In today's cyber-enabled smart grids, high penetration of uncertain renewables, purposeful manipulation of meter readings, and the need for wide-area situational awareness, call for fast, accurate, and robust power system state estimation. The least-absolute-value (LAV) estimator is known for its robustness relative to the weighted least-squares one. However, due to nonconvexity and non-smoothness, existing LAV solvers based on linear programming are typically slow and, hence, inadequate for real-time system monitoring. This paper, develops two novel algorithms for efficient LAV estimation, which draw from recent advances in composite optimization. The first is a deterministic linear proximal scheme that handles a sequence of (5 ~ 10 in general) convex quadratic problems, each efficiently solvable either via off-the-shelf toolboxes or through the alternating direction method of multipliers. Leveraging the sparse connectivity inherent to power networks, the second scheme is stochastic and updates only afew entries of the complex voltage state vector per iteration. In particular, when voltage magnitude and (re)active power flow measurements are used only, this number reduces to one or two regardless of the number of buses in the network. This computational complexity evidently scales well to largesize power systems. Furthermore, by carefully mini-batching the voltage and power flow measurements, accelerated implementation of the stochastic iterations becomes possible. The developed algorithms are numerically evaluated using a variety of benchmark power networks. Simulated tests corroborate that improved robustness can be attained at comparable or markedly reduced computation times for medium- or large-size networks relative to existing alternatives.