Integrated characteristic curves of the constant-pressure hydraulic power take-off in wave energy conversion

Electrical Power and Energy Systems 117 (2020) 105730

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Integrated characteristic curves of the constant-pressure hydraulic power take-off in wave energy conversion

T

Qijuan Chen, Xuhui Yue , Dazhou Geng, Donglin Yan, Wen Jiang ⁎

Ministry of Education Key Laboratory of Transients in Hydraulic Machinery, Wuhan University, 430072, China

ARTICLE INFO

ABSTRACT

Keywords: Integrated characteristic curves Constant-pressure hydraulic power take-off Floating-pendulum wave energy converter Efficiency test Fitting formula

Power take-off is an indispensable link in wave energy utilization, and its efficiency should be comprehensively investigated at both full load and part load for the effective conversion in variable wave conditions. However, up to now, the research about this issue is still scarce because of its complexity and difficulty. To overcome this obstacle, this paper studies the overall conversion efficiency of the constant-pressure hydraulic power take-off (CPHPTO) of a floating-pendulum wave energy converter, at a wide range of the system pressure, system flowrate and shaft speed, via the efficiency test and fitting formulas. Furthermore, the integrated characteristic curves, which consist of the characteristic curves and operating curves, are proposed to deal with the fourdimensional data obtained from the experiment or fitting. The results show that the integrated characteristic curves of the CPHPTO are available and play an important role in the optimal design and efficient operation. The stable operating region can also be defined when plotting the operating curves. In general, the integrated characteristic curves are suitable for the preliminary design and further optimization of the similar CPHPTOs for different kinds of oscillating-body wave energy converters.

1. Introduction The research on wave energy utilization was stimulated by the breakout of the oil crisis in the 1970s [1]. Nowadays, there have been over 1000 patents on wave energy converters (WECs) [1,2]. The WECs can be divided into oscillating-body WECs (OBWECs), overtopping WECs and oscillating water columns, according to different operating principles [2–4]. Among them, OBWECs drive the power equipment via the oscillation of one or several bodies in ocean waves, and have become the research hotspot especially for exploiting offshore wave energy [4]. Fan et al. [5] studied the OBWEC, which contains a buoy and a hydraulic transmission, combined with an offshore wind turbine. Results showed that the motor output torque control with a fuzzy controller contributes to WEC delivering the required electrical power. Rodríguez et al. [6] adopted the experimental-plus-numerical approach to estimate the performance of a CECO device, which utilizes the inclined motion of floating bodies to generate power. The control strategies for the OBWEC of a heaving buoy and a directly-driven permanent magnet linear motor were researched by Refs. [7,8]. Moreover Chen et al. [9] investigated the sealed-buoy WEC with an inner power unit, and concluded that the conversion efficiency in low wave energy density seas is slightly lower than that in high wave energy density seas.

Most of OBWECs adopt hydraulic power take-offs (HPTOs) to convert the mechanical energy captured from ocean waves into the electric energy, because HPTOs possess the features of the high torque at low frequency, fast frequency response and overload protection [10,11]. HPTOs can be organized into two types, i.e. variable pressure HPTOs (VPHPTOs) and constant pressure HPTOs (CPHPTOs) [12,13]. In VPHPTOs, the high-pressure oil, which is first produced by a hydraulic cylinder and then rectified by four check valves, is finally transmitted to the hydraulic motor connected to a generator for power generation. To enhance the performance of the wave-energy capture, the PTO force can be altered to the optimal magnitude according to the wave conditions, via the control of motor displacement. If the VPHPTO is a variable speed system, the shaft speed can also be controlled by tuning the generator torque. However, the disadvantage of VPHPTOs is obvious. Although the VPHPTO is designed for working at the ideal operating point with the peak efficiency, the wave-to-wave changes of wave energy make it probably operating at the part load with the low efficiency. For CPHPTOs, the produced unidirectional high-pressure oil from the hydraulic cylinder and four check valves, will be smoothed by highpressure accumulator first and then be delivered to a hydraulic motor. The high-pressure accumulator can temporarily store energy and partially filter out fluctuations of the system pressure and flow-rate [14].

Corresponding author. E-mail addresses: [email protected] (Q. Chen), [email protected] (X. Yue), [email protected] (D. Geng), [email protected] (D. Yan), [email protected] (W. Jiang). ⁎

https://doi.org/10.1016/j.ijepes.2019.105730 Received 6 July 2019; Received in revised form 17 October 2019; Accepted 21 November 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.

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Therefore, the main energy-conversion component, i.e. hydraulic motor, can work at a relatively stable condition with higher efficiency, compared with that in a VPHPTO. Nevertheless, CPHPTOs cannot quickly change its PTO force by adjusting motor displacement for the influence of the high-pressure accumulator. Many methods were proposed to improve the control performance of the CPHPTO. An optional approach is using control valves to replace check valves. Henderson [15] realized pseudo-continuous force control via the different combination of chambers in the control valve, while Babarit et al. [16] implemented declutching control by setting a bypass road and controlling the discrete action of the control valve. Moreover, Ricci et al. [17] added control accumulators between a cylinder and four check valves to increase the magnitude of the velocity in phase with the excitation force. Similar to Ref. [15], Hansen et al. [18] utilized a digital displacement cylinder, which can offer 27 force steps and four quadrant mode to achieve pseudo-continuous force control. Meanwhile, Lin et al. [10] advocated the force-adjusting technology based on a digital hydraulic cylinder group. Compared with VPHPTOs, CPHPTOs have wider uses in OBWECs, such as Pelamis [15], Wavestar [18], two-raft-type WEC [19,20], heaving buoy system [21,22] and different kinds of Pendulum-type WECs like SEAREV [23], Sharp Eagle Wanshan [24], inverse-pendulum WEC [25,26] and floating-pendulum WEC [27–29]. Floating-pendulum WECs can be installed on the offshore platform, the breakwater or the coast with a certain water depth. When the sea is still, the pendulums float on the water surface. However, if the waves come, they will oscillate around the hinge points to generate electricity. The losses in CPHPTOs cannot be neglected during the power generation of OBWECs. Excessive simplification of CPHPTOs could lead to the power gains and the incorrect design decisions [30]. Therefore, the efficiency should be carefully considered during the design and operation of CPHPTOs. Lasa et al. [31] tested the efficiency of a universal CPHPTO against different motion conditions and cylinder working areas. The results showed that the efficiency ranges from 69% to 80%. Choi et al. [32] researched the overall and partial efficiency of the CPHPTO for a 50 kW rotating-body-type WEC by experiment and found the maximum overall efficiency of 72.7%. Gasper et al. [33,34] modeled the pump efficiency with artificial neural networks and the

cylinder efficiency with Adaptive Neuro Fuzzy Inference System, and controlled speed to maintain 80% of the maximum displacement of the pump, where overall efficiency can attain 94.5%. However, the aforementioned efficiency tests [31,32] are based on few working conditions, and the proposed efficiency models [33,34] aim at the individual components not the whole CPHPTO. Actually, CPHPTO may still work at a wide range of the operating points with different system pressure p, system flow-rate q and shaft speed n, due to the changes of wave energy in irregular waves and the finite energystorage capacity of high-pressure accumulator. Therefore, to comprehensively study the CPHPTO efficiency, massive operating points with different p, q and n should be tested and the overall efficiency model should be given. Moreover, the obtained four-dimensional (4-D) data, which consist of (q, p , n) and the CPHPTO efficiency, are difficult to be displayed and analyzed without any handling. The paper focuses on the efficiency test for the CPHPTO at a series of the operating points of (q, p , n) and presents the fitting formula for the CPHPTO efficiency. Herein, we only take the CPHPTO of a floatingpendulum WEC as an example. To better analyze the experimental or fitting results, the integrated characteristic curves (ICCs) are introduced here. It is stimulated by the data processing of the energy experiment of the hydro-turbine model [35], in which there are also 4-D data composing of the water heads, flow-rates, rotational speeds and hydroturbine efficiency. The paper is described in the following structure. Section 2 introduces the research object, i.e. a particular CPHPTO for a floatingpendulum WEC, and simulates the dynamic process of the whole WEC in irregular waves. Section 3 provides the experimental setup for the efficiency test and gives the fitting formula of CPHPTO efficiency. Then theoretical ICCs with its achieving method and function are shown in Section 4. Afterwards the experimental and fitting results, as well as the actual ICCs drawn from the fitting formula are analyzed in Section 5. Finally, the corresponding conclusions are displayed in Section 6. 2. CPHPTO of a floating-pendulum WEC We focus on the case that the CPHPTO is attached on a floating pendulum (see Fig. 1). The pendulum connects rigidly to the gear

F G

C B

B

G D B

H

A-2

Gear and Rack

E

B

H

A-1

Lubricating Grease

Energy Capture Device Pendulum L-shaped Arm Fulcrum

Wave

Fig. 1. CPHPTO for a floating-pendulum WEC. The PTO consists of two single-acting single-rod hydraulic cylinders (A), four check valves (B), a high-pressure accumulator (C), a flow control valve (D), a variable axial-piston motor (E), a generator (F), a relief valve (G) and a low-pressure reservoir (H). 2

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through a L-shaped arm, whose knee point is hinged with a fulcrum. The pendulum swings around the fulcrum in ocean waves and delivers energy to the CPHPTO via the gear and rack. The gear and rack are lubricated by lubricating grease. The two hydraulic cylinders (A) are installed at the ends of the rack to convert the bidirectional energy into hydraulic energy via rodless chambers. The feature of this CPHPTO is that a flow control valve (D) and a relief valve (G) are utilized to relief the redundant flow-rate for the system protection when it faces the superior input power. The benefits of the layout of the energy capture device and hydraulic cylinders are as follows:

spectrum [37] and focus on the offshore wave energy resource assessment in East China Sea, i.e. significant wave height Hs = 1.5 3.5 m , wave energy period Te = 6 8 s [39]. Force from CPHPTO FPTO (t ) can be expressed as

FPTO(t ) =

Tm = cg

K (t

) l1 ( ) d + S3 l12 (t ) (1)

= l1 (Fe3 (t ) + FPTO (t ))

where mFP is the mass of floating pendulum , A33 the added mass at infinite frequency, (t ) the swing angle,S3 the hydrostatic restoring coefficient, Fe3 (t ) the wave excitation force, FPTO (t ) the force from CPHPTO, and K (t ) the impulse response function of

K ( ) = B33 ( ) + i (A33 ( )

A33 )

(2)

where A33 ( ) and B33 ( ) are the added mass and the radiation resistance against wave angular frequency . Herein state-space method (SSM) is adopted to replace the convolution term (I ) [36]:

I =

t

K (t

) l1 ( ) d

Y = AY + Bl1 (t ) I = CY

(3)

where Y is the state vector of the identified system; A , B and C are system matrices. Wave excitation force Fe3 (t ) [37,38] is calculated as m

Fe3 (t ) =

|fe3 ( j )| A ( j ) cos( j t + 2 rand()) j=1

(4)

where m is the number of frequency segments, fe3 the excitation-force coefficient, j the j-th angular frequency, and rand() a uniformly distributed random-number generator in the interval (0, 1). Here A ( j ) denotes the amplitude of the wave elevation at = j , and can be defined as

A ( j) =

2S ( j )

Ff ]

(6)

m

(7)

where Tm is the motor torque, cg the damping coefficient, and m the angular velocity of shaft. Simulation parameters of the floating-pendulum WEC are listed in Appendix B. Hydrodynamic parameters of the floating pendulum and K (t ) with its 8-order SSM approximation are displayed in Fig. 2. The hydrodynamic parameters are calculated by ANSYS AQWA [40]. Simulation results against the irregular waves of Hs = 1.5 m , Te = 6 s are shown in Figs. 3 and 4, and the power matric (Hs = 1.5 3.5 m , Te = 6 8 s ) of floating-pendulum WEC is given in Table 1. Fig. 3 shows that the system pressure is smoothed by the highpressure accumulator rather than changes from 0 to maximum value like the wave excitation force. The similar pressure curve of CPHPTO in irregular waves is also given by Ref. [19]. Herein the fluctuations of system pressure are more obvious than the cases in Ref. [19], because the smaller nominal volume (6.3 L VS 150 L) of the high-pressure accumulator limits its ability of energy storage and stabilization. Throughout the duration of simulation, we do not change the displacement of hydraulic motor and the damping coefficient of generator. Therefore, shaft speed, system flow-rate and system pressure are in proportion to each other, without consideration of the losses in CPHPTO. The phenomenon is displayed in Fig. 4. In the 3-D space of the shaft speed, system flow-rate and system pressure, the state vector varies closely to an oblique straight line. From Fig. 4, the state vector still changes between 4.5 and 28 MPa, 2.4–31.4 L/min and 44–573 r/ min, when the accumulator normally works. The main reason is that the wave energy changes wave to wave in irregular waves and the energystorage capacity of high-pressure accumulator is finite. Based on the simulations in irregular waves, we find that the CPHPTO may work at a wide range of the operating points. Hence it is necessary to investigate the efficiency at a wide range of the operating points and to optimize the design and operation according to the highefficiency region. The detailed methods and results of the efficiency test will be discussed in Sections 3–5.

To further study the dynamic characteristic and generated power of the floating-pendulum WEC in real waves, the dynamic simulation is implemented in MATLAB. The modeling procedures are as follows. Torque of the floating pendulum can be simplified as the product of the force in heave direction and the equivalent arm, i.e. the horizontal distance from the gravity center of pendulum to the fulcrum (l1) [28]. Regardless of the mass of L-shaped arm, the WEC dynamic equation can be given by t

p2 ) sc + Fend

where k t is the ratio between the horizontal distance from the gravity center of pendulum to the fulcrum and the vertical distance from the fulcrum to the engaging point of the gear and rack; p1 and p2 are the oil pressures at rodless chambers of A-1 and A-2, respectively; sc is the piston area; Fend denotes the force from the end-stop device; Ff means the friction force. The modeling of the dynamic equations of CPHPTO are based on Ref. [37], which has described the models of a hydraulic cylinder, four check valves, a high-pressure accumulator, a variable axial-piston motor, a generator, and a low-pressure reservoir in details. Herein we ignore the CPHPTO losses, especially for the losses at the two hydraulic cylinders, variable axial-piston motor and shaft, since the purpose of simulation is to preliminarily study the dynamic process of the model. The concrete losses of the CPHPTO should be tested by experiment. Moreover, the generator is simplified as a linear damping:

1) The strokes of cylinders can be shortened by rationally setting the ratio for the length of the two sides of the L-shaped arm. 2) In the general WECs, the cylinder bodies are articulated at the frame, and the ends of piston rods are hinged at the arms [18,27–29] or the pendulums/flaps [24–26]. Thereby the cylinder body and the corresponding piston rod will rotate around the hinge point when they do the relative motion. Herein the motion is simplified. The cylinder bodies are horizontally fixed to the frame, and the piston rods only do the left-or-right translation. 3) Symmetrical arrangement of the two hydraulic cylinders can throughout retain the horizontal motion of the rack and piston rods, and prevent the oblique motion that probably appears in the longterm operation with the existence of the only one hydraulic cylinder.

(mFP + A33 ) l12 ¨ (t ) + l1

1 [ (p1 kt

3. Methods 3.1. Experimental setup The experimental platform for the land test of the CPHPTO efficiency was established at Ministry of Education Key Laboratory of Transients in Hydraulic Machinery, Wuhan University, China. Fig. 5 shows the platform sketch. Here, a pump station (1), a 3-position 4-way

(5)

where S signifies one-sided wave spectrum and means the frequency component interval. Herein we select standard JONSWAP 3

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a

b

L=4.5m

b=1m

Wave direction

h=1.5m d=0.75m

c

d

Fig. 2. Hydrodynamic parameters and impulse response function of the floating pendulum. (a) Shape and dimensions of the floating pendulum. (b) Excitation-force coefficient. (c) Added mass and radiation resistance. (d) Impulse response function (IRF) and its 8-order SSM approximation.

direction-control valve (2) and a double-acting double-rod cylinder (3) were used to simulate the reciprocating motion of the rack in the waves. We only emulated the sinusoidal motion to easily obtain one fixed operating point and to reduce the difficulty of hydraulic test. The generator was replaced by the magnetic powder brake (15) with a current controller (14) for the convenience of motor-torque tuning during the test. The procedure of the efficiency test is shown in Fig. 6. During the test, the parameters of a certain operating condition (q, p , n) , i.e. the motor displacement Vm , the time-averaged system flow-rate q, system pressure p , shaft speed n and motor torque Tm , were calculated as well as CPHPTO efficiency t . First the operating condition (q, p , n) was altered by tuning the motor torque, the motor displacement and the input motion, i.e. the piston motion of single-acting single-rod hydraulic cylinders. Next the instantaneous piston displacement x , input forces F1 and F2 , q, p , n and Tm at each operating point were respectively measured by a displacement transducer (6), two bidirectional force transducers (5), a flow-rate transducer (11), a pressure transducer (12) and a torque-speed transducer (14). Then time-averaged values of q, p , n and Tm were derived from the instantaneous values, and the value of Vm was obtained via the relationship between control current and Vm . Afterwards, the time-averaged input power P¯i and the time-averaged output power P¯o were calculated as

b 1

P¯i =

F2i )(xi + 1 tb

b 1

P¯o =

(F1i

xi )

i=a

2 ni Tmi (ti + 1

i=a

tb

(8)

ta

ta

ti ) (9)

where t is the time, ta the beginning time, tb the end time, and physical variables with the subscript i (i + 1) means the instantaneous value at the i (i + 1)-th moment. Finally, the CPHPTO efficiency t was obtained by t

=

P¯o P¯i

(10)

The physical model of the experimental platform is indicated by Fig. 7. Both versions and parameters of the main components and sensors are displayed in Table 2. Among them, the parameters of the main components are consistent with the corresponding simulation parameters in Appendix B. Moreover, all main components of the CPHPTO were selected based on the designed output power Nr = 2000 W and the maximum system pressure pmax = 31.5 MPa for the designed sea state of Hs = 1.5 m ,Te = 6 s , since the simulation results 4

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a

b

c

d

Fig. 3. Simulation results against the irregular waves of Hs = 1.5 m , Te = 6 s (without consideration of the losses in CPHPTO). (a) Wave excitation force in heave direction. (b) Swing angle of the floating pendulum. (c) Piston displacement. (d) System pressure (red dashed line denotes pre-charge gas pressure). Table 1 Power matric of the floating (Hs = 1.5 3.5 m , Te = 6 8 s ). Power Matric [kW]

Significant wave height, Hs [m]

pendulum

wave

energy

converter

Energy period, Te [s]

3.5 3 2.5 2 1.5

6

6.5

7

7.5

8

10.73 8.16 5.92 3.93 2.26

10.07 7.49 5.32 3.52 2.01

9.97 7.33 5.09 3.26 1.86

9.65 7.03 4.81 3.03 1.7

9.24 6.7 4.57 2.87 1.63

3.2. Fitting formula Since the limitations of the testable system pressure, system flowrate and shaft speed, the fitting formulas of the CPHPTO efficiency t as well as the motor displacement Vm and motor torque Tm are required to reasonably expand the experiment results. The formulas recognize t as the product of cylinder efficiency and motor efficiency. It assumes that

Fig. 4. Variation of the state vector against the irregular waves of Hs = 1.5 m , Te = 6 s (without consideration of the losses in CPHPTO). Red part means the fully discharged mode of high-pressure accumulator [37,41]. Operating points of the maximum and minimum output power are marked in magenta against the normal working mode of high-pressure accumulator [37].

1) The losses in the check valves, the pipelines, the high-pressure accumulator and the low-pressure reservoir are small and can be neglected. 2) The oil gauge pressures at the low-pressure reservoir, the hydraulicmotor outlet and the expanded chamber of a cylinder are zeros, i.e. atmospheric pressure.

show that the CPHPTO without considering losses can output 2.26 kW at this case (see Table 1). Here, the test conditions are within the limits of 0–12 MPa, 3.3–20 L/min and 0–1000 r/min, respectively due to the restriction of the oil pressure from pump station (1), the measuring range of flow-rate transducer (11) and the measuring range of torquespeed transducer (14).

The cylinder efficiency is first described with the consideration of the mechanical-hydraulic loss and the volumetric loss. The mechanical5

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4-1 2 1

P

A

T

B

9

8-1

5-1 18

3 5-2 4-2

12 10

17

8-3

14

11

18

7-3 6

15

16

7-1

13

7-4

8-2

7-5 8-4

7-2 19

Fig. 5. Sketch of the experimental platform for the efficiency test of a CPHPTO. (1) a pump station, (2) a 3-position 4-way direction-control valve, (3) a double-acting double-rod cylinder, (4) two single-acting single-rod cylinders, (5) two bidirectional force transducers, (6) a displacement transducer, (7) five pressure gauges, (8) four check valves, (9) a high-pressure accumulator, (10) a flow control valve, (11) a flow-rate transducer, (12) a pressure transducer, (13) a variable axial-piston motor, (14) a torque-speed transducer, (15) a magnetic powder brake, (16) a current controller, (17) a relief valve, (18) a low-pressure reservoir, (19) a monitoring system consisting of a PC and a PLC.

hydraulic loss contains the friction loss between piston and cylinder, as well as the pressure loss at the cylinder port. The friction force Ff can be given by

established based on Jeong’s theory [42]. According to the theory, the leakages qm and the loss torque Tm of axial-piston motor are the functions of the pressure difference pm (here pm equals to p for the Assumption 2) between inlet and outlet as well as n:

(11)

Ff = Fc + bv

where Fc is the Coulomb friction force, b is the viscous coefficient and the piston velocity v can be regarded as

qc sc

v

(12)

(13)

=

pc sc (pc + pc ) sc + Ff

qc ) =

cm (pc ,

qc )· pc = pc + c1 +

cv (pc ,

Tm (p , n) =

m7

+

m8 p

+

m9 n

+

m10 p

q · c2 q q + c3 p +

=

p 2

q

(19) (20)

(21)

m15

j = 13, 14, 15 are also the motor parameters. Hence the

q

( qm + q~m ) n n3 m13 + m6 np + p 1 + (n / m14 ) m15

(

pVm 2

(22)

Tm n3 m13 + m6 np + p 1 + (n / m14 ) m15

m1 + m2 p + m3 n + m4 p + m5

n m7

+

m8 p

+

m9 n

+

m10 p

2

2 m11 n

+

+

p p m12 n

)

Furthermore, the theoretical motor efficiency PTO condition (q, p , n) is mt (q ,

c4 pq

m14 )

n

(17)

p, n) =

(

m7

+

m8 p

+

m9 n

+

At last, the fitting formula t (q ,

Next the efficiency model of variable axial-piston motor is 6

for one certain

n3 m13 + m6 np + p 1 + (n / m14 ) m15

q 2 n pq

(18)

mt

(23)

2 n T pq m

m1 + m2 p + m3 n + m4 p + m5

=1

Therefore, the theoretical cylinder efficiency ct for one certain PTO condition (q, p) can be approximately accessed by

p ct (q , p) = p + c1 +

mj,

Tm (q, p , n) =

qc )

c4 pc qc

+

m13

1 + (n /

m1 + m2 p + m3 n + m4 p + m5

=

(16)

qc · c2 qc qc + c3 pc +

p p m12 n

m4

q

which includes the compression loss ( pc qc ) and the leakage loss ( pc ). The parameters c3 and c4 are the scale coefficients. Overall, the instantaneous overall cylinder efficiency is ct (pc ,

+

+

Vm (q, p , n) =

(15)

c4 pc qc

2 m11 n

m3 n

formula Tm (q , p, n) of the motor torque are

Here parameters c1 and c2 denote Fc/ sc and k + b/ sc2 . The volumetric efficiency can be expressed by

qc cv (pc , qc ) = qc + c3 pc +

2

+

where

where pc is the pressure outside the cylinder port. According to Eqs. (11–13), Eq. (14) can be rewritten as the function of qc and pc : c2 qc

m6 np

m2 p

fitting formula Vm (q, p , n) of the motor displacement and the fitting

(14)

pc cm (pc , qc ) = pc + c1 +

+

+

q~m (p , n) =

where k is the coefficient relevant to the oil dynamic viscosity and the port size. The mechanical-hydraulic efficiency cm is cm

n3 m5 p

m1

, 12 are the motor parameters. The empirical Here mj, j = 1, 2, correction term q~m (p , n) is further introduced into the leakages to improve the fitting accuracy:

Here, qc is the cylinder output flow-rate. Since the oil in the slender port can be seen as the Poiseuille flow, the pressure loss pc is calculated as

pc = kqc

p +

qm (p , n) =

p , n) =

ct (q ,

p)·

mt (q ,

m10 p

2

t (q ,

p, n)

+

2 m11 n

+

p p m12 n

)

(24)

p , n) of CPHPTO efficiency is (25)

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RMSE =

1 m

m

(yk

yk ) 2

k=1

(27)

If RMSE approaches to 0, the fitting accuracy is higher [33]. 4. Integrated characteristic curves System flow-rate q, system pressure p and shaft speed n are the main operating parameters which determine the operating points and corresponding CPHPTO efficiency t , similarly to the water head, flow-rate and rotational speed of a hydro-turbine. Referring to ICCs of a hydroturbine, this section introduces the analogous curves. q is affected by different motions of cylinder piston at different wave conditions. So it is hard to be controlled directly. n can be adjusted via regulating the motor displacement Vm under one certain q, according to

n=

q

mv

(28)

Vm

where mv denotes the motor volumetric efficiency. The relationship between the motor torque Tm and Vm is given by

Tm =

pVm 2

mm

(29)

where mm is the motor mechanical-hydraulic efficiency. From Eq. (29), p can be adjusted via regulating Tm at a certain Vm . Therefore, the equi-displacement curves and equi-torque curves can also be obtained in the p-n plane under one certain q, except for the equi-efficiency curves. These curves form the characteristic curves of a CPHPTO. Furthermore, the operation of a CPHPTO is restricted by the maximum and minimum motor displacement and should be optimized for the sake of high efficiency and good stability. Hence the visible operating curves of the CPHPTO are also meaningful. Herein the operating curves attempt to instruct the CPHPTO to work around the designed system pressure pr and designed shaft speed nr , which locate at the high-efficiency region. Another reason to maintain p = pr is that if system pressure is mutative, the high-pressure accumulator with a fixed pre-charge pressure will be probably fully-discharged [37,41], resulting in the instability and inefficiency of the CPHPTO. In detail, the operating curves contain the equi-efficiency curves, maximum-flow-rate curve, minimum-flow-rate curve and output-limitation curve, and can be drawn in the q-N plane (N denotes the output power), based on the characteristic curves against different system flow-rates and the designed parameters of the CPHPTO. The characteristic curves and operating curves constitute the ICCs of a CPHPTO (see Fig. 8). The shape of the equi-displacement curves and equi-torque curves can be explained as below. Without considering efficiency, the relationship between p and n against a certain system flow-rate qct is given by

p=

Fig. 6. Flow chart for the efficiency test.

R2 = 1

k=1 m k=1

(yk

yk )2

(yk

y¯) 2

(30)

where Kp is the proportionality coefficient. From Eq. (30), p is proportional to n, while Kp rises as Tm increases. Therefore the curve Tm = Tm0 and the curve Tm = Tm1 (Tm0 > Tm1) are approximately the straight lines passing through the original point and the slope of the first curve is larger than that of the second one. Moreover, equi-torque curves are disjoint. Without consideration of the volumetric efficiency of a hydraulic motor, the relationship between Vm and n against qct is given by

The fitting accuracy of Eqs. (22,23,25) is evaluated by the determination coefficient R2 and root-mean-square error RMSE. R2 is calculated as m

2 Tm n = Kp n qct

(26)

Vm =

where yk is the k-th theoretical result derived from a fitting formula, y¯ the mean value of the experimental data, yk the k-th experimental datum and m the number of cases [33]. R2 lies at the range of 0–1. If R2 approaches to 1, the fitting accuracy is higher. RMSE can be given by

qct n

(31)

From Eq. (31), Vm is approximated as the reciprocal of n. Therefore the curve Vm = Vm0 and the curve Vm = Vm1 (Vm0 > Vm1) are approximately the straight lines parallel to p-axis and the first curve lies on the 7

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Flow control valve High-pressure accumulator Low-pressure reservoir

Directional-control valve 3 hydraulic cylinders

Layer 2

Relief valve 2 bidirectional force transducers Displacement transducer Hydraulic motor Torque-speed transducer

4 check valves

Magnetic powder brake

From pump station

Layer 2

Flow-rate transducer

To pump station

Pressure transducer Current controller

PC

PLC

Fig. 7. Physical model of the experimental platform. The pressure transducer and flow-rate transducer separately measuring system pressure and system flow-rate are arranged at Layer 2 of the frame.

Table 2 Versions and parameters of the main components and sensors. Name

Version

Parameters

Directional-control valve

4WREE10E-50-2XG24K31/F1V

Double-acting double-rod cylinder

/

Single-acting single-rod cylinder

/

Check valve High-pressure accumulator

S10A12.0B NXQ-A-6.3/31.5-L-Y

Relief valve Flow control valve

DBDH10G10B3152 2FRE10-40B25L

Variable axial-piston motor

A6V55EP22GP2

Low-pressure reservoir Magnetic powder brake

/ ZF100KB

Bidirectional force transducer Displacement transducer Flow-rate transducer Pressure transducer Torque-speed transducer

HCHZ-102 KTC-225MM LWGY-10BIC6/NE/NT HT/3403-17-C3.37C3 HCNJ-101

Maximum pressure: 31.5 MPa Maximum flow-rate: 180 L/min Maximum pressure: 31.5 MPa Piston diameter: 90 mm Rod diameter: 63 mm Stroke: 200 mm Maximum pressure: 31.5 MPa Piston diameter: 80 mm Piston area: 5E-3 m2 Rod diameter: 63 mm Stroke: 200 mm Maximum pressure: 31.5 MPa Maximum pressure: 31.5 MPa Nominal volume: 6.3 L Pre-charge gas pressure: 4.5 MPa Maximum cracking pressure: 31.5 MPa Maximum pressure: 31.5 MPa Adjustable flow-rate range: 0–25 L/min Maximum pressure: 35 MPa Maximum flow-rate: 206 L/min Maximum displacement (Vmax ): 54.8 mL/r Minimum displacement (Vmin or0.3Vmax ): 16.4 mL/r Volume: 175 L Torque range: 0–100 Nm Maximum power: 10 kW Force range: −100 kN- +100 kN Displacement range: 0–225 mm Flow-rate range: 3.3 L/min-20 L/min Pressure range: 0–25 MPa Torque range: 0–100 Nm Speed range: 0–1000 r/min

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Fig. 8. Sketch of the integrated characteristic curves for a CPHPTO. (a) Characteristic curves against a certain system flow-rate qct , which comprise equi-efficiency curves (black lines), equi-displacement curves (blue lines) and equi-torque curves (red lines). (b) Operating curves, which comprise equi-efficiency curves (black lines), a maximum-flow-rate curve (a red dashed line), a minimum-flow-rate curve (a red dashed line) and an output-limitation curve (a blue line). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

left side of the second one. Moreover, equi-displacement curves are disjoint. The achieving methods and roles of the ICCs are further mentioned in Subsections 4.1–4.3.

S3. Plot equi-efficiency curves against qct utilizing equi-displacement curves and equi-torque curves. Set several values of efficiency and plot corresponding equi-efficiency curves separately. We take curve t = t0 as an example (see Fig. 10). In particular, plot different efficiency curves against different Tm in the t n plane and different Vm in the p t plane utilizing the data from S1 and S2. Next, plot straight line t = t0 and intersecting the efficiency curves at different points e1 ~e4, f1 ~f4 . Afterwards, find out corresponding points e10 ~e40, f10 ~f40 in the p-n plane according to the corresponding n and Tm or p and Vm . Finally, connect all points with a smooth curve.

4.1. Achieving method of characteristic curves The characteristic curves against a certain system flow-rate qct can be obtained via three steps (see Fig. 9). S1. Plot equi-displacement curves against qct . Tune the piston motion to maintain q = qct . Then set several values of displacements and plot corresponding equi-displacement curves. In particular, equi-displacement curve against a certain displacement Vm0 can be available by maintaining Vm = Vm0 , firstly changing Tm to get different n, p and t , secondly drawing the corresponding points in the p-n plane and finally connecting all points with a smooth curve.

4.2. Achieving method of operating curves The operating curves can be also obtained via three steps (see Fig. 11). S1. Plot the maximum-flow-rate curve and the minimum-flow-rate curve.

S1. Plot equi-torque curves against qct . Tune the piston motion to maintain q = qct . Then set several values of torque and plot corresponding equi-torque curves. In particular, equi-torque curve against a certain mechanical torque Tm0 can be available by maintaining Tm = Tm0 , firstly changing Vm to get different n, p and t , secondly drawing the corresponding points in the p-n plane and finally connecting all points with a smooth curve.

p

The CPHPTO is supposed to operate around n = nr . Hence the range of q is limited by the maximum and minimum motor displacements, i.e. Vmax and Vmin , when n is set as nr . Regardless the volumetric efficiency of motor, the maximum system flow-rate qmax and minimum system flowrate qmin are constant and calculated as

p

n

Plot equi-displacement curves

p

Plot equi-torque curves

n

Fig. 9. Three steps to achieve characteristic curves for a certain system flow-rate qct . 9

n

Plot equi-efficiency curves

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Fig. 10. Achieving method of the curve

t

=

t0

in the p-n plane.

Fig. 11. Three steps to achieve operating curves.

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Fig. 12. Interpretations on the roles of integrated characteristic curves.

a

b

c

d

Fig. 13. Time-domain results against q = 17.5 L/min, p = 10 MPa and n = 524 r/min. (a) Referenced and actual displacements of the piston. (b) Input forces at two piston rods (positive values denote the compressive force and negative ones mean the tensile force). (c) Instantaneous system flow-rate and system pressure. (d) Instantaneous shaft speed and motor torque.

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a

b

c

d

Fig. 14. Comparison of experimental data with the fitting formula of motor displacement. (a) p = 6 MPa. (b) p = 8 MPa (c) p = 10 MPa. (d) p = 12 MPa. Herein red lines are fitting equi-displacement curves.

qmax = n r Vmax qmin = n r Vmin

efficiency and good stability. The first part is approximately the oblique line passing the original point of the q-N plane. The second part is in parallel with q-axis. The intersection point of the two parts is named designed operating point (Nr , qr ) , where qr means the designed system flow-rate. The drawing process is as bellow. Firstly figure out N (q, pr , n r ) of the point (nr , pr ) against each q by tests. Then find out corresponding points in the q-N plane and connect them with a smooth curve to get the first part of the output-limitation curve. Finally, plot the straight line N = Nr to get the second part.

(32)

Hence plot the straight lines q = qmax and q = qmin in parallel with the N-axis in the q-N plane. S2. Plot equi-efficiency curves against n = nr . First, plot several groups of characteristic curves against different q (qmin q qmax ). Next, plot a straight line n = nr at each group of characteristic curves, which intersects equi-efficiency curves at different points. Then calculate N for each point according to the formula N = 2 nTm . Afterwards, find out corresponding points in the q-N plane. Finally, connect all equi-efficiency points with a smooth curve in the qN plane.

4.3. Function of integrated characteristic curves The ICCs play a significant role in the optimal design and efficient operation of the CPHPTO. The details are shown in Fig. 12. Beforehand, we should set up the maximum system pressure pmax and determine Nr according to

S3. Plot the output-limitation curve.

Nr = Jmax Cw

The CPHPTO is supposed to operate around p = pr . Therefore, there exists the output power limitation Nmax determined by pr . Furthermore, the output power is also limited by the rated power of generator, i.e. the designed output power Nr of a CPHPTO. In brief, Nmax is stated as

Nmax =

N (q , pr , n r ) = pr q mt (q, pr , nr ), N Nr , N > Nr

(34)

where Jmax is the maximum wave-energy flux and Cw means the capture width. At first, all hydraulic components are preliminary designed to have the capability of operating at pmax and Nr . Herein, hydraulic motor can be selected based on Nr and pmax , while hydraulic cylinder can be designed according to

Nr (33)

sc

where N (q, pr , n r ) and mt (q , pr , n r ) are respectively the output power of the CPHPTO and the overall efficiency of the motor when p = pr , n = nr . The curve N = Nmax is named output-limitation curve. The CPHPTO should work along the output-limitation curve for high

xc

kt Fe 3max pmax Hs max kt

(35)

where k t is the ratio between the horizontal distance from the gravity center of pendulum to the fulcrum and the vertical distance from the 12

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a

b

c

d

Fig. 15. Comparison of experimental data with the fitting formula of motor torque. (a) p = 6 MPa. (b) p = 8 MPa (c) p = 10 MPa. (d) p = 12 MPa. Herein red lines are fitting equi-torque curves.

fulcrum to the engaging point of the gear and rack, Fe 3max the maximum wave excitation force in the heaving direction, x c the cylinder stroke and Hs max the maximum significant wave height. Besides the opening pressure of relief valve should be set as pmax to avoid the instantaneous condition of p > pmax . Then set up the test rig of the CPHPTO (the test rig exclude the generator and is similar to the structure in Fig. 5, implement efficiency test and plot characteristic curves. Afterwards, find the high-efficiency operating point for Nr and set it as the designed operating point (qr , pr , nr ) . Meanwhile, the generator can be chosen on the basis of the capacity equal to Nr and the rated speed equal to nr , the pre-charge pressure of high-pressure accumulator can be optimized as 80% of pr according to Ref. [43], and the set value of the flow control valve should be given as qr . Eventually, plot operating curves and guide operation. Since the absorbed power is generally lower than or equal to Nr , the CPHPTO can operate at the high-efficiency region under each q via constant speed control and constant pressure control to maintain n = nr and p = pr (see the first part of Eq. (33)). If N > Nr , system flowrate will be limited at qr via a flow control valve and a relief valve, because of the limitation of generator capacity (see the second part of Eq. (33)).

time with the amplitude of 90 mm and the period of 5.8 s, and motor displacement Vm is 31.2 mL/r. By adjusting the motor torque Tm , the instantaneous system flow-rate q, system pressure p and shaft speed n are respectively stabilized at 17.5 L/min, 10 MPa and 524 r/min with small fluctuations (variation range: 16.8–18.0 L/min, 9.6–10.4 MPa and 508–535 r/min). In the circumstance, the instantaneous Tm stays around 44.4 Nm. Moreover, input force at each piston rod appears as a square wave with the same period. The instantaneous negative values appear mainly due to the friction and piston-rod inertia during the expanding of the rodless chamber of each cylinder. 5.2. Comparison of experimental data with fitting formulas The operating points of q = 7.5, 9.5, 12.5, 15, 17.5, 19.5 L/min , p = 6, 8, 10, 12 MPa and n from 0 to 1000 r/min have been tested. The corresponding Vm , Tm and CPHPTO efficiency are shown in Figs. 14–16 respectively. To improve the visibility of experimental results, the data were extended by interpolation and revealed via cloud charts. The fitting results are also displayed via the red lines in each figure. From Figs. 14–16, the fitting results can reflect the changing trend of Vm , Tm and CPHPTO efficiency. Besides, Table 3 lists the performance assessments of the fitting. From Table 3, the fitting formulas have better effect due to the introduction of q~m . Since pmax = 31.5 MPa , Tm , Vm and CPHPTO efficiency can be forecasted up to p = 31.5 MPa by fitting formulas. The predicted results are displayed in Figs. 17–19 respectively based on the p-n plane, q-n plane and p-q plane. Fig. 17 is actually the characteristic curves against q = 15 L/min on

5. Results and discussion 5.1. Time-domain results for a certain (q, p, n) We take the time-domain results for a certain (q, p , n) as an example (see Fig. 13), to verify the existence of the stable operating points. The input motion of the rack is set as the sinusoidal function of 13

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a

b

c

d

Fig. 16. Comparison of experimental data with the fitting formula of CPHPTO efficiency. (a) p = 6 MPa. (b) p = 8 MPa (c) p = 10 MPa. (d) p = 12 MPa. Herein red lines are fitting equi-efficiency curves. Table 3 Overall assessments of fitting formulas.

Vm (without Vm Tm (without Tm t (without t

q~m ) q~m ) q~m )

R2

RSME

0.9536

0.05989

0.9926 0.8306

1.838 0.03074

0.9866 0.9847

0.9522

0.02854 2.619

0.01649

the p-n plane. Due to the limitation of the motor displacement, only the operating points with Vmin -Vmax , i.e. 0.3Vmax -Vmax , practically exist. To illustrate the curve appearances completely, other areas are also displayed. From Fig. 17, the shapes of equi-displacement curves, equitorque curves and equi-efficiency curves are similar to the theoretical curves in Fig. 8a. Firstly equi-torque curves pass through the original point, and the larger-slope curve has a greater torque. The curves will have larger positive slopes at the high-pressure-and-high-speed region, because of the low motor efficiency. Then due to the existence of the flow-rate loss, each equi-displacement curve is approximately a line with a large negative slope rather than a line completely parallel to paxis. And the curve with a large displacement lies on the left side of the one with a small displacement. Finally, the shape of an equi-efficiency curve is not standard oval.

Fig. 17. Expanded results in the p-n plane (q = 15 L/min). Herein blue lines are equi-displacement curves, red lines are equi-torque curves and black lines are equi-efficiency curves.

Fig. 18 expands the experimental results against p = 10 MPa on the q-n plane. In Fig. 18, the equi-torque curves are nearly parallel to equidisplacement curves, and the larger slope means the larger torque or

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5.3. Integrated characteristic curves from fitting results Herein ICCs of the 2 kW CPHPTO can be easily obtained according to the fitting results. The characteristic curves against different system flow-rates (q = 7.5, 10, 12.5, 15, 17.5, 20, 22.5, 25, 27.5 L/min) are first given in Fig. 20. Results reveal that the zone of t 75% commonly locates at q > 15 L/min, Vm > 0.5Vmax and p > 7.5 MPa . The curves of N = Nr = 2 kW are also plotted as the magenta lines. Herein, the highest-efficiency point of all magenta lines is found at q = 17.5 L/min , p = 8.5 MPa , n = 400 r/min via comparing all subgraphs in Fig. 20, where t = 74.8% . Therefore, we set the designed operating point at qr = 17.5 L/min , pr = 8.5 MPa and nr = 400 r/min . According to the designed operating points, the generator is chosen on the basis of the capacity equal to 2 kW (Nr ) and the rated speed equal to 400 r/min (nr ), the pre-charge pressure of high-pressure accumulator is optimized as 6.8 MPa (0.8pr ), and the set value of the flow control valve is given as 17.5 L/min (qr ). Therefore, the operating curves of the CPHPTO can be drawn via the three steps in Section 4.2. Results are shown in Fig. 21. The shapes of the equi-efficiency curves, maximumand minimum- flow-rate curves and output-limitation curve are similar to the theoretical curves in Fig. 8b. Each black equi-efficiency curve is v-shaped, and the curve with high efficiency lies at the upper side of that with low efficiency. The red dashed maximum- and minimumflow-rate curves are parallel to N-axis. The blue output-limitation curve contains two parts, i.e. Line p = pr (Nmax = N (q, pr , n r ) ) and Line Nmax = Nr . The operating points in Line p = pr possess higher efficiency, compared with other points against the same flow-rate or output power. If N > Nr , the practical output power and system flow-rate will be limited at (Nr , qr ) via a flow control valve and a relief valve. Magenta dashed Line Vm = Vmin and azure Line p = 0.8pr are also displayed. Due to the existence of volumetric efficiency, the actual operating points are limited above Line Vm = Vmin not the minimum-flow-rate curve. Moreover, available region of stable operating on the q-N plane is restricted by Lines p = 0.8pr , Vm = Vmin and q = qmax .

Fig. 18. Expanded results in the q-n plane (p = 10 MPa). Herein blue lines are equi-displacement curves, red lines are equi-torque curves, black lines are equiefficiency curves.

6. Conclusions This paper proposes ICCs to handle the 4-D data, which comprise the system pressures, system flow-rates, shaft speeds and CPHPTO efficiency obtained from the efficiency test for a CPHPTO of a floatingpendulum wave energy converter. In addition, the fitting formula for the efficiency of CPHPTO is established to compensate for the limitation of experimental conditions. The conclusions are as follows.

Fig. 19. Expanded results in the p-q plane (n = 500 r/min). Herein blue lines are equi-displacement curves, red lines are equi-torque curves, black lines are equi-efficiency curves.

1) ICCs are available and comprise the characteristic curves and operating curves. Characteristic curves, which consist of the equi-displacement curves with huge negative slopes, the equi-torque curves passing through the original point and the off-standardly oval equiefficiency curves, are plotted in the p-n plane. Meanwhile, operating curves, which contains the maximum- and minimum- flow-rate curves parallel to the N-axis, the v-shaped equi-efficiency curves and the output-limitation curve, are drawn in the q-N plane. 2) ICCs play an important role in the optimal design and efficient operation of the CPHPTO. The designed operating point (qr , pr , nr ) is derived from the characteristic curves and designed output power Nr . According to the designed operating point, the configurations of the high-pressure accumulator, flow control valve and generator can be optimized. Moreover working on the output-limitation curve of operating curves can guarantee the high CPHPTO efficiency and actual output power N Nr .

displacement. Each equi-efficiency curve will intersect equi-torque or equi-displacement curves at two points, which means the CPHPTO efficiency will rise first and then decline, if system flow-rate or shaft speed ascends along one equi-torque or equi-displacement curve. Fig. 19 describes the expanded results on the p-q plane (n = 500 r/ min). In Fig. 19, equi-displacement curves are approximately the large positive slope lines, and the curve with a large displacement lies on the right side of the one with a small displacement. Equi-torque curves are approximately the inverse-proportion curves, and the curve with a large torque lies on the upper side of the one with a small torque. The equiefficiency curves appear as the v-shaped curves rotating 90 degrees clockwise, and the curve with high efficiency lies on the right side of the one with low efficiency.

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Fig. 20. Characteristic curves against different system flow-rates. (a) q = 7.5 L/min. (b) q = 10 L/min. (c) q = 12.5 L/min. (d) q = 15 L/min. (e) q = 17.5 L/min. (f) q = 20 L/min. (g) q = 22.5 L/min. (h) q = 25 L/min. (i) q = 27.5 L/min. Herein blue lines are equi-displacement curves, red lines are equi-torque curves, black lines are equi-efficiency curves and magenta lines denote N = Nr = 2 kW .

3) In the q-N plane, the available stable operating points are within the area besieged by Lines p = 0.8pr , Vm = Vmin and q = qmax . 4) The formulas with an empirical correction term q~m (p , n) fit experimental results of the motor displacement, motor torque and CPHPTO efficiency better than those without q~m (p , n) . In the future, we will consider the practical operation of CPHPTO in real waves and research the feasible control strategy based on the operating curves. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments

Fig. 21. Operating curves (Nr = 2 kW , qr = 17.5 L/min , pr = 8.5 MPa and n r = 400 r/min ). Herein red dashed lines are the maximum-flow-rate and minimum-flow-rate curves, black lines are equi-efficiency curves, blue line is the output-limitation curve, magenta dashed line denotes Vm = Vmin = 16.4 mL/r , and azure line means p = 0.8pr .

The authors wish to acknowledge the support by the National Natural Science Foundation of China (Grant No. 51679171).

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Appendix A. Abbreviations Word (s)

Abbreviation

Wave energy converter Oscillating-body wave energy converter Hydraulic power take-off Variable-pressure hydraulic power take-off Constant-pressure hydraulic power take-off Integrated characteristic curves Impulse response function State-space method

WEC OBWEC HPTO VPHPTO CPHPTO ICCs IRF SSM

Appendix B. Simulation parameters Parameter

Symbol

Value

Unit

Length of the floating pendulum Width of the floating pendulum Height of the floating pendulum Corner radius of the floating pendulum Draft of the floating pendulum Mass of the floating pendulum Added mass at infinite frequency horizontal distance from the gravity center of pendulum to the fulcrum hydrostatic restoring coefficient Piston area of two single-acting single-rod hydraulic cylinders Oil type Oil dynamic viscosity Oil density Effective oil bulk modulus Pre-charge gas pressure of the high-pressure accumulator Nominal volume of the high-pressure accumulator Fractional displacement of the hydraulic motor Maximum displacement of the hydraulic motor Shaft moment of inertia Damping coefficient of the generator

L b h r d mFP A33 l1 S3 sc

4.5 1 1.5 1.5 0.75 3014.5 2100 10 43,213 5E-3 ISO VG 46 0.04 870 8E8 4.5E6 6.3 1 54.8 0.5 4

m m m m m kg kg m N/m m2

Vmax cg

Pas kg/m3 Pa Pa L mL/r kgm2 Nms

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