Study and Optimization of Shore-to-Ship Underwater Capacitive Power Transfer System Considering Parasitic Coupling

Abstract

The increasing trend towards electrification in transportation highlights the potential for electric ships and the demand for safe, rapid charging systems. Underwater capacitive power transfer (UCPT) is considered a suitable solution due to its high power density potential. This article presents research on and optimization of UCPT for shore-to-ship charging in freshwater environments. It proposes a design for an insulation coupler and explores the influence of parasitic capacitance in the environment. This study demonstrates how the ship and shore impact the coupler’s coupling coefficient and mutual capacitance, thereby affecting the power and efficiency. Through theoretical calculations and finite element analysis, a kW-level UCPT coupler is designed. Experimental verification under different cases confirms the efficiency and constant current output characteristics, with superior performance observed when the coupler is positioned further away from the ship or shore. This study showcases the potential of UCPT to provide reliable and efficient power supply for electric ships, while emphasizing the importance of considering environmental factors and parasitic effects in system design and operation. These findings contribute to advancing UCPT technology and offer insights for further optimization to enhance practical applicability.

1. Introduction

With the increasing global energy demand and growing environmental concerns, the electrification trend of transportation vehicles has become more evident. All-electric ships, which utilize batteries as the sole power source, are a type of electric vehicle. Compared to traditional fuel-powered ships, all-electric ships not only have lower pollution and energy consumption but also offer features such as quietness and comfort [1,2,3]. With advancements in key technologies for battery energy storage systems and motor drive systems, all-electric ships have encountered new development opportunities. They have broad prospects in areas such as urban passenger and sightseeing transportation, inland waterway freight transport, and lake surveillance and governance. However, the range of electric ships is limited due to the large size and weight of batteries, as well as the high initial investment. Fast and efficient charging has become an important factor in further promoting the development of electric ships.

Although traditional wired charging can achieve high power, it requires a significant amount of manpower to handle heavy cables and ensure precise alignment of charging interfaces. Moreover, plug connections with metal contacts are prone to leakage, wear, and oxidation, posing significant safety hazards. Additionally, the charging process is susceptible to environmental and weather conditions due to the difficulty of waterproofing the charging interface.

Wireless power transfer (WPT) technology addresses many of the drawbacks of traditional wired charging methods. WPT systems eliminate the need for cable connections between ships and shore, enabling safer and more convenient connection and disconnection. The charging process is not affected by environmental and weather conditions. Furthermore, the operational and maintenance costs are greatly reduced. Over the past decade, WPT has undergone revolutionary development, achieving significant improvements in transfer power, efficiency, and distance [4,5,6,7].

In the context of electric ship charging, WPT can be implemented above the water surface [8,9,10] or underwater [11,12,13]. Wireless transfer on the water surface obstructs navigation and cargo handling, and introduces additional safety issues. Even when wireless transfer is conducted above the water surface, the system still needs to be completely waterproof. Therefore, underwater wireless power transfer is a more suitable solution.

Based on their working principles, WPT systems can be classified into near-field and far-field types. The far-field method includes techniques such as microwave, laser, and ultrasonic power transfer. These methods can achieve long-range transmission, but the power they transfer is relatively low, making them unsuitable for electric ship charging applications. In contrast, near-field WPT technologies, including inductive power transfer (IPT) and capacitive power transfer (CPT), are more commonly used in scenarios requiring short-range, high-power transmission.

Although IPT has been more extensively studied, the complexity of the coupler structure presents challenges in terms of reliability and cost-effectiveness. IPT requires coils and ferrites to generate and guide magnetic fields [14,15,16,17]. Coils often use expensive Litz wire, and ferrites are heavy and fragile. Additionally, IPT is highly sensitive to metallic foreign objects, which limits its practicality in wireless charging scenarios for electric ships [18]. In contrast, CPT only requires a few metal plates to generate an electric field [19,20]. As a result, the coupler can easily achieve waterproofing and withstand high pressure in deep water. Furthermore, CPT is generally less sensitive to the presence of metallic foreign objects. This is because the electric field is more confined, and the interactions mainly lead to charge redistribution rather than generating significant eddy current heating, which improves safety and reliability [21,22,23]. Importantly, the permittivity of water is 81 times that of air, greatly increasing the coupling capacitance [24,25,26]. Therefore, underwater capacitive wireless power transfer (UCPT) has the potential to achieve high power density.

Previous UCPT research can be categorized into two types. The first type focuses on non-insulated couplers, which include bare metal plates. In [27,28], only milliwatt-level power and a transfer distance of 5 mm were achieved. In [29,30], a distance of 20 mm and a power of 400 W were realized. In [31], a stacked-type coupler was applied for underwater transfer, although its performance was inferior to that of a horizontal coupler. In conclusion, while freshwater can be used as the dielectric material in capacitive systems, it still possesses weak conductivity. In high-power applications, water losses increase, and insulation breakdown becomes an issue.

Insulated couplers are considered the second type of UCPT. In [32], an insulated coupler was applied to achieve long-distance transfer of 500 mm with a power of 219 W. Ref. [33] also provided the design of an insulated coupler and conducted low-power verification at a distance of 8 mm with a power of 47 W. In [34], only half of the coupler was insulated, which was reported to enhance performance, achieving a power transfer of 38.8 W and an efficiency of 34.85%. In [35], the UCPT power was increased to 1 kW, with an efficiency of 74%. In conclusion, insulated couplers mitigate issues such as electric shock, breakdown, and oxidation, demonstrating significant potential for high-power transmission.

However, previous studies mostly employed simple LC compensation networks and did not include rectifiers in the system [27,28,29,30,32,33]. Such low-level power can only be used for underwater sensors or autonomous underwater vehicles (AUVs) charging. For charging electric ships, even small vessels require power levels of at least kilowatts. Furthermore, previous research did not consider the influence of the UCPT coupler in the environment, especially the parasitic capacitance between the ship’s metallic hull and the shore.

This paper proposes the design of an insulated coupler. The modeling of the coupler considers not only coupling capacitance and cross capacitance but also the influence of parasitic capacitance caused by the ship’s hull and the shore. Subsequently, finite element analysis (FEA) is used to research the insulation and optimize the coupling coefficient. Finally, a high-order compensation network is employed in the UCPT system to achieve high-power transfer of 3.3 kW.

The key novel contributions of this work are threefold: (1) the development of a comprehensive UCPT coupler model that explicitly incorporates the parasitic effects of the ship’s metallic hull and the conductive shore, which are critical for practical design but often overlooked; (2) a systematic finite-element-analysis-based investigation and optimization of the coupler’s placement relative to the hull and shore to maximize the coupling coefficient; and (3) the experimental validation of a high-power UCPT system for electric ships, which demonstrates the performance gains achieved through the optimized coupler design and the use of a high-order M-M compensation network.

The organization of the rest of the paper is as follows: Section 2 introduces the structure of the UCPT system for electric ships. Section 3 presents the modeling and design of the coupler. Section 4 describes the design of the compensation network. Section 5 presents the experimental validation. Section 6 concludes the paper and discusses further implications.

2. Underwater Capacitive Power Transfer System for Electric Ships

The underwater capacitive power transfer system for electric ships is shown in Figure 1. The shore is considered as the primary side of the transfer, including the DC power supply, inverter, and compensation network. The ship is considered as the secondary side of the transfer, including the compensation network, rectifier, and battery load. Between the primary and secondary sides, the coupler is the core component of the power transfer, comprising four metal plates labeled P₁ to P₄. These plates are fixed to a mechanical arm to facilitate alignment and maintain a fixed transmission distance. Compared to CPT systems in air environments [36], the plates used for underwater CPT can be much smaller, enabling a simpler mechanical arm design. Additionally, the mechanical arm used for wireless charging can be integrated with the ship’s mooring system, thereby enhancing the overall system integration.

As shown in Figure 1, P₁ and P₂ are positioned on the shore, while P₃ and P₄ are embedded in the ship. With water as the dielectric, a coupling capacitance is formed between P₁ and P₃, as well as between P₂ and P₄. Therefore, the four plates form a complete circuit. The transfer of power is dominated by displacement current between the plates. Due to the small value of the coupling capacitance, compensation networks are necessary to compensate for the large reactance caused by the small capacitance. Another function of the compensation network is to elevate the plate voltage to achieve lower displacement current and reduce losses.

However, considering the practical charging environment, it is not sufficient to only consider the two coupling capacitances. On the one hand, there are additional parasitic capacitances in the system. As shown in Figure 2, these parasitic capacitances originate from the cross-coupling between the plates, the coupling between the same-side plates, the coupling between the plates and the metal hull of the ship, and the coupling between the plates and the shore. Some of these parasitic capacitances can even be larger than the coupling capacitance, so they cannot be neglected in the coupler modeling. On the other hand, since water has weak conductivity, all the coupling and parasitic capacitances should have a small conductance (or large resistance) in parallel.

Based on the above considerations, the model of the coupler is illustrated in Figure 3. The four power transfer plates are represented as P₁ − P₄. The shore, which is considered as the ground, has a certain conductivity and can be regarded as having equal potential everywhere. In modeling, the ground is generally treated as an independent plate [37,38]. The metal hull of the ship is treated as another independent plate. In Figure 3, the shore and the ship are denoted as P₅ and P₆, respectively. Using flat plates to represent the shore and the hull is a standard initial approach for capturing the main parasitic coupling effects without introducing excessive geometric complexity. Future work could adopt more realistic 3D structures for a more accurate representation.

According to Figure 3, the proposed model consists of 6 plates and includes a total of 15 admittances. Each admittance is composed of a capacitance and a resistance in parallel.

3. UCPT System for Electric Ships

3.1. Simplification and Equivalence of the Coupler Model

The full-admittance model of the coupler is illustrated in Figure 3. These 15 admittances can be expressed as follows:

$$Y_{mn}=j\omega C_{mn}+{\displaystyle \frac{1}{R_{mn}}}\begin{array}{c}\end{array}(m=1,2,3,4,5,6;\begin{array}{c}\end{array}n=1,2,3,4,5,6.)$$

(1)

Each admittance can be further represented by the coupling capacitance and quality factor. In a freshwater capacitive coupler, the quality factors of individual capacitors are approximately equal [32,39]. Therefore, the quality factor is defined as:

$$Q=\omega C_{mn}{R}_{mn}\begin{array}{c}\end{array}(m=1,2,3,4,5,6;\begin{array}{c}\end{array}n=1,2,3,4,5,6.)$$

(2)

Hence, Equation (1) can also be expressed as:

$$Y_{mn}=\omega C_{mn}\cdot (j+{\displaystyle \frac{1}{Q}})\begin{array}{c}\end{array}(m=1,2,3,4,5,6;\begin{array}{c}\end{array}n=1,2,3,4,5,6.)$$

(3)

Since P₁ and P₂ are connected to the transmitting circuit, P₃ and P₄ are connected to the receiving circuit, and P₅ and P₆ are not connected to any circuit, they can be divided into three ports. Port 1 consists of P₁ and P₂, Port 2 consists of P₃ and P₄, and Port 3 consists of P₅ and P₆. Thus, the complete admittance model of the coupler can be transformed into a three-port model, as shown in Figure 4. The voltages and currents at the three ports are denoted as V_a, I_a, V_b, I_b, V_c, and I_c, respectively. They satisfy the following expressions:

$$\left[\begin{array}{c}{I}_a\\ I_b\\ I_c\end{array}\right]=\left[\begin{array}{ccc}{Y}_a& -Y_{ab}& -Y_{ac}\\ -Y_{ab}& Y_b& -Y_{bc}\\ -Y_{ac}& -Y_{bc}& Y_c\end{array}\right]\cdot \left[\begin{array}{c}{V}_a\\ V_b\\ V_c\end{array}\right],$$

(4)

where Y_a, Y_b, and Y_c represent the self-admittance of each port, and Y_ab, Y_ac, and Y_bc represent the mutual admittance between ports. The nodal voltage method [40,41] can be employed to solve for these parameters. Therefore, the admittance parameters of the three-port model are expressed as:

$$\left\{\begin{array}{l}{Y}_a=Y_{12}+{\displaystyle \frac{(Y_{13}+Y_{14})(Y_{23}+Y_{24})}{Y_{13}+Y_{14}+Y_{23}+Y_{24}}}+{\displaystyle \frac{(Y_{15}+Y_{16})(Y_{25}+Y_{26})}{Y_{15}+Y_{16}+Y_{25}+Y_{26}}}\\ Y_b=Y_{34}+{\displaystyle \frac{(Y_{13}+Y_{23})(Y_{14}+Y_{24})}{Y_{13}+Y_{14}+Y_{23}+Y_{24}}}+{\displaystyle \frac{(Y_{35}+Y_{36})(Y_{45}+Y_{46})}{Y_{35}+Y_{36}+Y_{45}+Y_{46}}}\\ Y_c=Y_{56}+{\displaystyle \frac{(Y_{15}+Y_{25})(Y_{16}+Y_{26})}{Y_{15}+Y_{16}+Y_{25}+Y_{26}}}+{\displaystyle \frac{(Y_{35}+Y_{45})(Y_{36}+Y_{46})}{Y_{35}+Y_{36}+Y_{45}+Y_{46}}}\\ Y_{ab}={\displaystyle \frac{Y_{13}{Y}_{24}-Y_{14}{Y}_{23}}{Y_{13}+Y_{14}+Y_{23}+Y_{24}}}\\ Y_{ac}={\displaystyle \frac{Y_{15}{Y}_{26}-Y_{16}{Y}_{25}}{Y_{15}+Y_{16}+Y_{25}+Y_{26}}}\\ Y_{bc}={\displaystyle \frac{Y_{35}{Y}_{46}-Y_{36}{Y}_{45}}{Y_{35}+Y_{36}+Y_{45}+Y_{46}}}\end{array}\right..$$

(5)

As P₅ and P₆ are floating, i.e., I_c = 0, Figure 4 can be further simplified into a two-port model as shown in Figure 5. The voltages and currents at the two ports are denoted as V₁, I₁, V₂, and I₂, respectively. They satisfy the following expressions:

$$\left[\begin{array}{c}{I}_1\\ I_2\end{array}\right]=\left[\begin{array}{cc}{Y}_1& -Y_M\\ -Y_M& Y_2\end{array}\right]\cdot \left[\begin{array}{c}{V}_1\\ V_2\end{array}\right],$$

(6)

where Y₁ and Y₂ represent the self-admittance of each port, and Y_M represents the mutual admittance between ports. Similarly, the nodal voltage method [37,38] can be utilized to solve for these parameters. Therefore, the admittance parameters of the two-port model are expressed as:

$$\left\{\begin{array}{l}{Y}_M=Y_{ab}+{\displaystyle \frac{Y_{ac}{Y}_{bc}}{Y_c}}\\ Y_1=Y_a-{\displaystyle \frac{Y_{ac}^2}{Y_c}}\\ Y_2=Y_b-{\displaystyle \frac{Y_{bc}^2}{Y_c}}\end{array}\right..$$

(7)

Combining Equations (3), (5) and (7), the equivalent capacitance in the two-port model can be expressed as:

$$\left\{\begin{array}{l}{Y}_M=\omega C_M\cdot (j+{\displaystyle \frac{1}{Q}})\\ Y_1=\omega C_1\cdot (j+{\displaystyle \frac{1}{Q}})\\ Y_2=\omega C_2\cdot (j+{\displaystyle \frac{1}{Q}})\\ C_M={\displaystyle \frac{C_{13}{C}_{24}-C_{14}{C}_{23}}{C_{13}+C_{14}+C_{23}+C_{24}}}+{\displaystyle \frac{C_{ac}{C}_{bc}}{C_c}}\\ C_1=C_{12}+{\displaystyle \frac{(C_{13}+C_{14})(C_{23}+C_{24})}{C_{13}+C_{14}+C_{23}+C_{24}}}+{\displaystyle \frac{(C_{15}+C_{16})(C_{25}+C_{26})}{C_{15}+C_{16}+C_{25}+C_{26}}}-{\displaystyle \frac{C_{ac}^2}{C_c}}\\ C_2=C_{34}+{\displaystyle \frac{(C_{13}+C_{23})(C_{14}+C_{24})}{C_{13}+C_{14}+C_{23}+C_{24}}}+{\displaystyle \frac{(C_{35}+C_{36})(C_{45}+C_{46})}{C_{35}+C_{36}+C_{45}+C_{46}}}-{\displaystyle \frac{C_{bc}^2}{C_c}}\end{array}\right.$$

(8)

where

$$\left\{\begin{array}{l}{C}_{ac}={\displaystyle \frac{C_{15}{C}_{26}-C_{16}{C}_{25}}{C_{15}+C_{16}+C_{25}+C_{26}}}\\ C_{bc}={\displaystyle \frac{C_{35}{C}_{46}-C_{36}{C}_{45}}{C_{35}+C_{36}+C_{45}+C_{46}}}\\ C_c=C_{56}+{\displaystyle \frac{(C_{15}+C_{25})(C_{16}+C_{26})}{C_{15}+C_{16}+C_{25}+C_{26}}}+{\displaystyle \frac{(C_{35}+C_{45})(C_{36}+C_{46})}{C_{35}+C_{36}+C_{45}+C_{46}}}\end{array}\right.$$

(9)

C_M in Equation (8) is referred to as the mutual capacitance. C₁ and C₂ represent the self-capacitance of the primary and secondary sides, respectively. Thus, the coupling coefficient of the coupler is defined as follows:

$$k_C={\displaystyle \frac{C_M}{\sqrt{C_1\cdot C_2}}}$$

(10)

The definition of the coupling coefficient shown in Equation (10) is the most commonly used representation in CPT. It measures the strength of the useful capacitive coupling (C_M) relative to the parasitic coupling to the environment (C₁, C₂). A higher k_c value means that a larger proportion of the effective electric field in the charged state is used for power transmission across the gap, rather than dissipating into the surrounding environment, which is directly related to higher potential efficiency. Additionally, it forms a dual relationship with the definition in IPT [42].

The coupling coefficient serves as an important indicator for optimizing the CPT coupler. Previous studies have demonstrated a relationship between transfer efficiency and the coupling coefficient [21,42,43], as expressed as follows:

$$\eta ={\displaystyle \frac{k_C^2\cdot Q^2}{{(1+\sqrt{1+k_C^2\cdot Q^2})}^2}}$$

(11)

3.2. Coupler Dimension Configuration

Finite element analysis (FEA) is a commonly used method for designing coupler parameters [11,32,35,44]. In this work, a coupler model was established in Maxwell. The solution region was determined to be 1300 mm × 800 mm × 300 mm based on laboratory conditions, consistent with the dimensions of the water tank. Freshwater with a relative permittivity of 81 and a conductivity of 0.0002 S/m was filled in the solution region. This value was chosen to establish a performance baseline by mitigate ohmic losses in the water, thereby allowing the feasibility of the proposed high-power UCPT system and the effects of coupler optimization to be clearly isolated and validated [32,45]. The Solution type was set as AC conduction, and the analysis frequency was 1 MHz.

The proposed coupler model and its dimensions are shown in Figure 6 and Figure 7. P₁ − P₄ are power transfer plates, while P₅ and P₆ represent shore and ship sides, respectively. To simplify the design, P₁ − P₄ are all square aluminum plates with side lengths represented by l₁ − l₄. P₅ and P₆ are both rectangular aluminum plates with dimensions of l₅ × w₅ and l₆ × w₆, respectively. To approach reality as closely as possible, P₅ and P₆ need to fully cover P₁ − P₄. The distances between the edges of P₅ and P₆ and P₁ − P₄ are represented by l_e1 − l_e4. The horizontal distances between P₁ and P₂ and between P₃ and P₄ are l_s1 and l_s2, respectively. The vertical distances between P₁ and P₂ and between P₅ are d₁, while the vertical distances between P₃ and P₄ and between P₆ are d₂. The transfer distance is represented by d.

Considering laboratory conditions, the side length of the power transfer plates P₁ − P₄ is specified as 200 mm. Horizontal distances l_s1 and l_s2 are specified as 100 mm, while l_e1 − l_e4 are specified as 15 mm. Therefore, the dimensions of P₅ and P₆, which serve as shore and ship substitutes, are both 530 mm × 230 mm. The thickness of all aluminum plates is 2 mm. To ensure that the coupler can withstand high voltage for higher transfer power, all aluminum plates are covered with a 1 mm thick insulation layer. The insulation layer is made of FR4 material with a relative permittivity of 4.4. Therefore, the dimensions to be studied and optimized are the transfer distance d, as well as the distances d₁ and d₂ between the power transfer plates and the shore and ship, respectively.

To obtain the values of C_M, C₁, C₂, and k_C for each size of the coupler, the process shown in Figure 8 can be followed. First, a 3D model of the coupler must be created using FEA software. The FEA software used in this study is ANSYS Maxwell 2022 R2. Next, material properties, solution region, excitation sources, and other system parameters are set in Maxwell software. After setting up the parameters, a simulation is run to obtain the FEA results. From these results, the 15 admittance parameters (Y₁₂ to Y₅₆) are extracted. Using Equations (4)–(9), the values of C_M, C₁, and C₂ can be calculated. Finally, the coupling coefficient can be obtained using Equation (10).

3.3. Characteristics Analysis of Coupler

In a typical transfer scenario, the transfer distance is limited. In other words, the design objective is to achieve a specific transfer distance. In this work, the target transfer distance is set at 50 mm, and the impact of the distance between the coupler and the shore and ship on the transfer is investigated. This optimization process is carried out using FEA. By analyzing how the coupling coefficient k_C varies with the distances d₁, d₂, and d, the optimal placement of the coupler can be determined.

Figure 9 illustrates the FEA results of the equivalent self-capacitance, mutual capacitance, and coupling coefficient of the coupler as the distance between the coupler and the shore and ship varies when d = 50 mm. To simplify the analysis, let d₁ = d₂. As d₁ increases from 5 mm to 50 mm, the mutual capacitance increases from 132.32 pF to 158.22 pF, while the self-capacitance decreases from 621.74 pF to 458.85 pF. Since the transfer distance remains constant, the change in mutual capacitance is relatively small. However, the self-capacitance of the coupler is significantly affected by the ship and shore, resulting in an impact on the coupling coefficient. As d₁ increases from 5 mm to 50 mm, the coupling coefficient increases from 0.21 to 0.34. According to Equation (11), to maintain high transfer efficiency, the coupler needs to be positioned farther away from the ship and shore.

Figure 10 presents the FEA results of the equivalent self-capacitance, mutual capacitance, and coupling coefficient of the coupler as the transfer distance varies when d₁ = d₂ = 50 mm. As d increases from 10 mm to 100 mm, the mutual capacitance decreases from 320.54 pF to 90.15 pF, and the self-capacitance decreases from 584.65 pF to 414.938 pF. Therefore, both the mutual capacitance and self-capacitance decrease significantly with increasing transfer distance. However, due to the larger influence of the transfer distance on the mutual capacitance, the coupling coefficient also decreases significantly with increasing transfer distance. As d increases from 10 mm to 100 mm, the coupling coefficient decreases from 0.55 to 0.22. Consequently, achieving high-efficiency transfer at longer transfer distances becomes challenging. For d = 50 mm, the coupling coefficient is 0.34.

Based on the results shown in Figure 9 and Figure 10, it can be reasonably inferred that for relatively efficient underwater capacitive power transfer, the transfer distance should not be too far. Additionally, at a specific transfer distance, the coupler should not be positioned too close to the shore and ship.

In this work, d = 50 mm is chosen. This distance is selected to perform basic validation of the proposed coupler model and compensation network under high power levels, within the constraints of the current experimental conditions.

With d = 50 mm fixed, the configuration of d₁ = d₂ = 50 mm is the optimized solution for maximizing k_C. To further validate the optimization results obtained in this section, two operating conditions are used for comparative verification: case 1 with d₁ = d₂ = 10 mm, and case 2 with d₁ = d₂ = 50 mm. The subsequent sections will present the design and verification of the UCPT system for each of these cases.

4. Compensation Network Design

4.1. Compensation Network Structure

In the scenario of high-power wireless charging for ships, the load is a large-capacity battery, making high-power constant current charging desirable. The compensation network prioritizes the use of a constant current topology, with the S-S topology being the simplest and most widely used compensation network structure [46].

Figure 11 illustrates the UCPT system with an S-S compensation network. Here, Vin represents the DC voltage source. S₁ − S₄ are MOSFETs that form a full-bridge inverter. The inverter outputs voltage and current as U_1′ and I_1′, respectively. C_M, C₁, and C₂ denote the equivalent capacitance of the coupler. L_1′ and L_2′ represent the primary and secondary compensating inductors, respectively. D₁ − D₄ are diodes that form a full-bridge rectifier. The input voltage and current of the rectifier are U_2′ and I_2′, respectively. V_out represents a constant-voltage load, simulating the battery pack.

According to Ref. [42], the resonance condition can be expressed as:

$${\omega }^{\prime }=2\pi f’={\displaystyle \frac{1}{\sqrt{L_1^{\prime }{C}_1^{\prime }(1-k_C^2)}}}={\displaystyle \frac{1}{\sqrt{L_2^{\prime }{C}_2^{\prime }(1-k_C^2)}}}$$

(12)

where f′ denotes the inverter switching frequency and ω′ represents the angular frequency. Under resonance, the output current can be expressed as:

$$I_2^{\prime }=j\omega C_M({\displaystyle \frac{1}{k_C^2}}-1)\cdot U_1^{\prime }$$

(13)

This indicates that the output current is independent of the load, allowing constant current output to be achieved. The transfer power can be further expressed as:

$$P=\omega C_M({\displaystyle \frac{1}{k_C^2}}-1)\cdot U_1^{\prime }{U}_2^{’}$$

(14)

From Equation (14), in the case of a relatively small coupling capacitance C_M, to increase power, the coupling coefficient must be reduced. This implies a decrease in efficiency. Therefore, the basic S-S compensation network is not suitable for high-power transfer.

Figure 12 illustrates a UCPT system with an M-M compensation network. The only difference from Figure 11 lies in the compensation network. L_f1 and L₁ form a mutual inductance M₁ with a coupling coefficient of k_L1. L_f2 and L₂ form a mutual inductance M₂ with a coupling coefficient of k_L2. Their relationship can be expressed as:

$$\left\{\begin{array}{l}{M}_1=k_{L1}\sqrt{L_{f1}{L}_1}\\ M_2=k_{L2}\sqrt{L_{f2}{L}_2}\end{array}\right.$$

(15)

Hence, this is a higher-order compensation network. In comparison with other widely used higher-order compensation networks such as double-sided LCLC, LCL, CLL, etc., the M-M network offers higher integration and greater design flexibility.

In Figure 12, the output voltage and current of the inverter are denoted as U₁ and I₁, while the input voltage and current of the rectifier are represented as U₂ and I₂. According to Ref. [47], the resonance condition can be expressed as:

$$\omega =2\pi f={\displaystyle \frac{1}{\sqrt{L_{f1}{C}_{f1}}}}={\displaystyle \frac{1}{\sqrt{L_{f2}{C}_{f2}}}}={\displaystyle \frac{1}{\sqrt{L_1{C}_1(1-k_C^2)}}}={\displaystyle \frac{1}{\sqrt{L_2{C}_2(1-k_C^2)}}}$$

(16)

where f represents the inverter switching frequency and ω denotes the angular frequency. Under resonance, the output current can be expressed as:

$$I_2={\displaystyle \frac{j{U}_1}{{\omega }^3{M}_1{M}_2{C}_M(\frac{1}{k_C^2}-1)}}$$

(17)

This indicates that the output current is independent of the load. Therefore, the M-M compensation network also exhibits constant current output characteristics. The transfer power can be further expressed as:

$$P={\displaystyle \frac{U_1{U}_2}{{\omega }^3{M}_1{M}_2{C}_M(\frac{1}{k_C^2}-1)}}$$

(18)

From Equation (18), power enhancement can be achieved by adjusting M₁ and M₂, which are only related to the parameters of the compensation network. This means that achieving high power no longer necessitates sacrificing the coupling coefficient. This is why higher-order compensation networks are more suitable for high-power applications compared to the simple S-S topology.

4.2. Parameter Design

The design of the compensation network follows Figure 13. The target power is 3 kW. Considering the laboratory conditions, the primary-side power supply voltage and the secondary-side electronic load voltage are both set to 500 V. The inverter switching frequency is set to 1 MHz. The parameters related to the coupler, C_M, C₁, C₂, and k_C, are determined by the size of the coupler. The compensation network parameters to be determined are M₁ and M₂. According to (15), M₁ and M₂ are determined by k_L1, k_L2, L_f1, L_f2, L₁, and L₂. Here, k_L1 and k_L2 are specified as 0.4. L₁ and L₂ can be obtained based on the resonance condition given by (16). Next, combining (15) and (18) allows us to solve for L_f1 and L_f2. Finally, according to the resonance condition, C_f1 and C_f2 can be determined. It is worth noting that C_f1 should also be increased by 10% to achieve soft switching.

According to the analysis in Section 3, the coupler is divided into two cases for comparison and verification based on the distance from the coupler to the ship and the shore, namely, d₁ = 10 mm and d₁ = 50 mm. Therefore, the compensation network parameters corresponding to these two coupler sizes have been designed. The parameter selection is shown in

Table 1. Parameters of the UCPT Systems.

Designator	Parameter	Case 1 (d1 = 10 mm)	Case 2 (d1 = 50 mm)
Cf1	Source side capacitance	2.53 nF	1.71 nF
Cf2	Load side capacitance	2.53 nF	1.71 nF
Lf1	Source side inductance	9.99 uF	14.79 uH
Lf2	Load side inductance	9.99 uF	14.79 uH
L1	Primary side inductance	45.63 uF	62.65 uH
L2	Secondary side inductance	45.63 uF	62.65 uH
kL1	Inductive coupling coefficient	0.4	0.4
kL2	Inductive coupling coefficient	0.4	0.4
C1	Primary side capacitance	587.01 pF	458.85 pF
C2	Secondary side capacitance	587.01 pF	458.85 pF
CM	Coupling capacitance	136.79 pF	158.22 pF
kC	Capacitive coupling coefficient	0.23	0.34

.

5. Experimental Verification

5.1. Experimental Platform

According to the previous design, experimental platforms for both operating conditions of the UCPT system were established. Figure 14a,b, respectively, show the experimental setups for case 1 and case 2. The dimensions of the coupler were configured according to Section 3. The power transfer plate was composed of four 2 mm thick aluminum plates. Two 530 mm × 230 mm × 2 mm aluminum plates were used to represent the boat and the shore, respectively. All aluminum plates were covered with 1 mm thick insulation. The transfer distance was d = 50 mm. In case 1, d₁ = d₂ = 10 mm. In case 2, d₁ = d₂ = 50 mm. The coupler was placed in fresh water with a relative permittivity of 81 and a conductivity of 0.0002 S/m. The water tank had dimensions of 1300 mm × 800 mm × 300 mm.

On the primary side of the UCPT system, a DC power supply model IT6018D-1500 was used as the input. The inverter converted the DC power into high-frequency AC power with a frequency of 1 MHz. The inverter adopted a full-bridge structure, with IMZ120R045M1 SiC-MOSFET switch devices with a high voltage resistance of 1200 V/40 A and overcurrent capability. The on-state resistance was 45 mΩ, with low high-frequency loss.

On the secondary side of the UCPT system, an electronic load model IT6000C-800 was used as the load. To simulate the battery of the boat, the electronic load operated in constant voltage (CV) mode. The rectifier converted the high-frequency AC power into DC power. The rectifier adopted a full-bridge structure, with FFSH40120ADN SiC-Diode main power devices with a high voltage resistance of 1200 V/40 A and overcurrent capability. Without reverse recovery, it is suitable for 1 MHz high-frequency rectification.

The compensation network had an M-M topology. L_f1 and L₁ are resonant inductors on the primary side, wound with 0.04 mm × 4500 strands and 0.04 mm × 2000 strands of Litz wire, respectively. They have a coupling relationship, which is adjusted by a 3D-printed support plate and nylon screws. C_f1 is the resonant capacitor on the primary side, composed of multiple high-frequency film capacitors in series and parallel. Because the primary and secondary sides of the coupler are symmetrical, the compensation network is also designed to be symmetrical.

In addition, an oscilloscope model MSO46 was used to measure the voltage and current waveforms of the UCPT system, and a power analyzer model PA5000H was used to measure power and efficiency.

5.2. Experimental Results

To verify the optimization results, case 1 (suboptimal placement, low k_C) and case 2 (optimized placement, high k_C) were compared side by side.

Firstly, the UCPT system under case 1 was tested. Figure 15 presents the waveforms of the inverter’s output voltage and current, rectifier’s input voltage and current, and the driving signals of the inverter’s power devices. The system operated under rated conditions with both U_in and U_out at 500 V. The output power was 3.3 kW. From Figure 15, the stable sinusoidal waveforms indicating the system was operating at resonance. Additionally, I₁ slightly lagged behind U₁, indicating that the inverter’s power devices were operating in the zero voltage switching (ZVS) state.

Figure 16 shows the efficiency value captured by the power analyzer as 75.89%. It is worth noting that this is the DC-DC efficiency under rated conditions.

Subsequently, the electronic load voltage was set to 300 V, 400 V, and 500 V, respectively. The input DC power supply voltage was incremented from 100 V to 500 V in steps of 50 V. The efficiency vs. power curves under different load conditions were obtained, as shown in Figure 17. When U_out was 300 V, 400 V, and 500 V, the output power reached 2 kW, 2.7 kW, and 3.3 kW, respectively. In all three load conditions, an efficiency of over 75.9% was achieved. Additionally, it was observed that the efficiency increased as the load power increased. The observed trend is primarily attributable to the presence of fixed losses. Losses such as core losses in the compensation inductors, switching losses, gate drive power, and controller consumption remain relatively constant regardless of the power level. As the output power increases, these fixed losses constitute a smaller percentage of the total power processed, thereby leading to a higher overall system efficiency.

Next, the input DC power supply voltage was set to 300 V, 400 V, and 500 V, respectively. The load voltage was incremented from 100 V to 500 V in steps of 50 V. The load current vs. output power curves under different input voltages were obtained, as shown in Figure 18. It can be observed that the system exhibited a nearly invariant load current over a wide range of output power, demonstrating a robust constant-current output characteristic. Particularly, when the rated input voltage U_in was 500 V, the output power increased by 440%, while the output current decreased by only 11.6%.

Next, the UCPT system under case 2 was tested. A rated output power of 3.3 kW was achieved with U_in = U_out = 500 V. The voltage and current waveforms under rated conditions are shown in Figure 19. The power and efficiency are shown in Figure 20. The waveforms demonstrate that the system achieved resonance and operated in the ZVS state. The DC-DC efficiency was measured to be 78.88%, a significant improvement compared to case 1. These results are consistent with the previous analysis.

Subsequently, the electronic load voltage was set to 300 V, 400 V, and 500 V, respectively. The input DC power supply voltage was incremented from 100 V to 500 V in steps of 50 V. The efficiency vs. power curves under different load conditions were obtained, as shown in Figure 21. When U_out was 300 V, 400 V, and 500 V, the output power reached 2 kW, 2.7 kW, and 3.3 kW, respectively. Moreover, an efficiency of over 78% was achieved under all three load conditions.

Next, the input DC power supply voltage was set to 300 V, 400 V, and 500 V, respectively. The load voltage was incremented from 100 V to 500 V in steps of 50 V. The load current vs. output power curves under different input voltages were obtained, as shown in Figure 22. It can be observed that the system exhibited a nearly invariant load current over a wide range of output power, demonstrating a robust constant-current output characteristic. Particularly, when the rated input voltage U_in was 500 V, the output power increased by 469%, while the output current decreased by only 6.2%. These test results demonstrate that the UCPT system under case 2 performed well across a wide range of operating conditions.

To better compare the performance of the UCPT system under case 1 and case 2, the efficiency vs. power curves under rated load conditions for both cases are simultaneously presented in Figure 23. It can be observed that the efficiency increases with increasing output power. However, the efficiency of case 2 is consistently higher than that of case 1. Furthermore, the load current vs. output power curves under rated input conditions for both cases are simultaneously shown in Figure 24. It can be seen that case 2 exhibits better constant current characteristics compared to case 1.

All experimental results demonstrate that case 2 is more suitable for high-power UCPT than case 1. This is because the coupling coefficient and mutual capacitance of the coupler in case 2 are larger. It also means that case 2 is less affected by the parasitic effects of the hull and shore, which is consistent with the theoretical analysis and the coupler optimization results. In practical applications, it is necessary to separate the UCPT coupler from the ship and shore by an appropriate mechanical structure or non-metallic buffer to achieve optimal performance.

6. Conclusions and Discussions

This research successfully designed and validated a UCPT system, with the aim of providing a reliable power supply for electric ships. At the core of this system lies an insulated capacitive coupler, which utilizes four metal plates and water as a medium to facilitate power transfer through coupled capacitance. However, considering the environmental factors, there are additional parasitic capacitances, apart from the coupled capacitance, that can affect the overall transfer efficiency. Therefore, these parasitic effects have been accounted for in the coupler model, which comprises fifteen admittances. Through finite element analysis, the research investigated the influence of varying distances between the coupler and the shore or ship on parameters such as equivalent self-capacitance, mutual capacitance, and coupling coefficient. The analysis revealed that, to achieve relatively high-efficiency UCPT, the transfer distance should be moderate, and the distances between the coupler and the shore or ship should not be excessively close. The target transfer distance for this research was set at 50 mm, and two different cases were compared and validated based on the respective distances between the coupler and the ship or shore.

To enable high-power transfer, a high-order compensation network employing M-M topology was adopted. A comparison with a low-order compensation network was made to illustrate the reasons behind the high-power transfer capabilities of the former. Experimental results demonstrated that both scenarios could achieve an output power of 3.3 kW in a freshwater environment. However, the condition with a greater distance between the coupler and the hull or shore demonstrated better performance (higher efficiency and better constant current characteristics). This is a direct result of the coupler optimization process guided by our theoretical and finite element analysis work. Consequently, this study showcases the ability of the UCPT system to maintain efficient transfer under diverse operational conditions. With its promising applications, this system offers an effective and sustainable energy solution for electric ships.

However, despite the positive outcomes observed in this study, it is important to acknowledge certain limitations and challenges associated with the UCPT system.

Firstly, enhancing the practicality of the entire system will be the primary goal of future work. According to the UCPT system structure shown in Figure 1, the plates are mounted on a mechanical arm. The mechanical arm has multiple degrees of freedom. When not charging, the transmitter side of the coupler is attached to the shore, and the receiver side is attached to the ship. During charging, the two sides of the coupler are aligned via the mechanical arm. This allows the coupler to maintain a certain distance from the shore and the ship, thereby reducing parasitic effects. This study conducted basic power transmission research under laboratory conditions, and further practical research will be conducted in the future. Secondly, in practical applications, the complexity of the underwater environment and its impact on system performance need to be further considered. Natural water bodies have higher and more variable conductivity. Pollution can also cause further changes in conductivity. The investigation of system performance in such environments, including the development of mitigation strategies for the resultant ohmic losses, is the critical and logical next step for our research. Thirdly, future research endeavors should focus on exploring additional technological advancements and optimizations to enhance the reliability and applicability of the UCPT system.