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The factors affecting solar irradiance are usually complex and diverse, making it difficult to accurately predict the photovoltaic power generation. In this paper, an explainable recurrent neural network (ExRNN) algorithm is proposed based on deep recurrent neural network (RNN) and additive index model for solar irradiance forecasting problems. The proposed ExRNN is designed as an ante-hoc explainable algorithm with cyclic units by linearly combining single-feature models to learn explainable features of solar irradiances, and the ridge function is used as an activation function to extract and explain mapping correlations between meteorological features and solar irradiances. Furthermore, the RNN is used with memory characteristics to discover the time correlation hidden in the solar irradiance data sequence and retain the explainability. Therefore, the factors affecting solar irradiances can be quantified by the proposed ExRNN, and a legible explanation on the relationship between meteorological inputs and solar irradiances can be provided. Solar irradiance samples from Lyon France are used to evaluate the prediction accuracy and explainability of the proposed ExRNN.