Available online at ScienceDirect. Mathematical modelling and simulation of the behaviour of the steam turbine

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1 Available onle at ScienceDirect Procedia echnology 12 ( 2014 ) he 7 th International Conference Interdisciplarity Engeerg (INER-ENG 2013) Mathematical modellg and simulation of the behaviour of the steam turbe Mircea Dulau a, *, Dor Bica b a,b Department of Electrical and Computer Engeerg, Petru Maior University of îrgu-mureş, 1 N.Iorga st., Abstract Model simulations are becomg very important dynamic power system analyses. However, no mathematical system can exactly model a physical process. Based on mathematical models of the processes and design calculations, PC programs allow simulation and the determation of the control system performances. his paper presents the mathematical modellg of the steam turbe unit, developed based on the contuity equation. his model is used to determe the simulation diagram for the steam turbe with high, medium and low pressure sections. Usg Matlab/Simulk software facilities, have been simulated the behaviour of the shaft torque, dependg of the control valves openg, with uncerta parameters of the process, than the step response of the steam turbe, with load and proportional control algorithm he Authors. Published by Elsevier B.V he Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of Department of Electrical and Computer Engeerg, Faculty of Engeerg, Selection and peer-review under responsibility of the Petru Maior University of irgu Mures. Petru Maior University of îrgu Mureș. Keywords: steam turbe; mathematical modelg; power plant system; control system; 1. Introduction he steam turbe convert stored energy of high pressure and high temperature steam to rotary energy, which is turn converted to electrical energy by the generator. Each turbe section consists of a set of a movg blades attached to rotor and a set of stationary vanes which steam is accelerated to high velocity (Fig. 1). * Correspondg author. el.: address: mircea.dulau@g.upm.ro he Authors. Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and peer-review under responsibility of the Petru Maior University of irgu Mures. doi: /j.protcy

2 724 Mircea Dulau and Dor Bica / Procedia echnology 12 ( 2014 ) V F Steam Fig. 1. Steam turbe unit. he ketic energy of the high velocity steam is converted to shaft torque [2,4,5,9]. A large variety of steam turbes have been built, with respect to the capacity, application and desired performance. o crease the thermal efficiency applications, steam turbes consist of multistage steam expansion. 2. Mathematical modellg of steam turbe unit In many cases, the steam turbe models are simplified, many termediate variables are omitted and only map put variables to puts as led [2,3,9,10,12,13]. In these conditions, the put-put mathematical model (the transfer function) of a steam turbe from Fig. 1 and the expression for mechanical power developed by a turbe are based on the contuity equation: F dw d t d V F dt F (1) where: W is the weight of steam turbe [kg]; V volume of turbe [m 3 ]; density of steam [kg/m 3 ]; F steam mass flow rate [kg/s]; t time [sec.]. Assumg the flow of the turbe to be proportional to pressure the turbe [9]: F0 F P (2) P 0 where: P pressure of steam the turbe [kpa]; P 0 rated pressure; F 0 rated flow of turbe. With constant temperature the turbe: d dp dt dt P (3) From equations (1) (3), result the mathematical model: F F dp P0 df V V dt P P F dt 0 df dt (4)

3 Mircea Dulau and Dor Bica / Procedia echnology 12 ( 2014 ) df dt F F (5) and, after Laplace transform, the transfer function of a steam turbe unit: where: s 1 s s 1 F H s (6) F P0 V is the time constant [sec.]. F P 0 he turbe torque is proportional to the steam flow rate: m k F (7) where: k is a proportional constant. he change density of steam with respect to pressure P at a given temperature may be determed from tables [14]. 3. Block diagram of steam turbe configuration Dependg on the turbe configuration, gas units consist of high pressure (), medium pressure (MP) and low pressure () turbe sections (Fig. 2). A steam turbe is equipped with sets of valves: high pressure valves (V), re-heater valves (V) [15,17]. Boiler V pos V CP Shaft MP G V Condenser V pos Fig. 2. Steam turbe configuration. Steam enters to the section through the V and the let pipg. he exhaust steam is passed through the re-heater (). Re-heater section is a large heat exchanger, which has significant thermal capacity and steam mass storage. In the reheat type turbe, the steam upon leavg the section returns to the boiler, where is passed through a before returng to the MP section. he reheat steam flows to the MP turbe section through the V and the let pipg. he crossover pipg (CP) provides a path for the steam from MP section exhaust to the let. Based on Fig. 2 and equation (6), result the block diagram of the turbe configuration, used for simulation (Fig. 3), where [2,8,9,17]:

4 726 Mircea Dulau and Dor Bica / Procedia echnology 12 ( 2014 ) K V X H (s) H (s) X H (s) K orque m V pos Steam pressure Steam flow V X V K MP Fig. 3. Block diagram of steam turbe configuration. 1 H s (8) s 1 H s 1 (9) s 1 H s (10) s 1 he response of steam flow to a change V openg exhibits a time constant due to the changg time to the section. Usually, ( ) sec. he steam flow the MP and sections can change only with the build-up of pressure the re-heater volume. he re-heater holds a substantial amount of steam and the time sec. he steam flows to the sections, associated with the crossover pipg, and constant is express an additional time constant 0. 5 sec. he MP and sections generate ab (60 80) % of the total turbe power. he sum of the power fractions of the various turbes sections is [9,15,17]: K K K 1 (11) MP If <<, than is negligible comparison with, and a simplified transfer function of the turbe relatg perturbed values of the torque and V position ( Z ) may be written as follows: m V Z m V K 1 K Ks 1 s 1 s 1 s 1 s 1 s 1 (12) If the steam turbe is of a sgle reheat type, the transfer function may be approximated by: Z m V Ks 1 s 1 (13) No mathematical system can exactly model a physical process. ypically, the flow F, the pressure P, the density etc. are experimentally measured and/or calculated, leadg to the confidence tervals for time values and not just a sgle value:

5 Mircea Dulau and Dor Bica / Procedia echnology 12 ( 2014 ) nom nom (14) he values nom, nom are called nomal values and the, are called the maximum deviations from the nomal values [1,6,7,11,16]. As follow, practice, results many admissible transfer functions for this process, one for each possible combation of, the given tervals. 4. Simulation of the steam turbe configuration Based on block diagram of the steam turbe (Fig. 3) and used Matlab/Simulk facilities [1,6,16], result the shaft torque, as an put variable, dependg on the control valves position, V and V, as put variables, where 100% means fully open (Fig. 4, Fig. 5). V V V pos m V V orque V pos Steam turbe Fig. 4. Simulk block diagram of steam turbe configuration. Fig. 5. Shaft torque to different openg positions of: (a) V; (b) V. he V position (V pos) modulates the steam flow through the turbe for load control durg normal operation. he V with valve position (V pos) is normally used only for rapid control of turbe mechanical power the event of an overspeed. Fig. 6 shows the step response of the steam turbe, described by the simplified transfer functions (12) and (13). Some of parameters from these relationships have exact value (for example K 0. 3 ) and some of these varies with a specific range of values, defg the uncerta model of the process, as follow [9,15,17]: the nomal value of 7. 5 sec. and a range between (5 10) sec.; the nomal value of sec. and a range between ( ) sec. o fd how variations of the parameters from the nomal model affect the process, with Matlab techniques are generated a number of random samples of the uncerta parameters and plot the correspondg step responses [1,16]. he graphics from Fig. 6 show the same type of damped step response, with similar transient performances, so, order to simulate the behaviour of the steam turbe is adequate to use the transfer function from relation (13).

6 728 Mircea Dulau and Dor Bica / Procedia echnology 12 ( 2014 ) a b Fig. 6. Shaft torque: (a) nomal values; (b) random samples. It is consider that the steam turbe works with an isolated load, described by the transfer function: HL s 1 ( Ls), and a proportional algorithm (controller), feedback connection (Fig. 7). With simulation values: L 12 sec., 7. 5 sec. and K 0. 3 it is determed the values of tung parameter K C, for which the speed is stable and the step response is damped or critically damped. Based on characteristic equation of the close-loop system and values for critical dampg, any positive value, K C 0, will lead to the stable response, and 0 K C 0. 67, will lead to the (critically) damped stable response (Fig. 8). Reference K C Controller H (s) Steam turbe H L (s) Load Speed Fig. 7. Steam turbe closed-loop block diagram.

7 Mircea Dulau and Dor Bica / Procedia echnology 12 ( 2014 ) Fig. 8. Steam turbe closed-loop step response. 5. Conclusions Usually, the steam turbe transfer function is characterized by two time constants. In order to analyze the behaviour, simulation results show that it can use a first order transfer function, despite the variation of the steam turbe parameters. Largely, the dynamic response of a steam turbe is fluenced by two factors: entraed steam to high pressure turbe section and the storage action the re-heater. he dynamic response of a steam turbe can be related terms of changes steam valves openg (V position and V position). In order to meet the active power demand, puts to the generator (steam parameters) must be controlled, otherwise, the generator speed will vary. One of the control key refers to establish the conditions of the stable response, with steam turbe, generator and different type of controllers, close-loop connection. References [1] Balas GJ et al. Robust control toolbox. For use with Matlab. Version 3; he MathWorks; [2] Chaibakhsh A et al, Simulation modellg practice and theory p [3] Chen ZX et al. Modelg and simulation of steam turbe based on multi-modules high temperature gas-cooled reactor. IEEE 8 th World Congress on Intelligent Control and Automation (WCICA). Jan; 2010, p [4] Darie S et al. Production, transport and distribution of the electrical energy. Vol. I; Cluj Napoca echnical University Pres; [5] Dulău M. he automation of the contuous processes. Chemical and thermal processes. Petru Maior University Publishg House; [6] Dulău M et al. Behavioural study of a thermal process control under uncertaties. IEEE International Conference on Automation, Quality and estg, Robotics, ome I; Cluj-Napoca; p [7] Dumitrache I et al. Automatics. Romanian Academy Publishg House; [8] Inoue et al. A thermal power plant model for dynamic simulation of load frequency control. IEEE Power Systems Conference and Exposition. Atlanta; 2006; p [9] Kundur P. Power system stability and control. McGraw-Hill Professional; [10] Maslo K et al. Gas turbe model usg design of heat and power stations. IEEE Power ech Proceedgs. Vol. 4; Porto; [11] Nademi H et al. Robust controller design for governg steam turbe power generators. IEEE International Conference on Electrical Maches and Systems. okyo; 2009; p [12] Pan J et al. A new non-lear model of steam turbe unit for dynamic analysis of power system. IEEE International Conference on Power System echnology. Hangzhou; 2010; p [13] Soon KY et al. Validated models for gas turbes based on thermodynamic relationships. IEEE ransactions on Power Systems. Vol. 26; Issue 1; 2011; p [14] oganel M. hermodynamic tables for water and steam. Petru Maior University; [15] Yang et al. Parameter identification of steam turbe speed governor system. IEEE Power and Energy Engeerg Conference (APPEEC); Shanghai; 2012; p [16] ***. Matlab and simulk tutorial. he MathWorks; 2010; [17] ***. echnical documentation. Iernut thermoelectric power plant; 2010.