Steam turbine model Simulation Modelling Practice

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

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Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat

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, thermodynamic principles and semi-empirical equations. Then, the related parameters of developed 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 less than 0.3%. Ó 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 necessary in order to study the turbine transient dynamics. These models can be used for control system design synthesis, performing 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 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 * Corresponding author. Tel.: +98 21 886 748 41; fax: +98 21 886 747 48. E-mail address: [email protected] (A. Ghaffari). 1569-190X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2008.05.017

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

Nomenclature C D h h J k _ m m M p P Q q s t T Tr U V v W x

specific heat (kJ/kg K) droop characteristics (N m/rad/s) specific enthalpy (kJ/kg) absolute enthalpy (kJ/kmol) momentum of inertia (kg m2) index of expansion mass flow (kg/s) molecular weight (kg) inertia constant (kg m2/s) pressure (MPa) power (MW) heat transferred (MJ) flow (kg/s) entropy (kJ/kg K) time (s) temperature (°C) torque (N m) machine excitation voltage (V) terminal voltage (V) specific volume (m3/kg) power (MW) D-axis synchronous reactance (X)

Greek letters steam quality d rotor angle (rad) g efficiently q specific density (kg/m3) s time constant (s) x frequency (rad/s)

a

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 0 standard condition HP high pressure IP intermediate pressure LP low-pressure

that describe the system well in specific 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 identification approaches. System identification 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 [19–21]. 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–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 find


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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 optimization method for training and adaptation approaches [24–30]. 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 algorithms. 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 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 procedure 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 extractions, 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 generating 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 conditions, 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 535 °C and with 4.83 MPa 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 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.

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between system variables with a limited number of parameters and can be validated for the steam turbine by using real system 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 suitable 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:

K _ in ¼ pffiffiffiffiffiffi m T in

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2in p2out

ð1Þ

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

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2in p2out k¼ T in

ð2Þ

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 response 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 pressure–mass 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 input–output pressure is

pout 0:29475 ¼ 0:4s þ 1 pin

ð3Þ

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 pressure–temperature relation.

k1 T out pout ð k Þ ¼ T in pin

ð4Þ

where k ¼ 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:

_ in ðhin hout Þ ¼ gHP C p m _ in ðT in T out Þ W HP ¼ gHP m

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

Fig. 5. Responses of pressure model.

ð5Þ

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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,

k1 ! k1 ! pout ð k Þ pout ð k Þ _ in ðT in þ 273:15Þ 1 ¼ gHP C p m T in T in pin pin

_ in W HP ¼ gHP C p m

ð6Þ

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 input– output 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:

E¼

N 1X ðy yj Þ2 N j¼1 j

ð7Þ

where N is the number of entries used for training process. The optimized value for specific heat, C p , 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 turbine is about 85–90%. 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 K 1 ¼ K ¼ 520 and K 2 ¼ C p gHP =1000 ¼ 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:

s

dq _ in q ¼ ð1 bÞm dt

ð8Þ

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 employed 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 performance 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 regions, 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 specific entropy at these stages of turbines is necessary. The steam thermodynamic properties can be estimated in term of temperature 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, 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 properties 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:

Fðp; TÞ ¼ F s ðps ðTÞÞ þ RðTÞ ðp ps Þ

ð9Þ

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 properties in superheated conditions are presented as follows:

Fðp; TÞ ¼ F g ðpÞ þ Rðp; TÞ ðT T s Þ

ð10Þ

where 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 estimating 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 RðTÞ and RðP; TÞ are considered the 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 ¼ aðpÞb þ c where three parameters a, b and c are 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 ¼ aðp dÞ2 þ bðp dÞ þ c 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:

E¼

N 1 X jy yj j N j¼1 j

ð11Þ

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.

hðp; TÞ ¼ hf ðps ðTÞÞ þ 1:4

169 ðp ps Þ 369 T

ð12Þ

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 . In this 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 3

2.945 (233.91) 1.466 (197.58) 0.830 (171.85)

456.6 359 289.1

One phase One phase One phase

LP turbine

4 5 6 7

0.301 (133.63) 0.130 (105.80) 0.0459 0.0068

182.7 111.2 77.5 38.2

One phase Transient Two phases Two phases

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hf ¼ 44:12782275 ð1000pÞ0:60497549 þ 19:30060027 hf ¼ 194:57965086 ð100pÞ0:33190979 þ 1:73249442

hf ¼ 258:51219036 ð100pÞ

0:17513608

0:0038 MPa < p < 0:0068 MPa ð13Þ

0:0180 MPa < p < 0:0459 MPa

þ 11:83393526 0:0683 MPa < p < 0:13 MPa

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

2

3

4:5p 100 6 7 hðp; TÞ ¼ hg ðpÞ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 0:28 e0:008ðT162Þ 2:2255ðT T s Þ T 0:6 3 2 7:4529 10 T p

ð14Þ

It should be noted that in two-phase region, it could assume that T T s . 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 3.8–4.83 MPa.

hg ¼ 0:48465587 ð1000p 5Þ2 þ 6:47301169 ð1000p 5Þ þ 2560:91238452

hg ¼ 1:82709298 ð100p 2Þ2 þ 17:40365447 ð100p 2Þ þ 2606:680821285 2

hg ¼ 0:50205745 ð100p 12Þ þ 6:64525736 ð100p 12Þ þ 2679:80609322 hg ¼ 2:07829396 ð10p 3Þ2 þ 25:01448122 ð10p 3Þ þ 2704:84920557

0:0038 MPa < p < 0:0068 MPa 0:0180 MPa < p < 0:0459 MPa 0:0683 MPa < p < 0:13 MPa 0:195 MPa < p < 0:301 MPa

hg ¼ 0:49047808 ð10p 8Þ2 þ 10:48902998 ð10p 8Þ þ 2740:0576451

0:432 MPa < p < 0:830 MPa

hg ¼ 0:08217055 ð10p 29Þ2 þ 3:07429644 ð10p 29Þ þ 2816:82024234

1:471 MPa < p < 2:945 MPa

hg ¼ 0:21681424 ð10p 14:5Þ2 þ 6:13049409 ð10p 14:5Þ þ 2771:18901288 0:753 MPa < p < 1:466 MPa

hg ¼ 0:11673499 ð10p 48Þ2 þ 0:13784178 ð10p 48Þ þ 2862:43339472

2:388 MPa < p < 4:83 MPa ð15Þ

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 ¼ 0:27490714 ð1000pÞ0:60265499 0:22865089 0:0038 MPa < p < 0:0068 MPa sf ¼ 1:26673390 ð100pÞ0:17853959 0:61703122 sf ¼ 0:92671704 ð100pÞ0:14323925 0:03660477

ð16Þ

0:0180 MPa < p < 0:0459 MPa

0:0683 MPa < p < 0:13 MPa

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

2

3

1:2

0:004p 0:00006 7 6 sðp; TÞ ¼ sg ðpÞ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffi 4:125 106 T þ 0:00535ðT T s Þ p 5 1:1 2 3:025 10 ðT þ 46Þ p

ð17Þ

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 ¼ 8:83064734 0:12141594 ð1000pÞ0:77932806

0:0038 MPa < p < 0:0068 MPa

sg ¼ 9:0863247 0:96869236 ð100pÞ0:26139247

0:0180 MPa < p < 0:0459 MPa

sg ¼ 7:42364087 0:10328045 ð10pÞ1:27827923

0:195 MPa < p < 0:301 MPa

0:34246778

sg ¼ 8:36610497 0:45436108 ð100pÞ

0:0683 MPa < p < 0:13 MPa

ð18Þ

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 entered 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 section can be defined as a function of entered steam flow to turbine. As shown in Fig. 8, the input–output steam flow ratio at


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Table 2 Transfer function for steam pressure and temperature

IP turbine

Output

Temperature

Pressure

Extraction 1

0:8615 0:3sþ1 0:67736 0:7sþ1 0:5455 1:1sþ1 0:5466 1:4sþ1

0:6097 0:3sþ1 0:30352 0:7sþ1 0:1718 1:1sþ1 0:1718 1:4sþ1

0:6307 1:5sþ1 0:3828 1:7sþ1 0:2675 1:9sþ1 0:1219 2:1sþ1

0:3627 1:5sþ1 0:1566 1:7sþ1 0:0553 1:9sþ1 0:0082 2:1sþ1

Extraction 2 Extraction 3 LP turbine line LP turbine

Extraction 4 Extraction 5 Extraction 6 Extraction 7

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 ¼ sf þ x sfg ) x ¼

s0 sf sfg

ð19Þ

Then,

h ¼ hf þ x hfg

ð20Þ

The 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 T–S 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:

_ IP ðhIP hex1 Þ þ ðm _ IP m _ ex1 Þðhex1 hex2 Þ þ ðm _ IP m _ ex1 m _ ex2 Þðhex2 hex3 Þ W 0IP ¼ m

ð21Þ

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

W IP ¼ gIP W 0IP

ð22Þ

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

_ LP ðhLP hex4 Þ þ ðm _ LP m _ ex4 Þðhex4 hex5 Þ þ ðm _ LP m _ ex4 m _ ex5 Þðhex5 hex6 Þ W 0LP ¼ m _ ex4 m _ ex5 m _ ex6 Þðhex6 hex7 Þ _ LP m þðm

ð23Þ

_ LP ¼ m _ IP m _ ex1 m _ ex2 m _ ex3 then, where m

W LP ¼ gLP W 0LP

ð24Þ

The optimal values for efficiencies of IP and LP turbines are obtained 83.12% and 82.84%, respectively, which are fitting turbine 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 ¼ W HP þ W IP þ W LP

ð25Þ

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 model. 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 determined either from constructional data such as fuel and water steam specification, or by applying genetic algorithm techniques on the experimental data. The proposed equations for the temperature model is as follows:

dT out _ in ðT in T out þ B1 Þ þ B2 Þ _ fuel þ m ¼ K 2 ðK 1 m dt

ð26Þ

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 _ fuel m

¼

9:45039e 6 20s þ 1

ð27Þ

A modified version of the temperature model for the reheater sections is presented in Fig. 12. According to the mass accumulation 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 p0 _ in m _ out Þ ðm ¼ dt s mv

ð28Þ

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:

_ out _ out0 dm dp m ¼ dt 2ðpin0 pout0 Þ dt

ð29Þ

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 between 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].

DT out ¼

h out Þ h spray Þ _ m 1 ðh ðh in _ out þ in DT in in _ spray Dhout ¼ Dm Dm Cp _ out _ out _ out m Cp m Cpm

ð30Þ

_ in is inlet steam flow, hin is specific enthalpy of inlet steam hspray is specific enthalpy of water spray. The configurawhere m tion of reheater section is presented in Fig. 14.

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Fig. 11. IP and LP-turbine model.

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.

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Fig. 12. Temperature model for the reheater section.

_ out0 m . Fig. 13. Flow-pressure model K3 ¼ s P 0mv ; K4 ¼ 2ðPin0 Pout0 Þ

Table 3 Parameters for reheater model

Reheater a Reheater b

B1

B2

K1

K2

K3

K4

0.1706 0.1315

6.2893 15.723

2.41e3 3.77e4

1.06e2 1.06e2

0.0128 0.0128

36.0257 36.0257

Fig. 14. Reheater configuration.

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Fig. 15. Overall turbine and generator models.

Pm Pe ¼ M

d ðDxm Þ dt

ð31Þ

where M = J xm, 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 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,

ðx x0 Þ

1 þ ðTrm Trm0 Þ ¼ 0 D

ð32Þ

then,

Pm ffi Trm x0 ¼ P m0 ðx0 =DÞDx

ð33Þ

In 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 control and running at unregulated conditions, the torque-speed characteristics can be considered linear over a limited range as follows,

Trm ¼ P m =x

ð34Þ

For 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 ¼ ðUV=xÞ sinðdÞ

ð35Þ

The 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 unregulated 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 generator model, which is presented in Eq. (36).

PL ¼ P L0

x x0

2

ð36Þ

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% of nominal load is shown in Fig. 18 to illustrate the behavior of the turbine-generator system. Simulation results indicate that


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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

Enthalpy for Ext. 1 Entropy for Ext. 4

Max (jej)

Min (jej)

MAE

AAD (e)

R2 (e)

0. 8182 0.0095

2.213e4 1.101e4

0.4014 0.0041

1.0884e4 5.3533e4

0.9982 0.9977

Fig. 18. Response of the turbine-generator.

the response of the developed model is very close to the response of the real system such that the maximum difference between 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 defining the error as the dif-

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Fig. 19. Predicted values via real system response.

Table 5 Turbine modeling error

Power

Max (jej)

Min (jej)

MAE

AAD (e)

R2 (e)

1.6324

3.3156e5

0.8988

0.0026

0.9988

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 without 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–output operational data. In this paper, based on energy balance, thermodynamic state conversion and semi-empirical relations, 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 optimization method. Comparison between the responses of the turbine-generator model with the responses of real system validates the accuracy 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 operation 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

Population size Crossover rate Mutation rate Generations Selecting Reproduction

HP turbine

IP and LP turbine

Functions

20 0.7 0.1 50 Stochastic uniform Elite count: 2

50 0.7 0.1 100

100 0.8 0.2 2500

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.


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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 presented below.

ps ¼ ps ¼ ps ¼ ps ¼ ps ¼

Tþ57:0 5:602972 236:2315 Tþ28:0 4:778504 207:9248 Tþ5:0 4:304376 185:0779 Tþ16:0 4:460843 195:1819 Tþ50:0 4:960785 277:2963

89:965 C 6 T 6 139:781 C 139:781 C 6 T 6 203:622 C 203:622 C 6 T 6 299:40 C 299:407 C 6 T 6 355:636 C 355:636 C 6 T 6 373:253 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 0.070 to 21.85 MPa are presented as follows:

T s ¼ 236:2315p0:1784767 57:0

0:070 MPa 6 p 6 0:359 MPa

T s ¼ 158:0779p0:2323217 5:0

1:676 MPa 6 p 6 8:511 MPa

T s ¼ 227:2963p0:201581 50:00

17:690 MPa 6 p 6 21:850 MPa

T s ¼ 207:9248p0:2092705 28:0

0:359 MPa 6 p 6 1:676 MPa

T s ¼ 195:1819p0:2241729 16:00 8:511 MPa 6 p 6 17:690 MPa

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