# 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 a BN, each node has a different state, where each state corresponds to a certain probability. At the same time, a combination of different states of each parent node leading to different states of child nodes also has a certain probability. Accordingly, a CPT based on the nodes' states can be constructed between child nodes and parent nodes with polymorphic uncertainty. In general, conditional probability is determined by the corresponding functional relation, statistical probability, and subjective decision. To obtain the conditional probability relationship of the temperature rise, this study utilized the temperature rise theory and evidence from coupling experiments. (i) Conductor heating theory  
When a cable is connected to a constant-current circuit, its inner core conductor generates heat. Owing to the heat transfer, the temperature of the insulation layer will gradually increase with time, and the conductor insulation layer will dissipate heat into the environment around it. The dynamic process of heat dissipation is shown in Figure 4, where the circle indicates the cross-section of a cable. In Figure 4, the basic process of heat generation and dissipation by a conductor is presented, where the temperatures of the inner core and ambient are denoted by \(T_{c}\) and \(T_{0}\), respectively. The physical processes of heat generation and heat dissipation by a conductor follow the law of energy conservation. The temperature rise of the conductor  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Nodes** & **Interval** & **Meaning** & **Prior Probability** \\ \hline \multirow{8}{*}{Cable current (A)} & [0,6] & Low current state & 0.3 \\  & [6,10] & Normal & 0.6 \\  & [10,15] & Slight overload & 0.04 \\  & [15,20] & Moderate overload & 0.03 \\  & [20,25] & Severe overload & 0.02 \\  & [25,35] & Extremely server overload & 0.01 \\ \hline \multirow{4}{*}{Damage rate (\(\%\))} & 0 & No damage & 0.95 \\  & [0,50] & Moderate mechanical damage & 0.03 \\  & [50,75] & Moderate mechanical damage & 0.015 \\  & [75,100] & Serious mechanical damage & 0.015 \\ \hline \multirow{4}{*}{Aging degree (month)} & 0 & New cable & 0.50 \\  & [0,6] & Use 0–6 months & 0.35 \\  & [6, 7, 8, 9, 10, 11, 12, 13, 14,  
can be regarded as the difference between heat generation and dissipation, which can be expressed as:  
\[mC\frac{\text{d}T}{\text{d}t}=I^{2}R-K_{zh}A(T-T_{0}) \tag{4}\]  
where \(t\) is the time during which the current is applied, \(m\) is the linear density of the conductor, \(C\) denotes the specific heat capacity of the conductor, \(R\) is the core resistance per unit length, \(K_{zh}\) is the total heat transfer coefficient between the conductor insulation layer and external environment, \(A\) is the surface area of the conductor per unit length. The temperature rise function can be obtained by solving the first-order differential Equation (4), and the temperature difference, denoted by \(\Delta T\), is given by:  
\[\Delta T=(T-T_{0})=\frac{I^{2}R}{K_{zh}A}\bigg{[}1-\exp\left(-\frac{K_{zh}At}{ mC}\right)\bigg{]}+(T_{c}-T_{0})\exp\left(-\frac{K_{zh}At}{mC}\right) \tag{5}\]  
With \(T_{p}=\frac{I^{2}R}{K_{zh}A}\), \(T_{n}=(T_{c}-T_{0})\), \(T_{m}=\frac{mC}{K_{zh}A}\), the above Equation (5) can be written as follows:  
\[\Delta T=T_{p}\bigg{[}1-\exp\left(-\frac{t}{T_{m}}\right)\bigg{]}+T_{n}\exp \left(-\frac{t}{T_{m}}\right) \tag{6}\]  
Equation (6) gives the conductor temperature rise as a function of time, assuming ideal conditions, and given a certain current. There are three coefficient terms in the temperature rise function: temperature rise equilibrium coefficient \(T_{p}\), temperature difference coefficient \(T_{n}\), and time coefficient \(T_{m}\).