An Integrated EMT Small-Signal Stability Analysis Tool for Power Systems with High Inverter-Based Resource Penetration

II. EMT-LEVEL SMALL-SIGNAL MODEL FORMULATION

• The full-order model represents every generator, inverter, transmission line, and load with its own dynamic states.It is linearized about the operating point supplied by the extended power flow.Source: §II · p. 2

A. Device Modeling in a Common Reference Frame

• Each synchronous generator and inverter is described by nonlinear differential equations in its own d–q reference frame.The device state vector, terminal-bus voltage, common-frame frequency, and injected current are identified in the formulation.Source: A. Device Modeling in a Common… · p. 2
• The synchronous-generator model uses round-rotor machine dynamics with an IEEEG1 governor and lead–lag exciter.Its rotor and flux dynamics follow the standard formulation.Source: A. Device Modeling in a Common… · p. 2
• Grid-following and grid-forming inverters include phase-locked-loop or power-synchronization dynamics, controller loops, and LCL-filter states.Their full equations are given in.Source: A. Device Modeling in a Common… · p. 2
• Device models are rotated from their individual frames into a single common D–Q frame.The rotation uses the angle between the device frame and common frame.Source: A. Device Modeling in a Common… · p. 2
• The common-frame referral allows individual device models to be interconnected through the network.The angle δ is defined as the angle between the device frame and the common frame.Source: A. Device Modeling in a Common… · p. 2

B. Network and Load Modeling

• Each transmission line is a series R–L branch with inductor-current states, while each load is a constant-impedance R–L element with dynamic states.The line and load subsystems are linearized about the operating point and aggregated.Source: B. Network and Load Modeling · p. 2
• The aggregated line and load state-space model relates current deviations to terminal-voltage and common-frame-frequency deviations.The passage identifies the line and load current deviations and the corresponding state and input matrix coefficients.Source: B. Network and Load Modeling · p. 2

C. System Assembly and Linearization

• The extended power flow determines controller steady states as well as bus voltages and power injections, with system frequency as an additional variable.Buses are classified by connected-device control mode, and grid-forming buses regulate voltage and frequency.Source: C. System Assembly and Linearization · p. 2
• The network couples generator, line, and load subsystems, with interconnection matrices including virtual resistances for numerical conditioning.The system assembly substitutes the network relation into the device, line, and load models to eliminate terminal-voltage deviations.Source: C. System Assembly and Linearization · pp. 2–3
• The resulting state vector aggregates generator, inverter, line, and load states, and its matrix is the full-order EMT small-signal model used for analysis.The matrix is denoted A_sys.Source: C. System Assembly and Linearization · p. 3

D. Modal Analysis

• System behavior is characterized by the eigenvalues of A_sys, from which oscillation frequency and damping ratio are calculated.For a mode, frequency is based on the magnitude of its imaginary part, and damping ratio on its real and imaginary parts.Source: D. Modal Analysis · p. 3
• A mode is stable when its real part σ_i is negative, and the least-damped modes are of primary interest.The system is small-signal stable when every mode satisfies this condition.Source: D. Modal Analysis · p. 3
• Participation factors localize each mode to its dominant states and therefore to the responsible devices.The tool reports eigenvalue, oscillation, and participation-factor results.Source: D. Modal Analysis · p. 3

III. EMT-SSA TOOL ARCHITECTURE

• Four stages carry a case from standard input data to modal results: case-data import, model library, model assembly, and output production.The linearization and eigen-analysis operate on the assembled model.Source: §III · p. 3
• The modular stages pass data forward, allowing new device models or analysis routines to be added without disturbing the rest of the pipeline.Source: §III · p. 3
• Fig. 1 depicts EMT-SSA’s path from a standard PSS®E case to eigenvalue, oscillation, and participation-factor results.Source: Figure 1 · p. 3

A. Case Data Import

• The .raw and .dyr files are consolidated, mapped to library models, and preprocessed by merging same-bus machines and combining parallel circuits.Original PSS®E bus indexing and per-unit conventions are preserved to trace final states and modes to source-case buses and devices.Source: A. Case Data Import · p. 3

B. Model Library

• The library provides SG, GFL and GFM inverter variants, dynamic R–L transmission lines, and dynamic constant-impedance loads.The SG model includes an IEEEG1 governor and SEXS exciter; inverter variants include their specified control loops and LCL-filter states.Source: B. Model Library · p. 3
• Additional device models can be introduced without altering the surrounding stages.Source: B. Model Library · p. 4

C. Model Assembly

• The extended power flow solves steady state for both the power network and device controllers before model linearization.This makes controller-state initial values, as well as bus voltages and injections, consistent with the operating condition.Source: B. Model Library · p. 4
• Each imported component is instantiated and interconnected; inverters use individual virtual buses with short coupling branches, leaving the transmission topology unchanged.The assembled nonlinear model is linearized about the extended-power-flow operating point.Source: B. Model Library · p. 4

D. Output Production

• Eigenvalue analysis returns the full system-state-matrix spectrum and assesses stability from the rightmost eigenvalue.Source: D. Output Production · p. 4
• Oscillation analysis characterizes each oscillatory mode by frequency and damping ratio and identifies modes of interest.Source: D. Output Production · p. 4
• Participation-factor analysis identifies dominant states for each mode, linking it to a responsible device or network element.This provides root-cause information beyond a bare eigenvalue list.Source: D. Output Production · p. 4

IV. CASE STUDY

• The Kundur two-area case is used to check EMT-SSA output against well-understood behavior and PSS®E NEVA.Source: §IV · p. 4
• The case follows the tool workflow: extended power flow establishes the operating point before eigenvalue, oscillation, and participation-factor analyses.Source: §IV · p. 4
• The Kundur two-area system is a canonical benchmark for inter-area electromechanical oscillation.Source: §IV · p. 4

A. Kundur Two-Area System

• The benchmark has two areas, each with two synchronous generators, joined by a double-circuit tie, with loads placed near the tie.The standard parameter set exhibits a lightly damped inter-area mode.Source: A. Kundur Two-Area System · p. 4
• 7×10⁻⁵ p.u. maximum bus-voltage-magnitude difference and 0.06° maximum bus-angle difference separate EMT-SSA’s extended-power-flow solution from PSS®E’s.The bus-by-bus comparison required no case-specific tuning.Source: 1) Power Flow Validation · p. 4, Figure 3 · p. 4
• All eigenvalues have negative real parts, and the least-damped modes are low-frequency electromechanical modes; heavily damped modes cluster near ±377 rad/s (≈60 Hz).Fig. 4 shows the full spectrum and a low-frequency zoom.Source: 2) Eigenvalue Analysis · p. 4, Figure 4 · p. 4
• The inter-area mode near 0.70 Hz has damping ratio ζ = 0.035 and spans generators in both areas; local modes occur near 1.2 Hz, and EMT-SSA resolves heavily damped ≈60 Hz electromagnetic modes.Participation factors attribute the inter-area mode to SG@1, SG@3, and SG@4.Source: 3) Oscillation and Participation… · p. 5, Table I · p. 5
• Against PSS®E NEVA, electromechanical damping ratios are comparable, while modal frequencies differ by up to about 11% because EMT-SSA models loads as constant-impedance R–L elements.That load representation also produces a small number of spurious right-half-plane eigenvalues, identified as numerical artifacts and excluded from the reported spectrum.Source: 3) Oscillation and Participation… · p. 5, Table II · p. 5

• From a standard PSS®E case, EMT-SSA builds and linearizes a full-order EMT model, then analyzes its state matrix for eigenvalues, oscillations, and participation factors.Source: §V · p. 5
• On Kundur, EMT-SSA recovered the ~0.7-Hz inter-area and ~1.2-Hz local modes, plus heavily damped high-frequency electromagnetic modes absent from phasor analysis.Source: §V · p. 5
• Electromechanical damping ratios were comparable to PSS®E NEVA, while frequencies differed by up to about 11%, a gap traced to load representation.Source: §V · p. 5
• The extended power flow matched the PSS®E operating point within 7×10⁻⁵ p.u. in voltage magnitude and 0.06° in angle.Source: §V · p. 5
• Future work will broaden the device library and validate IBR-rich cases; a detailed large-scale modal study, including WECC 240-bus, is also planned.Source: §V · p. 5