# Wave Energy Converter Annual Energy Production Uncertainty Using Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Background

- The WEC deployment data-set consists of data collected during a WEC deployment. It is used to calculate the $\overline{L}$ matrix.
- The historic met-ocean data-set is a long-term (10+ year) parametric wave data-set representative of the WEC deployment location. It is used to calculate the $\overline{J}$ and $\overline{F}$ matrices.

## 3. Literature Review

- Measurement uncertainty derives from the inherent limitations of the wave measurement instrument (WMI).
- Temporal extrapolation uncertainty derives from using a limited duration data-set to estimate conditions over a much longer period.
- Spatial extrapolation uncertainty derives from any methods used to transfer wave data to the location of the WEC (such as a wave propagation model).
- Device performance uncertainty derives from the limitations and assumptions used in the performance characterization methodology.

- that the individual errors are uncorrelated and can be approximated as normal distributions.
- that the central limit theorem is applicable to multiplication of uncertain variables as well as addition (which is only valid for small uncertainty levels).

## 4. Methods

#### 4.1. Site Selection

#### 4.2. Wave Data Sources

#### 4.3. Wave Data Analysis

#### 4.4. WEC Simulations

#### 4.4.1. Self-Reacting Point Absorber

#### 4.4.2. Oscillating Wave Surge Converter

#### 4.5. Monte Carlo Simulations & Uncertainty Analysis

#### 4.5.1. Climate Uncertainty in the Historic Met-Ocean Data

#### 4.5.2. Sampling Uncertainty in the Historic Met-Ocean Data

#### 4.5.3. Modelling Uncertainty in the Historic Met-Ocean Data

#### 4.5.4. Climate Uncertainty in the WEC Deployment Performance Data

#### 4.5.5. Sampling Uncertainty in the WEC Deployment Performance

#### 4.5.6. Modelling Uncertainty in the WEC Deployment Performance Data

#### 4.5.7. Monte Carlo Simulation Procedure

## 5. Wave Data Analysis Results

#### 5.1. Wave Matrices

#### 5.2. Average Wave Spectrum

## 6. WEC Simulation Results

#### 6.1. SRPA Performance Matrix

#### 6.2. OWSC Performance Matrix

#### 6.3. Comparison of Performance Between WECs

#### 6.4. Comparison of Performance Locations

## 7. MAEP Uncertainty Assessment Results

#### 7.1. Distribution of MAEP

#### 7.2. Statistics of MAEP

#### 7.3. Assessment of Relative Contributions to MAEP Uncertainty

#### 7.4. Sensitivity to Data Length

#### 7.4.1. The SRPA

#### 7.4.2. The OWSC

#### 7.4.3. Discussion

## 8. Conclusions

- Wave occurrences at the UK Atlantic site are spread over a wide range of ${H}_{m0}$ and ${T}_{e}$ and the energy in the average wave spectrum is concentrated in swell frequencies with a narrow directional window.
- Wave occurrences at the North Sea site are concentrated at the lower ranges of ${H}_{m0}$ and ${T}_{e}$ and the energy in the average wave spectrum is spread through a wide swath of frequencies and directions.
- For the same sea-state, both WECs tends to operate with a higher capture length at the North Sea site compared to the Atlantic site.
- For the same sea-state, the OWSC tends to have a larger variability of capture length compared to the SRPA.
- For the same WEC, the capture length matrix for the simulated deployments is substantially different between the Atlantic and North Sea sites.

- Under most conditions, the MAEP populations are reasonably approximated by a normal distribution.
- MAEP estimates tend to under-predict MAEP due to missing data.
- Climate variability contributes most of the uncertainty to MAEP estimates.
- The MAEP of the SPRA is most sensitive to climate variability in the historic met-ocean data-set.
- The MAEP of the OWSC is most sensitive to climate variability in the WEC deployment data-set.
- The uncertainty in MAEP estimates vary significantly with the length of the historic met-ocean and WEC deployment data-sets used in the calculation.
- The uncertainty in MAEP estimates vary considerably between WECs and locations: variance in MAEP is higher for the OSWC and for the North Sea location.
- If a certain maximum level of uncertainty in MAEP is targeted, the minimum required lengths of the historic met-ocean and WEC deployment data-sets will be different for every WEC-location combination.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

${H}_{m0}$ | Significant wave height (m) |

${T}_{e}$ | Wave energy period (sec) |

P | Average WEC power output (kW) |

J | Wave power transport (kW/m) |

L | Capture length (m) |

g | Acceleration due to gravity (m/s^{2}) |

${C}_{g}$ | Wave group velocity (m/s) |

i | Linear index of a matrix bin |

$\overline{\xb7}$ | Overbar indicates that the enclosed parameters is a matrix of bin-averaged values |

S | The variance density (wave Spectrum) (m^{2}/Hz/rad) |

ω | Circular wave frequency (rad/s) |

θ | Wave direction (°) |

MAEP | Estimate of WEC mean annual energy production (MWh) |

u, v, w | Water particle velocity in the x, y and z directions (m/sec) |

${a}_{j,k}$ | Sinusoidal component amplitude (m) |

j | Index to bin location on the frequency axis of the wave spectrum |

k | Index to bin location on the direction axis of the wave spectrum |

${\sigma}_{MAEP}$ | Standard deviation of mean annual energy production (MWhr or %) |

μ | Mean estimate of mean annual energy production (MWhr) |

${P}_{x\%}$ | The x percentile of MEAP (MWhr) |

## Appendix A. SRPA Simulations

## Appendix B. OWSC Simulations

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^{1.}The approach may be extended into an arbitrary number of dimensions. In that case, the methodology is based on an “N-variate histogram”.^{2.}Data files for each simulated deployment are available for public download at www.cascadiacoast.com/projects under the Open Database License.

**Figure 4.**Wave matrices for UK offshore sites. Text indicates mean and standard deviation of wave power (${J}_{i}$, ${\sigma}_{J,i}$, in kW/m) and frequency of occurrence (${F}_{i}$ as a percentage). Color indicates average wave energy in MWhr/yr/m.

**Figure 5.**Average wave spectra at the UK off-shore sites. The polar angle axis indicates the direction of the spectral bin in nautical convention; the radial axis indicates the frequency of the spectral bin in Hz.

**Figure 6.**WEC performance matrices. Text indicates mean and standard deviation of capture length (${L}_{i}$, ${\sigma}_{L,i}$, in m) and frequency of occurrence (${F}_{i}$ as a percentage). Colour indicates average annual energy absorption in MWhr/yr.

**Figure 7.**Contours of ${\sigma}_{MAEP}$ as a percentage of the ‘true’ MAEP as a function of historic met-ocean and WEC deployment data length.

**Table 1.**Categorization of uncertainty in mean annual energy production estimates for wave energy converters [4].

Measurement | Temporal Extrapolation | Spatial Extrapolation | Device Performance |
---|---|---|---|

Instrument accuracy * | Historic resource estimation * | Model inputs * | Availability |

Measurement interference | Future resource variability * | Model error * | Array interactions |

Short-term data synthesis | Climate change | Perf. characterization * | |

Data quality & metadata | Electrical losses | ||

Perf. degradation | |||

Curtailment |

Location | Site | Lat | Lon | Depth | Ref. |
---|---|---|---|---|---|

Can. Pacific | Off-shore | 48.9° | −125.6° | 45 m | [11] |

Can. Pacific | Near-shore | 49.0° | −125.6° | 23 m | [11] |

Can. Atlantic | Off-shore | 46.5° | −55.6° | 142 m | [12] |

Can. Atlantic | Near-shore | 46.9° | −55.7° | 23 m | [13] |

UK Atlantic | Off-shore | 57.6° | −8.0° | 65 m | [14] |

UK Atlantic | Near-shore | 57.6° | −7.8° | 28 m | [14] |

UK North Sea | Off-shore | 57.4° | −1.5° | 83 m | [14] |

UK North Sea | Near-shore | 57.4° | −1.9° | 28 m | [14] |

Parameter | Value | Units |
---|---|---|

Draft | 35 | m |

Float Displacement | 201 | tonnes |

Reacting Body Displacement | 1644 | tonnes |

Float outer diameter | 14.75 | m |

Parameter | Value | Units |
---|---|---|

Flap width | 18 | m |

Flap thickness | 1.8 | m |

Flap rotational stiffness (about y) | 6.4 × 10^{6} | Nm/rad |

Flap mass moment of inertia (about y) | 2.05 × 10^{7} | kg · m^{2} |

**Table 5.**Results from the MCS for historic met-ocean data length of 10 years and WEC deployment length of 12 months. MAEP results in units of MW-hr.

Loc. | WEC | ‘True’ MAEP | MCS MAEP Estimates | ||||
---|---|---|---|---|---|---|---|

μ | σ | P_{50%} | P_{05%} | P_{95%} | |||

Atl | SRPA | 582 | 541 | 10 | 540 | 525 | 556 |

NS | SRPA | 311 | 286 | 10 | 286 | 270 | 303 |

Atl | OWSC | 992 | 937 | 32 | 938 | 884 | 990 |

NS | OWSC | 286 | 254 | 21 | 254 | 219 | 289 |

**Table 6.**Standard deviation of MAEP (${\sigma}_{MAEP}$) as a percentage of ‘true’ MAEP for a deployment length of 12 months and a met-ocean data length of 10 years. First 6 rows are evaluated based on a MCS with only a single uncertainty source enabled. The seventh row is evaluated based on a MCS with all six sources of uncertainty enabled simultaneously.

Data-Set | Uncertainty | SRPA | OWSC | SRPA | OWSC |
---|---|---|---|---|---|

Atl | Atl | NS | NS | ||

Met-ocean | Climate | 1.75 | 1.80 | 3.32 | 5.06 |

Met-ocean | Sample | 0.69 | 0.07 | 0.06 | 0.06 |

Met-ocean | Model | 0.23 | 0.24 | 0.25 | 0.31 |

Deployment | Climate | 0.58 | 2.63 | 1.29 | 6.14 |

Deployment | Sample | 0.20 | 0.58 | 0.27 | 1.15 |

Deployment | Model | 0.28 | 0.51 | 0.43 | 0.92 |

Both | All | 1.59 | 3.23 | 3.17 | 7.44 |

© 2016 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 license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hiles, C.E.; Beatty, S.J.; De Andres, A.
Wave Energy Converter Annual Energy Production Uncertainty Using Simulations. *J. Mar. Sci. Eng.* **2016**, *4*, 53.
https://doi.org/10.3390/jmse4030053

**AMA Style**

Hiles CE, Beatty SJ, De Andres A.
Wave Energy Converter Annual Energy Production Uncertainty Using Simulations. *Journal of Marine Science and Engineering*. 2016; 4(3):53.
https://doi.org/10.3390/jmse4030053

**Chicago/Turabian Style**

Hiles, Clayton E., Scott J. Beatty, and Adrian De Andres.
2016. "Wave Energy Converter Annual Energy Production Uncertainty Using Simulations" *Journal of Marine Science and Engineering* 4, no. 3: 53.
https://doi.org/10.3390/jmse4030053