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Verification and synthesis of co-simulation algorithms subject to algebraic loops and adaptive steps

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  • Special Issue: FMICS 2021
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Abstract

Simulation-based analyses are becoming increasingly vital for the development of cyber-physical systems. Co-simulation is one such technique, enabling the coupling of specialized simulation tools through an orchestration algorithm. The orchestrator describes how to coordinate the simulation of multiple simulation tools. The simulation result depends on the orchestration algorithm that must stabilize algebraic loops, choose the simulation resolution, and adhere to each simulation tool’s implementation. This paper describes how to verify that an orchestration algorithm respects all contracts related to the simulation tool’s implementation and how to synthesize such tailored orchestration algorithms. The approaches work for complex and adaptive co-simulation scenarios and have been applied to several real-world case studies.

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Notes

  1. The model is available online: https://github.com/INTO-CPS-Association/Scenario-Verifier.

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Acknowledgements

We are grateful to the Poul Due Jensen Foundation, which has supported the establishment of a new Centre for Digital Twin Technology at Aarhus University. We are would also like to thank the anonymous reviewers of the paper, who have provided valuable feedback on the paper.

Author information

Authors and Affiliations

  1. DIGIT, Department of Electrical and Computer Engineering, Aarhus University, Åbogade 34, 8000, Aarhus, Denmark

    Simon Thrane Hansen, Casper Thule, Cláudio Gomes & Frederik Madsen

  2. DIGIT, Department of Computer Science, Aarhus University, Åbogade 34, 8000, Aarhus, Denmark

    Jaco van de Pol

  3. DII, Department of Information Engineering, University of Pisa, Pisa, Italy

    Maurizio Palmieri

  4. Department of Information Engineering, Instituto Tecnologico de Aragon, Zaragoza, Spain

    Jesús Alfonso, José Ángel Castellanos & José Manuel Rodriguez

  5. Department of Mechanical Engineering, KU Leuven, Celestijnenlaan 300, 3001, Leuven, Belgium

    Emin Oguz Inci

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  1. Simon Thrane Hansen
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Appendices

Appendix A: Table of conventions

This appendix contains a table describing the notation used throughout the paper. Capitalized letters refer to sets, while lower case letters refer to a variable belonging to the set represented by the capitalized letter.

Convention

Description

\(U\)

All Inputs of the Scenario

\(U_{c}\)

Inputs of the SU c

\(Y\)

All Outputs of the Scenario

\(U_{c}\)

Outputs of the SU c

\(S_{}\)

States

\(s_{c}^{(t)}\)

State of SU c at time t

\(\mathcal {V_{T}}{}\)

Time stamped values of the type \(\mathcal {V}\times \mathbb {R}_{\ge 0}\)

\(s^{R}_{c}\)

The abstract state of SU c

t

Time t (\(t \in \mathbb {R}_{\ge 0}\))

H

Step duration H (\(H \in \mathbb {R}_{> 0}\))

\(L\)

Couplings between SUs

\(F\)

Feed-through constraints

\(R\)

Reactivity constraints

\(C\)

A set of SU identifiers

\(\mathcal {A}\)

Adaptations

\(M\)

A set of SUs that may reject a step duration

\(B\)

A set of SUs that must be backtracked

Appendix B: BNF grammar

The section presents the domain-specific language where user can describe co-simulation algorithms and scenarios for both simple, complex, and adaptive co-simulation scenario.

Examples of algorithms and scenarios described using the DSL are available online https://github.com/INTO-CPS-Association/Scenario-Verifier/tree/master/src/test/resources.

figure e
figure f

Appendix C: Algorithm of nested complex scenario

The co-simulation step of the scenario in Fig. 8a on page 26.

figure g

Appendix D: Parameters of the full vehicle model

Table 1 Parameters of the longitudinal vehicle model
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Table 2 Torque profile
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Table 3 Parameters of the vertical vehicle model
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Hansen, S.T., Thule, C., Gomes, C. et al. Verification and synthesis of co-simulation algorithms subject to algebraic loops and adaptive steps. Int J Softw Tools Technol Transfer 24, 999–1024 (2022). https://doi.org/10.1007/s10009-022-00686-8

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