Before probabilistic analysis, some measures are given for initial description such as minimum, maximum, mean, variance, skewness, kurtosis and coefficient of variation for wind speed data for all stations.
Table 3
Descriptive assessment of data taken from three stations
Stations
|
Min.\(\left(m{s}^{-1}\right)\)
|
Max.\(\left(m{s}^{-1}\right)\)
|
A.W.S\(\left(m{s}^{-1}\right)\)
|
Variance
|
Skewness
|
Kurtosis
|
C.V (%)
|
Sanghar
|
0.0010
|
20.6430
|
6.3100
|
9.1230
|
0.3074
|
2.7371
|
14.632
|
Sujawal
|
0.0346
|
16.5699
|
6.9743
|
6.2615
|
0.0215
|
3.0295
|
15.101
|
Umerkot
|
0.0126
|
16.9004
|
6.0967
|
7.5987
|
0.3802
|
2.9883
|
45.214
|
The average wind speed (A.W.S) for all observed stations is more than 6\(m{s}^{-1}\) while the observed maximum speed for all stations is more than 16\(m{s}^{-1}\). The calculated skewness for Sanghar, Sujawal and Umerkot stations is 0.3074, 0.0215 and 0.3802 respectively; therefore, all data sets are positively skewed. Moreover, variance, kurtosis and coefficient of variation (C.V) are expounded in Table 3 for all three stations.
It is known that the evidence related to failure rate can lead to an assortment of appropriate distribution. Therefore, we have utilized the total time on test (TTT) plots for the assessment of the failure rate. For constant failure rate, the TTT plot is diagonally straight. The convex shape of the TTT plots show decreasing failure rate while the concave shape of the TTT plots shows an increasing failure rate. Moreover, first convex and then concave shape TTT plots provide the bathtub behavior of failure rate. Since TTT plots are given in Fig. 2 for Sanghar, Sujawal and Umerkot stations show the concave shape thus all data sets contain increasing failure rate and the PBX distribution is appropriate for such kinds of data sets.
Table 4
Estimated parameters for PBX and competitor models at all stations
Stations
|
|
Competitor Models
|
PBX
|
GW
|
PL
|
W
|
BX
|
PLi
|
Li
|
Sanghar
|
\(a\)
|
0.68670
|
1.66970
|
19.4319
|
2.20922
|
1.13190
|
0.09995
|
0.28209
|
\(b\)
|
0.05404
|
1.95430
|
2.24453
|
7.12206
|
0.14861
|
1.55014
|
-
|
\(\theta\)
|
1.39017
|
0.01078
|
1527.22
|
-
|
-
|
-
|
-
|
Sujawal
|
\(a\)
|
0.85960
|
1.27720
|
7.45646
|
3.03933
|
1.91698
|
0.03217
|
0.25741
|
\(b\)
|
0.02672
|
2.65300
|
3.13242
|
7.79176
|
0.16288
|
2.05136
|
-
|
\(\theta\)
|
1.75608
|
0.00315
|
4124.01
|
-
|
-
|
-
|
-
|
Umerkot
|
\(a\)
|
0.94340
|
1.12310
|
42.1103
|
2.28708
|
1.27900
|
0.08712
|
0.29106
|
\(b\)
|
0.08308
|
2.21200
|
2.34454
|
6.87571
|
0.16127
|
1.65276
|
-
|
\(\theta\)
|
1.26082
|
0.01082
|
4112.55
|
-
|
-
|
-
|
-
|
A comparison was made to investigate the performances for PBX, GW, PL, W, BX, PLi and Li distributions at all coastal stations. Due to large sample data, we have utilized maximum likelihood estimation (MLE) for the estimation of parameters. The ML estimates for Sanghar under PBX distribution is computed as \(a=0.68670, b=0.05404\) and \(\theta =1.39017\). For Sujawal region estimated parameters for PBX distribution are \(a=0.85960, b=0.02672\) and \(\theta =1.75608\) and for Umerkot region estimated parameters for PBX distribution are \(a=0.94340, b=0.08308\) and \(\theta =1.26082\).
Table 5
Selection criteria for Sanghar, Sujawal and Umerkot stations
Stations
|
Model
|
Selection Criteria (Ranks)
|
Rank Totality
|
Rank
|
\(-2\mathcal{l}\)
|
\(AIC\)
|
\(KS\)
|
\({R}^{2}\)
|
\(Chi.sq\)
|
\(RMSE\)
|
Sanghar
|
PBX
|
261473.9(2)
|
261479.9(2)
|
0.0319(1)
|
0.9974(1)
|
0.00036(1)
|
0.0134(1)
|
8
|
1
|
GW
|
261458.1(1)
|
261464.1(1)
|
0.0339(2)
|
0.9972(2)
|
0.00162(2)
|
0.0139(2)
|
10
|
2
|
PL
|
262174.9(4)
|
262180.9(4)
|
0.0512(4)
|
0.9932(4)
|
0.00511(4)
|
0.0221(4)
|
24
|
4
|
W
|
261876.0(3)
|
261880.0(3)
|
0.0455(3)
|
0.9945(3)
|
0.00305(3)
|
0.0197(3)
|
18
|
3
|
BX
|
262208.4(5)
|
262212.4(5)
|
0.0532(5)
|
0.9926(5)
|
0.00674(6)
|
0.0228(5)
|
31
|
5
|
PLi
|
262876.1(6)
|
262880.1(6)
|
0.0561(6)
|
0.9914(6)
|
0.00551(5)
|
0.0249(6)
|
35
|
6
|
Li
|
280141.0(7)
|
280143.0(7)
|
0.1241(7)
|
0.8549(7)
|
0.03921(7)
|
0.0836(7)
|
42
|
7
|
Sujawal
|
PBX
|
226951.8(1)
|
226957.8(1)
|
0.0266(1)
|
0.9983(1)
|
0.00069(1)
|
0.0161(1)
|
6
|
1
|
GW
|
227373.7(3)
|
227379.7(3)
|
0.0354(4)
|
0.9874(6)
|
0.01185(6)
|
0.0282(5)
|
27
|
5
|
PL
|
227696.7(4)
|
227702.7(4)
|
0.0345(3)
|
0.9905(5)
|
0.00143(2)
|
0.0251(4)
|
22
|
3
|
W
|
227120.4(2)
|
227124.4(2)
|
0.0327(2)
|
0.9946(2)
|
0.00147(3)
|
0.0172(2)
|
13
|
2
|
BX
|
229412.4(6)
|
229416.4(6)
|
0.0467(6)
|
0.9844(3)
|
0.00531(5)
|
0.0320(6)
|
32
|
6
|
PLi
|
228117.4(5)
|
228121.4(5)
|
0.0411(5)
|
0.9927(4)
|
0.00352(4)
|
0.0223(3)
|
26
|
4
|
Li
|
264292.8(7)
|
264294.8(7)
|
0.2075(7)
|
0.6135(7)
|
0.04137(7)
|
0.1301(7)
|
42
|
7
|
Umerkot
|
PBX
|
251171.3(1)
|
251177.3(1)
|
0.0203(1)
|
0.9997(1)
|
0.00251(2)
|
0.0067(1)
|
7
|
1
|
GW
|
251179.6(2)
|
251185.6(2)
|
0.0222(2)
|
0.9991(2)
|
0.00151(1)
|
0.0076(2)
|
11
|
2
|
PL
|
251248.3(4)
|
251254.3(4)
|
0.0262(4)
|
0.9990(3)
|
0.01044(5)
|
0.0082(4)
|
24
|
4
|
W
|
251216.0(3)
|
251220.0(3)
|
0.0248(3)
|
0.9989(4)
|
0.00364(3)
|
0.0080(3)
|
19
|
3
|
BX
|
251581.1(5)
|
251585.1(5)
|
0.0364(6)
|
0.9971(6)
|
0.01196(6)
|
0.0140(6)
|
34
|
6
|
PLi
|
251695.7(6)
|
251699.7(6)
|
0.0358(5)
|
0.9977(5)
|
0.00685(4)
|
0.0126(5)
|
31
|
5
|
Li
|
273604.2(7)
|
273606.2(7)
|
0.1516(7)
|
0.8020(7)
|
0.05123(7)
|
0.0949(7)
|
42
|
7
|
The estimated parameters for all other competitor models are given in Table 4 for wind speed analysis. The performance of these estimated models are compared on the basis of numerous goodness of fit criteria (AIC, KS, RMSE, R2, and Adj. R2) defined in subsection 3.2. The findings for these measures are listed in Table 5 for all observed stations. It is known that the minimum value of -2 log-likelihood \(\left(-2\mathcal{l}\right)\), AIC, KS, \(Chi.sq\), and RMSE for any model corresponds to a better fitting model. On the other hand, the larger the value of R2 better would be the model for the given data. Therefore, we have computed and ranked the good of fit measures in Table 5 for all the considered models.
The results based on fitted PDF and CDF curves indicate that the PBX distribution has the best fit for Sanghar, Sujawal, and Umerkot stations as compared to the other wind speed distributions. Moreover, the GW distribution also has considerable fit as compared to PL, PLi, W, BX and Li distributions.
It is observed that the well-known and broadly accepted Weibull distribution for wind speed analysis has the least enactment as that of PBX and GW distribution and was not among the best-fitted distributions. Therefore, it is assumed that the W distribution could not be considered as the best wind speed distribution for all localities. In addition, the lowest performance and capability of the PLi and Li distributions has seen during the estimation of wind speed data for all stations. The results are given in Table 5 are also justified with graphical representation.
Consequently, Figs. 3–5 endorse the results given in Table 5 that the PBX distribution has a better fit than that of the other well-known wind speed distributions for Sanghar, Sujawal and Umerkot wind stations respectively. Figures 3–5 represent the fitted density curve, fitted distribution curve and PP plot for estimated PBX distribution for Sanghar, Sujawal and Umerkot wind station data sets respectively.
4.1 Power Density Error
Another selection criteria used for wind speed analysis was power density error (PE). This criterion was based on the relative difference of two average powers. One was mean power density (MPD) calculated from observations and other was mean power density (PDM) calculated from the model under study. The criterion was defined as
$$PE=\left|\frac{MPD-P{D}_{d}}{MPD}\right| \left(8\right)$$
The formula for \(MPD\) and \({PD}_{d}\) are defined respectively
$$MPD=\frac{1}{2m}\rho A\sum _{k=1}^{m}{z}_{i}^{3} \left(9\right)$$
Here, \(\rho =P/RT\) is air density (kg/m3) at sea level. The air density is proportional to surface pressure (P) and air temperature (T) while R is the proportionality gas constant with value\(287 (J/Kg)\). However, A represents turbine blade sweep area (m3), and d stands for the distribution of interest for example, PBX distribution.
A comparison was also made between coastal stations by using the power density error and results are listed in Table 6. Overall, it was found that the PBX and GW distribution had the least power density error as compared to the other competitor models for all the stations with less estimation cost and greater efficiency.