Modeling and Simulation of Hydraulic Power Take-Off Based on AQWA

Abstract

The AQWA software is often used to perform hydrodynamic analysis, and it is highly convenient for performing frequency domain simulations of Pelamis-like wave energy converters. However, hydraulic power take-off (PTO) must be simplified to a linear damping model or a Coulomb torque model when performing a time domain simulation. Although these simulation methods can reduce the computational complexity, they may not accurately reflect the energy capture characteristics of the hydraulic PTO. By analyzing system factors such as the flow and pressure of each branch of the hydraulic PTO, the output torque of the hydraulic cylinder to the buoy, and the electromagnetic torque of the generator, a relatively complete hydraulic PTO model is obtained, and the model is applied to AQWA using the FORTRAN language. Comparing and analyzing the simulation results of the linear damping model, the Coulomb torque model, and the hydraulic PTO, we found that the simulation results obtained by the linear damping model are quite different from those of the hydraulic PTO, while the torque characteristics, kinematic characteristics and energy capture characteristics of the Coulomb torque model are closer to those of the hydraulic PTO model. Therefore, it is more appropriate to simplify hydraulic PTO to a Coulomb torque model based on AQWA.

1. Introduction

Energy is one of the important material foundations for social and economic development in the world [1]. As a clean and renewable energy, wave energy has huge reserves and high energy density, and it has been highly valued by scholars and institutions from all walks of life in recent years [2,3,4,5,6]. In the research on wave energy power generation, various forms of wave energy converter have been successfully developed and applied [7,8,9,10,11,12,13,14]. The existing wave energy converters can be divided into three types, Attenuators, Point absorbers and Terminators. Representative Attenuators include the Pelamis wave energy converter in the United Kingdom [15]. This kind of converter ‘rides’ the waves in the direction parallel to the predominant wave. Point absorbers float up and down on or below the water surface. They are always of small size, like Ocean Power Technology’s Powerbuoy. Terminators, different from their counterparts, are located perpendicularly to the predominant wave direction, and intercept waves using their body. Salter’s Duck, developed at the University of Edinburgh, is an example of a Terminator.

The British Pelamis wave energy converter is highly adaptable to complex sea conditions and has good energy capture characteristics [15]. It was the first wave energy converter in the world that was able to generate electricity integrated to the power grid. With continuing development over several years, this type of wave energy converter has been a hot area of research. Thus, the wave energy converter we designed is a Pelamis-like wave energy converter. The layout of the device is shown in Figure 1. It consists of buoys and hydraulic PTO. The wave energy density is high, and is greatly affected by the environment and weather. Therefore, as transfer equipment for converting wave energy to electrical energy, hydraulic PTO requires sufficient power in order to be capable of converting unstable wave energy into stable electric energy. Hydraulic systems have the advantages of high force, good robustness, and good dynamic properties. They are suitable for low-frequency applications and are widely used in hydraulic PTO to form wave energy converters.

The Pelamis-like wave energy converter is fixed with an anchor chain cable in an area of the sea where waves are frequent. When the two adjacent buoys are subjected to waves, the relative motion is relative to the middle-hinged point. This motion contains three degrees of freedom, including surge, heave, and pitch. Only the energy generated by pitch motion can be captured by the hydraulic PTO. Therefore, only one degree of freedom, the pitch of the buoys, is considered in this paper. When a wave acts on a buoy in the positive direction, the energy-capturing hydraulic cylinder in the hydraulic PTO is driven to reciprocate and expand, and the corresponding ends of the two hydraulic cylinders discharge high-pressure oil. The high-pressure oil is rectified through the control valve group formed by the one-way valve and stored in the high-pressure accumulator. The PTO absorbs the kinetic energy of the pontoon’s pitch motion and converts it into hydraulic energy.

When the wave energy captured by hydraulic PTO is sufficient, the high-pressure accumulator exerts a stabilizing effect, and the oil directly powers the hydraulic motor, driving the generator to generate electricity. When the captured wave energy is insufficient, the high-pressure accumulator can provide oil to maintain the generator and convert hydraulic energy into electrical energy. The working principle of a Pelamis-like sea wave energy converter is shown in Figure 2.

There have been many studies on Pelamis-like wave energy capture devices. The software AQWA is a very powerful tool that can help researchers and developers to perform a wide range of fluid analyses [16,17]. Based on AQWA software, the wave capture energy conversion model of a five-section buoy was established by Jennifer Alane Eden of California State University. Considering different wave conditions and buoy sizes, the optimization of capture energy behaviors was carried out. The energy capture device model was scaled down by He et al. [18], and the influence of the hydrodynamic parameters of the floating body on its motion characteristics was analyzed. Based on the first-order Stokes wave theory and the Newtonian equation of motion, Zheng’s team at Tsinghua University analyzed the characteristics of wave energy capture under different wave periods and wave heights, and proposed a wave energy converter model suitable for the China sea area [19,20]. Luo of Harbin Institute of Technology analyzed system factors such as buoy size, flooding depth, wave period and damping based on AQWA, and obtained the specific relationship between the above system factors and the capture energy behaviors [21]. When these scholars simulated the hydraulic PTO based on AQWA, they simplified the hydraulic PTO into a linear damping system. This method greatly reduces the computational complexity, making the simulation process more convenient. However, it does not accurately reflect the behaviors of motion and energy capture of the hydraulic PTO. This problem needs further study.

To solve the above problem, system factors such as the flow and pressure of each branch of the hydraulic PTO, the output torque of the hydraulic cylinders on the buoys, and the electromagnetic torque of the generator are analyzed, and a more complete hydraulic PTO model is obtained, which is then applied to AQWA using the FORTRAN language. By analyzing the behaviors of motion and energy capture of hydraulic PTO, it is assumed that it can be temporarily simplified as a Coulomb torque model. In view of this conjecture, the Coulomb torque model was obtained on the basis of the newly established hydraulic PTO model, and the simulation results of the Coulomb torque model, the linear damping model, and the hydraulic PTO are compared. This paper is limited to the study of Pelamis-like wave energy converters composed of two buoys and a set of hydraulic PTO.

2. Modeling of Hydraulic PTO

2.1. Flow and Pressure of Each Branch

The working principle of hydraulic PTO is shown in Figure 3. When the buoy is subjected to waves and makes a pitch motion relative to the middle hinge point, the reciprocating motion of the hydraulic cylinders causes the corresponding chamber to pump out high-pressure oil. Assuming that the hydraulic cylinder has the speed shown in Figure 3, the oil in the high-pressure chamber of the hydraulic cylinder merges into the total high-pressure oil circuit, so the oil circuit pressures of the two high-pressure chambers are the same. The pressure change in the corresponding chamber of the hydraulic cylinder can be expressed as

$${\dot{p}}_{14}=\frac{{\beta }_e\left(q_1-q_2-A_4{v}_1+A_1{v}_2\right)}{V_1+V_4}$$

(1)

$${\dot{p}}_{23}=\frac{{\beta }_e\left(q_3-q_4-A_2{v}_2+A_3{v}_1\right)}{V_3+V_2}$$

(2)

where $V_1$, $V_2$, $V_3$ and $V_4$ are the volumes of each corresponding chamber of the hydraulic cylinders, $q_1$, $q_2$, $q_3$ and $q_4$ are the flows of the one-way valves, $A_1$, $A_2$, $A_3$ and $A_4$ are the action areas of each chamber of hydraulic cylinders, $p_{14}$ and $p_{23}$ are the pressures of the two chambers, respectively, $v_1$ and $v_2$ are the speeds of the hydraulic cylinders, and ${\beta }_e$ is oil bulk modulus.

After passing through the control valve group composed of four one-way valves, the flow through each one-way valve can be analyzed according to the pressure difference between the two ends of the one-way valve. The flow is expressed as

$$q_i=\left\{\begin{array}{ll}0& : p_1-p_2\le p_3\\ \frac{K_1\left(p_1-p_2-p_3\right)\sqrt{p_1-p_2}}{p_4-p_3}& : p_3<p_1-p_2\le p_4\\ K_2\sqrt{p_1-p_2}& : p_1-p_2>p_4\end{array}\right\}i=1, 2, 3, 4$$

(3)

where $p_1$, $p_2$, $p_3$ and $p_4$ represent the inlet pressure, outlet pressure, opening pressure and maximum opening pressure of the one-way valve, respectively, $K_1$ represents the flow coefficient when the one-way valve does not reach the maximum opening, $K_2$ is the flow coefficient when the one-way valve reaches maximum opening.

The flow into the high-pressure accumulator can be expressed as

$$q_A=q_2+q_4-q_m$$

(4)

The flow into the low-pressure accumulator can be expressed as

$$q_B=q_m-q_1-q_3$$

(5)

where $q_A$ is the flow into the high-pressure accumulator, $q_B$ is the flow into the low-pressure accumulator, and $q_m$ represents the flow into the hydraulic motor and can be expressed as

$$q_m=D_m{\omega }_m$$

(6)

where $D_m$ is the hydraulic motor displacement, and ${\omega }_m$ represents hydraulic motor speed.

The liquid volume of the high-pressure accumulator can be expressed as

$$V_A={\displaystyle {\int }_0^t{q}_{A}dt}+V_{A0}$$

(7)

The liquid volume of the low-pressure accumulator can be expressed as

$$V_B={\displaystyle {\int }_0^t{q}_{B}dt}+V_{B0}$$

(8)

where $V_{A0},V_{B0}$ represent the initial liquid volume of the two accumulators.

For the working frequency of the hydraulic PTO, the working process of the high-pressure accumulator and the low-pressure accumulator can be considered as an isentropic process. This can be expressed as

$$p_j=p_{j0}{(\frac{V_{j0}}{V_{j0}-V_j})}^{\gamma } j = A, B$$

(9)

where $p_j$ and $p_{j0}$ represent the pressure of the accumulator and the initial pressure, respectively, $V_j$ is the volume of the accumulator, $V_{j0}$ is the liquid volume at the initial time of the accumulator, and $\gamma$ is the adiabatic exponent of gas.

2.2. Output Force of Hydraulic Cylinder

The driving torque of the hydraulic motor can be expressed as

$$T_m=D_m(p_A-p_B)$$

(10)

where $T_m$ is the driving torque of the hydraulic motor.

The resistance torque of the generator is expressed by $T_e$, and the dynamic balance equation is

$${\dot{\omega }}_m{J}_m=T_m-T_e-{\stackrel{}{\omega }}_m{B}_m-T_c$$

(11)

where $B_m$ represents the rotational damping of hydraulic motor and couplings, $J_m$ is the rotational inertia of the hydraulic motor and couplings, $T_e$ is electromagnetic torque, and $T_c$ represents the Coulomb resistance torque of the hydraulic motor.

In the Cartesian coordinate system XOZ, the force of the two hydraulic cylinders on the buoys in the hydraulic PTO is equivalent to the force and torque on the mass center of the buoys. The force model of the buoys is shown in Figure 4.

Therefore, the forces of the upper and lower hydraulic cylinders on the buoys can be expressed as

$$\begin{array}{l}{F}_1=\left(p_{14}{A}_1-p_{23}{A}_2\right)\\ F_2=\left(p_{23}{A}_3-p_{14}{A}_4\right)\end{array}$$

(12)

The buoy centroid arms of force are expressed as

$$\begin{array}{l}{R}_1=l_4\cos \left(\frac{\left(\frac{\pi }{2}-{\theta }_1+{\theta }_2\right)}{2}\right)+\frac{\left(l_3+l\right)}{2}\sin \left(\frac{\left(-{\theta }_1+{\theta }_2\right)}{2}\right)\\ R_2=l_4\cos \left(\frac{\left(\frac{\pi }{2}-{\theta }_1+{\theta }_2\right)}{2}\right)+l_4\cos \left(\frac{\left(\frac{\pi }{2}+{\theta }_1-{\theta }_2\right)}{2}\right)-R_1\end{array}$$

(13)

where ${\theta }_1$ and ${\theta }_2$ represent the pitch angles of buoys 1 and 2, $R_1$ and $R_2$ represent the arms of the two hydraulic cylinders acting on the first buoy, respectively, $l$ is the length of a single buoy, $l_4$ is the distance from the hydraulic cylinder hinged point to the buoy hinged point, and $l_3$ is the distance between the two buoys when the buoys are horizontal.

The centroid torques of hydraulic cylinders to buoys are expressed as

$$\begin{array}{l}{M}_1=F_2{R}_2-F_1{R}_1\\ M_2=F_1{R}_1-F_2{R}_2\end{array}$$

(14)

where $M_1$ and $M_2$ are the torques of the two hydraulic cylinders against buoys 1 and 2.

2.3. Electromagnetic Torque of Permanent Magnet Synchronous Generator

The permanent magnet synchronous generator consists of a permanent magnet rotor and a stator. Assume that the air gap magnetic field distribution and back electromotive force of the generator are all sinusoidal. The magnetic saturation effect and eddy current loss are not considered in the process, and the effect of temperature is ignored [22,23,24], because these parameters have relatively little effect on the generator’s performance compared to others, like electric angular velocity. To build a model of these, complex nonlinear equations or the finite element method (FEM) would be required, which would greatly increase calculating time. In this work, we employ whole-system modeling instead of defining a precise model of the generator. Thus, temperature variation, magnetic saturation effect, and eddy current loss are not considered in our article.

The electromagnetic torque equation is expressed as

$$T_e=\frac{3}{2}{N}_P{i}_q\left. [\left(L_d-L_q\right)i_d+{\psi }_f\right]$$

(15)

where $i_d$ and $i_q$ represent the d-axis and q-axis fundamental components of the stator current respectively, $L_d$ and $L_q$ represent d-axis and q-axis inductances, ${\omega }_e$ is the electric angular velocity, $N_P$ represents the pole number of the generator rotor, and ${\psi }_f$ represents a permanent magnet flux linkage.

In the permanent magnet synchronous generator, the inductance values of $L_d$ and $L_q$ are small. When the generator speed is constant, the d-q two-phase current is also constant, and the inductance electro-motive force can be ignored. For pure resistive load, the above equation can be expressed as

$$T_e=\frac{3}{2}{N}_P{}^2{\psi }_f{}^2\frac{{\omega }_m}{R}$$

(16)

The above equation shows that the electromagnetic torque is proportional to the rotational speed in the steady state, and inversely proportional to the damping load. Therefore, the generator can be regarded as a damping load in the simulation, and the electromagnetic torque can be adjusted by adjusting the electrical resistance value.

3. Simulation of Hydraulic PTO

The AQWA software has the advantages of high computational accuracy and good coupling. It is particularly convenient for use in the frequency domain analysis of hydrodynamic systems, and has been widely used in the field of marine and ship engineering.

The cross-sections of the floating cylinders mentioned above are circular, and the parameters of the Pelamis-like wave energy converters are shown in

Table 1. Parameters of Pelamis-like wave energy converters.

Project	Parameter	Number	Unit
Buoy	Length (L)	10	m
	Diameter (d)	1	m
	Clearance (L3)	1	m
	Submergence depth (d3)	0.5	m
	Quality (M)	4025.166	kg
	Rotational inertia (IXX)	807.865	kg·m2
	Rotational inertia (IZZ)	36,180.897	kg·m2
Sea water	Water depth	250	m
	Density	1025	kg/m3
Wave	Wave height	0.4	m
	Period	3.62	s
Anchor chain	Original length	25.981	m
	Rigidity	1000	N/m
Hydraulic cylinder	Piston diameter Piston rod diameter	0.05 0.036	M m
High-pressure accumulator	Volume	10	L
Low-pressure accumulator	Volume	6.3	L
Hydraulic motor	Displacement	5	mL
Working pressure	Pressure	10–15	MPa
Action torque	Linear damping	100,000	ad/s
Action torque	Coulomb torque	17,000	N·m

.

On the basis of the simulation of hydraulic PTO using AQWA, first, the AQWA-LINE module’s simulation model for the hydraulic PTO is established. The equivalent additional mass matrix, equivalent radiation damping matrix, equivalent excitation force matrix and the stiffness matrix corresponding to different wave frequencies are obtained. By analyzing the three decomposition motions of the first buoy, the kinematic formula of the buoy can be expressed as

$$\left\{M+\right.\left.A\left(\infty \right)\right\}\left\{\ddot{X}\right.\left.\left(t\right)\right\}+{\displaystyle {\int }_{-\infty }^{t}R}\left(t-\tau \right)\left\{\dot{X}\right.\left.\left(t\right)\right\}d\tau +K_{hys}\left\{X\left(t\right)\right\}=\left\{F_{ex}\left(t\right)\right\}+\left\{F_{pto}\left(t\right)\right\}+\left\{F_m\left(t\right)\right\}$$

(17)

where $A\left(\infty \right)$ is the additional mass matrix, $\left\{X\left(t\right)\right\}$ denotes the generalized coordinates, $M$ is the mass matrix of buoy, $R\left(t\right)$ is the hysteresis function; $K_{hys}$ is the stiffness matrix of hydrostatic recovery, $\left\{F_{ex}\left(t\right)\right\}$ is the matrix of excitation force, and $\left\{F_{pto}\left(t\right)\right\}$ is the matrix of PTO force.

The mass matrix of the buoy can be expressed as

$$M=\left. [\begin{array}{ccc}m& 0& m{z}_G\\ 0& m& 0\\ m{z}_G& 0& I_5\end{array}\right]$$

(18)

where $m$ is the mass of the buoy, $z_G$ represents the gravity coordinates of the buoy, $I_5$ represents the moment of inertia of the buoy around the y-axis, and $\left\{F_{ex}\left(t\right)\right\}$ can be expressed as

$$\left\{F_{ex}\left(t\right)\right\}=\left. [\begin{array}{c}{F}_{ex}^1\\ F_{ex}^2\\ T_{ex}^3\end{array}\right]e^{-i\omega t}$$

(19)

where $F_{ex}^1$ is the force of the buoy in surge mode, $F_{ex}^2$ is the force in heave mode, and $T_{ex}^3$ represents the action torque of the buoy in pitch mode.

Then, the hydraulic PTO model is imported into the AQWA-NAUT module. The hydraulic PTO model cannot be directly opened in AQWA, and needs to be compiled using the FORTRAN computer language. The compiled program language is imported into the AQWA-NAUT time domain module using the external interface of the AQWA software. Finally, data reflecting the behaviors of motion and energy capture are extracted using AQWA to perform the simulation of hydraulic PTO. The simulation process of the wave energy converters based on AQWA is shown in Figure 5. In addition, two simplified processing models can be opened directly in order to perform the simulation in the AQWA calculation library.

4. Data Results

4.1. Comparison of PTO Torque

A comparison of the torques obtained using the three PTO models on the buoys is shown in Figure 6. It can be concluded that there is a deviation between the linear damping model and the hydraulic PTO model. The former can be categorized as being sine function fluctuation, while the latter is closer to the Coulomb torque. After the high-pressure oil is pumped out of the corresponding chamber of the upper and lower hydraulic cylinders, the accumulators play a stabilizing role. The oil pressure is relatively stable within a certain range, and the output forces of the hydraulic cylinders on the buoys are relatively constant. When the buoys are pitching, the changes in rotation angle and rotation angular velocity can be ignored, and the arms of force are relatively stable. When the buoys rotate in the same direction along the hinged point, the torque value of the hydraulic PTO changes very little, so it is closer to the Coulomb torque. When the pitch motion of the two buoys changes direction, the pressures of the oil reverse, and the output torque of the hydraulic PTO reverses and changes with the wave period.

The Coulomb torque model was simulated, and it was found that this processing method is worthy of further study. The motion characteristics and capture energy behaviors of the two simplified models and the hydraulic PTO model are further compared and analyzed below.

4.2. Motion Characteristics: Comparison of Pitch Angle and Angular Velocity

The comparison curves of relative pitch angle and the angular velocity of the buoys in the three PTO models are shown in Figure 7. It can be seen from the change diagram that when simplified as a linear damping system, the angle and angular velocity fluctuate up and down according to a sine function. At certain moments, the angles of the Coulomb torque model and the hydraulic PTO model remain relatively constant. At these points, the external wave force on the buoys is offset by the force of the hydraulic cylinders, and a balanced state is achieved. The angular velocity of the hydraulic PTO model is accompanied by small-range fluctuations, which is because although there is an accumulator to stabilize the pressure, the oil pressure still changes dynamically, while overall dynamic balance is maintained.

4.3. Behaviors of Capture Energy: Comparison of Capture Power

A comparison of the curves of the energy capture power of the three PTO models is presented in Figure 8. The average capture power is about 0.95 kW when PTO is simplified to a linear damping system, and the capture power is 1.26 kW when it is simplified to a Coulomb torque model. The power of hydraulic PTO is 1.13 kW. The relative error of the capture power of the linear damping model is 15.9%, and the relative error of the capture power of the Coulomb torque model is 11.5%. It can be seen that both methods are able to achieve the simplification effect, but the relative error of the Coulomb torque model is relatively smaller. Therefore, it is more appropriate to perform simplification using the Coulomb torque model when designing the hydraulic PTO at the beginning.

5. Conclusions and Discussion

A relatively complete hydraulic PTO model was obtained by analyzing system factors such as the flow and pressure of the hydraulic PTO branches, the output torque of the hydraulic cylinder to the buoy, and the electromagnetic torque of the generator in a Pelamis-like wave energy capture device. The model was applied in the AQWA software using the FORTRAN computer language.

When comparing the simulation results of the two simplified models with the hydraulic PTO model, the relative error of the capture power of the linear damping system model was found to be 15.9%, and the relative error of the capture power of the Coulomb torque model was found to be 11.5%, which shows that the Coulomb torque model is more suitable for the simplification of the hydraulic PTO model. Its torque characteristics, motion characteristics and energy capture characteristics are more accurately able to reflect hydraulic PTO, providing a new idea for simplifying the use of AQWA software when simulating hydraulic PTO.