## 4 Performance Assessment of Branch Power Change Estimation
### Influence of Systematic Gain Errors of Consumer MUT  
Figure 9 presents scatter plots of \(e_{I},\ e_{V},\ e_{P}\) estimations, which are obtained by solving (24). It indicates a good approximation precision.  
It can be seen from Figure 10a that the \(\overline{dP_{n}}\) estimation error is not influenced by the size of \(e_{P}\) increases in case of larger estimation error of \(e_{P}\) (see Figure 10b).  
Based on the applied test data set, the RMSE of \(\hat{e_{P}}\) was 0.10%, the RMSE of \(\hat{e_{V}}\) was 0.08%, and RMSE of \(\hat{e_{I}}\) was 0.07% (calculated from estimates shown in Figure 9). From the data shown in Figure 9 it was observed that the absolute error of the estimate of power measurement gain error was within \(\pm\)0.3% These performance metrics seem quite acceptable for practical needs because Class 1 m power (energy) measurement error should not exceed 1% and if consumer MUT power measurement error estimates are obtained with RMSE 0.10%. The rule for triggering detection of MUT non-compliance to standard requirements could be a fixed threshold-based or adaptive thresholding based on power gain error RMSE calculated after several consecutive estimations.  
Figure 10: Relative error of \(\overline{dP_{n}}\) versus \(e_{P}\) actual value (**a**) and versus absolute \(e_{P}\) estimation error of \(\Delta(\hat{e_{P}})\) (**b**). Every sample of relative error of \(\overline{dP_{n}}\) was obtained by averaging 10 estimates \(dP_{n}\) acquired at 10 different noise ensembles of random injected error. The training/testing data set is T1 from Table 7.  
Figure 9: Scatter plots of gain errors \(e_{P}\) (**a**), \(e_{V}\) (**b**), \(e_{I}\) (**c**) estimates. Test data: grid case no. 1 from Table 2, Meter class and random power error case M3 and Training data/Testing data case T3 from Table 7.  
RMSE of estimation of the relative error of \(\overline{dP_{n}}\) was 4.41% while observed relative error of PCC in the branch samples were within the \(\pm\)10% range (calculated from estimates shown in Figure 10b, which is superior compared to techniques that neglect the presence of PCC in the branch (neglect the presence means that relative error is 100%).  
In Figure 11, the influence of the resistance of the DG branch (grid cases in Table 2) on the absolute error of the estimation of gain error is investigated. Training was performed using the data set corresponding to grid case no. 1, whereas testing data set is indicated in the legend of plots. The training/testing procedure was repeated 10 times, each with different noise samples presenting random error. Ranges \([\overline{e}_{P,V,I}-\sigma(\overline{e}_{P,V,I}),\overline{e}_{P,V,I}+ \sigma(\overline{e}_{P,V,I})]\) are indicated in Figure 11 after calculating standard deviation of average \(\overline{e}_{P,V,I}\) from \(M=10\) estimates of \(e_{P}\), \(e_{V}\), and \(e_{I}\).  
It can be seen from results presented in Figures 11 and 12 that change of network losses compared to those present during models' training cause higher estimation errors of both gain error \(e_{P}\) and relative error of \(\overline{dP_{n}}\) estimation. Therefore, it should be acknowledged that the technique suggested for the DG branch PCC in response to consumer load power event is sensitive to the variations in distribution line power losses and the retraining of model (28) might be mandatory to retain better \(dP_{n}\) and low gain error estimation error.  
Figure 11: Mean and standard deviation of the mean of absolute error of gain error estimation (\(e_{P}\) (a), \(e_{V}\) (b), \(e_{I}\) (c)). Training was performed using data set corresponding to the T1 dataset of grid case no. 1 and T1 testing data set of the grid shown in the legend of the plot.