This paper develops deep reinforcement learning (DRL) algorithms for optimizing the operation of home energy system which consists of photovoltaic (PV) panels, battery energy storage system, and household appliances. Model-free DRL algorithms can efficiently handle the difficulty of energy system modeling and uncertainty of PV generation. However, discrete-continuous hybrid action space of the considered home energy system challenges existing DRL algorithms for either discrete actions or continuous actions. Thus, a mixed deep reinforcement learning (MDRL) algorithm is proposed, which integrates deep Q-learning (DQL) algorithm and deep deterministic policy gradient (DDPG) algorithm. The DQL algorithm deals with discrete actions, while the DDPG algorithm handles continuous actions. The MDRL algorithm learns optimal strategy by trial-and-error interactions with the environment. However, unsafe actions, which violate system constraints, can give rise to great cost. To handle such problem, a safe-MDRL algorithm is further proposed. Simulation studies demonstrate that the proposed MDRL algorithm can efficiently handle the challenge from discrete-continuous hybrid action space for home energy management. The proposed MDRL algorithm reduces the operation cost while maintaining the human thermal comfort by comparing with benchmark algorithms on the test dataset. Moreover, the safe-MDRL algorithm greatly reduces the loss of thermal comfort in the learning stage by the proposed MDRL algorithm.