# Design Options to Improve the Dynamic Behavior and the Control of Small H-Darrieus VAWTs
## 2 Background
### Guests Tracking  
The theme of VAWTs rotor gust tracking capability was approached analytically in a previous work by the author [39], where the relationships between inertia and rotor radius, inferred from datasheets of commercial and prototype turbines, were investigated. As shown in Figure 2, H-VAWTs inertia is, at least for small sizes, more than one order of magnitude higher than that of HAWTs for the very same diameter or swept area: as a consequence, VAWTs response times are much greater than those of their horizontal  
Figure 2: Moments of inertia of (H-type) commercial/prototype VAWTs and HAWT, according to diameter (**left**) and rotor swept area (**right**). axis counterparts, especially at low wind speeds, prompting a severe limitation to the efficient adoption of vertical axis rotor architectures within urban sites, which are generally characterized by a low wind potential, as well as rapidly varying wind conditions. Being that the present study is a natural continuation of the above-discussed work, its main features and results are here summarized; the complete model is fully detailed in [39]. To test the capability of a turbine to effectively adapt its rotational speed to satisfy a prescribed control strategy, two parameters were introduced and compared: the required rotor acceleration (RRA, defined as the acceleration required by the rotor to track a given gust) and the available rotor acceleration (ARA, defined as the maximum angular acceleration achieved when the rotor is free to accelerate under that gust). The required rotor acceleration (RRA) derives from the differentiation of the equation expressing the optimum tip speed ratio. \[\lambda_{opt}=\frac{\Omega R}{V_{\infty}}\]  
Therefore, for a given rotor radius, we have the following:  
\[RRA=\dot{\Omega}=\frac{\lambda_{opt}}{R}\frac{\partial V_{\infty}}{\partial t }=\frac{\lambda_{opt}}{R}\dot{V}_{\infty} \tag{2}\]  
where \(\dot{\Omega}\) is the rotor acceleration/deceleration, \(\lambda_{opt}\) is the design tip speed ratio that should be kept constant during normal operation to operate at maximum power coefficient, and \(\dot{V}_{\infty}\) is the gust acceleration. The available rotor acceleration (ARA) is defined on the basis of the rotor's allowable angular acceleration given by the mechanical balance of the power train as follows:  
\[N_{B}I^{\prime}\dot{\Omega}\simeq Q_{aero}-Q_{gen} \tag{3}\]  
Under the (conservative) hypothesis that the resistant torque \(Q_{gen}=0\), the acceleration of the rotor is equal to the following:  
\[ARA=\dot{\Omega}=\frac{Q_{aero}}{N_{B}I^{\prime}}\]  
where \(N_{B}\) is the number of rotor blades, \(I^{\prime}\) is the single blade inertia according to Equation (1), and the aerodynamic torque is \(Q_{aero}\) defined as follows [33]:  
\[Q_{aero}=\frac{\rho_{a}A_{D}V_{\infty}{}^{3}C_{P,MAX}(\lambda)}{2\Omega}. \tag{4}\]  
Moving from the definition of the latter parameters, a further synthetic index, the wind tracking index (\(WTI\), defined as the ratio between the \(ARA\) and the \(RRA\)) was introduced to assess the capability of the system to effectively track wind fluctuations:  
\[WTI=\frac{ARA}{RRA}. \tag{5}\]  
The condition that the available rotor acceleration is higher than the required one, or \(ARA>RRA\), implies that the condition \(WTI>\)1 has to be satisfied. It can be easily drawn that the \(WTI\), even if not capable of quantifying the exact amount of annual energy lost, due to the retarded response of the turbine, can be used to obtain an indication about the suitability of a given turbine architecture to adapt to given site characteristics. In the following, the maximization of this parameter is used to propose an H-VAWT design criterion. Figure 3a shows the record of gust cumulative probability occurrence sampled at the experimental test field of the University of Trento [9; 40; 41]. The curve derives from  
the interpolation experimental data. About 90% of the gusts presents an acceleration rate less than 2.5 m/s\({}^{2}\).