## 4 Results and Discussion
### Comparison of Error Distribution of SPAs and Their Influence on Performance of System
The error under the east-west axis tracking mode and polar axis tracking mode is obviously lower than the result under the north-south axis tracking mode.The variance of absolute error of the solar azimuth and altitude are given. The results show that in most cases, algorithms with lower accuracy have lager variance. The algorithms with the greatest variance are Cooper algorithm and Stine algorithm, and the value can reach 25 mrad\({}^{2}\). Table 2 also shows the calculation consumption of these algorithms, which is calculated by the method proposed by [PERSON] [9]. The average error of azimuth can be easily reduced to less than 1 mrad with about 450 calculation consumption. To evaluate the effect of tracking error on the efficiency of trough solar system, we need to take tracking error into the calculation program and calculate the efficiency at every moment. In Table 3, we take the Cooper and Woolf algorithm as an example and draw a comparison between the parabolic trough efficiency affected by error of SPA and the efficiency without tracking error on the spring equinox, summer solstice, autumn equinox and winter solstice under three tracking methods. Under the north-south axis tracking method, the tracking error range is from 0.0065 to 5.52 mrad. With the effect of tracking error, the efficiency decreased by 14.81% on the spring equinox, 0.18% on the summer solstice and 31.12% on the autumnal equinox. The tracking error of parabolic trough system under the three tracking modes is different in a year. Compared with the north-south axis tracking mode, the east-west axis tracking mode always has smaller tracking error, with the range from 0.52 to 5.61 mrad, especially in spring and autumn. The above results show that Cooper and Woolf algorithms have large errors and have a great impact on the trough system, which cannot meet the requirements of trough system tracking. See the results listed in Table 3. Due to the different calculation principles of the sun position algorithm, the errors of the same algorithm in different seasons are often different, but the changes of the tracking error and the sun position error in all tracking modes show a positive correlation trend. When these algorithms are applied in the tracking system, the tracking error caused by them will affect the performance of system. We make a comparison of two system with this algorithm and another system without tracking error under the north-south modes at the spring equinox, as shown in Figure 4. At the spring equinox, the system with the Spencer and Woolf algorithm had the lower performance, with an average efficiency of 0.32 with an average error of 9.05 mrad. At the same time, the efficiency of the system without tracking error was 0.3636. Therefore, 12% of efficiency is lost due to the error of the algorithm. Table 3 presents the average efficiency lost by the Spencer and Woolf algorithm  
Figure 4: Efficiency comparison of two parabolic trough system with tracking error generated by the Cooper and Woolf method and without tracking error condition, respectively, under NS tracking system at the spring equinox; NS: north–south axis tracking system. Figure 3: The probability density distribution of the calculation error of solar azimuth. (**a**) Solar position empirical algorithm. (**b**) Solar position complex algorithm. \begin{table}