A steam turbine dynamic model for full scope power plant simulators

Accepted Manuscript Research Paper A Steam Turbine Dynamic Model for Full Scope Power Plant Simulators Cesar Celis, Gustavo R.S. Pinto, Tairo Teixeira, Érica Xavier PII: DOI: Reference:

S1359-4311(17)30620-8 http://dx.doi.org/10.1016/j.applthermaleng.2017.03.131 ATE 10135

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

28 January 2017 17 March 2017 29 March 2017

Please cite this article as: C. Celis, G.R.S. Pinto, T. Teixeira, E. Xavier, A Steam Turbine Dynamic Model for Full Scope Power Plant Simulators, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/ j.applthermaleng.2017.03.131

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A Steam Turbine Dynamic Model for Full Scope Power Plant Simulators †

Cesar Celis , Gustavo R. S. Pinto, Tairo Teixeira GT2 Energia R. Hélio de Almeida, s/n, Sala 38, Cidade Universitária UFRJ, Rio de Janeiro, 21941-614, Brazil Érica Xavier [email protected]

Abstract

The development of a steam turbine dynamic model for full scope power plant simulators is described in this work. Model completeness is one of the main features of the model developed as it attempts to completely cover the operating envelope of steam turbines. For modelling purposes, distinct regions within the safe operating envelope of steam turbines are discriminated. Accordingly, the corresponding model formulation associated with each of the steam turbine-related operation modes accounted for is detailed. A particular emphasis is put on thermal and rotational inertia effects that are important during start up and shut down of steam turbines. The developed steam turbine dynamic model is utilized for simulating a 300 MW class steam turbine belonging to an existing combined cycle power plant. Comparisons between results obtained from such simulations and those characterizing the equipment actual operation are discussed. The agreement between model results and operating data is in general good, as in average the corresponding discrepancies are of the order of 2% or less. The outcomes from this work suggest that producing high fidelity results from steam turbine dynamic models requires accounting for a number of physical phenomena, primarily, those related to heat transfer processes. Keywords: Power plants, Steam turbines, Dynamic modelling, Full scope simulators

†

Corresponding author – Tel.: +55 21 99437 0723 E-mail address: [email protected] (C. Celis)

Nomenclature

1 Introduction Relative to the 2013 figures, global energy demand is expected to increase near onethird by 2040 [1]. This growth of the world energy demand will push even further the requirements for efficient power plants. Power plant performance then, in particular operational performance, will continue to be a crucial aspect with respect to environmental constraints and fuel costs. Two main points are worth highlighting. Firstly, unskilled actions of power plant operational personnel usually lead to deterioration of units’ efficiency and reliability [2], which may considerably affect the operating performance of power plants. Secondly, skilled and experienced people, capable of properly operating power plants are difficult to find in the market. This context promotes the need for the development of tools and procedures for preparing, through teaching and training, new workforces for power plants; as well for retraining experienced operators referring to new technologies, new control processes and new operational procedures in general [2]. One effective way of dealing with the referred need involves the use of computerbased simulators representing actual power plants. Power plant simulators have been around for a very long time – see for instance Zanobetti’s work [3] for a review and classification of early simulators utilized in power engineering. There are several types and classifications of power plant simulators. One particular type is the so-called full scope (full scale or full replica) simulators, which incorporate detailed computer models of whole power plants, including their control room along with all consoles, switches, keys, indicators and other human-machine interfaces [2]. In these simulators, the responses of the simulated power plant are identical, in both time and indication, to the responses received in the actual power plant control room under similar operating conditions [4]. For some of the full scope power plant simulators developed in the past the interested reader is referred to references [4-7]. Computer-based models for full scope power plant simulators are similar to those used in long-term dynamic simulations of power systems. There is an additional requirement to be fulfilled by the formers however, that is, they must be run in real time. This implies that computational cost is a primary issue in these models. In general, the more accurate the model, the more cost intensive it is. There has to be a trade-off then at some stage between model accuracy and computational cost. Kola et al. [8]

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discuss early power plant models of varying accuracy such as to allow the user tradingoff between accuracy and computation speed to achieve a desired power plant simulator performance. Lu [9] presents general steady-state and dynamic models for power plant components. More specifically, several dynamic models based on mass and energy conservation laws for single/two-phase water/steam volumes with and without heat transfer are described there. Steam turbines represent major components in power plants, especially, in combined cycle-based ones. Accordingly, several steam turbine dynamic models of varying complexity have been developed over the years. The development of these models has been carried out in different contexts, including: (i) combined cycle bottoming system responses to gas turbine exit varying conditions [10], (ii) transient behavior-focused analyses of combined cycle power plants (CCPPs) [11], (iii) dynamic characteristics modeling of biomass-fired steam cycle-based power plants [12], (iv) CCPPs start up procedures studies [13-16], (v) transient dynamics characterization of steam turbines sections [17], and (vi) non-linear modeling of steam turbine units [18]. More specifically, Kim et al. [10] and Shin et al. [11] describe steam turbine models featuring turbine sections treated as control volumes, where zero-dimensional mass and energy conservation equations are enforced. There, turbine sections pressure drops and isentropic efficiencies come from Stodola’s Ellipse law [19] and Spencer et al. [20], respectively. van Putten and Colonna [12] present a steam turbine model consisting of both a purely resistive fluid flow module and a storage solid one to account for the thermal capacity of turbine metal parts. The works by Alobaid et al. [13-16] describe commercial tools-based static and dynamic models developed for studying start up procedures of subcritical and supercritical heat recovery steam generators. In these models, the steam turbine is modeled using one-dimensional conservation equations of mass, momentum and energy. The turbine pressure and enthalpy drops, which are modeled in a similar fashion to [10,11], are added as source terms in the momentum and energy equations, respectively. Chaibakhsh and Ghaffari [17] discuss the development of different parametric models for steam turbine sections, based on energy balance, thermodynamic principles and semi-empirical relations. Either the associated model parameters are determined by empirical relations or adjusted by applying genetic algorithms. Jiang et al. [18] describe a non-linear steam turbine model developed using a genetic algorithms-

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based parameter identification method. The reasonable agreement between measure and simulation data indicates the accuracy of the proposed model. Using commercial tools, Benato et al. [21,22] develop and test dynamic models of combined cycle power plants. The focus in those works is predicting the power plants dynamic behavior during load variations, identifying the most stressed components, based mainly on steam pressure and temperatures, and estimating their lifetime reduction. Steam turbines are there modelled using the Stodola’s equation [19] and setting nominal operating conditions, as well as mechanical and isentropic efficiencies. Casella and Pretolani [23] model and simulate fast start up transients of combined cycle power plants. The aim in that work is reducing the power plant start up time while keeping the life-time consumption of the more critically stressed components, including the steam turbine, under control. The modeling of the steam turbine is similar to that used in [21,22]. Several other dynamic models for both steam turbines [24,28] and equipment closely related to, such as heat recovery steam generators [29-34], are available in literature. The interested reader may refer to the references provided for further details about some of these models. Furthermore, the most recent work by Alobaid et al. [35], presenting a comprehensive review of dynamic simulation, its development and application to various thermal power plants, is suggested. The referred review is divided in two parts, one dealing with the mathematical background required for modeling thermal power plants, and the other focused on the literature involving the application of dynamic simulation to specific energy system technologies, including combined cycle power plants. The present work describes the development of a steam turbine dynamic model for full scope power plant simulators. Model completeness constitutes one of the original contributions of this work. This is because the model developed here attempts to cover the whole operating envelope of steam turbines. Another original contribution of this work relates to the simultaneous use of both lumped steam and energy storage volumes. Indeed, for the modeling of steam turbines, similar steam storage volumes [21] and energy storage ones [12] have been separately utilized in the past. Nevertheless, to the best of our knowledge, no models have simultaneously included the type of steam and energy storage volumes used in this work. Accordingly, in Section 2 the main features of the dynamic model for steam turbines developed here are highlighted. This includes the corresponding formulation used for modeling particular regions of the steam turbine’s operating envelope. The use of the developed model to simulate a steam

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turbine belonging to an existing CCPP is discussed in Section 3. Finally, Section 4 highlights some of the main conclusions drawn from the results obtained. It is worth noticing that for modeling purposes an in-house modeling environment featuring several computational models is utilized in this work. These models focus on several power plant subsystems belonging to three different simulation domains, .i.e., thermo-fluid processes, electrical processes and control system-related processes. Such subsystems are both modeled according to the documentation of the particular power plant accounted for, and represented by dynamic-link libraries built from different programing languages. When building a full scope power simulator then, the referred subsystems are integrated within the modeling environment, which allows the flow of information among them and hence the influence of one subsystem on the others.

2 Steam turbine model The steam turbine dynamic model is developed in a context involving full scope power plant simulators. As such its applications involve real time simulations. Obtaining accurate results at low computational cost is thus one of the most desirable features of the model. It will be shown in Section 3 that the developed steam turbine dynamic model reasonably fulfills this accuracy-related requirement. The particular full scope power simulator accounted for in this work is an in-house tool developed right before the steam turbine model discussed here. Indeed, at some stage, there was an overlap between the simulator development and the development of some of the models (including the steam turbine one) utilized within the simulator. It is worth emphasizing that the referred in-house simulator follows a modeling approach similar to that described in [36,37]. Accordingly, flow and pressure networks, featuring generic potential (e.g. pressure) and flow (e.g. mass flow rate) variables [38], are utilized to characterize power plant thermo-fluid processes. This means that pressures and mass flow rates, for instance, are considered as bilaterally coupled variables (never enforced together on the system), whose product is a measure of the power transfer between the plant components and their surroundings [38]. The steam turbine modeled here constitutes one of these power plant components. Notice that the in-house full scope simulator is not the central point here. The main focus of this work is indeed on the development of a steam turbine dynamic model, usable in full scope power plant simulators.

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From a power plant simulator’s perspective, the steam turbine represents one of the accessories of the pressure and flow networks that imposes a pressure drop on the steam flow. Following [37] thus, at each time step, the steam turbine inlet mass flow rate and pressure distribution are obtained from the solution of the pressure and flow networks. The ST inlet temperature is in turn obtained from the upstream equipment exit conditions. Once known the ST inlet conditions, in terms of temperatures and mass flow rates, as well as pressures along it, the steam turbine dynamic model computes the flow conditions at the exit of each ST section. Additionally, the model also calculates other characteristic parameters such as steam turbine casing temperatures, power output and rotational speed. Other accessories that impose a pressure drop on the steam flow, such as control valves, are modeled within the flow and pressure networks. These accessories are not part then of the steam turbine model described here. The developed steam turbine dynamic model accounts for both steady and unsteady state operating conditions. Within the steam turbine safe operating envelope then, three distinct regions have been discriminated, (i) normal, (ii) start up/shut down and (iii) idle operation. The term ‘normal’ is used here to refer to the operation mode that englobes all steam turbine operating conditions other than those associated with start up, shut down and idle operation. This means that during ‘normal’ operation both steam turbine generator and power grid are synchronized. Idle operation has not been included in this work as it has no relevance in the context of power plant simulators. Therefore, the corresponding model formulation related to each of the two steam turbine operation modes accounted for is described below.

2.1 Normal operation As illustrated in Fig. 1, the steam turbine (ST) used as reference for modeling purposes comprises three sections, namely, high (HP), intermediate (IP) and low pressure (LP) sections. The last ST section (LP) is sub divided in turn in two ones, LP1 and LP2. Notice that the steam entering both IP and LP sections comes from two sources, the corresponding previous ST section and the heat recovery steam generator (HRSG). A three-pressure level with reheating HRSG is accounted for here. Although relatively simple, this configuration includes the essential components present in typical steam turbines found in power plants. Accordingly, as indicated above, the model inputs include steam mass flow rates and temperatures at Fig. 1 stations 1, 3 and 6, and

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pressures along the whole steam turbine. These mass flow rates and pressures are obtained from a Stodola-like equation, solved within the full scope simulator pressure and flow networks. More specifically, similarly to what is done in [17], a constant parameter relating mass flow rates and pressure drops across the steam turbine sections is fine-tuned using power plant actual operating data. During simulations then, this constant parameter is utilized by the simulator to determine the corresponding pressure drop associated to each mass flow rate. The steam temperatures used as inputs in the steam turbine model come in turn from the upstream equipment exit conditions. For the ST modeling a section-by-section approach is utilized instead of a more complicated stage-by-stage one. Accordingly, following Ordys et al. [39], each steam turbine section is represented by both (i) a lumped steam storage volume having dynamics, and (ii) a turbine section itself modeled by steady state relationships. The details of such a representation are illustrated in Fig. 2. The quasi-steady assumption used for modeling the turbine sections is based on the fact that the response time of a steam turbine is relatively short compared to other power system components, e.g. HRSG.

Fig. 1. Steam turbine diagram.

According to the approach used in this work, the steam storage volume is responsible for modeling the turbine section dynamics. This modeling process is carried out by solving mass and energy conservation equations [10,11], (1)

(2)

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along with an equation describing the section mass flow dynamics, between input and output, as a first order lag [39], (3) Pressure drops across the steam storage volumes are not accounted for in this work. The same occurs for steam kinetic energies. Momentum conservation equations are not required then in this work [11]. It is worth noticing that the steam turbine normal operation mode is characterized by a constant rotational speed. Rotational inertia effects are thus in this case negligible when compared to the corresponding ones characterizing the start up/shut down operation modes. Similarly, during normal operation, the influence on the results obtained of thermal inertia effects is small, when compared to what happens in the other operating modes accounted for in this work. For the sake of clarity thus, even though thermal and rotational inertia effects are also included in the normal operation mode, their discussion is postponed to Section 2.2 below. Accordingly, once specified the corresponding initial conditions, Eqs. (1)-(3) are integrated forward, in a time marching manner, in order to determine the subsequent system states. A first order Euler method has been utilized in this work and, as highlighted in literature [40], it has been proved an efficient algorithm for these purposes.

Steam Storage Volume

ST Section

Fig. 2. Generic representation of a steam turbine section for dynamic modeling.

As it can be seen from Fig. 2, the steam storage volume exit conditions define the entry conditions to the steam turbine section. Once known these entry conditions and using the hypotheses highlighted below, steady state relationships are used to estimate the steam exit conditions from each of the turbine sections considered. Accordingly, it is assumed [39,41] that the steam expansion in each turbine section follows a polytropic process of the form

. From the definition of polytropic (or small stage)

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efficiency [42],

, the infinitesimal enthalpy drop along the steam turbine

section is given by (using Maxwell’s relation

), (4)

The polytropic index is in turn related to the polytropic efficiency through the ratio of specific heats as follows [43],

. For a given turbine section

then, by integrating Eq. (4) between its inlet and outlet, the corresponding enthalpy drop is computed as (polytropic expansion) [41], (5) In order to use Eq. (5) to compute the enthalpy at a given turbine section exit one needs to know (i) the section entry conditions, (ii) the section outlet pressure, and (iii) the polytropic efficiency. At each time step, the section entry conditions are determined by solving the steam storage volume dynamics, Eqs. (1)-(3). The section outlet pressure is in turn taken from the solution of the pressure and flow networks at the previous time step. Finally it is assumed that the polytropic efficiency is constant and equal to the corresponding efficiency at design point. This hypothesis comes from the fact that, at a constant rotational speed, the stage efficiency mainly depends upon the enthalpy drop involved, which remains relatively constant during the normal operation mode, except in the last turbine stages [44]. The design point polytropic efficiency of each steam turbine section has been determined by tuning procedures using manufacturer data. From the section outlet pressure and enthalpy, Eq. (5), steam conditions at the turbine section exit are fully determined. Additionally, when required, the section isentropic efficiency is computed from [42,43],

(6)

This approach for computing the exit steam conditions is utilized in the three sections (HP, IP and LP) comprising the steam turbine modeled. Furthermore, the modeling approach of the other operation mode (start up/shut down) considered is similar to that described in this Section 2.1. For the sake of conciseness then, when

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discussing the model formulation corresponding to this second operation mode, only the main differences are highlighted.

2.2 Start up/shut down operation The modeling approach used for the steam turbine start up operation mode, which is applicable to cold, warm and hot start ups, is similar to that corresponding to the normal operation one. Even so there are two important differences that need to be emphasized. The referred differences relate to the thermal and rotational inertia effects that play a key role during this operation mode. Therefore these effects need to be properly accounted for in the steam turbine model such as to produce sound results. Indeed, thermal inertia effects appear because of the heat transfer processes present, for instance, during the warming-up of the turbine sections carried out as part of cold start up processes. Rotational (mechanical) inertia effects appear in turn as a consequence of bringing the steam turbine from a low rotational speed (rest or idle) to the speed corresponding to its normal operation. 2.2.1 Thermal inertia effects In this work thermal inertia effects are modeled using lumped energy storage volumes, similar to the steam storage ones utilized for modeling the turbine sections dynamics in normal operation mode. These energy storage volumes are then responsible for modeling the heat transfer processes occurring during the steam turbine start up. As it can be seen in Fig. 3, at each turbine section, the energy storage volume is placed upstream the corresponding steam storage one. This means that the steam storage volume entry conditions correspond to the exit conditions of the energy storage one. Once such entry conditions are determined, the section calculation procedure is similar to that utilized in the case of the normal operation mode (Section 2.1).

Energy Storage Volume

Steam Storage Volume

ST Section

Fig. 3. Steam turbine section scheme including a thermal storage volume for heat transfer.

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The particular ST modeled features HP and IP sections having two casings, an internal one that is particular to each section, and an external one shared by both HP and IP sections. The LP section has in turn one casing only. In order to take into account the interaction between the working fluid (steam) and the ST structure (casing), two heat transfer processes, namely conduction and convection, are modeled in the energy storage volumes. Radiative heat transfer is not accounted for in this work. Following [45,46], the heat transfer processes in the ST HP and IP sections are modeled according to the generic scheme shown in Fig. 4. In other to apply energy conservation laws thus, three different control volumes are utilized. One associated with the working fluid, and the other two surrounding each ST section casing. Accordingly, from an energy balance in each of these three control volumes, from the inner to the outer, the energy variation is given by, respectively, (7)

(8)

(9) In Eqs. (7)-(9),

represents the forced convection-based rate of heat transfer

between the working fluid and the ST section internal casing,

stands for the heat

transfer rate between internal and external casings through conduction, and

is the

heat transfer rate between the external casing and the ambient due to natural convection. Similarly,

and

represents, respectively, the conduction-based rate of heat

transfer between the internal casing and the adjacent upstream and downstream equipment. The referred heat transfer rates are given by, (10) (11)

(12)

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(13) The same formulation is used for modeling the ST LP section. Nevertheless, since this section features one casing only, there is no heat exchange between internal and external casings. Thus, only two heat transfer-related interactions, namely fluid-casing and casing-ambient, are accounted for in this case. Additionally, it is worth noticing that the HP section upstream and downstream equipment considered are, respectively, the steam intake valve and the IP section internal casing. In turn the IP section upstream and downstream equipment relate to, respectively, the HP internal casing and the steam exit pipe. In the case of the LP section, the upstream equipment are the LP internal casing and the HRSG steam pipe, whereas the downstream one is the power plant main condenser. Notice as well that in the case of the external casing there is no conductionbased heat transfer between this casing and adjacent equipment. This aspect is reflected in Eq. (9), which does not include any

related term.

Fig. 4. Scheme of the heat and mass transfer processes modeled in the energy storage volumes upstream the steam turbine HP and IP sections.

In order to use Eqs. (7)-(13) to model thermal inertia effects in the ST sections, it is necessary to determine first the overall heat transfer coefficients,

, between both the

steam and the ST section internal casing (Eq. (10)), and the external casing and the ambient (Eq. (12)). Following a classical approach [47], these coefficients are related to their corresponding total thermal resistances. By doing so what remains is the determination of the convective heat transfer or film coefficients, which in turn are computed from the corresponding Nusselt number

. In the case of the forced

convection describing the heat transfer processes between the steam and the ST sections

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internal casing, the Nusselt number is obtained from the Dittus-Boelter equation, [47]. The natural convection-related

, which is associated with the

heat transfer between the external casing and the ambient, is computed in turn from an expression of the form Prandtl

, Rayleigh

, where

. Notice that the Reynolds

and Grashof

,

numbers are determined from fluid and

flow properties, along with the corresponding equipment characteristic dimensions. Finally, following classical guidelines [47], the

and

exponents appearing in these

expressions are properly chosen such as to minimize the model prediction errors. 2.2.2 Mechanical inertia effects Regarding the rotational (mechanical) inertia effects, these are modeled applying momentum conservation on the ST shaft, as follows, (14) where

stands for rotational speed,

is the net power applied to the shaft, and

its

moment of inertia. It is worth noticing that rotational inertia is accounted when the equipment is not synchronized to the electric network only. That is when the electric network inertia effects on the rotational speed are not present. This is done because, when both ST and electric network are synchronized, the electrical generator behaves as an interface between the ST shaft and the electric network. In this situation, the electrical generator is responsible for the rotational speed definition. It is understood here that the electric network inertia is much higher than that of the turbine. When the steam turbine and the electrical network are synchronized thus, any ST mechanical power output variation is absorbed by the network, whereas the shaft rotational speed, still a consequence of the power balance on the electric generator, remains approximately constant. Finally, it is worth emphasizing that, once known the energy storage volume exit conditions, i.e., the conditions at the entry of the steam storage volume, the steam turbine section dynamics and exit conditions are computed following a similar approach to that used in the case of the normal operation mode (Section 2.1). Nevertheless, since both the steam flow and the rotational speed change significantly during the steam turbine start-up, the polytropic efficiency is expected to vary as well. In order to account for this aspect, several ST simulations were carried out varying the polytropic efficiency

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during different steam turbine start-ups, and it was found that the benefits in accuracy from such efficiency variations are small. The thermal effects that predominate in this ST operation mode seem to be the main responsible for this particular outcome. The constant polytropic efficiency used in the normal operation mode is also utilized then during the steam turbine start-up. Furthermore, the same modeling approach used for the steam turbine start up is utilized in the case of the shut down operation mode. The chief differences when shutting down the steam turbine relate to the direction of the heat transfer processes and the deceleration of the ST. In the next section the main results obtained from the use of the ST model developed in this work are presented and discussed.

3 Results and discussions In order to emphasize the capabilities of the ST dynamic model under different operating scenarios, two unrelated set of results are discussed in this section. The first one regards to casing temperatures along the ST, whereas the second one involves, in addition to power output and rotational speed, flow temperatures at each of the turbine sections exits. These results were obtained by both following the equations given in Section 2, and accounting for proper values for some of the main parameters characterizing the ST and the processes (e.g., flow dynamics, heat transfer, and polytropic expansion) occurring within it. A summary of the constant parameters whose values have been assumed/estimated in this work are highlighted Table 1. Table 1. Constant parameters accounted for in the simulation processes. Parameters Adjacent equipment, casing and ambient-related heat transfer areas Adjacent equipment, casing and ambient-related characteristic lengths Casing internal diameters Casing wall thicknesses Casing masses ST sections characteristic volumes ST sections characteristic times Adjacent equipment and casing specific heats Nusselt-related and coefficients

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Notice that Table 1 includes parameters associated with the ST hardware (areas, lengths, diameters, etc.), flow dynamics (volumes and times), and heat transfer processes (specific heats and Nusselt-related coefficients). Some of these parameters (heat transfer) have been assumed from classical literature, others (hardware specific) have been estimated from the geometry of the particular ST utilized for verification purposes, and the remaining ones (flow dynamics) have been chosen such as to produce the best results with the ST model developed here

3.1 Casing temperatures Casing temperatures obtained from the ST dynamic model developed are firstly compared to operating data belonging to a 300 MW class steam turbine used in an actual combined cycle power plant. From the three ST operating modes accounted for in this work, start up and shut down present the largest variations in casing temperatures. Accordingly, for assessing ST casing temperatures, a cold start up of the particular steam turbine accounted for is simulated and the obtained results are compared to those characterizing the steam turbine actual operation. Steam turbine start up procedures follow from time requirements imposed by heating up processes of specific parts of turbines such to avoid incurring in excessive material thermal stresses. Since manufacturer-specific know how is included in start up curves, these procedures vary from manufacturer to manufacturer. Therefore, in order to protect the proprietary data utilized in this work, only non-dimensional results are included in this section. Even though non-dimensional results may be difficult to follow, they are presented in this fashion here because of the confidentially issues associated with the sensitive data employed. Accordingly, Fig. 5 shows the evolution of both modeled and operating casing temperatures as a function of non-dimensional time. Notice that the reference temperature used for non-dimensionalizing these casing temperatures correspond to the maximum HP internal casing temperature characterizing the ST actual operation. From the results shown in Fig. 5, it is possible to notice first that model results agree well with operating data. The average discrepancies are indeed of the order of 0.2%. It is readily observed as well that both HP and IP internal casing temperatures are higher than the corresponding ones associated with both HP/IP external and LP casings. These results are expected because both steam temperatures at

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turbines later stages are lower than at first stages, and internal casings are in general hotter than external ones during cold start ups. The trends characterizing each of the casing temperatures evaluated here are a result of the particular start up schedule characterizing the steam turbine simulated in this work. For instance, as it can be seen from Fig. 5, when compared to the IP and LP casings, the HP internal casing is the one whose temperature varies the most. This occurs because the working fluid (steam) features the highest temperatures and mass flow rates at the entrance of the steam turbine. More specifically, Fig. 5 shows that at a non-dimensional time around 0.15, a small amount of steam is allowed to flow through the HP section, increasing slightly its internal casing temperature. Shortly before time 0.4, a first ST start up is attempted, resulting in a steep temperature rise. This first start up is aborted, as highlighted by the subsequent decrease in temperature, which leads to perform a second start up attempt right after time 0.4. As the second attempt is successful, a proper speed ramp is achieved. The effects of this process are clearly observed in the temperature rises characterizing both HP and IP internal casings, and the HP/IP external one. It is worth noticing that synchronization between the steam turbine generator and the power grid occurs in this case at time around 0.45. This point defines thus the end of the start up operation mode, and consequently the beginning of the normal operation one. As it will be further emphasized by the results shown in Section 3.2, the first part of the normal operation mode is usually characterized by large gradients in power output, which are accompanied by large gradients in casing temperatures at the first steam turbine sections (Fig. 5). Regarding the LP casing temperature behavior, it is seen that the associated temperature changes are relatively small. This happens because the steam temperatures at the LP section are significantly lower when compared to the corresponding HP and IP ones. Finally, it is worth noticing that the observed trends in the ST casing temperatures are mostly consequence of the steam mass flow rate modulation carried out, which intends to control both the shaft acceleration and the material thermal stresses.

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Fig. 5. Temporal evolution of ST casing temperatures.

3.2 Rotational speed, power output, and flow temperatures In addition to power output and rotational speed, the second set of results to be discussed involves flow temperatures at each of the turbine sections exits. As above (Section 3.1), only non-dimensional results are included in this section. Thus, Fig. 6 shows the temporal evolution of rotational speed and power output both modeled and operating data-related. The reference values utilized in this case for nondimensionalizing these parameters correspond to the steam turbine nominal rotational speed and power output. From Fig. 6 it is possible to see that both computed parameters follow the trends of the corresponding operational ones, even though rotational speed results seems to agree better with each other. Indeed the average discrepancies between

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the computed and operating rotational speeds are about 0.02%, whereas in the case of the power output these discrepancies are of de order of 2%. Notice that although some perceptible differences between data and model values are present in the case of the power output (Fig. 6b), in average, the corresponding discrepancies are relatively small (~2%). This happens because the number of computed points, i.e., the simulated period, is relatively large.

Fig. 6. Temporal evolution of rotational speed and power output.

The corresponding flow temperatures at each of the turbine sections exits are illustrated in Fig. 7. The reference value used for non-dimensionalizing these temperatures correspond to the maximum flow temperature at the HP section exit characterizing the ST actual operation. Similar to the results discussed previously, the agreement between the computed values and those associated with the actual operation are good, as overall the average discrepancies are of the order of 1% or less. From Fig. 7 it is readily seen that the HP section exit flow temperatures are higher than the corresponding IP section exit ones, and that these last ones are in turn higher than the corresponding LP section exit ones. These are expected results because the steam temperatures at the turbine later stages are lower than at the first stages.

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Fig. 7. Temporal evolution of ST sections exit flow temperatures.

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The results shown in Fig. 7 reveal as well that the IP section exit flow temperature takes longer to stabilize than the corresponding HP and LP section exit ones. Indeed the HP section exit flow temperature stabilizes almost at the same time as the power output does (Fig. 6b), emphasizing in this way the close correlation between these two parameters. The stabilization of the LP section exit flow temperature takes a little bit longer than that corresponding to the HP section exit one. The dynamics of the flow temperatures at each of the turbine sections exits is partially due to the particular schedule utilized to start up the steam turbine under discussion here. Accordingly, it is possible to see that the HP section exit temperature (Fig. 7a) oscillates when the turbine speed (Fig. 6a) increases to its nominal value. Such variations reflect the control system operation so as to avoid both excessive thermal and mechanical stresses on the ST casing and shaft. It is noticed as well that once the equipment is synchronized to the electrical network, which occurs at a non-dimensional time around 0.15 in this particular set of results, what follows is a power ramp until the desired power output is achieved. As observed in Fig. 6b thus, the early stages of the normal operation mode feature large gradients in power output. Concerning the LP section, it is observed that its exit flow temperature (Fig. 7c) varies slightly only. Such behavior is partially due to the steam condenser placed downstream the steam turbine, which is responsible for keeping a low-pressure level at the LP section exit, besides allowing a thermal interaction between the steam and the ambient. Finally, it is worth mentioning that owing to the proprietary nature of the information used in this work, it is not possible to disclose the control structures related to the particular steam turbine modeled here. However, it is important to observe that the steam turbine, as approached here, is a passive element as it responds to the inputs from other equipment and the conditions to which it is bound. Hence, the control system dictating

the

equipment’s

behavior,

by

observing

information

such

as

power/acceleration ramps and temperature change rates, actuates on the steam turbine. This actuation allows entering an acceptable amount of mass and energy into the equipment considering a trade-off between safety and operation flexibility. Such actions are mainly performed by the steam control valves, whose temperatures are also monitored so as to avoid excessive thermal stresses. Furthermore, during operating procedures, several criteria regarding the operating conditions throughout the steam line

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and the equipment on it are verified, and the steps undertaken are subject to the control system.

4 Conclusions In this work the development of a steam turbine dynamic model for full scope power plant simulators has been described. One of the main features of the model developed regards to its completeness as it attempts to cover the whole operating envelope of steam turbines. For modelling purposes, three distinct regions were discriminated within the ST safe operating envelope, (i) normal, (ii) start up/shut down, and (iii) idle operation. Accordingly, the corresponding model formulation associated with the first two ST operation modes accounted for has been detailed. A particular emphasis has been put on thermal and rotational inertia effects, which are important during the start up and shut down of steam turbines. Idle operation was neglected here because of its lack of relevance in power plant simulators. Once developed the steam turbine dynamic model has been used for simulating a 300 MW class steam turbine belonging to an existing combined cycle power plant. Several results have been obtained from such simulations, including those related to casing and flow temperatures, rotational speed and power output. The agreement between the computed results and those characterizing the equipment actual operation was in general good, as in average the corresponding discrepancies were of the order of 2% or less. Achieving this relatively low level of discrepancies between model results and operating data was possible thanks to the inclusion in the model formulation of several physical processes characterizing the actual operation of steam turbines. It is concluded thus that producing high fidelity results from steam dynamic models requires accounting for a number of physical phenomena, primarily, those related to heat transfer processes. A trade off needs to be achieved of course at some stage between fidelity of model results and computational cost involved.

5 Acknowledgments This work has been supported by the EDF Group (France).

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Highlights (A Steam Turbine Dynamic Model for Full Scope Power Plant Simulators)

   

A steam turbine dynamic model for full scope power plant simulators is developed Distinct regions within the operating envelope of steam turbines are discriminated Comparisons between model results and steam turbine operating data are carried out Accuracy of model predictions is good as related discrepancies are about 2% or less

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