We study the privacy-cost trade-off in a smart meter (SM) system with a renewable energy source (RES) and a finite-capacity rechargeable battery (RB). Privacy is measured by the mutual information rate between the energy demand and the energy received from the grid, where the latter also determines the cost, and hence, reported by the SM to the utility provider (UP). We consider a renewable energy generation process that fully charges the RB at random time instants, and its realization is assumed to be known also by the UP. We reformulate the problem as a Markov decision process (MDP), and solve it by dynamic programming (DP) to design battery charging and discharging policies that minimize a linear combination of the privacy leakage and energy cost. We also propose a lower bound and two alternative low-complexity energy management policies, one of which is shown numerically to perform close to the MDP solution.