*Article* **Power Quality Analysis of a Hybrid Microgrid-Based SVM Inverter-Fed Induction Motor Drive with Modulation Index Diversification**

**Subramanian Vasantharaj <sup>1</sup> , Vairavasundaram Indragandhi 1,\* , Mohan Bharathidasan <sup>1</sup> and Belqasem Aljafari <sup>2</sup>**


**\*** Correspondence: indragandhi.v@vit.ac.in

**Abstract:** The effects of varying modulation indices on the current and voltage harmonics of an induction motor (IM) powered by a three-phase space vector pulse-width modulation (SVM) inverter are presented in this research. The effects were examined using simulation and an experimental setup. IMs can be governed by an SVM inverter drive or a phase-angle control drive for applications that require varying speeds. The analysis of THD content in this study used the modulation index (MI), whose modification affects the harmonic content, and voltage-oriented control (VOC) with SVM in three-phase pulse-width modulation (PWM) inverters with fixed switching frequencies. The control technique relies on two cascaded feedback loops, one controlling the grid current and the other regulating the dc-link voltage to maintain the required level of dc-bus voltage. The control strategy was developed to transform between stationary (α–β) and synchronously rotating (d–q) coordinate systems. To test the viability of the suggested control technique, a 1-hp/3-phase/415-V experimental prototype system built on the DSPACE DS1104 platform was created, and the outcomes were evaluated with sinusoidal pulse-width modulation (SPWM).

**Keywords:** induction motor drive; dc-link voltage balancing; space vector modulation; switching loss; total harmonic distortion

## **1. Introduction**

Electric motors are used in many driving components worldwide, accounting for 40% to 50% of all electricity demand [1]. Seventy percent of the electricity needed to run industrial loads is provided by three-phase IMs [2]. Due to their appealing qualities, such as low cost, straightforward design, excellent reliability, and ease of maintenance, electric motor sales have climbed to 85% [3–5]. Although direct current (DC) motors are frequently seen in applications involving variable speed operations because of how simple it is to control torque and field flux with armature and field current, these motors possess the drawback of having a commutator and brushes that could cause corrosion and necessitate regular maintenance [6]. Due to their high output power, toughness, robustness, efficiency, affordability, capacity to tolerate hazardous or severe working situations, and ruggedness, IMs do not experience DC motor difficulties [7]. The abrupt variation in load, which uses a significant amount of electricity and raises the cost of energy, impacts the functioning of three-phase IMs [8]. To regulate speed and preserve high efficiency, a variable speed drive (VSD) may be employed [9]. Moreover, the controller and switching technique employed in VSD significantly impact the achievability of high efficiency and reliability for IMs [10,11]. Pulse-width modulation (PWM) techniques are often used to control the switches of voltage source inverters (VSIs), as well as the frequency and output voltage of IMs [12]. One of the better approaches for VSI is the SVM switching approach, which has reduced switching losses and the capacity to reduce the harmonic output signals generated by inverters [13].

**Citation:** Vasantharaj, S.; Indragandhi, V.; Bharathidasan, M.; Aljafari, B. Power Quality Analysis of a Hybrid Microgrid-Based SVM Inverter-Fed Induction Motor Drive with Modulation Index Diversification. *Energies* **2022**, *15*, 7916. https://doi.org/10.3390/en 15217916

Academic Editor: Adolfo Dannier

Received: 25 September 2022 Accepted: 21 October 2022 Published: 25 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Recent studies have proposed several control techniques for PWM rectifiers to enhance the input power factor and transform the input current into a sinusoidal waveform. Numerous PWM modulation techniques are in use, including sinusoidal PWM and the space vector method. Technically, the SVM methodology is the optimal modulation method overall. Compared to the conventional PWM approach, the harmonic distortion of current decreases with the SVM. The duty cycles of VSI switches have been determined via the SVM, using the space vector concept. The only things to have been digitally implemented are PWM modulators. The possibility of straightforward digital implementation and a large linear modulation range for line-to-line voltages are the distinguishing features of SVM. Despite the SVPWM technique's benefits over PWM, new techniques are continually being developed. The main objective [14] has been to find efficient methods to provide output voltages with low harmonic rates and less switch-level loss.

The voltage-oriented control (VOC) algorithm, which is well known among indirect power control algorithms that use current controllers and is equivalent to the field-oriented control (FOC) of the IM, is able to generate high dynamic and static performance by using internal current control loops and an outer voltage control loop. The basis of VOC is the orientation of the rotating synchronous reference frame with respect to the grid line voltage vector. The suggested VOC approach exhibits highly dynamic behavior, suitable output voltage, and a low input current THD [15,16].

The proper operation of the control system has generally been hampered by the hardware execution of the IM drive controller. Digital signal processors (DSPs) and field programmable gate arrays (FPGAs) are two integrated circuits that have been used extensively in control platforms [17,18]. Implementing a user-friendly control unit for online supervision and monitoring is now possible due to the presence of a graphical objectoriented package (Control Desk software) on a dSPACE system. The automation and automotive sectors both use the dSPACE system, which is extremely popular among controlling platforms. The development of PV standalone inverters uses the dSPACE system as a control platform as an additional application area. In this study, the SVM control method is demonstrated through a simulation to support actual inverter hardware. The Simpowersystems and dSPACE DS1104 blockset libraries were used in the simulation, performed in the Simulink/MATLAB environment.

The remaining sections of this article are organized as follows: A description of an IM drive is given in Section 2. Section 3 explains the VOC control for a three-phase VSI. A VSI using the SVM approach is introduced in Section 4. Section 5 contains the simulation and experimental discussion. Section 6 summarizes the research and presents the findings.

#### **2. Overview of the Induction Motor Drive**

The squirrel-cage three-phase IM is an asynchronous AC motor that operates on electromagnetic induction principles. The IM's primary purpose is to convert electrical energy into mechanical energy. A small air gap separates the stator and the rotor, the two components that make up the IM. The three-phase squirrel-cage IM [19–21] is the one most often used due to its insulated winding in both stator and rotor, which are formed of cast and solid bars with high conductivity, as shown in Figure 1.

For the rotor to rotate at synchronous speed (*ωsm* in rad/s), where *ωsm* = 2π*f* (rad/s), where *f* = synchronous frequency, three-phase voltages must first be applied to the stator winding. The stator currents produce a revolving magnetic field in the magnetic circuit, formed by the air gap between the stator and rotor cores. The number of poles (*P*) affects the mechanical synchronous speed (*ωsm* in rad/s). This is how the synchronous speed is determined [22].

$$
\omega\_{sm} = \frac{2\omega\_s}{P} \tag{1}
$$

$$N\_{sm} = \frac{120f}{P} \tag{2}$$

**Figure 1.** Three-phase IM cross-sectional model. **Figure 1.** Three-phase IM cross-sectional model.

#### For the rotor to rotate at synchronous speed (*ωsm* in rad/s), where *ωsm* = 2π*f* (rad/s), *Modelling of the Induction Motor*

where *f* = synchronous frequency, three-phase voltages must first be applied to the stator winding. The stator currents produce a revolving magnetic field in the magnetic circuit, formed by the air gap between the stator and rotor cores. The number of poles (*P*) affects the mechanical synchronous speed (*ωsm* in rad/s). This is how the synchronous speed is To depict the physical systems, the computer models the mathematical model of the three-phase IM and its driving system [4]. The control parameters of the models are highlighted in this three-phase IM modelling. For the three-phase IM, the magnetic connection between the stator and rotor voltage equations can be expressed as follows [23].

$$V\_{\rm as} = i\_{\rm as}r\_{\rm s} + \frac{d\lambda\_{\rm as}}{dt};\ V\_{\rm bs} = i\_{\rm bs}r\_{\rm s} + \frac{d\lambda\_{\rm bs}}{dt};\ V\_{\rm cs} = i\_{\rm cs}r\_{\rm s} + \frac{d\lambda\_{\rm cs}}{dt} \tag{3}$$

$$\cdots \qquad \cdots \qquad \cdots \qquad \cdots$$

$$V\_{ar} = i\_{ar}r\_r + \frac{d\lambda\_{ar}}{dt}; \ V\_{br} = i\_{br}r\_r + \frac{d\lambda\_{br}}{dt}; \ V\_{cr} = i\_{cr}r\_r + \frac{d\lambda\_{cr}}{dt} \tag{4}$$

$$V\_{ar} = 2 \dots \text{constant} \\ \text{where} \\ \vdots \\ V\_{ar} = 2 \dots \text{constant} \\ \text{and} \\ \text{cross} \\ \vdots \\ \dots \quad i\_{ar} = \gamma \dots \gamma \dots \dots$$

*2.1. Modelling of the Induction Motor*  where *Vas*, *Vbs*, *Vcs* = 3-ϕ stator voltages; *Var*, *Vbr*, *Vcr* = 3-ϕ rotor voltages; *ias*, *ibs*, *ics* = 3-ϕ stator currents; *iar*, *ibr*, *icr* = 3-ϕ rotor currents; *r<sup>s</sup>* = stator resistance; *r<sup>r</sup>* = rotor resistance; *λas*, *λbs*, *λcs* = flux linkages of the stator; and *λar*, *λbr*, *λcr* = flux linkages of the rotor.

To depict the physical systems, the computer models the mathematical model of the three-phase IM and its driving system [4]. The control parameters of the models are high-According to the winding currents and inductances, the flux linkages indicate the matrix formed between the rotor and stator windings, as depicted in the following matrix [24].

$$
\begin{bmatrix}
\lambda\_s^{abc} \\
\lambda\_r^{abc}
\end{bmatrix} = \begin{bmatrix}
L\_{ss}^{abc} & L\_{sr}^{abc} \\
L\_{rs}^{abc} & L\_{rr}^{abc}
\end{bmatrix} \begin{bmatrix}
i\_s^{abc} \\
i\_r^{dbc}
\end{bmatrix} \tag{5}
$$

 ௦ = ௦௦ + ; ௦ = ௦௦ <sup>+</sup> ; ௦ = ௦௦ <sup>+</sup> (3) = + ; = <sup>+</sup> ; = <sup>+</sup> (4) where *Vas*, *Vbs*, *Vcs* = 3-φ stator voltages; *Var*, *Vbr*, *Vcr* = 3-φ rotor voltages; *ias*, *ibs*, *ics* = 3-φ stator where *λ abc <sup>s</sup>* = - *<sup>λ</sup>as <sup>λ</sup>bs <sup>λ</sup>cs*<sup>T</sup> ,*i abc <sup>s</sup>* = - *<sup>i</sup>as <sup>i</sup>bs <sup>i</sup>cs*<sup>T</sup> ,*λ abc <sup>r</sup>* = - *<sup>λ</sup>ar <sup>λ</sup>br <sup>λ</sup>cr*<sup>T</sup> ,*i abc <sup>r</sup>* = - *<sup>i</sup>ar <sup>i</sup>br <sup>i</sup>cr<sup>T</sup>* , and superscript T = transpose of the array; *L abc ss* = stator-to-stator winding inductance; *L abc rr* = rotor-to-rotor winding inductance; and *L abc sr* = stator-to-rotor mutual inductance, which depends on the rotor angle *θ<sup>r</sup>* .

currents; *iar*, *ibr*, *icr* = 3-φ rotor currents; *rs* = stator resistance; *rr* = rotor resistance; *λas*, *λbs*, *λcs* = flux linkages of the stator; and *λar*, *λbr*, *λcr* = flux linkages of the rotor. According to the winding currents and inductances, the flux linkages indicate the matrix formed between the rotor and stator windings, as depicted in the following matrix [24]. ቈ ௦ = ቈ ௦௦ ௦ ௦ ቈ௦ (5) To solve and simplify the computation of the three-phase IM performance, contemporary research analyzes the transient and steady-state performance of three Ims with a direct-quadrature-zero (*dq*0) stationary motor model [21,25,26]. As it can accurately represent the real-time motor model, a *dq*0 reference frame is chosen in this study to create the motor simulation model. The relationship between the rotor *dq*0 and the *abc* stator axes is shown in Figure 2. The mechanical rotor speed is denoted by *ωrm*, and the reference frame speed is represented by *ω*. The transformation from *abc* to *dq*0 is analyzed using stationary and synchronously revolving frames. Similar to those typically used for supply networks, the stationary rotating frame has a speed frame that produces a value of *ω* = 0. When the

*Energies* **2022**

synchronous reference frame revolves in the same direction as the rotor revolution, the speed frame becomes *ω = ω<sup>s</sup>* . The matrix below contains their inverses and the transition from the *abc* to the *dq*0 reference frame [27].

$$
\begin{bmatrix} \mathbf{x}\_d \\ \mathbf{x}\_q \\ \mathbf{x}\_0 \end{bmatrix} = \frac{2}{3} \begin{bmatrix} \cos\left(\theta\right) & \cos\left(\theta - \frac{2\pi}{3}\right) & \cos\left(\theta + \frac{2\pi}{3}\right) \\ \sin\left(\theta\right) & \sin\left(\theta - \frac{2\pi}{3}\right) & \sin\left(\theta + \frac{2\pi}{3}\right) \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{bmatrix} \begin{bmatrix} \mathbf{x}\_d \\ \mathbf{x}\_b \\ \mathbf{x}\_c \end{bmatrix} = \begin{bmatrix} \mathbf{X}\_{dq0}(\theta) \end{bmatrix} \begin{bmatrix} \mathbf{x}\_d \\ \mathbf{x}\_b \\ \mathbf{x}\_c \end{bmatrix} \tag{6}
$$

$$
\begin{bmatrix} \mathbf{x}\_d \\ \mathbf{x}\_b \\ \mathbf{x}\_c \end{bmatrix} = \frac{2}{3} \begin{bmatrix} \cos\left(\theta\right) & \sin\left(\theta\right) & 1 \\ \cos\left(\theta - \frac{2\pi}{3}\right) & \sin\left(\theta - \frac{2\pi}{3}\right) & 1 \\ \cos\left(\theta + \frac{2\pi}{3}\right) & \sin\left(\theta + \frac{2\pi}{3}\right) & 1 \end{bmatrix} \begin{bmatrix} \mathbf{x}\_d \\ \mathbf{x}\_q \\ \mathbf{x}\_0 \end{bmatrix} = \begin{bmatrix} \mathbf{X}\_{dq0}(\theta) \end{bmatrix}^{-1} \begin{bmatrix} \mathbf{x}\_d \\ \mathbf{x}\_b \\ \mathbf{x}\_c \end{bmatrix} \tag{7}
$$

where variable *x* = the 3-ϕ IM's phase voltage, current, or flux linkage. The stator voltages are converted to the *dq*0 voltages into a matrix form that includes flux linkages, currents, and voltages of the *dq*0 reference frame to produce the *dq*0 voltages [28].

$$
\begin{bmatrix} v\_{ds} \\ v\_{qs} \\ v\_{0s} \end{bmatrix} = \frac{d\theta}{dt} \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \lambda\_{ds} \\ \lambda\_{qs} \\ \lambda\_{0s} \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \lambda\_{ds} \\ \lambda\_{qs} \\ \lambda\_{0s} \end{bmatrix} + r\_s \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} i\_{ds} \\ i\_{qs} \\ i\_{0s} \end{bmatrix} \tag{8}
$$

**Figure 2.** Equivalent circuits of the three-phase IM's *dq*0 stationary reference frames. **Figure 2.** Equivalent circuits of the three-phase IM's *dq*0 stationary reference frames.

**3. Voltage-Oriented Control**  The VOC technique for AC-DC converters is derived from FOC for IMs. FOC provides a quick, dynamic reaction due to the utilization of current control loops. The theoretical elements of the VOC approach used for grid-connected rectifiers have received ex-Similar to this, while transferring the voltages, currents, and flux linkages, the rotor voltages are converted to the *dq*0 frame into a matrix form that must be combined with the mechanical rotor angle *(θ* − *θrm)* to become *[Xdq*0*(θ – θrm)]* in order to achieve the subsequent equations shown below [29].

switching frequency, necessitating the use of an input filter at high-value parameters. To alleviate harmonic issues, the proposed method utilizes the VOC principle to regulate charging while maintaining low harmonic distortions to the grid. Figure 3 depicts VOC for AC-DC line-side converters. VOC operates most frequently in the two-phase zero and

$$
\begin{bmatrix} v\_{dr} \\ v\_{qr} \\ v\_{0r} \end{bmatrix} = \frac{d(\theta - \theta\_{rm})}{dt} \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \lambda\_{dr} \\ \lambda\_{qr} \\ \lambda\_{0r} \end{bmatrix} + \frac{d}{dt} \begin{bmatrix} \lambda\_{dr} \\ \lambda\_{qr} \\ \lambda\_{0r} \end{bmatrix} + r\_s \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} i\_{dr} \\ i\_{qr} \\ i\_{0r} \end{bmatrix} \tag{9}
$$

−1 2

1 √2

−() ()൨ ௦ఈ

Note that *Vsa*, *Vsb*, and *Vsc* = 3-φ source voltages in the *abc* domain, while *Vsα*, *Vsβ*, and *V*0 = source voltages in the *αβ*0 domain. *Vd*, *Vq*, and *V0* are source voltages of the *dq*0 domain, and *θ* = the operating phase of the power system. As shown in Figure 2, an equiva-

<sup>0</sup> √3

−1 2

1 √2 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤

 ௦ ௦ ௦

൩ (21)

௦ఉ<sup>൨</sup> (22)

<sup>2</sup> <sup>−</sup> √3 2

 ௦ఈ ௦ఉ 

> ௗ

 = <sup>ඨ</sup><sup>2</sup> 3 ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 1

lent transformation procedure is used to transform the 3-φ source current *isabc.*

1 √2

൨= () ()

The stator *abc* flux linkage is transformed to produce the stator *dq*0 flux linkages, which are stated as follows.

$$
\begin{bmatrix}
\lambda\_{\rm ds} \\
\lambda\_{\rm qs} \\
\lambda\_{\rm 0s}
\end{bmatrix} = \begin{bmatrix}
L\_{\rm ls} + \frac{3}{2}L\_{\rm ss} & 0 & 0 \\
0 & L\_{\rm ls} + \frac{3}{2}L\_{\rm ss} & 0 \\
0 & 0 & L\_{\rm ls}
\end{bmatrix} \begin{bmatrix}
i\_{\rm ds} \\
i\_{\rm qs} \\
i\_{\rm 0s}
\end{bmatrix} + \begin{bmatrix}
\frac{3}{2}L\_{\rm sr} & 0 & 0 \\
0 & \frac{3}{2}L\_{\rm sr} & 0 \\
0 & 0 & 0
\end{bmatrix} \begin{bmatrix}
i\_{\rm dr} \\
i\_{\rm qr} \\
i\_{\rm 0r}
\end{bmatrix} \tag{10}
$$

The rotor *dq*0 flux linkages are produced in a similar manner to the rotor *abc* flux linkage.

$$
\begin{bmatrix}
\lambda\_{dr} \\
\lambda\_{qr} \\
\lambda\_{0r}
\end{bmatrix} = \begin{bmatrix}
\frac{3}{2}L\_{sr} & 0 & 0 \\
0 & \frac{3}{2}L\_{sr} & 0 \\
0 & 0 & 0
\end{bmatrix} \begin{bmatrix}
i\_{ds} \\ i\_{qs} \\ i\_{0s}
\end{bmatrix} + \begin{bmatrix}
L\_{lr} + \frac{3}{2}L\_{rr} & 0 & 0 \\
0 & L\_{lr} + \frac{3}{2}L\_{rr} & 0 \\
0 & 0 & L\_{lr}
\end{bmatrix} \begin{bmatrix}
i\_{dr} \\ i\_{qr} \\ i\_{0r}
\end{bmatrix} \tag{11}
$$

The stator and rotor *dq*0 flux linkage relations can be stated as in Equation (12) [25], where *λ* 0 *dr*, *λ* 0 *qr* are the reference values on the stator side of the *dq* rotor flux linkages. This is based on Equations (10) and (11); the referred values on the stator side of the *dq* rotor currents are *i* 0 *dr*, *i* 0 *qr*.

$$
\begin{bmatrix}
\lambda\_{ds} \\
\lambda\_{qs} \\
\lambda\_{0s} \\
\lambda\_{dr}' \\
\lambda\_{qr}' \\
\lambda\_{0r}'
\end{bmatrix} = \begin{bmatrix}
L\_{ls} + L\_m & 0 & 0 & L\_m & 0 & 0 \\
0 & L\_{ls} + L\_m & 0 & 0 & L\_m & 0 \\
0 & 0 & L\_{ls} & 0 & 0 & 0 \\
L\_m & 0 & 0 & L\_{lr}' + L\_m & 0 & 0 \\
0 & L\_m & 0 & 0 & L\_{lr}' + L\_m & 0 \\
0 & 0 & 0 & 0 & 0 & L\_{lr}'
\end{bmatrix} \begin{bmatrix}
i\_{ds} \\
i\_{qs} \\
i\_{0s} \\
i\_{qr}' \\
i\_{qr}' \\
i\_{qr}'
\end{bmatrix} \tag{12}
$$

The governing equations [27] are produced by converting the final *dq*0 reference frame equations to the flux linkage, utilizing the formula *ψ* = *ωsλ*, and the inductance into reactance, using *x* = *ωsL*.

$$v\_{ds} = \frac{1}{\omega\_s} \frac{d\psi\_{ds}}{dt} + \frac{\omega}{\omega\_s} \psi\_{qs} + r\_s i\_{ds} \tag{13}$$

$$v\_{q\text{s}} = \frac{1}{\omega\_{\text{s}}} \frac{d\psi\_{q\text{s}}}{dt} + \frac{\omega}{\omega\_{\text{s}}} \psi\_{\text{ds}} + r\_{\text{s}} i\_{q\text{s}} \tag{14}$$

$$v\_{0s} = \frac{1}{\omega\_s} \frac{d\psi\_{0s}}{dt} + r\_s i\_{0s} \tag{15}$$

$$v\_{dr}' = \frac{1}{\omega\_s} \frac{d\psi\_{dr}'}{dt} + \left(\frac{\omega - \omega\_{rm}}{\omega\_s}\right) \psi\_{qr}' + r\_r' i\_{dr}' \tag{16}$$

$$v\_{qr}^{\prime} = \frac{1}{\omega\_{\rm s}} \frac{d\psi\_{qr}^{\prime}}{dt} - \left(\frac{\omega - \omega\_{rm}}{\omega\_{\rm s}}\right) \psi\_{dr}^{\prime} + r\_{r}^{\prime} i\_{qr}^{\prime} \tag{17}$$

$$v\_{0r}' = \frac{1}{\omega\_s} \frac{d\psi\_{0r}'}{dt} + r\_r' i\_{0r}' \tag{18}$$

$$
\begin{bmatrix}
\psi\_{ds} \\
\psi\_{qs} \\
\psi\_{0s} \\
\psi'\_{dr} \\
\psi'\_{qr} \\
\psi'\_{0r}
\end{bmatrix} = \begin{bmatrix}
\mathbf{x}\_{ls} + \mathbf{x}\_m & \mathbf{0} & \mathbf{0} & \mathbf{x}\_m & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{x}\_{ls} + L\_m & \mathbf{0} & \mathbf{0} & \mathbf{x}\_m & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{x}\_{ls} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\
\mathbf{x}\_m & \mathbf{0} & \mathbf{0} & \mathbf{x}'\_{lr} + \mathbf{x}\_m & \mathbf{0} & \mathbf{0} \\
\mathbf{0} & \mathbf{x}\_m & \mathbf{0} & \mathbf{0} & \mathbf{x}'\_{lr} + \mathbf{x}\_m & \mathbf{0} \\
\mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{x}'\_{lr}
\end{bmatrix} \begin{bmatrix}
i\_{ds} \\
i\_{qs} \\
i\_{dr} \\
i'\_{dr} \\
i'\_{qr} \\
i'\_{0r}
\end{bmatrix}.
\tag{19}
$$

The electromagnetic torque can manifest itself in the following ways:

$$\begin{array}{l} T\_{\rm em} = \frac{3}{2} \frac{P}{2\omega\_{\rm s}} \left( \psi\_{dr}' i\_{qr}' - \psi\_{qr}' i\_{dr}' \right) \\ = \frac{3}{2} \frac{P}{2\omega\_{\rm s}} \left( \psi\_{qs} i\_{ds} - \psi\_{ds} i\_{qs} \right) \\ = \frac{3}{2} \frac{P}{2\omega\_{\rm s}} \mathfrak{x}\_{m} \left( i\_{qr}' i\_{ds} - i\_{dr}' i\_{qs} \right) \end{array} \tag{20}$$

As depicted in Figure 2, the equivalent circuits of the *dq*0 stationary reference frames are generated by introducing Equation (19) into Equation (13) to Equation (18).

#### **3. Voltage-Oriented Control**

The VOC technique for AC-DC converters is derived from FOC for IMs. FOC provides a quick, dynamic reaction due to the utilization of current control loops. The theoretical elements of the VOC approach used for grid-connected rectifiers have received extensive coverage. The control system uses the PWM technique to ensure that the characteristics of the VOC control scheme are modified. It is possible to reduce the impact of interference (disturbances). By using the hysteresis pulse-width modulation approach, system solidity can be achieved. Power switching is stressed as a result of the fluctuating switching frequency, necessitating the use of an input filter at high-value parameters. To alleviate harmonic issues, the proposed method utilizes the VOC principle to regulate charging while maintaining low harmonic distortions to the grid. Figure 3 depicts VOC for AC-DC line-side converters. VOC operates most frequently in the two-phase zero and *dq*0 domains, where the Clarke and Park transformation matrices are employed.

$$
\begin{bmatrix} V\_{sa} \\ V\_{s\theta} \\ V\_0 \end{bmatrix} = \sqrt{\frac{2}{3}} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} V\_{sa} \\ V\_{sb} \\ V\_{sc} \end{bmatrix} \tag{21}
$$

$$
\begin{bmatrix} V\_d \\ V\_q \end{bmatrix} = \begin{bmatrix} \sin(\theta) & \cos(\theta) \\ -\cos(\theta) & \sin(\theta) \end{bmatrix} \begin{bmatrix} V\_{\text{st}} \\ V\_{\text{s\S}} \end{bmatrix} \tag{22}
$$

**Figure 3.** Overall control structure of VOC of a PWM inverter [30]. 33]. **Figure 3.** Overall control structure of VOC of a PWM inverter [30].

Based on the following techniques, steady-state errors are easily reduced by the proportional integral (PI) controllers by employing the transformation technique to convert

where and = gains of the PI controller, ௦ௗ and ௦ = source current in the *dq*0 domain, and ௦ௗ, and ௦, = reference signals for ௦ௗ and ௦, respectively. After obtaining the reference voltages ௗ, and ,, the gate switching pulses *Sabc*, which regulate the operation of the VOC inverter, are derived using the PWM switching approach

൨=() −()

The SVM approach, generally acknowledged as the optimum method for variable speed drive applications, creates PWM control signals in the 3-φ inverter. Compared to other PWM systems, this method provides an enhanced means of achieving a high output voltage, minimizing the harmonic output, and lowering switching losses. As a result, the SVM technique is confirmed as the best approach for reducing harmonic distortion [31–

() () ൨ ቂௗ,

, = ൫௦, − ௦൯+ න൫௦, − ௦൯ (24)

,<sup>ቃ</sup> (25)

and the inverse park transformation as given in Equation (25).

 ఈ, ఉ,

AC-side control variables into DC signals:

**4. SVM Switching Techniques** 

Note that *Vsa*, *Vsb*, and *Vsc* = 3-ϕ source voltages in the *abc* domain, while *Vsα*, *Vsβ*, and *V*<sup>0</sup> = source voltages in the *αβ*0 domain. *V<sup>d</sup>* , *Vq*, and *V<sup>0</sup>* are source voltages of the *dq*0 domain, and *θ* = the operating phase of the power system. As shown in Figure 2, an equivalent transformation procedure is used to transform the 3-ϕ source current *isabc.*

Based on the following techniques, steady-state errors are easily reduced by the proportional integral (PI) controllers by employing the transformation technique to convert AC-side control variables into DC signals:

$$v\_{d,ref} = \mathcal{K}\_p \left( \mathbf{i}\_{\rm sd,ref} - \mathbf{i}\_{\rm sd} \right) + \mathcal{K}\_i \int \left( \mathbf{i}\_{\rm sd,ref} - \mathbf{i}\_{\rm sd} \right) dt \tag{23}$$

$$v\_{q,ref} = K\_p \left( i\_{sq,ref} - i\_{sq} \right) + K\_i \int \left( i\_{sq,ref} - i\_{sq} \right) dt \tag{24}$$

where *K<sup>p</sup>* and *K<sup>i</sup>* = gains of the PI controller, *isd* and *isq* = source current in the *dq*0 domain, and *isd*,*re f* and *isq*,*re f* = reference signals for *isd* and *isq*, respectively. After obtaining the reference voltages *vd*,*re f* and *vq*,*re f* , the gate switching pulses *Sabc*, which regulate the operation of the VOC inverter, are derived using the PWM switching approach and the inverse park transformation as given in Equation (25).

$$
\begin{bmatrix} V\_{a,ref} \\ V\_{\beta,ref} \end{bmatrix} = \begin{bmatrix} \sin(\theta) & -\cos(\theta) \\ \cos(\theta) & \sin(\theta) \end{bmatrix} \begin{bmatrix} v\_{d,ref} \\ v\_{q,ref} \end{bmatrix} \tag{25}
$$

#### **4. SVM Switching Techniques**

The SVM approach, generally acknowledged as the optimum method for variable speed drive applications, creates PWM control signals in the 3-ϕ inverter. Compared to other PWM systems, this method provides an enhanced means of achieving a high output voltage, minimizing the harmonic output, and lowering switching losses. As a result, the SVM technique is confirmed as the best approach for reducing harmonic distortion [31–33].

$$
\begin{bmatrix} V\_{ab} \\ V\_{bc} \\ V\_{ca} \end{bmatrix} = V\_{dc} \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{bmatrix} \begin{bmatrix} s\_1 \\ s\_3 \\ s\_5 \end{bmatrix} \tag{26}
$$

$$
\begin{bmatrix} V\_{ab} \\ V\_{bc} \\ V\_{ca} \end{bmatrix} = \frac{V\_{dc}}{3} \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix} \begin{bmatrix} s\_1 \\ s\_3 \\ s\_5 \end{bmatrix} \tag{27}
$$

Eight switch variables can be created using the inverter's six IGBTs. Six switch variables—*V*1, *V*2, . . . , *V*6, and the last two—are zero vectors chosen for the three upper IGBT switches. The on and off patterns of the lower IGBT switches are the opposite of those of the higher switches. Figure 4 displays the eight switching vectors of the SVM [34].

The output waveform of the inverter is split into six hexagonal sectors, according to the SVM's working principle. The sector angle is 60◦ apart, and every sector is between two voltage space vectors (Figure 5) [35]. The SVM approach receives a 3-ϕ voltage (*Va*, *V<sup>b</sup>* , and *Vc*) with a 120◦ angle among the 2-ϕ and transforms it into 2-ϕ (*V<sup>α</sup>* and *Vβ*) with a 90◦ using Clark's transformation (Figure 6a).

To make the study of three-phase voltage more straightforward, 2-ϕ voltages (*V<sup>α</sup>* and *Vβ*) are used as part of a scientific transformation. The voltages are used to calculate the hexagon's voltage vector angle (*α*) and the reference voltage vector's magnitude (*Vref*). *Vref* is assumed as the magnitude of the *V<sup>α</sup>* and *V<sup>β</sup>* voltages, while α is the frequency of *V<sup>α</sup>* and *Vβ. Vref* and α are situated among the two neighboring non-zero and zero vectors. The following formulas can be used to calculate them [29].

$$
\begin{bmatrix} V\_a \\ V\_\beta \end{bmatrix} = \frac{2}{3} \begin{bmatrix} 1 & \frac{-1}{2} & \frac{-1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix} \begin{bmatrix} V\_a \\ V\_b \\ V\_c \end{bmatrix} \tag{28}
$$

$$\left| V\_{ref} \right| = \sqrt{V\_a^2 + V\_\beta^2} \tag{29}$$

൩ ଵ ଷ ହ

൩ ଵ ଷ ହ

൩ ଵ ଷ ହ

$$a = \tan^{-1} \frac{V\_{\beta}}{V\_a} \tag{30}$$

൩ (26)

൩ (26)

൩ (27)

Sector 1 contains the vectors that can be used to synthesize *Vref*, which is located here. switches. The on and off patterns of the lower IGBT switches are the opposite of those of the higher switches. Figure 4 displays the eight switching vectors of the SVM [34].

1 −1 0 0 1 −1 −1 0 1

Eight switch variables can be created using the inverter's six IGBTs. Six switch variables—*V*1, *V*2, ..., *V*6, and the last two—are zero vectors chosen for the three upper IGBT switches. The on and off patterns of the lower IGBT switches are the opposite of those of

2 −1 −1 −1 2 −1 −1 −1 2

1 −1 0 0 1 −1 −1 0 1

2 −1 −1

*Vb*, and *Vc*) with a 120° angle among the 2-φ and transforms it into 2-φ (*Vα* and *Vβ*) with a

**Figure 4.** Eight switching states for the inverter voltage vectors (*V*0 to *V*7). **Figure 4.** Eight switching states for the inverter voltage vectors (*V*<sup>0</sup> to *V*<sup>7</sup> ). 90° using Clark's transformation (Figure 6a).

*Energies* **2022**, *15*, x FOR PEER REVIEW 8 of 22

 

 

*Energies* **2022**, *15*, x FOR PEER REVIEW 8 of 22

 

  ൩=ௗ

the higher switches. Figure 4 displays the eight switching vectors of the SVM [34].

൩=ௗ

൩ = ௗ 3

**Figure 5. Figure 5.** Space vector diagram. Space vector diagram.

The relevant space vectors and time intervals (*T*1, *T*2, and *T*0) in sector 1 are depicted in Figure 6b. The duration of Vref is calculated by multiplying the reference voltage by the sampling time period, which is equivalent to the sum of the voltages times the time interval of the space vectors in the specified sector [36].

$$\int\_{0}^{T\_{\rm s}} \overline{\nabla}\_{ref} dt = \int\_{0}^{T\_{\rm 1}} \overline{\nabla}\_{1} dt + \int\_{T\_{1}}^{T\_{1} + T\_{2}} \overline{\nabla}\_{2} dt + \int\_{T\_{1} + T\_{2}}^{T\_{\rm s}} \overline{\nabla}\_{0} dt\\T\_{\rm s} = T\_{1} + T\_{2} + T\_{0} \tag{31}$$

where *Ts* = switching time, which is calculated by *Ts* = 1 *fs* , and *fs* = switching frequency.

lowing formulas can be used to calculate them [29].

**Figure 6.** (**a**) Space vector diagram. (**b**) Space of sector-1 between *V*1 and *V*2. **Figure 6.** (**a**) Space vector diagram. (**b**) Space of sector-1 between *V*<sup>1</sup> and *V*<sup>2</sup> .

்ೞ

 ఈ ൨ = <sup>2</sup> ⎡1 −1 2 −1 2 ⎤ Equation (30) illustrates how *V*<sup>0</sup> provides zero voltage to the output load. As a result, the equation is:

To make the study of three-phase voltage more straightforward, 2-φ voltages (*Vα* and *Vβ*) are used as part of a scientific transformation. The voltages are used to calculate the hexagon's voltage vector angle (*α*) and the reference voltage vector's magnitude (*Vref*). *Vref* is assumed as the magnitude of the *Vα* and *Vβ* voltages, while α is the frequency of *Vα* and *Vβ. Vref* and α are situated among the two neighboring non-zero and zero vectors. The fol-

$$T\_s \overline{V}\_{ref} = T\_1 \overline{V}\_1 + T\_2 \overline{V}\_2 \tag{32}$$
 
$$\text{Substituting into the } s\ell \text{ forms and the values constants}$$

⎣ 2 ⎦ หห = ටఈ <sup>ଶ</sup> + ఉ <sup>ଶ</sup> (29) When the values of *V*<sup>1</sup> and *V*<sup>2</sup> are substituted into the *αβ* frame and the voltage vectors are evaluated, the results are as follows [37].

$$T\_s \begin{vmatrix} V\_{ref} \end{vmatrix} \begin{bmatrix} \cos(\alpha) \\ \sin(\alpha) \end{bmatrix} = T\_1 \frac{2}{3} V\_{dc} \begin{bmatrix} 1 \\ 0 \end{bmatrix} + T\_2 \frac{2}{3} V\_{dc} \begin{bmatrix} \cos\frac{\pi}{3} \\ \sin\frac{\pi}{3} \end{bmatrix} \tag{33}$$

$$T\_1 = T\_s \frac{3}{2} \frac{\left| V\_{ref} \right|}{V\_{dc}} \frac{\sin\left(\frac{\pi}{3} - a\right)}{\sin\left(\frac{\pi}{3}\right)}\tag{34}$$

$$T\_2 = T\_s \frac{3}{2} \frac{\left| V\_{ref} \right|}{V\_{dc}} \frac{\sin(a)}{\sin\left(\frac{\pi}{3}\right)}\tag{35}$$

න ത = න ത ଵ + න ത ଶ + න ത ்భା்మ ்భ ௦ = ଵ + ଶ + (31) The relation between the magnitude of the reference voltage and the *dc* voltage value represented by the following equation is known as the MI for the SVM [38].

$$MI = \frac{\left| V\_{ref} \right|}{\frac{2}{\pi} V\_{dc}} \tag{36}$$

sult, the equation is: ௦ത = ଵത <sup>ଵ</sup> + ଶത ଶ (32) When the values of ଵ and ଶ are substituted into the *αβ* frame and the voltage vectors Equation (35) can be substituted into Equations (33) and (34) to determine the time duration in the other sectors (*n*), and 60 degrees with α can be used for each sector to obtain the result [39,40].

$$T\_1 = \frac{\sqrt{3}T\_s \Big| V\_{ref} \Big|}{V\_{dc}} \sin\left(\frac{n}{3}\pi - n\right) \tag{37}$$

$$T\_2 = \frac{\sqrt{3}T\_s \left| V\_{ref} \right|}{V\_{dc}} \sin \left( \alpha - \frac{n-1}{3} \pi \right) \tag{38}$$

$$T\_0 = T\_s - \left(T\_1 + T\_2\right) \tag{39}$$

Alternating the zero-vector sequence, the asymmetric sequence, the maximum current not switched sequence, and the right aligned sequence are the four different types of switching patterns. To reduce device switching frequency, all switching patterns must fulfill the following two requirements. Merely two switches in the same inverter leg are used to switch from one switching state to another. To minimize the switching frequency, one of the switches must be turned off if the other is activated. The least amount of

switching is necessary to move *Vref* from one sector to the next to minimize switching losses. The optimum strategy, according to research, is the symmetric sequence method, since it minimizes switching losses. The generation of the SVM signal and the inverter output voltages and a comparison of the duty ratio waveform's three signals with the triangle waveform is shown in Figure 7. This comparison assumes that *S* is ON if *VDutyRatio > Vtriangle*; otherwise, *S* is OFF. is shown in Figure 7. This comparison assumes that *S* is ON if *VDutyRatio > Vtriangle*; otherwise, *S* is OFF. Every switch in a bipolar switching scheme operates in opposition to the facing switch, as seen in the example where the triangle waveform and the *VTaDutyRatio* are compared to produce the PWM signal for the IGBT1 and the opposing IGBT4 in leg 1, which is identical to leg 2 and leg 3.

*Energies* **2022**, *15*, x FOR PEER REVIEW 10 of 22

ଵ = ௦

ଶ = ௦

represented by the following equation is known as the MI for the SVM [38].

ଵ <sup>=</sup> √3௦หห ௗ

ଶ <sup>=</sup> √3௦หห ௗ

3 2 หห ௗ

3 2 หห ௗ

The relation between the magnitude of the reference voltage and the *dc* voltage value

 = หห 2 ௗ

Equation (35) can be substituted into Equations (33) and (34) to determine the time duration in the other sectors (*n*), and 60 degrees with α can be used for each sector to

> (

( −

Alternating the zero-vector sequence, the asymmetric sequence, the maximum current not switched sequence, and the right aligned sequence are the four different types of switching patterns. To reduce device switching frequency, all switching patterns must fulfill the following two requirements. Merely two switches in the same inverter leg are used to switch from one switching state to another. To minimize the switching frequency, one of the switches must be turned off if the other is activated. The least amount of switching is necessary to move *Vref* from one sector to the next to minimize switching losses. The optimum strategy, according to research, is the symmetric sequence method, since it minimizes switching losses. The generation of the SVM signal and the inverter output voltages and a comparison of the duty ratio waveform's three signals with the triangle waveform

−1

= ௦ − (ଵ + ଶ) (39)

(

<sup>3</sup> − ) (

() (

<sup>3</sup>) (34)

<sup>3</sup>) (35)

<sup>3</sup> − ) (37)

<sup>3</sup> ) (38)

(36)

**Figure 7.** SVM waves for a three-phase inverter. **Figure 7.** SVM waves for a three-phase inverter.

Every switch in a bipolar switching scheme operates in opposition to the facing switch, as seen in the example where the triangle waveform and the *VTaDutyRatio* are compared to produce the PWM signal for the IGBT1 and the opposing IGBT4 in leg 1, which is identical to leg 2 and leg 3.
