The simulation of large scale nonlinear dynamical interconnected systems, as they arise in all modern engineering disciplines, is a usual task. Due to the high complexity of the considered systems, the principle of thinking in hierarchical structures is essential and common among engineers. Therefore, this contribution proposes an approach for the numerical simulation of large systems, which keeps the hierarchical system structure alive during the entire simulation process while simultaneously exploiting it for order reduction purposes. This is accomplished by embedding the trajectory piecewise linear order reduction scheme in a modified variant of the component connection modeling for building interconnected system structures. The application of this concept is demonstrated by means of a widely used benchmark example and a modified variant of it.

The design and operation of todays electrical systems, ranging from microelectronic circuits to power generation and distribution systems, is enabled by powerful simulation tools at hand. Models for such large scale systems are commonly built up hierarchically from smaller and less complex interconnected sub-systems. While this hierarchical point of view is the state-of-the-art, which is utilized within the modeling process itself, the actual simulation is performed by numerically integrating the resulting large scale system of nonlinear ordinary differential equations.

The same is true for model order reduction methods for linear systems,
(cf.

In contrast to this common practice, the simulation approach proposed in this
contribution makes use of the so called

Besides the mCCM itself, this contribution demonstrates, how the trajectory
piecewise-linear approach (TPWL)

In Sect.

The order reduction of linear dynamical systems of the form

As an example of such a method of computing a transformation matrix

A key benefit of the BT reduction scheme described above is the existence of
a maximum error bound

For the reduction of nonlinear dynamical systems of the form

Restricting the structure of the considered original nonlinear dynamical
system of order

While in the linear case, the system matrix of the reduced system of true
order

Over the last decades, several approaches have been developed in order to
find usable state transformations

One of these approaches is the TPWL order reduction scheme originally
developed by

In order to be able to evaluate the reduction transformations in

Since Eq. (

Also on an empirical basis, a procedure to select the linearization points

Even if there are many examples in the literature, in which the TPWL
(

Stages of the hierarchical model formulation process within the mCCM
framework;

Originally introduced by

Within the proposed mCCM, which is a structurally simplified variant of the
original CCM aiming towards hierarchical system formulations, the
interconnection process of individual components

In the first stage, the components

In the second formal step, the individual components

These are introduced in the third step by means of a linear algebraic

As depicted in Fig.

Schematic representation of the modular concept of CoSimMA.

As originally introduced in

In this section, first the application of the hierarchical modeling approach utilizing the mCCM framework is demonstrated by means of an example circuit. Afterwards, several reduced order models are generated by applying the TPWL order reduction scheme combined with the BT for reducing the intermediate linear systems.

Example system for the mCCM modeling and TPWL reduction;

The example circuit treated in this section originates from

In the original version of the considered example, the capacitor voltage

In order to demonstrate the severe influence of the choice of variables which
are to be preserved throughout the reduction process, a second scenario is
treated in the following. Therein, additionally to the fist capacitor voltage

Following the presented steps of the mCCM modeling procedure in
Sect.

The

Behavior of the weight functions

As described in Sect.

Comparison of the original model with two reduced variants for the case 1.

In this first reduction example the capacitor voltage

For the test trajectory resulting form zero initial conditions for all state
variables and a step excitation of

Reducing the order to

Comparison of the original model with two reduced variants for the case 2.

As a second case, a slightly modified variant of the original system in

The further reduction with

In order to clarify the severe differences in the reduction results between
case 1 and case 2, the Hankel singular values of the linearized and balanced
models of both cases are compared in Fig.

Comparison of the Hankel singular values

With the introduction of mCCM, a modified and structurally extended version of the component connection modeling by DeCarlo and Saeks, a useful framework for the convenient and efficient formulation of hierarchical interconnected dynamical systems had been presented and its advantageous properties were discussed in detail. Due to the rising computational complexity with an increasing number of components to be simulated, it is desirable to have advanced simulation techniques such as model order reduction methods in a usable form at hand. Therefore, it had been described, how such order reduction schemes can be embedded into the software framework CoSimMA, a fully functional simulator based on the mCCM formulation of interconnected dynamical systems. Two model order reduction schemes, namely the well established balanced truncation method for linear systems as well as the TPWL method for nonlinear systems were described and discussed with respect to their basic properties, advantages and disadvantages. Finally, a modeling, reduction and simulation study by means of a standard benchmark example had been presented and critically discussed with respect to the quality of the reduced models and the savings in computation time.

As a main result it had been shown, that the reducibility of a given system strongly depends on the choice of the output variables which are preserved throughout the reduction process. In the case of a well reducible system, considerable savings in computation time could be achieved while keeping the approximation error in moderate bounds. In the other case, only an insignificant reduction of the computation time or prohibitively large approximation errors result.

In summary, efforts considering not only order reduction techniques but also
efficient model formulations, cf.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Kleinheubacher Berichte 2017”. It is a result of the Kleinheubacher Tagung 2017, Miltenberg, Germany, 25–27 September 2017.

The authors would like to thank the Volkswagen Foundation for the financial support of this work within the project AMSES (Aggregated Models for the Simulation of Electromechanical Power Systems). Furthermore, the authors thank B. Eng. Philipp Eggers for providing parts of the simulation results of the presented example. The publication of this article was funded by the open-access fund of Leibniz Universität Hannover. Edited by: Jens Anders Reviewed by: two anonymous referees