# Modeling and Simulation of Flight Profile and Power Spectrum for Near-Space Solar-Powered UAV

^{*}

## Abstract

**:**

## 1. Introduction

_{2}soft pack lithium battery under the approximate vacuum pressure and found that the capacity decreased to 80% after 10 cycles. Mussa et al. [21] studied the effect of external pressure on the cycle life of lithium-ion batteries and found that the pressure had little effect on the initial capacity but had a significant impact on the impedance and cycle life.

## 2. Mathematical Physical Model

_{E}(m,S) is the energy balance function, f

_{M}(m,S) is the mass balance function, and other symbols are defined in [9].

^{2}, and W/S is 2.8 kg/m

^{2}, which is within the reasonable range of the solar-powered UAV according to the statistical data in [32]. In addition, according to the energy balance equation, 12 m

^{2}PV cells are enough to provide the energy required for 24 h continuous flight. As shown in Figure 2, part of the solar energy absorbed during the day is used for cruising, part is converted into gravitational potential energy through climbing, and the rest is stored in secondary batteries. At night, the UAV converts gravitational potential energy and electric energy into mechanical energy, thus completing 24 h uninterrupted flight.

#### 2.1. Aerodynamic Model

_{L}and C

_{D}are the lift coefficient and drag coefficient, respectively.

_{L}and C

_{D}are mainly affected by the airfoil, angle of attack α, and Reynolds number Re and are expressed by the following equations:

_{ij}and B

_{ij}(shown in Table 1 and Table 2) are determined through the least squares method (LSM).

#### 2.2. Energy System Model

#### 2.2.1. Photovoltaic Cells

_{sun}(t) is the solar spectral density, S

_{sc}is the power generation area of PV cells, η

_{sc}is the conversion efficiency of PV cells, η

_{MPPT}is the efficiency of MPPT modules, and κ is the solar incidence angle.

_{e}and α

_{a}are the solar altitude angle and the solar azimuth (Figure 6), respectively, which are expressed as

_{t}is the solar time angle, n

_{d}is the ordinal number of the date, and t

_{h}is the true solar time.

_{oc}, short circuit current I

_{sc}, and maximum power P

_{max}with time is expressed by the following equation [18]:

_{0}, I

_{0}, and P

_{0}are, respectively, the open circuit voltage, short circuit current, and maximum power of PV cells before irradiation, and ϕ is the electron fluence.

_{in}is the input power under specific radiation intensity, and FF is the filling factor of PV cells, which is about 0.8.

_{max}change curve in Figure 9 can also characterize the attenuation of the PV cell conversion efficiency. The performance retention rate of PV cells in the first 30 days is 99.9%, and the loss is almost negligible. However, the solar-powered UAV is oriented to non-stop flight for months or even years, so this model cannot be ignored in the simulation.

#### 2.2.2. Secondary Battery

_{B}is the battery capacity, and P

_{B}is the discharge power. In addition, in order to make the secondary battery have a higher cycle life, the discharge depth is set to 90%, which means that the SOC value is not allowed to be lower than 10%.

#### 2.3. Dynamic System Model

_{p}is affected by the forward ratio λ and the characteristic Reynolds number Re

_{p}and is calculated by the following equation:

_{ij}is obtained by the CFD method and corrected by the wind tunnel test data. The values of C

_{ij}are shown in Table 6.

_{p}are expressed as:

_{p}is the propeller diameter, n is the propeller speed, c

_{0.75R}is the local chord length at 75% radius of the blade, and μ is the aerodynamic viscosity coefficient.

## 3. Flight Profile and Power Spectrum Design

#### 3.1. Flight Profile Design

_{1}to t

_{4}cover all kinds of flight states of the solar-powered UAV, and the power consumption model is built based on this. There is a main DC bus connecting the batteries and all the subsystems including motors and equipment. Subsystems obtain electricity from the nearest location of the main DC bus, which means most of the energy loss is produced in the main DC bus; thus, the transfer efficiency from the battery to motors η

_{b-m}is very close to that from the battery to equipment η

_{b-e}. We use one symbol η

_{l}to represent them. The electrical equipment of the solar-powered UAV includes motor and airborne equipment, and the total power required is

_{eq}is the power of airborne equipment, η

_{DC}is the efficiency of the DC/DC voltage conversion module, η

_{l}is the cable transmission efficiency, and the required power of the motor is

_{m}is the motor efficiency, and η

_{p}is the propeller efficiency.

- (a)
- Level flight phase

_{P,h1}is the required power for level flight at h

_{1}altitude, C

_{L}

^{3/2}/C

_{D}is the aircraft endurance factor, and ρ

_{h}

_{1}is the atmospheric density at h

_{1}altitude.

_{1}, the available power of the system is equal to the total required power of the solar-powered UAV, expressed as

- (b)
- Climbing phase

_{2}, and the time t

_{2}is expressed as

_{1}to t

_{2}is divided into M equal-difference finite elements:

_{2}is expressed as

- (c)
- Powered gliding phase

_{3}, the required power of the airborne equipment and the available power are balanced, expressed as

_{2}to t

_{3}is divided into N equal-difference finite elements:

_{3}is finally expressed as

- (d)
- Unpowered gliding phase

_{P, h}is the required power for level flight at h altitude.

_{4}is expressed as

#### 3.2. Flight Strategy

_{SC}, the secondary battery capacity SOC, and the flight height H. Compared with the flight strategy of the solar-powered UAV proposed in the literature [27,28], the strategy shown in Figure 17 has the following advantages:

- (1)
- Maximizes solar energy utilization, which means the UAV will use the solar energy to climb as high as possible before dusk, instead of maintaining level flight after climbing to a certain height;
- (2)
- Tries to avoid cruising in the higher airspace, which means the UAV will use the gravitational potential energy to glide to the night flight altitude at the first time after dusk, instead of starting to slide at a fixed moment.

- (1)
- The remaining capacity of the secondary battery shall not be less than 10%;
- (2)
- Always meet the power demand of airborne equipment (including data link, flight control, navigation, etc.);
- (3)
- The charging and discharging power of each piece of equipment shall not exceed the allowable value.

## 4. Typical Flight Profile and Power Spectrum

## 5. Analysis

#### 5.1. Task Indicators

#### 5.1.1. Takeoff Time Window

#### 5.1.2. Flying Season

#### 5.1.3. Flight Latitude

#### 5.1.4. Takeoff Weight

#### 5.2. Performance Indicators

#### 5.2.1. Lift–Drag Ratio

#### 5.2.2. Secondary Battery Energy Density

## 6. Conclusions

- 1)
- When building the aerodynamic model, the influence of flexible photovoltaic film on the aerodynamic characteristics should be considered. The deformation of the film has little effect on the lift coefficient but will lead to a significant increase in the drag coefficient.
- 2)
- When establishing the flight strategy, the solar-powered UAV shall use light energy to climb to the highest possible height before sunset, and it shall avoid cruising at high altitude at night. These are the core means to maximize light utilization.
- 3)
- Too-early takeoff time will cause the secondary battery to be exhausted before the sun rises. Too-late takeoff time will lead to failure to climb to the specified altitude on the first day, thus affecting the night flight. The closer to the summer solstice, the wider the takeoff window.
- 4)
- In the Northern Hemisphere, in order to achieve a higher altitude, the solar-powered UAV should fly at a low latitude in winter and fly at an appropriate high latitude (30° N–45° N) in summer.
- 5)
- For every 1 kg increase in the total takeoff weight, the night height and the maximum flight height will decrease by about 0.2 km. When the total takeoff weight is more than 10% overweight, the whole aircraft will deviate from the design point more, and the efficiency of the power system will decrease.
- 6)
- Each time the lift-to-drag ratio decreases by 1, the night flight altitude decreases and the maximum flight altitude decreases by about 0.4 km. When the energy density of the secondary battery increases by 10 Wh/kg, the altitude will increase by 0.33 km.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**Electrical performance prediction of the GaInP/GaAs/Ge triple-junction PV cells served in the geosynchronous orbit.

**Figure 22.**Variation of remaining capacity of secondary battery with takeoff time during the first day of climb.

A_{ij} | (Re/10^{5})^{0} | (Re/10^{5})^{1} | (Re/10^{5})^{2} |
---|---|---|---|

α^{0} | 7.983 × 10^{−1} | 9.208 × 10^{−3} | −9.792 × 10^{−5} |

α^{1} | 5.898 × 10^{0} | 1.392 × 10^{−2} | 2.255 × 10^{−3} |

α^{2} | −7.246 × 10^{0} | 4.610 × 10^{−2} | −1.894 × 10^{−2} |

B_{ij} | (Re/10^{5})^{0} | (Re/10^{5})^{1} | (Re/10^{5})^{2} |
---|---|---|---|

α^{0} | 2.284 × 10^{−2} | −6.603 × 10^{−4} | 1.493 × 10^{−5} |

α^{1} | 1.403 × 10^{−1} | 2.108 × 10^{−4} | −8.493 × 10^{−5} |

α^{2} | 1.362 × 10^{0} | −6.438 × 10^{−2} | 2.983 × 10^{−3} |

Subsystems | Parameters | Value |
---|---|---|

PV cells | Initial conversion efficiency | 28% |

Power generation area | 12 m^{2} | |

Secondary batteries | Energy density | 350 Wh/kg |

Discharge depth | 90% | |

Maximum discharge rate | 0.2 C | |

Energy management modules | MPPT efficiency | 95% |

DC/DC module efficiency | 80% | |

Cable efficiency | 80% |

**Table 4.**Equivalent fluence per year for GaInP/GaAs/Ge PV cells irradiated by the solar protons and the radiation belt electrons in the geosynchronous orbit.

Electrical Parameters | Annual Equivalent Fluence of Solar Proton, cm^{−2} a^{−1} | Annual Equivalent Fluence of Electrons, cm^{−2} a^{−1} | Sum of Annual Equivalent Fluence of Proton and Electron, cm^{−2} a^{−1} |
---|---|---|---|

V_{oc} | 1.33 × 10^{10} | 1.02 × 10^{13} | 3.54 × 10^{13} |

I_{sc} | 1.44 × 10^{10} | 1.02 × 10^{13} | 1.52 × 10^{13} |

P_{max} | 1.33 × 10^{10} | 1.02 × 10^{13} | 2.35 × 10^{13} |

Parameters | Values |
---|---|

Number of motors | 2 |

Maximum output power of the motor | 560 W |

Propeller diameter | 1.4 m |

C_{ij} | (Re/10^{5})^{0} | (Re/10^{5})^{1} | (Re/10^{5})^{2} |
---|---|---|---|

λ^{0} | −2.481 × 10^{0} | 2.783 × 10^{0} | −1.818 × 10^{−1} |

λ^{1} | 6.882 × 10^{0} | −4.081 × 10^{0} | −1.432 × 10^{0} |

λ^{2} | −3.640 × 10^{0} | 8.042 × 10^{−1} | 2.200 × 10^{0} |

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**MDPI and ACS Style**

Zhang, L.; Ma, D.; Yang, M.; Yang, X.; Yu, Y. Modeling and Simulation of Flight Profile and Power Spectrum for Near-Space Solar-Powered UAV. *Aerospace* **2022**, *9*, 672.
https://doi.org/10.3390/aerospace9110672

**AMA Style**

Zhang L, Ma D, Yang M, Yang X, Yu Y. Modeling and Simulation of Flight Profile and Power Spectrum for Near-Space Solar-Powered UAV. *Aerospace*. 2022; 9(11):672.
https://doi.org/10.3390/aerospace9110672

**Chicago/Turabian Style**

Zhang, Liang, Dongli Ma, Muqing Yang, Xiaopeng Yang, and Yayun Yu. 2022. "Modeling and Simulation of Flight Profile and Power Spectrum for Near-Space Solar-Powered UAV" *Aerospace* 9, no. 11: 672.
https://doi.org/10.3390/aerospace9110672