# Design Options to Improve the Dynamic Behavior and the Control of Small H-Darrieus VAWTs
## 3 Wind Tracking Index Maximization Procedure  
Moving from the definition of the wind tracking index introduced in Section 2.3, a discussion can be promoted to draw some indications about the potential improvement of the design philosophy of VAWTs, aimed to operate in a gusty, typically low wind speed environment, such as, for instance, the urban one. This discussion deals with the issue of turbine acceleration/deceleration and disregards the analysis of the self-aligning capability of the machine. In fact, unlike their horizontal-axis counterparts, VAWTs do not need rotor orientation systems, as they are inherently panemone, which theoretically implies a better ability to gather energy in gusty environments with frequent changes in wind directions. The goal of the proposed analysis is to discuss a design procedure aimed to obtain better reaction capability in the operational range between cut-in and cut-out wind speeds, without entering in the discussion of self-starting behavior. The model is based on the well-known approach introduced by [PERSON] [31], which guides the designer throughout an educated process to the general purpose of an optimized project. Such an approach identifies both the performance to be maximized (or minimized) and equations for the variables set as design constraints (i.e., maximum allowable deflection, and limit mechanical resistance). The performance and constraints form a set of equations that, through substitutions and analytical manipulations, lead to a single performance equation of the following form:  
\[\text{P}=f_{1}(F_{1})\ \cdot f_{2}(F_{2})\ \cdot....f_{n}(F_{n})\]  
or  
\[\dfrac{\Delta P}{P}=\dfrac{\Delta f_{1}}{f_{1}}+\dfrac{\Delta f_{2}}{f_{2}}.... \cdot+\dfrac{\Delta f_{n}}{f_{n}}\]  
where \(f_{i}\) are separate functions of groups of parameters \(F_{i}\), called performance metrics, describing functional requirements, geometric data, material properties and so on. When the groups are separable and independent, maximization (or minimization) of the performance can be obtained by maximizing (or minimizing) singularly any group \(F_{i}\) appearing in the equation. For the present case, the maximization of the \(WTI\) performance starts from a suitable expression of such a parameter:  
\[WTI = Q_{zero}\dfrac{R}{N_{\text{B}}I^{\prime}\lambda_{opt}\hat{V}_{ \infty}} \tag{6}\] \[= \dfrac{1}{2}N_{\text{B}}\rho_{a}cH\mathcal{C}_{T}V_{\infty}^{2} \dfrac{R^{2}}{N_{\text{B}}I^{\prime}\lambda_{opt}\hat{V}_{\infty}}\] (7) \[= \dfrac{3}{4}\dfrac{\rho_{a}V_{\infty}^{2}}{V_{\infty}}\dfrac{C_{ T}}{\lambda_{opt}}\dfrac{cH}{m_{B}} \tag{8}\]  
Introducing the rotor solidity for an H-VAWT (\(\sigma=\dfrac{N_{\text{B}}cH}{2 RH}\)), Equation (8) can be rewritten as follows:  
\[WTI = \dfrac{3}{4}(A_{D})\ \dfrac{\rho_{a}V_{\infty}^{2}}{V_{\infty}}\ \left(\dfrac{C_{T}}{\lambda_{opt}}\right)\ \left(\dfrac{1}{m_{B}}\right)\ \left(\dfrac{\sigma}{N_{ \text{B}}}\right) \tag{9}\]being \(A_{D}=2 RH\). The solidity and the tip speed ratio are related to a general equation of the form \(\sigma=\alpha\lambda_{opt}^{-\beta}\), with \(\alpha\) and \(\beta\) constants. Figure 4 shows a collection of \(\sigma-\lambda_{opt}\) data for current and past commercial H-VAWTs, from which the constants \(\alpha\) and \(\beta\) can be deduced as best fit of an exponential low. Therefore, Equation (9) becomes the following:  
\[WT1 = \frac{3}{4}(A_{D})\ \frac{\rho_{a}V_{\infty}^{2}}{V_{\infty}}\left( \frac{\alpha C_{T}}{N_{B}\lambda_{opt}^{\beta+1}}\right)\ \left(\frac{1}{m_{B}}\right) \tag{10}\]  
To include the blade material properties in Equation (10), the blade mass equation can be written as follows:  
\[m_{B}=\rho_{B}\ A_{B}\ H \tag{11}\]  
To maximize \(WT1\), the mass cannot be as small as desired without compromising the blade structural integrity. It is seen clearly from Figure 1 that the H-VAWT blade is loaded mainly along the radial direction [42].