Region of Attraction

The Stable Equilibrium Point is the target state to which the system must converge after a disturbance. For a post-fault power system, this is the desired operating condition where rotor angles and speeds settle to acceptable values. The ROA is always defined with respect to a specific SEP. If the fault-on trajectory pushes the system state outside the ROA of the post-fault SEP, the generators will not return to synchronism. The SEP is found by solving the post-fault power flow equations, and its stability is verified by checking the eigenvalues of the linearized system Jacobian.

The stability boundary is the hypersurface that separates the ROA from the region of instability. In nonlinear power system dynamics, this boundary is composed of the stable manifolds of unstable equilibrium points (UEPs) that lie on the boundary. Trajectories starting inside the boundary converge to the SEP; those starting outside diverge, leading to loss of synchronism. The closest UEP is the unstable equilibrium point with the lowest energy on the boundary and is often used to approximate the ROA in direct methods like the transient energy function approach.

The Controlling UEP is the specific unstable equilibrium point whose stable manifold is intersected by the fault-on trajectory. It is the critical exit point that determines the stability boundary relevant to a particular fault. Identifying the correct CUEP is essential for accurate transient stability assessment using direct methods. Key characteristics:

• Depends on the fault location, type, and duration
• Determines the critical energy for the disturbance
• Computation requires solving the post-fault gradient system
• Incorrect CUEP identification leads to overly optimistic or pessimistic stability predictions

Lyapunov-based energy functions provide a direct analytical approach to estimate the ROA without time-domain simulation. The transient energy function (TEF) captures both kinetic energy (rotor speed deviations) and potential energy (rotor angle displacements and network magnetic effects). The ROA is approximated as the set of states where the total energy is less than the critical energy at the CUEP. This method yields the closest UEP or potential energy boundary surface (PEBS) approximations, enabling fast stability screening for contingency analysis.

For an n-generator system, the ROA exists in a (2n-1)-dimensional state space (n-1 relative rotor angles and n relative speed deviations). This high dimensionality makes exact ROA computation intractable for large systems. Practical approaches include:

• Model reduction via generator coherency grouping
• Projection onto critical machine subspaces
• Koopman operator theory to lift dynamics into a linear space
• Sum-of-squares (SOS) programming for polynomial Lyapunov function synthesis
• Machine learning classifiers trained on simulated boundary points

The size and shape of the ROA are significantly affected by system parameters and control devices. Positive damping from generator damper windings and Power System Stabilizers (PSS) expands the ROA by dissipating oscillatory energy. Fast-acting exciters with high ceiling voltages can increase the critical clearing time by boosting synchronizing torque during faults. Conversely, negative damping from poorly tuned controls or series compensation can shrink the ROA and create subsynchronous resonance risks. Grid-forming inverters also reshape the ROA in low-inertia systems by providing synthetic inertia.