# BN-RA: A Hybrid Model for Risk Analysis of Overload-Induced Early Cable Fires
## 3 Construction of a BN
### Determination of the BN Parameters
#### 3.3.2 CPT
In the limit of \(t\rightarrow\infty\) in Equation (6), the expression for the temperature difference is given by:  
\[\Delta T=T_{p}=\frac{I^{2}R}{K_{zh}A} \tag{7}\]  
When the heat generation rate of the inner core of the conductor equals the heat dissipation rate of the outer surface of the conductor insulation layer, the conductor is in a balanced state of heat generation and heat dissipation, and its temperature rise is the temperature rise balance coefficient \(T_{p}\), where the current \(I\) and the total heat transfer coefficient \(K_{zh}\) are two important factors. The total heat transfer coefficient for the conductor surface is affected by heat conduction, heat convection, and heat radiation on the conductor surface and environment. Therefore, changing the heat dissipation on the conductor surface, for example, owing to the conductor damage and/or aging, affects the total heat transfer coefficient on the conductor surface. According to Equation (7), after a sufficiently long time the conductor temperature will eventually stabilize at a certain value, which is called the equilibrium temperature for the specific current. The equilibrium temperature of a conductor can be determined in terms of the equilibrium temperature rise and ambient temperature, as follows:  
\[T_{balance}-T_{0}=\frac{I^{2}R}{K_{zh}A} \tag{8}\]  
\[T_{balance}=\frac{I^{2}R}{K_{zh}A}+T_{0}=T_{B}+T_{0} \tag{9}\]  
(ii) Coupling experiment of conductor temperature rise  
To explore the influence of mechanical damage, aging, and current on the cable temperature increase, an experiment was conducted to test the internal wiring single-core flexible conductor unshielded cable (60227 iec 05 BV) with a cross-sectional area of 0.75 mm\({}^{2}\). The equipment used in this experiment is shown in Figure 5. In particular, a grounding resistance tester, TSO6210 (Figure 5a), was used for providing a stable and large constant current. The temperature of the cable surface was measured using a 0.32 mm T-type thermocouple and recorded using an LR8431-30 data acquisition instrument (Figure 5b). A thermal aging test chamber (Figure 5c) provided a constant-temperature aging environment for aging cable samples. In this study, cables were tested for six currents (i.e., 6, 10, 13, 15, 18, and 20 A), three levels of mechanical damage (i.e., 50%, 75%, and 100%), and four aging states (i.e., 6, 12, 36, and 60 months). Meanwhile, cables under different damage degrees and aging time are tested under different loading currents, where the cable temperature rise is recorded. To accurately measure the cable temperature and calculate the cable temperature rise, three thermocouple measuring points (i.e., inlet, outlet, and middle end points) are fixed on the conductor, and another measuring point is arranged in the surrounding to record the ambient temperature nearby. For the damaged cable test group, it is necessary to add a temperature measurement point at the damaged conductor edge to monitor its temperature change. Finally, the temperature rise of the cable is obtained by calculating the difference value of cable equilibrium temperature and ambient temperature. Mechanical damage was modeled as partial insulation that affected heat dissipation, directly exposing the high-temperature core and causing cable overheating. Cable damage was quantified by the angle of the cable local damage; five examples that illustrate different exposed ratios are shown in Figure 6.  
In addition to the mechanical damage, aging affects the structure of the cable insulation layer and its insulation ability and resistance to high temperatures; aging causes cables to overheat easier. In aged cables, overload induces a sharp temperature rise. In this study, an aging model developed by [PERSON] [45], was used for modeling cables with different aging degrees.