Novel power-transformed statistical distributions in wind speed modeling

Abstract

Wind energy is one of the fastest-growing renewable power sources, and its accurate assessment requires statistical distributions capable of capturing the inherent variability in wind speed. This study introduces six power-transformed distributions, Generalized Power Generalized Weibull (GPGWD), Power Generalized Weibull (PGWD), Power Ishita (PID), Power Lindley (PLD), Power Akash (PAD), and Power Shanker (PSD), as new alternatives for wind energy modeling. Using wind speed data from Eskişehir, Türkiye, the proposed models were compared with common benchmark distributions. Results show that GPGWD and PGWD consistently outperform benchmarks, with GPGWD providing the highest accuracy, particularly in R², RMSE, and most KS and CHI tests; PGWD ranks second. Although PSD, PLD, and some other models perform well at stations with irregular or low wind regimes in metrics such as MAPE and PDE, overall findings indicate that GPGWD is the most flexible model across most stations.

Keywords

wind speed modeling power-transformed statistical distributions model selection criteria wind power estimation wind speed distribution

Introduction

Wind energy is currently one of the most rapidly growing sector in the energy market as an alternative to fossil fuels. However, in Türkiye, its potential is still underdeveloped. The proper utilization of wind energy systems depends on the accurate estimation of wind speed distribution and, consequently, wind power potential for specific locations. Numerous researchers have examined wind energy potential in different parts of the world (AlQdah et al., 2021; Bastin et al., 2023; Boopathi et al., 2023; Douiri, 2025; Ouahabi et al., 2017; Tizgui et al., 2019). For example, Pusat and Akkoyunlu (2018) evaluated the wind energy potential of several coastal regions of Türkiye using various statistical distributions, Dabbaghiyan et al. (2016) employed the Weibull distribution (WD) to estimate wind power generation in Bushehr, Iran, Ko et al. (2015) used both WD and Rayleigh distributions (RD) to assess wind energy potential in Chuuk State, Micronesia, similarly, Boopathi et al. (2021) investigated the wind potential of the coastal region of Tamil Nadu using WD.

Since wind speed is a random variable, selecting an appropriate probability distribution is essential for accurately calculating wind power. WD remains one of the most widely used models due to its flexibility and mathematical tractability (Bahraoui, 2022; Jowder, 2006). Nevertheless, several studies have demonstrated that WD may be inadequate for highly skewed, heavy-tailed, or multimodal wind speed distributions (Haq et al., 2020; Haq et al., 2022; Kantar et al., 2018; Lencastre et al., 2024). As a result, alternative distributions such as the Lognormal (LND), Gamma (GD), Inverse Gaussian (IGD), Lindley, Kumaraswamy, and mixture models have been proposed to better represent complex wind speed patterns.

Beyond these classical alternatives, more advanced models have also been introduced. These include the upper-truncated Weibull (Kantar and Usta, 2015), generalized Lindley distributions (Kantar et al., 2018), hierarchical mixture models (Hu et al., 2016), and the generalized extreme value distribution (Bahraoui, 2022). Additionally, high-parametric and hybrid approaches have gained attention for their improved fitting performance. For instance, Wang and Zhang (2024) proposed hybrid models integrating WD, the Normal distribution, and RD, while Gugliani et al. (2024) evaluated 14 continuous distributions across 17 sites in India. More recently, Haq et al. (2020) introduced the Power Lomax and length-biased exponential distributions for wind speed modeling. Furthermore, Lencastre et al. (2024) showed that Nakagami (NGD) and Rice distributions outperform WD in capturing wind speed histograms and corresponding power estimates.

In conclusion, findings from the literature indicate that no single probability distribution can adequately describe all types of wind speed data.

The evidence from literature suggests that there is no classical statistical distribution capable of modeling the variability of wind speed under different conditions.

Therefore, selecting the most suitable model remains a crucial step for achieving reliable wind power predictions.

The mean wind power density estimated from a fitted distribution is expressed as

P_{D} = \frac{1}{2} ρ A \int_{0}^{\infty} v^{3} f (v) d v = \frac{1}{2} ρ A μ_{3} .

(1)

Reference mean wind power density ( $P_{R E F}$ ) obtained from wind speed data (sample estimation) is given as follows:

P_{R E F} = \frac{1}{2} ρ A \sum_{i = 1}^{n} \frac{v_{i}^{3}}{n} .

(2)

Six power-transformed statistical distributions, Generalized Power Generalized Weibull (GPGWD), Power Generalized Weibull (PGWD), Power Ishita (PID), Power Akash (PAD), Power Lindley (PLD), and Power Shanker (PSD), are introduced for the first time to wind energy analysis. The term “Power” refers to a mathematical transformation applied to an existing distribution to enhance its flexibility and ability to capture diverse data characteristics. To evaluate the performance of these new models, six conventional wind speed distributions, RD, WD, LND, IGD, GD, and NGD, are also employed for comparison using six model selection criteria.

The proposed novel models are evaluated using wind speed data from six sub-provinces of Eskişehir, Türkiye, which represent diverse statistical characteristics. Data are measured at stations: Anadolu University, Beylikova, Çifteler, Eskişehir, Mihalıççık and Seyitgazi. According to most of the model selection criteria, GPGWD and PGWD are generally found to show the best overall fit among the 12 distributions analyzed. PID, PLD, PSD and PAD also show a respectable performance. These findings confirm the flexibility and robustness of the new power-transformed distributions, establishing them as useful tools for wind speed modeling and wind power potential estimation in various regional contexts.

The remainder of this paper is organized as follows: Section ‘Novel power-transformed statistical distributions’ introduces the proposed power-transformed statistical distributions. Section ‘Commonly used wind speed distributions’ briefly describes the wind speed distributions employed for comparative analysis. Section ‘Wind Data’ provides details on the wind data collected from the sub-provinces of Eskişehir, Türkiye. The parameter estimation methods and the model selection criteria used to evaluate distribution performance are presented in Sections ‘Parameter estimation methods’ and ‘Model selection criteria’, respectively. The results are presented and discussed in Section ‘Analysis and results’. Finally, Section ‘Conclusions and discussion’ concludes the paper with a summary and discussion of the main findings.

Novel power-transformed statistical distributions

This study introduces six novel power-transformed distributions, GPGWD, PGWD, PID, PLD, PSD, and PAD, to the filed of wind speed modeling. These models, previously unexamined in wind speed modeling, offer flexibility for capturing the stochastic characteristics of wind speed data, which typically exhibit skewness and heavy-tailed behavior. In this section, the pdf, cdf, and r-th raw statistical moments for each distribution are presented. In addition, wind power density is computed using the corresponding pdfs in order to evaluate wind power associated with each model.

Generalized Power Generalized Weibull (GPGWD)

Four parameters GPGWD was introduced by Selim (2018) as a new generalization of PGWD, and its statistical properties were extensively investigated. The pdf and cdf of the GPGWD are respectively given by

f_{G P G W D} (v) = c k p b v^{c - 1} {(1 + k v^{c})}^{p - 1} \exp (b [1 - {(1 + k v^{c})}^{p}]), b, c, k, p > 0, v > 0,

(3)

F_{G P G W D} (v) = 1 - \exp (b [1 - {(1 + k v^{c})}^{p}]), b, c, k, p > 0, v > 0 .

(4)

WD is obtained as a special case of the GPGWD when $k = b = 1$ . Furthermore, several well-known distributions such as RD, WD, PGWD, exponential, and Nadarajah–Haghighi distributions can be derived as its sub-models (see Selim, 2018 for details).

If $V$ follows the GPGWD, the r-th raw moment of $V$ for an integer value of $r ⁄ c$ is expressed as

μ_{r} = b e^{b} k^{- \frac{r}{c}} \sum_{j = 0}^{⌊ r / c ⌋} \frac{{(- 1)}^{j + r / c}}{b^{(\frac{j}{p}) + 1}} (\begin{array}{l} \frac{r}{c} \\ j \end{array}) Γ (\frac{j}{p} + 1, b),

(5)

where

⌊ r / c ⌋

denotes the greatest integer less than or equal to

r / c

Accordingly, the estimation of wind power density based on the GPGWD can be formulated as

P_{D_{G P G W D}} = \frac{1}{2} ρ A b e^{b} k^{- \frac{3}{c}} \sum_{j = 0}^{⌊ 3 / c ⌋} \frac{{(- 1)}^{j + 3 / c}}{p^{(\frac{j}{p}) + 1}} (\begin{array}{l} \frac{3}{c} \\ j \end{array}) Γ (\frac{j}{p} + 1, b) .

(6)

Power Generalized Weibull (PGWD)

As a generalization of the classical two-parameter WD, three-parameters PGWD was proposed by Bagdonavicius and Nikulin (2001). Its pdf and cdf are respectively given as follows:

f_{P G W D} (v) = k c p v^{c - 1} {(1 + k v^{c})}^{p - 1} \exp (1 - {(1 + k v^{c})}^{p}), k, c, p > 0, v > 0,

(7)

F_{P G W D} (v) = 1 - \exp (1 - {(1 + k v^{c})}^{p}), k, c, p > 0, v > 0 .

(8)

When

p = 1

, the PGWD reduces to the classical WD. This distribution can also be viewed as an extension of the Nadarajah–Haghighi and exponential distributions (Selim, 2018).

of The r-th raw moment of PGWD for integer values of $r / c$ is given as follows (Nikulin and Haghighi 2009):

μ_{r} = k^{- \frac{r}{c}} e^{1} \sum_{j = 0}^{⌊ r / c ⌋} \frac{{(- 1)}^{j + r / c}}{p^{(\frac{j}{p}) + 1}} (\begin{array}{l} \frac{r}{c} \\ j \end{array}) Γ (\frac{j}{p} + 1, 1) .

(9)

Accordingly, for $r = 3$ , power density estimation formula can be expressed as

P_{D_{P G W D}} = \frac{1}{2} ρ A e^{1} k^{- \frac{3}{c}} \sum_{j = 0}^{⌊ 3 / c ⌋} \frac{{(- 1)}^{j + 3 / c}}{p^{(\frac{j}{p}) + 1}} (\begin{array}{l} \frac{3}{c} \\ j \end{array}) Γ (\frac{j}{p} + 1, 1) .

(10)

Power Ishita distribution (PID)

Two parameters PID was introduced by Shukla and Shanker (2018) as a two-component mixture of an exponential distribution and a gamma distribution. The pdf and cdf are given as follows:

f_{P I D} (v) = \frac{c k^{3}}{k^{3} + 2} (k + v^{2 c}) v^{c - 1} \exp (- k v^{c}), k, c > 0, v > 0,

(11)

F_{P I D} (v) = 1 - [1 + \frac{k v^{c} (k v^{c} + 2)}{k^{3} + 2}] \exp (- k v^{c}), k, c > 0, v > 0 .

(12)

PID includes as special cases the Akash, Lindley (Haq et al., 2022), and exponential distributions. The pdf of PID can become negatively skewed, positively skewed, symmetrical, platykurtic, or mesokurtic.

The r-th raw moment of PID is given as follows:

μ_{r} = \frac{r Γ (\frac{r}{c})}{c^{3} k^{\frac{r}{c}}} \frac{(k^{3} c^{2} + (r + c) (r + 2 c))}{(k^{3} + 2)} .

(13)

The wind power density estimation via PID is given as follows:

P_{D_{P I D}} = \frac{1}{2} ρ A \frac{3 Γ (\frac{3}{c})}{c^{3} k^{\frac{3}{c}}} \frac{(k^{3} c^{2} + (3 + c) (3 + 2 c))}{(k^{3} + 2)} .

(14)

Power Akash distribution (PAD)

A two-parameter PAD was introduced by Shanker and Shukla (2017). Its pdf and cdf are respectively given as follows:

f_{P A D} (v) = \frac{{c k}^{3}}{k^{2} + 2} (1 + v^{2 c}) v^{c - 1} \exp (- k v^{c}), k, c > 0, v > 0,

(15)

F_{P A D} (v) = 1 - [1 + \frac{k v^{c} (k v^{c} + 2)}{k^{2} + 2}] \exp (- k v^{c}), k, c > 0, v > 0 .

(16)

PAD is also a two-component mixture model consisting of exponential distribution characterized by the scale parameter $k$ and a generalized gamma distribution with shape parameters three and scale parameter $k$ , combined with a specified mixing proportion (Shanker and Shukla 2017).

r-th raw moment of PAD is given as follows:

μ_{r} = \frac{r Γ (\frac{r}{c})}{c^{3} k^{\frac{r}{c}}} \frac{(k^{2} c^{2} + (r + c) (r + 2 c))}{(k^{2} + 2)} .

(17)

Wind power density estimation based on PAD is provided as follows:

P_{D_{P A D}} = \frac{1}{2} ρ A \frac{3 Γ (\frac{3}{c})}{c^{3} k^{\frac{3}{c}}} \frac{(k^{2} c^{2} + (3 + c) (3 + 2 c))}{(k^{2} + 2)} .

(18)

Power Lindley distribution (PLD)

Two parameters PLD was introduced and its properties were discussed in Ghitany et al. (2013). The formulas of pdf and cdf are provided as follows:

f_{P L D} (v) = \frac{c k^{2}}{k + 1} (1 + v^{c}) v^{c - 1} \exp (- k v^{c}), k, c > 0, v > 0,

(19)

F_{P L D} (v) = 1 - \exp (- k v^{c}) [1 + \frac{k v^{c}}{k + 1}], k, c > 0, v > 0 .

(20)

Haq et al. (2022) evaluated the potential of PLD for wind speed modeling as part of the Lindley distribution family. However, their analysis relied on limited evaluation criteria and excluded wind-power-based criteria. PLD is a two-component mixture of Weibull distribution (with shape $c$ and scale $k$ ), and a generalized gamma distribution (with shape parameters 2, $c$ and scale $k$ ) (Ghitany et al., 2013). This study extends the comparison by employing more comprehensive assessment metrics, offering a deeper understanding of the distribution’s performance in modeling wind speed.

The r-th raw moment of PLD is given by

μ_{r} = \frac{r Γ (\frac{r}{c} + 1) (c (k + 1) + r)}{c^{2} k^{\frac{r}{c}} (k + 1)} .

(21)

Therefore, wind power density estimation based on PLD is easily calculated as follows:

P_{D_{P L D}} = \frac{1}{2} ρ A (\frac{3 Γ (\frac{3}{c} + 1) (c (k + 1) + 3)}{c^{2} k^{\frac{3}{c}} (k + 1)}) .

(22)

Power Shanker distribution (PSD)

PSD is proposed by applying the exponential power transformation to the Shanker random variable, as introduced by Shanker and Shukla (2017). Its cdf and pdf are respectively defined as follows:

f_{P S D} (v) = \frac{c k^{2}}{k^{2} + 1} v^{c - 1} (k + v^{c}) \exp (- k v^{c}),

(23)

F_{P S D} (v) = 1 - [1 + \frac{k v^{c}}{k^{2} + 1}] \exp (- k v^{c}) .

(24)

The PSD can exhibit various shapes, including monotonically decreasing, positively skewed, negatively skewed, and symmetric forms. PSD is a two-component mixture of Weibull distribution and a generalized gamma distribution (Shanker and Shukla, 2017). It can also produce platykurtic, mesokurtic, or leptokurtic curves, depending on the parameter values.

The r-th raw moment of the distribution is given by

μ_{r} = \frac{r Γ (\frac{r}{c})}{c^{2} k^{\frac{r}{c}}} \frac{(k^{2} + r + c)}{(k^{2} + 1)} .

(25)

Accordingly, the wind power density based on the PSD can be easily obtained as

P_{D_{P S D}} = \frac{1}{2} ρ A \frac{3 Γ (\frac{3}{c})}{c^{2} k^{\frac{3}{c}}} \frac{(k^{2} + 3 + c)}{(k^{2} + 1)}

(26)

Commonly used wind speed distributions

To represent wind speed distributions, various probability distribution functions have been proposed in the literature, with RD, WD, LND, IGD, GD, and NGD being the most commonly applied. These distributions have been selected as benchmark distributions to assess the performance of the proposed models. The pdf, cdf, and P_D expressions of these benchmark distributions are provided in Table 1.

Table 1.

The pdf, cdf, and P_D of RD, WD, LND, IGD and GD (Rangaraj et al., 2024; Safari and Gasore, 2010; Zhou et al., 2010).

Dist	pdf	cdf	P _D
RD	$f_{R D} (v) = \frac{2 v}{k^{2}} \exp (- {(\frac{v}{k})}^{2})$	$F_{R D} (v) = 1 - \exp (- {(\frac{v}{k})}^{2})$	$\frac{1}{2} ρ A k^{3} Γ (1 + \frac{3}{2})$
WD	$f_{W D} (v) = \frac{c}{k} {(\frac{v}{k})}^{c - 1} \exp (- {(\frac{v}{k})}^{c})$	$F_{W D} (v) = 1 - \exp (- {(\frac{v}{k})}^{c})$	$\frac{1}{2} ρ A k^{3} Γ (1 + \frac{3}{c})$
LND	$f_{L N D} (v) = \frac{1}{v c \sqrt{2 π}} \exp (- \frac{{(l n v - k)}^{2}}{2 c^{2}})$	$F_{L N D} (v) = Φ (\frac{l n v - k}{c})$	$\frac{1}{2} ρ A \exp (3 k + 4.5 c^{2})$
IGD	$f_{I G D} (v) = \exp (- \frac{c {(v - k)}^{2}}{2 k^{2} v}) \sqrt{\frac{c}{2 π v^{3}}}$	$F_{L N D} (v) = Φ (v^{}) + \exp (\frac{2 c}{k}) Φ (- v^{ *})$	$\frac{1}{2} ρ A (k^{3} + \frac{3 k^{4}}{c} + \frac{{15 k}^{5}}{c^{2}})$
GD	$f_{G D} (v) = \frac{1}{Γ (c) k^{c}} v^{c - 1} \exp (- \frac{v}{k})$	$F_{G D} (v) = \frac{γ (c, \frac{v}{k})}{Γ (c)}$	$\frac{1}{2} ρ A k^{3} \frac{Γ (c + 3)}{Γ (c)}$
NGD	$f_{N G D} (v) = \frac{2 k^{k}}{Γ (k) c^{k}} v^{2 k - 1} \exp (- \frac{k v^{2}}{c})$	$F_{N G D} (v) = \frac{Γ (k, \frac{k v^{2}}{c})}{Γ (k)}$	$\frac{1}{2} ρ A \frac{Γ (k + \frac{3}{2})}{Γ (k)} {(\frac{c}{k})}^{\frac{3}{2}}$

Note: $Φ$ is the cdf of standard normal distribution, $v^{*} = \sqrt{\frac{c}{v}} (\frac{v}{k} - 1)$ and $v^{* *} = \sqrt{\frac{c}{v}} (\frac{v}{k} + 1)$

Wind data

To evaluate the performance of the proposed distributions, wind speed data were collected from meteorological stations located in sub-provinces of Eskişehir, which were selected to represent diverse statistical characteristics. The data were obtained from the Eskişehir Meteorology Directorate. Wind speeds were originally measured at a height of 10 m using cup anemometers installed to minimize surface friction effects and subsequently extrapolated to 100 m for further analysis. The empirical extrapolation to estimate wind speed at a height of 100 m was conducted using the following power-law relationship (Abbes and Belhadj, 2012):

\frac{v_{1}}{v_{2}} = {(\frac{h_{1}}{h_{2}})}^{m}

(27)

where

v_{1}

and

v_{2}

are the wind speeds at height

h_{1}

and

h_{2}

respectively, and

m

is the power-law exponent, taken as 0.14 following (Gualtieri, 2019).

The yearly descriptive statistics, along with the latitude and longitude information for the studied sub-provinces, are presented in Table 2. In addition, the locations of the meteorological stations in the sub-provinces are shown in Figure 1.

Table 2.

Geographic information of the selected stations in sub-provinces and descriptive statistics of wind speed (2019).

Stations in districts	Coordinates		Descriptive statistics
Stations in districts	Latitude (N)	Longitude (E)	Min	Max	Mean	Std	Std/Mean	Skewn	Kurto	n
Anadolu University	39.8083	30.5320	0.1	10.9	3.061	1.710	55.888	0.725	3.143	7639
Beylikova	39.7033	31.1930	0.1	10.3	2.346	1.587	67.641	1.261	4.393	8735
Çifteler	39.3659	31.0209	0.1	14.6	2.606	2.001	76.754	1.616	5.705	8544
Eskişehir	39.7656	30.5502	0.1	5.00	1.469	0.725	49.378	0.676	3.592	8586
Mihalıççık	39.8733	31.4889	0.1	11.2	2.771	1.634	58.959	0.578	3.086	8254
Seyitgazi	39.4511	30.7005	0.2	13.3	2.978	1.711	57.456	1.190	4.623	8593

Figure 1.

Locations of the meteorological stations in sub-provinces of Eskişehir.

From Table 2, Çifteler exhibits the highest standard deviation, skewness, and kurtosis, indicating high variability in wind speeds and the frequent occurrence of extreme values. Eskişehir has the lowest mean, standard deviation, and coefficient of variation, reflecting more stable wind speeds in the region. Anadolu University and Seyitgazi stand out with higher mean wind speeds, making them advantageous for wind energy potential. Mihalıççık, with the lowest skewness and kurtosis values, demonstrates a more balanced and symmetric distribution of wind speeds. These results reveal notable differences in data dynamics across the sub-provinces.

Parameter estimation methods

The literature commonly employs several methods for estimating the parameters of probability distributions, including the method of moments, maximum likelihood estimation (MLE), least squared method, etc. Among these approaches, MLE is widely preferred because it provides statistically reliable estimates in the case of large data sets, particularly due to its consistency and efficiency properties. In practice, MLE often relies on iterative numerical procedures, such as the Newton–Raphson or Fisher scoring methods, to maximize the likelihood function. However, these traditional algorithms may require high computational effort. In this study, parameter estimation within the MLE framework is carried out using the Nelder–Mead optimization method, which offers a simpler and more computationally efficient alternative.

Model selection criteria

This section presents the formulas for eight key criteria used to determine the best distributional model: the coefficient of determination (R²), Kolmogorov-Smirnov (KS), root mean square error (RMSE), Chi-Square (CHI), Akaike information criterion (AIC), Bayesian information criterion (BIC), mean absolute percentage error (MAPE), and probability density error (PDE). These formulas are provided in Table 3. A high R² value, close to 1, indicates an excellent fit, while lower values for the other criteria confirm the model’s goodness of fit.

Table 3.

The formulas of criteria for model evaluation.

Criteria	Formulas
R²	$1 - \frac{\sum_{i = 1}^{N} {(y_{i} - {\hat{y}}_{i})}^{2}}{\sum_{i = 1}^{N} {(y_{i} - \bar{y})}^{2}}$
RMSE	${(\frac{\sum_{i = 1}^{N} {(y_{i} - {\hat{y}}_{i})}^{2}}{N})}^{\frac{1}{2}}$
KS	${Max}_{1 \leq i \leq n} \| F_{n} (v_{i}) - F (v_{i}) \|$
CHI	$\frac{\sum_{i = 1}^{N} {(y_{i} - {\hat{y}}_{i})}^{2}}{N - d}$
AIC	$- 2 l o g L + 2 d$
BIC	$- 2 l o g L + d l o g (n)$
MAPE	$\frac{1}{N} \sum_{j = 1}^{N} \| \frac{y_{j} - {\hat{y}}_{j}}{y_{j}} \| \times 100$
PDE	$\| \frac{P_{R E F} - P_{D}}{P_{R E F}} \| \times 100$

These criteria collectively provide a comprehensive framework for selecting the most suitable model for wind speed data analysis; however, not all criteria consistently identify the same model as the best. Therefore, the final selection depends on a holistic evaluation of the results.

Analysis and results

This section evaluates the modeling performance of the proposed distributions—GPGWD, PGWD, PID, PAD, PLD, and PSD—using wind speed data measured in Eskişehir. All distributions were applied to model monthly and annual wind speed measurements, and the results were assessed using model selection criteria. The findings indicate that the proposed distributions accurately fit wind speed data and demonstrate strong modeling capabilities. Tables 4 –9 show that, when examined overall, PGWD and GPGWD stand out as the most effective models for wind speed data in terms of most criteria and demonstrate sufficient modeling performance. Additionally, PID, PAD, PLD, and PSD can be considered alternative models that show noteworthy and acceptable levels of performance.

Table 4.

Performance analysis results for Anadolu University Station.

	Distributions	$k$	$c$	$p$	$b$	R²	RMSE	KS	CHI	AIC	BIC	MAPE	PDE
2017	RD	3.6059	2.0000			0.96573	0.01607	0.04858	0.00028	30571.38	30578.39	34.46	3.19
	WD	3.5872	1.9552			0.96767	0.01568	0.04028	0.00029	30566.11	30580.12	32.27	2.33
	LND	0.9948	0.5917			0.95687	0.01901	0.04379	0.00042	30762.50	30776.51	185.28	48.43
	GD	0.9594	3.3039			0.97767	0.01311	0.03835	0.00020	30394.64	30408.65	35.71	3.45
	IGD	3.1698	7.5622			0.94547	0.02170	0.05835	0.00055	30935.20	30949.21	182.14	37.74
	NGD	13.0028	1.0024			0.96562	0.01609	0.04908	0.00030	30573.35	30587.36	34.60	3.24
	PID	1.2328	0.6392			0.94764	0.01983	0.05208	0.00046	30820.22	30834.23	29.24	2.31
	PAD	1.2714	0.5937			0.93932	0.02133	0.05095	0.00053	30907.27	30921.28	30.71	3.22
	PLD	1.5113	0.2805			0.96184	0.01694	0.04251	0.00033	30633.33	30647.34	29.23	2.57
	PSD	1.4185	0.3396			0.96886	0.01528	0.04264	0.00027	30537.89	30551.90	27.94	2.57
	PGWD	2.3611	2.4492	0.5467		0.97460	0.01390	0.03993	0.00025	30439.16	30460.17	29.43	1.40
	GPGWD	0.7394	4.7970	0.3578	0.0749	0.98965	0.00882	0.02733	0.00011	30273.34	30301.36	29.51	0.64
2018	RD	3.4453	2.0000			0.96361	0.01686	0.07702	0.00031	31227.02	31234.02	37.57	6.75
	WD	3.3148	1.7181			0.98539	0.01041	0.02031	0.00013	30894.85	30908.85	23.09	0.17
	LND	0.8605	0.7663			0.92131	0.02423	0.05693	0.00069	32739.63	32753.63	245.05	218.61
	GD	1.2417	2.3829			0.98454	0.01064	0.01862	0.00013	31133.19	31147.19	56.62	16.02
	IGD	2.9588	3.0907			0.76602	0.04379	0.134	0.00227	34889.62	34903.62	274.19	194.19
	NGD	11.8699	0.7895			0.97931	0.01244	0.02596	0.00018	30902.44	30916.45	24.18	1.43
	PID	1.1573	0.7355			0.96878	0.01512	0.03991	0.00027	30915.92	30929.93	22.35	1.23
	PAD	1.1884	0.6928			0.96268	0.01653	0.03917	0.00032	30946.9	30960.9	23.61	2.00
	PLD	1.3726	0.3595			0.98277	0.01125	0.02717	0.00015	30865.18	30879.19	21.54	0.04
	PSD	1.2860	0.4297			0.98721	0.00968	0.02403	0.00011	30866.08	30880.08	19.81	0.25
	PGWD	3.4607	1.6936	1.0513		0.98432	0.01080	0.02072	0.00015	30896.11	30917.12	23.45	0.16
	GPGWD	0.2794	1.3573	1.3117	0.0117	0.98217	0.01150	0.02471	0.00019	30880.87	30908.88	23.42	0.83
2019	RD	3.5060	2.0000			0.95755	0.0181	0.06122	0.00036	28749.35	28756.29	17.49	3.16
	WD	3.4519	1.8711			0.96776	0.0157	0.0369	0.0003	28693.39	28707.27	14.43	0.35
	LND	0.9342	0.6682			0.93965	0.02222	0.04821	0.0006	29793.64	29807.52	188.57	107.79
	GD	1.0673	2.8674			0.97738	0.01326	0.02864	0.00021	28789.35	28803.23	68.4	10.95
	IGD	3.0605	4.7339			0.86306	0.03391	0.09243	0.00141	31214.89	31228.78	216.25	103.22
	NGD	12.2922	0.9112			0.96426	0.01653	0.04185	0.00033	28706.95	28720.83	14.68	1.1
	PID	1.2042	0.6812			0.94211	0.02104	0.05021	0.00054	28888.45	28902.33	23.75	0.01
	PAD	1.2405	0.6359			0.93330	0.02256	0.04953	0.00062	28941.89	28955.77	20.89	0.93
	PLD	1.4611	0.3117			0.96087	0.01726	0.04012	0.00036	28734.13	28748.01	20.24	0.14
	PSD	1.3680	0.3766			0.96826	0.01557	0.0404	0.0003	28694.68	28708.56	27.44	0.01
	PGWD	2.9932	1.9806	0.8317		0.97181	0.01465	0.03574	0.0003	28686.96	28707.78	22.84	0.71
	GPGWD	1.0550	2.2166	0.7794	0.1424	0.97732	0.01314	0.03396	0.00027	28672.9	28700.67	23.73	1.10

Table 5.

Performance analysis results for Beylikova Station.

	Distributions	$k$	$c$	$p$	$b$	R²	RMSE	KS	CHI	AIC	BIC	MAPE	PDE
2017	RD	2.7939	2.0000			0.70807	0.05629	0.20812	0.00341	31109.60	31116.62	68.32	20.30
	WD	2.4200	1.3463			0.98845	0.01056	0.05123	0.00013	28453.27	28467.30	22.71	2.83
	LND	0.4707	0.8571			0.97681	0.01667	0.04425	0.00032	28551.70	28565.73	298.20	207.82
	GD	1.3044	1.6955			0.98507	0.01234	0.05311	0.00018	28329.72	28343.75	34.08	3.04
	IGD	2.2116	2.1282			0.96577	0.02147	0.06117	0.00054	28747.63	28761.66	301.44	118.77
	NGD	7.8059	0.5606			0.97742	0.01463	0.07393	0.00025	28769.75	28783.78	38.69	7.55
	PID	1.0429	1.0061			0.98249	0.01289	0.05872	0.00019	29009.66	29023.69	25.43	8.18
	PAD	1.0439	1.0045			0.98177	0.01315	0.06035	0.00020	29010.18	29024.21	25.50	7.93
	PLD	1.1299	0.6322			0.98739	0.01103	0.05266	0.00014	28598.38	28612.41	22.45	2.91
	PSD	1.0926	0.6862			0.98680	0.01131	0.05287	0.00015	28596.20	28610.23	22.92	2.86
	PGWD	1.2900	1.6876	0.5375		0.98428	0.01304	0.04952	0.00022	28350.89	28371.94	77.56	14.52
	GPGWD	0.2231	4.9357	0.2390	0.0674	0.99404	0.00807	0.03545	0.00009	28024.30	28052.36	45.11	8.70
2018	RD	2.6785	2.0000			0.76135	0.05259	0.21886	0.00298	29770.01	29777.02	71.24	25.13
	WD	2.3253	1.3821			0.98810	0.01129	0.06729	0.00015	27298.21	27312.23	34.41	13.24
	LND	0.4589	0.7804			0.99337	0.00998	0.03689	0.00012	26685.31	26699.33	78.45	79.99
	GD	1.1165	1.8892			0.99082	0.00998	0.06714	0.00012	26957.42	26971.44	28.11	13.41
	IGD	2.1093	2.5889			0.98953	0.01228	0.02691	0.00018	26715.24	26729.26	79.65	49.51
	NGD	7.1744	0.5886			0.96480	0.01964	0.10195	0.00045	27865.4	27879.42	49.85	14.37
	PID	1.0702	1.0181			0.97019	0.01783	0.07920	0.00037	28089.24	28103.26	42.55	16.94
	PAD	1.0678	1.0228			0.97044	0.01776	0.07895	0.00037	28088.77	28102.79	42.43	16.95
	PLD	1.1614	0.6437			0.98634	0.01207	0.06827	0.00017	27463.06	27477.08	34.73	14.26
	PSD	1.1289	0.6923			0.98678	0.01188	0.07002	0.00016	27432.04	27446.06	33.46	14.70
	PGWD	0.8109	2.5515	0.2817		0.99345	0.01016	0.03874	0.00013	26684.09	26705.12	53.84	29.68
	GPGWD	0.4015	4.0840	0.2578	0.2061	0.99785	0.00525	0.03032	0.00004	26501.69	26529.73	22.20	4.41
2019	RD	2.8318	2.0000			0.84577	0.04226	0.15715	0.00196	30682.42	30689.5	56.85	15.34
	WD	2.6328	1.5884			0.95559	0.02218	0.07038	0.00060	29750.89	29765.04	27.27	7.40
	LND	0.6325	0.6775			0.99278	0.01119	0.04424	0.00015	29040.79	29054.94	205.59	47.59
	GD	0.9669	2.4260			0.97904	0.01527	0.06485	0.00028	29289.57	29303.72	43.02	6.74
	IGD	2.3457	4.0821			0.99339	0.01059	0.03146	0.00014	29081.79	29095.94	209.53	34.45
	NGD	8.0194	0.7349			0.92935	0.02794	0.09610	0.00095	30067.83	30081.98	37.49	8.87
	PID	1.1474	0.8943			0.91712	0.03018	0.08462	0.00111	30564.97	30579.12	31.55	9.06
	PAD	1.1560	0.8787			0.91477	0.03060	0.08147	0.00114	30588.35	30602.5	31.45	9.57
	PLD	1.2991	0.5080			0.95102	0.02324	0.07049	0.00066	29918.43	29932.58	30.45	8.11
	PSD	1.2482	0.5665			0.95593	0.02203	0.07263	0.00059	29835.5	29849.65	26.88	6.87
	PGWD	0.9772	3.2479	0.2416		0.99132	0.01312	0.04137	0.00024	29005.17	29026.39	192.28	28.66
	GPGWD	0.5922	5.2224	0.2239	0.2363	0.99710	0.00605	0.03353	0.00006	28758.97	28787.27	94.14	6.68

Table 6.

Performance analysis results for Çifteler Station.

	Distributions	$k$	$c$	$p$	$b$	R²	RMSE	KS	CHI	AIC	BIC	MAPE	PDE
2017	RD	3.5332	2.0000			0.65652	0.05561	0.22829	0.00329	29798.91	29805.81	71.88	23.20
	WD	3.1458	1.4620			0.80552	0.04206	0.10070	0.00200	28212.50	28226.29	46.04	14.40
	LND	0.7934	0.6941			0.94677	0.02191	0.06910	0.00054	27014.05	27027.84	63.20	23.77
	GD	1.2846	2.1978			0.86421	0.03515	0.10403	0.00140	27691.15	27704.94	39.97	18.09
	IGD	2.8232	4.6799			0.96036	0.01907	0.05916	0.00041	26871.41	26885.21	59.39	14.99
	NGD	12.4834	0.6512			0.75644	0.04696	0.13649	0.00250	28739.76	28753.55	57.06	14.63
	PID	1.0358	0.8489			0.74410	0.04786	0.11681	0.00260	28912.67	28926.46	49.43	16.26
	PAD	1.0433	0.8326			0.73551	0.04861	0.11017	0.00268	28970.61	28984.40	49.96	17.02
	PLD	1.1868	0.4656			0.79650	0.04289	0.10025	0.00208	28339.30	28353.09	45.28	15.55
	PSD	1.1418	0.5215			0.81151	0.04136	0.10542	0.00194	28210.84	28224.64	43.34	15.57
	PGWD	0.9376	4.8398	0.1383		0.98771	0.01054	0.02946	0.00013	26752.68	26773.38	105.37	47.54
	GPGWD	0.8011	5.7166	0.1549	0.5000	0.97908	0.01392	0.04368	0.00025	26681.46	26709.05	57.42	15.44
2018	RD	3.4619	2.0000			0.64997	0.05555	0.25417	0.00328	33471.04	33478.03	76.12	27.08
	WD	2.9776	1.3724			0.85434	0.03575	0.09831	0.00145	30852.27	30866.25	40.39	16.56
	LND	0.7119	0.7649			0.96563	0.01735	0.04822	0.00034	29998.84	30012.83	87.33	55.70
	GD	1.4028	1.9253			0.90018	0.02964	0.09727	0.00100	30412.01	30426	35.23	19.31
	IGD	2.7008	3.2871			0.95855	0.01898	0.02192	0.00041	30465.87	30479.86	99.46	43.00
	NGD	11.9846	0.5852			0.80093	0.04179	0.13697	0.00198	31550.2	31564.19	56.92	16.46
	PID	1.0026	0.9022			0.79730	0.04185	0.11486	0.00198	31619.28	31633.27	45.91	19.24
	PAD	1.0065	0.8925			0.79230	0.04233	0.10924	0.00203	31652.38	31666.37	46.13	19.87
	PLD	1.1290	0.5195			0.84707	0.03653	0.09905	0.00151	30980.46	30994.44	39.95	17.82
	PSD	1.0898	0.5743			0.85823	0.03525	0.10322	0.00141	30877.28	30891.27	38.10	17.63
	PGWD	0.9753	3.0335	0.2293		0.97968	0.01326	0.04099	0.00021	29800.45	29821.43	76.59	27.38
	GPGWD	0.8926	3.1428	0.2457	0.7569	0.97739	0.01408	0.04464	0.00026	29795.5	29823.48	64.63	18.54
2019	RD	3.2856	2.0000			0.67252	0.05765	0.23753	0.00356	33911.58	33918.64	72.33	23.87
	WD	2.8950	1.4320			0.83792	0.04066	0.10037	0.00191	31788.18	31802.29	34.30	14.33
	LND	0.7016	0.7174			0.96107	0.01983	0.05776	0.00045	30562.97	30577.08	121.48	34.26
	GD	1.2404	2.1013			0.89067	0.03340	0.10028	0.00129	31231.09	31245.2	36.32	17.65
	IGD	2.6064	3.9014			0.96657	0.01851	0.04227	0.00040	30601.77	30615.87	124.84	24.17
	NGD	10.7953	0.6287			0.78889	0.04634	0.13594	0.00248	32436.63	32450.74	54.17	14.55
	PID	1.0386	0.8971			0.77719	0.04728	0.11514	0.00258	32674.58	32688.68	39.52	16.34
	PAD	1.0430	0.8868			0.77169	0.04783	0.10918	0.00264	32712.56	32726.67	39.84	17.01
	PLD	1.1739	0.5129			0.82989	0.04153	0.10034	0.00199	31947.08	31961.19	34.94	15.46
	PSD	1.1329	0.5677			0.84186	0.04012	0.10517	0.00186	31829.1	31843.21	34.30	14.53
	PGWD	0.9063	3.8575	0.1779		0.98482	0.01231	0.03472	0.00019	30351.3	30372.45	147.17	38.17
	GPGWD	0.8004	4.1750	0.1981	0.6228	0.97943	0.01448	0.04179	0.00029	30322.8	30351.05	112.99	19.07

Table 7.

Performance analysis results for Eskişehir Station.

	Distributions	$k$	$c$	$p$	$b$	R²	RMSE	KS	CHI	AIC	BIC	MAPE	PDE
2017	RD	1.7876	2.0000			0.95603	0.03789	0.03882	0.00164	15916.27	15923.14	45.40	0.82
	WD	1.8236	2.2110			0.98641	0.02152	0.04507	0.00062	15794.32	15808.07	45.69	2.71
	LND	0.3557	0.5272			0.96966	0.03091	0.03745	0.00127	16224.72	16238.47	118.97	34.79
	GD	0.3828	4.2172			0.99040	0.01733	0.02696	0.00040	15653.95	15667.70	34.01	1.87
	IGD	1.6143	4.9142			0.94839	0.03999	0.05516	0.00213	16540.34	16554.09	113.43	28.98
	NGD	3.1954	1.2506			0.99383	0.01439	0.04333	0.00028	15709.32	15723.06	44.01	3.71
	PID	1.6964	0.9425			0.98204	0.02422	0.06596	0.00078	16081.90	16095.65	50.97	4.37
	PAD	1.7020	0.9349			0.98047	0.02523	0.06379	0.00085	16089.94	16103.69	51.25	4.73
	PLD	1.8607	0.5696			0.98853	0.01963	0.04734	0.00051	15815.43	15829.18	46.00	3.74
	PSD	1.7958	0.6257			0.99133	0.01711	0.04748	0.00039	15780.50	15794.25	44.75	8.95
	PGWD	1.2842	2.8092	0.5549		0.99464	0.01296	0.02551	0.00027	15612.25	15632.87	36.38	1.62
	GPGWD	2.8367	2.6376	0.0014	2895.7291	0.99625	0.01059	0.02774	0.00022	15557.03	15584.53	27.33	1.66
2018	RD	1.6954	2.0000			0.94939	0.04431	0.04402	0.00236	18712.49	18719.56	26.94	0.71
	WD	1.7201	2.1486			0.97611	0.03088	0.05387	0.00143	18639.19	18653.32	33.48	1.95
	LND	0.2844	0.5647			0.93379	0.05045	0.04626	0.00382	19641.57	19655.71	110.8	53.20
	GD	0.3975	3.8311			0.97660	0.03033	0.02756	0.00138	18658.32	18672.46	20.92	5.36
	IGD	1.5228	3.8144			0.86844	0.06892	0.08037	0.00712	20411.91	20426.05	121.9	46.88
	NGD	2.8743	1.1650			0.98306	0.02600	0.05183	0.00101	18592.59	18606.72	32.27	2.47
	PID	1.6966	1.0005			0.96975	0.03426	0.07415	0.00176	18987.5	19001.64	40.94	3.39
	PAD	1.6932	1.0043			0.96899	0.03464	0.07245	0.00180	18987.22	19001.36	41.00	3.73
	PLD	1.8255	0.6347			0.97895	0.02888	0.05670	0.00125	18666.37	18680.51	34.52	2.76
	PSD	1.7720	0.6853			0.98196	0.02680	0.05617	0.00108	18640.24	18654.37	33.35	8.85
	PGWD	1.2834	2.5580	0.6297		0.98585	0.02366	0.03751	0.00112	18524.95	18546.15	20.72	0.74
	GPGWD	2.9936	2.4690	0.0049	951.6721	0.98996	0.01956	0.03511	0.00115	18487.51	18515.78	15.28	0.08
2019	RD	1.6385	2.0000			0.96728	0.03408	0.03170	0.00145	18114.27	18121.33	13.66	1.72
	WD	1.6593	2.1311			0.98907	0.01964	0.03980	0.00064	18059.77	18073.89	17.43	0.78
	LND	0.2397	0.5882			0.90501	0.05729	0.04759	0.00547	19371.17	19385.29	98.09	69.35
	GD	0.4074	3.6060			0.97186	0.03125	0.02994	0.00163	18263.52	18277.64	29.58	9.54
	IGD	1.4692	3.3411			0.81470	0.07899	0.08359	0.01040	20186.89	20201.01	108.8	59.94
	NGD	2.6848	1.1212			0.99030	0.01838	0.03700	0.00056	18047.45	18061.57	15.97	0.72
	PID	1.7135	1.0318			0.98848	0.02017	0.06129	0.00068	18312.47	18326.59	25.96	2.35
	PAD	1.7070	1.0401			0.98864	0.02000	0.06065	0.00067	18311.19	18325.31	25.83	2.46
	PLD	1.8209	0.6717			0.99172	0.01699	0.04324	0.00048	18083.72	18097.83	18.31	1.38
	PSD	1.7705	0.7200			0.99275	0.01584	0.04224	0.00042	18076.45	18090.57	17.40	8.18
	PGWD	1.4349	2.2881	0.8052		0.98992	0.01866	0.03260	0.00087	18042.55	18063.72	12.97	0.16
	GPGWD	4.6133	2.2550	0.0279	384.9182	0.99133	0.01718	0.03164	0.00147	18036.45	18064.68	10.81	0.02

Table 8.

Performance analysis results for Mihalıççık Station.

	Distributions	$k$	$c$	$p$	$b$	R²	RMSE	KS	CHI	AIC	BIC	MAPE	PDE
2017	RD	3.8678	2.0000			0.91346	0.02350	0.12063	0.00058	33940.94	33947.97	59.83	15.28
	WD	3.6617	1.6684			0.94461	0.01947	0.06619	0.00043	33383.30	33397.36	45.85	9.48
	LND	0.9804	0.6551			0.97625	0.01250	0.02042	0.00018	33018.36	33032.42	95.06	43.95
	GD	1.2207	2.6651			0.98289	0.01069	0.04353	0.00013	32931.68	32945.74	25.70	8.71
	IGD	3.2532	6.0074			0.95082	0.01818	0.04024	0.00037	33297.02	33311.08	95.28	32.89
	NGD	14.9600	0.7975			0.92953	0.02178	0.08745	0.00053	33636.83	33650.89	54.03	10.67
	PID	1.1150	0.7041			0.94053	0.01984	0.07296	0.00044	33596.87	33610.93	45.94	12.24
	PAD	1.1399	0.6672			0.92828	0.02174	0.07182	0.00053	33710.05	33724.11	48.20	13.03
	PLD	1.3251	0.3385			0.94794	0.01877	0.06342	0.00040	33399.80	33413.86	44.22	10.95
	PSD	1.2560	0.3977			0.96142	0.01622	0.06110	0.00030	33250.18	33264.24	40.09	10.97
	PGWD	1.7759	2.6583	0.3655		0.99088	0.00763	0.02271	0.00007	32835.44	32856.54	28.02	2.16
	GPGWD	1.7177	2.6803	0.3718	0.9143	0.99041	0.00782	0.02329	0.00008	32837.27	32865.39	27.26	1.77
2018	RD	3.9313	2.0000			0.88196	0.02673	0.14638	0.00075	35591.43	35598.5	68.14	18.45
	WD	3.6840	1.6328			0.92065	0.02259	0.07897	0.00056	34837.04	34851.17	55.06	12.65
	LND	0.9864	0.6441			0.97932	0.01123	0.01965	0.00014	33984.95	33999.08	80.67	25.92
	GD	1.2313	2.6592			0.96936	0.01395	0.06031	0.00022	34211.71	34225.84	42.64	14.55
	IGD	3.2742	6.3631			0.96231	0.01526	0.02813	0.00026	34122.44	34136.57	72.14	18.30
	NGD	15.4552	0.7785			0.90106	0.02505	0.10598	0.00069	35200.94	35215.07	62.93	13.50
	PID	1.1009	0.7098			0.91651	0.02288	0.08808	0.00058	35075.93	35090.06	56.45	15.95
	PAD	1.1221	0.6764			0.90287	0.02460	0.08570	0.00067	35217.5	35231.63	58.15	16.72
	PLD	1.3016	0.3456			0.92472	0.02191	0.07727	0.00053	34846.87	34861	54.40	14.52
	PSD	1.2387	0.4025			0.94054	0.01956	0.07659	0.00042	34653.4	34667.52	51.37	14.64
	PGWD	1.6023	3.0137	0.2964		0.98557	0.00934	0.02536	0.00010	33937.95	33959.15	42.31	2.96
	GPGWD	1.6146	3.0059	0.2949	1.0233	0.98577	0.00928	0.02526	0.00011	33939.94	33968.19	42.62	3.14
2019	RD	3.9960	2.0000			0.87399	0.02736	0.14635	0.00079	36465.41	36472.47	61.70	17.75
	WD	3.7058	1.5809			0.93906	0.01940	0.06571	0.00042	35473.91	35488.04	38.40	10.01
	LND	0.9717	0.7029			0.96031	0.01556	0.02537	0.00027	35293.55	35307.68	150.37	65.22
	GD	1.3906	2.3789			0.97208	0.01296	0.04760	0.00019	35096.59	35110.72	40.07	8.23
	IGD	3.3081	5.0575			0.92393	0.02188	0.05828	0.00054	35779.72	35793.86	162.76	49.03
	NGD	15.9683	0.7268			0.91672	0.02261	0.09207	0.00058	35807.63	35821.76	47.71	11.21
	PID	1.0677	0.7304			0.93256	0.02020	0.07664	0.00046	35752.63	35766.76	42.20	12.80
	PAD	1.0903	0.6945			0.92139	0.02176	0.07544	0.00053	35848.07	35862.21	42.81	13.60
	PLD	1.2610	0.3594			0.94181	0.01889	0.06554	0.00040	35508.89	35523.02	40.02	11.30
	PSD	1.1954	0.4205			0.95403	0.01681	0.06374	0.00032	35377.73	35391.86	39.51	11.00
	PGWD	1.8030	2.3888	0.3964		0.98042	0.01075	0.02752	0.00014	35017.86	35039.06	56.30	2.87
	GPGWD	1.6704	2.4262	0.4096	0.8310	0.97930	0.01105	0.02810	0.00016	35019.2	35047.47	53.87	2.05

Table 9.

Performance analysis results for Seyitgazi Station.

	Distributions	$k$	$c$	$p$	$b$	R²	RMSE	KS	CHI	AIC	BIC	MAPE	PDE
2017	RD	3.2498	2.0000			0.90332	0.02745	0.13225	0.00080	31328.01	31335.03	63.50	16.63
	WD	3.0251	1.5919			0.97165	0.01489	0.04791	0.00025	30454.01	30468.05	41.67	8.81
	LND	0.7689	0.6984			0.98964	0.00962	0.02317	0.00010	30235.57	30249.60	176.99	64.80
	GD	1.1305	2.3872			0.99333	0.00711	0.03620	0.00006	30119.50	30133.54	26.30	6.35
	IGD	2.6987	4.2944			0.97694	0.01409	0.03621	0.00022	30479.96	30494.00	169.99	46.17
	NGD	10.5611	0.7329			0.95447	0.01882	0.07241	0.00040	30735.44	30749.48	50.93	10.20
	PID	1.1179	0.8203			0.94671	0.02017	0.06315	0.00046	30953.63	30967.66	44.83	11.21
	PAD	1.1342	0.7931			0.94139	0.02114	0.06072	0.00051	30997.93	31011.97	45.94	11.82
	PLD	1.2896	0.4376			0.96882	0.01553	0.04826	0.00027	30547.71	30561.75	40.70	9.79
	PSD	1.2294	0.5000			0.97575	0.01369	0.04836	0.00021	30460.54	30474.58	37.42	10.02
	PGWD	1.4561	2.3909	0.3931		0.99759	0.00473	0.03144	0.00003	30079.10	30100.16	41.87	5.70
	GPGWD	0.6476	3.4097	0.3747	0.1750	0.99704	0.00474	0.01897	0.00003	29996.58	30024.66	26.71	0.37
2018	RD	3.3245	2.0000			0.91374	0.02768	0.11483	0.00082	32452.37	32459.45	53.86	15.51
	WD	3.1721	1.7096			0.94015	0.02387	0.06682	0.00066	32011.08	32025.23	45.86	10.85
	LND	0.8499	0.6189			0.99257	0.00843	0.01931	0.00008	31221.91	31236.06	27.61	24.15
	GD	0.9781	2.8745			0.98360	0.01230	0.04519	0.00017	31386.02	31400.16	34.76	12.11
	IGD	2.8117	6.0508			0.98454	0.01209	0.01902	0.00017	31302.91	31317.06	27.26	16.96
	NGD	11.0522	0.8391			0.92670	0.02612	0.08887	0.00079	32266.69	32280.84	50.19	12.02
	PID	1.1737	0.7537			0.91897	0.02714	0.07780	0.00085	32386.38	32400.52	47.42	13.69
	PAD	1.1934	0.7232			0.90834	0.02883	0.07448	0.00096	32495.02	32509.17	48.81	14.39
	PLD	1.3737	0.3813			0.93935	0.02382	0.06532	0.00065	32052.31	32066.46	45.22	12.53
	PSD	1.3095	0.4400			0.95245	0.02115	0.06457	0.00052	31885.46	31899.61	43.06	13.09
	PGWD	1.4381	3.0769	0.3044		0.99570	0.00632	0.02006	0.00005	31200.26	31221.48	18.07	2.66
	GPGWD	1.0899	3.5117	0.3265	0.4641	0.99344	0.00764	0.01948	0.00008	31183.8	31212.1	20.01	1.14
2019	RD	3.4345	2.0000			0.91208	0.02847	0.08958	0.00087	31681.23	31688.29	52.72	7.98
	WD	3.3723	1.8635			0.92287	0.02715	0.06702	0.00086	31602.49	31616.61	47.38	5.62
	LND	0.9313	0.5772			0.99383	0.00769	0.02340	0.00007	30948.14	30962.25	142.35	25.04
	GD	0.9071	3.2830			0.97219	0.01603	0.04984	0.00030	31047.36	31061.47	24.91	5.26
	IGD	2.9780	7.5880			0.99279	0.00844	0.01878	0.00008	30983.5	30997.62	130.42	19.11
	NGD	11.7960	0.9586			0.91448	0.02824	0.08296	0.00093	31673.13	31687.25	51.84	7.11
	PID	1.2232	0.6835			0.89968	0.03036	0.07815	0.00108	31892	31906.12	45.69	7.24
	PAD	1.2502	0.6471			0.88598	0.03237	0.07504	0.00122	32034.96	32049.08	49.02	8.02
	PLD	1.4656	0.3209			0.91970	0.02747	0.06630	0.00088	31653.13	31667.25	45.35	6.76
	PSD	1.3895	0.3780			0.93540	0.02463	0.06575	0.00071	31473.79	31487.91	40.18	7.24
	PGWD	1.5685	3.4150	0.2879		0.99335	0.00791	0.02547	0.00008	30912.8	30933.97	80.61	7.98
	GPGWD	0.9979	4.7831	0.2882	0.2548	0.99499	0.00676	0.02186	0.00006	30800.92	30829.16	24.05	1.28

At Anadolu University Station (Table 4), the GPGWD most closely describes the wind speed data for the three sample years and is the only model that outperforms all competitors in almost all goodness-of-fit criteria for 2017. The second-best performance varies between the GD and PGWD distributions, and depending on the evaluation criteria, both models can be considered strong alternatives. The other power-based distributions demonstrated acceptable performance, indicating their suitability for modeling wind speed data. For 2018, PSD and PLD distributions show an acceptable performance. PSD offers the best fit by achieving the lowest RMSE, KS, and CHI values and the highest R². PLD performs very similarly to PSD, providing a good fit with low AIC and BIC values and competitive error measures. PGWD and GPGWD represent strong alternatives. For the 2019 dataset, GPGWD stands out as the most suitable distribution for wind speed modeling by achieving the best results in key performance criteria such as R², RMSE, KS, CHI², AIC, and BIC. PGWD, on the other hand, can be considered a strong alternative as it performs similarly to GPGWD. The PSD and PLD also demonstrate acceptable modeling performance, particularly by yielding very low values in error metrics such as MAPE and PDE. Furthermore, other power-transformed distributions—PID, PAD, PLD, and PSD—generally provide reasonable and reliable results, positioning them as trustworthy options for modeling wind speed data.

According to the annual wind speed analysis results presented in Table 5 (Performance analysis results for Beylikova Station), for 2017, GPGWD stands out by demonstrating the best performance across all key criteria (R², RMSE, KS, CHI, AIC, BIC). Additionally, PGWD can be considered a strong alternative, showing performance close to GPGWD (R² = 0.98428, RMSE = 0.01304). PSD and PLD, on the other hand, offer low values in error metrics (MAPE and PDE), thus demonstrating acceptable performance in terms of modeling. For 2018 and 2019, GPGWD model consistently provides the best overall fit, with R² values exceeding 0.99 and the lowest RMSE, KS, and CHI, demonstrating its superior ability to represent wind speed distribution. Second-best performance is generally observed for the PGWD. PSD and PLD offer strong performance in terms of error metrics and are acceptable for modeling purposes. Other power-transformed distributions (PID, PAD) also generally provide reasonable and reliable results. For Beylikova Station, the weakest performance among the evaluated distributions is consistently observed for RD, which exhibits substantially lower R² values and higher RMSE and KS statistics across all years.

Based on the annual wind speed performance results obtained for Çifteler Station (Table 6), in the analysis conducted for 2017, the PGWD, and followed GPGWD demonstrate high performance across all main criteria (R², RMSE, KS, CHI). The PSD and PLD distributions showed acceptable modeling performance, particularly by providing low values in error metrics. The PID and PAD distributions also provide reasonable and reliable results. For 2018, PGWD again showed the best performance across all main criteria. GPGWD can be considered a strong alternative with results close to PGWD. PSD and PLD offered acceptable modeling performance thanks to their low values in error metrics, while PID and PAD distributions also showed generally reasonable and reliable performance. In the 2019 analyses, the PGWD distribution again showed superior performance in the main criteria, while GPGWD similarly demonstrated high performance and can be used as an alternative. The PSD and PLD distributions continued to offer low values in the error metrics, while the PID and PAD distributions showed reasonable and reliable performance.

For Eskişehir Station (Table 7), in the performance analysis conducted for 2017, the GPGWD distribution showed the highest performance across all main criteria. The PGWD distribution emerged as a strong alternative, yielding results close to those of GPGWD. The PSD and PLD distributions showed acceptable performance in terms of modeling, particularly by providing low values in error metrics (MAPE and PDE), while the PID and PAD distributions provided reasonable and reliable results. In the 2018 analyses, the GPGWD distribution continued to show superior performance across all main criteria. PGWD can again be considered a strong alternative. For 2019, the GPGWD distribution again showed the best performance in the main criteria. PGWD was a strong alternative with performance close to GPGWD. The PSD and PLD distributions offered reliable performance in terms of modeling with low error metrics, while the PID and PAD distributions also provided reasonable and acceptable performance.

Across all evaluation years (2017–2019) for Mihalıççık Station (Table 8), the PGWD and GPGWD distributions show the best performance in terms of all main criteria. Among the classical distributions, the GD distribution demonstrates consistently strong performance.

The performance assessment for Seyitgazi Station (Table 9) shows that the PGWD and GPGWD models consistently deliver the most accurate representation of the wind speed distribution across all years for almost all criteria. Among classical distributions, GD and WD also demonstrated strong performance, making them reliable alternatives. Among the classical distributions, GD and WD stood out as strong second options, particularly showing good performance in terms of PDE and R²

As a result, taking into account all performances, GPGWD and PGWD consistently provide the best fit for yearly wind speed data, demonstrating superior modeling capabilities across various stations. Other power alternative distributions in presented in this study, perform at an acceptable level, offering reliable results when compared to traditional models.

To enable a clearer comparison of the best-performing distributions, the two distributions with the highest performance for each station are also presented in separate Table 10 and Table 11, created for this purpose. These tables clearly show the distributions that perform best under the relevant criterion.

Table 10.

Top 1st and 2nd best-performing pdfs for Anadolu University, Beylikova and Çifteler Stations.

Stations	Year	Order	R²	RMSE	KS	CHI	AIC	BIC	MAPE	PDE
Anadolu University	2017	1	GPGWD (0.98965)	GPGWD (0.00882)	GPGWD (0.02733)	GPGWD (0.00011)	GPGWD (30273.34)	GPGWD (30301.36)	PSD (27.94)	GPGWD (0.64)
	2017	2	GD (0.97767)	GD (0.01311)	GD (0.03835)	GD (0.00020)	GD (30394.64)	GD (30408.65)	PLD (29.23)	PGWD (1.40)
	2018	1	PSD (0.98721)	PSD (0.00968)	GD (0.01862)	PSD (0.00011)	PLD (30865.18)	PLD (30879.19)	PSD (19.81)	PLD (0.04)
	2018	2	WD (0.98539)	WD (0.01041)	WD (0.02031)	WD (0.00013)	PSD (30866.08)	PSD (30880.08)	PLD (21.54)	PGWD (0.16)
	2019	1	GD (0.97738)	GPGWD (0.01314)	GD (0.02864)	GD (0.00021)	GPGWD (28672.90)	GPGWD (28700.67)	WD (14.43)	PSD (0.001)
	2019	2	GPGWD (0.97732)	GD (0.01326)	GPGWD (0.03396)	GPGWD (0.00027)	PGWD (28686.96)	WD (28707.27)	NGD (14.68)	PID (0.005)
Beylikova	2017	1	GPGWD (0.99404)	GPGWD (0.00807)	GPGWD (0.03545)	GPGWD (0.00009)	GPGWD (28024.30)	GPGWD (28052.36)	PLD (22.53)	WD (2.83)
	2017	2	WD (0.98845)	WD (0.01055)	LND (0.04424)	WD (0.00013)	GD (28329.71)	GD (28343.74)	WD (22.71)	GD (3.040)
	2018	1	GPGWD (0.99785)	GPGWD (0.00525)	GD (0.06713)	GPGWD (0.00004)	GPGWD (26501.69)	GPGWD (26529.73)	GPGWD (22.20)	GPGWD (4.41)
	2018	2	PGWD (0.99345)	GD (0.00997)	GPGWD (0.03031)	LND (0.00011)	PGWD (26684.09)	LND (26699.32)	GD (28.11)	WD (13.24)
	2019	1	GPGWD (0.99710)	GPGWD (0.00605)	IGD (0.03145)	GPGWD (0.00006)	GPGWD (28758.97)	GPGWD (28787.27)	WD (27.27)	GPGWD (6.68)
	2019	2	IGD (0.99339)	IGD (0.01059)	GPGWD (0.03352)	IGD (0.00013)	PGWD (2900.51)	PGWD (29026.39)	PLD (30.44)	GD (6.74)
Çifteler	2017	1	PGWD (0.98771)	PGWD (0.01054)	PGWD (0.02946)	PGWD (0.00013)	GPGWD (26681.46)	GPGWD (26709.05)	GD (39.96)	WD (14.39)
	2017	2	GPGWD (0.97908)	GPGWD (0.01392)	GPGWD (0.04368)	GPGWD (0.00025)	PGWD (26752.68)	PGWD (26773.38)	PSD (43.33)	NGD (14.62)
	2018	1	PGWD (0.97968)	PGWD (0.01326)	IGD (0.02192)	PGWD (0.00021)	GPGWD (29795.50)	PGWD (29821.43)	GD (35.23)	NGD (16.46)
	2018	2	GPGWD (0.97739)	GPGWD (0.01408)	PGWD (0.04098)	GPGWD (0.00026)	PGWD (29800.45)	GPGWD (29823.48)	PSD (38.09)	WD (16.56)
	2019	1	PGWD (0.98482)	PGWD (0.01231)	PGWD (0.03472)	PGWD (0.00019)	GPGWD (30322.80)	GPGWD (30351.05)	WD (34.29)	WD (14.33)
	2019	2	GPGWD (0.97943)	GPGWD (0.01448)	GPGWD (0.04179)	GPGWD (0.00029)	PGWD (30351.30)	PGWD (30372.45)	PSD (19.07)	NGD (14.54)

Table 11.

Top 1st and 2nd best-performing pdfs for Eskişehir, Mihalıççık and Seyitgazi Stations.

Stations	Year	Order	R²	RMSE	KS	CHI	AIC	BIC	MAPE	PDE
Eskişehir	2017	1	GPGWD (0.99625)	GPGWD (0.01059)	PGWD (0.02551)	GPGWD (0.00022)	GPGWD (15557.03)	GPGWD (15584.53)	GPGWD (27.33)	RD (0.82)
	2017	2	PGWD (0.99464)	PGWD (0.01296)	GD (0.02696)	PGWD (0.00027)	PGWD (15612.25)	PGWD (15632.87)	GD (34.011)	PGWD (1.61)
	2018	1	GPGWD (0.9899)	GPGWD (0.01958)	GD (0.0275)	NGD (0.00101)	GPGWD (18487.521)	GPGWD (18515.78)	GPGWD (15.28)	GPGWD (0.057)
	2018	2	PGWD (0.98585)	PGWD (0.02366)	PGWD (0.03751)	PGWD (0.00112)	PGWD (18524.95)	PGWD (18546.15)	PGWD (20.72)	PGWD (0.74)
	2019	1	PSD (0.99275)	PSD (0.01584)	GD (0.04224)	PSD (0.00041)	GPGWD (18036.45)	NGD (18061.56)	GPGWD (10.81)	GPGWD (0.014)
	2019	2	PLD (0.9917)	PDL (0.01698)	GPGWD (0.03164)	PLD (0.00048)	PGWD (18042.55)	PGWD (18063.72)	PGWD (12.97	PGWD (0.16)
Mihalıççık	2017	1	PGWD (0.99088)	PGWD (0.00763)	LND (0.02042)	PGWD (0.00007)	PGWD (32835.44)	PGWD (32856.54)	LND (25.703)	GPGWD (1.76)
	2017	2	GPGWD (0.99041)	GPGWD (0.00782)	PGWD (0.0227)	GPGWD (0.00008)	GPGWD (32837.27)	GPGWD (32865.39)	GPGWD (27.26)	PGWD (2.15)
	2018	1	GPGWD (0.98577)	GPGWD (0.00928)	LND (0.01965)	PGWD (0.00011)	PGWD (33939.94)	PGWD (33968.19)	PGWD (42.62)	PGWD (3.14)
	2018	2	PGWD (0.9855)	PGWD (0.00934)	GPGWD (0.02536)	GPGWD (0.00011)	PGWD (33939.94)	PGWD (33968.19)	PGWD (42.62)	PGWD (3.14)
	2019	1	PGWD (0.98042)	PGWD (0.01750)	LND (0.02536)	PGWD (0.00013)	PGWD (35017.86)	PGWD (35039.06)	WD (38.40)	PGWD (2.05)
	2019	2	GPGWD (0.97930)	GPGWD (0.01105)	PGWD (0.0275)	GPGWD (0.00016)	GPGWD (35019.20)	GPGWD (35047.47)	GPGWD (53.87)	PGWD (2.87)
Seyitgazi	2017	1	PGWD (0.99759)	PGWD (0.00473)	GPGWD (0.01897)	PGWD (0.00003)	GPGWD (29996.58)	GPGWD (30024.66)	GD (26.30)	GPGWD (0.37)
	2017	2	GPGWD (0.99704)	GPGWD (0.00474)	LND (0.02316)	GPGWD (0.00003)	PGWD (30079.10)	PGWD (30100.16)	GPGWD (26.70)	PGWD (5.70)
	2018	1	PGWD (0.99570)	PGWD (0.00632)	IGD (0.02006)	PGWD (0.00005)	GPGWD (31183.80)	GPGWD (31212.10)	PGWD (18.07)	GPGWD (1.14)
	2018	2	GPGWD (0.99344)	GPGWD (0.00764)	GPGWD (0.01948)	GPGWD (0.00008)	PGWD (31200.26)	PGWD (31221.48)	GPGWD (20.01)	PGWD (2.66)
	2019	1	GPGWD (0.99499)	GPGWD (0.00676)	IGD (0.01878)	GPGWD (0.00006)	GPGWD (30800.92)	GPGWD (30829.16)	GPGWD (24.05)	GPGWD (1.28)
	2019	2	PGWD (0.99335)	PGWD (0.00791)	GPGWD (0.02186)	LND (0.00006)	PGWD (30912.80)	PGWD (30933.97)	GD (24.91)	GD (5.26)

Overall, upon examining Table 10 and Table 11 and it is observed that the R² values of the PGWD and GPGWD distributions are generally high and the RMSE, KS, and AIC values are low for most years. However, other distributions except RD provide good performance in certain cases.

Additionally, performance analyses were carried out on both monthly and seasonal wind speed data, yielding consistent results. Furthermore, wind speed data were extrapolated to a height of 100 m, and comprehensive performance evaluations were conducted, demonstrating similar trends. These findings are available from the authors upon request for further scrutiny and verification.

The pdfs and histograms for each station, as illustrated in Figure 2, demonstrate the modeling efficiency of the PGWD and GPGWD. These visualizations clearly indicate that the proposed distribution provides a superior fit compared to other models, showcasing its robustness and effectiveness in capturing the underlying patterns of the wind speed data.

Figure 2.

Empirical histogram of wind speed observations for the study regions and the fitted distributions considered in this study.

Conclusions and discussion

This study introduces new distributions: GPGWD, PGWD, PID, PLD, PSD, and PAD, and provides a comprehensive evaluation of these distributions in comparison to well-known models across various stations in Eskişehir, Türkiye for 2017, 2018 and 2019. The results show that GPGWD and PGWD performed best across all years and stations. GPGWD, in particular, demonstrated strong performance in terms of R² and RMSE at stations such as Eskişehir and Çifteler, while also achieving the lowest values in most KS and CHI tests, showing consistent superior fit. PGWD, on the other hand, ranked first at stations such as Beylikova and Seyitgazi, demonstrating a performance very close to GPGWD and even competing with it at some locations. At stations such as Anadolu University and Eskişehir, PSD and PLD distributions yielded competitive results in certain metrics (especially MAPE and PDE for some years), with PSD even showing the best performance in R² and RMSE in some years. However, despite all these exceptional cases, the overall analysis shows that GPGWD is the distribution model least affected by geographical conditions and the most balanced and stable across all stations. These findings confirm the flexibility provided by power-transformed distributions in wind energy modeling and demonstrate that GPGWD can be used as a reference model. Moreover, the proposed distributions, particularly, GPGWD and PGWD can be incorporated into wind energy software packages, enabling improved modeling and analysis of wind speed characteristics.

Future research could focus on comparing estimators (robust/non-robust) for GPGWD and exploring its applications in other geographical regions. Futhermore, the development and evaluation of mixture extensions of GPGWD and PGWD can also be considered in future studies. Moreover, this distribution can also be used for kriging estimations within a Geographic Information System.

Footnotes

Acknowledgment

The authors would like to express their sincere gratitude to the Eskişehir Meteorology Directorate for providing the wind speed data used in this study. Their support and cooperation greatly contributed to the successful completion of this research.

ORCID iDs

İlhan Usta

Vural Yildirim

Yeliz Mert Kantar

Author contributions

Conceptualization, I.U.; methodology, I.U. and Y.M.K.; software, I.U.; validation, I.U. and V.Y.; formal analysis, I.U. and V.Y.; investigation, I.U. and V.Y.; data curation, I.U. and V.Y.; writing—original draft preparation, Y.M.K.; writing—review and editing, I.U., Y.M.K. and V.Y.; supervision, Y.M.K. All authors have read and agreed to the published version of the manuscript.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Use of AI technology

The authors used ChatGPT (OpenAI) for language editing and grammar checking. All responsibility for the final content rests with the authors.

Data Availability Statement

The data will be made available upon reasonable request.*

Appendix

References

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