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




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




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

16

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.

17

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.

18

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.

19

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


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


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


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


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




Length 4.90 m

Diameter 3.36 m


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.





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.





Length 1.40 m

Diameter 1.25 m


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

perfo

the i

The

to se

the m

wou

syste

DEVELOP

his chapter

ust advanc

emented in

Mo

first stage

btain the p

ponse mod

ormed with

nputs with

results of

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model iden

ld be expe

em (about

Figure 4

PMENT OF

r, the steps

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odel identif

in the imp

process m

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ach operati

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. 1 – Selecte

F THE PRO

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odel, in th

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periments a

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OPOSED A

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ter and inte

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ADVANCE

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These co

egrated wit

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he process

tion consid

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ds to a diff

performed

g because

rolled variabl

ED CONTR

velopment

ntrollers w

th the real

control bas

namic mod

s in Figure

dering step

nput range

Figure 4.2,

ferent step

in the rea

e of the hig

le for the ste

ROL STRA

t of the no

were propo

time optim

sed on line

del such a

4.1, step t

p in the se

of each in

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


, ,

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


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


(4.29)

68

1

, , , ,0

( ( | ) ) ( ( | ) )m

Tdes k u k u des k u k

j


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



(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


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


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



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



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

mod

5.1.4

The

deve

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The

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In

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Start: Star

ROMeo and

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dynamic si

nto the RO

SD task dev

hods. As a

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Figure 5.

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The sequ

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g the spe

e invokes m

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ence in ROM

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nt (scan v

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odel applic

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cation dev

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86

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ting model

s a typical

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ownloaded

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e

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p

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d

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





Optimizing input target weight Qu = diag ([0.01 10 0.1])


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





Optimizing input target weight Qu = diag ([0.1 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






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 (— — —),


Figure 6. 24 – Manipulated inputs (Second experiment), Economic MPC (— — —),


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 (— — —),


Figure 6. 26 – Economic function with penalization (Second experiment), Economic MPC (— — —),


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

integration of real time optimization (rto) and ......figure 4. 1 – selected manipulated and...

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