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Steamturbinemodel (3)

Steamturbinemodel (3)
Steamturbinemodel (3)

Steam turbine model

Ali Chaibakhsh,Ali Ghaffari *

Department of Mechanical Engineering,K.N.Toosi University of Technology,Pardis Street,Vanak Square,P.O.Box 19395-1999,Tehran,Iran

a r t i c l e i n f o Article history:

Received 9November 2007

Received in revised form 20May 2008Accepted 20May 2008

Available online 6June 2008

Keywords:Power plant Steam turbine

Mathematical model Genetic algorithm

Semi-empirical relations Experimental data

a b s t r a c t

In order to characterize the transient dynamics of steam turbines subsections,in this paper,nonlinear mathematical models are ?rst developed based on the energy balance,thermo-dynamic principles and semi-empirical equations.Then,the related parameters of devel-oped models are either determined by empirical relations or they are adjusted by applying genetic algorithms (GA)based on experimental data obtained from a complete set of ?eld experiments.In the intermediate and low-pressure turbines where,in the sub-cooled regions,steam variables deviate from prefect gas behavior,the thermodynamic characteristics are highly dependent on pressure and temperature of each region.Thus,nonlinear functions are developed to evaluate speci?c enthalpy and speci?c entropy at these stages of turbines.The parameters of proposed functions are individually adjusted for the operational range of each subsection by using genetic https://www.doczj.com/doc/088486612.html,parison between the responses of the overall turbine-generator model and the response of real plant indicates the accuracy and performance of the proposed models over wide range of operations.The simulation results show the validation of the developed model in term of more accurate and less deviation between the responses of the models and real system where errors of the proposed functions are less than 0.1%and the modeling error is less than 0.3%.

ó2008Elsevier B.V.All rights reserved.

1.Introduction

Over the past 100years,the steam turbines have been widely employed to power generating due to their ef?ciencies and costs.With respect to the capacity,application and desired performance,a different level of complexity is offered for the structure of steam turbines.For power plant applications,steam turbines generally have a complex feature and consist of multistage steam expansion to increase the thermal ef?ciency.It is always more dif?cult to predict the effects of proposed control system on the plant due to complexity of turbine structure.Therefore,developing nonlinear analytical models is nec-essary in order to study the turbine transient dynamics.These models can be used for control system design synthesis,per-forming real-time simulations and monitoring the desired states [1].Thus,no mathematical model can exactly describe such complicated processes and always there are inaccuracy in developed models due to un-modeled dynamics and parametric uncertainties [2,3].

A vast collection of models is developed for long-term dynamics of steam turbines [4–11].In many cases,the turbine models are such simpli?ed that they only map input variables to outputs,where many intermediate variables are omitted [12].The lack of accuracy in simpli?ed models emerges many dif?culties in control strategies and often,a satisfactory degree of precision is required to improve the overall control performance [13].

Identi?cation techniques are widely used to develop mathematical models based on the measured data obtained from real system performance in power plant applications where the developed models always comprise reasonable complexities

1569-190X/$-see front matter ó2008Elsevier B.V.All rights reserved.doi:10.1016/j.simpat.2008.05.017

*Corresponding author.Tel.:+982188674841;fax:+982188674748.E-mail address:ghaffari@kntu.ac.ir (A.Ghaffari).

Simulation Modelling Practice and Theory 16(2008)1145–1162

Contents lists available at ScienceDirect

Simulation Modelling Practice and Theory

j o ur na l h om e pa ge :w w w.e ls e v ie r.c o m/lo c at e/s im pa t

that describe the system well in speci?c operating conditions [14–18].Moreover,in large systems such as power plants,breaking major control loops when systems run at normal operating load conditions may put them in dangerous situations.Consequently,the system model should be developed by performing closed loop identi?cation approaches.System identi-?cation during normal operation without any external excitation or disruption would be an ideal target,but in many cases,using operating data for identi?cation faces limitations and external excitation is required [19–21].Assuming that paramet-ric models are available,in this case,using soft computing methods would be helpful in order to adjust model parameters over full range of input–output operational data.

Genetic algorithms (GA)have outstanding advantages over the conventional optimization methods,which allow them to seek globally for the optimal solution.It causes that a complete system model is not required and it will be possible to ?nd

Nomenclature C speci?c heat (kJ/kg K)

D droop characteristics (N m/rad/s)h speci?c enthalpy (kJ/kg) h absolute enthalpy (kJ/kmol)J momentum of inertia (kg m 2)k index of expansion _m mass ?ow (kg/s)

m molecular weight (kg)M inertia constant (kg m 2/s)p pressure (MPa)P power (MW)

Q heat transferred (MJ)q ?ow (kg/s)

s entropy (kJ/kg K)t time (s)

T temperature (°C)Tr torque (N m)

U machine excitation voltage (V)V terminal voltage (V)v speci?c volume (m 3/kg)W power (MW)

x

D-axis synchronous reactance (X )

Greek letters

a

steam quality

d rotor angl

e (rad)g ef?ciently q speci?c density (kg/m 3)s time constant (s)x frequency (rad/s)

Subscripts e electrical ex extraction f liquid phase fuel fuel g vapor phase in input m mechanical out output p constant pressure s saturation spray spray v constant volume w water 0standard condition HP high pressure IP intermediate pressure LP low-pressure

1146 A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–1162

parameters of the model with nonlinearities and complicated structures[22,23].In the recent years,genetic algorithms are investigated as potential solutions to obtain good estimation of the model parameters and are widely used as an optimiza-tion method for training and adaptation approaches[24–30].

In this paper,mathematical models are?rst developed for analysis of transient response of steam turbines subsections based on the energy balance,thermodynamic state conversion and semi-empirical equations.Then,the related parameters are either determined by empirical relations obtained from experimental data or they are adjusted by applying genetic algo-rithms.In the intermediate and low-pressure turbines where,in the sub-cooled regions,steam variables deviate from prefect gas behavior,the thermodynamic characteristics are highly dependent on pressure and temperature of each region.Thus nonlinear functions are developed to evaluate speci?c enthalpy and speci?c entropy at these stages of turbines.The param-eters of proposed functions are individually adjusted for the operational range of each subsection by using genetic algo-rithms as an optimization approach.Finally,the responses of the turbine and generator models are compared with the responses of the real plant in order to validate the accuracy and performance of the models over different operation conditions.

In the next section,a brief description of the plant turbine is presented.It consists of a general view of the steam turbine and its subsystems including their inputs and outputs.It follows by the analytical model development and the training pro-cedure of proposed models based on the experimental data.The next section presents the simulation results of this work by comparing the responses of the proposed model with the actual plant.The last section is the conclusion and suggestions for future studies.

2.System description

A steam turbine of a440MW power plant with once-through Benson type boiler is considered for the modeling approach. The steam turbine comprises high,intermediate and low-pressure sections.In addition,the system includes steam extrac-tions,feedwater heaters,moisture separators,and the related actuators.The turbine con?guration and steam conditions at extractions are shown in Fig.1.

The high-pressure superheated steam of the turbine is responsible for energy?ow and conversion results power gener-ating in the turbine stages.The superheated steam at535°C and18.6MPa pressure from main steam header is the input to the high-pressure(HP)turbine.The input steam pressure drops about0.5MPa by passing through the turbine chest system. The entered steam expands in the high-pressure turbine and is discharged into the cold reheater line.At the full load con-ditions,the output temperature and pressure of the high-pressure turbine is351°C and5.37MPa,respectively.The cold steam passes through moisture separator to become dry.The extracted moisture goes to HP heater and the cold steam for reheating is sent to reheat sections.The reheater consists of two sections and a de-superheating section is considered between them for controlling the outlet steam temperature.

The reheated steam at535°C and with4.83MPa pressure is fed to intermediate pressure(IP)turbine.Exhaust steam from IP-turbine for‘the last stage expansion is fed into the low-pressure(LP)turbine.The input temperature and pressure of the low-pressure turbine is289.7°C and0.83MPa,respectively.Extracted steam from?rst and second extractions of IP is sent to HP heater and de-aerator.Also,extracted steam from last IP and LP extractions are used for feedwater heating in a train of low-pressure heaters.The very low-pressure steam from the last extraction goes to main condenser to become cool and be used in generation loop again.

3.Turbine model development

The behavior of the subsystems can be captured in terms of the mass and energy conservation equations,semi-empirical relations and thermodynamic state conservation.The system dynamic is represented by a number of lumped models for each

subsections of turbine.There are many dynamic models for individual components,which are simple empirical

relations

Fig.1.Steam turbine con?guration and extraction.

A.Chaibakhsh,A.Ghaffari/Simulation Modelling Practice and Theory16(2008)1145–11621147

between system variables with a limited number of parameters and can be validated for the steam turbine by using real sys-tem responses.In addition,an optimization approach based on genetic algorithm is performed to estimate the unknown parameters of models with more complex structure based on experimental data.With the respect to model complexity,a suit-able ?tness function and optimization parameters are chosen for training process,which are presented in Appendix A .The models training process is performed by joining MATLAB Genetic Algorithm Toolbox and MATLAB Simulink .It makes it possible model training be performed on-line or based on recorded data in simulation space (Appendix B ).3.1.HP-turbine model

The high-pressure steam enters the turbine through a stage nozzle designed to increase its velocity.The pressure drop produced at the inlet nozzle of the turbine limits the mass ?ow through the turbine.A relationship between mass ?ow and the pressure drop across the HP turbine was developed by Stodola in 1927[31].The relationship was later modi?ed to include the effect of inlet temperature as follows:

_m

in ?K

??????T in

p ????????????????????

p 2in àp 2

out

q e1T

where K is a constant that can be obtained by the data taken from the turbine responses.Let k be de?ned as follows:

k ?

????????????????????p 2in àp 2

out

T in

s e2T

By plotting k via inlet mass ?ow rate based on the experimental data,the slope of linear ?tting is captured as K =520(Fig.2).Generally,Eq.(1)has a suf?cient accuracy where water steam is the working ?uid.A comparison between the model re-sponse and the experimental shown in Fig.3indicates the accuracy of the de?ned

constant.

Fig.2.Mass ?ow rate versos k

.

Fig.3.Response of pressure–mass ?ow model.

1148 A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–1162

The input output pressure relation for HP turbine based on experimental data is shown in Fig.4.It shows a quite linear relation with the slope of 0.29475.Noting that the time constant for HP turbines are normally between 0.1and 0.4s,here the time constant is measured to be about 0.4s and therefore the transfer function of the input–output pressure is

p out p in

?

0:29475

0:4s t1e3T

The time response of the proposed transfer function is shown in Fig.5.

To develop the dynamic model of HP turbine,the pressure,mass ?ow rate and temperature of steam at input and output of each section is required.The input and output relations for steam pressure and steam ?ow rate are de?ned in previous section.The steam temperature at turbine output can be captured in the terms of entered steam pressure and temperature.By assuming that the steam expansion in HP-turbine is an adiabatic and isentropic process,it is simple to estimate the steam temperature at discharge of HP turbine by using ideal gas pressure–temperature relation.

T out T in ?p out p in

k à1

k eT

e4T

where k ?C p =C v is the polytrophic expansion factor.

The energy equation for adiabatic expansion,which relates the power output to steam energy declining by passing through the HP turbine,is as follows:

W HP ?g HP á_m

in eh in àh out T?g HP áC p á_m in eT in àT out Te5

T

Fig.4.Pressure ratio of the HP-turbine cylinder input and

output.

Fig.5.Responses of pressure model.

A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–11621149

It is possible to de?ne the steam speci?c heat as a function of pressure and/or temperature.Here,for more simpli?cation,water steam is considered ideal gas.In addition,the turbine ef?ciency can be expressed as a function of the ratio of blade tip velocity to theoretical steam velocity.In this paper,turbine ef?ciency is considered as a constant value.Then,

W HP ?g HP áC p á_m

in T in àT in

p out in k à1eT

!?g HP áC p á_m in áeT in t273:15T1àp out in

k à1eT

!

e6T

The nonlinear model proposed for HP turbine is a parametric model with unknown parameters,which are associated with

ef?ciency and speci?c heat.These parameters can be de?ned by performing a training approach over a collection of input–output operational data.The model parameters adjustment is executed through a set of 650points of data and for transient and steady state conditions in the range of operation between 154and 440MW of load.The error E is given by the mean value of squared difference between the target output y *and model output y as follows:

E ?

1X N j ?1

ey ?j ày j T2

e7T

where N is the number of entries used for training process.

The optimized value for speci?c heat,C p ,in order to reach the best performance at different load conditions,is obtained 2.1581and consequently,the polytrophic expansion factor,k ,be equal to 1.2718.The ef?ciency of a well-designed HP tur-bine is about 85–90%.A fair comparison between the experimental data and the simulation results shows that the obtained HP turbine ef?ciency equal to 89.31%is good enough to ?t model responses on the real system responses.The proposed model for HP turbine is presented in Fig.6,where K 1?K ?520and K 2?C p ?g HP =1000?1:921?10à3.

The outlet steam from the HP turbine passes through the moisture separator to become dry.There are obvious advantages in inclusion of steam reheating and moisture separation in terms of improving low pressure exhaust wetness and need for less steam reheating.In this section,a considerable fraction of steam wetness is extracted which supplies the required steam for feedwater heating purpose at the HP heaters.The outlet ?ow from moisture separation is captured as follows:

s

d q

d t

?e1àb T_m

in àq e8T

where b is the fraction of moisture in output ?ow.In the technical documents,it is declared that the amount of liquid phase extracted as moisture form steam mixture is approximately 10%of total steam ?ow entered to HP turbine.3.2.IP and LP turbines model

The intermediate and low-pressure turbines have more complicated structure in where multiple extractions are em-ployed in order to increase the thermal ef?ciency of turbine.The steam pressure consecutively drops across the turbine stages.The condensation effect and steam conditions at extraction stages have considerable in?uences on the turbine per-formance and generated power.In this case,developing mathematical models,which are capable to evaluate the released energy from steam expansion in turbine stages,is recommended.At turbine extraction stages,where in the sub-cooled re-gions,steam variables deviate from prefect gas behavior and the thermodynamic characteristics are highly dependent on pressures and temperature of each region.Therefore,developing nonlinear functions to evaluate speci?c enthalpy and spe-ci?c entropy at these stages of turbines is necessary.The steam thermodynamic properties can be estimated in term of tem-perature and pressure as two independent variables.A variety of functions to give approximations of steam/water properties is presented,which are widely used in nuclear power plant applications [32–36]

.

Fig.6.HP-turbine model (B =273.15,K 1=520,K 2=1.927?10à3,f (u )=[u (1)/u (2)]0.2137).

1150 A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–1162

In1988,very simple formulations were presented by Garland and Hand to estimate the light water thermodynamic prop-erties for thermal-hydraulic systems analysis.In the proposed functions,saturation values of steam are used as the dominant terms in the approximation expressions.This causes that these functions have considerable accuracy at/or near saturation conditions.However,these functions are extended to be quite accurate even in the sub-cooled and superheated regions [37].The approximation functions for the thermodynamic properties in sub-cooled conditions are presented as follows:

Fep;TT?F sep seTTTtReTT?epàp sTe9T

where p

s is the steam pressure at saturation conditions.The proposed equations to estimate steam saturation pressure,p

s

,as

a function of temperature are listed in Appendix C.In addition,the approximation functions for the thermodynamic prop-erties in superheated conditions are presented as follows:

Fep;TT?F gepTtRep;TT?eTàT sTe10Twhere T s is the steam saturation temperature.The equations to evaluate steam saturation temperature,T s,as a function of steam pressure are presented in Appendix D.It is noted that,these functions are not able to cover the entire range of pressure changes and therefore the pressure range is divided into many sub-ranges.The proposed functions are quite suitable for esti-mating the water/steam thermodynamic properties;however,these functions are tuned for a given range from0.085MPa to 21.3MPa and they have not adequate accuracy for very low-pressure steam particularly for the extractions conditions.In this paper,it is recommended that these functions be tuned individually for each input and output and at desired operational ranges.It should be mentioned that pressure changes have signi?cant effects on the steam parameters and therefore,it is focused on adjusting the?rst term of functions,which depend on pressure and the functions ReTTand ReP;TTare considered the same as presented by Garland and Hand.

The working?uid at different turbine stages can be single or two phases.In this condition,it should be assumed that both phases of steam mixtures are in thermodynamic equilibrium and liquid and vapor phases are two separated phases.The steam conditions at each section are presented in Table1.The proposed functions for speci?c enthalpy for liquid phase and speci?c entropy in both liquid and vapor phases are de?ned by three parameters as follows:

F?aepTbtc

where three parameters a,b and c are adjusted for four different steam conditions at35,50,75and100%of load.In addition, the proposed function for speci?c enthalpy in vapor phase is de?ned by three parameters and one constant as follows: F?aepàdT2tbepàdTtc

Here,the constant d can be chosen manually with respect to pressure variation ranges.The error E is given by the mean value of absolute difference between the target output y*and model output y as follows:

E?1

N

X N

j?1

j y?

j

ày j je11T

where N is the number of entries used for training process.

3.2.1.Speci?c enthalpy,liquid phase

The following function is presented for estimating speci?c enthalpy of water in liquid phase.

hep;TT?h fep seTTTt1:4à

169

369àT

!

epàp sTe12T

As seen in Table1,the steam condition for extractions5,6and7are in two-phase region where it can assume that p%p

s

.In this condition,the speci?c enthalpy of steam for their ranges can be de?ned as a function of steam pressure.The functions listed below estimate the speci?c enthalpy of water in liquid phase,h(kJ/kg).

Table1

Steam condition at turbine extractions

Extraction No.Pressure(saturation temperature)Temperature(°C)Steam condition IP turbine1 2.945(233.91)456.6One phase

2 1.466(197.58)359One phase

30.830(171.85)289.1One phase

LP turbine40.301(133.63)182.7One phase

50.130(105.80)111.2Transient

60.045977.5Two phases

70.006838.2Two phases

A.Chaibakhsh,A.Ghaffari/Simulation Modelling Practice and Theory16(2008)1145–11621151

h f?44:12782275?e1000pT0:60497549t19:300600270:0038MPa

h f?194:57965086?e100pT0:33190979t1:732494420:0180MPa

h f?258:51219036?e100pT0:17513608t11:833935260:0683MPa

e13T

3.2.2.Speci?c enthalpy,vapor phase

The following function is presented for estimating speci?c enthalpy of water/steam in vapor phase.

hep;TT?h gepTà

4:5p

??????????????????????????????????????????????????

7:4529?10à0:6T3àp2

qt0:28?eà0:008eTà162Tà100Tà2:225

2

64

3

75eTàT

s

Te14T

It should be noted that in two-phase region,it could assume that T%T s.Therefore,speci?c enthalpy can be de?ned as a function of steam pressure.The functions listed below estimate the speci?c enthalpy of water/steam in vapor phase for the pressure range in3.8–4.83MPa.

h g?à0:48465587?e1000pà5T2t6:47301169?e1000pà5Tt2560:912384520:0038MPa

h g?à1:82709298?e100pà2T2t17:40365447?e100pà2Tt2606:6808212850:0180MPa

h g?0:50205745?e100pà12T2t6:64525736?e100pà12Tt2679:806093220:0683MPa

h g?à2:07829396?e10pà3T2t25:01448122?e10pà3Tt2704:849205570:195MPa

h g?à0:49047808?e10pà8T2t10:48902998?e10pà8Tt2740:05764510:432MPa

h g?à0:21681424?e10pà14:5T2t6:13049409?e10pà14:5Tt2771:189012880:753MPa

h g?à0:08217055?e10pà29T2t3:07429644?e10pà29Tt2816:820242341:471MPa

h g?à0:11673499?e10pà48T2t0:13784178?e10pà48Tt2862:433394722:388MPa

e15T3.2.3.Speci?c entropy,liquid phase

The optimized functions for estimating speci?c entropy of water/steam in liquid phase based on steam pressure where p%p

s

are as follows:

s f?0:27490714?e1000pT0:60265499à0:228650890:0038MPa

s f?1:26673390?e100pT0:17853959à0:617031220:0180MPa

s f?0:92671704?e100pT0:14323925à0:036604770:0683MPa

e16T

3.2.

4.Speci?c entropy,vapor phase

In addition,the following function is presented for estimating speci?c entropy of water/steam in vapor phase.

sep;TT?s gepTà

0:004p1:2

????????????????????????????????????????????????????????????

3:025?101:1eTt46T5àp2

qt0:00006

???pà4:125?10à6?Tt0:0053

2

64

3

75eTàT

s

Te17T

The optimized functions to evaluate the speci?c entropy water/steam of phase vapor in the pressure range between3.8kPa and0.301MPa.

s g?8:83064734à0:12141594?e1000pT0:779328060:0038MPa

s g?9:0863247à0:96869236?e100pT0:261392470:0180MPa

s g?8:36610497à0:45436108?e100pT0:342467780:0683MPa

s g?7:42364087à0:10328045?e10pT1:278279230:195MPa

e18T

The proposed functions are depending on pressure and temperature of the steam and these variables are necessary to be de?ned at deferent operational conditions.The steam temperature at each extraction stage is expressed as a function of en-tered steam temperature.The proposed transfer functions for steam temperature at extraction stages are presented in Table 2.It is possible to calculate the steam pressure at extractions as a function of the mass?ow through turbine stages.Here,it is recommended that the steam pressure be de?ned as a function of steam pressure entered to turbine.As shown in Fig.7,the pressure drops across the turbine stages are approximately linear and can be de?ned by?rst order transfer functions.The proposed transfer functions for steam pressure at extraction stages are presented in Table2.In addition,the mass?ow rate through the turbine stages is sequentially decreased as by subtracting extracted steam?ow.The extraction?ow at each sec-tion can be de?ned as a function of entered steam?ow to turbine.As shown in Fig.8,the input–output steam?ow ratio at 1152 A.Chaibakhsh,A.Ghaffari/Simulation Modelling Practice and Theory16(2008)1145–1162

Table 2

Fig.7.Steam pressure at extractions.

Fig.8.The steam ?ow rate at extractions.

A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–1162

1153

different load conditions,except 2nd extraction,are linear.However,considering a linear function of steam ?ow rate is good enough to ?t the model response on the real experimental data.

For the two-phase region,the enthalpy of the extracted steam is depending on its quality.By considering expansion of steam in extraction chamber is an adiabatic process;the steam quality can be captured base on the steam entropy as follows:

s 0?s f tx ás fg )x ?

s 0às f s fg

e19T

Then,

h ?h f tx áh fg e20T

The steam entropy at two-phase region (at ?fth,sixth and seventh extractions)is considered to be equal with steam entropy at fourth extraction (one-phase region).The proposed model for two-phase region is presented in Fig.9.The thermodynamic cycle for the steam turbine with seven extraction stages is shown in Fig.10.By considering steam expansion at turbine

stages

Fig.9.Enthalpy model for two-phase

region.

Fig.10.Power plant cycle T –S diagram.

1154 A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–1162

be an ideal process,the energy equations for steam expansion in turbine,which relates the power output to steam energy declining across turbine stages can be captured.Therefore,the work done in IP turbine can be captured as follows: W0

IP

?_m IPeh IPàh ex1Tte_m IPà_m ex1Teh ex1àh ex2Tte_m IPà_m ex1à_m ex2Teh ex2àh ex3Te21TNow,the performance index can be considered for IP turbine.

W IP?g IP W0IPe22TThe LP turbine consists of four extraction levels.The work done in the LP turbine can be captured as follows:

W0

LP

?_m LPeh LPàh ex4Tte_m LPà_m ex4Teh ex4àh ex5Tte_m LPà_m ex4à_m ex5Teh ex5àh ex6T

te_m LPà_m ex4à_m ex5à_m ex6Teh ex6àh ex7T

e23Twhere_m LP?_m IPà_m ex1à_m ex2à_m ex3then,

W LP?g LP W0LPe24TThe optimal values for ef?ciencies of IP and LP turbines are obtained83.12%and82.84%,respectively,which are?tting tur-bine model responses on the real system responses.The developed models for IP and LP turbines are presented in Fig.11.The overall generated mechanical power can be captured by summation of generated power in turbine stages as follows: P m?W HPtW IPtW LPe25T

3.3.Reheater model

Reheater section is a very large heat exchanger,which has signi?cant thermal capacity and steam mass storage.The reheater dynamics increase nonlinearity and time delay of the turbine and should take into account as a part of turbine mod-el.We have developed accurate Mathematical models for subsystems of a once through Benson type boiler based on the thermodynamics principles and energy balance,which are presented in[29,30].The parameters of these models are deter-mined either from constructional data such as fuel and water steam speci?cation,or by applying genetic algorithm tech-niques on the experimental data.The proposed equations for the temperature model is as follows:

d T out

d t

?K2eK1_m fuelt_m ineT inàT outtB1TtB2Te26TIn this model,the steam quality has signi?cant effects on output temperature and should be considered in related equations. The transfer function for fuel?ow rate and steam quality is as follows:

a _m fuel ?

9:45039eà6

20st1

e27T

A modi?ed version of the temperature model for the reheater sections is presented in Fig.12.According to the mass accu-mulation effects and by considering that the pressure loss due to change in?ow velocity is prevailing in the steam volume, the?ow-pressure model is presented as follows:

d p d t ?

p

sám

v

e_m inà_m outTe28T

A model for the mass?ow responding to steam pressure changes is proposed by Borsi[38].The swing of main steam?ow strictly relies on the change of steam pressure as follows:

d_m out

?

_m

out0

in0out0

d p

e29T

In Fig.13,the?ow-pressure model is presented.Generally,in power plants,the turbine inlet?ow is controlled by a governor or control valves to response to the grid frequency.Therefore,when this valve is acting,there is an interaction between steam pressure and?ow.When the control valve opening is completed,the pressure?uctuation is removed and the swing of steam?ow tends to zero.The adjusted parameters of the developed models are presented in Table3.

The reheater temperatures must be kept constant at speci?c temperature.The spray attemperators is implemented be-tween reheater sections to control outlet temperature.The attemperator has a relatively small volume and then its mass storage is negligible.In addition,it is considered that there is no pressure drop in this section.Then,the inlet temperature of the second reheater,T out is governed by the following equation[39].

D T out?1

C p

D h out?

e h inà h outT

C p _m out

D_m outt

_m

in

_m

out

D T inà

e h inà h sprayT

C p _m out

D_m spraye30T

where_m in is inlet steam?ow,h in is speci?c enthalpy of inlet steam h spray is speci?c enthalpy of water spray.The con?gura-tion of reheater section is presented in Fig.14.

A.Chaibakhsh,A.Ghaffari/Simulation Modelling Practice and Theory16(2008)1145–11621155

3.4.Generator model

The turbine-generator speed is described by the equation of motion of the machine rotor,which relates the system inertia to deference of the mechanical and electrical torque on the

rotor.

Fig.11.IP and LP-turbine model.

1156 A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–1162

Fig.12.Temperature model for the reheater

section.

Fig.13.Flow-pressure model K 3?P 0

s ám v ;K 4?_m out02eP in0

àP out0T

.Table 3

Parameters for reheater model

B 1

B 2

K 1

K 2

K 3K 4

Reheater a 0.1706 6.2893 2.41e à3

1.06e à20.012836.0257Reheater b

0.1315

15.723

3.77e à4

1.06e à2

0.0128

36.0257

Fig.14.Reheater con?guration.

A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–11621157

P m àP e ?M

d

eD x m Te31T

where M =J áx m,which is called inertia constant.In the steam turbines,the mechanical torqueses of the prime movers for large generators are function of speed.It is noted that the frequency control of a generator is generally investigated in two main situations.In the ?rst case,the generator is in the islanded operation and feeding load to the electrical grid.In this case,actions of the frequency control would be in steady state conditions,where the system is running as a regulated machine.In the regulated machines,the speed mechanism is responsible for the steam turbine throttle valves controlling.Therefore,in order to stabilize overall system,the frequency should be controlled with respect to the speed droop characteristics [40].The regulation equation is derived as follows,

ex àx 0T

1

D

teTr m àTr m0T?0e32T

then,

P m ?Tr m áx 0?P m0àex 0=D TD x

e33T

In the second case,the generator is part of a large interconnected system or be connected to an in?nite bus.In this case,the turbine controller regulates only the power,not the frequency.While the machine is not under an active governor con-trol and running at unregulated conditions,the torque-speed characteristics can be considered linear over a limited range as follows,

Tr m ?P m =x

e34T

For each case,the electrical power (P e )can be captured in term of terminal voltage (V ),machine excitation voltage (U ),direct axis synchronous reactance (x ),and the rotor angle (d )as follows,

P e ?eUV =x Tsin ed Te35T

The transient response of the machines are particularly investigated for turbine over-speed and load rejection conditions,where P e =0.It is noted that no difference is declared for the characteristics of transient and steady state conditions of unreg-ulated machines in the literature and therefore,Eq.(34)can be also used for the transient conditions [41].

In addition,it is recommended that the term of losses in rotating system be considered in Eq.(31)to complete the gen-erator model,which is presented in Eq.(36).

P L ?P L 0

x x 0

2

e36T

The proposed model for the turbine and generator is presented in Fig.15.4.Simulation results

In this section,responses of proposed functions for estimating the thermodynamic properties of water steam are ?rst compared with standard data,in order to show their accuracy.In this regard,the responses of proposed functions for speci?c enthalpy (extraction no.1)and speci?c entropy (extraction no.4)are presented as examples at different temperatures and pressures,which are shown in Figs.16and 17,respectively.In addition,we de?ne the error as the difference between the response of the proposed functions and standard values to evaluate the error functions.In Table 4the error functions are listed as;upper bound error Max(j e j ),lower bound error Min(j e j ),mean absolute error MAE,average absolute deviation AAD (e)and correlation coef?cient R 2(e).

The developed model for turbine is simulated by using Matlab Simulink .In order to validate the accuracy and performance of the developed model,a comparison between the responses of the proposed model and the responses of the real plant is performed.The load response in steady state and transient conditions over an operation range between 50%and 100%of nominal load is shown in Fig.18to illustrate the behavior of the turbine-generator system.Simulation results indicate

that

Fig.15.Overall turbine and generator models.

1158 A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–1162

the response of the developed model is very close to the response of the real system such that the maximum difference be-tween the response of the actual system and the proposed model is much less than 0.3%.The predicted values are plotted via real system response to subscribe the accuracy of developed models (Fig.19).In addition,by de?ning the error as the

dif-

Fig.16.Responses of enthalpy function at different temperatures and pressures (extraction

1).

Fig.17.Responses of entropy function at different temperatures and pressures (extraction 4).

Table 4

Thermodynamic property error function

Max (j e j )

Min (j e j )MAE AAD (e)R 2(e)Enthalpy for Ext.10.8182 2.213e à40.4014 1.0884e à40.9982Entropy for Ext.4

0.0095

1.101e à4

0.0041

5.3533e à4

0.9977

Fig.18.Response of the turbine-generator.

A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–11621159

ference between the response of the actual plant and the responses of the model,the error functions are evaluated in order to validate the accuracy of developed model,which are presented in Table 5.5.Conclusion

Developing nonlinear mathematical models based on system identi?cation approaches during normal operation with-out any external excitation or disruption is always a hard effort.Assuming that parametric models are available,in this case,using soft computing methods would be helpful in order to adjust model parameters over full range of input–out-put operational data.In this paper,based on energy balance,thermodynamic state conversion and semi-empirical rela-tions,different parametric models are developed for the steam turbine subsections.In this case,it is possible the model parameters are either determined by empirical relations or they are adjusted by applying genetic algorithms as optimi-zation method.

Comparison between the responses of the turbine-generator model with the responses of real system validates the accu-racy of the proposed model in steady state and transient conditions.The presented turbine-generator model can be used for control system design synthesis,performing real-time simulations and monitoring desired states in order to have safe oper-ation of a turbine-generator particularly during abnormal conditions such as load rejection or turbine over-speed.

The further model improvements will make the turbine-generator model proper to be used in emergency control system designing.

Appendix A.Optimization Parameters for GA

HP turbine

IP and LP turbine Functions Population size 2050100Crossover rate 0.70.70.8Mutation rate 0.10.10.2Generations 50

100

2500

Selecting

Stochastic uniform Reproduction

Elite count:2

Appendix B.Linking simulink and GA toolbox

It is mentioned that the MATLAB Simulink is able to read parameters from speci?ed location on disk.Here,an example for a model with two parameters is

presented.

Fig.19.Predicted values via real system response.

Table 5

Turbine modeling error

Max (j e j )

Min (j e j )MAE AAD (e)R 2(e)Power

1.6324

3.3156e à5

0.8988

0.0026

0.9988

1160 A.Chaibakhsh,A.Ghaffari /Simulation Modelling Practice and Theory 16(2008)1145–1162

Appendix C.Saturation pressure as a function of temperature

The proposed functions for estimating the steam saturation pressure,p s ,for the range of 89.965°C to 373.253°C are pre-sented below.

p s ?T t57:0236:2315

?

?5:602972

89:965 C 6T 6139:781 C p s ?T t28:0207:9248

??4:778504139:781 C 6T 6203:622 C p s ?T t5:0??4:304376203:622 C 6T 6299:40 C p s ?T t16:0??4:460843299:407 C 6T 6355:636 C p s ?

T t50:0??4:960785355:636 C 6T 6373:253 C

where,the modeling error is less than 0.02%[37].It should be noted it is not necessary to estimate saturation pressure for two-phase region.

Appendix D.Saturation temperature as a function of pressure

The proposed functions for estimating the steam saturation temperature,T s ,in the range of 0.070to 21.85MPa are pre-sented as follows:

T s ?236:2315p 0:1784767à57:00:070MPa 6p 60:359MPa T s ?207:9248p

0:2092705

à28:0

0:359MPa 6p 61:676MPa T s ?158:0779p 0:2323217à5:01:676MPa 6p 68:511MPa T s ?195:1819p 0:2241729à16:008:511MPa 6p 617:690MPa

T s ?227:2963p 0:201581à50:00

17:690MPa 6p 621:850MPa

where,the modeling error is less than 0.02%[37].It should be noted it is not necessary to estimate saturation temperature for two-phase region.References

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承揽合同范本3篇

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在承揽合同中,完成工作并交付工作成果的一方为承揽人;接受工作成果并支付报酬的一方称为定作人。以下是华律网小编为大家精心准备的:3篇承揽合同范本。欢迎参考阅读! 承揽合同范本一 经甲乙双方协商一致,签订本合同,共同信守。 一、甲方提供原料情况: 品(材料)名: 规格型号: 单位: 数量: 单价: 金额: 交货日期: 超欠幅度% 供应办法: 二、乙方加工产品的义务 1. 质量标准: 2. 质量检验及验收办法: 3. 包装要求及费用负担: 4. 交货(运输)办法及地点: 5. 货款共计人民币(大写): 6. 结算办法及期限: 7. 其他: 三、违约责任: 第一点:甲方的违约责任 (1) 变更产品品种、质量或包装的规格给乙方造成损失时,应负责赔偿乙方损失。 (2) 中途退货的,偿付乙方以退货部分货款总值%勺违约金。

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新视野大学英语 第三版 读写教程

新视野大学英语第三版读写教程1 十五选十 University students come from different parts of the country with various purposes. However, a closer look at their reasons for studying at the university will enable us to 1) them roughly into three groups: those who have a(n) 2) for learning, those who wish to 3) a bright future, and those who learn with no definite purpose. Firstly, there are many students who learn simply because they 4) their goal of learning. Some read a wealth of British and American novels because they are keenly interested in literature. Others sit in front of the computer screen, working on a new program, 5) day and night, because they find some computer programs 6) . and they dream of becoming a “Bill Gates” one day. Secondly, there are students who work hard mainly for a better and more 7) future. It seems that the majority of students fall into this group. After admission to the university, they read books after books to 8) knowledge from all of the resources which are 9) to them, and finally, to succeed in the future job market. Thirdly, there are still some students who learn without a clear goal. They take courses, finish homework, enjoy life on campus, but don’t want to 10) anything new or challenging. They have no idea what they will be doing after college. And they may end up with nothing in their lives. Parents and teenagers have different or even opposite things to worry about. For example, while a mother might have a hard time understanding why her teenagers' room is always a(n) 1) of dirty stuff, the teenagers are more worried about their next exams and may think it is 2) for their mother to insist on keeping a clean room. It is therefore important for you to 3) the differences and learn to communicate with your teenagers properly. 4) , your teenagers may say nothing and shut you out of their personal lives. Their refusal to talk with you may even create 5) stress in your life. Learning effective ways to communicate can 6) the situation of a difficult relationship, 7) the stress of your life, and lead to a friendly relationship with your teenagers. First, you should learn to discuss serious problems in daily conversations. So, important topics, such as driving a vehicle and building a(n) 8) relationship, could be dealt with through daily conversations. Second, learn to be an active listener. Many parents are so 9) with their work that they could hardly take some time for their 10) children. Spend your time listening carefully to what your children like to talk about, and make sure your children feel they are being taken seriously. This will increase the chances of good communication.

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