# 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
\[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]1  
Multiplying \(P_{1}\) from Equation (33) with \(P_{2}\) from Equation (34), an expression is found in Equation (41) to reduce the thermal coefficients. \[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]2  
Using Equation (41) to simplify Equation (40), an expression is found and shown in Equation (42) to find the thermal coefficient of the first loop in the thermal ladder network. Following the same procedure for the thermal coefficient \(T_{b}\) of the second loop, an expression is found and shown in Equation (43). \[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]3  
\[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]4  
With the thermal coefficients determined for each of the loops in the thermal ladder network, the total transient temperature response \(\theta_{t,i}(t)\) above the ambient temperature \(\theta_{A}\) is available. The total temperature transient is found as the function of three contributing transients that are due to the \(i\)th load step of the step functions \(W_{c,i}(t)\) and \(W_{l,i}(t)\), as shown in Equation (44). \[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]5  
The transient temperature response \(\theta_{c,i}(t)\), which is derived from Equation (28), describes the conductor (\(\theta_{c}\)) to cable surface (\(\theta_{os}\)) transient temperature response as a function of time in Equation (45). \[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]6  
IEC 60853-2 [8] proposes a method to determine the transient temperature rise and fall due to the impact of the surrounding conditions. The function is developed from an isotherm process between the centre of the conductor and the soil surface, as shown in \[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]7 The temperature response from the surroundings is given as an exponential integral function in Equation (46). \[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]8  
where \(-Ei(-x)\) is the exponential integral, defined as \(\int_{x}^{\infty}\frac{e^{-t}}{t}dt\), which can be developed in the series shown in Equation (47); \(d_{cc}^{\prime}\) is the distance from the centre of the conductor to the image of an adjacent conductor centre [m]. Figure 7 shows the relation between the burial depth \(L\) and the conductor centre to centre distance \(d_{cc}\) and an approximation to determine the distance from the centre of the conductor to the image of an adjacent conductor centre, given by Equation (48). \[\Delta\tau=\frac{\sum T\cdot\sum Q\cdot\left(\frac{\tau}{\sum T\cdot\sum Q} \right)\frac{1}{3}}{10^{125}}\qquad\left[\text{s}\right] \tag{26}\]9  
Figure 15:0  
To determine the real temperature gradient of the first part of a temperature response from the surroundings, an attainment factor is added. The attainment factor is computed as the ratio of the first part of the transient to the same segment in steady-state. This factor takes into account the heat dissipation from the centre of the conductor to the cable surface and is calculated by Equation (49) [9].