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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 byapplying genetic algorithms (GA) based on experimental data obtained from a completeset of field experiments. In the intermediate and low-pressure turbines where, in thesub-cooled regions, steam variables deviate from prefect gas behavior, the thermodynamiccharacteristics are highly dependent on pressure and temperature of each region. Thus,nonlinear functions are developed to evaluate specific enthalpy and specific entropy atthese stages of turbines.35485
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The parameters of proposed functions are inpidually adjustedfor the operational range of each subsection by using genetic algorithms. Comparisonbetween the responses of the overall turbine-generator model and the response of realplant indicates the accuracy and performance of the proposed models over wide rangeof operations. The simulation results show the validation of the developed model in termof more accurate and less deviation between the responses of the models and real systemwhere errors of the proposed functions are less than 0.1% and the modeling error is lessthan 0.3%. 1. IntroductionOver the past 100 years, the steam turbines have been widely employed to power generating due to their efficiencies andcosts. With respect to the capacity, application and desired performance, a different level of complexity is offered for thestructure of steam turbines. For power plant applications, steam turbines generally have a complex feature and consist ofmultistage steam expansion to increase the thermal efficiency. It is always more difficult to predict the effects of proposedcontrol 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 parametricuncertainties [2,3].A vast collection of models is developed for long-term dynamics of steam turbines [4–11]. In many cases, the turbinemodels are such simplified that they only map input variables to outputs, where many intermediate variables are omitted[12]. The lack of accuracy in simplifiedmodels emerges many difficulties in control strategies and often, a satisfactory degreeof 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 fromreal system performance in power plant applications where the developed models always comprise reasonable complexities 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 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 [19–21]. Assuming that paramet-ric models are available, in this case, using soft computing methods would be helpful in order to adjust model parametersover full range of input–output operational data.Genetic algorithms (GA) have outstanding advantages over the conventional optimization methods, which allow them toseek globally for the optimal solution. It causes that a complete system model is not required and it will be possible to findNomenclatureC 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 lettersa steam qualityd rotor angle (rad)g efficientlyq specific density (kg/m3)s time constant (s)x frequency (rad/s)Subscriptse 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 parameters of the model with nonlinearities and complicated structures [22,23]. In the recent years, genetic algorithms areinvestigated 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 first developed for analysis of transient response of steam turbines subsectionsbased on the energy balance, thermodynamic state conversion and semi-empirical equations. Then, the related parametersare 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, steamvariables deviate fromprefectgas behavior, the thermodynamic characteristics are highly dependent on pressure and temperature of each region. Thusnonlinear functions are developed to evaluate specific enthalpy and specific entropy at these stages of turbines.