## 4 Results and Discussion
### Comparison of Error Distribution of SPAs and Their Influence on Performance of System
In Table 4, we calculate its annual average optical efficiency while taking the annual average error of Cooper and Woolf algorithm as the fixed tracking error of parabolic trough system. Under the north-south axis tracking mode, the efficiency decreases by 7.14% because of tracking error, the east-west axis efficiency decreases by 3.34%, and the polar axis efficiency decreases by 3.03%. The tracking error of the north-south tracking mode is higher, and the error introduced is greater when we apply the average error to characterize the impact of the tracking error. In the north-south axis, east-west axis and polar axis tracking modes, the relative differences of efficiency obtained by using the average are 0.0116, 0.0071 and 0.0048, respectively. This shows that when the error of SPA is large, we replace the tracking error with a fixed average error, introducing a large calculation error, the maximum error is close to 2.75%. Cooper and Woolf algorithms have large error which is not suitable for trough systems, so this will not affect the application of average error calculation method in trough systems. Table 5 shows the comparison of efficiency with other SPAs under the north-south tracking mode. The result calculated with average error has error of less than 0.18% when the average error is less than 1.3 mrad, and the introduced error can be ignored. When the error is from 1.3 to 2.4 mrad, the difference between the efficiency calculated with the average error and the error at every moment is 0.18-0.74%, which will have a slight impact on the result. When the annual average tracking error of the parabolic trough system applying SPA reaches 4.2 mrad, the efficiency of the parabolic trough system is reduced by 5.42%. When the average error is applied, the efficiency is reduced by 2.69%, and the absolute error is 2.89%. The absolute error between those two calculation method is 3.65%, which obviously cannot be ignored. However, at the same time, the efficiency of trough solar system is greatly affected by tracking error, and this algorithm is not suitable for the trough system any more. The result above shows that when the SPA is suitable for the trough system, the average error is less than 2.5 mrad. The difference between the two calculation method is small. It is considerable to evaluate the influence of tracking error to performance of trough solar system by using average error instead in order to simplify the calculation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Tracking Methods**} & **Efficiency** & **Efficiency with** & **Average Annual** & **Efficiency** & **The Relative Difference** \\  & **without** & **Cooper and** & **Tracking Error** & **(mrad)** & **Average** & **Difference** \\  & & & & & \\ \hline North-South axis & 0.4538 & 0.4214 & 5.5 & 0.4330 & 2.75\% \\ East-West axis & 0.3447 & 0.3332 & 2.9 & 0.3403 & 2.13\% \\ Polar axis & 0.5617 & 0.56 & 2.7 & 0.5552 & 0.86\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average annual tracking error and efficiency of system with the Cooper and Woolf method under three different tracking methods, efficiency calculated with average tracking error and the relative difference between two results.