Accurate load forecasting is critical for electricity production, transmission, and maintenance. Deep learning (DL) model has replaced other classical models as the most popular prediction models. However, the deep prediction model requires users to provide a large amount of private electricity consumption data, which has potential privacy risks. Edge nodes can federally train a global model through aggregation using federated learning (FL). As a novel distributed machine learning (ML) technique, it only exchanges model parameters without sharing raw data. However, existing forecasting methods based on FL still face challenges from data heterogeneity and privacy disclosure. Accordingly, we propose a user-level load forecasting system based on personalized federated learning (PFL) to address these issues. The obtained personalized model outperforms the global model on local data. Further, we introduce a novel differential privacy (DP) algorithm in the proposed system to provide an additional privacy guarantee. Based on the principle of generative adversarial network (GAN), the algorithm achieves the balance between privacy and prediction accuracy throughout the game. We perform simulation experiments on the real-world dataset and the experimental results show that the proposed system can comply with the requirement for accuracy and privacy in real load forecasting scenarios.