Multi-connectivity ( $MC$ ) is proposed in Fifth-Generation mobile communications systems ( $5G$ ) to mitigate the deterioration of Quality-of-Service owing to line-of-sight blockage and lack of communication resources. The main idea of MC is to associate single user equipment with multiple network layers and multiple radio access technologies, simultaneously. In previous studies, MC is demonstrated to increase capacity, provide high reliability and decrease outage probability. However, for the first time in literature, in this letter, we incorporate MC into the optimization of the total power consumption in 5G Heterogeneous Cloud Radio Access Networks ( $HCRAN$ )s. Upon formulation of the problem as a binary integer linear programming problem, proving its NP-hardness, a heuristic algorithm is proposed consisting of linear programming relaxation and rounding and generalized assignment problem heuristic, which takes the outputs of the linear programming relaxation and rounding as inputs and checks if it further minimizes the power consumption. We analyze power consumption, time complexity and achievable rate of the proposed algorithm and verify its superiority over existing methods by simulations.