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Energy Conservation and Power Bonds in Co-Simulations: Non-Iterative Adaptive Step Size Control and Error Estimation

Severin Sadjina, Lars T. Kyllingstad, Eilif Pedersen, Stian Skjong

arXiv:1602.06434v2eess.SYcs.CEcs.DCcs.SY

TL;DR

Co-simulation needs accurate and stable coupling while preserving modularity, computational efficiency, and simulator confidentiality. The paper introduces power-bond energy residuals and the non-iterative ECCO method for adaptive macro step size control using current coupling data. In a quarter car benchmark, errors were reduced by up to 93% at no additional computational cost.

  • Problem

    Co-simulation faces stability and accuracy challenges, while many error-control methods require rollback or simulator-internal information unavailable in commercial or protected models.

    Source: A. Local Error Estimators · p. 5

  • Method

    ECCO uses power-bond energy residuals as a global error estimator and adaptive macro step size controller based only on current coupling data.

    Source: §I · p. 2, §VI · p. 12

  • Results

    Errors were reduced by between 80% and 93% at no additional computational cost in the quarter car benchmark’s alternative system reticulation.

    Source: C. Alternative System Reticulation · p. 10, §VI · p. 12

  • Takeaways & Limitations

    ECCO improves co-simulation accuracy and efficiency without rollback or simulator-internal information, supporting protection of sensitive model information.

    Source: B. Nonlinear Damping · p. 9, §VI · p. 12

Abstract

from arXiv · show

Here, we study the flow of energy between coupled simulators in a co-simulation environment using the concept of power bonds. We introduce energy residuals which are a direct expression of the coupling errors and hence the accuracy of co-simulation results. We propose a novel Energy-Conservation-based Co-Simulation method (ECCO) for adaptive macro step size control to improve accuracy and efficiency. In contrast to most other co-simulation algorithms, this method is non-iterative and only requires knowledge of the current coupling data. Consequently, it allows for significant speed ups and the protection of sensitive information contained within simulator models. A quarter car model with linear and nonlinear damping serves as a co-simulation benchmark and verifies the capabilities of the energy residual concept: Reductions in the errors of up to 93% are achieved at no additional computational cost.

I. INTRODUCTION

Co-simulation supports modular, parallel, and information-protective engineering simulation, but stability and accuracy remain difficult because common adaptive methods require rollback or simulator-internal data. The paper introduces power bonds and energy residuals as coupling-level tools for assessing and controlling these errors.

I. INTRODUCTION

  • Co-simulation enables independent and parallel modeling, domain-specific model reuse, simulation speedups, and protection of simulator-internal information.Source: §I · p. 1
  • Stability and accuracy are major challenges because macro step sizes are often chosen manually, while general-purpose error estimation and control remain lacking.Source: §I · p. 1
  • Most adaptive co-simulation methods require simulator-internal information, rollback, or both, limiting their use with commercial and information-sensitive simulators.Source: §I · p. 1
  • ECCO uses energy residuals to control co-simulation without iteration, rollback, or simulator-internal information, relying only on current coupling data.Source: §I · p. 2

A. Energy Continuity

B. The Power Bond in a Co-Simulation

In co-simulation, independently advancing simulators exchange extrapolated coupling inputs at discrete communication points. When the exchanged variables form a power bond, the simulators’ reported powers generally fail to balance, producing a measurable residual.

B. The Power Bond in a Co-Simulation

C. Residual Powers and Energies

Independent simulator integrations create residual powers and energies that quantify coupling-induced violations of energy conservation. These residuals directly alter the coupled system’s energy and provide a practical basis for error estimation using coupling data.

C. Residual Powers and Energies

III. ERROR ESTIMATION AND ADAPTIVE STEP SIZE CONTROL

The paper uses energy residuals as a global co-simulation error estimator and basis for adaptive macro step size control. This approach avoids repeating macro steps and requires no knowledge of simulator implementations.

III. ERROR ESTIMATION AND ADAPTIVE STEP SIZE CONTROL

  • Energy residuals enable a global error estimator and a non-iterative adaptive macro step size controller without repeated macro steps or simulator-implementation knowledge.Source: §III · p. 5

A. Local Error Estimators

Conventional co-simulation error estimators often compare multiple approximations, requiring rollback, re-stepping, or simulator-internal information. ECCO instead uses power-bond energy residuals, which directly represent coupling errors and provide exact average local power and energy errors.

A. Local Error Estimators

  • Conventional estimators compare multiple simulator approximations, but their implementation can require repeated macro steps and access to unavailable subsystem information.Examples include Richardson extrapolation, embedded methods, compound macro steps, and Milne’s device.Source: A. Local Error Estimators · p. 5
  • Rollback is required by these approaches, complicating practical use and particularly prohibiting applications such as Hardware-in-the-Loop.Re-stepping is also time-consuming and can reduce the efficiency gains of adaptive step size control.Source: A. Local Error Estimators · p. 5
  • Internal integration order and other simulator information may be unavailable or sensitive, limiting error estimation with commercial and industrial tools.The Busch et al. estimator is identified as an exception requiring neither re-stepping nor internal simulator information.Source: A. Local Error Estimators · p. 5

B. Error Estimation Based on Energy Conservation

  • Power-bond energy residuals directly express the energy incorrectly added to the coupled system and therefore serve as energy-based error estimators.Power residuals represent average local power errors, while energy residuals represent average local errors in transmitted energy.Source: B. Error Estimation Based on Energy… · p. 5
  • The residual-energy equations give exact average local errors irrespective of macro or micro step size and simulator implementation.The benchmark illustrates the relationship between power error and its residual estimate as macro step size varies.Source: B. Error Estimation Based on Energy… · p. 6, Figure 4 · p. 6
  • Energy-based metrics account directly for energy conservation and avoid disproportionate weighting when force, displacement, or velocity outputs have different numerical scales.The approach is applied to power-bond coupling and yields power-output error order O(∆t^(m+1)) under the stated extrapolation assumptions.Source: B. Error Estimation Based on Energy… · pp. 5–6

C. Adaptive Step Size Controller

ECCO turns residual energies into a scalar tolerance indicator and uses it to adapt the next macro step. The controller bounds step sizes and their rate of change while requiring neither repeated macro steps nor simulator-implementation knowledge.

C. Adaptive Step Size Controller

  • The scalar error indicator combines residual and transmitted energies across all power bonds with bond-specific energy scales and relative tolerances.Values at or below one indicate sufficiently small error relative to the configured tolerances; values above one indicate exceeded tolerances.Source: C. Adaptive Step Size Controller · p. 6
  • A PI controller uses the energy-based indicator to propose the next macro step size for accuracy-efficiency trade-offs.The controller uses integral and proportional gains with a safety factor, followed by user-defined step-size bounds and rate limits.Source: C. Adaptive Step Size Controller · p. 6
  • The controller constrains the next step with ∆t_min ≤ ∆t_i+1 ≤ ∆t_max and Θ_min ≤ ∆t_i+1/∆t_i ≤ Θ_max.Typical rate bounds are Θ_min ∈ [0.2, 0.5] and Θ_max ∈ [1.5, 5.0].Source: C. Adaptive Step Size Controller · p. 6
  • ECCO performs non-iterative error estimation and adaptive step control without repeating macro steps or requiring simulator-implementation information.This design is intended to improve co-simulation accuracy and efficiency while accommodating closed or protected simulator models.Source: C. Adaptive Step Size Controller · p. 6

D. Predictor/Corrector Error Estimator

The predictor/corrector estimator offers a non-iterative alternative by comparing polynomial predictions with corrected outputs. Compared with ECCO, it is less demanding on model interfaces but can be less reliable at larger steps and is sensitive to predictor order and output scaling.

D. Predictor/Corrector Error Estimator

  • Polynomial extrapolation predicts the next output from previously obtained values, and the difference from the corrected output supplies a local error estimate.A PI controller then uses a scalar error indicator to adapt the macro step size.Source: D. Predictor/Corrector Error Estimator · p. 7
  • ECCO gives the exact average local error in connected power outputs for any macro or micro step size, whereas predictor/corrector estimates may require relatively small macro steps.The predictor/corrector estimate is also generally sensitive to the polynomial order.Source: D. Predictor/Corrector Error Estimator · p. 7
  • The predictor/corrector method is less demanding than ECCO because it does not depend on power bonds, which are not yet used by most available models and tools.Its broader applicability follows from the status quo of existing co-simulation models and tools.Source: D. Predictor/Corrector Error Estimator · p. 7
  • ECCO’s global estimator is insensitive to simulator-output scaling, whereas the predictor/corrector estimator is not.This scaling sensitivity can create practical implementation difficulties for the predictor/corrector method.Source: D. Predictor/Corrector Error Estimator · p. 7

IV. CO-SIMULATION BENCHMARK MODEL

The quarter car model is adopted as a co-simulation benchmark, split into two simulators connected through a power bond. With a constant 1 ms macro step, the coupling introduces measurable residual energy.

  • The quarter car model is a mechanical interpretation of a linear 2-DOF oscillator and is modified to include nonlinear damping for benchmarking.Source: §IV · p. 7
  • The benchmark describes chassis and wheel displacements through spring-damper dynamics, with parameters controlling spring stiffness, damping, and damping-force nonlinearity.Source: §IV · p. 7
  • The co-simulation splits the model into S1 for the chassis and S2 for the remaining system, with coupling variables defining a power bond.Source: §IV · p. 8
  • 6.4 J of residual energy is incorrectly added with a constant macro step of ∆t = 1 ms, compared with 750 J initially stored and later dissipated.Source: §IV · p. 8

A. Adaptive Step Size Control

ECCO uses residual energy to select the next macro step and balance accuracy against computational efficiency. In the linear benchmark, adaptive control reduces errors at similar cost or reaches comparable error with fewer steps.

  • The controller uses a scalar residual-energy error indicator and PI control, with tolerance tuning governing the accuracy-efficiency balance.The energy scale is set to the system’s initial excitation E0 = 750 J.Source: A. Adaptive Step Size Control · p. 8
  • 70% reduction in mean absolute power error is achieved at approximately the same average macro step size and calculation time as constant stepping.The adaptive run uses tolerance r = 2.8 × 10−6 and average ∆t ≈1 ms.Source: A. Adaptive Step Size Control · p. 8, Table III · p. 8
  • Residual energy falls to approximately 1.6 J with adaptive control, only 25% of the 6.4 J accumulated using the constant-step simulation.Source: A. Adaptive Step Size Control · p. 8
  • With r = 3.1 × 10−5, adaptive control reaches |∆P| ≈1.3 W using about one-third of the total time steps required without the algorithm.Energy flows remain more accurately described than with constant step sizes.Source: A. Adaptive Step Size Control · p. 8

B. Nonlinear Damping

Nonlinear damping preserves the benefits of residual-energy-based adaptive control, while an alternative subsystem reticulation is less favorable but can be substantially improved by adaptation. Across these cases, reported error reductions reach 80%–93% without added computational cost.

B. Nonlinear Damping

  • B. Nonlinear Damping: the nonlinear benchmark returns to equilibrium faster than the linear case and is simulated only to t = 2 s.Source: B. Nonlinear Damping · pp. 8–9, Figure 9 · p. 9
  • B. Nonlinear Damping: 70% reductions in both mean absolute power error and wrongly added residual energy are achieved at about the same computational cost.Increasing tolerance preserves approximately the constant-step error while requiring about one-third as many time steps for |∆P| ≈4 W.Source: B. Nonlinear Damping · p. 9, Table V · p. 9

C. Alternative System Reticulation

V. COMPARISON TO PREDICTOR/CORRECTOR METHOD

The paper compares ECCO with a predictor/corrector controller across linear, nonlinear, and alternative-reticulation benchmarks. Predictor/corrector control reduces errors, but ECCO generally achieves larger energy-error reductions, subject to scaling and tuning considerations.

  • Predictor/corrector control reduces errors by 40%–55% versus constant step sizes across the reported benchmarks and by 40%–93% for alternative reticulation.Source: §V · p. 11, Table IX · p. 11, Table X · p. 11
  • ECCO generally produces more substantial reductions in energy errors, although both methods reach relatively high accuracy for the linear alternative-reticulation benchmark.Source: §V · p. 11, Table XII · p. 11
  • The predictor/corrector error indicator is distorted when spring-force outputs are much larger than velocity outputs, effectively disregarding one simulator’s contribution.Simulator-specific tolerances and weights can mitigate this issue but increase the number of tunable parameters to four.Source: §V · pp. 11–12
  • The predictor/corrector step-size selection is highly oscillatory in these benchmarks, likely because of PI-controller tuning and the output-scale sensitivity.Source: §V · p. 12

VI. CONCLUSION

The paper introduces power-bond energy residuals as coupling-error measures and uses them in ECCO, a non-iterative adaptive step-size method requiring only simulator input and output data. ECCO improves co-simulation accuracy and efficiency while avoiding rollback and simulator-internal information, though stability remains insufficiently addressed in general.

  • Energy residuals emerge from independent subsystem solves between communication points and directly alter total system energy and dynamic behavior.They are computable from simulator input and output values only and express coupling errors.Source: §VI · p. 12
  • ECCO uses energy residuals to adapt co-simulation step sizes, ensuring approximately the correct energy exchange and improving accuracy and efficiency over constant step sizes.The quarter car benchmark includes linear and nonlinear damping characteristics.Source: §VI · p. 12
  • Adaptive step-size control makes global accuracy less dependent on system reticulation and subsystem time-integration accuracy.Source: §VI · p. 12
  • ECCO avoids rollback and simulator-internal information, supporting sensitive-data protection and practical use with closed or commercial simulation tools.These restrictions commonly hinder other adaptive co-simulation algorithms.Source: §VI · p. 12
  • Residual energies represent local power-output errors exactly, irrespective of macro- or micro-step size, unlike predictor/corrector estimation that requires sufficiently small macro steps.Source: §VI · p. 12
  • Power bonds provide universal energy-flow connectivity across engineering domains, but their limited adoption can make alternative coupling methods more advantageous in practice.Source: §VI · p. 12
  • The paper only briefly addresses co-simulation stability, whose general treatment remains difficult across solver orders, coupling schemes, feed-through, and algebraic loops.Source: §VI · p. 12
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