## 3 Case Study and Results Analysis
### Case Study 1  
In this case, \"Qianpingxian-95\", which is a 220 kV high voltage line, is selected to performed the simulation. After the above preparation, apply the constructed model to verify its feasibility and robustness. Firstly, initialize the parameters of QFA. Let the maximum iteration number \(Maxgen=200\), population number \(PopNum=40\), Sparks number determination constant \(M=100\), explosion radius determination constant \(\hat{R}=150\), the border of parameter \(C\) be \([2^{-5}\,2^{10}]\), the border of parameter \(\sigma\) be \([2^{-5}\,2^{5}]\), the upper and lower limits of searching space of fireworks individual be \(V_{up}=512\) and \(V_{down}=-512\), respectively, and mutation rate \(P_{id}=0.05\). Then use the steps of QFA to optimize the parameters of w-SVM and obtain \(C=18.3516\), \(\sigma=0.031402\). Finally, we predict the icing thickness of the testing sample after putting the optimal parameters into the w-SVM regression model. Figure 5 shows the optimization process of the quantum fireworks algorithm (QFA). As we can see, the proposed model obtains the optimal value when the iteration number is 35, and the optimal value is 0.11; this illustrates that the proposed algorithm can obtain the global optimum with a fast convergence speed. Figure 6 shows the forecasting results of the proposed method. This paper also  
Figure 4: Original data charts of “Yangtongdian”. For further analysis, this paper will use the relative error (RE), the mean absolute percentage error (MAPE), mean square error (MSE) and average absolute error (AAE) to evaluate the results of prediction. \[ \begin{split}& RE=\left|\frac{Y_{i}-Y^{\prime}_{i}}{Y_{i}}\right| \times 100\%\ \ MAPE=\frac{1}{n}\left|\frac{Y_{i}-Y^{\prime}_{i}}{Y_{i}}\right|\times 1 00\%\\ & MSE=\frac{1}{n}\sum\limits_{i=1}^{n}\left(Y_{i}-Y^{\prime}_{i} \right)^{2}\ \ AAE=\frac{1}{n}(\sum\limits_{i=1}^{n}|Y_{i}-Y^{\prime}_{i}|)/(\frac{1}{n} \sum\limits_{i=1}^{n}Y_{i})\end{split}  \tag{48}\]  
where \(Y_{i}\) is the actual value of power load; \(Y^{\prime}_{i}\) is the forecasting value of power load; \(i=1,2,\cdots,n\). The relative errors (RE) value of the QFA-w-SVM, FA-w-SVM, PSO-SVM, SVM and MLR models are shown in Figure 8. It can be clearly seen that the RE curve of QFA-w-SVM is the lowest among the other four models, which demonstrates that the accuracy of the proposed algorithm is much higher than other mentioned algorithms. The RE curve of the MLR model is the highest, and this demonstrates the forecasting results based on the MLR model are not satisfied here. The relative error range [\(-3\%\),\(+3\%\)] is always regarded as a standard to evaluate the performance of a forecasting model. As we can see from Figure 7, 35 points of QFA-w-SVM means 85% forcasting points are in the range [\(-3\%\),\(+3\%\)], and only 6 forecasting points are out of this scope. In the model of FA-w-SVM, only 8 out of 41 points are in the range [\(-3\%\),\(+3\%\)], which means that 78% of forecasting points are out of the scope. In addition, only 5 forecasting points of PSO-SVM and one forecasting point of SVM are in the scope of [\(-3\%\),\(+3\%\)]; none of the forecasting points of the MLR model are in the scope. These results demonstrate that the QFA-w-SVM model has a better performance in icing forecasting when compared with other models. In addition, the maximum and minimum relative errors (_Max_RE and _Min_RE) can also reflect the forecasting accuracy of icing forecasting models. Firstly, both the _Max_RE and _Min_RE values of combined models are smaller than those of single models, which proves the optimization algorithm can facilitate improving accuracy of w-SVM through finding optimal parameters. Secondly, the _Max_RE and _Min_RE values of QFA-w-SVM model are 3.61% and 0.89%, respectively, both of which are the smallest among the five icing forecasting models, which illustrates that QFA-w-SVM has a better nonlinear fitting ability in sample training.