# Toward Low Time Fragmentation of Equipment: A Double-DQN Based TT&C Task Planning Approach
## 3 Mathematical Model  
During the execution of the TT&C task, the transceiver equipment between the satellite and the ground station can only perform one task at a time. Therefore, there may be conflicts in the simultaneous scheduling of different visible windows. For any two visible windows \(w^{k}_{i,j},w^{r}_{m,n}\in W\), they conflict with each other if both of the following conditions are satisfied:  
1. There is time overlap between visible windows \(w^{k}_{i,j}\) and \(w^{r}_{m,n}\), i.e., \(ws^{r}_{m,n}<ws^{k}_{i,j}<we^{r}_{m,n}\) or \(ws^{r}_{m,n}<we^{k}_{i,j}<we^{r}_{m,n}\). 2. Windows \(w^{k}_{i,j}\) and \(w^{r}_{m,n}\) occupy the same satellite or TT&C equipment, i.e., \(i=m\) or \(j=n\). As shown in Figure 3, the start time of window \(w^{1}_{2,1}\) is before the end time of window \(w^{1}_{1,1}\), and the end time of window \(w^{1}_{2,1}\) is after the start time of window \(w^{2}_{1,1}\). The overlapping part of the two windows together uses the same equipment to cause a conflict. A TT&C task of a satellite requires multiple ascending or descending orbit TT&C windows. The combination of visible windows that meet the requirements constitutes a feasible scheme for the TT&C task. In this paper, \(y^{\alpha}_{i}\) represents the \(\alpha\)-th feasible scheme of task \(om_{i}\), so the set of feasible schemes for the task \(om_{i}\) is \(Y_{i}=\left\{y^{1}_{i},y^{2}_{i},\ldots,y^{\alpha}_{i},\ldots\right\}\), and the set of feasible schemes for all TT&C tasks is \(Y=\underset{1\leq i\leq N}{\cup}Y_{i}\). For example, the window requirement \(wr_{i}\) of task \(om_{i}\) is two ascending orbit windows and two descending orbit windows. The \(\alpha\)-th feasible scheme of TT&C task \(om_{i}\) is \(y^{\alpha}_{i}=\left\{w^{\alpha,1}_{i},w^{\alpha,2}_{i},w^{\alpha,3}_{i},w^{ \alpha,4}_{i}\right\}\), and \(w^{\alpha,1}_{i},w^{\alpha,2}_{i},w^{\alpha,3}_{i},w^{\alpha,A}_{i}\in W\), where \(w^{\alpha,1}_{i}\), \(w^{\alpha,1}_{i}\), \(w^{\alpha,3}_{i}\), and \(w^{\alpha,A}_{i}\) are, respectively, the first ascending orbit window, the first descending orbit window, the second ascending orbit window, and the second descending orbit window in the \(\alpha\)-th scheme of the TT&C task. Figure 3: Conflicts among visible windows. Figure 2: Time fragments. Since different visible windows between satellites and TT&C equipment may conflict, there may also be conflicts between different feasible schemes. If different feasible schemes contain the same window or if there is a conflict between their visible windows, there is a conflict between the feasible schemes. Therefore, for any two feasible schemes \(y_{i}^{\alpha},y_{j}^{\beta}\in Y\), a conflict between them must meet one of the following two conditions:  
(1) \(\exists w_{i,j}^{k}\in W\) has \(w_{i,j}^{k}\in y_{i}^{\alpha}\) and \(w_{i,j}^{k}\in y_{j}^{\beta}\). (2) \(\exists w_{i,j}^{k}\in y_{i}^{\alpha},w_{m,n}^{\alpha}\in y_{j}^{\beta}\), there is a conflict between window \(w_{i,j}^{k}\) and window \(w_{m,n}^{\alpha}\). Use \(O\big{(}y_{i}^{\alpha}\big{)}\) to represent the set of all feasible schemes that conflict with \(y_{i}^{\alpha}\). The Boolean variable \(x_{i}^{\alpha}=\{0,1\}\) is used to indicate whether the \(\alpha\)-th feasible scheme of task \(om_{i}\) is scheduled. Specifically, \(x_{i}^{\alpha}=1\) means that the \(\alpha\)-th feasible scheme of task \(om_{i}\) is scheduled. Otherwise, \(x_{i}^{\alpha}=0\) means that the \(\alpha\)-th feasible scheme of task \(om_{i}\) is not scheduling. The goal of TT&C task planning is to allocate appropriate time windows for all TT&C tasks within the planning horizon so that the total revenue of TT&C tasks is maximized and the time fragments generated by them are minimized. The following conditions must be met when planning the TT&C tasks:  
(1) A TT&C task can only be completed once by a TT&C feasible scheme. (2) A TT&C scheme and its conflicting scheme cannot be scheduling at the same time.