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This paper proposes two new approaches for transient stability analysis and they have the new ideas. These approaches use Athay's PEBS. The first method is called CTSA that it's complex simulation and Athay's PEBS. CTSA is more reliable than all existing methods that use unstable equilibrium point (UEP) as it doesn't use any convergence methods. POMP's method follows the point of maximum potential energy on post-fault system trajectory that this point is approximated to Taylor's expansion second orders. This paper presents a detailed implementation of a fast and accurate method for Available Transfer Capability (ATC) calculations. The authors use two new methods for termination criteria in ATC calculations. A novel formulation of the ATC problem has been adopted based on full AC power flow solution with matrix operations to incorporate the effects of voltage limits, and voltage collapse. This program written by MATLAB and does not use any do loop, then this power flow program is fivefold faster than any program in MATLAB. The ideas are demonstrated on 4. 7 and 30 bus IEEE systems.