# Optimisation of Segregation Distances between Electric Cable Bundles Embedded in a Structure
## 3 Segregation Distance Optimisation Problem with EMC Constraints  
The objective in this article was to determine the smallest value of the distance \(d\), denoted as \(d_{opt}\), under the constraints that:  
1. \(d_{opt}\in[d_{\min},d_{\max}]\);
2. \(G(d_{opt},l,h_{A},h_{B})<0\) for any \(l\), \(h_{A}\), and \(h_{B}\);
3. For every value \(d>d_{opt}\) and for any \(l\), \(h_{A}\), and \(h_{B}\), \(G(d,l,h_{A},h_{B})<0\).  
This definition of the optimal segregation distance \(d_{opt}\) is in fact very conservative. As \(G\) is not a monotonic decreasing function with \(d\), our objective was to find the maximum of different minima \(d\) that satisfy the constraints. For a given use-case, it may be possible to find a distance \(d^{\prime}\) lower than \(d_{opt}\) that will be EM compatible, but if such a distance \(d^{\prime}\) exists, by definition, we can also find a distance \(d^{\prime\prime}\) with \(d^{\prime}<d^{\prime\prime}<d_{opt}\) such that \(d^{\prime\prime}\) does not lead to an EM-compatible configuration. Moreover, these conditions are computationally hard to verify since the \(G\) function has to be evaluated for every combination of \(l\), \(h_{A}\), and \(h_{B}\) at a given distance \(d\). We thus propose to replace these constraints with probabilistic ones. For this purpose, we now consider that the length \(l\) and the heights \(h_{A}\) and \(h_{B}\) can be represented by three random variables. We thus define three independent uniform randomvariables \(L\), \(H_{A}\), and \(H_{B}\), respectively, between the intervals \([l_{\min},l_{\max}]\) and \([h_{\min},h_{\max}]\). The associated random vector \((L,H_{A},H_{B})\) is denoted as \(\mathbf{X}\) with a probability distribution \(f_{\mathbf{X}}\). The probability \(\mathbb{P}(G(d,\mathbf{X})<0)\) defines the probability that bundles \(A\) and \(B\) are EM compatible at a distance \(d\). Then, the previous minimisation problem can thus be rewritten in the following way as we seek to evaluate \(d_{opt}\) with:  
\[d_{opt}=\underset{d\in[d_{\min}d_{\max}]}{\text{argmin}}d\quad \text{ such as:}\quad\mathbb{P}(G(d,\mathbf{X})<0)=1 \tag{2}\] \[\mathbb{P}(G(d,\mathbf{X})<0)=1,\;\forall d>d_{opt} \tag{3}\]  
However, even in this case, it can be quite difficult from a computational point of view to verify that the EMC constraints are respected. Moreover, it is probable that the resulting \(d_{opt}\) will be too conservative. For this reason, the probability constraint on \(\mathbb{P}\big{(}G(d_{opt},\mathbf{X})<0\big{)}\) can be relaxed so that if the probability to be EM compatible at the distance \(d_{opt}\) is greater than \(1-\alpha\), then we consider that EMC constraints to be respected. The value \(\alpha\) is defined by the end-user and is typically set to 0.05 in practice. The optimisation problem formulation we evaluate here is now defined by the following equations:  
\[d_{opt}=\underset{d\in[d_{\min},d_{\max}]}{\text{argmin}}d\quad \text{ such as:}\quad\mathbb{P}(G(d,\mathbf{X})<0)>1-\alpha \tag{4}\] \[\mathbb{P}(G(d,\mathbf{X})<0)>1-\alpha,\;\forall d>d_{opt} \tag{5}\]  
The sought distance \(d_{opt}\) is the lowest distance so that the bundles \(A\) and \(B\) are EM compatible with probability \(1-\alpha\) and for any distance \(d\) greater than \(d_{opt}\), the bundles \(A\) and \(B\) are still EM compatible with a probability at least equal to \(1-\alpha\).