Multi-scale process model description by generalized hierarchical CPN models. E. Németh, K. M. Hangos

Transcription

1 Multi-scale process model description by generalized ierarcical CPN models E. Német, K. M. Hangos Researc Report SCL-002/2004

2 Researc Report SCL-002/ Contents 1 Introduction 2 2 Coloured Petri nets (CPNs) Coloured Petri nets Generalized ierarcical coloured Petri nets Introduction to ierarcical coloured Petri nets Formal definition of generalized ierarcical coloured Petri nets Instances Examples for te CPN ierarcies Substitution nodes Invocation nodes Fusion sets Case study Te countercurrent eat excanger: models of various levels of detail Te single cell model of te HE Cascade model of te HE Te CPN models of te countercurrent eat excanger Single cell models Cascade models Conclusion and future work 18 List of Figures 1 Substitution transition Substitution place Substitution place-transition Invocation transition Fusion sets Te diagram of te single cell countercurrent eat excanger Te cascade model of te eat excanger Te diagram of a eat excanger cell Te ierarcical CPN model of one cell eat excanger Te ierarcical CPN model of te cascade eat excanger Te CPN model of te cascade eat excanger after te substitutions

3 Researc Report SCL-002/ Abstract In tis paper we sow ow to extend coloured Petri nets (CPNs) wit a ierarcy concept. Te basic idea beind using ierarcical CPNs is to allow te modeller to construct a large model by using a number of small CPNs wic are related to eac oter in a well-defined way. We discuss seven different ways to relate submodels to eac oter, and tey are illustrated by examples. Te ierarcy constructions can be used for a simple countercurrent eat excanger. 1 Introduction Tis paper purposes a generalized definition of ierarcical coloured Petri nets (CPNs) for multi-scale process modelling purposes. It sows ow a set of subnets (called pages) can be related to eac oter in suc a way tat tey togeter constitute a single model [1]. Te basic idea is to allow te modeller to describe a set of submodels wic all contribute to a larger model, in wic te submodels interact wit eac oter in a well-defined way. Tis idea is well-known from oter kind of artificial languages. Te purpose is to break down te complexity of te large model by dividing it into a number of submodels. Te idea is most easily explained by proposing te notion of a substitution node, wic is a place or a transition or a combination of a place and a transition related to a submodel. Usually, te submodel totally replaces te substitution node and te surrounding arcs. Te paper describes 7 different ways to relate submodels to eac oter. Tese 7 ierarcy constructions are not at all independent and it is often possible to coose between tem, in order to fulfil a certain modelling goal. Tee proposed 7 ierarcy constructions are as follows: substitution transition, substitution place, substitution place-transition pair, invocation transition, invocation place, fusion place, fusion transition. Te concepts and notions are illustrated by using a simple process example: a countercurrent eat excanger model. 2 Coloured Petri nets (CPNs) In tis section, we summarize te basic notations found in te literature about coloured Petri nets and generalized ierarcical coloured Petri nets. For first, te general non-ierarcical coloured case will be considered, wile te second part concerns wit te various ierarcical Petri nets.

4 Researc Report SCL-002/ Coloured Petri nets Coloured Petri nets (CPNs) [2] belong to te area of discrete event system metodology. A CPN is well known for its capability in modelling discrete event systems. Te structure of a Petri net is a bipartite directed grap describing te structure of a discrete event system, wile te dynamics of te system is described by te execution of te Petri net. A Petri net is coloured if te tokens are distinguisable. According to te formal definition of CPNs [3] a coloured Petri net model is a nine-tuple satisfying te following requirements: CPN = (Σ, P, T, A, N, C, G, E, IN) (i) Σ is a finite set of non-empty types, called colour sets (ii) P is a finite set of places (iii) T is a finite set of transitions (iv) A is a finite set of arcs suc tat P T = P A = T A = (v) N : A P T T P is a node function (vi) C : P Σ is a colour function (vii) G is a guard function. It is defined from T into expressions suc tat t T : [Type(G(t)) = Bool Type(V ar(g(t))) Σ] (viii) E is an arc function. It is defined from A into expressions suc tat a A : [Type(E(a)) = C(p(s)) MS Type(V ar(e(a))) Σ] were p(a) is te place of N(a) and C MS denotes te set of all multi-sets over C (ix) IN is an initialization function. It is defined from P into expressions suc tat p P : [Type(IN(p)) = C(p(s)) MS V ar(in(p)) = ] were: Type(expr) denotes te type of an expression, V ar(expr) denotes te set of variables in an expression, C(p) MS denotes a multi-set over C(p). A binding of a transition t is a function b defined on V ar(t), suc tat: (i) v V ar(t) : b(v) Type(v), (ii) G(t) <b> denotes te evaluation of te guard expression G(t) in te binding b. A token element is a pair (p, c) were p P and c C(p). A binding element is a pair (t, b) were t T and b B(t). By B(t) denotes te set of all bindings for t. Te set of all token elements is denoted by TE wile te set of all binding elements is denoted by BE. A marking is a multi-set over TE wile a step is a non-empty and finite multi-set over BE. Te initial marking M 0 is te marking wic is obtained by evaluating te initialization expressions.

5 Researc Report SCL-002/ A transition is enabled if eac of its input places contain te multi-set specified by te input arc inscription (possibly in conjunction wit te guard), and te guard evaluates to true. Wen a transition is enabled it may occur, and tis means tat te tokens are removed from te input places and added to te output places of te occurring transitions. Te number and colour of te tokens are determined by te arc expressions, evaluated for te occurring bindings. A finite occurrence sequence is a sequence of markings and steps: M 1 [Y 1 M 2 [Y 2 M 3... M n [Y n M n+1, suc tat n N. A marking M is reacable from a marking M if and only if exists a finite occurrence sequence aving M as start marking and M as end marking, i.e., if and only if for some n N tere exists a sequence of steps Y 1 Y 2...Y n suc tat: M [Y 1 Y 2...Y n M. We ten also say tat M is reacable from M in n step. Te set of markings wic are reacable from M is denoted by [M. 2.2 Generalized ierarcical coloured Petri nets In tis subsection we sall see ow non-ierarcical CPNs can be extended to ierarcical nets, i.e., ow it is possible to construct a large CPN by combining a number of smaller nets by different several ierarcical constructions Introduction to ierarcical coloured Petri nets CPNs [3, 4] are capable of describing te dynamic beaviour of process systems and andle te ierarcy. Te basic idea beind using ierarcical CPNs [1, 5] is to allow te modeller to construct a large model by using a number of small CPNs wic are related to eac oter in a well-defined way. At one level, we want to give a simple description of te modelled activity witout aving to consider internal details about ow it is carried out. Moreover, we want to be able to integrate te detailed specification wit more crude descriptions and tis integration must be done in suc a way tat it is meaningful to speak about te beaviour of te complete net. Pages and teir instances We want to relate individual CPNs to nodes, wic are members of oter CPNs, and tis means tat our description will contain a set of non-ierarcical CPNs, wic we sall call pages. A diagram is a set of related non-ierarcical CPNs, called pages. A page may ave many different page instances. Tese page instances will ave teir own private markings, wic are independent of te markings of te oter instances (in a similar way tat eac procedure call as its own private copies of te local variables in te procedure). Eac ierarcical inscription in a CPN tells us te identity of te subpage, i.e. te page wic contains te detailed description of te activity modelled by te corresponding substitution node. Eac substitution node is said to be a supernode (of te corresponding subpage) wile te page of substitution node is a superpage (of te corresponding subpage). Page ierarcy To give an overview of te set of pages, we use a page ierarcy grap. Tis is a directed grap wic contains a node for eac page and an arc for eac direct superpage-subpage relationsip. Eac node is inscribed by te name of te corresponding page, wile eac arc is inscribed wit te names of te corresponding substitution or invocation nodes. Ellipse sape indicates tat all supernodes must be places, box sape tat tey must be transitions, exagon sape tat tey must be place-transition pair and rounded box sape tat tere is no restriction. Global fusion sets

6 Researc Report SCL-002/ are indicated in te page ierarcy, but page and instance fusion sets are not represented in te page ierarcy, because tey involve only a single page. Substitution nodes Te idea of substitution nodes is to allow te user to relate a node (and its surrounding arcs) to a more complex CPN, called subnet, wic usually gives a more precise and detailed description of te activity represented by te substitution nodes. Eac subnet as a number of places called port nodes and tey constitute te interface wit wic te subnet communicates wit its surroundings. A substitution node as some input nodes and some output nodes called input and output socket nodes, respectively. To specify te relationsip between a substitution node and its subnet, we must describe ow te port nodes of te subnet are related to te socket nodes of te substitution node. Tis is done by providing a port assignment. Wen a port node is assigned to a socket node, te two nodes become identical. Troug te input and output ports te subnet can be in communication wit its surroundings. Invocation nodes In contrast to substitution nodes, te invocation nodes are not substituted by teir subpage. Tis means tat tey can occur and eac of teir occurrences triggers te creation of a new instance of te subpage. Tese subpage instances are executed concurrently wit te oter page instances in te model, until some specified exit condition is reaced. Te termination of a recursive substitution is usually triggered by execution of te last statement or by an explicit exit statement. Te execution is terminated te first time an exit transition occurs or an exit place receives a token. Eac invocation node can ave any page as a subpage. Tis means tat te invocation ierarcy is allowed to contain circular (i.e. recursive) dependencies wile te substitution ierarcy is demanded to be acyclic (to avoid infinite substitution). Fusion sets Te main idea beind fusion is to allow te modeller conceptually to fold a set of nodes into a single node witout grapically aving to represent tem as a single object. A fusion is obtained by defining a fusion set containing an arbitrary number of places or an arbitrary number of transitions. Te nodes tat participate in suc a fusion set may belong to a single page or to several different pages. Tere are tree different kind of fusion sets: global fusion sets are allowed to ave members from many different pages, page fusion sets and instance fusion sets only ave members from a single page. A page fusion unifies all te instances of its nodes (independently of te page instance at wic te node instance appear). An instance fusion set only identifies node instances tat belong to te same page instance Formal definition of generalized ierarcical coloured Petri nets A ierarcical CPN consists of a set of subnets. Eac subnet s S is a non-ierarcical CPN, i.e., a tuple: (Σ s, P s, T s, A s, N s, C s, G s, E s, IN s ).

7 Researc Report SCL-002/ In ere we must to note tat a substitution or invocation node(s) sould not be considered as a normal node(s). Tese are special elements signifying wit special tags. Wen we talk about te elements of te entire ierarcical CPN, we use te following notations: Σ = s S Σ s, P = s S P s, T = s S T s, A = s S A s, were it sould be noted tat te sets of colour sets usually ave common elements, wile te sets of net elements (P s, T s, A s ) are required to be disjoint. X = P T is te set of nodes. X [X X S ] maps eac node x to te set of its surrounding nodes, i.e., te nodes tat are connected to x by an arc: X(x) = {x X a A : [N(a) = (x, x ) N(a) = (x, x)]}. In [X X S ] maps eac node x to te set of its input nodes, i.e., te nodes tat are connected to x by an arc: In(x) = {x X a A : N(a) = (x, x)}. Out [X X S ] maps eac node x to te set of its output nodes, i.e., te nodes tat are connected to x by an arc: Out(x) = {x X a A : N(a) = (x, x )}. ST(x 1, x 2 ), x 1 P, x 2 T is te socket type function, wic maps from pairs of socket nodes and substitution nodes into {in,out,i/o}. in if x 1 (In(x 2 ) Out(x 2 )), ST(x 1, x 2 ) = out if x 1 (Out(x 2 ) In(x 2 )), i/o if x 1 (In(x 2 ) Out(x 2 )) A generalized ierarcical coloured Petri-net is ten a tuple satisfying te following requirements: HCPN = (S, SN, SA, PN, PT, PA, FS, FT, PP) (i) S is a finite set of pages (subnets) suc tat: Eac page s S is a non-ierarcical CPN: (Σ s, P s, T s, A s, N s, C s, G s, E s, IN s ). Te sets of net elements are pairwise disjoint: s 1, s 2 S : [s 1 s 2 (P s1 T s1 A s1 ) (P s2 T s2 A s2 ) = ]. (ii) SN T P (T P) (P T) is a set of substitution nodes. (iii) SA is a page assignment function. It is defined from SN into S suc tat: 1 Wen we allow invocation nodes: page is a subpage of itself: {s 0 s 1...s n S n N + s 0 = s n k N + : s k SA(SN sk 1 )}. Wen we do NOT allow invocation nodes: No page is a subpage of itself: {s 0 s 1...s n S n N + s 0 = s n k 1...n : s k SA(SN sk 1 )} =. (iv) PN P T is a set of port nodes. (v) P T is a port type function. It is defined from P N into {in, out, i/o}. 1 S denotes all finite sequences wit element from S.

8 Researc Report SCL-002/ (vi) PA is a port assignment function. It is defined from SN into binary relations suc tat: Socket nodes are related to port nodes : x SN : PA(x) X(x) PN SA(x). Socket nodes are of te correct type: x SN : (x 1, x 2 ) PA(x) : [PT(x 2 ) {in, out, i/o} ST(x 1, x) = PT(x 2 )]. Related nodes ave identical colour sets, equivalent initialization expressions, equivalent guards, equivalent arc expressions: (a) t SN T : (p 1, p 2 ) PA(t) : [C(p 1 ) = C(p 2 ) I(p 1 ) <> = I(p 2 ) <> ]. (b) p SN P : (t 1, t 2 ) PA(p) : [G(t 1 ) = G(t 2 ) A(t 1, p) = A(t 2, p) if PT(t 2 ) = in, A(p, t 1 ) = A(p, t 2 ) if PT(t 2 ) = out, ]. PT(t 2 ) = in PT(t 2 ) = out if PT(t 2 ) = i/o. (c) (p, t) SN and (t, p) SN : for t te (a) is required and for p te (b) is required. (vii) FS P s is a finite set of fusion sets suc tat: Members of fusion set ave identical colour sets and equivalent initialization expressions: fs FS : p 1, p 2 fs : [C(p 1 ) = C(p 2 ) I(p 1 ) <> = I(p 2 ) <> ]. (viii) FT is a fusion type function. It is defined from fusion sets into {global, page, instance} suc tat: Page and instance fusion sets belong to a single page: fs FS : [FT(fs) global s S : fs P s ]. (ix) PP S MS is a multi-set of prime pages Instances A page s S may ave many different page instances. Te set of page instances of a page s S is te set SI s of all triples (s, n, x 1 x 2...x m ) tat satisfy te following requirements: (i) s PP n 1... PP(s ). (ii) x 1 x 2...x m is a sequence of substitution nodes, wit m N, suc tat: m = 0 s = s m > 0 (x 1 SN s [k 2...m x k SN SA(xk 1 )] SA(x m ) = s). Page instances were te tird component is te empty sequence are said to be prime page instances, wile all oters are secondary page instances. Wen a page as several page instances, tese eac ave teir own instances of te corresponding places, transitions and arcs. However, it sould be noted tat substitution nodes and teir surrounding arcs do not ave instances because tey are replaced by instances of te corresponding direct subpages. Te set of place instances of a page s S is te set PI s of all pair (p, id) tat satisfy te following requirements: (i) p P s \ SN s. (ii) id SI s.

9 Researc Report SCL-002/ Some of te place instances are related to eac oter, because of te fusion sets and because of te port assignments. Two place instances (p 1, id 1 ) and (p 2, id 2 ) are related by a fusion set fs FS iff te following conditions are fulfilled: Te two original places must bot belong to fs, i.e. p 1, p 2 fs. Wen fs is an instance fusion set, te two place instances must belong to te same page instance, i.e. id 1 = id 2. Wen fs is a global fusion set or a page fusion set, tere is no restriction on te relation between id 1 and id 2. Analogously, two place instances (p 1, id 1 ) and (p 2, id 2 ) are related by te port assignment of a substitution transition t SN iff te following conditions are fulfilled: Te two original places must be related by te port assignment, i.e. (p 1, p 2 ) PA(t). Te page instance id 2 = (s 2, n 2, tt 2 ) of te port node p 2 must not be a prime page instance because of te existance of te substitution transition t on te page instance id 1 = (s 1, n 1, tt 1 ) of te socket node p 1. Tis means tat te two page instance must originate from te same prime page instance, i.e. tat (s 1, n 1 ) = (s 2, n 2 )- Moreover, id 2 must ave te same sequence of substitution nodes as id 1, expect tat t as been added, i.e. tt 1 t = tt 2, were denotes concatenation of sequences. Te set of transition instances of a page s S is te set TI s of all pair (t, id) tat satisfy te following requirements: (i) t T s \ SN s. (ii) id SI s. Te set of arc instances of a page s S is te set AI s of all pair (a, id) tat satisfy te following requirements: (i) a A s \ A(SN s ). (ii) id SI s. Eac place instance, transition instance and arc instance is said to belong to te page instance in its second component. We define token elements, binding elements, markings, steps, initial markings, reacability and occurrence sequences analogously to te corresponding concepts for non-ierarcical CPNs. 2.3 Examples for te CPN ierarcies In te following we sow some illustrative examples for te different types of CPN ierarcies. We ave omitted most of te oter net inscriptions, e.g. te initial markings, since we are focusing more on te net structure tan on te details of colour sets, arc expressions and guards, etc.

10 Researc Report SCL-002/ Substitution nodes Te relationsip between eac of te substitution nodes and te corresponding subpages is defined by te inscription next to te tag ( = Hierarcy + Substitution). Tis inscription tells te name of te subpage and it describes ow eac of te nodes surrounding te compound node is assigned to one of te border nodes of te subpage. Te interfaces of subnets are defined by te B-tags (B = Border) and te inscription (in, out, i/o) next to tem. Substitution transition Te page in te left part of Fig. 1 represents a supernet, and te subnet of te substituted transition an be see in te rigt part. Below it is sown ow te ierarcical CPN in Fig. 1 is represented as a many-tuple. To save space (and time) we don t give te tuple-definition of te individual pages. SuperPage#1 SubPage#2 PA B in PF B in PC B in PA PB PC T1 T2 T SubPage#2 PB PF PE PG PH PD PE PD B out T3 PG B out Figure 1: Substitution transition Te tuple HCPN = (S, SN, SA, PN, PT, PA, FS, FT, PP) of te ierarcical CPN sown in Fig. 1 is defined as below: (i) S = {SuperP age#1, SubP age#2} (ii) SN = {T@SuperPage#1} (iii) SA(x) = SubPage#2 if x = T@SuperPage#1. (iv) PN = {PA@SubPage#2, PF@SubPage#2, PC@SubPage#2, PD@SubPage#2, PG@SubPage#2} { in if x {PA@SubPage#2, PF@SubPage#2, PC@SubPage#2} (v) PT(x) = out if x {PD@SubPage#2, PG@SubPage#2}. (vi) PA(x) = { (PA@SuperPage#1, PA@SubPage#2), (PB@SuperPage#1, PF@SubPage#2), (P C@SuperP age#1, P C@SubP age#2), (P D@SuperP age#1, P D@SubP age#2), (PE@SuperPage#1, PG@SubPage#2) } if x = T@SuperPage#1. (vii) FS = (viii) FT(fs) = global (ix) PP = 1`SuperPage#1. for all fs FS Substitution place Te page in te left part of Fig. 2 represents a supernet, and te subnet of te substituted place can be seen in te rigt part. Below we sow ow te ierarcical CPN in Fig. 2 is represented as a many-tuple. To save space (and time) we don t give te tuple-definition of te

11 Researc Report SCL-002/ SuperPage#3 SubPage#4 PI PJ PK B in B in T1 T2 T3 B in T1 T2 T3 PL PM P SubPage#4 T6 T7 T4 T5 PN PO PL T4 B out T5 B out Figure 2: Substitution place individual pages. Te tuple HCPN = (S, SN, SA, PN, PT, PA, FS, FT, PP) of te ierarcical CPN sown in Fig. 2 is defined as below: (i) S = {SuperP age#3, SubP age#4} (ii) SN = {P@SuperPage#3} (iii) SA(x) = SubPage#4 if x = P@SuperPage#3. (iv) PN = {T1@SubPage#4, T2@SubPage#4, T3@SubPage#4, T4@SubPage#4, T5@SubPage#4} { in if x {T1@SubPage#4, T2@SubPage#4, T3@SubPage#4} (v) PT(x) = out if x {T4@SubPage#4, T5@SubPage#4}. (vi) PA(x) = { (T1@SuperPage#3, T1@SubPage#4), (T2@SuperPage#3, T2@SubPage#4), (T 3@SuperP age#3, T 3@SubP age#4), (T 4@SuperP age#3, T 4@SubP age#4), (T5@SuperPage#3, T5@SubPage#4) } if x = P@SuperPage#3. (vii) FS = (viii) FT(fs) = global (ix) PP = 1`SuperPage#3. for all fs FS Substitution place-transition pair Te page in te left part of Fig. 3 presents a supernet, and te subnet of te substituted place-transition pair can be seen in te rigt part. Te exagon represents te substituted place-transition pair as a composite node. Te order of te place and te transition in te pair depends on te input and output node types (i.e. wic are places and wic are transitions). Below it is sown ow te ierarcical CPN in Fig. 3 represented as a many-tuple. Te tuple HCPN = (S, SN, SA, PN, PT, PA, FS, FT, PP) of te ierarcical CPN sown in Fig. 3 is defined as below: (i) S = {SuperP age#5, SubP age#6} (ii) SN = {(T1, P1)@SuperPage#5, (T2, P2)@SuperPage#5} (iii) SA(x) = SubPage#6 if x {(T1, P1)@SuperPage#5, (T2, P2)@SuperPage#5, T2@SubPage#8}. (iv) PN = {P3@SubPage#6, P4@SubPage#6, T5@SubPage#6, T6@SubPage#6} { in if x {P3@SubPage#6, P4@SubPage#6} (v) PT(x) = out if x {T5@SubPage#6, T6@SubPage#6}.

12 Researc Report SCL-002/ SuperPage#5 SubPage#6 Pa T1 P1 Ta Pb P3 B in T3 T5 B out Pd SubPage#6 Pa P3 P2 P4 Ta T5 T2 T6 P2 T2 SubPage#6 Pc P3 P1 P4 Tb T5 T1 T6 Pc P4 B in T4 PF T6 B out Tb Figure 3: Substitution place-transition (vi) PA(x) = {(P a@superp age#5, P 3@SubP age#6), (P 2@SuperP age#5, P 4@SubP age#6) (T a@superp age#5, T 5@SubP age#6), (T 2@SuperP age#5, T 6@SubP age#6)} if x = (T1, P1)@SuperPage#5 {(P c@superp age#5, P 3@SubP age#6), (P 1@SuperP age#5, P 4@SubP age#6) (T b@superp age#5, T 5@SubP age#6), (T 1@SuperP age#5, T 6@SubP age#6)} if x = (T2, P2)@SuperPage#5 (vii) FS = (viii) FT(fs) = global for all fs FS (ix) PP = 1`SuperPage#5. In ere we note tat we do not allow a substitution node to be a neigbor of anoter substitution node because ten it would be impossible to construct an equivalent non-ierarcical CPN by te metods defined above. To solve cases wen two substitution nodes come too close togeter, tey can always be separated by inserting an extra place and an extra transitions between tem Invocation nodes Te invocation nodes are distinguisable by te HI-tags (HI = Hierarcy + Invocation) and te inscription next to tem specifies te subpage and te port assignment. Invocation transition Te page in te left part of Fig. 4 presents a supernet, and te subnet of te invocation transition can be seen in te rigt part. Below we sow ow te ierarcical CPN in Fig. 3 is represented as a many-tuple. Te tuple HCPN = (S, SN, SA, PN, PT, PA, FS, FT, PP) of te ierarcical CPN sown in Fig. 4 is defined as below: (i) S = {SuperP age#7, SubP age#8} (ii) SN = {T@SuperPage#7, T1@SubPage#8, T2@SubPage#8} (iii) SA(x) = SubPage#8 (iv) P N = {Start@SubP age#8, Stop@SubP age#8} { in if x {Start@SubPage#8} (v) PT(x) = out if x {Stop@SubP age#8}. if x {T@SuperPage#7, T1@SubPage#8, T2@SubPage#8}.

13 Researc Report SCL-002/ P1 P2 SuperPage#7 B in Start T3 SubPage#8 B out Stop T1 T4 P3 Pa Pb T5 T P4 HI SubPage#8 P3 Start P4 Stop HI SubPage#8 Pa Start Pc Stop T1 Pc T2 Pd HI SubPage#8 Pb Start Pd Stop Figure 4: Invocation transition (vi) PA(x) = {(P 3@SuperP age#7, Start@SubP age#8), (P 4@SuperP age#7, Stop@SubP age#8)} if x = T@SuperP age#7 {(P a@subp age#8, Start@SubP age#8), (P c@subrp age#8, Stop@SubP age#8)} if x = T 1@SubP age#8 {(P b@subp age#8, Start@SubP age#8), (P d@subrp age#8, Stop@SubP age#8)} if x = T 2@SubP age#8 (vii) FS = (viii) FT(fs) = global for all fs FS (ix) PP = 1`SuperPage#7. Invocation place Exactly te same set of concepts as above for te invocation transition applies to te invocation place Fusion sets Place fusion Tis idea is illustrated in Fig. 5 were te left CPN as a fusion set called A. Tis fusion set contains te fusion set members A1 and A2 wic are distinguisable by te FP-tags (FP = Fusion + Page). FusA is a page fusion set and tis means tat it is allowed to ave only fusion set members from a single page in te diagram. Let us assume tat tis page as only one instance. Ten te equivalent CPN net is sown in Fig. 5. It is obtained by merging A1 into A2. Intuitively, tis semantics means tat te places A1 and A2 sare te same marking. Now let us consider te case were te page of FusA as more tan one page instances. Ten we ave two possibilities. Eiter we can merge all instances of all fusion set members into a single conceptual node FI-tags (FI = Fusion + Instance), or we can merge tem into a node for eac instance wit FP-tags. Finally we allow global fusion sets wit FG-tags (FG = Fusion + Global). Tis allows fusion set members from all pages in te diagram and all instances of tese nodes are merged into a single conceptual node. Tis means tat a page fusion set is a special case of global fusion set. Te part of te tuple HCPN = (S, SN, SA, PN, PT, PA, FS, FT, PP) of te ierarcical CPN sown in Fig. 1 is defined as below: (i) S = {..., SubPage#10,...}.

14 Researc Report SCL-002/ SubPage#10 SubPage#10 Before unification of fusion nodes After unification of fusion nodes FusA T1 T4 FusA T1 T4 FP FP A1 T2 B A2 A T2 B T3 T5 T3 T5 Figure 5: Fusion sets (vii) F S = {{A1@SubP age#10, A2@SubP age#10}} (viii) FT(fs) = SubPage#10. for all fs FS Te members of a fusion set must be comparable to eac oter. For places tis means tat tey must ave te same colour set and te same initial marking. It also means tat tey eiter all must be ordinary places, all be substitution places or all be invocation places, and in te two latter cases tey must all ave te same subpage. Fusion of substitution places are useful, wen we want to apply te same instance at several locations in te diagram. Transition fusion Exactly te same set of concepts as above for place fusion applies to transition fusion. For transitions we do not demand tat te guards are identical. Instead, we form te conjunction of te guards. Te members of a transition fusion set must eiter all be ordinary transitions, all be substitution transitions or all be invocation transitions. In te two latter cases tey must all ave te same subpage. In addition, it is not allowed to use global and page fusion for transitions, wic appear on subpages of invocation transitions (or on subpages of suc pages). 3 Case study In te following example a simple multi-scale process model will be used to demonstrate te top-down model building by using generalized ierarcical coloured Petri net models. 3.1 Te countercurrent eat excanger: models of various levels of detail Te single cell model of te HE Consider a countercurrent eat excanger (HE) sown in Fig. 6, were te cold liquid stream is being eated by a ot liquid stream. On te top level te eat excanger is modelled by a single pair of perfectly stirred lumps forming a so called eat excanger cell. Eac cell consists of two perfectly stirred tanks wit in- and outflows. Te two tanks are connected by a eat transfer area between tem.

15 Researc Report SCL-002/ ( in) Heat transfer ( in) Figure 6: Te diagram of te single cell countercurrent eat excanger Modelling assumptions Te overall mass (volume) of te liquids on bot sides is constant. No diffusion takes place. No eat lost to te surroundings. Heat transfer coefficients are constant. Specific eats and densities are constant. Bot liquids are in plug flow. Te eat excanger is described as a CSTR pair of ot and cold liquid volumes. Balance volumes We consider perfectly mixed balance volumes wit equal oldups for eac of te ot and cold sides. Te subscripts and c denote te ot and cold sides respectively. Model Equations: Variables ( (t), (t)), 0 t (1) were (t) and (t) is te ot and te cold side temperature in te tank and t is time. Energy balances for te ot side d (t) dt = F ( ) T (in) V KA ( (t) (t)) (2) c p ρ V T (in) (t) is te ot liquid inlet temperature to te eat excanger. Energy balances for te cold side d (t) = F ( c c dt V c T (in) ) (t) (t) (in) (t) is te cold liquid inlet temperature to te eat excanger. KA c pc ρ c V c ( (t) (t)) (3) Here F and F c are te flowrates, V and V c are te volumes, A is te eat transfer area, c p and c pc are te specific eats, ρ and ρ c are te densities, K is te eat transfer coefficient.

16 Researc Report SCL-002/ Cascade model of te HE If we want to refine te model, te eat excanger is divided into n equal parts, and it gives te cascade model of te HE (see in Fig. 7). Te cascade model consists of eat excanger (HE) cells sown in Fig. 8. (0) (1) (2) (1) (1) (2) (2) ( n) ot side ( n 1) T ( n) ( n) (1) (1) (2) (2) (3) Heat transfer T ( n) c ( n) ( n 1) + cold side Figure 7: Te cascade model of te eat excanger (i-1) v V (i) (i) A (i) (i) v (i) v c V c (i) (i) Q (i) (i-1) v c Figure 8: Te diagram of a eat excanger cell Modelling assumptions Replace te last item of te modelling assumptions wit te following item: Te eat excanger is described as a sequence of n CSTR pairs of ot and cold liquid volumes (n = 3). Balance volumes We consider tree perfectly mixed balance volumes wit equal oldups for eac of te ot and cold sides. Model Equations: Variables ( T (k) ) (k) (t), (t), k = 1, 2, 3, 0 t (4) were T (k) (k) (t) and (t) is te ot and te cold side temperature in te kt tank pair respectively and t is time.

17 Researc Report SCL-002/ Energy balances for te ot side dt (k) (t) dt = F (k) ( T (k 1) V ) T (k) K(k) A (k) c (k) p ρ (k) V (k) ( T (k) ) (k) k = 1, 2, 3, T (0) (i) (t) = T (t) (5) T (i) (t) is te ot liquid inlet temperature to te eat excanger. Energy balances for te cold side d (k) (t) dt = F (k) ( c (k+1) V c ) (k) K(k) A (k) c (k) p c ρ c (k) V c (k) ( T (k) c ) T (k) k = 1, 2, 3, T (4) (i) (t) = (t) (6) T (i) c (t) is te cold liquid inlet temperature to te eat excanger. Note tat te cold stream flows in te direction of descending volume indices. Initial conditions T (k) (0) = f(k) 1, k = 1, 2, 3 (7) were te values of f (k) 1, k = 1, 2, 3 and f (k) 2, k = 1, 2, 3 are given. (k) (0) = f (k) 2, k = 1, 2, 3 (8) 3.2 Te CPN models of te countercurrent eat excanger Single cell models First consider a single cell ierarcical CPN model in Fig. 9. Te SuperPage#1 as prime page of te single cell HE model is divided into two main parts: te ot side and te cold side parts wic are connected. Te left part of Fig. 9 sows te top level of te one cell HE, te rigt part sows te subpages HotCell#2 and ColdCell#3 of te substitution place-transition pairs. Subnets denoted by exagons contain te ot and cold side cell details, respectively. We use substitution place-transition pairs for ierarcical decomposition. Te ot or cold side balance volume is described by a node containing its state variable value, and te incoming transitions realized te applications of input data. Te outgoing arcs serve te information to te surrounding about te balance volume Cascade models Te next step is to divide te single cell model into a n-cell model. Te CPN realization can be seen in Fig. 10. Te prime page of te cascade CPN model is te page SuperPage_Cascade#3, wic is similar to te prime page of te single cell model, but te substitution subnets are canged. Te new substitution subnets SubP age_casc_ot#4 and SubP age_casc_cold#5 in Fig. 10 describe te division of balance volumes of te single cell model. Tese inserted subnets do not describe te cell details. Te ot and cold cell details are substituted to te corresponding substitution nodes by using subnets HotCell#2 and ColdCell#3 (see in Fig. 9).

18 Researc Report SCL-002/ HotCell#2 B in B out T (in) t1 t4 SuperPage#1 B in T B out (in) t t8 (out) T (2) t2 t5 t3 HotCell#2 (in) T (in) T (2) t8 t4 t c t5 ColdCell#2 (in) T (in) T (2) t9 t4 t t5 B in ColdCell#3 B out T (in) t1 t4 (out) t9 t c (in) B in T B out T (2) t2 t5 t3 Figure 9: Te ierarcical CPN model of one cell eat excanger SuperPage_Cascade#3 (in) t t8 (out) SubPage_Casc_ot#4 SubPage_Casc_cold#5 (out) t9 t c (in) SubPage_Casc_ot#4 (in) (1) (1) HotCell#2 (2) (2) HotCell#2 (n) (n) HotCell#2 SubPage_Casc_cold#5 (1) (1) ColdCell#3 (2) (2) ColdCell#3 (n) (n) ColdCell#3 (in) Figure 10: Te ierarcical CPN model of te cascade eat excanger Finally, te wole CPN model of te cascade HE can be see in Fig. 11 after te substitutions of te corresponding subnets.

19 Researc Report SCL-002/ (0) (1) (2) (n) (n+1) (0) (1) (2) (n) (n+1) Figure 11: Te CPN model of te cascade eat excanger after te substitutions 4 Conclusion and future work We ave proposed extension of coloured petri nets (CPNs) by ierarcy concept in seven ways: substitution transition, substitution place, substitution place-transition pair, invocation transition, invocation place, fusion place, fusion transition. Hierarcical CPN is formulated and illustrated by examples. Te presented experiments are very difficult and rater complex, but we believe tat our work will form te base for multi-scale process description for diagnostics.

20 Researc Report SCL-002/ References [1] Huber P., K. Jensen, and R.M. Sapiro. Hierarcies in coloured petri nets. Lecture Notes in Computer Science 483, Advances in Petri Nets 1990, pages , [2] Kurt Jensen and Grzegorz Rosenberg. Hig-level Petri nets: Teory and Application. Springer- Verlag, [3] Kurt Jensen. Coloured Petri Nets. Basic Concepts, Analysis Metods and Practical Use Volume 1. Springer-Verlag, [4] Kurt Jensen. Coloured Petri Nets. Basic Concepts, Analysis Metods and Practical Use Volume 2. Springer-Verlag, [5] Lakos C. A. Te role of substitution places in erarcical coloured petri nets. Tecnical Report TR93-7, Computer Science Department, University of Tasmania, 1993.