Modeling and Control Simulation of Power Converters in Automotive Applications

Abstract

This research introduces a model-based approach for the analysis and control of an onboard charger (OBC) system for contemporary electrified vehicles. The primary objective is to integrate the modeling of SiC/GaN MOSFETs electrothermal behaviors into a unified simulation framework. The motivation behind this project stems from the fact that existing literature often relies on finite element method (FEM) software to examine thermal dynamics, necessitating the development of complex models through partial derivative equations. Such intricate models are computationally demanding, making it difficult to integrate them with circuit equations in the same virtual environment. As a result, lengthy wait periods and a lack of communication between the electrothermal models limit the thorough study that can be conducted during the design stage. The selected case study for examination is a modular 1$\varphi$ (single phase) onboard computer (OBC). This system comprises a dual active bridge (DAB) type DC/DC converter, which is positioned after a totem pole power factor correction (PFC) AC/DC converter. Specifically, the focus is directed toward a 7 kW onboard computer (OBC) utilizing high-voltage SiC/GaN MOSFETs to ensure optimal efficiency and performance. A systematic approach is presented for the assessment and selection of electronic components, employing circuit models for the totem pole power factor correction (PFC) and dual active bridge (DAB) converter. These models are employed in simulations closely mimicking real-world scenarios. Furthermore, rigorous testing of the generated models is conducted across a spectrum of real-world operating conditions to validate the stability of the implemented control algorithms. The validation process is bolstered by a comprehensive exploration of parametric variations relative to the nominal case. Notably, each simulation adheres to the recommended operational limits of the selected components and devices. Detailed data sheets encompassing electrothermal properties are provided for contextual reference.

1. Introduction

1.1. Motivations

In recent years, the commercial panorama of electrified cars has experienced unparalleled growth. The increasing need for extremely efficient electric and hybrid cars is a modern challenge that is driving important advances in science. It is noteworthy that the creation of resilient electronic power flow systems has become a central area of interest for innovation [1,2]. For a variety of compelling reasons, it is crucial to use mathematical models and run simulations when designing in-vehicle electronics systems, such as power electronics, onboard mechatronics, electric drives, and power electronics. Design optimization: Using mathematical models and simulations, it is possible to assess several design configurations and choose the most promising one that will provide optimal performance in terms of effectiveness, dependability, and durability over an extended period of time. This approach can prevent expensive design mistakes and shorten the time needed to produce a product from start to finish [3,4].

Cost reduction: The adoption of mathematical models and simulations obviates the necessity of creating physical prototypes to test various design setups. Consequently, this leads to a substantial reduction in product development costs and expedites the time required to bring the product to the market [5,6,7]. Performance improvement: Power electronic converters may have their performance carefully assessed under a range of operating situations, including variations in temperature or load, by using mathematical models and simulations. This makes it possible to adjust converter control and guarantees top performance in a variety of operating conditions [8]. Safety assurance: Through the use of mathematical models and simulations, it is possible to evaluate power electronic converter behavior in the event of overload or failure. This opens the door for the installation of strong defenses, guaranteeing the security of both drivers and passengers [9,10,11,12,13,14]. The use of mathematical models and a model-based design (MBD) approach is essential to increase the performance of power electronic systems and to analyze the solution space, including studying the effects of the variation of circuit parameters, the effects of choices and simplified hypotheses in the design, and the robustness of control algorithms [15,16] against unwanted effects such as measurement noise or transduction errors [17,18]. In the literature, there are many concrete examples of the development and analysis of power electronic systems and electric drives for automotive applications and the mechatronics industry. For example, the authors of [19,20] use the MBD approach very extensively to improve the performance of the control of electric drives in the presence of the Cogging effect, analyzing the robustness and profiling the computational complexity in simulation. In [21,22], an analysis of the computational complexity of advanced control algorithms for power electronic systems based on SW MBD tools is proposed to ensure integration with real-time constraints in embedded platforms. In [23,24,25,26], the use of the MBD approach for the design and integration of mechatronic systems up to the implementation phases is proposed. In other works, the authors show how to use MBD and simulation environments massively, to compensate for the lack of experimental data and address battery health estimation problems. Works such as [27] show how simulations can create scalable models of SiC/GaN devices not yet on the market, combining experimental characterization measurements of such devices with complex and scalable models, to be integrated into more complex electronic systems. Essentially, the use of mathematical models and simulations in power electronic converter designs for automotive applications helps to ensure the highest level of user and vehicle safety while also improving product performance and reducing development costs [28,29,30,31]. During these advancements, the growing significance of accurately modeling and simulating complex systems has become evident. A comprehensive understanding of the intricacies inherent in electric battery charging is particularly crucial for the design of electric vehicles. Consequently, an exceedingly detailed model is presented, encompassing the electrothermal dynamics of an onboard computer (OBC) with a modular design [32,33,34,35]. Developing a precise simulation model for onboard electronic systems is a critical first step in creating innovative, effective gadgets. Model development is often oriented toward fine-grained attention on certain aspects, such as voltage and current behavior, thermal patterns, or the subtleties of component magnetism. Still, this method restricts the scope for a thorough analysis of important scenarios, and ‘destructive’ testing is, thus, required to investigate difficult operational combinations [36,37,38,39]. By utilizing a multifaceted model that encompasses many components, these constraints are addressed. While it is understood that not every physical component can be precisely represented using traditional software methods, finding a balance between model complexity and other factors is still crucial. This balance makes it easier to quickly and efficiently assess a variety of scenarios and operational circumstances [40,41,42,43]. Hence, the proposed operating protocol advocates the development of a simulation model that integrates two essential aspects: the circuit dynamics and the thermal conductivity of switching components. This approach aims to bridge the divide between comprehensive modeling and practical implementation, utilizing the 1$\varphi$ OBC circuit configuration as a case study.

This initiative is poised to usher in a new era of innovation in the realm of electronic power systems for modern automobiles.

1.2. Related Works

In [44], the description of a unidirectional onboard charger (OBC) configuration includes modeling and design aspects. This configuration incorporates a step-up DC/DC converter following an AC/DC diode rectifier stage to achieve power factor correction (PFC). The system connects to the vehicle’s battery through a phase-shifted full-bridge (PSFB) type DC/DC converter. The control algorithm relies on electrical circuit modeling to primarily define transfer functions. In contrast, the current work incorporates a model that integrates the complexities of a dual active bridge (DAB) circuit and a totem pole circuit into the control algorithm. Additionally, a detailed examination of the thermal behavior of SiC/GaN MOSFETs, a critical aspect absent from the cited paper, is included. In contrast, ref. [45] introduces an alternative control approach employing a step-up converter following a diode rectifier for the AC/DC conversion stage. However, their simulation experiments primarily concentrate on diverse operating scenarios. In contrast, the present study integrates the intrinsic properties of the switching components and conducts thermal evaluations during simulations, employing a more comprehensive modeling method for onboard chargers (OBCs). Subsequently, in [46], a thermal study of a ∼7 kW onboard charger (OBC) is conducted using a multi-stage output DC/DC converter and finite element method software. The circuit analysis is based on a rudimentary linear approximation, while the thermal analysis is thorough. In contrast to the benefits of the proposed model-based design (MBD) approach, the disadvantage of segregating dynamic and thermal simulations into separate software environments, resulting in heightened computing costs, remains significant, particularly for predictive diagnostics. Similarly, unlike the thorough method employed in this study, ref. [47] does not consider the thermal properties of the components and instead focuses on modeling the electrical behavior of the onboard charger (OBC) at lower power and voltage outputs. Furthermore, neither ref. [48] nor ref. [49] includes comprehensive modeling of the thermal behavior of electronic components, an essential aspect integrated into the suggested framework in this study; moreover, they do not provide specific design details of the OBC device. Similar to this, ref. [50] employs an MBD technique for the validation of control techniques in onboard chargers (OBCs). In contrast to the all-inclusive technique applied in this study, it solely addresses the analysis of electrical behavior, disregarding thermal behavior. Likewise, refs. [51,52] primarily concentrate on energy efficiency and simulations without incorporating a comprehensive model of the thermal behavior of electronic devices, another crucial aspect of the suggested work. Based on the literature study, it is evident that the majority of published research focuses on the dynamics of electrical quantities, ignoring the important connection to the thermal behavior of the SiC/GaN MOSFETs in the energy conversion stages. Furthermore, even energy-efficient works neglect to represent temperature elements, which are essential for confirming the physical boundaries of simulated electronic devices, including the transistors that make up electronic converters.

1.3. Authors’ Contributions

This work represents a significant advancement in the design of electronic power systems for electric and hybrid cars. Advanced methodologies have been introduced in the design of onboard chargers (OBCs), addressing the electrothermal behavior of SiC/GaN MOSFETs. Utilizing modern 900 V GaN and 1200 V SiC technologies, a comprehensive system has been developed to ensure optimal efficiency and performance in a 7 kW OBC environment. The systematic approach to electrical component selection has contributed to a deeper understanding of interactions between complex circuit models and realistic simulations.

The rigorous validation process, encompassing various operating scenarios and robustness evaluations, has yielded an innovative outcome, opening new perspectives in research and establishing a solid foundation for the future development of electronic power systems in the automotive industry.

2. Electrothermal Modeling and Control Algorithm

This section provides a detailed description of operational concepts and control algorithms in energy conversion and explains the OBC circuit being considered. A thorough evaluation of the thermal performance of the switching components is suggested to verify the physical limitations of the transistors simulated in Simulink/Simscape (MATLAB 2020b version in our work). The general configuration of the OBC circuit is shown in Figure 1. It should be noted that the suggested model-based design (MBD) technique is easily adaptable to different circuit topologies, including three-phase circuits, even with only one single-phase connection to the power supply.

2.1. AC/DC Converter Model and Control

The study examines the totem pole configuration, a specific type of AC/DC converter with PFC [53,54,55,56,57,58,59,60,61,62,63]. In Figure 2, there are two SiC transistors ($S_{D_1}$, $S_{D_2}$) and two GaN transistors ($Q_1$, $Q_2$). The SiC components rectify the current at line frequency, replacing traditional diodes, while the GaN components perform fast switching at frequencies above 100 kHz. The circuit operates in two modes, depending on the polarization of the AC line voltage, as shown in the schematic in Figure 3. The activation of $S_{D_1}$ and the deactivation of $S_{D_2}$ together with the GaNs define the positive AC half-cycle. The modulation logic handles the negative AC half-cycle, with $S_{D_2}$ active, by deactivating $S_{D_1}$ and activating the GaNs. This cyclic interaction between phases produces the desired PFC effect, aligning the current with the AC supply voltage and eliminating reactive power. The SiCs handle the rectification, while the GaNs orchestrate the switching to obtain the correct voltage on the DC bus. Figure 4 shows a schematic of the complex control system of a totem pole-type AC/DC converter. The control is divided into various functional blocks that are crucial for the operation of the system. A second-order low-pass filter (LPF) separates the in-line and 90-degree offset components of the AC voltage. A phase-locked loop (PLL) calculates the phase compensation required to balance the current in the totem pole input connection with the supply voltage. The feedback controller mediator corrects errors between the actual voltage, $V_o$, and the reference, $V_{ref}$, for the DC bus; it also handles the AC $I_{ac}$.

The modulation system, based on an inverse calculation mechanism, synchronizes the generation of the required AC voltage $V_{ac}^{*}$ and the phase compensation $\omega$ to ensure the effective operation of the system.

$$\begin{array}{c}{V}_Q=\left[2\left(\frac{{\widehat{\omega }}_g}{s^2+{{\widehat{\omega }}^2}_g}\right)\left(\frac{{\omega }_r}{s^2+{\omega }_r^2}\right)+\frac{s}{s^2+{{\widehat{\omega }}^2}_g}\right]V_g\\ \\ {\widehat{\omega }}_g=\left(P+\frac{I}{s}+D\frac{Ns}{s+N}\right)V_Q\end{array}$$

(1)

Figure 5 highlights the importance of the second-order low-pass filter to obtain a signal with a 90-degree phase shift relative to the source voltage. This phase shift allows the phase-locked loop (PLL) block to accurately calculate the frequency required to synchronize the main voltage and current in the totem pole, achieving the desired PFC effect. The accurate measurement of the voltage on the DC bus depends on the control logic of the voltage and current feedback. Next, Figure 6 illustrates the internal configuration of the PLL block and how it processes input signals, such as $\alpha$ and $\beta$ from the LPF block. The result of this complex block is translated into carefully calculated trigonometric functions related to the $\omega$ pulsation, which are fundamental for coordinating the complex dynamics of the system. Equation (1) shows the transfer functions of $V_Q$ and the estimated network frequency/pulse, which are the outputs of the PLL system.

The signal processing system is detailed in Figure 6. It is based on the Park transformation of the input signals to obtain components along the direct and quadrature axes. The goal is to cancel the quadrature component to achieve the desired PFC effect in the totem pole. A PID controller corrects any residual errors. This complex process identifies the impulse needed to control the current in the totem pole and, after integration, produces the angle for the Park transformation. The sine and cosine trigonometric functions are crucial in the feedback control block. Figure 7 depicts the control logic governing the voltage on the DC bus and the current in the totem pole. Employing PWM modulation concepts, the objective is to simulate the reverse engineering of an inverter designed for conventional DC-AC conversion but operating with active devices. The external control loop, supported by a PID system, stabilizes voltage errors by calculating the magnitude of the current at the totem pole. With the modulus and the cosine function, adapted to the AC voltage, the reference value for the internal current control is obtained. In the internal control, the proportional-resonant (PR) controller is preferred over the traditional PID to handle sinusoidal systems.

It combines proportional and resonant, offering good performance and simplicity of implementation, especially in active rectifiers, where standard PI/PID controllers may have limitations.

$${V_g}^{ref}=\left[\left(P+\frac{I}{s}+D\frac{Ns}{s+N}\right)\left({V_o}^{ref}-V_o\right)\frac{s}{s^2+{{\widehat{\omega }}^2}_g}-I_g\right]\left(K_p+\frac{K_{r}s}{s^2+{{\omega }_r}^2}\right)+V_g$$

(2)

Equation (2) offers a mathematical representation of the active rectifier controller, linked to the schematic illustration above. The $\omega$ resonant pulse plays a vital role by providing optimized gain precisely tuned to the resonant frequency. This crucial feature allows the proportional-resonant (PR) controller to effectively reduce steady-state recurring errors when following or canceling sinusoidal signals; this is consistent with the general principles of the internal model. Functionally, the PR controller’s output is carefully designed to counteract any disturbances encountered and meet precise DC voltage amplitude requirements. It is inserted as an additive or subtracted component with respect to the input AC voltage. A detailed analysis of the modulation block reveals a precise division into two parts, each of which serves the modulation needs of the circuit. The first part deals with pulse width modulation (PWM) methods, precisely regulating the branch composed of GaN-type MOSFETs. On the other hand, the second part generates a “high” signal that is precisely timed to coincide with the positive half-wave of the cosine signal, carefully calculated in the previous blocks.

2.2. DC/DC Converter Model and Control

The dual active bridge (DAB) converter, illustrated in Figure 8, is designed with two active bridges, one on the primary side and one on the secondary side of a high-frequency transformer [64,65,66,67,68,69,70,71,72,73]. Its operation can be described in the following steps:

1

Primary side (DC-AC conversion): The first inverter converts the direct voltage into a high-frequency alternating voltage. The duty cycle $D_1$ regulates the primary voltage $V_1^{sw}$. This voltage is applied to the transformer.

2

Power transfer: The high-frequency alternating voltage flows through the transformer, generating an alternating current on the secondary side and transferring the power through the inductance.

3

Secondary side (AC-DC conversion): The second inverter converts the high-frequency alternating current into direct voltage. The duty cycle $D_2$ regulates the secondary voltage $V_2^{sw}$.

4

Regulation and control: Duty cycles and phase shifts are adjusted to control power flow and output voltage. Control occurs through modulation, which determines the duration and synchronization of the work cycles.

Phase shift modulation is a key control technique in DAB, regulating the power transmission between the primary and secondary sides [74,75,76,77,78,79,80,81]. The process can be understood through the following concepts:

-

Phase and work cycles: The phase $\varphi$ represents the time shift between the voltages $V_1^{sw}$ and $V_2^{sw}$. The phase shift approach often requires that $D_1=D_2=D=0.5$, simplifying the control to just one parameter, $\varphi$.

-

Power equation: The power equation (see Equation (3)) relates the phase shift $\varphi$ to the power transfer between the primary and secondary sides. Maximum power transfer occurs when $\varphi =\pi /2$.

$$\begin{array}{c}P=\frac{n{V}_1{V}_2}{2{\pi }^2{f}_{sw}{L}_{tot}}\varphi \left(1-\frac{\varphi }{\pi }\right)\\ \\ {\varphi }^{*}=\frac{\pi }{2}\left(1-\sqrt{1-\frac{8P^{*}{f}_{sw}{L}_{tot}}{n{V}_1{V}_2^{*}}}\right)\end{array}$$

(3)

-

Advantages and limitations: Phase shift modulation is simple and requires only one control parameter. However, it has limitations such as a narrow working range and higher currents in the transformer.

The concept of the phase shifting modulation is illustrated in Figure 9. In summary, DAB offers flexible bidirectional power control through the transformer, with phase-shift modulation as the key mechanism for efficient control of power flow.

$$\begin{array}{c}{t}_{\varphi }^{*}=\left(P_v+\frac{I_v}{s}\right)\left[\left(P_i+\frac{I_i}{s}\right)\left({V_2}^{ref}-V_2\right)-I_2\right]\frac{T_{sw}}{2\pi }\\ \\ PW{M}_2=PW{M}_1(t-t_{\varphi }^{*})\end{array}$$

(4)

As schematized in Figure 10, the objective of the feedback control system is to automatically obtain the temporal shift between the modulation signal of the first stage $PW{M}_1$ and the one that manages the second conversion stage $PW{M}_2$. In particular, the control algorithm must manage the voltage request ${V_2}^{ref}$, and manage the current feedback; this process obtains the phase shift ${\varphi }^{*}$ between the primary and secondary voltage of the transformer. In Equation (4), the control expressions are explicitly reported.

2.3. Switching Device Thermal Modeling and Power Losses

The transition from the physical device of a power MOSFET to its equivalent circuit model, based on the Foster model for thermal modeling, involves the representation of the thermal layers that extend from the junction (J) to the surrounding environment (A) (as schematized in Figure 11). Imagine the MOSFET as a system of stacked thermal layers. The junction (J) is the internal region where heat is generated due to power losses. Surrounding it is the case layer (C), which is the body of the MOSFET that wraps around the junction. The case is connected to the heat sink (S), an external component that absorbs heat from the case. Finally, the heat sink is in contact with the environment (A), which is the surrounding temperature influencing the heat dissipation process. The Foster model simplifies this thermal complexity by using thermal resistors to connect the layers. Thermal resistances, such as $R_{thJC}$, $R_{thCS}$, and $R_{thSA}$, act as thermal bridges between layers, allowing heat to flow through the model, analogous to the flow of current in an electrical circuit. The conceptual step then involves assigning thermal resistances to the different thermal layers, reflecting the propagation of heat through the MOSFET during its operation. The Foster model equations describe the change in temperature in each of the layers in response to heat flow. The utilization of this equivalent circuit representation enables simulations to comprehend the thermal behavior of the MOSFET under diverse operating conditions. In summary, Foster’s model simplifies the intricate thermal behavior of the MOSFET, facilitating the analysis and simulation of device heating in real-world scenarios.

3. Technical Specifications and Components: Sizing and Selection

In this section, the rationale employed for selecting the values of passive and active circuit components is presented, referencing the design specifications for an onboard charger (OBC) device for electric vehicles. In this study, reference is made to the commercial solution presented by Infineon [82], and for completeness, the specific operational techniques are detailed. The technical specifications of the active rectifier are outlined in

Table 1. Active rectifier key specifications.

Parameters	Specifications
	Single Phase
	Voltage: $90–265V_{rms}$
Input	AC frequency: $f_g=50–60$ Hz
	Current: $32A_{rms}$@240 V
	Power Factor: ≥0.99
	PFC output: $400V_{dc}$ (typical)
Output	Max output power: 7.4 kW@400$V_{dc}$
	peak efficiency: ${\eta }_{max}=0.985$
	PFC stage for HV Lion Battery OBC
	switching frequency: $f_{sw}=120$ kHz
Performance	isolation: reinforced
	input AC sensing
	PFC output voltage sensing
	over-temperature protection
	short-circuit-protection
Protection	over-current protection
	under-voltage protection
	over-voltage protection

, while

Table 2. Dual Active Bridge Key Specifications.

Parameters	Specifications
Prim. voltage	$400–450V_{dc}$
Sec. voltage	$250–450V_{dc}$
Power rating forward	$7.4$ kW
Output current	20 A
Efficiency	${\eta }_{max}=0.98$
Switching frequency	$f_{sw}\in [200;800]$ kHz
	resonance at $f_{res}=500$ kHz

provides the characteristics of the DC/DC converter. The first step is to size the circuit components of the LCL filter input to the active rectifier. For such a filter, its main objective involves attenuating high-frequency harmonics with respect to the mains frequency; it aims to meet the specifications for current ripple at the input of the controlled rectifier (the part consists of the active switching components). The maximum value of the filter capacitance can be derived by Equation (5).

The value must take into account the reactive power $Q_{max}$ associated with a desired power factor value $PF$, as well as take into account the value of the AC supply voltage $V_g$ (rms) and the supply frequency $f_g$.

$$C_{f,max}=\frac{Q_{max}}{2\pi f_g{V_g}^2}=\frac{tan\left[arc\left(cos(1-PF)\right)\right]P_{in}}{2\pi f_g{V_g}^2}\cong 1.1\mathrm{mF}$$

(5)

Ensuring that the resonant frequency of the LCL filter meets the constraint in Equation (6), it must necessarily be much higher than the supply frequency, and much lower than the switching frequency of the rectifier stage.

$$\begin{array}{c}{f}_g<<f_{LCL}<<f_{sw}\\ \\ f_{LCL}^{lower}=10f_g=500\mathrm{Hz}\\ \\ f_{LCL}^{upper}=f_{sw}/10=10\mathrm{kHz}\end{array}⟶f_{LCL}=\frac{f_{LCL}^{lower}+f_{LCL}^{upper}}{2}\cong 5\mathrm{kHz}$$

(6)

From the circuit analysis, it is possible to derive the expression of the filter resonant frequency as an explicit function of the circuit components, and derive the value of the series inductance of the LCL filter, as in Equation (7).

$$f_{LCL}=\frac{1}{\sqrt{2\pi L_f{C}_f}}⟶L_{f,min}=\frac{1}{2\pi C_{f,max}{f}_{LCL}^2}\cong 5.8\mathsf{\mu }\mathrm{H}$$

(7)

The relationships reported above are obviously useful as guidelines for choosing the values of the components, but on their own, they do not represent systematic and reliable procedures. In fact, starting from the values obtained, a simulation analysis was carried out to find the values that make the dynamic behavior of the filter connected to the AC/DC converter as satisfactory as possible, in compliance with the specifications. This involved calculating the PF and the ripple (in terms of THD) for each combination of $L_f$ and $C_f$ values. The values chosen at the end of the evaluation were $L_f=190\mathsf{\mu }\mathrm{H}$ and $C_f=525\mathsf{\mu }\mathrm{F}$. Regarding the selection of active components using SiC/GaN technology, for each of the three power conversion stages, practical considerations must be made that guarantee the safety of the device itself, and they must be in line with the technical specifications. For the choice of switching devices, the most relevant aspects concern the maximum voltage between the drain and source $V_{ds}$, the maximum current absorbed by the drain channel $I_d$, and the switching frequency $f_{sw}$. Once commercial devices that respect these three fundamental aspects are identified, it will be necessary to select the device that can be driven with the gate density that can be provided via a switching system, possibly equipped with a gate driver unit (GDU); this device should also dissipate the lowest possible power per joule effect and have the lowest-declared resistance, $R_{d{s}_{\left(on\right)}}$.

$$\begin{array}{cc}\hfill V_{ds,max}& \ge (1+{\alpha }_v)V_{dc,max}\hfill \\ \hfill I_{d,max}& \ge (1+{\alpha }_i)\frac{P_{out,max}}{V_{g,min}}\hfill \end{array}$$

(8)

Referring to the conditions in Equation (8), it is possible to obtain the “minimum” values for the search for marketed devices, in terms of the maximum drain source voltage and maximum drain current. Coefficients ${\alpha }_v$ and ${\alpha }_i$ represent robustness parameters, and serve to make the selection of components “oversized”. Obviously, these parameters cannot be too high to avoid excessive sizing compared to the design specifications of the converter. The above choice refers to the devices for the PFC AC/DC (SiC) stage and the DC/AC (GaN) module of the DAB converter. Assuming that the AC/AC conversion by the transformer from galvanic isolation is as ideal as possible, for power conservation, GaN devices could be selected for smaller voltages and higher currents (consistent with the transformer gain), and to meet the input–output voltage specifications of the DAB itself. For the PFC active rectifier and the initial DC/AC conversion phase of the DAB, the minimum considered AC supply voltage is $V_{g,min}=110V_{rms}$, while the maximum value for the DC-bus voltage is $V_{dc,max}$. By incorporating safety coefficients of $10\%$, the minimum thresholds for component selections are derived: $V_{ds,max}\simeq 900$ V and $I_{d,max}\simeq 60$ A. These GaN devices are regarded as suitable for the second AC/DC conversion phase of the DAB converter, although in this instance, the $V_{ds,max}$ constraint may be reduced.

4. Software-in-the-Loop Simulation Analysis

In the article, the utilization of a C-caller in Simulink for implementing code equivalent to the previously explained controller models will be described. This approach enables the translation of transfer functions into executable code, leveraging the discretization process to obtain recursive functions. The C-caller in Simulink serves as a bridge connecting the Simulink modeling environment and code written in the C language. This enables the integration of more complex control models, derived from transfer functions, within the Simulink environment, facilitating more advanced simulations. The process begins with extracting transfer functions from control models. These functions are then translated into C code, incorporating the discretization process to suit the needs of the simulation in Simulink. Discretization is critical when adapting the continuum model to the characteristics of Simulink discrete-time simulation. Once the equivalent C code is obtained, the C-caller in Simulink is used to integrate this implementation into the overall model. This component allows Simulink to call C code during simulations, allowing the implemented control functions to be executed (see Figure 12).

The integration enables the evaluation of the system performance in greater detail, taking into account the effects of the implemented controllers. Furthermore, employing the C-caller allows flexibility to experiment with different control implementations directly within the Simulink environment, streamlining the analysis and optimization of the control system.

4.1. AC/DC Converter Simulation Analysis

The response of the AC/DC converter to a voltage step of magnitude $V_{ref}=400$ V is shown in Figure 13. In particular, the DC bus’s transient voltage is compared with various AC source circumstances. The following specifications are typical of single-phase AC power supply standards: (i) $V_g=230V_{rms}$ & $f_g=50$ Hz (standard in the EU); and (ii) $V_g=110V_{rms}$ & $f_g=60$ Hz (standard in the USA). Under both power supply scenarios, it is evident that the control system can manage the demand with a comparatively small steady-state mean error. As anticipated, the voltage transients are noticeably different, nevertheless. Since the controller must make up for the AC/DC conversion from drastically different absolute voltage levels, the overshoot is greater in the case of lower AC voltage. The transient voltage simulation results are shown in Figure 14, Figure 15, Figure 16 and Figure 17. All of these results show comparable trends, and the intended DC-bus voltages are $V_{ref}=500$ V, $V_{ref}=600$ V, $V_{ref}=700$ V, and $V_{ref}=800$ V, respectively. It is crucial to remember that the simulations are run by first setting beginning conditions for the voltage $V_C\left(0\right)$ of the active rectifier’s output capacitor, which is always set to be around 50 V lower than the given $V_{ref}$ value.

Furthermore, it can be observed that the difference between the AC voltage instances that meet EU and USA requirements becomes less significant as the needed DC-bus voltage rises. This pattern highlights the role that greater DC-bus voltage requirements play in mitigating the effects of the differences in AC voltage between EU and USA standards. The current $I_L\left(t\right)$ behavior under the EU-standard AC supply is depicted in Figure 18, which overlays the scenarios with the various needed DC-bus voltages.

The current behavior is clearly influenced by the fluctuating $V_{ref}$ it makes sense, based on the form of the current, in that the total harmonic distortion (THD) varies noticeably at lower DC-bus voltages. The alignment of $I_L$ with $V_g$, however, is still mostly steady despite the fluctuation in $V_{ref}$. This illustrates the control mechanism’s resilience and accomplishes the desired power factor adjustment goal. It is evident that the transient voltage behavior also has an indirect impact on the current’s adjustment to align with the supply voltage. Specifically, for $V_{ref}=400$ V, the effective current is in line with $V_g$, and its effective value only makes sense in relation to the needed power flow when the transient voltage becomes closer to the end of the steady-state operation (see Figure 19).

Figure 20 displays the outcomes of a comparable study, with the current profiles corresponding to simulations run with the supply circumstances, $V_g/f_g=110V_{rms}/60$ Hz. It is immediately noticed that the inductor current dynamics are far less sensitive to alignment with $V_g$ with this setup.

The transient voltage response for the scenario when $V_{ref}=600$ V is shown in Figure 20, which also provides a robustness analysis of the variance in the beginning voltage of the active rectifier’s output capacitor. While this change affects the transient voltage directly, the behavior is essentially the same, especially in the steady-state operation.

The transient current behavior as it aligns with $V_g$, as seen in Figure 21, is more sensitive to this fluctuation. The inductor current exhibits a high peak before aligning with $V_g$ due to the substantial voltage differential during the transient, which is ascribed to $V_C\left(0\right)$. Since the DC/DC module would be linked to the battery during a bench experiment, which naturally imposes a continuous voltage (depending on its state of charge), it is also important to note that some starting circumstances are just simulated. Therefore, the current peak seen in this circumstance might be caused by the voltage differential applied across the AC/DC converter’s input inductor. Such an analysis may be used to validate design decisions or to develop the project’s hardware and software monitoring and preventive logic.

Figure 22 shows an additional potential robustness study, showing the impact of changing the AC/DC converter’s output capacitor value. The difference in the voltage ripple is undoubtedly the most noticeable consequence. Considering the capacitor’s characteristic to store charge and, consequently, energy, the rationale holds. Consequently, a larger capacitance can absorb a greater quantity of energy related to oscillations, leading to a reduction in the voltage ripple. As no discernible impact of the supply voltage on current alignment was identified, this aspect is not presented in the study.

4.2. Dual Active Bridge Simulation Analysis

As the input voltage $V_1$ supplied by the AC/DC converter varies within the operational range of interest $V_1\in [200;500]$, Figure 23 shows the analysis of the output voltage transient of the DC/DC converter for a control voltage of $V_{ref}=250$ V (battery voltage connected to the second stage of the DAB conversion). Figure 24 and Figure 25 also show a completely similar analysis. The step response gravitates toward an (almost) linear first-order transient for lower voltage levels and displays second-order behavior for higher $V_1$ voltage levels under all operating circumstances that have been investigated and demonstrated. The more DAB output voltages used, the more obvious this discovery becomes. A thorough examination of the robustness of the phase-shifting control method is provided in the previously stated Figure 26. This study takes into account possible fluctuations in the values of circuit components, with a particular emphasis on the variations in leakage inductance and output capacitor. The analysis is given for the intermediate working state, with the goal of reaching a steady battery voltage of 350 V, to guarantee clarity and comprehensiveness. It is clear that adjustments to the values of both circuit components affect the transient behavior more than the steady-state value, with the residual voltage ripple being partially impacted.

It was found that, in particular, a greater value for the inductance tends to enhance the voltage step response’s overshoot, whereas a value that is too low makes the dynamic response resemble a first-order linear system, which also affects the steady-state error. It is evident that the output capacitance influences the maximum overshoot and the settling time, but it appears to have little effect on the steady-state value (at least within the examined range of variation). This allows the desired voltage step response to be achieved with almost no steady-state error.

4.3. Switching Devices Thermal Simulation Analysis

The thermal behavior of the switching components of the previously examined AC/DC converter is depicted in Figure 27, with reference to the “intermediate” working state, which calls for an AC supply of 230 V & 50 Hz and a DC bus of 650 V. One branch of the active rectifier operates at a high switching frequency, while the other operates at the grid frequency, as was previously mentioned in relation to converter operation. Since the two branches are typically powered by the same current, their conduction losses are comparable; in contrast, the high-frequency branch has much larger conduction losses. The simulation results align perfectly with this observation.

Consequently, for the two branches operating at distinct frequencies, a stable state value of $T_{j_{HP,final}}\cong 75$ [°C] is identified for the high-frequency branch devices, while a regime value of $T_{j_{LP,final}}\cong 61$ [°C] is observed for the grid-frequency devices within the branch. Additionally, two transients with differing rise times are identified for each branch. The thermal transient of the SiC-integrated devices is displayed in Figure 28 for the chosen DAB converter. The switching losses (as well as those for switching) differ greatly because, on average, the primary and secondary stage branches are not crossed by the same current due to the galvanic isolation transformer. Specifically, the average current flowing through the active rectifier’s components crosses the first conversion stage (DC/AC).

In fact, the operating temperature of the SiC primary components is comparable to that of the rectifier’s high-frequency components, with a steady state average value of $T_{j_p,final}\cong 82$ [°C]. Since the resonance frequency is higher than that of high-frequency components, the switching losses weigh much more, leading to overheating at higher speeds. On the other hand, because the transformer is employed to decrease the effective voltage value, a greater current passes through the components of the AC/DC secondary circuit, making them susceptible to noticeably higher conduction losses.

The average junction temperature in secondary SiC devices has a steady-state value of $T_{j_s,final}\cong 109$ [°C]. It is noteworthy that the steady-state values are within the manufacturer’s stated operating limitations and are consistent with the design and size decisions made during the early modeling stages.

4.4. Battery Charging Scenario

To fully validate the proposed OBC model, in this section, we show the simulation results obtained by applying three different charging protocols of a Li-ion battery. The simulations carried out refer to the use of a model in the Simscape environment (MathWorks), which, starting from high-level specifications, identifies the intrinsic characteristics of the battery, taking into account the chemistry. The Li-ion battery models made available by the simulation environment are created by integrating the following Equation (9):

$$\begin{array}{cc}\hfill V_{battery}& =\left\{\begin{array}{c}{V}_{b,0}-R{I}_b-V_{Q,d}+V_{exp}\mathrm{in}\mathrm{discharge}\mathrm{mode}\hfill \\ V_{b,0}-R{I}_b-V_{Q,c}+A{e}^{-Bq}\mathrm{in}\mathrm{discharge}\mathrm{mode}\hfill \end{array}\right.\hfill \\ \hfill V_{Q,d}& =\frac{KQ}{Q-q}\left(q+I^{*}t\right)\hfill \\ \hfill V_{Q,c}& =\frac{KQ}{Q-0.1q}{I}^{*}t-\frac{KQ}{Q-q}q\hfill \\ \hfill \frac{d{V}_{exp}}{dt}& =B|I_b|\left(-V_{exp}+Asign\left(I_b\right)\right)\hfill \end{array}$$

(9)

where $V_b$ denotes the measured battery voltage charge/discharge) [V]; $V_{b,0}$ denotes the battery’s constant voltage [V]; K denotes the polarization constant [V/Ah]; Q denotes the battery capacity [Ah]; $q={\int }_0^t{I}_b\left(\tau \right)d\tau$ the battery charge [A]; A denotes the exponential zone amplitude [V]; $I^{*}$ denotes the filtered current [A]; B denotes the exp. zone time constant inverse; R denotes the internal resistance [$\Omega$]; and $I_b$ denotes the measured battery current [A]. A realistic characteristic inherent to the discharge behavior of the battery for the different currents supplied is automatically generated, and the nominal characteristic is explained by highlighting the nominal and exponential zones (see Figure 29). Regarding the proposed simulation results, the data relating to the simulated battery are presented in

Table 3. Battery parameters of Simscape model.

Characteristic	Value
Maximum capacity	150 [Ah]
Cut-off voltage	187.5 [V]
Fully charged voltage	291 [V]
Nominal discharge current	65.2 [A]
Internal resistance	16.67 [m$\Omega$]
Capacity at nominal voltage	135.65 [Ah]
Exp. zone voltage	270 [V]
Exp. zone capacity	7.37 [Ah]

. Figure 30 shows the different behaviors of the battery in terms of the voltage and charging current supplied by the electronic converters, in three different modes.

In particular, the results relate to the following charging protocols:

• Constant current protocol (CCP): In this case, the nominal voltage of the battery is taken as reference and the current value is imposed to obtain the desired output power.

  $$I_{ref,CCP}=\frac{P_{out}^{*}}{V_{b,nominal}}$$

  (10)
• Adaptive current protocol (ACP): In this case, the current is calculated based on the instantaneous battery voltage to keep the power flow between the OBC and the battery constant.

  $$I_{ref,ACP}\left(t\right)=\frac{P_{out}^{*}}{V_b\left(t\right)}$$

  (11)
• Pulse current protocol (PCP): In this case, a current greater than that necessary to obtain a constant desired power flow is used, providing the battery with current pulses followed by zero-current phases.

  $$I_{ref,PCP}=2\frac{P_{out}^{*}}{V_{b,min}}PWM(0.1\mathrm{Hz},50\%d.c.)$$

  (12)

As expected from engineering practices, the PCP protocol provides the fastest charging, compared to the other two cases, for the same power supplied to the battery for charging (see Figure 31).

This is due to the intrinsic behavior of the battery, which is more reactive in the high current phase than it is when it discharges in the zero current phase (in fact, batteries are systems with memory). Obviously, the aim of this work is not to propose an efficient charging protocol, but to show that the design and control systems obtained in simulation for the two converters individually can provide realistic information on the behavior of the Li-ion battery, with typical specifications of automotive applications.

5. Final Discussion and Conclusions

In the pursuit of a scientific article, simulations were conducted to evaluate the performance of control algorithms and the sizing of passive components within the framework of an onboard charger (OBC) power electronic system. The investigation comprised separate simulations for the AC/DC conversion stage, implemented via a single-phase PFC circuit. Notably, a detailed analysis of the voltage step response for various DC-link reference values was undertaken, yielding noteworthy outcomes. For $V_{ref}=400$ V, the steady-state error was 0, with settling times ranging between 1.36 and 1.65 s, contingent on the AC voltage being 230 V/50 Hz or 110 V/60 Hz. Similarly, for $V_{ref}=550$ V, a steady-state error of 0 was achieved, with nearly identical settling times for both AC input configurations (approximately 1.2 s). For $V_{ref}=600$ V, $V_{ref}=700$ V, and $V_{ref}=800$ V, behaviors in both AC input configurations were practically overlapping, exhibiting a steady-state error trending toward 0 and a voltage ripple consistently below $2\%$. The evaluation of the AC/DC converter included an assessment of the power factor (PF), reaching approximately $97\%$ after the voltage transient for 230 V/50 Hz AC input and even reaching $98\%$ for 110 V/60 Hz. This performance aligns with market supplier declarations and surpasses the declared design in some industrial products. Furthermore, in the 230 V/50 Hz configuration, current harmonic distortion was lower compared to the 110 V/60 Hz configuration, never exceeding $5\%$. Robustness analysis of the feedback system concerning variations in circuit parameters within a $20\%$ range from the nominal case demonstrated acceptable performance degradation. This involved maintaining a voltage ripple consistently below $2.5\%$, a current ripple below $10\%$, a steady-state error below $2\%$, a PF consistently above $90\%$, and total harmonic distortion (THD) consistently below $6.5\%$. A parallel analysis was performed on the DC/DC conversion stage, employing a DAB converter. The output voltage response, evaluated at a constant output power of 7 KW, yielded distinct performances for various $V_{ref}$ values. For $V_{ref}=250$ V, there was a maximum overshoot of $15\%$, with zero steady-state error and a maximum settling time of 3.2 ms. Similarly, for $V_{ref}=350$ V, a maximum overshoot of $15\%$ was observed, along with zero steady-state error and a settling time below 6 ms. For $V_{ref}=450$ V, the overshoot was below $10\%$, the steady-state error was below $1\%$, and the settling time was below 52.5 ms. These performances were assessed while varying the input voltage at the DC/DC primary. Additionally, feedback robustness was evaluated concerning variations in passive parameters within a $50\%$ uncertainty range from the nominal value, revealing slight performance degradation limited to a worst-case $25\%$ overshoot. Thermal performances were also scrutinized to ensure that the junction temperature of switching components remained within the operational limits declared in the supplier data sheets. This comprehensive investigation into the intricacies of OBC system design and control, encompassing both AC/DC and DC/DC conversion stages, sets the stage for future advancements. The study highlights the adaptability and resilience of control algorithms, emphasizing the critical importance of precision in component sizing and algorithmic design. The findings pave the way for optimizing OBC efficiency and reliability through advanced control strategies and thermal management. Future developments could explore the implementation of feedback from thermal models, leveraging adaptive or predictive controllers for enhanced precision. However, careful consideration of computational challenges, especially in resource-limited embedded systems, is imperative for practical implementation in automotive environments.

The balance between control precision and computational practicality must be carefully evaluated in ongoing efforts to enhance OBC performance and contribute to the continued evolution of electric vehicle technology.

In conclusion, our comprehensive investigation into the intricacies of the onboard charger (OBC) system design and control, covering both AC/DC and DC/DC conversion stages, aligns with the contemporary emphasis on nonlinear circuits and systems. The nonlinear nature of power systems is becoming increasingly crucial in the age of complexity, with a growing focus on advanced power plant technologies. Our study underscores the adaptability and resilience of control algorithms, emphasizing the critical importance of precision in component sizing and algorithmic design within the context of nonlinear systems. The findings not only contribute to the current state of knowledge in power electronics but also align with the broader trends in nonlinear technology highlighted in the referenced editorial. As we navigate the nonlinear landscape of power systems, our results showcase the robustness of the OBC system under various conditions, demonstrating acceptable performance degradation within a specified parameter range. This aligns with the challenges posed by nonlinear dynamics in power systems, as discussed in the editorial [83]. Moreover, our study paves the way for future advancements by highlighting the potential for optimizing OBC efficiency and reliability through advanced control strategies and thermal management. The editorial emphasizes the multidisciplinary nature of research in nonlinear technology, mirroring our approach that considers both AC/DC and DC/DC conversion stages. Looking ahead, future developments could explore the implementation of feedback from thermal models, leveraging adaptive or predictive controllers for enhanced precision. However, as the editorial suggests, careful consideration of computational challenges, especially in resource-limited embedded systems, is imperative for practical implementation in automotive environments. The delicate balance between control precision and computational practicality must be thoroughly evaluated in ongoing efforts to enhance OBC performance and contribute to the continued evolution of electric vehicle technology. In summary, our study not only adds valuable insights to the field of OBC design and control but also aligns with the overarching themes of nonlinear technology discussed in the referenced editorial. By recognizing the interconnected nature of nonlinear components and systems, our research contributes to the ongoing dialogue on advanced power plant technologies, paving the way for future interdisciplinary research in the realm of nonlinear power systems.