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An Analytical Review of Model Order Reduction Methodologies of Converter-Dominated Power Systems for Stability Studies
Goran Grdenić, Marco Fraccaro, Josipa-Pina Milišić
TL;DR
Converter-dominated systems combine overlapping timescales with stability phenomena at frequencies that can exceed conventional model validity, complicating accurate and efficient reduction. This review synthesizes MOR methods and their trade-offs, concluding that no single method suits every objective and that high-frequency interactions remain a modeling challenge.
Problem
Converter controls and electromagnetic network modes create multi-timescale dynamics and high-frequency stability phenomena that challenge accurate reduced models, while frequency-lifted full-order models can become very large.
Source: §1.2 · p. 9, §6 · p. 41Method
The paper reviews and classifies timescale-based, projection-based, operator-based, data-driven, and machine-learning MOR methods, discussing their equations, advantages, and limitations.
Source: §1.3 · p. 10Results
No single MOR method fits all objectives: modal, balanced, Krylov, and data-driven approaches offer different trade-offs in interpretability, error or stability guarantees, scalability, and applicability to nonlinear systems.
Source: §6 · pp. 39–40Takeaways & Limitations
Future MOR methods for converter-rich systems should seek computational efficiency while preserving harmonic interactions and multi-frequency dynamics.
Source: §6 · p. 41The reviewed studies make only limited contributions to supersynchronous converter-stability issues, and many use converter and line models with limited frequency-domain validity.
Source: §6 · p. 40
Abstract
from arXiv · showhide
The rapid integration of renewable energy sources is transforming modern electric power systems into converter-dominated networks characterized by complex multi-timescale dynamics and new stability interactions over a wide frequency range. Accurate analysis of these phenomena requires high-fidelity models of transmission networks and power electronic converters, which often result in large-scale differential-algebraic systems with significant computational burden. This paper provides an analytical review of model order reduction techniques for converter-rich power systems, with particular emphasis on the key equations employed in the reduction process. Advantages and limitations of commonly used strategies, including timescale-based singular perturbation, projection-based, operator-based, data-driven, and machine-learning based methods, are discussed. The synthesis highlights key challenges related to harmonic interactions and multi-frequency dynamics, and outlines directions for the development of computationally efficient yet dynamically accurate reduced-order models for future power systems.
1. Model order reduction
The paper frames model order reduction (MOR) as constructing a lower-dimensional model that preserves the original system’s input–output behavior within a prescribed error tolerance while reducing computational complexity. Converter-dominated systems complicate this task through overlapping timescales and higher-frequency stability interactions; the review classifies methods for addressing it.
1. Model order reduction
- A reduced model lowers the number of dynamic states while aiming to preserve input–output behavior within a user-defined error tolerance and reduce computational complexity.The full system is represented by nonlinear differential–algebraic equations with dynamic states, network algebraic variables, inputs, parameters, and outputs.Source: §1 · p. 6
1.1. Singular perturbation viewpoint for power-system DAE models
- In singular perturbation, fast dynamics are replaced in the singular limit by algebraic constraints, and their states are represented implicitly as functions of slow states.Substituting the fast-state mapping into the slow equations yields the reduced-order model; this is residualization rather than simply omitting fast states.Source: §1.1 · pp. 7–8
1.2. Multi-timescale nature of PED-based power systems
- Converter controls create overlapping timescales, so time-scale separation alone may not identify slow and fast states, while slow-only models risk missing high-frequency stability interactions.Outer control-loop timescales can resemble electromechanical dynamics, and inner current-control timescales can resemble electromagnetic network modes.Source: §1.2 · pp. 9–10
1.3. Overview of the literature
- The review classifies MOR into timescale-based singular perturbation, projection-based, operator-based and data-driven, and machine-learning methods; it notes that classification is non-unique and methods may be combined.Projection methods, particularly modal-analysis methods, are the most common in the reviewed literature; non-intrusive methods can use measurements when system equations are protected.Source: §1.3 · p. 10
2. Timescale-based singular perturbation
Timescale-based singular perturbation reduces power-system models by replacing sufficiently fast differential equations with algebraic constraints. The reviewed applications include grid-forming converters and converter-based microgrids, with some approaches imposing specific model or stability conditions.
2. Timescale-based singular perturbation
- The classical small-parameter approach orders differential equations by characteristic timescale and replaces sufficiently fast dynamics with algebraic constraints to form a reduced slow subsystem.The reviewed grid-forming-converter application omits grid and fast converter dynamics; another large-disturbance study uses simplified converter models that exclude inner control loops, reactive-power outer control, and filter dynamics.Source: §2 · p. 11
- For converter-based microgrids, the boundary layer model captures fast dynamics, and the authors prove full-model input-to-state stability if it is globally asymptotically stable and the reduced model is input-to-state stable.The stated stability result is conditional on both properties.Source: §2 · p. 12
- A state-matrix singular-perturbation method uses cosine similarity to select transitions between reduced-model options, but was applied only to a grid-forming converter model.Source: §2 · p. 12
3. Projection based methods
The review identifies modal reduction, balanced reduction, Krylov subspace methods, and proper orthogonal decomposition (POD) as the main projection-based techniques. The first three use linearized models, while POD can operate directly on nonlinear systems.
3. Projection based methods
- Modal reduction, balanced reduction, and Krylov methods use linearized models, whereas POD can be applied directly to nonlinear systems without prior linearization.Modal reduction is the most widely adopted of these methods in the reviewed literature.Source: §3 · p. 12
3.1. Modal reduction
Modal reduction linearizes the nonlinear system at an operating point and uses eigenvalue and participation analyses to identify influential states and modes. The review also describes iterative reduction for overlapping timescales and group-based residualization that removes selected modal groups while limiting shifts in the remaining spectrum.
3.1. Modal reduction
- Modal reduction begins by linearizing the nonlinear power-system model around a steady-state operating point, then uses the state matrix eigenvalues to characterize dynamics and small-signal stability.Small-signal stability requires negative real parts for all system eigenvalues.Source: §3.1 · pp. 12–13
3.1.1. Participation factor analysis
- Participation factor analysis identifies state variables that contribute most to individual modes, helping locate sources of poorly damped oscillations in large models.Studies use participation-based classifications to retain coupling states and dominant fast dynamics or to support computationally efficient reachability analysis under uncertainty.Source: §3.1.1 · pp. 13–14
3.1.2. Iterative construction of SP models
- When interacting timescales prevent direct slow–fast decomposition, iterative modal reduction repeatedly identifies and reduces the fastest participating states until the dynamics have a two-timescale structure.The procedure repeats eigenvalue analysis after each reduction because the reduction changes the system’s spectral properties.Source: §3.1.2 · pp. 14–15
3.1.3. Participation-based grouping and adaptive residualization
- Participation-based grouping retains states with the largest participation factors until a threshold is exceeded, then residualizes whole state groups, removing associated eigenvalues with limited shifts in the remaining spectrum.For transmission-line reduction, interaction modes are eigenmodes with significant participation from both line and power-electronic-device states; participation thresholds trade computational efficiency against modeling accuracy.Source: §3.1.3 · pp. 15–16
3.1.4. Model-stiffness-oriented reduction and modal decoupling
The reviewed methods reduce stiffness or preserve selected dynamics by separating timescales, decoupling modes, or retaining subsystem interactions. First-order and three-time-scale singular-perturbation variants account for some fast dynamics, while component connection methods preserve interaction modes.
3.1.4. Model-stiffness-oriented reduction and modal decoupling
- [24] modal decoupling transforms the linearized system into independent modal coordinates, then retains prescribed slow eigenmodes to reduce stiffness and improve computational efficiency.The method uses the Jordan canonical form, which gives the participation-factor matrix a block-diagonal structure.Source: §3.1.4 · pp. 16–17
3.1.5. Influence of fast dynamics: zero- and first-order singular perturbation models
- [25] first-order singular perturbation retains a correction for fast-dynamic influence, so a stable zero-order model does not guarantee first-order stability.In inverter-based microgrids, high transmission-line resistance-to-reactance ratios strengthen coupling between network electromagnetic dynamics and converter controls.Source: §3.1.5 · pp. 17–19
3.1.6. Three-time-scale SP reduction
- [27] three-time-scale reduction retains slow dynamics explicitly, approximates fast states with first-order singular perturbation, and neglects very fast states using zero-order approximation.Iterative participation-factor analysis refines the time-scale separation and identifies dynamically dominant states.Source: §3.1.6 · p. 19
3.1.7. Component connection methods
- [28] component connection methods identify interaction modes, residualize low-participation states within subsystems, then assemble the reduced system.Interaction modes are selected using eigenvalue proximity and eigenvalue displacement criteria.Source: §3.1.7 · pp. 19–20
3.1.8. Residue-based modal reduction
These modal-based approaches identify and retain dynamics considered important to the system response, using residues, segmented operating-point models, or transformations that preserve selected system characteristics.
3.1.8. Residue-based modal reduction
- [29] residue-based reduction partitions interconnected microgrids into slow and fast subsystems, retains fast states important to dominant modes, and eliminates the other fast states.The residue indicator combines modal residue, reflecting controllability and observability, with modal decay rate.Source: §3.1.8 · p. 21
3.1.9. Segmented linearization
- [32, 33] segmented linearization builds multiple local models along a fault trajectory and uses dominance degree and generalized participation factors to distinguish state importance.The generalized participation factor combines classical participation factors with dominance degree.Source: §3.1.9 · p. 22
3.1.10. Other modal-based MOR techniques
- Transfer-based MOR eliminates zeros and poles associated with fast dynamics while preserving their influence via open-loop gain; approximates large-system eigenvalues using smaller double-infeed subsystems and block-diagonalizes Kron-reduced admittance.The methodology is demonstrated with converters in grid-following and grid-forming operation modes.Source: §3.1.10 · p. 23
3.2. Balanced reduction
Balanced reduction ranks states by their joint controllability and observability, then truncates or residualizes states with low Hankel singular values. Variants tailor this assessment to selected frequencies, time intervals, subsystems, or initial-condition responses.
3.2. Balanced reduction
- Balanced reduction can truncate states with small Hankel singular values while offering an a priori error bound, but its transformation generally removes physical state interpretation.The bound remains small when discarded Hankel singular values decay sufficiently rapidly; loss of interpretability can make the method unsuitable when structured reduction or subsystem preservation is required.Source: §3.2 · pp. 24–25
3.2.1. Combining Hankel singular values and participation factors
- [36–38] combined Gramian-based state relevance measures with modal indicators or participation factors to guide reduction, balancing computational efficiency with dynamic fidelity.The review notes that balanced truncation can require fewer states, but does not preserve their physical interpretability.Source: §3.2.1 · p. 25
3.2.2. Frequency-limited Gramians
- [39] frequency-limited Gramians retain states whose input-output participation exceeds a threshold over a selected frequency interval; the method was demonstrated on a modified IEEE 39-bus grid with two converters.[40] frequency-weighted residualization was tested on systems without converters, while applied frequency-weighted and frequency-limited methods to DFIG-based wind plants.Source: §3.2.2 · pp. 25–26
3.2.3. Time-weighted Gramians
- [42] time-weighted Gramians emphasize behavior in a specified interval, such as transient response, rather than over an infinite time horizon.The method was used to reduce a DFIG-based wind power plant model.Source: §3.2.3 · p. 26
3.2.4. Cross-Gramian
- [43] the cross-Gramian combines controllability and observability into one measure, reducing an external subsystem while leaving the target synchronous generator and its power system stabilizer unaltered.The method was demonstrated for classical synchronous-generator systems with poorly damped inter-area oscillations.Source: §3.2.4 · p. 27
3.2.5. Specialized balanced truncation methods
- [44, 45] specialized balanced truncation separates forced and unforced responses for systems with nonzero initial conditions, while another approach preserves PLL dynamics and reduces the remaining system.The PLL-focused method was applied to a system of 20 DFIG wind turbines.Source: §3.2.5 · pp. 27–28
3.3. Krylov subspace methods
Krylov methods construct lower-dimensional models by matching transfer-function moments around selected expansion points; variants adapt interpolation, preserve stability or circuit structure, and use data-driven excitation. A comparison reports a trade-off between broad-band accuracy and stability preservation versus computational efficiency.
3.3. Krylov subspace methods
- [47] balanced reduction outperformed Krylov for accuracy over a broad frequency range and reduced-model stability preservation, while Krylov was more computationally efficient but credible only over a narrow band and without a stability guarantee.The comparison used an IEEE 37-bus grid with inverter-based generation.Source: §3.3 · pp. 28–29
3.3.1. Iterative Rational Krylov Algorithm
- [48] IRKA updates interpolation points iteratively to satisfy first-order H2-optimal approximation conditions; uses heuristic multiobjective optimization to preserve damping ratios and frequencies of poorly damped modes.Unlike Arnoldi's fixed expansion points, IRKA updates shifts to improve approximation accuracy.Source: §3.3.1 · p. 29
3.3.2. Extended Krylov subspace method
- [51] an extended Krylov subspace improved Lyapunov-equation solution efficiency and supported balanced realization for unstable systems, applied to 39-bus and 145-bus grids without PEDs.The approach was used to obtain reduced models for tuning a synchronous-generator power system stabilizer.Source: §3.3.2 · p. 30
3.3.3. Parametric Krylov method
- [52] parametric Krylov reduction matches the model across frequency and parameter values, using an adaptive method that considers convergence error, cumulative error, and projection-matrix rank.The methodology was applied to converter-based systems of different sizes and assessed stability across a wide frequency range.Source: §3.3.3 · pp. 30–31
3.3.4. Stability-preserving Krylov method
- [53] stability-preserving Krylov projection uses distinct left and right projections, with the left projection satisfying a Lyapunov stability condition and the right matching moments.Source: §3.3.4 · p. 31
3.3.5. Structure-preserving Krylov method
- [54] SPRIM reduces passive RLC elements in a dynamic-phasor distribution-network model while representing inverters, generators, and other components through boundary conditions.The reduced model preserves a structure typical of RLC circuits.Source: §3.3.5 · pp. 31–32
3.3.6. Two-sided moment matching
- [55] two-sided moment matching excites the full model at interpolation points and imposes conditions on both inputs and outputs before constructing the reduced model via a Sylvester equation.The method was tested on a large wind farm connected to the IEEE 14-bus system.Source: §3.3.6 · p. 32
3.4. Proper orthogonal decomposition
Proper orthogonal decomposition (POD) builds a reduced model from simulated state snapshots, retaining the state directions associated with the largest eigenvalues. Structure-preserving and balanced POD variants are also described for power-system models.
3.4. Proper orthogonal decomposition
- POD forms a snapshot matrix from system states sampled over time, then decomposes XXT and retains the r state variables associated with the largest eigenvalues.The number of time steps m should be much larger than the number of state variables n; other variables are truncated or residualized.Source: §3.4 · pp. 32–33
- Structure-preserving POD separately reduces dynamic and algebraic variables, while balanced POD approximates controllability and observability matrices from input-output response snapshots.The cited balanced POD method showed higher accuracy than balanced truncation, and both methods were tested on IEEE 39-bus and 2000-bus systems with grid-forming PV plants.Source: §3.4 · p. 33
4. Operator-based and data-driven methods
This section surveys operator-based and data-driven reductions, including Koopman and Mori-Zwanzig formulations, frequency-domain system identification, and right-tangential interpolation.
4.1. Koopman operator
- The Koopman approach transforms nonlinear dynamics into a linear system in observable space, then retains relevant modes through eigenvalue decomposition of the operator.The operator is calculated from system simulation data.Source: §4.1 · p. 34
4.2. Mori-Zwanzig formulation with Dynamic Mode Decomposition
- Combining Mori-Zwanzig with Dynamic Mode Decomposition enables parallel-in-time simulation of converter-dominated microgrids and significantly reduces transient simulation time.Mori-Zwanzig represents preserved-state dynamics alongside eliminated-state effects and noise; DMD is then applied to simulation data.Source: §4.2 · pp. 34–35
4.3. Frequency-domain system identification
- Frequency-domain identification uses impedance/admittance fitting and transfer-function reduction, while Loewner realization applies SVD to reduce large-scale and black-box models.The cited approaches include vector/matrix fitting with state-space assembly operators, frequency-domain transfer-function reduction, and truncation of directions with low singular values.Source: §4.3 · pp. 35–36
4.4. Right-tangential interpolation
- Right-tangential interpolation derives moments from input-output data organized in Hankel matrices, and is theoretically linked to Krylov subspace methods.A separate cited method combines intrusive nonlinear moment matching with non-intrusive DMD, but does not address converter-dominated systems.Source: §4.4 · p. 36
5. ML-based dynamic representation
Machine-learning approaches span reduced-order dynamical representations and dynamic equivalents, with distinctions between them not always clear-cut.
5. ML-based dynamic representation
- ML qualifies as classical model order reduction when it constructs a low-dimensional latent representation of essential dynamics; input-output fitting alone is more aptly a surrogate or dynamic equivalent.The distinction is not always clear-cut, and some recent approaches combine both paradigms.Source: §5 · p. 37
5.1. ML-based MOR
- An autoencoder-LSTM model outperformed linear projection methods on two test systems, while Neural ODE and physics-aware approaches address renewable-rich systems and external-system equivalents.The IEEE 118-bus and 1354-bus tests lacked PEDs; the cited physics-aware external-system method also did not consider inverter-based generation.Source: §5.1 · pp. 37–38
5.2. ML-based dynamic equivalents
- Dynamic equivalents use DNNs, RNNs, and LSTMs with boundary or historical measurements, while other work models inverter-interfaced DERs and combines neural networks with physical constraints.Transient and fault data improved modeling accuracy in one cited microgrid study; feedforward networks have also been used to compensate residual errors in reduced models.Source: §5.2 · pp. 38–39
6. Discussion and future perspectives
The review finds no single reduction method suited to every objective, and identifies limited treatment of supersynchronous converter-stability issues alongside frequency-validity limits in full-order models.
6. Discussion and future perspectives
- The reviewed literature offers varied methods, and no single reduction technique fits all objectives.Method choice depends on the desired properties of the reduced-order model.Source: §6 · p. 39
- Modal reduction offers physical interpretability; balanced reduction provides stability preservation and theoretical error limits; Krylov methods scale to large models, while data-driven performance depends on training-data quality.Balanced reduction can be computationally inefficient for large systems because it requires solving Lyapunov equations, and Krylov error bounds and stability guarantees are harder to evaluate.Source: §6 · p. 40
- MOR methods are generally mature for conventional dq-based models, but few contributions address supersynchronous converter-stability issues, and commonly used converter and line models have limited frequency validity.Capturing higher-frequency interactions may require frequency-lifted converter models and distributed, frequency-dependent line models, which increase full-order model size because each retained harmonic adds variables.Source: §6 · pp. 40–41
- Future MOR research should seek computational efficiency while preserving harmonic interactions and multi-frequency dynamics in converter-dominated systems.Frequency-lifted approaches can produce immense models, while power systems are operated closer to the stability margin and require fast, reliable security assessments.Source: §6 · p. 41