## 3 Uncertainty Modeling Techniques
### Latin Hypercube Sampling (LHS)  
LHS is a statistical method that improves upon random sampling by ensuring that the entire range of possible values is evenly sampled across all dimensions. This methodis highly efficient for estimating uncertainties in simulations with fewer samples and is commonly used in engineering and risk analysis. The uncertainty quantification of nuclear steam supply systems, particularly small pressurized water reactors (SPWRs) with once-through steam generators (OTSGs), has gained significant attention. A nonlinear mathematical model incorporating the reactor core, OTSG, and pressurizer models is established in [27]. A control strategy maintaining a constant reactor core coolant temperature and secondary-side outlet pressure of the OTSG was adopted. Leveraging LHS and statistical methods, they developed a quantitative platform for parameter uncertainty, showcasing its application on the primary-side flow rate of the OTSG under variable loads and step disturbances. Results indicated acceptable maximum uncertainties for SPWR critical output parameters. In [28], an overview of uncertainty and sensitivity analysis techniques is provided, emphasizing LHS within Monte Carlo analysis. The LHS has been applied in power systems to enhance reliability assessments and optimize system designs. In [29], LHS is integrated into an extendable Latin hypercube importance sampling (ELHIS) method to address the computational challenges of evaluating the reliability of power systems with renewable energy integration. By combining LHS with importance sampling, and leveraging cross entropy theory and Gaussian mixture models, the method improves the accuracy of sampling and reduces computational costs. The ELHIS approach is tested on a power system with real data from wind farms and photovoltaic stations, demonstrating faster and more efficient reliability assessments compared to traditional importance sampling methods. In [30], LHS is used alongside Grey Relational Analysis (GRA) for the optimization of a battery thermal management system. This study applies LHS to explore the design parameters of a liquid cooling system, such as mass flow rate and channel dimensions, to minimize pressure drop and temperature, while maximizing convective heat transfer. The optimized design, achieved through LHS and GRA, shows a significant improvement in cooling performance. LHS is a method that draws on the principles of stratified sampling. In LHS, stratification is applied solely to the marginal distributions. This involves dividing each random variable \(X\) into \(N\) consecutive intervals, each with an equal probability as defined by the cumulative distribution function \(F(X)\). To facilitate this, the unit interval [0,1] is split into \(N\) equal segments, each with a probability of \(1/N\). The boundaries of these segments are marked by lower (\(\phi_{k-1}\)) and upper (\(\phi_{k}\)) bounds, where the upper bound can be calculated as follows [31]:  
\[\phi_{k}=\frac{k}{N},\quad k=1,\dots,N \tag{31}\]  
The bounds for each interval across the values of \(X\) can be determined by the inverse of \(F(X)\) as follows:  
\[\xi_{k}=F^{-1}(\phi_{k}) \tag{32}\]  
One scenario, \(x_{k}\), must be chosen within each interval, ensuring that \(x_{k}\in(\xi_{k-1},\xi_{k})\). There are different variations of LHS, including random, mean, and median LHS, which differ in how the samples are selected within each probability interval. In median LHS, the sampling probabilities for each interval are defined as follows:  
\[P=(p_{1},p_{2},\dots,p_{k},\dots,p_{N}),\quad p_{k}=\frac{k-0.5}{N} \tag{33}\]  
The samples are then selected using the inverse function of these probabilities as follows:  
\[x_{k}=F^{-1}(p_{k}) \tag{34}\]  
For mean LHS, the method focuses on selecting the mean within each interval, which necessitates a numerical integration of the probability density function \(f(X)\) to compute these means.