2.1.2. Quantile-Mapping Calibration

The cup anemometers installed on the buildings of the University of Basque Country in Eibar enabled the development of a preliminary calibration methodology based on quantile-matching techniques that were used previously by the authors for wind energy and wave energy [25–27]. In the scientific literature, different calibration or bias correction techniques have been developed and compared for the analysis of several parameters, such as temperature and precipitation (see [28–30]). Data from models and reanalysis are compared with observations. In the present study, a simple but effective statistical procedure based on quantile mapping was used.

For this approach, several other terms can be found in the literature: "probability mapping" [31], "quantile-quantile mapping" [32,33], "statistical downscaling" [34], and "histogram equalization" [35]. With this general approach, empirical quantile-mapping bias correction was applied to calibrate ERA5 versus an anemometer in the building. In [36], the same procedure was used for estimating wind energy trends. To summarize, this method of calibration or bias correction is fundamentally statistical, and the idea is to match values with the same quantile in two empirical probability distributions: the one to be calibrated (ERA5), and the one that is the basis for the calibration (anemometer). Figure 2 illustrates the main aspects of this calibration procedure, including the intersection periods and the concept of applying the transference function.

For this paper, the authors obtained an eight-month, 10-min data series, which was filtered every 6 h to match the ERA5 reanalysis for a 10-year period (1-h time resolution, in this case). Thus, there were around 34,500 cases in the anemometer time series and around 87,600 cases in the ERA5 series. Taking 1-h data for both series in the intersection period resulted in 5390 cases, from which the correlation was measured and the subsequent calibration transference function was generated.

Having determined the average wind speed *U* after calibration on the corresponding facade and considering the typical shape parameter of the Weibull distribution (Rayleigh distribution, *k* = 2), the corresponding scale parameter can be obtained:

$$
\mathfrak{c} = \overline{\mathfrak{U}}/\Gamma(1 + 1/k). \tag{1}
$$

Then, the cumulative distribution function and the fraction of time between two wind speeds are determined:

$$F(\mathcal{U}) = 1 - \exp\left(-(\mathcal{U}/\mathfrak{c})^k\right) \tag{2}$$

and the augmentation factor *AF* of the PAGV (the ratio between the outlet and inlet free wind speed) can be incorporated into the *c* parameter [9] because it is proportional to the average wind speed *U*.

**Figure 2.** Calibration procedure and periods of the ERA5 and anemometer data.

The value of *AF* can only be based on a virtual definition of the outlet velocity of the flux, since the complex interaction between the diffusive flux in the exterior part of the vanes and the motion of the rotor do not permit a simplistic application of the Venturi effect according to the relation between the capture width of the free wind *U* and the outlet width. However, the optimum tip speed ratio (*TSRopt*) at which the power coefficient *Cp* is maximized is directly related to the outlet effective velocity (*Uout*), because it is well known that *TSRopt*0 ≈ 1/3 for a drag turbine without augmentation techniques [9]. Due to the Magnus and lift effects in the Savonius rotor, this value can reach 0.4–0.5. Therefore, the augmen<sup>t</sup> of *TSRopt* should be similar to *AF* considering an effective *Uout* at the position of the rotor in relation to the blade tip speed *Vtip*. Being *VAtip* the tip speed in the augmented rotor and *TSR<sup>A</sup>* the tip speed ratio in the augmented rotor, the hypothesis is that the *TSR* should be the same for the conventional Savonius and for the augmentation technique if the outlet velocity *AF* · *U* is the reference wind speed:

$$TSR = \frac{V\_{tip}}{\mathcal{U}} = \frac{V\_{tip}^A}{AF \cdot \mathcal{U}} \Rightarrow V\_{tip}^A = AF \cdot V\_{tip} \tag{3}$$

However, the *TSR* with augmentation *TSR<sup>A</sup>* should be defined with respect to the free wind speed: *VAtip U*. Therefore,

$$TSR^A = AF \cdot TSR \Rightarrow AF = \frac{TSR^A}{TSR} \tag{4}$$

Consequently, *AF* can be also computed using the ratio of the *TSR* with augmentation versus the *TSR* without augmentation.

On the other hand, experiment using nozzles by Shika et al. [37] have shown that *AF* can be 4 or even 5 measuring directly *Uout* at the position of the rotor for *U* between 0.6 and 0.9 m/s. This relevant increment for low free wind speed is very interesting for our purpose, since reducing significantly the cut-in speed of the rotor. Furthermore, these *AF*s ensure a grea<sup>t</sup> quantity of working hours at rated power, as shown below.
