Uncertainty Analysis of Power System Components

based on Stochastic Response Surfaces C. Bastiaensen, W. Deprez, E. Haesen, J. Driesen, R. Belmans

AbstractThe output of a power system analysis mostly

requires extensive knowledge and correct handling of input uncertainties. Analytical approaches often depend on simplified models whereas Monte Carlo based sampling methods are often time consuming. This paper presents an uncertainty analysis based on a limited number of well chosen samples which are used to model a stochastic response surface, based on a polynomial of independent standard normal random variables. To illustrate the approach a torque estimation model for induction machines is studied. The model requires different machine parameters as input. The analysis consists of three steps: the characterization of the input uncertainty, the uncertainty propagation and the characterization of the output uncertainty. First a simple sensitivity testing method is performed after which different sampling based methods are compared. The specific advantages of the stochastic response surface method over the Monte Carlo and Latin Hypercube sampling method are shown.

Index Terms-- Sampling Methods, Stochastic Response Surface Method, Torque Estimation, Induction Machines

I. INTRODUCTION N literature on probabilistic power system analysis three approaches can often be distinguished, i.e.

Analytical Probabilistic Analysis uses a linearization around an equilibrium point. Assumptions like normality and independence of uncertainties relax calculations significantly. Strongly non-linear systems are not well suited for this approach. Also the assumption of independence is often erroneous; Monte Carlo based Probabilistic Analysis performs numerous evaluations of the system for input samples. This allows taking into account all possible probability density functions and correlation structures in a straightforward manner. Variance reduction methods to ensure a rapid deduction of

C. Bastiaensen, Department of Electrical Engineering, University of Leuven, Kasteelpark Aremberg, 10, 3001 Leuven, Belgium, Phone: +32 16 18 79, Fax: +39 16 32 19 85, E-mail: cindy.bastiaensen@esat.kuleuven.be

W. Deprez, Department of Electrical Engineering, University of Leuven, E-mail: wim.deprez@esat.kuleuven.be

E. Haesen, Department of Electrical Engineering, University of Leuven, E-mail: edwin.haesen@esat.kuleuven.be

J. Driesen, Department of Electrical Engineering, University of Leuven, E-mail: johan.driesen@esat.kuleuven.be

R. Belmans, Department of Electrical Engineering, University of Leuven, E-mail: ronnie.belmans@esat.kuleuven.be

performance characteristics moments is crucial; Interval Arithmetic and in extension Fuzzy Analysis handle levels of possibility of occurrence of system inputs. The use of interval arithmetics is still based on a single system assessment adapted to arithmetic operators for intervals instead of single values. The often used trapezoidal fuzzy variables represent two levels of possibility for which the same interval arithmetic rules can be applied. This paper describes the case of reliable torque estimations of induction machines. The stochastic nature of this analysis is illustrated from measurement, through processing, to output estimation. An efficient method for the response analysis is presented based on stochastic response surfaces. The input random variables are first expressed in terms of independent standard normal distributed random variables, either exact or as an approximation of a series expansion. Then the output variable is approximated through a series expansion of these standard random variables. The series expansion is known as Polynomial Chaos expansion and is composed by Hermite Polynomials. To obtain the unknown coefficients in the expansion the deterministic model is used with a limited number of well chosen samples. A schematic overview of the stochastic response surface method (SRSM) is shown in Figure 1. The schematic overview for the conventional sampling based methods is given for comparison in Figure 2. It is the authors opinion that a similar approach can be used in many other fields of probabilistic power system analysis.

Fig. 1. Schematic overview of the SRSM

I

Fig. 2. Schematic overview of conventional sampling method

The structure of the paper is as follows. In the second section the torque estimation model is briefly discussed and the necessary input parameters for the model are deduced. In the third section the probability distribution functions of the input parameters are deduced from measurement datasets. In the fourth section a simple sensitivity testing method is performed. As sample points the mean value and some error bound of the input parameters are taken. This gives qualitative and quantitative information regarding variation in the model output due to variations in the input. In the fifth section different sampling based methods are compared. In the Monte Carlo and Latin Hypercube sampling methods sample points are chosen from the input distributions. From the results of the successive model runs the distribution of the output is obtained. A third sampling based method is the stochastic response surface method. The output is expressed as a series expansion in normal random variables. The number of samples compared to the conventional sampling methods is reduced significantly, since the model runs are only dedicated to obtain the unknown coefficients in the expansion. The machine used in lab experiments is an off-the-shelf 7.5 kW premium efficiency two pole induction machine with a rated speed of 2925 rpm.

II. TORQUE ESTIMATION MODEL Reliable torque measurement equipment is expensive and often also calibration and maintenance are required. In some applications, due to a variety of reasons, the overhead of torque measurement devices is not desired or not optimal. Sometimes there is also no space to equip the application with such systems. In most cases it is sufficient to make an accurate estimation of the torque. Many expressions for the electromechanical torque of an induction machine can be found [1]. Based on basic formulae requiring electrical quantities and sometimes mechanical speed, the mechanical torque can be estimated. In this paper, it is illustrated that such torque estimation expressions could be used to replace mechanical torque measurement devices. There are several expressions that describe the electromechanical torque of an induction machine [1]. However, they all originate from the same general description of three-phase ac machines, namely the stator and rotor voltage equations with Rs and Rr representing the resistance of the stator and the resistance of the rotor windings (stator referred) respectively:

, , , , , ,ssa sb sc sa sb sc sa sb scd

Rdt

u i = + (1)

, , , , , ,rra rb rc ra rb rc ra rb rcd

Rdt

u i = + (2) with ira, irb and irc the phase currents in the rotor, isa, isb and isc the phase currents in the stator, ura, urb and urc the rotor phase

voltages, usa, usb and usc the stator phase voltages, ra, rb and rc the rotor flux and sa, sb and sc the stator flux. These equations can be simplified according to established transformations in electrical engineering [1], i.e. the transformation of rotor and stator variables to the -reference frame (Clarke transformation) [4] and the transformation to the dq-reference frame (Park transformation) [5]. Depending on the chosen reference frame, this can lead to different expressions for the torque. The expression for the torque, obtained according to a reference frame rotating with the rotor flux, becomes:

2

( ) h hel rq rd rd rq rd sq sqr r

L LT p i i p i p i i

L L = = = (3)

with p the number of pole pairs, i the magnetizing current, isq the stator current in the q-axis, Lh the magnetizing inductance and Lr the rotor leakage inductance. The developed torque estimation model is based on this expression. The inputs for the model are two currents isa and isb and the mechanical rotor speed. The angular velocity of the rotor flux and the magnetizing current are calculated from the rotor equations:

rsd

r

diLi i

R dt

+ = (4)

sqrr

r

iRL i

= + (5) with Rr the rotor resistance, r the rotor speed and the synchronous speed. As can be noticed from equations (3) to (5), the torque estimation model requires three important machine parameters: the magnetizing inductance Lh, the rotor resistance Rr and the rotor leakage inductance Lr from the single phase T-equivalent circuit representation of induction machines. The elements of this equivalent scheme can be obtained by basic, straightforward calculations based on a short circuit and a no-load test [2]-[3]. However, in order to obtain correct torque estimates, skin effect in the rotor bars [6]-[7] and, to a minor extent, temperature corrections should be taken into account. Such a correction is provided in the proposal for an IEC standard [3] for the determination of the quantities of equivalent circuit diagrams. To obtain the motor parameters necessary as input for the torque estimation model, the conditions prescribed by the norm are observed. From the different motor tests, performed according to the norm, the motor parameters can be determined. By repeating the motor tests several times, a dataset for each parameter is obtained. From these sets the input uncertainties are characterized.

III. CHARACTERIZATION OF INPUT UNCERTAINTIES The datasets obtained from the measurements according to the norm are analyzed in this section. First the parameters are handled separately. The appropriate distribution of each parameter is determined. Secondly, each set of a motor parameter is used as input for the model and the distribution of the set of outputs is determined. This is performed to verify

the results of the propagation methods discussed in the fifth section.

A. Model input: distributions The tests necessary to calculate the motor parameters according to the IEC norm are repeated 55 times. Although the IEC norm already implements corrections for analysis of rated power situations, measurement errors are still present. This results in a data set of 55 points for each parameter. From the dataset it is possible to estimate the probability distribution function. Since the normal distribution is the most obvious distribution, the data sets are first tested on normality. Therefore a graphical method as well as a numerical method is performed. The normal probability plot and the normal probability plot correlation coefficient respectively are appropriate for determining whether or not a data set comes from a population with approximately a normal distribution. The normal probability plot is a plot of the ordered data points Xi versus the normal order statistic medians Yi. Since the data are plotted against a theoretical normal distribution, the points should form an approximate straight line in case of normality. Substantial deviations from straightness are considered evidence against normality of the distribution. To measure the linearity of a normal probability plot the normal probability plot correlation coefficient is defined as the product moment correlation coefficient between the ordered data and the order statistic medians from a normal distribution.

2 2

( ) ( )( , )

( ) ( )

i i

i i

X X Y YCorr X Y

X X Y Y

=

(6)

The correlation coefficient of the normal probability plot can be compared to a table of critical values to provide a formal test of the hypothesis that the data come from a normal distribution. The test statistic is the correlation coefficient of the points that make up a normal probability plot. This test statistic is compared with the critical value in the table. The values in this table were determined from simulation studies by Filliben [12]. If the test statistic is less than the tabulated value, the null hypothesis that the data comes from a population with a normal distribution is rejected. The two methods described above are used to determine whether the data sets of the motor parameters are normal distributed. The normal probability plots are shown in Figure 3 for the motor parameters Lr, Rr and Lh. The following conclusions can be made from the plots. The points on the Lr plot show a strongly linear pattern. There are only minor deviations from the line fit to the points on the probability plot. This indicates that the normal distribution is a good model for this data set. The points on the Rr plot shows a reasonably linear pattern in the centre of the data. However, the tails, particularly the lower tail, show departures from the fitted line. It can not be concluded exclusively that the normal distribution is a good model for this data set. This should be verified by the correlation coefficient. The same phenomenon appears in the normal probability plot of the data points for Lh,

the first few and the last few points show a marked departure from the reference fitted line. The non-linearity of the normal probability plot also shows up in the S-like pattern in the center of the data set. In this case it can be reasonably concluded that the normal distribution can be improved upon as a model for these data.

Fig. 3. Normal probability plot for Lr, Rr and Lh

These findings are verified by the normal probability plot correlation coefficients. Table 1 lists the coefficients for the parameters. Since perfect normality implies perfect correlation (i.e. a correlation value of 1), one wants to reject normality for correlation values that are too low. Therefore, the correlation coefficients should be compared to the values in the table made up by Filliben [12]. At the 0.5% and 5% significance level, the critical values are respectively 0.962 and 0.978. Since all the coefficients for the parameters are greater than 0.962, the null hypothesis that the data sets come from a population with a normal distribution can not be rejected at the 0.5% significance level. However at the 5% significance level the correlation coefficient of the Lh dataset is lower than the critical value.The null hypothesis that the data comes from a normal population is rejected.

TABLE I CORRELATION COEFFICIENTS

PARAMETER CORRELATION COEFFICIENT

Lr 0.99246 Rr 0.98024 Lh 0.96473

The above methods give a decisive answer about the first two parameters. The normal distribution provides a good model for the data. For the last dataset the normal distribution can be improved upon as a model for these data. However, since the influence of this motor parameter is limited, the distribution is assumed normal in this paper. Figure 4 shows the cumulative probability of the dataset. All the data points are located in the 99% confidence bounds of the normal fit. The assumption of normality seems to be a plausible choice.

Fig. 4. Cumulative probability plot for Lh

B. Model output: distribution Now the distribution of the motor parameters is known, the distribution of the torque dataset is determined. The results will be used to verify results from the uncertainty propagation methods in the next section. Figure 5 shows the normal probability plot. The plot bears a close resemblance to the plot

of the motor parameter Rr. This is plausible since Rr has the largest influence on the torque. It can be concluded that the torque dataset is normal distributed. This is also verified by the correlation coefficient of 0.98462 of the line fit to the probability plot. Since the distribution of the datasets of the motor parameters and the torque are normal, the mean and variance are easy to determine. The values are listed in Table 2.

TABLE II MEAN AND VARIANCE OF NORMAL DISTRIBUTED PARAMETERS AND OUTPUT

MEAN STANDARD

DEVIATION Lr 0.00398 0.000084 Rr 0.32292 0.004183 Lh 0.14595 0.001210 Tel 20.9468 0.258589

Fig. 5. Normal probability plot for Tel

IV. SENSITIVITY TESTING To understand the behavior of the computer model and to determine the input parameters for which it is desirable to have more accurate values, a sensitivity analysis is important. The sensitivity of the torque calculation to inaccuracies or changes of the motor parameters is analyzed. First the influence of the parameters on the torque calculation is simulated by assuming the parameters are independent. From this analysis it can be concluded which motor parameter has the largest influence on the torque calculation. In reality the motor parameters are dependent and therefore the analysis is repeated for the dataset of dependent parameters obtained from the measurements. The principal component analysis is used for this purpose. This analysis shows again which parameter has the largest influence on the calculated torque, but also indicates the interdependency of the parameters and how the influence on the torque is changed by this interdependency. The influence of a deviation on the independent motor parameters is shown in Figure 6 by changing in turn the motor parameters. The deviation on the motor parameters is plotted

on the x-axis, while the corresponding torque is plotted on the y-axis. The order of the parameters according to the sensitivity of the calculated torque is: Rr > Lh > Lr. A deviation of 5% on the rotor resistance leads to a deviation of 4.25% on the electromechanical torque, while a deviation of 5% of the rotor leakage inductance or the magnetizing inductance causes smaller torque deviations. The motor parameter Rr influences motor performance the most and therefore an accurate determination of this parameter is important. This supports the importance of the accurate consideration of the rotor bar height and material to take into account the skin effect, as discussed in the previous section.

Fig. 6. Influence of the independent motor parameters

Up till now the motor parameters are examined separately. As the parameters are not independent, the parameters show some correlation. To visualize the dependency of the parameters the principal component analysis [16] is performed. PCA is a technique that reduces multidimensional datasets to lower dimensions. The simplification of the dataset is performed by an orthogonal linear transformation which transforms the data to a new coordinate system. The first principal component indicates the largest variance of the original data; the second component corresponds with the largest variance in the subspace orthogonal to the first component, etc Results indicate that third and following components are negligible. Therefore 2D plots can be made which simplifies the analysis of the data. PCA starts with the set [Lr, Lh, Rr, Tel]. Figure 7 shows the data mapped into the new coordinate system, the interdependency of the motor parameters and the influence on the calculated torque. One can conclude that Rr has the largest influence, while the influence of Lr is almost negligible and opposite. The influence of Lh is small and in the same direction as Rr. These results correspond to previous results of the independent influence of the parameters. However from Figure 7 it is clear that the parameters are dependent. So it is not realistic to change the parameters in turn. The interdependency of the motor parameters can not be neglected when analyzing the influence on the torque. This interdependency should also be taken into account in the sampling methods described in the next session. This can be done by determining the covariance matrix of the datasets.

Fig. 7. Principal component analysis

In the sensitivity testing method the model response is studied for selected model input parameter combinations. The input parameters are not handled as random variables. As sample points only the mean value and some error bound of the input parameters are taken. This method does not provide adequate information about the nature of the output uncertainty. Especially as the probability density functions of the input parameters are known, this method ignores the available information.

V. PROPAGATION OF UNCERTAINTIES In this section different sampling based methods are compared. In all methods the model runs for a set of sampling points. The model results at these sampling points are used to establish a relationship between the inputs and the output. The sampling based methods discussed are the Monte Carlo and Latin Hypercube sampling method and the stochastic response surface method.

A. Monte Carlo and Latin Hypercube sampling In the Monte Carlo and Latin Hypercube sampling method the input parameters are handled as random variables [8]-[10]. The sample points are chosen from the input distributions. From the results of the successive model runs the distribution of the output is obtained. From [8] it is clear that the number of samples, the sampling method and the choice of random generator are important for the Monte Carlo method. According to [9] information derived from the output of a Monte Carlo sampling should not be presented as constant, but as random variable that depend on the sampling method, the number of samples taken and the random number generator. As in [8] and [9], the influence of these effects is quantitatively examined by considering two different sampling methods namely random sampling and Latin hypercube sampling, and two different numbers of samples, 55 and 200. The tests are performed 5 times to assess the impact of different sets produced by the random generator. The sampling methods performed are random sampling and Latin Hypercube sampling. Simple random sampling involves repeatedly forming random vectors of input parameters from prescribed probability distributions, namely random vectors from the multivariate normal distribution. Random samples

are taken without taking into account previously generated sample points. It is not necessary to determine beforehand the number of sample points. In Latin Hypercube Sampling it is necessary to decide how many sample points one wants to take. Suppose the number of sample points is n, the range of each variable is divided in n non overlapping intervals of equal probability. One value from each interval is selected at random. The n values obtained for the first variable are combined in a random manner with the n values obtained for the second variable. These n pairs are combined in a random manner with the n values of the third variable, to form the input vectors of the model. This way of sampling ensures a full coverage of the range of each variable by maximally stratifying each marginal distribution. However, whenever the variables are dependent, combining variables can not be done completely in a random manner. The pairing is restricted due to the correlation among the variables. Iman and Conover [11] proposed a method by restricting the way the variables are paired based on the rank correlation of some target values. Here the method proposed by Stein is used. The method is based on the rank of a target multivariate distribution. For more details see [12]. Figure 8 shows the results for the two sampling methods, two different sample sizes and for 5 different sets produced by the random generator. For each sample the mean and standard deviation is displayed. From the random sampling plots it is clear that the predicted values of the mean from the 55 samples vary much more widely over the 5 sets as compared to the 200 sample results. It is also immediately clear from the figure that for the same number of samples the random sampling results vary much more widely than the Latin hypercube results do. Even for a small sample size the means of the sample results approach the earlier determined torque mean.

B. Stochastic response surface method The stochastic response surface method (SRSM) can be seen as an extension of the classical response surface method (RSM). Therefore the response surface method is briefly discussed first. The response surface method gives insight into the relationship between multiple input parameters and an output variable. This is realized by considering the output variable as a polynomial function of the input parameters. The approximated model is called the response surface model. The RSM is described extensively in [13] and consists in essence of three steps. The first step is the determination of the different input parameters of the model. If the set of input parameters is large, the most important parameters need to be selected. This selection can be done for example by a principal component method as in the previous section. The second step is making several runs of the computer model to estimate the values of the unknown coefficients in the expansion. The input sets are chosen according to a specific response surface design, for example the central composite design, see Figure 9. In the last step a polynomial response surface is fitted for the output variable as a function of the input parameters. The least square method is used for this purpose.

Fig. 8. Mean and Standard deviation for Random Sampling and Latin

Hypercube Sampling at 55 and 200 samples

For the torque estimation model discussed in the previous section the second order model is:

0 1 1 2 2 3 3 12 1 2 13 12 2 2

23 2 3 11 1 22 2 33 3

elT a a x a x a x a x x a x xa x x a x a x a x

= + + + + ++ + + +

3 (7)

The input parameters are referred to respectively as x1, x2 and x3. The various response surface methods proposed in the literature differ only in the terms retained in the polynomial expression, for example with or without cross terms, and the selection of the input parameter sets for the determination of the unknown coefficients. The fitting points have to be chosen in a consistent way in order to get independent equations. Once the coefficients are known, the expression is an approximation for the computer model and can be used to characterize the output parameter. However, this method can not be used to analyze the uncertainty of the output. For this purpose the method should be adapted.

Fig. 9. Central composite design

The stochastic response surface method can be seen as an extension of the classical response surface method, since the inputs are handled as random variables. The method is extensively described in [14]. The different steps in the method are discussed briefly and applied to the torque estimation model. As a first step in the SRSM the input random variables are expressed in terms of independent standard normal distributed random variables, either exact or as an approximation of a series expansion. Then the output variable is approximated through a series expansion of these standard random variables. The series expansion is known as Polynomial Chaos expansion and is composed by Hermite Polynomials [15]:

1

1 1 1 2 1 2

1 1 2

1 2

1 2 3 1 2 3

1 2 3

0 1 21 1 1

31 1 1

( ) ( , )

( , , ) ...

in n

i i i i i ii i i

i in

i i i i i ii i i

y a a T a T

a T

= = =

= = =

= + +

+ +

(8)

Tp(i1, i2, , ip) represents the Hermite polynomial of degree p, y represents the model output and the coefficients ai are constants to be determined. To determine the unknown coefficients in the expansion, the collocation method proposed in [14] is used. This method selects for each term in the series expansion a corresponding

collocation point. For each term the values of the involving variables are set to a root of the higher order Hermite Polynomial and the non involving variables are set to zero. The number of sample points obtained exceeds the number of unknowns. The system of equations is solved using the singular value decomposition. The number of model runs is limited to the number of sample points. The SRSM is now applied to the torque estimation model discussed in section 2. First the three input parameters are transformed to standard random variables i (i= 1, 2, 3). Second the torque is approximated by a second order Polynomial Chaos expansion. The number of unknown coefficients to be determined is 10. After the determination the torque is completely expressed as function of the input variables:

0 1 1 2 2 3 3

2 2 24 1 5 2 6 3

7 1 2 8 1 3 9 2 3

( 1) ( 1) ( 1elT a a a aa a aa a a

= + + +)+ + +

+ + + (9)

The values of the coefficients ai (i=1,,10) are shown in Table 3. The statistical properties of the torque can be easily deduced from this expression. By taking a large number of samples of the standard random variables, the probability distribution can be calculated using standard methods. The first and second moment of the output variable are calculated through:

,1

1el

N

Ti

TN

=

= el i (10) 2

,1

1 (1el el

N

T el ii

TN

=

= 2)T (11)

TABLE III COEFFICIENTS OF THE EXPANSION

COEFFICIENT VALUE

a0 20.9468 a1 0.22773 a2 0.10912 a3 -0.05663 a4 -0.00047 a5 0.00038 a6 -0.00007 a7 -0.00011 a8 0.00001 a9 0.00004

In order to test the accuracy of the functional approximation, the calculations are repeated for a higher order of the Polynomial Chaos expansion. The resulting probability distribution function is compared to the previous one to check the convergence. The results are also compared with the Monte Carlo simulations performed earlier to ensure the approximation converges to the real distribution. Figure 10

shows the similarity of the second and third order Polynomial Chaos expansion and the Monte Carlo sampling method for 10000 samples. The time necessary to attain these results (Pentium IV, 3.20 GHz) varies from 12 seconds for the second order Polynomial Chaos expansion, 45 seconds for the third order Polynomial Chaos expansion to 2173 seconds for the Monte Carlo simulations. For the Monte Carlo simulation it is necessary to run the model for each sample, while for the SRSM it is only necessary to run the model to determine the unknown coefficients in the approximation. Once the functional approximation is known, the model is replaced by this formula. The number of simulations for the SRSM compared to the number of simulations of the Monte Carlo simulation proves the efficiency of the method. This can be concluded only in case the number of random variables is limited and in case the order of the Polynomial Chaos expansion can be kept low.

Fig. 10. Comparison of second and third order Polynomial Chaos expansion

and Monte Carlo simulations

VI. CONCLUSIONS In this paper an uncertainty analysis of a torque estimation model is performed. After the characterization of the input parameters, a simple sensitivity test is performed to achieve a better understanding of the behavior of the model. To provide adequate information about the nature of the output uncertainty three sampling based methods are compared. From Monte Carlo and Latin Hypercube sampling method the distribution of the output can be deduced, however these methods are computationally expensive. It is proven that the stochastic response surface method is more efficient to analyze the torque estimation model. It is the authors strong opinion other branches in probabilistic power system analysis can benefit as well from the presented approach.

VII. REFERENCES [1] G. Terrde, Electrical Drives and Control Techniques,

Leuven/Belgium: Acco 2004 [2] IEEE Standard Test Procedure for Polyphase Induction Motors and

Generators, IEEE Power Engineering Society 1996, IEEE Std 112-1996, New York, NY/USA

[3] IEC Rotating Electrical Machines - Part 28: Test methods for determining quantities of equivalent circuit diagrams for three-phase low-voltage cage induction motors, 60034-28, Ed.1 (to be published)

[4] P. Vas, Electrical Machines and Drives: a space vector theory approach, New York/USA: Oxford University Press, 1992

[5] R. H. Park, Two-reaction Theory of Synchronous Machines, Trans AIEE, 1929

[6] J. Langheim, Modeling of Rotorbars with Skin Effect for Dynamic Simulation of Induction Machines, Industry Applications Society Annual Meeting, Conference Record 1-5, Oct. 1989, pp. 38 44, Vol.1

[7] O. I. Okoro, Steady State Analysis of Squirrel-Cage Induction Machine with Skin-Effect, The Pacific Journal of Science and Technology 5 , 2004 No. 2

[8] M.D.Mckay, R.J.Beckman, W.J.Conover, A Comparison of Three Methods for Selecting Values in the Analysis of Output from a Computer Code, Technometrics, Vol. 21, No.2. (May 1979), pp. 239-245

[9] Vicente J. Romero, Effect on Initial Seed and Number of Samples on Simple-Random and Latin-Hypercube Monte Carlo Probabilities (confidence Interval Considerations), Specialty Conference on Probabilistic Mechanics and Structural Reliability, 2000

[10] M. Stein, Large Sample Properties of Simulations Using Latin Hypercube Sampling, Technometrics, V. 29, No.2, p.143, 1987

[11] R.L. Iman, W.J. Conover, A Distribution Free Approach to Inducing Rank Correlations Among Input Variables, Communications in Statistics, Part B, Simulation and Computation, 11, pp. 311-334

[12] J.J. Filliben, The probability Plot Correlation Coefficient Test for Normality, Technometrics, Vol. 17, No. 1 Feb. 1975, pp. 111-117

[13] R.H. Myers, D.C. Montgomery, Response Surface Methodology, John Wiley & Sons, INC., 1995

[14] S.S. Isukapalli, A. Roy, P.G. Georgopoulos, Stochastic Response Surface Methods (SRSMs) for Uncertainty Propagation: Application to Environmental and Biological Systems, Risk Analysis, Vol. 18, No. 3, 1998

[15] R.G. Ghanem, P.D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991

[16] D. N. Lawley, A. E. Maxwell, Factor Analysis as a statistical method, London Butterworths, 1971