Learned image compression has exhibited promising compression performance, but variable bitrates over a wide range remain a challenge. State-of-the-art variable rate methods compromise the loss of model performance and require numerous additional parameters. In this paper, we present a Quantization-error-aware Variable Rate Framework (QVRF) that utilizes a univariate quantization regulator a to achieve wide-range variable rates within a single model. Specifically, QVRF defines a quantization regulator vector coupled with predefined Lagrange multipliers to control quantization error of all latent representation for discrete variable rates. Additionally, a reparameterization method makes QVRF compatible with round quantizer and integer entropy coding. Exhaustive experiments demonstrate that existing fixed-rate VAE-based methods equipped with QVRF can achieve wide-range continuous variable rates within a single model without significant performance degradation. Furthermore, QVRF outperforms contemporary variable-rate methods in rate-distortion performance with minimal additional parameters. The code is available at https://github.com/bytedance/QRAF.