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Accurately predicting carbon trading prices using deep learning models can help enterprises understand the operational mechanisms and regulations of the carbon market. This is crucial for expanding the industries covered by the carbon market and ensuring its stable and healthy development. To ensure the accuracy and reliability of the predictions in practical applications, it is important to evaluate the model's robustness. In this paper, we built models with different parameters to predict carbon trading prices, and proposed models with high accuracy and robustness. The accuracy of the models was assessed using traditional survey indicators. The robustness of the CNN-LSTM model was compared to that of the LSTM model using Z-scores. The CNN-LSTM model with the best prediction performance was compared to a single LSTM model, resulting in a 9% reduction in MSE and a 0.0133 shortening of the Z-score range. Furthermore, the CNN-LSTM model achieved a level of accuracy comparable to other popular models such as CEEMDAN, Boosting, and GRU. It also demonstrated a training speed improvement of at least 40% compared to the aforementioned methods. These results suggest that the CNN-LSTM enhances model resilience. Moreover, the practicality of using Z-score to evaluate model robustness is confirmed. • Deep learning by combined CNN-LSTM networks for carbon price forecasting. • Hyperparameter tuning of deep learning models for improving forecast accuracy. • CNN-LSTM models outperform single CNN models and single LSTM models. • Performance metric of deep learning for abrupt changes in carbon price variation. [ABSTRACT FROM AUTHOR]