Research on a DC–DC Converter and Its Advanced Control Strategy Applied to the Integrated Energy System of Marine Breeding Platforms

Abstract

The deep-sea aquaculture industry will become one of the important pillars of the future marine economy. However, the application of clean energy in the new scenario needs to be strengthened for platform operation. For this kind of renewable-energy distributed-generation system, an energy storage system is essential. A bidirectional DC–DC converter is essential for distributed power generation systems. It connects a variety of renewable energy sources with energy storage cells. A high-gain bidirectional Cuk circuit with zero ripple is proposed in the paper. It is characterized by a simple structure, zero ripple, low voltage stress of semiconductor power devices, and high voltage gain. A passivity-based control with linear active disturbance rejection is proposed to solve the problems of the large steady-state error. The zero steady-state error, strong robustness, and whole-range stability have been obtained for the proposed control strategy. Finally, a simulation was carried out. A 100 W, 48 V/400 V prototype was built to verify the validity of the theoretical analysis for the proposed circuit. The improved passivity-based control strategy was verified to solve the contradiction between rapidity and overshoot. It can be realized to improve the dynamic performance of the proposed converter and achieve robust control.

1. Introduction

The ocean is known as the “blue granary”. Most of the traditional marine fisheries are extensive aquaculture in bays or shallow beaches. Now, with the rapid development of marine fisheries, the aquaculture mode has changed from “light blue” to “deep blue” [1]. The profound marine aquaculture industry will become one of the important pillars of the future marine economy. However, most of the existing deep-water aquaculture platforms only use fuel engines as energy devices, with high noise, many pollutants, and amplified harmful gas emissions. Moreover, with the development of the deep-sea and offshore aquaculture industry, the energy required for platform operation is growing continuously, and the consumption of traditional fossil energy is huge. The massive use of oil will lead to marine environmental pollution, restricting its sustainable development [2]. Therefore, attention should be given to the development and utilization of clean energy such as solar energy and wind energy. The integrated energy system of an aquaculture platform is subject to the intermittent strong impact of clean energy, which easily causes untimely energy regulation and unsafe offshore production. The safe, efficient, and green integrated energy utilization of aquaculture platforms is facing new theoretical and technical challenges [3].

The development of distributed power generation systems is a resolution to mitigate high fossil-fuel dependency, especially in the deep-sea aquaculture industry where the possibility of grid connection is still uncertain. Most of them are heavily dependent on transported fossil fuels. Therefore, installing a distributed power generation system for these aquaculture industries is seen as very promising. The distributed power generation system and its power conversion are interrelated systems. A suitable power converter system is designed based on the output voltage or current amplitude at the generator side and to match the conditions at the load. This process ensures a smooth power transfer from generator side to the loads. Among them, a bidirectional DC–DC converter is essential for distributed power-generation systems. It connects the bus terminal with energy storage cells such as batteries or supercapacitors, and realizes the management and rational utilization of system energy [4,5].

For renewable-energy power-generation systems on deep-sea aquaculture platforms, a DC microgrid mode is often used because of its simple structure, no phase synchronization, reactive power loss, and harmonics [6]. Additionally, the DC bus voltage is usually high. However, the output voltage of batteries is usually low. An additional battery equalization circuit is required if the output port voltage is increased by the battery series. It increases unnecessary costs and reduces system reliability. In addition, the load of the breeding platform includes the mixer, feeder, household electricity, and so on. These devices start and stop frequently with different needs in the process of breeding. It causes the power load to fluctuate constantly. Hence, the bidirectional converters require high voltage gain and high efficiency. Advanced control strategy is indispensable, which makes the system maintain stable and fast dynamic response. However, there are few high-gain bidirectional converters and their advanced control strategies are necessary in the marine field. This paper is a good supplement to the research gap and needs.

In recent years, to improve the voltage gain and the conversion efficiency of the converter, some bidirectional high-gain DC–DC converters have been proposed. For example, [7] proposes a quadratic bidirectional buck/boost circuit with the same gain as the basic cascade type. However, there are many semiconductor devices in this circuit, and the voltage stress of semiconductor devices is large on the high-voltage side. As described in [8], cascading different bidirectional converters by multiplexing the switching tubes increases the gain and reduces the number of switches, but switching the circuit causes high voltage stress on the output side. In addition, networks such as switched capacitors [9], switched inductors [10], coupled inductors [11,12], and Z-sources [13] are introduced to increase the range in voltage gain for the converter. Among them, a family of pure switched-capacitor, bidirectional high-gain DC–DC circuits is proposed in [9]. It can enable higher power density due to the absence of magnetic components. However, its output voltage can only be an integer multiple of the input voltage, and there is a current spike on the capacitor. Based on a bidirectional DC–DC circuit, a switching inductor unit is introduced [10], which limits the current spikes of the pure switched capacitor. However, the voltage stress of switching is large on the high-voltage side. To increase the voltage gain, a composite nonisolated, bidirectional buck/boost circuit is proposed by multiplexing the inductor as the primary winding of the flyback converter [11]. However, the current waveform of this topology is a square wave on the low-voltage side, and the current ripple on the low-voltage side is large. Accordingly, the tapped inductor is introduced into bidirectional quadratic DC–DC converters to improve the converter gain and reduce the current ripple on the low-voltage side [12], but there is also the problem of the large voltage stress of the switching on the high-voltage side. In addition, [13] introduces a Z-source network into the bidirectional buck/boost converter, improving the voltage-gain range and reliability of the converter. However, it increases the current stress of the main switch. These studies are mainly concentrated on systems connected to a large power grid. However, owing to the particularity of the marine environment, current research on high-gain bidirectional circuits is minimal.

In addition, because load variations and external disturbances are often uncertain and fluctuating in the marine environment, the traditional control strategy cannot guarantee the dynamic response of the system over a wide range of input and load variations, as well as making it difficult to further improve system performance. Passive-based control was successfully applied to DC–DC converters, as described in [14]. This scheme is a globally stable control strategy with simple implementation and strong robustness. However, there are steady-state output errors. The amount of injection damping affects the effectiveness of the control and there is no single design method for optimal damping. To analyze the effect of injection damping, the range of injection damping was analyzed and a genetic algorithm was used to find the optimal injection damping [15], but the calculation process is complicated. To reduce the steady-state output error, the parasitic resistance of the device and the on-state voltage drop of the semiconductor device are considered in the modeling process [16]. However, it is not conducive to real-time control due to its computational complexity. A novel passive-based control is proposed by combining it with PI control [17]. Zero steady-state error was achieved by obtaining the current reference through the voltage outer loop, but there is an irreconcilable conflict between rapidity and overshoot. In addition, the proposed control strategy described in [18] reduces the jitter under sliding mode control and suppresses the voltage overshoot under passive-based control by combining sliding mode control with passive control. However, there are still steady-state errors, narrow-amplitude vibration, and a large output ripple in this control. A passivity-based control strategy with a nonlinear disturbance observer is proposed in [19], where the effects of parameter uncertainty and system disturbances are compensated by a nonlinear disturbance observer. It reduces the steady-state error under typical passive control and improves the control robustness of the system, but the implementation of the existing observer is more complicated. Moreover, IDA-PBC has been proposed in [20] as a control technique for designing high-performance nonlinear controllers for systems described by port-Hamiltonian models. This method achieves stabilization by rendering the system passive with respect to a desired storage function and injecting damping. The development of IDA-PBC control law via modification in interconnection and damping matrices for the rectifier is discussed in [21,22].

In the marine environment, the current research on high-gain bidirectional circuits and their control strategy is minimal. A bidirectional high-gain Cuk converter with zero-ripple is proposed in this paper. It uses switched-capacitor gain units to increase the voltage gain range. The EL model of the proposed converter is developed from the global energy perspective based on the dynamic modeling method. Then, based on this EL model, passivity-based control with linear active disturbance rejection is proposed to solve the problem of large steady-state errors that are caused by model errors and disturbances in the bidirectional high-gain converter using typical passivity-based control. The control strategy is characterized by strong robustness and global stability. Afterward, the proposed converter and control strategy are specifically analyzed in the paper, and the voltage gain relationship, ripple characteristics, and voltage stress characteristics are analyzed in detail. Finally, the important parameters of the proposed converter and control strategy are designed and analyzed in this paper. As described herein, a simulation was carried out and a 100W, 48V/400V prototype was built to verify the validity of the theoretical analysis of the proposed circuit. The simulation and experimental results show that the proposed circuit achieves high voltage gain, low voltage stress, and zero ripple. The proposed control method improves the dynamic performance of the converter and achieves robust control and zero steady-state error.

The remainder of this paper is organized as follows. Section 2 provides the proposed converter’s operation process and principle. In Section 3, the improved control strategy is formulated. The simulation and experimental results are presented in Section 4. Discussion is provided in Section 5. Finally, Section 6 concludes this paper.

2. Bidirectional High-Gain Cuk Converter

2.1. Topology of Circuit

By combining basic a bidirectional Cuk circuit with a gain cell, a bidirectional high-gain Cuk converter with zero ripple is proposed in this paper. The basic bidirectional Cuk is shown in Figure 1, and the proposed converter is shown in Figure 2. The circuit consists of V₁, V₂, L₁, L₂, S_B1, S_B2, C_B, and the gain network of C_2n−1, C_2n, S_2n−1, and S_2n, where the diodes D_B1, D_B2, D₁, …, D_2n are parasitic body diodes corresponding to power switching.

2.2. Operating Principle of the Converter

We take the bidirectional high-gain Cuk circuit with a basic gain network as an example to illustrate the operating principle and operating process for the proposed converter, and its equivalent circuit in step-up/step-down mode is shown in Figure 3 and Figure 4. The proposed circuit operates with V₁ as the high-voltage side and V₂ as the low-voltage side. This circuit exists in two electrical energy conversion directions, such as step-up direction conversion and step-down direction conversion. When the converter works in the step-up mode, the low-voltage side provides energy to the high-voltage side, and S_B2, D_B1, D₁, and D₂ are in working condition. When the converter works in the step-down mode, the high-voltage side provides energy to the low-voltage side, and S_B1, S₁, S₂, and D_B2 are in working condition.

To simplify the analysis, the following assumptions are made: (1) the voltage across the capacitor is constant at steady state; (2) all devices are ideal devices and the effect of parasitic parameters is not considered; (3) the circuit operates in CCM mode.

2.2.1. Step-Up Mode

At steady state, there are two operating stages in one duty cycle. The equivalent circuit of each switching stage and its main operating waveforms are shown in Figure 3 and Figure 4, where v_{gs_SB2} is the drive signal for S_B2. As shown in the figures, i_L1, i_L2 is the current flowing through inductor L₁, L₂; i_C1, i_C2, i_CB is the current flowing through capacitor C₁, C₂, C_B; i_SB2, i_D1, i_D2, i_DB1 is the current flowing through S_B2, D₁, D₂, D_B1; and T_s is a switching cycle. The operating process of the circuit is as follows.

Mode 1 (t₁–t₂): The equivalent operating circuit in this stage is shown in Figure 3a. At the moment of t₁, S_B2 and D₂ are turned on, diodes D₁ and D_B1 are turned off. The low-voltage side charges inductor L₁. Capacitor C₁ and low-voltage side charges C₂, and the low-voltage side, capacitor C₁, and capacitor C_B are connected in series to charge inductor L₂ and supply the high-voltage side. The inductor current reaches its maximum value until moment t₂. In this mode, the voltage and current of the inductor L₁ are:

$$\{\begin{array}{l}{v}_{L1\_1}=V_2=V_{C2}-V_{C1}\hfill \\ \begin{array}{l}{i}_{L1\_1}(t)=I_2-\frac{v_{L1\_1}{D}_{up}}{2L_1{f}_{\mathrm{s}}}+\frac{v_{L1\_1}}{L_1}t\\ \Delta i_{L1\_1}=\frac{v_{L1\_1}{D}_{up}}{L_1{f}_{\mathrm{s}}}\end{array}\hfill \end{array}$$

(1)

The voltage and current of inductor L₂ are:

$$\{\begin{array}{l}{v}_{L2\_1}=V_{C2}+V_{C\mathrm{B}}-V_1\hfill \\ \begin{array}{l}{i}_{L2\_1}(t)=I_1-\frac{v_{L2\_1}{D}_{up}}{2L_2{f}_{\mathrm{s}}}+\frac{v_{L2\_1}}{L_2}t\\ \Delta i_{L2\_1}=\frac{v_{L2\_1}{D}_{up}}{L_1{f}_{\mathrm{s}}}\end{array}\hfill \end{array}$$

(2)

where I₂ and I₁ are the average values of the input current on the low-voltage side and the output current on the high-voltage side, respectively. V_C1 is the voltage across capacitor C₁, V_C2 is the voltage across capacitor C₂, and V_CB is the voltage across capacitor C_B.

Mode 2 (t₂–t₃): The equivalent operating circuit in this stage is shown in Figure 3b. At the moment of t₂, S_B2 and D₂ are turned off, diodes D₁ and D_B1 are turned on. Inductor L₁ charges capacitor C₁. The low-voltage side and inductor L₁ charge C_B. The low-voltage side, inductor L₁, and capacitor C₂ are connected in series to supply the high-voltage side. In this mode, the voltage and current of the inductor L₁ are:

$$\{\begin{array}{l}{v}_{L1\_2}=V_2-V_{C\mathrm{B}}=-V_{C1}\hfill \\ \begin{array}{l}{i}_{L1\_2}(t)=I_2-\frac{v_{L1\_2}(1-D_{up})}{2L_1{f}_{\mathrm{s}}}+\frac{v_{L1\_2}}{L_1}(t-D_{up}{T}_{\mathrm{s}})\\ \Delta i_{L1\_2}=\frac{v_{L1\_2}(1-D_{up})}{L_1{f}_{\mathrm{s}}}\end{array}\hfill \end{array}$$

(3)

The voltage and current of the inductor L₂ are:

$$\{\begin{array}{l}{v}_{L2\_2}=V_2+V_{C1}+V_{C\mathrm{B}}-V_1=V_{C2}+V_{C\mathrm{B}}-V_1\hfill \\ \begin{array}{l}{i}_{L2\_2}(t)=I_1-\frac{v_{L2\_2}(1-D_{up})}{2L_2{f}_{\mathrm{s}}}+\frac{v_{L2\_2}}{L_2}(t-D_{up}{T}_{\mathrm{s}})\\ \Delta i_{L2\_2}=\frac{v_{L2\_1}(1-D_{up})}{L_1{f}_{\mathrm{s}}}\end{array}\hfill \end{array}$$

(4)

The inductor current reaches its minimum value until moment t₃. This stage ends when the switching S_B2 turns on at moment t₃.

2.2.2. Step-Down Mode

At steady state, there are two operating stages in one duty cycle. The equivalent circuit of each switching stage and its main operating waveforms are shown in Figure 5 and Figure 6, where v_{gs_SB1} is the drive signal for S_B1, v_{gs_S1} is the drive signal for S_B1, and v_{gs_S2} is the drive signal for S₂. As shown in the figures, i_L1, i_L2 is the current flowing through inductor L₁, L₂; i_C1, i_C2, i_CB is the current flowing through capacitor C₁, C₂, C_B; i_SB2, i_D1, i_D2, i_DB1 is the current flowing through S_B2, D₁, D₂, D_B1; and T_s is a switching cycle. The operating process of the circuit is as follows.

Mode 1 (t₁–t₂): The equivalent operating circuit in this stage is shown in Figure 5a. At the moment of t₁, S₂ and D_B1 are turned on, capacitor C₁ charges inductor L₁. The high-voltage side V₁ charges capacitor C₂, L₁, L₂ and the low-voltage side. Capacitor C_B supplies the low-voltage side and L₁. The inductor current reaches its maximum value until moment t₂. In this mode, the voltage and current of the inductor L₁ are:

$$\{\begin{array}{l}{v}_{L1\_1}=V_{C1}=V_{CB}-V_2\\ i_{L1\_1}(t)=I_2-\frac{v_{L1\_1}{D}_{down}}{2L_1{f}_{\mathrm{s}}}+\frac{v_{L1\_1}}{L_1}t\\ \Delta i_{L1\_2}=\frac{v_{L1\_1}{D}_{down}}{L_1{f}_{\mathrm{s}}}\end{array}\begin{array}{l}\hfill \\ \hfill \end{array}$$

(5)

The voltage and current of the inductor L₂ are:

$$\{\begin{array}{l}{v}_{L2\_1}=V_1-V_{C2}-V_{C\mathrm{B}}\hfill \\ \begin{array}{l}{i}_{L2\_1}(t)=I_1-\frac{v_{L2\_1}{D}_{down}}{2L_2{f}_{\mathrm{s}}}+\frac{v_{L2\_1}}{L_2}t\\ \Delta i_{L2\_1}=\frac{v_{L2\_1}{D}_{down}}{L_2{f}_{\mathrm{s}}}\end{array}\hfill \end{array}$$

(6)

Mode 2 (t₂–t₃): The equivalent operating circuit in this stage is shown in Figure 5b. At the moment of t₂, S_B1 and S₁ are turned off, S₂ and D_B1 are turned on. Inductor L₁ charges the low-voltage side. Capacitor C₂ charges C₁ and the low-voltage side. The high-voltage side V₁ charges capacitor C₁, C_B, and the low-voltage side. In this mode, the voltage and current of the inductor L₁ are:

$$\{\begin{array}{l}{v}_{L1\_2}=-V_2=V_{C1}-V_{C2}\hfill \\ \begin{array}{l}{i}_{L1\_2}(t)=I_2+\frac{v_{L1\_2}{D}_{down}}{2L_1{f}_{\mathrm{s}}}+\frac{v_{L1\_2}}{L_1}(t-D_{down}{T}_{\mathrm{s}})\\ \Delta i_{L1\_2}=\frac{v_{L1\_2}{D}_{down}}{L_1{f}_{\mathrm{s}}}\end{array}\hfill \end{array}$$

(7)

The voltage and current of the inductor L₂ are:

$$\{\begin{array}{l}{v}_{L2\_2}=V_1-(V_2+V_{C1}+V_{C\mathrm{B}})=V_1-V_{C2}-V_{C\mathrm{B}}{}_1\hfill \\ \begin{array}{l}{i}_{L2\_2}(t)=I_1+\frac{v_{L2\_2}{D}_{down}}{2L_2{f}_{\mathrm{s}}}+\frac{v_{L2\_2}}{L_2}(t-D_{down}{T}_{\mathrm{s}})\\ \Delta i_{L2\_2}=\frac{v_{L2\_2}{D}_{down}}{L_2{f}_{\mathrm{s}}}\end{array}\hfill \end{array}$$

(8)

The inductor current reaches its minimum value until moment t₃. This stage ends when the switching S_B1 and S₁ turns on at moment t₃.

2.3. Steady-State Characterization

2.3.1. Voltage Gain

• Step-up mode

At steady state, the inductance characteristics of the proposed converter satisfy the volt–second balance relation, and the average value of the intermediate capacitance–voltage can be obtained from Equations (1) and (3).

$$\{\begin{array}{l}{V}_{C1}=\frac{D_{up}}{1-D_{up}}{V}_2\hfill \\ V_{C2}=\frac{1}{1-D_{up}}{V}_2\hfill \\ V_{CB}=\frac{1}{1-D_{up}}{V}_2\hfill \\ V_1=V_{C2}+V_{CB}=\frac{2}{1-D_{up}}{V}_2\hfill \end{array}$$

(9)

The voltage gain of the bidirectional high-gain Cuk circuit with the basic gain network in step-up mode is:

$$M_{up}=\frac{V_1}{V_2}=\frac{2}{1-D_{up}}$$

(10)

Similarly, by analyzing the operating principle of the bidirectional high-gain Cuk circuit with multigain network, we can obtain the voltage gain in step-up mode as follows.

$$M_{up}=\frac{V_1}{V_2}=\frac{1+n}{1-D_{up}}$$

(11)

where n is the number of gain cells.

2.

Step-down mode

Similarly, by analyzing the operating principle, we can obtain the voltage gain of bidirectional high-gain Cuk circuit with basic gain network in step-down mode as follows.

$$M_{down}=\frac{V_2}{V_1}=\frac{D_{down}}{2}$$

(12)

The voltage gain of bidirectional high-gain Cuk circuit with multigain network in step-down mode:

$$M_{down}=\frac{V_2}{V_1}=\frac{D_{down}}{1+n}$$

(13)

2.3.2. Zero Ripple Characteristics

• Step-up mode

At steady state, the inductance characteristics of L₂ in step-up mode satisfy the volt–second balance relation. It is also known that the voltage across inductor L₂ in one switching cycle is the same from Equations (2) and (4). The voltage of L₂ is:

$$\begin{array}{l}{v}_{L2\_1}=v_{L2\_2}=V_{C2}+V_{CB}-V_1\\ =\frac{1}{1-D_{up}}{V}_2+\frac{1}{1-D_{up}}{V}_2-\frac{2}{1-D_{up}}{V}_2=0\end{array}$$

(14)

Therefore, the voltage across the inductor L₂ is constantly zero, and it is known that the instantaneous the current of L₂ is:

$$i_{L2}(t)=I_1$$

(15)

Therefore, the proposed converter achieves zero current ripple on the high-voltage side.

2.

Step-down mode

Similarly, analyzing the operating principle, the proposed converter achieves zero current ripple on the high-voltage side.

2.3.3. Voltage Stress of the Power Device

• Step-up mode

Analyzing the operation of the converter, the voltage stress of S_B2 is:

$$v_{\mathrm{SB}2\_\max }=V_{CB}=\frac{1}{1-D_{up}}{V}_2=\frac{V_1}{2}$$

(16)

The voltage stress of D₁, D₂, and D_B1 is:

$$v_{\mathrm{D}1.\max }=v_{\mathrm{D}2.\max }=v_{\mathrm{DB}1.\max }=\frac{1}{1-D_{up}}{V}_2=\frac{V_1}{2}$$

(17)

Similarly, at the step-up mode, the voltage stress of all power devices in the bidirectional high-gain Cuk circuit with multigain network is:

$$v_{S.\max }=\frac{V_1}{1+n}$$

(18)

2.

Step-down mode

The voltage stress of S₁, S₂, and S_B1 is:

$$v_{\mathrm{S}1.\max }=v_{\mathrm{S}2.\max }=v_{\mathrm{SB}1\_\max }=\frac{1}{D_{down}}{V}_2=\frac{V_1}{2}$$

(19)

The voltage stress of D_B2 is:

$$v_{\mathrm{DB}2\_\max }=V_{CB}=\frac{1}{D_{down}}{V}_2$$

(20)

Similarly, at the step-down mode, the voltage stress of all power devices in the bidirectional high-gain Cuk circuit with multigain network is:

$$v_{S.\max }=\frac{V_1}{1+n}$$

(21)

2.3.4. Characteristic Comparison

The characteristic comparison of the proposed converter with the counterparts is shown in

Table 1. Characteristic comparison of the proposed converter with its main competitors.

Bidirectional Converter	Max. Voltage Stress of Switches	Efficiency	Structural Complexity	NS	Ripple	Voltage Gain
						Step-Up	Step-Down
Buck/Boost Converter	VH	-	simple	2	large	$\frac{1}{1-D}$	D
Converter described in [23]	VH	88.9~92.3% (250 W)	complex	4	large	$\frac{1}{{\left(1-D\right)}^2}$	D2
Converter in [24] (one-cell)	$\frac{V_L+V_H}{2}$	-	complex	3	low	$\frac{1+D}{1-D}$	$\frac{D}{2-D}$
Proposed Converter	$\frac{1}{1+n}\cdot V_H$	91.5~94.8% (100 W, two-cell)	complex	4	low	$\frac{1+n}{1-D}$	$\frac{D}{1+n}$

. The conventional buck/boost converter can achieve bidirectional power flows while employing the fewest number of power switches, but the converter’s conversion ratio range is limited. The converter in reference [12] has a high step-up/step-down conversion ratio, but the voltage stress is high. Compared with the converters in reference [12], the converter’s voltage stress in reference [23] has been improved, but the converter’s conversion ratio range is limited. It can be seen that the proposed converter achieves a high and wide voltage-gain range. The voltage stress is reduced further, and the ripple is low.

3. Passivity-Based Control with LADRC

For the converter shown in Figure 2, the Euler–Lagrange equations are used to build the EL model of the bidirectional high-gain Cuk circuit [24]. The inductive charge and capacitive charge in the circuit are chosen as the generalized coordinates q of the proposed converter. Thus, the kinetic energy $K(\dot{\mathit{q}},\mathit{q})$, potential energy $G(\mathit{q})$, and dissipation function $W(\dot{\mathit{q}})$ of the circuit are obtained, and the external force F_q for each energy component is obtained. Then, the EL equation of the circuit can be obtained [25]:

$$\{\begin{array}{l}E(\dot{\mathit{q}},\mathit{q})=K(\dot{\mathit{q}},\mathit{q})-G(\mathit{q})\hfill \\ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial E(\dot{\mathit{q}},\mathit{q})}{\partial \dot{\mathit{q}}}\right)-\frac{\partial E(\dot{\mathit{q}},\mathit{q})}{\partial \mathit{q}}+\frac{\partial W(\dot{\mathit{q}})}{\partial \dot{\mathit{q}}}={\mathit{F}}_q\hfill \end{array}$$

(22)

Using the switching function μ(t), the EL equation for each stage is obtained as follows.

$${\mathit{D}}_B\cdot \dot{\mathit{x}}+{\mathit{J}}_B\cdot \mathit{x}+{\mathit{R}}_B\cdot \mathit{x}=\mathit{F}$$

(23)

The respective matrices in step-up mode are:

$$\begin{array}{l}{\mathit{J}}_B=\left[\begin{array}{cccccc}0& 0& 1& -\mu (t)& 0& 0\\ 0& 0& 0& -1& -1& 1\\ -1& 0& 0& 0& 0& 0\\ \mu (t)& 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\end{array}\right]\\ {\mathit{D}}_B=\mathrm{diag}[L_1,L_2,C_1,C_2,C_B,C_{o1}]\\ x=[x_1,x_2,x_3,x_4,x_5,x_6]\\ \mathit{F}={[0,0,-1,\mu (t),1-\mu (t),0]}^T{i}_2\\ {\mathit{R}}_B=\mathrm{diag}[0,0,0,0,0,\frac{1}{R_{L1}}]\end{array}$$

(24)

where i₂ is the input current on the low-voltage side, $x_1={\dot{q}}_{L1}$, $x_2={\dot{q}}_{L2}$, $x_3=q_{C1}/C_1$, $x_4=q_{C2}/C_2$, $x_5=q_{CB}/C_B$, and $x_6=q_{Co1}/C_{o1}$.

The respective matrices in step-down mode are:

$$\begin{array}{l}{\mathit{J}}_B=\left[\begin{array}{cccccc}0& 0& -1& 1-\mu (t)& 0& 0\\ 0& 0& 0& 1& 1& 0\\ 1& 0& 0& 0& 0& 0\\ -[1-\mu (t)]& -1& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]\\ {\mathit{D}}_B=\mathrm{diag}[L_1,L_2,C_1,C_2,C_B,C_{o2}]\\ x=[x_1,x_2,x_3,x_4,x_5,x_6]\\ \mathit{F}={[0,1,k,-[1-\mu (t)]k,-\mu (t)k,k]}^T{V}_1\\ {\mathit{R}}_B=\mathrm{diag}[0,0,0,0,0,\frac{1}{R_{L2}}]\end{array}$$

(25)

3.1. Design of the Passive Controller

Assuming the desired state variable is x₀, the system error function is x_e = x − x₀. To make it converge quickly to the expectation, the damping factor ${\mathit{R}}_a=\mathrm{diag}[R_a,R_a,R_a,R_a,R_a,1/R_a]$ is introduced. Bringing it into Equation (23), the error equation is obtained as:

$${\mathit{D}}_B\cdot {\dot{\mathit{x}}}_e+{\mathit{J}}_B\cdot {\mathit{x}}_e+({\mathit{R}}_B+{\mathit{R}}_a)\cdot {\mathit{x}}_e=\mathit{F}-({\mathit{D}}_B\cdot {\dot{\mathit{x}}}_0+{\mathit{J}}_B\cdot {\mathit{x}}_0+{\mathit{R}}_B\cdot {\mathit{x}}_0)+{\mathit{R}}_a{\mathit{x}}_e$$

(26)

Letting Equation (26) equal zero, the passive controller can be obtained as:

$$\mathit{F}-({\mathit{D}}_B\cdot {\dot{\mathit{x}}}_0+{\mathit{J}}_B\cdot {\mathit{x}}_0+{\mathit{R}}_B\cdot {\mathit{x}}_0)+{\mathit{R}}_a{\mathit{x}}_e=0$$

(27)

Define the switching function of the circuit’s operating mode as:

$$\varphi (t)=\{\begin{array}{c}1 t\in \mathrm{when} \mathrm{the} \mathrm{converter} \mathrm{operates} \mathrm{in} \mathrm{step}-\mathrm{up} \mathrm{mode}\\ 0 t\in \mathrm{when} \mathrm{the} \mathrm{converter} \mathrm{operates} \mathrm{in} \mathrm{step}-\mathrm{down} \mathrm{mode}\end{array}$$

(28)

Therefore, according to the first line of Equation (27), and combined with Equations (24) and (25), the passivity-based control law of the bidirectional high-gain Cuk circuit with basic gain network can be obtained as:

$$d(t)=\varphi (t)[1-\frac{2V_2}{V_1}-R_a\frac{x_1(t)-i_{L1}^{*}}{\frac{V_1}{2}}]+[1-\varphi (t)][\frac{2V_2}{V_1}-R_a\frac{x_1(t)-i_{L1}^{*}}{\frac{V_1}{2}}]$$

(29)

3.2. Design of LADRC

Auto-disturbance rejection control (ADRC) is a control technology that does not rely on the mathematical model of the controlled object. It can automatically detect and compensate the internal and external disturbances of the controlled object. The control object can acquire a good control effect when it encounters uncertain disturbance or parameter changes, and it has strong adaptability and robustness. However, there are many parameters of ADRC that have no physical significance. It is difficult to adjust. In view of this situation, linear ADRC is introduced in this paper, which not only has the advantages of good robustness of ADRC, but also avoids the disadvantages of many adjustable parameters, difficult adjustment, and large calculation.

Assuming that the devices of converters are all ideal, the input power of the bidirectional high-gain Cuk circuit with basic gain network is equal to the output power in step-up mode.

$$V_2\cdot i_{L1}^{*}=V_1\cdot (C_{o1}\frac{\mathrm{d}{v}_1}{\mathrm{d}t}+i_1)$$

(30)

Then, Equation (30) is organized into the standard form of active disturbance rejection control:

$$\frac{\mathrm{d}{v}_1}{\mathrm{d}t}=\frac{1}{C_{o1}}\cdot \frac{V_2}{V_1}\cdot i_{L1}^{*}+\frac{1}{C_{o1}}\cdot i_1=b_o\cdot w+h$$

(31)

where $b_o=V_2/(C_{o1}{V}_1)$, $w=i_{L1}^{*}$, $h=i_1/C_{o1}$.

Since Equation (27) is a first-order equation, the first-order LADRC can be used to realize the control of voltage. Figure 7 shows the control block diagram of the passivity-based control with LADRC, where LESO is the first-order linear extended observer. LESO is used to eliminate unknown items and disturbances in the system. z₁ and z₂ are the outputs of the observer, V₁* is reference voltage on the high-voltage side, and i_L1* is reference current of the inductor L₁. The corresponding equations are as follows.

$$\{\begin{array}{l}{\dot{z}}_1=z_2+{\beta }_1(v_1-z_1)+b_{o}y\hfill \\ {\dot{z}}_2={\beta }_2(v_1-z_1)\hfill \end{array}$$

(32)

The law of linear feedback control is:

$$\{\begin{array}{l}y=\frac{h_o-z_2}{b_o}\hfill \\ h_o=k_p(V_1^{*}-z_1)\hfill \end{array}$$

(33)

Similarly, the first-order LADRC controller in step-down mode can be obtained. This is not repeated here, because its form is the same as step-up mode.

Figure 7. The passivity-based control with LADRC.

Figure 7. The passivity-based control with LADRC.

4. Results

4.1. Simulation Results

To verify the effectiveness of the proposed converter in this paper, a simulation model was built in Matlab/Simulink 2021(b). The simulation model parameters are V₂ = 48 V(36–60 V); V₁ = 400 V(300–450 V); P_o = 100 W; f_s = 50 kHz; L₁ = 1.2 mH; L₂ = 1.2 mH; C₁–C₄; C_B = 20 uF; and C_o1=100 uF.

4.1.1. Voltage and Current Waveform of Steady State

At steady state, the output voltage V₁ is set at 400 V in the step-up mode, and the voltage and current simulation waveform of the main device is shown in Figure 8, where v_gs2 is the waveform of the drive signal; v_SB2 is the voltage across S_B2; i_SB2 is the current waveform flowing through S_B2; v_C1–v_C2 and v_CB are the corresponding capacitor voltage waveforms; and v_D1–v_D2 and v_DB1 are the corresponding voltage waveforms across the diode.

The output voltage V₂ is set at 48 V in the step-down mode, and the voltage and current simulation waveform of the main device are shown in Figure 9, where v_gs1 and v_gs2 are the waveforms of the drive signal; v_SB1 is the voltage across S_B1; i_SB1 is the current waveform flowing through S_B1; v_C1–v_C2 and v_CB are the corresponding capacitor voltage waveforms; v_S1–v_S2 is the voltage waveform of S₁, S₂; and v_DB2 is the voltage waveform of D_B2.

With the above analysis, it can be seen that the simulation waveforms are consistent with the theoretical analysis, and the voltage stress of its semiconductor device is half of the voltage on the high-voltage side.

4.1.2. Ripple Characteristics

In step-up mode, the simulation waveforms i_L1 and i_L2 of inductors L₁ and L₂ with different input voltages and different numbers of gain cells are shown in Figure 10. In step-down mode, these waveforms are shown in Figure 11. It is seen that the current waveform of i_L2 is a straight line, after adding the gain unit. It achieves zero current ripple on the high-voltage side.

4.1.3. Dynamic Characteristics

To verify the effectiveness of the proposed control strategy, the simulation of the passivity-based control with LADRC is carried out in this paper, and it is compared with the typical passivity-based control and the passivity-based control with PI.

In step-up mode, the variation in v₁ during the load step at V₂ = 48 V is shown in Figure 12. In step-down mode, the variation in v₂ during the load step at V₁ = 400 V is shown in Figure 13.

From Figure 12 and Figure 13, it can be seen that when the power supply or load fluctuates, the output voltage of PBC has steady-state error, and the other two control output voltages have no error. In addition, when the power supply or load fluctuates, the output voltage overshoot of PBC is the largest, followed by PBC with PI, and the proposed control strategy is the smallest. Similarly, PBC control recovery time is the longest, followed by PBC with PI, and the proposed control strategy is the smallest.

4.2. Experimental Results

A 100 W 48 V/400 V prototype was built to verify the validity of the theoretical analysis of the proposed converter. The prototype of the proposed converter is shown in Figure 14; the parameters of the experimental prototype are shown in

Table 2. Parameters of an experimental prototype.

Parameters	Value	Parameters	Value
Po/W	100	C1–C2	20 uF/630 V
V1/V	400 (300–450)	CB	20 uF/630 V
V2/V	48 (36–60)	Co1	100 uF/450 V
L1/mH	1.2	Co2	100 uF/160 V
L2/mH	1.2	SB1–SB2	SVD12N65T
fs/kHz	50	S1–S2	FCP099N65S3

.

4.2.1. Voltage and Current Waveform of Steady State

At V₂ = 48 V and P_o = 100 W, the steady-state waveform of the prototype in step-up mode is shown in Figure 15, where v_SB2 is the voltage across S_B2; i_SB2 is the current waveform flowing through S_B2; v_C1–v_C2 and v_CB are the corresponding capacitor voltage waveforms; v_D1–v_D2 and v_DB1 are the corresponding voltage waveforms across the diode; and i_L1 and i_L2 are the current waveforms flowing through L₁ and L₂.

At V₁ = 400V and P_o = 100 W, the steady-state waveform of the prototype in step-down mode is shown in Figure 16, where v_o is the output voltage; v_gs1 and v_gs2 are the waveforms of the drive signal; v_SB1 is the voltage across S_B1; i_SB1 is the current waveform flowing through S_B1; v_C1–v_C2 and v_CB are the corresponding capacitor voltage waveforms; v_S1–v_S2 is the corresponding voltage waveform across the switching, and v_DB2 is the voltage waveform across D_B2; and i_L1 and i_L2 are the current waveforms flowing through L₁ and L₂.

As can be seen from Figure 15, the duty cycle of S_B2 under the step-up mode is 0.76, and the output is stabilized to 399.7 V. The voltage stress of S_B2 is 223 V, and the voltage stress of diodes D₁, D₂, and D_B1 are 202, 202, and 211 V. Similarly, the duty cycle of S_B1 under the step-down mode is 0.24, and the output is stabilized to 48.2 V, as shown in Figure 16. The corresponding voltage stresses of S₁, S₂, and S_B1 are 208, 208, and 213 V, and the voltage stress of D_B2 is 212 V.

4.2.2. Ripple Characteristics

In step-up mode, the stable experimental waveforms i_L1 and i_L2 of inductors L₁ and L₂ with different input voltages and different number of gain cells are shown in Figure 17. In step-down mode, these waveforms are shown in Figure 18. It is seen that the current waveform of i_L2 is a straight line, after adding the gain unit. It achieves zero current ripple on the high-voltage side.

4.2.3. Efficiency Characteristics

The efficiency curves of the prototype operating in both directions are shown in Figure 19, where the horizontal coordinate is the output power P_o and the vertical coordinate is the efficiency η. As the number of cells increases, the efficiency of the proposed converter is higher.

It can be seen from Figure 19 that the efficiency of the two units is the highest. Although the period increases with the increase in the number of units, the stress of the switch tube decreases significantly, and the total loss decreases.

4.2.4. Dynamic Characteristics

When v₂ jumps between 36 and 60 V, the dynamic response waveforms in step-up mode are shown in Figure 20. When v₂ varies in the ranges 36–48 V and 36–60 V, the data for dynamic performance are shown in

Table 3. The dynamic performance comparison during the v₂ step in step-up mode.

	PBC		PBC with PI		PBC with LADRC
Input voltage	Overshoot	Recovery time	Overshoot	Input voltage	Overshoot	Recovery time
36–48 V	6.83V	120 ms	4.75 V	86 ms	4.37 V	19 ms
48–36 V	−4.1V	154 ms	−1.94 V	80 ms	−1.38 V	40 ms
36–60 V	9.5V	192 ms	5.7 V	98 ms	4.74 V	46 ms
60–36 V	−5.5V	213 ms	−2.14 V	103 ms	−1.79 V	89 ms

.

When i₁ jumps between 0.08 and 0.25, the data for dynamic performance are shown in

Table 4. The dynamic performance comparison during the v₂ step in step-up mode.

	PBC		PBC with PI		PBC with LADRC
Input voltage	Overshoot	Recovery time	Overshoot	Input voltage	Overshoot	Recovery time
0.12–0.25 A	−2.31 V	39 ms	−2.34 V	87 ms	−1.04 V	19 ms
0.25–0.12 A	2.1 V	53 ms	2.41 V	83 ms	0.73 V	23 ms
0.08–0.25 A	−3.94 V	78 ms	−4 V	98 ms	−1.21 V	63 ms
0.25–0.08 A	3.69 V	84 ms	3.65 V	104 ms	0.94 V	72 ms

.

When v₁ jumps between 300 and 400 V, the dynamic response waveforms in step-down mode are shown in Figure 21. When v₁ varies in the ranges 300–400 and 400–450 V, the data for dynamic performance are shown in

Table 5. The dynamic performance comparison during the v₂ step in step-down mode.

	PBC		PBC with PI		PBC with LADRC
Input voltage	Overshoot	Recovery time	Overshoot	Recovery time	Overshoot	Recovery time
300–400 V	2.11 V	83 ms	2.23 V	56 ms	0.56 V	39 ms
400–300 V	−1.07 V	108 ms	−0.94 V	85 ms	−0.43 V	66 ms
400–450 V	2.11 V	83 ms	2.23 V	56 ms	0.56 V	39 ms
450–400 V	−1.07 V	108 ms	−0.94 V	85 ms	−0.43 V	66 ms

.

For the changes in input voltage and output load, it can be seen from Figure 20 and Figure 21 that the circuit has less voltage overshoot and shorter recovery time when applying the proposed control strategy described herein. Additionally, the proposed control strategy described in this paper demonstrates better performance in the dynamic process and achieves zero steady-state error.

When i₂ jumps between 0.63 and 2.08 A, the data for dynamic performance are shown in

Table 6. The dynamic performance comparison during i₂ step in step-down mode.

	PBC		PBC with PI		PBC with LADRC
Input voltage	Overshoot	Recovery time	Overshoot	Input voltage	Overshoot	Recovery time
1.04–2.08 A	−0.93 V	58 ms	−1.05 V	37 ms	−0.24 V	26 ms
2.08–1.04 A	0.72 V	66 ms	1 V	34 ms	0.31 V	29 ms
0.63–2.08 A	−2.17 V	75 ms	−2.08 V	54 ms	−0.62 V	38 ms
2.08–0.63 A	2.32 V	87 ms	1.81 V	53 ms	0.7 V	42 ms

.

5. Discussion

In this paper, progress is made in two aspects. First, a novel bidirectional converter topology is proposed. It has high voltage gain and low ripple. Compared with results reported in [23], it has higher efficiency. These advantages make it more suitable for application in a marine environment. Secondly, the proposed control strategy has strong adaptability and robustness. Compared with PBC or PBC with PI, it has better dynamic performance. It is very suitable for a marine environment because power supply or load is prone to large-range fluctuations. This paper achieves some results, but some aspects need further improvement: (1) Combine PBC with some intelligent optimization algorithms, such as neural network algorithm and genetic algorithm, to further improve the steady-state accuracy and dynamic performance of the control system. (2) Carry out engineering research for marine environment applications, such as waterproofing and anticorrosion. (3) Verifying the validity of bidirectional converter topology and its control strategy will be mainly completed in the laboratory. As a future research direction, we intend to direct this study to a practical case either carried out in situ or at one of the locations in the study.

6. Conclusions

Most deep-sea aquaculture industries heavily depend on transported fossil fuels. Thus, installing a distributed power generation system for these aquaculture industries is seen as very promising. A bidirectional DC–DC converter is essential for distributed power generation systems. It connects a variety of renewable energy sources with energy storage cells, and the DC bus voltage is usually high. The output voltage of batteries is usually low. Therefore, the high efficiency, low ripple, high voltage gain topology and its control robustness and stability of the converters will be the development trend, when it works in a marine environment. Thus, the zero-ripple bidirectional high-gain Cuk circuit proposed in the paper improves the voltage gain capability of the circuit by incorporating an extended switched-capacitor gain cell. This converter is characterized by simple structure, zero ripple, and low voltage stress of semiconductor power devices, and high voltage gain in both step-down and step-up modes. At the same time, passivity-based control with linear active disturbance rejection is proposed. This control strategy ensures that the system is stable when the power supply end or load end fluctuates in a large range. Additionally, it solves the problem of large steady-state errors that are caused by model errors and disturbances in the bidirectional high-gain converter that uses typical passivity-based control. The proposed control strategy solves the contradiction between rapidity and overshoot, which avoids the side effects of integral feedback. Compared with the PBC and PBC with PI, the proposed control strategy has better dynamic performance, such as smaller voltage overshoot and shorter recovery time. The simulation was carried out and a 100 W, 48 V/400 V prototype was built. The simulation and experimental results verify the validity of the theoretical analysis of the proposed circuit. The results show that the proposed control strategy has excellent control performance.