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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)


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


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


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:


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.


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)


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)


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.


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)


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


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


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)


−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)


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


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.


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


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


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


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


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


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


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


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


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


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.


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


−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


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


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.


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.


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]


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.


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]


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


↑ 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.


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


)

× 100 =

((1.1)2 × (q2s ∗ psd)|surge


)

× 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.


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)


−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)


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.


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).


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


−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).


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


+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


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


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.


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.


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



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



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.


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.


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.


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


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.


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.


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)


20: V(j)← V(j-1)-1

21: end for


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))


38: V(j)← V(j-1)+1

39: end for


41: V(j)← V(j-1)-1

42: end for

43: else


44: d←round(igv(k − 1))+70

45: l←80+d

46: V←zeros(l+1,1)

47: V(1)←round(igv(k − 1))


49: V(j)← V(j-1)-1

50: end for


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=simfcn(igv 1+IGV mpc)-IGV mpc;


case run

case runen, du: igv 1=igv 1+4;





case run1

case run1en, du: igv 1=igv 1-1;




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.

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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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