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In today's transmission networks (TNs), the sudden failure of some generation units (GUs) from the generation cycle can cause some problems such as the imbalance between generation and demand-side (consumption) leading to increasing the cost of operating the power system. On the other hand, distributed generators (DGs) in distribution networks (DNs) can be a relatively good alternative to-the-disconnected GUs with their ability to react quickly. Therefore, robust cooperation-for integrated TN&DN is essential. However, it is impossible to solve the cooperation of TN&DN problem without considering the AC optimal power flow (ACOPF), the operators' independence, and the information privacy. Thus, in this paper, a linearized ACOPF model for the robust cooperation of TN&DN problem is presented along with an efficient hierarchical decentralized solution method to solve the robust cooperation of integrated TN&DN in which the information-privacy-is-maintained-for both network operators. In this paper, the numerical simulations are analyzed for standard IEEE 30/118-bus transmission and IEEE 33/69-bus distribution networks. Finally, the paper validates through simulations the accuracy and computational-efficiency-of the proposed robust optimization problem and hierarchical decentralized method. [ABSTRACT FROM AUTHOR]