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The electric power grid is a crucial infrastructure in our society and is always a target of malicious users and attackers. In this paper, we first introduce the concept of unidentifiable attack, in which the control center cannot identify the attack even though it detects its presence. Thus, the control center cannot obtain deterministic state estimates, since there may have several feasible cases and the control center cannot simply favor one over the others. Given an unidentifiable attack, we present algorithms to enumerate all feasible cases, and propose an optimization strategy from the perspective of the control center to deal with an unidentifiable attack. Furthermore, we propose a heuristic algorithm from the view of an attacker to find good attack regions such that the number of meters required to compromise is as few as possible. We also formulate the problem that how to distinguish all feasible cases if the control center has some limited resources to verify some meters, and solve it with standard algorithms. Finally, we briefly evaluate and validate our enumerating algorithms and optimization strategy.