Stability & control
Transient stability, swing equation, equal-area criterion, state estimation, EMS workflow.
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).
Y-bus and power flow
Bus admittance matrix, slack / PV / PQ bus classification, nonlinear power-flow equations, and the Newton-Raphson / Fast Decoupled / DC solution methods. Foundation of SCADA, planning, and electricity markets.
Power-system state estimation — WLS, observability, bad data, PMU
WLS minimises J(x) = Σ (z_i − h_i(x))² / σ_i² via Gauss-Newton: (Hᵀ R⁻¹ H) Δx = Hᵀ R⁻¹ r. State = |V|, θ at every bus (n = 2N−1). Observable iff rank(H) = n. Bad-data: χ²(m−n) test + |r_N,i| > 3 normalized residual. PMU (IEEE C37.118) measures V and θ directly → linear estimator. EMS workflow: SCADA + PMU → SE → CA → SCED → market clearing. Vendors: GE iDM / Siemens Spectrum / ABB-Hitachi.
Optimal Power Flow & SCED — LMPs, DAM/RTM markets
OPF: min Σ c_i·P_g,i s.t. power balance + V band + line MVA + gen caps. DC OPF (linear, <1 s) vs AC OPF (nonlinear, 10 s-5 min). SCOPF adds N-1 contingencies via Benders / iterative screening. LMP_k = ∂J/∂P_k = energy + congestion (PTDF·μ_line) + losses. Two-settlement: DAM (unit commitment MILP) + RTM (SCED every 5-15 min). Frontier: stochastic, chance-constrained, AC convex relaxations (SDP/SOCP), ML-warm-start, FERC 2222 DER aggregation.
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.
Reliability indices — SAIDI / SAIFI / CAIDI / MAIFI / LOLE / EUE
Distribution (IEEE 1366): SAIFI = Σ N_i/N_total (sustained ≥ 5 min); SAIDI = Σ (U_i·N_i)/N_total; CAIDI = SAIDI/SAIFI; MAIFI for momentary. Typical US: SAIDI 60-300 min/yr; SAIFI 0.5-2.0. MED β-method excludes storms. Generation adequacy (NERC): LOLE ≤ 0.1 days/yr (1-in-10); EUE in MWh; PRM 15-20%. Capacity accreditation via ELCC. EEI benchmarking + PUC PBR. PSPS, microgrids, DERMS, resilience.