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N-1 static security analysis (SSA) is an important method for power system stability analysis that requires solving N alternating-current power flows (ACPF) for a system with N elements to obtain strictly accurate results. Past researches have shown the potential of accelerating these calculations using an iterative solver with graphics processing unit (GPU). This paper proposes a GPU-based N-1 SSA algorithm with the preconditioned conjugate gradient (PCG) method. First, a shared preconditioner is selected to accelerate preprocessing of the iterative method for fast decoupled power flow (FDPF) in N-1 SSA. Second, it proposes a GPU-based batch-PCG solver, which packages a massive number of PCG subtasks into a large-scale problem to achieve a higher degree of parallelism and better coalesced memory accesses. Finally, the paper presents a novel GPU-accelerated batch-PCG solution for N-1 SSA. Case studies on a practical 10828-bus system show that the GPU-based N-1 SSA algorithm with the batch-PCG solver is 4.90 times faster than a sequential algorithm on an 8-core CPU. This demonstrates the potential of the GPU-based high-performance SSA solution with the PCG method under a batch framework.