**1. Introduction**

Ocean waves store a tremendous potential that is still untapped. The main reason why wave energy has not been yet exploited is the difficulty to economically harvest energy from ocean waves. Different technological solutions have been suggested in recent decades [1], but none of these technologies has achieved yet economical viability to compete in the energy market with other more established energy sources. The Mutriku wave energy plant, based on the oscillating water column principle, is one of the examples of wave energy converters (WECs), which is the first wave energy plant connected to the electricity grid. Despite its aim to be an experimental plant, the plant has achieved a capacity factor (CF) of 11% in the first years of operation [2], showing WECs can be an additional alternative to reduce greenhouse emissions.

In the way towards designing technically and economically viable WECs, there are several aspects to be improved, such as the survivability of the devices, the cost reduction in the construction stage, minimisation of maintenance operations, energy maximisation via control strategies and design of efficient power take-off (PTO) systems. A precise assessment of the wave energy resource is also essential to accurately design WECs and maximise the power extraction from ocean waves. Wave energy converters, based on different working principles and deployed in different locations, are analysed in [3], where the CF of each WEC varies significantly depending on the wave climate in each location.

In particular, the variability of the resource, both inter- and intra-annual variability, is a critical aspect that can affect the design of the WEC. Intra-annual variations of the resource have been analysed in different spatial scales [4–6]. However, historical long-term variations of the wave climate have been often neglected. The authors have studied these long-term variations in the Bay of Biscay and off the west coast of Ireland in [7] and [8], respectively, analysing wave trends over the 20th century and the influence of these trends in the performance of specific WECs. In both locations, positive trends have been found, with wave height (*Hs*), peak period (*Tp*) and wave energy flux (*WEF*) increasing significantly over the 20th century.

The evaluation of the wave energy potential in a specific location usually involves a combination of observations (buoys and satellites), reanalyses and, sometimes, numerical modelling with models such as SWAN or Mike 21. Using these techniques, several studies have recently focused on different parts of the American continent for wave energy potential assessment [9–11]. The Chilean coast is one of the interesting areas in South America for the implementation of WEC farms, with an estimated resource of about 165 GW along its 5000 km [12]. This background has encouraged researchers to study the potential of specific converters at specific locations [13].

Chile and the west coast of Ireland are considered paradigmatic examples, with respect to their wave energy resource. Both present high mean wave power, but the wave resource off the west coast of Ireland is highly variable, while the resource off the Chilean coast is more consistent. This contrast is shown in [14], based on measurements collected by several buoys around the world, via the coefficient of variation (*COV*), which is given as follows,

$$COV = \frac{\sigma(WEF)}{\overline{WEF}},$$

where *σ*(*WEF*) is the standard deviation (SD) of the *WEF* and *WEF* is the average of the *WEF*.

Ringwood and Brandle [14] showed that the maximum *COV* for Chile is 0.9, which is observed in the south of the Chilean coast, where the *WEF* is about 121 kW/m. In contrast, the *COV* for Ireland is 1.8 (twice as high as the maximum *COV* for Chile), where the mean wave power is lower than for the southern Chilean coast: 95 kW/m. Therefore, Chile shows a high energetic and consistent wave resource, which is important in the process to harness and produce energy from ocean waves.

The most recent validation of wave energy assessment in Chile covers the 1989–2013 period [15], where a third generation wave model is used, validated against buoy measurements, to generate three-hourly sea-state parameters (*Hs*-*Tp* pairs). These data are then propagated from offshore to nearshore locations by means of the SWAM model, to characterise the wave climate along the entire coast. Validation results show good agreement between modelled and measured data, with errors of less than 10%. Wave power and seasonal variability in Chile increase with the latitude, which fluctuates between 20 and 35 kW/m in the areas close to shore, and where the most energetic sea-states happen in winter. These results and others [16] show lower wave resource potential, about 5–10 kW/m lower, compared to previous studies, which suggests that previous assessments may have overestimated the wave resource [17].

However, these studies have been developed within a period of 30–40 years, and, thus, historical trends of wave energy are not considered. In the present study, the objective was to analyse the long-term variability of the wave energy resource in Chile, studying the wave energy resource in Chile over the 20th century and comparing the results from the Chilean coast to other locations. Following the recommendations by the World Meteorological Organization [18,19], reliable estimation of climate variables requires at least 30 years of data. These data may be obtained by means of different techniques: buoy measurements [20], observations from ships [21,22], satellite altimeter [23,24] or models and reanalysis datasets [5,25–31]. The latter method was used in the present study, using the ERA20 reanalysis from the European Centre for Medium-Range Weather Forecasts (ECMWF), calibrated with the ERA-Interim (ERAI) reanalysis.

Among the different techniques to estimate climate variables, the satellite altimeter has considerably improved in the recent decades, generating very interesting data for long-term wind or wave trends. For instance, Young et al. [24] presented a study of wind speed and wave height trends in a global scale, which is the only study, to the best of authors' knowledge, that covers the same area of study analysed in the present paper (the Chilean shoreline). However, the time period analysed in [24] is shorter, 23 years between 1985 and 2008, and only the trends of wave heights are analysed, which may be insufficient to draw strong conclusions on the historical wave energy trends, since the influence of the wave period is demonstrated to be essential [7]. Young et al. [24] showed a significant statistical trend for wave heights, analysing mean monthly values in the east of Africa, in the east of North America and the west of South America (the Chilean coast and south-eastern Pacific Ocean). Furthermore, the increase of wind speed is very important in Chile, due to the effect of a hot spot (a positive trend of 15%/decade) in the middle of the Pacific Ocean at equatorial latitudes. Thus, according to [24], wave height trends are large in Chile, showing a positive trend between 0% and 0.25% per year, meaning that a typical wave of 2 m height would increase about 5 cm per decade.

A similar analysis is carried out in [30,31], where the global wave energy resource is assessed via the ERA40 reanalysis, the previous version of the ERAI reanalysis used in this paper. Wave trends in [30,31] are calculated by means of a linear regression. As for the study carried out in [24], positive wave trends are found in [30,31] for the Chilean coast, where these trends are also shown to increase with the latitude. More precisely, wave trends of about 1 kW/m/decade are found in the north of the Chilean coast, while trends of 2 kW/m/decade are observed in the south, which implies about 3.3%/decade for a mean wave power of 60kW/m. Additionally, other studies, such as [32], also show long-term wave energy resource variations in several areas.

Therefore, a precise characterisation of the resource requires the analysis of long-term variations, including a time-evolving description of the resource, to accurately understand the resource in which the WEC is deployed. However, wave energy resource assessment studies commonly rely on recent past data (typically, between 10 and 30 years of past data), and analyse the resource as a static element, even though it has been widely demonstrated that the ocean is a highly dynamic environment [7,14,30,31]. The objective of this paper is to fill this gap for the Chilean coast, analysing the long-term trends of wave energy over the 20th century. In addition, the impact of these variations is evaluated on a realistic point absorber (PA) type WEC, similar to the real Corpower device.

The remainder of the paper is as follows: Section 2 presents the datasets and the methodology to calibrate and study the wave trend, Section 3 describes the realistic WEC and the hydrodynamic model employed to evaluate its power absorption, Section 4 shows the results related to the resource variations and the power absorptions, Section 5 discusses the results and Section 6 presents the conclusions of the study.

#### **2. Wave Data and Methodology**

#### *2.1. Wave Data*

Two different sources of wave data are used in this paper. On the one hand, two reanalyses of the ECMWF are employed and, on the other hand, wave data collected via buoy measurements is used. Further information about both ECMWF reanalyses and buoy measurements is given in Sections 2.1.1 and 2.1.2.

#### 2.1.1. ERA20C and ERA-Interim Reanalyses

The two reanalyses of the ECMWF used in this study are the ERA20C and ERA-Interim reanalyses (ERA20 and ERAI henceforth). The ERA20 reanalysis is ECMWF's first atmospheric reanalysis, which covers the whole 20th century. In the ERA20 reanalysis, observations of surface pressure and surface marine winds are assimilated [33] by means of a 4D-Var analysis. More observations are available, the more reliable are the data generated via the ERA20 reanalysis [34]. Consequently, the data provided by the ERA20 reanalysis are more accurate in the northern hemisphere, although they have also been used for the study of historical wave trends and coastal evolution in different locations of the southern hemisphere [35–37]. The spatial resolution of the ERA20 reanalysis is approximately 125 km and wave parameters can be obtained three-hourly [34].

The ERAI reanalysis is also a global reanalysis, but only covers the time period since 1979. The wave model implemented in the ERAI reanalysis is the Wave Modelling Project (WAM) approach [38], which reduces the error in the wave period assimilation, compared to previous ECMWF reanalyses, such as the ERA40 [39]. The ERAI reanalysis also assimilates data via a 4D-Var method, but using a 75 km spatial resolution and providing wave parameters every six hours [40].

Therefore, the ERA20 reanalysis was calibrated against the ERAI reanalysis, providing more precise wave data for the whole 20th century. Due to the discrepancy of the temporal resolution between ERA20 and ERAI, only six-hourly data were used in the calibration and the calibration was carried out in the intersection period between both reanalyses, which spans 32 years from 1979 to 2010.

#### 2.1.2. Buoy Measurements

Buoy data used in the present study were provided by the SHOA (Spanish acronym of the Chilean Navy Hydrographic and Oceanographic Service). Data from two specific locations were used, i.e., Iquique and Valparaiso, depicted in Figure 1 together with the complete study area, from the Magellan Strait to the Peruvian border in the southern Pacific Ocean.

Table 1 shows the characteristics of the two buoys: the exact position (longitude, latitude), the distance to the nearest gridpoint in the reanalyses and the time period in which data were collected.



**Figure 1.** The complete area of study and the exact location of the two buoys used in this study.

#### *2.2. Methodology*

#### 2.2.1. Computation of the Wave Energy Flux

Although the *WEF* is directly obtained from the parameters available in the reanalysis, it can be given as a function of *Hs* and the energy period (*Te*) as follows [41],

$$WEF = 0.49 H\_s^2 T\_\ell. \tag{2}$$

However, the wave period data provided by SHOA include only *Tp* for our validation. Therefore, to calculate the *WEF* at the buoy, Equation (2) needs to be adapted by including a correction coefficient between *Te* and *Tp*, similar to the *α* = *Te*/*Tp* coefficient described in [42]. According to this study, for a standard JONSWAP spectrum, *α* reaches 0.90. We took into account the correction factor *α* for *Tp* given by the reanalysis data in the validation procedure at the buoy's nearest gridpoint. The power matrix of the WECs is given also for *Tp* and this period was therefore used for the analysis of the production of our device.

On the other hand, Wave Period Ratio (WPR) described in [7,8,41] which relates the energy period to the average zero-crossing period was obtained for the trend computation of wave period. The WPR changes depending on the area of study, which, in the case of Iquique and Valparaiso, lies between 0.8 and 1. Furthermore, mean wave period and average zero-crossing period are very similar, within an error of 10% for monthly means values used for the trends. In any case, this kind of scale factors does not affect the results of the trends, since they constitute the relative slopes of the absolute values. Thus, *Tm* is the chosen period for the representation of maps, because it is one of the most frequently used parameters for the description of the wave period.

#### 2.2.2. Directional Quantile-Matching Calibration

The calibration or bias correction is usually referred to as the classified quantile-matching method, although it has also been named in the literature as probability mapping [43], quantile-quantile mapping [44,45], statistical downscaling [46] or histogram equalization [47]. This calibration method is commonly used in the literature to calibrate temperature, precipitation, wind speed or other parameters [48–52].

The classified quantile-matching method presented in this paper allows for a more sophisticated classified calibration, where the relevance of each variable in the calibration can be considered. Hence, instead of using a single transfer function for the whole time-series, as in the previous studies of the authors [7,8], different transfer functions are used for each time-interval. In the case of wind speed, the variable used for the classification can be directionality, using four main wind directions (northeast, southeast, northwest and southwest). That way, a different transfer function is obtained for each direction interval, for a total of four transfer functions.

Another option is selecting irregular directional intervals to match the specific characteristics, i.e. the predominant wind directions, of the location under analysis. This type of classification is known as wind rose bias correction [53], which is an interesting approach for wave resources, due to their directional characterisation. The closer to the shore, the more defined this directionality is, which in the case of the Chilean coast, is significantly dominated by the western waves. More specifically, the predominant wave direction in Iquique and Valparaiso, illustrated by the wave roses depicted in Figure 2, is the southwest direction. Therefore, the strong directionality of the wave resource in Iquique and Valparaiso justifies the use of the classified quantile-matching technique for the calibration, using wave direction as the variable for the classification.

**Figure 2.** Wave roses of the two buoys: (**a**) Iquique; and (**b**) Valparaiso.

To consider the directionality of the wave resources, seven transfer functions for seven intervals of wave direction, following the wave roses in Figure 2, have been created. Hence, the northeast, southeast and northwest quadrants are represented by a single interval of 90◦, while the southeast quadrant is divided into four different intervals of 22.5◦. Although the comparative of different types of distributions is out of the scope of this paper, the seven-interval distribution has been compared via proof and error to other distribution schemes, such as four quadrant or eight octant regular

distributions, where the seven-interval distribution scheme provides the best results. In any case, the visual intuition provided by the wind roses is enough to justify the decision.

In previous studies by the authors in the Bay of Biscay and west coast of Ireland [7,8], no classification technique is used in the calibration, and, consequently, this directionally-classified quantile-matching technique can be considered as a novel contribution of this paper in the context of wave energy. To the best of authors' knowledge, this is the first time that a wave rose bias correction is used, analogous to the aforementioned "wind rose bias correction".

This method is a statistical method that matches the values with the same quantile between the ERA20 and ERAI reanalyses. Hence, by calibrating the ERAI reanalysis against the ERA20 reanalysis, using the directionally-classified quantile-matching technique, directionally-calibrated wave data (dcERA20) can be obtained. The process is divided into the following steps:

