Energy Conversion and Management 66 (2013) 234–245

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Energy Conversion and Management

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Optimal control strategy for minimization of exergy destructionin boiler superheater

Tapan K. Ray a, Ranjan Ganguly b, Amitava Gupta b,⇑a PMI, NTPC Limited, Noida 201 301, Indiab Department of Power Engineering, Jadavpur University, Kolkata 700 098, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 November 2011Received in revised form 23 October 2012Accepted 24 October 2012

Keywords:Boiler superheaterExergyLinear Quadratic RegulatorState space modelSteam power plantSteam temperature control

0196-8904/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.enconman.2012.10.013

⇑ Corresponding author. Tel.: +91 33 23355813; faxE-mail address: amitg@pe.jusl.ac.in (A. Gupta).

Steam temperatures in large capacity boilers of modern electric power stations are maintained closelyaround the design specification by spraying water in the superheater (SH) attemperator to ensure safeand efficient operation and long plant life. Although the process of attemperation involves exergydestruction, and optimal controllers have previously been proposed for steam temperature control, priorstudies on such controllers have not considered exergy as an important parameter. Exergy analysis of atwo-stage SH attemperator with real time operation parameters in a 500 MWe pulverized fuel firedpower plant pinpoints the avenues for optimization that is beyond the scope of the traditional First-Law based analysis. Strategies to minimize exergy destruction by suitably varying the proportions ofstage I and stage II spray flows are established. Further, a MATLAB-SIMULINK-based model is developedand optimal control strategies are devised for SH steam temperature control following a Linear QuadraticRegulator (LQR) approach. Variation of the process parameters and the exergy destructions during thetransient operations of the attemperator under stipulated disturbances have been analyzed using themodel, with different values of the controller parameters. Guidelines are formulated for the spray flowcontroller tuning so that the total exergy destructions during the system transients are minimized.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In a large capacity pulverized fuel fired power plant, boiler out-let main steam (MS) temperatures tend to vary due to fluctuationsin operating conditions like variation in fuel characteristics fromdesign values, odd mill combination, part load operation or otheroff-design conditions. MS temperatures higher than the specifiedlimit can cause damages to superheater (SH) tubes and turbine in-let stages [1]. Low MS temperature reduces the mean temperatureof heat addition and affects the cycle efficiency. Steam tempera-tures must be kept precisely within small deviations around theirset values in order to ensure long life, high efficiency, high avail-ability and safe operation of the plant. Although several techniquesof flame management (e.g., through burner tilting mechanism inCombustion Engineering [2] or presently the Alstom design), andheat transfer management (e.g., through the control of second passgas distribution in the Babcock and Wilcox (B&W) design boiler[3]) are in vogue, a fast and reliable control of steam temperatureswould warrant attemperation (water spray in the steam line) ofsuperheated steam to improve follow-up to a load fluctuationand a fuel change. Large capacity high-pressure temperature

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boilers are generally equipped with two stages of attemperatorsthat facilitate mixing of desuperheating water in the stream ofsuperheated steam to control MS temperature. The attemperatorspray water is taken from the boiler feed pump discharge (i.e.,bypassing the high pressure regenerative feed water heaters) andthus it enters into the boiler at a lower temperature than the mainstream (feed water) from the turbine cycle. An increase in attem-perator spray water flow rate diminishes the extent of regenerativefeed heating and brings down the mean temperature of heat addi-tion from external source (due to increase in the amount of heataddition at low temperature) to the cycle. Deterioration of cycleheat rate attributable to higher spray water flow rate or lowerMS temperature is traditionally evaluated by knowing the MS tem-perature and the total spray water flow rates.

A conventional performance analysis based on the First Law ofthermodynamics merely serves as a necessary tool for accountingenergy flows during the process [4–9]. Attemperation is a highlyirreversible process and causes irretrievable loss of exergy [9]. Ex-tent of exergy destruction is also a strong function of temperaturedifference between two fluids (the temperature differences be-tween superheated steam and spray water are not same in the firstand second stage) mixing in the attemperators. However, a tradi-tional analysis in the present context fails to predict any deviationin the heat rate if different proportions of first and second stagesprays are used, as long as the total spray flow rate and the MS

Nomenclature

As heat transfer surface area of SH (m2)Cp Sp. heat of SH tube metal (kJ/kg K)Cpg Sp. heat of flue gas (kJ/kg K)h Sp. enthalpy (kJ/kg)_I exergy destruction or irreversibility rate (kW)M total metal mass of SH (kg)_m mass flow rate (kg/s)

p SH steam pressure (bar)_Qgm heat transfer rate from flue gas to tube metal (kW)_Qmf heat transfer rate from tube metal to fluid (steam) (kW)S Mean total heat transfer area of SH (m2)T main steam (MS) temperature (�C)Tm SH mean temperature (�C)Vg volume of flue gas space in SH section (m3)VS volume of steam flow passage in SH section (m3)

Greek symbolsagmc heat transfer coefficient from flue gas to metal (kW/

m2 K)

ams heat transfer coefficient from metal to steam (kW/m2 K)e specific flow exergy (kJ/kg)qg mean density of flue gas in SH section (kg/m3)qgavg

average density of flue gas in SH section (kg/m3)qs mean density of fluid in SH (kg/m3)qSavg

average density of steam in SH section (kg/m3)/ the average non-flow bulk exergy of the content (kJ)ugavg

average non-flow specific exergy of flue gas in SH sec-tion (kJ/kg)

uSavgaverage non-flow specific exergy of steam in SH section(kJ/kg)

Subscriptsa attemperator spray waterg flue gasi inlet steamo outlet steam

T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245 235

temperature remain unchanged. On the contrary, exergy analysis(which follows the Second Law of thermodynamics) clearly pin-points that exergy destruction varies significantly when spray flowdistributions between stages are altered. For example, a highertemperature difference between steam and spray water in stageII attemperator results in larger exergy destruction than that instage I for similar flow rates. Therefore, it is intuitive that sprayflow controller action to counter the effect of different perturbationsignificantly influence exergy destruction during the transientperiods. The amount spray flow is directly related to the exergy de-stroyed during the attemperation process, which indicates the lostwork that could have been produced in the turbine had there beenno attemperation. Therefore, the objective of a spray flow control-ler should also include minimization of spray flow, while keepingthe process parameters within limit. Thus, the approach adoptedin this paper is a minimization of spray flow with pre-defined dis-tribution criteria between the two superheater stages while limit-ing the excursion of the controlled variables from their regulatedvalues. Optimal state feedback controllers are known to achievethis by associating different weights to states (and hence theirdeviation) and inputs.

Several researchers have conducted modeling and simulationto improve the exergy efficiency of the energy systems. Ahmadiet al. [10] performed a thermodynamic modeling of a combinedheat and power system in order to maximize plant exergy effi-ciency and minimize the fuel consumption and related overallenvironmental impact. Barzegar Avval et al. [11] modeled a gasturbine power plant with preheater and carried out multiobjec-tive optimization considering exergetic, economic and environ-mental aspects. Superheater steam temperature control hasbeen studied by quite a few researchers [12–16] but mostly withthe objective of designing robust controllers which address theuncertainties in SH spray flow control. An optimal controller pro-posed by Nakamura and Uchida [17] uses autoregressive plantmodels identified at different load levels using actual plant dataand it has been demonstrated that the optimal controller so de-signed is able to limit deviations in steam temperatures withinmuch smaller values compared to conventional Proportional plusIntegral plus Derivative (PID) controllers. The underlyingassumption here [17] is that the parameters in the state equa-tion matrix and the feedback gain matrix vary linearly between

two load levels. However, none of these studies include the exer-gy destruction in SH steam attemperators and in spray flow con-trol action during the transient periods due to any perturbationin the boiler.

In this paper, the approach followed in the design of an optimalcontroller is similar to the approach presented by Nakamura andUchida [17] except for the fact that statistical system identificationusing plant data is replaced by physical model of the superheaterbased on principles of conservation of mass and energy as put for-ward by Gupta et al. [18].

The present model is used to design the spray flow controller inthe 500 MWe operating plant in order to minimize exergy destruc-tion during the transient states (by curbing steam temperaturesexcursions around its set values without using excessive spray).The controller presented in the study acts as a regulator and notas a tracking controller. Effect of variation of the weighing matrices(in the cost function associated with the control action) on theexergy destruction is investigated. It may be mentioned that thefocus of the work presented in this paper is to study exergydestruction using an optimal controller that aims to minimizethe attemperator spray flow while limiting excursion in MS tem-perature for given input disturbances.

The specific contributions of this paper are summarized asfollows:

� Methodologies for evaluation of exergy destructions in a two-stage SH attemperators are developed.� Exergy analysis of attemperators during plant operation under

different proportions of stage I and stage II spray flows are con-ducted with real time operation parameters in a 500 MWe pul-verized fuel fired power plant to demonstrate that such analysiscan pinpoint the avenues for optimization that is otherwiseimpossible to achieve using the traditional First-Law basedanalysis.� A MATLAB-SIMULINK-based model is developed and an optimal

controller for SH steam temperature control is devised follow-ing LQR approach with an aim to study exergy destructions dur-ing the transient periods.� Exergy destruction computation methodologies are also formu-

lated for the entire SH section during transient states with thespray flow controller in loop.

236 T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245

� The variation of the process parameters and the exergy destruc-tions during the transient operations under given disturbancesare investigated for different values of controller parameters.� Guidelines for the fine tuning of the spray flow controller in the

operating plant are formulated in order to minimize the totalexergy destruction during the system transients. To the knowl-edge of the authors, such attempt of integrating exergy aspectin optimal control strategy has not been reported yet in theliterature.

2. Description of the process

2.1. System and governing equations

Fig. 1 shows the relative position of SH sections in flue gas pathand the arrangement of attemperator blocks in the 500 MWesteam power plant in which the present study is carried out. In or-der to control the MS temperature by facilitating mixing of waterin the stream of SH steam, the boiler is equipped with stage I (be-tween Primary SH and Platen SH) and stage II (between Platen SHand Final SH) attemperators (as shown in Fig. 1). Steam flowsthrough the Primary, Platen and Final SH in sequence, interspacedby two stages of attemperators, while the flue gas flows throughPlaten, Final and then Primary SH.

For the 500 MWe KWU turbine cycle of the plant, the conden-sate is taken sequentially through three indirect contact (surfacetype) low pressure regenerative feedwater heaters (LP heaters 1through 3), one direct contact type feedwater heater (deaerator),and two indirect contact (surface type) high pressure regenerativefeedwater heaters (HP heaters 5 and 6) to the boiler. The boilerfeed pump (BFP) takes suction from the deaerator and dischargesfeedwater (normal stream) through both the HP heaters 5 and 6to the boiler. The rated final feed water (flow to economizer inthe boiler) temperature after the HP heater 6 is 254 �C while theattemperator spray water temperature is 165 �C [19]. This is be-cause the attemperator spray water is taken from BFP dischargeand it does not pass through the HP heaters. Therefore, an increasein attemperator spray water flow rate diminishes the extent ofregenerative feed heating effectiveness and brings down the meantemperature of heat supply (due to increase in the amount of heataddition at low temperature) to the cycle and thus results in poor

Fig. 1. Relative position of Primary, Platen and Final SH in the flue gas path showing the Svolume of SH section. Mass flow rates and specific enthalpies are denoted by _m and h. Substreams respectively.

cycle efficiency. An increase in SH attemperation spray flow rate by10 kg/s results in an increase in cycle heat rate by 1.7 kcal/kW hand effect of reduction in MS temperature by 10 �C is cycle heatrate rise of 8.1 kcal/kW h [20].

The mass and energy balance equations for the SH and theattemperators in the boiler are formulated following the methodsand using the terms as in Gupta et al. [18]:

_mi � _mo þ _ma ¼ VddtðqsÞ ð1Þ

_mihi � _moho þ _maha þ _Q mf ¼ VddtðqshoÞ ð2Þ

_Qgm � _Q mf ¼ MCpddtðTmÞ ð3Þ

_mgðhgi � hgoÞ � _Q gm ¼ VgqgddtðhgoÞ ð4Þ

Expressing the temperature as a function of fluid enthalpy andpressure, the heat transfer rate from metal to fluid ( _Q mf ) and fromflue gas to metal ( _Q gm) are expressed as

_Qmf ¼ amsSðTm � 0:5f ðh0; p0Þ � 0:5f ðhi;piÞÞ; ð5Þ

_Qgm ¼ agmcAs 0:5hgi

Cpg� 0:5

hgo

Cpg� Tm

� �: ð6Þ

2.2. Staging of SH spray

Staging of SH spray is done within the constraints of two majorpractical requirements: (i) the spray should not be done too early(or at too low a degree of superheat) in the line such that the localsteam temperature falls to the saturation temperature, (ii) thespray should not be done at the exit of SH so that the spray waterhas a chance of carryover to the turbine. The first constraint limitsthe spray flow at the first stage attemperator, since otherwise thereis a possibility that the steam temperature after first stage ofattemperator may fall below the saturation temperature, leadingto two-phase flow through the remaining SH section. Such asituation is to be avoided, since SH tubes are not designed for

H steam and attemperator spray flows. Dotted (. . .. . .) rectangle indicates the controlscripts g, a, i and o indicate gas flow, attemperator spray water flow, inlet and outlet

T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245 237

two-phase flow, and the evaporation in super heater tubes maylead to deposition of solid impurities in SH section. A specifiedminimum degree of superheat of 10 �C at the end of first stage ofattemperator is to be maintained to provide desired level of oper-ational flexibility. The second constraint is more easily met byplacing the second stage of attemperator before the Final SH, andcontrolling the overall spray flow rate. However, remaining withinthese two constraints, there is still room for adjusting the first andsecond stage spray flow rates such that the overall exergy destruc-tions in the two attemperator sections are minimized. One of theobjectives of the present study is to analyze, by appropriately dis-tributing the total spray (as computed by the optimal controller)between stages subject to the above constraints, the extent towhich exergy destruction rate can be minimized in attemperatorblocks.

3. Exergy analysis methodology for SH and attemperators

Exergy of a system is the maximum useful work that it can pro-duce while interacting with the surrounding, eventually coming toan equilibrium with it [4]. Exergy can be of two types: chemicalexergy and thermomechanical exergy. The chemical exergy is asso-ciated with the departure of the chemical composition of a systemfrom its chemical equilibrium composition, while thermomechan-ical exergy comprises of the physical exergy (due to its thermody-namic state) and mechanical exergy due to potential and kineticenergies [21]. Exergy can be reckoned as a measure of the qualityof energy. Unlike energy, exergy is not conserved, but it is de-stroyed in any spontaneous process that is irreversible in nature.Exergy destruction during such a process is proportional to the en-tropy generation in it due to irreversibilities. Heat and mass trans-fer in steam generator entail entropy generation through severalphenomena, viz, fluid friction, heat and mass transfers across finitegradients of temperature and concentration, spontaneous chemicalreaction, etc. Since no significant change in the chemical composi-tion of the flue gas, and kinetic and potential energies of the fluidsoccur in the superheater section, exergy analysis across it will in-volve only the physical exergy. The irretrievable loss of exergy inthe boiler SH is mainly attributed to heat transfer from hot fluegas to steam and mixing of low temperature spray water with hightemperature steam in SH attemperators. The rate of exergydestruction (_I) in SH section is expressed by the following equation,

Xi

ei _mi �X

e

ee _me �d/dt¼ _I ð7Þ

where _mi denote mass flow rates of incoming fluids (SH inlet steam,inlet flue gas and spray water), _me denote mass flow rates of outgo-ing fluids (MS and exit flue gas) and / denotes the average bulknon-flow exergy of the content. The rate of change of average bulkexergy of the content (gas and steam) for the control volume of SHsection is represented by d/

dt .Specific flow exergy (e) is computed [22] by the following equa-

tion (ignoring the changes in kinetic and potential energies).

e ¼ ðh� T0sÞ � ðh0 � T0s0Þ ð8Þ

The pressure and temperature of the reference environmentconsidered are P0 = 101.325 kPa and T0 = 298.15 K. Specific flowexergy (e) is evaluated by using the exergy calculation softwaredeveloped in Ray et al. [23]. In the software, add-in library func-tions are integrated in a spreadsheet to obtain the thermodynamicproperties of water and steam. The library functions use a steamtable based on the scientific formulation of International Associa-tion for the Properties of Water and Steam, Industrial Formulation1997 (IAPWS-IF97).

For all the inlet and exit fluid streams h and T are obtained asfunctions of time (t) by using the controller developed here andexergy (e) is computed as functions of t for the transient periodby using the exergy calculation software [23], in which the ther-modynamic property values (h and s) are obtained from the corre-sponding thermodynamic parameters (p and T) of the streams.Exergy flow rate is obtained as the product of mass flow rate andthe specific flow exergy (e) of the stream. The exergy destructionrate in each SH attemperator block is derived as the difference ofsum of exergy flow rates associated with the incoming fluidstreams (SH steam inlet and spray flow) and exergy flow ratesassociated with the exit fluid stream (attemperator outlet SHsteam).

Denoting the net advective exergy influx, i.e.,Piei _mi �

Pe

ee _me ¼ c, Eq. (7) can be expressed as

c� d/dt¼ _I ð9Þ

Integrating Eq. (9) w. r. t t from initial (b: the beginning of per-turbation) to final (f: steady state after the transient condition),

Z f

bcdt � ½/�fb ¼

Z f

b

_Idt ð10Þ

Numerically integratingR f

b cdt,

Z f

b

_Idt ¼X

cDt � ½/f � /b� ð11Þ

where the net exergy flow (P

cDt) is obtained by using the Trape-zoidal Rule for integration and, /b and /f are average initial and finalbulk exergy values of the content (before and after the transientstate).

The average bulk exergy (kJ) of the content (/) is computed byusing the following relation

/ ¼ qSavgVSuSavg

þ qgavgVgugavg

ð12Þ

where VS and Vg are space for steam and gas in SH section. qSavgand

qgavgis average density of steam and gas in SH section. uSavg

andugavg

are average non-flow specific exergy (kJ/kg) of steam andgas respectively in SH section. The methods followed in computa-tion of flue gas exergy at the inlet and exit of SH section are similarto that explicated by Hajabdollahi et al. [21]. After inlet and outletparameters are obtained non-flow specific exergy values for steamand gas at inlet and outlet are computed by using the exergy calcu-lation software [23] and then, uSavg

and ugavgare derived as average

of inlet and outlet.The transient exergy destruction (kJ) for the SH control volume

(I ¼R f

b_Idt) is computed by using Eq. (11) and the steady state exer-

gy destruction rates (kW) in attemperator blocks are computed byusing Eq. (7) while neglecting the exergy change of the fluid withinthe attemperator block, i.e., d/

dt ¼ 0.

4. State space model of SH

A linear system with n states, m inputs and p outputs can berepresented using the general state variable formulation by the fol-lowing set of equations

X ¼ AXþ BU ð13Þ

Y ¼ CXþ DU ð14Þ

where X = [x0, x1, . . ., xn]T represents the set of state variables,U = [u0, u1, . . ., um]T represents the set of inputs and Y = [y0, y1, -. . ., yp]T represents the set of output variables. The matricesA, B, C, D have usual significance [24].

Fig. 2. MIMO control scheme with the control law assumed to be U = �KX.

238 T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245

The SH for a coal-fired thermal power plant modeled by Guptaet al. [18] is used in the present work for development of the opti-mal controller. Since the SH represented by Eqs. (1)–(6) is a non-linear system, it is linearized about an equilibrium point definedby X0 ¼ ½hoss ; hgoss

; Tmss �T and U0 ¼ ½miss ;hiss ;mgss

;hgiss;ma;ha�T , where

steady state condition is indicated by subscript ss, the system canbe expressed in state-variable formulation as

_~X ¼ A~Xþ B~U ð15Þ

~Y ¼ C~X ð16Þ

by replacing X ¼ ~X;U ¼ ~U;Y ¼ ~Y and modifying the matrices A, Busing Jacobian. The matrix C remains unchanged since the relation-ship between ~Y and ~X remains same as the relationship between ~Yand ~X after linearization and the matrix D = [0] in the present case,since none of the input variables directly affect the outputs. Thus,for the present system, ~X ¼ ½Dho;Dhgo;DTm�T , ~U ¼ ½Dmi;Dhi;Dmg ;

Dhgi;Dma;Dha�T and ~Y ¼ ½Dho;Dhgo;DTm�T represent the state-variables, inputs and outputs, respectively, and the D-terms indi-cate variation of the respective parameters about their steady-statevalues which is taken as the equilibrium point. The matrices A and Bare evaluated for the state-space formulation with Eqs. (1)–(6),which leads to

A11 ¼ � ðmiþmaÞVqs

� 0:5� amsSVqs

dTdh jp¼p0

h¼h0

; A12 ¼ 0; A13 ¼ amsSVqs

;

A21 ¼ 0; A22 ¼ �0:5� mg

Vgqg� agmcS

Vgqg Cpg; A23 ¼ agmc S

Vgqg;

A31 ¼ 0:5� amsSMCp

dTdh jp¼p0

h¼h0

; A32 ¼ 0:5� agmc SMCpCpg

; A33 ¼ � agmcSMCp� amsS

MCp;

and the elements of matrix B are:

B11 ¼ðhi�hoÞ

Vqs; B12 ¼ mi

Vqs� 0:5� amsS

Vqs

dTdh jh¼hi

p¼pi

; B13 ¼ 0;

B14 ¼ 0; B15 ¼ ðha�hoÞVqs

; B16 ¼ moVqs

;

B21 ¼ 0; B22 ¼ 0; B23 ¼ðhgi�hgoÞ

Vgqg; B24 ¼ mg

Vgqg� 0:5� agmcS

Vgqg Cpg;

B25 ¼ 0; B26 ¼ 0;

B31 ¼ 0; B32 ¼ 0:5� amsSMCp

dTdh jh¼hi

p¼pi

; B33 ¼ 0; B34 ¼ 0:5� agmcSMCP�Cpg

;

B35 ¼ 0; B36 ¼ 0;

The C matrix is defined as C = I3 meaning that all state variablesare output variables. It is next attempted to design an optimal con-troller using LQR approach [25]. An infinite time linear regulator isassumed with the system remaining invariant within the zone ofperturbation DU around the equilibrium point. Thus the cost func-tion associated with the control action becomes represented in thegeneral form [25]

JX ¼12

Z 1

0

~XTðtÞQ ~XðtÞdt þZ 1

0

~UTðtÞR ~UðtÞdt� �

; ð17Þ

where Q and R are the weighing matrices. Q is a [3 � 3] positivesemi-definite matrix while R is [6 � 6] positive definite matrix.For the present case Q = CTC is assumed while for matrices

Q ¼1 0 00 1 00 0 1

264

375 and R ¼

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

2666666664

3777777775: ð18Þ

These values are referred to as nominal values of Q and Rthroughout the rest of the paper. The idea is to design Full State-Feedback Controller (FSFC) satisfying a control law of the form

~U ¼ �K~X which minimizes JX. The performance index JX representsan energy term and minimizing it means that the total energy ofthe control loop is minimized. Since JX is minimum, it is finiteand the Eq. (17) being an infinite integral indicates that the closedloop system is at least asymptotically stable. Further, since boththe state vector ~XðtÞ and the control input ~UðtÞ are weighted andincorporated within JX, which is minimized, neither can be toolarge.

The system described by the governing equations is linearizedaround an operating point with actual data from the plant as pre-sented in Gupta et al. [18]. The basic assumption in Nakamura andUchida [17] that the state equation matrix and feedback gainparameters change linearly with power levels remains valid inthe present context. Therefore, design of the controller is restrictedto one power level only in the present work that studies exergydestruction with an optimal controller.

The SH model is considered in state-space as formulated inGupta et al. [18] and the equilibrium values of the parametersand the physical dimensions correspond to those of the500 MWe plant operating at rated load. A full state-feedback is as-sumed with nominal values of the weighing matrices Q and R andthese weighing matrices are varied so as to focus more on temper-ature regulation and spray flow minimization, respectively.

If the system is controllable, then for a given disturbance e.g. inthe form of a change in the reference input, the controller modu-lates the inputs ~U to change the state of the system X. As men-tioned before, using optimal control methodology, it is attemptedto minimize the cost function JX for a given Q and R specified bythe designer with a control law ~U ¼ �K~X where K is the optimalstate-feedback controller gain matrix to be designed. The signifi-cance of the term X(t)TQX(t) in Eq. (17) indicates the desire to con-trol the state close to the equilibrium. While for a linear systemthis can always be done in a finite time, the inputs ~U may be verylarge, thus posing constraints on design of practical control sys-tems. The term U(t)TRU(t) limits the input by penalizing the costfunction for large inputs. The choice of Q and R allows the designerto construct the cost function with varying weights associated withcontrol of state excursions ~X and the modulation of inputs ~U re-quired for achieving the change in the state of the system. Forexample, in the present context, if the objective is to minimizethe deviation in MS temperature only, then following a distur-bance, a controller would use maximum spray flow rate. This cor-responds to a case where the elements of Q are much larger thanthose of R. On the other hand, if the objective is to minimize thespray flow rates, the elements of Q should be much less than theelements of R. The choice of Q, R thus, allows the designer to opti-mize between the two objectives viz. regulation of MS temperatureand limiting the modulated inputs and thus the spray flow. Theindividual elements of the Q, R matrices may also be varied withrespect to other elements of the same matrix to indicate differentimportance for different states and inputs. The optimal controllerthus designed with a specified value of Q, R leads to a multiple-in-put multiple-output (MIMO) control scheme as shown in Fig. 2.

Since C = I3, for the present case, the state feedback controller isalso an output feedback controller, with the enthalpies Dho and

T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245 239

Dhgo derived from the temperature and pressure of MS and flue gasrespectively. Thus minimizing excursions in ho and hgo minimizestemperature excursions around set values.

To investigate the response of the system under the action ofthe controller, the system requires modulation of various inputs,for example, enthalpy of incoming steam which involves additionaldynamics. In the present paper, these dynamics are represented byappropriate approximate subsystems for simplicity. All these sub-systems are assumed to be delay dominated First Order Plus TimeDelay (FOPTD) systems identified from practical test data usingMATLAB system identification toolbox.

5. Results and analysis

5.1. SH and optimal controller

The MATLAB-SIMULINK-based model shown in Fig. 3 representsthe SH modeled using Eqs. (1)–(6), controlled with a state feedbackcontroller represented by Fig. 2. The model in Fig. 3 is a linearizedmodel of a SH linearized about an equilibrium point defined by thesteady state plant operation parameters summarized in Table 1.

The SH spray flow controller is designed to act to counter the ef-fect of different perturbations. In a typical power plant boiler, suchdisturbance can originate due to several reasons. Two typical dis-turbance conditions that are most commonly encountered duringthe plant operation are tripping of a running pulverizer andputting an additional pulverizer in service. Here, the actions of

Fig. 3. MATLAB-SIMULINK-based mo

the controller are investigated under the following two transientconditions: (a) an increase in flue gas flow rate by 50 kg/s (this ex-tent of perturbation is inflicted in the plant when an additionalpulverizer is taken in service with the minimum allowablethroughput of the pulverizer) and (b) a reduction in flue gas flowrate by 111 kg/s (such interruption occurs when a pulverizer run-ning with rated coal flow trips). The exergy destructions under dif-ferent control actions during these transient periods are analyzed.The input to the controller in Fig. 3 is the set of state variables~U ¼ ½Dmi;Dhi;Dmg ;Dhgi;Dma;Dha�T and the output is the set~X ¼ ½Dho;Dhgo;DTm�T . This disturbance causes a deviation ~Yð¼ ~XÞand the controller responds by modulating ~U. The optimal state-feedback controller regulates ~X for a given disturbance while suit-ably modulating the inputs ~U so that the cost function defined inEq. (17) is minimum.

With Q at its nominal value (see Eq. (18)), all states have equalweights implying that the controller assigns equal priority to con-trolling excursions of all states. Similarly, nominal R (see Eq. (18))ensures that equal priority is assigned to modulation of all inputs.The controller with nominal values of Q and R thus assigns equalimportance to regulating ~X (and hence ~Y) and minimizing ~U i.e.it tends to emphasize equally on controlling the MS temperatureand minimizing the attemperator spray flow.

Now if Q is replaced by aQ where the scalar a > 1, keeping Rfixed, the controller shall tend to put more emphasis on controlof ~X for a deviation than minimization of ~U. On the other hand, ifR is replaced by aR, keeping Q fixed, the controller will tend toput more emphasis on minimization of ~U rather than control of

del of the SH and the controller.

Table 1Steady state values of the parameters for the SH section.

_mi (kg/s) _mo (kg/s) _ma (kg/s) _mg (kg/s) p (bar) T (�C)378.7 427.8 49.1 592.8 170.0 540.0

hi (kJ/kg) ho (kJ/kg) ha (kJ/kg) hgi (kJ/kg) hgo (kJ/kg) S (m2)2513.9 3387.8 707.9 1625.6 845.3 58.7

M (kg) agmc (kW/m2 K) ams (kW/m2 K) qs (kg/m3) qg (kg/m3) VS (m3)433.1 30.0 140.0 114.0 0.9 0.2

Vg (m3) Cp (kJ/kg K) Cpg (kJ/kg K) _Qgm (kW) _Qmf (kW)62.1 0.6 1.6 462532.9 462532.9

240 T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245

~X [25]. Thus, for a given disturbance, different values of Q and R arelikely to produce different optimal controllers with varying ~XðtÞand ~UðtÞ. Exergy destruction for these controllers under transientand steady state conditions are also expected to be different dueto the reasons that spray flows and temperature excursions are dif-ferent for different control actions. The exergy destruction withstate-feedback controllers using different combinations of Q andR is presented in Sub-section 5.4.

It is important to note that the optimal controller presented inthis paper modulates the attemperator spray flow and exergydestruction tends to decrease with decrease in spray flow. How-ever, as mentioned before, for a given spray flow, the exergydestruction also depends on the distribution of the flow betweenthe SH attemperator stages and this dependence is presented inthe next Sub-section.

5.2. Minimizing exergy destruction by proper distribution of sprayflows between stages

The foregoing analysis shows how the controller alters the totalspray flow and adjusts the MS temperature under an operationaldisturbance, e.g., a step change in load. Once the required totalspray flow rate for a given operating condition is obtained fromthe controller model, it is attempted to investigate how distribut-ing this total spray flow rate between the SH attemperator stagesaffects exergy destruction in the attemperator blocks (see Fig. 1).In order to keep MS temperature within specified limit, attemper-ation spray flow requirement varies under different operating sit-uations. The variation in requirement of spray flow rate is largelyattributed to operational constraints imposed by wide fluctuationof coal characteristics (mainly heat value and volatile matter con-tent of the coal) and other off-design or deviated operating condi-tions like part-load operation, problems in second pass flue gasbiasing damper control, improper distribution of air due to prob-lems in the furnace secondary air damper control (SADC) mecha-nism etc. For the 500 MWe plant considered in this paper, prioroperational experience has shown that the total SH spray flow rategenerally varies between a low value of 45.4 kg/s to a high value of56.9 kg/s. These two typical values of total spray flow rates aretherefore considered for analysis in order to establish strategiesfor proper distribution of sprays between stages for minimizationof exergy destruction subject to maintaining a minimum degreeof superheat (10 �C) after stage I attemperator that provides oper-ational flexibility and avoids two-phase flow in SH tubes. The sprayflow rates between stage I and stage II are varied, keeping the totalspray flow rate at a fixed value, and the exergy destruction ratesare computed following the mass and energy balance equations(Eqs. (1)–(6)) and the exergy evaluation methods described in Sec-tion 3. For a total spray flow rate of 45.4 kg/s, the operating param-eters for the control volume of attemperators, energy and exergyflow rates computed by using thermodynamic property valuesare shown in Fig. 4a (for stage I spray flow rate of 22.1 kg/s andstage II spray flow rate of 23.3 kg/s) and Fig. 4b (for stage I spray

flow rate 30.1 kg/s and stage II spray flow rate of 15.3 kg/s). Understeady operating condition, the exergy destruction rates are com-puted as the difference of exergy inflow and outflow rates for thecontrol volume encompassing each attemperator. For the condi-tions described in Fig. 4a, the exergy destruction rate in stage Iattemperator is 2.951 MW and stage II attemperator is4.135 MW, while for the conditions depicted in Fig. 4b the exergydestruction rates are 3.799 MW and 2.778 MW in the stage I andII, respectively. Thus, the total exergy destructions correspondingto flow parameters shown in Fig. 4a and b are different(7.086 MW and 6.577 MW, respectively), even though the totalspray flow rate is same in both the cases. For total SH spray flowrates of 45.4 kg/s and 56.9 kg/s, the exergy destruction rates inthe attemperator blocks are summarized in Table 2a and b. Thevariation in exergy destructions rates for stage I and stage II attem-perator, and the sum of these two are shown as functions of thestage I SH spray flow rate in Fig. 5a (for a total SH spray flow rateof 45.4 kg/s) and Fig. 5b (for a total SH spray flow rate of 59.6 kg/s).The total exergy destruction values both in Table 2a and b clearlyindicate that with the increase in stage I spray flow rate, the totalexergy destruction reduces for the same total SH spray flow rate.Fig. 5a and b indicate that as the distribution of spray flows varybetween the stages, the change in exergy destruction rates in stageI and stage II differ. With an increase in relative proportion of stageI spray flow rate, the exergy destruction in stage I attemperator in-creases, while that in the stage II attemperator decreases (seeFig. 5). However, these changes are such that the incremental riseof exergy destruction rate in stage I is lower than the decrease inexergy destruction rate in stage II attemperator.

A higher temperature difference between steam and spraywater in stage II attemperator results in larger exergy destructionthan that in stage I for similar flow rates. Thus, exergy destructionis minimized by increasing the stage I spray flow rate for a giventotal SH spray flow rate. However, the fraction of total spray flowin stage I cannot be increased to unity since that might reducethe degree of superheat of the steam at the end of stage I attemper-ator below the permissible limit of 10 �C (the first constraint asmentioned in Sub-section 2.2). Degree of SH values after stage Iattemperator as shown in Table 2a and b are kept above 10 �C. Thisconstraint is imposed in the plant in order to provide the desiredlevel of operational flexibility to eliminate the possibility of two-phase flow through the SH section.

Apart from the exergy benefit, there are a few other practicaladvantages of spraying more in the stage I attemperator. If the lar-ger part of total SH spray is supplied in the stage I attemperator, itleads to higher steam flow rate through the Platen SH section andthus reduces the possibilities of Platen SH tube metal temperatureexcursion.

5.3. Simulation results with different optimal controllers

The actions of the controller are investigated here under cases(a) and (b) as stated in Sub-section 5.1 and the exergy destructions

(a)

(b)Fig. 4. Energy and exergy flows with operation parameters to and from stage I and stage II attemperators for a total spray flow rate 45.4 kg/s. (a) Stage I spray flow rate22.1 kg/s and stage II spray flow rate 23.3 kg/s. (b) Stage I spray flow rate 30.1 kg/s and stage II spray flow rate 15.3 kg/s.

Table 2Exergy destruction rates for different spray flow distributions between stages.

Spray flow rates (kg/s) Exergy destruction rates in attemperator blocks (kW) Degree of SH after St I attemperator

St I St II Total St I St II Total (�C)

(a) Total spray flow rate 45.4 kg/s22.1 23.4 45.4 2951 4135 7086 14.224.3 21.1 45.4 3213 3751 6964 13.025.4 20.0 45.4 3313 3570 6882 12.428.2 17.3 45.4 3589 3106 6695 11.030.1 15.3 45.4 3799 2778 6577 10.1

(b) Total spray flow rate 56.9 kg/s23.6 33.3 56.9 3193 6394 9587 15.024.4 32.5 56.9 3293 6215 9508 14.525.8 31.1 56.9 3422 5945 9367 13.730.6 26.4 56.9 3945 5000 8945 11.132.8 24.2 56.9 4159 4551 8710 10.0

T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245 241

under different control actions during these transient periods areanalyzed in the Sub-section to follow. With the disturbance duringunit load ramp up (increase in flue gas flow rate by 50 kg/s) asmentioned in case (a), the system is studied with different optimal

controllers obtained by varying Q and R matrices. Deviations in MStemperatures during the transient period following the perturba-

2000

3000

4000

5000

6000

7000

8000

22.1 24.3 25.4 28.2 30.1

Stage I Stage II Total

Stage I spray flow rate (kg/s)

Exe

rgy

dest

ruct

ion

rate

(kW

)

(a)

3000

4000

5000

6000

7000

8000

9000

10000

23.6 24.4 25.8 30.6 32.8

Stage I Stage II Total

Stage I spray flow rate (kg/s)

Exe

rgy

dest

ruct

ion

rate

(kW

)

(b) Total Spray Flow = 56.9 kg/s

Total Spray Flow = 45.4 kg/s

Fig. 5. Variation of stage wise and total exergy destruction rates for different stage Ispray flow rates, when the total spray flow rate remains (a) 45.4 kg/s, and (b)56.9 kg/s. Stage I spray flow is capped so that the degree of superheat after Stage Idoes not drop below 10 �C.

Fig. 6. (a) Temperature and (b) spray flow rate deviations for Q = 1 andR = 1, 2, . . ., 10, due to control action to counter the effect of increase in gas flowrate by 50 kg/s.

242 T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245

tion are plotted in Fig. 6a for Q = 1. The value of R is varied from 1to 10, implying that the controller assigns progressively moreweightage (with varying R) to the regulation of ~UðtÞ. Physically, alarger value of R implies that the minimization of additional sprayflow is given higher priority over the excursion of MS temperatureby the optimal controller. The spray flow rate deviations during thetransient period are plotted in Fig. 6b for Q = 1 and R = 1, 2, . . ., 10.MS temperature deviations curves shown in Fig. 6a indicate that asthe optimal controllers progressively assign more weightage tominimization of additional spray, (i.e., high value of R) it leads tohigher temperature excursion (the peak temperature deviationfor R = 10 is �2.9 �C as opposed to 2.6 �C for R = 1). This is due tothe reason that for higher values of R, the increase in spray flowrates are low as shown in Fig. 6b (for R = 10, peak deviation inspray flow rate is 1.23 kg/s while for R = 1, it is 3.69 kg/s).

Fig. 6a indicates that the steady-state temperature deviationsare �1.4 �C for R = 1 and �1.9 �C for R = 10. This happens whenthe spray flow rates also stabilizes at higher than the initial steadyvalues, i.e., at 3.67 kg/s for R = 1 and 1.23 kg/s for R = 10 (Fig. 6b). Ahigh value of R results in lower consumption of spray water, andmay therefore be energetically favorable (since lower spray flowleads to reduction in heat rate), but it does not quite serve the pur-pose of limiting steam temperature excursion. Therefore, keepingQ = 1 and varying values of R can not control the effect of a largerdegree of perturbations and it is not a suitable option for thecontroller.

Next, the system is studied for R = 1 and Q = 1, 2, . . ., 10 underthe same disturbance, i.e., an increase of flue gas flow rate by50 kg/s. The MS temperatures and the spray flow rate deviationsduring the transient period are shown in Fig. 7a and b respectively.MS temperature deviation curves in Fig. 7a indicate that the tem-perature excursion reduces with higher value of Q, but the re-

sponse time remains almost unchanged. The peak temperaturedeviation is found to be 2.6 �C for Q = 1 and 2.0 �C for Q = 10. Thelower temperature excursion (for high value of Q) is attributed tohigher rise in spray flow rate, which is shown in Fig. 7b. The max-imum deviation in spray flow rate for Q = 1 is 3.69 kg/s while thatfor Q = 10 is 3.86 kg/s. Although the transient peak temperatures atR = 1 are different for different values of Q, the steady state tem-peratures are the same (�1.4 �C higher than the steady tempera-ture before perturbation). This trend can be explained from thefact that the deviation in spray flow rates for R = 1 and differentvalues of Q also converge to the same steady value of �3.68 kg/safter the transients are over (see Fig. 7b).

Figs. 6 and 7 show that the LQR controller produces expectedresults under a disturbance in the form of a sudden load ramp-up. Spray flow is minimized when controller assigns moreweightage to spray minimization with simultaneous increase insteady-state excursion of MS temperature. Since exergy has notbeen expressed as a state of the system, the LQR controller cannotbe directly used as an optimal controller that minimizes exergydestruction against excursion of MS temperature and spray flow.Thus, it is necessary to separately analyze the performance of theLQR controller following an exergy destruction consideration,which is presented in the following Sub-section.

5.4. Controller performance and incremental exergy destruction duringtransient states

Any perturbation in the boiler heat transfer from hot flue gas tosteam in SH section takes place at varying temperatures, and leads

Fig. 7. (a) Temperature and (b) spray flow rate deviations for R = 1 andQ = 1, 2, . . ., 10, due to control action to counter the effect of increase in gas flowrate by 50 kg/s.

Fig. 8. Incremental exergy destruction during the transient period, when the gasflow rate increases by 50 kg/s.

Fig. 9. Incremental exergy destruction during the transient period, when the gasflow rate reduces by 111 kg/s.

T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245 243

to a change in SH spray flow rate to arrest MS temperature over-shoot or undershoot. The transient disturbance of MS temperature,and steam, gas and spray water flow rates impact the additionalexergy destruction rates during the transient period. After the sim-ulation results are obtained, the incremental exergy destructions(I ¼

R fb

_Idt) in the transient period are computed by following themethodologies explained in the Section 3. The incremental exergydestruction values for several options of the controller parametersare evaluated from the pre-disturbance steady state to the post-

disturbance steady condition. The value of additional exergydestructions (I) during the transient period computed for R = 1and Q = 1, 2, . . ., 10 are plotted for the two cases as stated in Sub-section 5.1: Fig. 8 for case (a) (i.e., when gas the flow rate increasesby 50 kg/s) and Fig. 9 for case (b) (i.e., when gas flow rate reducesby 111 kg/s).

Fig. 8 shows that in case (a), for R = 1, the incremental exergydestruction in transient period rises with the increase in value ofQ (e.g., for Q = 1, I = 237.6 MJ while for Q = 10, I = 264.2 MJ). Thishappens mainly due to the fact that higher Q results in faster con-trol action (rapid rise in spray flow rate to counter the effect of risein gas flow rate, see Fig. 7b). Thus, although the temperature over-shoot is better managed at high Q, larger spray flow leads to higherincremental exergy destruction. On the other hand, in case (b), alarger value of Q implies that the controller action is faster, i.e.,the spray flow rate decreases at a higher rate to counter the fallof SH steam temperature.

5.5. Comparison of transient response with actual plant data

Comparison is made with the transient response of SH steamtemperature and spray flow rate of the actual plant for a similardisturbance viz., a step rise in flue gas rate. The pulverized coalfired B&W design boiler (manufactured by Ansaldo Componenti,Italy) of this 500 MWe sub-critical unit is equipped with op-posed-wall firing system. It has four elevations of coal burners with6 burners in each elevation from each pulverizer (burners of pul-verizers A, B, C and D, sequentially from bottom to top in the frontwater wall of the furnace and burners of pulverizers E, F, G and H,sequentially from bottom to top in the rear wall, i.e., 24 burners inthe front wall and 24 burners in the rear wall). While taking anadditional pulverizer in service, the impact on MS temperaturedeviation and change in spray flow rate differs significantlydepending on which pulverizer is put in service. A 50 kg/s step riseof flue gas flow rate is observed when the pulverizer G is taken inservice.

A comparison of simulation results for a representative case ofR = 1, Q = 5 and plant data (deviation in spray flow rates and thedeviation in MS temperatures) during the transient conditionsdue to taking pulverizer G in service are shown in Fig. 10a and b.As can be seen from the figures, the transient response of MS tem-perature and spray flow rate of the actual plant match closely withthe simulated response using the model, though the long-term re-sponse of the actual plant differs from the model prediction. This isdue to the fact that in the actual plant, the control action is not only

0.0

1.0

2.0

3.0

4.0

5.0

0 50 100 150 200

Plant data

R=1, Q=5

t (s)

Δ

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 50 100 150 200

Plant data

R=1, Q=5

t (s)

Δ

R = 1, Q =5 R = 1, Q =5

(a) (b)

Fig. 10. Comparison of the initial 200 s of transient responses in (a) SH steam temperature and (b) spray flow rate deviations, as computed from the model for R = 1 and Q = 5,with the actual plant data during a control action to counter the effect of increase in flue gas flow rate by 50 kg/s.

244 T.K. Ray et al. / Energy Conversion and Management 66 (2013) 234–245

achieved by the spray (which forms the immediate action), but alsothrough several other factors (e.g., through the second pass gasbiasing dampers control in this case). The maximum temperaturedeviation in the plant is 2.4 �C, which is higher than that for thesimulation (2.0 �C), when the weighing matrices are chosen asR = 1, Q = 5 (see Fig. 10a). Higher temperature excursion in the ac-tual plant is observed in spite of higher peak value in spray flowrate deviation (4.6 kg/s, in contrast to 3.8 kg/s in the model). Thisis mainly attributed to the slightly delayed increase in spray flowrate (see Fig. 10b), Also, a higher peak in spray flow rate deviationin the actual plant results in relatively sharp dip in MS temperaturedeviation immediately after the occurrence of peak temperature.Thus, the model offers a lower peak temperature deviation and alower maximum spray flow rate deviation. This would result in re-duced exergy destruction during the transient periods from themodel. The additional exergy destruction due to control action ofthe proposed model for R = 1, Q = 5 as shown in Fig. 8 is259.3 MJ, whereas that calculated for the actual plant during thetransient period is 262.6 MJ.

5.6. Formulation of guidelines for fine tuning the controller

The matrices Q and R for the control action are chosen so thatthe total exergy destruction over the duration of the transient isminimized and the temperature deviation remain within the spec-ified limit. A quicker control action and rapid reduction in sprayflow rate leads to less incremental exergy destruction during thetransient period, which is shown in Fig. 9 (for Q = 1, I = 398.0 MJand for, Q = 10, I = 285.3 MJ). For case (a), as shown in Fig. 8, it ap-pears that the low value of Q is exergetically beneficial but themaximum SH temperature deviation (DT) rises significantly (asshown in Fig. 7a for Q below 4. Similarly, for case (b), the reductionin exergy destruction is marginal for Q beyond 6 (see Fig. 9). Be-sides, a higher value of Q entails a faster decrease in spray flow rateunder such a transient, and may result in practical difficulties of SHtemperature control. For the case of a sudden reduction in the fluegas flow rate in the boiler, the proportion of radiative heat transferincreases in comparison to the convective heat transfer. This oftenleads to steep SH temperature rise following the initial undershootright after the gas flow rate reduction over the SH. The condition isexacerbated when the attemperation controller reduces the sprayflow rapidly during this transient, viz., when large Q value is usedfor the controller. Thus, a more rapid rate of decrease in spray flowrate (for higher Q values) poses problem in SH temperature man-agement. Figs. 8 and 9 provides a qualitative basis for choosingthe appropriate values of the weighing matrices for the controllerfrom the exergy point of view, over and above the practical opera-tion issues for proper tuning of controller.

The analysis highlights the implication of the SH spray controlstrategies on the exergy destruction. The saving in exergy destruc-

tion in an optimized control action during the plant operation isequivalent to a saving the equivalent amount of electrical energy[7]. For a given tariff of electricity this can be directly linked withcost. However, a more accurate economic evaluation needs to bedone in future to compare the cost of the saved exergy againstthe cost of implementing the control action following the recom-mendations of this paper.

6. Conclusions

A MATLAB-SIMULINK-based model developed and used to de-vise optimal control strategies following LQR approach. Actionsof the optimal controller under transient disturbances are investi-gated for different values of the controller parameters. The processparameter deviations and exergy destructions are evaluated in or-der to guide the fine-tuning of SH spray controller in the operatingplant to minimize the exergy destructions during the transientstates while keeping the steam temperature excursions accuratelywithin small deviations around their set values.

While traditional First-Law analysis fails to predict any devia-tion in the performance for different proportion of stage I and stageII spray flows in a two stage SH attemperator, exergy analyses con-ducted with real time plant operation data unambiguously portraythat the exergy destruction rates vary significantly with changes inthe distributions of spray flows even if the total spray flow rate re-mains unaltered. By prudently varying the distributions (increas-ing spray flow rate in stage I subject to maintaining a minimumdegree of superheat at stage I attemperator outlet and reducingthe spray flow rate in stage II by the same amount), exergy destruc-tions in SH attemperators have been minimized.

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