## 2 Modeling  
### Solar Radiation Model  
The radiation received by the PV arrays primarily consists of direct solar radiation and scattered solar radiation. [33] The direct radiation intensity at the cruising altitude of the airship can be expressed as [34]  
\[I_{d0}=\tau_{h}\cdot\left(\frac{1+e_{e}\cdot\cos(\lambda_{e})}{1-e_{e}^{2}} \right)\cdot 1367 \tag{1}\]  
where \(e_{e}\) is the orbital eccentricity of Earth, with a value of 0.016708, \(\tau_{h}\) is the atmospheric transmissivity, and \(\lambda_{e}\) is the true anomaly, which can be given by  
\[\lambda_{e}=\theta_{day}+2e_{e}\cdot\sin(\theta_{day})+1.25e_{e}^{2}\cdot\sin (2\theta_{day}) \tag{2}\]  
where \(\theta_{day}\) is the solar day angle, calculated as  
\[\theta_{day}=\frac{(d_{n}-N_{r})}{365.2422} \tag{3}\]  
where \(d_{n}\) is the day number in a year, and \(N_{r}\) is the correction term of the day number. The atmospheric transmissivity \(\tau_{h}\) can be defined as [35]  
\[\tau_{h}=0.5\cdot\left(e^{-0.65-amr}+e^{-0.095 amr}\right) \tag{4}\]  
where \(amr\) is the air mass ratio, which can be formulated as  
\[amr=\left(\frac{p_{h}}{p_{0}}\right)\cdot\left[\sqrt{1229+\left(614\cdot\sin( \theta_{ele})\right)^{2}}-614\cdot\sin(\theta_{ele})\right] \tag{5}\]  
where \(p_{h}\) and \(p_{0}\) are the atmospheric pressure at the cruising altitude and the sea level, respectively, and \(\theta_{ele}\) is the solar elevation angle, given by  
\[\theta_{ele}=\arcsin(\sin(\theta_{dec})\sin(\Phi)+\cos(\theta_{dec})\cos(\Phi )\cos(\theta_{hour})) \tag{6}\]  
where \(\Phi\) is the local latitude, \(\theta_{hour}\) is the solar hour angle, and \(\theta_{dec}\) is the solar declination, which can be expressed as  
\[\theta_{dec}=0.3723+23.2567\cdot\sin(\theta_{day})+0.1149\cdot\sin(2\cdot \theta_{day})-0.1712\cdot\sin(3\cdot\theta_{day}) \tag{7}\]  
\[-0.758\cdot\cos(\theta_{day})+0.3656\cdot\cos(2\cdot\theta_{day})+0.0201\cdot \cos(3\cdot\theta_{day})\]  
\[\theta_{hour}=(time+e_{t}/60-12)\cdot 15 \tag{8}\]where \(e_{t}\) is the time correction term. The scattered radiation intensity at the airship's cruising altitude \(I_{dh}\) can be calculated as  
\[I_{dh}=0.5\cdot\sin(\theta_{ele})\cdot\frac{amr(1-\tau_{h})}{amr-1.41\cdot\tau_{ h}}\cdot\left(\frac{1+e_{e}\cdot\cos(\lambda_{e})}{1-e_{e}^{2}}\right)\cdot I_{h0} \tag{9}\]  
### Solar Radiation Model  
The angle between PV modules and the solar irradiance vector is closely associated with their placement, thus requiring the development of a solar irradiance distribution model for PV arrays. Divide the PV array into \(m\times n\) units, with the axial direction divided into \(m\) segments and the radial direction divided into \(n\) segments, and the central angle is \(d\theta\), as shown in \[\lambda_{e}=\theta_{day}+2e_{e}\cdot\sin(\theta_{day})+1.25e_{e}^{2}\cdot\sin (2\theta_{day}) \tag{2}\]0 The unit normal vector of each PV module is \(\overrightarrow{n}_{ij}=\left(n_{ijx},n_{ijy},n_{ijz}\right)\). The envelope of the airship can be considered a planar curve obtained by rotating around a symmetric axis. The equation of the control surface can be expressed as  
\[\lambda_{e}=\theta_{day}+2e_{e}\cdot\sin(\theta_{day})+1.25e_{e}^{2}\cdot\sin (2\theta_{day}) \tag{2}\]1  
where \(x,y,\) and \(z\) represent the three axis coordinates of the airship. In this paper, we utilize the GNVR-50 airship, and its envelope equation can be given by  
\[\left\{ \begin{aligned} z=\sqrt{0.25D^{2}\cdot\left(1-\frac{(x-1.25D )^{2}}{1.5625D^{2}}\right)}&& 0\leq x<1.25D\\ z=\sqrt{16D^{2}-(x-1.25D)^{2}-3.5D^{2}}&& 1.25D\leq x<2.875D\\ z=\sqrt{0.1373D\cdot\left(1.7998D-(x-1.25D)\right)}&& 2.875D\leq x<3.05D\end{aligned} \right. \tag{11}\]  
where \(D\) is the airship's maximum diameter. To determine the angle between the PV module and the irradiation vector, the module normal vector and the direct radiation unit vector \(\overrightarrow{n}_{s}\) in the inertial coordinate need to be calculated.