In this paper, we discuss the structure preserving iterative solutions of large scale sparse projected periodic discrete-time algebraic Lyapunov equations which arise in periodic state feedback control problems and in model reduction of periodic descriptor systems. We extend the idea of computing the generalized inverses of periodic discrete-time descriptor systems using the left and right deflating projectors associated with the eigenstructures of the periodic singular matrix pairs. The computed periodic inverses are then used in the Smith iterative method to compute the iterative solutions of the projected periodic discrete-time algebraic Lyapunov equations. A low-rank version of this method is also presented, which can be used to compute low-rank approximations to the solutions of projected periodic discrete-time algebraic Lyapunov equations. Moreover, we consider an application of the Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.