As a domain-dependent prior knowledge, side information has been introduced into Robust Principal Component Analysis (RPCA) to alleviate its degenerate or suboptimal performance in some real applications. It has recently realized that the natural structural information can be better retained if the observed data is kept in the original tensor form rather than matricizing it or other order reduction means. Hence, studies on RPCA of tensor version have attracted more and more attentions. To share the merits from both direct tensor modeling and side information, we propose three models to deal with the problem of Tensor RPCA with side information based on tensor Singular Value Decomposition (t-SVD). To solve these models, we develop an efficient algorithm with convergence guarantee using the well-known alternating direction method of multiplier. Extensive experimental studies on both synthetic and real-world tensor data have been carried out to demonstrate the superiority of the proposed models over several other state-of-the-arts. Our code is released at https://github.com/zsj9509/TPCPSF.