This paper addresses eco-driving of an electric vehicle driving in a hilly terrain under stochastic wind speed uncertainty. The eco-driving problem has been formulated as an optimisation problem, subject to road and traffic information. To enhance the computational efficiency, the dimension of the formulated problem has been reduced by appending trip time dynamics to the problem objective, which is facilitated by necessary Pontryagin's Maximum Principle conditions. To cope with the wind speed uncertainty, stochastic dynamic programming has been applied to solve the problem. Moreover, soft constraints on speed limits (kinetic energy) have been considered in the problem by enforcing sharp penalties in the objective. To benchmark the results, a deterministic controller has also been obtained with the aim of investigating possible constraints violations due to the wind speed uncertainty. For the proposed stochastic controller the optimised speed trajectories always remain within the limits and the violation on the trip time limit is only 8%. On the other hand, the speed and trip time constraints violations for the deterministic controller are 21% and 25%, respectively.