In power industries, it is a common practice to use an equivalent model of the external power network in an optimization model of the internal power network in order to protect data privacy. However, existing equivalent models cannot accurately consider the operational constraints of an external power network. In this paper, a unified equivalent model is proposed to precisely capture the transfer capacity region of tie lines. The operational constraints of the external power network are preserved via multi-parametric programming. The operational costs of the external power network are formulated as an explicit function of power transmission on tie lines. This unified equivalent model can consider both AC and DC tie lines. The intuitive difference between the transfer capacity of AC and DC tie lines is revealed from a new perspective. On this basis, a market model for the internal power network with the equivalence of the operational constraints for the external power network is presented. Two typical trading modes are considered: the bulk sale mode and the direct power purchase mode. The exact convex relaxation formulation of the market model is derived under the direct power purchase mode.