## 5 Assessment of Different Sailing Strategies
### Downwind&Upwind  
For the strategy Downwind&Upwind, the SailREC System, as depicted schematically in Figure 7, sails downwind for a specified distance and then reverts the moving direction or turns and drives straight upwind until it reaches the start point. In the downwind phase, in which the wind turbine is operating, the resulting thrust force drives the SailREC System forward while the hull resistance slows down the forward motion, and--to increase the effective wind speed at the wind turbine--the driving speed can be further reduced by means of a water brake, providing additional hydrodynamic drag. In the upwind phase, the wind turbine is stopped, the water brake is released, and the active propulsion system is used to drive back against the wind; however, opposing forces due to hydrodynamic drag as well as aerodynamic drag on the parked wind turbine are again present.  
This Downwind&Upwind sailing strategy might not be an ideal one since wind energy can only be captured for a fraction of the time of one cycle. However, it is easy to calculate the energy yield for this strategy and, hence, obtain a lower limit for the possible energy performance.  
For the evaluation of the effective power, a steady-state condition is considered, meaning that the velocity of the SailREC System in the direction of travel (\(v_{\text{drift}}\)) is constant and, hence, its derivative (\(\dot{v}_{\text{drift}}\)) is zero, and it is assumed that all other parts of the SailREC System except for the wind turbine rotor plane, the hull, and the brake have no aero- or hydrodynamic drag, respectively. In the following, the downwind and upwind phases are first considered separately. In the downwind phase, the equilibrium of the forces on the  
Figure 6: Use of electric propulsion in the SailREC System.  
wind turbine (\(F_{\rm Thrust}\)), hull (\(F_{\rm Hull}\)), and brake (\(F_{\rm Brake}\)), as presented in Equation (1), with the total mass of the SailREC System (\(m_{\rm System}\)), is used as the calculation basis.  
\[\sum F=m_{\rm System}\vartheta_{\rm drift}=F_{\rm Thrust}-F_{\rm Hull}-F_{\rm Brake }\xlonge  
remainder of the power generated by the wind turbine (\(R_{\text{WindTurbine}}\)) during the downwind phase less the power required by the propulsion system (\(P_{\text{Propulsion}}\)) during the upwind phase.  
\[P_{\text{effective}}=\frac{t}{2t}P_{\text{WindTurbine}}-\frac{t}{2t}P_{\text{ Propulsion}}. \tag{9}\]  
The power of a wind turbine is computed following Equation (10) [36].  
\[P_{\text{WindTurbine}}=\frac{1}{2}\rho_{\text{Air}}A_{rot}4a(1-a)^{2}(v_{\text{ wind}}-v_{\text{drift}})^{3}. \tag{10}\]  
The distance of travel within each phase is set to be \(s\). For simplicity, it is assumed that the SailREC System drives back at the same drift speed as in the downwind phase. Hence, the pure power required by the propulsion system (\(\dot{P}_{\text{Propulsion}}\)) can be determined according to Equation (11).  
\[\dot{P}_{\text{Propulsion}}=F_{\text{Propulsion}}\frac{s}{t}=F_{\text{ Propulsion}}v_{\text{drift}}. \tag{11}\]  
The SailREC System would use the energy of the fuel produced from the wind energy captured during the downwind phase to drive back.--There are also other possibilities with higher efficiency, such as superconductors or flywheels, that could be used for this purpose. As, however, this is a worst-case calculation, the round-trip efficiency of the produced fuel is used.--Thus, a round-trip efficiency (\(\eta_{\text{RoundTrip}}\)) is introduced to the pure propulsion power calculation (Equation (12)).  
\[P_{\text{Propulsion}}=\frac{1}{\eta_{\text{RoundTrip}}}\dot{P}_{\text{ Propulsion}}. \tag{12}\]  
Finally, the effective power of the Downwind&Upwind strategy, using Equations (8)-(10) and (12), is computed as summarised in Equation (13).  
\[P_{\text{effective}}=\rho_{\text{Air}}A_{\text{rot}}a(1-a)^{2}(v_{\text{wind}}-v _{\text{drift}})^{3}-\frac{1}{4\eta_{\text{RoundTrip}}}\rho_{\text{Water}}A_{ \text{Wettted}}\text{C}_{\text{T}}v_{\text{drift}}^{3}. \tag{13}\]