Comparative Study of Discretization Methods for Non-Ideal Proportional-Resonant Controllers in Voltage Regulation of Three-Phase Four-Wire Converters with Vehicle-to-Home Mode

Abstract

Vehicle-to-home (V2H) technology enables electric vehicles (EVs) to supply power to residential loads, offering enhanced energy self-sufficiency and backup capabilities. Accurate voltage regulation is essential in such systems, especially under nonlinear and time-varying load conditions. The control method for three-phase four-wire (3P4W) converters plays a vital role in addressing these challenges. In the control configuration of such systems, the non-ideal proportional-resonant (PR) controller stands out due to its ability to reject periodic disturbances. However, the comprehensive study on the discretization of this controller for digital implementation in 3P4W systems has not been available in the literature to date. This paper presents a comparative study of several discretization methods for non-ideal PR controllers. The continuous-time complete transfer function of the integral term of non-ideal PR controllers is discretized using techniques such as Forward Euler, Backward Euler, Tustin, Zero-Order Hold, and Impulse Invariance. Additionally, the discretization methods based on two discrete integrators for the non-ideal PR controller, such as Forward Euler and Backward Euler, Backward Euler and Backward Euler plus computational delay, and Tustin and Tustin, are also evaluated. In the MATLAB/Simulink platform, through evaluating the performance of the non-ideal PR controllers, which are discretized using the above discretization methods, in controlling the output voltage of the 3P4W converter in the V2H application under nonlinear load scenarios, including substantial and sudden changes in load, the discretization method Backward Euler and Backward Euler plus delay is recommended.

1. Introduction

The rapid adoption of electric vehicles (EVs) has introduced new possibilities in energy management, particularly through bidirectional power exchange with the grid (vehicle-to-grid—V2G) [1,2], with the load (vehicle-to-load—V2L) [3,4], or with residential/commercial systems (vehicle-to-home, V2H) [5,6]. Among these, V2H has gained attention due to its ability to enhance energy self-sufficiency, especially when combined with rooftop solar systems [7,8]. More importantly, V2H provides backup power during grid outages, making it a crucial feature for a reliable energy supply [9].

A key component in V2H systems is the three-phase four-wire (3P4W) converter, which facilitates effective voltage regulation while addressing challenges posed by unbalanced loads and nonlinearities [10,11]. By providing a neutral current path, the 3P4W converter helps reduce the total harmonic distortion (THD) [12] and enables compatibility with diverse load conditions [13].

However, achieving stable and accurate voltage regulation in 3P4W converters remains a significant challenge, especially under dynamic operating conditions. The most commonly used controllers for voltage regulation in such systems are the proportional-integral (PI) [14] and proportional-resonant (PR) [15] controllers.

The PI controller features a simple structure and is easy to tune, with the capability to eliminate steady-state errors. Nevertheless, it performs best when regulating DC quantities. As such, PI controllers are typically implemented in the rotating dq0 reference frames via Park transformations. This approach, while effective, introduces additional complexity, particularly in the presence of cross-coupling between the d-axis and q-axis.

To overcome these limitations, the PR controller has been suggested. Operating directly in the stationary αβ frame, the PR controller can regulate AC signals without requiring coordinate transformation. Furthermore, it offers superior tracking performance and the rejection of periodic disturbances [16], making it a highly attractive solution for AC control applications. PR controllers have demonstrated effectiveness in various domains such as wind turbine generators [17], grid-connected PV inverters [18], and active power filters [19].

Despite these advantages, ideal PR controllers are highly sensitive to frequency deviations. While it offers theoretically infinite gain at the resonant frequency, which is beneficial for perfect tracking, it may lead to stability concerns in practical systems [20]. To mitigate this problem, adaptive PR controllers have been proposed, in which the resonant frequency is continuously adjusted based on the estimated grid frequency using Phase-Locked-Loop (PLL) [21] or Frequency-Locked-Loop (FLL) [22] techniques. Although effective, these adaptive approaches require higher computational effort and introduce additional implementation complexity.

As a simpler alternative, the non-ideal PR controller has been introduced. By incorporating a damping term into the ideal PR structure [23], the infinite gain at the resonant frequency is limited, offering improved robustness and making it more suitable for digital implementation. Although non-ideal PR controllers can improve system stability, their digital implementation poses significant challenges. The discretization process, which converts continuous-time controllers into discrete-time representations, introduces errors that can degrade control accuracy and stability [24]. Selecting an appropriate discretization method is thus crucial to preserving the desired frequency response and performance of the PR controller in real-world applications. It is noted that to date, a thorough investigation into the discretization of non-ideal PR controllers has not yet been conducted.

In this paper, we present a comparative study of several popular discretization methods for non-ideal PR controllers employed in the control system of a 3P4W converter in V2H applications. The discretization methods examined for discretizing the transfer function of the integral term of the non-ideal PR controller include Forward Euler, Backward Euler, Tustin, Impulse Invariance, and Zero-Order Hold. Additionally, the discretization methods based on two discrete integrators for the non-ideal PR controller [24], such as Forward Euler and Backward Euler, Backward Euler and Backward Euler plus computational delay, and Tustin and Tustin, are also considered. To evaluate the effectiveness of these discretization methods, the discretized non-ideal PR controllers are used for the regulation of the output voltage of 3P4W converters operating in the V2H mode, in which the achievement of accurate phase-voltage control is crucial. In the scenarios for testing, nonlinear residential and commercial loads, such as computers, refrigerators, air conditioners, etc., can be applied. Special attention is given to dynamic events, such as the sudden disconnection of these loads, which are typical in practical V2H applications and pose significant challenges for controller stability and voltage quality. In the MATLAB (https://www.mathworks.com/products/matlab.html, accessed on 15 June 2025)/Simulink (https://www.mathworks.com/products/simulink.html, accessed on 15 June 2025) platform, by analyzing the voltage control performance of the non-ideal PR controller, which is discretized using the above discretization methods, an appropriate discretization method will be recommended.

2. Proportional-Resonant Controller

2.1. Ideal Proportional-Resonant Controller

The proportional-resonant (PR) controller is widely used in AC control applications. It is designed to achieve zero steady-state errors for sinusoidal references by providing infinite gain at the resonant frequency [25]. The transfer function of an ideal PR controller can be represented as

$$G_{PR\_ideal}(s)=K_p+K_r{\displaystyle \frac{2s}{s^2+{\omega }_o^2}}$$

(1)

It consists of the integrator and proportion terms that can be adjusted independently, where K_p is the proportional gain, K_r is the integral gain of the resonant regulator, and ${\omega }_o$ is the resonant angular frequency.

Figure 1a presents the Bode plot of the transfer function for the ideal PR controller, illustrating that the controller achieves infinite gain at the resonance frequency while providing no gain at other frequencies, which helps eliminate disturbances caused by nonlinear loads. As the PR controller generates infinite gain at the resonant frequency, even a slight deviation in the reference signal’s frequency can cause significant errors in tracking the reference signal. Therefore, a non-ideal PR controller is introduced [15].

2.2. Non-Ideal Proportional-Resonant Controller

The transfer function of the non-ideal PR controller is defined as

$$G_{PR\_non-ideal}(s)=K_p+K_r{\displaystyle \frac{2{\omega }_{c}s}{s^2+2{\omega }_{c}s+{\omega }_o^2}}=K_p+K_{r}R(s)$$

(2)

where K_p is the proportional gain, K_r is the integral gain of the resonant regulator, ${\omega }_o$ is the resonant angular frequency, ${\omega }_c$ is the resonance bandwidth of the controller, and R(s) is the resonant term.

Figure 1b illustrates the Bode plot of the transfer function for the non-ideal PR controller, where K_p = 1, K_r = 100, ${\omega }_o$ = $100\pi$ (rad/s), and ${\omega }_c$ = 8 (rad/s). The non-ideal PR controller exhibits a finite gain at the resonant frequency, but the gain is high enough to minimize small steady-state errors [20].

Figure 2a–c present the Bode plots of a non-ideal PR controller with different parameter settings.

Based on Figure 2a–c, proportional gain K_p significantly influences system stability parameters (phase margin, gain margin, and bandwidth). Meanwhile, K_r primarily governs the characteristics of the resonance peak, such as its width and height.

3. Discretization Implementations of Non-Ideal Proportional-Resonant Controllers

3.1. Discretization of Continuous-Time Complete Transfer Function

An essential step in the implementation of digital resonant controllers is the discretization process. As mentioned in the Introduction Section, a comprehensive discretization process of a non-ideal PR controller is not currently available in the literature. Such a process will be provided in detail in this section.

The discretization methods listed in

Table 1. Relations for discretizing R(s) by different methods.

Discretization Method	Equivalence	Notation
Forward Euler	$s={\displaystyle \frac{z-1}{T_s}}$	Rf(z)
Backward Euler	$s={\displaystyle \frac{z-1}{z{T}_s}}$	Rb(z)
Tustin	$s={\displaystyle \frac{2}{T_s}}{\displaystyle \frac{z-1}{z+1}}$	Rt(z)
Zero-Order Hold	$X(z)=(1-z^{-1})Z\{L^{-1}\{{\displaystyle \frac{X(s)}{s}}\}\}$	Rzoh(z)
Impulse Invariance	$X(z)=Z\{L^{-1}\{X(s)\}\}$	Rimp(z)

are applied to R(s), resulting in the discrete-time mathematical expressions shown in

Table 2. Discretized Z-domain transfer functions of R(s) using methods from Table 1.

Discretization Method	Discretized R(s)
Forward Euler	$R_f(z)=2{\omega }_c{T}_s{\displaystyle \frac{z-1}{z^2+z(2{\omega }_c{T}_s-2)+(1-2{\omega }_c{T}_s+{\omega }_o^2{T}_s^2)}}$
Backward Euler	$R_b(z)=2{\omega }_c{T}_s{\displaystyle \frac{z^2-z}{z^2(1+2{\omega }_c{T}_s+{\omega }_o^2{T}_s^2)-z(2{\omega }_c{T}_s+2)+1}}$
Tustin	$R_t(z)=4{\omega }_c{T}_s{\displaystyle \frac{z^2-1}{z^2(4+4{\omega }_c{T}_s+{\omega }_o^2{T}_s^2)+z(2{\omega }_o^2{T}_s^2-8)+(4-4{\omega }_c{T}_s+{\omega }_o^2{T}_s^2)}}$
Zero-Order Hold	$R_{zoh}(z)={\displaystyle \frac{2{\omega }_c}{\sqrt{{\omega }_o^2-{\omega }_c^2}}}{\displaystyle \frac{(z-1)e^{-{\omega }_c{T}_s}\sin (T_s\sqrt{{\omega }_o^2-{\omega }_c^2})}{z^2-2z{e}^{-{\omega }_c{T}_s}\cos (T_s\sqrt{{\omega }_o^2-{\omega }_c^2})+e^{-2{\omega }_c{T}_s}}}$
Impulse Invariance	$\begin{array}{l}{R}_{imp}(z)={\displaystyle \frac{2z{\omega }_c{T}_s}{z^2-2z{e}^{-{\omega }_c{T}_s}\cos (T_s\sqrt{{\omega }_o^2-{\omega }_c^2})+e^{-2{\omega }_c{T}_s}}}.\\ [z-e^{-{\omega }_c{T}_s}\cos (T_s\sqrt{{\omega }_o^2-{\omega }_c^2})-{\displaystyle \frac{{\omega }_c{e}^{-{\omega }_c{T}_s}}{\sqrt{{\omega }_o^2-{\omega }_c^2}}}\sin (T_s\sqrt{{\omega }_o^2-{\omega }_c^2})]\end{array}$

, where $T_s$ is the controller’s sampling period, and $f_s=1/T_s$ is the sampling rate. Detailed derivations can be found in Appendix A.

Figure 3a–e compare the frequency response of a resonant controller R(s) designed for the fundamental frequency, using the discretization methods presented in Table 2: ${\omega }_o$ = $100\pi$ (rad/s), ${\omega }_c$ = 1 (rad/s), and $f_s$ = 20 kHz.

Based on Figure 3a, it can be observed that the amplitude gain of R_f(z) with the Forward Euler discretization method at the fundamental frequency is approximately 3.3 dB lower than that of R(s), and its phase response exhibits a significant deviation from R(s) at this frequency. In the frequency range from 60 Hz to 600 Hz, the phase deviation remains within 5 deg. From 600 Hz onward, the phase of R_f(z) lags increasingly compared to that of R(s).

The transfer function R_b(z) with the Backward Euler discretization method demonstrates the greatest amplitude gain discrepancy compared to R(s) at the fundamental frequency, specifically around 11 dB lower, as illustrated in Figure 3b. The phase plot of R_b(z) also differs from R(s) around the fundamental frequency; specifically, at 50 Hz, R_b(z) leads R(s) by 0.18 degrees. As the frequency increases, this phase lead becomes more significant. It can be said that although the discretization of 11dB of R(s) with the Backward Euler method is quite straightforward, the magnitude attenuation of R_b(z) can pose a risk of introducing steady-state errors, negatively impacting the output signal’s quality.

When examining Figure 3c–e, no significant differences in magnitude response at the fundamental frequency are observed. However, R_t(z) with the Tustin discretization method exhibits a phase lag of 0.18 degrees compared to R(s), and R_zoh(z) with the Zero-Order Hold discretization method shows a lag of 0.44 degrees, while R_imp(z) with the Impulse-Invariant discretization method does not show substantial deviation from R(s). At the 11th harmonic, the discrete transfer function obtained using the Tustin method shows a resonance peak shift of approximately 1.5 Hz and a phase lag of around 28 degrees compared to the continuous-domain transfer function. Although the Zero-Order Hold method does not introduce a resonance peak shift, R_zoh(z) exhibits a magnitude attenuation of 0.01 dB and a phase lag of 5 degrees relative to R(s). The Impulse-Invariant method also does not exhibit significant differences between the continuous and discrete transfer functions. It is worth noting that although the Bode plots of the discretized transfer functions with Tustin, Zero-Order Hold, and Impulse-Invariant methods can offer negligible amplitude attenuation and small phase deviations at the fundamental frequency. Practical challenges may arise from their complicated transfer functions; for example, the trigonometric functions in the discretized transfer functions with the Zero-Order Hold and Impulse-Invariant methods usually introduce a greater computational burden.

3.2. Discretization Using Discrete Integrators

Apart from the aforementioned discretization of the continuous-time complete transfer function, transfer function G_PR(s) can be converted from the s-domain to the z-domain by discretizing two integrators contained within R(s) [24], as shown in Figure 4a. This approach is simpler and more straightforward to implement as it does not require complicated transformations or computations.

Various implementation methods have been introduced to discretize the direct and feedback integrators in Figure 4a. Based on [26], the schemes Forward Euler and Backward Euler, Backward Euler and Backward Euler plus delay, and Tustin and Tustin, of which their applications to non-ideal PR controllers have not been available in the literature so far, are chosen in this section for analysis and evaluation.

The first recommendation is to discretize the direct integrator in the scheme of Figure 4a using the Forward Euler discretization method and the feedback integrator using the Backward Euler discretization method, as shown in Figure 4b. Additionally, both integrators can be discretized using the Backward Euler discretization method, with a one-step delay added to the feedback line, as illustrated in Figure 4c. For the last recommendation, the Tustin discretization method can be applied to both integrators, as shown in Figure 4d [26]. It is worth mentioning that employing the Tustin discretization method for both integrators may introduce implementation challenges due to the presence of algebraic loops [26]. To address this issue, the mathematical expression of the relationship between the variables in the diagram in Figure 4d needs to be reformulated, as provided in Appendix B.

The discretized transfer functions corresponding to these methods in Figure 4b–d are shown as follows, respectively:

$$R_{fb}(z)=2{\omega }_c{T}_s{\displaystyle \frac{z-1}{z^2+z({\omega }_o^2{T}_s^2+2{\omega }_c{T}_s-2)+(1-2{\omega }_c{T}_s)}}$$

(3)

$$R_{bb}(z)=2{\omega }_c{T}_s{\displaystyle \frac{z^2-z}{z^2(1+2{\omega }_c{T}_s)-z(2{\omega }_c{T}_s+2-{\omega }_o^2{T}_s^2)+1}}$$

(4)

$$R_{tt}(z)=4{\omega }_c{T}_s{\displaystyle \frac{z^2-1}{z^2(4+4{\omega }_c{T}_s+{\omega }_o^2{T}_s^2)+z(2{\omega }_o^2{T}_s^2-8)+(4-4{\omega }_c{T}_s+{\omega }_o^2{T}_s^2)}}$$

(5)

It can be observed from Equation (5) that with the discretization of two integrators using the Tustin method illustrated in Figure 4d, the discretized transfer function $R_{tt}(z)$ remains identical to that obtained by discretizing the continuous transfer function with the Tustin method $R_t(z)$, as mentioned in Table 2.

Figure 5a,b compare the frequency response of a resonant controller R(s) designed for the fundamental frequency, using the discretization methods presented in Figure 4b,c at ${\omega }_o$ = $100\pi$ (rad/s), ${\omega }_c$ = 1 (rad/s), and $f_s$ = 20 kHz.

As shown in Figure 5a,b, both methods do not exhibit any significant differences in the amplitude response at the fundamental frequency. In terms of phase response, the R_fb(_z) with the Forward Euler and Backward Euler discretization method produces a phase lead of 0.14 degrees relative to R(s), while the R_bb(z) with the Backward Euler and Backward Euler plus delay discretization method introduces a phase lag of 0.32 degrees. At the 11th harmonic, both methods exhibit a resonance peak shift of approximately 0.7 Hz. At this resonance frequency, R_bb(z) leads R(s) by approximately 10 degrees, whereas R_bb(z) lags R(s) by 19 degrees.

Discretizing the integrators using the Zero-Order Hold method yields similar results to the Forward Euler method’s results, while the Impulse-Invariant method produces results identical to those of the Backward Euler method [24]. Therefore, the Zero-Order Hold and Impulse-Invariant discretization methods are not considered in the analysis above nor in Figure 5.

From Table 2 and Equations (3) and (4), the pole loci of transfer functions corresponding to various discretization methods are illustrated in Figure 6.

The pole locations of the discretized transfer functions in Table 2 and Equations (3) and (4) are illustrated in Figure 6. The Forward Euler method maps the poles outside the unit circle, while the Backward Euler method maps them toward the origin, resulting in a damping factor different from zero [24]. As the frequency increases (e.g., the 11th harmonic), the poles shift further away from the unit circle. These effects lead to further amplitude attenuation. Due to these limitations, the Forward Euler and Backward Euler methods are not considered in the three-phase four-wire system simulation presented in Section 4. In contrast, the Tustin, Zero-Order Hold, Impulse-Invariant, Forward Euler and Backward Euler, and Backward Euler and Backward Euler plus delay methods offer poles close to the unit circle, which corresponds to a near-zero-damping factor [24].

4. Simulation Results

To validate the correctness of the above analysis, the discretized non-ideal PR controller was implemented for the output voltage regulation of a three-phase four-wire converter in an electric vehicle charging station operating in the V2H mode.

Conventional 3P4W DC/AC converters often use a split DC-link capacitor to balance the neutral-point voltage [27]. While effective, this requires large capacitance [27]. Another approach adds a fourth leg to control the neutral current [28], reducing the capacitor’s size but increasing control complexity. To address these drawbacks, the 3P4W converter with an Independently Controlled Neutral Module (ICNM) is introduced. It enables efficient neutral-point voltage control without affecting output voltage regulation.

The configuration of a 3P4W converter with ICNM is shown in Figure 7, in which, besides the conventional three-phase three-wire inverter, an additional leg with split capacitors and a neutral inductor forms the ICNM—indicated by the blue dashed frame. This design decouples neutral-point voltage control from output voltage control, improving system flexibility and performance.

This paper focuses solely on achieving symmetric sinusoidal phase-to-phase voltages. Therefore, the control of the ICNM will not be addressed. The control structure for voltage generation, illustrated in Figure 7, comprises an external voltage control loop and an internal current control loop. In this control strategy, the primary objective is to control the inverter to deliver balanced three-phase load voltages with low harmonic distortion [29]. To achieve this, the voltage loop employs a non-ideal PR controller to maintain the desired output voltage, while the internal current loop uses a simple proportional controller (P controller) to generate the PWM reference signal [29]. In this example, the aforementioned discretization methods are used for the non-ideal PR controller in the voltage control loop highlighted in yellow in Figure 7. The simulations were conducted in the MATLAB/Simulink environment.

Table 3. Controller parameters for 3P4W converter with ICNM.

Parameters	Value	Parameters	Value
Kpu	0.37	${\omega }_c$	1 rad/s
Kru	150	Kpi	78
${\omega }_o$	$100\pi$ rad/s

lists the system’s control parameters, where K_pu and K_ru are the proportional and integral gains of the non-ideal PR controller, respectively, and K_pi is the proportional gain of the P controller. These control parameters are calculated based on the methodologies presented in [16,29]. They are designed such that the system’s bandwidth f_bv falls within the range of $10f_o\le f_{bv}\le f_{sw}/10$. The objective is to ensure the system’s stability while achieving sufficiently fast dynamic responses. The 3P4W converter is connected to a nonlinear load, which is framed by the red dashed line in Figure 7, and its parameters are listed in

Table 4. System parameters.

Parameters	Values
Phase-A load	Ra = 20 $\Omega$, La = 20 mH
Phase-B load	Rb = 20 $\Omega$, Lb = 20 mH
Phase-C load	Rc = 20 $\Omega$, Lc = 20 mH
Inductor of LC filter	Lf = 0.4 mH
Capacitor of LC filter	Cf = 20 $\mathsf{\mu }\mathrm{F}$
Split-capacitor of ICNM	C1N= C2N = 1000 $\mathsf{\mu }\mathrm{F}$
Neutral inductor of ICNM	LN = 2.8 mH
DC voltage	VDC = 650 V
Switching frequency	fsw = 20 kHz

.

The simulation results with the utilization of non-ideal PR controllers discretized based on equations in Table 2 under the steady-state operating conditions of the three-phase four-wire system, are shown in Figure 8, Figure 9 and Figure 10. At t = 4 s, the phase-A load is disconnected from the system, causing the phase-A current to drop to zero.

Figure 8, Figure 9 and Figure 10 illustrates the voltage regulation performance of the 3P4W converter with ICNM with the use of a non-ideal PR controller discretized by the discretization methods discussed in Section 3.1. Signal ${\delta }_A\left(t\right)$ represents the input to the non-ideal PR controller for phase-A, while signal ${\gamma }_A\left(t\right)$ denotes its output, as shown in Figure 7.

As shown in Figure 8b, Figure 9b and Figure 10b, the discretization of non-ideal PR controllers using the Tustin, Zero-Order Hold, and Impulse-Invariant methods all preserve a sinusoidal output voltage waveform with a THD of approximately 4%. The corresponding output currents exhibit THDs of approximately 15%, as illustrated in Figure 8c, Figure 9c, and Figure 10c. Moreover, the Bode plots of R(s) discretized based on these discretization methods show no significant deviation from those in the continuous-time domain. In particular, the output signal ${\gamma }_A\left(t\right)$of the non-ideal PR controller effectively amplifies the input signal ${\delta }_A\left(t\right)$at the fundamental frequency of 50 Hz and remains in phase with phase-A’s voltage. The Forward Euler and Backward Euler methods are not considered in this analysis, as previously discussed in Section 3.2.

The simulation results for the discretization of the non-ideal PR controller using two discrete integrators under the steady-state operating conditions of the three-phase four-wire system are shown in Figure 11, Figure 12 and Figure 13. At t = 4 s, phase-A’s load is disconnected from the system, causing phase-A’s current to drop to zero.

Figure 11, Figure 12 and Figure 13 illustrate the voltage regulation performance of the 3P4W converter with ICNM with the employment of non-ideal PR controllers discretized using the two discrete integrators introduced in Section 3.2. Based on the simulation results, the output voltage maintains a sinusoidal waveform at 50 Hz. The controller’s output signal ${\gamma }_A\left(t\right)$ remains in phase with phase-A’s voltage. Moreover, these discretization methods yield voltage and current THD levels comparable to those obtained by the approaches discussed in Section 3.1.

Table 5. THDs of phase-A voltage and phase-B current of a 3P4W converter with ICNM corresponding to different discretization methods.

Method	Voltage THD (%)	Current THD (%)
Tustin	4.10	15.18
Zero-Order Hold	4.02	15.26
Impulse Invariance	4.07	15.36
Forward Euler and Backward Euler	4.06	15.18
Backward Euler and Backward Euler plus delay	4.01	15.27
Tustin and Tustin	4.07	15.22

summarizes the THDs of the phase-A voltage and phase-B current of the three-phase four-wire system corresponding to the different discretization methods in Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. For the primary objective of achieving the lowest three-phase output voltage THD, coupled with the ease of experimental implementation, the Backward Euler and Backward Euler plus delay discretization method is recommended.

5. Conclusions

This paper has provided a detailed analysis of the discretization process for the non-ideal PR controller employed in the 3P4W converter with ICNM for V2H applications, in which precise phase-voltage control is essential. The continuous-time complete transfer function of the integral term of the non-ideal PR controller was discretized using various methods, such as Forward Euler, Backward Euler, Tustin, Impulse-Invariant, and Zero-Order Hold methods. The alternative approach for discretization using two discrete integrators for the non-ideal PR controller, such as Forward Euler and Backward Euler, Backward Euler and Backward Euler plus delay, and Tustin and Tustin methods, was shown as well. In the MATLAB/Simulink platform, through evaluating the voltage regulation performance of 3P4W converters with ICNM with the use of a non-ideal PR controller discretized by various discretization methods, it is illustrated that, among the examined methods, the Backward Euler and Backward Euler plus delay discretization method can produce the lowest THDs of phase-A’s voltage, i.e., 4.01%, under operating conditions where there are significant and rapid changes in nonlinear loads. Therefore, this method, i.e., Backward Euler and Backward Euler plus delay, is recommended. To further validate the effectiveness of these discretization methods in generating high-quality three-phase voltages for the 3P4W converter with ICNM, an experimental system will be set up for future studies. In this experimental system, the nonlinear loads specified in this paper will be considered, and the mentioned discretization methods will be evaluated in terms of voltage THDs, computational load, and other relevant performance metrics.