# Dynamic Rating of Three-Core XLPE Submarine Cables for Offshore Wind Farms
## 2 Dynamic Cable Rating
### Thermal Cable Analysis of Transient Conditions
#### 2.3.1 Thermal Ladder Network Construction  
To analyse the transient temperature response in the centre of the conductor, it is essential to thermally model the cable with high accuracy. The cable is transformed from the three-core cable into a single-core equivalent cable with the same thermal properties to simplify the equivalent thermal circuit. According to IEC 60853-2 [8], transients greater than 1 h are assumed to be long duration transients, which usually is the case, of an offshore wind farm. Therefore, this method is based on long transients and is intended for the use of long duration temperature transients only. The ladder network method (LNM) is only applicable in cases where the thermal parameters do not change as a function of temperature variations, as it solves heat flow equations using the superposition principle, which is used exclusively for linear systems. Figure 9 shows a quarter of the cross-section of the equivalent single-core cable, with thermal resistances represented on the \(x\)-axis below the respective layer and thermal capacitances represented on the left-hand side of the \(y\)-axis beside the respective layer. To visually demonstrate the modelling of the total thermal ladder network representing a three-core XLPE submarine cable, the ladder network is divided into three sections, each surrounding a thermal resistance. Representing the insulation layer and the screens surrounding it, the thermal resistances are divided by three, \(T_{1}/3\), to represent the three internal parts of the cable in parallel. The thermal capacitance of the insulation \(Q_{i}\) is divided into two non-equal lumped parameters, distributed in parallel on each side of the thermal resistance \(T_{1}/3\). Figure 10 shows the thermal representation of the dielectric layers, where [PERSON]'s coefficient, shown in Equation (17), is applied to determine the allocation of the thermal capacitance \(Q_{i}\) more accurately [9]. Figure 9: Quarter cross-sectional area of a single-core XLPE SL-type equivalent submarine cable with thermal resistances \(T\) and thermal capacitances \(Q\) represented along the axes. In order to improve the accuracy in the use of lumped parameters, the thermal insulation capacitance \(Q_{i}\) is distributed between the single-core equivalent conductor diameter \(D_{c}^{*}\) given in Equation (18) and the external diameter of insulation \(D_{i}\) using [PERSON]'s coefficient \(p\)[9], shown in Equation (17), for long duration transients. \[p=\frac{1}{2\cdot ln\left(\frac{D_{i}}{D_{c}^{*}}\right)}-\frac{1}{\left(\frac {D_{i}}{D_{c}^{*}}\right)^{2}-1} \tag{17}\]  
As mentioned before, due to the asymmetry of the internal parts of the three-core cable shown in Figure 8, the allocation of thermal resistance \(T_{1}\) has to be defined as an equivalent single-core conductor diameter \(D_{c}^{*}\) dissipating the same losses as that determined by Equation (18). \[\frac{T_{1}}{3}=\frac{\rho}{2\cdot\pi}\cdot ln\left(\frac{D_{i}}{D_{c}^{*}} \right)\qquad\left[\mathrm{K}\cdot\mathrm{m}/\mathrm{W}\right]\qquad\quad \rightarrow\qquad\qquad D_{c}^{*}=D_{i}\cdot e^{\frac{-2\cdot\pi\cdot T_{1}}{ \rho\cdot 3}}\qquad\left[\mathrm{m}\right] \tag{18}\]  
Representing the sheaths around the thermal resistance \(T_{2}\), the metallic sheath and anti-corrosion sheath are distributed on each side of \(T_{2}\), as they do not have the same specific heat capacity coefficient \(c_{m}\). Due to the asymmetry of the internal part of the cable, the thermal resistance \(T_{2}\) is calculated using the geometric factor \(G(X_{s})\) to include the fillers in the thermal resistance [10]. Estimations of the single-core diameter of the fillers are made on the basis of the assumption that each layer is occupied by insulation, and this means that the thermal capacitance is separated into two equal capacitances because the diameter varies in the original three-core cable. Figure 11 shows the distribution of the sheath parameters.