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Model-Based Control Techniques
for Centrifugal Compressors
Toufik Bentaleb Ph.D Thesis in Information Engineering and Science
University of Siena
UNIVERSITA DEGLI STUDI DI SIENA
DIPARTIMENTO DI INGEGNERIA DELL’INFORMAZIONE
Model-Based Control Techniques
for Centrifugal Compressors
Tesi di Dottorato di
Toufik Bentaleb
Advisor: Prof. Andrea Garulli
Siena, February 2015
DOTTORATO DI RICERCA IN INGEGNERIA DELL’INFORMAZIONE
− CICLO XXVII −
Model-Based Control Techniques for Centrifugal Compressors
Toufik Bentaleb,
© Ph.D. Thesis, University of Siena,
February, 2015.
This research was financially supported by GE Oil & Gas Nuovo Pignone Florence.
ACKNOWLEDGMENTS
First of all, I would like to thank all those who have shared something with me during these
years. It is thanks to each one of you that I have managed to grow as much as I have
done. Among all of you, I am very grateful to Professor Andrea Garulli, who has been the
best guide I could have ever had. He has taught me so many things, both consciously and
unconsciously, that I will be in debt with him for a very long time. Without him I would not
have ever imagined that studying optimal control systems can also be funny.
I enjoyed the time in Siena with Alessandro, Mirko and Donato who formed with me a
group of people that I will hardly forget. Moreover, I want to thank people from GE Nuovo
Pignone of Florence, L. Giovanardi, A. Cacitti, S. De Franciscis, D. Galeotti, and M. Pasquotti.
I would like to express my gratitude to my family, especially my parents, for their love and
support. Finally, I would like to thanks my wife, Mira, who made everything possible. She
sacrificed a lot to keep me on track in finishing this study taking all the duties in the family,
caring about our two babies unsparing her efforts. Her true love has helped me through all
the hard periods.
Siena
February 27, 2015
xi
Contents
Preface ix
Acknowledgements x
Glossary xii
1 Introduction 1
1.1 Turbomachines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Centrifugal Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Gas Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Compressor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Compressor Surge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 The Goal of this Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Control Techniques 11
2.1 Proportional Integral Control with Anti-Windup . . . . . . . . . . . . . . . . . 11
2.2 Linear Quadratic Regulator Optimal Control . . . . . . . . . . . . . . . . . . . 12
2.3 Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Modeling of Gas Compression Plant 19
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Upstream Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.2 Downstream Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Mixer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Orifice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Cooler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6.1 Anti-Surge Valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
v
3.6.2 Upstream and Downstream Valves . . . . . . . . . . . . . . . . . . . . 30
3.7 Centrifugal Gas Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7.1 Compressor Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7.2 Difference between the accuracy of the interpolation methods . . . . . 40
3.7.3 Performance evaluation of the centrifugal compressor . . . . . . . . . 41
3.8 Gas Turbine Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.9 Linearization of the Gas Compression System . . . . . . . . . . . . . . . . . . 45
4 Control Techniques for Pressure Regulation 49
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 The Discharge Pressure Regulation Problem . . . . . . . . . . . . . . . . . . . 49
4.2.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Implementation of Anti-Windup Proportional-Integral (PI) Control . . 51
4.2.3 Implementation of LQI . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.4 Implementation of MPC . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Comparison of Multivariable Control Schemes . . . . . . . . . . . . . . . . . . 54
5 Surge Prevention 61
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Centrifugal Compressor Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.1 Compressor Map in (Hp vs Qv) Coordinates . . . . . . . . . . . . . . . 63
5.2.2 Compressor Map in Invariant Coordinates . . . . . . . . . . . . . . . . 64
5.3 Distance to Surge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3.1 Distance to surge in the (Rc − 1 vs q2s × psd) coordinates . . . . . . . . 66
5.3.2 Distance to surge in the (hr vs q2s) coordinates . . . . . . . . . . . . . . 67
5.4 MPC Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Results Analysis and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5.1 Case study I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5.2 Case study II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5.3 Case study III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.6 Comparison of Different MPC-Based Control Schemes . . . . . . . . . . . . . . 78
5.7 Noise Rejection and Chattering Avoidance . . . . . . . . . . . . . . . . . . . . 85
6 Fuel Consumption Optimization 87
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Fuel Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Fuel Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Open-Loop Fuel Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4.1 Fuel Minimization by Acting on Inlet Guide Vane at Steady State . . . 90
6.4.2 Fuel Minimization by Acting on Rotational Speed at Steady State . . . 91
6.5 Closed-Loop Fuel Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5.1 Closed-Loop Fuel Optimization Local Search . . . . . . . . . . . . . . . 93
6.5.1.1 Case Study I . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5.1.2 Case Study II . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5.2 Closed-Loop Fuel Optimization Global Search . . . . . . . . . . . . . . 102
6.5.2.1 MPC Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5.2.2 Case Study I . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.5.2.3 Case Study II . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7 Conclusions and Future Research 109
A Finite-State Machine 111
A.1 Finite-State Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Bibliography 115
PREFACE
This dissertation is submitted for the degree of Doctor of Philosophy at the University of
Siena (Universit degli Studi di Siena). The research described herein was conducted under
the supervision of Professor Andrea Garulli in the Department of Dipartimento di Ingegneria
dell’Informazione e Scienze Matematiche (DIISM), Universita’ degli Studi di Siena, during
the period January 2012 through February 2015. It has been financed by GE Oil & Gas
Nuovo Pignone Florence.
During my studies I have visited GE Oil & Gas - Nuovo Pignone many times for guidance
and feedback. I am grateful to my advisors S. De Franciscis and Dr A. Cacitti at GE Oil &
Gas, who have followed up my work monthly through meetings. De Franciscis’s expertise
has been very helpful for my understanding of natural gas compression systems.
Part of this work has been presented in the following publications:
T. Bentaleb, A. Cacitti, S. De Franciscis, A. Garulli, Multivariable Control for Regulating High
Pressure Centrifugal Compressor with Variable Speed and IGV, 2014 IEEE International Con-
ference on Control Applications (CCA) Part of 2014 IEEE Multi-conference on Systems and
Control (MSC), October 8-10, 2014, pages 486-491, Antibes, France.
T. Bentaleb, A. Cacitti, S. De Franciscis, A. Garulli, Model Predictive Control for Pressure
Regulation and Surge Prevention in Centrifugal Compressors, submitted to the European
Control Conference ECC15, July 15-17, 2015, Linz, Austria.
Toufik Bentaleb
February, 2015
ix
Glossary
Notation and Symbols
N speed, [RPM ]
P power, [kW/h]
p pressure, [bara]
∆p difference pressure in the orifice, [bara]
w flow rate, [kg/s]
T temperature, [K]
Ta ambient temperature, [K]
Ru molar gas constant (universal gas constant) ≈ 8.134, [J/(mol.K)]
cp specific heat capacity wit p= const.,cp =kvRu
kv − 1
cv specific heat capacity with v= const.,cv =Ru
kv − 1kv isentropic volume exponent (specific heat ratio) (kv = cp/cv), [.]
Rc compression ratio (Rc = pd/ps), [.]
Z compressibility factor, [1/MPa]
Zavg average compressibility factor (Zavg = (Zs + Zd)/2), [1/MPa]
V specific volume, [m3]
MW gas-molecular weight, [g/mol]
A orifice constant, [.]
ρ fluid density, [kg/m3]
u2 impeller tip speed, [m/s]
dh head, [J ]
α inlet guide vane, [◦]
Hp polytropic head, [kJ/kg]
Qv volumetric flow rate, [m3/s]
D impeller diameter, [m]
Ψ pressure coefficient, [.]
M mach number, [.]
ηp polytropic efficiency, [.]
Hhot effective head, [kJ/kg]
ϕhr heat rate, [kJ/kWh]
a sound speed, [m/s]
utr travel of the control valve, [%]
ηGT gas turbine efficiency
Z field of integer numbers
R field of real numbers
xiii
Subscripts
1 upstream 2 downstream
avg average c compressor
d discharge mech mechanical losses
s suction o orifice
v volume sd design conditions
in input to compression stage out output from compression stage
r recycle hr heat rate
i integral b back calculation
p proportional
List of Acronyms
ASV Anti-Surge Valve
CCLib Centrifugal Compressor Libraries
PGT Power Gas Turbine
SAC Standard Annular Combustor
DLE Dry Low Emissions
LTI Linear Time-Invariant
LPV Linear Parameter Varying
RPM Revolutions Per Minute
IGV Inlet Guide Vane
LQG Linear Quadratic Gaussian
LQI Linear-Quadratic-Integral
QP Quadratic Programming
SISO Single-Input Single-Output
MIMO Multi-Input Multi-Output
PI Proportional Integral
MPC Model Predictive Control
Chapter 1
Introduction
The general topic of this thesis is control of turbomachinery. In particular, it deals with mod-
eling and control techniques for variable speed centrifugal gas compression systems. In this
introduction, first the operation principles of centrifugal compressors and gas turbines are
presented in Section 1.1. Section 1.2 describes the performance curves of the compressors.
The main characteristics of the surge phenomenon are discussed in Section 1.3. Finally, the
main contributions of this research and the outline of the thesis are summarized in Section
1.4 and 1.5, respectively.
Figure 1.1: Applications of Turbomachinery.
1.1 Turbomachines
Any device that extracts energy from or imparts energy to a continuously moving stream of
fluid (liquid or gas), by using rotating blades, can be called a Turbomachine. Turbomachin-
ery is the generic name of turbine, compressor, fan, blower and pump machines. A turbine
machine is a device that extracts energy from the fluid, while the other machines deliver
2 1. Introduction
energy to the fluid. Turbomachines are used in a wide variety of applications, the primary
ones being electrical power generation, pipeline transportation, aircraft propulsion and ve-
hicular propulsion for civilian and military use. The classification of turbomachinery can be
tackled from different viewpoints:
• Energy transfer direction
– Absorbing Power: Compressors, pumps
– Producing Power: Turbines, fluid motors
• Fluid density change
– Hydraulic Machines (constant density): Pumps, Fans
– Thermal Machines (variable density): Compressors, gas turbines
Figure 1.1 shows four important applications of turbomachinery: gas turbine propulsion
for aeroplanes; wind turbine for electricity production; stream turbine for power production
or mechanical driver; centrifugal compressor for gas compression. Two types of turboma-
chines are considered in this thesis: centrifugal compressors and gas turbines.
1.1.1 Centrifugal Compressors
Centrifugal Compressor Axial Compressor
Rotary Compressor
Reciprocating Compressor
2 2 2 23 3 3 34 4 4 45 5 56 6 67 7 7
100
100
1000
1000
10,000
10,000
100,000
100,000
500
5000
50,000
SP
EE
D(R
PM
)
SUCTION FLOW (m3/hr)
Figure 1.2: Operating range of centrifugal compressor compared with other types of compressors.
1.1. Turbomachines 3
The word ”Compressor” may have different meanings in different domains. One of them
is ”Gas Compressor”, which we are considering in this work. Gas compressors are defined
as follows in the Encyclopedia Britannica:
A compressor is a device for increasing the pressure of a gas by mechani-
cally decreasing its volume. Air is the most frequently compressed gas but
natural gas, oxygen, nitrogen, and other industrially important gases are
also compressed.
There are two primary categories of compressors: positive displacement and dynamic
compressors. There are two subcategories in positive displacement: reciprocating and ro-
tary. Dynamic compressors also present two sub-categories: axial and centrifugal. Fig-
ure 1.2 shows the operating range of the four types of gas compressors, expressed in terms
of rotational speed versus suction flow. Centrifugal compressors are also known as radial
compressors, turbocompressors or continuous flow compressors.
(a) (b)
(c) (d)
Intake
Intake
IntakeIntake
IntakeIntake
Discharge
Discharge
Discharge
Discharge
Discharge
1 st1 st
2 nd2 nd
Rotation
Rotation
Figure 1.3: Centrifugal compressors (courtesy of Dresser-Rand): (a) Straight-through (inline) cen-
trifugal compressor cross section, (b) Inline (compound) centrifugal compressor cross section, (c)
Back-to-back centrifugal compressor cross section, and (d) Dual-flow centrifugal compressor cross
section.
The major components of various centrifugal compressor flowpath configurations are
illustrated in Figure 1.3. These centrifugal compressors are used for compressing large
amounts of gas. Centrifugal compressors have three basic components: an impeller, a dif-
fuser and a volute casing. Large capacity centrifugal compressors may have two impellers
or stages in the same casing.
4 1. Introduction
Discharge
Suction
IGV
Impeller
Diffuser
Volute
Figure 1.4: Centrifugal compressor.
Centrifugal compressors are usually driven by electricity motors. However, open-drivers
centrifugal compressors are also available for applications with stream turbines, gas tur-
bines or other types of engines. The impeller is a rotating circular disk with curved blades.
As the impeller rotates, it moves the gas from the suction opening in its center into the
diffuser, using centrifugal force. The impeller forces the flow to spin faster and faster. The
gas enters the suction at relatively low velocity and leaves the outer edge of the impeller at
a high velocity: this means that the impeller transfers its rotational energy to the gas. To
achieve the desired pressure increase or compression, the gas must be slowed down, con-
verting its velocity pressure to static pressure. Here is when the diffuser comes in. As high
velocity gas moves radially out through the diffuser, the flow area increases, thus slowing
the gas and increasing the static pressure. Some centrifugal compressors have diffusers with
vanes or pipes which change the flow direction and further slows the gas. The volute shape
casing collects the high pressure gas from the diffuser and sends it to the discharge of the
compressor (see Figure 1.4).
Figure 1.5: Centrifugal compressor with three different IGV opening.
Inlet Guide Vane (IGV) is a device used to control the inlet flow to the centrifugal com-
pressor. IGV is usually mounted at the inlet to the first stage is used to control centrifugal
compressor capacity, this movable vanes are located in suction opening (see Figure 1.5).
With vanes turned fully open the compressor, produces full capacity. As the vanes closes,
1.1. Turbomachines 5
the gas direction is changed and the compressor reduces its capacity. In addition, capacity
control in centrifugal compressors can also be pursued by changing the rotating speed.
1.1.2 Gas Turbines
Air InletHot exhaust gase
Compressor
Combustion chamber
Turbine
Duct
Power turbine
Second shaft
First shaft
Figure 1.6: PGT25 SAC Two Shaft Gas Turbine [1].
Gas turbines are defined as follows in the Encyclopedia Britannica:
A Gas-turbine engine is any internal-combustion engine employing a gas
as the working fluid used to turn a turbine. The term also is convention-
ally used to describe a complete internal-combustion engine consisting of
at least a compressor, a combustion chamber and a turbine.
In pipeline applications, the gas turbine is used to drive a centrifugal compressor train. A
gas turbine, also called combustion turbine, is a heat engine that converts chemical energy
from the fuel into heat energy, which in turn can be transformed into mechanical energy.
The gas turbine can be divided into four parts:
• air compressor;
• combustion chamber;
• power turbine;
• actuators (fuel valve, steam and bleed valves, inlet guide vane, variable stator vane).
6 1. Introduction
In a gas turbine engine, the air passes through the intake by a compressor which is
attached to the main shaft. At the end of the shaft the turbine is attached. Air passed
through the intake by the compressor will increase its pressure and temperature as passes
through the compressor stages of the engine. Air leaves the compressor and passes to the
diffuser which converts the velocity energy to pressure energy. On leaving the diffuser, air
passes into combustion chambers which are used for combustion and cooling. After passing
through the combustion chamber, the hot gases enter the nozzle guide vanes, which direct
them at increased velocity on to turbine blades, causing the turbine disks, or stages, to
rotate. This in turn forces the compressor to bring the air in and compress it. In the case of
the PGT25 gas turbine, a power turbine placed after the turbine stages, with a second shaft
to drive a centrifugal compressor or an electricity generator. Between the turbine stages and
the power turbine there is no mechanical connection, i.e. each one has different velocity,
which in between the hot exhaust gas passes through the duct to the power turbine (see
Figure 1.6).
1.2 Compressor Control
Usually the performance of a compressor is described by suitable curves provided by ven-
dors. A typical example of such curves shows differential pressure versus inlet flow rate
at certain fixed conditions. Since the compressor will be connected to other devices, this
curves are not sufficient to evaluate the compressor performance. In fact, an industrial plant
provides many different inlet operating conditions to the compressor. The operation of the
centrifugal compressor is usually described by three operating parameters: polytropic head,
flow rate and rotating speed. Figure 1.7 shows the performance map of centrifugal com-
pressor. It plots polytropic head (Hp) as a function of volumetric flow (Qv) and rotational
speed, which is related to the mechanical proprieties and dependent to gas properties. The
definition of these quantities and their relationships with the compressor characteristics are
reported in Chapter 5.2.1.
1.3 Compressor Surge
Dynamic turbomachines are subject to nonlinear phenomena of different natures: aerody-
namic (surge and rotating stall), aeroelastic (flutter) and combustion that may not function
properly. Surge is characterized by oscillations of mass flow and pressure. This phenomenon
is therefore highly undesirable for the compressor and gas compression system. The objec-
tives of a surge control system can be divided in two types [2]:
• Surge avoidance: the control system has to avoid surge phenomenon preventing the
occurrence of the instabilities: this is the industrial standard;
• Active surge control: when surge oscillations develop, the control system must stop
surge oscillations very rapidly.
1.3. Compressor Surge 7
Poly
tropic
Head
(Hp)
Flow rate (Qv)
N
Load or throttle
Surgeline
line
line
Choke or Stonewall
Speedline
Figure 1.7: A schematic representation of a compressor map.
Mathematical models that describe the dynamic behavior of turbocompressors during
surge and rotating stall are available. Several researchers have contributed to the evolu-
tion of these models, starting with Moore and Greitzer [3], who developed a mathematical
model for the turbine, capable of explaining the its nonlinear behaviour. Van Helvoirt [4]
gives an overview of literature on compression system modeling from year 1955 to year
2000 in particular, the reader is referred to [3, 5–9] for an extensive treatment. Active surge
control has received increasing attention in recent years. Most of the literature concerns
control techniques for an analytical model of the surge phenomenon, which has been de-
veloped by Greizer and Moore [3, 6]. Several other models of this nonlinear behavior can
be found in the literature (see e.g. [10–22]). A wide variety of control techniques have
been considered, including linear control [16, 23–27], Adaptive control [25, 28–33], Active
Control Techniques [34], Nonlinear H2 and H∞ [24, 35], backstepping [15, 36], fuzzy logic
[37–41], and sliding mode control [42, 43].
This thesis will be focused on surge prevention, rather then on active surge control, for
two main reasons. The first is that the aim is to design the primary control systems which
works in standard conditions, while it is common practice in industry to have a complemen-
tary protection system taking care of emergency situations such as recovery from surge or
stall. Secondly, the industrial plant model considered in this work is a simulation model
with many non analytical components, which would prevent from applying control tech-
niques based on purely analytical models, such as the Greizer-Moore model.
8 1. Introduction
1.4 The Goal of this Research
The objective of the control system of a centrifugal compressor is to keep the primary process
variable (for example, suction/discharge pressure or mass flow rate) at a desired set point
level and to track the set point as quickly as possible whenever a process disturbance oc-
curs. At the same time, the operating point must be kept within the safe or acceptable train
operating envelope, considering limits such as surge or stonewall. Moreover, limitations on
speed, inlet guide vane, pressure and power, must be kept into account. The project focuses
on the modeling and control of a gas turbine and centrifugal gas compressor in natural gas
pipeline transportation systems. The main objective of this thesis is to design and imple-
ment multivariable controllers that regulate the centrifugal gas compressor and minimize
fuel consumption of the gas turbine. In particular, the first aim is to design a multivariable
controller that regulates the compressor discharge pressure as faster as possible, while pre-
venting surge, by acting on the rotation speed, the inlet guide vane and the anti surge valve
(ASV). A further contribution, is the development of a control system that minimizes fuel
consumption of the gas turbine at steady state. In order to achieve this, the following sub
goals are necessary:
• Developing a model of a multi-speed centrifugal gas compressor equipped with inlet
guide vane, from real data.
• Constructing and implementing a dynamic model of a compression plant which con-
tains the centrifugal compressor, a gas turbine, two volumes, a cooler, two actuator
valves at the downstream and upstream, and a recycle valve.
• Designing a multi-variable controller tracking a reference value of the discharge pres-
sure of the centrifugal compressor by acting only on rotational speed and inlet guide
vane.
• Extending the model of the plant by adding the recycle flow to prevent the compressor
from entering into surge.
1.5 Thesis Organization
This Thesis reports the results of the performed research activity and is organized as follows:
Chapter 2 describes the control techniques considered in the thesis. First, it presents a Pro-
portional Integral (PI) controller with anti-windup. Then, detailed description of Linear
Quadratic Gaussian (LQG) optimal control with integral action for MIMO feedback design is
shown. Finally, a Model Predictive Control (MPC) algorithm for linear discrete-time systems
is introduced.
Chapter 3 introduces and explains the gas compression system components. The centrifugal
compressor model includes the effects of the inlet guide vane. Then, a full system model is
1.5. Thesis Organization 9
developed, reproducing the dynamic behaviour of a gas compression plant.
Chapter 4 describes the implementation of the MIMO (multivariable control) and PI con-
trollers for the plant, in order to regulate the discharge pressure as quickly as possible after
a process disturbance. The input constraints are considered. The results show that MIMO
controllers are able to reject disturbances in the upstream and downstream valves, while
tracking the desired reference pressure within shorter time with respect to PI SISO control.
Chapter 5 describes the recycle compression system with anti-surge valve and the proposed
surge prevention strategy.
Chapter 6: presents four different strategies for minimizing the fuel consumption of the gas
turbine at steady-state. Two open-loop approaches coupled with the MPC controller, are
• Fuel optimisation by acting on inlet guide vane at steady state, without measuring the
fuel consumption of the gas turbine;
• Fuel optimisation by acting on rotational speed at steady state, without measuring the
fuel consumption of the gas turbine.
Two closed-loop approaches coupled with the MPC controller, are
• Fuel optimisation by acting on inlet guide vane at steady state, without measuring the
fuel consumption of the gas turbine;
• Fuel optimisation by acting on inlet guide vane at steady state, using logic control law,
with measuring the fuel consumption of the gas turbine.
In chapter 7, the main contributions of this Thesis are summarized and discussed. On-
going work and future research directions are also highlighted.
Chapter 2
Control Techniques
Abstract
This chapter briefly reviews the control techniques which are used in this thesis. First, a Pro-
portional Integral (PI) controller with anti-windup is presented. Then, the Linear Quadratic
Gaussian (LQG) optimal control with Integral action for MIMO feedback design is described.
Finally, a Linear Model Predictive Control (MPC) algorithm for linear discrete-time system is
presented.
2.1 Proportional Integral Control with Anti-Windup
Proportional-integral control is a feedback control technique widely used in industrial con-
trol systems. PI controller with anti-windup design based on the back calculation method
is adopted in this thesis. The usefulness of anti-windup logic to prevent input saturation
has been shown in many applications (see e.g. [44], [45], [46], [47], and [48] for more
details). PI control is a standard control technique, which is commonly used to track a ref-
erence signal. In this section, we will see briefly the description of the PI controller with
anti-windup protection. The basic PI controller is described by:
u(t) = K
(
e(t) +1
Ti
∫ t
0
e(τ)dτ
)
(2.1)
where u is the control signal, e is the control error (e = ysp − y), y is the measured process
variable and ysp is the set point (i.e. reference value). The control signal is thus a sum of two
terms: the P-term (which is proportional to the error) and the I-term (which is the integral
of the error). The controller parameters are the proportional gain K and the integral time
Ti.
There are different form of the nonlinearities contained in real-world control systems,
which make the control more difficult. Saturations are a typical example. For instance,
each gas turbine model covers a speed range, which in turn limits the operating points. The
control saturation occurs when the control signal reaches the maximum or minimum limit
of the speed. This phenomenon makes the feedback-loop inaccurate, because the speed
remains on its maximum or minimum limit independently of the plant output. In this case,
the feedback loop is broken and the system runs in open loop. However, the controller with
12 2. Control Techniques
plant
∫
().dt
eysp u
et
v y
1
Tt
Kp
Ki
actuator+
+
+
+
+
-
-
-
PI
anti-windup
Figure 2.1: Tracking anti-windup, back calculation strategy
integral action is still active, thus the error continues to be integrated. This phenomenon is
called windup. There are several ways to prevent the integrator windup, and may be mainly
divided into three categories which include the back tracking calculation, the conditional
integration, and the limited integrator schemes [49, 50]. In this work, the back-calculation
anti-windup method, which is the most popular technique, is applied. As an illustration
we consider the PI controller in Figure 2.1. Saturation is dangerous because it breaks the
control loop. The protection against integral windup, consists of an additional feedback-
loop. The error et(t), which is the difference between the input and output of the actuator,
is added to the input of the integrator with a gain 1/Tt. The anti-windup loop has no effect
when there is no saturation (et(t) = 0). The error is defined as
et(t) = v(t)− u(t) (2.2)
where u(t) = sat(v(t)), and the saturation function sat is a static nonlinearity which is
defined by
sat(v(t)) =
umin if v(t) < umin
v(t) if umin ≤ v(t) ≤ umax
umax if v(t) > umax
(2.3)
The tuning parameters of the PI controller are:
• The proportional gain Kp = K;
• The integral gain Ki = K/Ti;
• The back-calculation coefficient Kb = 1/Tt;
2.2 Linear Quadratic Regulator Optimal Control
In this section we describe linear quadratic (LQ) optimal control for MIMO feedback design.
In order to perform output tracking of a non-zero reference signal ysp, an integral action is
added to classic LQ optimal control, by suitable augmenting the original state-space system.
2.2. Linear Quadratic Regulator Optimal Control 13
Since complete state information is not available, it is necessary to use a Linear Quadratic
Gaussian (LQG) control scheme, which includes Kalman filter estimation of the state vector.
Linear Quadratic control with Integral action (LQI) is an optimal control technique which
aims at minimizing a quadratic function of the state and input variables. Kalman filtering
techniques are used to estimate the state vector, when some of the states are not accessible.
Ensuring offset-free reference tracking by integrating the error between the reference signal
and the output signal, is done by augmenting the system with the reference error integration
state and then minimize the augmented state in the controller cost function. The complete
control scheme is depicted in Figure 2.2.
plant∫
eysp uv y
Kalman filter
K
actuator
+
+++
-
-
LQI
uss
yss
xi
x
Figure 2.2: The block diagram of the closed loop system using LQI controller with anti-windup con-
trol.
The linearized model around the stationary points xss and uss is
δx = Aδx+Bδu (2.4)
δy = Cδx
where
δx = x− x0, (2.5)
δu = u− u0.
The discrete-time model of linearized model (2.4) is
δxk+1 = A δxk + B δuk (2.6)
δyk = C δxk,
Assume the system is described by the discrete-time model (2.6) (to simplify notation,
hereafter we omit the symbol δ before the variable names). Thus, we can rewrite the
discrete-time model as
{xk+1 = A xk + B uk
yk = C xk,(2.7)
14 2. Control Techniques
where A ∈ Rn×n, B ∈ Rn×m and C ∈ Rp×m are state space matrices.
Define the output tracking error as ek = ysp,k − yk = ysp,k − yk and the error sum as
xi,k+1 = xi,k + Ts ek, (2.8)
By defining the augmented state as ξk = [xk, xi,k]T , one gets the augmented system equa-
tions[
xk+1
xi,k+1
]
=
[
A 0
−TsC 1
][
xk
xi,k
]
+
[
B
0
]
uk +
[
0
Ts
]
ysp,k (2.9)
The aim is to define a state feedback control law
uk = −K ξk (2.10)
minimizing the quadratic cost function
J =
∞∑
k=0
{ξTk Q ξk + uT
k R uk
}. (2.11)
The classic LQ control theory provides the way to compute the state feedback matrix K by
solving a Riccati equation [51–53]. When the state vector is not fully measurable, xk is
replaced by the corresponding estimate xk provided by the Kalman filter. The augmented
state is hence ξk = [xk, xi,k]T . The presence of the integral action ensures that the output
yk tracks the reference command ysp,k asymptotically (for more detailed about using linear
quadratic regulator with integrator function see [54]).
2.3 Model Predictive Control
Model Predictive Control (MPC) is widely used in industry because it can handle multivari-
able control problems naturally with safety constraints. MPC is a control strategy in which
the current control action is obtained by solving on-line, at each sampling instant, a finite
horizon open-loop optimal control problem, using the current state of the plant as the initial
state; the optimization yields an optimal control sequence and the first control input of this
sequence is applied to the plant. The use of model predictive control for centrifugal com-
pressors has been considered and investigated by several works in the literature. In [55],
a linear MPC scheme has been designed for anti-surge control of a plant with two com-
pressors. In [56–58], nonlinear MPC formulations have been proposed for different plant
families. Some works use both the compressor rotational speed and the position of an Anti-
Surge Valve (ASV), also called recycle valve, for anti-surge control, see e.g. [59]. However,
in all these studies the Inlet Guide Vane (IGV) is not used as a further degree of freedom
of the control system. In centrifugal compressors, IGV is typically used to modify the mass
flow rate without acting on the rotational speed or compressor ratio [60].
Until recently, industrial applications of MPC have mostly relied on linear dynamic mod-
els even though most processes are nonlinear [61], [62]. Although there is an increasing
2.3. Model Predictive Control 15
research interest in nonlinear MPC ([63], [64], [65], [66]), most of this literature is dedi-
cated to systems described by analytical models, while our reference applications contains
non analytical parts (e.g. look-up tables). For these reasons, Linear MPC (LMPC) is em-
ployed, based on a suitable linearised model of the plant.
The main idea behind MPC is illustrated in Figure 2.3. At each time instant k ∈ Z0+,
where Z0+ is the set of nonnegative integers, the controller solves an optimal control prob-
lem over a finite prediction horizon [k, k+np]. To limit the number of optimization variables,
the control input may be allowed to change over a shorter control horizon [k, k + nm], with
nm < np, and then kept constant from the optimal input sequence is applied to the plant.
Then, a new optimal control problem is solved at time k + 1 and so on.
k + nm k + npk + 1
uk+1
uk
k
Re-optimal inputtrajectory (time k + 1)
Optimal inputtrajectory (time k)Closed-loop input
Closed-loop state
(measured)
Past Future/prediction
State (forecast)
Desired set-point
Control horizon nm
Prediction horizon np
Figure 2.3: The basic concept of MPC.
In this section, we give a brief review of the basic setup of linear model predictive control.
Consider the discrete-time linear time-invariant system (2.7) with n state variables in vector
x ∈ Rn, m inputs in u ∈ Rm and p inputs in y ∈ Rp. Its evolution is described by
{xk+1 = A xk + B uk,
yk = C xk.(2.12)
where A ∈ Rn×n, B ∈ Rn×m and C ∈ Rp×m are state space matrices. Then xk+2 and
16 2. Control Techniques
yk+1 can be writing in terms of xk+1 and uk and then substitute from (Eq. 2.12)
{xk+2 = A xk+1 + B uk+1 = A2 xk + AB uk + B uk+1,
yk+1 = C xk+1 = CA xk + CB uk.(2.13)
For the other future time steps a similar procedure can be done. The resulting time
series can be expressed in a convenient block matrix form. For the predicted system states
we obtain the following:
xk+1
xk+2
...
xk+np
︸ ︷︷ ︸
xk+1,np
=
A
A2
...
Anp
︸ ︷︷ ︸
Px
xk +
B 0 · · ·
AB B · · ·...
.... . .
Anp−1B Anp−2B · · ·
︸ ︷︷ ︸
Hx
uk
uk+1
...
uk+np−1
︸ ︷︷ ︸
uk,np−1
(2.14)
We get a similar expression for the system outputs.
yk
yk+1
yk+2...
yk+np−1
︸ ︷︷ ︸
yk,np−1
=
C
CA
CA2
...
CAnp−1
︸ ︷︷ ︸
P
xk +
0 0 0 · · ·
CB CB 0 · · ·
CAB 0 0 · · ·...
......
. . .
CAnp−2B CAnp−3B CAnp−4B · · ·
︸ ︷︷ ︸
H
uk,np−1.
(2.15)
We call the matrices Px, Hx, P and H state and output prediction matrices respectively.
Obtaining a prediction of future system evolution is now straightforward. The only thing we
need is an input time series uk,np−1 and an initial state vector xk. The prediction of system
state and system output is now easily computed as
xk+1,np= Pxxk + Hxuk,np−1 and yk,np−1 = Pxk + Huk,np−1. (2.16)
The objective is to find at each time k the optimal solution to the quadratic control
problem
minu(k|k),...,∆u(nm−1+k|k)
{np−1∑
i=0
(nu∑
j=1
∣∣wu
i,j [uj(k + i|k)− utarget,j ]∣∣2
+
nu∑
j=1
∣∣w∆u
i,j ∆uj(k + i|k)∣∣2+
ny∑
j=1
|wyi+1,j [yj(k + i+ 1|k)
−ysp,j(k + i+ 1)]|2
)
+ ρǫǫ2
}
(2.17)
2.3. Model Predictive Control 17
Subject to:
umini ≤ u(k + i|k) ≤ umax
i
∆umini ≤ ∆u(k + i|k) ≤ ∆umax
i , i = 0, . . . , np − 1
−ǫ+ ymini ≤ y(k + i+ 1|k) ≤ ymax
i + ǫ
∆u(k + j|k) = 0, j = nm, . . . , np
ǫ ≥ 0
Here in (2.17), ∗(k + i|k) denotes the value predicted at time k for time (k + i). This
predictive value is compared by using the expressions (2.15)-(2.16). wui,j is the input weight,
w∆ui,j is the input increment weight and wy
i+1,j is the output weight. utarget,j is the target
setpoint for the manipulated variables. r(k) is the current sample of the output reference.
The cost function is minimized by a Quadratic Programming (QP) solver. Schmid et al. [67]
describe the algorithm of the solver. When the state vector is not fully measurable, xk|k is
replaced in the MPC equations by the corresponding estimate xk|k, which the linear mean
square error estimate of xk|k, based on the input and output measurements up to time k,
delivered by the standard Kalman filter.
Chapter 3
Modeling of Gas Compression Plant
Abstract
This chapter presents a compression system modeling for natural gas transportation via
pipeline in industrial plants. The model is divided into several section: turbomachines, vol-
umes, upstream/downstream valve, cooler, antisurge valve, and mixer. The turbomachines
part models the variable speed centrifugal gas compressors and gas turbines. The compres-
sor model is then augmented to include an inlet guide vane control system. The augmented
compressor model is validated by means of experimental data, taken from a real centrifugal
compressor.
3.1 Introduction
A dynamic model of a gas compression plant is a key tool for simulation and validation
of control systems. The model is based on both first principles and on experimental data,
i.e. the dynamic performance is determined by the compressor geometry and by the exper-
imentally determined characteristic performance curves. A schematic presentation of the
overall model structure of the compression plant is shown in Figure 3.1. The industrial com-
pression recycling plant includes a variable speed centrifugal compressor, driven by a gas
turbine. The compressor is equipped with an adjustable IGV and an ASV which allows gas
to recycle from compressor discharge to inlet. Two volumes are present, one at suction (V 1)
and another one at discharge of the compressor (V 2). They are equipped with an upstream
and downstream valves (UV and DV , respectively) to simulate different load conditions.
The plant also contains a cooler located after the second volume. Closing or opening of these
valves cause sudden variations in the flow at the upstream and downstream of the plant.
These events represent disturbances that affect the compression system due, for example, to
changes that occur along the pipeline. The plant includes several transmitters, measuring
the following variables: rotational speed (ST), inlet guide vane position (ZT), anti-surge
valve position (FY), temperature (TT), pressure (PT), and differential pressure across the
suction orifice (FT). The model also includes a cooler for gas cooling. By changing the po-
sition of the IGV, it is possible to modify the compressor characteristics, thus enlarging the
region of feasible operating conditions. In other words, the use of IGV as an additional con-
trol variable provides a significant enhancement of the authority of the control system, both
in terms of performance and for surge prevention.
20 3. Modeling of Gas Compression Plant
V1
V2
FY
FT
PT
PT
ST
TT
TT
ZT DOWNSTREAM
UPSTREAM
VALVE
VALVE
COOLEROutlet
Inlet
ASVN
IGV
pd
Td
ps Ts ∆po,s
Figure 3.1: Schematic description of a gas compression plant
The block diagram shown in Figure 3.2 gives the main components or modules of the
compression system analysed in this study, where the input/output variables are defined in
the following sections together with the description of each module.
3.2 Volumes
Volume (called the plenum) is used in the gas compression system to stabilize the pressure
fluctuation. In the volume the pressure is not static because the fluid is moving, then the
pressure is dynamic.
3.2.1 Upstream Volume
The mass balance of the upstream volume (V1) in Figure 3.1 is described by
V1 ρ1 = wm − wc (3.1)
where V1 is the upstream volume, ρ is the density, wm is the mass flow rate entering into the
volume (V1), wc is the mass flow rate through the compressor.
3.2. Volumes 21
Ts
Ts
Ts
UV
DV
Com
pre
ssor
Upstre
am
Dow
nstre
am
An
tisurg
eValv
e
Valv
eValv
eO
rifice
Gas
Tu
rbin
e
Mix
er
Volu
me
1Volu
me
2C
oole
r
NN
P
wc
wc
wc
wr
wr
wout
wout
win
wm
IGV
pd
pd
Td
Td
Ta
ps
pr
pr
Tr
Tr
Tr
Tcd
Tin
Tin
pvpv
pv
pv
pin
pout
∆po,s
AS
V
Fc
Figure 3.2: Block diagram of the plant model
22 3. Modeling of Gas Compression Plant
By further assuming ideal gas and that entropy does not change (isentropic process), the
pressure differential is given by
dpv = a21 dρ1 (3.2)
where a1 is the acoustic velocity (speed of sound) which is the speed at which an infinites-
imally small pressure wave (sound wave) propagates through a fluid. Since the process
experienced by the fluid as a sound wave passes through it is an isentropic process. The
speed of sound in an ideal gas is then given by
a1 =
√
kv Z R Ts
MW(3.3)
where kv is isentropic volume exponent, Z is the compressibility factor, Ts is the temperature
of the flow at compressor suction (see Eq. (3.8)), R is the molar gas constant ≈ 8.134
[J.mol−1.K−1], and MW is the molecular weight. The mass balance of the upstream volume
can now be developed from combining the Eqs. (3.1) and (3.2)
dpvdt
=a21V1
(wm − wc) (3.4)
where the pressure unit of pv is in [MPa], and when we convert it to [bara] the Eq. (3.4)
becomesdpvdt
= 0.01a21V1
(wm − wc) (3.5)
where the initial pressure at the volume (V1) is pv(0) = pv,0. The implementation of the
Eq. (3.5) in Simulink as shown in Figure 3.3.
3.2.2 Downstream Volume
The mass balance of the downstream volume (V2) can be described by
dpddt
= 0.01a22V2
[wc − (wout + wr)] (3.6)
where wc is mass flow through the compressor, wr is the recycle flow, wout is the flow coming
out of the plant, Td is the discharge temperature of the compressor. The initial pressure at
the first volume is pd(0) = pd,0. The acoustic velocity a2 in the second volume is calculated
according to
a2 =
√
kv Z R Td
MW(3.7)
where Td is the discharge temperature of the compressor (see Eq. (3.40)).
3.3. Mixer 23
wm[kg/s]
wc[kg/s]
Ts[K]
pv[bara]
Volume 1 1
kv
kPa�>bara
0.01
1
Saturation1
Saturation
8.314
MW
1
s
xo
3
2
1
wm[kg/s]
wc[kg/s]
Ts[K]
pv[bara]pv0
pv,0[bara]
R
MW
Z
V1
V1
kv
Figure 3.3: Simulink model of the upstream volume (V1).
3.3 Mixer
In general, the mixer is a system for mixing two or more incoming flows with different
temperature; the outcome flow is the sum of all incoming flows, with temperature being the
average of the incoming flows. In the gas compression plant, mixer is a system for mixing
flows in the stage. In Figure 3.2, the mixer is used for mixing the flow entering the stage and
the cooler flow back through the surge vane; then the output flow of the mixer goes to first
volume in the stage. The outcome temperature is the weighted average of the temperature
of mixed flows.
win
wr
Tin
Tr
wm
Ts
Mixer
Mixer
I�e��ei�����
u1
[��i��
[��i��
A �
[��i��
[��i��
[��i��
[��i��
3
4
1
2
[��i��
[��i��
[��i��
[��i��
[��i��
[��i��
I� A��i�S������e�
�i�
�i�����
�i�
�i�A��i�
I� A��i�S������e�
�i�
�i�
����
A��i�
��e�
Me��ry
Me��e 2
1
win[kg/s]
wr[kg/s]
Tin[K]
Tr[K]
wm[kg/s]
Ts[K]
Figure 3.4: Simulink model of the mixer.
In the mixer model depicted in Figure 3.4, win, wr, Tin and Tr are the mass of the inlet
flow, the recycle flow, the temperature of the inlet flow, and the temperature of the recycle
flow, respectively. The outputs wm and Ts are the mass and temperature of the output flow
are given by:
24 3. Modeling of Gas Compression Plant
wm = win + wr
Ts =Tin win + Tr wr
wm
(3.8)
In case of zero input flows:
• output temperature maintains the last value;
• at initialization (last value missing) output temperature is calculated as the mean of
the two input temperatures.
3.4 Orifice
Industrial plants usually employ devices such as orifice, nozzle and Venturi meters to calcu-
late the mass flow, based on the Bernoulli principle. In this study, the flow measurement is
based on the installation of a flow restriction named ’orifice’. In the gas compression model,
the orifice is a device sensor to measure the differential pressure transmitter through it. The
downstream pressure of any discrete flow element is calculated by the following law
wc = A
√
(pv − ps) pv MW
Z R Ts
(3.9)
where A is the cross-sectional area of the pipe.
In the gas compression model the inputs are the flow wc, input pressure pv, and temper-
ature Ts; The outputs are the outlet pressure ps and pressure difference ∆po,s, which are
given by the following equations
ps = pv − sgn(wc)w2
c Ts Z R
pv A2 MW(3.10)
∆po,s = pv − ps (3.11)
The orifice is implemented in Simulink as shown in Figure 3.5.
3.5 Cooler
Gas coolers are placed after the compressors and used for removing the heat of discharged
compression gas. During recycling of the flow using anti-surge valve, without gas cooler, the
discharge flow temperature rises more and more which may cause damages to the machine.
The function of a cooler is to regulate the temperature to a certain value. The model of the
cooler system estimates the output temperature according to the following equation
Tr = min(Td, Tcd) (3.12)
3.6. Valves 25
wc
Ts
pv
ps
∆ po,s
Suction Orifice
Orifice
2
1
8.314
30:Prod
3
2
1
PaPaPaPaPaPaPaPaPaPaPaPaPa
bara
barabarabarabarabarabarabarabarabarabarabarabarabara
k orifice
barakPa0.01
100
k
kkkkkkkkkkkkk
R
Z
Z
MW
MW [mol/s]
wc[kg/s]
Ts[K]
pv[bara] ps[bara]
∆ po,s[bara]
Figure 3.5: Simulink model of the orifice.
where Tcd is the cooler output demand temperature and we see that if the inlet temperature
is lower than the demand one, the outlet temperature is equal to inlet temperature. The
cooling system has a variable orifice plate to estimate the pressure losses in a perfect cooler.
The output pressure of the cooler is calculated as follows
pr = pd − sgn(wout)
[
Td2w2
out Z R
pd A2 MW
]
(3.13)
Td2 =min(Td, Tcd) + Td
2(3.14)
The cooler is implemented in Simulink as shown in Figure 3.6.
wout
Td
pd
pr
Tr
Cooler
Cooler
2
1
3
2
1
Orifice
Losses
0.5
min
Tm[K] Td2
wout[kg/s]wout pr
pd
Tr [K]
pr [bara]
pd [bara]
Tcd [K]
T2d
Figure 3.6: Simulink model of the cooler.
3.6 Valves
A valve is a device that regulates, directs or controls the flow of a fluid by opening, closing,
or partially obstructing various passageways. Valves are used for critical applications in
oil and gas, for instance for compressor protection (anti-surge control valves, hot by-pass
valves, large size control valves, etc), for oil pipeline (discharge pressure control valves),
etc. Figure 3.7 shows an antisurge valve.
26 3. Modeling of Gas Compression Plant
Figure 3.7: Anti-surge valve [1]
3.6.1 Anti-Surge Valve
In this study, we use the anti-surge valve to prevent surge and to improve the performance of
the control system. Moreover, we employ two large size valves to model disturbances at the
upstream and downstream of the plant (see Figure 3.1). Each valve is modeled to calculate
the flow rate which passes through it. For instance, the recycle flow (wr) in the anti-surge
valve is calculated as a function of the input pressure (pr), the output pressure (pv), the
input temperature (Tr), and the valve stroke or travel (utr). The valve is implemented in
Simulink as shown in Figure 3.10.
[a63gd]
[a26gd]
[Pin]
[Was]
1
1
Valve
AS Actuator
Actuator
Valve
Masoneilan
pr
pv
Tr
travel travelChoked
ASVcmd [%]
emerg [bool]
wr
wr
Figure 3.8: Simulink model of the valve with actuator.
Figure 3.9 shows the valve actuator block which is represents the model of the valve
actuator. The transfer function of the valve actuator is
G(s) =1
Ts+ 1(3.15)
3.6. Valves 27
3
2
1
0
0
2
1
replacements
y0=01
s+1
travel [%open]emerg [bool]
cmd [%]cmd [%]
100
100
100
0.5
0.5133 o
133 c>
>
security
security
security
Figure 3.9: Simulink model of the valve actuator.
where T = 1s is the time constant.
2
1
u(1)�0.148*u(1)^3
1
1.5
1.25998��4
2.8
14.50377
R�lationalOp�rator
max
120�Prod
4
3
2
1
0���
1���
replacements
wr [kg/s]
pr
pv
Tr
MW
MW
Z
Z sqrt
sqrt
sqrt
[inv]
[inv]
inv
inv
sqrt
sqrt
sqrt
min
lbs/hr - kg/s
max y
y
FL
288
29
MWair
travel [%]
|u|
Choked
<
>0
bar -> psia
Cv,max
Gf
G
CV max
Cv
Fw
Hw
S
Figure 3.10: Simulink model of the valves.
The operation of the valve is described by distinguishing two cases:
• case 1: if pv = pr, then the mass flow is equal to zero.
• case 2: pv 6= pr
The mass flow wr is given by the following equation
wr = +2.8 (1.25998 10−4) Cv,max FL Cv Fw Hw
√
Gf Z (3.16)
28 3. Modeling of Gas Compression Plant
where FL is the liquid critical pressure ratio factor, or critical flow factor, which is considered
as a constant value equal to 0.95 (more detailed see [68]), Z is compressibility factor, Gf
is the specific gravity at flowing temperature, Cv is valve flow coefficient at the given valve
position, or percentage plug rotation, or valve travel (utr). The valve travel changes between
0 to 100%, and the control valve is limited between 0 to 1. Figure 3.11 shows the control
valve Cv as a function of valve travel utr. Fw and Hw are functions of pr and pv (see right
Figure 3.10). The function Fw is given by
0 20 40 60 80 1000
10
20
30
40
50
60
70
80
90
100
% Stroke (utr)
%C
on
trol
valv
eC
v
Figure 3.11: Valve control Cv versus travel
Fw = S − 0.148 S3 (3.17)
where
S = min
(
1.63
FL
√
|pr − pv|
max (pv, 1), 1.5
)
(3.18)
and if
(
1.63
FL
√
|pr − pv|
max (pv, 1)
)
> 1.5 then the valve is in choke.
The function Hw is given by
Hw = 14.50377 max (pv, 1) (3.19)
3.6. Valves 29
where the value 14.50377 is to convert the pressure from [bara] to [psia].
The specific gravity at flowing temperature is given by the following equation
Gf = G288
Tr
(3.20)
where G is the specific gas gravity and is given by the following equation
G =MWair
MW(3.21)
where the molecular weight of the air MWair = 29 [g/mol].
0 2 4 6 8 10 12
0
10
20
30
40
50
60
70
80
90
100
time [s]
ASV,utr
an
dw
r[%
]
ASVutrwr
Figure 3.12: Flow variation during opening of the anti-surge valve
• case 3: if pv > pr then
Herein, the mass flow wr is equal to
wr = −2.8 (1.25998 10−4) Cv,max FL Cv Fw Hw
√
Gf Z (3.22)
where Cv and Hw have the same formulas as in the previous case, the function Fw is given
by the following equation
Fw = S − 0.148 S3 (3.23)
30 3. Modeling of Gas Compression Plant
where
S = min
(
1.63
FL
√
|pr − pv|
max (pr, 1), 1.5
)
(3.24)
and if
(
1.63
FL
√
|pr − pv|
max (pr, 1)
)
> 1.5 then the valve in choke.
The antisurge valve considered in this work has a time delay at the input utr of 0.33s,
and a time delay at the output wr of 0.67s. The flow characteristics is shown in Figure 3.12.
The flow is plotted as a function of time, when the ASV opens quickly from 0 to 100%
in 2s. The valve is tested for a constant value of pd =121.14 [bara], pv =72.055 [bara],
Tr =315.575 [K] and 100% wr = 205.2 [kg/s].
3.6.2 Upstream and Downstream Valves
The model of the upstream/downstream valve is the same as antisurge valve model, the
only difference is in the size and control valve Cv = 100 utr (see Figure 3.13). The valve is
tested for a constant value of pd =121.14 [bara], pv =72.055 [bara], Tr =315.575 [K] and
100% win,out = 1250.3 [kg/s].
0 2 4 6 8 10 12
0
10
20
30
40
50
60
70
80
90
100
time [s]
UV
/DV,utr
an
dw
r[%
]
UV/DVutrwr
Figure 3.13: Flow characteristics in upstream/downstream valve
3.7. Centrifugal Gas Compressor 31
3.7 Centrifugal Gas Compressor
In this section, the gas compressor model is introduced.
3.7.1 Compressor Modeling
The model of the centrifugal compressor is developed to calculate the discharge tempera-
ture, flow rate, volumetric flow, compressor power, and some other variable states which
are discussed later. Figure 3.14 represents the Simulink model of the dimensional centrifu-
gal compressor block. The inputs of the compressor block are IGV , Ts, ps, pd, N and gas
composition. The main outputs considered in the model are discharge temperature Td, mass
flow wc , power P , and polytropic head Hp, which is used to define the characteristic curve
of the compressor.
4
3
2
1 IGV [deg.]
IGV [deg.]
ps [bara]
ps [bara]
pd [bara]
pd [bara]
Ts [K]
Ts [K]
N [RPM]
N [RPM]
C NumC Num
C CodesC Codes
C QuantsC Quants
Centrifugal Compressor
with IGV
rev. 2.0
CC Adimensional IGV rev. 2.0
Td [K] Td [K]
Qv [m3/s] Qv [m3/s]
wc [kW] wc [kW]
P [kJ/kg] P [kJ/kg]
Hp [kJ/kg] Hp [kJ/kg]
kv
nm1 n1nm1 n1
surge surge
choke choke
m [.]
ηp [.]
Figure 3.14: Simulink model of the variable speed centrifugal compressor.
These variable states are computed under different operating conditions, according to
the flow chart given in Figure 3.15. The compressibility factor Z, the molecular weight MW ,
the volumetric exponent of real isentropic kv, the temperature exponent of real isentropic
kt, the speed of sound as, the partial derivative of Z with respect to (w.r.t.) temperature, at
pressure constant at suction Xs, the partial derivative of Z w.r.t. pressure, at temperature
constant at suction Ys, the partial derivative of Z w.r.t. temperature, at pressure constant at
discharge Xd, the partial derivative of Z w.r.t. temperature, at pressure constant at discharge
Yd, variable states are calculated by using Clac80 Simulink block. The inputs are the vector
with the properties of the mixture, pressure and temperature variable states. Clac80 block
is a Simulink implementation of the Nuovo Pignone, programmed by PASCAL language. The
state equation used in this block is Benedict-Webb-Rubin (B.W.R.) Starling equation, and the
calculation based on polytropic compression (more detailed see [69]). The Benedict-Webb-
Rubin Starling equation of state is semiempirical relationship.
32 3. Modeling of Gas Compression Plant
Yes No
Adjust ηp
Adjust wc
Stop !
Set: ps, pd, Ts, N ,
composition
Guess: wc
Calculate: MW , Z, as,
Xs, Ys, Xd, Yd,
kv,s, kt,s, kv,d, kt,d
Calculate: m,n− 1
n, ρ
Calculate: Qv , U2
Calculate: M , φ
Calculate: ηp, τ
Calculate: Hp
Calculate: p
Calculate: wc, Qv
Calculate: P , Td
Is p = pd?
Figure 3.15: Flow chart of the centrifugal compressor.
Volumetric mass flow: Volumetric mass flow Qv is the flow of volume of fluid through a
surface per unit time. It is can be defined as
Qv = wc/ρ (3.25)
where wc is mass flow and ρ mass density, the later being given by the gas state equation:
ρ =100 ps MW
Ts Z R(3.26)
3.7. Centrifugal Gas Compressor 33
Impeller Tip Speed: Impeller speed U2 is one of the most crucial process variables that
affect the properties of the centrifugal compressor because there is a direct relationship
between impeller tip speed, velocity, pressure, and flow. As the impeller tip speed increases,
velocity increases. As velocity increases, pressure increases. As pressure increases, flow
increases. The equation used for calculating impeller tip speed is
U2 = N D π/60 (3.27)
where N is rotation speed and D is impeller diameter.
Flow Coefficient: Flow coefficient φ is the dimensionless flow, which is converted from
the dimensional flow. It can be calculated by using the following equation
φ =4 Qv
π U2 D2(3.28)
Machine Mach Number: Mach number M is a dimensionless quantity representing the
ratio between the machine tip speed and the velocity of sound in the reference conditions:
M = U2/as (3.29)
Pressure Coefficient: The pressure coefficient or head coefficient τ is a dimensionless
number, is equal to polytropic head divided by Euler’s head, can be expressed as
τ =Hp
(1gU22
) (3.30)
The pressure coefficient is important in most fluid flow applications, and is calculated as a
function of the three variables φ, IGV , and M as follows
τ = τ(φ, IGV, M) (3.31)
Linear interpolation has been used to obtain the pressure coefficient τ . Figure 3.16 shows
the experimental data taken from a real centrifugal compressor. In the figure, it’s clear that
the inlet guide vane and the mach numbers effect on the pressure coefficient for different
flow coefficient value. The red, green and black curves are corresponding the values of mach
numbers M equal to 0.707, 0.643 and 0.450, respectively. These curves are measured for
eight angular positions of IGV from -70 to 10 degree.
Polytropic efficiency: The polytropic efficiency ηp is the ratio of the polytropic head Hp
to the gas work input. The polytropic efficiency is a function of φ, IGV and M . In this
work, we estimated this function from real data provided by GE Gas & Oil Nuovo Pignone
manufacturing company of the centrifugal compressor. The real data of ηp are not available
for all values of φ for different values of IGV and M (see Figure 3.17). In the Figure 3.18,
the magenta straight lines represent the real data. The missing data ηp(φ) for fixed values
34 3. Modeling of Gas Compression Plant
Pre
ssu
reco
effi
cien
t(τ
)
Flow coefficient (φ)
IGV ↓
M ↓ {0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
0.026 0.035 0.044 0.053 0.062 0.071 0.08
Figure 3.16: Real data of the head coefficient τ
of IGV and M are estimated using linear interpolation with linear extrapolation extend
mode. The extrapolation is the extension of ηp data beyond the range of the measurements.
We compared the linear extrapolation approach with other methods, such as cubic spline
and polynomial interpolations. To estimate the intermediate values of ηp, we apply a linear
interpolation (more detailed see Section 3.7.2) with respect to IGV , M and φ where
ηp = ηp(IGV, M, φ) (3.32)
Polytropic Head: Polytropic head Hp is the reversible work required to compress a unit
mass of gas by a polytropic process from the inlet total pressure and temperature to the
discharge total pressure and temperature.
Hp =τ ηp U2
2
1000(3.33)
Discharge Pressure: Discharge pressure is the total gas pressure (Static Pressure plus
Velocity Pressure) at the discharge flange of the compressor. The discharge pressure p in the
3.7. Centrifugal Gas Compressor 35
Poly
tropic
head
(ηp)
Flow coefficient (φ)
0.9
0.85
0.8
0.75
0.7
0.65
0.026 0.035 0.044 0.053 0.062 0.071 0.08
Figure 3.17: Real data of the polytropic efficiency ηp
centrifugal compressor can be calculated as follows:
p =
ps
(
1 +n− 1
n.Hp MW
Z R Ts
)( nn−1
)
ifn− 1
n≥ 10−3
ps e
(
1+Hp MW
Z R Ts
)
ifn− 1
n< 10−3
(3.34)
wheren− 1
n=
n−1n
∣∣s+ n−1
n
∣∣d
2(3.35)
where
n− 1
n
∣∣∣∣s
=
[
kv,s − 1
kv,s.
1ηp
+Xs
1 +Xs
]
+
[1− Ys
1 +Xs
.
(
1−1
ηp
)]
(3.36)
and
n− 1
n
∣∣∣∣d
=
[
kv,d − 1
kv,d.
1ηp
+Xd
1 +Xd
]
+
[1− Yd
1 +Xd
.
(
1−1
ηp
)]
(3.37)
Mass Flow: The pressures and the temperatures at the suction and discharge of the com-
pressor are different. However, the exit mass flow is equal to the inlet mass flow. The mass
flow wc at suction/discharge of the compressor is modeled by the dynamic system
d
dtwc =
wc
Ts
[(pdpd
)conv
− 1
]
. (3.38)
36 3. Modeling of Gas Compression Plant
−60−40
−200
0.02
0.04
0.06
0.080.45
0.5
0.55
0.6
0.65
0.7
0.2 0.3 0.4 0.5 0.6 0.7 0.8
IGVM
φ
Figure 3.18: Polytropic efficiency ηp estimation (see color) and straight lines of the real data
where Ts is sample time, conv is the convergence factor, pd the desired compressor discharge
pressure, pd the actual compressor discharge pressure, and the initial flow is wc,0.
Volumetric flow:
Qv =wc
ρ(3.39)
Discharge Temperature: The discharge temperature is defined by
Td = Ts ·
(pdps
)m
(3.40)
The exponent m is given by
m =ms +md
2(3.41)
3.7. Centrifugal Gas Compressor 37
where
ms =kt,s − 1
kt,s.
Xs
ηp
1 +Xs
(3.42)
and
md =kt,d − 1
kt,d.
Xd
ηp
1 +Xd
(3.43)
Compressor Power: The total shaft power P of the compressor is measured as
P =wc Hp
ηp+ Pmech (3.44)
The losses in the centrifugal gas compressor are almost of the same types as those in a
centrifugal pump. The losses can be classified in different types, which are influenced by
each other:
1. Mechanical losses;
2. Aerodynamic losses;
3. Losses caused by leakages.
5500 5600 5700 5800 5900 6000 6100 6200 6300 640040
42
44
46
48
50
52
54
Gas Turbine Speed N , (RPM)
Mech
an
ical
Loss
esPm
ech,(k
W)
Figure 3.19: Mechanical losses versus rotating speed
We consider only the mechanical losses Pmech in our simulation. Mechanical losses are
due to the friction between mechanical parts of the machine. The losses of mechanical
38 3. Modeling of Gas Compression Plant
energy due to friction has been determined experimentally for some operating points. The
intermediate points are calculated using online linear interpolation. For most centrifugal
compressors, mechanical losses are relatively small. Figure 3.19 shows the result of linear
interpolation of the mechanical losses dependent on rotating speed. Changing the inlet
guide vane angle can modify the compressor power. Figure 3.20 shows the influence of IGV
angle variation on compressor power consumption in certain conditions. This will be one of
!"0 !60 !#0 !40 !30 !20 !10 0 101.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4x 10
4
Inlet Guide Vane (IGV), [deg.]
Com
pre
ssor
Pow
er
(P),
[kW
]
Figure 3.20: The influence of IGV control on required compressor power.
the key features that will be exploited in the design of the compressor control systems, in
next chapters.
Molecular Weight: Natural gas is a naturally occurring gas mixture, consisting of gas
composition like methane (C1H4), ethane (C2H6), propane (C3H8), butane (C4H10), and
so on. The molecular weight of the natural gas mixture that consists of components such as
C1, C2, C3 and C4 is defined by the equation
MW =∑
yiMi (3.45)
where
• MW apparent molecular weight of gas mixture;
• yi mole fraction of gas component i;
• Mi molecular weight of the gas component i.
3.7. Centrifugal Gas Compressor 39
We can calculate the apparent molecular weight of a natural gas mixture that has 85%
methane, 9% ethane, 4% propane, and 2% normal butane as shown below:
Component Mole Fraction Molecular Weight
C1 85 16.01
C2 9 30.10
C3 4 44.10
C4 2 58.10
Total 100
MW = (0.85 ∗ 16.01) + (0.09 ∗ 30.1) + (0.04 ∗ 44.1) + (0.02 ∗ 58.1) = 19.24
Therefore, the apparent molecular weight of the gas mixture is 19.24. Table 3.1 shows types
of natural gas used to test and validate the developed compressor model.
Table 3.1: Gas Mixture Composition
ITEM COMPOSITION SYMBEL GAS (1) GAS (2) GAS (3)
1 METHANE CH4 87.665 79.487 89.870
2 ETHANE C2H6 6.984 15.550 6.500
3 PROPANE C3H8 2.754 3.230 1.900
5 N-BUTANE C4H10 0.655 0.470 0.350
4 I-BUTANE C3H7.CH3 0.327 0.270 0.250
7 N-PENTANE C5H12 0.055 0.030 0.130
6 I-PENTANE C4H9.CH3 0.062 0.030 0.000
8 N-HEXANE C6H14 0.029 0.053 0.000
9 N-HEPTANE C7H16 0.014 0.000 0.000
10 N-OCTANE C8H18 0.004 0.000 0.000
11 HELIUM HE 0.000 0.000 0.000
12 HYDROGEN H2 0.000 0.000 0.000
13 NITROGEN N2 0.775 0.440 0.500
14 HYDROGEN SULFIDE H2S 0.000 0.000 0.000
15 CARBON DIOXIDE CO2 0.674 0.440 0.500
16 WATER H2O 0.001 0.000 0.000
17 TOTAL 100 100 100
40 3. Modeling of Gas Compression Plant
3.7.2 Difference between the accuracy of the interpolation methods
To compare between the interpolation methods used to estimate the head coefficient τ and
the polytropic efficiency ηp, we use the percentage error between validation data and the
simulation results. Data used for comparison represent 16 different operating points of
the real compressor for different constant speeds, IGV opening, and gas proprieties (see
Table 3.2). In the comparison, we will use only two state variables: mass flow and required
compressor power. The percent error of the mass flow is
δ = |wc − wc
wc
| × 100 (3.46)
where the real value wc is taken from real data, while its approximation wc is obtained using
one of the methods above. A same formula used for power comparison by using P instead
wc. Figure 3.21, shows the percent error of the mass flow of each method. The green dark
bars in the figure represent the linear interpolation, the light green bars represent the spline
interpolation only with respect to φ, while the yellow bars represent the spline interpolation
with respect to φ, IGV and M . In general, the last approximation method is more accurate
than the other methods. Figure 3.22 shows the percent error of the compressor power P .
Also in this case, the third method of interpolation gives usually better results, especially
when the error is larger than 0.5%.
Table 3.2: Validation data of the centrifugal compressor in different operating points
Operating point IGV N Qv Td P
SH. 7 0 6176.8 463.614 315.65 29656
SH. 17 -62 6109.2 258.367 317.65 18126
SH. 27 -40 6012.7 310.041 316.95 21125
SH. 37 0 5867.9 361.714 315.55 23080
SH. 47 0 6012.7 413.388 315.55 26331
SH. 57 0 5840.4 346 315.75 22217
SH. 62 -62 5757.9 273.928 315.65 17076
SH. 72 -40 5666.2 328.714 315.05 19894
SH. 82 0 5531 383.5 313.85 21746
SH. 92 0 5666.2 438.285 313.85 24797
SH. 102 15 6321.5 417.305 318.75 29577
SH. 107 -62 6331.2 246.923 319.05 18654
SH. 117 -44 6263.6 296.308 318.45 21868
SH. 127 0 6089.9 345.692 316.95 23842
SH. 137 0 6225 395.077 316.75 27056
SH. 147 15 5964.5 413.388 315.65 26424
3.7. Centrifugal Gas Compressor 41
0 2 4 6 8 10 12 14 160
0.5
1
1.5
2
2.5
LinearSpline3D Spline
Operating points
Perc
en
terr
or
of
the
mass
flow
Figure 3.21: Percent error of the polytropic efficiency for different operating points
0 2 4 6 8 10 12 14 160
0.5
1
1.5
2
2.5
LinearSpline3D Spline
Operating points
Perc
en
terr
or
of
the
com
pre
ssor
pow
er
Figure 3.22: Percent error of the compressor power for different operating points
3.7.3 Performance evaluation of the centrifugal compressor
Usually the performance curves of the compressor provided by a vendor are given as dif-
ferential pressure versus inlet mass flow at certain fixed conditions. Since the compressor
42 3. Modeling of Gas Compression Plant
will be connected to other devices, like volumes at pipelines, this curves are not sufficient
to evaluate the compressor performance. The overall plant system requires the compressor
to work in many different operating conditions. Nevertheless, the polytropic head gives a
sufficiently accurate evaluation of the compressor performance.
The flow limits [wc,min, wc,max] are calculated using dynamic look-up table, and are
functions of the geometry of the compressor, density of the gas, and shaft speed. The func-
tions are denoted by
wc,min = wc,min(M, ρ, U2, D, IGV ) (3.47)
wc,max = wc,max(M, ρ, U2, D, IGV ) (3.48)
Thus the dynamic instabilities, surge and choke, are defined as following:
{surge if wc < wc,min
choke if wc > wc,max
(3.49)
where the surge zone presents unstable operations and the choke zone, also known as
stonewall, is characterized by low efficiency.
4238
6054
665966596659
Hp
(kJ/
kg)
Qv (m3/s)
01 2 3 4 5 6 7 8 9
10
10 11 12
20
30
40
50
60
70
80
90
100
Figure 3.23: Performance map of centrifugal compressor
Figure 3.23 shows the compressor map which plots polytropic head (Hp) as a function of
volumetric flow (Qv) and rotational speed. The limits of the volumetric flow are calculated
by using Eq. (3.39). The map depends on mechanical proprieties and is independent on gas
proprieties. Figure 3.24 shows the performance map of a multispeed compressor system at
different inlet guide vane (IGV) angles opening.
3.8. Gas Turbine Models 43
↓ IGVH
p(k
J/kg)
Qv (m3/s)
01 2 3 4 5 6 7 8 9
10
10 11 12
20
30
40
50
60
70
80
90
100
Figure 3.24: Performance maps of a multispeed gas centrifugal compressor system at different inlet
guide vane (IGV) angles.
3.8 Gas Turbine Models
Gas turbine modeling has been addressed in many studies [70–74]. In [75], a detailed
review of the typical gas turbine performance maps is provided. In this study, the gas turbine
steady-state operating maps are used to determine the engine heat rate at a given ambient
temperature, as a function of compressor power and turbine speed. The rotational speed
can be varied from 70% to 105% of the nominal turbine speed. The ambient temperature
is assumed to be 15◦C. We used the experimental data from the following four power gas
turbines:
• PGT25 SAC
• PGT25 DLE
• PGT25+ SAC
• PGT25+ DLE
where SAC means Standard Annular Combustor, and DLE means Dry Low Emissions. These
types of gas turbines are used for mechanical drive applications. Table 3.3 shows the specifi-
cations of each gas turbine at ambient temperature 15oC. For instance in gas turbine PGT25
SAC, the 6500 rpm design permits direct coupling with the driven equipment.
The power gas turbine (PGT25) is a highly efficient machine for mechanical and genera-
tor drive applications. PGT25 gas turbine consists of General Electric LM2500 aeroderivative
gas generator coupled with a power turbine designed by GE’s Oil & Gas [1]. Figure 3.25
shows the effect of compressor speed and power on heat rate base load. The heat rate is
44 3. Modeling of Gas Compression Plant
Table 3.3: Gas turbines
Model Continuous Duty Heat Rate RPM on the
Power kW kJ/kWh load side
PGT25 SAC 23269 9905 6500
PGT25+ SAC 31373 8978 6100
PGT25 DLE 23262 9958 6500
PGT25+ DLE 31077 9007 6100
40
60
80
100 5060
7080
90100
1
1.1
1.2
1.3
1.4
1.5
1.6
x 104
Output Shaft Speed [%]
Heat rate maps for PGT25 SAC gas turbine at 15°C
baseload shaft power [%]
Hea
t rat
e [k
J/kW
h]
4050
6070
8090
10050
60
70
80
90
100
1
1.2
1.4
1.6
1.8
x 104
Output Shaft Speed [\%
]
Heat rate maps for PGT25 DLE gas turbine at 15°C
baseload shaft power [%]
Hea
t rat
e [k
J/kW
h]
(a) (b)
Figure 3.25: Steady-state heat rate: (a) PGT25 SAC gas turbine; (b) PGT25 DLE gas turbine
defined as ”the energy input to a system divided by the electricity generated. The PGT25+
SAC gas turbine consists of a LM2500+ GE aeroderivative gas generator (updated version of
the LM2500 gas generator with the addition of a zero stage to the axial compressor) coupled
with a 6100 rpm power turbine [1]. Figure 3.26 shows the heat rate data of the PGT25+
gas turbine.
The total fuel consumption of a gas turbine can be generated by interpolation of heat
rate (HR) maps. Turbine maps depend on the ambient temperature and need to be known
for the off-design calculations. The fuel consumption Fc of the whole system is given by the
relationship:
Fc = P ϕhr (3.50)
where P is the power required for the compressor, and ϕhr is the heat rate.
3.9. Linearization of the Gas Compression System 45
40
60
80
10050
6070
8090
100
0.9
1
1.1
1.2
1.3
x 104
Output Shaft Speed [%
]
Heat rate maps for PGT25+ SAC gas turbine at 15°C
baseload shaft power [%]
Hea
t rat
e [k
J/kW
h]
4050
6070
8090
100
50
60
70
80
90
100
1
1.2
1.4
x 104
Out
put S
haft
Spee
d [%
]
Heat rate maps for PGT25+ DLE gas turbine at 15°C
baseload shaft power [%]
Hea
t rat
e [k
J/kW
h]
(a) (b)
Figure 3.26: Steady-state heat rate: (a) PGT25+ SAC gas turbine; (b) PGT25+ DLE gas turbine
3.9 Linearization of the Gas Compression System
The overall dynamic model of the plant in Figure 3.1 includes the mass flow dynamics,
the models of speed and IGV actuators, the dynamics of the suction and discharge volumes
and dynamic of the antisurge valve. Model-based control techniques usually require an
analytical model of the plant to be controlled: this is not the case for the considered plant,
which contains lookup tables for the turbine and compressor maps. The standard approach
used in these cases is to generate an approximate model which is used only for the purpose
of designing the controller. The model does not have to be accurate enough to reproduce the
behaviour of the system: it just has to capture the dominant dynamics which are relevant to
control design. Most of the times, linearized models around the considered operating point
are employed. This greatly simplifies the control design procedure. In order to get a linear
approximation of the plant nonlinear model, the function linearize of the Matlab/Simulink
Control Toolbox has been applied to the Simulink model of the plant. The linearization
routine uses a Jacobian scheme to generate LTI state space models. The linearized dynamic
model for the discharge pressure, distance to surge, and compressor power as a function of
inlet guide vane, speed, and antisurge valve is
46 3. Modeling of Gas Compression Plant
x =
−9.1191 2.6402 16.1229 250.5368 −97.0842 −1.9394
0 −1.0000 0 0 0 0
0 0 −1.0000 0 0 0
−0.0261 0 0 −0.1355 0 0.0077
0.0235 0.0003 0.0017 0.0102 −0.1294 −0.0069
0 0 0 0 0 −1.0000
x+
0 0 0
0 1 0
1 0 0
0 0 0
0 0 0
0 0 1
u
y =
0 0 0 0 1.0000 0
0.9695 0.0016 −1.4942 0.6626 −4.1893 0.0218
0.0749 0.0435 0.1834 1.0532 0 −0.0128
x
where u contains the variance of inlet guide vane angle, speed rotational, and antisurge
valve angle, y the discharge pressure, percentage of distance to surge, and compressor
power. The eigenvalues λ of system matrix A are found to be
λ = {−7.9949, − 1.2610, − 0.1281, − 1, − 1, − 1}.
The matrix A has full rank, which indicates the system is observable and controllable.
Meanwhile, the eigenvalues of the linearized dynamic model are shown in Eq. (3.9) for the
operating point. All negative real parts are found for the values, which means the system is
asymptotically stable.
Figure 3.27 shows the bode diagram of the frequency response of a dynamic system
model. The plot displays the magnitude (in dB) and phase (in degrees) of the system re-
sponse as a function of frequency. Figure 3.28 compares the behaviour of the nonlinear
plant and of the corresponding linearized model, when they are inserted in a feedback loop
with an MPC control scheme (see Section 2.3). To this purpose, the linearised model is dis-
cretized with sampling time Ts= 0.04 sec. In the considered test, the control has to regulate
the discharge pressure to the reference value yref1 = 117 bara, starting from an initial con-
dition which is equal to the steady state one. It can be observed that the output behaviours
are very similar, although the resulting control inputs u1 and u2 are different at steady state.
This suggests that the linearised model capture the essential dynamics of the plant, at least
for control design purposes.
3.9. Linearization of the Gas Compression System 47
$400
$200
0
$3%0
0
3%0
$200
$100
0
100
$&20
0
&20
1440
$200
$100
0
100
10'
$3%0
0
3%0
&20
100
10'
100
10'
IGV N AS
pd
pd
δs
δs
PP
Frequency (rad/s)
Bode Diagram
Magn
itu
de
(dB
);
Ph
ase
(deg)
Figure 3.27: bode diagram
48 3. Modeling of Gas Compression Plant
117
118
119
120
121
122
NonlinearLinearizedReference
−60
−40
−20
0
NonlinearLinearized
120
140
160
180
200
5900
5950
6000
6050
NonlinearLinearized
0 10 20 3080
85
90
95
NonlinearLinearized
0 10 20 300
10
20
30
40
NonlinearLinearized
NonlinearLinearized
IG
VN
AS
pd
δs
P
Outputs Inputs
timetime
Figure 3.28: Nonlinear vs Linear model.
Chapter 4
Control Techniques for Pressure Regulation
Abstract
This chapter considers the development of a multi-variable control system for a class of cen-
trifugal compressors, which exploit as control signals both the rotational speed and the Inlet
Guide Vane (IGV). Linear Quadratic Gaussian control with Integral action (LQGI) and Model
Predictive Control (MPC) are investigated. The LQGI and MPC controllers are compared to a
standard proportional integral (PI) controller, to regulate the discharge pressure of the com-
pressor. The control algorithms are simulated and compared in different operating scenarios.
Results demonstrate that the proposed multivariabe control schemes provide better perfor-
mance than the single-loop PI controller, thus motivating the use of IGV for control purposes.
Part of the material of this chapter is based on [76].
4.1 Introduction
High-pressure multistage centrifugal compressors are an essential part of the process ma-
chinery in the oil and gas industry across a wide variety of applications. Centrifugal com-
pressors in a connected process system are very sensitive to changes in the inlet conditions
[77], such as the suction pressure, the temperature, as well as the inlet gas density. This type
of gas compressors requires quick response and reliable control systems to increase their ca-
pacity. Process regulation (for example discharge pressure control) is usually performed by
acting on the shaft speed. More recently, a variable Inlet Guide Vane (IGV) has been used
for control purposes. The IGV system allows wide capacity control of the centrifugal com-
pressor with reduced energy losses. In centrifugal compressors, IGV is used to control the
mass flow rate with negligible change in compressor ratio and shaft speed [60]. When the
turbine speed, antisurge valve and the IGV are used altogether to regulate the process, a
multivariable control system is necessary. This is the subject of the present chapter.
This chapter is structured as follows: in Section 4.2 the PI, LQGI, and MPC control
schemes, and the details about their implementation are presented. Section 4.3 compares
the performances of the above control schemes in several different case studies.
4.2 The Discharge Pressure Regulation Problem
Hereafter, two types of multivariable control, namely a linear quadratic regulator (LQR)
and a model predictive controller (MPC), are presented, and compared to a standard PI
50 4. Control Techniques for Pressure Regulation
controller based on pressure regulation. The LQR is widely and effectively used in many
industrial applications [78, 79], including control of the compressor stations for natural gas
pipelines [80, 81]. Since complete state information is not available, it is necessary to use a
LQG control scheme, which includes Kalman filter estimation of the state vector. The LQG
scheme includes an integral action to compensate the reference tracking error. In recent
years, MPC has confirmed itself as a successful approach to multivariable control due to
its advantages over traditional controllers [66]. In particular, it is widely employed in the
oil and gas industry to deal with power plant control [55, 59, 82]. Although there is an
increasing research interest in nonlinear MPC [63–65] most of this literature is dedicated
to systems described by analytical models, while the model of the reference application
considered in this thesis contains non analytical parts (e.g. look-up tables). Hence, in this
study linear MPC is employed, based on a linearized model of the plant.
4.2.1 Simulation Setup
The plant model is similar to that depicted in Figure 3.1 without considering the antisurge
valve which is always close and only one output (y1 = p2). Model parameters are shown in
Table 4.1 for a specific operating point considered in the simulations.
Table 4.1: Plant parameters at design condition
Parameter Value
Upstream valve UV (%) 47.926
Downstream valve DV (%) 90.441
Inlet pressure of the plant pin (bar-a) 103.1
Outlet pressure of the plant pout (bar-a) 70
Compressor suction pressure ps (bar-a) 71.578
Compressor discharge pressure pd (bar-a) 121.177
Rotational speed N (rpm) 6053.264
Inlet guide vane IGV (deg) 0.00
Flow rate wc (kg/s) 460.386
Compressor inlet temp. Ts (K) 275.65
Molecular weight MW (g/mol) 18.591
Power P (kW) 18126
Ambient temperature Ta (◦C) 15
Speed of sound as (m/s) 340
Volume V1 (m3) 80
Volume V2 (m3) 80
The input values at the operating point considered in Table 4.1 are uss1 = 0 [deg] and
uss2 = 6053.264 [rpm]. The system discharge pressure is yss = 121.27 [bara] and the corre-
sponding state vector is equal to xss = [454.8877 6053.264 0 74.1093 121.27]T . The linearized
4.2. The Discharge Pressure Regulation Problem 51
dynamic model for the discharge pressure as a function of inlet guide vane and speed is
x =
−15.7 2.6327 17.028 270.9 −96.8
0 −1.00 0 0 0
0 0 −1.00 0 0
−0.026 0 0 −0.18 0
0.023 0.0003 0.0017 0.01 −0.13
x+
0 0
0 1
1 0
0 0
0 0
u
y =[0 0 0 0 1
]x
where u contains the variance of inlet guide vane angle, speed rotational, and antisurge
valve angle, y the discharge pressure, percentage of distance to surge, and compressor
power. The eigenvalues λ of system matrix A are found to be
λ = {−15.0729, − 0.7986, − 0.1379, − 1, − 1}.
4.2.2 Implementation of Anti-Windup Proportional-Integral (PI) Con-
trol
From a preliminary open loop analysis, it has been observed that the plant achieves the min-
imum fuel consumption at the desired reference output pressure, for values of IGV always
between 0 and 10 [deg] (see section 6.3). Therefore, we decided to set u1(t) = 5 [deg], ∀t,
and use a SISO PI controller on the rotational speed u2(t), to regulate the discharge pressure
y(t) to its reference value. Results with this approach will be used as a baseline solution,
for comparison with MIMO control techniques. The simulations are performed with control
actuator saturation, anti-windup logic, and rate limiter at the output of the controller. The
PI control scheme is shown in Figure 4.1.
The PI controller parameters are presented in the Table 4.2. The proportional and inte-
gral gains and the anti-windup back calculation coefficient have been obtained via trial-and-
error techniques.
52 4. Control Techniques for Pressure Regulation
Figure 4.1: Simulink model of feedback PI controller model and above its PI controller model
Table 4.2: Parameters for simulation
Parameter Value
Proportional gain 220
Integral gain 20
Initial conditions: Integrator at t=0 6025.28
Upper saturation limit 6405
Lower saturation limit 4270
Anti-windup method: back calculation 1
with back-calculation coefficient (kb)
4.2.3 Implementation of LQI
The tuning parameters of the LQ controller are the matrices Q and R in the cost function
(2.11). In the simulations, following values have been used
Q = diag([0 0 0 0 30 15])
R = diag([10 0.1]),
where diag(v) denotes a diagonal matrix, with diagonal v. Notice that by acting on the
matrix R one can tune the relative influence on the control action of compressor speed and
IGV opening.
By solving the LQR problem, one obtains the gain matrix
K =
[0.0010 0.0043 0.0270 1.6542 3.1866 −0.6575
0.0166 0.0692 0.4322 26.5168 51.0039 −10.4958
]
(4.1)
The full LQI control scheme is depicted in Figure 4.2.
4.2. The Discharge Pressure Regulation Problem 53
Figure 4.2: The Plant with LQI Controller
4.2.4 Implementation of MPC
For the MPC controller described in Section 2.3, the values of the tuning parameters adopted
in the simulations are reported in Table 4.3. The cost function is minimized by using the
quadratic programming solver provided by the Model Predictive Control Toolbox for Matlab
[83]. The weights have been obtained via trial-and-error, the horizon lengths have been
Table 4.3: Parameters for MPC controller
Parameter Value
[p;m] [2; 5]
[w∆u1 ;w∆u2 ;wy] [5; .1; 30]
[∆umin1 ; ∆umax
1 ; ∆umin2 ; ∆umax
2 ] [−4; 4;−15; 15]
[umin1 ;umax
1 ;umin2 ;umax
2 ] [−70; 10; 4270; 6405]
chosen to keep a moderate computational burden. The constraints on input and output
signals in the optimization problem (2.17) have been set according to the plant limitations.
In particular, the inlet guide vane is constrained not to operate beyond fully opened and
fully closed (saturation) i.e. −70 < u1(t) <10 [deg], while the rotational speed must satisfy
4270 < u2(t) <6405 [rpm]. Similarly, MPC can handle constraints on the rate of variation of
the control variables. This is realized by specifying bounds on the maximum move size per
sample period ∆ui(t) = ui(t)− ui(t− 1). For instance, the IGV from minimum to maximum
requires 20 [s] to move from completely opened to completely closed. When using a sample
time of Ts = 1 [s], this means that the move per sample may not exceed −4 < ∆u1(t) < 4
[deg], while the second input which is the speed variation of the gas turbine must satisfy
54 4. Control Techniques for Pressure Regulation
−15 < ∆u2(t) <15 [rpm/s].
In this study, the power of the turbine, is not allowed to exceed the bounds between 30%
and 100% of its nominal power. It is worth observing that the opening/closing range of
the suction and discharge valves are limited by the turbine max power, or by the speed
limitations, when tracking the reference pressure. The MPC control scheme is depicted in
Figure 4.3.
Figure 4.3: Simulink model of MPC control of a plant with input saturation.
4.3 Comparison of Multivariable Control Schemes
This section provides the evaluation of the PI controller and MIMO controllers (MPC and
LQI), within different scenarios. The initial values of the state variables used in the simu-
lation are presented in Table 4.4: they correspond to the operating point of SH.07 in [84].
This operating point was obtained as a steady state condition in our model by setting the
upstream valve to UV = 47.9257% and the downstream valve to DV = 90.44%, and will be
referred to in the following as operating point ”A”. We have changed the upstream valve to
UV = 65% and we have called this new operating condition ”B”. The operating point ”C”
was set by fixing UV = 47.9257% and DV = 65%. ”A”, ”B” and ”C” operating conditions are
not referred to the operating point in the compressor map but they represent the position of
the upstream and downstream valves.
Scenario I
In the first simulation test, we act on the discharge valve, slowly moving the operating
point from A to B and then back to A, while controlling discharge pressure at set point.
Figure 4.4 depicts the results for the PI controller (blue) and for the MPC (green) while
the red curve corresponds to the LQI controller. The left top plot shows the opening and
closing of the the discharge valve. The left middle plot shows the discharge pressure. The
left bottom plot shows the manipulated IGV. The right top plot depicts the flow rate. The
right middle plot shows the fuel consumption. The right bottom plot shows the rotational
speed.
Scenario II
4.3. Comparison of Multivariable Control Schemes 55
Table 4.4: Parameters of the starting operating point used in the simulations for MPC and LQI.
Variables OP. A
Speed (N) 6053.264 [rpm]
Inlet guide vane (IGV ) 0 [deg]
Inlet pressure (ps) 71.578 [bara]
Discharge pressure (pd) 121.177 [bara]
Flow rate (wc) 463.614 [kg/s]
0 100 200 300 400 500
70
80
90
Dow
nstr
eam
Val
ve [\
%]
0 100 200 300 400 500
350
400
450
500
Flo
w R
ate
[kg/
s]
0 100 200 300 400 500
120
122
124
Dis
char
ge P
ress
ure
[bar
a]
0 100 200 300 400 5001.5
2
2.5
x 108
Fue
l Con
sum
ptio
n [k
J/h]
0 100 200 300 400 500
−60
−40
−20
0
Inle
t Gui
de V
ane
[deg
.]
Time [s]0 100 200 300 400 500
4500
5000
5500
6000
Time [s]
Spe
ed [r
pm]
Figure 4.4: Simulation results: case I.
In the second simulation scenario, we act on the suction valve (closing), slowly moving
the operating point from A to C, while controlling discharge pressure at set point. We then
repeat back from C to A. Figure 4.5 shows the process responses to an increase of upstream
valve at 50s to 150s from 47.926% to 65% than decrease at 300s to return to initial value
at 400s. Figure 4.6 shows the result when weight of the IGV in the LQI controller is equal to
100 instead of 10. This significantly reduces the transient of the IGV signal, but it does not
improve the control performance, which remains similar to that of the PI controller.
Scenario III
56 4. Control Techniques for Pressure Regulation
0 100 200 300 400 50045
50
55
60
65U
pstr
eam
val
ve [\
%]
0 100 200 300 400 500455
460
465
470
475
Flo
w r
ate
[kg/
s]
0 100 200 300 400 500120.5
121
121.5
122
Dis
char
ge p
ress
ure
[bar
a]
0 100 200 300 400 500
2.2
2.4
2.6
x 108
Fue
l con
sum
ptio
n [k
J/h]
0 100 200 300 400 500
−40
−20
0
Inle
t gui
de v
ane
[deg
.]
Time [s]0 100 200 300 400 500
5000
5500
6000
Time [s]
Rot
atio
nal s
peed
[rpm
]
Figure 4.5: Simulation results: case II-a.
In the third simulation, we consider a process upset: the discharge valve changes follow-
ing a fixed slope rate faster than those considered in cases 1 and 2. Valve closing occurs in
30 seconds and the limits are determined so that to prevent crossing the surge control line
(surge prevention is not considered here within the control scheme: it will be addressed in
Chapter 5). Figure 4.7 shows the process responses to a decrease of downstream valve at
50 seconds and its effect on the flow rate, fuel consumption, and discharge pressure for all
three control schemes. In this case, it is clear that a single loop control (PI) performs poorly
compared to both MPC and LQI.
Scenario IV
In the fourth simulation, the downstream valve undergo a step change. Figure 4.8 shows
the results when the discharge valve goes from 90.44% to 65%, Figure 4.9 shows on the left
the operating point in the performance map of the centrifugal compressor, while on the
right the operating point in the heat rate map of the gas turbine and its trajectories in the
both maps. It can be observed that in the four scenario, the combined use of IGV and
rotational speed allows one to regulate the discharge pressure to the desired reference value
in a much faster way and with much more limited transients. This demonstrates that the
variation of inlet guide vane can improve the regulation of the discharge pressure, compared
to controlling only the rotational speed.
Scenario V
4.3. Comparison of Multivariable Control Schemes 57
0 100 200 300 400 50045
50
55
60
65U
pstr
eam
val
ve [%
]
0 100 200 300 400 500455
460
465
470
475
Flo
w r
ate
[kg/
s]
0 100 200 300 400 500
120.5
121
121.5
122
Dis
char
ge p
ress
ure
[bar
a]
0 100 200 300 400 500
2.2
2.4
2.6
x 108
Fue
l con
sum
ptio
n [k
J/h]
0 100 200 300 400 500−10
−5
0
5
Inle
t gui
de v
ane
[deg
.]
Time [s]0 100 200 300 400 500
5000
5500
6000
Time [s]
Rot
atio
nal s
peed
[rpm
]
Figure 4.6: Simulation results: case II-b.
In the fourth simulation, the upstream valve undergo a step change. Figure 4.10 shows
the results when we simulate a step change on the suction valve from 47.93% to 65%.
Figure 4.11 show on the left the operating point in the performance map of the centrifugal
compressor, while on the right the operating point in the heat rate map of the gas turbine
and its trajectories in the both maps.
In this scenario, it can be observed the same remark as previous case where the combined
use of IGV and rotational speed allows one to regulate the discharge pressure to the desired
reference value in a much faster way and with much more limited transients. This demon-
strates that the variation of inlet guide vane can improve the regulation of the discharge
pressure, compared to controlling only the rotational speed. The improvement provided by
the MIMO MPC controller with respect to the SISO PI one is much more evident in cases I, II
and III, when the change in the suction or discharge valve is much faster. On the other hand,
it is observed that when fast decrease of the flow rate occurs, the IGV tends to be driven to
the lowest admissible values in order to achieve a faster regulation of the output signal.
58 4. Control Techniques for Pressure Regulation
0 50 100 150 200 250
70
80
90
Dow
nstr
eam
Val
ve [\
%]
0 50 100 150 200 250
350
400
450
Flo
w R
ate
[kg/
s]
0 50 100 150 200 250120
125
130
135
140
Dis
char
ge P
ress
ure
[bar
a]
0 50 100 150 200 2501.5
2
2.5
x 108
Fue
l Con
sum
ptio
n [k
J/h]
0 50 100 150 200 250
−60
−40
−20
0
Inle
t Gui
de V
ane
[deg
.]
Time [s]0 50 100 150 200 250
4500
5000
5500
6000
Time [s]
Spe
ed [r
pm]
Figure 4.7: Simulation results: case III.
4.3. Comparison of Multivariable Control Schemes 59
0 50 100 150 200
70
80
90
Dow
nstr
eam
Val
ve [\
%]
0 50 100 150 200
350
400
450
Flo
w R
ate
[kg/
s]0 50 100 150 200
120
125
130
135
140
Dis
char
ge P
ress
ure
[bar
a]
0 50 100 150 2001.5
2
2.5
x 108
Fue
l Con
sum
ptio
n [k
J/h]
0 50 100 150 200
−60
−40
−20
0
Inle
t Gui
de V
ane
[deg
.]
Time [s]0 50 100 150 200
4500
5000
5500
6000
Time [s]
Spe
ed [r
pm]
Figure 4.8: Simulation results: case IV with step change on discharge valve.
1 2 3 4 5 6 7 8 9 10 11 120
10
20
30
40
50
60
70
80
90
100
4290
6128
674167416741
IGV= −69.5451
Hp
[kJ/
kg]
Qv [m3/s]
4238
6054
665966596659
4050
6070
8090
100
50
60
70
80
90
100
0.9
1
1.1
1.2
1.3
1.4
1.5
x 104
Baseload Shaft Power [%]
Real: Fc=1.64892e+008 [kJ/h] Optimal: Fc=1.64892e+008 [kJ/h]
Output Shaft Speed [%]
Hea
t Rat
e [k
J/kW
h]
Figure 4.9: Case IV, acting on downstream valve: the compressor performance map (left) and the gas
turbine HR map (right) The black line is the trajectory of the operating point.
60 4. Control Techniques for Pressure Regulation
0 50 100 150 200
50
55
60
65
Ups
trea
m v
alve
[\%
]
0 50 100 150 200420
440
460
480
500
520
Flo
w r
ate
[kg/
s]0 50 100 150 200
120
125
130
Dis
char
ge p
ress
ure
[bar
a]
0 50 100 150 2002
2.2
2.4
2.6
2.8x 10
8
Fue
l con
sum
ptio
n [k
J/h]
0 50 100 150 200
−60
−40
−20
0
Inle
t gui
de v
ane
[deg
.]
Time [s]0 50 100 150 200
5000
5500
6000
Time [s]
Rot
atio
nal s
peed
[rpm
]
Figure 4.10: Simulation results: case V with changing on suction valve.
1 2 3 4 5 6 7 8 9 10 11 120
10
20
30
40
50
60
70
80
90
100
4273
6104
671467146714
IGV= −59.828
Hp
[kJ/
kg]
Qv [m3/s]
4238
6054
665966596659
4050
6070
8090
100
50
60
70
80
90
100
0.9
1
1.1
1.2
1.3
1.4
1.5
x 104
Baseload Shaft Power [%]
Real: Fc=2.31680e+008 [kJ/h] Optimal: Fc=2.31680e+008 [kJ/h]
Output Shaft Speed [%]
Hea
t Rat
e [k
J/kW
h]
Figure 4.11: Case V, acting on upstream valve: the compressor performance map (left) and the gas
turbine HR map (right). The black line is the trajectory of the operating point.
Chapter 5
Surge Prevention
Abstract
This chapter deals with multivariable model predictive control scheme for surge prevention
in centrifugal compressors. The main novelty of the proposed approach is that three control
inputs are considered: the rotational speed of the compressor, an anti-surge valve for gas
recycle and the inlet guide vane, whose variations allow one to significantly enlarge the op-
erating region of the compressor and hence to enhance the authority of the control system.
Surge prevention is achieved by including in the model an output variable accounting for the
distance of the operating point from the surge limit. Such distance is defined on a compressor
performance map which is invariant to changes in the inlet conditions, and thus its compu-
tation requires only standard pressure measurements available from the plant. Numerical
simulations show that the proposed control system is able to meet the desired specifications,
in the presence of different types of disturbances occurring along the pipeline. Subsequently,
the antisurge valve is considered as an additional control variable and we compare the per-
formance of the SIMO (only N control variable), TIMO (N and ASV control variables), and
MIMO (IGV, N and ASV control variables) linear model predictive controllers, within different
scenarios. Noise rejection and chattering avoidance will be treated for disturbances.
5.1 Introduction
The objective of the control system of a centrifugal compressor is to keep the primary process
variable (for example, suction/discharge pressure or mass flow rate) at a desired set point
level and to track the set point as quickly as possible whenever a process disturbance occurs.
At the same time, the operating point must be kept within the safe or acceptable train
operating envelope, considering limits such as surge or stonewall. Moreover, limitations
on speed, inlet guide vane, pressure and power, must be kept into account. The reference
application considered in this paper is that of a plant for natural gas transportation, in
which the centrifugal compressor is required to maintain a prescribed discharge pressure,
no matter of the variations occurring upstream or downstream along the pipeline.
The use of Model Predictive Control (MPC) for centrifugal compressors has been consid-
ered and investigated by several works in the literature. Cortinovis et L. [85] have design
a linear MPC scheme for anti-surge control, where the inputs of the linear model are cho-
sen as normalized torque and normalized recycle valve opening as deviations from actual
conditions, and the outputs represent the normalized deviations from nominal conditions of
discharge pressure and distance to surge. In [55], a linear MPC scheme has been designed
62 5. Surge Prevention
for anti-surge control of a plant with two compressors. In [56–58], nonlinear MPC formula-
tions have been proposed for different plant families. Some works use both the compressor
rotational speed and the position of an Anti-Surge Valve (ASV), also called recycle valve, for
anti-surge control, see e.g. [30, 59, 86]. However, in all these studies the Inlet Guide Vane
(IGV) is not used as a further degree of freedom of the control system. In centrifugal com-
pressors, IGV is typically used to modify the mass flow rate without acting on the rotational
speed or compressor ratio [60].
We propose an MPC scheme which uses three control variables: the rotational speed,
the IGV opening angle and the position of the ASV. The control system has to pursue three
main objectives: i) the controller has to maintain the discharge pressure at a desired set
point; ii) some process-limiting variables (such as rotational speed, inlet guide vane and
power) must be maintained within given ranges; iii) the control system has to prevent surge,
without sacrificing energy efficiency or system capacity. A novel feature of the proposed MPC
approach is that a suitable distance to surge, which can be computed by using only pressure
measurements available from standard plant transmitters, is included in the model as an
additional output to be used for surge prevention. The performance of the control scheme
is tested via numerical simulations on the model in Chapter 3.
In this study we consider three objectives of the compressor control system. First, the
controller has to maintain the discharge pressure at a desired set point level. Second objec-
tive is to keep some process-limiting variables (such as rotational speed, inlet guide vane,
and power) within safe or acceptable ranges. Last objective is the surge protection i.e. pre-
venting surge-induced compressor damage and process upsets without sacrificing energy
efficiency or system capacity. In the following, we will focus on how to implement the surge
prevention control on the industrial compression systems.
This chapter is structured as follows: in Section 5.2 the compressor performance maps
are introduced and their use for surge prevention is explained. Several methods for defining
the distance to surge of the compressor system are discussed in Section 5.3. In Section 5.4,
we introduce an MPC control scheme exploiting the further control authority provided by the
anti surge valve. Section 5.5 provides the evaluation of the MIMO (IGV, N and ASV control
variables) linear model predictive controllers, within different scenarios. In Section 5.6, the
behavior of the SIMO (only N control variable), TIMO (N and ASV control variables), and
MIMO (IGV, N and ASV control variables) linear model predictive controllers is evaluated
within different scenarios. In Section 5.7, noise rejection and chattering avoidance will be
treated for disturbances.
5.2 Centrifugal Compressor Maps
In this section, the compressor performance maps are introduced and their use for surge
prevention is explained. In the literature, many different coordinate combinations for the
analysis of the compressor for antisurge control have been proposed [87–89]. Only a few
of these are invariant to changes in the inlet conditions, such as molecular weight and inlet
temperature. In this work, we will focus on three coordinate systems which are integrated
5.2. Centrifugal Compressor Maps 63
for analysis of the industrial compressors. The coordinate system can be described simply
as (Y vs X) which can be any pair of coordinates system, for instance (Hp vs Qv). The
(Hp vs Qv) coordinates system allows a number of advantages but is not suitable for surge
control due to the unavailable real-time measurement of some parameters such as molecular
weight. Then we will focus on the coordinate system (hr vs q2s) which is invariant to change
in polytropic exponent σ. Later on, the nearly invariant systems (Rc − 1 vs q2s) and (Rc − 1
vs q2s × psd) will be discussed.
5.2.1 Compressor Map in (Hp vs Qv) Coordinates
One of the typical Compressor Map (or, Compressor Performance Map) is presented in (Hp
vs Qv) coordinates of polytropic head Hp and volumetric flow in suction Qv. This coordi-
nates are not easy to use in the antisurge control system. This is due to the reason that the
polytropic head cannot be measured directly, but it must be calculated as a function of fluid
properties and several measurable process variables. The polytropic head is defined as
Hp =Zav ×R× Ts
MW×
Rσc − 1
σ(5.1)
where the subscripts (s) and (av) mean at compressor suction and average, respectively.
The polytropic head is a function only of the actual volumetric flow in suction. A simpli-
fied equation for the volumetric flow Qv is
Qv = A×
√
∆po,s × Zs ×R× Ts
ps ×MW(5.2)
where A is the orifice coefficient, R is the specific gas constant (R = Ru/MW ), Ru is the
universal gas constant, MW is molecular Weight of the gas, Rc is the pressure ratio (pd/ps),
∆po,s is the differential pressure across orifice plate in suction, Ts is the temperature of the
gas in suction, Zav is the average compressibility factor (Zs +Zd)/2, and σ is the polytropic
exponent and is defined by
σ =
(n− 1
n
)
=
(k − 1
k × η
)
(5.3)
where η is the polytropic efficiency, n is the polytropic index or number of moles, and the
gas k-value k is the ratio of specific heats (cp/cv), cp is the specific heat at constant pressure,
and cv specific heat at constant volume.
The centrifugal compressor operating behavior can be well presented in the Compressor
Map (Figure 5.1). The useable section of the map relating to dynamic compressors is limited
by the resistance lines (Surge Limit Line SLL, Surge Control Line SCL, and choke line ”or
stonewall line”) and the maximum permissible rotational speed of the compressor. In steady
state, The operating point OP of the compressor is always located at a point of intersection
of the line of constant speed and the line of constant resistance [89], where the system
resistance line is based on the downstream piping and equipment. The Surge Margin SM
64 5. Surge Prevention
↑ IGV
↓ IGV
SM
Min speed
Max speed
SLLSCL
Choke lineSpeed lines
Resistance line
OPds
Hp
[kJ/
kg]
Qv [m3/s]
Figure 5.1: Hp vs Qv centrifugal compressor performance map.
is the distance from SCL to SLL (usually, about 8-12% depending on the importance of the
compressor). The distance ds is the relative distance from OP to SLL (expressed in percent).
The problem with the use of (Hp vs Qv) coordinates in the control system is that mea-
suring the property of gas (such as, molecular weight (MW), specific heat ratio (k) and
compressibility factor (Z)) and the specific gas composition in mole percents requires the
use of modern and expensive equipments. Only temperatures and pressures in Eq. (5.1) can
be measured immediately by use of available industrial transmitters.
5.2.2 Compressor Map in Invariant Coordinates
As noted above, the compressor performance map in the coordinate system (Hp vs Qv) is not
useful for the purpose of control, because its curves are given for unique suction conditions.
But in practice, the inlet conditions are not constant. There are many ways to obtain an
invariant coordinate system. Two of those are presented here. Form Eq. (5.1) and (5.2), the
ratio of Hp to Q2v can be computed without measuring the molecular weight. If we assume
5.2. Centrifugal Compressor Maps 65
that compressibility effects are negligible, we can show that
Hp
Q2v
∝Rσ
c −1σ
× R×Zav×Ts
MW∆po,s
ps× R×Zs×Ts
MW
≈hr
q2s(5.4)
where the average compressibility (Zav = (Zs + Zd)/2)1, the Reduced polytropic head (hr)
and Reduced flow rate in suction (qs) are defined as
hr =Rσ
c − 1
σ(5.5)
q2s =∆po,sps
(5.6)
All of these process variables are easily measured except the polytropic exponent (σ).
However, this variable can be determined indirectly by using the following well known
relationship between the temperature and compression ratios for polytropic processes
Rθ = Rσc (5.7)
where Rθ is the temperature ratio across the compressor.
From Eq. (5.7), the polytropic exponent σ can be calculated by using only available
pressure and temperature signals
σ =log(Rθ)
log(Rc)=
log(
Td
Ts
)
log(
pd
ps
) (5.8)
Figure 5.2 shows the SLL for different value of the molecular weight, in the (Hp vs Qv)
and (hr vs q2s) coordinates.
The second coordinate system is pressure ratio Rc versus reduced flow rate in suction
(qs). We assume that the isentropic exponent k does not vary significantly, so we can com-
bine a linear function of Rcwith q2s as following
Rc − 1
q2s=
pd
ps− 1
∆po,s
ps
(5.9)
Note that the gas composition does not affect these coordinates, and the reduced flow
rate can be calculated by using an orifice device at suction and at discharge to measure the
differential pressure ∆po,s and ∆po,d, respectively. It should be pointed out these coordi-
nates are nearly invariant, although the term invariant is used.
The coordinate systems (Rc vs q2s) is easy to obtain from previous coordinates, by just
summing 1 to the numerator in the fraction of the Eq. (5.9), as
1The subscript (s) and (d) mean at suction and discharge of the compressor, respectively.
66 5. Surge Prevention
NOT Invariant coordinates (Hp vs Qv) Invariant coordinates (hr vs q2s)H
p[k
J/kg]
Qv [m3/s]
MW=17.761MW=18.591
MW=19.688
hr
q2s
Figure 5.2: Representation of the SLL in (Hp vs Qv) and (hr vs q2s) coordinates
Rc
q2s=
pd
ps
∆po,s
ps
(5.10)
Then to obtain the coordinate system (Rc − 1 vs q2s × psd) we need just to multiply the
denominator of that equation by the compressor inlet pressure at design point psd, as
Rc − 1
q2s × psd=
pd
ps− 1
∆po,s
ps× psd
(5.11)
The advantage of using the (hr vs q2s) with respect to (Rc vs q2s), or (Rc − 1 vs q2s × psd)
coordinate systems is that the polytropic exponent σ gives a more accurate representation
of gas composition changes. The disadvantage of the (hr vs q2s) coordinate system is that it
requires a larger number of transmitter devices.
5.3 Distance to Surge
In this section we present several methods for defining the distance to surge of the com-
pressor system. First, we will consider a method in the (Rc − 1 vs q2s × psd) coordinates,
which is the one that will be adopted in all the test performed in this work. Then, we will
also review other definitions, in the (hr vs q2s) coordinates, which are also employed in real
world applications.
5.3.1 Distance to surge in the (Rc − 1 vs q2s × psd) coordinates
In this study, we choose the (Rc − 1 vs q2s × psd) coordinate system to compute the distance
to surge. The distance to surge or antisurge control variable, δs, is defined as
5.3. Distance to Surge 67
δs =
((q2s ∗ psd)|OP
(q2s ∗ psd)|surge− 1
)
× 100 =
((q2s ∗ psd)|OP
F(Rc − 1, α)− 1
)
× 100. (5.12)
where the function F(Rc − 1, α) models the distance of the surge point SP, defined as the
point on the SLL with the same value of Rc − 1 as the OP. Notice that the position of the SP
on the SLL depends only on Rc − 1, while changes in IGV position modify also the position
of the SLL in the map. Figure 5.3 shows the SLL for a constant value of the IGV and the
operating point with its corresponding surge point.
SP OP
(q2
s ∗ psd)|surge (q2
s ∗ psd)|OP
(Rc−
1)|
OP
=(R
c−
1)|
surge
SLL
SCL
b1
Figure 5.3: Compressor performance map (Rc−1 vs q2s ×psd) showing SLL, SCL, operating point and
the corresponding surge point.
A safety margin of 10% in terms of qs at the SCL corresponds to δs = b1, where
b1 =
((q2s ∗ psd)|SCL
(q2s ∗ psd)|surge− 1
)
× 100 =
((1.1)2 × (q2s ∗ psd)|surge
(q2s ∗ psd)|surge− 1
)
× 100 = 21%. (5.13)
5.3.2 Distance to surge in the (hr vs q2s) coordinates
Although in this study we use the definition of distance to surge in the (Rc − 1 vs q2s ×
psd) coordinate system described above, for the sake of completeness we hereafter review
alternative methods for computing the distance to surge in the (hr vs q2s) coordinates.
68 5. Surge Prevention
In order to compare the operating point to the corresponding surge point, we must know
their locations in the map. The location of the operating point is known because the com-
pressor is instrumented for this purpose. Each operating point has only one corresponding
surge limit point. For a fixed value of hr, then we can describe the operating point in the
coordinates (hr vs q2s) by the slope of a line passing through the origin and the operating
point, which is called Operating Point Line (OPL). Similarly, we consider the slope of a line
connecting the origin to the surge limit point (SPL, see Figure 5.4). These slopes can be
written as
b1
N1NN2
N3
OP
SP
SLL
Red
uce
dh
ead
Reduced flow rate
SPL
SCL
OPL
hr
=hr| S
LL
q2s |OPq
2s |SLL
o
N1NN2
N3
OP
SP
SLL
Reduced flow rate
SPL
SCL
OPL
hr
hr| S
LL
q2s |OPq
2s |SLL
o
(a) (b)
Figure 5.4: Measurement of distance to surge: (a) for constant reduced head, (b) for constant speed.
M =Y
X. (5.14)
A comparison between the location of the operating point and the surge limit is required.
One way to compare quantities is to take their ratio. Here we divide the slope corresponding
to the operating point by that of the surge limit as
Ss =MOP
MSP
=(Y/X)OP
(Y/X)SP
. (5.15)
The value of the slope MOP of the line from the origin to the operating point in the
coordinates (q2s vs hr) is
MOP =hr
q2s. (5.16)
The ratio in the Eq. (5.15) is then used to calculate the deviation, δs, between operating
point and the Surge Limit. The ratio of the slopes is one when the operating point is on the
Surge Limit, so the deviation (”distance to surge”) is
δs = 1−(Y/X)OP
(Y/X)SP
. (5.17)
5.3. Distance to Surge 69
−70 −60−50 −40 −30 −20
−10 0 104500
5000
5500
6000200
250
300
350
400
450
500
Speed N [rpm]
Inlet guide vane α [deg.]
F(α
,N)
Figure 5.5: The plot of the function F(α,N).
It is clear from Eq. (5.17) that if the slope of the operating point is bigger than the slope
of the surge point, the deviation δs is positive and the compressor is operating in the stable
region.
The are several possible approaches to defining the distance δs of the operating point
from its corresponding point on the Surge Limit Line because there are an infinite number
of points on the Surge Limit Line. Two approaches are used and compared.
In the first approach, we assume that the point on the surge limit line and the operating
point lie on the same horizontal, as shown in Figure 5.4-a. Therefore, YSP = YOP , which
simplifies Eq. (5.17) to
δs = 1−(X)SP
(X)OP
. (5.18)
The relative distance can be calculated by using the theoretical compressor map provided
by the compressor manufacturer. We consider this relative distance as a real relative distance
(δs) because is a function of the suction conditions and is defined as
δs = 1−(X)SP
(X)OP
= 1−[qs]
2SP
[qs]2OP
. (5.19)
70 5. Surge Prevention
Figure 5.4-a shows the operating point (OP), the slope line of the operating point (OPL),
the deviation b1 of the slope of the surge control line (SCL), and the surge limit point (SP)
with its slope line (SPL) obtained with this approach.
0 20 40 60 80 100 120 140 160 180 2000.25
0.3
0.35
0.4
0.45
0.5
0.55
Time [s]
Rel
ativ
e di
stan
ce [−
]
Real relative distanceApproximate relative distance
Figure 5.6: Approximate relative distance from surge limit using F(α,N) function.
In this approach, the deviation from surge is modeled by an experimentally determined
function of rotational speed (N) and guide vane position (α), which are manipulated vari-
ables. Then, we consider the slope of the SPL line as a reference. The slope of the SPL line is
also the value of the relative distance to the surge line and can approximated as a function of
α and N , in order to compensate for any variation of rotational speed, compressor efficiency,
inlet conditions or gas composition in the controller. We denote the resulting approximate
relative distance by δs and we define it as
δs = 1−MOP
MSP
= 1−hr
q2s × F(α,N). (5.20)
where
MSP = F(α,N). (5.21)
Figure 5.4-b shows an example of the position of the surge point for fixed rotational
speed (N) and inlet guide vane position (α). Note that, the change in IGV position affects
the SLL: the SLL in Figure 5.4-b is depicted only for a fixed inlet guide vane position (α).
The effect of changes in IGV position on the SLL is discussed in the next section.
5.3. Distance to Surge 71
The function F(α,N) is obtained, for each N and α, by moving the operating point
towards the surge limit line by acting on the pressure at suction (ps), until the operating
point reaches the SLL. As shown in Figure 5.4-b, the surge point is unique for constant
rotational speed and inlet guide vane: this is due to the reason that the surge limit line
is invariant to the suction condition. Figure 5.5 plots the function F(α,N) for all possible
values of N and α.
Figure 5.6 reports the relative distance δs from surge limit (blue curve) and the relative
distance δs from surge limit (green curve), for a simulation example. When both relative
distances are near or less than 0.3, the difference between δs and δs is negligible.
The Surge Control Line is defined as a line on the right of the Surge Limit Line by a
”safe distance”, into the stable operating region. The distance between these two lines is the
safety margin b1.
b1
δc
δc <0
δc
=0
δc>
0
OP
SP
SPL
SCL
OPL
Red
uce
dh
ead
Reduced flow rate
δs = 1 −MOP
MSLL
o
Figure 5.7: Surge Point Line (SPL), Surge Control Line (SCL), and Deviation.
δc = δs − b1 (5.22)
Equation (5.22) Expresses deviation of the operating point from the SCL. The deviation is
zero (δc = 0) when the operating point is located on the SCL; it becomes progressively more
negative (δc < 0) as the operating point crosses the SCL and moves further into the surge
zone, and deviation is becoming progressively more positive (δc > 0) as the operating point
is located in the normal operating zone and moving away from surge (see Figure 5.7).
72 5. Surge Prevention
The value of the Surge Margin (SM) is usually assumed to be equal to 10%, i.e. the flow
rate at the surge control line is equal to 1.1 times the flow at surge limit line. The value 1.1
becomes 1.21 when using reduced coordinates, in which the flow is squared. Therefore, the
safety margin b1 is equal to δs when the operating point is on the SCL and hence it is equal
to
b1 = 1−Xsp
[Xop]SCL
= 1−Xsp
(1.1)2Xsp
= 0.1736. (5.23)
5.4 MPC Controller Design
In this section, we introduce an MPC control scheme exploiting the further control authority
provided by the anti surge valve, in order to achieve a twofold objective: output pressure
regulation and surge prevention. The MPC scheme is an extension of the MPC controller
described in Section 2.3 and therefore the same notation will be adopted. In standard
regulation, the goal is to reduce or eliminate the error between the controlled variable and
its set point; no matter the error is positive or negative. In contrast, the objective of anti-
surge control is to keep the controlled variable to one side (right side) of the surge limit line;
the deviations to the other side must be prevented [90]. Thus, antisurge control is usually
based on opening and closing the surge control valve to maintain the operating point to the
right of the surge control line (soft constraint), without allowing deviations to the left of the
surge limit line (hard constraint). This type of constraints can be easily enforced within an
MPC control scheme. In the following, we list the objective function, the tuning parameters
and constraints of the MPC scheme (the numerical values are those employed in all the
simulations).
Objective function:
minu(k|k),...,u(k+p−1|k)
{m−1∑
i=0
(3∑
h=1
∣∣∣wuhuh(k + i|k)
∣∣∣
2
+3∑
h=1
∣∣∣w∆uh∆uh(k + i|k)
∣∣∣
2
+
3∑
h=1
∣∣∣wyh
[yh(k + i+ 1|k)− ysph
]∣∣∣
2)}
(5.24)
subject to:
yminh ≤ yh(k + i+ 1|k) ≤ ymax
h
uminh ≤ uh(k + i|k) ≤ umax
h
∆uminh ≤ ∆uh(k + i|k) ≤ ∆umax
h
∆uh(k + j|k) = 0
for h = 1, 2, 3, i = 0, . . . ,m− 1, j = p, . . . ,m− 1
In Eq. (5.24), y(k + i|k) denotes the predicted output value at time (k + i), based on the
information available at time k, while u(k+ i|k) is the input value at time k+ i, for the input
sequence starting at time k, and ∆u(k + i|k) = u(k + i+ 1|k)− u(k + i|k).
Tuning parameters:
5.5. Results Analysis and Discussion 73
• Sampling time Ts = 0.04 s.
• Prediction horizon length m = 125.
• Control horizon length p = 2.
• Constant input weights: wu1 = 0 weight on control input IGV, wu2 = 0 weight on
control input N, and wu3 = 1 weight on control input ASV.
• Constant input increment weights: w∆u1 = 5 weight on control input IGV, w∆u2 = 10
weight on control input N, and w∆u3 = 1 weight on control input ASV (only the abso-
lute values of ASV is penalized, to drive it back to zero when the system is regulated
at steady state).
• Constant output weights: wy1 = 30 weight on the discharge pressure, wy2 = 0 weight
on the relative distance of the operating point from surge, and wy3 = 0 weight on
compressor power (only the ouput pressure is regulated; the other outputs are used
only to enforce constraints on them, see below).
The input constraints:
• Inlet guide vane: −70 < u1 < 10 [deg] and −4 < ∆u1 < 4 [deg/s].
• Rotational speed: 4270 < u2 < 6405 [rpm] and −15 < ∆u2 < 15 [rpm/s].
• Anti-surge valve: 0 < u3 < 100 [%] and −5 < ∆u2 < 50 [%].
The output constraints and set-points:
• Discharge pressure: ysp1 = yss1 = 121.177 [bara].
• Distance to surge: y2 > ymin2 =
((1.1)2 ×Xsp
Xsp
− 1
)
× 100 = 21 [·].
• Compressor power: 30 < y3 < 100 [%].
Notice that here we don’t use the soft constraint on the surge distance, because we consider
the steady operation of the system far from surge. In some application, if the compressor is
operating close to the surge line the soft constraint is needed.
5.5 Results Analysis and Discussion
This section provides the evaluation of the MIMO (IGV, N and ASV control variables) linear
model predictive controllers, within different scenarios.
74 5. Surge Prevention
5.5.1 Case study I
In this test, we move slowly the operating point towards the surge area by acting on up-
stream and downstream valves as shown in Figure 5.8. The results of the simulation are
presented in Figure 5.9, which shows the three inputs (IGV position, rotational speed, and
ASV position) and the three outputs of the system, namely the discharge pressure, the dis-
tance to surge and the compressor power. The distance to surge is reported in terms of ds,
as defined in the (Hp vs Qv) map (and not in terms of the actual y2 = δs used by the MPC)
because this is the standard practice in industrial applications. The control system is able to
regulate the output pressure with negligible tracking error, while keeping all the input and
output variables within the prescribed limitations and the operating point faraway from the
surge control line. It can be observed that a moderate opening of the anti-surge valve occurs
to improve pressure regulation, when both IGV opening and rotational speed are decreasing
at the maximum admissible rate.
5.5.2 Case study II
The second simulation is conducted starting from the same operating point; the only dif-
ference is in the profile of the closing upstream/downstream valve, which undergo a step
change instead of a slow closing (see Figure 5.10). This sudden closing of the valves causes
a large upset, which quickly moves the operating point close to the surge limit line. Without
control, the operating point would cross the surge limit line in less than 5 [s]. Figure 5.11
reports the resulting evolution of the input and output variables. Clearly, in this case the
predictive controller detects that the distance to surge is rapidly decreasing and reacts with
a fast opening of the anti-surge valve. This causes the quick rebound of the operating point
throughout the time interval from 20 [s] to 27 [s]. After that, being IGV saturated at -70
[deg] and shaft speed decaying at the maximum admissible rate, the closure of the ASV is
used to guarantee tracking of the reference discharge pressure.
5.5.3 Case study III
In the third simulation, the upstream and downstream valves change following a fixed slope
rate faster than those considered in case study I. Valves closing occurs in 15 seconds and
their final values are determined so that the operating point crosses the surge control line.
Figure 5.12 shows the opening and closing of both valves (upstream and downstream). Fig-
ure 5.13 shows the effect of surge control. The oscillation of the operating point throughout
the time from 20 [s] to 40 [s] is due to small opening and closing of the anti-surge valve.
Also in this case, it can be remarked that the MIMO MPC does a good job, the MPC uses the
IGV and ASV first to avoid surge and then regulates the pressure. When the IGV is saturated
at -70, the controller uses the speed N . At 120 seconds, the compressor is at steady state
because the discharge pressure is regulated and the antisurge valve is closed.
5.5. Results Analysis and Discussion 75
0 50 100 150 200 25020
30
40
50
60
70
80
90
100
time [s]
Valv
e[%
]UV
DV
Figure 5.8: Case study I. Opening/closing of upstream and downstream valves.
−60
−40
−20
0
121
121.2
121.4
121.6
121.8
122
5950
6000
6050
6100
0
10
20
30
40
50
SLLSCL
0 50 100 150 200 250−1
0
1
2
3
4
0 50 100 150 200 25050
60
70
80
90
100
time [s]time [s]
pssd
pd
IGV
NASV
pd
δ sP
Inputs Outputs
Figure 5.9: Case study I. Control signals for MPC control with surge control.
76 5. Surge Prevention
0 50 100 150 20020
30
40
50
60
70
80
90
100
time [s]
Valv
e[%
]
UV
DV
Figure 5.10: Case study II. Opening/closing of upstream and downstream valves.
−60
−40
−20
0
122
124
126
128
130
132
4500
5000
5500
6000
0
20
40
60
80
100
SLLSCL
0 50 100 150 2000
20
40
60
80
100
0 50 100 150 200
40
60
80
100
time [s]time [s]
pssd
pd
IGV
NASV
pd
δ sP
Inputs Outputs
Figure 5.11: Case study II. Control signals for MPC control with surge control.
5.5. Results Analysis and Discussion 77
0 50 100 150 200 25020
30
40
50
60
70
80
90
100
time [s]
Valv
e[%
]
UV
DV
Figure 5.12: Case study III. Opening/closing of upstream and downstream valves.
−60
−40
−20
0
121
121.2
121.4
121.6
121.8
122
5950
6000
6050
6100
0
10
20
30
40
50
SLLSCL
0 50 100 150 200−1
0
1
2
3
4
0 50 100 150 20050
60
70
80
90
100
time [s]time [s]
pssd
pd
IGV
NASV
pd
δ sP
Inputs Outputs
Figure 5.13: Case study III. Control signals for MPC control with surge control.
78 5. Surge Prevention
5.6 Comparison of Different MPC-Based Control Schemes
This section provides the evaluation of the SIMO (only N control variable), TIMO (N and
ASV control variables), and MIMO (IGV, N and ASV control variables) linear model predic-
tive controllers, within different scenarios. The controlled variables are discharge pressure
pd, distance to surge δs, and compressor power P . The overall model structure of the com-
pression plant used in this study is shown in Figure 3.1. The simulations shown in this
section are different to those shown in section 4.2, because here we use three MPC con-
trollers with different control variables while in the section 4.2 we compared between three
different controllers (PI, LQI, and MPC). The starting operating point was obtained as a
steady state condition in our model by setting the upstream valve to UV = 44.53% and the
downstream valve to DV = 90.72%, which is a little bit different to initial operating point in
Chapter 4, and will be referred to in the following as operating point A. We have changed
the upstream valve to UV = 65% and we have called this new operating condition B. The
operating point C was set by fixing UV = 44.53% and DV = 65%. Five simulation tests have
been carried out with these initial configurations. Two first simulations have a duration of
500 seconds, and the rest have a duration of 200 seconds.
First scenario
In the first simulation test, we act on the discharge valve same as described in Case I
Section 4.3. Figure 5.14 shows the closing and opening of the discharge valve. Figure 5.15
depicts the simulation results for the MIMO MPC (blue) and for the TIMO MPC (green)
while the red curve represents SIMO MPC. Such a figure is composed by six plots, which
represent the inputs (left) and outputs (right). The three left plots represent (from top to
bottom) inlet guide vane IGV , rotational speed N , and antisurge valve ASV .
Second scenario
In the second simulation test, we act on the upstream valve same as described in Case
II Section 4.3. . Figure 5.17 shows the process responses to an increase of upstream valve
at 50s to 150s from 44.53% to 65% than decrease at 300s to return to initial value at 400s
(see Figure 5.16). In this case the use of IGV not only guarantees a better tracking of the
discharge pressure, but it also avoids the opening of the ASV for regulation purposes.
Third scenario
In the third simulation, we consider a process upset: the discharge valve changes follow-
ing a fixed slope rate faster than those considered in cases 1 and 2. Valve closing occurs in
50 seconds and the limits are determined so that to prevent crossing the surge control line
(see Figure 5.18). Figure 5.19 shows the effect on the discharge pressure, distance to surge,
and compressor power for all three control schemes. In this case, it is clear that a SIMO
MPC performs poorly compared to TIMO MPC, while MIMO MPC gives better results.
Fourth and fifth scenarios
In the fourth and fifth simulation, the valves undergo a step change (see Figures 5.20
and 5.22). Figure 5.21 shows the results when the discharge valve goes from 90.72% to
65%, while Figure 5.23 shows the results when we simulate a step change on the suction
valve from 44.53% to 65%. It can be observed that in all the five scenarios, the combined use
of IGV, rotational speed, and antisurge valve allows one to regulate the discharge pressure
5.6. Comparison of Different MPC-Based Control Schemes 79
to the desired reference value in a much faster way and with much more limited transients.
This demonstrates that the variation of inlet guide vane and antisurge valve can improve the
regulation of the discharge pressure, compared to controlling only the rotational speed. The
improvement provided by the MIMO and TIMO MPC controllers with respect to the SISO
MPC one is much more evident in cases III, IV, and V, when the change in the suction or
discharge valve is much faster. On the other hand, it is observed that when fast decrease of
the flow rate occurs, the IGV tends to be driven to the lowest admissible values in order to
achieve a faster regulation of the output signal, while ASV goes to the maximum.
80 5. Surge Prevention
0 100 200 300 400 50020
30
40
50
60
70
80
90
100
UVDV
time [s]
UV/D
V[%
]
Figure 5.14: Case study I: upstream and downstream valves opening.
−60
−40
−20
0
118
120
122
124
126
4500
5000
5500
6000
0
20
40
60
80
100
SLLSCL
0 100 200 300 400 5000
20
40
60
80
100
0 100 200 300 400 500
40
60
80
100
time [s]time [s]
IGV
NASV
pd
δ sP
Inputs Outputs
Figure 5.15: Simulation results: case I.
5.6. Comparison of Different MPC-Based Control Schemes 81
0 100 200 300 400 50020
30
40
50
60
70
80
90
100
UVDV
time [s]
UV/D
V[%
]
Figure 5.16: Case study II: upstream and downstream valves opening.
−60
−40
−20
0
118
120
122
124
126
4500
5000
5500
6000
0
50
100
150
SLLSCL
0 100 200 300 400 5000
20
40
60
80
100
0 100 200 300 400 500
40
60
80
100
time [s]time [s]
IGV
NASV
pd
δ sP
Inputs Outputs
Figure 5.17: Simulation results: case II.
82 5. Surge Prevention
0 50 100 150 20020
30
40
50
60
70
80
90
100
UVDV
time [s]
UV/D
V[%
]
Figure 5.18: Case study III: upstream and downstream valves opening.
−60
−40
−20
0
120
125
130
135
4500
5000
5500
6000
0
20
40
60
80
100
SLLSCL
0 50 100 150 2000
20
40
60
80
100
0 50 100 150 200
40
60
80
100
time [s]time [s]
IGV
NASV
pd
δ sP
Inputs Outputs
Figure 5.19: Simulation results: case III.
5.6. Comparison of Different MPC-Based Control Schemes 83
0 50 100 150 20020
30
40
50
60
70
80
90
100
UVDV
time [s]
UV/D
V[%
]
Figure 5.20: Case study IV: upstream and downstream valves opening.
−60
−40
−20
0
120
125
130
135
140
4500
5000
5500
6000
0
20
40
60
80
100
SLLSCL
0 50 100 150 2000
20
40
60
80
100
0 50 100 150 200
40
60
80
100
time [s]time [s]
IGV
NASV
pd
δ sP
Inputs Outputs
Figure 5.21: Simulation results: case IV.
84 5. Surge Prevention
0 50 100 150 20020
30
40
50
60
70
80
90
100
UVDV
time [s]
UV/D
V[%
]
Figure 5.22: Case study V: upstream and downstream valves opening.
−60
−40
−20
0
120
125
130
135
140
4500
5000
5500
6000
0
50
100
150
SLLSCL
0 50 100 150 2000
20
40
60
80
100
0 50 100 150 200
40
60
80
100
time [s]time [s]
IGV
NASV
pd
δ sP
Inputs Outputs
Figure 5.23: Simulation results: case V.
5.7. Noise Rejection and Chattering Avoidance 85
5.7 Noise Rejection and Chattering Avoidance
When output pressure measurements are affected by sensor noise, the proposed MPC scheme
might be affected by high frequency components in the control signals, thus resulting in un-
desirable chattering of IGV and ASV. In this section we give an example of how to avoid
chattering phenomena and reducing noise at output of the plant. We add a low-pass fil-
ter after the discharge pressure sensor. The first-order low-pass filter can be described in
Laplace notation as:Output
Input= K
1
τs+ 1(5.25)
where s is the Laplace transform variable, tau is the filter time constant, and K is the filter
passband gain. Figure 5.24 shows the simulation of the nonlinear dynamics of the gas
compression plant with low-pass filter.
u(t)y(t)
r(t)
To Workspace1
u_mpc
To Workspace
y
Set−point
r0
U Y
U Y
U Y
Outputs/References
Nonlinear Plant
IGV
N
ASV
UV
DV
P2
d
P
MPC Controller
MPC mv
mo
ref
Lowpass filter
1
s+1 y0= 0.0
Input
Disturbance
d
Band−LimitedWhite Noise
Figure 5.24: Simulation of the nonlinear dynamics of the gas compression plant with low-pass filter.
The example shown in the case study is a critical case where the operating point goes
towards surge many times. Figure 5.25 shows the result of the simulation without (top
graph) and with (bottom graph) the low-pass filter. MPC controller used in both simulations
has the same tuning. The blue curve is the discharge pressure without considering the
measurement noise, while the red curve is the one with noise. Figure 5.26 (top graph)
represents the antisurge valve opening and closing before and after adding the low-pass
filter. The bottom graph shows the IGV opening and closing before and after adding the
low-pass filter. It can be observed that we can avoid the chattering phenomenon in the ASV
and IGV by adding the low-pass filter at pressure feedback loop.
86 5. Surge Prevention
118
119
120
121
122
123
124
125
pdref
pd
pd with noise
0 20 40 60 80 100 120 140 160 180 200117
118
119
120
121
122
123
124
125
126
pdref
pd
pd with noise
pd
(wit
hou
tLow
-pass
Fil
ter)
pd
(wit
hLow
-pass
Fil
ter)
time
Figure 5.25: Result of noise reduction in discharge pressure
−10
0
10
20
30
40
50
60
ASV
min
ASV without Low−pass FilterASV with Low−pass Filter
0 20 40 60 80 100 120 140 160 180 200−70
−60
−50
−40
−30
−20
−10
0
10
IGV
min
IGV without Low−pass FilterIGV with Low−pass Filter
ASV
[%]
IGV
[deg]
time
Figure 5.26: Result of avoiding chattering in antisurge valve and IGV
Chapter 6
Fuel Consumption Optimization
Abstract
In this chapter, we address the problem of minimizing the fuel consumption incurred by
compressor station with one-stage compressor train in a natural gas pipeline transmission
system. Four fuel optimization strategies have proposed. Two open-loop approaches based
on model predictive control by acting on inlet guide vane and speed at steady state. In the
two other approaches, each one has two components: outer loop aims at minimising the fuel
consumption at steady state, and inner loop based on MPC to regulate the plant.
6.1 Introduction
Natural gas is transported from the wellhead to the final customers through pipeline net-
work systems. To maintain the continuous flow of natural gas in pipeline network systems,
many gas compressor stations are installed. Most of those compressor stations are either
parallel or multistage. At each station, the lost energy in pipeline is periodically restored
and this typically consumes about 3-5% of the transported gas. Therefore, the problem of
minimizing the fuel consumption of each individual compressor unit in a pipeline network
is of tremendous importance. This can be done by improving the whole compression plant
itself, for instance by developing more efficient gas turbines and compressors, and also by
choosing good controlling strategies. There many of research works on the fuel cost min-
imization of steady-state gas pipeline networks, i.e. on a set of compressor stations, see
for instance, Tabkhi et al. [91] and Christo et al. [92]. Kurz et al. [93] have studies the
relationship between the compressor running speed and the power turbine optimum speed
at the required given compressor load, in order to maximize the efficiency of the compres-
sion plant. Habibvand et al. [94] have proposed a method to optimize the compressors’ fuel
consumption through manipulating the compressors’ parameters as well as the operating
condition parameters of the turbines and the air coolers within a gas compression station
unit, by using Genetic Algorithms (GA). However, in all these studies the inlet guide vane in
the compressor is not used as a further degree of freedom of the control system for the opti-
misation of the gas turbine fuel consumption. In this chapter, open and closed loop control
schemes are proposed, whose objective is to minimize fuel consumption for the plant model
considered in Section 3. Two open-loop approaches by acting on inlet guide vane and speed
at steady state, respectively, are investigated. In the two closed-loop approaches, are outer
loop aims at minimising the fuel consumption at steady state, while an inner loop based on
MPC to regulate the plant.
88 6. Fuel Consumption Optimization
This chapter starts by introducing the fuel maps in Section 6.2, followed by a few case
studies of fuel consumption for different pressures in Section 6.3. In Section 6.4, two differ-
ent open-loop fuel optimization schemes are proposed while two closed-loop fuel optimiza-
tion schemes are presented in Section 6.5.
6.2 Fuel Map
The experimental data of the gas turbines is represented by the heat rate maps (see Sec-
tion 3.8). The heat rate map depends on the required useful compressor power P and
current compressor speed N . The fuel map contains measured data of heat rate at differ-
ent working points in the gas turbine. The relationship between gas turbine power P , fuel
consumption Fc, heat rate ϕhr and gas turbine efficiency ηGT is
Fc = P × ϕhr =P
ηGT
(6.1)
It can be expressed as a function dependent on the values of P and N as follows
Fc = P × f(P,N) (6.2)
Figure 6.1 shows the heat rate and the fuel consumption data of the PGT25 SAC gas
turbine. The fueling rate (mass flow per hour) is expressed on the z-axis and is dependent
on N and P .
PowerSpeed
Hea
t Rat
e
PowerSpeed
Fue
l con
sum
ptio
n
Figure 6.1: PGT25+ SAC gas turbine: Steady-state of (left) heat rate map; (right) fuel consumption
map
6.3 Fuel Consumption
In this section, we address the problem of minimizing the fuel consumption of a compressor
station with one-stage compressor train in a natural gas pipeline transmission system. As a
6.4. Open-Loop Fuel Optimization 89
primary study, we have compared fuel consumption Fc for steady state values of rotational
speed N and inlet guide vane IGV for which the required output pressure prefd is main-
tained. Figure 6.2 shows different curves of the fuel consumption for different values of the
discharge pressure, as a function of speed. The blue curve represents the result when dis-
charge pressure is equal to 116 (bara), the curve directly above represents the result when
discharge pressure is equal to 117 (bara), and so on. Figure 6.3 shows different curves of
the fuel consumption for different values of the discharge pressure, as a function of IGV. It
can be observed that there are indeed cases in which these curves present local minima, but
the most common situation is that in which minimum fuel consumption is achieved for the
lowest admissible speed value and for the maximum admissible IGV opening.
4000 4500 5000 5500 6000 65001.6
1.8
2
2.2
2.4
2.6
2.8x 10
8
116117118119120121122123124125126127
N (rpm)
Fc
(kJ/
h)
Figure 6.2: Result of fuel consumption versus speed for different discharge pressure
6.4 Open-Loop Fuel Optimization
In this section, for the purpose of minimizing the fuel consumption, we propose an open-
loop strategy. The main problem with a closed-loop fuel optimization strategy is that fuel
consumption is not easy to measure in real applications. This value can be measured only
by using expensive equipments. Therefore, hereafter we provide two open-loop approaches,
which are applied to the plant introduced in Section 3.1, coupled with the MPC controller
for output pressure regulation presented in Section 4.2.1.
90 6. Fuel Consumption Optimization
−70 −60 −50 −40 −30 −20 −10 0 101.6
1.8
2
2.2
2.4
2.6
2.8x 10
8
116117118119120121122123124125126127
IGV (deg.)
Fc
(kJ/
h)
Figure 6.3: Result of fuel consumption versus IGV for different discharge pressure
6.4.1 Fuel Minimization by Acting on Inlet Guide Vane at Steady State
Based on the static studies of the compressor with gas turbine in Section 6.3, showing that
the optimal fuel consumption occurs when the inlet guide vane position lies in the interval
0 ≤ α ≤ 10, we act on IGV at steady state in order to drive its value within this interval.
Nevertheless, we still want the MPC closed-loop control to be able to regulate the output
pressure. Therefore, we modify the model that has been used for designing the MPC (see
Section 4.2.1), to incorporate the actual value of IGV as an output variable and we force it
to track a value between 0 and 10 whenever the output pressure is at steady state. This can
be done in the MPC framework by suitably augmenting the state space model and adding a
term to the cost function, accounting for the difference between the actual value of IGV and
its desired value at steady state.
By defining the augmented state as x(k) = [x(k) x(k)]T and the output as y(k) =
6.4. Open-Loop Fuel Optimization 91
[y(k) y(k)]T , one gets the augmented system equations1
[δx
δ ˙x
]
=
[A 0
0 −1
] [δx
δx
]
+
[B
1 0
]
δu (6.3)
[δy
δy
]
=
[C 0
0 1
] [δx
δx
]
.
The MPC objective function is modified as
minu(k|k),...,u(k+p−1|k)
{2∑
h=1
m−1∑
i=0
∣∣∣w∆uh∆uh(k + i|k)
∣∣∣
2
+∣∣∣wyh
[yh(k + i+ 1|k)− ysph
]∣∣∣
2
(6.4)
+∣∣∣wy[y(k + i+ 1|k)− ysp
]∣∣∣
2}
(6.5)
subject to:
yminh ≤ yh(k + i+ 1|k) ≤ ymax
h
uminh ≤ uh(k + i|k) ≤ umax
h
∆uminh ≤ ∆uh(k + i|k) ≤ ∆umax
h
∆uh(k + j|k) = 0
for h = 1, 2, i = 0, . . . ,m− 1, j = p, . . . ,m− 1.
The MPC controller is implemented for the model represented in Eq. (6.3) using the same
configuration as in Section 4.2.4. The set point of the new output variable, representing the
actual value of the IGV, is ysp = 5; the corresponding output weight is set to wy = 0.2.
Figure 6.4 shows an example of an application of the proposed fuel consumption reduc-
tion strategy by acting on IGV. It can be observed that, as expected from the steady state
analysis in Section 6.3, the fuel consumption decreases monotonically as IGV increases from
-60 to 5 degrees. Notice also that the MPC controller compensates the change of IGV by
suitably acting on the speed N , in order to keep the output pressure to the desired reference
value.
6.4.2 Fuel Minimization by Acting on Rotational Speed at Steady State
The second method uses the same approach as in the previous section, by acting on the
rotational speed N instead of IGV, to reduce the fuel consumption. The model is augmented
by adding a new output variable, which correspond to the actual value of N , one gets the
augmented system equations
1We did not choose the simpler extension δy = δu1 because the Model Predictive Control Toolbox does not
allow a direct feedthrough from the manipulated variables u(k) to the output vector y(k).
92 6. Fuel Consumption Optimization
0 200 400 600 800 1000
40
60
80
100
0 200 400 600 800 1000
450
500
550
0 200 400 600 800 1000120
121
122
0 200 400 600 800 100080
81
82
0 200 400 600 800 1000
2.2
2.3
2.4x 10
8
0 200 400 600 800 1000
−60
−40
−20
0
0 200 400 600 800 1000
4500
5000
5500
6000
0 200 400 600 800 10000
50
100
Upstream valve
Downstream valve
Posi
tion
[%]
wc
[kg/s
]
pd
[bara
]
ps
[bara
]
Fc
[kJ/
h]
IGV
[deg]
N[r
pm
]
ASV
[%]
Time [s]Time [s]
Figure 6.4: Open loop fuel consumption reduction by acting on IGV.
[δx
δ ˙x
]
=
[A 0
0 −1
] [δx
δx
]
+
[B
0 1
]
δu (6.6)
[δy
δy
]
=
[C 0
0 1
] [δx
δx
]
.
The chosen values for the set point and the constant output weight are ysp = 4270 and
wy = 0.005, respectively. Figure 6.5 shows an example of the open loop fuel reduction
strategy based on N . As expected, the results are pretty similar to those in Figure 6.4. A
possible motivation for the fact that lower consumption values are always obtained for N
close to its lower bound can be as follows. The heat rate data that we have is a function
of power and speed (for given ambient temperature). From Eq. (6.2) we obtain the fuel
consumption map as a function of power and speed. Figure 6.1 shows the heat rate map
(left) and the corresponding fuel consumption map (right). As it can be observed, although
the former is not monotonic, the latter takes lower values as speed decreases. Clearly, since
by acting on IGV the power changes, the combined action of pressure regulation and fuel
optimization may lead to the presence of local minima along the operating point trajectory.
Nevertheless, at least in the examples considered in this thesis, it is apparent that driving
the speed to its lower bound, always leads to lower values of fuel consumption.
6.5. Closed-Loop Fuel Optimization 93
0 200 400 600 800 1000
40
60
80
100
0 200 400 600 800 1000
450
500
550
0 200 400 600 800 1000120
121
122
0 200 400 600 800 100080
81
82
0 200 400 600 800 1000
2.2
2.3
2.4x 10
8
0 200 400 600 800 1000
−60
−40
−20
0
0 200 400 600 800 1000
4500
5000
5500
6000
0 200 400 600 800 10000
50
100
Upstream valve
Downstream valve
Posi
tion
[%]
wc
[kg/s
]
pd
[bara
]
ps
[bara
]
Fc
[kJ/
h]
IGV
[deg]
N[r
pm
]
ASV
[%]
Time [s]Time [s]
Figure 6.5: Open loop fuel consumption reduction by acting on N.
6.5 Closed-Loop Fuel Optimization
6.5.1 Closed-Loop Fuel Optimization Local Search
In this Section, we propose a closed-loop approach to address fuel optimisation. To that
end, the controller has two components: in the inner loop, a model predictive controller
for pressure regulation and antisurge control; in the outer loop, a controller which aims at
minimising the fuel consumption at steady state, see Figure 6.6. In the inner loop, the model
predictive controller presented in Section 5.4 uses the speed, the inlet guide vane and the
anti-surge valve, for surge prevention and pressure regulation. In the outer loop, the inlet
guide vane position is slowly changed to reduce the fuel consumption of the turbine, while
the MPC controller adjusts the speed and, if needed, the anti-surge valve, to maintain the
discharge pressure at the reference value. The key idea is that the outer loop operates at
a much slower rate with respect to the inner loop, because its objective is to minimize fuel
consumption at steady state. When the output pressure undergoes a significant change, the
outer loop does not operate and the IGV is kept equal to that provided by the MPC controller
until the pressure is regulated again.
The fuel optimisation strategy is illustrated by the flow chart reported in Figure 6.7.
When regulating the output pressure, one has IGV (t) = IGVmpc(t), and the algorithm will
94 6. Fuel Consumption Optimization
+ut
1MPC mv
mo
r,f
Fu,- Consum.tion Contro--,r /GV
P2
Fc
/GV_m.c
r,f
2
/n
1
2G34m8c
N
AS
IGV
IGVNASV
p2δsPFc
p2,refδs,refPref
Figure 6.6: The Matlab/Simulink blocks of the mixed controller.
Start
IGV(t)=IGVmpc(t) (1)
IGV(t)=IGV(t-T)+δ
Fc(t)≤Fc(t-T)
No Dist.
Fc(t)>Fc(t-T)
No Dist.
(1)
Dist.
No Dist.
IGV(t)=IGV(t-T)+δ
(1)
Dist.
Fc(t)≤Fc(t-T)
No Dist.
Fc(t)>Fc(t-T)
No Dist. (1)
Dist.
IGV(t)=IGV(t-T)-ǫ
χ=IGV(t)+ǫ Fc(t)≤Fc(t-T)
No Dist.
Fc(t)>Fc(t-T)
No Dist.
IGV(t)=χ
(1)
Dist.
(1)
Dist.
Fc(t)≤Fc(t-T)
No Dist.
Fc(t)>Fc(t-T)
No Dist.
IGV(t)=IGV(t-T)-δ
(1)
Dist.
IGV(t)=IGV(t-T)+ǫ
χ=IGV(t)-ǫ
Fc(t)≤Fc(t-T)
No Dist.
Fc(t)>Fc(t-T)
Figure 6.7: Steady-state fuel optimization strategy
return to this condition whenever there is a disturbance that deviates the output pressure
from its reference value. If the output pressure is at its steady state value, the controller
will increase IGV (t) by a predefined step δ, and than check for fuel consumption after T
6.5. Closed-Loop Fuel Optimization 95
seconds (where T is large enough for the system to reach steady state after the IGV change).
If fuel decreases (i.e., Fc(t) < Fc(t − T )), IGV (t) is increased again by δ, otherwise it is
decreased by the same quantity. When the fuel consumption does not decrease anymore, a
finer tuning of IGV (t) is performed with the same strategy, but with a smaller step ǫ < δ,
and only in the opposite sense with respect to the last IGV change: this ensures that a
steady state value of IGV is always reached. The algorithm can be easily implemented in
Simulink by means of a finite-state machine (FSM). More details about the implementation
are reported in Appendix A.
In general, the proposed strategy guarantees to achieve a local minimum of the fuel
consumption with respect to the IGV. A similar strategy can be applied by optimizing with
respect to the speed N and letting IGV regulate the output pressure through the MPC con-
troller. In general, the computation of the global minimum remains an open problem, due
to the fact that the fuel consumption at steady state might be a non convex function of IGV
and N (this happens, e.g., when using PGT25 DLE as a gas turbine model).
6.5.1.1 Case Study I
In all the simulations, we set the parameters δ = 4 [deg] and ǫ = 1 [deg] in Figure 6.7, rep-
resenting the size of the changes in IGV at each step during the coarse and fine optimization,
respectively.
This example considers the PGT25+ DLE gas turbine and starts from the operating point
reported in SH:07. Two output pressure abrupt changes are enforced: the downstream valve
is quick by closed, from 90.44% to 75% at 15 sec, then it is quickly opened, from 75% to
90.44% at t = 600s, while the upstream valve always remains at 47.93%. The simulation
time is about 1200s.
The change on the downstream valve makes the MPC controller drive the IGV command
from 0 [deg] to approximately -70 [deg] in order to regulate the discharge pressure. Simi-
larly, also the speed N is changed and the output pressure reaches the reference value around
t = 50s. The surge valve is always closed because the operating point does not reach the
surge control line and acting on N and IGV is sufficient to regulate the output pressure. From
the preliminary static analysis in open loop, it is known that the optimal fuel consumption
is reached when IGV ≈ 5 [deg]. Hence, we expect the fuel minimisation control to drive
IGV towards that value.
Figure 6.8 shows the curve of the upstream/downstream (UV/DV) valve position, the
opening closing of the inlet guide vane (IGV) blades, the rotational speed (N), the discharge
pressure (P2), and fuel consumption (Fc) for the gas turbine, with and without the fuel
controller. The blue curves show the result of the controller without fuel optimization,
while the green curves show the result of the controller with fuel optimization. The results
show that after t ≈ 50s, the controller increases IGV as expected, while the MPC adjusts
N to keep the output pressure constant. The value of IGV settles around -10 [deg], with a
fuel saving around 10% with respect to the case without fuel optimisation, from t ≈ 350s
onwards. At t = 350s the value of IGV does not change because the turbine reached the
speed lower bound of 4270 RPM. Then, at t = 600s the second output pressure disturbance
96 6. Fuel Consumption Optimization
occurs and it is quickly compensated by the MPC. Notice that the compensation takes a
little longer for the case with fuel optimization, due to the different operating conditions at
which the compressor is working when the disturbance occurs. Once the pressure has been
regulated again, the fuel minimization controller is re-activated, but it has no significant
effect, meaning that the achieved value of IGV (around 10 deg) corresponds to a local
minimum of the fuel consumption for the current operating conditions.
We present other numerical simulations starting from the same operating points, but
with different changes in the upstream or downstream valve. Figure 6.9 shows an exam-
ple of fuel saving when several changes on the downstream valve are enforced. It should
be observed that whenever IGV takes low values and the pressure is regulated, the fuel
optimization algorithm increases IGV thus achieving significant reduction in the fuel con-
sumption. Figure 6.10 shows an example in which we act only on downstream valve. Once
again, the fuel optimizer allows one to achieve lower values of fuel consumption by acting
on the IGV.
6.5.1.2 Case Study II
In this case study, we choose a different starting operating point, which is on top of the
step in the turbine heat rate map (see Figure 6.11). The aim is to test whether the fuel
optimization algorithm is able to move away from this condition, because crossing the step in
the heat rate map should lead to a significant reduction in the fuel consumption. Therefore,
we start with an output pressure equal to the desired one and we do not move the upstream
and downstream valves, so that changes in N and IGV are only driven by the Finite-State
Machine optimizing fuel consumption. Figure 6.11 shows the results of this test. It can
be noticed that fuel consumption is indeed reduced, until IGV settles around -60. The
trajectory of the operating point on the heat rate map (black line in Figure 6.12) shows that
the operating point has crossed the step as expected, but afterwards has stopped (red point
on the map).
In order to understand what happened, another test starting from the same initial con-
dition has been preformed, in which IGV is moved slowly from -70 to 10 (see Figures 6.13-
6.14). It can be noticed that when IGV goes over -50, the fuel consumption starts going
down again, thus proving that the one reached in the previous test was only a local mini-
mum (this can be noticed also from the plot of fuel consumption itself). The global minimum
is achieved for IGV approximately equal to 5 deg and N very close to its lower bound. This
is due to the fact that although the heat rate map grows at lower power values, the overall
fuel consumption goes down just due to the lower values of power (see the red point in
Figure 6.14 which corresponds to the final operating point at IGV=10). In other words,
it seems that along the trajectory of the operating point the power goes down faster with
respect to the growth of the heat rate.
6.5. Closed-Loop Fuel Optimization 97
0 200 400 600 800 1000 1200
50
100
posi
tion
[%]
UVDV
0 200 400 600 800 1000 1200
−60−40−20
0
IGV
[deg
.]
(1)(2)
0 200 400 600 800 1000 1200
500055006000
N [r
pm]
(1)(2)
0 200 400 600 800 1000 1200110
120
130
P2
[bar
a]
(1)(2)
0 200 400 600 800 1000 12002
2.22.42.62.8
x 108
Time [s]
Fc
[kJ/
h]
(1)(2)
Figure 6.8: Case study I(a): Comparison between result of the controller with (green) and without
(blue) fuel optimization.
98 6. Fuel Consumption Optimization
0 200 400 600 800 1000 1200
50
100
posi
tion
[%]
UVDV
0 200 400 600 800 1000 1200
−60−40−20
0
IGV
[deg
.]
(1)(2)
0 200 400 600 800 1000 1200
500055006000
N [r
pm]
(1)(2)
0 200 400 600 800 1000 1200110
120
130
P2
[bar
a]
(1)(2)
0 200 400 600 800 1000 12002
2.22.42.62.8
x 108
Time [s]
Fc
[kJ/
h]
(1)(2)
Figure 6.9: Case study I(b): Comparison between result of the controller with (green) and without
(blue) fuel optimization.
6.5. Closed-Loop Fuel Optimization 99
0 200 400 600 800 1000 1200
50
100
posi
tion
[%]
UVDV
0 200 400 600 800 1000 1200
−60−40−20
0
IGV
[deg
.]
(1)(2)
0 200 400 600 800 1000 1200
500055006000
N [r
pm]
(1)(2)
0 200 400 600 800 1000 1200110
120
130
P2
[bar
a]
(1)(2)
0 200 400 600 800 1000 12002
2.22.42.62.8
x 108
Time [s]
Fc
[kJ/
h]
(1)(2)
Figure 6.10: Case study I(c): Comparison between result of the controller with (green) and without
(blue) fuel optimization.
100 6. Fuel Consumption Optimization
0 50 100 150 200
40
60
80
100po
sitio
n [%
]
Upstream valveDownstream valve
0 50 100 150 200250
300
350
400
450
Wc
[kg/
s]
Flow Rate
0 50 100 150 200
120
125
130
135
P2
[bar
a]
Discharge Pressure
0 50 100 150 20070
80
90
P1
[bar
a]
Suction Pressure
0 50 100 150 200
1.5
2
2.5
x 108
Fc
[kJ/
h]
Fuel Consumption
0 50 100 150 200
−60
−40
−20
0
IGV
[deg
.]
Inlet guide valve
0 50 100 150 2004500
5000
5500
6000
Time [s]
N [r
pm]
GT drive speed
0 50 100 150 2000
50
100
Time [s]
AS
V [%
]
Anti−surge valve
Figure 6.11: Case study II(a): fuel optimization with constant opening of the upstream and down-
stream valves.
40
50
60
70
80
90
100
50
60
70
80
90
100
1
1.5
x 104
Speed
Power
Hea
t Rat
e
Figure 6.12: Case study II(a): Trajectory of the operating point on heat rate map.
6.5. Closed-Loop Fuel Optimization 101
0 50 100 150 200 250
40
60
80
100
posi
tion
[%]
Upstream valveDownstream valve
0 50 100 150 200 250250
300
350
400
450
Wc
[kg/
s]
Flow Rate
0 50 100 150 200 250115
120
125
130
P2
[bar
a]
Discharge Pressure
0 50 100 150 200 250
86
88
90
92
P1
[bar
a]
Suction Pressure
0 50 100 150 200 250
1.6
1.8
2x 10
8
Fc
[kJ/
h]
Fuel Consumption
0 50 100 150 200 250
−60
−40
−20
0
IGV
[deg
.]
Inlet guide valve
0 50 100 150 200 250
4500
5000
5500
Time [s]
N [r
pm]
GT drive speed
0 50 100 150 200 2500
50
100
Time [s]
AS
V [%
]
Anti−surge valve
Figure 6.13: Case study II(b): IGV transition from -70 to 10 deg, with constant opening of the
upstream and downstream valves.
4050
6070
8090
100
50
60
70
80
90
100
0.9
1
1.1
1.2
1.3
1.4
1.5
x 104
PowerSpeed
Hea
t Rat
e
Figure 6.14: Case study II(b): Trajectory of the operating point on heat rate map.
102 6. Fuel Consumption Optimization
6.5.2 Closed-Loop Fuel Optimization Global Search
In this Section we modify the MPC controller presented in Section 5.4 and propose an al-
ternative closed-loop approach. Here, we span the range of IGV admissible values at steady
state to find the optimal fuel consumption. First, we augment the model used in MPC con-
troller by adding a new output to regulate IGV, to a desired steady state value. The new
controller has two components: in the inner loop, a model predictive controller for pressure
regulation and antisurge control; in the outer loop, a controller which aims at minimising
the fuel consumption in steady state, see Figure 6.15.
u(t)y(t)
r(t)
igv
To Workspace2
Fc
To Workspace1
u_mpc
To Workspace
y
Set−point
r0
U Y
U Y
U Y
Outputs/References
OPT−Feul
IGV
Feul
Move−IGV
Nonlinear Plant
IGV
N
ASV
UV
DV
P2
d
P
Fc
MPC Controller
MPC mv
mo
ref
Input
Disturbance
d
Figure 6.15: Simulation of the nonlinear dynamics of the gas compression plant with fuel optimiza-
tion.
6.5.2.1 MPC Controller
By defining the augmented output as y = [y y]T , one gets the augmented system equations
{x = A x +B u,
y = C x.(6.7)
where
C =
0 0 0 0 1.0000 0
0.9695 0.0016 −1.4942 0.6626 −4.1893 0.0218
0.0749 0.0435 0.1834 1.0532 0 −0.0128
0 0 1 0 0 0
.
The above continuous-time model is converted to discrete time with sampling time 0.04s,
then is used in MPC Controller. The weights in the new MPC controller are the same as those
in Section 5.4, while the weight in the forth output is wy4 = 0.5.
6.5. Closed-Loop Fuel Optimization 103
The main idea of the fuel optimisation strategy is shown in Algorithm 1. At steady state,
the controller starts searching for optimal fuel consumption. The controller starts working
when c = 1. At this moment, step = 1 and the output of the controller igv is equal to the first
control variable of MPC (i.e. igv = IGV ), while the optimal fuel consumption is Fc = Fc
and its corresponding value of IGV is igv = IGV . Then, step, Fc, igv remain constant if c
stays equal to 1. The controller starts searching for optimal fuel consumption if c changes
to 0. The flag variable c should remain constant and equal to 0 as long as there are no
disturbances. It changes to c = 1 whenever there is a transient. To find the optimal fuel, the
controller selects the shortest path for the change of igv, by taking the current value of the
IGV and calculate the distance to 10 and to -70 and choose the nearest. If the current value
of IGV is closer to 10, then igv moves from IGV to 10 after that goes to -70 (as shown
in Figure 6.16-red path), otherwise it chooses the green path. In each step, the controller
keeps memory of the minimal value of fuel consumption Fc and its corresponding value of
IGV (igv = IGV ).Thus, the controller selects the value of IGV where the fuel consumption
is optimal. The MPC controller moves IGV to its desired value igv, taking in to consideration
the pressure tracking and all limits in the system.
u1(k) 1098· · ·· · ·-68-69-70
Figure 6.16: Path of igv.
Next sections are devoted to show some simulation studies carried out to analyze the
effectiveness of the controller proposed. Two simulation tests have been preformed with
two different initial conditions.
6.5.2.2 Case Study I
The first test has been simulated during t = 2000 seconds (≈ 33 min). The simulation results
are shown in Figure 6.17. Such a figure is composed by three plots, which represent (from
top to bottom) the fuel consumption, inlet guide vane, and rotational speed. In the inlet
guide vane graph, the black line refers to igv that MPC controller consider as a reference
to its output IGV (red curve). The rotational speed plot shows the control variable N that
MPC delivers to regulate pressure during the change of IGV . After spanning the whole
range of IGV values from -70 to 10, the fuel optimizer selects the minimum fuel condition
at IGV ≈ 5 and drive the system to that value. Notice that in this case one has IGV = igv
throughout the whole simulation, because all spanned igv values are feasible (i.e. pressure
can be regulated and all process conditions are satisfied).
6.5.2.3 Case Study II
The second test has been simulated during t = 3000 seconds (≈ 50 min). As can be observed
in Figure 6.18, at the beginning of the simulation, the MPC controller is able to move N to
104 6. Fuel Consumption Optimization
2.15
2.2
2.25
2.3
2.35
2.4x 10
8
−70
−60
−50
−40
−30
−20
−10
0
10
0 200 400 600 800 1000 1200 1400 1600 1800 2000
4500
5000
5500
6000
NFc
IGV
igvIGV
time
Figure 6.17: Simulation test 1 - results. From top to bottom: fuel consumption; inlet guide vane;
rotational speed.
meet all limits and maintain the discharge pressure equal to 121.14 (bara) while tracking
igv. At 950 seconds the compressor power reached the maximum power of the gas turbine
(see top graph in Figure 6.18), MPC controller stopped changing its control variables while
the controller of the fuel consumption optimisation goes on decreasing the value of igv to
find the optimal fuel. Then, at 2270 seconds the output IGV of MPC starts tracking again
the igv reference value to stop at igv = 5 deg. which is the value of IGV where the optimal
fuel consumption was achieved.
6.5. Closed-Loop Fuel Optimization 105
94
96
98
100
2.7
2.75
2.8
x 108
−60
−40
−20
0
0 500 1000 1500 2000 2500 3000
6000
6100
6200
6300
PN
Fc
IGV
igvIGV
time
Figure 6.18: Simulation test 2 - results. From top to bottom: power; fuel consumption; inlet guide
vane; rotational speed.
106 6. Fuel Consumption Optimization
Algorithm 1 AUTONOMOUS FUEL OPTIMIZATION
Require: s(k) integer value (0 or 1)
1: if s(k)=1 then
2: step(k)← 1
3: d← 10-IGV(k)
4: if d ≤ 40 then
5: l← 80+d
6: V← zeros(l+1,1)
7: V(1)← IGV(k)
8: for j← 2 to d+1 do
9: V(j)← V(j-1)+1
10: end for
11: for j← d+2 to l+1 do
12: V(j)← V(j-1)-1
13: end for
14: else
15: d←IGV(k)+70
16: l←80+d
17: V←zeros(l+1,1)
18: V(1)←IGV(k)
19: for j← 2 to d+1 do
20: V(j)← V(j-1)-1
21: end for
22: for j← d+2 to l+1 do
23: V(j)← V(j-1)+1
24: end for
25: end if
26: Fc(k)← Fc(k)
27: igv(k)← IGV(k)
28: igv(k)← IGV(k)
29: igv(k)← igv(k)
30: else
31: step(k)← step(k − 1)+1
32: d← 10-round(igv(k − 1))
33: if d ≤ 40 then
34: l=80+round(d)
35: V=zeros(l+1,1)
36: V(1)=round(igv(k − 1))
37: for j← 2 to d+1 do
38: V(j)← V(j-1)+1
39: end for
40: for j← d+2 to l+1 do
41: V(j)← V(j-1)-1
42: end for
43: else
6.5. Closed-Loop Fuel Optimization 107
44: d←round(igv(k − 1))+70
45: l←80+d
46: V←zeros(l+1,1)
47: V(1)←round(igv(k − 1))
48: for j← 2 to d+1 do
49: V(j)← V(j-1)-1
50: end for
51: for j← d+2 to l+1 do
52: V(j)← V(j-1)+1
53: end for
54: end if
55: if step(k − 1) ¡ l then
56: igv(k)← V(step(k))
57: if Fc(k − 1) ¡ Fc(k) then
58: Fc(k)← Fc(k − 1)
59: igv(k)← igv(k − 1)
60: else
61: Fc(k)← Fc(k)
62: igv(k)← igv(k − 1)
63: end if
64: else
65: igv(k)← igv(k − 1)
66: igv(k)← igv(k − 1)
67: Fc(k)← Fc(k − 1)
68: end if
69: igv(k)← igv(k − 1)
70: end if
//ROUND is a Matlab function, it rounds the elements of X to the nearest integers;
71: return igv(k), step(k), Fc(k), igv(k), igv(k)
Chapter 7
Conclusions and Future Research
This chapter contains a summary of the work presented in this thesis as well as some ideas
for future work in control of gas compression plants. The basic conclusions that can be
drawn from this work can be summarized as follows:
• The centrifugal compressor model has been extended to include the effects of inlet
guide vane, then a gas compression system model has been developed reproducing
the dynamic behaviour of a plant.
• Model-based multivariate controllers have been implemented for the reference plant
model in order to regulate the discharge pressure as quickly as possible after a process
disturbance. The input constraints are considered. This study demonstrates that the
variation of IGV can improve the regulation of the discharge pressure, compared to
controlling only the rotational speed. The results showed that MIMO controls per-
formed well on disturbance in the upstream and downstream valves, tracking the
desired reference pressure within shorter time compared to standard PI control.
• A recycle compression system with anti-surge valve has been developed and an appro-
priate surge prevention strategy has been proposed. Three coordinate systems have
been considered for the analysis of the compressor for antisurge control, which are in-
variant to changes in the compressor inlet conditions. A new prevention surge control
strategy has been derived to keep the operating point within the safe train operating
envelope, i.e. in the right of the surge control line, considering limits such as surge,
speed, inlet guide vane and compressor power. The control strategy is based on an
MPC approach. Numerical simulations has shown that the proposed approach is able
to meet the requirements in terms of both discharge pressure regulation and surge
prevention, under different types of variations in the upstream and downstream con-
ditions along the pipeline. In particular, it has been observed that the anti-surge valve,
besides guaranteeing protection from surge when a critical process upset occurs, is
also useful in the pressure regulation task, especially when the other control variables
are saturated or they already reached their maximum rate of variation.
• For the purpose of minimizing the fuel consumption at steady-state, we have proposed
four strategies. Two open-loop approaches coupled with the MPC controller, are
110 7. Conclusions and Future Research
– Fuel optimisation by acting on inlet guide vane at steady state, without measuring
the fuel consumption of the gas turbine;
– Fuel optimisation by acting on rotational speed at steady state, without measur-
ing the fuel consumption of the gas turbine.
Two closed-loop approaches coupled with the MPC controller, are
– Fuel optimisation by acting on inlet guide vane at steady state, without measuring
the fuel consumption of the gas turbine;
– Fuel optimisation by acting on inlet guide vane at steady state, using logic control
law, with measuring the fuel consumption of the gas turbine.
In the simulations we observed that the two first methods in the later strategy drives
the system to the minimum fuel operating conditions, for both SAC and DRY aeroderiva-
tive gas turbines. This is confirmed by the calculated maps of the fuel consumption.
While the latter method is expensive and insures that the gas turbine works in mini-
mum fuel consumption.
Future work will concern: the tuning of MPC and compared on different types of plants;
the use of explicit MPC formulations for reducing the computational effort; the investiga-
tion of LPV models for describing the plant dynamics under different operating conditions;
sensitivity analysis (temperature, gas conditions). The disturbances or changing of the set
point may cause crossing of the surge line, active surge control is an approach to stabilize
the compressor in the left side of the surge line and is also a topic of future work. Another
extensions of the work could use other actuators, for surge control, for instance blow off
valve, close coupled valve (CCV), movable plenum wall, gas injection or a piston actuation.
Appendix A
Finite-State Machine
In this appendix the Finite-State Machine used in the fuel consumption controller (see Fig-
ure 6.6) is described in detailed, previously presented in Section 6.5.1.
A.1 Finite-State Machine
Cart1
IGV
Cart
IGV
;<=>?@B
;<=>?@B
IGV
Fc 1
swtch
swtch
igv
1
z
|u|du/dt
IGV MPC
IGV MPC
Rate Limiter (2)
Rate Limiter (2)
Rate Limiter (3)P2
Fc
Fc
1
1
2
2
2
3
Unit DelayConvert
Convert
Figure A.1: Model of fuel consumption controller in Simulink.
Figure A.1 shows the overall block diagram of the fuel saving control. Whereas, the
first, second, and third inputs are represented by the discharge pressure, fuel consumption,
and the IGV command of the MPC controller (IGVmpc), respectively. Then, the output of the
controller is the IGV which is computed by two finite state machines (igv2) plus the IGVmpc.
IGV (t) = igv2(t) + igvmpc(t) (A.1)
where igv2 is the IGV computed by the second finite state machine Chart (2), the Simulink
awakens this Stateflow block at the rate 0.04s, can be written as,
igv2(t) =
{
igv1(t) if ∆ P2 ≤ 0.1
0 if ∆ P2 > 0.1(A.2)
112 A. Finite-State Machine
Rate Limiter (1) is the rate limiter dynamic block, which is limits the rising and falling
rates of the output signal of the finite state machine, where the derivative of the decreasing
and increasing input signals not less than -0.1 and more than 0.1, respectively. While the
second rate limiter Rate Limiter (2) is limits its output derivative between (∆IGV ∈
[−4, 4]). The third rate limiter Rate Limiter (3) represents the final output increment
(∆IGV ∈ [−4, 4]) of the IGV in the controller. The Zero-Order Hold block holds its input
for the period of Tsf [s] i.e. it gives the fuel consumption at Fc(t− Tsf ) at each Tsf [s] and
the input of the finite state machine Fc 1 is changed each that period.
The fuel consumption controller is implemented in Simulink as shown in Figure A.1. The
small FSM operates at fast rate (small sampling time), in order to react quickly if there is any
disturbance on the plant and force the IGV signal to be equal to the IGV value commanded by
the MPC controller. When this happens, the fuel minimizing controller stops for a prescribed
time (in our simulations set to about 0.04s, see Figure 6.8). The small FSM contains three
modes as shown in Figure A.2. The big FSM in Figure A.1 contains many modes as shown
in Figure A.3. In the swtch0/swtch1 mode the algorithm activates/deactivates the fuel
minimisation as following
Algorithm ⇐=
{
deactivate, if swtch = 0
activate, if swtch = 1(A.3)
strt
entry, during: igv 2=0;
Wait1
entry, during: igv 2=igv 1;
swtch=1;
Wait
entry, during: igv 2=0;
swtch=0;
[delta P2≤0.1]
[delta P2≤0.1]
[delta P2>0.1]
[delta P2≤0.1]
[delta P2>0.1]
Figure A.2: Logic diagram using Finite-State Machine.
A.1. Finite-State Machine 113
In the rising mode, the inlet guide vane position is increases by a predefined step δ (here
set to 4 [deg]). In the falling mode, the position is decreases by the same step. Then, the
IGV command input is
igv1(t) =
{
igv1(t− Tsf ) + δ, if rising mode
igv1(t− Tsf )− δ, if falling mode(A.4)
where igv1(t) denotes the IGV variation proposed at time t, while igv1(t− Tsf ) denotes the
IGV variation that was proposed at time t − Tsf [s]. Here Tsf denotes the sampling time
of the fuel optimisation module, which must be typically large enough to allow the system
to reach steady state after the last IGV variation. The igv1 will keep changing until the fuel
consumption decreases in either one of the two modes. After that, a finer IGV adjustment is
performed according to
igv1(t) =
{
igv1(t− Tsf )− ǫ, if rising mode
igv1(t− Tsf ) + ǫ, if falling mode(A.5)
The Simulink function simfcn is a saturation block to limit the IGV. The Stateflow truth
table functions Imfcn role is
igv1(t) =
{
igv1(t), if ∆ P2 ≤ 0.1 [bara]
0, if ∆ P2 > 0.1 [bara](A.6)
114 A. Finite-State Machine
swtch0
rising
case str
case stren, du: igv 1=igv 1+4;
igv 1=Infcn(delta P2,igv 1);
igv 1=Infcn(delta P2,igv 1);
igv 1=simfcn(igv 1+IGV mpc)-IGV mpc;
igv 1=simfcn(igv 1+IGV mpc)-IGV mpc;
case run
case runen, du: igv 1=igv 1+4;
igv 1=Infcn(delta P2,igv 1);
igv 1=Infcn(delta P2,igv 1);
igv 1=simfcn(igv 1+IGV mpc)-IGV mpc;
igv 1=simfcn(igv 1+IGV mpc)-IGV mpc;
case run1
case run1en, du: igv 1=igv 1-1;
igv 1=Infcn(delta P2,igv 1);
igv 1=Infcn(delta P2,igv 1);
igv 1=simfcn(igv 1+IGV mpc)-IGV mpc;
igv 1=simfcn(igv 1+IGV mpc)-IGV mpc;igv 1=igv 1+1;
case stpen, du: igv 1=igv;
[Fc≤Fc 1]
[Fc≤Fc 1]
[delta P2>0.1]
[delta P2>0.1]
[delta P2>0.1]
[delta P2>0.1]
[delta P2>0.1]
[Fc≤Fc 1]
[Fc≤Fc 1]
[Fc≤Fc 1]
[Fc≤Fc 1]
[delta P2≤0.1]
[delta P2≤0.1]
[delta P2≤0.1]
[delta P2≤0.1]
[delta P2≤0.1]
[Fc>Fc 1]
[Fc>Fc 1]
[Fc>Fc 1]
[Fc>Fc 1]
[Fc>Fc 1]
[Fc>Fc 1]
[swtch==0][swtch==1]{igv 1=0}
swtch1entry, during: igv 1=0;
rising1
en, du: igv 1=igv 1-4;
en, du: igv 1=igv 1-4;
en, du: igv 1=igv 1+1;
igv 1=igv 1-1;
Simulink FcnOut=simfcn(In)
truthtableOut=Imfcn(In,In2)
Figure A.3: Logic diagram using Finite-State Machine.
Bibliography
[1] PGT25 Gas Turbine. http://www.ge-energy.com/products_and_services/products/gas_
turbines_aeroderivative/pgt25_gas_turbine.jsp.
[2] N. Tauveron. Plant control to avoid surge development in the case of a pipe rupture in a direct
cycle htgr. Nuclear Engineering and Design, 238(11):2925 – 2934, 2008.
[3] E.M. Greitzer. Surge and rotating stall in axial flow compressors, part i: Theoretical compression
system model. In ASME J. Eng. for Power, volume 98, pages 190–198, April 1976.
[4] J. Van Helvoirt. Centrifugal Compressor Surge Modeling and Identification for Control,. PhD thesis,
Eindhoven University of Technology, 2007.
[5] E.M. Greitzer. Surge and rotating stall in axial flow compressors, part ii: Theoretical compression
system model. In ASME J. Eng. for Power, volume 98, pages 199–217, April 1976.
[6] F.K. Moore and E.M. Greitzer. A theory of post-stall transients in axial compression systems, part
i: Development of equations. In ASME J. Eng. for Gas Turbines and Power, volume 108, pages
68–76, January 1986.
[7] E.M. Greitzer and F.K. Moore. A theory of post-stall transients in axial compression systems, part
ii: Applications. In ASME J. Eng. for Gas Turbines and Power, volume 108, pages 231–240, April
1986.
[8] J.T. Gravdahl and O. Egeland. A moore-greitzer axial compressor model spool dynamics. In Proc.
36th IEEE Conf. on Decision and Control, pages 4714–4719, San Diego, CA, USA., 1997.
[9] J.T. Gravdahl and O. Egeland. Centrifugal compressor surge and speed control. In IEEE TRANS-
ACTIONS ON CONTROL SYSTEMS TECHNOLOGY,, volume 7, pages 567–579, sep 1999.
[10] E. H. Abed, P. K. Houpt, and W. M. Hosny. Bifurcation analysis of surge and rotating stall in axial
flow compressors. In American Control Conference, 1990, pages 2239 –2246, may 1990.
[11] D-C. Liaw, R. A. Adomaitis, and E. H. Abed. Two-parameter bifurcation analysis of axial flow
compressor dynamics. In American Control Conference, 1991, pages 2955 –2960, june 1991.
[12] A. Sparks and G. Gu. Control of compressor rotating stall without distributed sensing using bi-
furcation stabilization. In American Control Conference, 1997. Proceedings of the 1997, volume 6,
pages 3716 –3720 vol.6, jun 1997.
116 BIBLIOGRAPHY
[13] C. Beltax, G. Gux, A. Sparksy, and S. Banday. Rotating stall control for axial flow compressors.
In 37th IEEE Conference on Decision and Contol, Tampa, Florida, December 1998.
[14] X. Chen, G. Gu, P. Martin, and K. Zhou. Rotating stall control via bifurcation stabilization.
Automatica, 34(4):437 – 443, 1998.
[15] M. Krstic, D. Fontaine, P.V. Kokotovic, and J.D. Paduano. Useful nonlinearities and global stabi-
lization of bifurcations in a model of jet engine surge and stall. Automatic Control, IEEE Transac-
tions on, 43(12):1739 –1745, dec 1998.
[16] G. Gu, A. Sparks, and S.S. Banda. An overview of rotating stall and surge control for axial flow
compressors. Control Systems Technology, IEEE Transactions on, 7(6):639 –647, nov 1999.
[17] Y. Wang and R.M. Murray. Bifurcation control of rotating stall with actuator magnitude and rate
limits: Part i model reduction and qualitative dynamics. Automatica, 38(4):597 – 610, 2002.
[18] Y. Wang, S. Yeung, and Murray R.M. Bifurcation control of rotating stall with actuator magni-
tude and rate limits: Part ii control synthesis and comparison with experiments. Automatica,
38(4):611 – 625, 2002.
[19] T. Yang and X. Chen. Local gain of bifurcation stabilization. Automatica, 44(10):2614 – 2620,
2008.
[20] P. Chen and H. Qin. Bifurcation control of rotating stall in axial flow compressors via dynamic
output feedback. In Intelligent Control and Automation (WCICA), 2010 8th World Congress on,
pages 2919 –2924, july 2010.
[21] P. Chen, H. Qin, Y. Wang, and Y. Hong. Bifurcation stabilization of nonlinear systems by dynamic
output feedback with application to rotating stall control. SCIENCE CHINA Information Sciences,
55:200–213, 2012. 10.1007/s11432-011-4325-5.
[22] D-C. Liaw and S-T. Chang. Bifurcation analysis of a centrifugal compressor. In Systems, Man,
and Cybernetics (SMC), 2011 IEEE International Conference on, pages 1538 –1543, oct. 2011.
[23] Y-W. Liang, D-C. Liaw, and Y-C. Wu. Application of FIDF to the detection of unstable behaviors
in compression systems. In Decision and Control, 2000. Proceedings of the 39th IEEE Conference
on, volume 2, pages 1347 –1352 vol.2, 2000.
[24] F. Bruzelius and C. Breitholtz. Gain scheduling via affine linear parameter-varying systems and
hscr; infin; synthesis. In Decision and Control, 2001. Proceedings of the 40th IEEE Conference on,
volume 3, pages 2386 –2391 vol.3, 2001.
[25] K.B. Ariyur and M. Krstic. Slope seeking: a generalization of extremum seeking. In INTER-
NATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING Int. J. Adapt. Control
Signal Process. 2004; 18:1-22 (DOI: 10.1002/acs.777), 2003.
[26] J. Van Helvoirt, B. de Jager, M. Steinbuch, and J. Smeulers. Stability parameter identification for
a centrifugal compression system. In Decision and Control, 2004. CDC. 43rd IEEE Conference on,
volume 4, pages 3400 – 3405 Vol.4, dec. 2004.
[27] J. Van Helvoirt, B. de Jager, M. Steinbuch, and J. Smeulers. Modeling and identification of
centrifugal compressor dynamics with approximate realizations. In Control Applications, 2005.
CCA 2005. Proceedings of 2005 IEEE Conference on, pages 1441 –1447, aug. 2005.
BIBLIOGRAPHY 117
[28] K.B. Ariyur and M. Krstic. Slope seeking and application to compressor instability control. In
Decision and Control, 2002, Proceedings of the 41st IEEE Conference on, volume 4, pages 3690 –
3697, dec. 2002.
[29] F. Tyan and S-C. Lee. An adaptive control for rotating stall and surge of jet engines - a function
approximation approach. In Decision and Control, 2005 and 2005 European Control Conference.
CDC-ECC ’05. 44th IEEE Conference on, pages 5498 – 5503, dec. 2005.
[30] N. Daroogheh, M.R. Jahed-Motlagh, and M.T.H. Beheshti. Robust adaptive control of surge
instability in a centrifugal compressor with variable speed. In American Control Conference (ACC),
2010, pages 5056 –5061, 30 2010-july 2 2010.
[31] Y. Chengzhi and W. Cong. Persistency of excitation and performance of deterministic learning.
In Control Conference (CCC), 2011 30th Chinese, pages 2764 –2771, july 2011.
[32] V.R. Ravi, T. Thyagarajan, and M.M. Darshini. A multiple model adaptive control strategy for
model predictive controller for interacting non linear systems. In Process Automation, Control
and Computing (PACC), 2011 International Conference on, pages 1 –8, july 2011.
[33] G.S. Copeland, I. G. Kevrekidis, and R. Rico-Martinez. Adaptive detection of instabilities and
nonlinear analysis of a reduced-order model for flutter and rotating stall in turbomachinery. In
Proceedings of the 1999 IEEE International Conference on Control Applications, 1999., volume 2,
pages 1146–1150, 1999.
[34] R.L. Behnken, R. D’Andrea, and R.M. Murray. Control of rotating stall in a low-speed axial flow
compressor using pulsed air injection: modeling, simulations, and experimental validation. In
Decision and Control, 1995., Proceedings of the 34th IEEE Conference on, volume 3, pages 3056
–3061 vol.3, dec 1995.
[35] M. Hardt, J.W. Helton, and K. Kreutz-Delgado. Numerical solution of nonlinear hscr;2 and hscr;
infin; control problems with application to jet engine compressors. In IEEE TRANSACTIONS ON
CONTROL SYSTEMS TECHNOLOGY,, volume 8, pages 98 –111, jan 2000.
[36] A. Banaszuk, H.A. Hauksson, and I. Mezic. Control of stall and surge in compressors based on
moore-greitzer pde model. In Control Applications, 1997., Proceedings of the 1997 IEEE Interna-
tional Conference on, pages 677 –682, oct. 1997.
[37] F. Laaouad, A. Hafaifa, and K. Laroussi. A nonlinear model for a turbo compressor using fuzzy
logic approach. In Proceedings of the 6th WSEAS Int. Conf. on Electronics, Hardware, Wireless and
Optical Communications,, Corfu Island, Greece,, February 16-19, 2007.
[38] R. S. Shehata, H. A. Abdullah, and F. F.G. Areed. Variable structure surge control for constant
speed centrifugal compressors. Control Engineering Practice, 17(7):815 – 833, 2009.
[39] R. Malhotra, N. Singh, and Y. Singh. An efficient fuzzy-ga flow control of turbine compres-
sor system: A process control case study. International Journal of Advancements in Computing
Technology, 2(3):128–139, October 2010.
[40] A. Hafaifa, K. Laroussi, and F. Laaouad. Robust fuzzy fault detection and isolation approach
applied to surge in centrifugal compressor modeling and control. Fuzzy Information and Engi-
neering, 2:49–73, 2010. 10.1007/s12543-010-0037-6.
118 BIBLIOGRAPHY
[41] A. Hafaifa, F. Laaouad, and K. Laroussi. A numerical structural approach to surge detection and
isolation in compression systems using fuzzy logic controller. International Journal of Control,
Automation and Systems, 9:69–79, 2011. 10.1007/s12555-011-0109-3.
[42] G. Bartolini, A. Muntoni, A. Pisano, and E. Usai. Compressor surge active control via throttle and
ccv actuators. a second-order sliding-mode approach. In Variable Structure Systems, 2008. VSS
’08. International Workshop on, pages 274 –279, june 2008.
[43] L. Wen, L. Gao, and Y. Dai. Research on system modeling and control of turbine-driven centrifu-
gal compressor. In Industrial Electronics and Applications (ICIEA), 2011 6th IEEE Conference on,
pages 2090 –2095, june 2011.
[44] C. Bohn and D.P. Atherton. An analysis package comparing pid anti-windup strategies. Control
Systems, IEEE, 15(2):34–40, 1995.
[45] S. Mandarapu, S. Lolla, and M.V. Suresh Kumar. Digital pi controller using anti-wind-up mecha-
nism for a speed controlled electric drive system. In International Journal of Innovative Technology
and Exploring Engineering (IJITEE), volume 3, pages 2278–3075, June 2013.
[46] M. S. Neto Antnio, P. Damo Thaise, and A. R. Coelho Antonio. Laboratory essay with online
back-calculation anti-windup scheme for a mtg system. In IFAC Conference on Advances in PID
Control (PID’12), Brescia (Italy), March 28-30 2012.
[47] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained model predictive control:
Stability and optimality. Automatica, 36(6):789 – 814, 2000.
[48] M. Yang, L. Niu, and XU. Dianguo. Antiwindup design for the speed loop pi controller of a pmsm
servo system. In Turkish Journal of Electrical Engineering & Computer Sciences, volume 21, pages
1318–1327, September 2013.
[49] L. Rundqwist. Anti-Reset Windup for PID Controllers,. PhD thesis, Department of Automatic
Control, Lund Institute of Technology, Lund, Sweden, 1991.
[50] X-L. Li, J-G. Park, and H-B. Shin. Comparison and evaluation of anti-windup pi controllers.
Journal of power electronics (JPE), 11(1):45–50, January 2011.
[51] A. K. Madhavapeddi Sesha and M. Abdelrahman. Linear-quadratic-integral controller for a solid
oxide fuel cell. In System Theory (SSST), 2010 42nd Southeastern Symposium on, pages 90–95,
2010.
[52] Y. Fushun and L. Yongqing. The linear quadratic integral optimal variable structure control for
linear systems with delay. In TENCON ’93. Proceedings. Computer, Communication, Control and
Power Engineering.1993 IEEE Region 10 Conference on, pages 375–378 vol.4, 1993.
[53] T. Ueno and T. Azuma. A design of state predictive lqi controllers for a networked control system.
In ICCAS-SICE, 2009, pages 2192–2195, 2009.
[54] P. C. Young and J. C. Willems. An approach to the linear multivariable servomechanism problem.
International Journal of Control, 15(5):961–979, May 1972.
[55] J.P.M. Smeulers, W.J. Bouman, and H.A. van Essen. Model predictive control of compressor in-
stallations. In Proceedings of the Conference on Compressors and their systems,, I Mech E, London,
1999.
BIBLIOGRAPHY 119
[56] T.A. Johansen. On multi-parametric nonlinear programming and explicit nonlinear model pre-
dictive control. In IEEE Conf. Decision and Control, volume 3, pages 2768–2773, Las Vegas, NV,,
2002.
[57] P.J.H. Zillinger Molenaar. Model predictive compressor surge control. Master’s thesis, Technische
Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Technology
Group, Eindhoven, August 2007.
[58] A. Grancharova, T. A. Johansen, and P. Tndel. Computational aspects of approximate explicit
nonlinear model predictive control assessment and future directions of NMPC. In Springer Ver-
lag,, page 181192, 2007.
[59] A. Cortinovis, D. Pareschi, M. Mercangez, and Th. Besselmann. Model predictive anti-surge
control of centrifugal compressors with variable-speed drives. In IFAC Workshop on Automatic
Control in Offshore Oil and Gas Production, pages 251–256, Norway, May 31 - June 1 2012.
[60] A. Mohseni, R. A. Van den Braembussche, J. R. Seume, and E. Goldhahn. Novel igv designs
for centrifugal compressors and their interaction with the impeller. Journal of Turbomachinery,
134:021006 (8 pages), June 22 2011.
[61] S. J. Qin and T. A. Badgwell. An overview of industrial model predictive control technology. In
Fifth International Conference on Chemical Process control, pages 232–256, 1997.
[62] A. Bemporad and M. Morari. Robust model predictive control: A survey. In A. Garulli and
A. Tesi, editors, Robustness in identification and control, volume 245 of Lecture Notes in Control
and Information Sciences, pages 207–226. Springer London, 1999.
[63] R.K. Al Seyab and Y. Cao. Nonlinear model predictive control for the alstom gasifier. Journal of
Process Control, 16(8):795 – 808, 2006.
[64] J. Prakash and R. Senthil. Design of observer based nonlinear model predictive controller for a
continuous stirred tank reactor. Journal of Process Control, 18(5):504 – 514, 2008.
[65] B. Chaudhuri, R. Majumder, and B.C. Pal. Application of multiple-model adaptive control strat-
egy for robust damping of interarea oscillations in power system. IEEE Transactions on Control
Systems Technology, 12(5):727–736, 2004.
[66] S. J. Qin and T. A. Badgwell. A survey of industrial model predictive control technology. In
Control Engineering Practice 11, pages 733–764, 2003.
[67] C. Schmid and L.T. Biegler. Quadratic programming methods for reduced hessian SQP. Comput-
ers & Chemical Engineering, 18(9):817–832, 1994.
[68] ISA-75.01.01-2007 (60534-2-1 Mod). Flow equations for sizing control valves, 2007.
[69] F.M. DI and F.A. DI. Antisurge protection method for centrifugal compressors, June 27 2013.
WO Patent App. PCT/IB2012/053,187.
[70] F. P. Mello, Ahner D. J., P. M. Anderson, J. H. Doudna, P. Kundur, L. M. Richardson, G. Tandy,
C. W. Taylor, and F. Van de Meulebroeke. Dynamic models for combined cycle plants in power
system studies. Power Systems, IEEE Transactions on, 9(3):1698–1708, Aug 1994.
120 BIBLIOGRAPHY
[71] M. Nagpal, A. Moshref, G. K. Morison, and P. Kundur. Experience with testing and modeling of
gas turbines. In Power Engineering Society Winter Meeting, 2001. IEEE, volume 2, pages 652–656
vol.2, 2001.
[72] K. Kunitomi, A. Kurita, H. Okamoto, Y. Tada, S. Ihara, P. Pourbeik, W. W. Price, A.B. Leirbukt, and
J. J. Sanchez-Gasca. Modeling frequency dependency of gas turbine output. In Power Engineering
Society Winter Meeting, 2001. IEEE, volume 2, pages 678–683 vol.2, 2001.
[73] H. Iki, M. Komatsu, Y. Uriu, M. Watanabe, and Y. Mitani. The development of a model of
the industrial gas turbine generator on the analysis tool for the power system dynamics. In
Universities Power Engineering Conference, 2007. UPEC 2007. 42nd International, pages 148–151,
Sept 2007.
[74] S. Massucco, A. Pitto, and F. Silvestro. A gas turbine model for studies on distributed generation
penetration into distribution networks. Power Systems, IEEE Transactions on, 26(3):992–999,
2011.
[75] R. Kurz and K. Brun. Gas turbine performance-what makes the map. In 29th Texas A&M Turbo-
machinery Symposium, pages 247–262, Houston, TX, USA, Sept. 2000.
[76] T. Bentaleb, A. Cacitti, S. De Franciscis, and A. Garulli. Multivariable control for regulating high
pressure centrifugal compressor with variable speed and IGV. In 2014 IEEE Multi-conference on
Systems and Control (MSC), pages 486–491, Antibes, France, Oct. 8-10, 2014.
[77] S. Golden, S. Fulton, and D. Hanson. Understanding centrifugal compressor performance in a
connected process system. In Petroleum Technology Quarterly,, Spring 2002.
[78] S. Varigonda, J. Eborn, and S.A. Bortoff. Multivariable control design for the water gas shift reac-
tor in a fuel processor. In American Control Conference, 2004. Proceedings of the 2004, volume 1,
pages 840–844 vol.1, June 2004.
[79] T. Goya, E. Omine, T. Senjyu, M. Tokudome, A. Yona, N. Urasaki, T. Funabashi, and Chul-Hwan
Kim. Torsional torque suppression of decentralized generators using lqr obsever with parameter
identification. In IEEE International Symposium on Industrial Electronics, 2009. ISIE 2009, pages
2109–2114, July 2009.
[80] R. E. Stillwagon. Economic aspects of electrically driven compressor stations for natural gas
pipelines. IEEE Transactions on Industry Applications, IA-11(2):240–245, March 1975.
[81] N. Uddin and J.T. Gravdahl. Piston-actuated active surge control of centrifugal compressor
including integral action. In 11th International Conference on Control, Automation and Systems
(ICCAS), pages 991–996, Oct 2011.
[82] H. Ghorbani, A. Ghaffari, and M. Rahnama. Constrained model predictive control implementa-
tion for a heavy-duty gas turbine power plant. In WSEAS Transactions on System and Control,
2008.
[83] A. Bemporad, M. Morari, and R. N. Lawrence. Model Predictive Control Toolbox for use with
Matlab. In Mathworks, 2005.
[84] Operating points file, 2013.
BIBLIOGRAPHY 121
[85] A. Cortinovis, H.J. Ferreau, D. Lewandowski, and M. Mercangoz. Safe and efficient operation of
centrifugal compressors using linearized MPC. In 53rd IEEE Conference on Decision and Control,
Los Angeles, California, USA, December 15-17 2014.
[86] L. Wen, L. Gao, and Y. Dai. Research on system modeling and control of turbine-driven centrifu-
gal compressor. In Industrial Electronics and Applications (ICIEA), 2011 6th IEEE Conference on,
pages 2090–2095, 2011.
[87] B. W. Batson. Invariant coordinate systems for compressor control. In International gas turbine
and aeroengine congress & Exhibition, pages 1–16, Birmingham, UK, June 10-13 1996.
[88] H.P. Bloch. Apractical guide to compressor technology. In B. Narasimhan, S. Mallapragada, and
M.D. Porter, editors, Apractical guide to compressor Technology. McGraw-Hill Companies, 1996.
[89] S. Mirsky, J. M. Whirter, W. Jacobson, M. Zaghloul, and D. Tiscornia. Development and design
of antisurge and performance control systems for centrifugal compressors. In Proceedings of the
Forty-Second Turbomachinery Symposium, Houston, Texas, October 1-3, 2012.
[90] P. Heinz. A Practical Guide to Compressor Technology. Wiley-Interscience; 2 edition (September
14, 2006), 2006.
[91] F. Tabkhi, L. Pibouleau, G. Hernandez-Rodriguez, C.e Azzaro-Pantel, and S. Domenech. Improv-
ing the performance of natural gas pipeline networks fuel consumption minimization problems.
AIChE Journal, 56 (n4):946–964, 2010.
[92] A. T. Christo, G. M. Jubini, J. Arica, and M. M. Azeredo. An approximation for the fuel cost min-
imization of the steady-state gas pipeline network. In XIV International Conference on Industrial
Engineering and Operations Management, Rio de Janeiro, Brazil, 13-16 October 2008.
[93] R. Rainer Kurz, M. Lubomirsky, and K. Brun. Gas compressor station economic optimization.
International Journal of Rotating Machinery, 2012:9, 2012.
[94] D. Habibvand and B.M. Behbahani. Using genetic algorithm for fuel consumption optimization
of a natural gas transmission compressor station. International Journal of Computer Applications,
43:1–6, 2012.
Abstract of the Thesis:
The general topic of this thesis is control of turbomachinery. In particular,
it deals with modeling and control techniques for variable speed
centrifugal gas compression systems. In the thesis, several multi-variable
control systems are studied and validated on a gas compression plant
model, based on real data from a class of industrial centrifugal
compressors. The main novelty of the proposed approaches is that three
control inputs are considered: the rotational speed of the compressor,
an anti-surge valve for gas recycle and the inlet guide vane, whose
variations allow one to significantly enlarge the operating region of the
compressor and hence to enhance the authority of the control system.
Surge prevention is achieved by including in the model an output
variable accounting for the distance of the operating point from the
surge limit. Such distance is defined on a compressor performance map
which is invariant to changes in the inlet conditions, and thus its
computation requires only standard pressure and temperature
measurements available from the plant. The thesis also addressed to
the problem of minimizing the fuel consumption incurred by compressor.
The Ph.D. School
of Information
Engineering and
Science of the
University of
Siena is a
school aiming at
e d u c a t i n g
scholars in a number of fields of
research in the areas of
Information Engineering and
Mathematics. The Ph.D. School of
Information Engineering is part of
the Santa Chiara High School of
the University of Siena. A Scientific
Committee of external experts
recognized Ph.D. Schools
belonging to Santa Chiara as
excellent, according to their
degree of internationalization,
their research, and educational.
DI
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