# Distribution System State Estimation Using Model-Optimized Neural Networks
## 5 Results and Discussion  
Shown in Tables 1 and 2 are the errors in state estimate per state estimator and subgrid. As can be observed, the LSTM networks performed better than the conventional WLS-SE, largely due to the learning capability of neural networks and to the relative simplicity of the network and data. Another important note is that the error metrics increase steadily as the target sparsity increases, all while the estimation time decreases. This is largely due to the fact that there are less weights to hold information and less weights to actually multiply. It is also important to note that the speed up for each method may or may not be accurately presented because this is only tested for one PC unit equipped with an Intel Core i5, 8 GB RAM, and a hard disk drive. However, it can be inferred that with better equipment, such as a GPU with support for Nvidia's CUDA cores which are optimized for neural network operations, the estimation time will decrease. This event will be more pronounced when the methodology is subject to large, practical system configurations, that is, networks with hundreds or thousands of buses, tens or hundreds of measuring devices, and of course, millions of \(Y_{bus}\) entries. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**State Estimator**} & **Voltage** & **Voltage** & **Voltage** & **Voltage** & **Estimation** \\  & **Magnitude** & **Angle** & **Magnitude** & **Angle** & **Time (s)** \\ \hline Conventional & \multirow{2}{*}{0.0128} & \multirow{2}{*}{0.1691} & \multirow{2}{*}{0.0151} & \multirow{2}{*}{0.2498} & \multirow{2}{*}{3.5828} \\ \cline{1-1} \cline{5-5} \cline{5-5} \  
Shown in Figures 9-17 are the graphical representations of the state estimation errors against the true system states. This is where we can see individually how the methods perform and at which points they could possibly fail. One possible trend that can be inferred is that non-pruned networks tend to under predict, which is not the case. The under and over prediction of the model is highly dependent on the sequence of data it has seen during training. Recall that the data are always shuffled every training step in a session and that the weights that will be pruned are nearly random and are at the mercy of the scheduler and its gradients. However, this does not mean that there is absolutely no definite and concrete answer. One of the possible reasons for this is the modeled (or anticipated) errors in the measurement units, which is why it tends to under (or over) predict. What is certain, however, is that a neural network's aim is always to generalize the dataset; thus, the testing methodology exhausts the dataset. Therefore, by consequence, the error metrics are good enough approximations of their performances. The file sizes of the neural networks after compression are 837 kB for the normal and modified LSTM, 380 kB for the 25% pruned LSTM, 375 kB for the 50% pruned LSTM, and 370 kB for the 75% pruned LSTM. The reasons behind the compression are outside the scope of this paper, but loosely put, it is due to the compression algorithms and its fundamental limits and tradeoffs, i.e., processing time, memory, chunk size, etc. [41]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**State Estimator** & \
\begin{tabular}{c} **Voltage Magnitude** \\ **(MAE)** \\ \end{tabular} &
\begin{tabular}{c} **Voltage Magnitude** \\ **(RMSE)** \\ \end{tabular} & **Estimation Time (s)** \\ \hline Conventional WLS & 0.0035 & 0.0036 & 0.9689 \\ Normal LSTM & \(-\)0.0005 & 0.0005 & 0.3043 \\ Modified LSTM & 0.0002 & 0.0003 & 0.2989 \\
25\%-Pruned LSTM & 0.0008 & 0.0008 & 0.2863 \\
50\%-Pruned LSTM & 0.0013 & 0.0015 & 0.2814 \\
75\%-Pruned LSTM & \(-\)0.0017 & 0.0020 & 0.2797 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of errors in state estimates for each DC subgrid state estimator. Figure 9: Conventional weighted least square state estimation results: (a) AC voltage magnitudes, and (b) AC voltage angles. \(y\)-axis corresponds to estimated bus voltage in (p.u.