## 2 The Uncertainty Models of Renewable Energy and Load Demand
### The Objective Function of the Robust Control Model for Automatic Load Commutation Devices  
Radial distribution networks are commonly structured to adopt a radial configuration for zone power supply, which poses a challenge in the planning of such networks. Therefore, selecting appropriate power flow equations that are specifically designed for calculating radial distribution networks is imperative. In this work, the Distflow model for branch power flow [21] is utilized to describe the power flow in radial distribution networks. Accounting for the uncertain factors of the distribution system, an enhanced Distflow model for branch power flow in three-phase unbalanced radial distribution networks can be represented as follows:  
For any node \(j\) at time \(t\),  
\[\left\{ \begin{array}{l}\sum\limits_{i\in m(j)}\left[P_{ij}^{\varphi,t}-r_{ ij}^{\varphi}\frac{\left(P_{ij}^{\varphi,t}\right)^{2}+\left(Q_{ij}^{\varphi,t} \right)^{2}}{\left(V_{i}^{\varphi,t}\right)^{2}}\right]=\sum\limits_{k\in n(j) }P_{jk}^{\varphi,t}+P_{j}^{\varphi,t}\\ \sum\limits_{i\in m(j)}\left[Q_{ij}^{\varphi,t}-X_{ij}^{\varphi}\frac{\left(P_ {ij}^{\varphi,t}\right)^{2}+\left(Q_{ij}^{\varphi,t}\right)^{2}}{\left(V_{i} ^{\varphi,t}\right)^{2}}\right]=\sum\limits_{k\in n(j)}Q_{jk}^{\varphi,t}+Q_{ j}^{\varphi,t}\\ P_{j}^{\varphi,t}=PU_{j,DG}^{\varphi,t}-PU_{j,L}^{\varphi,t}\\ Q_{j}^{\varphi,t}=QU_{j,DG}^{\varphi,t}-QU_{j,L}^{\varphi,t}\end{array} \right. \tag{9}\]  
For the branch \(ij\) at time \(t\),  
\[\left(V_{i}^{\varphi,t}\right)^{2}-\left(V_{j}^{\varphi,t}\right)^{2}=2\left( P_{ij}^{\varphi,t_{ij}}+Q_{ij}^{\varphi,t,\varphi}\right)-\left[\left(r_{ij}^{ \varphi}\right)^{2}+\left(x_{ij}^{\varphi}\right)^{2}\right]\frac{\left(P_{ ij}^{\varphi,t}\right)^{2}+\left(Q_{ij}^{\varphi,t}\right)^{2}}{\left(V_{i}^{ \varphi,t}\right)^{2}} \tag{10}\]  
where \(\varphi\in\{A,B,C\}\), \(m(j)\) denotes the set of initial nodes of the branch in the grid with node \(j\) as the terminal node; \(n(j)\) denotes the set of terminal nodes of the branch in the grid network with node \(j\) as the initial node; \(P_{ij}^{\varphi,t}\) and \(Q_{ij}^{\varphi,t}\) are the active and reactive power of each phase of the initial end of branch \(ij\), respectively; \(r_{ij}^{\varphi}\) and \(x_{ij}^{\varphi}\) are the three-phase branch resistance and reactance of branch \(ij\), respectively; \(V_{i}^{\varphi,t}\) is the voltage amplitude of each phase of node \(i\); \(P_{j}^{\varphi,t}\) and \(Q_{j}^{\varphi,t}\) are the net active and reactive power injections of each phase at node \(j\), respectively; \(PU_{j,DG}^{\varphi,t}\) and \(QU_{j,DG}^{\varphi,t}\) are the renewable energy active and reactive power values at node \(j\), respectively; and \(PU_{j,L}^{\varphi,t}\) and \(QU_{j,L}^{\varphi,t}\) are the load active and reactive power demands, respectively. According to the Distflow model for branch power flow, node voltage and power flow from the source node to the terminal node of the distribution network can be incrementally calculated. To accommodate the uncertainty of renewable energy and load demand, this paper aims to minimize the total line loss for three-phase systems to ensure maximum profit in worst-case scenarios. In addition, considering the impact of frequent switching actions of load commutation switches on their service life, this paper incorporates the cost of switching actions into the objective function. Therefore, a robust control model for automatic load commutation devices is developed in this paper, which accounts for uncertainties in both source and load, with an objective function that seeks to minimize the total line loss for three-phase systems and costs associated with commutation switch actions.