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,

hilal.bahlawan@unife.it, michele.pinelli@unife.it, pier.ruggero.spina@unife.it, mauro.venturini@unife.it b Dipartimento di Ingegneria Industriale, Università degli Studi di Parma, Parma, Italy,

mirko.morini@unipr.it * 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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