# An Evaluation of Optimization Algorithms for the Optimal Selection of GNSS Satellite Subsets
## 1 Introduction  
The Global Navigation Satellite System (GNSS) is a system of satellites that provides positioning, navigation, and timing (PNT) services and consists of GPS (Global Positioning System, USA), GLONASS (Global Navigation Satellite System, Russian), Galileo (European Union), BDS (BeiDou Navigation Satellite System, China), and other regional navigation satellite systems. Due to the recent development in GNSS satellite systems, the number of satellites available in space reached 114 in January 2024: 31 GPS [1], 24 GLONASS, 24 Galileo, and 35 BeiDou satellites [2]. As a result, the number of satellites simultaneously visible from a single location, depending on the geographic location, can reach or exceed30 [3]. The potential of using such a diverse and large number of GNSS satellites can improve the accuracy and availability of GNSS solutions for positioning, navigation, and timing (PNT) services. However, it also has limitations. More specifically, the number of computations required for GNSS positioning grows as the number of used GNSS satellites increases. Compared to 10 GNSS satellites, the required time for processing the position based on 20 and 30 GNSS satellites increases by a factor of 3 and 6, respectively [4]. In addition, GNSS positioning accuracy generally increases with the number of available satellites, but practically no further enhancement is achieved after a certain number of satellites [5; 6]. This was also revealed by [PERSON] et al. [7] using zero-baseline GNSS measurements. The accuracy of GNSS positioning did not improve with the decrease in geometric dilution of precision (GDOP). Moreover, increasing the number of GNSS satellites cannot always guarantee a higher-quality sustained position. The positioning derived from multi-satellite system (multi-GNSS) integration is not of the highest possible accuracy due to introduced errors from various error sources. Some representative examples of error sources are multipath error [8] and inter-system bias [7], even under relative positioning [9]. Also, one of the main requirements in many GNSS monitoring applications is to limit as much as possible the GNSS data size for more sustainable and reliable GNSS data transmission, which makes it important to choose only optimal satellites which will lead to an accurate and reliable GNSS solution. Therefore, GNSS satellite selection is required to not only improve the GNSS's performance but also to make the application of GNSS technology more sustainable. GNSS satellite selection is the process of choosing the right GNSS satellite combination from the available GNSS satellites, aiming for better positioning quality with a lower computational load [10]. Various satellite selection criteria have been used for selecting satellites, including satellite elevation angle [11; 12; 13], satellite signal quality [14; 15], GDOP [3; 16; 17], and weighted GDOP (WGDOP) [18; 19]. Since GNSS accuracy is affected by both satellite geometry (i.e., GDOP) and satellite signal quality, the WGDOP, which is based on the satellite signal quality and the satellite geometry (GDOP), can be considered as the most suitable criterion for choosing satellites. The optimal satellite subset based on WGDOP (for subsets with a specific number of satellites) can be determined simply and accurately by checking the WGDOP value for all possible satellite subsets (of the same number of satellites) and then considering the satellite subset with the lowest WGDOP value as the optimal one. This technique is known as the traditional method (TM). However, due to the required matrix inversion, the TM can be time-consuming as it calculates the WGDOP value of all possible satellite subsets, especially if the number of available tracked GNSS satellites is large. To reduce this computational burden, various techniques including the closed-form solution [18; 19], maximum volume [20], artificial neural network (ANN) [21], and optimisation algorithms [22; 23] have been used to select optimal satellite subsets based on WGDOP.