Optimization of operating conditions for steam turbine using an artificial neural network inverse

Applied Thermal Engineering xxx (2014) 1e10

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Optimization of operating conditions for steam turbine using an artificial neural network inverse ndez b, Victor Salazar b Y.El. Hamzaoui a, J.A. Rodríguez b, *, J.A. Herna rez (UACJ), n, CU, Universidad Auto noma de Ciudad Jua Instituto de Ingeniería y Tecnología, Dpto. Ingeniería El ectrica y Computacio rez, Chihuahua, Mexico Av. Del Charro # 450 Norte, CP 32310, AP 1594-D Ciudad Jua b n en Ingeniería y Ciencias Aplicadas (CIICAp-UAEM), Av. Universidad #1001, Col Chamilpa, CP 62209 Cuernavaca, Morelos, Mexico Centro de Investigacio a

h i g h l i g h t s
The failure assessment in blades is optimized using artificial neural network inverse (ANNi).
(ANNi) is a very effective modeling the useful life in blades of steam turbines.
Failure assessment in blades is optimized using artificial neural network inverse.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 June 2014 Accepted 23 September 2014 Available online xxx

The useful life (UL) of the failure assessment in blades of steam turbines is optimized using the artificial intelligence. The objective of this paper is to develop an integrated approach using artificial neural network inverse (ANNi) coupling with a Nelder Mead optimization method to estimate the resonance stress when the UL of the blades is required. The proposed method ANNi is a new tool which inverts the artificial neural network (ANN). Firstly, It is necessary to build the artificial neural network (ANN) that simulates the output parameter (UL). ANN's model is constituted of feedforward network with one hidden layer to calculate the output of the process when input parameters are well known, then inverting ANN. The ANNi could be used as a tool to estimate the optimal unknown parameter required (resonance stress). Very low percentage of error and short computing time are precise and efficient, make this methodology (ANNi) attractive to be applied for control on line the UL of the system and constitutes a very promising framework for finding set of “good solutions”. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Inverse neural network Optimal parameters Optimization Steam turbine failure Life cycle assessment in blades

1. Introduction Steam turbines have many applications in various industrial sectors. However, by common experience blade failures are the main origin of operational breakdowns in these machines, causing great economic lost in turbo machinery industry. The turbines are designed to work in stable operation condition [1e4]. Nevertheless, failure in blades has been present after a short time period of work. These failures commonly attributed to resonance stress of the blades at different stages to certain excitation frequencies. The expense of downtime and repair is about the millions of dollars [5]. The useful life (UL) is a very important variable for determining the

* Corresponding author. Tel.: þ52 7772677638. E-mail address: [email protected] (J.A. Rodríguez).

performance of steam turbines [6]. Therefore, the critical components which determine the useful life of the turbine should be evaluated to determine the rehabilitation or replacement of them. The critical components are the blades of steam turbines [7]. Most of the existing analytical models used to predict the useful life of the failure assessment in blades of steam turbines are based on analysis using analysis of vibrations for the construction of the diagram of Campbell, which shows the natural frequencies of the blades like a function of the speed of the rotor (RPM) [8]. These models do not provide reliable predictions for useful life (UL). This is caused by the complexity of solving the equations that involve the radiant energy balance, the spatial distribution of the absorbed radiation, mass transfer, and the mechanisms of steam turbines [5]. Moreover, in the light of the rapid development witnessed by the modern world in different fields of knowledge, science and technology, due to the increased speed of complexity of the system, in response to the issues requiring urgent attention of the people, in

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order to check for access with high performance and reliability. In recent years, computer science and artificial intelligence have received increasing attention proving effectiveness in solving many issues and outstanding challenges [6]. Artificial intelligence has been used in a wide range of fields including industrial application system has already proved successful, both on the technical level and as reservoirs of expertise [7,8]. Therefore in practice an artificial intelligence tools such as an artificial neural network (ANN) can provide a new approach to process without take into account any previous assumptions [5]. ANN is a collection of interconnecting computational elements which function like neurons in biological brain. It has the ability to model processes by learning from input and output data, without mathematical knowledge of the process. The (UL) could be calculated on-line, when the input variables are well known in blades of steam turbines. Indeed, the problem is that this (UL) computed by ANN is not ideal in the system, and therefore it is necessary that its input variables are well known when a given (UL) is required, that means ‘Find the effect of this cause’. However, the term of neural network inverse can be misleading sometimes. However, within the scope of this paper is to show the new strategy which is proposed in our current paper based on ANN model about the use of artificial neural network (ANN) for modeling the useful life of the failure assessment in blades of steam turbines. The main idea is coupling the neural network model with optimization algorithms to estimate the adequate value of a selected input to obtain the desired output. On the other hand, to understand very well the difference between inverse and optimization problem, Marcelo J. Colaço and others [9] presented basic concepts of inverse and optimization problems, and illustrated the inverse and optimization problems are conceptually different, despite their similarities. According to Marcelo J. Colaço et al., inverse problems are concerned with the identification of unknown quantities appearing in the mathematical formulation of physical problems, by using measurements of the system response. On the other hand, optimization problems generally deal with the minimization or maximization of a certain objective or cost function, in order to find design variables that will result on desired state variables [9e15]. The problem was developed through inverting an artificial neural network (ANNi) to estimate the optimum input variables on a required (UL) in the system. However, artificial neural networks inverse (ANNi) have been used successfully in different applications in which an output desired parameter is selected and then the input parameters values are determined [6e8,16e29]. The proposed method ANNi is a new tool which inverts artificial neural network (ANN) and it uses an optimization method to find the optimum parameter value (or unknown parameter) for given required conditions in the process. In order to do so, first, it is necessary to build the artificial neural network (ANN) model that simulates the output parameters of the failure assessment in blades of steam turbines is constituted of a feedforward network with one hidden layer to simulate output, considering one or more wellknown input parameters of the process. LevenbergeMarquardt

OutputðkÞ ¼ yfkg ¼ purelin

" S X s¼1

LWfk;sg $ tansig

K X

Fig. 1. General neural network inverse model.

learning algorithm, hyperbolic tangent sigmoid transfer-function, linear transfer-function and several neurons in the hidden layer (due to the complexity of the process) are considered in the built model. As soon as the model was validated, the second step was to invert the model. With the required output and some input parameters it is possible to calculate the unknown input parameters. However, it is important to note that the analytical solution with one neuron in the hidden layer neural model exists, and it is described in Section 4. Nevertheless, in the case that a proposed ANN model has more than one neuron in the hidden layer it is necessary to use an optimization method. On the other hand, in many cases, when an optimal output is required, the optimal input parameters are unknown, that's why, for this reason, we found that the inverse artificial neural network (ANNi) is a fundamental strategy to estimate the optimal operation condition. Rodriguez et al., have performed the sensitivity analysis to show which parameters have the most influence on (UL) [4] in order to optimize them by means of inverse neural network (ANNi). The paper is organized as follows second section will give an overview about artificial neural network inverse (ANNi), third section will discuss the Nelder Mead method, fourth section is devoted to the optimization approach applied on ANNi, then the fifth section is assigned to the results and discussions. Whilst, the comparative study is showed in Section 6. Finally, the conclusions on this work are drawn.

2. Artificial neural network inverse (ANNi) A general neural network is shown in Fig. 1 which is constituted by hyperbolic tangent (tanh) or sigmoid function (tansig) in the hidden layer and linear transfer functions in the output layer. Then the output is given by,

!!# IWðs;kÞ $InðkÞ þ b1ðsÞ

! þ b2ðkÞ

(1)

k¼1

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Y.El. Hamzaoui et al. / Applied Thermal Engineering xxx (2014) 1e10

3

According to the definition of purelin and tansig functions are used, y(k¼2) is given by,

yð2Þ ¼

13

0

2

B S 6 X B 6 6LWð2;sÞ $B B 6 @ s¼1 4 1 þ exp

2 2

PR

r¼1 IWðs;rÞ $pr þ b1ðsÞ

C7 C7 7 !!  1C C7 þ b2ð2Þ A5

(2)

Let p(r¼3) be the input to be estimated and k ¼ 2 the required output y(2) then

2 yð2Þ

S S 6 X X 6 6 ¼ b2ð2Þ  LWð2;sÞ þ 6 s¼1 s¼1 4

3 2LWð2;sÞ 1 þ exp

 2 IWðs;3Þ $p3 þ

PR

rs3 IWðs;rs3Þ $pfrs3g þ b1ðsÞ

where subscripts s is the number of neurons in the hidden layer; subscripts r is the number of neurons in the input layer; l is the number of neurons in output layer; S is the number of neurons in the hidden layer; R is the number of input; Tansig is the hyperbolic tangent sigmoid transfer function; Purelin is linear transfer function; and IW, LW and b1s, b2l are the input and output weights and the biases, respectively. The Eq. (4) is to be minimized to zero to find the optimal input(s) parameter(s) in a general ANN, in this case, x is the p3 value to be computed to zero by an optimization method.

7 7 !!7 7 5

method approximately finds a local optimal solution with N variables when the objective function varies smoothly. NeldereMead generates a new test position by extrapolating the behavior of the objective function measured at each test point arranged as a simplex. Then, the algorithm chooses to replace one of these test points with the new test point. Thereby, a new simplex is generated with a single evaluation of the objective. The numerical algorithm of the NeldereMead simplex method has been described in detail by Nelder and Mead [30].

2 f ðxÞ ¼ b2ð2Þ 

S X

LWð2;sÞ  yð2Þ þ

s¼1

S 6 X 6 6 6 s¼1 4 1 þ exp

(3)

3 2$LWð2;sÞ  2 IWðs;3Þ $x þ

Therefore, optimization can be done using the NeldereMead method.

PR

rs3 IWðs;rs3ÞÞ $pðrs3Þ þ b1s

7 7 !!7 7 5

(4)

4. Optimization approach 4.1. Neural network learning

3. NeldereMead method The NeldereMead method is a generally used nonlinear optimization algorithm. This method is a numerical method to minimize to zero an objective function in a multi-dimensional space. This algorithm is a direct search method that does not use numerical or analytic gradient [27]. It attends to minimize a scalarvalued nonlinear function of n real variables using only function values, without any derivative information. The method uses the concept of simplex, which is a polyhedron of N þ 1 in N dimensions. Simplices are a line, a triangle and tetrahedron in one-, two-, and three-dimensional space, respectively, and so forth [28]. The

A learning (or training) algorithm is defined as a procedure that consists of adjusting the coefficients (weights and biases) of a network, to minimize an error function (usually a quadratic one) between the network outputs, for a given set of inputs, and the correct (already known) outputs as shown in Fig. 2. If smooth nonlinearities are used, the gradient of the error function can be computed by the classical backpropagation procedure. To determine the best backpropagation training algorithm, ten backpropagation algorithms were studied. In addition, three neurons were used in the hidden layer for all backpropagation algorithms. Table 1 shows a comparison of different backpropagation training

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Fig. 2. Numerical procedure used for the ANN learning process, and the iterative architecture used by the model to predict the UL of the failure assessment in blades of steam turbines (S is the number of the neuron in the hidden layer).

Consequently, RMSE and R2 were used as the error function which measures the performance of the network. Therefore, the network having minimum RMSE and maximum R2 was selected the best ANN model. More detail about neural network training was already described by Rodriguez et al. [4].

algorithms. LevenbergeMarquardt backpropagation training algorithm could have smaller mean square error (RMSE), on the other hand, we found training with Levenberg Marquardt algorithm can run smoothly in computer with lower expanded memory specification (EMS), and the training time is quickly, than the other backpropagation algorithms. Because, the LevenbergeMarquardt algorithm was designed to approach second order training speed without having to compute the Hessian matrix. However, the performance of the ANN model was statistically measured by the root mean square error (RMSE) and regression coefficient (R2), which are calculated with the experimental values and network predictions as illustrated in Figs. 3 and 4. These calculations are used as a criterion for model adequacy, obtained as follows:

RMSE ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u  2 u P N u u n¼1 yn;pred  yn;exp t

In the case of one neuron in the hidden layer. The analytical solution is performed as follow:
If tansig and purelin are considered as the hyperbolic tangent sigmoid and linear transfer function in the hidden layer and output layer, and k ¼ 1.

1

0

Outð1Þ ¼ LWð1;1Þ $@

(5)

N

2

  1A þ b2   1 þ exp  2$ IWð1;rÞ $pr þ b1 (7)

2 yn;pred  yn;exp n¼1 R2 ¼ 1  2 PN  n¼1 yn;exp  ym PN

4.2. Inverse neural network considering one neuron in the hidden layer in ANN model



This can be transformed into:

(6)

1 2 @   1A   Outð1Þ  b2 ¼ LWð1;1Þ $ 1 þ exp  2$ IWð1;rÞ $pr þ b1 0

where N is the number of data points, yn,pred is the network prediction, yn,exp is the experimental response, ym is the average of actual values and n is an index of data.

(8)

Table 1 Comparison of 10 backpropagation algorithms with five neurons in the hidden layer. Backpropagation algorithm

Function

Root mean square error (RMSE)

Epoch

Correlation coefficient (R2)

Best linear equation

LevenbergeMarquardt backpropagation Batch gradient descent Batch gradient descent with momentum PolakeRibiere conjugate gradient backpropagation Scaled conjugate gradient backpropagation BFGS quasi-Newton backpropagation PowelleBeale conjugate gradient backpropagation One step secant backpropagation FletchereReeves conjugate gradient backpropagation Variable learning rate backpropagation

trainlm traingd traingdm traincgp trainscg trainbfg traincgb trainoss traincgf traingdx

0.00235005 0.01657932 0.01982303 0.03267017 0.44944913 0.48619630 0.50820237 0.02753381 0.01756329 0.02039637

1000 2000 2000 2000 2000 2000 2000 2000 2000 2000

0.990 0.988 0.987 0.979 0.974 0.971 0.965 0.782 0.725 0.718

Y Y Y Y Y Y Y Y Y Y

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

0.990X þ 0.306 0.986X þ 0.927 0.988X þ 0.837 0.957X þ 2.53 1.020X  0.783 0.982X þ 1.23 0.960X þ 2.03 0.617Xþ45.3 0.425X þ 34.8 0.386X þ 38

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5

Fig. 3. Schematic methodology.

0

1 2$LWð1;1Þ   LWð1;1Þ A   Outð1Þ  b2 ¼ @ 1 þ exp  2$ IWð1;rÞ $pr þ b1

 LW   ð1;1Þ  Outð1Þ þ b2 ¼ exp  2 IWð1;1Þ $p1 þ b1 Outð1Þ þ LWð1;1Þ  b2

(13)

(9) Outð1Þ  b2 þ LWð1;1Þ

2$LWð1;1Þ    ¼ 1 þ exp  2$ IWð1;rÞ $pr þ b1

  LWð1;1Þ  Outð1Þ þ b2 2 IWð1;1Þ $p1 þ b1 ¼ ln Outð1Þ þ LWð1;1Þ  b2

! (14)

(10)    1 þ exp  2$ IWð1;rÞ $pr þ b1 ¼

LWð1;1Þ  Outð1Þ þ b2 1 IWð1;1Þ $p1 þ b1 ¼  ln 2 Outð1Þ þ LWð1;1Þ  b2

2LWð1;1Þ Outð1Þ  b2 þ LWð1;1Þ (11)

 LW   ð1;1Þ  Outð1Þ þ b2 ¼ exp  2 IWð1;rÞ $pr þ b1 Outð1Þ þ LWð1;1Þ  b2 As k ¼ 1, R ¼ 1, So

(12)

LWð1;1Þ  Outð1Þ þ b2 1 IWð1;1Þ $p1 ¼  ln 2 Outð1Þ þ LWð1;1Þ  b2

! (15)

!  b1

(16)

Let p{r¼1} would be the input parameter to be calculated when one output parameter is required. Then:

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  exp  IWð1;1Þ $p1 þ b1 ¼

LWð1;1Þ 1 Outð1Þ  b2

 IWð1;1Þ p1 ¼ ln

p1 ¼

Fig. 4. Performance plots of ANN during training, validating and testing of the network.

LWð1;1Þ  Outð1Þ þ b2 1 p1 ¼  ln 2$IWð1;1Þ Outð1Þ þ LWð1;1Þ  b2

! 

b1 IWð1;1Þ

(17)

 LWð1;1Þ  1  b1 Outð1Þ  b2

(21)

  LWð1;1Þ b1  ln Outð1Þb 1 2

IWð1;1Þ

(22)

(23)

Therefore, according to Eq. (1), that it is possible to simulate the outputs values, when input parameters are well known. However, in many cases, the problem is that ANN predicted output values which are not satisfactory in the system, and therefore, it is necessary that its inputs variables are well known when giving a required or satisfactory output. Consequently, the new control strategy which is proposed here using ANN model applied to energy systems. The proposed strategy uses an inverse of neural network and the NeldereMead optimization algorithm to find the optimal input values for the required output value. Then in this ANNi methodology, as mentioned above, the required output value is well-known.

5. Results and discussion
If logsig and purelin are considered as the logistic tangent sigmoid and linear transfer function in the hidden layer and output layer, and k ¼ 1

0 Outð1Þ ¼ LWð1;1Þ @ Outð1Þ  b2 ¼

1 1

A þ b2   1 þ exp  IWð1;1Þ p1 þ b1

LWð1;1Þ   1 þ exp  IWð1;1Þ $p1 þ b1

  1 þ exp  IWð1;1Þ p1 þ b1 ¼

LWð1;1Þ Outð1Þ  b2

Fig. 5. The cracked leading edge of the L-0 blade.

(18)

(19)

(20)

The experimental set up used in this work has been previously described in detail elsewhere [4]. Failures of turbine blade usually initiate at the zone of high stress concentration which occur in metallurgical discontinuities or where corrosion is present or even in regions of excessive wear. During operation of the turbine, the cracks are frequently caused by erosion, corrosion or small imperfections and then propagate into the fracture. However, these imperfections increase the fatigue stress concentration factor and of course the stresses themselves. Turbine inspection revealed that sets of 10 blades failure of the L-0 in low pressure stage of a 110 MW steam turbine were illustrated within 15 cm from the root. The blades are significantly affected by a crack practically is a fracture observed in a blade root is shown in Fig. 5. The L-0 stage had 110 blades of 0.6 m in length with groups of 10 blades. The blades of each group are connected in their top end by a shroud and two wires as shown in Fig. 6. In addition, a visual inspection combined with a revision of the turbine operation history was carried out into system description and experimental data. Furthermore, a turbine

Fig. 6. Discrete models of blade group for the calculation of natural frequencies.

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flow analysis was also conducted together with verification of the blade dimensions and tolerances. The natural frequencies and vibration modes analysis of the 10 blades group and the stress analysis of blades were realized in experimental mode and a finite element program called ANSYS [2,31e34]. The operation history of the turbine was carefully studied and revised. The turbine was operated by approximately 1800 h in intermittent mode, with a record of 650 start-ups, during a period of 5 years with at least five replicates of information acquisition to ensure the verification of measurements a database about 2500 samples was obtained According to the work developed by Garcia [31], Kubiak [32] and Rodríguez [34]. Rodriguez et al. [5] proposed a neural network model which demonstrating an efficiency of 99% in predicting useful life (UL) of the failure assessment in blades of steam turbines. This developed ANN model has three neurons in the hidden layer (21 weights and 4 biases) and considering 6 inputs parameters (resonance stress, frequency ratio, dynamic stress, damping, fatigue strength, mean stress). The proposed equation developed by Rodriguez et al. [5] is

2 UL ¼

  2LWð1;1Þ 2LWð1;2Þ 2LWð1;3Þ UL ¼ þ þ 1 þ eX1 1 þ eX2 1 þ eX3    LWð1;1Þ þ LWð1;2Þ þ LWð1;3Þ þ b2ð1Þ

(25)

where

 X1 ¼ 2 IWð1;1Þ V1 þ IWð1;2Þ V2 þ IWð1;3Þ V3 þ IWð1;4Þ V4  þ IWð1;5Þ V5 þ IWð1;6Þ V6 þ b1ð1Þ

(26)

 X2 ¼ 2 IWð2;1Þ V1 þ IWð2;2Þ V2 þ IWð2;3Þ V3 þ IWð2;4Þ V4  þ IWð2;5Þ V5 þ IWð2;6Þ V6 þ b1ð2Þ

(27)

13

0

B S 6 X B 6 6LWð1;sÞ $B B 6 @ s¼1 4 1 þ exp

7

 2$

PR

2 

r¼1

IWðs;rÞ $pr þ b1ðsÞ

C7 C7 7 !!  1C C7 þ b2  A5

where the number of neurons in the input layer is 6. According to the model given by Eq. ( 24), it is possible to simulate the useful life of the failure assessment in blades of steam turbines, while the input parameters are well known. Since we found, that the resonance stress is the most influential parameter [5]. Therefore, it is important to know in this process, what optimal resonance stress is needed for a required UL (input number 1). Consequently, we have developed a strategy to estimate the optimal resonance stress in the failure assessment process about blades of steam turbines from the inverse artificial neural network (ANNi). In the meantime, Hern andez et al. [23] and El-Hamzaoui et al. [18] have been applied ANNi with ANN model prediction in order to different processes of R2 ¼ 0.915 and R2 ¼ 0.986, respectively. The authors mentioned that it is possible to use ANN model with predictions of R2 > 0.985 for implementing ANNi. In our case the ANN model is of R ¼ 0.99. Consequently, we believe that it is possible to developed ANNi in this process. The results have been showed that the UL's error evaluation between the experimental and simulated by ANNi is 0.7%. As we can see very small. In addition, the neural network model developed by Rodriguez et al. has an efficiency of 99%. Therefore, with this motivation mentioned above, we can use this model to perform ANNi model. The proposed method (ANNi) inverts the artificial neural network. Then we have the following equation that calculates UL during the failure assessment process in blades of steam turbines. The key information (optimal performance) for useful life of the failure assessment in blades of steam turbines, when controlling the required output is to know the optimal input parameters. An inverted ANN could be considered as a model based method of supervisory control, the control action in which the unknown input parameters are obtained by solving an on line optimization problem for the desired output. The inverse of the artificial neural network deduced from Eq. (24) is the following Eq. (25) that calculates UL in the system.

(24)

 X3 ¼ 2 IWð3;1Þ V1 þ IWð3;2Þ V2 þ IWð3;3Þ V3 þ IWð3;4Þ V4  þ IWð3;5Þ V5 þ IWð3;6Þ V6 þ b1ð3Þ

(28)

V1 ¼ Resonance stress [MPa] V2 ¼ Frequency ratio V3 ¼ Dynamic stress [MPa] V4 ¼ Damping V5 ¼ Fatigue strength [MPa] V6 ¼ Mean stress [MPa] UL ¼ Useful Life [Min] At this step, we have obtained the function which has to be optimized to get the optimal input parameter:

f ðV1 Þ ¼ A þ

2LWð1;1Þ 1 þ eX11 3:58V1

þ

2LWð1;2Þ 1 þ eX22 34:1V1

þ

2LWð1;3Þ 1 þ eX33 þ5:5V3 (29)

Table 2 Adjustable parameters obtained (weights and bias) in the proposed model with S ¼ 3, K ¼ 6. IW(s,k)

Wo(s) b1(s) b2

Wi(1,1) 1.79 Wi(2,1) 17.05 Wi(3,1) 2.75 Wo(1) 0.19 b1(1) 123.17 0.29

Wi(1,2) 1.14 Wi(2,2) 7.45 Wi(3,2) 0.17 Wo(2) 0.13 b1(2) 17.15

Wi(1,3) 2.53 Wi(2,3) 43.37 Wi(3,3) 3.58 Wo(3) 0.81 b1(3) 6.39

Wi(1,4) 1.21 Wi(2,4) 5.38 Wi(3,4) 0.59

Wi(1,5) 2.37 Wi(2,5) 27.57 Wi(3,5) 0.69

Wi(1,6) 1.17 Wi(2,6) 10.29 Wi(3,6) 12.76

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Table 3 Some samples of the experimental and simulated information of the system. Test number Input Mean stress Frequency ratio Dynamic stress Damping Fatigue strength Resonance stress_Exp Output [Useful Life]EXP [Useful Life]SIMANN

500

1000

2000

3000

4000

5000

6000

7000

8000

9000

557.12 0.91 24.12 0.018 1295.36 244

655.97 1.052 35.41 0.020 967.85 221

483.77 1.017 101.75 0.0201 1186.33 208

562.34 0.902 31.66 0.023 977.33 208

602.87 0.916 30.184 0.0188 923.95 211.5

500.71 1.029 85.985 0.0201 1070.33 216

634.83 0.966 60.516 0.0185 971.45 219

492.83 0.889 21.95 0.0201 1128.079 221

673.60 1.036 60.59 0.0207 883.3121 221.5

566.39 0.937 32.159 0.0198 1038.347 220

4.67Eþ11 4.67Eþ9

4.27Eþ17 4.27Eþ15

4.7855Eþ10 4.7855Eþ8

5.38Eþ14 5.38Eþ12

6.87Eþ14 6.87Eþ12

3.19Eþ14 3.19Eþ12

1.54Eþ14 1.54Eþ12

4.18Eþ14 4.18Eþ12

6.03Eþ12 6.03Eþ10

3.04Eþ13 3.04Eþ11

where

A ¼ UL  b2ð1Þ þ LWð1;1Þ þ LWð1;2Þ þ LWð1;3Þ

(30)

 X11 ¼ 2 IWð1;2Þ V2 þ IWð1;3Þ V3 þ IWð1;4Þ V4 þ IWð1;5Þ V5  þ IWð1;6Þ V6 þ b1ð1Þ

(31)

 X22 ¼ 2 IWð2;2Þ V2 þ IWð2;3Þ V3 þ IWð2;4Þ V4 þ IWð2;5Þ V5  þ IWð2;6Þ V6 þ b1ð2Þ

(32)

 X33 ¼ 2 IWð3;2Þ V2 þ IWð3;3Þ V3 þ IWð3;4Þ V4 þ IWð3;5Þ V5  þ IWð3;6Þ V6 þ b1ð3Þ

(33)

The weights (IW and LW) and biases (b1 and b2) of ANN's model are showed in Table 2 and the input parameters are reported by Rodriguez et al. [4] in order to minimize to zero the Eq. (29), an optimization method (NeldereMead Simplex Method) is used to calculate the resonance stress (V1). Where, the Input of ANN are frequency ratio; dynamic stress; damping; fatigue strength; mean

stress. Whilst, the required output is the useful life. However, the resonance stress will be estimated by ANNi. The optimization method finds the minimum of a scalar function of several variables, starting at an initial estimate. However, this is generally referred to as unconstrained nonlinear optimization. However, the optimization method starts at the initial value and finds a local minimum V1 described in Eq. (29). On the other hand, Table 3, shows some samples of the experimental and simulated information of the system as shown in Fig. 7. An example of this application is shown to calculate the required resonance stress (V1) considering the experimental data reported by Rodriguez et al. [34,35], which values are obtained from experimental test database showing in test number 1000, thus only, we want to calculate the resonance stress value (V1): Mean stress (V6) ¼ 655.97 [MPa]; Frequency ratio (V2) ¼ 1.052; Dynamic stress (V3) ¼ 35.41 [MPa]; Damping (V4) ¼ 0.020; Fatigue strength (V5) ¼ 967.85 [MPa]; Resonance stress (V1) ¼ ? And an output value, UL ¼ 4.27E þ 17 [Min] According to the weights and biases of Table 1 and optimization method of the NeldereMead, it is possible to calculate the optimum Resonance Stress (V1), which is: Resonance Stress ANNi ¼ V1ANNi ¼ 224 [MPa].

Fig. 7. Architecture of the artificial neural network inverse for determining the optimum resonance stress.

Please cite this article in press as: Y.E. Hamzaoui, et al., Optimization of operating conditions for steam turbine using an artificial neural network inverse, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.065

Y.El. Hamzaoui et al. / Applied Thermal Engineering xxx (2014) 1e10

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Fig. 8. Experimental resonance stress vs resonance stress by ANNi.

However, to validate this value, a test was performed with different data to optimize the resonance stress in different conditions to demonstrate the feasibility of this method about ANNi. However, the simulation outcomes were then compared with experimental data in order to check the accuracy of ANNi. This error is given by:

Err ¼ 100

jExp  Simj Exp

(34)

That means Resonance Stress estimated by ANNi is compared to experimental resonance stress of the experimental data using Eq (34). In Table 2, within the test number 1000, the experimental value of Resonance Stress is V1EXP ¼ 221 [MPa]. Whilst, the error is given by Eq (34). Therefore, in this case, an error of 1.5% is obtained which is very acceptable. On the other hand, the elapsed time to calculate this Resonance Stress from this methodology (ANNi with Nelder Mead Simplex) is only 35.17 s. It's seem that this time is good enough to control the process. In addition, Fig. 8, illustrates that there is a good agreement between the experimental resonance stress and resonance stress estimated by ANNi. In the meantime, the fitting quality is so good. It has been an outstandingly successful models in estimating experimental results by ANNi.

6. Comparative results The remarkable thing is that, according to Fig. 9, there is good agreement between the predicted values for useful life of the failure assessment in blades of steam turbines by ANN and ANNi models with experimental data. Indeed, it has been an outstandingly successful models in predicting the experimental results. Consequently, the UL error between the experimental and simulated by ANNi is 0.7%. These models: artificial neural network (ANN) and artificial neural network inverse (ANNi) prove to be very effective in modeling the useful life of the failure assessment in blades of steam turbines. The smaller RMSE and larger R2 mean better performance [5]. However, the performance of the ANN and ANNi on modeling UL of the failure assessment in blades of steam turbines is presented in Fig. 9, where the two models are trained using the same training datasets and validated by the same datasets (fresh data). In practice, however, the calculation required for system are so complicated, that's why, all the calculations were carried out on LINUX system, Intel D CPU 2.80 Ghz, 2.99 GB of RAM. According to Fig. 9, we can distinguish the following results: The ANN model has smaller error for datasets than the ANNi (about 0.85%). In this

Fig. 9. UL versus number of test patterns for failure assessment of steam turbines.

Please cite this article in press as: Y.E. Hamzaoui, et al., Optimization of operating conditions for steam turbine using an artificial neural network inverse, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.065

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Y.El. Hamzaoui et al. / Applied Thermal Engineering xxx (2014) 1e10

way, the ANN achieves better performances than the ANNi model. Therefore, ANN is a good choice for modeling the useful life of the failure assessment in blades of steam turbines. It is believed also, that ANN and ANNi could be used to handle many other types of problems about the failure assessment in blades of steam turbines. 7. Conclusions The useful life (UL) of the failure assessment in blades of steam turbines is optimized using artificial neural network inverse in order to calculate an ideal input value from an ideal (UL) and taking into account the above well known input values excepting required input value as resonance stress. Then, NeldereMead method is applied in the inverse problem to optimize the optimal operating condition is tested for a single parameter. Thanks to this method, it is possible to find any unknown input variable on line in the engineering failure analysis in blades of steam turbines. Indeed, it is very important to note that the elapsed time to calculate the optimum input parameter is only a few seconds (<40 s), thus it is feasible to get optimal parameters on line and is sufficiently suitable to direct control of steam turbines. Briefly, ANNi integrated with NeldereMead method significantly reduced the computational time with better convergence for optimal solution for useful life of the failure assessment in blades of steam turbines. Despite its successes, ANNi, is still in its infancy. It's part of the future. In a way it's amazing we have done so much with so little, and we have barely begun. However, if there are many input parameters to be found (solution to multi-parameter problems) then NeldereMead method couldn't be able to solve the optimization problem. It would be recommended to use another advanced techniques for solving optimization problem, such as genetic algorithms (GAs) and particle swarm optimization (PSO).

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Acknowledgements [22]

J.A. Rodríguez, expresses his gratitude to CONACYT for the project with title is: Experimental and numerical study for evaluation of reliability and life estimation of turbine blades under resonance conditions [In Spanish] whit reference number: 156757. Y.El. Hamzaoui, expresses his gratitude to PROMEP for financial support (Project PROMEP/103.5/13/7073.UACJ-PTC-289).

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Please cite this article in press as: Y.E. Hamzaoui, et al., Optimization of operating conditions for steam turbine using an artificial neural network inverse, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.065