The present work proposes an original reduced order model for prediction of passive reduction of structural vibration by means of shunted piezoelectric patches. The problem consists of an elastic structure with surface-mounted piezoelectric patches. The piezoelectric elements are connected to a resonant shunt circuits in order to damp specific resonant frequencies of the structure. An efficient electromechanical finite element formulation for the dynamic analysis of the problem is first presented. The classical modal superposition techniques using the system eigenvectors with all patches short-circuited or open-circuited are then recalled. An advanced reduced order models using two new modal projection bases able to solve the problem at lower cost are developed: (i) the combined basis formed by both the short-circuited and open-circuited modes, and (ii) the coupled basis formed by the electromechanical modes that take into account the effect of the inductances of the electrical shunt circuits. Various numerical and experimental results are presented in order to validate and illustrate the efficiency of the proposed new finite element reduced order formulations.