Assessment of Wind Power Density Based on Weibull Distribution in Region of Junin, Peru

This paper appraises the accuracy of methods for calculating wind power density (WPD), by comparing measurement values to the shape and scale parameters of the Weibull distribution (WD). For the estimation of WD parameters, the Graphical method (GP), Empirical method of Justus (EMJ), Empirical method of Lysen (EML), Energy pattern factor method (EPF), and Maximum likelihood method (ML) are used. The accuracy of each method was evaluated via multiple metrics: Mean absolute bias error (MABE), Mean absolute percentage error (MAPE), Root mean square error (RMSE), Relative root mean square error (RRMSE), Correlation coefficient (R), and Index of agreement (IA). The study’s objective is to select the most suitable methods to evaluate the WD parameters (k and c) for calculating WDP in four meteorological stations located in Junin-Peru: Comas, Huasahuasi, Junin, and Yantac. According to the statistical index results, the ML, EMJ, and EML methods are the most accurate for each station, however, it is important to note that the methods do not perform equally well in all stations, presumably the graphical conditions and external factors play a major role.


Introduction
Renewable power capacity is increasing at a faster annual rate than all fossil fuels combined. In 2016, green energy comprised nearly 62% of the total power-generating capacity in the world, and more and more countries are using this technology [1]. According to renewable energy statistics, onshore wind energy in Section 3. Finally, the conclusion is presented in Section 4.

Weibull Distribution
Numerous probability distribution functions are available to evaluate wind speed frequency distributions. These functions include the Weibull, Rayleigh, Gamma, Pearson (V), Inverse Gaussian, and Log-normal functions [10]. The evaluation of wind energy is based on wind speed measurements. However, these are often variable, so it is useful to conduct the wind speed modelling as a function of certain parameters. The WD is widely accepted in the literature for the evaluation of wind energy [3]. The probability density of the WD is mathematically described with three parameters. With the minimal speed considered near zero, the distribution is then expressed as [9]: where k-shape parameter(dimensionless), c-scale parameters (m/s) and ν -wind speed (m/s) for WD, k and c parameters are often used to characterize wind regimes because it has been found to provide a good fit with measured wind data. The cumulative function is defined [9]: According to the literature, several methods are used to estimate the WD Parameters, the application of which is related to geographical conditions among other external factors [3]- [19]. The Graphical Method pursues the minimum last square error in the fitting process between measures and linear equations of the form y ax b = + . The Weibull parameters using this method can be calculated as [4] [5] [19]: In the group of empirical methods, the methods of Justus and Lysen are used. The method of Justus was formulated by C. G. Justus [20]; this is an experimental approach using a wind-powered generator system characterized by cut-in, rated, and cut-out speed to estimate the power output. The parameters of the Weibull are shown in [4] [18] [20] 1.086 where σ -standard deviation of wind speed (m/s), ν -mean wind speed (m/s).
The method of Lysen uses experimental data to approximate the Gamma  ( Γ ) by the next expression, as discussed in [21]. For the calculation of shape factor, the method of Justus is used.
The method of Energy Pattern Factor, uses pf E (aerodynamic parameter design) to calculate the k parameter, as discussed in [4] [18] [19]. The c parameter is calculated using the Justus formulation.
Finally the method of Maximum Likehood is used. For this method, the likelihood function is used to estimate the Weibull parameters where n is the number of observation, after several iterations the parameters are obtained; the mathematical expressions for the method are discussed in [4] [17] [19].
The method for estimating wind power density is based on energy balances and Bernoulli's equation, the process of extracting the kinetic energy of the wind can be explained. The power per surface is dependent on the cube of wind speed and surface evaluation and ρ -air density (kg/m 3 ). The measurement of WDP is obtained through a calculation as: WDP is a measure of the energy that can be produced by wind turbine in a specific location. This measure can be calculated using the WD [4]:

Study Case: Region of Junin
The onshore wind-generated energy in Peru is projected to be 372 MW by the end of 2018 [2]. Since 2014, economic strategies have been developed by the Ministry of Energy and Mines to increase renewable energy capacity-for example, through energy auctions for solar and wind energy. Additionally, to incrementalize the massive increase in the use of electricity, and with the objec-tive of increasing the rural electrification coefficient, domestic photovoltaic panels and mini wind turbines have been installed in rural areas. However, there is either no information about the renewable energy potential of new projects or else the existing information is outdated. The focus of this study is the Junín region; this region is one of the most important in Peru when it comes to the economic development of the country. Because of the productivity of this region, it represents a large percentage of Peruvian energy demand. To date, however, no feasibility studies for projects using renewable energy sources such as wind and solar have been carried out. This study uses information provided by the National Meteorology and Hydrology Service of Peru (SENAMHI). SENAMHI indicates that there are 22 stations throughout the Junín region; situated 10 m up from the ground level, these stations have all been installed since 2003. This paper uses the wind measurements from four of these stations, in four areas with the highest average wind speed. These are: Comas, Huasahuasi, Junín, and Yantac. We use historical information from 2012 to 2017, basing our analysis on SENAMHI measurements that were made daily at 13:00 h. Table 1 shows the main characteristics of the meteorological stations in Junin-Peru. The Andes are the highest mountains in the world. These peaks modify the climate of the region that hosts them, and several areas within Junín do have a modified climate. For example, the coastal areas have a typical summer climate, whereas the areas near the Andes that have a rainy and cloudy summer climate. Both types of weather occur during the first four months of the year.

Evaluation of Weibull Parameters
The authors reviewed previous literature about methods used to estimate the WD parameters (shape and scale) and WDP. The data obtained from the meteorological station of Junin were subjected to these methods for purposes of evaluation. The calculus of indicators and data processing were performed using the scripts in Matlab (2017b). In daily analysis, historical information from 2012 to 2017 were used, with one measurement per day. The WD parameters were calculated using five methods: EMJ, EPF, ML, GP, and EML.

Results
This section describes the approaches used first to estimate and then to evaluate the k and c parameters for the four meteorological stations in the Junin region of Peru. The objective is to find the most convenient method to estimate the accuracy of the WD, with statistical indicators being used for the performance evaluation of the methods.

Estimation of the Weibull Parameters
The k and c parameters were estimated using five methods: EMJ, EPF, ML, GP,    Table 4.

Estimation of Wind Power Density
This section estimates the WDP with the chosen methods, as a function of the results from an analysis based on statistical indicators. For a more detailed analysis, the use of wind turbine characteristics would be used. Table 5    except the Yantac station, for which data from 2014 to 2017 were used. The values of the calculation using measurements and using the WD are closely correlated. This conclusion is confirmed by the high correlation revealed through the R and IA coefficients.

Conclusions
This paper has investigated five well-know methods for estimating the Weibull distribution parameters and wind power density, statistical metrics are used to find the most convenient method to estimate the accuracy of the Weibull distribution. The evaluation was carried out in four locations in region of Junin-Peru with data from SENAMHI for the years 2012-2017. The following conclusion is made: • According to the results of the statistical indicators, the ML, EMJ and EML proved to be the most suitable methods for calculating wind power density in each of the four locations in region of Junin. • The wind power density calculated using experimental data and Weibull distribution parameter has a difference less or equal than 1%. • The Weibull distribution considering the shape and scale parameters presents accuracy calculation of the wind power density, independently of the three suitable methods found by the authors (ML, EMJ and EML), the results are very close to each other.