The energy-efficient power allocation (PA) of distributed antenna system using multiple receive antennas is studied over spatially correlated Rayleigh channels. A new energy efficiency (EE) defined as the ratio of the average transmission rate to the consumed total power is presented. According to this definition, subject to the maximum power constraint, an optimal PA scheme is developed by maximising the EE. It is shown that the solution of PA in the constrained EE optimisation does exist and is unique. A Newton method with iterative algorithm is proposed to attain the PA. To avoid iterative calculation of the optimal scheme, a suboptimal PA scheme is derived by means of the Lambert function. As a result, closed-form PA is obtained. The developed schemes only need the statistical channel information and large scale information, and thus they have lower overhead and better robustness. Moreover, they may include the ones over spatially independent channel as special cases. The theoretical EE is further derived for performance evaluation, and the resultant closed-form expression is obtained. Computer simulation indicates that the theoretical EE agrees well the simulation result, and the suboptimal PA scheme can achieve the EE value near to the optimal one with lower complexity.