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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 R¨ unzi

1st Examiner: Prof. Dr.-Ing. Kai Michels 2nd Examiner: Prof. Dr.-Ing. Bernd Orlik

Universität Bremen

Acknowledgment It gives me great pleasure in expressing my sincere gratitude to everyone who have supported and contributed into making this thesis possible. I would like first to acknowledge my direct supervisor Dipl.-Ing. Simon R¨ unzi for his enthusiasm, inspiration and huge efforts to explain things clearly and simply. His in-depth knowledge regarding the CHP plant in Munich, related to his PhD research, was quite helpful and beneficial for my work. Furthermore I would like to thank my examiner Prof. Dr.-Ing Kai Michels for offering me the project which have evolved over the course of time into an interesting thesis topic. I wish also to address their constructive criticism following initial review of the thesis. My appreciation for SWM Services GmbH, specially Mr. Julian Niedermeier for his willingness to perform experiments on the plant, its priceless valuable information contributed significantly to improve my understanding of the real process. I wish to acknowledge the scholarship support provided by the Katholischer Akademischer Ausl¨ ander-Dienst (KAAD). In particular I am very grateful to Dr. Christina Pfestroff as I do believe that my master studies in Germany wouldn’t have been possible without her guidance when applying for the scholarship. I thank as well Prof. Dr.-Ing Rainer Laur, Mr. Hans Landsberg, Mr. Raphael Nabholz and Mrs. Claudia Dillmann for their continuous follow-up and 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 controller designed using LQ and pole placement methods to optimize pressure and water level control performance of a drum-boiler unit that belongs to a 450 MW CHP plant in Germany. The ˚ Astr¨ om-Bell nonlinear model is initially built within MATLAB/Simulink environment, later enlarged to include the process PID-controllers and control valves regulating mass flow rates before being validated against data measurements with very rich excitation. The concluded simulation results adopting the newly proposed control strategy shows that the suggested multivariable control technique outperforms the existing PID-controller in many aspects improving the control performance significantly and yielding much tighter reference value tracking during load changes. Keywords: drum-boiler level control; optimal control; multivariable feedback control; power plants simulation

Contents

Contents 1. Introduction 2. Process modelling 2.1. Combined cycle process overview . . . . . . . 2.1.1. Gas turbine . . . . . . . . . . . . . . . 2.1.2. Heat recovery steam generator . . . . 2.1.3. Steam turbine . . . . . . . . . . . . . 2.1.4. Surface condenser . . . . . . . . . . . 2.2. Steam generation process description . . . . . 2.2.1. Drum-boiler mass and energy balance 2.2.2. Drum-boiler nonlinear state equations 2.2.3. Mass flow control valve . . . . . . . . 2.2.4. Process PID-controller . . . . . . . . . 2.3. MATALB/Simulink model . . . . . . . . . . . 2.3.1. Drum-boiler model . . . . . . . . . . . 2.3.2. Control valve and actuator model . . . 2.3.3. Process PID-controller model . . . . .

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3. Process analysis and validation 3.1. Theoretical overview . . . . . . . . . . . . . . . . . . 3.1.1. Concept of stability . . . . . . . . . . . . . . 3.1.2. Linearization . . . . . . . . . . . . . . . . . . 3.1.3. Poles and zeros . . . . . . . . . . . . . . . . . 3.2. Stability analysis . . . . . . . . . . . . . . . . . . . . 3.2.1. Linear state-space model . . . . . . . . . . . 3.2.2. I/O pole-zero plot . . . . . . . . . . . . . . . 3.3. Open loop step response . . . . . . . . . . . . . . . . 3.3.1. Change of gas turbine electrical output power 3.3.2. Change of butterfly valve position . . . . . . 3.3.3. Change of feedwater control valve position . . 3.4. Validation . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Assumptions . . . . . . . . . . . . . . . . . .

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Contents 3.4.2. Comparison with measurement data . . . . . . . . . . . 3.5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 4. Process optimization 4.1. Concept of state-feedback control . . . . 4.1.1. Controllability and observability 4.1.2. Observer-based control . . . . . . 4.1.3. PI-based state-feedback control . 4.2. Controller design methods . . . . . . . . 4.2.1. Pole placement method . . . . . 4.2.2. Linear-Quadratic method . . . . 4.3. Observer-based state-feedback controller 4.3.1. Riccati controller . . . . . . . . . 4.3.2. Luenberger observer . . . . . . . 4.4. Simulation results . . . . . . . . . . . .

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5. Conclusion and future work A. Appendix A.1. Nomenclature . . . . . . . . . . . . . . . A.2. MATLAB Control System Toolbox . . . A.2.1. Linear analysis functions . . . . . A.2.2. Controller design functions . . . A.3. MATLAB script . . . . . . . . . . . . . A.3.1. Drum-boiler model . . . . . . . . A.3.2. Controller design . . . . . . . . . A.4. Heat engines . . . . . . . . . . . . . . . A.4.1. Brayton cycle . . . . . . . . . . . A.4.2. Rankine cycle . . . . . . . . . . . A.5. Non-minimum phase systems . . . . . . A.6. Integral anti-windup control . . . . . . . A.7. Drum-boiler state equations coefficients A.8. Operator interface . . . . . . . . . . . .

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B. List of Figures

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C. List of Tables

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D. Bibliography

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1. Introduction

1. Introduction Energy market deregulation and integration of renewable energy resources into the electrical grid have led to dramatic changes in the power industry which escalated rapidly new challenges that have to be met by conventional power plants. Such evolution caused a noticeable process modification regarding how power plants operate, as they should become more flexible to fulfill their load requirements which are more frequent nowadays. The process controllers have to be designed in a way which can simultaneously fulfill the load demand as soon as possible while at the same time bearing in mind safety and life span of the plant crucial elements. One common challenge is control of steam drum-boiler units handling supply of the steam turbine continuously with steam at high pressure and temperature. The controller should maintain drum pressure and water level within acceptable ranges for all operating conditions. If the level exceeds upper limits, water would be carried over to the superheater or the turbine leading to outage in either of the turbine or the boiler. Surpassing lower limits would cause overheating of the water wall tube resulting in serious tube rupture and severe damage. Drum level control in particular is quite tough due to the process physical phenomena known as shrink/swell of steam bubbles under the water level which causes the system to react with an initial inverse response known as a nonminimum phase behaviour. Classical control design methods using 2-element or 3-element PID-controllers can behave fairly well to compensate such effect. However as the process is quite complicated, dealing with several input variables to regulate each process variable separately might end up with bad parameter tuning and poor level performance observed during load changes, eventually leading the boiler unit to trip or even worse cause emergency shutdown of the power plant. It is stated that about 30% of the emergency shutdowns in French pressurized water reactors (PWR) plants were caused by poor level control of a steam drum-boiler unit [21]. An ongoing research project is taking place at the moment in collaboration with Stadtwerke M¨ unchen GmbH - Munich City Utilities (SWM) in regards with the process PID-controllers of the low pressure drum-boiler unit,

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1. Introduction located within the combined cycle plant GuD 2, short for Gas-und-DampfKombikraftwerk at Heizkraftwerk S¨ ud (HKW) - combined heat and power (CHP) facility. The main objective is drum level and pressure closed loop performance optimization which have been reported to behave very poorly under huge load changes taking place frequently following energy deregulation in Germany. The thesis is presented as follows, initially the complete process is briefly introduced before being simplified to highlight the significant elements dominating the steam generation process which are mainly focused on during modelling procedure. Derivation of the differential equations is carried out for each established featured element to develop a mathematical model capable of capturing most of the system nonlinearities and later on suitable for model-based control. The model parameterized and implemented within MATLAB/Simulink environment will be subjected to a detailed analysis by examining stability, simulating the model open loop response and validating the closed loop against data measurements from the plant. The investigation concluded results will offer a good insight into the system inner dynamics and shall inspect the model ability to catch the plant dynamical behaviour for a wide spectrum of operating conditions. In the end, the proposed control strategy is addressed. First, state-feedback control concept and the numerous methods which applies it shall be briefly described to illustrate their applicability and major difference between them. The most convenient and suitable approach shall be employed to compute the state-feedback and observer gain matrices. Finally, simulation results of the process utilizing the newly designed observer-based state-feedback controller is presented for various sequences to ensure stability of the optimized closed loop.

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2. Process modelling

2. Process modelling HKW S¨ ud plant is classified as a combined cycle cogeneration plant, it can handle concurrent production of electrical power and useful heat utilizing a class of sustainable integrated technologies progressively being used. Cogeneration plants reduce thermal and mechanical losses, harmful carbon dioxide (CO2 ) emissions and more importantly increases the overall plant efficiency to approximately 81% in comparison to stand alone plants which don’t exceed 45%. The German government is planning to double its share of CHP plants from approximately 12% to 25% by 2020, as part of the Integrated Energy and Climate Protection Program (IECPP) [8]. GuD 2 at HKW S¨ ud manages electrical power generation by combining both Brayton and Rankine thermodynamic theoretical cycles (A.4) [12] [20] using gas and steam turbines. Exhaust gas emitted from the gas turbine can be reused as the heat source for steam production required to operate the steam turbine, therefore more useful energy can be extracted, supplying additional electricity to the grid. Further energy can by even withdrawn from the low pressure steam leaving the turbine when condensed using a heat exchanger where the low temperature steam released can be utilized for district heating or water desalination. In this chapter, the overall combined cycle process is being narrowed down to draw the focus on one particular key element within the plant. The process is further simplified in order to spotlight primarily our aim interest which is the steam production using the low pressure drum-boiler unit along side with its process PID-controllers.

2.1. Combined cycle process overview GuD 2 at HKW s¨ ud plant combined cycle principle is shown in figure (2.1), it consists of the following main elements briefly described 2 1 1 1

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

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2.1. Combined cycle process overview

Low pressure water

Waste heat to atmosphere

Electricl power

Cooling water

Condenser Steam turbine Superheated steam Feedwater pump

Fresh air Electricl power Gas turbine Exhaust heat

Heat Recovery Steam Generator

Gas

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 on the Brayton open cycle (A.4.1). Continuous fresh air is compressed then mixed with the supplied natural gas before being burned inside the combustion chamber at around 1124◦ C. The hot compressed air expands within the turbine driving its blades which eventually turns the generator shaft producing electrical power and the exhaust low pressure gas leaving the turbine at 535◦ C is used as the heating source for the HRSG. Gas turbines typically have capacities between 500 kW and 250 MW.

2.1.2. Heat recovery steam generator HRSG acts as a heat exchanger between exhaust heat supplied from the gas turbine and the liquid/vapour mixture circulating into finned tubes through 3 heat exchangers highlighted in figure (2.2) where additional firing can take place if necessary. Production of high pressure steam is carried out using high and low pressure drum-boiler units according to the following process. 1. Economizer stage Water fed by the pump supplied to the drum inlet

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2.1. Combined cycle process overview is preheated in order to reduce energy consumption. 2. Evaporator stage Due to the gravity water flows down through a downcomer-riser closed loop producing saturated steam which flows along the riser tubes before being collected and fed back into the drum. 3. Superheater stage The saturated steam flows through the water level till it exits upon reaching the drum outlet. Then it is reheated one more time 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 pressure and temperature superheated steam enters the turbine converting thermal energy into rotational mechanical energy capable of moving its blades and generator shaft producing additional electricity. The steam losing most of its temperature during the conversion process is collected and fed into the condenser. Steam turbines typically have capacities between 50 kW and 250 MW.

2.1.4. Surface condenser Also known as water-cooled shell and tube heat exchanger, it installed at the turbine outlet handling the last phase of the combined cycle by condensing the exhaust steam to achieve maximum attainable efficiency. Water is used to carry off waste heat from the steam due to its availability, high specific thermal capacity and heat transfer properties.

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2.2. Steam generation process description

2.2. Steam generation process description The differential equations describing dynamics associated with the drumboiler, feedwater and steam regulating valves with their actuators, in addition to the existing process PID-controllers shall be explained and addressed in details throughout the following sections. The simplified process relevant to our analysis concerning steam generation using the drum-boiler unit is illustrated1 in figure (2.3). Supplied inflow from the feedwater pump is regulated using one control valve. As for the steam flow rate leaving the drum, it can be regulated using five valves2 connected in parallel with distinctive construction and functionalities. Water tank control valve always kept opened at a predefined position Bypass valve A butterfly valve handling supply of heat to city districts Security valve for safety matters when the drum pressure exceeds limits Steam turbine control valve feeds the steam turbine Condenser control valve bypasses the steam turbine feeding directly the condenser

1

3

Outflow - qs

1. 2. 3. 4. 5.

Feedwater Tank 2 4

5

Inflow - qf

Downcomer-riser loop

Exhaust heat - Q

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 the drum. Its complex geometry, number of riser and downcomer tubes and spe1 2

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

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2.2. Steam generation process description cially the two phase flow modelling attempt is usually quite complicated requiring typically usage of partial differential equations. In literature there exists a lot of research papers that were devoted into developing relatively simple physical models [2] [7] [13] [14]. In particular the well developed ˚ Astr¨ om - Bell model3 is being considered. The majority of the system attitude can be captured through a 4th order nonlinear model by means of defining mass flow and energy balance with the help a physical mechanism introduced under the following elementary assumptions. Steam demand (to downstream) Drum Upper void (saturated steam)

Sat. steam

Internal Separation Device

Mixture from riser

Steam rises

Steam-water Condensation

Feedwater

Sat. water

Steam-water mixture

Sat. steam

Upper collecting header Water boils and flows upward

Downcomer

Riser

Heat from hot medium

Lower distribution header

Figure 2.4.: Schematic diagram of the downcomer-riser circulation loop [13] Most of the system parts will be under thermal equilibrium due to their direct contact with saturated liquid/vapour mixture. The energy stored in the mixture is either absorbed or released quickly following drum pressure changes, meaning that various metal parts of the system would adapt their temperatures in the same manner. This agrees with experimental observation which have proven that the difference between both temperatures is very small, thus a detailed representation 3

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

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2.2. Steam generation process description 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. It describes the drum pressure dynamical behavior quite well by simply computing properties of liquid/vapour mixture using steam tables. Condensation of the steam within the drum causes the coupling between the drum pressure P and water total volume Vwt . d (ρs Vst + ρw Vwt ) = qf − qs dt d (ρs hs Vst + ρw hw Vwt − P Vwt + mt cp tsat ) = Q˙ + qf hf w − qs hs dt

(2.1)

Distribution of steam along the riser tubes was carried out using a lumped model which represents the energy and mass balance caused by the naturally circulated downcomer-riser closed loop as seen in equation (2.2). The steam mass fraction αr assumed to vary linearly from the inlet to the outlet of the riser is characterized in response to changes in the downcomer qdc , riser qr and heat Q˙ flow rates respectively. d [ρs α ¯ v Vr + ρw (1 − α ¯ v )Vr ] = qdc − qr dt d [ρs hs α ¯ v Vr + ρw hw (1 − α ¯ v )Vr − P Vr + mr Cp tsat ] dt = Q˙ + qdc hw − qr (hw + αr hc )

(2.2)

The empirical equation (2.3) resulted from various attempts to fit with the experimental data. It defines mass balance of the steam bubbles under the water level in terms of condensation flow qcd and steam flow through the liquid surface qsd driven by density difference of the mixture and momentum of the flow qr entering through the riser tubes. It can capture most of the process dynamics by proper parameterizations of residence time of steam inside the ◦ and empirical drum Td , the bubbles steam volume at hypothetical situation4 Vsd coefficient β correspondingly. d (ρs Vsd ) = αr qr − qcd − qsd dt ρs ◦ qsd = (Vsd − Vsd ) + αr qdc + αr β(qdc − qr ) Td 4

(2.3)

Theoretical state that assumes no condensation of steam inside the drum

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2.2. Steam generation process description

2.2.2. Drum-boiler nonlinear state equations To derive a state model, chosen state variables should have a good physical interpretation. Drum pressure P is obviously chosen as it describes the total energy of the system. The accumulation of water related to total water volume Vwt 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 the water, thus estimating the level. The resulting nonlinear state-space model would 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 flow rates are manipulated to control primarily the drum water level and pressure respectively, whereas the heat flow rate Q˙ is rather considered as a model input disturbance z due to the fact that its amount is associated with the gas turbine exhaust heat which in return corresponds to its electrical output power as we elaborated concisely the combined cycle working principle in section (2.1). On the contrary heat flow rate becomes a control variable in thermal plants as it can be regulated directly by adjusting the boiler firing rate. Arrangement of the mass and energy balance differential equations was carried out in order to derive the algebraic state equations. The liquid/vapour mixture properties time derivative in terms of the drum pressure are calculated using the coefficients enm provided in appendix (A.7). dP dt dVwt dt dαr dt dVsd dt

e12 Q˙ + qf (e12 hf w − e22 ) − qs (e22 − e12 hs ) e12 e21 − e11 e22   1 dP = qf − qs − e11 e12 dt   1 dP ˙ Q − αr hc qdc − e31 = e33 dt   h f w − hw 1 ρs ◦ dP dαr = − e43 (Vsd − Vsd ) − qf − e41 e44 Td hc dt dt =

(2.4) (2.5) (2.6) (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 loop in a single equation and equation (2.7) considers only the mass balance of steam bubbles under water level. The interesting feature of this model is that the states can be grouped in the form ((P, Vwt ), αr , Vsd ), where each term can be computed separately in a nested manner treating the system as 2nd , 3rd or 4th order according to modelling requirements.

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2.2. Steam generation process description

2.2.3. Mass flow control valve The process concerned with regulation of feedwater flow rate supplied from the pump can be simplified and highlighted as seen in figure (2.5). The flow is computed with the aid of the nonlinear equation (2.8) essentially used by mechanical engineers to size their valves and meet mass flow requirements. The pressure drop ΔP across the valve would be the difference between the feedwater pump and drum pressures, xf is its percentage opening ranging from 0% to 100% and finally Kv is the valve sizing coefficient.

p1

H100 H0

p2

t1 Q W

Figure 2.5.: Flow through control valve for liquid service [22] The dynamics related to regulation of steam flow rate are quite complicated where additional considerations have to be taken care of when compared to feedwater mainly due to the difference in properties between both. One good approximation to describe the flow rate through a control valve meeting practical 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 into account to distinguish between saturated and superheated steam. √ Kvf · ρw · ΔP (2.8) qf = xf · 3600 Kvs · Z · m qs = x s · (2.9) 3600 Clearly the valve position value would vary according to the type of valve being used. The inherent flow characteristic depicted in figure (2.6) highlight the comparison between the commonly used control valve demonstrating that mass flow rate for the same opening position and pressure drop across it is obviously altered according to the category it belongs to. Examining halfway opened linear, butterfly and relief valves correspondingly,

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2.2. Steam generation process description undoubtedly the butterfly valve would supply approximately one-third of the total amount provided using the linear valve while the relief valve employed for safety precautions would grant roughly twice the flow afforded by the linear valve. The actuators used to operate the control valve handle the positioning imposed by the controller using electrical motors with 3 basic states which are opening, closing or holding the same opening percentage. The rate of opening/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 of PID-controller algorithm [3] commonly known in industry as series (2.10) and parallel (2.11). It is mainly introduced to identify the controller realization and not to describe it, since the algorithms are identical to one another where the overall transfer behaviour from the controller input to the output is always the same, regardless of how the derivative action is being handled by different manufactures.     1 1 + s · Td u(s) = Kp · 1 + · (2.10) e(s) s · Ti 1 + s · Tf   1 s · Td u(s) = Kp · 1 + + (2.11) e(s) s · Ti 1 + s · Tf

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2.2. Steam generation process description The controller parameter Kp corresponds to the proportional gain element, Ti and Td represent the time constants assigned to the integrator and derivative elements respectively and finally the time constant Tf relates to the filter frequency applied on the derivative term which is mandatory from practical and theoretical aspects. Practically the measurement sensors produce noise at high frequency further getting amplified due to the derivative action leading to very large unusable controller output. The additional low pass filter pole is placed in a manner which attenuates high frequency noise. From the theoretical point of view, the PID-controller transfer function without the additional pole cannot be realized since the nominator would be higher than the dominator. The series algorithm -still being adopted in digital controllers- was first introduced within the early analog controllers, which were realized using electrical circuits or pneumatic elements. Its corresponding transfer function can be represented easily in the frequency domain where the poles and zeros correspond to the inverse of the corner frequencies. In the parallel form referred to as non-interacting, the unity feedforward signal and derivative action predict the error at the moment assigned by its time constant Td . The integrator intends to eliminate the error between reference and process output completely where the resulting action from both is modified afterwards using the proportional gain Kp . It is worth mentioning that such minor difference in implementation would have a major impact when attempting to tune the controller parameters using analytical methods such as Cohen-Coon or Lambda since they can only be applied on the parallel algorithm. One notable problem using PID-controller is integrator windup (A.6) necessitating usage of an anti-windup mechanism to prevent the integral element from growing up further as soon as the controller output hits the saturation limits entering the nonlinear region. This would occur when the control signal exceeds the predefined physical boundaries related to the control valve opening range and allowable amount of mass flow rate which can be supplied. The drum pressure and water level PID-controllers adopts the parallel algorithm and their set values are always kept constant regardless of the supplied amount of heat flow rate. Each output is controlled with its separate control loop without considering any sort of coupling or interaction between both outputs.

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2.3. MATALB/Simulink model Water level and pressure control Drum level control can be realized using 3 different industry-standard strategies with typical application for each noted as single, two and three-element control respectively [1]. The numbered term corresponds to the set of measurements being utilized to control the system. GuD 2 at HKW S¨ ud currently implements the 2-element structure which employees a cascaded control architecture using level and feedwater flow rate as process variables. Such strategy is useful as it addresses disturbance imposed on the level and improves set point response performance when compared to 1-element control. When directly controlling the level it isn’t enough for the controller by itself to directly open or close the valve since it have to decide as well whether it should be feeding more or less feedwater into the drum. By considering the feedwater flow rate as well, the outer loop compares the current level with the specified reference and the computed error signal generates using the PID-controller a new set value for feedwater flow rate. The inner loop examines the current flow with the amount established by the outer loop in order to adjust accordingly the control valve percentage opening using a PI-controller. During normal operation the pressure is regulated by modifying the linear or butterfly valves’ position using identical control loop structures consisting of a simple feedback loop which compares the reference value with the drum actual pressure. The error is subjected to unity negative gain that ensures an inverse response to the valve position where its value is altered using a PI-controller equipped with a dead zone. The control valve should open if the drum pressure increases to relief the pressure inside. The same holds if it drops, where the required action is steam valve closure, thus increasing the pressure within the drum and restoring it back to the defined set value. The dead zone guarantees a region of zero output causing the PI-controller to hold its previous state as it’s only allowed to react when the error signal exceeds certain limits.

2.3. MATALB/Simulink model The complete physical model is realized within MATLAB/Simulink environment carrying out direct computation of the differential and algebraic state equations describing the process elaborated in section (2.2). The parameters were either extracted from the construction data and control schemes of GuD2 at HKW S¨ ud plant or estimated following their unavailability.

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2.3. MATALB/Simulink model

2.3.1. Drum-boiler model A simple MATLAB code was written to implement directly the established algebraic state equations. The Simulink model uses the user-defined MATLAB function 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 order to calculate the state variables. They are integrated before being fed back again as shown in figure (2.7). States 1

1 s

State variables (dx/dt)

Heat flow rate (z) 2

Model inputs

1

Pressure (bar)

Mass flow rates (u)

Pressure (bar) 2

Level (mm) States (x)

Drum-Boiler m-file

Level (mm)

Figure 2.7.: Simulink model of the Drum-boiler unit The liquid/vapour mixture properties are calculated using functions of water properties and derivatives based on the International Association on Properties of Water and Steam (IAPWS). These functions were realized using two implemented MATLAB functions which are XSteam [10] and IAPWS-IF97 [19]. The partial derivatives of water and steam densities with pressure were approximated linearly within the drum pressure operating range as they weren’t implemented in both of the above mentioned functions. The riser and downcomer tubes volumes Vr /Vdc is computed by knowledge of their cylindrical pipe length and cross sectional area. The downcomer-riser closed loop consists of two tubes that belong to the downcomer supplying water to the evaporator, the saturated steam is fed back into the drum through six collectors related to the riser. The drum oval shape was approximated and considered as a cylinder whose volume Vd would be calculated similarly. The drum area Ad is associated with the water surface area assumed to be constant even though it would vary according to the level. The downcomer area Adc is the pipe cross sectional area already obtained while computing the volume. Parameters related to empirical equations or lumped models such as the friction coefficient in downcomer-rise loop K, empirical coefficient β and the 5

Three different subsystems were constructed separately block-by-block in Simulink during early development of the model

19

2.3. MATALB/Simulink model ◦ were quite hard to obtain, therefore were either kept hypothetical volume Vsd constant or scaled down relative to the ˚ Astr¨ om-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 chapter in section (3.4). It have a huge influence on the overall behaviour which is reasonably expected since it can be interpreted as the time constant of water and steam separation process. Simulation results shows that a residence time higher than 6 sec leads to closed loop instability when utilizing the existing PID-controllers.

Drum-boiler construction data Variable Drum Downcomer Riser Volume 20.204 m3 0.9 m3 20 m3 Mass8 1363 kg 580 kg 1300 kg 2 2 Area 14.7 m 0.0637 m Drum-boiler model parameters Residence time in drum 4s Hypothetical volume 2 m3 Friction coefficient 25 Empirical 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 first specified in order to compute initial values. This allows computation of the necessary heat flow rate Q˙ that preservers energy balance. A primary simulation can run once these values are assigned as the model drives by itself the variables αr and Vsd to steady state by solving equations (2.12). Finally, the total volume Vwt is the amount required to keep the water at the relative zero level.  2ρw Adc (ρw − ρs )g α ¯ v Vr Q˙ = qs hs − qf hf w = αr hc  K   (2.12) ρw ρs ρ w − ρs α ¯v = ln 1 + αr 1− ρw − ρs (ρw − ρs )αr ρs

6

◦ K and β were kept the same, whereas Vsd is chosen as a rule of thumb Changed within the range [ 2 sec - 6 sec ] until the model closed loop behaviour matched the plant real measurements 8 Total mass including the evaporator mt = 98888 kg 7

20

2.3. MATALB/Simulink model

2.3.2. Control valve and actuator model The actuator dealing with the positioning is modeled as a 1st order lag element whose integrator saturation corresponds to the valve position ranging between 0% and 100%. The discontinuous rate limiter block afforded by Simulink library is used to model the motor rate of opening and closing. Finally, the amount of feedwater flow rate varies according to the pressure drop across the valve and its opening percentage as illustrated in equation (2.8). The pressure ratio across the valve was estimated not to exceed 0.7, therefore the head loss m according to steam service tables would be 0.96 [22]. The dimensionless compressibility factor Z is treated as a function of the superheated steam pressure and density. Therefore, the amount of steam flowing through a control valve can be rewritten as described in equation (2.13).  Z = 14.2 ρs P √ (2.13) Kvs xs ρs P qs = 13.6 3600 The feedwater and steam turbine control valves sizing coefficients were obtained directly from their corresponding data sheets. Unfortunately data about the butterfly valve was missing, its sizing had to be estimated using table charts from [24] and its inherent flow characteristic curve was simplified as linear. Such assumption is still very plausible as the valve in the real process never opens beyond 30%.

1

Pdrop (bar)

Pressure drop 2 Postion Set Point

qf (kg/s)

1 s Actuator

qf (Kg/s)

xf (%)

Electrical Motor

1

Feedwater Control Valve

Figure 2.8.: Simulink model of the control valve combined with its actuator Control valves and actuators Variable Feedwater Steam turbine 3 Sizing coefficient 20.368 mhr 364 kg hr Rate of opening ± 3.333 %s ± 0.166 %s

Butterfly 1363 kg hr ± 0.555 %s

Table 2.2.: Control valve and actuator parameters

21

2.3. MATALB/Simulink model

2.3.3. Process PID-controller model Simulink continuous PID block offers functionalities which meets exactly our needs, thus a separate realization wasn’t required regarding implementation of the process controllers. It can simulate the non-interacting PID algorithm according to equation (2.14)9 , provide output saturation when required, reset integrators and more importantly equipping an anti-windup mechanism.   Ki N ·s u(s) = Kp · 1 + + Td · (2.14) e(s) s s+N The anti-windup can be handled using the back-calculation method or a logical clamping circuit. Back-calculation feedback loop when employed attempts to discharge the PID-Controller internal integrator when the controller hits specified saturation limits by proper tuning of the highlighted coefficient Kb as shown in figure (2.9) [3].

1 Error

1 s

Ki Integrator gain

Integrator

1

Kp Proportional gain

Saturation

Controller output

Kb Td

Back-calculation coefficient

N

Derivative gain

Filter coefficient 1 s Filter

Figure 2.9.: Simulink model of a parallel PID-controller equipped with an antiwindup mechanism PID-Controller parameters Controller Kp Ti Td Level 0.05 300 100 Feedwater valve 2.3 25 Pressure10 1.8 12 -

Tf 50 -

Table 2.3.: PID-controller parameters 9

Ki = T1i and N = T1f 10 Pressure 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 out keeping in mind future plans and design considerations. First, essential theoretical aspects required to effectively analyze the system are briefly covered. Then stability of the process is examined analytically by linearizing the nonlinear model at various operating conditions in order to predict its open loop behavior. Later on, the expected behaviour shall be addressed in details by conducting several simulation scenarios. Finally, the closed loop response is validated against real measured data from GuD 2 at HKW S¨ ud.

3.1. Theoretical overview 3.1.1. Concept of stability Stability of linear systems can be roughly summarized as follows, a system output will be limited and restricted for any applied bounded input referred to as Bounded-Input Bounded-Output (BIBO) stability. Examining stability of linear systems is fairly simple and straightforward as they can be described either as transfer function or in state-space form, thus stability can be determined by direct computation and graphical visualisations of its eigenvalues within the complex plane. Furthermore, stability of the controlled closed loop system can be predicated by merely inspecting the system in open loop while applying well established methods such as the Nyquist criterion. On the other hand, stability analysis for nonlinear systems is relatively complicated and requires a high level of mathematical understanding since further matters have to be considered. The analysis should address stability of equilibrium 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 variables 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 as operating points - have a finite number associated with the solution of equa-

23

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

3.1.2. Linearization For simplicity the intended stability analysis shall be performed by linearization of the nonlinear model at typical operating points of the drum whose state algebraic equations can be summarized into the generalized description shown in equation (3.2). x˙ = f (x, u) (3.2) y = g(x, u) 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 and feedforward matrices respectively. This will come in handy when attempting to optimize the controller since algorithm execution of the proposed strategy requires these matrices. In addition they would reduce the nonlinear state equations complexity offering a rather simplified overview of the states and inputs dominating the process outputs. The open loop response of both linear and nonlinear models should be compared to inspect if both still match, therefore answering the crucial question concerned with the linearized model reliability during design of the optimal controller. x˙ = Ax + Bu + Gz

(3.3)

y = Cx + Du

The matrices are computed with the help of Taylor series approximation neglecting quadratic and higher order terms (3.4). The method intuitive basis is that a smooth curve differs very little from its tangent line as long as the variable doesn’t wander from the point of tangency.   ∂fi  ∂fi  , bij = aij = ∂xj xR ,u0 ∂uj xR ,u0   (3.4) ∂gi  ∂gi  cij = , d = ij ∂xj  ∂uj  xR ,u0

xR ,u0

3.1.3. Poles and zeros System zeros affects only shape of the output which can lead to minimum or non-minimum phase behaviour according to their position within the complex

24

3.1. Theoretical overview plane (A.5) [9] [23]. Alternatively poles determine stability as they are directly associated with the system eigenvalues. That’s why inspection of the system poles and zeros is quite efficient while analysing and predicting the system response. If the eigenvalues are located on the left-hand side (LHS) of the complex plane the states will converge to zero stabilizing over the course of time. However if located on the right-hand side (RHS) the states will keep growing due to the exponential product as depicted throughout equation (3.5) where ci are constants coefficients related to the solution of the homogenous differential equation describing dynamics of the system. y(t) =

n

ci eλi t

(3.5)

i=1

We still need to define the relationship between the zeros, poles and eigenvalues, in addition understand how it differs when comparing Multiple-input Multiple-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 exactly equivalent to the characteristic polynomial evaluated by solving equation (3.7), that why all poles corresponds to the system eigenvalues λ. G(s) =

(s − z1 )(s − z2 )...(s − zn ) N (s) = D(s) (s − p1 )(s − p2 )...(s − pn ) . det(λI − A) = 0

(3.6) (3.7)

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

1

Only 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 of decoupling 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 even worse if the uncontrollable eigenvalue is located on the RHS of the complex plane because there no way to stabilize the plant with its current setup using any control technique.   sI − A −B P (s) = (3.8) C 0

3.2. Stability analysis The drum-boiler unit stability can be easily comprehended from the physical point of view with basic understanding of the drum mass and energy balance equations discussed during modelling in section (2.2.1). For example, if the outflow leaving the drum is less than the amount which is supplied by the inflow then water will start filling the drum and vice-versa. Alternatively while assuming constant mass flow rate, if additional firing takes place by the HRSG providing more heat leading the drum temperature to rise and causing the pressure to build up in return, thus reaching dangerous limits which will cause explosion of the drum ultimately. Even though this shortened explanation could be enough, additional analytically driven investigations needs to conducted by performing a stability analysis to extend our understanding of the expected behaviour and establish the 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) using MATALB Control System ToolboxTM linear time-invariant (LTI) functions dedicated for continuous systems time-domain analysis (A.2). They cover the drum operating range whose upper limit is specified by the maximum amount of saturated steam allowed to flow through the pipes upon exiting the drum. The input B and disturbance G matrices show that the dominant inputs which affects the drum pressure P dynamics are heat Q˙ and steam qs flow rates as expected. The water total volume Vwt is obviously affected mainly by the mass flow rates. Steam mass fraction αr depends heavily on condensation enthalpy 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 seen that it’s associated with all states and input variables as it follows an empirical

26

3.2. Stability analysis equation derived through continuous observation of the process to capture the drum complicated dynamics. ⎡ ⎤ −7.148e-5 1.051e-14 0 0 ⎢−5.094e-11 −9.723e-21 ⎥ 0 0 ⎥ A=⎢ ⎣ 1.021e-9 ⎦ −2.684e-21 −0.1827 0 1.752e-6 7.424e-18 −297.3 −0.3333 ⎡ ⎡ ⎤ ⎤ −21.09 −216.6 103 ⎢ 0.001085 −0.001255⎥ ⎢7.341e-5⎥ ⎢ ⎥ ⎥ B=⎢ G = ⎣ 5.386e-6 ⎣7.904e-5⎦ 5.534e-5 ⎦ −0.01486 0.2023 0.07526   1e-5 0 0 0 C= −4.86e-4 68.027 2035 68.027 States

Operation Low Medium High

P (bar) 5.5 5.5 5.5

2

3

Vwt (m ) 21.501 20.391 19.736

αr (%) 0.0098 0.0138 0.0178

Inputs 3

Vsd (m ) 1.378 1.067 0.756

˙ Q(MW) 13.8473 20.771 27.6947

qf ( kg s ) 6 9 12

qs ( kg s ) 6 9 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 the transfer behaviour from inputs to the water level shows that all four eigenvalues appear as poles. The first three are always located at − T1d and the origin3 , associated with the drum pressure, water volume and dynamics of the steam bubbles under water level respectively. The last pole which depends on the qdc , it relates to the steam dynamics flowing operating point is situated at − hec33 through the riser tubes. It keeps shifting to the left along the negative real axis towards infinity as long as the load increases. This was quite expected from our basic understanding regarding steam generation working principle using the drum-boiler. Higher loads require more electrical power generated by the gas turbine which in return provides additional heat to the riser tubes, thus accelerating conversion process of feedwater into steam within the naturally circulated downcomer-riser loop. If an enormous amount of heat is supplied the pole keeps approaching negative infinity, 2 3

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

27

3.2. Stability analysis when inspected in the complex plane, since the conversion shall take place instantaneously. Since no compensation of eigenvalues have occurred, all invariant zeros are classified as transfer zeros. In addition one can stay assured that the system is completely controllable because any eigenvalue can be influenced by affecting its corresponding pole. The transfer zeros which are located on the right hand side (RHS) of the complex plane have been anticipated earlier from experimental observation and physical understanding. They are directly correlated with the shrink and swell physical phenomena leading the system to react in a nonminimum phase behaviour. In particular zeros related to the transfer behavior from steam flow rate to water level are very close to the origin when compared with zeros linked to feedwater and heat flow rates transfer functions respectively as seen in figure (3.1). Therefore we should be definitely expecting a significant difference in regards with amplitude of the water level initial inverse response when stimulated by the input variables. This shall verified in the next section concerned mainly with the open loop response to a step input. Heat flow rate Q (MW) 1

3

0
1


0.28

2

1 23 1

1


0.21


0.14

2

3


0.07

0

0.07


0.07

0

0.07


0.07

0

0.07

Flow rate qf (kg/s) 1 0
1

3

2
0.28

1

3

2

1


0.21


0.14 Flow rate qs (kg/s)

1

3

0
1


0.28


0.21

2

1

3 21


0.14 Real axis

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

28

3.3. Open loop step response

3.3. Open loop step response The open loop response shall be studied by simulating the model4 consisting of the drum-boiler unit and control valves5 when subjected to a step change of the gas turbine electrical output power and control valves position respectively. One input at a time is stimulated using a step function while the other inputs remain 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 electrical power equivalent to a decrease of 20 MW. The amount of heat flow rate Q˙ supplied to the drum, required as an input to run the simulation, is assumed to vary instantaneously following the change of the gas turbine power. Butterfly and feedwater control valves positions were kept constant. The pressure P starts decreasing following the declination of heat flow rate associated with gas turbine output power. It affects the amount of steam flow rate qs leaving the drum as the valve position haven’t changed. On the other hand, the pressure drop across the feedwater valve starts building up since the feedwater pump pressure is kept constant, hence causing more feedwater qf to flow into the drum. The water total volume Vwt initially decreases due to evaporation caused by sudden pressure drop before incrementing eventually following the increase of feedwater. The steam mass fraction αr in the riser tubes immediately steps down once the heat supplied is smaller than its initial state then keeps sliding down gradually as the amount of water being vaporized by the evaporator within the downcomer-riser loop was reduced. The level response l depends on a combination of complicated dynamics related to distribution of water and steam. The step-like change of steam mass fraction αr leads to the initial undershoot as the quantity of steam bubbles fed back to the drum rapidly drops. The swelling effect is then noticed once the pressure starts to decrease resulting in steam bubbles expansion causing the level to rise. Finally following this transient effect, water keeps filling in the drum due mass imbalance where feedwater supplied to drum inlet is much higher than the steam leaving from the outlet.

4 5

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

29

3.3. Open loop step response

3.3.2. Change of butterfly valve position Figure (3.3) shows the step response of the system due to opening of the butterfly valve equivalent to 10% while heat flow rate and position of feedwater control valve were kept constant. The steam flow rate qs rapidly increases according to the valve rate of opening after the unexpected rapid change in valve positions. The opening of the valve relieves the pressure inside the drum and hence it starts decreasing. Once the valve reaches its designated opening percentage, the pressure P starts dominating the behaviour of the steam flow rate thus reducing the amount of steam leaving the drum because both are related to each other. Feedwater flow rate qf increments due to the increased pressure drop across the feedwater valve. The water total volume Vwt decreases for two reasons; one is the evaporation caused by the pressure drop and the other being the relatively high difference in mass flow rates. The steam mass fraction αr steps up once the pressure have decreased then starts sliding gradually until it approaches its original state following the transient effect occurring to downcomer qdc and riser qr flow rates. Finally the level l initial inverse response is caused by the bubbles swelling and volume expansion 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 the feedwater control valve equivalent to 10% while the heat flow rate and position of butterfly control valve were kept constant. The feedwater qf drops in step fashion since the control valve reaches its designated position very quickly with its fast rate of opening/closing. The decrease of cold feedwater fed into the drum increases its temperature which in return affects the pressure allowing more steam qs to leave the drum. The mass balance inflow and outflow was disturbed within the drum, therefore the water 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 sliding gradually upwards towards its initial state. The sudden drop of feedwater flow rate resulted as expected in the level l initial inverse response corresponding to the predicated system non-minimum phase behaviour.

30

3.3. Open loop step response Pressure (bar)

Flow rate qs (Kg/s) 9

5.5

0.013

8.5

0.012

5

4.5

8

0

100

200

7.5

0.011 0

Level (mm) 50

100

200

0.01

0

Flow rate qf (Kg/s) 9.2

100

200

Volume Vwt (m3) 20.8 20.7

9 0

20.6 8.8


50

Steam quality (%) 0.014

0

100

200

8.6

20.5 0

100

200

20.4

0

100

200

Figure 3.2.: Open loop response for a step change equivalent to decrease of 20 MW of the gas turbine electrical output power

Pressure (bar)

Flow rate qs (Kg/s) 12

5.5

11

0.014

10

5

0.0135

9 4.5

Steam quality (%) 0.0145

0

100

200

8

0

Level (mm) 100 50

100

200

0.013

0

Flow rate qf (Kg/s)

100

200

Volume Vwt (m3)

9.2

20.6

9

20.4

8.8

20.2

0 0

100

200

8.6

0

100

200

20

0

100

200

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

31

3.4. Validation r esuu es(ba) e7

Flow(e) ts(qu(bKg/u7

4.P4

8.84

4.P

8.8

4.44

8.94

4.4

8.9

4.54

0

100

200

8.P4

0.0135

0

Lsvsl(bmm7

100

200

0.0133

0

20.5

9

20.2


20

P

20

100

200

4

200

20.P


10

0

100

Vol ms(Vwt(bm37

8


30

0

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

10

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

16.8 0

100

200

0

100

200

Figure 3.4.: Open loop response for a step change equivalent to 10% closing of feedwater 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 capture the real process dynamics at various operating conditions. The complete model with 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 in reality, yet can be predicted from the gas turbine electrical output power which 1 MW . The transfer function relating changes as ramp function with a slope of 12 s st both is assumed to be 1 order lag element whose time constant was identified τ = 280 s assuming that the supplied heat behaviour is directly associated with the evaporator temperature. The feedwater valve position is always kept half-way opened in the plant without considering the amount of feedwater which flows through it. Therefore the pressure drop across the valve should increase or decrease accordingly to preserve such condition which is achieved using he feedwater pump controller. 6

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

32

3.4. Validation Implementation of the controller was neglected for simplicity and it is assumed 1 bar followed by that the pump output would vary ramp-wise with a slope of 60 s PT1 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 regulate the steam flow rate therefore the pressure control loop will consist only of the corresponding PI-controller. Additionally steam flow rate measurement from the plant doesn’t take into account the amount supplied back to the feedwater tank which was assumed to be approximately around 1 kgs . The corresponding valve is usually kept open at a predefined position and its contribution to control the drum pressure can be neglected. However as the drum-boiler model requires the total flow rate which leaves the drum as an input variable, the estimated amount flowing into the feedwater tank is simply related to the drum pressure dynamics with a low pass transfer behaviour which is similarly chosen as another rule of thumb. The pressure loop PI-controller proportional gain Kp was adjusted from 1.8 to 4. The model closed loop performance improved and matched much better the measurements when compared to its initial value when observing the simulations results. This is due to several factors discussed when concluding the chapter in section (3.5).

3.4.2. Comparison with measurement data The pressure controller senses the pressure decrease within the drum as less heat is being supplied as shown in figure (3.6), thus it tries to close the butterfly valve to restore pressure back to its set point. Once the valve starts closing, the water level l drops due to the shirking effect of steam bubbles. It experiences an undershoot followed by an overshoot since the cascade controller is simultaneously trying to restore the level back to its set point and to reestablish energy balance for the drum as well as mass balance for inflow and outflow. Figure (3.7) illustrates the comparison considering gas turbine power increase. The controllers react on the pressure rise within the drum caused by the additional heat supplied, therefore opening the corresponding valve to relief drum pressure allowing more steam to leave from the drum outlet in the process. The water level l increases due to steam bubbles swelling, yet a smaller overshoot is observed since the change of electrical power is less when compared to the previous scenario, thus in return permit the controllers to settle and drive the process back to steady-state faster.

33

Level Reference value

0

Feedwater pump

Pressure Reference value

5.5

PID(s)

15s+1

1

Low pass

Level PID Controller Kp = 0.05 Ti = 1/300 Td =50 Tv = 100

Saturation

Dead Zone -0.04 to 0.04

Actuator xf

1 s

Actuator xs

1 s

Electrical Motor 10 % by 3 sec

Heat flow rate (Watt)

Electrical Motor 10% by 18sec

280s+1

1e+006

Butterfly valve

qs (kg/s)

qf (kg/s)

Feedwater valve

xf (%)

Pdrop (bar)

Low pass Gain watt to MW

xs (%)

Figure 3.5.: Simulink validation model

qf PI Controller Kp = 2 Ti = 1/25

PI(s)

qs PI control Kp = 4 Ti = 1/15

PI(s)

P_drum (bar)

Steam to feedwater tank 5s+1

0.18182

Level (mm)

P (Bar)

Drum-boiler model

qf (Kg/s)

Q (W)

qs (Kg/s)

3.4. Validation

34

3.4. Validation Pressure (bar)

Valve position xs (%)

Flow rate qs (Kg/s) 8

20

5.6

7

5.5

6

15

5.4

5

5.3 5.2

0

2000

4000

10

0

Level (mm)

2000

4

4000

0

Valve position xf (%)

2000

4000

Power Q (MW)

100 50 0
50
100

0

2000

80

100

60

90

40

80

4000

0

2000

4000

0

2000

4000

Figure 3.6.: Comparison between model (dashed) and plant data (solid line) for a decrease of the gas turbine electrical output power equivalent to 20 MW r esuu es(r (ba) e1

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

4.P

2q

4.6

q P.4

26

4.4

P 6.4

23

4.3 0

Kpng (e) % s(/ u(bL mfu1 q.4

2400

5400

0

l svsp(p(bo o 1

2400

5400

6

0

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

2400

5400

r ng se(Q(bMW1

P0

40

90

60

V4

40

0

30


V4

50


40

V0

0

2400

5400

q4 q0 0

2400

5400

0

2400

5400

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

35

3.5. Concluding remarks

3.5. Concluding remarks In the end of the process comprehensive study, we shall wrap up the analysis and summarize results as follows. The resulting linearized model of the drum-boiler unit predicted the open loop response of the dynamic realistic nonlinear system quite well and can be relied on safely in regards with intended future plans. The I/O pole-zero plot assures full controllability of the system as all of its eigenvalues are appearing as poles. It clarified as well how the shrink and swell phenomena is associated with the transfer zeros located on the RHS of the complex plane leading to an initial inverse response which should be handled by the PID process controllers. The identified main problem with the existing level control can be outlined as follows Assuming drop of exhaust heat provided to the drum leading to decrease of pressure and water level. The feedwater control valve supplies more water through the inlet yet unintentionally contributing into additional drop of the level From the physical point of view this takes place since the cold water fed into the drum decrease its temperature and as result its pressure as well The pressure controller tries to close the steam valve even more to track the set value leading eventually to further water drop due to steam bubbles shirking Clearly the level control isn’t considering the initial inverse response identified while examining the system open loop behaviour The comparison results show that the model can capture the drum dynamics to a great extent. However, a relatively small deviation from real measurements and slightly faster response specially regarding pressure and level is still noticeable. The error arises due to the uncertainty of some model parameters such as Td and K, in addition to the suggested assumptions necessary in order to simplify 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 require an observer gain, whose design shall be discussed in section (4.3.2), to correct the states based upon the difference between real measurements and model outputs. Figure (3.8) shows the closed loop behaviour for different values of the pressure controller proportional gain. As mentioned earlier in the initial assumptions (3.4.1), the gain doesn’t correspond to the current real value and had to

36

3.5. Concluding remarks be increased so that the simulation results match the plant measurements. The drum pressure drops more when the gain isn’t changed and as a consequence it causes higher overshoot of the level due to steam bubbles swelling. The reasons behind such modification which as seen in the previous simulation results improved the model overall performance could be summarized throughout the following The treatment of the bypass butterfly valve as a linear valve, therefore in reality the valve opens more to allow flow of additional steam, yet in the model it doesn’t open with the same percentage due to the different nature of both valves, which was already been clarified in figure (2.6) Neglecting storage of low pressure steam within the pipes and superheater in the current model, as a result, introduction of an additional state variable7 might be necessary Pressure P (bar)

Flow rate qs (kg/s) 8

5.6

7 6

5.4

5 4

5.2 0

1000

2000

3000

4000

0

1000

Level l (mm)

2000

3000

4000

Flow rate qf (kg/s) 14

100 50

10

0 6


50
100

0

1000

2000

3000

4000

2

0

1000

2000

3000

4000

Figure 3.8.: Comparison between model and plant data (solid line) for a decrease of the gas turbine electrical output power equivalent to 20 MW when Kp = 1.8 (dotted dashed) and Kp = 5 (dashed)

7

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

37

4. Process optimization

4. Process optimization The concluded results brought to our attention during system modelling and analysis suggests that an optimization of the process is achievable using a multivariable control technique. The strategy would account for synergy between feedwater and steam flow rates instead of just decoupling the MIMO system into several coupled SISO systems regulated by their own noninteracting control loops. A state-feedback controller is suggested in order to consider the internal variables of the system instead of the process outputs, therefore accounting for additional aspects which were discarded using the classical control methodology. The inner dynamics of the drum-boiler unit correspond to the developed nonlinear model state variables which were defined in section (2.2.2). The control concept shall be addressed presenting the available control methods and algorithms applying the approach while highlighting advantages and disadvantages for each.

4.1. Concept of state-feedback control For a continuous linear system described in state-space form (3.3) whose states are available for feedback, it can be subjected to a multidimensional proportional gain element F compared with the reference value to compute the actuating variables u according to the control law (4.1) where the resulting system matrix would be A − BF . The designed state-feedback matrix would place the poles of the closed loop system in a desired position within the complex plane, thus directly influencing its rise and settling time, damping and transient oscillations. u = −F x

(4.1)

4.1.1. Controllability and observability The concept of controllability was briefly hinted to while examining the system poles and zeros in section (3.1.3). The term was introduced to investigate whether the actuating variable are perfectly able to drive the system from any

38

4.1. Concept of state-feedback control initial state to the desired state. Alternatively observability was proposed to examine ability of estimating the system states from a set of available measurements. The terms are dual to each other, thus any criterion or control design method can be applied for both by adjusting A  AT and B  C T . State-space description offers the opportunity to investigate both properties analytically by simply inspecting the system, input and output matrices respectively using Kalman, Hautus or Gilbert criterions [18]. Kalman criteria only investigate controllability/observability of the entire system without mentioning any particular eigenvalue. Fortunately Hautus and Gilbert criterions can identify the non-controllable/observable eigenvalue, thus allowing the control designer 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 the fact that they are most likely hard to be measured in the real process. The difference between the real and estimated output is subjected to the so-called Luenberger 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 computation of a feedback matrix L which modifies the system matrix into A − LC. The observer gain is designed in such a manner that ensures convergence of the estimation error to zero which allows usage of the corrected states for feedback.

C

B A Luenberger Observer

L C

B A

Figure 4.1.: State estimation using Luenberger observer

39

4.1. Concept of state-feedback control

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

-I

B

C A -F

State Controller

Figure 4.2.: PI-based state-feedback control structure The previous control law have to be slightly modified considering error signals as additional states h. The newly computed state-feedback matrix K would consist of 3 parameters F , I and P affecting the states, integrated error signals and reference tracker correspondingly. The tunable parameters are assigned upon computation of the feedback matrix K. By default I is uniquely defined as it corresponds to the left-hand side of the matrix. However F and P can be freely selected due to the underdeterministic nature of equation (4.2). Ignoring P element would lead to a simple I-controller, while setting F to zero isn’t relevant when attempting to design a state controller, besides, this would normally introduce an unsolvable equation1 (4.3).   x u = [P C − F , −I] = −Kx (4.2) h P = −KC −1 1

(4.3)

A solution might exist under very special certain conditions

40

4.2. Controller design methods

4.2. Controller design methods 4.2.1. Pole placement method Several algorithms do exist to apply the pole placement method which directly assigns 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 practice with real systems due to the huge amount of freedom offered. Additionally it doesn’t take into consideration the limitations imposed from the real actuating variables and no clear guidelines do exist regarding how and where the poles should be placed. Finally predicting the system dynamical behaviour by just positioning the poles is a complicated task and the design process requires usually tedious work even for low order systems before satisfactory results can be achieved. Ackermann’s formula The algorithm is regarded as a standard design procedure which computes the unique state-feedback f row vector or observer gain l column vector using the formula (4.4) [18] where n is the system order, p are the coefficients of characteristic polynomial calculated while defining positions of the eigenvalues for the closed loop and finally t is the last column/row obtained from the computed controllability/observability matrix inverse. Unfortunately such method cannot be applied for MIMO systems as the algorithm requires inversion of controllability/observability matrix which is only attainable with square matrices with full rank. f = t1 (p0 + p1 A + ... + pn−1 An−1 + An ) l = (p0 + p1 A + ... + pn−1 An−1 + An )t1

(4.4)

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

41

4.2. Controller design methods

4.2.2. Linear-Quadratic method Optimal control theory handles issues related to computation of a control law for a given system bearing in mind that certain optimality criterion has to be achieved, mainly focusing on how to operate a dynamical process at minimum cost. The calculus of variations maximum principle formulated an abstract framework which describes the optimal control problem trying to minimize the cost function (4.5) subjected to a dynamical linear system with zero initial conditions where ψ and L are the terminal cost and lagrangian2 respectively.  tf J = ψ(x(tf ), tf ) + L(x(t), u(t), t)dt (4.5) t0 x˙ = f (x(t), u(t), t) LQ method is a significant result of the theory that manages problems associated with quadratic performance criteria for state-space systems. The method algorithm computes a Linear-Quadratic Regulator/Estimator (LQR/LQE) referred to as Riccati controller when designing a state-feedback matrix or Kalman filter when attempting to estimate states of a real system. Riccati controller The Riccati controller allows trade off between regulation performance and control effort compared with the pole placement method. It’s regarded as a robust controller since it attains infinite marginal gain and offers a phase margin δ ≥ 60◦ [15] which is aligned with practical guidelines for control system design. The resulting optimal feedback gain should drive the closed-loop system without external input from any initial state to the zero state minimizing the cost function described by equation (4.6).  ∞ (xT (t)Qx(t) + uT (t)Ru(t))dt (4.6) J= 0

Q and R matrices are positive definite matrices assigned as weighting factors for the course of states and input variables. Faster convergence of a particular state towards zero should increase its equivalent coefficient inside the matrix Q. If a slower response of the actuating variables is preferred to lower the energy consumption and minimize control effort then coefficients of R matrix have to be chosen larger. 2

Function that summarizes dynamics of the system

42

4.2. Controller design methods The control law (4.1) tries to minimize the quality function (4.6) whose optimal feedback matrix F requires the symmetrical positive definite matrix P 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−1 B T P AT P + P A − P BR−1 B T P + Q = 0

(4.7)

Choosing the values of Q and R matrices in principle is similar to tuning of PID-controller parameters where the weighting matrices are varied until a satisfactory response is reached. Tuning a Riccati controller first carries investigations concerning the range of values for each system state and actuating variable. Once these limits are established, an initial guess can take place by constructing diagonal matrices with the normalized values as seen in (4.8). ⎡ ⎡ ⎤ ⎤ r11 · · · 0 q11 · · · 0 ⎢ ⎢ .. ⎥ .. ⎥ R = ⎣ ... . . . Q = ⎣ ... . . . . ⎦ . ⎦ · · · qnn 0 · · · rnn 1 1 qii =  max , rii =  max , i = 1, 2, ..., n xi ui 0

(4.8)

Kalman filter The Kalman filter optimizes the estimation of the system states using a series of process measurements. It takes into account the input w and measurement v noises assumed to be random unbiased white noise. It constructs an optimal state estimator that minimizes the cost function (4.9) where E() calculates the expected value based on the assumed random noise. J = E(eT W e)

(4.9)

The observer gain L computed by solving the modified MARE (4.10) attempts to minimize the difference between estimated and real states considering noise influence on the process. The weighting matrices Q and R aren’t considered as punishing factors anymore as they define intensities of the expected process noise. Choice of their values usually starts with identity matrices as initial guess then adjusted repeatedly until a decent estimation is achieved. L = P C T R−1 AP + P T A − P C T R−1 CP + 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 advantages over the pole placement method. The existing control valves PI-control loops would remain the same, leading to smooth implementation of the proposed structure when performed at the distributed control system (DCS) of the 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 to generate the mass flow rates signals required to stabilize the process. These signals can be treated as the set values for their corresponding control valves handled by their own actual controllers, therefore further modifications in the plant control schemes would be skipped. Setting the drum-boiler model states limits were straight forward and helped performing the initial simulation. The drum pressure limit would be 9 bar as specified by construction data, the error between reference and actual measurement shouldn’t exceed 0.3 bar. The water level maximum allowable deviation from set point isn’t allowed to surpass ±150 mm. The remaining states and actuating variables were kept at unity. The main challenge was system stabilization considering the limitations and constraints imposed by the control valves position range and fixed rate of opening/closing. Several simulation took place varying mainly r11 , r22 , q56 and q66 associated with mass flow rates, pressure and level error signals respectively until good results were obtained with the weighting matrices provided alongside with the MATLAB script to calculate the state-feedback matrix K in appendix (A.3.2). The tunable parameters from equation (4.2) were assigned as follows, the integrator 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-feedback matrix F doesn’t require any modifications since it’s described by the first four columns when solving equation (4.2).   7.1296e-5 21.786 −319.78 0.29002 F = −1.5083e-4 −3.8327 36.181 −1.2566e-2 

1.0356e-2 −1.4135e-3 I= 0.18248 2.6739e-5 3



P should be a 2-by-2 identity matrix in the real process, since the proportional gain of the 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 the error between measured data and model outputs noticed during validation in section (3.4). The observer gain isn’t calculated using Kalman filter because efficient practice of the method requires continuous testing of the estimator alongside the real process. Further, the optimal choice of the filter weighting matrices depend on reliable prediction of the process noise which is only guaranteed by regular observation. Pole placement method is applied despite its disadvantages mentioned in section (4.2.1) with respect to the following practical consideration. The observer response must be faster than the closed loop employing the state-feedback controller because the estimation error have to decay to zero causing 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 the state-feedback controller poles A − BF . This would guide us in a certain way to assign its position assuring suitable and decent estimation. The observer poles are provided within the MATLAB script to calculate the matrix L in appendix (A.3.2). 

T

992.85 1.0584e-2 2.6068e-4 0.47076 L= 2.2837e-16 2.1728e-4 3.2667e-6 6.0672e-3

4.4. Simulation results The simulated optimized system performance will be shown in the following figures. First we shall examine the estimated states when employing the observer to check if the new poles associated with A − LC introduce noise into the system. Then the model behaviour with state correction is validated against new measurement data with very rich excitation covering the drumboiler operating range. Therefore the observer stability can be investigated, providing a good indication of the proposed control strategy applicability since the states are crucial for feedback. Later on, comparison between both controllers is addressed to check if the newly proposed state-feedback controller did handle efficiently the main problems identified with the current controller causing its poor performance which was discussed when concluding the process analysis in section (3.5). Further, the states and flow rates at different load conditions are inspected to ensure stability of state-feedback matrix K for the drum operating range.

45

4.4. Simulation results States estimation The observer pole placement was successful as shown in figure (4.3) since the estimated states noise is almost negligible therefore they can be efficiently fed back to the state-feedback controller matrix. Observer performance subjected to perturbations in gas turbine electrical output power Significant improvement of the model closed loop response when combined with the observer was achieved. Both outputs are almost matching perfectly when compared to the simulation conducted during primary validation in section (3.4). Figure (4.4) shows the comparison between the observer and real process when electrical power of the gas turbine was switched in between 120 MW and 80 MW for approximately one hour. Even for different initial conditions as seen in figure (4.5), the observer gain was still being able to adjust the states and accordingly the process controllers to track the real output. Finally, the noticeable error which occurred due to model uncertainty and assumptions discussed in section (3.4.1) was eliminated when adopting the observer gain as illustrated in figure (4.6). State controller performance subjected perturbations in gas turbine electrical output power The drum pressure and water level were vastly enhanced when analyzing both behaviours depicted in figure (4.7). The level maximum peak overshoot/undershoot didn’t exceeds ±100 mm during transients and the pressure never surpasses the safety limits which might lead to operation of the security safety valve. Obviously the steam flow rate performance is the same using both controllers but the feedwater flow rate behaviour was modified in a way which boosted the overall closed loop performance. This is no surprise and should have been expected following process analysis which diagnosed the drum level cascade controller and highlighted its particular weakness. The optimal state controller was smart enough paying attention to the initial inverse response and shrink/swell physical phenomena by considering the inner dynamics of the system instead of the output. It clearly solved one of the main problems reported by the plant engineers.

46

4.4. Simulation results State controller performance at different loading conditions One should note that we designed the optimal state controller feedback matrix using a linearized model which is normally valid for one particular operating point. Therefore we have to investigate if it is still able to stabilize the system at various loading conditions and whether the closed loop response is still tolerable in regards with the requirements we assigned while weighting the matrices Q and 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 illustrated in table (3.2.1). Obviously the level drop was much higher at low load as the feedback gain matrix wasn’t computed in order to optimize this particular operation, nevertheless we still have a decent better response when compared with the existing process PID-controllers. Figure (4.9) shows how nicely the mass flows set points considers the limitations imposed by the control valves opening and closing rates allowing feedwater and steam flow rates to track them smoothly. Water volume Vwt (kg/s) 22 20 0

1000

2000

0

1000

0

1000

3000 4000 Steam quality (%)

5000

6000

7000

2000 3000 4000 5000 Steam volume under water level Vsd (m3)

6000

7000

2000

6000

7000

0.015

0.01

1.7 0.95 0.2

3000

4000

5000

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

47

4.4. Simulation results

Pressue P (bar)

Flow rate qs (Kg/s)

5.8

10

5.6

8

5.4

6

5.2 5

4 0

2000

4000

6000

0

Level l (mm)

2000

4000

6000

Flow rate qf (Kg/s)

200 100

15

0

10


100

5


200 0

2000

4000

6000

0

0

2000

4000

6000

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

Flow rate qs (Kg/s)

5.6

8

5.5 7

5.4 5.3 0

1000

2000

3000

6

0

Level (mm)

2000

3000

Flow rate qf (Kg/s)

60

12

40 20

10

0

8


20
40

1000

6 0

1000

2000

3000

0

1000

2000

3000

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

48

4.4. Simulation results Pressure P (bar)

Flow rate qs (Kg/s) 8

5.6

6

5.4 5.2

0

1500

3000

4500

4

0

Level (mm)

1500

3000

4500

Flow rate qf (Kg/s)

100

15

50 10

0
50
100

5 0

1500

3000

4500

0

1500

3000

4500

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

Flow rate qs (kg/s)

5.8

10

5.6

8

5.4

6

5.2 5

4 0

2000

4000

6000

0

Level (mm)

2000

4000

6000

Flow rate qf (kg/s) 15

100 0

10


100

5


200 0

2000

4000

6000

0

0

2000

4000

6000

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

49

4.4. Simulation results Pressure (bar)

Flow rate qs (kg/s) 10

5.5 5.45

5

5.4 5.35

0

500

1000

1500

0

0

Level (mm)

500

1000

1500

Flow rate qf (Kg/s) 12 10 8 6 4 2

0
50
100 0

500

1000

1500

0

500

1000

1500

Figure 4.8.: Model closed loop response using the PI-based state-feedback controller for high (dashed), medium (solid line) and low load (dotted dashed) Flow rates qf,qs (kg/s)

Water Volume Vwt (m3)

10 20.8

9

20.6

8

20.4

7

20.2 0

500

1000

1500

0

Steam quality (%)

1000

1500

Steam bubbles volume Vsd (m3)

0.016

1.3

0.014

1.2 1.1

0.012 0.01

500

1 0

500

1000

1500

0.9

0

500

1000

1500

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

50

Flow rates set value

Existing PI controllers

Observer-based State-feedback controller

Mass flow control valves

-F

Drum-boiler model

L

Plant

Figure 4.10.: Block diagram of the proposed multivariable feedback control strategy

-I

Set value tracking

4.4. Simulation results

51

5. Conclusion and future work

5. Conclusion and future work A mathematical nonlinear model which describes the dynamical process of steam generation using steam drum-boiler units including its control valves and process PID-controllers was fitted into to a real drum-boiler unit which corresponds to 450 MW CHP in Munich, in order to analyse the pressure and water level control performance which was reported to behave very poorly during transients corresponding to huge load changes. The model was implemented within MATLAB/Simulink environment and examined intensively throughout various scenarios with very rich excitation from the plant covering a wide operating range to ensure its validity and reliability. Further it pointed out very clearly the main drawbacks of the existing control strategy employed to stabilize the process. Stability analysis was conducted by linearization at typical operating points of the drum-boiler unit. It predicts the plant open loop response quite well and clarifies the reason behind its non-minimum behaviour which is associated with the steam bubbles shrink/swell physical phenomena. A multivariable feedback control strategy is proposed in order to optimize the process using a PI-based state-feedback controller designed using LQ method ensuring steady-state accuracy and set value tracking. Additionally an observer gain which guarantees correct estimation of the state variables required for feedback is realized using pole placement method. Simulation results shows that the state-feedback controller outperforms the PID-control in terms of control behaviour and performance. Unfortunately the complete control structure which combines both the statefeedback and observer together cannot be examined at the moment within the simulation environment because the observer gains requires new measurements from the plant while being handled by the proposed control strategy. In the near future, the nonlinear model shall be realized within the real plant Distributed Control System (DCS) of GuD 2 at HKW S¨ ud to act as an observer of the process, thus offering in return a great opportunity to test and examine the model more closely before being combined with the suggested state-feedback controller.

52

A. Appendix

A. Appendix A.1. Nomenclature Symbol A Cp h Kv m P Q˙ q Tsat V V◦ Td ρ

Unit m2 J/kg K J/kg kg/hr kg Pascal W kg/s ◦C m3 m3 s kg/m3

Description Area Metal specific heat capacity Specific enthalpy Valve sizing coefficient Mass Pressure Heat flow rate Mass flow rate Saturation temperature Volume Volume in hypothetical situation Residence time of steam in drum Density

Table A.1.: Physical units

Symbol K m x Z αr α¯v β ζ

Description Friction coefficient Head loss Valve opening percentage Compressibility factor Steam-mass fraction Steam-volume fraction Empirical coefficient Normalized length

Table A.2.: Dimensionless units

Symbol c d dc fw r t s w

Description Condensation Drum Downcomer Feedwater Riser Total Steam 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 and tuning of linear controllers. In this section, the tools adopted in the thesis will be briefly featured.

A.2.1. Linear analysis functions The embedded functions can be utilized using the graphical user interface ”LTI viewer” or MATLAB command window.

linio defines the linearization input/output points operspec specifies operating point requirements for states, inputs and outputs findop computes steady-state operating point meeting predefined specifications linearize performs linear approximation of a non-linear model pzplot computes poles and zeros of a dynamic linear system and plot 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 in the complex plane lqr computes an optimal state-feedback controller given the statespace 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 caldP dVwt dαr dVsd culate the states derivatives x˙ = dt , dt , dt , dt 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 data Vd = 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 downcomer Td = 3; %Residence time of steam in drum(s) Beta = 0.3 ; %Empirical coefficient Vsd0 = 2; %Steam bubbles volume in the hypothetical situation (m3) Cp = 550; %Metal specific heat capacity (Pascal.m3/Kg.K) %% Liquid/Vapour mixture properties P = P*1e−5; %Pascal to Bar %% Temperature Tfw = 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) %% Density rhoV = XSteam ('rhoV p',P); %Steam (Kg/m3) rhoL = XSteam ('rhoL p',P); %Water (Kg/m3) %Partial derivative with pressure drhoL 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 Enthalpy hfW = XSteam('hL T',Tfw) *1e3; %Feedwater (J/Kg)

55

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

(J/Kg) (J/Kg)

* 1e−3; %Water (J/Kg.Pa) * 1e−3; %Steam (J/Kg.Pa)

%% Coefficients Values Eta = (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; %% e11 e12 e21 e22 e32

e33 e42

e43 e44

State equations coefficients = rhoL − rhoV; = Vwt*drhoL dP + (Vt−Vwt)*drhoV dP; = rhoL*hL − rhoV*hV; = Vwt*(hL*drhoL dP + rhoL* dhL dP) + (Vt−Vwt)*(hV*drhoV dP + ... rhoV* dhV dP) − Vt + mt*Cp* dT Sat dP; = (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; = ((1−Alpha)*rhoV + Alpha*rhoL)*hC*Vr*dAlphaV dAlpha; = 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); = Alpha*(1+Beta)*(rhoV−rhoL)*Vr*dAlphaV dAlpha; = rhoV;

%% State variables dP 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 outputs y = [ dP dt dVwt dt dAlpha dt dVsd dt Level ]; end

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A.3. MATLAB script

A.3.2. Controller design The following m.file was used to obtain a linearized model in state-state form at a common operating point to compute the state-feedback and observer matrices gains using LQ and pole placement methods. % Specify the model name model = 'Drum Model'; % Create ios(5) = ios(4) = ios(3) = ios(2) = ios(1) =

the linearization I/O as specified in Linearize Model linio('Drum Model/Drum model',2,'out'); linio('Drum Model/Drum model',1,'out'); linio('Drum Model/qs (Kg//s)',1,'in'); linio('Drum Model/qf (Kg//s)',1,'in'); linio('Drum Model/Q (MW)',1,'in');

% Create the operating point specification object opspec = operspec(model); % Set the constraints in the model opspec.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 model op = 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 observer Q = [ 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 L K = lqi(Drum ss,Q,R); L = place(Drum ss.A',Drum ss.C',P)';

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A.4. Heat engines

A.4. Heat engines Heat engine plays an essential role in electrical power generation as they convert the thermal energy into mechanical energy required to drive the generator shaft producing electricity. The heat engine can be a closed or open loop system, which involves typically four thermodynamic basic processes shown in figure (A.1). It converts the state of the working fluid into another before returning it to its original state. The processes are compression, heat addition, expansion and heat rejection, each can 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 the system and no work done. Isentropic At constant entropy, reversible adiabatic conditions

Heat addition

Compression

Closed system

Expansion

Heat rejection Fluid Exhaust heat

Open 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 the thermodynamics and heat cycle for the operation of the gas turbine.

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A.4. Heat engines Process 1-2 Fresh air is being supplied for an open cycle, as for a closed one it’s drawn back from the turbine to a compressor increasing its pressure in an adiabatic compression process. Process 2-3 The compressed air is mixed with fuel or natural gas before being 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, in an adiabatic expansion process. Process 4-1 Finally, heat rejection to the surrounding atmosphere takes place at constant pressure. T 3

q in

p

2

co ns t.

3

q in

=

P

s=

s

=

4

t. ns co

co ns t.

1 q out P-v Diagram

4

2 1

v

t. ns co = p

q out

T-s Diagram

s

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 describe closed cycle heat conversion into mechanical energy using two phase working fluids that drive a steam turbine blades producing electricity. Process 1-B High pressure water is pre-heated at constant pressure at the economizer stage, until it reaches its boiling point converting it to water-vapour mixture. Process B-2 A second heating phase takes place using evaporator and superheater, to convert the converting the mixture into superheated steam. Process 2-3 The vapour at high pressure and temperature enters the turbine, where vapour energy is converted into mechanical work driving the turbine blades.

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A.5. Non-minimum phase systems Process 3-4 Vapour leaving the turbine at low pressure and temperature is condensed, converting it into wet saturated water. Process 4-1 Saturated water is pumped back, feeding the boiler at high pressure, 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 lefthand 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 restricted within 0◦ to −90◦ degrees for a given amplitude response when being examined in frequency domain. If a stable plant have one or more zeros in the right hand side (RHS) of the complex plane, then phase shift range is always greater than −90◦ . Such systems are known as non-minimum phase, where an inverse response always exists, leading to an initial overshoot or undershoot delaying the output behaviour. Assume a Single-Input, Single-Output (SISO) plant whose closed loop transfer function is assumed to be internally stable and given by equation (A.1), only one zero lies in RHS of the complex plane for the sake of simplicity. A step input 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-loop zero z0 lies within of the region of convergence (ROC) of Y (s), which yields the unilateral laplace transform to (A.3).

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A.6. Integral anti-windup control Since the output signal has an initial value y(0) = 0, final value y(∞) = 1 and its area under the curve evaluated by the integral in equation (A.3) is equal to 0, then this implies that the output signal y(t) must take negative values over time. (s − z0 ) Y (s) = R(s) s + p0 (s − z0 ) Y (s) = Gcl (s)R(s) = s(2s + p0 − z0 )  ∞ y(t)e−z0 t dt = Y (s)|s=z0 = Y (z0 ) =

Gcl (s) =

0

(A.1) (A.2) (z0 − z0 ) =0 s(2s + z0 − z0 )

(A.3)

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

Bode Diagram

1.2 Magnitude (dB)

1 0.8 0.6

Phase (deg)

0.4 0.2 0
0.2

0

5

10

15

20

0
20
40 180 135 90 45 0
45
90
2 10


1

10

0

10

1

10

2

10

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

A.6. Integral anti-windup control Most PID-controllers in practical applications are equipped with a nonlinear saturation element which saturates the controller output once it attains a certain values, imposed by the physical limitations of the actuators. The nonlin-

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A.6. Integral anti-windup control earity might lead to integral windup. Such occurring phenomena take place when the PID-controller integral element builds up and accumulates the error signal even if the controller output is saturated, it might degrade the controller performance or even lead to closed loop instability if neglected. Consider a simple linear motor positioner modeled as an integrator, whose input and output are velocity and position respectively. Clearly, the motor velocity will be physically limited according to its type and manufacturer, therefore the equipped PI-controller output should be limited when used to control the closed loop. The controller parameters weren’t tuned since this is merely an explanatory example focused on effects of windup phenomena and saturation limit is set to be ±0.3 ms associated with the motor allowable maximum speed. Position (m)

Velocity (m/s) 0.4

4

0.2

3 0 2
0.2

1 0

0

10

20

30

40

50


0.4

0

10

20

30

40

50

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 positioner while equipping an anti-windup mechanism using back-calculation method and compares it when no anti-windup is utilized. The position set point changed to 3 m, due to this large switch, the controller tries to track the reference value as fast as possible, however it was limited by the motor velocity upper limit. Without anti-windup, the integrator element output keeps growing and the motor position would require more time to reach the steady-state. On the other hand, while equipping an anti-windup, the back-calculation gain starts discharging the PI-controller integrators and prevents it from building up once the controller output is saturated, therefore vastly improving the output settling time while eliminating undesired overshoots.

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A.7. Drum-boiler state equations coefficients

A.7. Drum-boiler state equations coefficients ∂ρw ∂ρs + Vst ∂p ∂p e12 = ρw − ρs     ∂ρw ∂hw ∂ρs ∂hs ∂tsat + ρw + ρs e21 = Vwt hw + Vst hs − Vt + mt Cp ∂p ∂p ∂p ∂p ∂p e22 = ρw hw − ρs hs     ∂hw ∂ρw ∂hs ∂ρs − α r hc + (1 − αr )hc e31 = ρw (1 − α ¯ v )Vr + ρs α ¯ v Vr + ∂p ∂p ∂p ∂p ∂α ¯v ∂tsat − Vr + mr Cp (ρs + (ρw − ρs )αr ))hc Vr ∂p ∂p ∂α ¯v e33 = ((1 − αr )ρs + αr ρw )hc Vr ∂p   ∂ρs ∂ρs ∂ρw ∂α ¯v e41 = Vsd + αr (1 + β)Vr α + (1 − α ¯v ) + (ρs − ρw ) + ¯v ∂p ∂p ∂p ∂p   ∂hs ∂hw ∂tsat 1 + ρw Vwd − Vsd − Vwd + md Cp ρs Vsd hc ∂p ∂p ∂p ∂α ¯v e43 = αr (1 + β)(ρw + ρs )V r ∂p  2ρw Adc (ρw − ρs )g α ¯ v Vr qdc = K   h w − hf w ∂hw ∂hs ∂tsat dP 1 + ρs Vst − Vt + mt Cp qct = qf + ρw Vwt hc hc ∂p ∂p ∂p dt   ∂ρs ∂ α¯v dαr ∂ρw ∂α ¯ v dP qr = qdc − Vr α + (1 − α ¯v ) + (ρw − ρs ) ¯v + (ρw − ρs )Vr ∂p ∂p ∂p dt ∂αr dt    ρw ρ w − ρs ρs α ¯v = ln 1 + αr 1− ρ w − ρs (ρw − ρs )αr ρs (ρw − ρs ) ζ = αr ρs   1 ∂α ¯v ρw ln(1 + ζ) − = ∂αr ρs ζ ζ 1+ζ    1 ρs + ρw ∂ρs ∂ρw ∂α ¯v ρw = − ρ − ln(1 + ζ) ρ 1 + w s ∂p (ρw − ρs )2 ∂p ∂p ρs (1 + ζ) ζρs e11 = Vwt

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A.8. Operator interface

A.8. Operator interface

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

64

A.8. Operator interface

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

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B. List of Figures

B. List of Figures 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

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

3.1. Pole-zero plot of the linearized models at low (1), medium (2) and high (3) load . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Open loop response for a step change equivalent to decrease of 20 MW of the gas turbine electrical output power . . . . . . . . 3.3. Open loop response for a step change equivalent to 10% opening of butterfly valve position . . . . . . . . . . . . . . . . . . . . . 3.4. Open loop response for a step change equivalent to 10% closing of feedwater control valve position . . . . . . . . . . . . . . . . 3.5. Simulink validation model . . . . . . . . . . . . . . . . . . . . . 3.6. Comparison between model (dashed) and plant data (solid line) for a decrease of the gas turbine electrical output power equivalent to 20 MW . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Comparison between model (dashed) and plant data (solid line) for an increase of the gas turbine electrical output power equivalent to 10 MW . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Comparison between model and plant data (solid line) for a decrease of the gas turbine electrical output power equivalent to 20 MW when Kp = 1.8 (dotted dashed) and Kp = 5 (dashed) . 4.1. State estimation using Luenberger observer . . . . . . . . . . . 4.2. PI-based state-feedback control structure . . . . . . . . . . . .

9 10 11 12 15 16 19 21 22 28 31 31 32 34

35

35

37 39 40

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B. List of Figures 4.3. Estimated states using the observer for perturbations in gas turbine electrical output power . . . . . . . . . . . . . . . . . . . . 4.4. Comparison between state observer (dashed) and plant (solid line) for perturbations in gas turbine electrical output power . 4.5. Comparison between state observer (dashed) and plant (solid line) for a decrease of the gas turbine electrical output power equivalent to 10 MW . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Comparison between state observer (dashed) and plant (solid line) for a decrease of the gas turbine electrical output power equivalent to 20 MW . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Comparison between PI-based state-feedback controller (dashed) and plant (solid line) for perturbations in gas turbine electrical output power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Model closed loop response using the PI-based state-feedback controller for high (dashed), medium (solid line) and low load (dotted dashed) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9. States and input variables behaviour using the PI-based statefeedback controller at medium load . . . . . . . . . . . . . . . . 4.10. Block diagram of the proposed multivariable feedback control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. A.2. A.3. A.4.

Heat engine typical closed/open loop heat cycle . . . . . . . . . T-S and P-V diagram of a typical ideal Brayton cycle . . . . . T-S diagram of a typical ideal Rankine cycle . . . . . . . . . . Step response and bode plot for a minimum (solid line) and a non-minimum (dashed) system . . . . . . . . . . . . . . . . . . A.5. Motor velocity and position behaviours with (dashed) and without (solid line) anti-windup . . . . . . . . . . . . . . . . . . . . A.6. Screenshot of the drum-boiler unit in the real process DCS . . A.7. Screenshot of the low pressure steam distribution network in the real process DCS . . . . . . . . . . . . . . . . . . . . . . . . . .

47 48

48

49

49

50 50 51 58 59 60 61 62 64 65

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C. List of Tables

C. List of Tables 2.1. Drum-boiler model parameters . . . . . . . . . . . . . . . . . . 2.2. Control valve and actuator parameters . . . . . . . . . . . . . . 2.3. PID-controller parameters . . . . . . . . . . . . . . . . . . . . .

20 21 22

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

27

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

53 53 53

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D. Bibliography

D. Bibliography [1] Asea Brown Boveri: Drum Level Control Systems in the Process Industries, 1997. ABB Download Center. [2] ˚ Astr¨ om, Karl Johan und Rodney D. Bell: Drum Boiler Dynamics. Automatica, 36:363–378, M¨ arz 2000. [3] ˚ Astr¨ om, K.J. und T. H¨ agglund: PID Controllers - Theory Design and Tuning. International Society of America, 1995, ISBN 9781556175169. [4] Ellis, G.: Observers in Control Systems - A Practical Guide. Academic Press, 2002, ISBN 9780122374722. [5] Emami, A. und P. Van Dooren: Computation of zeros of linear multivariable systems. Automatical, 18:412–430, 1982. [6] Falb, P. L. und W. A. Wolovich: On the decoupling of multivariable systems. Proc. JACC, Philadelphia, Pennsylvanial, 41:791–796, 1967. [7] Flynn, M.E. und M.J. O Malley: A drum Boiler Model for Long Term Power 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 and steam, 2006. MATLAB Central File Exchange. [11] Kautsky, J., N.K. Nichols und P. Van Dooren: Robust Pole Assignment in Linear State Feedback. International Journal of Control, 41:1129–1155, 1985.

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D. Bibliography [12] Kehlhofer, R., B. Rukes, F. Hannemann und F.X. Stirnimann: CombinedCycle Gas and Steam Turbine Power Plants. PennWell, 2009, ISBN 9781593701680. [13] Kim, H. und S. Choi: A model on water level dynamics in natural circulation drum-type boilers. International Communications in Heat and Mass Transfer, 32:786 – 796, 2005. [14] Kim, T.S., D.K. Lee und S.T. Ro: Analysis of thermal stress evolution in 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 engineering 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 Transactions on Military Electronics, 8:74–80, 1964. [17] Luenberger, D. G.: Observers for multivariable systems. IEEE Transactions on Automatic Control, 11:190–197, 1966. [18] Michels, K.: Regelungstechnik (Vorlesungsmanuskript). Institut f¨ ur Automatisierungstechnik, Universit¨ at Bremen, 2013. [19] Mikofski, Mark: IAPWS IF97 functional form with no slip, 2012. MATLAB Central File Exchange. [20] Moran, M.J. und H.N. Shapiro: Fundamentals of engineering thermodynamics. John Wiley and Sons Inc., New York, NY, 2009. [21] Parry, A., Petetrot J. F. und M. J Vivier: Recent progress in sg level control in french pwr plants. British Nuclear Energy Society, Seiten 81–88, 1995. [22] Samson AG: Application Notes for Valve Sizing, 2012. Samson Product Documentation. [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 of Marine Vehicles (Lecture Notes). Massachusetts Institute of Technology, 2003. MIT OpenCourseWare.

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