## 3 Object System by Using HS Theory
### HS Searching Procedure
In this step, the termination of a computation process is allowed before the final conclusion, in accordance with specified termination criterion (maximum number of searches = 2000). If not, repeat Steps 3 and 4. **4. Statistical Methods**  
To assess the accuracy and suitability of the models, numerous statistical methods are used to compare the reference optimum tilt angle obtained by the ergodic method and the optimum tilt angle calculated by the PSO and HS algorithms. PSO is a stochastic technique for exploring the search space for optimization [34]. It emulates the social behavior of a natural swarm. With the term \"swarm\", a population of individuals (particles) is indicated. Each swarm particle follows a path in the solution space. The motion of each particle is governed by a list of constraints based on ecology (i.e., natural swarm behavior) [35]. In the present study, PSO is used to compare the solution quality with the HS algorithm. The model equations are evaluated using mean percentage error (MPE), mean absolute percentage error (MAPE), mean absolute bias error (MABE), and root mean square error (RMSE), which are defined as below [36; 37; 38; 39; 40; 41; 42]:  
\[MPE=\frac{1}{N}{\sum\limits_{i=1}^{N}}\bigg{(}\frac{\beta_{ic}-\beta_{im}}{ \beta_{im}}\cdot 100\bigg{)}, \tag{21}\]  
\[MAPE=\frac{1}{N}{\sum\limits_{i=1}^{N}}\bigg{(}\bigg{|}\frac{\beta_{ic}-\beta_ {im}}{\beta_{im}}\bigg{|}\cdot 100\bigg{)}, \tag{22}\]  
\[MABE=\sum\limits_{i=1}^{N}\lvert\beta_{ic}-\beta_{im}\rvert/N, \tag{23}\]  
\[RMSE=\sqrt{\sum\limits_{i=1}^{N}\left(\beta_{ic}-\beta_{im}\right)^{2}/N}, \tag{24}\]  
where \(\beta_{ic}\) and \(\beta_{im}\) are the \(i\)th calculated and standard optimum tilt angles, respectively; \(N\) is the total number of observations; and \(\beta_{ca}\) and \(\beta_{\rm ma}\) are the average of the calculated and standard values, respectively. **5. Results and Discussions**  
The optimum tilt angles and orientations of PV modules for six stations in different climates are established using the HS method, accounting for the climate and latitude of each site. The six Chinese cities selected for study are Sanya, Shanghai, Zhengzhou, Harbin, Mohe, and Lhasa. In practical applications, PV modules do not operate under a standard condition. During the sun's movements, the solar irradiance accumulates each hour angle with the corresponding extraterrestrial radiation. Based on Equations (1)-(11), the monthly mean daily extraterrestrial radiation data were determined accordingly; Figure 2 displays the results from January to December in Shanghai. Taking the result of March as an example, in Figure 2c, \(I_{dm}\) on a tilted surface varies from 0 to over 35 MJ/m\({}^{2}\). Apart from south-facing orientations, the solar radiation decreases gradually with an increasing incline angle from horizontal to vertical surfaces. The maximum value is observed at the inclined angles between 20\({}^{\circ}\) and 40\({}^{\circ}\), with the azimuth angles from 150\({}^{\circ}\) to 210\({}^{\circ}\). The figure also indicates that \(I_{d}\) is quite symmetrical with respect to due south (\(\gamma=180^{\circ}\)). As the result of the sun path in Shanghai, the sun is visible for most of the day throughout the year; therefore, solar collection is greatest among all of the surfaces. Since the east-facing surfaces and west-facing surfaces face the sun in the morning and afternoon, respectively, the solar radiations at these two orientations are very close to each other. Due to the shortest period facing the direct sun, the north-facing planes receive the least solar radiation. Figure 2: _Cont._  
From Figure 2, the optimum azimuth angles are \(0^{\circ}\) (\(360^{\circ}\)) or \(180^{\circ}\) in every month. Therefore, when referring to the optimum results, the azimuth angle data are simplified to the sign in the discussion below: a positive sign means due south, and a negative sign means due north. Table 2 demonstrates the values of \(\beta_{\text{opt}}\) obtained using the ergodic method, the PSO and the HS algorithm.