## 3 Relevant Research in Wind Farm Design and Optimization
### Electromagnetic Interference (EMI)
#### 3.2.5 Uncertainty and Risk Modeling  
The works presented by [PERSON] _et al._[135, 310, 318] were the first to introduce the concept of uncertainty and risk within the scope of the WFDO problem. [PERSON] _et al._ pointed out that energy future prices, future variable costs, discount rate (e.g., accounting the time value of money) and wind resource estimation are the main factors contributing to the overall uncertainty in the profitability of wind farm projects. [PERSON] _et al._ studied the effects of the uncertainty on the wind resource (_i.e.,_ the wind direction and wind speed). They developed a probabilistic optimization methodology, which attempted to mimic the effects of both short and long-term variations in wind resource, where a set of possible scenarios and their probability of occurrence were taken into account during the calculation of the project expected NPV, as shown in Equation (58):  
\[\mathrm{E}[\mathrm{NPV}]=\sum\
olimits_{q=1}^{m}\mathrm{NPV}_{q}p_{q} \tag{58}\]  
where \(\mathrm{E}[\mathrm{NPV}]\) is the expected net present value of the wind farm, and \(p_{q}\) denotes the probability of occurrence of the scenario \(q\). An additional criterion was proposed, incorporating concepts of utility theory (UT), in which a utility function, as shown in Equation (59), was used to express the level of risk the wind farm designer is willing to assume given the different possible scenarios:  
\[\mathrm{u}\big{(}\mathrm{NPV}(x_{F})\big{)}= \begin{cases}\frac{1-\mathrm{e}^{ \frac{\mathrm{NPV}(x_{F})-\mathrm{NPV}_{\mathrm{min}}}{\rho_{R}}}}{1-\mathrm{e }^{\frac{\mathrm{NPV}_{\mathrm{max}-\mathrm{NPV}_{\mathrm{min}}}}{\rho_{R}}}},if\ \rho_{\mathrm{R}}\
eq\infty\\ \frac{\mathrm{NPV}(x_{F})-\mathrm{NPV}_{\mathrm{min}}}{\mathrm{NPV}_{\mathrm{ max}}-\mathrm{NPV}_{\mathrm{min}}},if\ \rho_{\mathrm{R}}\to\infty\end{cases}  \tag{59}\]  
where \(u\) is the utility value corresponding to \(\mathrm{NPV}(x_{F})\), \(x_{F}\) is the wind farm configuration, and \(\rho_{\mathrm{R}}\) is the parameter of risk tolerance, which allows modeling the risk attitude of the wind farm designer (e.g., \(\rho_{\mathrm{R}}<0\) increases risk level). \(\mathrm{NPV}_{\mathrm{min}}\) is the minimum value of NPV that the wind farm designer would be willing to accept to undertake the project. Therefore, the expected value of the utility function can be calculated using Equation (60):  
\[\mathrm{E}[\mathrm{u}(\mathrm{NPV})]=\sum\
olimits_{q=1}^{m}\mathrm{u}(\mathrm{ NPV}_{q})p_{q} \tag{60}\]  
[PERSON] _et al._[135] concluded that by increasing the risk attitude, the optimized wind farm layouts are more sensitive to changes in wind resource, however, the expected profitability increases. In addition, they concluded that the risk analysis (considering conservative risk attitude levels) led to wind farm configurations that are less sensitive to uncertainty in contrast with common deterministic approaches. Another interesting approach was provided by [PERSON] _et al._[134], who considered the effects of the uncertainty on the WT power curve and the uncertainty on the wind resource (wind speed and direction) during the calculation of the \(\mathrm{AEP}_{\mathrm{Farm}}\). The uncertainty in the \(\mathrm{AEP}_{\mathrm{Farm}}\) was translated into uncertainty in the NPV of the project, which was the maximized objective metric during the optimization procedure. They employed an embedded Monte Carlo simulation method to obtain the probability distribution of the NPV of the project (_i.e._, the approach maximized the NPV for a chosen certainty level). The input variables were given as probability density functions instead of expected values, which were assumed uncorrelated, to the objective function during the optimization process. The standard uncertainty of the experimental WT electrical power curve was assumed to follow a normal distribution along the discretized wind speed bins.