Quantised State System Simulation in Dymola/Modelica using the
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1. The following sections will offer more insight into the reasons for this specific block structure In accordance with the PowerDEVS implemen tation ModelicaDEVS event ports connectors consist of four variables representing the coefficients to the first three terms constant linear and quadratic of the function s Taylor series expansion and a Boolean value that indicates whether a block is currently sending an event Dense output can then be approximated as Yout Yo oe al t tae Tyo t Daa whereby the coefficient of the quadratic term of the Taylor series y2 yval 3 is only used by the third order accurate method QSS3 whereas the linear term yj yVal 2 is used by QSS2 and QSS3 Let us now consider a small example in order to gain increased insight into the role of the Boolean variable of the port Let us assume a two block system consist ing of block A and block B where the only input port of block B is connected to the only output port of block A Every block features a variable dext accompanied The Modelica Association 76 by an equation dext where uEvent is the Boolean component of the con nector that represents an input event Suppose now that block A produces an output event at time t 3 At this precise instant it updates its output vector with the appropriate values the coefficients of the Taylor series and sets A yEvent to true uEvent when dint then yVal 1l new2. the lambda function has to be executed prior to the internal transition The variable yEvent has to be true in the exact instant when an internal transition is executed and false otherwise This behaviour is obtained by using the Modelica edge operator There is one particular situation that can occur in a model that requires special attention let us assume two connected blocks where both block A and block B have to execute an internal transition simultaneously Figure 4 Whereas block A simply t 1 t 2 sigma 4 sigma 3 t 5 dint t 5 sala Figure 4 Concurrent events at block B executes its internal transition block B is confronted with the problem of concurrent events from block A it receives an external event but at the same time it was about to undergo its own internal transition Which event should be processed first This question is to be answered by the priority settings between the two blocks In our simple two block example there are only two possible priority orderings with the following consequences either block A is prior to block B and block A will produce the output event before block B executes the internal transition block B will first execute an external transition triggered by the external event it received from block A or block B 1s prior The Modelica Association to block A such that block B will first undergo its internal transition and receive the external event right afterwards when A w
3. 2005 He recently returned to his home country of Switzerland Dr Cellier s main scientific interests concern modeling and simulation methodologies and the design of advanced software systems for simulation computer aided modeling and computer aided design Dr Cellier has authored or co authored more than 200 technical publications and he has edited several books He published a textbook on Continuous System Modeling in 1991 and a second textbook on Continuous System Simulation in 2006 both with Springer Verlag New York He served as general chair or program chair of many international conferences and serves currently as president of the Society for Modeling and Simulation International Modelica 2006 September 4 5
4. the correct transition functions describing the system Fortunately complex systems can be broken down into simpler submodels that are easier to handle The fact that DEVS is closed under coupling 2 makes such an approach viable Figure 2 illustrates this concept the model N consists of two coupled atomic models M and Mp N can be said to wrap M and M and is indistinguishable from the outside from an atomic model Modelica 2006 September 4 5 Quantised State System Simulation in Dymola Modelica Using the DEVS Formalism Figure 2 Coupled DEVS models 2 2 2 Quantised State Systems For a system to be representable by a DEVS model it must exhibit an input output behaviour that is de scribable by a sequence of events In other words the DEVS formalism is able to model any system with piecewise constant input output trajectories since piecewise constant trajectories can be described by events 2 Continuous state variables are being quantised Con sider the following system represented by the state space description x t f x t u t where x t is the state vector and u t is the input vector 1 e a piecewise constant function The cor responding quantised state system has the following form x t f a u t t where q t is the componentwise quantised version of the original state vector x t A simple quantisation function could be q t floor x t Unfortunately the transformation of a
5. Quantised State System Simulation in Dymola Modelica Using the DEVS Formalism Quantised State System Simulation in Dymola Modelica using the DEVS Formalism Tamara Beltrame VTT Industrial Systems PO Box 1000 VM3 02150 Espoo Finland Tamara Beltrame vtt fi Abstract Continuous time systems can be converted to discrete event descriptions using the Quantised State Systems QSS formalism Hence it is possible to simulate continuous time systems using a discrete event simu lation tool such as a simulation engine based on the DEVS formalism A new Dymola library ModelicaDEVS was devel oped that implements the DEVS formalism DEVS has been shown to be efficient for the simu lation of systems exhibiting frequent switching opera tions such as flyback converters ModelicaDEVS con tains a number of basic components that can be used to carry out DEVS simulations of physical systems Furthermore it is also possible with some restric tions to combine the two simulation types of Mod elicaDEVS and Dymola discrete event and discrete time simulation and create hybrid models that contain ModelicaDEVS as well as standard Dymola compo nents Keywords DEVS formalism Quantised State Sys tems Event Based Simulation Numerical Integration 1 Introduction Since Dymola Modelica was primarily designed to deal with continuous physical problems numerical integration 1s central to its operation and therefore the search for new a
6. are important in the context of hybrid modelling In the context of a pure discrete event simulation these algorithms are an overkill For example in a pure discrete event simulation there is no need for iteration after each event to determine a new consistent set of initial conditions In Dymola many variables are being stored internally in order to allow LSODAR to integrate continuous state equations correctly across discontinuities In a pure discrete event simulation variables need to be stored for output only Modelica 2006 September 4 5 T Beltrame F E Cellier 4 2 Mixed Systems Mixed systems contain both Dymola and Modelica blocks Figure 9 shows an example of a simple elec trical circuit modelled in Dymola and in a mixed ver sion with a ModelicaDEVS capacitor Figure 10 illus Resistor Resistor R 5 Capacitor1 Gr ape d1 Ground ConstantCurrent ConstantCurrent Figure 9 Two versions Dymola and Dy mola ModelicaDEVS of a simple electrical circuit trates the implementation of the ModelicaDEVS ca pacitor On its outside this block looks like a normal electrical Dymola component but internally it consists of ModelicaDEVS blocks that model the behaviour of a capacitor The Gain block multiplies the incom ing signal by the value of where C is specified by a parameter and passes it on to the Interpolator Taken as a whole the ModelicaDEVS blocks consti tute nothing more than the well kn
7. ary val ues Qo and Q are the upper and the lower saturation values and is the width of the hysteresis window Figure 3 shows a quantisation function with uniform quantisation intervals q t Q GT Qo Figure 3 Quantisation function with hysteresis 2 The QSS described above is a first order approxima tion of the real system trajectory Kofman however has also introduced second and third order approx imations that may reduce the error made by the approximation These systems are referred to as QSS2 7 and QSS3 9 respectively 3 ModelicaDEVS The average block of the ModelicaDEVS library ex hibits the following basic structure Modelica 2006 September 4 5 T Beltrame F E Cellier 1 block SampleBlock 2 extends ModelicaDEVS InterfacesS 3 3 parameter Real 4 5 protected 6 discrete Real lastTime start 0 7 discrete Real sigma start 8 Real e 9 Boolean dext 10 Boolean dint 11 other variable declarations 12 13 equation 14 dext ukEvent 15 dint time gt pre lastTime pre sigma 16 lay when dint then 18 Yaw pita 19 VivaZe Jere 20 yVal 3 s 21 end when 22 yEvent edge dint 23 24 when dint dext then 25 e time pre lastTime 26 if dint then 27 internal transition behaviour 28 else 29 external transition behaviour 30 end if 31 lastTime time 32 end when 33 34 end SampleBlock
8. at least one Dymola capacitor inductor using the Modelica der opera tor The flyback converter of Section 4 1 where the capacitor in the secondary winding is replaced by an equivalent ModelicaDEVS capacitor may serve as an Modelica 2006 September 4 5 Quantised State System Simulation in Dymola Modelica Using the DEVS Formalism Resistorl y 14 4014 Resistor vy 14 4437 V time 1 56 1 2 3 SimplecircutDbymola Resistor w simplecircuthtixed Resistor Figure 11 Standard Dymola blue and mixed red simulation of the simple electrical circuit Figure 9 example of a hybrid system Note that the ModelicaDEVS capacitor applies numer ical differentiation in order not to obtain DAE index reduction error messages see previous section 10 2 0 000 0 001 O 002 FlybackConverterDymoala iL FlybackConverterMixedMumerical iL Figure 12 Standard Dymola blue and mixed red simulation of the flyback converter Figure 12 shows the output of the mixed simulation compared to the result of the standard Dymola simula tion Just as it was the case with the simpler example of Section 4 2 the output of the hybrid simulation dif fers only slightly from the Dymola simulation Thus it is also possible to perform not only accurate mixed simulations but also hybrid simulations gt Note that due to the numerical differentiation used in the Sam plerTimeNumerical block the result
9. ch in turn is faster than ModelicaDEVS First it needs to be remarked that standard Dymola simulates this model very efficiently The switching BooleanPulse block leads to time events only whereas the diode should lead to state events Yet this is not the case Switching at the input leads immediately to a switch ing of the diode as well Since Dymola iterates after each event to determine a consistent set of initial conditions the switching of the diode is accomplished at once without need of first triggering a state event Second the model is quite trivial The execution time is almost entirely dictated by the number of time events handled What happens in between events is harmless in comparison Standard Dymola performs exactly one time event per switching In contrast ModelicaDEVS performs considerably more time events Time events take here the role of integration steps Figure 8 shows the constant term of the Taylor series expansion of the load voltage as a function of time for QSSI and QSS3 QSS1 requires a new time event as soon as the constant output no longer represents the true output whereas QSS3 requires an event only when the second order accurate Taylor series expansion no longer approximates the true output QSS1 requires roughly eight times as many events as QSS3 and is therefore between five and six times slower Yet even QSS3 requires roughly three times as many events as standard Dymola In addition the Mod
10. continuous system into its discrete counterpart by applying an arbitrarily chosen quantisation function can yield an illegitimate model Thus the quantisation function has to be chosen carefully such that it prevents the system from switching states with an infinite fequency This property can be achieved by adding hysteresis to the quantisation function 6 which leads to the notion of a Quantised State System QSS as introduced by Kofman 6 providing legitimate models that can be simulated by the DEVS formalism A hysteretic quantisation function is defined as follows 2 Let Q Qo Q1 Q be a set of real numbers where Ox lt Ox with 1 lt k lt r Let Q Definition 2 A DEVS model is said to be legitimate if it cannot perform an infinite number of transitions in a finite interval of time Illustrative examples of illegitimate models can be found in 2 and 6 The Modelica Association be the set of piecewise continuous trajectories and let x Q be a continuous trajectory The mapping b Q Q is a hysteretic quantisation function if the trajectory g b x satisfies Or if t t a Ok 1 if x t Qk Aql QkA Ak lt r Ok if x t Qk E q t Q A Ak lt 0 q t otherwise and 0 if x to lt Qo m lt r ifx to gt Q j if Q lt x to lt Qj41 The discrete values Q and the distance Q 1 Qk usually constant are called the quantisation levels and the quantum respectively The bound
11. de available so far in ModelicaDEVS are geared towards the simulation of continuous systems using QSS algorithms References 1 Beltrame T 2006 Design and Development of a Dymola Modelica Library for Discrete Event oriented Systems using DEVS Methodology MS Thesis Institute of Computational Science ETH Zurich Switzerland 2 Cellier FE and E Kofman 2006 Continuous System Simulation Springer Verlag New York 3 Dynasim AB 2006 Dymola Users Manual Version 6 0 Lund Sweden 4 Fritzson P 2004 Principles of Object Oriented Modeling and Simulation with Model ica 2 1 Wiley Interscience New York Glaser J S FE Cellier and A F Witulski 1995 Object Oriented Switching Power Con 5 Modelica 2006 September 4 5 T Beltrame F E Cellier verter Modeling Using Dymola With Event Handling Proc OOS 95 SCS Object Oriented Simulation Conference Las Vegas NV pp 141 146 6 Kofman E and S Junco 2001 Quantised State Systems A DEVS Approach for Contin uous Systems Simulation Transactions of SCS 18 3 pp 123 132 7 Kofman E A Second Order Approximation for DEVS Simulation of Continuous Systems Sim ulation 78 2 pp 76 89 8 Kofman E M Lapadula and E Pagliero 2003 PowerDEVS A DEVS based Environ ment for Hybrid System Modeling and Simula tion Technical Report LSD0306 LSD Univer sidad Naci
12. elicaDEVS model contains roughly three times as many variables as the standard Dymola model All of these variables are being stored at every event Consequently QSS3 is roughly nine times slower than standard Dymola The Modelica Association 79 20 0 SAE ie wt 2 0E 4 Resistor_u DEVS Resistor_u DEVS 2 5E 4 Figure 8 QSS3 simulation vs QSS1 simulation Yet QSS3 in PowerDEVS is roughly three times faster than standard Dymola for comparable ac curacy A comparison between PowerDEVS and ModelicaDEVS is not straightforward PowerDEVS implements Zeigler s hierarchical simulator 12 whereas ModelicaDEVS operates on simultaneous equations and synchronous information flow 10 Consequently PowerDEVS suffers from requiring message passing to implement the communication between blocks but enjoys the advantage of only having to process those equations that are directly involved with the event being handled In contrast ModelicaDEVS needs to visit all equations of all blocks whenever an event takes place Which vari ables are to be updated in each case is decided by Boolean expressions associated with the various when statements Yet the true difference in speed has probably more to do with the event handling itself Dymola has been designed for optimal speed in the simulation of continuous models and for optimal robustness in handling hybrid models The algorithms implemented in Dymola for robust event handling
13. es feature described in the Dymola User s Manual 3 functions for the first and second derivatives have been inserted into the Interpolator but due to unknown reasons this did not resolve the issue either In order to be able to perform mixed simulations nonetheless another trick has been applied supple mentary to the standard ModelicaDEVS SamplerTime block that uses the Modelica der operator an ad ditional block has been programmed the Sampler TimeNumerical block avoids the problem caused by the der operator by means of the delay function that is used to differentiate the input variable numer ically Instead of the first and second derivatives of the input signal the SamplerTimeNumerical returns a numerical approximation Du delay pre u D D2u delay pre u 2 D yVal 1l pre u yVal 2 if method gt 1 then pre u Du D else 0 yVal 3 if method gt 2 then pre u 2 Du D2u D D else 0 Using the new sampler block the mixed simulation could be carried out without any problems and the re sults differ only slightly from the simulation with con ventional Dymola components see Figure 11 4 3 Hybrid Systems Hybrid systems contain mixed integration methods standard Modelica integrators and ModelicaDEVS In tegrator blocks An example of a hybrid system is for instance an electrical circuit with at least one ModelicaDEVS capacitor inductor using the Model icaDEVS Integrator block and
14. esenting the external transition is due to a rule of the Modelica language specification that states that equations in different when statements may be evaluated simul taneously Hence if there are two when statements each containing an expression for a variable X X is considered overdetermined This circumstance would cause a syntactical problem with variables that have to be updated both during the internal and the time gt pre lastTime ukEvent Modelica 2006 September 4 5 Quantised State System Simulation in Dymola Modelica Using the DEVS Formalism external transition and thus would have to appear in both when statements For this reason we need to have a when statement that is active if either dint or dext becomes true Subsequently an additional discrimination 1s done within the when statement determining whether it was an internal dint is true or an external transition dext 1s true that made the when statement become active and as the case may be updating the variables with the appropriate value The lambda function is executed right before an internal transition Lines 17 22 of the block basic structure code beginning of Section 3 constitute the typical lambda function part containing a when statement and a separate instruction for the yEvent variable The right hand side of the equations in the lambda function normally depends on pre values of the used variables This is due to the fact that
15. he flyback converter model built in ModelicaDEVS The structure of this block diagram is also valid for the PowerDEVS model world Q351 Q552 Q553 Figure 7 The ModelicaDEVS flyback converter Table 1 provides the average simulation CPU time for a simulation of 0 002 seconds of the flyback con verter model in standard Dymola ModelicaDEVS and PowerDEVS respectively The Dymola and Mod elicaDEVS model were simulated setting the numeri cal integration method to LSODAR Testing has been carried out on an IntelCeleron 2 6 GHz Laptop with 256MB RAM The resulting CPU time may vary from 3For more details on the causalising process in the flyback con verter example see 1 4 Although ModelicaDEVS does not make use of LSODAR di rectly the event handling behaviour of Dymola is somewhat influ enced by the selection of the numerical integration algorithm Modelica 2006 September 4 5 Quantised State System Simulation in Dymola Modelica Using the DEVS Formalism one computer system to another but the relative order ing 1s expected to remain the same Table 1 Execution efficiency comparison CPU time result time s events points Dymola 738 QSS1 3 55 6363 11829 M DEVS QSS2 0 688 2299 QSS3 0 656 2164 QSS1 0 064 N A P DEVS QSS2 0 019 N A QSS3 0 018 N A Table 1 shows a clear ordering of the three different systems in terms of performance PowerDEVS is faster than Dymola whi
16. ill be allowed to execute its internal transition The problem of block priorities can be solved in two ways by an explicit absolute ordering of all components in a model e g a list or by letting every block determine itself whether it processes the external or the internal event first in case both of them occur simultaneously ModelicaDEVS implements the latter approach As can be seen in the block basic structure code internal transitions take always priority over external transitions line 26 the code checks first whether dint 1s true The reason for this choice is quite simple As internal events are processed before external events and since internal events are accompanied by output events the variable yEvent can be computed as a function of dint alone If we were to force external events to be processed before internal events we would need to make sure that yEvent is only set true in the case that the internal event is not accompanied by a simultaneous external event Thus yEvent would now be a function of both dint and dext Yet dext is a function of uEvent Thus if ModelicaDEVS blocks were connected in a circular fashion as this is often the case an algebraic loop in discrete Boolean variables would be created which would get the Dymola compiler into trouble By forcing the internal events to always take pref erence over external events ModelicaDEVS blocks can be interconnected in an arbitrary fashion withou
17. is not as accurate as if analyt ical differentiation had been used However the accuracy is suf ficient for most purposes and also adjustable through selection of the width parameter D The Modelica Association 5 Conclusions A new Dymola Modelica library implementing a number of Quantised State System QSS simulation algorithms has been presented ModelicaDEVS dupli cates the capabilities of PowerDEVS The graphical user interfaces of both tools are practically identical However the underlying simulators are very different Whereas PowerDEVS implements Zeigler s hierar chical DEVS simulator ModelicaDEVS operates on simultaneous equations and synchronous information flows The embedding of ModelicaDEVS within the Dy mola Modelica environment enables users to mix DEVS models with other modelling methodologies that are supported by Dymola and for which Dymola offers software libraries Unfortunately ModelicaDEVS is much less efficient in run time performance than PowerDEVS The loss of run time efficiency is probably caused by Dymola s 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 modelling blocks that have been ma
18. lgorithms that may improve the efficiency of simulation runs is justified Toward the end of the nineties a new approach for numerical integration by a discrete event formalism has been developed by Zeigler et al 13 given the fact that all computer based continuous system simulations have to undergo a discretisation of one form or another as digital machines are not able to process raw continuous signals the basic idea The Modelica Association Francois E Cellier Institute of Computational Science ETH Zurich 8092 Zurich Switzerland FCellier Inf ETHZ CH of the new integration approach was to replace the discretisation of time by a quantisation of state The DEVS formalism turned out to be particularly well suited for implementing such a state quantisation approach given that it is not limited to a finite number of system states which is in contrast to many other discrete event simulation techniques The Quantised State Systems QSS introduced by Kofman 6 in 2001 improved the original quantised state approach of Zeigler by avoiding the problem of ever creating illegitimate models and hence gave rise to efficient DEVS simulation of large and complex systems The simulation of a continuous system by a dis crete DEVS model comes with several benefits When using discretisation of time variables have to be updated synchronously Thus the time steps have to be chosen according to the variable that changes the fas
19. onal de Rosario Argentina 9 Kofman E A Third Order Discrete Event Method for Continuous System Simulation 36 2 Latin American Applied Research pp 101 108 10 Otter M H Emqvist and S E Mattsson 1999 Hybrid Modeling in Modelica Based on the Synchronous Data Flow Principle CACSD 99 IEEE Symposium on Computer Aided Control System Design Hawaii pp 151 157 11 Zeigler B P 1976 Theory of Modeling and Simulation John Wiley amp Sons New York 12 Zeigler B P 1984 Multifacetted Modelling and Discrete Event Simulation Academic Press London 13 Zeigler B P and J S Lee 1998 The ory of Quantized Systems Formal Basis for DEVS HLA_ Distributed Simulation Environ ment SPIE Proceedings Vol 3369 pp 49 58 The Modelica Association 82 Biographies Tamara Beltrame received her MS degree in computer science from the Swiss Federal Institute of Technology ETH Zurich in 2006 She recently started working at VTT Finland where she deals with problems of simulation aided automation testing Francois E Cellier received his BS degree in electrical engineering in 1972 his MS degree in automatic control in 1973 and his PhD de gree in technical sciences in 1979 all from the Swiss Federal Institute of Technology ETH Zurich Dr Cellier worked at the University of Arizona as professor of Electrical and Computer Engi neering from 1984 until
20. output value 1 yVal 2 new output value 2 yVal 3 new output value 3 end when yEvent edge dint Still at time t 3 block B notices that now B uEvent has become true note that B uEvent because the two blocks are connected and therefore dext has become true also Consequently Block B is executing its external transition 4 A yEvent A DEVS model must contain code to perform internal and external transitions as well as execute the time advance and output functions at the appropriate instants All of these functions have to be explicitly or implicitly present in the ModelicaDEVS blocks The time advance function is normally represented by a variable sigma It is a popular trick in DEVS to represent the current value of the time advance function by sigma 2 The internal transition 1s executed when dint 1s true An internal transition depends only on sigma Hence the value of dint can be calculated as dint pre sigma where last Time holds the time of the last execution of a transition internal or external The external transition is executed when dext 1s true The variable dext is defined as follows dext The internal and external transitions are represented by a when statement The reason for packing the internal and external transitions into a single when statement instead of having two separate when statements one representing the internal transition and the lambda function the other one repr
21. own capacitor for mula v 4 f i dt Sampler Time1 Gain Figure 10 The internal structure of the Modeli caDEVS capacitor Unfortunately it is not as straightforward as it may seem at first glance to replace a component from the Dymola standard electrical library by its Modeli caDEVS equivalent since the electrical components do not assume a certain data flow direction they are described by acausal equations whereas the Model icaDEVS components do DEVS components feature input and output ports the ModelicaDEVS capacitor must turn acausal equations into causal ones It as sumes the capacitive current i to be given and hence computes the capacitive voltage v Note that such a ca pacitor would not work anymore correctly if we were to connect it to a voltage source instead of a current source An even more severe problem is caused by the Sam plerTime block applying the der operator to the sig nal that it receives through its input port The Modelica Association 80 du der u when sample start period then yVal 1 u yVal 2 if method gt 1 then du else yVal 3 i1f method gt 2 then der du end when else 0 Given that the input of the SamplerTime block de pends algebraically on the output of the Interpolator in the DEVS capacitor Dymola would have to differ entiate discrete variables which it is unable to do An attempt to solve this problem was made using Dy mola s User specified Derivativ
22. s the set of states The variable e indicates the amount of time the system has already spent in the current state dez s e x is the external transition that is executed after an The Modelica Association 74 external event has been received djn s is the internal transition that is executed as soon as the system has spent in its current state the time indicated by the time advance function ta s 1s the so called time advance function that indicates how much time has to pass until the system undergoes the next internal transition The time advance function is often represented by variable which holds the value for the amount of time that the system has to remain in its current state in the absence of external events The A function is the output function It is executed prior to performing an internal transition External transitions do not produce output Figure illustrates the functioning of a DEVS model the system receives input top graph at certain time instants changes its states according to the internal and external transitions center graph and produces output bottom graph Ta r 1 f i od int 83 53 dint 51 Y r 1 59g ext 52 T1 ta 84 j yo Al sg wy A s Figure 1 Trajectories in a DEVS model In theory DEVS models can describe arbitrarily complex systems The only drawback is that the more complex the system is the more difficult it will be to set up
23. t ever creating algebraic loops in the Boolean event indication variables Note that since Dymola Modelica is already aimed at object oriented modelling which includes the reuse of multi component models as parts of larger models the issue of hierarchically coupled models did not require any special treatment in ModelicaDEVS Dymola can trigger two types of events state events that require iteration to locate the event time and time events that make Dymola jump directly to the point in time when the time event takes place The only expressions responsible for activating the when statements in the models namely dext uEvent and dint time gt pre lastTime pre Sigma both trigger time events and hence avoid the computa tionally more expensive state events Modelica 2006 September 4 5 T Beltrame F E Cellier An earlier version of ModelicaDEVS used an ap proach that triggered mostly state events Inspired by the book of Fritzson 4 a number of small modifica tions have been applied that converted all state events to time events Performance comparisons carried out between the two versions showed that the time event approach is roughly four times faster than an equiva lent approach triggering state events 4 Results 4 1 Efficiency In order to compare the run time efficiency of Mod elicaDEVS to other simulation software systems PowerDEVS and standard Dymola a system with frequent swi
24. tching operations was modelled using each of the three tools PowerDEVS ModelicaDEVS and Dymola and the execution times of the three codes were compared against each other The chosen system is the flyback converter example presented in 5 The flyback converter can be used to transform a given input voltage to a different output voltage It belongs to the group of DC DC converters A very simple electrical circuit with a voltage source connected to the primary winding of the converter and a load to its secondary winding looks as shown in Figure 5 BooleanPulse1 IdealDiode IdealClosingSwitch abeyoAjeysuoy Hopnpuj 9 3002 1 p4oyoedes 40psIS38y Ground Ground2 Figure 5 The flyback converter in Dymola Figure 6 shows the first two milliseconds of a simula tion run of the flyback converter circuit given in Figure 5 The rapid switching 1s a result of the high switching rate of the ideal switch The flyback converter is described by a set of acausal equations in Dymola However in order to be able to model the flyback converter in either ModelicaDEVS The Modelica Association 78 30 20 10 0 0 000 0 001 0 002 Resistor w W Resistor J 4 Figure 6 The flyback converter output or PowerDEVS the behaviour of the converter needs to be converted to a causalised block diagram which then can be modelled using component models of the PowerDEVS ModelicaDEVS libraries Figure 7 shows t
25. test otherwise a change in that variable could be missed In a large system where probably very slow but also very fast variables are present this is critical to computation time since the slow variables have to be updated way too often The DEVS formalism however allows for asynchronous variable updates whereby the computational costs can be reduced significantly every variable updates at its own speed there is no need anymore for an adaptation to the fastest one in order not to miss important develop ments between time steps This property could be extremely useful in stiff systems that exhibit widely spread eigenvalues 1 e that feature mixed slow and fast variables The DEVS formalism is very well suited for problems with frequent switching operations such as electrical Note that this is not true for methods with dense output How ever the above statement holds for the majority of today s integra tion methods since they rarely make use of dense output Modelica 2006 September 4 5 T Beltrame F E Cellier power systems Given that the problem of iteration at discontinuities does not apply anymore it even allows for real time simulation For hybrid systems with continuous time discrete time and discrete event parts a discrete event method provides a unified simulation framework discrete time methods can be seen as a particular case of discrete event methods 6 and continuous time parts can be
26. transformed in a straightforward manner to discrete time discrete event systems When using the QSS approach of Kofman in order to transform a continuous system into a corresponding discrete system there exists a closed formula for the global error bound 2 which allows a mathematical analysis of the simulation Since the mid seventies when Zeigler introduced the DEVS formalism 11 there have emerged several DEVS implementations most of them designed to simulate discrete systems However one simula tion modelling software system aimed at simulating continuous systems is PowerDEVS 8 it provides a library consisting of block diagram component models that can be used for modelling any system described by ODE s DAE s thereby allowing for the simulation of continuous systems The implementation of ModelicaDEVS has been kept close to the PowerDEVS simulation soft ware Hence ModelicaDEVS can be considered a re implementation of PowerDEVS in Modelica 2 Continuous with DEVS System Simulation 2 1 The DEVS Formalism The DEVS formalism has been introduced by Zeigler in 1976 11 It was the first methodology designed for discrete event system simulation that is based on system theory A DEVS model has the following structure M X Y S Sint S Set 5 X A s fa s where the variables have the following meaning see also 2 Chapter 11 X represents all possible inputs Y represents the outputs and S i
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