## 5 Comparison of Forecasting Results by Different Forecasting Models

Table 1: Optimal parameters’ values and forecasting results by using rolling-MFO-GM (1,1). \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Year**} & \multicolumn{2}{c}{**LSR**} & \multicolumn{2}{c}{**GM (L,l)**} & \multicolumn{2}{c}{**FOA-GM (L,l)**} & \multicolumn{2}{c}{**MFO-GM (L,l)**} & \multicolumn{2}{c}{**Rolling-LSR**} & \multicolumn{2}{c}{**Rolling-GM (L,l)**} & \multicolumn{2}{c}{**Rolling-FOA (l,l)**} \\ \cline{2-13}  & \(a\) & \(b\) & \(a\) & \(b\) & \(a\) & \(b\) & \(a\) & \(b\) & \(a\) & \(b\) & \(a\) & \(b\) & \(a\) & \(b\) \\ \hline
2010 & & & & & & & & & & & & & & & & \\
2011 & & & & & & & & & & & & & & & & \\
2012 & 128.8292 & 39.4175 & \(-\)0.1821 & 313.8648 & 0.1808 & 307.2826 & \(-\)0.2137 & 241.6629 & 162.1653 & 1938.8000 & \(-\)0.1554 & 512.346 & \(-\)0.2041 & 316.606 \\
2013 & & & & & & & & & & & & & \\
2014 & & & & & & & & & & & & & \\  & & & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters’ values determined by different compared forecasting models. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Year**} & \multicolumn{2}{c}{**Actual**} & \multicolumn{2}{c}{**LSR**} & \multicolumn{2}{c}{**GM (L,l)**} & \multicolumn{2}{c}{**FOA-GM (L,l)**} & \multicolumn{2}{c}{**MFO-GM (L,l)**} & \multicolumn{2}{c}{**Rolling-LSR**} & \multicolumn{2}{c}{**Rolling-GM (L,l)**} & \multicolumn{2}{c}{**Rolling-FOA (l,l)**} \\ \cline{2-13}  & **Value** & \multicolumn{2}{c}{**Forecasting**} & \multicolumn{2}{c}{**The Gap**} & \multicolumn{2}{c}{**Forecasting**} & \multicolumn{2}{c}{**The Gap**} & \multicolumn{2}{c}{**Forecasting**} & \multicolumn{2}{c}{**The Gap**} & \multicolumn{2}{c}{**Forecasting**} & \multicolumn{2}{c}{**The Gap**} & \multicolumn{2}{c}{**Forecasting**} & \multicolumn{2}{c}{**The Gap**} & \multicolumn{2}{c}{**Forecasting**} & \multicolumn{2}{c}{**The Gap**} \\ \hline
2010 & & & & & & & & & & & & & & & \\
2011 & & & & & & & & & & & & & & & & \\
2012 & & & & & & & & & & & & & & & \\
2010 & & & & & & & & & & & & & & & \\
2011 & & & & & & & & & & & & & & & \\
2012 & & & & & & & & & & & & & & & \\
2014 & & & & & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The forecasting results of different compared models. For Rolling-GM (1,1), the forecasting procedure is illustrated in Figure 5. We use the electricity consumption from 2001 to 2009 as the input sequence of GM (1,1), and the first group of parameters \(a\) and \(b\) can be obtained, then the electricity consumption of 2010 can be calculated. After that, the data of 2002-2010 are utilized as input sequence, and the second group of parameters \(a\) and \(b\) can be obtained, then the electricity consumption of 2011 can be forecasted. The rest can be done in the same manner. Finally, five groups of parameters \(a\) and \(b\) and the forecasted electricity consumption of 2010-2014 can be obtained (see Table 3). For Rolling-FOA-GM (1,1), the procedure of prediction is the same as Rolling-MFO-GM (1,1). The only difference between these two models is the meta-heuristic algorithm employed to optimize the parameters \(a\) and \(b\) of GM (1,1). The five groups of optimized values of parameters \(a\) and \(b\) obtained by FOA in each iteration are listed in Table 2. The forecasting results are listed in Table 3.  
The forecasting results of electricity consumption of Inner Mongolia from 2010 to 2014, utilizing the different models above, are illustrated in Figure 9.  
As can be seen in Figure 9, the models with rolling mechanism show a better forecasting performance than the models without rolling mechanism. Specially, the forecasted annual electricity consumption using GM (1,1), FOA-GM (1,1) and MFO-GM (1,1) are much higher than actual electricity consumption, while the forecasting results of Rolling-LSR, Rolling-GM (1,1), Rolling-FOA-GM (1,1), and Rolling-MFO-GM (1,1) are much closer to actual values.