# 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 represent the cable's outer covering surrounding the thermal resistance \(T_{3}\), the armour and the outer serving layer must be distributed around. As the armour is a metallic layer, the thermal capacitance \(Q_{a}\) is placed in the original position, and the thermal capacitance of the outer serving \(Q_{os}\) is divided into two capacitances by [PERSON]'s coefficient \(p^{\prime}\) in Equation (19) to allocate the thermal capacitance distribution of the outer covering [8]. Figure 12 shows the distributed parameters of the armour and the outer covering. \[p^{\prime}=\frac{1}{2\cdot ln\left(\frac{D_{os}}{D_{a}}\right)}-\frac{1}{ \left(\frac{D_{os}}{D_{a}}\right)^{2}-1} \tag{19}\]  
Figure 11: Transient representation of thermal parameters for sheath. Figure 10: Transient representation of thermal parameters for insulation. By developing lumped parameters for each part of the cable, a three-core XLPE submarine cable is thermally represented by connecting the models of Figures 10-12, as shown in Figure 13.  
To represent the cable in operation, the thermal ladder network must include the cable losses. Using the Cigre two-loop method [8] to reduce the mathematical complexity of the circuit analysis, the final thermal ladder network including power losses can be derived as shown in Figure 14, where \(\theta_{c}\) is the denotation of the conductor temperature and \(\theta_{os}\) represents the temperature at the outer serving. According to IEC 60853-2 [8], the apparent thermal resistances in the ladder network can be defined as shown in Equations (20) and (21), and the apparent thermal capacitances are defined in Equations (22) and (23). The cable losses are equally found; cable conductor loss \(W_{c}\) is determined by Equation (24), and the total internal cable losses \(W_{I}\), including dielectric, sheath and armour losses, are determined by Equation (25). \[T_{A}=T_{1}/3\qquad[\mathrm{K}\cdot\mathrm{m}/\mathrm{W}] \tag{20}\]  
\[T_{B}=T_{2}+T_{3}\qquad[\mathrm{K}\cdot\mathrm{m}/\mathrm{W}] \tag{21}\]  
\[Q_{A}=Q_{c}+p\cdot Q_{i}\qquad[\mathrm{J}/\mathrm{m}\cdot\mathrm{K}] \tag{22}\]  
\[ \begin{split} Q_{B}&=(1-p)\cdot Q_{i}+\left(\frac{T _{2}+T_{3}}{T_{2}+T_{3}}\right)^{2}\cdot Q_{ms}+\left(\frac{T_{2}+T_{3}}{T_{2} +T_{3}}\right)^{2}\cdot\frac{Q_{f}}{2}+\left(\frac{T_{3}}{T_{2}+T_{3}}\right)^ {2}\cdot Q_{acs}+\\ &\qquad\left(\frac{T_{3}}{T_{2}+T_{3}}\right)^{2}\cdot\frac{Q_{ f}}{2}+\left(\frac{T_{3}}{T_{2}+T_{3}}\right)^{2}\cdot Q_{a}+\left(\frac{T_{3}}{T_{2}+T_{3}} \right)^{2}\cdot p^{\prime}\cdot Q_{os}\qquad[\mathrm{J}/\mathrm{m}\cdot \mathrm{K}]\end{split}  \tag{23}\]  
Figure 14: Reduced thermal ladder network representing a three-core XLPE submarine cable. Figure 12: Transient representation of thermal parameters for outer covering. Figure 13: Thermal three-core XLPE submarine cable ladder network. \[W_{c}(t)=3\cdot\left[I_{a}(t)^{2}\cdot R_{ac}\right]\qquad\left[\text{W}/\text{m}\right] \tag{24}\]  
\[W_{I}(t)=W_{c}(t)\cdot\left[1+\lambda_{1}+\lambda_{2}\right]+3\cdot W_{d}\qquad \left[\text{W}/\text{m}\right] \tag{25}\]