# Dynamic Rating of Three-Core XLPE Submarine Cables for Offshore Wind Farms
## 2 Dynamic Cable Rating
### Determination of Cable Parameters
#### 2.2.2 Determination of Thermal Resistances  
To determine the cable rating, the thermal resistances are needed. IEC 60287-2-1 [11] defines the thermal resistances \(T_{1}\), \(T_{2}\), \(T_{3}\) and \(T_{4}\), where \(T_{1}\) is the resistivity between the conductor and metallic sheath/screen, representing the layers \(t_{2-5}\) in Figure 2; \(T_{2}\) is between the metallic sheath and armour, representing the layers \(t_{7-8}\) in Figure 2; \(T_{3}\) is between the armour, representing the layer \(t_{10}\) in Figure 2 and surroundings; and \(T_{4}\) represents the surroundings illustrated in Figure 7. To simplify the calculation, the following assumptions are made:  
* Metallic layers are neglected, as their thermal resistance is negligible compared with that of poly-composite materials.
* The optical fibre inside the fillers is assumed to have no thermal impact.
* Swelling tape is assumed to be a part of the insulation since its thickness is small, and it is assumed to have the same thermal resistivity.
* The bedding material is assumed to be the same material as that composing the anti-corrosion sheath.  
Figure 8 shows a static thermal representation of a three-core cable based on the cable construction in Figure 2, where the electrical resistances are equivalent to the thermal resistances, as shown in Figures 5 and 6, and define the ability of the materials to impede heat flow. In Figure 8, \(\theta_{c}\) is the conductor temperature, \(\theta_{os}\) is the outer covering (cable surface) temperature and \(\theta_{A}\) is the ambient temperature of the surroundings. As the cable is a three-core cable, and \(T_{1}\) represents the resistance of the inner cables, the circuit is represented with \(T_{1}\) as three resistances in parallel.  
The thermal resistances shown in Figure 8 are determined by Equations (9)-(12) [11], where the thermal resistivity \(\rho\) for commonly used cable materials and surrounding seabed conditions is defined in Table 1, \(t\) is the thickness of the respective cable layers, \(D_{c}\) is the conductor diameter, \(D_{a}\) is the outer diameter of the armour, \(D_{os}\) is the outer cable diameter, \(G(X_{s})\) is a geometric factor as a function of thickness of material between sheaths and armour and \(X_{s}\) expressed as a fraction of the outer diameter of the sheath in Equation (13).  
\[T_{1}=\frac{\rho}{2\pi}\cdot\ln\left(1+\frac{2\cdot t_{2-5}}{D_{c}}\right)\qquad \left[\mathrm{K}\cdot\mathrm{m}/\mathrm{W}\right] \tag{9}\]  
\[T_{2}=\frac{\rho}{6\pi}\cdot G(X_{s})\qquad\left[\mathrm{K}\cdot\mathrm{m}/ \mathrm{W}\right] \tag{10}\]  
\[T_{3}=\frac{\rho}{2\pi}\cdot\ln\left(1+\frac{2\cdot t_{10}}{D_{a}}\right) \qquad\left[\mathrm{K}\cdot\mathrm{m}/\mathrm{W}\right] \tag{11}\]  
Figure 8: Static representation of thermal resistances in a three-core XLE SL-type submarine cable, including power losses located in the respective layers and temperature denotation.  
Figure 7: Surrounding cable to soil surface dimensions and mirror of adjacent cable dimensions; representation of the surrounding thermal resistance \(T_{4}\) (the drawing is not to scale).  
\[T_{4}=\frac{\rho}{2\pi}\cdot\ln\left(\frac{2\cdot L}{D_{os}}+\sqrt{\left(\frac{2 \cdot L}{D_{os}}\right)^{2}-1}\right)\qquad[\text{K}\cdot\text{m}/\text{W}] \tag{12}\]  
The geometric factor is found as the function \(G(X_{s})\) and is determined by Equation (14) [11].  
\[X_{s}=\frac{t_{7-8}}{D_{s}} \tag{13}\]  
\[G(X_{s})=\begin{cases}2\pi\cdot(0.000202380+2.03214\cdot X_{s}-21.6667\cdot X _{s}^{2})&\text{if }0<X_{s}\leq 0.03\\ 2\pi\cdot(0.0126529+1.101\cdot X_{s}-4.59737\cdot X_{s}^{2}+11.5093\cdot X_{s} ^{3})&\text{if }0.03<X_{s}\leq 0.15\end{cases} \tag{14}\]