## 4 Result Experiment
### Definition of Aging Condition Index (ACI)  
To evaluate the aging condition and estimate the life of a PP insulated cable subjected to mechanical aging, aging indices were developed. Equation (5) represents the index based on tensile strength, while Equation (6) pertains to elongation at break. The aging index is expressed as a normalized value between 0 and 1, where 1 indicates an undegraded, pristine condition, and 0 represents a fully aged state. The critical threshold values of 12.5 MPa for tensile strength and 350% for elongation were selected based on the minimum mechanical property requirements specified in relevant standards for cable-grade insulation materials. The threshold values for tensile strength (12.5 MPa) and elongation at break (350%) were determined based on the Korean industrial standard SPS-C-KWS-501-7524:2022 [18], established by the Korea Electric Wire Industry Cooperative. When the measured property falls below these thresholds, the material is considered mechanically failed, and the degradation index is accordingly set to 0.  
\[\textit{ACI}_{\textit{Tensile strength}}(T,\ t)=\left\{\begin{array}{ccc} \frac{z\left(T,t\right)-Z_{\textit{Minimum}}}{Z_{0}-Z_{\textit{Minimum}}}= \frac{z\left(T,t\right)-12.5}{20.94-12.5}&if&z(T,t)>12.5\\ 0&if&z(T,\ t)\leq 12.5\end{array}\right. \tag{5}\]  
where z(T,t) denotes the tensile strength (in MPa) as a function of temperature and time. The z\({}_{0}\) (initial tensile strength) is 20.94 MPa, and the failure threshold is defined as z\({}_{\text{minimum}}\) (12.5 MPa), which corresponds to the minimum mechanical requirement for the material to be qualified for cable insulation. When the tensile strength falls below this value, the material is regarded as not meeting the minimum specification criteria, and the aging index is accordingly set to zero.  
\[\textit{ACI}_{\textit{elongation}}(T,\ t)=\left\{\begin{array}{ccc}\frac{z \left(T,t\right)-Z_{\textit{Minimum}}}{Z_{0}-Z_{\textit{Minimum}}}=\frac{z \left(T,t\right)-350}{997.75-350}&if&z(T,t)>350\\ 0&if&z(T,\ t)\leq 350\end{array}\right. \tag{6}\]  
In this case, z(T,t) denotes the elongation at break (in %) over time and temperature. The initial elongation is assumed to be 997.75% as z\({}_{0}\), and the requirement of the criterion is set to 350% as z\({}_{\text{minimum}}\), which is the minimum acceptable value for maintaining mechanical integrity. The index reflects the loss of flexibility and ductility of the insulation material during aging.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Classification**} & \multicolumn{2}{c}{**Aging Model: P(T,t) = \(-\)A \(\times\) exp(T/B) \(\times\) exp(t/C) + TP\({}_{0}\)**} & \multirow{2}{*}{**Note**} \\ \cline{2-2} \cline{4-6}  & **Maximum** & & & & **Aging Coefficient** & **Initial** \\  & **Reduction** & & & **Values** & **R\({}^{2}\)** \\  & **A** & **1/B (Temp)** & **1/C (Time)** & **TP\({}_{0}\)** & & \\ \hline Tensile strength & 0.46 & 0.011831519 & 0.000391489 & 21.4 & 0.845 \\ Elongation & 12.43 & 0.013933398 & 0.000521521 & 1010.18 & 0.834 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Detailed parameters of the extended aging model and aging coefficients.