Realization-preserving model reduction of object-oriented models using energy-based metrics

Cyber-physical modelling in Modelica with model-reduction techniques

In this paper are discussed model order reduction and simplification techniques (e.g.; metrics used by these techniques for rankings of elements) which are applicable to wide range of Modelica models built from already available libraries. Modelica models are translated to set of differential-algebraic equations and for the latter there are numerous tools for model order reduction already available. However; these tools are not designed for helping users understand the model’s behavior and the reduced model may be hard to understand by the user because the structure of the original model is lost. Hierarchical decomposition of the model must be presereved and if the model is developed with a graphical schematics then elements (nodes) of the schematics must be ranked. Therefore we adapted energy-based metrics used in ranking of bond-graphs’ elements to much more losely defined Modelica’s schematics; so they can be used complementary with ranking methods that work with equations.

We describe the model reduction software developed for the control and systems library SLICOT. Besides a powerful collection of Fortran 77 routines implementing the last algorithmic developments for several well-known balancing related methods, we also describe model reduction tools developed to facilitate the usage of SLICOT routines in user friendly environments like MATLAB or Scilab. Extensive testing of the implemented tools has been done using both special benchmark problems as well as models of several complex industrial plants. Testing results and performance comparisons show the superiority of SLICOT model reduction tools over existing model reduction software.

Modelica is a new object-oriented language for modeling of large, complex, and heterogeneous physical systems. It is designed to allow for convenient and efficient modeling and simulation of the differential-algebraic equations (DAEs) systems. However, in many engineering fields, partial differential equations (PDEs) are applied. So it is necessary to be able to handle PDE models in Modelica. In this paper, we present how the Modelica language can be extended to support object-oriented modeling with partial differential equations. The description of PDE models in Modelica is introduced, which uses a natural way to describe the PDE models so that the extending language is easier to be understood and more convenient to be used. The prototype implementation included external translator and processor is described. An example modeled with PDE in Modelica and solved using the prototype is presented.

The paper briefly describes the Modelica model of a cubic shaped room with one window. The 'physical' model was then implemented as a Modelica (Dymola) block in MATLAB-Simulink environment. Simulink was used for the realisation of different control schemes, which were 'manually' and 'automatically' optimised. The experiments show that the synergetic combination of MATLAB-Simulink and Dymola-Modelica environments is an efficient and powerful approach giving the possibility to realise several important goals: realisation preserving modelling in Modelica, efficient simulation with Simulink and many possibilities for control system design and optimisation using basic MATLAB and appropriate MATLAB toolboxes. However the experiences with Modelica modelling taught us that Modelica models become rather complex and therefore model reduction techniques in order to obtain usable and efficient models are desired. The last part of the paper briefly describes some research activities in this area and also our contributions.

A comparative assessment of efficient uncertainty analysis techniques for environmental fate and transport models: application to the FACT model

This paper presents a simulation study of an optimal control approach being applied to control the volume trajectory of a heart's ventricle using a ventricular assist device (VAD). Usually VADs are used for heart failure therapy. However, in this study a novel VAD application within the context of animal trials will be investigated. Preload (the volume of blood within the ventricle just before the contraction) is an important determinant of heart function and therefore an interesting quantity to be manipulated in trials. To overcome limitations of classical preload manipulation techniques we propose to use a VAD to manipulate preload. The aim of this study will be to find the input trajectories for the VAD which are required to precisely hit specific preload values. Two different test scenarios were defined: the first scenario aims at resembling a traditional preload reduction technique. The second scenario defines a preload progression which is not possible to realize with traditional preload manipulation techniques. Besides the test scenarios, a comparison to experimentally obtained data is drawn. Object-oriented Modelica models of a simplified cardiovascular system and the VAD were developed to formulate a nonlinear optimal control problem. A fully open-source software tool-chain was used to automatically discretize the problem into a collocation scheme and to generate the necessary inputs for the nonlinear optimization solver. This tool-chain includes the software packages JModelica, CasADi and Ipopt. Object-oriented modeling in combination with the automated tool-chain allow for easy and fast modifications of the underlying models as well as the introduction of new objectives. Hence, the presented approach can easily be modified to be applied to other scopes of VAD and cardiovascular research.

An approach to agent-based modeling with Modelica

The complexity of mathematical models in biology has rendered model reduction an essential tool in the quantitative biologist's toolkit. For stochastic reaction networks described using the Chemical Master Equation, commonly used methods include time-scale separation, Linear Mapping Approximation, and state-space lumping. Despite the success of these techniques, they appear to be rather disparate, and at present, no general-purpose approach to model reduction for stochastic reaction networks is known. In this paper, we show that most common model reduction approaches for the Chemical Master Equation can be seen as minimizing a well-known information-theoretic quantity between the full model and its reduction, the Kullback-Leibler divergence defined on the space of trajectories. This allows us to recast the task of model reduction as a variational problem that can be tackled using standard numerical optimization approaches. In addition, we derive general expressions for propensities of a reduced system that generalize those found using classical methods. We show that the Kullback-Leibler divergence is a useful metric to assess model discrepancy and to compare different model reduction techniques using three examples from the literature: an autoregulatory feedback loop, the Michaelis-Menten enzyme system, and a genetic oscillator.

Model reduction techniques are simplification methods based on mathematical approaches employed to realize reduced models for the original high order systems. Some existing classical model reduction techniques for multivariable system are considered and compared for their performances. Interlacing property and coefficients matching (IPCM) method gives overall minimum integral square error (ISE), integral absolute error (IAE) and integral time absolute error (ITAE) values compared to other methods. Though the IPCM method is efficient, it may not guarantee for minimization of all objective functions simultaneously. In this paper, model reduction approach based on objectives like ISE, IAE and ITAE using multi-objective differential evolution (MODE) method is proposed for reducing the numerator and the denominator is reduced by interlacing property. MODE method minimizes the small, normal and large errors persisting for long time between original and reduced models. This multi-objective approach is applied for model reduction of 10th order multivariable linear time invariant power system model. Simulation results are demonstrated for single and multi-objective model reduction and compared with multi-objective particle swarm optimization (MOPSO) method to prove the validity of proposed MODE technique.

<div class="section abstract"><div class="htmlview paragraph">Increasing demand for simulation accuracy often leads to increased finite element model complexity, which in turn, results in higher computational costs. As a provision, component mode synthesis approaches are employed to approximate the system response by using dynamic substructuring and model reduction techniques in linear systems. However, the use of available model reduction techniques in nonlinear problems containing the contact type of nonlinearities remains an interesting topic. In this paper, the application of a component mode synthesis method in squeak and rattle nonlinear simulation has been investigated. Critical regions for squeak and rattle of the side door model of a passenger car were modelled by nonlinear contact definition in finite element simulation. Craig-Bampton model reduction method was employed to substructure the finite element model while keeping the nonlinear contacts in the model. The model response was evaluated using the modal assurance criterion, frequency response analysis and contact force magnitude in comparison with the baseline model. Results showed that a great reduction in computational time (about 98%) can be achieved while the accuracy of the system response was maintained at an acceptable range for the intended application for squeak and rattle simulation. Although the prediction of impact events in time was done accurately, the contact force magnitude was estimated with average error of 2.5% to 22%, compared with the baseline results. The outcomes of the study show that to empower squeak and rattle prediction by including contact interfaces in finite element simulations, implementation of the model reduction approach can compensate the simulation cost.</div></div>

District heating grids (DHG) face the problem of seasonal variability and peak demands, e.g., there is a high demand for heat in winter and a low demand in summer. Operators of district heating networks and energy suppliers are therefore seeking solutions to store heat and access it at peak times, thus reducing the use of carbon-emitting heat sources. Aquifers are well suited for heat storage because of high storage capacity relative to the surface occupancy and reasonably high recovery efficiency, if geological suitable. In this study, the potential of Aquifer Thermal Energy Storages (ATES) contribution in decreasing seasonal peak heat loads is investigated by numerical modelling. Each part of the overall system (surface and subsurface) is modelled in their corresponding simulation environment. They are numerically coupled with a novel approach, allowing for highest accuracy in both subsystems. The District Heating Grid is modelled in the object-oriented modelling language Modelica including the Open-Source library MoSDH. Using the Functional Mock-Up Interface (FMI) the Modelica models are exported to executable Functional Mock-Up Units (FMU). The groundwater flow and heat transport processes are modelled in the finite-element software FEFLOW. By developing a C++ Plug-In for FEFLOW the FMU is imported and dynamically co-simulated. This approach allows for easier adjustments in both subsystems and more coupling options to existing models and softwares.  Through numerical analysis different hydrological and geological scenarios of the ATES and different operational cycles are investigated to determine the long-term efficiency and capacity of the storage. This modelling approach can be used to develop strategies for the operation of the ATES as well as to evaluate in advance whether geological conditions are suitable for the particular network situation.

This paper shows the various steps a simulation tool has to perform to create a real-time-capable model from a Modelica model. Reduction techniques are often necessary for complex models to meet the realtime requirements. For non-linear models with discontinuities no automatic methods of model reduction are known. The analysis methods supporting developers in identifying critical model parts are explained by means of an illustrating example model.

Model reduction techniques such as the proper orthogonal decomposition (POD) method can be used to reduce the computational effort of simulations with a very large number of degrees-of-freedom. The reduction of finite element models including elastoplastic material behavior is still far from being trivial and inevitably leads to problems of efficiency and accuracy. One aim of the present paper is to combine the two methods--sub-structuring and POD-based model reduction--to overcome these difficulties. A typical field of application where plasticity dominates the material behavior are forming processes. The second aim of the paper is to investigate the applicability of the new approach in this context. The presented combined approach selective POD (SPOD) where the reduction is only applied in sub-domains with approximately elastic behavior shows higher accuracy than the general POD method. In this paper, the SPOD method is extended by an adaptive method of sub-structuring (A-SPOD) in which the sub-domain where model reduction is applied is determined automatically. While achieving errors of the same magnitude as by means of SPOD, the computational effort can be reduced significantly by using the A-SPOD approach.

A method of model reduction for reducing a higher order transfer function to its lower order model is developed based on the shifted Legendre function approximation. The expansion coefficients of the shifted Legendre series which represents the approximate responses of transfer functions are computed by the recursive formula, via operational matrix approach. The significances of applying shifted Legendre function to model reduction problem are that the method is simple, straightforward and the computational results obtained are accurate as well as the final time of the control system can be adjustable without any restriction. Based on the model reduction technique, a new algebraic method is proposed for the design of a feedback control system to satisfy specifications. Illustrative examples are given and satisfactory results are obtained.