This thesis investigates the reliable operation of residential electricity distribution networks under stochastic loads arising from household demand, renewable energy, and electric vehicles. Because real-world systems are governed by nonlinear power flow equations, their analysis is complex. Linear approximations are often used for tractability, but their accuracy and implications require systematic evaluation. The work combines analytical methods, stochastic simulation, and large deviations theory to compare nonlinear and linear models.nnThe first part (Chapters 2–3) studies nonlinear Bus Injection and Distflow models, emphasizing feasibility regions—power consumptions that satisfy equations and voltage constraints. Chapter 2 develops polyhedral restrictions and a sequential optimization method for tree networks, while Chapter 3 analyzes the asymptotic behavior of feasibility regions in line networks as size increases.nnThe second part (Chapters 4–6) compares linearized and nonlinear models for performance. Chapter 4 applies queuing models to assess stability in line networks. Chapter 5 extends to tree networks, introducing a duality-based heuristic for feasibility checks and simulations to compare efficiency and capacity. Chapter 6 focuses on rare events, designing importance sampling strategies to estimate probabilities of voltage violations using both linear and nonlinear models.nnOverall, the thesis clarifies trade-offs between nonlinear accuracy and linear simplicity in managing uncertainty in distribution networks.