Quantized State System Simulation in Dymola/Modelica Using the DEVS Formalism

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Quantized State System Simulation in
Dymola/ModelicaUsing the DEVS Formalism

• Tamara Beltrame
• VTT, Industrial Systems
• PO Box 1000, VM3
• 02150 Espoo, Finland
• Tamara.Beltrame_at_vtt.fi

François E. Cellier Inst. of Computational
Science ETH Zurich 8092 Zurich,
Switzerland FCellier_at_Inf.ETHZ.CH
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Abstract

• A tool has been created for the simulation of
  highly discontinuous models in Dymola/Modelica
  that is based on Zeiglers DEVS (Discrete Event
  Systems) formalism.
• ModelicaDEVS provides a reimplementation of the
  PowerDEVS software, a tool developed for the
  simulation of physical systems that is based on
  Kofmans QSS (Quantized State Systems) algorithms.

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Overview

• Motivation
• The DEVS Formalism
• Quantized State Systems
• ModelicaDEVS
• Simulation Results
• Conclusions

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Motivation

• When simulating continuous models on a digital
  computer, a quantization must take place, as
  digital computers can only perform a finite
  number of computations within any finite time
  span.
• Usually, it is the time axis that is being
  discretized, i.e., we are asking ourselves given
  the current state at time t, what value shall
  the state vector assume at time t h ?
• Yet, it is also possible to discretize the state
  axes, i.e., we can ask the question given the
  current value of the quantized state variable qi
  , what is the earliest time when the variable
  will assume a value of qi ?qi ?

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Motivation 2

• Numerical simulation algorithms can be based on
  either paradigm.
• Whereas almost all simulation algorithms
  currently on the market are based on the former
  approach, the QSS techniques are based on the
  latter.
• The QSS approach has some distinct advantages
• The method is naturally asynchronous.
• QSS simulations can be more easily combined with
  discrete events.
• Most state events turn into time events in QSS.
• A QSS simulation of an analytically stable system
  is always stable.
• QSS offers a global error bound.
• It is possible to design implicit algorithms
  that are not truly implicit.

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The DEVS Formalism

• DEVS was introduced by B. Zeigler in 1976.
• DEVS is one among several discrete-event
  simulation methodologies. Other discrete-event
  techniques include Petri nets, finite state
  machines, Markov chains, ...
• Specialty DEVS models work with an infinite
  number of states. This is useful for numerical
  integration.

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The DEVS Formalism 2
Atomic Models

• Accept an input trajectory (external event),
  generate an output trajectory.
• Definition M ltX, Y, S, dint, dext, ?, tagt
• X set of inputs
• S set of possible states
• Y set of outputs
• dext external transition
• ta time-advance function, often represented by
  s
• dint internal transition
• ? output function

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The DEVS Formalism 3
Atomic Models (cont.)
Example
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The DEVS Formalism 4
Coupled Models

• DEVS is closed under coupling.
• Useful to split a complex model into simpler
  models.
• The dynamics of the coupled model N
• Evaluate the atomic model d that is the next one
  to execute an internal transition. Let tn be the
  time when the transition has to take place.
• Advance the simulation time to t tn and let d
  execute the internal transition.
• Forward the output of d to all connected atomic
  models and let them execute their external
  transitions.

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The DEVS Formalism 5
Hierarchical Models

• Reuse of coupled models as atomic models.
• The actual task of N is to wrap Ma and Mb, in
  order to make them look like they were one single
  model.
• The coupled model N features the same transitions
  as an atomic model, but the transitions of N
  depend on the transitions of its sub-models.

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Quantized State Systems (QSS)

• QSS operate on piecewise constant input and
  output trajectories.
• Systems with piecewise constant trajectories can
  be simulated using the DEVS formalism.
• QSS use a quantization function to transform a
  continuous system into a system with piecewise
  constant input and output trajectories.
• The quantization function is hysteretic in order
  to avoid illegitimate models.
• Illegitimate models perform an infinite number of
  transitions in a finite time interval.

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Quantized State Systems 2
Hysteretic Quantization Function

• A quantization function maps real numbers x(t)
  into a discrete set of real values q(t).
• Problem x(t) -sign(q(t))
• A hysteretic quantization function prevents
  infinite oscillations from occurring within a
  single time step.

.
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Quantized State Systems 3
Discretization of a Continuous System
.

• Conventional continuous system x(t) f(x(t),
  u(t), t)
• Quantized continuous system ?(t) f(q(t),
  u(t), t)
• Example x(t) -x(t) 10e(t - 1.76)
• Used quantization function q(t) floor(?(t))
• ? ?(t) -floor(?(t)) 10e(t - 1.76)
• ? ?(t) -q(t) 10e(t - 1.76)
• q(t) is a piecewise constant, linear or quadratic
  function.
• QSS1 ? uses constant function.
• QSS2 ? uses linear function.
• QSS3 ? uses quadratic function.

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.
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Quantized State Systems 4
Discretization of a Continuous System (cont.)
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The PowerDEVS Simulator

• PowerDEVS is written in C ? sequential
  variable updates.
• The software employs a hierarchical simulation
  scheme.
• Coordinators represent coupled models, simulators
  represent atomic models.
• Coordinators contain simulators or other
  coordinators.
• Coordinators control the interaction between
  their children.
  ? Components on the same level do not communicate
  with each other, but only with their parent
  coordinator.

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The ModelicaDEVS Simulator

• ModelicaDEVS operates on the synchronous data
  flow principle.
• All equations are constantly monitored.
• Whether a variable gets updated or not must be
  determined by Boolean variables.

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Atomic Models in ModelicaDEVS

• ModelicaDEVS models have one or more input ports
  and one output port.
• ModelicaDEVS signals/events consist of the
  following values
• Coefficients of Taylor series up to second order
  of the current function value.
• Boolean value. Indicates the creation of an
  event.
• Input event uVal1, uVal2, uVal3 and
  uEvent.
• Output event yVal1, yVal2, yVal3 and
  yEvent.
• Components have two Boolean variables dint and
  dext ...
• dint true ? execute internal transition.
• dext true ? execute external transition.
• and two real-valued variables lastTime and
  sigma.
• lastTime stores the time of the last event.
• sigma stores the amount of time that has to
  elapse before the next internal transition takes
  place.

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Coupled Models in ModelicaDEVS

• Communication between blocks
• When block A executes its internal transition
  (dint true), it sends an output to block B
  (yEvent true).
• When block B receives an event (uEvent true),
  it executes its external transition.

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Coupled Models in ModelicaDEVS 2

• Benefits of the Dymola simulator
• The dynamics of coupled models are still
  determined by their sub-models.
• Performs the same loop as defined by the DEVS
  formalism ...
• ... but the evaluation of d is done implicitly
  by Modelicas concept of simultaneous equation
  evaluation.
• Coupled models are handled implicitly by the
  Dymola Simulator.

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Hierarchic Models in ModelicaDEVS

• A hierarchic model contains a component that
  consists of other components (sub-models).
• Sub-models just add a number of equations to the
  model equation pool ? no special treatment
  required.
• Hierarchic models are handled implicitly by the
  Dymola Simulator.

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The Flyback Converter
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The Flyback Converter 2
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The Flyback Converter 3
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The Flyback Converter 4
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The Flyback Converter 5
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Mixed Systems
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The Flyback Converter 6
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Conclusions

• A new Dymola/Modelica library implementing a
  number of Quantized State System (QSS) simulation
  algorithms has been presented. ModelicaDEVS
  duplicates the capabilities of PowerDEVS. The
  graphical user interfaces of both tools are
  practically identical. However, the underlying
  simulators are very different. Whereas PowerDEVS
  implements Zeiglers hierarchical DEVS simulator,
  ModelicaDEVS operates on simultaneous equations
  and synchronous information flows.
• The embedding of ModelicaDEVS within the Dymola/
  Modelica environment enables users to mix DEVS
  models with other modeling methodologies that are
  supported by Dymola and for which Dymola offers
  software libraries.

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Conclusions 2

• Unfortunately, ModelicaDEVS is much less
  efficient in run-time performance than PowerDEVS.
  The loss of run-time efficiency is probably
  caused by Dymolas event handling algorithms that
  have been designed for optimal robustness in the
  context of hybrid system simulation rather than
  run-time efficiency in the context of pure
  discrete-event system simulation.
• Although ModelicaDEVS offers a full
  implementation of a DEVS kernel and can therefore
  be used for the simulation of arbitrary
  discrete-event systems, the modeling blocks that
  have been made available so far in ModelicaDEVS
  are geared towards the simulation of continuous
  systems using QSS algorithms.

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The End