Block Principal Component Analysis (Block PCA) of a data matrix A, where loadings Z are determined by maximization of the froebonius norm of AZ over unit norm orthogonal loadings, is difficult to use for the design of sparse PCA by l1 regularization, due to the difficulty of taking care of both the orthogonality constraint on loadings and the non differentiable l1 penalty. Our objective in this paper is to relax the orthogonality constraint on loadings by introducing new objective functions expvar(Y) which measure the part of the variance of the data matrix A explained by correlated components Y = AZ. So we propose first a comprehensive study of mathematical and numerical properties of expvar(Y) for two existing definitions Zou et al. [2006], Shen and Huang [2008] and four new definitions. Then we show that only two of these explained variance are fit to use as objective function in block PCA formulations rid of orthogonality constraints.