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During the last 15 years, the regulatory framework for the wholesale sector of the electricity industry has been replaced in many countries by market competition. In such an environment, electric firms compete through both spot market bidding and bilateral contract trading. Firms have to consider their forward contract positions when they make spot market decisions. In this paper, we propose a linear asymmetric supply function equilibrium (SFE) model with transmission constraints to develop firms' optimal bidding strategies considering forward contracts. The characteristics of firms' behaviors are analyzed under the induced equilibrium conditions. The model is applied to the real-time energy market of the Electric Reliability Council of Texas (ERCOT). The effects of forward contracts on the ERCOT market are evaluated. It is shown that the model is able to capture features of bidding behaviors in the ERCOT market.