Power system functionality is determined on the basis of the power system state estimation (PSSE). Thus, corruption of the PSSE may lead to severe consequences, such as financial losses, maintenance damage, and disruptions in electricity distribution. Classical bad data detection (BDD) methods, developed to ensure PSSE reliability, are unable to detect well-designed attacks, named unobservable false data injection (FDI) attacks. In this paper, we develop novel structural-constrained methods for the detection of unobservable FDI attacks, the identification of the attacked buses' locations, and PSSE under the presence of such attacks. The proposed methods are based on formulating structural, sparse constraints on both the attack and the system loads. First, we exploit these constraints in order to compose an appropriate model selection problem. Then, we develop the associated generalized information criterion (GIC) for this problem. However, for large networks, the GIC method's computational complexity grows exponentially with the network size. Thus, based on the proposed structural and sparse constraints, we develop two novel low-complexity methods for unobservable FDI attack identification: 1) a modification of the state-of-the-art orthogonal matching pursuit (OMP); and 2) a method that utilizes the graph Markovian property in power systems, i.e. the second-neighbor relationship between the power data at the system's buses. The methods' performance is evaluated on a IEEE-30 bus test case system.