# Multi-Satellite Observation-Relay Transmission-Downloading Coupling Scheduling Method
## 2 Problem Description
### Improvement of Observation Solution Local Search Based on Satellite Cluster
#### 3.3.2 Satellite Cluster Profit-State Evaluation Function  
In artificial potential field methods, robots are guided towards their goal and avoid obstacles by defining an attractive potential field \(U_{att}\), a repulsive potential field \(U_{rep}\), as well as attractive force \(F_{att}\) and repulsive force \(F_{rep}\). Referring to the idea of the artificial potential field method, the objective function attraction term and the local optimal repulsive term are defined to guide the tabu search direction. During the tabu search process, when there is no neighborhood solution superior to the historical optimal solution, the satellite cluster profit-state evaluation function is used to evaluate the neighborhood solutions and select a neighborhood to escape from local optima. The satellite cluster profit-state evaluation function is described as  
\[V_{O}=V_{att}+\sum\limits_{k=1}^{n_{r}}V_{repElectrick}+V_{repDatatrans}, \tag{15}\]  
where \(V_{att}=\sum\limits_{i=1}^{n_{t}}\left(v_{i}\sum\limits_{k=1}^{n_{r}}\sum \limits_{m=1}^{n_{t}}z_{ikm}\right)\) represents the attraction term for the search direction, \(\sum\limits_{k=1}^{n_{r}}V_{repElectrick}\) is the electricity repulsion term for the search direction, and \(V_{repDatatrans}\) is the data transmission repulsion term for the search direction. \(V_{repElectrick}\) is calculated as  
\[V_{repElectrick}=\left\{-\xi_{W}\frac{\hat{W}_{DisChargDetNet}-W_{DisChargDetCur} }{W_{Naring}}\quad W_{\text{kminCur}}<W_{Warning}\right., \tag{16}\]where \(W_{DisChargedCur}\) represents the current solution's electric energy consumption of satellite \(k\), \(\hat{W}_{DisChargedNext}\) is an estimated electric energy consumption value of satellite \(k\) for a neighborhood solution, \(W_{Waming}\) is the electric energy warning value, \(W_{KminCur}\) represents the minimum available electric energy of satellite \(k\) during the scheduling period in the current solution, and \(\xi_{W}\) is the affect coefficient of the electricity repulsion term. The purpose of this term is to guide the local search direction towards solutions where a satellite has more remaining electric energy when the satellite's minimum available electric energy is in a warning state. The calculation method for the estimated electric energy consumption value \(\hat{W}_{DisChargedNext}\) is expressed as  
\[\hat{W}_{DisChargedNext}=\sum\limits_{i=1}^{n_{t}}x_{kiNext}\bigg{(}P_{Camk} \Delta t_{o}+P_{GroundDataTransk}\frac{R_{Camk}\Delta t_{o}}{R_{DataTransk}} \bigg{)}+\sum P_{Attitudek}\Delta t_{Attitudek}, \tag{17}\]  
where the first term represents the estimated electric energy required for observation and data downloading of satellite \(k\) in the neighborhood solution, and the second term represents the estimated attitude maneuver electric energy consumption for the observations. \(\Delta t_{Attitudek}\) represents the total attitude maneuver time, which can be calculated from the satellite's observation sequence. The calculation method can be referenced in reference [54]. The data transmission repulsion term for the search direction \(V_{repDatatrans}\) can be expressed as  
\[V_{repDatatrans}=\left\{-\xi_{D}\frac{\sum\limits_{k=1}^{n_{r}}\sum\limits_{m= 1}^{n_{dashm}}\sum\limits_{q=1}^{n_{dashm}}Flag_{kmpCut}\Delta t_{dubsmCur}- \sum\limits_{k=1}^{n_{r}}\sum\limits_{m=1}^{n_{r}}\sum\limits_{q=1}^{n_{dashm} }Flag_{kmpNext}\Delta t_{dubsmNext}}{\Delta t_{d}}\right.\quad\left.\exists k, \Delta t_{dRemainKur}<\Delta t_{DWarning}\right. \tag{18}\]  
where \(\Delta t_{DWarning}\) denotes a preset warning state for remaining data transmission time, \(\Delta t_{dubsmq}=t_{dubsmq}-t_{dubsmq}\) represents the duration of the satellite-ground data transmission window, \(Flag_{kmpq}\) stands for a flag indicating the validity of the ground station, \(\Delta t_{dRemaink}\) is the available data transmission time in the last segment of satellite \(k\)'s satellite-ground data transmission time window, and \(\xi_{D}\) is the affect coefficient of the data transmission repulsion term.