This paper proposes an evolutionary game-theoretic model of massive distributed renewable energy deployment in order to shed light on the self-organization sustainable developments of renewable energies in distribution networks towards low-carbon targets. Since neighboring buses can interact in terms of energy exchanges, the return matrices of individual buses in the evolutionary game are established based on profiles of loads and renewable energy generation. More specifically, an evolutionary strategy is proposed based on the return matrices for individual buses to determine whether or not to deploy renewable energies in the next round of the game. Then, a dynamic model is derived for analyzing the renewable energy penetration rate in the distribution network throughout the multi-round evolutionary game. In theory, this model can reveal the self-organization process of renewable energy deployment in the distribution network. With this model, the distribution network operator would be aided in designing the incentives for buses deploying renewable energies toward a pre-defined low-carbon target. Numerical results on an actual 141-bus system and a synthetic 2000-bus system have demonstrated the validity and efficiency of the proposed model.