MU-SYNTHESIS CONTROL OF THE EBR-I1 PRIMARY REACTOR SYSTEM

Michael A. Power and Robert M. Edwards Nuclear Engineering Department The Pennsylvania State University

University Park, PA 16802

Eric Dean Argonne National Laboratory - West

P.O. Box 2528 Idaho Falls, ID 83403-2528

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The 9th Power Plant Dynamics, Control and Testing Symposium Hyatt Regency Hotel, Knoxville, Tennessee

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F

t

MU-SYNTHESIS CONTROL OF THE EBR-II PRIMARY REACTOR SYSTEM

Michael A. Power and Robert M. Edwards Nuclear Engineering Department The Pennsylvania State University

University Park, PA 16802

Eric Dean Argonne National Laboratory - West

P.O. Box 2528 Idaho Falls, Idaho 83403-2528

ABSTRACT

A robust p-synthesis controller for the EBR-11 reactor primary system and the DSNP simulations verifying the excellent robustness of the controller over a wide range of power are

presented.

INTRODUCTION

Significant advanced control research has been conducted at EBR-11 (Experimental

Breeder Reactor-n), a full-scale experimental reactor facility capable of generating 62.5 M W of

thermal power and 19 MW of electric power. During 1986, dramatic tests demonstrated the

inherent safety of EBR-II reactor.[ll In addition to safety, tight control of reactor power is

desired. The p-synthesis controller presented in this work provides tighter control than the

existing Automatic Control Rod Drive System (ACRDS)[*I which uses a gain-scheduled proportional control method. The p-synthesis controller guarantees robust performance of EBR-

II reactor power using the D-K[3941 iteration technique to reduce p (one divided by the worst case

multiplicative destabilizing perturbation). When p < 1 over all frequencies, the closed-loop

system achieves robust performance. The peak value of p is less than one (0.89) for the second

D-K iteration.

MATHEMATICAL MODEL OF THE EBR-11 SYSTEM

The assumed dynamical model of the reactor primary system for the p-synthesis

controller is comprised of the prompt neutron kinetics, the delayed neutron kinetics, the fuel and

coolant temperature dynamics, and the control rod dynamics. The required model is further

simplified by lumping the delayed neutron kinetics into one group. This introduces uncertainties

between the nominal plant model and the system to be controlled. The EBR-II zero-power

kinetics are modelled as,

and

--nr P + -cr P + A A

hn, - kr

where

nr(t) = relative neutron density (neutrons per cm3)

c,(t) = relative delayed neutron precursor density (per cm3) h = effective delayed neutron decay constant (sec-l)

P - - effective delayed neutron fraction A = prompt neutron lifetime (seconds)

P - - total reactivity (Akik).

The additional fuel and coolant temperature dynamics associated with high reactor power

levels are modeled using conservation of energy laws,

- - - dT f dt

f P i2 i2 - On, - -Tf + -T, Pf 2Pf Pf

(3)

and

sz (2M + Q) e nr + -Tf - TC

(1 - ffP, - - dTC dt P C P C 2PC

where

Tf

TC

Te c2

Pc

Pf M ff

PO

average reactor fuel temperature ("C)

average coolant temperature ("C) exit temperature of coolant ("C) heat transfer coefficient between fuel and coolant (MW/OC)

total heat capacity of reactor coolant (MW sec/"C)

total heat capacity of fuel and structure (MW sec/"C)

mass flow rate times heat capacity (MW/"C)

fraction of reactor power deposited in fuel

design power level (MW) at 100%.

Equations (3) and (4) model the time-dependency of reactor fuel and coolant temperatures.

Experimental results have shown that the actuator dynamics, represented by a simple pole

located at lo&, can be ignored since the dynamics lie outside the control bandwidth of the plant.

The dynamics associated with control rod reactivity are modeled by

where

- - dPmd - rod reactivity insertion rate dt

gr - - Zr =

reactivity inserted per unit length (W per fraction of core)

control rod speed (fraction of core length per second)

The total reactivity inserted into the core is the summation of the control rod reactivity dynamics

and the fuel and coolant temperature dynamics,

P

where

af

a, Tf

TC

Prod

PTc

PTf

temperature coefficient of reactivity for fuel (Akk per "C) temperature coefficient of reactivity for coolant (Akk per "C)

temperature of fuel ("C) temperature of coolant ("C) reactivity inserted into the core due to rod position (Ak/k)

reactivity inserted into the core due to coolant temperature (Akk) reactivity inserted into the core due to fuel temperature (Akk).

Equation (6) provides the total reactivity input for Equation (1).

For controller design purposes, Equations (1) through (5) are represented as,

P A

h

--

ffPO Clf

-h 0

Q -- 0 2Pf

Q

P C

-- 0

0 0

nroac 2A

0

-(2M + Q)

2Pc

0

- nrO A

0

0

0

0

C - - [l O O O O ] ,

[Ol . - D -

(7)

Table 1 defines the parameters for the nominal state space model given by Equations (7) through

(10). EBR-I1 is more precisely modeled with six reactivity feedback paths, but to simplify controller design, the six feedback paths are lumped into two feedback paths--one for coolant induced reactivity a, and one for fuel induced reactivity af.

Table 1

Parameters for Controller Design

Po = 62.5 Mw

ff = 0.9997

h = 0.075 (sec-1)

pf = 0.20 (MW-sec/"C)

i2 = 0.686 (MW/OC)

nfl= 1.0

f3 = 0.00670269 (AI&)

gr = 0.670269 (Ak/k)

= 0.065 (MW-sec/"C)

M = 0.613 (MW/"C)

Ctf = -3.175E-6 (AkMOC) = -1.078E-5 (Ak/W"C)

ROBUST CONTROL THEORY

A robust controller achieves acceptable performance, accommodates model uncertainty

and allows for design constraints. Some forms of acceptable performance are fast settling time

and good steady-state tracking error. Model uncertainty occurs when the nominal model used for

control (such as described in the previous section) does not model all the dynamics of the

process, mismodels the dynamics, or does not account for time-varying parameters. Design constraints may appear as disturbance rejections and actuator efforts.[S] The p-synthesis

controller developed by Doyle, et al. [3341, achieves acceptable performance and accommodates

model uncertainty by quantifying the performance specification and the uncertainty specification

into a frequency dependent weighting function and by using induced norms. The structure of p-synthesis controller developed by Doyle is shown in Figure 1.

Figure 1 - Layout of the System

In Figure 1, r is the reference input, e is the error signal, the uncertainty A is the allowed

perturbation in G, and K is the forward-loop feedback controller. The plant G is a linear time-

invariant (LTI) transfer function which relates the system input u to the system output y. The

perturbed plant G is a function of the nominal plant Go and the plant uncertainty. The frequency dependent weighting function Wdel weights the plant perturbation A, accounting for unmodelled

dynamics, nonlinear perturbations, and time-varying parameters.

Robust stability of G, nominal performance of Go, and robust performance of G are

addressed by equations (1 1) through (16). The plant G can be represented by

to account for the allowed perturbation in Go. Applying the small-gain theorem to the system in Figure 1 indicates that the plant G is robustly stable against the perturbation A when

where

closed-loop transfer function.

Nominal performance (good disturbance rejection of d and good steady-state tracking error of r)

for Go is achieved when

IlWpSIlm < 1

such that lie112 SI. If some sensitivity function S is known to provide good performance, then Wp

can be used to shape the plant S such that

Equation (14) indicates that the plant sensitivity function can be kept below Wp-l. For example,

if a 1% steady state error is desired, then the weight WP-l(O) < 0.01.

Simultaneously achieving robust stability, equation ( 12), and nominal performance, equation

(14), yields robust performance of G. For all plants G, a necessary and sufficient condition for

robust performance is given by

Since equation (15) is currently intractable, p can be used to synthesize a controller which

achieves robust performance. The mathematical definition for p is given by

where

- - CL M =

B(A) =

1 min{B(A): AED, det(1-MA) = 0)

complex structured singular value

closed loop transfer matrix, M E Cmn

maximum singular value of A.

APPLICATION OF MU-SYNTHESIS CONTROL TO EBR-I1

To synthesize the controller, the nominal plant model Go, the performance weighting

function Wp, and the uncertainty weighting function Wdel must be specified. The D-K iteration technique uses these specifications and a sequence of convex optimizations to reduce ~ A ( M ) . If

~ A ( M ) < 1 for a given iteration, then the closed-loop system achieves robust performance.

The performance weighting function for EBR-11 is given by

s + 1 3s + 0.01’

Figure 2, which plots the inverse of the performance weighting function, indicates a steady-state

tracking error of Wp-l(0) = 0.01 or 1%. At low frequencies, the disturbance is attenuated by a

factor of 100.[61

Uncertainty in the primary plant dynamics is due to the assumed linear point kinetics

equations and the lumped parameter model of the primary system. Figure 3 is used to generate

the multiplicative uncertainty

and plots three open-loop transfer functions. The solid line represents the nominal plant Go(s)

for n d = 0.75, the dashed line represents the open-loop transfer function for n d = 1.0, and the

dashed-dotted line represents the open-loop transfer function for n d = 0.5. The Bode plot having

the greatest distance from Go(s) is designated as Gl(s).

Figure 4 illustrates the multiplicative uncertainty. The dashed-dotted line in Figure 4 represents the plant uncertainty between the nominal model Go and the plant model for nd = 0.5.

The dashed line represents the uncertainty between the nominal model and the plant model for

nd = 1 .O. The solid line represents the first order transfer function

lo0Os + 0.1 lo0Os + 5

W,(S) = 0.35

which simplifies the multiplicative uncertainty description. Also, the solid line covers the plant

uncertainty generated between the nominal model and any transfer function with nd = [OS, 1.01.

The uncertainty for Wdel is relatively small at low frequencies, but attains a maximum at very

large frequencies. Figure 5 is a p plot for the second D-K iteration. This plot attains a maximum of p = 0.89

at 0.4 radshec and yields a multiplicative margin to instability of U0.89. Since the maximum

value of p is less than one, the closed-loop system achieves robust performance.

n a

10

0

-10

-20

-30

Inverse of Performance Weighting Function / . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, . , . . . , . , . , . * . , . , . . . , . , . , .

........

........ . . , . . . . . , . . . * . * . . . , . . . . . . . , . , . * . . . . . , . . .

........

. -. . , ~ .._ I . , . . . . . , . . . I Y

...... . * ..,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I . e , , * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . . a . * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.............

._.."._.__._. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ::> :a. ....... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.....................

.....................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...,,.. ....... ,- . .,., ,,,, ....... , ... ,..., .,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

3 10-2 10-1 100 101 102 103

Frequency (raddsec)

Figure 2 - Magnitude of Inverse Weighting Function, Wp-l

40

20

0

-20

a -40

-60

-80

-100

10-2 10-1 100 101

Frequency (raddsec)

102

-

. I .

. .-

. .

..

103

Figure 3- Open-Loop Transfer Function for nd = 0.5 (-.), 0.75 (-), 1 .O (--)

c Multiplicative Uncertainty -3

-10

-15

-20

-25 e3

-30

-35

. . . . . . . * . . . . . . . . . . .......... * . . * . .

. . . I . . . . .

: 4 , . I/-:-

. / 1 1 j . .

1 : : :

/: ....... I ; / ; ; ...... -... 1 : : : . , . . . . . . * .

1

0.8

4 0.6 E! -a 5 0.4 on

0.2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t.,.. * I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.............. I ..,, * . , . . . . . . . . . . ................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e . . , . * . * . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P A%. a.. _e

d 1 . C ... 2. .* . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... ....., . . , -.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... .....-. .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................... ................ ".L..I.. ................... ____._.<.. .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . I . . , * . . , , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . =__.--i- i'& I C . > . .................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,.-.,-. ................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - I - . ....... .._ . ....-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . , .., ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .._.A_. .......... .._.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . . , . . , ,.,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I ... L .......................... ::.:::*.*.. ..:. ..........:. ... 1 .......... :.: 2 : : L ' . . . ... ..:. .... t..'..... . ..:. .............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10-2 10-1 100 101 102

Frequency (radshec)

Figure 4 - Multiplicative Uncertainty Due to Variation in n d

MU Plot for Second D-K Iteration

103

.._

..

...............

I . . . . . . , . . . , . . . . . a ,

10-2 10-1 100

Frequency (raddsec)

101 102

Figure 5- Complex Structured Singular Value, p

DSNP SIMULATION RESULTS

The Dynamic Simulator for Nuclear Power Plants (DSNP)[7] simulation uses a high- order, three-pin distributed parameter model of the reactor whereas the p-synthesis controller

uses a fifth-order, lumped parameter model of the reactor. Figure 6 evaluates the performance of the p-synthesis controller. The figure shows three step changes in relative reactor power level.

The step change for n d = 0.3 is the dotted line, the step change for n d = 0.75 is the solid line,

and the step change for n d = 1.2 is the dashed line. The dashed and solid line simulations are

almost identical. The simulations for the n d = 0.3,0.75 and 1.2 cases are stable, well-damped,

and demonstrate a steady-state tracking error of less than 1%, consequently predicting a closed-

loop system which achieves robust performance for all plants n d = c0.3, 1.21.

DSNP Simulation of Reactor Power 1.02

1

0.98

0.96

0.94

0.92

0.9

0.88 0 10 20 30 40 50 60 70 80

Time (sec)

Figure 6 - Step Change Response of p-Synthesis Controller From a Normalized Relative Reactor Power Level of 1.0 to 0.9 for nd = 0.3 (.), 0.75 (-), 1.2 (--)

CONCLUSIONS

The p-analysis and synthesis toolbox from Matlab creates a 1.1 controller that attains a

maximum p = 0.89. Since 1.1 e 1 over all frequencies, robust performance is guaranteed. Robust

performance occurs when the closed-loop system for all plants in the uncertainty description

simultaneously achieves robust stability in the presence of plant perturbations and achieves

nominal performance in the form of excellent steady-state tracking error. The DSNP simulations of the EBR-II reactor primary system verify the robust performance of the p-synthesis controller

within the designed operating band of relative reactor power from 50% to 100% and also down

to 30% and up to 120%.

.

REFERENCES

[I] W.H. Planchon, et al., Implications of the EBR-11 Inherent Safety Demonstration Test, Nuclear Engineering and Design, Volume 101, No. 1, p.75, (April 1987).

[2] H.A. Larson, et. al. "Installation Of Automatic Control At Experimental Breeder Reactor 11, Nuclear Technology, Vol. 70, pp. 167-179, (August 1985).

John C. Doyle Keith Glover, Pramod P. Khargonekar and Bruce A. Francis, "State-Space Solutions to Standard H2 and H, Control Problems," ZEEE Transactions on Automatic Control, Vol. 34, No. 8, pp. 831-847, (August 1989).

131

141 "p-Analysis and Synthesis Toolbox", The Mathworks, Sherborn, Massachusetts, 1993.

[SI Sznaier, Mario, "Lecture Notes from EE597B - Robust Control", The Pennsylvania State University, (Spring 1993).

[61 C. K. Weng, R. M. Edwards, A. Ray, "Robust Wide Range Control Of Nuclear Reactors Using The Feedforward-Feedback Concept," Nuclear Science and Engineering, 1 17: 177- 185, (July 1994).

D. Saphier, The Simulation Language of DSNP, ANL-CT-77-20, Rev. 02, Argonne National Laboratory, (1978).