# Mdp1 _sciences_
## 2 Methods and Data
### Forecast-Based Curtailment (FBC)
#### 2.1.1 Ideal Forecasts  
In order to illustrate the idea we will first derive an expression for the FBC correction in the case of ideal forecasts, i.e., \(\hat{P}_{n}=P_{n}\) and \(\hat{P}_{n+1}=P_{n+1}\). In the case of an _incompliance event_, i.e., \(y_{p}<-a_{p}\), the criterion for curtailing power at \(t_{n}\) is for the _average true slope_\(\frac{1}{2}(x_{p}+y_{p})\) to be larger than the critical slope \(-a_{p}\). Then the expression for the adjusted power \(P_{n}^{\prime}\) at \(t_{n}\) becomes  
\[P_{n}^{\prime}=\left\{\begin{array}{ll}P_{n+1}+a_{p}&\text{if }P_{m}\geq 0\\ P_{n}&\text{if }P_{m}<0,\end{array}\right. \tag{3}\]  
where the _curtailment margin_\(P_{m}=\frac{1}{2}(x_{p}+y_{p})+a_{p}\) has been introduced. Equation (3), applicable to the case \(y_{p}<-a_{p}\), can be restated in a compact form in terms of the power changes \(x_{p}\) and \(y_{p}\):  
\[x_{p}^{\prime}=x_{p}+(y_{p}+a)\theta(x_{p}+y_{p}+2a_{p}), \tag{4}\]  
where \(\theta(z)\) is Heaviside's theta function. This expression can be easily extended to the case of arbitrary \(y\)-values by applying the factor \(1-\theta(y_{p}+a_{p})\). Taking into account that a decrease in \(x_{p}\) translates into a corresponding increase in \(y_{p}\), we have  
\[\begin{array}{l}x_{p}^{\prime}=x_{p}-\Delta x_{p}=x_{p}+[(y_{p}+a_{p})\theta (x_{p}+y_{p}+2a_{p})](1-\theta(y_{p}+a_{p}))\\ y_{p}^{\prime}=y_{p}+\Delta x_{p}=y_{p}-[(y_{p}+a_{p})\theta(x_{p}+y_{p}+2a_{p} )](1-\theta(y_{p}+a_{p}))\end{array}, \tag{5}\]  
where the positive quantity \(\Delta x_{p}=-[(y_{p}+a_{p})\theta(x_{p}+y_{p}+2a_{p})](1-\theta(y_{p}+a_{p}))\) has been introduced. Equation (5) allows for a straightforward generation of a FBC-processed time series for the case of ideal forecasts (\(\hat{x}_{p}=x_{p}\) and \(\hat{y}_{p}=y_{p}\)). Analytical expressions for the FBC _compliance fraction_\(\rho_{\text{FBC}}\)  
\[\rho_{\text{FBC}}=\frac{N(y_{p}<-a_{p}\wedge y_{p}^{\prime}>-a_{p})}{N_{\text {all}}} \tag{6}\]  
and the _natural compliance_ fraction  
\[\rho_{0}=\frac{N(y_{p}\geq-a_{p})}{N_{\text{all}}} \tag{7}\]  
Figure 1: Basic idea of forecast-based curtailment: (**a**) Three events on the output power time line; \(P_{n-1}\) is the last measured event, \(P_{n}\) and \(P_{n+1}\) lie in the future. (**b**) Compliance areas in the \(x\)-\(y\)-plane, where the \(x_{p}=P_{n}-P_{n-1}\) and \(y_{p}=P_{n+1}-P_{n}\).  
were derived in Reference [20] for simple and generalized Laplace-type, as well as Cauchy-Lorentz power change distributions, under the assumption of vanishing correlation between subsequent power changes, and their validity was demonstrated with numerical simulations. \(N(.)\) is the number of events fulfilling the logical condition in brackets, and \(N_{\text{all}}\) is the total number of registered events (power changes). Laplace distributions were found to be a good fit to empirical 1-min power change distributions of an operating wind farm. It is also convenient to introduce the _FBC efficiency_\(\eta_{\text{FBC}}\) by  
\[\eta_{\text{FBC}}=\frac{\rho_{\text{FBC}}}{1-\rho_{0}}, \tag{8}\]  
which identifies the degree to which incompliance events are mitigated by the FBC method. In Reference [20], analytical expression for the FBC _curtailment cost_, i.e., the energy not served, were also derived. The energy cost of curtailing wind power based on forecasts for a given critical or _tolerable_ slope \(a\) was found to be very small, i.e., around two orders of magnitude smaller than the cost for _flat curtailment_, i.e., the permanent reduction of output power to a certain fraction of the potential output power at any given time step.