The volatility of cloud movement introduces significant uncertainty in short-term solar power forecasting, which can complicate the operation of modern power systems. This work proposes a specialized convolutional neural network (CNN) "SUNSET" to predict 15-min ahead minutely-averaged PV output. The model is characterized by its usage of hybrid input, temporal history and strong regularization. On a 1-year database, the "baseline" model achieves 16.3% forecast skill in cloudy conditions and 15.7% in all weather conditions, relative to a smart persistence forecast. Optimal input and output configurations are explored and suggestions are given. In terms of input, both sky images and PV output history are found to be crucial. Output-wise, training against PV output significantly outperforms using clear sky index (CSI). Careful down-sampling can reduce the training time by as much as 83% without affecting accuracy. For lag term configurations, using the same length of history as the forecast horizon is a good heuristic, while using slightly shorter history yielded a modest 0.5-0.9% improvement in this case. To ensure reproducibility and facilitate future works, the code base of this work is available at Github/YuchiSun/SUNSET.