Volt-var control (VVC) is essentially a non-convex optimization problem due to the non-convexity of power flow (PF) constraints, resulting in the difficulty in obtaining the optimum without convexity conversion. The existing second-order cone method for the convexity conversion often leads to a sharp increase in PF constraints and optimization variables, which in turn increases the optimization difficulty or even leads to optimization failure. This paper first proposes a deterministic VVC method based on convex deep learning power flow (DLPF). This method uses the input convex neural network (ICNN) to establish a single convex mapping between state parameters and node voltage to complete the convexity conversion while the optimization variables only correspond to reactive power equipment, which can ensure the global optimum with extremely fast computation speed. To cope with the impact brought by the uncertainty of distributed energy and omit the additional worst scenario search of traditional robust VVC, this paper proposes robust VVC method based on convex deep learning interval power flow (DLIPF), which continues to adopt ICNN to establish another convex mapping between state parameters and node voltage interval. Combining DLIPF with DLPF, this method decreases the modeling and optimization difficulty of robust VVC significantly. Test results on 30-bus, 118-bus, and 200-bus systems prove the correctness and rapidity of the proposed methods.