# Optimal Tilt Angle and Orientation of Photovoltaic Modules Using HS Algorithm in Different Climates of China

^{1}

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## Abstract

**:**

## 1. Introduction

_{opt}) and optimum azimuth angle (γ

_{opt}) of solar collectors. Dixit [6] used the artificial neural network (ANN) estimator taking the H

_{g}, φ and E

_{L}of the site as inputs, to estimate the optimum tilt angle almost instantaneously while testing. Gopinathan [7] represented the optimum slope and the azimuth angles with the anisotropic model for South Africa, calculating the daily radiation at various slopes and orientations, thus obtaining β

_{opt}and γ

_{opt}. The past few decades have seen an increased interest in general-purpose “black-box” optimization algorithms that have drawn inspiration from optimization processes that occur in nature in large part [8,9,10,11,12,13,14]. In Ref. [15], an approach combining the orthogonal array experimental technique and an ant direction hybrid differential evolution algorithm (ADHDEOA) was presented for determining the tilt angle for PV modules. In Ref. [16], the tilt angle was changed from −20° to 90° in a step size of 0.1°, and the corresponding value of maximum global solar radiation for a specific period is defined as the optimal tilt angle. In Ref. [17], a particle-swarm optimization method with nonlinear time-varying evolution (PSO-NTVE) was proposed, by which the tilt angle of PV modules were determined for Taiwan. The calculation can be formulated as an optimization problem for maximizing the output electrical energy of the modules. From previous applications, the defect of particle swarm optimization (PSO) prematurity makes it easy to fall into a local optimum; thus, it is necessary to select other algorithms to attempt this research.

_{opt}and γ

_{opt}by calculating the extraterrestrial solar radiation at various tilt angles and azimuth angles. The ergodic results are used as standard values for comparison. Next, the calculation of β

_{opt}and γ

_{opt}is formulated as an optimization problem. Then, the HS method is employed to determine the optimal angles based on the established objective function and constraints. Finally, the HS and PSO results are compared with reference values obtained from the ergodic algorithm through the most widely used statistical methods. Comparisons show that the HS is an accurate and reliable method for the PV module to determine the tilt angle and orientation.

## 2. Mathematical Model

_{opt}and γ

_{opt}. There are various methods to classify climates in China [23,24,25]. In this paper, several methods are applied to solar energy collection optimization on the tilted surfaces for six stations (Sanya, Shanghai, Zhengzhou, Harbin, Mohe, and Lhasa) in different climate zones (tropical zone [TZ], subtropical zone [SZ], warm temperate zone [WTZ], mid temperate zone [MTZ], cold temperate zone [CTZ], and Tibetan plateau zone [TPZ]) in China [26]. General layout of the six major climates across China is shown in Figure 1. General information about the selected six typical stations is shown in Table 1.

#### 2.1. Julian Day (JD)

#### 2.2. Solar Declination (δ)

#### 2.3. Angle Incidence (θ)

#### 2.4. Sunrise and Sunset Hour Angle

_{r}(ω

_{s}) on a south-facing (γ = 0) tilt surface, one can use the following formulas [32]:

_{r}(ω

_{s}) for the inclined surface is given by [31]:

#### 2.5. Extraterrestrial Solar Radiation

_{c}is the solar constant (=1367 W/m

^{2}).

_{d}) is calculated from (10):

_{m}) can be calculated from Equation (11):

_{1}and n

_{2}are the JD number of the first day and the last day of a month, respectively. I

_{di}is the extraterrestrial solar radiation on a tilted surface of the day, which has the JD number of i.

## 3. Object System by Using HS Theory

#### 3.1. Objective Function

_{m}:

#### 3.2. Constraints

- Tilt angles$${\beta}_{\mathrm{min}}\le \beta \le {\beta}_{\mathrm{max}},$$
_{min}and β_{max}are the lower and the upper values of β. In this optimization problem, β_{min}and β_{max}are set as 0° and 90°, respectively. - Azimuth angles$${\gamma}_{\mathrm{min}}\le \gamma \le {\gamma}_{\mathrm{max}},$$
_{min}and γ_{max}are the lower and the upper value of γ. In this optimization problem, γ_{min}and γ_{max}are set as 0° and 360°, respectively.

#### 3.3. HS Searching Procedure

_{opt}and γ

_{opt}[13] is as follows:

^{HMS}). Values of the other design variables (γ′) can be chosen in the same way. Here, the algorithm chooses the new value with HMCR = 0.9:

## 4. Statistical Methods

_{ic}and β

_{im}are the ith calculated and standard optimum tilt angles, respectively; N is the total number of observations; and β

_{ca}and β

_{ma}are the average of the calculated and standard values, respectively.

## 5. Results and Discussions

_{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° and 40°, with the azimuth angles from 150° to 210°. The figure also indicates that I

_{d}is quite symmetrical with respect to due south (γ = 180°). 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.

_{opt}obtained using the ergodic method, the PSO and the HS algorithm. In the ergodic results, β

_{opt}values from Sanya, Shanghai, Zhengzhou, Harbin, Mohe, and Lhasa ranges from −18.1°(Jun) to 49.9°(Dec), from −7.6°(Jun) to 61.4°(Dec), from 5.5°(Jun) to 64.3°(Dec), from 12.6°(Jun) to 73.7°(Dec), from 16.6°(Jun) to 80.0°(Dec), and from −8.9°(Jun) to 59.9°(Dec), respectively. The values of β

_{opt}are negative from May to July. The results also show that γ

_{opt}values obtained with the HS method are exactly the same as those obtained with the PSO method and ergodic method, and that the trend of β

_{opt}values in each month is roughly the same.

_{opt}obtained with the HS method is clearly higher than that obtained using the PSO method and methods, taking β to be equal to Φ and Φ ± 15°. Hence, the tilt angle obtained using Φ for the PV modules should not be adopted for different places to obtain the maximum overall solar energy. However, in Table 3, the results of the HS method are closest to the standard values. These data show that the HS method can be used as a substitute for the ergodic method.

## 6. Conclusions

- In most cases, the best orientation is due south (optimum azimuth angle, 180°) in the selected cities. Except when the azimuth angle equals 180°, the extraterrestrial solar radiation decreases as the tilt angle increases.
- The optimum tilt angle increases during the winter months and reaches a maximum in December for all of the stations. To enhance the energy collected by the panel, if possible, the tilt angle should be changed once a month.
- According to MPE, MAPE, MABE, and RMSE, errors with the HS method are less than those with PSO. Moreover, the extraterrestrial solar radiation of HS is larger than that of PSO. The application of HS performs better in the search for β
_{opt}. - The proposed approach, the HS method, provides an accurate and simple alternative to the ergodic method. The experimental results of the HS method are very close to the standard values.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**General layout of the six major climates across China. TZ = tropical zone; SZ = subtropical zone; WTZ = warm temperate zone; MTZ = mid temperate zone; CTZ = cold temperate zone; TPZ = Tibetan plateau zone.

**Figure 2.**Monthly daily extraterrestrial solar radiation for various tilted angles and orientations in Shanghai: (

**a**) Jan; (

**b**) Feb; (

**c**) Mar; (

**d**) Apr; (

**e**) May; (

**f**) Jun; (

**g**) Jul; (

**h**) Aug; (

**i**) Sep; (

**j**) Oct; (

**k**) Nov; and, (

**l**) Dec.

Climate | Location | Latitude (Φ) | Longitude (E) | Elevation (m) |
---|---|---|---|---|

TZ | Sanya | 18°14′ | 109°31′ | 5.9 |

SZ | Shanghai | 31°24′ | 121°29′ | 6 |

WTZ | Zhengzhou | 34°43′ | 113°39′ | 110.4 |

MTZ | Harbin | 45°45′ | 126°46′ | 142.3 |

CTZ | Mohe | 53°28′ | 122°31′ | 433 |

TPZ | Lhasa | 29°40′ | 91°08′ | 3648.7 |

**Table 2.**Optimum tilt angles of each station obtained by ergodic, particle swarm optimization (PSO), and harmony search (HS) methods.

Station | Method | Month | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | ||

Sanya (TZ) | Ergodic | 47.2 | 37.7 | 21.8 | 3.6 | −11.7 | −18.1 | −15.2 | 3.1 | 15.1 | 32.9 | 45.0 | 49.9 |

PSO | 47.45 | 37.17 | 22.23 | 3.74 | −11.88 | −18.14 | −14.67 | 3.28 | 15.60 | 32.66 | 45.25 | 49.78 | |

HS | 47.24 | 37.73 | 21.90 | 3.67 | −11.76 | −18.12 | −15.22 | 3.15 | 15.11 | 32.95 | 45.08 | 49.86 | |

Shanghai (SZ) | Ergodic | 59.0 | 50.1 | 34.9 | 16.6 | 5.3 | −7.6 | 3.8 | 11.1 | 28.3 | 45.6 | 57.0 | 61.4 |

PSO | 58.19 | 49.17 | 33.89 | 15.22 | 5.49 | −8.32 | 4.02 | 11.35 | 28.17 | 44.33 | 56.26 | 60.51 | |

HS | 58.99 | 50.18 | 35.00 | 16.56 | 5.33 | −7.58 | 3.81 | 11.07 | 28.26 | 45.66 | 56.96 | 61.40 | |

Zhengzhou (WTZ) | Ergodic | 61.9 | 53.2 | 38.2 | 19.8 | 8.0 | 5.5 | 6.4 | 14.2 | 31.6 | 48.8 | 59.9 | 64.3 |

PSO | 61.95 | 53.06 | 38.36 | 19.37 | 7.74 | 5.45 | 6.31 | 13.89 | 31.52 | 48.80 | 59.83 | 64.19 | |

HS | 61.93 | 53.20 | 38.29 | 19.80 | 8.01 | 5.52 | 6.42 | 14.13 | 31.58 | 48.85 | 59.88 | 64.28 | |

Harbin (MTZ) | Ergodic | 71.5 | 63.5 | 49.1 | 30.8 | 16.5 | 12.6 | 14.0 | 24.7 | 42.6 | 59.3 | 69.7 | 73.7 |

PSO | 71.59 | 63.46 | 49.83 | 30.67 | 16.15 | 12.82 | 13.96 | 23.67 | 42.69 | 59.14 | 69.41 | 73.58 | |

HS | 71.52 | 63.65 | 49.23 | 30.49 | 16.56 | 12.62 | 13.96 | 24.73 | 42.58 | 59.29 | 69.47 | 73.68 | |

Mohe (CTZ) | Ergodic | 77.9 | 70.5 | 56.8 | 38.7 | 23.0 | 16.6 | 19.2 | 32.7 | 50.3 | 66.5 | 76.3 | 80.0 |

PSO | 77.69 | 70.33 | 56.90 | 38.47 | 22.12 | 16.57 | 19.20 | 31.20 | 50.60 | 66.30 | 76.48 | 80.04 | |

HS | 77.90 | 70.51 | 56.86 | 38.57 | 23.06 | 16.59 | 19.16 | 32.76 | 50.33 | 66.47 | 76.34 | 80.00 | |

Lhasa (TPZ) | Ergodic | 57.5 | 48.6 | 33.3 | 14.9 | 3.8 | −8.9 | −5.7 | 9.5 | 26.5 | 44.0 | 55.4 | 59.9 |

PSO | 57.56 | 48.73 | 33.14 | 14.97 | 3.58 | −8.62 | −5.64 | 8.99 | 26.48 | 44.18 | 55.31 | 59.98 | |

HS | 57.46 | 48.54 | 33.28 | 14.92 | 3.88 | −8.91 | −5.69 | 9.55 | 26.53 | 43.99 | 55.44 | 59.89 |

Month | Ergodic Method | PSO Method | HS Method | β_{opt} = Φ | β_{opt} = Φ + 15(°) | β_{opt} = Φ − 15(°) | |||
---|---|---|---|---|---|---|---|---|---|

β_{opt}(°) | I(MJ/m^{2}) | β_{opt}(°) | I(MJ/m^{2}) | β_{opt}(°) | I(MJ/m^{2}) | ||||

Jan | 59.0 | 1100.86 | 58.19 | 1100.76 | 58.99 | 1100.86 | 975.89 | 1074.49 | 810.79 |

Feb | 50.1 | 1046.94 | 49.17 | 1046.80 | 50.18 | 1046.94 | 991.57 | 1044.74 | 870.83 |

Mar | 34.9 | 1182.78 | 33.89 | 1182.58 | 35 | 1182.78 | 1180.54 | 1159.16 | 1121.48 |

Apr | 16.6 | 1156.53 | 15.22 | 1156.21 | 16.56 | 1156.53 | 1118.21 | 1004.55 | 1156.52 |

May | 5.3 | 1213.51 | 5.49 | 1213.50 | 5.33 | 1213.51 | 1070.68 | 887.24 | 1184.17 |

Jun | −7.6 | 1180.48 | −8.32 | 1180.38 | −7.58 | 1180.48 | 982.83 | 780.58 | 1122.34 |

Jul | 3.8 | 1215.11 | 4.02 | 1215.10 | 3.81 | 1215.11 | 1042.77 | 845.05 | 1173.14 |

Aug | 11.1 | 1205.31 | 11.35 | 1205.29 | 11.07 | 1205.31 | 1129.09 | 983.04 | 1199.82 |

Sep | 28.3 | 1146.41 | 28.17 | 1146.41 | 28.26 | 1146.41 | 1144.69 | 1089.48 | 1122.01 |

Oct | 45.6 | 1169.21 | 44.33 | 1168.94 | 45.66 | 1169.21 | 1133.67 | 1169.09 | 1020.99 |

Nov | 57.0 | 1083.85 | 56.26 | 1083.77 | 56.96 | 1083.85 | 977.85 | 1065.52 | 823.54 |

Dec | 61.4 | 1078.14 | 60.51 | 1078.01 | 61.4 | 1078.14 | 933.68 | 1041.40 | 762.34 |

**Table 4.**Statistical indicators (mean percentage error (MPE), mean absolute percentage error (MAPE), mean absolute bias error (MABE), root mean square error (RMSE)) of PSO and HS at six different climatic stations.

Methods | Statistical Indicators | Sanya | Shanghai | Zhengzhou | Harbin | Mohe | Lhasa |
---|---|---|---|---|---|---|---|

PSO | MPE | 0.9968 | 0.0560 | −0.8521 | −0.3617 | −0.7451 | −1.2242 |

MAPE | 1.9739 | 3.4607 | 0.9353 | 0.9567 | 0.9215 | 1.4549 | |

MABE | 0.2825 | 0.7117 | 0.1458 | 0.2742 | 0.3200 | 0.1550 | |

RMSE | 0.3266 | 0.8206 | 0.1894 | 0.4005 | 0.5264 | 0.2018 | |

HS | MPE | 0.4376 | 0.0337 | 0.0472 | −0.0450 | 0.0022 | 0.2163 |

MAPE | 0.4510 | 0.2008 | 0.1507 | 0.2405 | 0.1105 | 0.2943 | |

MABE | 0.0475 | 0.0383 | 0.0292 | 0.0867 | 0.0392 | 0.0317 | |

RMSE | 0.0539 | 0.0476 | 0.0393 | 0.1279 | 0.0524 | 0.0385 |

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**MDPI and ACS Style**

Guo, M.; Zang, H.; Gao, S.; Chen, T.; Xiao, J.; Cheng, L.; Wei, Z.; Sun, G.
Optimal Tilt Angle and Orientation of Photovoltaic Modules Using HS Algorithm in Different Climates of China. *Appl. Sci.* **2017**, *7*, 1028.
https://doi.org/10.3390/app7101028

**AMA Style**

Guo M, Zang H, Gao S, Chen T, Xiao J, Cheng L, Wei Z, Sun G.
Optimal Tilt Angle and Orientation of Photovoltaic Modules Using HS Algorithm in Different Climates of China. *Applied Sciences*. 2017; 7(10):1028.
https://doi.org/10.3390/app7101028

**Chicago/Turabian Style**

Guo, Mian, Haixiang Zang, Shengyu Gao, Tingji Chen, Jing Xiao, Lexiang Cheng, Zhinong Wei, and Guoqiang Sun.
2017. "Optimal Tilt Angle and Orientation of Photovoltaic Modules Using HS Algorithm in Different Climates of China" *Applied Sciences* 7, no. 10: 1028.
https://doi.org/10.3390/app7101028