MBST exhibits unstable operation at 2250 r/min, it is tested at 1500 r/min instead. The load torque is maintained at around 1.8 N · m. To maintain the speed at a constant value while capturing the experimental results using the ControlDesk, the demanded torque is generated by an outer PI-speed loop with gains of K P = 0.006843 and K I = 0.78046. For better efficiency, the maximum torque per ampere principle is used to calculate the reference stator-flux amplitude from the demanded torque [14].
From Figs. 16 and 17, it can be observed that the proposed VSST-based method provides a significant reduction in torque ripple and current distortions compared to the other pulse selectors as it avoids selecting the AVVs that give high deviations in torque during steady-state operation, as discussed in Section IV. Although the application of MBST for DTC causes less torque ripple than BST and AST, it produces higher flux ripple. The ZST improves both the torque and flux ripples compared with BST and AST because of the more frequently employed ZVV, as can be noticed from the vector-index waveform. Fig. 18 presents a quantitative evaluation for the steadystate performance of DTC under different pulse selectors in terms of torque ripple, flux ripple, average switching frequency, and total harmonic distortion of the stator current (THD I ). The torque ripple of the proposed VSST is significantly less than that of BST, MBST, AST, and ZST by averages of 46%, 44%, 48%, and 41%, respectively. Furthermore, the proposed method achieves the lowest flux ripple, current distortions, and average switching frequency that are respectively reduced by 16%, 19%, and 37%, on average, compared with the other methods. On the other hand, MBST can mitigate the torque ripple at low speed (750 r/min) by about 6% with 16% lower average switching frequency compared to BST, but the stator flux ripple and the current distortion increase by 24% and 4%, respectively. These results match well with the performance analysis and the simulation results presented in Sections III-C and V-B, respectively.
To further verify the superiority of the proposed method, it is compared with the BST-based pulse selector with different  values of hysteresis band limits of the torque regulator (±h T e ), as shown in Fig. 19. In this test, the load torque is set to 1.8 N · m, and the speed is maintained around 750 r/min. From the vector index x in Fig. 19, the frequency of employing the ZVV (0 or 7) increases when the h T e is increased from 2% to 10% of the rated torque, resulting in reduced torque ripple and current distortion with lower average switching frequency. As the proposed strategy applies ZVV at the look-up table state (ε T e =↓) instead of AVV, it can hit the maximum possible frequency of employing the ZVVs. Consequently, the proposed VSST scheme can always provide a lower torque ripple with reduced average switching frequency than BST regardless of the value of the torque-hysteresis band limits (±h T e ).

B. DYNAMIC PERFORMANCE TESTS
The torque dynamic responses of DTC under different pulse selectors are also investigated and compared, as shown in Fig. 20. For these experiments, the motor is operated in torque control mode with a fixed load toque at around 1.6 N · m. The demanded torque is stepped up from 0 to 2 N · m to test the torque dynamic performance during start up. Then, once the velocity hits the value of 1500 r/min, the demanded torque is stepped down from 2 to −2 N · m to investigate the torque behavior during speed reversal. The corresponding speed response and phase current of the PMSM for that test scenario are illustrated to the bottom of the torque response. The zoomed zones 1 and 2 in Fig. 20 show the rise and fall time of the torque responses.
In Fig. 20, it can be noticed that BST, AST, ZST, and the proposed VSST achieve fast torque response with a rise time of 0.2 ms, approximately. On the other hand, the torque dynamic response is slightly slower for MBST (almost 0.27 ms) as it uses the sector definition n 2 (7) where the AVVs give lower deviations in torque than that with n 1 (6) [see Fig. 7]. When the demanded torque was changed from 2 to −2 N · m, the falling time of the torque  for all the employed pulse selectors was around 0.27 ms except ZST, the torque response of which needed about 0.55 ms to match the demanded value. Moreover, during speed reversal, the ZST causes system instability when the operating speed reaches zero, as shown in Fig. 20(d), because the torque variation direction of the ZVVs reverses [see Fig. 16(c)]. Instead, the proposed VSST provides faster response during torque-demand decay (because it employs only the AVVs during the dynamic state), and it can achieve smooth speed reversal, as can be noticed in Fig. 20(e). After speed reversal, it can satisfy a steady-state performance identical to that during the forward operation by changing the switching-table structure. These results confirm well with the performance analysis and the simulation results presented in Sections III-C and V-C, respectively, and show that the proposed VSST-based pulse selection strategy can keep the fast dynamic performance of DTC with smooth speed reversal and reduced torque and flux ripples.

C. DISCUSSION
The criteria of pulse selection without considering the application of ZVVs in DTC, such as the AST-based method, results in excessive torque and flux ripples with high average switching frequency. Therefore, by employing three-level torque hysteresis regulators in both BST and MBST, the torque ripple and average switching frequency can be reduced by increasing how frequently the ZVVs are selected. However, the latter results in increased flux ripple, instability at high-speed operation, and degraded dynamic torque response because it uses a sector definition n 2 (7) where the switching pulses give high deviations in flux amplitude, low torque deviations, and they fail to increase the torque at high-speed conditions. Although applying a ZVV at a single table state in ZST can also reduce the torque ripple, the speed reversal is not allowed because the torque deviations produced by ZVV reverse with the rotational direction. The proposed VSST strategy overcomes all these performance issues due to its flexibility of pulse selection with the operating state and direction of rotation. It achieves the lowest torque and flux ripples with reduced average switching frequency while retaining the fast dynamic response of DTC. Table 3 summarizes the comparative evaluation of the DTC performance under the different pulse selectors discussed in this work. The superiority of a particular pulse selector for a particular performance index is ranked such that illustrates the highest superiority.

VII. CONCLUSION
In this article, a generalized performance analysis using a high-fidelity FEA-based model is presented to investigate the impact of the pulse selection criteria on the DTC performance of three-phase PMSM drives, including nonlinear machine behavior such as magnetic saturation. Different pulse selectors are theoretically analyzed to identify the performance issues for both steady-state and dynamic operating conditions. By considering the operating state in the pulse selection criteria, the torque and flux ripples can be minimized while maintaining robustness and fast torque dynamics. Therefore, a VSST-based pulse selector is proposed, where the voltage vectors that produce relatively low deviations in torque and flux are employed during steady-state operation, while those of them that give high variation rates are applied during system dynamics. A detailed comparative evaluation for the DTC performance under different pulse selectors is presented using both simulations and experiments, which conform well with the theoretical analysis and verify the effectiveness of the proposed strategy.
The FEA-based model can be utilized for performance investigation of DTC considering other nonlinear characteristics, such as thermal or cogging effects or other control scenarios like different fault conditions. These can help identify new methodologies for pulse selection in DTC, as will be further demonstrated in future studies.

APPENDIX
The electrical parameters of the machines used for simulation and experimental verification in this work are presented in Table 4.