IM Fed by Three-Level Inverter under DTC Strategy Combined with Sliding Mode Theory

Abstract

The classical direct torque control (CDTC) of the induction motor (IM) drive is characterized by high ripples in the stator flux and the electromagnetic torque waveforms due to the use of hysteresis comparators. Furthermore, the motor speed in this control strategy is ensured through a proportional integral (PI) regulator, due to its simple structure. Nonetheless, this controller is sensitive to load disturbances. Hence, it is not robust against parameter variance, which can degrade the motor performance. To overcome this deficiency, many endeavors have been conducted in the literature to ensure a high dynamic response of the motor in all speed ranges, with minimum flux and torque undulations. Thus, the DTC of an IM associated with a three-level inverter based on sliding mode (SM) flux, torque and speed controllers was adopted to substitute the hysteresis comparators and the traditional PI regulator, since the SM speed controller is able to prevail against external disturbances. The second contribution of this manuscript is to develop the proposed DTC_SM approach using the Xilinx System Generator (XSG) in order to implement it on a field programmable gate array (FPGA) Virtex 5 on account of its ability to adopt parallel processing. The hardware co-simulation results verify clearly the merits of the suggested modified DTC strategy.

1. Introduction

Induction motors (IM) are nowadays the most used machines in the industrial area due to their reasonable cost, good performance, simple structure and plain control [1]. Therefore, numerous control strategies have been proposed to satisfy the industry’s growing demand. Within this trend, the direct torque control (DTC) approach, initially introduced by Takahashi and Depenbrock in the middle of the 1980s [2], has gained extensive interest. It has been touted as the first field-oriented control competitor since it is less sensitive to machine parameter variations and ensures the precise and fast dynamics of the torque. The basic concept of this strategy is to control, independently and directly, the electromagnetic torque and the stator flux through two hysteresis comparators. However, the latter systematize the motor into a variable switching frequency range, which definitely engenders fluctuations in flux and torque waveforms [3]. Among the several prevalent methods developed to enhance the CDTC strategy is the use of the space vector modulation (SVM) technique [4,5,6]. Nevertheless, the DTC-SVM algorithm is somewhat complex when compared to CDTC, and it is sensitive to the machine parameter variations and external disturbances [3]. A further improvement in the motor torque response is attained by using intelligent techniques such as fuzzy logic and neural networks. However, the switching frequency in this approach remains variable, which does not satisfy the requirement for torque and flux ripple reduction. To enhance the CDTC performance, another method is suggested, built on multilevel inverters instead of the standard two-level one. This scheme offers a greater number of active voltage vectors able to minimize the torque ripples [7]. Other attempts to improve the DTC strategy have been elaborated, notably combining the DTC with predictive control [8], where the optimized switching state is selected in a manner in which torque ripples are significantly minimized. However, the structure of the algorithm is mathematically complex because it needs the discretization of the power converter.

In most aforementioned cases, flux and torque ripples have been noticeably decreased but the robustness of the motor control is neglected. Thence, solutions adopting the sliding mode (SM) approach become a promising alternative. In fact, it ensures not only a fast dynamic response, stability and robustness against parameter variation, but also reduced flux and torque undulation, as well as a low current total harmonic distortion (THD) rate.

For this reason, unlike the previously published work, the main contribution and novelty of this paper is to combine several approaches that ensure the improvement of the classical DTC control to enable the IM to operate with minimum flux and torque ripples in all speed ranges. The proposed DTC-SM method in this paper is established by substituting the conventional two-level inverter feeding the motor with a three-level one. Indeed, the semiconductor switches in the three-level neutral point clamped (NPC) inverter are subjected to a lower voltage change rate (dv/dt), which causes the harmonic losses on the motor side to be significantly lower. Furthermore, the flux and torque hysteresis comparators in the suggested DTC strategy are replaced with sliding mode controllers. Consequently, and through this work, torque and flux oscillations are notably reduced. Likewise, the tuning of the PI speed controller is delicate because its parameters are strongly related to the IM parameters. It is also sensitive to any load torque disturbance, which can arguably degrade the machine performance. As an auspicious solution, this research work seeks to replace the classical PI speed controller with a sliding mode one in view of its robustness against external disturbances [9]. The second part of this manuscript is devoted to implementing the developed control strategy in a digital platform.

Ordinarily, two hardware design solutions are exploited when controlling the IM drive. The first one involves digital signal processors (DSPs) and microcontrollers (STM32, etc.) [10,11]. The second category is the field programmable gate array (FPGA). Nowadays, the first software family is almost excluded because of the sequential treatment of its processors, and then they became able to cope with the market needs. In fact, the processing speed of DSP and µcontrollers relies on the complexity of the implemented control system, which can heavily increase the computational development. Moreover, the DSP computing speed imposes a limit for the inverter switching frequency that must not be exceeded. Accordingly, the control algorithm’s performance is affected [12]. To remedy the above-mentioned limitations, the FPGA board is proposed to guarantee faster executions [13]. Indeed, this software target is chosen as a promising alternative to implement the DTC strategy with a shorter processing time due to its parallel execution, flexibility and large computing capability. To configure an FPGA, various digital design languages are deployed, namely Verilog and the Very-High-speed integrated circuits Hardware Description Language (VHDL). However, programming Verilog and VHDL requires extensive skills and a large amount of development time. To address this dilemma, the Xilinx System Generator (XSG) tool is used in this paper to generate automatically the bitstream file required for FPGA configuration, and so this toolbox enables designers to test and validate the effectiveness of the control algorithm without deteriorating the real system [14]. This method is called “hardware co-simulation” or “hardware in the loop” (HIL).

In this work, the DTC-SM of the IM drive fed by a three-level NPC inverter is developed. This control strategy is based on three sliding mode controllers dedicated to torque, flux and motor speed, instead of hysteresis and PI regulators. To validate the performance of the proposed method, hardware-in-the-loop simulations are carried out between MATLAB/Simulink and FPGA Virtex 5.

The previously published research works either deal with modeling the standard DTC-SM strategy (based on a two-level inverter) with XSG or with implementing the DTC-SM strategy on an FPGA using VHDL code, which require advanced programming skills. The literature has not yet taken into account the HIL simulation of DTC-SM based on a three-level inverter using the XSG toolbox.

The rest of this paper is organized as follows. Section 2 is devoted to developing the DTC strategy of the IM drive controlled with a three-level inverter. The sliding mode flux, torque and speed controllers are detailed in Section 3. A hardware co-simulation of the modified DTC approach implemented in the FPGA Virtex 5 board is presented in Section 4. Finally, the main conclusions are drawn in the Section 5.

Table 1. Review of some proposed methods to improve DTC strategy.

Ref.	Proposed Method	Motor Type	Converter Type	Current THD	Support	Programming Language
[10]	DTC based on twelve sectors and modified hysteresis controllers	Three-phase IM	Two-level inverter	19.92%	DSP TMS320F28069M	C
[15]	DTC based on a specific selection of the inverter switching states	Five-phase IM	Three-level five-phase inverter	11.59%	DSP TMS320F28377S	C
[16]	DTC based on intelligent technique	Three-phase IM	Two-level inverter	5.882%	FPGA Virtex 5	XSG
[17]	DTC with sliding mode controller and observer	Four-phase switched reluctance motor	Asymmetrical half-bridge	-	dsPACE 1401	C
[18]	Supertwisting sliding mode DTC	Three-phase IM	Two-level inverter	-	TMS320F28335 eZdsp platform	C++
This work	DTC based on sliding mode theory	Three-phase IM	Three-level NPC inverter	4.19%	FPGA Virtex 5	XSG

provides a summary of the main contributions of this study in comparison to other publications. The proposed improved DTC method, the type of motor, the type of inverter driving the motor, the current THD rate, the digital platform for hardware implementation and the programming language were the six criteria considered to conduct this brief review.

2. DTC of IM Associated with Three-Level NPC Inverter

2.1. IM Modeling

The first step in designing the DTC strategy is to build the mathematical model of the IM in the stationary reference frame (α, β), as introduced in Equation (1) [19]:

$$\{\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\mathrm{X}=\left[\mathrm{A}\right]\mathrm{X}+\left[\mathrm{B}\right]\mathrm{U}\\ \mathrm{Y}=\mathrm{C}\mathrm{X}\end{array}$$

(1)

where X is the state vector defined as $\mathrm{X}=\left[\begin{array}{l}{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }}\\ {\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}\\ {\mathsf{\phi }}_{\mathrm{s}\mathsf{\alpha }}\\ {\mathsf{\phi }}_{\mathrm{s}\mathsf{\beta }}\end{array}\right]$.

U and Y are the state control and output vectors determined as $\mathrm{U}=\left[\begin{array}{l}{\mathrm{V}}_{\mathrm{s}\mathsf{\alpha }}\\ {\mathrm{V}}_{\mathrm{s}\mathsf{\beta }}\end{array}\right]$, $\mathrm{Y}=\left[\begin{array}{l}{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }}\\ {\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}\end{array}\right]$.

[A], [B] and [C] are three matrices given in the following form:

$$\left[\mathrm{A}\right]=\left(\begin{array}{cccc}-(\frac{{\mathrm{R}}_{\mathrm{s}}}{\mathsf{\sigma }{\mathrm{R}}_{\mathrm{r}}}+\frac{{\mathrm{R}}_{\mathrm{r}}}{\mathsf{\sigma }{\mathrm{L}}_{\mathrm{r}}})& -\mathsf{\omega }& \frac{{\mathrm{R}}_{\mathrm{r}}}{\mathsf{\sigma }{\mathrm{L}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{r}}}& \frac{\mathsf{\omega }}{\mathsf{\sigma }{\mathrm{L}}_{\mathrm{s}}}\\ \mathsf{\omega }& -(\frac{{\mathrm{R}}_{\mathrm{s}}}{\mathsf{\sigma }{\mathrm{R}}_{\mathrm{r}}}+\frac{{\mathrm{R}}_{\mathrm{r}}}{\mathsf{\sigma }{\mathrm{L}}_{\mathrm{r}}})& \frac{\mathsf{\omega }}{\mathsf{\sigma }{\mathrm{L}}_{\mathrm{s}}}& \frac{{\mathrm{R}}_{\mathrm{r}}}{\mathsf{\sigma }{\mathrm{L}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{r}}}\\ -{\mathrm{R}}_{\mathrm{s}}& 0& 0& 0\\ 0& -{\mathrm{R}}_{\mathrm{s}}& 0& 0\end{array}\right);\left[\mathrm{B}\right]=\left[\begin{array}{cc}\frac{1}{\mathsf{\sigma }\hspace{0.17em}{\mathrm{L}}_{\mathrm{s}}}& 0\\ 0& \frac{1}{\mathsf{\sigma }\hspace{0.17em}{\mathrm{L}}_{\mathrm{s}}}\\ 1& 0\\ 0& 1\end{array}\right];\left[\mathrm{C}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\\ 0& 0\end{array}\right]$$

R_s, R_r and ω are the stator resistance, the rotor resistance and the motor speed. Likewise, L_s and L_r are the stator and rotor inductances. $\mathsf{\sigma }=1-\frac{{\mathrm{M}}_{\mathrm{s}\mathrm{r}}{}^2}{{\mathrm{L}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{r}}}$ is the Blondel coefficient, where ${\mathrm{M}}_{\mathrm{s}\mathrm{r}}$ represents the mutual inductance.

The equation describing the mechanical behavior of the IM is established hereafter [20]:

$$\mathrm{J}\hspace{0.17em}\frac{\mathrm{d}}{\mathrm{d}\hspace{0.17em}\mathrm{t}}\mathsf{\Omega }\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\mathrm{T}{\hspace{0.17em}}_{\mathrm{e}\mathrm{m}}-\mathrm{f}\hspace{0.17em}\mathsf{\Omega }\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}{\mathrm{T}}_{\mathrm{l}}$$

(2)

where $\mathrm{T}{\hspace{0.17em}}_{\mathrm{e}\mathrm{m}}$, ${\mathrm{T}}_{\mathrm{l}}$, J and f are, respectively, the electromagnetic torque, the load torque, the inertia moment and the viscous friction coefficient.

The motor’s parameters are presented in Appendix A.

2.2. DTC Principle

The fundamental concept of the DTC approach, as originally introduced in the 1980s, is based on the direct and independent control of both stator flux and electromagnetic torque through selecting a suitable voltage vector that should be applied to the inverter switches [21]. The basic scheme of this control strategy is illustrated in Figure 1.

Referring to Figure 1, the DTC strategy is mainly composed of three blocks.

Torque and flux estimator: The torque can be estimated based on Equation (3):

$${\mathrm{T}}_{\mathrm{e}\mathrm{m}}=\frac{3}{2}{\mathrm{n}}_{\mathrm{p}}\hspace{0.17em}({\mathsf{\phi }}_{\mathrm{s}\mathsf{\alpha }}\hspace{0.17em}{\mathrm{i}}_{\mathrm{s}\mathsf{\beta }}-{\mathsf{\phi }}_{\mathrm{s}\mathsf{\beta }}\hspace{0.17em}{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }})$$

(3)

where n_p is the pair pole number.

Likewise, the stator flux components are estimated as given in Equation (4):

$$\{\begin{array}{l}{\mathsf{\phi }}_{\mathrm{s}\mathsf{\alpha }}={\displaystyle \int ({\mathrm{V}}_{\mathrm{s}\mathsf{\alpha }}-{\mathrm{R}}_{\mathrm{s}}\hspace{0.17em}{\mathrm{i}}_{\mathrm{s}\mathsf{\alpha }})\hspace{0.17em}}\mathrm{d}\mathrm{t}\\ {\mathsf{\phi }}_{\mathrm{s}\mathsf{\beta }}={\displaystyle \int ({\mathrm{V}}_{\mathrm{s}\mathsf{\beta }}-{\mathrm{R}}_{\mathrm{s}}\hspace{0.17em}{\mathrm{i}}_{\mathrm{s}\mathsf{\beta }})\hspace{0.17em}}\mathrm{d}\mathrm{t}\end{array}$$

(4)

Therefore, the module of the stator flux as well as its corresponding angle are evaluated as

$$\{\begin{array}{l}\left|{\mathsf{\phi }}_{\mathrm{s}}\right|=\sqrt{{\mathsf{\phi }}_{\mathrm{s}\mathsf{\alpha }}²+{\mathsf{\phi }}_{\mathrm{s}\mathsf{\beta }}²}\\ {\theta }_{\mathrm{s}}=\arg \hspace{0.17em}({\mathsf{\phi }}_{\mathrm{s}})\end{array}$$

(5)

Torque and flux hysteresis comparators: Once the torque and stator flux are estimated, they are compared to their reference values. Indeed, the obtained errors present the inputs of two hysteresis controllers. These latter produce, in their outputs, two logical rules ${\mathrm{K}}_{\mathsf{\phi }}$ and ${\mathrm{K}}_{\mathrm{T}\mathrm{e}\mathrm{m}}$.

Selecting table: The output voltage vector of the three-level inverter, given in Equation (6), is chosen based on a switching table. This table needs as inputs ${\mathrm{K}}_{\mathsf{\phi }}$, ${\mathrm{K}}_{\mathrm{T}\mathrm{e}\mathrm{m}}$ and the sector where the stator flux is located [22].

$${\mathrm{V}}_{\mathrm{s}}=\sqrt{\frac{2}{3}}\hspace{0.17em}{\mathrm{V}}_{\mathrm{d}\mathrm{c}}({\mathrm{S}}_{\mathrm{a}}+{\mathrm{S}}_{\mathrm{b}}.{\mathrm{e}}^{\mathrm{j}\frac{2\mathsf{\pi }}{3}}+{\mathrm{S}}_{\mathrm{c}}.{\mathrm{e}}^{\mathrm{j}-\frac{2\mathsf{\pi }}{3}})$$

(6)

S_a, S_b and S_c are the inverter switching states.

The paper [23] provides a thorough explanation of the DTC control approach for additional details.

2.3. Three-Level NPC Inverter

Continuous developments in the field of power electronics have prompted researchers to use three-level inverter topologies in controlling IM drives. The increase in level number can resolve efficiently torque and flux ripples, as well as current distortion, in the DTC strategy when compared to conventional ones [24,25].

Figure 2 illustrates the general scheme of the three-level NPC inverter. The number of switching states in this case is $3^3=27$. Among them are three zero vectors and 24 active vectors.

3. Proposed DTC Based on Sliding Mode Approach

In the DTC of the three-level inverter-fed IM drive, ripples are notably reduced when compared to the classical control approach. However, the voltage vector selection in this strategy is mainly based on the hysteresis controllers, which are responsible for torque and stator flux undulations. Furthermore, the speed control loop is settled by a PI regulator, known by its sensitivity to parameter variations. To surmount the above-mentioned deficiencies, the stator flux, torque and motor speed are controlled by sliding mode functions [26]. Indeed, the strength characteristics of the SM regulators are a fast response, robustness against parameter uncertainties and simple hardware implementation [27,28,29,30]. The design of SMC hinges on two steps. The first one is the determination of the switching surface and the second one relies on designing the control law in such a manner as to maintain the system trajectory toward the sliding surface [31].

The SM structure, shown in Figure 3, is the most commonly used one for electrical machine drives owing to the relay function, which is considered as a suitable control for power electronic converters.

The control law expression is provided by

$$\mathrm{U}={\mathrm{U}}_{\mathrm{e}\mathrm{q}}+{\mathrm{U}}_{\mathrm{n}}$$

(7)

where U_eq is the equivalent command, which is a continuous function used to maintain the controlled variable on the sliding surface. It is obtained when the invariance conditions of the sliding surface are satisfied.

$$\{\begin{array}{l}\mathrm{S}=0\\ \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\mathrm{S}=0\end{array}$$

(8)

U_n is the switching term based on the sign function of the sliding surface.

3.1. Speed Integral Sliding Mode Controller

The speed regulation loop is designed to generate the electromagnetic toque reference and to provide a fast response and good dynamics. The speed tracking error considered in some research works is defined as

$${\mathrm{e}}_{\mathsf{\Omega }}={\mathsf{\Omega }}_{\mathrm{r}\mathrm{e}\mathrm{f}}-\mathsf{\Omega }$$

(9)

where Ω_ref denotes the motor speed reference.

To provide a faster dynamic response with good accuracy, an integral sliding surface is suggested to establish the control law, as formulated in Equation (10):

$${\mathrm{S}}_{\mathsf{\Omega }}={\mathrm{e}}_{\mathsf{\Omega }}+{\mathsf{\lambda }}_{\mathsf{\Omega }}{\displaystyle \int {\mathrm{e}}_{\mathsf{\Omega }}\hspace{0.17em}\mathrm{d}\mathrm{t}}$$

(10)

${\mathsf{\lambda }}_{\mathsf{\Omega }}$ is a positive constant.

Its derivative is therefore

$${\dot{\mathrm{S}}}_{\mathsf{\Omega }}={\dot{\mathsf{\Omega }}}_{\mathrm{r}\mathrm{e}\mathrm{f}}-\dot{\mathsf{\Omega }}+\mathsf{\lambda }\left({\mathsf{\Omega }}_{\mathrm{r}\mathrm{e}\mathrm{f}}-\mathsf{\Omega }\right)$$

(11)

When substituting Equation (2) into (10), and assuming that there is convergence to the sliding surface (${\dot{\mathrm{S}}}_{\mathsf{\Omega }}=0$), we obtain the equation below:

$$\frac{1}{\mathrm{J}}{\mathrm{Tem}}_{\mathrm{eq}}={\dot{\mathsf{\Omega }}}_{\mathrm{r}\mathrm{e}\mathrm{f}}+\frac{1}{\mathrm{J}}\mathrm{T}{\hspace{0.17em}}_{\mathrm{l}}+\frac{\mathrm{f}}{\mathrm{J}}\mathsf{\Omega }+{\mathsf{\lambda }}_{\mathsf{\Omega }}\left({\mathsf{\Omega }}_{\mathrm{r}\mathrm{e}\mathrm{f}}-\mathsf{\Omega }\right)$$

(12)

Based on Equation (7), the expression of the reference torque is written as

$${\mathrm{Tem}}_{\mathrm{ref}}=\underset{{\mathrm{Tem}}_{\mathrm{eq}}}{\underbrace{\mathrm{J}\left({\dot{\mathsf{\Omega }}}_{\mathrm{r}\mathrm{e}\mathrm{f}}+\frac{1}{\mathrm{J}}{\mathrm{T}}_{\mathrm{l}}+\frac{\mathrm{f}}{\mathrm{J}}\mathsf{\Omega }+{\mathsf{\lambda }}_{\mathsf{\Omega }}\left({\mathsf{\Omega }}_{\mathrm{r}\mathrm{e}\mathrm{f}}-\mathsf{\Omega }\right)\right)}}+\underset{{\mathrm{Tem}}_{\mathrm{n}}}{\underbrace{{\mathrm{K}}_{\mathsf{\Omega }}\left|{\mathrm{S}}_{\mathsf{\Omega }}\right|\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left({\mathrm{S}}_{\mathsf{\Omega }}\right)}}$$

(13)

${\mathrm{K}}_{\mathsf{\Omega }}$ is a positive gain, 0 < α < 1, and sign(.) denotes the sign function.

The diagram corresponding to the implementation of this speed regulation loop based on the variable structure control is given in Figure 4.

Stability proof: Once the design of the sliding control laws is settled, it is necessary to verify the system’s stability, which can be assessed in terms of the Lyapunov function as follows:

$$\mathrm{V}=\frac{1}{2}{\mathrm{S}}^2$$

(14)

If the derivative of this function is defined as negative, then the system trajectory is driven to the sliding mode surface and remains on it,

$$\dot{\mathrm{V}}=\mathrm{S}\hspace{0.17em}\dot{\mathrm{S}}<0$$

(15)

Then, replacing expression (12) in (13), we obtain the following equation:

$$\dot{\mathrm{S}}=-\frac{{\mathrm{k}}_{\mathsf{\Omega }}}{\mathrm{J}}\left|{\mathrm{S}}_{\mathsf{\Omega }}\right|\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\hspace{0.17em}({\mathrm{S}}_{\mathsf{\Omega }})<0$$

(16)

Subsequently, the stability condition is approved.

3.2. Sliding Mode PI Controller of Flux and Torque

Using Equations (3) and (4), and assuming that the direction of the stator flux vector is aligned with the d component, which means that ${\mathsf{\phi }}_{\mathrm{s}\mathrm{q}}=0$, we can write the following equation:

$$\{\begin{array}{l}{\mathrm{V}}_{\mathrm{s}}={\mathrm{V}}_{\mathrm{s}\mathrm{d}}+\mathrm{j}{\mathrm{V}}_{\mathrm{s}\mathrm{q}}\hspace{0.17em}={\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{s}}+\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{\mathsf{\phi }}_{\mathrm{s}}+\mathrm{j}{\mathsf{\omega }}_{\mathsf{\phi }\mathrm{s}}{\mathsf{\phi }}_{\mathrm{s}}\hspace{0.17em}=\underset{{\mathrm{V}}_{\mathrm{s}\mathrm{d}}}{\underbrace{{\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{s}\mathrm{d}}+\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{\mathsf{\phi }}_{\mathrm{s}}}}+\underset{{\mathrm{V}}_{\mathrm{s}\mathrm{q}=0}}{\underbrace{{\mathrm{R}}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{s}\mathrm{q}}+{\mathsf{\omega }}_{\mathsf{\phi }\mathrm{s}}{\mathsf{\phi }}_{\mathrm{s}}}}\\ {\mathrm{T}}_{\mathrm{e}\mathrm{m}}=\frac{3}{2}\mathrm{p}\hspace{0.17em}\hspace{0.17em}{\mathsf{\phi }}_{\mathrm{s}}{\mathrm{i}}_{\mathrm{s}\mathrm{q}}=\frac{3}{2}\mathrm{p}\hspace{0.17em}\hspace{0.17em}{\mathsf{\phi }}_{\mathrm{s}}\frac{\left({\mathrm{V}}_{\mathrm{s}\mathrm{q}}-{\mathsf{\omega }}_{\mathsf{\phi }\mathrm{s}}{\mathsf{\phi }}_{\mathrm{s}}\right)}{{\mathrm{R}}_{\mathrm{s}}}\end{array}$$

(17)

where ${\mathsf{\omega }}_{\mathsf{\phi }\mathrm{s}}$ is the angular speed of the stator flux.

According to the aforementioned equations, flux and torque control can be accomplished using the direct and quadrature stator voltage components, respectively (V_sd and V_sq). The two selected sliding surfaces are ${\mathrm{S}}_{\mathsf{\phi }}$ and ${\mathrm{S}}_{\mathrm{T}\mathrm{e}\mathrm{m}}$. While ${\mathrm{S}}_{\mathrm{T}\mathrm{e}\mathrm{m}}$ represents the sliding surface of the electromagnetic torque, ${\mathrm{S}}_{\mathsf{\phi }}$ is defined for the stator flux error to control the direct component of voltage V_s.

Equation (18) provides the corresponding sliding surfaces, given below:

$$\{\begin{array}{l}{\mathrm{S}}_{\mathrm{T}\mathrm{e}\mathrm{m}}={\mathrm{e}}_{\mathrm{T}\mathrm{e}\mathrm{m}}+{\mathrm{k}}_{\mathrm{T}\mathrm{e}\mathrm{m}}{\displaystyle \int {\mathrm{e}}_{\mathrm{T}\mathrm{e}\mathrm{m}}\hspace{0.17em}\mathrm{d}\mathrm{t}}\\ {\mathrm{S}}_{\mathsf{\phi }}={\mathrm{e}}_{\mathsf{\phi }}+{\mathrm{k}}_{\mathsf{\phi }}{\displaystyle \int {\mathrm{e}}_{\mathsf{\phi }}\hspace{0.17em}\mathrm{d}\mathrm{t}}\end{array}$$

(18)

${\mathrm{k}}_{\mathrm{T}\mathrm{e}\mathrm{m}}$ and ${\mathrm{k}}_{\mathsf{\phi }}$ are two positive constants. ${\mathrm{e}}_{\mathrm{T}\mathrm{e}\mathrm{m}}$ and ${\mathrm{e}}_{\mathsf{\phi }}$ are, respectively, the torque and flux errors, defined as

$$\{\begin{array}{l}{\mathrm{e}}_{\mathrm{Tem}}={\mathrm{Tem}}_{\mathrm{r}\mathrm{e}\mathrm{f}}-\mathrm{T}\mathrm{e}\mathrm{m}\\ {\mathrm{e}}_{\mathsf{\phi }}={\mathsf{\phi }}_{\mathrm{r}\mathrm{e}\mathrm{f}}-\mathsf{\phi }\end{array}$$

(19)

Hence, we proceed to write the following equation:

$$\{\begin{array}{l}{\mathrm{V}}_{\mathrm{s}\mathrm{d}}=\left({\mathrm{K}}_{\mathrm{p}\mathsf{\phi }\mathrm{s}}+\frac{{\mathrm{K}}_{\mathrm{I}\hspace{0.17em}\mathsf{\phi }\mathrm{s}}}{\mathrm{S}}\right)\mathrm{s}\mathrm{a}\mathrm{t}\left({\mathrm{S}}_1\right)\\ {\mathrm{V}}_{\mathrm{s}\mathrm{q}}=\left({\mathrm{K}}_{\mathrm{p}\mathrm{T}\mathrm{e}\mathrm{m}}+\frac{{\mathrm{K}}_{\mathrm{I}\hspace{0.17em}\mathrm{T}\mathrm{e}\mathrm{m}}}{\mathrm{S}}\right)\mathrm{s}\mathrm{a}\mathrm{t}\left({\mathrm{S}}_2\right)+{\mathsf{\omega }}_{\mathsf{\phi }\mathrm{s}}\end{array}$$

(20)

where ${\mathrm{K}}_{\mathrm{p}\mathsf{\phi }\mathrm{s}}$, ${\mathrm{K}}_{\mathrm{p}\mathrm{T}\mathrm{e}\mathrm{m}}$, ${\mathrm{K}}_{\mathrm{I}\hspace{0.17em}\mathsf{\phi }\mathrm{s}}$ and ${\mathrm{K}}_{\mathrm{I}\hspace{0.17em}\mathrm{T}\mathrm{e}\mathrm{m}}$ are the PI regulator gains that must be properly chosen to satisfy the condition of stability using the Lyapunov criterion.

The proposed flux and torque variable structure controller (VSC) is depicted in Figure 5.

The major problem of the SM controllers is the chattering phenomenon, due to the discontinuous nature of the control law caused by an infinite commutation around the sliding surface [26]. To solve this problem, the sign function is replaced by a smoother one (Sat function), as illustrated in Figure 6.

3.3. Simulation Results

To substantiate the effectiveness of the proposed DTC-SM control strategy and evaluate the designed control system’s dynamic response, we have modeled it in Matlab/Simulink software. The applied reference speed profile of the given system is depicted in Figure 7.

To test the robustness of the suggested control approach, sudden electromagnetic load torques are applied in each speed range, as given in Figure 8.

In the startup, the machine is not loaded until t = 0.4 s. Then, from t = 0.4 s to t = 1 s, we apply a load torque of 10 N·m, which represents the motor’s nominal electromagnetic torque. From t = 1 s to t = 1.5 s, the IM operates again without any load. Finally, a load torque of 5 N·m is suddenly applied at a low speed range.

The following figures illustrate the comparative analysis of the DTC of the three-level inverter-fed IM (based on hysteresis flux and torque controllers and PI speed regulator) and DTC_SM (based on SM torque, flux and speed controllers).

Figure 9a,b illustrate, respectively, the stator flux trajectory in the (α, β) frame. It can be noted that when using both strategies, a fully circular trajectory was guaranteed. This demonstrates that the flux vector amplitude is kept constant throughout all operation stages. However, compared to the DTC technique, the suggested DTC-SM delivers a finer and smoother trajectory, which confirms the reliability of this control approach. Figure 9c,d represent magnified views of the stator current waveforms of the IM controlled with the DTC and DTC-SM strategies, respectively. It is clear that in the case of the DTC-SM, the harmonics content was less important, with THDi = 4.19%, in comparison to the case of DTC, which was characterized by THDi = 26.68%. The torque responses and the stator current modules of DTC (red line) and DTC-SM (blue line) with the same load scenario are illustrated in Figure 9e,f. It can be clearly perceived that the proposed strategy has reduced torque and current ripple levels when compared to the DTC control method. In fact, the torque ripple range of the DTC technique is approximately ∆_Tem = 0.4 N·m, while that of the proposed DTC-SM is neglected, which confirms the effectiveness of the suggested control technique in terms of torque and current ripple reduction.

The figure exhibiting the motor speed with the DTC of the three-level inverter-fed IM drive and DTC based on sliding mode theory, with a change in load level during operation, is illustrated in Figure 10.

The speed responses of the machine with both developed control techniques converge towards their reference values. However, during the load torque’s sudden application (Figure 8), we recorded speed drops, which are given in

Table 2. Speed drop of the IM under both control strategies in different operating ranges.

	At t = 0.4 s	At t = 1 s	At t = 1.5 s
Speed drop with DTC strategy under load variation	29.7 rad/s	13.8 rad/s	4.8 rad/s
Speed drop with DTC-SM strategy under load variation	0.4 rad/s	0.4 rad/s	0.07 rad/s

.

In the case of the DTC strategy, the motor speed under external load introduction of 10 N·m at t = 0.4 s displays a drop of approximately 29.7 rad/s, while the speed curve in DTC-SM is not significantly affected by the load torque disturbance (very slight drop of 0.4 rad/s). Likewise, when a load torque of 5 N·m is applied at a low speed range at t = 1.5 s, the speed drop with DTC is around 4.8 rad/s, while in DTC-SM, the speed curve closely follows the reference value.

It is obvious from Figure 10 and Table 2 that, in various operating ranges, when the load torque scenario is applied, as shown in Figure 8, the speed curve generated by the IM under the DTC-SM approach thoroughly tracks the reference curve with a disregarded disturbance, before regaining immediately its reference value. Furthermore, it can also be observed that there exists faster dynamics and better speed reference tracking with the suggested control technique. Therefore, the proposed DTC-SM method has better speed robustness against load perturbation and a better dynamic response when compared to the DTC method.

4. HIL Simulation of DTC-SM Strategy

The first step to implement the developed control strategy on the FPGA board is to design it under the Matlab/Simulink environment using the XSG toolbox. Thus, this section is devoted to describing the software configuration of the suggested approach based on XSG blocks. Nonetheless, many operations are not directly available, mainly the signal generators and trigonometric functions. To address this constraint, we can mathematically approximate the desired expression by exploiting the toolbox’s basic elements [14].

The co-simulation method is used to validate the control strategy performance and optimize the time required for experimental test execution. The fundamental step of the HIL consists of generating a JTAG block that substitutes the previously designed control architecture. This block is automatically created between the “gateway in” and “gateway out” resources. Concerning the data type, we used the fixed point format instead of the floating point, because it requires a longer execution time, especially for complex mathematical functions. The diagram in Figure 11 explains the approach to hardware co-simulation.

4.1. XSG Design of DTC Based on Sliding Mode Approach of IM Supplied by Three-Level Inverter

Exploiting the basic elements of the XSG toolbox, the DTC of the IM built on the sliding mode approach is as depicted in Figure 12. By comparing the design functionality to the reference model behavior displayed in the Simulink interface, this library enables us to validate the design functionality via digital simulation. The communication between the proposed architecture under XSG and the association of the IM/inverter is conferred through the “gateway in” and “gateway out” blocks.

The architecture of the coordinate transformation $(\mathrm{d},\mathrm{q})$ to $(\mathsf{\alpha },\mathsf{\beta })$ frame is illustrated in Figure 13. This block is based on trigonometric functions that are not directly available in the XSG toolbox library. Therefore, we have resorted to an ROM unit. The accuracy of the generated value depends on the internal parameters, such as the size of the ROM, the step size and the gain coefficient in the input module.

Likewise, to settle the angle of the reference stator vector as well as its module, the arc-tangent function existing in the CORDIC block of the XSG library produces a computation error, which leads to inaccurate results. To address this deficiency, we chose to use ROM modules. The flux angle and module designed using the XSG toolbox are outlined in Figure 14.

4.2. Software in-the-Loop Simulation Results

To show the performance of the DTC-SM, a simulation study is carried out using the XSG toolbox, giving the following results depicted in Figure 15, which presents the motor speed, the electromagnetic torque and the stator flux magnitude.

The same scenario of load torque variation and reference speed evolution, described in Figure 8 and Figure 9, is applied. As can be observed, the speed profiles in Figure 15a follow the reference value quite closely, with only a few minor deviations at the moment of load torque application. Figure 15b–d show the evolution of the electromagnetic torque, the stator flux module and the current magnitude under the DTC-SM strategy for both the Simulink and XSG models. It can be claimed that the obtained results using XSG tools are very similar to those obtained by numerical simulation in terms of speed tracking, electromagnetic torque response and stator flux and current aspects, which confirms the reliability and effectiveness of the XSG modeling in designing architectures on FPGA.

The error rates between quantities modeled in Simulink and those modeled with XSG tools are presented in

Table 3. Error rate collection between quantities modeled in Simulink and those modeled with XSG.

	Speed	Torque	Flux	Current
Errors between Simulink and XSG	0.066%	1%	0.022%	0.69%

. These promising results allow us to proceed further with the hardware implementation of the suggested control technique.

The next step is to proceed to the experimental validation of the proposed control strategy. Hence, the test bench conceived in our research laboratory, based on a three-phase IM fed with a three-level NPC inverter and controlled through an FPGA board in closed loop mode, is presented in Figure 16.

5. Conclusions

This study proposed a DTC strategy based on sliding mode theory for a high-performance IM controlled with a three-level NPC inverter. The suggested control paradigm is designed by combining the SM algorithm dedicated to flux, torque and speed loops with the merits of the multi-level inverter. Hence, the control structure solves not only the problem of high flux and torque ripples caused by the discrete nature of hysteresis regulators, but also improves the robustness and stability of the system. Additionally, in various speed operating ranges, the suggested DTC-SM strategy’s robustness against load torque perturbation has been extensively demonstrated.

Once the conceived control system is verified through simulations, the XSG modeling enables the implementation of the whole design by loading the generated bitstream file on the FPGA board and, thus, a hardware co-simulation is launched.

Overall, the simulation results of this study offer a very appealing and prospective control technique for high-performance AC drives. For future studies, the proposed methodology will be tested on an experimental setup. Moreover, the robust control strategy can be further improved by other variants, such as using a sliding mode observer to ensure sensorless control, or exploiting the DTC-SM in electric vehicle traction applications, as well as in solar- and PV-powered water pumping systems.