This paper investigates the problem of a delay-dependent stability analysis in power systems, where the communication delay is modeled as a continuous, differentiable, aperiodic, and bounded function with constrained derivatives. First, a unified load frequency control system model is formulated that explicitly incorporates time-varying delays. Then, to more accurately address the delay characteristics, a monotonic interval partitioning strategy is introduced, which divides the delay trajectory into multiple segments based on its increasing or decreasing behavior. Each segment is associated with corresponding local extremal points of the delay function. Furthermore, to more effectively exploit the delay information, a new Lyapunov-Krasovskii functional (LKF) is developed, which integrates system states related to the local extremal values of the delay in each subinterval. By leveraging this LKF together with a convex combination method, less conservative stability conditions are derived. Finally, numerical case studies are presented to demonstrate the proposed approach's capability of enlarging the admissible delay bounds and validating its improved performance.