Electricity forecasting has important implications for the key decisions in modern electricity systems, ranging from power generation, transmission, distribution and so on. In the literature, traditional statistic approaches, machine-learning methods and deep learning (e.g., recurrent neural network) based models are utilized to model the trends and patterns in electricity time-series data. However, they are restricted either by their deterministic forms or by independence in probabilistic assumptions -- thereby neglecting the uncertainty or significant correlations between distributions of electricity data. Ignoring these, in turn, may yield error accumulation, especially when relying on historical data and aiming at multi-step prediction. To overcome these, we propose a novel method named Probabilistic Electricity Forecasting (PrEF) by proposing a non-linear neural state space model (SSM) and incorporating copula-augmented mechanism into that, which can learn uncertainty-dependencies knowledge and understand interactive relationships between various factors from large-scale electricity time-series data. Our method distinguishes itself from existing models by its traceable inference procedure and its capability of providing high-quality probabilistic distribution predictions. Extensive experiments on two real-world electricity datasets demonstrate that our method consistently outperforms the alternatives.