Design of a Controller for Supercapacitor’s Bidirectional High-Gain Interleaved Converter

Abstract

This study focuses on the mathematical modeling, control design, and analysis of an interleaved bidirectional high-voltage-gain DC-DC converter for energy management in supercapacitors. The state of the art is reviewed, with an emphasis on research related to DC-DC converters and energy storage systems. The characteristics and modeling of the supercapacitors are thoroughly analyzed. The converter’s operation in both buck and boost modes is described, detailing its operating stages, design parameters, and component sizing. The modeling accounts for the dynamics of the converter in both operational modes. PI controllers and compensation techniques were implemented to ensure the desired performance and meet the design criteria. Simulations were conducted using PSIM software, version 2023.1, with a power flow of 1 kW, a 48 V DC bus (buck mode), and a 162 V supercapacitor module (boost mode), operating at 500 kHz. The performance of the controllers was evaluated during both the charging and discharging processes of the supercapacitor, analyzing the dynamic response and behavior in the continuous mode, even in the presence of system disturbances.

1. Introduction

1.1. General Considerations

Achieving sustainability goals remains a key priority for institutions and governments worldwide. In this regard, clean technologies offer a viable pathway for meeting the targets established by the 2015 Paris Agreement [1]. Among the available clean energy options, photovoltaic (PV) solar power has experienced substantial growth in recent years, with approximately 1 terawatt (TW) of capacity added to the global energy mix since 2015 [2]. However, despite this rapid expansion, most renewable energy sources still lack firm capacity due to their intermittent nature, which limits their ability to function as standalone energy solutions. To address this limitation, hybrid systems that combine multiple energy sources with energy storage technologies—such as batteries and supercapacitors—have emerged as viable solutions to enhance energy autonomy and improve system resilience [3].

To ensure the effective deployment of renewable energy, it is essential to expand storage capacity to offset its intermittency. It is projected that by 2030, global battery storage capacity will reach approximately 200 gigawatts—equivalent to about 25% of the total installed PV capacity expected for the same year [2].

Hybrid energy storage systems (HESS), which integrate high-energy-density devices (e.g., lithium-ion batteries) with high-power-density devices (e.g., supercapacitors), have gained traction as promising solutions for dynamic and high-performance applications [4]. The implementation of these systems is made possible through microgrids, which can be categorized as alternating current (AC), direct current (DC), or hybrid (AC/DC) configurations [4]. In addition to mitigating intermittency, energy storage systems enable the capture of surplus energy generated during the day, thereby ensuring balanced and reliable energy supply for both on-grid and off-grid operations [5,6].

Microgrids play a crucial role in integrating energy storage technologies. Their flexibility to operate in AC, DC, or hybrid configurations has led to widespread adoption in on-grid and off-grid settings, enabling the management of variable generation, enhancing system reliability, and reducing dependence on fossil fuels [7]. Within this context, bidirectional DC-DC converters are fundamental components for interfacing and managing energy storage devices. These converters facilitate two-way energy transfer between the storage units and the DC bus, adjusting voltage levels and supporting both charging and discharging operations efficiently. Moreover, due to their rapid dynamic response and high-power density, supercapacitors have been increasingly utilized as effective complements to batteries in hybrid systems, particularly in applications requiring short-duration energy buffering [8]. The performance of such systems is highly influenced by the converter’s topology and control strategy, especially under variable load conditions and fluctuating renewable energy profiles [9,10].

Recent advancements in electrochemical energy storage have underscored the potential of optimized converter architectures tailored to supercapacitor applications. For example, Ref. [11] investigates a zinc-based hybrid supercapacitor with enhanced thermal and mechanical properties. Meanwhile, Refs. [12,13] explore aqueous zinc-ion batteries, whose benefits—such as dendrite suppression and improved cycling performance—are highly relevant for hybrid storage systems incorporating bidirectional converters in microgrids, uninterruptible power supplies (UPS), and electric mobility applications.

This paper presents the implementation of a bidirectional DC-DC converter for supercapacitor-based energy storage. The proposed converter design aims to address current challenges in renewable energy integration by offering a high-efficiency, fast-response solution suited for microgrid and hybrid system applications. The following section reviews the state of the art, laying the groundwork for the proposed implementation in renewable energy systems.

1.2. Literature Review

The earliest studies on supercapacitors and batteries date back to the 1990s [14,15], when supercapacitors were already seen as a promising enhancement to battery storage due to their high-power density, long lifespan, and efficiency. Although early research primarily focused on electric vehicle applications, interest in hybrid energy storage systems (HESS) has grown substantially in recent years, particularly with the integration of control techniques based on computational intelligence [16,17,18]. A comprehensive review of reports, articles, and documents reveals real-world implementations of HESS in several countries, including Germany [19], the United States of America [20], Spain, Portugal [21], Iran [22], and Australia [23]. These systems serve various purposes, such as peak shaving, load shifting, solar smoothing, frequency regulation, grid stabilization, and renewable energy integration. Each system is tailored to specific requirements in terms of capacity, voltage, and power.

The choice of a suitable HESS topology depends on application-specific demands and factors such as cost, scalability, efficiency, and operational lifespan. The most common configurations are classified as passive, semi-active, and fully active topologies [24,25]. While topology refers to the system’s physical configuration, the control strategy defines how devices are managed to optimize overall performance. As such, topology imposes functional constraints that must be considered during control design to ensure effective energy management. Furthermore, control strategies must adapt to the characteristics of renewable sources, especially in systems powered by intermittent generation like solar PV and wind.

Converter topology is another key aspect in power electronics design. The selection depends on various factors such as the intended application, cost, efficiency targets, and voltage/current specifications [26]. DC-DC converters are broadly categorized into isolated and non-isolated types. Isolated topologies offer higher voltage gains and enhanced safety due to galvanic isolation, making them suitable for applications requiring voltage regulation under variable conditions, such as grid-tied renewable energy systems [27]. Common isolated DC-DC converter topologies include the isolated bidirectional buck–boost converter, the dual active bridge, the isolated bidirectional Cuk and Sepic/Zeta converters, the bidirectional push–pull, and the forward converter [27,28,29].

In contrast, non-isolated topologies lack galvanic isolation, resulting in lower weight but also reduced voltage gain. As reviewed in [27], non-isolated converter topologies can be grouped into eight categories, including buck, basic boost, Cuk, cascaded, switched-capacitor, and interleaved structures for renewable and storage applications [30,31,32]. Recent advancements have introduced customized converter designs to suit specific use cases. For example, Ref. [33] proposes a high-voltage converter with extended gain and a continuous conduction mode by integrating switched inductors and capacitors, achieving high efficiency and zero-voltage switching (ZVS). In contrast, Ref. [34] introduces a reconfigurable converter tailored for hybrid energy storage that integrates batteries and supercapacitors. While it offers design flexibility and compatibility with various storage media, it also faces challenges such as limited supercapacitor voltage and increased control complexity—illustrating the trade-offs of flexible architectures. Similarly, Ref. [35] presents a bidirectional converter featuring current-sharing control and coupled inductors aimed at minimizing power losses and enhancing efficiency.

Control strategies are further explored in [24,36]. Specifically, Refs. [37,38] employ filter-based strategies in wind and solar applications using both supercapacitors and batteries. In [38], the combined use of supercapacitors and batteries is shown to extend system lifespan by up to 19% compared to battery-only setups. Similarly, Ref. [37] applies this strategy to a solar PV system, demonstrating effective voltage regulation, reduced battery current stress, and enhanced battery longevity under different load and generation conditions. Additional diversity in control strategies is illustrated in [33], which applies soft-switching techniques to increase efficiency and minimize switching losses—especially beneficial for high-gain applications such as microgrids. In contrast, Ref. [35] emphasizes power flow regulation and storage management through current-sharing and coupled inductor techniques, achieving high efficiency in grid-connected scenarios. Likewise, Ref. [39] investigates multiphase interleaved bidirectional converters with integrated supercapacitors, employing duty cycle optimization to reduce ripple and filtering requirements, particularly for microgrid and EV contexts. Meanwhile, Ref. [40] presents a switched-inductor buck–boost topology offering multiple operation modes and high voltage conversion ratios, validated through experimental results.

The review presented in [41] provides an overarching analysis of trends in bidirectional and multiport converters, focusing on soft-switching and hybrid storage integration. Rather than emphasizing specific topologies, the study highlights shared design challenges, such as component complexity, conduction, and switching losses. Additional strategies based on predictive and optimization techniques are discussed in [42,43,44].

To facilitate converter selection for hybrid energy storage systems (HESS) in renewable integration and microgrids,

Table 1. Comparison of topologies and control strategies for bidirectional DC-DC converters in energy storage systems.

Paper	Converter Topology	Control Strategy	Applications	Observations
[31]	Dual boost with four switches and passive components.	PWM for power and voltage regulation.	Fuel cells, batteries, and renewable energy systems.	High efficiency and voltage gain; simple design, but more components and higher cost.
[47]	Quasi Z-source DC-DC converter.	MPPT using Perturb and Observe (P&O) algorithm.	Photovoltaic systems with supercapacitor energy storage.	Efficient PV–supercapacitor interface; sensitive to rapid irradiance changes.
[48]	Separate converters for battery and supercapacitor.	Frequency-based control with PI for duty cycle adjustment.	Power regulation, grid support, and energy backup.	Flexible and stable; requires precise measurements and involves higher complexity.
[49]	Supercapacitor + C2P2C DC-DC converter.	Active cell balancing by modulating charging/discharging.	Electric vehicles and storage systems.	Improved voltage balance; fewer components, but requires accurate parameter estimation.
[50]	Bidirectional DC-DC converter (boost/buck mode) with ultracapacitor	Adaptive PI-based control.	Hybrid EVs, microgrids, backup systems.	Reduces settling time (10–50%); complex design and sensitive to parameter tuning.
[46]	Boost converter with series-connected supercapacitors	Open-loop vs. closed-loop with PI controller.	Renewables, electric mobility, motor drive systems.	Stable voltage; needs accurate control to prevent voltage drop.
[45]	Inverter-based three-phase converter.	Frequency estimation and dual-mode control.	Rubber-tyred gantry crane (RTGC) for container handling in ports	The system achieves approximately 35% energy savings and over 40% reduction in engine emissions. Limitations include the need for precise control and reliance on simulations for validation
[51]	Four-phase interleaved bidirectional converter.	DSP-based unified control.	High-power EVs and fuel cell systems.	Smooth buck/boost transitions with ZVS; requires high sampling and careful design.

summarizes key high-gain DC-DC converters. Supercapacitor-based systems stand out for their fast response and energy savings, achieving up to 35% fuel reduction in RTGC cranes [45]. However, they face challenges like current ripple [45] and control sensitivity. Adaptive PI control [46] helps mitigate stability issues, particularly right-hand plane (RHP) zero effects. While other solutions like dual-boost [31] and quasi Z-source converters [47] support fuel cells and PVs, supercapacitors offer an efficient and robust alternative for high-power, dynamic energy applications.

In summary, while previous studies have focused on general classifications and theoretical comparisons of converter topologies and control strategies, this work distinguishes itself by proposing a practical and targeted design for a supercapacitor’s energy management. It introduces a non-isolated interleaved bidirectional DC-DC converter with high voltage gain specifically tailored for supercapacitor-based energy management. Additionally, it incorporates supercapacitor-tailored dynamic modeling and practical design parameters, forming the foundation for the development of a control strategy that is validated through computational simulations in the subsequent sections.

The proposed controller can respond quickly during charging and discharging operation of a supercapacitor, maintaining the voltage and current overshoot under low levels compared to the converter presented in Table 1. This fact is possible due to the integration of the supercapacitor’s model with the bidirectional interleaved high-gain converter’s dynamic modeling.

1.3. Motivation and Contribution

The implementation of hybrid energy storage systems (HESS) offers a solution to enhance the reliability, continuity, and quality of the energy supply, considering the intermittent nature of renewable energy and the need for uninterrupted service to consumers. This paper addresses the need for efficient and controllable energy storage systems, particularly in hybrid applications, where supercapacitors are becoming increasingly crucial due to their ability to rapidly supply and absorb energy.

The primary objective of this paper is to model and develop control strategies for the high-voltage-gain bidirectional buck–boost converter topology for the energy management of supercapacitors coupled in hybrid storage systems. This topology, originally presented by [30] to deal with resistive loads, was tailored to manage the charge and discharge operation of a supercapacitor and then enable power transfer between two sources with significant voltage differences while minimizing efficiency losses in the system. The system schematic is shown in Figure 1, where the converter works in different modes during charging and discharging.

2. Supercapacitors

Supercapacitors, also known as electric double-layer capacitors (EDLCs) or ultracapacitors, are energy storage devices first described by Helmholtz in the mid-19th century, based on the electric double-layer phenomenon. In 1966, the Standard Oil Company patented the first supercapacitor. A decade later, in 1978, NEC (Nippon Electric Company) licensed, developed, and marketed a supercapacitor product for use as a memory backup device. The first military-grade supercapacitors were developed by the Pinnacle Research Institute (PRI) in 1982. In 1992, Maxwell Laboratories took over the development of supercapacitors for hybrid electric vehicles [52,53,54].

In 1853, Hermann Helmholtz explained that when a voltage is applied between two electrodes immersed in a conductive liquid, no current flows until a certain limiting voltage is reached [55]. A supercapacitor is an electrochemical capacitor composed of two porous conductive electrodes coated with activated carbon, along with a separator that prevents direct electrical contact between the electrodes while allowing the free passage of ions. This assembly is immersed in a highly conductive electrolyte, providing a high concentration of mobile ions [52,54,55]. An example of the internal structure of the supercapacitor is shown in Figure 2.

The method of charging and discharging a supercapacitor depends on the interaction between the power converter and the supercapacitor itself. These processes can be categorized into four modes: constant resistance mode, constant current mode, constant voltage mode, and power mode. However, the constant voltage mode is not applicable due to the inherent characteristics of the supercapacitor [52]. In general, the charging process occurs through the movement of ions across the carbon surface, while the discharging process involves the reverse movement of the ions [53,55].

Supercapacitors can be classified into three main types: electrochemical double-layer supercapacitors, pseudo-supercapacitors, and hybrid capacitors. Each type is distinguished by its specific method of energy storage [52,53,56].

2.1. Supercapacitor Modeling

In supercapacitor systems, modeling is crucial for system sizing, condition management, design, and control synthesis. Numerous models are available in the literature for various applications, based on electrical, mathematical, chemical, thermal, self-discharge, dynamic structures, aging, and artificial intelligence characteristics [54,57]. The following models are commonly used to represent supercapacitor behavior: electrochemical models, equivalent circuit models, intelligent models, self-discharge models, and thermal models [54].

For this article, the equivalent circuit model was employed. Specifically, a simple RC model was used to develop the dynamic model, while the transmission line model was utilized in computer simulations with PSIM software. The developed models and the study are applicable exclusively to double-layer electrochemical supercapacitors. After a thorough literature review, these models were specifically developed for this type of supercapacitor, which is the most widely used in energy storage systems and various other applications.

2.1.1. Simple RC Model

The simplest equivalent circuit model is the series RC circuit, which captures only the instantaneous dynamic behavior and represents the external characteristics of the supercapacitor, as shown in Figure 3. One of the advantages of this model is its simplicity in parameter adjustment and its accuracy in calculating charge and discharge processes. However, a key limitation is that it does not account for variations in the supercapacitor’s performance over time, making it unsuitable for simulating the supercapacitor’s working principle in detail [52,54,58].

2.1.2. Transmission Line Model

The transmission line model is designed to simulate the equivalent resistances and the electrostatic double-layer capacitance inherent to each pore in the electrode material, utilizing nonlinear resistances and capacitances. This model captures both the transient and long-term behavior of the supercapacitor, as illustrated in Figure 4. Furthermore, the transmission line model offers high accuracy across a broad frequency range. As the number of branches increases, both the order and accuracy of the model improve; however, this also makes the identification of parameters more complex [54,58,59].

The circuit shown above is nonlinear, as the capacitances depend on the voltage applied to the circuit terminals. A simplified representation of this dynamic model is provided in Equation (1) [54,58,59].

$$C\left(V_C\right)=C_O+K_V{V}_C$$

(1)

3. Bidirectional DC-DC Converters for Supercapacitors

The defining feature of bidirectional DC-DC converters is their ability to enable power flow in both directions. This bidirectional power transfer is achieved through the switching operation of semiconductor devices, which allows current to flow either way. These converters are widely used in supercapacitor-based storage systems and electric vehicles, as they enable precise control of energy flow between different power sources.

3.1. Bidirectional DC-DC Converter with High Voltage Gain

In [30], a bidirectional DC-DC converter with high voltage gain is proposed. The topology comprises four switches and diodes—labeled S1, S2, S3, and S4—along with two inductors and three capacitors. In each power flow direction, two switches function as active power switches, while the other two act as synchronous rectifiers. The converter essentially integrates two conventional boost converters, and its voltage gain in both buck and boost modes is superior to that of the traditional bidirectional buck–boost converter. The converter proposed in [30] is illustrated in Figure 5.

3.1.1. Buck Mode

In the voltage step-down mode, switches S3 and S4 operate as power switches, while S1 and S2 function as synchronous rectifiers. The current flow in this mode is illustrated in Figure 6. During the time interval [t₀, t₁], S3 and S4 are turned on, and S1 and S2 are turned off. In this phase, energy from the supercapacitor is transferred to inductor L1, while capacitor C discharges into inductor L2 and output capacitor CL, as shown in Figure 6a.

During the interval [t₁, t₂], switches S1 and S2 are turned on, while S3 and S4 are turned off. In this phase, inductor L1 is demagnetized, transferring energy to capacitors C and CL. Simultaneously, inductor L2 discharges into capacitor CL, thereby supplying energy to the DC bus, as illustrated in Figure 6b.

Based on the analysis of the previous circuits, [30] defines the static gain for the buck mode as:

$${\displaystyle \frac{V_{DC}}{V_{SC}}}=G_{VCCM (step-down)}=D^2$$

(2)

where D is defined as the duty cycle.

The minimum inductance values for the step-down mode are defined by Equations (3) and (4):

$${\mathrm{L}}_1\ge {\displaystyle \frac{{(1-\mathrm{D}}^2){\mathrm{R}}_{DC}}{2{\mathrm{D}}^2{\mathrm{f}}_{\mathrm{s}}}}$$

(3)

$${\mathrm{L}}_2\ge {\displaystyle \frac{{\mathrm{R}}_{DC}}{2{\mathrm{f}}_{\mathrm{s}}}}$$

(4)

where $f_s$ is the switching frequency and $R_{DC}$ is the DC bus load.

3.1.2. Boost Mode

In the voltage step-up mode, switches S1 and S2 act as power switches, while S3 and S4 function as synchronous rectifiers. The power flow during the voltage step-up mode is illustrated in Figure 7. During the interval [t₀, t₁], switches S1 and S2 are activated, and S3 and S4 are turned off. In this state, the DC bus supplies energy to inductor L2, causing inductor L1 to become magnetized. Additionally, the energy stored in capacitor C is transferred, and capacitor CH discharges into the supercapacitor, as shown in Figure 7a.

In the interval [t₁, t₂], switches S1 and S2 are turned off while switches S3 and S4 are turned on. Capacitor C is charged by the DC bus, and the energy stored in inductor L2 is transferred to capacitor CH, which is charged by the DC bus along with the energy stored in inductor L1, as shown in Figure 7b.

Based on the analysis of the previous circuits, the static gain for the boost mode is:

$${\displaystyle \frac{V_{SC}}{V_{DC}}}=G_{VCCM (step-up)}={\displaystyle \frac{1}{{(1-D)}^2}}$$

(5)

For the step-up mode, the minimum inductance values are determined by Equations (6) and (7):

$${\mathrm{L}}_1\ge {\displaystyle \frac{\mathrm{D}(2-\mathrm{D}){(1-\mathrm{D})}^2{\mathrm{R}}_{SC}}{2{\mathrm{f}}_{\mathrm{s}}}}$$

(6)

$${\mathrm{L}}_2\ge {\displaystyle \frac{{{(1-\mathrm{D})}^4\mathrm{R}}_{SC}}{2{\mathrm{f}}_{\mathrm{s}}}}$$

(7)

where $R_{SC}$ is the supercapacitor load.

3.1.3. Design of a High-Voltage-Gain Interleaved Bidirectional Buck–Boost Converter

The design of the interleaved bidirectional buck–boost converter with high voltage gain, intended for supercapacitor energy management, was based on the parameters outlined in

Table 2. Project parameters.

Parameter	Specification
Average supercapacitor voltage	$V_{SC}$$=162 \mathrm{V}$
Supercapacitor module capacitance	$C_{SC}$$=62 \mathrm{F}$
Supercapacitor module	SKELMOD 162V62F
Average DC bus voltage	$V_{DC}$$=48 \mathrm{V}$
Average output power	$P_o$$=1 \mathrm{k}\mathrm{W}$
Switching frequency current ripple across inductors	$f_s$$=500 \mathrm{k}\mathrm{H}\mathrm{z}$
Voltage ripple across DC bus	$∆I_L$$=5\%I_L$
Voltage ripple across supercapacitor	$∆V_{DC}$$=1\%V_{DC}$
Semiconductor switches	$∆V_{SC}$$=2\%V_{SC}$

.

The bidirectional high-gain converter was selected for this application due to the substantial voltage difference between the commercial supercapacitor module ($V_{SC}$ = $162 \mathrm{V}$) and the DC bus ($V_{DC}$= $48 \mathrm{V}$). In such conditions, a converter capable of achieving a high voltage gain is crucial to maintaining system efficiency. Conventional topologies commonly employed in supercapacitor-based applications generally lack this high-gain capability. Both simulations and previous studies have shown that using these conventional alternatives often leads to a significant reduction in overall system efficiency.

The selection of a 1 kW dispatch power is based on the specific requirements of the target system, in which the converter will be integrated. This converter will operate within a microgrid that incorporates photovoltaic generation and a hybrid energy storage system comprising a battery bank and supercapacitors. The microgrid is designed to supply electricity to a small, remote community in the Amazon region of Brazil. Operating in conjunction with the supercapacitors, the bidirectional high-gain converter is intended to deliver up to 1 kW of power in response to load demands.

4. Dynamic Modeling of the Interleaved Buck–Boost Converter with High Voltage Gain

The bidirectional high-gain interleaved converter topology proposed in was used to design the appropriate controller to manage the energy flow between the electrical grid and the supercapacitor module. This converter was designed to deal with a maximum output power of $P_o$ = $1 \mathrm{k}\mathrm{W}$, since it must be capable of supplying a reasonable amount of power to mitigate voltage fluctuations and short-period demand.

Since DC-DC converters are inherently nonlinear systems, they are often modeled by the state-space averaging approach; this method determines the weighted average of the converter’s operating modes over a switching cycle.

Before presenting the state variables and the equations corresponding to each operating mode,

Table 3. Variables used.

Variable	Specification
$D$	Duty cycle
$C$	Central capacitance
$V_C$	Central capacitor voltage
$L_1$$\mathrm{and} L_2$	Inductances
$R_{L1}$$\mathrm{and} R_{L2}$	Parasitic resistance of inductors
$I_{L_1}$$\mathrm{and} I_{L_2}$	Inductor current
$R_{DC}$	DC bus resistance
$C_L$	DC bus output capacitance
$r_{C_L}$	DC bus output capacitor resistance
$V_{C_L}$	Output capacitor voltage
$R_{DON1}, R_{DON2}, R_{DON3}$$\mathrm{and} R_{DON4}$	Source-drain resistances
$V_{SC}$	Supercapacitor voltage
$C_H$	Supercapacitor output capacitance
$r_{C_H}$	Supercapacitor output capacitor resistance
$R_{SC}$	Calculated supercapacitor resistance
$C_{SC}$	Supercapacitor capacitance
$V_{C_{SC}}$	Output voltage on the supercapacitor

summarizes all the variables used in the formulation of the model, along with their respective definitions.

4.1. Buck Operating Mode

For the high-voltage-gain bidirectional converter, separate models were developed for each operating mode—buck and boost—based on the respective state variables. The state variables for the buck mode are presented in the equations below.

$$L_1{\displaystyle \frac{{di}_{L_1}\left(t\right)}{dt}}=v_{L_1}(t)$$

(8)

$$L_2{\displaystyle \frac{{di}_{L_2}\left(t\right)}{dt}}=v_{L_2}(t)$$

(9)

$$C{\displaystyle \frac{{dv}_C\left(t\right)}{dt}}=i_C(t)$$

(10)

$$C_L{\displaystyle \frac{{dv}_{C_L}\left(t\right)}{dt}}=i_{C_L}(t)$$

(11)

The linearized small-signal models were derived from the previously defined state variables and the corresponding circuit configurations for each operating mode. Linearization involves approximating a nonlinear system around a specific operating point under small perturbations. In DC-DC converters, small-signal analysis is conducted by introducing small disturbances into the state variables near the steady-state operating point. The linearized state-space model for buck mode operation is presented in the equations below.

$$\begin{gathered} \left[\begin{array}{c}{\displaystyle \frac{d\widetilde{i_{L_1}}\left(t\right)}{dt}}\\ {\displaystyle \frac{d\widetilde{i_{L_2}}\left(t\right)}{dt}}\\ {\displaystyle \frac{d\widetilde{v_C}\left(t\right)}{dt}}\\ {\displaystyle \frac{d\widetilde{v_{C_L}}\left(t\right)}{dt}}\end{array}\right]=\left[\begin{array}{cccc}{\displaystyle \frac{K_1}{L_1}}& {\displaystyle \frac{K_2}{L_1}}& {\displaystyle \frac{\left(D-1\right)}{L_1}}& {\displaystyle \frac{K_3}{L_1}}\\ {\displaystyle \frac{K_4}{L_2}}& {\displaystyle \frac{K_5}{L_2}}& -{\displaystyle \frac{D}{L_2}}& {\displaystyle \frac{K_6}{L_2}}\\ {\displaystyle \frac{\left(1-D\right)}{C}}& -{\displaystyle \frac{D}{C}}& 0& 0\\ {\displaystyle \frac{R_{DC}}{\left(R_{DC}+r_{C_L}\right)·C_L}}& {\displaystyle \frac{R_{DC}}{\left(R_{CC}+r_{C_L}\right)·C_L}}& 0& -{\displaystyle \frac{1}{\left(R_{DC}+r_{C_L}\right)·C_L}}\end{array}\right]·\left[\begin{array}{c}\widetilde{i_{L_1}}\\ \widetilde{i_{L_2}}\\ \widetilde{v_C}\\ \widetilde{v_{C_L}}\end{array}\right]+ \\ \left[\begin{array}{cc}{\displaystyle \frac{D}{L_1}}& {\displaystyle \frac{K_7}{L_1}}\\ 0& {\displaystyle \frac{K_8}{L_2}}\\ 0& {\displaystyle \frac{\left({-I}_{L_1}-I_{L_2}\right)}{C}}\\ 0& 0\end{array}\right]·\left[\begin{array}{c}\widetilde{v_{SC}}\\ \tilde{d}\end{array}\right] \end{gathered}$$

(12)

$$\left[\widetilde{v_o}\right]=\left[\begin{array}{cccc}{\displaystyle \frac{R_{DC}·r_{C_L}}{\left(R_{DC}+r_{C_{DC}}\right)}}& {\displaystyle \frac{R_{DC}·r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}}& 0& {\displaystyle \frac{R_{CC}}{\left(R_{DC}+r_{C_L}\right)}}\end{array}\right]·\left[\begin{array}{c}\widetilde{i_{L_1}}\\ \widetilde{i_{L_2}}\\ \widetilde{v_C}\\ \widetilde{v_{C_L}}\end{array}\right]$$

(13)

where

$$\begin{gathered} K_1=-D·\left(R_{DON4}+R_{L1}+R_{DC}\right)+{\displaystyle \frac{{R_{DC}}^2}{\left(R_{DC}+r_{C_L}\right)}}·D+\left(D-1\right)·\left(R_{DON1}+R_{DON2}+ \\ {R}_{L1}\right)+{\displaystyle \frac{R_{DON2}·R_{DC}}{\left(R_{DC}+r_{C_L}\right)}}·(1-D)+(D-1)·{\displaystyle \frac{R_{DC}·r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}} \end{gathered}$$

$$\begin{gathered} K_2={\displaystyle \frac{{R_{DC}}^2}{\left(R_{DC}+r_{C_L}\right)}}·D+R_{DON2}·\left(D-1\right)+{\displaystyle \frac{R_{DON2}·R_{DC}}{\left(R_{DC}+r_{C_L}\right)}}·\left(1-D\right)-R_{DC}·D+(D-1)· \\ {\displaystyle \frac{R_{DC}·r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}} \end{gathered}$$

$$K_3=-{\displaystyle \frac{R_{DC}}{\left(R_{DC}+r_{C_L}\right)}}·D+\left(D-1\right)+\left(D-1\right)·{\displaystyle \frac{R_{DON2}}{\left(R_{DC}+r_{C_L}\right)}}+(1-D)·{\displaystyle \frac{r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}}$$

$$K_4=\left(D-1\right)-{\displaystyle \frac{R_{DC}·r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}}+{\displaystyle \frac{R_{DON2}·R_{DC}}{\left(R_{DC}+r_{C_L}\right)}}·(1-D)$$

$$K_5=-\left(R_{DON3}+R_{L2}\right)·D+\left(D-1\right)·\left(R_{DON2}+R_{L2}\right)-{\displaystyle \frac{R_{DC}·r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}}+{\displaystyle \frac{R_{DON2}·R_{DC}}{\left(R_{DC}+r_{C_L}\right)}}·(1-D)$$

$$K_6={\displaystyle \frac{r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}}+{\displaystyle \frac{R_{DON2}}{\left(R_{DC}+r_{C_L}\right)}}·\left(D-1\right)-1$$

$$\begin{gathered} K_7=V_{SC}-I_{L_1}·\left(R_{DON4}+R_{L1}+R_{DC}\right)+{\displaystyle \frac{{R_{DC}}^2}{\left(R_{DC}+r_{C_L}\right)}}·\left(I_{L_1}+I_{L_2}\right)-{\displaystyle \frac{R_{DC}}{\left(R_{DC}+r_{C_L}\right)}}·V_{C_L}- \\ {I}_{L_2}·R_{DC}+I_{L_1}·\left(R_{DON1}+R_{DON2}+R_{L1}\right)+I_{L_2}·R_{DON2}+V_{C_L}+V_C-{\displaystyle \frac{R_{DON2}·R_{DC}}{\left(R_{CC}+r_{C_L}\right)}}· \\ \left(I_{L_1}+I_{L_2}\right)+{\displaystyle \frac{R_{DON2}}{\left(R_{DC}+r_{C_L}\right)}}·V_{C_L}+{\displaystyle \frac{R_{DC}·r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}}\left(I_{L_1}+I_{L_2}\right)-{\displaystyle \frac{r_{C_L}}{\left(R_{DC}+r_{C_L}\right)}}·V_{C_L} \end{gathered}$$

$$\begin{gathered} K_8=V_C-I_{L_2}·\left(R_{DON3}+R_{L2}\right)+I_{L_2}·\left(R_{DON2}+R_{L2}\right)+I_{L_1}·R_{DON2}-{\displaystyle \frac{R_{DON2}·R_{DC}}{\left(R_{DC}+r_{C_L}\right)}}· \\ \left(I_{L_1}+I_{L_2}\right)+{\displaystyle \frac{R_{DON2}}{\left(R_{DC}+r_{C_L}\right)}}·V_{C_L} \end{gathered}$$

The average small-signal models in the S-plane for the buck mode were defined. Based on the design parameters and the values calculated in the previous chapter for the operating mode and considering $\widetilde{v_{SC}}=0$, the transfer function representing the system ${\displaystyle \frac{\widetilde{v_o}}{\tilde{d}}}$ was obtained:

$${\displaystyle \frac{\widetilde{v_o}}{\tilde{d}}}={\displaystyle \frac{1728s^3+3.455\times {10}^{10}{s}^2-1.669\times {10}^{14}s+1.188\times {10}^{19}}{s^4+1.725\times {10}^4{s}^3+6.386\times {10}^8{s}^2+7.887\times {10}^{12}s+6.687\times {10}^{16}}}=G_{vod}(s)$$

(14)

4.2. Boost Operating Mode

To improve accuracy, the supercapacitor is modeled as a resistor and capacitor in series. In boost mode, the state variables are described by the following equations.

$$L_1{\displaystyle \frac{{di}_{L_1}(t)}{dt}}=v_{L_1}(t)$$

(15)

$$L_2{\displaystyle \frac{{di}_{L_2}(t)}{dt}}=v_{L_2}(t)$$

(16)

$$C{\displaystyle \frac{{dv}_C(t)}{dt}}=i_C(t)$$

(17)

$$C_H{\displaystyle \frac{{dv}_{C_H}(t)}{dt}}=i_{C_H}(t)$$

(18)

$$C_{SC}{\displaystyle \frac{{dv}_{C_{SC}}\left(t\right)}{dt}}=i_{C_{SC}}(t)$$

(19)

Equations (20) and (21) are the linearized state-space model for the boost mode.

$$\begin{gathered} \left[\begin{array}{c}{\displaystyle \frac{d\widetilde{i_{L_1}}\left(t\right)}{dt}}\\ {\displaystyle \frac{d\widetilde{i_{L_2}}\left(t\right)}{dt}}\\ {\displaystyle \frac{d\widetilde{v_C}\left(t\right)}{dt}}\\ {\displaystyle \frac{d\widetilde{v_{C_H}}\left(t\right)}{dt}}\\ {\displaystyle \frac{d\widetilde{v_{C_{SC}}}\left(t\right)}{dt}}\end{array}\right]=\left[\begin{array}{ccccc}{\displaystyle \frac{K_9}{L_1}}& -{\displaystyle \frac{D·R_{DON2}}{L_1}}& {\displaystyle \frac{D}{L_1}}& {\displaystyle \frac{K_{10}}{L_1}}& -{\displaystyle \frac{r_{C_H}}{\left(R_{SC}+r_{C_H}\right)·L_1}}\\ -{\displaystyle \frac{D·R_{DON2}}{L_2}}& {\displaystyle \frac{K_{11}}{L_2}}& {\displaystyle \frac{\left(D-1\right)}{L_2}}& 0& 0\\ -{\displaystyle \frac{D}{C}}& {\displaystyle \frac{\left(1-D\right)}{C}}& 0& 0& 0\\ {\displaystyle \frac{(1-D)·R_{SC}}{\left(R_{SC}+r_{C_H}\right)·C_H}}& 0& 0& -{\displaystyle \frac{1}{\left(R_{SC}+r_{C_H}\right)·C_H}}& {\displaystyle \frac{1}{\left(R_{SC}+r_{C_H}\right)·C_H}}\\ {\displaystyle \frac{(1-D)·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)·C_{SC}}}& 0& 0& {\displaystyle \frac{1}{\left(R_{SC}+r_{C_H}\right)·C_{SC}}}& -{\displaystyle \frac{1}{\left(R_{SC}+r_{C_H}\right)·C_{SC}}}\end{array}\right]· \\ \left[\begin{array}{c}\widetilde{i_{L_1}}\\ \widetilde{i_{L_2}}\\ \widetilde{v_C}\\ \widetilde{v_{C_H}}\\ \widetilde{v_{C_{SC}}}\end{array}\right]+\left[\begin{array}{cc}{\displaystyle \frac{1}{L_1}}& {\displaystyle \frac{K_{12}}{L_1}}\\ {\displaystyle \frac{1}{L_2}}& {\displaystyle \frac{K_{13}}{L_2}}\\ 0& {\displaystyle \frac{-\left(I_{L_1}+I_{L_2}\right)}{C}}\\ 0& -{\displaystyle \frac{I_{L_1}·R_{SC}}{\left(R_{SC}+r_{C_H}\right)·C_H}}\\ 0& -{\displaystyle \frac{I_{L_1}·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}\end{array}\right]·\left[\begin{array}{c}\widetilde{v_{DC}}\\ \tilde{d}\end{array}\right] \end{gathered}$$

(20)

$$\begin{gathered} \left[\begin{array}{c}\widetilde{v_o}\\ \widetilde{i_o}\end{array}\right]=\left[\begin{array}{ccccc}{\displaystyle \frac{(1-D)·R_{SC}·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}& 0& 0& {\displaystyle \frac{R_{SC}}{\left(R_{SC}+r_{C_H}\right)}}& {\displaystyle \frac{r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}\\ {\displaystyle \frac{(1-D)·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}& 0& 0& {\displaystyle \frac{1}{\left(R_{SC}+r_{C_H}\right)}}& -{\displaystyle \frac{1}{\left(R_{SC}+r_{C_H}\right)}}\end{array}\right]·\left[\begin{array}{c}\widetilde{i_{L_1}}\\ \widetilde{i_{L_2}}\\ \widetilde{v_C}\\ \widetilde{v_{C_H}}\\ \widetilde{v_{C_{SC}}}\end{array}\right]+ \\ \left[\begin{array}{cc}0& -{\displaystyle \frac{I_{L_1}·R_{SC}·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}\\ 0& -{\displaystyle \frac{I_{L_1}·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}\end{array}\right]\left[\begin{array}{c}\widetilde{v_{DC}}\\ \tilde{d}\end{array}\right] \end{gathered}$$

(21)

where

$$\begin{gathered} K_9=D·\left(R_{L1}+R_{DON4}\right)+D·{\displaystyle \frac{R_{SC}·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}-D·\left(R_{L1}+R_{DON1}+R_{DON2}\right)- \\ \left(R_{L1}+R_{DON4}\right)-{\displaystyle \frac{R_{SC}·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}} \end{gathered}$$

$$K_{10}=D·{\displaystyle \frac{R_{SC}}{\left(R_{SC}+r_{C_H}\right)}}-{\displaystyle \frac{R_{SC}}{\left(R_{SC}+r_{C_H}\right)}}$$

$$K_{11}=D·\left(R_{L2}+R_{DON3}\right)-D·\left(R_{L2}+R_{DON2}\right)-\left(R_{L2}+R_{DON3}\right)$$

$$\begin{gathered} K_{12}=I_{L_1}·\left((R_{L1}+R_{DON4})+{\displaystyle \frac{R_{SC}·r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}-\left(R_{L1}+R_{DON1}+R_{DON2}\right)\right)-I_{L_2}· \\ {R}_{DON2}+V_C+{\displaystyle \frac{R_{SC}}{\left(R_{SC}+r_{C_H}\right)}}·V_{C_H}+{\displaystyle \frac{r_{C_H}}{\left(R_{SC}+r_{C_H}\right)}}·V_{C_{SC}} \end{gathered}$$

$$K_{13}=-I_{L_1}·R_{DON2}+I_{L_2}·\left((R_{L2}+R_{DON3})-\left(R_{L2}+R_{DON2}\right)\right)+V_C$$

Similar to the buck mode, the average small-signal models in the s-plane were defined for the boost mode. The transfer function for ${\displaystyle \frac{\widetilde{i_o}}{\tilde{d}}}$ is derived from the design parameters and the calculated values, as expressed in the following equation:

$${\displaystyle \frac{\widetilde{i_o}}{\tilde{d}}}={\displaystyle \frac{-13.3s^5+1.81\times {10}^5{s}^4-3.857\times {10}^9{s}^4+5.39\times {10}^9{s}^2+1.129\times {10}^{18}s-12.81}{s^5+7112s^4+3.687\times {10}^8{s}^3+2.505\times {10}^{12}{s}^2+5.091\times {10}^{14}s-1.24\times {10}^{13}}}=G_{iod}(s)$$

(22)

5. Controller Design

In some systems, the closed-loop poles can be shifted to the desired position by adjusting the system gains. If the desired response is not achieved, it may be necessary to design a controller or compensator. Introducing this controller alters the system’s dynamics with the aim of correcting any undesirable behaviors. Based on the dynamic models of the two defined converter operating modes, the controller can be designed.

The values used in the controller design are derived from the equations presented for the converter and the actual component values.

5.1. Buck Operating Mode: Results

Table 4. Parameters used to design the control for buck operation.

Parameter	Specification
Rated power	$P_o$$=1 \mathrm{k}\mathrm{W}$
Input voltage	$V_{SC}$$=162 \mathrm{V}$
Output voltage	$V_{CC}$$=48 \mathrm{V}$
Maximum cyclic ratio	$D_{Buck}=0.55$
Switching frequency	$f_s$$=500 \mathrm{k}\mathrm{H}\mathrm{z}$
$\mathrm{Inductance} L_1$	$L_1=300 \mathsf{\mu }\mathrm{H}$
$\mathrm{Inductance} L_2$	$L_2=100 \mathsf{\mu }\mathrm{H}$
Central capacitance	$C=10 \mathsf{\mu }\mathrm{F}$
Output capacitance	$C_{DC}=50 \mathsf{\mu }\mathrm{F}$
Load resistance	$R_{DC}$$=2.304$
Series resistance of inductors	$R_L$$=4 \mathrm{m}\mathsf{\Omega }$
Output capacitor series resistance	$R_C$$=1 \mathrm{m}\mathsf{\Omega }$
Source-drain resistor	$R_{DS\left(ON\right)}$$=15 \mathrm{m}\mathsf{\Omega }$

presents the parameters of the buck converter. It is important to note that, to achieve realistic results, commercial values have been used for the components.

The voltage control is based on the difference between the reference voltage value $V_{ref}$ and the feedback signal $v_o$, multiplied by the voltage gain $H_v\left(s\right)$, resulting in a voltage error $e_v$. This error is used as the input signal for the controller (compensator) $C_v\left(s\right)$. The output of this process generates a control signal u, which, when compared to the carrier, produces a PWM signal $d$. The diagram corresponding to current control is shown in Figure 8.

The uncompensated plant $G_{vod}\left(s\right)$ exhibited a non-zero steady-state error and unstable closed-loop poles. To enhance the output voltage performance—targeted to remain around 48 V—a PI controller was designed. This controller introduces an open-loop pole to eliminate the steady-state error and shifts the closed-loop poles to stabilize the system.

The transfer function of the designed PI controller, considering the plant $G_{vod}\left(s\right)$, is presented in Equation (23). The controller ensures zero steady-state error and relocates the control poles to the left half of the complex plane, thereby stabilizing the system.

$$PI={\displaystyle \frac{1.27\times {10}^{-6}s+0.004001}{0.0003183s}}$$

(23)

Analysis of Results—Buck Operation

The Bode plot and the geometric root locus, obtained after the applied adjustments, are shown in Figure 9 and Figure 10, respectively.

In Figure 9, the Bode plot reveals key insights into the dynamic behavior and stability of the system (PI controller and converter) during the buck operation (discharging mode). A distinct resonance peak is observed around 10⁴ rad/s in the magnitude response, indicating the presence of lightly damped poles. The phase response exhibits a sharp drop near this frequency, crossing −180°, while the corresponding gain remains below 0 dB, ensuring a positive gain margin and suggesting that the system is stable. However, the phase margin, measured at the 0 dB crossover frequency (approximately 6 × 10³ rad/s), appears to be relatively low—around 20° to 30°—which may lead to underdamped transient behavior.

Moreover, the Bode plot depicts that the implemented control strategy successfully reduced the gain crossover frequency compared to the system without the PI controller, which yielded to a reduction in the noise and oscillations.

In Figure 10, it is shown that the controller introduced a zero into the bidirectional interleaved high-gain converter transfer function and, thus, maintained its operation in the stable zone.

The step response resulting from the voltage control applied to the buck converter is presented in Figure 11. In response to the step input—produced by the action of the controller designed for step-down mode operation—a rise time of 0.0011 s was observed, indicating an extremely fast response and high dynamic performance. The settling time, defined as the period required for the output to remain within a specified tolerance band around the steady-state value, was 0.0023 s, demonstrating the system’s rapid convergence to stability. In this mode of operation, the response is overdamped, as no overshoot was recorded; the output smoothly reached the final value with an overshoot of 0%. These results confirm that the system exhibits stable and accurate dynamic behavior with excellent transient performance.

To validate the developed model and controller for the buck converter, computer simulations were carried out using PSIM^TM software. The parameters listed in Table 4 were applied during the simulations. The output voltage $V_o$ and output current $I_o$ obtained from the simulations are shown in Figure 12.

Figure 12 shows that the output voltage and current reach a steady state within 0.005 s, exhibiting minimal ripple on the DC bus. The output voltage stabilizes precisely at $V_o=48 \mathrm{V}$, while the output current is $I_o=20.83 \mathrm{A}$, resulting in an output power of $P_o=1 \mathrm{k}\mathrm{W}$.

One of the design objectives of the converter was to minimize the stress on the inductors. Therefore, the design parameters were selected accordingly. Based on the parameters listed in Table 2 and the implemented control strategy, the current profiles of the inductors were obtained, as illustrated in Figure 13.

The inductor currents $I_{L_1}$ and $I_{L_2}$ reach steady-state behavior in approximately 0.015 s. After the transient period, the average current through $I_{L_1}$ is 11.37 A, while $I_{L_2}$ averages 9.46 A. The current ripple in both inductors is low—approximately 0.4 A—which remains within the design parameters, as illustrated in Figure 14.

.

5.2. Boost Operating Mode

Table 5. Parameters used to design the control for boost operation.

Parameter	Specification
Rated power	$P_o$$=1 \mathrm{k}\mathrm{W}$
Input voltage	$V_{CC}$$=48 \mathrm{V}$
Output voltage	$V_{SC}$$=162 \mathrm{V}$
Maximum cyclic ratio	$D_{Boost}=0.45$
Switching frequency	$f_s$$=500 \mathrm{k}\mathrm{H}\mathrm{z}$
$\mathrm{Inductance} L_1$	$L_1=300 \mathsf{\mu }\mathrm{H}$
$\mathrm{Inductance} L_2$	$L_2=100 \mathsf{\mu }\mathrm{H}$
Central capacitance	$C=10 \mathsf{\mu }\mathrm{F}$
Output capacitance	$C_H=3.5 \mathrm{m}\mathrm{F}$
Supercapacitor capacitance	$C_{SC}=62 \mathrm{F}$
Load resistance adopted	$R_{SC}=26.244 \mathsf{\Omega }$
Series resistance of the inductors	$R_L$$=4 \mathrm{m}\mathsf{\Omega }$
Output capacitor series resistance	$R_C$$=1 \mathrm{m}\mathsf{\Omega }$
Source-drain resistor	$R_{DS\left(ON\right)}$$=15 \mathrm{m}\mathsf{\Omega }$

presents the parameters of the converter operating in the boost direction.

Similar to voltage control, current control is achieved by the difference between the current reference value $I_{ref}$ and the feedback signal $i_o$, multiplied by the gain $H_i\left(s\right)$, which in this case equals 1. The difference results in a current error signal $e_i$, which is used as the input signal for the control block $C_i\left(s\right)$. This block can represent both a controller and a compensator. The product generates a control signal u, which, when compared with the carrier, produces a PWM signal $d$. This signal is then used to control the switches. This control method is particularly well suited for converters operating at high switching frequencies. The diagram for the current control is illustrated in Figure 15.

The uncompensated $G_{iod}\left(s\right)$ plant exhibited a non-zero steady-state error and unstable closed-loop poles. A PI controller was initially designed to eliminate the steady-state error. However, unlike the buck converter, the PI controller alone was insufficient to resolve all the issues in the $G_{iod}\left(s\right)$ plant. To address the system’s instability, it was necessary to design a phase advance compensator in combination with the PI controller. This combination not only eliminated the steady-state error but also relocated the unstable poles to the left half-plane, stabilizing the system.

The transfer function for the PI controller designed for the $G_{iod}\left(s\right)$ plant is shown in Equation (24).

$$PI={\displaystyle \frac{1.829\times {10}^{-6}s+0.00057}{0.003183s}}$$

(24)

The transfer function corresponding to the calculated phase advance compensator is given by Equation (25).

$$G_{comp.}(s)={\displaystyle \frac{0.5s+187.1}{s+263.73}}$$

(25)

Finally, the transfer function obtained by the product of the PI controller and phase advance compensator is given by Equation (26).

$$C_i(s)={\displaystyle \frac{9.145\times {10}^{-7}{\mathrm{s}}^2+0.6296\times {10}^{-3}\mathrm{s}+0.1075}{3.183\times {10}^{-3}{\mathrm{s}}^2+0.8395\mathrm{s}}}$$

(26)

Analysis of Results—Boost Operation

The Bode plot and the geometric root locus, obtained after implementing the PI controller and the phase advance compensator, are shown in Figure 16 and Figure 17, respectively.

In Figure 16, The Bode plot of the system composed of the PI controller, compensator, and converter reveals critical aspects of the system’s frequency-domain behavior. The magnitude plot shows a high gain at low frequencies—above 100 dB—indicating excellent tracking and disturbance rejection capabilities in a steady state. The gain remains flat over a wide mid-frequency range, before gradually decreasing, suggesting the presence of dominant low-frequency poles and effective bandwidth limitation. At high frequencies, the sharp drop in gain reflects appropriate attenuation of noise and high-frequency disturbances. The phase plot exhibits a consistent phase lead above 500° in the low- to mid-frequency range, likely due to the PI controller and compensator action. A rapid phase decline occurs near 10⁵ rad/s, which indicates the system’s stability boundary. The plot overall confirms that the controller–compensator combination ensures strong low-frequency performance while maintaining attenuation and stability at higher frequencies.

In Figure 17, the zeros introduced by the PI controller and phase advance compensator have shifted the poles, which were previously located in the right half-plane, to the left half-plane. This fact contributed to the system’s stabilization during the supercapacitor charging mode.

The step response resulting from the control applied to the boost converter is shown in Figure 18. By analyzing the closed-loop response, the dynamic behavior of the converter operating in boost mode under the action of the designed controller was evaluated. The system exhibited a rise time of 0.0072 s, reflecting high agility during the initial transition. The settling time was measured at 0.0229 s. Additionally, the response was underdamped, with an overshoot of 6.63%, indicating moderate oscillation before stabilization. Based on these parameters, it can be concluded that the controller implemented in the high-voltage-gain DC-DC converter provides satisfactory dynamic performance, characterized by a fast response, minimal overshoot, and efficient stabilization—demonstrating the controller’s effectiveness for this mode of operation.

Computer simulations were conducted using PSIM^TM software to validate the PI controller model and compensator designed for the converter operating in boost mode. The parameters listed in Table 5 were used for the simulations. The output voltage ($V_o$) and output current ($I_o$) obtained from the PSIM simulations for the boost converter are shown in Figure 19.

Figure 19 illustrates the charging voltage and current behavior of the supercapacitor from the 48 V DC bus. Given that the initial voltage of the supercapacitor is 0 V, upon system startup, the supercapacitor voltage peaks at nearly 80 V, after which it drops to 50 V and remains there for 7 s. It is worth mentioning that the controller designed for the bidirectional interleaved high-gain converter was able to reduce this non-controlling period to 7 s compared to the converters discussed in the Section 1.2, which were greater than 100 s.

Prior to actual charging, there is a transient phase lasting approximately 0.7 s, followed by the charging process, which lasts around 20 s. The transient period can be mitigated by fine tuning in the converter’s controller, adjusting the phase margin of the compensator.

Regarding the output current during the charging process, the control strategies developed ensure charging with a constant current of $I_o=6 \mathrm{A}$. The transient phase observed in the voltage $V_o$ coincides with the current $I_o$, occurring within the same time interval and duration. The final output power transferred from the DC bus to the supercapacitor is $P_o=1.015 \mathrm{k}\mathrm{W}$. In addition, the currents in inductors $I_{L_1}$ and $I_{L_2}$ exhibit minimal ripple in the steady state, in accordance with the design parameters outlined in Table 2. The waveforms of the currents through the inductors $I_{L_1}$ and $I_{L_2}$ are shown in Figure 20.

6. Conclusions

Hybrid systems are crucial for ensuring the stability and reliability of renewable energy sources, and bidirectional DC-DC converters play a pivotal role in this context. This study explored the use of supercapacitors in a high-voltage-gain bidirectional buck–boost converter, which helps reduce the size and weight of components while minimizing stress on the inductors. Dynamic state-space modeling, accounting for losses, provided a more realistic representation of the system. For control, PI controllers and a phase advance compensator were implemented to maintain stability on the DC bus and ensure a constant current during supercapacitor charging. The simulations demonstrated the effectiveness of control in handling disturbances, confirming the feasibility of using supercapacitors for efficient energy management in hybrid systems.

In addition to these contributions, the proposed system stands out by integrating realistic modeling with practical design parameters and a concrete control structure, distinguishing it from previous studies that often focused on generalized topologies or theoretical approaches. This initial phase has been validated through comprehensive computational simulations using PSIM software. Future work will involve the experimental implementation of the system under real-world operating conditions to further evaluate its performance and confirm its practical applicability in hybrid energy storage environments.