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Modular Model Order Reduction for Inverter-Based Power Systems with Stability and Accuracy Guarantees

Hanqing Zhang, Pudong Ge, Mohammad Fahim Shakib, Adrià Junyent-Ferré, Balarko Chaudhuri, Luuk Poort, Lars A. L. Janssen, Giordano Scarciotti

arXiv:2610.02477v1eess.SYcs.SY

TL;DR

Large interconnected power grids pose computational challenges, and existing methods lack guarantees of accuracy and stability for the overall interconnected reduced model. This paper proposes modular, independent sub-grid reduction with local specifications derived from a global requirement. The framework is reported to reduce model order from 1304 to 143 within the predefined accuracy specification while preserving stability.

  • Problem

    Existing reductions can be computationally demanding for multi-area grids, and the authors report no prior power-system MOR method guaranteeing both user-defined accuracy and stability of the overall interconnected reduced model.

    Source: §I · p. 1

  • Method

    The framework independently reduces regional sub-grids while preserving their physical interconnections and translates a global accuracy requirement into local specifications.

    Source: §I · p. 2

  • Results

    The interconnected model order falls from 1304 to 143 within the predefined accuracy specification, and its dominant pole is −0.083 versus −0.087 for the original model.

    Source: B. Modular Model Order Reduction… · p. 8, C. Stability Preservation and Dominant… · p. 8

  • Takeaways & Limitations

    Satisfying the local specifications guarantees overall accuracy and stability after reconnection, supporting reliable dynamic simulations of the entire grid.

    Source: §I · p. 2

Abstract

from arXiv · show

This paper proposes a modular model order reduction framework for large-scale power grids integrated with renewable energy sources. The proposed framework has the advantage of reducing individual sub-areas independently while retaining the power grid interconnection structure. Moreover, given a user-defined accuracy specification on the overall reduced-order interconnected power grid, corresponding target specifications for the individual reduced sub-areas are determined: satisfaction of such individual specifications guarantees the overall user-defined accuracy specification, while also preserving the stability of the overall reduced-order interconnected power grid. The proposed framework is validated on a modified IEEE 118-bus system (where synchronous generators have been replaced by inverter-based resources) by assessing the accuracy specification satisfaction of the overall reduced-order interconnected power grid, in terms of frequency response, stability, active power, and sub-synchronous behaviour when injected with disturbances.

I. INTRODUCTION

Inverter-based resources increase power-grid modeling complexity, while existing reduction approaches struggle with the computational demands of multi-area grids and do not guarantee accuracy and stability for the interconnected reduced model. The paper introduces a modular framework that translates overall accuracy requirements into local specifications and guarantees system-level accuracy and stability.

I. INTRODUCTION

  • Inverter-based resources raise simulation complexity through their scale, fast control timescales, and sub-synchronous phenomena, motivating model order reduction.Balanced truncation, and moment matching, are among the reduction techniques adopted to address the computational burden.Source: §I · p. 1
  • Because modern power grids span independently managed regional subsystems, reducing the whole interconnected system in one pass remains computationally demanding: MOR resources scale with the overall system order, not individual subsystems.Available power-system MOR methods usually treat the external area as one lumped region, neglecting its multi-area structure, –.Source: §I · p. 1
  • The proposed framework preserves regional sub-grid boundaries and tie-line and boundary input-output meanings, keeping the reduced model physically interpretable.It adapts methods from and to a dynamical interconnection that includes the unreduced study area.Source: §I · p. 2
  • A mathematical formulation quantifies error propagation between regional sub-grids and the overall grid, translating a global frequency-weighted accuracy requirement into local specifications.The method combines the Loewner framework with a greedy algorithm for sub-grid reductions, including cases where only black-box frequency scans are available.Source: §I · p. 2
  • The framework guarantees stability and a user-defined accuracy specification after reduced external-area sub-grids are reconnected to the unreduced study area.The framework translates the overall requirement into local sub-grid specifications whose satisfaction yields those system-level guarantees.Source: §I · p. 2

II. PRELIMINARIES AND PROBLEM STATEMENT

  • The preliminaries formulate the interconnected grid using an upper linear fractional transformation and set out the accuracy-specification derivation and problem statement.The paper organizes these preliminaries as the modular model formulation, accuracy analysis, and formal problem statement.Source: §II · p. 2

A. Modular Model Description

The model represents the unreduced study area and decoupled external-area sub-grids as an upper LFT connected through a dynamic interface. Reduction replaces each external subsystem independently while retaining the interface and interconnection structure.

A. Modular Model Description

  • The grid is linearized around an operating point and represented as a study-area subsystem G0 interconnected with external-area subsystems Gj.External inputs represent nodal disturbances and external outputs represent states of interest.Source: A. Modular Model Description · p. 2
  • The external-area transfer functions are collected in block-diagonal form, while a dynamic interface aggregates their interconnections and the study-area dynamics.The interface K takes external-area outputs and external inputs, and produces external-area inputs and external outputs.Source: A. Modular Model Description · pp. 2–3
  • The full-order grid is represented as an upper LFT of the decoupled external-area model and the dynamic interface.The interface is dynamic because it incorporates the unchanged study area, unlike the static interface in.Source: A. Modular Model Description · pp. 2–3
  • Each external-area subsystem is reduced independently with its input-output channels preserved, then reassembled using the same interface and interconnection relationships.This modular process is intended to retain stability and satisfy user-defined accuracy specifications without sharing full model information among sub-grids.Source: A. Modular Model Description · p. 3

B. Unified Representation of Reduction Accuracy Specifications

The section defines frequency-weighted accuracy specifications for the overall grid and its external-area subsystems. These specifications support relating local reduction performance to the interconnected system’s performance.

B. Unified Representation of Reduction Accuracy Specifications

  • The global ROM accuracy specification bounds the weighted interconnected-grid reduction error using Wc and Vc.The weights scale the error; the formulation unifies allowed reduction errors.Source: B. Unified Representation of Reduction… · p. 3
  • Choosing Wc and Vc sets channel-specific error bounds across frequency ranges, including stricter requirements for critical oscillations.The weights can prioritize inter-area or sub-synchronous oscillations while relaxing constraints at less relevant frequencies.Source: B. Unified Representation of Reduction… · p. 4
  • Each external-area subsystem Gj has its own ROM error dynamics and weighted accuracy specification using Wj and Vj.The subsystem specifications provide the local counterparts to the global specification.Source: B. Unified Representation of Reduction… · p. 4

C. Problem Statement

The problem is to reduce external-area subsystems independently while retaining their interconnection with the unreduced study area. Their local accuracy specifications must ensure the interconnected ROM meets the global accuracy specification and preserves internal stability.

C. Problem Statement

  • The interconnected grid is assumed well-posed and internally stable, so it is computationally tractable and its internal states remain bounded under port perturbations.This is the stated assumption for the modular MOR problem.Source: C. Problem Statement · p. 4
  • Given a global accuracy specification, the framework must find local subsystem specifications that guarantee the reassembled ROM meets that specification and preserves internal stability.Each subsystem ROM is required to satisfy its derived local specification.Source: C. Problem Statement · p. 4
  • The objective is to reduce external-area subsystems independently and reassemble them with the unreduced study area while preserving the interconnection structure.The resulting interconnected ROM must satisfy the prescribed global accuracy and stability requirements.Source: C. Problem Statement · p. 4

III. MODULAR MODEL ORDER REDUCTION FRAMEWORK

The framework constructively solves the two modular MOR problems: translating global accuracy requirements into local specifications, then constructing subsystem ROMs that satisfy them.

III. MODULAR MODEL ORDER REDUCTION FRAMEWORK

  • The framework relates overall-grid and external-area accuracy specifications to determine admissible local specifications.This relationship addresses Problem 1.Source: §III · p. 4
  • It proposes MOR methods to construct external-area ROMs that satisfy their specified accuracy requirements.This addresses Problem 2.Source: §III · p. 4
  • The framework uses an upper-LFT reformulation of the power-grid model together with the reduction accuracy specification established earlier.These are the stated basis for the constructive solution.Source: §III · p. 4

A. Top-Down Accuracy Specifications Translation

The top-down translation represents overall reduction error through a nominal propagation system and local accuracy blocks. A sufficient condition then links subsystem specifications to global accuracy and internal stability.

A. Top-Down Accuracy Specifications Translation

  • The nominal system isolates fixed, known grid and interface dynamics to describe how subsystem accuracy specifications propagate into overall reduction error.The reduction-error diagram represents the overall error as an upper LFT of the nominal system and subsystem accuracy blocks.Source: A. Top-Down Accuracy Specifications… · pp. 4–5, Figure 3 · p. 5
  • The nominal system is weighted by W and V and augmented with a fictitious block Δc to relate global and subsystem accuracy specifications.The global weights and fictitious block are shown in the weighted error-dynamics diagram.Source: A. Top-Down Accuracy Specifications… · p. 5, Figure 4 · p. 5
  • If the stated frequency-domain condition holds for all ω, satisfying every external-area ROM specification ensures the interconnected ROM meets the global specification and preserves internal stability.The theorem provides a sufficient condition for this implication.Source: A. Top-Down Accuracy Specifications… · p. 5
  • The theorem’s condition is checked on a fine frequency grid using D-K iteration.The relationship can also be used bottom-up to derive a global specification from known subsystem specifications.Source: A. Top-Down Accuracy Specifications… · p. 5

B. Model Order Reduction Based on Accuracy Specifications

The framework supports conventional reduction when subsystem models are known and data-driven reduction when only frequency-response data are available. Its greedy Loewner approach iteratively targets the largest weighted error until the local accuracy specification is met.

B. Model Order Reduction Based on Accuracy Specifications

  • When a subsystem model is known, balanced truncation can be applied iteratively, with frequency-weighted balanced truncation also available.The reduced-model order is increased until its weighted error satisfies the subsystem specification across the frequency grid.Source: B. Model Order Reduction Based on… · p. 5
  • The greedy Loewner method adds frequency-response samples at peak-related frequencies, then iteratively samples the frequency with the largest weighted error until the subsystem accuracy specification is met.It reflects unstable poles without changing their associated residues when constructing each reduced model.Source: B. Model Order Reduction Based on… · p. 6
  • When only subsystem frequency-response data are available, the data-driven method increases reduced-model complexity until the weighted error is below one across the frequency grid.The paper describes the method as a greedy heuristic that reduces dominant peaks in the frequency-weighted reduction error.Source: B. Model Order Reduction Based on… · p. 6

C. Overview of Modular Model Order Reduction Framework

The framework translates an overall accuracy requirement into local specifications for external-area sub-grids, reduces those sub-grids independently, and reconnects them through the preserved interface. When each local specification is met, the interconnected reduced model is guaranteed to satisfy the overall accuracy requirement and preserve internal stability.

C. Overview of Modular Model Order Reduction Framework

  • The framework represents the grid as an LFT, derives each external-area ROM’s specification from the global requirement, reduces sub-grids independently, and reconnects them through the preserved interface.The unreduced study area remains part of the reassembled interconnection.Source: C. Overview of Modular Model Order… · p. 6
  • Meeting each derived external-area specification guarantees that the reassembled interconnected model satisfies the prescribed overall accuracy specification and preserves internal stability.The framework derives and enforces subsystem specifications from the requirement on the overall interconnection.Source: C. Overview of Modular Model Order… · p. 6
  • The benchmark partition separates a study area from an external area comprising two sub-grids.The figure marks the study area grey and the two external sub-grids red and green.Source: Figure 5 · p. 6

IV. APPLICATION TO IEEE 118-BUS BENCHMARK MODEL

The case study applies the modular framework to a modified IEEE 118-bus benchmark partitioned into a study area and external subsystems. The benchmark is linearized at a preset operating point and reformulated as an interconnected model.

IV. APPLICATION TO IEEE 118-BUS BENCHMARK MODEL

  • The benchmark is linearized at a preset operating point, with the study area retained in detail and the external areas designated for reduction.The study area is of much lower order than the external areas.Source: A. Power Grid Model Reformulation · p. 6

A. Power Grid Model Reformulation

  • Boundary voltages and currents in the dq-frame serve as subsystem inputs and outputs, while the overall model uses GFL 24’s active-power reference and actual active-power output.The selected overall input is the active power reference; the selected output is actual active power output.Source: A. Power Grid Model Reformulation · p. 6
  • The decoupled external-area model is collected and the dynamic interface is determined to reformulate the benchmark as an upper LFT.The interface is constructed using the paper’s stated relations.Source: A. Power Grid Model Reformulation · p. 6

B. Modular Model Order Reduction Implementation

The implementation derives subsystem accuracy specifications from a frequency-dependent global requirement, then applies different reduction methods to the external sub-grids. The resulting ROM meets the overall specification, with the benchmark’s model order reduced substantially.

B. Modular Model Order Reduction Implementation

  • The global frequency-dependent specification uses an envelope based on a fraction of the grid response magnitude plus an offset, allowing higher relative error at higher frequencies.The purple region in Fig. 6 depicts the admissible magnitude-response area.Source: B. Modular Model Order Reduction… · p. 7
  • The method solves for local specifications using a bilinear optimization problem, with diagonal positive weights assigning bounds to individual input-output pairs.The paper notes that D-K iteration can be used for this calculation.Source: B. Modular Model Order Reduction… · p. 7
  • The implementation uses frequency-weighted balanced truncation for G1 and the greedy Loewner method for G2, based on their respective model and data availability.Fitted G1 weighting functions are bounded below by the specified weights, making the fitted requirement conservative while guaranteeing specification satisfaction on the frequency grid.Source: B. Modular Model Order Reduction… · p. 7
  • The overall interconnected model is reduced from order 1305 to 143 (89.04% reduction).G1 is reduced from order 367 to 61 (83.38% reduction), and G2 from order 904 to 48 (94.69% reduction).Source: B. Modular Model Order Reduction… · p. 7
  • The overall ROM magnitude response lies within the user-defined accuracy envelope, and the external-area ROMs satisfy their derived specifications.The reported subsystem result covers all 164 input-output pairs, although only two responses are plotted.Source: B. Modular Model Order Reduction… · p. 7, Figure 6 · p. 7
  • The reduced interconnected benchmark preserves internal stability.The paper reports this alongside the achieved overall reduction specification.Source: B. Modular Model Order Reduction… · p. 8

C. Stability Preservation and Dominant Behaviours

The proposed framework yields a stable interconnected ROM with dominant poles close to the full model. Under two disturbances, reduced models reproduce key grid outputs with small errors.

C. Stability Preservation and Dominant Behaviours

  • The framework’s interconnected ROM stays stable, with dominant pole −0.083 versus −0.087 for the FOM, while the vector-fitting interconnected surrogate has unstable poles.The vector-fitting external-area surrogates themselves preserve subsystem stability.Source: C. Stability Preservation and Dominant… · p. 8, Figure 8 · p. 8

D. Time-Domain Validation: Oscillatory Behaviour under Disturbances

E. Comparison of Computation Time

The reduced external-area model cuts simulation time while retaining accuracy and stability guarantees. The paper also compares subsystem reduction methods and concludes with the framework’s modular workflow.

E. Comparison of Computation Time

  • 0.13 s: the external-area ROM connected to the linear study area simulates 1 s roughly 44 times faster than the external-area FOM connection and almost 865 times faster than the original nonlinear grid.With the nonlinear study area, the ROM connection takes 8.63 s, reducing time by roughly 35% versus the FOM connection and roughly 93% versus the original nonlinear grid.Source: Table II · p. 9, E. Comparison of Computation Time · p. 9

F. Comparison of Different Model Order Reduction Techniques

  • The greedy Loewner framework achieves relatively small frequency-weighted system errors at intermediate subsystem order and the smallest weighted H∞ error, while extracting ROMs faster than BT and FWBT, especially for G2.The paper attributes the extraction-time advantage to the data-driven approach, compared with the Gramian computations required by BT and FWBT.Source: F. Comparison of Different Model Order… · p. 10, Table III · p. 10

V. CONCLUSION

  • External-area subsystems are reduced independently to assigned specifications, then reconnected to the unreduced study area, preserving interconnection structure while meeting accuracy and stability guarantees.The conclusion reports that the resulting model requires less simulation time.Source: §V · p. 10
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