## 4 Case Studies  
### Data  
The solar irradiance data used in this paper is measured Global Horizontal Irradiance data from a grid-connected PV plant situated in Hohhot, Inner Mongolia, China (111.12\({}^{\circ}\) E, 40.73\({}^{\circ}\) N, 5 MW) from 12:00 a.m. to 11:30 p.m. with 30 min sampling interval. The time range of measuring data is from 1 January 2012 to 31 December 2012, with 310 days of available data. Figure 6 shows a photo of the grid-connected PV farm. Figure 5: Factors influencing classification accuracy. Figure 6: The grid-connected photovoltaic (PV) farm used in the case study in this work. The extraterrestrial solar irradiance is calculated by the following formula [61]:  
\[G_{e}=G_{sc}\bigg{(}1+0.33\cos\frac{360n}{365}\bigg{)}(\cos\delta\cos\phi\cos \omega+\sin\delta\sin\phi), \tag{4}\]  
where \(G_{sc}\) is solar constant (1368 W/m\({}^{2}\)), \(n\) is the date sequence number in one year, \(n\in[1,365]\) is the date sequence number in one year, \(\phi\), \(\delta\) is solar declination, is latitude and \(\omega\) is solar hour angle. The weather status refers to the meteorological environment at the individual PV plant and can be classified into four generalized weather classes (GWC) named A, B, C and D in light of the meteorological characteristics. The typical weather patterns of the GWC A, B, C, and D can be described as: sunny, cloudy, showers, and torrential rains. By making use of the data of extraterrestrial and surface solar irradiance over a whole day, six indices reflecting the solar irradiance characteristics of different weather status are defined and calculated according to our previous study [8]:  
\[F_{1}=\frac{\sum\limits_{i=1}^{N+1}(G_{s,i-1}+G_{s,i})}{\sum\limits_{i=1}^{N+ 1}(G_{e,i-1}+G_{e,i})} \tag{5}\]  
Normalized root mean square deviation:  
\[F_{2}=\sqrt{\frac{1}{N}\sum\limits_{i=1}^{N}\left(\frac{G_{e,i}}{\max\limits_{ i=1,\ldots,N}\{G_{e,i}\}}\times 100-\frac{G_{s,i}}{\max\limits_{i=1,\ldots,N}\{G_{s,i }\}}\times 100\right)^{2}}, \tag{6}\]  
Maximum value of third-order derivative of solar irradiance difference:  
\[F_{3}=\max\Big{[}\
abla^{3}(G_{e}-G_{s})\Big{]}, \tag{7}\]  
Ratio of maximum solar irradiance:  
\[F_{4}=\frac{\max\limits_{i=1,\ldots,N}\{G_{s,i}\}}{\max\limits_{i=1,\ldots,N}\{ G_{e,i}\}}, \tag{8}\]  
Variance of solar irradiance difference:  
\[F_{5}=\frac{1}{N}\sum\limits_{i=1}^{N}\left[G_{s,i}-\left(\frac{1}{N}\sum \limits_{i=1}^{N}G_{s,i}\right)\right]^{2}, \tag{9}\]  
Inconsistency coefficient:  
\[ \begin{array}{l}F_{6}=\sum\limits_{i=2}^{N}f_{i}(G_{s,i},G_{e,i})\\ \text{where}\\ f_{i}(G_{s,i},G_{e,i})=\left\{\begin{array}{l}1\;\;if\;\;(G_{s,i}-G_{s,i-1 })(G_{e,i}-G_{e,i-1})<0\\ 0\;\;if\;\;(G_{s,i}-G_{s,i-1})(G_{e,i}-G_{e,i-1})\geq 0\end{array}\right.\end{array} . \tag{10}\]  
\(G_{s,i}\) is the discretely sampled data of surface solar irradiance, \(i=1,2,\ldots,N\), \(N\) is the number of sample points during one day, \(G_{e,i}\) is the corresponding extraterrestrial solar irradiance with the same fixed time interval of \(G_{s}\), \(G_{s,0}=G_{s,N+1}=G_{e,0}=G_{e,N+1}=0\). Function \(\
abla^{3}(G_{e}-G_{s})\) indicates the third-order derivative of solar irradiance difference between extraterrestrial and surface solar irradiance. The input feature vectors for the classification model are determined as \(F_{in}=\{F_{1},F_{2},F_{3},F_{4},F_{5},F_{6}\}\) according to Formulas (5-10). The output of the model is \(L_{out}=A,B,C\) or \(D\). The resulting quantities of each GWC data bin are 24, 119, 148 and 19 as shown in Figure 7.  
Typical surface irradiance curves (the red solid lines) with corresponding extraterrestrial irradiance curves (the blue dotted lines) of the four GWCs are illustrated in Figure 8.  
Figure 8: Typical solar irradiance curves of four GWCs. (**a**) Solar irradiance curves of GWC A; (**b**) Solar irradiance curves of GWC B; (**c**) Solar irradiance curves of GWC C; (**d**) Solar irradiance curves of GWC D.  
Figure 7: The quantities of data in each generalized weather classes (GWC) weather class.