See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/233200767 Modelling and simulation of steam turbine processes: Individual models for individual tasks Article in Mathematical and Computer Modelling of Dynamical Systems · December 2008 DOI: 10.1080/13873950802384001 CITATIONS READS 20 1,882 1 author: Gerta Zimmer Siemens 29 PUBLICATIONS 299 CITATIONS SEE PROFILE All content following this page was uploaded by Gerta Zimmer on 21 December 2020. The user has requested enhancement of the downloaded file. Journal: Author: Title: Article Type: Mathematical and Computer Modelling of Dynamical Systems G. Zimmer Modelling and Simulation of Steam Turbine Processes full paper Modelling and Simulation of Steam Turbine Processes: Individual Models for Individual Tasks GERTA ZIMMER* Siemens Power Generation, P11M3, Postbox 101755, D-45466 Mülheim, Germany Abstract: Within power plants several physical, chemical, and mechanical processes are conducted to transfer the energy, stored in fossil fuel, into electrical energy. This energy conversion is divided into several stages. Hitherto the largest conventional power plants employ steam turbines as prime movers to drive a generator. Hence a steam turbine is one module to convert heat energy into mechanical energy. And thus it is one link in the chain of energy conversions with the aim of generating electrical energy. Today, steam turbine industry faces numerous challenges concerning efficiency, commissioning time, start-up times, operation, availability, safety, cost effectiveness, etc. Many of these tasks can be supported by simulating the transient operational behavior of the turbine in advance. For example the commissioning time can be shortened if the turbine controllers are initialized with well-tuned pre-set parameters, cost effectiveness can be increased by setting aside unnecessary devices and exactly determining material specifications, safety may be increased by predicting the impacts of failures and thus taking the necessary precautions. Different tasks require different details regarding the employed turbine simulation model. Thus, the turbine controller may be well tuned with less complex simulation models of turbine, generator and electrical grid, whereas detailed studies of failures, mainly the transient behavior which mal lead to serious damages, may require detailed modeling of the turbine-internal thermodynamic processes. Here a brief overview of models which simulate the transient thermodynamic behavior of a steam turbine is presented. Three different approaches will be introduced and compared with respect to different operating situations. Here special attention is directed towards the time dependence of critical states, mainly turbine speed and pressure development in certain areas. The first model is based on a simple, linear approach and is suitable of giving a quick overview. The second one incorporates more details and is useful if the operating point is close to the design point. Finally, the last model incorporates mass and energy balances as well as the major nonlinearities. Hence it depicts the turbine behavior over a large range of operating points. Keywords: steam turbine, transient behavior, thermodynamic model, dynamic simulation AMS Subject Classification: 00A69, 37M05, 68U20, 80A20, 93A30, 93C15, 93C83 * Corresponding author. E-mail: gerta.zimmer@siemens.com Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes 00A69 General applied mathematics 37M05 Simulation 68U20 Simulation 80A20 Heat and mass transfer, heat flow 93A30 Mathematical modeling (models of systems, model-matching, etc.) 93C15 Systems governed by ordinary differential equations 93C83 Control problems involving computers (process control) 1. Introduction A steam turbine is a module to convert heat energy into mechanical energy. The principal task in operating a steam turbine is to convert the energy of hot steam into rotational energy. Thus it is one link in the chain of energy conversions with the aim of generating electrical energy. In order to produce electrical energy, a series of energy conversions are conducted in fossil-fired power plants. Firstly, chemical energy stored in the form of fuel and oxygen is transformed into thermal energy by the process of combustion. In a second step, the heat energy is used to produce superheated steam where the (potential) energy is stored as steam enthalpy. Turbines are used to transform this potential energy into mechanical energy. Here, hot steam with up to 300 bar and 600°C is condensed to approximately 0,03 bar and 25°C. The released energy is transformed into rotational energy, which in turn drives a generator. Finally, the generator transforms mechanical energy into electrical energy which is available as electric current. Today, the steam turbine industry faces numerous challenges concerning efficiency, commissioning time, start-up times, operation, availability, safety, cost effectiveness, etc. Many of these tasks can be supported by simulating the operational behaviour of the turbine in advance. For example, the commissioning time can be shortened if the turbine controllers are initialized with well-tuned pre-set parameters, cost effectiveness can be increased by setting aside unnecessary devices and exactly determining material specifications, safety may be increased by predicting the impacts of failures and thus taking the necessary precautions. Different tasks require different details regarding the employed turbine simulation model. Thus, the turbine controller may be well tuned with less complex simulation models of turbine, generator and electrical grid, whereas detailed studies of failures may require detailed modelling of the turbine-internal thermodynamic processes. In many applications it is necessary to study the transient behaviour. For example, at a load reception, the turbine speed rises and has to be intercepted through closing the steam admitting control valves. In this situation, it is important to know the speed transient in order to determine the necessary reaction time of the protection system, as well as the resulting peak, the maximum overspeed. Whereas the transient behaviour of a steam turbines rotational and electrical devices have long been thoroughly studied, thermodynamic properties of a steam turbine are mainly investigated at steady state operating points. Only a small number of publications deal with the modelling of transient thermodynamic behaviour, see [1,2,3] e.g. 2. General Set-up In general, a steam power plant consists of a steam generator (or boiler), the steam turbine itself and a condenser. Steam with high pressure is produced in the boiler by heating water, which was previously Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes compressed by feedwater pumps, and is then continuously expanded in the turbine until it reaches condenser pressure and enthalpy. Power transfer from steam to the rotating shaft, and thus to the generator, is realized by leading the steam alternately through guide wheels, which are fixed to the casing and rotating wheels, attached to the shaft. A guide wheel together with the corresponding rotating wheel is called a stage. Due to the ample increase in volume during the expansion, it is not desirable to conduct the complete expansion in a single turbine. Therefore, in general, several stages are combined to a turbine section. Steam transition from one section to another gives the opportunity to spread the steam to two flows or to several identical sections. In general, expansion is divided into high- (HP), intermediate- (IP) and low-pressure (LP) sections. In order to increase efficiency, a re-heater is located between the high- and intermediate pressure sections. At the entrance of the high- as well as of the intermediate-pressure section, the admittance of steam may be controlled by appropriate control valves. Finally, the steam turbine controller, or governor, and the protection system supervise the interactions between demands, internal states and admissible limits. Figure 1 gives an overview of the elements involved. steam generator reheater boiler electric grid direction of flow control valve HP LP IP generator turbine governor p condenser p n P 'external requirements' Figure 1: Elements of a steam turbine power plant - general set-up All other parts of a steam power plant, e.g. the remaining devices of the steam-water-cycle, boiler control, generator control and electrical devices, will be neglected in this context. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes From a control point of view, the steam turbine can be regarded as a multi dimensional control system whose inputs are the lift of the main steam valve, hMV, and reheat valve, hRV, respectively. Outputs are some internal pressure values pi, the shaft velocity (i.e. the turbine speed n) and the forcing torque tf. The later is not directly measurable. Instead, the power output P has to be utilized. The thermodynamic state of the steam at the steam generator outlet as well as inside the condenser together with the grid frequency depicts the external coupling. The focus of the subsequent investigation is to determine the necessary level of detail for a model of the thermodynamic part of the turbine, i.e. the expansion sections between the boiler an the condenser. Firstly, a general frame was developed to model the boiler, condenser, shaft, generator and turbine governor. Secondly, differently detailed models of the thermodynamic parts were developed and connected to the said frame. Thus, the effects of the different thermodynamic models can be studied individually. Roughly, a steam turbine generates power with the Clausius-Rankine-Process, expanding steam from high temperature ϑin and high pressure pin to low temperature ϑout and low pressure pout. This expansion is associated with a loss of specific enthalpy, Δh = hin - hout . The thermodynamic power output of a turbine, Pth, is directly proportional to the steam mass flow m& and the enthalpy drop Δh Pth = Δh m& . Ideal expansion is isentropic, i.e. without any change in the entropy s, see figure 2. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes 100bar 300bar 4000 30bar 3bar 10bar h (kJ/kg) 600°C 1bar 3600 500°C 400°C 0.3 bar 3200 300°C 0.1 bar 200°C 2800 0.05 bar 100°C 0.02 bar x=1 x=0.95 2400 x=0.9 x=0.8 6 x=0.85 7 8 S (kJ/kg.K) Figure 2: Expansion in a steam turbine; - - - ideal expansion; ________ real expansion Real expansion is polytropic, i.e. with an increase in entropy. Hence Pth = Δh poly m& = η Δhis m& . Thus, the ‘real’ thermodynamic power output is less than the theoretical power output, the decrease is described by the efficiency η. 3. Simulation Models Since the effects of the different thermodynamic models with respect to predicting the transient behavior of a steam turbine shall be studied in this paper, the following description of the simulation models is divided into the thermodynamic description of the turbine, and into the rest of the power plant. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes 3.1. Thermodynamic Turbine Models In order to meet different needs in simulation complexity, three turbine models with different degrees of complexity were developed. All thermodynamic turbine models are designed as storage-throttle-systems. Hence, the internal volume of a turbine is divided into several (steam-) storages which are linked together by throttle devices. Thus, pipes, valves, and turbine sections are regarded as throttles, whereas the volumes of the devices represent the storage. This general principle is sketched in figure 3. throttle storage steam mass flow Figure 3: Turbine as a storage-throttle-system The simplest approach describes the turbine as a linear system, see [1], [3] e.g. An extension takes the nonlinearities which due to changes of thermodynamic states into account, but still abides the ideal-gasbehaviour assumption [2]. Finally, the most rigorous model thoroughly considers mass and energy flows as well as water-steam-properties in detail. 3.1.1. Linear Turbine Model Firstly, each turbine section is described through linear relationships. A storage is simply viewed as a steam mass storage, hence (3.1) m ( t ) = ∫ (m& in (τ ) − m& out (τ ) ) d τ . Furthermore, it is assumed that the pressure in a storage is proportional to the stored mass of steam: p(t ) = K M m(t ) = K M ∫ (m& in (τ ) − m& out (τ ) ) dτ (3.2) (3.2) holds true for ideal gases. Considering the overall turbine, it is widely known that - in steady state operation - the steam mass flow through the turbine is proportional to the pressure at the first stage (inlet) and that , except at low load conditions, the power output is almost proportional to the steam mass flow through the turbine. Generalizing this observation to a turbine section, the following equations are derived to describe the pressure and the steam mass flow through a turbine section. (3.3) m& out (t ) = K p p (t ) Assumptions (3.2) and (3.3) hold true in a wide set of steady state operating points. However, the respective proportional constants depend on the particularly chosen operating point. Considering a control valve, steam mass flow through the valve not only depends on the pressure in front of the control valve, but also on the valve lift and on the pressure behind the valve. Since the valves considered here are always in front of a turbine section, equation (3.3) holds true for the turbine section subsequent to the respective valve. Furthermore, it is known (see (3.8), next subsection) that at fixed valve lift and fixed pressure ratio π = pb/pa, the steam mass flow m& is directly proportional to the pressure pa in front of the valve. Therefore, the steam mass flow can be approximated by the product of the pressure in front, pa, and a function of the valve lift hV: Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes m& V (t ) = p a (t ) * char (hV ) (3.4) The function “char(hV)“ is called the “valve characteristic”. It depends on the mechanical design of the valve as well as on the thermodynamic range of application. A typical valve characteristic for a control valve is displayed in figure 4. Figure 4: Typical control valve characteristic In general, control valves operate with super critical as well as with sub critical pressure ratios. Figure 4 shows a linear behaviour in the lower part. This is related to a super critical pressure ratio and, consequently, a critical discharge through the valve. At valve lifts above 25% to 30% pressure ratio as well as discharge through the valve are sub critical. Hence comparatively large changes in the valve lift result into relatively small changes with respect to the resulting steam mass flow. Obviously, the simple modelling approach does not consider any effect associated with change of temperature and heat flow, respectively. Rewriting the derived equations as block diagrams results in the plant described by figure 5. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes Figure 5: Linear simulation model for a HP-IP-LP turbine Control inputs of the plant are the valve lifts hMS and hRS, output is the power P. The states of the linear model (3.1), (3.3) correspond to the stored steam masses. 3.1.2. Non Linear Turbine Model In a second approach, the idea of the storage being a mass storage according to (3.1) is maintained, whereas the linearity in steam mass flow is abandoned. Instead, steam mass flow is modeled according to the thermodynamic standard description, see e.g. [4]. Thus, it is distinguished between flow through a pipe, a turbine (section) and a valve. Let p0, v0, and m& 0 denote the steam parameters and mass flow, respectively, at a given design condition. Assuming ideal gas properties and using T0 for the absolute temperature at design condition, the equation pv p 0 v 0 = T T0 (3.5) is obtained. Hence, with T = T0 pv = p 0 v 0 holds true. Steam mass flow through a pipe segment is described by m& = m& 0 v 0 ,b p b − p a vb (p 0 ,b − p0,a ) sign( pb − p a ) (3.6) (3.7) With (3.6) , (3.7) is simplified to m& = m& 0 pa pa − pb sign( pa − pb ) . p0 ,a ( p0 ,a − p0 ,b ) Let A and A0 denote the free area and the maximum free area respectively of a valve and let π 0 = p 0 ,b p0 ,a denote the design pressure ratio. With the isentropic exponent κ = κ(pa,va), the critical pressure ratio is determined through κ ⎛ 2 ⎞ κ −1 ε cr = ⎜ ⎟ , ⎝ κ + 1⎠ and thus steam mass flow through a valve is described by Author: Title: m& = m& 0 G. Zimmer Modelling and Simulation of Steam Turbine Processes A A0 2κ κ −1 pa va 2κ 0 κ 0 −1 p0,a v0,a ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 ⎛ pb ⎞ κ ⎛ pb ⎞ ⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎝ pa ⎠ ⎝ pa ⎠ 2 π 0 κ0 − π 0 κ +1 κ κ 0 +1 κ0 , pb ≥ ε cr pa , pb < ε cr pa κ +1 (ε cr )κ − (ε cr ) κ 2 π0 2 κ0 −π0 κ 0 +1 κ0 Assuming κ ≈ κ0 and using (3.5) once again, steam mass flow through a valve may be simplified to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ A pa ⎪ m& = m& 0 ⎨ A0 p0 ,a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 ⎛ pb ⎞ κ ⎛ p b ⎜⎜ ⎟⎟ − ⎜⎜ ⎝ pa ⎠ ⎝ pa 2 π 0 κ0 − π 0 ⎞ ⎟⎟ ⎠ κ +1 κ κ 0 +1 κ0 , pb ≥ ε cr pa . κ +1 (ε cr )κ − (ε cr ) κ 2 2 π 0 κ0 − π 0 (3.8) κ 0 +1 κ0 , pb < ε cr pa According to [4], steam mass flow through a turbine section is, given by p 0 ,a v0 ,a m& = m& 0 p a va (p (p − pb2 ) 2 a 2 0 ,a − p02,b ) , or, once again utilising (3.6), by (p m& = m& 0 (p 2 a 2 0 ,a − pb2 ) . − p02,b ) (3.9) Power output of a turbine section is the product of steam mass flow m& and enthalpy-drop Δh which in turn is the product of the isenthalpic enthalpy drop and the efficiency, Δh = Δhs η. Using ideal gas theory once again κ −1 ⎞ ⎛ ⎜ ⎛ pb ⎞ κ ⎟ ⎜ ⎟ . p v 1− ΔhS = κ − 1 a a ⎜⎜ ⎜⎝ pa ⎟⎠ ⎟⎟ ⎠ ⎝ κ Assuming constant efficiency η ≈ η0, κ ≈ κ0, and using (3.5) once again, Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes κ 0 −1 ⎛ ⎜ ⎛ pb ⎞ κ 0 ⎜ 1 − ⎜⎜ ⎟⎟ ⎝ pa ⎠ Δh = Δh0 ⎜⎜ κ 0 −1 ⎜ ⎛ p 0 ,b ⎞ κ 0 ⎟ ⎜⎜ 1 − ⎜⎜ ⎟ p 0 , a ⎠ ⎝ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎟ ⎠ (3.10) is obtained. Thus with (3.9) and (3.10) the power output P is obtained as (p P = m& Δh = P0 (p 2 a 2 0 ,a − pb2 ) − p02,b ) ⎛p ⎞ 1 − ⎜⎜ b ⎟⎟ ⎝ pa ⎠ ⎛p 1 − ⎜⎜ 0,b ⎝ p0,a κ 0 −1 κ0 ⎞ ⎟ ⎟ ⎠ κ 0 −1 κ0 . Using the equations derived above, a non linear simulation model for a HP-IP-LP turbine in form of a block diagram is developed as depicted in figure 6. Figure 6: Non linear simulation model of a HP-IP-LP turbine The steam flow through the re-heater was modelled as the flow through a pipe. Consequently, transient effects of heat transfer are neglected. Note that the derived system has the same control inputs hMS and hRS, and output P as has the linear system, depicted in figure 5. The states of the non linear model also correspond to the stored steam masses. Coupling to the environment is given through the steam generator pressure and the condenser pressure. Thus through the condenser pressure the non linear model features one more ‘input’ than the linear model. 3.1.3. Rigorous Turbine Model As soon as operating conditions are considered which entail changes in temperature, the above models will not be capable of providing reliable predictions. Hence a turbine model is required which is capable of precisely describing the thermodynamics inside the turbine over the whole range of relevant operating points. Since the existing turbine models [1,4] did not meet the requirements, a rigorous model was developed by SIEMENS (patent pending). This rigorous model uses the non linear mass flow equations (3.7) to (3.10), and considers mass as well as energy storage capacities. Hence, the mass balance (3.1) is augmented an energy balance: t m(t ) = m 0 + ∫ (m& in (τ ) − m& out (τ ) ) dτ 0 t U (t ) = U 0 + ∫( 0 ) Q& (τ ) + m& in (τ )hin (τ ) − m& out (τ )hout (τ ) dτ (3.11) Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes In particular, the re-heater is modelled more precisely by considering the heat-flow Q& which is required to lift a steam mass flow m& from the enthalpy ha at the entrance to the enthalpy hb at the exit. Accordingly, this heat flow is equal to Q& = m& (h a − hb ) . V U Since the specific volume is equal to v = and the specific internal energy equals u = , a complete m m description of the state of the steam inside the storage is obtained. That means, by using the steam-watertables, the properties p, v, h, x, e.t.c. are directly accessible. Hence, the storage model features mass flows as well as energy flows, in terms of m& h or Q& , as inputs and delivers the steams thermodynamic state as the output. Instead of continuing to apply the simplified equations for steam mass flow through a pipe, a valve or a turbine section, the rigorous descriptions of the throttle elements are enhanced by featuring the exact equations (3.7), (3.8) and (3.10). Additionally, power output of a turbine can be rendered more precisely by using Δh poly = η Δhis with Δhis = ha − hb ,is = ha − h( p b , s ( p a , ha ) ) . The isentropic enthalpy drop Δhis, which corresponds to the pressure drop from pa to pb, is directly accessed via the steam water tables. Thus, opposite to storages, throttle elements require the steam states of the neighbouring storages as inputs and provide mass- and energy flows as outputs. Hence, although having different internal realizations and supplying mode details, the simulation model for the rigorous turbine description, depicted in figure 6, looks very similar to the nonlinear simulation model in figure 7. Figure 7: Rigorous simulation model of a HP-IP-LP turbine Another advantage is that the rigorous description allows for modelling condensation and evaporation without further efforts. Condensation occurs whenever energy is extracted below the condensation level, i.e. enthalpy drops below h”(p). On the other hand, whenever energy is supplied, water starts to evaporate as soon as the enthalpy rises above h’(p). When h”(p) is exceeded, evaporation is completed. 3.2. Remaining Components All remaining components of the water-steam cycle and the turbine-generator, such as boiler, condenser, turbine shaft, generator, as well as the controller- and protection-system, will be described by fairly simple models. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes 3.2.1. Boiler and Condenser Models In this context, a boiler as well as a condenser is simply a constant, namely a pressure or a constant steam state, providing pressure and enthalpy. 3.2.2. Mechanical Devices: Shaft and Valve Model A momentum balanced system yields the equation to determine the variation of angular velocity of the shaft dω Θ =∑ M f − ∑ M br . (3.12) dt P Using ω = 2π n and M f = , ω dn 1 ⎛ P ⎞ = − ∑ M br ⎟ ⎜ dt Θ2π ⎝ 2π n ⎠ is obtained. The braking torque Mbr consists of the frictional torque of the shaft and the torque which is due to the generator load. The friction depends on the speed n and is determined by a plant specific characteristic. The mechanical part of a valve is simply described by a limited integrator. The value of the integral corresponds to the valve lift. The integrand consequently corresponds to the velocity, which in turn is determined by the valve controller. The valves free area is determined by a characteristic h2A(hV), which maps the lift to the corresponding discharge area.: d hV = K (hd (t ) − hV (t ) ) dt AV = h 2 A(hV ) The characteristic depends on the mechanical layout and is individual for each valve. In general, it is tried to construct a valve with a linear characteristic, i.e. A = KV*h. 3.2.3. Electrical devices: Generator and Electrical Grid Models A simple generator model considers the slip between turbine speed n and grid frequency f to determine the polar wheel angle ε. The generators braking torque MG,br is the sum of a damping torque, which is proportional to the slip σ = n-f, a forcing torque, which is proportional to the sine of the polar wheel angle, and an internal loss, which depends on the speed n. The corresponding torque MG,f onto the electrical grid is the braking torque without the internal losses (see [5]) ε = ∫ (n(τ ) − f (τ ) ) dτ M G ,br = K 1 sin(ε ) + K 2 ( n − f ) + K 3 n M G , f = K 1 sin(ε ) + K 2 ( n − f ) The electrical grid is either modeled by a constant frequency f, or by a system analogously to (3.12), which would slowly react to unbalance in forcing and braking torques, i.e. a system with a huge inertia. The latter model is used to study a turbine controllers response to the demand of frequency support in the electrical grid. 3.2.4. Turbine Governor and Protection System The turbine governor controls the speed of the turbine as well as the power output. In principle, it operates in two different modes: ramp-up and load. In the first mode, the turbine governor solely operates as a speed controller. In the second mode, the turbine speed is Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes mainly fixed by the frequency of the electrical grid. Here the governor operates on a characteristic, combining design speed and design load. That is, the controller mainly responds to the demands of the load dispatcher, but also reacts automatically to maintain the design frequency of the electrical grid. In both modes, the governor is designed as a PI-Controller. In general, a protection system is designed to issue a trip-signal once a monitored mode leaves an admissible area. A turbine, however, features yet another device, a failure mode detection system. This system is mainly designed to detect a load rejection. That is, the turbine as well as the generator are operating properly, but the electric connection between the generator and the electrical grid has been lost. In this case, the generator still supplies electrical energy to operate the turbine auxiliary systems, e.g. the coal mills and the pumps. The governor has then to switch to “speed control” since the turbine speed is no longer fixed by the frequency of the electrical grid. 3.3. Complete Power Plant Model A complete power plant, consisting of a steam turbine, controller, valve actuators, shaft, generator and grid coupling is modelled according to figure 8: Figure 8: Simulation model of a HP-IP-LP power plant in SIMULINK Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes For simplicity, the water-steam system has been neglected. Thus the condenser, which in itself represents a complex, dynamic system, is not modelled but give as a constant steam state, i.e. pressure and enthalpy, the same holds true for the steam generator. The complete feed water system in between is omitted. The sub-model “HP-IP-LP turbine” can be given by either of the three previously discussed thermodynamic turbine models. The remaining components, turbine controller, shaft, valve actuators e.t.c., remain the same, regardless of the thermodynamic details of the utilised turbine model. The model, lined out in figure 8, thus may be used to study the impacts of the degree of thermodynamic details. 4. Investigated Operation and Failure Modes The three outlined approaches to describe turbine thermodynamics as well as their range of application will be exemplified by reference turbine models, each model built with the outlined level of detail. The surrounding elements, shaft, generator, grid, turbine governor, and control valves, will be the same for all three models. Different operating conditions and failure modes, respectively, will be considered and simulated with the three differently detailed turbine models. If available, the simulation results will be compared to actual plant data. The level of detail which is required for standard tasks as well as some necessary failure investigations, will be discussed. 4.1. Operational Modes During commissioning, numerous tests are conducted and documented. Here the records of three standard commissioning tests of a certain HP-IP-LP steam power plant were selected to compare simulation data to turbine data. The utilized commissioning records were provided in form of monitoring charts. Those plots not only depict the subsequently analysed data, but also numerous other variables. Figures 8, 10 and 12 which provide the plant data are cut-outs of the original plot. Unfortunately, the resulting quality turned out to be somewhat poor. 4.1.1. Turbine Governor Test: Speed Control Firstly, a turbine governor test with respect to speed control is investigated. The test is conducted by following a trajectory with two steps, nominal speed to 102% speed, and then back to nominal speed. Figure 9 depicts details of data obtained during commissioning of a power plant. The recorded values of turbine speed and valve lift will be compared to data obtained by simulating the according process. Since the data are only available as a printout, the quality of the figure is a bit poor and some other data, which are not relevant in this context, are visible as well. In order to enhance understandability, the employed data were bolded. Since the axes of the original diagram had to be cut, some relevant values were specified. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes 1/s 3060 turbine speed % 15 main steam valve lift 14.6 % 10 3000 6.8 % 5 200 220 240 260 280 300 320 s 200 220 240 260 280 300 320 s Figure 9: Turbine governor test: speed control (plant data); left (in red) turbine speed, right (in black) corresponding main steam valve lift Figure 10 shows the results of the turbine test procedure and simulation with the three different turbine models. Figure 10: Turbine governor test: speed control, simulation results As can be seen on the left hand side of figure 10, the linear model follows the speed set points, but mirrors the details very poorly, whereas both the non linear and the rigorous model display the real plant behaviour regarding overshoot and settling time almost perfectly. In fact, no difference between the speed of the non linear and the rigorous model is visible. The fact that all three models follow the speed set-point very well is not astonishing; the speed set-point is the controlled variable and hence included in an according feed-back loop. Regarding the valve lift, the results of the linear model are obviously out of range. This is mainly due to the fact that most of the assumptions made for the linear model do not hold at low load conditions. Minor differences are noticeable in valve lift for the non linear and the rigorous approach, but the correspondence of stationary value and overshoot regarding is still very good in both cases. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes 4.1.2. Turbine Governor Test: Ramp-up Another test for the speed controller is the ramp-up of the turbine from turning speed to design speed. Here the turbine speed has to follow a ramp. The allowable transient is in the range of 100 – 900 rev/min². Normally, the gradient is set to values between 500 and 700 rev/min². Figures 11 and 12 depict a turbine ramp-up from 800 rev/min to design speed of 3000 rev/min, i.e. 50 Hz. Figure 11 gives details of data, recorded at commissioning. Thus the same drawbacks as already encountered for figure 9 have to be faced. 1/s 3000 2000 main steam valve lift turbine speed % 10 9.1 % 6.8 % 5 1000 100 120 140 160 180 200 220 s 0 100 120 140 160 180 200s Figure 11: Turbine governor test: ramp-up (plant data) left (in red) turbine speed, right (in black) corresponding main steam valve lift Figure 12: Turbine governor test: ramp-up, simulation In this case, all three simulation models follow the ramp very well. This is not surprising, since the speed is the controlled state component and the controller acts upon the individual models (virtual) measurement. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes Once again, it can be noticed that the simulation results of the valve lift which were obtained with the linear model do not match the real data very well. In contrast, the nonlinear and the rigorous model display fairly accurate quantitative predictions of the valve lift as well. 4.1.3. Turbine Governor Test: Load rejection A very important and crucial test for the turbine governor is a load rejection. Here the turbine is run at partial or full load and the connection to the electrical grid is (at test conditions!) deliberately shut off. Consequently, the mechanical system is no longer in an equilibrium with respect to torque. Since the braking torque of the grid is lost, the shaft will accelerate. Now, the turbine controller should intercept the turbine at nominal speed, either idling (no load at all) or with house load, i.e. the generator is still running and supplies electricity to the electrical devices of the turbine and the plants auxiliary systems. Figure 13 depicts plant data of a load rejection from 66% load to idling with an excited generator. Once again, the diagram depicts details of a commissioning printout. Thus the drawbacks which were already addressed in the context of figures 9 and 11 apply here, too. 1/s 3111 /s = 103,7 % 3120 turbine speed 3060 15,2 s 3000 100 106 112 118 s Figure 13: Load rejection, plant data Unfortunately, the available monitoring chart does not depict the time horizon until the speed is permanently back to nominal speed. As the in the previous sections, the commissioning scenario was simulated with all three types of simulation models. The results are depicted in figure 14. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes Figure 14: Load rejection, simulation results Simulation results which display a load rejection from 66% load to idling (with a still exited generator) once again display a fairly good match between simulation data and plant data during the first 20 seconds after separation from the electrical grid. The linear model displays a maximum speed of 105%, which overestimates the real speed by approximately 1%. The non linear model overestimates the maximum speed by roughly 0,4%. An almost perfect match is reached with the rigorous model. Not only is the maximum speed met exactly, but also the time when maximum speed (2,1 sec) is obtained, as well as the time when the turbine speed is firstly back to nominal (synchronous) speed (15,3 sec), are almost identical for plant data and simulated data. In all cases investigated so far, the linear model roughly displays the turbine behaviour as long as the controlled outputs, i.e. speed and/or power output are concerned. Internal quantities, valve lift for example, are poorly mirrored with respect to the absolute values, but are still more or less correct with respect to the direction and to the direction of change of the valve movement. By contrast, the non linear as well as the rigorous model not only depict the turbine behaviour, but also simulate the corresponding valve lift fairly precisely. The standard turbine operation modes discussed above obey the assumption of isothermal conditions. Hence, it is fairly reasonable that the nonlinear and the rigorous model display very similar results since the investigated cases in fact operate at almost isothermal conditions. 4.2. Failure Modes Secondly, some turbine failure modes shall be investigated. Failures modes with a reasonable probability of occurrence have to be anticipated and, consequently, sufficient measures have to be provided to protect the turbine and the adjacent devices in case the failure really occurs. Since the majority of possible failures never occur, there is no adequate data base to compare the simulation results to plant data. 4.2.1. Pressure protection Pressure protection will shut down the turbine once a critical pressure (with respect to a certain device) is reached. In this context, simulation results are used to identify necessary Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes protection measures beforehand. One protection measure is to predict the resulting maximum pressure and design the devices accordingly. An other measure would be to determine a sufficiently small automation delay, which ensures that the pressure remains below an admissible level. In general, there is a certain pressure and temperature drop along the steam path. Main steam, which enters the HP section with 260 bar and 540°C, will be expanded to around 60 bar and 350°C. Now it shall be assumed that the main steam valves start to open while the cold end of the re-heat system, that is the piping system leading to the boiler, is blocked. Thus, the main steam supply is operating and hence main steam is supplied to the HP section, whereas the discharge is blocked except for the sealing leakages. Hence, the HP section as well as the reheat section, which is designed for a significantly lower pressure than the main steam generation section of the boiler, will be supplied with high pressure steam. Consequently, the pressure will rise until pressure protection will issue a shut down, and the main steam valves are closed as well. Figure 15 displays the simulated rise in pressure at the exhaust of the HP-section. Figure 15: Simulated operation of the pressure protection system, development of the HP-exhaust pressure Again, it can be seen that the nonlinear and the rigorous model display a similar behaviour with respect to the dynamic pressure development in the first seconds, whereas the linear model is completely out of range. Concerning the long term development, certain differences between the non linear model and the rigorous model become obvious. The calculated pressure for the non linear models starts to decrease approximately three seconds after the pressure protection tripped the turbine. The pressure calculated with the rigorous model increases for approximately three minutes. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes The reason for the different development becomes obvious if the temperature development in the HPexhaust is considered as well. The non linear model is designed for isothermal processes, whereas the temperature in the HP-exhaust rises significantly above the design temperature. See figure 16 to view the temperature increase inside the cold reheat. Figure 16: Simulated operation of the pressure protection system, development of the HP-exhaust temperature Furthermore, it is investigated how different delay times in the automation system will result in different maximum pressures. In order to do so, the above described set-up was run for the three different models utilizing various delay times. Figure 17 depicts the resulting pressure for the tree different models in dependence of the applied delay time. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes Figure 17: Simulated operation of the pressure protection system, development of the HP-exhaust pressure, depending on the automation delay time The allowable pressure limit with respect to the piping material is 47 bar. Thus, relying on the results of the linear model, one would be forced to operate with an overall delay time of less than 50 ms. Due to the 55 ms reaction time of the stop valve’s servo, this it impossible. Relying on the results of the simulation results of the non-linear model, an overall reaction time of 325 ms would be sufficient. The simulation outputs of the rigorous model resulted more conservatively into a required maximum delay time of less than 300 ms. Ignoring the requirements of the linear model, the overall delay time of the considered plant was restricted to 255 ms. With the restriction of the fixed stop valve reaction time of 55 ms, this was achieved by using a pressure transducer with 100 ms delay, a fast automation cycle, accounting for another 100 ms. it can be seen that the resulting maximum pressure rises by 2 bar when the delay time is raised by 100 ms. Hence, the material of the turbine casing and piping determines the required delay time. If the piping system sustains 50 bar, 200 ms reaction time is sufficient. If the material limits are reached at 47 bar, a faster reaction, and thus the shorter delay time, is necessary. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes 4.2.2. Evaporation Finally, some effects of evaporation will be studied. Here a turbine with steam extractions to condensation pre-heaters is considered. The condensation occurs at a pressure which is slightly below the pressure of the extraction point. Consequently, a pressure drop inside the turbine could cause a reversal in the direction of flow and, subsequently, evaporation out of the condensate vessels. Thus the turbine might be accelerated by the reversed extraction steam. In general, non return valves protect the turbine against steam mass flow out of the condensate vessels and thus against undesirable acceleration. Fast closing of the control valves, which happen at load rejection for example, is accompanied by a pressure drop. It was stated in paragraph 4.1.3. that the turbine governor should be capable of intercepting the turbine at nominal speed. If steam evaporating out of the pre heaters is supplied to the turbine, it might gain additional momentum and subsequently accelerate to trip speed or, even worse, to over-speed. If the turbine controller fails, the turbine is protected by the “over-speed protection system”, which automatically issues a turbine trip if the speed exceeds trip speed, which is usually set to 110% nominal speed. The over-speed protection system is designed to ensure that the turbine speed remains below overspeed, which is usually 120% of nominal speed. Hence, those extractions which might be critical with respect to the maximum allowable speed have to be identified and protected through redundant non return valves. Since the rigorous model is the only one which can handle the effects of condensation and evaporation without further efforts, the subsequently displayed calculations were only conducted with the rigorous turbine model. Figure 18: Comparative study: simulated malfunctioning of extraction non return valves (NRV) left: interception by turbine governor; right: turbine trip at trip speed (110% of nominal speed) In figure 18 the effects of malfunction of non return valves with subsequent evaporation are studied. The solid line shows the speed development during load rejection. On the left hand side the speed development with a properly operating control system, on the right hand side the resulting speed development without control, but over-speed-trip at 110% speed, is shown. Under the assumption that the non return valve in the 150 bar extraction line (high pressure section) fails, it can be seen (dashed lines) that the resulting maximum speed is obviously higher than the maximum speed with properly operating non return valves. Hence in both cases, the situation does not become critical. The maximum speed still remains below trip speed (110%) if the control system operates properly, and below over-speed (120% of nominal speed) in either case. Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes The dotted line mirrors the turbine behaviour if the non return valve in the 20 bar extraction line fails. It is clearly seen that the maximum speed exceeds trip speed in the case where the controller operates properly. If the controller fails and a trip is issued at trip speed, the maximum speed would be higher than over-speed, which is undesirable. Hence, proper operation of the 20 bar line non return valve is mandatory. Consequently, this valve has to be provided redundantly, i.e. two consecutive non return valves have to be used. 5. Summary The preceding sections gave an overview of different approaches to model the transient thermodynamic and mechanical behaviour of a steam turbine. Three different approaches were presented and compared to plant data gained at standard tests during commissioning. For standard situations it could be seen that all three models simulate the turbine behaviour quite well, although differences concerning the quantitative description were noticeable. Thus, the linear model displays the turbine behaviour, but already fails to make sufficient precise prediction of the development with respect to valve lift and pressure. The non linear as well as the rigorous model mirror the thermodynamics very well as long as isothermal operating conditions are sustained. For non isothermal operating conditions, only the rigorous model is suitable for simulating the steam turbine. Subsequently, it was outlined how situations which might occur due to failure modes or malfunctioning of a single device may be anticipated and evaluated. Hence, if during the turbine development phase simulation predicts that additional precautions have to be taken, those can be included in the design at an early stage already. That is significantly more economical than adding additional features, such as a second non return valve, or exchanging devices with stronger material, or adapting the automation system for faster control reaction. If quantitative prediction is required in an operation mode which is accompanied with heat transfer or a change in temperature, it became obvious that only the rigorous model provided reliable results. Thus, the linear model may be used if a quick overview of the turbine behaviour is required and no quantitative predictions with respect to the thermodynamic performance are needed. The nonlinear model additionally simulates the thermodynamic behaviour as well as the behaviour of other devices such as valve lift, turbine speed etc as long as the isothermal assumption is fulfilled. In case an exact prediction of a transient operation or a failure mode which is accompanied with a change in temperature is needed, only the rigorous approach will deliver reliable simulations. 6. Nomenclature 6.1. Abbreviations HP IP LP NRV high pressure intermediate pressure low pressure non return valve Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes 6.2. Notation h h m m& p P Q& T t u U v V ε κ η π ϑ ω Θ specific Enthalpy (thermodynamical), [kJ/kg] lift (mecanical), [m] mass, [kg] mass flow, [kg/s] pressure, [bar] power, [MW] energy flow absolute temperature (thermodynamic), [K] time, [s] specific energy, [kJ/kg] energy, [kJ] specific volume, [m³/kg] volume,[m³] pole wheel angle, [rad] insentropic exponent efficiency pressure ratio temperature, [°C] angular velocity, [rad/s] inertia 6.3. Indices a b cr 0 in out in front of a section after a section critical design value flow into a section flow out of a section References [1] Mathias, G., 1991, Zur Nachbildung anlagenseitiger Störungen bei Dampfturbosätzen, BWK, 43, (9), 403-416. [2] Ordys, A. W., Pike, A. W., Johnson, M. A., Katebi, R. M., Grimble, M. J., 1994, Modelling and Simulation of Power Generation Plants, (London: Springer Lecture Notes). [3] Teichmann, W., 1983, Angewandte Anlagenautomatisierung, (Berlin: VEB Verlag Technik). [4] Traupel, W., 1988, Thermische Turbomaschinen I und II, (Berlin, Heidelberg, New York: Springer). [5] Weedy, B. M., Cory, B. J., 1999, Electrical Power Systems, (Chichester, New York: John Wiley & Sons) Author: Title: G. Zimmer Modelling and Simulation of Steam Turbine Processes Erschienen: Author: Gerta Zimmer a Affiliation: a Siemens Power Generation, M lheim, Germany DOI: 10.1080/13873950802384001 Publication Frequency: 6 issues per year Published in: Mathematical and Computer Modelling of Dynamical Systems, Volume 14, Issue 6 December 2008 , pages 469 - 493 Subjects: Analysis - Mathematics; Applied Mechanics; Dynamical Control Systems; Dynamical Systems; Mathematical Modelling; Mathematics & Statistics for Engineers; Simulation & Modeling; View publication stats