## 2 Problem Description and Formulation
### Formulation
#### 2.3.2 Energy Consumption Cost(EC)  
The energy consumption cost includes two part: the cost in attitude maneuver (\(EC_{1}\)) and the cost in data download (\(EC_{2}\)). The \(EC_{1}\) and \(EC_{2}\) EC refers to the on-board energy required from issuing a downlink request to completing the mission. \(EC_{1}\) represents the energy consumption of the satellite maneuvering its attitude to communicate with the ground. Additionally, \(EC_{2}\) is the energy consumption of satellite data transmission. The energy consumption cost can be calculated by,  
\[EC=EC_{1}+EC_{2} \tag{3}\]  
where  
\[EC_{1}=\sum_{s\in S}\sum_{r\in R_{s}}\omega_{EC_{1}}\times u_{sr}\]  
\[EC_{2}=\sum_{i\in S}\sum_{k\in G}\omega_{EC_{2}}\times x_{ijk}\times DT(R_{s}^{ r})\]  
The mathematical model of RC-HVRPTW is defined as follows.  
\[Minimize(C)=DC+EC \tag{4}\]subject to:  
\[\sum_{j\in S,i\
eq j}\sum_{k\text{ }inG}x_{ijk}\leq 1 \tag{5}\] \[\sum_{s\in S}\sum_{r\in\mathcal{R}_{s}}\sum_{g\in G}AT(R_{s}^{r}) \leq ET(tw_{sg})\] (6) \[\sum_{r\in R}\sum_{g\in G}y_{sr}\geq ST(tw_{sg})\] (7) \[\sum_{r\in R}\sum_{g\in G}y_{sr}\leq ET(tw_{sg})\] (8) \[\sum_{r\in R}\sum_{g\in G}DR(R_{s}^{r})\leq(ET(tw_{sg})-ST(tw_{ sg}))\] (9) \[\sum_{s\in S}\sum_{r\in\mathcal{R}_{s}}\omega_{EC_{1}}\times u_{ sr}\leq Max\_Energy\] (10) \[DT(R_{s}^{r})+ST(tw_{sg})\leq ET(tw_{sg}),for\ \forall s\] (11) \[\sum_{s\in S}\mathcal{R}_{s}(0)=0\] (12) \[\sum_{s\in S}\mathcal{R}_{s}(\mathcal{R}_{s}+1)=0 \tag{13}\]  
Equation (4) is the optimization objective function. Equation (5) means that observation is made for a target at most once. Equation (6) stands for that means that the end time of each access window is greater than the start time. Equations (7) and (8) require the service start time for each request. Equation (10) requires energy consumption to be within the total energy range on-board. Equation (11) means that only when there is a request longer than the duration can be served. Equations (12) and (13) represents that the mission execution plan of each ground station has a dummy request at the beginning and end Tables 1 and 2.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Description** \\ \hline \(s\)\(in\)\(1,2,\cdots,S\) & Satellite set \\ \hline \(g\)\(in\)\(1,2,\cdots,G\) & Ground station set \\ \hline \(r\)\(in\)\(1,2,\cdots,R\) & Downlink request set, \(R_{r}=(id_{r},sat_{r},tArrival_{r},tDur_{r})\) \\ \hline \(a\)\(in\)\(1,2,\cdots,A\) & Access window set, \(A_{a}=(id_{a},gs_{a},sat_{a},tBega_{a},tEnd_{a})\) \\ \hline \(id_{r}\) & id of downlink request \(r\) \\ \hline \(sat_{r}\) & satellite number \(sat_{r}\) for request \(r\) \\ \hline \(tArrival_{r}\) & the arrival time of request \(r\) \\ \hline \(tDur_{r}\) & the requirement time of request \(r\) \\ \hline \(id_{a}\) & id of downlink request \(r\) \\ \hline \(gs_{a}\) & ground station number \(gs_{a}\) for access window \(a\) \\ \hline \(sat_{a}\) & satellite number \(sat_{a}\) for access window \(a\) \\ \hline \(tBega_{a}\) & the start time of access window \(a\) \\ \hline \(tEnd_{a}\) & the end time of access window \(a\) \\ \hline \(H\) & scheduling horizon length \\ \hline \(tw_{sg}^{n}\) & \(n\)th access time window in \(g\) and \(s\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters.