## 2 Mathematical Models
### Owfcclp  
The wind turbines (WTs) are widely spread in the open sea area. In order to collect the power generated from each WT economically, the cable connection layout should be optimized to make the electrical system cost effective. Since the power losses along the cables reduce the energy reached at the point of common coupling and thus influence the revenue for the wind farm owner, this part of revenue loss should be counted in the cable connection layout and cable type selection optimization process which formulates the OWFCCLP. In this paper, the objective function is set up by considering both the CAPEX and OPEX. The CAPEX of the offshore wind farm includes the cable cost and the cable trenching cost, while the costs of the energy losses in the internal network are considered as the OPEX. The overall target is to minimize the total cost to ensure the best return on investment for investors during the entire operational lifetime of the wind farm. The mathematical model of the total cost can be expressed as  
\[C_{Total} =\min(Z_{Trench}+Z_{Cable}+Z_{EnergyLoss}), \tag{1}\]  
where \(Z_{Trench}\) means the total trenching cost of the internal array cables, \(Z_{Cable}\) is the investment on cables and \(Z_{EnergyLoss}\) represents the cost of energy losses associated with the array cables in the internal network during the entire lifetime of the wind farm. The detailed expressions of each cost variable can be seen in the following:  
\[Z_{Trench} =C_{Trench}\cdot(\sum_{k=1}^{K}\sum_{m=1}^{M}\sum_{p=1}^{N}L_{m}g_ {C_{T},k,m,p}+\sum_{k=1}^{K}\sum_{m=1}^{M}\sum_{m^{\prime}=1}^{M},\sum_{p=1}^{ N}L^{\prime}_{m,m^{\prime}}g^{\prime}_{C_{T},k,m,m^{\prime},p}), \tag{2}\] \[m^{\prime} \
eq m\]  
It can be seen in Equation (2) that \(Z_{Trench}\) is related to the total length of cables. Similarly, it can be calculated according to the cable connection layout and the cable size for each path, which can be written as follows:  
\[Z_{Cable} =3(\sum_{k=1}^{K}\sum_{m=1}^{M}\sum_{p=1}^{N}C_{p}L_{m}g_{C_{T},k,m,p}+ \sum_{k=1}^{K}\sum_{m=1}^{M}\sum_{m=1}^{M}\sum_{ \begin{array}{c}m^{\prime}= 1,\\ m^{\prime}\
eq m\end{array} }^{N}\sum_{p=1}^{N}C_{p}L^{\prime}_{m,m^{\prime}}g^ {\prime}_{C_{T},k,m,m^{\prime},p}), \tag{3}\] \[m^{\prime} \
eq m\]  
The power losses of the cables are related to the resistance and are proportional to the square of the passed current. The expression of power losses is expressed as  
\[Z_{EnergyLoss,Unit}=(I_{rated})^{2}\cdot(\sum_{k=1}^{K}\sum_{m=1}^{M}\sum_{p=1 }^{N}(O_{C_{T},k,m,p})^{2}L_{m}R_{p}g_{C_{T},k,m,p}+\sum_{k=1}^{K}\sum_{m=1}^ {M}\sum_{ \begin{array}{c}m^{\prime}=1,\\ m^{\prime}\
eq m\end{array} }^{N}\sum_{p=1}^{N}(O^{\prime}_{C_{T},k,m,m^{ \prime},p})^{2}L^{\prime}_{m,m^{\prime}}R_{p}g^{\prime}_{C_{T},k,m,m^{\prime}, p}), \tag{4}\]  
where \(I_{rated}\) is the rated current of the WT:  
\[I_{rated}=\frac{P_{rated}}{\sqrt{3}\cos\mu I_{rated}}, \tag{5}\]Furthermore, considering the duration time of the peak energy loss and interest rate of each year, \(Z_{EnergyLoss}\) can be written as  
\[Z_{EnergyLoss}=3\sum\limits_{\tau=1}^{T}{(1+D)^{\tau}\tau_{\Delta}C_{EnergyLoss}Z _{EnergyLoss,Unit}}, \tag{6}\]  
It should be noticed that, for the protection of the electrical equipment, the maximum current flowing in each cable is capped by an upper limit, which means the number of WTs carried by a specified cable is limited. A feasible cable connection layout must strictly meet the maximum load flow constraint; in other words, the actual current going through each cable should be no more than \(I_{max,p}\), which formulates a new constraint as follows:  
\[ \begin{array}{l}I_{rateal}(O_{G_{T},k,m,p}g_{G_{T},k,m,p}+\sum\limits_{ \begin{array}{c}m^{\prime}\\ m^{\prime}=1,\end{array}}^{M}O_{G_{T},k,m,m^{\prime},p}^{\prime}g_{G_{T},k,m,m ^{\prime},p}^{\prime})\leq I_{max,p},\\ \begin{array}{c}m^{\prime}\
eq m\\ \forall k=1,2,\cdots,K,\forall m=1,2,\cdots,M,\;\forall m=1,2,\cdots,M,\; \forall p=1,2,\cdots,N,\end{array}\end{array}