## 2 The Uncertainty Models of Renewable Energy and Load Demand
### The Constraints of the Robust Control Model for Automatic Load Commutation Devices  
(1) Constraints on power balance  
In the robust control model of automatic load commutation considering the uncertainties of source and load, the power balance constraints are shown as Equations (1) and (3). (2) Security constraints on node voltage and branch current  
To ensure the secure and stable operation of a low-voltage distribution network that considers the uncertainty of renewable energy and load demand, the node voltage and branch current must meet the following constraints  
\[\left\{ \begin{array}{l}V_{i}^{\varphi,\min}\leq V_{i}^{\varphi}\leq V_{i}^{ \varphi,\max}\\ I_{ij}^{\varphi}\leq I_{ij}^{\varphi,\max}\end{array} \right. \tag{12}\]  
where \(V_{i}^{\varphi,\min}\) and \(V_{i}^{\varphi,\max}\) denote, respectively, the \(\varphi\)-phase minimum and maximum voltage limits for node \(i\), and \(I_{i}^{\varphi,\max}\) is the limit of the overload critical current of the \(\varphi\)-phase branch. (3) Constraint on the output power of root node  
The uncertainty of load demand and renewable energy will cause power fluctuations in the distribution network and adversely affect the transmission network, so the gateway exchange power at the root node of the distribution network must meet the following constraints:  
\[\left\{ \begin{array}{l}P_{0}^{\varphi,\min}\leq P_{0}^{\varphi}\leq P_{0}^{ \varphi,\max}\\ Q_{0}^{\varphi,\min}\leq Q_{0}^{\varphi}\leq Q_{0}^{\varphi,\max}\end{array} \right. \tag{13}\]  
where \(P_{0}^{\varphi}\) and \(Q_{0}^{\varphi}\), respectively, represent the active power and reactive power of phase \(\varphi\) flowing into the distribution network from the root node, and \(P_{0}^{\varphi,\min}\), \(P_{0}^{\varphi,\max}\), \(Q_{0}^{\varphi,\min}\), and \(Q_{0}^{\varphi,\max}\) are, respectively, the upper and lower limits of the active power and reactive power at the junctions of phase \(\varphi\). (4) Constraint on the number of commutation switch operations  
In order to minimize the impact of the commutation switch operation on its service life and to ensure the normal operation of the switch during the optimization period, the number of commutation switch operations should meet the following constraint:  
\[\sum_{t\in T}\sum_{i\in F}a_{i,t}< \tag{14}\]  
where \(S_{\max}\) is the maximum number of all commutation switch operations during the optimization period. Constraints on the power fluctuations of renewable energy and load demand considering the uncertainties  
\[\left\{ \begin{array}{l}P_{i,DG}^{\varphi,t}=PU_{i,DG}^{\varphi,t}\\ Q_{i,DG}^{\varphi,t}=PU_{i,DG}^{\varphi,t}\tan\gamma^{\prime}\\ PU_{i,DG}^{\varphi,t}\in\left[P_{i,tG}^{\varphi,t}-PD_{i,DG}^{\varphi,t},P_{i,DG}^{\varphi,t}+PD_{i,DG}^{\varphi,t}\right]\\ QU_{i,DG}^{\varphi,t}\in\left[Q_{i,DG}^{\varphi,t}-QD_{i,DG}^{\varphi,t},Q_{ i,DG}^{\varphi,t}+QD_{i,DG}^{\varphi,t}\right]\\ PU_{i,L}^{\varphi,t}\in\left[P_{i,L}^{\varphi,t}-PD_{i,L}^{\varphi,t},P_{i,t}^{\varphi,t}+PD_{i,L}^{\varphi,t}\right]\\ QU_{i,L}^{\varphi,t}\in\left[Q_{i,L}^{\varphi,t}-QD_{i,L}^{\varphi,t},Q_{i,L }^{\varphi,t}+QD_{i,L}^{\varphi,t}\right]\\ \sum\limits_{k\in n(i)}\left|\varepsilon_{i,DC}^{\varphi,t}\right|\leq\Gamma_ {DG}N_{DG}\\ \sum\limits_{k\in n(i)}\left|\varepsilon_{i,L}^{\varphi,t}\right|\leq\Gamma_ {L}N_{L}\end{array} \right.