Robust Controllers Design for Load Frequency

! " ! # ! !!

$ %

! # & ' %

!

( )

! *' %

+,+ ' -

. /) ' $001% 23 ' 456 7338- 9 % : ;

!"# !!$

! "

! " !#!$%

&' %

!"#$ % "

! & ' & ( )*! &

" "*)+,-!

.%

/ %

$ ! . 0,)1 % "

! & ' & ( " "0,)1

Robust Controllers Design for Load Frequency Control in Power Systems

By Asst. Prof Ibraheem Kasim Ibraheem Prof. Rami A. Maher

Electrical Engineering Department College of Engineering University of Baghdad

7RP\SDUHQWV IRUWKHLU ORYHDQGXQGHUVWDQGLQJ

dgement Acknowledgement

This work is the conclusion of three years of research at the Department of Electrical Engineering of the University of Baghdad. Many people have helped us over the past three years and it is our great pleasure to take this opportunity to express my gratitude to them all. First and foremost, we would like to thank the late Prof. Dr. Ismail A. Mohammed for his supervision and constant support which started from the first year of the postgraduate study. We have benefited from his serious research attitude and without his guidance, suggestions, continuous encouragement, and fruitful contributions, this book FRXOGQ¶W KDYH EHHQ finished. Secondly, we are greatly indebted to the administration of the college of engineering for their financial support. This support allowed us to get recent and valuable textbooks which I benefited much from them in our book. On a more personal note, our IDPLO\¶V ORYH DQG VXSSRUW have been consistent and hugely appreciated. Our deepest and heartfelt gratitude, love, admiration and respect for our parents, who have never lost faith in us and have given us an endless patience. Their encouragement and care throughout our entire phase of study are unforgettable. For their we supplicate the Almighty to protect them and give them the reward that they deserve. Finally, we would like to thank all our relatives, friends, and colleagues for their help, understanding, and support.

i

Abstract Preface The demands of internal stability, optimal transient and steady state response, and minimal interaction makes the design burden of the controllers and the filters used for disturbance and noise rejection so huge. Therefore the optimal solution based on conventional methods is difficult to achieve. The proposed approach in this book is to apply the theories of optimal robust control based on H-norms on modern power systems, which in turns are complex, nonlinear, MIMO, and exhibits strong time variations over wide range and they are subjected to noises, and load disturbances. The proposed approach ensures internal stability, satisfying both frequency and time domains requirements, and obtaining minimal performance H-norm of the closed-loop system in one burden. In the current state-of-the-art there are two approaches according to the systems description; namely the state space and the polynomial representations and hence the solutions will have different features in the suboptimal sense for the individual characteristics. The work in this book is essentially composed of two parts. In the first part, an H robust controller is designed to replace the conventional governor of the steam turbine of the power system to regulate the frequency of the power grid. The controller is synthesized using both the state-space and the polynomial approaches with time variations, neglected dynamics, and constant main steam pressure are considered in the design. In the second part, a case of varying pressure of the produced main steam is considered and hence the boiler dynamics have been taken into account in the design of the robust governor. Issues like time-variations, nonlinearity, and dynamic couplings of the boiler-turbine unit are addressed in the design of the H robust controller for this unit. Also the comparison between the two approaches indicates that although both approaches give the same results for the same weighting filters, ii

the polynomial approach is more flexible and adjustable to obtain optimal solution. The simulations have been done using MATLAB 7 software with the aid of the Polynomial Toolbox. Finally, the simulations results show that the overall system output performance can be improved using our proposed H robust controllers.

Ibraheem K. Ibraheem & Rami A. Maher Baghdad, Iraq

iii

List of Contents List of Contents Acknowledgement

i

Abstract

ii

List of Contents

iv

List of Symbols

vii

List of Acronyms

ix

Chapter One

1

Introduction

1.1 Introduction

1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2 Uncertainty and Robust Control . . . . . . . . . . . . . . . . . . . . . . .

4

1.3 Robust Control: Polynomial and State-Space Approaches

. . . . . . . . .

7

1.4 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.5 Book Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.6 Book Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Chapter Two

Robust Control and HTechniques

14

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.2 Singular Values and H-Norm

15

. . . . . . . . . . . . . . . . . . . . . . .

2.3 Performance Specifications and Limitations

. . . . . . . . . . . . . . . .

16

2.4 Weighted Performance and Selection of Weighting Filters . . . . . . . . .

19

2.5 H Control Problem Formulation: A State-Space Approach . . . . . . . .

22

2.5.1 H Suboptimal Controller Design Procedure

. . . . . . . . . . . .

23

2.6 Mixed Sensitivity H Control . . . . . . . . . . . . . . . . . . . . . . . .

25

2.7 H Control Problem Formulation: A Polynomial Approach . . . . . . . . .

26

2.7.1 Frequency Response Shaping

. . . . . . . . . . . . . . . . . . . .

28

. . . . . . . . . . . . . . . . . . .

29

2.7.3 Partial Pole Placement . . . . . . . . . . . . . . . . . . . . . . . .

30

2.8 The Polynomial Solution of the Standard H Problem . . . . . . . . . . .

32

2.8.1 Spectral Factorization (SF) . . . . . . . . . . . . . . . . . . . . . .

34

2.7.2 Selection of the Shaping Filters

iv

v

2.8.2 Polynomial H Controller Design Procedure

. . . . . . . . . . . .

35

2.9 Controller Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . .

38

Chapter Three Modeling of Steam-Turbine and Boiler-Turbine Systems . . .

40

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

3.2 Steam Turbines and Governing Systems

. . . . . . . . . . . . . . . . . .

41

3.3 Modeling of Steam Turbine and Speed Governing System . . . . . . . . .

42

3.4 Modeling of Boiler-Turbine System . . . . . . . . . . . . . . . . . . . . .

51

3.4.1 Boiler -Turbine Control Mode . . . . . . . . . . . . . . . . . . . .

51

3.4.2 Nonlinear Model of the Drum-Boiler-Turbine System

54

. . . . . . .

3.4.3 Linearization of the Nonlinear Model . . . . . . . . . . . . . . . . Chapter Four

55

Uncertainty Modeling and Robustness Tools . . . . . . . . . .

59

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.2 Model Uncertainty

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.3 Uncertainty Representation . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.4 Obtaining the Weight for Complex Uncertainty

. . . . . . . . . . . . . .

62

. . . . . . . . . . . . . . . . . . . . . . . .

63

4.5 Poles on the Imaginary Axis

4.6 Robustness Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

4.7 Requirements for Stability and Performance . . . . . . . . . . . . . . . .

66

4.8 High frequency Roll-off and Integrating action

68

. . . . . . . . . . . . . .

4.9 Robust Controller Design and Assessment . . . . . . . . . . . . . . . . .

70

4.10 Proposed Configuration for LFC of Steam Turbine

. . . . . . . . . . . .

71

4.10.1 Design Problem Formulation: A State-Space Approach . . . . . .

71

4.10.2 Design Problem Formulation: A Polynomial Approach . . . . . . 4.11 Proposed Control System for Boiler-Turbine Unit

74

. . . . . . . . . . . . .

76

4.11.1 Representing Uncertainty by Complex Perturbations . . . . . . . .

79

4.11.2 Control System Interconnection: A State-Space Approach . . . . .

81

4.11.3 Control System Interconnection: A Polynomial Approach . . . . .

83

4.12 Simulation of the Closed-Loop System . . . . . . . . . . . . . . . . . . .

84

Chapter Five

Robust Controllers Design for Uncertain Steam Turbine and Boiler-Turbine Systems using H Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85

5.2 Controller Design for Steam Turbine System: A State-Space Approach . .

86

5.2.1 H Design without Modeling Uncertainty . . . . . . . . . . . . . .

86

5.1 Introduction

vi H Design with Modeling Uncertainty . . . . . . . . . . . . . . .

87

5.2.2.1 Selection of the Weighting Filters . . . . . . . . . . . . . . . . .

90

5.3 Controller Design for Steam Turbine System: A Polynomial Approach . .

92

5.2.2

5.3.1

H Design without Modeling uncertainty

5.3.2

H Design with Modeling Uncertainty . . . . . . . . . . . . . . .

95

5.4 Robust Controller Design for Boiler-Turbine System . . . . . . . . . . . .

98

5.4.1

. . . . . . . . . . . . .

93

Design Specifications of Boiler-Turbine System . . . . . . . . . .

98

5.4.2 Derivation of the Performance Weighting filter WP(s) . . . . . . .

99

5.4.3 Derivation of the Uncertainty weighting filter WI(s) . . . . . . . . 101 5.5 Analysis and Simulations Results

. . . . . . . . . . . . . . . . . . . . . 104

5.5.1 Simulations and Results for Steam Turbine H Design without Uncertainty: A State-Space Approach . . . . . . . . . . . . . . . . . . . 105 5.5.2 Simulations and Results for Steam Turbine H Design with Uncertainty: A State-Space Approach . . . . . . . . . . . . . . . . . . . 106 5.5.3 Simulations and Results for Steam Turbine H Design without Uncertainty: A Polynomial Approach . . . . . . . . . . . . . . . . . . . 116 5.5.4 Simulations and Results for Steam Turbine H Design with Uncertainty: A Polynomial Approach . . . . . . . . . . . . . . . . . . . 120 5.5.5 Comparison between State-Space and Polynomial Methods . . . . . 126 5.5.6 Simulations and Results for Boiler-Turbine H Design Chapter Six

. . . . . . . 132

Conclusions and Directions for Future Research . . . . . . . . . . 143

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2 Directions for Future Research . . . . . . . . . . . . . . . . . . . . . . . 145 Appendix A References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 149

vii

List of of Symbols Symbols List transpose of matrix A

AT

A ( jZ)

complex conjugate transpose of the matrix A( jZ ) at each frequency, that is A ( jZ)

AT ( jZ)

C mun

complex matrices of dimension mu n

Fl ( P, K )

lower linear fractional transformation

Fu (', P)

upper linear fractional transformation subspace containing all uniformly square integrable and

H2

analytic functions in the open right-half plane.

Hf

subspace containing all analytic and bounded transfer functions in the open right-half plane.

I

identity matrix of compatible diemsnion

R

set of real numbers

Re(x)

real value of the complex number x

X

state vector of the linearized state-space system

Xo

vector X at the nominal operating point diagonal concatenation of the two matrices X and Y ,

diag ( X , Y )

ªX

0º

that is diag ( X , Y ) « » ¬ 0 Y¼

inf (G(s))

most lower bound of the function G(s)

j

imaginary unit, i.e. j

1

sup f (Z )

supremum or the least upper bound of f (Z) over Z

x

derivative of the continuous variable x

G

modulus (or magnitude) of G C mun

Z

G

f

H-infinity norm of G

H

very small number

infinity

viii ¨

complex perturbaWLRQ ZLWK -norm bounded by 1, i.e ǻ

f

d1

P

structured singular value

Oi (G )

i-th eignvalues of the matrix G

U (G)

maximum eignvalue of the matrix G

V i (G )

i-th singular values of the matrix G

V

maximum singular values of the matrix G

V

minimum singular values of the matrix G

VH

hankel singular value

belongs to

|

approximately equal to

d

less than or equal to

t

greater than or equal to

much less than

!!

much greater than

!

greater than

less than

implies

is equivalent to

for all

ǻ

defined as

ªA Bº «C D» ¼ ¬

shorthand for state-space realization of C(sI A) 1 B D

WP (s )

performance Weighting filter

ZB

Bandwidth of the closed-loop system

J

solution of the H-infinity optimization

Dcl

Characteristic polynomial of the closed-loop system.

ix

List of Acronyms AGC

Automatic Generation Control

ANFIS

Adaptive Neuro-Fuzzy Inference System

COT

Continually Online Trained

CV

Control Valves

FLC

Fuzzy Logic Control

GA

Genetic Algorithm

HP

High Pressure

HSV

Hankel Singular Value

IP

Intermediate Pressure

IV

Intercept valves

LFC

Load Frequency Control

LHP

Left Half Plane

LMI

Linear Matrix Inequalities

LP

Low Pressure

LQG

Linear Quadratic Gaussian

LQR

Linear Quadratic Control

MD

Machine Dynamics

MIMO

Multi Input Multi Output

MLP

Multi-Layer Perceptron

MSV

Main inlet Stop Valves

NN

Neural Network

NP

Nominal Performance

NS

Nominal Stability

PID

Proportional Integral Derivative

PMF

Polynomial Matrix Fraction

PWR

Pressurized Water Reactor

RH

Reheater

x

RHP

Right Half Plane

RP

Robust Performance

RS

Robust stability

RSV

Reheater Stop Valves

SF

Spectral Factorization

SG

Speed Governor

SISO

Single Input Single Output

SM

Servomotor

OSOFLC

Online Self-Organizing Fuzzy Logic Controller

SR

Speed Relay

SRP

Speed Reference Position

TD

Temporal Difference

TDOF

Two-Degree-Of-Freedom

CHAPTER ONE CHAPTER ONE INTRODUCTION INTRODUCTION

1.1 Introduction Since WKH LQGXVWULDO UHYROXWLRQ PDQ¶V GHPDQG IRU DQG FRQVXPSWLRQRI energy has increased steadily. A major portion of the energy needs of a modern society is supplied in the form of electrical energy. Industrially developed societies need an ever-increasing supply of electrical power, and the demand has been doubling every ten years. Very complex power systems have been built to satisfy this increasing demand. This vast enterprise of supplying electrical energy presents many engineering problems that provide the engineer with a variety of challenges [1]. The planning, construction, and operation of such systems become H[FHHGLQJO\ FRPSOH[ 6RPH RI WKH SUREOHPV VWLPXODWH WKH HQJLQHHU¶V managerial talents; others tax his knowledge and experience in system design. An intensive research has been done in the field of electrical power production and many control methods have been developed to coordinate the operation of the different components that constitute the power system. Among the newly invented control methods is the robust control which aims to find a control mechanism that governs the operation of the controlled system even with the existence of the system uncertainty.

1

1.2 Background and Motivation

2

1.2 Background and Motivation Modern power systems are highly complex and non-linear and their operating conditions can vary significantly over a wide range. Power system stability can be defined as that property of a power system that enables it to remain in state of operating equilibrium under normal operating conditions and to regain an acceptable state of equilibrium after being subjected to a disturbance. The increasing size of generating units, sudden changes in load or generation, or short circuits on transmission lines are the main causes affecting small signal stability of power systems [1, 2]. The ability of a power system to maintain stability depends to a large extent on the controls available on the system to damp the electromechanical oscillations. Hence, the study and design of controls are very important. Thus a properly designed and operated power system should, therefore, meet the following fundamental requirements: 1. The system must be able to meet the continuously changing load demand. 2. The system should supply energy at minimum cost. 3. The quality of power supply must meet certain minimum standard requirements with regard to the following factors: x Constancy of frequency. x Constancy of voltage. x Level of reliability. The most universal method of electric generation is accomplished using thermal generation, and the most common machine for this production is the steam turbines. In the world over 85% of all generation is powered by steamturbine-driven generators. The size of these generating units has increased over time, with largest units now being over 1200 MW. The steam used in electric production is produced in steam generators or boilers using either

1.3 Background and Motivation

3

fossil or nuclear fuels as primary energy sources [2]. The boiler-turbine unit is a highly nonlinear, strongly coupled complex multivariable and timevarying system. It consists of equipments such as boiler, turbine, generator, network and load [3]. Ideally, the load must be fed at constant voltage and frequency at all times. In practical terms this means that both voltage and frequency must be KHOGZLWKLQFORVHWROHUDQFHVVRWKDWWKHFRQVXPHU¶VHTXLSPHQWVPD\RSHUDWH satisfactorily. For example, reduction of the system frequency of only a few hertz may lead to stalling of the motor loads on the system [2]. Subsequent large change in the system frequency might cause cascade trips of the generating power plants and results in system breakup. Thus it can be accurately stated that the power system operator or automatic control system must maintain a very high standard of continuous electrical service. Electric power systems consist of a number of control areas, which generate power to match the power demand. However, poor balancing between generated power and demand can cause the system frequency to deviate away from the nominal value, and create inadvertent power exchanges between control areas. To avoid such a situation, Load Frequency Controllers (LFC) are designed and implemented to automatically balance between generated power and the demand power in each control area [1, 4, 5]. On the other hand, steam generation system (boiler) is very crucial part of most power plants. In order to hold the frequency stable and to control disturbances, the boiler component must be investigated for possible behavior that might be detrimental to system damping [2]. Boiler-turbine unit supplies high pressure steam to rotate the turbine in thermal electric power generation. The purpose of the boiler-turbine unit is to meet the load demand of electric power while maintaining the pressure andwater level in the drum within tolerance. This results in a MIMO boilerturbine system. Hence, a good multivariable control system is required for

1.3 Uncertainty and Robust Control

4

such a unit. Particular attention has been devoted to drum water level control since it is an important problem for nuclear control as well as conventional plants. It is stated that about 30% of the emergency shutdowns in Pressurized Water Reactor (PWR) are caused by poor level control of the steam water level. One reason is that the control problem is difficult because of the complicated shrink and swell dynamics. This creates a non-minimum phase behavior which changes with the operating point

1.3 Uncertainty and Robust Control Central to the development of feedback control theory has been the notion of uncertainty. Uncertainty refers to the differences or errors between models and reality. It can be classified into two categories: (a) Discrepancy between the physical plant and the mathematical model used for controller design. (b) Unmeasured noises and disturbances that act on the physical plant. A good model should be simple enough to facilitate design, yet complex enough to give the engineer confidence that designs based on the model will work on the true plant. Feedback is used to desensitize the control system from the effect of both these types of uncertainty. Care must be exercised, however, as feedback in the presence of an uncertain plant can easily lead to instability if due consideration is not given to the way in which this uncertainty modifies the system behavior. Classical control techniques were developed with both these uncertainty types in mind. Graphical techniques capable of dealing with single-input single-output plants were the primary tool and quickly found wide use in practice. These techniques accounts for uncertainty intuitively and unfortunately, are somewhat ad-hoc and often require a lot of iteration and much intuition on the part of the designer. Also, they do not provide easy answers WR IXQGDPHQWDO TXHVWLRQV VXFK DV ³:KDW LV WKH DFKLHYDEOH


5

performance?´. These techniques have also been found to be rather difficult to apply on complex plants, and they do not easily generalize to multivariable systems. The classical control period pave the way to the so-called Modern Control era which saw the development of optimization techniques that were more able to deal with performance and existence issues. Of these, the Linear Quadratic Gaussian technique (LQG) proved to be very popular as it dealt with multivariable plants in an elegant way [6]. Unfortunately, LQG optimal control did not address uncertainty and hence provides no guaranteed stability margins. Not surprisingly, considerable research effort subsequently went into the development of design techniques which were based on optimization principles but which allowed robustness properties to be built into the controller directly. The result of this effort is a comprehensive theory [6, 7] which has its origins in the seminal paper by Zames [8] and which has come to be known as H f control theory. The H f control problem synthesizes a controller which internally stabilizes the feedback interconnection and minimizes the H f -norm from exogenous inputs to regulated outputs. This H f -norm has direct robustness interpretations in terms of the small gain

theorem and also satisfies the performance objectives of minimizing the energy in the output for the worst-case bounded-energy input. There now exist a number of elegant solutions to this problem using a wide variety of mathematical techniques. These range from the early operator-theoretic approaches [7] to the more recent state-space procedures [9] and LMI techniques [10].


6

Point (a) mentioned early in this section can be further classified into two types, namely: (1) parametric uncertainty, and (2) unstructured uncertainty (i.e. unmodeled dynamics). A typical example of the first kind is the variations of steam turbine system parameters (e.g. time constants and gain factors) due to environmental changes. Second example, is the boiler-turbine unit, where its parameters are highly depends on the operating point at which the system is working. Depending on the equilibrium point, their measurement is only possible up to or less large amount of uncertainty. Typical sources of unmodeled dynamics are when certain cross-couplings in MIMO system are neglected; only simplified actuator or sensor models are available, neglected nonlinearities in the modeling, and deliberate reduced-order models. Figure 1.1 summarizes the above. Modern robust control paradigms, such as standard H f problem, H f loop-shaping, and P -synbook, have a lot to offer in that they provide systematic procedures for obtaining sensible controllers that meet performance objectives and guarantee robustness against model uncertainty.

Other controlled signals

Parameter Variations

Disturbance

Reference signals Unmodeled Dynamics

Tracking Errors

System Interconnection

Unmodeled Disturbances Noise

Controller

Figure 1.1: General system Interconnection


7

In a standard H f problem, for example, the designer specifies performance weights to reflect the desired closed-loop performance objectives in different frequency regions and a controller is synthesized to give robust performance guarantees. Similar thing holds for H f loop-shaping and P -synbook. It is clear, however, that although these techniques are very

systematic and give controllers which perform sensibly, fundamental questions such DV ³:KDW LV WKH DFKLHYDEOH SHUIRUPDQFH"´ DUH not addressed since the success of these paradigms hinges strongly on the designer being able to specify performance weights or loop-shaping weights which meet the specifications. The design of these weights is non-trivial and may be timeconsuming for complex plants.

1.4 Robust Control: Polynomial and State-Space Approaches Both polynomial and state-space modeling methods are considered in this book, since both have their own merits in particular industrial applications. The polynomial systems approach to frequency domain modeling and design is preferred, since this is more appropriate for numerical algorithm development than the state-space based methods [11]. The advantages of the polynomial approach to optimal control have many advantages [11]: 1. Engineers often have a better feel for the transfer function models of systems than for state-space representations. 2. Noise and disturbances are easier to characterize in the frequency domain. 3. If the plant is identified from system measurements, it will normally be obtained in transfer function form. 4. Stability and robustness properties of linear-time invariant systems are much easier to determine in the frequency domain.


8

5. Self-tuning and adaptive system often involve a straightforward extension of basic polynomial control design results. 6. Manipulation of total systems is often simpler in the frequency domain (for example forming cascade systems). The state-space approach has become particularly dominant in particular industries such as the aerospace industry. Some of the advantages are as follows [11]: 1. There is a common belief that state-space methods have a significant advantage for calculations involving large systems. 2. Physical models of systems are normally obtained in terms of nonlinear or ordinary differential equations, and linear state-space models can therefore be linked to the underlying physical quantities involved. 3. The ability to estimate state variables using a state-space based Kalman filter is often valuable for control, monitoring and diagnostic change. 4. There are more commercially available software packages for control synbook and design, using state-equation models than for frequency domain approach. 5. Some control design methods naturally involve a state-space equation structure. 6. State-space models often provide a convenient mechanism for scheduling the controller as the nonlinear system operating points vary.

1.5 Literature Survey

9

1.5 Literature Survey Many investigations in the area of LFC problem of interconnected power systems have been reported in the past five decades. The conventional control systems have been successful to some extent. One of the conventional control strategies for the load frequency control is the PI or the most common PID. Many speed governors have been designed based on PID techniques with different philosophies because of its simplicity and ease of implementation [12, 13, 14, 15]. The incapability of the PID to cope with time variations of the system parameters and the poor performance that it introduces especially in the existence of nonlinearities that is already inherent in the system stimulated many researchers to design a more reliable and intelligent controllers than the PID like Fuzzy Logic Control (FLC), Neural Networks (NNs), and Genetic Algorithms (GA) based controllers [16, 17, 18, 19, 20]. Fuzzy sliding mode controller for LFC has been designed by Q. P. Ha in [16] to account for the V\VWHP¶VSDUDPHWHUVYDULDWLRQVDQGWKHJRYHUQRUEDFNODVK$QRWKHUGLIILFXOW\ with the PID design is the selection of its gains. Guihua Han et al. [17] presented a nonlinear self-adaptive fuzzy PID controller adopting fuzzy rule and inference to adjust PID parameters online. The self-tuning fuzzy PID controller regulates the parameters of the PID by fuzzy inference online according to different errors and its rate of change. As the operating point of a power system and its parameter changes continuously, a fixed controller may no longer be suitable in all operating conditions.

In order to take parametric uncertainties into account when

designing load frequency controller for power system, [18, 19] designed a new NN-based LFC. Mehran Rashidi et al. in [18] made use of the capability of the MLP neural network based on Temporal Difference (TD) learning of the design. While in [19] Gaesh et al. presented the design of two


10

separate Continually Online Trained (COT) neurocontrollers for excitation and turbine control of a turbogenerator. Genetic Algorithm (GA) is a global search optimization technique. Dulpichet et al. [20] used GA for tuning the control parameters of the Proportional-Integral (PI) control subject to the H f constraints in terms of LMI. Combinations of the approaches mentioned above have been studied by many researchers. A load frequency controller in [21, 22] has been proposed for power system based on fuzzy gain scheduling approach. In this approach, a fuzzy system is used to adaptively decide the integral controller. To reduce the fuzzy rules, the fuzzy system is designed automatically by GA. Modern control techniques have been reported in [23, 24] in which a load frequency controller for power systems has been designed using LQR techniques. During the last ten years, researcher made intensive research regarding the use of H f techniques to solve many crucial problems in the power systems. H. Bevrani et al. [25] addressed a new method to design a PI based LFC controller taking into account the communication time delays. These time delays are considered as model uncertainties and modeled as unstructured multiplicative uncertainty and solved via H 2 H f

mixed

sensitivity approach. In [26] Adirak et al. investigated the design problem or robust load frequency controller using LMI methods for solving the H f control problem. Again the LMI technique has been used to damp the power system oscillations [27]. On the other hand, boiler-turbine is strongly time varying nonlinear system and has been studied extensively by many researchers from many perspectives. Intelligent control has been applied successfully to the boilerturbine unit [28, 29, 30, 31]. Jingfang Wan et al. [28] studied a scheme of


11

coordinated control system using PID neural network to control boiler-turbine system through the use of feedforward MLP neural network. Boiler-turbine controller design based on Adaptive Neuro-Fuzzy Inference System (ANFIS) has been studied in [29, 30]. It is based on designing a set of controllers using LQR to generate the training data for the ANFIS. Un-Chul Moon et al. in [31] presented an application of an Online SelfOrganizing Fuzzy Logic Controller (OSOFLC) to the boiler-turbine unit without a need in making control rules. Instead, control rules are generated using the history of input-output data. Rober Dimeo et al. in [32] proposed an approach to design a genetic algorithm (GA) based PI and LQR controllers for boiler-turbine system where GA is used to train both the PI gains and the weight of the state-feedback matrix of the LQR. Nonlinear control scheme has been proposed by Yu Daren et al. [33] which tried to design a nonlinear coordinated MIMO controller to nonlinear boiler-turbine system by applying feedback linearization technique to the nonlinear control of steam pressure and power output of a boiler-turbine generating unit. H f design has been exploited in the control design of boiler-turbine

system. Since the H f design can handle the uncertainties, time variations, and disturbances and noises anywhere in the system, robust controllers have been designed and tested [34, 35, 36, 37, 38]. H f loop-shaping controller has been analyzed and synthesized by [34, 36, 38] and a proposed method is introduced to convert it into PID with the exception that in [38] Wen Tan et al. used a gap metric as a distance measure. Chang-Sun Hwang et al. in [35] designed a TDOF ( H 2 H f ) controller for the boiler-turbine unit. Dong Wan Kim et al. [37] presented a method for H f control design based on GA. GA is used in the determination of the weighting functions that reflect the desired performance specifications.

1.6 Book Objectives

12

1.6 Book Objectives The aims of this book are summarized in the following: 1. Study and analysis of the uncertain behavior of the steam turbine system by identifying the uncertain parameters in the plant which have a GLUHFW HIIHFW RQ WKH V\VWHP¶V SHUIRUPDQFH DQG representing these parameters together with the neglected dynamics of the plant by a single complex perturbation from which the uncertainty weight will be derived. 2. Synthesizing a suboptimal H f robust controller for the steam turbine system using a state-space approach and testing the robust stability and performance of the system with the existence of the quantified uncertainty. 3. Designing an optimal H f robust controller for the steam turbine system using a polynomial approach and evaluating the system stability and performance under the same considered uncertainty but with additive representation. 4. Evaluating the state-space and the polynomial approaches by making a comparative study between them in time and frequency domains. 5. Designing a suboptimal multivariable H f robust controller for boilerturbine system based on the linearized model using both approaches and testing the designed robust controller on the linearized system.

1.7 Book Organization Objectives

13

1.7 Book Organization This book consists of six chapters. In chapter one an introduction to the research project, the research motivation, and scope of the research are given. In chapter two, material concerning robust control using both statespace and polynomial approaches is given to facilitate understanding. Firstly, state-space analysis and design is introduced followed by polynomial approach. The weighted mixed sensitivity optimization is discussed and a formula for the central controller is presented. A concise description about controller reduction is included at the end of this chapter. Chapter three describes the mathematical model for both steam turbine and boiler-turbine system. Time domain response of a closed-loop system with the conventional speed governor is depicted. Also, time domain response of the open-loop transfer function of the boiler-turbine system is illustrated. In chapter four the necessary robustness analysis and tools are introduced to clarify the design procedure. Methods of uncertainty representation and high frequency roll-off are explained. Problem formulation for steam turbine and boiler systems has been introduced using both approaches. In chapter five the design specifications for each system are listed. Two case studies are investigated for steam turbine and boiler-turbine systems and solved using both approaches. The simulations and results for steam turbine system are considered here. A comparative study between state-space and polynomial approaches is presented. Also, this chapter contains the simulations results concerning boiler-turbine control design for the linearized model. The conclusions and potential directions for future research are included in chapter six.

CHAPTER THREE

CHAPTER TWO ROBUST CONTROL AND H-INFINITY ROBUST COTECHNIQUES NTROL AND H-INFINITY TECHNIQUES

2.1 Introduction The last two decades witnessed an intensive research in robust control designs. The term robust stressed the importance of developing methods which maintained closed-loop stability and performance not only for the nominal model of the controlled plant but also for a set of plants including the invariable discrepancy between the nominal model and the true plant. The H f optimization approach, being developed and still an active research area, has been shown to be an effective and efficient robust design method for both linear and nonlinear control systems. It solves in general the robust

stabilization

problems

and

nominal

performance

designs.

Consequently a definition of robust control could be stated as: Design a controller such that some level of performance of the controlled system is guaranteed irrespective of changes in the plant dynamics within a predefined class [39, 40]. This chapter describes the robust controller design within the context of two methodologies: the state-space approach and the polynomial approach. Starting with the state-space approach, the formulation of the robust design

14

2.2 Singular Values and H-Norm

15

problem into such a minimization problem and how to find the solution will be discussed. In the second part of this chapter the robust design and its solution using the polynomial approach will be presented.

2.2 Singular Values and H-Norm The singular values of a complex matrix G C mun , denoted V i (G ) , are the k largest nonnegative square roots of the eignvalues of G G where k

min{n, m} [39, 40]. Thus,

Oi (G G)

V i ( s)

i 1,2,......,k

(2.1)

It is usually assumed that the singular values are ordered such that V i t V i 1 . Thus,

V

V 1 (G )

(2.2)

V

V k (G)

(2.3)

For transfer function matrix G(s) the V for SISO systems V

s

are function of frequency and

V { Bode plot of G(s) . Consider the system in figure

2.1, with transfer function G(s) and impulse response g (t ) . To evaluate the system performance, the relevant system norm should be evaluated. w

G(s)

z

Figure 2.1: System G(s).

For a scalar stable transfer function G(s) , the H f -norm is simply the peak value of G ( jZ ) as a function of frequency (i.e. maximum of Bode plot) [6, 7, 39, 40, 41]: G(s)

f

sup G ( jZ )

(2.4)

Ȧ

Where sup is the supremum or the least upper bound, it is the same as max value and it is used here instead of max because the maximum may only be approached as Z o f and may therefore not actually be achieved. The H stands for Hardy Space which is a functional space containing all analytic and


16

bounded transfer functions in the RHP and the symbol f implies that it is designed to accomplish minmax restrictions in the frequency domain. For a linear stable transfer function matrix G(s) , the H f -norm is defined as follows [41, 42]: G (s)

f

sup V G ( jZ )

(2.5)

Z

The computation of the H f -norm is complicated and requires a search in the frequency domain. The H f -norm can be obtained graphically. To get an estimate, set up a fine grid of frequency points {Z1 ,......,Z n } , then an estimate for G (s )

f

is [43]: max V G ( jZk )

1dk d n

This value is usually read directly from a bode diagram of the singular value plot. The H f -norm can be computed in state-space more efficiently [9].

2.3 Performance Specifications and Limitations The shaping of a multivariable transfer function is based on the idea that a satisfactory definition of gain (range of gain) for matrix transfer function is given by the singular values of the transfer function [41]. By multivariable transfer function shaping, therefore it is meant that the shaping of singular values of appropriately specified transfer functions such as the loop transfer function or possibly one or more closed-loop transfer functions [40, 41]. To see how this can be done consider the one-degree-of-freedom configuration shown in figure 2.2. d r

¦

K(s)

u

G(s)

+

+

c

¦

+

¦ +

n

Figure 2.2: One-degree-of-freedom feedback configuration.


17

Define the following three transfer functions: S

c d

1 1 L

R

u r

K 1 L

KS

I T

c r

T S

(2.6)

(2.7)

I S

L (1 L)

(2.8)

Where L GK . In terms of sensitivity S and the complementary sensitivity T , the following relationships can be derived [43, 44]: c(s) T (s)r (s) S (s)d (s) T (s)n(s)

u ( s)

K (s)S (s)>r (s) n(s) d (s)@

(2.9) (2.10)

From the sensitivity S ( I L) 1 , it can be seen that V (S ) | 1 / V ( L) at frequencies where V (L) is much larger than 1. Furthermore, from T

L( I L) 1 it follows that V (T ) | V ( L) at frequencies where V (L) is small.

The above two relationships (2.9)-(2.10) determine several closed-loop and open-loop objectives, in addition to the requirement that K (s) stabilizes G(s) , namely [40, 41, 43, 44]: 1. For distubance rejection make 1 V (S ) t WP or V (GK ) large; valid for freuqnecies at which V (GK ) !! 1 , where WP is the performance bound. 2. For noise attenuation make V (T ) small or V (GK ) small; valid for frequencies at which V (GK ) small. 3. For input usage (control ) energy reduction make V (KS ) or V (K ) small; valid for frequencies at which V (GK ) 1 . 4. For refercene tracking make V (T ) | V (T ) small or V (GK ) large. 5. For robust stability in the presence of an additive perturbation Gp

G ǻ make V (KS ) or V (K ) small; valid for frequencies at which

V (GK ) 1 .


18

6. For robust stability in the presence of a multiplicative perturbation Gp

G( I ǻ)

make V (T ) d WI1

or V (GK )

small; valid for

frequencies at which V (GK ) 1 , where WI1 is the robustness bound. The closed loop requirements 1 to 6 cannot all be satisfied simultaneously. Feedback design is therefore a tradeoff over frequency of conflicting objectives. This is not always as difficult as it sounds because the frequency ranges over which the objectives are important can be quite different [41, 42, 43]. Typically, the open loop requirements 1 and 4 are valid and important at low frequencies, 0 d Z d Zl d Z B , where Z B is the bandwidth of the system defined on page 20. While 2, 3, 5 and 6 are conditions which are valid and important at high frequencies, Z B d Z h d Z d f [41, 43, 44]. These are illustrated in figure 2.3.

Magnitude

V (L)

WP

V (T )

Zh

Performance Bound

Zl

Z 1 V (S )

Robustness Bound

W I1

V (L)

Figure 2.3: Design Trade-offs for the multivariable loop transfer function L= GK.

2.4 Weighted Performance and Selection of Weighting Filters

19

2.4 Weighted Performance and Selection of Weighting Filters The performance objectives of the feedback system can usually be specified in terms of requirements on the sensitivity S and /or complementary sensitivity T functions or in terms of some other closed loop transfer functions like control sensitivity function R as defined in (2.6)-(2.8). Sensitivity S is a very good indicator of closed loop performance both for SISO and MIMO systems. The disturbance is typically a low-frequency signal, and therefore it will be successfully rejected if the maximum singular value V of the sensitivity function S is made small over the same low frequencies. To do this one could select a weighting filter WP (s) that has lowpass filter characteristics with bandwidth equal to the bandwidth of the disturbance. The optimal robust control problem is then solved by finding a stabilizing controller that minimizes W P S f . The maximum singular value of complementary sensitivity T needs to be minimized as large as possible at high frequencies to attenuate the noise and to account for the unstructured uncertainties that appear at that frequency range. To achieve this goal a weighting filter W I (s ) which has a high magnitude at high frequencies and low at low frequencies is chosen such that WI T

f

is minimized. Finally, the control sensitivity R should be kept at low

values to limit the magnitude of the control signal u to prevent saturation of the actuators. To get this done, the control sensitivity R has to be reshaped with a weighting filters such that WU R

f

is minimized in the specified

frequency range. The minimization process to the closed-loop transfer functions is achieved at different frequency regions especially for S and T to keep the equation S T

I valid over the entire frequency range. This can be done via

the selection of corresponding weighting filters with different frequency characteristics as explained above.


20

From the above discussion the requirements for the performance becomes: WP S

f

d 1 V ( S ( jZ )) d V (WP1 ( jZ ))

WI T

f

d 1 V (T ( jZ )) d V (WI1 ( jZ ))

WU R

f

(2.11)

d 1 V ( R( jZ )) d V (WU1 ( jZ ))

The selection of the weighting filters is not an easy task for a specific design problem and often involves ad hoc, and fine tuning. It is very hard to give a general formula for the weighting filters that will work in every case. Nevertheless, general guidelines for the selection of weighting filters will be introduced here that can help in choosing an initial weights [40, 43]. Suppose that a typical specifications in terms of S are given such as the minimum bandwidth frequency is Z B (defined as the frequency where S ( jZ ) crosses 0.707 from below), maximum tracking error as H , and maximum peak magnitude of S to be M P , i.e. S ( jZ ) f d M P . Mathematically, these time domain specifications may be captured in the frequency domain by an upper bound 1 / WP on the magnitude of S , where W P is weighting filter to be selected such that the above specifications are satisfied. An asymptotic plot of a typical upper bound 1 / WP is shown in figure 2.4. The filter illustrated may be represented by [40, 41, 42, 43, 45]:

1 / WP

MP

ZB 1

S ( jZ )

Z w

H

Figure 2.4: Practical performance weight W P and desired S .


WP ( s)

21

s / M P ZB s H ZB

(2.12)

As can be seen from figure 2.4, that 1 / WP is equal to H (typically H | 0 ) at low frequencies, is equal to M P at high frequencies, and the

asymptote crosses 1 at

the frequency Z B , which is approximately the

bandwidth requirement. The selection of the control weighting filter WU follows similarly from the preceding discussion by considering the control signal equation (2.10). The magnitude of KS in the low frequency range is essentially limited by the allowable cost of control effort and the saturation limit of the actuators; hence, in general, the maximum gain M U of KS can be fairly large while the high frequency gain is essentially limited by the controller bandwidth Z BC and the sensor noise frequencies. Ideally, it is preferable to roll off as fast as possible beyond control bandwidth so that the high frequency noises are attenuated as much as possible. An asymptotic plot for 1 / WU similar to that of 1 / WP is shown in figure 2.5. The weighting filter WU (s) can be expressed as [40, 43, 45]:

WU ( s)

s Z BC / M U H s Z BC

(2.13)

1 / WU

MU | KS ( jZ ) |

1

Z BC

Z w

H

Figure 2.5: Practical performance weight WU and desired SK.

2.5 H Control Problem Formulation: A State-Space Approach

22

2.5 H Control Problem Formulation: A State-Space Approach There are many ways in which feedback problems can be casted as H f optimization problem. It is very useful therefore to have a standard problem formulation into which any particular problem may be manipulated. Such a general formulation is afforded by the general configuration shown in figure 2.6. w

z

P u

y

K Figure 2.6: General control configuration.

$OVRWKHV\VWHPLQWKHILJXUHDERYHLVFDOOHG³îEORFNSUREOHP´DQG is described by: ª wº P( s ) « » ¬u ¼

ªzº « y» ¬ ¼

u

ª P11 ( s) P12 ( s) º ª wº « P ( s) P ( s)» « u » ¬ 21 ¼¬ ¼ 22

K ( s) y

(2.14) (2.15)

With state-space realization of the generalized plant P given by: P( s )

ǻ

ªA «C « 1 ¬«C2

B1 D11 D21

B2 º D12 »» D22 ¼»

(2.16)

The signals are: u the control variables, y the measured variables, w the exogenous signals such as disturbances d and reference commands r , and z the so-FDOOHG³HUURU´VLJQDOVZKLFKDUHWREHPLQLPL]HGLQVRPHVHQVH to meet the control objectives. The closed-loop transfer function from w to z is given by the Linear Fractional Transformation (LFT) [41, 42, 43, 45]: z

Tzw w

Fl ( P, K ) w

(2.17)

Where Tzw

Fl ( P, K )

P11 P12 K ( I - P22 K ) -1 P21

(2.18)


23

H f control involve the minimization of the H f norm of Fl ( P, K )

which will be investigated next. Firstly, some remarks about the algorithms used to solve such problems have to be clarified. The most general, widely available and widely used algorithms for H f control problems are based on the state-space solutions in [9, 40, 43, 45]. The H f solution based on statespace requires solving two Riccati equations and give controllers K (s) of state dimension equal to that of the generalized plant P(s) . The following assumptions are typically made in the H f problems [40, 41, 43, 45]: 1. ( A, B1 ) is controllable and (C1 , A) is observable. 2. ( A, B2 ) is stabilizable and (C 2 , A) is detectible . 3. D11

0 and D22

4. D12

ª0 º « I » and D21 ¬ ¼

5. D12T C1

0.

>0

T 0 and B1 D21

The assumption D22

I @.

0. 0 simplifies the formula in H f algorithms and is

made without loss of generality, since a substitution K D

K ( I D22 K ) 1

gives the H f controller K (s) when D22 z 0 [41, 43, 45]. Lastly, it should be said that the H f algorithms, in general, find a suboptimal controller. That is, for a specified J a stabilizing controller is found for which Fl ( P, K )

f

J . If

an optimal controller is required then the algorithm can be used iteratively, reducing J until the minimum is reached within a given tolerance.

2.5.1 H Suboptimal Controller Design Procedure With reference to the general control configuration of figure 2.6, the standard H f suboptimal control problem is to find all stabilizing controllers K (s) which minimize:

Tzw

f

Fl ( P, K ) f

max V ( Fl ( P, K )( jw)) Z

(2.19)


24

The H f -norm has several interpretations in terms of performance. One is that it minimizes the peak of the maximum singular value of Fl ( P( jZ ), K ( jZ )) . In practice, it is often computationally (and theoretically)

simpler to design a suboptimal one (i.e. one close to the optimal ones in the sense of the H f norm). Let J min be the minimum value of Fl ( P, K ) f over all stabilizing controllers K (s) . Then the H f suboptimal control problem is given a J ! J min , find all stabilizing controllers K (s) such that: Tzw

f

J

(2.20)

This can be solved efficiently using the algorithm of [9] and by reducing J iteratively, an optimal solution is approached. The algorithm is summarized below with all the simplifying assumptions. For the general control configuration of figure 2.6 described by (2.14)(2.16), with assumptions (1) to (5) in the previous section, there exist a stabilizing controller K (s) such that Fl ( P, K ) f J if and only if [43, 45, 46]: (i) X f t 0 is a solution to the following algebraic Riccati equation: AT X f X f A C1T C1 X f (J 2 B1 B1T B2 B2T ) X f

0

(2.21)

Such that Re(Oi [ A (J 2 B1 B1T B2 B2T ) X f ]) 0, i ; and (ii) Yf t 0 is a solution to the following algebraic Riccati equation: AYf Yf AT B1 B1T Yf (J 2 C1T C1 C2T C2 )Yf

0

(2.22)

Such that Re(Oi [ A Yf (J 2 C1T C1 C2T C2 )]) 0, i ; and (iii) U ( X fYf ) J 2 Moreover, when these conditions hold, one such controller K (s) is given with the following state-space representation [40, 42, 44, 46]: ªA K sub ( s ) ǻ « f ¬ Ff

Z f Lf º 0 »¼

(2.23)

Af

A J 2 B1 B1T X f B2 Ff Z f Lf C2

Ff

B2T X f , Lf

Yf C2T , Z f

( I J 2Yf X f ) 1

2.5 Mixed Sensitivity H Control

25

2.5.2 Mixed Sensitivity H Control Mixed sensitivity is the name given to transfer function shaping problems in which the sensitivity function S ( I GK ) 1 is shaped along with one or more other closed-loop transfer functions

such as KS or the

complementary sensitivity function T I S . Consider for example a regulation problem in which it is required to reject the disturbance d entering at the plant output. Assume that the noise is negligible and the peak of the control signal u needs to be limited to avoid saturation of the actuators. Therefore, for this problem it is necessary minimize S to reject the disturbance at the output of the system since the transfer function from d to the output c (figure 2.2) is S and minimizing KS to limit the control energy used since KS is the transfer function from r to u . Augmented Plant P(s) w=d

G

+

¦

+

-

z1

WU

z2

¦ +

u

WP

r=0

y

K Figure 2.7: S/KS mixed sensitivity optimization

Including KS in the mixed sensitivity minimization is also important for robust stability against unstructured uncertainty modeled as additive perturbation [41, 42, 43, 45, 46]. It is not difficult from figure 2.7 to show that z1 WP S w and z2

WU KS w as required and to determine the elements of the generalized

plant P as: P11

ªWP º « 0 » , ¬ ¼

P12

P21

I ,

P22

ªWP G º « W » ¬ U¼ G

(2.24)

2.6 H Control Problem Formulation: A Polynomial Approach

26

Where the partitioning is such that: ª z1 º «z » « 2» ¬« y ¼»

ª P11 ( s) P12 ( s) º ª wº « P ( s) P ( s)» « u » 22 ¬ 21 ¼¬ ¼

(2.25)

and Tzw

Fl ( P, K )

ª WP S º «W KS » ¬ U ¼

(2.26)

2.6 H Control Problem Formulation: A Polynomial Approach The importance of the famous standard H f optimal regulator problem as a tool for practical control system design is widely accepted and wellunderstood. This book presents frequency domain solution using the polynomial approach of the standard H f problem that is based on polynomial matrix techniques. Various solutions of the standard problem exist. The most important is the so-FDOOHG ³WZR 5LFFDWL HTXDWLRQ´ VROXWLRQ >9]. It relies on a state space representation of the problem and requires the solution of two indefinite algebraic Riccati equations. Implementations on various computational platforms ([44, 47] document two instances) are available. There are various reasons to consider the polynomial solution as an alternative to the state space solution [48, 49]: 1. The state-space formulation implicitly requires the generalized plant P(s) to have a proper transfer matrix. Practical design methods based

on H f typically involve improper frequency dependent weighting filters; however, resulting in an improper generalized plant P(s) . 2. The state-space solution does not appear to be a very good tool for studying optimal rather than the usual suboptimal solutions. As a matter of fact very little has been published on optimal solutions [48].


27

3. The state-space solution involves a number of technical assumptions on the generalized plant P(s) that impose rank conditions, including certain rank conditions on the imaginary axis. These conditions are irrelevant for control system design, and limit the applicability of the standard problem. The polynomial approach presented in this book bypasses the properness and rank condition problems, and provides access to the study of optimal solutions. Moreover, it makes very clear that the solution of the H f problem no matter which approach is used is based on spectral factorization discussed later. Consider figure 2.8 which defines the standard H f problem whose generalized plant has the transfer function matrix given as: w

z2 W2 r

y

¦

K

u

V +

G

+

¦

W1

z1

-

Figure 2.8: Mixed Sensitivity Configuration.

ª z1 º «z » « 2» ¬« y ¼»

ª P11 «P ¬ 21

P12 º ª wº P22 »¼ «¬ u »¼

ªW1V « 0 « ¬« V

W1G º ª wº W2 »» « » u G ¼» ¬ ¼

(2.27)

The mixed sensitivity problem is the problem of minimizing the H f norm of the following transfer function matrix: Tzw

ª W1SV º « W RV » ¬ 2 ¼

(2.28)

With suitably chosen weighting filters matrices W1 , W2 and V a suitably chosen shaping matrix. In the SISO case the H f -norm of Tzw is given by: Tzw

f

2

sup W1 ( jZ )S ( jZ )V ( jZ ) W2 ( jZ ) R( jZ )V ( jZ )

fZ f

2

(2.29)


28

The closed-loop transfer function matrix from w to z

>z1

z 2 @ is T

precisely the function Tzw whose norm has to be minimized. There are many ways to solve this problem but only the polynomial solution allows the generalized plant P(s) to have improper transfer matrix [50].

2.6.1 Frequency Response Shaping The mixed sensitivity problem may be used for simultaneously shaping the sensitivity S and control sensitivity R functions. The reason is that the solution of the mixed sensitivity problem often has the equalizing property. This property implies that the frequency dependent function [49], 2

W1 ( jZ ) S ( jZ )V ( jZ ) W2 ( jZ ) R( jZ )V ( jZ )

2

(2.30)

whose peak value is minimized, is actually constant [51]. If the constant is denoted as J 2 , with J nonnegative, then it immediately follows from, 2

W1 ( jZ ) S ( jZ )V ( jZ ) W2 ( jZ ) R( jZ )V ( jZ )

2

J2

(2.31)

That for the optimal solution, 2

W1 ( jZ ) S ( jZ )V ( jZ ) d J 2

Z R

(2.32)

Z R

(2.33)

,

Z R

(2.34)

,

Z R

(2.35)

2

W2 ( jZ ) R( jZ )V ( jZ ) d J 2

Hence, S ( jZ ) d R ( jZ ) d

J W1 ( jZ )V ( jZ )

J W2 ( jZ )V ( jZ )

By choosing the functions W1 , W2 , and V correctly the functions S and R may be made small in appropriate frequency regions. This is also true if the

optimal solution does not have the equalizing property. If the weighting functions are suitably chosen (in particular, with W1V large at low frequencies and W2V large at high frequencies), then often the solution of the mixed sensitivity problem has the property that the first term


29

of the criterion dominates at low frequencies and the second at high frequencies, that is [52]: W1 ( jZ ) S ( jZ )V ( jZ )

2

dominates at low frequencies

W2 ( jZ ) R( jZ )V ( jZ )

2

J2

(2.36)

dominates at high frequencies

As a result, S ( jZ ) |

R ( jZ ) |

J W1 ( jZ )V ( jZ )

J W2 ( jZ )V ( jZ )

,

for Z small

(2.37)

,

for Z high

(2.38)

This result allows quite effective control over the shape of the sensitivity S and control sensitivity functions, and, hence, over the performance of the feedback system. Because the H f -norm involves the supremum, the frequency response shaping based on minimization of the f -norm is more direct than for the H 2 optimization methods [49, 51, 52].

2.6.2 Selection of the Shaping Filters The weighting filters are very important design tools in the H f optimization as they reflect the design requirements decided by the designer. So, carefully chosen weighting (shaping) filters results in an H f robust controller that excellently fits the design specifications. In the polynomial solution the weighting filters can be made improper or they may have an imaginary axis pole (free integrator) without violating the design procedure in contrast to state-space design techniques which can be applied to a problem with only proper weighting filters. Furthermore, statespace approach does not allow the weighting filters to have a pole on the imaginary axis since it violates the very first assumption for the H f control theory (assumptions 1 & 2 on page 23) [49, 50, 52]. The weighting filter W1 has the following general form [49, 52]:


W1 ( s)

30

sB s

(2.39)

Where B is the frequency at which the effect of the integrating action is extended. Choosing W1 in this form is necessary for providing an integrating action (see section 4.8). While the general form for W2 is given by [49, 52]: W2 ( s)

h(1 rs)

(2.40)

Where h and r are nonnegative constants to be selected carefully such that h z 0 . Then for high frequencies the magnitude of W2 asymptotically behaves as h if r

0 , and as ( hr Z ) if r z 0 . Hence, if r

0 this results in

a proper but not strictly proper controller transfer function K (s) . While making r z 0 results in a strictly proper H f controller K (s) . Finally, choosing W2 in this form is important to make the H f controller has high frequency roll-off as will be clear in chapter four.

2.6.3 Partial Pole Placement There is a further important property of the solution of the mixed sensitivity problem that needs to be discussed before considering the polynomial H f design procedure. This involves a pole cancellation phenomenon. The equalizing property of section 2.7.1 implies that [49]: 2

W1 ( s)W1 ( s) S ( s) S (s)V ( s)V ( s) W2 ( s)W2 ( s) R( s) R( s)V ( s)V (s) 2

J 2 (2.41)

for all s in the complex plane. The transfer function of the plant G and the weighting filters W1 , W2 and V can be written in polynomial (rational ) form as [49, 50, 52]: G

N , D

W1

A1 , W2 B1

A2 B2

, V

M E

(2.42)


31

with all the numerators and the denominators are polynomials. If also the controller transfer function K (s) is represented in polynomial form as: K

Y X

(2.43)

Then it easily follows that: S

DX , R DX NY

DY DX NY

(2.44)

The denominator, Dcl

DX NY

(2.45)

is the closed loop characteristic polynomial of the feedback system. Substituting S and R as given by (2.44) into (2.41) to get easily [49, 52], D D M M ( A1 A1 B2 B2 X X A2 A2 B1 B1 Y Y ) E E A1 A1 B2 B2 Dcl Dcl

J2

(2.46)

If A is any rational or polynomial function then A is defined by A (s) AT (s) . Since the right-hand side of (2.46) is constant, all polynomial

factors in the numerator of the rational function on the left cancel against corresponding factors in the denominator. In particular, the factor D D cancels [49]. By choosing the denominator polynomial E of V equal to the plant denominator polynomial D , so that V

M / D , with this special choice of the

denominator of V , the polynomial E cancels against D in the left-hand side of (2.46), so that the open-loop poles of D do not reappear as closed-loop poles [52]. All this means that by letting: V

M D

(2.47)

where the polynomial M has the same degree as the denominator polynomial D of the plant G(s), the open-loop poles (the roots of D ) are reassigned to the

2.7 The Polynomial Solution of the Standard H Problem

32

locations of the roots of M . By suitably choosing the remaining weighting filters W1 and W2 these roots may often be arranged to be the dominant poles. This technique, known as partial pole placement [49, 50, 52] allows further control over the design. It is very useful in designing for a specified bandwidth and good time response properties.

2.7 The Polynomial Solution of the Standard H Problem The polynomial solution (frequency domain) version of the continuoustime H f optimal regulator problem will be studied. Figure 2.6 depicts the situation with the generalized plant P given by (2.14) with individual entries (P11,..) given by (2.27) interconnected with the feedback compensator u

Ky .

It is desired to minimize the H f -norm of the closed-loop transfer matrix Tzw : Tzw

Fl ( P, K )

P11 P12 K ( I - P22 K ) -1 P21

(2.48)

With respect to all compensators K (s) that stabilize the closed-loop system. The following considerations will be assumed about P(s) [48, 49]: 1. P(s) is rational but not necessarily proper. P11 has dimensions m1 u k1 , P12 has dimensions m1 u k 2 , P21 has dimensions m2 u k1 , and P22 has

dimensions m2 u k 2 . 2. P12 ( s ) has full normal column rank and P21 has full normal row rank. This implies that k1 t m2 and m1 t k 2 . These assumptions are much less restrictive than those usually found in the state-space based H f -literature [9, 43, 53]. They allow improper weighting matrices in the mixed sensitivity problem, which is important for obtaining high frequency roll-off [52]. Throughout the rest of the book that the generalized plant P(s) has left and right coprime Polynomial Matrix Fraction (PMF) representations [48,49]: P

D 1 N

N D 1

Respectively, with the partitionings

(2.49)


m1 >D1

D

D

m2 D2 @ ,

ª D1 º k1 , « » ¬ D2 ¼ k 2

N

N

33

k1 k 2 >N 1 N 2 @

(2.50)

ª N1 º m1 « » ¬ N 2 ¼ m2

(2.51)

For more information about how to obtain the PMF for a given system with certain state-space matrices Â, B, C , and D` refer to [54]. The PMF representation of the controller K (s) is given by: K

with X

X 1Y

and Y

(2.52)

left coprime stable rational matrices (possibly

polynomial). Recall that the closed-loop transfer matrix is given by (2.48), then the H f optimal problem can be reformulated as: The problem of minimizing the H f -norm as defined by (2.5) of the closed-loop transfer function matrix Tzw defined in (2.48) with respect to all stable rational X and Y such that the closed loop system Tzw is stable. That is [48, 49, 50, 52]: Tzw

Jo

f

(2.53)

Where, Jo

^

inf Tzw

f

| K ( s)

X 1Y is stabilizab le

`

(2.54)

While the H f suboptimal problem can be stated as: find all the stable rational X and Y such that the H f -norm of closed loop transfer function matrix Tzw is less than J where J t J o a nonnegative number. That is [48, 49]: Tzw

f

dJ

(2.55)

The above equation (2.55) is equivalent to either of the following statements [48, 49]: 1. Tzw Tzw d J 2 I on the imaginary axis. 2. TzwTzw d J 2 I on the imaginary axis.


34

Substituting Tzw in (2.48) into one of the above inequalities results in K

X 1Y that achieves Tzw

f

d J for the standard problem with J t J o a

nonnegative number, if and only if,

>X

ªX º Y @ J « » t 0 ¬Y ¼

(2.56)

Where J is the rational matrix [48, 49, 55]: J

ª0 «P ¬ 21

I º ª P11 P11 J 2 I « P22 »¼ ¬ P12 P11

1

P11 P12 º ª 0 » « P12 P12 ¼ ¬ I

P21 º » P22 ¼

(2.57)

And the formula for the J1 may be obtained at the cost of some not inconsiderable algebra as [48, 49, 55]: J1

ª P12 « ¬0

1

P22 º ª I - J - 2 P11 P11 J 2 P11 P21 º ª P12 » »« J 2 P21 P21 ¼ «¬ P22 I ¼ ¬ J 2 P21 P11

0º I »¼

(2.58)

2.7.1 Spectral Factorization (SF) A nonsingular Hermitian1 rational matrix has a spectral factorization if there exists a nonsingular square rational matrix Z such that both Z and its inverse Z 1 have all their poles in the closed left-half complex plane, and as defined below is Hermitian on the imaginary axis [52, 56],

Z JZ

(2.59)

With J a signature matrix2 and Z is called a spectral factor. J is the signature matrix of on the imaginary axis except at the poles and zeros of on the imaginary axis. That is, J

diag ( I , - I ) with the dimension of I

equal to the number of positive eigenvalues and that of I equal to the number of negative eigenvalues. Hence, a necessary condition for the 1

A rational matrix Ȇ is Hermitian if Ȇ* Ȇ. A matrix J is a signature matrix if it is of the form J = diag (I,íI ), with the two unit matrices not necessarily of the same dimension.

2


35

existence of a spectral factorization is that except in finitely many points has constant numbers of positive and negative eigenvalues on the imaginary axis (that is, has constant inertia on the imaginary axis). This condition is also sufficient for the existence of a spectral factorization. The factorization

Z JZ , if it exists, is not at all unique. If Z is a

spectral factor then UZ is also a spectral factor, where U is any rational or constant matrix such that U JU

J . The matrix U is said to be J -unitary.

Sometimes it is appropriate to consider the spectral cofactorization:

ZJZ

(2.60)

With Z again square such that both Z and Z 1 have all their poles in the left-half complex plane, and J a signature matrix. The spectral factorization of (2.60) of a biproper3 Hermitian rational matrix is canonical if the spectral factor Z is biproper.

2.7.2 Polynomial H Controller Design Procedure With the polynomial matrices defined in (2.49)-(2.51) it follows that (2.57)-(2.58) can be expressed in polynomial form as [48, 49]: J

ª D2 º 2

« » N1 N1 J D1 D1 ¬ N2 ¼

J1

ª N 2 º D1 D1 J 2 N1 N1 «

» D 2¼ ¬

> D 1

>N

2

N 2

2

D2 @

1

@

(2.61)

(2.62)

The spectral factorization of the rational matrix J or its inverse may be reduced to two successive polynomial factorizations. Various algorithms for polynomial factorization are described by [56], including one that is based on state space methods.

3

Ȇ LVELSURSHULIERWKȆ DQGLWVLQYHUVHȆ-1 are proper.


36

The computational steps for the H f controller with J t J o are summarized below [48, 49]: 1. First do the polynomial spectral cofactorization: D1 D1 J 2 N1 N1

QJ J c QJ

(2.63)

Such that the square polynomial matrix QJ is Hurwitz. 2. Next, do the left-to-right conversion: QJ1 >N 2

D 2 @ ǻ J ȁ J1

(2.64)

This amounts to the solution of the linear polynomial matrix equation:

>N 2

QJ ǻ J

D 2 @ ȁJ

(2.65)

For the polynomial matrices ǻ J and ȁ J . 3. Do the second polynomial spectral factorization: ǻ J J c ǻ J

ī J J ī J

(2.66)

With ī square and Hurwitz (i.e. all of its eignvalues lie in the LHP). The desired factorization then is J1 MJ

Z J1

M J J M J , with

ī J ȁ J-1

(2.67)

4. All of the suboptimal H f controllers are given by:

>X

Y@

>A

B@ M J

(2.68)

With A a k 2 u k 2 nonsingular square rational and B a k 2 u m2 rational, both with all their poles in the closed left-half plane and such that: AA t BB

on the imaginary axis.

(2.69)

There are many matrices of stable rational functions A and B that satisfy (2.71). An obvious choice is A

I ,B

0 . The resulting H f controller

will be called the central solution. The central solution as defined here is not


37

at all unique, because as seen in section 2.8.1 the spectral factor Z J is not unique. The procedure mentioned above can be iterated with respect to J to obtain optimal solutions as shown in figure 2.9. Optimal solutions are obtained by a binary search procedure where the algorithm will be initiated with J

J max and starts decreasing J as long as the resulting suboptimal

controller is stabilizing controller until J

J min

J o at which the resulting

controller is optimal and stabilizing.

Start &RPSXWHȖo, or guess a lower bound Choose a value of Ȗ!Ȗo

Is SF of ɉȖ Exist

Decrease Ȗ

No

Increase Ȗ

Yes Compute the central compensator K(s)

Is K(s) stabilizing

No

Yes No

Is K(s) close to optimal Yes End

Figure 2.9: Flow chart for Search Procedure.

2.8 Controller Order Reduction

38

2.8 Controller Order Reduction The order of the controllers designed using for instance H f optimization approach or P -method, is higher than, or at least equal, to that of the plant. On the other hand, in the implementation of controllers, high order controllers will lead to high cost, difficult commissioning, poor reliability and potential problems in maintenance. Lower order controllers are always welcomed by practicing control engineers. Hence, how to obtain a low order controller from a high order plant is an important and interesting task, and is the subject of the present section. In control theory, eigenvalues define system stability, whereas Hankel Singular Values (HSV) GHILQH WKH ³energy´ RI HDFK VWDWH LQ WKH V\VWHP Keeping larger energy states of a system preserves most of its characteristics in terms of stability, frequency, and time responses. Model reduction techniques presented here are all based on the Hankel singular values of a system. They can achieve a reduced-order model that preserves the majority of the system characteristics. Mathematically, given a stable state-space system ( A , B , C , and D ), its HSV are defined as: VH

Oi (PQ)

(2.70)

where P and Q are controllability and observability grammians satisfying: AP PAT

BB T

(2.71)

AT Q QA

C T C

(2.72)

As will be discussed in the next chapter, robust control theory quantifies a system uncertainty as either additive or multiplicative types. These controller reduction routines are also categorized into two groups: Additive

and

Multiplicative

Error

types.

In

other

words,

some

3.8 Controller Order Reduction

39

controller reduction routines produce a reduced-order controller K r of the original controller K with a bound on the error K K r f , the peak gain across frequency. Others produce a reduced-order controller with a bound on the relative error K 1 ( K K r ) . These theoretical bounds are based on the f

³WDLOV´RIWKH+SV of the model, i.e. 1. Additive Error Bound:

K Kr

n

f

d 2¦ V H i

(2.73)

k 1

Where V H i are denoted the i-th Hankel singular value of the original controller K . 2. Multiplicative Error Bound: K 1 ( K K r )

n

f

d (1 2V H i ( 1 V H i V H i )) 1 2

(2.74)

k 1

Where V H i are denoted the i-th Hankel singular value of the phase matrix of K .

CHAPTER TWO CHAPTER TH REE MODELING OF STEAM –TURBINE AND MODELING OF STEAM-TURBINE AND BOUILER-TURBINE SYSTEMS BOILER-TURBINE SYSTEMS

3.1 Introduction The primary sources of electrical energy supplied by utilities are the kinetic energy of water and the thermal energy derived from fossil fuels and nuclear fission. The prime movers convert these sources of energy into mechanical energy, that is, in turn converted into electrical energy by synchronous generators. The prime mover governing systems provide a means of controlling power and frequency, a function commonly referred to as Load Frequency Control (LFC) or Automatic Generation Control (AGC). Figure 3.1 portrays the schematic diagram of the load frequency control of synchronous machine. In this chapter the mathematical model of the system will be presented. It consists of two parts; the first part describes fully the steam turbine system and the mathematical model of this system. Time domain response of a closed-loop system with the conventional speed governor is depicted. The second part describes the boiler-turbine system and the linearization process at seven operating points. Also, time domain response of the open loop system is illustrated.

40

3.2 Steam Turbines and Governing Systems

41

3.2 Steam Turbines and Governing Systems A steam turbine converts stored energy of high pressure and high temperature steam into mechanical energy, which is in turn converted into electrical energy by the generator. The heat source for the boiler supplying the steam may be a nuclear reactor or a furnace fired by fossil fuel (coal, oil, or gas) [1, 57]. Steam turbines normally consist of two or more turbine sections or cylinders

coupled

in series. Most units placed in service in recent years

have been of the tandem-compound design. In tandem-compound, the sections are all on one shaft, with a single generator. Typical configuration of tandem-compound steam turbine with single reheat is shown in figure 3.2.

Zs

Set Point

Turbine

Control valve

To Load

G

Load Frequency Control (LFC)

Frequency Feedback

Figure 3.1: Schematic diagram of speed governor of the synchronous machine.

Crossover

Reheater

From boiler CV

IV Steam chest

HP

Shaft

IP

LP

LP

LP

To condenser Figure 3.2: Tandem-compound single reheat steam turbine Configuration.

LP

3.3 Modeling of Steam Turbine and Speed Governing System

42

Fossil fuelled units consist of High Pressure (HP), Intermediate Pressure (IP), and Low Pressure (LP) turbine sections. They may be either reheat or non-reheat type. In reheat type turbine, the steam upon leaving the HP section returns to the boiler, where it is passed through a reheater (RH) before returning to the IP section. Reheating improves efficiency. Large steam turbines for fossil fuelled are equipped with four sets of valves [1, 2, 5, 57]: x Main Inlet Stop Valves (MSVs). x Intercept Valves (IVs). x Reheater Stop Valves (RSVs). x Control Valves (CVs). The MSVs are not normally used for control of speed and load, they are primarily emergency trip valves and they need not be modeled in system studies. The RSVs is either fully open or fully closed and serves the function of shutting off the steam supply to the IP section in the event the unit experiences shut-down such as in an overspeed trip operation. During normal operation both MSV and RSV are fully open. The IV shuts off the steam to the IP turbine in case of loss of load. The CVs or governor valves modulate the steam flow through the turbine during normal operation (i.e. light changes of load). CVs as well as IVs are responsive to overspeed following a sudden loss of electrical load.

3.3 Modeling of Steam Turbine and Speed Governing System A typical mechanical-hydraulic speed governing system consists of a Speed Governor (SG), a Speed Relay (SR), a Hydraulic Servomotor (SM), and Governor-Controller Valves (CVs) which are functionally related as in figure 2.3-a [58, 59].


Speed-control mechanism

Governor Speed changer Position

Speed Relay

Speed Governor Position

Speed Governor

Governor Controller valves (CVs)

Servo Motor

43

Valve Position

Speed

(a) Position limit

Speed Reference Position (SRP)

¦

1 sTSR

¦ -

-

+

-

Zr

SG

1 Valve Position (¨ PV) s

1 TSM

¦

(b)

Figure 3.3: Mechanical-hydraulic speed-governing system for steam turbines, (a) Functional block diagram, (b) Approximate mathematical representation.

Figure 3.3-b shows an approximate linear mathematical model [59]. In steam turbine-generator system, the governing is accomplished by a speed transducer, a comparator, and one or more force-stroke amplifiers. Figure 3.4 depicts the complete system block diagram of a steam turbine generator [2].

Disturbance Load Power Relay Position

Position Error

Speed Reference Position ¦ (SRP)

Speed Relay

Valve Position Servo Motor

Mechanical Power

Steam Turbine

+

Accelerating Power Machine ¦ Dynamics

-

Conventional Governor Speed governor Position

Speed Disturbance Spectrum

Speed governor

Figure 3.4: Block diagram of steam turbine control system.

+ +

¦

Speed


44

It is interesting to mention some considerations that will be adopted in studying the steam turbine model throughout this thesis: 1. Small signal stability is assumed, i.e. frequency excursions are small. 2. Dead band and hysteresis are negligible. 3. The maximum load limit is fixed at PVmax. 4. The thermal generating unit is feeding an isolated load which means that only one generator is considered in the simulation with only one load that is attached to the infinite bus bar. 5. Main steam pressure is assumed to be at rated and an invariant, i.e. the response of the steam generator (boiler is not considered to be important). The transfer function of each individual block is derived as follows:

Speed Governor (SG): The speed governor in the figure above is a speed transducer, the output of which is typically the position (stroke) of a rod that is therefore proportional to speed. This stroke is mechanically compared to a preset reference position to give a position error proportional to the speed error. The signal SRP is obtained from the governor speed changer and is determined by the AGC system. It represents a composite load and speed reference and is assumed constant over the interval of a stability study.

Normally SG is

defined as [58, 60]: SG

1 R

(3.1)

Where R is the steady-state speed regulation. The value of R determines the steady-state speed load characteristic of the generating unit and it can be expressed as: Percent R

percent speed or frequency change u 100 percent power output change

§ Z NL Z FL · ¨¨ ¸¸ u 100 ZO © ¹

(3.2)


45

Where, Z NL = Steady-state speed at no load. Z FL = Steady-state speed at full load. Z O = Nominal or rated speed.

For example, a 5 % droop or regulation means that a 5 % frequency deviation causes 100 % change in valve position or power output [2, 58, 61]. The speed transducer is the heart of the governor system and may be a mechanical, hydraulic, or electrical device. It must measure shaft speed and provide an output signal in an appropriate form (position, pressure, or voltage) for comparison against the reference, and the subsequent amplification of the error.

Speed Relay (SR): The force that controls the position error is small and must be amplified in both force and stoke. This is the purpose of the speed relay (SR). Speed relay is represented by an integrator with time constant TSR and direct feedback as shown in figure 3.3-b. The transfer function of the speed relay is found to be:

S R ( s)

1 sTSR 1 1 sTSR

1 sTSR 1

(3.3)

Servo Motor (SM): A second amplifier that is used to amplify the valve stroke is the servomotor. The servomotor is represented by an integrator and time constant TSM and direct feedback. It moves the valves and is physically large,

particularly on large units as shown in figure 3.3-b. Position limits are also indicated and may correspond to wide-open valves or the setting of a load limiter. The transfer function of the servomotor S M ( s ) is given below.


46

PV max

S M (s )

{

Speed relay output

1

(¨ PV)

sTSM 1 PV min

Figure 3.5: Servomotor equivalent block.

Steam Turbine (ST): Steam enters the HP section through the control valves and the inlet piping. The housing for the control valves is called the steam chest. A substantial amount of steam is stored in the chest. The HP exhaust steam is passed through a reheater. The reheat steam flows into the IP turbine section through the IV. The crossover piping provides a path for the steam from IP section exhaust to the LP inlet. As said earlier, the control valves modulate the steam flow through the turbine for load/frequency control during normal operation. The response of steam flow to a change in control valve opening exhibits time constant TCH due to the charging time of the steam chest and the inlet piping to the HP section. The IP and LP sections generate about 70 % of the total turbine power. The steam flow in the IP and LP sections can change only with the buildup of pressure in the reheater volume. The reheater holds a substantial amount of steam and the time constants TRH associated with it is very long. The steam flow into the LP sections experiences an additional time constant TCO associated with the crossover piping. Figure 3.6 shows the block diagram representation of the tandemcompound reheat turbine [1, 2, 5, 62]. The model accounts for the effects of inlet steam chest, reheater, and crossover piping.


47 FHP

HP pressure

ûPV

1 sTCH 1 Steam chest

¦

-

HP flow

1 sTRH Reheater

1 sTCO 1

FHP

+

+

¦

ûPm

+

Crossover

FHP Figure 3.6: Block diagram representation of tandem compound single reheat steam turbine.

It is important to keep in mind that the parameter values in the block diagram are all in per unit. In this system, CV position is 1.0 pu when fully open and the sum of the power fractions of the various turbine sections ( FHP FIP FLP ) is equal to one [1, 2, 5]. From figure 3.6 the complete transfer function of the turbine S T ( s ) relating perturbed values of the turbine power Pm and control valve position PV may be written as follows: S T ( s)

ûPm ûPV

( FHP ( sTCO 1)( sTRH 1) FIP ( sTCO 1) FLP ) ( sTCH 1)( sTCO 1)( sTRH 1)

(3.4)

Machine Dynamics (MD): When there is a load change, it is reflected instantaneously as a change in the electrical power output Pe of the generator. This causes a mismatch between the mechanical power Pm and the electrical power Pe and the result is an accelerated power Pa which in turn results in speed variations as determined by the swing equation defined in the following block diagram (figure 3.7-a). In general, power system loads are a composite of a variety of electrical devices. For resistive loads, such as lighting and heating loads, the electrical power is independent of frequency. In the case of motor loads, such as fans and pumps, the electrical power changes with frequency due to changes in


48

motor speed. The overall frequency-dependent characteristic of a composite load may be expressed as [1, 2, 63]: ûPL K D ûZ r

ûPe

(3.5)

Where, ûPL = non-frequency-sensitive load change.

K D ûZ r = frequency-sensitive load change. K D = load-damping constant. ûPm +

¦

ûPa

1 sTM

-

ûZ r

ûPm +

¦ -

ûPa -

ûPL

ûPe (a)

1 sTM

ûZ r

KD (b)

Figure 3.7: Transfer function relating speed and power, (a) without load damping, (b) with load damping.

The damping constant is expressed as a percent change in load for one percent change in frequency. Typical values of K D are 1 to 2 percent. A values of K D

2 means that 1 % change in frequency would cause a 2 %

change in load [1, 2, 5, 60, 61, 63]. The system block diagram including the effect of load damping is shown in figure 3.7-b. In this case ûPL is called input load disturbance.

Speed Disturbance Spectrum (SD) Small signal stability may be associated with rotor angle oscillations or a single generator or a single plant against the rest of the power system. Such oscillations are called local plant mode oscillations. Usually the local plant mode oscillations have frequencies f o listed in table 3.1. Then the speed disturbance model WD 3 ( s ) can be found as: WD 3 ( s )

1 s 2Sf o

(3.6)


49

The model of the complete steam turbine system has been derived with typical values of parameters of the model shown in figure 3.4. This model is applicable to a tandem-compound single reheat turbine of fossil-fuelled units are listed in table 3.1 [2, 59].

Table 3.1: Description of the Steam Turbine System Parameters. Parameter

Value

TM

Description Damping factor = torque (pu) / speed (pu) Mechanical starting time

FIP

IP turbine power fraction

0.4

-

FLP

LP turbine power fraction

0.3

-

FHP

HP turbine power fraction

0.3

-

TCO

Crossover time constant

0.4

sec

TSR

Speed relay time constant

0.1

sec

KD

Unit

2

pu

8

sec

TSM

Servomotor time constant

0.2

TCH

Steam chest time constant

0.25

sec sec

TRH

Reheater time constant

7

sec

PV max

Maximum valve position

1

pu

PV min

Minimum valve position

0

pu

Speed disturbance bandwidth

0.5-2

Hz

fo

It is of particular interest to stress that the system parameters ( Pe , Pm , ..etc) in figure 3.4 are all in per unit (pu) form with exception of the time constants which are expressed in seconds [1]. Figure 3.8 shows a time domain response for the output speed deviation ûZr and mechanical power ûPm with the conventional speed governor 1 / R

and in comparison with the PID speed governor due to 0.08 pu load disturbance ûPL . As can be seen from the figure, with the conventional governor the speed deviation ûZr has a non-zero steady state error as compared to PID speed governor in which the speed deviation ûZ r does go.

3.4 Modeling of Boiler-Turbine System

50

Speed deviation #ORNKVWFG¨r(pu)

EQPXGPVKQPCNIQXGTPQT

2+&IQXGTPQT

6KOG UGE

(a)

#ORNKVWFG

Mechanical power ¨Pm(pu)

%QPXGPVKQPCNIQXGTPQT 2+&IQXGTPQT

6KOG UGE

(b) Figure 3.8: Transient response of the closed loop steam turbine system, (a) ûZ r , (b) ûPm due to 0.08 pu load disturbance ûPL

to zero with settling time not less than 30 seconds. The mechanical power ûPm reaches the new load demand in the case of PID speed governor in about

31 seconds while it settles at 0.073 with the conventional governor within 7 seconds.


51

3.4 Modeling of Boiler-Turbine System The expansion of power system interconnection has necessitated more precise control in order to hold the frequency stable and to control disturbances. Thus system components that are usually thought of as quite slow in response must be investigated for possible behavior that might be detrimental to system damping. The steam generator (boiler) is such component [1, 2]. The control of the steam generator and turbine in a power plant are nearly always considered to be a single control system. This is true because the two units, generator and turbine, operate together to provide a given power output and, since limited energy storage is possible in the boilerturbine system, the two subsystems must operate in unison under both steadystate and transient conditions [64].

3.4.1 Boiler-Turbine Control Mode A drum-boiler-turbine unit will be considered in this thesis which employs a large drum as a reservoir for fluid that is at an evaporation temperature. A drum-boiler-turbine system works in the boiler-following control mode as shown in figure 3.9.

Pumping &firing rate control Pressure measurement Boiler

Load demand Turbine control valves Turbine

Figure 3.9: The boiler-following unit control mode.

G


52

In this mode of control, the control function is divided such that the governor responds directly to changes in load demand. The response is an immediate change in generator load due to a change in turbine valve position and the resulting steam flow rate. The boiler “follows” this change and must not only catch up to the new load level, but also must account for the energy borrowed or stored in the boiler at the time the change was initiated. This type of control mode responds quickly, utilizes the stored boiler energy effectively, and is generally stable. It has the disadvantage that the pressure restoration is slow and the control is nonlinear [1, 2, 5]. A simplified sketch of a drum-boiler-turbine system is given in figure 3.10. The principal components modeled are: the furnace, drum, riser, downcomer, superheater [2]. In such a system, the drum serves as a reservoir of thermal energy that can supply limited amounts of steam to satisfy sudden increases in load demand. It also serves as storage reservoir to receive energy following a sudden load rejection. The drum serves as a buffer between the turbinegenerator system and boiler-firing system [2, 64]. The heat supplied to the risers causes boiling. Gravity forces the saturated steam to rise causing a circulation in the riser-drum-downcomer loop. Feed-water is supplied to the drum and saturated steam is taken from the drum to the superheater and the turbine. The steam generated in the waterwalls is separated from the water at the drum. Normal water level is maintained near the centerline of the drum, which is a critical variable for both plant protection and equipment safety [64, 65].


53

Figure 2.10: A drum-type boiler arrangement [2].

A dramatic decrease in this level may uncover boiler tubes, allowing them to become overheated and damaged. An increase in this level may interfere with the process of separating moisture from steam within the drum, thus reducing boiler efficiency and carrying moisture into the process or turbine.


54

3.4.2 Nonlinear Model of the Drum-Boiler-Turbine System In order to apply advanced model-based control concepts, it is necessary to have an adequate mathematical model of the boiler-turbine unit. The boiler-turbine used in this thesis was first derived in 1987 by Bell and Åström and has been adopted by many researchers in their designs [34, 35, 38]. The model is third order, nonlinear MIMO system, time-varying, and with hard constraints and rate limits imposed on the actuators. The model is assumed as a real plant among various nonlinear models for the boiler-turbine system, that is a 160 MW oil-fired drum-type boiler-turbine-generator model. The nonlinear state-space equation of this boiler unit is given as follows [66]: x1

0.0018 u 2 x19 / 8 0.9 u1 0.15 u 3

(3.7)

x 2

(0.073 u 2 0.016) x

0.1x 2

(3.8)

x 3

[141 u 3 (1.1u 2 0.19) x1 ] / 85

(3.9)

y1

x1

(3.10)

y2

x2

(3.11)

y3

0.05(0.13073x3 100 a cs

9/8 1

qc 67.975) 9

(3.12)

Where, a cs

(1 - 0.001538 x3 )(0.8 x1 25.6) x3 (1.0394 0.0012304 x1 )

(3.13)

qc

(0.854 u 2 0.147) x1 45.59 u1 2.514 u 3 2.096

(3.14)

The three state variables x1 , x 2 , and x3 are drum steam pressure (kg / cm2), electric power (MW), and steam water fluid density (kg / m3), respectively. The three outputs y1 , y 2 and y 3 are steam pressure x1 , electric output x 2 and drum water level deviation (in meters), respectively. The drum water level y 3 is calculated using two algebraic equations a cs and q c which are the steam quality (mass ratio) and the evaporation rate (kg/sec), respectively.


55

The control inputs u1 , u 2 , and u 3 are normalized valve positions of fuel flow, produced main steam, and feedwater flow, respectively. Due to actuator limitations, the control inputs are subject to the following constraints [37, 65]: 0 d ui d 1 , i

(3.15)

1,2,3,

0.007 d u1 d 0.007,

2 d u 2 d 0.02,

(3.16)

0.05 d u 3 d 0.05.

3.4.3 Linearization of the Nonlinear Model In most cases of designing boiler-turbine control systems, it is assumed that the exact theoretical model is given; therefore, the linearization of the nonlinear model is necessary to design the linear H f controller. With this assumption, the first step towards the design of boiler-turbine control system is to derive a linearized model for the nonlinear boiler-turbine system dynamics. A linearized model is obtained from a truncated Taylor series expansion of the nonlinear equations. The nonlinear dynamics are of the form: dX (t ) dt Y (t )

Xo

>x

o 1

x 2o

of x

(3.17)

g ( X (t ) ,U (t ))

>x1

Where X linearization

f ( X (t ) ,U (t ))

the

@

o T 3

x3 @ , U T

x2

system and

Uo

(3.18)

>u1 about

>u

o 1

u 3 @ , and Y T

u2

the

u 2o

following linear system state-space matrices:

u

@

o T 3

> y1

y2

y 3 @ .The

nominal

operating

requires

calculating

T

point the


wf wX

A

wg wX

C

56

B ( X o ,U o )

wf wU

( X o ,U o )

(3.19)

wg wU

D ( X o ,U o )

( X o ,U o )

The resulting linearized state-space system is: dX dt

(3.20) C X DU

Y

Where

A X BU

X

X X o ,Y

Y Y o , and U

U U o . The state-space

matrices of the linearized system are:

A

ª 0.002025 u o x o 1 / 8 2 1 « o o1 / 8 «(0.082125 u 2 0.018) x1 « 0.002235 0.012941 u o 2 ¬«

ª0.9 0.0018 x o 9 / 8 1 « 9/8 B «0 0.073 x1o « 0 0.012941 x o 1 «¬

C

0º » 0.1 0 » 0 0» ¼» 0

0.15º » 0 » 0 »» ¼

ª º 1 0 0 « » « » 0 1 0 « wa w a cs » 0.05 w q c cs «(5 ) 0 (0.065 5 )» 9 w x1 w x3 »¼ «¬ w x1

0 ª 0 0 D «« 0 «¬0.2533 0.00474 x1o

º 0 »» 0.014»¼

(3.21)

(3.22)

(3.23)

0

(3.24)

Table 3.2 shows the various operating points of the boiler-turbine system. The state-space matrices of the system at other operating points are included in appendix A.


57

Table 3.2: Typical operating points of the boiler system. Parameter

Operating points #1

#2

#3

#4

x1o

75.6

86.4

97.2

108

x2o

15.27 36.65 50.52 66.65 85.06 105.8 128.9

o 3

#5

#6

#7

118.8 129.6 140.4

x

299.6 342.4 385.2 428

470.8 513.6 556.4

u1o

0.156 0.209 0.271 0.34

0.418 0.505 0.6

u2o

0.483 0.552 0.621 0.69

0.759 0.828 0.897

u3o

0.183 0.256 0.34

0.435 0.543 0.663 0.793

With the assumption that the system is at equilibrium at the third operating point (see Table 3.2), then the constant state-space matrices ( A, B, C , D ) with X o

>97.2

50.52 385.2@ and U o T

>0.271

0.621 0.34@ at T

this point are given as follows:

A

0 0º ª 0.00222 « 0.05847 0.1 0» , « » 0 0»¼ ¬« 0.0058

C

0 0 ª 1 º « 0 », D 1 0 « » ¬«0.00713 0 00.004625¼»

B

0.31 0.15 º ª0.9 «0 12.573 0 » « » ¬« 0 0 1.2578 01.6588¼» 0 0 º ª 0 « 0 0 0 »» « ¬«0.25327 0.4611 0.014¼»

A simple algebraic operating gives the transfer function matrix as follows: Y (s)

>C ( sI A) 1B D @U ( s )

G3 ( s ) U ( s )

(3.25)

0.9 - 0.31 - 0.15 ª º « » (s+0.00222) (s+0.00222) (s+0.00222) « » 12.573 (s+0.0007784) 0.052623 - 0.0087705 G3 (s) « » (s+0.1) (s+0.00222) (s+0.1) (s+0.00222) (s+0.1) (s+0.00222) «0.25327 (s+0.03067) (s- 0.003109) 0.4611 (s+0.0006304) (s- 0.01582) - 0.014 (s+0.003183) (s- 0.4726)» « » s (s+0.00222) s (s+0.00222) s (s+0.00222) ¬ ¼

The input-output relationship of the transfer function matrix G3(s) for boiler-turbine unit is depicted in figure 3.10.

3.4 Modeling of Boiler-Turbine System y1 Drum Pressure y2 Electrical Output

G11 ( s) G12 ( s) G12 ( s) G21 ( s) G22 ( s) G23 ( s) G31 ( s) G32 ( s) G33 ( s)

Fuel Flow Valve u1 Steam Valve

58

u2

Feedwater Flow u3 Valve

y3 Drum Water Level

Figure 3.10: Open-loop transfer function of the boiler-turbine system.

The open-loop response (i.e. y1 , y 2 and y 3 ) of the linearized boilerturbine system with the above transfer function matrix in response to a unit step change in the manipulated variables (i.e. u1 , u 2 , and u 3 ) is shown in figure 3.11. (TQO(WGN(NQY

(TQO5VGCO%QPVTQN

(TQO(GGFYCVGT(NQY

&TWO2TGUUWTG

2QYGT1WVRWV

&TWO9CVGT.GXGN

6KOG UGE

Figure 3.11: Step response of the open-loop linearized transfer function matrix G3(s) of the boiler-turbine system at 3rd operating point.

As can be seen from figure 3.11 the system is open-loop unstable. For a uint step reference command of a fuel flow, the drum water level decreases indinitely. Samething holds for the other channels, the drum water level increases without limit in response to a unit step command in feedwater.

CHAPTER FOUR

UNCERTAINTY MODELING AND

ROBUSTNESS TOOLS

4.1 Introduction A mathematical model of any real system is always just an approximation of the true, physical reality of the system dynamics. In this chapter, a special theory regarding uncertainty subject and how dynamic perturbations are usually described is intentionally added so that they can be well considered in system robustness analysis and design. Also, this chapter presents varieties of tools that are necessary in the design of the H f controller. These tools handle several problems that may appear during the design H f controller, like a plant or weighting filter that has a pole on the imaginary axis or achieving integrating action and high frequency roll-off to the designed controller. Techniques will be discussed in this chapter WRDGGUHVVWKHVHLVVXHV7HVWVIRUFKHFNLQJWKHIHHGEDFNV\VWHP¶VVWDELOLW\DQd performance with and without plant uncertainties will be introduced for SISO and MIMO systems. Finally, incorporating the uncertainty modeling into the design procedure and addressing the above mentioned issues will be considered by presenting the mixed sensitivity configuration for both the steam and Boiler turbine systems.

59

4.2 Model Uncertainty

60 60

4.2 Model Uncertainty A control system is robust if it is insensitive to differences between the actual system and the mathematical model of the system which was used to design the controller [51]. To account for model uncertainty, it will be assumed that the dynamics behavior of a plant is described not by a single linear-timeinvariant model but by a set of possible linear time-invariant models, sometimes denoted as the uncertainty set. The following notation will be adopted in the rest of the book: x - a set of possible perturbed plant models. x Gnom (s) nominal plant model (without uncertainty). x G P (s) particular perturbed plant model. A norm bounded uncertainty description will be used where the set is generated by allowing H f norm-bounded stable perturbations ǻ to the nominal plant G nom (s ) where ǻ denotes a normalized perturbation with H f -norm less than or equal to 1. Uncertainty in the plant model may have several origins [51, 53, 55]: (1) there are always parameters in the linear model which are only known approximately or vary due to changes in the operating point, (2) measurement devices have imperfections, (3) at high frequencies even the structure and the model order are unknown, and the uncertainty will always exceed 100% at some frequency, (4) even when a very detailed model is available one may choose to work with a simpler (low order) nominal model and represent he neglected dynamics as uncertainty. The various sources of model uncertainty mentioned may be grouped into two main classes [51, 55, 56]: 1. Parametric (Real) Uncertainty: here the structure of the model is known, but some of the parameters are uncertain. Parametric uncertainty is quantified by assuming each uncertain parameter is bounded within some region >D min , D max @. That is, the parameter sets of the form [51]:

4.3 Uncertainty Representation

DP

61 61

D (1 rD ǻ

(4.1)

where D is the mean parameter value and rD is the relative uncertainty in the parameter defined below: D

ǻ

D min D max

(4.2)

2 ǻ

rD

(D max D min ) 2

(4.3)

D

While ǻ is any scalar such that ǻ d 1. Parametric uncertainty is sometimes called structured uncertainty as it models the uncertainty in a structured manner.

1. Dynamic (Neglected Dynamics) Uncertainty: here the model is in error because of missing dynamics, usually at high frequencies, either through deliberate neglect or because of a lack of understanding of the physical process. Any model of a real system will contain this source of uncertainty. Consider a set of plants GP ( s )

Go ( s) f ( s)

(4.4)

Where Go (s) is a fixed (and certain) plant. It is required to neglect the term f (s) (which may be fixed or may be an uncertain set f , and represent G P (s) by multiplicative uncertainty (described below) with a nominal model Gnom

Go .

4.3 Uncertainty Representation In many cases, it is preferable to lump the various sources of dynamic uncertainty into one of the two forms [51, 55, 56]: x Multiplicative uncertainty: this often the most preferable uncertainty form and defined as: I :

GP ( s )

Gnom ( s )( I WI ( s )ǻ I ( s )) ;

ǻI

f

d1

(4.5)

Where I is the identity matrix and ' I (s) is any stable transfer function which at each frequency ' I ( jZ ) d 1 and the subscript I in ' I ( jZ) d 1 GHQRWHV³LQSXW´

4.4 Obtaining the Weight for Complex Uncertainty

62 62

EXWIRU6,62V\VWHPVLWGRHVQ¶WPDWWHUZKHWKHUWKHSHUWXUEDWLRQLVFRQVLGHUHG at the input or at the output of the plant while it does matter in MIMO systems. x Additive uncertainty: this kind of uncertainty is represented by the following equation: A :

G P (s)

Gnom ( s ) W A ( s )ǻ A ( s ) ;

ǻA

f

d1

(4.6)

Both representations are shown in figure 4.1.

Figure 4.1: Plant with, (a) Multiplicative, (b) Additive, Uncertainty.

4.4 Obtaining the Weight for Complex Uncertainty In terms of quantifying uncertainty arising from neglected dynamics, the frequency domain approach H f does not seem to have much competition (when compared with other norms). Parametric uncertainty is also often represented by complex perturbation. This has the advantage of simplifying analysis and especially controller synbook. For example, it is simple to replace the real perturbations, 1 ǻ 1 by a complex perturbation ǻ jZ ) d 1. Consider a set

of possible plants resulting, for example, from

parametric uncertainty as in (4.1) or from a neglected dynamics as in (4.4). It is required to describe this set of plants by a single (lumped) complex perturbation, ǻ A or ǻ I . This complex uncertainty description may be generated as follows [51]:

4.5 Poles on the Imaginary Axis

63 63

1. Select a nominal model G nom (s ) . 2. Additive uncertainty: at each frequency find the smallest radius l A (Z ) which includes all the possible plants . That is, l A (Z )

max V GP ( jZ ) Gnom ( jZ )

GP

(4.7)

If a rational transfer function weight, WA (s ) , then it must be chosen to cover the set, so W A ( jZ ) t l A (Z )

Z

(4.8)

Usually W A (s) is of a low order to simplify the controller design. Furthermore, an objective of frequency domain uncertainty is usually to represent uncertainty in a simple straightforward manner. 3. Multiplicative (Relative) Uncertainty: This is often the mostly uncertainty form used, l I (Z ) is expressed as: l I (Z )

§ G ( jZ ) G nom ( jZ ) max V ¨¨ P G nom ( jZ ) ©

GP

· ¸ ¸ ¹

(4.9)

With a rational weight, WI ( jZ ) t l I (Z )

Z

(4.10)

4.5 Poles on the Imaginary Axis One of the drawbacks of the H f design procedure based on a state-space approach is that it does not allow the plant or the weighting filter to have a pole on the jZ -axis. When given real plants having poles on the jZ -D[LVLQ³ 2 u 2 EORFN´ PL[HG VHQVLWLYLW\ SUREOHPV WKH\ EHFRPH ]HURV RI P12 or P21 in generalized plants and the corresponding H f control problems become nonstandard. To get around this problem, the following steps have to be done: 1. Shift the eignvalues of the plant G (s) that have one or more poles on the jZ -axis to the right by a small number H . This will make the generalized plant

free from any pole on jZ -axis. Then, the standard H f techniques based on state-space approach can be applied easily.

4.6 Robustness Analysis

64 64

2. Apply the H f design procedure to get the H f controller. 3. Back shift the eigenvalues of the resulting controller from step (2) to the left by the same number H . There are many methods other than the above mentioned procedure, interested reader is referred to [53, 67]. Plants or weighting filters with poles on jZ -axis presents no problem with the H f design using the polynomial approach since this approach handles this situation implicitly within the design algorithm.

4.6 Robustness Analysis The performance of a nominally-stable uncertain system model will generally degrade for specific values of its uncertain elements. Moreover, the maximum possible degradation increases as the uncertain elements are allowed to further and further deviate from their nominal values. The graph in figure 4.2 shows the typical tradeoff curve between allowable deviation of uncertain elements from their nominal values and the worst-case degradation in system performance [54]. Here, system performance is characterized by system gain (e.g., peak magnitude on Bode plot). Interpreting the system as the relationship mapping disturbances/commands to errors, small system gains are desirable, and large gains are undesirable. The heavy blue line represents the maximum system gain due to uncertainty of various sizes (the horizontal axis) and is called the system performance degradation curve. It is monotonically increasing. Two robustness measures will be defined [54]: x Robust Stability Margin: The robust stability margin, StabMarg, is the size of the smallest deviation from nominal of the uncertain elements that leads to system instability. System instability is equivalent to the system gain becoming arbitrarily large, and hence characterized by the vertical line tangent to the asymptotic behavior of the performance degradation curve.

DĂǆŝŵƵŵ^ǇƐƚĞŵ'ĂŝŶŽǀĞƌhŶĐĞƌƚĂŝŶƚǇ

4.6 Robustness Analysis

65 65

^ǇƐƚĞŵ'ĂŝŶĂƐůĂƌŐĞ ĂƐϭ͘ϳϮŝĨƵŶĐĞƌƚĂŝŶ ĞůĞŵĞŶƚƐĐĂŶĚĞǀŝĂƚĞ ĨƌŽŵƚŚĞŝƌŶŽŵŝŶĂů ǀĂůƵĞƐďǇϭ͘ϱƵŶŝƚƐ͘ hŶĐĞƌƚĂŝŶƚǇůĞǀĞůĂƚ ǁŚŝĐŚƐǇƐƚĞŵĐĂŶ ďĞĐŽŵĞƵŶƐƚĂďůĞ

EŽŵŝŶĂů ^ǇƐƚĞŵ

^ƚĂďDĂƌŐсϭ͘ϵ

ŽƵŶĚŽŶEŽƌŵĂůŝǌĞĚhŶĐĞƌƚĂŝŶƚǇ

Figure 4.2: Maximum System Gain due to varying amounts of uncertainty.

DĂǆŝŵƵŵ^ǇƐƚĞŵ'ĂŝŶŽǀĞƌhŶĐĞƌƚĂŝŶƚǇ

^ǇƐƚĞŵƉĞƌĨŽƌŵĂŶĐĞ ĚĞŐƌĂĚĂƚŝŽŶ ĐƵƌǀĞ ǇсϭͬǆĐƵƌǀĞŝŶ ƵŶĐĞƌƚĂŝŶƚǇƐŝǌĞ WĞƌĨDĂƌŐсϬ͘ϴϴ

ŽƵŶĚŽŶEŽƌŵĂůŝǌĞĚhŶĐĞƌƚĂŝŶƚǇ

Figure 4.3: Robust Performance Margin.

x Robust Performance Margin [54]: The hyperbola is used to define the performance margin. Systems whose performance degradation curve intersects KLJKRQWKHJUHHQOLQHILJXUH UHSUHVHQW³QRQ-robustly-SHUIRUPLQJV\VWHPV´ in that very small deviations of the uncertain elements from their nominal values can result in very large system gains. Conversely, an intersection low on WKHK\SHUERODUHSUHVHQWV³UREXVWO\-performing V\VWHPV´.

66 66

4.7 Requirements for Stability and Performance

The point where the system performance degradation curve crosses the green line in figure 4.3 is used as a scalar measure of the robustness of a system to uncertainty. The horizontal coordinate of the crossing point is the robust performance margin, PerfMarg, see figure 4.3 [54].

4.7 Requirements for Stability and Performance So far it has been discussed how to represent the uncertainty mathematically. In this section conditions which will ensure the system stability and performance for all perturbations in the uncertainty set will be studied in addition to conditions of nominal stability and performance. Consider the uncertain feedback system in figure 4.1-a when there is multiplicative (relative) uncertainty of magnitude W I ( jZ ) . The generalized plant P which has inputs >u ǻ

w u @ and outputs > yǻ

z2

y @. By inspecting

figure 4.1-a (remember to break the loop before and after K ), P is given by:

P

ª P11 «P ¬ 21

P12 º P22 »¼

0 WI º ª 0 «W G W W G » P P » « P «¬ G I G »¼

;ϰ͘ϭϭͿ

Next it is required to derive the closed loop transfer function matrix T zw corresponding to figure 4.4-a. First, partition P to be compatible with K , as indicated in (4.11) and then find Tzw

Fl ( P, K ) using (2.18) and the result is

given by [51]:

Figure 4.4: (a) General control configuration, (b) N¨-structure for robust performance analysis.

67 67

4.7 Requirements for Stability and Performance

ª yǻ º ªuǻ º « z » Tzw « w » , Tzw ¬ ¼ ¬ 1¼

ª WI T «W GS ¬ P

WI KS º WP S »¼

(4.12)

Four conditions will be derived below to verify Robust Stability (RS), Nominal Stability (NS), Nominal Performance (NP), robust performance (RP) of the feedback system [50, 51, 53, 55]: 1. Robust Stability (RS): To determine the stability of the uncertain feedback system when there is multiplicative uncertainty of magnitude W I ( jZ ) the transfer function from u ǻ to y ǻ has to be determined. From entry (1,1) in (4.12) it is found as:

M

WI K (1 GK ) 1 K

WI T

(4.13)

As shown in figure 4.5 which has to be minimized to ensure the feedback system is robustly stable. That is, RS WI T 1 T 1 WI

(4.14)

2. Nominal Stability (NS): The nominal ( ǻ I

0 ) feedback system is nominally stable if T zw is

internally stable, and the system is internally stable if the four transfer functions (the four entries in (4.12)) are S , SG , KS and T are all stable. NS Tzw internally stable S , SG , KS , and T are stable

(4.15)

3. Nominal Performance (NP): To derive a condition for the nominal performance ( ǻ I

0 , the transfer

function from w to z1 in figure 4.4-a has to be determined. From entry (2,2) it is found to be S , the sensitivity function. So the condition for NP is: NP WP S 1 S 1 WP

(4.16)

68 68


4. Robust Performance (RP): For robust performance it is required that the performance condition of (4.16) to be satisfied for all possible plants in , i.e. including the worse-case uncertainty. In symbols this condition can be written as: RP WP S P 1

RP Tzw

S P , Z

(4.17)

(1 G P K ) 1 . In other words,

Where S P f

1 max WP S P WI T 1 , Z Z

(4.18)

Corresponding conditions for the polynomial approach have been derived and included in table 4.1. Also, similar conditions have been derived for the both approaches in the case of additive uncertainty instead of multiplicative one for both SISO and MIMO cases as shown in table 4.1.

4.8 High frequency Roll-off and Integrating action Consider the feedback system shown in figure 2.2. In order to let the output c track the reference signal r as well as excellent low-frequency disturbance attenuation, its required that K has to contain an integrator (i.e. K (s) has a pole at s

0 ). If the plant G KDV³QDWXUDO´LQWHJUDWLQJDFWLRQ that

is, G has a pole at 0, then no special provisions are needed. In the absence of natural integrating action, one may introduce integrating action by letting the product W1V have a pole at 0 for the polynomial approach and by letting W P have a pole at0 for the state-space approach.This forces the sensitivity function S to have a zero at 0, because otherwise (3.30) is unbounded at Z 0 [59]. On the other hand, high frequency roll-off in the compensator transfer function K and the control sensitivity function R may be pre-assigned by suitably choosing the high-frequency behavior of the weighting function W2 or WU . If W2 (s)V (s) or WU (s) is of order s m as s approaches f then K and R have

a high-frequency roll-off of m dB/decade. Normally V (s ) is chosen biproper such that 9 . Hence, to achieve a roll-off of m dB/decade it is necessary that W2(s) or WU(s) behaves as sm for large s [59].


69 69

4.9 Robust Controller Design and Assessment

70 70

4.9 Robust Controller Design and Assessment Robust control based on H f techniques involves many steps before, during, and after the controller design. These steps can be divided into two phases: Design phase and assessment phase. In the design phase, the first thing to do is to obtain the augmented plant. This can be achieved via the selection of the properly designed weighting filters and combining these filters with the closed loop transfer functions of the original plant. Secondly, an H f controller would be designed by calculating an upper bound for gamma through one of the iterative procedures like bisection algorithm for solving the H f problem. This upper bound ensures the sufficient conditions to obtain the stabilizing controller K (s) . Next, it is necessary to obtain the closed loop transfer function Tzw which maps the set of inputs to an outputs and to check whether the H f -norm of Tzw less than gamma (i.e. Tzw

f

J ).

Finally, the final controller may be passed through order reduction process if it's of high order since high order controllers are not desirable from the practical perspective. In the assessment phase, it consists of two major sessions, the first is evaluating the frequency response results, which include singular values plot of the open and closed loop system, compensated and uncompensated loop gains. These results indicate whether the control objectives have been met or not. It also includes the sensitivity S , complementary sensitivity T , or control sensitivity R plots for checking how the corresponding weighting filter bounds it. The second session is verifying the responses in the time domain. This contains plotting step responses for the outputs, control efforts, and tracking errors to see the satisfaction of the design requirements.

71 71

4.10 Proposed Configuration for LFC of Steam Turbine System

The results of the assessment phase, evaluates how robust the closed loop system is and it may involves reselection some of the weighting filters until the fulfillment of a certain design requirements is achieved.

4.10 Proposed Configuration for LFC of Steam Turbine System A different configuration for the problem of LFC of steam turbine has been proposed in this book as shown in figure 4.6.

:KHUH LQ WKH SURSRVHG FRQILJXUDWLRQ WKH FRQWUROOHU JRYHUĿQRU) is placed in the feed forward path in contrast to the conventional governor in which the controller is positioned in the feed backward path. In what follows the above proposed configuration together with the robustness tools discussed in the previous sections will be used to setup the problem within the framework of the H f design methodology using the state-space and the polynomial methods.

4.10.1 Design Problem Formulation: A State - Space Approach Consider the block diagram of the proposed load frequency controller of steam turbine as shown in figure 4.6. By substituting system parameters values listed in table 3.1 into the transfer functions derived in section 2.3, the corresponding transfer function of each block can be found as follows [1]: Machine Dynamics Model: MD( s)

1 8s 2

(4.19)


72 72

Steam Turbine Model: (s + 2.32) (s + 0.5131) (s + 3.333) (s + 2.5) (s + 0.1429)

ST (s)

(4.20)

Servo Motor Model: 1 0.2s 1

SM (s)

(4.21)

Speed relay Model: SR( s)

1 0.1s 1

(4.22)

Load Disturbance Model: WD 2 ( s ) 1

(4.23)

dŚŝƐ ŝƐ ĐĂůůĞĚ ĐŽŶƐƚĂŶƚ ůŽĂĚ ĚŝƐƚƵƌďĂŶĐĞ ĂŶĚ ŝƚ ŚĂƐ ĞĨĨĞĐƚ Ăƚ Ăůů ĨƌĞƋƵĞŶĐŝĞƐ ĚĞƐƉŝƚĞ ƚŚĞ ƐƉĞĞĚ ĚŝƐƚƵƌďĂŶĐĞ ǁŚŝĐŚ ŚĂƐ ůŝŵŝƚĞĚ ďĂŶĚǁŝĚƚŚ ŽĨ ĂďŽƵƚϬ͘ϱ,ǌĚĞĨŝŶĞĚďĞůŽǁ͘ Speed Disturbance Model: WD3 ( s)

1 s S

(4.24)

Now, Define G1 ( s) and G2 ( s) as follows: G1 ( s) SR( s) SM ( s) ST ( s) ( s + 2.32) ( s + 0.5131) (0.1s 1) (0.2s 1) ( s + 3.333) ( s + 2.5) ( s + 0.1429)

G2 (s) MD(s)

1 (8s 2)

G ( s ) G1 ( s ) G2 ( s )

(4.25)

(4.26) (4.27)

The block diagram of the augmented plant including the performance and uncertainty weighting filters together with plant is shown in figure 4.7,

73 73


where an output multiplicative uncertainty is assumed in the system as indicated by the uncertainty weight W I (s ) and the input d 1 . The exogenous input vector contains: the uncertainty output d1 , the load disturbance signal d 2 , and the speed disturbance signal d 3 , while the output vector consists of z1 , z 2 , and z 3 which represents the weighted output frequency deviations, the filtered control signal, and the uncertainty input respectively. In symbolic notations: z

ª z1 º ª d1 º « z » , w «d » , « 2» « 2» «¬ d 3 ¼» ¬« z3 ¼»

ªzº « y» ¬ ¼

ª wº P( s ) « » ¬u ¼

ª P11 ( s) P12 ( s) º ª wº « P ( s) P ( s)» « u » ¬ 21 ¼¬ ¼ 22

(4.28)

According to these definitions, the open loop generalized plant can be obtained as:

P( s)

ª 0 « 0 « «W p G2 « ¬ - G2

0 0 WI G1 º 0 0 WU »» : W p G2 W pWD 2 W p G1G2 » » G2 - WD 2 G1G2 ¼

ªA «C « 1 «¬C2

B1 D11 D21

B2 º D12 »» D22 »¼

;ϰ͘ϮϵͿ

The control input u , and the measured output y are related to the K (s) as follows: u ( s) K ( s) y ( s)

(4.30)

74 74


The weighted closed loop transfer function matrix T zw is given by: ª z1 º ª d1 º « z » T «d » « 2 » zw « 2 » «¬ z3 »¼ «¬ d 3 »¼

WI T ª WI T « W K G S W KG S U 2 2 « U «¬ WP G2 S WP G2 S

WI G1 KS WD1 º ª d1 º WU KS WD1 »» ««d 2 »» WP SWD1 »¼ «¬ d 3 »¼

(4.31)

It is obtained by connecting the measured output y with the control input u through the controller K (s) . Now the problem has been set up and the last step to do before the H f controller design is the selection of the weighting filters which is highly depends on the application and usually the optimum set of weighting filters is reached by an ad hoc procedure. This will be handled in chapter 5. Two cases studies of this system will be demonstrated in the next chapter, H f controller design with modeling uncertainty, and H f controller design

without any modeling uncertainty.

4.10.2 Design Problem Formulation: A Polynomial Approach This section deals with same problem dealt with in the previous section but this time using the frequency domain solution of the standard H f problem. This solution will be implemented in terms of polynomial matrix manipulations. Again, consider the block diagram of the proposed LFC configuration of steam turbine shown in figure 4.6 where the open loop transfer function G(s) will be written in a compact form: u ( s) K ( s) y ( s) G ( s)

G1 ( s) * G2 ( s)

(4.32)

N1 ( s) N 2 ( s) D1 ( s) D2 ( s)

N ( s) D( s )

0.3(0.4s 1)(7 s 1) 0.4(0.4s 1) 0.3 (0.1s 1)(0.2s 1)(0.3s 1)(0.4s 1)(7 s 1)(8s 2)

(4.33)

Figure 4.8 represents the mixed sensitivity problem for steam turbine plant which includes the performance shaping filters ( V (s ) and W1 (s) ) and the uncertainty filters W2 (s) where additive uncertainty is used to compensate for

75 75


neglected dynamics which is represented as unstructured uncertainty through W2 (s) . The exogenous input d generates the disturbance v after passing

through a shaping filter with transfer function V (s ) . The control error z has two components, z1 the control system output after passing shaping filter with transfer function W1 (s) . The second component z 2 is the plant input u after passing through a shaping filter with transfer function W2 (s) . Where the numerators and denominators are (scalar) polynomials. Note that G (s ) and V (s ) have the same denominator D (s ) . The design of the shaping

filters is highly depends on the model at hand and certain considerations may be taken in the design of these shaping filters like uncertainty, high frequency roll-off, and integral control. G(s)

N (s) , V (s) D( s)

M ( s) D( s )

,

W1 ( s)

A1 ( s) B1 ( s)

, W2 (s)

A2 (s) B2 (s)

Figure 4.8 defines the standard problem whose generalized plant has the following transfer function matrix:

>z @

ª z1 º «z » ¬ 2¼

ªzº « y» ¬ ¼

ªd º P« » ¬u ¼

76 76

4.11 Proposed Control System for Boiler-Turbine Unit

P

ªW1V W1 G º « 0 W2 »» « «¬ - V - G »¼

ª P11 P12 º « P 21 P 22» ¬ ¼

D -1 N

>D1

D2 @ >N1 1

N2 @

ª A1 M «BD « 1 « 0 « « M « ¬« D

A1 N º B1 D » » A2 » B2 » N» » D ¼»

;ϰ͘ϯϰͿ

The mixed sensitivity problem schematized above is the problem of minimizing the H f - norm of the closed loop transfer function matrix: Tzw ( s)

ª W1 S V º «W R V » ¬ 2 ¼

ªz º , where « 1 » ¬ z2 ¼

Tzw ( s) >d @

;ϰ͘ϯϱͿ

Now the problem can be stated as: Design a stabilizing controller such that the H-infinity norm Tzw (s ) f of the above closed loop transfer function is minimized,i.e. Tzw (s ) f d J . Based on the forgoing analysis, two cases will be investigated in details that illustrate the effective features of the polynomial approach over that of the state-space as will be seen in the next chapter.

4.11 Proposed Control System for Boiler-Turbine Unit In this section, the design of the proposed H f control system for multivariable boiler-turbine system which has the robust stability and satisfactory command tracking performance will be considered based on the proposed scheme as shown in figure 4.9. As seen in chapter two, the plant to be controlled is highly nonlinear, coupled, and time-variant which makes the controller design difficult. Since most of the available theories in the field of the robust control deal with linear models only. Starting from this point, the first step in the design of the robust controllers for such kind of plants is to derive the linearized model. The linearized model for this plant has been derived at many operating points as seen in chapter two. Despite the severe nonlinearity that the plant has, it will be


77 77

shown in this section that by careful choice of the weighting filters a single linear controller works well for the specific operating range even with the nonlinear model. The block diagram of the closed-loop system of the Boiler-turbine system with the proposed controller is shown in figure 4.9, where the plant G(s) is a MIMO transfer function matrix ( 3 3 ) and the proposed controller transfer function matrix denoted as K (s) needs to be designed such that the design requirements (listed in chapter five ) have to be satisfied as large possible. The output vector c

>c1 , c2 , c3 @ represents the controlled variables drum pressure,

electrical output, and the drum water level respectively. The vector d

>d1 , d 2 , d 3 @ is the disturbance vector at the output of each channel. While,

the vector r

>r1 , r2 , r3 @ is the input command vector (i.e. set-point or reference

vector) for the controlled output vector c mentioned above.

It will be assumed that the operating point varies from 77% to 144% of the nominal operating point. Seven operating points have been assumed for system operation and the values of variables at these seven operating points are given in table 3.2. The system state-space matrices Âi , Bi , C i , Di ` for the seven operating points are derived in chapter three and appendix A.


78 78

After detailed analysis to the boiler-turbine system at various operating points, it is found that the nominal operating point at which the H f design will be carried out is the third operating point (i.e. G3 ( s) ) which corresponds to the third column in table 3.2. The transfer function matrix G(s) of the system at this nominal operating point is given by: 0.9 - 0.31 - 0.15 º ª » « ( s +0.00222) ( s +0.00222) ( s +0.00222) « » 12.573 ( s +0.0007784) 0.052623 - 0.0087705 G3 ( s) « » ( s +0.1) ( s +0.00222) ( s +0.1) ( s +0.00222) ( s +0.1) ( s +0.00222) « 0.25327 ( s +0.03067) ( s- 0.003109) 0.4611 ( s +0.0006304) ( s- 0.01582) - 0.014 ( s +0.003183) ( s- 0.4726)» « » s ( s +0.00222) s (s+0.00222) s ( s +0.00222) ¼ ¬

As can be seen from the transfer function matrix, the system has a pure pole on the imaginary axis which is not allowed within context of the H f design procedure using state-space approach. So, a modification to the design procedure will be carried out in light of modified procedure outlined in section 4.7 to handle this problem. After making the linearization to the nonlinear plant at different operating points and selecting one of these linearized plants to apply the H f design on it, the next step is to map the time ±variation of the plant into uncertainty and deriving the corresponding uncertainty weight W I (s ) for that uncertainty. Derivation of the performance weighting filter is considered in the next section.

79 79


4.11.1 Representing Uncertainty by Complex Perturbations The uncertainty weighting filter will be derived based on the fact that the system at hand is time-varying as its operating point varies over several load conditions, pressure of the drum, and fluid density as indicated in table 2.2. To investigate how extensive the model varies with the system operating conditions. Frequency response of the plant of the open loop system from the controller output ( u ) to the plant output ( c ) are illustrated in figure 4.10 in which the seven linearized models ( G1 ( s),......,G7 ( s) ) are shown. Notice that since the plant is MIMO ( 3

3 ), then each transfer function matrix has three singular values. The maximum singular value V (G( jw)) has been shown in figure 4.10 for each transfer function matrix.

From the figure above, one can notice that the frequency responses of the seven transfer functions matrices are very similar even though their parametric models are significantly different.


80 80

Such similarities in the frequency domain suggest that it might be possible to select a single transfer function matrix as the nominal transfer function matrix Gnom(s) of the system, and represent the other transfer functions matrices as variations around this nominal one as model uncertainties. The advantage of such modeling approach is that suboptimal robust controller can be easily designed for the entire system operating range. A proposed method will be presented here which tries to lump all the parameter variations into single (lumped) complex perturbation and then deriving an uncertainty weighting filter WI that best reflects all the parameter variations in the system. This complex perturbation has been represented as input multiplicative uncertainty as follows: GP

Go ( I WI 'I ) ;

'I

f

d1

(4.36)

Since the uncertainty appears as an unstructured multiplication term, ( 'I ) is known as the unstructured multiplicative uncertainty. So, to derive a meaningful uncertainty model, first of all the relative modeling errors among the seven transfer functions matrices should be calculated by: loj (Z )

§ (G j ( jZ ) Gnom ( jZ ) · ¸; ¸ Gnom ( jZ ) © ¹

V ¨¨

(4.37)

Where V is the maximum singular value and j 1,2,4,5,6,7 . As can be seen operating point number 3 is not included this is because the nominal model will be taken at this point as discussed next. The nominal transfer function Gnom(s) = Go(s) should be the one which gives the overall smallest modeling error.

In the current situation, it is found that G3 ( jZ ) provides the best compromise compared with the rest of the six functions. In fact this can be readily seen from figure 4.10. Once the nominal transfer function matrix is chosen, the uncertainty model bound can be calculated using (4.10) such that: l I (Z) t max(loj (Z)) ; Z and j 1,2,4,5,6,7

(4.38)

WI ( jZ ) t l I (Z )

(4.39)

Z

81 81


4.11.2 Control System Interconnection: A State-Space Approach The block diagram of the closed loop system, which includes the nominal boiler-turbine linearized model, the feedback structure, and the controller, as well as the weighting filters reflecting the model uncertainty and the performance objectives, is depicted in figure 4.11.

This can be drawn in expanded form as shown below:

82 82


Where, ª d1 º «d » ͖ z « 2» «¬ d 3 »¼

d

ª z1 º « z » ͖ z c « 2» «¬ z 3 »¼

ª z1c º « z c » ͖ z cc « 2» «¬ z 3c »¼

0 0 º ª wI 1 ( s ) « 0 0 »» wI 2 ( s ) WP ( s ) « «¬ 0 0 wI 3 ( s)»¼

WI ( s )

ª z cº «z» ¬ ¼

0 0 º ª wP1 ( s) « 0 0 »» wP 2 ( s ) « «¬ 0 0 wP 3 ( s)»¼

The robust controller design can be formulated as a |S/T| mixed sensitivity suboptimal H f design as will be shown below. The open loop generalized plant P(s) of the system can be obtained from figure: ª z ccº « y» ¬ ¼

ª z cº «z» « » «¬ y »¼

ªd º P( s ) « » ¬u ¼

WI ( s)Go ( s) º ª 0 «W ( s) W ( s)G ( s)» ªd º P o » «u » « P «¬ 0 G o ( s) »¼ ¬ ¼

(4.40)

While the closed loop weighted transfer function Tzw (s ) is given by: ªzº « z c» ¬ ¼

Tzw ( s ) >d @

ª W P ( s ) S ( s )º « W ( s )T ( s) » >d @ ¬ ¼ I

(4.41)

For the sake of simplicity the negative sign and dependent variable s will be removed from the closed loop transfer function. Then it becomes: Tzw

ªWP S º «W T » ¬ I ¼

(4.42)

Then H f suboptimal design is to find a stabilizing controller K (s) such that the H f -norm of the closed loop transfer function Tzw (s ) is less than a given positive integer J , i.e., Fl ( P, K )

Where,

f

Tzw

f

max V (Tzw ) w

ªWP S º ») J ¬WI T ¼

max V ( « w

(4.43)

83 83


J ! J min

min

K ( s ) stabilisin g

Tzw

(4.44)

f

4.11.3 Control System Interconnection: A Polynomial Approach The polynomial approach based interconnected closed loop system which includes the nominal model, the controller, as well as the weighting filters reflecting the model uncertainty and the performance objectives, is the same as depicted in figure 4.8. The weighting filters in the H f solution of the boiler-turbine system using polynomial approach are expressed as:

V

ª ( s a1 ) ( s B) « ( s 0.00222) s « « 0 « « « 0 «¬

W1 ( s)

0 s 2 a 2 s a3 ( s B) s ( s 0.1)(2 0.00222) 0

ª1 0 0º «0 1 0 » , » « «¬0 0 1»¼

º » » » 0 » » 2 s a 4 s a5 » s( s 0.00222) »¼ 0

;ϰ͘ϰϱͿ

0 º ªc1 s 0 » «0 c s 0 2 » « «¬ 0 0 c3 (1 r3 s)»¼

W2 ( s )

Or in alternative form:

V

ª ( s a1 ) « ( s 0.00222) « « 0 « « « 0 «¬

W1 ( s )

ªs B « s « « 0 « « 0 «¬

0 s 2 a 2 s a3 ( s 0.1)(2 0.00222) 0

0 sB s 0

º 0» » 0» , W2 ( s) » 1» »¼

º » » » 0 » » s 2 a 4 s a5 » s ( s 0.00222) »¼ 0

0 ªc1 s 0 º «0 c s » 0 2 « » «¬ 0 0 c3 (1 r3 s)»¼

;ϰ͘ϰϲͿ

4.12 Simulation of the Closed-Loop System

84 84

Where the parameters (a1, a2, a3, a4, a5, c1, c2, c3, r3) have to be determined during the design phase as will be clear in the next chapter. The difference between the above two sets is that in (4.45) the integrating action is achieved through introducing a pole at s 0 in the entries (1,1) and (2,2) of the transfer function matrix V (s) . While in the second set (4.46) the integrating action is achieved by introducing the same pole but this time in the corresponding entries of W1 ( s) .

4.12 Simulation of the Closed-Loop System As mentioned in section 4.8, following the design phase is the evaluation of the closed loop system with the designed controller which is very important stage since through which that the designer will know whether the design specifications have been met or not. So, it is worthy to mention that all the simulations that will be carried out in chapter five concerning steam turbine system will be applied on the following closed loop feedback system.

Where, G1 (s) = SR(s) * SM (s) * ST (s) , G2 (s) = MD(s) . ǻPL = Load disturbance signal, ǻPm

= Output Mechanical power of the turbine,

ǻZ r

= Output frequency deviation ( ǻZr

d3

= Speed disturbance signal,

u

= Control signal (output of the governor),

0 if ǻPL

0 ),

While simulations regarding the boiler turbine unit will be carried out on closed loop feedback system shown in figure 4.9.

CHAPTER FIVE

ROBUST CONTROLLERS DESIGN FOR UNCERTAIN STEAM-TURBINE AND BOILER-TURBINE SYSTEMS

USING G H TECHNIQUES

5.1 Introduction This chapter is divided into two parts, in the first part the proposed methods for the robust controllers (optimal and suboptimal) of the steam turbines and boiler systems based on H f techniques will be presented. Statespace and polynomial approaches have been adopted in the design procedure of the robust H f controllers. For steam turbine system, different schemes for modeling the uncertainty in the system are exploited in the design procedure with different configurations of control systems. For boiler-turbine system which is a nonlinear MIMO system a linear H f controller has been designed based on the H f control configuration already set up in the previous chapter. The second part of this chapter contains the simulations and the results of the closed-loop system to illustrate the behavior of the robust controllers on the linear system of the steam turbine unit. Also it includes the simulations and the results of the boiler-turbine unit to show the abilities of H f robust controller on the linearized system. Simulations regarding the controller order reduction and variable load conditions are carried out to show how robust the designed MIMO H f controller is with different requirements.

85

5.2 H Controller Design for Steam Turbine System: A State-Space Approach

86

5.2 H Controller Design for Steam Turbine System: A StateSpace Approach This section summarizes the design procedure of the H f robust controller design for the steam turbine system based on state-space techniques in which the system will be represented by four system matrices Â, B, C, D` . For steam turbine system, the design requirements that have to be satisfied are summarized as follows: x Time-domain requirements: 1. Steady-state error less than 0.2 % . 2. Good and swift disturbance rejection. 3. The closed-loop system is stable at all operating conditions. 4. Undershoot of the output frequency deviation is as minimum as possible. 5. Control signal ( u ) lies in the range [0,1] in response to step load change. x Frequency-domain requirements: 1. Phase margin > 40 o . 2. Gain margin > 10 dB. 3. Crossover frequency about 1 rad /sec. 4. High frequency roll-off to the controller.

5.2.1 H Design without Model Uncertainty At first, our H f robust controller will be designed directly on the detailed open loop transfer function of the plant. In this case the following issues will be considered: 1. ǻ 0 WI

0 Gnom

Gdet .

2. All the system parameters are constant. So, no parametric perturbations are assumed.


87

3. After a thorough analysis, the weighting functions are found to be: WP ( s)

0.5s 0.2 , s 2 10 6

WU ( s )

s 12.5 10 4 s 100

Now, the problem can be stated as follows: find a stabilizing controller K (s) that makes H f -norm of the closed-loop transfer function Tzw (s ) less than

J (i.e. Tzw (s ) J ).

Following the design procedure in section 3.5.1, a robust suboptimal H f controller is obtained within eight iterations with the following results: x Minimum value of gamma achieved is J

0.9882 .

x The controller is suboptimal, strictly proper, and of 8 th order: 6

K (s)

0.0049472( s + 10 )( s + 10)(s + 5)(s + 3.333)(s + 2.5)(s + 0.2554)(s + 0.1429) ( s + 58.04)(s + 2.386)(s + 0.4391)(s + 2 10

-6

2

2

)( s + 23.02s + 132.5)(s 8.215s + 19.78)

x The H f -norm of the closed loop transfer function is Tzw ( s )

0.988155.

5.2.2 H Design with Model Uncertainty In the previous section, the H f robust controller has been designed and the controller provides very good results both in time and frequency domains as can be seen in section 5.5.1. But the parameter variations have not been taken into consideration in the design process as parameter variations are a common phenomenon in many practical applications, one of them is the steam turbine system. Moreover, the controller was of 8th order which is somewhat high to implement directly into the practice. It will be shown shortly in this section that the order of the suboptimal H-infinity controller designed in the previous section can be reduced by two. At the same time it is still preserving the stability conditions and satisfying the specification requirements.


88

This can be done by designing the H f robust controller for the nominal model Gnom (s) rather than the detailed one Gdet . Usually the nominal model Gnom (s) is of a transfer function which is simpler than the detailed Gdet or

perturbed one. One possible proposed choice of the nominal model for the steam turbine system is: Gnom (s)

G1nom ( s) G2 nom ( s)

( FHP TRH s 1) 1 (TCH s 1) (TRH s 1) (TM s + K D )

(5.1)

Where the dynamics that represents the speed relay SR(s) , servo motor SM (s) , and the crossover piping in the turbine transfer TCO have been neglected

as they have very small time constants. These neglected high frequency dynamics will be represented as an output multiplicative uncertainty in the system, where the detailed model will be replaced by a nominal model Gnom (s) together with uncertainty filter W I (s ) usually of finite and low order. In addition, real perturbations will be added to the system by allowing specific parameters to vary in their ranges. In this model two parameters are assumed to be uncertain. With the aid of table 2.1, the characteristics of these two uncertain parameters are listed in table 5.1. At this end the nominal model has been chosen based on simplification in the detailed model only. With perturbations added to the two parameters mentioned above, the nominal model becomes: G1nom ( s) G2 nom ( s) Gnom (s)

(1 + sFHP T RH ) (1 sT CH ) (1 sT RH ) 1 (TM s + K D ) G1nom ( s) G2 nom ( s)

(5.2)

(5.3)

(2.1s 1) 1 (0.25s 1) (7 s 1) (8s + 2)

(5.4)

5.2 H Controller Design for Steam Turbine System: A State-Space Approach Table 5.1: Specifications of uncertain parameters.

89

Parameter

Nominal symbol

Nominal Value/sec

Range/sec

Steam Chest time ( TCH )

T CH

0.25

0.1-0.4

Reheat Time ( T RH )

T RH

7

3-11

Where T RH , T CH are the nominal values of their respective variables. Their values can be obtained from table 5.1 above. While the perturbed model GP (s ) is given by:

GP ( s )

0.3(0.4s 1)(TRH s 1) 0.4(0.4s 1) 0.3 1 1 1 0.1s 1 0.2s 1 (TCH s 1)(0.4s 1)(TRH s 1) 8s 2

(5.5)

Where TCH and TRH are uncertain, their ranges are defined in table 5.1 above. Based on the above analysis, the next step is to design the uncertainty filter W I (s ) which represents the neglected dynamics as well as the variations in TCH and T RH . The uncertainties of these two parameters will contribute in the design of the uncertainty weighting filter W I (s ) as explained next. Now, the H f design procedure will be carried out on the nominal plant Gnom (s) with the uncertainty filter W I (s ) that compensates for the modeling

errors (neglected dynamics plus parameter perturbations) rather than on the perturbed model G P (s ) . With output multiplicative uncertainty has been incorporated into the system, the open-loop generalized plant will be represented by (4.29), while the weighted transfer function of the closed loop system is given by (4.31). Weighted mixed sensitivity structure depicted in figure (4.7) will be adopted for the solution of H f optimization.


90

5.2.2.1 Selection of the Weighting Filters Two kinds of weighting filters have been designed in the H f optimization process. These are: 1. Performance weighting filters: Two performance weighting filters have been used. The performance weighting filter WP (s) reflects the requirements of disturbance rejection. While the second weighting filter WU (s) , is included to limit the value of the control signal ( u ) to avoid the saturation of the actuators. After deep analysis it is found that the following weighting filters are suitable for the problem at hand: WP ( s )

s 0.29 s 2.9 10 6

,

WU ( s)

1 10

2. Uncertainty weighting filter: This weighting filter represents the neglected dynamics in the model in addition to the parametric uncertainties in the variables TCH and TRH . According to formula (4.9), G P (s ) and G nom (s ) are given as in (5.4)-(5.5) respectively, and

is the set of the uncertain plants and

is defined as follows:

{G P (TRH , TCH ); TRH , TCH }

(5.6)

Where: TRH {3, 5, 7, 8, 10, 11} TCH {0.1, 0.15, 0.2, 0.25, 0.3, 0.4}

The corresponding relative errors (GP Gnom ) / Gnom are shown as a function of frequency for the 62 36 resulting G P 's in figure 5.1. In figure 5.1 the perturbations are represented by the dotted blue lines and three uncertainty filters are drawn in the same figure. According to Eq. (4.10), at every frequency point ( Z ) each of the uncertainty filters (the black, green, and red) must lie above all the dotted blue lines to include all of the possible plants.

91

5.2 H Controller Design for Steam Turbine System: A State-Space Approach 2

10

WI ( s ) 1

1

10

WI ( s ) 2

Magnitude

0

10

WI ( s)

-1

3

10

-2

10

-3

10

-4

-3

10

-2

10

10

-1

0

10 10 Frequency (rad/sec)

1

10

2

10

Figure 5.1: Relative errors for 36 plants combinations of TRH and TCH.

It can be noticed that the relative error (GP Gnom ) / Gnom is very small at low frequencies; it exceeds 1 at about Z1 3.715 (rad/sec), while the value 1.18 is attained at Z 2 1.96 (rad/sec). This means that uncertainty is very low at the low frequencies and grows up until it becomes 100 % at Z1 , and 118 % is reached at Z 2 . As seen all of these filters satisfy Eq. (4.10) and are qualified to represent the unstructured uncertainty in the system together with parametric uncertainties in TCH and TRH . After a detailed analysis to the uncertain plant and with the aid of Eq.s (4.9)-(4.10) the uncertainty weighting filter W I1 ( s ) is chosen and given by: WI ( s) WI ( s) 1

4500 s + 1067 s + 3333

Up to this point all the weighting filters have been designed and statespace matrices of the augmented plant P(s) are given below:

5.3 H Controller Design for Steam Turbine System: A Polynomial Approach

ª 3321 « 144.7 « « - 92.96 A « « 25.18 « - 48.2 « ¬ - 68.89

B2

204.3 165.9 274.2 11.39 7.904 14.1 - 6.202 0.6944 5.122 1.83 - 1.516 - 2.696 - 3.502 2.088 4.048 - 5.005 2.983 5.861

ª- 0.4832º « 20.56 » « » « - 84.28 » « » « - 78.58 » « 88.83 » « » ¬ 2.925 ¼

C1

C2

331 78.15º - 16.84 - 3.939 »» - 6.349 - 2.403 » »B 2.902 0.8079 » 1 - 5.128 - 1.151 » » - 7.078 - 1.793 ¼

ª0.08059 0.001152 - 0.001152 º « - 1.125 0.04899 - 0.04899 » » « «- 0.7412 0.00689 - 0.00689 » » « 0.1747 - 0.1747 » « 0.2241 «- 0.2353 0.1568 - 0.1568 » » « 0.2136 ¼ ¬- 0.2709 - 0.2136

ª0.03447 - 0.6536 - 0.426 0.145 - 0.1389 0.4384º « 0 0 0 0 0 0 »» « 16.29 - 13.22 - 21.85 26.38 6.23 »¼ ¬« 264.6

>0

92

ª0º

0 0.3536 0 0 1@ D12 ««0.1»» D21 ¬« 0 ¼»

>0

D11

ª0 0 0 º «0 0 0 » « » «¬0 0 0»¼

0 0@ D22

>0@

Applying the H design procedure to the above generalized plant P(s) , the optimization process ends with the following: x J min Tzw ( s)

0.9631 and the H f -norm of the closed-loop transfer function is

0.963054 .

x The H f controller is suboptimal, strictly proper, and of 6 th order: K (s)

41176293.2898 ( s + 3333) ( s + 4) ( s + 2.975) ( s + 0.2588) ( s + 0.1429) 4 4 s ( s + 5.41 10 ) ( s + 6.499 10 ) ( s + 66.89) ( s + 0.959) ( s + 0.4825)

5.3 H Controller Design for Steam Turbine System: A Polynomial Approach This section presents the results of the H f robust controller design for the same plant dealt with it in the previous section but this time using polynomial approach. Based on the forgoing analysis, two cases will be investigated in details that illustrate the effective features of the polynomial approach over that of the state-space as will be seen later.


93

5.3.1 HDesign without Model Uncertainty In this mode of design the nominal plant represents the real plant itself. Since uncertainty has been ignored ǻ WKHQWKHIROORZLQJDUHWUXH 1. Both W1 ( s ) and W2 ( s ) act as performance shaping filters. 2. System parameters are assumed invariant during system operation. 3. The two performance filters W1 ( s ) and W2 ( s ) have been designed to reflect the performance requirements. These filters are: W1 ( s)

sB s

,

W2 (s)

s 1 (1 ) 75 6

(5.7)

Where B is a factor usually taken as the bandwidth of the system. Since our selection to the bandwidth is 1 then B 1 comes naturally. Consider the shaping filter V (s) , consider the transfer function of V (s) : V (s)

M ( s) D( s )

sc

( s 3.5)( s 5.5)( s 1 / TG )( s 1 / TCH )( s 2 s 2 1) (0.1s 1)(0.2s 1)(0.3s 1)(0.4s 1)(7s 1)(8s 2)

(5.8)

With M (s) a polynomial of degree 6 (same order of D(s) ) to be chosen, M (s) should be selected such that the roots of M (s) reappear as closed-loop

poles. Furthermore, to obtain a good time response corresponding to bandwidth 1, which does not suffer from sluggishness or excessive overshoot, six dominant poles will be placed, two of them will be chosen as a second order Butterworth configuration with radius that is the pole pair is ( 2 2)(1 r j ) . The choice of the radius 1 comes in line with our decision to choose the closedloop bandwidth equal to 1. Two from the remaining four poles of M (s) will be chosen as 1 / TCH and 1 / TG . This means that the open loop faraway pole at 5 will be left in place in addition to the pole at 3.333 . Where sc is a scaling factor, it is introduced to make normalization to M (s ) such that W1 (f)V (f) 1 , this means sc TSRTGTCH TRH TCOTM

0.0672.


94

The open-loop generalized plant is given by (4.34) while the closed- loop weighted transfer function Tzw is expressed by (4.35). with r 1 / 6 the sensitivity function S has a little extra peaking while starting at the frequency 6 rad/sec the complementary sensitivity function T rolls off at a rate of 100 dB/decade, control sensitivity rolls off with slope 20 dB/decade. The polynomial matrix fraction for this case is found to be:

D1

ª s 10 7 « « 0 « 0 ¬

N2

0 º ª « 0.013 + 0.0022s » N 1 » « «¬- 1 - 2.4s - 0.84s 2 »¼

0º » 1» 0»¼

D2

s 1 ª º « » 0 « » 2 3 4 5 6 «¬2 + 24 s + 79 s + 64 s + 21s + 2.9s + 0.13s »¼

0 ª º « » 0 « » 2 2 2 3 4 5 6 «¬- 43 - 10 s 1.2 10 s 65s 18s 2.5s 0.13s »¼

The optimal solution is found with optimum gamma J opt 1.76888 . The corresponding optimal H f controller: K ( s)

656.5882 ( s + 11.4) ( s + 5) ( s + 3.333) ( s + 2.428) ( s 2 + 0.9779s + 0.3414) s ( s + 2.327) ( s + 0.489) ( s 2 + 18.56s + 96.8) ( s 2 + 5.605s + 36.42)

Figure 5.2 shows the frequency response of the closed-loop transfer function Tzw (s ) ((4.35)). As seen in figure 5.2 the equalizing property is almost realized and the H f - norm can be obtained from the figure as the maximum of the frequency response which in this case is Tzw ( s ) f 1.76888 .


95

50 40 30

Magnitude (dB)

20 10 0 -10 -20 -30 -40 -50 -3 10

-2

10

-1

0

10

1

10

10

2

10

3

10

Frequency (rad/sec)

Figure 5.2: Frequency response of the closed-loop weighted transfer function Tzw (s).

5.3.2 HDesign with Model Uncertainty To account for the neglected dynamic as well as the parametric uncertainties inherently exist in the system, uncertainty needs to be considered during the design stage of the H f controller. The neglected dynamics and the parameter variations studied in section 5.2.2 will be reconsidered here, but this time with different representation, it will be dealt with using the polynomial approach. The unstructured uncertainty will be represented as additive uncertainty (4.6) instead of multiplicative one. The shaping filter W2 ( s ) will represent the neglected dynamics in the system in addition to the parameter variations in the two variables T RH and TCH . At the same time, this filter still acts as a shaping filter to the control sensitivity function which in turn improve the performance of the system and providing high frequency roll-off.


96

In this case, the nominal plant is given by (5.4) while the perturbed plant is given by (5.5). The ranges of the time-varying parameters are given in table 5.1. Finally, the set of uncertain plants as defined by the variations of the two parameters T RH and TCH over their ranges is given by (5.6). The corresponding additive errors (GP Gnom ) are plotted in figure 5.3 below.

W2 ( s )

-1

10

Magnitude

-2

10

-3

10

-4

10

-2

10

0

10 Frequency (rad/sec)

2

10

Figure 5.3: Additive errors for 36 plants combinations of TRH and TCH.

The filter that has been proposed for this kind of uncertainty is: W2 (s) W2c(s) rolloff (s)

0.272 (s + 0.0002941) (s + 80) (s + 0.08) 160

Where W2c(s) is the filter that reflects the unstructured and parametric perturbations. While the function rolloff (s) 1 2 (1 s 80) provides high frequency roll-off. Thus the uncertainty weight is improper. As can be seen from the figure, the filter exactly fits the additive perturbations and covers it


97

entirely. Hence this filter is qualified to represent the uncertainty in the system. The shaping filter that is used to shape the sensitivity function is W1 ( s ) and takes the following form: W1 ( s)

s+B s

The bandwidth of the closed-loop system will be set to about 0.85. So, without any doubt that B 0.85 to limit the effect of disturbance up to the bandwidth of the closed-loop system. Finally, the disturbance shaping filter V (s) will take the form: V ( s)

M nom ( s) Dnom ( s)

M ( s) (0.25s 1)(7 s 1)(8s 2)

where Dnom (s ) is the denominator of the nominal plant G nom (s ) and is of order 3 , and M nom (s ) a polynomial of degree 3 (same order as D(s) ). The polynomial M nom (s ) will be determined such that two of them as a second order Butterworth configuration pole pair with radius 0.85 will be placed at - 0.5992 r j 0.5992 . Again the choice of radius 0.85 is in accordance with the

selected bandwidth which in this case 0.85 . The third root will be chosen as -2. This makes M nom (s ) look like: M nom ( s) ( s 2)( 0.72 + 1.2s + s 2 )

So, V (s) can be written as: V ( s)

M nom ( s) Dnom ( s)

( s 2)( 0.72 + 1.2s + s 2 ) (0.25s 1)(7 s 1)(8s 2)

According to V (s ) , W1 (s) , and W2 (s) mentioned above and in reference to (4.34), the polynomial matrix fraction of the extended plant (4.34) can be calculated as follows:

98

5.4 Robust Controller Design for Boiler-Turbine System

D1

N1

ª10 -10 + s « 0 « « 0 ¬

0 º » 1 + 13s » 0 »¼

,

0 º ª » , « 0 » « 2 3 ¬«- 20 - 44 s - 45s - 14 s ¼»

D2

s 0.85 ª º « » 0 « » «¬2 + 23s + 62 s 2 + 14 s 3 »¼

N2

0 ª º «5 10 4 1.7 s 0.021s 2 » « » - 1 - 2.1s ¬« ¼»

The results of the polynomial solution to the H f controller design are: x gamma optimum J opt

2.8887 .

x The frequency response of the wiehgted closed loop transfer function H (s) is approximately constant with respect to time with H f -norm of Tzw ( s )

f

2.8887 .

x The H f robust controller is optimal with transfer function given as: K ( s)

1594.2197 ( s + 4.172) ( s + 0.08) ( s 2 + 0.6475s + 0.1349) s ( s + 79.96) ( s + 3.301) ( s 2 + 0.3868s + 0.04608)

5.4 Robust Controller Design for Boiler-Turbine System The design of the H f control system for multivariable boiler-turbine system which has the robust stability and satisfactory command tracking performance will be considered in this section.

5.4.1 Design Specifications of the Boiler-Turbine System The closed-loop system performance specifications that should be considered in the design are: x Time-domain requirements: for a unit step command to the first input channel at t

0 , the first channel output c1 (tracking) and the outputs c 2 and c3

(interaction) of the other channels should satisfy: 1. Rise time for c1 is less than 2.85 sec. 2. Settling time of c1 no more than 5 sec.


99

3. Steady-state errors of c1 is less than 0.4% ; 4. Output disturbance suppression as large as possible at the output channel; 5. c 2 and c 3 should be less than 10% for all t ; 6. Overshoot at the channels output should be as minimum as possible. 7. Control signals ( u1 , u 2 , u3 ) are constrained to lie in the interval [0,1] . Correspondingly, similar requirements should be met for a unit step command at the second and third input channels just with different settling and rise times, that are close to ( 7 sec and 4 sec) for c 2 and ( 10 sec and 4 sec) for c 3 respectively.

x Frequency Domain requirements: 1. Crossover frequency of the loop gain, 0.05 d Zc d 20 . 2. High frequency roll-off to the open loop gain. 3. The controller must have an integrating action.

5.4.2 Derivation of the Performance Weighting Filter WP (s) In robust control designs, the design specifications are usually converted into appropriate weighting functions. These weightings functions are then combined with the closed-loop system transfer functions, which will later be used in the H f optimization process. The output disturbance in the boiler-turbine system has a dominant frequency which is not exceeding 0.05 rad/sec that lies in the low frequency band. Therefore, to eliminate this disturbance as well as the steady-state error of the closed-loop system, the sensitivity function S needs to be as small as possible in the low frequency region that is below 0.05 rad/sec. From figure 4.11: z ( s) WP ( s) S ( s) d ( s)

100


Thus, the weighting function WP (s ) that is used to shape the sensitivity has to be large at the same frequency region to ensure a satisfactory results in response to disturbance signals. One of the weighting filters WP (s ) that achieves the above requirements is: ª 0.5263 s + 0.45 0 « s + 0.00171 « 0.5263 s + 0.45 WP (s) « 0 s + 0.00171 « « 0 0 «¬

º » » » 0 » 0.5263 s + 0.45 » s + 0.00171 »¼ 0

While the corresponding weighting filter W1 ( s ) and the disturbance shaping filter V (s) that are used in the H f solution using the polynomial approach are obtained by substituting the following proposed values in (4.46): a1

0.00222, a 2

c1

1 , c2 90

W1 ( s)

a4

2 , a3

a5

1, B 1,

1 , c3 100

1 , r1 110

r2

r3

ªs 1 « s « « 0 « « 0 «¬

1 50

º 0» s 1 » 0» s » 0 1» »¼ 0

While the disturbance shaping filter V (s) is obtained by:

V

ª « 0 «1 s 2 2s 1 « «0 ( s 0.1)(2 0.00222) « «0 0 «¬

º » » » 0 » » 2 s 2s 1 » s( s 0.00222) »¼ 0

And finally the uncertainty weighting filter W2 ( s ) is given as:


101

s ª1 º 0 0 « 90 (1 50 ) » « » 1 s » (1 ) 0 0 W2 ( s ) « 100 50 « » 1 s » « (1 )» 0 0 «¬ 110 50 ¼

5.4.3 Derivation of the Uncertainty Weighting Filter W I (s ) In light of section 4.11.1, the complex perturbation loj (Z ) and hence the uncertainty bound l I (Z ) will be also of size 3u 3 and its frequency content can be represented by three singular values in the frequency domain. For the purpose of illustration and starting from (4.37), the maximum singular values of V (loj (Z)) and l I (Z ) (equation (4.38)) are shown in figure 5.4. Six complex perturbations have been plotted ( V (lo1 (Z ) ) « V (lo6 (Z )) . These correspond to the models that are considered varying around the nominal one. As can be seen from figure 5.4 the uncertainty bound l I (Z ) covers all the perturbations over the entire frequency range. According to this, and taking into account the dominant frequency of the modeling errors which is always high and assumed as 20 rad /sec, the weighting filter has been proposed and are drawn in the same figure. This weighting filter can be expressed as: º ª 1.282 s + 0.97 0 0 » « 0.005128 s + 1 » « 1.282 s + 0.97 » 0 0 WI (s) « 0.005128 s + 1 » « 1.282 s + 0.97 » « 0 0 «¬ 0.005128 s + 1»¼

As one can see from the figure 5.4, the uncertainty bound l I (Z ) covers all of the perturbations at all frequencies. Also, the filter wI (s) covers the uncertainty bound l I (Z ) over the entire frequency band as predicted. The

102


corresponding uncertainty weighting filter that is used in the polynomial solution of H f solution to the boiler-turbine problem is given in the previous section.

ʍ;ůdata1 Žϭ;ʘͿͿ ʍ;ůdata2 ŽϮ;ʘͿͿ ʍ;ůdata3 Žϰ;ʘͿͿ ʍ;ůdata4 Žϱ;ʘͿͿ ʍ;ůdata5 Žϲ;ʘͿͿ ʍ;ůdata6 Žϳ;ʘͿͿ ǁdata7 /;ʘͿ

1

10

Magnitude

ʍ;ů/;ʘͿͿ

ǁ/;ʘͿ

0

10

-1

10

-6

-4

10

10

-2

0


2

10

4

10

Figure 5.4: Uncertainty modeling corresponding to six variable load conditions.

Up to this point, the last thing to be done before completing the H f design procedure is to shift the eignvalues of the matrix A of the nominal openloop transfer function matrix G nom (s ) as it has a pole that lies on the imaginary axis which is not allowed in the H f design using state-space approach. The eigvalues of the system before and after shifting them to the right by small number are: O1, 2,3

{0, - 0.1, - 0.00222}

Onew1, 2,3

Onew1, 2,3 0.001 {0.001,-0.099,-0.00122}

Based on the above analysis, the new matrix A that produces these new eignvalues becomes:

103


Aold

ª- 0.00222 0 0º « 0.0585 - 0.1 0» « » «¬ - 0.0058 0 0»¼

ª- 0.00122 « 0.0585 « «¬ - 0.0058

Anew

0 0 º - 0.099 0 »» 0 0.001»¼

According to this, state-space realization of the open-loop generalized plant P(s) can be obtained as follows:

A

B1

C1

C2

ª - 0.00122 « 0.05847 « « - 0.0058 « - 0.5 « « 0 « «- 0.003565 « 256 « 0 « « « 1.825 ¬

0 - 0.099 0 0 - 0.5 0 0 256

0 0 0.001 0 0 - 0.002312 0 0

0 0 0 - 0.00171 0 0 0 0

0 0 0 0 - 0.00171 0 0 0

0 0 0 0 0 - 0.00171 0 0

0 0 0 0 0 0 - 195 0

0 0 0 0 0 0 0 - 195

0

1.184

0

0

0

0

0

0 0º ª0 «0 0 0 »» « «0 0 0» » « 0» « 0 .5 0 « 0 0.5 0 » » « 0 0 .5 » «0 «0 0 0» » « 0 0» «0 «0 0 0 »¼ ¬

B2

- 0.31 ª 0.9 « 0 12.57 « « 0 - 1.258 « 0 « 0 « 0 0 « «- 0.1266 - 0.2306 « 0 0 « 0 « 0 « 64.84 118 ¬

- 0.15 º 0 »» 1.659 » » 0 » 0 » » 0.007 » 0 » » 0 » - 3.584»¼

D11

ª 0 « 0 « « 0 « «0.5263 « 0 « ¬ 0

189.7 0 0 0 0 0 0 0 º ª 250 « 0 250 0 0 0 0 0 189 . 7 0 »» « « 1.782 0 1.156 0 0 0 0 0 189.7 » « » 0 0 0.8982 0 0 0 0 0 » « 0.5263 « 0.5263 0 0 0 0.8982 0 0 0 0 » « » 0 0 0 0.8982 0 0 0 ¼ 0.002434 ¬ 0.003753

-1 ª « 0 « «¬ - 0.00713

0 -1 0

0 0 - 0.004625

D 22

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 ª 1 « 0 1 « ¬«- 0.2533 - 0.4611

0º 0»» 0»¼

D12

0 º 0 »» 0 » » 0 » 0 » » 0 » 0 » » 0 » - 195» » ¼

0 0 º 0 0 »» 0 0 » » 0 0 » 0.5263 0 » » 0 0.5263 ¼

D21

0 ª 0 « 0 0 « « 63.32 115.3 « 0 « 0 « 0 0 « ¬- 0.1333 - 0.2427

ª1 «0 « ¬«0

0 1 0

0º 0»» 1»¼

0 º » 0 » - 3.5 » » 0 » » 0 » 0.007368 ¼

0 º 0 »» 0.014¼»

`The problem has been solved via both the state-space and the polynomial approaches. In the polynomial approach numerical difficulties

104

5.5 Analysis and Simulation Results

emerged since the polynomial spectral factorizations fails because >D1 N1 @ and

>N 2

D2 @ do not have full rank on the imaginary axis.

The H f optimization process using the state-space approach when applied to the above generalized plant ends with the following results: x The achievable minimum value of gamma is J 1.1335 . x The H f -norm of the closed-loop transfer function Tzw (s ) is Tzw ( s )

f

1.133411 .

x The H f controller is suboptimal and of order 8 with the elements K ij (s) for i K11 ( s)

K 21 ( s )

K 31 ( s )

K12 ( s )

K 22 ( s )

K 32 ( s)

K13 ( s)

K 23 ( s )

K33 ( s)

j 1,..,3 given as: 2315.3336 ( s - 1.076 10 6 ) ( s + 195)( s - 0.4954)(s + 0.002663)( s + 0.0008583) ( s + 1.533 10 6 )( s + 2.473 10 4 )( s + 30.53)(s + 0.2721)( s + 0.002664) 2 - 2171.3465 ( s + 195) ( s + 27.04) ( s + 0.2721) ( s + 0.002663)( s 2 - 45.32s + 1.091 10 4 ) (s + 1.533 10 6 )(s + 2.473 10 4 ) (s + 30.53)(s + 30.47)(s + 0.2721)(s + 0.002664) 2

- 29628.8099 ( s + 1.521 10 6 )( s + 195)( s + 0.01834)( s + 0.002663)( s + 2.869 10 -5 ) ( s + 1.533 10 6 )( s + 2.473 10 4 )( s + 30.53)( s + 0.2721)( s + 0.002664) 2 - 25967.315 ( s + 4.002 104 )( s + 195) ( s + 30.51)( s + 0.09994)( s - 0.04429)( s + 0.002663) ( s + 1.533 106 )( s + 2.473 104 )( s + 30.53)( s + 30.47)( s + 0.2721)( s + 0.002664) 2

14169.4696 ( s + 2.532 10 4 ) ( s + 195) ( s + 0.1) ( s + 0.002663) ( s + 1.533 10 6 ) ( s + 2.473 10 4 ) ( s + 30.47) ( s + 0.002664) 2

- 3085.6722 ( s + 2.264 10 6 )( s + 195) ( s + 0.09993)( s - 0.01063 ( s + 0.002663) ( s + 1.533 10 6 )( s + 2.473 10 4 )( s + 30.53)( s + 0.2721)( s + 0.002664) 2

4.0531 10-6 ( s + 7.441 106 ) ( s + 9.628 104 ) ( s + 1.344 104 ) ( s + 195) ( s + 30.54) ( s + 0.002007) ( s + 1.533 106 ) ( s + 2.473 104 ) ( s + 44.18) ( s + 30.53) ( s + 0.2721) ( s + 0.002664) 7.379 10-6 ( s - 1.712 105 ) ( s + 3.904 104 ) ( s + 195) ( s + 30.52) ( s + 5.876) ( s - 0.2208)( s + 0.1143) u ( s + 0.002663) ( s + 1.533 106 ) ( s + 2.473 104 ) ( s + 44.18) ( s + 30.53) ( s + 30.47) ( s + 0.2721)( s + 0.002664) 2

2.2404 10-7 ( s-7.233 107 ) ( s +1.654 107 ) ( s +8733) ( s +195) ( s +30.55) ( s +0.002007) ( s +1.533 106 ) ( s +2.473 104 ) ( s +44.18) ( s +30.53) ( s +0.2721)(s +0.002664)

5.5 Analysis and Simulation Results The performance evaluations for both steam turbine and boiler-turbine systems with the H f robust controller in the closed-loop will be discussed for different cases as follows.

105


5.5.1 Simulations and Results for Steam Turbine H Design without Uncertainty: A State-Space Approach The denominator of the robust controller designed above, indicates that the integral action already achieved. Since the uncertainty has been neglected in this case, just the nominal stability and nominal performance will be evaluated. All the poles of closed-loop system (i.e. zeros of 1 K (s)G(s) ) lie in the left- half s-plane which indicates nominal stability. Nominal performance has been evaluated by plotting the singular values of the individual entries of Tzw (s) (Eq. 4.31) with respect to frequency and showed good frequency

coverage as shown in figure 5.5. Output response of the frequency deviation is shown in figure 5.6, which illustrates good disturbance rejection with setting time about 23 sec with zero steady-state error and maximum undershoot of about 0.0172.

Control effort requirement has been achieved and speed

disturbance is damped very well.

20

0

Magnitude (dB)

-20

-40

-60 Control Sensitivity(R) Performance Bound (1/Wu) Sensitivity(S) Performance Bound(1/Wp)

-80

-100 -5

10

0

5

10

10

Frequency (rad/sec)

Figure 5.5: Singular value plots for S with its bound 1/WP and R with its bound 1/ WU.

106


0 -0.002 -0.004

-0.006

Amplitude

-0.008 -0.01 -0.012 -0.014 -0.016 -0.018

0

5

10

15

20

25

30

35

40

Time (sec)

Figure 5.6: System output response disturbance (

to step of 0.08 pu of load

).

In summary, the H f controller design concerning performance can be considered as successful for this problem.

5.5.2 Simulations and Results for Steam Turbine H Design with Uncertainty: A State-Space Approach In this section the time and frequency domain simulations have been carried out on the closed-loop configuration of the steam turbine system as shown in figure 4.13 where K (s) is the designed + controller. The frequency response of the compensated system G(s) K (s) is drawn in figure 5.7. The controller has an integrating action as planned for which tends to damp the low frequency disturbances as will be clear in the time responses simulations. Also the controller has a high frequency roll-off of one dB/decade. o The controller gives a very good phase margin about 72.42 with 3.908 rad/sec

phase crossover frequency and gain margin about 35 dB with gain crossover frequency of 0.26(rad/sec).

107


The tests for Nominal Stability (NS) and Nominal Performance (NP) should be carried out in addition to Robust Stability (RS) and Robust Performance (RP). Nominal stability (NS): since the open-loop system and the controller are stable transfer functions then it is sufficient to check the roots of 1 G(s) K (s) . the closed-loop poles are: 6.499 10 4 , 5.41 10 4 , 66.89 , 10.06 , 4.951, 3.321 , 2.492 ,

0.6691 , 0.2777 , 0.1729 , 0.37715 r j 0.1796 . As seen all the poles lie in

the left-half plane. Hence nominal stability is guaranteed. Nominal performance (NP): firstly this test is conducted by computing the singular values of sensitivity function S of the closed-loop system and comparing it with the bound 1 WPWD3 . Secondly, Comparing the singular values of the control sensitivity R with its performance bound 1 WU WD3 as indicated by entries (1,1) and (2,1) in (4.31). Figure 5.9 confirms that the nominal performance condition has been satisfied. As seen the bound covers the control Sensitivity with wide gap between them. Robust stability (RS): Since the uncertainty in the system is unstructured and is modeled in multiplicative form, the verification of the robust stability is performed by plotting the singular values of the complementary sensitivity function T and comparing it with the frequency response of the bound 1 WI (entry (3, 3) in (4.31)). Both robust stability and nominal performance tests are illustrated in figure 5.8 and figure 5.9. The first figure shows a good coverage for the Sensitivity S and the complementary sensitivity T functions by their bounds, at the same time the sensitivity functions keeps a smooth shape with very little resonant peak. Very sharp roll-off to the complementary function is achieved by the H f controller which is an indication that the high frequency

108


unstructured uncertainty is highly suppressed and hence robust stability is achieved. Robust performance (RP): the inequality PD[ WP S WD3 WI T 1 must be held to achieve robust RP. This is checked graphically as shown in figure 5.10. As illustrated in the figure, the robust performance is satisfied evidently. In fact, as the system under study is SISO, then the above figure can be concluded from figures 5.8 and 5.9, i.e. RP is automatically satisfied when the subobjectives of NP and RS are satisfied. This is special case just for SISO systems. The above results are summarized in tables 5.2 and 5.3, where margins for both robust stability and performance have been calculated for different transfer functions. These are: 1. Complementary Sensitivity function (T). 2. Sensitivity function (S). 3. Load transfer functions (LTF) defined as the transfer functions from d 2 ( s ) to 'Z r (s) ; i.e. 'Z r ( s) d 2 ( s)

LTF ( s)

G2 ( s) S ( s) .

The critical values listed in table 5.2 represent the values of the uncertain elements associated with the intersection of the performance degradation curve and the y 1 x curve (figure 4.3). As it is clear from table 5.2 that the H f norm of the corresponding transfer function in the first column when the uncertain parameters are equal to the critical values is approximately equal to that values obtained from the hyperbola curve in the system performance degradation curve (compare the values of the fifth and sixth columns). Also, the destabilizing values listed in table 5.3 represents the values of uncertain elements, closest to nominal, such that when jointly combined, lead to instability.

109


Figure 5.11 illustrates the output frequency deviation ǻZ r when the system is subjected to a severe speed disturbance of d 2

0.3 pu where the

uncertain parameters ( TCH and TRH ) are equal to typical, critical, and destabilizing values. As a sanity check, it would be useful to compare between the stability and performance margins in tables 5.2 and 5.3, which verifies that the robust performance margin is less than the robust stability margin (it always will be the case) and this in turns verifies the correctness of the above calculations. To check the stability of the system against the variations of the system parameters, the two uncertain parameters has been set to two extreme values as depicted in figure 5.12 where the transient behavior of the output speed deviations due to load disturbance step of 'PL

0.02 pu is plotted. As seen in

the figure, the system exhibits clear performance degradation due to the parametric perturbations of the uncertain variables. But, evidently the system ensures stability even when the uncertain parameters vary over their entire range. The nature of the generating unit with reheat turbine when subjected to 0.04 pu step change in load ǻPL (load disturbance) is illustrated in figure 5.13.

These responses have been computed by using the typical parameters tabulated in chapter two. Values shown are in per unit of the step change. As can be seen in the figure, the increase in PL causes the frequency to decay at a rate determined by the inertia of rotor. As the speed drops, the turbine mechanical power begins to increase. This in turns causes a reduction in the rate of decrease of speed with undershoot of 8.5 10 3 , and then an increase in speed when the turbine power is in excess of the load power(this starts beyond 10.7 sec). The speed will ultimately return to its reference within

value

23 sec settling time and steady-state error of about 0.17% and the

110


steady-state turbine power is increased by an amount equal to the additional load. Lastly, a controller order reduction can be considered where the order of the designed H f controller is of 6th order. An additive error controller reduction method will be applied after plotting the Hankel singular values of the controller as shown in figure 5.14. The reduced order controller is given as: K r ( s)

41176291.2039 ( s + 3333) ( s + 5.995) ( s + 0.1593) ( s + 6.499 10 4 ) ( s + 5.41 10 4 ) ( s + 67.17) ( s + 2.862 10-5 )

If further reduction is needed then one can discard the fourth state. Although this does not affect stability, it affects performance. To determine 6

whether the theoretical error bound is satisfied, the error bound 2 ¦V i is 5

calculated as 2.7963 and K Kr

f

equals to 1.9031 which is less than the

bound (equation (2.73)). Hence the controller reduction process is successful. To compare the performance of the reduced order controller K r (s ) with the original one K (s) , a simulation has been carried out in time domain that shows the transient response of the output frequency deviation ǻZr ( s) when both load disturbance of ( 'PL

0.05 ) and speed disturbance ( d 2

0.04 pu) are

applied to the system simultaneously with the full order K (s) and reduced order K r (s ) controllers as shown in figure 5.15. Table 5.2: Performance margin calculations.

Transfer function (Tf)

Critical Performance frequency margin (rad/sec)

Critical values of the uncertain elements ǻ

TCHU

Tf f

ǻ

f

at critical Values

S

0.9886

0.7667

0.9886 0.3982 3.0454 1.01153

1.01178

LTF

1.04035

0.7667

1.0403 0.4060 11.161 0.96153

0.96162

TRHU

Tf

111

5.5 Analysis and Simulation Results Table 5.3: Stability margin calculations.

Transfer Stability function margin

ǻ

TCHU

TRHU

Destabilizing Closed loop poles

1.089

0.70094

1.3152

0.4472 1.7394

±ʁ0.70094

S

1.0813

0.7667

1.3293

0.4493 1.6833

±ʁ 0.7667

LTF

1.0778

0.7667

1.3293

0.4493 1.6833

±ʁ 0.7667

Destabilizing values of the uncertain elements

T

Destabilizing frequency rad/sec

0

Magnitude (dB)

-200 -400 -600

-800 0

-90

Magnitude (deg)

Compensated System K(s)*G(s) Open loop system G(s)

-180 -270 -360 -450

-2

10

0

10

2

4

10

10

8

10

Frequency (rad/sec)

Figure 5.7: Frequency Response of the open loop system.

6

10

10

10

112


10

0

-10

-20

Magnitude (dB)

-30 -40 -50

-60

Sensitivity (S) WΎtϯͿ Performance bound ϭͬ;t 1/(Wp*Wd1) Complementary (T) / Robustness bound ϭͬt (1/Wi)

-70

-80

-4

10

-2

0

10

2

10

10

4

10

Frequency (rad/sec)

Figure 5.8: Singular values of S, T, and their Performance bounds.

Control Sesitivity (R) Performance bound ϭͬ;t (1/Wu*Wd1) hΎtϯͿ

100

80

Magnitude (dB)

60

40

20

0

-20 -5

10

0

5

10

10

Frequency (rad/sec)

Figure 5.9: Singular values of R and its bound.

113

5.5 Analysis and Simulation Results 1

10

Magnitude (dB)

0

10

-1

10

WP S WD3 WI T

-2

10

-4

-2

10

10

0

2

10

Frequency (rad/sec) (rad/sec)

4

10

10

Figure 5.10: Robust performance test.

0.4

Output response at typical values Output response at destablizing values Output response at critical values

0.3

0.2

ŵƉůŝƚƵĚĞ

0.1

0

-0.1

-0.2

-0.3

0

5

10

15

20

25

30

Time (sec)

Figure 5.11: System performance degradation.

35

40

114


-3

1

0

-1

x 10

T RH

-2

Amplitude

0.4

Typical values T RH 7, TCH 0.3

-3

11, TCH

-4 -5

-6

-7

-8

-9

T RH

0

10

20

3, TCH

0.1

30

40

50

60

Time (sec)

Figure 5.12: Response of ǻZ r for extreme values of the uncertain parameters TRH and TCH .

0.06

Output frequency deviation Control effort Mechanical power Valve position

0.05

0.04

Amplitude

0.03

0.02

0.01

0

-0.01

0

5

10

15

20

25

30

35

40

Time (sec)

Figure 5.13: Transient response of the closed loop system to a small VWHSLQFUHDVH¨PL.

115


10000

9000

8000

7000

6000

abs

20

5000

15

10

4000

5

3000

0

2000

1000

0

4

1

2

5

3

6

4

5

6

Order

Figure 5.14: Hankel singular values of the

controller.

0.04

0.035

0.025

0.02

Amplitude

Reduced order controller Kr(s)

0.015 0.01 0.005 0

-0.005

-0.01

Orginial controller K(s)

0.03

0

5

10

15

20 25 Time (sec)

Figure 5.15: Transient response of disturbances are applied to the system.

30

35

40

with both load and speed

116


5.5.3 Simulations and Results for Steam Turbine H Design without Uncertainty: A Polynomial Approach The frequency response of the compensated open loop system G(s) K (s) gives very good results concerning the phase margin about 37 o and

gain margin 8.3 dB as shown in the nyquist plot in figure 5.16. The crossover frequency is 1.4 (rad/sec) while the phase crossover frequency is 3.58 (rad/sec) which is higher than that of the uncompensated open loop system which is 3 (rad/sec). The singular values of the sensitivity, complementary sensitivity, and control sensitivity functions together with their performance bounds are depicted in figure 5.17. The performance bound 1 W1V is exactly coincident on the sensitivity function S which in this case indicates that the sensitivity functions has been well reshaped, as can be seen it has a smooth shape with little resonant peak near the cutoff frequency. Hence nominal performance (NP) is achieved. Plotting the frequency response of T and comparing it with the bound G W2V is exactly the same as checking the condition (2.35) since R( s) T ( s) G ( s) . The bound G W2V

accurately fits the complementary

sensitivity function and also it has a regular shape with little peak. The controller resulted in the following closed-loop system poles :9.47107 r j1.6648 , 3.3796 r j 3.4707 , 5.5 , 5 , 3.5 3.3333 , 2.3348 , 0.526

, 0.7071 r j 0.7071 , 0.8928 . All of the closed-loop poles lie in the left half of the complex plane. Hence, the controller ensures the nominal stability (NS). They include the pre-assigned poles as planned for it. As can be seen, the last four poles dominate the rest. Simulations concerning system performance in the time domain are conducted in the following. The output frequency deviation in response to step change in the speed disturbance of about

0.04 pu is shown in

figure 5.18. while the control effort ( u ) response when the system is subjected

117


to load disturbance of 0.08 pu is shown in figure 5.19 which indicates that the control signal remain below its bound 1 for all t as required. A said earlier the bandwidth of the closed-loop system has been set to 1 rad/sec. This requirement has been achieved as indicated in figure 5.17. A curve that illustrates the effect of the closed-loop bandwidth on the speed of the system is shown in figure 5.20. As clear in the figure, as the bandwidth of system gets larger, the response of the output frequency deviation gets faster.

2

1.5

1

Imaginary Axis

0.5 0 -0.5

-1 -1.5 -2 -4

-3.5

-3

-2.5

-2

-1.5

-0.5

0

Figure 5.16: Nyquist Plot of the compensated open-loop system.

-1

Real Axis

0.5

118


2

10

0

10

-2

10

Magnitude

-4

10

-6

10

-8

10

Sensitivity (S) 1/|W1V| Complementary (T) |G/W2V|

-10

10

-12

10

-3

10

-2

10

-1

10

0

1


2

3

10

10

Figure 5.17: Singular values of S and T functions with their performance bounds.

0.04

0.03

0.02

Amplitude

0.01

0

-0.01

-0.02

0

5

10

15

Time (sec)

Figure 5.18: Transient response of

due to

pu.

119


0.4

0.35

0.3

0.25

Amplitude

0.2 0.15

0.1

0.05

0

0

2

4

6

8

10

12

14

16

18

20

Time (sec)

Figure 5.19: Transient response of u due to step change of .

-3

3

1

0

-1

Amplitude

-2

-3

-4

-5

-6

B=1.111 B=1 B=1.25 B=0.9

2

x 10

-7

0

5

10

15

Time (sec)

Figure 5.20: Relationship between the bandwidth of the closed loop system (B) and .

120


5.5.4 Simulations and Results for Steam Turbine H Design with Uncertainty: A Polynomial Approach The frequency response of the compensated system is the same as shown in figure 5.16 with the phase margin decreased to 32o while the gain margin rose to 11 dB. The singular values of the sensitivity, complementary, and control sensitivity functions together with their bounds are depicted in the figures 5.21. As evident from the figure, the sensitivity function at low frequencies satisfy (2.34), while at high frequencies the control sensitivity function satisfy (2.35) which indicates that the design made quite effective control over the shape of the sensitivity and control sensitivity functions and hence over the performance of the closed-loop system. The transient response of the output frequency deviation due to a step change of 0.04 pu in the load is shown in figure 5.22. Also the mechanical power is drawn in the same figure. The settling time of the output frequency deviation 'Z r is about 10.4 sec with undershoot of about 0.0041 . While the overshoot in the mechanical power response is about 48% . Time response of the control effort ( u ) is illustrated in figure 5.22 with little peaking during the transient but settles down at a value within its allowable range [0,1] as intended by the deisgn. The response approximately has the same shape as that of the valve position which confirms the fact that the dynamics of the speed relay (SR) as well as the servo motor (SM) are almost negligible. Nominal staility(NS) is achieved since all of the closed-loop poles of the nominal plant lie in the open left-half plane. Nominal performance(NP) is guaranteed via shaping the closed-loop transfer functions by the filters W1 ( s ) and W2 ( s ) so that the performance bounds 1 W1V

and

1 W2V

covers the frequency response of the sensitivity S and

control senstivity functions as done in figure5.21.

121


Robust stability(RS) can be checked by plotting the frequency response of the of the bound 1 W2V and that of the control sensitivity R . as shown in figure 5.21, the frequency response of R lie below the bound 1 W2V . So, this RS condition is satisfied. And finally since RS and NP tests have been passed, then Robust Performance (RP) will be satisifed consequently. Load test is very important for checking the robust stability of the closedloop system. Assume the situation of a variable load condtion, where the load signal ǻPL represents the load demand that is variable with time as shown in figure 5.23. Figure 5.23-a shows the mechanical power in response to same load demand. The output response of the freqeucny deviation ǻZ r is shown in Figure 5.23-b. It is clear from figure 5.23 that the controller and hence the closed loop system has the capability to follow the variable load demand while at the same time the system suppresses the load disturbance even with variable step changes up and down as can be seen in figure 5.23-b. with settling time of 10 sec. To show the ability of the controller against parametric varaitions, consider figure 5.24. where the response of the output frequency devaition is plotted when a speed distrubance of 0.04 pu has been applied to the system. As can be seen stability against the varaitions is obviously guaranteed, of course, with clear degradation in the performance of the system. Effect of the bandwidth on system UHVSRQVH¶VVSHHGLVWKHVDPHDVLQWKH previous section (figure 5.20).

So, the designed controller is considered

successful. Consider now the reduction of controller order. After balancing of the controller followed by optimal hankel approximation, its order is reduced to

122


two. Further reduction of the controller leads to deterioration of the closed-loop system dynamics. The transfer function of the reduced order controller is given by: K r ( s)

118.5328 ( s + 0.1763) s ( s + 3.541)

The time domain simulation of the output frequency deviation of the closed loop system with full order and reduced order controllers when load step change of 0.04 pu is applied to the system is illustrated in figure 5.25. While The frequency responses of the full order K (s) and reduced order K r (s ) controllers are shown in figure 5.26. Finally, it is of particular interset to end this secion with some of the observations regarding the polynomial apporach: 1. By experience it is found that when the pre-assgined poles (roots of M (s) ) are located far from the imajinary axis, the system becomes more faster. But this leads to high control effort ( u ) to be introduced by the controller and this sometimes causes actuator saturation. So, it is preferable to avoid saturation as large as possible and keeping faster response at the same time. 2. There is direct effect presented by the coeffieciets h and r of the filter W2 ( s ) . As h increases the system gets slower while increasing r leads to high

gain and phase margin. Also increasing h and r leads to high value of gamma which is sensitive to h more than to r . Finally, as the bandwith of the system B becomes higher, the system gets faster and the converse is true.

123


2

10

1

10

Magnitude

0

10

-1

10

Sensitivity (S)

-2

10

Bound |1/W1V| Control Sensitivity (R)

-3

10

Bound |1/W2V|

0


Figure 5.21: Singular values plots of

and

with their bounds.

0.16

0.12

0.1

Amplitude

Valve position Mechanical power Control effort output frequency deviation

0.14

0.08

0.06

0.04

0.02

0 0

5

10 Time (sec)

Figure 5.22: Transient response of the steam turbine generating unit.

15

124


0.04

Mechanical power Load demand

0.035

0.03

0.025 Amplitude

0.02 0.015

0.01

0.005

0

0

5

10

15 Time (sec)

30

0.03

0.025

0.02

Amplitude

25

;ĂͿ

20

Output frequency deviation Load demand

0.015 0.01

0.005

0 -0.005

0

5

10

15 Time(sec)

20

25

30

;ďͿ

Figure 5.23: Transient response of : (a) variable load demand.

, (b)

in response to

125

5.5 Analysis and Simulation Results 0.04

TRH= 3, TCH=0.1 typical data (TRH =7, TCH =0.3) TRH=11, TCH= 0.4

0.03

0.02

Amplitude

0.01

0

-0.01

-0.02

-0.03

0

5

10

15

20

25

30

Time (sec)

Figure 5.24: Transient Response of

due to parameter variations.

-3

2

x 10

Full order K(s)

Reduced order Kr(s)

1

0

Amplitude

-1 -2

-3 -4 -5

0

5

10

15

20

25

Time (sec)


GXHWR¨PL=0.04 pu.

30

126


80

60

Magnitude (dB)

Phase (deg)

20 0

-40 0

40

-20

Full order controller K(s) Reduced order controller Kr(s)

-45

-90 -3 10

-2

10

-1

0

10

1

10

10

2

10

3

10

4

10

Frequency (rad/sec)

Figure 5.26: Frequency Response of

controller.

5.5.5 Comparison between State-Space and Polynomial Methods This section summarizes the results of the prevoius sections through a comparison between the state-space and the poylnomial approaches. The comparison will be extended to include time domain performance, freuqency doamain performance, controller order, uncertainty representation, weighting filters, optimality ans suboptimality. This will be repsresneted through tables and curves with reasons of similarities and dissimilarities. As will be clear through discussion: 1. Concerning time domain performance, the polynomial approach is evidently faster than the state-space approach as shown in figure 5.27, where the transient response of the output frequency deviation 'Zr is drawn using both approaches when the system is subjected to sudden increase in the load signal of 0.04 pu (i.e. ǻPL

0.04 ).

127


As shown in the figure, the polynomial approach has less settling time ( 10.4 sec) as compared to state-space approach ( 20.3 sec) with overshoot that

exist in the polynomial approach response of peak 1.48 10 3 at 4.7 sec while the undershoot in the response of the polynomial is approximately half to that using the state-space approach. This means that the polynomial approach satisfy the design specifications (item 1 and 2 in the time domain specifications) better than the state-space approaches. Both of them settle down to almost zero steady-state error. The reason behind this improved and fast response lies in the added flexibility to the design using the polynomial approach, whereas partial poleplacement process enables the designer to get some control on the location of some of the poles of the closed-loop system and the ability to determine nature of the response through placing the poles in different place in the open LHP. Table 5.4 shows the settling time and undershoot values for the signal ǻZ r using both approaches for ǻPL

0.04 pu.

2. Controller order, the order of the resulting controller using state-space approach equals to the order of the original plant plus the orders of the weighting filters (performance and uncertainty filters) that are used to generate the extended plant. While in the polynomial approach controller order equals to the order of the original plant plus the order of inverse of the weighting filter W2 ( s ) (i.e. W 12 (s) ).

Hence, it can be concluded that the order of the controller designed using the polynomial approach is always less than the order of the controller designed using the state-space approach at least by one.

If W2 (s) was constant

(i.e. W2 ( s ) 1 c ), then the order reduction will be two. This is if the uncertainty was neglected, while in case that the uncertainty has been taken

128


under consideration in the design, then the design will be carried out on the nominal plant. Hence, the reduction in the order will be at least two. Furthermore, If W2 ( s ) was constant (i.e. W2 ( s) 1 c ), then the gain in the reduction will be three, see table 5.5. Moreover, the zeros of the controller designed using the state-space approach includes the poles of the weighting filter WU (s) and W I (s ) , and the stable open-loop poles of the original plant G(s) . While its poles includes the poles of WP (s ) . By using the polynomial approach, the poles of the controller includes the poles of W1 ( s) . But its zeros involves the zeros of M (s) that represent the open-loop poles of the original plant G(s) . In table 5.5, the orders of the H controllers with their formulas for steam turbine system are shown. 3. Frequency domain specifications, regarding the design requirements of the system in the frequency domain, the polynomial approach exhibits obvious reductions in the gain and phase margins from their corresponding values using state-space approach. Table 5.6 lists the values using both approaches. As it is shown in table 5.6, the phase and gain margins of the polynomial approach are within the allowed range, where the safety range of the gain margin for most of the practical system is GM t 6 dB . While for the phase margin 30R d I d 80R . 4. Increasing or decreasing the bandwidth is more flexible with the polynomial approach than the state-space approach. The reason behind this is that the closed-loop bandwidth is tightly related to radius of the desired complex conjugates poles that are pre-assigned through a partial pole placement process. While in the state-space approach, the bandwidth of the system is determined through the selection of the frequency Z B which is the cutoff frequency of the filter W P . This frequency is a very sensitive

129


parameter since selecting Z B should not violate the condition of nominal performance (4.16). 5. Integral control can be designed easily in the polynomial approach and much easier than using the state-space approach. This is because the state-space method doesn't accept a pole on jZ -axis in the weighing function or the plant, while there is not such limitation in the polynomial approach. Furthermore, high frequency roll-off occurs by using improper weighting filters in the design procedure. This can occur in the polynomial approach by using improper W 2 ( s ) which cannot occur in the state-space approach.

130


0.03

0.025

Polynomial approach 0.02

0.015

Amplitude

0.01 0.005

0

-0.005

State-space approach

-0.01 -0.015

0

5

10

15

20

25

30

35

40

Time (sec)

;ĂͿ

-3

2

x 10

0

Amplitude

-2

Polynomial approach State-space approach

-4

-6

-8

-10

0

5

10

15

20

25

30

40

;ďͿ


in response to, (a) Speed

disturbance step change, (b) Load disturbance step change.

35

Time (sec)

131


Table 5.4: Time Domain specifications of the closed loop system.

State-Space Approach

Parameter

Polynomial Approach

Without With Without With uncertainty uncertainty uncertainty uncertainty

Settling time /sec

23.7

20.3

7.06

10.4

Undershoot

8.58*10-3

8.46*10-3

5.56*10-3

4.27*10-3

7DEOH2UGHUVRI+FRQWUROOHUIRUGLIIHUHQWDSSURDFKHV

State-Space Approach Parameter

Without uncertainty

With uncertainty

ord(G)+ord(WP)+ord(WU)+ord(WI)

Polynomial Approach Without uncertainty

With uncertainty

ord(G)+ord(1/W2)

Formula

Order

8

6

7

5

Order after reduction

5

4

3

2

Table 5.6: Frequency domain specifications of the open loop system.

State-Space Approach

Parameter

Without With Without uncertainty uncertainty uncertainty 78 o

72 o

37 o

32 o

Gain Margin /dB

25

56

8.3

11

2.68

3.91

3.58

2.375

0.227

0.26

1.4

0.965

With uncertainty

Phase Margin

Phase crossover frequency rad/sec Gain crossover frequency rad/sec

Polynomial Approach

132


5.5.6 Simulations and Results for Boiler-Turbine HDesign It is necessary to investigate the stability and performance of the closedloop system with the designed controller K (s) in both time and frequency domains. Figure 5.28 shows the singular values of the closed-loop system Tzw (s ) . Since it is of size 6 3 , there are three (nonzero) singular values at each

frequency. As shown in the figure, two of these three singular values are highly coincident on each other. With this designed controller the conditions NS, NP, RS, and RP can be checked easily. NS Test: With G(s) as given in (5.5) and the designed controller K (s) presented in section 5.4, it is found S , and T are stable since all poles of the closed-loop system lie in the open LHP, so the system is nominally stable. NP Test: Since the H f - norm of the closed loop system is close to one, the condition for nominal performance V WP S 1 is satisfied in this case. This may be checked by plotting the maximum singular value of the sensitivity function and comparing it with that of the inverse of the performance weighting filter WP (s ) . The result of the comparison is shown in figure 5.29. It can be seen that the sensitivity function S lies below 1 W P over the whole frequency region. RS Test: robust stability test V WI T 1 is conducted by drawing the maximum singular values plots of the complementary sensitivity function T and the uncertainty filter W I together on the same frequency grid. As shown in figure 5.29. It is evident that the system with this controller K (s) achieves robust stability since 1 WI exactly fits T at low frequencies and still covers it even at high frequencies. Hence, RS has been satisfied. It is interesting to observe from figure 5.29 that no resonant peaks have been found in both the sensitivity S and the complementary sensitivity T functions and they have smooth shapes which indicate that their transient

133


responses will be free from any oscillations. Also, S is low in the frequency region [0,0.05] where the disturbance is active which means satisfactory disturbance rejection obtained at each output channel (see figure 5.33). While the complementary rolls off quickly before 20 rad/sec to indicate that the modeling errors and measurement noise have been attenuated. RP Test: the robust performance of the closed-loop system may be tested by means of singular values plot of V WI T V WP S 1 with respect to the frequency as shown in figure 5.30. It is clear from the figure that the closedloop system with K (s) fails to satisfy the robust performance criterion. This confirms that for MIMO systems even that the subobjectives of NP and RS have been satisfied, and then it does not imply that RP is achieved. With respect to the robust performance, this means in the present case that the size of the perturbations must be limited up to certain bound at which the system becomes on the edge of robust performance. The frequency response of the system with and without compensation is shown in figure 5.31. As can be seen in the figure, the crossover frequency is 1 which lies in the interval [0.05,20] as required. Also, the compensated system G(s) K (s) rolls off at higher rates than the uncompensated one. Finally, the

controller provides the system with integral control which is shown in the figure as the compensated system rolls off at low frequencies with 20 dB/decade. Transient responses of the drum pressure, electrical output, and drum water level in response to unit step command signals ( r1 , r2 , r3 ) sequentially applied to the system at three different operating points G1 ( s), G3 ( s), and G 7 ( s ) are shown in figure 5.32. As can be seen in the figure, the controller K (s) achieves all of the design specifications mentioned early. The decoupling property is highly evident and the steady state error is about 0.4% .

134


The disturbance at each output channel is well rejected with settling time is around 5 sec as shown in figure 5.33, whereas the output response of the drum pressure (first channel) has been plotted against time when the system is subjected to a unit step disturbance signal d [1,0,0] . Consider now the reduction of the controller order. As indicated in the results of the H f optimization, the controller is of order 9 . It would be good for implementation if the order could be reduced while essentially keeping an acceptable performance. After balancing the controller and neglecting the small hankel singular values its order is reduced to 6 without loosing too much performance. In Figure 5.34 the transient response of the electrical output when a step change in the electrical output command is imposed on the system with the full order and reduced order controllers is shown. As can be concluded from the figure, stability is ensured with the reduced order controller with little performance degradation. Finally, the load test is very important issues in MIMO system, where the system will be subjected to practical command signals other than steps of units in magnitude. The following signals have been applied to the input and disturbance channels: r1 (t ) 108 (120 - 108) ^ u (t - 200) - u (t - 300)` 20 u (t - 300) r2 (t ) 66.65 (120 - 66.65) û (t - 500) - u (t - 650)`

40 û (t - 650) - u (t - 680)` 65 u (t 680) r3 (t ) 6 u (t 250) d1 (t ) - 5 u (t 100) d 2 (t ) 40 u (t 400) d 3 (t ) 3 u (t 600)

;ϱ͘ϵͿ

;ϱ͘ϭϬͿ

The responses of the outputs to these test signals are shown in figures 5.35. As can be seen in the figure, the effect of the disturbance is almost negligible at the output channels.

135


Control signals ( u1 , u 2 , u 3 ) responses when step change in the electrical output is applied at the 6th operating point are illustrated in figure 5.36. All of the control signals ( u ) lie at the steady state within the range [0,1] as required.

136


1.2

1.1

1

Magnitude

0.9 0.8 0.7

0.6 0.5 -2

10

0

2


10

Figure 5.28: Singular values plot of the weighted closed loop Tzw(s).

0

10

ʍ;ϭͬt data1 WͿ data2 /Ϳ ʍ;ϭͬt

Magnitude

data3 ʍ;d;ʘͿͿ

-1

10

data4

ʍ;^;ʘͿͿ

-2

10

-3

10

-4

10

-3

10

-2

10

-1

10

0


1

10

2

10

3

10

Figure 5.29: Nominal performance and robust stability conditions.

4

10

137


Magnitude

0

10

-5

0

10

5


10

300

'ϯ;ƐͿ 200

100

Magnitude (dB)

0

-100

-200

-300

-400

-10

10

-5

0

10

10

5

10

Figure 5.31: Frequency response plot of the open loop system L=G*K.

10

10

Frequency (rad/sec)

138


139


0.8

Magnitude

0.6

ŽͬƉϭ;ĚƌƵŵƉƌĞƐƐƵƌĞͿ 0.4

ŽͬƉϮ;ĞůĞĐƚƌŝĐĂůŽƵƚƉƵƚͿ

0.2

ĂŶĚŽͬƉϯ;ĚƌƵŵǁĂƚĞƌůĞǀĞůͿ

0

-0.2

0

2

4

6

8

10 12 Time /sec))

14

16

18

20

Figure 5.33: Disturbance rejection at the output channels due to step disturbance d= [1,0,0].

1

Full order controller K(s)

Magnitude

0.8

Reduced order controller Kr(s)

0.6

0.4

0.2

0

0

5

10 Time /sec)

Figure 5.34: Transient response of the Drum Pressure.

15

140


130

125

ƌƵŵƉƌĞƐƐƵƌĞ;ĞǀĞů;ŵͿ

6 5 4 3

-4

x 10 3.5 3

2

2.5 2 1.5 1 0.5

1

0

-1

0 -0.5 100

0

100

200

110

120

130

140

150 160 Time (sec)

170

180

190

200

300 400 Time (sec)

500

600

700

;ĐͿ

Figure 5.35: Transient response of the outputs: (a) c1,(b) c2, (c) c3 in response to (5.9) and (5.10).

0.05

0

ŽŶƚƌŽůƐŝŐŶĂů;ƵϭͿ

-0.05 -0

-0.1

-0.15

-0.2

0

2

4

6

8

10

12

14

Time ((sec)

;ĂͿ

Figure 5.36: Continued

16

18

20

142

5.5 Analysis and Simulation Results 0.06

0.05

ŽŶƚƌŽůƐŝŐŶĂů;ƵϮͿ

0.04

0.03

0.02

0.01

0

0

2

4

6

8

10

12

14

16

18

20

12

14

16

18

20

Time (sec)

;ďͿ

0.5

0

ŽŶƚƌŽůƐŝŐŶĂů;ƵϯͿ

-0.5

-1

0

2

4

6

8

10 Time (sec)

;ĐͿ

Figure 5.36: Transient responses of the signals (u1, u2, and u3) due to unit step in the electrical output reference command (r2).

Chapter C HAPTERFive SIX CONCLUSIONS AND DIRECTIONS

Robust Controllers Design for Uncertain FOR FUTURE RESEARCH Power Systems using H-infinity echniques 6.1 Conclusions LFC is an important mechanism in power systems, and by which a balance between power generation and demand is satisfied. In this work LFC is achieved through design of a robust controller using H f control theory to improve the system robustness, disturbance rejection, and tracking the command signal. The state-space and the polynomial approaches are employed to design a robust governor that solves the LFC problem. The proposed H f robust controller has been designed and the following points have been concluded: 1. Both state-space and polynomial approaches result in a proposed robust controller which ensures both robust stability and robust performance of the closed-loop system against parametric variations and neglected dynamics for the steam turbine system (see figures 5.8, 5.12, 5.17, 5.24). 2. Both state-space and polynomial approaches satisfy time domain specifications but it is important to stress that better results, with respect to the transient response characteristics (undershoot, settling time, rise time) are obtained by using the polynomial approach (see figure 5.27).

143

6.1 Conclusions

144

3. The state-space approach achieves slightly better results in the frequency domain than the polynomial approach. 4. Problem formulation and satisfying the design requirements is much easier to do with the polynomial approach than with the state-space one. 5. The H f robust controller designed by using the two approaches for the boiler-turbine unit satisfies the nominal performance and robust stability conditions while it fails to ensure the robust performance condition. 6. The designed H f robust MIMO controller applied on the boiler-turbine system gives very good time and frequency domain results which means that the design requirements are almost satisfied with the designed robust controller (see figures 5.29, 5.31, 5.33, and 5.35). 7. Good disturbance attenuation requires sufficiently large closed-loop bandwidth. This may, however, lead to difficulties in achieving robust stability and robust performance of the closed-loop system in the presence of parametric variations in the system as in the case of the boiler-turbine unit. 8. The H f robust MIMO controller designed for the boiler-turbine unit introduces a very good decoupling action to the closed-loop system (see figure 5.32). 9. The order of the resulting H f controller is high, thus required order reduction to preserve the closed-loop performance and robustness while making the implementation of the controller easier and more reliable, knowing that polynomial approach results in robust controller of lower order than the state-space approach. 10. Finding an appropriate weighting functions is a crucial step in H robust control designs. It usually involves trials-and-errors. Design experience and knowledge of the plants will help in choosing weighting functions.

6.2 Directions for Future Research

145

11. Integral control and high frequency roll-off are already met in the designed robust H f controller for both steam-turbine and boiler-turbine systems. Furthermore, integral control can be achieved in the polynomial approach much easier than in the state-space approach.

6.2 Directions for Future Research Although the main problems in this book are solved successfully, several areas can be further investigated: 1. Solving the LFC problem for multimachine power systems in which the power systems consist of several interconnected control areas where each one is responsible for its native load. Interarea oscillations modes have to be considered during the design of the robust governor. 2. Designing a microprocessor based digital controller for solving the LFC problem in power systems. The digital controller can be made robust by including the uncertainty into account during the design. Then for implementation issue, a microprocessor based system has to be developed to achieve this purpose. 3. Designing a P -optimal MIMO controller for boiler-turbine system to achieve robust stability as well as robust performance conditions against V\VWHP¶VRSHUDWLQJSRLQWFKDQJH 4. Developing an optimization algorithm for finding the optimum set of performance weighting functions that are used in the H f design techniques. 5. Applying the nonlinear H f theory directly to the nonlinear boiler-turbine system. In this case the design procedure will deal with the nonlinear dynamics of the system without a need to make any linearization to the nonlinear model and the result is a nonlinear H f robust controller.

Appendix A

146

Appendix A Appendix A State-Space Matrices of the Linearized Boiler System at Seven Operating Points This appendix contains the state-space matrices ( A, B, C , and D ) of the linearized model of the boiler-turbine system at seven operating points as listed in table 2.2. According the data listed in that table the state-space matrices at each operating point are given below:

first operating point

A

0 0º ª 0.001679 « 0.00372 0 . 1 0»» , « 0 0»¼ ¬« 0.004

C

1 0 0 º ª », D « 0 1 0 » « ¬«0.009512 0 0.004483¼»

B

ª0.9 0.23367 0.15 º «0 9.4767 0 »» « ¬« 0 0 0.9783 1.6588¼» 0 0 º ª 0 « 0 0 0 »» « ¬«0.25327 0.3586 0.014¼»

Second operating point

0 0º ª 0.00195 « 0.1 0»» , A « 0.0477 «¬ 0.0049 0 0»¼

C

B

1 0 0 º ª « », D 0 1 0 « » «¬0.008154 0 0.004547 »¼

ª0.9 0.27155 0.15 º «0 11.0128 0 »» « «¬ 0 1.11811 1.6588»¼

0 0 º ª 0 « 0 0 0 »» « «¬0.25327 0.40992 0.014»¼

Appendix A

147

Third operating point

A

0 0º ª 0.00222 « 0.05847 0.1 0» , » « «¬ 0.0058 0 0»¼

C

0 0 º ª 1 « 0 », D 1 0 « » ¬«0.00713 0 00.004625¼»

B

0.31 0.15 º ª0.9 «0 12 . 573 0 »» « «¬ 0 0 1.2578 01.6588»¼ 0 0 º ª 0 « 0 0 0 »» « ¬«0.25327 0.4611 0.014¼»

Fourth operating point

0 0º ª 0.0025 « A « 0.06942 0.1 0»» , B «¬ 0.00669 0 0»¼

C

0 0 º ª 1 « 0 1 0 »» , D « «¬0.00634 0 0.0047 »¼

0.15 º ª0.9 0.349 «0 14.1554 0 »» « «¬ 0 0 1.3976 1.6588»¼

0 0 º ª 0 « 0 0 0 »» « «¬0.25327 0.5124 0.014»¼

Fifth operating point

0 0º ª 0.00279 A «« 0.08055 0.1 0»» , «¬ 0.007587 0 0»¼

B

ª0.9 0.3885 0.15 º «0 15.7576 0 »» « «¬ 0 0 1.5374 1.6588»¼

Appendix A

C

148

1 0 0 ª º « », D 0 1 0 « » «¬0.005722 0 0.004782 »¼

0 0 º ª 0 « 0 0 0 »» « «¬0.25327 0.5636 0.014»¼

Sixth operating point

A

0 0º ª 0.00307 « 0.09184 0.1 0» , « » «¬ 0.00848 0 0»¼

C

0 0 ª 1 º « 0 », D 1 0 « » ¬«0.00522 0 0.004854 ¼»

B

ª0.9 0.4285 0.15 º «0 17.3781 0 »» « «¬ 0 0 1.6771 1.6588»¼ 0 0 º ª 0 « 0 0 0 »» « ¬«0.25327 0.61488 0.014¼»

Seventh operating point

0 0º ª 0.00337 « A « 0.10327 0.1 0»» , B «¬ 0.00937 0 0»¼ C

0 0 º ª 1 « 0 1 0 »» , D « «¬0.00481 0 0.00492»¼

ª0.9 0.4688 0.15 º « 0 19.0156 0 »» « «¬ 0 0.9783 1.6588»¼ 0 0 º ª 0 « 0 0 0 »» « «¬0.253 0.6661 0.014»¼

References

149

References Bibliography [1] .XQGXU 3 ³3RZHU 6\VWHP 6WDELOLW\ DQG &RQWURO´ 0F*UDZ-Hill Inc., 1994. [2] $QGHUVRQ 3 0 DQG )RXDG $ $ ³3RZHU 6\VWHP &RQWURO DQG 6WDELOLW\´-RKQ:LOH\DQG6RQV,QF [3] Kim W. G., Moon U. C., Lee S. C., and Lee K. Y., ³Application of dynamic matrix control to a boiler-turbine s\VWHP´ ,(((, Power Engineering Society General Meeting, Vol. 2, pp. 1595-1600, June 12-16, 2005. [4] Khodabakhshian A. and Golbon 1³5REXVWload frequency controller design for hydro power s\VWHPV´ ,((( Conference on Control Applications, Toronto, Canada, pp. 1510-1515, August 28-31, 2005. [5] Hadi S. ³3RZHU6\VWHPDQDO\VLV´0cGraw-Hill Inc., 1999. [6] *UHHQ0DQG/LPHEHHU'-1³/LQHDU5REXVW&RQWURO´3UHQWLFH-Hall Inc., 1995. [7] )UDQFLV%$³$FRXUVHLQ H f Control TKHRU\´Volume 88 of Lecture Notes in Control and Information Sciences, Springer-Verlag, 1st edition, 1987. [8] =DPHV * ³)HHGEDFN DQG optimal sensitivity: Model reference transformations, multiplicative semi-norms and approximate iQYHUVHV´ IEEE Transactions on Automatic Control, Vol. 26, pp. 301±320, 1981. [9] Doyle J. C., Glover K., Khargonekar P. P., and Francis B. A., ³6WDWHspace solutions to the standard H 2 and H f control pUREOHPV´ ,((( Transactions on Automatic Control, Vol. 34, No. 8, pp. 831±847, 1989.

References

150

[10] *DKLQHW33DQG$SNDULDQ3³$OLQHDUPDWUL[LQHTXDOLW\DSSURDFKWR H &RQWURO´,QWHUQDWLRQDO-RXUQDORI5REXVWDQG1RQOLQHDU&RQWURO, pp. 421-448, 4(1994). [11] *ULPEOH 0 - ³,QGXVWULDO &RQWURO 6\VWHPV 'HVLJQ´ -RKQ :LOH\ Sons Ltd, Chichester, UK, 2001. [12] 5HUNSUHHGDSRQJ ' DQG )HOLDFKL $ ³3, gain scheduler for load frequency control using splines tHFKQLTXHV´ IEEE Proceedings of the 35th Southeastern Symposium on System Theory, pp. 259-263,16-18 March 2003. [13] .KRGDEDNKVKLDQ $ DQG *ROERQ 1 ³8QLILHG 3,' design for load frequency cRQWURO´,((( Conference on Control Applications, Taipei, Taiwan, pp. 1627-1632, September 2-4, 2004. [14] Moon Y. H., Ryu H. S., &KRL%.DQG&KR%+³0RGLILHG3,'load frequency control with the consideration of valve position lLPLWV´ IEEE, Power Engineering Society Winter Meeting, NY, USA, Vol. 1, pp. 701-706, January 31- February 4, 1999. [15] 0RRQ