## 2 Power Kite Generator System Modeling  
The operation of a wind-powered pumping kite generator presented in Figure 1 involves several phases to harness and efficiently convert wind energy. In the generation phase, the kite is maneuvered to rise in a crosswind motion, tracing an eight-trajectory pattern, which maximizes the extraction of energy from high-altitude winds [22]. Once the tether reaches its maximum length, the system transitions to the recovery phase. During recovery, the kite movement shifts to a non-crosswind motion, which is more effective in this phase [19], and the tether is reeled in consuming energy. This retraction prepares the kite for the next phase, forming a continuous cycle of alternating generation and recovery phases. The mechanical energy generated during the generation phase is transmitted via the tether and converted into electrical energy by an electrical generator on the ground. The forces acting on the kite are given by  
\[F=F_{grav}+F_{app}+F_{aer}+F_{trac} \tag{1}\]  
where \(F_{grav}\) is the gravity force, \(F_{app}\) is the apparent force, and \(F_{aer}\) is the aerodynamic force, which depends on the effective wind speed \(W_{e}\):  
Figure 1: Simplified power conversion system for a grid-connected PKGS. \[W_{e}=W_{0}-W_{a} \tag{2}\]  
where \(W_{0}\) is the wind speed, and \(W_{a}\) is the kite speed. The aerodynamic force comprises two fundamental components: the lift force \(F_{L}\) and the drag force \(F_{D}\), defined by [33]. \[F_{L}=\frac{1}{2}\rho_{air}AC_{L}|W_{e}|^{2} \tag{3}\]  
\[F_{D}=\frac{1}{2}\rho_{air}AC_{D}|W_{e}|^{2} \tag{4}\]  
where \(A\) is the kite surface, \(\rho_{air}\) is the air density, \(C_{L}\) is the lift coefficient, and \(C_{D}\) is the drag coefficient. For the simplicity of analysis, it is assumed that inertial and apparent forces are negligible compared to aerodynamic forces. In addition, both the weight and the drag force of the tether are also neglected. Then, using a procedure similar to that described in [15; 17; 33; 34], the traction force \(F_{trac}\) applied by the tether during the generation phase is found:  
\[F_{trac}=\frac{1}{2}\rho_{air}AC_{L}E^{2}\bigg{(}1+\frac{1}{E^{2}}\bigg{)}^{3 /2}|W_{e,r}|^{2} \tag{5}\]  
where \(E\) is the aerodynamic efficiency coefficient of the kite (\(E=C_{L}/C_{D}\)), and \(W_{e,r}\) is the effective wind speed along the cable direction. Posing the X axis aligned with wind speed, \(W_{e,r}\) can be defined as  
\[W_{e,r}=W_{0}sin(\theta)cos(\varphi)-i^{trac} \tag{6}\]  
where \(\theta\) denotes the polar angle, \(\varphi\) represents the azimuth angle, and \(i^{trac}\) is the kite speed along the cable direction. Additionally, the power generated can be provided via  
\[P_{trac}=\frac{1}{2}\rho_{air}AC_{L}E^{2}\bigg{(}1+\frac{1}{E^{2}}\bigg{)}^{3 /2}(W_{0}sin(\theta)cos(\varphi)-i^{trac})^{2}r^{trac} \tag{7}\]  
The derivative of \(P_{trac}\) is calculated with respect to \(i^{trac}\) in order to obtain the optimal instantaneous value of the tether's reeling out speed:  
\[\frac{dP_{trac}}{d\dot{r}^{trac}}=\frac{1}{2}\rho_{air}AC_{L}E^{2}\bigg{(}1+ \frac{1}{E^{2}}\bigg{)}^{3/2}\big{[}-2(W_{0}sin(\theta)cos(\varphi)-i^{trac}) \dot{r}^{trac}+(W_{0}sin(\theta)cos(\varphi)-i^{trac})^{2}\bigg{]} \tag{8}\]  
By solving for \(\dot{r}^{trac}\) and setting the derivative to zero. There are two crucial points identified. Since one of them (\(W_{0}\sin(\theta)\cos(\varphi)\)) sets the power to zero, the optimal tether reeling out speed value is  
\[i^{trac}_{opt}=\frac{W_{0}sin(\theta)cos(\varphi)}{3} \tag{9}\]  
During the recovery phase, a wing glide maneuver is employed. The kite is controlled to align parallel to the tether, minimizing its aerodynamic lift, so the cable can be wound back fast with minimal energy consumption [35]. Additionally, in order to improve system efficiency during this phase, a higher reeling-in speed is used.