Improved Dynamic Performance of Average-Value Modelled Active Front-End Rectifiers

Abstract

Active front-end (AFE) rectifiers have become widely employed in power systems to achieve unity power factor and harmonic mitigations. The typical modeling approaches applied for AFE rectifiers in the literature mostly relied on two baselines: the detailed model and the time-average model. The former approach deals with the switching element model (SEM), which leads to significant harmonics in currents with distorted waveforms. The latter approach uses the average-value model (AVM) to overcome the currents’ harmonics as well as provide fast responses. However, even the AVM baseline has shown problems during the starting stage (lack of control signals) and over the dead-time periods, which causes serious issues in the implementation process. This paper presents an improved dynamic AVM for AFE rectifiers by precisely considering the issues mentioned above, along with the practical starting procedure and desirable initialization. The studied AFE rectifier is developed using the voltage-oriented control (VOC) technique based on the different modeling methodologies, including SEM, Conventional AVM, and the proposed AVM. The performance of all models is analyzed and compared using simulation results with MATLAB/Simulink R2023a Function blocks for all the algorithm parts and SimScape elements for the electrical circuit model. The simulation results illustrate that the performance of the proposed AVM approach can closely resemble the behavior of the SEM baseline with low harmonic distortion. To evaluate the performance of the proposed model, several case studies are investigated to verify the AFE rectifier operation, regarding mostly the total harmonic distortion (THD) wherein the THD percentages are improved to 4.78 and 2.5 from 5.14 and 2.78 for low- and high-power loads, respectively.

1. Introduction

The power systems have undergone a huge transformation due to the development of power converters and renewable energy resources [1]. Power quality challenges, the widespread employing of high-order filters, and the overall decrease in the impact of the power grid due to distributed generation are the main changes that have caused this transition [2]. To address these challenges, many approaches have been investigated to guarantee the proper control and operation of power systems equipped with power converters. Among these solutions, smart transformers, also called power electronic transformers, have been widely applied to interface medium- and low-voltage systems through fully controllable power electronic devices [3]. Similarly, power electronic transformations have played a key role in the onboard systems for transportation applications, aircraft, and shipboard [4,5]. Active rectifiers, as a first stage in those applications, have become the most attractive approaches towards smart power systems.

Active front-end (AFE) rectifiers have been widely used in medium-to-high-power adjustable speed drives, high-voltage DC systems, electric vehicles, and traction battery chargers to achieve regenerative operation and meet energy efficiency and harmonic requirements. In many applications where improved harmonic distortion is demanded, conventional front-end devices have been replaced by AFE rectifiers [6]. In addition, the AFE rectifiers can improve the reactive power balance by operating at a near unity power factor or by compensating the inductive/capacitive impact of other loads in the vicinity. Moreover, the fast transient feature of the new generation of AFE rectifiers enables superior ride-through capability during system disturbances.

Several marine vessels have utilized DC power systems for their simple controllability, high efficiency, and high reliability based on AFE rectifiers and synchronous generators [7]. AFE rectifiers have been employed in high-power electric vehicle (EV) charging stations, and those vehicles are promising future transportation systems. AFE rectifiers reduce the distortions produced via the operation of the EV charging infrastructures by minimizing harmonic disturbances and operating close to the unity power factor (PF) [8,9]. In [10], an AFE rectifier with a PF correction (PFC) capability is proposed for power quality improvement in telecom applications using a sensorless voltage control strategy to tackle harmonic issues and boost overall performance. Modular multilevel converter-based AFE rectifier topology is studied in [11] under unbalanced power grid conditions to improve the steady-state performance of the system that feeds the variable-speed motor drive. Variable-speed motor drives are a massive part of the grid-connected loads in distribution networks that use either conventional diode bridge rectifiers along with inverters or AFE rectifiers to adjust the operating frequency [12]. Recently, modular power converters with AFE rectifiers have been extensively utilized by motor drives to improve their harmonic characteristics and raise the overall power quality. In [13], a modular AFE rectifier with a common DC link is extended for multidrive configuration to mitigate harmonic distortions based on the electronic phase shifting technique. To elevate the modular AFE rectifier behavior, it is useful to separate the rectifier module from the drive unit and consider it as the main interface circuit between the AC power grid and the motor drive system’s DC link and deploy a robust and feasible control strategy.

The typical control techniques developed for the AFE rectifiers are voltage-oriented control (VOC) [14], virtual flux-based strategies [15], and direct power control (DPC) [16]. Several other control approaches have been proposed for the AFE converters, including predictive hybrid pulse–width modulation (PWM) [17], predictive duty cycle control [18], multicarrier-based PWM [19], synchronous space vector modulation [20], proportional–integral (PI) controller with anti-windup scheme [21], and digital control with multiloop tuning approaches [22]. Dynamic transient simulation of AFE rectifiers is a powerful item widely developed by many researchers to evaluate the effect of switched loads on power grids. Although several transient simulation tools with extensive elements can be used to develop a detailed switching model of the AFE converter, such detailed schemes contain heavy computational burdens and may cause long simulation times for large distribution networks with a large number of interfaced loads. Dynamic modeling, particularly the average-value model (AVM), is a powerful method that averages or mitigates the high-order harmonics of the AFE rectifier while maintaining the capability to precisely estimate the system’s dynamic behavior [23]. Dynamic AVM approach, as a precise alternative to the switching PWM models, has been widely developed for the line-commutated AFE rectifiers, wherein slower (non-switched) dynamic functions are acquainted via related AC and DC sides variables based on their average-value equations instead of the individual discrete switching functions [24,25,26,27,28,29]. Consequently, the AVM approaches are efficient and fast for system-level simulation studies. An AVM-based line-commutated AC-DC converter is investigated in [30] when the AC power grid is in unbalanced conditions. The studied AVM methodology relies on the system parameters to achieve fast simulations for the converter. The AVM approaches can be divided into analytical and parametric schemes, wherein the parametric AVM includes high accuracy and is very promising for electromagnetic transient simulation studies.

For AVM of AFE rectifiers that operate based on conventional PWM strategies, the models are typically performed by rotating reference frame operators [31,32,33]. In [34], the double Fourier series of PWM was studied to figure out the dead-time effect by inserting a correction value. These results extended to clarify the dead-time impact on the conventional PWM signals [35]. However, all these approaches can only be performed in a switching function baseline and cannot be employed in a time-average model. In [36], an improved dynamic model of an AFE converter is studied using the calculation of the duty cycle impact and considering the distortion by modeling the voltage drop on the semiconductor switching devices as an error of the duty cycle. However, those methods ignore the effect of duty cycle distortion over the zero crossing points of the current when they are being altered to null. Moreover, this voltage drop on the switching devices does not directly distort the DC link current and, consequently, should not be considered a duty cycle distortion. Another technique presented in [37] for modeling a motor drive is adding the effect of the dead time to the modulation space vectors; however, this approach did not consider the impact of voltage drops on switching devices. The overall dead-time definition procedure, as well as its impact on output current and voltage distortions, are comprehensively explained in [38]. The definition in [38] is the well-known two-level distortion model based on the duration of the switch’s conduction mode within one period. The study [39] extended the conventional definition of the two-level dead-time distortion approximation and improved it to a three-level distortion model by dividing the duty cycle of phase output voltage into three separate levels based on the maximum peak ripple of phase currents along with the switching frequency amount. However, it still suffers from the zero-crossing harmonics in currents.

This article provides improved AVM (IAVM) for the AFE rectifiers based on the VOC approach to enhance the performance of the time-average simulation using several dynamic case studies, including startup and load variations. To validate the accuracy of the studied time-average approach, the result of the proposed IAVM simulation was compared with the standard AVM (SAVM) and the switching element model (SEM) simulations based on the same control strategy. While some SAVM-based AFE rectifiers have been developed to overcome the aforementioned issues related to the converters, some challenges have remained unsolved, such as starting stage without command signals from the controllers, the impact of inrush currents, the effect of dead-time distortion, and fully considering the voltage drop on switching devices. To address these issues, an improved AVM distortion model is proposed with the following contributions:

• Practical SEM, SAVM, and IAVM simulation models are built for the AFE rectifiers;
• The AFE rectifiers are initialized in proper ways to implement the startup procedure in the absence of command control signals;
• The inrush current is limited during the startup period using a pre-charge circuit;
• A robust closed-loop control strategy is proposed to keep the unity power factor;
• The dead-time impact on the rectifier is improved by the proposed advanced AVM;
• The effect of switching devices’ voltage drop is accurately considered in the models;
• The operations of the models are compared to validate the accuracy of the proposed IAVM performance.

Section 2 describes different topologies for the three-phase AFE rectifiers. Section 3 discusses the proposed time average model for the AFE topology as well as its VOC implementation. Section 4 illustrates the simulation results of the case studies. Section 5 concludes this research work and provides some outlooks on AFE rectifiers.

2. AFE Rectifier Models

As explained in Section 1, three-phase rectifiers are widely utilized in power system applications as front-end converters because of their reliability and simplicity. Therefore, this section explores the mathematical model of the AFE rectifiers with both switched models based on PWM switching strategies and a dynamic time-average model.

2.1. Switched (PWM) Model

Figure 1 depicts the physical topology of the three-phase AFE rectifier, i.e., the SEM model, based on the VOC scheme. The control strategy can be simulated by software-based function modules and then implemented by hardware tools in digital processors with related requirements, including analog-to-digital (A/D) and digital-to-analog (D/A) converters. The dynamic performance of the AFE rectifier topology is practically measured by a dynamic signal analyzer to verify the responses of the rectifier. As shown in Figure 1b, an AFE rectifier, regardless of the switching or average-value model, utilizes a cascaded control approach with outer and inner closed loops. The VOC scheme, as one of the common cascade control methods, uses the outer control loop to regulate the rectifier DC link voltage value via a classic PI controller. The output of the DC link voltage controller provides the reference of the d-axis component of the current vector, which is required to maintain the AFE rectifier output DC voltage at a desired value. To achieve unity PF, another control loop is required to keep the q-axis current at zero. Accordingly, these DC signals, as d- and q-axis current references, are delivered to the inner voltage control loops to regulate the rectifier’s voltage vectors in the d–q axis.

In the SEM approach, the active semiconductor switches of the AFE rectifier are considered to be in on/off states to connect or disconnect the DC link voltage to the output load based on the controlled time discontinuous signals. The AC side dynamic requirements of the AFE rectifier with the SEM approach can be developed by Kirchhoff’s voltage law (KVL) equations [40] as follows:

$$\left[\begin{array}{l}{V}_{sa}\\ V_{sb}\\ V_{sc}\end{array}\right]=L_s\frac{d}{dt}\left[\begin{array}{l}{i}_{sa}\\ i_{sb}\\ i_{sc}\end{array}\right]+\left[\begin{array}{l}{S}_a\\ S_b\\ S_c\end{array}\right]V_{dc}-\left[\begin{array}{l}{v}_{nN}\\ v_{nN}\\ v_{nN}\end{array}\right]$$

(1)

$$v_{nN}=\frac{1}{3}(S_a+S_b+S_c)V_{dc}$$

(2)

$$S_a{i}_{sa}+S_b{i}_{sb}+S_c{i}_{sc}=C_{dc}\frac{d{V}_{dc}}{dt}+i_L$$

(3)

where $V_{sa}$, $V_{sb}$, and $V_{sc}$ are the AC side phase voltages, and $i_{sa}$, $i_{sb}$, and $i_{sc}$ represent the line currents. $S_a$, $S_b$, and $S_c$ denote the switching states of the upper switching devices, i.e., $S_{aH}$, $S_{bH}$, and $S_{cH}$ based on Figure 1a, which are equal to 1 when the switch is closed and 0 when the switch is open. The lower switches of the rectifier legs, i.e., $S_{aL}$, $S_{bL}$, and $S_{cL}$ are switched complementarily with a specific amount of dead-time applied to gating signals of the upper and lower switches in the same leg. The DC link voltage and capacitor are denoted by $V_{dc}$ and $C_{dc}$, respectively, and $i_L$ is the rectifier output current.

To employ the VOC scheme, Equations (1) and (3) are transformed into the d–q reference frame as follows:

$$\frac{d}{dt}{i}_q=\frac{1}{L_s}(V_q-S_q{V}_{dc}-{\omega }_s{L}_s{i}_d)$$

(4)

$$\frac{d}{dt}{i}_d=\frac{1}{L_s}(V_d-S_d{V}_{dc}+{\omega }_s{L}_s{i}_q)$$

(5)

$$\frac{d}{dt}{V}_{dc}=\frac{1}{C_{dc}}\left[\frac{3}{2}(S_q{i}_q+S_d{i}_d)-i_L\right]$$

(6)

where $V_d$ and $V_q$ represent the phase voltages in the d–q frame, $i_d$ and $i_q$ are the line currents in the d–q frame, $S_d$ and $S_q$ are the switching states in the d–q frame, and ${\omega }_s$ is the power grid angular frequency synchronized to the utility by a software phase-locked loop (PLL).

To acquire the desired DC link voltage $V_{dc}$, the switching states $S_d$ and $S_q$ should be determined properly, then passed on three-phase values and transformed into the space vector PWM (SVPWM) module. To achieve this goal, according to Figure 1b, the power grid voltages in the d–q axis are added to the output signals of the current controllers $V_q^{*}$ and $V_d^{*}$, along with decoupling terms to provide reference switching states, $S_q^{*}$ and $S_d^{*}$, as follows:

$$S_q^{*}=\left(V_q-{\omega }_s{L}_s{i}_d-V_q^{*}\right)/V_{dc}$$

(7)

$$S_d^{*}=\left(V_d+{\omega }_s{L}_s{i}_q-V_d^{*}\right)/V_{dc}$$

(8)

It should be noted that the VOC scheme requires three AC current sensors, two AC voltage sensors, and one DC voltage sensor. In addition, a PLL is demanded to attain the power grid voltage angle, which is used in the Parke transformation.

2.2. Dynamic Average-Value Model

The standard dynamic average-value model, i.e., SAVM, of the AFE rectifier includes controlled current sources to replace the switching elements and modulation indexes ranging from 0 to 1 instead of switching states. The dynamic model of the AFE rectifier under the synchronous reference frame can be obtained by replacing the time-discontinuous switch states $S_d$ and $S_q$ with time-continuous functions named modulation indexes $M_d$ and $M_q$ in Equations (4)–(6) as follows:

$$\frac{d}{dt}{i}_q=\frac{1}{L_s}(V_q-M_q{V}_{dc}-{\omega }_s{L}_s{i}_d)$$

(9)

$$\frac{d}{dt}{i}_d=\frac{1}{L_s}(V_d-M_d{V}_{dc}+{\omega }_s{L}_s{i}_q)$$

(10)

$$\frac{d}{dt}{V}_{dc}=\frac{1}{C_{dc}}\left[\frac{3}{2}(M_q{i}_q+M_d{i}_d)-i_L\right]$$

(11)

where these time-continuous modulation indexes are, namely, the duty cycles of the phases’ switching devices and can be expressed as follows:

$$M_x=\frac{1}{T_{sw}}{\displaystyle {\int }_{t-T_{sw}}^t{S}_x(\tau )d\tau } , x\in \{a,b,c\}$$

(12)

wherein $T_{sw}$ is the one switching period.

Figure 2 shows the overall block diagram of the SAVM for the AFE rectifiers, along with their controller structure. The dynamics explained in Equations (9)–(11) are developed on the right-hand side of Figure 2b with the SAVM of the AFE rectifier block diagram. The DC link voltage regulation and the d–q axis currents’ controls are implemented by the VOC block diagrams illustrated on the left-hand side of Figure 2b. The difference between the desired DC link voltage $V_{dc}^{*}$ and the actual $V_{dc}$, as an error command, is fed into a PI regulator to control the DC link voltage using the proportional gain ($K_{pv}$) and integral gain ($K_{iv}$). To maintain the DC link voltage at a desired level as well as receive power from the AC grid at unity PF, the output of the DC voltage PI regulator is multiplied by the grid voltages in the d–q axis to accordingly produce the current references in the d–q terms $I_d^{*}$ and $I_q^{*}$, which are in perfect match with the grid phase currents. To accurately consider the load impact on the dynamic model, the load current $i_L$ is multiplied by a load coefficient $K_L$, which is determined by the type and magnitude of the output load, and then applied to the q-axis current control loop as a feedforward term. The current control loops employ the proportional (P) regulator with a gain $K_{pi}$ to generate the modulation indexes in the d–q axis $M_d$ and $M_q$ as the switching duty cycles as expressed below:

$$M_q=\left(V_q-{\omega }_s{\widehat{L}}_s{i}_d-(I_q^{*}-i_q)\right)/V_{dc}$$

(13)

$$M_d=\left(V_d+{\omega }_s{\widehat{L}}_s{i}_q-(I_d^{*}-i_d)\right)/V_{dc}$$

(14)

where ${\widehat{L}}_s$ is the approximated value of the grid side filter inductor $L_s$, multiplied by the grid angular frequency and current terms to compensate the cross-coupling terms of ${\omega }_s{L}_s{i}_q$ and ${\omega }_s{L}_s{i}_d$ in the AFE rectifier dynamic model on the right-hand of Figure 2b. To perfectly decouple the current control loops as well as simplify the overall controller implementation, the ${\widehat{L}}_s$ value can be considered the same as the $L_s$ practical amount [41].

3. Proposed Improved AVM

As discussed earlier, the AVM-based AFE rectifiers provide fast simulation running time with less memory usage. However, using linear equations for modeling and integrating them with an ideal switching process leads to impractical voltage and current waveforms with zero distortions. Another drawback of the SAVM approach in comparison with the SEM baseline is the lack of the startup procedure simulation ability because it cannot offer the switching signals in the off status before enabling the controller module. To address those issues, an improved AVM, the IAVM, is proposed for AFE rectifiers to mitigate the dead-time impacts on total harmonic distortion (THD) of currents and precisely start the simulation with an auxiliary circuit model. Figure 3 illustrates the complete structure of the proposed IAVM, including an enhanced distortion model to reduce the dead-time harmonics and auxiliary circuit for the startup process. In this approach, the AFE rectifier is modeled as a controlled current source $I_{dc}$, which is calculated as

$$I_{dc}(t)=D_{abc}^T\times I_{sabc}=\left[D_a(t) D_b(t) D_c(t)\right] \left[\begin{array}{l}{i}_{sa}(t)\\ i_{sb}(t)\\ i_{sc}(t)\end{array}\right]$$

(15)

where $D_{abc}$ is the vector of the proposed enhanced phases’ duty cycles. Accordingly, the $V_{dc}$ can be obtained as

$$\frac{d}{dt}{V}_{dc}=\frac{1}{C_{dc}}\left(I_{dc}-i_L\right)$$

(16)

Each block of the proposed IAVM approach along with $D_{abc}$ will be clarified in the following subsections.

3.1. Enhanced Dead-Time Distortion Model

Dead-time of switching devices causes zero crossing distortion on phase currents’ waveforms of the three-phase AFE rectifiers due to its impact on the deviation of the duty cycle of phase voltages (namely $D_{abc}$) from the command duty cycles produced by controller output ($M_{abc}$ in Equation (12)). This deviation is a result of phase currents’ signs that compel rectifier leg output voltages to be equal to either DC link voltage or zero during dead-time ($t_d$) narrow intervals, correspondingly leading to different $D_{abc}$ from the command duty cycle $M_{abc}$ as follows:

$$D_x=\{\begin{array}{l}{M}_x-t_d/T_{sw} , i_{sx}>0\\ M_x+t_d/T_{sw} , i_{sx}<0\end{array} , x\in \{a,b,c\}$$

(17)

The duty cycle definition in Equation (17), as a conventional two-level distortion model, causes higher harmonics on phases’ currents, which leads to a high THD. To enhance the definition of the dead-time distortion, one effective solution is dividing the $D_{abc}$ into more than two general levels to tackle the current zero crossing distortions. This target can be achieved by considering a ripple current to allocate relevant duty cycles instead of zero. To estimate the ripple current ($I_p$) for each rectifier phase leg, the difference between peak-to-peak ripples ($\Delta I_{pp}$) is demanded to be precisely calculated based on the AFE rectifier parameters and variables as follows [42,43]:

$$\Delta I_{pp,x}=\frac{V_{dc}{T}_{sw}}{2L_s}\times \frac{M_x}{\sqrt{3}} , x\in \{a,b,c\}$$

(18)

here, the modulation index $M_x$ is considered in the range of 0 to $1/\sqrt{3}$ because of utilizing the SVPWM scheme instead of sinusoidal PWM. From $\Delta I_{pp}$, the ripple current $I_p$ for each phase current can be approximated as $\Delta I_{pp}/2$.

According to the ripple current, an improved duty cycle based on an enhanced dead-time distortion model, which is approximated in five levels, is defined to reduce the THD of IAVM as follows [43]:

$$D_x=\{\begin{array}{ll}{M}_x-T_d,& i_{sx}>I_p\\ M_x-0.5T_d,& 0.5I_p\le i_{sx}\le I_p\\ M_x,& -0.5I_p\le i_{sx}\le 0.5I_p\\ M_x+0.5T_d,& -I_p\le i_{sx}\le -0.5I_p\\ M_x+T_d,& i_{sx}<-I_p\end{array} , x\in \{a,b,c\}$$

(19)

where $T_d$ is the total time delay over the $T_{sw}$ for each switching period based on the IGBTs turn-on time ($t_{on}$) and turn-off time ($t_{off}$), which can be calculated as

$$T_d=(t_d+t_{on}-t_{off})/T_{sw}$$

(20)

To avoid the gradient transition between turning duty cycle levels ON and OFF, introduced in Equation (15) by active switching devices, an additional three levels are presented in this study using $\pm 0.5I_p$ to mitigate the dead-time distortion on currents. Since the dead-time intervals are too short in practice, these half-point values are feasible approximations. The studied five-level dead-time distortion model is illustrated in Figure 4 and integrates the ripple current calculator with the enhanced duty cycle model. A limiter is also added to the model to maintain the outputs of duty cycles $D_{abc}$ in a range between 0 and 1 [43].

3.2. Improved Startup Procedure

The practical startup circuit model, as the main part of this IAVM approach, is implemented to enhance the proposed IAVM performance by precisely taking into account the voltage drops of the switching devices (e.g., IGBTs and the freewheeling diodes) and employing a standard three-phase diode rectifier with an enable switch.

According to Figure 3, the total voltage drop ($V_{dr}$) of each phase leg of the AFE rectifier on the IGBTs ($V_S$) and diodes ($V_D$) is implemented as a controlled voltage source. To accurately calculate the $V_{dr}$ as a function of voltages duty cycles $D_{abc}$ and phase currents polarities, the forward voltages of the IGBT switches ($V_{fS}$) and diodes ($V_{fD}$) are required, along with their internal resistances $r_S$ and $r_D$, respectively. Consequently, the total voltage drop of each phase $V_{dr,x}$ can be calculated as [43,44]

$$V_{dr,x}=\{\begin{array}{l}{D}_x{V}_D+(1-D_x)V_S , i_{sx}>0\\ -D_x{V}_S-(1-D_x)V_D , i_{sx}<0\end{array}$$

(21)

where

$$\{\begin{array}{l}{V}_s=r_s\left|i_{sx}\right|+V_{fS}\\ V_D=r_D\left|i_{sx}\right|+V_{fD}\end{array}$$

(22)

As the last part of the IAVM approach, the three-phase standard diode rectifier is added, along with the enable switch (shown by the EN switch in Figure 3), to simulate the AFE rectifier before connecting to the VOC controller. The reason behind this is that during the AVM-based rectifier startup for real hardware test of AFE, there are no control signals to enable the IGBT switches ON and OFF. In other words, IAVM will short-circuit the three phases during startup. To avoid this issue, an enabled switch will disconnect the IAVM circuit before the VOC controller is initiated. As a result, the proposed IAVM baseline operation is similar to the SEM before the controller is connected and is verified by the simulation results in the following section.

4. Results

To verify the performance of the proposed methodologies, a grid-connected AFE rectifier system with a PFC module, whose nominal circuit parameters as well as controller gains are listed in

Table 1. Main parameters of the studied AFE rectifier.

AFE Parameters
IGBT Module	Vdc [V]	Vf (IGBT || D) [V]	rS || rD [mΩ]	Cdc [mF]	Tsw [s]	td [s]
Fs400R07A1E3	600	1.5	1	4.7	1 × 10−4	2 × 10−6
Ac Side Parameters
Vgrid (L-L) [Vrms]	Fs [Hz]	Ls [mH]	Cs [μF]	Vout [V]
380	50	10	5	600
Controller Gains
DC voltage controller			Kp = 2.6 || Ki = 20.8
d-axis current controller			Kp = 31.72 || Ki = 157.44
q-axis current controller			Kp = 31.72 || Ki = 157.44

, is considered for implementation. Simulation studies are conducted in MATLAB/Simulink R2023a environment to demonstrate the validity of the detailed switching model, along with the electrical circuit utilized in all mentioned approaches, including SEM, SAVM, and IAVM. The proposed IAVM baseline and SAVM are implemented by MATLAB S-Functions for all the blocks and the SimScape Tools for all electrical circuits employed in the SEM baseline. The tested AFE rectifier topology is shown in Figure 5, wherein a pre-charging relay is used for soft-starting the system.

Several case studies are implemented to compare the operation of the proposed IAVM with the other two algorithms in the conditions of startup and transient behaviors discussed in the following subsections.

4.1. Case 1: Startup Performance

In this study, the transient behavior of the IAVM is tested during the startup conditions under the zero load (open circuit output) to validate the starting procedure performance in comparison with the SAVM and SEM approaches. Figure 6 shows the simulation results under startup transient to start and bring the AFE rectifier DC link voltage from 0 to the desired value of 600 V with zero active power and, consequently, zero phase currents. The operation procedure of this case study includes four stages. During the first stage (from starting to 0.2 s, as shown in Figure 6a), the so-called pre-charging stage, the DC link capacitor $C_{dc}$ is slowly charged from 0 to a safe limit of around 80% of the peak value of input line to line grid voltage $\sqrt{2}{V}_{grid(L-L)}$ (around 440 V) through three pre-charge resistors for each phase. All IGBT devices are OFF during this stage, and pre-charging resistors in series with the boost inductors $L_s$ are connected to the output capacitor $C_{dc}$ via an auxiliary three-phase diode rectifier to limit the current. During the second stage, from 0.2 s to 0.3 s, the bypass relay 1 short-circuits the pre-charging resistors; then, the capacitor $C_{dc}$ is normally charged to $\sqrt{2}{V}_{grid(L-L)}$ using the inductors $L_s$ and the auxiliary diode rectifier. It should be noted that during these two stages, the VOC controller is not connected to the AFE rectifier; thus, control signals are not yet applied to the rectifier. When $C_{dc}$ the voltage reaches over 85% of the desired DC link voltage (600 V), the VOC controller is switched ON and starts the third stage. In the third stage, the auxiliary diode rectifier is disconnected, and the controller is connected to the AFE rectifier by considering the total voltage drop $V_{dr}$ in the IAVM algorithm. During this stage, the controlled d–q axis currents are restricted to 50 A for soft starting, and then at the fourth stage (at 0.35 s), the current upper limit is set to 80 A, allowing a nominal current to flow. According to Figure 6a, by considering this four-stage procedure, the DC link voltage can safely reach its desired value without producing a huge inrush current. The simulation waveforms of $V_{dc}$ for all three baselines closely match each other except for some differences among them during the third stage when the first current limiter is disabled. The simulation results of three-phase grid currents in Figure 6b illustrate the operational differences between the proposed IAVM and the other two approaches as current ripples during no-load conditions. It is observed that the IAVM-based AFE rectifier has lower ripples than the SAVM baseline and significantly lower than the SEM baseline. This is because of the development of enhanced duty cycles for the IAVM to reduce the zero crossing current ripples, as shown in Figure 6c in the upper waveform for the d-axis current.

To better reveal the studied startup performance differences, some simulations are carried out without startup stages, as shown in Figure 7. From Figure 7a, the phase current peak value is very high when there is no pre-charge relay; however, using the pre-charge relay limits the starting current at a lower level than the nominal current. In addition, without the pre-charging procedure, the controlled d–q-axis currents contain excessive overshoot and undershoot, which is more than 40 A for q-axis current undershoot.

4.2. Case 2: Load Responses

The AFE rectifier steady-state current waveforms for the three mentioned baselines are illustrated in Figure 8 for operating power of 3.6 kW and 7.2 kW. Despite Case 1, where the two AVM baselines have similar current waveforms, this case provides different current results, especially at lower power levels because of the longer range of the ripple current $\pm I_p$. From Figure 8a, for output power 3.6 kW, the current waveform for the SEM approach depicts significant ripples due to switching elements at the high switching frequency. For the SAVM approach, some ripples are still shown because of the conventional two-level duty cycle; however, the IAVM baseline shows better current waveforms where such significant ripples no longer exist. These variances are precisely shown in Figure 8c by their d-axis currents. Similar behaviors are also illustrated by the current waveforms at the output power of 7.2 kW, although the ripple distortions in this study are not as significant as in the previous study with the power level of 3.6 kW.

Table 2. Harmonic comparisons for the AFE rectifier models during the steady-state operation.

AFE Rectifier Model	SEM	SAVM	IAVM
THD% for 3.6 kW load	5.14	4.96	4.78
THD% for 7.2 kW load	2.55	2.78	2.5

summarizes the harmonic distortion results of this case study for both output power levels with THD %, reporting that the IAVM current waveforms include the lowest harmonic distortion for both power rates. It is depicted that at a high-power rate, the harmonic distortion is decreased dramatically for all the models, and THD evaluation is almost the same for the SEM and IAVM baselines. Therefore, the best performance of the proposed IAVM approach can be obtained for applications with low and medium power demands.

4.3. Case 3: Transient Performance

This study deals with performance comparisons between the physical SEM and time average model IAVM AFE rectifier during load changes and step transient conditions. Figure 9 shows the rectifier responses for the mentioned baselines during step changes in output resistive loads to ensure the proposed model’s smooth operation. For a fixed DC link voltage reference of 600 V, Figure 9a depicts that both baselines track the desired voltage, although the SEM contains more fluctuations than the IAVM. However, the main difference is illustrated by the controlled d–q axis currents, especially with d-axis current ripples, as shown in Figure 9d. Those significant fluctuations in the d-axis current for the SEM baseline cause more harmonic distortions as well as high switching losses.

Finally, to verify the unity PF, A-phase voltage and current are simulated for different load amounts using the IAVM baseline, as shown in Figure 10. It is clear that both voltage and current waveforms are in perfect angle match, which leads to proper PFC over low and high output powers.

5. Conclusions

This article presents a useful methodology to implement dynamic AVM of AFE rectifiers under VOC-based controllers. Unlike the SEM baselines, the use of AVM approaches can significantly decrease the computational burden (which is about 200 times faster than a SEM baseline) while maintaining the transient performance of these topologies. An enhanced AVM baseline is proposed by precisely considering the dead-time distortions and total voltage drops on switching devices during various operating circumstances. Based on the comparative studies, it has been illustrated that such an improved model includes fewer harmonic distortions and lower current ripples. A good part of the proposed model is the practical startup circuit, which leads to the smoothest operation of the AFE rectifier. In addition, it is shown that the IAVM has low THD because of employing the enhanced five-level duty cycle scheme by developing the conventional two-level dead-time distortion model with maximum peak ripple of phase currents. It is evaluated that the proposed IAVM can be an accurate alternative for the actual SEM-based AFE rectifier, which can be used in industries to develop software. Furthermore, the proposed system can be utilized for modern plug-in electric vehicle battery chargers because of its quick response and lower current distortions.