Steam Turbine Model

8/3/2019 Steam Turbine Model

1/18

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 9 November 2007

Received in revised form 20 May 2008

Accepted 20 May 2008

Available online 6 June 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 first 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 field 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 specific enthalpy and specific entropy at

these stages of turbines. The parameters of proposed functions are individually adjusted

for the operational range of each subsection by using genetic algorithms. Comparison

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 lessthan 0.3%.

2008 Elsevier B.V. All rights reserved.

1. Introduction

Over the past 100 years, the steam turbines have been widely employed to power generating due to their efficiencies 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 efficiency. It is always more difficult 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 suchcomplicated 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 [411]. In many cases, the turbine

models are such simplified that they only map input variables to outputs, where many intermediate variables are omitted

[12]. The lack of accuracy in simplified models emerges many difficulties in control strategies and often, a satisfactory degree

of precision is required to improve the overall control performance [13].

Identification 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 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.simpat.2008.05.017

* Corresponding author. Tel.: +98 21 886 748 41; fax: +98 21 886 747 48.

E-mail address: [email protected] (A. Ghaffari).

Simulation Modelling Practice and Theory 16 (2008) 11451162

Contents lists available at ScienceDirect

Simulation Modelling Practice and Theory

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / s i m p a t
mailto:[email protected]://www.sciencedirect.com/science/journal/1569190Xhttp://www.elsevier.com/locate/simpathttp://www.elsevier.com/locate/simpathttp://www.sciencedirect.com/science/journal/1569190Xmailto:[email protected]


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that describe the system well in specific operating conditions [1418]. 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 identification approaches. System identi-

fication during normal operation without any external excitation or disruption would be an ideal target, but in many cases,

using operating data for identification faces limitations and external excitation is required [1921]. 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 inputoutput 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 find

Nomenclature

C specific heat (kJ/kg K)D droop characteristics (N m/rad/s)h specific enthalpy (kJ/kg)h absolute enthalpy (kJ/kmol)

J momentum of inertia (kg m2)

k index of expansion_m mass flow (kg/s)m molecular weight (kg)M inertia constant (kg m2/s)p pressure (MPa)P power (MW)Q heat transferred (MJ)q flow (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 specific volume (m3/kg)W power (MW)x D-axis synchronous reactance (X)

Greek letters

a steam qualityd rotor angle (rad)g efficientlyq specific density (kg/m3)s time constant (s)x frequency (rad/s)

Subscripts

e electricalex extractionf liquid phasefuel fuelg vapor phasein inputm mechanicalout outputp constant pressures saturationspray sprayv constant volumew water0 standard conditionHP high pressureIP intermediate pressureLP low-pressure

1146 A. Chaibakhsh, A. Ghaffari / Simulation Modelling Practice and Theory 16 (2008) 11451162


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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 [2430].

In this paper, mathematical models are first 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 specific enthalpy and specific 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 a 440 MW 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 configuration and steam conditions at

extractions are shown in Fig. 1.

The high-pressure superheated steam of the turbine is responsible for energy flow and conversion results power gener-

ating in the turbine stages. The superheated steam at 535 C and 18.6 MPa pressure from main steam header is the input to

the high-pressure (HP) turbine. The input steam pressure drops about 0.5 MPa 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 is 351 C and 5.37 MPa, 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 at 535C and with 4.83 MPa pressure is fed to intermediate pressure (IP) turbine. Exhaust steam fromIP-turbine for the last stage expansion is fed into the low-pressure (LP) turbine. The input temperature and pressure of the

low-pressure turbine is 289.7 C and 0.83 MPa, respectively. Extracted steam from first 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 configuration and extraction.

A. Chaibakhsh, A. Ghaffari / Simulation Modelling Practice and Theory 16 (2008) 11451162 1147


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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

parametersof models with more complex structure based on experimental data. With the respect to model complexity, a suit-

able fitness 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 flow through the turbine. A relationship between mass flow

and the pressure drop across the HP turbine was developed by Stodola in 1927 [31]. The relationship was later modified

to include the effect of inlet temperature as follows:

_min KffiffiffiffiffiffiTin

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p2in p2outq

1

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

k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p2in p2outTin

s2

By plotting k via inlet mass flow rate based on the experimental data, the slope of linear fitting is captured as K= 520 (Fig. 2).Generally, Eq. (1) has a sufficient accuracy where water steam is the working fluid. A comparison between the model re-

sponse and the experimental shown in Fig. 3 indicates the accuracy of the defined constant.

Fig. 2. Mass flow rate versos k.

Fig. 3. Response of pressuremass flow model.



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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 0.1 and 0.4 s, here the

time constant is measured to be about 0.4 s and therefore the transfer function of the inputoutput pressure is

poutpin

0:294750:4s 1 3

The time response of the proposed transfer function is shown in Fig. 5.To develop the dynamic model of HP turbine, the pressure, mass flow rate and temperature of steam at input and output

of each section is required. The input and output relations for steam pressure and steam flow rate are defined 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 pressuretemperature relation.

ToutTin

poutpin

k1k

4

where k Cp=Cv 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:

WHP gHP _minhin hout gHP Cp _minTin Tout 5

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

Fig. 5. Responses of pressure model.



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It is possible to define the steam specific heat as a function of pressure and/or temperature. Here, for more simplification,

water steam is considered ideal gas. In addition, the turbine efficiency can be expressed as a function of the ratio of blade

tip velocity to theoretical steam velocity. In this paper, turbine efficiency is considered as a constant value. Then,

WHP gHP Cp _min Tin Tinpoutpin

k1k

gHP Cp _min Tin 273:15 1 poutpin

k1k

6

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

efficiency and specific heat. These parameters can be defined by performing a training approach over a collection of inputoutput operational data. The model parameters adjustment is executed through a set of 650 points of data and for transient

and steady state conditions in the range of operation between 154 and 440 MW 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 1N

XNj1

yj yj2 7

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

The optimized value for specific heat, Cp, in order to reach the best performance at different load conditions, is obtained

2.1581 and consequently, the polytrophic expansion factor, k, be equal to 1.2718. The efficiency of a well-designed HP tur-

bine is about 8590%. A fair comparison between the experimental data and the simulation results shows that the obtained

HP turbine efficiency equal to 89.31% is good enough to fit model responses on the real system responses. The proposed

model for HP turbine is presented in Fig. 6, where K1 K 520 and K2 Cp gHP=1000 1:921 103

.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 flow from moisture separation is captured as follows:

sdq

dt 1 b _min q 8

where b is the fraction of moisture in output flow. 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 flow 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 efficiency of turbine. The steam pressure consecutively drops across the turbine

stages. The condensation effect and steam conditions at extraction stages have considerable influences 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 specific enthalpy and spe-

cific 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 [3236].

Fig. 6. HP-turbine model (B = 273.15, K1 = 520, K2 = 1.927 103, f(u) = [u(1)/u(2)]0.2137).



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In 1988, 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:

Fp; T FspsT RT p ps 9

where ps is the steam pressure at saturation conditions. The proposed equations to estimate steam saturation pressure, ps, 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:

Fp; T Fgp Rp; T T Ts 10

where Ts is the steam saturation temperature. The equations to evaluate steam saturation temperature, Ts, 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 from 0.085 MPa to

21.3 MPa 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 significant effects on the steam parameters and therefore, it is

focused on adjusting the first term of functions, which depend on pressure and the functions RT and RP; T are consideredthe same as presented by Garland and Hand.

The working fluid 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 Table 1. The proposed functions for specific enthalpy for liquid phase

and specific entropy in both liquid and vapor phases are defined by three parameters as follows:

F apb c

where three parameters a, b and care adjusted for four different steam conditions at 35, 50, 75 and 100% of load. In addition,

the proposed function for specific enthalpy in vapor phase is defined by three parameters and one constant as follows:

F ap d2 bp d c

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

E 1N

XNj1

jyj yjj 11

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

3.2.1. Specific enthalpy, liquid phase

The following function is presented for estimating specific enthalpy of water in liquid phase.

hp; T hfpsT 1:4 169

369 T !

p ps 12

As seen in Table 1, the steam condition for extractions 5, 6 and 7 are in two-phase region where it can assume that p % ps. Inthis condition, the specific enthalpy of steam for their ranges can be defined as a function of steam pressure. The functions

listed below estimate the specific enthalpy of water in liquid phase, h (kJ/kg).

Table 1

Steam condition at turbine extractions

Extraction No. Pressure (saturation temperature) Temperature (C) Steam condition

IP turbine 1 2.945 (233.91) 456.6 One phase

2 1.466 (197.58) 359 One phase

3 0.830 (171.85) 289.1 One phase

LP turbine 4 0.301 (133.63) 182.7 One phase

5 0.130 (105.80) 111.2 Transient

6 0.0459 77.5 Two phases

7 0.0068 38.2 Two phases



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hf 44:12782275 1000p0:60497549 19:30060027 0:0038 MPa < p < 0:0068 MPahf 194:57965086 100p0:33190979 1:73249442 0:0180 MPa < p < 0:0459 MPahf 258:51219036 100p0:17513608 11:83393526 0:0683 MPa < p < 0:13 MPa

13

3.2.2. Specific enthalpy, vapor phase

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

hp; T hgp 4:5pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi7:4529 100:6T3 p2

q 0:28 e0:008T162 100T

2:225264

375T Ts 14

It should be noted that in two-phase region, it could assume that T% Ts. Therefore, specific enthalpy can be defined as a

function of steam pressure. The functions listed below estimate the specific enthalpy of water/steam in vapor phase for

the pressure range in 3.84.83 MPa.

hg 0:48465587 1000p 52 6:47301169 1000p 5 2560:91238452 0:0038 MPa < p < 0:0068 MPahg 1:82709298 100p 22 17:40365447 100p 2 2606:680821285 0:0180 MPa < p < 0:0459 MPahg 0:50205745 100p 122 6:64525736 100p 12 2679:80609322 0:0683 MPa < p < 0:13 MPahg 2:07829396 10p 32 25:01448122 10p 3 2704:84920557 0:195 MPa < p < 0:301 MPahg 0:49047808 10p 82 10:48902998 10p 8 2740:0576451 0:432 MPa < p < 0:830 MPahg 0:21681424 10p 14:52 6:13049409 10p 14:5 2771:18901288 0:753 MPa < p < 1:466 MPahg 0:08217055 10p 292 3:07429644 10p 29 2816:82024234 1:471 MPa < p < 2:945 MPahg 0:11673499 10p 482 0:13784178 10p 48 2862:43339472 2:388 MPa < p < 4:83 MPa

15

3.2.3. Specific entropy, liquid phase

The optimized functions for estimating specific entropy of water/steam in liquid phase based on steam pressure where

p % ps are as follows:

sf 0:27490714 1000p0:60265499 0:22865089 0:0038 MPa < p < 0:0068 MPa

sf 1:26673390 100p0:17853959

0:61703122 0:0180 MPa < p < 0:0459 MPasf 0:92671704 100p0:14323925 0:03660477 0:0683 MPa < p < 0:13 MPa

16

3.2.4. Specific entropy, vapor phase

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

sp; T sgp 0:004p1:2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3:025 101:1T 465 p2q 0:00006ffiffiffi

pp 4:125 106 T 0:0053

264

375T Ts 17

The optimized functions to evaluate the specific entropy water/steam of phase vapor in the pressure range between 3.8 kPa

and 0.301 MPa.

sg 8:83064734 0:12141594 1000p0:77932806

0:0038 MPa < p < 0:0068 MPasg 9:0863247 0:96869236 100p0:26139247 0:0180 MPa < p < 0:0459 MPasg 8:36610497 0:45436108 100p0:34246778 0:0683 MPa < p < 0:13 MPasg 7:42364087 0:10328045 10p1:27827923 0:195 MPa < p < 0:301 MPa

18

The proposed functions are depending on pressure and temperature of the steam and these variables are necessary to be

defined 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 flow through turbine stages. Here, it is

recommended that the steam pressure be defined 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 defined by first order transfer functions. The

proposed transfer functions for steam pressure at extraction stages are presented in Table 2. In addition, the mass flow rate

through the turbine stages is sequentially decreased as by subtracting extracted steam flow. The extraction flow at each sec-

tion can be defined as a function of entered steam flow to turbine. As shown in Fig. 8, the inputoutput steam flow ratio at



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Table 2

Transfer function for steam pressure and temperature

Output Temperature Pressure

IP turbine Extraction 1 0:86150:3s10:60970:3s1

Extraction 2 0:677360:7s10:303520:7s1

Extraction 3 0:54551:1s10:17181:1s1

LP turbine line 0:54661:4s10:17181:4s1

LP turbine Extraction 4 0:63071:5s1 0:36271:5s1

Extraction 5 0:38281:7s10:15661:7s1

Extraction 6 0:26751:9s10:05531:9s1

Extraction 7 0:12192:1s10:00822:1s1

Fig. 7. Steam pressure at extractions.

Fig. 8. The steam flow rate at extractions.



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different load conditions, except 2nd extraction, are linear. However, considering a linear function of steam flow rate is good

enough to fit 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:

s0 sf x sfg ) x s0 sfsfg

19

Then,

h hf x hfg 20The steam entropy at two-phase region (at fifth, 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 TS diagram.



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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:

W0IP _mIPhIP hex1 _mIP _mex1hex1 hex2 _mIP _mex1 _mex2hex2 hex3 21

Now, the performance index can be considered for IP turbine.

WIP gIPW0IP 22

The LP turbine consists of four extraction levels. The work done in the LP turbine can be captured as follows:

W0LP _mLPhLP hex4 _mLP _mex4hex4 hex5 _mLP _mex4 _mex5hex5 hex6 _mLP _mex4 _mex5 _mex6hex6 hex7

23

where _mLP _mIP _mex1 _mex2 _mex3 then,

WLP gLPW0LP 24The optimal values for efficiencies of IP and LP turbines are obtained 83.12% and 82.84%, respectively, which are fitting 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:

Pm WHP WIP WLP 25

3.3. Reheater model

Reheater section is a very large heat exchanger, which has significant 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 specification, or by applying genetic algorithm tech-

niques on the experimental data. The proposed equations for the temperature model is as follows:

dToutdt

K2K1 _mfuel _minTin Tout B1 B2 26

In this model, the steam quality has significant effects on output temperature and should be considered in related equations.

The transfer function for fuel flow rate and steam quality is as follows:

a_mfuel

9:45039e 620s 1 27

A modified 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 flow velocity is prevailing in the steam volume,

the flow-pressure model is presented as follows:

dp

dt p0s mv

_min _mout 28

A model for the mass flow responding to steam pressure changes is proposed by Borsi [38]. The swing of main steam flow

strictly relies on the change of steam pressure as follows:

d _mout

dt

_mout0

2pin0 pout0

dp

dt 29

In Fig. 13, the flow-pressure model is presented. Generally, in power plants, the turbine inlet flow 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 flow. When the control valve opening is completed, the pressure fluctuation is removed and the swing

of steam flow tends to zero. The adjusted parameters of the developed models are presented in Table 3.

The reheater temperatures must be kept constant at specific 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, Tout is governed by the following equation [39].

DTout 1CpDhout

hin houtCp _mout

D _mout _min_mout

DTin hin hspray

Cp _moutD _mspray 30

where _min

is inlet steam flow, hin

is specific enthalpy of inlet steam hspray

is specific enthalpy of water spray. The configura-

tion of reheater section is presented in Fig. 14.



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3.4. Generator model

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

Fig. 11. IP and LP-turbine model.



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Fig. 12. Temperature model for the reheater section.

Fig. 13. Flow-pressure model K3 P0s mv ; K4 _mout0

2Pin0Pout0

.

Table 3

Parameters for reheater model

B1 B2 K1 K2 K3 K4

Reheater a 0.1706 6.2893 2.41e3 1.06e2 0.0128 36.0257

Reheater b 0.1315 15.723 3.77e4 1.06e2 0.0128 36.0257

Fig. 14. Reheater configuration.



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Pm Pe M ddt

Dxm 31

where M= Jxm, which is called inertia constant. In the steam turbines, the mechanical torqueses of the prime movers forlarge generators are function of speed. It is noted that the frequency control of a generator is generally investigated in two

main situations. In the first 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,

xx0 1D

Trm Trm0 0 32

then,

Pm ffi Trm x0 Pm0 x0=DDx 33In the second case, the generator is part of a large interconnected system or be connected to an infinite 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,

Trm Pm=x 34For each case, the electrical power (Pe) 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,

Pe UV=x sind 35The transient response of the machines are particularly investigated for turbine over-speed and load rejection conditions,

where Pe = 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).

PL PL0 xx0

236

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 first

compared with standard data, in order to show their accuracy. In this regard, the responses of proposed functions for specific

enthalpy (extraction no. 1) and specific entropy (extraction no. 4) are presented as examples at different temperatures and

pressures, which are shown in Figs. 16 and 17, respectively. In addition, we define the error as the difference between the

response of the proposed functions and standard values to evaluate the error functions. In Table 4 the error functions are

listed as; upper bound error Max(jej), lower bound error Min(jej), mean absolute error MAE, average absolute deviation

AAD (e) and correlation coefficient R2(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% ofnominal load is shown in Fig. 18 to illustrate the behavior of the turbine-generator system. Simulation results indicate that

Fig. 15. Overall turbine and generator models.



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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 viareal system response to subscribe the accuracy of developed models ( Fig. 19). In addition, by defining 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 (jej) Min (jej) MAE AAD (e) R2 (e)

Enthalpy for Ext. 1 0. 8182 2.213e4 0.4014 1.0884e4 0.9982

Entropy for Ext. 4 0.0095 1.101e4 0.0041 5.3533e4 0.9977

Fig. 18. Response of the turbine-generator.



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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 identification 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 inputout-

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 20 50 100

Crossover rate 0.7 0.7 0.8

Mutation rate 0.1 0.1 0.2

Generations 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 specified 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 (jej) Min (jej) MAE AAD (e) R2 (e)

Power 1.6324 3.3156e5 0.8988 0.0026 0.9988



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Appendix C. Saturation pressure as a function of temperature

The proposed functions for estimating the steam saturation pressure, ps, for the range of 89.965 C to 373.253 C are pre-

sented below.

ps T57:0236:2315 5:602972

89:965 C 6 T6 139:781 C

ps T28:0207:9248 4:778504

139:781 C 6 T6 203:622 C

ps T5:0185:0779 4:304376

203:622 C 6 T6 299:40 C

ps T16:0195:1819 4:460843

299:407 C 6 T6 355:636 C

ps T50:0277:2963 4:960785

355:636C 6 T6 373:253C

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, Ts, in the range of 0.070 to 21.85 MPa are pre-

sented as follows:

Ts 236:2315p0:1784767 57:0 0:070 MPa 6p 6 0:359 MPaTs 207:9248p0:2092705 28:0 0:359 MPa 6p 6 1:676 MPaTs 158:0779p0:2323217 5:0 1:676 MPa 6p 6 8:511 MPaTs 195:1819p0:2241729 16:00 8:511 MPa 6p 6 17:690 MPaTs 227:2963p0:201581 50:00 17:690 MPa 6p 6 21:850 MPa

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.

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