# Optimizing Wave Overtopping Energy Converters by ANN Modelling: Evaluating the Overtopping Rate Forecasting as the First Step

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{h}is the hydrodynamic power, and $f$ is a factor that comprises several efficiency related factors, as electrical, mechanical, or relative to the electrical energy transmission.

## 2. Materials and Methods

#### 2.1. The Data Base and Its Component Parameters

_{0}there were N

_{0}waves falling upon the structure, of heights and periods (H

_{i}, T

_{i}), where each wave produces a certain volume of overflow V

_{i}(H

_{i}, T

_{i}), the overtopping rate can be defined as:

^{3}/s/m or m

^{2}/s); ${N}_{0}$ is the number of total waves; ${H}_{i,},\text{}{T}_{i}$ are the height and period of every i wave that fall upon the structure (m; s); ${V}_{i}\left({H}_{i,},\text{}{T}_{i}\right)$ is the overflow volume produced by each wave of the wave series, per unit length; and ${t}_{0}={{\displaystyle \sum}}_{i=1}^{{N}_{0}}{T}_{i}$ is the duration of the record of waves in a storm (s).

_{c}). Other factors that also influence, although to a lesser extent, would be the bottom slope, the depth at the foot of the structure, the wind (direction and intensity), the wave grouping, the run-up interference, and so on. So, any database related to the quantification of that phenomenon will necessarily be enriched by data of this nature.

_{m}

_{0 deep}; H

_{m}

_{0}), the mean spectral wave period offshore and at the toe (T

_{m}

_{−1,0}

_{deep}; T

_{m−}

_{1,0}), the average and peak period offshore (T

_{m}

_{,deep}; T

_{p}

_{,deep}), the peak period at toe of the structure (T

_{p}

_{,0}), and the wave incidence angle (β). The structural parameters, considered to geometrically define the structure: the water depth offshore (h

_{deep}), the water depth in front of the structure (h), the water depth at the toe of the structure (h

_{t}), the width of the toe berm (B

_{t}), the width of the berm (B), the berm submergence (h

_{b}), the slope of the structure downward of the berm (cotα

_{d}), the slope of the structure upward of the berm (cotα

_{u}), the average co-tangent, considering the contribution or not of the berm and the slope (cotα

_{incl}; cotα

_{excl}), the slope of the berm (tanα

_{b}), the crest freeboard of the structure (R

_{c}), the armour crest freeboard of the structure (A

_{c}) and the crest width of the structure (G

_{c}), and finally among the structural parameters, those related with the armour elements characterization, like the permeability/roughness factor of the armour layer (γ

_{f}) or the size of the structure elements along the slope (D).

#### 2.2. Artificial Neural Network Models

#### 2.2.1. Multilayer Perceptron

^{m}, output data y ∈ R, the weight matrix being V ∈ R

^{1xn}, W ∈ R

^{nxm}, and where the vector of bias terms b ∈ R

^{n}, where m is the dimension of the input space, n is the number of neurons in the hidden layer, and R is the set of real numbers. More detailed information for the MLP networks can be found in Haykin (1991) [43].

#### 2.2.2. Kohonen Neural Network

_{j}between the input vector X and the vector of synaptic weights W

_{j}:

_{J}(t + 1) for time t + 1, and h

_{j}(t) is the neighborhood function for time t.

#### 2.3. Pre-Processing of Data

#### 2.3.1. Data Escalation

#### 2.3.2. Debugging the Data

#### 2.3.3. Dimensionality Reduction

#### 2.4. Proposed Models

_{u}, cotα

_{incl}, h, h

_{t}, h

_{b}) meet the hypothesis of equality of means, but additionally, none of them meet the hypothesis of equality in variances.

- Model I: Corresponds to an ANN model for the definition of which, all the available patterns have been used after the debugging and the dimensionality reduction process.
- Model II: Involves a division of the input pattern space into two distinct groups or clusters. The first of them trained with data from laboratory tests, and the second with tests from prototypes.

## 3. Discussion

#### 3.1. Obtaining the Reduced Dimension of the Input Vector

- B and B
_{h}: referred to the dimensions of the berm (0.999) - h and h
_{t}: referred to depth at the toe of the structure or the submergence of the toe (0.91) - T
_{m t}and T_{m1 t}: period values at the toe of the structure (0.63) - D and γ
_{f}: variables related to the size and roughness factors (0.777) - R
_{c}and A_{c}: variables relative to freeboard (0.860) - Cotα
_{incl}, cotα_{excl}, cotα_{d}: variables related to the slope geometry (0.828 to 0.921)

_{excl}. And the third component (F3) explains 12.64% of the total variance and is dominated by the variables cotα

_{incl}and A

_{c}.

_{m}

_{1.0t}; T

_{p t}; T

_{m t}), with a high positive correlation between them, a trivial matter already detected in the correlation matrix, which at least allows reducing their number, in any case keeping only one of them. Another group with a strong positive correlation is composed of those variables related to the geometric characterization of the slope (cotα

_{incl}; cotα

_{excl}; cotα

_{d}) on which it will act in a similar way. The same procedure can be carried out with the variables relative to the width of the berm and its horizontal projection (B; B

_{h}), and from which it is inferred that only variable B will be preserved. The grouping of variables in the correlation circle, in the projection of both axes F1 and F2, seems to determine the lack of correlation between the variables that make up the most obvious groupings, with similar direct cosines, such as those determined by cotα

_{incl}, cotα

_{excl}, cotαd, and those like: cotα

_{u}, tanα

_{B}, H

_{m0 t}. The foregoing leads to considering that both groupings of variables must be present in the input space, although with the particular restrictions indicated previously for some of them. The spectral wave steepness variable (H

_{m}

_{0 t}/L

_{m}

_{1 t}), negatively correlated with the freeboard variables (R

_{c}, A

_{c}), should be kept as above. Finally, the strong link between the width of the crest and the characteristic size of the protection elements in the breakwater is clearly reflected along with its strong link with the F2 axis.

_{excl}, cotα

_{incl}, cotα

_{d}). A reason why the information contributed by them can be redundant, and informs which two of them should be discarded. However, it is noteworthy that another of the variables related to that group of parameters, which refers to the cotangent of the slope of the structure in the part of the slope above the berm (cotα

_{u}), presents a projection pattern that is notoriously different from the previous ones, but without a distinctive response in the component plane, so it should not be taken into account.

_{f}) and the mean diameter (D). The comparison of both planes shows the existence of a negative correlation between them, and thus the greater sizes, the lower the roughness factor. This relationship is evidence in the existing empirical knowledge and taken into account [32,41], but it is comforting to confirm that the ANN is capable of detecting it as well. Both parameters should be preserved a priori.

_{h}), and that is also preserved across the data range. Therefore, only one of them should be selected, discarding the other.

_{t}), or with the width of the crest, however, this has not been detected at the data base analyzed for the crest width of the structure, so this supposed relationship will be discarded. While the existence of a partial correlation, at least in a region of the projection plane, between the variables of the berm width and the width of the bench (see Figure 5) is detected, which indicates that some of the breakwaters that have been tested have been designed with a theoretical pattern that relates both variables. The foregoing forces not discarding these variables, but to keep them in the input space, since this relationship is partial in the sample space it is necessary to preserve that differentiation.

_{t}). In this case, as expected, its correlation is direct or positive. However, the lack of correlation between both variables and the berm (h

_{b}) is also striking, therefore, following the above reasoning, at least two of them should be maintained, discarding the third of them.

#### 3.2. Model Selection

^{−5}) and model II (3.82 × 10

^{−5}), with the known exception that the MSE is not an absolute statistic, but a relative one [64].

#### 3.3. Sensitivity Analysis

_{c}), an issue that is confirmed by PCA analysis. It is noteworthy that a variable closely related to it, the other freeboard parameter, the crest height with respect to swl (A

_{c}), is quite far, in terms of significance, from the parameter R

_{c}. This fact is relevant since in some works [59] it has been determined that the scale effect seems to depend a lot on the superior geometry of the breakwater. This results in many more significant associated effects on small overtopping rates, which are incidentally, also the most numerous in the database. Given this, and to try to mitigate these effects as much as possible, some authors propose the dimensionless of these variables [25,30,59].

_{incl}) is considered [42,59]. Similarly of interest, the wave steepness (H

_{m}

_{0}/L

_{m}

_{−1}) is highlighted. In addition to these, are both parameters related to roughness (and in essence, to the porosity of the mantle) where their close relationship with overtopping is already known empirically [32], and which in turn have a substantial dependence on the dimensionless freeboard (R

_{c}/H

_{m}

_{0}).

_{m}

_{o t}) is presented since, as Van der Meer cites [48], this relationship is strongly related to the overtopping phenomenon, and shows the existence of poor representability in the ranges greater than 50°.

_{c}/H

_{s}relationships [71] are highly desirable. These will generally be associated with low crested structures, specifically with R

_{c}/H

_{s}lower than 1. It would be desirable that the training sample be well represented in this range, as does happen, and is shown in the following figure (Figure 11a). Checking the model for those exceedance rates corresponding to the range in the previously mentioned sample of 100 extra cases (corresponding to a total of 57 cases), the result is encouraging, with values of the correlation coefficient greater than 0.98, as shown in the Figure 11b.

## 4. Conclusions

_{m}

_{0 toe}= 1 m). Subsequently, the entire data base was subjected to an extensive process of exploration, debugging, and dimensionality reduction, until an optimized input pattern in the ANN model was obtained. Using only 15 of the 34 initial features, sufficient relevant information was used to train a model with generalization skills and high predictive efficiency. This preliminary phase derives substantial conclusions such as:

- It is worth noting the lack of homogeneity in the database due to its diverse origin, where the existence of data at different scales forces the adoption of a data scaling procedure, which introduces uncertainty into the model. It is concluded that this lack of homogeneity is masked in the final model by the significant difference in sample size.
- Relevant effects associated to the scale are quoted, especially in what concerns to the superior geometry of the breakwater. WEC devices that are located in existing structures, where power generation capacity is combined with defensive capacity, that have small magnitudes of the overtopping rate, and which incidentally are the most common, are especially sensitive to these effects.
- Linked with the above conclusion, a new and more appropriate transformation of the inputs must be proposed that minimizes the observed heteroscedasticity effects in this range of overtopping rates.
- The present work shows the suitability of multivariate statistical techniques, and specifically the Mahalanobis distance, for the detection of outliers, and also the Principal Component Analysis for the reduction of the dimension of the input vector, a task shared with the Kohonen Self Organizing Maps application.

_{c}/H

_{s}relationship the higher overtopping rate, therefore this criterion will allow future models to be developed and trained in that specific range of patterns.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- IEA-OES (International Energy Agency-Ocean Energy Systems). Annual Report 2006; Technical Report; IEA-OES: Paris, France, 2006. [Google Scholar]
- Esteban, M.D.; Espada, J.M.; Ortega, J.M.; López-Gutiérrez, J.S.; Negro, V. What about Marine Renewable Energies in Spain? J. Mar. Sci. Eng.
**2019**, 7, 249. [Google Scholar] [CrossRef] [Green Version] - Breeze, P. Marine Power Generation Technologies. In Power Generation Technologies, 3rd ed.; Elsevier Ltd.: Oxford, UK, 2019; pp. 323–349. [Google Scholar]
- Zhou, Z.; Benbouzid, M.; Charpentier, J.F.; Scuiller, F.; Tang, T. A Review of Energy Storage Technologies for Marine Current Energy Systems. Renew. Sustain. Energy Rev.
**2013**, 18, 390–400. [Google Scholar] [CrossRef] [Green Version] - Lewis, M.; Neill, S.; Robins, P.; Hashemi, M. Resource assessment for future generations of tidal-stream energy arrays. Energy
**2015**, 83, 403–415. [Google Scholar] [CrossRef] [Green Version] - Lavi, A. Ocean thermal energy conversion: A general introduction. Energy
**1980**, 5, 469–480. [Google Scholar] [CrossRef] - Seyfried, C.; Palko, H.; Dubbs, L. Potential local environmental impacts of salinity gradient energy: A review. Renew. Sustain. Energy Rev.
**2019**, 102, 111–120. [Google Scholar] [CrossRef] - Falcão, A.F.O. Wave energy utilization: A review of the technologies. Renew. Energy
**2010**, 14, 899–918. [Google Scholar] [CrossRef] - Babarit, A.; Bull, D.; Dykes, K.; Malins, R.; Nielsen, K.; Costello, R.; Roberts, J.; Bittencourt Ferreira, C.; Kennedy, B.; Weber, J. Stakeholder requirements for commercially successful wave energy converter farms. Renew. Energy
**2017**, 113, 742–755. [Google Scholar] [CrossRef] - IDAE. Evaluación del Potencial de la Energía de las olas; Technical Report; IDEA: Madrid, Spain, 2011. [Google Scholar]
- Margheritini, L.; Vicinanza, D.; Frigaard, P. SSG wave energy converter: Design, reliability and hydraulic performance of an innovative overtopping device. Renew. Energy
**2009**, 34, 1371–1380. [Google Scholar] [CrossRef] - Astariz, S.; Iglesias, G. The economics of wave energy: A review. Renew. Sustain. Energy Rev.
**2015**, 45, 397–408. [Google Scholar] [CrossRef] - Reguero, B.G.; Losada, I.J.; Méndez, F.J. A global wave power resource and it seasonal, interannual and long-term variability. Appl. Energy
**2015**, 148, 366–380. [Google Scholar] [CrossRef] - Shields, M.A.; Payne, A.I.L. Strategic Sectoral Planning for Offshore Renewable Energy in Scotland. In Marine Renewable Energy Technology and Environmental Interactions, 1st ed.; Shields, M.A., Payne, A.I.L., Eds.; Springer: Berlin, Germany, 2014; pp. 141–152. [Google Scholar]
- Aderinto, T.; Li, H. Ocean Wave Energy Converters: Status and challenges. Energies
**2018**, 11, 1250. [Google Scholar] [CrossRef] [Green Version] - The Wave Dragon Technology. Wave Dragon Web Site. Available online: http://www.wavedragon.net/?option=com_content&task=view&id=4&Itemid=35 (accessed on 1 June 2020).
- Foteinis, S.; Tsoutsos, T. Strategies to improve sustainability and offset the initial high capital expenditure of wave energy converters (WECs). Renew. Sustain. Energy Rev.
**2017**, 70, 775–785. [Google Scholar] [CrossRef] - Contestabile, P.; Iuppa, C.; Di Lauro, E.; Cavallaro, L.; Andersen, T.L.; Vicinanza, D. Wave loadings acting on innovative rubble mound breakwater for overtopping wave energy conversion. Coast. Eng.
**2017**, 122, 60–74. [Google Scholar] [CrossRef] - Iuppa, C.; Contestabile, P.; Cavallaro, L.; Foti, E.; Vicinanza, D. Hydraulic Performance of an Innovative Breakwater for Overtopping Wave Energy Conversion. Sustainnabily
**2016**, 8, 1226. [Google Scholar] [CrossRef] [Green Version] - Vicinanza, D.; Nørgaard, J.H.; Contestabile, P.; Andersen, T.L. Wave loadings acting on Overtopping Breakwater for Energy Conversion. J. Coast. Res.
**2013**, 65, 1669–1674. [Google Scholar] [CrossRef] - Cuadra, L.; Salcedo-Sanz, S.; Nieto-Borge, J.C.; Alexandre, E.; Rodríguez, G. Computational intelligence in wave energy: Comprehensive review and case study. Renew. Sustain. Energy Rev.
**2016**, 58, 1223–1246. [Google Scholar] [CrossRef] - Govindaraju, R.S.; Rao, A.R. Artificial Neural Networks in Hydrology, 1st ed.; Water Science and Technology Library, Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000; pp. 1–329. [Google Scholar]
- Bishop, C.M. Neural Networks. In Pattern Recognition and Machine Learning, 3rd ed.; Springer: New York, NY, USA, 2006; pp. 225–290. [Google Scholar]
- Van Gent, M.R.A.; Van den Boogaard, H.F.P.; Pozueta, B.; Medina, J.R. Neural network modelling of wave overtopping at coastal structures. Coast. Eng.
**2007**, 54, 586–593. [Google Scholar] [CrossRef] [Green Version] - EurOtop Manual. EurOtop-Wave Overtopping of Sea Defenses and Related Structures. An Overtopping Manual Largely Based on European Research, but for Worldwide Application, 2nd ed. 2018. p. 320. Available online: https://www.overtopping-manual.com (accessed on 24 May 2020).
- Formentin, S.M.; Zanuttigh, B. A methodological approach for the development and verification of artificial neural networks based on application to wave-structure interaction processes. Coast. Eng. J.
**2018**, 60, 1–20. [Google Scholar] [CrossRef] - Verhaeghe, H.; De Rouck, J.; Van der Meer, J.W. Combined classifier–quantifier model: A 2-phases neural model for prediction of wave overtopping at coastal structures. Coast. Eng.
**2008**, 55, 357–374. [Google Scholar] [CrossRef] - Peixó, J.; Van Oosten, R.P. Wave Transmission at Various Types of Low-Crested Structures Using Neural Networks. Master’s Thesis, Delft University of Technology, Delft, The Netherlands, 2005. [Google Scholar]
- Panizzo, A.; Briganti, R. Analysis of wave transmission behind low-crested breakwaters using neural networks. Coast. Eng.
**2007**, 54, 643–656. [Google Scholar] [CrossRef] - Formentin, S.M.; Zanuttigh, B.; Van der Meer, J.W.A. Neural Network Tool for Predicting Wave Reflection, Overtopping and Transmission. Coast. Eng.
**2017**, 59, 1–31. [Google Scholar] [CrossRef] - Zanuttigh, B.; Formentin, S.M.; Briganti, R. A neural network for the prediction of wave reflection from coastal and harbour structures. Coast. Eng.
**2013**, 80, 49–67. [Google Scholar] [CrossRef] - Molines, J.; Medina, J.R. Explicit Wave-Overtopping Formula for Mound Breakwaters with Crown Walls Using CLASH Neural Network–Derived Data. J. Waterw. Port Coast. Ocean Eng.
**2016**, 142, 1–13. [Google Scholar] [CrossRef] [Green Version] - Deo, M.C. Artificial neural networks in coastal and ocean engineering. Indian J. Geo-Mar. Sci.
**2010**, 34, 589–596. [Google Scholar] - Najafzadeh, M.; Barani, G.A.; Hessami-Kermani, M.R. Group method of data handling to predict scour depth around vertical piles under regular waves. Sci. Iran.
**2013**, 20, 406–413. [Google Scholar] - Azamathulla, H.M.; Zakaria, N.A. Prediction of scour below submerged pipeline crossing a river using ANN. Water Sci. Technol.
**2011**, 63, 2225–2230. [Google Scholar] [CrossRef] - Ayoubloo, M.K.; Etemad-Shahidi, A.; Mahjoobi, J. Evaluation of regular wave scour around a circular pile using data mining approaches. Appl. Ocean Res.
**2010**, 32, 34–39. [Google Scholar] [CrossRef] [Green Version] - Bateni, S.M.; Borghei, S.M.; Jeng, D.S. Neural network and neuro-fuzzy assessments for scour depth around bridge piers. Eng. Appl. Artif. Intell.
**2007**, 20, 401–414. [Google Scholar] [CrossRef] - López, I.; Aragonés, L.; Villacampa, Y.; Satorre, R. Modelling the cross-shore beach profiles of sandy beaches with Posidonia oceanica using artificial neural networks: Murcia (Spain) as study case. Appl. Ocean Res.
**2018**, 74, 205–216. [Google Scholar] [CrossRef] - Negro, V.; Varela, O. Comportamiento funcional, reflexión, transmisión, y amortiguación. Remonte, descenso y rebase. In Diseño de Diques Rompeolas, 2nd ed.; Colegio de Ingenieros de Caminos Canales y Puertos: Madrid, Spain, 2010; pp. 267–304. [Google Scholar]
- Rodríguez, A.M.; Sánchez, J.F.; Gutiérrez, R.; Negro, V. Overtopping of harbour breakwaters: A comparison of semi-empirical equations, neural networks, and physical model tests. J. Hydraul. Res.
**2015**, 53, 1–14. [Google Scholar] [CrossRef] [Green Version] - Verhaeghe, H.; Van der Mer, J.W.; Steendam, G.J.; Besley, P.; Franco, L.; Van Gent, M.R.A. Wave overtopping database as the starting point for a neural network prediction method. In Coastal Structures; ASCE: Portland, OR, USA, 2003; pp. 418–429. [Google Scholar]
- De Rouck, J.; Geeraerts, J. CLASH—D46: Final Report; Full Scientific and Technical Report; Gent University: Gent, Belgium, 2005. [Google Scholar]
- Haykin, S. Neural Networks: A Comprehensive Foundation, 2nd ed.; Pearson Prentice Hall: Upper Saddle River, NJ, USA, 1999; pp. 1–823. [Google Scholar]
- Kohonen, T. Self-Organized Formation of Topologically Correct Feature Maps. Biol. Cybern.
**1982**, 43, 59–69. [Google Scholar] [CrossRef] - Kohonen, T. Self-Organizing Map, 3rd ed.; Springer: New York, NY, USA, 2001; pp. 1–501. [Google Scholar]
- Van der Meer, J.W.; Van Gent, M.R.A.; Pozueta, B.; Steendam, G.J.; Medina, J.R. Applications of a neural network to predict wave overtopping at coastal structures. In Proceedings of the International Conference on Coastlines, Structures and Breakwaters, London, UK, 20–22 April 2005; ICE Thomas Telford: London, UK, 2005; pp. 259–268. [Google Scholar]
- Buckingham, E. On Physically Similar Systems: Illustrations of the Use of Dimensional Equations. Phys. Rev.
**1914**, 4, 345–376. [Google Scholar] [CrossRef] - Van der Meer, J.W.; Verhaeghe, H.; Steendam, G.J. The new wave overtopping database for coastal structures. Coast. Eng.
**2009**, 56, 108–120. [Google Scholar] [CrossRef] - Azme, K.; Zuhaymi, I.; Haron, K. The Effects of Outliers Data on Neural Network Performance. J. Appl. Sci.
**2005**, 5, 1394–1398. [Google Scholar] - Maier, H.R.; Dandy, G.C. Neural networks for prediction and forecasting of water resources variables: A review of modelling issues and applications. Environ. Model. Softw.
**2000**, 15, 101–124. [Google Scholar] [CrossRef] - Peña, D. Análisis de Datos Multivariantes, 1st ed.; McGraw-Hill: Madrid, Spain, 2002; pp. 133–170. [Google Scholar]
- Aggarwal, C.C. Outlier Analysis, 1st ed.; Springer International Publishing: Cham, Switzerland, 2017; pp. 1–422. [Google Scholar]
- Jolliffe, I.T. Principal Component Analysis, 2nd ed.; Springer: New York, NY, USA, 2002; pp. 111–147. [Google Scholar]
- Vesanto, J.; Alhoniemi, E. Clustering of the Self-Organizing Maps. IEEE Trans. Neural Netw.
**2000**, 11, 586–600. [Google Scholar] [CrossRef] - Ultsch, A.; Siemon, H.P. Kohonen’s Self Organizing Feature Maps for Exploratory Data Analysis. In Proceedings of the INNC’90, Paris, France, 9–13 July 1990; Kluwer: Dordrecht, The Netherlands, 1990; pp. 305–308. [Google Scholar]
- Pozueta, B.; Van Gent, M.; Van der Boogaard, H.; Medina, J. Neural network modelling of wave overtopping at coastal structures. World Scientific. In Proceedings of the 29th International Conference on Coastal Engeniering, Lisbon, Portugal, 19–24 September 2004; pp. 4275–4287. [Google Scholar]
- Kortenhaus, A.; Oumeraci, H.; Geeraerts, J.; De Rouck, J.; Medina, J.R.; González-Escrivá, J.A. Laboratory effects and further uncertainties associated with wave overtopping measurements. World Scientific. In Proceedings of the 29th International Conference on Coastal Engeniering, Lisbon, Portugal, 19–24 September 2004; Volume 4, pp. 4456–4468. [Google Scholar]
- Franco, L.; Geeraerts, J.; Briganti, R.; Willems, M.; Bellotti, G.; De Rouck, J. Prototype measurements and small-scale model tests of wave overtopping at shallow rubble-mound breakwaters: The Ostia-Rome yacht harbour case. Coast. Eng.
**2009**, 56, 154–165. [Google Scholar] [CrossRef] - Andersen, L.; Burcharth, H.T.; Gironella, X. Comparison of new large and small scale overtopping tests for rubble mound breakwaters. Coast. Eng.
**2011**, 58, 351–373. [Google Scholar] [CrossRef] - Romano, A.; Bellotti, G.; Briganti, R.; Franco, L. Uncertainties in the physical modelling of the wave overtopping over a rubble mound breakwater: The role of the seeding number and of the test duration. Coast. Eng.
**2015**, 103, 15–21. [Google Scholar] [CrossRef] - Williams, H.E.; Briganti, R.; Romano, A.; Dodd, N. Experimental analysis of wave overtopping: A new small scale laboratory dataset for the assessment of uncertainty for smooth sloped and vertical coastal structures. J. Mar. Sci. Eng.
**2019**, 7, 217. [Google Scholar] [CrossRef] [Green Version] - Bowden, G.J.; Maier, H.R. Optimal division of data for neural network models in water resources applications. Water Resour. Res.
**2002**, 3, 1611–1619. [Google Scholar] [CrossRef] [Green Version] - Hornik, K.; Stinchcombre, M.; White, H. Multilayer feedforward networks are universal approximators. Neural Netw.
**1989**, 2, 359–366. [Google Scholar] [CrossRef] - García-Bartual, R.L. Redes Neuronales Artificiales en Ingeniería Hidráulica y Medio Ambiental: Fundamentos; Technical University of Valencia: Valencia, Spain, 2005. [Google Scholar]
- Minns, A.W.; Hall, M.J. Artificial neural networks as rainfall-runoff models. Hydrol. Sci. J.
**1996**, 41, 399–417. [Google Scholar] [CrossRef] - Hagan, M.T.; Demuth, H.B.; Beale, M.H.; De Jesús, O. Neural Network Design, 2nd ed.; Martin Hagan: Stillwater, MN, USA, 2014; pp. 36–60. [Google Scholar]
- Levenberg, K. A method for the solution of certain problem in least squares. Q. Appl. Math.
**1944**, 2, 164–168. [Google Scholar] [CrossRef] [Green Version] - Marquart, D.W. An algorithm for least-squares Estimation of Nonlinear Parameters. J. Soc. Ind. Appl. Math.
**1963**, 11, 431–441. [Google Scholar] [CrossRef] - Zanuttigh, B.; Formentin, S.M.; Van der Meer, J.W. Prediction of extreme and tolerable wave overtopping discharges through an advanced neural network. Ocean Eng.
**2016**, 127, 7–22. [Google Scholar] [CrossRef] - Kofoed, J.P.; Frigaard, P.; Friis-Madsen, E.; Sørensen, H.C. Prototype testing of the wave energy converter wave dragon. Renew. Energy
**2006**, 31, 181–189. [Google Scholar] [CrossRef] [Green Version] - Tedd, J.; Kofoed, J.P.P. Measurements of overtopping flow time series on the Wave Dragon, wave energy converter. Renew. Energy
**2009**, 34, 711–717. [Google Scholar] [CrossRef] - Flood, I.; Kartam, N. Neural networks in civil engineering, I: Principles and understanding. J. Comput. Civil Eng.
**1994**, 8, 131–148. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the initial descriptive parameters of the model, representing a generic model of a coastal structure.

**Figure 2.**Spatial interpretation of the PCA technique. The projection on the new axes PCA1 and PCA2 maximizes the information provided by the multivariate variable in terms of variance.

**Figure 3.**Dimensionality reduction using the PCA technique: (

**a**) Correlation circle on the F1–F2 plane; (

**b**) Correlation circle on the F1–F3 plane.

**Figure 4.**Correlation circle in the first two principal components with variable elimination. In blue colored appear those variables that are candidates to be eliminated.

**Figure 6.**Results on the proposed models. Correlation graphs are shown for each one and correspond to the test subset: (

**a**) Model I; (

**b**) Model II.1; (

**b**) Model II.2. In each figure the dashed line corresponds to a perfect fit. Additionally, a linear adjustment of the results is shown for each model (only for test subset).

**Figure 7.**Results for the selected model: (

**a**) Graphical fit of the residual histogram to a normal distribution; (

**b**) Scatter plot of the model residual versus outputs.

**Figure 8.**Extra-validation results on the selected model: (

**a**) Correlation graph on the seawall models; (

**b**) Correlation graph on the slope breakwaters models.

**Figure 9.**Sensibility analysis. The histogram represents the contribution of each variables of the model input space in terms of significance of the r

_{s}ratio.

**Figure 10.**Analysis of the model sample: (

**a**) Scatterplot of the parameter β vs. H

_{m}

_{o t}; (

**b**) Histogram of the S

_{m}

_{−1,0}parameter (spectral wave steepness), with adjustment of a normal probability density function.

**Figure 11.**Results on the selected model for the optimum specific range for wave overtopping conversion: (

**a**) Vertical freeboard (R

_{c}) vs. overtopping rate relationship in the model sample; (

**b**) Correlation graph for the R

_{c}/H

_{s}< 1 specific range.

**Table 1.**Statistical parameters of the scaled input–output patterns of the database used for training of the ANN, after the debugging process (scaled to H

_{m}

_{0 t}= 1 m).

Variable | Min | Max | Mean | Std. Deviation |
---|---|---|---|---|

H_{m}_{0 toe}/L_{m}_{−1} (−) | 0.004 | 0.113 | 0.046 | 0.014 |

H_{m}_{0 toe} (m) | 0.280 | 3.765 | 1.013 | 0.142 |

T_{p toe} (s) | 2.993 | 108.138 | 5.314 | 3.111 |

T_{m}_{−1,0t} (s) | 2.721 | 69.942 | 4.940 | 2.500 |

T_{m toe} (s) | 1.441 | 20.466 | 4.299 | 1.224 |

cotα_{u} (−) | −5.000 | 100.000 | 2.860 | 7.946 |

cotα_{d} (−) | 0.000 | 7.000 | 1.872 | 1.351 |

cotα_{incl} (−) | −1.347 | 11.299 | 2.166 | 1.692 |

cotα_{excl} (−) | −1.347 | 8.144 | 1.845 | 1.474 |

tanα_{B} (−) | 0.000 | 0.101 | 0.002 | 0.010 |

B_{h} (m) | 0.000 | 34.188 | 1.042 | 2.511 |

B_{t} (m) | 0.000 | 19.231 | 0.743 | 1.832 |

B (m) | 0.000 | 34.188 | 1.055 | 2.521 |

Β (◦) | 0.000 | 80.000 | 3.126 | 10.469 |

h (m) | 0.375 | 30.000 | 4.251 | 3.215 |

h_{t} (m) | 0.375 | 23.434 | 3.800 | 2.952 |

h_{b} (m) | −2.652 | 7.826 | 0.070 | 0.675 |

D (m) | 0.000 | 2.370 | 0.195 | 0.249 |

γ_{f} (−) | 0.380 | 1.000 | 0.739 | 0.277 |

R_{c} (m) | 0.000 | 13.675 | 1.594 | 1.089 |

A_{c} (m) | −5.242 | 13.675 | 1.506 | 1.069 |

G_{c} (m) | 0.000 | 15.170 | 0.988 | 1.443 |

q (m^{2}/s) | 0.000 | 0.320 | 0.012 | 0.032 |

**Table 2.**p-valor results on the bilateral F-Snedecor and t-Student contrast (confidence level α = 0.05). In bold, those values that allow accepting the null hypothesis of equality of means or variances have been highlighted.

Variable | Variances | Means | Variable | Variances | Means |
---|---|---|---|---|---|

H_{m}_{0 toe}/L_{m}_{−1} | <0.0001 | 0.005 | B | <0.0001 | <0.0001 |

H_{m}_{0 toe} | <0.0001 | <0.0001 | β | 0.027 | <0.0001 |

T_{p toe} | <0.0001 | <0.0001 | h | <0.0001 | 0.218 |

T_{m}_{−1,0t} | <0.0001 | <0.0001 | h_{t} | <0.0001 | 0.286 |

T_{m toe} | 0.974 | <0.0001 | h_{b} | <0.0001 | 0.382 |

cotα_{u} | <0.0001 | 0.286 | D | <0.0001 | <0.0001 |

cotα_{d} | <0.0001 | 0.003 | ꙋ_{f} | 0.351 | <0.0001 |

cotα_{incl} | 0.044 | 0.589 | R_{c} | <0.0001 | <0.0001 |

cotα_{excl} | <0.0001 | 0.005 | A_{c} | <0.0001 | <0.0001 |

tanα_{B} | <0.0001 | 0.025 | G_{c} | <0.0001 | <0.0001 |

B_{h} | <0.0001 | <0.0001 | q | <0.0001 | <0.0001 |

B_{t} | <0.0001 | <0.0001 |

Variable | Pearson Coef. r | Variable | Pearson Coef. r |
---|---|---|---|

H_{m}_{0 toe/}L_{m}_{−1} | −0.097 | cotα_{excl} | −0.028 |

β | −0.100 | cotα_{incl} | −0.067 |

h | 0.068 | γf | 0.306 |

H_{m}_{0}_{toe} | −0.033 | D | −0.253 |

T_{p toe} | 0.007 | R_{c} | −0.322 |

T_{m toe} | 0.058 | B | −0.058 |

T_{m}_{1,0t} | 0.031 | h_{b} | 0.067 |

h_{t} | 0.107 | tanαB | 0.043 |

B_{t} | −0.098 | B_{h} | −0.059 |

cotα_{d} | −0.022 | A_{c} | −0.312 |

cotα_{u} | −0.012 | G_{c} | −0.212 |

Variable | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 |
---|---|---|---|---|---|---|---|---|

H_{m}_{0 toe/}L_{m}_{1 0 t} | −0.603 | −0.254 | 0.305 | −0.292 | −0.319 | 0.197 | −0.117 | 0.220 |

β | −0.020 | −0.153 | −0.058 | −0.195 | 0.302 | 0.131 | −0.107 | −0.050 |

h | 0.273 | 0.631 | −0.145 | −0.153 | 0.575 | 0.259 | 0.166 | −0.041 |

H_{m}_{0 toe} | 0.222 | −0.184 | −0.388 | −0.271 | −0.003 | 0.089 | 0.017 | 0.683 |

T_{p toe} | 0.499 | −0.090 | −0.358 | 0.334 | −0.115 | −0.304 | −0.118 | −0.073 |

T_{m toe} | 0.597 | −0.096 | −0.402 | 0.319 | 0.063 | −0.325 | 0.089 | 0.067 |

T_{m}_{1,0t} | 0.563 | −0.083 | −0.370 | 0.391 | −0.078 | −0.352 | −0.119 | −0.072 |

h_{t} | 0.338 | 0.575 | 0.011 | −0.186 | 0.612 | 0.201 | 0.042 | −0.060 |

B_{t} | −0.127 | 0.152 | −0.446 | −0.028 | 0.001 | 0.219 | 0.328 | 0.158 |

cotα_{d} | 0.617 | −0.445 | 0.414 | −0.109 | 0.197 | 0.233 | −0.273 | 0.015 |

cotα_{u} | 0.195 | −0.273 | 0.169 | 0.180 | 0.003 | 0.270 | 0.614 | −0.184 |

cotα_{excl} | 0.591 | −0.589 | 0.408 | −0.025 | 0.163 | 0.233 | −0.104 | 0.039 |

cotα_{incl} | 0.638 | −0.442 | 0.527 | −0.178 | −0.017 | 0.068 | −0.013 | 0.010 |

γf | 0.188 | 0.457 | 0.280 | 0.581 | −0.239 | 0.305 | −0.040 | 0.172 |

D | −0.012 | −0.444 | −0.489 | −0.532 | 0.224 | −0.195 | 0.089 | 0.137 |

R_{c} | 0.422 | 0.213 | −0.489 | −0.185 | −0.425 | 0.431 | −0.109 | −0.006 |

B | 0.367 | 0.461 | 0.379 | −0.481 | −0.303 | −0.329 | 0.142 | −0.029 |

h_{b} | 0.192 | 0.528 | 0.205 | −0.346 | 0.017 | −0.205 | −0.232 | 0.208 |

tanα_{B} | 0.247 | −0.141 | 0.277 | 0.127 | −0.233 | −0.013 | 0.589 | 0.231 |

B_{h} | 0.360 | 0.467 | 0.371 | −0.484 | −0.297 | −0.330 | 0.128 | −0.038 |

A_{c} | 0.421 | 0.142 | −0.512 | −0.175 | −0.417 | 0.433 | −0.107 | −0.027 |

G_{c} | −0.022 | −0.410 | −0.286 | −0.440 | 0.111 | −0.249 | 0.183 | −0.082 |

q | −0.049 | 0.183 | 0.261 | 0.457 | 0.303 | −0.207 | −0.016 | 0.518 |

Eigenvalue | 3.471 | 3.116 | 2.908 | 2.387 | 1.786 | 1.596 | 1.165 | 1.021 |

% Explained | 15.093 | 13.550 | 12.641 | 10.380 | 7.766 | 6.938 | 5.063 | 4.439 |

Cumulative | 15.093 | 28.643 | 41.284 | 51.665 | 59.431 | 66.369 | 71.432 | 75.871 |

Parameter ^{1} | Description | Unit |
---|---|---|

H_{m}_{0 toe}/L_{m}_{−1} | Dimensionless spectral wave steepness | - |

β | Wave incidence angle | ◦ |

T_{m t} | Average period at the toe of the structure | s |

H_{m}_{0 toe} | Significant spectral wave height at the foot of the structure | m |

h | Water depth at the structure toe | m |

h_{t} | Toe submergence | m |

B_{t} | Toe width | m |

cotα_{incl} | Average co-tangent of the slope of the structure, considering the contribution of the berm | - |

γ_{f} | Roughness/permeability factor for the structure | - |

B | Berm width | m |

D | Size of the structure elements along the slope | m |

h_{b} | Berm submergence | m |

R_{c} | Crest freeboard of the structure respect to sea water level (swl) | m |

A_{c} | Armour crest freeboard respect to swl | m |

G_{c} | Crest width | m |

^{1}Windward side.

**Table 6.**Model I: Results in the test subset based on the number of neurons in the hidden layer. MSE and r statistic.

N° ud. | 5 | 10 | 15 | 20 | 25 | 30 | 35 |
---|---|---|---|---|---|---|---|

MSE | 7.02 × 10^{−5} | 6.75 × 10^{−5} | 5.69 × 10^{−5} | 5.72 × 10^{−5} | 3.82 × 10^{−5} | 5.06 × 10^{−5} | 7.08 × 10^{−5} |

r | 96.51 | 96.28 | 97.33 | 97.17 | 98.17 | 97.68 | 97.28 |

**Table 7.**Model II.I: Results in the test subset based on the number of neurons in the hidden layer. MSE and r statistic.

N° ud. | 5 | 10 | 15 | 20 | 25 | 30 | 35 |
---|---|---|---|---|---|---|---|

MSE | 7.02 × 10^{−5} | 6.71 × 10^{−5} | 8.43 × 10^{−5} | 5.86 × 10^{−5} | 3.82 × 10^{−5} | 6.28 × 10^{−5} | 6.98 × 10^{−5} |

r | 96.98 | 96.24 | 96.31 | 97.37 | 96.13 | 97.28 | 96.81 |

**Table 8.**Model II.II: Results in the test subset based on the number of neurons in the hidden layer. MSE and r statistic.

N° ud. | 5 | 10 | 15 | 20 | 25 | 30 | 35 |
---|---|---|---|---|---|---|---|

MSE | 1.68 × 10^{−7} | 1.82 × 10^{−7} | 3.00 × 10^{−7} | 3.88 × 10^{−8} | 1.45 × 10^{−8} | 1.46 × 10^{−7} | 1.02 × 10^{−7} |

r | 84.05 | 63.10 | 66.10 | 84.51 | 75.33 | 74.66 | 74.52 |

Statistics | Chi-Squared | Kolmogorov-Smirnov |
---|---|---|

Empirical Parameter | 3.922 × 10^{15} | 0.161 |

Theoretical Parameter | 72.15 | |

p-value (bilateral) | <0.0001 | <0.0001 |

Confidence interval (α) | 0.05 | 0.05 |

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**MDPI and ACS Style**

Oliver, J.M.; Esteban, M.D.; López-Gutiérrez, J.-S.; Negro, V.; Neves, M.G.
Optimizing Wave Overtopping Energy Converters by ANN Modelling: Evaluating the Overtopping Rate Forecasting as the First Step. *Sustainability* **2021**, *13*, 1483.
https://doi.org/10.3390/su13031483

**AMA Style**

Oliver JM, Esteban MD, López-Gutiérrez J-S, Negro V, Neves MG.
Optimizing Wave Overtopping Energy Converters by ANN Modelling: Evaluating the Overtopping Rate Forecasting as the First Step. *Sustainability*. 2021; 13(3):1483.
https://doi.org/10.3390/su13031483

**Chicago/Turabian Style**

Oliver, José Manuel, Maria Dolores Esteban, José-Santos López-Gutiérrez, Vicente Negro, and Maria Graça Neves.
2021. "Optimizing Wave Overtopping Energy Converters by ANN Modelling: Evaluating the Overtopping Rate Forecasting as the First Step" *Sustainability* 13, no. 3: 1483.
https://doi.org/10.3390/su13031483