# Discrete Displacement Hydraulic Power Take-Off System for the Wavestar Wave Energy Converter

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Different embodiments of Wave Energy Converters (WECs), capturing wave energy using buoyant oscillating bodies.

**Figure 2.**(

**a**) a Wavestar absorber; (

**b**) velocities and PTO load in typical production waves; and (

**c**), instantaneous power during two minutes production for a single absorber.

**Figure 3.**(

**a**) the definition of Power Take-Off (PTO) system; (

**b**–

**d**), different fluid power based transmission in PTO systems.

^{2}[6]. Resultantly, a linear generator for a Wavestar float of 5 m diameter, requiring a load force of 400 kN, would be equivalent to a generator with 16–20 m

^{2}of active air-gap surface. In [7], the weight of active magnetic material alone (copper, iron laminations, magnets and back-iron) is estimated to be 1500 kg/m

^{2}. Neglecting the required support structure, the material requirements still is 24–30 tons, rendering the solution infeasible. Resultantly, effort is put into using a transmission combined with more conventional generators. Mechanical transmissions have been explored. However, these would be too massive, as a gearing ratio, which is a factor of 10 higher than a wind turbine transmission, is required, along with handling a bidirectional input.

**Figure 4.**(

**a**) the discrete PTO system based on discrete displacement control of a multi-chambered cylinder (Discrete Displacement Cylinder (DDC)-system); (

**b**) the forces, ${F}_{\mathrm{PTO}}$, the PTO may produce; and (

**c**) how ${F}_{\mathrm{PTO}}$ is discretely varied using the DDC-system during a wave.

**Figure 5.**(

**a**) full-scale Wavestar C5 prototype with two floats [22]; (

**b**) Wavestar C-concept with 20 floats total; (

**c**) Wavestar SC-concept integration with wind turbine.

## 2. Methods

^{®}. Using this approach, the model still ends up having more than 600 states, but is able to simulate about a factor of 4 slower than real time on a reasonable work station. The reasonable execution time is important, as the model presented in the paper is used for optimizing the design.

## 3. PTO Layout

- Low pressure line, ${p}_{\mathrm{L}}$: 10 bar–30 bar;
- Mid-(intermediate) pressure line, ${p}_{\mathrm{M}}$: 70 bar–170 bar;
- High pressure line, ${p}_{\mathrm{H}}$: 150 bar–320 bar.

#### 3.1. Hydraulic Motors and Generators

^{3}/rev hydraulic motor, operating the generator at a torque of 1193 Nm at a pressure of 300 bar. The hydraulic motors are fixed displacement bent-axis motors, whose efficiency is about 95% in these operating conditions.

#### 3.1.1. Pressure Line System and Accumulators

**Figure 6.**(

**a**) an overview of the PTO concept for the Wavestar WEC in Figure 5b; (

**b**) the DDC-system, consisting of a multi-chambered cylinder with integrated shifting manifold for discrete throttle-less force control; (

**c**) a sketch of the shifting manifold; and (

**d**) the illustration of the 27 available forces.

^{3}/rev motor running at 1500 RPM for one minute. The low pressure accumulator batteries each consists of 10 × 50 L accumulators, the mid-pressure batteries of 4 × 50 L accumulators and the high pressure of 16 × 50 L.

#### 3.1.2. DDC

**Figure 7.**(

**a**) an example of how pressure and chamber connection may change during a wave; (

**b**) the available cylinder forces. The numbers 1, 2 and 3 relate to which pressure is in the different cylinder areas in (a) (1 = ${p}_{\mathrm{L}}$, 2 = ${p}_{\mathrm{M}}$, 3 = ${p}_{\mathrm{H}}$).

## 4. Modeling

^{®}Simulink

^{®}and solved with a Runge-Kutta solver, running at a fixed step time of 0.5 ms.

## 5. Wave and Float Model

#### 5.1. Absorber Model

**Figure 10.**(

**a**) definition of variables of the Wavestar absorber; (

**b**) some dimensions for the case study; and (

**c**) the moment arm of the cylinder.

#### 5.1.1. Single Absorber Model

#### 5.1.2. Multi-Absorber Model

- By diffracting the incident wave;
- By radiating waves, interacting with neighboring absorbers.

#### 5.1.3. Model Parameters

**Figure 11.**The wave excitation force filter impulse responses, ${h}_{\mathrm{ext}}$, for different floats. (

**a**) is for wave direction, ${\eta}_{\mathrm{w}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{0}^{\circ}$; (

**b**) is for wave direction, ${\eta}_{\mathrm{w}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}-{45}^{\circ}$; and (

**c**), the wave “measurement” point, float numbering and wave angle, ${\theta}_{\mathrm{w}}$, are defined.

**Figure 12.**(

**a**) the impulse response function, ${k}_{\mathrm{r}}\left(t\right)$, being approximated; and (

**b**) Bode diagram of Equation (12), ${\theta}_{\mathrm{arm}}\left(s\right)/{\tau}_{\mathrm{ext}}\left(s\right)$.

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

Inertia of arm and float (with ballast water) | ${J}_{\mathrm{mech}}$ | 2.45 $\times {10}^{6}$ | [kgm^{2}] |

Hydrostatic restoring torque coefficient | ${k}_{\mathrm{res}}$ | 14.0 $\times {10}^{6}$ | [Nm/rad] |

Added-inertia ${J}_{\mathrm{add}}\left(\omega \right)$ for $\omega \to \infty $ | ${J}_{\mathrm{add},\infty}$ | 1.32 $\times {10}^{6}$ | [kgm^{2}] |

Transfer-function coefficients for ${K}_{\mathrm{r}}\left(s\right)$: | |||

$\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">{b}_{0},{b}_{1},\cdots ,{b}_{5}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">0.0001,\phantom{\rule{1.0pt}{0ex}}0.0144,\phantom{\rule{1.0pt}{0ex}}0.624,\phantom{\rule{1.0pt}{0ex}}8.16,\phantom{\rule{1.0pt}{0ex}}13.1,\phantom{\rule{1.0pt}{0ex}}1.44$ | |||

$\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">{a}_{0},{a}_{1},\cdots ,{a}_{5}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\left(\right)open="("\; close=")">0.0010,\phantom{\rule{1.0pt}{0ex}}0.0906,\phantom{\rule{1.0pt}{0ex}}1.67,\phantom{\rule{1.0pt}{0ex}}6.31,\phantom{\rule{1.0pt}{0ex}}13.3,\phantom{\rule{1.0pt}{0ex}}9.18$ |

## 6. Absorption System

#### 6.1. Discrete Displacement Cylinder—DDC

**Figure 13.**(

**a**) diagram of DDC-system, consisting of cylinder, manifold with nine on/off valves, accumulators and pipe lines. Fitting are indicated, where ξ denotes the fitting resistance coefficient; (

**b**) illustration of hose model; and (

**c**) illustration for combined valve and hose model.

#### 6.1.1. Cylinder

#### 6.1.2. Transmission Line Model

#### 6.1.3. Manifold with Valves and Cylinder Pipe Connection

#### 6.1.4. Manifold Accumulators

h : | Heat exchange coefficient between gas and environment | ${A}_{\mathrm{w}}$ : | Wall area |

${m}_{\mathrm{g}}$ : | Gas mass | ${T}_{\mathrm{w}}$ : | Wall temperature |

${v}_{\mathrm{g}}$ : | Gas specific volume (${V}_{\mathrm{g}}/{m}_{\mathrm{g}}$) | ${V}_{\mathrm{g}}$ : | Gas volume |

T : | Gas temperature | ${p}_{\mathrm{a}}$ : | Gas pressure |

${c}_{\mathrm{v}}$ : | Gas specific heat at constant volume | R : | Ideal gas constant |

#### 6.1.5. Hoses from Manifold to Pressure Lines

**Table 2.**Values used for the float module, where “Par.” and “Val.” abbreviate parameter and value, respectively. * hose connection length is 3 m for float 1–10 and 5 m for float 11–20.

Par. | Val. | Par. | Val. | Par. | Val. | Par. | Val. | Par. | Val. |
---|---|---|---|---|---|---|---|---|---|

${A}_{1}$ [cm^{2}]: | 111 | ${\eta}_{\mathrm{c}}$ [-]: | 0.97 | ρ [kg/m^{3}]: | 900 | ${d}_{\mathrm{A}1}$ [in]: | 1.5 | ${l}_{\mathrm{A}1}$ [m]: | 2.0 |

${A}_{2}$ [cm^{2}]: | 196 | ${x}_{\mathrm{c},\mathrm{max}}$ [m]: | 3.0 | ν [m^{2}s]: | 26 $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{-6}$ | ${d}_{\mathrm{A}2}$ [in]: | 1.5 | ${l}_{\mathrm{A}2}$ [m]: | 0.40 |

${A}_{3}$ [cm^{2}]: | 72 | ${\xi}_{\mathrm{c}}$ [-]: | 1.0 | ${\xi}_{\mathrm{A}1}$ [-]: | 1.2 | ${d}_{\mathrm{A}3}$ [in]: | 1.5 | ${l}_{\mathrm{A}3}$ [m]: | 0.40 |

${A}_{\mathrm{a}}$ [cm^{2}]: | 7.9 | ${A}_{\text{o1}}$ [cm^{2}]: | 2.8 | ${A}_{\text{o2}}$ [cm^{2}]: | 7.7 | ${A}_{\text{o3}}$ [cm^{2}]: | 2.8 | ${t}_{\mathrm{v}}$ [ms]: | 12 |

${p}_{\text{0ML}}$ [bar]: | 10 | ${p}_{\text{0MM}}$ [bar]: | 70 | ${p}_{\text{0MH}}$ [bar]: | 145 | ${V}_{\text{AM}}$ [L]: | 3.6 | ||

${T}_{\text{W}}$ [° C]: | 50 | ${T}_{\text{0}}$ [° C]: | 50 | R [J/kg/K]: | 276 | ${c}_{\mathrm{v}}$ [J/kg/K]: | 760 | ${\tau}_{\text{aH}}$ [s] | 15 |

${d}_{\text{ML}}$ [in]: | 2.0 | ${l}_{\text{ML}}$ [m]: | 3, 5 * | ${d}_{\text{MM}}$ [in]: | 2.0 | ${l}_{\text{MM}}$ [m]: | 3, 5 * | ${\tau}_{\text{aM}}$ [s] | 8 |

${d}_{\mathrm{MH}}$ [in]: | 2.0 | ${l}_{\mathrm{MH}}$ [m]: | 3, 5 * | ${\xi}_{\text{Min}}$ [-]: | 0.6 | ${\xi}_{\mathrm{M}}$ [-]: | 1.3 | ${\tau}_{\text{aL}}$ [s] | 4.5 |

#### 6.2. Pressure Line System

**Table 3.**Parameter values used for the pressure line system, where “Par.” and “Val.” abbreviate parameter and value, respectively.

Par. | Val. | Par. | Val. | Par. | Val. | Par. | Val. | Val. | Par. |
---|---|---|---|---|---|---|---|---|---|

${A}_{\text{A}}$ [cm^{2}]: | 11.4 | ${d}_{\text{P}}$ [in]: | 2.0 | ${l}_{\text{P}}$ [m]: | 6 | ${\xi}_{\text{P1}}$ [-]: | 1.14 | ${\xi}_{\text{P2}}$ [-]: | 1.14 |

${p}_{\text{0L}}$ [bar]: | 10 | ${p}_{\text{0M}}$ [bar]: | 70 | ${p}_{\mathrm{MH}}$ [bar]: | 145 | ${V}_{\text{0H}}$ [L]: | 50 | ${V}_{\text{0M}}$ [L]: | 50 |

${\tau}_{\text{AL}}$ [s]: | 23 | ${\tau}_{\text{AM}}$ [s]: | 34 | ${\tau}_{\text{AH}}$ [s]: | 50 | ${V}_{\text{0L}}$ [L]: | 50 |

#### 6.3. Generation System

#### 6.3.1. Hydraulic Motor

**Figure 17.**(

**a**) efficiency plots (from datasheet) of a Sauer-Danfoss 250 cc Series 51-1 bent-axis motor at full displacement; (

**b**) the efficiency of the modelled motor; and (

**c**) the efficiency of a generator when inverter control is configured for optimal efficiency.

#### 6.3.2. Generator and Inverter

**Figure 18.**(

**a**) Delta-connected induction motor; (

**b**) per phase equivalent circuit; and (

**c**) efficiency as a function of load relative to rated torque at 50 Hz and 400 V.

**Figure 19.**(

**a**) torque control of the generator and inverter model; and (

**b**) the efficiency map used for the inverter.

**Table 4.**Parameter values used for the hydraulic motor, generator and inverter system, where “Par.” and “Val.” abbreviate parameter and value, respectively.

Par. | Val. | Par. | Val. | Par. | Val. | Par. | Val. |
---|---|---|---|---|---|---|---|

${C}_{\tau 1}$ [Nm] : | 11.22 | ${C}_{\tau 2}$ [Nm/Pa]: | 0.17$\xb7{10}^{-6}$ | ${C}_{\tau 3}$ [Nms]: | 0.0085 | ${C}_{\tau 4}$ [Nms^{2}]: | 0.68$\xb7{10}^{-3}$ |

${C}_{Q1}$ [m^{3}/s/Pa]: | 5.4$\xb7{10}^{-12}$ | ${D}_{\omega}$ [m^{3}/rad]: | 39.79$\xb7{10}^{-6}$ | ${J}_{\mathrm{M}}$ [kgm^{2}] : | 0.048 |

## 7. PTO Control

#### 7.1. Wave Power Extraction Algorithm

**Figure 20.**(

**a**) float control (20 parallel system), handling the manifold control to track the generated force reference to extract wave power; and (

**b**) system control for controlling generation and pressure lines.

**Figure 21.**(

**a**) average extracted power measurements for a single absorber of the prototype in Figure 5a [22]. Each point is a 10 min measurement; (

**b**) comparison of power extraction of linear damping and reaction as a function of efficiency in a sea state with ${H}_{\mathrm{m},0}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}1.75\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\text{m}$ and ${T}_{\mathrm{p}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4.5\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathrm{s}$ [11]; and (

**c**), yearly production of a single absorber at the prototype site [28] as a function of PTO torque limitation and efficiency [11].

**Figure 22.**(

**a**) model for reactive control optimization [8]; and (

**b**) control parameters for two sea states as a function of PTO power conversion efficiency, ${\eta}_{\mathrm{PTO}}$.

**Figure 23.**Power matrices of average produced power in [kW]. The matrices are shown for a continuous torque control and a discrete control with $\pm 4$ force values.

#### 7.2. DDC Control—Force Shifting Algorithm (FSA)

**Figure 24.**(

**a**) illustration of the shifting loses due to compression; (

**b**) illustration of the force shifting algorithm; and (

**c**) the opening and closing procedure of the valves during shifting.

#### 7.3. System Control

- Avoid the high pressure accumulator storage from depletion or saturation;
- Keep the mid-pressure line floating between high and low pressure;
- Ensure as steady a power production as possible, while satisfying 1 and 2;
- Choose the proper number of generators for a given sea state;
- Reduce power absorption when full load capacity is reached.

^{3}/rad]. As ${p}_{\text{avg,expected}}$ is the absorber power, the efficiency from cylinder to power out of the generator, ${\eta}_{\text{to-Gn}}$, is required to calculate the average generator power. The maximum allowed speed is set according to 180 kW per generator, which is 15% overload. The lowest speed is set to 400 RPM.

## 8. Results and Discussion

#### 8.1. Overall System Performance

**Figure 26.**(

**a**) PTO simulation results for Wavestar C5 for sea state 1, ${H}_{\mathrm{m},0}$ = 1.00 m, ${T}_{\mathrm{p}}$ = 4.5 s; and (

**b**) PTO simulation results for Wavestar C5 for sea state 2, ${H}_{\mathrm{m},0}$ = 1.75 m, ${T}_{\mathrm{p}}$ = 5.5 s.

**Figure 27.**PTO simulation results for Wavestar C5 for sea state 3, ${H}_{\mathrm{m},0}$ = 2.50 m, ${T}_{\mathrm{p}}$ = 6.5 s.

**Figure 28.**(

**a**) the definition of average power input and output of the different subsystems; and (

**b**) how the difference in stored energy is calculated.

**Table 5.**Efficiencies of subsystems in Figure 28. SS1, SS2 and SS3 are abbreviations of the three tested sea states (1, 2, 3).

- Sea state 1: ${P}_{\text{ext,\u2212}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.57\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\text{kW}$ and ${P}_{\text{ext,+}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}7.5\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\text{kW}$: Ratio of reactive and real power: $c\phantom{\rule{-0.166667em}{0ex}}={\textstyle \frac{{P}_{\text{ext,+}}}{{P}_{\text{ext,\u2212}}}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.075$;
- Sea state 2: ${P}_{\text{ext,\u2212}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}2.5\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\text{kW}$ and ${P}_{\text{ext,+}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}19.5\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\text{kW}$: Ratio of reactive and real power: $c\phantom{\rule{-0.166667em}{0ex}}={\textstyle \frac{{P}_{\text{ext,+}}}{{P}_{\text{ext,\u2212}}}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.088$;
- Sea state 3: ${P}_{\text{ext,\u2212}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}3.1\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\text{kW}$ and ${P}_{\text{ext,+}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}49.0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\text{kW}$: Ratio of reactive and real power: $c\phantom{\rule{-0.166667em}{0ex}}={\textstyle \frac{{P}_{\text{ext,+}}}{{P}_{\text{ext,\u2212}}}}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.063$.

#### 8.2. Control System Performance

## 9. Conclusions

- Handling peak power input, which is a factor of 12 higher than mean power, while maintaining component efficiency—the DDC maintains above 90% in these conditions, and the remaining components are above 94% in efficiency, each;
- Full controllability of load force on absorbers and four quadrant mode—the DDC offers 27 force steps and four quadrant mode;
- Incorporating a short-term storage for supplying reactive power—reactive power is only processed by DDCs and accumulators;
- Incorporating an efficient energy storage for power smoothing—generators are operated independent of the wave absorption; energy storage for operating generators at 1500 RPM for one minute;
- Maintaining PTO efficiency in small waves when operating at 15% of full load capacity—total efficiency maintained above 70% in all sea states;
- Being able to reduce power absorption when full load capacity is reached—the DDCs reduce absorption when the WEC reaches full load;
- Being scalable to future multi-MW systems.

^{®}technology. These variable displacement also shows high part load efficiency, leading to the option of operating at a fixed generator speed of the 1500 RPM, thereby potentially removing the power converters.

## Conflicts of Interest

## Nomenclature:

${\beta}_{\mathrm{eff}}$ | Effective bulk modulus of a volume of fluid | [Pa] |

$\Delta p$ | Pressure difference across motor ports | [Pa] |

${\eta}_{\mathrm{w}}$ | Wave height | [m] |

${\eta}_{\mathrm{c}}$ | Efficiency of cylinder | [-] |

${\eta}_{\mathrm{PTO}}$ | PTO conversion efficiency from input to output and vice versa | [-] |

${\eta}_{\text{tot}}$ | Ratio of input and output energy of the PTO | [-] |

${\eta}_{\text{DCC}}$ | Average conversion efficiency of DCC | [-] |

γ | Control input to reduce power absorption | [-] |

ν | Kinematic viscosity of fluid | [m^{2}/s] |

${\theta}_{\mathrm{arm}}$ | Angular position of arm | [rad] |

${\theta}_{\mathrm{w}}$ | Approach angle of incoming waves | [rad] |

ρ | Density of hydraulic fluid | [kg/m^{3}] |

σ | Variance | [-] |

${\tau}_{\mathrm{a}}$ | Thermal time constant of accumulator | [s] |

${\tau}_{\text{aH}}$ | Thermal time constant of accumulator | [s] |

${\tau}_{\mathrm{Arch}}$ | Torque on arm due to Archimedes force | [Nm] |

${\tau}_{\mathrm{G}}$ | Torque on arm due to gravity on float and arm | [Nm] |

${\tau}_{\mathrm{ext}}$ | Torque on arm due to wave excitation | [Nm] |

${\tau}_{\mathrm{rad}}$ | Torque on arm due to wave radiation | [Nm] |

${\tau}_{\text{AL}}$ | Thermal time constant of low pressure acc. (accumulator), energy storage | [s] |

${\tau}_{\text{AM}}$ | Thermal time constant of mid pressure acc., energy storage | [s] |

${\tau}_{\text{AH}}$ | Thermal time constant of high pressure acc., energy storage | [s] |

${\tau}_{\text{aL}}$ | Thermal time constant of low pressure acc., manifold | [s] |

${\tau}_{\text{aM}}$ | Thermal time constant of mid pressure acc., manifold | [s] |

${\tau}_{\text{aH}}$ | Thermal time constant of high pressure acc., manifold | [s] |

${\tau}_{\mathrm{M}}$ | Hydraulic motor output torque | [Nm] |

${\tau}_{\mathrm{PTO}}$ | Applied PTO load torque | [Nm] |

${\omega}_{\mathrm{arm}}$ | Angular velocity of float arm | [rad/s] |

${\omega}_{\text{charge}}$ | Charge motor speed | [rad/s] |

${\omega}_{\mathrm{M}}$ | Angular velocity of hydraulic motors and generators | [rad/s] |

${\xi}_{\mathrm{A}1}$ | Fitting loss coefficient at chamber 1 inlet | [-] |

${\xi}_{\text{Min}}$ | Fitting coefficient in connections from manifold to pressure line | [-] |

${\xi}_{\mathrm{M}}$ | Fitting loss coefficient for internal connections in manifold | [-] |

${\xi}_{\text{P1}}$, ${\xi}_{\text{P2}}$ | Fitting loss coefficients for fitting in pressure lines | [-] |

${A}_{\text{A}}$ | Opening area of accumulator inlet, energy storage | [m^{2}] |

${A}_{1}$ | Piston area of chamber 1 | [m^{2}] |

${A}_{2}$ | Piston area of chamber 2 | [m^{2}] |

${A}_{3}$ | Piston area of chamber 3 | [m^{2}] |

${A}_{\mathrm{a}}$ | Opening area of accumulator inlet, manifold | [m^{2}] |

${A}_{\text{A}}$ | Opening area of accumulator inlet, energy storage | [m^{2}] |

${A}_{\text{o1}}$ | Opening area of valves to chamber 1 | [m^{2}] |

${A}_{\text{o2}}$ | Opening area of valves to chamber 2 | [m^{2}] |

${A}_{\text{o3}}$ | Opening area of valves to chamber 3 | [m^{2}] |

${B}_{\mathrm{PTO}}$ | Damping coefficient of PTO load torque | [kgm^{2}/s] |

${c}_{\tau 1}$,${c}_{\tau 2}$,${c}_{\tau 3}$,${c}_{\tau 4}$ | Friction coefficients for hydraulic motor | [Nm],[Nm/Pa],[Nm/(rad/s)],[Nm/(rad/s)^{2}] |

${c}_{\text{Q1}}$ | Flow loss coefficient for hydraulic motor | [(m^{3}/s)/Pa] |

${c}_{\mathrm{v}}$ | Gas specific heat at constant volume | [J/(kg K)] |

${D}_{\omega}$ | Main hyd. motor displacement | [m^{3}] |

${D}_{\omega \text{charge}}$ | Charge motor displacement | [m^{3}/rad] |

${d}_{\mathrm{a}}$ | Cylinder moment arm | [m] |

${d}_{\mathrm{A}1}$ | Diameter of hose to cylinder chamber 1 | [m] |

${d}_{\mathrm{A}2}$ | Diameter of hose to cylinder chamber 2 | [m] |

${d}_{\mathrm{A}3}$ | Diameter of hose to cylinder chamber 3 | [m] |

${d}_{\text{ML}}$ | Inner diameter of low pressure connection, manifold to main lines | [m] |

${d}_{\text{MM}}$ | Inner diameter of mid pressure connection, manifold to main lines | [m] |

${d}_{\mathrm{MH}}$ | Inner diameter of high pressure connection, manifold to main lines | [m] |

${d}_{\text{P}}$ | Inner diameter of pressure lines | [m] |

${F}_{\mathrm{fric},\mathrm{c}}$ | Cylinder friction force | [N] |

${F}_{\mathrm{PTO}}$, ${F}_{\mathrm{c}}$ | Force applied by PTO cylinder | [N] |

f | Frequency | [Hz] |

${H}_{\mathrm{s}}$, ${H}_{\mathrm{m},0}$ | significant wave height | [m] |

${h}_{\mathrm{ext}}$ | Impulse response relating ${\eta}_{\mathrm{w}}$ and ${\tau}_{\mathrm{ext}}$ | [Nm/(sm)] |

${J}_{\mathrm{M}}$ | Hyd. motor inertia | [kgm^{2}] |

${J}_{\mathrm{mech}}$ | Mass moment of inertia of float and arm | [kgm^{2}] |

${J}_{\mathrm{add}}$ | Added mass moment of inertia | [kgm^{2}] |

${J}_{\mathrm{add},\infty}$ | Added inertia of float for oscillation frequency going to infinity | [kgm^{2}] |

${k}_{\mathrm{r}}$ | Radiation force impulse response function | [Nm] |

${k}_{\mathrm{res}}$ | Stiffness coefficient of linearized hydrostatic restoring torque | [Nm/rad] |

${k}_{\mathrm{PTO}}$ | Virtual spring coefficient emulated by PTO | [Nm/rad] |

${l}_{\text{P}}$ | Length of of pressure lines between manifold nodes | [m] |

${l}_{\mathrm{A}1}$ | Length of hose to chamber 1 | [m] |

${l}_{\mathrm{A}2}$ | Length of hose to chamber 2 | [m] |

${l}_{\mathrm{A}3}$ | Length of hose to chamber 3 | [m] |

${l}_{\text{ML}}$ | Length of low pressure hose from manifold to pressure lines | [m] |

${l}_{\text{MM}}$ | Length of mid pressure hose from manifold to pressure lines | [m] |

${l}_{\mathrm{MH}}$ | Length of high pressure hose from manifold to pressure lines | [m] |

${P}_{\text{out}}$ | Instantaneous electrical power output of PTO | [W] |

${P}_{\mathrm{ext}}$, ${P}_{\text{har}}$ | Instantaneous absorbed/extracted power | [W] |

${p}_{\mathrm{L}}$ | Pressure in low pressure line | [Pa] |

${p}_{\mathrm{M}}$ | Pressure in intermediate line | [Pa] |

${p}_{\mathrm{H}}$ | Pressure in high pressure line | [Pa] |

${p}_{\mathrm{f}}$ | Total pressure drop of line with fittings and hoses | [Pa] |

${p}_{\lambda}$ | Pressure drop of line | [Pa] |

${p}_{\xi}$ | Pressure drop of fitting | [Pa] |

${p}_{\text{0ML}}$ | Pre-charge pressure of low pressure acc. manifold | [Pa] |

${p}_{\text{0MM}}$ | Pre-charge pressure of mid pressure acc. manifold | [Pa] |

${p}_{\text{0MH}}$ | Pre-charge pressure of high pressure acc. manifold | [Pa] |

${p}_{\text{0L}}$ | Pre-charge pressure of low pressure acc., energy storage | [Pa] |

${p}_{\text{0M}}$ | Pre-charge pressure of mid pressure acc., energy storage | [Pa] |

${p}_{\text{0H}}$ | Pre-charge pressure of high pressure acc., energy storage | [Pa] |

${Q}_{\text{vxx}}$ | Valve flows in manifold | [m^{3}/s] |

R | Ideal gas constant | [kgm^{2}/(s^{2}K)] |

Re | Reynolds number | [-] |

S | Power density spectrum of sea state | [m^{2}/Hz] |

${T}_{\text{P}}$ | Peak wave period | [s] |

T | Gas temperature | [K] |

${T}_{\mathrm{w}}$ | Accumulator wall temperature | [K] |

${t}_{\mathrm{v}}$ | Valve opening and closing time | [s] |

${u}_{\mathrm{v}}$ | Valve opening reference | [-] |

${V}_{\mathrm{g}}$ | Current gas volume in accumulator | [m^{3}] |

${V}_{\text{aM}}$ | Sizes of manifold accumulators | [m^{3}] |

${v}_{\mathrm{c}}$ | Cylinder piston velocity | [m/s] |

${x}_{\mathrm{c}}$ | Current cylinder stroke | [m] |

${x}_{\mathrm{c},\mathrm{max}}$ | Max stroke length of cylinder | [m] |

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**MDPI and ACS Style**

Hansen, R.H.; Kramer, M.M.; Vidal, E.
Discrete Displacement Hydraulic Power Take-Off System for the Wavestar Wave Energy Converter. *Energies* **2013**, *6*, 4001-4044.
https://doi.org/10.3390/en6084001

**AMA Style**

Hansen RH, Kramer MM, Vidal E.
Discrete Displacement Hydraulic Power Take-Off System for the Wavestar Wave Energy Converter. *Energies*. 2013; 6(8):4001-4044.
https://doi.org/10.3390/en6084001

**Chicago/Turabian Style**

Hansen, Rico H., Morten M. Kramer, and Enrique Vidal.
2013. "Discrete Displacement Hydraulic Power Take-Off System for the Wavestar Wave Energy Converter" *Energies* 6, no. 8: 4001-4044.
https://doi.org/10.3390/en6084001