Computational speed and global optimality are a key need for pratical algorithms of the OPF problem. Recently, we proposed a tight-and-cheap conic relaxation for the ACOPF problem that offers a favourable trade-off between the standard second-order cone and the standard semidefinite relaxations for large-scale meshed networks in terms of optimality gap and computation time. In this paper, we show theoretically and numerically that this relaxation can be exact and can provide a global optimal solution for the ACOPF problem. Thereafter, we propose a multi-period tight-and-cheap relaxation for the multi-period ACOPF problem. Computational experiments using MATPOWER test cases with up to 500 buses show that this new relaxation is promising for real-life applications.