Coupled trajectory optimization and tuning of tracking controllers for parafoil generator

ABSTRACT

The high-altitude wind power generation (HAWPG) parafoil is an innovative generator with remarkable environmental benefits. Due to the unique trajectory shape, there is an inherent coupling between the pitch and yaw channels, which imposes severe difficulties in the trajectory optimization and tracking controller design. To deal with this intrinsic problem, the trajectory optimization and tracking of the HAWPG parafoil are provided in a comprehensive manner. By using the pseudospectral method (PSM), the parametric optimal trajectory, which has the maximum net power generation, can be obtained. To satisfactorily track this trajectory, well-tuned trajectory controllers are needed to ensure the desired generation performance for the strong coupling dynamics between the two channels. The deep deterministic policy gradient (DDPG) algorithm is applied to the joint tuning of two conventional PID controllers for strongly coupled nonlinear dynamics. A reasonable cost function is established in this process. The mappings between the errors and control gains for the two channels can then be obtained. The gain tuning results are interpretable and well match the empirical knowledge. The simulations are carried out and the comparisons are performed with regard to multiple uncertainties and perturbations. The effectiveness of the proposed method can be illustrated.

KEYWORDS:

• High altitude wind power generation (HAWPG)
• pseudospectral method (PSM)
• deep reinforcement learning (DRL)
• deep deterministic policy gradient (DDPG)

Nomenclature

$\theta$	=	pitch angle in Figure 1a
$\phi$	=	yaw angle in Figure 1a
$r$	=	cable length in Figure 1a
$\dot{r}$	=	cable changing rate
$\ddot{r}$	=	cable acceleration
$A$	=	parafoil area
$A_c$	=	area of the line on the effective wind
$m$	=	parafoil mass
$\rho$	=	air density
${\rho }_c$	=	cable density
$d_c$	=	cable diameter
$H$	=	parafoil realistic height
$H_{\mathrm{r}\mathrm{e}\mathrm{f}}$	=	reference height
$W_{\mathrm{r}\mathrm{e}\mathrm{f}}$	=	reference wind velocity
$C_{\mathrm{L}}$	=	parafoil coefficient of lift
$C_{\mathrm{D}}$	=	parafoil coefficient of drag
$C_{\mathrm{c},\mathrm{D}}$	=	cable coefficient of drag
$E$	=	$\frac{C_L}{C_D}$
$\mathrm{\Delta }l$	=	cable length difference at the two ends of the traction parafoil in Figure 1b
$d$	=	:length of the wingspan in Figure 1b
$\psi$	=	$=arcsin\left(\mathrm{\Delta }l/d\right)$ in Figure 1b
$\mathrm{\Delta }\alpha$	=	angle between the effective wind velocity and the $\left(e_{\theta },e_{\phi }\right)$ plane in Figure 1c
${\mathbf{W}}_{\mathrm{e}}$	=	effective wind velocity
${\mathbf{W}}_{\mathrm{e}}^{\mathrm{p}}$	=	projection of the effective wind velocity on the $\left(e_{\theta },e_{\phi }\right)$ plane
${\mathbf{e}}_0$	=	$\hspace{0.17em}={\mathbf{e}}_r\times {\mathbf{e}}_{\mathrm{w}}$
${\mathbf{e}}_{\mathrm{w}}$	=	$\frac{{\mathrm{w}}_{\mathrm{p}}^{\mathrm{e}}}{\left|{\mathrm{w}}_{\mathrm{p}}^{\mathrm{e}}\right|}$
${\mathbf{F}}^{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$	=	parafoil and cable weight
${\mathbf{F}}^{\mathrm{a}\mathrm{e}\mathrm{r}}$	=	parafoil aerodynamic force
${\mathbf{F}}^{\mathrm{c},\mathrm{a}\mathrm{e}\mathrm{r}}$	=	cable aerodynamic force
${\mathbf{F}}^{\mathrm{a}\mathrm{p}\mathrm{p}}$	=	apparent force
$F^{\mathrm{c},\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}}$	=	cable traction force
$F_{\mathrm{L}}^{\mathrm{a}\mathrm{e}\mathrm{r}}$	=	aerodynamic lift force
$F_{\mathrm{D}}^{\mathrm{a}\mathrm{e}\mathrm{r}}$	=	aerodynamic drag force

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [62073177, 61973175, 52175038]