[MUSIC] Welcome back, in this video we will test the different methods presented in form video for two specific locations in France. And afterwards, we will relate those modeling results to the wind energy power availability. We have chosen two stations with significantly different goodness fit scores. Melun is situated in the eastern suburbs of Paris, while Orange, in southern France, in the Rhone valley. At Melun, the east ground is too peaked to be fitted with a variable distribution model. This is in fact a common behavior at most stations. However, the fit is still reasonable with respect to the observations. Conversely, the Orange station data distribution presents a bimodal shape that the viable distributions are not able to reproduce. The question is, where do these two modes come from? In this region, a strong northerly wind, called mistral, blows about one-third of the year without strong distinction between summer and winter. When the strong regional wind does not blow, it is replaced by weaker and more isotropic winds. Variable distribution is by construction a unimodal distribution and cannot fit the observed data. After those examples, it is clear that alternatives to the variable distributions are needed. The derivation of alternative distributions is based on the recently developed super-statistics theory. And the simplest way of implementing super-statistics is to superimpose two different local dynamics. We introduce here the Rayleigh-Rice distribution, based on the observations for the valleys in southern France, having two main regimes. Both of which can be described by a particular bivariate normal distribution. The first regime driven by a weak isotropic flow, which statistics is well described by a Rayleigh distribution. Second regime, driven by a Mistral valley flow, which statistics can be well described with a Rice distribution. The resulting distribution is the sum of the two distributions condition to the absence and presence of the channel flow occurrence. In the equation, we can see a prevailing wind term corresponding to second regime contribution, and a random flow term driven by the first regime. The parameter alpha is here, the weight corresponding to the occurrence of channeled-flow events. This model can be applied also to the combination of the weak isotropic regime in a sustained prevailing flow, as is the case in northern France with prevailing strong westerlies. It is, however, more difficult to fit because it has four parameters, especially because of the non-linear effect of the alpha parameter that modulates the respective weights of the Rayleigh and Rice distributions. The four parameters are alpha, mu, and sigmas. I0 is the modified Bessel function of the first kind and zero order. In the figure at Orange, we have a shouldered histogram where the Weibull is not well distributed. Only the Rayleigh-Rice distribution is capable of fitting the two peaks. At the end of block two, we presented the Cramér-von Mises score for the Weibull distribution in some of sites in France. And this figure summarizes the performances of the Rayleigh-Rice distribution, so we can compare both cases. As we can observe, it is doing similar or better than Wiebull distribution on the center of distributions at all stations. And surprisingly, in other areas without bi-modal distribution, the Rayleigh-Rice also brings some improvement. All the bi-modal distributions have been proposed, one of the most popular is the bi-modal Weibull distribution with various configurations, shown here with an example of a site in China. You can see that all tested bi-modal distributions are better suited than the Weibull distribution to fit the observed distribution. In the former block, we introduced the notion of wind power. It is the energy available per unit of time and area. In the following it will be convenient for us to use E to refer to the wind power. Here we have the wind power obtained from statistical methods from our district set of data or from a model generic curve. And the easy way to compare both values, the relative error, and let's use it in both data set examples, Melun and Orange. In the figures, the discrete wind power densities formula in Orange were computed and the fitting curves are superimposed using the full methods we've been comparing. As we can see, the Rayleigh-Rice curves work the best for both cases in terms of shape, but it tends to slightly overestimate the integral. At Melun, all three variable fits tend to underestimate the power at very high wind speed. At Orange, the opposite happens, there is a large overestimation of power at strong winds. This will have strong consequences when we discuss about wind energy production that we'll be seeing later. Here we have included the relative errors showing the deviation between the integrals of each curve and the real data. The computation of wind power is very sensitive to the adjustment of the right tail of the distributions since the very high winds values, once skewed, have an important weight despite their low frequency. The Weibull distribution, especially the maximum likelihood estimation fit in red, tends to underestimate the frequency of these very high winds and therefore underestimate E. This is very visible in Melun. The shape of the curve is less well fitted despite the integral is well captured when using the moments method in blue and the WAsP methods, violet. Indeed, both methods for fitting the Weibull make no error, since the energy content of the distribution is fixed to the observed energy content by construction. All right, at Orange we can see in the figure that the Weibull distribution, whatever the fitting method, overestimates the probability and therefore the contribution to the energy of winds above 15 meter per second. As the distribution has two peaks, the Weibull distribution persists through both. It underestimates the wind frequency at the two peaks and overestimates the winds in between the two peaks and for the very high winds beyond the second peak. With the moments and WAsP methods, this overestimation is smaller and balanced by the underestimation in the range 8 to 15 meter per second corresponding to the second peak. In the case of the maximum likelihood estimate method, it is not completely balanced. And it leads to an overestimation of the energy by more than 10%. The figure maps the relative error on the wind energy, delta E, at 89 stations in France for the Weibull distribution feat by the maximum likelihood method, and for the Rayleigh-Rice distribution. Indeed, both methods, moments and the WAsP method, when fitting the Weibull curve, make no errors, since the energy content of the distribution is fixed to the observed energy content by construction. With the Weibull distribution fitted by maximum likelihood, the errors are low in the northeastern region, but much larger in the southern region. The absolute errors are above 5% at 32 stations and above 10% at 10 stations, a maximum being 29%. The energy is almost always underestimated except in some places in the valleys of southeastern region. With the Rayleigh-Rice distribution, the errors are in general very low with some exceptions. The absolute errors are below 3% at 80% of the stations. Only ten stations are above 5%, including two above 10. Now, we have assess for wind energy resource but our main interest is still the wind energy production and there is a major difference in assessing the wind energy resource and the wind energy production. In order to compute the energy production, from now on represented by a p, we need a power curve, and this power curve depends on the wind turbine used. As an example, here we have represented a power curve for a Vestas wind turbine of 19 meter diameter and two megawatt nominal power. The figure shows in red the wind energy resource and in blue the wind energy production using the Vestas wind turbine characteristics. Let's focus on the power production curve, the blue one. This curve shows that there is a cut in wind speed above which the turbine blades start to rotate, and the wind turbine effectively produces energy. As the wind speed rises above the cut-in speed, the level of electrical output power rises rapidly. However, typically somewhere between 12 and 17 meter per second, the power output reaches the limit that the electrical generator is capable of producing. This limit to this generator output is called the rated power output, and the wind speed at which it is reached is called the rated output wind speed. At higher wind speeds, the design of the turbine is arranged to limit the power to this maximum level, and there is no further rise in the output power. As the speed increases above the rate output wind speed, the forces on the turbine structure continue to rise, and at some point there is a risk of damage to the rotor. As a result, a braking system is employed to bring the rotor to a standstill. This is called the the cut-out wind speed and is usually around 25 meter per second. When it comes to the estimation of the energy produced from a wind turbine, the very high winds are not so important, since the power output is constant between the rated wind speed and the cut-out wind speed of the wind turbine. As we have applied statistical studies on the wind power available, studies on the power production might be of most interest as well. We will have a real set of data from which a real data set of power production will be computed and a fitted curve modeling it. We will then compute the relative error associated to this modeling. Taking our two stations from Melun and Orange, here we present the power outputs normalized by the rated power, showing the power output. An underestimation of the very high winds is associated with an overestimation of winds around the rated wind speed, which have the largest contribution to the production. This is why we get opposite errors on the wind energy, and energy production with the viable fit by the maximum likelihood estimate. With the WAsP method, the Weibull fit is better adjusted to the high winds. We can see clearly that there is less underestimation on the winds over eight meter per second, and less overestimation for the winds below eight meter per second, than with the two other methods. Therefore, the errors in production are much reduced in both cases. In the bimodal cases, the overestimation of the very high winds is associated with an underestimation of middle high winds, and therefore an underestimation of the energy yield. This is particularly critical at Orange, where the winds having the largest weight in production are exactly around the second peak, largely underestimated by the Weibull distribution. When it comes to the power output, there are errors with all four distributions even when there was no error, [FOREIGN]. When it comes to the power output, there are errors with all four distributions, even when there was no error on the energy. Indeed, the fact that there is no error on the energy does not mean that the wind speed histogram is well fitted by the theoretical probability density function. There may be positive and negative errors balancing one another when integrating of the whole distribution. Since the power curve is not a linear function of the wind speed cubed, the balanced is different for the production calculations, and this may lead to larger errors. For the Weibull distribution fitted by maximum likelihood, the error on the production is always of the same order of magnitude, but of opposite sign as the error on the wind energy. The mean absolute error is 5.2%. Two-thirds of the stations have an absolute error above 3%, the maximum error is 32%. For the method of moments, the errors are similar to these four maximum likelihood estimate. The spatial pattern is the same, the values are just slightly lower. With the WAsP method, on the contrary, the errors are low. The mean absolute error is only 1.7%, only 13 stations have an absolute error above 3%, and the maximum error is 9%. Finally, with the Rayleigh-Rice distribution, we find very low errors with a slight bias towards overestimation. The behavior is similar everywhere. All the errors are between 0.2% and 3.1%, the mean being 1.4%. So in this lecture we have explained the importance of evaluating the errors in production instead of only the energy, such as is done for example to produce wind atlas. Indeed, a perfect fit on the energy does not guarantee at all that the distribution really fits the observation. It may come from the cancellation of opposite errors, and may lead to large errors in the production due to the non linear effect of the power curve. And I have tried to show it in real cases. I hope this lecture helps you to better understand how wind speed statistics is used in the energy field. Thank you.