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PROCEEDINGS OF ECOS 2016 - THE 29TH INTERNATIONAL CONFERENCE ON EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS
JUNE 19-23, 2016, PORTOROŽ, SLOVENIA
SET-UP OF A ROBUST NARX MODEL SIMULATOR OF GAS TURBINE START-UP
Hilal Bahlawana, Mirko Morinib, Michele Pinellia,
Pier Ruggero Spina and Mauro Venturinia*
a Dipartimento di Ingegneria, Università degli Studi di Ferrara, Ferrara, Italy,
[email protected], [email protected], [email protected], [email protected] b Dipartimento di Ingegneria Industriale, Università degli Studi di Parma, Parma, Italy,
[email protected] * Corresponding author
Abstract:
This paper documents the set-up and validation of nonlinear autoregressive exogenous (NARX) models of a heavy-duty single-shaft gas turbine. The considered gas turbine is a General Electric PG 9351FA located in Italy. The data used for model training are time series data sets of five different maneuvers taken experimentally during the start-up procedure. The trained NARX models are subsequently used to predict other five experimental data sets and comparisons are made among the outputs of the models and the corresponding measured data. Therefore, this paper addresses the challenge of setting up robust NARX models, by means of an accurate selection of training data sets and a sensitivity analysis on the number of neurons. Moreover, a new performance function for the training process is defined to weigh more the most rapid transients. The final aim of the developed simulation models is the set-up of a powerful and easy-to-build simulation tool which can be used for both control logic tuning and gas turbine diagnostics, characterized by good generalization capability.
Keywords:
Gas turbine, Neural Networks, Start-up, Dynamics.
1. Introduction
To address the challenge of modeling gas turbine (GT) start-up, white-box and black-box models can
be considered as the two main categories of system models. Gray-box models may also be used as a
combination of the above mentioned methods. White-box models are usually used with coupled and
dynamics equations, thermodynamic relations, energy balance and linearization methods [1]. Instead,
black-box models, such as nonlinear autoregressive models with exogenous inputs (NARX), are used
when there is not enough information about the physics of the system or when is not possible to access
complex dynamic equations [1].
To setup a black-box model, different types of neural networks can be trained on the basis of the
different system parameters within its operating range. Before selecting the most suitable modeling
approach, it is necessary to perform an analysis of the entire system (in this paper, a gas turbine)
which includes the measurable parameters, monitoring system, condition and reliability of the
sensors, system history recording, data system accessibility, performance curves availability,
technical characteristics. Clearly, the selected modeling approach should be consistent with research
expectations.
Recently, artificial neural networks (ANNs) have been widely employed for modeling and simulation
of industrial systems. As a model based on data, ANN is one of the most significant black-box
modeling methods. It includes different approaches, among which the nonlinear autoregressive with
exogenous inputs (NARX). As a recurrent neural network, a NARX model has the ability to capture
the dynamics of complex systems such as GTs. Such a simulator can be used for the design and
production optimization of gas turbines, as well as the operation and maintenance of GTs.
Different types of gas turbines have been modeled so far by using ANNs. The model under
consideration may be a micro gas turbine, an aero gas turbine or an industrial power plant gas turbine
including heavy-duty industrial gas turbines, which is the subject of this study. In the area of black-
box models, specifically constructed for heavy-duty gas turbines, one can refer to the research
activities carried out by Lazzaretto and Toffolo [2], Ogaji et al. [3], Arriagada et al. [4], Basso et al.
[5], Bettocchi et al. [6-7], Spina and Venturini [8], Simani and Patton [9], Yoru et al. [10], Fast et al.
[11-12], Fast and Palme [13] and Fast [14]. As can be seen from the literature review, most of the
black-box models were built to model the steady-state operation of a gas turbine systems, i.e after
they have passed the start-up phase and have reached the stationary phase. Whereas, there is a limited
number of studies related to modeling GTs during start-up.
Asgari et al. [15] presented NARX models for simulation of a heavy duty single shaft gas turbine
during the start-up procedure. The NARX models were capable of capturing system dynamics during
start-up operation of the gas turbine and predicting the GT behavior. Also in [16], the authors explored
the transient behavior of a heavy-duty gas turbine by developing and comparing two different
approaches, a physics-based and black box approach by using Simulink and NARX models
respectively. The results showed that both Simulink and NARX models are capable of satisfactory
prediction. Dominiczak et al. [17] considered NARX model of a steam turbine to perform an on-line
prediction of temperature and stress in steam turbine components. Vatani et al. [18] developed two
methodologies, i.e. NARX neural network and recurrent neural network (RNN), for degradation
prognosis and health monitoring of gas turbine engines using dynamic neural networks. They showed
that the prediction errors of NARX neural networks are lower compared to the RNN architecture. The
state of a gas turbine engine in aircraft was estimated by Lakshmi et al. [19] using neural network
algorithms. Sina et al. [20] designed and developed a fault detection and isolation scheme for aircraft
gas turbine by using dynamic neural networks. They found that their proposed scheme represents a
promising tool for aircraft engine diagnostics and health monitoring. Tsoutsanis et al. [21] proposed
and developed a novel scheme for prognostics of industrial gas turbines operating under dynamic
conditions. They presented the results of their methodology for detecting and predicting the
performance of gas turbine components.
Many other white-box and black-box methods for simulating the transient behavior of gas turbine
individual components (such as compressors) were presented in literature. The authors also
contributed to this field of research. For instance, one can refer to the neural network techniques used
by Venturini [22, 23] to explore the transient behavior of compressors. Similar studies were carried
out by Venturini [24] and Morini et al. [25, 26] by using white-box methods.
The importance of GTs start-up procedure and its direct effect on the performance and life time is a
strong motivation for researchers to work in this field. There are still many unforeseen events during
start-up process which can arise from complex dynamics of gas turbines. Therefore, many research
activities should be still carried out to fill the existing information gaps in the literature and to develop
such black-box models.
For this reason, in this study, the authors model the start-up process of a heavy duty gas turbine by
using NARX models. Modeling and simulation are performed on the basis of experimental time series
data sets obtained from a heavy duty GT. The use of this modeling approach, which uses as input
only the variables at antecedent time step (i.e. no information on the current time step is required),
allows the setting up of a powerful simulation tool that can also be used for the real time control and
diagnostics of GT sensors. Particular attention is paid in this paper to the training methodology and
the optimal selection of training data.
1.1 – Gas turbine specifications
The gas turbine modelled in this paper is the General Electric PG 9351FA. It is a heavy-duty single-
shaft gas turbine used for power generation. The main specifications are summarized in Table 1.
1.2 – Gas turbine start-up
The start-up period is the gas turbine operating time until steady state combustion conditions are
reached. GTs require an external source as an electric or Diesel motor to start working. The starter is
used until the GT speed reaches a certain percentage of the design speed. GT start up procedure can
be divided into four stages, including dry cranking, purging, light-off and acceleration to idle [27, 28].
During dry cranking, the motor shaft is rotated by the starter system without any fuel supply. In
purging phase, residual fuel from previous operation or failed start attempts is purged out of the fuel
system. In this phase, the rotational speed is kept at a constant value with a proper mass flow rate
through the combustion chamber, the turbine and the heat recovery steam generator. In the light-off
phase, the fuel is supplied to the combustor and igniters are excited. This causes a local start ignition
within the combustor, followed by light-around of all the burners. At the end, in the acceleration to
idle phase, the fuel mass flow rate is further increased and the rotational speed increases towards the
idle value.
1.3– Available field data
The data sets used for the set-up and validation of the NARX models were taken experimentally
during several start-up maneuvers and cover the whole operating range and conditions of the gas
turbine during start-up. Moreover, they consider all the conditions relating to this type of transient
operation (e.g. bleed valve opening, IGV control, etc.).
In general, the data sets during start-up can be categorized as follows:
cold start-up: the gas turbine was shut down some days before start-up;
warm start-up: the gas turbine was shut down some hours before start-up;
hot start-up: the gas turbine was shut down just few hours or less before start-up.
Table 1. Gas turbine design specifications
GT type GE 9351FA
Number of shafts 1
Rotational speed 3000 rpm
Pressure ratio 15.8
TIT 1327 °C
TOT 599 °C
Air flow rate 648 kg/s
Power 259.5 MW
Heat rate 9643 kJ/kWh
Efficiency 37.3 %
In this paper, the available data sets refer to cold start-up. The measured time series data sets which
were used for NARX model training are called TR1, TR2, TR3, TR4 and TR5. They cover the whole
operating range of the gas turbine during the cold start-up. They were selected in such a manner that
the resulting model can generalize well for GT simulation during cold start-up. In fact, TE1, TE2,
TE3, TE4 and TE5 are the data sets which are used to test and validate the trained models and also
refer to cold start-up. More details concerning the operating range for the two input parameters of the
available data sets are shown in Table 2. It should be noticed that data sampling frequency is 1 s.
It can be noted from Table 2 that the operating range of all input and output parameters of the testing
data sets is included within the operating range of the training data sets. Moreover, even though the
trends of all maneuvers are similar, the rate of change of the rotational speed to reach the full
speed/no-load condition is different. Therefore, to set up a robust NARX model, the rate of change
of the fuel mass flow rate of the maneuvers used for testing (TE1 through TE5) is included between
that of TR1 and TR2 (most rapid variation) and TR5 (slowest variation), as shown in Figure 1.
2. NARX model development A NARX model is an autoregressive, recurrent and dynamic neural network with exogenous inputs.
It includes feedback connections which build-up different network layers and therefore it is mainly
used in time series modelling. The following equation relates NARX model output to its inputs:
Table 2. Experimental time series data sets
Data sets Number of data
n
Operational range of the inputs
T01, K Mf, kg/s
TR1 486 [286.5; 290.9] [0.94; 4.86]
TR2 419 [289.8; 292.0] [0.73; 4.86]
TR3 494 [305.4; 308.2] [0.28; 4.79]
TR4 487 [298.2; 299.8] [0.33; 4.74]
TR5 635 [282.6; 287.0] [0.49; 4.92]
TE1 397 [298.7; 299.8] [0.52; 4.77]
TE2 408 [299.8; 300.9] [0.49; 4.79]
TE3 524 [294.3; 297.6] [0.44; 4.85]
TE4 538 [295.9; 299.3] [0.37; 4.82]
TE5 548 [292.0; 293.7] [0.45; 4.90]
Fig. 1. Trend over time of the fuel mass flow rate
0
1
2
3
4
5
6
0 180 360 540
Fu
el
ma
ss f
low
ra
te (
kg
/s)
Time (s)
TR1
TR2
TR3
TR4
TR5
TE1
TE2
TE3
TE4
TE5
TE1
TE2
TE3
TE4
TE5
TR1
TR2
TR3
TR4TR5
)]n),…,y(t),y(tn),…,x(ty(t)=f[x(t yx 11 , (1)
Where y is the output variable and x is an exogenous input. From (1), it can be seen that the value of
the output signal y at time t is regressed on previous values of the same output y (back to the time
point t-ny) and exogenous input signals x (back to the time point t-nx). It is important to highlight that
both output and exogenous inputs are evaluated at antecedent time step, i.e. no information on the
current time step is required. For this reason, NARX models are mainly used for modelling nonlinear
dynamic systems [29]. In addition, they are also used for non-linear filtering of noisy input signals.
In this study, the NARX models were created by using the Neural Network Toolbox of MATLAB.
To train the NARX models, five measured time series data (TR1, TR2, TR3, TR4 and TR5) were
selected. The training data sets were selected in such a manner that the resulting NARX model can
generalize well in the operating range of the GT during the cold start-up. Therefore, the training data
sets cover the overall operating range of the GT and contain the maximum and minimum values of
all the inputs and outputs.
The optimal number of neurons in the hidden layer was identified by means of a sensitivity analysis
which considered 6, 12 and 18 neurons. Different values of initial weights can lead to various
solutions. To avoid local minimum problems, several networks were trained starting from different
random initial weights. Each time the initial random weights are generated by using standard uniform
distribution between -0.5 and 0.5, so that all sigmoid units are in their linear region. Otherwise, nodes
can be initialized into flat regions of the sigmoid causing very small gradients. Therefore, for each
number of neurons, an optimal set of initial weights was identified.
According to [29], (2) reports the function (mean square error - mse) which is commonly adopted for
defining neural network performance during training:
2
1
2
1
11
n
i
imi
n
i
i yyn
en
mseF , (2)
where n is the number of data in each data set.
Instead, in this study, NARX models were trained by modifying the performance function reported
in (2) to the form expressed in (3), so that each term can be properly weighted:
2
1
2
2
1
2 11
n
i
imii
n
i
iiw yywn
ewn
F , (3)
In this paper, an error weight function (EWF) defined in (4):
*
*)(
;1
,...,3,2,1 ;99.0*
tiw
tiwEWF
i
it
i (4)
EWF follows an exponential trend until it reaches the value of 1, as shown in Fig. 2. Different error
weight functions were analyzed preliminarily. The function in (4) was selected so that the errors in
the last part of each training sequence (i.e. when steady-state conditions are almost reached)
contribute more to the weighted performance function defined in (3) than the errors in the initial part
of each training sequence (i.e. when purely transient conditions are occurring). The time point t* in
(4) was tuned by considering mainly the output T04. In fact, in Fig. 2, the trend of T04 is superimposed
to that of the EWF to show that, for every training data set of T04, t* falls in correspondence of the
time point when T04 starts to vary rapidly and subsequently tends to its steady-state value. The final
values of t* are reported in Table 3.
Once the sensitivity analysis allowed the identification of the optimal number of neurons and the
optimal set of initial weights were identified, the NARX models with the best training performance
were also identified and tested against five additional time series data sets (TE1, TE2, TE3, TE4,
TE5) separately.
The parameter which was used to compare the measured data (ym) to NARX model predictions (y) is
the root mean square relative error (RMSE) defined according to (5):
2
1
1
n
i mi
imi
y
yy
nRMSE , (5)
where n is the number of data in each data set.
Figure 3 shows the open-loop and closed-loop structure of a NARX model. The training of all NARX
models was made by considering an open loop structure. Instead, when used for simulation and
testing, a closed loop structure was adopted. This means that the NARX model is fed with the values
estimated by itself at antecedent time steps.
As shown in Fig. 4, each NARX model has a multi-input single-output structure, which consists of
two inputs and one output. The input variables x(t) are the compressor inlet temperature T01 and the
fuel mass flow rate Mf. The output variables y(t) are the turbine outlet temperature T04, the compressor
outlet temperature T02, the compressor pressure ratio PrC and the rotational speed N.
Figure. 2. Trend over time of the error weight function
Figure. 3. Structure of a single NARX model: Figure. 4. Block diagram of the complete
a) open-loop, b) closed-loop. NARX model.
300
400
500
600
700
800
0
0,2
0,4
0,6
0,8
1
Erro
r w
eig
ht
(-)
Time (s)
Error weight function T04
1 207 486 1 247 419 1 317 494 1 335 487 1 395 635
Tu
rb
ine
ou
tlet
tem
pera
ture T
04
,(k
)
TR1 TR2 TR3 TR4 TR5
T04
NARX_T04
NARX_T02
NARX_PrC
NARX_N
T04
T02
PrC
T01
Mf
(a)
(b)
N
Data set t*
TR1 207
TR2 247
TR3 317
TR4 335
TR5 395
Table 3. Time point t* in (4)
NARX models were trained by using the Bayesian regularization training algorithm (trainbr), one
hidden layer and tapped delay lines with two delays (1:2) both for inputs x(t) and output y(t). This
means that the previous time steps are (t-1) and (t-2), according to the analysis performed in [15].
The two initial values (at time points t = 1 s and t = 2 s) of the two inputs and four outputs were
calculated as in (6):
2,1;5;)(
)(1
tkk
tt
k
j
TRj
average
. (6)
As it can be seen, the initial values at time steps 1 s and 2 s were selected to be the average of the
corresponding values in the five (k=5) transients used for training (TR1 through TR5). This was a
pragmatic approach, since these values are always known to NARX user. These values are reported
in Table 4.
NARX model tuning and training In order to identify the optimal number of neurons in the hidden layer, a sensitivity analysis was made
by considering NARX models with 6, 12 and 18 neurons. Moreover, for every number of neurons,
the training was repeated several times by initializing the neural network with different initial weights
as initial conditions. It should be noticed that the RMSE calculated over all the training data sets was
considered the only target parameter, since the training time was similar in all cases (less than one
minute). Furthermore, to account for the initial delay of the NARX model to work correctly, the
values corresponding to the first ten seconds of each data set were not used for RMSE calculation.
Figure 5 shows the training performance of the NARX models as a function of the number of neurons.
Table 4. Inputs and outputs averaged values
Input variables x Output variables y
Time T01,av (K) Mf,av (kg/s) T04,av (K) T02,av (K) PrC,av (-) Nav (rpm)
1 s 293.9 0.583 501.6 309.1 1.179 683.0
2 s 293.9 0.585 506.6 309.4 1.182 690.4
Figure. 5. Sensitivity analysis on the number of neurons in the hidden layer.
0
5
10
15
20
25
30
35
6 12 18
RM
SE
(%
)
Neurons number
best training performance
NARX model, T04
0
5
10
15
20
25
30
35
6 12 18
RM
SE
(%
)
Neurons number
best training performance
NARX model, T02
0
5
10
15
20
25
30
35
6 12 18
RM
SE
(%
)
Neurons number
best training performance
NARX model, PrC
0
5
10
15
20
25
30
35
6 12 18
RM
SE
(%
)
Neurons number
best training performance
NARX model, N
It can be seen that different initial conditions with the same number of neurons can lead to different
RMSE values, mainly for the outputs which are calculated with a higher RMSE (i.e. PrC and N).
Otherwise, if the RMSE on training data is lower (i.e. good accuracy for T04 and T02), the influence
of initial weights is lower.
It can be seen that twelve neurons is the optimal number for T04, T02, PrC and N with RMSE values of
1.6%, 0.6%, 1.9% and 12.2% respectively. Thus, training is clearly satisfactory for T04, T02, PrC,
while, also in the best case, the RMSE value of N is high (12.2%) since the NARX model cannot
reproduce the initial part of the maneuvers TR4 and TR5 (see Fig. 6), which are in practice at quasi
steady-state conditions. This heavily affects the RMSE value. In fact, by neglecting the first 24 s of
TR4 and 60 s of TR5, the RMSE value calculated on all training maneuvers decrease from 12.2% to
5.9%. However, these initial transients contain the minimum values of T04, T02 and N and their
presence in the training data sets should increase the robustness of the NARX models, since they can
also “learn” system stationary behavior. For these reasons, the RMSE values obtained by using twelve
neurons, were considered all-in-all acceptable and the training phase was considered satisfactory.
Figure 6 shows the trend of the four output parameters for the training maneuvers TR1, TR2, TR3,
TR4 and TR5, both measured on the considered GT and predicted by the NARX model which allowed
the best performance (i.e. with 12 neurons).
Figure. 6. Variations of turbine outlet temperature T04, compressor outlet temperature T02,
compressor pressure ratio PrC and rotational speed N for TR1, TR2, TR3, TR4 and TR5.
300
400
500
600
700
800
900
0 180 360 540
Tu
rb
ine o
utl
et
tem
pera
ture, T
04
(K
)
Time (s)
Measured value
NARX model TR1
RMSE=0.79 %
300
400
500
600
700
800
900
0 180 360 540Time (s)
TR2
RMSE=0.81 %
300
400
500
600
700
800
900
0 180 360 540
Time (s)
TR3
RMSE=0.88 %
300
400
500
600
700
800
900
0 180 360 540Time (s)
TR4
RMSE=0.82 %
300
400
500
600
700
800
900
0 180 360 540
Time (s)
TR5
RMSE=3.08 %
250
300
350
400
450
500
550
600
0 180 360 540
Com
press
or o
utl
et
tem
peratu
re, T
02
(K)
Time (s)
Measured value
NARX model TR1
RMSE=0.40 %
250
300
350
400
450
500
550
600
0 180 360 540Time (s)
TR2
RMSE=0.58 %
250
300
350
400
450
500
550
600
0 180 360 540
Time (s)
TR3
RMSE=0.51 %
250
300
350
400
450
500
550
600
0 180 360 540Time (s)
TR4
RMSE=3.36 %
250
300
350
400
450
500
550
600
0 180 360 540Time (s)
TR5
RMSE=0.80 %
1
2
3
4
5
6
7
8
9
0 180 360 540
Com
press
or p
ress
ure r
ati
o, P
rC
(-)
Time (s)
Measured value
NARX model TR1
RMSE=1.38 %
1
2
3
4
5
6
7
8
9
0 180 360 540Time (s)
TR2
RMSE=2.15 %
1
2
3
4
5
6
7
8
9
0 180 360 540
Time (s)
TR3
RMSE=1.70 %
1
2
3
4
5
6
7
8
9
0 180 360 540Time (s)
TR4
RMSE=2.37 %
1
2
3
4
5
6
7
8
9
0 180 360 540Time (s)
TR5
RMSE=1.95 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540
Rota
tion
al sp
eed
, N
(rp
m)
Time (s)
Measured value
NARX model TR1
RMSE=2.18 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540
Time (s)
TR2
RMSE=3.67 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540
Time (s)
TR3
RMSE=2.05 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540Time (s)
TR4
RMSE=13.13 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540
Time (s)
TR5
RMSE=22.35 %
It should be noted that, during the training phase, the training data sets were provided to the NARX
models in sequence, since all data have to be supplied contemporarily for training. Instead, the results
shown in Fig. 6 were obtained by simulating these maneuvers one by one. It can be observed that
significant deviations between measured and predicted values occur at the start of every maneuver.
This can be explained by the fact that the iterated prediction was performed by starting from two
averaged initial conditions (see Table 4). Then, the NARX model requires few seconds to stabilize
and reproduce the behavior of the gas turbine. In all cases, the most significant deviation occurs in
TR5. This is due to the fact that the difference between the measured value and the first predicted
value in TR5 is maximum and the NARX model requires more time to stabilize. Figure 6 also shows
that the NARX models can follow the physical behavior when the trend remains at quasi steady-state
condition. Moreover, they tend to smooth the too rapid variations, as clearly highlighted for T04.
NARX model testing The trained NARX models were tested against five additional experimental data sets (namely TE1,
TE2, TE3, TE4 and TE5) and the results are summarized in Fig. 7. The iterated prediction was
performed by using a closed-loop structure, by starting from two averaged initial conditions (see
Table 4). Also in this case, to account for the initial delay of the NARX model to work correctly, the
values corresponding to the first ten seconds of each data set are not used for RMSE calculation.
It can be observed that RMSE values for T04, T02 and PrC (see Fig. 7) are slightly higher than the
corresponding values obtained during training (see Fig. 5). This means that the trained NARX models
have good generalization capability. The most significant achievement is represented by the fact that
the RMSE values of N are even lower than the ones obtained during training and twelve neurons in
the hidden layer actually represent the best solution. More in detail, by considering the NARX models
with twelve neurons, it can be observed that the RMSE values are acceptable, i.e. 3.0% for T04, 0.8%
for T02, 3.2% for PrC and 2.7% for N. It should be noted that the RMSE of N calculated on the testing
maneuvers (2.7%) is decreased by about 78% compared to the one obtained on the training maneuvers
(12.2%).
Figure 8 shows the trend-over-time of TE1, TE2, TE3, TE4 and TE5. In all cases, the trends of the
field data and NARX predictions are very similar. This means that the NARX models can follow the
changes in GT parameters, even though they are subject to significant changes. Furthermore, it can
be clearly seen that the quasi steady-state operation in the last minutes of each maneuvers can be also
reproduced accurately. The most significant deviations between the measured and predicted values
occur in the maneuver TE5 with an RMSE value of 6.1 % for T04, 1.2 % for T02, 4.8 % for PrC and
5.3 % for N. As already discussed for the maneuver TR5 of the training data sets, this is due to the
fact that the difference between the measured value and the predicted value in correspondence of the
Figure. 7. NARX model performance against testing data sets.
0
1
2
3
4
5
6
RM
SE
(%
)
T04
6 Neurons 12 Neurons 18 Neurons
T02 PrCN
two initial time steps in TE5 is maximum, and the NARX model requires more time to stabilize. For
example, the first predicted value by the NARX model of T04 is 511 K, while at the same time point
the measured values for the maneuvers TE1, TE2, TE3, TE4 and TE5 of T04, are 544.2 K, 534.3 K,
510.4 K, 508.2 K and 344.8 K, respectively.
In any case, the overall deviation can still be acceptable. Therefore, it can be concluded that the
NARX models reproduce the five testing data sets TE1, TE2, TE3, TE4 and TE5 accurately and can
generalize well in the operating range of the GT during cold start-up. Finally, it should be noticed
that, on average, the NARX models developed in this paper can reproduce the GT dynamic behavior
more accurately (i.e. with a lower RMSE) than the NARX models presented by the same authors in
previous work [15], thanks to the optimization of the training phase performed in this paper.
Conclusions This paper presented a methodology for the development of NARX neural network models, in order
to simulate the dynamic behavior of a heavy-duty gas turbine during start-up. The NARX models
were trained by using five measured time-series data sets taken during cold start-up maneuvers. For
validation purposes, the resulting models were tested against other five experimental data sets, also
taken during cold start-up maneuvers.
Figure. 8. Variations of turbine outlet temperature T04, compressor outlet temperature T02,
compressor pressure ratio PrC and rotational speed N for the testing maneuvers.
300
400
500
600
700
800
900
0 180 360 540
Tu
rb
ine o
utl
et
tem
pera
ture, T
04
(K
)
Time (s)
Measured value
NARX model TE1
RMSE=0.98 %
300
400
500
600
700
800
900
0 180 360 540
Time (s)
TE2
RMSE=0.69 %
300
400
500
600
700
800
900
0 180 360 540
Time (s)
TE3
RMSE=1.05 %
300
400
500
600
700
800
900
0 180 360 540
Time (s)
TE4
RMSE=1.04 %
300
400
500
600
700
800
900
0 180 360 540
Time (s)
TE5
RMSE=6.11 %
250
300
350
400
450
500
550
600
0 180 360 540
Co
mp
ress
or o
utl
et
tem
pera
ture, T
02
(K)
Time (s)
Measured value
NARX model TE1
RMSE=0.90 %
250
300
350
400
450
500
550
600
0 180 360 540
Time (s)
TE2
RMSE=0.41 %
250
300
350
400
450
500
550
600
0 180 360 540Time (s)
TE3
RMSE=0.48 %
250
300
350
400
450
500
550
600
0 180 360 540Time (s)
TE4
RMSE=0.31 %
250
300
350
400
450
500
550
600
0 180 360 540Time (s)
TE5
RMSE=1.20 %
1
2
3
4
5
6
7
8
9
0 180 360 540
Com
press
or p
ress
ure r
ati
o, P
rC
(-)
Time (s)
Measured value
NARX model
TE1RMSE=3.80 %
1
2
3
4
5
6
7
8
9
0 180 360 540Time (s)
TE2
RMSE=2.54 %
1
2
3
4
5
6
7
8
9
0 180 360 540
Time (s)
TE3
RMSE=2.08 %
1
2
3
4
5
6
7
8
9
0 180 360 540Time (s)
TE4
RMSE=2.03 %
1
2
3
4
5
6
7
8
9
0 180 360 540Time (s)
TE5
RMSE=4.78 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540
Rota
tion
al sp
eed
, N
(rp
m)
Time (s)
Measured value
NARX model TE1
RMSE=1.65 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540Time (s)
TE2
RMSE=0.76 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540
Time (s)
TE3
RMSE=1.83 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540Time (s)
TE4
RMSE=1.01 %
0
500
1000
1500
2000
2500
3000
3500
0 180 360 540Time (s)
TE5
RMSE=5.27 %
During the training phase, the neural network performance index was modified, so that every term
can be weighted properly by means of an exponential error weight function. A sensitivity analysis on
the number of neurons in the hidden layer and on the initial weights was also performed. The results
indicated that NARX models with twelve neurons in the hidden layer (each of them with two inputs
and one output) provide better results.
In all simulations (during both training and testing), the most significant deviations occurred at the
beginning of maneuvers. In spite of that, the RMSE values during the training phase were satisfactory,
i.e. below 2% for three outputs and equal to 12.2% only for the rotational speed. Anyway, this latter
value drops to 5.9% by neglecting the first 60 s of two maneuvers. During the testing phase, the
RMSE values proved acceptable, i.e. in the range 0.8% - 3.2%, i.e. also the RMSE value on the
rotational speed was considerably lower than for training data.
In summary, the NARX model developed in this paper proved very robust to reproduce the unsteady
behavior and capture system dynamics. Moreover, the steps followed to achieve the final optimal
tuning were tracked and discussed.
Nomenclature EWF Error weight function
f NARX model transfer function
M mass flow rate, kg/s
n number
N rotational speed, rpm
p stagnation pressure, kPa
PrC compressor pressure ratio
RMSE root mean square relative error
mse mean square error
ϕ variable parameter
w weight coefficient
t time, s
T temperature, K
TIT turbine inlet temperature, °C
TOT turbine outlet temperature, °C
x externally determined input variable
y output variable
nx input delays
ny output delays
Subscripts and superscripts
* reference time point in (4)
01 compressor inlet section
02 compressor outlet section
04 turbine outlet section
f fuel
m measured
x externally determined input variable
y output variable
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