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RAPTOR: RAndom-projection Physics-informed Transient sOlveR

Petros Ellinas, Benjamin Vilmann, Spyros Chatzivasileiadis, Johanna Vorwerk

arXiv:2609.29714v1eess.SYcs.LGcs.MScs.NAcs.SYmath.NA

TL;DR

RAPTOR addresses the difficulty of solving stiff, multi-timescale power-system DAEs efficiently and reliably. It represents each interval’s trajectory with fixed Gaussian RBFs and solves for their combination; across tested benchmarks, it reports accurate results and speedups including 16.17× over an accuracy-qualified Radau configuration on IEEE 118.

  • Problem

    Stiff, multi-timescale power-system DAEs can make timestep-by-timestep nonlinear solves sensitive to initial estimates, equation scaling, and algebraic consistency.

    Source: §II · p. 2

  • Method

    RAPTOR uses fixed Gaussian RBFs with broad and localized shapes to represent interval trajectories, solving for their combination to satisfy the governing equations.

    Source: §I · p. 2, §V · p. 10

  • Results

    Across tested RMS, networked DAE, and EMT benchmarks, RAPTOR meets prescribed accuracy requirements, including a 16.17× speedup over the accuracy-qualified Radau configuration on IEEE 118.

    Source: §V · p. 10

  • Takeaways & Limitations

    RAPTOR’s numerical advantage is most relevant when short fast transients are embedded within slower dynamics, rather than from stiffness alone.

    Source: §V · p. 10

    The reported results use a custom research implementation of RAPTOR, while established time-domain solvers have benefited from decades of algorithmic and software optimization.

    Source: §V · p. 10

Abstract

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The complexity of time-domain simulation of modern power systems has increased significantly because converter-based resources introduce control dynamics that must be simulated alongside slower system-level and fast electromagnetic dynamics. The resulting wide range of timescales may force classical time-domain solvers to use small timesteps, complicate the solution of nonlinear equations at each timestep, and reduce solver reliability under strongly nonlinear and multi-timescale transient conditions. This paper introduces RAPTOR, a first-of-its-kind time-domain simulation framework for power system dynamic simulations. At its core, RAPTOR introduces a new integration technique that represents the unknown trajectory of hybrid differential-algebraic equations (DAEs) over a time interval using a physics-informed random-projection neural network (PIRPNN) built from fixed Gaussian radial basis functions (RBFs). A nonlinear solver, e.g., Newton-Raphson, determines how to combine the fixed RBFs so that the resulting trajectory satisfies the governing equations. By combining RBFs with broad and localized shapes, RAPTOR can represent complex multi-timescale behavior over comparatively long time intervals. This enables larger effective simulation advances, fewer overall nonlinear solver iterations, and faster integration over long time horizons. Numerical studies on stiff RMS models, IEEE 9-, 14-, 39-, 57-, and 118-bus RMS simulation benchmarks, and EMT test cases show that RAPTOR accurately finds solutions with substantially fewer sequential simulation advances while offering superior accuracy-runtime performance. With speedups of over 10x in certain cases, these results showcase RAPTOR's potential to outperform long-standing and widely used integration methods, such as the Radau method and the trapezoidal rule.

I. INTRODUCTION

RAPTOR targets the numerical cost and convergence challenges of simulating stiff, multi-timescale power-system DAEs without simplifying their physical equations. It uses fixed Gaussian RBFs to represent interval-wide trajectories and reports accurate benchmark solutions with fewer advances and speedups exceeding 10× in difficult cases.

I. INTRODUCTION

  • Converter controls and electrical dynamics add faster timescales to slower system dynamics, making detailed RMS and EMT studies increasingly need to retain several timescales.Simplifying models can remove fast behavior that materially affects the transient response; the paper instead targets faster numerical solution without changing physical equations.Source: §I · p. 1

II. CONCEPTUAL FOUNDATION OF RAPTOR

  • Stiff multi-timescale DAEs can make timestep-by-timestep nonlinear solves sensitive to initial estimates, equation scaling, and algebraic consistency, while smaller timesteps raise computational cost.The conceptual foundation identifies poor initial guesses, anisotropic Jacobians, algebraic coupling, and switching-related discontinuities as challenges to convergence and reliability.Source: §I · p. 1, §II · p. 2

I. INTRODUCTION

  • Time-spectral collocation solves for a continuous state function over an interval, but polynomial representations may need many collocation points for multi-timescale dynamics, enlarging the nonlinear solve.The introduction presents power-system-scale time-spectral integration as not previously explored to the authors’ knowledge.Source: §I · p. 1
  • RAPTOR represents hybrid DAE trajectories with fixed Gaussian RBFs whose broad and localized shapes capture slow evolution and short transients.The nonlinear solver combines the RBFs to satisfy governing equations; initialization, algebraic-variable enforcement, interval control, and event handling complete the solver workflow.Source: §I · p. 2
  • Speedups exceeding 10× in difficult transient cases accompany accurate solutions with substantially fewer accepted advances across RMS, IEEE system, and EMT benchmarks.The comparisons are against classic integration methods established in the power-system domain.Source: §I · p. 2

II. CONCEPTUAL FOUNDATION OF RAPTOR

  • RAPTOR combines broad and localized RBFs into a continuous interval trajectory and enforces governing equations at multiple collocation points, unlike successive local timestep solves.The conceptual illustration contrasts the approaches; its iteration traces are schematic, not measured counts.Source: §II · p. 3, Figure 1 · p. 3

III. RAPTOR: MATHEMATICAL FORMULATION

RAPTOR constructs a differential-state trajectory for each interval with a PIRPNN and adjusts its coefficients to reduce the DAE residual. Accepted interval endpoints seed subsequent intervals; rejected intervals are shortened and solved again.

III. RAPTOR: MATHEMATICAL FORMULATION

  • For each interval, RAPTOR builds a PIRPNN trajectory with fixed, uniformly centered RBFs whose physics-informed shape parameters are sampled before the nonlinear solve.The solve adjusts the RBF coefficients to minimize the DAE residual at collocation points while enforcing the initial condition exactly.Source: §III · p. 3
  • An interval is accepted when its equation residual meets the criterion; otherwise it is rejected, shortened, and solved again before its endpoint can seed the next interval.The paper distinguishes a RAPTOR interval from a comparison integrator’s timestep and uses accepted advance for either accepted step type.Source: §III · p. 3
  • A PIRPNN parameterizes only the current interval’s unknown trajectory; RAPTOR recomputes its coefficients for each new interval rather than using it to predict future trajectories.The complete trajectory is assembled from successively solved interval trajectories.Source: §III · p. 3

A. Solving a Single RAPTOR Interval

A RAPTOR interval represents each differential-state trajectory as a fixed Gaussian-RBF expansion whose coefficients are solved for that interval. The representation makes the interval’s initial condition exact by construction.

1) PIRPNN Trajectory Representation:

  • Each differential state has a PIRPNN representation over one RAPTOR interval, formed from a fixed set of Gaussian RBFs with coefficients determined for that interval.The state-specific coefficient vectors are collected in w; these coefficients are the only PIRPNN quantities changed by the nonlinear interval solve.Source: 1) PIRPNN Trajectory Representation · pp. 3–4
  • RBF centers locate each basis function within the interval, while its shape parameter controls localization: small values give broad functions and large values give narrow ones.The centers are distributed over normalized local time s, which maps the interval to [0, 1].Source: 1) PIRPNN Trajectory Representation · p. 4
  • The factor s makes the PIRPNN correction vanish at the interval start, so the represented trajectory satisfies xw(0) = x(tk) for any coefficient values.This enforces the initial condition exactly.Source: 1) PIRPNN Trajectory Representation · p. 4

2) Algebraic States Enforcement:

RAPTOR optimizes the differential-state trajectory and recovers algebraic variables from their constraints at collocation points. It solves a weighted residual problem over the interval with a modified Newton method.

2) Algebraic States Enforcement:

  • RAPTOR parameterizes differential states independently but obtains algebraic variables by solving the network and component constraint equations at each collocation point.For index-1 DAEs, local recovery follows when the algebraic equations are smooth and their Jacobian with respect to algebraic variables is nonsingular.Source: 2) Algebraic States Enforcement · p. 4
  • At each collocation point, RAPTOR recovers algebraic variables and uses them to evaluate the differential-equation mismatch for the represented trajectory.The residual evaluation proceeds from the trajectory value to the algebraic solve and then to the differential mismatch.Source: 2) Algebraic States Enforcement · p. 4

3) Nonlinear Interval Solve:

  • The interval solve chooses RBF coefficients to reduce equation mismatch at a finite set of collocation points, which are internal checks rather than accepted timesteps.The points can be selected independently of the RBF count and centers.Source: 3) Nonlinear Interval Solve · p. 4
  • A modified Newton (Chord) method reuses the residual Jacobian and its factorization across coefficient updates, refreshing them when convergence deteriorates.Overlapping RBFs can make coefficients non-unique, so rank-aware least-squares methods or regularization handle the resulting solve.Source: 3) Nonlinear Interval Solve · p. 4

4) PIRPNN Basis Design:

RAPTOR places Gaussian RBF centers across normalized intervals and selects their localization using both interval length and local power-system dynamics. This basis design provides functions for representing the time scales present in each interval.

4) PIRPNN Basis Design:

  • The RBF parameters use normalized time s in [0, 1], with centers placed uniformly to cover each interval while separating basis density from localization.Basis density and localization together determine the available temporal resolution.Source: 4) PIRPNN Basis Design · p. 4
  • The admissible shape range scales with interval length, while faster local dynamics increase the available RBF localization.The range is presented as a physics-informed design rule for the RBF basis.Source: 4) PIRPNN Basis Design · p. 5
  • RAPTOR samples Gaussian RBF shape parameters using a range that reflects both the proposed interval length and local power-system dynamics.The local Jacobian’s eigenvalues estimate the fastest dynamic time scale, motivating the localization range.Source: 4) PIRPNN Basis Design · p. 5

B. Building the Simulation Over the Full Time Horizon

RAPTOR builds a full simulation by repeatedly solving intervals, carrying each interval’s initial state forward. Its coefficient initialization combines derivative information with, when suitable, coefficients from the preceding interval.

B. Building the Simulation Over the Full Time Horizon

1) Coefficient Initialization:

  • Initialization matches the trajectory derivatives to the known start derivative and a predicted end derivative based on a forward Euler state prediction.The fixed basis makes the represented trajectory derivative linear in the coefficient vector.Source: 1) Coefficient Initialization · p. 5
  • When the operating segment remains smooth, RAPTOR biases new coefficients toward the previous interval’s accepted coefficients, weakening or resetting that continuation after discrete events.A regularized linear system combines the available terms, and its solution initializes the nonlinear solver.Source: 1) Coefficient Initialization · p. 6

2) Residual-Based Interval Acceptance:

RAPTOR evaluates an endpoint DAE residual indicator to decide whether to accept an interval or retry it at a shorter length.

2) Residual-Based Interval Acceptance:

  • An interval is accepted when err_k < 1; when err_k > 1, RAPTOR rejects it, shortens the interval, and repeats the solve.After acceptance, the endpoint state becomes the initial condition for the next interval.Source: 2) Residual-Based Interval Acceptance · p. 6
  • The endpoint indicator measures differential-equation mismatch normalized by user-defined absolute and relative tolerances atol and rtol.The numerator is the differential-equation mismatch at the interval endpoint.Source: 2) Residual-Based Interval Acceptance · p. 6
  • Unlike classical adaptive ODE step acceptance based on local truncation error, RAPTOR bases the decision on the DAE residual..Source: 2) Residual-Based Interval Acceptance · p. 6

3) Adaptive Interval-Length Selection:

RAPTOR adapts interval lengths with a PI-type controller after accepted intervals and shrinks and retries rejected intervals. Discrete events terminate the current smooth trajectory and trigger reinitialization on the updated system.

3) Adaptive Interval-Length Selection:

  • The PI controller uses the current error to react to the present interval and the previous error to smooth interval-length changes.For the first accepted interval, RAPTOR sets the unavailable previous error equal to the current error.Source: 3) Adaptive Interval-Length Selection · p. 6
  • After acceptance, a safety factor and minimum and maximum growth limits constrain the next interval length.The update aims to enlarge intervals when error is well below one and keep them similar when error is near one.Source: 3) Adaptive Interval-Length Selection · p. 6
  • A rejected interval does not advance the state; RAPTOR reduces its length and retries the same interval.The rejection shrink factor controls the immediate reduction.Source: 3) Adaptive Interval-Length Selection · p. 6
  • At a discrete event, RAPTOR ends the interval, updates the active state and algebraic equations, computes a consistent post-event state, and starts a new trajectory.Events include faults, breaker actions, and limiter activations.Source: 3) Adaptive Interval-Length Selection · p. 6
  • Before previous error information is available, RAPTOR selects its first interval length using a standard adaptive-ODE starting-step heuristic..Source: 3) Adaptive Interval-Length Selection · p. 6

IV. RESULTS AND PERFORMANCE ANALYSIS

RAPTOR is evaluated across RMS and EMT cases selected to span stiffness, multiscale behavior, oscillations, and fast electromagnetic transients. The setup compares it with established solvers on identical systems, using empirically selected accuracy/runtime configurations.

IV. RESULTS AND PERFORMANCE ANALYSIS

  • The suite spans RMS and EMT regimes, including converter and transmission-line models, IEEE systems, and synchronous-machine and inverter EMT models; very-large-system scalability is left for future work.The authors emphasize consistent performance across diverse numerical regimes because no single case captures all relevant challenges.Source: §IV · p. 6

A. Experimental and Evaluation Setup

  • Experiments use one CPU core, exclude one-time JAX compilation and warm-up, and solve RAPTOR intervals with a modified Newton (Chord) method.The method reuses the residual Jacobian and its factorization across coefficient updates.Source: A. Experimental and Evaluation Setup · p. 7
  • RMS component-model errors are measured against a high-accuracy Kvaerno–5 reference, while EMT results are averaged across 10 disturbance events.RMS results are averaged over 10 initial-condition instances; reported Best Time and Best MAE configurations use empirically selected solver settings.Source: A. Experimental and Evaluation Setup · p. 7
  • RAPTOR is compared with representative stiff and nonstiff solvers, and the physical system and control parameters are held fixed across methods for each benchmark.RAPTOR and fixed-step EMT trapezoidal solvers are custom implementations; conventional solvers use SciPy or Diffrax implementations.Source: A. Experimental and Evaluation Setup · p. 7

B. RMS Benchmarks: Semi-Explicit ODE/DAE Models

Across the RMS benchmarks, RAPTOR maintains high accuracy with relatively few accepted intervals, including on IEEE systems up to 118 buses. On the cascaded line, it compares favorably in accuracy at competitive runtime, though RBF count affects the trade-off.

1) Cascaded M-Section Line Benchmark:

  • RAPTOR configurations show significantly lower error than other variants on the cascaded M-section line, while Tsit5 and Dopri5 fail in 80% of tested initial conditions.Among completed explicit runs, errors are generally substantially larger; Fig. 3 presents the speed–accuracy trade-off.Source: Figure 3 · p. 7, 1) Cascaded M-Section Line Benchmark · p. 8
  • On the cascaded line, RAPTOR completes all trials and keeps accepted-advance counts within a comparatively narrow range, but does not attain the lowest runtime or fewest advances.Configurations with runtime similar to RAPTOR have significantly larger errors; Fig. 4 plots accepted advances/timesteps against runtime.Source: Figure 4 · p. 7, 1) Cascaded M-Section Line Benchmark · p. 8
  • The cascaded line includes fast oscillatory modes up to approximately 7 kHz, and RAPTOR closely reproduces reference current and voltage trajectories over 50–58 ms.Fig. 5 compares the reference solution with RAPTOR’s best-MAE configuration.Source: Figure 5 · p. 7, 1) Cascaded M-Section Line Benchmark · pp. 7–8
  • RAPTOR-8 achieves the highest accuracy at runtime similar to RAPTOR-6, whereas RAPTOR-10 takes longer and is less accurate than RAPTOR-6.Too few RBFs can limit trajectory flexibility, while extra RBFs enlarge the nonlinear coefficient problem without necessarily adding useful resolution.Source: 1) Cascaded M-Section Line Benchmark · p. 8

C. Classic RMS Power System Benchmarks: IEEE 9-, 14-, 39-,

  • 22–32 intervals and errors of 10^-7–10^-8 characterize RAPTOR across the IEEE 9-, 14-, 39-, 57-, and 118-bus cases; on 118 buses it is 16.17× faster than accuracy-qualified Radau and 1.12× faster than LSODA.The largest case completes within 0.312 s; RAPTOR’s runtime increases with DAE order, while accepted-advance counts do not generally increase with system size.Source: Table I · p. 8, C. Classic RMS Power System… · p. 8
  • The IEEE benchmark table reports RAPTOR state and bus-voltage MAEs and speedups relative to each solver’s fastest accuracy-qualified configuration.The table covers a 1 s simulation horizon.Source: Table I · p. 8
  • For the IEEE RMS cases, runtime comparisons use the fastest tested configuration meeting the normalized state-error, bus-voltage MAE, and maximum KCL-residual requirements.The same disturbance sequence is applied across systems, including changes to motor torque, inverter active-power reference, and synchronous-machine AVR voltage reference.Source: C. Classic RMS Power System… · p. 8
  • The EMT figure compares RAPTOR trajectories with a fine simultaneous trapezoidal reference at common evaluation times.For one operating condition with a switching event at 50 ms, RAPTOR reproduces the reference despite using a small number of accepted intervals.Source: Figure 6 · p. 9

D. EMT Benchmarks: Trapezoidal and Delayed-Control Comparisons

The EMT comparisons test RAPTOR against custom fixed-step trapezoidal solvers, including a synchronous-machine switching event. In that benchmark, RAPTOR matches the fine reference with substantially longer effective intervals and can provide continuous trajectory values within each interval.

D. EMT Benchmarks: Trapezoidal and Delayed-Control Comparisons

  • RAPTOR is compared with custom fixed-step implicit trapezoidal solvers, including a fine simultaneous reference and a grid-following-inverter delayed-control version.The delayed-control version uses the previous timestep’s controller output, introducing a one-step delay; reported runtimes are from custom implementations, not the PSCAD executable.Source: D. EMT Benchmarks: Trapezoidal and… · p. 9

1) EMT Synchronous Machine Benchmark:

  • For the synchronous-machine test, an infinite-bus voltage source switches off at 0.05 s in a model with 8 differential states and 11 algebraic states.The reference uses implicit trapezoidal integration with a full Newton solve at each micro-step, at ∆t = 10−6 s and ∆t = 10−7 s.Source: 1) EMT Synchronous Machine Benchmark · p. 9
  • RAPTOR closely matches the fine simultaneous trapezoidal trajectory with few accepted intervals, and each interval’s PIRPNN provides continuous state values at any desired time without extra integration or nonlinear solves.The interval endpoints mark validity ranges, not the only times when the solution is available.Source: 1) EMT Synchronous Machine Benchmark · p. 9
  • Runtime below 0.5 s and last-point maximum error of 2.3 × 10−6; RAPTOR gains over an order of magnitude over the finer ∆t = 10−7 s trapezoidal reference.The comparison supports accurate reproduction of the benchmark’s current and voltage waveforms over substantially longer effective intervals.Source: 1) EMT Synchronous Machine Benchmark · p. 9

2) EMT Grid-Following Inverter:

Across RMS, networked DAE, and EMT benchmarks, RAPTOR meets the prescribed accuracy requirements and reports speedups over the tested baselines. In the grid-following inverter test, it captures fast electrical and slower controller responses while outperforming both trapezoidal comparisons.

2) EMT Grid-Following Inverter:

  • For the inverter test with two consecutive disturbances, RAPTOR closely agrees with the fine fully coupled trapezoidal reference and captures both fast electrical response and slower controller dynamics with high accuracy.The manuscript presents one example trajectory and selected states; additional evaluation and tests are in the supplementary material.Source: 2) EMT Grid-Following Inverter · p. 9
  • Using 360 accepted intervals on average, RAPTOR completes the 2 s inverter simulation in 1.6 s, approximately 2.2× faster than delayed-control trapezoidal and 18.2× faster than the fine fully coupled reference.RAPTOR evaluates electromagnetic and network equations using current controller values, avoiding the delayed-control formulation’s one-step controller delay.Source: 2) EMT Grid-Following Inverter · p. 10

V. CONCLUSION

  • RAPTOR represents stiff power-system ODEs and index-1 DAEs using fixed Gaussian RBFs with broad and localized shapes, whose coefficients a nonlinear solver determines at collocation points.The solver also uses physics-guided random shape selection, coefficient initialization, residual-based interval acceptance, interval-length control, advanced linear algebra, and event-aligned restart.Source: §V · p. 10
  • The conclusion identifies short fast transients embedded in slower dynamics as a harder regime than stiffness alone, motivating interval-wide trajectories built from broad and localized temporal functions.The paper notes that mature implicit methods can already be effective for small stiff systems with smooth trajectories.Source: §V · p. 10
  • RAPTOR reports 16.17× speedup over accuracy-qualified Radau on IEEE 118, approximately 2.2× over delayed-control trapezoidal, and 18.2× over the fine fully coupled reference while meeting prescribed accuracy requirements.The reported results span tested RMS, networked DAE, and EMT benchmarks; RAPTOR is a custom research implementation, and further work will target per-interval cost.Source: §V · p. 10
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