In this paper, we propose a simplified cubic polynomial R-D model with corresponding rate control methods for Versatile Video Coding (VVC) intra frame coding. First, we explore the rate-distortion (R-D) characteristics of VVC intra coding. By comparing several potential R-D modeling approaches, a new intra coding R-D model has been proposed based on the simplified cubic polynomial function. Subsequently, we derive the corresponding $R-\lambda$ model and introduce a complexity measurement to improve the performance of intra frame rate control. Furthermore, we propose a Coding Tree Unit (CTU)-level rate control method based on the newly proposed R-D model and further develop a pre-compression-based approach on this basis. Experimental results show that the proposed method can achieve 1.87% and 0.55% bit rate reduction for All-Intra (AI) and Random-Access (RA) configurations over the original rate control in VVC Test Model (VTM), while the computational complexity increment is negligible. Meanwhile, the enhanced bit rate accuracy from rate control has been observed in the proposed methods.