# 1D CNN Based Detection and Localisation of Defective Droppers in Railway Catenary
## 3 Simulations and Damage Classification
### Dropper Damage Categorisation  
Dropper damage is expected to be categorised and labelled for further classification. It was found in [7] that the impact of defective droppers on the quality of the contact force varies significantly across different positions. Therefore, instead of using the damage degree, the impact level on the contact force will be used for dropper condition categorisation, which can intuitively represent the impact of defective droppers on pantograph-catenary dynamics.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Output** & **Simulation Results** & **EN 50318 Requirements** \\ \hline Mean force (N) & 116.65 & 110–120 \\ Force standard deviation (N) & 27.25 & 26–31 \\ Maximum force (N) & 185.94 & 175–210 \\ Minimum force (N) & 54.31 & 50–75 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Model validation at 250 km/h.  
Figure 2: Static impact on the contact wire state (**a**) the displacement (**b**) the stiffness.  
To quantify the impact, three key indicators of the contact force (i.e., standard deviation (\(std\)) maximum (\(max\)) and minimum (\(min\))) are considered. They reflect the degree of mechanical wear, possibility of contact loss and contact robustness. The impact index, introduced in Equation (4), quantifies the impact of defective droppers on the contact quality. It finds the greatest deterioration in the force standard deviation, maximum, and minimum by comparing the contact force signal of the damaged catenary (\(D\in\ \{D_{std}\), \(D_{min}\), \(D_{max}\}\)) with the nominal result (\(N\in\{N_{std}\), \(N_{min}\), \(N_{max}\}\)).  
\[I=\text{max}\bigg{(}\frac{D_{std}-N_{std}}{D_{std}},\frac{D_{max}-N_{max}}{D_{ max}},\frac{N_{min}-D_{min}}{D_{min}}\bigg{)}\times 100\% \tag{4}\]  
Figure 4 shows the impact index of each dropper's defect with damage degree ranging from 100% to 0%. It is clear that each dropper's damage has a distinctive impact on the contact performance. Regarding the index, the damage degree of each dropper is categorised into two risk categories, low risk (\(I\geq 1\%\)) and high risk (\(I\geq 3\%\)). Once a damage degree is labelled, a worse degree will not be categorised as healthier than that for conservatism. The damage categorisation is summarised in Table 3. For example, if the damage of the fifth dropper is more than 27% of the nominal value, the condition will be labelled as low risk and if it is more than 62.5%, it will be labelled as high risk.  
Moreover, Gaussian white noise filtered up to 20 Hz (the interested frequency range associated with the contact force) will be added to include the measurement error. According to the requirement for the measurements of the pantograph-catenary dynamic interaction, the maximum error of the measurement system should be less than 10% of the measured value [37]. After adding noise into the raw contact force result, the maximum error of all data is 8.88% at 20 dB noise power whereas it is 16.49% at 25 dB. Therefore, a noise power of no more than 25 dB is worthwhile to be considered.  
Figure 3: Dynamic impact on contact force.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Category**} & \multicolumn{8}{c}{**Damage Degree (\%)**} \\ \cline{2-10}  & **D1** & **D2** & **D3** & **D4** & **D5** & **D6** & **D7** & **D8** & **D9** \\ \hline Healthy (I \(<\) 1\%) & \(<\)51 & \(<\)56.5 & \(<\)57.5 & \(<\)72 & \(<\)27 & \(<\)81.5 & \(<\)41 & \(<\)82.5 & \(<\)45 \\ Low risk (1\% \(\leq\) I \(<\) 3\%) & 51–68 & 56.5–88 & 57.5–100 & \(\geq\)72 & 27–62.5 & 81.5–89.5 & 41–93 & 82.5–99 & 45–68.5 \\ High risk (\(I\geq 3\%\)) & \(\geq\)68 & \(\geq\)88 & \(\geq\)100 & None & \(\geq\)62.5 & \(\geq\)89.5 & \(\geq\)93 & \(\geq\)99 & \(\geq\)68.5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Dropper damage categorisation (D = dropper no.).