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The increasing penetration of electric vehicles (EV) and fast charging stations (FCS) is tightly coupling the operation of power and transportation systems. In this context, the characterization of the EV flows and charging demand in response to varying traffic conditions and coordinated optimization strategies play a vital role. Previous work on the computation of stochastic traffic user equilibrium (TUE) involve non-linearities in the traffic link and FCS congestion representations and are generally inefficient in dealing with the multi-source uncertainties associated with the operating conditions of the traffic network (TN) and power distribution network (PDN). To address this, this paper proposed a novel deep learning (DL) based surrogate modeling method, leveraging the strength of edge-conditioned convolutional network (ECCN) and deep belief network (DBN). ECCN enables automatic extraction of spatial dependencies, taking into account both node and edge features characterizing the operation of TN. DBN leverages the value of the extracted features and achieves an accurate mapping between the latter to the EV charging demand and EV flows in the TUE, while adaptively generalizing to the multi-dimensional uncertainties. Case studies on three test systems of different scales (including a real-world case involving the matched TN and PDN of Nanjing city) demonstrate that the proposed surrogate model achieves a higher solution accuracy with respect to the state-of-the-art DL-based methods, and exhibits favorable computational performance. Quantitative results also corroborate the benefits brought by the proposed coordinated spatial optimization of EV flows and charging demand on the operation of both TN and PDN. • Coordinated optimization of EV flows and charging demand is investigated. • A novel deep learning based surrogate modeling framework is proposed. • Edge-conditioned convolutional network is employed to extract spatial dependencies. • Deep belief network forecasts the EV charging demand with generalization capability. • Significant techno-economic benefits for the operation of network nexus. [ABSTRACT FROM AUTHOR]