## 2 Modeling of Offshore Oil and Gas Platform Cluster System
### Cluster-Division Principle Based on Voltage and Loss-Sensitivity Index  
The partition of a power-system cluster is based on the electrical distance between each node and the slack bus, and the nodes with similar distance to the slack bus are classified into one category. Therefore, the entire distribution network is partitioned into multiple voltage-control clusters, as shown in Figure 3. The nodes in each cluster have close electrical coupling. For the control of distribution network node voltage and loss, after the cluster is partitioned, it can be more effective; at the same time, it can also more effectively improve the matching of the timing of power and voltage stability within each cluster.  
Figure 3: Cluster regional structure.  
Electrical distance is based on voltage and loss-sensitivity-matrix analysis, which is used to measure the coupling closeness of voltage changes between nodes. The output form of DG connected to the distribution network is mainly active power; the 10 kV submarine cable has small geometric distance and close connection and the reactance is smaller than that of the overhead line of the same grade. Therefore, its resistance and reactance are not much different. Therefore, based on the voltage-active power-sensitivity matrix and the loss-sensitivity matrix, the control ability of the active power of node \(m\) to the voltage and loss of each node can be expressed by the column vector \(V_{Sm}\) of the voltage-active power-sensitivity matrix and the column vector \(L_{Sm}\) of the loss-sensitivity matrix:  
\[\left\{\begin{array}{l}VS_{m}=\left[\begin{array}{cccc}\frac{\partial U_{ 2}}{\partial T_{lm}}&\frac{\partial U_{3}}{\partial T_{lm}}&\ldots&\frac{ \partial U_{n}}{\partial T_{lm}}\\ LS_{m}=\frac{\partial T_{\rm{min}}}{\partial T_{lm}}\end{array}\right]^{ \sf T},\right. \tag{22}\]  
In Equation (22), the derivative of the power of each node with respect to node \(m\) can be regarded as a dimension of spatial coordinates. Matrix \(V_{Sm}\) reflects the relationship between voltage and active power and \(L_{Sm}\) reflects the relationship between loss and active power. The specific solving process of electrical distance is shown in the following equations:  
\[x_{vmk}=\frac{VS_{mk}-VS_{\rm{min}}}{VS_{\rm{max}}-VS_{\rm{min}}}, \tag{23}\]  
\[x_{lm}=\frac{LS_{m}-LS_{\rm{min}}}{LS_{\rm{max}}-LS_{\rm{min}}}, \tag{24}\]  
\[d_{ij}=\sqrt{p(\left|x_{vi2}-x_{vj2}\right|^{2}+\ldots+\left|x_{vik}-x_{vjk} \right|^{2})+q\left|x_{li}-x_{lj}\right|^{2}}, \tag{25}\]  
where \(VS_{\rm{min}}\) and \(VS_{\rm{max}}\) are the minimum and maximum values of voltage-active power sensitivity matrix elements, respectively; \(LS_{\rm{min}}\) and \(LS_{\rm{max}}\) are the minimum and maximum values of loss-sensitivity matrix elements, respectively; \(x_{\rm{vmk}}\) is the \(k\)-th spatial coordinate of node \(m\) after the normalization of voltage-sensitivity matrix elements; \(x_{lm}\) is the spatial coordinate of node \(m\) after the normalization of loss-sensitivity matrix elements; \(d_{ij}\) is the electrical distance between nodes \(i\) and \(j\); \(p\) and \(q\) are the weight coefficients of different indicators; and \(p+q=1\). Equations (23) and (24) unify the data of voltage sensitivity and loss sensitivity to one standard through minimum-maximum normalization transformation and combined with the Euclidean distance calculation method of Equation (25), the size of the electrical distance between nodes \(i\) and \(j\) can be solved when the degrees of influence of loss sensitivity and voltage sensitivity are different.  
Nodes with relatively close electrical distance have a similar influence of active power on voltage and loss, and these nodes can be grouped into a sub-cluster that can be controlled with one voltage controller. According to Equations (15) and (19), the factors affecting the electrical distance mainly include the impedance of the line, the voltage amplitude of the node, and the injected power of the node.