# An Evaluation of Optimization Algorithms for the Optimal Selection of GNSS Satellite Subsets
## 5 Results and Discussion
### Multi-GNSS Optimal Satellites
However, ABC was relatively more accurate in selecting the optimal 15 satellites than 8 and 12 satellites using the first parameter setting (Figure 15). The accuracy of choosing the optimal 8-, 12-, and 15-satellite constellations was 84%, 79.5%, and 84.7%, respectively. Since similar parameter values were used for the three satellite constellation sizes, it is the initial randomly generated solution by the ABC algorithm which influences the produced solutions [47]. In terms of speed, the TM required an overall longer processing time to find the optimal satellites than the ABC algorithm, as the required time for the TM increased significantly with the satellite constellation size (which increased from 4 to 15), since the TM checks all possible satellite combinations, which increase with the size of the satellite constellation. It should be noted that in the cases of 12- and 15-GNSS-satellite constellations, the time required by the TM exceeded 150 min (i.e., >30 s per epoch). \begin{table}
\begin{tabular}{c c c c} \hline
**Setting** & **maxIter** & **pSize** & **Limit** \\ \hline
1 & 100 & 100 & D \(\times\) pSize \\ \hline
2 & 100 & 200 & D \(\times\) pSize \\ \hline
3 & 200 & 200 & D \(\times\) pSize \\ \hline \end{tabular}
\end{table} 
Table 2: ABC parameter values. On the other hand, the speed of ABC varied based on the parameters' settings. In the case of the first set of parameters (maxIter and pSize equal to 100), the ABC algorithm required approximately the same processing time (i.e., about 4.3 min; 0.9 s per epoch) for the four cases of satellite constellations (4, 8, 12, and 15), indicating that the number of satellite combinations does not affect the speed of the ABC algorithm. For the other two settings, an increased processing time was observed, caused by the population size/number of iterations, which was increased to 200. However, a slight difference in the required processing time for the various satellite constellations (4, 8, 12, and 15) can be explained by the adjustment of satellite duplication in the solutions. The total number of GPS, GLONASS, and Galileo satellites reached 31, providing more than 300 million possible satellite combinations (Figure 10). However, ABC provides good results regardless of the setting used. According to Figure 15, ABC achieved a selection accuracy of 100% using setting 3, and its selection accuracy was no less than 80% with other settings. Furthermore, in all cases, the fitness (CNR-WGDOP) of the mis-selected optimal GNSS satellite constellations differed by no more than 0.0063 (i.e., settings and constellation size) from those of the actual optimal GNSS satellite constellations determined by the TM (Figure 16). The largest difference between the quality (i.e., CNR-WGDOP) of ABC satellite selection and the actual selection (TM) occurred in the selection of the four optimal satellites in setting 1. Figure 17 shows the sky plots of the satellites selected by ABC setting 1 and the TM at epochs 20 and 88, which are the epochs with the largest difference in the CNR-WGDOP values between ABC setting 1 and the TM (0.0059 and 0.0063, respectively). In epoch 20, the selection of ABC setting 1 differed from the TM in one satellite; the two different satellites of the ABC and TM are of similar elevation angles and roughly symmetrical with respect to the azimuth. In epoch 88, ABC setting 1 selected three satellites that were different from those of the TM. Similar to epoch 20, the six different satellites of the ABC and TM were of a similar elevation angle and roughly symmetrical with respect to the azimuth. Hence, even for the epochs where ABC's satellite selection was not actually optimal, its geometry was very close to the actual (as defined by the TM), as shown in Figure 17. As a result, the ABC satellite subset had a very similar CNR-WGDOP value compared to the actual optimal satellite subset.