# Dynamic Rating of Three-Core XLPE Submarine Cables for Offshore Wind Farms
## 2 Dynamic Cable Rating
### Thermal Cable Analysis of Transient Conditions
#### 2.3.2 Transient Temperature Response to a Step Function  
To evaluate the transient temperature behaviour in the centre of the conductor resulting from an applied step function, it is necessary to model the thermal temperature response mathematically. A step function is made to reduce the number of calculations needed. Figure 15 illustrates a current step function \(I_{s}\) with a total number of \(i\) 4 h constant load steps \(\Delta\tau\). \(I_{s}\) is the average value of the actual flowing current \(I_{a}\) for each step during the current profile segment illustrated in Figure 3.  
To determine an appropriate length for the constant time-steps \(\Delta\tau\), an empirical calculation was proposed by [PERSON] [9] as given by Equation (26). This method was proposed at a time when computing time meant a lot; nowadays, the computing time is not an issue. Considering wind-based load profiles, it is not recommendable to use time-steps \(\Delta\tau\) greater than 4 times the data resolution, as it does not represent the behaviour of the wind. \[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]  
In this case, the step function has to represent the dynamic power losses, \(W_{c}(t)\) and \(W_{I}(t)\), inside the cable, as shown in Figure 14 and determined by Equations (24) and (25). Fourier transform of the unit impulse [9] as the transfer function is used to represent the response of the network [9]. The transfer function \(H(s)\) of the two-loop network shown in Figure 6 can be represented by the ratio between the conductor temperature \(\theta_{c}\) and the cable power losses \(W_{n}\), according to Equation (27). Figure 15: Step function construction of a wind-based load profile. \[H(s)=\frac{\theta_{c}(s)}{W_{n}(s)}=\frac{1}{s\cdot Q_{A}+\frac{1}{T_{A}+\frac{1}{s \cdot Q_{B}+\frac{1}{T_{B}}}}}\]  
\[=\frac{(T_{A}+T_{B})+s\cdot T_{A}\cdot T_{B}\cdot Q_{B}}{1+s\cdot(T_{A}\cdot Q_ {A}+T_{B}\cdot Q_{B}+T_{B}\cdot Q_{A})+s^{2}\cdot T_{A}\cdot Q_{A}\cdot T_{B} \cdot Q_{B}} \tag{27}\]  
where \(\theta_{c}(s)\) and \(W_{n}(s)\) are polynomial of the transfer function obtained by [PERSON]'s current law of the two-loop network; \(s\) is the complex transfer function variable. Considering any node \(h\) in the thermal ladder network, the temperature response \(\theta_{h}(t)\) as function of time can be expressed as shown in Equation (28) [8]. \[\theta_{h}(t)=W_{c}\cdot\sum_{j=1}^{2}T_{hj}\cdot\left(1-e^{P_{j}\cdot t} \right)\qquad[\text{K}] \tag{28}\]  
where \(P_{j}\) is the time constant \([\text{s}^{-1}]\); \(T_{hj}\) is a thermal coefficient \([\text{K}\cdot\text{m}/\text{W}]\); \(t\) is time \([\text{s}]\); \(h\) is the node index; \(J\) is the loop index (1, 2); \(P_{J}\) and \(T_{hj}\) are obtained from poles and zeros of the transfer function given in Equation (27). The coefficient \(T_{hj}\) is given by Equation (29), developed by [PERSON] [8]. \[T_{hj}=-\frac{a_{(2-h)h}}{b_{2}}\cdot\frac{\prod_{k=1}^{2-h}(Z_{kh}-P_{J})}{P_ {J}\cdot\prod_{k=1\delta k\
eq h}^{2}(P_{k}-P_{J})} \tag{29}\]  
where \(a_{(2-h)h}\) is a coefficient of the numerator equation; \(b_{2}\) is the first coefficient of the denominator equation  
\[N_{0}=T_{A}\cdot Q_{A}\cdot T_{B}\cdot Q_{B} \tag{36}\]  
From Equation (27), \(a_{(2-h)h}\) can be found as shown in Equation (37) for loop one, and \(b_{2}\) is found in Equation (38). Thus, the first part of Equation (29) can be found as shown in Equation (39). \[a_{(2-1)1}=a_{11}=T_{A}\cdot T_{B}\cdot Q_{B} \tag{37}\]  
\[b_{2}=T_{A}\cdot T_{B}\cdot Q_{A}\cdot Q_{B} \tag{38}\]  
\[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]0  
The thermal coefficient \(T_{a}\) from Equation (29) can be found as shown in Equation (40).