978-1-4799-5538-1/14/$31.00 ©2014 IEEE

Neural Networks and Wavelet Transform in Waveform Approximation

Paul Farago, Gabriel Oltean, Laura-Nicoleta Ivanciu Bases of Electronics Department

Technical University of Cluj-Napoca Cluj-Napoca, Romania

Gabriel.Oltean@bel.utcluj.ro

Abstract—To fully analyze the time response of a complex system, in order to discover its critical operation points, the output waveform (under all conceivable conditions) needs to be generated. Using conventional methods as physical experiments or detailed simulations can be prohibitive from the resources point of view (time, equipment). The challenge is to generate the waveform by its numerous time samples as a function of different operating conditions described by a set of parameters. In this paper, we propose a fast to evaluate, but also accurate model that approximates the waveforms, as a reliable substitute for complex physical experiments or overwhelming system simulations. Our proposed model consists of two stages. In the first stage, a previously trained artificial neural network produces some coefficients standing for “primary” coefficients of a wavelet transform. In the second stage, an inverse wavelet transform generates all the time samples of the expected waveform, using a fusion between the “primary” coefficients and some “secondary” coefficients previously extracted from the nominal waveform in the family. The test results for a number of 100 different combinations of three waveform parameters show that our model is a reliable one, featuring high accuracy and generalization capabilities, as well as high computation speed.

Keywords—waveform approximation; wavelet transform; neural network; coefficients selection.

I. INTRODUCTION Designing and testing a robust complex system usually

requires the analysis of its time response under different operating conditions, or, if possible, under all conceivable conditions. These operating conditions can be described by a set of parameters whose values can be generated in any possible combinations, keeping each parameter in its own admissible range of variation. So, for each combination of parameters the system can generate a different time response, or a different output waveform. Due to very large costs (time and equipment), it is almost impossible to fully investigate the waveform for any possible combination of the input parameters, under all allowed operating conditions for a physical system. The costs can be reduced by using different simulation environment that can accurately model the system under test. However, detailed low level simulation may imply a lot of time for each run (for each parameter combination that generates only one output waveform). So,

once again, an extensive analysis for any possible combination in the parameter space introduces a lot of overhead.

The idea proposed in this paper is to build a model that is not only cheap and fast to evaluate, but also an accurate substitute for complex physical experiments or long system simulations. The model should approximate the output waveform for any possible combination of input parameters.

Because the problem falls into the category of curve fitting approaches, a very good candidate to be used in our model is a feedforward artificial neural network (ANN). Some of the main features that recommend the artificial neural networks are [1], [2], [3], [4]: ANNs are universal approximators that can learn data by example and can approximate any nonlinear multi-variable function with any desired accuracy, ANNs are nonlinear input-output mapping, ANNs can generalize, no prior assumption of the model form is required, ANN can often correctly infer the unseen part of a population even if the sample data contain some noisy information.

The literature offers multiple examples of using ANNs to address different aspects in waveform processing. A Hopfield neural network is applied in [5] for ECG signal modeling and noise reduction. The algorithm retrieves a pattern stored in memory in response to the presentation of an incomplete or noisy version of that pattern. Automatic detection of spikes in electroencephalograms (EEG) can be solved using neural network as it is presented in [6]. Back Propagation Neural Network and Radial Basis Function Neural Network are used to develop behavioral models of an RF power amplifier, the predicted output signal corresponding to sampling points of the amplifier output waveform value [7]. In [8], an ANN is used for detection and classification of electrical disturbances in three-phase systems.

In [9], a neural network provides a means of determining a degree of belief for each identified disturbance waveform in Power System. Three types of neural networks (multilayer perceptron, radial basis function and wavenet) are used in [10] to estimate the feedback signal for a vector controlled induction motor drive.

To approximate periodic, exponential and piecewise

continuous functions for one or two input variables, radial basis function neural network and wavelet neural network are used in [4]. In [11] the configuration of an ANN (number of hidden units in the hidden layer, transfer function to use at the hidden layer, and transfer function to use at the output layer) is optimized such that the network’s approximation error for signal approximation problems is minimized. Three different signals were considered there: Boolean XOR function, sinusoidal signal, and a signal representing the activity measurements on a server system. It is worth to mention that for this class of function approximations, the neural network should generate the value of the function for each value(s) of the variable(s) presented to its input(s), so the output signal is generated point by point.

Another related field of applications for artificial neural networks is the one of forecasting models, which has enjoyed fruitful applications in forecasting social, economic, engineering, foreign exchange, stock problems, etc [3]. In this particular case of forecasting model, we have to deal with time series, so the task for the neural network is to perform a nonlinear functional mapping from past observations to the future value ty [3]:

tpttt wyyfy ε+= −− ),...,,( 1 (1)

where yt is the function value at time period t, p is the number of inputs, w is a vector of all network parameters, εt is the random error at time period t and f(⋅) is a nonlinear function determined by the network structure and connection weights. Thus, the neural network is equivalent to a nonlinear auto-regressive model.

On the contrary, the problem to be solved in this paper is different from the above presented forecasting one, through the fact that our model has to generate the entire output waveform for a certain combination of the input parameters. In other words, we should simultaneously generate all the time samples of the waveform, so we need a model for a nonlinear multidimensional and multivariable function. Our model performs a functional nonlinear mapping from a vector of input parameters to the vector of time samples of the waveform:

( ]...,,,[ 21 Qsss ) )],...,,,([ 21 wpppf N= (2)

where ]...,,,[ 21 Qsss is the output vector of the time samples of the waveform, Q is the number of time samples,

]...,,,[ 21 Nppp is the input parameter vector, N is the number of parameters, and w is a vector of all network parameters.

To capture the characteristics of certain set of waveforms, common and differentiating, we have to use a quite high sample frequency that will generate thousands or even many more data points (time samples). If the neural network has to generate all these time samples, this would lead to an ANN with a lot of neurons in its output layer, which is not

practical, from the point of view of network training resources (time, computer memory, size of the training data set) and also form the point of view of implementing and simulating such a huge ANN.

To reduce the problem dimensionality (very large number of neurons in the output layer of the ANN) we propose a solution inspired from the pattern recognition domain, where in order to recognize (classify) a pattern we have to first describe the pattern by a reduced set of representative features [12], [13].

To extract representative waveform features, ANNs are usually combined with mathematical analysis, such as Fourier and wavelet transforms, for the generation of signal features which serve as inputs for the neural network [9], [11], [14], [15], [16]. In this paper we chose the wavelet transform, because according to [17], [18] it is an improved version of the Fourier transform. The coefficients of the wavelet transform contain both time and frequency information, while the coefficients of the Fourier transform preserve only frequency information.

The discrete wavelet decomposition generates a number of coefficients equal or greater than with the number of time samples. In order to reduce the dimensionality of our problem, we will select only a small fraction of the coefficients, the most significant (“primary”) ones to be used as outputs of our ANN.

The proposed model for waveform approximation contains an ANN that will generate the primary coefficients corresponding to a discrete wavelet transform, as a function of a certain parameter combination. These coefficients will be used in conjunction with the secondary coefficients of the discrete wavelet decomposition of the nominal waveform of the family to reconstruct the approximated waveform using the inverse discrete wavelet transform.

II. STRUCTURE OF THE MODEL FOR WAVEFORM APPROXIMATION

The model used for waveform approximation has to rapidly generate an accurate approximation of the output waveform (by its time samples) for any combination of the waveform parameters. So, at the top level, the model consists of a black box (Fig. 1.), having as inputs the parameter vector ]...,,,[ 21 Npppp = , where N is the number of parameters, and as outputs the vector of the time samples of the waveform ]...,,,[ 21 Qssss = , where Q is the number of time samples.

WAVEFORM APPROXIMATION BASED ON ANNs AND

WAVELET TRANSFORMS

Np

pp

2

1

Qs

ss

2

1

Fig. 1. General model for waveform prediction.

The problem to be solved can be formulated as: Given a particular (new) combination of input

parameters, ]...,,,[ **2

*1

*Npppp = , generate the

corresponding waveform by its time samples ]...,,,[ **

2*1

*Qssss = .

As discussed in Section I, it is not possible to use an ANN to generate all the time samples of the waveform, because of the size of the necessary network. To reduce the dimensionality of the problem we will drastically decrease the number of outputs by selecting as outputs only the primary coefficients from the discrete wavelet decomposition.

The wavelet transform is a tool that cuts data or functions or operators into different frequency components, and then studies each component with a resolution matched to its scale [19].

Our waveform is broken down into many lower resolution components using discrete multilevel wavelet decomposition (DMWD). At each level of decomposition the signal is decomposed into low and high frequencies. Due to the decomposition process the input signal must be a multiple of L2 , where L is the number of decomposition levels [18], [20].

For many signals, the low-frequency content is the most important part. It is what gives the signal its identity. The high-frequency content, on the other hand, imparts flavor or nuance. In wavelet analysis, we speak of approximations (cA – approximation coefficients) and details (cD–details coefficients). The approximations are the high-scale, low-frequency components of the signal. The details are the low-scale, high-frequency components [18].

For example if one chooses L=5 and the waveform is described by KL2 samples, the number of coefficients on each level are the ones listed in Table I.

TABLE I. NUMBER OF COEFFICIENTS ON EACH LEVEL, FOR L=5

Now, we have to select the most important (“primary”) coefficients that characterize the waveform. In our proposed approach we use a global threshold and we select as primary coefficients only the coefficients whose magnitude is greater or equal than the threshold. To maintain the generality, we will not impose a certain value of the threshold, (that is dependent of the waveform family), but rather we impose a

fraction from the total number of coefficients to be selected. For example we can use 5% of the total number of the coefficients to be selected in descending order of their magnitude. Because we know that the approximation coefficients are high-scale, low frequency components of the signal, containing information about the shape of the signal, it is necessary to select all these coefficients, so the resulting threshold should allow the selection of all approximation coefficients, plus some “important” (according with their magnitude) details coefficients. In accordance with the numbers of coefficients in Table 1, we have to select more than K coefficients as primary ones.

The block diagram of the model able to generate the approximation of the waveform for every combination of input parameters is presented in Fig. 2.

The ANN takes as input the actual combination of input parameters ( *p ) and generates the corresponding vector of

primary coefficients *PC . The COEFFICIENTS FUSION block

performs the fusion between the vector of coefficients of the nominal waveform NomC and the vector of primary

coefficients for the actual waveform *PC . This is done by

replacing every primary coefficient in NomC with its

counterpart from *PC as indicated in the vector of indices of

primary coefficients NomCI _ . The resulting vector

coefficients *C is used to generate the approximation of the output waveform, *s , using the DMWR block.

III. MODEL DEVELOPMENT To be able to accurately generate primary coefficients, the ANN should previously be trained using a set of numerical data. This set of numerical data has to contain the relations between each parameter combinations and associated primary coefficients in the wavelet transform of the signal. To build the data set it is necessary to collect those data for a certain number of parameter combinations. The diagram illustrating this process is presented in Fig. 3.

We are starting with the nominal waveform, represented by its time samples Noms , that corresponds to a combination of nominal (reference) values of the parameters Nomp . The nominal waveform suffers a DMWD that generates the wavelet coefficients, NomC . We store NomC for later use in waveform approximation (see Fig. 2). Then, we select the “primary” coefficients by using the global threshold method, but we are interested not in the value, but rather in the

Level Coefficients

Type Number

5 Approximation cA5 Details cD5

KKKL ==− 05 22 KKKL ==− 05 22

4 Details cD4 KKKL 222 14 ==−

3 Details cD3 KKKL 422 23 ==−

2 Details cD2 KKKL 822 32 ==−

1 Details cD1 KKKL 1622 41 ==−

Fig. 2. Block diagram of the model for waveform approximation.

*p ANN

*sCOEFFICIENTS

FUSION *PC

NomC

NomCI _MDWR

*C

positions (indices) of these global coefficients, NomCI _ , that are stored to be used both in generating the training data set and later in waveform approximation (see Fig. 2).

To generate a reliable training dataset first of all it is necessary to generate a pool of parameter combinations. This happens in the block “Generate parameter combinations”, where for each parameter a range of variation around nominal values are considered.

Generating good parameter combinations, or, in other words, the design of experiment, is a truly tedious task. For the design of the experiment we need to generate those parameter combinations that fill the parameter space in order to encompass all the regions in the parameter space. According to [21] a good experimental design (ED) is essential to simultaneously reduce the possible effect of noise and bias error. It is recommended to construct an experimental design by combining multiple techniques for design experiment to reduce the risk of using a poor ED. In our approach we consider the Latin hypercube sampling (LHS) as the primary design experiment technique. But LHS designs may leave out the boundary and the final model may lead to large extrapolation errors. To avoid this we will generate our experimental design by mixing the LHS design with a 2 level full factorial design, as explained below:

1. Latin Hypercube sampling is used to generate a specified number of randomly distributed values for each parameter, these values being randomly permuted to obtain different parameter combinations.

2. Two-level full-factorial design is used to generate all possible parameter combinations, considering only the extreme values (minimum and maximum) for all parameters.

Details on this design of experiments can be found for example in [21], [22], [23].

The result of this process is the matrix of parameters P. The number of rows represents the number of input parameters that are varied (N), while the number of columns is given by the number of combinations of the input parameters (M).

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

NMNN

M

M

ppp

pppppp

P

21

22221

11211

(3)

For each parameter combination in matrix P the corresponding waveforms is obtained, S being the set of all waveforms, represented by their time sample. The set of wavelet coefficient SC results by applying a DMWD operation for every waveform in S. In accordance with the position of the primary coefficients given by the vector NomCI _ , the matrix of primary coefficients SPC for the entire set of waveform is created:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

QMQQ

M

M

s

pcpcpc

pcpcpcpcpcpc

PC

21

22221

11211

(4)

In the matrix SPC , each column represents the vector of primary coefficients for one waveform, M being the number

Nominal waveform ( Nomp + Noms )

Generate parameter combinations

Generate waveforms

DMWD

Extract primary coefficients

Store P

Store SPC

DMWD

Select primary coefficients and extract

their indices

Store NomC

Store NomCI _

NompNoms

P

S

SC

SPC

NomC

NomCI _

Fig. 3. Data set generation for ANN training.

of waveforms while Q represents the number of primary coefficients.

Now the necessary data set for building and training the ANN is ready to use. Matrix P contains the inputs data for the ANN, while matrix SPC contains the output data (targets). The neural network will have as many input as the number of parameters (N) and a number of outputs (neurons in the output layer) equal to the number of primary coefficients (Q).

IV. IMPLEMENTATION AND EXPERIMENTAL RESULTS To test and validate our proposed method, we considered a certain family of waveforms, generated in Matlab, for a number of three variable parameters. Our family of waveforms is not a random one, but it simulates the time response of an RC (resistor and capacitor) type circuit, when the circuit is driven by a rectangular pulse. We choose this kind of waveform because it is a very common waveform of interest in analog electronic circuits for a broad area of applications especially in command and control circuits. The parameters are:

- pulse amplitude with a variation of ±20% around the nominal value; - pulse delay with a variation of ±0.1% from the pulse duration; - time constant of the RC circuit (for signal rising), with a variation of 25% around nominal value. An illustration of the waveform family containing a number of only 30 waveforms is presented in Fig. 4. Each waveform is represented by 6210 time samples.

For the multilevel wavelet decomposition we used level 5 decomposition with db2 wavelet. This results in a number of 6220 coefficients, out of which 196 are for approximation, while the remaining ones are for details. Imposing a percent of 5% of coefficients to be selected as primary coefficients, for the nominal waveform the value of the global threshold was found to be 0.00022804 and 311

coefficients were selected as primary ones. This translates into a substantial data dimensionality reduction, by a factor of 20 (from 6210 to 311), which ensures the avoidance of “not enough memory” issues or very long training time.

In accordance with the data set generation presented in Section III, we generate a number of 300 parameter combinations, meaning that we have a set of 300 data pairs for training the ANN.

As the problem to be solved falls into the category of function approximation, a fitting artificial neural network is used with the following topology:

• 3 inputs - equals the number of input parameters; • 311 outputs – equals the number of output (primary

coefficients), the neurons in the output layer having a linear activation function;

• 1 hidden layer with 25 neurons having a sigmoid activation function; the number of hidden layers and the number of neurons was chosen via trial and error, in order to obtain a trade-off between network complexity and accuracy.

The structure of the ANN is presented in Fig. 5, where b is the bias vector in each layer and w is the connection weights matrix between layers. For the training procedure, the full data set (300 data pairs) is split into three data subsets: training subset (80% of the data set), validation subset (10% of the data set) and testing subset (10% of the data set). The training subset is used to train the neural network, adapting the neurons weights and biases. The validation subset supervises the training, detecting an overfitting phenomenon. Finally, the testing subset measures the performance of the neural network, inasmuch as that is not at all involved in the training process.

Fig. 6. Performance validation graph.

0 100 200 300 400 500 600 700 800 90010

-5

10-4

10-3

10-2

10-1

100

101

Best Validation Performance is 3.669e-05 at epoch 973

Mea

n S

qu

ared

Err

or

(mse

)

974 Epochs

Train

ValidationTest

Best

Fig. 5. ANN structure.

Fig. 4. Family of 30 waveforms sampled in 6210 points.

0 1000 2000 3000 4000 5000 6000

0

0.2

0.4

0.6

0.8

1

1.2

Family of 30 waveforms

sample index

Y

The performance validation graph is presented in Figure 6. This figure illustrates the evolution of the mean squared error within the three subsets, over the duration of the training. In the first 150 training epochs one can see a steep improvement (reduction) of the errors in all data subsets. Then, the training enters the phase of “fine tuning” continuously improving the performance. At the end of the training (973 epochs), the errors in all data subsets are very close to zero: 4.1810⋅10-5 for training, 3.669⋅10-5 for validation, 4.0111⋅10-5 for test, and 4.1128⋅10-5 for the entire data set.

The resulted trained neural network presents extremely good fitting performances in all data subsets. These can be seen by inspecting the results for linear regression of outputs of the neural network (predicted values) relative to the targets (as reference values) presented in Fig.7. A measure of the ‘goodness of fit’ is the regression value, R. The linear regression equation is:

Output = a⋅ Target + b (5)

where a is the slope of regression fit and b is the offset of regression fit. An ideal fit (network outputs match the targets exactly) means R=1, a=1 and b=0.

The regression value R is 1 in all data subsets, the slope of the regression fit a is 0.9999 in all subsets, while the offset b have very low values in all subsets (1.56⋅10-4 for the training subset, 1.1231⋅10-4 for the validation subset and 2.2544⋅10-4 for the testing subset). All our values can be considered as being ideal for a perfect fit.

To increase the generality and confidence degree of our model, we generate a completely independent set of 100 new data points (parameter combinations) using the LHS technique once again. All these 100 brand new parameter

combinations are subdued to the modeling procedure to generate the corresponding waveform. The quality of the prediction is evaluated using the mean squared error (MSE) (the difference between the estimator – generated waveform and what is estimated – reference waveform) for each waveform:

MSE= ∑=

−K

k

rk

gk ss

K 1

2||1 (6)

where K is the number of time samples, gks is the kth

generated time sample, and rks is the kth reference time

sample.

The resulted MSEs are plotted in Fig. 8. All these values are very small, lying in the range of [8.466·10-7; 4.878·10-6]. The maximum MSE values (4.878·10-6) occur for waveform number 7.

For this waveform (number 7), Fig. 9 presents the

primary coefficients generated by the neural network in

Fig. 9. Predicted and reference primary coefficients for waveform 7 (the waveform that generates maximum MSE).

0 50 100 150 200 250 300-1

0

1

2

3

4

5

6

7

index of primary coefficients

prim

ary

coef

ficie

nts

reference coefficientspredicted coefficients

Fig. 7. Results of linear regression analysis for fitting performance

0 2 4 6

0

1

2

3

4

5

6

Target

Ou

tpu

t ~=

1*T

arg

et +

0.0

0016

Training: R=1

Data

FitY = T

0 2 4 6

0

1

2

3

4

5

6

Target

Ou

tpu

t ~=

1*T

arg

et +

0.0

0011

Validation: R=1

Data

FitY = T

0 2 4 6

0

1

2

3

4

5

6

Target

Ou

tpu

t ~=

1*T

arg

et +

0.0

0023

Test: R=1

Data

FitY = T

0 2 4 6

0

1

2

3

4

5

6

Target

Ou

tpu

t ~=

1*T

arg

et +

0.0

0016

All: R=1

Data

FitY = T

Fig. 8. MSE for 100 predicted waveforms.

0 20 40 60 80 1000

1

2

3

4

5x 10

-6

X= 7Y= 4.8775e-06

waveform index

MS

E

comparison with the primary coefficients resulted after DMWD of the reference waveform. It is easy to see that the predicted coefficients are almost identical with the reference ones, with only few small differences. The top 5 coefficients with the maximum difference are presented in Table II.

TABLE II. COEFFICIENTS COMPARISON

The reference waveform and the predicted waveform for the analyzed case are presented in Fig. 10. Using the nominal representation scale, the two waveforms can barely be distinguished, so our method of waveform approximation proves to be highly reliable. Anyway, for a better understanding of the source of errors that leads to the highest value of MSE (from that 100 waveforms), Fig. 11 presents three “dilated” regions for reference and predicted waveforms. Fig. 11. a) and b) refer to regions with maximum nonlinearities of the waveform: samples 1470 - 1530, where the signal changes from an almost constant value (around 0) to an almost +∞ slope variation, respectively samples 3965 – 4020, where the signal changes from an almost constant value of 1.192 to an almost -∞ slope variation. For region a), the maximum errors are encountered for samples 1490 (0.082 predicted vs. 0.1327 reference) and 1560 (0.3857 predicted vs. 0.3333 reference). For region b), the maximum errors happens for samples 3986 (1.177 predicted vs. 1.163 reference) and 3981 (1.181 predicted vs. 1.192 reference). In Fig 11.c) one can see the differences in the region from sample 2200 to sample 3000, where the signal should stay constant to a value of 1.192. Despite this “constant value” region, we encounter some errors, but they

are very small: for example, some of the highest are 1.193 predicted vs. 1.192 reference or 1.191 predicted vs. 1.192 reference.

The results obtained here are compared, from the accuracy point of view, with the results presented in [24] where some gravitational waveforms are predicted using a surrogate model. The error used in [24] is e:

∑=

−=L

iisi ththte

1

2|);();(|Δ λλ (7)

where );( λith is the ith time sample of the reference (fiducial) waveform, );( λis th is the ith time sample of the predicted waveform by the surrogate model, λ represents the waveform parametrization, L is the number of time samples, and

)1/()(Δ minmax −−= Lttt .

It is easily to see that the two errors (MSE) in this paper and e in [24] are similar, so comparing them makes sense. While in [24] the error e, for different waveforms predictions, is in the range of [8⋅10-9; 10-5], our MSE is in the range [8.466·10-7; 4.878·10-6]. The accuracy of our model is similar or even superior to the one obtained in [24], confirming the quality of our solution.

V. CONCLUSIONS This paper describes a model based on neural networks and wavelet decomposition and reconstruction to approximate waveforms with respect to different waveform parameter combinations. Our model features high accuracy and generalization capabilities, as well as high speed. For a

Index Reference value

Predicted value

Difference (predicted-reference)

144 -0.3030 -0.4009 -0.0980 188 0.0097 -0.0620 -0.0717 187 -0.1280 -0.0609 +0.0671 145 0.0804 0.1360 +0.0556 189 0.0156 0.0402 +0.0245

Fig. 11. Comparison between reference and predicted waveform for three distinct regions: a) samples 1470 – 1530; b) samples 3965 – 4020; c) samples 2200 – 3000.

1470 1480 1490 1500 1510 1520 1530-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

sample index

Y

Reference

Predicted

3970 3980 3990 4000 4010 4020

1

1.05

1.1

1.15

1.2

sample index

Y

Reference

Predicted

2200 2300 2400 2500 2600 2700 2800 2900 30001.191

1.1915

1.192

1.1925

1.193

1.1935

1.194

sample indexY

Reference

Predicted

a) a) b)

c)

0 1000 2000 3000 4000 5000 6000

0

0.2

0.4

0.6

0.8

1

1.2

sample index

Y

Reference

Predicted

Fig. 10. Reference and predicted waveform (that generates maximum MSE).

set of 100 new arbitrary generated parameter combinations, completely independent of the parameter combinations used to develop the model, the MSE was found to lie in the range of [8.466·10-7; 4.878·10-6]. Also, the time necessary to generate all these 100 waveforms was only 8.3s on an common computer (Duo CPU @ 3 GHz, and 2 GB RAM). Hence, this model can be used for extensive analyses of a complex system response under all allowed operating condition to identify for example the worst case of system operation for different criteria.

ACKNOWLEDGMENT This paper was supported by the Post-Doctoral

Programme POSDRU/159/1.5/S/137516, project co-funded from European Social Fund through the Human Resources Sectorial Operational Program 2007-2013.

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