## 3 Discussion
### Integrated Model of Metering Error under Current and Temperature  
The mean values and standard deviation of metering error under all power coefficients and currents are listed in Tables 4 and 5. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
\begin{tabular}{c} **Power** \\ **Coeff-** \\ **cient** \\ \end{tabular} & \multirow{2}{*}{**Current**} & \multicolumn{8}{c}{**Mean Values of Metering Error (\%)**} \\ \cline{3-11}  & & \(-\)**15 \({}^{\circ}\)C** & \(-\)**10 \({}^{\circ}\)C** & \(-\)5 \({}^{\circ}\)C** & **0 \({}^{\circ}\)C** & **5 \({}^{\circ}\)C** & **10 \({}^{\circ}\)C** & **15 \({}^{\circ}\)C** & **20 \({}^{\circ}\)C** & **25 \({}^{\circ}\)C** & **28 \({}^{\circ}\)C** & **32 \({}^{\circ}\)C** \\ \hline \multirow{4}{*}{1} & Ib & 0.1404 & 0.1248 & 0.1034 & 0.1033 & 0.0889 & 0.1243 & 0.0535 & 0.0518 & 0.0541 & 0.0077 & 0.0317 \\  & 0.51b & 0.0620 & 0.0671 & 0.0604 & 0.0739 & 0.0490 & 0.0536 & 0.0278 & 0.0188 & 0.0518 & 0.0092 & 0.0287 \\  & 0.11b & 0.0731 & 0.0477 & 0.0198 & 0.0501 & 0.0219 & 0.0683 & 0.0495 & 0.0287 & 0.0107 & 0.0280 & 0.0140 \\  & 0.051b & 0.0771 & 0.0821 & 0.0799 & 0.0774 & 0.0884 & 0.0785 & 0.0649 & 0.0704 & 0.0598 & 0.0583 & 0.0542 \\ \hline \multirow{4}{*}{0.8C} & Ib & 0.0967 & 0.0938 & 0.0954 & 0.1029 & 0.0411 & 0.0875 & 0.0290 & \(-\)0.0353 & 0.0283 & 0.0067 & \(-\)0.0048 \\  & 0.51b & 0.0501 & 0.0548 & 0.0343 & 0.0342 & 0.0153 & 0.0757 & 0.0162 & 0.0182 & \(-\)0.0216 & 0.0394 & \(-\)0.0057 \\  & 0.11b & 0.0559 & 0.0592 & 0.0435 & 0.0712 & 0.0551 & 0.0634 & \(-\)0.0194 & 0.0594 & 0.0359 & 0.0007 & \(-\)0.0268 \\ \hline \multirow{4}{*}{0.5L} & Ib & 0.0607 & 0.0812 & 0.0480 & 0.0052 & 0.0677 & 0.0008 & 0.0162 & \(-\)0.0077 & \(-\)0.0089 & \(-\)0.0210 & 0.0376 \\  & 0.51b & 0.0295 & 0.0370 & 0.0200 & \(-\)0.0213 & 0.0618 & \(-\)0.0218 & 0.0277 & 0.0112 & 0.0364 & \(-\)0.0373 & 0.0369 \\ \cline{1-1}  & 0.11b & 0.0435 & 0.0861 & 0.0429 & \(-\)0.0146 & 0.0437 & \(-\)0.0185 & 0.0205 & 0.0695 & 0.0186 & \(-\)0.0442 & 0.0301 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mean values of metering error under different temperatures. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\Omega\) & \(\psi\) & \(P_{\Omega,\psi,2}\) & \(P_{\Omega,\psi,1}\) & \(P_{\Omega,\psi,0}\) & \(\mathcal{SSE}\) & \(R^{2}\) \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} \(\Omega=0\), \\ Mean \\ \end{tabular} } & 1.0 & \(-1.9780\times 10^{-5}\) & \(-0.0021\) & 0.1068 & 0.0032 & 0.8291 \\  & 0.8C & \(-1.4090\times 10^{-5}\) & \(-0.0023\) & 0.0758 & 0.0068 & 0.7098 \\  & 0.5L & \(4.5830\times 10^{-5}\) & \(-0.0023\) & 0.0328 & 0.0057 & 0.5346 \\ \hline \multirow{4}{*}{
\begin{tabular}{c} \(\Omega=1\), \\ \(3\sigma\) \\ \end{tabular} } & 1.0 & 0.0002 & \(-0.0053\) & 0.2257 & 0.0084 & 0.7523 \\  & 0.8C & 0.0002 & \(-0.0066\) & 0.2091 & 0.0086 & 0.8345 \\ \cline{1-1}  & 0.5L & 0.0002 & \(-0.0085\) & 0.2133 & 0.0160 & 0.8234 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter estimates for measurement error/temperature model. The mean value and standard deviation of the metering error presented a nonlinear trend with increasing temperature as shown in Figure 9a,b and with increasing current as shown in Figures 6b and 7b, respectively. So, a binary quadratic polynomial was employed to fit the joint influences of current and temperature on the mean and standard deviation of the measurement error. The model was written as:  
\[\gamma\Omega(T,I)=\left[ \begin{array}{c}I^{2}\\ I\\ 1\end{array} \right]^{\prime}\left[ \begin{array}{cc}w_{c\ell,\Omega}&w_{b, \Omega}&w_{a,\Omega}\\ u_{c,\Omega}&u_{b,\Omega}&u_{a,\Omega}\\ v_{c,\Omega}&v_{b,\Omega}&v_{a,\Omega}\end{array} \right]\left[ \begin{array}{ c}T^{2}\\ T\\ 1\end{array} \right], \tag{4}\]  
where \(\gamma_{\Omega}\)(\(T\),\(I\)) is the mean function and the standard deviation function about the random variable of temperature \(T\) and current \(I\) when \(\Omega=0\) and \(\Omega=1\).