DESIGN OF SPECIFICALLY OPTIMAL FILTERSFOR NUCLEAR REACTOR SYSTEMS

Philipp F. SchweizerWestinghouse Electric CorporationResearch and Development Center

Pittsburgh, Pa. 15235

Abstract

This paper presents a design procedure for select-ing the optimal parameters of a filter used in extract-ing true signals from noisy reactor measurements. Theequations for the optimal filter are presented and aprocedure is given for approximating the optimal filterwith a simpler and less expensive structure. A designexample demonstrates how the method may be applied toa reactor protection system.

Introduction

This paper presents a procedure for designing afilter which may be used in extracting true signalsfrom noisy reactor measurements. Figure 1 depicts apossible application of this method in a reactor pro-tection system. The protection system combines measuredprocess variables such as reactor coolant temperatures,nuclear flux and coolant pressure and from thesedetermines the state of the reactor according to aspecific law. A signal is usually generated whichcontinuously presents the difference between the reactorstate and a maximum or minimum allowable state such thatthe reactor remains within design limits. This signalis defined as the protection system margin.

Measurements of the reactor variables (i.e. coolanttemperature, pressure, nuclear flux) are corrupted withnoise and are subject to process and measurement instru-ment drift. The use of these noisy and drifting measure-ments to determine protection 8ystem margin (definedhere as the raw margin) usually results in an inaccurateobservation. To assure that a conservative margin ismaintained, the reactor operating state is changed toprovide additional margin. This usually results inlower reactor operating powers.

It is desirable to have a filter which operates onthe raw margin signal (Figure 1) and produces a signalwhich represents the true margin as nearly as possible.This filter, if optimal, would be capable of exactlyseparating the raw margin signal into components causedby noise, drift and the true margin.

The use of a filter of this type in actual reactoroperation would result in 1) safer operation because ofa better knowledge of the true protection margin or2) increased power with the same protection marginavailable as without the filter and 3) fewer spuriousreactor trips because of noise "spikes".

This paper presents equations which specify afilter which is optimal in the sense that it minimizesthe error in the estimate of the true reactor protectionsystem margin. These equations for the general linearsystem have previously been developed in referenceland2. The exact implementation of these equations onan on-site computer or special purpose digital equip-ment may not be necessary if some degradation in filterperformance is acceptable. A design procedure is devel-oped which produces filter equations which are an approx-imation to the optimal filter equations and which havereduced implementation costs. The filter resultingfrom these equations is defined as Specifically Optimal(S.O.F., Specifically Optimal Filter).

The term "design" is used in this paper in thepreliminary sense. The final filter design results ina set of equations and an algorithm for obtaining theequation parameters and solutions. To achieve a work-able design these equations and algorithms must beeither programmed on an on-site computer or implementedin circuit hardware.

The remaining sections of this paper define theproblem mathematically (Problem Statement), presentthe optimal filter equations (Optimal Filter Solution)and develop approximate filter equations with reducedimplementation requirements (Design Procedure For TheSpecifically Optimal Filter). An example is presentedwhich illustrates the method (Application).

Problem Statement

This section presents the equations for the opti-mal filter as they pertain to the reactor protectionsystem. The work presented here is a specific appli-cation of the more general treatment given in

19,2References1 2.The dynamics of the reactor protection system may

be described by the following equations.

xl (k) = A1 x1 (k-l)

x2 (k) = A2 x2 (k-l)(1)(2)

where

x1 (k-l) defines the reactor protection marginat a time specified by k-l.

x2 (k-l) defines the instrumentation drift ata time specified by k-l.

A1 defines the margin time constant

A2 defines the drift time cons tant

Equations (1) and (2) are given in discrete formwith discrete time interval k. Since the final algo-rithm will most likely require digital implementationthe discrete form, from the initial problem formulation,is convenient. The discrete interval size is selectedbased on the dynamics of the protection system.

In the practical design problem the designer may(most likely) want to model the reactor protection mar-gin x1(k) more exactly by including the dependence onmost process variables. The methods presented here areapplicable to a more detailed model. However, the mar-gin is modeled as a simple first order equation so thatthe techniques presented are not obscured by the addi-tional notational complexity. The more general n dim-ensional linear case is developed in the Appendix.

Measurements made on the reactor system to deter-mine the raw margin are represented by

z(k) - h1 x1(k) + x2(k) + v(k) (3)

where

z(k) is the measured margin at time definedby k

x1(k) is the true reactor margin at time k

x2(k) is the drift component of the measure-ment at time k

v(k) is a random noise corrupting themeasured margin

h is the gain of the measurement channel

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The problem is to develop a design procedure forspecifying a filter which has measurement inputs z(k)and has as its output response an estimate of the truereactor protection margin x1(k) and the instrumentdrift x2(k). This filter is to be determined such thatthe estimates x (k) and x (k) are optimal in the sensethat they minimize the foilowing expression

E [(x(k) - x(k))' (x(k) - x(k))]

K1(k) =

K2(k) +(4)

A1 p11(k-l) + A1A2 P12(k-1) (7)

A1 p1(k-1) + 2 A A2p 2(k-1) + A2p22(k-l) + R

A2 P22(k-1)

2 P22A1 p1(k-1)

(8)1 2 P12 (k1)

+ 2 A1A2p12 (k-1) + A2p22(k-1) + R

where

E(-] designates the expected values of termsinside the brackets

x(k) = col (x1(k), x2(k)) is the vectorrepresentation of the estimates

x(k) = col (x1(k)o x2(k)) is the vectorrepresentation of the true reactor state(margin and drift)

denotes the vector transpose

Optimal Filter Solution

General optimal filtering theory has been develop-ed in referencesl 2 which is applicable to the reactorproblem of concern in this paper. Using other methodssimilar filter equations are also presented inreference3.

The detailed derivation of the optimal filterequations are presented in the Appendix. The resultingequations are applied to the specific problem definedin the problem statement section of this paper.

The equations which the optimal filter must satisfyare determined by applying Appendix Equation (A5) whichyields

xl(k) = Alx1(k-l) + K1(k) [z(k) - hjA1x-(k-1)

-A2x2(k-l)] (5)

x2(k) = A2x2(k-1) + K2(k) [z(k) - hAixi(k-1)

where* *

K (k) and KI2(k) are the optimal gains at time(A) for the filter equations (5) and (6)

pll(k-l), P12(k-1), and P22(k-l) are the elementsof the errors covariance matrix at time k

R is the variance of the random noise corrupt-ing the reactor measurement.

The covariance terms used in (7) and (8) are deter-mined by applying Appendix Equation (A12) which yields

p11(k) = A2p(k-l) -

(A4p2(k-l) + 2A1A2p3p(k-l)p 2(k-l) + A2A2p2 (k-1))2plI121 12 21

A2p1(k-1) + 2A A2p12(k-l) + A2p22(k-1) + R

P12 (k) = A1A2p12 (k-1) -

(A3A2p11(k-l)p12(k-l) + A2A2p11(k-1)P22(kOl) +12pll 12 1 2P11~~~~

A2A2 2 (k-1) + A A3 (k-1) p(k-1))1 2Pl2 1212 P22

A1 p1(k-1) + 2AjA2p1(k-1) + A2p2(k-1) + R(10)

-A2x2 (k-l)] (6)

where

1 (k) and x-2(k) are the estimates at timesdefined by k

K1(k) and K2(k) are the active filter gains ata time defined by k

The estimates satisfy the linear differenceequations (5) and (6) which relate the estimates at afuture time (k) to the present estimate plus a cor-rection term. The correction term is composed of atime dependent gain (K1(k) or K2(k)) multiplied by thedifference between the measurement at time (k) and thebest estimate of that measurement using values fromtime k-l.

To obtain optimal estimates from (5) and (6)(denoted by x*(k)), optimal values for the gains(denoted by K*(k)) must be used. These can be deter-mined using Appendix Equation (All) which yields

P22(k) = A2 p22 (k-l) -

(A1 2p2(k- 1) + 2A A2p 2(k-1)p22(k-1) + A4p22(k-1))2 2(Alpl(k- 1) + 2AIA2p12(k-1) + A2p22(k-l) + R

(11)

Using equations (5) through (11) the optimal filtermay be determined. Inspection of these equations showsthat to solve both equations (5) and (6) and equations(9) through (11) initial conditions (pii(O), P12(0o)P22(0), x1(O), x2(0)) must be known. tfectively, onemust assume an initial estimate or initial value forthe covariances.

The structure of equations (5) through (11) allowsthe designer to implement the optimal filter in an on-line manner. For example, given a time k, the co-variance of the error may be determined from Equations(9) through (11) from a knowledge of error covarianceat the past time (k-l). Using the covariance at time k-lthe optimal filter gains at time k may be determinedfrom Equation (7) and (8). Using these gains theestimate for the margin and drift may be determinedfrom Equations (5) and (6). If these computations andsignal transfers can be accomplished in "a faster than

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real time mode" then the filter may be implemented-directly. However, this continuous computation mayrequire a specially dedicated device or extra burdenon an existing plant computer. In the followingsection some approximations to the above will beinvestigated with the goal to alleviate computationalburden.

Design Procedure For The

Specifically Optimal Filter

The basic investigation to be conducted here is;what degradation in filter performance occurs when thegains in the estimator equations (5) and (6) are notchanged at each k time interval. Since the computa-tional burden in implementing the filter is propor-tional to the "frequency of equation update" somereduction in instrumentation costs may be achievedthrough less frequent changing of filter gains inEquations (5) and (6).

The problem is formulated as a tradeoff ininstrumentation costs versus the cost of adding reactorprotection margin (Figures 3 and 4). This may bequantitatively expressed as

C = C1(ui) + C2(ui) (12)

where

C = net cost benefit through use of a filter

C1(ui) = cost of instrumentation

C2(ui) = cost benefit of reduced margin

ui = update interval which is defined as thenumber of k time intervals that elapse be-tween changing filter gains

C1 and C2 are determined from

k+T1 . z T ( Ix(i) - yi(i) + Ix2(i) - y2(i)|) (13)

i=k

C2 IC(ui.) (14)

y2(k) = A2y2(k-1) + G2(k) [z(k) - Alyl(k-1) -

A2y2 (k-l)] (16)

where

G1(k) and G2 (k) are non-optimal gains which aredetermined by updating these gains with theoptimal gains at delayed times

G1(k) = K1(si + n x ui)

G2(k) = K2(si + n x ui)

(17)

(18)

for

si + n(ui + 1) < k < (n + 1) (ui + 1)where

si = starting interval for the filter

n = number of update intervals that have occurredsince si

ui = update interval as previously defined

K1 (.) and K2 () are optimal filter gains at atime defined by (si - n x ui)

Using the above definitions and equations theproblem of determining the specifically optimal filterreduces to determining an optimal update interval ui*which gives a minimum cost defined by (12) with con-straints defined by Equations (13) through (16).

This optimal update interval may be determined byusing a procedure which investigates the first differ-ence of the cost, equation (12) ,which is expressed as

AC= AC AuiAui (19)

To minimize cost the increment defined by (19)should be negative which can be achieved by selecting

Au = -e -ACAui

where

x;(i) and x(i) are the optimal estimates formargin and grift at a time defined by i

yl(i) and y2(i) are the estimates for margin anddrift using gain values in equations (5) and (6)which are non-optimal

T is a fixed time period over which the averageabsolute deviation in (13) is computed

(-) denotes the absolute magnitude

IC(ui) represents the instrumentation costs as afunction of the update interval ui (seeFigures 3 and 4).

y = scalar weight for cost benefit

The estimates y1 and Y2 are determined from

y1(k) = Alyl(k-l) + G1(k) I z(k) - A1y1(k-l) -

A2y2 (k-l1)(15)

where

£ is determined such that the change in updateinterval is equal to one k interval

Using the above definitions and equations thedesign procedure for the specifically optimal filter(S.O.F.) may be delineated as follows.

1. Calculate the oitimal gains K (k), K2 (k) andestimates xl, x2, from equations (5) through(11) using initial variances and estimatesfor design problem of interest.

2. Select initial update interval ui.3. Calculate Specifically Optimal Gains (17) (18).4. Calculate Specifically Optimal Filter (SOF)

Estimates (15) (16).5. Compute new update interval ui using (20).6. Compute increment in cost using new update

interval (19) (13) (14).7. If increment is negative repeat steps 3 through

6-, otherwise stop.

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(20)

Applications

A simplified design problem is considered todemonstrate the method. This example is not intendedas an actual reactor design. It is noteworthy thatthe reactor design problem would most likely be morecomplex because of the more detailed model for reactorprotection margin. However, the method demonstratedcould be applied in exactly the same manner.

For the curves shown in Figures 2 and 3 thesystem parametes assumed were: margin time constantAl = 0.95, drift time constant A2 = 0.995 The initialestimate of the drift and margin were assumed zerowhile the initial covariance terms were assumed as

P1l(0) = 12100, p12(0) - -2200, P 2(°) = 400. Thiscorresponds to a practical case s nce at the initiationof the filter calculations the system would be cali-brated and initial variances would be known moreaccurately than at any other time. It is noteworthy,however, that even if the initial covariances are inerror (less than 20%) the filter will converge to theoptimal estimate although the convergence is slower.

Figure 2 shows protection margin versus time forthree cases 1) system without filter 2) system withOptimal Filter and 3) system with SOF. The truemargin is also shown for this example. Comparison ofthe curves as the transient dies out shows that thesavings in margin using the optimal filter isapproximately 10% while the savings using SOF is 8%.

Figure 3 and 4 show the tradeoff in costs ob-tained by using the SOF rather than the optimal filter.

Instrumentation Costs, Cost Savings Because ofMargin Reduction and Net Cost Savings are plotted versusfilter update interval. With ui = 10 the filter isoptimal. Figure 3 which expresses the costs for thedesign example, shows that the optimum design policy(maximum net savings) would be to implement the optimalfilter. Figure 4 shows estimated costs for a more de-tailed model of reactor margin (6 or 10 equations).Because of the increase complexity the optimum designpolicy would be to implement an SOF with an updateinterval of 30 secs.

Summary

A procedure has been developed for designing a

filter which can be used in extracting true signalsfrom noise reactor measurements. The filter is optimalin the sense that it minimizes the expected value ofthe measurement error for a specific design. Thedesign procedure is given in algorithm form. Atechnique is developed and demonstrated for evaluatingthe effectiveness of this design procedure for a

specific application.

Appendix

The equations which may be used for the generalreactor system model and which were used for obtainingthe optimal filter equations in the paper are developedin this section. The development parallels that givenin Reference 2.

The system dynamics are defined by

x(k) = A x(k-l)

where

(Al)

x(k) is an n dimensional vector describingthe model state at time k

x(k-l) is similarly defined at time (k-l)

A is system fundamental matrix

The system measurements are defined by

z(k) = Hx(k) + v(k) (A2)

where

z(k) is an m dimensional vector of reactormeasurements

H is a constant m x n matrix

v(k) is an m dimensional random noise vector

The noise vector v(k) is assumed to have thefollowing properties

covariance matrix defined

(A3)E [v(k), v(k)'] = R(k)

E (v(k), x0) - 0

The structure for the optimal filter is defined by

x(k) = A x(k-l) + K(k) [z(k) - HA x(k-l) ](A5)

where

x(k), x(k-l) are estimates of x(k)

K(k) nxn matrix of filter gains to be determined

The K(k) matrix is to be determined such that the

following is minimized

E [(x(k) - x(k))' (x(k) - x(k))]

where

E [-] denotes the expected value

Define the estimation error as

x(k) = x(k) - x(k)

(A6)

(A7)

The covariance matrix of the error estimates is

defined by

P(k) = E [x(k), xt (k) ] (A8)

P(k-l) = E [x(k-l),x'(k-l)]

Substituting (A5) and (Al) into (A7) yields

x(k) = [Ai(k-l) + K(k) [z(k) - HA x(k-l)]]- A x(k-l)

(A9)

Substituting (A9) into (A8) and using (A3) yields

P(k) - (I - K(k) H) A P(k-l)A'(I - K(k) H)'

+ K(k) R(k) K' (k) (AlO)

where

E [v(k),v'(k-l)] = 0

Using the matrix minimum principle (Reference 4)oi a similar technique (Reference 2) the gain matrix

K (k) (* optimal) which minimizes the trace of matrix

P(k) (trace of P(k) is equation A6) defined by (AlO) is

K (k) = PI(k-l) Hs [H P (k-l) H' + R(k)] 1 (All)

where

-1[] denotes inverse matrix

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(A4)

P1(k-l) 5 A P(k-l) A'

Substituting (All) into (AlO) and simplifyingyields

P(k) = P1(k-l) - K(k) H P1(k-l) (A12)

Equations (A12) (All) and (A9) are the equationsfor the optimal filter for a system modeled by (All)with measurements given by (A2).

References

3. Sage, A. P., Optimum Systems Control, Prentice-Hall Inc., Chap. 9 and 15, 1968.

4. Athans, M., and Tse, E., "A Direct Derivation ofthe Optimal Linear Filter Using the MaximumPrinciple", IEEE Trans. on Auto. Control, Vol AC-12, pp 690, Dec. 1967.

$ 1.0

1. Kalmam, R. E., "A New Approach to Linear Filteringand Prediction Problems", Trans. ASME, Series D,Journal Basic Eng., Vol. 82, pp 34-45, 1960.

0.5

2. Sorenson, H. W., "Kalman Filtering Techniques",Advances in Control Systems, Edited by C. T.Leondes, Vol. 3, Academic Press, pp 219, 1966. Cost

(NormalizedDollars)

-0.5

I I".V Process /Gains --10ss (At k

Kfk) &Gl k) True

MarginIEstimates

Measuremensystem

Optimal or Specifically Optimal Filter

C1 =Cost Benefit in Margin SavingsC -Net Cost SavingsC2 -Instrumentation Costs

9~~~~~ 1

20 30 4) 50 60 70 80 90 ui(sacs)

Fig. 3-Instrumentabtion costs, margin benefit and net costssaving versus update interval (design example)

Fig. I -Functional diagram of rector protection system with filter

$1.0

0.5

Cost(Normalized

Dollars)

Measured Margin-simulated, z (kl

I I I a

100 210 300 400 500Time, seconds

C1 = Cost Bendit in Margin SavingsC2= Instrumentation Costs

-C =Net Cost Savings

io b io 50 ( to lb 90 ui(secsl

Fig. 2 -Protection margin versus time with and without filters Fig. 4-Instrumentation costs, margin benefit and net costssaving versus update interval fcosts estimated for detailed

margin model)

407

and

ProtectionMargin, 20S

10

-5.0