## 3 Discussion
### Integrated Model of Metering Error under Current and Temperature
The model parameters are \(w_{a,\Omega}\), \(u_{a,\Omega}\), \(v_{a,\Omega}\), \(w_{b,\Omega}\), \(u_{b,\Omega}\), \(v_{b,\Omega}\), \(w_{c,\Omega}\), \(u_{c,\Omega}\), \(v_{c,\Omega}\). The estimates of the model parameters were computed using Matlab tools and are presented in Table 6, the fitting results are plotted in Figure 10. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(\Omega\)} & \multirow{2}{*}{**Parameters**} & \multicolumn{3}{c}{**Power Coefficient**} \\ \cline{3-5}  & & **1.0** & **0.8C** & **0.5L** \\ \hline \multirow{8}{*}{\(\Omega=0\), Mean} & \(w_{a,0}\) & 0.1450 & 0.1284 & 0.0747 \\  & \(u_{a,0}\) & \(-0.1095\) & \(-0.1214\) & \(-0.0787\) \\  & \(v_{a,0}\) & 0.0719 & 0.0688 & 0.0368 \\  & \(w_{b,0}\) & \(-0.0010\) & \(-0.0024\) & \(-0.0057\) \\  & \(u_{b,0}\) & \(-0.0007\) & 0.0006 & 0.0054 \\  & \(v_{b,0}\) & \(-0.0003\) & \(-0.0005\) & \(-0.0020\) \\  & \(w_{c,0}\) & \(-5.4070\times 10^{-5}\) & \(-0.0001\) & \(7.5650\times 10^{-5}\) \\  & \(u_{c,0}\) & \(5.2410\times 10^{-5}\) & 0.0002 & \(-6.7290\times 10^{-5}\) \\  & \(v_{c,0}\) & \(-1.8450\times 10^{-5}\) & \(-6.9960\times 10^{-5}\) & \(3.7470\times 10^{-5}\) \\  & _SSE_ & 0.0134 & 0.0176 & 0.0269 \\  & _R-square_ & 0.6863 & 0.6161 & 0.2803 \\ \hline \multirow{8}{*}{\(\Omega=1\), \(3\sigma\)} & \(w_{a,1}\) & \(-0.1093\) & 0.0631 & 0.1031 \\  & \(u_{a,1}\) & 0.1346 & \(-0.1017\) & \(-0.1729\) \\  & \(v_{a,1}\) & 0.1999 & 0.2477 & 0.2813 \\  & \(w_{b,1}\) & 0.0015 & 0.0051 & 0.0001 \\  & \(u_{b,1}\) & \(-0.0010\) & \(-0.0081\) & \(-0.0037\) \\  & \(w_{b,1}\) & \(-0.0058\) & \(-0.0037\) & \(-0.0049\) \\  & \(w_{c,1}\) & \(-0.0001\) & \(-0.0003\) & \(7.4560\times 10^{-5}\) \\  & \(u_{c,1}\) & 0.0002 & 0.0004 & \(1.1790\times 10^{-5}\) \\  & \(v_{c,1}\) & \(8.9980\times 10^{-5}\) & \(6.0890\times 10^{-5}\) & 0.0001 \\  & _SSE_ & 0.0534 & 0.0584 & 0.1234 \\  & \(R\)-square_ & 0.7646 & 0.6514 & 0.5888 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model parameter estimates. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Power** \\ **Coeffi-** \\ **cient** \\ \end{tabular} } & \multirow{2}{*}{**Current**} & \multicolumn{8}{c}{**3\(\sigma\) of Metering Error (\%)**} \\ \cline{3-14}  & & \(-\)**15 °C** & \(-\)**10 °C** & \(-\)**5 °C** & **0 °C** & **5 °C** & **10 °C** & **15 °C** & **20 °C** & **25 °C** & **28 °C** & **32 °C** \\ \hline \multirow{4}{*}{\(1\)} & Ib & 0.3291 & 0.3061 & 0.2616 & 0.2281 & 0.1765 & 0.2435 & 0.1407 & 0.1718 & 0.2332 & 0.1738 & 0.2308 \\  & 0.51b & 0.3259 & 0.3213 & 0.2896 & 0.3060 & 0.1876 & 0.1923 & 0.1599 & 0.1144 & 0.2460 & 0.1786 & 0.2279 \\  & 0.11b & 0.3329 & 0.2973 & 0.2736 & 0.2249 & 0.1722 & 0.1850 & 0.2218 & 0.0989 & 0.1361 & 0.1680 & 0.1663 \\  & 0.051b & 0.2777 & 0.2616 & 0.2439 & 0.2018 & 0.2499 & 0.1455 & 0.0569 & 0.0730 & 0.0994 & 0.1283 & 0.1256 \\ \hline \multirow{4}{*}{\(0.8C\)} & Ib & 0.3179 & 0.2956 & 0.2829 & 0.2464 & 0.1550 & 0.1848 & 0.1016 & 0.1290 & 0.1464 & 0.1932 & 0.1887 \\  & 0.51b & 0.3198 & 0.3164 & 0.2672 & 0.2141 & 0.1835 & 0.2348 & 0.0807 & 0.1239 & 0.1380 & 0.2521 & 0.1826 \\  & 0.11b & 0.3193 & 0.2916 & 0.2688 & 0.2810 & 0.1604 & 0.1592 & 0.3250 & 0.1252 & 0.1605 & 0.1835 & 0.2166 \\ \hline \multirow{4}{*}{\(0.5L\)} & Ib & 0.3453 & 0.3400 & 0.2750 & 0.2451 & 0.2068 & 0.1873 & 0.0590 & 0.0900 & 0.1144 & 0.1466 & 0.2118 \\  & 0.51b & 0.3226 & 0.3109 & 0.2793 & 0.2544 & 0.2226 & 0.2143 & 0.0783 & 0.0789 & 0.1556 & 0.1776 & 0.1915 \\ \cline{1-1}  & 0.11b & 0.3546 & 0.3122 & 0.3021 & 0.3557 & 0.2131 & 0.3593 & 0.0799 & 0.0925 & 0.2031 & 0.3854 & 0.1881 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: \(3\sigma\) of metering error under different temperatures. As shown in Figure 10, the metering error was severely affected by current and environmental temperature. For power coefficient 1.0, as presented in Figure 10a, the effects of temperature on metering error were more severe under the large current (\(l_{\text{b}}\)) than under the small current (0.1 \(l_{\text{b}}\)).