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Comparison between Thermodynamic Model and Neural Network Model Approach

S. Dettori, V. Colla, Scuola Superiore Sant’Anna, TeCIP Institute, PERCRO Laboratory, Via Alamanni 13D, Ghezzano, Pisa, ItalyG. Salerno, A. Signorini, GE Oil & Gas, via F.Matteucci 2, Firenze, Italy

AbstractIn this work, two different approaches – an iterative thermodynamic method and a neural network model – are proposed and compared for the modeling of steam turbines. The iterative thermodynamic model can predict steam mass flow, pressure, temperature, enthalpy and power on each turbine drum. The NN model can predict the generated mechanical power. Both models have been trained and validated on a massive dataset created through the internal sizing design tool, which contains the turbine geometrical and mechanical data. Further validation tests have been successfully executed by exploiting field data coming from a solar power plant where a high-pressure and a low-pressure turbine were installed.

1. IntroductionMonitoring and controlling steam turbines is challenging, especially when power generation occurs under variable operating conditions. That is what happens with steam turbines whose operation involves variable steam production profiles (steam turbine discontinuous power generation). One strength of accurate steam turbine modeling is that it can be applied with phenomena (such as aging, efficiency degradation, windage, etc.) that can’t be directly monitored through sensors. In this case, system modeling and the application of advanced tools, such as Artificial Intelligence techniques, are a way to estimate and monitor the physical characteristics of the phenomena being scrutinized.

Advanced steam turbine modeling takes three main approaches. The first is completely based on thermodynamic models. The second is based on a black box model that often shows little or no linkage with the represented system of phenomenon (such as a neural network). The third, based on hybrid models, is a combination of the first two approaches and can include automatic procedures to determine or tune some parameters of a physical model derived from experimental data. Here are some examples:

In particular, the work presented in [1] proposes a modified thermodynamic model for the prediction of thermal performances of back pressure steam turbines with single and multiple extractions; in [2] energy balance, thermodynamic state conversion and semi-empirical relations are used for a nonlinear modeling of a multiple extraction turbine, in [3] a steam turbine nonlinear dynamic model is formulated from approximation of fundamental thermodynamic equations; In [4] a nonlinear mathematical model based on energy balance, thermodynamic principles and semi-empirical equations is used in order to characterize the transient dynamics of steam turbines sections, tuned by applying genetic algorithms on field data. In [5] a model

integrating thermodynamic laws and semi-empirical equations is used in order to estimate the generated power and the condition of the exhaust steam. In [6] a thermodynamic based model is used for steam turbine efficiency monitoring, instead of classical graphical method. The work proposed in [7], proposes two different approaches for the modeling of a power plant, including steam turbines, in particular a model based on energy and mass balance and a neural network approach.

Hence, this work proposes some turbomachinery models that address different factors, such as how a degradation prediction algorithm can tell how turbine features will evolve. These models identify how certain characteristics vary from data that is measurable on the operating field. They are powered by suitable monitoring and control algorithms that are implemented directly on the PLC and meet its stringent requirements for memory usage and computational speed.

We are proposing two different approaches for this work: a hybrid technique that integrates physical/thermodynamic laws and can predict mass flow, pressure, temperature, enthalpy and power on each steam turbine drum for a given set of boundary condition; and a neural network approach that can predict the generated mechanical power.

Besides the introduction, this paper has four main sections. Section 2 describes the physical equations applied to identify the thermodynamic model. Section 3 develops the neural network approach. Section 4 describes the strategy for defining the dataset used to train and build the models and the dataset used for model validation, and presents the results of both modeling approaches when compared with field data. Finally, section 5 discusses conclusions and some consideration for future research.

2Comparison between Thermodynamic Model and Neural Network Model Approach

inlet steam condition, inlet steam valve stroke, mass extractions, extraction pressures and condenser pressure. The mass flow of each drum can be related to the inlet and outlet conditions by using the Stodola equation for convergent/divergent nozzle; this approach is well documented in literature and is based on the concept that the mass flow is constant in steady state condition and the energy absorbed by the turbine is reflected on the turbine as ΔP. The simplified Stodola equation is expressed in Eq. (1):

(1)

The coefficient k can be retrieved to exploit the data-driven approach described in the previous section. Since the identification of the k constant is performed for a wide range of operating conditions it is not possible to use just one k; instead, it must vary according to the operating condition. For these reasons, we have introduced a third order function:

(2)

The sto coefficients are not constant, but they are dependent on the rotational speed and the stroke of the inlet steam valve; these two dependencies introduce matched non linearities that express the sto coefficients as lookup tables.

(3)

An example of the identification of the coefficients performed is presented in Figure 2.

Figure 2: Example of Stodola Equation Fit

Knowing the sto coefficients and the inlet conditions of each drum (pressure, temperature and mass flow) makes it possible to estimate the exit pressure of the drum:

(4)

It’s possible to evaluate some other quantities (such as enthalpy and entropy) by exploiting the steam tables’ functions described in the “IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam” [1]. This table shows the outputs that must be evaluated in the thermodynamic model.

2. The iterative thermodynamic model

2.1 IntroductionWe have developed the thermodynamic model proposed in this article by trying to balance three different characteristics:

1. Accuracy: The model must be accurate to match the requirements of a particular simulation.

2. Flexibility: Different types of turbine arrangements must be modeled and the setup time must be as small as possible.

3. Simplicity: The model must be simple and have low computational requirements so that it can be “translated” into the PLC language.

For these reasons, the model has been developed by using physics-based and data-driven approaches. The physics-based approach implements simple equations (Stodola, Efficiency and Power) that reproduce the real behavior of the machine but also must be accurately tuned. A data-driven approach has been chosen to tune the above-mentioned equations by exploiting data from a GE internal sizing tool that captures geometrical and mechanical parameters of the turbine that allow high-accuracy evaluations.

Finally, the thermodynamic model is integrated with a steam turbine mechanical model to perform dynamic simulations, estimating not only the operating condition of the steam turbine but also factors such as turbine speed, actual generated power and power losses.

2.2 Parametric EquationsThe parametric equations proposed in this section are applied to “similar” parts of the steam turbine; this means that a set of similar blades (a.k.a., the drum) can be modeled together and with just one “set of equations” (see Figure 1 for more details).

Figure 1: Impulse, Reaction and Condensing Drum

For each drum of the steam turbine, it is possible to estimate the fundamental parameters (mass flow, pressure, temperature, enthalpy and power) for a given set of boundary conditions:


Finally, it’s possible to evaluate the outlet temperature and the vapor quality by exploiting the steam tables functions [1].

OutputsInputs for steam tables

Description

TOUT POUT , HOUT Temperature at the outlet of the drum is evaluated by knowing outlet pressure and enthalpy.

XOUT POUT , HOUT Vapor quality at the outlet of the drum is evaluated by knowing outlet pressure and enthalpy.

ρOUT POUT , HOUT Steam density at the outlet of the drum is evaluated by knowing outlet pressure and temperature.

Table 2: Outputs Evaluated with Steam Characteristics Tables

By knowing the mass flow and the enthalpy drop on each drum of the steam turbine, it’s possible to estimate the power generated by applying Eq. (9); the kinetic energy contributions also have been introduced in this equation.

(9)

Where:

(10)

(11)

Assuming that the flow is exactly axial at the inlet of each drum, it’s possible to evaluate the inlet speed by exploiting the mass flow conservation law (Eq. (12)).

(12)

The formula changes slightly for the impulse drum because the number of blades hit by the steam changes with the stroke of the inlet steam valve.

(13)

The flow at the exit cannot be considered axial because of the exit angle of the last rotoric blade; this geometrical parameter and the mass flow conservation law are exploited to evaluate the exit speed (Eq. (14) and Figure 4).

(14)

OutputsInputs for steam tables

Description

HIN PIN , TIN Enthalpy at the inlet condition can be evaluated by knowing the pressure and temperature at the inlet of the drum.

SIN PIN , HIN Entropy at the inlet condition can be evaluated by the knowing pressure and enthalpy at the inlet of the drum.

H ISO OUT POUT , SIN Steam enthalpy that happens at

the outlet condition through an isentropic expansion can be evaluated by knowing the pressure at the outlet condition (evaluated with Eq. (3)) and the entropy at the inlet of the drum.

ρIN PIN , TIN Steam density at the inlet condition can be evaluated by knowing the pressure and temperature at the inlet of the drum.

Table 1: Outputs Evaluated with Steam Tables

Knowing the isentropic exit enthalpy makes it possible to estimate the real exit enthalpy by exploiting the efficiency equation.

(5)

The efficiency function is identified by exploiting the internal sizing tool, with the same approach used for the Stodola equation. The data coming from the tool showed that the relationship between the isentropic ΔH and the real ΔH is not linear, but can be approximated with a third order function.

(6)

(7)

The coefficients are dependent upon the rotational speed, and this dependency introduces non- linearities that are matched by expressing the coefficients as lookup tables.

(8)

Figure 3 shows how the identification of the coefficients has been performed.

Figure 3: Example of Efficiency of Equation Fit


flow with some of the boundary conditions (see Eq. (16)). Applying this concept makes it possible to summarize the iteration process in three steps:

Definition:

(15)

1. Check if the estimated exit pressure matches the actual exit pressure; if it does, the current mass flow is correct; if it doesn’t, go to the next step.

2. If the estimated exit pressure coming from Eq. (4) is a complex number, it means that the current mass flow is too high; in this case, lower it by 5%. If the estimated exit pressure coming from Eq. (4) is a real number, apply Eq. (16).

(16)

3. As soon as the error changes the sign ( apply the following condition:

(17)

4. If the error on pressure is lower than a predefined threshold (usually of the exit pressure), the mass flow is correct

Figure 4: Scheme of Velocities of the Last Rotoric Blades (1=inlet, 2=outlet, c=absolute speed, w=relative speed, u=tangential speed).

The condensing drum exit speed is evaluated with functions that are proprietary to GE; these functions estimate the exit kinetic energy of the steam, taking into account the changing of the exit rotoric angle along the height of the blade.

It’s necessary to note that the pressure at the last drum exit of the turbine is a given boundary condition (for example, given by the condenser), and the Stodola equation (Eq. (2)) is redundant. For this reason, an iterative approach has been implemented; in fact, the mass flow at the inlet is changed to match the actual exit pressure with the one coming from the thermodynamic model (see logical flow example of Figure 5).

The logic that changes the mass flow at each iteration is based on Eq. (1); in fact, it’s possible to assume that, for each operating condition, a exists that relates the mass

Figure 5: Iteration Logical Flow


Figure 7: Neural Network to Predict Each Drum Power

Figure 8: Neural Network to Predict the Overall Turbine Power

Figure 6 gives an example of the iteration process.

The iteration process shown in Figure 6 is applicable for turbines with one boundary condition on exit pressure. If steam turbines are arranged with one or two controlled extractions, then additional boundary conditions at the exit of the controlled drums shall be considered; in this case, the iteration process is slightly different and not described in this paper, since the turbines under analysis here have no controlled extractions.

3. The neural network approachThe neural network (NN) approach – in particular, the feed forward neural network – is proposed for modeling the non-linear relation between steam features and the generated mechanical power or thermodynamic characteristics of the steam in the turbine drums.

Two different neural networks have been developed, with different aims:

1. A neural network that can predict the power on each steam turbine drum

2. A neural network that can predict the overall turbine power

Figure 6: Iteration Process


A dataset of operating conditions has to be created for the thermodynamic model and the artificial neural network. For this purpose, the GE internal sizing tool, which contains the turbine geometrical and mechanical data, has been interrogated on a wide range of operating conditions (boundary conditions); these conditions have been identified by exploiting requisition documents in which the main boundary condition for a particular steam turbine is defined. The interrogation procedure generates a database that has been divided into two datasets: the former, the training dataset, is created by using 80% of the original dataset, randomly sampled. It is directly used to identify the coefficients in the parametric equations described in section 3 for the iterative thermodynamic model, and the train of the neural networks weights. The latter is used as a validation dataset and is created using the remaining 20% of the original dataset.

Moreover, one full day of field data has been used to test the accuracy of the two approaches. The tests have been made by using the field data as boundary condition for the GE internal sizing tool; the outputs of this tool will be used to evaluate the accuracy of the two models. The database of the HP turbine contains 4,320 different conditions that are a vector combination of the inputs listed in Table 3: HP Turbine Database.

The database of the LP turbine contains 12,150 different conditions that are a vector combination of the inputs listed in Table 4. The dataset includes the operating condition and the off-design conditions. The parameters identification process for the thermodynamic model showed it to be necessary to purge the training dataset of extremely off-design conditions that the machine is very unlikely to experience. For example, in Figure 9, all points obtained by the interrogation procedure are represented in blue, while the nominal or reasonable off-design condition that the turbine may experience is represented in red.

Input DescriptionN. Operating Points

Ranges

TIN Inlet Steam Temperature

3 358-396 [°C]

PIN Inlet Steam Pressure 8 12-111 [BarA]

POUT Exit Steam Pressure 5 2.5-21 [BarA]

RPM Turbine Rotational Speed

6 3500-6500 [RPM]

Stroke Inlet Steam Valve Stroke

3 2-4 [n° opened valves]

M1 Mass Flow Bleeding 2 0-5 [kg/s]

Total 4320 = 3 . 8 . 5 . 6 . 3 . 2

Table 3: HP Turbine Database

To define the architecture of the hidden layer (number of neurons), an optimization study has been performed with the intention of minimizing the NN error committed on the test set (the experimental optimization is reported in section Error! Reference source not found.).

The final architecture of the neural network is comprised of an input layer, a hidden layer with N neurons, and the output layer.

The used transfer function of hidden layer is the hyperbolic tangent, defined as:

whereas the transfer function of the output is linear.

The neural network training was performed by using the Neural Network Toolbox (MATLAB 8.0), specifically, the Levenberg-Marquardt back-propagation, an algorithm that is optimal for NN training that features several hundred connections between neurons. The target function to be minimized was the mean squared error between the prediction and the desired output.

The training set of the NN was defined as a couple input/output:

• For the net1: The input Inet1, is comprised of Inlet Steam

Pressure Pin and Temperature Tin, bleedings or extractions mass flow [Mextr1 ... Mextrn], and output pressure Pout; the output Onet1

is comprised of the power WDi of each drum, with i = 1, … ,m drums; The k-th input and output elements of the training dataset is defined as:

• For the net2: The input Inet2 is comprised of Inlet Steam

Pressure Pin and Temperature Tin, bleedings or extractions mass flow [Mextr1 ... Mextrn], and output pressure Pout; the output Onet2

is comprised of the overall power;

4. Validation test campaign

4.1 Dataset selection strategy The results campaign has been performed on two different steam turbines:

1. High Pressure (HP) steam turbine with impulse stage and three reaction drums

2. Low Pressure (LP) steam turbine (no impulse stage), with four reaction drums and a condenser


4.2 Results on estimation of real data gathered from field

This section reports the results of the tests of the thermodynamic and neural network models on real field data. The inlet steam pressure, temperature, and outlet pressure were collected from site the during one day of activity with a sampling time of 0.5 seconds. Five indexes have been computed to evaluate the accuracy of the models: Mean Error (Mean e), Max Error (Max e), Mean Percent Error (Mean e%), Max Percent Error (Max e%) and Standard Deviation.

As mentioned in section 3, the number of neurons for each neural network has been defined in two different ways by an optimization study. In the first, the best neural network architecture is selected by minimizing the median error on the validation dataset; in the second, the optimal neural network has been defined to obtain a good compromise between the average error on the test (field) dataset, the minimization of maximum % error obtained with the network, and a good standard deviation. The possibility of using field data as feedback for the optimization study doesn’t imply additional computation time, but rather a selection within a set of already trained neural networks.

• The optimal NN net1 to predict the power of each HP Turbine drum yielded 15 neurons with the field dataset, and 30 with the validation dataset.

• The optimal NN net1 to predict the power of each LP Turbine drum yielded 10 neurons with the field dataset, and 15 with the validation dataset.

• The optimal NN net2 to predict the overall power of the HP Turbine yielded 15 neurons with field the dataset, and 25 with the validation dataset.

• The optimal NN net2 to predict the overall power of the LP Turbine yielded 20 neurons with the field dataset, and 20 with the validation dataset.

In Table 5 and 6, which report the results for the HP and LP turbines, both models’ output in terms of power are compared with the computed power that the GE internal sizing tool associates to the collected field operating points. In particular, two different overall power errors are shown for the thermodynamic model. The former overall data (∑W) has been evaluated by using the entire field dataset. The latter (∑W*) has been evaluated by discarding all the anomaly conditions that the model can identify (e.g., windage detection on the condensing drum). Two different overall power errors also are shown for the neural network approach. The former has been evaluated by using the best NN-chosen analyzing results on the validation dataset, while the latter has been evaluated by using the best NN-chosen analyzing results on field data.

Figures 12-14 show the daily profile of the power generation of the HP and LP turbines and the power predicted by the neural network models. Figure 13-Figure 14 show the daily profile of the power generation of the HP and LP turbines and the power predicted by the thermodynamic model.

Results show that both approaches can predict the generated power with few errors and good accuracy as soon as the power is above 9 MW for the HP turbine and 27vMW for the LP turbine. This is because the non-linearity of the steam inlet valve is simplified in the identification of the model, and the comparison

Input DescriptionN. Operating Points

Ranges

TIN Inlet Steam Temperature

5 330-390 [°C]

m.

IN Inlet Mass Flow 10 5-50 [kg/s]

POUT Exit Steam Pressure 3 0.04-0.08 [BarA]

RPM Turbine Rotational Speed

1 3000 [RPM]





Total 12150 = 5 . 10 . 35

Table 4: LP Turbine Database

Figure 9: Effect of Data Filtering

The filtering rules to clean the training dataset characterizing extremely off-design conditions are:

• Mach number: If the Mach number at the exit of each drum exceeds a specified limit the point is deleted from the training set.

• Windage: If the pressure ratio on each drum (calculated as exit pressure over inlet pressure) is close to one, then the drum is ventilating (windage).

• Diffuser: Steam turbines with a condenser usually have the diffuser; when the steam exit axial velocity reaches a limit the diffuser works out of nominal condition.

The validation dataset (used to validate the tuning of the equations) doesn’t have to be filtered to evaluate the response of the model in the high off-design conditions described above and its capability to capture such rare events.

The dataset for the neural network training is not cleared from extremely off-design conditions. High non-linearities can be fitted with this approach.


with an error below 0.6% and 1%, respectively, on the HP and LP turbines, if the capability of the thermodynamic model to detect anomaly conditions like windage is exploited. By discarding those points since they will not match a correct power estimation, the accuracy on the LP turbine improves to about 0.4%.

between models and assessment on the accuracy of estimation concentrates on the data reported where the inlet valves are fully open. In that portion of data, only the variation of the inlet steam pressure and temperature produce the variation of power. This allows an easier comparison and evaluation of results. The thermodynamic model can estimate the power delivered to load

Figure 10: Power Estimations for HP Turbine with t1. (Left) Overall predicted power and % absolute error. (Right) Power of each drum and relative % absolute error

Figure 11: Power Estimations for LP Turbine with t1. (Left) Overall predicted power and % absolute error. (Right) Power of each drum and relative % absolute error

Figure 12: Overall Power Estimations for HP and LP Turbine with t2. (Left) HP turbine overall predicted power and % absolute error. (Right) LP turbine overall predicted power and % absolute error


Model SectionIndex

Mean |e| [kW] Max |e| [kW] Mean |e| % Max |e| % Std [kW]

Thermodynamic model

Impulse 31 54 3 15 7.2

D1 16 38 0.63 2.5 7.6

D2 15 45 0.3 2.7 9.5

D3 42 112 1.2 10 28

∑W 71 137 0.59 2.8 30

NN Model net1 Impulse 8.7 29 0.89 8.6 4.8

D1 7.3 19 0.33 1.5 8.6

D2 48 82 1.2 5.5 46

D3 31 83 0.98 6.4 35

∑W 800 1654 8.2 41 828

∑W* 29 81 0.26 2.2 31

NN Model net2 Overall W 159 908 2.2 20 227

Overall W* 18 70 0.18 1.9 20

Table 5: Test Results on HP Turbine Models Vs. Field Data

Figure 13: Power Estimations for HP Turbine with Thermodynamic Model. (Left) Overall predicted power and % absolute error. (Right) Power of each drum and relative % absolute error

Figure 14: Power Estimations for LP Turbine with Thermodynamic Model. (Left) Overall predicted power and % absolute error. (Right) Power of each drum and relative % absolute error


Model SectionIndex

Mean |e| [kW] Max |e| [kW] Mean |e| % Max |e| % Std [kW]

Thermodynamic model

D1 2.5 6.5 0.06 0.6 1.5

D2 10 40 0.21 1.1 5.8

D3 12 78 0.3 4.7 12

D4 74 1157 1.8 30 125

Condensing drum 96 1150 12 920 155

∑W 161 1400 0.92 20 184

∑W* 114 250 0.4 1.2 48

NN Model net1 D1 25 64 0.74 3.7 13.3

D2 30 54 0.57 1.9 8.2

D3 51 66 0.95 2.0 6.6

D4 24 618 0.74 16 60

Condensing drum 68 715 7.4 1200 73

∑W 93 947 0.44 7.1 75

∑W* 44 546 0.27 4.2 63

NN Model net2 Overall W 87 1470 0.40 11 76

Overall W* 47 714 0.28 5.3 77

Table 6: Test Results on LP Turbine Models Vs. Field Data

5. Conclusion and future worksIn this work, two different approaches have been proposed to model a generic steam turbine: a hybrid thermodynamic model and a neural network black box approach. Both models have been trained and validated on a massive dataset created through the internal sizing design tool – which contains the turbine geometrical and mechanical data – and then tested on one full day of field data. The performances obtained for both models are promising and demonstrate the possibility of using these approaches on steam turbine monitoring that includes extremely off-design conditions.

On one hand, if the developed thermodynamic model can recognize inconvenient situations such as windage of the last blade stage – since it can estimate internal data where no instrument can be placed, like steam pressure, temperature or steam speed – then it is an iterative algorithm, which may not be suitable to compute a future implementation on a turbine controller platform. It produces a very large amount of data on the turbine’s internal state, which can be exploited by an off-line monitoring tool. This approach can be downscaled to model the turbine expansion field instead of just a single drum’s behavior. This could yield a less accurate but also less complex model.

On the other hand, neural network models have excellent results for the prediction of power, since they can use real data for fast selection of the best net within a set of already trained and built neural networks. The neural network can be implemented on typical turbine controller platforms without much effort and can be re-tuned online. Since the complexity of the neural network increases with the number of parameters

that you want to identify, it also increases the probability that it will give diverging results if an actual condition of the turbine is not included in the training dataset. Hence, it’s preferable to select only one or a very few parameters to be identified through a single network.

References1 Medina-Flores, José Martín, and Martín Picón-Núñez. “Modelling the

power production of single and multiple extraction steam turbines.” Chemical Engineering Science 65.9 (2010): 2811-2820.

2 Luo, Xianglong, et al. “Modeling and optimization of a utility system containing multiple extractions steam turbines.” Energy 36.5 (2011): 3501-3512.

3 Ray, Asok. “Dynamic modelling of power plant turbines for controller design.” Applied Mathematical Modelling 4.2 (1980): 109-112.

4 Chaibakhsh, Ali, and Ali Ghaffari. “Steam turbine model.” Simulation Modelling Practice and Theory 16.9 (2008): 1145-1162.

5 Varbanov, P. S., S. Doyle, and R. Smith. “Modelling and optimization of utility systems.” Chemical Engineering Research and Design 82.5 (2004): 561-578.

6 Zaleta-Aguilar, Alejandro, et al. “Thermodynamic characterization of the power loss factor in steam turbines.” Energy conversion and management 43.17 (2002): 2369-2378.

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Comparison between Thermodynamic Model and Neural Network Model Approach

By S. Dettori, V. Colla, G. Salerno, A. Signorini

©2016 General Electric Company and its Affiliates

GEA32786 (10/2016)

DOWNSTREAM TECHNOLOGY SOLUTIONS PRODUCTS ......thermodynamic model. 2. The iterative thermodynamic...

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