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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.

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Reference [

This paper addresses the WDP accuracy calculation in four locations in Junín, Peru, with the aim of establishing reliable methods for calculating wind resources (their available energy potential) in each area. It’s important to mention that there are no prior studies of WDP in these areas. In the present study, five methods are used to calculate shape and scale variables and statistical indicators to determine the accuracy of WDP. The methods are GP, EMJ, EML, EPF, and ML.

The paper is organized as follows. In Section 2, the methods for estimating WD parameters and WDP are shown, also the four stations used as data sources for this paper, and the wind speed measurements are described. The analysis of results considering k and c parameters and statistics methods is presented in Section 3. Finally, the conclusion is presented in Section 4.

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 [

f ( v ) = k c ( ν c ) k − 1 exp [ − ( ν c ) k ] (1)

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 [

F ( v ) = 1 − exp [ − ( ν c ) k ] (2)

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 [

ln { ln [ 1 − F ( ν ) ] } = k ln ( ν ) − k ln ( c ) (3)

In the group of empirical methods, the methods of Justus and Lysen are used. The method of Justus was formulated by C. G. Justus [

k = ( σ ν ¯ ) − 1.086 (4)

where σ -standard deviation of wind speed (m/s), ν ¯ -mean wind speed (m/s).

c = ν ¯ Γ ( 1 + 1 k ) (5)

The method of Lysen uses experimental data to approximate the Gamma function ( Γ ) by the next expression, as discussed in [

c = ν ¯ ( 0.568 + 0.4333 k ) − 1 k (6)

The method of Energy Pattern Factor, uses E p f (aerodynamic parameter design) to calculate the k parameter, as discussed in [

E p f = Γ ( 1 + 3 k ) Γ 3 ( 1 + 1 k ) (7)

k = 1 + 3.69 ( E p f ) 2 (8)

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 [

k = [ ∑ i = 1 n ν i k ln ( ν i ) ∑ i = 1 n ν i k − ∑ i = 1 n ln ( ν i ) n ] − 1 (9)

c = [ ∑ i = 1 n ν i k n ] 1 k (10)

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:

P ¯ v = 1 2 ρ v ¯ 3 (11)

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 [

P ¯ v = 1 2 ρ c 3 Γ ( 1 + 3 k ) (12)

The onshore wind-generated energy in Peru is projected to be 372 MW by the end of 2018 [

Station | Latitude | Longitude | Altitude (m.a.s.l.) | Mean Speed (m/s) | Town |
---|---|---|---|---|---|

Acopalca | 11˚55'39'' | 75˚06'59'' | 3839 | 2.7 | Huancayo |

Chacapalca | 11˚43'58'' | 75˚45'21'' | 3752 | 2.3 | Chacapalca |

Comas | 11˚11'44'' | 75˚07'45'' | 3640 | 6.1 | Comas |

Huasahuasi | 11˚15'42'' | 75˚37'15'' | 2750 | 6.7 | Huasahuasi |

Huayao | 12˚02'18'' | 75˚20'17'' | 3360 | 2.6 | Huanchac |

Huaytapallana | 11˚55'36'' | 75˚03'42'' | 4584 | 1.7 | Huancayo |

Ingenio | 11˚52'51'' | 75˚17'16'' | 3390 | 2.9 | Santa Rosa de Ocopa |

Jauja | 11˚47'12'' | 75˚29'13'' | 3378 | 2.8 | Jauja |

Junin | 11˚11'08'' | 75˚59'20'' | 4120 | 8.2 | Junin |

La Oroya | 11˚34'07'' | 75˚57'34'' | 3910 | 3.2 | Santa Rosa de Saco |

Laive | 12˚15'08'' | 75˚21'19'' | 3860 | 5.2 | Yanacancha |

Marcapomacocha | 11˚24'16'' | 76˚19'30'' | 4447 | 2.8 | Marcapomacocha |

Puerto Ocopa | 11˚08'01'' | 74˚15'01'' | 690 | 3.1 | Rio Tambo |

Ricran | 11˚32'22'' | 75˚31'26'' | 3820 | 4.3 | Ricran |

San Juan de Jarpa | 12˚07'30'' | 75˚25'55'' | 3600 | 2.1 | San Juan de Jarpa |

Santa Ana | 12˚00'15'' | 75˚13'15'' | 3395 | 5.4 | El Tambo |

Satipo | 11˚14'01'' | 74˚42'01'' | 1370 | 2.8 | Satipo |

Tarma | 11˚23'49'' | 75˚41'25'' | 3000 | 5.7 | Tarma |

Viques | 12˚09'47'' | 75˚14'07'' | 3186 | 4 | Viques |

Yantac | 11˚11'20'' | 76˚24'16'' | 4617 | 6.3 | Marcapomacocha |

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.

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.

In order to determine the relative precision of each of the five methods the statistical indicators were used to assess the accuracy of each of them. The k and c variables were used to calculate the WDP in comparison with the measured data, MAPE and MABE indicators show the general degree of error of each of the methods; for a more precise diagnostic, RMSE is used. RMSE gives the model’s precision as a result of contrasting the Weibull function and the measured data [^{2}. Correlation coefficient R is a statistical index that calculates the strength of the linkage between the WDP calculated by the WD and the WDP calculated by measures; this index is in the range of −1 to 1. The IA or Willmont index is a normalized metric used to assess the model prediction error [

The results of statistical indicators for each meteorological station show the convenience methods for the estimation of parameters. These selected methods are evaluated in more detail in what follows. The evolution of the k and c parameters is tracked over a 72-month period to determine the fluctuation of the parameters, with the average value per month being used for this analysis.

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.

The k and c parameters were estimated using five methods: EMJ, EPF, ML, GP, and EML. For the Yantac station, the data were collected from 2014 (April) to 2017 (December), but for all the other stations, data were collected from 2012 (January) to 2017 (December). In the case of Comas station, the Weibull parameters are calculated using a total of 2191 wind measurements; for the k parameter, the EPF method shows 16% variation with respect to mean values, hence this method is likely not accurate enough to estimate the WD parameters; by way of comparison, all the methods used to estimate the c parameter show variations of less than 3% with respect to the mean value. For Huasahuasi Station, 2190 wind measurements were used to estimate the Weibull parameters; for the k parameter, the GP method shows 9% variation with respect to mean values; all the methods used to estimate the c parameter show variations of less than 1% with respect to the mean value. In the case of Junin Station, the Weibull parameters were calculated using 2191 wind measurements; in this case all results for both parameters show variation of less than 1% with respect to the mean value. Finally for Yantac Station the parameters of the WD were estimated with 1404 wind measurements. In this case, the shape parameter in the GP method shows 17% variation with respect to mean values and the EPF method shows 11% variation. All methods used to estimate the c parameter show variations of less than 1% with respect to the mean value. The results of the five methods for all station are shown in

Station | Variable | Method | ||||
---|---|---|---|---|---|---|

GP | EMJ | EML | EPF | ML | ||

Comas | k | 4.539 | 4.424 | 4.424 | 3.619 | 4.570 |

c | 6.736 | 6.771 | 6.767 | 6.848 | 6.758 | |

Huasahuasi | k | 2.800 | 3.153 | 3.153 | 3.033 | 3.213 |

c | 7.394 | 7.337 | 7.336 | 7.351 | 7.311 | |

Junin | k | 3.836 | 3.703 | 3.703 | 3.290 | 3.690 |

c | 8.844 | 8.891 | 8.887 | 8.946 | 8.886 | |

Yantac | k | 4.175 | 3.547 | 3.547 | 3.160 | 3.401 |

c | 7.180 | 7.308 | 7.305 | 7.351 | 7.313 |

Station | Method | Index | |||||
---|---|---|---|---|---|---|---|

MAPE (%) | MABE (W/m^{2}) | RMSE (W/m^{2}) | RRMSE (%) | R | IA | ||

Comas | GP | 0.010 | 0.578 | 1.272 | 19.299 | 0.996 | 0.970 |

EMJ | 0.012 | 0.522 | 1.132 | 17.175 | 0.997 | 0.973 | |

EML | 0.011 | 0.516 | 1.114 | 16.913 | 0.997 | 0.973 | |

EPF | 0.010 | 1.651 | 3.435 | 52.124 | 0.967 | 0.915 | |

ML | 0.014 | 0.620 | 1.368 | 20.755 | 0.996 | 0.968 | |

Huasahuasi | GP | 0.006 | 1.955 | 3.422 | 37.802 | 0.974 | 0.916 |

EMJ | 0.001 | 0.347 | 0.656 | 7.242 | 0.999 | 0.985 | |

EML | 0.002 | 0.343 | 0.647 | 7.150 | 0.999 | 0.985 | |

EPF | 0.003 | 0.819 | 1.482 | 16.370 | 0.995 | 0.965 | |

ML | 0.004 | 0.207 | 0.347 | 3.836 | 1.000 | 0.991 | |

Junin | GP | 0.004 | 0.910 | 1.691 | 10.991 | 0.998 | 0.977 |

EMJ | 0.006 | 0.738 | 1.292 | 8.400 | 0.999 | 0.981 | |

EML | 0.005 | 0.727 | 1.264 | 8.218 | 0.999 | 0.981 | |

EPF | 0.003 | 2.625 | 4.540 | 29.506 | 0.982 | 0.932 | |

ML | 0.005 | 0.725 | 1.265 | 8.223 | 0.999 | 0.981 | |

Yantac | GP | 0.004 | 1.767 | 3.612 | 41.838 | 0.974 | 0.925 |

EMJ | 0.004 | 0.361 | 0.687 | 7.962 | 0.999 | 0.985 | |

EML | 0.004 | 0.355 | 0.671 | 7.776 | 0.999 | 0.985 | |

EPF | 0.004 | 1.508 | 2.814 | 32.599 | 0.982 | 0.935 | |

ML | 0.003 | 0.616 | 1.222 | 14.155 | 0.997 | 0.974 |

with EMJ, EML, and ML when it comes to calculating the WD parameters; the values of 8.4, 8.218, and 8.223 for RRMSE indicate excellent model precision. Finally for Yantac station we found that the highest agreement was attained when the EMJ and EML methods were used to estimate the WD parameters, the values of 7.962 and 7.776 for RRMSE indicate excellent model precision. All good and excellent ratings are according to the criteria as shown in [

The calculations by year were used to develop the analysis more fully. In the case of Comas station EMJ and EML method were chosen, the evolution in the data patterning over 72 months were evaluated for the k and c parameters using both methods; these parameters have similar values; our findings confirm that the EMJ and EML methods are the most accurate for WDP calculations. For Husahuasi station the analysis was used for a more detailed assessment of the ML, EMJ, and EML methods; the three methods show the same tendencies for both parameters over time. Overall, we found that ML, EMJ, and EML methods have the greatest accuracy for WDP calculation. In the case of Junin station the analysis was used for the EMJ, EML, and ML methods to develop the analysis more fully; the three methods reveal the same tendencies for both parameters; overall, we found that the ML, EMJ, and EML methods have the greatest accuracy for WDP calculation. Finally for Yantac station the analysis was used for the EMJ and EML methods to develop the analysis more fully; the evolution from year to year was evaluated for the k and c parameters using both methods; the three methods show the same tendencies for both parameters; overall, we found that EMJ and EML methods have the most accuracy for WDP calculation. For all stations the evaluation results by year are displayed in

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.

Station | Method | Variable | Year | |||||
---|---|---|---|---|---|---|---|---|

2012 | 2013 | 2014 | 2015 | 2016 | 2017 | |||

Comas | EMJ | k | 4.2138 | 4.5722 | 4.0236 | 4.5059 | 4.8256 | 4.9984 |

EML | k | 4.2138 | 4.5722 | 4.0236 | 4.5059 | 4.8256 | 4.9984 | |

EMJ | c | 6.8914 | 6.5627 | 6.2910 | 6.6252 | 7.1475 | 7.0839 | |

EML | c | 6.8878 | 6.5589 | 6.2880 | 6.6214 | 7.1432 | 7.0795 | |

Huasahuasi | ML | k | 3.7308 | 3.3100 | 3.2020 | 3.2005 | 3.1670 | 2.9125 |

EMJ | k | 3.6450 | 3.2284 | 3.1237 | 3.1392 | 3.1155 | 2.8654 | |

EML | k | 3.6450 | 3.2284 | 3.1237 | 3.1392 | 3.1155 | 2.8654 | |

ML | c | 7.3031 | 6.8387 | 7.4429 | 7.4633 | 7.5288 | 7.2677 | |

EMJ | c | 7.3255 | 6.8643 | 7.4683 | 7.4850 | 7.5529 | 7.3100 | |

EML | c | 7.3227 | 6.8627 | 7.4670 | 7.4836 | 7.5516 | 7.3096 | |

Junin | ML | k | 3.6756 | 3.9960 | 4.1782 | 4.0218 | 3.6271 | 3.2081 |

EMJ | k | 3.5805 | 3.8716 | 4.1788 | 4.1683 | 3.7692 | 3.2326 | |

EML | k | 3.5805 | 3.8716 | 4.1788 | 4.1683 | 3.7692 | 3.2326 | |

ML | c | 9.6353 | 9.1121 | 8.9848 | 8.7853 | 8.5872 | 8.1208 | |

EMJ | c | 9.6479 | 9.1271 | 8.9940 | 8.7842 | 8.5865 | 8.1204 | |

EML | c | 9.6445 | 9.1230 | 8.9894 | 8.7797 | 8.5829 | 8.1186 | |

Yantac | EMJ | k | - | - | 2.9940 | 3.3970 | 3.9044 | 4.8229 |

EML | k | - | - | 2.9940 | 3.3970 | 3.9044 | 4.8229 | |

EMJ | c | - | - | 6.2656 | 7.2340 | 7.9236 | 7.4933 | |

EML | c | - | - | 6.2648 | 7.2319 | 7.9200 | 7.4887 |

lightaqua Station | Method | WPD - m | WPD - d |
---|---|---|---|

Comas | EMJ | 28.495 | 28.662 |

EML | 28.495 | 28.615 | |

Huasahuasi | ML | 38.952 | 38.837 |

Junin | EMJ | 66.944 | 67.009 |

EML | 66.944 | 66.929 | |

ML | 66.944 | 66.973 | |

Yantac | EMJ | 60.009 | 59.402 |

EML | 60.009 | 59.339 |

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.

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.

The authors thank the National University of the Center of Peru for support in carrying out this research.

The authors declare no conflicts of interest regarding the publication of this paper.

Galarza, J., Condezo, D., Camayo, B. and Mucha, E. (2020) Assessment of Wind Power Density Based on Weibull Distribution in Region of Junin, Peru. Energy and Power Engineering, 12, 16-27. https://doi.org/10.4236/epe.2020.121002