H. Power system stability
Transient stability (equal-area criterion), voltage stability (PV curve), frequency response (AGC), small-signal stability (PSS).
Lessons
Transient stability — equal-area criterion
Swing equation H·δ̈ = P_m − P_e. Pre/during/post-fault P-δ curves. Equal-area criterion A1 = A2. Critical clearing time t_cr. Methods to improve stability (fast clearing, fast valving, PSS).
Voltage stability — P-V curve, nose point, collapse cascade
P-V curve: V vs P at fixed pf; nose at dP/dV = 0 = max loadability. Margin = P_max − P_op > 5-10% normal, 2.5-5% post-contingency (NERC TPL-001). Cascade 1-10 min: V drops → OLTC boosts → more Q drawn → V drops more → induction-motor I rises → gen excitation limit → collapse. CPF (continuation power flow) traces past the nose. STATCOM > cap bank (Q ∝ V² collapses). PRC-022 UVLS, IEEE 1547 DER, DSA real-time monitoring.
Frequency response & AGC — inertia, droop, ACE, three response layers
Swing eq: 2H·dω/dt = ΔP_pu → ROCOF = ΔP/(2·H_sys·S_base). Four layers: inertial (0-5 s) → primary droop ΔP_i=(1/R_i)·Δf_pu, R=5% (1-30 s) → secondary AGC drives ACE=ΔP_tie+β·Δf to 0 (1-15 min) → tertiary SCED (15 min+). High-IBR grids (H_sys=1-2 s): 3-5× higher ROCOF; mitigate with sync condensers + GFM inverters + FFR batteries. NERC BAL-001/002/003 + PRC-006 UFLS + PRC-024 + IEEE 1547.
Small-signal stability & PSS — eigenvalues, modes, wide-area damping
Linearise DAEs → A · Δx = Δẋ → eigenvalues λ = σ ± jω → damping ζ = −σ/sqrt(σ²+ω²); target ζ > 5-10%. Three rotor modes: local (1-3 Hz), inter-area (0.1-0.7 Hz, most problematic), torsional (10-50 Hz, SSR/SSCI). PSS: K·washout·lead-lag·limiter on AVR ref → boosts damping 2-5% → 15-25%. Tune via participation factors + residues. WADC for inter-area via PMU + HVDC/FACTS modulation. IBR: SSCI (Texas 2009), grid-forming inverters per IEEE 2800.