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Power grid fault diagnosis methods based on analytical models possess advantages such as logical rigor, strong interpretability, and high applicability. However, the existing design of their objective functions is presented as a 0–1 integer nonlinear programming model, making it difficult to approximate the optimal solution. To address this issue, this paper proposes an improvement to the expression of the second backup protection expectation involved in the diagnostic model. Thus, the order of the objective function is successfully reduced, transforming the original 0–1 integer nonlinear programming model into a constrained 0–1 integer linear programming model, which can be efficiently solved by typical commercial solvers based on a linear integer programming framework. Numerical results demonstrate that the solution time of the proposed linearized model is lower than that of existing nonlinear programming models, and the diagnostic accuracy of the proposed model is higher than that of other state-of-the-art methods.