A Constant-Pressure Hydraulic PTO System for a Wave Energy Converter Based on a Hydraulic Transformer and Multi-Chamber Cylinder

Abstract

This paper presents a constant-pressure hydraulic PTO system that can convert stored pressure energy into electrical energy at a stable speed through hydraulic motors and generators. A multi-chamber cylinder can be connected to the main power generation circuit by check valves, and the motor displacement can be controlled by a fuzzy controller to maintain the main power generation circuit under stable pressure. The hydraulic transformer can control the forces applied to the floater. The hydrodynamic parameters of the floater are calculated by AQWA, and the optimal PTO damping of the hydraulic system is analyzed as the target of transformer control. MATLAB/Simulink and AMESim are used to carry out the co-simulation. Three kinds of wave elevation time-series for the specific state are designed for the simulation. In the co-simulation, three approaches are carried out for the simulation including no control strategy, fuzzy control with a fixed transformer ratio, and fuzzy control with a variable transformer ratio. Under the fuzzy control with a fixed transformer ratio, the floater displacement and captured energy do not increase significantly, but the oil pressure fluctuation is very stable, which indicates that the fuzzy controller maintains the stability of the main power circuit. While under fuzzy control with a variable transformer ratio, the power generation is not larger than those under no control strategy or fuzzy control with a fixed transformer ratio, which proves that this hydraulic transformer concept is less efficient.

1. Introduction

In the context of a resource crisis and environmental pollution, the development of clean resources is the only way out of the current predicament. The ocean covers more than 70% of the Earth and provides a lot of wave energy. It is estimated that the global ocean wave power potential is about 10 TW [1]. Wave energy is expected to replace fossil fuels to contribute to the future electricity supply [2]. However, extracting this energy is technically and economically challenging, and equipment development is still at an early stage [3,4,5]. According to their working principle, existing WECs can be divided into three categories: oscillating water column, oscillating bodies, and overtopping devices [6]. Oscillating bodies are referred to in some contexts as wave-activated bodies [7]. Compared with other types, oscillating body WECs have been widely studied because of their high efficiency and small dimensions. This paper therefore focuses on the oscillating body WEC characterized by a heaving response in the water.

Currently, WEC researchers and developers have proposed a variety of PTO systems, such as air/hydraulic turbine based, direct-mechanical drive systems, direct-electrical drive, and hydraulic systems. Among them, the hydraulic system is ideal for low-frequency high-power density waves, with an efficiency of up to 90% in typical WECs. Consequently, the hydraulic system is the most suitable PTO system for WECs [8,9,10]. However, simplicity is an important factor that should be taken into consideration when designing a hydraulic PTO system. Complex systems involve a large number of components, which not only reduces the efficiency, but also increases the installation difficulty as well as the operation and maintenance costs. Hence, a successful PTO system not only has the characteristics of reliability, standardization, simplicity, and scalability, but also works with high and stable operation efficiencies in various sea states. The aim of this paper is to propose a novel hydraulic system to meet these design requirements.

In Section 2, the latest technology of the hydraulic PTO system is reviewed, and the proposed PTO system including some main components and a system control method is introduced as well. In Section 3, AQWA is used to calculate the hydrodynamic parameters of the floater to analyze the optimal PTO damping of the hydraulic system. Then, a hydrodynamic calculation model is built in Matlab/Simulink. In Section 4, a hydraulic system model is built to perform a joint simulation of Matlab/Simulink and AMEsim under the PM irregular sequence wave. In Section 5, based on the fuzzy control method, the pressure of the main power generation circuit is controlled, and the PTO damping of the system is regulated through the hydraulic transformer, and finally, the joint simulation under the control strategy is carried out. Section 6 is the conclusion of this paper.

2. Hydraulic PTO System

2.1. Multi-Concepts of PTO

In general, the classification of hydraulic PTO systems is difficult. According to [8] [10,11], the hydraulic PTO system is mainly divided into the variable-pressure system and the constant-pressure system. A variable-pressure system is assumed to be simpler than a constant-pressure system [8]. The conventional variable-pressure system can be described as double-acting with a double-rod hydraulic cylinder that is directly connected with the bi-directional variable-displacement hydraulic motor, which is connected with the generator to achieve power generation. Therefore, it is the simplest and most economical PTO system.

However, without using any accumulator to store energy, the inconsistent fluid pressure cannot drive the generator output smoothly. The improved variable-pressure system overcomes these deficiencies, as shown in Figure 1a [11,12,13]. This concept increases a reservoir module. This module includes four directional check valves, a low-pressure accumulator (LPA), a relief valve, and a booster pump connected to the LPA. This concept adds directional check valves, a low-pressure accumulator, a relief valve, and a booster pump to form a reservoir module. The check valves control the flow of fluid in the pipeline, and the relief valve is used to prevent the components of the PTO from overpressure. The LPA is mainly used to avoid cavitation accidents on the low-pressure side while the booster pump is used to supplement the flow.

Although the variable-pressure system is simple, it is inferior to the constant-pressure system in terms of the control efficiency and flexibility, and its efficiency can reach more than 90% [10]. The reason for achieving such excellent results is that the concept can flexibly implement advanced control strategies. However, implementing flexible control makes the concept more complex and expensive than a variable-pressure system [17]. Categories of constant-pressure systems mainly include concepts based on two check valves (Figure 1b), four check valves (Figure 1c), directional control valves (Figure 1d), variable displacement hydraulic pump/motor (Figure 1e), or hydraulic transformers (Figure 1f).

As presented in Figure 1b [14], in order to correct the flow difference between the hydraulic cylinder and the hydraulic motor during cylinder stretching, the large chamber of the hydraulic cylinder is connected to a simple two-check valve system. The small actuator chamber is directly connected to the tank, where the energy is not used, and the system adopts a pressure relief valve to prevent the entire oil system from overpressure. During the upward movement, the piston forces the high-pressure fluid through the check valve into the large actuator chamber and the high-pressure accumulator (HPA). The hydraulic motor is directly connected to a generator, and the hydraulic motor converts hydraulic energy into electricity. As the piston moves down, the low-pressure fluid directly flows from the oil tank into the small actuator chamber through the other check-valve back. In this system, only the energy of the double-acting with a single rod hydraulic actuator moving upward is utilized with part of the energy wasted.

In order to solve this shortcoming, the concepts of the hydraulic PTO system based on four check valves is presented in Figure 1c [15]. The double-rod double-acting hydraulic actuator solves the problem of unbalanced fluid pressure in the actuator chamber. For the four-check rectifier system, only two check valves operate during the upward movement of the hydraulic actuator, while the remaining two check valves operate during downward movement. In order to solve the problem of hydraulic fluid leakage in the line, the concept uses an accumulator instead of a booster pump, which requires an external power source [11]. In addition, pressure relief valves are designed to protect hydraulic components.

A new system is proposed to improve the controllability of the hydraulic PTO system, which is based on the directional control valve as presented in Figure 1d [13]. In the rectification module, a single directional control valve is used to replace the four-valve system. A three-position four-way directional valve is added to the hydraulic PTO system. Its input port is directly connected to the two cavities of the hydraulic actuator chamber. Under the working condition, its two output ends are connected to the high-pressure and low-pressure lines of the system, respectively.

Variable displacement hydraulic pumps/motors with four operating modes are used in the concept of another constant-pressure system that is presented in Figure 1e [16]. In this concept, the actuator uses a symmetrical hydraulic cylinder to generate two-way flow through a three-dimensional four-way valve connected to the hydraulic pump/motor. The hydraulic pump/motor is adjusted with the amount of incoming flow to keep the generator at a fixed speed. If the direction of fluid flow changes, the hydraulic motor will also transform at an opposite angle. Therefore, the hydraulic system realized a reversible transmission with a continuously variable gearing ratio between the floater and generator. The booster pump is used to supplement the leakage flow of the system, and set the flushing and safety valve module to flush the dirt particles in the system to play a cooling role. By controlling the displacement of the hydraulic pump/motor, the differential pressure of the cylinder may be controlled, thereby also implementing continuous force control of the cylinder. The system is capable of four-quadrant control. The pump can be used as a rectifier for hydraulic power, and when it is connected to a second pump, the arrangement is called a hydraulic transformer.

The concept is based on a hydraulic transformer as presented in Figure 1f, where a four-quadrant mode pump acts as a rectification module, which is an architectural evolution from Figure 1e [10]. The bidirectional fluid flow of the hydraulic actuator is converted into a one-way high-speed rotation through the hydrostatic circuit. Pressure control controls the force applied to the WEC device by adjusting the swash plate angle of the hydraulic transformer through the generator circuit. This is an advanced hydraulic power factor corrector that not only corrects bidirectional flow, but also controls the hydraulics of the PTO system.

In summary, many PTO systems have been proposed to maximize power generation. However, as far as the author knows, there is no research in the literature that uses a multi-chamber cylinder with a hydraulic transformer to extract power. Therefore, this research work takes into account the characteristics of the constant pressure power take-off, and puts forward the concept of the constant pressure power take-off.

2.2. PTO Concept

The aim of this paper is to capture more wave energy by improving the hydraulic PTO system, and study whether the hydraulic transformer has a positive effect on the improvement of efficiency. To achieve this target, the concept integrates modules 1, 2, and 5, 6, as well as other necessary components, as shown in Figure 2.

The aim of the multi-chamber cylinder (1) (with four chambers) is to extract energy from the waves. The two chambers of the medium are connected to the hydraulic transformer circuit whereas another two chambers are connected to the generator circuit. The two circuits are combined with a module of the hydraulic transformer (5, 6). The function of the rectifier device (2) is to rectify the two-way flow into a one-way flow, which is transmitted to the hydraulic motor to drive the generator to generate electricity. The generator (10) is coupled with the hydraulic motor, transforming the rotating mechanical energy into electric energy.

The hydraulic system is made up of central, common, and closed-circuit oil supplies (i and j), which collect energy from all hydraulic units and deliver it to all hydraulic units, through these systems to achieve the goal of the array. It collects all hydraulic units that reach the goal of the array. However, the prerequisite for achieving this goal is to keep the system oil circulation pressure constant. This can be achieved by controlling the displacement of unit 7 and accumulator 8. In this condition, every system connected through the common lines (i and j) with the same pressure, resulting in throttling pressure losses and minimized compressibility.

One purpose of this concept is to investigate whether the efficiency of energy acquisition can be improved by using hydraulic transformers. Because the generator circuit pressure is constant, the pressure of the medium two chambers can be adjusted to change the movement of the floater by controlling the displacement ratio of the motor and pump in the hydraulic transformer. In this case, the floater can be controlled to extract more energy and improve the efficiency of the PTO concept.

2.3. Hydraulic Transformer Multi-Chamber Cylinder

Unit 5 and unit 6 (Figure 2) are combined with a module of the hydraulic transformer. In this paper, the analysis is based on the assumption that the hydraulic transformer does not have any energy loss during operation. So, the driving torque of unit 5 can be calculated by:

$$T_5=\frac{\Delta P_5{V}_5}{2\mathsf{\pi }}=\frac{V_{5max}}{2\mathsf{\pi }}\frac{{\beta }_5}{{\beta }_{5max}}\Delta P_5$$

(1)

Unit 6 rotates with unit 5, so it will generate resistance torque, which has the form of:

$$T_6=\frac{\Delta P_6{V}_6}{2\mathsf{\pi }}=\frac{V_{6max}}{2\mathsf{\pi }}\frac{{\beta }_6}{{\beta }_{6max}}\Delta P_6$$

(2)

When the hydraulic transformer is stable, the difference between the driving torque and the resistance torque is zero. With the back pressure of units 5 and 6 considered as zero, the transformer ratio can be determined with the development of Equations (1) and (2), which gives:

$$\lambda =\frac{P_5}{P_6}=\frac{V_5}{V_6}=\frac{{\beta }_{6max}}{{\beta }_{5max}}\frac{{\beta }_6}{{\beta }_5}\propto \frac{{\beta }_6}{{\beta }_5}$$

(3)

where $\Delta P_5$ and $\Delta P_6$ are the pressure differentials of units 5 and 6;$V_{5max}$ and $V_{6max}$ are their max displacement; ${\beta }_{5max}$ and ${\beta }_{6max}$ are their max swash plate inclination; $V_5$ and $V_6$ are their real displacement; ${\beta }_5$ and ${\beta }_6$ are their real swash plate inclination; and $P_5$ and $P_6$ are the pressure of their entrance.

Then, unit 6 is connected to a steady circuit, so the pressure in the medium two chambers of the cylinder can be controlled by adjusting the number of λ. It indicates that the movement of the floater can be controlled to extract max energy.

2.4. Multi-Chamber Cylinder

In this analysis, the multi-chamber cylinder is assumed similar to the Wavestar Wave Energy Converter (WEC), which makes use of three of the four chambers [18], while in this paper, four chambers are used as presented in Figure 3. So, the force in the hydraulic cylinder has the form of:

$$F_{PTO}=P_{A1}{A}_1-P_{A2}{A}_2+P_{A3}{A}_3-P_{A4}{A}_4$$

(4)

The pressure of the generator circuit is constant and one side of the hydraulic is also connected to the generator circuit, so according to Equation (3), the force of the hydraulic $F_{PTO}$ can be simplified:

$$F_{PTO}=\begin{array}{c}{P}_{constant}{A}_1+\lambda P_{constant}{A}_3 v>0\\ P_{constant}{A}_4+\lambda P_{constant}{A}_2 v<0\end{array}$$

(5)

where $P_{constant}$ is the pressure of the generator circuit.

Then, $\lambda$ can be controlled to provide adequate force resolution and extract as much wave power as possible.

3. Hydrodynamics of the Point Floater WEC

3.1. Dynamic Equation

A circular cylindrical-shaped floater is used in the analysis. The heave motion of the floater is used for the dynamic analysis of the system. The motion of the floater obeys the Newton second law, hence the time-domain dynamic equation has the form of [19]:

$$\left[m+A_{33}(\infty )\right]\ddot{x}\left(t\right)+{{\displaystyle \int }}_{-\infty }^{t}K\left(t-\tau \right)\dot{x}\left(\tau \right)d\tau +S_{3}x\left(t\right)=F_{e3}\left(t\right)+F_{PTO}\left(t\right)$$

(6)

where $m$ is the mass of a heaving cylinder, which is assumed to be equal to the mass of the displaced water volume. $A_{33}(\infty )$ is the added mass at infinite frequency, as presented in Figure 4c; $S_3$ is the hydrostatic restoring coefficient; $F_{e3}\left(t\right)$ is the wave excitation force; $F_{PTO}\left(t\right)$ is the linear hydraulic damping force of PTO systems; and $K\left(t\right)$ is the impulse response function (IRF) calculated by:

$$K\left(t\right)=\frac{2}{\mathsf{\pi }}{{\displaystyle \int }}_0^{\infty }{B}_{33}\left(\omega \right)\cos \left(\omega t\right)$$

(7)

where $B_{33}\left(\omega \right)$ is the radiation resistance against the wave angular frequency $\omega$.

$F_{e3}\left(t\right)$ can be calculated by [20,21]:

$$F_{e3}\left(t\right)={\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}\left|f_{e3}\left({\omega }_{\mathrm{j}}\right)\right|A\left({\omega }_{\mathrm{j}}\right)\cos \left({\omega }_{\mathrm{j}}t+2\mathsf{\pi }rand()\right)$$

(8)

where n is the number of frequency segments; ${\omega }_{\mathrm{j}}$ is the j-th angular frequency; rand () is a uniformly distributed random-number generator in the interval (0, 1); $f_{e3}\left({\omega }_{\mathrm{j}}\right)$ is the excitation-force coefficient; and $A\left({\omega }_{\mathrm{j}}\right)$ denotes the amplitude of the wave elevation when $\omega$ is equal to ${\omega }_{\mathrm{j}}$:

$$A\left({\omega }_{\mathrm{j}}\right)=\sqrt{2S_{\omega }\left({\omega }_{\mathrm{j}}\right)\Delta \omega }$$

(9)

where $S_{\omega }$ is one certain wave spectrum in terms of the angular frequency; $\Delta \omega$ signifies the difference between the adjacent angular frequencies.

$F_{PTO}\left(t\right)$ can be described as:

$$F_{PTO}\left(t\right)=C_{PTO}\dot{x}\left(t\right)$$

(10)

where $C_{PTO}$ is the power absorption (damping) coefficient.

In the heaving motion system, the damping force is the main force for absorbing wave energy. In a cycle of waves, the average power absorbed by the hydraulic PTO system can be described as:

$$\overline{P_{PTO}}\left(\omega \right)=\frac{1}{2}{C}_{PTO}\frac{F_{e3}{}^2}{{\left(C_{PTO}+B_{33}\left(\omega \right)\right)}^2}\frac{1}{1+\frac{1}{4{\delta }_{\mathrm{n}}{}^2}{\left(1-{\left(\frac{\omega }{{\omega }_{\mathrm{n}}}\right)}^2\right)}^2}$$

(11)

where ${\delta }_{\mathrm{n}}{}^2$ is the damping factor of the floater.

When $\omega$ is equal to ${\omega }_{\mathrm{n}}$, resonance of the floater occurs. The amplitude of the floater reaches the maximum and the energy capture performance is the optimum. So, the $\overline{P_{PTO}}\left(\omega \right)$ reaches the maximum. $\overline{P_{PTO}}\left(\omega \right)$ can be described as:

$$\overline{P_{PTO}}\left({\omega }_{\mathrm{n}}\right)=\frac{1}{2}\frac{C_{PTO}{F}_{e3}{}^2}{{\left(C_{PTO}+B_c\left(\omega \right)\right)}^2}$$

(12)

When $C_{PTO}$ is 0 and $\overline{P_{PTO}}$ is 0, the PTO system does not absorb energy. When $C_{PTO}\to \infty$ and $\overline{P_{PTO}}$ is 0, the floater hardly moves, and the input power of the PTO system is almost 0. When $0<C_{PTO}<\infty$, $\overline{P_{PTO}}\left({\omega }_{\mathrm{n}}\right)$ is the function of $C_{PTO}$ and the PTO system has a maximum value. When $\frac{\partial \overline{P_{PTO}}}{\partial C_{PTO}}$ is 0, $\overline{P_{PTO}}\left({\omega }_{\mathrm{n}}\right)$ reaches the maximum. The maximum output damping conditions of the PTO system can be described as [21]:

$$C_{PTO}=B_{33}$$

(13)

$E_{har}\left(t\right)$ represents the energy harvested by the floater from the waves [22]:

$$E_{har}\left(t\right)={{\displaystyle \int }}_0^t{F}_{PTO}\left(t\right)\dot{x}\left(t\right)dt$$

(14)

3.2. Hydrodynamics Performance Simulation

The hydrodynamic analysis software AQWA is used to calculate the hydrodynamic performance of the floater. The main parameters of the floater are shown in

Table 1. Design parameters of the floater.

Parameter	Value
Diameter $D/\mathrm{m}$	5
Height $H/\mathrm{m}$	2
Draft $Dr/\mathrm{m}$	1
Center of gravity $\left(x,y,z\right)/\mathrm{m}$	(−0.451, 0, 0)
Weight $m/\mathrm{Kg}$	20,125
Moment of inertia $Ixx/{\mathrm{Kgm}}^2$	39,518.8
Moment of inertia $Iyy/{\mathrm{Kgm}}^2$	39,518.8
Moment of inertia $Iyy/{\mathrm{Kgm}}^2$	62,851.2

. The moment of inertia presented is calculated relative to the center of gravity of the body. Here, the sea-water density ${\rho }_w$ is $1025\mathrm{kg}/{\mathrm{m}}^3$ and the gravity acceleration $\mathrm{g}$ is $9.81\mathrm{m}/{\mathrm{s}}^2$. The frequency-domain numerical results of the floater are shown in Figure 4.

It can be seen that the heave response of the floater is more severe in the low frequency stage. The value of the maximum heaving RAO of the floater is 1.746 mm at 2.08 rad/s. As the frequency increases, the added mass of the floater $A_{33}(\infty )$ decreases first and then increases, reaching the minimum value of 23,082 kg when the wave frequency is 2.576 rad/s. The excitation-force coefficient $f_{e3}\left({\omega }_{\mathrm{j}}\right)$ shows an obvious downward trend, which is close to 0 N/mm after 3.897 rad/s. The maximum radiation damping of the floater $B_{33}\left(\omega \right)$ is 17.65 Ns/mm at 1.696 rad/s. A peculiar behavior is presented in $B_{33}\left(\omega \right)$ at the neighborhood of 3.567 rad/s in that $B_{33}\left(\omega \right)$ suddenly drops to −4.35 Ns/mm, and the radiation damping becomes almost 0 Ns/mm as the wave frequency increases further. The variation pattern of $B_{33}\left(\omega \right)$ does not decrease smoothly but reaches these oscillations. This phenomenon is related to AQWA’s solution of the waterline integral term [23].

Then, hydrodynamic time-domain calculations of three regular waves with different wave heights, S1 ($H_s$ is $0.5\mathrm{m}$), S2 ($H_s$ is $0.75\mathrm{m}$), S3 ($H_s$ is $1\mathrm{m}$), under different periods are performed. In this paper, regular waves between periods of 2 and 10 s are calculated at intervals of 0.5 s to obtain the velocity of the floater under different wave conditions. Figure 5 shows the velocity–time curves of three different wave heights in the stable period for a wave period of 5 s. As seen from Figure 5, the maximum velocity of the floater for $H_s$ is 1, 0.75, and 0.5 m for 606, 461.4, and 309.6 mm/s, respectively.

According to Equation (10), the optimal damping force of the PTO can be derived by the multiplication of the heave velocity in the stable period of the floater under different regular waves with the optimal PTO damping. Figure 6 depicts the maximum value of the PTO damping force. As seen from the figure, when the wave period is 3 s, $F_{PTO}$ achieves the maximum value. At this period, the wave frequency is 2.08 rad/s, and it can be seen from Figure 4a that the motion response of the floater is the most intense at this frequency. When $H_s$ is 1 m, the maximum value of $F_{PTO}$ reaches 16,110 N, and when $H_s$ is 0.75 and 0.5 m, $F_{PTO}$ is 12,090 and 8407 N, respectively. As the period increases to 10 s, $F_{PTO}$ tends to 0. Obviously, the larger the wave, the greater the requirements of the floater on the optimal damping force of the PTO to maximize energy absorption.

3.3. Hydrodynamic Model

The hydrodynamic parameters obtained from the above calculation are used as the design input of the Simulink model to build a hydrodynamic model for co-simulation with AMESim. Simulink implementation of the dynamic equation of the heaving floater is shown in Figure 7.

4. Co-Simulation

In order to investigate the system operation, the co-simulation in this section is based on these assumptions: Fluid pressure is uniform, whereas fluid density, temperature, and pressure are constant. The co-simulation in the next section is also based on these assumptions.

4.1. Wave Model

The irregular waves vary in frequency, amplitude, and direction from time to time. Thus, real sea waves are investigated in a statistical way using so-called power spectra [17]. In this study, three sea states were analyzed ((A: $H_s$ is $0.5\mathrm{m}$, $T_p$ is $4.5\mathrm{s}$), (B: $H_s$ is $0.75\mathrm{m}$, $T_p$ is $5\mathrm{s}$), (C: $H_s$ is $1\mathrm{m}$, $T_p$ is $5.5\mathrm{s}$)).

$S\left(\omega \right)$ is the wave amplitude spectrum, as presented in Figure 8:

$$S\left(\omega \right)=5{\mathsf{\pi }}^4\frac{H_s^2}{T_p^4}\frac{1}{{\omega }^5}exp\left(\frac{-20{\mathsf{\pi }}^4}{{\omega }^4{T}_p^4}\right)$$

(15)

where $H_s$ is the significant wave height;$T_p$ is the peak period of the spectrum.

$S\left(\omega \right)$ can be used to extract the individual wave components given by i:

$${\eta }_{\mathrm{i}}\left(t\right)=\sqrt{2\Delta \omega ·s\left({\omega }_{\mathrm{i}}\right)} \cos \left(\omega t+{\phi }_{\mathrm{i}}\right)$$

(16)

The time series of irregular surface elevation is established by stacking wave components:

$$\eta \left(t\right)={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\sqrt{2\Delta \omega ·s\left({\omega }_{\mathrm{i}}\right)} \cos \left(\omega t+{\phi }_{\mathrm{i}}\right)$$

(17)

Here, the ${\phi }_{\mathrm{i}}$ is a random phase for each component.

Wave elevation time-series were created for a 40 s simulation time period based on the previous formulation and are shown in Figure 9.

4.2. Full Model Design

In order to prove the effectiveness of the hydraulic system, a complete model without a control strategy is established in the co-simulation environment, which is the combination of AMESim and MATLAB/Simulink software, as shown in Figure 10.

4.3. Simulation

In the co-simulation of the hydraulic system without a control strategy, the wave elevation time-series for a specific state in Figure 9 is adopted, and the calculation time of each specific state is 100 s. The transformation ratio was set to a constant value of 0.35. The displacement and energy capture efficiency of the floater under three kinds of irregular waves are calculated. The transformer ratio and motor displacement are not controlled, and they are all set to constants. The simulation results are presented in Figure 11 and Figure 12.

It is seen from Figure 11 that the maximum displacement of the floater at the sea state C during 20 and 60 s is about −0.9133 m, with the sea state A and B at −0.7171 and −0.3125 m, respectively. Figure 12 compares the absorbed energy of the floater at three different sea states during the computation time of 100 s. At 100 s, the absorbed energy of sea state C reaches 366 kW, while the absorbed energy of sea states A and B are 241.8 and 112.8 kW, respectively. It is worth noting that the more severe the sea conditions, the stronger the floater’s motion response, and the more energy harvested.

5. PTO Control Strategies

To study the energy harvest efficiency of the PTO system, the whole model of the system with control strategy is established and the fuzzy controller is designed. The control performance of the fuzzy controller with a fixed transformer ratio is evaluated by using the wave elevation time-series for a specific state. Then, the energy capture efficiency under the dual control of variable transformer ratio and the fuzzy controller is calculated to verify the optimization effect of the transformer on system energy capture.

5.1. Motor Displacement Control

The purpose of this task is to adjust the inlet pressure of unit 7 (Figure 2) according to the pressure measured in the pipeline by adjusting the angle of the swash plate [24]. An adaptive controller based on intelligence, such as fuzzy, is suitable for this case. As presented in Figure 13, the control was optimized to reduce the error $\left(\mathrm{e}\right)$ between the reference $P_{ref}$ and the real pressure $P_{real}$.

As discussed above, there are two inputs and one output. One of the input pressures errors $\mathrm{e}$ are forced into the range from −1 to 1 by using suitable scaling factors. Another input $\dot{\mathrm{e}}$ is forced into the range from −1 to 1 in the same way. The final fuzzy output is the pump control signal $u_m$, which is forced into the range from 0 (minimum motor displacement) to 1 (maximum motor displacement) to turn the motor swash angle.

In this research, for each of the input variables, seven membership functions are used, namely, “NB”, “NM”, “NS”, “ZERO”, “PS”, “PM”, and “PB”. For the output variable, four membership functions are used, namely “ZERO”, “PS”, “PM”, and “PB” for smooth tuning consumption. Details of the fuzzy input as well as output membership functions are shown in Figure 13. So, 49 if-then laws set were employed to the inputs (error and its first derivate), which are presented in

Table 2. Rule table of the motor displacement—fuzzy controller.

${\mathit{u}}_{\mathit{m}}$		e
		NB	NM	NS	ZERO	PS	PM	PB
de	NB	ZERO	PS	ZERO	PS	PS	PM	PB
	NM	ZERO	PS	ZERO	PS	PS	PM	PB
	NS	ZERO	ZERO	ZERO	ZERO	PS	PM	PB
	ZERO	ZERO	ZERO	ZERO	ZERO	PS	PM	PB
	PS	ZERO	ZERO	PS	PS	PM	PM	PB
	PM	ZERO	PS	PS	PS	PM	PB	PB
	PB	ZERO	PS	PS	PM	PM	PB	PB

. Then, the corresponding model can be designed as shown in Figure 14.

According to the input and output variables and the rule set established in Table 2, the control signal corresponding to the wave fluctuation can be obtained to adjust the slope of the pump.

5.2. Parameter Control of Hydraulic Transformer

The energy harvested by the system can be improved by controlling the PTO force ($F_{PTO}\left(t\right)$). In this study, $F_{PTO}\left(t\right)$ is composed of the pressure of the power generation circuit and the hydraulic transformer circuit (Equation (5)). For simplicity, the $F_{PTO}\left(t\right)$ can be written as:

$$F_{PTO}\left(t\right)=F_1+\lambda F_2$$

(18)

where the coefficients $F_1$ and $F_2$ are constant values.

The optimal coefficient $\lambda$ of Equation (18) is closely related to the dynamic of the system. When the value of $\lambda$ increases, the increase of $F_{PTO}\left(t\right)$ causes the transformer circuit to absorb a large number of pressure peaks, resulting in a more stable circuit pressure. However, it will also stop the hydraulic cylinder from moving in the case of small waves. In order to find the optimal parameter $\lambda$, the efficiency of PTO ${\eta }_{PTO}$, the harvested energy and the pressure stability of the generated circuit $p_v$ should be taken into account. In the calculation, the stability of pressure can be evaluated by the change of pressure:

$$\begin{gathered} \left(\lambda \right)=\underset{\lambda }{\max }{\omega }_1\frac{1}{T_{sim}}{{\displaystyle \int }}_0^{T_{sim}}{F}_{PTO}\dot{x}\left(t\right)dt+{\omega }_2{\eta }_{PTO}+{\omega }_3{p}_v \\ \left(s,t\right)F_{PTO}\le F_{max} \end{gathered}$$

(19)

where:

$${\eta }_{PTO}=\frac{\frac{1}{T_{sim}}{{\displaystyle \int }}_0^{T_{sim}}{Q}_m{P}_{m}dt}{\frac{1}{T_{sim}}{{\displaystyle \int }}_0^{T_{sim}}{F}_{PTO}\dot{x}\left(t\right)dt}$$

(20)

Here, ${\omega }_1$, ${\omega }_2$, ${\omega }_3$ are weighting coefficients; $Q_m$ is the flow of unit 7; and $P_m$ is the pressure of the unit 7.

Equation (19) is solved by using batch processing. The PTO efficiency, pressure stability of the power generation circuit, and the expected power outputs are all related to the optimal $\lambda$.

5.3. Full Model Design

In order to verify the effectiveness of the hydraulic control system, a system model with a designed motor displacement fuzzy controller was established in a co-simulation environment. The complete model with a control strategy is shown in Figure 15.

5.4. Simulation

The pressure of the main power generation circuit in the system, $P_{ref}$, is set as 45 bar. The diameter of each piston in the multi-chamber hydraulic cylinder is 65 mm, and the diameter of the piston rod is 55 mm. The transformer controls the transformer ratio to meet the optimal damping force of PTO, and the optimal damping force of PTO calculated above is used as the control target in this simulation. The irregular wave adopts the wave elevation time-series for specific states in Figure 9, and the calculation time of each specific state is 100 s.

5.4.1. Fuzzy Control with a Fixed Transformer Ratio

In the co-simulation of a hydraulic system with a control strategy, the displacement of the motor is controlled to ensure that the pressure of the main power generation circuit is constant, while the transformation ratio still adopts the constant value of 0.35. The system pressure simulation results of the main generation circuit are shown in Figure 16. It can be seen from Figure 16 that the system pressure of the main generation circuit fluctuates less than that without control after fuzzy control, and the system is more stable. It indicates that fuzzy control has a beneficial effect on the pressure stability of the main power circuit.

5.4.2. Fuzzy Control with a Variable Transformer Ratio

This section carries out the co-simulation of fuzzy control with a variable transformer ratio. The floater displacement, energy capture efficiency, and generating capacity of the generator with an irregular wave as input are calculated respectively. The simulation results are shown in Figure 17, Figure 18 and Figure 19.

Under the same wave condition, the displacement results of the different control methods are almost the same. Compared with the fuzzy control with a fixed variable ratio or no control strategy, the energy capture efficiency and power generation capacity of the fuzzy control with a variable ratio are not significantly improved. As shown in Figure 18, the meaningful wave height $H_s$ is $0.5\mathrm{m}$, the energy capture efficiency of the floater in the 100th second is $1.14\times {10}^5$ W under dual control; it increases to $3.5\times {10}^5$ W when $H_s$ is $1\mathrm{m}$. It shows that the energy capture efficiency of the floater grows with the increase of the wave strength, and the power generation capacity of the generator also follows the same law.

As shown in Figure 19, under the same sea condition, dual control will reduce the power generation efficiency. When $H_s$ is $1\mathrm{m}$, the power generation capacity of the generator is about $2.7\times {10}^5$ W without control, and the power generation capacity is close to $2.2\times {10}^5$ W under dual control. It can be seen that the fuzzy control is mainly to maintain the stability of the main power generation circuit, and the fuzzy control with a variable transformer ratio has no obvious advantage in energy capture. The main reason is that the two pumps of the hydraulic transformer work back-to-back, so the performance of the pumps depends on each other. Although the hydraulic transformer can achieve phase control of the floater, its efficiency is not significantly improved, and even counterproductive. This is consistent with the exposition in [8].

6. Conclusions

This paper reviews the main components of the existing PTOs hydraulic system, and proposes a constant-pressure hydraulic PTO system for the wave energy converter based on a hydraulic transformer and multi-chamber cylinder, which can convert the stored pressure energy into electrical energy at a stable speed through hydraulic motors and generators. The multiple chambers of the hydraulic cylinder can be connected to the main power generation circuit by checking valves, and the fuzzy controller is used to control the motor displacement. Hence, the main power generation circuit can be maintained at a stable pressure.

The hydrodynamic calculation results of the floater by AQWA show that with the increase of the frequency, the excitation-force coefficient shows an obvious downward trend, which is close to 0 N/mm after 3.897 rad/s. When the wave frequency is near 2 rad/s, the added mass reaches the minimum value of 23,082 kg, but the maximum motion response of the floater goes up to 1.746 mm. The maximum radiation damping of the floater $B_{33}\left(\omega \right)$ is 17.65 Ns/mm at 1.696 rad/s. The radiation damping becomes almost 0 Ns/mm as the wave frequency increases further. The variation pattern of $B_{33}\left(\omega \right)$ does not decrease smoothly but reaches oscillations. This phenomenon is related to AQWA’s solution of the waterline integral term.

Three kinds of wave elevation time-series for the specific state were designed for simulation. In the co-simulation, three kinds of calculations were carried out, including no control strategy, fuzzy control with a fixed transformer ratio, and fuzzy control with a variable transformer ratio. Under the fuzzy control, the floater displacement and captured energy do not increase significantly, but the oil pressure fluctuation is very stable, indicating that the fuzzy controller is mainly to maintain the stability of the main power circuit. Under the variable ratio fuzzy control, the power generation is not greater than the power generation under no control strategy or variable ratio fuzzy control, which proves that this hydraulic transformer concept is less efficient.