Abstract: The alternating direction method of multipliers with linear convergence is generally used in distributed optimal dispatch of integrated electricity and gas systems. However, this method usually requires many iterations to solve the subproblems and is not computationally efficient. In addition, the alternating direction method of multipliers typically requires a coordination center to update the Lagrange multipliers. To this end, this paper proposes a fully distributed superlinear convergent optimal dispatch method without a coordination center to deal with the above two issues. The proposed method is based on the fully distributed interior point conjugate gradient method, which decouples the correction equations based on the interior point method and inherits the superlinear convergence of the interior point method. In this paper, the decoupled correction equations are processed to ensure that the coefficient matrix is symmetric positive definite so that they can be solved efficiently by the conjugate gradient method. The distributed conjugate gradient method is used to solve the decoupled modified equations in a fully distributed manner to avoid the introduction of a coordination center. Finally, the efficiency of the fully distributed interior point conjugate gradient method is verified in three systems of different sizes.