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Energy conservation and power bonds in co-simulations: non-iterative adaptive step size control and error estimation

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Abstract

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.

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Notes

  1. Note that we use the word ‘simulator’ throughout in the sense of a subsimulator: a mathematical model of a subsystem coupled to other such models of subsystems to form a full model of a total system.

  2. This is true if we assume an ideal bond that does not include physical energy sources or sinks. This is well justified in a co-simulation setting where (relevant) energy creation and dissipation ought to be accounted for inside subsystems. Our discussion is not limited by this choice, however, and it is fully possible to work with non-ideal bonds.

  3. Note that some coupling schemes interpolate input values. In the Gauss–Seidel iteration pattern, for example, one simulator is stepped first using input value extrapolation, and thereafter, the second simulator is stepped using interpolation of the other system’s outputs.

  4. The functions \(\mathbf {f}\) and \(\mathbf {g}\) may also depend on time explicitly.

  5. Note that the sign convention for the residual power is different than the one used for energy fluxes encountered so far. A positive residual power means that the energy of the total system increases.

  6. For a more comprehensive overview see, for example, Ref. [8] and references therein.

  7. S\(_1\) can be solved exactly with constant extrapolation of the input \(F_\text {c}(t) = F_\text {c}(t_i)\) for \(t \in (t_i,t_{i+1}]\).

  8. Setting \(k_\text {P} = 0\) and substantially lowering \(k_\text {I}\), for example, seems to be a much better choice to configure the PI-controller in this case.

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Acknowledgements

This work was funded by the Research Council of Norway (Grant Number 225322), and the industrial partners in the ViProMa project consortium (VARD, Rolls-Royce Marine, and DNV GL). We are grateful for their financial support.

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Authors and Affiliations

  1. Department of Marine Technology, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Severin Sadjina, Stian Skjong & Eilif Pedersen

  2. SINTEF Fisheries and Aquaculture, 7465, Trondheim, Norway

    Lars T. Kyllingstad

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  1. Severin Sadjina
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  2. Lars T. Kyllingstad
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  4. Eilif Pedersen
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Correspondence to Severin Sadjina.

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Sadjina, S., Kyllingstad, L.T., Skjong, S. et al. Energy conservation and power bonds in co-simulations: non-iterative adaptive step size control and error estimation. Engineering with Computers 33, 607–620 (2017). https://doi.org/10.1007/s00366-016-0492-8

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