Linear fuzzy clustering-induced local PCA can have better interpretability than statistic-induced ones. The interpretability and stability have been demonstrated to be improved by considering cluster separation in the FCM clustering context. In this paper, with the goal of further improving the interpretability and stability of local PCA models, a novel linear fuzzy clustering model is proposed by introducing the cluster separation principle. Besides line-shape prototypes, 2-D plane-like linear prototypes and other linear varieties are adopted in conjunction with multi-dimensional local principal component scores. In order to consider mutual separation of linear cluster prototypes, the standard PCA criterion of point-wise information loss is replaced with the element-wise lower-rank approximation measure. Then, the novel clustering criterion is optimized by minimizing within-cluster errors and by maximizing intra-cluster PC score deviations and inter-cluster separation. Experimental results demonstrate that the proposed algorithm is useful for improving the initialization sensitivity of linear clustering with multi-dimensional prototypes.