# Time Series RUL Estimation of Medium Voltage Connectors to Ease Predictive Maintenance Plans
## 6 Results  
This section evaluates the behavior and accuracy of the proposed approach for estimating the RUL based on the experimental results carried out in the electrical loop, including the seven analyzed connectors. The experimental results were obtained by means of the accelerated degradation process induced by the heat cycle tests detailed in Section 5. After running several models, some conclusions can be stated. First, in this problem the inclusion of the I or integral term makes the results poorer, because the prediction that is based only on the finite differences (of order \(d\)) of the measured instantaneous resistance does not provide useful information to the model. Therefore, for the presented case study, ARMA models present better performances compared to the ARIMA models. Second, the AR term is always required, and the MA term can help to improve the performance of the AR part. Table 1 shows the estimated RUL for the 7 considered electrical connectors when fitted by means of the AR (\(p\) = 3) and ARMA (\(p\) = 2, \(q\) = 2) models, as formulated in (4) and (8), respectively. Each of the columns corresponds to a different time horizon, for which the training stage of the model and the forward RUL estimation has been carried out. The RUL of each connector has been inferred in three different instants of time, which are summarized in Table 2. Note that not all the analyzed connectors have reached their EOL in the degradation experiment. The AR (3) with \(p\) = 3 and the ARMA (2,2) with \(p\) = 2 and \(q\) = 2 have been revealed as the most reliable and adequate modes combining both ease of use in terms of computational efficiency as well RUL estimation accuracy. Figures 5 and 6 show, respectively, the results regarding the AR (3) and the ARMA (2,2) models, which consider 500 past data points, accounting for the last 4 h of past data. Each of the seven rows in Figures 5 and 6 correspond to each of the seven studied connectors, whereas the three columns correspond to different time horizons. According to the observed degradation trajectory of the resistance for each connector, three prediction instants have been defined, as summarized in Table 2. The graphs \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**-**} & **RUL (h)** & **Time 1** & **Time 2** & **Time 3** & **Measured** \\ \hline \multirow{2}{*}{Connector \#1} & AR & 18.4 & 28.1 & 29.8 & \multirow{2}{*}{31.5} \\  & ARMA & 18.6 & 28.5 & 29.9 & \\ \hline \multirow{2}{*}{Connector \#2} & AR & 65.9 & 93.4 & 112.3 & \multirow{2}{*}{53.7} \\  & ARMA & 69.7 & 87.9 & 79.3 & \\ \hline \multirow{2}{*}{Connector \#3} & AR & 53.7 & 106.1 & 147.7 & \multirow{2}{*}{n/a} \\  & ARMA & 58.8 & 98.6 & 129 & \\ \hline \multirow{2}{*}{Connector \#4} & AR & 60.2 & 102.7 & - & \multirow{2}{*}{n/a} \\  & ARMA & 58.2 & 105.2 & - & \\ \hline \multirow{2}{*}{Connector \#5} & AR & 87.2 & 137.0 & - & \multirow{2}{*}{n/a} \\  & ARMA & 83.1 & 182.5 & - & \\ \hline \multirow{2}{*}{Connector \#6} & AR & 68 & 60.8 & 64.9 & \multirow{2}{*}{50.3} \\  & ARMA & 69.7 & 59.8 & 59.4 & \\ \hline \multirow{2}{*}{Connector \#7} & AR & 160.9 & 99.5 & 125.5 & \multirow{2}{*}{n/a} \\  & ARMA & 197.3 & 140.7 & 140.3 & \\ \hline \end{tabular} n/a indicates that the model prediction does not cross the established threshold, so under the 40% criterion, the connector has not achieved the EOL and therefore, no RUL can be inferred.  
\end{table} 
Table 1: Resulting remaining useful life (RUL) estimations for the seven electrical connectors under study considering autoregressive (AR) (\(p\) = 3) and autoregressive integrated moving average (ARMA) (\(p\) = \(q\) = 2) models. RUL estimations are carried out at three different service points according to the 40% degradation criterion.