The location of solar parks far from load areas may lead to transmission congestion and thus solar curtailment for secure system operation. Battery energy storage (BES) Train as mobile storage can transmit solar energy from site to load centers using a transport network while relieving lines from congestion. Therefore, stochastic security-constrained unit commitment (SCUC) with BES Train is modeled in this paper under solar uncertainty and N-1 critical line contingency. Time-space network models transportation constraints of BES Train routing problem. To reduce the computational burden, this paper proposes modeling of line outage distribution factor (LODF) for DC power flow that reduces the nonzero coefficients of post-contingency network constraints. The benders decomposition technique simplifies stochastic optimization for the mixed-integer linear programming (MILP) problem. BES Train integrated modified IEEE reliability test and 118-bus system is investigated as a case study. Simulations evaluate the effect of BES Train, solar uncertainty and critical line contingency state with LODF on operation costs, BES Train scheduling, solar curtailment, congestion and computational time. The proposed model shows that BES Train is an economical option for large solar integration to reduce transmission congestion and curtail solar power effectively.