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1. pend on the marking of P1 gt gt be j Sy 1 2 1 2 m P1 can gt 2 4 A a 3 5 m P1 a i N gt gt Figure 12 Input of arc weights arcWeightOut 52 Proceedings of the 9 International Modelica Conference September 3 5 2012 Munich Germany DOI 10 3384 ecp1207647 Session 1A Hybrid Modeling Transitions can also be provided with additional conditions that have to be satisfied to permit the ac tivation The condition firingCon time gt 9 7 causes that the transition cannot be activated as long as time is less than 9 7 Figure 13 shows two continuous Petri nets Transi tion T1 has a maximum speed function which de pends on the makings of P1 and P2 The input of this function to the property dialog or as modification equation is performed by the expression maximumSpeed 0 75 P1 t P2 t whereby P1 t and P2 t accesses the marks of P1 and P2 respectively Transition T2 has a maximum speed function that depends on time and can be en tered by the expression if time lt 6 5 then 2 6 else 1 7 maximumSpeed T2 gt Fa 2 26 time lt 6 5 gt LT time gt 6 5 E Figure 13 Input of maximum speed functions Based on the current markings of the places it is checked in the transition model by an algorithmic procedure if the transition can become active Dis crete transitions wait then as long as the delay is passed and stochastic2. condition false expres sion reStartTokens reStartMarks When the reStart condi tion is fulfilled the mark ing is set to reStartTo kens Marks scalar 0 The input of enabling probabilities as vector is demonstrated by Figure 11 Place P1 is connected to the transitions T1 T2 and T3 and the connection to T1 is indexed by 1 the connection to T2 is indexed by 2 and the connection to T3 is indexed by 3 Thus the corresponding connect equations are connect Pl outTransition 1l Tl inPlaces 1 connect Pl outTransition 2 T2 inPlaces 1 connect Pl outTransition 3 T3 inPlaces 1 The enabling probabilities 0 3 for T1 0 25 for T2 and 0 45 for T3 have to be entered by the parameter vector enablingProboOut 0 3 0 25 0 45 DOI 10 3384 ecp1207647 Proceedings of the 9 International Modelica Conference 51 September 3 5 2012 Munich Germany PNlib An Advanced Petri Net Library for Hybrid Process Modeling Figure 11 Input of enabling probabilities The main process in the place model is the recal culation of the marking after firing a connected tran sition In the case of the discrete place model this is realized by the discrete equation when tokeninout or pre reStart then t if tokeninout then pre t firingSumIn firingSumOut else reStartTokens end when whereby pre t accesses the marking t immediate ly before the transitions fire To this amount the arc
3. 5 The places are able to contain a non negative integer number of tokens and can be provided with non negative integer minimum and maximum capacities Furthermore the transitions are timed with fixed or stochastic delays The third library called StateGraph is based on Grafcharts which combines the function chart for malism of Grafcet with the hierarchical states of Statecharts 6 The StateGraph library is part of the Modelica standard library and was developed by Ot ter et al 7 The relationships between the mentioned con cepts are displayed in Figure 1 To enable modeling of different systems with Petri nets in Modelica the existing libraries have to be extended by the follow ing aspects Transfer of the discrete Petri net concept to a con tinuous one Support of edges with functional weightings Support of test inhibitor and read arcs Support of different conflict resolutions ran dom decisions Combination of discrete and continuous Petri net elements to hybrid Petri nets 2 Extended Hybrid Petri Nets The extended Hybrid Petri Net KHPN formalism comprises three different processes called transi tions discrete stochastic and continuous transition two different states called places discrete and con tinuous places and four different arcs normal in hibitor test and read arcs The icons of the formal ism are shown in Figure 2 Discrete places contain a non negative in
4. Alla H Continuous petri nets Pro ceedings of 8th European Workshop on Ap plication and Theory of Petri nets 275 294 1987 3 David R Alla H On Hybrid Petri Nets Dis crete Event Dynamic Systems Theory and Applications 11 9 40 2001 4 Mosterman P J Otter M Elmqvist H Mod eling Petri nets as local constraint equations for hybrid systems using Modelica Proceed ings of SCS Summer Simulation Confer ence 3 14 319 1998 5 Fabricius S M Extensions to the Petri Net Library in Modelica ETH Zurich Switzer land 2001 6 Johnsson C Arz n K E Grafchart and grafcet A comparison between two graphical languages aimed for sequential control appli cations Preprints 14th World Congress of IFAC A 19 24 1999 7 Otter M rz n K E Dressler I StateGraph a Modelica library for hierarchical state ma chines Proceedings of 4th International Modelica Conference 21 33 2005 8 Prof S Hybrid Modeling and Optimization of Biological Processes Bielefeld Germany PhD thesis in preparation Faculty of Tech nology Bielefeld University Germany 2012 9 Dynasim AB Dymola Dynamic Modeling Laboratory User Manual Volume 2 Lund Sweden 2010 56 Proceedings of the 9 International Modelica Conference DOI September 3 5 2012 Munich Germany 10 3384 ecp1207647
5. Germany PNlib An Advanced Petri Net Library for Hybrid Process Modeling Matlab Simulink This is realized with the aid of a Dymola interface in Simulink and a set of Matlab m files utilities 9 Time 1 Time 3 Figure 15 Animation of an xHPN model All markings which should be available in Matlab have to be declared with the prefix output on the highest level This is achieved by creating a connect or of the output connector at the top of the place icon In the case of discrete places it is an orange IntegerOutput connector and in the case of con tinuous places it is a blue RealOutput connector In Figure 15 the markings of P1 P3 P5 and P6 are available in Matlab 5 Application The PNlib is so powerful but also so universal and generic that it is an ideal all round tool for model ing and simulation of nearly all kinds of processes such as business processes production processes logistic processes work flows traffic flows data flows multi processor systems communication pro tocols and functional principals This section gives an overview of the different application fields using the PNlib Three selected examples e Modeling a Senseo coffee machine e Modeling a printing process and e Modeling a business process are part of the PNlib and should demonstrate the huge application field Additionally the application of
6. Petri Net Library for Hybrid Process Modeling elica to get the PNlib work with it because some 10 Prof S Bachmann B Hybrid Modelling and Modelica features are not supported so far Process Optimization of Biological Systems Moreover the xHPN formalism as well as the MATHMOD Conference Wien Austria PNlib will be extended by fuzzy logic e g 11 and 2012 the color concept e g 12 to enhance the range of 11 Chen S Ke J Chang J Knowledge represen application fields further tation using fuzzy Petri nets Knowledge and Furthermore the PNlib is already connected to Data Engineering IEEE Transactions on VANESA an open source tool for visualization and 2 3 311 319 1990 analysis of works n order 1 enable modeling 12 Jensen K Coloured petri nets Petri nets cen editing visualization and animation of xHPN mod l dels and their properties 248 299 els by an easy to use interface 13 This connection a Were es las Berli is ie lb 1987 will be further improved Peed ee A 13 Pro S Janowski S J Bachmann B Kalt schmidt C Kaltschmidt B PNlib A Model ica Library for Simulation of Biological Sys References tems based on Extended Hybrid Petri Nets 1 Petri C A Kommunikation mit Automaten ord Tnlemauional Workshop on Biological PhD thesis Rheinisch Westfalisches Institut Processes amp Petri Nets accepted Hamburg fiir Instrumentelle Mathematik Bonn Ger Cemnany gate many 1962 2 David R
7. weight sum of all firing input transitions is added and the arc weight sum of all firing output transitions is subtracted from it Additionally the tokens are reset to reStartTokens when the user defined condition reStart becomes true The marking of continuous places can change continuously as well as discretely This is imple mented by the following construct der t conMarkChange when disMarksInOut then reinit t t disMarkChange end when when reStart then reinit t reStartMarks end when whereby the der operator access the derivative of the marking t according to time The continuous mark change is performed by a differential equation while the discrete mark change is performed by the reinit operator within a discrete equation This operator causes a re initialization of the continuous marking every time when a connected discrete tran sition fires Additionally the marking is re initialized by reStartMarks when the condition reStart becomes true 3 3 Transitions The parameters of transitions are summarized in Ta ble 2 Thereby it has to be distinguished between the following input types scalar vector scalar function vector function and condition expression The input of arc weights as vectors in the transition model and not at the respective arcs is necessary due to the fact that connections cannot be provided with properties according to the Modelica Specification 3 2 Table 2 Parameters and modification possibil
8. PNlib An Advanced Petri Net Library for Hybrid Process Modeling Sabrina ProB Bernhard Bachmann University of Applied Sciences Department of Engineering and Mathematics Am Stadtholz 24 33609 Bielefeld sabrina pross fh bielefeld de Bernhard bachmann fh bielefeld de Abstract A new Petri net library called PNlib is presented to enable graphical hierarchical modeling hybrid simu lation and animation of processes in life sciences technical applications among others In order to model these most different processes a new power ful and universally usable mathematical modeling concept xHPN extended Hybrid Petri Net has been established This formalism is used as specifi cation for the PNlib Petri Net library realized by the object oriented modeling language Modelica The application of the new environment is demon strated by three selected examples The first example demonstrates the representation of functional princi ples by a model of a Senseo coffee machine and the second one is a model of a printing production pro cess The third example presents the applicability of modeling business processes All models are provid ed as application cases in the library Keywords Petri nets hybrid modeling xHPN pro cess modeling 1 Introduction The Petri net formalism was first introduced by Carl Adam Petri in 1962 for modeling and visualization of concurrency parallelism synchronization re source sharing
9. and non determinism 1 A Petri net is a graph with two different kinds of nodes called transitions and places thereby places and transi tions are connected by arcs Every place in a Petri net can contain a non negative integer number of tokens These tokens initiate transitions to fire ac cording to specific conditions These firings lead to changes of the tokens in the places In the recent years Petri nets with their various extensions are becoming increasingly popular They have been proven to be a universal graphical model ing concept for representing different systems in nearly all degrees of abstraction They support the qualitative modeling approach as well as the quanti tative one Furthermore the processes can be mod eled discretely as well as continuously refer to 2 In addition discrete and continuous processes can also be combined within a Petri net model to so called hybrid Petri nets first introduced by David and Alla 3 The Petri net formalism with all its ex tensions is so powerful that nearly all other formal isms are included Hence only one formalism is needed regardless of the approach qualitative vs quantitative discrete vs continuous vs hybrid de terministic vs stochastic which is appropriate for the respective system The Petri net formalism is easy to understand for researchers from different dis ciplines It is an ideal way for intuitive representing and communicating data and new knowledg
10. ble firing of transitions test arc the marking of the place must be smaller to enable firing inhibitor arc gers if connected to discrete places non negative real values otherwise normalArc If yes is chosen then the arc is also a normal arc to change the marking by firing called double arc choice no scalar no or yes 4 Animation and Connection to Matlab Simulink A possibility to represent the simulation results of an xHPN model is an animation Thereby several set tings can be made in the property dialog of the set tings box These settings are global and thus affect all components of the Petri net model By using the prefixes inner and outer it is achieved that the set tings are common to all Petri net components of a model An animation offers a way to analyze the marking evolutions of large and complex xHPNs Figure 15 shows four selected points in time of the animation of an xHPN example All display and an imation options are realized with the DynamicSe lect annotation To simulate the established xHPN model several times with different parameter settings and use the arising simulation results for parameter estimation sensitivity analysis deterministic and stochastic hy brid simulation or process optimization 8 the Modelica models in Dymola are connected to DOI 10 3384 ecp1207647 Proceedings of the 9 International Modelica Conference 53 September 3 5 2012 Munich
11. brary called PNlib enables the modeling of extended hybrid Petri Nets KHPN It comprises e adiscrete PD and a continuous place PC e a discrete TD a stochastic TS and a continu ous transitions TC and e atest TA an inhibitor IA and a read arc RA O QO on on ee sp Figure 9 Component icons of the PNlib The main package PNlib is divided into the fol lowing sub packages e Interfaces contains the connectors of the Petri net component models e Blocks contains blocks with specific procedures that are used in the Petri net component models e Functions contains functions with specific algo rithmic procedures which are used in the Petri net component models e Constants contains constants which are used in the Petri net component models e Models contains several examples and offers the possibility to structure further Petri net models Additionally the package contains the component settings which enables the setting of global parame ters for the display and the animation of Petri net models 50 Proceedings of the 9 International Modelica Conference September 3 5 2012 Munich Germany DOI 10 3384 ecp1207647 Session 1A Hybrid Modeling Places transitions and arcs are represented by the icons depicted in Figure 9 Thereby the discrete place is represented by a circle and the continuous place by a double circle The transitions are boxes which are black for discrete transition
12. can occur when a discrete place has not enough marks to enable all connected con tinuous transitions This is solved by prioritization of the involved transitions type 4 conflict see Figure 7 Visitor enters toilet T1 P1 Visitor e Water in P4 _ tank a gt P5 Water in j bowl AX A T6 Sewer Figure 8 Hybrid modeling of a flush toilet with the aid of xHPN formalism Figure 8 shows an example of hybrid modeling by the xHPN formalism The model represents a flush toilet A visitor enters the toilet thereby the time between two visitors is not exactly known so that it is modeled by a stochastic transition with an exponentially distributed delay T1 The visitor P1 pushes T2 the lever P2 which lifts the flush valve flapper P3 Then the water can flow T5 from the tank P4 to the bowl P5 and afterwards to the sewer T6 When the water flows to the bowl the float P6 sinks in the toilet tank If the float falls below a specific level inhibitory arc the tank fill valve P7 is opened T7 and new water can flow T9 into the tank This causes also that the float ris es and when a specific level is reached test arc the tank fill valve is closed T8 If the lever has re turned to its starting position the flush valve flapper sinks back to the bottom T4 and no water can flow into the bowl anymore 3 PNlib The advanced Petri Net li
13. cur between a continuous place and two or more continuous transitions when the input speed is not sufficient to fire all output transitions with the respective speed or when the output speed is not sufficient to fire all input transi tions with the respective speed type 2 conflict see Figure 5 This conflict is solved by sharing the speeds proportional to the assigned maximum speeds cf 8 aN ATN pr M p2 8 6 t T1 i gt a3 Wee A 4 1 1 PN E SASA 1 gt 42 eo bo d d2 2 Figure 6 Example of a type 3 conflict at time 0 T1 be comes active and fires continuously At time 2 the delay of T2 is passed and it becomes firable At this point in time P3 has a conflict because it cannot fire tokens in T1 and T2 simultaneously Hence T2 takes priority over T1 and fires DOI 10 3384 ecp1207647 Proceedings of the 9 International Modelica Conference 49 September 3 5 2012 Munich Germany PNlib An Advanced Petri Net Library for Hybrid Process Modeling If a conflict occurs between a place and continu ous as well as discrete stochastic transitions the dis crete stochastic transitions take always priority over the continuous transitions type 3 conflict see Fig ure 6 Figure 7 Example of a type 4 conflict at time 0 P3 can either enable T1 or T2 but not both simultaneously This conflict can be solved by prioritization of the transitions A last conflict
14. e How and when maculation occurs What are the causes and how can maculation be prevented e How much paper is need for the particular order e How long does the order take Orders Exemplars Maculation Duration 2 31887 9623 Paper 49812 7223 orders meters on role 45 9 S oy maculation Figure 17 Model of a printing process on the highest lev el The PNlib can also be used for modeling and simu lating business processes A business processes de scribes a sequence of activities or tasks which have to be carry out in order to achieve a particular busi ness goal e g a service or product for a particular customer Figure 18 shows a small part of a business process model The major advantages of this ap proach are 1 the hierarchical structure which pro vides a compact and clear view of the processes on the highest level and 2 the simulation and anima tion option which enable analyzing and optimizing of the processes A possible question may arise in this juncture is how much employees are needed to accomplish the requests and orders of the customers or simple how the profit can be maximized All ques tions of this kind can be answered by simulating the model with different parameter settings om Figure 18 Part of a business process model 6 Conclusions A powerful Petri net environment has been devel oped for graphical hi
15. e of mechanisms and processes Furthermore Petri nets allow hierarchical structuring of models and there fore offer the possibility of different detailed views for every observer of the model capacities test arcs inhibitor arcs read arcs functions delays random delays maximum speeds conditions priorities probabilities n o z c y a p E E S z ava suogenbg Dieiqabjy pue jeiuas yiqg jo WJS S Continuous Petri Net Discrete Petri Net Hybrid Petri Net hybrid DAE Figure 1 Relationships between the different formalisms There are already three Petri net libraries availa ble on the Modelica homepage www modelica org The first was developed by Mosterman et al and enables the modeling of a restricted class of discrete DOI 10 3384 ecp1207647 Proceedings of the 9 International Modelica Conference 47 September 3 5 2012 Munich Germany PNlib An Advanced Petri Net Library for Hybrid Process Modeling Petri nets called normal Petri nets 4 The places of normal Petri nets can only contain zero or one token Additionally all arcs have the weight one and exter nal signals initiate the firing of transitions If a con flict occurs between two or more transitions the transition with the highest priority fires Hence only deterministic behavior is represented by this kind of Petri net The second Petri net library is an extension of the previous one and was developed by Fabricius
16. erarchical modeling and hybrid simulation as well as animation of processes from most different application fields Thereby the math ematical modeling concept xHPN serves as specifi cation for performing a hybrid simulation The XHPN elements are modeled object oriented by dis crete differential and algebraic equations in the Modelica language This allows an easy way to maintain extend and modify the components Moreover the connection to Matlab Simulink of fers the whole Matlab power for post processing the simulation results of Modelica models The Matlab based tool AMMod Analysis of Modelica Models provides already several mathematical methods for data pre processing relationship analysis parameter estimation sensitivity analysis deterministic and stochastic hybrid simulation and process optimiza tion 10 The application of the new Petri net simulation environment has been demonstrated by a model of a Senseo coffee machine a model of a printing pro cess and a model of a business process All models show the applicability of the xHPN formalism as well as graphical hierarchical modeling and hybrid simulation with the PNlib A future goal is to provide an open source Petri net simulation tool This demands a further devel opment of the open source Modelica tool OpenMod DOI 10 3384 ecp1207647 Proceedings of the 9 International Modelica Conference 55 September 3 5 2012 Munich Germany PNlib An Advanced
17. ica Conference September 3 5 2012 Munich Germany DOI 10 3384 ecp1207647 Session 1A Hybrid Modeling necting a continuous place to a discrete transition However the conversion process is always per formed by discrete transitions discrete places can only influence the time when continuous transitions fire but their marking cannot be changed during the continuous firing process Figure 3 shows examples of these two basic principles e TI can only fire when P1 has more than zero marks and P3 has at least one mark influence e T2 can only fire when P4 has at least one mark and P6 has at least 5 4 marks influence e T3 fires by removing one mark from P7 and add ing 1 8 marks to P8 conversion e T4 fires by removing 0 8 marks from P9 and add ing one mark to P10 conversion Wie fo VAP ON VATE RN LEN l Iep Ti 1 13 Pag 1 1 gt P5 Say See e ud 4 1 1 5 4 5 P N Pa ae y Se 2 0 2 5 3 0 1 5 Time Figure 3 Basic concepts of hybrid Petri nets and marking evolution of places P7 and P8 achieved by firing T3 with a delay of 1 of the bottom left Petri net It is important to mention that a discrete transition fires always in a discrete manner by removing and adding marks after a delay is passed regardless of whether a discrete or a continuous place is connected to it However a continuous transition fires always by a continuous flow so that a d
18. iscrete place can only be connected to continuous transition if it is input as well as output of the transition with arcs of same weight In this way continuous transitions can only be influenced by discrete places but discrete mark ings cannot be changed by continuous firing Several conflicts can occur when the places have to enable their connected active transitions Possibly a discrete place or a continuous place connected to discrete transitions has not enough marks to enable all discrete output transitions simultaneously or can not receive marks from all active input transitions due to the maximum capacity Then a conflict arises that has to be resolved type 1 conflict see Figure 4 m EO Te Figure 4 Example of a type 1 conflict P1 has not enough tokens to fire T1 and T2 simultaneously This can be either done by providing the transi tions with priorities or probabilities In the first case a deterministic process decides which place enables which transition and in the second case the enabling is performed at random thereby transitions assigned with a high probability are chosen preferentially VY T1 1 gt T2 1 gt v2 7 5 V4 ne YN A TA 1 gt Sey v3 6 Ew mal PN EE Je S 4 v4 1 Figure 5 Example of a type 2 conflict the input speed of P2 and P3 is not sufficient to fire T5 and T6 with the de termined speed Another conflict can oc
19. ities of dis crete d stochastic s and continuous c transitions Name Type Part Default Description of Allowed delay scalar d 1 Delay of timed non negative transitions real values h scalaror s 1 Hazard function scalar non negative to determine the function real values characteristic value of exponen tial distribution maximumSpeed scalaror c 1 Maximum speed scalar non negative function real values arcWeightIn vectoror d s c 1 Weights of input vector non negative arcs function integers d s non negative real values c arcWeightOut vectoror d s c 1 Weights of output vector non negative arcs function integers d s non negative real values c firingCon condition d s c true Firing condition expression Boolean con dition expres sion The input is demonstrated by the following ex amples Figure 12 shows a discrete Petri net The indices of the connections are written at the arcs within square brackets e g the connection P1 gt T1 has the input index 1 and T1 P5 has the output index 3 The input of the arc weights dis played after the indices to property dialog or as mod ification equation is performed by the vector func tions arcWeightIn 2 P1 t 4 and 2 ly o Pl th whereby the expression P1 t accesses the current marking of P1 Thus the weights of the arcs P1 gt T1 and T1 gt P5 are functions which de
20. ns fire by removing the arc weight from all input places and adding the arc weight to all output places On the contrary the firing of continuous transitions takes place as a continuous flow determined by the firing speed which can depend functionally on markings and or time Places and transitions are connected by normal arcs which are weighted by non negative integers and real numbers respectively But also functions can be written at the arcs depending on the current markings of the places and or time Places can also be connected to transitions by test inhibitor and read arcs Then their markings do not change during the firing process In the case of test and inhibitor arcs the markings are only read to influence the time of firing while read arcs only indicate the usage of the marking in the transition e g for firing condi tions or speed functions If a place is connected to a transition by a test arc the marking of the place must be greater than the arc weight to enable firing If a place is connected to a transition by an inhibitor arc the marking of the place must be less than the arc weight to enable firing In both cases the markings of the places are not changed by firing The conversion of a discrete to a continuous marking is realized by connecting a discrete transi tion to a continuous place and the conversion from a continuous to a discrete marking is realized by con 48 Proceedings of the 9 International Model
21. s black with a white triangle for stochastic transitions and white for continuous transitions The test arc is represented by a dashed arc the inhibitor arc by an arc with a white circle at its end and the read arc by an arc with a black square at its end 3 1 Connectors The PNlib contains four different connectors PlaceOut PlaceIn TransitionOut and Tran sitionIn The connectors PlaceOut and PlaceIn are part of place models and connect them to output and input transitions respectively Similar Transi tionOut and TransitionIn are connectors of the transition model and connect them to output and in put places respectively Figure 10 shows which con nector belongs to which Petri net component model PlaceiIn Transitionin on TO PlaceOut TransitionOut Figure 10 Connectors of the PNlib The connectors of the Petri net component models are vectors to enable the connection to an arbitrary number of input and output components Therefore the dimension parameters nIn and nOut are declared in the place and transition models with the con nectorSizing annotation 3 2 Places The parameters of places are summarized in Table 1 If the type 1 conflict is resolved by priorities the corresponding priorities of the transitions are given by the indices of the connections i e the transition connected to the place with the index 1 has also the priority 1 the transition connec
22. ted to the place with the index 2 has also the priority 2 etc Otherwise if the probabilistic enabling type is chosen the corre sponding probabilities for the transitions have to be entered as a vector Thereby the first vector element corresponds to the connection with the index 1 the second to the connection with the index 2 etc The input of enabling probabilities as vectors in the place model and not at the corresponding arcs is neces sary due to the fact that properties cannot be as signed to connections according to the Modelica Specification 3 2 Table 1 Parameters and modification possibilities of dis crete d and continuous c places Name Default Description Type startTokens startMarks Marking at the beginning of the simulation scalar 0 minTokens minMarks Minimum capacity scalar 0 maxTokens maxMarks Maximum capacity scalar infinite choice scalar enablingType Type of enabling if type 1 conflicts occur the priorities are defined by the connection indices and the probabilities by the variables ena blingProbIn Out Priority enablingProbIn Enabling probabilities of input transitions vector fill 1 nIn nIn enablingProbOut Enabling probabilities of output transitions vector fill 1 mOut nOut N scalar Amount of levels for sto chastic simulation settings 1 N restart Condition for resetting the marking to reStartTokens Marks
23. teger quan tity called tokens or marks while continuous plac es contain a non negative real quantity These marks initiate transitions to fire according to specific condi tions and the firings lead to changes of the marks in the connected places Discrete transitions are provided with delays and firing conditions and fire first when the associated delay is passed and the conditions are fulfilled The se fixed delays can be replaced by exponentially dis tributed random variables then the corresponding transition is called stochastic transition Thereby the characteristic parameter of the exponential dis tribution can depend functionally on the markings of several places and is recalculated at each point in time when the respective transition becomes active or when one or more markings of involved places change Based on the characteristic parameter the next putative firing time t time Exp A of the transition can be evaluated and it fires when this point in time is reached XHPN Extended Hybrid Petri Nets Places Transitions time discrete process event time discrete state integer quantity continuous state stochastic process real quantity random event 3 gt normal arc continuous process flow oO inhibitor arc gt testarc e readarc Figure 2 Icons of the xHPN formalism Both discrete and stochastic transitio
24. the PNlib for modeling biological processes is shown in 10 USER INSERT PAD SENSEO MACHINE 1 A inserting e y hg i puffer 2 71 I Coffee Cup Water Tank water_tank REFILL WATER WATER TANK Figure 16 Hierarchical model of a Senseo coffee machine and simulation results A model of a Senseo coffee machine is presented The main feature of a Senseo coffee machine is that the coffee is placed in the machine in a pre portioned form by so called coffee pads One pad is generally used to make one cup of coffee 125 ml and two pads reach for two cups at 125 ml or one big cup at 250 ml After a warm up time of about 60 seconds and the insertion of a coffee pad the coffee can be made In this warm up phase the water is heated at 90 C and then pressed with a pressure of about 1 4 bar within 40 seconds through the pad In contrast to a normal coffee machine that boils the water continuously and transports it by its own buoyancy hot bubbles up into the filter the Senseo machine heats a portion of water completely in a heating chamber and pumps it then
25. through the pad To ensure that the heating chamber in the machine is always filled with water a float is placed in the 54 Proceedings of the 9 International Modelica Conference September 3 5 2012 Munich Germany DOI 10 3384 ecp1207647 Session 1A Hybrid Modeling removable water tank which allows measuring the mini mal capacity If the minimum level is exceeded the heater is turned off If there is sufficient water level the next portion of water is heated directly after the scalding and filling These functional principles are represented by the hierarchically structured model shown in Figure 16 and also some simulation results Addi tionally a detailed description of the model can be found in the PNlib The applicability of the PNlib for modeling pro duction processes is shown by a model of a printing process It is also modeled hierarchically to provide a compact and clear view on the highest level contain ing all important facts see Figure 17 The process starts with paper on a role and ends with printed leaf lets for supermarkets During the process misprints also called maculation could occur due to several reasons If the worker at the printing machine detects these misprints he presses a button and all incorrect exemplars are transferred outward When the macu lation is over he presses the button again and the process is continued With the help of this model several new insights can be detected e g
26. transitions wait till the next putative firing time is reached Based on this infor mation the places enable some of the active transi tion to fire At this point several conflicts can occur which have to be resolved appropriately by the methods mentioned in 8 to get a successful and reliable simulation When a transition is enabled by all its connected places it is firable and reports this via the connector variable fire to the connected plac es The places recalculate then their markings based on this information 3 4 Arcs xHPNs comprise four different kinds of arcs normal test inhibitor and read arc The Modelica language do not support the assignment of properties to arcs that are generated by connect equations Due to that fact test inhibitor and read arcs are realized by component models which are interposed between places and transitions see Figure 14 the normal arc is simply generated by the connect equation Test and inhibitor arc can be normal arcs simultaneously C e i m 7 gt E T2 o gt Ep 14 oe gt Figure 14 Modeling of normal top left test bottom left inhibitor top right and read arcs bottom right with the PNlib Table 3 Parameters and modification possibilities of test and inhibitor arcs read arcs have no parameters Name Type Default Description Allowed testValue scalar 1 The marking of the place non negative inte must be greater to ena
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