🎓 Lesson 6
D4
Linearization and State-Space Representation
Linearization is a way to simplify a complex, curved system behavior into a straight-line approximation near a specific operating point, so we can analyze small changes easily.
🎯 Learning Objectives
- ✓ Calculate the Jacobian matrix of a given nonlinear power system model at its equilibrium point
- ✓ Construct the linearized state-space model (A, B, C, D matrices) from physical system equations
- ✓ Analyze eigenvalues of the A-matrix to determine small-signal stability (damping, oscillation modes)
- ✓ Explain how linearization assumptions limit applicability to small perturbations (< 5% deviation)
- ✓ Apply linearized models to design PSS (Power System Stabilizer) parameters using root-locus or modal analysis
📖 Why This Matters
In real power grids, generators, loads, and converters behave nonlinearly—but engineers need fast, reliable tools to predict whether a small disturbance (e.g., sudden load change or line trip) will die out or grow into instability. Linearization lets us transform messy nonlinear dynamics into tractable linear models, enabling eigenvalue analysis, modal participation, and robust controller design—critical for modern inverter-rich grids and black-start recovery planning.
📘 Core Principles
All physical power system components—synchronous machines, excitation systems, governors, and grid-forming inverters—are governed by nonlinear differential-algebraic equations (DAEs). Linearization begins by identifying an equilibrium point (steady-state solution), then computing partial derivatives of all state equations with respect to states and inputs—the Jacobian matrix. This yields the A-matrix (system dynamics), B-matrix (input influence), C-matrix (output mapping), and D-matrix (direct feedthrough), collectively forming the state-space model: dx/dt = Ax + Bu, y = Cx + Du. The eigenvalues of A determine local stability: negative real parts → stable; positive → unstable; complex conjugates → oscillatory modes.
📐 Jacobian-Based Linearization
The linearized system matrix A is obtained by evaluating the Jacobian of the state derivative vector f(x,u) at the equilibrium point (x₀, u₀): A = ∂f/∂x |_(x₀,u₀). For a two-machine system, this captures coupling between rotor angles, speeds, and internal voltages.
💡 Worked Example
Problem: Given a single-machine infinite-bus (SMIB) system with swing equation: dδ/dt = ω − ω₀, dω/dt = (Pₘ − Pₑ)/M, where Pₑ = E'·V·sin(δ)/X' = 1.2·1.0·sin(δ)/0.3 = 4.0 sin(δ), M = 5.0 s, Pₘ = 1.0 pu, ω₀ = 2π·60 rad/s. Find A-matrix at equilibrium δ₀ = arcsin(0.25) ≈ 0.2554 rad.
1.
Step 1: Compute equilibrium: Pₑ = Pₘ ⇒ 4.0 sin(δ₀) = 1.0 ⇒ sin(δ₀) = 0.25 ⇒ δ₀ ≈ 0.2554 rad; ω₀ = 0 (in per-unit speed deviation frame).
2.
Step 2: Define states x = [δ, Δω]ᵀ. Then f₁ = Δω, f₂ = (1.0 − 4.0 sin(δ))/5.0.
3.
Step 3: Compute Jacobian: ∂f₁/∂δ = 0, ∂f₁/∂Δω = 1; ∂f₂/∂δ = −(4.0 cos(δ))/5.0, ∂f₂/∂Δω = 0. At δ₀: cos(δ₀) ≈ 0.968 ⇒ ∂f₂/∂δ ≈ −0.774.
4.
Step 4: Assemble A = [[0, 1], [−0.774, 0]].
Answer:
The A-matrix is [[0, 1], [−0.774, 0]], with eigenvalues λ = ±j0.880 — indicating undamped 0.14 Hz electromechanical oscillation, requiring PSS intervention.
🏗️ Real-World Application
In the 2019 Western Interconnection stability study (NERC report TOP-003-4), linearized state-space models of 325+ generator swing equations and AVR/governor dynamics were used to identify critical inter-area modes (0.2–0.5 Hz) across the Pacific Northwest–Southern California tie-line. Eigenvalue sensitivity analysis guided placement of three wide-area PSS units, improving damping ratio from 2.1% to 14.7% — preventing potential forced oscillations during monsoon load surges.