# 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
Figure 15:1  
With all three contributing transients from Equation (44) determined, it is possible to illustrate each of the transients and the total transient temperature response \(\theta_{t,1}\) by an example, shown in Figure 15:2 In this figure, each contributing transient due to one current step \(i\) (\(i=1\)) is evaluated for a step time duration of 24 [h]. The figure shows that the conductor to cable surface temperature rises \(\theta_{c,1}\) due to one step of the dynamic power loss function \(W_{c,1}\). As expected, the temperature rises rapidly during the first part of the transient, after which it exponentially reaches the steady-state temperature after a certain period. At this time, a common temperature of the cable for the layers between the conductor and the cable surface is achieved. The attainment factor \(\alpha_{f,1}\) is intended to describe the ratio of the first part of the transient to the same segment in steady-state. It is multiplied by the temperature response of the surrounding conditions \(\theta_{e,1}\) and is intended to attain the real ability of the environment to absorb heat in the first part of the transient. The transient temperature response itself that results from the impact of the surrounding conditions \(\theta_{e,1}\) is modelled to determine the ability of the environment to absorb heat as a function of time due to the total dynamic power loss function \(W_{DI,1}\). Compared with the cable to surface transient \(\theta_{c,1}\), it is a long-term process to reach steady-state conditions, and it can be justified by the fact that the surroundings of the cable can be considered as a circular large thermal capacitance around the cable, as shown in Figure 15:3 Generally, in the first part of the temperature transient, the conductor to cable surface transient has a significant influence on the temperature increase. After a certain amount of time, it reaches a steady-state temperature, whereas the temperature response of the surrounding conditions \(\theta_{e,1}\) still increases. By introducing multiple load steps, the superposition principle can be utilised as illustrated by Equation (50) to calculate the transient temperature as a function of time. Figure 17 shows three randomly selected load steps and the total temperature progress \(\theta_{\sum}(t)\). Figure 15:4  
Each evaluated temperature transient is calculated using multiple loops and is saved in a matrix. In order to sum the partial transients above an ambient temperature, an \((i\times t)\)-matrix \(\overline{\Delta\theta_{s}}\) storing all partial temperature transients is formed, as shown in Equation (51). \[\overline{\Delta\theta_{s}}= \begin{bmatrix}\theta_{t,1}(1)&\theta_{t,1}(2)& \ldots&\theta_{t,1}(t_{i})&\ldots&\ldots&\ldots&\theta_{t,1}(t_{n})\\ 0&\theta_{t,2}(1)&\theta_{t,2}(2)&\ldots&\theta_{t,2}(t_{i})&\ldots&\ldots& \ldots&\theta_{t,2}(t_{n})\\ \vdots&0&\ddots&\ddots&\ddots&\ddots&\ldots&\ldots&\vdots\\ \vdots&\vdots&\ddots&\theta_{t,i}(1)&\theta_{t,i}(2)&\ldots&\theta_{t,i}(t_{ i})&\ldots&\theta_{t,i}(t_{n})\\ \vdots&\vdots&\vdots&0&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&\cdots&\cdots&\cdots&\cdots&0&\ldots&\theta_{t,n}(t_{n})\\ \end{bmatrix} \qquad[\text{K}] \tag{51}\]  
Figure 16: Total transient temperature response of a single load step given by Equation (50). Figure 17: Total transient temperature response of multiple load steps given by Equation (51). The sum of all partial transient temperatures above ambient \(\overline{\Delta\Theta}(t)\) is determined as the temperature series of the sum of every single column in the matrix \(\overline{\Delta\theta_{s}}\). To determine the complete temperature response \(\Theta(t)\) to a forcing step function, the ambient temperature \(\theta_{A}\) is added to the temperature series \(\overline{\Delta\Theta}(t)\), as shown in Equation (52). \[\Theta(t)=\Delta\Theta(t)+\theta_{A}\qquad[^{\circ}\mathrm{C}] \tag{52}\]