# A Novel Virtual Power Plant Uncertainty Modeling Framework Using Unscented Transform
## 3 Unscented Transform (UT)  
Power-grid uncertainties are gradually increasing, making safe and stable operation a challenge and constraining accurate estimates of variables during VPP operation. Therefore, it is necessary to recognize the sources of these uncertainties and to select the appropriate means of description.  
It is important to have accurate forecasts of demands and generations with the unpredictability of their uncertainties, so that the forecasting-error impact on system operating performance is minimized [21; 22].  
The integration of VPPs in the market occurs with the need to consider uncertainties without being overlooked. One solution for managing renewable energy uncertainties and demands is using the Unscented Transform (UT), in which the uncertainties can be modeled by one or more random variables for multiple sources of uncertainty.  
The UT consists of a discrete approximation method of the continuous probability-density function (PDF), ensuring that the two distributions have statistically the same properties, being even easier to implement and using the same order of calculations as linearization [23]. For the realization of the transform, one of the prerequisites is that the discrete distribution has the same moments as the continuous distribution.  
The UT can effectively model uncertainties and decision-making risks. It was developed with the intuition that it is easier to approximate a probability of distribution than to approximate a function or transformation of an arbitrary nonlinear function. The basic principle of the UT is the approximation of the PDF by a set of selected points, called sigma points, and their associated weights. The statistics of mean, variance, and other moments of the mapping will be available from a weighting of these sigma points [23; 24].  
As such, the UT is a powerful tool for obtaining statistics from a distribution that has undergone a linear or nonlinear process [25]. It can be viewed either as a nonlinear mapping expansion, a discrete approximation of the continuous probability-density function by a discrete distribution, or even with Gauss-quadrature integration.  
an interval \(I\). This gives the weights, which are important for finding the associated sigma points.  
The sigma points \(S_{i}\) are determined by calculating the roots of the polynomial \(\pi_{m}(x)\), with the values of \(w_{i}\) found by determining the roots \(x_{1},x_{2},\ldots x_{m}\), of the polynomial by the following equation:  
\[\pi_{m}(x)=\sum_{i=0}^{m}w_{i}x^{m-1} \tag{7}\]  
With this nonlinear transformation defined by the UT, it is possible to obtain an approximation of the moments of \(w(\hat{u})\) from the moments of \(w_{i}\). The UT makes the discrete distribution have the same moments as the continuous distribution after nonlinear transformation or mapping, according to the following equation [25]:  
\[E_{d}\{\hat{u}^{k}\}=\int_{-\infty}^{+\infty}\hat{u}^{k}w(\hat{u})d\hat{u}= \sum_{i}w_{i}S_{i}^{k} \tag{8}\]  
where:  
\(E_{d}\)--expected value of the discrete distribution;  
\(\hat{u}\)--the set of random variables with known probability distribution;  
\(\int_{-\infty}^{+\infty}\hat{u}^{k}w(\hat{u})d\hat{u}\)--the expected value of the continuous distribution;  
\(S_{i}\)--UT sigma points;  
\(k\)--desired approach;  
\(w_{i}\)--defined as UT weights, besides being the discrete probability density function.  
The use of the simple moving average method where the forecast is the average of the most recent \(N\) observations of the \(X\) series, as can be observed in Equation (9), is necessary.  
\[X_{t}=\frac{1}{N}\sum_{i=1}^{N}X_{t-1} \tag{9}\]