Received: July 05, 2011 / Accepted: August 02, 2011 / Published: February 15, 2012.
　　Abstract: The standard methodology of derivation VVER engineering factors is based on representation of analyzed power peaking in the linear form of random factors and on presumption about their normal probability density function (PDF). The goal of this paper is to present the derivation of locally defined VVER power peaking engineering factors which involve new parameter: a mean. Engineering factors definition from endpoints of uncertainty tolerance interval is recommended. An approach (95%/95%) for the normal PDF is discussed in detail, the relation to present standard uncertainty methodology of power distribution is found and problem of optimality in tolerance factor finding including limitation of sample size is discussed. On the bases of statistically based uncertainty kinf analysis for linear model has been shown that multivariate outputs vector of power peaking has nearly normal PDF independently on the PDF character of multivariate input vector of relatively small dimension (lower than number of fuel assemblies (FAs) in 1/6 core symmetry). Finally, the development of the methodological part of the engineering factors for VVER-1000 design macrocode MOBY-DICK is described and their calculation direct on the bases of experimental data (neutron detectors measurements) of Temelin NPP and Volgodonsk NPP has been performed.
　　Key words: VVER core, power distribution, probability density function, tolerance interval and tolerance factor, engineering factors.
　　1. Introduction
　　The problem of the power distribution uncertainties is in general a multidimensional problem of random vector and problem of multidimensional probability density function (PDF). The paper presents development of engineering factors methodology applied to core power distribution and discusses some problems of used procedure.
　　In Section 2, it is shown that derivation of engineering factors is based on representation of analyzed power peaking in the linear form of random factors and on presumption about their normal PDF. It is possible to calculate power distribution engineering factors for each core place (locally) when their definition is based on statistics of relative deviations. In such case it is possible define engineering factors for partial volume of core or directly for power peaking.
　　Engineering factors definition from so-called endpoints of uncertainty tolerance interval is recommended in Section 3. Approach (95%/95%) is explained and discussed in detail and is pointed out that the term “probability content” is key term (point) in the definition of tolerance interval. It is possible to determine engineering factors from so-called endpoints of uncertainty tolerance interval of relative deviations which is defined by tabulated tolerance factors and standard deviations. It is shown that tolerance factors represent also uncertainty which is formally uncertainty of sample mean and uncertainty of sample standard deviations. In Appendix the derivation of one sided tolerance factor to explain mathematical background of the “probability content” and its uncertainty like uncertainty of distribution function is presented. The problem of optimality in tolerance factor finding including limitation size of the sample can give answer to the question how to use tolerance factors for small number observations.
　　Final formulation of engineering factors (Section 4) involves new parameter-mean (bias) and a new conservative criterion (safety margin) according which only the positive mean is used into account. This mean simplifies the problem with normalization of power distribution for partial part of core or in the case of developing engineering factors directly from self powered neutron detector (SPD) currents.
　　The statistically based uncertainty analysis via kinf perturbations was realized in linear model with conclusion that multivariate outputs vector of power peaking has nearly normal PDF independently on the PDF character of input multivariate vector from under small number of its dimension (10 in Section 5). This fact helps overcome problems with some mechanical tolerances which character of PDF is not correctly defined and also opens question about the application of normal PDF approaches (e.g., tolerance factors(95%/95%)) in statistically based uncertainty analysis of power peaking.
　　Final Section 6 is dealing with application of presented local engineering factors methodology for calculation of methodical factors of Temelin NPP and Volgodonsk NPP core loaded with TVSA fuel. The calculation was performed direct on the bases of SPD predicted/measured currents ratios with subtracting the variability of SPD in convolution.
　　2. Basic Formula of the Engineering Factors in Reactor Practice
　　In present VVER reactor practice [1] the engineering factors for power distribution uncertainties are defined as one factor for both methodological and mechanical deviations in common.
　　Let to correct the power distribution in place
　　Δ is deviation(error) Δ caused by the mechanical uncertainties, Q is integral power of reactor, ΔQ is error in determination of reactor power.
　　Since the base for engineering factor determination can be expressed
　　(2) where for the next processing on the statistical level are important relative deviations caused by the methodological and mechanical uncertainties separately. Shortly say, we suppose sepearability between methodological and mechanical dependences. In the next for simplification in recording the r dependence was formally omitted.
　　The safety analysis uses detailed power distribution represented by the peaking factors from which more important is the relative linear power of fuel rod QK
　　In presented methodology it is supposed that the separability of the power distribution (and also peaking factors) has the following form
　　(4) where Kk is relative linear power of fuel rod inside FA, Kz is relative axial power distribution, Kμ is relative integral power of fuel rod inside FA .
　　Thus for power distribution (3) and (4) we obtain
　　(5) where y = Q, ΔH and both relative deviations in Eq. (5), i.e., methodological and mechanical, can be represented in the following formwhere x = M, E.
　　The methodological errors ΔM are determined mainly from the comparisons calculated and monitored powers with consideration of the measurements errors(variability of measured detectors). The mechanical errors ΔE are determined from the presumption of linear dependence of powers on the mechanical uncertainties.
　　Until now we have not defined statistical procedure. We see that in this final stage the engineering factor is transformed into linear form (see Appendix), which can be statistically processed after some conditions. In practice the errors are calculated like deviations Δ of the normalized powers [1] or directly from relative powers [2].
　　Within statistical processing we should define sample for which will be calculated statistical parameters (mean and standard deviations). In the case of the core we can define sample of partial core volume or time of operation. Obviously power distribution engineering factors are performed for whole core or for positions linear power limit (dependent on the fuel rod burn up) checking should be used more conservative engineering factors reflecting higher relative power deviations of FAs positioned on the edge of the core.
　　Abovementioned limiting of the engineering factors to the part of the core is usually applied only for its methodological component. Application of the same procedure for mechanical component is in principle also possible, but not up to now standard (see below).
　　3. Estimation of Uncertainty Interval
　　3.1 Definition of the Tolerance Interval
　　If random variable x has normal PDF with distribution function F(x), the task to find limit U and L of x such that probability p of F(x) at least C is inside interval (L,U) with confidence of (1-α) leads to solving equation [3]
　　(8)
　　The value C is usually called “probability content”(Fig. 1). Sometimes we refer to (L, U) as a tolerance interval.
　　The first application of methodology of tolerance intervals was applied by Wilks [4] in 1941 for determination of minimum size of random variables with non-parameter distribution function. This work initiated other works, namely for normal PDF, and in each of this works the key role played so-called“probability content”, which represents an integral of probability within some interval (tolerance interval). 3.2 Introducing a Tolerance Factor Methodology into Normal PDF Representation
　　The modification of the limits L and U in Eq. (8) for normal statistics N(μ, σ2) was presented by Wald-Wolfowitz in 1946. In practice is tolerance interval expressed (for symmetrical PDF and not known variance and mean) for (1 – C) = α = 0.05 as
　　(
　　are called endpoints of tolerance
　　interval (9).
　　3.3 Tolerance Factors and Standard VVER Engineering Factor Methodology
　　Tolerance factor T has different definitions for one-sided and both-sided interval, known and unknown parameters and also values for some other options e.g., minimizing of tolerance interval, maximizing of content etc.. In Table 1, three types(Cases) of the one sided tolerance interval for normal PDF are presented. As an example and for understanding what is “probability content” and its confidence, the derivation of tolerance factor of Case 3 is presented in Appendix. The three definitions in Table 1 differ by yes/not knowing parameters (μ, σ). Present VVER standard methodology derivation of power peaking factors [1] is based on absolute power deviations (difference calculated-measured power) with normalized integral power to the same value and calculated standard deviation for a whole core volume. This corresponds to the Case 2 in Table 1,
　　μ, but all calculations in Ref. [1] supposes interval according Case 1, i.e., both parameters μ andσ are known and μc represents quantile of normal PDF. Calculations in section 6 suppose that both parameters(μ, σ) are unknown and tolerance parameters are calculated like in Case 3. In Fig. 2 are depicted all three Cases and compared with t and uc quantile dependent on the probability content C and number of observations (size) n (only the visible points are table values). Further Fig. 2 indicates that for greater n Case 2 and Case 1 (present praxis) are close curves. But for small number of observations differences between Case 1 and other Cases is not negligible.
　　From Table 1, it is seen that present VVER practice(Case 1) represents probability content C estimation without evaluation errors caused limited value of n, and Cases 2 and 3 are the possible estimation of these error. 3.4 Are Tolerance Factors Unambiguously Defined?
　　Eq. (8) of tolerance interval is exact. Only practical application of evaluation tolerance factors (simplified formulae) has several authors (formulas) some of which are depicted in Table 2. But, as you see, differences between authors are not important.
　　The interesting question is: How is (95%/95%) statistics in safety documentation realized? For example in statistically based safety analysis [5] is used in full scale (together with Wilks [4]), but in subcriticality assessment [6] and errors in isotopic concentrations prediction (assays) the application is modified due to the problems with regression and small number of experiments.
　　3.5 Does Exist Optimal (Minimal) Allowed Value n for(95%/95%) Tolerance Interval in Normal Statistics N(μ,σ)?
　　For lower values of n than 10 width of tolerance interval according Case 3 (and also Case 2) will be grater than tolerance interval from Case 1. This problem touches optimality of number of information, i.e., definition of n0 as a minimum size for which tolerance factor give credible information about e.g.,(95%/95%) statistics. It is interesting that to find such n0 for normal PDF is in some sense more complicated than for non-parametric statistic [4]. We have to define additional criteria (stems from the variability of the probability content of tolerance interval), some of which have no practical application, because depend on parameters (μ, σ). Nevertheless the simplest additional criteria will give the value of n0 for one-sided and two-sided approximation lower than for Wilks. Thus one-sided tolerance factor has value
　　. The one-sided tolerance factor is more sensitive to n than two-sided one and for some criteria the minimal allowed value n0 for one-sided could be greater than for two-sided.
　　Optimality of the tolerance factors is closely bounded with the robustness of tolerance factors which is broadly studied in the last years.
　　4. Final Formulation of the Engineering Factors
　　From philosophy of the interval estimate presented in Section 3 the uncertainty factor of power distribution can be defined in the form
　　(11b) where NMx,=.
　　Introducing these representations in Eqs. (2), (5), (6),(7) and (10) we obtain the engineering factor for fuel rod linear power
　　In Eqs. (12) and (13), T define endpoints of uncertainty interval according chapter 3. From the character of linear form (see Appendix) the factor T can be defined individually for each random variable.
　　Foregoing engineering factors are evaluated with correction on the mean which represents systematic deviation (bias). This correction is important namely in the case that engineering factor is determined for part of the core or directly from SPD. In analogy with the subcriticality safety calculation philosophy we can also apply so called safety margin for application mean values (bias) in engineering factors (12) and (13) in the following sense: 0
　　μ= for 0≥μ
　　In general the reality 0≠μ stems from the fact that engineering factors methodology presented in this paper are calculated from relative deviations of local powers which is not reflected in the standard core integral power normalization.
　　But in general it can be theoretically demonstrated that the population which have 0≠μand standard deviation σ, for 0≠μ will have standard distributionσ~ and
　　σ
　　5. Statistically Based Uncertainty Analysis Applied to Kq
　　The statistically based uncertainty analysis is a good tool for analyzing non-parametric statistics problems. In last year, it was applied in safety analysis e.g., “Best Estimate Methods Uncertainty and Sensitivity Evaluations” [8] - realization best estimate calculations licensed in statistics (95%/95%). Also for detail isotopic uncertainty assessment was used this method with consideration of non-linearity effects [5].
　　In next the statistically based uncertainty methodology supposes that uncertainties of cross-sections can be represented by uncertainties of kinf and linear dependences of qKΔon infkΔ, i.e., the first order perturbation is valid . Then for one sixth of core, which contains 59 FAs, a statistically based uncertainty analysis can be realized for the vector equationΔis vector which elements are infkperturbation, one for each FA, and M)
　　is sensitivity (59 × 59) matrix. The elements of sensitivity matrix were calculated from direct calculations by macrocode MOBY-DICK for 5 years cycle of VVER440 (NPP Dukovany).
　　In our analysis it is supposed that matrix M) transforms input multivariate vector
　　Δ: uniform (UNI), triangular (LIN, line linearly increasing from left to right), normal (NOR) and voluntary (VOL-like crank shaft) (Fig. 3). The presented analysis is in some sense important because in general we are not sure about the input probability distribution, namely in the case of mechanical deviations.
　　The number of variants of vector
　　Δwas relatively great (850,000), to get correct statistical results according Eq. (17). In normal statistically based uncertainty analysis (e.g., GRS [5 or 8]) are demanded small counts and input PDF quantifying the uncertainty is generated usually by quantile function (if exists). The results (statistical parameters mean μ and σ ofΔ) of three typical FAs are presented in Table 3: No. 3 (close to the core centrum), No. 6 (inside core) and No. 9 (close to the edge of core). The first conclusion from Table 3 is that for the some PDF (in our case LIN) the mean μ can be significantly dependent on the position of FA in core and its value can be comparable with σ.
　　The calculated variants of Table 3 were for PDF comparison transformed into standard normal distribution by
　　N) 1 , 0 ( (17) and graphical presentation is in Fig. 4 (dimension of input vector is 59).
　　Important is dependence of outputs vectorΔ. Fig. 5 depicted this dependence for multivariate input PDF LIN (the number of perturbed FAs is written in legend and number of counts of each curve (input vector) was 850,000). The results in Fig. 5 show that the outputΔ PDF of FA No. 3 (for other FAs are results similar) is independent on the PDF character of input multivariate vector for small number of its dimension(e.g., more than 10 FAs).
　　Finally, we can conclude that very probably the PDF of power peaking will be close to normal PDF independent of character PDF of mechanical tolerances.
　　6. Development of the NPP Temelin Engineering Factors for Macrocode MOBY-DICK
　　The SKODA JS in the frame of calculations for SAR of NPP Temelin [9] formulates engineering factors of macrocode MOBY-DICK (MD1000). At present in principle (i.e., also for Dukovany NPP) all mechanical uncertainties in the present MD1000 engineering factor methodology are taken from fuel documentation of TVEL and only methodological factors are in agreement with standardized documentation of the MD1000.
　　Methodology of engineering factors for MD1000, i.e., for VVER1000, presented in this paper is also presentation of application of formulas from Section 4.
　　Suppose that Qik is the power in position i and at time k. The Sections 2 and 4 are recommended to calculate methodological factors from relative power deviations i.e.,
　　Qik = (Qeik – Qvik)/Qvik = Qeik /Qvik – 1 (18) where Qvik is calculated value and Qeik is measured value.
　　In this paper, for VVER1000 the engineering factors for maximum were derived for Kv > 1.2 and Kq > 1.0. Main reason of this higher conservatism, in comparison with Ref. [1], is that SPD do not cover the whole core volume. In Temelin NPP monitoring system BEACON the SPD variability is determined for SPD strings in each fuel assembly (KNI) and individual SPD and its value is for one SPD σvar = 0.0186 and KNI σvar = 0.0123.
　　Direct comparison of the measured (Ieik) and predicted (Ivik) SPD currents (predicted means calculated from MD1000 power distribution) is used for determination relative power deviations (18)
　　?Qik≈ Ieik/Ivik – 1 (19)
　　The variability of the SPD detector is included into standard deviation of coarse power distributions by formulaσis variability SPD (in next will be used values for KNIσvar = 0.012 and for SPD σvar = 0.019).
　　Some simplification, which should be taken into account in final comparisons, is in using the same values of SPD variability for Temelin NPP and Volgodonsk NPP.
　　In Eq. (19) it is supposed that technological parameters and SPD (Rh) burnup (on which is dependent quality of interpretation SPD formula) calculated and real are identical.
　　In the case of KNI statistics we are calculating only ratio of arithmetical summation of SPD currents measured to predicted 1
　　It can be supposed that the Eq. (21) better validate of integral FA power in the place of measurements than interpolated and integrated currents in axial direction.
　　Determination of the standard deviations of relative radial distributions of the fuel rods power inside fuel assemblies was performed from comparisons MD1000 versus Monte Carlo MCNP calculations on three models: MINICORE (central model with 7 FAs), MIDICORE (the core edge model with 10 FAs in 60°symmetrical sector) and 30DEG (the full core geometry in 30° symmetrical sector). In Ref. [9], it was shown that MD1000 versus MCNP comparison gives standard deviation of KKΔ lower than 0.7%. Suppose, we have normal distribution, the maximum error will be 2.1% (for 0.99% probability). The abovementioned models also showed that MD1000 underestimates fuel rod powers in the majority peaking positions and then according margin (14) we can conservatively suppose0= have shown that the peaking relative values kkKK /Δare not greater than 2% and final conservative conclusion agreed with recommend value from Ref. [1] of standard deviation of kkKK /Δno more than 2%. This value was used also for Kμ at present.
　　Table 4 is presented comparison of the standard deviations σ of peaking factors from SPD analysis of Temelin NPP and Volgodonsk NPP and Ref. [1]. From Table 4, it can be seen that the standard deviation σ of MD1000 is for KQ lower than σ of Ref. [1] and for Kq is on the comparable level. This better results of MD1000 concerns only presented comparisons in which the global conservative view is not included.
　　7. Conclusions
　　The goal of this paper was to present basic principles of the power distribution engineering factor methodology and exhibit same specific qualities which are not standardly discussed.
　　Derivation of engineering factors can be comprised into the following two steps: transforming dependence of limiting parameter into linear form of random variables and then application error processing on the level of normal PDF approaches.
　　This process is leading to calculation of statistical parameters (mean and standard deviations) of relative power deviations. It was shown that engineering factors can be defined locally and after choosing appropriate sample in variants for assessment of power peaking uncertainties.
　　The uncertainty interval estimate was included into engineering factor formulations, and so-called “safety margin” was suggested to apply for systematic deviations (bias).
　　An approach (95%/95%) was discussed in detail and was pointed out that term “probability content” is key term in the definition of tolerance interval. It was shown that at some conditions the tolerance factors represent uncertainty standardly used in present engineering factors methodology.
　　The statistically based uncertainty analysis applied to FΔH have shown (on the level of linearized model) that non-normal and non-parametric PDF of statistically independent random variables is on the level the output FΔH deviations transformed into PDF close to the normal PDF
　　The presented local engineering factors methodology was applied for calculation of methodical factors of Temelin NPP and Volgodonsk NPP core loaded with TVSA fuel. The calculation was performed direct on the bases of SPD predicted/measured currents ratios with subtracting the variability of SPD in convolution. Thus resulted differences in engineering factors MD1000 versus Ref. [1] are caused by the differences of used macrocodes, experience of organizations (number and quality of validated data sets) and last but not least applied methodology of engineering factor calculation.
　　References
　　[1] SAR, VVER1000-NPP Temelin for TVSA-T fuel,(internal report), 2009.
　　[2] J. ?varny, Contribution to the power distribution methodology uncertainties assessment, 18th Symposium of AER, Eger, Hungary, October, 2008.
　　[3] M. Jílek, The statistical tolernace intervals, Praque, 1988.
　　[4] S.S. Wilks, Determination of sample sizes for setting tolerance limits, Bull. of the American Mathematical Society 12 (1941) 91-96.
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　　[6] J.J. Lichtenwalter, S.M. Bowman, M.D. Dehart, Criticality Benchmark Guide for Light-Water, Reactor Fuel on Transportation and Storage Packages, Rep. NUREG/CR-6361 (ORNL/TM-13211), Washington, DC, 1977.
　　[7] J. Like?, J. Laga, Basic Statistical Tables, SNTL, Praque 1978.
　　[8] NEA/CSNI/R, 2007, 4.
　　[9] Validation of macrocode MOBY-DICK-1000, Ae 12938/Dok Rev.0, (internal SKODA report), unpublished.
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　　Appendix [7, 10]
　　Example of Derivation of One Sided Tolerance Interval for Case 3 in Table 1. Let have random choice x from normal distribution ),(2σμ
　　N
　　, where μ and σ are unknown parameters. We are selecting tolerance limits in the form
　　?∞ (A1) is one-sided uncertainty interval.
　　Tolerance factor T is finding from condition
　　(A3) Then the tolerance factor T is determined as with confidence of (1 - α) is valid (A1), i.e., )1 ()(α?=≥CzPand)
　　(A5) is a tolerance factor (which result was at first presented by Proschan in 1953).
　　The same procedure of derivation of tolerance factor can be realized also for other Cases of Table 1.
　　From Eqs. (A2) and (A3), it is seen that tolerance interval concerns only “probability content” and uncertainty of its determination. In other words into tolerance factor is included uncertainty of inequality (A1), or uncertainty of distribution function Φ which stems from the fact that parameters
　　σ represent only approximation of correct μ and σ values. So Eq. (8) is confidence statement about the unknown distribution function F.
　　Characteristics of the linear form
　　Suppose that random vector
　　and correlation ),(ξξicorris defined like
　　For mutually independent or non-correlated random variables iξthe simple formula for variability is valid