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This paper is devoted to reduce the conservatism of distributionally robust optimization with moments information. Since the optimal solution of distributionally robust optimization is required to be feasible for all uncertain distributions in a given ambiguity distribution set and so the conservatism of the optimal solution is inevitable. To address this issue, we introduce the globalized distributionally robust counterpart (GDRC) which allows constraint violations controlled by functional distance of the true distribution to the inner uncertainty distribution set. We obtain the deterministic equivalent forms for several GDRCs under the moment-based framework. To be specific, we show the deterministic equivalent system of inequalities for the GDRCs under second order moment information with a separable distance function and a jointly convex distance function, respectively. Moreover, the feasible set of the system is convex. We also develop the deterministic equivalent inequality for the GDRC under first order moment and support information. The computationally tractable examples are presented for these GDRCs. A numerical tests of a portfolio optimization problem is given to show the efficiency of our methods and the results demonstrate that the globalized distributionally robust solutions is non-conservative and flexible compared to the distributionally robust solutions.