## 4 Results and Discussion
### Comparison of Error Distribution of SPAs and Their Influence on Performance of System  
Evaluating the influence of errors of SPAs on the efficiency of trough system requires the calculation of the error of the algorithm at every moment, which usually takes considerable time and effort. Different SPAs have different calculation principles and therefore have different error distributions, taking one year as the cycle. We consider calculating the average error of SPAs and use it to evaluate the effort of error of SPA. In this paper, we choose SPAs with different accuracies and evaluate the influence of error of those SPAs on the efficiency of system by calculating the tracking error caused by those algorithms under different tracking modes [7; 9; 13; 14; 15; 16; 17; 18; 19; 20]. The algorithms proposed in some studies only include the calculation of solar declination or equation of time. In this paper, we combine those algorithms with similar errors to calculate the tracking error, such as the solar declination method proposed by [PERSON] [14] and equation of time method proposed by [PERSON] [13], the solar declination method pro \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Data** \\ \hline Focus length \(f_{0}\) & 1.7 m \\ Half trough width \(w\) & 3 m \\ Length of trough \(L\) & 100 m \\ Radius of receiver tube \(r_{0}\) & 0.035 m \\ Radius of envelope of receiver \(R\) & 0.055 m \\ Operation temperature of receiver \(T\) & 400 \({}^{\circ}\)C \\ Transverse optical error \(\theta_{\perp}\) & 6 mrad \\ Longitudinal optical error \(\theta_{\parallel}\) & 6 mrad \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter of the typical parabolic trough solar collector. Figure 1: Comparison between predicted result and experimental data; (**a**) The intercept factor of parabolic trough solar system calculated by program and SolTrace under different tracking error. (**b**) The efficiency of parabolic trough solar system calculated by program and SolTrace under different tracking error. posed by [PERSON] [19] and equation of time method proposed by [PERSON] [20], and the solar declination method proposed by [PERSON] [17] and the equation of time method proposed by [PERSON] [16]. Algorithms mentioned in this paper also include those proposed by [PERSON] [7], [PERSON] [9] and [PERSON] [15]. We assume those algorithms are used by tracking systems of the trough solar system and calculate tracking error generated by those algorithms. As shown in Figures 2 and 3, in order to clearly display the error range and distribution of various algorithms within a year, we calculate the distribution of error on solar altitude and error on solar azimuth in comparison with the result obtained at different times in 2020 with those SPA methods. For every day, 160 times are computed, from 7:12 am to 4:48 pm solar time, every 3.6 min. The latitude of calculated position is 37.1 and the longitude is \(-\)2.36. The maximum errors of solar altitude and solar azimuth angle of the empirical algorithm are bigger than 1\({}^{\circ}\), and the maximum errors of the sophisticated algorithms are less than 0.05\({}^{\circ}\). For all algorithms, the azimuth error is generally greater than the altitude error. Table 2 shows these SPAs and the annual average absolute error of azimuth \(\gamma\), Zenith angle \(\alpha\) and hour angle \(\omega\), the corresponding annual average absolute tracking error is also given. Among those algorithms, the algorithm proposed by [PERSON] has the lowest error. The average azimuth error is only 0.012 mrad, and the tracking error is less than 0.03 mrad, which is negligible for the line focusing parabolic trough system. The tracking error under polar axis tracking mode is approximately equal to the time angle error, and the tracking error under east-west axis and north-south axis mode can be calculated using the formula proposed in Section 2.2.