## 3 Uncertainty Modeling Techniques
### Info-Gap Decision Theory (IGDT)  
IGDT has been increasingly applied in power systems to address uncertainties in various contexts. In [58], IGDT is used for day-ahead generation scheduling, helping mitigate uncertainties introduced by large-scale PV integration, where output variability affects system voltage and frequency regulation. The IGDT-based model provides a robust confidence region for decision-making under uncertainty. Similarly, in [59], IGDT is applied to peer-to-peer energy trading, incorporating EVs and renewable energy sources. The IGDT framework supports optimal risk management for both risk-averse and risk-seeking prosumers by modeling uncertain energy demand, improving the robustness of energy sharing strategies. In a study focusing on optimal allocation of clean energy resources in distribution networks [60], IGDT is employed to minimize power losses and enhance reliability amid generation and load uncertainties. The approach utilizes a risk aversion strategy, comparing favorably with Monte Carlo simulation. In the context of energy conservation, [61] employs IGDT to address the uncertainty of the DR programs aimed at reducing energy use. The theory enables the evaluation of DR programs based on their robustness to uncertainty, rather than relying solely on predicted outcomes, thus facilitating more reliable decision-making. Reference [62] applies IGDT to develop optimal bidding strategies for thermal units in day-ahead power markets. The method accounts for market price uncertainty, allowing generation stations to optimize their profits by adjusting their strategies based on their risk tolerance. Across these studies, IGDT proves to be a versatile  
Figure 7: Comparison of traditional optimization and RO. tool for managing uncertainty in power systems, from generation scheduling and energy trading to conservation efforts and market participation. The uncertainty in IGDT is modeled as an information gap between what is known and what needs to be known for making a well-informed decision. The uncertainty model can be expressed as  
\[U(q,\alpha)=\left\{q^{\prime}\::d\big{(}q,q^{\prime}\big{)}\ \leq\alpha\right\} \tag{55}\]  
where \(q\) is the nominal estimate of the uncertain parameter, \(q^{\prime}\) represents the actual but unknown value of the parameter, \(\alpha\) is the information-gap parameter representing the horizon of uncertainty, and \(d(.,. )\) is a distance measure quantifying the disparity between the nominal and actual values. The decision model in IGDT is typically framed around a performance or utility function \(f(x,q)\), where \(x\) denotes the decision variables and \(q\) denotes the uncertain parameters. The objective is to find a decision \(x\) that maximizes the robustness against the worst-case scenario within the horizon of uncertainty: \(\underset{x\in X_{d}}{\min}\underset{q^{\prime}\in U(q,\alpha)}{\max}f(x,q^{ \prime})\). This formulation seeks to maximize the minimum utility, ensuring that the decision remains viable under the worst deviations from the nominal scenario within the defined uncertainty horizon. The robustness function \(\hat{\alpha}(x,\rho)\) is defined as the largest horizon of uncertainty for which the worst-case outcome is still above a certain threshold (\(\rho\)) and can be shown as follows:  
\[\hat{\alpha}(x,\rho)=\max\left\{\alpha\ :\ \underset{q^{\prime}\in U(q, \alpha)}{\min}f\big{(}x,q^{\prime}\big{)}\ \geq\rho\right\} \tag{56}\]  
On the other hand, the opportunity function quantifies the potential for better-than-expected outcomes if the uncertainties resolve favorably. The opportunity function can be expressed as follows:  
\[\hat{\beta}(x,\tau)=\min\left\{\beta\ :\ \underset{q^{\prime}\in U(q, \beta)}{\max}f\big{(}x,q^{\prime}\big{)}\geq\tau\right\} \tag{57}\]  
where \(\tau\) represents a desired level of performance higher than \(\rho\). Table 1 shows the different applications of the uncertainty modeling techniques in the power systems.