# Control of Wave Energy Converters with Discrete Displacement Hydraulic Power Take-Off Units

^{*}

## Abstract

**:**

## 1. Introduction and Background

#### 1.1. Constant Pressure Configuration

#### 1.2. Variable Pressure

#### 1.3. Variable–Constant Pressure

## 2. WEC Dynamics

## 3. The Hydraulic PTO System

#### 3.1. The Hydraulic Cylinder

#### 3.2. The Hoses

#### 3.3. The Directional Valves

#### 3.4. The Pressure Accumulators

#### 3.5. The Hydraulic Motor

## 4. The Control Algorithm

#### 4.1. The Buoy Control Method

#### 4.2. The Force-Shifting Algorithm

## 5. Simulation Tool

#### 5.1. Wave Model

#### 5.2. The System Losses

## 6. Simulation Results

## 7. Discussion

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

ymbol | Meaning | I Units |

${A}_{hose}$ | Area of the hose | m${}^{2}$ |

${A}_{v}$ | Instantaneous opening area of the valve | m${}^{2}$ |

${A}_{0}$ | Maximum opening area of the valve | m${}^{2}$ |

$A1$ | Piston area of chamber 1 | m${}^{2}$ |

$A2$ | Piston area of chamber 2 | m${}^{2}$ |

$A3$ | Piston area of chamber 3 | m${}^{2}$ |

$A4$ | Piston area of chamber 4 | m${}^{2}$ |

B | Feedback controller gain on the angular velocity | N.m.s.rad${}^{-1}$ |

${C}_{d}$ | Valve discharge coefficient | - |

${C}_{Q1}$ | Flow loss coefficient of the hydraulic motor | m${}^{3}$.s${}^{-1}$.Pa${}^{-1}$ |

${C}_{v}$ | Gas specific heat at constant volume | J.(kg.K)${}^{-1}$ |

$CWR$ | Capture width ratio | - |

${d}_{hose}$ | Diameter of the hose | m |

D | Characteristic dimension of the buoy | m |

${D}_{M}$ | Total hydraulic motor displacement | m${}^{3}$ |

${D}_{w}$ | Hydraulic motor displacement | m${}^{3}$ |

${F}_{c}$ | Force applied by the cylinder | N |

${F}_{fric}$ | Friction force of the cylinder | N |

${F}_{ref}$ | Reference control force | N |

${h}_{ex}$ | Wave excitation torque impulse response function | N.m |

${h}_{r}$ | Radiation torque impulse response function | N.m |

${H}_{m0}$ | Wave significant height | m |

${J}_{rigid}$ | Rigid body inertia | kg.m${}^{2}$ |

${J}_{\infty}$ | Moment of the added mass | kg.m${}^{2}$ |

${k}_{gen}$ | Number of generators | - |

${K}_{res}$ | Hydrostatic stiffness coefficient | N.m.rad${}^{-1}$ |

K | Feedback controller gain on the angular position | N.m.rad${}^{-1}$ |

${l}_{hose}$ | Length of the hose | m |

${p}_{acc}$ | Pressure in the accumulator | Pa |

${p}_{avg,exp}$ | Expected average power output | W |

${p}_{A1}$ | Pressure in chamber 1 | Pa |

${p}_{A2}$ | Pressure in chamber 2 | Pa |

${p}_{A3}$ | Pressure in chamber 3 | Pa |

${p}_{A4}$ | Pressure in chamber 4 | Pa |

${p}_{f}$ | Pressure drop across the hose | Pa |

${p}_{H}$ | Pressure of the high pressure accumulator | Pa |

${p}_{L}$ | Pressure of the low pressure accumulator | Pa |

${p}_{\zeta}$ | Pressure drop of the fitting | Pa |

${p}_{\lambda}$ | Pressure drop across a straight pipe/hose | Pa |

${P}_{actuator}$ | Actuator power extraction | W |

${P}_{ave}$ | Average extracted power | W |

${P}_{gen}$ | Generator power output | W |

${P}_{M}$ | Motor power output | W |

${P}_{w}$ | Wave energy transport | W.m${}^{-1}$ |

${Q}_{acc}$ | Inlet flow to the accumulator | m${}^{3}$.s${}^{-1}$ |

${Q}_{A1}$ | Inlet flow to chamber 1 | m${}^{3}$.s${}^{-1}$ |

${Q}_{A2}$ | Inlet flow to chamber 2 | m${}^{3}$.s${}^{-1}$ |

${Q}_{A3}$ | Inlet flow to chamber 3 | m${}^{3}$.s${}^{-1}$ |

${Q}_{A4}$ | Inlet flow to chamber 4 | m${}^{3}$.s${}^{-1}$ |

${Q}_{in}$ | Inlet flow of the hose | m${}^{3}$.s${}^{-1}$ |

${Q}_{out}$ | Outlet flow of the hose | m${}^{3}$.s${}^{-1}$ |

R | Ideal gas constant | kg.m${}^{2}$.(s${}^{2}$.K)${}^{-1}$ |

$Re$ | Reynold’s number | - |

S | Wave spectral density | m${}^{2}$.s.rad${}^{-1}$ |

${t}_{v}$ | Valve opening and closing time | s |

T | Gas temperature | K |

${T}_{w}$ | Hydraulic accumulator wall temperature | K |

${u}_{v}$ | Valve opening and closing signal | - |

${v}_{c}$ | Instantaneous piston velocity | m.s${}^{-1}$ |

${v}_{out}$ | Velocity of the outlet flow of the hose | m.s${}^{-1}$ |

${V}_{a0}$ | Accumulator size | m${}^{3}$ |

${V}_{ext}$ | Accumulator external volume of the pipeline | m${}^{3}$ |

${V}_{g}$ | Accumulator gas volume | m${}^{3}$ |

${V}_{0,A1}$ | External volume of the connecting hose to chamber 1 | m${}^{3}$ |

${V}_{0,A2}$ | External volume of the connecting hose to chamber 2 | m${}^{3}$ |

${V}_{0,A3}$ | External volume of the connecting hose to chamber 3 | m${}^{3}$ |

${V}_{0,A4}$ | External volume of the connecting hose to chamber 4 | m${}^{3}$ |

${x}_{c}$ | Instantaneous stroke of the cylinder | m |

${x}_{c,max}$ | Maximum stroke of the cylinder | m |

$\beta $ | Effective bulk modulus of the fluid | Pa |

$\zeta $ | Fitting loss coefficient | - |

$\eta $ | Wave elevation | m |

${\eta}_{c}$ | Cylinder efficiency | - |

${\eta}_{out}$ | Electricity generation efficiency | - |

$\theta $ | Angular position of the arm | rad |

$\nu $ | Kinematic viscosity of the fluid | m${}^{2}$.s${}^{-1}$ |

$\rho $ | Fluid density | kg.m${}^{-3}$ |

${\tau}_{a}$ | Accumulator thermal time constant | s |

${\tau}_{ex}$ | Wave excitation torque | N.m |

${\tau}_{PTO}$ | Power take-off torque | N.m |

${\tau}_{rad}$ | Torque due to the radiation wave | N.m |

$\varphi $ | Random phase shift | rad |

$\psi $ | Motor speed control coefficient | - |

$\omega $ | Wave frequency | rad.s${}^{-1}$ |

${\omega}_{M}$ | Angular velocity of the hydraulic motor | rad.s${}^{-1}$ |

${\omega}_{p}$ | Peak frequency of the wave | rad.s${}^{-1}$ |

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**Figure 5.**When accounting for displacement constraints, some unconstrained methods harvest less energy. PD: proportional-derivative; SA: singular arc; PDC3: proportional-derivative complex conjugate control; SB: shape-based; MPC: model predictive control; PS: pseudo-spectral.

**Figure 9.**The pressure loss of the hose which has a 1-m length and $3.81\times {10}^{-2}$ m diameter with different flow rates across the hose.

**Figure 11.**The friction force of the cylinder with different velocities when the cylinder force is 100 kN.

**Figure 12.**The energy extracted accounting for displacement and force constraints, including the hydraulic system dynamics model.

Symbol | Value | Unit |
---|---|---|

${J}_{tot}$ | $3.8\times {10}^{6}$ | kg m${}^{2}$ |

${K}_{res}$ | $14\times {10}^{6}$ | Nm/rad |

The transfer function ${H}_{r}(s)$ | ||

$({b}_{0},{b}_{1},\dots ,{b}_{5})$ | * | |

$({a}_{0},{a}_{1},\dots ,{a}_{5})$ | ** | |

The transfer function ${H}_{ex}(s)$ | ||

$({b}_{0},{b}_{1})$ | *** | |

$({a}_{0},{a}_{1},\dots ,{a}_{4})$ | **** |

Symbol | Value | Unit |
---|---|---|

Length of the arms | ||

${l}_{2}$ | 3 | m |

${l}_{3}$ | $2.6$ | m |

${l}_{4}$ | $1.6$ | m |

Length of the hoses C2M | ||

${l}_{A1}$ | 1 | m |

${l}_{A2}$ | 1 | m |

${l}_{A3}$ | 1 | m |

${l}_{A4}$ | 1 | m |

Diameter of the hoses C2M | ||

${d}_{A1}$ | $1.5$ | in |

${d}_{A2}$ | $1.5$ | in |

${d}_{A3}$ | $1.5$ | in |

${d}_{A4}$ | $1.5$ | in |

Maximum stroke | ||

${x}_{c,max}$ | 3 | m |

Area of the chambers | ||

${A}_{1}$ | $113.4\times {10}^{-4}$ | m${}^{2}$ |

${A}_{2}$ | $32.55\times {10}^{-4}$ | m${}^{2}$ |

${A}_{3}$ | $80.85\times {10}^{-4}$ | m${}^{2}$ |

${A}_{4}$ | $162.75\times {10}^{-4}$ | m${}^{2}$ |

Max Area of the valves | ||

${A}_{01}$ | $1.6\times {10}^{-4}$ | m${}^{2}$ |

${A}_{02}$ | $1.6\times {10}^{-4}$ | m${}^{2}$ |

${A}_{03}$ | $1.6\times {10}^{-4}$ | m${}^{2}$ |

${A}_{04}$ | $1.6\times {10}^{-4}$ | m${}^{2}$ |

Accumulator size | ||

${V}_{a0}$ | $100\times {10}^{-3}$ | m${}^{3}$ |

Pressure drop coef | ||

${\zeta}_{M}$ | $1.3$ | |

${\zeta}_{C}$ | 1 | |

Specific time constant S | ||

${\tau}_{l}$ | 23 | s |

${\tau}_{h}$ | 34 | s |

Initial pressure of the accumulators | ||

${p}_{a,l}$ | 20 | bar |

${p}_{a,h}$ | 130 | bar |

Initial angle | ||

${\alpha}_{0}$ | $1.0821$ | rad |

Control parameters | ||

K | $-9.16\times {10}^{6}$ | Nm/rad |

B | $4.4\times {10}^{6}$ | Nms/rad |

Valve opening time | ||

${t}_{v}$ | $30\times {10}^{-3}$ | s |

Wall temperature | ||

${T}_{w}$ | 50 | ^{o}C |

Ideal gas constant | ||

R | 276 | J/kg/K |

Gas specific heat at constant volume | ||

${C}_{v}$ | 760 | J/kg/K |

Motor displacement | ||

${D}_{w}$ | 100 | cc/rev |

Flow loss coefficient | ||

${C}_{Q1}$ | $5.4\times {10}^{-12}$ | m${}^{3}$/s/Pa |

Fluid bulk modulus | ||

$\beta $ | $1.5\times {10}^{9}$ | Pa |

Symbol | Value | Unit |
---|---|---|

The SA controller | ||

${F}_{PTO,max}$ | 3705 | kN |

${x}_{c,max}-{x}_{c,min}$ | $3.2$ | m |

${P}_{ave}$ | $35.11$ | kW |

The PD controller | ||

${F}_{PTO,max}$ | 1119 | kN |

${x}_{c,max}-{x}_{c,min}$ | $1.1$ | m |

${P}_{ave}$ | $21.00$ | kW |

The PDC3 controller | ||

${F}_{PTO,max}$ | 1404 | kN |

${x}_{c,max}-{x}_{c,min}$ | $1.6$ | m |

${P}_{ave}$ | $21.08$ | kW |

Symbol | Value | Unit |
---|---|---|

The SA controller | ||

${F}_{PTO,max}$ | 215 | kN |

${x}_{c,max}-{x}_{c,min}$ | $0.96$ | m |

${P}_{ave}$ | $13.49$ | kW |

The PD controller | ||

${F}_{PTO,max}$ | 215 | kN |

${x}_{c,max}-{x}_{c,min}$ | $1.1$ | m |

${P}_{ave}$ | $18.26$ | kW |

The PDC3 controller | ||

${F}_{PTO,max}$ | 215 | kN |

${x}_{c,max}-{x}_{c,min}$ | $1.2$ | m |

${P}_{ave}$ | $13.32$ | kW |

The SB controller | ||

${F}_{PTO,max}$ | 215 | kN |

${x}_{c,max}-{x}_{c,min}$ | $0.8$ | m |

${P}_{ave}$ | $16.02$ | kW |

The MPC controller | ||

${F}_{PTO,max}$ | 215 | kN |

${x}_{c,max}-{x}_{c,min}$ | $1.1$ | m |

${P}_{ave}$ | $18.37$ | kW |

The PS controller | ||

${F}_{PTO,max}$ | 215 | kN |

${x}_{c,max}-{x}_{c,min}$ | $0.90$ | m |

${P}_{ave}$ | $13.22$ | kW |

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## Share and Cite

**MDPI and ACS Style**

Zou, S.; Abdelkhalik, O.
Control of Wave Energy Converters with Discrete Displacement Hydraulic Power Take-Off Units. *J. Mar. Sci. Eng.* **2018**, *6*, 31.
https://doi.org/10.3390/jmse6020031

**AMA Style**

Zou S, Abdelkhalik O.
Control of Wave Energy Converters with Discrete Displacement Hydraulic Power Take-Off Units. *Journal of Marine Science and Engineering*. 2018; 6(2):31.
https://doi.org/10.3390/jmse6020031

**Chicago/Turabian Style**

Zou, Shangyan, and Ossama Abdelkhalik.
2018. "Control of Wave Energy Converters with Discrete Displacement Hydraulic Power Take-Off Units" *Journal of Marine Science and Engineering* 6, no. 2: 31.
https://doi.org/10.3390/jmse6020031