Annals of Nuclear Energy 53 (2013) 447–457

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Annals of Nuclear Energy

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On the estimation of the unknown reactivity coefficients in a CANDU reactor

Lobat Tayebi a, Daryoosh Vashaee b,⇑a Helmerich Advanced Technology Research Center, School of Materials Science and Engineering, Oklahoma State University, OK 74106, USAb Helmerich Advanced Technology Research Center, School of Electrical and Computer Engineering, Oklahoma State University, OK 74106, USA

a r t i c l e i n f o

Article history:Received 25 July 2011Received in revised form 6 July 2012Accepted 6 July 2012Available online 1 December 2012

Keywords:CANDU reactorParameter estimationSpace-time kineticsCoolant void

0306-4549/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.anucene.2012.07.025

⇑ Corresponding author. Tel.: +1 918 594 8017; faxE-mail address: daryoosh.vashaee@okstate.edu (D

a b s t r a c t

A space-time kinetics based inverse architecture method is suggested to analyze the reactivity varia-tions associated with power excursions in a generic CANDU reactor. It is intended to provide diagnosistools to gain enhanced control thereby ensuring safe operation of the plant. A methodology for analyz-ing the data available from the in core flux detectors and extracting the unknown reactivity coefficientsis presented. The proposed system uses a reference model in conjunction with an optimal estimator.The reference model is composed of a state space representation of the space-time dynamics of neu-tron flux in the core, based on modal expansion approximation, and a time domain optimal estimatorfilter. We investigated three different estimation techniques based on recursive prediction errormethod (RPEM), dual extended Kalman filter (DEKF), and joint extended Kalman filter (JEKF). Wecompared their applicability to the estimation of coolant-void dynamic reactivity in loss-of-coolantaccident in a CANDU reactor. The state equations also include the characteristics of the detectorresponses. The thermal hydraulic models were not included in the calculations. Two different typesof detectors are considered in this analysis, the over prompt responsive Platinum detector of thereactor shutdown systems, and the under delayed responsive Vanadium detector of the flux mappingsystem.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

While the concept of parameter estimation based on Kalman fil-tering is not new, application in the area of reactor dynamics re-mains little explored. The concept is especially under-exploited inlarge core reactor applications, like CANDU reactors, where consid-eration of spatial variation of flux is essential in analyzing the reac-tor power dynamics. The availability of advanced computationaltools and microprocessors has facilitated the implementation ofthese techniques in many other similar engineering applications.Thus it is possible to use such models based on optimal estimationtechniques to estimate vital unknown parameters such as reactivitycoefficients. Knowledge of these coefficients plays a key role in ana-lyzing reactor accidents, testing reactor shutdown systems, andlicensing safety control systems. The coefficient estimates must ex-hibit certain desirable characteristics. They must be unbiased, suchthat their expected value is the same as that of the parameter beingestimated. They must have a low variance, so that the error covari-ance is less than or equal to any other unbiased estimate. They mustalso be consistent, to allow for convergence of the estimatedparameter to its true value as the number of measurementsincreases. The Kalman filter provides a simple means of estimating

ll rights reserved.

: +1 270 897 1179.. Vashaee).

immeasurable parameters of the power plant, and these estimatesare optimal as the variance of the error between the estimated andthe real parameter value is at minimum and cannot be improvedwith any other filter. These estimates can be used to improve theknowledge of the plant and its operation. Previous treatments ofthis topic (Paratte et al., 2006; Tayebi and Vashaee, 2005; Sastre,1960; Lawrence and Bullock, 1970) based on state space descriptionof reactor dynamics, have been generally applied to a point reactor,i.e., spatial variation of neutron flux in the reactor core has been ne-glected. This is not an accurate approximation in the case of CANDUreactors where the space kinetics of flux can alter reactivity coeffi-cients significantly. Adding spatial information requires an enor-mous increase in the number of state variables, equivalent tomultiplying the number of the states in a point reactor model bythe number of mesh points defined in the core. This can greatly in-crease the computational cost making the method unfit for practi-cal use. We hereby develop a reference model that combines theexpansive approximation of spatial flux descriptions with state-space parameter estimation techniques to reduce computationalcosts. We will specifically adopt the modal expansion descriptionof the flux. This provides a powerful and cost-effective tool thatcan provide an adequately accurate estimation of the importantreactivity coefficients. CANDU reactors use more than one type ofdetector to read fluxes where each type of detector can have differ-ent response dynamics. Therefore, the measured neutron flux is not

2

6 7

448 L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457

in fact the true value of the instantaneous flux, as the measurementmay be delayed, attenuated, or over amplified. It is essential thatthe model correct the variations caused by the detectors to producean accurate estimation of the reactivity coefficients.

2. Model identification

The algorithms based on optimal estimation techniques aregaining popularity in the nuclear field. Fig. 1 shows the block dia-gram for the calculations that take place in the execution of thealgorithms to extract the unknown reactivity coefficients. Hereu(r, t) and u

_ðr; tÞ are the actual and the estimated space functions,

respectively. d(i, t) is the ith detector’s reading at time t. The esti-mator chosen for this application is linked with a mathematicalmodel that simulates the space time kinetics of the reactor. Thedetectors’ readings are the input data to the estimator modulewhere the unknown reactivity coefficients are predicted and cor-rected to generate similar observations from the detectors.

2.1. Reactor model

To model the space-time dynamic response of the reactor, weused the modal expansion method, which is widely used in the nu-clear field. The basic idea of the modal expansion method is toapproximate the unknown space and time related neutron fluxfunction by a linear combination of known space functions withtime-dependent coefficients. Therefore, the modal expansionmethod includes two steps: (1) define the space functions (2) de-rive the time-dependent coefficients. In this formalism, the spacefunction is given by:

uðr; tÞ ¼XM

m¼1

wgmðrÞamðtÞ ð1Þ

In which wgnðrÞ are the lambda modes of a reference core, i.e., the

eigenfunctions of the static neutron balance equations for the non-perturbed reactor for the successive eigenvalues. It contains thespace, r, and energy group, g, dependence. The time-dependentcoefficients, am(t), can be described as (Gold, 1990):

dam

dt¼ ðqscm þ qmm � bÞam

KmþPM

n–mqmnan

KmþX

j

kjcmj

dcmj

dt¼ �kjcmj þ

bj

Kmam m ¼ 1; 2; . . . M

ð2Þ

In which am is the prompt neutron modal amplitude; cm is thedelayed neutron modal amplitude; b is the prompt neutron frac-tion; bj is the delayed precursor fraction of group j; kj is the delayedprecursor decay constant; qmn is the modal reactivity; and M is thenumber of eigenmodes.

qscm is subcritical reactivity of mode m defined by:

qscm � 1� ‘m

where ‘m is the eigenvalue for the m-th harmonic mode (Gold,1990).

Fig. 1. Calculation procedure for the estimation of the plant’s unknown parameters.

Km is prompt neutron generation time defined by:

Km �wg

mjwgm

� �f

v wgmjmR

sf jw

gm

D EIn which v is the neutron speed, m is the fission yield, and Rs

f is thesteady state macroscopic neutron fission cross-section.

The modal reactivity qmn is related to the effective incrementalcross-section for the perturbation as follows:

qmn �hwmjðDtRf0

� DRa þ DDXÞdðrÞjwnihwmjtRf0

jwmið3Þ

where X is the operator defining the product of gradients, i.e., hwm-

jXjwni = hrwmjr wni; tRf is the fission cross section; Ra is theabsorption cross section; D is the neutron diffusion coefficient;and d(r) is the delta function.

2.2. Virtual reactor system

To demonstrate the concepts for resolving the uncertainty inour parameter estimation code and model, a generic tool that isbased on a modal expansion solution of the two-dimensional neu-tron transport equation in the core is used. This acts as a virtualsystem simulating the neutron flux out from a real reactor core.

The virtual system consists of a modal description of the space-time kinetics of the reactor core. In order to apply the referencemodel method to a trip event the following scenario was consid-ered. This problem is a revisit of the benchmark activity presentedin McDonnell (1977).

The transient starts with a uniform loss-of-coolant that resultsin the increase of coolant void reactivity and is followed by anasymmetric insertion of shutoff rods which results in an increasein the thermal absorption cross section in one-half of the core.

Fig. 2 shows the layout of the reactor, which consists of inner(regions 11 and 12) and outer fueled regions surrounded by aD2O reflector (regions 1, 2, 3, 4).

The first four thermal modes are calculated using a finite ele-ment solution of the eigenvalue equations:

�r �Drun þ Run ¼ ‘nMun ð4Þ

where within a two-group approximation:

/n ¼w1

n

w2n

� �; D ¼ D1 0

0 D2

� �; R ¼ R1 þ R12 0

�R12 R2

� �; M

¼ m1Rf 1 m2Rf 2

0 0

� �ð5Þ

where w1;2n is the scalar neutron flux in energy groups 1(fast) and 2

(thermal); D1;2 is the diffusion coefficients for the group 1 and 2;

11 12 8 3

9

4

1 5

10

780 cm

780

cm

x

y

Fig. 2. Reactor core configuration.

x (cm) y (cm) y (cm)

y (cm)y (cm)

Fundamental mode: 018.1=λ

0200

400600

800

0

500

10000

0.05

0.1Second mode (first azimuthal): 036.1=λ

0

500

1000

0200

400600

8000

0.05

0.1

0.15

0.2

Third mode (second azimuthal): 036.1=λ

0200

400600

800

0

500

10000

0.05

0.1

0.15

0.2Third mode: 066.1=λ

0 200 400 600 8000

500

10000

0.05

0.1

0.15

0.2

x (cm)

x (cm)

x (cm)

Fig. 3. Spatial density distribution of the thermal neutron flux for the eigenmodes one to four. k is the corresponding eigenvalue.

Table 1Lattice parameters.

Region Groupg

Dg

(cm)Rg (cm�1) tRf g (cm�1) R1?2 (cm�1)

1, 2, 3, 4 1 1.310 1.018 � 10�2 0.0 1.018 � 10�2

2 0.8695 2.117 � 10�4 0.05, 6, 7, 8, 9,

101 1.264 8.154 � 10�3 0.0 7.368 � 10�3

2 0.6328 4.014 � 10�3 4.523 � 10�3

11, 12 1 1.264 8.154 � 10�3 0.0 7.368 � 10�3

2 0.9328 4.100 � 10�3 4.462 � 10�3

Table 2Delayed neutron data.

bj kj (s�1)

4.170 � 10�4 0.012441.457 � 10�3 0.030631.339 � 10�3 0.113903.339 � 10�3 0.307908.970 � 10�4 1.19803.200 � 10�4 3.2120

L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457 449

R12 is the transfer cross section from group 1 to 2; R1,2 is the ther-mal absorption cross section for the group 1 and 2; m1,2Rf1,2 is thegroup neutron production cross sections; and ‘n is the eigenvalue.

Eq. (4) is solved in the MATLAB environment for a two-dimen-sional CANDU reference core. As examples, Fig. 3 shows the calcu-lation results for the thermal neutron flux distributions for the firstfour modes.

The two-group cell parameters and other data for the problemare given in Table 1 and 2. Six groups of delayed neutrons wereused in the simulation. The initiating perturbation is representedby an exponential decay decrease in the thermal absorption crosssection, R2, in regions 5, 6, 10, and 11 of the form: R2 -= �5.8667 � 10�5(1 � e�t/0.4) cm�1

At t = 0.6 shutoff rods are inserted that results in an increaseDR2 = 1.23 � 10�4 cm�1 progressively in regions 2, 4, 6, 7, 9, 10,11, and 12 from top to bottom. This simulates the insertion of shut-off rods at a constant velocity of 520 cm/s in the y direction. Thismovement is perpendicular to the x axis and moves from region2 to 4. During this transient significant flux distortion and delayedneutron holdback occurs.

The results are calculated in terms of total and regional fissionyields versus time:

Zmi

tX

f 2

u2ð�r; tÞdm,Z

mi

tX

f 2

u2ð�r;0Þdm

where mi is the area of region i. The calculated time dependent ther-mal flux in the y direction at x = 375 cm is plotted in Fig. 4-left.

Fig. 4-right also shows the calculated relative power in regions5 and 12, and the total core.

2.3. Detector response

CANDU reactors are equipped with two separate safety shut-down systems (SDS-1 & SDS-2) which detect an emergency situa-tion and actuate the safety systems. Each of these systems consistsof both out-of-core ion chambers and in-core self-powered detec-tors. The larger number of detectors configured in the core are tooptimize sensitivity to local high power. Fast-response platinumdetectors are used for the measurement of the neutron power toactivate SDS1 and SDS2. In these detectors, the output current isinitially generated from three different sources: 3% from beta

Table 3Detectors’ response parameters.

Prompt over response detectors Delayed under responsedetectors

cP 1.05 0.914ci �0.01 �0.004 �0.033 �0.0015 0.016 0.014 0.044 0.012si 95 1538 13,333 333,333 30 250 2440 160,000

0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

t (s)

Rel

ativ

e po

wer Total

Region 12

Region 5

Fig. 4. Variation of thermal flux (left) and relative power (right) versus time.

450 L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457

decay, 60% from neutron capture gammas, and 40% from externalgammas. The detector reading approaches the actual neutronpower after a fast over shoot, i.e., 103%. Therefore, the detectors’signal is often called over prompt.

The CANDU 6 also uses readings from 102 in-core Vanadiumdetectors placed at various positions in the core to synthesize the3-dimensional flux distribution. The output current in Vanadiumdetectors is almost all generated by neutron captured beta decayproducing a delayed response that approaches the actual neutronpower from smaller values. Hence, these detectors are often calledunder delayed.

Since the detector readings are not equal to the actual neutronpower in the core, it is necessary to consider the detector dynamicsin the model. The detector dynamics can be modeled as:

xðtÞ ¼ cPuðrd; tÞ þX4

i¼1

DiðtÞsi

ð6Þ

d(t) is the output signal of the detector, si is a time constant, /(rd, t)is the amplitude of the neutron flux at the detector’s location rd attime t, cp is the coefficient of the prompt response, and Di is calcu-lated from:

@DiðtÞ@t

¼ �DiðtÞsiþ ciuðrd; tÞ i ¼ 1; . . . ;4 ð7Þ

Table 3 lists the parameters used in the model for the prompt overresponse (Platinum) and the delayed under response (Vanadium)detectors.

2.4. State space model

The time domain estimator requires a reactor model in thestate-space form. For this purpose, we adopt the prompt and de-layed neutron modal amplitudes for the state variables since theirtime derivatives are included in the modal model. Therefore, thestate x vector is defined as:

x ¼ ½½am�½c1�½c2� � � � ½cM��T½ðMþJMÞ�1� ð8Þ

Superscript T means the transpose of the vector or matrix, where,

½am� ¼ ½a1 a2 . . . aM �TðM�1Þ

½cm� ¼ ½c1m c2m . . . cJM�TðJ�1Þ m ¼ 1;2; . . . Mð9Þ

Therefore the state space model can be obtained as:

_x ¼ f ðx; ½am�; ½cm�;qmnÞ ¼ Adð½am�; ½cm�;qmnÞx ð10Þ

where Ad([am], [cm], qmn) is the transition matrix. Using Eq. (2), wecan write Ad as:check the PDF for Eq. (11)

Ad¼

qK

� �ðM�MÞ ½k1�ðM�JÞ � �� ½kM �ðM�JÞ

aK1

h iðM�MÞ

� � � aKM

h iðM�MÞ

bK1

h iðJ�MÞ

½~k�ðJ�JÞ � �� ½0�ðJ�JÞ ½0�ðJ�MÞ � � � ½0�ðJ�MÞ

..

. ... . .

. ... ..

.

bKM

h iðJ�MÞ

½0�ðJ�JÞ � �� ½~k�ðJ�JÞ ½0�ðJ�MÞ � � � ½0�ðJ�MÞ

2666666664

3777777775

ð11Þ

J is the number of delayed neutron precursor groups and M is thenumber of eigenmodes. The matrix elements are defined as follows:

½q=K�¼

ðqsc1þq11�bÞK1

q12K1

�� � q1MK1

q21K2

ðqsc2þq22�bÞK2

�� � q2MK2

..

. ...

�� � ...

qM1KM

�� � ðqscMþqMM�bÞKM

266666664

377777775ðM�MÞ

½ki�¼ ith row :

0 �� � 0... ..

.

k1 k2 �� � kJ

..

. ...

0 �� � 0

266666664

377777775ðM�JÞ

;

½b=Kj�¼

0 � � � b1 � � � 0

..

.b2

..

.

..

. ... ..

.

0 � � � bJ � � � 0

2666664

3777775

jth column

ðJ�MÞ;

½~k�¼

�k1 0 �� � 0

0 �k2...

..

. . ..

00 � �� 0 �kJ

2666664

3777775ðJ�JÞ

;

½a=Ki�¼ ith row :

0 0 � �� 0... ..

. ...

a1=Ki a2=Ki � �� aM=Ki

..

. ... ..

.

0 0 � �� 0

266666664

377777775ðM�MÞ

ð12Þ

Fig. 5. State space system model.

L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457 451

Therefore, the discrete model of the reference model in the statespace representation can be written as:

xkþ1 ¼ f ðxk;qmn;wkÞ ð13Þ

where

f ðxk;qmn;wkÞ ¼ f1þ DtAdð½am�; ½cm�;qmnÞgxk þ Gwk ð14Þ

xk and xk+1 refer to x(tk) and x(tk + Dt) respectively. Dt is the sam-pling period, and wk is the system noise assumed to be a zero meanwhite Gaussian noise independent of xk, and with covariance Q. G isthe Jacobian matrix of partial derivatives of f with respect to w, i.e.,a unit matrix for our problem. If we define:

A ¼ 1þ DtAdð½am�; ½cm�;qmnÞ ð15Þ

We have:

xkþ1 ¼ Axk þ Gwk ð16Þ

Let the available discrete time measurements be modeled as:

ykþ1 ¼ cPzkþ1 þX4

i¼1

Di;kþ1

siþ tkþ1 ð17Þ

where

Di;kþ1 ¼ 1� Dtsi

� Di;k þ cizkDt ð18Þ

tk+1 is the measurement noise that is assumed to be zero meanwhite Gaussian independent of xk+1 and wk+1, with covariance R.zk+1 is the amplitude of the flux at the detectors’ locations at timetk+1. In the modal expansion description we can write:

zðrd; tkþ1Þ ¼XM

m¼1

wmðrdÞamðtkþ1Þ ð19Þ

or

zkþ1 ¼ Hxkþ1 ð20Þ

with:

H ¼

w1ðr1Þ w2ðr1Þ � � � wMðr1Þ 0 � � � 0

w1ðr2Þ w2ðr2Þ wMðr2Þ ... . .

. ...

..

. . .. ..

.

w1ðrdÞ w2ðrdÞ � � � wMðrdÞ 0 � � � 0

2666664

3777775½d�ðMþJMÞ�

ð21Þ

Eq. (21) indicates that the neutron flux is measured at d locations inthe core. Using Eq. (20) in Eqs. (17) and (18), the equation relatingthe measurement to the state vector is:

ykþ1 ¼ cPHxkþ1 þX4

i¼1

Di;kþ1

siþ tkþ1 ð22Þ

where

Di;kþ1 ¼ BiDi;k þ ciHxkDt ð23Þ

where Bi = (1 � Dt/si) Eqs. (16) and (22) define the state space mod-el of the system and will be used in the estimator module. The sys-tem model is shown in Fig. 5.

Now that the complete system model is obtained, an estimationalgorithm can be applied.

3. Comparison of different estimators

The particular estimator to be used will depend on characteris-tics of the process to be ‘observed’. We investigated three differenttechniques based on recursive prediction error method (RPEM),dual extended Kalman filter (DEKF), and joint extended Kalman fil-

ter (JEKF). There are different approximations in each method thatchanges the accuracy of the estimations for the reactivity coeffi-cients under study.

3.1. Dual extended Kalman filter (DEKF)

Among various estimation techniques, the well-known ex-tended Kalman filter (EKF) is one of the most widely used in appli-cations. A historical survey of the development of Kalman filteringcan be found in Anderson and Moore (1977). The extended Kalmanfilter (EKF) provides an efficient method for generating approxi-mate maximum likelihood estimates of the state of a discretetime-nonlinear dynamical system. The filter involves a recursiveprocedure to optimally combine noisy observations with predic-tions from the known dynamic model. For the specific benchmarkactivity of LOCA, we apply a DEKF filter for estimating both thestates of the core, xk, and the unknown parameters, which arethe void reactivity coefficients, qmn, simultaneously, given onlynoisy measurements of the flux at certain points in the core. Tobe more specific, we consider the problem of learning both the hid-den states xk and parameters qmn of a discrete-time nonlineardynamical system:

xkþ1 ¼ f ðxk;qmn; qmn;wkÞ ð24Þykþ1 ¼ Hxkþ1 þ tk ð25Þ

qmn consists of the sum of all the known reactivity components andqmn are the void reactivity coefficients to be estimated. In order tosimplify the algorithm for this case, we have assumed an idealdetector that measures the real time amplitude of the flux. Whenthe clean state is not available, a dual estimation approach is re-quired. Dual EKF algorithm combines the Kalman state and reactiv-ity filters, where essentially two EKFs are run concurrently. At everytime step, an EKF state filter estimates the state using the currentreactivity estimates qmn while the EKF reactivity filter estimatesthe reactivities using the current state estimate xk. The system isshown schematically in Fig. 6, where the scalar observation yk isconsidered as one of the states. Thus, we only need to consider esti-mating the parameters associated with a single nonlinear function f.The top EKF generates state estimates and requires qmn;k�1 for thetime update. The bottom EKF generates void reactivity estimates,qmn;k�1, and requires xk�1 for the measurement update.

The dual EKF equations for this system are summarized as:

1. Initialize x0; Px;0; qmn;0; Pqmn ;0

Time-update equations of the extended Kalman filter:

Fig. 6. The dual extended Kalman filter.

452 L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457

2. q�mn;k ¼ qmn;k�1

3. P�qmn ;k ¼ Pqmn ;k�1 þ Q k�1

For the state filter:4. x�k ¼ f ðx�k�1;qmn;k�1; qmn;k�1;0Þ5. P�x;k ¼ AkPx;k�1AT

k þ Q k�1

The measurement-update equations for the state filter:6. Kx;k ¼ P�x;kHT

kðHkP�x;kHTk þ Rx

k�1

7. xk ¼ x�k þ Kx;kðzk � Hx�k Þ8. Px;k ¼ ðI � Kx;kHkÞP�x;k

For the reactivity filter:9. K qmn

k ¼ P�qmn ;kðHqmnk Þ

TðHqmnk P�qmn ;kðH

qmnk Þ

T þ Rqmnk Þ

�1

10. qmn;k ¼ q�mn;k þ Kmn;kðzk � Hx�k Þ11. Repeat steps 2 to 10 for k = 1 to Nt, where Nt is the number of

time steps.

In which:

Q k ¼ wkwTk

� �Rx

k ¼ tktTk

� �R

qmn;k

k ¼ ðqmn;k � qvoidmn;kÞðqmn;k � qvoid

mn;kÞT

D E ð26Þ

and:

Ak�1 �@f ðx; q�mn;kÞ

@x

xk�1

;

Hqmnk � � @ðzk � Hx�k Þ

@qmn¼ H

@x�k@qmn

qmn¼q�

mn;k

ð27Þ

Px,k and Pqmn ;k are the variances of the error in xk and qmn;k, respec-tively. Rqmn;k

k is the noise covariance and is taken as a constant diag-onal matrix. It, in fact, cancels out of the algorithm and can be setarbitrarily (e.g., Rqmn;k

k ¼ 0:5I). Rxk and Qk are the covariance of wk

and tk which are additive measurement noise and innovation noisecovariance, respectively. The innovation covariance affects the con-vergence rate and tracking performance. Convergence is oftenquicker for larger innovation covariance. Here, it is set to an arbi-trary diagonal value of Qk = 0.1I, which implies an independenceassumption on the parameters.

3.2. Recursive prediction error model (RPEM)

In recursive prediction error method (RPEM) the Kalman gain istreated as a parameter to be estimated (Ljung, 1979; Ljung, 1999).In this method, the unknown parameter vector qmn may enter thestate and measurement transition matrices. However, it is as-sumed that the matrix entries are differentiable with respect to

qmn. The RPEM algorithm applied to the problem under study isas follows:

1. Initialize x0; P0, cPrediction error or innovations error:

2. ek ¼ zk � Hxk

Recursive egressions for C and q:3. Ck ¼ Ck�1 þ ck 1kS�1

k 1Tk � Ck�1

h i4. qmn;k ¼ qmn;k�1 þ ckC

�1k 1kS�1

k ek

The measurement-update equations:5. Kk ¼ AkPkHT

k S�1k

6. Pkþ1 ¼ AkPkATk þ Qk � KkSkKT

k

7. Sk ¼ HkPkHTk þ Rk

8. xkþ1 ¼ Akxk þ Kkek

9. zkþ1 ¼ Hxkþ1

10. PðiÞkþ1¼ @@qmni

AkPkATk þAkP

ðiÞk AT

k þAkPk@AT

k@qmniþ @Q k

@qmni

h�jðiÞt SkKT

k �KkrðiÞk KTk �KkSkðjðiÞk Þ

T �

11. rðiÞk ¼ @@qmni

HkPkHTk þ HkP

ðiÞk HT

k þ HkPk@HT

k@qmniþ @Rk

@qmni

h i12. jðiÞk ¼ @

@qmniAkPkHT

k þ AkPðiÞk HT

k þ FkPk@HT

k@qmni

h i� S�1

k � KkrðiÞk S�1k

13. ith column of M�k ¼ @

@qmniHkðqmniÞxðtÞ þ jðiÞk ek i ¼ 1;2; . . . d

14. M�k ¼ Mðqmn;k; xk;qmn;kÞ þ jkek

15. Wkþ1 ¼ ½Ak � KkHk�Wk þM�k

16. 1kþ1 ¼WTkþ1HT

k

17. Repeat steps 2 to 16 for k = 1 to N

where

Qk ¼ wkwTk

� �; Rk ¼ tktT

k

� �ð28Þ

C is often called search direction matrix, and c represents amagnitude of correction and is chosen as a trade-off betweentracking capability and noise insensitivity. C, S, and 1 are initializedto arbitrary values at beginning. Here, we set C = I/c and S = 0.

3.3. Joint extended Kalman filter (JEKF)

The dual EKF algorithm represents a decoupled type of ap-proach, in which separate state-space representations are used toestimate xk and qmn;k. An alternative direct approach is given bythe joint EKF which generates simultaneous maximum a posterioriprobability (MAP) estimates of xk and qmn;k. This is accomplished bydefining a new joint state-space representation with an augmentedstate vector:

gk ¼ ½xk qmn;k�T ð29Þ

In this approach the dual EKF is modified to minimize a joint costfunction. Given the measurementzk, maximizing a probability den-sity function of Pgk jyk

is equivalent to maximizing Pxkqmn;k jyk(Haykin,

2001). Hence, the MAP optimal estimate (Kay, 1998) of gk will con-tain the optimal values of xk and qmn;k. Running the single EKF withthis state vector provides a sequential estimation algorithm. For thispurpose, similar to Eq. (15), we define a new state transition matrixas:

H ¼ I þ DtHdð½am�; ½cm�; qmnÞ ð30Þ

where

Hd ¼½Ad�ðMþJMÞ�ðMþJMþM2Þ

½0�M2�ðMþJMþM2Þ

" #ð31Þ

We further assume a zeroth order dynamics for the void reactivitycoefficients, i.e., q

�mn ¼ 0. This approximation may increase the esti-

mation error when the reactivity coefficient experiences a fast

Time (s)

Det

ecto

r rea

ding

(AU

)

0 0.5 1 1.5 2 2.5-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 100 200 300 400 500 600 7000

100

200

300

400

500

600

700

X (cm)

Y(c

m)

Fig. 7. Detectors layout (left) and a sample detector response (right).

L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457 453

transient. We will remove this constraint later in the model (see Eq.(34)). The measurement transition matrix is also revised as:

H ! ½½H�d�ðMþJMÞ ½0�d�M2 � ð32Þ

The joint EKF algorithm is summarized as follows:

1. Initialize g0; P0

Time-update equations:2. g�k ¼ f ðg�k�1;qmn;k�1; qmn;k�1;0Þ3. P�k ¼ HkPk�1H

Tk þ Q k�1

The measurement-update equations:4. Kk ¼ P�k HT

kðHkP�k HTk þ RkÞ�1

5. gk ¼ g�k þ Kkðzk � Hg�k Þ6. Pk ¼ ðI � KkHkÞP�k

Repeat steps 2 to 6 for k = 1 to N.Note that because the gradient of f(z) with respect to qmn is ta-

ken with the other elements (namely, xk) fixed, it will not involverecursive derivatives of xk with respect to qmn. This fact is citedin Ljung (1979) and Ljung and Söderstrom (1983) as a potentialsource of convergence problems for the joint EKF. Additional re-sults and citations in Nelson and Stear (1967) uphold the difficul-ties of the approach, although the cause of divergence is linkedthere to the linearization of the coupled system, rather than thelack of recurrent derivatives. Although the use of recurrent deriva-tives is suggested in Ljung (1979) and Ljung and Söderstrom(1983), there is no theoretical justification for this. In fact, for thesystem under study, it is shown that the joint EKF provides a moreaccurate alternative to the dual EKF for sequential optimal estima-tion of the void reactivity coefficients.

3.4. Estimation results

We use the data measured in thirteen detectors distributed inthe core as shown in Fig. 7-left. At this point, it is assumed thatthe detectors are ideal and their signal is modeled with the virtualsystem described in Section 2.

In this transient, the event starts by uniform loss of coolant;hence, the neutron flux can be dominantly described by low orderharmonic modes. Here, we used the first five harmonic modes forthe nominal core configuration. Taking into account of more num-ber of modes did not produce noticeable difference. Since the high-er order modes are not excited in this transient, the number ofdetectors is not as important as their location and ability tomeasure the excited mode shapes. Therefore, systematic errorsdue to systematic loss of detector signals are not affecting the

estimation results. In this case study, our model calculations showthat adding more detectors does not change the final results. Theoutput from the modeled detector of the operating reactor consistsof both true neutronic signal and random noise. The random noisecomes from random neutronic fluctuations in the core and theelectronic hardware.

The effect of the noise component at any particular detector atany particular time is an independent random phenomenon. Therandom error component of the detector signal is completelyunpredictable and cannot be filtered out using an electronic circuit.The effect of random detector error is simulated by adding a set ofindependent random errors to the thermal flux amplitude at thedetectors locations. The distribution of the random error in eachset is a zero mean white Gaussian distribution with a specifieddeviation. To evaluate the robustness of the estimators, an exces-sively large measurement noise with amplitude of about 20% ofthe peak value of the thermal flux at the center of the core is as-sumed for all the detectors. In reality noise levels are below 10%.The modeled detector reading for the one located at the center ofthe core is shown in Fig. 7-right.

Detector readings are finally input to the reference modelwhere the reactivity components are to be extracted. RPEM, DEKF,and JEKF estimators are applied to the measurements from thethirteen detectors, and the value of the modal amplitudes andthe coolant void reactivity coefficients during the LOCA are simul-taneously estimated. Fig. 8 shows the estimated modal amplitudefor the first three modes. The accuracy of the estimated modalamplitudes is almost similar using each of the three estimators.

Fig. 9 shows the coolant void reactivity estimations. The shut-off rod reactivities are assumed to be as known input to the estima-tors and the coolant void reactivities are left as the unknownparameters to be estimated. Dynamic reactivity shown in Fig. 9-right is defined as (Rozon, 1998):

qvoidDynamic ¼ qvoid

11 þXM

m¼1

qvoid1m

am

a1ð33Þ

qvoid11 is the self-coupling reactivity of the fundamental mode due to

the coolant void perturbation effect. In these calculations, eightthermal modes have been used, which were generated as staticeigenfunctions of the unperturbed reactor core (i.e., k modes) aspreviously described.

The fit using JEKF is obviously good considering the large ampli-tude of the measurement noise shown in Fig. 7. However, the esti-mations using DEKF and RPEM show a large deviation from theactual values. For the case of RPEM, the initial lag in reactivityestimation and the afterward accumulation of errors results in a

0 0.5 1 1.5 2 2.5-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (s)

Ampl

itude

s (A

U)

Fig. 8. Estimation of the modal amplitudes. Solid curves are the actual amplitudesand dotted curves are the estimations.

454 L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457

large deviation from the actual value after a short period of time.DEKF estimations follow the shape of the reactivity coefficients;however, there exists a large attenuation in the estimated qmn. Thisis due to the fact that the lower EKF in Fig. 6 presumes a static reac-tivity behavior, which forces slow variations in qmn. The measure-ment updates are not able to overcome this drive and thereremains a constant lag in the estimations. For the case of JEKF,the initial estimations of the reactivity coefficients also have a timedelay that is due to the zeroth order approximation of the reactiv-ity variations made in the EKF. The filter makes a history and up-dates the reactivity coefficient to a higher value. The delay, andthus the estimated values, is improved after about one secondwhen the reactivity variation is relaxed.

4. JEKF and smoother

It is possible to improve the accuracy of the estimation by theuse of a smoother combined with the JEKF. The use of a smootheris possible since the estimation process is done post-facto. Thesmoother allows better overall estimates and much improved per-formance in the estimation of the reactivity coefficients. Thesmoother accomplishes this by utilizing state and measurementdata from the entire time period to improve the estimate at eachpoint in time. The JEKF filter alone is not able to use future datato produce estimates. Suppose that we are given a set of data overthe time interval 0 < k 6 N. Smoothing is a non-real-time operation

0 0.5 1 1.5 2 2.5-15

-10

-5

0

5

10

RealJEKF

DEKF

RPEM

Time (s)

Fig. 9. Estimation of the void self-coupling reactivity for the fi

in that it involves estimation of the state gk for 0 < k 6 N, using allthe available data, past as well as future.

However, in our problem, we will assume that the final time Nis fixed. To determine the optimum state estimates gk for 0 < k 6 Nm we need to account for past data zj defined by 0 < j 6 k, and fu-ture data zj defined by k < j 6 N. The estimation pertaining to thepast data, which we refer to as forward filteringtheory, is the al-ready discussed JEKF process. To deal with the issue of state esti-mation pertaining to the future data, we use backward filtering,which starts at the final time N and runs backwards. Let gf

k andgb

k denote the augmented state estimates obtained from the for-ward and backward recursions, respectively. Given these two esti-mates, the next issue to be considered is how to combine them intoan overall smoothed estimate gk, which accounts for data over theentire time interval. Note that the symbol gk used for the smoothedestimate here is not to be confused with the filtered (i.e., a poste-riori) estimate used in the JEKF. The approach is based on theRauch–Tung–Striebel (RTS) smoother (Lewis, 1986).

Moreover, in the single JEKF model, we assumed the void reac-tivity coefficients are static, whereas their actual values vary withtime. Therefore, for a more accurate estimation of these coeffi-cients, the essential feature of their dynamics must be also mod-eled in the estimator.

For this purpose, the estimations can be improved with assum-ing a shape function for the variation of the reactivity coefficientsunder study. We assumed the following dynamics for the reactivitycomponents:

@qmn

@t¼ gðtÞqmnðtÞ þwk ð34Þ

where g(t) is a function that has a similar kinetics as the reactivitycoefficients under study. Similar to Eq. (15), we define a new statetransition matrix as:

N ¼ I þ DtNdð½am�; ½cm�;qmn; gðtÞÞ ð35Þ

where

Nd ¼½Ad�ðMþJMÞ�ðMþJMþM2Þ

½0�M2�ðMþJMÞ gðtÞIM2�M2

" #ð36Þ

I is the unit matrix. The JEKF/RTS smoother algorithm is summa-rized as follows:

Forward filter:

1. Starting value gf0; P

f0

2. gf�k ¼ f ðgf�

k�1;qmn;k�1; qmn;k�1;0Þ3. Pf�

k ¼ NkPfk�1N

Tk þ Q k�1

4. Kfk ¼ Pf�

k HTkðHkPf�

k HTk þ RkÞ�1

-15

-10

-5

0

5

10

RealJEKF

DEKF

RPEM

0 0.5 1 1.5 2 2.5

Time (s)

rst mode (left) and of the void dynamic reactivity (right).

Time (s)

Det

ecto

r rea

ding

(AU

)

0 0.5 1 1.5 2 2.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Detectorsreading

Exact

Time (s)

Det

ecto

r rea

ding

(AU

)

0 0.5 1 1.5 2 2.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Detectorsreading

Exact

Fig. 11. Over (left) and under (right) response detector output (dotted curves) and the actual values (solid curve).

0 0.5 1 1.5 2 2.5-1

0

1

2

3

4

5

6

7

8

Real

Forward sweep

Combined sweeps

Real

Forward sweep

Time (s) Time (s)0 0.5 1 1.5 2 2.5-1

0

1

2

3

4

5

6

7

8

9

Real

Forwardsweep

Combined sweeps

Fig. 10. Estimation of the void self-coupling reactivity for the first mode (left) and of the void dynamic reactivity (right).

L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457 455

5. gfk ¼ gf�

k þ Kfkðzk � Hxf�

k Þ

6. Pfk ¼ ðI � Kf

kHkÞPf�k

7. Repeat steps 2 to 6 for k = 1 to NBackward smoother:

8. Initializing value gN ¼ gfn; and PN ¼ Pf

N

9. Fk ¼ PfkN

Tkþ1;k Pf�

kþ1

h i�1

10. Pk ¼ Pfk � FkðPf�

kþ1 � Pkþ1ÞFTk

11. gk ¼ gfk þ Fkðgkþ1 � gf�

kþ1Þ12. Repeat steps 9 to 11 for k = N�1 to 1

In the forward filtering a zeroth order dynamics is assumed asbefore; however, in the backward sweep, from the fact that voidreactivity saturates at about 10mk in CANDU reactor, we assumedg(t) = a /[exp (at) � 1]. a is a constant that can be added as an extraunknown parameter to the state vector to be estimated by JEKF.However, it turned out that its value is not affecting the reactivityestimations. We assumed an arbitrary constant value of a = 1 in thepresent calculations.

Fig. 10 shows the estimation results for the same reactivitytransient described in the previous section. Comparing with thereactivity estimations from a single JEKF in Fig. 9, we notice a sig-nificant improvement in reducing the estimation error. While theJEKF estimator shows a maximum error of about 3 mk, the RTSsmoother has a maximum error of about 1 mk for q11 and an al-most negligible error for the dynamic reactivity. The improvementis especially significant at the initial transition time where the JEKF

shows about 100% error whereas the error in JEKF/RTS estimationis trivial.

We now investigate the effects of the detector response time inour model. We add the state space equations related to the detec-tor dynamics to our model as depicted in Fig. 5. We only considerthe JEKF/RTS estimator as it showed a superior performance com-pared with DEKF and RPEM. We will consider two types of detec-tors: under responsive Vanadium detectors used for fluxmapping and over response Platinum detectors available fromthe reactor shutdown systems.

The measurement equation is:

ykþ1 ¼ cPHxkþ1 þX4

i¼1

Di;kþ1

siþ tkþ1 ð37Þ

where Di,k+1 is according to Eq. (23). Fig. 11 shows the responsefrom one of the Inconel detectors (one with Platinum emitter andone with Vanadium emitter). We again assume a similar detectorconfiguration as in Fig. 7-left. The simulated responses from thir-teen detectors are input to the JEKF/RTS estimator where the voidreactivity coefficients are estimated. We add an independent zeromean white Gaussian noise to the output of each detector signalas before. Figs. 12 and 13 show the estimated void self-couplingreactivity for the fundamental mode and the dynamic reactivityversus time. It is seen that in both cases of over and under responsedetectors the model is able to estimate the reactivity coefficientswith a similar accuracy as the case of ideal detectors.

In a CANDU reactor, there are data available from both over andunder response detectors. Therefore, it is essential that the model

Time (s)

Real

Combinedsweeps

0 0.5 1 1.5 2 2.5-1

0

1

2

3

4

5

6

7

8

Forward sweep

Time (s)0 0.5 1 1.5 2 2.5

-1

0

1

2

3

4

5

6

7

8

9Real

Forward sweep

Combined sweeps

Fig. 12. Estimation of the void self-coupling reactivity of the first mode (left) and the void dynamic reactivity (right) using the measured data by over response detectors.

Over responsive detector

Under responsive detector

Fig. 14. Detectors layout (left) and a sample detector response (right).

Time (s)0 0.5 1 1.5 2 2.5-1

0

1

2

3

4

5

6

7

8

Real

Forward sweep

Combined sweeps

Time (s)0 0.5 1 1.5 2 2.5-1

0

1

2

3

4

5

6

7

8

9

Real

Forward sweep

Combined sweeps

Fig. 13. Estimation of the void self-coupling reactivity for the first mode (left) and of the void dynamic reactivity (right) using the measured data by under response detectors.

456 L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457

be able to use a mixture of measurements data from both types ofthe detectors.

For this test case, the performance of the JEKF/RTS state andparameter estimation algorithm is examined for the similar prob-lem of estimating void reactivity coefficients in a loss of coolantaccident. The presented model gives the possibility to define a sep-arate dynamic response for each detector. As an example, we con-sidered measurement data available from a mixture of over andunder response detectors as shown in Fig. 14. Fig. 15 presents

the results from the estimator module for this case. As shown inFigs. 10, 12, 13, and 15, the results from the estimator are very sim-ilar to the actual values especially for the important case of thevoid dynamic reactivity; thus validating the inverse space-timedynamics method based on JEKF/RTS method presented in thispaper.

The state and parameter estimation technique applied to thereactor dynamics presented here can provide us with extremelyuseful information. For example, similar calculations can be

Time (s)0 0.5 1 1.5 2 2.5-1

0

1

2

3

4

5

6

7

8

Real

Forward sweep

Combined sweeps

Time (s)0 0.5 1 1.5 2 2.5-1

0

1

2

3

4

5

6

7

8

9Real

Forward sweep

Combined sweeps

Fig. 15. Estimation of the void self-coupling reactivity for the first mode (left) and of the void dynamic reactivity (right) using a mixture of measured data from both over andunder response detectors.

L. Tayebi, D. Vashaee / Annals of Nuclear Energy 53 (2013) 447–457 457

performed for other reactivity coefficients. In a power rundowntest, the method can be used to extract the reactivity coefficientsrelated to shutoff rods. But in this case, it is actually simpler todo and interpret the data, as there is dominantly one reactivitytransient involved.

5. Conclusion

This method demonstrates better estimation of space-timedynamics of reactivity in a CANDU core than previous attempts.Different methods were investigated to improve the accuracy ofspecific nonlinear state and parameter estimation applied to lossof coolant accidents. It was shown that an approach combiningthe joint extended Kalman filter (JEKF) and Rauch–Tung–Striebel(RTS) methods yields more accurate estimations than the dual ex-tended Kalman filter and recursive prediction error method. Thenew method also provides much smoother results. To accomplishaccurate estimation of void reactivity, the backward sweep of theRTS smoother is assumed to possess exponential dynamics. The re-sults obtained are independent of the assumed amplitude and timeconstant of the sweep. The detector dynamics are also consideredin the same estimator. It is demonstrated that the estimator re-mains stable and accurate, and is independent of the detectordynamics as well. The proposed model is flexible and can be usedto define different dynamic responses for each detector while pre-serving the accuracy of estimation.

Acknowledgments

The Authors gratefully acknowledge Professors J.C. Luxat andW.J. Garland, of McMaster University, for helpful discussions dur-ing the conduct of this work.

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