COMPUTER AIDED MODELLING USING COMPUTER SCIENCE METHODS

E. Németh1,2, R. Lakner2, K. M. Hangos1,2, A. Leitold3

1Systems and Control Laboratory, Computer and Automation Research Institute

HAS, Budapest, Hungary, http://www.sztaki.hu/scl/PCRG 2Department of Computer Science, University of Veszprém, Veszprém, Hungary,

http://www.dcs.vein.hu/CICS 3Department of Mathematics and Computing, University of Veszprém, Veszprém,

Hungary

Computer aided modelling tools Process model

structured knowledge collection model elements:

balance volumes extensive quantities balance equations transport mechanisms constitutive equations

mathematical elements: differential and algebraic equations differential and algebraic variables

further classification of the variables: defined by an equation defined as constant defined as unspecified (design) variable

Assumptions: two phases (vapour, liquid) single component phase equilibrium feed and output flows heating

1

2

( ) ( )

( ) ( )

,

,

v

l

vv e lv

lel l lv

e vlv vl l

cslv vl

v v

l l

dM V Edt

dML F E

dtdU V h Q E hdt

dUL h F h Q E h Q

dtQ u u A T T

E k k A P P

h f T P

h f T P

V

F

L

QeE

Q

Model editor - model building interactive intelligent interface assumption-driven model building procedure result: process model in canonical form

Model editor - model simplifying syntax and semantics of modelling assumptions:

additional mathematical relationships or constraints described formally by triplet: model-element/relation/keyword

effects of assumptions on the process model: formal simplification and algebraic transformations forward reasoning

Model editor - assumption retrieving given: a detailed and a simplified process model question: simplification assumption sequences forward reasoning with iterative deepening search

Structural analysis of dynamic lumped process models

The structural analysis includes the determination of the degree of freedom, structural solvability, differential index and the dynamic degrees of freedom.

Basic notions Representation of algebraic equations: standard form

yi = fi (x, u), i = 1, …, M where: X = x1 ,…, xN set of unknowns,

uk = gk (x, u), k = 1, …, K U = u1 ,…, uK set of unknowns,

Y = y1 ,…, yM set of parameters.

The model is structurally solvable if the Jacobian matrix J(x,u) is non-singular.

Representation graph: Vertex-set: X Y U;

Arc-set: corresponds to the model equations.

Menger-type linking: a set of vertex-disjoint directed paths from a vertex in X to a vertex in Y. If the number of paths = |X|= |Y | complete linking

Linkage theorem (Murota 1987): Standard form model with a generality assumption is structurally solvable there exists a Menger-type complete linking on the representation graph.

Representation of dynamic process models described by DAEs

Dynamic representation graph: sequence of static graphs corresponding to each time step of numerical integration.

Steps of structural analysis using the representation graph Rewrite the model into standard form, create the representation graph. Assignment of types to vertices according to the model specification. Reduction of the representation graph implicit part of the model. Analysis of the reduced graph:

determination of the differential index using the structure of the graph, structural decomposition computational path.

In case of higher index models: modification of model to obtain a structurally solvable model form.

Advices on how to improve the computational properties of the model by modifying its form or its specification.

x

x1

x2

xn

x'

t

x

x1

x2

xn

x'

t+h

x’ = f(x1,…, xn) arcs: correspond to the structure of the differential equation, arcs: correspond to the applied numerical solution method (here: first order, single-step, explicit solution method).

Main results: The differential index of the investigated dynamic lumped model M is equal to one there

exists a Menger-type complete linking on the reduced graph. The structure of the representation graph is suitable for determination of the differential index

in case of higher index models. Important properties of representation graph are independent of the assumption whether a

single step first order or higher order, or a single step implicit numerical method is used for the solution of differential equations the analysis method is numerical method independent.

Example: A simple liquid systemThe standard form of the model:

M = M’ U= U’

M’= –L + F U’= –LhL + FhF + Q

hL= U M hL*= f1(TL, p)

hF= f2(TF, pF) s= hL – h L*, s = 0

L = f3(M)

F

hF, TF, pF

p

Q

L, hL

TL

Specification 1.

Given: F, TF, pF, Q,

as function of time

M0, U0, p

as constants

To be calculated:

M, U, TL and L

as function of time

Specification 2.

Given: F, TF, pF, TL,

as function of time

M0, U0, p

as constants

To be calculated:

M, U, Q and L

as function of time

Reduced graph:

There is a Menger-type complete linking on the graph.

differential index = 1

There is no Menger-type complete linking on the graph.

differential index > 1

M

t

F

U

TL

p

TF

<S>

pF

<S>

Q

<S*>

<S>

L

hL

hL*

hF

U'

M'

s<G>

<S>

<S*>

<S>

TL s<G>

hL*

M

t

F

U

TL

p<S>

TF

<S>

pF

<S>

Q

<S*>

<S>

<S*>

L

hL

hL*

hF U'

M'

s<G>

<S>

underspecified

overspecified

General strategies (the order in which the model is constructed) bottom-up top-down concurrent

Approaches to integrating partial models into a multiscale model (how the partial models at different scales are linked together) multidomain embedded paralel serial

simplification transformation one-way coupling

simultaneous

Multiscale process modelling by coloured Petri nets (CPNs)

flow of information between the scales

(a) Multidomain (b) Multidomain(with interface zone)

(c) Embedded

(d) Parallel (e) Serial (f) Simultaneous

Broken lines refer to regions where the balance volumes overlap.

M

I

M M

M1

2

Simple multiscale model of a heat exchanger (cascade model)

Heat transfer

hot side

cold side

( )n(1) (2)(0)

hT (1)hT (2)

hT ( 1)nhT ( )n

hT

( 1)ncT ( )n

cT(3)cT(2)

cT(1)cT

(1)hT

(1)cT

(2)hT

(2)cT

( )nhT

( )ncT

( )inhT

( )incT

Heat transfer

( )( )

( ) ( )

( ) ( )( )

( ) ( )

( )

( )

h

c

kinh h

ch h hk kph h h

k kinc c

c c c hk kc p c c

Knn

nn

dT t F AT T T Tdt nV c V

dT t F K AT T T Tdt V c V

( ) ( ) ( )( 1) ( ) ( ) ( )

( ) ( ) ( ) ( )

(0) ( )

( ) ( ) ( )( 1) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

1,2,3, ( ) ( )

( )

h

c

k k kk k k kh h

ch h hk k k kph h h

ih h

k k kk k k kc c

c c c hk k k kc p c c

dT t F K AT T T Tdt V c V

k n T t T t

dT t F K AT T T Tdt V c V

( 1) ( ) 1,2,3, ( ) ( )n ic ck n T t T t

Energy balances for the hot side:

Energy balances for the cold side:

hThT

cTcT

hot side

cold side

Multiscale CPN model of the heat exchanger

T c

T h( in )

T c( o ut ) T c

( in )

T h( o ut )

T h( 0 )

T c( 0 ) T c

( 1 ) T c( 2 )

T h( 1 ) T h

( 2 ) T h( n ) T h

( n + 1 )

T c( n + 1 )T c

( n )

_

_

T h

References Hangos, K.M. and Cameron, I.T., 2001: Process Modelling and Model Analysis. Academic Press, London, pp. 1-543. Lakner, R., Hangos, K.M. and Cameron, I.T., 1999, An assumption-driven case sensitive model editor. Computer and

Chemical Engineering (Supplement), 23 S695-698. Hangos, K.M. and Cameron, I.T., 2001: A Formal Representation of Assumptions in Process Modelling. Computers

and Chemical Engineering, 25 237-255. Lakner, R. and Hangos, K.M., 2001, Intelligent assumption retrieval from process models by model-based reasoning.

Engineering of Intelligent Systems (Lecture Notes in Artificial Intelligence), 2070 145-154. A. Leitold, K.M. Hangos, 2001: Structural Solvability Analysis of Dynamic Process Models, Computers and Chemical

Engineering, 25 1633-1646. Leitold, A. and Hangos, K.M, 2002: Effect of Steady State Assumption on the Structural Solvability of Dynamic

Process Models, Hung. J. of Ind. Chem. 30 1 61-71. A. Leitold, K.M. Hangos, 2004: Numerical Method Independent Structural Solvability Analysis of DAE Models Models,

submitted to System Analysis Modelling Simulation Németh, E., Lakner, R., Hangos, K.M. and Cameron, I.T., 2003: Hierarchical CPN model-based diagnosis using

HAZOP knowledge, Technical report of the Systems and Control Laboratory SCL-009/2003. Budapest, MTA SZTAKI. Ingram, G.D., Cameron, I.T. and Hangos, K.M, 2004: Classification and analysis of integrating frameworks in

multiscale modelling, Chemical Engineering Science