Drum-boilercontrolperformance …elib.suub.uni-bremen.de/edocs/00104228-1.pdf ·...

Universität Bremen

Master Thesis

Drum-boiler control performance

optimization using an observer-based

state-feedback controller within

MATLAB/Simulink environment

Ahmed Elguindy

December 11, 2013

Tutor:

Dipl.-Ing. Simon Runzi

1st Examiner:

Prof. Dr.-Ing. Kai Michels

2nd Examiner:

Prof. Dr.-Ing. Bernd Orlik

Acknowledgment

It gives me great pleasure in expressing my sincere gratitude to everyone whohave supported and contributed into making this thesis possible.

I would like first to acknowledge my direct supervisor Dipl.-Ing. Simon Runzifor his enthusiasm, inspiration and huge efforts to explain things clearly andsimply. His in-depth knowledge regarding the CHP plant in Munich, related tohis PhD research, was quite helpful and beneficial for my work. Furthermore Iwould like to thank my examiner Prof. Dr.-Ing Kai Michels for offering me theproject which have evolved over the course of time into an interesting thesistopic. I wish also to address their constructive criticism following initial reviewof the thesis.

My appreciation for SWM Services GmbH, specially Mr. Julian Niedermeierfor his willingness to perform experiments on the plant, its priceless valuableinformation contributed significantly to improve my understanding of the realprocess.

I wish to acknowledge the scholarship support provided by the KatholischerAkademischer Auslander-Dienst (KAAD). In particular I am very grateful toDr. Christina Pfestroff as I do believe that my master studies in Germanywouldn’t have been possible without her guidance when applying for the schol-arship. I thank as well Prof. Dr.-Ing Rainer Laur, Mr. Hans Landsberg, Mr.Raphael Nabholz and Mrs. Claudia Dillmann for their continuous follow-upand assistance.

Lastly and most importantly, I dedicate this thesis to my parents who raised,supported, taught and loved me throughout my entire life.

Abstract

This thesis presents the development of an observer-based state-feedback con-troller designed using LQ and pole placement methods to optimize pressureand water level control performance of a drum-boiler unit that belongs to a450MW CHP plant in Germany. The Astrom-Bell nonlinear model is initiallybuilt within MATLAB/Simulink environment, later enlarged to include theprocess PID-controllers and control valves regulating mass flow rates beforebeing validated against data measurements with very rich excitation. The con-cluded simulation results adopting the newly proposed control strategy showsthat the suggested multivariable control technique outperforms the existingPID-controller in many aspects improving the control performance significantlyand yielding much tighter reference value tracking during load changes.

Keywords: drum-boiler level control; optimal control; multivariable feedbackcontrol; power plants simulation

Contents

Contents

1. Introduction 6

2. Process modelling 82.1. Combined cycle process overview . . . . . . . . . . . . . . . . . 8

2.1.1. Gas turbine . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.2. Heat recovery steam generator . . . . . . . . . . . . . . 92.1.3. Steam turbine . . . . . . . . . . . . . . . . . . . . . . . 102.1.4. Surface condenser . . . . . . . . . . . . . . . . . . . . . 10

2.2. Steam generation process description . . . . . . . . . . . . . . . 112.2.1. Drum-boiler mass and energy balance . . . . . . . . . . 112.2.2. Drum-boiler nonlinear state equations . . . . . . . . . . 142.2.3. Mass flow control valve . . . . . . . . . . . . . . . . . . 152.2.4. Process PID-controller . . . . . . . . . . . . . . . . . . . 16

2.3. MATALB/Simulink model . . . . . . . . . . . . . . . . . . . . . 182.3.1. Drum-boiler model . . . . . . . . . . . . . . . . . . . . . 192.3.2. Control valve and actuator model . . . . . . . . . . . . . 212.3.3. Process PID-controller model . . . . . . . . . . . . . . . 22

3. Process analysis and validation 233.1. Theoretical overview . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1. Concept of stability . . . . . . . . . . . . . . . . . . . . 233.1.2. Linearization . . . . . . . . . . . . . . . . . . . . . . . . 243.1.3. Poles and zeros . . . . . . . . . . . . . . . . . . . . . . . 24

3.2. Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1. Linear state-space model . . . . . . . . . . . . . . . . . 263.2.2. I/O pole-zero plot . . . . . . . . . . . . . . . . . . . . . 27

3.3. Open loop step response . . . . . . . . . . . . . . . . . . . . . . 293.3.1. Change of gas turbine electrical output power . . . . . . 293.3.2. Change of butterfly valve position . . . . . . . . . . . . 303.3.3. Change of feedwater control valve position . . . . . . . . 30

3.4. Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.1. Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 32

4

Contents

3.4.2. Comparison with measurement data . . . . . . . . . . . 333.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 36

4. Process optimization 384.1. Concept of state-feedback control . . . . . . . . . . . . . . . . . 38

4.1.1. Controllability and observability . . . . . . . . . . . . . 384.1.2. Observer-based control . . . . . . . . . . . . . . . . . . . 394.1.3. PI-based state-feedback control . . . . . . . . . . . . . . 40

4.2. Controller design methods . . . . . . . . . . . . . . . . . . . . . 414.2.1. Pole placement method . . . . . . . . . . . . . . . . . . 414.2.2. Linear-Quadratic method . . . . . . . . . . . . . . . . . 42

4.3. Observer-based state-feedback controller design . . . . . . . . . 444.3.1. Riccati controller . . . . . . . . . . . . . . . . . . . . . . 444.3.2. Luenberger observer . . . . . . . . . . . . . . . . . . . . 45

4.4. Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . 45

5. Conclusion and future work 52

A. Appendix 53A.1. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.2. MATLAB Control System Toolbox . . . . . . . . . . . . . . . . 54

A.2.1. Linear analysis functions . . . . . . . . . . . . . . . . . . 54A.2.2. Controller design functions . . . . . . . . . . . . . . . . 54

A.3. MATLAB script . . . . . . . . . . . . . . . . . . . . . . . . . . 55A.3.1. Drum-boiler model . . . . . . . . . . . . . . . . . . . . . 55A.3.2. Controller design . . . . . . . . . . . . . . . . . . . . . . 57

A.4. Heat engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.4.1. Brayton cycle . . . . . . . . . . . . . . . . . . . . . . . . 58A.4.2. Rankine cycle . . . . . . . . . . . . . . . . . . . . . . . . 59

A.5. Non-minimum phase systems . . . . . . . . . . . . . . . . . . . 60A.6. Integral anti-windup control . . . . . . . . . . . . . . . . . . . . 61A.7. Drum-boiler state equations coefficients . . . . . . . . . . . . . 63A.8. Operator interface . . . . . . . . . . . . . . . . . . . . . . . . . 64

B. List of Figures 66

C. List of Tables 68

D. Bibliography 69

5

1. Introduction

1. Introduction

Energy market deregulation and integration of renewable energy resourcesinto the electrical grid have led to dramatic changes in the power industry whichescalated rapidly new challenges that have to be met by conventional powerplants. Such evolution caused a noticeable process modification regarding howpower plants operate, as they should become more flexible to fulfill their loadrequirements which are more frequent nowadays. The process controllers haveto be designed in a way which can simultaneously fulfill the load demand assoon as possible while at the same time bearing in mind safety and life span ofthe plant crucial elements.

One common challenge is control of steam drum-boiler units handling supplyof the steam turbine continuously with steam at high pressure and temperature.The controller should maintain drum pressure and water level within acceptableranges for all operating conditions. If the level exceeds upper limits, waterwould be carried over to the superheater or the turbine leading to outagein either of the turbine or the boiler. Surpassing lower limits would causeoverheating of the water wall tube resulting in serious tube rupture and severedamage.

Drum level control in particular is quite tough due to the process physicalphenomena known as shrink/swell of steam bubbles under the water level whichcauses the system to react with an initial inverse response known as a non-minimum phase behaviour.

Classical control design methods using 2-element or 3-element PID-controllerscan behave fairly well to compensate such effect. However as the process isquite complicated, dealing with several input variables to regulate each processvariable separately might end up with bad parameter tuning and poor levelperformance observed during load changes, eventually leading the boiler unitto trip or even worse cause emergency shutdown of the power plant. It is statedthat about 30% of the emergency shutdowns in French pressurized water reac-tors (PWR) plants were caused by poor level control of a steam drum-boilerunit [21].

An ongoing research project is taking place at the moment in collabora-tion with Stadtwerke Munchen GmbH - Munich City Utilities (SWM) in re-gards with the process PID-controllers of the low pressure drum-boiler unit,

6

1. Introduction

located within the combined cycle plant GuD 2, short for Gas-und-Dampf-Kombikraftwerk at Heizkraftwerk Sud (HKW) - combined heat and power(CHP) facility. The main objective is drum level and pressure closed loopperformance optimization which have been reported to behave very poorly un-der huge load changes taking place frequently following energy deregulation inGermany.

The thesis is presented as follows, initially the complete process is brieflyintroduced before being simplified to highlight the significant elements domi-nating the steam generation process which are mainly focused on during mod-elling procedure. Derivation of the differential equations is carried out for eachestablished featured element to develop a mathematical model capable of cap-turing most of the system nonlinearities and later on suitable for model-basedcontrol.

The model parameterized and implemented within MATLAB/Simulink en-vironment will be subjected to a detailed analysis by examining stability, simu-lating the model open loop response and validating the closed loop against datameasurements from the plant. The investigation concluded results will offer agood insight into the system inner dynamics and shall inspect the model abil-ity to catch the plant dynamical behaviour for a wide spectrum of operatingconditions.

In the end, the proposed control strategy is addressed. First, state-feedbackcontrol concept and the numerous methods which applies it shall be brieflydescribed to illustrate their applicability and major difference between them.The most convenient and suitable approach shall be employed to compute thestate-feedback and observer gain matrices. Finally, simulation results of theprocess utilizing the newly designed observer-based state-feedback controller ispresented for various sequences to ensure stability of the optimized closed loop.

7

2. Process modelling

2. Process modelling

HKW Sud plant is classified as a combined cycle cogeneration plant, it canhandle concurrent production of electrical power and useful heat utilizing aclass of sustainable integrated technologies progressively being used.

Cogeneration plants reduce thermal and mechanical losses, harmful carbondioxide (CO2) emissions and more importantly increases the overall plant effi-ciency to approximately 81% in comparison to stand alone plants which don’texceed 45%. The German government is planning to double its share of CHPplants from approximately 12% to 25% by 2020, as part of the IntegratedEnergy and Climate Protection Program (IECPP) [8].

GuD 2 at HKW Sud manages electrical power generation by combining bothBrayton and Rankine thermodynamic theoretical cycles (A.4) [12] [20] usinggas and steam turbines. Exhaust gas emitted from the gas turbine can bereused as the heat source for steam production required to operate the steamturbine, therefore more useful energy can be extracted, supplying additionalelectricity to the grid.

Further energy can by even withdrawn from the low pressure steam leavingthe turbine when condensed using a heat exchanger where the low temperaturesteam released can be utilized for district heating or water desalination.

In this chapter, the overall combined cycle process is being narrowed downto draw the focus on one particular key element within the plant. The processis further simplified in order to spotlight primarily our aim interest which isthe steam production using the low pressure drum-boiler unit along side withits process PID-controllers.

2.1. Combined cycle process overview

GuD 2 at HKW sud plant combined cycle principle is shown in figure (2.1),it consists of the following main elements briefly described

2 General Electric gas turbine units producing a total of 278MW1 Heat recovery steam generator equipped with supplementary firing1 Alstom steam turbine unit producing additional 139MW1 heat-exchange surface condenser supported with an auxiliary unit

8


Electricl power

Electricl power

Fresh air

Gas

Superheated steam

Heat Recovery Steam Generator

Low pressure water

Condenser

Steam turbine

Gas turbine

Feedwater pump

Exhaust heat

Cooling water

Waste heat to atmosphere

Figure 2.1.: Combined cycle working principle

2.1.1. Gas turbine

The combined cycle starts at the gas turbine unit whose process is based onthe Brayton open cycle (A.4.1). Continuous fresh air is compressed then mixedwith the supplied natural gas before being burned inside the combustion cham-ber at around 1124◦C. The hot compressed air expands within the turbinedriving its blades which eventually turns the generator shaft producing elec-trical power and the exhaust low pressure gas leaving the turbine at 535◦C isused as the heating source for the HRSG. Gas turbines typically have capacitiesbetween 500 kW and 250MW.

2.1.2. Heat recovery steam generator

HRSG acts as a heat exchanger between exhaust heat supplied from the gasturbine and the liquid/vapour mixture circulating into finned tubes through3 heat exchangers highlighted in figure (2.2) where additional firing can takeplace if necessary. Production of high pressure steam is carried out using highand low pressure drum-boiler units according to the following process.

1. Economizer stage Water fed by the pump supplied to the drum inlet

9


is preheated in order to reduce energy consumption.

2. Evaporator stage Due to the gravity water flows down through adowncomer-riser closed loop producing saturated steam which flows alongthe riser tubes before being collected and fed back into the drum.

3. Superheater stage The saturated steam flows through the water leveltill it exits upon reaching the drum outlet. Then it is reheated one moretime producing superheated steam supplied the turbine

Figure 2.2.: Heat Recovery Steam Generator (HRSG) [14]

2.1.3. Steam turbine

The theory of operation is based on the Rankine cycle (A.4.2) where high pres-sure and temperature superheated steam enters the turbine converting thermalenergy into rotational mechanical energy capable of moving its blades and gen-erator shaft producing additional electricity. The steam losing most of its tem-perature during the conversion process is collected and fed into the condenser.Steam turbines typically have capacities between 50 kW and 250MW.

2.1.4. Surface condenser

Also known as water-cooled shell and tube heat exchanger, it installed at theturbine outlet handling the last phase of the combined cycle by condensingthe exhaust steam to achieve maximum attainable efficiency. Water is used tocarry off waste heat from the steam due to its availability, high specific thermalcapacity and heat transfer properties.

10

2.2. Steam generation process description


The differential equations describing dynamics associated with the drum-boiler, feedwater and steam regulating valves with their actuators, in additionto the existing process PID-controllers shall be explained and addressed indetails throughout the following sections.

The simplified process relevant to our analysis concerning steam generationusing the drum-boiler unit is illustrated1 in figure (2.3). Supplied inflow fromthe feedwater pump is regulated using one control valve. As for the steamflow rate leaving the drum, it can be regulated using five valves2 connected inparallel with distinctive construction and functionalities.

1. Water tank control valve always kept opened at a predefined position

2. Bypass valve A butterfly valve handling supply of heat to city districts

3. Security valve for safety matters when the drum pressure exceeds limits

4. Steam turbine control valve feeds the steam turbine

5. Condenser control valve bypasses the steam turbine feeding directlythe condenser

Out

flow

- qs

Exhaust heat - Q

Downcomer-riser loop

Inflow - qf

Feedwater Tank1 3

2

4 5

Figure 2.3.: Schematic diagram of the low pressure steam generation process

2.2.1. Drum-boiler mass and energy balance

Figure (2.4) illustrates the detailed process of steam generation within thedrum. Its complex geometry, number of riser and downcomer tubes and spe-

1Process PID-controllers are excluded2In steady state only one valve is operational while the others are closed

11


cially the two phase flow modelling attempt is usually quite complicated requir-ing typically usage of partial differential equations. In literature there existsa lot of research papers that were devoted into developing relatively simplephysical models [2] [7] [13] [14].

In particular the well developed Astrom - Bell model3 is being considered.The majority of the system attitude can be captured through a 4th order non-linear model by means of defining mass flow and energy balance with the helpa physical mechanism introduced under the following elementary assumptions.

Upper void(saturated steam)

Sat.steam

Stea

m-w

ater

mix

ture

Downcomer

Riser

Lower distribution header

Upper collecting header

Drum

Feedwater

Steam demand(to downstream)

InternalSeparationDevice Mixture

from riser

Steam-water

Sat. water

Sat. steam

Condensation

Steam rises

Water boilsand flowsupward

Heat fromhot medium

Figure 2.4.: Schematic diagram of the downcomer-riser circulation loop [13]

Most of the system parts will be under thermal equilibrium due to theirdirect contact with saturated liquid/vapour mixture. The energy stored in themixture is either absorbed or released quickly following drum pressure changes,meaning that various metal parts of the system would adapt their temperaturesin the same manner.

This agrees with experimental observation which have proven that the differ-ence between both temperatures is very small, thus a detailed representation

3Part of an ongoing research project which started back in the early seventies

12


of the temperature distribution within the metal isn’t necessary.Equations (2.1) presents the mass and global energy balance for the drum

in terms of feedwater qf , steam qs and heat Q flow rates respectively. Itdescribes the drum pressure dynamical behavior quite well by simply computingproperties of liquid/vapour mixture using steam tables. Condensation of thesteam within the drum causes the coupling between the drum pressure P andwater total volume Vwt.

d

dt(ρsVst + ρwVwt) = qf − qs

d

dt(ρshsVst + ρwhwVwt − PVwt +mtcptsat) = Q+ qfhfw − qshs

(2.1)

Distribution of steam along the riser tubes was carried out using a lumpedmodel which represents the energy and mass balance caused by the naturallycirculated downcomer-riser closed loop as seen in equation (2.2). The steammass fraction αr assumed to vary linearly from the inlet to the outlet of theriser is characterized in response to changes in the downcomer qdc, riser qr andheat Q flow rates respectively.

d

dt[ρsαvVr + ρw(1− αv)Vr] = qdc − qr

d

dt[ρshsαvVr + ρwhw(1− αv)Vr − PVr +mrCptsat]

= Q+ qdchw − qr(hw + αrhc)

(2.2)

The empirical equation (2.3) resulted from various attempts to fit with theexperimental data. It defines mass balance of the steam bubbles under thewater level in terms of condensation flow qcd and steam flow through the liquidsurface qsd driven by density difference of the mixture and momentum of theflow qr entering through the riser tubes. It can capture most of the processdynamics by proper parameterizations of residence time of steam inside thedrum Td, the bubbles steam volume at hypothetical situation4 V ◦

sd and empiricalcoefficient β correspondingly.

d

dt(ρsVsd) = αrqr − qcd − qsd

qsd =ρsTd

(Vsd − V ◦sd) + αrqdc + αrβ(qdc − qr)

(2.3)

4Theoretical state that assumes no condensation of steam inside the drum

13


2.2.2. Drum-boiler nonlinear state equations

To derive a state model, chosen state variables should have a good physicalinterpretation. Drum pressure P is obviously chosen as it describes the totalenergy of the system. The accumulation of water related to total water volumeVwt in the system is selected since it represents the storage of mass.Steam quality αr in the riser tubes and steam bubbles volume under the

liquid level Vsd are chosen as well to describe distribution of steam under thewater, thus estimating the level. The resulting nonlinear state-space modelwould be a 4th order system whose states are x = [ P , Vwt, αr, Vsd ].

The model actuating variables are u = [ qf , qs ], the feedwater and steam flowrates are manipulated to control primarily the drum water level and pressurerespectively, whereas the heat flow rate Q is rather considered as a model inputdisturbance z due to the fact that its amount is associated with the gas turbineexhaust heat which in return corresponds to its electrical output power as weelaborated concisely the combined cycle working principle in section (2.1). Onthe contrary heat flow rate becomes a control variable in thermal plants as itcan be regulated directly by adjusting the boiler firing rate.

Arrangement of the mass and energy balance differential equations was car-ried out in order to derive the algebraic state equations. The liquid/vapourmixture properties time derivative in terms of the drum pressure are calculatedusing the coefficients enm provided in appendix (A.7).

dP

dt=

e12Q+ qf (e12hfw − e22)− qs(e22 − e12hs)

e12e21 − e11e22(2.4)

dVwt

dt=

1

e12

[qf − qs − e11

dP

dt

](2.5)

dαr

dt=

1

e33

[Q− αrhcqdc − e31

dP

dt

](2.6)

dVsd

dt=

1

e44

[ρsTd

(V ◦sd − Vsd)− qf

hfw − hwhc

− e41dP

dt− e43

dαr

dt

](2.7)

Equations (2.4), (2.5) rearrange the drum mass and energy balance, equation(2.6) combines the mass and energy balance of the downcomer-riser closed loopin a single equation and equation (2.7) considers only the mass balance of steambubbles under water level. The interesting feature of this model is that thestates can be grouped in the form ((P, Vwt), αr, Vsd), where each term can becomputed separately in a nested manner treating the system as 2nd, 3rd or 4th

order according to modelling requirements.

14


2.2.3. Mass flow control valve

The process concerned with regulation of feedwater flow rate supplied fromthe pump can be simplified and highlighted as seen in figure (2.5). The flowis computed with the aid of the nonlinear equation (2.8) essentially used bymechanical engineers to size their valves and meet mass flow requirements.The pressure drop ΔP across the valve would be the difference between thefeedwater pump and drum pressures, xf is its percentage opening ranging from0% to 100% and finally Kv is the valve sizing coefficient.

p1 p2

Q W

H100H0

t1

Figure 2.5.: Flow through control valve for liquid service [22]

The dynamics related to regulation of steam flow rate are quite complicatedwhere additional considerations have to be taken care of when compared tofeedwater mainly due to the difference in properties between both. One goodapproximation to describe the flow rate through a control valve meeting prac-tical needs can be achieved using equation (2.9) where P is the drum pressure,the head loss coefficient m and the compressibility factor Z are taken intoaccount to distinguish between saturated and superheated steam.

qf = xf · Kvf · ρw · √ΔP

3600(2.8)

qs = xs · Kvs · Z ·m3600

(2.9)

Clearly the valve position value would vary according to the type of valvebeing used. The inherent flow characteristic depicted in figure (2.6) highlightthe comparison between the commonly used control valve demonstrating thatmass flow rate for the same opening position and pressure drop across it isobviously altered according to the category it belongs to.

Examining halfway opened linear, butterfly and relief valves correspondingly,

15


undoubtedly the butterfly valve would supply approximately one-third of thetotal amount provided using the linear valve while the relief valve employed forsafety precautions would grant roughly twice the flow afforded by the linearvalve.

The actuators used to operate the control valve handle the positioning im-posed by the controller using electrical motors with 3 basic states which areopening, closing or holding the same opening percentage. The rate of open-ing/closing is correlated to the motor maximum speed.

Figure 2.6.: Inherent flow characteristics of typical control valves [24]

2.2.4. Process PID-controller

There exists two major classifications in regards with implementation ofPID-controller algorithm [3] commonly known in industry as series (2.10) andparallel (2.11). It is mainly introduced to identify the controller realizationand not to describe it, since the algorithms are identical to one another wherethe overall transfer behaviour from the controller input to the output is alwaysthe same, regardless of how the derivative action is being handled by differentmanufactures.

u(s)

e(s)= Kp ·

[1 +

1

s · Ti

]·[1 + s · Td

1 + s · Tf

](2.10)

u(s)

e(s)= Kp ·

[1 +

1

s · Ti+

s · Td

1 + s · Tf

](2.11)

16


The controller parameter Kp corresponds to the proportional gain element,Ti and Td represent the time constants assigned to the integrator and deriva-tive elements respectively and finally the time constant Tf relates to the filterfrequency applied on the derivative term which is mandatory from practicaland theoretical aspects.

Practically the measurement sensors produce noise at high frequency furthergetting amplified due to the derivative action leading to very large unusablecontroller output. The additional low pass filter pole is placed in a mannerwhich attenuates high frequency noise.

From the theoretical point of view, the PID-controller transfer function with-out the additional pole cannot be realized since the nominator would be higherthan the dominator.

The series algorithm -still being adopted in digital controllers- was first intro-duced within the early analog controllers, which were realized using electricalcircuits or pneumatic elements. Its corresponding transfer function can be rep-resented easily in the frequency domain where the poles and zeros correspondto the inverse of the corner frequencies.

In the parallel form referred to as non-interacting, the unity feedforwardsignal and derivative action predict the error at the moment assigned by its timeconstant Td. The integrator intends to eliminate the error between referenceand process output completely where the resulting action from both is modifiedafterwards using the proportional gain Kp.

It is worth mentioning that such minor difference in implementation wouldhave a major impact when attempting to tune the controller parameters usinganalytical methods such as Cohen-Coon or Lambda since they can only beapplied on the parallel algorithm.

One notable problem using PID-controller is integrator windup (A.6) neces-sitating usage of an anti-windup mechanism to prevent the integral elementfrom growing up further as soon as the controller output hits the saturationlimits entering the nonlinear region. This would occur when the control signalexceeds the predefined physical boundaries related to the control valve openingrange and allowable amount of mass flow rate which can be supplied.

The drum pressure and water level PID-controllers adopts the parallel algo-rithm and their set values are always kept constant regardless of the suppliedamount of heat flow rate. Each output is controlled with its separate con-trol loop without considering any sort of coupling or interaction between bothoutputs.

17

2.3. MATALB/Simulink model

Water level and pressure control

Drum level control can be realized using 3 different industry-standard strate-gies with typical application for each noted as single, two and three-elementcontrol respectively [1]. The numbered term corresponds to the set of mea-surements being utilized to control the system. GuD 2 at HKW Sud currentlyimplements the 2-element structure which employees a cascaded control archi-tecture using level and feedwater flow rate as process variables.

Such strategy is useful as it addresses disturbance imposed on the level andimproves set point response performance when compared to 1-element control.When directly controlling the level it isn’t enough for the controller by itselfto directly open or close the valve since it have to decide as well whether itshould be feeding more or less feedwater into the drum.

By considering the feedwater flow rate as well, the outer loop comparesthe current level with the specified reference and the computed error signalgenerates using the PID-controller a new set value for feedwater flow rate. Theinner loop examines the current flow with the amount established by the outerloop in order to adjust accordingly the control valve percentage opening usinga PI-controller.

During normal operation the pressure is regulated by modifying the linear orbutterfly valves’ position using identical control loop structures consisting of asimple feedback loop which compares the reference value with the drum actualpressure. The error is subjected to unity negative gain that ensures an inverseresponse to the valve position where its value is altered using a PI-controllerequipped with a dead zone.

The control valve should open if the drum pressure increases to relief thepressure inside. The same holds if it drops, where the required action is steamvalve closure, thus increasing the pressure within the drum and restoring itback to the defined set value. The dead zone guarantees a region of zerooutput causing the PI-controller to hold its previous state as it’s only allowedto react when the error signal exceeds certain limits.


The complete physical model is realized within MATLAB/Simulink environ-ment carrying out direct computation of the differential and algebraic stateequations describing the process elaborated in section (2.2). The parameterswere either extracted from the construction data and control schemes of GuD2at HKW Sud plant or estimated following their unavailability.

18


2.3.1. Drum-boiler model

A simple MATLAB code was written to implement directly the establishedalgebraic state equations. The Simulink model uses the user-defined MATLABfunction block to manage the developed script5 provided in appendix (A.3.1).It requires input of the states, heat and mass flow rates current values in orderto calculate the state variables. They are integrated before being fed backagain as shown in figure (2.7).

Level (mm)

2Pressure (bar)

1

States1s

Drum-Boiler m-file

Model inputs

State variables (dx/dt)

Pressure (bar)

Level (mm)

Mass flow rates (u)

2Heat flow rate (z)

1

States (x)

Figure 2.7.: Simulink model of the Drum-boiler unit

The liquid/vapour mixture properties are calculated using functions of waterproperties and derivatives based on the International Association on Propertiesof Water and Steam (IAPWS). These functions were realized using two im-plemented MATLAB functions which are XSteam [10] and IAPWS-IF97 [19].The partial derivatives of water and steam densities with pressure were ap-proximated linearly within the drum pressure operating range as they weren’timplemented in both of the above mentioned functions.

The riser and downcomer tubes volumes Vr/Vdc is computed by knowledgeof their cylindrical pipe length and cross sectional area. The downcomer-riserclosed loop consists of two tubes that belong to the downcomer supplying waterto the evaporator, the saturated steam is fed back into the drum through sixcollectors related to the riser. The drum oval shape was approximated andconsidered as a cylinder whose volume Vd would be calculated similarly. Thedrum area Ad is associated with the water surface area assumed to be constanteven though it would vary according to the level. The downcomer area Adc isthe pipe cross sectional area already obtained while computing the volume.

Parameters related to empirical equations or lumped models such as thefriction coefficient in downcomer-rise loop K, empirical coefficient β and the

5Three different subsystems were constructed separately block-by-block in Simulink duringearly development of the model

19


hypothetical volume V ◦sd were quite hard to obtain, therefore were either kept

constant or scaled down relative to the Astrom-Bell model6.The residence time of steam within the drum Td was identified7 following

system analysis which shall be illustrated in details throughout the next chapterin section (3.4). It have a huge influence on the overall behaviour which isreasonably expected since it can be interpreted as the time constant of waterand steam separation process. Simulation results shows that a residence timehigher than 6 sec leads to closed loop instability when utilizing the existingPID-controllers.

Drum-boiler construction data

Variable Drum Downcomer Riser

Volume 20.204m3 0.9m3 20m3

Mass8 1363kg 580kg 1300kgArea 14.7m2 0.0637m2 -

Drum-boiler model parameters

Residence time in drum 4sHypothetical volume 2m3

Friction coefficient 25Empirical coefficient 0.3

Table 2.1.: Drum-boiler model parameters

The amount of mass flow rates qf and qs at a given pressure P are firstspecified in order to compute initial values. This allows computation of thenecessary heat flow rate Q that preservers energy balance. A primary simula-tion can run once these values are assigned as the model drives by itself thevariables αr and Vsd to steady state by solving equations (2.12). Finally, thetotal volume Vwt is the amount required to keep the water at the relative zerolevel.

Q = qshs − qfhfw = αrhc

√2ρwAdc(ρw − ρs)gαvVr

K

αv =ρw

ρw − ρs

(1− ρs

(ρw − ρs)αrln

(1 +

ρw − ρsρs

αr

)) (2.12)

6K and β were kept the same, whereas V ◦sd is chosen as a rule of thumb

7Changed within the range [ 2 sec - 6 sec ] until the model closed loop behaviour matchedthe plant real measurements

8Total mass including the evaporator mt = 98888kg

20


2.3.2. Control valve and actuator model

The actuator dealing with the positioning is modeled as a 1st order lag elementwhose integrator saturation corresponds to the valve position ranging between0% and 100%. The discontinuous rate limiter block afforded by Simulink libraryis used to model the motor rate of opening and closing. Finally, the amount offeedwater flow rate varies according to the pressure drop across the valve andits opening percentage as illustrated in equation (2.8).

The pressure ratio across the valve was estimated not to exceed 0.7, thereforethe head loss m according to steam service tables would be 0.96 [22]. The di-mensionless compressibility factor Z is treated as a function of the superheatedsteam pressure and density. Therefore, the amount of steam flowing through acontrol valve can be rewritten as described in equation (2.13).

Z = 14.2√ρsP

qs = 13.6Kvsxs

√ρsP

3600

(2.13)

The feedwater and steam turbine control valves sizing coefficients were ob-tained directly from their corresponding data sheets. Unfortunately data aboutthe butterfly valve was missing, its sizing had to be estimated using table chartsfrom [24] and its inherent flow characteristic curve was simplified as linear. Suchassumption is still very plausible as the valve in the real process never opensbeyond 30%.

qf (Kg/s)

1

Feedwater Control Valve

Pdrop (bar)

xf (%)

qf (kg/s)

Electrical MotorActuator

1s

PostionSet Point

2

Pressure drop

1

Figure 2.8.: Simulink model of the control valve combined with its actuator

Control valves and actuators

Variable Feedwater Steam turbine Butterfly

Sizing coefficient 20.368 m3

hr 364 kghr 1363 kg

hr

Rate of opening ± 3.333 %s ± 0.166 %

s ± 0.555 %s

Table 2.2.: Control valve and actuator parameters

21


2.3.3. Process PID-controller model

Simulink continuous PID block offers functionalities which meets exactly ourneeds, thus a separate realization wasn’t required regarding implementationof the process controllers. It can simulate the non-interacting PID algorithmaccording to equation (2.14)9, provide output saturation when required, resetintegrators and more importantly equipping an anti-windup mechanism.

u(s)

e(s)= Kp ·

[1 +

Ki

s+ Td · N · s

s+N

](2.14)

The anti-windup can be handled using the back-calculation method or a log-ical clamping circuit. Back-calculation feedback loop when employed attemptsto discharge the PID-Controller internal integrator when the controller hitsspecified saturation limits by proper tuning of the highlighted coefficient Kb asshown in figure (2.9) [3].

Controlleroutput

1

SaturationProportional gain

Kp

Integrator gain

Ki

Integrator

1s

Filter coefficient

N

Filter

1s

Derivative gain

TdBack-calculation

coefficient

Kb

Error

1

Figure 2.9.: Simulink model of a parallel PID-controller equipped with an anti-windup mechanism

PID-Controller parameters

Controller Kp Ti Td Tf

Level 0.05 300 100 50Feedwater valve 2.3 25 - -Pressure10 1.8 12 - -

Table 2.3.: PID-controller parameters

9Ki =1Ti

and N = 1Tf

10Pressure control is carried on during model validation using only the bypass butterfly valve

22

3. Process analysis and validation

3. Process analysis and validation

In this chapter a detailed analysis of the drum-boiler unit shall be carried outkeeping in mind future plans and design considerations. First, essential the-oretical aspects required to effectively analyze the system are briefly covered.Then stability of the process is examined analytically by linearizing the non-linear model at various operating conditions in order to predict its open loopbehavior. Later on, the expected behaviour shall be addressed in details byconducting several simulation scenarios. Finally, the closed loop response isvalidated against real measured data from GuD 2 at HKW Sud.

3.1. Theoretical overview

3.1.1. Concept of stability

Stability of linear systems can be roughly summarized as follows, a systemoutput will be limited and restricted for any applied bounded input referredto as Bounded-Input Bounded-Output (BIBO) stability. Examining stabilityof linear systems is fairly simple and straightforward as they can be describedeither as transfer function or in state-space form, thus stability can be de-termined by direct computation and graphical visualisations of its eigenvalueswithin the complex plane. Furthermore, stability of the controlled closed loopsystem can be predicated by merely inspecting the system in open loop whileapplying well established methods such as the Nyquist criterion.

On the other hand, stability analysis for nonlinear systems is relatively com-plicated and requires a high level of mathematical understanding since furthermatters have to be considered. The analysis should address stability of equi-librium points, known as position of rest xR, instead of the overall system.Steady state takes place for a constant input u0 if and only if the state vari-ables remains constant as defined in equation (3.1).

x = f(xR,u0) = 0 (3.1)

Obviously nonlinear systems positions of rest - referred to from now on asoperating points - have a finite number associated with the solution of equa-

23


tion (3.1), hence requires a more generalized definition offered by Lyapunov[18] discriminating between different stability forms for each operating pointsclassified as stable, asymptotically stable and unstable.

3.1.2. Linearization

For simplicity the intended stability analysis shall be performed by lineariza-tion of the nonlinear model at typical operating points of the drum whose statealgebraic equations can be summarized into the generalized description shownin equation (3.2).

x = f(x,u)

y = g(x,u)(3.2)

The resulting linearized model can be described in state-space form (3.3)where A, B, G, C and D are the system, input, disturbance, output andfeedforward matrices respectively. This will come in handy when attemptingto optimize the controller since algorithm execution of the proposed strategyrequires these matrices. In addition they would reduce the nonlinear stateequations complexity offering a rather simplified overview of the states andinputs dominating the process outputs. The open loop response of both lin-ear and nonlinear models should be compared to inspect if both still match,therefore answering the crucial question concerned with the linearized modelreliability during design of the optimal controller.

x = Ax+Bu+Gz

y = Cx+Du(3.3)

The matrices are computed with the help of Taylor series approximationneglecting quadratic and higher order terms (3.4). The method intuitive basisis that a smooth curve differs very little from its tangent line as long as thevariable doesn’t wander from the point of tangency.

aij =∂fi∂xj

∣∣∣∣xR,u0

, bij =∂fi∂uj

∣∣∣∣xR,u0

cij =∂gi∂xj

∣∣∣∣xR,u0

, dij =∂gi∂uj

∣∣∣∣xR,u0

(3.4)

3.1.3. Poles and zeros

System zeros affects only shape of the output which can lead to minimum ornon-minimum phase behaviour according to their position within the complex

24


plane (A.5) [9] [23]. Alternatively poles determine stability as they are directlyassociated with the system eigenvalues. That’s why inspection of the systempoles and zeros is quite efficient while analysing and predicting the systemresponse.

If the eigenvalues are located on the left-hand side (LHS) of the complexplane the states will converge to zero stabilizing over the course of time. How-ever if located on the right-hand side (RHS) the states will keep growing dueto the exponential product as depicted throughout equation (3.5) where ciare constants coefficients related to the solution of the homogenous differentialequation describing dynamics of the system.

y(t) =n∑

i=1

cieλit (3.5)

We still need to define the relationship between the zeros, poles and eigen-values, in addition understand how it differs when comparing Multiple-inputMultiple-Output (MIMO) systems to Single-Input Single-Output (SISO).

SISO systems

Commonly input-output (I/O) behavior is presented using transfer functions(3.6) where zeros zi and poles pi are simply the roots of the numerator N(s)and dominator D(s) respectively. The transfer function dominator is exactlyequivalent to the characteristic polynomial evaluated by solving equation (3.7),that why all poles corresponds to the system eigenvalues λ.

G(s) =N(s)

D(s)=

(s− z1)(s− z2)...(s− zn)

(s− p1)(s− p2)...(s− pn)(3.6)

det(λI −A).= 0 (3.7)

MIMO systems

Zeros in multivariable systems do play an additional role besides affectingsystem shape and performance since it might gravely influence the ability tofully control the system [5] [6]. They are redefined with the help of Rosenbrockmatrix which benefits from the state-space description distinguishing betweentransfer and decoupling zeros. The complete set consisting of both known asinvariant zeros1 is examined by computing the rank of the matrix (3.8). Not

1Only under the assumption that feedforward matrix D = 0

25

3.2. Stability analysis

all of the system eigenvalues necessary appear as poles due to existence ofdecoupling zeros compensating poles in I/O transfer functions Gij(s).If this occurs in a system, it would be impossible to fully control the system

since some eigenvalues are no longer influenced by the controller. It gets evenworse if the uncontrollable eigenvalue is located on the RHS of the complexplane because there no way to stabilize the plant with its current setup usingany control technique.

P (s) =

[sI −A −B

C 0

](3.8)


The drum-boiler unit stability can be easily comprehended from the physicalpoint of view with basic understanding of the drum mass and energy balanceequations discussed during modelling in section (2.2.1). For example, if theoutflow leaving the drum is less than the amount which is supplied by theinflow then water will start filling the drum and vice-versa. Alternatively whileassuming constant mass flow rate, if additional firing takes place by the HRSGproviding more heat leading the drum temperature to rise and causing thepressure to build up in return, thus reaching dangerous limits which will causeexplosion of the drum ultimately.

Even though this shortened explanation could be enough, additional ana-lytically driven investigations needs to conducted by performing a stabilityanalysis to extend our understanding of the expected behaviour and establishthe foundation necessary for the controller design.

3.2.1. Linear state-space model

The model is linearized at 3 operating points shown in table (3.2.1) us-ing MATALB Control System ToolboxTM linear time-invariant (LTI) functionsdedicated for continuous systems time-domain analysis (A.2). They cover thedrum operating range whose upper limit is specified by the maximum amountof saturated steam allowed to flow through the pipes upon exiting the drum.

The input B and disturbance G matrices show that the dominant inputswhich affects the drum pressure P dynamics are heat Q and steam qs flowrates as expected. The water total volume Vwt is obviously affected mainly bythe mass flow rates. Steam mass fraction αr depends heavily on condensationenthalpy hc, downcomer qdc and heat Q flow rates according to state equation(2.6). As for the steam bubbles volume Vsd in the water level, it can be seenthat it’s associated with all states and input variables as it follows an empirical

26


equation derived through continuous observation of the process to capture thedrum complicated dynamics.

A =

⎡⎢⎢⎣−7.148e-5 1.051e-14 0 0−5.094e-11 −9.723e-21 0 01.021e-9 −2.684e-21 −0.1827 01.752e-6 7.424e-18 −297.3 −0.3333

⎤⎥⎥⎦

B =

⎡⎢⎢⎣

−21.09 −216.60.001085 −0.0012555.386e-6 5.534e-5−0.01486 0.2023

⎤⎥⎥⎦ G =

⎡⎢⎢⎣

1037.341e-57.904e-50.07526

⎤⎥⎥⎦

C =

[1e-5 0 0 0

−4.86e-4 68.027 2035 68.027

]

OperationStates Inputs

P (bar)2 Vwt(m3) αr(%) Vsd(m

3) Q(MW) qf (

kgs ) qs(

kgs )

Low 5.5 21.501 0.0098 1.378 13.8473 6 6Medium 5.5 20.391 0.0138 1.067 20.771 9 9High 5.5 19.736 0.0178 0.756 27.6947 12 12

Table 3.1.: Drum-boiler operating points for low, medium and high load

3.2.2. I/O pole-zero plot

The input-output pole-zero map illustrated in figure (3.1) concerning thetransfer behaviour from inputs to the water level shows that all four eigenvaluesappear as poles. The first three are always located at − 1

Tdand the origin3,

associated with the drum pressure, water volume and dynamics of the steambubbles under water level respectively. The last pole which depends on theoperating point is situated at −hcqdc

e33, it relates to the steam dynamics flowing

through the riser tubes. It keeps shifting to the left along the negative real axistowards infinity as long as the load increases.

This was quite expected from our basic understanding regarding steam gen-eration working principle using the drum-boiler. Higher loads require moreelectrical power generated by the gas turbine which in return provides addi-tional heat to the riser tubes, thus accelerating conversion process of feedwaterinto steam within the naturally circulated downcomer-riser loop. If an enor-mous amount of heat is supplied the pole keeps approaching negative infinity,

2The indicated pressure through the thesis is the absolute pressure3Assuming constant residence time of steam within the drum

27


when inspected in the complex plane, since the conversion shall take placeinstantaneously.

Since no compensation of eigenvalues have occurred, all invariant zeros areclassified as transfer zeros. In addition one can stay assured that the system iscompletely controllable because any eigenvalue can be influenced by affecting itscorresponding pole. The transfer zeros which are located on the right hand side(RHS) of the complex plane have been anticipated earlier from experimentalobservation and physical understanding. They are directly correlated with theshrink and swell physical phenomena leading the system to react in a non-minimum phase behaviour.

In particular zeros related to the transfer behavior from steam flow rate towater level are very close to the origin when compared with zeros linked tofeedwater and heat flow rates transfer functions respectively as seen in figure(3.1). Therefore we should be definitely expecting a significant difference inregards with amplitude of the water level initial inverse response when stimu-lated by the input variables. This shall verified in the next section concernedmainly with the open loop response to a step input.

0.28 0.21 0.14 0.07 0 0.071

0

1Flow rate qf (kg/s)

0.28 0.21 0.14 0.07 0 0.071

0

1Heat flow rate Q (MW)

0.28 0.21 0.14 0.07 0 0.071

0

1

Real axis

Flow rate qs (kg/s)

3 2 1

3 2 1

3 2 1 321

123

1 23

123

Figure 3.1.: Pole-zero plot of the linearized models at low (1), medium (2) andhigh (3) load

28

3.3. Open loop step response


The open loop response shall be studied by simulating the model4 consistingof the drum-boiler unit and control valves5 when subjected to a step change ofthe gas turbine electrical output power and control valves position respectively.One input at a time is stimulated using a step function while the other inputsremain intact. The mass flow rates would vary according to pressure dynamics.

3.3.1. Change of gas turbine electrical output power

Figure (3.2) shows the response to a step input of gas turbine output electricalpower equivalent to a decrease of 20MW. The amount of heat flow rate Qsupplied to the drum, required as an input to run the simulation, is assumedto vary instantaneously following the change of the gas turbine power. Butterflyand feedwater control valves positions were kept constant.

The pressure P starts decreasing following the declination of heat flow rateassociated with gas turbine output power. It affects the amount of steam flowrate qs leaving the drum as the valve position haven’t changed. On the otherhand, the pressure drop across the feedwater valve starts building up since thefeedwater pump pressure is kept constant, hence causing more feedwater qf toflow into the drum.

The water total volume Vwt initially decreases due to evaporation caused bysudden pressure drop before incrementing eventually following the increase offeedwater. The steam mass fraction αr in the riser tubes immediately stepsdown once the heat supplied is smaller than its initial state then keeps slidingdown gradually as the amount of water being vaporized by the evaporatorwithin the downcomer-riser loop was reduced.

The level response l depends on a combination of complicated dynamicsrelated to distribution of water and steam. The step-like change of steam massfraction αr leads to the initial undershoot as the quantity of steam bubblesfed back to the drum rapidly drops. The swelling effect is then noticed oncethe pressure starts to decrease resulting in steam bubbles expansion causingthe level to rise. Finally following this transient effect, water keeps filling inthe drum due mass imbalance where feedwater supplied to drum inlet is muchhigher than the steam leaving from the outlet.

4All simulations were conducted in Simulink using a fixed step size of 1 s5In [2] the open loop response considers only the drum-boiler unit

29


3.3.2. Change of butterfly valve position

Figure (3.3) shows the step response of the system due to opening of thebutterfly valve equivalent to 10% while heat flow rate and position of feedwatercontrol valve were kept constant.

The steam flow rate qs rapidly increases according to the valve rate of open-ing after the unexpected rapid change in valve positions. The opening of thevalve relieves the pressure inside the drum and hence it starts decreasing. Oncethe valve reaches its designated opening percentage, the pressure P starts dom-inating the behaviour of the steam flow rate thus reducing the amount of steamleaving the drum because both are related to each other. Feedwater flow rateqf increments due to the increased pressure drop across the feedwater valve.The water total volume Vwt decreases for two reasons; one is the evaporationcaused by the pressure drop and the other being the relatively high differencein mass flow rates.

The steam mass fraction αr steps up once the pressure have decreased thenstarts sliding gradually until it approaches its original state following the tran-sient effect occurring to downcomer qdc and riser qr flow rates. Finally thelevel l initial inverse response is caused by the bubbles swelling and volumeexpansion then it falls constantly due to mass imbalance.

3.3.3. Change of feedwater control valve position

Figure (3.4) shows the step response of the system due to closing of thefeedwater control valve equivalent to 10% while the heat flow rate and positionof butterfly control valve were kept constant.

The feedwater qf drops in step fashion since the control valve reaches itsdesignated position very quickly with its fast rate of opening/closing. Thedecrease of cold feedwater fed into the drum increases its temperature whichin return affects the pressure allowing more steam qs to leave the drum. Themass balance inflow and outflow was disturbed within the drum, therefore thewater total volume Vwt declines at high rate.The steam mass fraction αr behaviour is similar to the open loop response

of the steam control valve initially dropping following pressure rise then slidinggradually upwards towards its initial state. The sudden drop of feedwater flowrate resulted as expected in the level l initial inverse response corresponding tothe predicated system non-minimum phase behaviour.

30


0 100 2004.5

5

5.5

Pressure (bar)

0 100 2007.5

8

8.5

9Flow rate qs (Kg/s)

0 100 2000.01

0.011

0.012

0.013

0.014Steam quality (%)

0 100 20050

0

50Level (mm)

0 100 2008.6

8.8

9

9.2Flow rate qf (Kg/s)

0 100 20020.4

20.5

20.6

20.7

20.8Volume Vwt (m3)

Figure 3.2.: Open loop response for a step change equivalent to decrease of20MW of the gas turbine electrical output power

0 100 2004.5

5

5.5

Pressure (bar)

0 100 2008

9

10

11

12Flow rate qs (Kg/s)

0 100 2000.013

0.0135

0.014


0 100 200

0

50

100

Level (mm)

0 100 2008.6

8.8

9

9.2Flow rate qf (Kg/s)

0 100 20020

20.2

20.4

20.6Volume Vwt (m3)

Figure 3.3.: Open loop response for a step change equivalent to 10% openingof butterfly valve position

31

3.4. Validation

0 100 2004.54

4.4

4.44

4.P

4.P4r esuu es(ba) e7

0 100 2008.P4

8.9

8.94

8.8

8.84Flow(e) ts(qu(bKg/u7

0 100 2000.0133

0.0135

0.0135Sts) m(q ) lity(b%7

0 100 20030

20

10

0

10

Lsvsl(bmm7

0 100 2004

P

9

8

6Flow(e) ts(qf(bKg/u7

0 100 20016.8

20

20.2

20.5

20.P

Vol ms(Vwt(bm37

Figure 3.4.: Open loop response for a step change equivalent to 10% closing offeedwater control valve position

3.4. Validation

The system closed loop response will be validated and examined against data6

from the real plant for different scenarios to experiment its ability to capturethe real process dynamics at various operating conditions. The complete modelwith the PID-controllers is shown in figure (3.5).

3.4.1. Assumptions

The heat flow rate required as an input of the model cannot be measured inreality, yet can be predicted from the gas turbine electrical output power whichchanges as ramp function with a slope of 1

12MWs. The transfer function relating

both is assumed to be 1st order lag element whose time constant was identifiedτ = 280 s assuming that the supplied heat behaviour is directly associated withthe evaporator temperature.

The feedwater valve position is always kept half-way opened in the plantwithout considering the amount of feedwater which flows through it. Thereforethe pressure drop across the valve should increase or decrease accordingly topreserve such condition which is achieved using he feedwater pump controller.

6The measurements of the plant are being filtered and sampled at a rate of 1Hz

32

3.4. Validation

Implementation of the controller was neglected for simplicity and it is assumedthat the pump output would vary ramp-wise with a slope of 1

60bars

followed byPT1 element with a time constant τ = 15 s chosen as a rule of thumb.

The tests conducted in the plant were using only the butterfly valve to reg-ulate the steam flow rate therefore the pressure control loop will consist onlyof the corresponding PI-controller. Additionally steam flow rate measurementfrom the plant doesn’t take into account the amount supplied back to thefeedwater tank which was assumed to be approximately around 1 kg

s.

The corresponding valve is usually kept open at a predefined position andits contribution to control the drum pressure can be neglected. However asthe drum-boiler model requires the total flow rate which leaves the drum as aninput variable, the estimated amount flowing into the feedwater tank is simplyrelated to the drum pressure dynamics with a low pass transfer behaviour whichis similarly chosen as another rule of thumb.

The pressure loop PI-controller proportional gain Kp was adjusted from 1.8to 4. The model closed loop performance improved and matched much betterthe measurements when compared to its initial value when observing the sim-ulations results. This is due to several factors discussed when concluding thechapter in section (3.5).

3.4.2. Comparison with measurement data

The pressure controller senses the pressure decrease within the drum as lessheat is being supplied as shown in figure (3.6), thus it tries to close the butterflyvalve to restore pressure back to its set point.

Once the valve starts closing, the water level l drops due to the shirkingeffect of steam bubbles. It experiences an undershoot followed by an overshootsince the cascade controller is simultaneously trying to restore the level backto its set point and to reestablish energy balance for the drum as well as massbalance for inflow and outflow.

Figure (3.7) illustrates the comparison considering gas turbine power in-crease. The controllers react on the pressure rise within the drum caused bythe additional heat supplied, therefore opening the corresponding valve to re-lief drum pressure allowing more steam to leave from the drum outlet in theprocess.

The water level l increases due to steam bubbles swelling, yet a smallerovershoot is observed since the change of electrical power is less when comparedto the previous scenario, thus in return permit the controllers to settle and drivethe process back to steady-state faster.

33

3.4. Validation

qs P

I con

trol

Kp =

4 T

i = 1

/15

PI(s

)

qf P

I Con

trolle

rKp

= 2

Ti =

1/2

5

PI(s

)

Stea

m to

feed

wat

erta

nk

0.18

182

5s+1

Satu

ratio

n

Pres

sure

Ref

eren

ce v

alue

5.5

Low

pas

sG

ain

wat

t to

MW

1e+0

06

280s

+1

Low

pas

s

1

15s+

1

Leve

l PID

Con

trolle

rKp

= 0

.05

Ti =

1/3

00Td

=50

Tv

= 10

0

PID

(s)

Leve

lR

efer

ence

val

ue

0

Hea

t flo

w ra

te (W

att)

Feed

wat

er v

alve

Pdro

p (b

ar)

xf (%

)

qf (k

g/s)

Feed

wat

erpu

mp

Elec

trica

l Mot

or10

% b

y 18

sec

Elec

trica

l Mot

or10

% b

y 3

sec

Dru

m-b

oile

r mod

el

qs (K

g/s)

Q (W

)

qf (K

g/s)

P (B

ar)

Leve

l (m

m)

Dea

d Zo

ne-0

.04

to 0

.04

Butte

rfly

valv

e

P_dr

um (b

ar)

xs (%

)

qs (k

g/s)

Actu

ator

xs

1 s

Actu

ator

xf

1 s

Figure

3.5.:Sim

ulinkvalidationmodel

34

3.4. Validation

0 2000 40005.2

5.3

5.4

5.5

5.6

Pressure (bar)

0 2000 400010

15

20

Valve position xs (%)

0 2000 40004

5

6

7

8

Flow rate qs (Kg/s)

0 2000 4000100

50

0

50

100Level (mm)

0 2000 4000

40

60

80

Valve position xf (%)

0 2000 4000

80

90

100

Power Q (MW)

Figure 3.6.: Comparison between model (dashed) and plant data (solid line)for a decrease of the gas turbine electrical output power equivalentto 20MW

0 2400 5400

4.3

4.4

4.6

4.P

r esuu es(r (ba) e1

0 2400 540040

V4

0

V4

40

l svsp(p(bo o 1

0 2400 5400V0

50

30

40

60

P0i ) pvs(t nux%xn7(8F(bw 1

0 2400 5400

23

26

2q

i ) pvs(t nux%xn7(8u(bw 1

0 2400 54006

6.4

P

P.4

q

q.4Kpng (e) %s(/ u(bLmfu1

0 2400 5400

q0

q4

90

r ng se(Q(bMW1

Figure 3.7.: Comparison between model (dashed) and plant data (solid line) foran increase of the gas turbine electrical output power equivalentto 10MW

35

3.5. Concluding remarks


In the end of the process comprehensive study, we shall wrap up the analysisand summarize results as follows.

The resulting linearized model of the drum-boiler unit predicted the openloop response of the dynamic realistic nonlinear system quite well and can berelied on safely in regards with intended future plans.

The I/O pole-zero plot assures full controllability of the system as all of itseigenvalues are appearing as poles. It clarified as well how the shrink and swellphenomena is associated with the transfer zeros located on the RHS of thecomplex plane leading to an initial inverse response which should be handledby the PID process controllers.

The identified main problem with the existing level control can be outlinedas follows

Assuming drop of exhaust heat provided to the drum leading to decreaseof pressure and water level. The feedwater control valve supplies morewater through the inlet yet unintentionally contributing into additionaldrop of the levelFrom the physical point of view this takes place since the cold water fedinto the drum decrease its temperature and as result its pressure as wellThe pressure controller tries to close the steam valve even more to trackthe set value leading eventually to further water drop due to steam bub-bles shirkingClearly the level control isn’t considering the initial inverse response iden-tified while examining the system open loop behaviour

The comparison results show that the model can capture the drum dynamicsto a great extent. However, a relatively small deviation from real measurementsand slightly faster response specially regarding pressure and level is still notice-able. The error arises due to the uncertainty of some model parameters suchas Td and K, in addition to the suggested assumptions necessary in order tosimplify the complete process simulation.

Nevertheless, the model current attitude can be regarded as satisfactory,bearing in mind that several control loops were omitted for simplification.Sooner or later, a realization within the real plant would ultimately requirean observer gain, whose design shall be discussed in section (4.3.2), to correctthe states based upon the difference between real measurements and modeloutputs.

Figure (3.8) shows the closed loop behaviour for different values of the pres-sure controller proportional gain. As mentioned earlier in the initial assump-tions (3.4.1), the gain doesn’t correspond to the current real value and had to

36


be increased so that the simulation results match the plant measurements. Thedrum pressure drops more when the gain isn’t changed and as a consequenceit causes higher overshoot of the level due to steam bubbles swelling.

The reasons behind such modification which as seen in the previous simu-lation results improved the model overall performance could be summarizedthroughout the following

The treatment of the bypass butterfly valve as a linear valve, thereforein reality the valve opens more to allow flow of additional steam, yet inthe model it doesn’t open with the same percentage due to the differentnature of both valves, which was already been clarified in figure (2.6)Neglecting storage of low pressure steam within the pipes and superheaterin the current model, as a result, introduction of an additional statevariable7 might be necessary

0 1000 2000 3000 4000

5.2

5.4

5.6

Pressure P (bar)

0 1000 2000 3000 4000

45678

Flow rate qs (kg/s)

0 1000 2000 3000 4000100

50

0

50

100

Level l (mm)

0 1000 2000 3000 40002

6

10

14

Flow rate qf (kg/s)

Figure 3.8.: Comparison between model and plant data (solid line) for a de-crease of the gas turbine electrical output power equivalent to20MW when Kp = 1.8 (dotted dashed) and Kp = 5 (dashed)

7System identification shows that a 5th order system matches better the plant measurementswhen compared to the current 4th order model

37

4. Process optimization

4. Process optimization

The concluded results brought to our attention during system modellingand analysis suggests that an optimization of the process is achievable usinga multivariable control technique. The strategy would account for synergybetween feedwater and steam flow rates instead of just decoupling the MIMOsystem into several coupled SISO systems regulated by their own noninteractingcontrol loops.

A state-feedback controller is suggested in order to consider the internalvariables of the system instead of the process outputs, therefore accounting foradditional aspects which were discarded using the classical control methodol-ogy. The inner dynamics of the drum-boiler unit correspond to the developednonlinear model state variables which were defined in section (2.2.2).

The control concept shall be addressed presenting the available control meth-ods and algorithms applying the approach while highlighting advantages anddisadvantages for each.

4.1. Concept of state-feedback control

For a continuous linear system described in state-space form (3.3) whosestates are available for feedback, it can be subjected to a multidimensionalproportional gain element F compared with the reference value to computethe actuating variables u according to the control law (4.1) where the resultingsystem matrix would be A−BF .

The designed state-feedback matrix would place the poles of the closed loopsystem in a desired position within the complex plane, thus directly influencingits rise and settling time, damping and transient oscillations.

u = −Fx (4.1)

4.1.1. Controllability and observability

The concept of controllability was briefly hinted to while examining the sys-tem poles and zeros in section (3.1.3). The term was introduced to investigatewhether the actuating variable are perfectly able to drive the system from any

38


initial state to the desired state. Alternatively observability was proposed toexamine ability of estimating the system states from a set of available measure-ments. The terms are dual to each other, thus any criterion or control designmethod can be applied for both by adjusting A � AT and B � CT .

State-space description offers the opportunity to investigate both propertiesanalytically by simply inspecting the system, input and output matrices respec-tively using Kalman, Hautus or Gilbert criterions [18]. Kalman criteria onlyinvestigate controllability/observability of the entire system without mention-ing any particular eigenvalue. Fortunately Hautus and Gilbert criterions canidentify the non-controllable/observable eigenvalue, thus allowing the controldesigner to adjust the plant structure accordingly if necessary.

4.1.2. Observer-based control

The states have to be estimated with the help of a plant model due to thefact that they are most likely hard to be measured in the real process. Thedifference between the real and estimated output is subjected to the so-calledLuenberger observer gain matrix [4] [16] [17] before being fed back to the model,thus correcting the states and matching the reality as much as possible.

The newly established objective similarly to state-feedback is computationof a feedback matrix L which modifies the system matrix into A − LC. Theobserver gain is designed in such a manner that ensures convergence of theestimation error to zero which allows usage of the corrected states for feedback.

C

A

B

A

CB

L

Luenberger Observer

Figure 4.1.: State estimation using Luenberger observer

39


4.1.3. PI-based state-feedback control

One major disadvantage with the basic state-feedback structure is lack ofreference value tracking even without existence of external disturbance due tothe absence of an integrator element. An enlargement of the basic structureto integrate the error is mandatory to ensure that the output can follow theset value, thus guaranteeing steady-state accuracy. Low control speed draw-back would arise similar to the classical I-controller which necessitates furtherenlargement by incorporating a supplementary P-controller which improvescontrol performance. The complete enlarged PI-based state-feedback controlstructure is shown in figure (4.2).

-F

B-I

P

-

A

C

State Controller

Figure 4.2.: PI-based state-feedback control structure

The previous control law have to be slightly modified considering error signalsas additional states h. The newly computed state-feedback matrix K wouldconsist of 3 parameters F , I and P affecting the states, integrated error signalsand reference tracker correspondingly.

The tunable parameters are assigned upon computation of the feedback ma-trix K. By default I is uniquely defined as it corresponds to the left-handside of the matrix. However F and P can be freely selected due to the un-derdeterministic nature of equation (4.2). Ignoring P element would lead to asimple I-controller, while setting F to zero isn’t relevant when attempting todesign a state controller, besides, this would normally introduce an unsolvableequation1 (4.3).

u = [PC − F ,−I]

[xh

]= −Kx′ (4.2)

P = −KC−1 (4.3)

1A solution might exist under very special certain conditions

40

4.2. Controller design methods


4.2.1. Pole placement method

Several algorithms do exist to apply the pole placement method which directlyassigns poles of the closed loop at the desired positions chosen by the designer.The method however is quite hard and time consuming when applied in practicewith real systems due to the huge amount of freedom offered.

Additionally it doesn’t take into consideration the limitations imposed fromthe real actuating variables and no clear guidelines do exist regarding how andwhere the poles should be placed. Finally predicting the system dynamicalbehaviour by just positioning the poles is a complicated task and the designprocess requires usually tedious work even for low order systems before satis-factory results can be achieved.

Ackermann’s formula

The algorithm is regarded as a standard design procedure which computesthe unique state-feedback f row vector or observer gain l column vector usingthe formula (4.4) [18] where n is the system order, p are the coefficients ofcharacteristic polynomial calculated while defining positions of the eigenvaluesfor the closed loop and finally t is the last column/row obtained from thecomputed controllability/observability matrix inverse.

Unfortunately such method cannot be applied for MIMO systems as the al-gorithm requires inversion of controllability/observability matrix which is onlyattainable with square matrices with full rank.

f = t1(p0 + p1A+ ...+ pn−1An−1 +An)

l = (p0 + p1A+ ...+ pn−1An−1 +An)t1

(4.4)

Kautsky, Nichols, and Van Dooren algorithm

The major difference which should be pointed out when designing a state con-troller for a MIMO system is that the state and observer feedback gain matricesare no longer unique, therefore offering an extra degree of freedom for design-ers. The algorithm [11] tries to find a solution which improves robustness ofthe resulting state controller by computation the matrix F and estimating inan iterative manner how closely are the eigenvalues of the closed loop systemmatrix A−BF from the desired position.

41


4.2.2. Linear-Quadratic method

Optimal control theory handles issues related to computation of a control lawfor a given system bearing in mind that certain optimality criterion has to beachieved, mainly focusing on how to operate a dynamical process at minimumcost. The calculus of variations maximum principle formulated an abstractframework which describes the optimal control problem trying to minimize thecost function (4.5) subjected to a dynamical linear system with zero initialconditions where ψ and L are the terminal cost and lagrangian2 respectively.

J = ψ(x(tf ), tf ) +

∫ tf

t0

L(x(t),u(t), t)dt

x = f(x(t),u(t), t)

(4.5)

LQ method is a significant result of the theory that manages problems associ-ated with quadratic performance criteria for state-space systems. The methodalgorithm computes a Linear-Quadratic Regulator/Estimator (LQR/LQE) re-ferred to as Riccati controller when designing a state-feedback matrix or Kalmanfilter when attempting to estimate states of a real system.

Riccati controller

The Riccati controller allows trade off between regulation performance andcontrol effort compared with the pole placement method. It’s regarded asa robust controller since it attains infinite marginal gain and offers a phasemargin δ ≥ 60◦ [15] which is aligned with practical guidelines for control systemdesign. The resulting optimal feedback gain should drive the closed-loop systemwithout external input from any initial state to the zero state minimizing thecost function described by equation (4.6).

J =

∫ ∞

0(xT (t)Qx(t) + uT (t)Ru(t))dt (4.6)

Q and R matrices are positive definite matrices assigned as weighting factorsfor the course of states and input variables. Faster convergence of a particularstate towards zero should increase its equivalent coefficient inside the matrixQ. If a slower response of the actuating variables is preferred to lower theenergy consumption and minimize control effort then coefficients of R matrixhave to be chosen larger.

2Function that summarizes dynamics of the system

42


The control law (4.1) tries to minimize the quality function (4.6) whoseoptimal feedback matrix F requires the symmetrical positive definite matrixP resulting from solution of the Matrix Algebraic Riccati Equation (MARE)(4.7) and hence the reason why a LQ regulator is named after Riccati.

F = R−1BTP

ATP + PA− PBR−1BTP +Q = 0(4.7)

Choosing the values of Q and R matrices in principle is similar to tuningof PID-controller parameters where the weighting matrices are varied until asatisfactory response is reached. Tuning a Riccati controller first carries inves-tigations concerning the range of values for each system state and actuatingvariable. Once these limits are established, an initial guess can take place byconstructing diagonal matrices with the normalized values as seen in (4.8).

Q =

⎡⎢⎣q11 · · · 0...

. . ....

0 · · · qnn

⎤⎥⎦ R =

⎡⎢⎣r11 · · · 0...

. . ....

0 · · · rnn

⎤⎥⎦

qii =1√xmaxi

, rii =1√umaxi

, i = 1, 2, ..., n

(4.8)

Kalman filter

The Kalman filter optimizes the estimation of the system states using a seriesof process measurements. It takes into account the input w and measurementv noises assumed to be random unbiased white noise. It constructs an optimalstate estimator that minimizes the cost function (4.9) where E() calculates theexpected value based on the assumed random noise.

J = E(eTWe) (4.9)

The observer gain L computed by solving the modified MARE (4.10) at-tempts to minimize the difference between estimated and real states consideringnoise influence on the process. The weighting matrices Q and R aren’t con-sidered as punishing factors anymore as they define intensities of the expectedprocess noise. Choice of their values usually starts with identity matrices asinitial guess then adjusted repeatedly until a decent estimation is achieved.

L = PCTR−1

AP + P TA− PCTR−1CP +Q = 0(4.10)

43

4.3. Observer-based state-feedback controller design

4.3. Observer-based state-feedback controller design

4.3.1. Riccati controller

The state-feedback matrix K is computed using LQ method due to its advan-tages over the pole placement method. The existing control valves PI-controlloops would remain the same, leading to smooth implementation of the pro-posed structure when performed at the distributed control system (DCS) ofthe real plant as seen in the block diagram shown in figure (4.10).

This idea was suggested since the state controller would be only trying togenerate the mass flow rates signals required to stabilize the process. Thesesignals can be treated as the set values for their corresponding control valveshandled by their own actual controllers, therefore further modifications in theplant control schemes would be skipped.

Setting the drum-boiler model states limits were straight forward and helpedperforming the initial simulation. The drum pressure limit would be 9 bar asspecified by construction data, the error between reference and actual measure-ment shouldn’t exceed 0.3 bar. The water level maximum allowable deviationfrom set point isn’t allowed to surpass ±150mm. The remaining states andactuating variables were kept at unity.

The main challenge was system stabilization considering the limitations andconstraints imposed by the control valves position range and fixed rate of open-ing/closing. Several simulation took place varying mainly r11, r22, q56 and q66associated with mass flow rates, pressure and level error signals respectivelyuntil good results were obtained with the weighting matrices provided along-side with the MATLAB script to calculate the state-feedback matrix K inappendix (A.3.2).

The tunable parameters from equation (4.2) were assigned as follows, theintegrator gain Q as mentioned is uniquely defined by the last two columns.The proportional gain P was set to zero3 and as a result the state-feedbackmatrix F doesn’t require any modifications since it’s described by the first fourcolumns when solving equation (4.2).

F =

[7.1296e-5 21.786 −319.78 0.29002−1.5083e-4 −3.8327 36.181 −1.2566e-2

]

I =

[1.0356e-2 −1.4135e-30.18248 2.6739e-5

]

3 P should be a 2-by-2 identity matrix in the real process, since the proportional gain ofthe PID-controller reacts on the sum of all actions and not the error signal

44

4.4. Simulation results

4.3.2. Luenberger observer

The drum-boiler unit nonlinear model is already estimating drum-boiler unit,however only lacking the observer gain. It shall be designed to eliminate theerror between measured data and model outputs noticed during validation insection (3.4). The observer gain isn’t calculated using Kalman filter becauseefficient practice of the method requires continuous testing of the estimatoralongside the real process. Further, the optimal choice of the filter weight-ing matrices depend on reliable prediction of the process noise which is onlyguaranteed by regular observation. Pole placement method is applied despiteits disadvantages mentioned in section (4.2.1) with respect to the followingpractical consideration.

The observer response must be faster than the closed loop employing thestate-feedback controller because the estimation error have to decay to zerocausing the state variables to converge before the states can be used for control.As a rule of thumb, the observer slowest pole A−LC should be faster than thestate-feedback controller poles A−BF . This would guide us in a certain wayto assign its position assuring suitable and decent estimation. The observerpoles are provided within the MATLAB script to calculate the matrix L inappendix (A.3.2).

L =

[992.85 1.0584e-2 2.6068e-4 0.47076

2.2837e-16 2.1728e-4 3.2667e-6 6.0672e-3

]T


The simulated optimized system performance will be shown in the follow-ing figures. First we shall examine the estimated states when employing theobserver to check if the new poles associated with A − LC introduce noiseinto the system. Then the model behaviour with state correction is validatedagainst new measurement data with very rich excitation covering the drum-boiler operating range. Therefore the observer stability can be investigated,providing a good indication of the proposed control strategy applicability sincethe states are crucial for feedback.

Later on, comparison between both controllers is addressed to check if thenewly proposed state-feedback controller did handle efficiently the main prob-lems identified with the current controller causing its poor performance whichwas discussed when concluding the process analysis in section (3.5). Further,the states and flow rates at different load conditions are inspected to ensurestability of state-feedback matrix K for the drum operating range.

45


States estimation

The observer pole placement was successful as shown in figure (4.3) since theestimated states noise is almost negligible therefore they can be efficiently fedback to the state-feedback controller matrix.

Observer performance subjected to perturbations in gas turbine electricaloutput power

Significant improvement of the model closed loop response when combinedwith the observer was achieved. Both outputs are almost matching perfectlywhen compared to the simulation conducted during primary validation in sec-tion (3.4). Figure (4.4) shows the comparison between the observer and realprocess when electrical power of the gas turbine was switched in between120MW and 80MW for approximately one hour.

Even for different initial conditions as seen in figure (4.5), the observer gainwas still being able to adjust the states and accordingly the process controllersto track the real output. Finally, the noticeable error which occurred due tomodel uncertainty and assumptions discussed in section (3.4.1) was eliminatedwhen adopting the observer gain as illustrated in figure (4.6).

State controller performance subjected perturbations in gas turbineelectrical output power

The drum pressure and water level were vastly enhanced when analyzing bothbehaviours depicted in figure (4.7). The level maximum peak overshoot/un-dershoot didn’t exceeds ±100mm during transients and the pressure neversurpasses the safety limits which might lead to operation of the security safetyvalve. Obviously the steam flow rate performance is the same using both con-trollers but the feedwater flow rate behaviour was modified in a way whichboosted the overall closed loop performance.

This is no surprise and should have been expected following process analysiswhich diagnosed the drum level cascade controller and highlighted its particularweakness. The optimal state controller was smart enough paying attention tothe initial inverse response and shrink/swell physical phenomena by consideringthe inner dynamics of the system instead of the output. It clearly solved oneof the main problems reported by the plant engineers.

46


State controller performance at different loading conditions

One should note that we designed the optimal state controller feedback matrixusing a linearized model which is normally valid for one particular operatingpoint. Therefore we have to investigate if it is still able to stabilize the system atvarious loading conditions and whether the closed loop response is still tolerablein regards with the requirements we assigned while weighting the matrices Qand R.Figure (4.8) illustrates the performance of the drum pressure and level for the

same loading conditions utilized to linearize the model, which were illustratedin table (3.2.1). Obviously the level drop was much higher at low load asthe feedback gain matrix wasn’t computed in order to optimize this particularoperation, nevertheless we still have a decent better response when comparedwith the existing process PID-controllers. Figure (4.9) shows how nicely themass flows set points considers the limitations imposed by the control valvesopening and closing rates allowing feedwater and steam flow rates to trackthem smoothly.

0 1000 2000 3000 4000 5000 6000 7000

20

22

Water volume Vwt (kg/s)

0 1000 2000 3000 4000 5000 6000 70000.01

0.015

Steam quality (%)

0 1000 2000 3000 4000 5000 6000 70000.2

0.95

1.7Steam volume under water level Vsd (m3)

Figure 4.3.: Estimated states using the observer for perturbations in gas tur-bine electrical output power

47


0 2000 4000 60005

5.2

5.4

5.6

5.8

Pressue P (bar)

0 2000 4000 6000

4

6

8

10

Flow rate qs (Kg/s)

0 2000 4000 6000

100

200

100

200

0

Level l (mm)

0 2000 4000 60000

5

10

15

Flow rate qf (Kg/s)

Figure 4.4.: Comparison between state observer (dashed) and plant (solid line)for perturbations in gas turbine electrical output power

0 1000 2000 30006

8

10

12

Flow rate qf (Kg/s)

0 1000 2000 30004020

0204060

Level (mm)0 1000 2000 3000

6

7

8

Flow rate qs (Kg/s)

0 1000 2000 3000

5.3

5.4

5.5

5.6

Pressure (bar)

Figure 4.5.: Comparison between state observer (dashed) and plant (solid line)for a decrease of the gas turbine electrical output power equivalentto 10MW

48


0 1500 3000 45005.2

5.4

5.6

Pressure P (bar)

0 1500 3000 45004

6

8

Flow rate qs (Kg/s)

0 1500 3000 4500100

50

0

50

100Level (mm)

0 1500 3000 4500

5

10

15Flow rate qf (Kg/s)

Figure 4.6.: Comparison between state observer (dashed) and plant (solid line)for a decrease of the gas turbine electrical output power equivalentto 20MW

0 2000 4000 60005

5.2

5.4

5.6

5.8Pressure (bar)

0 2000 4000 6000

4

6

8

10

Flow rate qs (kg/s)

0 2000 4000 6000

200

100

0

100

Level (mm)

0 2000 4000 60000

5

10

15

Flow rate qf (kg/s)

Figure 4.7.: Comparison between PI-based state-feedback controller (dashed)and plant (solid line) for perturbations in gas turbine electricaloutput power

49


0 500 1000 15005.35

5.4

5.45

5.5

Pressure (bar)

0 500 1000 15000

5

10Flow rate qs (kg/s)

0 500 1000 1500

100

50

0

Level (mm)

0 500 1000 15002468

1012

Flow rate qf (Kg/s)

Figure 4.8.: Model closed loop response using the PI-based state-feedback con-troller for high (dashed), medium (solid line) and low load (dotteddashed)

0 500 1000 1500

7

8

9

10Flow rates qf,qs (kg/s)

0 500 1000 1500

20.2

20.4

20.6

20.8

Water Volume Vwt (m3)

0 500 1000 15000.01

0.012

0.014


0 500 1000 15000.9

1

1.1

1.2

1.3

Steam bubbles volume Vsd (m3)

Figure 4.9.: States and input variables behaviour using the PI-based state-feedback controller at medium load

50


Dru

m-b

oile

r m

odel

Plan

t

L -F

-IEx

istin

g PI

cont

rolle

rsM

ass f

low

cont

rol v

alve

s

Flow

rate

s se

t val

ue

Obs

erve

r-bas

ed

Stat

e-fe

edba

ck c

ontro

ller

Set v

alue

trac

king

Figure

4.10.:Block

diagram

oftheproposed

multivariable

feed

back

controlstrategy

51

5. Conclusion and future work

5. Conclusion and future work

A mathematical nonlinear model which describes the dynamical process ofsteam generation using steam drum-boiler units including its control valvesand process PID-controllers was fitted into to a real drum-boiler unit whichcorresponds to 450MW CHP in Munich, in order to analyse the pressure andwater level control performance which was reported to behave very poorlyduring transients corresponding to huge load changes.

The model was implemented within MATLAB/Simulink environment andexamined intensively throughout various scenarios with very rich excitationfrom the plant covering a wide operating range to ensure its validity and reli-ability. Further it pointed out very clearly the main drawbacks of the existingcontrol strategy employed to stabilize the process.

Stability analysis was conducted by linearization at typical operating pointsof the drum-boiler unit. It predicts the plant open loop response quite welland clarifies the reason behind its non-minimum behaviour which is associatedwith the steam bubbles shrink/swell physical phenomena.

A multivariable feedback control strategy is proposed in order to optimize theprocess using a PI-based state-feedback controller designed using LQ methodensuring steady-state accuracy and set value tracking. Additionally an ob-server gain which guarantees correct estimation of the state variables requiredfor feedback is realized using pole placement method. Simulation results showsthat the state-feedback controller outperforms the PID-control in terms of con-trol behaviour and performance.

Unfortunately the complete control structure which combines both the state-feedback and observer together cannot be examined at the moment within thesimulation environment because the observer gains requires new measurementsfrom the plant while being handled by the proposed control strategy.

In the near future, the nonlinear model shall be realized within the realplant Distributed Control System (DCS) of GuD 2 at HKW Sud to act asan observer of the process, thus offering in return a great opportunity to testand examine the model more closely before being combined with the suggestedstate-feedback controller.

52

A. Appendix

A. Appendix

A.1. Nomenclature

Symbol Unit Description

A m2 AreaCp J/kg K Metal specific heat capacityh J/kg Specific enthalpyKv kg/hr Valve sizing coefficientm kg MassP Pascal Pressure

Q W Heat flow rateq kg/s Mass flow rate

Tsat◦C Saturation temperature

V m3 VolumeV ◦ m3 Volume in hypothetical situationTd s Residence time of steam in drumρ kg/m3 Density

Table A.1.: Physical units

Symbol Description

K Friction coefficientm Head lossx Valve opening percentageZ Compressibility factorαr Steam-mass fractionαv Steam-volume fractionβ Empirical coefficientζ Normalized length

Table A.2.: Dimensionless units

Symbol Description

c Condensationd Drumdc Downcomerfw Feedwaterr Risert Totals Steamw Water

Table A.3.: Subscripts

53

A.2. MATLAB Control System Toolbox

A.2. MATLAB Control System Toolbox

Control System ToolboxTM offers various functionality to design, analysis andtuning of linear controllers. In this section, the tools adopted in the thesis willbe briefly featured.

A.2.1. Linear analysis functions

The embedded functions can be utilized using the graphical user interface ”LTIviewer” or MATLAB command window.

linio defines the linearization input/output points

operspec specifies operating point requirements for states, inputsand outputs

findop computes steady-state operating point meeting predefinedspecifications

linearize performs linear approximation of a non-linear model

pzplot computes poles and zeros of a dynamic linear system andplot them in the complex plane

tzero computes invariant zeros of a linear MIMO system

A.2.2. Controller design functions

The functions are only accessible using MATLAB command window.

ss creates state-space model given the system matrices

eig computes eigenvalues for a system

ctrb computes the controllability matrix for state-space model

obsv computes the observability matrix for state-space model

place places the desired closed-loop poles at a desired position inthe complex plane

lqr computes an optimal state-feedback controller given the state-space model and weighting matrices

lqi computes an I-based optimal state-feedback controller

54

A.3. MATLAB script

A.3. MATLAB script

A.3.1. Drum-boiler model

The following function implements the state algebraic equations (2.2.2) to cal-

culate the states derivatives x =[dPdt ,

dVwtdt , dαr

dt ,dVsddt

]and the level l.

function y = DrumBoiler SWM (Q,qf,qs,P,Vwt,Alpha,Vsd)%% Model inputs% Q = Amount of heat flow rate added to the system (Watt)% qf = Feedwater flow rate (Kg/s)% qs = Steam flow rate (Kg/s)% P = Pressure (Pascal)% Vwt = Water total volume (m3)% Alpha = Steam quality (%)% Vsd = Steam bubbles volume under water level (m3)

%% Drum−boiler parameters and construction dataVd = 20.204; %Drum volume (m3)Vr = 20; %Drum riser volume (m3)Vdc = 0.9; %Drum downcomer volume (m3)Vt = Vd + Vr + Vdc; %Total drum volume (m3)Ad = 14.7; %Drum area (m2)Adc = 0.0637; %Downcomer area (m2)mr = 1300; %Riser mass (Kg)md = 1363; %Drum mass (Kg)mt = mr+md+98888; %Total metal mass (kg)K = 25; %Friction coefficient in downcomerTd = 3; %Residence time of steam in drum(s)Beta = 0.3 ; %Empirical coefficientVsd0 = 2; %Steam bubbles volume in the hypothetical situation (m3)Cp = 550; %Metal specific heat capacity (Pascal.m3/Kg.K)

%% Liquid/Vapour mixture propertiesP = P*1e−5; %Pascal to Bar

%% TemperatureTfw = 104; %Feedwater (C)T Sat = XSteam('Tsat p',P); %Saturation (C)dT Sat dP = IAPWS IF97('dTsatdpsat p',P*0.1) * 1e−6; %(K/Pa)%% DensityrhoV = XSteam ('rhoV p',P); %Steam (Kg/m3)rhoL = XSteam ('rhoL p',P); %Water (Kg/m3)%Partial derivative with pressuredrhoL dP = (2*P*0.0148 − 3.7836) * 1e−5; %Water (Kg/J)drhoV dP = (2*P*0.0010 + 0.4450) * 1e−5; %Steam (Kg/J)%% Specific EnthalpyhfW = XSteam('hL T',Tfw) *1e3; %Feedwater (J/Kg)

55

A.3. MATLAB script

hL = XSteam ('hL p',P) * 1e3; %Water (J/Kg)hV = XSteam ('hV p',P) * 1e3; %Steam (J/Kg)hC = hV−hL; %Condensation (J/Kg)%Partial derivative with pressuredhL dP = IAPWS IF97('dhLdp p',P*0.1) * 1e−3; %Water (J/Kg.Pa)dhV dP = IAPWS IF97('dhVdp p',P*0.1) * 1e−3; %Steam (J/Kg.Pa)

%% Coefficients ValuesEta = (Alpha*(rhoL−rhoV))/rhoV;AlphaV = (rhoL / (rhoL−rhoV)) * (1 − ((rhoV/((rhoL−rhoV)*Alpha)) ...

* log(1+(((rhoL−rhoV)*Alpha)/rhoV))));dAlphaV dP = (1/((rhoL−rhoV)ˆ2))*(rhoL*drhoV dP − ...

rhoV*drhoL dP)*(1 + (rhoL/(rhoV *(1+Eta))) − ...(((rhoV+rhoL)*log(1+Eta))/(rhoV*Eta)));

dAlphaV dAlpha = (rhoL/(rhoV*Eta))*(((1/Eta)*log(1+Eta)) − ...(1/(1+Eta)));

qdc = sqrt((2*rhoL*Adc*(rhoL−rhoV)*9.81*AlphaV*Vr)/K);Vwd = Vwt − Vdc − (1−AlphaV)*Vr;

%% State equations coefficientse11 = rhoL − rhoV;e12 = Vwt*drhoL dP + (Vt−Vwt)*drhoV dP;e21 = rhoL*hL − rhoV*hV;e22 = Vwt*(hL*drhoL dP + rhoL*dhL dP) + (Vt−Vwt)*(hV*drhoV dP + ...

rhoV*dhV dP) − Vt + mt*Cp*dT Sat dP;e32 = (rhoL*dhL dP − Alpha*hC*drhoL dP)*(1−AlphaV)*Vr + ...

((1−Alpha)*hC*drhoV dP + rhoV*dhV dP)*AlphaV*Vr + (rhoV + ...(rhoL−rhoV)*Alpha)*hC*Vr*dAlphaV dP − Vr + mr*Cp*dT Sat dP;

e33 = ((1−Alpha)*rhoV + Alpha*rhoL)*hC*Vr*dAlphaV dAlpha;e42 = Vsd*drhoV dP + (1/hC)*(rhoV*Vsd*dhV dP + rhoL*Vwd*dhL dP − ...

Vsd − Vwd + md*Cp*dT Sat dP) + ...Alpha*Vr*(1+Beta)*(AlphaV*drhoV dP + (1−AlphaV)*drhoL dP + ...(rhoV−rhoL)*dAlphaV dP);

e43 = Alpha*(1+Beta)*(rhoV−rhoL)*Vr*dAlphaV dAlpha;e44 = rhoV;

%% State variablesdP dt = (1/(e11*e22 − e12*e21))*(e11*Q + qf*(hfW*e11 − e21) + ...

qs*(e21 − hV*e11));dVwt dt = (1/(e11*e22 − e12*e21))*(qf*(e22 − hfW*e12) + ...

qs*(hV*e12 − e22) − e12*Q);dAlpha dt = (1/e33)*(Q − Alpha*hC*qdc − e32*dP dt);dVsd dt = (1/e44)*(((rhoV/Td)*(Vsd0−Vsd)) + ((hfW − hL)*qf/hC) − ...

e42*dP dt − e43*dAlpha dt);Level = (Vwd+Vsd)/Ad −0.875;

%% Model outputsy = [ dP dt dVwt dt dAlpha dt dVsd dt Level ];end

56

A.3. MATLAB script

A.3.2. Controller design

The following m.file was used to obtain a linearized model in state-state form ata common operating point to compute the state-feedback and observer matricesgains using LQ and pole placement methods.

% Specify the model namemodel = 'Drum Model';

% Create the linearization I/O as specified in Linearize Modelios(5) = linio('Drum Model/Drum model',2,'out');ios(4) = linio('Drum Model/Drum model',1,'out');ios(3) = linio('Drum Model/qs (Kg//s)',1,'in');ios(2) = linio('Drum Model/qf (Kg//s)',1,'in');ios(1) = linio('Drum Model/Q (MW)',1,'in');

% Create the operating point specification objectopspec = operspec(model);

% Set the constraints in the modelopspec.Inputs(2).u = 9; % qf (Kg/s)opspec.Inputs(2).Known = true;opspec.Outputs(1).y = 5.5; % P (bar)opspec.Outputs(1).Known = true;opspec.Outputs(2).y = 0; % l (mm)opspec.Outputs(2).Known = true;

% Perform the operating point search% Linearize the modelop = findop(model,opspec,opt);sys = linearize(model,op,ios);

% Create state−space model excluding Q(MW)Drum ss = ss(sys.A,sys.B(:,2:3),sys.C,sys.D(:,2:3));

% Assign weighting matrices Q and R% Assign pole position for the observerQ = [ 1.23e−12 0 0 0 0 0;

0 1 0 0 0 0;0 0 1 0 0 0;0 0 0 0.25 0 0;0 0 0 0 5 0;0 0 0 0 0 1e−4 ];

R = [ 50 0; 0 150 ];P = [ −0.03 −0.03 −0.2 −0.9];

% Compute optimal state−feedback controller K and observer gain LK = lqi(Drum ss,Q,R);L = place(Drum ss.A',Drum ss.C',P)';

57

A.4. Heat engines

A.4. Heat engines

Heat engine plays an essential role in electrical power generation as they con-vert the thermal energy into mechanical energy required to drive the generatorshaft producing electricity.

The heat engine can be a closed or open loop system, which involves typicallyfour thermodynamic basic processes shown in figure (A.1). It converts the stateof the working fluid into another before returning it to its original state. Theprocesses are compression, heat addition, expansion and heat rejection, eachcan be carried out under one or more of the following conditions:

Isothermal At constant temperature

Isobaric At constant pressure

Isometric / Isochoric / Iso-volumetric At constant volume

Adiabatic At constant entropy, no heat is added or removed from thesystem and no work done.

Isentropic At constant entropy, reversible adiabatic conditions

Heat addition

Expansion

Heat rejection

Compression

Exhaustheat

FluidOpen system

Closed system

Figure A.1.: Heat engine typical closed/open loop heat cycle

A.4.1. Brayton cycle

The cycle shown in figure (A.2) is mathematical model used describe thethermodynamics and heat cycle for the operation of the gas turbine.

58

A.4. Heat engines

Process 1-2 Fresh air is being supplied for an open cycle, as for a closedone it’s drawn back from the turbine to a compressor increasing its pres-sure in an adiabatic compression process.

Process 2-3 The compressed air is mixed with fuel or natural gas beforebeing burned inside the combustion chamber at constant pressure.

Process 3-4 The heated pressurized air is supplied to the heat engine,where it’s allowed to expand through the turbine driving its blades, inan adiabatic expansion process.

Process 4-1 Finally, heat rejection to the surrounding atmosphere takesplace at constant pressure.

P

v

q in

q out

s = const.

s = const.

1

2 3

4

P-v Diagram

T

s

q in

q outp = const.

p = co

nst.

1

2

3

4

T-s Diagram

Figure A.2.: T-S and P-V diagram of a typical ideal Brayton cycle

A.4.2. Rankine cycle

The cycle shown in figure (A.3) is a mathematical model used to describeclosed cycle heat conversion into mechanical energy using two phase workingfluids that drive a steam turbine blades producing electricity.

Process 1-B High pressure water is pre-heated at constant pressure atthe economizer stage, until it reaches its boiling point converting it towater-vapour mixture.

Process B-2 A second heating phase takes place using evaporator andsuperheater, to convert the converting the mixture into superheated steam.

Process 2-3 The vapour at high pressure and temperature enters theturbine, where vapour energy is converted into mechanical work drivingthe turbine blades.

59

A.5. Non-minimum phase systems

Process 3-4 Vapour leaving the turbine at low pressure and temperatureis condensed, converting it into wet saturated water.

Process 4-1 Saturated water is pumped back, feeding the boiler at highpressure, where the cycle is repeated.

Figure A.3.: T-S diagram of a typical ideal Rankine cycle

A.5. Non-minimum phase systems

A plant whose poles and zeros are real numbers and located within the left-hand side (LHS) of the complex plane is known as a minimum phase system.This is due to the fact that the phase shift have a minimum range restrictedwithin 0◦ to −90◦ degrees for a given amplitude response when being examinedin frequency domain. If a stable plant have one or more zeros in the right handside (RHS) of the complex plane, then phase shift range is always greaterthan −90◦. Such systems are known as non-minimum phase, where an inverseresponse always exists, leading to an initial overshoot or undershoot delayingthe output behaviour.

Assume a Single-Input, Single-Output (SISO) plant whose closed loop trans-fer function is assumed to be internally stable and given by equation (A.1), onlyone zero lies in RHS of the complex plane for the sake of simplicity. A step in-put is applied to the plant closed loop and the output Y (s) is given by equation(A.2). Due to the assumption that the plant is internally stable, the open-loopzero z0 lies within of the region of convergence (ROC) of Y (s), which yieldsthe unilateral laplace transform to (A.3).

60

A.6. Integral anti-windup control

Since the output signal has an initial value y(0) = 0, final value y(∞) = 1and its area under the curve evaluated by the integral in equation (A.3) is equalto 0, then this implies that the output signal y(t) must take negative valuesover time.

Gcl(s) =Y (s)

R(s)=

(s− z0)

s+ p0(A.1)

Y (s) = Gcl(s)R(s) =(s− z0)

s(2s+ p0 − z0)(A.2)

∫ ∞

0y(t)e−z0tdt = Y (s)|s=z0 = Y (z0) =

(z0 − z0)

s(2s+ z0 − z0)= 0 (A.3)

Figure (A.4) illustrates the step and bode responses of a simple 2nd orderminimum and non-minimum phase plant. The zero of the first plant lies in theLHS leading to the normal expected step response, however, as it’s is shifted tothe RHS for the second plant, there exists an undershoot in the initial responseand delay in overall response, caused by the phase shift difference as seen inthe bode plot.

0 5 10 15 200.2

0

0.2

0.4

0.6

0.8

1

1.2Step response

40

20

0

Mag

nitu

de (d

B)

102

101

100

101

102

90450

4590

135180

Phas

e (d

eg)

Bode Diagram

Figure A.4.: Step response and bode plot for a minimum (solid line) and anon-minimum (dashed) system


Most PID-controllers in practical applications are equipped with a nonlinearsaturation element which saturates the controller output once it attains a cer-tain values, imposed by the physical limitations of the actuators. The nonlin-

61


earity might lead to integral windup.Such occurring phenomena take place when the PID-controller integral ele-

ment builds up and accumulates the error signal even if the controller outputis saturated, it might degrade the controller performance or even lead to closedloop instability if neglected.

Consider a simple linear motor positioner modeled as an integrator, whoseinput and output are velocity and position respectively. Clearly, the motor ve-locity will be physically limited according to its type and manufacturer, there-fore the equipped PI-controller output should be limited when used to controlthe closed loop. The controller parameters weren’t tuned since this is merely anexplanatory example focused on effects of windup phenomena and saturationlimit is set to be ±0.3m

s associated with the motor allowable maximum speed.

0 10 20 30 40 500.4

0.2

0

0.2

0.4Velocity (m/s)

0 10 20 30 40 500

1

2

3

4

Position (m)

Figure A.5.: Motor velocity and position behaviours with (dashed) and without(solid line) anti-windup

Figure (A.6) illustrates the closed-loop behaviour of the motor positionerwhile equipping an anti-windup mechanism using back-calculation method andcompares it when no anti-windup is utilized. The position set point changedto 3m, due to this large switch, the controller tries to track the reference valueas fast as possible, however it was limited by the motor velocity upper limit.Without anti-windup, the integrator element output keeps growing and themotor position would require more time to reach the steady-state.

On the other hand, while equipping an anti-windup, the back-calculation gainstarts discharging the PI-controller integrators and prevents it from building uponce the controller output is saturated, therefore vastly improving the outputsettling time while eliminating undesired overshoots.

62

A.7. Drum-boiler state equations coefficients

A.7. Drum-boiler state equations coefficients

e11 = Vwt∂ρw∂p

+ Vst∂ρs∂p

e12 = ρw − ρs

e21 = Vwt

(hw

∂ρw∂p

+ ρw∂hw∂p

)+ Vst

(hs

∂ρs∂p

+ ρs∂hs∂p

)− Vt +mtCp

∂tsat∂p

e22 = ρwhw − ρshs

e31 =

(ρw

∂hw∂p

− αrhc∂ρw∂p

)(1− αv)Vr +

(ρs

∂hs∂p

+ (1− αr)hc∂ρs∂p

)αvVr+

(ρs + (ρw − ρs)αr))hcVr∂αv

∂p− Vr +mrCp

∂tsat∂p

e33 = ((1− αr)ρs + αrρw)hcVr∂αv

∂p

e41 = Vsd∂ρs∂p

+ αr(1 + β)Vr

(αv

∂ρs∂p

+ (1− αv)∂ρw∂p

+ (ρs − ρw) +∂αv

∂p

)

1

hc

(ρsVsd

∂hs∂p

+ ρwVwd∂hw∂p

− Vsd − Vwd +mdCp∂tsat∂p

)

e43 = αr(1 + β)(ρw + ρs)V r∂αv

∂p

qdc =

√2ρwAdc(ρw − ρs)gαvVr

K

qct =hw − hfw

hcqf +

1

hc

(ρwVwt

∂hw∂p

+ ρsVst∂hs∂p

− Vt +mtCp∂tsat∂p

)dP

dt

qr = qdc − Vr

(αv

∂ρs∂p

+ (1− αv)∂ρw∂p

+ (ρw − ρs)∂αv

∂p

dP

dt

)+ (ρw − ρs)Vr

∂αv

∂αr

dαr

dt

αv =ρw

ρw − ρs

(1− ρs

(ρw − ρs)αrln

(1 +

ρw − ρsρs

αr

))

ζ = αr(ρw − ρs)

ρs∂αv

∂αr=

ρwρsζ

(ln(1 + ζ)

ζ− 1

1 + ζ

)

∂αv

∂p=

1

(ρw − ρs)2

(ρw

∂ρs∂p

− ρs∂ρw∂p

)(1 +

ρwρs(1 + ζ)

− ρs + ρwζρs

ln(1 + ζ)

)

63

A.8. Operator interface


Figure A.6.: Screenshot of the drum-boiler unit in the real process DCS

64


Figure A.7.: Screenshot of the low pressure steam distribution network in thereal process DCS

65

B. List of Figures

B. List of Figures

2.1. Combined cycle working principle . . . . . . . . . . . . . . . . . 92.2. Heat Recovery Steam Generator (HRSG) [14] . . . . . . . . . . 102.3. Schematic diagram of the low pressure steam generation process 112.4. Schematic diagram of the downcomer-riser circulation loop [13] 122.5. Flow through control valve for liquid service [22] . . . . . . . . 152.6. Inherent flow characteristics of typical control valves [24] . . . . 162.7. Simulink model of the Drum-boiler unit . . . . . . . . . . . . . 192.8. Simulink model of the control valve combined with its actuator 212.9. Simulink model of a parallel PID-controller equipped with an

anti-windup mechanism . . . . . . . . . . . . . . . . . . . . . . 22

3.1. Pole-zero plot of the linearized models at low (1), medium (2)and high (3) load . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2. Open loop response for a step change equivalent to decrease of20MW of the gas turbine electrical output power . . . . . . . . 31

3.3. Open loop response for a step change equivalent to 10% openingof butterfly valve position . . . . . . . . . . . . . . . . . . . . . 31

3.4. Open loop response for a step change equivalent to 10% closingof feedwater control valve position . . . . . . . . . . . . . . . . 32

3.5. Simulink validation model . . . . . . . . . . . . . . . . . . . . . 343.6. Comparison between model (dashed) and plant data (solid line)

for a decrease of the gas turbine electrical output power equiva-lent to 20MW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.7. Comparison between model (dashed) and plant data (solid line)for an increase of the gas turbine electrical output power equiv-alent to 10MW . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.8. Comparison between model and plant data (solid line) for a de-crease of the gas turbine electrical output power equivalent to20MW when Kp = 1.8 (dotted dashed) and Kp = 5 (dashed) . 37

4.1. State estimation using Luenberger observer . . . . . . . . . . . 394.2. PI-based state-feedback control structure . . . . . . . . . . . . 40

66

B. List of Figures

4.3. Estimated states using the observer for perturbations in gas tur-bine electrical output power . . . . . . . . . . . . . . . . . . . . 47

4.4. Comparison between state observer (dashed) and plant (solidline) for perturbations in gas turbine electrical output power . 48

4.5. Comparison between state observer (dashed) and plant (solidline) for a decrease of the gas turbine electrical output powerequivalent to 10MW . . . . . . . . . . . . . . . . . . . . . . . . 48

4.6. Comparison between state observer (dashed) and plant (solidline) for a decrease of the gas turbine electrical output powerequivalent to 20MW . . . . . . . . . . . . . . . . . . . . . . . . 49

4.7. Comparison between PI-based state-feedback controller (dashed)and plant (solid line) for perturbations in gas turbine electricaloutput power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.8. Model closed loop response using the PI-based state-feedbackcontroller for high (dashed), medium (solid line) and low load(dotted dashed) . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.9. States and input variables behaviour using the PI-based state-feedback controller at medium load . . . . . . . . . . . . . . . . 50

4.10. Block diagram of the proposed multivariable feedback controlstrategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

A.1. Heat engine typical closed/open loop heat cycle . . . . . . . . . 58A.2. T-S and P-V diagram of a typical ideal Brayton cycle . . . . . 59A.3. T-S diagram of a typical ideal Rankine cycle . . . . . . . . . . 60A.4. Step response and bode plot for a minimum (solid line) and a

non-minimum (dashed) system . . . . . . . . . . . . . . . . . . 61A.5. Motor velocity and position behaviours with (dashed) and with-

out (solid line) anti-windup . . . . . . . . . . . . . . . . . . . . 62A.6. Screenshot of the drum-boiler unit in the real process DCS . . 64A.7. Screenshot of the low pressure steam distribution network in the

real process DCS . . . . . . . . . . . . . . . . . . . . . . . . . . 65

67

C. List of Tables

C. List of Tables

2.1. Drum-boiler model parameters . . . . . . . . . . . . . . . . . . 202.2. Control valve and actuator parameters . . . . . . . . . . . . . . 212.3. PID-controller parameters . . . . . . . . . . . . . . . . . . . . . 22

3.1. Drum-boiler operating points for low, medium and high load . 27

A.1. Physical units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.2. Dimensionless units . . . . . . . . . . . . . . . . . . . . . . . . . 53A.3. Subscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

68

D. Bibliography

D. Bibliography

[1] Asea Brown Boveri: Drum Level Control Systems in the Process Industries,1997. ABB Download Center.

[2] Astrom, Karl Johan und Rodney D. Bell: Drum Boiler Dynamics. Auto-matica, 36:363–378, Marz 2000.

[3] Astrom, K.J. und T. Hagglund: PID Controllers - Theory Design andTuning. International Society of America, 1995, ISBN 9781556175169.

[4] Ellis, G.: Observers in Control Systems - A Practical Guide. AcademicPress, 2002, ISBN 9780122374722.

[5] Emami, A. und P. Van Dooren: Computation of zeros of linear multiva-riable systems. Automatical, 18:412–430, 1982.

[6] Falb, P. L. und W. A. Wolovich: On the decoupling of multivariable sys-tems. Proc. JACC, Philadelphia, Pennsylvanial, 41:791–796, 1967.

[7] Flynn, M.E. und M.J. O Malley: A drum Boiler Model for Long TermPower System Dynamic Simulation. IEEE Transaction Power System,14(1):209–217, 1999.

[8] G., Westner und Madlener R.: Development of Cogeneration in Germany:A Dynamic Portfolio Analysis Based on the New Regulatory Framework.FCN Working Paper, 2009.

[9] Goodwin, G. C., S. F. Graebe und M. E. Salgado: Control System Design.Prentice Hall, Upper Saddle River, New Jersey, international Auflage,2001.

[10] Holmgren, Magnus: X Steam, Thermodynamic properties of water andsteam, 2006. MATLAB Central File Exchange.

[11] Kautsky, J., N.K. Nichols und P. Van Dooren: Robust Pole Assignmentin Linear State Feedback. International Journal of Control, 41:1129–1155,1985.

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[12] Kehlhofer, R., B. Rukes, F. Hannemann und F.X. Stirnimann: Combined-Cycle Gas and Steam Turbine Power Plants. PennWell, 2009,ISBN 9781593701680.

[13] Kim, H. und S. Choi: A model on water level dynamics in natural circula-tion drum-type boilers. International Communications in Heat and MassTransfer, 32:786 – 796, 2005.

[14] Kim, T.S., D.K. Lee und S.T. Ro: Analysis of thermal stress evolutionin the steam drum during start-up of a heat recovery steam generator.Applied Thermal Engineering, 20(11):977 – 992, 2000, ISSN 1359-4311.

[15] Levine, William S.: The control handbook. The electrical enginee-ring handbook series. CRC Press New York, Boca Raton (Fl.), 1996,ISBN 0-8493-8570-9.

[16] Luenberger, D. G.: Observing the State of a Linear System. IEEE Tran-sactions on Military Electronics, 8:74–80, 1964.

[17] Luenberger, D. G.: Observers for multivariable systems. IEEE Transacti-ons on Automatic Control, 11:190–197, 1966.

[18] Michels, K.: Regelungstechnik (Vorlesungsmanuskript). Institut fur Auto-matisierungstechnik, Universitat Bremen, 2013.

[19] Mikofski, Mark: IAPWS IF97 functional form with no slip, 2012. MAT-LAB Central File Exchange.

[20] Moran, M.J. und H.N. Shapiro: Fundamentals of engineering thermodyna-mics. John Wiley and Sons Inc., New York, NY, 2009.

[21] Parry, A., Petetrot J. F. und M. J Vivier: Recent progress in sg level controlin french pwr plants. British Nuclear Energy Society, Seiten 81–88, 1995.

[22] Samson AG: Application Notes for Valve Sizing, 2012. Samson ProductDocumentation.

[23] Skogestad, Sigurd und Ian Postlethwaite: Multivariable Feedback Control:Analysis and Design. John Wiley & Sons, 2005, ISBN 0470011688.

[24] Spirax-sarco: The Steam and Condensate Loop Book, 2011.

[25] Triantafyllou, Michael S. und Franz S. Hover: Maneuvering and Control ofMarine Vehicles (Lecture Notes). Massachusetts Institute of Technology,2003. MIT OpenCourseWare.

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