In this paper, a serial distributed model predictive control (MPC) approach for connected automated vehicles (CAVS) is developed with local stability (disturbance dissipation over time) and multi-criteria string stability (disturbance attenuation through a vehicular string). Two string stability criteria are considered within the proposed MPC: (i) the l_∞-norm string stability criterion for attenuation of the maximum disturbance magnitude and (ii) l₂-norm string stability criterion for attenuation of disturbance energy. The l_∞-norm string stability is achieved by formulating constraints within the MPC based on the future states of the leading CAV, and the l₂-norm string stability is achieved by proper weight matrix tuning over a robust positive invariant set. For rigor, mathematical proofs for asymptotical local stability and multi-criteria string stability are provided. Simulation experiments verify that the distributed serial MPC proposed in this study is effective for disturbance attenuation and performs better than traditional MPC without stability guarantee.