# Design and Performance Evaluation of an Enclosed Inertial Wave Energy Converter with a Nonlinear Stiffness Mechanism

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## Abstract

**:**

## 1. Introduction

## 2. Schematic Design and Modeling

#### 2.1. Schematic Design

#### 2.2. Mathematical Modeling

**A**,

**B**, and

**C**are constant coefficient matrices of the state space model that can be determined using the realization theory approach.

_{j}is a point on the magnetic pole surface ${S}_{j}$ of the outer sector magnet ($j=3$,4). $\overrightarrow{{r}_{ij}}$ is the position vector between points ${P}_{i}$ and ${P}_{j}$, and ${z}_{ij}$ is the axial component of $\overrightarrow{{r}_{ij}}$. The superscripts (r) and (a) denote repulsive and attractive magnets, respectively. ${B}_{r}$ is the residual magnetic flux density. ${\mu}_{0}$ is the vacuum permeability.

#### 2.3. Software Application and Configuration

#### 2.4. Validation of the Time-Domain Model

## 3. Results and Discussion

#### 3.1. Mechanical Property Analysis of NSM

#### 3.1.1. Structural Parameters

#### 3.1.2. Magnetic Force and Stiffness of NSM

#### 3.2. Effect of Nonlinear Stiffness Mechanism (NSM)

#### 3.2.1. Motion Response Analysis

#### 3.2.2. Output Power Analysis

#### 3.3. Effect of Mass Body and Linear Spring

#### 3.4. Influence of Hydraulic PTO Parameters

#### 3.4.1. Effect of Pre-Charged Pressure of Accumulator

#### 3.4.2. Effect of the Initial Gas Volume of the Accumulator

#### 3.4.3. Effect of Diameter of Throttle Valve

#### 3.4.4. Effect of Displacement of the Hydraulic Motor and Load Resistance

## 4. Conclusions

- (1)
- Compared to linear EIWEC, the introduction of NSM increased the motion response of nonlinear EIWEC. The large amplitude relative motion provided favorable conditions for energy conversion.
- (2)
- The nonlinear negative stiffness property of the NSM reduced the intrinsic frequency and broadened the frequency bandwidth of the EIWEC. The effective frequency band shifted to a lower frequency range, and the output power of the nonlinear EIWEC was considerably enhanced.
- (3)
- Within the typical wave frequency range, the output power of the nonlinear EIWEC is insensitive to changes in the wave frequencies and linear spring stiffness. This not only increases the robustness of the system but also reduces the design difficulty of the linear spring.
- (4)
- Increasing the mass of the internal mass body and decreasing the stiffness of the linear spring will have a positive effect on enhancing the power performance of the system.
- (5)
- Choosing an accumulator with lower pre-charged pressure and larger gas initial volume can fully absorb the pressure and flow pulsations in the hydraulic PTO system, which is favorable to the smooth power output.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Coyle, E.D.; Simmons, R.A. Understanding the Global Energy Crisis; Purdue University Press: West Lafayette, IN, USA, 2014; ISBN 1557536619. [Google Scholar]
- Bodansky, D. The United Nations framework convention on climate change: A commentary. Yale J. Int’l L.
**1993**, 18, 451. [Google Scholar] - Schleussner, C.; Rogelj, J.; Schaeffer, M.; Lissner, T.; Licker, R.; Fischer, E.M.; Knutti, R.; Levermann, A.; Frieler, K.; Hare, W. Science and policy characteristics of the Paris Agreement temperature goal. Nat. Clim. Chang.
**2016**, 6, 827–835. [Google Scholar] [CrossRef] - Davis, M.; Moronkeji, A.; Ahiduzzaman, M.; Kumar, A. Assessment of renewable energy transition pathways for a fossil fuel-dependent electricity-producing jurisdiction. Energy Sustain. Dev.
**2020**, 59, 243–261. [Google Scholar] [CrossRef] - Vidal-Amaro, J.J.; Østergaard, P.A.; Sheinbaum-Pardo, C. Optimal energy mix for transitioning from fossil fuels to renewable energy sources—The case of the Mexican electricity system. Appl. Energy
**2015**, 150, 80–96. [Google Scholar] [CrossRef] - Halkos, G.E.; Gkampoura, E. Reviewing usage, potentials, and limitations of renewable energy sources. Energies
**2020**, 13, 2906. [Google Scholar] [CrossRef] - Guo, B.; Wang, T.; Jin, S.; Duan, S.; Yang, K.; Zhao, Y. A review of point absorber wave energy converters. J. Mar. Sci. Eng.
**2022**, 10, 1534. [Google Scholar] [CrossRef] - Yue, W.; Wang, Z.; Ding, W.; Sheng, S.; Zhang, Y.; Huang, Z.; Wang, W. Feasibility of Co-locating wave energy converters with offshore aquaculture: The Pioneering case study of China’s Penghu platform. Ocean Eng.
**2023**, 288, 116039. [Google Scholar] [CrossRef] - Clemente, D.; Rosa-Santos, P.; Ferradosa, T.; Taveira-Pinto, F. Wave energy conversion energizing offshore aquaculture: Prospects along the Portuguese coastline. Renew. Energy
**2023**, 204, 347–358. [Google Scholar] [CrossRef] - Clemente, D.; Rosa-Santos, P.; Taveira-Pinto, F. On the potential synergies and applications of wave energy converters: A review. Renew. Sustain. Energy Rev.
**2021**, 135, 110162. [Google Scholar] [CrossRef] - Gallutia, D.; Fard, M.T.; Soto, M.G.; He, J. Recent advances in wave energy conversion systems: From wave theory to devices and control strategies. Ocean Eng.
**2022**, 252, 111105. [Google Scholar] [CrossRef] - Tiron, R.; Mallon, F.; Dias, F.; Reynaud, E.G. The challenging life of wave energy devices at sea: A few points to consider. Renew. Sustain. Energy Rev.
**2015**, 43, 1263–1272. [Google Scholar] [CrossRef] - Xue, G.; Qin, J.; Zhang, Z.; Huang, S.; Liu, Y. Experimental Investigation of Mooring Performance and Energy-Harvesting Performance of Eccentric Rotor Wave Energy Converter. J. Mar. Sci. Eng.
**2022**, 10, 1774. [Google Scholar] [CrossRef] - Cordonnier, J.; Gorintin, F.; De Cagny, A.; Clément, A.H.; Babarit, A. SEAREV: Case study of the development of a wave energy converter. Renew. Energy
**2015**, 80, 40–52. [Google Scholar] [CrossRef] - Pozzi, N.; Bracco, G.; Passione, B.; Sirigu, S.A.; Mattiazzo, G. PeWEC: Experimental validation of wave to PTO numerical model. Ocean Eng.
**2018**, 167, 114–129. [Google Scholar] [CrossRef] - Khedkar, K.; Nangia, N.; Thirumalaisamy, R.; Bhalla, A.P.S. The inertial sea wave energy converter (ISWEC) technology: Device-physics, multiphase modeling and simulations. Ocean Eng.
**2021**, 229, 108879. [Google Scholar] [CrossRef] - The Penguin Wave Energy Converter. Available online: https://wello.eu/the-penguin-2/ (accessed on 9 January 2024).
- Clemente, D.; Rosa-Santos, P.; Taveira-Pinto, F.; Martins, P. Influence of platform design and power take-off characteristics on the performance of the E-Motions wave energy converter. Energy Conv. Manag.
**2021**, 244, 114481. [Google Scholar] [CrossRef] - Scarcity of Resources, Climate Change. Available online: https://seaturns.com/ (accessed on 9 January 2024).
- Crowley, S.; Porter, R.; Taunton, D.J.; Wilson, P.A. Modelling of the WITT wave energy converter. Renew. Energy
**2018**, 115, 159–174. [Google Scholar] [CrossRef] - Maheen, M.H.; Yang, Y. Wave energy converters with rigid hull encapsulation: A review. Sustain. Energy Technol. Assess.
**2023**, 57, 103273. [Google Scholar] - Clemente, D.; Rosa-Santos, P.; Taveira-Pinto, F.; Martins, P. Experimental performance assessment of geometric hull designs for the E-Motions wave energy converter. Ocean Eng.
**2022**, 260, 111962. [Google Scholar] [CrossRef] - Ding, W.; Wang, K.; Mao, Z.; Cao, H. Layout optimization of an inertial energy harvester for miniature underwater mooring platforms. Mar. Struct.
**2020**, 69, 102681. [Google Scholar] [CrossRef] - Chen, Z.; Zhang, L.; Yeung, R.W. Analysis and optimization of a Dual Mass-Spring-Damper (DMSD) wave-energy convertor with variable resonance capability. Renew. Energy
**2019**, 131, 1060–1072. [Google Scholar] [CrossRef] - Falnes, J.; Kurniawan, A. Ocean Waves and Oscillating Systems: Linear Interactions Including Wave-Energy Extraction; Cambridge University Press: Cambridge, UK, 2020; ISBN 1108481663. [Google Scholar]
- Khasawneh, M.A.; Daqaq, M.F. Response behavior of bi-stable point wave energy absorbers under harmonic wave excitations. Nonlinear Dyn.
**2022**, 109, 371–391. [Google Scholar] [CrossRef] - Khasawneh, M.A.; Daqaq, M.F. Experimental assessment of the performance of a bi-stable point wave energy absorber under harmonic incident waves. Ocean Eng.
**2023**, 280, 114494. [Google Scholar] [CrossRef] - Qin, J.; Zhang, Z.; Zhang, Y.; Huang, S.; Liu, Y.; Xue, G. Design and performance evaluation of novel magnetic tristable wave energy converter. Ocean Eng.
**2023**, 285, 115424. [Google Scholar] [CrossRef] - Kurniawan, A.; Greaves, D.; Chaplin, J. Wave energy devices with compressible volumes. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2014**, 470, 20140559. [Google Scholar] [CrossRef] [PubMed] - Pecher, A.; Kofoed, J.P. Handbook of Ocean Wave Energy; Springer Nature: Berlin, Germany, 2017. [Google Scholar]
- Bracco, G.; Canale, M.; Cerone, V. Optimizing energy production of an inertial sea wave energy converter via model predictive control. Control Eng. Pract.
**2020**, 96, 104299. [Google Scholar] [CrossRef] - Vissio, G.; Valério, D.; Bracco, G.; Beirão, P.; Pozzi, N.; Mattiazzo, G. ISWEC linear quadratic regulator oscillating control. Renew. Energy
**2017**, 103, 372–382. [Google Scholar] [CrossRef] - Salcedo, F.; Ruiz-Minguela, P.; Rodriguez, R.; Ricci, P.; Santos, M. Oceantec: Sea trials of a quarter scale prototype. In Proceedings of the 8th European Wave and Tidal Energy Conference, Uppsala, Sweden, 7–10 September 2009; pp. 460–465. [Google Scholar]
- Sirigu, S.A.; Bracco, G.; Bonfanti, M.; Dafnakis, P.; Mattiazzo, G. On-board sea state estimation method validation based on measured floater motion. IFAC-PapersOnLine
**2018**, 51, 68–73. [Google Scholar] [CrossRef] - Scapolan, M.; Tehrani, M.G.; Bonisoli, E. Energy harvesting using parametric resonant system due to time-varying damping. Mech. Syst. Signal Proc.
**2016**, 79, 149–165. [Google Scholar] [CrossRef] - Giorgi, G.; Faedo, N. Performance enhancement of a vibration energy harvester via harmonic time-varying damping: A pseudospectral-based approach. Mech. Syst. Signal Proc.
**2022**, 165, 108331. [Google Scholar] [CrossRef] - Yurchenko, D.; Alevras, P. Dynamics of the N-pendulum and its application to a wave energy converter concept. Int. J. Dyn. Control
**2013**, 1, 290–299. [Google Scholar] [CrossRef] - Giorgi, G. Embedding parametric resonance in a 2:1 wave energy converter to get a broader bandwidth. Renew. Energy
**2023**, 222, 119928. [Google Scholar] [CrossRef] - Schubert, B.W.; Sergiienko, N.Y.; Cazzolato, B.S.; Robertson, W.S.; Ghayesh, M.H. The true potential of nonlinear stiffness for point absorbing wave energy converters. Ocean Eng.
**2022**, 245, 110342. [Google Scholar] [CrossRef] - Guo, B.; Ringwood, J.V. Non-linear modeling of a vibro-impact wave energy converter. IEEE Trans. Sustain. Energy
**2020**, 12, 492–500. [Google Scholar] [CrossRef] - Zhang, H.; Xi, R.; Xu, D.; Wang, K.; Shi, Q.; Zhao, H.; Wu, B. Efficiency enhancement of a point wave energy converter with a magnetic bistable mechanism. Energy
**2019**, 181, 1152–1165. [Google Scholar] [CrossRef] - Zhang, X.; Tian, X.; Xiao, L.; Li, X.; Chen, L. Application of an adaptive bistable power capture mechanism to a point absorber wave energy converter. Appl. Energy
**2018**, 228, 450–467. [Google Scholar] [CrossRef] - Liu, B.; Yi, H.; Levi, C.; Estefen, S.F.; Wu, Z.; Duan, M. Improved bistable mechanism for wave energy harvesting. Ocean Eng.
**2021**, 232, 109139. [Google Scholar] [CrossRef] - Wang, L.; Tang, H.; Wu, Y. On a submerged wave energy converter with snap-through power take-off. Appl. Ocean Res.
**2018**, 80, 24–36. [Google Scholar] [CrossRef] - Todalshaug, J.H.; ásgeirsson, G.S.; Hjálmarsson, E.; Maillet, J.; Möller, P.; Pires, P.; Guérinel, M.; Lopes, M. Tank testing of an inherently phase-controlled wave energy converter. Int. J. Mar. Energy
**2016**, 15, 68–84. [Google Scholar] [CrossRef] - Corpower. Available online: https://www.corpowerocean.com/ (accessed on 9 January 2024).
- Li, M.; Jing, X. A bistable X-structured electromagnetic wave energy converter with a novel mechanical-motion-rectifier: Design, analysis, and experimental tests. Energy Conv. Manag.
**2021**, 244, 114466. [Google Scholar] [CrossRef] - Wang, Y.; Huang, S.; Xue, G.; Liu, Y. Influence of Hydraulic PTO Parameters on Power Capture and Motion Response of a Floating Wind-Wave Hybrid System. J. Mar. Sci. Eng.
**2022**, 10, 1660. [Google Scholar] [CrossRef] - Choi, K.; Yang, D.; Park, S.; Cho, B. Design and performance test of hydraulic PTO for wave energy converter. Int. J. Precis. Eng. Manuf.
**2012**, 13, 795–801. [Google Scholar] [CrossRef] - Gao, H.; Xiao, J. Effects of power take-off parameters and harvester shape on wave energy extraction and output of a hydraulic conversion system. Appl. Energy
**2021**, 299, 117278. [Google Scholar] [CrossRef] - Liu, C.; Hu, M.; Zhao, Z.; Zeng, Y.; Gao, W.; Chen, J.; Yan, H.; Zhang, J.; Yang, Q.; Bao, G. Latching control of a raft-type wave energy converter with a hydraulic power take-off system. Ocean Eng.
**2021**, 236, 109512. [Google Scholar] [CrossRef] - Xue, G.; Zhang, Z.; Qin, J.; Huang, S.; Liu, Y. Control Parameters Optimization of Accumulator in Hydraulic Power Take-Off System for Eccentric Rotating Wave Energy Converter. J. Mar. Sci. Eng.
**2023**, 11, 792. [Google Scholar] [CrossRef] - Zhang, N.; Zhang, X.; Xiao, L.; Wei, H.; Chen, W. Evaluation of long-term power capture performance of a bistable point absorber wave energy converter in South China Sea. Ocean Eng.
**2021**, 237, 109338. [Google Scholar] [CrossRef] - Newman, J.N. The exciting forces on fixed bodies in waves. J. Ship Res.
**1962**, 6, 10–17. [Google Scholar] [CrossRef] - Cummins, W.E. The Impulse Response Function and Ship Motions; David Taylor Model Basin: Washington, DC, USA, 1962. [Google Scholar]
- Ogden, D.; Ruehl, K.; Yu, Y.; Keester, A.; Forbush, D.; Leon, J.; Tom, N. Review of WEC-Sim development and applications. Int. Mar. Energy J.
**2022**, 5, 5–9. [Google Scholar] [CrossRef] - Tampier, G.; Grueter, L. Hydrodynamic analysis of a heaving wave energy converter. Int. J. Mar. Energy
**2017**, 19, 304–318. [Google Scholar] [CrossRef]

**Figure 1.**Nonlinear enclosed inertial wave energy converter (nonlinear EIWEC). (

**a**) Working scenario, (

**b**) internal structure.

**Figure 2.**Working schematic of nonlinear EIWEC. (

**a**) Hydraulic PTO unit, (

**b**) nonlinear stiffness mechanism (NSM).

**Figure 5.**Equivalent magnetic charge model for coaxial inner and outer sector magnets. (

**a**) Front view of the magnetic charge model, (

**b**) Top view of the magnetic charge model.

**Figure 7.**Comparison of RAO in heave obtained from the present time-domain model and simulation and experimental results reported by Tampier and Grueter (2017) [57].

**Figure 9.**RAOs of the buoy, mass body, and relative motion. (

**a**–

**c**) $\u2206l=1\mathrm{m};$ (

**d**–

**f**) $\u2206l=2\mathrm{m};$ (

**g**–

**i**) $\u2206l=3\mathrm{m}$. Simulation conditions: ${m}_{2}$ = 6670 kg, ${D}_{m}$ = 80 cc/rev, $R$ = 50 Ω, ${p}_{pre}$ = 30 bar, and ${V}_{g0}$ = 40 L.

**Figure 10.**Phase trajectories of the motion response of the buoy and the internal mass body for nonlinear EIWECs with different spring stiffnesses when the wave frequencies are 2.3 rad/s, 2.4 rad/s, and 2.5 rad/s, respectively. (

**a**,

**b**) $\u2206l=1\mathrm{m};$ (

**c**,

**d**) $\u2206l=2\mathrm{m}$.

**Figure 11.**Phase trajectories of the buoys of linear and nonlinear EIWEC for different stiffness conditions when the wave height is 1 m and the frequency is 1 rad/s. (

**a**) $\u2206l=1\mathrm{m}$; (

**b**) $\u2206l=2\mathrm{m}$; (

**c**) $\u2206l=3\mathrm{m}$.

**Figure 12.**Distribution of generation power in the ($\omega $, $\u2206l$) space. (

**a**,

**c**): Linear EIWEC; (

**b**,

**d**) nonlinear EIWEC. Simulation conditions: ${m}_{2}$ = 6670 kg, ${D}_{m}$= 80 cc/rev, $R$ = 50 Ω, ${p}_{pre}$ = 30 bar, and ${V}_{g0}$ = 40 L.

**Figure 13.**Output power of nonlinear EIWECs as a function of $\gamma $ and $\u2206l$ under typical wave period conditions. (

**a**) T = 5 s, (

**b**) T = 7.5 s, (

**c**) T = 10 s, (

**d**) T = 12.5 s, and (

**e**) T = 15 s.

**Figure 14.**Average output power ${P}_{m}$ and power standard deviation $\mathsf{\sigma}$ as a function of the pre-charged pressure of accumulator ${p}_{pre}$. (

**a**) Average output power, (

**b**) power standard deviation.

**Figure 15.**Time-varying curves corresponding to the pre-charged pressures of 20 bar and 60 bar when the wave height is 1 m and the period is 7 s. (

**a**) accumulator inlet pressure ${q}_{ac}$, (

**b**) instantaneous power ${P}_{i}$, and (

**c**) PTO force ${F}_{PTO}$.

**Figure 16.**Variation in the average output power and power standard deviation with the initial gas volume of accumulator ${V}_{g0}$. (

**a**) Average output power, (

**b**) power standard deviation.

**Figure 17.**Average output power of the nonlinear EIWEC as a function of throttle diameter ${D}_{v}$. Simulation conditions: ${m}_{2}$ = 6670 kg, ${D}_{m}$ = 80 cc/rev, $R$ = 50 Ω, ${p}_{pre}$ = 25 bar, and ${V}_{g0}$ = 40 L.

**Figure 18.**Variation in the output power of nonlinear EIWEC with wave frequency under different displacements of the hydraulic motor and load resistance conditions. (

**a**) 10 Ω, (

**b**) 30 Ω, (

**c**) 50 Ω, and (

**d**) 70 Ω. Simulation conditions: ${m}_{2}$ = 6670 kg, ${p}_{pre}$ = 25 bar, and ${V}_{g0}$ = 40 L.

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

Inner diameter of inner magnetic rings (m) | ${R}_{1}$ | 0.2 |

External diameter of inner magnetic rings (m) | ${R}_{2}$ | 0.5 |

Inner diameter of outer magnetic rings (m) | ${R}_{3}$ | 0.6 |

External diameter of outer magnetic rings (m) | ${R}_{4}$ | 1.0 |

Center angle of attractive magnet ($\xb0)$ | $\mathsf{\alpha}$ | 18 |

Center angle of repulsive magnet ($\xb0)$ | 90° $-\mathsf{\alpha}$ | 72 |

Height difference between attractive and repulsive magnets (m) | $\u2206h$ | 0.08 |

Height of inner magnetic ring (m) | ${h}_{1}$ | 0.15 |

Permeability of vacuum (H/m) | ${\mu}_{0}$ | $4\pi \times {10}^{-7}$ |

Residual magnetic flux density (T) | ${B}_{r}$ | 1.25 |

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## Share and Cite

**MDPI and ACS Style**

Qin, J.; Zhang, Z.; Song, X.; Huang, S.; Liu, Y.; Xue, G.
Design and Performance Evaluation of an Enclosed Inertial Wave Energy Converter with a Nonlinear Stiffness Mechanism. *J. Mar. Sci. Eng.* **2024**, *12*, 191.
https://doi.org/10.3390/jmse12010191

**AMA Style**

Qin J, Zhang Z, Song X, Huang S, Liu Y, Xue G.
Design and Performance Evaluation of an Enclosed Inertial Wave Energy Converter with a Nonlinear Stiffness Mechanism. *Journal of Marine Science and Engineering*. 2024; 12(1):191.
https://doi.org/10.3390/jmse12010191

**Chicago/Turabian Style**

Qin, Jian, Zhenquan Zhang, Xuening Song, Shuting Huang, Yanjun Liu, and Gang Xue.
2024. "Design and Performance Evaluation of an Enclosed Inertial Wave Energy Converter with a Nonlinear Stiffness Mechanism" *Journal of Marine Science and Engineering* 12, no. 1: 191.
https://doi.org/10.3390/jmse12010191