The accuracy of power system state estimation (PSSE), its robustness against bad data and the speed of its algorithm are crucial to economic and secure system operation. On the other hand, observability and redundancy considerations mandate PSSE to take advantage of traditional supervisory control and data acquisition (SCADA) measurements along with available phasor measurement unit (PMU) measurements. This set of hybrid PMU/SCADA inputs has traditionally made the problem formulation non-linear, and hence time-consuming to solve due to the iterative process of solution. This study addresses the foregoing challenges by proposing a novel linear least-absolute-value (LAV) estimation, without the need for an initial guess of the system state. The linearity of the proposed PSSE formulation is guaranteed regardless of whether PMU-only, SCADA-only or hybrid SCADA/PMU measurements are utilised. This facilitates the fast and non-iterative solution of the LAV estimation of system state based on linear programming. The LAV estimator outperforms the weighted-least-squares estimator in dealing with erroneous measurements, by automatically rejecting bad data of any size. An extensive number of simulation studies carried out on test systems of different sizes confirm the superiorities of the proposed method in comparison with other existing PSSE methods.