# Wave-to-Wire Power Maximization Control for All-Electric Wave Energy Converters with Non-Ideal Power Take-Off

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling an All-Electric Wave Energy Converter

#### 2.1. Hydrodynamic Simulation

_{m}), linear restoring coefficient or stiffness (k

_{c}), linear damping (b

_{d}) and the radiation force (F

_{rad}) are used in the point absorber model to form the mass-spring-damper system. Radiation damping and added mass as a function of the wave frequency input are given in Figure 4. The constant hydrodynamic parameters are given in Table 1. As part of this work, a simple cylinder design was considered as a point absorber. This is due to the fact that the main aim of this research is to demonstrate a control process to maximize wave-to-wire power through the control of the electrical generator. Optimizing the structure of the point absorber was deemed outside the main scope of this study.

#### 2.2. Point Absorber Model and System Constraints

_{pa}is the acceleration and v

_{pa}the velocity of the point absorber. F

_{rad}is the force as a result of wave radiation and is estimated using a State Space model. The parameters of the State Space model are taken from the hydrodynamic simulation and are based on the convolution integral formulation of the radiation impulse response function (K

_{r}) of the point absorber and is shown in Equation (2).

_{m}), linear damping (b

_{d}) and stiffness (k

_{c}) are considered constants and their values are given in Table 1, along with point absorber parameters. Linear damping generates a linear damping force (F

_{bd}) and stiffness generates a buoyancy force (F

_{kc}). The F

_{exc}is generated by the wave resource model, and the F

_{pto}is generated by the PMLG, which is discussed in detail in Section 2.3.

_{pa}) to its maximum value (pa

_{zmax}). The ES system operates only when the z

_{pa}has reached 90% of the pa

_{zmax}, and generates the end-stop force (F

_{ES}), which slows down the translator. The F

_{ES}is generated by taking into account that the end-stop spring has a damping coefficient (b

_{ES}) and a stiffness coefficient (k

_{ES}). The values for the end-stop system are given in Table 1. The second constraint implemented is the electrical power limitation mechanism. In order to avoid overrated devices, keep the cost of the device low, and avoid excessive peaks of current that could cause damage to the WEC, a power limitation mechanism is essential. In this research paper, the power limitation mechanism is based on a simple braking force (F

_{brake}) that is generated when the instantaneous electrical power generated from the PMLG (P

_{gen}) is larger than the rated power of the device (P

_{rated}). The power difference is converted to a damping coefficient (b

_{db}), which can be implemented in reality by using standard brakes. The conversion of the power difference to b

_{db}can be modelled either with a linear relationship or with a proportional-integral (PI) controller. For this study, using a power limitation mechanism is of crucial importance. To maximize the electrical power delivered to the grid, it should to be taken into account that sudden peaks in wave power may have to be limited, and therefore will not result in an increase in electrical generation. It has to be noted that even though the optimized controllers developed in this research paper take into account the power limitation for power production the same does not apply for the displacement constraint. Further development of the system will focus on making the controllers aware of the displacement constraint and avoid the end-stop system slamming forces. The PMLG and the parameters of the electrical generator are discussed in Section 2.3 and shown in Table 2, respectively.

#### 2.3. The Permanent Magnet Linear Generator

_{pto}, which is used in the mechanical mass-spring-damper system. The PMLG is modelled using Equations (3)–(5) [28,30]:

_{t}is the number of turns around a tooth, Φ is the flux in the tooth and τ

_{p}is the pole pitch. As shown in (3), the EMF of the PMLG is a function of the displacement of the translator, which is taken from the point absorber model discussed in Section 2.2. The EMF is transferred to the SimPowerSystems physical modelling environment, and the voltage at the terminals of the PMLG is acquired by taking into account the stator resistance (Rs) and inductance (Ls). The F

_{pto}can be calculated using Equation (4), where I is the peak current and the current leads the EMF with an angle φ. The generation of the three-phase currents is given in Equation (5) and is a function of displacement.

#### 2.4. Voltage Oriented Controller and Power Transmission

_{abc}* with the inverse park transformation by using the angle at the point of measurement θ

_{g}. Figure 6 depicts the block diagram of the VOC scheme as it was implemented in MATLAB/Simulink.

## 3. Control Strategies for Wave-to-Wire Power Maximization

- The reference F
_{pto}controller; and - The pulse generator for the VSC.

_{pto}*), which forces the point absorber model to move based on the requirements of the control strategy. Depending on the control strategy, F

_{pto}* can bring, for example, the point absorber velocity into phase with the excitation force from the waves to maximize hydrodynamic power, or can only control the damping to maximize the power transfer without any phase control. As mentioned in Section 2, the W2W model is based on physical modelling of the mechanical and electrical quantities. This means that the F

_{pto}* generated by the controller is not directly applied to the point absorber model which is a common practice. The F

_{pto}* from the control strategy is used as an input to generate pulses for the generator-side VSC. The pulses that control the generator-side VSC will force the PMLG to produce the appropriate PTO force. The block diagram of the generator-side control strategy and the pulse generator are depicted in Figure 7. The pulse generator for the generator-side VSC is based on the zero d-axis current (ZDC) controller for PMLG.

_{pto}*, z

_{pa}, the reference d-axis current (i

_{ds}*), and the measured three-phase current at the terminals of the PMLG (i

_{abc}). As shown in Figure 7b, the displacement is converted to electrical angle, and the F

_{pto}* is converted to reference q-axis current (i

_{qs}*). Reference currents, i

_{ds}* and i

_{qs}*, are compared to actually measured currents to produce errors. The current errors are fed to PI controllers to generate the appropriate reference voltage signals V

_{q}* and V

_{d}* to minimize the current error. The reference voltages are used as input to a PWM scheme for pulse generation for the generator-side VSC.

_{pto}controllers will be presented theoretically and tested for their ability to deliver maximum power to the grid. These control strategies are the real damping, reactive spring-damping and velocity controller. For all three different reference F

_{pto}controllers the ZDC controller remains the same as a pulse generator for the generator-side VSC.

#### 3.1. Real Damping Controller

_{p}to calculate the damping and added mass coefficients would lead to improved wave energy capture, but this would require continuous frequency measurements of the incoming waves. Using a single frequency value at which most energy exists for a specific wave climate will allow most of the wave energy for this specific climate to be efficiently converted to mechanical energy, but less energy will be converted overall compared to the use of a variable wave frequency. Simulation results of the W2W WEC system developed using the RdC method are given in Figure 8.

_{s}) 1.45 m and peak energy period (T

_{e}) 6 s. The wave power density for this spectrum in deep water can be calculated using Equation (8) in kW/m.

_{pto}closely follows the ${F}_{pto}^{RdC}$, but they are not identical. Some differences appear when the force is above 12 kN. This may be the result of a sudden wave elevation peak. Figure 8c shows the synchronization of the velocity of the point absorber and wave excitation force. The RdC method does not provide any phase control, and therefore the synchronization is based only on the point absorber’s natural frequency and the incoming wave frequencies. The point absorber’s natural frequency is given in Table 1 as 1.8083 rad/s, which is larger than the 1.0467 rad/s peak energy frequency of the sea state used. For the calculation of the ${F}_{pto}^{RdC}$ using Equation (7), ω

_{p}was equal to 1.0467 rad/s, which is the peak energy frequency of the sea state used. Figure 8d presents the power output at different stages of the system. The peak instantaneous electrical generator power output (P

_{gen}) is 19 kW, whereas the average values for the hydrodynamic power (${P}_{hydro}^{avg}$), generator power (${P}_{gen}^{avg}$) and grid power (${P}_{grid}^{avg}$) are 1.9147 kW, 1.6909 kW and 1.4701 kW, respectively. This leads to a PMLG efficiency (${\eta}_{PMLG}$) of 88.3% and electrical system efficiency (generator output to grid) of 86.9% (${\eta}_{Ele}$). If we assume that the power of the incoming waves is P

_{W}, then the hydrodynamic efficiency (${\eta}_{hydro}$) can also be calculated as being 10.1%. The W2W efficiency (${\eta}_{W2W}$) is 7.8%. The different efficiencies can be calculated using Equation (9).

#### Power Maximization Process for the Real Damping Controller

- The constant hydrodynamic parameters in the calculation of the reference PTO force may not be accurate.
- The objective is to increase the power exported to the grid and not necessarily the power captured by the waves. The behavior of the electrical generator can change the losses in the electrical system.
- The mechanical and electrical constrains have to be included in the generation of the reference PTO force.
- Each WEC is installed for a specific wave climate. The reference PTO force has to be modified to reflect power maximization for this specific wave climate.

#### 3.2. Reactive Spring Damping Controller

_{p}of the sea state and does not include an inertia term in the calculation of the reference PTO force ${F}_{pto}^{RsdC}$. The reference PTO force calculation for the RsdC is described in Equation (11).

_{pto}generated by the PMLG. The PMLG PTO force does not follow the reference signal in all the cases. Several factors can affect this performance including PMLG design parameters such as the phase inductance. An additional observation is that F

_{pto}has much higher values compared to Figure 8b. Regarding phase synchronization between the wave excitation force and the velocity of the point absorber, it is evident that the RsdC affects the ${v}_{pa}$ phase, especially when compared to Figure 8c. Apart from the phase control, the amplitude control of the RsdC is significant as well. The velocity of the point absorber reaches 0.6 m/s even at low excitation force amplitudes. Finally, the power at different stages can be seen in Figure 10d. The peak instantaneous electrical generator power output (P

_{gen}) is 23.8 kW, whereas the average values for the hydrodynamic power (${P}_{hydro}^{avg}$), generator power (${P}_{gen}^{avg}$) and grid power (${P}_{grid}^{avg}$) are 2.6549 kW, 2.2537 kW and 2.0247 kW, respectively. This leads to ${\eta}_{PMLG}$ = 84.9%, ${\eta}_{Ele}$ = 89.8%, ${\eta}_{hydro}$ = 14.1% and ${\eta}_{W2G}$ = 10.7%. The different efficiencies are calculated using Equation (9). In addition, reactive power flow can be seen in Figure 10d. The peak negative power at the generator terminals is around 2 kW, and the average negative power in a 150 s simulation is 680 W. By implementing the RsdC, total power to the grid is increased by 38% compared to RdC, despite the negative power consumed by the generator to perform phase control. In the following section, the RsdC is optimized in a similar way to the RdC in order to maximize the power exported to the grid.

#### Power Maximization Process for the Reactive Spring Damping Controller

- The constant hydrodynamic parameters in the calculation of the reference PTO force may not be accurate.
- The objective is to increase the power exported to the grid, and not necessarily the power captured by the waves. The behavior of the electrical generator can change the losses in the electrical system.
- The mechanical and electrical constraints have to be included in the generation of the reference PTO force.
- Each WEC is installed in a specific wave climate. The reference PTO force has to be modified to reflect power maximization for this specific wave climate.
- Reactive power control can bring the point absorber into phase with the wave excitation force for maximum power extraction from the waves, but may lead to excessive power being consumed by the WEC. The amount of reactive power has to be controlled through optimization.

#### 3.3. Velocity Controller

_{linear}approach uses the maximum velocity of the point absorber (1 m/s) for all the normalized excitation force input points. Therefore, at rated excitation force, the optimum velocity will be the maximum, and a linear approach is used for the rest of excitation force input points. The v

_{constant}approach for the Look-up table aims at keeping the velocity of the point absorber near the maximum velocity, despite the low excitation force input. Therefore, at low ${F}_{exc}^{N}$, the velocity magnitude is high, and at high ${F}_{exc}^{N}$, the velocity magnitude tends to 1 m/s. The W2W WEC with the v

_{linear}VelC is simulated under the same resource as the previous controllers, and the results are presented in Figure 14.

_{linear}VelC and v

_{constant}VelC, along with all the options presented in this paper, is given in Section 3.4.

#### Power Maximization Process for the Velocity Controller

- The objective is to increase the power exported to the grid, and not necessarily the power captured by the waves. The behavior of the electrical generator can change the losses in the electrical system.
- The mechanical and electrical constrains have to be included in the generation of the reference PTO force.
- Each WEC is installed in a specific wave climate. The Look-up table has to be modified to maximize power for the specific wave climate.
- Reactive power can bring the point absorber into phase with the wave excitation force for maximum power extraction from the waves, but may lead to excessive power consumption by the WEC. The amount of reactive power has to be controlled through optimization.

#### 3.4. Summary of Controller Options

_{grid}has the highest W2W efficiency with an average of 2.498 kW for a 150 s simulation. The VelC v

_{grid}achieved more than 5% extra efficiency, and an additional 1 kW power at the grid compared to the RdC Reference. High efficiencies were also achieved when the VelC v

_{constant}, RsdC Gradient Descent and RsdC Grid Search were used.

## 4. Test Cases in a Realistic Environment

#### 4.1. Dominant Operation Sea State

_{grid}controllers.

_{zmax}, and F

_{ES}was zero in all cases. On the other hand, the power limitation system was enabled in the VelC v

_{grid}only at 630 s, 733 s, 998 s and 1371 s, in order to keep the instantaneous power output below 30 kW. Despite the power limitation system being enabled, the VelC v

_{grid}achieved the lowest ${P}_{gen}^{ratio}$ and the highest W2G efficiency.

#### 4.2. High-Energy Sea State

_{zmax}. A summary of the results of the simulations presented in Figure 18 is given in Table 10. The peak-to-average ratio is kept a lot lower, almost half, compared to the averaged conditions tested in Section 4.1, and the efficiencies of all three controllers are significantly higher. Comparing the different controllers, it is observed that the trend is the same despite the fact that the controller settings are for averaged conditions. The VelC v

_{grid}controller has the highest W2G efficiency and the lowest peak-to-average power ratio. The RsdC Gradient Descent Controller achieves a good efficiency, 0.78% more compared to the RdC Gradient Descent, but has the highest peak-to-average power ratio.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Ruud, K.; Frank, N. Wave Energy Technology Brief; International Renewable Energy Agency (IRENA): Abu Dhabi, UAE, 2014. [Google Scholar]
- World Energy Council. World Energy Council World Energy Resources | 2016; World Energy Council: London, UK, 2016; Volume 24. [Google Scholar]
- Al Shami, E.; Zhang, R.; Wang, X. Point absorber wave energy harvesters: A review of recent developments. Energies
**2019**, 12, 47. [Google Scholar] [CrossRef] - Aderinto, T.; Li, H. Ocean Wave energy converters: Status and challenges. Energies
**2018**, 11, 1250. [Google Scholar] [CrossRef] - Mueller, M.A. Electrical generators for direct drive wave energy converters. IEE Proc. Gener. Transm. Distrib.
**2002**, 149, 446–456. [Google Scholar] [CrossRef] - Mendes, R.; Calado, M.; Mariano, S. Maximum Power Point Tracking for a Point Absorber Device with a Tubular Linear Switched Reluctance Generator. Energies
**2018**, 11, 2192. [Google Scholar] [CrossRef] - Penalba, M.; Ringwood, J.V. A high-fidelity wave-to-wire model for wave energy converters. Renew. Energy
**2019**, 134, 367–378. [Google Scholar] [CrossRef] - Balitsky, P.; Quartier, N.; Verao Fernandez, G.; Stratigaki, V.; Troch, P. Analyzing the Near-Field Effects and the Power Production of an Array of Heaving Cylindrical WECs and OSWECs Using a Coupled Hydrodynamic-PTO Model. Energies
**2018**, 11, 3489. [Google Scholar] [CrossRef] - Tedeschi, E.; Santos-Mugica, M. Modeling and Control of a Wave Energy Farm Including Energy Storage for Power Quality Enhancement: The Bimep Case Study. IEEE Trans. Power Syst.
**2014**, 29, 1489–1497. [Google Scholar] [CrossRef] - Tedeschi, E.; Molinas, M. Tunable Control Strategy for Wave Energy Converters With Limited Power Takeoff Rating. IEEE Trans. Ind. Electron.
**2012**, 59, 3838–3846. [Google Scholar] [CrossRef] - Sjolte, J.; Sandvik, C.M.; Tedeschi, E.; Molinas, M. Exploring the potential for increased production from the wave energy converter lifesaver by reactive control. Energies
**2013**, 6, 3706–3733. [Google Scholar] [CrossRef] - Ringwood, J.V.; Bacelli, G.; Fusco, F. Energy-Maximizing Control of Wave-Energy Converters: The Development of Control System Technology to Optimize Their Operation. IEEE Control Syst.
**2014**, 34, 30–55. [Google Scholar] - Li, B. Reaction Force Control Implementation of a Linear Generator in Irregular Waves for a Wave Power System. Ph.D. Thesis, The University of Edinburgh, Edinburgh, Scotland, 2012. [Google Scholar]
- Shek, J.K.H.; Macpherson, D.E.; Mueller, M.A.; Xiang, J. Reaction force control of a linear electrical generator for direct drive wave energy conversion. IET Renew. Power Gener.
**2007**, 1, 17–24. [Google Scholar] [CrossRef] [Green Version] - Garcia-Rosa, P.B.; Ringwood, J.V.; Fosso, O.; Molinas, M. The impact of time-frequency estimation methods on the performance of wave energy converters under passive and reactive control. IEEE Trans. Sustain. Energy
**2018**, PP, 1. [Google Scholar] [CrossRef] - Park, J.S.; Gu, B.-G.; Kim, J.R.; Cho, I.H.; Jeong, I.; Lee, J. Active Phase Control for Maximum Power Point Tracking of a Linear Wave Generator. IEEE Trans. Power Electron.
**2017**, 32, 7651–7662. [Google Scholar] [CrossRef] - Faedo, N.; Olaya, S.; Ringwood, J.V. Optimal control, MPC and MPC-like algorithms for wave energy systems: An overview. IFAC J. Syst. Control
**2017**, 1, 37–56. [Google Scholar] [CrossRef] [Green Version] - Fusco, F.; Ringwood, J.V. A Simple and Effective Real-Time Controller for Wave Energy Converters. IEEE Trans. Sustain. Energy
**2013**, 4, 21–30. [Google Scholar] [CrossRef] - Ammar, R.; Trabelsi, M.; Mimouni, M.F.; Ben Ahmed, H.; Benbouzid, M. Flux weakening control of PMSG based on direct wave energy converter systems. In Proceedings of the IEEE 2017 International Conference on Green Energy Conversion Systems (GECS), Hammamet, Tunisia, 23–25 March 2017; pp. 1–7. [Google Scholar]
- O’Sullivan, A.C.M.; Lightbody, G. Co-design of a wave energy converter using constrained predictive control. Renew. Energy
**2017**, 102, 142–156. [Google Scholar] [CrossRef] - Falcão, A.F.O.; Henriques, J.C.C. Effect of non-ideal power take-off efficiency on performance of single- and two-body reactively controlled wave energy converters. J. Ocean Eng. Mar. Energy
**2015**, 1, 273–286. [Google Scholar] [CrossRef] [Green Version] - Genest, R.; Bonnefoy, F.; Clément, A.H.; Babarit, A. Effect of non-ideal power take-off on the energy absorption of a reactively controlled one degree of freedom wave energy converter. Appl. Ocean Res.
**2014**, 48, 236–243. [Google Scholar] [CrossRef] [Green Version] - Hansen, R.H. Design and Control of the PowerTake-Off System for a Wave Energy Converter with Multiple Absorbers. Ph.D. Thesis, Aalborg University, Esbjerg Aalborg, Denmark, 2013. [Google Scholar]
- Wang, L.; Isberg, J.; Tedeschi, E. Review of control strategies for wave energy conversion systems and their validation: The wave-to-wire approach. Renew. Sustain. Energy Rev.
**2018**, 81, 366–379. [Google Scholar] [CrossRef] - Penalba, M.; Cortajarena, J.A.; Ringwood, J.V. Validating a wave-to-wire model for a wave energy converter-part II: The electrical system. Energies
**2017**, 10, 1002. [Google Scholar] [CrossRef] - Kovaltchouk, T.; Rongère, F.; Primot, M.; Aubry, J.; Ben Ahmed, H.; Multon, B. Model Predictive Control of a Direct Wave Energy Converter Constrained by the Electrical Chain using an Energetic Approach. In Proceedings of the European Wave and Tidal Energy Conference, Nantes, France, 6–11 September 2015. [Google Scholar]
- Xiao, X.; Huang, X.; Kang, Q. A Hill-Climbing-Method-Based Maximum-Power-Point-Tracking Strategy for Direct-Drive Wave Energy Converters. IEEE Trans. Ind. Electron.
**2016**, 63, 257–267. [Google Scholar] [CrossRef] - Sousounis, M.C.; Gan, L.K.; Kiprakis, A.E.; Shek, J.K.H. Direct drive wave energy array with offshore energy storage supplying off-grid residential load. IET Renew. Power Gener.
**2017**, 11, 1081–1088. [Google Scholar] [CrossRef] - Crozier, R. RenewNet Foundry 1.6. Available online: https://sourceforge.net/projects/rnfoundry/ (accessed on 18 March 2019).
- Polinder, H.; Slootweg, J.G.; Hoeijmakers, M.J.; Compter, J.C. Modelling of a linear PM machine including magnetic saturation and end effects: Maximum force to current ratio. IEEE Trans. Ind. Appl.
**2003**, 39, 1681–1688. [Google Scholar] [CrossRef] - Wu, B.; Lang, Y.; Zargari, N.; Kouro, S. Power Converters in Wind Energy Conversion Systems. In Power Conversion and Control of Wind Energy Systems; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2011; pp. 87–152. [Google Scholar]
- Wan, Y.; Fan, C.; Zhang, J.; Meng, J.; Dai, Y.; Li, L.; Sun, W.; Zhou, P.; Wang, J.; Zhang, X. Wave energy resource assessment off the coast of China around the Zhoushan Islands. Energies
**2017**, 10, 1320. [Google Scholar] [CrossRef]

**Figure 1.**The conversion of power at each stage for different concepts of PTO systems in point absorber WECs.

**Figure 2.**Wave-to-wire WEC model developed primarily in MATLAB/Simulink. The hydrodynamic simulations were performed in NEMOH. The model incorporates a DD All-Electric PTO system, system constraints, power transmission and grid connection.

**Figure 3.**Mesh design for the submerged part of a cylindrical buoy generated by EWST to perform NEMOH hydrodynamic simulation. (

**a**) NEMOH mesh of the cylindrical buoy. The center of gravity (CoG) is visible; (

**b**) Refined mesh of the cylindrical buoy with the normal pointing out. The center of buoyancy (CoB) is visible.

**Figure 4.**Hydrodynamic simulation outputs as a function of wave frequency for the cylindrical buoy of Figure 3: (

**a**) Added mass; (

**b**) Radiation damping.

**Figure 5.**(

**a**) Block diagram of the mechanical mass-spring-damper model of the single buoy point absorber described in (1) developed in MATLAB/Simulink; (

**b**) Diagram of the WEC and PTO configuration. All forces are acting on the heaving buoy (one degree of freedom).

**Figure 6.**Block diagram of the Voltage Oriented Controller as it was implemented in MATLAB/Simulink. The point of measurement for the electrical quantities is at the low voltage side of the power transmission transformer after the grid-side filter.

**Figure 8.**Simulation results using the RdC method: (

**a**) Wave resource used as input; (

**b**) Reference and actual PTO force; (

**c**) Phase comparison between the velocity and wave excitation force; (

**d**) Instantaneous generator power and average hydrodynamic, generator and grid power.

**Figure 9.**Grid search optimization of ${b}^{RdC}$ and ${k}_{c}^{RdC}$ to maximize (

**a**) hydrodynamic power ${P}_{hydro}^{avg}$; (

**b**) PMLG power ${P}_{gen}^{avg}$; and (

**c**) power exported to the grid ${P}_{grid}^{avg}$.

**Figure 10.**Simulation results using the RsdC method. (

**a**) Wave resource used as input; (

**b**) Reference and actual PTO force; (

**c**) Phase comparison between the velocity and wave excitation force; (

**d**) Instantaneous generator power and average hydrodynamic, generator and grid power.

**Figure 11.**Grid search optimization of ${b}^{RsdC}$ and ${k}_{c}^{RsdC}$ to maximize: (

**a**) Hydrodynamic power ${P}_{hydro}^{avg}$; (

**b**) PMLG power ${P}_{gen}^{avg}$; (

**c**) Power exported to the grid ${P}_{grid}^{avg}$; (

**d**) Power exported to the grid with overlapped peak power points of hydrodynamic and PMLG output.

**Figure 14.**Simulation results using the v

_{linear}VelC. (

**a**) Wave resource used as input; (

**b**) Reference and actual PTO force; (

**c**) Phase comparison between the velocity and wave excitation force; (

**d**) Instantaneous generator power and average hydrodynamic, generator and grid power.

**Figure 15.**Optimized Look-up table, v

_{grid}, of the VelC for maximum ${P}_{grid}^{avg}$ in a specific wave climate. Comparison of v

_{grid}with v

_{linear}and v

_{constant}.

**Figure 16.**Results from a 1500 s simulation. (

**a**) Wave resource input generated from JONSWAP spectrum with significant wave height 1.45 m and energy period 6 s; (

**b**) Instantaneous generator power and averaged grid power for the RdC Gradient Descent case; (

**c**) Instantaneous generator power and averaged grid power for the RsdC Gradient Descent case; (

**d**) Instantaneous generator power and averaged grid power for the VelC v

_{grid}case.

**Figure 17.**Spectral density of the wave resource used for the optimization of the controllers (blue line) and for the realistic simulation (red line).

**Figure 18.**Results from a 1500 s simulation for a high-energy sea state. (

**a**) Wave resource input generated from JONSWAP spectrum with significant wave height 1.95 m and energy period 7 s; (

**b**) Instantaneous generator power and averaged grid power for the RdC Gradient Descent case; (

**c**) Instantaneous generator power and averaged grid power for the RsdC Gradient Descent case; (

**d**) Instantaneous generator power and averaged grid power for the VelC v

_{grid}case.

Symbol | Quantity | Value | Units |
---|---|---|---|

M | Mass | 10,630 | kg |

A_{m} | Added mass at 5 rad/s | 2354 | kg |

b_{d} | Linear damping | 133 | Ns/m |

k_{c} | Stiffness | 34,760 | N/m |

w_{n} | Natural frequency | 1.8083 | rad/s |

pa_{r} | Buoy radius | 1.05 | m |

pa_{h} | Buoy height | 4 | m |

${F}_{rated}$ | Rated mechanical force | 30,000 | N |

b_{ES} | End-stop damping | 45 | Ns/m |

k_{ES} | End-stop stiffness | 11,650 | N/m |

pa_{zmax} | Translator maximum displacement | 1.5 | m |

Symbol | Quantity | Value | Units |
---|---|---|---|

P_{rated} | PMLG rated power | 30,000 | W |

N_{t} | Number of turns around a tooth | 250 | - |

τ_{p} | Pole pitch | 0.1 | m |

Φ | Flux in the tooth | 0.1073 | Wb |

Rs | Stator resistance per phase | 2.9667 | Ω |

Ls | Stator inductance per phase | 0.0789 | H |

Symbol | Quantity | Value | Units |
---|---|---|---|

f_{g} | Grid side frequency | 50 | Hz |

${\omega}_{g}$ | Grid side angular frequency | 314 | rad/s |

V_{dc}* | DC link reference voltage | 800 | V |

Q_{g}* | Reactive power reference | 0 | var |

${L}_{g}$ | Line inductance | 0.0019 | H |

R_{on} | IGBT on-state resistance | 0.01493 | Ω |

V_{ce} | IGBT collector emitter saturation voltage | 1.9 | V |

Symbol | Quantity | Value | Units |
---|---|---|---|

P_{gt} | Transformer nominal power | 40,000 | W |

V_{ygt} | Wye line RMS nominal voltage | 400 | V |

V_{Dgt} | Delta line RMS nominal voltage | 11,000 | V |

R_{ygt} | Wye resistance | 0.0058 | pu |

L_{ygt} | Wye inductance | 0.0612 | pu |

R_{Dgt} | Delta resistance | 0.0038 | pu |

L_{Dgt} | Delta inductance | 0.0151 | pu |

l_{C} | Cable length | 3 | km |

R_{C} | Cable resistance per km | 0.223 | Ω/km |

L_{C} | Cable inductance per km | 0.43 | mH/km |

C_{C} | Cable capacitance per km | 0.24 | μF/km |

Method | ${\mathit{b}}^{\mathit{R}\mathit{d}\mathit{C}}(\mathbf{Ns}/\mathbf{m})$ | ${\mathit{k}}_{\mathit{c}}^{\mathit{R}\mathit{d}\mathit{C}}(\mathbf{N}/\mathbf{m})$ | ${\mathit{P}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}^{\mathit{a}\mathit{v}\mathit{g}}\left(\mathbf{kW}\right)$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{a}\mathit{v}\mathit{g}}\left(\mathbf{kW}\right)$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}}$ | ${\mathit{P}}_{\mathit{g}\mathit{r}\mathit{i}\mathit{d}}^{\mathit{a}\mathit{v}\mathit{g}}\left(\mathbf{kW}\right)$ | ${\mathit{\eta}}_{\mathit{W}2\mathit{W}}(\%)$ |
---|---|---|---|---|---|---|---|

Reference | 274 | 34,756 | 1.915 | 1.691 | 11.05 | 1.470 | 7.77 |

Grid Search | 0 | 14,050 | 2.007 | 1.747 | 11.78 | 1.565 | 8.26 |

Gradient Descent | 0.07 | 14,144 | 2.009 | 1.748 | 11.76 | 1.566 | 8.28 |

Method | ${\mathit{b}}^{\mathit{R}\mathit{s}\mathit{d}\mathit{C}}(\mathbf{Ns}/\mathbf{m})$ | ${\mathit{k}}_{\mathit{c}}^{\mathit{R}\mathit{s}\mathit{d}\mathit{C}}(\mathbf{N}/\mathbf{m})$ | ${\mathit{P}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}^{\mathit{a}\mathit{v}\mathit{g}}\left(\mathbf{kW}\right)$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}}$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{a}\mathit{v}\mathit{g}}\left(\mathbf{kW}\right)$ | ${\mathit{P}}_{\mathit{g}\mathit{r}\mathit{i}\mathit{d}}^{\mathit{a}\mathit{v}\mathit{g}}\left(\mathbf{kW}\right)$ | ${\mathit{\eta}}_{\mathit{W}2\mathit{G}}(\%)$ |
---|---|---|---|---|---|---|---|

Reference | 274 | 34,756 | 2.655 | 10.52 | 2.254 | 2.025 | 10.7 |

Grid Search ${P}_{hydro}^{avg}$ | −20,000 | 48,000 | 3.088 | 11.58 | 2.444 | 2.235 | 11.8 |

Grid Search ${P}_{grid}^{avg}$ | −3000 | 51,000 | 3.043 | 10.78 | 2.499 | 2.272 | 12.0 |

Gradient Descent | −5742 | 49,452 | 3.049 | 10.84 | 2.501 | 2.277 | 12.1 |

Controller | Variation | ${\mathit{P}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}}$ | ${\mathit{P}}_{\mathit{g}\mathit{r}\mathit{i}\mathit{d}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{\eta}}_{\mathit{W}2\mathit{W}}(\%)$ |
---|---|---|---|---|---|---|

VelC | v_{linear} | 2.314 | 2.038 | 10.21 | 1.829 | 9.67 |

VelC | v_{constant} | 3.214 | 2.623 | 9.56 | 2.460 | 12.99 |

VelC | v_{grid} | 3.353 | 2.647 | 10.22 | 2.489 | 13.15 |

Controller | Variation | ${\mathit{P}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}^{\mathit{a}\mathit{v}\mathit{g}}$ (kW) | ${\mathit{\eta}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}$ (%) | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{a}\mathit{v}\mathit{g}}$ (kW) | ${\mathit{P}}_{\mathit{g}\mathit{r}\mathit{i}\mathit{d}}^{\mathit{a}\mathit{v}\mathit{g}}$ (kW) | ${\mathit{\eta}}_{\mathit{W}\mathbf{2}\mathit{W}}$ (%) | ${\mathit{\eta}}_{\mathit{W}\mathbf{2}\mathit{W}}^{\mathit{D}\mathit{i}\mathit{f}\mathit{f}}$ (%) |
---|---|---|---|---|---|---|---|

RdC | Reference | 1.915 | 10.10 | 1.691 | 1.470 | 7.77 | 0 |

RdC | Grid Search | 2.007 | 10.53 | 1.747 | 1.565 | 8.26 | +0.49 |

RdC | Gradient Descent | 2.009 | 10.61 | 1.748 | 1.566 | 8.28 | +0.51 |

RsdC | Reference | 2.655 | 14.03 | 2.254 | 2.025 | 10.70 | +2.93 |

RsdC | Grid Search | 3.043 | 16.08 | 2.499 | 2.272 | 12.01 | +4.24 |

RsdC | Gradient Descent | 3.049 | 16.11 | 2.501 | 2.277 | 12.03 | +4.26 |

VelC | v_{linear} | 2.314 | 12.23 | 2.038 | 1.829 | 9.67 | +1.99 |

VelC | v_{constant} | 3.214 | 16.98 | 2.623 | 2.460 | 12.99 | +5.22 |

VelC | v_{grid} | 3.353 | 17.72 | 2.647 | 2.489 | 13.15 | +5.38 |

Controller | Variation | ${\mathit{P}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{\eta}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}(\%)$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}}$ | ${\mathit{P}}_{\mathit{g}\mathit{r}\mathit{i}\mathit{d}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{\eta}}_{\mathit{W}2\mathit{G}}(\%)$ | ${\mathit{\eta}}_{\mathit{W}2\mathit{G}}^{\mathit{D}\mathit{i}\mathit{f}\mathit{f}}(\%)$ |
---|---|---|---|---|---|---|---|---|

RdC | Gradient Descent | 1.954 | 10.32 | 1.708 | 13.05 | 1.514 | 8.01 | 0 |

RsdC | Gradient Descent | 2.758 | 14.58 | 2.244 | 13.26 | 2.005 | 10.59 | +2.58 |

VelC | v_{grid} | 2.999 | 15.85 | 2.415 | 12.82 | 2.234 | 11.81 | +3.80 |

**Table 10.**Summarized results from the 1500 s high-energy low occurrence wave simulation for RdC, RsdC and VelC.

Controller | Variation | ${\mathit{P}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{\eta}}_{\mathit{h}\mathit{y}\mathit{d}\mathit{r}\mathit{o}}(\%)$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{P}}_{\mathit{g}\mathit{e}\mathit{n}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}}$ | ${\mathit{P}}_{\mathit{g}\mathit{r}\mathit{i}\mathit{d}}^{\mathit{a}\mathit{v}\mathit{g}}(\mathbf{kW})$ | ${\mathit{\eta}}_{\mathit{W}2\mathit{G}}(\%)$ | ${\mathit{\eta}}_{\mathit{W}2\mathit{G}}^{\mathit{D}\mathit{i}\mathit{f}\mathit{f}}(\%)$ |
---|---|---|---|---|---|---|---|---|

RdC | Gradient Descent | 8.714 | 22.99 | 7.189 | 5.59 | 7.011 | 18.49 | 0 |

RsdC | Gradient Descent | 9.969 | 26.30 | 7.533 | 5.61 | 7.305 | 19.27 | +0.78 |

VelC | v_{grid} | 10.114 | 26.68 | 7.739 | 5.48 | 7.559 | 19.94 | +1.45 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sousounis, M.C.; Shek, J.
Wave-to-Wire Power Maximization Control for All-Electric Wave Energy Converters with Non-Ideal Power Take-Off. *Energies* **2019**, *12*, 2948.
https://doi.org/10.3390/en12152948

**AMA Style**

Sousounis MC, Shek J.
Wave-to-Wire Power Maximization Control for All-Electric Wave Energy Converters with Non-Ideal Power Take-Off. *Energies*. 2019; 12(15):2948.
https://doi.org/10.3390/en12152948

**Chicago/Turabian Style**

Sousounis, Marios Charilaos, and Jonathan Shek.
2019. "Wave-to-Wire Power Maximization Control for All-Electric Wave Energy Converters with Non-Ideal Power Take-Off" *Energies* 12, no. 15: 2948.
https://doi.org/10.3390/en12152948