Mathematical Analysis of Dynamic Process Models

Transcription

Index, inputs and interconnectivity
Index, inputs and interconnectivity
PROEFSCHRIFT
ter verkrijging van de graad doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof.dr.ir. J.T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 4 december 2006 om 12:30 uur
door
Aloysius Thomas More Judi SOETJAHJO
geboren te Malang, Indonesia
werktuigbouwkundige ingenieur
Dit proefschrift is goedgekeurd door de promotoren:
Prof. ir. O.H. Bosgra
Prof. ir. J. Grievink
Toegevoegd promotor:
Dr.S. Dijkstra
Samenstelling promotiecommissie:
Rector Magnificus,
Prof. ir. O.H. Bosgra,
Prof. ir. J. Grievink,
Dr. S. Dijkstra,
Prof. dr. ir. P.M.J. Van den Hof,
Prof. dr. ir. G. van Straten,
Prof. dr. ir. A.C.P.M. Backx,
Dr. ir. P.J.T. Verheijen,
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ISBN: 90 – 8559 – 255 – 0
voorzitter
Technische Universiteit Delft, promotor
Technische Universiteit Delft, promotor
Technische Universiteit Delft, toegevoegd promotor
Technische Universiteit Delft
Universiteit Wageningen
Technische Universiteit Eindhoven
Technische Universiteit Delft
Dedicated to my mother : Lia Indriawati
Terima kasih atas doa-doamu, Tuhan beserta kita
Ya Bapaku dalam surga, aku bersyukur padaMu
Pimpinlah aku selalu untuk membagi kasihMu,
kebijaksanaanMu and pengetahuanMu
Preface
This thesis is the result of my research experiences during the period of 1992 to 1997
at the Delft University of Technology, on dynamic modelling of a complex
interconnected DAE (differential Algebraic Equations) of a Coal Gasification
Combined Cycle Plant using Speed-Up (Aspen Tech). This thesis describes specific
mathematical problems, namely a high index problem and consistent (re-) initialisation
problem that I have encountered during modelling of a process and during
interconnecting of process sub models. The literature references used are dated until
year 2003.
At this time, I wish to express my gratitude to all the people who have contributed to
this thesis, although unfortunately I am able to mention just a few! Firstly I would like
to thank my promoters Prof. O.H. Bosgra and Prof. J. Grievink, and my co-promoter,
Dr. S. Dijkstra for providing a valuable research project. Their support, advice &
encouragement even after my resignation from university, ensured that I did not give
up on completing this book. Thank you for opening the door to an incredible
educational journey for which I am thankful! I am grateful to Dr. P.J.T.Verheijen for
our discussions and his critical remarks regarding the presentation of my work. Thank
you also to Prof. P.M.J. van den Hof, Prof. G. van Straten, and Prof. A.C.P.M. Backx
for serving on the thesis committee and for their valuable comments.
Secondly, I would like to thank SBM (Single Buoy Mooring) Chief Operating Officer,
Dick van der Zee and GustoMSC Head of the System Engineering Department, Bertus
Bernhard, both of whom have always supported me and actively encouraged me to
finish this thesis in conjunction with my daily work and responsibilities at GustoMSC.
Thank you to Andries Mastenbroek and Paul Spoeltman for their understanding and
support.
Thirdly, I would like to thank for all my colleagues and students at the Delft
University of Technology, all of whom supported me during my research period; Peter
Valk, Arie Korving, Gideon Go, Agung Hermawan, Habieb, Trias Hermanu, and
Folmer de Haan. My special thanks to Angela Hernandez from GustoMSC for her
assistance with linguistic and layout corrections.
Finally, thank you to my dearest friend Ton Peters. Thank you for your unconditional
support and your everlasting care.
Delft, December 2006
A.T.M. Judi Soetjahjo
i
Summary
Process modelling plays an important role in process engineering and operation. The
process models are to be used for: designing of process equipment and process control,
prediction of process behaviour and optimisation of product properties during the
process operation.
Usually, a complex process model consists of interconnected sub models, i.e. as a
composite network of sub models in different layers with increasing detail of model
components. Modelling of each sub system as a lumped system described with first
principles equations, i.e. balance and constitutive equations, gives a set of first order
non-linear DAE (Differential Algebraic Equations) model.
A consequence of modelling with DAE is the possibility to get a high index and/or a
consistent (re-) initialisation problem. An (differential) index of DAE model can be
easily defined as the number of differentiations required to transform the original DAE
model to an explicit ODE model. A high index occurs if this required differentiation
has to be done more than once.
A high index problem can be caused by: the wrong choice of input variables, internal
state constraint equations (e.g. reaction equilibrium, phase equilibrium), or sub models
interconnection.
A consistent (re-) initialisation problem means that it is not possible to set arbitrary
values of the accumulation/storage variables at any time. A high index model and a
low index model with non minimum differential variables always pose a consistent
(re-) initialisation problem.
Basically, solving a high index problem needs differentiation, instead of integration
only. Besides that most numerical algorithms do not cover a differentiation algorithm,
it is not practical to solve an interconnected high index DAE models, because a high
index DAE poses also a consistent (re-) initialisation problem. A modeller can get
unexpected behaviour on the defined/chosen accumulation/storage variables; i.e. these
storage variables can ‘jump’ in their time response. When modelling a large scale
interconnected system one should avoid a high index DAE model.
There have been many researches carried out to solve a high index DAE model. The
distinction of the work in this thesis compared with others is that the detection of a
high index problem and an index reduction techniques is developed from the original
theory of Kronecker and Rosenbrock [107], i.e. a strict system equivalence operation.
A strict system equivalence operation performs the required differentiations while
maintaining the input-output behaviour of the system. The theory of Kronecker and
Rosenbrock [107] is extended for a first order nonlinear DAE model by using the
structural information of the equations, rather than the exact numerical value.
This thesis shows, for a first order DAE process model, that it is possible to find a
procedure (based on a strict system equivalence operation) for a high index reduction,
which does not apply complicated non-linear algebraic manipulations. With this
procedure the original model equations and variables will not be destroyed.
ii
Moreover for a given DAE model it is possible to detect which input variables set will
not raise a high index problem, provided that there are no state constraints equations.
These state constraints equations can be detected by any choice of an input variables
set which give the maximum number of state variables of the model.
Interconnection of DAE sub models requires information on possible input variables of
each sub model. This input variables set does not have to be unique. Interconnection of
sub models will not raise a high index problem, when there is no state constraint
equation arising due to the interconnection, i.e. the interconnected low index models
resulting to a low index composite model.
Despite of advantage on applicability of the structural method, for a rare case it is
possible to have an analytical cancellation problem on a high index detection; i.e. the
structural algorithm shows that the DAE has a high index; but analytically the DAE
does not have a high index. A combination of symbolic and structural approaches can
solve this problem and the theory in this thesis can be expanded to such application.
Moreover the extension of the high index theory for a distributed system, PDAE
(Partial Differential Algebraic Equations) is necessary, since a state constraint
problems can arise either in a time domain, or in a spatial domain or a combination of
them.
iii
Table of Contents
Preface ………………………………………………………………..
Summary ……………………………………………………………..
i
ii
Chapter 1: Introduction
1.1.
1.2.
1.3.
1.4.
1.5.
Background .…………………………………………………
Motivation ..…………………………………………………
Related works ………………………………………… ……
Problem definition and scope …………………………………
Approach and outline of the thesis ……………………………
1
2
7
9
10
Chapter 2: Mathematical Properties of a First Order DAE model
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
Introduction …………………………………………………..
Kronecker Canonical Form …………………………………...
Low index criterion of a DAE model ………………………….
Calculation of the index of a DAE model ……………….…….
Solving a consistent initialisation and a re-initialisation problem
Index reduction of a DAE model ………………………………
Conclusions ……………………………………………………
13
14
18
19
21
27
38
Chapter 3: Structural Properties of a First Order DAE model
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
Introduction …………………………………………………..
−1
……………….
Structure of system matrices E , T ( s ), T ( s )
Structural properties of a first order non linear DAE model ….
Genericity of structural properties ……………………………
Examples ……………………………………………………..
Conclusions …………………………………………………..
39
40
44
49
54
61
Chapter 4: Application of Low Index Detection and High Index Reduction
of Non-linear DAE Process models
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
Introduction …………………………………………………..
High index reduction through variables differentiation ……….
High index reduction through equations differentiation ……….
Consistent re-initialisation problem …………………………..
High index problem due to interconnection …………………..
Conclusions …………………………………………………..
63
65
77
84
89
94
Chapter 5: Input Variables Assignments of a DAE Model
5.1.
5.2.
5.3.
Introduction …………………………………………………... 95
Model entities and variables classification …………………… 98
Low index input variables assignment …………………….….. 105
iv
Table of Contents
5.4.
5.5.
Maximum dynamic degree input variables assignment ………………... 118
Conclusions ……………………………………………………………. 126
Chapter 6: Interconnection of DAE Models
6.1.
6.2.
6.3.
6.4.
6.5.
Introduction …………………………………………………………….
Process model ports …………………………………………………….
Process model connectors ………………………………………………
Interconnection of sub models ………………………………………….
Conclusions …………………………………………………………….
127
128
133
138
152
Chapter 7: Conclusions and Recommendations
7.1.
7.2.
Conclusions ……………………………………………………………. 153
Recommendations ……………………………………………………… 155
Appendix A: Structural Frame Work ………………………………………… 157
Appendix B: Graph Algorithm ……………………………………………….. 161
Appendix C: Modelling of an ICGCC Plant …………………………………. 171
List of symbols and abbreviations …………………………………………….. 183
References ……………………………………………………………………… 185
Samenvatting …………………………………………………………………… vi
Curriculum vitae ……………………………………………………………….. viii
v
Chapter 1: Introduction
Design engineering practices always require process models as a predicting
and calculation tool. The model modularisation, interconnectivity and
transparency are essential in model building and model re-use. The
mathematical models of process result mostly into a set of interconnected
Differential Algebraic Equations (DAE). The research motivation of this thesis
is analysis of the mathematical process model structure and its
interconnectivity.
“A system is our ‘projection’ of a structure or an ordering of ‘mechanisms’”
1.1. Background
Mathematical models are used in many areas of science or engineering to understand
the system’s behaviour and/or design of specific system behaviour. This model
represents internal behaviour of a system subject to the purpose, assumptions, and
simplifications of the modeller. The system internal behaviour is generally described
with balances and other constitutive equations.
A system can be anything of a single part, organ, or mechanism or a set of
interconnected parts, organs, and/or any physical process, that may perform a
particular function or composite functions. A dynamic system is a system where its
internal properties or quantities undergo a progress of change in time. A mathematical
description of a dynamic system is called a mathematical (dynamic) model.
A mathematical (dynamic) model (Willems [131]) is a mathematical description of a
dynamic system (for a given boundary) that consists of aspects: time and behaviour
equations. Behaviour B of a dynamic system manifests to a set of signal trajectories
z (t ) ∈ R n on some time interval t ∈ (a b); a, b ∈ R , satisfied behaviour
equations,
{
}
B = z (t ) : R → R n Behaviour equations are satisfied
(1.1)
Modelling of a complex process consists of interconnection of sub-models each
represented by a set of behaviour equations (1.1).
Prior to 1960 – 1980’s, it was common practice for modelling of a system behaviour
by a set of Ordinary Differential Equations (ODE) with fixed chosen input variables,
x = f ( x , u, t ) ,
(1.2)
where f : R n + k +1 → R n , x , x ∈ R n are the model solutions, u ∈ R k are the input
variables, and t ∈ T denotes the time at some interval T ⊂ R .
1
Chapter 1 : Introduction
Simulation of an ODE model was done by translating it into a Computer Language as
interconnection of mathematical library operations, from which it was difficult to
recognise the original set of equations, difficult to modify the model, and difficult to
re-use the model for other purposes.
Nowadays mathematical process modelling and simulation are based on balance
equations and constitutive equations resulting into a set of Differential Algebraic
Equations (DAE), given by :
F ( z(t ), z (t ), t ) = 0 ,
(1.3a)
or F ( z (t ), z (t ), u (t ), t ) = 0 ,
(1.3b)
where F : V → R m , V is open in R 2 n +1 is a vector defining the model; z : T → R n
denotes its solution, with m ≤ n and t ∈ T denoting the time at some interval T ⊂ R ,
is a common approach, where the modeller always has the transparent original DAEs
model in the simulation engine. This has many advantages for re-usability or model
modification.
1.2. Motivation
1.2.1. Industrial needs
The development of numerical solving methods and computer hardware in the last
decennium make it possible to do large-scale mathematical model
simulation/calculation even with a personal computer.
Process model simulation on the process industry consists of two main items, namely:
the process model building and the process model re-use. On the process model
building the modeller needs to make the model equations. For the process model re-use
the modeller uses the existing made models and interconnect them with other models
when it is necessary.
Commercial software such as Aspen [5] and Hysis [61] built a lot of standard model
libraries, which can be re-used and interconnected for designing complex processes.
Also there is a tendency to migrate the software programmes to open concepts, where
different model libraries built in different programmes can be re-used and
interconnected.
Nowadays engineering design activities include integration of processes from different
physical domains, e.g.: electrical, chemical, physical, and mechanical processes.
Engineers need to be able to build, re-use and interconnect mathematical models from
different physical domains into one common simulation platform.
Despite the fast developments in different fields of software programming and
hardware availability, industrial practice shows that there are still several problems to
be solved on dynamic model building and model interconnection. The available
2
simulation tools on the market give different ways for model representations, model
aggregation and model interconnectivity of processes from different physical domains.
The dynamic modelling process and representation should be standardised in
universal-ways. The standardisation in this context also brings a ‘bridge’ between
Conceptual Modelling Process and its translation into Computer Language for solving
the problems (see Figure 1.1.). Dynamic process model becomes transparent when the
‘gap’ between conceptual and simulation language is not too large, which makes life
easier for a lot of engineers to communicate with each other, re-using models from
different sources/authors, and doing maintenance or modification of the model when
required. On the computation level the computer model should be translated to be able
to solve with a numerical engine.
Conceptual Mathematical Dynamic
Model
Model building
Computer/Simulation Language
Translation
Simulation
Numerical Language Translation
And Computation
Figure 1.1 Modelling and Simulation stages.
In conclusion, the industries application fields need:
- an open modelling platform,
- universal modelling standardisation,
- flexibility on model modularisation, and interconnectivity,
- flexibility on model re-use and modification (model transparency), and
- easy use and fast model building.
1.2.2. Academic research
The mathematical modelling of physical systems consists of the decomposition of
complex systems and composition/interconnection of sub-models. This topological
decomposition (see figure 1.2) is necessary for several reasons:
1. The real physical processes exist of interconnected sub-systems.
2. The decomposition and composition of sub systems are human tools to
‘create’ a new system.
3
Model
Topological structure
assumptions
Topology Decomposition
Interconnection
Sub-model
Sub-sub-model
Elementary model
Elementary behaviour
Interconnection
Interconnection
Behaviour
assumptions
Figure 1.2. Model decomposition
Preisig [104] defines that an elementary model/system is a finite volume body or a
finite volume single-phase system. If the elementary model/system has spatial uniform
physical properties then this system is called a lumped model/system. If the physical
properties are not uniform then it is called a distributed model/system.
Figure 1.3. shows an example of a topological decomposition of a distillation system.
Part A of this figure gives schematically that this distillation system is a composite of
interconnected models (a distillation column, a condenser, a reboiler, several control
valves, and controllers). Part B gives schematically that the distillation column consists
of interconnected distillation trays sub-model and finally part C shows that a tray submodel consists of interconnected two elementary phases (liquid and vapour phase),
where mass and energy are transferred through the interface.
4
Condensor
Distilation
Column
Vapour
flow
Liquid
flow
Interface mass & energy transfer
N i,L
FIC
Ni,V
EL,exc
E
V,exc
Control
Valve
vapour phase
liquid phase
Reboiler
interface
A. Interconnected models
of a distillation system
B. Interconnected tray submodels
of a distillation column
C. Interconnected elementary liquid &
vapour phases in a distillation tray
Figure 1.3. A topological decomposition of a distillation model
Basically the modeller will make two kinds of assumptions during the modelling of a
dynamic system, namely:
- the topological assumptions
- the behavioural assumptions
The topological assumption on the process model decomposition or composition of
lumped models describe:
- the interconnection structure of the elementary models, and
- the geometry of the elementary models.
The set up of the behavioural equations within an elementary model depends on the
behavioural assumptions made by the modeller. Basically there are two types of
behavioural assumptions (Marquardt [85]), namely:
1. The primitive behavioural assumptions:
The primitive behavioural assumptions are the first assumptions that should be made
by the modeller to decide the relevant physical balance equations or balance
mechanisms within the system, e.g. component mass balances and energy balance.
These must form independent balance equations; therefore the accumulation or storage
terms in these balance equations should be also independent. The primitive
balance/storage variables are principally the state variables of the model.
2. The constitutive behavioural assumptions:
The second behavioural assumptions, made after the set up of the primitive balance
equations, are the constitutive behavioural assumptions. These assumptions describe
e.g.:
5
-
the transport behaviour assumptions
the reaction behaviour assumptions
the physical properties behaviour assumptions
other behavioural simplification assumptions
The constitutive assumptions must also include the validity range of the used
equations.
A sub set of behaviour equations (1.1) for a mathematical model representation based
on the balance equations and constitutive equations are given on a set of first order
differential algebraic equations (DAE (1.3a), (1.3b)).
In the last decade it is has been realised that DAEs have two essential different
properties than sets of Ordinary Differential Equations (ODEs), namely the term ‘high
index’ and ‘consistent (re-) initialisation problem’ of DAE model.
A high index DAE model
What is a high index DAE model ? A high index DAE model is, roughly speaking, if
in the model equations there are (hidden) state constraint equations or better to say,
constrained differential variables equations. A high index DAE process model may
cause conflict with the primitive assumptions of the modeller about the choice of
storage variables and related model equations. These storage variables are no longer
independent.
In some cases the reaction rate, or the mass transfer rate, or the energy transfer rate are
not known or to be considered as the phases or the components are in equilibrium,
examples: applying a thermodynamic equilibrium or reaction equilibrium assumptions.
These assumptions will result to non-independency of the phases or the components.
Basically there are three causes of a high index DAE, namely:
a) internal states constraints, e.g.:
- introduction of behavioural constraint equation, e.g.: reaction or phase
equilibrium
- introduction of geometrical constraint equation, e.g.: volume constraint on
multiphase fluid system
- choice of coordinate system, e.g. pendulum with cartesian coordinate
(Mattsson [86])
b) non causal relation of the chosen ‘input’ variables of the model,
c) interconnection of DAE sub models
Finding an input variables set for a given DAE model (1.3a), which result to a low
index DAE model is sometimes not easy. The wrong choice of input variables set can
result in a high index model (A.Lefkopoulos [74,75]).
It is known that a high index DAE model is numerically difficult to solve. The
problems on numerical solving of a high index model are:
- break down of the numerical solver error control, e.g. the error control of
implicit numerical solver backward formula (Brenan, Petzold, Campbell [17],
Bujakiewicz [21]) increases for a high index DAE model.
6
-
finding consistent initial conditions and solving a consistent re-initialisation
problem (B. Leimkuhler et.al [76], C.C. Pantelides [98])
DAE models interconnection
A complex mathematical model consists of interconnection of sub-models. The
interconnection is the way of ‘transferring goods’ between sub models, ‘the goods’
herein are e.g.: information, material, energy, and momentum. These are generally
given in the interconnected sub models variables, e.g.: mass flow, pressure, mass
fraction, specific enthalpy, heat flow, temperature, entropy, position, speed, force,
electric current, and electric voltage.
The interconnection equations can act as constraint equations for the state variables
such that the interconnected DAE sub-models gives a high index model.
The interconnections between DAE sub-models are important during model building
and a model replacement. The questions here are how to interconnect DAE sub-models
and to find the external input signals, such that the interconnected DAE model does
not have a high index structure.
Thus it is important to recognize whether a DAE model has a high index and to detect
what is the cause of this high index problem, i.e.:
- to detect if the state variables constraints are due to internal constraint
equations within the model, or due to the choice of input variables, or due to
the interconnection of sub models,
- to detect which equations result in state variables constraints,
- to know if the pre-assumed storage variables are independent,
- to look for the solving method for the high index model, either by symbolic
‘index’ reduction or special numerical handling
Moreover for interconnecting for DAE sub models, it is important to know:
- which input variables sets will not result in a high index model, and
- how to interconnect DAE sub models that it will not result in a high index
composite model.
1.3. Related works
A lot of research has been done to explore properties of a DAE model (1.3b). The
related works are briefly described in this section.
1.3.1. The index of a DAE model
The index property of a linear DAE model was analysed by Rosenbrock [107], G.C.
Verghesse, B.C. Levy and T. Kailath [127], Van Dooren [32], J. Demmel and B.
Kagstrom [31]. The theory of Rosenbrock [107] is very nice, since it is based on an
equivalence system operation. This system operation performs manipulation on the
internal variables and equations of the model, but resulting to an equivalent inputs
outputs behaviour. The equivalent system operations are done by algebraic
manipulations and differentiations on the model equations and/or variables. These are
7
to calculate the index and reduce the index of a linear DAE model. The disadvantages
of these operations are practically difficult to perform and destroy the original
variables and equations of the model.
Luenberger [81] did an attempt to perform an index reduction of a linear DAE model
which includes only row manipulations, i.e. model equations differentiation and
variables elimination and substitution.
The works of C.W. Gear, L.R. Petzold [44], and Unger, et.al [125, 126] on index
detection of non linear DAE model have the root from the work of Luenberger [81].
These algebraic manipulations usually difficult to apply for non linear DAE model and
destroy the original character of the equations. The Gear’s algorithm and Unger’s
algorithm are practically suitable to calculate the index of a DAE model.
The work of Pantelides [98] is based on the solving of a consistent initialisation
problem of a DAE model. This algorithm does not involve the structural eliminations
and substitutions of variables on the equations. It determines only the minimal subset
of the model equations that must be differentiated of which impose constraints on the
initial conditions. The disadvantage of this algorithm is possible introducing
unnecessary equation differentiation. Mattsson and Söderlind [86] developed a
symbolic index reduction technique base on the algorithm of Pantelides for finding the
necessary equations to be differentiated.
Bujakiewics [21] proposed a numerical solution of a high index model by numerical
scaling of the error control using the information from the DAE model structure,
without performing any change on the original model equations. The index of a DAE
model can be calculated using the structural information of the model.
Since a high index and a consistent initialisation problems are different, we want to
have an algorithm that can detect a high index and a consistent initialisation problem,
calculates the index of the model, and gives proposals to perform an index reduction.
The index reduction procedure shall result in an equivalent model. This algorithm
should not perform complicated non linear algebraic manipulations, which can destroy
the original form of the model equations and model variables.
1.3.2. The interconnection of DAE sub models
The work on the linear models interconnection was done by H.H. Rosenbrock, A.C.
Pugh [109]. This work is limited to linear state space models with fixed input and
output variables.
Breedveld [15, 16] gives the concept of bilaterally coupled models interconnection,
i.e. that sub models are interconnected through pair of ‘effort’ and ‘flow’ variables.
Marquardt [89] uses almost the same concept with bilaterally coupled connection, i.e.
the interconnection between sub models is given by the fluxing relations. This fluxing
relation calculates the ‘flow’ variable as result of the differences of the ‘flux’ or
‘effort’ variables values. The extension of this work is recently done by B.Maschke, A.
van der Schaft [90] for the interconnection of mechanical systems. The new concept
of the work of Maschke and A. van der Schaft is formulation of interconnection
8
equations without pre knowledge of the directionality of the input and outputs
variables on the interconnected sub models.
In the process modelling, the interconnected sub models can consists of model
equations and/or encapsulated sub-routines (e.g. physical properties routines).
Moreover the interconnected variables can be bilaterally coupled variables and/or non
bilaterally coupled variables. Due to the complexity of interconnected DAE sub
models in process modelling, we want to avoid a high index problem due to
interconnection. Therefore we need to examine what is the necessary condition to get
interconnected low index sub models.
1.3.3. Other related works of a DAE model
a. Numerical solving methods of a DAE model
Research on the numerical solving methods of DAE are done for example by:
K.E. Brenan, S.L. Campbell and L.R. Petzold [17], E. Hairer, C. Lubich, and M.
Roche [54], Wijckmans [133].
b. Process modelling and simulation using DAE models
Research on the process modelling and simulation using DAE are done for example
by: P.I. Barton [6] and C.C. Pantelides [99], Mattsson and
Söderlind [86], Nilsson [93], Marquardt [85].
1.4. Problem definition and scope
Despite the work that has been done around properties of a DAE model, in my opinion
the following academic items are still ongoing discussion:
1. How to detect a high index, calculate the index and perform index reduction of
a DAE model based only on the DAE model structure ?
2. How to assign input variables to a DAE model and what is the necessary
conditions to avoid a high index model as a result of the interconnections of
low index DAE sub-models ?
These questions give motivation for my research of:
Mathematical analysis of dynamic process models – Index, inputs and
interconnectivity
with main contributions being:
- to clarify the index detection, index calculation, and proposing index reduction
based on the mathematical structure of DAE model,
- to find input variables for a DAE model, and
- to avoid a high index problem due to interconnection of DAE sub-models.
M.R. Westerweele [135] works emphasize on the process modelling process, i.e. how
to avoid a high index in modelling process. Additional equations (restrictive type)
9
might be necessary to be added, when during modelling a high index structure arises.
The work presented in this thesis comes from different view, namely how to analyze a
given mathematical model structure of a process model and to reduce a high index
from a generic system theoritical approach.
This thesis forms an extension of the works that have been done by
H.H. Rosenbrock [107] and Bujakiewics [21], i.e. derived a method to detect high
index, perform index reduction, finding input variables based on Kronecker form for
non linear first order DAE model.
The scope will be limited to analyse the structural properties of a first order DAE
model for a lumped system description. The DAE model here is the result of
modelling using the balance and constitutive equations. The issue of a first order DAE
solvability is also outside the scope of this thesis; since the structural approach can not
detect independency of DAE equations. The existence of a local consistent initial
conditions, as it will be described in Chapter 2, gives only a local necessary condition
for an existence of a solution.
1.5. Approach and outline of the thesis
The first question in my thesis will be answered in Chapter 2 and Chapter 3.
Chapter 4 will describe several examples of high index model detection and reduction,
based on the theory developed in Chapter 2 and Chapter 3. The second question in my
thesis will be answered in Chapter 5 and Chapter 6, namely it describes a method for
finding input variables of a DAE model and gives a sub-models interconnect ability
condition.
A model diagnosis tool is developed by use of a structural approach to help the
modeller during the model building phase to avoid a high index problem.
The outline of this thesis is as follows:
Chapter 2: Mathematical properties of a first order DAE model
This chapter gives as a short description of equivalence system operations to calculate
the index of a DAE model and to reduce the index of a DAE model. Further easier
methods are derived for a low index detection, index calculation and detection of a
consistent (re-) initialisation problem. Moreover this chapter also presents the
conjunctures of high index reduction techniques on a non linear first order DAE model
and strict system equivalence operation on a linear DAE model.
Chapter 3: Structural properties of a first order DAE model
This chapter describes the structural properties of a first order non linear DAE model
based on the theory developed in Chapter 2. Some definitions of mathematical
structure are given, where properties such as structural determinant, structural rank,
structural low index criterion, and structural index calculation are derived. The
structural theory is used to analyse the properties of a first order non linear DAE
model.
10
Chapter 4: Application of low index detection and high index reduction of
nonlinear DAE process models
This chapter describes several examples of high index dynamic process models. The
high index detection and the high index reduction techniques derived in Chapter 2 are
applied in this chapter. Moreover this chapter also gives the difference between a DAE
model with a consistent (re-) initialisation problem and the high index DAE.
Chapter 5: Input variables assignment of a first order DAE model
Modelling with a DAE allows different input sets assignment, which result in low
index systems. This chapter describes a structural approach of modelling diagnose tool
/ algorithm for finding a low index inputs assignments or for detection of state
constraint equations.
Chapter 6: Interconnection of DAE models
This chapter includes definition of model ports, model connector, interconnection, and
interconnect ability structure between sub-models. Based on the previous chapters, it
will be derived a necessary condition for interconnection of low index sub models that
result to a low index interconnected model.
Chapter 7: Conclusions and recommendations
This chapter includes the main conclusions of my research and recommendations for
future work.
Appendix A : Structural frame work
This appendix describes the structural frame work for development of the theories in
Chapter 3.
Appendix B : Graph algorithm
This appendix describes the principal of a graph algorithm for implementations on the
theories in Chapter 3 and Chapter 5.
Appendix C : Modelling of an ICGCC plant
11
12
Chapter 2: Mathematical Properties of a First Order DAE
model
The system matrices E, ( Es − A) , and ( Es − A) −1 of a DAE model,
F ( z (t ), z (t ), u (t ), t ) = 0 , with E =
∂F
∂F
and A =
contain all system
∂ z
∂z
properties of the DAE, i.e.: index, consistent re-initialisation, state and nonstate variables.
‘A change of a physical system structure may result in ‘an impulsive
behaviour’”
2.1.
Introduction
In Chapter 1 we have seen that mathematical modelling of a physical system leads into
interconnected independent first order differential algebraic equations with the general
form (1.3b):
F ( z (t ), z (t ), u (t ), t ) = 0
(1.3b)
In the last decade much research has been carried out into the properties of differential
algebraic equations. Historically the properties of linear time invariant DAE models
have been analysed (e.g.: Rosenbrock [107], Verghese et.al [127], Van den Weiden
[128]). In the 1980’s analysis of a non-linear DAE began as a research topic in
modelling and numerical solving areas (e.g.: Brenan, Petzold, Campbell [17], Gear
[44], Marquardt [87],[88], Pantelides [97],[98]).
Rosenbrock [107] shows the important properties of a linear DAE model (i.e. index of
DAE, consistent initialisation, index reduction) via equivalence system operations to
transform the DAE into a Kronecker Canonical Form. Practically this transformation is
difficult to carry out.
The objectives of this chapter are to derive easier methods for detecting a high index
problem, calculating the index, and detecting a consistent (re-) initialisation problem
based on the information of the system matrices of a DAE model (1.3b),
F ( z , z , u , t ) = 0 . Those system matrices are E , ( Es − A) , and (Es − A) −1 , where
E=
∂F
∂F
and A =
.
∂ z
∂z
Further this chapter describes proposals of index reduction techniques of a DAE model
without performing non linear algebraic manipulations. This is to retain the original
form of the model equations and model variables. The index reduction techniques will
be derived in conjunction with an equivalence system operation (Rosenbrock [107]).
13
Chapter 2 : Mathematical properties of first order DAE model
This chapter consists of seven sections. Section 2.2 briefly describes the Kronecker
Canonical From (KCF) transformation, to know the index of a DAE model and to
perform an index reduction. Section 2.3 describes methods to detect if a DAE model
has a high index problem. Section 2.4 gives a method to calculate the index of a DAE
model. Section 2.5 describes a consistent (re-) initialisation problem of a DAE model.
Section 2.6. gives procedures to reduce the index of a DAE model and finally, section
2.7 gives some conclusions.
2.2. Kronecker Canonical Form
Let a non-linear input-output DAE model be given by the equations:
F ( z (t ), z (t ), u , t ) = 0
y = g ( z, t)
(2.1a, 2.1b)
where t ∈ I denotes the time on some interval I ⊂ R ; the function F : V → R m , V is
open in R 2 m + k +1 is a vector space defining the model; the solution variables and the
input variables z : I → R m , u : I → R k . Moreover the output functions g : W → Y ,
where W and Y ⊆ R m +1 with the output variables y : I → R n .
The index of DAE model (2.1) is defined as follows:
Definition 2.1A. Index κ of a DAE model (Brenan, Campbell, Petzold [17],
Bujakiewicz [21])
Given a DAE (2.1a): F ( z (t ), z (t ), u (t ), t ) = 0
The index κ of a DAE is the minimum number of times that the complete or a part of
equation (2.1a) must be differentiated with respect to t, such that the system equations:
F ( z (t ), z (t ), u (t ), t ) = 0
d
F ( z (t ), z (t ), u (t ), t ) = 0
dt
#
d (κ )
F ( z (t ), z (t ), u (t ), t ) = 0
dt (κ )
can be transformed into an explicit ODE, i.e. z as a continuous function of z ,
u, u1 ,..., u ( l ) and t, by algebraic manipulations. (note: l ≤ κ ).
A DAE model with a (differential) index greater than one is called a high index DAE
otherwise it is a low index DAE.
14
Definition 2.1B. Perturbation Index p of a DAE model (Hairer et.al [54])
Equation (2.1a) has a perturbation index p if p is the smallest integer such that, for
all functions z p (t ) being solutions of the DAE perturbed by δ u (t )
F ( z p (t ), z p (t ), u (t ), t ) = δ u (t )
(2.1c)
There exists a bound on the difference between z (t ) and z p (t ) :
(
)
z (t ) − z p (t ) ≤ C z (0) − z p (0) + max δ u (ξ ) + " + max δ u ( p −1) (ξ ) (2.1d)
0 ≤ξ ≤t
0 ≤ξ ≤ t
whenever the expression on the right-hand side is sufficient small.
Gear [46] has shown that κ ≤ p ≤ κ + 1 . Moreover Gear derived that κ = p for DAE
having an integral form of:
F ( z , z , u ) =
df ( z )
dt
− g ( z ) − h (u ) = 0
(2.1e)
which includes also DAE of the form (2.2).
Since in this thesis, we will analyse the properties of a DAE based on the form of (2.2),
then we will not make difference between differential index and perturbation index.
We call only the ‘index’ of a DAE (2.2).
The definition (2.1) is closely related with a strict system equivalence operations of a
linear implicit DAE (Rosenbrock [107]). To understand, suppose
ξ 0 = ( z 0 , z 0 , u 0 , t0 )T lies on a solution manifold of (2.1a) at time t0 ∈ I and suppose
(2.1a) is locally time independent. Without losing generality for analysis purpose we
choose ( z 0 , z 0 ) = (0, 0) . Linearisation of (2.1a) at ξ gives:
0
E z(t ) = Az (t ) + Bu(t )
(2.2)
where:
E=
∂F
ξ ,
∂ z o
e j
∂F
ξ ,
∂z o
e j
A=−
B=−
∂F
ξ
∂u 0
e j
After Laplace transformation we get the solution of (2.2):
b g bE z
z ( s ) = T ( s)
where: T ( s ) = ⎡⎣t ( s )ij ⎤⎦
m
1
[107]). Matrix
−1
0
+ Bu( s)
g
(2.3)
= [ Es − A] is called the system matrix of (2.2) (Rosenbrock
( Eλ − A) is
called pencil of (2.2) (Brenan, Campbell and Petzold
[17]).
Let T(s) be non-singular, then there is a constant pre-multiplying matrix M and a
constant post-multiplying matrix N (i.e. algebraic manipulations on the DAE model
equations and variables) that transform (2.2) to a Kronecker Canonical Form (KCF).
15
This equivalence system operation is also called a restricted system equivalence
operation, (Rosenbrock [107]). This is given as follows:
MT ( s ) NN −1 z ( s ) = MENN −1 z 0 + MBu ( s )
(2.4a)
⇒ MT ( s) N zˆ ( s) = MEN zˆ 0 + uˆ ( s )
where
MT ( s) N =
LMsI − A
N 0
r
r
OP
Q
LM
N
Ir
0
, MEN =
I m− r + sJ
0
u( s) = MBu( s)
0
J m− r
OP,
Q
z( s) = N −1 z ( s),
with J = block diag( J1 ,..., J q ) ; where:
LM0 1
%
J =M
MM
N0 "
LsI − A
⇒M
N 0
i
r
r
OP
P
1P
P
0Q
0
%
0
0
I m− r + sJ
, i = 1,2,..., q
(2.4b)
OPFG z (s)IJ = LM I 0 OPFG z IJ + FG u (s)IJ
QH z (s)K N 0 J QH z K H u (s)K
1
r
2
10
1
20
2
(2.5)
Each of Ji has a size ji and the largest max( ji ) = j is called the degree or index of
nilpotency (J). For j ≥ 1 the DAE (2.4) is called a high index DAE, otherwise it has a
low index. For j = 0 the DAE (2.2) is called an index one DAE.
In the time domain the first row of (2.5), the ordinary state space part, can be written
as:
zˆ1 (t ) = Ar zˆ1 (t ) + uˆ1 (t ) , zˆ1 (t0 ) = zˆ10
(2.6)
The solution of the second row can be done by pre- or post-multiplication with
unimodulair polynomial matrix M(s) or N(s) (i.e. algebraic manipulations and
differentiations on model equations and variables). This equivalence system operation
is called as a strict system equivalence operation (Rosenbrock [107]):
M ( s) I m− r + sJ z 2 ( s) = M ( s) J z 20 + M ( s)u 2 ( s)
or
I m− r + sJ N ( s) z 2 ( s) = N ( s) J z 20 + N ( s)u 2 ( s)
where M ( s) = N ( s) = Diag [ I + sJi ]−1
and every block entry of M ( s), N ( s) has a form of:
16
(2.7a)
[ I + sJi ]−1
LM1
=M
MM #
N0
− s " ( −1) ji −1 s ji −1
% %
−s
0 1
"
1
OP
PP
PQ
(2.7b)
in the time domain this can be written as:
j −1
j −1
i =0
i =0
j
j
i =0
i =0
⇒ z 2 (t ) = ∑ ( −1) i J iδ ( i ) J z 20 + ∑ ( −1) i J i u 2 (t )
(i )
(2.8a)
and
(i )
⇒ z 2 (t ) = ∑ ( −1) i J iδ ( i ) J z 20 + ∑ ( −1) i J i u 2 (t )
(2.8b)
Where : δ (i ) = i-th time derivate of impulse function.
The formulas (2.7a,b) show that for j ≥ 1 , the solution of the second row of (2.5)
requires row or column differentiations, where this solution (2.8a,b) may introduce
impulsive behaviour in case of z 20 ≠ 0 and may require differentiation of ‘input’
variables if u 2 (t ) ≠ 0 . Thus for j ≥ 1 the DAE (2.2) requires j times differentiations
to be able to transform (2.2) into an ODE with zˆ1 (t ), zˆ 2 (t ) as functions of
j
(i )
zˆ1 (t ), uˆ1 (t ) and ∑ (−1)i J i uˆ 2 (t ) (see definition 2.1).
i =0
For j=0, i.e. an index one DAE, the solution (2.2) becomes:
zˆ1 (t ) = Ar zˆ1 (t ) + uˆ1 (t )
, zˆ1 (t0 ) = zˆ10
zˆ 2 (t ) = uˆ 2 (t )
Lemma 2.2
Consider (2.4a,b); then j- degree of nilpotency of J equals to the (differential) index
κ.
Proof: to solve (2.5) it needs ( j − 1) differentiations (see 2.7b), and one extra
differentiation to calculate z( t ) (see: (2.8a,b)). It follows that the (differential) index
κ of (2.2) equals to j-degree of nilpotency of J.
With this KCF transformation, we can calculate the index of a DAE model as given on
(2.5) and perform an index reduction through a strict system equivalence operation as
given on (2.7). The problems on these equivalence system operations are:
- difficult to get the transformation matrices M and N to bring to Kronecker
Canonical Form
- the model equations and variables are changed
17
2.3. Low index criterion of a DAE model
In case if we only want to know if a DAE model has a low index, the following
Lemmas 2.3 or 2.4 give an easy computation procedure:
Lemma 2.3 Low Index Criterion - 1
Consider a DAE model (2.2):
T ( s) z ( s) = Bu( s)
(2.3)
where T ( s ) = [ Es − A] and let det T ( s) ≠ 0 ,i.e. non singular, then the DAE (2.3) has
a low index if and only if
deg det T ( s) = rank E
(2.9)
Proof:
Given a DAE (2.3), T(s) and E by a restricted system equivalence operation
transformed to (2.5), T '( s), E ' ; where:
⎡ sI − Ar
T '( s ) = ⎢ r
⎣ 0
⎤
,
I m − r + sJ ⎥⎦
0
⎡I
E'= ⎢ r
⎣0
0⎤
J ⎥⎦
J is defined as in (2.7b). Since deg det I n − r + sJ = 0 , we have
deg det T ( s) ≡ deg det T '( s) = deg det sI r − Ar = r.
Further rank E ≡ rank E ' = rank
LM I 0 OP = r + rank J , but since DAE model has a
N 0 JQ
r
low index if and only if rank J = 0 , we have rank E = r if and only if the DAE model
has a low index.
When the system matrix T(s) is not in KCF, then r can be calculated from
deg det T ( s) , as follows :
S is the set of all
)ij ⎥⎦
⎢⎣(
permutation σ of numbers 1, 2,..., m , then the determinant of T ( s ) , det T ( s ) is
Given a square matrix T ( s ) = ⎡ tij ( s ) ⎤ ∈ R m×m [ s ] and
defined (J.G. Broida [18], P. Lancaster [72]) as:
det T ( s ) := ∑ sign(σ )t1σ1 ( s )t2σ 2 ( s )...tmσ m ( s )
σ ∈S
m
= ∑ sign(σ )∏ tiσ i ( s )
σ ∈S
i =1
where sign(σ ) equals –1 for σ odd and 1 for σ even.
18
(2.10a)
Then
r = deg det T ( s )
m
(2.10b)
= deg( ∑ sign(σ )∏ tiσ i ( s ))
σ ∈S
i =1
Lemma 2.4. Low Index Criterion – 2
Consider a DAE model with the following equations:
z1 (t ) = A11 z1 (t ) + A12 z2 (t ) + B1u1 (t )
(2.11)
0 = A21 z1 (t ) + A22 z2 (t ) + B2 u2 (t )
This means that:
⎡I
E=⎢ r
⎣0
0⎤
, and
0 ⎥⎦
⎡A
A = ⎢ 11
⎣ A21
A12 ⎤
A22 ⎥⎦
then (2.11) has a low index if and only if A22 is non-singular.
Proof: rank E = r and
deg det T ( s) ≡ deg det
LMsI − A
N A
r
11
21
A22 is non-singular.
OP
Q
A12
= deg sI r − A11 A22 equals to r if only if
A22
b
g
2.4. Calculation of the index of a DAE model
The index of a DAE (2.2) can be calculated more simply with the following Lemma
2.5, 2.6.
Lemma 2.5
Consider a DAE (2.2) after KCF transformation is represented as DAE (2.5) then the
index of the DAE (2.2) is
max ij {0, (power of s of
( I m−r + sJ )
−1
) + 1}
Proof:
from (2.5) and (2.7b), we see after a KCF transformation that the max power of s in
Es − A
−1
is s j−1 determined by the largest :
19
[ I + sJ ]−1
LM1
=M
MM #
N0
− s " ( −1) j −1 s j −1
% %
0 1
−s
"
1
OP
PP
PQ
where det I + sJ = 1 and j is the degree of nilpotency or index of DAE (2.2).
The following Lemma 2.6 is derived to calculate the index of DAE (2.2) from its
fractional system matrix inverse, since in most cases transformation of a DAE (2.2)
into a KCF is not easy.
Lemma 2.6.
Consider a DAE (2.2) with its system matrix T ( s) = Es − A and its fractional
polynomial matrix inverse:
(
)
⎡
⎤
= ⎢ tij−1 ( s ) ⎥
ij ⎦
⎣
⎡⎛ det S ( s ) ⎞ ⎤
ji
= ⎢⎜
⎟ ⎥
⎢⎝ det T ( s ) ⎠ij ⎥
⎣
⎦
i, j = 1,..., m
−1
T −1 ( s ) = [ Es − A]
(2.12)
where sub-matrix Sij ( x ) of matrix T ∈ R m×m [ s ] is m × m matrix obtained by placing
zeros in the i-th row and in j-th column of T ( s ) and replacing the ij-th element with 1
(J.G. Broida [18], P. Lancaster [72]):
⎛ t11 ( s ) " 0 " t1m ( s ) ⎞
⎜
⎟
#
# ⎟
⎜ #
Sij ( s ) = ⎜ 0
0 ⎟
" 1ij "
⎜
⎟
#
# ⎟
⎜ #
⎜ t ( s) " 0 " t ( s) ⎟
mm
⎝ m1
⎠
then the index of DAE (2.2) or the index of T ( s) is given by:
index (T ( s)) = maxij {0,(deg det S ji ( s) − deg det T ( s) + 1)}
Proof.
From Kronecker Canonical Form transformation we get:
−1
0
⎡ sI r − Ar
⎤ = Nˆ Es − A −1 Mˆ ,
[
]
I m − r + sJ ⎥⎦
⎢⎣ 0
20
N −1 = Nˆ , M −1 = Mˆ
(2.13)
⇒ [ Es − A] = N ⎡
⎣⎢
−1
−1
sI r − Ar
0
⎤ M
0
I m − r + sJ ⎦⎥
⎡⎛ det S ji ⎞ ⎤
−1
−1
⇒ ⎢⎜
⎟ ⎥ = N1 ( sI r − Ar ) M 1 + N 2 ( I m −r + sJ ) M 2 ,
⎢⎣⎝ det( Es − A) ⎠ij ⎥⎦
M
M = ⎡ 1⎤
⎢⎣ M 2 ⎥⎦
−1
⇒ ⎡⎢( det S ji ( s) ) ⎤⎥ = P( s) + det( Es − A) −1.N 2 ( I m − r + sJ ) M 2
ij ⎦
⎣
N = [ N1 N 2 ] ,
⇒ ⎡⎢( det S ji ( s ) ) ⎤⎥ = ⎡⎢( pij ( s ) ) ⎤⎥ + det( Es − A) −1. ⎡⎢( kij ( s) ) ⎤⎥
ij ⎦
ij ⎦
ij ⎦
⎣
⎣
⎣
where: deg pij ( s) < deg det ( sE − A)
it follows:
{
}
deg det S ji ( s ) = max deg pij ( s ), deg det( Es − A) + deg kij ( s )
⇒ deg det S ji ( s ) = deg det( Es − A) + deg kij ( s )
⇒ deg kij ( s ) = deg det S ji ( s ) − deg det( Es − A)
Since the index of DAE (2.2) is equal to
max ij {0, (power of s of
( I m−r + sJ )
−1
) + 1} (Lemma 2.5), then it follows:
index (T ( s)) = maxij {0,(deg det S ji ( s) − deg det T ( s) + 1)}
2.5. Solving a consistent initialisation and a re-initialisation
problem
Initial conditions and input variables are required to solve a DAE (2.2). The following
gives a definition of consistent initial conditions and solving of a consistent reinitialisation problem.
In this section we will that a DAE model will have a problem for the solving a
consistent (re-) initialisation, when it has a high index or it has non minimum state
variables. Moreover, we will see also that a low index DAE model can have non
minimum state variables, which results to a problem for solving a consistent (re-)
initialisation.
2.5.1. Solving a consistent initialisation problem
Definition 2.7 Consistent Initial Conditions
Given a DAE (2.2) and its input signals u (t0 ) , then ( z (t0 ), z (t0 )) is a Consistent
Initial Conditions of (2.2) if and only if it satisfies:
21
a) E z (t0 ) = Az (t0 ) + Bu (t0 )
(2.14a)
b) z (t 0 ) lies on the solution trajectories of DAE (2.2).
(2.14b)
or
The Kronecker Canonical Form transformation of DAE model (2.2) (see equation 2.5)
decomposes the system into a r-dimensional ordinary state space part and a (m-r)
dimensional algebraic part, such that the system has r-arbitrary initial conditions
z10 and (m-r) fixed initial conditions of z 20 = 0 to avoid impulsive behaviour.
In other words, the original differential variables z (t ) are transformed and
decomposed into r- state variables z1 (t ) and (m-r) algebraic variables (non-state)
z 2 (t ) .
For given input variables u (t0 ) at time t = t0 the m equations (2.14a) have m+r’
variables ( z (t0 ), z (t0 )) , where r’ is the column rank of E. The column rank of E gives
the number of differential variables on the DAE (2.2). From the KCF it follows,
Corollary 2.8
model (2.2) with r ' differential variables, where
r ' = column rank ( E ) . There are only r-variables of ( z (t0 )) that can be assigned
For
a
given
DAE
arbitrarily to compute a set of consistent initialisation values ( z (t0 ), z (t0 )) of DAE
(2.2), where r = deg det T ( s ) and r ≤ r ' . This number r is called the ‘dynamic
degree of freedom’ or the ‘system order’ of the DAE (2.2).
Proof:
See the KCF transformation, where :
r = deg det T ( s ) = deg det ⎡
⎢⎣
= deg det [ sI r − Ar ]
≡r
sI r − Ar
0
⎤
0
I m − r + sJ ⎥⎦
⎡ Ir
⎣0
and column rank of E ≡ r ' = column rank ⎢
0⎤
= r + rank ( J ) ≥ r
J ⎥⎦
For a given low index DAE with a minimum state variables, i.e.
Column rank( E ) = r ' = r , then this gives:
22
Corollary 2.9
Given a low index DAE of the form (2.2) with colum rank ( E ) = rank ( E ) , r ' = r ,
then there are r-differential variables of ( z (t0 )) that can be assigned arbitrarily to
compute a set of consistent initialisation values ( z (t0 ), z (t0 )) of DAE (2.2), where
r = deg det T ( s ) .
We see from Corollary 2.8 and 2.9 that the degree of freedom to assign initial values
z (to ) arbitrarily is given by r = deg det T ( s ) . This means that not all of the variables
z (to ) can be assigned arbitrarily to calculate consistent initial values ( z (t0 ), z (t0 )) for
a high index DAE model or a low index DAE model with non minimum state
variables.
Solving Consistent Initialisation Problem (P.I. Barton [7])
Usually a Consistent Initialisation Problem is solved as an algebraic problem to satisfy
criterion (2.14a) on definition 2.7, i.e. for the existence of a solution. The following
gives a sufficient condition for solving of a consistent initialisation problem.
Consider a DAE of the form:
z1 (t ) = A11 z1 (t ) + A12 z2 (t ) + B1u1 (t )
0 = A21 z1 (t ) + A22 z2 (t ) + B2 u2 (t )
(2.11)
Solving of a general consistent initialisation problem of (2.11) at a given time
t = t0 ∈ I ⊂ R and given input variables u (t0 ) ∈ R k is formulated as a solving set of
algebraic non-linear equations for the unknown vector ( z1 (t0 ), z1 (t0 ), z 2 (t0 ) ) :
z1 (t0 ) = A11 z1 (t0 ) + A12 z2 (t0 ) + B1u1 (t0 )
0 = A21 z1 (t0 ) + A22 z2 (t0 ) + B2 u2 (t0 )
(2.15a,b,c)
A30 z1 (t0 ) = A31 z1 (t0 ) + A32 z2 (t0 ) + B13u1 (t0 ) + B23u2 (t0 )
Where: A30 z1 (t0 ) = A31 z1 (t0 ) + A32 z2 (t0 ) + B13u1 (t0 ) + B23 u2 (t0 ) are r independent
initial equations.
⎡ −I
⎢ 0
If the matrix
⎢
⎢⎣ − A30
A11
A21
A31
A12 ⎤
A22 ⎥⎥
is non singular then a set of initial
A32 ⎥⎦ ( m + r *) x ( m + r *)
conditions ( z1 (t0 ), z1 (t0 ), z 2 (t0 ) ) can be calculated.
23
Specific case 1
Usually one gives a set of z1i (t0 ) = z1i 0 ,
i = 1,..., r to compute initial condition
vector ( z1 (t0 ), z1 (t0 ), z 2 (t0 ) ) . This is a special case of an initialisation equations set
(2.15c) and the consistent initialisation problem is formulated as algebraic solving of
the following equations:
z1 (t0 ) = A11 z1 (t0 ) + A12 z2 (t0 ) + B1u1 (t0 )
0 = A21 z1 (t0 ) + A22 z2 (t0 ) + B2 u2 (t0 )
z1i (t0 ) = z1i 0 ,
i = 1,..., r
⎡ − I A11 A12 ⎤
⎢0 A
if the matrix
A22 ⎥⎥ or A22 is non-singular, then a set of initial
21
⎢
⎢⎣ 0
0 ⎥⎦
I
conditions ( z1 (t0 ), z1 (t0 ), z 2 (t0 ) ) can be calculated.
We see here (as given on Lemma 2.4 and Corollary 2.9) that solving a consistent
initialisation problem by specifying r-independent initial conditions ( z (t0 )) can be
only applied for a low index DAE model with minimum state variables.
Specific case 2:
Corollary 2.10. Steady state solution is a consistent initial condition (Kroner et.al
[68]):
Given a DAE (2.10) with a given input variables u s . The DAE (2.10) will have a
b g b g
L A A OP is non-singular
≡M
NA A Q
steady state solution of z s , z s = 0, z s if and only if:
T ( s)
s= 0
11
12
21
22
Since a steady state solution of a DAE lies on the solution trajectory, then
a steady state solution, z s , z s = 0, z s , is a consistent initial condition of a given
b
g b g
DAE for the given input variable values u s .
2.5.2. Solving a consistent re-initialisation problem
Obviously system behaviour can be subjected to discontinuities of non-state variables
or input signals. Below we define a consistent re-initialisation condition.
24
Definition 2.11. Consistent Re-initialisation Condition
Given a DAE (2.11) of the form :
z1 (t ) = A11 z1 (t ) + A12 z2 (t ) + B1u1 (t )
0 = A21 z1 (t ) + A22 z2 (t ) + B2 u2 (t )
at t + = lim t 0 + ε , for given non-state variable (discontinuities) as:
ε →0
u( t ) =
RS u(t ),
Tu(t ),
0
+
t = t0
t = t + = t0 + ε
⎧ z 2 (t0 ),
⎩ z 2 (t+ ),
and/or z 2 (t ) = ⎨
t = t0
t = t + = t0 + ε
(note u (t ), z 2 (t ) do not need to be continuous and differentiable)
then vector ( z1 (t+ ), z1 (t+ ), z 2 (t+ ) ) is called Consistent Re-initialisation Condition of
(2.11) for given input signals ( u (t+ ) ) under the condition of C 1 continuity of the state
variables, i.e.:
z1 (t0 ) = z1 (t+ )
if and only if it satisfies:
z1 (t+ ) = A11 z1 (t+ ) + A12 z2 (t+ ) + B1u1 (t+ )
0 = A21 z1 (t+ ) + A22 z2 (t+ ) + B2 u2 (t+ )
From definition 2.11, it follows :
Corollary 2.12. Consistent Re-initialisation Criterion - 1
DAE (2.11) has the property of consistent re-initialisation if and only if:
A22 is non-singular or the DAE model (2.11) has a low index with minimum state
variables.
From corollary 2.9, it follows:
Corollary 2.13. Consistent Re-initialisation Criterion – 2
Consider a DAE with given matrices E and T ( s) has the property of consistent reinitialisation if and only if :
The column rank of E (or number of differential variables) =
deg det T ( s)
25
For a high index DAE the number of differential variables (the column rank (E)) is
greater than deg det T ( s) . This means that a high index does not have a consistent reinitialisation property.
Note:
In some cases an index one DAE model does not have a consistent re-initialisation
property, since the column rank of E (or number of differential variables) >
deg det T ( s) . This is called a low index DAE model with non minimum state
variables.
Process modelling based on the balance and constitutive equations always gives a
DAE model which has the integral form, i.e.:
F ( z , z , u ) =
df ( z )
dt
− g ( z ) − h (u ) = 0
(2.16a)
These equations are usually represented as non minimal state variables and can be
transformed/written into a minimal state representation as follows:
x = g ( z ) + h (u )
(2.16b)
0 = x − f (z)
An index one DAE model given with non minimum states representation can be
transformed into an index one DAE model with minimum state representation, without
performing additional differentiation.
Brull and Pallaske [20] give an example of an index one DAE, which does not satisfy
consistent re-initialisation criterion, given in the following:
Consider an index one DAE of the form:
T11 ( x1 , x 2 ) x 1 + T12 ( x1 , x 2 ) x 2 = f ( x1 , x 2 , u)
0 = g ( x1 , x 2 , u)
(2.17a)
can be written on an index one DAE , and satisfies consistent re-initialisation criterion
as follows:
z = f ( x1 , x 2 , u)
0 = g ( x1 , x 2 , u)
(2.17b)
0 = p( x 1 , x 2 ) − z
where:
p ( x1 , x 2 ) = T11 ( x1 , x 2 ) x 1 + T12 ( x1 , x 2 ) x 2 and
∂p
∂ x1
26
= T11 ( x1 , x 2 );
∂p
∂ x2
= T12 ( x1 , x 2 )
2.6.
Index reduction of a DAE model
There are 3 main difficulties in solving a high index DAE:
1. it may require differentiation operators (see 2.8.a,b),
2. the algebraic solving consistent (re-) initialisation problem (see corollary
2.13),
3. the break down of error control on standard multistep implicit numerical
methods (Brenan, et.al. [17])
Besides that, the degree of freedom of the state variables having C 1 continuity in a
high index model is less than the primitive state variable or ‘storage’ variables as
assumed by the modeller.
Bujakiewicz [21] proposed a modified multistep implicit Backward Difference
Formulae (BDF), where ‘numerical differentiation’ is implemented for the error
control. This is done by ‘scaling’ of error control by the information from the matrix
( Es − A) −1 . The drawbacks of this method are:
1. We do not know the numerical stability and performance of the method for
large scale DAE models.
2. This numerical solving technique may not be easy and can be time consuming
for a large scale DAE model.
3. Solving algebraic consistent (re-) initialisation problem remains problematic.
Another method to solve high index DAE models is through symbolic index reduction
to find a low index DAE model representation. Several advantages why we want to get
low index DAE models, are:
1. the problem on algebraic solving of a consistent (re-) initialisation does not
exist
2. to be able to use a standard (multistep) implicit numerical solving method
3. to know information about the state variables in the model
The extra information is obtained from the index reduction steps, namely:
1. the (hidden) states constraint equations and
2. to alert the modeller in case the chosen ‘inputs’ variables imply non causal
behaviour
In the following we give two methods of index reduction namely: modified row strict
system equivalence and modified column strict system equivalence operation, but first
we are going to examine the strict system equivalence operation for an input output
linear DAE model that is given as follows:
E z = Az + Bu
y = Cz
(2.18a)
or in Laplace domain with zero initial values gives:
T ( s) z ( s) = Bu( s)
y ( s) = Cz ( s)
where: T ( s) = ( Es − A)
(2.18b)
27
Rosenbrock [107], Van Den Weiden [128] transform (2.18b) into a Kronecker
Canonical Form (see also (2.4a)):
MT ( s) N z( s) = MBu( s)
y ( s) = CN z( s)
(2.19)
where: z = N z and:
MT ( s ) N = ⎡
⎢⎣
sI r − Ar
0
⎤
0
I m − r + sJ ⎥⎦
The relation (2.18b) can be written as:
0
− B1 ⎤ ⎡ zˆ1 ( s ) ⎤ ⎡ 0 ⎤
⎡ sI r − Ar
I m − r + sJ − B2 ⎥ ⎢ zˆ 2 ( s ) ⎥ = ⎢ 0 ⎥
⎢ 0
⎢⎣ C1
C2
0 ⎥⎦ ⎣⎢ u ( s ) ⎦⎥ ⎢⎣ y ( s ) ⎥⎦
where: MB =
LM B OP,
NB Q
1
(2.20)
CN = C1 C2
2
Moreover Rosenbrock [107] and Van der Weiden [128] introduced the term “input
decoupling zeros at infinity”, when B2 = 0 and “ output decoupling zeros at infinity”,
when C2 = 0 . On the “input decoupling zeros at infinity” the strict system equivalence
operation can be performed with the equations/rows differentiation and algebraic
manipulation, without introducing differentiation on the input variables. On the
“output decoupling zeros at infinity” the strict system equivalence operation can be
performed with columns/variables differentiation and algebraic manipulation, without
introducing unnecessary differentiation on the input variables. To test if there exists
“input and/or output decoupling zeros at infinity” we can examine the maximum
power of s on the entries of the following matrix:
i ) p1 = max{0, power of s in ( Es − A) −1}
ii ) p2 = max{0, power of s in ( Es − A) −1 B}
(2.21)
iii ) p3 = max{0, power of s in C( Es − A) −1}
when p2 < p1 then we have “input decoupling zeros at infinity” and for p3 < p1 we
have “output decoupling zeros at infinity”
2.6.1. Index reductions through equation differentiations/modified row
strict system equivalence operation
In this section we will give an index reduction technique based on the analysis of KCF
transformation properties (see 2.4 to 2.8). Figure 2.1 again schematically shows the
28
KCF transformation and required differentiations to get a low index presentation of a
DAE model:
Given DAE
KCF
algebraic
manipulations
( Es − A)
(r.s.e)
(2.4)
LMsI − A
N 0
r
A low index DAE
0
I + sJ
OP ‘differentiations’ LMsI − A 0OP
N 0 IQ
Q (s.s.e)
r
(2.7)
Figure 2.1 A schematic of an index reduction via a KCF transformation
r.s.e = restricted system equivalence operation
s.s.e = strict system equivalence operation
Generally it is not easy to bring a DAE to a Kronecker Canonical Form and for a nonlinear DAE practically it is even more difficult. Thus we prefer to perform index
reduction in original DAE structure, schematically given on figure 2.2.
Given DAE differentiations
( Es − A)
algebraic
Extended manipulations
DAE
model
extracting:
-
A low index DAE
state &
non-state var
Figure 2.2. A schematic of an index reduction without a KCF transformation
The idea on figure 2.2 is to minimize changes on the model equations by first
performing the required differentiation and last algebraic manipulations to extract state
and non-state variables. To perform this transformation, there are two questions:
1. How to find the required differentiation based on information of the system
matrix Es − A ?
2. How to abstract state and non-state variables ?
Answer question 1:
See (2.12) the required row differentiations are given on column-wise power of s on
the matrix:
⎡ nij ( s ) ⎤
( Es − A)−1 = ⎢
⎥ , where nij ( s ) = det S ji ( s )
⎣ det( Es − A) ⎦
29
which is equivalent (2.7b) of abstracting required differentiation from the column of
matrix: I + sJ
−1
Example 2.1:
Given an index 3 DAE :
z1 (t ) + z2 (t ) = 0
z2 (t ) + z3 (t ) = 0
(2.22)
z3 ( t ) = 0
LM1
with system matrix b Es − Ag = 0
MM
N0
OP
PP
Q
s 0
1 s and det( Es − A) = 1
0 1
Information of the required differentiations are abstracted from the column of the
⎡1 − s s 2 ⎤
n
(
s
)
⎢
⎥
⎡
⎤
ij
−1
matrix: ( Es − A ) = ⎢
⎥ = ⎢0 1 − s ⎥
⎣ 1 ⎦ ⎢0 0 1 ⎥
⎢⎣
⎥⎦
namely: 2-times differentiation of 3-rd equation (given on the power of s on
3-rd column) and once differentiation of 2-nd equation (given on the power of s on 2
column).
The strict system equivalence operation, results in a low index DAE, and is
schematically given as follows:
- (a) subtract from the first row the once differentiated second row
- (b) add to result of (a) the twice differentiated third row
- (c) subtract from the second row the once differentiated third row
LM1
MM0
N0
OP LM1
s P ~ M0
1PQ MN0
OP L1
s P ~ MM0
1 PQ MN0
OP
0P
1PQ
s 0
0 − s2
0 0
1
0
1
0
1
0
(2.23)
or the operation (a),(b),(c) can be done by first performing differentiations then
algebraic manipulation, schematically given as follows:
30
(1)
( 2)
1 s 0
0 1 s
(3)
(3 )
0 0 1
-1
1
-1
0 0 s
(2 )
(3)
0 s s
2
(2.24)
0 0 s2
where:
(2 ) = once differentiation of equation 2
= once differentiation of equation 3
(3)
= twice differentiation of equation 3
(3)
Answer question 2:
From the strict system equivalence operation above, we can make a conjuncture that
non-state variables can be deduced from differentiated parts of the DAE, resulting in a
low index DAE. The following gives a rule of extracting non-state variables for index
reduction based on how a strict system equivalence operation works.
Conjuncture 2.14a: Extracting non-state variables
Given a high index DAE model : F ( z, z , u , t ) = 0 denotes with F and its
extended high index DAE model of the form:
LM F
MM D # f
MM D f
N
(1)
1
(k)
where:
k
OP
PP = 0
PP
Q
(2.25)
F = original high index DAE model
D ( k ) f k = a partition of extended set (2.25) formed by differentiation of part of
the equations of F . These partitions are ordered from the lowest
degree of differentiation required of the sub-set of equations on F to
the highest one. The information of required equations to be
differentiated is from the columns of the matrix [T ( s )]−1 = ( Es − A) −1
31
e.g.: on the example 2.1:
ch
F 2 I 0
:G J →
H 3K 0
D (1) f 1: 3 → 0 0 s
D( 2) f
2
s s2
0 s2
Suppose each partition of extended DAE equations, D ( k ) f k , has p equations and
contains q variables (size ( pxq ) ), where qi ≤ q are differentiated variables, then
from this differentiated partition p non-state variables can be chosen from qi
L ∂( D f
with M
MN ∂z
(k )
variables z d
k
d
)
OP is non-singular.
PQ
The resulting extended DAE equations with chosen non-state variables are strict
system equivalence with the original DAE, since algebraic manipulation on the strict
system equivalence operation equals to the elimination of these non-state variables
from extended DAE equations.
Application of the operation above on example 2.1 is given as follows:
(a) from : D ( 2 ) f 2 :
F 2 I → 0
GH 3JK 0
state variables :
ch
(b) from D (1) f : 3 → 0 0
1
s s2
0 s2
z2′d , z3′′d
s
we choose : second and third variables as non-
we choose : third variable as non-state variable:
z3′d
Thus an extended equivalent low index model is:
z1 + z2′ d = 0
z2 + z3′d = 0
z3 = 0
z3′d = 0
z2′ d + z3′′d = 0
z3′′d = 0
(2.26)
and elimination of all non-state variables will give:
z1 = 0
z2 = 0
z3 = 0
(2.27)
as expected from a strict system equivalence operation.
More examples of high index problem and reduction are given in the Chapter 4.
32
Notes:
1. The difference in the result given above with the algorithm as proposed by
Mattsson and Soderlind [86] is:
a. the required differentiations are directly derived from information of the
−1
system matrix, Es − A .
b. the result above gives the direct conjuncture with strict system equivalence
operation for index reduction as described in section 2.2.
2. Since strict system equivalence is not unique the choice of non-state variables from
extended DAE equations may also not be unique.
3. The advantages of the DAE index reduction above are:
a. variables elimination is not required, where for a non-linear DAE system
variables elimination is not easy
b. physical DAE equations structure does not change
4. Index reductions through equations differentiation is recommended if we have
“input decoupling zeros at infinity”
2.6.2. Index reductions through variables differentiation/modified column
In some cases we have a DAE with “output decoupling zeros at infinity”,
Example 2.2:
z1 = − z1 + z2 − 2 z3 + u
z2 = z1 − 2 z2 + z3
0 = z1 + z2 + u
(2.28)
y1 = z1
y2 = z2
if the DAE (2.28) contains variable z3 , which is not of interest or does not appear in
the output of the system, we can write (2.28) in:
LMs + 1
MM −−11
MM 1
MN 0
OPL
PPMM
PPMM
PQN
LM
MM
MM
MN
OP
PP
PP
PQ
0
−1 2 −1
z1
s + 2 −1 0
0
z2
0 −1
−1
= 0 ,
z3
0
0 0
y1
u
1
0 0
y2
OP
PP
PQ
LMs + 1
T ( s ) = M −1
MN −1
OP
PP
Q
−1 2
s + 2 −1
−1
0
(2.29)
with:
33
( Es − A) −1
and
LM− 1
s+4
M
1
=M
MM ss ++ 43
MN s + 4
2
s+4
2
s+4
s+2
s+4
−
⎡ 2s + 4 ⎤
⎢
⎥
⎢ s+4 ⎥
−s
⎢
⎥
( Es − A)−1 B = ⎢
,
s+4 ⎥
⎢
⎥
⎢ s 2 + 2s − 2 ⎥
⎢
⎥
⎣ s+4 ⎦
⎡ 2s + 4 ⎤
⎢ s+4 ⎥
C ( Es − A)−1 B = ⎢
⎥
⎢ −s ⎥
⎣⎢ s + 4 ⎦⎥
3 + 2s
s+4
s −1
s+4
s2 + 3s + 1
s+4
−
OP
PP
PP
PQ
⎡ −1
⎢s + 4
C ( Es − A)−1 = ⎢
⎢ 1
⎢⎣ s + 4
−2
s+4
2
s+4
−3 − 2s ⎤
s+4 ⎥
⎥
s −1 ⎥
s + 4 ⎥⎦
(2.30)
The DAE (2.28) has a system order (a dynamic degree of freedom) = 1,
index = 2 and 1 “output decoupling zero at infinity”.
We can perform restricted system equivalence operation with matrices:
⎡1 2 0 ⎤
M = ⎢⎢ 0 1 0 ⎥⎥ ,
⎢⎣ 0 0 1 ⎥⎦
N = I3×3
(2.31)
this transforms(2.29) into:
LMs − 1
MM −−11
MM 1
MN 0
OPL z O LM 0 OP
PPMMz PP = MM 00 PP,
PPMMz PP MM y PP
PQN u Q MN y PQ
3 + 2 s 0 −1
s + 2 −1 0
−1
0 −1
0
0 0
1
0 0
1
2
3
1
2
LMs + 2 −1OP
N −1 0 Q
This has a form as the left side
lower part of KCF
s
A strict system equivalence operation can be performed by multiplying the 3rd column
with s and by adding to the 2nd column, this resulting in:
34
LMs − 1
MM −−11
MM 1
MN 0
OPL z O LM 0 OP
PPMMz PP = MM 00 PP,
PPMMz PP MM y PP
PQN u Q MN y PQ
3 + 2 s 0 −1
−1 0
2
−1
0 −1
0
0 0
1
0 0
or:
1
2
C ( sE − A) −1
3
1
LM 2s + 4 OP
B = M s+4 P
MN s−+s4 PQ
(2.32)
2
z4 = z1 − 3z2 + u
z1 + 2 z2 = z1 − 3z2 + u
0 = z1 − 2 z2 + z3
0 = z1 − 2 z2 + z3
or
0 = z1 + z2 + u
0 = z1 + z2 + u
0 = z1 + 2 z2 − z4
(2.33)
y1 = z1
y2 = z2
The DAE equation set (2.32), (2.33) are strictly system equivalent with (2.28). The
DAE (2.33) has index one. Application of a high index reduction of DAE (2.28) using
the method from section 2.6.1 will give unnecessary differentiation of the input
variables, since it requires differentiating the 3rd equation of (2.28).
Again figure 2.3 gives schematically the steps on the high index reduction, and
example 2.2 shows how this operates.
Given DAE algebraic
manipulations
( Es − A)
DAE
with
some variables
transformation
‘differentiation’
A low index DAE
extracting:
- state &
- non-state var
Figure 2.3. A schematic of an index reduction without a KCF transformation
The question that arises is:
- How to find the transformation matrix M ?
35
Conjucture 2.14b:
Consider a high index input output DAE model of the form:
z1 = A11 z1 + A12 z2 + A13 z3 + B1u
0 = A21 z1 + A22 z2 + B2 u
0 = A31 z1 + B3 u
y = C1 C2 0 z1 z2 z3
LM(sI − A )
T ( s) = M − A
MN − A
n×n
11
21
T
31
− A12 − A13
− A22
0
0
0
OP
PP is non - singular
Q
(2.34)
with variables z 3 as “output decoupling zeros at infinity”, then there exists a constant
matrix M , which performs a restricted system equivalence operation on T ( s)
resulting in:
LM P (s)
P ( s)
B =M
MM − A
N −A
P12 ( s) 0
11
P22 ( s) I
− A22 0
21
M T ( s)
21
0
31
0
OP
B P
B P
P
B Q
B11*
*
12
*
2
*
3
n
B
m
(2.35)
s
P11 ( s), P12 ( s), P21 ( s), P22 ( s) are first order polynomials in s and
MB = B11*
B12*
B2*
T
B3* . The sizes of the rows of these polynomials are given
with n and m.
The column operation of a strict system equivalence are given with arrows
(2.35), which transforms (2.35) to:
LM P (s)
MM Q
MN −− AA
11
21
21
31
P12 ( s) 0
Q22
− A22
0
I
0
0
OP
B P
B P
P
B Q
on
B11*
*
12
*
2
*
3
(2.36)
The entries of constant matrix M can be constructed from identity matrix I having
the same size as the system matrix T ( s) and replacing the first m-rows with column
vector of Ker ( A13 ) , i.e.:
36
mb g b g b gr
R|F a I F a I F a
|G a J G a J G a
= SG J , G J ,", G
||GG # JJ GG # JJ GG
TH a K H a K H a
Ker ( A13 ) = a1 , a 2 ,..., a m
11
21
m1
12
22
m2
1n
2n
mn
I U|
JJ |
JJ V|
K |W
(2.37)
is given as follows:
LM( Ker ( A ))
0 I "
M =M
MM # % 1
N0 " 0
T
13
LM a
0O M #
0P M a
P=
0P M
M
1PQ M
MN
11
m1
OP
PP
PP
PP
Q
(2.38)
i = 1,..., m
(2.39)
" a1n
" "
" amn
0
0
0
0
I ( n−m )×( n−m )
0
#
%
1
0
0
"
0
1
Each new differential equation constructed of a linear combination from the entries of
column vector Ker ( A13 ) will remove variables z 3 , i.e:
ba g z = ba g ( A z + A
T
i
T
1
i
11 1
z + B1 u),
12 2
Therefore a restricted system equivalence operation on the DAE (2.34) using matrix
M as in (2.38) will result in (2.35), where the first m rows on the z 3 entries (3rd
column) are zero. This makes it possible to eliminate the s terms on the following (nm) rows through a strict system equivalence operation (see (2.35) and (2.36)). In
Chapter 4 we will use this kind of operations to do the high index reduction without
performing unnecessary differentiation on the equations.
Notes:
1. In case we have variables that have to be differentiated, that are in the
differential equations and not in non-linear of the algebraic equations, we are
able to apply the column wise strict system equivalence as described above.
2. Another advantage of this technique besides avoiding unnecessary equations
differentiation is: only the first m equations are transformed by the restricted
system equivalence and the other remaining equations are unchanged. The nm unchanged differential equations can easily be replaced by steady state
equations, since the s term / differential variable will disappear through strict
system equivalence operation. This implies that the method can be easily
applied for non-linear DAE model as long as the given n-differential equations
are linear and the remaining algebraic equations can be non-linear algebraic
relations.
37
2.7.
Conclusions
The objectives of this chapter are to derive easy methods: to detect a high index
problem, to detect a consistent (re-) initialisation problem, to calculate index of a DAE
model, and to propose index reduction techniques without non linear algebraic
manipulations. The index reduction should not destroy the original model equations
and variables. The method should not use complicated algebraic manipulations, e.g. by
transformation to a Kronecker Canonical Form.
Given DAE model, F ( z , z , u , t ) = 0 , with the matrices E =
∂F
∂F
,A=
. The high
∂ z
∂z
index problem of the DAE model can easily be detected by the use of information of
the system matrices, ( Es − A), E only. A DAE model has a low index if
rank ( E ) = deg det( Es − A) .
The dynamic degree of freedom r of a DAE model, i.e. the number of state variables
that can be set arbitrarily to solve a consistent (re-) initialisation problem, can be
calculated from r = deg det( Es − A) .
Problem on finding consistent (re-) initialisation condition means that it is not possible
to set all values of the state variables arbitrarily to solve the consistent initial
conditions of a given DAE model. This problem is caused by a high index problem or
non minimum state variables on an index one problem.
An index reduction of a DAE model can be performed using information from
( Es − A) −1 for the required differentiation and extracting non-state variables from the
differentiated part of original DAE. It is demonstrated that this operation is a strict
system equivalent.
There are two kinds of the high index reduction techniques, namely strict system
equivalence operation through equations differentiation (Conjuncture 2.14a) and strict
system equivalence operation through variables differentiation (Conjuncture 2.14b).
Both proposed methods will preserve the original model equations and variables.
38
Chapter 3: Structural Properties of a First Order DAE
model
The structural information of system matrices E , ( Es − A), ( Es − A) −1 can be
used to analyse the structural properties (dynamic degree of freedom, index,
and required differentiation for index reduction) of a first order non linear
DAE model.
‘Every system has a structure’
3.1.
Introduction
This chapter gives the notions of structural properties of a first order non-linear DAE
model,
F ( z (t ), z (t ), u (t ), t ) = 0
(1.3b)
for the low index criterion, index calculation, and the required row/equations
differentiations for index reduction.
The objectives of this chapter are to analyse the properties of a DAE model based on
the structural information on the matrices : struc( E ) , struc(( Es − A)) ,
struc( Es − A) −1 , instead of the exact numerical values of the system matrices. The
∂F
and
structural matrices struc( E ) and struc( A) contain structural information of
∂ z
∂F
respectively. These structural properties will be described in conjunction with the
∂z
properties of a DAE model derived in chapter 2. Moreover this chapter will explore
the bounds of the exact numerical properties of a DAE model compared with its
structural properties.
The definitions of structural theory to be used in this chapter are adopted from ChingTai Lin [28] and Unger [126]. This is shortly described in Appendix A. The algorithm
to implement the structural theory in this chapter is using the graph algorithm
developed by Bujakiewics [21]. The principal of a graph algorithm is described in
Appendix B.
The structural approach chosen in this section is different with the structural approach
done by C.C. Pantelides [98] and Unger et.al. [126]. Our structural approach is based
on the structural analysis of matrices ( Es − A), ( Es − A) −1 and matrix E as given in
Chapter 2. Pantelides’ algorithm [98] is used to determine the required differentiation
to allow consistent initialisation of high index DAE. The drawback of Pantelides’
39
Chapter 3 : Structural properties of first order DAE model
algorithm is that it can give unnecessary differentiation (see example 3.3 in section
3.5). Unger et.al [126] developed structural index detection and calculation of
structural dynamic degree of freedom based on the Gear index transformation
algorithm (C.W. Gear [45]). The drawback of Gear’s approach is the algebraic row
manipulation, where the required equation differentiation on the original equation set
is difficult to recognize.
This chapter consists of six sections. Section 3.2 describes the definitions of the
structure of the system matrices E , ( Es − A), ( Es − A) −1 . Section 3.3 derives the
structural properties of a DAE model. Section 3.4 examines the genericity of the
structural properties. Section 3.5 gives several examples and section 3.6 gives the
conclusions of this chapter.
3.2.
Structure of system matrices E, T(s), T(s)-1
In this section first we will define and derive the structural properties of system
matrices struc( E ), struc (T ( s )), struc (T −1 ( s )) in relation with the system matrices
E , T ( s ), T −1 ( s ) of a DAE model (1.3b).
Suppose a given DAE (1.3) F ( z (t ), z (t ), u (t ), t ) = 0 with its system matrices :
E=
∂F
∂F
and A =
respectively. The following matrices are defined:
∂ z
∂z
3.2.1. Structure of system matrix E
Definition 3.1. Structural rank of a matrix E
Consider a constant matrix with struc E = ⎡⎣ ( struc eij )ij ⎤⎦ ,
E ∈ R m×m , and:
⎧ o , if eij ≡ 0
struc eij = ⎨ 0
⎩ x , otherwise
(3.1)
Let S E ⊂ R m×m be the set of all possible numerical representation of Ei of struc E ,
then the maximum of all values rank Ei is called the structural rank of E , i.e:
rank struc E := max Ei ∈SE rank Ei
(3.2)
Corollary 3.2.
rank E ≤ rank struc E
Note: Corollary 3.2. is the consequence of definition 3.1.
40
(3.3)
Consider the definition of rank (see J.G. Broida [18]). Suppose Eγγ ∈ Rγ ×γ is a minor
with order γ generated from E ∈ R m×m with 1 ≤ γ ≤ m and Sγ be the set of all
permutation σ of {1,2,..., γ } , then a γ × γ minor Eγγ satisfies
det Eγγ =
∑
σ ∈Sγ
γ
sign(σ )∏ eiσ i ≠ 0
(3.4)
i =1
then follows,
Corollary 3.3
rank E = η ⇔ ∃ Eηη with det Eηη ≠ 0 ∧
∀ Eηη** with η *>η , det Eηη** = 0
Now if struc E is known then let Γγγ is it’s minor with order γ and satisfies
det Γγγ ≠ o , then follows,
Corollary 3.4
η
η
rank struc E = η ⇔ ∃ Γη
with det Γη
≠o ∧
η*
*
∀ Γη
η * with η *>η , det Γη * = o
3.2.2. Structure of system matrix T(s)
Definition 3.5. Structure of polynomial matrix T(s)
The structure of polynomial matrix T ( s ) ∈ R m×m [ s ] is defined as
strucT ( s ) = ⎡⎣( struc tij ( s ))ij ⎤⎦
i = 1, 2,..., m,
j = 1, 2,..., m,
⎧
where : struc tij ( s ) := ⎨ k
⎩s
(3.5)
0 if tij ( s ) ≡ 0
if tij ( s ) = a0 + a1s + ... + ak s k
41
Definition 3.6. Structural determinant of polynomial matrix T(s)
( )ij ⎦⎥
⎣⎢
permutation σ of numbers 1, 2,..., m , then the determinant of T ( s ) , det T ( s ) is
Consider a square matrix T ( s ) = ⎡ tij ( s ) ⎤ ∈ R m×m [ s ] and suppose S is the set of all
defined (e.g. J.G. Broida [18], P. Lancaster [72]) as:
det T ( s ) := ∑ sign(σ )t1σ1 ( s )t2σ 2 ( s )...tmσ m ( s )
σ ∈S
(2.10a)
m
= ∑ sign(σ )∏ tiσ i ( s )
σ ∈S
i =1
where sign(σ ) equals –1 if σ odd, and 1 if σ even.
Then the determinant of the structure of matrix T ( s ) is defined as:
m
det strucT ( s ) = ∑ sign(σ )∏ struc tiσ i ( s )
σ ∈S
(3.6a)
i =1
Suppose that struc T ( s ) is given, T ( s ) ∈ R m×m [ s ] , where the zero polynomials
T ( s ) are known. Since the terms of determinant containing zero polynomials must be
zero also, we may use the set:
Γ = {σ : struc tiσ i ( s ) ≠ o,i =1,2,...,m}
instead of S. Further since struc(− a) = struc(+ a ) then (3.6a) can be written as:
m
det strucT ( s ) = ∑∏ struc tiσ i ( s )
(3.6b)
σ ∈Γ i =1
As we know from Chapter 2 that the information of the degree of determinant of
system matrix of a DAE model, i.e.: deg det T ( s ) is used for the low index criterion
(2.9), to calculate the index of a DAE model (2.13), and to calculate the dynamic
degree of freedom of a DAE model to solve a consistent re-initialisation problem
(Corollary 2.8).
Since we want to analyse the properties of the system matrix by using the information
of struc T ( s ) , instead of the matrix T ( s ) itself, the following relations for
deg det struc T ( s) are derived by using the Definition 3.6 and Corollary A.3 (see
appendix A).
Corollary 3.7.
m
deg det struc T ( s ) = maxσ ∈Γ ∑ deg tiσ i ( s )
i =1
Proof:
42
(3.7)
det struc T ( s ) =
m
∑ ∏ struc tiσ
σ ∈Γ i =1
=
m
∑ ∏s
i
(s)
deg(tiσ i ( s ))
σ ∈Γ i =1
=
∑
⎛ m deg(t ( s )) ⎞
⎜∑
⎟
iσ i
⎠
s⎝ i =1
σ ∈Γ
=
⎛m
⎞
maxσ ∈Γ ⎜ ∑ deg(tiσ i ( s )) ⎟
i
=
1
⎝
⎠
s
m
⇒ deg det struc T ( s ) = maxσ ∈Γ ∑ deg tiσ i ( s )
i =1
and the following Lemma 3.8. gives the relation between deg det T ( s ) and
deg det struc T ( s) .
Lemma 3.8.
Let T ( s ) ∈ R m×m [ s ] be non-singular, then
0 ≤ deg det T ( s ) ≤ deg det struc T ( s )
(3.8)
Proof:
Since T ( s ) ∈ R m×m [ s ] is non-singular then det T ( s) ≠ 0 ⇒ deg det T ( s ) ≥ 0 , further
from (A.5) it follows that :
deg det T ( s ) = deg
∑
σ ∈Γ
m
sign(σ )∏ tiσ i ( s )
i =1
m
⎛
⎞
≤ deg ∑ struc ⎜ sign(σ )∏ tiσ i ( s ) ⎟
⎜
⎟
σ ∈Γ
i =1
⎝
⎠
m
⎛
⎞
= deg ∑ ⎜ sign(σ ) struc∏ tiσ i ( s ) ⎟
⎜
⎟
σ ∈Γ ⎝
i =1
⎠
(since struc (p (s ).q ( s ))=struc p (s ).struc q (s ) then
m
⎞
⎛
σ
sign
(
)
struc
t
(
s
)
⎟
⎜
∑⎜
∏
iσ i
⎟
σ ∈Γ ⎝
i =1
⎠
= deg det struc T ( s )
= deg
43
3.2.3. Structure of system matrix T-1(s)
Definition 3.9. Structural matrix inverse ( struc T ( s )) −1
Consider T ( s ) ∈ R m×m [ s ] , the structural matrix inverse of T ( s ) is defined as:
(
( struc T ( s )) −1 := ( struc tij−1 ( s ))ij
)
⎛ ⎛ det struc S ( s ) ⎞ ⎞
ji
= ⎜⎜
⎟ ⎟,
⎜ ⎝ det struc T ( s ) ⎠ ⎟
ij ⎠
⎝
i = 1, 2,..., m
j = 1, 2,..., m
(3.9)
where the relation for the sub-matrix struc ( Sij ( s )) as given on (2.12), is obtained by
placing zeros in the i-th row and in j-th column of struc(T ( s )) and replacing the ij-th
element with xo or 1 :
⎛ struc(t11 ( s )) " 0 " struc(t1m ( s )) ⎞
⎜
⎟
#
#
#
⎜
⎟
⎟
" xij0 "
struc( Sij ( s )) = ⎜
0
0
⎜
⎟
#
#
#
⎜
⎟
⎜ struc(t ( s )) " 0 " struc(t ( s )) ⎟
m1
mm
⎝
⎠
3.3.
Structural properties of a first order non linear DAE model
3.3.1. Structural low index criterion
The following are given:
A DAE model (1.3b) : F ( z (t ), z (t ), u (t ), t ) = 0 and its structural matrices
⎛ ∂F ⎞
struc( E ) = struc ⎜
⎟,
⎝ ∂ z ⎠
⎛ ∂F
∂F ⎞
struc(T ( s )) = struc( Es − A) = struc ⎜
s−
as
∂z ⎟⎠
⎝ ∂ z
defined in (3.1) and (3.3). Suppose the deg det struc(T ( s )) can be calculated as given
in Corollary 3.7. Further the rank ( struc( E )) can be calculated following Definition
3.1.
It follows from the low index criterion –1 (Lemma 2.3);
44
Corollary 3.10. Structural Low Index Criterion
The DAE (1.3) has a structurally low index if:
deg det struc(T ( s )) = rank struc( E )
(3.10)
3.3.2. Structural system matrix inverse
In Chapter 2 we see that the inverse of the system matrix of a DAE,
−1
T −1 ( s ) = [ Es − A] , can be used to calculate the index of a DAE (section 2.4) and to
know the required differentiations on the DAE equations for index reduction (section
2.6).
In this section we will show the relation between the structural information of matrix
inverse of a DAE ( struc (T ( s ) )
−1
and the inverse of the system matrix itself T −1 ( s ) .
The following is given:
A DAE model (1.3) : F ( z (t ), z (t ), u (t ), t ) = 0 and its structural matrices
⎛ ∂F ⎞
struc( E ) = struc ⎜
⎟,
⎝ ∂ z ⎠
⎛ ∂F
∂F ⎞
struc(T ( s )) = struc( Es − A) = struc ⎜
s−
as
∂z ⎟⎠
⎝ ∂ z
defined in (3.1) and (3.3). Suppose: the matrices struc( Sij ( s )) can be constructed base
on the Definition 3.9, each deg det struc( Sij ( s )) and deg det struc(T ( s )) can be
calculated using Definition 3.6 and Corollary 3.7.
Then with Lemma 2.6 (see Chapter 2), it follows :
Corollary 3.11. Structural index of a DAE
index( struc T ( s)) = max ij {0, (deg det struc S ji ( s ) − deg det struc T ( s ) + 1)}
(3.11)
The following will show the bounds for a fractional polynomial entry of matrix
T −1 ( s ) and the index of a DAE model (1.3b).
Given the fractional polynomial and the structural matrix inverse of the system matrix
of a DAE model (1.3b):
45
(
)
⎡
⎤
= ⎢ tij−1 ( s ) ⎥
ij ⎦
⎣
⎡⎛ det S ( s ) ⎞ ⎤
ji
= ⎢⎜
⎟ ⎥,
⎢⎝ det T ( s ) ⎠ij ⎥
⎣
⎦
i, j = 1,..., m
−1
T −1 ( s ) = [ Es − A]
(
struc T −1 ( s ) := ( struc tij−1 ( s ))ij
(2.12)
)
ji
= ⎜⎜
⎟ ⎟,
⎜ ⎝ det struc T ( s) ⎠ ⎟
ij ⎠
⎝
(3.9)
i, j = 1, 2,..., m
From (2.12) and (3.9) we see that:
deg(t −1 ( s )) ≡ (deg det S ji ( s ) − deg det T ( s ))
ij
and
deg( struc t −1 ( s )) ≡ (deg det struc S ji ( s) − deg det struc T ( s ))
ij
Following Lemma 3.8, we have:
0 ≤ deg det S ji ( s) ≤ deg det struc S ji ( s ) and 0 ≤ deg det T ( s ) ≤ deg det struc T ( s )
Then this result in the following inequalities:
− deg det struc T ( s ) < deg(tij−1 ( s )) ≤ deg det struc S ji ( s )
and
− deg det struc T ( s ) < deg( struc tij−1 ( s )) ≤ deg det struc S ji ( s )
It follows,
Corollary 3.12
a) The upper bound of degree of a polynomial entry of system matrix inverse
deg(t −1 ( s )) is deg det struc S ji ( s ) ,
ij
46
b) deg( struc tij−1 ( s )) gives the ‘best’ estimate of deg(tij−1 ( s )) .
Special cases: for deg det struc S ji ( s ) ≥ deg det struc T ( s )
From Lemma 3.8, we have:
0 ≤ deg det S ji ( s ) ≤ deg det struc S ji ( s ) and
0 ≤ deg det T ( s ) ≤ deg det struc T ( s )
since deg det struc S ji ( s ) ≥ deg det struc T ( s ) , for:
a) deg det struc S ji ( s ) = deg det S ji ( s ) and deg det struc T ( s ) = deg det T ( s ) ,
then
deg det S ji ( s ) − deg det T ( s ) = deg det struc S ji ( s ) − deg det struc T ( s )
or deg(t −1 ( s )) = deg( struc t −1 ( s )) .
ij
ij
b) deg det struc S ji ( s ) > deg det S ji ( s ) and deg det struc T ( s ) > deg det T ( s )
then deg det S ji ( s ) − deg det T ( s ) < deg det struc S ji ( s ) − deg det struc T ( s )
or deg(t −1 ( s )) < deg( struc t −1 ( s )) .
ij
ij
c) deg det struc S ji ( s ) > deg det S ji ( s ) and deg det struc T ( s ) = deg det T ( s )
then deg det S ji ( s ) − deg det T ( s ) < deg det struc S ji ( s ) − deg det struc T ( s )
or deg(t −1 ( s )) < deg( struc t −1 ( s )) .
ij
ij
d) deg det struc S ji ( s ) = deg det S ji ( s ) and deg det struc T ( s ) > deg det T ( s )
then deg det S ji ( s ) − deg det T ( s ) > deg det struc S ji ( s ) − deg det struc T ( s )
or deg(t −1 ( s )) > deg( struc t −1 ( s )) .
ij
ij
It seems that in most cases the degree of structural polynomial entry
deg( struc t −1 ( s )) gives the upper bound of deg(t −1 ( s )) , except in the case (d).
ij
ij
It follows,
Corollary 3.13
The degree of a structural polynomial entry of system matrix inverse
deg( struc t −1 ( s )) gives in the most cases the upper bound of the degree of the
ij
47
polynomial entry deg(t −1 ( s )) for deg det struc T ( s ) ≥ deg det T ( s ) and
ij
deg det struc S ji ( s ) ≥ deg det struc T ( s ) .
The index of a DAE model is given in the Lemma 2.6 (Chapter 2) as:
index(T ( s )) = max ij {0, (deg det S ji ( s ) − deg det T ( s ) + 1)} , then from Corollary
3.12 and Corollary 3.13 it follows :
Corollary 3.14
The upper bound of the index of a DAE model (1.3b) is
index(T ( s )) ≤ max ij {0, (deg det struc S ji ( s ) + 1)}
From Chapter 2, section 2.6.1, we see that the required equations differentiation to
reduce the index of a DAE model are given on the maximum column-wise power of s
on the polynomial matrix T −1 ( s ) = ⎡ (t −1 ( s ))ij ⎤ .
⎢⎣
⎥⎦
ij
The relation (2.12) gives that
⎛ det S ji ( s ) ⎞
tij−1 ( s) = ⎜
⎟ . Here follows that an equation differentiation is only required
⎝ det T ( s ) ⎠
when deg det S ji ( s ) > deg det T ( s ) . It follows from the Corollary 3.12 and
Corollary 3.13, that:
Corollary 3.15.
The column-wise maximum degree of the structural polynomial entries of system
m
matrix inverse, max ⎛⎜ deg( struc t −1 ( s )) ⎞⎟
i =1 j ⎝
ij
⎠
is
the ‘best’ estimate for required
amount of differentiation of j-th equation for index reduction of a DAE model.
3.3.3. Structural system order
In Chapter 2, Corollary 2.8 gives that for a given DAE (2.2) with system matrix T ( s ) ,
there are only r-variables of ( z (t0 )) that can be assigned arbitrarily to compute a set of
consistent initial values of the DAE, ( z (t0 ), z (t0 )) , where r = deg det T ( s ) .This is
called the ‘dynamic degree of freedom’ or the ‘system order’ of the DAE.
For a non-linear first order DAE model (1.3), where the structural information of the
system matrix is given, struc(T ( s )) , the structural dynamic degree of freedom or
structural system order of the DAE can be calculated (see Corollary 3.7.), as:
r = deg det struc(T ( s)) .
48
From Lemma 3.8. is given that : 0 ≤ deg det T ( s ) ≤ deg det struc T ( s ) . Here
follows,
Corollary 3.16. Structural dynamic degree of freedom
The structural system order or structural dynamic degree of freedom of a DAE (1.3)
( r ) gives the upper bound of the system order or dynamic degree of freedom of a DAE
(1.3) ( r ), i.e.: 0 ≤ r ≤ r .
3.4.
Genericity of structural properties
Previous sections 3.1, 3.2, 3.3. show that the structural properties of system matrices,
i.e. rank ( struc( E )) , structural index, degree of structural polynomial of system
matrix inverse, and structural system order form the upper bounds of the exact values
of corresponding properties of the system matrices. Since we deal with the properties
of non-linear DAE models, in most cases the result of the structural analysis is the
same as with the exact values, i.e. only for a few numerical coincidences or for a few
model structures then the analysis from structural properties gives the upper bound
values. Therefore in this section we introduce the structural definiteness and genericity
aspects of the structural properties of a DAE model.
The term ‘definiteness’ of a structural property is to be defined that this structural
property is equal to the corresponding exact numerical property. The term ‘genericity’
is introduced by Wonham [134], where a property will only fail for limited cases.
3.4.1. Structural definiteness of system matrices
Definition 3.17. Matrix rank structural definiteness
Suppose E ∈ R m×m and rank struc E = γ , where γ ∈ ` + (positive integer) then
the rank of E is said to be structurally definite:
⇔ rank E = rank struc E
=γ
Definition 3.18. System order structural definiteness
Consider a DAE with m × m system matrix T ( s) with its structural system matrix
struc T ( s) . This DAE has the system order r = deg det T ( s) and structural system
order r = deg det struc T ( s ) . If r = r , the system order has a structurally definite
system order.
It follows from Corollary 3.11., Definition 3.17., Definition 3.18.,
Corollary 3.19. Low index structural definiteness
Consider a DAE with m × m system matrices T ( s ) and E has a structurally definite
low index if:
49
a) rank E is structurally definite
and
b) system order of T ( s) is structurally definite
and
c) deg det struc T ( s) = rank struc E
Definition 3.20. System index structural definiteness
Consider a DAE with m × m system matrix T ( s) and structural system matrix
struc T ( s) , then the index of the DAE is said to be structurally definite if:
index(T ( s )) = index( struc T ( s))
Definition 3.21. Structural definiteness of the degree of a fractional polynomial entry
of T −1 ( s )
Consider a DAE with m × m system matrix inverse T −1 ( s ) and structural system
matrix inverse struc T −1 ( s ) , where :
(
)
⎡
⎤
= ⎢ tij−1 ( s ) ⎥
ij ⎦
⎣
⎡⎛ det S ( s ) ⎞ ⎤
ji
= ⎢⎜
⎟ ⎥,
⎢⎝ det T ( s ) ⎠ij ⎥
⎣
⎦
i, j = 1,..., m
−1
T −1 ( s ) = [ Es − A]
and
(
struc T −1 ( s ) := ( struc tij−1 ( s ))ij
(2.12)
)
ji
= ⎜⎜
⎟ ⎟,
⎜ ⎝ det struc T ( s) ⎠ ⎟
ij ⎠
⎝
i, j = 1, 2,..., m
(3.9)
Then the degree of a fractional polynomial entry of T −1 ( s ) , i.e. deg(t −1 ( s )) , to be
ij
structurally definite if:
deg(t −1 ( s )) = deg( struc t −1 ( s ))
ij
50
ij
Corollary 3.22.
Suppose t ij−1 ( s ) and ( struc t −1 ( s )) is an inverse fractional polynomial entry matrix
ij
of T −1 ( s ) and struc T −1 ( s ) of a DAE (1.3) respectively, then this inverse fractional
polynomial entry is structurally definite if:
(deg det S ji ( s ) = deg det struc S ji ( s )) ∧ (deg det T ( s ) = deg det struc T ( s ))
Proof:
deg(t −1 ( s )) ≡ (deg det S ji ( s ) − deg det T ( s )) .
ij
Since 0 ≤ deg det Sij ( s ) ≤ deg det struc Sij ( s ) and
0 ≤ deg det T ( s ) ≤ deg det struc T ( s ) and, then
(deg det S ji ( s ) − deg det struc T ( s )) ≤ deg(t −1 ( s)) ≤ (deg det struc S ji ( s ) − deg det T ( s ))
ij
where the equality sign is only applicable when :
(deg det S ji ( s ) = deg det struc S ji ( s )) ∧ (deg det T ( s ) = deg det struc T ( s ))
3.4.2. Genericity aspects
Definition 3.23. Genericity (Wonham [134])
Suppose that p ∈ R n is a vector of parameters of the problem we are interested in,
and that a given property Π holds for large sets of values p . Further a variety
V ∈ R n is defined as
n
s
V = p : φ i ( p) = 0, i = i ,2,..., k ,
where φ i ( p) are a finite number of polynomial functions. V is called proper if
V ≠ R n and nontrivial if V ≠ ∅ . Then property Π is called generic if there exists a
proper variety V such that Π fails only for points p ∈V .
In this section we will show that the structural definiteness properties of system order,
system index, rank of a constant matrix, and low index criterion are generic.
51
Lemma 3.24. Genericity of structural system properties
a) Structural definiteness of system order is a generic property
b) Structural definiteness of the difference of the degrees of fractional polynomial
entry of T −1 ( s) , i.e. deg tij−1 ( s ) is a generic property
c) Structural definiteness of rank of a constant matrix is a generic property
d) Structural definiteness of low index criterion is a generic property
Proof:
(
⎣⎢
)
a) Consider T ( s ) = ⎡ tij ( s ) ⎤ ∈ R m×m [ s ] and let
ij
⎦⎥
n
Γr = {σ ∈ Γ:deg sign(σ )∏ tiσ i ( s) = r } and α ij be the coefficient of the
i =1
highest order term of tij ( x ) . Further let A = {α iσ i :σ ∈ Γr } , and let α be a
vector of element of A , then we can define a polynomial function:
φ (α ) =
∑
σ ∈Γ r
m
sign(σ )∏ α iσ i
i =1
From the definition of Γr we can distinguish two cases:
i) Γr = 1, i.e. element of the set Γr is one, then there exists only one permutation
result to the highest structural degree.
ii) Γr ≥ 2, i.e. elements of the set Γr are more than one, then there exists more than
one permutation result to the highest structural degree.
ad i) It means:
m
sign(σ ) ∏ αiσ i
i =1
≠0
σ ∈Γ r
m
⇔ deg sign(σ )∏ tiσ i ( s )
=r
i =1
σ ∈Γ r
from which follows that the system order is structurally definite.
ad ii) Notice that φ (α ) = 0 if and only if
∑
σ ∈Γ r
m
sign(σ )∏ α iσ i = 0
⇔ deg
i =1
∑
m
sign(σ )∏ tiσ i ( s ) < r
σ ∈Γ r
i =1
from which follows that the system order is not structurally definite. Thus if we define
V = {α : φ (α ) = 0} , (which is proper and non-trivial) then the structural system
order definiteness property fails only for points α ∈V . This means for a given
52
struc T ( s ) ∈ R m×m [ s ] the structural system order definiteness fails only for
“unlucky” coefficient values of the highest order of tij ( s) of the matrix set of
T ( s ) ∈ R m×m [ s ] having the same structure as struc T ( s ) ∈ R m×m [ s ] . Therefore this
property is a generic property.
b) to prove that a fractional polynomial entry of structural matrix inverse is a generic
property, we consider Corollary 3.11.:
(
)
deg tij−1 ( s) := deg det S ji ( s ) − deg det T ( s )
and the same as in Lemma 3.24a for the polynomial matrix
S ji ( s ) = ⎡⎢( s ji ( s ) ) ⎤⎥ ∈ R m×m [ s ] , we define set
ij ⎦
⎣
m
Γ s = {σ ∈ Γ : deg sign(σ )∏ siσ i ( s ) = deg det struc S ji ( s )} and β ij be the
i =1
coefficient of the highest order term of s ji ( s ) . Further let B = {β iσ i :σ ∈ Γs } , we can
define a polynomial function:
m
m
⎞⎛
⎜
⎟
sign
(
σ
)
α
sign
(
σ
)
∑
∏ iσ i ⎟ ⎜ ∑
∏ βiσ i
⎜ σ ∈Γ
i =1
i =1
r
⎝
⎠⎝ σ ∈Γ s
⎛
φ (α , β ) = ⎜
⎞
⎟
⎟
⎠
The difference of degree of the fractional polynomial entries of T −1 ( s) is not
structurally definite only for some points (α , β ) such that φ (α , β ) = 0 . This
⎛
m
⎞
sign
(
σ
)
α
∑
∏ iσ i ⎟⎟ = 0 or
⎜ σ ∈Γ
i =1
r
⎝
⎠
polynomial φ (α , β ) shows that either ⎜
⎛
m
⎜ ∑ sign(σ )∏ βiσ
i
⎜ σ ∈Γ
i =1
s
⎝
(
⎞
⎟ = 0 will give a structurally indefiniteness of the fractional
⎟
⎠
)
polynomial deg tij−1 ( s ) as given in Corollary 3.22.
c) the following gives the proof of the generic property of the rank of constant matrix.
It resembles much to the proof of Lemma 3.24a. Consider a constant matrix
E ∈ R m×m and let Γ((ηη)) is a set of largest minors of struc E have structural rank η
with Γ((ηη)) = {Γηη :σ ∈ Γη ,det Γηη ≠ 0} . Further let A = {eiσ : σ ∈ Γη } and let e be
i
a vector of the elements of A (see (3.4)) then we can define a function:
φ (e) =
∑
σ ∈Γη
η
sign(σ )∏ eiσ i
i =1
again we can distinguish between:
53
i) Γ((ηη)) = 1 , there exists only one permutation which results in the largest minor
Γηη , with det Γηη ≠ o
ii) Γ((ηη)) ≥ 2 , there exists more than one permutation which results in the largest
minor Γηη , with det Γηη ≠ o
ad i) The rank is structurally definite
ad ii) Notice that φ (e) = 0 if and only if
∑
σ ∈Γη
η
sign(σ )∏ eiσ i = 0
i =1
If we define V = {e : φ (e) = 0} , (which is proper and non trivial) then the rank of a
constant matrix is not structurally definite only for some points e ∈ V . Again this
means for a given struc E ∈ R m×m the structural rank definiteness fails only for
“unlucky” values of the entries of the matrix E ∈ R m×m having the same structure as
struc E ∈ R m×m . Therefore this property is a generic property.
d) to prove the genericity property of structural low index criterion, let’s define a
function :
m
⎛
⎜
φ (α , e) = ∑ sign(σ )∏ α iσ i
⎜ σ ∈Γ
i =1
r
⎝
m
⎞⎛
⎟ ⎜ ∑ sign(σ )∏ eiσ
i
⎟ ⎜ σ ∈Γ
i =1
η
⎠⎝
⎞
⎟
⎟
⎠
from the proof of Lemma 3.24a and Lemma 3.24c, it is obvious that the low index
criterion is not structurally definite only for some points (α , e) such that φ (α , e) = 0 .
3.5.
Examples
The following examples give results of all 3 methods (structural method, Pantelides’s
algorithm and Unger’s algorithm).
Example 3.1:
Consider the following DAE:
z1 + z2 = − z1 + z2
0 = z1 − z2 + z3
0 = − z1 − 2 z2 + u
with matrices:
54
where u = f (t ) is a given input variable
L
E=M
MM
N
1
1
0
0
0
0
0
0
0
LM
OP
PP, struc E = MM
Q
N
x
0
x
0
o
o
o
o
o
o
o
OP L
PP, A = MMM
Q N
−1
1
0
1
−1
1
−1
−2
0
LM
OP
PP, struc A = MM
Q
N
x
0
x
0
o
x
0
x
0
0
x
0
x
0
x
o
OP
PP
Q
and :
⎡s
⎡s +1 s −1 0 ⎤
⎢
T ( s ) = [ Es − A] = ⎢⎢ −1
1
−1⎥⎥ , struc T ( s ) = ⎢ xo
⎢ o
⎢⎣ 1
2
0 ⎥⎦
⎣⎢ x
s
xo
xo
0⎤
⎥
xo ⎥
⎥
0 ⎦⎥
A symbolic analysis using Maple 8 gives results:
A1) det( sE − A) = s + 3
It means (following Corollary 2.8):
r (dynamic degree of freedom)=deg det T ( s )
= deg det( Es − A)
=1
rank ( E ) = 1 . It follows from Lemma 2.3 that the DAE has a low index.
B1)
⎡ 2
⎢ s+3
⎢
1
( Es − A)−1 = ⎢ −
⎢ s+3
⎢ 3
⎢−
⎣ s+3
0
0
−1
⎤
s + 3⎥
⎥
s +1
⎥
s+3 ⎥
⎥
2s
⎥
s+3 ⎦
−
s
It means (following Lemma 2.6) that the index of the DAE is 1.
C1) Column rank (E) =2 and the DAE dynamic degree of freedom = 1; thus this DAE
does not satisfy a consistent re-initialisation problem (see Chapter 2, Corollary 2.12).
This DAE will be satisfy a consistent re-initialisation problem by algebraic
manipulation or algebraic transformation given as follows:
z4 = − z1 + z2
0 = z1 + z2 − z4
0 = z1 − z2 + z3
0 = − z1 − 2 z2 + u
55
The structural analysis gives the following results:
A2) det struc( sE − A) = s
r (structural dynamic degree of freedom)=deg det struc T ( s )
= deg det struc( Es − A)
=1
rank struc E = 1 . It follows from Corollary 3.10 that the DAE has a structural low
index.
B2)
⎡⎛ det struc S ( s ) ⎞ ⎤
ji
struc T −1 ( s ) = ⎢⎜
⎟ ⎥ , for i, j = 1,...,3 , where:
⎢⎝ det struc T ( s ) ⎠ij ⎥
⎣
⎦
⎡1 o s ⎤
⎡ det struc S ji ( s ) ⎤ = ⎢1 o s ⎥ , det struc T ( s ) = s , with xo ≡ 1 ,
⎣
⎦ ⎢
⎥
⎢⎣1 s s ⎥⎦
then follows:
struc ( Es − A) −1
LM1s
1
=M
MM1s
Ns
OP
PP .
PQ
0
1
0
1
1
1
This means that the DAE has a structurally index one.
This results renders with the symbolic analysis above.
Unger’s algorithm gives: a structural index = 1 and the number of dynamic degree of
freedom = 1
Pantelides’ algorithm gives: the third equation needs to be differentiated, which also
results in unnecessary required differentiation of the input signal u .
Example 3.2:
Consider the following DAE:
z1 = − z1 + z2 − 2 z3 + u
z2 = − z1 − 2 z2 + z3 + u where u = f (t ) is a given input variable
0 = z1 + z2
56
with matrices:
⎡1
E = ⎢0
⎢
⎣⎢ 0
0
1
0
⎡ x0
⎢
0 ⎥ , struc E = ⎢ o
⎥
⎢
⎥
0⎦
⎢⎣ o
0⎤
o
x
0
o
o⎤
⎥
o⎥ ,
⎥
o
⎥⎦
⎡ −1
A = ⎢ −1
⎢
⎣⎢ 1
1
⎡ x0
⎢
0
1 ⎥ , struc A = ⎢ x
⎥
⎢ 0
⎥
0⎦
⎢⎣ x
−2 ⎤
−2
1
x
x
x
0
0
0
⎤
⎥
0
x ⎥
⎥
o⎥
⎦
x
0
and
⎡s
2⎤
⎡ s + 1 −1
⎢
⎢
⎥
T ( s ) = [ Es − A] = ⎢ 1
s + 2 −1⎥ , struc T ( s ) = ⎢ xo
⎢
⎢⎣ −1
−1
0 ⎥⎦
⎢ xo
⎣
xo
s
xo
xo ⎤
⎥
o⎥
x
⎥
0⎥
⎦
A symbolic analysis using Maple 8 gives results:
A1) det( sE − A) = s
=1
rank ( E ) = 2 . It follows from Lemma 2.3 that the DAE does not have a low index.
B1)
⎡ −1
⎢ s
⎢
−1 ⎢ 1
( Es − A) =
⎢ s
⎢
⎢s +1
⎢⎣ s
−2
s
2
s
s+2
s
⎤
⎥
s
⎥
s+3
⎥
⎥
s
⎥
2
s + 3s + 3 ⎥
⎥⎦
s
−
2s + 3
It means (following Lemma 2.6) that the index of the DAE is 2. Following the
Conjuncture 2.14a (Chapter 2, section 2.6.1) the 3rd equation needs to be differentiated
once to reduce the index of the DAE and by choosing z2 as an algebraic variable, i.e.
z2 → z '2 , then the following index one DAE can be constructed:
57
z1 = − z1 + z2 − 2 z3 + u
z '2 = − z1 − 2 z2 + z3 + u
0 = z1 + z2
0 = z1 + z '2
A2) det struc ( sE − A) = s
=1
rank struc E = 2 . It follows from Corollary 3.10 that the DAE does not have a
structural low index.
B2)
ji
struc T −1 ( s ) = ⎢⎜
⎟ ⎥ , for i, j = 1,...,3 , where:
⎣
⎦
⎡1 1 s ⎤
⎢
⎥
⎡ det struc S ji ( s ) ⎤ = ⎢1 1 s ⎥ , det struc T ( s ) = s , with xo ≡ 1 ,
⎣
⎦
⎢
2⎥
⎣s s s ⎦
then follows:
⎡1
⎢s
⎢
1
struc ( Es − A)−1 = ⎢
⎢s
⎢1
⎢
⎣
1
s
1
s
1
⎤
⎥
⎥
1⎥ .
⎥
⎥
s
⎥
⎦
1
This means that the DAE has a structurally index two.
This result renders with the symbolic analysis above and the 3rd equation need to be
differentiated once to perform index reduction.
Unger’s algorithm gives: the structural index = 2 and the number of dynamic degree of
freedom = 1.
58
Pantelides’ algorithm gives: to allow a consistent initialisation problem the third
equation needs to be differentiated once.
The structural approach has the advantages of a more efficient computational effort
than a symbolic approach (Bujakiewics [21], Unger et.al [126]), and in some cases
symbolic analysis is not feasible, e.g. the models use external program routines.
Despite the advantages of this structural approach it still fails for some “unlucky
values” or indefiniteness of the properties. This case is shown by the following
example 3.3.
Example 3.3:
(This example shows the difference between Pantelides’ algorithm [98] and our
algorithm)
Consider a DAE:
z1 + z2 = − z1
z1 + z2 = − z2 + z3
0 = − z1 − 2 z2 + u
where u = f (t ) is a given input variable and
⎡1
E = ⎢1
⎢
⎢⎣ 0
1
1
0
⎡ x0
⎢
0
0 ⎥ , struc E = ⎢ x
⎥
⎢
0⎥
⎦
⎢⎣ o
0⎤
x
x
0
0
o
o⎤
⎥
o⎥ ,
⎥
o
⎥⎦
⎡ −1
A=⎢0
⎢
⎢⎣ −1
0
−1
−2
⎡ x0
⎢
1 ⎥ , struc A = ⎢ o
⎥
⎢ 0
0⎥
⎦
⎢⎣ x
0⎤
o
x
x
0
0
⎤
⎥
0
x ⎥
⎥
o⎥
⎦
o
and
⎡s
0⎤
⎡s +1 s
⎢
T ( s ) = [ Es − A] = ⎢⎢ s
s + 1 −1⎥⎥ , struc T ( s ) = ⎢ s
⎢ o
⎢⎣ 1
2
0 ⎥⎦
⎣⎢ x
s
s
xo
0⎤
⎥
xo ⎥
⎥
0 ⎦⎥
A symbolic analysis using Maple 8 gives the following results:
A1) det( sE − A) = s + 2
=1
rank ( E ) = 1 . It follows from Lemma 2.3 that the DAE has a low index.
59
B1)
⎡ 2
⎢s + 2
⎢
−1
( Es − A)−1 = ⎢
⎢s + 2
⎢ s −1
⎢
⎣s+3
0
⎤
⎥
⎥
s +1
⎥
s+2 ⎥
⎥
2s + 1
⎥
s+2 ⎦
−s
s+2
0
−1
It means (following Lemma 2.6) that the index of the DAE is 1.
C1) Column rank (E) =2 and the DAE dynamic degree of freedom = 1; thus this DAE
does not satisfy a consistent re-initialisation problem (see Chapter 2, Corollary 2.12).
This DAE will satisfy a consistent re-initialisation problem by algebraic manipulation
or algebraic transformation given as follows:
z4 = − z1
z4 = − z2 + z3
0 = − z1 − 2 z2 + u
0 = z1 + z2 − z4
A2) det struc ( sE − A) = s
=1
so the system order is structurally definite, but rank E is not structurally definite
since rank struc E = 2 while rank E = 1 . The structural algorithm shows that the
DAE does not have a structurally low index.
B2)
ji
struc T −1 ( s ) = ⎢⎜
⎟ ⎥ , for i, j = 1,...,3 , where:
⎣
⎦
⎡1 o s ⎤
⎢
⎥
⎡ det struc S ji ( s ) ⎤ = ⎢1 o s ⎥ , det struc T ( s ) = s , with xo ≡ 1 ,
⎣
⎦
⎢
2⎥
⎣s s s ⎦
then follows:
60
⎡1
⎢s
⎢1
struc ( Es − A)−1 = ⎢
s
⎢
⎢⎣ 1
0
0
1
1⎤
⎥
⎥
1 .
⎥
⎥
s
⎥⎦
This means that the DAE has a structurally index two and the third equation needs to
be differentiated to perform index reduction.
Unger’s algorithm gives results as structurally index two problem with one dynamic
degree of freedom.
Pantelides’ algorithm gives results to allow a consistent initialisation problem the third
equation needs to be differentiated once.
3.6.
Conclusions
The objectives of this Chapter are to analyse the properties of a DAE model ,
F ( z , z , u , t ) = 0 , based on the structural information of the matrices struc( E ),
struc( Es − A), struc( Es − A) −1 instead of the exact numerical values of these
system matrices. This structural analysis results in a meaningful result based only on a
⎛ ∂F ⎞
⎟,
⎝ ∂ z ⎠
priori pattern of the matrices E and A , i.e. struc( E ) = struc ⎜
⎛ ∂F ⎞
struc( A) = struc ⎜
⎟ , where it may be that the exact numerical values take more
⎝ ∂z ⎠
computational effort or as not even feasible.
A high index problem of a DAE model can be detected from the structural information
of the system matrices, struc( E ), struc( Es − A) . The DAE model has a low index if
rank ( struc( E )) = deg det struc( Es − A) .
The analysis of the dynamic degrees of freedom to allow a consistent (re-)
initialisation of a DAE model based on the structural information, i.e.
r = deg det struc( Es − A) , gives the upper bound of the exact values.
The analysis of the index of a DAE model base on the structural information of the
matrix struc( Es − A) −1 gives mostly the upper bound of the exact values. The
required differentiations for an index reduction of a DAE model can also be derived
from the information of the matrix struc( Es − A) −1 .
61
Analysis of the properties of a DAE model based on the structural information of the
system matrix ( Es − A) and matrix E provides generic properties for the low index
criterion, dynamic degree of freedom (system order), and the required equation
differentiation for high index reduction.
Despite of advantage on the applicability of the structural method, there is still a
problem of analytical cancellation (the “unlucky values”), which cannot be detected. A
possible combination of symbolic and structural analysis may form a solution in the
future.
62
Chapter 4: Application of Low Index Detection and High
Index Reduction of Non-linear DAE Process
models
The DAE model structure of a given system can be changed from a low index
to a high index structure due to a change of its behavioural description, new
assumption or interconnection. The high index structure arises due to
constraint equations of its state variables. This states constraint can be caused
by behavioural constraint, e.g. equilibrium and / or by topological constraint,
e.g. interconnection. Strict system equivalence operation for a high index
reduction shows that the internal representations of a physical system are nonunique for the same inputs outputs behaviour.
“Any (complex) system is ‘controllable’ if its structure is recognized”
4.1.
Introduction
Mathematical modelling of a physical system is always subjected to assumptions.
These assumptions are made to reduce complexity of a process modelling. The model
assumptions can be distinguished in two types: behavioural assumptions and
topological assumptions. The behavioural assumptions include “the unmodelled
phenomena” and “the unmodelled quantities” within the system, when:
- the contribution of some phenomena or quantities are negligible, e.g:
contribution of potential and kinetic energy on the energy balance equations
are neglected.
- the dynamic change of some phenomena or quantities are negligible, e.g:
(pseudo) steady state assumptions of some phenomena, ignoring dynamics of
some quantities due to “fast” dynamic change of these quantities in
comparison with other quantities, phase and/or reaction equilibrium
- the information of some phenomena or quantities are not available, e.g: non
availability of: the reaction rate relation, mass transfer rate coefficient.
The topological assumptions include
“the unmodelled structure/parts”, “the
unmodelled geometry” of the system and the topological system decomposition itself,
example:
- ideally mixed fluid system
- pipe connections between unit operations in a system are not modelled
- three dimensional system is modelled as one dimensional system
The behavioural assumptions can be closely related to the topological assumptions, see
example in section 4.3. This example shows that a contradiction on system
assumptions can result in a high index model.
Our approach of physical modelling is based on the physical balance equations and its
constitutive equations. The set up of balance equations assumes that “the
accumulating/storage” quantities are independent state variables of the system. The
independency of the state variables on the balance equations, we call the “primitive
assumption” of the model.
De Groot and Mazur [52, page 85] recommended
63
Chapter 4 : Application of a High Index Detection and Reduction
describing the state variables of the system with the extensive variables (mass, energy,
impulse, etc) rather than the intensive variables (concentration, temperature, pressure):
“Extensive variables have been chosen to describe the states of the system because of
the difficulties encountered if one tries to introduce from the start intensive
thermodynamic state variables for non-equilibrium states”.
The intensive variables of the system can be derived from extensive variable relations
(as additional constitutive equations in the system description). The examples for the
problem indicated by De Groot and Mazur [52] as a consistent re-initialisation
problem will be given in the section 4.4.
Sometimes the primitive assumption on the independency of the extensive state
variables of the system is “disturbed” by behavioural and/or topological assumptions,
which can lead to “states constraint / states dependency”. Mathematically we call this
phenomenon a high index problem. The result of this “constraint” is that the original
independency of given r number state variables is reduced to less than r number new
state variables of the system. Mathematically we call the number independent state
variables as the dynamic degree of freedom of the model. Section 4.5 shows that the
interconnection between two subsystems can cause a high index problem.
This chapter describes several examples of high index problems on the process
modelling. We will use the mathematical theory derived on Chapter 2 and Chapter 3
for detection of a low index DAE model, the calculation of the structural index of a
DAE model. We will also use index reduction techniques (see Chapter 2 section 2.6.1
and section 2.6.2.) to transform a high index to an index one model. Further the
consistent re-initialisation criterion (see Chapter 2 section 2.5.2) is to be used to check
if a DAE model has minimum state variables.
Most of the models in this chapter are adopted from existing literature, some of them
are made for example purposes only. A lot of constitute equations are presented only
as “a function of variables”, example: a mass flow Fout through a valve is a function of
its inlet pressure Pin and its outlet pressure Pout , presented as: Fin = f ( Pin , Pout ) ,
instead of: Fin = Cv Pin − Pout , where Cv is the valve parameter. This is done for two
reasons, first our structural approach neither requires the exact analytical functions nor
the numerical values, second, to reduce the complex presentation of a system
description, where we need only its structural information.
This chapter is divided into five sections. In section 4.2, examples for column
wise/variable differentiation for high index reduction are presented through examples
of a chemical and a phase equilibrium model. In section 4.3, examples for row
wise/equation differentiation for high index reduction are given. Section 4.4 presents
examples for a non-consistent re-initialisation problem and its solution. Section 4.5
shows an example for a high index problem due to interconnection.
64
4.2.
High Index reduction through variables differentiation
The high index reduction technique as described in section 2.6.2, i.e. Conjuncture
2.14b is mentioned for a high index reduction of a DAE model, where some of the
internal variables are “not interested as outputs of the model” and can be removed
from the model equations. The strict system equivalence operation is done by variable
differentiations. It must be noted that the steady state equations as a result of the strict
system equivalence operation through variables differentiation do not contribute the
input-output behaviour. Therefore these equations may be removed from the final
model equations. The value of the removed variables from the balance equations may
not be derived from the steady state relations as the result of a strict system
equivalence operation, the real value of this variable can only be derived from index
reduction through equations differentiation, see example in section 4.4.
The following examples of the steam-reforming reactor and the distillation tray model
show the technique presented in Section 2.6.2., Conjuncture 2.14b.
4.2.1. Reaction equilibrium model (steam reforming reactor model)
In a catalytic methane-steam reforming reactor synthese gas (mixture mainly of CO
and H2) is produced, as the feedstock of a lot of chemical processes, e.g.: ammonia
synthesis and hydro-cracking process or as feedstock for a fuel cell. Schematically
such a catalytic methane-steam reforming reactor is shown in figure 5.1. The major
feed of the reactor is methane and steam and the product is a mixture of
CH4 , CO, CO2 , H2 , H2O . The main reactions are:
a. methane-steam reaction (see reaction scheme 4.1a)
b. water-gas shift reaction (see reaction scheme 4.1b)
The system boundary in figure 4.1 given with dash line and model variables are:
Fin, Fout = inlet and outlet mass flow
Pin,Pout = inlet and outlet pressure
hin, hout = inlet and outlet specific enthalpy
xi,in,xi,out = inlet and outlet gas mass fraction with i=CH4,CO,CO2,H2,H2O
Qadd,Qloss = additional and loss of heat
P, T= gas pressure and temperature within the reactor
Fout, Pout, hout,
i = CH4,CO,CO2,H2,H2O
Qadd
Qloss
Fin, Pin, hin, xi,in
Figure 4.1. A catalytic methane-steam reforming reactor
65
r
CH4
CH4 + H2O ⎯ ⎯
→ 3H2 + CO
(4.1a, 4.1b)
r
shift
CO + H2O ⎯ ⎯
→ CO2 + H2
rCH4 = kCH4
F P. x
GH
CH4 ,out
−
I
(T ) JK
xCO ,out . xH2 ,out . P 3
xH2O . KCH 4
(4.1c, 4.1d)
rshift = f1 ( xCO , xCO2 , xH2 , xH2O , T , P)
The reaction rates of both reactions are given on the equations (4.1c) and (4.1d). The
kinetic reaction rate (4.1c) is adopted from Take et.al [119] for methane-steam
reforming reaction with a Ni-Al2O3 catalyst.
In most cases the water-gas shift reaction rate relation is not available and is assumed
to be taking place on equilibrium condition. Herewith this reaction is assumed as “fast”
dynamic reaching its equilibrium condition. This replaces the reaction rates given in
the reaction schemes (4.1a) and (4.1b) with the following reaction schemes ((4.1a) and
(4.2a)):
r
CH4
CH4 + H2O ⎯ ⎯
→ 3H2 + CO
(4.1a, 4.2a)
K
shift
CO + H2O ←⎯⎯
→ CO2 + H2
where the equilibrium relation of water-gas shift is:
Kshift (T ) =
xCO2 ,out . xH2 ,out
(4.2b)
xCO ,out . xH2O ,out
The model equations are given with the system’s assumptions:
-
one “well-mixed” gas phase in the reactor,
neglecting contribution of kinetic and potential energy,
When the component on the model are given with:
i = CH4 , CO, CO2 , H2 , H2 O;
j = CH4 , shift
and stoichiometric reactions coefficient matrix
ϑ ij
L −1
=M
N0
OP
Q
1 0 3 −1
−1 1 1 −1
T
(4.3)
The component mass balance, energy balance and its constitutive system equations are
given as follows:
66
2
dMi
= Fin xi ,in − Fout xi ,out + Veff ∑ ϑ ij rj
dt
j =1
(4.4 - 4.8)
dEh
= Fin hin − Fout hout + Qadd − Qloss
dt
(4.9)
5
M T = ∑ Mi ,
Mi = xi ,out M T
(4.10-4.15)
i =1
M T = ρ.Veff ,
E = M T . hout − P.Veff
ρ = f1 (T , P, xi ,out ),
hout = f 2 (T , P, xi ,out )
(4.16,4.17)
Fin = f3 ( Pin − P),
Fout = f 4 ( P − Pout )
⎛
xCO ,out .xH 2 ,out .P 3 ⎞
r1 := rCH 4 = kCH 4 ⎜ P.xCH 4 ,out −
⎟
⎜
xH 2O .K CH 4 (T ) ⎟⎠
⎝
xCO2 ,out . xH2 ,out
Kshift (T ) =
xCO ,out . x H2O ,out
(4.20,4.21)
(4.18,4.19)
(4.1c)
(4.2b)
In this equations set r2 := rshift is replaced by (4.2b), hence a high index problem
occurs.
where:
Mi , M T = components mass and total mass hold up
Eh = thermodynamic energy hold up
ρ = gas mixture density
xi = mass fraction of component i
kCH4 , KCH4 (T ) = kinetic rate constant / parameter and the temperature dependent
equilibrium constant for methane-steam reaction
Kshift (T ) = the temperature dependent equilibrium constant for water-gas shift
reaction
Veff = effective void volume of the reactor (as a model parameter)
This model is given with 20 differential algebraic equations ((4.4 – 4.21), (4.1c)
(4.2b)) with 20 unknowns, namely:
M T , Eh , P, T , ρ , hout , Fin , Fout , rCH 4 , rshift , M i , xi, out ,
and the chosen input variables are:
u = ( xi ,in , hin , Pin , Pout , Qadd , Qloss ) T , where
i = CH4 , CO, CO2 , H2 , H2O .
Consider the DAE model equations above as F ( z, z , u) = 0 with model variables:
z = ( M i , Eh , xi,out , hout , rCH 4 , rshift , M T , T , ρ , P, Fin , Fout )T , then we can
construct the structural system matrix struc ( Es − A) , where matrices
E = struc
∂F
∂F
and A = struc
.
∂ z
∂z
67
The result of structural analysis gives:
a) rank struc E = 6
b) deg det struc( sE − A) = 5 , means that the number of dynamic degree of freedom
of this model is 5
c) the maximum power of s on the structural matrix ( Es − A) −1 = 1 . This means that
the DAE model has structural index = 2.
The maximum power of s on the structural matrix ( Es − A) −1 = 1 are on the 14th row
of ( Es − A) −1 with column numbers: 7,8,9,10,11,12,13,14,15,16,20.
This means that if a high index reduction can be done by:
i)
the equations differentiation, where we need to differentiate once the
equations: (4.10), (4.11), (4.12), (4.13), (4.14), (4.15), (4.16), (4.17),
(4.18), (4.19), (4.2b) and then chose the state or non-state variables (as
described in Chapter 2, section 2.6.1.), or
ii)
the variable differentiation, where we need to do restricted system
equivalence operation on the DAE model such that the 14th variable
(rshift ) is vanished from the differential equations and perform strict
system equivalence operation. This operation can be chosen when we are
not interested in the variable rshift as the output of system.
If we choose the second possibility (ii) then we can apply a restricted system
equivalence operation as described in Chapter 2 – section 2.6.2. The entry of premultiplying matrix M is as follows,
Given the stoichiometric coefficients of the rshift on the balance equations, as
uninterested variable of the system output is
a = [ 0 −1 1 1 −1] ; then Ker (a) is:
⎧⎛ 1 ⎞ ⎛ 0 ⎞ ⎛ 0 ⎞ ⎛ 0 ⎞ ⎫
⎪⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪
⎪⎪⎜ 0 ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎪⎪
Ker [ 0 −1 1 1 −1] = ⎨⎜ 0 ⎟ , ⎜ 0 ⎟ , ⎜ 1 ⎟ , ⎜ 0 ⎟ ⎬
⎪⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ ⎪
⎪⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪
⎪⎩⎜⎝ 0 ⎟⎠ ⎜⎝ −1⎟⎠ ⎜⎝ 1 ⎟⎠ ⎜⎝ 1 ⎟⎠ ⎪⎭
and the matrix M is:
68
LM1
MM00
MM0
M=
MM0
MM0
MM0#
N
OP
PP
P
0P
0P
P
0P
#P
P
1PQ
1
0
0 0 " 0
0 0 −1 0 " 0
1 0 1 0 " 0
0
0 1
1
0 "
0
0
0 0
0 0
1
0
0 "
1 "
0
0 0
%
"
0
(Ker [0 –1 1 1 –1])T
20× 20
Perform pre-multiplication of the model equations ((4.4 – 4.21), (4.1c) (4.2b)) with
this matrix M (restricted system equivalence operation), gives the following set of
equations:
dM CH4
= Fin xCH4 ,in − Fout xCH4 ,out − Veff . rCH4
dt
dM A
= Fin ( xCO ,in − x H2 O ,in ) − Fout ( xCO ,out − x H2 O ,out ) + Veff .2rCH4
dt
dM B
= Fin ( xCO2 ,in + x H2 O ,in ) − Fout ( xCO2 ,out + x H2 O ,out ) − Veff . rCH4
dt
dM C
= Fin ( x H2 ,in + x H2 O ,in ) − Fout ( x H2 ,out + x H2 O ,out ) + Veff .2rCH4
dt
dM H2 O
= Fin x H2 O ,in − Fout x H2 O ,out − Veff . rCH4 − Veff . rshift
dt
dE
dt
(4.22 – 4.26a, 4.27)
M A = M CO − M H2O ;
M B = M CO2 + M H2 O ;
M C = M H2 + M H 2 O
(4.28 – 4.30)
and equations ((4.10 – 4.21), (4.1c) (4.2b)).
Since the variable rshift is not interesting as an output of the system, we can perform a
strict system equivalence operation to remove
dM H2O
dt
from equation (4.26a) (see
Chapter 2 - Conjuncture 2.14b). Replace this equation (4.26a) with the new one
(4.26b) for the new equivalent index one with 23 differential algebraic equations for
the methane-steam reforming reactor, given as follows:
69
dM CH4
= Fin xCH4 ,in − Fout xCH4 ,out − Veff . rCH4
dt
dM A
= Fin ( xCO ,in − x H2 O ,in ) − Fout ( xCO ,out − x H2 O ,out ) + Veff .2rCH4
dt
dM B
= Fin ( xCO2 ,in + x H2 O ,in ) − Fout ( xCO2 ,out + x H2 O ,out ) − Veff . rCH4
dt
dM C
= Fin ( x H2 ,in + x H2 O ,in ) − Fout ( x H2 ,out + x H2O ,out ) + Veff .2rCH4
dt
0 = Fin x H2 O ,in − Fout x H2 O ,out − Veff . rCH4 − Veff . rshift
dE
dt
(4.22 – 4.26b,4.27)
M A = M CO − M H2O ;
M B = M CO2 + M H2 O ;
M C = M H2 + M H 2 O
(4.28 – 4.30)
and equations (4.10 – 4.21), (4.1c), (4.2b).
for 23 unknowns variables:
M T , M A , M B , M C , E , P, T , ρ , hout , Fin , Fout , rCH4 , rshift , Mi , xi ,out ,
i = CH4 , CO, CO2 , H2 , H2O
Moreover the equation (4.26b) can be removed from the model equations without any
effect for model inputs-outputs behaviour, such that we also remove the variable rshift
from the model description. The final model equations are given by 22 index one
differential algebraic equations (4.22 – 4.25,4.27), (4.28 – 4.30), (4.10 – 4.21), (4.1c),
(4.2b) for 22 unknowns variables:
M T , M A , M B , M C , E , P, T , ρ , hout , Fin , Fout , rCH4 , Mi , xi ,out ,
i = CH4 , CO, CO2 , H2 , H2O
having 5 dynamic degrees of freedom. The index one model equations also satisfy the
consistent re-initialisation criterion (see Chapter 2 – Corollary 2.12). The number of
differential variables are equals to the dynamic degrees of freedom. We may specify
these
5
number
dynamic
degree
of
freedom
variables,
e.g:
M CH4 (0), M A (0), M B (0), M C (0), E (0) to calculate the remaining variables:
M T (0), P(0), T (0), ρ (0), hout (0), Fin (0), Fout (0), rCH4 (0), Mi (0), xi ,out (0),
dM CH4 (0) dM A (0) dM B (0) dM C (0) dE (0)
,
,
,
,
dt
dt
dt
dt
dt
as consistent (re-)initialisation conditions of the model equations.
70
4.2.2. Non reacting phase equilibrium distillation tray model
There have been a lot of works done on the dynamic modelling of phase equilibrium
tray distillation column, e.g. Luyben [82], Gallun and Holland [42], Gani [43], Aspen
Tech [5,118], Brooks [19]. Some of these models made by oversimplification of the
physical behaviour within the system, e.g: equal vapour flow rate through all the trays,
and/or steady state assumption of the energy balance equation. Ranzi et.al. [106]
shows that model prediction using the steady state heat balance equation gives less
accurate solution than using the dynamic heat balance equation. Pantelides [97] and
Ponton [103] give a systematic approach for a phase equilibrium tray modelling to
avoid a high index problem. Krishna, Taylor and Kooijman [64, 66,120] describe the
modelling of non-equilibrium phase for a tray distillation column.
In this section we will show the index reduction technique for a phase equilibrium
distillation tray model in conjunction with a strict system equivalence operation as
given in Chapter 2 section 2.6.2. In the following figure 4.2 (a) gives a schematic of a
tray distillation column and figure 4.2 (b) presents a tray of this unit operation.
entrainmen
liquid
vapour
weepin
(b)
(a)
Figure 4.2. A tray distillation column
Moreover figure 4.3 (a) gives a schematic representation of a distillation tray model,
with the following assumptions:
- well mixed liquid phase and vapour phase
- neglecting tray weeping and entraintment
- only one vapour phase within system boundary
- no heat loss through system boundary
- neglecting contribution of kinetic and potential energy
71
and figure 4.3 (b) gives a schematic mass transfer and energy transfer through phase
interface.
Vout,yi,out,hvout,P
Lin,xi,in,hl,in
Ni,
Aef
ZL
Ni,
EL,e
Ev,e
vapour phase
liquid phase
Vin,yi,in,hv,in,Pv,i
Lout,xi,out,hl,
(a)
interface
(b)
Figure 4.3
where:
Vin, Vout = vapour inlet and outlet mass flow rate
yi,in, yi,out = inlet and outlet mass fraction of component i in vapour phase
hv,in, hv,out = vapour inlet and outlet specific enthalpy
Pv,in, Pv,out = vapour inlet and outlet pressure
Lin, Lout = liquid inlet and outlet mass flow rate
xi,in, xi,out = inlet and outlet mass fraction of componet i in liquid phase
hl,in, hl,out = liquid inlet and outlet specific enthalpy
Ni,L,Ni,V = mass transfer of component i from the liquid (vapour) phase to the vapour
(liquid) phase
EL,exc, EV,exc = interfacial energy transfer
Non equilibrium model
Krishna and Taylor [120] give the relations for interfacial mass and energy transfer:
Ni , L = ki ,l ai ,l ( xi ,itf − xi ,out ) + N L xi ,out
Ni ,V = ki ,v ai ,v ( yi ,out − yi ,itf ) + NV yi ,out
i = 1,2,..., n (components)
E L ,exc = α l al (Titf − Tl ,out ) + N Lhl ,out
EV ,exc = α v av (Tv ,out − Titf ) + NV hv ,out
(4.31 – 4.35)
where the first part of the right hand side of the equations (4.31 – 4.35) are the transfer
due to diffusion and the second one is due to convection. Further:
72
n
n
N L = ∑ Ni , L
;
i =1
Ni , L = Ni ,V
NV = ∑ N i ,V
(4.36, 4.37)
i =1
;
i = 1,2,..., n
(4.38, 4.39)
E L ,exc = EV ,exc
0 = Ki ,itf ( xi ,itf , yi ,itf , Titf , P)
i = 1,2,..., n
;
(4.40)
where:
ki ,l ai ,l , ki ,v ai ,v ,α l al , α v av = mass and energy transfer coefficients
N L , NV = total liquid / vapour mass transfer through the phase interface
Ki ,itf = interface equilibrium relation
xi ,itf , yi ,itf = liquid / vapour mass fraction of component i on phase interface
Titf = temperature on phase interface.
The interface transport behaviour is described with 4 N + 5 non-linear independent
algebraic equations with 4 N + 5 unknowns, namely:
Ni , L , N i ,V , E L ,exc , EV ,exc , N L , NV , xi ,itf , yi ,itf , Titf and the input variables of these
algebraic equations are xi ,out , yi ,out , hl ,out , hv ,out , Tl ,out , Tv ,out , P .
Equilibrium Model
In some cases the interfacial transport parameters are not available or the interface
transport phenomenon are modelled with N + 1 phase equilibrium equations given as
follows:
0 = Ki ,eq ( xi ,out , yi ,out , Tv ,out , P )
;
i = 1,2,..., n(components)
0 = Tv ,out − Tl ,out
(4.41, 4.42)
The component mass balance and energy balance of a tray are given as:
dMi , L
= Lin xi ,in − Lout xi ,out + Ni , L
dt
dMi ,V
= Vin yi ,in − Vout yi ,out + N i ,V
dt
i = 1,2,... n(components)
(4.43 - 4.46)
dE L
= Linhl ,in − Lout hl ,out + E L ,exc
dt
dEV
= VinhV ,in − Vout hV ,out + EV ,exc
dt
Since N i , L + N i ,V = 0 and E L ,exc + EV ,exc = 0 , then we can write (4.43 – 4.46) as:
73
dMi , L
= Lin xi ,in − Lout xi ,out + Ni , L
dt
dMi ,V
= Vin yi ,in − Vout yi ,out − N i , L
dt
(4.47 – 4.50)
dE L
= Linhl ,in − Lout hl ,out + E L ,exc
dt
dEV
= VinhV ,in − Vout hV ,out − E L ,exc
dt
and the constitutive equations are given as follows:
n
M L = ∑ Mi , L
n
MV = ∑ Mi ,V
;
i =1
Mi , L = xi ,out M L
(4.51, 4.52)
i =1
;
Mi ,V = yi ,out MV
;
E L = M L hl ,out − PVL
;
EV = MV hv ,out − PVV
M L = ρ LVL
; VL = Aeff . Z L
MV = ρVVV
Vtray = VL + VV
Pv ,out = P ;
Pl ,out = P
Lout = f1 ( Z L )
Vin = f 2 ( P, Pv ,in , Z L )
ρ L = f 3 (Tl ,out , Pl ,out , xi ,out ) ; ρV = f 4 (Tv ,out , Pv ,out , yi ,out )
hl ,out = f 5 (Tl ,out , Pl ,out , xi ,out ) ; hv ,out = f 6 (Tv ,out , Pv ,out , yi ,out )
(4.53, 4.54)
(4.55, 4.56)
(4.57, 4.58)
(4.59)
(4.60)
(4.61, 4.62)
(4.63)
(4.64)
(4.65, 4.66)
(4.67, 4.68)
the equations (4.63) and (4.64) give the mass transport relations through the system
boundary.
The phase equilibrium equations of the tray model are given by 5 N + 19 differential
algebraic equations (4.47 – 4.68), with 5 N + 19 unknown variables:
Mi , L , Mi ,V , E L , EV , xi ,out , yi ,out , N i , L , E L ,exc , M L , MV ,VL ,VV , P, Z L , Pv ,out , Pl ,out ,
Tl ,out , Tv ,out , hl ,out , hv ,out , ρ L , ρV
where:
Mi , L , Mi ,V = liquid / vapour phase mass hold up of component i
M L , MV = total liquid / vapour phase mass hold up
E L , EV = liquid / vapour phase energy hold up
VL ,VV = liquid / vapour phase volume
ρ L , ρV = liquid / vapour phase density
74
P, Pl ,out = equilibrium pressure and liquid outlet pressure
with the model parameters are:
Vtray = total tray volume within the system boundary
Aeff = tray area
and the input variables are: Lin ,Vout , xi ,in , yi ,in , hl ,in , hv ,in , Pv ,in . The model parameters
are the tray volume and tray area, Vtray , Aeff .
Structural analysis of this tray model gives that this model forms an index two DAE
model, with the number of the dynamic degree of freedom being N + 1 . The index
reduction can be done through vanishing the variables N i , L from the component
balance equations (4.47) and vanishing the variable E L ,exc from the energy balance
equation (4.49). Again this operation can be done if these two variables are not
interested as the outputs of the model. The operation performs through premultiplication of the system equations with the following matrix:
LM1
MM0#
MM0
MM0
M = M0
MM #
MM0
MM0#
MN0
" 0 −1 0 "
0
0 " 0 −1 0
%
% %
" 0 1 0 " 0 −1 0
0 1 0 "
"
"
0 1 0
"
0
1
% % %
0 1
"
"
0
0
"
"
"
0
1
0
OP
PP
P
0P
0P
P
#P
PP
0P
0P
P
0P
P
IQ
0
#
"
−1
1
0
N
N
( 5 N +19 ) × ( 5 N +19 )
Pre-multiplication of the system equations with the first N-rows of matrix M results
in linear combinations of liquid phase and vapour phase mass balance, where the
variables N i , L are removed. This is also done for the energy balance equation, see
matrix M on the (2N+1)th row. Other equations are intact.
After strict system equivalence operation to remove the remaining
d (.)
terms from
dt
the component balance equations (4.48) and the energy balance equation (4.50)
through variable differentiation, results in the following equivalent set of index one
differential algebraic equations:
75
dMi , M
= Lin xi ,in + Vin yi ,in − Lout xi ,out − Vout yi ,out
dt
0 = Vin yi ,in − Vout yi ,out − Ni , L
(4.69 – 4.72)
dE M
= Linhl ,in + Vinhv ,in − Lout hl ,out − Vout hv ,out
dt
0 = VinhV ,in − Vout hV ,out − E L ,exc
Mi , M = Mi , L + Mi ,V
;
E M = E L + EV
(4.73,4.74)
and equations (4.41,4.42,4.51 – 4.68).
Analogue with the example in Section 4.2.1; the variables N i , L , E L ,exc are not
interesting as outputs of the model so the equations (4.70) and (4.72) can be removed
without any change in its inputs-outputs behaviour. This gives index one differential
algebraic model equations, which satisfy the consistent
(re-) initialisation criterion, since the number of differential variables is equal to the
dynamic degrees of freedom, N + 1 . This gives the same result as proposed by Ponton
[103].
The high index reduction for a high index problem on modelling of multiphase
reactive equilibrium system can be done with the same procedure as presented above.
Different to the approach presented above, Aspen Tech [5,118] gives a model relations
/ library for a distillation tray with further simplifications, namely:
-
neglecting of the change of vapour hold up, therefore the tray pressure
dynamic is neglected
neglecting the contribution of pressure and volume variation on the
(thermodynamic) energy balance
This gives the following index one set of 3 N + 14 differential algebraic equations:
dMi , L
= Lin xi ,in + Vin yi ,in − Lout xi ,out − Vout yi ,out
dt
dE L
= Linhl ,in + Vinhv ,in − Lout hl ,out − Vout hv ,out
dt
n
M L = ∑ Mi , L
i =1
Mi , L = xi ,out M L ,
E L = M L hl ,out
76
(4.75, 4.76)
n
;
1 = ∑ yi ,out
(4.77, 4.78)
(4.79)
i =1
(4.80)
M L = ρ LVL
; VL = Aeff . Z L
Pv ,out = P ;
Pl ,out = P
(4.81)
Lout = f1 ( Z L )
Vin = f 2 ( P, Pv ,in , Z L )
ρ L = f 3 (Tl ,out , Pl ,out , xi ,out )
hl ,out = f 4 (Tl ,out , Pl ,out , xi ,out )
(4.83)
(4.84)
(4.82)
(4.85)
hv ,out = f 5 (Tv ,out , Pv ,out , yi ,out )
;
0 = Ki ,eq ( xi ,out , yi ,out , Tv ,out , P )
;
0 = Tv ,out − Tl ,out
(4.86,4.87)
(4.88,4.89)
for 3 N + 14 unknown variables :
Mi , L , xi ,out , yi ,out , M L , E L , hl ,out , hv ,out , ρ L ,VL , Z L , Tl ,out , Tv ,out , Pl ,out , Pv ,out , P, Lout ,Vin for
the following input variables : Lin , xi ,in , yi ,in , hl ,in , hv ,in , Pv ,in ,Vout and the model
parameter is the tray area Aeff .
4.3.
High index reduction through equations differentiation
In some cases a strict system equivalence operation through variable differentiation for
a high index reduction is not possible, e.g.: the high index reduction on the pendulum
model (Mattsson and Soderlind [86]). The following example also gives a case, where
equation differentiations are required for the index reduction.
4.3.1. Two phases incompressible fluid system with volume constraint
Consider two phases of incompressible liquid-liquid flows through a constant volume
static mixer (figure 4.4). Phase-1 is denoted with “A” and phase-2 is denoted with “B”.
Moreover the volume fractions of A and B are given by ε A , ε B . The mass balance
equations and other constitutive equations are given as follows,
Pin,mixer
Pout,mixer
Fin,A,Pin,A
Fout,Pout
Fin,B,Pin,B
Figure 4.4
77
dε A
= Fin , A − Fout , A
dt
dε
ρ B .Vmixer . B = Fin ,B − Fout ,B
dt
ρ A .Vmixer .
εA +εB =1
Fout , A = ρ Aε A Ftot
;
Fout ,B = ρ Bε B Ftot
(4.90, 4.91a)
(4.92)
(4.93, 4.94)
Fout = f1 ( Pout ,mixer , Pout )
Fin , A = f 2 ( Pin , A , Pin ,mixer )
;
Fin , B = f 3 ( Pin , B , Pin ,mixer )
2
Pin ,mixer − Pout ,mixer = f 4 ( Fout
)
(4.95 – 4.98)
where:
Fin,A, Fin,B = inlet mass flow rate of A and B
FoutA, Fout,B = outlet mass flow rate of A and B
Fout = total outlet mass flow rate
Pin,A, Pin,B = inlet pressure of fluid A and fluid B
Pin,mixer, Pout, mixer = inlet and outlet pressure of the mixer
Pout = outlet pressure
ρ A , ρ B = density of fluid A and fluid B
ε A , ε B = volume fraction of fluid A and fluid B
Vmixer = volume of the mixer
This model consists of 9 differential algebraic equations set with 9 unknowns:
ε A , ε B , Fin , A , Fin ,B , Fout , A , Fout ,B , Fout , Pin ,mixer , Pout ,mixer , with the input variables:
Pin , A , Pin ,B , Pout and the model parameters: ρ A , ρ B ,Vmixer
The structural analysis gives:
a) rank struc E = 2 , deg det struc( sE − A) = 1
b) the maximum power of s in the structural matrix ( Es − A) −1 = 1 . It means that this
DAE model has a structural index of two
c) The 3rd column on the 3rd to 9th rows of matrix ( Es − A) −1 shows that the maximum
power of s is one. This means that when we want to do a high index reduction then
we need to differentiate the third equation.
We may chose
d
d
ε B → ε ′B or
ε A → ε ′A as non-state variable.
dt
dt
Suppose we chose the first one, then this gives results as an index one model with 10
dε A
= Fin , A − Fout , A
dt
ρ B .Vmixer .ε ′B = Fin ,B − Fout ,B
ρ A .Vmixer .
78
(4.90, 4.91b)
εA +εB =1
Fout , A = ρ Aε A Ftot
;
(4.92)
(4.93, 4.94)
Fout ,B = ρ Bε B Ftot
Fout = f1 ( Pout ,mixer , Pout )
Fin , A = f 2 ( Pin , A , Pin ,mixer )
;
Fin ,B = f 3 ( Pin ,B , Pin ,mixer )
(4.95 – 4.98)
2
Pin ,mixer − Pout ,mixer = f 4 ( Fout
)
d
ε A + ε B′ = 0
dt
(4.99)
with the 10 unknowns: ε A , ε B , ε ′B , Fin , A , Fin ,B , Fout , A , Fout , B , Fout , Pin ,mixer , Pout ,mixer
4.3.2. Condenser model
This paragraph describes the dynamic model of a one phase totally condenser unit, as
schematically given in the figure 4.5:
Vin, Pin, hvin
Lout, Pout, hlout
Qcool
Figure 4.5.
where:
Vin, Lout = inlet of vapour mass flow rate and outlet of liquid mass flow rate
Pin, Pout = inlet and outlet pressure
hvin, hlout = vapour specific enthalpy inlet and liquid specific enthalpy outlet
Qcool = additional cooling energy
The assumptions for this model are:
- each phase (vapour and liquid) is well mixed
- no heat loss to the environment
- the condensation process takes place in thermodynamically equilibrium
79
As it has been derived in the previous example of the distillation tray model, that the
system is in a phase equilibrium, then the balance equations of each phase is not
independent. The balance and constitutive equations of the condenser are given as
follows:
dM T
= Vin − Lout
dt
dET
= Vinhvin − Lout hlout − Qcool
dt
M T = M L + MV
;
M L = ρ LVL
VL + VV = Vcond
(4.100, 4.101)
;
ET = E L + EV
;
E L = M Lhlout − PVL
ρ L = f (T , P)
;
ρV = f ( T , P )
P = f (T )
hlout = f (T , P)
Vin = f ( Pin , P)
;
;
hV = f (T , P)
Lout = f (VL , P, Pout )
MV = ρVVV
;
(4.102 – 4.104)
(4.105)
EV = MV hv − PVV
(4.106 – 4.108)
(4.109 – 4.110)
(4.111)
(4.112, 4.113)
(4.114, 4.115)
where:
M L , MV = liquid and vapour mass hold up
M T , ET = total mass and energy hold up
ρ L , ρV = liquid and vapour density
VL ,VV = liquid and vapour volume
hV = vapour phase specific enthalpy
P, T = equilibrium pressure and temperature
These are 16 differential algebraic equations with 16 unknown variables:
M T , ET , M L , MV , E L , EV ,VL ,VV , ρ L , ρV , hlout , hv , T , P,Vin , Lout and the input variables
are hv ,in , Pin , Pout , Qcool . The volume of condenser Vcond is given as the model
parameter.
Structural analysis of this model gives:
a) rank struc E = 2 , deg det struc( Es − A) = 2 . This means the model has two
number of dynamic degree of freedom and it does not have a high index, i.e. the
index of the DAE model is one.
b) column rank E = 2 and since the number of dynamic degree of freedom is two,
then this DAE model satisfies consistent re-initialisation criterion.
In the following we will see that splitting the condenser model into two subsystems,
vapour phase and liquid phase, will result in a high index model. Figure 4.6 shows the
original system (figure 4.6a) is decomposed into two subsystems (figure 4.6b)
80
Vin, Pin, hvin
Vin, Pin, hvin
S-I
Qcool
VC
LIout, PIout, hIlout
S-II
Qcool
Lout, Pout, hlout
Lout, Pout, hlout
(a)
(b)
Figure 4.6.
The system hold up is decomposed into two subsystems. The vapour phase is on
subsystem S-I and the liquid phase is on subsystem S-II.
Modeling of sub system S-I, with the assumption that liquid phase hold up is
neglected, gives the following 10 differential algebraic equations:
S-I:
dMV
I
= Vin − Lout
dt
(4.116, 4.117)
dEV
I
I
= Vinhv ,in − Lout hl ,out − Qcool
dt
MV = ρVVC
;
EV = MV hv − PVC
(4.118, 4.119)
ρV = f 1 ( T , P )
(for ideal gas P = ρV RT )
(4.120)
B
P = f 2 (T )
(e.g. P = A.exp
; A, B, C = physical property constants)
T +C
FG
H
IJ
K
(4.121)
(4.122)
I
out
P =P
hv = f 3 (T , P )
;
hlI,out = f 4 (T , P )
(4.123, 4.124)
Vin = f 5 ( Pin , P)
(4.125)
I
out
I
l ,out
with 10 unknowns : MV , EV , L , h
, hv , ρV , P , P, T ,Vin and the input variables
I
out
are : hv ,in , Pin , Qcool and the model parameter is the volume of condenser VC .
Where:
I
Lout
, hlI,out , PoutI = liquid outlet mass flow, specific enthalpy and pressure from
subsystem S-I
81
Modeling of sub system S-II, with the assumption that the vapour hold up is neglected,
gives the following 7 differential algebraic equations:
dM L
I
= Lout
− Lout
dt
dE L
I
= Lout
hlI,out − Lout hl ,out
dt
M L = ρ LVL
;
E L = M L hl ,out − PoutIVL
(4.128, 4.129)
ρ L = f (T , PoutI )
(4.130, 4.131)
;
hl ,out = f (T , PoutI )
Lout = f ( PoutI ,VL , Pout )
(4.126a, 4.127)
(4.132)
with 7 unknowns : M L , E L ,VL , ρ L , hl ,out , T , Lout and the input variables are :
I
Lout
, hlI,out , PoutI , Pout .
Structural analysis for the subsystem S-II gives an index one DAE model, but for the
subsystem S-I gives:
a) rank struc( E ) = 2 and deg det struc( Es − A) = 1 , and
b) the structural index of the DAE model for subsystem S-I is two
c) Index reduction requires differentiation of the following equations (4.118-4.121)
(4.123) gives results of:
D(1) F =
d (1) F
dρ dMV
: 0 = VC
−
dt
dt
dt
dMV
dh
dP dEV
0 = hv
+ MV v − VC
−
dt
dt
dt
dt
dT df1 (T , P) dP df1 (T , P) dρV
0=
+
−
dt
dT
dt
dP
dt
dT df 2 (T ) dP
0=
+
dt dT
dt
dT df 3 (T , P) dP df 3 (T , P ) dhv
0=
+
−
dt
dP
dt
dt
dT
(4.133a – 4.137a)
following Conjuncture 2.14a (see Chapter 2) we have to chose 5 non-state variables.
Table 4.1 gives 6 possibilities to chose the state and non-state variables,
82
No. State variables
1
EV
MV
hv
2
3
4
ρV
5
P
T
6
Non state variables
MV , hv , ρV , P , T
hv , ρV , P, T , EV
ρV , P , T , EV , MV
P, T , EV , MV , hv
T , EV , MV , hv , ρV
EV , MV , hv , ρV , P
Table 4.1
e.g. for the choice of the 1st possibility, we replace all
dx
or x
dt
x′
; x = MV , hv , ρV , P , T with algebraic (non-state / dummy) variables
∂D(1) F
; x = MV , hv , ρV , P, T where
is non-singular, such that the sub system
∂ x′
S-I is given by the following 15 differential algebraic equations having index one :
I
MV′ = Vin − Lout
dEV
I
= Vin hv ,in − Lout
hlI,out − Qcool
dt
(4.126b, 4.127)
0 = ρ ′VC − MV′
0 = MV′ hv + MV hv′ − P ′VC − EV
∂f 1 ( T , P )
∂f ( T , P )
+ P′ 1
− ρV′
∂T
∂P
∂f ( T )
+ P′
0 = T′ 2
∂T
∂f ( T , P )
∂f ( T , P )
+ P′ 3
− hv′
0 = T′ 3
∂T
∂P
0 = T′
(4.133b – 4.137b)
and equations (4.128 – 4.132), with 15 unknowns variables
I
MV , EV , Lout
, hlI,out , hv , ρV , PoutI , P, T ,Vin and MV′ , hv′ , ρV′ , P′, T ′
This example shows a contradictory assumption will result in a high index model. The
assumption that system is on thermodynamically phase’s equilibrium (4.121), but there
is only one phase on subsystem S-I. This example also shows that the high index
reduction can lead to the necessity of differentiation of physical properties relations
(4.120, 4.121, 4.123), where practically it is not always possible, i.e. if these physical
properties relations are given as external subroutines then it is not easy to perform this
differentiation.
Another possibility to avoid a high index model of such condenser model as presented
in subsystem S-I is through steady state simplification, i.e. Aspen Tech [5, 118] model
83
library of the condenser model. This is not a strict system equivalence with the original
system description.
4.4.
Consistent re-initialisation problem
This paragraph presents several examples for index one DAE models, which can suffer
consistent re-initialisation problem, i.e. not satisfying consistent re-initialisation
criterion (see Chapter 2 – Corollary 2.12).
4.4.1. Intensive variables as state variables
Fin ,Pin, hin
P0
VL
Fout, Pout, hout
Figure 4.7.
Fin, Fout = inlet and outlet liquid mass flow rate
Pin, Pout = inlet and outlet pressure
hin, hout = inlet and outlet specific liquid enthalpy
P0 = ambient pressure
VL = liquid volume hold up
Figure 4.7 shows a simply ideally mixed liquid in a tank given by the following index
one model with 8 differential algebraic equations:
dM L
= Fin − Fout
dt
(4.138, 4.139)
dE L
= Finhin − Fout hout
dt
M L = ρ LVL
;
E L = M L hout − PV
(4.140, 4.141)
0 L
ρ L = f1 (T ) ; hout = f 2 (T )
(4.142, 4.143)
Fin = f 3 ( Pin , P0 ) ;
Fout = f 4 ( P0 ,VL , Pout )
(4.144, 4.145)
with 8 unknowns variables: M L , E L , hout , ρ L ,VL , T , Fin , Fout and input variables are
hin , Pin , Pout , P0 , where:
M L , E L = mass and energy hold up
ρ L , T = liquid density and temperature
Previously it is stated that the use of intensive variable can lead to solving of a
consistent (re-) initialisation problem. This is demonstrated in the following.
84
In the model equations (4.138-4.139) we chose the extensive variable M L , E L as our
state variables. Representing these model equations with intensive variables as its state
variables, e.g.:
dρ L
dV
+ ρ L L = Fin − Fout
dt
dt
dρ
dV
dh
dV
VL hout L + ρ L hout L + VL ρ L out − P0 L = Finhin − Fout hout
dt
dt
dt
dt
ρ L = f1 (T ) ; hout = f 2 (T )
Fin = f 3 ( Pin , P0 ) ;
Fout = f 4 ( P0 ,VL , Pout )
VL
(4.146, 4.147)
(4.148, 4.149)
(4.150-4.151)
This gives 6 differential algebraic equations with 6 unknown variables
ρ L ,VL , hout , T , Fin , Fout .
Structural analysis of DAE (4.146 – 4.151) gives:
a) rank struc E = 2 and deg det struc ( Es − A) = 2 . This means that the DAE
model (4.146 – 4.151) has no high index structure and the number of dynamic degree
of freedom is two.
b) column rank struc E = 3 (for variables ρ L , VL , hout ) can be interpreted that the
number of the dynamic degree of freedom for consistent (re-) initialisation is three, but
it is not. The dynamic degree of freedom of the DAE model is two.
Clearly this example shows the problem of consistent (re-) initialisation can be caused
through the choice of the state variable, i.e. the intensive variables instead of the
extensive variables.
Note:
The consistent re-initialisation problem can also occur for the previous example on
section 4.2.1 and on section 4.2.2, namely:
a) In the example 4.2.1, if we replace:
dM A
dM CO dM H2O
with
−
dt
dt
dt
dM B
dM CO dM H2O
with
+
dt
dt
dt
dM H2 dM H2O
dM C
with
+
dt
dt
dt
(see (5.23))
(see (5.24))
(see (5.25))
b) In the example 5.2.2, if we replace:
dMi , M
dMi , L dMi ,V
with
+
dt
dt
dt
dE M
dE L dEV
with
+
dt
dt
dt
(see (5.69))
(see (5.70))
85
4.4.2. Tubular reactor ethane cracking model
In the following we describe a non-isothermal tubular reactor ethane-cracking model
adopted from (Froment [39], van Goethem [51]). The thermal cracking of
hydrocarbons is carried out in long coils inside a gas-fired furnace. Figure 4.8
schematically shows such a reactor. The feed of the reactor is mainly ethane C2 H6 and
steam H2 O . The feed steam can contain some other impurities, e.g. C2 H4 , C3 H6 . The
main products of this cracking unit are ethylene C2 H2 and propylene C3 H6 and other
side products as hydrogen H2 , methane CH4 , etc. The products of this cracking unit
are the building blocks of the petrochemical industry.
The cracking reactor scheme of this craking process are:
1. C2 H6 ⎯
⎯r1 → C2 H4 + H2
r2
⎯
→ C2 H6
2. C2 H4 + H2 ⎯
r3
⎯
→ C3 H8 + CH4
3. 2C2 H6 ⎯
r4
⎯
→ C2 H2 + CH4
4. C3 H6 ⎯
r5
⎯
→ C3 H6
5. C2 H2 + CH4 ⎯
r6
⎯
→ C4 H6
6. C2 H2 + C2 H4 ⎯
r7
⎯
→ C3 H6 + CH4
7. C2 H4 + C2 H6 ⎯
dt=pipe internal diameter
z=0; Fj(0),T(0),P(0)
heat flux,
q(z)
j=CH4,C2H2,C2H4,C2H6,C3H6,C3H8,C4H6,H2,H2O
z=L; Fj(L),T(L),P(L)
rb= pipe bending radius
Figure 4.8. Schematic configuration of ethane cracking unit
86
where:
Fj(0), T(0), P(0) = mol feed flow rate of component j, temperature and pressure at the
entry of the reactor tube
Fj(L), T(L), P(L) = mol product flow rate of componet j, temperature and pressure at
the outlet of the reactor tube
q(z) = heat flux supply to the reactor
The steady state place independent balance and consititutive equations are given as
follows:
dFj
dz
Ft Cp
−G
=
πdt2
4
7
∑α
j = 1,2,...,8
;
r
ij i
;
i = 1,2,...,7
i =1
dT
πd 2 7
= q ( z )πdt + t ∑ ( − ΔHi )ri
dz
4 i =1
FG
H
IJ
K
du dP
2 f N bend f bend
−
=
+
ρ g u2
dz dz
dt
2L
(4.152, 4.153, 4.154a)
9
Ft = ∑ Fj
M m Ft = G
;
;
ri = f i1 ( Fj , Ft , P, T )
( − ΔHi ) = f i 2 (T )
Cp = f15 (T )
(4.155, 4.156)
4
M m Pu = GRT
i =1
PM m = ρ g RT
πd
2
t
i = 1,2,...,7
;
;
(4.157, 4.158)
;
j = 1,2,...8
i = 1,2,...,7
(4.159, 4.160, 4.161)
These are 29 equations with 29 unknowns variables:
Fj ( j = 1,2,...,8), T , P, u, ρ g , Ft , M m , ri (i = 1,2,...,7),( − ΔHi )(i = 1,2,...,7), Cp
and model input variables are: F9 , q ( z ), G , where :
α ij = stoichiometric coefficient of i-th reaction for the component j
ri = molar reaction rate per unit of volume of i-th reaction
Ft = total molar flow rate
C p = molar specific heat capacity of the gas mixture
( −ΔHi ) = molar heat of i-th reaction
T , P = temperature and pressure of gas mixture in the reactor tube
G = superficial mass flow velocity
u = gas velocity
f , f bend = tube and tube bending friction factor
Nbend = number of bendings of reactor tube
M m = mean molecular weight
ρ g = density of gas mixture
R = gas constant
87
The first three sets of equations (4.152, 4.153, 4.154) give the component molar
balance, the energy balance and the impulse balance equations. The equations (4.1554.156) give the relations for the total molar flow, and equation (4.157) is the ideal gas
relation. The equation (4.158) is obtained from the substitution of equation (4.157) to
the following equation:
u=
G
(4.162)
ρg
and the equations (4.159) are the kinetic reaction rate equations, where each reaction
rate here is only presented as a function of Fj , Ft , P, T . The details of these reaction
rate equations can be found in Van Goethem [51].
Van Goethem [51] solved this model by the use of an implicit integration algorithm in
gPROMS [50], where the model equations are detected to have a high index structure,
but our structural analysis of this model gives results:
a) rank struc E = 10 and deg det struc ( Es − A) = 10 , which means that the model
has a structural index one with the number of dynamic degree of freedom is 10.
b) column rank struc E = 11 , which means that the model does not satisfy consistent
(re-) initialisation criterion (see Chapter 2 – Corollary 2.12)
This is an example of a solving of consistent (re-) initialisation problem that has been
detected as a high index problem, but it is not.
A consistent re-initialisation model structure can be made through the following
transformation:
dFj
dz
Ft Cp
=
πdt2
4
7
∑α
r
ij i
j = 1,2,...,8
;
;
i = 1,2,...,7
i =1
πd 2 7
dT
= q ( z )πd t + t ∑ ( − ΔHi )ri
4 i =1
dz
FG
H
IJ
K
2 f N bend f bend
dPu
=
+
ρ g u2
2L
dz
dt
Pu = − Gu − P
(4.152, 4.153, 4.154b, 4.154c)
together with equations (4.155 – 4.161) give an index one DAE model having
consistent re-initialisation structure, where the number of dynamic degree of freedom
equals to the number of differential variables, i.e. 10.
Although the following differentations are not required, Froment [39] presented the
solution as the ‘index reduction’ through Pantelides’ algorithm for a consistent
initialisation problem [98]. It needs to differentiate the following equations (4.155)
(4.156), (4.158). This gives result to :
88
(1)
D(1) F =
dF
dP
dM m
du
dT
: M mu
+ Pu
+ M m P − GR
=0
dz
dz
dz
dz
dz
8
dF
dFt
−∑ j = 0
dz j =1 dz
Mm
dFt
dM m
+ Ft
=0
dz
dz
The following 3 variables are chosen as “dummy/algebraic” z ′ variables, where
D(1) F
d z′
is non − singular :
du
→ u′
dz
;
dFt
→ Ft ′
dz
;
dM m
→ M m′
dz
where an index one consistent (re-) initialisation model is given by the following 32
dFj
dz
Ft Cp
− Gu′ −
=
πdt2
4
7
∑α
r
ij i
j = 1,2,...,8
;
;
i = 1,2,...,7
i =1
dT
πd 2 7
= q ( z )πdt + t ∑ ( − ΔHi )ri
dz
4 i =1
FG
H
IJ
K
2 f N bend f bend
dP
ρ g u2
=
+
2L
dz
dt
(4.152, 4.153, 4.154d)
dP
dT
M mu
+ PuM m′ + M m Pu′ − GR
=0
dz
dz
8
dF
Ft ′− ∑ j = 0
j =1 dz
M m Ft ′+ Ft M m′ = 0
(4.155’, 4.156’, 4.158’)
together with equations (4.155 – 4.161) and having 32 unknown variables
Fj ( j = 1,2,...,8), T , P, u, ρ g , Ft , M m , ri (i = 1,2,...,7),( − ΔHi )(i = 1,2,...,7), Cp
and
u′, Ft ′, M m′ .
4.5.
High index problem due to interconnection
The following gives analysis of a high index problem due to interconnection of two
ideally mixed tanks, as given in the figure 4.9. This example gives that a high index
problem can arise, when inputs of the interconnected subsystems are not matched (see
Chapter 6).
89
The system boundary of each subsystem in the figure 4.9 is given with dashed lines
and the model equations of each subsystem and its interconnection equations are also
shown in the figure 4.9.
Subsystem S-I is described by 4 index one differential algebraic equations, with 4
unknowns variables M1 , H1 , P1 , Fin and the input variables are Pin , Fout 1 .
Subsystem S-II is described by 4 index one differential algebraic equations, with 4
unknowns variables M 2 , H2 , P2 , Fout and the input variables are Pout , Fin 2 .
A1=area tank 1
A2=area tank 2
S-I
S-II
H1
Fin, Pin
H2
P1
Fout1, P1
S-I :
dM1
= Fin − Fout
dt
M1 = ρA1 H1
P1 = ρgH1
Fin = f1 ( Pin , P1
Fout, Pout
P2
Fin2, P2
S-II :
interconnection :
Fout 1 = Fin 2
P1 = P2
dM 2
dt
M2
P2
Fout
= Fin 2 − Fout
= ρA2 H2
= ρgH2
= f 2 ( P2 , Pout
Figure 4.9. Two interconnected ideally mixed tanks
Compound system S-I + S-II is described by 10 differential algebraic equations, where
two of them are additional interconnection equations, to solve 10 unknowns variables
M1 , H1 , P1 , Fin , M 2 , H2 , P2 , Fout 1 , Fin 2 , Fout . The input variables are Pin , Pout . The DAE
model equations are given as follows:
dM1
= Fin − Fout1
dt
M1 = ρ A1H1
P1 = ρ gH1
Fin = f1 ( Pin , P1 )
90
(4.163a, 4.164, 4.165, 4.166)
dM 2
= Fin 2 − Fout
dt
M 2 = ρ A2 H 2
P2 = ρ gH 2
(4.167a, 4.168, 4.169, 4.170)
Fout = f 2 ( P2 , Pout )
Fout1 = Fin 2
(4.171, 4.172)
P1 = P2
The structural analysis of this interconnected sub-models gives:
a) rank struc E = 2 and deg det struc( sE − A) = 1 that means that the model has a
high index with the number of dynamic degree of freedom is one.
b) Analyzing the structural information from struc ( Es − A) −1 gives that this DAE
model has index two.
In the following different ways for the high index reductions will be shown:
(i) Based on the information of the matrix struc ( Es − A) −1 , a high index reduction
can be done through differentiation of the equations (4.164),(4.165),(4.168),
(4.169), (4.172), gives:
dM1
dH
− ρA1 1 = 0
dt
dt
dP1
dH1
− ρg
=0
dt
dt
dP1 dP2
−
=0
dt
dt
;
;
dM 2
dH
− ρA2 2 = 0
dt
dt
dP2
dH2
− ρg
=0
dt
dt
and the following 5 variables z ′ can be chosen as non-state variables (i.e. algebraic
variables), where
d (1) F
d z′
dM 2
→ M 2′
dt
dH1
→ H1′
dt
;
is non − singular :
;
dH2
→ H2′
dt
;
dP1
→ P1′
dt
;
dP2
→ P2′
dt
where an index one consistent (re-) initialisation model is given by the following 15
91
dM1
= Fin − Fout1
dt
M1 = ρ A1H1
P1 = ρ gH1
Fin = f1 ( Pin , P1 )
M 2' = Fin 2 − Fout
M 2 = ρ A2 H 2
P2 = ρ gH 2
Fout1 = Fin 2
P1 = P2
dM1
− ρA1 H1′ = 0
dt
P1′− ρgH1′ = 0
(4.163a, 4.164, 4.165, 4.166, 4.167b,
4.168, 4.169, 4.170, 4.171, 4.172)
;
M 2′ − ρA2 H2′ = 0
;
P2′ − ρgH2′ = 0
(4.173 – 4.177)
P1′− P2′ = 0
or by substitution and elimination of all non-state variables, we get the following index
one DAE model:
dM1
= Fin − Fout1
dt
M1 = ρ A1H1
P1 = ρ gH1
Fin = f1 ( Pin , P1 )
A2
( Fin − Fout1 ) = Fin 2 − Fout
A1
M 2 = ρ A2 H 2
P2 = ρ gH 2
(4.163a, 4.164, 4.165, 4.166, 4.167c,
4.168, 4.169, 4.170, 4.171, 4.172)
Fout1 = Fin 2
P1 = P2
Note:
Substitute equation (4.171), Fout 1 = Fin 2 , to equation (4.167c), we get:
Fout 1 = Fin 2 =
92
b
1
A2 Fin + A1 Fout
A1 + A2
g
(4.173)
ii) Another method for the high index reduction can be done by performing variable
transformation as explained in section 4.2. This is done by replacing the first
equation (4.163) with the combination linear of equation (4.163a) and (4.167) that
removes variable Fout1 . This gives the following:
dM t
= Fin − Fout
dt
dM 2
= Fin 2 − Fout
dt
M1 = ρA1 H1 ;
M 2 = ρA2 H2
P1 = ρgH1 ;
P2 = ρgH2
Fout 1 = Fin 2
;
Fin = f ( Pin , P1 )
M t = M1 + M 2
P1 = P2
;
Fout = f ( P2 , Pout )
a) when we are not interested on variable Fin 2 , Fout 1 as outputs of the system then we
can perform column wise strict system equivalence operation resulting in an index
one DAE model:
dM t
= Fin − Fout
dt
0 = Fin 2 − Fout
M1 = ρA1 H1 ;
M 2 = ρA2 H2
P1 = ρgH1 ;
P2 = ρgH2
;
Fout 1 = Fin 2
P1 = P2
Fin = f1 ( Pin , P1 ) ;
M t = M1 + M 2
Note: The equations 0 = Fin 2 − Fout and Fout1 = Fin 2 do not have any contributions
on inputs outputs behaviour. Therefore these two equations may be removed from the
equation set above.
b) but when we are interested in variable Fin 2 , Fout 1 as outputs of the system, we may
not perform column wise strict system equivalence operation as described above.
The index of the DAE model can be reduced through differentiation of equations
(4.165), (4.166), (4.167), (4.168), (4.170), (4.173), and choose the variables
M1′, M 2′ , H1′, H2′ , P1′, P2′ are non-state variables, then we get the equivalent index
one model equations:
93
dM t
= Fin − Fout
dt
M 2′ = Fin 2 − Fout
M1 = ρA1 H1 ;
M 2 = ρA2 H2
P1 = ρgH1 ;
P2 = ρgH2
;
Fout 1 = Fin 2
P1 = P2
Fin = f1 ( Pin , P1 ) ;
M t = M1 + M 2
M1′ − ρA1 H1′ = 0
P1′− ρgH1′ = 0
;
;
P1′− P2′ = 0
;
M 2′ − ρA2 H2′ = 0
P2′− ρgH2′ = 0
dM t
− M1′ − M 2′ = 0
dt
Substitution and elimination of the non-state variables give the result:
dM t
= Fin − Fout
dt
( A1 + A2 ) Fin 2 = A2 Fin + A1 Fout
M1 = ρA1 H1 ;
M 2 = ρA2 H2
P1 = ρgH1 ;
P2 = ρgH2
;
Fout 1 = Fin 2
P1 = P2
Fin = f1 ( Pin , P1 ) ;
M t = M1 + M 2
In this example, we see that the resulting low index DAE model from a strict system
equivalence operation, i.e. an index reduction, is non unique.
4.6.
Conclusions
The objectives of this Chapter are to show several examples of a high index model.
The examples show that a high index problem can be caused due to internal states
constraint equations or due to sub models interconnection.
A consistent (re-) initialisation problem on a low index model is caused due to a non
minimum state variables presentation. It is recommended to use physical extensive
variables as the state variables.
A high index reduction can be done through equation differentiations or variable
differentiations, called a strict system equivalence operation. The resulting low index
model due to a strict system equivalence operation can be non unique.
94
Chapter 5: Input Variables Assignment of a DAE Model
Input variables of a DAE model are sets of possible model variables
acting from the model boundaries, such that the DAE has a low index.
The constraining equations for a high index DAE model are invariant
under the choice of input variables resulting to the highest model’s
degree of freedom.
“A dissonance is always recognisable......”
5.1.
Introduction
This Chapter describes the choice of input signals to a DAE model for their role in
interconnection between DAE sub models. In Chapter 2,3 and 4 we have seen that a
dynamic process model leads to a first order non-linear DAE given by the equation
(1.3b): F ( z (t ), z (t ), u (t ), t ) = 0 .
It was thought that the choice of input variables u (t ) for a process model is straight
forward. A schematic of external influences through the model boundaries into or
acting to the processes, as given on figure 5.1.
u3
Model boundary
u5
u1
u2
Model
u6
u4
Figure 5.1. Model input variables.
In the process modelling activity it became clear that the choice of input variables for a
DAE model is not always straight forward. The reasons for this are:
a) Most modelling activity is more implicit behavioural or model equations
description driven. It is not about the relation of signals paths or variables within
the equations and how to solve these equations in relation to the external variables.
b) The equations within a DAE model can be implicit. Therefore sometimes the input
variables are not easy to recognise from the model equations.
c) The input and output variables are dependent on the model equations within the
model itself. Arbitrary signals/variables can not be chosen for input variables of a
model.
95
Chapter 5 : Input Variables Assignment …
d) The input variables for a model can not be arbitrary assigned for a given set of
model equations, see the following example 5.1. Lefkopoulos and Stadtherr
[74,75] also pointed out that the wrong choice of input variables can result in a
high index DAE model.
e) Sometimes the directionality of input variables of a DAE model in relation with its
interconnection with other DAE model is not fixed, see example 5.2.
Example 5.1. A simplified ideal gas buffer models
Fin,Pin
Fout,Pout
Figure 5.2. A simplified gas buffer model
The model equations are: Where:
dM
= Fin − Fout
dt
PV
M=
RT
Pin = P
Pout = P
Fin = inlet gas mass flow
Fout = outlet gas mass flow
Pin = buffer gas inlet pressure
Pout = buffer gas outlet pressure
M = buffer gas mass hold up
P = buffer gas pressure
V = buffer volume (model parameter)
R = buffer gas constant (model parameter)
T = buffer gas temperature (assume as a model parameter)
These are 4 model equations with 6 variables: Fin , Pin , Fout , Pout , M , P . There are 2
variables must be assigned as a model input variables. It seems that they can not be
assigned arbitrarily, i.e. Fin , Fout must be the input variables for this model.
Example 5.2.
Figure 5.3 gives schematically interconnected 2 gas buffer models with inlet and outlet
pressure drop equations represented in the figure as inlet and outlet ‘valve models’.
There are two possible directionalities of pressure flow variables P, F between the
outlet of the buffer 1 and the inlet of the buffer 2.
96
Model boundary
Model boundary
P
P
F
F
Figure 5.3. Interconnection directionalities of two gas buffer models
Interconnection of a DAE model with another DAE model depends on the possible
input variables assignments to these DAE sub-models. From a model-user point of
view, it is very useful when the information of possible input variables assignment to a
model is available. Moreover we have already seen on the section 4.5 (Chapter 4), that
interconnections between low index sub models can still lead to a high index model.
A modeller expects that each storage variable on each sub model has always at least
C1 continuity, i.e. that there is no storage variable that gives a ‘jump’ or an
‘impulsive’ behaviour due to changes in the model variables values, other than the
change on a sudden change on the storage variable itself. Therefore the modeller
expects a preservation of the amount of dynamic degree of freedom in the storage
variables on each sub model where sub-models are interconnected. In other words, we
want that interconnection of low index sub models should result in a low index model.
Thus the essentials questions to be answered in this chapter are:
a) what are the input variables of a DAE model ?
b) how can we choose the input variables sets ?
c) can we always choose a set of input variables of a DAE model to give a low
index DAE ?
to answers those questions this Chapter 5 is divided into the following sections:
- Section 5.2 describes the DAE model entities and variable classifications. This
section also gives the definition of input variables to a DAE model.
- Section 5.3 describes a method for finding input variables sets, namely the low
index input variables sets assignment for a DAE model.
- Section 5.4 describes if there is no input variables set that gives a low index
DAE, then we will see that the state constraint equations are invariant under
97
-
the choice of a maximum degree input variables sets assignment for a DAE
model.
Section 5.5 gives conclusions of this chapter.
Finding the input variables of a DAE model is done by using the structural theory in
Chapter 3 and extending the graph algorithm as described in Appendix B.
5.2.
Model entities and variables classification
5.2.1. Model entities
P.I. Barton [6] and Nilsson [93] describe that principally there are 5 model entities
(see figure 5.4), namely:
1. Model parameters as the time invariant quantities that are not, under any
circumstances, determined from the results of a calculation.
2. Model variables as the quantities describing time dependent behaviour of a
model.
3. Model equations as the behavioural description that determines the time
trajectories of the model variables.
4. Model initial values. The model initial values are the chosen consistent set of
initial values at time t0 of a model. (note: for partial differential (algebraic)
equations the model boundary values are also required)
5. Model ports or connections are model boundaries. Each model port consists of
the chosen sub set of the model variables to interact with its environment.
MODEL
C
o
n
n
e
c
t
i
o
n
s
Parameters
Variables
Initial & Boundary
values
Equations
C
o
n
n
e
c
t
i
o
n
s
Figure 5.4. Schematic of model entities
Usually the DAE model equations are given as:
F ( z (t ), z (t ), t ) = 0
(1.3a)
or
F ( x (t ), x (t ), w(t ), t ) = 0
98
(5.1)
Where: z (t ) = [ x(t )
( a b)
w(t ) ] , t ∈(a b), denotes time on some interval
; a, b ∈ R ; z (t ) ∈ R n , z (t ) ∈ R n , F : R 2 n +1 → R m , n, m ∈ I
\0,
m≤n.
The DAE model’s consistent initial values ( z (to ), z (to )) are mostly calculated by
algebraic solving of the equations by providing r values of arbitrary chosen initial
conditions (see Chapter 2, section 2.5).
Principally the model interconnections are the chosen set of variables by the modeller
to interconnect or interact the model with its environment. Process modelling
environments like Aspen [5], gProms [50], Hysis [61] use several classes of model
connections. Examples:
- stream model connection, includes information of flow, pressure, components,
temperature, enthalpy
- heat or power connection, includes information of heat flow or power from or
into a model
A subset of these variables within a model connection can be the input variables of the
model.
5.2.2. Model variable classifications
Consider a DAE model F ( z (t ), z (t ), t ) = 0 (1.3a) or it can be written as
F ( x (t ), x (t ), w(t ), t ) = 0
where:
(5.1)
b g
x(t ), x(t ) ∈ R r , w(t ) ∈ R p , t ∈ ( a b ) , a, b ∈ R denotes time on interval a b and
F : R × R × R → R , with m < (r + p )
2r
p
m
With the variables in this DAE model (5.1) are:
1. x (t ) as the state variables and
2. w(t ) as the non-state variables
5.2.2.1. State variables x (t )
Willems [131] introduced the term system memory for understanding the term ‘state’
of a system in the time domain,
Definition 5.1. System memory (Willems [131])
A system has memory if there are C1 continuous or stepwise continuous internal
signals x (t ) ⊆ z (t ) where the value of each variable x (t o ) at time t o can be
reconstructed from the value of x (t ) , where t < t o .
99
We call these variables as physical memory variables or state variables.
Mathematically the memory-effect of the system behaviour is given by a difference or
differential variable.
The states of the system are not unique, i.e. new state variables can be defined by state
variables transformations. However starting with the primitive balance equations, we
can choose the accumulation terms as the primitive state variables of the system. The
accumulation or storage variables in the balance equations are the memory of the
system,
Proposition 5.2. Storage variable.
A storage variable of a system given in the balance equation is a state variable of the
physical system. Since we expect that a storage property of a system at time to
depends only on the past value at t < t o and must be C1 continue at ∀t ∈ (a
b)
De Groot and Mazur [52], and Preisig [104] recommend using the fundamental
extensive quantities as the common starting point of the choice of the state variables in
the balance equations. This will give the minimal physical state realisation of the
model.
As shown in Chapter 2 and Chapter 3, mathematically we can calculate the number of
dynamic degrees of freedom of a system, i.e. the number of independent state
variables, is given as deg det struc ( Es − A) . Our primitive assumption of
independent balance equations gives that from the physical modelling point of view,
the number of dynamic degrees of freedom is equal to the number of balance equations
in the system. This applies when there is no state constraints equation in the system or
if the model has a low index.
5.2.2.2. Non-state variables w(t )
The non-state variables of a model can be distinguished into:
-
external variables, i.e. input variables. These are the variables determined by
model equations and that can be chosen as the variables from the external
acting to the model.
internal model variables: the model variables which are derived from the state
variables and/or external variables.
Input variables
The following will define input variables of a DAE model.
Definition 5.3. Autonomous and non-autonomous physical process model
Consider a DAE model (1.3a): F ( z (t ), z (t ), t ) = 0
b), denotes time on some interval (a b) ; a, b ∈ R ;
z (t ) ∈ R , z (t ) ∈ R n and F : R 2 n +1 → R m , n, m ∈ I
\0, m ≤ n .
Where: t ∈(a
n
100
An autonomous physical process model or an autonomous continuous model
behaviour is given by equation (1.3a) with m=n, if a model behaviour at time
to , z (to ), will determine a future behaviour z (t ), for t ≥ to .
A non-autonomous physical process model is given by equation (1.3a) where m < n .
There are k = n − m variables that can be chosen as input variables. The amount of
k-variables is called degree of freedom (DOF) of (1.3a)
Definition 5.4. Input variable
Consider a model S given by a DAE model (1.3a) or (5.1):
F ( z (t ), z (t ), t ) = 0 or F ( x (t ), x (t ), w(t ), t ) = 0
where: t ∈ ( a b);
a, b ∈ R denotes time on time interval (a b) ,
z (t ) = x (t ) w(t ) , x (t ) = state variables , z (t ) ∈ R n , z (t ) ∈ R n and
F : R 2 n +1 → R m ; n, m ∈ I \ 0,
m < n.
Then input variables u(t ) of S are possible sets of k = n − m system variables,
u( t ) ⊆ z ( t ) ,
Under conditions that:
a. ui (t ),
i = 1,..., k may have a free form (e.g. pulse, step, polynomial)
b. The resulting DAE , with a chosen set of input variables u(t ) :
F ( z (t ), z (t ), u (t ), t ) = 0
u (t ) = f (t )
l
q
or with w(t ) = v (t ), u(t ) :
F ( x (t ), x (t ), v (t ), u(t ), t ) = 0
u( t ) = f ( t )
is a fully determinated DAE (number of equations = number of variables.
then this fully determined DAE together with its initial conditions will determine
the behaviour of the model.
c. At each time instant t ∈(a b) , the property of physical storage variable,
proposition 5.2 holds, i.e. pre-assumed physical storage variable is C1 continue
on t ∈(a b)
d. x (t ) and v (t ) do not anticipate the chosen u(t ) (see b.) for t ∈(a b) , and
behaviour equation (b) is satisfied
101
We call u(t ) as (causal) input variables, when conditions a,b,c,d in the definition 5.4.
are fulfilled.
A set of causal input variables for a DAE model (1.3a) also gives a low index DAE
model (1.3b). Therefore in the following we will derive how to detect a low index
input variables set.
Consider an under determined DAE model (1.3a) F ( z (t ), z (t ), t ) = 0 with its system
matrix written as:
T (s) z = 0
, with
(
(5.2)
)
T ( s ) = Es − A , and E , A ∈ R m×n , m < n
Then there is k = m − n variables that must be set for (5.2) to become a fully
determined DAE. If we assign these a variable from (5.2) as time function equation:
zi = ui (t ) then we can represent (5.2) as:
LMT~(s)OPz = F 0 I
MN μ PQ GH u JK
(5.3)
T
i
i
where μ i ∈ R n is a vector for which the i-th element is 1, and all other elements are 0.
Obviously, to get a fully determined DAE (i.e. a square matrix) we have to assign time
functions for k variables.
If the choice of these k-time functions variables is free from the internal properties of
the system, we may have a high index DAE or the resulting system may even be
inconsistent. Therefore it makes sense to define a low index input assignment as
follows,
Definition 5.5. A low index input assignment
Given a DAE model (1.3a) F ( z (t ), z (t ), t ) = 0 with m equations and n variables z , if
the set of k = m − n variables can be set as external independent time functions
variables where the resulting DAE is consistent and has a low index, then these k
variables are called ‘a low index input variables set / assignment’ for the given DAE
model.
Example 5.3
Fin, Pin
P
Figure 5.5.Input variables of a tank model
102
Fout, Pout
Consider a liquid tank with valves as shown in the figure 5.5. The DAE model of this
system is:
dM
= ( Fin − Fout )
dt
A
M= P
g
Fin = f1 ( Pin , P )
Fout = f 2 ( P, Pout )
Where :
M = mass hold up within the tank
P = hydraulic pressure at the bottom of the tank
Pin = inlet pressure
Pout = outlet pressure
f1 = function describing inlet valve model
f 2 = function describing outlet valve model
A = tank area (model parameter)
g = gravity acceleration constant (model parameter)
These are 4 differential algebraic equations with 6 unknowns: M , P, Pin , Pout , Fin , Fout .
Thus two from these six variables should be chosen results to a fully determined DAE
model. Table 5.1 shows the 15 possibilities to assign the ‘input’ variables to the model,
No
1
2
3
4
5
6
7
8
Variable set
M, P
M, Pin
M, Pout
M, Fin
M, Fout
P, Pin
P, Pout
P, Fin
No
9
10
11
12
13
14
15
Variable set
P, Fout
Pin, Pout
Pin, Fin
Pin, Fout
Pout, Fin
Pout, Fout
Fin, Fout
Table 5.1. Possible ‘input’ variables
In this example, we can show that only the variable sets as shown in table 5.2 give the
low index assignment, while others are not. In particular the set { M , P} as ‘input’
variables give a non-consistent DAE.
No
10
12
13
15
Variable set
Pin, Pout
Pin, Fout
Pout, Fin
Fin, Fout
Table 5.2. Low index input variables sets
103
Example 5.3. shows also that:
1. The causal or low index input variables set is not trivially given by all
variables with indices “in” , e.g. {Fin , Pin } is not a causal input set for this
model.
2. The causal or low index input variables set is not necessary to be unique
3. There are two variables, which may not be chosen as an input variable of the
model namely: M or P , since it will give a high index DAE model. We call
later these variables the “compelled internal variables”
The notion of finding a low index input variables set of a given DAE model motivates
us to make an algorithm to do this work. This will described in the section 5.3.
Several definitions for model variable classification are given as follows,
Consider a DAE model (1.3a), F ( z (t ), z (t ), t ) = 0 , is represented by S , with m
equations and n variables z , where m < n and let Z = {zi : i = 1,2,..., n} . Further, let
U = {all distinct low index input variables sets assignments of S}
then we can define:
Internal variables
Definition 5.6. Internal variables
An internal variable is a variable, which is derived from the state variables and/or
external/input variables of a DAE model.
Definition 5.7. Compelled internal variables
A variable c ∈ Z is called a compelled internal variable if c ∉ U ∀ U ∈ U .
Compelled internals are those variables that ensure the resulting DAE model to have a
high index if chosen to be as input variables. These are in particular the state
variables and all other variables that are as a function of state variables only, e.g. the
intensive variables.
Definition 5.8. Necessary input variables
A variable u ∈ Z is called a necessary input variable if u ∈ U ∀ U ∈ U .
In other words, necessary input variables are those variables that are elements of all
low index assignments of an under determined DAE model. This variable is externally
acting independently to the system.
Definition 5.9. Optional input variables
A variable is called an optional input variable if it is neither necessary, nor compelled,
nor internal variable. This variable can act externally independent on the system.
Definition 5.10. Output variables
For a given fully determined DAE model, any variable zi ∈ Z can be chosen as an
output of the model.
104
5.3. Low index input variables assignment
5.3.1. Finding a low index input variables set
The next proposition is used for finding a low index input assignment for the given
DAE model using the Structural Low Index Criterion developed in Chapter 3,
Corollary 3.10. The implementation of finding a low index input assignment is by
using the Lemma B.1 and Lemma B.2 as given in Appendix B.
Proposition 5.11. A low index input assignment of a DAE model
Consider an underdetermined DAE model (1.3a) F ( z (t ), z (t ), t ) = 0
or it can be
written in its system matrix equation: ⎡⎣ Es − A⎤⎦ z = 0 , where E , A ∈ R m×n , m < n .
We can extend this underdetermined system matrix, such that the new system matrix is
given as:
⎡ Es − A 0 ⎤
T (s) = ⎢
⎥
K⎦
⎣ In
(5.4)
where : T ( s ) ∈ R n*× n* ( s ), n* = m + n and the second row of T ( s) , I n
K gives
additional equations of the form: zi = ud1 + ud2 + + udm , i = 1,2,..., n ; the set these
equations is denoted as F . Further ud1 , ud2 ,..., udm are m new dummy variables
and K ∈ R n×m with struc K has no zero elements.
Let T ( s) be structurally non-singular, and GT is the graph obtained from T ( s) as
defined in Lemma B.2. The vertices of bipartite graph GT are
V1 (GT ) = { f1 , f 2 ,..., f m ,..., f n , f n+1 , f n+m} and V2 (GT ) = {z1 , z2 ,..., zn , ud1 , ud2 ,..., udm } .
Let M MW be the result of a Maximum Weighted Matching algorithm on GT .
Further let GE be the graph obtained from E as defined in Lemma B.1 and let M MC
be the result of a Maximum Cardinality Matching algorithm on GE . Then each M MW
gives rise to a low index input variables set assignment U ( M MW ) ,
l
q
U( M MW ) = zi : ∃( f m+i , zi ) ∈ M MW , i = 1,.., n
with W ( MW ) = M MC (see Lemma B.1 and Lemma B.2).
(5.5a)
(5.5b)
Note:
Since T ( s) is structurally non-singular it follows that M MW = m + n , and hence
there exists a permutation σ ∈Γ , such that ( f i , zσ i ) ∈ M MW .
105
Let F is the new n equations set forms by zi = ud1 + ud2 + + udm , i = 1,2,..., n .
For each M MW found, a set of exactly m matched edges of vertices
FD ⊂ F = { f n +1, f n + 2 ,..., f n + m } ⊂ V1 (GT ) with all m vertices of
D = {ud1 , ud2 ,..., udm } ⊂ V2 (GT ) which can be removed, since
D = {ud1 , ud2 ,..., udm } ⊂ V2 (GT ) are only dummy variables. So, if we delete all edges
corresponding to FD . The remaining k = n − m equations vertices FUD = F ⊃ FD are
matched with k variables of zi which can be assigned as input variables.
Since the removed edges FD had zero weight, the remaining edges still have a
maximum weighted matching. This gives a low index input variables set assignment
with W ( MW ) = M MC (see Lemma B.1 and Lemma B.2). The original
underdetermined DAE model and the found low index input assignment can be
represented as (5.3):
LMT~(s)OPz = F 0 I
MN μ PQ GH u JK
T
i
(5.3)
i
Example 5.4.
Consider an under determinated DAE model:
z1 = − z1 + z2 − z3
z2 = − z2 + z5
(5.6)
0 = z2 + z4 + z5
With 3 equations and 5 variables z1 , z2 , z3 , z4 , z5 . Two of these variables must be
assigned as input variables. To detect a low index input variables set assignment, we
construct additional equations with additional dummy variables given as follows (see
Proposition 5.11):
f1: z1 = − z1 + z2 − z3
f 2 : z2 = − z2 + z5
f 3:0 = z2 + z4 + z5
f 4 : z1 = ud1 + ud2 + ud3
f 5: z2 = ud1 + ud2 + ud3
f 6: z3 = ud1 + ud2 + ud3
f 7 : z4 = ud1 + ud2 + ud3
f 8: z5 = ud1 + ud2 + ud3
The system matrix of this DAE model can be written as (see (5.4)):
106
(5.7)
⎡s + 1
⎢ 0
⎢
⎢ 0
⎢ 1
T (s) = ⎢
⎢ 0
⎢
⎢ 0
⎢ 0
⎢
⎣ 0
−1
1
0
0
0
0
s +1
0
0
−1
0
0
−1
0
−1
−1
0
0
0
0
0
0
−1
−1
1
0
0
0
−1
−1
0
1
0
0
−1
−1
0
0
1
0
−1
−1
0
0
0
1
−1
−1
0⎤
⎥
⎥
0⎥
⎥
−1
⎥
−1⎥
⎥
−1⎥
−1⎥
⎥
−1⎦
0
(5.8)
To find a low index input variables set of this DAE from equations (5.7) or the system
matrix (5.8), we can construct the graph of the DAE with its edges ( f i , zi ),( f i , udi ) , as
shown in the figure 5.6. The thick lines are a result of maximum weighted matching.
The value “1” on the edges are the weighted edges, other edges are zero weighted.
1
z1
f1
f2
1
z2
f3
z3
f4
z4
f5
z5
f6
ud1
f7
ud2
f8
ud3
Figure 5.6. A maximum weighted matching graph of the DAE (5.8)
From a maximum weighted matching graph figure 5.6 of DAE (5.6, 5.7) and following
the Proposition 5.11, the equations FD = { f 4 , f5 , f 7 } matched to vertices ud1 , ud2 , ud3
can be removed. The variables z3 , z5 can be chosen as input variables. Since the
cardinality of maximum cardinality matching of matrix E is two and the weight of
maximum weighted matching of (5.8) is also two, then following Corollary 3.10 this
input variables set is a low index input assignment. The DAE (5.7) can be written as a
consistent and fully determined low index DAE:
107
z1 = − z1 + z2 − z3
z2 = − z2 + z5
0 = z2 + z4 + z5
z3 = u1 (t )
z5 = u2 (t )
(5.9)
Another maximum weighted matching graph of DAE (5.7) can be found and result in a
consistent and fully determined low index DAE of (5.7) is:
z1 = − z1 + z2 − z3
z2 = − z2 + z5
0 = z2 + z4 + z5
z3 = u1 (t )
(5.10)
z4 = u2 (t )
In this example 5.4 shows that there are two sets of the low index input assignment for
this model, namely: U1 = {z3 = u (t ), z5 = u (t )} and U 2 = {z3 = u (t ), z4 = u (t )} or
set of all low index input variables sets is U = {{z3 , z5 },{z3 , z4 }} .
5.3.2. Finding the necessary input variables
To find the set of all low index assignments, we need the notion of the necessary input
variables as defined in section 5.2.2 (Definition 5.8). Clearly that all necessary inputs
following the
are elements of U( M MW ) = zi : ∃( f m+i , zi ) ∈ M MW , i = 1,.., n
Proposition 5.11.
l
q
We can check for each u ∈U whether it is a necessary input or not by deleting the
corresponding equation from I n K of (5.4) and removing one of the introduced
dummy variables. This gives new matrix:
T * ( s) =
LMsE − A
NI
n −1
0
K*
OP
Q
(5.11)
where K * ∈ R ( n −1)×( m −1) and the dummy variable set is {ud1 , ud2 ,..., udm−1 } .
If no new maximum weighted matching can be found for the graph of T * ( s) with a
maximum cardinality matching of E then the deleted variable, which was assigned as
an input variable, is a necessary input.
108
Example 5.5.
Consider the DAE from example 5.4,
f1: z1 = − z1 + z2 − z3
f 2 : z2 = − z2 + z5
f 3:0 = z2 + z4 + z5
f 4 : z1 = ud1 + ud2 + ud3
f 5: z2 = ud1 + ud2 + ud3
(5.7)
f 6: z3 = ud1 + ud2 + ud3
f 7 : z4 = ud1 + ud2 + ud3
f 8: z5 = ud1 + ud2 + ud3
We found that a low index input assignment set for this DAE is
U1 = {z3 = u1 (t ), z5 = u2 (t )} . To check whether z5 = u2 (t ) is a necessary input, then
we delete the equation f 8 and the dummy variable ud3 to get:
f1: z1 = − z1 + z2 − z3
f 2 : z2 = − z2 + z5
f 3:0 = z2 + z4 + z5
f 4 : z1 = ud1 + ud2
(5.12)
f 5: z2 = ud1 + ud2
f 6: z3 = ud1 + ud2
f 7 : z4 = ud1 + ud2
The graph for this new set DAE (5.12) and the found maximum weighted matching
with the maximum cardinality matching of its E matrix is given in the figure 5.7.
From a maximum weighted matching graph figure 5.7 of the DAE (5.12) and
following Proposition 5.11, the equations FD = { f 4 , f 5} matched to vertices ud1 , ud2
can be removed and the variables z3 , z4 can be chosen as input variables. In this
example we see that z5 = u2 (t ) is not a necessary input variable. Further we can show
that z3 = u1 (t ) is a necessary input variable, since no maximum weighted matching
with maximum cardinality of its state variable (matrix E ) can be found through
deletion of equation f 6 and removing one dummy variable ud3 from the DAE (5.7).
Remarks: To do a necessary input variable check, we have to start from complete
T ( s) for each u ∈U
109
f1
1
f2
1
z1
z2
f3
z3
f4
z4
f5
z5
f6
ud1
f7
ud2
Figure 5.7.Maximum weighted matching graph of the DAE (5.12)
5.3.3. Finding the compelled internal variables
To check if zi is a compelled internal variable, we search a maximum weighting
matching with the maximum cardinality of its matrix E of the graph of T ( s) with zi
as input variable. If a match cannot be found, then zi is a compelled internal, else it is
not.
Note: zi = u(t ) , u(t ) is not a variable of DAE, but an external chosen value.
Therefore this equation has only one vertex zi .
Example 5.6.
Consider the DAE of the example 5.4:
f1: z1 = − z1 + z2 − z3
f 2 : z2 = − z2 + z5
f 3:0 = z2 + z4 + z5
(5.6)
Suppose we will examine whether z1 is a compelled internal or not, then we examine a
maximum weighted matching for z1 = u(t ) , the DAE with additional new dummy
variables are given as follows:
110
f1: z1 = − z1 + z2 − z3
f 2 : z2 = − z2 + z5
f 3:0 = z2 + z4 + z5
f 4 : z1 = u(t )
f 5: z2 = ud1 + ud2 + ud3
(5.13)
f 6: z3 = ud1 + ud2 + ud3
f 7 : z4 = ud1 + ud2 + ud3
f 8: z5 = ud1 + ud2 + ud3
where the equation f 4 is only connected to the variable z1 , i.e. z1 = u1 (t ) to be chosen
as an input variable (see figure 5.8).
The resulting maximum weighted matching graph is given in the figure 5.8. The found
input variables are z1 = u1 (t ), z5 = u2 (t ) and the weight of this maximum weighted
matching is one. Since the cardinality of the maximum cardinality matching of its
matrix E is two. Then this matching is not a low index assignment, thus the variable
z1 is a compelled internal variable.
f1
f2
z1
1
z2
f3
z3
f4
z4
f5
z5
f6
ud1
f7
ud2
f8
ud3
Figure 5.8. a maximum weighted matching of the DAE (5.13)
111
5.3.4. Finding all low index input variables assignments
In section 5.3.2.we have shown how to find the necessary input variables. Let’s call
the set of all necessary input variables as U N0 . This set of the necessary input
variables is a sub set of any found low index input assignment, i.e.: U N0 ⊆ U( M MW ) .
This means to find all low index input assignment we need to find all optional (low
index) input variables, which can be done in the following:
Proposition 5.12. : Finding all low index input assignment
Consider an underdetermined DAE model (1.3a):
S:
F ( z (t ), z (t ), t ) = 0
with m equations and n variables, m < n . Suppose with the Proposition 5.11, we can
find a set of k = n − m low index input variables U0 ( M MW ) , where k1 ≤ k of these
input variables are the necessary input variables, given with the set
U N0 ⊆ U0 ( M MW ) , e.g.: {zi = ui (t ), zi +1 = ui +1 (t ),..., zi + k1 = ui + k1 (t )} . The remaining
found input variables are optional input variables belongs to the matching U0 ( M MW )
is denoted as U O0 = U 0 ( M MW ) \ U N0 .
To find all optional input variables for the low index input variables sets of the DAE
(1.3a), we construct an search branch algorithm, as follows:
1. A subsequence finding new low index input assignment with setting one of the
optional input variables as an input variable:
F ( z (t ), z (t ), t ) = 0 with m equations and
Consider an undetermined DAE set: S :
n variables, m < n . This DAE S becomes fully determined by having a low index
input assignment U 0 = U N0 ∪ U O0 with the number of elements of U N0 is k1 . When
UO0 ≠ ∅ then we can make an underdetermined DAE:
S1* : F ( z (t ), z (t ), t ) = 0
∪ { zi = ui (t ), i = 1,..., k1} ∪ z j = u j (t ), z j ∈ UO0
zi ∈U N0
(5.14)
The equations set (5.14) consists of m + k1 +1 equations with n variables. Let
UC1 = {zi = u(t ), i = 1,... k1 , zi ∈ U N0} ∪ z j ∈ UO0 be the set of chosen input variables
for new branch DAE S1 and further we can extend S1* with n − ( k1 + 1) new equations
of the remaining variables, where each of this equation is given as, zk =
m
∑u
i =1
are dummy variables, such that we can reconstruct:
112
di
, udi
⎡ Es − A
T (s) = ⎢
⎣⎢ I n − ( k1 +1)
0⎤
⎥
K ⎦⎥
and apply Proposition 5.11 to find a new low index input assignment set U1* consists
of
l1 = (n − m) − ( k1 + 1) variables. The complete low index input variables
assignment for S1 is U1 = UC1 ∪ U1* . We can perform test on this branch S1 which
elements of U1 are necessary input variables or optional input variables due to the
choice of U C1 as a subset of U1 , i.e. U1 = U N1 ∪ U O1 . The same procedure of
finding new low index assignment for the next branch can be applied, using
information from U N1 , UO1 and so on. Therefore we called this method as the
subsequence finding new low index input assignment with setting one of the optional
input variables from the previous branch as an input variable..
2. A subsequence finding new low index input assignment with setting one of the
optional input variables, chosen on the other branch, as an internal variable
F ( z (t ), z (t ), t ) = 0 with m equations and
Consider an undetermined DAE set: S :
n variables, m < n . This DAE S becomes fully determined by having a low index
input assignment U 0 = U N0 ∪ U O0 with the number of elements of U N0 is k1 . When
UO0 ≠ ∅ then we can make an underdetermined DAE:
∪ { zi = ui (t ), i = 1,..., k1}
S2* : F ( z (t ), z (t ), t ) = 0
zi ∈U N0
\ z j = u j (t ), z j ∈ U O0
(5.15)
The equations set (5.15) consists of m + k1 equations with n variables, further
we can construct:
⎡ Es − A 0 ⎤
T (s) = ⎢
⎥
⎣⎢ I n − k1 K ⎥⎦
where on the second row of T ( s) , i.e. I n− k1
K , are the equations of remaining
n − k1 variables (which are not chosen as input variables) as functions of dummy
variables, i.e. zk =
m
∑u
i =1
di
. Since the variable z j must be excluded to be an input
variable (chosen to be an internal variable), then we need to delete the corresponding
equation of z j on the I n− k1 K and removing one dummy variable, gives:
113
⎡ Es − A
T * (s) = ⎢
⎢⎣ I n − k1 −1
where K * ∈ R
0 ⎤
⎥
K * ⎥⎦
( n − k1 −1)×( m −1)
and the dummy variable set is {ud1 , ud2 ,..., udm−1 } .
Applying Proposition 5.11 to find a new low index input assignment set U*2 consists of
l2 = (n − m) − k1 variables for S2 .
Let U C2 = {zi = ui (t ), i = 1,...k1 , zi ∈ U N0 } be the set of chosen input variables for
new branch DAE S2 , then the complete low index input assignment for S2
is U 2 = UC2 ∪ U*2 .
Moreover we can perform a test on this branch S2 which elements of U 2 are
necessary input variables or optional input variables, i.e. U2 = U N2 ∪ UO2 . Also from
this branch if U O2 ≠ ∅ , then the same procedure of finding new low index assignment
for the next branch can be applied, using information from U N2 , UO2 and so on.
Therefore we called this method as the subsequence finding new low index input
assignment with setting one of the optional input variables from the previous branch
as an internal variable.
Each step when its optional input set is not empty will create two branches and finding
new low index input assignment with one branch performs (1) and the other branch
performs (2) as described above.
This method to find all low index input assignment sets of a DAE model is
implemented by Y.G.G.J. Go [48] in DIAG as an extension of the graph algorithm of
Bujakiewics [21].
Example 5.7:
Consider the following underdetermined DAE, 3 equations with 6 variables:
S :
z1 = − z1 + z2 − z3
z2 = − z2 + z5 + z6
0 = z2 + z4 + z6
(5.16)
DIAG gives the following family tree of the DAE with low index assignments:
114
S0
S1
S11
S2
S12
Level 1
Level 2
Figure 5.9.
where:
a. S0 = S ∪ U N0 = {z3 = u(t )} ∪ UO0 = {z4 = u(t ), z5 = u(t )}
Level 1:
b. S1 = S ∪ ( U N0 = {z3 = u(t )}) ∪ {z4 = u(t )} ∪ {z5 = u(t )} ,
where {z3 = u(t )} ∪ {z4 = u(t )} are set as the input variables of S1 and {z5 = u(t )}
is found by finding a low index assignment (as described above in Proposition 5.12).
Testing on the found additional input variable set {z5 = u(t )} gives the result that this
variable is an optional input for S1 , i.e. U O1 = {z5} .
c. S2 = S ∪ ( U N0 = {z3 = u(t )}) ∪ {z5 = u(t ), z6 = u(t )}
{z3 = u(t )} is set as the necessary input variable of S2 and
{z5 = u(t ), z6 = u(t )} is found by finding a low index assignment, with a requirement
that the z4 becomes an internal variable of S2 . Testing on the found additional input
variables {z5 = u(t ), z6 = u(t )} gives the result that these variables are necessary input
variables for S2 . Thus the optional input variable set of S2 , U O2 = ∅ . Thus S2 is an
where
end node.
Level 2:
d. S11 = S ∪ {z3 = u(t ), z4 = u(t )} ∪ {z5 = u(t )}
where {z3 = u(t )} ∪ {z4 = u(t )} is already set as an input variable from the previous
node S1 and {z5 = u(t )} is an additional setting of an input variable from the optional
input set of S1 . Since the node S1 has only one element of optional input variable
then S11 is an end node.
115
e. S12 = S ∪ {z3 = u(t ), z4 = u(t )} ∪ {z6 = u(t )}
where {z3 = u(t )} ∪ {z4 = u(t )} is already set as an input variable from the previous
node S1 and {z5 = u(t )} is the input variable {z6 = u(t )} is found by a low index
assignment, with a requirement that the z5 becomes an internal variable of S12 . This
node is also an end node.
Collecting all low index input assignments for the DAE S from the end nodes, gives
the following sets:
U2 = {z3 , z5 , z6}
U11 = {z3 , z4 , z5}
U12 = {z3 , z4 , z6}
The searching algorithm implemented on DIAG will remedy any possible combination
of input variables resulting in a high index structure (when in the DAE model there is
no constraining states equations), even the DAE model has only optional input
variables, i.e. the necessary input variables set U N0 = ∅ , the following example 5.8
shows this case.
Example 5.8:
Consider an underdetermined DAE model, with 3 equations and 5 variables:
z1 = − z1 + z2 + z3
0 = z1 − z2 + z3 − z5
0 = z1 + z2 − z4
(5.17)
DIAG found the following 5 low index input assignments:
U = {{z3 , z4 },{z3 , z4 },{z3 , z5},{z5 , z2 },{z5 , z4 }}
None of these input variables are a necessary input variable, but none of the found sets
give {z2 , z4 } as a low index input assignment, since this will lead to a high index
model.
5.3.5. Example: finding low index input assignments of a binary tray
distillation column model
In this section we will demonstrate the application for finding all low index input
assignments using the tray model presented in Chapter 4. The DIAG algorithm needs
only the structural information of the model, namely: the equation number and the
116
variables on the corresponding equation. The operator “$” is used to replace
d
(.) in
dt
the original DAE equation. This is to give the weight “1” for the maximum weighted
matching of the graph of the DAE model.
A binary tray distillation column model
Consider the behavioural equations of a binary tray distillation column model consist
of the equations (4.69, 4.71, 4.73, 4.74, 4.51 – 4.68, 4.41, 4.42). In this case, for binary
components, the equation (4.69) consists of two component balance equations. The
equations (4.70) and (4.72) are removed, since the variables N i , L , E L ,exc are vanish
from the outputs of the model. The structural information of these equations is written
as:
Input file (structural information):
.eqn f1
.eqn f 2
.eqn f3
.eqn f 4
.eqn f5
= {$M1, M ,Lin ,x1,in ,Vin ,y1,in ,Lout ,x1,out ,Vout ,y1,out }
= {$M 2, M ,Lin ,x2,in ,Vin ,y2,in ,Lout ,x2,out ,Vout ,y2,out }
= {$ EM ,Lin ,hl ,in ,Vin ,hv,in ,Lout ,hl ,out ,Vout ,hv,out }
= {M1,M , M1, L , M1,V }
= {M 2,M , M 2, L , M 2,V }
.eqn
.eqn
.eqn
.eqn
.eqn
= {EM , EL , EV }
= {M L , M1, L , M 2, L }
= {MV , M1,V , M 2,V }
= {M1,L , M L , x1, out }
= {M 2,L , M L , x2, out }
f6
f7
f8
f9
f10
. eqn f 11
. eqn f 12
. eqn f 13
. eqn f 14
. eqn f 15
. eqn f 16
. eqn f 17
. eqn f 18
. eqn f 19
. eqn f 20
. eqn f 21
. eqn f 22
. eqn f 23
. eqn f 24
. eqn f 25
. eqn f 26
. eqn f 27
. eqn f 28
. eqn f 29
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{ M 1,V , M V , y1,out }
{ M 2 ,V , M V , y 2 ,out }
{E L , M L , hl ,out , P,V L }
{EV , M V , hv ,out , P,VV }
{ M L , ρ L ,V L }
{V L , Z L }
{ M V , ρ V ,VV }
{V L ,VV }
{ Pv ,out , P}
{ Pl ,out , P}
{ Lout , Z L }
{Vin , P , Pv ,in , Z L }
{ρ L , Tl ,out , Pl ,out , x1,out , x 2 ,out }
{ρ V , Tv ,out , Pv ,out , y1,out , y 2 ,out }
{hl ,out , Tl ,out , Pl ,out , x1,out , x 2 ,out }
{hv ,out , Tv ,out , Pv ,out , y1,out , y 2 ,out }
{x1,out , x 2 ,out , y1,out , y 2 ,out , Tv ,out , P}
{x1,out , x 2 ,out , y1,out , y 2 ,out , Tv ,out , P}
{Tv ,out , Tl ,out }
and the output of algorithm for finding low index assignment is as follows:
117
OUTPUT
Problem name
Number of set variables
Number of unknowns
Number of equations
= tray.dgn
=0
= 38
= 29
Degree of freedom
=9
Structurally consistent? Yes
Structural rank E = 3
Structural order (dyn.degree of freedom) = 3
Structural index
Low
Low index input assignment:
Assignment [1]: Lin ,Vout , x1,in , x2 ,in , y1,in , y2 ,in , hl ,in , hv ,in , Pv ,in
Assignment [2]: Lin ,Vout , x1,in , x2 ,in , y1,in , y2 ,in , hl ,in , hv ,in ,Vin
Necessary input variables: Lin , Vout , x1,in , x2,in , y1,in , y2,in , hl ,in , hv,in
Compelled internal variables:
M1, M , M 2 , M , M1, L , M1,V , M 2 , L , M 2 ,V , M L , MV , E L , EV , E M , x1,out , x2 ,out , y1,out , y2 ,out ,
VL ,VV , Z L , ρ L , ρV , P , Pl ,out , Pv ,out , hl ,out , hv ,out , Tl ,out , Tv ,out , Lout
Optional input variables: Pv,in , Vin
A wrong choice of input variables for this binary tray distillation column model will
result to a high index model. An example of a such input variables set is:
Lin , Pv,out , x1,in , x2,in , y1,in , y2,in , hl ,in , hv,in , Pv,in . The high index model is caused
due to the choice of Pv,out as an input variable.
5.4.
Maximum dynamic degree input variables assignment
5.4.1. Finding maximum degree input variables assignment
In case the given DAE model has constraining state equations (a high index structure)
then there is no possibility of finding a low index input assignment. In such DAE
model structure, we define the following:
Definition 5.13. A maximum dynamic degree input assignment
Given a DAE model (1.3a): F ( z (t ), z (t ), t ) = 0 with m equations and n variables z , if
the set of k = m − n variables can be set as external independent time functions
variables where the resulting DAE is consistent and has a maximum dynamic degree
of freedom, then these k variables are called a maximum dynamic degree assignment
for the given DAE.
118
Hence, it follows that:
Corollary 5.14.
A low index input assignment is a maximum dynamic degree assignment, i.e. if the low
index criterion for the found input assignment is satisfied.
Further we define a constraining state equations set as follows:
Definition 5.15. A constraining state equations set
A constraining state equations set is a subset of equations in the DAE model by which
the state variables are constrained, i.e. the DAE model has always a high index
structure for any choice of maximum dynamic degree input variables set.
From definition 5.15, we can say,
Proposition 5.16.
If a DAE model has a constraining equations set, then this equations set is invariant
for any maximum dynamic degree input variables assignment.
The detection of a maximum dynamic degree assignment is quite similar with the
detection method to find a low index input assignment. The difference is only that the
low index criterion is not included. This gives by the following:
Proposition 5.17. A maximum dynamic degree input assignment
Consider an underdetermined DAE model (1.3a) F ( z (t ), z (t ), t ) = 0 or can be
written as ⎡⎣ Es − A⎤⎦ z = 0 , where E , A ∈ R m×n , m < n . Let we can extend this
underdetermined system matrix, such that the new system matrix is given as:
⎡ Es − A 0 ⎤
T (s) = ⎢
⎥
K⎦
⎣ In
where : T ( s ) ∈ R n*× n* ( s ), n* = m + n and the second row of T ( s) , I n
additional equations of the form: zi = ud1 + ud2 +
K gives
+ udm , i = 1,2,..., n ; the set these
equations is denoted as F . Further ud1 , ud2 ,..., udm are m new dummy variables
and K ∈ R n×m with struc K has no zero elements.
Let T ( s) be structurally non-singular, and GT is the graph obtained from T ( s) as
defined in Lemma B.2. The vertices of bipartite graph GT are
V1 (GT ) = { f1 , f 2 ,..., f m ,..., f n , f n+1 , f n+m} and V2 (GT ) = {z1 , z2 ,..., zn , ud1 , ud2 ,..., udm } .
Let M MW is the result of a Maximum Weighted Matching algorithm on GT ,
then each M MW gives rise to a maximum dynamic degree input assignment:
l
q
U md ( M MW ) = zi : ∃( f m+i , zi ) ∈ M MW , i = 1,.., n
(5.18a)
119
with W ( MW ) =
max
i ,i =1,..., M MC
{ M MC − i}
(5.18b)
The method to find all maximum degree input assignments for a DAE model having
constraining equations is also quite analogue with the method for finding all low index
input assignments. Again the difference is only the criterion used; for finding all
maximum dynamic degree input assignments, we used the maximum weighted
matching criterion (5.18a, 5.18b).
Example 5.9:
Consider the following DAE model:
f1 : z1 = − x1 + x2 − x3 + x5
f 2 : z2 = − x2 − x4 − x7
f3 : z3 = x2 − x3 + x4 + x6
f 4 : 0 = x1 − z1
(5.19)
f5 : 0 = x2 − z2
f 6 : 0 = x3 − z3
f7 : 0 = x1 + x2
f8 : 0 = x1 + x3 + x7
f9 : 0 = x7 + x8 − x9
This DAE model consists of 9 equations ( f1, f 2 ,..., f9 ) and 12 unknowns
( z1, z2 , z3 , x1, x2 , x3 ,..., x9 ) . There are 3 variables that must be assigned as input
variables of the model. Finding input variables give the following result:
OUTPUT
Problem name
Number of unknowns
Number of equations
= example.dgn
=0
= 12
=9
Degree of freedom
=3
Structural rank E
=3
Structural index
High Index
Maximum dynamic degree input assignment:
Assignment [1]: x6 , x4 , x8
120
With these 4 maximum dynamic degree input assignments, we will analyse the
struc( Es − A) and struc( Es − A) −1 . Figure (a) shows the struc( Es − A) for input
assignment [1]: x6 , x4 , x8 , figure (b) for input assignment [2]: x6 , x4 , x9 , figure (c) for
input assignment [3]: x6 , x5 , x8 and figure (d) for input assignment [4]: x6 , x5 , x9 .
struc( Es − A) =
z1
z2
z3
x1
x2
x3
x5
x7
x9
z1
z2
z3
x1
x2
x3
x5
x7
x8
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
1
o
o
1
o
o
1
1
o
1
1
1
o
1
o
1
o
o
1
o
1
o
o
1
o
1
o
1
o
o
o
o
o
o
o
o
o
1
o
o
o
o
o
1
1
o
o
o
o
o
o
o
o
1
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
1
o
o
1
o
o
1
1
o
1
1
1
o
1
o
1
o
o
1
o
1
o
o
1
o
1
o
1
o
o
o
o
o
o
o
o
o
1
o
o
o
o
o
1
1
o
o
o
o
o
o
o
o
1
(a)
(b)
z1
z2
z3
x1
x2
x3
x4
x7
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
1
o
o
1
o
o
1
1
o
1
1
1
o
1
o
1
o
o
1
o
1
o
o
1
o
1
o
o
1
1
o
o
o
o
o
o
o
1
o
o
o
o
o
1
1
x9
o
o
o
o
o
o
o
o
1
z1
z2
z3
x1
x2
x3
x4
x7
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
1
o
o
1
o
o
1
1
o
1
1
1
o
1
o
1
o
o
1
o
1
o
o
1
o
1
o
o
1
1
o
o
o
o
o
o
o
1
o
o
o
o
o
1
1
(c)
Figure 5.10. struc( Es − A) of the example 5.9.
x8
o
o
o
o
o
o
o
o
1
(d)
The figures (5.10a) and (5.10b) show struc( Es − A) −1 , where for input assignment [1]
and [2] is represented by the figure (5.11a), since they have the same structure of
121
struc( Es − A) and the same case for input assignment [3] and [4] is represented by
the figure (5.11b).
From figure (5.11) we can see that for any maximum dynamic degree input assignment
the equations need to be differentiated (or constraining state equations) to perform a
high index reduction is invariant, namely the 4th, 5th and 7th equations, given on the
power of s on the 4th, 5th and 7th column of the matrix struc( Es − A) −1 .
struc( Es − A) −1 =
z1
z2
z3
x1
o s -1 s -2 1
x2
1
x3
s −1
x5
x7
1
x9 (or x8 )
s -1 o
z1
z2
z3
x1
x2
x3
x4
x7
x9 (or x8 )
s -1 s -2 s -2 s -1 s -1 s −1 s -1 s -2 o
o s -1 s -2 o s −1 s -1 s -1 s −1 o
s -1 s -2 s -2
1
1
o s -2 s -1 o s −1 s −1 s -2 s -2 o
s -1 s -1 s -1
1
1 s −1
o s -1 s -2 o
1
s -1
s -1 o
s -1 s -2 s -2
1 s -1 s -1 s -1 s -2 o
o s −1 s -2 o
1
s -1 s -1 s -1 o
s -1 s −2 s -2
1 s -1 s -1
1
s -2 o
o s -2 s −1 o s -1
1
s -2 s -2 o
s -1 s -1 s −1
1
1
1
1
s -1 o
1
1
1
s -1
1 s −2 o
1
s -1 o
s -1 s
s
1
s
1
o
1
s -1
s
s
1
s
1
o
o s -1 s -1 o
1
1
1
1
o
s -1 s -1 s -1
1
1
1
1
1
o
o s -1 s -1 o
1
1
1
1
1
s -1 s -1 s -1
1
1
1
1
1
1
(a)
1
(b)
Figure 5.11. struc( Es − A) −1 of the example 5.9.
Moreover if we suppress x4 to be an input variable, but as an internal model variable,
then the maximum dynamic degree input assignments are only given by the
assignment [3] and assignment [4]. For these assignments the information of matrix
struc( Es − A) shows a structurally singularity of variable x4 on the lower right part,
as given in the figure (5.12). On such structure it is possible to do a high index
reduction through variable differentiation, as described in Chapter 2, if the variable
x4 is not interested as an output variable of the model.
122
struc( Es − A) =
z1
z2
z3
x1
x2
x3
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
o
o
o
s
o
o
1
o
o
o
1
o
o
1
o
o
1
1
o
1
1
1
o
1
o
1
o
o
1
o
1
o
o
1
o
1
o
x4
o
1
1
o
o
o
o
o
o
x7
o
1
o
o
o
o
o
1
1
x8 (or x9 )
o
o
o
o
o
o
o
o
1
Structurally singular variable
Figure 5.12. Structural singular variable detection of example 5.9
5.4.2. Example: finding maximum dynamic degree input assignments for
a steam reformer model
In this section we will demonstrate the application of DIAG for finding maximum
degree input assignments using the high index methane steam reforming reactor model
presented in Chapter 4. The presented description is as exactly as the input file to
DIAG, namely only the equation numbers and the structural information of these
equations and the operator “$” is used to replace
d
(.) . A high index methane steamdt
reforming reactor model is given by the equations (5.4 – 5.21, 5.1c, 5.2b). The
structural analysis for input variables detection using DIAG gives the following result:
OUTPUT
Problem name
Number of unknowns
Number of equations
= methsteam.dgn
=0
= 30
= 20
Degree of freedom
= 10
Structurally consistent?
Yes
Structural rank E
=6
Structural order (dyn.deg. of freedom) = 5
Structural index:
High Index
Assignment [1]: xCH4 ,in , xCO ,in , xCO2 ,in , xH2 ,in , xH2O ,in , hin , Qadd , Qloss , Fin , Fout
Assignment [2]: xCH4 ,in , xCO ,in , xCO2 ,in , xH2 ,in , xH2O ,in , hin , Qadd , Qloss , Fin , Pout
Assignment [3]: xCH4 ,in , xCO ,in , xCO2 ,in , xH2 ,in , xH2O ,in , hin , Qadd , Qloss , Fout , Pin
Assignment [4]: xCH4 ,in , xCO ,in , xCO2 ,in , xH2 ,in , xH2O ,in , hin , Qadd , Qloss , Pin , Pout
123
Assignment [5]: xCH4 ,in , xCO ,in , xCO2 ,in , xH2 ,in , xH2O ,in , hin , Qadd , Qloss , Fin , rshift
Assignment [6]: xCH4 ,in , xCO ,in , xCO2 ,in , x H2 ,in , x H2O ,in , hin , Qadd , Qloss , rshift , Pout
Assignment [7]: xCH4 ,in , xCO ,in , xCO2 ,in , x H2 ,in , x H2O ,in , hin , Qadd , Qloss , Fout , rshift
Assignment [8]: xCH4 ,in , xCO ,in , xCO2 ,in , xH2 ,in , xH2O ,in , hin , Qadd , Qloss , Pin , rshift
If we suppress the variable rshift as an input variable of the model then we have 4
maximum dynamic degree input assignments [1 – 4]. Suppose we choose input
assignment [2].
The struc( Es − A) is given in the figure (5.13). Here we can see that there is
structural singularity on the rshift . This structural singularity is invariant for input
assignment [1 – 4]. This means that we are able to perform column / variable
differentiation for a index reduction if the variable rshift is not interested as a model
output (see Chapter 4).
Moreover the struc( Es − A) −1 is given on the figure (5.14), where the constraining
states are detected on the 7th to 16th and 20th equations, which are analogue as detected
with input assignment [4] used as the example in Chapter 4.
124
struc( Es − A) =
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 s o o o o o
2 o s o o o o
3 o o s o o o
4 o o o s o o
1 o o
o
o
o
1
o
o
o
o
o
o
1
o 1 o
o
o
o
1
1
o
o
o
o
o
1
o o 1
o
o
o
o
1
o
o
o
o
o
1
o o o
1
o
o
1
1
o
o
o
o
o
1
o o o o
s o o o
o
o
1
o
o
1
1
o
1
o
o
o
o
o
o
o
o
o
o
o
1
1
o o o o
o
o
o
o
o
1
o
o
o
o
o
o 1 o o
o
o
o
o
o
1
o
o
o
o
o
o 1 o
o
o
o
o
o
1
o
o
o
o
o
o o 1
o
o
o
o
o
1
o
o
o
o
o
o o o
1
o
o
o
o
1
o
o
o
o
o
o o o
o
1
o
o
o
1
o
o
o
o
o
o o o o o
o
o
o
o
o
1
o
1
o
o
o
o 1 o o o
o o 1 1 1
o
1
o
1
1
o
o
o
o
o
1
o
o
1
o
1
1
1
o
o
o
o
o o 1 1 1
1
1
1
o
o
o
1
o
1
o
o
o o o
o
o
o
o
o
o
o
o
1
1
o
o o o
o
o
o
o
o
o
o
o
1
o
1
1 1 o
1
1
o
1
o
o
1
o
1
o
o
o 1 1
1
1
o
o
o
o
1
o
o
o
o
5 o o o o s
6 o o o o o
7 1 1 1 1 1
8 1 o o o o
9 o 1 o o o o
10 o o 1 o o o
11 o o o 1 o o
12 o o o o 1 o
13 o o o o
14 o o o o
15 o o o o
16 o o o o
17 o o o o o o
18 o o o o o o
19 o o o o o o
20 o o o o o o
Structurally singular variable
Figure 5.13. struc( Es − A) of the methane steam reforming reactor
Model variables:
1 = M CH4 ,2 = M CO ,3 = M CO2 ,4 = M H2 ,5 = M H2 O ,6 = E ,7 = xCH4 ,8 = xCO ,9 = xCO2 ,10 = x H2 ,
11 = x H2 O ,12 = hout ,13 = rCH4 ,14 = rshift ,15 = M T ,16 = T ,17 = ρ ,18 = P,19 = Pin ,20 = Fout
Input variables: xCH4 ,in , xCO ,in , xCO2 ,in , xH2 ,in , xH2O ,in , hin , Qadd , Qloss , Fin , Pout
125
1
2
1 s
−1
2 s
−1
3 s
−1
4 s
−1
5 s
−1
3
s
−2
s
−1
s
−1
s
−1
s
−1
6 s
−2
7 s
−1
8 s
−1
9 s
−1
10 s
−1
11 s
−1
s
−1
13 s
−1
12
14
1
15 s
−1
16 s
−1
s
−1
18 s
−1
s
−1
20 s
−1
17
19
s
−2
s
−1
s
−1
s
−1
s
−1
s
−2
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
4
s
−2
s
−1
s
−1
s
−1
s
−1
s
−2
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
5
6
s
−2
s
−1
s
−1
s
−1
s
−1
s
−2
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
s
−1
s
−1
s
−1
s
−1
s
−1
8
11
12
13
14
15
−1
−1
−1
−1
−1
16 17 18
s
−1
s
−1
1
1
1
1
1
1
1
1
1
1
s
−1
1
1
1
1
1
1
1
1
1
1
s
−2
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
1
s
−1
s
−1
s
−1
s
−1
s
−1
s
−1
1
1
1
s
s
s
1
−1
s
1
−1
s
−1
s
1
−1
1
−1
1
1
1
s
s
−1
1
1
1
s
1
1
1
1
s
1
1
1
1
1
1
1
1
1
s
1
1
1
1
s
1
1
−1
−1
−1
s
s
−1
10
−1
s
−1
9
s
1
−1
7
−1
1
1
1
s
1
1
s
−1
1
1
1
s
1
1
s
−1
1
1
1
s
−1
1
1
s
−1
1
1
1
s
−1
1
1
1
1
1
1
1
1
1
1
s
−1
1
1
1
1
1
1
1
1
1
1
s
−1
s
−1
1
1
1
1
1
1
1
1
1
1
1
s
−1
s
−1
s
−1
0
s
−1
0
s
−1
0
0
0
0
0
0
s
s
−1
1
s
−1
1
s
−1
1
s
−1
1
s
−1
s
−2
s
−1
s
−1
1
s
−1
s
−1
1
s
−1
1
s
0 s
s
−1
1
0 s
−1
s
−1
1
0 s
−1
s
−1
1
−1
s
−1
1
1
1
1
1
1
1
1
0 s
s
s
s
s
s
s
s
s
s
s
0
s
s
−1
s
−1
1
1
1
1
1
1
1
1
1
s
−1
1
1
1
1
1
1
1
1
1
s
−1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0 s
0 s
−1
0 s
−1
s
−1
1
0 s
−1
s
−1
1
1
1 s
−1
s
−1
1
s
−1
1
1
1
1
1
0
1
s
−1
1
s
−1
1
1
Conclusions
DAE model variables can be classified as the necessary input variables, the optional
input variables, and the compelled internal variables.
The input variables of a DAE model are determined by its internal DAE model
equations. The input variables are those model variables that utilise the maximum
dynamic degrees of the model and rise to causal input-output relations. A low index
input variables set is causal input variables for a DAE model. The low index input
variables set for a DAE model does not need to be a unique set.
The constraining state equations within a DAE model are invariant for all possible
input variables sets resulting in the maximum dynamic degree of freedom of the DAE
model.
126
s
−1
Figure 5.14. struc( Es − A) −1 of the methane steam reforming reactor
5.5.
s
−1
0
1
−1
s
−1
1
s
20
−1
1
−1
19
−1
−1
Chapter 6: Interconnection of DAE models
Interconnection of DAE sub models consist of the interconnected variables on
the model ports and the model connectors. The interconnected variables can
be a pair of bilaterally coupled variables or a non bilaterally coupled
variables. The model connector is a linear equality equations. The
interconnected variables of DAE sub models must be matched for the variable
type, variable unit of measurement, and its directionality. A matched
interconnection of low index DAE sub models resulting to a low index DAE
model.
‘A connector has no behavioural description’
6.1.
Introduction
Most kinds of modelling and simulation approaches for (complex) physical systems
are done by interconnecting sub-models. This has several advantages: it is seen as a
‘natural’ modelling of interconnected sub systems of a physical process, it can re-use
modular models of sub systems, and it may reduce the complexity of a physical
modelling process. Although modularisation and interconnection between sub models
have been implemented in different modelling and simulation software packages there
are several essential different concepts about: what is a model connector and how to
interconnect sub models.
There are two views of the model connectors. Marquardt [89] defines a connector as
a ‘fluxing’ relation (see figure 6.1). The definition of a connector as a ‘fluxing’
relation is analogue with the approach of the bilateral coupling of sub models
(Breedveld [15,16]). In this approach a model connector is described with a behaviour
equation, i.e. a fluxing relation. Another definition of a model connector, e.g. in Aspen
[5], gProms [50] and Hysis [61], is given as an ‘equality’ relation. A model
connector relates a variable from one sub model to the other variable from other sub
model in an equality equation.
Interconnection in several of the modelling and simulation tools require sub models
with a fixed set of input and output variables, by which the sub models can be
interconnected to each other, e.g. Simulink [113], and others not, e.g. Aspen [5],
gProms [50], Hysis [61].The disadvantage of modelling with fixed input and output
variables is the possibility of an unnecessary requirement to introduce a dynamic
component between two interconnected sub models, i.e. when algebraic loops occur.
The disadvantage of without knowing the matching of possible input and output
variables of interconnected sub models is a potential arising of a high index problem
due to interconnection.
127
Chapter 6 : Interconnection of DAE models
Sub-system 1
x1
x2
φ1
φ2
Sub-system 2
state information
flux information
fluxing relation, φ1,2:=f(x1,x2)
“......complementary to a device, a connection responds to state information with flux
information.”(Marquardt [89])
Figure 6.1. Schematic representation of a fluxing connection (Marquardt [89])
Recently Aspen [5] and Hysis [61] applied a pressure and flow checker on the
interconnection of sub models. The modeller should choose if the total model is a
pressure driven or a flow driven. A pressure driven model means that the pressure will
be an external/input variable specified outside the model boundary. A flow driven
model means that the flow will be an external/input variable outside the model
boundary. Aspen [5] and Hysis [61] provide a check through the interconnected sub
models when the pressure and flow variables can be solved without pressure states
constraint, i.e. the pressure variables on two interconnected sub models are not the
compelled variables. This approach forms a part of the solution to avoid a high index
problem on a pressure-flow interconnected sub models.
Motivated by the different definitions of model connectors and the necessity to avoid a
high index problem due to interconnection of sub models, this Chapter 6 will describe
an approach for model connection definition and interconnection matching of sub
systems. The goal is to get a match of interconnected structural low index sub models,
where the resulting total model has a structurally low index. This Chapter is outlined
as follows, section 6.2. describes some type of model ports and their variables
classifications, section 6.3 gives a description of model connectors, section 6.4. gives
aspects on interconnection of sub models, and section 6.5 gives some conclusions.
6.2.
Process model ports
6.2.1. Process model ports
A process model port or model (variables) connection is a model boundary where the
chosen sub set of the model variables passes through this boundary to interact with its
environment. A process model can have several model ports, where different groups of
variables go through a model port. Figure 6.2 schematically shows a process model
with several model ports.
128
Port 3
Po
rt
4
Port 5
Port 1
2
rt
o
P
Model
Po
rt
8
Port 7
6
rt
o
P
Figure 6.2 Schematic model ports
A model port can be classified into the physical domain of variables through this port.
Example of such classifications are.:
a) Material port, e.g. contains information of : material flow, material
temperature, material pressure, material component fraction, material specific
enthalpy.
b) Mechanical port, e.g. contains information of : force, acceleration, velocity,
position, torque, angular acceleration, angular velocity, angular position.
c) Energy port, e.g. contains information of : power, heat flow, entropy flow,
temperature.
d) Information/signal port, e.g. contains information for the controller input and
output variables.
Moreover if necessary, each port can be indicated as an ‘inlet’ or ‘outlet’ port, e.g. an
inlet material port, an outlet material port. Figure 6.3 schematically shows an example
of a model with inlet and outlet material ports.
Inlet material port
Outlet material port
Fin (mass flow inlet)
Pin (pressure inlet)
Tin (temperature inlet)
Fout (mass flow outlet)
Model
hin (specific enthalpy inlet)
Pout (pressure outlet)
Tout (temperature outlet)
hout (specific enthalpy outlet)
Figure 6.3. Example of a model with inlet and outlet material ports
6.2.2. Model variables classifications on a model port
Mathematically the variables through a model port can be distinguished into two
classes (O.Bosgra [13], Breedveld [15, 16]), namely:
a) pair bilaterally coupled variables
b) non bilaterally coupled variables
129
The bilaterally coupled variables consist of a pair ‘flow (f)’ and ‘effort (e)’ quantities
with different directionalities (i.e. one as input and other as output) through a model
port. The non bilateral coupled variables do not form a pair of variables going through
a model port.
A model port can consist of a pair of bilaterally coupled variables only or non
bilaterally coupled variables only or a combination of both type of variables. Table 6.1
shows the possible pairs of ‘flow (f)’ and ‘effort (e)’ for different physical domains,
adopted from Breedveld [15, 16].
Physical Domain
Hydraulic/pneumatic/material
flow
Mass transfer
Flow (f) variable
F (flow rate)
Effort (e) variable
P (pressure)
x (mass fraction)
Mechanical translation
Mechanical rotation
Thermal/energy transfer
N (mass fraction flow
rate)
v (velocity)
ω (angular velocity)
T (temperature)
Electromagnetic
Chemical
i (current)
FN (diffusion flow)
f (force)
Γ (torque)
S (entropy flow), Q
(heat flow)
V (voltage)
μ (chemical potential)
Table 6.1. Bilaterally coupled flow and effort variables
The directionalities of the ‘flow’ and ‘effort’ variables on a model port depends on the
transport equation(s) described within the model corresponding to the particular model
port. A model can have a model port with:
- a ‘flow’ variable as an input variable and an ‘effort’ variable as an output
variable. We can call this ‘a flow driven model port’, or
- an ‘effort’ variable as an input variable and a ‘flow’ variable as an output
variable. We can call this ‘an effort driven port’, or
- either a flow driven or an effort driven port, i.e. the directionalities of the flow
and effort variables are not fixed (as long as one is an input and the other is an
output variable).
Mathematically a DAE model variables can be classified into (see Chapter 5):
a) compelled internal variables, i.e. treating them as inputs will lead to a high
index model,
b) optional input variables, i.e. these variables can be chosen as input variables
and don’t have an affect on the index of the model,
c) necessary input variables, i.e. these variables are the necessary input variables,
i.e. as elements for all low index input assignment of the model.
The following example 6.1 of a binary tray distillation model gives the relation
between the physical and mathematical model variables classification.
130
Example 6.1. Variable classifications on model ports of a binary tray model
Given a binary tray distillation column model as described in Chapter 4, section 4.2.2.
consists of equations (4.69, 4.71, 4.73, 4.74, 4.51 – 4.68, 4.41, 4.42). We have seen in
Chapter 5, section 5.3.5 that this model gives 2 sets of a low index input assignment:
Assignment [1]: Lin ,Vout , x1,in , x2 ,in , y1,in , y2 ,in , hl ,in , hv ,in , Pv ,in
Assignment [2]: Lin ,Vout , x1,in , x2 ,in , y1,in , y2 ,in , hl ,in , hv ,in ,Vin
The binary tray model has the following variables classifications:
a) Necessary input variables: Lin , Vout , x1,in , x2,in , y1,in , y2,in , hl ,in , hv,in
b) Compelled internals:
M1, M , M 2 , M , M1, L , M1,V , M 2 , L , M 2 ,V , M L , MV , E L , EV , E M , x1,out , x2 ,out , y1,out , y2 ,out ,
VL ,VV , Z L , ρ L , ρV , P , Pl ,out , Pv ,out , hl ,out , hv ,out , Tl ,out , Tv ,out , Lout
Optional input variables: Pv,in , Vin
Figure 6.4a and 6.4b show the 4 material ports of the binary tray distillation model,
with the corresponding variables classifications through these ports.
(*)
Pv,out
Vapour material
outlet port
(**)
Vout
y1,out
y2,out
(**)
hv,out
Lin
x1,in
x2,in
hl,in
x2,out
hl,out
A binary tray distillation model
Vapour material
inlet port
Pv,in
Vin
(*)
y1,in
y2,in
hv,in
Lout
(**)
(*) = Bilaterally coupled variables
(**) = Non bilaterally coupled variables
x1,out
(**)
u
= Necessary input variable
u
= Optional input variable
z
= Compelled internal variable
Figure 6.4a. A binary tray model ports
131
(*)
Pv,out
Vapour material
outlet port
(**)
Vout
y1,out
y2,out
(**)
hv,out
Lin
x1,in
x2,in
hl,in
Liquid material
inlet port
A binary tray distillation model
Vapour material
inlet port
Vin
Pv,in
y1,in
(*)
y2,in
hv,in
Lout
(**)
x1,out
x2,out
Liquid material
outlet port
hl,out
(**)
(*) = Bilaterally coupled variables
(**) = Non bilaterally coupled variables
u
= Necessary input variable
u
= Optional input variable
z
= Compelled internal variable
Figure 6.4b. A binary tray model ports
The vapour material outlet port is an example of a flow driven model port and the
vapour material inlet port is either a flow or pressure (effort) driven model port.
The combination of optional input variables can be on one model port, but it can also
relate to another model port. This depends on the model equations.
Example 6.2. A valve model
Given a simplified valve model:
Fout = Cv . ( Pin − Pout
Fin = Fout
For the choice of a flow driven inlet port then its outlet port must be a pressure driven
port (see figure 6.5a). For the choice of a pressure driven inlet port then its outlet port
can be either pressure driven or flow driven (see figure 6.5b).
Pin
Pout
Pin
Pout
Fin
Fout
Fin
Fout
Pin
Pout
Fin
Fout
Figure 6.5a
Figure 6.5b
In principal for a given DAE model, we can make model port’s sets which include
information of the variable classifications on each model port. This will be very useful
for the re-use and interconnection of a model with the other models.
132
6.3.
Process model connectors
6.3.1. Process model connectors
The function of a process model connector is only to pass information or quantity from
(one) sub model(s) to other sub model(s). There is no process of change of this
quantity on the model connector itself. Therefore a model connector does not describe
any behaviour of a system. Mathematically interconnection of sub models can be
formulated as follows,
Proposition 6.1. a model connector
Given N physical sub models, each of them can be described with a DAE sub model
i = 1...N .
Fi ( zi (t ), zi (t ), t ) = 0 ,
with
Suppose
zic ,
with
(1.3a):
zic ⊂ zi ,
i = 1...N , are the subset of interconnected variables for each sub model,
then the model connector between the DAE sub models can be expressed as ‘a physical
consistent equality equation’ :
Lzic = 0
(6.1)
where:
a) physical consistent means that:
- the interconnected variables are of the same type
- the interconnected variables have the same unit of measurement
- the interconnected variables have the same variable value range
b) the matrix L is having entries 1,-1,0 depending on the interconnected
variables zic
A model connector can have various manifestations. Figure 6.6. shows schematic of a
model connector. This kind of model connector can be called ‘a N-branches
summation model connector’.
Moreover figure 6.7. shows an application of a model connector as shown in figure
6.6. This is a flow distribution system supplied by 2 pumps in parallel to three separate
pipes.
133
Sub model
1
Sub model
2
Z1
Zi+1
Z2
Zi+2
Sub model
i+1
∑
Zi
ZN
Sub model
N
Sub model
i
Note: zi + 2 represents an external signal acting on a model connector
Figure 6.6. N branches summation model connector
Fin _ valve1
Valve 1
Fout _ pump1
Fin _ valve 2
Pump 1
Valve 2
Fout _ pump 2
Fin _ valve3
Pump 2
Valve 3
Figure 6.7. Example of a process model connector
The interconnection equation for the flow variables is given by the equation:
Fout _ pump1 + Fout _ pump 2 = Fin _ valve1 + Fin _ valve 2 + Fin _ valve3
From Proposition 6.1, it can derived 3 types of model connectors described here
follows,
Proposition 6.2. Equality interconnection
N interconnected DAE sub models of (1.3a) has an equality type interconnection when
the interconnected variables zic have the same quantity or value, i.e.:
z1 = z2 = ... = zi ,
i = 1,...N with i is the sub model number and zi is the
interconnected variable between the sub models.
This type of model connector is called ‘a N-branches equality connector’.
Schematically the equality interconnection is given on figure 6.8.
134
Sub model
1
Z1
=
Z2
=
Z3
=
ZN
Sub model
2
Sub model
3
Sub model
N
Figure 6.8. N branches equality model connector
The variables on the interconnected sub models can be stacked as a vector of variables.
This is to be applied on the equality stream connection in Aspen [5]. Figure 6.9.
schematically shows the vector variables interconnection between two sub models,
where zij are the interconnected variables with i = for sub model 1 or 2 and j the
variable from corresponding sub model.
Sub model
1
z11
z12
z13
z1j
=
=
=
=
=
=
z21
z22
z23
Sub model
2
z2j
Figure 6.9. A vector equality model connector
Each equality connector forms ‘a 2-branches node connector’.
The second and third type of process model connectors derived from Proposition 6.1 is
a mixer or a splitter connector,
Proposition 6.3. A Mixer or a splitter connector
N interconnected DAE sub models of (1.3a) have a mixer or a splitter type
interconnection when an interconnected variable zic from a sub model is composed of
sum of other interconnected variables zic from other sub models, i.e.:
0 = z1 + z2 + ... + zi −1 − zi + zi +1 + ... + z N with i is the sub model number and zi
is the interconnected variable between the sub models.
135
A mixer connector and a splitter connector are schematically given on figure 6.10a.
and figure 6.10b. respectively.
Sub model
1
Sub model
2
Z1
Z2
Zi
∑
Sub model
i
Sub model
i
Zi
Z1
Sub model
1
Z2
Sub model
2
∑
ZN
Sub model
N
Sub model
N
ZN
Figure 6.10a. A mixer connector
Figure 6.10b. A splitter connector
The Proposition 6.1. gives a framework of a process model connector, where
physically there is no process of change within this connector. In some cases there are
confusions between a model connector and a sub model description, e.g. between a
mixer connector and a mixer model. In a mixer model the property of material to be
mixed can change, example the specific enthalpy of the outlet stream from a mixer
model is changed with respect to each inlet stream specific enthalpy. Thus a mixer
model should not be categorized as a model connector.
6.3.2. Process model connectors for bilaterally coupled variables
In section 6.2.2. is described that bilaterally coupled model port variables consists of a
pair ‘effort (e)’ and ‘flow (f)’ variables. Sometimes the ‘effort’ variable is called as the
‘potential’ variable. Each branch of a model connector, connecting bilaterally coupled
sub models, shall also consist of this pair ‘effort’ and ‘flow’ variables. The relations on
a bilaterally coupled model connector follows the ‘Generalized Kirchoff’s 1st Law’,
namely:
‘on a node with N branches, the sum of ‘flow’ variables is zero’
This follows the model connector relation (6.1), i.e. : Lzc = 0 , and the potential or
effort variables value on a given node is equal to all directions.
The schematic figures of bilaterally coupled model connectors are shown in figure
6.11 and figure 6.12a, 6.12b. The variables ei , i = 1,..., N represent the interconnected
‘effort’ variables and the variables fi , i = 1,..., N are the interconnected ‘flow’
variables. On each branch of a bilaterally coupled model connector, the ‘effort’ and
‘flow’ variables shall not have the same direction.
136
Sub model
1
f1
e1
Sub model
2
f2
e2
ei+1
fi+2
Lf = 0
∑
ei+2
=
fN
fi
Sub model
i
Sub model
i+1
fi+1
eN
ei
Sub model
N
e1 = e2 = …= eN
Figure 6.11. A bilaterally coupled model connector
ein
fin
=
=
eout
ein
fout
fin
=
=
eout
fout
Figure 6.12a. Equality bilaterally coupled connectors
fout1
fin
ein
∑
=
eout1
fin1
ein1
fin2
fout2
eout2
ein2
foutn
eoutn
finn
einn
∑
=
fout
eout
Figure 6.12b. A possible directionalities on a splitter and a mixer bilaterally coupled
connectors
Since the potential variables values on a node/model connector is equal to all
directions, i.e.: e1 = e2 = ... = eN , then it follows that:
Proposition 6.4. Directionality of effort variables on a model connector
For a node or model connector with N branches, then the effort variables quantities
are determined only by the effort value of one of the branches.
Figure 6.12b shows the examples schematic of Proposition 6.4. The variables
ein = eout1 = ... = eoutn = ek and eout = ein1 = ... = einn = ek , i.e. all effort variables
on the node can be replaced with one variable ek .
137
6.4.
Interconnection of sub models
6.4.1. Mathematical formulation of interconnected sub models
Suppose we have an interconnected DAE sub models as shown on figure 6.13; where
the input variables are chosen by the modeller, u = [ u1
u1
u2
Sub model
1
Sub model
2
z1c
z2c
zic
ui
Sub model
i+1
zi +1c
u2
ui
T
uc ] .
ui +1
uc
L
zNc
Sub model
i
Sub model
N
uN
Figure 6.13. Interconnected sub models
Mathematical formulation of interconnected DAE sub models as shown in figure 6.13
can be formulated as follows,
Proposition 6.5. Interconnected DAE sub models – 1
Consider N underdetermined DAE sub-models. Each sub-model is described with a
DAE model (1.3a) having system matrix Ti ( s ) , with mi equations and ni variables
T
z i , mi < ni . Let z = ( z1 z 2 " z i z N ) is a vector of n = n1 + n2 +"+ n N
model variables and m = m1 + m2 +" mN is the total number of model equations.
Each DAE sub model has a low index input assignment.
Suppose there are nuT variables that can be specified as external input variables
u = [ ui
T
uc ] . With nu is the number of specified variables ui (the external input
variables acting on these interconnected DAE sub models) and nuc is the number of
specified variables uc (the external input variables acting on the model connectors),
then nuT = nu + nuc . The number of total external input variables is
nuT = n − m − mc , where mc is the number of the model connector equations.
138
Let mc equations of Lz c + uc = 0 be the set of model connector equations for these
interconnected sub models with nc variables z c ⊂ z \ u and nuc variables uc .
T
Suppose z = [ zi
zc ] , then we can write these interconnected sub-models as:
⎡ Pi ( s ) Pc ( s ) ⎤ ⎡ zi ⎤ ⎡ Bi
=
⎢ 0
L ⎥⎦ ⎢⎣ zc ⎥⎦ ⎢⎣ 0
⎣
0 ⎤ ⎡ ui ⎤
I ⎥⎦ ⎢⎣ uc ⎥⎦
(6.2)
The relation (6.2) consist of m + mc equations and m + mc variables, where it applies
that n − nuT = m + mc . Pi ( s ), Pc ( s ) are polynomial function of ‘s’. The structural
index property of the DAE (6.2) can be analysed using the theory as described in
Chapter 3. Example:
index( struc Tc ( s)) = max ij {0, (deg det struc S ji ( s ) − deg det struc T ( s ) + 1)}
⎡ Pi ( s ) Pc ( s ) ⎤
L ⎥⎦
⎣ 0
where : Tc ( s ) = ⎢
Example 6.3. an interconnected valves and a gas buffer model
Figure 6.14 shows interconnected 2 valves models and 1 gas buffer model. They are
interconnected with model connectors as shown L1 and L2. The model ports and a
connector port are given as: 1,2,2’,3,4,5,6,7. Each port connected variables Fi , Pi (gas
mass flow rate and pressure) from the sub models, where i=1,..., 7 according to the
ports number. Thus the interconnected sub models have 17 variables, consisting of 16
ports variables Fi , Pi and 1 pressure hold up variable from the gas buffer. The
simplified interconnected model equations are given as follows:
Gas buffer:
Valve 1:
Connector L1:
dP R.T
=
( F2 ' − F3 − F4 )
dt
V
P = P2 '
F2 = Cv1 P1 - P2
F2 = F2 '
P2 = P2 '
Connector L 2 :
P = P3
P = P4
F1 = F2
Valve 2:
F2 = Cv2 P5 - P6
F5 = F6
F4 = F5 + F7
P4 = P5
P4 = P7
The model parameters are: R, T , V , Cv1, Cv 2 . There are in total 13 model equations
with 17 variables. Thus we need to specify 4 input variables on the model boundaries
to solve these interconnected sub models.
139
3
F3
P3
1
F1
P1
2
valve1
F2
P2
L1
2'
F2'
P2'
Gas buffer
5
4
F4
P4
L2
F5
P5
6
valve2
F6
P6
7
F7
P7
Figure 6.14. Interconnection of valves and a gas buffer
Using the structural graph algorithm as described in Chapter 3, we can test that the
following input variables sets will give a structural low index interconnected DAE sub
models: ( F3 , F7 , P1, P6 ) or (F3 , F7 , F1, F6 ) or (F3 , F7 , F1, P6 ) or
(F3 , F7 , P1, F6 ) .
6.4.2. Finding input signals for interconnected DAE sub models
Theoretically we can find input signals for interconnected DAE sub models by using
the extension of Proposition 5.11, i.e. by adding the interconnection equations and
searching to the input variables of the interconnected sub models. This is described as
follows:
Proposition 6.6. Interconnected DAE sub-models-2
Consider N underdetermined sub-models. Each sub-model is described with a DAE
model (1.3a) having system matrix Ti ( s ) , with mi equations and ni variables z i ,
mi < ni . Let z = ( z1
T
z 2 " z i z N ) is a vector of n = n1 + n2 +"+ n N model
variables and m = m1 + m2 +" mN is the total number of model equations. Further
let Lz c = 0 is the set of model connector equations for this system with mc equations
and nc variables z c ⊂ z ; where mc ≤ n − m . We can write these interconnected submodels as:
140
m
mc
n
⎡ diag (Ti ( s ))
⎢
⎢
⎢
0
⎢
⎢
⎢⎣
In
0 ⎤⎛ z ⎞
⎥⎜ ⎟
⎥⎜ ⎟
0 ⎥ ⎜ zc ⎟ = 0
⎥⎜ ⎟
⎥⎜ ⎟
K ⎥⎦ ⎜⎝ u d ⎟⎠
0
L
0
n
nc
(6.3)
m+mc-nc
with:
⎡ diag (Ti ( s ))
⎢
⎢
0
Tc ( s ) = ⎢
⎢
⎢
⎢⎣
In
0⎤
⎥
⎥
0 ⎥,
⎥
⎥
K ⎥⎦
0
L
0
Ti ( s ) = ( Ei s − Ai )
the second row gives the additional interconnection equations and the third row of
, where
(6.3) gives additional equations of the form zi = ud + ud + " + ud
1
2
m + mc − nc
ud1 , ud2 ," , udm+m −n are new dummy variables.
c c
Analogue with Proposition 5.11 in Chapter 5 (for finding a index input variables set),
let GT is the graph obtained from Tc ( s ) and let M MW is the result of a maximum
weighted matching algorithm, then each of M MW gives rise to a maximum dynamic
degree input assignment.
Further let GEi be the graph for Ei and let M MCi be the result of a maximum
cardinality-matching algorithm on GEi , then this input assignments is a low index
input assignment if W ( MW ) =
N
∑M
i =1
MC i
.
Thus if a low index input assignment set can be found for the interconnected sub
models, then the model connector Lz c = 0 does not form a constraint for the state
variables.
Suppose for the individual DAE sub model we can find a low index input assignment,
but for the interconnected DAE sub models there is no low index input assignment set,
then part or all of the model connector of Lz c = 0 forms constraint equations for the
state variables on the interconnected DAE models.
141
Example 6.4. Interconnected tank model
This example gives an illustration of the states constraint equations due to
interconnection equations. The interconnected tanks model in Chapter 4, section 4.5 is
used. The model equations are given (4.163a – 4.172) as follows:
Tank 1:
dM1
= Fin − Fout1
dt
M1 = ρ A1H1
P1 = ρ gH1
Fin = f1 ( Pin , P1 )
(4.163a)
(4.164)
(4.165)
(4.166)
Tank 2:
dM 2
= Fin 2 − Fout
dt
M 2 = ρ A2 H 2
P2 = ρ gH 2
(4.167a)
(4.168)
(4.169)
(4.170)
Model connectors:
Fout1 = Fin 2 (model connector equation)
(4.171)
P1 = P2
(4.172)
(model connector equation)
Each tank model has the following variables classifications for a low index input
assignment,
Tank model 1 :
- Compelled internal variables = {M1, P1, H1}
-
Optional input variables = {Fin , Pin }
-
Necessary input variables = {Fout1}
Tank model 2 :
- Compelled internal variables = {M 2 , P2 , H 2 }
-
Optional input variables = {Fout , Pout }
-
Necessary input variables = {Fin 2 }
Applying Proposition 6.4. and using the graph algorithm as described in
Chapter 5 for finding index input sets of the interconnected tank models, give the
following results:
142
Output:
Problem name
Number of unknowns
Number of equations
= tanks.dgn
=0
= 12
= 10
Degree of freedom
=2
Structural rank E
=2
Structural index : High Index
Assignment [1]: Fin , Fout
Assignment [2]: Fin , Pout
Assignment [3]: Pin , Fout
Assignment [4]: Pin , Pout
We see that no low index input assignment can be found, i.e. the interconnected submodels has an internally high index or the interconnection equation introduces a high
index structure. As we expected, if the interconnection equations are the cause of the
high index structure then for any maximum dynamic degree input assignment the
interconnection equations belong to the required equations to be differentiated for a
high index reduction. To demonstrate this case we chose the input assignment [1] for
the input variables. The structure of struc( Es − A) and struc( Es − A) −1 are given as
follows:
struc( Es − A) =
1
2
3
4
5
6
7
8
9
10
1
s
1
o
o
o
o
o
o
o
o
2
o
o
o
o
s
1
o
o
o
o
3
o
1
1
o
o
o
o
o
o
o
4
o
o
o
o
o
1
1
o
o
o
5
o
o
1
1
o
o
o
o
o
1
6
o
o
o
o
o
o
1
1
o
1
7
1
o
o
o
o
o
o
o
1
o
8
o
o
o
o
1
o
o
o
1
o
9
o
o
o
1
o
o
o
o
o
o
10
o
o
o
o
o
o
o
1
o
o
1
2
3
4
5
6
7
8
9
10
1
s-1
s-1
s-1
s-1
s-1
s-1
1
1
s-1
s-1
2
1
1
1
1
1
1
s
s
1
1
3
1
1
1
1
1
1
s
s
1
1
4
o
o
o
o
o
o
o
o
1
o
5
s-1
s-1
s-1
s-1
s-1
s-1
1
1
s-1
s-1
6
1
1
1
1
1
1
s
s
1
1
7
1
1
1
1
1
1
s
s
1
1
8
o
o
o
o
o
o
o
o
o
1
9 10
s-1 1
s-1 1
s-1 1
s-1 1
s-1 1
s-1 1
1 s
1 s
s-1 1
s-1 1
where column wise of struc( Es − A) gives the following variables:
1 = M1 ,2 = M 2 ,3 = H1 ,4 = H2 ,5 = P1 ,6 = P2 ,7 = Fout 1 ,8 = Fin 2 ,9 = Pin ,10 = Pout
The result of the matrix struc( Es − A) −1 shows that the 2nd, 3rd, 6th, 7th, 10th model
equations (i.e. equations (4.164), (4.165), (4.168), (4.169), (4.172)) need to be
differentiated for a high index reduction, which is the same as we found in Chapter 4,
section 4.5 for a different input assignment. We also see that the interconnection
equation 10th or (4.172) belongs to the set of differentiated equations for a high index
143
reduction. Since each tank model has a low index structure then in this case the
interconnection causes the high index structure.
Implementation of Proposition 6.6. for finding input signals face a practical problem
for large scale interconnected DAE sub models (e.g. 200 equations with 220 variables)
that consist of more than 90% algebraic variables. The algorithm will find a lot of
possible input assignments sets, with some of the chosen variables are algebraic
internal variables. In the following section, it will be described a matching
interconnection of DAE sub models by the use of possible information of model ports
variables classifications. The remaining possible input variables at the boundaries of
interconnected sub models can be chosen for the input variables to solve the model.
Additional structural check following Proposition 6.5 can be done to ensure that the
chosen input variables set and the interconnection does not give a structural high index
model.
6.4.3. Interconnection matching
For each DAE sub model, we can find all possible low index input assignment sets and
proceed variables classifications on each model port. Thus for each model there are
sets of possible input variables on its model ports.
Proposition 6.7. Interconnection matching
Interconnection matching of given interconnected DAE sub models means that for
each model connector of the form Lzc = 0 does not form a constraint equation for the
states or compelled internal variables of the interconnected model ports of the DAE
sub models.
This means that:
a) a matching connector for ‘a N-branches summation model connector’ shall
have one connector output or one (optional) input variable to a DAE sub
model (see figure 6.15a),
b) a matching connector for ‘a N-branches equality model connector’ shall have
only one connector input or one (compelled) output variable from a DAE sub
model (see figure 6.15b).
Sub model
1
Sub model
2
Z1
Z2
Zi+2
∑
Zi
Sub model
i
Sub model
i+1
Zi+1
ZN
Sub model
N
Figure 6.15a. a summation connector with zi +1 as an (optional) input variable
144
Sub model
1
Sub model
2
Z1
Zi+1
Z2
Zi+2
Sub model
i+1
=
Zi
Sub model
i
ZN
Sub model
N
Figure 6.15b. an equality connector with zi +1 as a model output variable
When all N sub models have interconnected optional input/output variables, e.g. all N
sub models are valve models, then the choice for the interconnection matching are free.
Example 6.5.
See figure 6.14. Given interconnection of 2 valves and a gas buffer models. The
interconnected variables on the model connector L2 are P4 , P5 , P7 , F4 , F5 , F7 .
Since P4 = P , the pressure on the buffer model is a state variable, then P4 is a
compelled internal variable. Therefore the pressure on this connector L2 will be
determined by P4 , i.e. no other pressure variable can be set.
Besides the Proposition 6.7, practically the interconnection matching of sub models
shall also satisfy the following criteria for the interconnected variables:
- the variables shall have the same variable type (e.g. real, integer)
- the variables shall have the same unit of measurement
- the variables shall have the same variable range (this is to check the validity
range of the interconnected sub models)
Since the interconnected sub models have low index input assignment sets and they are
match interconnected, i.e. there will be no state variables constraint equations due to
the interconnection, then the total interconnected sub models shall have a structural
low index input assignment set.
Proposition 6.8. Low index interconnected sub models
A model consisting of interconnected low index sub-models will have a low index
assignment if the interconnected connector’s are matched.
The input variables of interconnected DAE sub models are the remaining low index
free input variables on the model boundaries or on the model connectors.
145
Example 6.6. Interconnection matching of binary trays of distillation column models
See figure 6.3a and figure 6.3b on the example 6.1. We can make a match
interconnection between two trays of distillation column as shown in the figure 6.16a
and figure 6.16b.
Pv,out
Vout
y1,out
y2,out
hv,out
Lin
x1,in
x2,in
hl,in
A binary tray-2 distillation model
Pv,in
=
Pv,out
Vin
=
Vout
y1,in
=
y1,out
y2,in
=
y2,out
hv,in
=
hv,out
Lout
x1,out
x2,out
hl,out
=
Lin
=
x1,in
=
x2,in
=
hl,in
x2,out
hl,out
Pv,in
Vin
y1,in
y2,in
hv,in
Lout
x1,out
Figure 6.16a. An interconnection matching of 2 trays of distillation column
Pv,out
Vout
y1,out
y2,out
hv,out
Lin
x1,in
x2,in
hl,in
Pv,in
=
Pv,out
Vin
=
Vout
y1,in
=
y1,out
y2,in
=
y2,out
hv,in
=
hv,out
Lout
x1,out
x2,out
hl,out
=
Lin
=
x1,in
=
x2,in
=
hl,in
x2,out
hl,out
Pv,in
Vin
y1,in
y2,in
hv,in
Lout
x1,out
Figure 6.16b. An interconnection matching of 2 trays of distillation column
146
6.4.4. Example of application on sub-models interconnection of a
distillation column model
Vin, Pvin
Qcond
Lreflux
Total Condensor
Model
Ltop, Pouttop
Tray Model
Fin, z1in,z2in,hfin
Feed Tray Model
Qreboil
Reboiler Model
Lbot, Poutbot
Figure 6.17. A schematic of a distillation column model
Tray model and feed tray model:
The tray model used in this example is as given in Chapter 4, section 4.2.2. This is
applied for a binary components system. The feed tray model is made through a
slightly small modification of this tray model on the components and energy balances.
It consists of 29 equations with 42 variables, given as follows:
dM1m
= Fin z1in + Lin x1in + Vin y1in − Lout x1out − Vout y1out
dt
dM 2m
= Fin z2in + Lin x2in + Vin y2in − Lout x2out − Vout y2out
dt
dEm
= Fin h fin + Lin hlin + Vin hvin − Lout hlout − Vout hvout
dt
M1m = M1L + M1V ;
M 2m = M 2 L + M 2V
Em = EL + EV
M L = M1L + M 2 L ;
MV = M1V + M 2V
M1L = M L x1out ;
M 2 L = M L x2out
M1V = M L y1out ;
M 2V = M L y2out
147
EL = M L hlout − PVL ;
EV = MV hvout − PVV
M L = ρ LVL ;
MV = ρV VV
VL + VV = Vtray
VL = Aeff .Z L
Plout = P;
Pvout = P
Lout = f1 ( Z L )
Vin = f 2 ( P, Pvin , Z L )
ρ L = f3 (Tlout , Plout , x1out , x2out );
hlout = f5 (Tlout , Plout , x1out , x2out );
ρV = f 4 ( Tvout , Pvout , y1out , y2out )
hvout = f 6 ( Tvout , Pvout , y1out , y2out )
0 = K1 ( x1out , x2out , y1out , y2out , Tvout , P );
0 = K 2 ( x1out , x2out , y1out , y2out , Tvout , P)
Tlout = Tvout
The low index input assignments of this model are:
Assignment [1]: Fin , z1in , z2in , h fin , Lin , x1in , x2in , hlin ,Vout , y1in , y2in , hvin , Pvin
Assignment [2]: Fin , z1in , z2in , h fin , Lin , x1in , x2in , hlin ,Vout , y1in , y2in , hvin ,Vin
Total Condenser Model:
The model in this section presents a binary components total condenser model as an
extension of the condenser model in Chapter 5, section 5.3.1. The last two equations
are the valve equations for the vapour flow inlet and top liquid product flow. The
model consists of 29 equations with 35 unknowns, as follows:
dM1m
= Vin y1in − Lreflux x1out − Ltop x1out
dt
dM 2m
= Vin y2in − Lreflux x2out − Ltop x2out
dt
dEm
= Vin hvin − Lreflux hlout − Ltop hlout − Qcond
dt
M1m = M1L + M1V ;
M 2m = M 2 L + M 2V
Em = EL + EV
M L = M1L + M 2 L ;
M1L = M L x1out ;
M1V = M L y1;
148
MV = M1V + M 2V
M 2 L = M L x2out
M 2V = M L y2
EL = M L hlout − PLVL ;
EV = MV hV − PVV
M L = ρ LVL ;
MV = ρV VV
VL + VV = Vcond
Z L = f1 (VL )
PL = f 2 ( P, Z L )
Lreflux = f3 ( PL )
ρ L = f 4 (Tlout , PL , x1out , x2out );
ρV = f5 ( Tv , P, y1, y2 )
hlout = f6 (Tlout , PL , x1out , x2out );
0 = K1 ( x1out , x2out , y1, y2 , Tv , P);
Tlout = Tv
Vin = f 8 ( Pvin , P);
hV = f 7 ( Tv , P, y1, y2 )
0 = K 2 ( x1out , x2out , y1, y2 , Tv , P)
Ltop = f 9 ( Pouttop , PL )
The low index input assignments for this model are given as follows:
Assignment [1]: y1in , y2in , hvin , Qcond , Pouttop , Pvin
Assignment [2]: y1in , y2in , hvin , Qcond , Pouttop ,Vin
Assignment [3]: y1in , y2in , hvin , Qcond , Ltop ,Vin
Assignment [4]: y1in , y2in , hvin , Qcond , Ltop , Pvin
Reboiler Model
The following set of differential algebraic equations describes a reboiler model for
binary components for a distillation column, consists of 27 equations with 34
unknowns:
dM1m
= Lin x1in − Vout y1out − Lbot x1out
dt
dM 2m
= Lin x2in − Vout y2out − Lbot x2out
dt
dEm
= Lin hlin − Vout hvout − Lbot hlout + Qreboil
dt
M1m = M1L + M1V ;
M 2m = M 2 L + M 2V
Em = EL + EV
M L = M1L + M 2 L ;
MV = M1V + M 2V
M1L = M L x1out ;
M 2 L = M L x2out
M1V = M L y1out ;
M 2V = M L y2out
EL = M L hlout − PLVL ;
EV = MV hvout − PvoutVV
M L = ρ LVL ;
MV = ρV VV
149
VL + VV = Vreboil
Z L = f1 (VL )
PL = f 2 ( P, Z L )
ρ L = f3 (Tlout , PL , x1out , x2out );
ρV = f 4 ( Tvout , Pvout , y1out , y2out )
hlout = f5 (Tlout , PL , x1out , x2out ); hvout = f 6 ( Tvout , Pvout , y1out , y2out )
0 = K1 ( x1out , x2out , y1out , y2out , Tvout , Pvout ); 0 = K 2 ( x1out , x2out , y1out , y2out , Tvout , Pvout )
Tlout = Tvout
Lbot = f 7 ( PL , Poutbot )
The low index input assignments for this model are:
Assignment [1]: Lin , x1in , x2in , hlin ,Vout , Qreboil , Poutbot
Assignment [2]: Lin , x1in , x2in , hlin ,Vout , Qreboil , Lbot
The following shows that by the choice of assignments of each model, a complete
distillation column model can be built from matching of sub-models interconnection,
as given in figure 6.18.
150
An interconnection matching on a distillation column model:
Pout,top Ltop
Qcond
x1,out x2,out hl,out
Condenser
Pv,in
=
Pv,out
Vin
=
Vout
y1,in y2,in hv,in
=
=
=
y1,out y2,out hv,out
Lout
=
Lin
=
=
=
x1,in x2,in hl,in
Lout
=
Lin
=
=
=
x1,in x2,in hl,in
Lout
=
Lin
=
=
=
x1,in x2,in hl,in
Lout
=
Lin
=
=
=
x1,in x2,in hl,in
Poutbot Lbot
Tray-N
Pv,in
=
Pv,out
Fin
Vin
=
Vout
z1in
Feed tray
z2in
hfin
y1,in y2,in hv,in
=
=
=
Pv,in
=
Pv,out
Vin
=
Vout
y1,in y2,in hv,in
=
=
=
Tray-1
Pv,in
=
Pv,out
Qreboiler
Vin
=
Vout
y1,in y2,in hv,in
=
=
=
Reboiler
Figure 6.18. Interconnected sub models on a tray distillation column model
151
6.5.
Conclusions
The objective of this Chapter is to investigate when interconnection of DAE sub
models will not result in a high index problem. There are three important main
ingredients for interconnection of DAE sub models, namely: model ports, model
connectors, and interconnection matching.
Model ports are the model boundaries. Each model port consists of the chosen model
variables to interact with its environment. Model connectors are non behavioural
equality equations, which relates variables from a model port to other variables on
other models ports.
The knowledge of model ports and their variable classifications for each DAE sub
model are important for interconnecting and re-use of DAE sub models. Application
modelling and simulation programs should give information about the variable
classifications on the model ports.
Interconnection matching of structural low index DAE sub models shall result to a
structural low index model. A matching interconnection means that there is no state
constraint equation introduced by the interconnection equation. Application modelling
and simulation programs should give information when a model connector forms a
non-matching connection.
152
Chapter 7: Conclusions and Recommendations
7.1. Conclusions
Usually, a complex process model consists of interconnected sub models. Modelling of
each sub model based on balance and constitutive equations gives a first order nonlinear differential algebraic equations (DAE) model. The consequences of modelling
with DAE are the possibility to get a high index and/or a consistent (re-) initialisation
problem.
A high index problem arises due to the state constraint equations within the DAE
model. A consistent (re-) initialisation problem means that it is not possible to set
arbitrary values to the state or accumulation variables at any time. A high index model
and a low index model with non-minimum state variables pose a consistent (re-)
initialisation problem. A high index problem can be caused by: non-causal choice of
the input variables, internal state constraints, and sub models interconnection. In
process modelling a low index DAE model with a consistent (re-) initialisation
problem can be caused by the choice of intensive variables as the state variables
instead of the extensive variables. Apart from the possibility that the state
independency of a DAE model can conflict with the intention of the modeller, a high
index DAE model is numerically difficult to solve.
The proposed method of process modelling is to avoid a high index and/or consistent
(re-) initialisation problem are consistently derived from Rosenbrock and Kronecker
[107]. The approach to do this is to develop procedures to detect the problems and to
reduce the index during process modelling.
The strategy followed in this research has four features. First, examining the
mathematical structure of a DAE model that gives a high index and/or a consistent (re) initialisation problem and deriving easier methods to solve the problems. Second, to
apply the proposed methods to a first order non-linear DAE model. Third, to develop a
method for searching low index input variables of a DAE model and to detect
possible state variables constraint equations. Finally to give the necessary condition so
that the interconnection of sub models will not lead to a high index problem.
The results of the research shows that it is possible to implement algorithms in the
process modelling environment to detect a high index problem, to detect a consistent
(re-) initialisation problem, to calculate the index of a DAE model and to reduce the
index of a DAE model without non-linear algebraic manipulations. The index
reduction without non-linear algebraic manipulations will preserve the original DAE
models and variables. Moreover, it is possible to detect when sub models
interconnection will lead to a high index problem.
The conclusions related to detection of a high index and consistent (re-) initialisation
problem, to calculation of the index and execution of an index reduction in process
modelling are summarized here as follows:
153
Chapter 7 : Conclusions and Recommendations
1.
2.
Solving a high index DAE model needs differentiation manipulations. This thesis
describes high index reduction techniques for a non-linear first order DAE model
in conjunction with the strict system equivalence operation on the linear DAE
model (Rosenbrock [107]). A high index model reduction can require equations
differentiation or variables differentiation.
The proposed procedures for a high index reduction do not need to apply
complicated non-linear algebraic manipulations. The original model equations
and variables will not be destroyed. The information to gather which variables or
equations need to be differentiated for a high index reduction is derived from the
structural information of the system matrix ( Es − A) or ( Es − A)−1 .
The conclusions related to finding input variables of a DAE model and the requirement
of DAE sub models interconnection are summarized here as follows:
3.
4.
5.
6.
7.
The input variables of a first order non-linear DAE model are determined by its
internal DAE model equations. The input variables are those model variables that
utilise the maximum dynamic degree of the model and give rise to causal inputoutput relations. Low index input variables are causal input variables of a DAE
model.
The sets of low index input variables of a model are possibly non unique. This
thesis describes a procedure to find low index inputs sets assignment for a first
order non-linear DAE model with no internal state constraint equations. When a
first order non-linear DAE model has internal state constraints equations, then
these equations can be detected for any maximum degree inputs set assignment.
In this thesis a graph algorithm is used to analyse the process model structure and
input variables assignment. Actually the process-modelling environment must
assist the modeller to make a well-structured process model. In our perspective
the graph algorithm can support this activity. A graph algorithm can be used to
assist the modeller in the book keeping of process model equations and variables
and warning the modeller for a possible state constraints condition during the
model building.
The interconnection of low index DAE sub models shall be such that no state
constraint is introduced by interconnection, i.e. resulting in a low index
composite model. The interconnected input-output variables of sub models are
matched.
A model-building tool can be developed to collect all inputs sets for a low index
assignment of a DAE model. Process modelling environment should provide
information to the user to be able to get low index interconnected models
matching.
Process model building activities can be resumed into the following steps:
a) Set up model behaviour equations,
b) Checking if there are states constraint equations by finding a maximum
degree inputs assignment set,
c) Execute index reduction if it is required,
d) Checking consistent re-initialisation problem,
e) Execute state transformation if it is required,
f) Finding low index inputs assignment sets.
154
A DAE model diagnosis tool during a process model building is required for:
a) Detection of internal states constraints equations (a high index model),
b) Proposing a high index reduction,
c) Finding: low index input variables sets, optional input variables and
compelled internal variables,
d) Detection of a consistent re-initialisation problem.
The theory and algorithm described in this thesis can be imbedded in the process
modelling tools.
7.2. Recommendations
1.
This thesis describes only the mathematical structure and interconnectivity of
lumped systems given by first order non-linear DAE models. We believe that the
phenomenon of the ‘index’ problem is not only a time domain aspect, but also it
exists in the spatial domain. Campbell and Marszalek [24] presented examples of
the ‘index’ problem in the spatial domain, therefore further research on the
mathematical structure and interconnectivity of process models described with
partial differential algebraic equations is required (PDAE).
2.
The recent development of powerful symbolic manipulation software makes it
possible to make a model building diagnosis tool using symbolic manipulator
software instead of using the structural approach. Using symbolic manipulation
tool in model building might improve the ‘unlucky’ values for a high-index
detection with structural analysis.
3.
To be able to re-use and interconnect low index models made from different
modelling environments, then it is necessary to develop standardisation on the
model attributes, especially for the model ports, and the model connectors. The
users need also to get information about which sets of input variables for the
model will give a low index model.
155
156
Appendix A: Structural Frame Work
A.1. Structure
Ching-Tai Lin [28] introduced the concept of “structure” for a linear time-invariant
system. He distinguishes between entries of matrices that are known with 100 percent
precision – called fixed entries (usually only zeros) and those that are known only
approximately. If two equally sized matrices have all their fixed entries on the same
positions, then they are said to have the same structure.
Unger et.al. [126] formulated a definition of structure also:
Definition A.1 (Unger et.al. [126]): Structure.
Whenever there is a numerical (or symbolical) representation of operands in R such
that a certain operation in R yields a nonzero result then the result of the
corresponding structural representation of this operation is ∗. Conversely, if there is
no such numerical (or symbolical) representation then the structural result is 0.
Next we use 'o' to represent operands that are exactly zero with 100 percent certainty.
All other operands, including those with a very small value or even a value of zero (0)
– but not with 100 percent certainty – are represented by ‘∗’. The operands ‘∗’ can be
extended with information of the structure of polynomial entries.
A.2. Structural Operation
The structural framework herein will be used to analyse the structure of polynomial
⎡ p ( s ) ⎤ , i = 1, 2,..., m , j = 1, 2,..., m ,
⎢⎣ ij
ij ⎥⎦
where pij ( s ) ∈ \[ s ] and R[ s ] denote the set of all polynomial p : R → R ,
matrices P ( s ) =
(
)
p ( s ) = a0 + a1s + " + ak s k , a0 ," , ak , s ∈ R, ak ≠ 0, k ∈ N ,
(A.1)
with k is the degree of polynomial p ( s ) . Consider also p ( s ) = 0 as a polynomial on
R[ s ] .
For this purpose let’s consider the algebraic system ( Λ ,+ ,.) , where
Λ = {o, s 0 , s1, s 2 ,...} and where the binary operators of addition and multiplication
are defined as follows:
157
Appendix A : Structural Frame Work
addition +
:Λ × Λ → Λ
o+ sj = s j
o + o=o
si + o = si
si + s j = s max{i, j}
:Λ × Λ → Λ
multiplication .
o.o=o
si .o = o
o.s j = o
si .s j = si + j
Furthermore, if a ∈Λ , then let the unitary operators +:Λ → Λ and −:Λ → Λ be
defined as + a = a and − a = a respectively.
Definition A.2. Structure of a polynomial function
Suppose p, q ∈ R[ s ] are polynomials with degree k , l ,
p ( s ) = a0 + a1s + " + ak s k , a0 ," , ak , s ∈ R, ak ≠ 0, k ∈ N
q ( s ) = b0 + b1s + " + bl s l , b0 ," , bl , s ∈ R, bl ≠ 0, l ∈ N
or
p( s) = 0
q( s) = 0
(has no degree)
then, the operator struc : R[ s ] → Λ is defined as:
⎧⎪ o if p( x) ≡ 0
struc p( s) := ⎨ k
otherwise
⎪⎩ s
(A.2)
struc p ( s )
is called the structural representation of p( s ) and the structural
degree of p( x ) is defined as:
deg struc p ( s ) = k , ∀p ( s ) ∈ R[ s ] \{0}
(A.3)
and
struc( p ( s) + q( s )) = struc p( s)+struc q ( s )
struc( p ( s).q ( s )) = struc p ( s ).struc q ( s )
Example A.1:
Given p ( s ) = −2 s3 − 7 s 2 + s + 1 . Its structural representation is
struc p( s) = s3
158
and deg struc p ( s ) = 3
(A.4)
Corollary A.3.
Suppose p, q ∈ R[ s ], deg p ( s ) = k , deg q ( s ) = l
p ( s ) = pk s k + pk −1s k −1 + ... + p0
q ( s ) = ql sl + ql −1sl −1 + ... + q0
then :
deg( p ( s)q( s )) = deg( struc p( s).struc q ( s ))
= deg struc p( s ) + deg struc q ( s )
(A.5)
and if ( p ( s ) + q( s ) ≠ 0 and k ≠ l ) then
deg( p ( s) + q( s )) = deg( struc p( s ) + struc q ( s))
= max{deg struc p( s), deg struc q( s)}
(A.6)
and if (( p ( s ) + q ( s ) ≠ 0) ∧ (k = l ) ∧ ( pk + ql ≠ 0)) then
deg( p( s ) + q( s )) = k
(A.7)
and if (( p ( s ) + q ( s ) ≠ 0) ∧ (k = l ) ∧ ( pk + ql = 0)) then
deg( p ( s ) + q ( s )) < k
(A.8)
159
160
Appendix B: Graph Algorithm
B.1. Introduction
This appendix describes shortly the graph algorithm as implemented by Bujakiewitcz
[21] and Y.G.G.J.Go [48] to calculate the structural properties of a DAE model, i.e.
the structural rank of E , the structural dynamic degree of freedom
deg det struc( Es − A) , and the required differentiations for index reduction from
matrix struc( Es − A) −1 . This is for a given DAE model of the form :
⎛ ∂F ⎞
F ( z (t ), z (t ), u (t ), t ) = 0 ; where struc E = struc ⎜
⎟ and
⎝ ∂z ⎠
⎛ ∂F ⎞
struc A = struc ⎜
⎟.
⎝ ∂z ⎠
The definitions of structural aspects are given on the Chapter 3 and Appendix A. The
graph algorithm here is used to relates the structure between equations and variables
in a DAE model. The implementation of the graph algorithm is using graph library of
LEDA (Naeher, et.al. [91]).
For further reading of the graph theory and algorithm can be found in Bondy [11],
Bujakiewics [21], G. Chartrand [27] and Naeher et.al. [91].
B.2. Graph terminology
A graph G of a set of m DAE equations consists of a finite nonempty set of vertices
V(G) and set of edges E(G). The vertices are ordered on one side the equation number,
fi (i = 1...m) ∈ V1 (G ) , and other side are the variables z j ( j = 1...m) ∈ V2 (G ) .
An edge ei = ( fi , z j ) ∈ E (G ) joins vertices fi , z j ∈ V (G ) . The connected vertices
fi , z j ∈ V (G ) are called adjacent in G.
A graph G is called a bipartite graph if its vertices set V(G) can be portioned into two
non-empty subset V1 (G ) = { f1, f 2 ,..., f m } and V2 (G ) = {z1, z2 ,..., zm } and every
f i ∈V1 (G ) and zi ∈V2 (G ) , there is no edge between
two vertices from V1 (G ) or between two vertices from V2 (G ) .
edge ei ∈ E (G ) joins vertices
161
Appendix B : Graph Algorithm
B.3. Maximum cardinality matching
A subset of edges M (G ) ⊆ E (G ) is called matching if there is no two edges of
M (G ) sharing the same vertex. The set of all subset-matched edges is denoted as
S M ( G ) . Further a matching of graph G is denoted as M .
A cardinality of a set A is defined as the number of elements in A and is denoted as
A . Therefore the cardinality of matching edges M (G ) is defined as the number of
edges in M (G ) and is denoted as
M (G ) . A matching edges of maximum
cardinality in G is called the maximum cardinality matching (set), M MC , where:
M MC = max M ( G )∈S M ( G ) M (G )
(B.1)
If a graph G with cardinality of its vertices V (G ) = p has a matching of cardinality
p
then such a matching is called a perfect matching, M MP , which has a
2
perfect matching set M MP . Since every matching of G has at most p/2 edges, a
M (G ) =
perfect matching is a maximum matching.
Lemma B.1 (Y.G.G.J.Go [48])
(
⎣⎢
Consider a constant matrix with struc E = ⎡ struc eij
struc eij =
RS o, if e ≡ 0
Tx , otherwise
)
ij
⎤ , E ∈ R m×m and:
⎦⎥
ij
0
Let the bipartite graph G is defined as follows:
a) Let V (G ) = 2m
b) Let V (G ) be split up into two distinct sets:
V1 (G ) = { fi ∈ V (G ) : i = 1, 2,..., m} ≡ equation vertices
V2 (G ) = {z j ∈ V (G ) : j = 1, 2,..., m} ≡ variable vertices
c) Let E (G ) = {( f i , z j ): struc eij = x 0 }
Now, let the resulting matching of the Maximum Cardinality Matching set
algorithm on the constant matrix E be M MC , then we have:
rank struc E = M MC
162
(B.2)
Proof:
Let rank struc E = γ . It follows from Corollary 3.4 (Chapter 3) that there is a
permutation σ of {1,2,..., γ } which result to:
∃Γγγ with det Γγγ ≠ o ∧
Γγγ** with γ * > γ , det Γγγ** = o
this is equivalent with if there are matching edges of equation vertices and
variable vertices due to the same permutation σ of {1,2,..., γ } :
∃{( f σ i , zσ i ) ∈ M (G ), i = 1,2,...γ } ∧
∃/ {( f σ i , zσ i ) ∈ M (G ), i = 1,2,...γ *} with γ * > γ
since it is a maximum matching, then it can be written as:
∃{( f σ i , zσ i ) ∈ M MC , i = 1,2,...γ } with its cardinality : M MC = γ .
It follows that: rank struc E = M MC .
Example B.1 (see Chapter 3: Example 3.3)
Consider a DAE model as given on example 3.3. with
⎡ x0
⎡1 1 0 ⎤
⎢
E = ⎢⎢1 1 0 ⎥⎥ , struc( E ) = ⎢ x 0
⎢
⎢⎣0 0 0 ⎥⎦
⎢o
⎣
x0
x0
o
o⎤
⎥
o⎥
⎥
o⎥
⎦
Figure B.1 shows two matching graphs of struc( E ) , where:
V1 = { f1, f 2 , f3}, V2 = {z1, z2 , z3} and E (G ) = {e1, e2 , e3 , e4 } .
The cardinality of graph edges E (G ) = 4 .
= {m1, m4 } or M MC 2 = {m2 , m3} , with
= 2.
the maximum cardinality matching M MC = M MC = M MC
1
2
The maximum matching set is M MC
1
Thus : rank struc( E ) = M MC = 2 .
163
f1
m1 or e1
e3
e2
f2
m4 or e4
f3
z1
f1
z2
f2
z3
f3
e1
m3 or e3
m2 or e2
e4
z1
z2
z3
Matching-1
Matching-2
Figure B.1. Maximum Matching Graph of struc( E )
B.4. Maximum weighted matching
A weighted graph G of is a graph obtained from assigning a weight w(ei ) to
ei ∈ E (G ) . The weight W (G ) of weighted graph G is defined as
∑ w( e )
W (G ) =
(B.3)
i
ei ∈E ( G )
A weighted matching set in a weighted graph of G is denoted as M w and a maximum
(cardinality) weighted matching set in a weighted graph of G , denoted as M MW , is a
matching set in for which the weight of weighted graph is maximum, denoted as,
b
M MW : = arg W M MW
g
b g
= arg max M w ∈S Mw W M w
(B.4)
∑ w( e )
= arg max M w ∈S Mw
i
ei ∈E ( G )
Since a perfect matching is a maximum matching we can also define,
Definition B.2. (Bujakiewicz [21])
A maximum weighted matching set M MW of graph G is one of the perfect matching of
G which maximizes the sum of weights of its edges, is denoted as:
RS
|T
M MW : = M w ∈ M MP :max M w ∈S MP
U
W
∑ w(e )V|
i
ei ∈E ( G )
(B.5)
The weight of the maximum weighted matching, is denoted as:
W ( M MW ) := max M w ∈S MP
164
∑
ei ∈E (G )
w(ei )
(B.6)
Therefore: the cardinality of the maximum weighted matching set is equal to the
cardinality of the perfect matching set, M MW = M MP , and
W ( M MW ) ≥ W ( M MP ) .
(B.7)
Corollary B.3. (Y.G.G.J.Go [48])
Consider structural matrix struc T ( s) = struc ( Es − A) = (tij ( s))ij ,
⎢⎣(
with struc E = ⎡ struc eij
)
ij
⎤ , E ∈ R m×m , struc A = ⎡( struc a ) ⎤ , A ∈ R m×m
ij ij ⎥
⎥⎦
⎢⎣
⎦
and:
struc eij =
RS o, if e ≡ 0
Tx , otherwise
ij
0
struc aij =
RS o, if a ≡ 0
Tx , otherwise
ij
0
Further suppose the weighted bipartite graph be defined as follows:
V (G ) = 2m
b) V (G ) be split up into two equally sized subsets:
V1 (G ) = { fi : i = 1, 2,..., m} ≡ equation vertices
a)
V2 (G ) = {z j : j = 1, 2,..., m} ≡ variable vertices
E (G ) = {( f i , z j ): struc eij = x 0 ∨ struc aij = x 0}
d) and let the weighting w: E (G ) → N be defined as,
c)
⎧⎪1,
w( x 0 ) = ⎨
⎪⎩0,
∀x0 ∈ E (W (G ))
∀x 0 ∉ E (W (G ))
where E (W (G )) = {( f i , z j ) ∈ E ( G ): struc eij = x 0 } .
Suppose M MW denotes the resulting matching set of the Maximum Weight
Matching algorithm, then:
i) T ( s) is structurally singular ⇔ M MW < m, and
ii) if T ( s) is structurally non-singular then
(B.8)
r = deg det struc T ( s ) = W ( M MW )
(B.9)
Proof:
i) a) The “if” part can be proven by contradiction statement, namely:
165
Let M MW < m and suppose that T ( s) is structurally non-singular, then we
have:
T ( s) is structurally non-singular
⇔ ∃σ ∈ Γ such that ( fi , zσ i ) ∈ E (G ),
i = 1, 2,..., m
⇔ ∃ perfect matching set M MP ∈ S M
with cardinality M MP = m and following the Definition B.2:
M MW = M MP = m . From which follows the contradiction with the
assumption that M MW < m .
b) The “if only” part follows by the same way, while letting T ( s) be
structurally non-singular and supposing M MW < m .
ii) Let T ( s)
be structurally non-singular, then for each matching set
M (G ) ∈ M MW there exists a structural permutation σ ∈Γ such that
( f i , zσ i ) ∈ M (G ) for i = 1, 2,..., m .
Furthermore deg ti ,σ ( s ) = w(( fi , zσ )) , and following Corollary 3.7 (see
i
i
Chapter 3)
m
r = deg det struc T ( s ) = maxσ ∈Γ ∑ deg ti ,σ i ( s ) , hence:
i =1
m
r = maxσ ∈Γ ∑ w(( fi , zσ i ))
=
∑
ei ∈M MW
i =1
w(ei )
= W ( M MW )
The degree of determinant structural matrix T ( s ) is equal to the weights of maximum
weighted matching of the graph of T ( s ) .
Given a DAE with :
⎡s
2⎤
⎡ s + 1 −1
⎢
⎢
⎥
T ( s ) = [ Es − A] = ⎢ 1
s + 2 −1⎥ , struc T ( s ) = ⎢ x o
⎢
⎢⎣ −1
0 ⎥⎦
−1
⎢ xo
⎣
166
xo
s
xo
xo ⎤
⎥
o⎥
x
⎥
0⎥
⎦
Figure B.2. shows the maximum weighted matching sets of graph of struc(T ( s )) . The
weighted edge is shown with value = 1. According to (B.9) in Corollary B.2 that:
r = deg det struc T ( s )
=
∑
ei ∈M MW
w(ei )
= W ( M MW )
=1
Thus the structural dynamic degree of freedom of this DAE = 1, and
det struc T ( s ) = s r
=s
f1
1
z1
f1
f2
z2
f2
f3
z3
f3
Max.weighted matching-1
z1
1
z2
z3
Max.weighted matching-2
Figure B.2. Maximum weighted matching
B.5. Calculation of struc(T-1(s))
Following the Definition 3.9 (Chapter 3):
Consider T ( s ) ∈ R m×m [ s ] , the structural matrix inverse of T ( s ) is defined as:
(
( struc T ( s ))−1 := ( struc tij−1 ( s ))ij
)
ji
= ⎜⎜
⎟ ⎟,
⎜ ⎝ det struc T ( s ) ⎠ ⎟
ij ⎠
⎝
i = 1, 2,..., m
j = 1, 2,..., m
(3.9)
167
where the relation for the sub-matrix struc ( Sij ( s )) as given on (2.12), is obtained by
placing zeros in the i-th row and in j-th column of struc (T ( s )) and replacing the ij-th
element with xo or 1 :
⎛ struc(t11 ( s )) " 0 " struc(t1m ( s )) ⎞
⎜
⎟
#
#
#
⎜
⎟
⎟
struc( Sij ( s )) = ⎜
0
0
" xij0 "
⎜
⎟
#
#
#
⎜
⎟
⎜ struc(t ( s )) " 0 " struc(t ( s )) ⎟
m1
mm
⎝
⎠
The same procedure to calculate det struc T ( s ) as described in section B.2; we can
calculate each entry of det struc S ji ( s ) .
Given a DAE with :
⎡s
2⎤
⎡ s + 1 −1
⎢
⎢
⎥
T ( s ) = [ Es − A] = ⎢ 1
s + 2 −1⎥ , struc T ( s ) = ⎢ x o
⎢
⎢⎣ −1
−1
0 ⎥⎦
⎢ xo
⎣
xo
s
xo
xo ⎤ ⎡ s 1 1 ⎤
⎥
o⎥ ⎢
x = ⎢1 s 1 ⎥⎥
⎥
0 ⎥ ⎢⎣1 1 0 ⎥⎦
⎦
with det struc T ( s ) = s ;
Suppose we are going to calculate det struc S33 ( s ) , where :
⎡ s 1 0⎤
struc S33 ( s ) = ⎢⎢1 s 0 ⎥⎥
⎢⎣0 0 1 ⎥⎦
Figure B.3. shows the maximum weighted matching for the graph of S33 ( s ) , where
rs 33 = deg det struc S33 ( s ) = 2 . Thus :
det struc S33 ( s ) = s rs
= s2
It follows that :
168
33
det struc S33 ( s )
struc tij−1 ( s ) =
det struc T ( s )
s2
s
=s
=
f1
f2
1
z1
1
z2
f3
z3
Max.weighted matching
S33(s)
Figure B.3. Maximum weighted matching of S33 ( s )
We can calculate the other fractional polynomial entries of the matrix struc(T −1 ( s ))
of the DAE on the example 3.2. above, resulting to the following:
⎡1
⎢s
⎢
1
−1
struc T ( s ) = ⎢
⎢s
⎢1
⎢
⎣
1
s
1
s
1
⎤
⎥
⎥
1⎥ .
⎥
⎥
s
⎥
⎦
1
169
170
Appendix C: Modelling of an ICGCC Plant
We have tried to apply the structural analysis technique and interconnection matching
as presented in this thesis for the dynamic modelling of a complex interconnected
index-one sub-models, namely the dynamic modelling of integrated coal gasification
combined cycle (ICGCC) plant, as schematically given in the following figure C.1.
The modelling strategy will be briefly described in the section C.2. Further this section
is not mention to analyse the dynamic behaviour and control strategy of an ICGCC.
Analysis of the dynamic behaviour and control strategy of an ICGCC plant can be
found e.g. in P.Schoen [112].
Air
Separation
Plant
Oxygen
Quench
gas
Coal
Coal
Gasifier
Syngas
Cooler
Nitrogen
Gas
Cleaning
Saturator
Steam
Water
Heat Recovery
Steam Generator
Flue
gas
Syn
gas
Water
Air
Steam
Electric
Power
Steam Turbine
Gas Turbine
Air
Compressor
Figure C.1. An Integrated Coal Gasification Combined Cycle Plant
(P.Schoen [112])
At the time of the research SpeedUp [118] is chosen for the modelling and simulation
environment for the example problem, since it is supported with enhanced numerical
solver and physical properties database. Although through the process modelling
exercises we have faced with a lot of consistent initialisation problems, index problem
and interconnection problem. The index problem rises as internally states constraint
problem or as the result of interconnection. Moreover the index problem is also caused
due to wrong choice of the input variables. This is partially because SpeedUp [118]
library models does not give any hints about the choice of input variables. Therefore
based on the theoretical results of this thesis we hope that in the near future the
171
Appendix C : Modeling of an ICGCC Plant
conceptual object oriented modelling for interconnection matching of index-one submodels and a better diagnosis environment for index problem will be available in the
process modelling tools.
In the following we will describe briefly about the ICGCC process, modelling
approach and show some simulation results.
C.1.
Process description of an Integrated Coal Gasification
Combined Cycle Plant
Coal still gives major contribution as the raw material of the electrical production. This
is caused by the large worldwide reserves, constant price and the suitability for the fuel
of a large-scale electrical power generation plant
(> 100 MW). An integrated coal gasification combined cycle (ICGCC) plant offers a
better efficiency and low emissions compared with conventional coal combustion
electrical power plant. In the early 1990’s several demonstration ICGCC plants were
built worldwide. One such plant is at Buggenum, the Netherlands. This plant has been
in operation since 1996 to produce electricity of 250 MW for the Netherlands
electricity-net. Schematically the ICGCC Buggenum is shown in the figure C.1. This
consists of the Shell coal gasifier reactor, the syngas cooler, the gas cleaning unit, the
air separation plant, the heat recovery steam generator and the steam and gas turbines
(Demkolec [30], P. Schoen [112])
The production of electricity begins when fine pulverised coal together with steam
feed and oxygen feed from the air separation plant are burned and gasified in a reactor
at about 1600 oC under a pressure of about 28 bar. The membrane pipes wall cooling is
installed for keeping the reactor wall at temperature and the produced steam is
integrated in the steam cycle. The produced gas from the gasifier is called the syngas,
which mainly consists of 60 % CO, 25 % H2 and the remaining 15 % are N2, H2O,
CO2, CH4, COS and H2S. This hot syngas in quenched with cold recycled gas at the
top of the reactor to a temperature of about 900 oC. After leaving the gasifier, the
product gas flows through the syngas cooler unit where further cooling occurs until a
temperature of about 235 oC is achieved. The heat released in the syngas cooler is used
to produce medium pressure steam, which is integrated with the steam cycle. The coal
gasifier and syngas cooler plant is shown in the figure C.2.
After the syngas cooler the product gas is cleaned in the gas cleaning unit (see figure
C.3). The COS is converted to H2S in a catalytic converter unit and almost all H2S is
removed in the H2S absorber unit. To lowered the adiabatic flame temperature of
syngas combustion in the gas turbine, before the combustion the cleaned syngas is premixed with dilute nitrogen (nitrogen gas product from air separation plant) and
saturated with water in a saturator. The gas outlet from the saturator is used as the fuel
of gas turbine to produce electricity. The heat of flue gas leaving the gas turbine is
recovered in the heat recovery steam generator, which produces mainly high and
medium pressure steam for the steam turbines. The saturator unit, heat recovery steam
generator and the turbines are schematically shown on the figure C.4.
A cryogenic air distillation plant produces the oxygen needed for the gasification of
coal in the gasifier (see figure C.5). This plant mainly consists of an integrated
condenser-reboiler double columns distillation unit. The high pressure section (bottom
172
column) operates at about 7 – 10.5 bar and the low pressure section (upper column)
operates at about 1.8 – 3 bar. The products of this plant are mainly oxygen feed for the
gasifier (with purity about 95 %), dilute nitrogen for the saturator (with oxygen
impurity of less then 1%). The air compressor of the gas turbine supplies the air feed
needed for this air separation plant.
P. Schoen has done the dynamic modelling and control study of the ICGCC plant
described in section in ACSL [2]. The problem was encountered in ACSL
programming is incapability to solve implicit interconnected DAE models. Therefore
the ICGCC model is re-modelled in the SpeedUp [118]. The description of the ICGCC
sub-models can be found in F.R. Habieb [56], A.Hermawan [57], T. Hermanu [58],
H.J. de Lange [73]. The integrated model of the ICGCC consists of about 6500
differential algebraic equations with about 900 state variables. Moreover due to the
confidentially of the real plant data, the models are made based on data on the open
literatures.
173
Steam to
Heat Recovery
Steam generator
PC
Recycle
Booster Compressor
Oxygen
(GOX)
FRC
Coal
FRC
Steam
FRC
Water
Dry Solid Removal
Unit
Syngas to
Gas Cleaning
Unit
Figure C.2. A Schematic flowsheet of a Shell Coal gasifier and Syngas cooler unit (T.
Hermanu [58], P.Schoen [112], Van der Burgt [23])
174
PC
Cooler
Lean H2S
absorbent
Syngas to
Saturator
Wet Cleaning Unit
Water
H2S
Absorber
COS →H2S
Converter
Syngas from
Syngas-cooler
FlashTank
PC
Steam
Rich H2S
absorbent
Slop water
Slop water
Figure C.3. A Schematic flowsheet of Syngas cleaning plant (Demkolec [30], P.
Schoen [112])
Flue gas
LC
Heat Recovery Steam Generator
Feed
Water
Economiser
Saturator
Syngas from
Gas Cleaning unit
Evaporator
Steam from
Syngas-cooler
Nitrogen (DGAN) from
Air Separation Plant
Superheater
Reheater
Water
SC
Fuel heater
Air
P
Electric
Generator
MP/LP Steam
Turbine
To Condenser
HP Steam
Turbine
Gas Turbine
Air Compressor
Compressed Air
to Air Separation Plant
Figure C.4. A Schematic process flowsheet of the saturator unit and STEG unit (H.J.
de Lange [73], P. Schoen [112])
175
Oxygen (GOX)
to Gasifier
Nitrogen (DGAN)
to Saturator
PC
LP
column
LC
Subcooler
FRC
Main Heat
Exchanger
Pure Nitrogen
Reboiler
Condenser
(PGAN)
HP
column
DPC
Subcooler
LC
mol
sieve
Air
from Air Compressor
Figure C.5. A schematic process flowsheet of an air separation plant
(J.M. Abrardo et.al [1], Demkolec [30], A. Hermawan [57])
C.2.
Modeling strategy
The goals of the dynamic modelling of the ICGCC are to make a model, which can be
used for prediction of the steady state behaviour and dynamic interactions, and to
study the control strategies. The outline of modelling strategy is given as follows:
1. Topological decomposition:
The plant model topological decomposition is done in 4 levels:
a) Sub-plant level: the ICGCC is decomposed into 4 functional units
compounds, namely: the syngas plant (see figure C.2), the gas
176
cleaning plant (see figure C.3), the combined cycle plant (see figure
C.4) and the air separation plant (see figure C.5)
b) Unit level: each sub-plant level is decomposed in several unit
operations, e.g. the syngas plant consists of the gasifier unit, the
syngas cooling unit, the dry solid removal unit and the recycle
compressor.
c) Sub-unit level: each unit operation can be decomposed into several
sub-units, e.g. the syngas cooler consists of economisers, boilers and
superheaters.
d) Phase level: a phase ψ is defined as a non-decomposable or
elementary model aggregate, since all intensive properties on this
phase ψ are to be considered spatial uniformly. An example on the
modelling of the syngas cooler, each heat exchanger wall is assumed
to have one uniform temperature. Modelling of spatial uniform phase
will result in a set of differential algebraic equations.
Through the sub-models decomposition, indirectly we have defined the system
boundaries of each sub-model.
2. Set up of model connectors (or flowsheet set up)
The following step on the modelling approach is to define the interaction variables
through the sub models, by defining the model connectors. These model
connectors can be grouped in several classes:
a) Material model connector, e.g. this includes the information of mass
flow, pressure, temperature, mol fraction and specific enthalpy
b) Energy model connector, e.g. this includes the information of heat and
temperature
c) Information model connector, e.g. this includes one variable
connection for the controller.
Further in this step, we prepare which part of the sub models will interact through
bilaterally coupling connectors or not.
3. Defining Overall Unit of Measurements
The overall Unit of Measurements is decided before the set-up of the model
equations in each sub-model, e.g. SI unit of measurements
4. Sub-model set-up
The sub-model set-up is in principally composed of the following steps:
a) Choice of the interesting/important physical mechanisms described
within the system (i.e. behavioural assumptions). In general this
results to a set of DAE’s consists of the balance and constitutive
equations (model equations entity).
b) Collecting interconnection variables on model ports
177
c) Collecting and defining the type and range of model variables (model
variables entity).
d) Collecting the model parameters (model parameters entity).
e) Collecting the model initial values (model initial values entity)
f) Defining the required model unit conversion factors if required.
5. Sub-model mathematical properties check
The mathematical properties check of each sub-model, includes:
a)
b)
c)
d)
Equations independency check
The structural consistent (re-) initialisation and index check
Collecting sub-model input assignments
Checking if the proposed bilaterally coupling for the sub model can be
achieved
6. Sub-model stand alone simulation test and evaluation
The model behaviour and robustness must be tested and evaluated for the steady
state and dynamic behaviour in its defined model applicability range.
7. Sub-models interconnection matching, testing and evaluation
The sub-models interconnection matching, testing and evaluation are the last step
on the strategy of modelling of interconnected sub-models. In the most cases this
interconnection matching, testing and evaluation are done in several steps as ‘a
growing’ interconnected sub-models.
C.3.
Examples of the simulation results
In the following we will show several model simulation results. The dynamic
responses are the open-loop responses, namely when only the liquid hold-ups are
controlled.
178
The coal gasifier – steady state performance:
0.1
0.05
11.9 %
11.9 %
0.04
0.03
CO2
mole
fraction
0.08
H2O/Coal percentage
0.06
0.04
0.01
0.02
0.75
0.8
0.85
0.9
O2 (95% purity)/Coal mass flow
0.95
0
0.7
1
4.8 %
2.4 %
0.75
0.8
0.85
0.9
0.95
1
x 10 4
1.4
0.34
Syngas Low Heating Value
H2O/Coal percentage
0.32
1.35
11.9 %
9.5 %
7.1 %
4.8 %
0.3 H2
0.28
7.1 %
H2O
mole
fraction
0.02
0
0.7
9.5 %
H2O/Coal percentage
9.5 %
7.1 %
4.8 %
2.4 %
mole
fraction
1.3
H2O/Coal percentage
1.25
2.4 %
2.4 %
4.8 %
7.1 %
LHV [KJ/Kg]
1.2
1.15
0.24
0.22
0.7
9.5 %
11.9 %
0.26
1.1
1.05
0.75
0.8
0.85
0.9
0.95
0.7
1
0.68
0.75
0.8
0.85
0.9
0.95
1
O2 (95 % purity)/Coal mass flow
520
LHV * Syngas mass flow [MW]
510
11.9 %
0.66
2.4 %
CO mole
0.64
fraction
0.62
0.6
500
9.5 %
4.8 %
490
7.1 %
7.1 %
480
9.5 %
H2O/Coal percentage
4.8 %
470
11.9 %
H2O/Coal percentage
2.4 %
460
450
0.58
0.7
0.75
0.8
0.85
0.9
0.95
1
440
0.7
0.75
0.8
0.85
0.9
0.95
1
Figure C.6. Some steady state sensitivity performance of the Coal Gasifier
179
The air separation sensitivity performance:
Figure C.7. The air separation plant sensitivity performance
The open loop step response on the oxygen mass flow to the gasifier
The following figures C.8 show the effects of the oxygen mass flow step response on
oxygen product purity of the air separation plant and the figure C.9 show the effects on
the gasifier syngas compositions, temperature, pressure and gas turbine power.
Figure C.8. Step response on the oxygen mass flow to the gasifier
180
Figure C.9. Open loop effects on the oxygen step response
181
182
List of symbols and abbreviations
t
c
u
v
w
x
y
z
z
Time
Compelled internal variables of a DAE model
Input variables of a DAE model
Algebraic variables of a DAE model
Non state variables of a DAE model
State variables of a DAE model
Output variables of a DAE model
A DAE model variables
Differential variables of a DAE model
F ( z (t ), z (t ), t ) = 0
F ( z (t ), z (t ), u (t ), t ) = 0
∂F
∂F
E=
,A=
∂ z
∂z
First Order Differential Algebraic Equations
First Order Differential Algebraic Equations
Matrices of the first derivates of a DAE model
D(k ) fk
k-th derivates to the variable z of a subset of the DAE model
G
V (G )
E (G )
equations
A graph of a DAE model
A set of vertices of a graph
A set of edges of a graph
A
Cardinality of the matrix A
M MC
M MP
M MW
A maximum cardinality matching set
A perfect matching set
U
U
UN
Set of all low index input variables assignment sets
A set of low index input variables assignment
A set of necessary input variables of a DAE model
UO
A set of optional input variables of a DAE model
r.s.e
s.s.e
BDF
DAE
KCF
ODE
restricted system equivalence operation
Backward Difference Formulae
Differential Algebraic Equations
Kronecker Canonical Form
Ordinary Differential Equations
A maximum weighted matching set
183
List of symbols and abbreviations
184
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Samenvatting
Proces modellering speelt een belangrijke rol in de proces engineering en de
bedrijfsvoering van processen. Procesmodellen worden gebruikt voor het ontwerpen
van procesapparatuur en procesregelingen, het voorspellen van procesgedragingen en
de optimalisatie van producteigenschappen bij de bedrijfsvoering van processen.
Gewoonlijk bestaat een procesmodel uit meerdere, onderling met elkaar verbonden,
submodellen, namelijk uit een netwerk van submodellen, opgebouwd uit verschillende
lagen, met op elke volgende laag een toenemende detaillering van de
modelbeschrijving. Modellering van elk afzonderlijk subsysteem wordt gepresenteerd
door een 1e orde niet lineair DAE (Differential Algebraic Equations).
Een consequentie van proces modellering met een DAE model is de kans op een hoog
index en/of een consistent (re-) initialisatie probleem. Een index van een DAE model
kan eenvoudig worden samengevat als het aantal keren van differentiëren welke nodig
zijn om het originele DAE model te transformeren naar een expliciet ODE model. Een
hoge index ontstaat indien deze differentiatie meer dan een keer dient te worden
uitgevoerd.
Een hoog index probleem kan worden veroorzaakt door een verkeerde keuze van de
ingangsvariabelen, interne afhankelijkheid tussen de toestandsvariabelen (bijvoorbeeld
door evenwicht reactie vergelijking, of door een fasen evenwicht vergelijking) of door
interconnectie van verschillende submodellen.
Een consistent (re-) initialisatie probleem betekent dat het niet mogelijk is om een
willekeurige waarde te geven voor de opslag/toestand variabelen. Een hoog index
model en een laag index model met niet minimale differentiële variabelen geeft altijd
een consistent (re-) initialisatie probleem.
In principe vraagt de oplossing van een hoog index probleem om differentiatie in
plaats van alleen maar integratie. Afgezien van het feit dat de meeste numerieke
algoritmes niet beschikken over een differentiatie algoritme, is het niet praktisch om
wederzijds verbonden hoog index modellen op te lossen omdat een hoog index DAE
zich ook manifesteert als een consistent (re-) initialisatie probleem. Een modelbouwer
kan te maken krijgen met onvoorspelbare gedragingen van het model op de
gedefinieerde of gekozen opgeslag/toestand variabelen, namelijk deze opslag/toestand
variabelen kunnen een ‘sprong’ maken in hun tijd respons. Wanneer men een
grootschalig verbonden systemen modelleert dient men een hoog index DAE model te
vermijden.
Er is veel onderzoek gedaan naar het oplossen van een hoog index DAE model. Het
verschil van het werk in deze thesis in vergelijking met dat van anderen is het feit dat
het vaststellen van een hoog index probleem en index reductie technieken worden
ontwikkeld vanuit de originele theorie van Kronecker en Rosenbrock [107], namelijk
de strikte systeem equivalente operatie. Deze strikte systeem equivalente operatie
voert de gewenste differentiatie met behoud van het ingang uitgang gedrag van het
systeem. De theorie van Kronecker en Rosenbrock [107] is uitgebreid voor een 1e orde
vi
Samenvatting
niet lineair DAE model door gebruik te maken van de structurele informatie van de
vergelijkingen in plaats van de exacte numerieke variabelen.
Deze thesis laat zien dat het voor een 1e orde DAE procesmodel mogelijk is een
procedure te vinden (gebaseerd op een strikte systeem equivalent operatie) voor een
hoog index reductie welke geen gebruik maakt van niet lineaire algebraïsche
bewerkingen. Door toepassing van deze methodiek worden de originele
modelvergelijkingen en variabelen niet aangetast.
Bovendien is het mogelijk voor een gegeven DAE model vast te stellen welke input
variabelen geen hoog index probleem veroorzaken ervan uitgaande dat er geen
toestand begrenzende vergelijkingen aanwezig zijn. Deze toestand begrenzende
vergelijkingen kunnen worden gedetecteerd door elke gekozen input variabelen, welke
geeft een maximale aantal toestandsvariabelen van het model.
Interconnectie van DAE sub modellen vraagt om informatie over mogelijke
ingangsvariabelen van elk afzonderlijk submodel. Deze ingangsvariabelen zijn niet per
definitie uniek. Interconnecties veroorzaken geen hoogindex probleem als er geen
toestand begrenzende vergelijking ontstaat door deze verbinding, dit wil zeggen dat
laag index modellen resulteren in laag index samengestelde modellen.
Ondanks de voordelen van toepassing van deze methode (gebruik maken alleen van de
structurele informatie), komt het in uitzonderlijke gevallen voor dat een exact
analytische methode geen hoge index geeft, maar structurele methode wel. Een
combinatie van een symbolische en gestructureerde benadering kan dit probleem
oplossen en de theorie in deze thesis kan worden gebruikt bij dergelijke toepassingen.
Verder is onderzoek noodzakelijk van de hoge index theorie voor gedistribueerde
systemen PDAE (Partial Differential Algenbraic Equations) omdat een toestand
begrenzingprobleem zich kan manifesteren in een tijdgebied of een ruimtelijk gebied
of in een combinatie van beide.
vii
Curriculum Vitae
A.T.M. Judi Soetjahjo was born in Malang, Indonesia, 21 January 1964. He obtained
his MSc. in Mechanical Engineering – Process Control and Dynamics at Delft
Technical University, The Netherlands in 1992. It was here that he commenced his
extensive research on dynamic process modelling until his departure in 1997. He is
currently employed as a Senior (Principal) Engineer for an International Offshore Oil
and Gas Engineering Company, GustoMSC Bv., based in The Netherlands. His most
current assignment is as the Department Head of Electrical and Instrumentation at
SBM (Single Buoy Mooring) Malaysia SDN.BHD, (Sister Company of GustoMSC
Bv.)
viii

Mathematical Analysis of Dynamic Process Models

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