# Power Generation Calculation Model and Validation of Solar Array on Stratospheric Airships

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## Abstract

**:**

## 1. Introduction

- Based on the actual stratospheric airship solar array layout model, a calculation model of real-time solar radiation power received by the stratospheric airship solar array is obtained by comprehensively considering kinematic factors such as airship flight attitude, position, time, and date.
- We construct a high-precision calculation model for real-time power generation for the stratospheric airship solar array. Model corrections are conducted for the case where the photovoltaic conversion efficiency of solar cells varies with the radiation incidence angle and the case of the constant-voltage charging stage in the energy storage battery pack.
- The calculation results of the constructed model are compared and analyzed with the power generation measured in an actual flight test, and the accuracy of the model calculation is validated.

## 2. Basic Calculation Model for Power Generation

#### 2.1. Model Variables Associated with Flight Date

#### 2.2. The Relative Positions of the Airship and the Sun

#### 2.3. Radiation Model

^{2}; $\hspace{1em}{\mathrm{H}}_{\mathrm{t}}\hspace{1em}$ $=11\hspace{0.17em}000$ m; $\hspace{1em}{\mathrm{R}}_{\mathrm{b}}\hspace{1em}$ $=287.052\hspace{0.17em}87$ m

^{2}/K·s

^{2}; $\hspace{1em}{\mathrm{T}}_{\mathrm{b}}\hspace{1em}$ $=216.65$ K.

## 3. Model Correction

#### 3.1. Angular Loss Correction

#### 3.2. Model Correction for an Energy Storage Battery Pack in Constant-Voltage Charging Stage

## 4. Experimental Validation

^{−5}s, which can be applied to large-scale iterative optimization algorithms or fast simulation models.

## 5. Conclusions

- Correcting for angular loss and optimizing the storage battery pack during the constant-voltage charging stage substantially improved the accuracy of the overall power generation calculation model. This refinement reduced the normalized root-mean-square error from 12.17% to 2.59%.
- The model developed in this study demonstrates low computational complexity, making it suitable for integration into iterative optimization algorithms and rapid simulation models.
- Comparative analysis of the model calculation results across multiple flight dates with corresponding experimental data demonstrated the newly established power generation model’s commendable computational precision and robust generalization capabilities, evidenced by an average normalized root-mean-square error of 2.47%.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbols | |||

${A}_{g}^{b}$ | The coordinate conversion matrix from the ground coordinate system to the body-fixed coordinate system. | ${t}_{sr}$ | Sunrise time |

${D}_{solar}$ | The duration of sunlight | ${t}_{LAT}$ | Local apparent solar time |

${E}_{0}$ | The Sun–Earth distance correction factor | ${t}_{ss}$ | Sunset time |

${E}_{t}$ | The deviation of the solar day from 24 h | ${\mathrm{V}}_{\mathrm{charge}}\hspace{1em}$ | The charging voltage of the storage battery packs |

$H$ | The flight altitude | $\gamma $ | The solar azimuth angle |

${I}_{0}$ | The intensity of the extraterrestrial normal solar radiation | $\delta $ | The solar declination angle |

${\mathrm{I}}_{\mathrm{c}}$ | The solar constant | ${\eta}_{0}$ | The photovoltaic conversion efficiency under standard conditions |

${I}_{charge}$ | The charging current of storage battery packs | ${\eta}_{\mathrm{c}}$ | The efficiency of the power controller |

${I}_{D}$ | The intensity of direct solar radiation | ${\eta}_{pv}$ | The photovoltaic conversion efficiency |

${L}_{S}$ | The longitude of the standard meridian for the local time zone | $\theta $ | The pitch angle of the airship |

${L}_{solar}$ | The solar day duration | ${\theta}_{in}$ | The solar radiation incidence angle |

${L}_{L}$ | The local longitude | ${\theta}_{day}$ | The day angle |

${N}_{i}$ | The normal outward vector of the mesh grid | ${\lambda}_{m}$ | The air mass |

${P}_{a}$ | The atmospheric pressure at flight altitude | $\mu $ | The photovoltaic conversion efficiency decay factor |

${P}_{charge}\hspace{1em}$ | The charging power | ${\tau}_{atm}$ | The atmospheric transmittance |

${P}_{con}$ | The power generation | $\varphi $ | The roll angle of the airship |

${P}_{gen}$ | The power consumption | $\phi $ | The flight latitude |

${S}_{b}$ | The coordinates of the solar direction vector in the body-fixed coordinate system | $\psi $ | The yaw angle of the airship |

${S}_{g}$ | The coordinates of the solar direction vector in the ground coordinate system | $\omega $ | The solar hour angle |

${t}_{LST}$ | Local standard time | ${\omega}_{ss}$ | The sunset hour angle at flight altitude |

Abbreviations | |||

AU | Astronomical unit | ||

HAP | High-altitude platform | ||

MPP | Maximum power point | ||

MPPT | Maximum power point tracking | ||

nRMSE | Normalized root-mean-square error | ||

PCEDF | Photovoltaic conversion efficiency decay factor | ||

SOC | State of charge |

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**Figure 3.**Solar radiation flux calculation model: (

**a**) A render graph of an airship with a solar array on its upper surface; (

**b**) The meshing of the curved solar array on the case-study airship; (

**c**) Model of a grid cell receiving solar radiation.

**Figure 8.**Measured data from the actual flight: (

**a**) Pitch and roll; (

**b**) Yaw; (

**c**) Relative Position; (

**d**) Height; (

**e**) Power Consumption; (

**f**) SOC.

**Figure 9.**Comparison between the basic and corrected model outputs and the sampled data: (

**a**) Comparison of the model output with the sampled data for a constant value of photovoltaic conversion efficiency; (

**b**) Comparison of model output and sampled data after angular loss correction; (

**c**) Comparison of model output and sampled data after constant-charging stage correction and angular loss correction.

**Figure 10.**Comparison of power generation model calculation results and measured results from different flight dates: (

**a**) 7 September; (

**b**) 23 September; (

**c**) 29 September.

Parameter | Value |
---|---|

Open-Circuit Voltage | 48 V |

Short-Circuit Current | 6 A |

Area | 1 m^{2} |

Photovoltaic Conversion Efficiency | 18% |

Cell Type | Heterojunction with Intrinsic Thin Layer |

Model | 8:00–10:00 and 18:00–20:00 nRMSE | 13:00–16:00 nRMSE | 24 h Overall nRMSE |
---|---|---|---|

Basic model | 13.03% | 27.24% | 12.17% |

Angular loss correction only | 4.73% | 27.11% | 10.62% |

After correction for both angular loss and constant-voltage charging stage | 4.73% | 3.59% | 2.59% |

Item | Configuration |
---|---|

CPU | AMD Ryzen 9 6900HS 3.30 GHz |

RAM | 32.0 GB |

Operating System | Windows 11 22H2 |

Programming Environment | MATLAB R2022A |

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**MDPI and ACS Style**

Song, K.; Li, Z.; Zhang, Y.; Wang, X.; Xu, G.; Zhang, X.
Power Generation Calculation Model and Validation of Solar Array on Stratospheric Airships. *Energies* **2023**, *16*, 7106.
https://doi.org/10.3390/en16207106

**AMA Style**

Song K, Li Z, Zhang Y, Wang X, Xu G, Zhang X.
Power Generation Calculation Model and Validation of Solar Array on Stratospheric Airships. *Energies*. 2023; 16(20):7106.
https://doi.org/10.3390/en16207106

**Chicago/Turabian Style**

Song, Kaiyin, Zhaojie Li, Yanlei Zhang, Xuwei Wang, Guoning Xu, and Xiaojun Zhang.
2023. "Power Generation Calculation Model and Validation of Solar Array on Stratospheric Airships" *Energies* 16, no. 20: 7106.
https://doi.org/10.3390/en16207106