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Power inspection plays an important role in ensuring the normal operation of the power grid. However, inspection of transmission lines in an unoccupied area is time-consuming and labor-intensive. Recently, unmanned aerial vehicle (UAV) inspection has attracted remarkable attention in the space-ground collaborative smart grid, where UAVs are able to provide full converge of patrol points on transmission lines without the limitation of communication and manpower. Nevertheless, how to schedule UAVs to traverse numerous, dispersed target nodes in a vast area with the least cost (e.g., time consumption and total distance) has rarely been studied. In this paper, we focus on this challenging and practical issue which can be considered as a family of vehicle routing problems (VRPs) with regard to different constraints, and propose a Diverse Trajectory-driven Deep Reinforcement Learning (DT-DRL) approach with encoder-decoder scheme to tackle it. First, we bring in a threshold unit in our encoder for better state representation. Secondly, we realize that the already visited nodes have no impact on future decisions, and then devise a dynamic-aware context embedding which removes irrelevant nodes to trace the current graph. Finally, we introduce multiply decoders with identical structure but unshared parameters, and design a Kullback-Leibler divergence based regular term to enforce decoders to output diverse trajectories, which expands the search space and enhances the routing performance. Comprehensive experiments on five types of routing problems show that our approach consistently outperforms both DRL and heuristic methods by a clear margin.