# An Evaluation of Optimization Algorithms for the Optimal Selection of GNSS Satellite Subsets
## 3 Methodology of Selecting Tracked GNSS Satellites Based on Optimization Algorithms
The optimization algorithms were applied to select the optimal GNSS satellite subset for specific time intervals where the change in the GNSS satellite position (azimuth and elevation) was adequate to affect the satellite constellation geometry. For the given location of the GNSS station (i.e., Nottingham Geospatial Institute, Nottingham) and available GNSS satellites (i.e., GPS, Galileo, GLONASS), the required time for each GNSS satellite to move on its orbit by one degree was analyzed. Figure 6 displays the required time of each available GNSS satellite to move by one degree, where the maximum required time reaches up to 3 min. In previous studies [23; 61], a 10 min satellite selection interval was used. However, in the current study, the satellite selection interval was set to 5 min to reduce the computational load of satellite selection while maintaining a large number of epochs for the satellite selection. Hence, we selected optimal satellite subsets for 288 epochs, corresponding to 1 epoch per 5 min interval, for the 24 h GNSS dataset. The GNSS satellite selection becomes more challenging as the number of possible satellite combinations increases, which occurs when there is a large number of available GNSS satellites in relation to the number of satellites in the subset. This can be described more clearly with Equation (4):  
\[c_{r}^{n}=\frac{n!}{r!(n-r)!} \tag{4}\]  
where \(c\) is the number of possible GNSS satellite combinations, expressed as a factorial function of \(n\) (the number of available GNSS satellites), and \(r\) (the number of GNSS satellites in the subset). Hence, by increasing the number of available satellites, the number of possible satellite combinations increase dramatically, making the evaluation of the five optimization algorithms more complex. In this study, satellite data were collected from multiple satellite  
Figure 6: Time period of satellites’ mean movement by one degree, considering satellite azimuth and elevation angles. systems (GPS, GLONASS, and Galileo). Therefore, the number of available satellites can easily exceed 20, resulting in a large number of possible satellite combinations, making satellite selection more difficult. Thus, the optimization of the GNSS satellite subset was performed in two stages: (i) for a GPS-only satellite constellation and (ii) for a multi-GNSS satellite constellation (GPS, GLONASS, and Galileo). Based on these two case studies, we evaluated how the performance of the optimization algorithms was affected by the number of GNSS satellites. In the first stage, the performance of the five optimization algorithms was evaluated by optimizing GPS subsets of different sizes to change the selection conditions. Based on the first stage of evaluation, the optimization algorithm with the best performance, in terms of satellite selection accuracy and computation time requirement, was chosen. In the second stage, only the best-performing optimization algorithm was applied and evaluated to optimize different-sized multi-GNSS satellite subsets under more challenging conditions as the number of available GNSS satellites was significantly high, resulting in a large number of possible satellite constellations. In addition, at the second stage, a parametric analysis of the best-performing optimization algorithm was conducted to evaluate the impact of the various settings, such as the number of iterations and the population size, on the performance of the optimization algorithm. In all the tests, the algorithms' accuracy in satellite selection was evaluated by comparing their selection with the actual optimal satellite subsets, which were identified by TM.