A Novel Hill Climbing-Golden Section Search Maximum Energy Efficiency Tracking Method for Wireless Power Transfer Systems in Unmanned Underwater Vehicles

Abstract

Efficiency has always been one of the most critical indicators for evaluating wireless power transfer (WPT) systems. To achieve fast maximum energy efficiency tracking (MEET), this paper provides an innovative control method utilizing the hill climbing-golden section search (HC-GSS) method of an LCC-S compensated WPT system. The receiver side includes a buck-boost converter that regulates the output current or voltage to meet output requirements. In the meantime, the buck-boost converter on the transmitter side is managed by the HC-GSS approach for MEET by minimizing the input power under the premise of output stability. Compared with the conventional P&O method, the HC-GSS method can eliminate the trade-off between the oscillation and convergence rate because it is designed for different system stages. In this WPT system, there is no need for direct communication between the transmitter and receiver. Therefore, the system is potentially cheaper to implement and does not suffer from annoying communication delays, which are prevalent in underwater environments for unmanned underwater vehicles’ (UUV) WPT systems. Both the simulation and experiment results show that this method can improve the efficiency of the WPT system without communication. The proposed method remains valid with coupler displacement as it does not include the mutual inductance of the system.

1. Introduction

Unmanned underwater vehicles (UUV) are becoming an increasingly important tool for human exploration and exploitation of the oceans, and the way they are replenished with power has been a major factor limiting their further development [1,2]. To solve the power supply of UUV, wireless power transfer (WPT) technology has received widespread attention. Compared with the traditional power replenishment methods, WPT can effectively avoid short circuits, electric leakage, and other safety hazards caused by metal electrode contact [3,4,5], and thus has high application value in complex environments such as implantable medical devices [6] and mines. When used in underwater environments, it can significantly improve the safety, reliability, convenience, and covertness of UUV charging [7,8,9].

Contrary to convenient real-time wireless electromagnetic wave communication in the air, the different temperatures, salinity, and conductivity of the underwater environment will cause different degrees of attenuation of electromagnetic waves, resulting in a long-time delay and poor stability of the communication system [10], thus deteriorating control performance. In addition, the size and cost of the WPT system will increase with the addition of communication systems. For this reason, the underwater WPT system should avoid real-time communication as much as possible to reduce the cost, size, and complexity of the system and improve its stability [11].

Efficiency has always been one of the key metrics of the energy system, and WPT systems are no exception. The review [12] makes a strict distinction between the principles of maximum power transfer and maximum energy efficiency in the WPT system. However, many factors, such as coupling coefficient, load resistance, and environmental conditions, will affect the efficiency of the WPT system. Improving system efficiency has been an important research topic. Much research has been devoted to improving the efficiency of WPT systems, mainly including coupler optimization [13,14], frequency tracking [15,16], passive component tuning [17,18,19], and DC–DC converter control [20,21,22,23,24,25,26]. Coupler optimization improves efficiency by designing the structure or parameters of the magnetic coupler [13,14], but this method adds complexity to the coupler and increases manufacturing costs. Frequency tracking achieves efficiency optimization by adjusting the switching frequency of the inverter [15]. However, when the switching frequency of the inverter is not equal to the resonant frequency, the reactive power will become large, thus reducing the system efficiency [16]. Passive component adjustment achieves MEET by dynamically adjusting the inductance or capacitance values of the WPT systems. In the literature [17], dual switch-controlled capacitors were introduced in series-compensated WPT systems for impedance matching to optimize efficiency. Zhao et al. [18] use an LCC/S-S compensation to maintain efficiency above 80% in an ultrawide coupling variation range. In [19], an inductor is parallelly connected with a diode full-bridge rectifier cascaded with a buck converter serving as a tuning circuit to improve efficiency. However, adding additional components inside the WPT system reduces its stability. DC–DC converter control improves efficiency by matching the optimal resistance of the WPT system, which has high stability. In [20], both the primary and secondary sides adopt the buck-boost converter in order to maintain a constant voltage output and to optimize efficiency through the use of wireless communication. Reference [21] describes switching the mode of a dual full bridge converter between full bridge and half bridge for half impedance matching to improve efficiency. This method is simple and effective but does not track the point of highest efficiency. In [22], the perturbation and observation (P&O) method is first used to pursue the maximum energy efficiency of the WPT system by minimizing the DC input power. As the search process occurs only at the transmitter side, the method does not necessitate any wireless communication between transmitter side and receiver side. After this, P&O methods based on direct current minimization [23] and inverter operating frequency [24] are presented for the MEET. The advantages of the P&O method are its simplicity, ease of implementation, and no real-time communication required. Nevertheless, such methodologies suffer from persistent oscillations and convergence speed trade-offs. To solve this dilemma, Zhao et al. [25] implemented MEET based on the variable-step P&O algorithm to minimize the input direct current using a primary-side buck converter. Recently, reference [26] proposed an adaptive perturbation step-size to remove the trade-off between the oscillation and convergence rate in the conventional P&O method. The P&O method and the hill climbing algorithm are basically implemented in the same way, starting from a point in the solution space and moving to neighboring points until it reaches a locally optimal solution that cannot be moved. Both methods are greedy algorithms, as they only compare with the previous solution. The time complexity of a typical greedy algorithm is often linear or close to linear, that is, O(n), where n is the dimension of the solution space [27].

To reduce the time complexity of the MEET algorithm to improve the speed of tracking the maximum efficiency point, this paper improves the hill climbing algorithm. The current solution not only compares with the previous solution but also with the latter one. If the current solution is better than both the previous and the latter, the interval in which the optimal solution is located can be determined, and then the MEET is carried out by the golden section search algorithm. In science, the time complexity of the golden section search algorithm is typically O(log n), where n is the dimension of the solution space [28]. Therefore, the speed of MEET can be improved to some extent by using the hill climbing-golden section search (HC-GSS) method and effectively avoiding the oscillations that eventually occur with the conventional P&O method proposed in [22,23,24,25]. The WPT system in this paper uses LCC-S compensation, where the secondary side buck-boost is used to maintain the voltage or current required by the load and the primary side buck-boost is used to achieve MEET by minimizing the input power. The MEET uses the HC-GSS method, which optimizes the efficiency within the controllable voltage range of the primary buck-boost. Since the MEET is achieved by a search method, underwater real-time communication can be avoided in this work.

The organization of this paper is as follows: Section 2 analyzes the existence and uniqueness of the optimal efficiency point and its corresponding inverter input voltage using the LCC-S compensated WPT system as an example. Section 3 introduces the proposed HC-GSS method for MEET, which is divided into the optimal solution existence judgment stage, the improved hill climbing stage, and the golden section search stage. The proposed method achieves fast maximum efficiency tracking while avoiding the final oscillations of conventional P&O methods. Simulation and experiment results are analyzed in Section 4, and this study is concluded in Section 5.

2. Existence and Uniqueness of Optimal Efficiency Point and Inverter Input Voltage

The schematic diagram of the LCC-S compensated WPT system is shown in Figure 1, which consists of a primary buck-boost, inverter, LCC-S compensated WPT, rectifier, and secondary buck-boost. In Figure 1, Vbattery is the supply battery voltage; L1 and C1 are the energy storage components of the primary buck-boost converter; Lp and Ls are the self-inductance of the primary and secondary windings; Rp and Rs are the corresponding primary and secondary winding resistances; M is the mutual inductance of the two windings; Lps, Cpp, and Cps denote primary-side compensating components for compensating the large leakage inductance; Css denotes secondary-side compensating capacitor; C3 is the filter capacitor of the rectifier; L2 and C2 are the energy storage components of the secondary buck-boost converter; and R is the resistance of the load.

In this work, a buck-boost converter on the secondary side is used to regulate the output to maintain a constant output voltage Vout or constant output current Iout required by the load. Then, the input voltage Vin (the root-mean-square (RMS) value of vin) of the primary resonator is adjusted iteratively (decreased or increased) only by the primary-side buck-boost until the output power of the battery reaches a minimum point. For a certain output power, attaining the minimum input power is indicative of reaching the maximum energy-efficient operating point. The drive signal of the inverter is kept constant throughout the operation with a duty cycle of 50% and a phase shift angle of 180°, which is used to ensure that the ZVS operation of the inverter is achieved over the entire load range [29]. Furthermore, no real-time communication is required between Controller 1 and Controller 2 on the primary and secondary sides, respectively.

The secondary-side buck-boost converter has two modes of operation: continuous conduction mode (CCM) and discontinuous conduction mode (DCM), which depends on the value of inductance L2. If $L_2\ge {\displaystyle \frac{D_2\left(1-D_2\right)T_s{V}_{c3}R}{2V_{out}}}$, the buck-boost converter will operate in CCM; otherwise, if $L_2<{\displaystyle \frac{D_2\left(1-D_2\right)T_s{V}_{c3}R}{2V_{out}}}$, the buck-boost converter will operate in DCM. Where D2 is the duty ratio of switch Ss, Ts is the switching period, Vc3 is the average value of vc3, and Vout is the output voltage across the load resistor R.

In actual operation, the output voltage vc2 of the system is regulated to the required value Vout. In CCM and DCMs, the equivalent impedance Req1 and voltage vc3 are as follows:

$$\begin{array}{l}CCM:\left\{\begin{array}{c}{R}_{eq1}={\displaystyle \frac{{\left(1-D_2\right)}^2}{D_2^2}}R\\ v_{c3}=\frac{1-D_2}{D_2}{v}_{c2}=\frac{1-D_2}{D_2}{V}_{out}\end{array}\right.\\ DCM:\left\{\begin{array}{c}{R}_{eq1}={\displaystyle \frac{2L}{D_2^{2}T}}\\ v_{c3}=\frac{1}{D_2}\sqrt{\frac{2L}{RT}}{v}_{c2}=\frac{1}{D_2}\sqrt{\frac{2L}{RT}}{V}_{out}\end{array}\right.\end{array}$$

(1)

Since the resonant tank is tuned to the fundamental, all higher harmonics can be neglected according to reference [23]. The equivalent impedance Req2 and voltage vc2 can therefore be further derived from Equation (1) with the inclusion of the diode rectifier and filter capacitor C3.

$$\begin{array}{l}CCM:\left\{\begin{array}{c}{R}_{eq2}=\frac{8}{{\pi }^2}{R}_{eq1}=\frac{8}{{\pi }^2}\frac{{\left(1-D_2\right)}^2}{D_2^2}R\\ V_1=\frac{1}{\left(0.9~\sqrt{2}\right)}{V}_{c3}=\frac{1}{\left(0.9~\sqrt{2}\right)}\frac{1-D_2}{D_2}{V}_{c2}=\frac{1}{\left(0.9~\sqrt{2}\right)}\frac{1-D_2}{D_2}{V}_{out}\end{array}\right.\\ DCM:\left\{\begin{array}{c}{R}_{eq2}=\frac{8}{{\pi }^2}{R}_{eq1}=\frac{8}{{\pi }^2}\frac{2L}{D_2^{2}T}\\ V_1=\frac{1}{\left(0.9~\sqrt{2}\right)}{V}_{c3}=\frac{1}{\left(0.9~\sqrt{2}\right)}\frac{1}{D_2}\sqrt{\frac{2L}{RT}}{V}_{c2}=\frac{1}{\left(0.9~\sqrt{2}\right)}\frac{1}{D_2}\sqrt{\frac{2L}{RT}}{V}_{out}\end{array}\right.\end{array}$$

(2)

In Equation (2), V1 is the RMS value of v1, Vc3 and Vc2 are the average values of vc3 and vc2, respectively. Since vc3 and vc2 are DC voltages, it is generally accepted that Vc3 = vc3 and Vc2 = vc2.

The simplified schematic diagram of the equivalent impedance of the LCC-S compensated WPT system is shown in Figure 2, where Vin and V1 are the RMS values of vin and v1, respectively.

According to Kirchhoff’s law, the equations in Figure 2 are as follows:

$$\left\{\begin{array}{c}j\omega L_{ps}{I}_{in}+{\displaystyle \frac{I_{cpp}}{j\omega C_{pp}}}=V_{in}\\ I_{cpp}+I_p=I_{in}\\ {\displaystyle \frac{I_{cpp}}{j\omega C_{pp}}}=\left({\displaystyle \frac{1}{j\omega C_{ps}}}+j\omega L_p+R_p\right)I_p+j\omega M{I}_s\\ \left(R_s+{\displaystyle \frac{1}{j\omega C_{ss}}}+j\omega L_s+R_{eq2}\right)I_s+j\omega M{I}_p=0\end{array}\right.$$

(3)

where ω is the operating angular frequency. The resonance condition of the system in Figure 2 is as follows:

$$\left\{\begin{array}{c}j{\omega }_0{L}_{ps}+{\displaystyle \frac{1}{j{\omega }_0{C}_{pp}}}=0\\ j{\omega }_0{L}_p+\frac{1}{j{\omega }_0{C}_{ps}}+\frac{1}{j{\omega }_0{C}_{pp}}=0\\ j{\omega }_0{L}_s+\frac{1}{j{\omega }_0{C}_{ss}}=0\end{array}\right.$$

(4)

where ω0 is defined as the resonance angular frequency. When the WPT system operates at the resonance frequency, the output voltage and power transfer efficiency can be calculated as follows:

$$V_1=\frac{M{R}_{eq2}}{\left(R_s+R_{eq2}\right)L_{ps}}{V}_{in}$$

(5)

$$\eta =\frac{{\omega }_0^2{M}^2{R}_{eq2}}{{\left(R_s+R_{eq2}\right)}^2\left(R_p+\frac{{\omega }_0^2{M}^2}{R_s + R_{eq2}}\right)}$$

(6)

To analyze the existence and uniqueness of the optimal efficiency point and its corresponding inverter input voltage, we first differentiate η concerning Req2 and equate the differential function to zero.

$$\frac{\partial \eta }{\partial R_{eq2}}=\frac{{\omega }_0^2{M}^2[-R_{eq2}^2{R}_p+R_s(R_p{R}_s+{\omega }_0^2{M}^2)]}{{\left(R_s+R_{eq2}\right)}^2{\left(R_p{R}_s+R_p{R}_{eq2}+{\omega }_0^2{M}^2\right)}^2}=0\to R_{eq2}=\pm R_s\sqrt{1+\frac{{\omega }_0^2{M}^2}{R_p{R}_s}}$$

(7)

Since the resistance can only be positive, the extreme point, or the stationary point, of the efficiency can be at $R_{eq2}=R_s\sqrt{1+\frac{{\omega }_0^2{M}^2}{R_p{R}_s}}$. In order to determine whether it is an extreme point or a stationary point, then we differentiate the second order of η concerning Req2.

$$\frac{{\partial }^2\eta }{{\partial }^2{R}_{eq2}}=-\frac{2{\omega }_0^2{M}^2[-R_{eq2}^3{R}_p^2+3R_{eq2}{R}_p{R}_s(R_p{R}_s+{\omega }_0^2{M}^2)+R_s(2R_p^2{R}_s^2+3{\omega }_0^2{M}^2{R}_p{R}_s+{\omega }_0^4{M}^4)]}{{\left(R_s+R_{eq2}\right)}^3{[R_p{R}_s+R_p{R}_{eq2}+{\omega }_0^2{M}^2]}^3}$$

(8)

Then, we can obtain the following:

$$\eta ″(R_{eq2}=R_s\sqrt{1+\frac{{\omega }_0^2{M}^2}{R_p{R}_s}})=-\frac{2{\omega }_0^2{M}^2[2\sqrt{R_p}{R}_s^{3/2}{(R_p{R}_s+{\omega }_0^2{M}^2)}^{3/2}+R_s(2R_p^2{R}_s^2+3{\omega }_0^2{M}^2{R}_p{R}_s+{\omega }_0^4{M}^4)]}{{\left(R_s+R_s\sqrt{1+\frac{{\omega }_0^2{M}^2}{R_p{R}_s}}\right)}^3{[R_p{R}_s+{\omega }_0^2{M}^2+\sqrt{R_p}\sqrt{R_s}\sqrt{R_p{R}_s+{\omega }_0^2{M}^2}]}^3}$$

(9)

Obviously, $\eta ″(R_{eq2}=R_s\sqrt{1+\frac{{\omega }_0^2{M}^2}{R_p{R}_s}})<0$. Therefore, the efficiency of the WPT system in Figure 1 has a maximum value at the point $R_{eq2-opt-\eta }=R_s\sqrt{1+\frac{{\omega }_0^2{M}^2}{R_p{R}_s}}$. Therefore, the optimal efficiency point exists and is unique, depending only on the parameters of the WPT system.

By letting Req2 = Req2-opt-η in Equation (2), we obtain the relation between the optimal duty cycle D2_-opt-η and Req2-opt-η, rounding off the negative root since the duty cycle is between 0 and 1. The final optimal value of the duty cycle for the secondary side buck-boost converter is as follows:

$$\begin{array}{l}CCM:\frac{1-D_{2-opt-\eta }}{D_{2-opt-\eta }}=\sqrt{\frac{{\pi }^2}{8}\frac{R_{eq2-opt-\eta }}{R}}\\ DCM: D_{2-opt-\eta }=\sqrt{\frac{8}{{\pi }^2}\frac{2L}{T{R}_{eq2-opt-\eta }}}\end{array}$$

(10)

From Equations (2), (5), and (10), we can deduce that the relationship between Vin, Vout, and D2_-opt-η.

$$\begin{array}{l}CCM:\frac{\frac{1}{\left(0.9~\sqrt{2}\right)}\frac{1-D_{2-opt-\eta }}{D_{2-opt-\eta }}{V}_{out}}{V_{in}}=\frac{M\frac{8}{{\pi }^2}\frac{{\left(1-D_{2-opt-\eta }\right)}^2}{D_{2-opt-\eta }^2}R}{\left(R_s+\frac{8}{{\pi }^2}\frac{{\left(1-D_{2-opt-\eta }\right)}^2}{D_{2-opt-\eta }^2}R\right)L_{ps}}\\ DCM:\frac{\frac{1}{\left(0.9~\sqrt{2}\right)}\frac{1}{D_{2-opt-\eta }}\sqrt{\frac{2L}{RT}}{V}_{out}}{V_{in}}=\frac{M\frac{8}{{\pi }^2}\frac{2L}{D_{2-opt-\eta }^{2}T}}{\left(R_s+\frac{8}{{\pi }^2}\frac{2L}{D_{2-opt-\eta }^{2}T}\right)L_{ps}}\end{array}$$

(11)

Since the output voltage Vout is normally constant specified by the load, there is a definite function relationship between Vin and D2_-opt-η. In addition, D2_-opt-η is determined by the unique Req2-opt-η, and a buck-boost converter can only operate in either CCM or DCM. Therefore, there is a unique point of Vin that makes the system in Figure 1 achieve maximum energy efficiency transfer, so the optimal inverter input voltage also exists and is unique.

3. Searching for the Optimal Inverter Input Voltage for MEET Using the HC-GSS Method

In Section 2, we analyzed the existence and uniqueness of the optimal efficiency point and its corresponding inverter input voltage under stable output voltage and constant load resistance. The analysis indicates that the relationship between Vin and the maximum efficiency point is too complex to be solved. However, in the steady state, the system’s output power can be regarded as constant. Consequently, the search algorithm can identify the global maximum efficiency point by adjusting the inverter input voltage to achieve minimum input power.

In this section, the conventional P&O method for MEET is briefly described. Then, the improved HC-GSS method is introduced.

3.1. Conventional P&O Method for MEET

Figure 3 illustrates the mechanism for running the conventional P&O MEET method in the WPT system. When the system is in regions A and C, the input power Pin can be continuously reduced by increasing or decreasing the inverter input voltage value by perturbation step ΔV. Upon entering region B, the state of Vin undergoes a three-point oscillation between points P2, Pmin, and P4, indicating that the optimal inverter input voltage for the MEET has been identified. During operation, the change in the inverter input voltage is controlled by the duty cycle of the primary side buck-boost converter.

The detailed implementation principle of the conventional P&O method is shown in Figure 4. It is clear that the method is independent of the system parameters and achieves MEET. However, the ultimate point of optimal efficiency is a three-point oscillation in the region B of two times the size of the perturbation step ΔV. Thus, the choice of ΔV requires a trade-off between convergence speed and oscillations, which is not conducive to the stability of the system.

According to Figure 4, the P&O method and the hill climbing algorithm are implemented in the same way, starting from a point in the solution space and moving to neighboring points until it reaches a locally optimal solution that cannot be moved. During MEET on the WPT system, the locally optimal solution is the globally optimal solution due to the existence and uniqueness of the maximum efficiency point.

3.2. HC-GSS Method for MEET

In order to improve the speed of MEET and avoid the conflict between convergence speed and steady state performance of conventional P&O methods, an HC-GSS method is proposed in this work. In this method, the search process of MEET is divided into three stages: optimal solution existence judgment stage, improved hill climbing stage, and golden section search stage. The search algorithm starts at the maximum output voltage of the primary side buck-boost and gradually reduces the output voltage during the search to ensure output power stability as a priority.

3.2.1. Optimal Solution Existence Judgment Stage

Since the stable output voltage range of the primary side buck-boost controller is a finite interval, the interval may be to the left of the optimal efficiency point, to the right of the optimal efficiency point, or to contain the optimal efficiency point, as shown in Figure 5. Therefore, it is necessary to determine whether the interval of the primary side buck-boost controllable output voltage contains the optimal solution before the MEET is performed.

In the optimal solution existence judgment stage, the primary side buck-boost starts from the maximum output voltage V(1) = Vinmax and decreases twice with the minimum resolution, i.e., V(2) = Vinmax − ΔV and V(3) = Vinmax – 2 × ΔV, and calculates the corresponding input powers Pin(1), Pin(2), and Pin(3), respectively.

If Pin(1) < Pin(2) < Pin(3) belong to Case A in Figure 5, the maximum output voltage of the primary side buck-boost converter is the optimum point of the system efficiency, and the MEET is terminated; otherwise, it is necessary to proceed to stages 2 and 3.

3.2.2. Improved Hill Climbing Stage

In this stage, we improve the hill climbing algorithm; the current solution, Pin(k), not only compares with the previous solution, Pin(k − 1), but also compares with the latter one, Pin(k + 1). In order to increase the speed of MEET, the step size of the voltage change can take a larger value, i.e., V(k) = V(k − 1) – n × ΔV, where ΔV is the minimum resolution of the output voltage of the primary side buck-boost converter.

During the operation of the improved hill-climbing search algorithm, the maximum efficiency point is included in the output voltage range of the primary-side buck-boost converter, as shown in Case B in Figure 5. In this situation, if Pin(k + 1) > Pin(k) < Pin(k − 1), the interval in which the optimal solution is located is [V(k + 1), V(k − 1)]; if Pin(k + 1) > Pin(k) > Pin(k − 1), the interval in which the optimal solution is located is [V(k), V(k − 1)]; else if Pin(k) = Pin(k − 1), the interval of the optimal solution is [V(k), V(k − 1)]. Figure 6 shows the iterative process for the optimal solution interval in the improved hill climbing stage.

During the search, if there is always Pin(k + 1) < Pin(k) < Pin(k − 1) until V(k + 1) ≤ Vinmin, which belongs to Case C in Figure 5, the minimum output voltage of the primary side buck-boost converter is the optimum point of the system efficiency, and the MEET is completed.

3.2.3. Golden Section Search Stage

After the improved hill climbing stage, we obtain the interval in which the optimal solution exists. The next step is to find the optimal solution using the golden section search algorithm.

The golden section search is an exact line search algorithm. When using the golden section search method, the second point and the third point form the golden section point of the initial interval, as shown in Figure 7. When the optimal solution interval is from the first point to the third point, the second point is just the golden section of the new interval; when the optimal solution interval is from the second point to the fourth point, the third point is just the golden section of the new interval. Therefore, the golden section point of the initial interval can be reused in the next iteration, greatly reducing the number of trials.

3.2.4. Flowchart of the HC-GSS Method

The flow chart of the HC-GSS method is shown in Figure 8, which is divided into three main sections, each of which is briefly described in detail below.

Firstly, in the optimal solution existence judgment stage, the search starts from the maximum output voltage of the primary side buck-boost to ensure output power stability as a priority. Set V(1) = Vinmax and decreases twice with minimum resolution, i.e., V(2) = V(1) − ΔV and V(3) = V(2) − ΔV, and calculates the corresponding input powers Pin(1), Pin(2), and Pin(3), respectively. If Pin(1) < Pin(2) < Pin(3), Vopt = Vinmax is obtained, and the MEET is terminated; otherwise, enter the improved hill climbing stage.

Secondly, in the improved hill climbing stage, the step size of the voltage change can take a larger value to increase the speed of MEET, i.e., V(k) = V(k − 1) − n × ΔV. At the same time, the current solution Pin(k) not only compares with the previous Pin(k − 1) but also compares with the latter Pin(k + 1). If there is always Pin(k + 1) < Pin(k) < Pin(k − 1) until V(k) ≤ Vinmin, Vopt = Vinmin is obtained and the MEET is terminated; else the interval where the optimal solution is located is obtained.

Finally, in the golden section search stage, the optimal solution Vopt = (a + b)/2 is searched by the golden section search algorithm.

Compared with the conventional P&O method, the proposed HC-GSS method can achieve a faster MEET. This is because the step size of the voltage change can take a larger value in the stage of the improved hill climbing stage, and the time complexity of the golden section search algorithm is lower than the P&O method. In addition, the proposed HC-GSS method can effectively avoid the conventional P&O method of three-point oscillation.

4. Simulation, Experiment, and Analysis

In Section 3, a detailed description of the proposed HC-GSS method is given. In this section, we first compare the HC-GSS method without MEET and the conventional P&O method through simulation, then analyze and illustrate the related results. Finally, the feasibility of the proposed method is verified through experiments.

4.1. Effectiveness of the HC-GSS Method

The Simulink model is built according to Figure 1, and the relevant parameters in the model are shown in

Table 1. Parameters of the Simulink model.

Parameter	Value	Parameter	Value
VBattery	48 V	Output range of the primary buck-boost	24–72 V
ω0	2π × 120 kHz	L1	1000 μH
C1	100 μF	Lps	11.5 μH
Cpp	153.8 nF	Cps	65.6 nF
Lp	38.74 μH	Rp	96 mΩ
Ls	4.09 μH	Rs	26 mΩ
Css	431 nF	C3	100 μF
L2	1000 μH	C2	100 μF
Vout	24 V	R	5, 7.5, 10, 12.5, 15 Ω
k	0.45, 0.50

. Without MEET, the primary side buck-boost converter is removed, and the inverter is connected directly to the battery. During the HC-GSS MEET method, the Simulink model interacts with the .m file; the .m file reads the Pin output of the Simulink model and modifies the Vin-Ref according to the process shown in Figure 8.

In order to validate the generality of the proposed HC-GSS method, we have varied the values of the load resistance R and the coupling coefficient k and carried out simulations under different operating conditions, where R = 5, 7.5, 10, 12.5, 15 Ω and k = 0.45, 0.50.

Table 2. Results of k = 0.45 and R = 5, 7.5, 10, 12.5, 15 Ω.

k = 0.45, Vout = 24 V	Without MEET	With HC-GSS Method
R = 5 Ω, Pout = 115.2 W	Vin = 48 V, Pin = 142.57 W	Vin = 56.33 V, Pin = 140.91 W
R = 7.5 Ω, Pout = 76.8 W	Vin = 48 V, Pin = 95.21 W	Vin = 46.52 V, Pin = 95.26 W
R = 10 Ω, Pout = 57.6 W	Vin = 48 V, Pin = 72.62 W	Vin = 43.53 V, Pin = 72.14 W
R = 12.5 Ω, Pout = 46.08 W	Vin = 48 V, Pin = 59.21 W	Vin = 39.55 V, Pin = 58.33 W
R = 15 Ω, Pout = 38.4 W	Vin = 48 V, Pin = 50.25 W	Vin = 35.25 V, Pin = 49.01 W

shows the results of k = 0.45 and R = 5, 7.5, 10, 12.5, 15 Ω, since Vout is controlled by the secondary side buck-boost converter at 24 V, the corresponding output power Pout for different resistances R can be calculated from Ohm’s law under steady-state conditions.

When without MEET, the inverter is connected directly to the battery, so that Vin is always equal to the battery voltage, i.e., 48 V. When the HC-GSS is used for MEET, it can be seen that there is a change in the inverter input voltage Vin, and at the same time, the system input power Pin is reduced.

Table 3. Results of k = 0.50 and R = 5, 7.5, 10, 12.5, 15 Ω.

k = 0.50, Vout = 24 V	Without MEET	With HC-GSS Method
R = 5 Ω, Pout = 115.2 W	Vin = 48 V, Pin = 140.69 W	Vin = 52.93 V, Pin = 139.79 W
R = 7.5 Ω, Pout = 76.8 W	Vin = 48 V, Pin = 94.61 W	Vin = 44.40 V, Pin = 94.37 W
R = 10 Ω, Pout = 57.6 W	Vin = 48 V, Pin = 72.10 W	Vin = 38.65 V, Pin = 71.56 W
R = 12.5 Ω, Pout = 46.08 W	Vin = 48 V, Pin = 58.80 W	Vin = 37.61 V, Pin = 57.92 W
R = 15 Ω, Pout = 38.4 W	Vin = 48 V, Pin = 50.09 W	Vin = 32.40 V, Pin = 48.62 W

shows the results of k = 0.50 and R = 5, 7.5, 10, 12.5, 15 Ω, same as Table 2, the system input power Pin also reduced compared to without MEET. Furthermore, by increasing the coupling coefficient, the input power can be further reduced compared to k = 0.45.

Figure 9 shows the curves of system efficiency with load resistance, coupling coefficient, and presence or absence of MEET. As can be seen, the system efficiency decreases as the load resistance increases, which is because the system efficiency is related to the load resistance as shown in Equation (6). The system efficiency with the HC-GSS method is higher than that without MEET, under the coupling coefficients of 0.45 and 0.50, so the HC-GSS method can be verified for efficiency optimization. In addition, it can also be shown that the proposed method is still effective in the case of coupling coefficient changes, i.e., coupler offsets.

4.2. Superiority of the HC-GSS Method over the Conventional P&O Method

After verifying the effectiveness of the HC-GSS method for MEET, we compare it with the most commonly used conventional P&O method to investigate the differences between the two in terms of final efficiency and speed of convergence. The relevant parameters in the Simulink model are shown in Table 1, and the reference output voltage of the primary side buck-boost is set according to the HC-GSS method and the conventional P&O method, respectively.

In the HC-GSS method, the step size of the voltage Vin change is the same as in the conventional P&O method, i.e., ΔV, in the optimal solution existence judgment stage. Whereas in the improved hill climbing stage, the step size of the voltage Vin change is taken as (Vinmax − Vinmin)/10 to accelerate the speed of the MEET.

Figure 10 shows the curves of the input power Pin versus the number of iterations for the HC-GSS and the conventional P&O methods, respectively, implemented with a coupling coefficient k = 0.45, 0.50, and load resistance R = 5, 7.5, 10, 12.5, 15 Ω.

In Figure 10a,b, it can clearly be seen that there is a three-point oscillation present in the final stages of the conventional P&O method, while the proposed HC-GSS method can effectively avoid the oscillation. In addition, for the same number of iterations in Figure 10, the HC-GSS method has a smaller Pin than the conventional P&O method, which means higher efficiency.

Table 4. Stable number of iterations and ultimate efficiency of k = 0.45 and R = 5, 7.5, 10, 12.5, 15 Ω.

k = 0.45, Vout = 24 V	HC-GSS Method		Conventional P&O Method
	Stable Number of Iterations	Ultimate Efficiency	Stable Number of Iterations	Ultimate Efficiency
R = 5 Ω, Pout = 115.2 W	18	81.75%	16	81.61%
R = 7.5 Ω, Pout = 76.8 W	20	80.62%	25	80.65%
R = 10 Ω, Pout = 57.6 W	15	79.83%	32	79.76%
R = 12.5 Ω, Pout = 46.08 W	15	79.05%	36	78.95%
R = 15 Ω, Pout = 38.4 W	15	78.34%	37	77.82%

and

Table 5. Stable number of iterations and ultimate efficiency of k = 0.50 and R = 5, 7.5, 10, 12.5, 15 Ω.

k = 0.50, Vout = 24 V	HC-GSS Method		Conventional P&O Method
	Stable Number of Iterations	Ultimate Efficiency	Stable Number of Iterations	Ultimate Efficiency
R = 5 Ω, Pout = 115.2 W	19	82.39%	18	82.07%
R = 7.5 Ω, Pout = 76.8 W	21	81.37%	28	81.18%
R = 10 Ω, Pout = 57.6 W	16	80.48%	32	80.36%
R = 12.5 Ω, Pout = 46.08 W	10	79.62%	32	78.94%
R = 15 Ω, Pout = 38.4 W	15	78.97%	37	78.87%

show the number of stable iterations and final efficiencies of the HC-GSS Method and Conventional P&O Method under different coupling coefficients and load conditions. As can be seen from Table 4 and Table 5, there is little difference in the ultimate efficiency between the HC-GSS method and the conventional P&O method, but the proposed HC-GSS method achieves stabilization faster in most cases.

In summary, the HC-GSS method has a fast convergence rate in most cases. At the same time, it effectively avoids the oscillations that eventually occur with the conventional P&O method.

4.3. Consistency Analysis between Simulations and Experiments

In order to verify the results of the above analyses, we built the experimental prototype shown in Figure 11. The relevant parameters of individual components in the experiment are consistent with the parameters of the Simulink simulation model in Table 1, and the coupling coefficient k in the experimental prototype is 0.45.

The voltage regulation ability of the secondary-side buck-boost converter was first tested. Figure 12a,b show the output waveforms of the converter when the input voltage is increased and decreased by 10 V with R = 15 Ω and Vout = 24 V, respectively. The converter adopts the PI control method with a rise and fall regulation time of about 300 ms and overshoots of 6 V and 4 V, respectively. The shorter the secondary side regulation time, the more generations can be iterated at the same time, and the faster the MEET.

Then, the MEET effects of the conventional P&O method and the HC-GSS method were verified under load conditions of R = 10 Ω, as shown in Figure 13a,b. The ΔV of the conventional P&O method is 2 V, and the ΔV of the improved hill climbing stage in the HC-GSS method is 4.8 V. It can be seen that the HC-GSS method converges faster and avoids final oscillations.

Using the DC power supply to simulate a 48 V battery connected directly to the inverter, i.e., without MEET, the input power of the system is 73.44 W at R = 10 Ω. The input power of the system is higher when MEET is not used compared to that using the HC-GSS method (72.5 W), and therefore, there is an improvement in the efficiency of the system with the proposed method.

The main reason for the decrease in the efficiency of the system in the prototype experiments is the parasitic resistance of the wires in the experiments, but the proposed HC-GSS method achieves fast MEET in both theory and experiments and avoids the oscillation in the final stage.

5. Conclusions

In this paper, we took the LCC-S compensated WPT system as an example. Through theoretical calculations, we obtain that there exists a correspondence between the inverter optimal input voltage and the maximum efficiency point of the proposed communication-free WPT system when the output current or voltage is determined. Considering that the inverter needs to work in a weak inductive switching state in the real system, the relationship between the optimal input voltage and the maximum efficiency point is too complex to be solved. We propose a primary-side MEET search optimization algorithm based on the HC-GSS method. Simulation and experimental results show that the proposed HC-GSS method can achieve a faster MEET compared with the conventional P&O method. At the same time, it effectively avoids the oscillations that eventually occur with the conventional P&O method and is still applicable when the coupler is offset. As the proposed HC-GSS method does not require real-time communication, it avoids the different degrees of attenuation of electromagnetic waves caused by the different temperatures, salinity, and conductivity of the underwater environment, which results in long delay and poor stability of the communication system and improves the stability of the underwater WPT system to a certain extent.