INTEGRATION OF REAL TIME OPTIMIZATION (RTO) AND ......Figure 4. 1 – Selected manipulated and...
Transcript of INTEGRATION OF REAL TIME OPTIMIZATION (RTO) AND ......Figure 4. 1 – Selected manipulated and...
ALDO IGNACIO HINOJOSA CALVO
INTEGRATION OF REAL TIME OPTIMIZATION (RTO) AND MODEL PREDICTIVE CONTROL (MPC) OF AN INDUSTRIAL
PROPYLENE/PROPANE SPLITTER
São Paulo
2015
ALDO IGNACIO HINOJOSA CALVO
INTEGRATION OF REAL TIME OPTIMIZATION (RTO) AND MODEL PREDICTIVE CONTROL (MPC) OF AN INDUSTRIAL
PROPYLENE/PROPANE SPLITTER
Tese apresentada à Escola Politécnica da Universidade de São Paulo para obtenção do título de Doutor em Ciências
Orientador: Prof. Dr. Darci Odloak
São Paulo
2015
ALDO IGNACIO HINOJOSA CALVO
INTEGRATION OF REAL TIME OPTIMIZATION (RTO) AND MODEL PREDICTIVE CONTROL (MPC) OF AN INDUSTRIAL
PROPYLENE/PROPANE SPLITTER
Tese apresentada à Escola Politécnica da Universidade de São Paulo para obtenção do título de Doutor em Ciências
Área de Concentração: Engenharia Química
Orientador: Prof. Dr. Darci Odloak
São Paulo
2015
ACKNOWLEDGMENTS (In portuguese)
Primeiro agradecer a Deus pela vida e a oportunidade de completar um sonho e
recobrar a saúde. A minha esposa Rosita e toda minha família na Bolívia pelo apoio
incondicional durante todos esses anos de estudo, por torcer e confiar sempre em
mim.
Ao Prof. Darci Odloak pelo conhecimento transmitido, comentários, sugestões,
amizade, por confiar em mim e me dar a oportunidade de trabalhar com controle
preditivo de processos.
À Profa. Rita Alves pelo primeiro contato, apoio e pela amizade desde que cheguei a
São Paulo.
Ao pessoal da Invensys por facilitar a licença acadêmica para o uso de software para
a realização deste trabalho e pelas dicas no uso correto do software, em especial,
Erika Fernandez, Rubens Rejowski e Neliana Azacón.
À minha banca examinadora da qualificação e pessoal da Petrobras pela informação
e criticas construtivas para realizar este projeto, em especial, Dr. Antonio Carlos
Zanin (CETAI), Prof. Galo Le Roux e Eng. Carlos Henrique (RECAP).
Aos Professores do Departamento de Engenharia Química por transmitir os
conhecimentos e ferramentas para concluir este trabalho.
Aos meus amigos da SJB, Eduardo, Nedher, Juancho, Ana, Roxana, Sandra pelos
bons momentos que compartilhamos e por estar comigo nos momentos mais
difíceis.
Aos amigos do Laboratório de Simulação e Controle de Processos (LSCP) pela
amizade e pelas conversas na copa do Bloco 21, em especial, Brunos, André, Daniel
(Dorminhoco), Ricardo, Álvaro, Zés e Marion.
Ás agências de fomento CAPES, FUNDESPA e CNPq pelo apoio financeiro.
The die is cast and you can’t restart or change
the past, but if given only one more chance,
could you carve the way?
I’m sick of being afraid and living by these
mistakes that I have made, but I’ll change that
with these hands of mine.
Believing in something more, I’ll carve a path
through that rusted doorway. There’s still more
that’s still worth fighting for, so take aim and don't
wait or hesitate.
(SAO)
RESUMO
O propósito desta Tese é realizar o estudo da implementação do controle avançado do tipo controle preditivo baseado em modelo (MPC) e otimização em tempo real (RTO) em uma unidade de processo industrial usando como ferramentas softwares comerciais de simulação e otimização de processos. As soluções propostas podem ser consideradas como estratégias de integração entre RTO e MPC de uma e duas camadas.
Na estratégia de duas camadas, a camada superior que considera um modelo rigoroso não linear do processo computa e envia targets otimizantes à camada dinâmica do MPC, que computa as ações de controle necessárias para alcançar esses targets e estabilizar o processo. Na estratégia de uma camada, mais conhecida como MPC econômico, temos a inclusão do gradiente da função econômica na função custo do controlador preditivo.
Ambas as estratégias foram estudadas e suas implementações na coluna de destilação de propeno/propano com integração energética da unidade de produção de propeno da refinaria de Capuava da Petrobras foram simuladas. Este estudo foi realizado em varias etapas. Primeiro, uma simulação dinâmica do processo foi realizada usando o simulador dinâmico SimSci Dynsim® para ser usada como uma planta virtual que também foi usada para a identificação dos modelos usados nos controladores preditivos. Segundo, os algoritmos de controle avançado foram desenvolvidos em Matlab® baseados no controlador preditivo de horizonte infinito (IHMPC), no controlador preditivo robusto (RIHMPC) e no MPC econômico. Terceiro, o algoritmo de RTO foi desenvolvido no pacote de otimização em tempo real Simsci ROMeo®, onde o modelo rigoroso não linear do processo foi implantado incluindo as etapas de simulação, reconciliação de dados e otimização. Quarto, modificações e adaptações dos algoritmos e rotinas desenvolvidas foram feitas para permitir a comunicação de dados em tempo real usando o protocolo de transferência de dados OPC entre Matlab®, Simsci Dynsim® e Simsci ROMeo®. Finalmente, foram desenvolvidos o sequenciamento e automação dos algoritmos tanto para leitura e escritura de dados, assim como, para a rotina do RTO.
Para todas as estratégias propostas nesta Tese, foram incluídos exemplos de simulação representativos onde se pode evidenciar a estabilidade e convergência das estratégias propostas, chegando-se à conclusão que as estruturas propostas de RTO/MPC podem ser implementadas no sistema real.
Palavras-Chave: Controle de processos. Otimização. Simulação dinâmica. Unidade
de produção de propeno.
ABSTRACT
The aim of this Thesis is to study the implementation of advanced control, specifically, Model Predictive Control (MPC) and real time optimization (RTO) in an industrial process system using tools such as commercial software for process simulation and optimization. The proposed solutions can be considered as integration strategies of RTO and MPC with one and two layers.
In the two layer approach, the upper layer that considers a rigorous non-linear steady-state model of the process computes optimizing targets that are sent to the dynamic layer that are based on the MPC, which computes the necessary control actions to reach those targets and stabilize the process system. In the one layer strategy, also called as Economic MPC, the gradient of the economic function is included in the cost function of the predictive controller.
Both strategies were studied and their implementation in the energy-recovery propylene/propane splitter system of the propylene production unit at the Capuava Refinery of Petrobras was simulated. In order to accomplish this objective, the work was developed in several steps. Firstly, a dynamic simulation of the process was built in the dynamic simulator Simsci Dynsim® so that it could be used as a virtual plant in which the model identification could also be performed. Secondly, the advanced control algorithms were developed in Matlab® based on the Infinite Horizon Model Predictive Control (IHMPC), the robust predictive controller (RIHMPC) and the Economic MPC. Thirdly, the RTO algorithm was developed in the real-time optimization package Simsci ROMeo®, where the non-linear rigorous model of the process was built including the stages of simulation, data reconciliation and optimization. Fourthly, modifications and adaptation of the developed algorithms and routines were included to allow the real-time data communication considering the OPC data transfer protocol between Matlab®, Dynsim® and ROMeo®. Finally, a sequence of algorithms was developed and automated for data reading and writing, as well as, for the RTO sequence.
For all the strategies developed in this Thesis, representative simulation examples were presented in order to show the closed-loop stability and convergence of the proposed approaches, leading to the conclusion that the proposed RTO/MPC structures can be implemented in the real system.
Keywords: Process control. Optimization. Dynamic simulation. Propylene production
unit.
LIST OF FIGURES
Figure 2. 1 – Two-layer hierarchical structure without RTO ....................................... 22 Figure 2. 2 – Two-layer hierarchical structure with RTO ............................................ 23 Figure 2. 3 – Three-layer hierarchical structure with RTO .......................................... 23 Figure 2. 4 – One-layer hierarchical structure (Economic MPC) ................................ 25 Figure 2. 5 – ROMeo’s internal modules (ROMeo User Guide, 2012) ....................... 27 Figure 3. 1 – Schematic representation of the Propylene/Propane splitter ................ 33 Figure 3. 2 – Snapshot summary of the PP Splitter in Dynsim ................................... 47 Figure 3. 3 – PFD of the PP Splitter in Dynsim ........................................................... 48 Figure 4. 1 – Selected manipulated and controlled variable for step response test ... 49 Figure 4. 2 – Step responses of the PP splitter at different operating points ............. 50 Figure 5. 1 – PFD of the PP splitter in Simulation mode in ROMeo ........................... 79 Figure 5. 2 – PFD of the PP splitter in Data reconciliation mode in ROMeo .............. 81 Figure 5. 3 – SSD task in ROMeo RTS ...................................................................... 86 Figure 5. 4 – Model Sequence in ROMeo RTS .......................................................... 86 Figure 5. 5 –Sensitivity analysis tool in ROMeo OPS ................................................. 88 Figure 6. 1 – Two layer structure strategy .................................................................. 89 Figure 6. 2 – Controlled outputs IHMPC (First experiment) ........................................ 93 Figure 6. 3 – Manipulated inputs IHMPC (First experiment) ....................................... 94 Figure 6. 4 – Economic function IHMPC (First experiment) ....................................... 94 Figure 6. 5 – Economic function IHMPC with penalization (First experiment) ........... 95 Figure 6. 6 – Controlled outputs IHMPC (Second experiment) .................................. 96 Figure 6. 7 – Manipulated inputs IHMPC (Second experiment) ................................. 96 Figure 6. 8 – Economic function IHMPC (Second experiment) .................................. 97 Figure 6. 9 – Economic function IHMPC with penalization (Second experiment) ...... 97 Figure 6. 10 – Controlled outputs (First experiment) .................................................. 99 Figure 6. 11 – Manipulated inputs (First experiment) ............................................... 100 Figure 6. 12 – Economic function (First experiment) ................................................ 100 Figure 6. 13 – Economic function with penalization (First experiment) .................... 101 Figure 6. 14 – Controlled outputs (Second experiment) ........................................... 102 Figure 6. 15 – Manipulated inputs (Second experiment) .......................................... 103 Figure 6. 16 – Economic function (Second experiment) ........................................... 103 Figure 6. 17 – Economic function with penalization (Second experiment) ............... 104 Figure 6. 18 – One layer structure solution ............................................................... 105 Figure 6. 19 – Controlled outputs (First experiment) ................................................ 106 Figure 6. 20 – Manipulated inputs (First experiment) ............................................... 106 Figure 6. 21 – Economic function (First experiment) ................................................ 107 Figure 6. 22 – Economic function with penalization (First experiment) .................... 107 Figure 6. 23 – Controlled outputs (Second experiment) ........................................... 109 Figure 6. 24 – Manipulated inputs (Second experiment) .......................................... 109 Figure 6. 25 – Economic function (Second experiment) ........................................... 110 Figure 6. 26 – Economic function with penalization (Second experiment) ............... 110
LIST OF TABLES
Table 3. 1 – Typical feed composition of propylene/propane splitter ......................... 32 Table 3. 2 – Column T-03 description ........................................................................ 34 Table 3. 3 – Heat exchangers M-01 A/B description .................................................. 35 Table 3. 4 – Heat exchanger M-02 description ........................................................... 36 Table 3. 5 – Heat exchangers M-03 A/B description .................................................. 36 Table 3. 6 – Heat exchanger M-04 description ........................................................... 37 Table 3. 7 – Drum O-01 description ............................................................................ 37 Table 3. 8 – Separator O-02 description ..................................................................... 38 Table 3. 9 – Drum O-03 description ............................................................................ 38 Table 3. 10 – Drum O-04 description .......................................................................... 39 Table 3. 11 – Valve coefficients (Cv) .......................................................................... 42 Table 3. 12 – Overall heat transfer coefficients .......................................................... 42 Table 3. 13 – Compressor V-01 curves ...................................................................... 43 Table 3. 14 – Dynamic equipment data T-03 .............................................................. 43 Table 3. 15 – Main PID controllers used in the propylene/propane splitter ................ 44 Table 3. 16 – Main PID controller tuning ..................................................................... 45 Table 4. 1 – Different operating conditions of the PP splitter...................................... 50 Table 4. 2 – Transfer function models of the PP splitter ............................................. 52 Table 6. 1 – Output zones of the propylene/propane splitter ...................................... 90 Table 6. 2 – Input constraints of the propylene/propane splitter ................................. 90 Table 6. 3 – Feed molar composition (Disturbance) ................................................... 91 Table 6. 4 – IHMPC-OPOM tuning parameters .......................................................... 92 Table 6. 5 – IHMPC-Realignment model maximum input moves ............................... 92 Table 6. 6 – IHMPC-Realignment model tuning parameters ...................................... 93 Table 6. 7 – Robust MPC tuning parameters .............................................................. 99 Table 6. 8 – Economic MPC tuning parameters ....................................................... 105
NOMENCLATURE
Acronyms and Abbreviations
ROMeo Rigorous Online Modeling with equation-based optimization
Dynsim Dynamic Simulation
MPC Model Predictive Control
RTO Real Time Optimization
IHMPC Infinite Horizon Model Predictive Control
RIHMPC Robust Infinite Horizon Model Predictive Control
DOF Degrees of Freedom
PFD Process Flow Diagram
QP Quadratic Programming
LP Linear Programming
DMC Dynamic Matrix Control
LDMC Linear Dynamic Matrix Controller
MATLAB Matrix Laboratory
DCS Distributed Control System
NLP Non Linear Programming
OPC OLE for Process Control
OMPC Optimizing Model Predictive Control
PID Proportional, Integrative and Derivative
OPOM Output Predictive Oriented Model
PP Propylene/Propane
SSD Steady State Detection
DataRec Data Reconciliation
SICON Control System
APC Advanced Process Control
SimSci Simulation Science
PID Proportional Integral and Derivative
DA Data Acquisition
EDI External Data Interface
Roman Symbols
0 Null matrix of any dimension
A State transition matrix
A Auxiliary matrix used in the state and output prediction
B Matrix that relates system inputs and states
B Auxiliary matrix used in the calculation of output prediction
dlB Matrix that relates the control actions with the state component dx
slB Matrix that relates the control actions with the state component sx
C Matrix that relates the states to the system outputs
0,i jd Gain of the transfer function ,i jG
, ,di j kd k-nth residual of the transfer function ,i jG
dD Matrix that concentrates all , ,di j kd
e Error between the real and estimated state
feco Economic function of the system
F Matrix with the dynamic of the stable modes
( )G s Transfer function that represents the system to be controlled
m Control horizon
p Prediction horizon
nI Identity matrix of dimension n
nuI Auxiliary matrix used in the formulation of input constraints
nyI Auxiliary matrix used in computation of the cost function contribution of
the deviation between outputs and control zones
J Auxiliary matrix to relate control actions and components dlB
k Actual discrete time
,i jK Gain of transfer functions , ( )i jG s
FK Gain of the Kalman filter
L Gain of the state corrector used in the realigned IHMPC
na Transfer function orders of , ( )i jG s
nd Dimension of component dx ( nd nu ny na )
dN ( sN ) Matrix used to extract component dx ( sx )
nu Number of system inputs
ny Number of system outputs
uQ Weight in the controller objective function of the deviation between the
inputs and the optimizing targets
yQ Weight in the controller objective function on the deviation between the
outputs and their control zones
, ,i j kr kth pole of the transfer function , ( )i jG s
R Weight used in the controller cost function for the suppression of
control actions
, ,i u yS S S Weight matrix in the controller cost function of the slack variables
, , ,, ,i k u k y k
t Time
T Sampling time
u Vector of system inputs
desu Optimizing targets for the system inputs
maxu Upper constraint of the system inputs
minu Lower constraint if the system inputs
kV Total cost of the controller objective function at time k
x Vector containing the system states
dx Vector that computes the evolution of the system stable modes
sx System output prediction at steady-state
y Vector of the system outputs
( | )y k i k Prediction at time k of the system output at time k i
miny Lower limit of the system output control zone
maxy Upper limit of the system output control zone
spy Set-point for the system output
,sp ky Set-point computed by the controller at time k
lz State that stores the control actions implemented in the l previous
sample instants
Greek Symbols
Slack variable
,u k Slack variable for the deviation between the system inputs and
the optimizing targets
,y k Slack variable for the deviation between the system outputs and
the computed set-points
u Input move (Control action)
ku Vector of control actions computed for all control horizon m
maxu Maximum admissible input move
maxU Vector containing the input moves for all the control horizon m
ecof Gradient of the economic function feco
,i j Time delay for the transfer function , ( )i jG s
max Biggest time delay of the system transfer function , ( )i jG s
, ,i j k kth coefficient of the partial fraction expansion of transfer function
, ( )i jG s
Auxiliary matrix for the construction of
Matrix that relates outputs with state components dx
TABLE OF CONTENTS
1 INTRODUCTION ........................................................................................ 16
1.1 Advanced control and Real-time optimization ........................................... 16
1.2 Motivation ................................................................................................... 18
1.3 Objectives .................................................................................................. 19
1.4 Organization of the thesis .......................................................................... 19
1.5 Publications ................................................................................................ 20
1.5.1 Published paper .......................................................................................... 20
1.5.2 Submitted paper ......................................................................................... 20
1.5.3 Participation in conferences ....................................................................... 20
1.5.4 Awards ........................................................................................................ 21
2 LITERATURE REVIEW ............................................................................. 22
2.1 Model predictive control and Real time Optimization ................................ 22
2.2 Process simulation ..................................................................................... 26
2.2.1 Steady-state modeling and optimization using ROMeo® ........................... 27
2.2.2 Dynamic simulation .................................................................................... 28
2.3 Real time data communication................................................................... 29
2.3.1 Open platform communications (OPC) ...................................................... 29
2.3.2 MATLAB OPC toolbox ................................................................................ 30
3 HIGH PURITY DISTILLATION PROCESS AND DYNAMIC SIMULATION DEVELOPMENT ..................................................................................................... 32
3.1 Process description .................................................................................... 32
3.2 Equipment description ............................................................................... 34
3.2.1 Depropenizer splitter (T-03) ....................................................................... 34
3.2.2 Propylene compressor (V-01) .................................................................... 35
3.2.3 Heat exchangers ........................................................................................ 35
3.2.3.1 Reboilers M-01 A/B .................................................................................... 35
3.2.3.2 Reboiler M-02 ............................................................................................. 35
3.2.3.3 Cooler M-03 A/B ......................................................................................... 36
3.2.3.4 Cooler M-04 ................................................................................................ 36
3.2.4 Drums and separators ................................................................................ 37
3.2.4.1 Drum O-01 .................................................................................................. 37
3.2.4.2 Separator O-02 ........................................................................................... 37
3.2.4.3 Drum O-03 .................................................................................................. 38
3.2.4.4 Drum O-04 .................................................................................................. 39
3.3 Existing multivariable advanced controller (LDMC) ................................... 39
3.3.1 Controlled variables (Outputs) .................................................................... 39
3.3.2 Manipulated variables (Inputs) ................................................................... 40
3.4 Dynamic simulation of the PP Splitter ........................................................ 41
3.4.1 Equipment description for the dynamic simulation ..................................... 41
3.4.2 Valve coefficients ....................................................................................... 42
3.4.3 Heat transfer coefficients ........................................................................... 42
3.4.4 Curve of the heat pump compressor ......................................................... 43
3.4.5 Main dimensions of the Propylene/Propane splitter .................................. 43
3.4.6 Regulatory level PID control loop strategies and tuning ............................ 44
3.4.7 Initialization and convergence ................................................................... 46
3.4.8 PFD of the propylene/propane splitter in Dynsim ...................................... 47
4 DEVELOPMENT OF THE PROPOSED ADVANCED CONTROL STRATEGY ............................................................................................................. 49
4.1 Model identification .................................................................................... 49
4.2 The output prediction oriented model (OPOM) .......................................... 52
4.3 Realigned model of the propylene/propane splitter ................................... 55
4.4 IHMPC with zone control and optimizing targets ....................................... 62
4.4.1 The nominal IHMPC with OPOM ............................................................... 62
4.4.2 The nominal IHMPC with the realigned model .......................................... 65
4.5 Robust IHMPC with multi-model uncertainty ............................................. 69
4.6 The one layer Economic MPC ................................................................... 72
5 REAL TIME OPTIMIZATION DEVELOPMENT ........................................ 77
5.1 The two-layer RTO strategy based on ROMeo ......................................... 77
5.1.1 ROMeo’s simulation mode ......................................................................... 78
5.1.2 ROMeo’s data reconciliation mode (DataRec) .......................................... 80
5.1.3 ROMeo’s optimization mode ...................................................................... 82
5.1.4 On-line sequence algorithm ....................................................................... 83
5.1.4.1 Steady State Detection (SSD) ................................................................... 83
5.1.4.2 Model Sequence ........................................................................................ 86
5.2 One layer structure strategy ....................................................................... 87
6 RESULTS ................................................................................................... 89
6.1 Two-layer structure of the RTO/MPC integration ....................................... 89
6.1.1 Nominal case .............................................................................................. 91
6.1.1.1 IHMPC using OPOM .................................................................................. 92
6.1.1.2 IHMPC using the realignment model ......................................................... 92
6.1.1.3 Nominal IHMPC results .............................................................................. 93
6.1.2 Robust case ................................................................................................ 98
6.2 One layer structure (Economic MPC) ...................................................... 104
7 CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK .................. 112
References ........................................................................................................... 114
Appendix A – Development of the data communication interface ................ 119
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1 INTRODUCTION
1.1 Advanced control and Real-time optimization
The high complexity of chemical and petroleum processes, the need of maximizing
the economic profit of the plant, strong market competition, operational constraints
and environmental safety regulations make necessary the adoption of advanced
process control (APC) and real-time optimization (RTO) strategies. Therefore, one of
the key challenges in the process industry is how to best control and stabilize the
plant while looking for the most profitable operating point. Then, Model Predictive
Control (MPC), which is an advanced control standard in the oil refining industry, is
frequently implemented as one of the layers of a control structure where a Real Time
Optimization algorithm – laying in an upper layer of this structure – defines optimal
targets for some of the inputs and outputs (Kassmann et al., 2000). Several
examples of successful MPC implementations are documented in the literature in the
last 30 years, such as Cutler and Hawkins (1987), Carrapiço et al. (2009) and
Pinheiro et al. (2012). Most of the MPC applications in the industry are based on the
step-response model of the process as in the seminal application of MPC also called
Dynamic Matrix Control (DMC) developed by Cutler and Ramaker (1980). Despite of
the good performance of the step-response-based MPC, it does not guarantee
nominal stability, because of the finite output prediction horizon. Also, as it uses the
step-response coefficients of the process, the state of the model is non-minimal,
which means that a state with smaller dimension could be obtained.
As a result, MPC approaches based on state-space system representation that allow
the use of infinite prediction horizon have been proposed. Rodrigues and Odloak
(2003) developed a minimal order state-space representation for stable and
integrating systems, which is based on the step response of transfer function models.
The method was extended by Carrapiço and Odloak (2005) for time delayed systems
but the proposed model shown to be not always observable. More recently, to
circumvent this problem, Santoro and Odloak (2012) developed a new space-state
representation that is still equivalent to the step response but preserves observability.
This new space-state model is particularly suited to the implementation of the Infinite
Horizon Model Predictive Controller (IHMPC) with zone control and optimizing targets
for stable, integrating and time-delayed systems with guaranteed nominal stability.
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The lack of robust stability is still one of the weaknesses of the available model
predictive controllers that are usually implemented in industry. A robust controller is
able to maintain closed-loop stability at different operating conditions. Typically, at
each operating point the process system can be represented by a different linear
model. This case is called the multi-plant uncertainty by Qin and Badgwell (2003).
Then, a Robust Infinite Horizon Model Predictive Controller (RIHMPC) can be
proposed in which a set of models is used to represent the uncertain system and the
objective is to produce a control strategy that is stable for every single model of that
set (Badgwell, 1997; Lee and Cooley, 2000). In a similar way, polytopic uncertainty
can also be considered where the true model is assumed to be the convex
combination of a finite set of models that represent the vertices of a polytope
(Kothare et al., 1996; Wan and Kothare, 2002). These ideas can also be extended to
result in various strategies for implementing infinite horizon robust controllers that
allow integration with the RTO layer (Alvarez and Odloak, 2010; González and
Odloak, 2011).
Nevertheless, the design, tuning and implementation of the IHMPC or RIHMPC
integrated with RTO would be time and cost consuming without the use of dynamic
simulation. Steady-state and dynamic simulation based on first principles is a mature
technology. As plant designs are becoming more complex, integrated and interactive,
they tend to constitute a challenge to the design of a structure for the control of the
dynamic behavior of the plant (Svrcek et al. 2000). Commercial process simulators
are typically used to understand the process dynamics and interactions as well as to
the evaluation and tuning of control strategies before implementation. Then, various
control practitioners have adopted the use of dynamic simulation as an alternative to
plant testing so that the required performance information of the dynamic process
can be obtained. The advantages of conducting a step test on a dynamic simulation
instead of on the real plant are obvious. As no plant test is required, effort can be
minimized especially for processes with many variables and/or long settling times
(Alsop and Ferrer, 2004). Once all the required data is collected, it is possible to use
it for model identification and advanced control implementation, such as the DMC
multivariable controller implementation by Alsop and Ferrer (2006). Here, these ideas
will be extended to study the implementation of IHMPC and RIHMPC integrated with
RTO in a real process system of an oil refinery.
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This thesis presents the study of the implementation of advanced process control and
optimization, using rigorous process simulation software, in a Propylene/Propane
(PP) splitter of the Propylene Production Unit of the Capuava Refinery (RECAP),
PETROBRAS. In this study, it will be used an economic function of the process that
must be maximized so that profit would increase. In this way, the integration of RTO
and MPC will be developed considering two different hierarchical structures, the so-
called multi-layer approach, and the one layer approach also called Economic MPC.
1.2 Motivation
Nowadays, the refining and petrochemical industries are investing large amounts of
money in order to optimize their processes and to maximize profit. One basic
requirement to optimize a process is the implementation of a Model Predictive
Controller (MPC) that employs a linear dynamic model and can stabilize the process
considering constraints. Another requirement to optimize the process system is the
implementation of real-time optimization (RTO) that employs a steady-state non-
linear rigorous process model, and aims to calculate the optimal operation point of
the process that maximizes or minimizes an economical criterion.
As the propylene production unit produces high-purity propylene (99.5% polymer
grade), it justifies the energetic integration through the use of vapor recompression
technology, which makes the process dynamically more complex. Also, the selected
process system is well-instrumented with Proportional Integral and Derivative
controllers (PID) and composition analyzers; consequently, it is a perfect candidate to
the implementation of new advanced controllers and real time optimization.
Nevertheless, studying and implementing MPC and RTO strategies in the real plant
or process would be extremely expensive in terms of economical and time resources.
Therefore, the main motivation for this thesis began with the possibility of the use of a
rigorous dynamic simulation of the process as a virtual plant and, then, studying the
implementation of different MPC algorithms and RTO strategies. As a consequence,
no real plant test would be required for model identification and the study of these
implementations would be much easier.
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1.3 Objectives
General Objective
The main objective of this research is to develop and compare strategies for the
integration of RTO with the advanced control layer for the Propylene/Propane splitter
of the Refinery of Capuava (RECAP), PETROBRAS.
Specific Objectives
Developing a rigorous dynamic simulation of the plant using SimSci
Dynsim software
Identifying the process linear dynamic models of the plant at various
operating points
Developing the Infinite Horizon Model Predictive Control (IHMPC)
algorithm with zone control and optimizing targets
Developing the Robust Infinite Horizon Model Predictive Control
(RIHMPC) algorithm
Developing the Economic MPC algorithm
Developing the Real Time Optimization strategy for the plant using
SimSci ROMeo
Developing the communication interface and integration of the
advanced process packages with the advanced control algorithms
Integrating the RTO with the MPC algorithms using the two and one
layer approaches
Evaluating the economic gain of this implementation
1.4 Organization of the thesis
This thesis consists of seven chapters, including this introduction, divided as follows:
Chapter 2 presents the literature review, considering the MPC and RTO integration
strategies, process simulation and real-time data communication between the
different software modules. The third chapter describes the process and equipments
of the PP splitter. It also describes the current advanced control algorithm, which is
performed in that unit. It is presented the dynamic simulation developed using SimSci
Dynsim®, including the dynamic equipment data, regulatory (PID) control strategies,
20
initialization procedure and convergence analysis. Chapter 4 describes the proposed
advanced control algorithms, which were developed in MATLAB® and include the
IHMPC, RIHMPC and the Economic MPC. In chapter 5, it is described the RTO
algorithm that was developed in SimSci ROMeo® for the PP splitter, including all the
steps that are necessary for a real implementation. In chapter 6, the results of the
simulation tests are shown, using the proposed APC and RTO structures and
integration. Finally, chapter 7 presents the conclusions and discusses the main
contributions of this thesis, as well as the future perspectives of this work in terms of
a possible continuity.
1.5 Publications
Some of the results of this Thesis were submitted to the following journals and
conferences.
1.5.1 Published paper
Hinojosa A. I., Odloak D. Study of the implementation of a Robust MPC in a
Propylene/Propane splitter using rigorous dynamic simulation. Canadian
Journal of Chemical Engineering, 92(7), 2014.
1.5.2 Submitted paper
Hinojosa A. I., Capron B. D. O., Odloak D. Realigned Model Predictive Control
of a propylene distillation column. Accepted for publication. Brazilian Journal of
Chemical Engineering, 2015.
1.5.3 Participation in conferences
Hinojosa A. I., Odloak D. Novo conceito APC: Estudo na unidade de propeno
da RECAP. Refinery Wide Optimization by Invensys and Petrobras, RWO.
São Paulo, Brazil. May 14, 2013.
Hinojosa A. I., Capron B. D. O., Odloak D. Study of Infinite Horizon MPC
Implementation with Non-Minimal State Space Feedback in a Propylene
Production Unit using Dynamic Process Simulation. AIChE Annual Meeting.
San Francisco, CA, USA. November 03 – 08, 2013.
21
Hinojosa A. I., Odloak D. Robust Model Predictive Control Extension for
Integrating Systems with Time Delay, Optimizing Targets and Zone Control.
17th Congress and Exhibition of Automation, Instrumentation and Systems,
Brazil Automation, ISA. São Paulo, Brazil. November 05 – 07, 2013.
Hinojosa A. I., Odloak D. Using Dynsim® to study the implementation of
advanced control in a Propylene/Propane Splitter. 10th Symposium on
Dynamics and Control of Process Systems, DYCOPS. Mumbai, India.
December 18 – 20, 2013.
Hinojosa A. I., Odloak D. Robust multi-model predictive controller of a crude oil
distillation unit. 21st International Congress of Chemical and Process
Engineering, CHISA. Prague, Czech Republic. August 24 – 27, 2014.
1.5.4 Awards
Best presentation award, Process Optimization and Control I Session, 10th
Symposium on Dynamics and Control of Process Systems, DYCOPS.
Mumbai, India. December 18 – 20, 2013. For the paper contribution “Using
Dynsim® to study the implementation of advanced control in a
Propylene/Propane Splitter”.
22
2 LITERATURE REVIEW
2.1 Model predictive control and Real time Optimization
There are several strategies and structures to integrate and to implement real-time
optimization (RTO) and model predictive control (MPC). The classical approach
corresponds to the multi-layer structure in which RTO and MPC are executed in
different layers of the control structure. When the real-time optimization based on a
rigorous process model is not present in the control structure, one way to optimize
the process is to use a simplified linear economic function in a linear optimization
layer, which solves a linear programming (LP) or quadratic programming (QP). Using
this structure as shown in Figure 2.1, the upper layer sends approximated optimizing
targets to the inputs and/or outputs of the advanced control layer. These targets are
based on the predicted steady-state and the inputs are either minimized or
maximized while the process constraints are satisfied. The sampling time related with
the linear optimization algorithm is of the order of minutes, usually 1 minute in the oil
refining industry.
Figure 2. 1 – Two-layer hierarchical structure without RTO
When there is a RTO layer, possible hierarchical structures are shown in Figures 2.2
and 2.3. In the first case, there is a two-layer structure in which the upper RTO layer
sends optimizing targets to the advanced control layer (Marlin and Hrymak, 1996). In
this case, the RTO uses a rigorous steady-state non-linear model of the process,
which is executed with a sampling time of hours or even days. The advanced control
layer, which uses a linear dynamic model, is designed to drive the system to those
23
optimizing targets. Since, those targets could be unreachable for the advanced
controller; the system will be driven to an operating point as near as possible to the
point defined through the optimizing targets. The controller is executed with a
sampling time of minutes, usually 1 minute in the oil refining systems.
In the second case, the so-called three-layer structure is shown in Figure 2.3 (Rotava
and Zanin, 2005). The main difference in comparison with the two-layer structure is
the inclusion of a linear optimization layer, whose objective is to make the linear
dynamic model of the controller compatible with the non-linear model of the RTO
layer. Then, the linear optimization layer is formulated such that the difference
between the optimizing targets of this layer and the calculated RTO targets is
minimized.
Figure 2. 2 – Two-layer hierarchical structure with RTO
Figure 2. 3 – Three-layer hierarchical structure with RTO
24
The drawback of the structures presented above is that the RTO employs complex
stationary non-linear models to perform the optimization and has a sampling time
much larger than the sampling time of the controller layer. As a consequence, the
economic set-points (optimizing targets) calculated by the RTO may be inconsistent
with the model of the dynamic layer, producing in this way problems that go from
unreachability of the targets to poor economic performances (Alamo et al., 2012,
2014). As a result, a proper strategy to unify these (probable competing) objectives
becomes highly desirable from an operating point of view.
First, Zanin et al. (2002) proposed the inclusion of an economic function term (feco) in
the advanced controller cost function, producing what was called as optimizing
controller. This approach was tested by simulation and implemented in the Fluid
Catalytic Cracking (FCC) process presented in Moro and Odloak (1995). The main
disadvantage of this strategy is that the optimization problem is a non-linear one,
which becomes difficult to solve within the controller sampling time. It requires a high
computational effort and does not guarantee a global optimum.
To circumvent the problem of dealing with a non-linear optimizing problem (NLP), De
Souza et al. (2010) proposed a simplified version of the optimizing controller where
the gradient or reduced gradient – depending of constraint violation – of the
economic function was included in the controller’s cost function instead of directly
including the economic function. Then, the control objective becomes to zero the
reduced gradient of the economic objective while maintaining the system outputs
inside their control zones. Because of the use of a finite prediction horizon for the
controller outputs and the presence of the economic optimization component, there
could be some constraint violation. Then, at each sampling time, the predicted values
of the controlled variables were checked, in order to confirm that there are no
violations of the constraints. Depending of the existence of any violation of the output
bounds, additional constraints were included in the control problem or inputs were
removed from the calculation of the economic gradient. With such approach, the
integrated control/optimization problem became a quadratic programming (QP) that
could be solved with any of the available QP solvers, instead of a NLP solver as in
the previous approach. Simulations results with the FCC system presented in Moro
and Odloak (1995) showed that this strategy produces almost the same economic
25
benefit as the one with the full economic function inside the control cost, and could
be implemented in the real system.
The good simulation results obtained by De Souza et al. (2010) motivated Porfírio
and Odloak (2011) to implement this approach in an industrial toluene distillation
column. In this case, a rigorous steady-state distillation model is included in the
controller and it is used in the computation of the gradient of the economic objective
as can be observed in Figure 2. 4. Although this method was restricted to the case
where the economic function to be minimized is convex, practical results showed that
the approach is efficient and robust for several economic objectives of the toluene
system. Moreover, this controller remained in continuous operation since its
implementation in the Petrobras Control System (SICON).
ecof
u
Figure 2. 4 – One-layer hierarchical structure (Economic MPC)
Esturilio (2012) extended the previous approach to the infinite output prediction
horizon case using the Output Prediction Oriented Model (OPOM), which is a state-
space system representation that exactly emulates the step response. Also, It was
presented a discussion that showed that the consideration of an infinite output
horizon in the controller is sufficient to avoid that the economic optimization term
forces the controlled variables out of their constraints. Then, in this way, it is not
necessary to consider the reduced gradient of the economic function. This extended
approach was tested by simulation in an industrial ammonia reactor. Simulation
results, with or without disturbances, showed that the optimizing model predictive
controller (OMPC) had a satisfactory performance, and it can be tested and applied
in the real system.
26
More recently, Alamo et al. (2012) presented a MPC controller that also integrates
RTO in the same control problem, in such a way that the controller cost function
includes the gradient of the economic objective cost. However, instead of applying to
the system the optimal solution of the approximated problem, they propose to apply
the convex combination of a feasible solution and the approximated solution.
Therefore, a sub-optimal MPC strategy that only requires a QP solver was obtained,
and they show that the strategy ensures recursive feasibility and convergence to the
optimal steady-state in the economic sense. This approach was tested by simulation
in a simplified version of the FCC unit, and the simulation results showed that the
proposed algorithm has a good performance and can be tested using dynamic
simulation in order to prove its applicability in real systems. In the present work, the
approach of Alamo et al. (2012) will be implemented in the propene/propane splitter
and compared to the conventional multi-layer approach.
2.2 Process simulation
Nowadays, process simulation has become a common tool in the chemical process
industry and it is widely adopted for the design and optimization of processes. It is a
model-based representation of the unit operations of the process. Therefore, the
process simulation software describes the process systems through flow diagrams
where unit operations are connected using material and energy streams. The
process simulation may describe the steady-state behavior of the system as well as
the dynamic response to process disturbances. A possible benefit of using steady-
state and dynamic commercial process simulation software is to allow a better
judgment of the plant operating conditions in terms of the economy and productivity
of the plant, as well as the improvement of the control system (Bezzo et al., 2004).
Between the various worldwide companies that offers simulation software, there are
Aspen Technology (Aspen Tech) and Schneider Electric. Aspen Tech is one of the
largest software companies focused on optimizing process manufacturing. Among its
various solutions, there is Aspen RTO for real time optimization and Aspen Hysys
Dynamics for dynamic simulation of industrial processes. Schneider Electric is a
global technology company, commercializes software for simulation and optimization
of several processes. Invensys was taken and integrated by Schneider Electric in
2014. Among its global automation supplies and solutions, there is Simsci ROMeo®
(Rigorous Online Modeling with equation-based optimization) for advanced online
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virtually any type of customized unit operation in the process plant model using
equations.
2.2.1.2 Data reconciliation mode
In the second step, the abstract model is brought into harmony with the actual
operating conditions of the process. This is achieved by reconciling redundant and
sometimes inconsistent measurements using already well-established algorithms for
evaluating the validity of observed process data. Based on reconciled observed data,
the process model unit specifications and parameters are modified and adjusted to
make the process model conform even more closely to observed reality.
2.2.1.3 Optimization mode
In the third step, monetary values are assigned to pertinent process variables and
controller set-points are adjusted to maximize the economics of the overall process.
Typical assignments of monetary values would be prices of plant utilities, feed and
product materials.
2.2.2 Dynamic simulation
Dynamic simulation software is based on rigorous first-principle process models,
accurate and detailed thermodynamics to match operating conditions. The main
advantages of using dynamic simulation are:
No real plant tests are required
No risks for the real plant and no waste of products
Understanding of the interactions and dynamics of the process
APC implementation effort is minimized
Tuning of APC and PID controllers can be done more quickly and
without risking the plant safety
Dynamic simulation is most commonly used for process projects, evaluation and
validation of regulatory control systems and operator training. In this way, it can also
be used as a virtual plant in order to study APC and RTO implementations.
29
2.3 Real time data communication
In the petroleum process industry, it is essential to have high fidelity and real time
data. This data refers to all measured process variables and estimated parameters of
the process unit. The process data is obtained from measurement instruments such
as pressure gauges, thermometers, flow-meters and others. In order to collect and
observe this data in real time, with the information delivered immediately after
collection, it is needed a communication interface between the equipments that
contain the information and the computer memory that stores that information. This
interface could be achieved using the object linking and embedding technology
(OLE) developed by Microsoft.
This facility allows the linking and embedding of documents and other objects, and it
is an evolution of the dynamic data exchange (DDE). Its primary use is for managing
compound documents, but it is also used for transferring data between different
applications. There are other facilities for communication interfaces developed by
Microsoft such as COM (Component Object Model) and DCOM (Distributed
Component Object Model).
2.3.1 Open platform communications (OPC)
The OPC specification is based on the OLE, COM and DCOM facilities and it
involves the best characteristics of these technologies. At the beginning, it was called
OLE for process control as it defines a standard set of objects, interfaces and
methods for using in process control and manufacturing automation applications to
facilitate interoperability. The most common OPC specification is the OPC Data
Access (OPC DA), which is used to read and write real time data.
OPC was designed to provide a common bridge for Windows based software
applications and process control hardware. It is a hardware and software interface
standard using client and server modes. It offers a general standard mechanism for
client’s and server’s data communication. So, it makes easier the integration of
hardware and software of different manufacturers, and it offers an effective solution
for real time communication between PC and process devices (Lieping et al., 2007).
30
2.3.2 MATLAB OPC toolbox
MATLAB 7.0 and its latter versions integrate the OPC Toolbox, which is a function
module to expand the MATLAB numerical calculations environment. It implements
the object-oriented hierarchy and OPC server communication method by using OPC
data access standard. It provides a method to read or write OPC data through
accessing the OPC server directly in the MATLAB environment. By utilizing the OPC
Toolbox, it is possible to create the OPC customer application programming quite
easily in order to build the communication between MATLAB and the OPC server and
to perform a fast raw data analysis, measurement and control.
OPC Toolbox software implements a hierarchical object-oriented approach for
communicating with OPC servers using the OPC Data Access and Historical Data
Access Standards. Using the toolbox functions, it is possible to create OPC Data
Access (DA) and Historical Data Access (HDA) client objects, which represents the
connection between MATLAB and an OPC server. Using the properties of the client
objects, various aspects of the communication link can be controlled, such as time
out periods, connection status, and storage of events associated with the client.
Once a connection to an OPC DA server is established, Data Access Group objects
(dagroup objects) are created to represent collections of OPC Data Access items,
which can be read and written depending of the user objectives. To work with the
acquired data, it must be brought it into the MATLAB workspace. When the records
are acquired, the toolbox stores them in a memory buffer or on the disk.
Every server item on the OPC server has three properties that describe the status of
the device or memory location associated with that server item:
Value — The Value of the server item is the last value that the OPC server
stored for that particular item. The value in the cac0he is updated whenever
the server reads new values from the device. The server reads values from
the device at the update rate specified by the DA group object's Update Rate
property, and only when the item and group are both active.
Quality — The Quality of the server item is a string that represents information
about how well the cache value matches the device value. The Quality is
made up of two parts: a major quality, which can be 'Good', 'Bad', or
31
'Uncertain', and a minor quality, which describes the reason for the major
quality.
Time Stamp — The Time Stamp of a server item represents the most recent
time that the server assessed the information or the device set the Value and
Quality properties of that server item.
OPC Toolbox software provides access to the Value, Quality, and Time Stamp
properties of a server item through the properties of the data item object associated
with that server item.
32
3 HIGH PURITY DISTILLATION PROCESS AND DYNAMIC SIMULATION
DEVELOPMENT
3.1 Process description
The propylene splitter studied here is part of an industrial propylene production unit
from the Petrobras Capuava Refinery (RECAP) located at Mauá, São Paulo, Brazil.
This process was designed to produce 145 000 ton/year of propylene polymer grade
with high purity (99.5% molar at least). This production unit consists of three
distillation columns: depropanizer, deethanizer and depropenizer. The bottom liquid
product from the deethanizer (T-02) is mixed with similar composition streams
(Propint) of other refineries to feed the propylene/propane splitter (T-03). That is the
reason for using the feed flow-rate as a manipulated variable in the advanced control
strategy as it will be shown in subsection 3.4.2. The propylene/propane splitter
operates at a pressure of 9 kgf/cm2 (gauge) and is equipped with 157 valve trays.
In this section, which is schematically represented in Fig. 3.1, propylene is separated
from propane which also carries other hydrocarbons with four atoms of carbon. A
typical feed composition of column T-03 is shown in Table 3.1. The propylene stream
is produced as the top stream of the splitter and is sold to a nearby petrochemical
plant, and the propane stream obtained as the bottom product is stored in propane
spheres and sold as liquefied petroleum gas (LPG).
Table 3. 1 – Typical feed composition of propylene/propane splitter
Component % molar fraction
Ethane 0.0102
Propylene 64.41
Propane 34.77
i-Butene 0.337
1-Butene 0.061
Cis-2-Butene 0.0334
Trans-2-Butene 0.0334
1,3-Butadiene 0.012
i-Butane 0.298
Butane 0.0334
33
The distillation system studied here includes an energy recovery system through the
use of heat integration. As it can be observed from Fig. 3.1, there is a vapor
recompression system, which has become the standard heat pump technology in
distillation systems. Energy savings of about 50% have been reported in high-purity
separation processes (Bruinsma and Spoelstra, 2010). In this way, compressor V-01
increases the pressure and temperature of the vapor leaving the top of the column to
about 7 kgf/cm2 and 30°C, and this recompressed vapor is condensed in the reboilers
M-01 A/B and M-02 at the bottom section of the column. This column presents three
bottom reboilers that work in parallel. Reboilers M-01 A/B are vertical, while reboiler
M-02, which was included in a revamp project of the system, is horizontal. All of them
present variable exposed heat transfer area that depends on the condensed liquid
level in drums O-03 and O-04.
Figure 3. 1 – Schematic representation of the Propylene/Propane splitter
34
The top column product is sent to the compressor suction drum (O-01) and after that,
the vapor phase stream is compressed in the propylene compressor (V-01). It
increases the pressure until 16.2 kgf/cm2 and the temperature approximately up to
50°C in order to allow the exchange of heat in the column reboilers (M-01A/B and M-
02). This process stream is divided in three different streams, one of them goes to
the M-01A/B reboilers and is collected in the drum O-03. After that, it is cooled to 35
°C in the water cooler M-04; the second stream goes to reboiler M-02 and is
collected in the drum O-04, and the last stream goes to the water cooler M-03A/B so
that column’s top pressure could be controlled.
The outlet liquid products of M-04, M-02 and M-03A/B are mixed together and sent to
the reflux separator (O-02). The O-02 liquid product is divided into two streams, one
of them is sent back to the column as a reflux, and the other stream is the propylene
product that is pumped to the customer storage spheres.
3.2 Equipment description
In this section, the main equipments of the Propylene/Propane splitter are described
for a better understanding of the process and process operating conditions.
3.2.1 Depropenizer splitter (T-03)
This column, where the separation between propylene and propane is carried out,
contains 157 trays. The high number of trays is typical in this type of process
because of the difficult separation between propylene and propane and the desired
high purity of the propylene product. Some of the process conditions of this column
are describe in Table 3.2.
Table 3. 2 – Column T-03 description
Description Value Unit of measurement
Top operation pressure 9 kgf/cm2 g
Top operation temperature 18.9 °C
Bottom operation temperature 28.9 °C
Number of trays 157 (valve trays)
35
3.2.2 Propylene compressor (V-01)
This compressor is the heart of the energy integration system used in this process. It
increases the pressure up to 16.2 kgf/cm2 where the propylene temperature is,
approximately, 50.4 °C. This compressed stream is cooled in the bottom column
reboilers and in the water cooler M-03 A/B. The compressor runs at a fixed rotation
speed of 7250 rpm.
3.2.3 Heat exchangers
In this process unit, there are four heat exchangers that are described below, some
of them are used as reboilers and others as water coolers.
3.2.3.1 Reboilers M-01 A/B
These heat exchangers are the reboilers of the T-03 splitter. The heat transferred
from the compressed propylene to the bottom liquid product (propane) enables the
vaporization of propylene, therefore, minimizing the loss of propylene in the bottom
stream. These reboilers are vertical and connected in parallel. The required
information about these reboilers is described in Table 3.3:
Table 3. 3 – Heat exchangers M-01 A/B description
Description Value
Number of passes 1 Orientation Vertical
Number of tubes 6829 Outside tube
diameter 15.9 mm Tube length 6096 mm
Shell side Tube side Product Propylene Propane
Inlet temperature 50.4 °C 28.9 °C Outlet temperature 39.3 °C 29.3°C
3.2.3.2 Reboiler M-02
This heat exchanger was installed as the result of a revamp project in the propylene
production unit. In order to increase the propylene production, it was necessary to
increase the heat transfer capacity in the bottom section of the propylene column. As
a result, this new equipment was installed and kept in operation since then. It has the
36
same purpose as reboilers M-01 A/B, but it has difference characteristics as shown in
Table 3.4:
Table 3. 4 – Heat exchanger M-02 description
Description Value
Number of passes 2 Orientation Horizontal
Number of tubes 1978 Outside tube
diameter 15.87 mm Tube length 6000 mm
Shell side Tube side Product Propane Propylene
Inlet temperature 28.9 °C 50.4 °C Outlet temperature 29.3 °C 39.3 °C
3.2.3.3 Cooler M-03 A/B
These water coolers are connected in parallel and cool the compressed propylene
stream in order to allow the control of the column pressure at 9.0kgf/cm2. The
temperature is decreased down to 35 °C and after that it goes to the reflux separator
O-02. One of these equipments has the description given in Table 3.5:
Table 3. 5 – Heat exchangers M-03 A/B description
Description Value
Number of passes 2Orientation Horizontal
Number of tubes 1268 Outside tube
diameter 19 mm Tube length 6096 mm
Shell side Tube side Product Propylene Cooling water
Inlet temperature 50.4 °C 18.5 °C Outlet temperature 35 °C 30 °C
3.2.3.4 Cooler M-04
In this water cooler, the liquid product stream from O-03 is also cooled to 35°C and
sent to the reflux separator O-02. The required details of this heat exchanger are
given in Table 3.6:
37
Table 3. 6 – Heat exchanger M-04 description
Description Value
Number of passes 2 Orientation Horizontal
Number of tubes 1050 Outside tube
diameter 19 mm Tube length 6096 mm
Shell side Tube side Product Propylene Cooling water
Inlet temperature 39.3 °C 18.5 °C Outlet temperature 35 °C 30 °C
3.2.4 Drums and separators
The propylene/propane separation system has a number of auxiliary drums and
separators that need to be included in the dynamic simulation of the system.
3.2.4.1 Drum O-01
This is the suction drum of compressor V-01, which is also fed with the vapor phase
stream of O-02. Its main objective is to avoid that any liquid to be carried to the
compressor, because it would damage the compressor. The process conditions and
dimension of this drum are given in Table 3.7.
Table 3. 7 – Drum O-01 description
Description Value Unit of measurement
Operation pressure 9.20 kgf/cm2 g
Operation temperature 18.2 °C
Length 4.90 m
Diameter 3.36 m
Orientation Vertical -
3.2.4.2 Separator O-02
This separator also called reflux drum, receives liquid streams from M-03A/B, M-04
and M-02, and O-01. Because of the pressure drop inside this equipment and valves,
38
there is a partial vaporization of the liquid stream and some vapor is generated and
sent to the compressor suction drum O-01. The outlet liquid stream of this separator
is divided in two, one returns to the column as the reflux and the other goes to the
propylene storage sphere. The conditions and dimension of this drum are given in
Table 3.8.
Table 3. 8 – Separator O-02 description
Description Value Unit of measurement
Operation pressure 9.20 kgf/cm2 g
Operation temperature 18.2 °C
Length 4.90 m
Diameter 3.36 m
Orientation Vertical -
3.2.4.3 Drum O-03
This drum is located below reboilers M-01 A/B, in order to establish a liquid level that
modifies the heat transfer area of these reboilers. Then, its main objective is to allow
a reduction or increase of the reboiler’s heat exchange area. The details about this
drum are given in Table 3.9.
Table 3. 9 – Drum O-03 description
Description Value Unit of measurement
Operation pressure 15.8 kgf/cm2 g
Operation temperature 39.3 °C
Length 1.70 m
Diameter 0.85 m
Orientation Horizontal -
39
3.2.4.4 Drum O-04
This drum is located below reboiler M-02 in order to establish a liquid level that
modifies the heat transfer area of the reboiler. The required details for the simulation
of this drum are given in Table 3.10.
Table 3. 10 – Drum O-04 description
Description Value Unit of measurement
Operation pressure 15.8 kgf/cm2 g
Operation temperature 39.3 °C
Length 1.40 m
Diameter 1.25 m
Orientation Vertical -
3.3 Existing multivariable advanced controller (LDMC)
In order to maintain the propylene product specification and to minimize the loss of
propylene to the propane product stream, it has been already implemented a
conventional advanced controller of the DMC type (LDMC - Linear Dynamic Matrix
Controller). This controller is based on the step response coefficients of the plant
and, although it has shown a good performance from the viewpoint of keeping the
controlled variables inside their control zones, it does not have a guarantee of
stability and it is not integrated with a RTO algorithm. The existing controller
considers a control structure where there are three manipulated variables to control
another three outputs. The control structure is described in the next section.
3.3.1 Controlled variables (Outputs)
The existing advanced control structure aims at the control of the following variables:
Liquid level on heat exchangers M-01 A/B and O-03 (LC-5)
This controlled variable (y1) directly reflects the heat transfer area of
reboilers M-01A/B of the propylene/propane distillation column, because in
these reboilers, the area that really counts for the heat transfer is the area
of the tubes not submerged in the subcooled liquid. For example, for a
40
liquid level of 80%, the heat transfer area corresponds to only 20% of the
total area. The total height considered for the level controller (LC-5) is the
sum of the reboiler height and drum O-03 height, because the reboiler is
exactly above drum O-03. This controlled variable is used to guarantee a
minimum level for pumping and a maximum liquid level for process safety.
Propane molar composition in the propylene stream (AC-1)
This controlled variable (y2) indicates the quality of the propylene
product that must have a molar composition of at least 99.5%. This means
that the molar percentage of propane in the propylene stream must have a
maximum value of 0.5%, which is measured through a composition
analyzer.
Propylene molar composition in the propane stream (AC-2)
This controlled variable (y3) corresponds to the amount of propylene
that is lost to the propane stream. In the conventional advanced control
structure, this loss of propylene is limited to an upper bound of 8% of
propylene in the propane stream, but it should be limited to an upper
bound of about 2% for an optimal operation of the process.
3.3.2 Manipulated variables (Inputs)
The existing controller manipulates the following variables in order to maintain the
controlled variables defined in the previous section inside their respective control
zones.
Heat pump flow rate set-point (FC-3)
This process variable set-point (u1) affects mainly the loss of propylene in
the propane stream and the level of liquid in the drum O-03. This liquid
level should not be less than 10% in order to guarantee the reflux pump
will work properly.
41
Feed flow rate set-point (FC-1)
This process variable set-point (u2) affects the three controlled outputs and
can be manipulated in order to maintain all the controlled variables inside
their respective control zones. However, this variable should be mainly
manipulated to maximize the propylene production and to achieve the
scheduled propylene production.
Reflux flow rate set-point (FC-2)
The reflux flow rate set-point (u3) is manipulated mainly to attain the high
purity of the propylene product stream. It has a strong influence on that
controlled variable. The controller tends to manipulate the column reflux to
obtain a product of at least 99.5% molar of propylene.
3.4 Dynamic simulation of the PP Splitter
The dynamic simulation of the propylene distillation column was developed using the
software SimSci Dynsim. The idea is to consider the rigorous dynamic model as the
virtual plant so that the advanced control implementation, controller tuning and model
identification will be developed and tested without any cost (Dynsim User Guide,
2012). In order to represent a realistic operating scenario, all the regulatory PID
control loops will be included in the simulation besides the advanced control and real
time optimization algorithms. This dynamic simulation will also be useful to identify
the linear dynamic models at different operating points, which will be used in the
IHMPC and RIHMPC.
3.4.1 Equipment description for the dynamic simulation
To build up the dynamic simulation, a number of details are required for each
equipment involved in the simulation. The details of the most important equipment of
the propylene/propane splitter are presented in Tables 3.11 to 3.16, which were
obtained from process data sheets and project description of the propylene
production unit of the Capuava Refinery, Petrobras.
42
3.4.2 Valve coefficients
The valve coefficients (Cv) of the main control valves of the system must be provided
so that the dynamic simulation can be built up.
Table 3. 11 – Valve coefficients (Cv)
Valve Tag Cv
FV-1 47
FV-2 780
FV-3 500
FV-4 46
FV-5 26
FV-6 80
3.4.3 Heat transfer coefficients
The heat transfer coefficients of the four heat exchangers of the Propylene/propane
system are shown in Table 3.12.
Table 3. 12 – Overall heat transfer coefficients
Heat Exchanger tag U (kcal/hr-m2-°C)
M-01 A/B 541
M-03 A/B 439
M-04 631
M-02 923
43
3.4.4 Curve of the heat pump compressor
Table 3. 13 – Compressor V-01 curves
Flow rate (m3/hr) Head (kJ/kg) Efficiency (%)
13 000 28.449 81
13 500 28.2 82
14 000 27.958 84
15 000 27.7623 845
16 000 27.369 85
17 000 25.9965 845
18 000 25.0155 83
19 000 23.0535 80
20 000 19.62 73
3.4.5 Main dimensions of the Propylene/Propane splitter
Table 3. 14 – Dynamic equipment data T-03
Description Value Units
Tray spacing 0.45 m
Sump diameter 4.2 m
Sump height 2.55 m
Column diameter 4.2 m
Column height 76.55 m
44
3.4.6 Regulatory level PID control loop strategies and tuning
Table 3. 15 – Main PID controllers used in the propylene/propane splitter
PID controller Function
FC-1 Feed flow rate to column T-03
PC-1 Pressure at the top of T-03
LC-1 Liquid level at the bottom of T-03
FC-2 Reflux flow rate to column T-03
PC-2 Pressure at the outlet of compressor V-01 (Relief)
LC-2 Liquid level in the knockout drum O-01
FC-3 Total vapor flow rate through the heat pump
LC-3 Upper-liquid level in separator O-02
LC-3A Lower-liquid level in separator O-02
FC-4 Propylene flow rate to storage
LC-4 Liquid level in drum O-04 and reboiler M-02
FC-5 Propane flow rate to storage
LC-5 Liquid level in drum O-03 and reboilers M-01 A/B
FC-6 Flow rate through water cooler M-03
FC-7 Outlet flow rate of drum O-04
AC-1 Propane molar composition in the propylene product stream
AC-2 Propylene molar composition in the propane product stream
Considering the PID controllers listed in Table 3.15, the operation of the main
regulatory control loops can be summarized as follows: FC-5 and LC-1 are cascaded
such that the liquid level at the bottom of T-03 is maintained at 1.9 m. In the same
45
way, LC-4 and FC-7 are cascaded such that the liquid level in drum O-04 is kept at
65%. Also, PC-1 and FC-6 are cascaded such that FC-6 set-point is manipulated to
keep the column top pressure at 9.0 kgf/cm2. There is an override control strategy
involving controllers LC-3, LC-3A and FC-4. The resulting signal of the high selector
between LC-3 and FC-4 is sent to a low selector between this signal and the output
of LC-3A so that propylene liquid level in separator O-02 is held between its upper
and lower limits. As the propylene product purity is important, there is a composition
analyzer (AC-1) that cascades FC-2 so that reflux flow rate set-point changes
depending of the AC-1 output. Finally, in order to minimize the loss of propylene to
the bottom product of T-03, there is a composition analyzer (AC-2) that cascades FC-
3, the resulting signal is sent to a low selector with LC-5 output so that a minimum
liquid level in drum O-03 will be maintained. The tuning parameters of the PID
controllers are summarized in Table 3.16.
Table 3. 16 – Main PID controller tuning
PID controller tag Proportional gain Integral reset time (min)
FC-1 0.5 2
LC-1 2 4.5
FC-5 0.4 0.6
AC-2 0.05 420
LC-5 1 3
LC-4 0.9 20
FC-3 0.25 2
FC-6 0.5 1
PC-1 15 10
FC-2 0.15 0.45
FC-4 0.2 8
46
PID controller tag Proportional gain Integral reset time (min)
LC-2 2 1
LC-3 1.5 2
LC-3A 1 0.5
PC-2 20 10
FC-7 0.6 1
AC-1 0.08 360
3.4.7 Initialization and convergence
The initialization or convergence to an initial steady-state of the dynamic simulation of
the PP splitter is difficult and complex because most of the PID regulatory would be
in manual mode and process operation knowledge is needed to stabilize the process
system. In Dynsim, there are algorithms for the initialization at a converged known
steady-state. Initially, it is necessary to estimate the composition and flow rates of the
top and bottom products. Then, the operation of the plant should be simulated with
the PID regulatory controllers in manual mode until the pressure of the column and
the liquid levels inside the vessels are near the normal operation values. Then, PID
controllers can be switched on to the automatic mode.
It is easy to realize why the convergence to any steady-state is slow since the
stabilizing time of the propylene process is very large (about 20 – 30 hours). As a
result, many attempts (about 30) were necessary until the desired steady-state was
attained. The easiest way to save the system states in Dynsim is trough snapshots
that are similar to an instantaneous photo that can be recovered whenever it is
necessary. An example of the snapshots taken from the dynamic simulation can be
seen in Figure 3.2.
3.4.8
The
in Dy
8
complete
ynsim is sh
Figure 3.
PFD of
process flo
hown in Fig
. 2 – Snapsh
the propy
ow-sheet d
gure 3.3.
hot summary
ylene/prop
diagram (P
of the PP Sp
pane splitt
PFD) of the
plitter in Dyn
ter in Dyns
PP splitte
nsim
sim
er process d
47
developed
7
d
FFigure 3. 3 – PFD of the PPP Splitter inn Dynsim
488
4 D
In th
robu
imple
4.1
The
to ob
resp
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The
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DEVELOP
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ust advanc
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Figure 4
PMENT OF
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odel identif
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. 1 – Selecte
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ylene splitt
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odel, in th
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OPOSED A
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ADVANCE
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th the real
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ROL STRA
t of the no
were propo
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sed on line
del such a
4.1, step t
p in the se
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where it i
p response
l industrial
h settling t
ep response t
49
ATEGY
ominal and
osed to be
mization.
ear MPC is
s the step
tests were
et-points of
put.
s possible
e model. If
l system it
time of the
test
9
d
e
s
p
e
f
e
f
t
e
y1
y2
y3
The
follow
cond
capa
capa
proc
Pro
Feed
T
Fig
three diff
wing oper
dition; Poin
acity; Poin
acity as de
cess history
ocess varia
d flow rate
Top columnpressure
u1
ure 4. 2 – St
ferent ope
ating cond
nt 1 corres
nt 2 corres
efined in Ta
y database
Table 4
able
(u2) T
n kg
tep response
erating po
ditions: Po
sponds to
sponds to
able 4.1. T
e of the las
. 1 – Differen
UOM
Ton./h
gf/cm2g
u2
es of the PP
ints prese
int 3 corre
an operat
an operat
These oper
st two year
nt operating c
OperatPoint
29.51
9.0
2
splitter at dif
ented in F
esponds to
ting condit
ting point
rating poin
rs.
conditions of
ing 1
1
fferent opera
Figure 4.2
o the most
ion with a
with an in
ts were se
f the PP split
OperatingPoint 2
32.3
9.0
u3
ting points
correspo
t common
reduced p
ncreased p
elected bas
tter
g OpP
50
nd to the
operating
production
production
sed on the
perating Point 3
30.0
9.0
0
e
g
n
n
e
51
Process variable UOM Operating
Point 1 Operating
Point 2 Operating
Point 3
Reflux flow rate (u3)
Ton./h 255.8 273.02 268.0
Reflux temperature
°C 18.9 18.99 19.5
Compressor outlet temperature
°C 50.4 48.9 50.01
Reboiler M-01 A/B liquid level (y1)
% 51.9 49.6 42.1
Total heat pump flow rate (u1)
Ton./h 281.28 289.13 302.0
Propylene product flow rate
Ton./h 17.79 19.48 18.45
Propane molar composition in
Propylene stream (y2)
% 0.35 0.436 0.5
Propane product flow rate
Ton./h 11.72 12.82 11.55
Propylene molar composition in
Propane stream (y3)
% 8.0 7.54 1.0
So, in the proposed robust controller, the dynamics of the non-linear distillation
system will be approximately represented through three linear models that constitute
the multi-model set on which the robust controller will be based. The third model
will be used to implement the nominal IHMPC, which is based on the nominal model.
Once the step response coefficients were obtained for each excitation step, there
was used an autoregressive exogenous model (ARX) to obtain the transfer function
that reproduces the step response. The parameters corresponding to each of these
identified models can be seen in Table 4.2 that shows the three transfer function
models, where the time constants are in minutes.
52
Table 4. 2 – Transfer function models of the PP splitter
5 6 6 6
1 2 5 2 5 2 5
5
2
1.035 0.1303 0.145
29 1 118.7 1 118.6 1
5.589 10 2.73 10 1.496 10 3.141 10 exp( 111 )
0.02156 7.439 10 0.01688 9.857 10 0.6 0.01355 7.557 10
0.0003027 3.267 10
0.2 0.01034
s s s
s sG
s s s s s s
s
s s
5
5 2 4 2 5
5 6
2 2
0.000135 1.942 10
4.756 10 1.4 0.04418 4.425 10 0.0149 5.535 10
1.237 0.18exp( 3 ) 0.0928exp( 3 )
32.2 1 138.79 1 126.54 1
( 5.361 10 2.51 10 )exp( 7 )
0.0225 6.87
s s s s
s s
s s s
s sG
s s
6 6
5 2 5 2 5
5 5
2 5 2 4 2
1.252 10 exp( 4 ) 2.843 10 exp( 92 )
3 10 0.01439 9.732 10 0.6 0.01465 7.883 10
0.0003005 3.039 10 0.000122exp( 3 ) 2.11 10 exp( 7 )
0.2 0.0129 3.846 10 1.4 0.054 6.75 10 0.0141
s s
s s s s
s s s
s s s s s
5
5 6 6 6
3 2 5 2 5
3.956 10
1.44 0.1684exp( 6 ) 0.1821exp( 10 )
31.5 1 159 1 119.8 1
( 7.657 10 3.69 10 )exp( 7 ) 2.094 10 exp( 4 ) 3.501 10 exp(
0.017156 6.439 10 0.01776 9.857 10
s
s s
s s s
s s sG
s s s s
2 5
5 5
2 5 2 4 2 5
98 )
0.6 0.013 6.257 10
0.0003423 3.704 10 0.0001589 1.99 10 exp( 9 )
0.2 0.0142 4.986 10 1.4 0.0423 4.194 10 0.0184 5.66 10
s
s s
s s
s s s s s s
4.2 The output prediction oriented model (OPOM)
To be applied in the controller considered here, the transfer function models defined
in Table 4.2, have to be translated into a more suitable state space form. For this
purpose, consider a system with nu inputs and ny outputs, and assume that this
system can be represented by a transfer function model ( )G s :
1,1 1,
,1 ,
( ) ( )( ) ,
( ) ( )
nu
ny ny nu
G s G sG s
G s G s
Also, to simplify, assume that the poles relating any input uj to any output yi, are non-
repeated and na is the transfer function order , ( )i jG s that can be represented as
follows:
,, ,0 , ,1 , ,,
, ,1 , ,2 , ,
( )( )( ) ( )
i j
nbsi j i j i j nb
i ji j i j i j na
b b s b sG s e
s r s r s r
The step response of these transfer functions can be obtained using the partial
fraction expansion:
53
, , , ,
0, , , ,1 , , ,
, 2, ,1 , ,
( )( ) i j i j i j i j
d d is s s si j i j i j i j na i j
i ji j i j na
G s d d d dS s e e e e
s s s r s r s
and, a state space realization that is equivalent to the step response model of the
system, also designated Output Predictive Oriented Model (OPOM) and originally
presented in Rodrigues and Odloak (2003) and extended by Santoro and Odloak
(2012) for the time delayed system, can be obtained. This model form was used here
to represent the propylene/ propane splitter. For this purpose, let us designate θmax
as the maximum dead time of the system and let the transfer function order na be the
same for any input and output. Then, a state space model that represents the
propylene distillation column, which has only stable poles can be written as follows:
( 1) ( ) ( )
( ) ( )
x k Ax k B u k
y k Cx k
max max
max max
max max
1 2 1
1 2 1
1 1
2 2
0( 1) ( )
( 1) ( )0
( 1) ( )0 0 0 0 0 0( 1) ( )0 0 0 0 0
( 1) ( )0 0 0 0 0
s s s ss sny
d dd d d d
nu
nu
I B B B Bx k x k
x k x kF B B B B
z k z k
z k z kI
z k z kI
0
0
( )0
0
s
d
nu
B
B
Iu k
(4.1)
max
1
2
( )
( )
( )( ) 0 0 0
( )
( )
s
d
ny
x k
x k
z ky k I
z k
z k
(4.2)
where,
maxmax 1, , , , , , ,nx s ny d ny nu na nux nd nu ny na nx ny nd nu x x z z
The main advantage of adopting the model described above is that each component
of the state vector xd can be associated with a particular dynamic mode. In the state
space model in Equation (4.1), the state component xs corresponds to the predicted
output steady state, xd corresponds to the dynamic modes, and, considering a stable
system, they tend to zero when the system approaches a steady state. For the case
of non-repeated poles F is a diagonal matrix with components of the form it re where
54
ir is a pole of the system and t is the sampling period. The upper right block of
matrix A is included to account for the time delay of the system.
Matrices slB , with max1, ,l can be computed as follows:
If , ,i jl
then ,
0sl i j
B
If , ,i jl then 0
,,
sl i ji j
B d
Construction of matrices dlB is a little more subtle. If there are no dead times (l = 0)
then 0d dB D FN , where matrices dD and N are computed as follows:
, nd nu
JJ
N ny N
J
1 0 0
1 0 0
0 1 00
0 1 0
0 0 1
0 0 1
na
naJ
na
nu na nuJ
1,1,1 1,1, 1, ,1 1, , ,1,1 ,1, , ,1 , ,diagd d d d d d d d dna nu nu na ny ny na ny nu ny nu naD d d d d d d d d
d nd ndD
1,1,1 1, ,1 1, , ,1,1 ,1, , ,1 , ,1,1,diag nu nu na ny ny na ny nu ny nu nanat r t r t r t r t r t r t rt rF e e e e e e e e
nd ndF
55
1,1,1
1,1,
1,2,1
1,2,
1, ,1
1, ,
,1,1
, ,
1,1,1
1,1,
1,2,1
1,2,
1, ,1
1, ,
,1,1
, ,
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
na
na
nu
nu na
ny
ny nu na
rd
rdna
rd
rdna
drd
nu
rdnu na
rdny
rdny nu na
d e
d e
d e
d e
D FNd e
d e
d e
d e
, d nd nuD FN
Alternatively, if l ≠ 0, then each matrix dlB would have the same dimension as dD FN
where those elements corresponding to transfer functions with a dead time different
from l are replaced with zeros.
Finally, matrix is defined as follows
Φ 0
Ψ
0 Φ
, Ψ ny nd , Φ 1 1 , Φ nu na
4.3 Realigned model (RM) of the propylene/propane splitter
An alternative formulation of the state space model equivalent to the model defined in
Table 4.2 can be obtained following the approach presented in Maciejowski (2002).
The realigned model is based on the following differences equation:
1 1
( ) ( ) ( )na nb
i ii i
y k a y k i b u k i
(4.3)
Where, na is the number of system poles and nb is the number of system zeros, and
ia , ib are appropriate dimension matrixes constructed using the difference Equation
(4.3).
56
In practice, this model could be obtained by discretizing the transfer function model of
Table 4.2 or straight from the plant step tests and trough model identification
techniques. In the present work, the transfer functions were discretized to obtain the
model.
The model defined in Equation (4.3) corresponds to the following state-space representation:
( 1) ( ) ( )
( ) ( )
x k Ax k B u k
y k Cx k
(4.4)
Where,
0y uA A
AI
, uBB
I
, y uC C C (4.5)
1 1 2 1
( 1) ( 1)
0 0 0
0 0 0
0 0 0
ny na na na
ny
ny na ny nanyy
ny
I a a a a a a
I
IA
I
(4.6a)
2 1
( 1) ( 1)
0 0 0
0 0 0
0 0 0
nb nb
ny na nu nbu
b b b
A
(4.6b)
1
( 1) ( 1) ( 1) ( 1)
0 0 0
0 00 0, ,
0 00 0
nu
nuny na nu nu nb nu nb nu nb nuu
nu
b I
IB I I
I
(4.7)
( 1)0 0 ny ny nay nyC I , ( 1)0 0 0 nu nb
uC (4.8)
57
The state x at time k is defined as:
( )( ) , ( 1) ( 1)
( )y nx
u
x kx k nx na ny nb nu
x k
(4.9)
where,
( ) ( 1) ( 1) ( )TT T T T
yx y k y k y k na y k na
( ) ( 1) ( 2) ( 1)TT T T
ux k u k u k u k nb
(4.10)
The main advantage of using the model described above is that the partition of the
state defined in Equations (4.9) and (4.10) is convenient in order to separate the
state components related to the system output at past sampling steps from those
related to the past inputs. Also, since the model is written in terms of the input
increment, model defined in Equation (4.4) contains the modes of model defined in
Equation (4.3) plus ny integrating modes. In this way, it is possible that the controller
using this state-space system representation has a better performance when
considering a real situation in which non-measured disturbances are introduced into
the system. Therefore, the controller is able to reject faster the disturbance as well as
to converge faster to the real state in case of model mismatch between the real plant
behavior and the linear model considered in the controller.
Matrix A in model Equation (4.4) has the following property:
Property 1. Matrix A is rank deficient. Furthermore,
( ) ( ) , 1 1
( ) ( 1) ( 1) , 1
n
n
rank A nx n nu n nb
rank A nx nb nu na ny n nb
The last equality implies: 1 1n n nb nbA A A for 1n nb , where
0.
0 0yA
A
58
It is well-known that any integrating mode cannot be allowed to proceed over an
unbounded time interval without control action. Therefore, in order to develop an
MPC based on model Equation (4.4) with infinite prediction horizon, first it is needed
to find a state transformation that makes explicit the stable and integrating parts of
the plant.
One solution could have been to adopt the eigenvalue-eigenvector Jordan
decomposition:
com com dAV V A
where dA is a block diagonal matrix (Jordan canonical form) that makes explicit the
different dynamic modes of the system, and the columns of comV are the
eigenvectors, or generalized eigenvectors, of A.
However, since the realigned model defined in Equation (4.5) is rank deficient
(property 1), comV is not invertible. As a result, it would not possible to recover the
original states from the transformed states and the main advantage of the realigned
model (i.e., the avoidance of an observer) is lost.
Nevertheless, it is possible to find states along the prediction horizon where a
similarity transformation can be performed (i.e. comV is invertible). To define these
states, let us consider the following sequence of input moves:
( ), ( 1), , ( 1),0,u k u k u k m
Taking into account this input moves sequence and Property 1, the open-loop state
predictions at time instants beyond the control horizon m, and computed at time k,
can be written as follows:
59
1 1
( | ) ( ) ( 1) ( 1)
( 1| ) ( | )
( 1| ) ( 1) 0 ( 1) ( 2)
( ' | ) ( | ) ( | )
( ' | ) ( ' ) 0 0 0 , 1
TT T Ty
TT T Ty
nb j j nb
TTy
x k m k x k m u k m u k m nb
x k m k Ax k m k
x k m k x k m u k m u k m nb
x k m j k A x k m k A A x k m k
x k m j k x k m j j
Where, ' 1m m nb . This means that beyond step 'k m , the predictions of the last
( 1)nb nu state components will be null and that B also does not affect the evolution
of the state as the input moves are assumed to be null beyond 'k m .
In this scenario, consider the following transformation:
y dA V VA
where the similarity transformation matrix V is now full rank since the modeled
system is supposed to have ( 1)na ny poles. Matrix dA is again a block diagonal
matrix.
Consider now the following augmented matrices:
( 1)
0nx na nyV
V
and 1 ( 1)0 na ny nxinV V
Then the following equality holds:
0
0
nstnst
d in nst st stst
VFA VA V V V
VF
(4.11)
Where, the columns of nstV and stV span the integrating and stable subspaces of the
system, respectively. Also nst nns nnsF is the state block diagonal matrix
corresponding to the integrating modes and nst nns nnsF is the state block diagonal
matrix corresponding to the stable modes of the system ( nns is the total number of
integrating modes and ns is the total number of stable modes)
60
The transformation defined in Equation (4.11) has the following property related to
the state matrix A of the model defined in Equation (4.4):
Property 2. 1 1n n nb nbin d inV A A V A I, for 1n nb
Remark 1. For the predicted states where the similarity transformation is possible, the
two system representations: ( 1) ( )dz k A z k and ( 1) ( )x k Ax k are equivalent. In
fact, for any time instant ( ' ), 0k m j j :
( ' | ) ( ' | ),jdx k m j k A x k m k
and using Equation (4.11), it is obtained:
( ' | ) ( ' | )
( ' | ) ( ' | )
jd in
jd
x k m j k VA V x k m k
z k m j k A z k m k
Where, ( ) ( ), ( ) nzinz k V x k z k and ( 1)nz na ny .
Now, the integrating and stable states of the transformed model that is equivalent to
model Equation (4.4) can be computed as
( )( ) ( ) ( )
( )
nstnznst
instst
Vz kz k V x k x k
Vz k
In addition, the system output can be expressed as
( )( ) ,
( )
nst
d st
z ky k C
z k
Where, , , ,nst nns st nsdC CV z z nz nns ns .
Model ( 1) ( )dz k A z k may have a different structure depending on the system. In the
case of systems with nun non-repeated integrating modes, matrix nstF can be written
as:
61
,0nynst nns nns
nun
I DF
I
Where, nns ny nun , nyI and nuI are identity matrices, corresponding to the
integrating modes coming from the model incremental form and coming from the
plant, respectively, and, ny nunD is a particular matrix, where , 1i j for i j and
, 0i j for i j . It is assumed, for the sake of simplicity, that max( , ).nun ny nu
In addition, the state component nstz can be decomposed as ( )
`,( )
inst
un
z kz
z k
and the
transformation matrix nstV as inst
un
VV
V
. The state ( )i nyz k corresponds to the
integrating states related to the incremental model (4.1), ( )nun nunz k corresponds to
the actual integrating modes of the system.
Now, for the system defined above, similarity transformation (4.11) has the following
properties.
Property 3
1 1( )n nst n nb nbnst nstV A F V A
1 1and ( )n st n nb nb
st stV A F V A for all 1n nb
Property 4
1 for all 1n nbun unV A V A n nb
Property 5
If it is defined an output set-point spy , then the set-point of the system represented in
(4.4) can be defined as 0 0T Tsp sp spx y y .
Furthermore, it can also be defined a set-point to the transformed state as sp spinz V x
which satisfies:
62
,
,
,
0
0
i sp spi
sp un sp
st sp
z V x
z z
z
and , 1i sp i spdz C y
4.4 IHMPC with zone control and optimizing targets
In most of the advanced control MPC applications, the outputs need only to be kept
inside specified ranges (zones) instead of at exact values. The control zone strategy
is then usually implemented in applications where the exact values of the controlled
outputs are not required, as long as these outputs remain inside a range with
specified limits. To handle the zone control strategy, it is necessary an appropriate
strategy to the output error penalization in the usual MPC cost function. Typically, the
output weight is made equal to zero when the system output is inside the range and
the output weight assumes a value that is not null when the outputs predictions are
outside their control zones. While the outputs predictions are inside the zones, the
inputs are free to be moved towards their optimum values. Another strategy to
implement the output control zone is to assume that the output set-points are free to
be moved inside the control zones. In the latter approach, the set-points become
additional decision variables of the control problem.
4.4.1 The nominal IHMPC with OPOM
Based on the work of González and Odloak (2009), and taking into account that the
propylene/propane splitter is an open loop stable system, it is considered the
following cost function, which is based on the nominal model:
( | ) ( | ), , , ,0
( | ) ( | ), , , ,0
1( | ) ( | ) , , , ,0
TV y k j k y Q y k j k yyk sp k y k sp k y kj
Tu k j k u Q u k j k uudes k u k des k u kj
Tm T Tu k j k R u k j k S Sy uy k y k u k u kj
(4.12)
where |u k j k is the control move computed at time k to be applied at time k+j,
m is the control or input horizon, Qy, Qu, R, Sy, Su, are positive weighting matrices of
appropriate dimension, ysp,k and udes,k are the output set point and input optimizing
target, respectively. The output target ysp,k becomes a computed set point when the
63
output has no optimizing target and consequently the output is controlled within a
zone. This cost explicitly incorporates an input deviation penalty that tries to
accommodate the system into an optimal economic stationary point. The slack
variables ,y k and ,u k eliminate any infeasibility of the control problem. It can be
shown that the cost defined in Equation (4.12) will be bounded only if the following
constraints are included in the control problem:
max , ,| 0ssp k y kx k m k y
. ,1| 0des k u ku k m k u
(4.13a)
(4.13b)
The IHMPC also includes constraints related with the actuator bounds and moves:
min max
min max
| 0,1, , 1
| ,
u u k j k u j m
u u k j k u j
(4.14)
Since the proposed controller considers the existence of input targets, udes, constraint
(4.13b) means that the input target shall be reached at the end of the control horizon.
However, in order to assure the feasibility of control problem that is solved by the
controller at each sample time, it cannot be imposed the exact value of the inputs at
the end of the control horizon, and instead a relaxed constraint will be used.
The slack variable ,u k , by definition, is unrestricted and guarantees feasibility of
Equation (14.3b) in any condition. The use of this slack variable is heavily penalized
in the objective function to prevent the appearance of offset in the desired input,
which corresponds to , 0u k .
As explained before, there are not fixed set points for the outputs as in the majority of
the MPC formulations. It will be considered a control zone in which the controlled
output variables must remain. As a result, when the value of set-point ysp,k is not a
parameter defined by the optimization layer, it becomes a decision variable of the
control problem. A constraint must be imposed on this set-point and corresponds to
the definition of control zone:
min , maxsp ky y y (4.15)
64
Constraint (4.13a) means that, it is desired that the predicted values of the outputs at
steady-state be equal to the set-points. As it is not always possible to attain this
target after a finite number of steps, there exists a analogous procedure as used in
constraint (4.13b), that is, the inclusion of slack variables, ,y k , to guarantee the
feasibility of the control problem.
Nevertheless, the system has time delays and it is necessary to wait m+ max time
intervals until the last control action affects the output with the largest time delay.
In the case of systems with time delay, the term corresponding to the infinite output
error in the cost Vk can be separated in two parts: the first part goes from the current
time k to the end of the control horizon plus the largest time delay, maxk m , while
the second part goes from time max 1k m to infinity. This is so because beyond
the control horizon there are no control actions that will be implemented and
consequently, the infinite series can be reduced to a single terminal cost term. As a
result, the cost defined in Equation (4.12) can be developed as follows:
max
max
, , , ,0
, , , ,1
1
, , , ,0
, ,
( | ) ( | )
( | ) ( | )
( | ) ( | )
( | ) ( | )
Tm
k sp k y k y sp k y kj
T
sp k y k y sp k y kj m
Tm
des k u k u des k u kj
T
des k u k u dej m
V y k j k y Q y k j k y
y k j k y Q y k j k y
u k j k u Q u k j k u
u k j k u Q u k j k u
, ,
1
, , , ,0
( | ) ( | )
s k u k
TmT T
y k y y k u k u u kj
u k j k R u k j k S S
(4.16)
Finally, the objective function of the controller could be defined as:
65
, , , ,, , , ,min
k y k u k i k sp kk
u yV
Subject to:
min max
min max
,
min , max
max , ,
1, ,
1, ,
1 0
0
b
des u k
sp k
ssp k y k
u u k j u j m
u u k j u j m
u k m u
y y y
x k m y
(4.17)
It is easy to show that the problem defined in Equation (4.17) can be characterized as
a quadratic programming because all constraints are linear and the objective function
is quadratic. The advantage of solving a quadratic programming is the good
robustness of the commercial solvers and the guarantee that the optimum solution is
the global optimum because of the convexity of the problem.
4.4.2 The nominal IHMPC with the realigned model
The optimization problem that defines the MPC based on the nominal realignment
model that is proposed here is the same as the problem solved by the IHMPC based
on OPOM, with some particularities related with the realignment model (González et
al., 2009 and González and Odloak, 2009).
For this purpose, consider the control cost of the IHMPC in its original form (without
the slacks):
0
( ( | ) ) ( ( | ) )sp T spk
j
V Cx k j k y Q Cx k j k y
1
, ,0 0
( | ) ( | ) ( ( | ) ) ( ( | ) )m
T Tdes k u des k
j j
u k j k R u k j k u k j k u Q u k j k u
(4.18)
Then, González and Odloak (2009) have shown that for the infinite sum:
0
( ( | ) ) ( ( | ) )sp T sp
j
Cx k j k y Q Cx k j k y
to be bounded, it is necessary to include the following constraint in the control
problem:
66
,( ') 0, ' 1,nst nst spz k m z m m nb (4.19)
Constraint (4.19) can also be expressed in terms of the current state ( )x k as follows:
' ' ,( ) 0m nst spnst nst aug kV A x k V B u z (4.20)
Where,
' 1 ' 2 '
'
, ( | ) ( 1| ) 0...0T
m m T Taug k
m m
B A B A B B u u k k u k m k
(4.21)
In addition, as in the IHMPC based on OPOM, for the third term on the right hand
side of Equation (4.18) to be bounded, it is necessary to add the following constraint:
,( 1) 0des ku k m u (4.22)
Since, in most cases there are constraints on the manipulated variables, constraints
(4.19) and (4.22) may happen to be unfeasible at some time instant and
consequently can make the optimization problem unfeasible. Thus following
(Gonzáles et al., 2007), the region where the controller is feasible is enlarged by
including slack variables in the control problem. The cost defined in Equation (4.18) is
then extended as follows:
0
1
0
, , , ,0
, ,
( ( | ) ( , )) ( ( | ) ( , ))
( | ) ( | )
( ( | ) ) ( ( | ) )
sp T spk y
j
mT
j
Tdes k u k u des k u k
j
nstT nst Tk y k u k u u k
V Cx k j k y CV k j Q Cx k j k y CV k j
u k j k R u k j k
u k j k u Q u k j k u
S S
(4.23)
Where,
( , )( , )
0 0
nst nstj nzk
d
k jk j A
67
As in the IHMPC described in the previous section nstk e ,u k are slack variables
introduced to enlarge the domain of attraction of the proposed controller. yS and uS
are positive weighting matrices of appropriate dimension.
As a result of this modification, the terminal constraint defined in Equations (4.19)
and (4.22) becomes:
,( ') ( , ') 0nst nst sp nstz k m z k m (4.24)
, ,( 1) 0des k u ku k m u (4.25)
Constraint (4.24) can be expressed in terms of the current state x as follows:
' ' , '( ) ( ) 0m nst sp nst m nstnst nst aug k kV A x k V B u z F (4.26)
To these terminal constraints, one has to add the constraints related to the input and
the input increment:
( | ) , 1,..., 1u k j k U j m (4.27)
Where,
max max
min max
0
( | )
( | )( 1) ( | )
j
i
u u k j k u
U u k j ku u k u k i k u
In addition, the constraint that defines the boundaries of output zones must be
added:
min maxspy y y (4.28)
If constraint (4.25) is satisfied, then the third term on the right hand side of Equation
(4.18) can be written as follows:
, , , ,0
( ( | ) ) ( ( | ) )Tdes k u k u des k u k
j
u k j k u Q u k j k u
(4.29)
68
1
, , , ,0
( ( | ) ) ( ( | ) )m
Tdes k u k u des k u k
j
u k j k u Q u k j k u
Now the first term of the right hand side of (4.18) can be developed as follows:
0
' 1
0
'
( ( | ) ( , )) ( ( | ) ( , ))
( ( | ) ( , )) ( ( | ) ( , ))
( ( | ) ( , )) ( ( | ) ( , ))
sp T spy
j
msp T sp
yj
sp T spy
j m
Cx k j k y CV k j Q Cx k j k y CV k j
Cx k j k y CV k j Q Cx k j k y CV k j
Cx k j k y CV k j Q Cx k j k y CV k j
(4.30)
Considering constraint (4.26), the second term on the right hand side of (4.30) can be
developed as follows:
'
'
'
,
( ( | ) ( , )) ( ( | ) ( , ))
( ( | ) ( , )) ( ( | ) ( , ))
( ( | ) ( , )) ( ( | ) ( , ))
( ) ( ( ' | )
sp T sp
j m
sp T T sp
j m
sp T T sp
j m
nst i nst nst s
Cx k j k y CV k j Q Cx k j k y CV k j
x k j k x V k j C QC x k j k x V k j
Vz k j k Vz V k j C QC Vz k j k Vz V k j
F z k m k z
0
0
0
,
0
( , ))
( ) ( ' | )
( ) ( ( ' | ) ( , ))
( ) ( ' | )
( ' | ) ( )
T
p nst
st i sti
nst i nst nst sp nstT T
st i st
Tst T st i Tst
i
k m
F z k m k
F z k m k z k mV C QCVF z k m k
z k m k F V
( ) ( ' | )T st i ststC QCV F z k m k
(4.31)
Finally the infinite sum developed in (4.30) can be written as follows:
'
( ( | ) ( , )) ( ( | ) ( , ))
( ' | ) ( ' | )
sp T sp
j m
st T st
Cx k j k y CV k j Q Cx k j k y CV k j
z k m k Pz k m k
(4.32)
Where, P is computed through the solution to the following Lyapunov equation:
TT T st stst stP V C QCV F PF
69
Considering (4.29) and (4.32) the objective cost defined in Equation (4.23) can be
rewritten as follows:
' 1
0
1 1
, , , ,0 0
, ,
( ( | ) ( , )) ( ( | ) ( , ))
( | ) ( | ) ( ( | ) ) ( ( | ) )
( ' | ) ( ' | )
msp T sp
k yj
m mT T
des k u k u des k u kj j
st T st nstT nst Tk y k u k u u k
V Cx k j k y CV k j Q Cx k j k y CV k j
u k j k R u k j k u k j k u Q u k j k u
z k m k Pz k m k S S
(4.33)
Which represents a bounded quantity.
Making use of the results above, the extended infinite horizon MPC for zone control
strategy is obtained from the solution to the following optimization problem.
,
' 1
, , , 0
1 1
, , , ,0 0
min ( ( | ) ( , )) ( ( | ) ( , ))
( | ) ( | ) ( ( | ) ) ( ( | ) )
( ' | ) ( ' | )
sp nstk k u kk
msp T sp
k yu y j
m mT T
b b b des k u k u b des k u kj j
st T st ik
V Cx k j k y CV k j Q Cx k j k y CV k j
u k j k R u k j k u k j k u Q u k j k u
z k m k Pz k m k
, ,T i i T
k u k u u kS S
Subject to Equations 4.25 to 4.28.
4.5 Robust IHMPC with multi-model uncertainty
It can be shown that the controller resulting from the solution to the problem defined
in Equation (4.17) is nominally stable. This means that if the model considered in the
controller is the true model of the plant, the resulting closed loop is stable and if the
input targets are reachable, the manipulated inputs will be driven to their targets while
the controlled outputs tend to a steady-state inside their control zones. This property
is followed by a better performance of IHMPC in comparison with the conventional
MPC with finite output horizon. However, for nonlinear systems or when there is a
mismatch between the model considered in the controller and the true model of the
plant, the performance of the IHMPC may not be as good as it would be expected. As
already shown in Figure 4.2, a linear model that represents the propylene distillation
column at a given operating condition will change significantly depending on the
operating point of the system. Then, the consideration of a single nominal model in
70
the IHMPC may not be an adequate strategy and some improvements should be
included in the control strategy when model uncertainty is present.
There are several ways to represent model uncertainty in model predictive control,
but the most practical way is the multi-model uncertainty where it is considered a set
of possible plant models and the real plant is unknown, but it is known that the
real plant is one of the components of this set. Therefore, it can be defined the set of
possible plant models as 1, , L where each n corresponds to a particular
model. This approach is particularly suitable to a nonlinear process system that can
be operated at different operating points that correspond to different product
specifications, market economic conditions, unknown disturbances, etc. In this case,
each model n represents the true system only locally around an operating point.
Badgwell (1997) developed a robust linear quadratic regulator for stable systems with
the multi-plant uncertainty. Later, Odloak (2004) extended the method to the output
tracking of stable systems considering the same kind of model uncertainty. These
strategies include a new constraint that prevents the plant cost function from
increasing at successive time steps. More recently, González and Odloak (2011)
presented an extension of the method to the zone control of time delayed systems
with input targets. Therefore, by considering the multi-plant uncertainty, it is assumed
that each model is represented by a set of parameters defined as
, , , , 1, ,s dn n n n nB B F n L . Then, for systems without integrating poles, it can be
defined, for each model n , the following cost function:
( | ) ( | ), , , ,0
( | ) ( | ), , , ,1
1( |) ( | )
, , , ,0
TpV y k j k y Q y k j k yn n n n y n n nk sp k y k sp k y kj
Ty k j k y Q y k j k yn n n y n n nsp k y k sp k y kj p
Tmu k j u Q u k j k uudes k u k des k u kj
( | ) ( | ), , , ,
1( | ) ( | )
, , , ,0
Tu k j k u Q u k j k uudes k u k des k u kj m
Tm T Tu k j k R u k j k S Sn y n uy k y k u k u kj
(4.34)
71
Following the same steps as in case of the nominal system, it can be concluded that
the cost defined in Equation (4.34) will be bounded only if the control actions, set
points and slack variables are such that Equation (4.14), Equation (4.13a) and
Equation (4.13b) are satisfied. In this case, Equation (4.13a) can be written as
follows:
max , ,| 0 1, ,n
sn sp k n y k nx k m k y n L
(4.35)
where max |n
snx k m k
is the output prediction at steady-state corresponding to
model n .
The optimization problem that defines the robust IHMPC adopted here can be
defined as follows:
, , ,, , ,min
k sp k N y k N u kk Nu y
V
subject to Eq. (4.25), Eq. (4.35) and
min max
min max
min , max
| ; 0,1, , 1
| ; 0,1, , 1
; 1, ,sp k n
u u k j k u j m
u u k j k u j m
y y y n L
(4.36)
, , , , , ,, , , , , , , , ; 1, ,k k sp k n y k n u k n k k sp k n y k n u k nV u y V u y n L (4.37)
where, N corresponds to the most probable model and, in Equation (4.37), it is
assumed that if * * *1 , 1 , 1 , 1, , ,k sp k n y k n u ku y is the optimum solution to the control
problem defined in Equation (4.36) at time step k-1, then:
* *| 1 2 | 1 0 ;TT T
ku u k k u k m k *, , 1sp k n sp k ny y
and ,y k n , ,u k are such that
max, ,( ) 0p
s n n n k ny sp k ny y kN A x k A Co u I y I (4.38)
,( 1) 0Tnu k des nu u ku k I u u I (4.39)
Stability of the closed-loop system with the controller defined above is achieved by
imposing the non-increasing cost constraints Equation (4.37) in order to prevent the
72
cost corresponding to the true plant to increase. The inclusion of these constraints,
which are non-linear, turns the control problem a non-linear program and a NLP
solver will be required for its solution.
4.6 The one layer Economic MPC
Consider a system described by a linear time-invariant discrete time model, which is
subject to hard constraints.
( ) ( ) ( )( ) , ( ) , ( )
x k 1 A x k B u kx k X u k U u k U k 0
(4.40)
For all k 0 , where nX and mU .
Also, consider the following economic function, such that the optimal steady state, xs,
satisfies:
argmin ( , )
. . SS
xx f x ps eco
s t x X
(4.41)
Where, feco(x, p) defines an economic cost function and p is a parameter that takes
into account prices, costs and production goals. One hypothesis about this function is
that it is convex in x and twice differentiable. To comply with the real industrial cases,
it is assumed that this economic function is non-linear and its evaluation takes a
significant computation time, provided that it is based on a rigorous non-linear
stationary model of the real plant. ssX represents the set of admissible stationary
states that could be defined as:
|ss ssx X x WX
which is a convex set in the equilibrium subspace Wss.
Then, the controller cost function could be defined as:
( , ; ) ( , ) ( , )dynN N ss ssV x p u V x u V x p (4.42)
Where,
73
( , ) ( ) ( ) ( )
( , ) ( , )
N 1
ss ssk 0 k N
ss ss ss
2 2 2dynV x u x k x u k x k xQ R QNV x p f x peco
(4.43)
Finally, the optimization problem to be solved by the Economic MPC is given by:
min ( , ; )ˆ
. . ( ),( ) ( ) ( ), ,...,( ) , ( ) , ( ) ,...,
N
0
ss ssX
V x p uu
st x x 0x k 1 Ax k B u k k 0 N 1x k X u k U u k U k 0 N 1x
(4.44)
Given that the economic function is not easy to solve, mainly when large dimension
processes are considered, then, in many real applications the available
computational power would not be sufficient to solve this problem at each sampling
time of the control system. In this context, instead of directly solving the complex one-
layer problem, the convex combination of an easy to obtain feasible solution and an
approximated optimal solution could be used to obtain a decreasing cost.
Consequently, the gradient of the economic cost, feco, instead of the cost itself, is
used to produce the approximated cost appNV and the approximated optimal solution
through the solution to the following problem (Alamo et al., 2012):
min ( , ; )*
. . ( ),( ) ( ) ( ), ,...,( ) , ( ) , ( ) ,...,
appN
0
ss ssX
V x p uu
s t x x 0x k 1 Ax k B u k k 0 N 1x k X u k U u k U k 0 N 1x
(4.45)
Where, the approximated cost is given by:
ˆ ˆ( , ; ) ; ( , , )
ˆˆ ˆ( , , )
ˆ
app dynN N ss ss ss
ss ss
ss ss ss
ss ss
V x p u V x u V x u p
x xV x u p
u u
(4.46)
And ˆ ˆ( , , )ss ss ssV x u p represents the gradient of Vss w.r.t. (x, u), evaluated at the point
, )ˆ ˆ( ss ssx u . This approximated solution is suboptimal (in the transient) with respect to
the optimal solution of cost function VN and hence its direct application into the MPC
scheme does not guarantee convergence nor a recursive feasibility of the closed-
loop system.
74
To circumvent this limitation, consider a parameterized family of feasible solutions,
given by the convex combination of the feasible solution u and the approximated
optimal solution *u :
,
*ˆ( ; ) ( ) ( ) ( )*ˆ( ; ) ( ) ( ) ( )
0 1
u k 1 u k u k
x k 1 x k x k
(4.47)
Now, there can be defined the following performance indexes which are the original
cost and approximated cost functions parameterized in .
( )
( )
( ) ( ; ) ( ) ( ; )
ˆ ( ( ), ( ), )
( ) ( ; ) ( ) ( ; )
ˆ ˆ( , , )
ˆˆ ˆ( , , )
ˆ
ss
ssss ss
ss
ss ss ss
ss ss
ss ss ss
ss ss
2 2V x k x u kQ R
k 0
V x u p2 2
V x k x u kg Q Rk 0V x u p
x xV x u p
u u
(4.48)
Lemma: If ˆ ˆ( , ) ( *, *)x u x u , then
( ) ( )V 1 V 0g g (4.49)
Next, the following theorem is presented.
Theorem: The following hold:
i. The pair ,x u , for every ,0 1 , provides a feasible solution to the
problem defined in Equation (4.44).
ii. If ˆ( ) ( )V 1 V 0 Vg g , then there exists ,0 1 such that
ˆ( ) ( )V V 0 V
The first part of the theorem presented above means that any convex combination of
the feasible and the approximated optimal solutions results in a feasible solution to
the optimization problem defined in Equation (4.44). The second part of the theorem
means that for every , the pair ,x u provides not only a feasible
solution to the original problem, but also an improved original cost when compared
75
with the original feasible solution. Provided that the sampling time of the process is
large enough, the solution ,x u can be iteratively improved within the
current sampling time.
Finally, the following algorithm is proposed for the implementation of the Economic
MPC:
Algorithm
At each sampling time:
1. Compute the feasible solution ˆ ˆ,x u to problem , ,V x p uN defined in
Equation (4.44), using the shifted solution applied to the system at
sampling time k – 1. If the current time is k = 0, compute the feasible
solution ˆ ˆ,x u by solving the reduced problem ,dynV x uN .
2. Compute the gradient of the economic cost function ,ssV x pss defined
in Equation (4.45) at the predicted steady-state, ,V x pss ss .
3. Compute an approximated optimal solution by solving , ,appNV x p u .
4. Compute the parameter value , such that the theorem is attended.
5. From the obtained solution, extract the first action and implement it in
the real system.
Advantages of the Economic MPC:
The controller implementation requires the solution of just one QP.
There is no need to compute the Hessian of feco, provided that an
heuristic procedure can be used to compute the parameter to be
used in the convex combination.
The controller remains feasible under any change of the economic
objective.
The controller ensures convergence to the point that minimizes the
economic function feco.
76
5.7 State observer
All the controllers proposed in this Thesis assume that the state x is known at time
step k and can be used for the computation of the output predictions. This state is
either measured, as in the realignment model, or must be estimated from the plant
dynamic measurements as in the case of the OPOM model. As the model considered
in the controller is a linear one, it will always be different of the real non-linear model
of the system, and consequently, the state estimate will not be exact.
When considering a state observer, one of the key aspects to be taken into account
is that the observer has to guarantee the asymptotic convergence of the error
between the estimated state and the real state to zero. Nevertheless, other requisites
could be added such as an optimal performance for the expected plant noise.
Here, the Kalman filter will be included in the control strategies based on the OPOM
model. In this case the filter must estimate the full state x, using a feedback
relationship of the estimation error as follows:
ˆ ˆ ˆ( 1) ( ) ( ) ( )F tx k A x k B u k K y C x k
For a practical implementation of this filter, the noise covariances are to be known or
considered as tuning parameters of the filter. Here, these parameters are considered
to be:
0.05
0.05
ny nyny
nx nxnx
V I
W I
Then, the gain FK of the Kalman filter gain can be computed as:
1T T T TP APA W APC CPC V CPA
1( )( )T T
FK APC CPC V
77
5 REAL TIME OPTIMIZATION DEVELOPMENT
5.1 The two-layer RTO strategy based on ROMeo
Here, it is studied the RTO algorithm and the sequence of optimizing actions
provided by ROMeo Online Performance System (OPS) and ROMeo Real Time
System (RTS), respectively. ROMeo represents a new generation of commercial
process softwares designed to help maximizing the profitability of the refining and
petrochemical processes. ROMeo is a unified process modeling software with both
off-line simulation and on-line optimization capabilities. ROMeo also has an equation-
based modeling engine based on flexible algebraic modeling language that utilizes
advanced optimization solution techniques and proven thermodynamic methods in
open, object-oriented, client/server architecture to ensure adequate support and
performance (ROMeo’s user guide, 2012).
Variables in ROMeo are classified into the following three categories depending of
their attributes:
Fixed/independent: These are the unit or process specifications in
the simulation, data reconciliation and optimization modes, and are
also known as specification variables.
Free/independent: They correspond to the controller set-points
determined in the optimization mode or reconciled measurement
values computed in the data reconciliation mode.
Free/dependent: These are remaining model variables whose values
are determined by the solver, they are also known as solution
variables.
It is important to note that the variable attributes are automatically changed by
ROMeo when a mode change occurs for a particular variable.
The general non-linear programming problem that is solved by ROMeo in the
different modes can be defined as follows:
78
( ) ( )
subject to :
( ) 0
( ) 0
where
a vector of realnumbers,consistingof fixed/independent,
free/independent and free/dependent variables
( ) a linear/non-linear objectivefunction
( ) 0 modelequat
Maximize minimize f x
g x
h x
x
f x
g x
ions with linear andnon-linear equalityconstraints
( ) 0 variablesimplebounds inequalitiesh x
(5.1)
5.1.1 ROMeo’s simulation mode
In simulation mode, ROMeo solves the material and energy balance equations and
the phase equilibrium relations for all the units and streams in the flow-sheet. Unlike
the common sequential-modular steady-state simulators, ROMeo uses an open-
equation solver to simultaneously solve the model and constraints equations. This
approach can solve more efficiently the large problems with recycles that are typical
in online plant optimization. When using ROMeo to develop a simulation, it is
important to follow the steps outlined below (ROMeo user guide, 2012):
Define the units of measure (UOM) set
Select the components and create component slates
Define the thermodynamic calculation methods
Build the PFD and supply the operating conditions
Generate initial estimates
Run the simulation and analyze the results
Update the simulation with new initial values
In simulation mode, the objective function f(x) and bounds h(x) are removed from the
problem defined in Equation (5.1) and only the model equations g(x) are considered.
Once the simulation of the Propylene/Propane splitter is concluded, it is possible to
simulate various steady-state operating points in order to validate the process model.
In Figure 5.1, it is shown the process flow-sheet diagram of the PP splitter in
ROMeo’s simulation mode. In this diagram, it is also included some heat-exchanger
linkers that facilitate the simulation of the two sides of the shell and tube heat
exchangers.
Figure 5. 1 – PFD of thee PP splitter in Simmulation mode inn ROMeo
79
80
5.1.2 ROMeo’s data reconciliation mode (DataRec)
Real-time process data are subject to random errors due to measurement noise and
gross errors due to faulty equipment and miscalibration. Data reconciliation can turn
process measurements into consistent and reliable information that can be used to
improve and optimize the plant operation and help management. The data reconciliation
package of ROMeo increases the accuracy of the plant measurements while ensuring
that they conform to the mass and energy conservation laws. Then, statistical tests are
performed by ROMeo based on the properties of normally distributed random variables.
When all the measurements and tuning parameters are included in the flow-sheet, it is
possible to add a multivariable controller (MVC) so that it takes into account the
controlled and manipulated variables as well as their bound constraints (ROMeo User
Guide, 2012). The PFD of the propylene/propane splitter can be observed in Figure 5.2.
For each measurement (scan variable xi,scan) that is imported from the process (in this
case from Dynsim), a corresponding measurement unit must be configured in ROMeo
(xi). This measurement unit adds a single equation to the overall equation set g(x) as
follows:
,x x offseti i scan i (5.2)
Written in open equation form, it becomes:
( ) ,res i x x offseti i scan i (5.3)
The optimization problem to be solved in DataRec is to minimize the weighted sum of
squares of the offsets:
2
where
is thesummation of all measurement unit variables
reconciled scannedMeas
Meas
Minimize x x
(5.4)
Figure 5. 2 – PFD of the PP splitter in Data reeconciliation modde in ROMeo
81
82
5.1.3 ROMeo’s optimization mode
In optimization mode, ROMeo’s goal is to maximize the process cash flow (i.e.,
currency/time). This economic objective function can have contributions from sources
and sinks, mechanical equipments that produce or consume power and others. In
this mode, ROMeo automatically let the set-points of all controllers free to become
the optimization variables. Therefore, the optimum operation point of the plant is
computed such that the economic is maximized.
Then, computed input optimizing targets ,des ku are defined by the RTO layer, which is
based on a rigorous steady-state simulation of the distillation process and computes
the optimum operation point of the plant that maximizes the following economical
function, which was defined in ROMeo:
1 1 1
product streams feed streams utilities
eco i i i i i ii i i
f PPS PFR PFS FFR PU UC
(5.5)
where,
PPS is the price of the product [$/ton], PFR is the product flow rate [ton/h], PFS is the
price of the feedstock [$/ton], FFR is the feed flow rate [ton/h], PU is the price of
electricity [$/kW-h], UC is electricity consumption [kW-h/h]
The economic function defined in Equation (5.5) is maximized producing the optimum
input and/or output targets subject to the following constraints:
The rigorous steady-state model that relates the system inputs and measured
disturbances to the outputs.
Lower and upper bounds to the input targets.
Lower and upper bounds to the controlled outputs.
In optimization mode, the measurement equations, which were added while in the
configuration of DataRec are retained, but their variable attributes are modified such
that the offset is fixed at the value calculated during Data Reconciliation mode when
the offset was one of the decision variables. Then, the offset becomes a fixed
adjustment of the model variable, whose value is determined by the Solver to close
the model equations, compared to a calculated scan variable. This calculated value
of the scan variable becomes an estimate of what the actual measurement should be
83
to match the current model solution. This is useful information, particularly when a
bound needs to be placed on a variable that is related with a process measurement.
Also in optimization mode, free/independent variables are made available to the
solver to maximize the economic function. The solution algorithm adjusts these
free/independent variables simultaneously with the model dependent variables to
satisfy the equality and inequality constraints. Typically, these free/independent
variables correspond to set-points to the existing controllers that were already
implemented in the process system. This is a convenient way to configure the
free/independent variables for optimization.
5.1.4 On-line sequence algorithm
The on-line sequence algorithm was developed using ROMeo Real-time System
(RTS), which is a sequence development environment with easy-communication
access to ROMeo’s OPS application models through the use of the External Data
interface (EDI) subsystem. In order to accomplish a successful on-line
implementation, the algorithm was developed using the two available types of
sequences in ROMeo RTS, the generic and model sequences (ROMeo RTS user
guide, 2012).
5.1.4.1 Steady State Detection (SSD)
The steady state detection (SSD) task, which is a generic sequence, was
implemented so that it monitors the values of a selected set of process
measurements to determine whether the plant is operating at steady-state. This task
is important because reaching a steady-state is usually a prerequisite to be attained
before the optimization of the plant can be triggered.
For each period, the SSD task takes the measurements of each selected point and
saves them for statistical analysis. The length of a period is defined by the execution
schedule of the sequence. In order to execute this task, a minimum percentage of the
steady value is defined as a cutoff for determining if the unit is unsteady or not.
Each time it runs, the SSD Task computes the corresponding percentage of the
steady value indicating whether the unit is steady or unsteady. The plant is
considered steady if the percentage-steady value (the average of the percentage-
steady values of all the points examined) is greater than or equal to the minimum
84
established percentage-steady. Once the SSD Task determines whether the plant is
steady or unsteady, you can select one of two statistical methods available to
determine the level of steady state of the system. In the present work, Method 1 was
selected and is briefly described as follows:
Method 1: In this method the user provides tolerances as it compares the mean and
variance of two halves of the data set, to determine whether a given data set
corresponds to a steady state (ROMeo RTS user guide, 2012).
The SSD task holds two Value Sets for each configured measurement. Each Value
Set contains n values of scans, their sums, and their sum squared values. The scan
values for each point come from the External Data Interface (EDI). The SSD task
performs on each value set the calculations shown in Equations (5.6-5.9).
kn
ii 1
kk
v
nMean
(5.6)
k
k
n
n i2 i 1i
i 1 kk
k
2 vv
n
n 1Variance
(5.7)
k kVarianceStdDev (5.8)
2
k pk 1
n n
(5.9)
The SSD Task uses these values and Equation (5.10) to calculate a Standard
Distribution curve to test the level of significance.
1 2Mean Mean ToleranceMeanDifference (5.10)
Before calculating the degrees of freedom, the SSD task uses Equations (5.11) and
(5.12) to compare the ratio between the variances of the data on these intervals to
the absolute value of 90% confidence level of degrees of freedom of these two
intervals.
, ( , )11 2
2
VarianceF 90 n 1 n 1
Variance (5.11)
, ( , )22 1
1
VarianceF 90 n 1 n 1
Variance (5.12)
If the test passes, the SSD task assumes that there is a significant difference
between the two variances, so the SSD task uses Equation (5.13) to calculate the
85
degrees of freedom and uses Equation (5.14) to calculate the t value as shown in
Equation (5.18). 2
1 2
1 2
2 2
1 2
1 2
1 2
Variance Variancen n
DOF 2Variance Variance
n n 1n 1 n 1
(5.13)
1 2
1 2
MeanDifferencetVariance Variance
n n
(5.14)
If the test fails, the SSD Task assumes that there is no significant difference between
the two variances and therefore the variances are equal, as described in Equations
(5.15-5.18).
Variance Variance1 2 (5.15)
DegreesOfFreedom n n 21 2 (5.16)
1 1 2 2Variance n 1 Variance n 1
DegreesOfFreedomOverallStdDev
(5.17)
1 2
MeanDifferencet1 1OverallStdDev n n
(5.18)
The SSD Task calculates the maximum permissible variation MAXT for 95% probability
level using the standard distribution curve shown in Equation (5.19).
( )MAX MAX 95T T DegreesOfFreedom (5.19)
If the t value is less than the permissible value (that is, MAXT ), then the SSD task
increments the steady count by one. The SSD task calculates the Percent Steady for
each point as the ratio of the steady count and the minimum steady count. It
compares the t value to MAXT as shown in the following algorithm:
MAX
MAX
If t T Steady StateSteadyCount SteadyCount 1
else t T Unsteady StateSteadyCount 0
Figu
alrea
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86
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, from the
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e
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87
Data Reconciliation Pre-processor: Stops the sequence according to
unit or subunit unsteady, down or implementing criteria, screens
measurements and changes to DataRec mode.
Solve DataRec: Solves the case with prepared DataRec case input.
DataRec Review: Checks good fit criterion.
Optimization pre-processor: It prepares the model application for
optimization calculations and it specifies criteria for stopping the
sequence.
Solve Optimization: Solves the case with the prepared optimization
case input using calculation mode as previously selected.
Output: Shows output successful solution results, uploads optimization
targets and exports them to data source.
5.2 One layer structure strategy
In order to implement the Economic MPC, it is necessary to evaluate the gradient of
the economic function with respect to the system inputs, at each predicted steady-
state. Therefore, the sensitivity analysis of ROMeo OPS is used to compute the
gradient. The sensitivity analysis calculates how the changes in the fixed variables
affect the free variables, where the model equation is written as follows:
, 0free fixed
F X Y
Then, the sensitivity analysis tool reports:
ˆ
ˆX Response Free
Cause FixedY
(5.20)
In Equation (5.20), X and Y are subsets of the flow-sheet’s free (response) and
fixed (cause) variables, respectively. It is necessary to specify which cause and
which response variables should be included in the analysis. This calculation requires
only about the time of a single solver iteration. Figure 5.5 shows the sensitivity tool
utility where the response and cause variables are selected.
Figuure 5. 5 –Sennsitivity analyysis tool in ROMeo OPS
88
8
89
6 RESULTS
In order to automate the data reading and writing, from and to the dynamic
simulation, it was adopted the timer function of Matlab and following the sequence:
First, the data from Dynsim is read every 5sec and sent to Matlab where the average
of last 12 data points is computed. Next, the MPC algorithm is run with a sampling
time equal to 1 minute and new values of the control inputs are computed. These
inputs are the set-points to the dynamic simulation regulatory PID controllers and are
sent to Dynsim through the OPC interface already described. In addition, the transfer
of data from ROMeo to Dynsim is done through the export function of OPC EDI, in
the same way as the reading of data was done using the import and download
functions using ROMeo RTS sequences.
6.1 Two-layer structure of the RTO/MPC integration
The implementation of the control and optimization structure presented in Figure 6.1
on the PP splitter described in Chapter 3 was tested through simulation experiments
considering the nominal MPC defined in section 4.4 and the robust controller with
multiple model uncertainty defined in section 4.5.
Figure 6. 1 – Two layer structure strategy
90
The output zones and input constraints, as well as the maximum input increments,
considered in the simulations presented here are shown in Tables 6.1 and 6.2,
according to the controlled and manipulated variables defined previously in
subsections 3.3.1 and 3.3.2.
Table 6. 1 – Output zones of the propylene/propane splitter
Output Description ymin ymax
y1 (% level) Liquid level on heat exchangers
M-01 A/B and O-03 4 80
y2 (%molar) Propane molar composition in
the propylene stream 0 0.45
y3 (%molar) Propylene molar composition in
the propane stream 0 2
Table 6. 2 – Input constraints of the propylene/propane splitter
Input Description ∆umax umin umax
u1 (ton/h) Heat pump flow rate set-point 0.15 220 350
u2 (ton/h) Feed flow rate set-point 0.02 10 45
u3 (ton/h) Reflux flow rate set-point 0.13 200 320
First, the closed loop simulation began at the steady-state corresponding to the
operating point 3 as defined in Table 4.1, which corresponds to input u0 = [302 30
268] and to output y0 = [42 0.5 1] that are read from the dynamic simulation at the
beginning of the test. The initial steady-state corresponds to feco = 14 300 $/h. Then
with the assumed market conditions and the available composition of the feed stream
to the propylene distillation system, ROMeo computes a new optimum operating
point and defines the optimum targets to the MPC. These input targets are udes =
[330 34 294.8], which corresponds to an increase of the feed flow rate while the heat
pump and reflux flow rates are minimized in the attempt to use the minimum values
that maximize the economic function and do not violate any constraint. At this new
operating point the value of the economic function is feco = 16 000 $/h.
91
The second simulation experiment started when the plant had already stabilized at
time 1800min and a disturbance, which was unknown to the controller, was
introduced in the feed stream of propylene/propane splitter. The new feed molar
composition represented in Table 6.3 is significantly different from the initial feed
composition represented in Table 3.1.
Table 6. 3 – Feed molar composition (Disturbance)
Component % molar fraction
Ethane 0.0118
Propylene 50.37
Propane 48.69
i-Butene 0.3895
1-Butene 0.0705
Cis-2-Butene 0.0386
Trans-2-Butene 0.0386
1,3-Butadiene 0.0134
i-Butane 0.3442
n-Butane 0.0386
Assuming that this disturbance was known by ROMeo, a new optimum steady state
was calculated and corresponds to the following input targets udes = [325.7 34
295.4]. Analogously to the first simulation case, the controllers should be able to
reject the disturbance and drive the distillation column to the new optimum operating
point.
6.1.1 Nominal case
The results for the nominal IHMPC using the two state-space system
representations, presented in this Thesis, are shown in this section. The nominal
model used for these controllers is 1G . Also, the tuning parameters adopted for
each nominal controller are presented.
92
6.1.1.1 IHMPC using OPOM
Results of the simulation experiments on the PP splitter with the nominal IHMPC
using the OPOM model representation are shown in Figures 6.2 – 6.9 in blue line.
The input constraints and output control zones are defined in tables 6.1 and 6.2
respectively. The tuning parameters adopted to test this controller are shown in Table
6.4.
Table 6. 4 – IHMPC-OPOM tuning parameters
Sampling time T = 1 min.
Control horizon m = 3
Input suppression weight R = diag ([0.5 3 0.5])
Controlled output weight Qy = diag ([6 25 2])
Optimizing input target weight Qu = diag ([0.1 10 1])
Slack variables y weight Sy = 710 * diag ([1 10 1])
Slack variables u weight Su = 410 * diag ([0.1 100 1])
6.1.1.2 IHMPC using the realignment model
Results of the simulation experiments of the PP splitter with the nominal IHMPC
using a realignment model representation are shown in Figures 6.2 – 6.9 in red lines.
Because the system poles are near to one and some transfer functions are of second
order, the realignment model was built with a sampling time of ten minutes.
Consequently, the maximum input moves were modified and new tuning parameters
were adopted to test this controller as can be seen in Tables 6.5 and 6.6.
Table 6. 5 – IHMPC-Realignment model maximum input moves
Input ∆umax
u1 (ton/h) 1.5
u2 (ton/h) 0.2
u3 (ton/h) 1.3
93
Table 6. 6 – IHMPC-Realignment model tuning parameters
Sampling time T = 10 min.
Control horizon m = 3
Input suppression weight R = diag ([0.5 3 0.5])
Controlled output weight Qy = diag ([6 25 2])
Optimizing input target weight Qu = diag ([0.01 10 0.1])
Slack variables y weight Sy = 1010 * diag ([1 1 1])
Slack variables u weight Su = 610 * diag ([0.01 100 0.1])
6.1.1.3 Nominal IHMPC results
In order to compare the performance of both nominal controllers, the simulation
results of the IHMPC based on the OPOM were plotted in blue lines and the IHMPC
based on the realignment model were plotted in red line in both experiments. The
control zones are represented by the black dashed lines, and the input optimizing
targets are shown in green lines.
Figure 6. 2 – Controlled outputs IHMPC (First experiment), OPOM (— — —), Realigned (— — —)
0 200 400 600 800 1000 1200 1400 1600 180020
30
40
50
nT (min)
y1
0 200 400 600 800 1000 1200 1400 1600 18000
0.2
0.4
nT (min)
y2
0 200 400 600 800 1000 1200 1400 1600 18000
2
4
nT (min)
y3
94
Figure 6. 3 – Manipulated inputs IHMPC (First experiment), OPOM (— — —), Realigned (— — —)
Figure 6. 4 – Economic function IHMPC (First experiment), OPOM (— — —), Realigned (— — —)
0 200 400 600 800 1000 1200 1400 1600 1800300
320
340
nT (min)
u1
0 200 400 600 800 1000 1200 1400 1600 180030
32
34
nT (min)
u2
0 200 400 600 800 1000 1200 1400 1600 1800260
280
300
nT (min)
u3
0 200 400 600 800 1000 1200 1400 1600 18001.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65x 10
4
nT (min)
f eco (
$/h
)
95
Figure 6. 5 – Economic function IHMPC with penalization (First experiment), OPOM (— — —),
Realigned (— — —)
It is clear from the simulation results presented above that the nominal IHMPC seems
to show a reasonable performance, but the high-non-linearity of the propylene
distillation column may not be properly dealt with by this controller if strict
specifications of product quality are enforced. This is evidenced by the long period of
time that the controlled variables may remain outside the control zones when a
change of the operating point is required for both controllers. Nevertheless, the
IHMPC using the realignment model has a better performance compared with the
one using the OPOM representation, since it shows faster response to drive the
system output y3 to its control zone, but the most important controlled variable y2
seems to have almost the same performance for both controllers. Figure 6.4 shows
the instantaneous economic functions of both controllers without any penalization in
the products price due to the violation of the propylene product quality reflected in y2.
In Figure 6.5, it is plotted the instantaneous economic function using a penalization of
propylene price (lower price) due to the violation of the quality specification. It is also
plotted the accumulated economic function that considers the economic profit
obtained in the whole simulation time interval for both controllers.
Once the system is stabilized at a new steady-state, the feed composition
disturbance is introduced into the system, which is unknown to the controller and new
optimizing targets are computed and sent to the controller. The new optimizing
0 500 1000 1500-5000
0
5000
10000
15000
20000
nT (min)
f eco (
$/h
)
0 500 1000 1500-5
0
5
10
15
20x 10
4
nT (min)
f eco (
$)
96
targets correspond to udes = [325.7 34 295.4]. The results corresponding to this
experiment can be observed in Figures 6.6 to 6.9.
Figure 6. 6 – Controlled outputs IHMPC (Second experiment), OPOM (— — —), Realigned (— — —)
Figure 6. 7 – Manipulated inputs IHMPC (Second experiment), OPOM (— — —), Realigned (— — —)
1800 2000 2200 2400 2600 2800 3000 3200 3400 360020
25
30
nT (min)
y1
1800 2000 2200 2400 2600 2800 3000 3200 3400 36000
0.5
nT (min)
y2
1800 2000 2200 2400 2600 2800 3000 3200 3400 36000
1
2
nT (min)
y3
1800 2000 2200 2400 2600 2800 3000 3200 3400 3600320
325
330
335
nT (min)
u1
1800 2000 2200 2400 2600 2800 3000 3200 3400 360033.8
34
34.2
nT (min)
u2
1800 2000 2200 2400 2600 2800 3000 3200 3400 3600294
296
298
nT (min)
u3
97
Figure 6. 8 – Economic function IHMPC (Second experiment), OPOM (— — —), Realigned (— — —)
Figure 6. 9 – Economic function IHMPC with penalization (Second experiment), OPOM (— — —),
Realigned (— — —)
From these figures, it is easy to conclude that both controllers are able to stabilize the
plant and to maintain the controlled variables inside their respective zones, while the
manipulated variables are driven to their respective targets. The responses are also
quite slow and the system takes about 30h to reach the new steady state. In Figure
6.6, it is clear that the output y2 remains all the simulation time inside its control zone
when the controller is based on the realignment model, while with the OPOM model,
1800 2000 2200 2400 2600 2800 3000 3200 3400 36001.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65x 10
4
nT (min)
f eco (
$/h
)
2000 2500 3000 3500
-2000
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
nT (min)
f eco (
$/h
)
2000 2500 3000 35000
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
5
nT (min)
f eco (
$)
98
the controller allows the controlled variable to remain outside the control zone for a
large period of time. The manipulated inputs in Fig. 6.7 are driven to their respective
optimizing targets, it is easy to realize that the IHMPC based on the realigned model
is faster in stabilizing the system at that new targets. Figure 6.8 and 6.9 show the
instantaneous and accumulated economic function for the IHMPC based on the
OPOM (blue line) and based on the realignment model (red line). It is also shown the
effect of using or not using penalization in the propylene product price for the case of
violation of quality specification of controlled variable y2. For this simulation
experiment, there is an economic benefit of about US$ 1.9 x105 when the IHMPC is
based on the realignment model in comparison to the IHMPC based on the OPOM
model.
6.1.2 Robust case
At this point, it should be interesting to note that the new operating point
corresponding to the optimum economic point is quite far from the initial operating
point where the nominal model was obtained. Certainly, model uncertainty is
significant at this operating point when only the nominal model is included in the
IHMPC. Then, there is motivation to investigate if the performance of the advanced
control structure based on the RIHMPC would be better than the performance of the
IHMPC.
The robust IHMPC was tested in the distillation column assuming 1G as the most
probable model and including the other two models represented in Table 4.2 to
account for the multi-model uncertainty. The system starts from the same steady-
state as in the previous simulation case with IHMPC and the optimum targets defined
by ROMeo are also the same as in case of the nominal IHMPC, as well as the input
and output bounds. The tuning parameters adopted to test this controller are shown
in Table 6.7.
99
Table 6. 7 – Robust MPC tuning parameters
Sampling time T = 1 min.
Control horizon m = 3
Input suppression weight R = diag ([0.5 3 0.5])
Controlled output weight Qy = diag ([6 25 2])
Optimizing input target weight Qu = diag ([0.1 10 1])
Slack variables y weight Sy = 710 * diag ([1 10 1])
Slack variables u weight Su = 410 * diag ([0.1 100 1])
Figure 6. 10 – Controlled outputs (First experiment), IHMPC (— — —), RIHMPC (— — —)
0 200 400 600 800 1000 1200 1400 1600 180020
30
40
50
nT (min)
y1
0 200 400 600 800 1000 1200 1400 1600 18000
0.2
0.4
nT (min)
y2
0 200 400 600 800 1000 1200 1400 1600 18000
2
4
6
nT (min)
y3
100
Figure 6. 11 – Manipulated inputs (First experiment), IHMPC (— — —), RIHMPC (— — —)
Figure 6. 12 – Economic function (First experiment), IHMPC (— — —), RIHMPC (— — —)
0 200 400 600 800 1000 1200 1400 1600 1800300
320
340
nT (min)
u1
0 200 400 600 800 1000 1200 1400 1600 180030
32
34
36
nT (min)
u2
0 200 400 600 800 1000 1200 1400 1600 1800260
280
300
nT (min)
u3
0 200 400 600 800 1000 1200 1400 1600 18001.1
1.2
1.3
1.4
1.5
1.6
1.7x 10
4
nT (min)
f eco (
$/h
)
101
Figure 6. 13 – Economic function with penalization (First experiment), IHMPC (— — —), RIHMPC (—
— —)
Figure 6.10 compares the system outputs with RIHMPC (dashed red line) alongside
the responses with IHMPC (dashed blue line), and the output zones (dashed black
line) for the total simulation time (1800 min). It is clear that the two controllers have
acceptable performances, but when comparing the IHMPC and RIHMPC
performances, it is easy to see that the robust controller has a better performance
than the IHMPC with the nominal model. The robust controller drives the controlled
variables to their control zones faster than the nominal IHMPC. As a result, with the
robust controller, the controlled output y2, which corresponds to the most important
process specification of the propylene/propane splitter and that started at a point
outside its control zone, was brought to inside the control zone very rapidly and
maintained almost all the simulation time in its control zone. However, with the
IHMPC, y2 violates the upper bound of its control zone for a significant period of time.
Figure 6.11 compares the calculated inputs for the two controllers and it also shown
the optimizing input targets (green line) calculated by ROMeo. It is clear, that both
controllers are capable of driving the propylene/propane splitter to the optimum
operating point, but for the RIHMPC, the inputs u1 (heat pump flow rate) and u3
(reflux flow rate) are driven faster to their targets, which corresponds to a better
transient economic performance as it is shown in Figure 6.13. Observe that the
estimated accumulated economic benefit for the strategy based on the robust
0 500 1000 1500-5000
0
5000
10000
15000
20000
nT (min)
f eco (
$/h
)
0 500 1000 1500-1
0
1
2
3
4
5x 10
5
nT (min)
f eco (
$)
102
controller is nearly US$ 200x103 higher than the benefit for the strategy based on the
nominal controller.
It was evidenced by the simulation results presented above that the nominal IHMPC
seems to show a reasonable performance, but the high-non-linearity of the propylene
distillation column may not be properly dealt with by this controller if strict
specifications of product quality are enforced. This is evidenced by the long period of
time that the controlled variables may remain outside the control zones when a
change of the operating point is required. While the controlled outputs are outside
their control zones, the propylene product stream will be out of specification and
cannot be sent to be commercialized. Meanwhile, the robust controller showed a
better performance and does not seem to be significantly affected by the process
nonlinearities since it considers plant models at different operating points and is able
to stabilize the plant faster than the nominal controller.
The second simulation experiment started when the plant was already stabilized at
time 1800 min and a disturbance, which was unknown by the controller, was
introduced in the feed of the PP splitter.
Figure 6. 14 – Controlled outputs (Second experiment), IHMPC (— — —), RIHMPC (— — —)
1800 2000 2200 2400 2600 2800 3000 3200 3400 360020
25
30
nT (min)
y1
1800 2000 2200 2400 2600 2800 3000 3200 3400 36000
0.2
0.4
nT (min)
y2
1800 2000 2200 2400 2600 2800 3000 3200 3400 36000
1
2
nT (min)
y3
103
Figure 6. 15 – Manipulated inputs (Second experiment), IHMPC (— — —), RIHMPC (— — —)
Figure 6. 16 – Economic function (Second experiment), IHMPC (— — —), RIHMPC (— — —)
1800 2000 2200 2400 2600 2800 3000 3200 3400 3600324
326
328
330
nT (min)
u1
1800 2000 2200 2400 2600 2800 3000 3200 3400 360033.8
34
34.2
nT (min)
u2
1800 2000 2200 2400 2600 2800 3000 3200 3400 3600294
295
296
297
nT (min)
u3
1800 2000 2200 2400 2600 2800 3000 3200 3400 36001.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65
1.7x 10
4
nT (min)
f eco (
$/h
)
104
Figure 6. 17 – Economic function with penalization (Second experiment), IHMPC (— — —),
RIHMPC (— — —)
When comparing the IHMPC and RIHMPC performances, it is easy to realize that, as
in the first simulation experiment, the robust controller has a better performance than
the nominal IHMPC, because the controlled variables respond faster and were kept
inside the control zones more efficiently. For instance, controlled output y2, which
started the simulation at a point at the upper bound of its zone control, tends to leave
the control zone for a period of time, but it was brought back to the zone by the
RIHMPC faster than by the IHMPC. Figure 6.15 compares the calculated inputs for
the two controllers. It is also plotted the optimizing input targets (green line)
calculated by ROMeo and corresponding to the feed column composition shown in
Table 6.3. The consequence of the new feed composition is that the flow rate of the
propylene product has to be decreased while the propane product flow rate is
increased. Then, the economic function decreases but the optimizer tries to maximize
the propylene production, which represents the most valued product as shown in
Figures 6.16 and 6.17. As in the first simulation case, the robust controller has a
better economic performance, which corresponds to an accumulated benefit nearly
US$ 70x103 higher than the nominal controller.
6.2 One layer structure (Economic MPC)
At this point, it should be interesting to compare the RTO/MPC hierarchical structures
presented in Figures 6.1 and 6.9. Here, the motivation is to investigate and compare
the performances of both structures by performing simulation experiments in the PP
2000 2500 3000 3500-5000
0
5000
10000
15000
20000
nT (min)
f eco (
$/h
)
2000 2500 3000 35000
0.5
1
1.5
2
2.5
3x 10
5
nT (min)
f eco (
$)
105
splitter. In Figure 6.18, it is presented the one-layer structure that was used for the
study of the implementation of the gradient-based MPC.
ecof
u
ˆssu
Figure 6. 18 – One layer structure solution
The system starts from the same steady-state as in the previous simulation case with
IHMPC and the optimum targets defined by ROMeo are also the same as in case of
the nominal IHMPC, as well as the input and output bounds. The tuning parameters
that were considered for the Economic MPC are presented in Table 6.8.
Table 6. 8 – Economic MPC tuning parameters
Sampling time T = 1 min.
Control horizon m = 7
Input suppression weight R = diag ([0.5 3 0.5])
Controlled output weight Qy = diag ([60 250 1])
Slack variables y weight Sy = 1010 * diag ([1 1 1])
In order to compare the performances of the two RTO/MPC integration approaches
based on the nominal controllers, the simulation results of the IHMPC using the two-
layer structure was plotted in blue dashed line while the Economic IHMPC (One-layer
structure), which is a gradient-based MPC, was plotted in red dashed line for the first
simulation experiment. The control zones are represented by the black dashed lines,
and the input optimizing targets of the two layer integration approach are shown in
green dashed lines.
106
Figure 6. 19 – Controlled outputs (First experiment), Economic MPC (— — —), Two-layer (— — —)
Figure 6. 20 – Manipulated inputs (First experiment), Economic MPC (— — —), Two-layer (— — —)
0 500 1000 1500 2000 2500 3000 350020
30
40
50
nT(min)
y1
0 500 1000 1500 2000 2500 3000 35000
0.2
0.4
nT(min)
y2
0 500 1000 1500 2000 2500 3000 35000
1
2
3
4
nT(min)
y3
0 500 1000 1500 2000 2500 3000 3500300
320
340
nT (min)
u1
0 500 1000 1500 2000 2500 3000 350030
32
34
nT (min)
u2
0 500 1000 1500 2000 2500 3000 3500260
280
300
nT (min)
u3
107
Figure 6. 21 – Economic function (First experiment), Economic MPC (— — —), Two-layer (— — —)
Figure 6. 22 – Economic function with penalization (First experiment), Economic MPC (— — —),
Two-layer (— — —)
Figure 6.19 shows the system outputs for the two-layer RTO/IHMPC (blue line)
alongside the responses for the Economic MPC (red line), and the output zones
(dashed line) for the simulation time of 3500 min. It is clear that the two controllers
have acceptable performances, but when comparing both performances, it is easy to
see that the two-layer IHMPC stabilizes the system faster (1800 min) while the
0 500 1000 1500 2000 2500 3000 35001.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6
1.65x 10
4
nT (min)
f eco (
$/h
)
0 1000 2000 3000-5000
0
5000
10000
15000
20000
nT (min)
f eco (
$/h
)
0 1000 2000 3000-2
0
2
4
6
8x 10
5
nT (min)
f eco (
$)
108
Economic MPC requires almost 3500 min. Nonetheless, in the Economic MPC the
controlled output y2, which corresponds to the most important process specification of
the propylene/propane splitter and started at a point outside its control zone, was
brought to inside the control zone very rapidly and was maintained almost all the
simulation time in its control zone. However, with the two-layer IHMPC, y2 violates
the upper bound of its control zone for a significant period of time. Figure 6.20
compares the calculated inputs for each of the controllers and it also shows the
optimizing input targets (green line) calculated by ROMeo for the two-layer strategy.
It is clear, that both controllers are capable of driving the propylene/propane splitter to
the optimum operating point. However, the Economic MPC shows a better transient
economic performance as can be seen from Figures 6.21 and 6.22. Observe that the
estimated accumulated economic benefit for the strategy based on the Economic
MPC is nearly US$ 1x105 higher than the benefit resulting from the strategy based on
the two-layer approach.
It was evidenced by the simulation results presented above that the RTO/MPC
integration with the two layer approach or the Economic MPC based on the nominal
IHMPC seem to show reasonable and acceptable performances, but the high-non-
linearity of the propylene distillation column may not be properly dealt with by this
controller if strict specifications on product quality are enforced. This is emphasized
by the long period of time that the controlled variables remained outside of the control
zones when a change of the operating point was required. While the controlled
outputs are outside their control zones, the propylene product will be out of
specification and cannot commercialized as high purity propylene. Instead, it will be
degraded to a lower value product as the liquefied petroleum gas (LPG). Despite of
this, the Economic MPC showed a good performance and was capable of bringing
and maintaining the controlled variables inside their respective control zones, but it
was not able to stabilize the plant faster than the two-layer approach.
Next, a new simulation experiment was performed in order to compare the two-layer
and one-layer approaches when a non-measured disturbance on the feed
composition, as shown in Table 6.3 reaches the column. This disturbance was
introduced when the plant was already at the new steady state.
109
Figure 6. 23 – Controlled outputs (Second experiment), Economic MPC (— — —),
Two-layer (— — —)
Figure 6. 24 – Manipulated inputs (Second experiment), Economic MPC (— — —),
Two-layer (— — —)
3600 3800 4000 4200 4400 4600 4800 5000 520020
25
30
nT (min)
y1
3600 3800 4000 4200 4400 4600 4800 5000 52000
0.5
nT (min)
y2
3600 3800 4000 4200 4400 4600 4800 5000 52000
1
2
nT (min)
y3
3600 3800 4000 4200 4400 4600 4800 5000 5200
326
328
330
nT (min)
u1
3600 3800 4000 4200 4400 4600 4800 5000 520033.9
34
34.1
nT (min)
u2
3600 3800 4000 4200 4400 4600 4800 5000 5200294
296
298
nT (min)
u3
110
Figure 6. 25 – Economic function (Second experiment), Economic MPC (— — —),
Two-layer (— — —)
Figure 6. 26 – Economic function with penalization (Second experiment), Economic MPC (— — —),
Two-layer (— — —)
When comparing the performances of the Economic MPC and the two-layer strategy
based on the nominal controller, it is easy to realize that, as in the first simulation
experiment, the Economic MPC controller has a better performance, because the
controlled variables respond faster and are kept inside the control zones more
3600 3800 4000 4200 4400 4600 4800 5000 52001.25
1.3
1.35
1.4
1.45
1.5
1.55
1.6x 10
4
nT (min)
f eco (
$/h
)
3500 4000 4500 5000-5000
0
5000
10000
15000
20000
nT (min)
f eco (
$/h
)
3500 4000 4500 50000
0.5
1
1.5
2
2.5
3x 10
5
nT (min)
f eco (
$)
111
efficiently. For instance, the controlled output y2, tends to leave the control zone for a
period of time, but it was brought back to the zone by the Economic MPC faster than
by the MPC using the two-layer approach. Figure 6.24 compares the calculated
inputs for the two controllers and also shows the optimizing input targets (green line)
calculated by ROMeo for the two-layer strategy and corresponding to the feed
column composition shown in Table 6.3. As a consequence of the new feed
composition, the flow rate of the propylene product has to be decreased while the
propane product flow rate is increased. Then, the economic function decreases but
the optimizer tries to maximize the propylene production, which represents the most
valued product as shown in Figure 6.25. As in the first simulation case, the Economic
MPC has a better economic performance, which corresponds to an accumulated
benefit nearly US$ 0.5x105 higher than the controller using a two-layer approach as
shown in Figure 6.26.
112
7 CONCLUSIONS AND DIRECTIONS FOR FUTURE WORK
In this thesis, it was studied the implementation of two advanced control and
optimization strategies based on the nominal IHMPC and on the robust IHMPC in a
propylene production unit. The study was based on the commercial dynamic
simulation software Dynsim® integrated to the real time process optimizer ROMeo®
and the real time facilities of Matlab.
The two-layer hierarchical structure for the integration of RTO and MPC showed to
have an acceptable performance for the nominal and robust controller. First, it was
compared the performances of the IHMPC using OPOM and realignment models to
represent the system, and both demonstrated almost the same performance for the
first experiment where the feed flow rate was increased. Nevertheless, the
realignment model-based IHMPC showed a better performance for the case of non-
measured disturbances (second simulation experiment), as it was able to stabilize
the system faster than the OPOM-based IHMPC, because the realignment model
states are built with the past inputs and outputs.
The robust control structure assumed that model uncertainty can be represented as a
discrete set of models (multi-model uncertainty), each one corresponding to a
possible operating point of the system. Representative simulation examples were
presented, leading to the conclusion that the robust controller shows a better
performance when compared with the nominal one. Since this better performance
was achieved with a reduced set of only three models to represent the propylene
system, it is possible to conclude that the implementation of the control structure
based on the robust IHMPC in the real process is feasible in terms of the resulting
computer time. The NLP problem that is solved in the robust controller takes only a
small fraction of the required sampling time.
The gradient-based IHMPC also called as Economic MPC was studied and
implemented in the PP splitter. It showed an acceptable performance when
compared with the two-layer strategy. The Economic MPC was able to stabilize the
system near the optimal operating point of the system previously calculated by
ROMeo. The controller was difficult to tune since there was a conflict between the
weight of the controlled variables y2 and y3 and the gradient component weight of the
manipulated variable u2. As a consequence, the velocity of the system to be driven to
113
the optimal operating point is highly related with the behavior of the controlled
variables. Then, the controller tends to drive the system to the optimal operating point
more cautiously when any controlled variable is outside its control zone. The
Economic MPC had a better performance in terms of product specification but it
requires a settling time that is almost twice the settling time of the two-layer
integration strategy for the feed flow-rate simulation experiment. In the second
simulation experiment, in which a feed composition disturbance was introduced into
the system, the Economic MPC was able to reject the disturbance, to stabilize the
process and to drive the system to the new optimal operating point. In this case, the
controller response was faster than in the first experiment, but allowed the controlled
variable y2 to remain outside its control zone for a short period of the simulation time
due to the unknown disturbance. The performance of the one-layer approach was
acceptable in the first simulation experiment and the controller was faster to stabilize
the system in the second simulation experiment when compared with the two-layer
strategy.
It is interesting and useful to study, test and implement novel advanced control
algorithms integrated with real-time optimization, because the prototype of a dynamic
simulation (virtual plant) communicating with the real-time optimization and advanced
control algorithms offers a wide spectrum of opportunities and advantages before the
implementation of the control strategy into the real system such as eliminating plant
risks and tests that would be time consuming and expensive. Furthermore, this
combination is not usual in the academic research area, and constitutes an important
achievement.
It would be interesting to identify a larger number of models for the robust controller
and to implement and schedule more disturbed scenarios to test the performance of
the proposed strategies. Also bias and noise could be introduced into the
measurement instruments so that the virtual plant behavior would be as close as
possible to the real one. Other interesting idea would be to implement and study this
prototype when communicating with the Petrobras’ Control System (SICON) to test
advanced controllers and identify models in PP splitters of different refineries. This
idea could be used and applied to any dynamically simulated unit.
114
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Appendix A – Development of the data communication interface
In the simulation implemented here, the advanced control strategy was to be
executed in a Matlab platform, the real time optimization was to be performed in the
steady-state rigorous simulator and optimizer ROMeo and the real plant was to be
simulated in Dynsim. Then, to coordinate these programs, it was necessary to
develop a real-time data communication system. However, there is available in
Dynsim a communication interface based on OPC technology that allows the real-
time data transfer between Dynsim, MATLAB and ROMeo. This OPC facility was
designed to provide a common bridge for Windows based software applications and
process control hardware. To obtain a successful communication, there must be at
least one OPC server and one or various OPC clients.
Interface between Dynsim and MATLAB
The Data communication interface between the commercial process softwares used
in this Thesis, was based on the OPC server (OPC Gateway) of Dynsim and the
OPC client (OPC DA) of Matlab. The data communication interface developed here is
shown in Figure 1, where it is indicated that the advanced control algorithms were
developed in Matlab and the dynamic simulation was performed in Dynsim. As a
result, the values of selected process variables that are necessary to run the
controller are available for reading and writing in both softwares.
Figure 1 – Matlab and Dynsim OPC interface
Interface between ROMeo and Dynsim
Analogously to the previous section, a Data communication interface was developed
to allow the communication of data necessary to the steady state optimization of the
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process system. For this purpose, it was used the OPC server (OPC Gateway) of
Dynsim and the OPC client (OPC EDI) of ROMeo. The data communication interface
could be appreciated as shown in Figure 2, in which the steady-state simulation and
optimization was developed using ROMeo and the dynamic simulation was
developed using Dynsim. As a result, the selected variables values are available for
reading and writing in both softwares.
Figure 2 – ROMeo and Dynsim OPC interface
Interface MATLAB-Dynsim-ROMeo
In this communication structure, the OPC server is the OPC Gateway, which is part
of Dynsim and the OPC clients are the OPC DA, which is part of the OPC toolbox of
MATLAB, and the OPC EDI (External Data Interface) of ROMeo. A representative
scheme of this communication structure is shown in Figure 3. Moreover, in order to
implement the control action in the dynamic simulation at each time interval that
depends on the sampling time of the advanced controller, the code was adapted to
use the timer function of MATLAB.
Figure 3 – Matlab, Dynsim and ROMeo OPC interface