In this paper, we propose a rate control (RC) scheme with a linear model in versatile video coding (VVC) intercoding, our method introduces a compromise global Lagrange multiplier to minimize the distortion induced under group-of-pictures (GOP)- and frame-level bit budget constraints. To obtain the optimal solution, the corresponding problem is transformed into a convex optimization problem. Then, the Karush-Kuhn-Tucker (KKT) conditions are used to obtain the optimal quantization parameter (QP). The experimental results show that our RC algorithm achieves better coding efficiency, a better RC effect, and higher subjective quality than the default algorithm in VVC Test Model (VTM) 17.0 and other state-of-the-art algorithms.