2.2.4. Wave Resource Maps

The historical evolution of the resource is also represented by means of maps, showing wave trends of the entire area of study. The trends were computed using the non-parametric Theil-Sen [55,56] method, which fits a line using the median of the slopes. The significance of the trend at each grid point was evaluated at a 95% confidence level using bootstrap resampling with 1000 samples.

Thus, different variables are illustrated using maps:

1. The average *Hs*, *Tm* and *WEF* values for the entire area of study, based on the ERAI reanalysis, which provides a picture of the wave resource in the recent decades. In addition, the map with the average *WEF* is useful to identify the highest energetic locations (see Section 4.2.1).


#### **3. Hydrodynamic Modelling**

Several different WECs have been suggested in the literature to extract energy from ocean waves. Based on their working principles, all WEC can be categorised into four main groups [1]: overtopping devices, oscillating water column devices, oscillating wave surge converters and PAs. Although none of the prototypes suggested so far has yet achieved commercial viability, a large part of the most developed prototypes are PAs, such as the CETO [57], Seabased [58] and the Corpower [59] devices. The PA used in this study is inspired by the Corpower device, referred in the following as the cPA, illustrated in Figure 3a, adapted from [60]. Further details about the characteristics of the WEC implemented in this study are provided in [8].

(**a**)

**Figure 3.** The illustration (**a**); and the power-matrix (**b**) of the Corpower device.

Power production capabilities of the cPA for a wide range of sea-states, represented using irregular-wave time-series based on the JOSNWAP spectrum [61], were estimated via numerical simulation, computing the power-matrix depicted in Figure 3b. The behaviour of the cPA in this study was evaluated by analysing its motion in two degrees of freedom, as in [59], following the linear Cummins' equation [62] as follows,

$$\dot{\chi}(M+\mu\_{\infty})\ddot{\mathbf{x}}(t) = F\_{\rm ex} - K\_{\rm H}\mathbf{x} - \int\_{0}^{t} K\_{\rm rad}(t-\tau)\dot{\mathbf{x}}(\tau)dt + F\_{\rm vis} + F\_{\rm PTO} + F\_{\rm MOO} + F\_{\rm EndStop} \tag{3}$$

where *M* and *μ*∞ are the mass and infinite added-mass matrices; *x*, *x*˙ and *x*¨ are the displacement, velocity and acceleration of the device, respectively; *Fex* is the excitation force; *KH* is the hydrostatic stiffness; *Krad* is the radiation impulse response; *FPTO* is the PTO force; *FMOO* is the mooring force; and *FEndStop* is the force that reproduces the end-stop effect of the PTO mechanism. The only nonlinearity included in the model is the viscous force (*Fvisc*), which is modelled using a quadratic damping based on a Morison-like equation [63] as follows,

$$F\_{\text{visc}} = \frac{1}{2} \rho C\_D A\_D (\dot{\mathbf{x}} - V\_0) |\dot{\mathbf{x}} - V\_0| \tag{4}$$

where *ρ* is the water density, *CD* is drag coefficient, *AD* is the characteristic area of the WEC and *V*<sup>0</sup> is the velocity of the undisturbed water flow.

The objective of the present study was to study the impact of wave energy resource variations, assessing the power produced by the cPA over the 20th century off the Chilean coast. Nonlinear effects, such as nonlinear Froude–Krylov forces, are shown to be essential for accurately estimating power production capabilities of PAs [64] and, more specifically, for the Corpower device [65]. However, since the authors are only interested in the relative values of the comparison, the mathematical model based on the linear Cummins's equation is considered adequate.

Similarly, the need for including the most relevant PTO dynamics, losses and constraints to accurately estimate power production capabilities of a WEC is demonstrated in [66]. Nevertheless, the present paper focuses on the impact of wave energy resource variations, for which analysing power absorption is found sufficient. However, constraints of the PTO system significantly affect the power absorption of a WEC. Therefore, three main constraints usually included by any PTO system, i.e., displacement, velocity and force constraints, are also considered, similar to [67].

#### **4. Results**

#### *4.1. Evaluation Versus Buoys*

Figure 4a,b illustrates the Taylor Diagrams for the Iquique and Valparaiso locations, respectively, where Pearson's correlation, the RMSE and the SD between the ERA20, ERAI and dcERA20 reanalyses are shown. The correlation of the dcERA20 reanalysis is shown to be very similar to that of the original ERA20, meaning that the correction did not improve the correlation. The correlation of the ERAI reanalysis, which is the upper limit for the calibration, is shown to be about 0.9 in both locations, while the ERA20 and dcERA20 show correlations of about 0.7–0.8. However, the dcERA20 reanalysis shows reasonable improvement in RMSE and SD. The RMSE for the ERA20 reanalysis is above 10 kW/m for Iquique, while reducing below that threshold after calibration. The RMSE reduction is even more significant for Valparaiso, reducing the *WEF* from 20 kW/m to 15 kW/m. In the case of the SD ratio, which shows the ratio between SDs of the wave models and buoy measurements, the calibration corrects the SD from 4/14 to 11/14 in Iquique, and from 6/22 to 18/22 in Valparaiso, which mean an improvement of over 50% in both cases. Hence, despite the low impact of the calibration on the correlation, the calibration is shown to significantly improve the wave data, approaching the more reliable ERAI reanalysis.

**Figure 4.** Taylor Diagrams for Iquique (**a**) and Valparaiso (**b**) buoys.

Analysing other metrics described in Section 2.2.3, the effect of the calibration is even more evident, as shown in Tables 2 and 3. In Iquique, the ERA20 reanalysis underestimates the *WEF*, compared to the observations, which is corrected in the dcERA20 reanalysis, as presented in Table 2. This underestimation results in a negative bias of the ERA20 and a MAPE of only 37%. In contrast, the MAPE of the dcERA20 is halved due to the calibration. Important underestimation of the *COV* is also shown in Table 2 for the ERA20 reanalysis, which is remarkably corrected by the calibration.


**Table 2.** Mean *WEF*, bias and MAPE metrics for Iquique.

In the case of Valparaiso, the improvement of wave data due to the directional calibration method is even more important, as shown in Table 3, reaching mean *WEF*, *COV*, bias and MAPE values very close to the ERAI reanalysis, which sets the upper limit of the calibration. The mean *WEF* is improved substantially, reducing the MAPE from 56% to 4% and correcting the strong underestimation of the ERA20 reanalysis. The bias is also significantly reduced, from −17.6 to −1.2, and the *COV* given by the the dcERA20 reanalysis is identical to the *COV* obtained from the ERAI reanalysis. It should be noted that the *COV*s obtained from the buoy measurements in Iquique and Valparaiso are similar to the *COV* metrics presented in [14] for the same area, although data from different buoys were used: 0.6 in northern Chile [14], which is slightly lower than the 0.69 observed in this paper for Iquique; and 0.8 in central Chile [14], slightly higher than the 0.7 observed in this study for Valparaiso.


**Table 3.** Mean *WEF*, bias and MAPE metrics for Valparaiso.

#### *4.2. Representation of Maps in the Study Area*

#### 4.2.1. Mean Values

Figure 5a–c shows, respectively, the mean values of *Hs*, *Tm* and *WEF* obtained from the ERAI reanalysis between 1979 and 2010. Mean *Hs* reaches very significant values in the south (up to 3.5 m) and progressively diminishes towards the north, with a minimum mean *Hs* of 1.7 m shown in the north of the country. The case of the *Tm* is exactly the opposite, where the mean *Tm* increases towards the north, with areas of long wave periods (up to 10 s) in the north and relatively short periods (about 7 s) in the south. Since the *WEF* is proportional to the square of *Hs*, as shown in Equation (2), the *WEF* follows the same spatial distribution as the *Hs*, meaning that wave power is highest in the south of the country. The results of the nearshore *WEF*, as shown in Figure 5c, are consistent with the recent study by Lucero et al. [15].

The map that illustrates the *COV* along the Chilean coast is shown in Figure 5d, where the spatial distribution of the *COV* is similar to that shown in [14], with *COV* values increasing towards the south. Additionally, one can observe a relation between the most energetic area in the south and the highest *COV*, and the decrease of both parameters towards the North. Consequently, the two locations under analysis in this paper (Iquique and Valparaiso) show reasonably low *COV* values (about 0.6). Note that, in the map illustrated in Figure 5d, and the maps shown in the following sections, purple cells with an **x** symbol mean that there is no significant variation at a 95% confidence level in that location.

**Figure 5.** *Cont.*

**Figure 5.** Mean values of *Hs* (**a**), *Tm* (**b**), *WEF* (**c**) and *COV* (**d**) in the study area.

According to the spatial distribution of the mean *WEF* and the *COV*, Valparaiso shows interesting characteristics for the deployment of a WEC and, therefore, wave data from the dcERA20 reanalysis for the closest gridpoint to Valparaiso were used to evaluate historical wave resource variations and their impact on the power production of the cPA presented in Section 3.

#### 4.2.2. Decadal Wave Trends

Decadal trends of the wave resource in Valparaiso are shown in Figure 6, where the evolution of the *WEF*, *Hs* and *Tp* are given in kW/m/decade, cm/decade and centiseconds per decade (cs/decade), respectively. The *Hs* is shown to increase slightly in the central and northern latitudes of Chile (1 cs/decade), while more significant increases (up to 5 cs/decade) are observed in the southern latitudes, as illustrated in Figure 6a. A similar pattern is also observed for *WEF* variations, as shown in Figure 6c, where the *WEF* increases up to 2 kW/m/decade. In contrast, variations of the mean *Tp* seem to be negligible in the north of the Chilean coast, while the *Tp* increases significantly (up to 4 cs/decade) in the central and southern latitudes. These results are consistent with the results shown in [24] for the *Hs* and in [30] with respect to the *WEF*. Trends of the *COV* are also studied over the 20th century, but are not shown in Figure 6, because the results do not show any significant variation at a 95% confidence level.

(**c**) Non-seasonal *WEF* trends

**Figure 6.** Trends of *Hs* (cm/decade) (**a**), *Tm* (cs/decade) (**b**) and *WEF* (kW/m/decade) (**c**) in the study area.

#### 4.2.3. Seasonal Wave Energy Trends

The calibration of the ERA20 reanalysis, described in Section 2.2, can also be classified according to the seasonal variations, creating a transfer function for each season, referred to as the seasonally-calibrated ERA20 (scERA20). Figure 7a–d shows the four maps corresponding to the decadal *WEF* trends for autumn, winter, spring and summer, respectively, along the Chilean coast. The results obtained from the scERA20 show a relevant hot spot in the south of Chile, where the wave trend is particularly strong in autumn and winter (up to 2.5 kW/m/decade). This wave trend is still positive in the central and northern latitudes of the Chilean coast, although the wave trend is slightly weaker (about 0.5 kW/m/decade). Wave resource variations are slightly different in spring, where the positive wave trends can be observed in the south (up to 2 kW/m/decade) and north (about 0.5 kW/m/decade) of the Chilean coast. However, no significant variations are observed in the central latitudes, where the resource is consistent all over the 20th century, as illustrated in Figure 7c. Finally, resource variations over the 20th century are negligible in summer, as shown in Figure 7d.

**Figure 7.** *Cont.*

**Figure 7.** Wave energy trends for autumn (**a**), winter (**b**), spring (**c**) and summer (**d**).

#### *4.3. Wave Trends and Power Production in Valparaiso*

#### 4.3.1. Wave Resource Variations

According to Figure 6, which is created using wave data from the dcERA20 model, decadal *Hs*, *Tm* and *WEF* trends are approximately 1.2 cm/decade, 2. 5 cs/decade and 0.4 kW/m/decade, respectively. A more detailed study, analysing each do-decade of the 20th century separately, shows that the mean *WEF* in the first do-decade, i.e., 1900–1920, was about 19.1 kW/m/decade. Hence, the decadal *WEF* increase of 0.4 kW/m shown in Figure 6c corresponds to a decadal increase of about 2% (0.39/19.1 × 100 = 2%).

The scatter-diagram of the resource in this first do-decade of the 20th century is illustrated in Figure 8a, where the most frequent *Hs* and *Tp* are shown to be 1.5 m and 10.5 s, respectively. Wave trends of the next do-decades are shown as relative variations (in percentage) with respect to the wave resource in the first do-decade, as shown in Figure 8b, where the decadal increase of 2% is also illustrated. This general trend is consistent with other studies carried out in the same area [24,30]. However, as depicted in Figure 8b, the inter-decadal trends over the 20th century are highly irregular, with significant increases in some do-decades, between 1920 and 1960, for example, and a strong reduction in others, such as between 1960 and 1980.

The bars in Figure 8 show the increase of *WEF*, *Hs* and *Tp* in percentage at each do-decade, with respect to the first do-decade. These bars show a progressive increase of the wave period all over the century, while the *WEF* and *Hs* show more irregular patterns, with significant increases in the first two do-decades, and a very strong decrease in the fourth do-decade. The *WEF* and *Hs* increase again in the last do-decade. Despite the irregular variations in the different do-decades, wave energy resource variations are significantly more consistent compared to the progressive strong increases detected in the Bay of Biscay and west coast of Ireland in [7] and [8], respectively.

**Figure 8.** The scatter-diagram of Valparaiso using the data for the first do-decade of the 20th century (1900–1920) (**a**) and the variations of the resource over the century (**b**).

(**b**)

#### 4.3.2. Impact on Wave Energy Absorption

The power absorbed from ocean waves by the cPA is assessed using the annual mean power production (AMPP) metric. Figure 9 illustrates the variation of the AMPP over the 20th century, if the cPA were deployed in Valparaiso. AMPP variations follow the same pattern as the *WEF* variations, illustrated in Figure 8. However, AMPP variations are lower than resource variations. An increase of 6% in AMPP is observed in the second do-decade, with respect to the first do-decade, while the resource increases by 8%. This difference is even larger for the third do-decade, with an increase of over 12% in *WEF*, while the increase in AMPP is almost identical to the second do-decade (6%). The reason the AMPP does not increase with the *WEF* may be the variation of the mean *Tp* between the second and third do-decades. While *WEF* increases in the second do-decade, the variation of the mean *Tp* is negative, meaning that the resource shows lower wave periods, getting closer to the natural period of the cPA (5.2 s). In contrast, mean *Tp* variation in the third do-decade is positive, which means that the mean *Tp* moves away from the natural period of the cPA and, as a consequence, limits the increase in AMPP. Likewise, *WEF* variation is very similar in the second and fifth do-decades, as shown in Figure 8, while the AMPP increase depicted in Figure 9 is significantly lower in the fifth do-decade. The only difference between resource variations in the second and the fifth do-decades is again the mean *Tp*, which is significantly larger in the fifth do-decade. Consequently, the same

increase in *WEF* does not involve the same increase in AMPP, illustrating the relevance of wave period variations.

**Figure 9.** AMPP variation over the 20th century in Valparaiso (Chile) and Galway bay (Ireland).

#### **5. Discussion**

The wave trend results of the calibrated wave model presented in this paper for the Chilean coast are consistent with previous studies presented in the literature, either based on data from satellite altimeter [24] or reanalyses [30,31]. The particular case of Valparaiso presented in this study shows an overall *WEF* increase of 2%/decade over the last century, while previous studies show increments of 0–3.3% from the north to the south of Chile. In any case, this general slope is important only for the validation of the calibration, since the barplot presented in Figure 8 shows that resource variations are quite irregular over the century, including relatively strong decreases between the third and the fourth do-decades. This irregular trend, without a clear uniform variation profile, contrasts the almost linear increment profile observed in the Atlantic Ocean, more specifically, in the Bay of Biscay [7] and the west coast of Ireland [8].

This non-uniform behaviour in trends over the different decades suggests that the causes of the detected variations may be complex and multiple. Although these kind of variations are often attributed to climate change [9], different, relatively unknown global mechanisms may also play an important role.

The comparison of the wave resources in the Chilean coast and the west coast of Ireland is a judicious comparison, due to their similar wave power, as shown in [14]. However, an important difference between Chile and Ireland, also pointed out in [14], is the *COV*, which is significantly greater in Ireland. In fact, results for the *COV* presented in [14] are similar to those presented in the present study, where a longer time-period is analysed. Hence, the wave resource in Chile is shown to be significantly more consistent than the wave resource in the west coast of Ireland. However, the impact of a more consistent resource on the AMPP of WECs is not analysed in [14]. Figure 9 compares the variations of the AMPP for the same cPA in the Chilean coast and the west coast of Ireland, where AMPP variations are shown to be significantly lower in Chile, as expected. It should be noted that Valparaiso is located in the centre of Chile, while the variability of the resource (as the *WEF*) increases most in the southern latitudes [14].

Apart from the differences in resource variations and their impact on WECs' power production capabilities, the frequency of storms is another important factor in the process of selecting the optimum location to install a WEC farm. Storms imply that WECs are shifted into survivability mode in order to protect them from structural damages, which directly affects the cost of the structure. A simple method to quantify the frequency of storms is using a maximum *Hs* value (*HMAX <sup>s</sup>* ) that delimits the operational space of the WEC, above which the WEC shifts into survivability mode. The same method is used in [68] or [8], among others. The *HMAX <sup>s</sup>* = 4 m value is found to be quite restrictive in [8], where the AMPP is shown to be drastically reduced, up to 50%, compared to the unlimited case. The events where the wave resource exceeds the *HMAX <sup>s</sup>* limit are referred to as off-limit events, and the frequency of these off-limit events is shown in Figure 10a. These off-limit events represent 20% of the resource off the west coast of Ireland at the beginning of the 20th century, which has significantly increased, up to almost 30%, by the end of the century. In contrast, using the same *HMAX <sup>s</sup>* limitation, the off-limit events represent less than 1% of the resource in Valparaiso, which remains quite constant over the whole century.

**Figure 10.** Occurrence of the off-limit sea-states (**a**) and its impact on the AMPP of the Corpower device (**b**) over the 20th century in Valparaiso (Chile) and Galway bay (Ireland).

The impact of this difference in the frequency of off-limit events on the AMPP is shown in Figure 10b, where the AMPP is shown to decrease up to a 50% off the west coast of Ireland, while only decreasing by 1% in Valparaiso. Hence, the significantly more consistent and mild wave resource in Chile represents a more attractive location, compared to Ireland, for the implementation of wave energy projects, allowing for cost minimization and facilitating deployment or maintenance operations.

The Southern areas of the Pacific Ocean lack detailed data coverage for reanalyses during the early parts of the 20th century. Even in periods relatively close in time (1985–2012), the lack of a dense coverage by observations over scarce data regions of the Pacific implies that the computations of surface heat fluxes over the Eastern Pacific show there the highest errors [69]. In long reanalysis spanning back in time to early instrumental periods, such as is the case with ERA-20CM or 20CRA, observations poorly constrain the reanalyses during early 20th Century. This is evident in the higher spread of the forcing by HadISST during 1900 versus 2000 in ERA20-CM or the lack of closure of the surface energy balance during early years over that area even though SSTs are prescribed [70]. In the case of 20CRA, there appear lacking trends of the Pacific Circulation or the Pacific-North American (PNA) pattern [71]. The smaller number of observations over the southern Pacific during early 20th

Century [34] affects the ability of the ERA-20C reanalysis to simulate a realistic PNA index and other atmospheric indices before 1940. Since ocean waves are produced by atmospheric forcing at the surface, the fact that these problems due to data coverage of the early period have already been identified in the literature point that the use of bias correction techniques such as the ones used in this paper might be more important in Southern Pacific areas than in others. However, since different bias correction techniques are available [72,73], it will be interesting in the future to compare the results presented in this paper with the ones from complementary techniques.

#### **6. Conclusions**

Results of the ERA20 reanalysis of the European Centre for Medium-Range Weather Forecasts has been shown unreliable when comparing to buoy measurements. However, the directional calibration, based on the ERAI reanalysis, presented in this paper, is shown to significantly improve the results of the ERA20 reanalysis. This calibration provides reasonably reliable wave data for the whole 20th century, which allows the study of long-term wave resource variations off the Chilean coast.

Positive wave trends over the 20th century off the Chilean coast are detected in the present study, with the southern coast showing the most significant variations. However, in contrast to other locations in the Atlantic Ocean, the positive wave trend in the Chilean coast is reasonably irregular, with the strong increases of the wave resource in some do-decades and significant reductions in others. These irregular variations indicate that a straightforward attribution of these changes to climate change may be misleading and, thus, further research is needed to establish the driving mechanisms behind these trends. In addition, although long-term variations are significant, about 2kW/m/decade over the 20th century, inter- and intra-annual variations, represented by the coefficient of variation, are shown to be significantly lower than in other locations in the Atlantic Ocean.

These resource variations over the 20th century also affect the the power absorption of wave energy converters, although variations on annual mean power production do not exactly agree with the variations of the wave energy resource. Similar to the wave energy resource variations, annual mean power production variations are significantly lower in the Chilean coast than off the west coast of Ireland. In addition, the frequency of storms, for which the wave energy converters shift to survivability mode, is significantly lower in the Chilean coast (always lower than 1%), compared to the west cost of Ireland (up to 30%). This directly affects the annual mean power production of wave energy converters, and the design of different aspects of the wave energy converters, such as mooring lines, foundations and the structure.

Hence, the selection of the optimum location for the implementation of a wave energy converter farm should consider both short- and long-term variations of the wave energy resource where the farm is planned to be installed.

**Author Contributions:** Conceptualization, A.U. and M.P.; Methodology, A.U. and M.P.; Software, A.U., M.P. and A.R.; Validation, A.U. and A.R.; Investigation, A.U., M.P., G. I., J.S., and J. R.; Writing—Original Draft Preparation, A.U.; Writing—Review and Editing, all authors; Supervision, all authors; Project Administration, G.I. and J.R.; and Funding Acquisition, G.I., J.S. and J.R.

**Funding:** The authors with the Centre for Ocean Energy Research in Maynooth University are supported by Science Foundation Ireland under Grant No. 13/IA/1886. It is also supported by grant CGL2016-76561-R, MINECO/ERDF, UE. Additional funding was received from the University of Basque Country (UPV/EHU, GIU17/002).

**Acknowledgments:** The buoy data have been kindly provided by the SHOA (Chilean Navy Hydrographic and Oceanographic Service): http://www.shoa.cl/php/inicio.php.

**Conflicts of Interest:** The authors declare no conflict of interest.

*Energies* **2018**, *11*, 2289
