# Energy Consumption Model for Drilling Processes Based on Cutting Force
## 4 Results and Discussions
### Model Comparison  
In order to prove the validity and accuracy of the new model, a comparative study was used to compare it with the three existing models (the form of each model is listed in Table 4): Model-1 [10], Model-2 [11], and Model-3 [19]--all of which were established based on the same machine tool, and with the same data from our experiments. First of all, the coefficients of the three models were calculated by the data from Experiment-I which are shown in Table 4:  
Then, the data collected from Experiment I and Experiment II were used to compare the proposed power model, \(P\) *, with the three types of models. Table 5 shows the comparison results based on the data of Experiment-I, where \(P(W)\) is the measured total power. As can be seen in Table 5, on average, the proposed model prediction error was 1.13% which is the smallest error in the comparison experiments. Model 1 and Model 2 (based on the MRR) had prediction errors of 3.09% and 2.39%, respectively, and their prediction errors were greater than the 2.09% of Model 3 (based on cutting force). This indicates that the MRR was not sufficient to fully characterize the total energy consumption compared to the cutting force. Although Model 3 also gave acceptable predictions, the accuracy was not as good as the proposed model which may be due to the large influence of idle power on total power, while Model 3 ignores this. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Power Form** & & **Power Expression** \\ \hline Model-1 & \(P\) = \(C_{0}MRR+C_{1}\) & \(P^{1}\) = \(5.749 MRR+1393.6\) \\ Model-2 & \(P\) = \(k_{0}MRR+k_{1}n+k_{2}\) & \(P^{2}\) = \(5.801 MRR+0.334n+1170.5\) \\ Model-3 & \(P\) = \(C_{0}P_{cutting}+C_{1}\) & \(P^{3}\) = \(1.404P_{cutting}+1237.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Three comparison models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(No.\) & \(P\) (W) & \(P\) * (W) & \(P\) 1 (W) & \(P^{2}\) (W) & \(P^{3}\) (W) \\ \hline
1 & 1625.0 & 1587.5 & 1635.6 & 1547.0 & 1606.7 \\
2 & 1679.0 & 1645.2 & 1726.3 & 1638.0 & 1687.3 \\
3 & 1744.6 & 1703.1 & 1817.1 & 1729.1 & 1750.6 \\
4 & 1781.4 & 1761.1 & 1907.8 & 1820.2 & 1860.1 \\
5 & 1750.0 & 1749.6 & 1726.3 & 1688.1 & 1745.9 \\
6 & 1837.4 & 1829.0 & 1851.1 & 1813.4 & 1843.4 \\
7 & 1937.0 & 1908.6 & 1975.9 & 1938.6 & 1964.7 \\
8 & 2001.8 & 1988.4 & 2100.7 & 2063.8 & 2096.2 \\
9 & 1906.1 & 1925.3 & 1817.1 & 1829.3 & 1830.4 \\
10 & 1988.8 & 2026.4 & 1975.9 & 1988.7 & 1985.6 \\
11 & 2117.4 & 2127.7 & 2134.7 & 2148.1 & 2205.7 \\
12 & 2217.9 & 2229.2 & 2293.5 & 2307.4 & 2265.8 \\
13 & 2067.9 & 21 In addition, Figure 8 shows the specific prediction error of each model under 16 sets of experimental parameters. The maximum values of the fluctuation range of Models 1, 2, and 3 were approximately 8%, 5%, and 5%, respectively. The maximum value of the proposed model fluctuation range was about 2%. It can be seen that the fluctuation of the prediction error of the other three models was significantly higher than that of the proposed model which clearly shows the stability of the proposed model prediction. Similarly, Table 6 lists the predicted powers of our model \(P\) * and the three benchmarks \(P^{1},P^{2}\), and \(P^{3}\) against the actual power consumption \(P\). From the table, it can be found that the proposed model had the smallest prediction error (1.02%), and the fluctuation of error was smaller as well (see Figure 9). For Model-1, the errors of the No. 1 and 4 experiments were relatively large as compared with that of the No. 2 and 3. For the second benchmark, since Model-2 considered the influence of spindle speed on energy consumption on the basis of Model-1, the prediction error (1.73%) was much smaller than that of Model-1, but the error was also larger than ours. For Model-3, the prediction accuracy was equivalent to that of Model-2. Model-3 used the \(P_{cutting}\) to predict energy consumption, but since the \(P_{iulle}\) at different spindle speeds was not considered, there was still no higher prediction accuracy (1.77%) than ours.