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3.2.1. Specific enthalpy, liquid phaseThe following function is presented for estimating specific enthalpy of water in liquid phase.hðp; TÞ¼ hf ðpsðTÞÞ þ 1:4 169369 T ðp psÞð12ÞAs 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 functionslisted below estimate the specific enthalpy of water in liquid phase, h (kJ/kg). be an ideal process, the energy equations for steam expansion in turbine, which relates the power output to steam energydeclining across turbine stages can be captured. Therefore, the work done in IP turbine can be captured as follows:W0IP ¼ _ mIPðhIP hex1Þþð _ mIP _ mex1Þðhex1 hex2Þþð _ mIP _ mex1 _ mex2Þðhex2 hex3Þð21ÞNow, the performance index can be considered for IP turbine.WIP ¼ gIPW0IP ð22ÞThe LP turbine consists of four extraction levels. The work done in the LP turbine can be captured as follows:W0LP ¼ _ mLPðhLP hex4Þþð _ mLP _ mex4Þðhex4 hex5Þþð _ mLP _ mex4 _ mex5Þðhex5 hex6Þþð _ mLP _ mex4 _ mex5 _ mex6Þðhex6 hex7Þð23Þwhere _ mLP ¼ _ mIP _ mex1 _ mex2 _ mex3 then,WLP ¼ gLPW0LP ð24ÞThe optimal values for efficiencies of IP and LP turbines are obtained 83.12% and 82.84%, respectively, which are fitting tur-binemodel responses on the real systemresponses. The developedmodels for IP and LP turbines are presented in Fig. 11. Theoverall generated mechanical power can be captured by summation of generated power in turbine stages as follows:Pm ¼ WHP þWIP þWLP ð25Þ3.3. Reheater modelReheater section is a very large heat exchanger, which has significant thermal capacity and steam mass storage.
Thereheater 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 thethermodynamics 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:dToutdt¼ K2ðK1 _ mfuel þ _ minðTin Tout þ B1Þþ B2Þð26ÞIn 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_ mfuel¼ 9:45039e 620s þ 1ð27ÞA 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:dpdt¼ p0s mvð _ min _ moutÞ ð28ÞA model for the mass flow responding to steam pressure changes is proposed by Borsi [38]. The swing of main steam flowstrictly relies on the change of steam pressure as follows:d _ moutdt¼_ mout02ðpin0 pout0Þdpdtð29ÞIn Fig. 13, the flow-pressure model is presented. Generally, in power plants, the turbine inlet flow is controlled by a governoror control valves to response to the grid frequency. Therefore, when this valve is acting, there is an interaction betweensteam pressure and flow. When the control valve opening is completed, the pressure fluctuation is removed and the swingof 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 massstorage is negligible. In addition, it is considered that there is no pressure drop in this section. Then, the inlet temperatureof the second reheater, Tout is governed by the following equation [39].DTout ¼ 1CpDhout ¼ ð hin houtÞCp _ moutD _ mout þ _ min _ moutDTin ð hin hsprayÞCp _ moutD _ mspray ð30Þwhere _ 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. where M = J xm,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 twomain 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. Inthe regulated machines, the speed mechanism is responsible for the steam turbine throttle valves controlling. Therefore, inorder to stabilize overall system, the frequency should be controlled with respect to the speed droop characteristics [40]. Theregulation equation is derived as follows,ðx x0Þ1D þðTrm Trm0Þ¼ 0 ð32Þthen,Pm ffi Trm x0 ¼ Pm0 ðx0=DÞDx ð33ÞIn 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 asfollows,Trm ¼ Pm=x ð34ÞFor 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 ¼ðUV=xÞ sinðdÞ ð35ÞThe 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 ¼ PL0xx0 2ð36ÞThe proposed model for the turbine and generator is presented in Fig. 15.4. Simulation resultsIn this section, responses of proposed functions for estimating the thermodynamic properties of water steam are firstcompared with standard data, in order to show their accuracy. In this regard, the responses of proposed functions for specificenthalpy (extraction no. 1) and specific entropy (extraction no. 4) are presented as examples at different temperatures andpressures, which are shown in Figs. 16 and 17, respectively. In addition, we define the error as the difference between theresponse of the proposed functions and standard values to evaluate the error functions. In Table 4 the error functions arelisted as; upper bound error Max(jej), lower bound error Min(jej), mean absolute error MAE, average absolute deviationAAD (e) and correlation coefficient R2(e).The developed model for turbine is simulated by using Matlab Simulink. In order to validate the accuracy and performanceof the developed model, a comparison between the responses of the proposed model and the responses of the real plant isperformed. 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 ference between the response of the actual plant and the responses of themodel, the error functions are evaluated in order tovalidate the accuracy of developed model, which are presented in Table 5.5. ConclusionDeveloping 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 thiscase, 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 modelparameters 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 forcontrol 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 systemdesigning. Appendix C. Saturation pressure as a function of temperatureThe 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 ¼ Tþ57:0236:2315 5:60297289:965 C 6 T 6 139:781 Cps ¼ Tþ28:0207:9248 4:778504139:781 C 6 T 6 203:622 Cps ¼ Tþ5:0185:0779 4:304376203:622 C 6 T 6 299:40 Cps ¼ Tþ16:0195:1819 4:460843299:407 C 6 T 6 355:636 Cps ¼ Tþ50:0277:2963 4:960785355:636 C 6 T 6 373:253 Cwhere, the modeling error is less than 0.02% [37]. It should be noted it is not necessary to estimate saturation pressure fortwo-phase region.Appendix D. Saturation temperature as a function of pressure