Small-signal stability & PSS — eigenvalues, modes, wide-area damping

Reference notes

Small-signal stability is the ability of a power system to remain in synchronism following small disturbances. Method: linearize nonlinear differential-algebraic equations around the operating point; analyze eigenvalues of the state matrix. Damping ratio ζ < 5% is concerning; ζ < 0 is unstable. Power System Stabilizer (PSS) is the primary remediation device, in production since the 1960s. Wide-Area Damping Control (WADC) addresses inter-area modes that single PSS cannot fully damp. High-IBR grids face new instability classes (SSCI, SSTI).

Linearization and eigenvalue analysis

• Linearize the nonlinear differential-algebraic equations (DAEs) around the operating point.
• State matrix A typically has thousands of dimensions for a real-system model.
• Each eigenvalue is the system's natural mode:
  • Real eigenvalues: first-order modes (no oscillation). Stable if σ < 0.
  • Complex-conjugate pairs: second-order oscillatory modes.
• Natural frequency: f_n,k = ω_k / (2π) Hz.
• Damping ratio: ζ_k = −σ_k / sqrt(σ_k² + ω_k²).
• Target: ζ > 5-10% for all modes in normal operation; ζ < 0 (positive σ) is unstable.

Rotor mode categories

Historical examples

• 1970-71 Mohave Power Plant: SSR (sub-synchronous resonance) with a series-compensated 500 kV line destroyed turbine-generator shafts.
• 1996 WSCC August blackout: 0.7 Hz inter-area mode between Pacific Northwest and California became unstable, cascaded to system breakup.
• Northern Europe 0.3 Hz mode: between Nordic and Continental Europe via HVDC links.
• 2009 Texas Lubbock-Brownsville SSCI: DFIG wind farms + series-capacitor-compensated 345 kV line caused 25 Hz oscillations; multiple converter and line trips.

Power System Stabilizer (PSS)

• Adds a supplementary signal to the AVR (Automatic Voltage Regulator) voltage reference, modulating field excitation to provide a DAMPING TORQUE proportional to rotor speed deviation.
• Input signal: rotor speed deviation Δω, or accelerating power (ΔP_mech − ΔP_elec), or both via dual-input PSS.
• Transfer function: K · washout · two lead-lag stages · output limiter
  • Gain K_PSS: 10-30 typical.
  • Washout T_w = 5-20 s — high-pass filter; PSS responds to TRANSIENT speed changes only (removes DC bias).
  • Lead-lag stages T_1-T_4 = 0.05-1 s — compensate phase shift through AVR-excitation system at the target mode frequency.
  • Output limiter ±0.05-0.15 pu voltage — prevents AVR saturation.
• Boosts rotor-mode damping from 2-5% (no PSS) to 15-25% (well-tuned PSS).
• Standards: IEEE Std 421.5 defines PSS models — PSS1A (single-input), PSS2B (dual-input), PSS4B (multi-band, damps multiple modes simultaneously).
• Vendors: GE, Siemens, ABB, Hitachi, Basler — every major excitation supplier offers PSS-equipped digital AVR platforms.

PSS tuning workflow

1. Identify problematic modes: eigenvalue analysis flags modes with ζ < 5%.
2. Compute participation factors: p_ki = (right-eigenvector)_i × (left-eigenvector)_i for each generator i and mode k. Generators with p_ki > 0.1 are PSS-installation candidates.
3. Compute residues: R_ki = sensitivity of mode k's eigenvalue to a feedback control loop at generator i. Magnitude = max damping improvement; angle = required phase compensation.
4. Design PSS: choose K_PSS, T_w, and lead-lag T_1-T_4 to produce the required phase compensation at the mode frequency.
5. Verify: eigenvalue analysis on modified A matrix; time-domain simulation.
6. Field-test: inject small step or harmonic into AVR setpoint; record rotor swing response; verify damping matches design.

Tools

• PSS/E SSAT (Small-Signal Analysis Tool, Siemens)
• GE PSLF NEVA
• DSATools SSAT (Powertech Labs)
• DIgSILENT PowerFactory
• PowerWorld TSAT
• Mudpack (academic, University of Illinois)

Wide-Area Damping Control (WADC)

• Addresses INTER-AREA modes that no single local PSS can fully damp.
• Architecture:
  • PMUs at multiple buses across the inter-area corridor stream synchronized phasors at 30-60 Hz (IEEE C37.118.2).
  • Central controller computes the inter-area mode (FFT, Prony, modal-tracking algorithms) and applies a damping control law.
  • Modulates HVDC link power orders, FACTS device setpoints (SVC, STATCOM, TCSC), or PSS reference signals.
• Demonstrations:
  • Pacific DC Intertie (California-Oregon corridor) — HVDC modulation.
  • China Southern Grid — wide-area damping at HVDC links.
  • Brazil — wide-area system control.
  • ENTSO-E European interconnection — inter-area damping studies.
  • ERCOT — prototype PMU-based oscillation detection.
• Communication delay: 10-50 ms for fiber-optic substation-to-control-center links.

IBR-induced instability classes (new for high-renewable grids)

• SSCI — Sub-Synchronous Control Interaction: DFIG wind farms + series-capacitor-compensated lines → oscillations at sub-synchronous frequencies (10-50 Hz). 2009 Texas Lubbock-Brownsville incident. ERCOT now requires SSR-specific studies for any new series-capacitor near IBR concentrations.
• SSTI — Sub-Synchronous Torsional Interaction: HVDC link or FACTS device interaction with nearby turbine-generator shaft modes.
• Weak-grid PLL instability: grid-following inverters' PLL (phase-locked loop) interacts with grid impedance on weak grids (SCR < 3), producing 1-10 Hz oscillations.
• PLL-PLL interaction: multiple IBRs on the same bus interact through their PLL dynamics, destabilizing voltage at 10s-100s of Hz.

Grid-following vs Grid-forming inverters

• Grid-forming pilots: AEMO South Australia (Hornsdale Power Reserve battery), Hawaiian Electric Oahu, Eirgrid SOEF mode, ERCOT pilots.

Analysis tools for IBR stability

• Extended small-signal models including IBR control loops.
• Impedance-based stability analysis — Nyquist criterion on grid-IBR impedance ratio.
• EMT (Electro-Magnetic Transient) simulation in PSCAD or RTDS for millisecond-scale dynamics.
• Frequency-domain analysis for harmonic-stability issues in high-IBR grids.

Standards

• IEEE Std 421.5 — Excitation system models (PSS1A, PSS2B, PSS4B).
• IEEE C37.118.1 / C37.118.2 — Synchrophasor measurement and transmission.
• IEEE 2800-2022 — IBR interconnection requirements.
• IEEE P2800.2 — Grid-forming-specific requirements (in development).
• NERC PRC-024 — Generator off-nominal frequency capability.