# Applications of Probabilistic Forecasting in Smart Grids: A Review
## 6 Decision Making under Uncertainties
### Chance-Constrained Programming  
Chance-constrained optimization problems aim to give an optimization constraint the possibility to be satisfied up to a specified level. It can be formulated as follows:  
\[\begin{split}\min_{x}&\mathbb{E}(f(x,\omega))\\ & Subject\ to:\\ & g(x,\omega)=0\\ &\Pr\left[h(x,\omega)\leq 0\right]\geq\eta\end{split} \tag{17}\]  
where Pr refers to probability and \(\eta\) indicates the confidence level [120]. In this regard, \(\eta\) should be selected between 0 and 1. According to (16), constraint \(h(x,\omega)\leq 0\) needs to be satisfied up to \(\eta\) level. In other words, operators or designers that use the chance-constrained method ensure that \(h(x,\omega)\leq 0\) will be satisfied in \((\eta\times 100)\%\) of scenarios.  
Chance-constrained programming has a wide range of applications in smart grids. Here are some examples:  
1. Operations of renewable-based systems to guarantee a certain level of reliability/security/flexibility
2. Planning of distribution/transmission networks in a way to guarantee a certain level of reliability/security/flexibility
3. Determining system reserves to guarantee a certain level of reliability/security/flexibility  
Table 8 summarizes some recent works that adopt chance-constrained programming methods to capture uncertainties.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ref.** & **Uncertain Parameters** & \
\begin{tabular}{c} **Methods to Capture** \\ **Uncertainties** \\ \end{tabular} &
\begin{tabular}{c} **Objective** \\ \end{tabular} \\ \hline
[121] & \begin{tabular}{c} Electricity prices and \\ PV generation \\ \end{tabular} & \
\begin{tabular}{c} Chance-constrained \\ programming \\ \end{tabular} &
\begin{tabular}{c} Minimizing the costs of energy trading between the power grid and \\ microgrids \\ \({}^{+}\) \\ Minimizing the fuel costs of fuel-based power generation and boiler units \\ \end{tabular} \\ \hline
[122] & \begin{tabular}{c} PV generation \\ \end{tabular} & \
\begin{tabular}{c} Chance-constrained \\ programming \\ \end{tabular} &
\begin{tabular}{c} The study aims to integrate renewable energy as much as possible by \\ minimizing the hybrid system’s power curtailment \\ \({}^{+}\) \\ Maximizing the renewable power generation injected into the system \\ \end{tabular} \\ \hline
[123] & \begin{tabular}{c} Loads, market prices, \\ renewable generation \\ \end{tabular} & \
\begin{tabular}{c} Chance-constrained \\ programming \\ \end{tabular} &
\begin{tabular}{c} Minimizing the costs of power and gas purchased \\ \({}^{+}\) \\ Minimizing the operation costs of microgrid’s CHP units and micro turbine \\ \end{tabular} \\ \hline
[124] & \begin{tabular}{c} Loads, wind generation \\ \end{tabular} & \
\begin{tabular}{c} Chance-constrained \\ programming \\ \end{tabular} &
\begin{tabular}{c} Minimizing the costs of buying power from thermal units \\ \({}^{+}\) \\ Minimizing the costs of buying spinning reserves from generators as well as \\ demand-response resources \\ \end{tabular} \\ \hline
[125] & \begin{tabular}{c} Renewable generations \\ \end{tabular} & \
\begin{tabular}{c} Chance-constrained \\ programming \\ \end{tabular} &
\begin{tabular}{c} Minimizing the costs of buying power from the upstream grid \\ \({}^{+}\) \\ Maximizing the revenue obtained from selling electricity to the \\ upstream grid \\ \end{tabular} \\ \hline
[126] & \begin{tabular}{c} Operational modes of \\ the microgrids \\ \end{tabular} & \
\begin{tabular}{c} Chance-constrained \\ stochastic conic \\ programming \\ \end{tabular} &
\begin{tabular}{c} Solving multi-site microgrids’ investment problem and microgrid \\ dual-mode network operations by minimizing microgrid operation and \\ maintenance costs, microgrid electricity transaction costs, its network loss \\ costs, and microgrid load curtailment costs \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 8: An overview of recent papers using chance-constrained programming for decision making in a smart grid environment.