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Abstract—The installation of a new Instrumentation and Control (I&C) system for the TRIGA MARK II reactor at University of Pavia has recently been completed in order to assure a safe and continuous reactor operation for the future. The intervention involved nearly the whole I&C system and required a channel-by-channel component substitution. One of the most sensitive part of the intervention concerned the Automatic Reactor Power Controller (ARPC) which permits to keep the reactor at an operator-selected power level acting on the control rod devoted to the fine regulation of system reactivity. This controller installed can be set up using different control logics: currently the system is working in relay mode. The main goal of the work presented in this paper is to set up a Proportional-Integral-Derivative (PID) configuration of the new controller installed on the TRIGA reactor of Pavia so as to optimize the response to system perturbations. Index Terms — Fission Reactor Control and Monitoring, TRIGA

I. INTRODUCTION HE TRIGA Mark II reactor at Applied Nuclear Energy Laboratory of University of Pavia has recently been

equipped with a new Automatic Reactor Power Controller capable of different control logics. Currently the system is working on relay mode. In order to study the best configuration of a PID logic, the development of a non linear dynamic model of the reactor was necessary.

The TRIGA Mark II is a 250kW thermal pool type reactor moderated and cooled by light water in natural circulation (in core). Fuel consists of a uniform mixture of uranium (8%wt enriched 20% wt in 235-U), hydrogen (1% wt) and zirconium (91% wt). This particular composition provides the fuel with an effective moderation property strongly dependent on fuel temperature due to the discretised energy levels of hydrogen in the ZrH molecular structure. The reactivity control system

A. Borio di Tigliole and G. Magrotti are with the Laboratorio Energia

Nucleare Applicata (L.E.N.A.) of the University of Pavia – Via Aselli 41 – 27100 Italy.

A. Cammi and V. Memoli are with Politecnico di Milano, Department of Energy, Nuclear Engineering Division (CeSNEF) – Via Ponzio 34/3 – 20133 Milano, Italy.

A. Gadan is with the Instrumentation and Control Department of the National Atomic Energy Comission of Argentina and Theoretical and Nuclear Physics Department of the University of Pavia - Italy.

consists of three control rods. The “regulating” rod provides the possibility to finely change the reactivity and is the one on which the controller acts. The model was validated by comparing the experimental results of the behavior of the

power of the reactor controlled by the ARPC in relay mode.

The paper is organized as follows: in the first section the dynamic model is presented; in the second section, the comparison of the experimental results is shown; in the last section the optimization of the PID controller is discussed.

II. THE MODEL

A. Neutronics The neutron-kinetics model adopted in the present work is

based on the neutron diffusion equation with one energy group considering one group of delayed neutron precursors; thus, the governing equations are the following [1], [2]:

)()()()1()( tCtntnk

dttdn λβ +−−=

(1)

)()()( tCtnkdt

tdC λβ −=

(2)

The right term of equation (1) which describes the neutron

balance accounts for the prompt fission neutron generation, the neutron loss due to absorption and leakage and the delayed neutrons coming from the decay of the precursors. The neutron density, n, is related to the power system according to the relation , being n0 and P0 the steady values of the

neutron density and power. The equation (2) gives the rate of precursor concentration which depends on the precursor formation during the fissions and the decay of the precursors characterized by the decay constant λ. If the reactivity ρ, the mean neutron generation time , and the normalized quantities are introduced, (1) and (2) can be

rewritten as:

(3))()())(()( tttdt

td ηβψβρψΛ

+Λ

−=

(4)))()(()( tt

dttd ηψλη

−= I

Study of a New Automatic Reactor Power Control for the

TRIGA Mark II Reactor at University of Pavia A. Borio di Tigliole, A. Cammi, Mario A. Gadan, G. Magrotti, V. Memoli

T

n this sub-model the reactivity ρ(t) is the input variable, whereas ψ and η are the state variables.

B. TRIGA Thermo-hydraulics The thermal hydraulic model provides a description of the

fuel and coolant average temperatures (Tf, Tc) with the strong assumption to neglect the propagation velocity of the temperature front in the coolant (also called zero dimensional approach). Thus the coolant temperature is given by the mean value between the inlet and outlet core temperature:

(5)2

inoutc

TTT

+=

The heat transfer between the coolant and the fuel has been

modeled using a global heat transfer coefficient K which accounts for the thermal conduction in the fuel rod and cladding, and the convective heat transfer between cladding and coolant. The parameter K also contains information regarding the system geometry which strongly affects the heat transfer. Thus the energy balance equations of fuel and coolant are given by:

(6))(0 cff

ff TTKPdt

dTCM −−=

(7))()( inoutccfc

cc TTCTTKdt

dTCM −Γ−−=

These equations have been implemented in the model introducing the time constants of the fuel and the coolant given by:

(8)KCM ff

f =τ

(9)KCM cc

c =τ

The natural circulation equation has been modeled using

the Bussinesq approximation [6] which provides the relation between the coolant mass flow rate and the core inlet and outlet temperature of the coolant:

(10))()( 021

2 LTTgL inout −=⋅+Γ νρξξ

The coefficient ξ1 describes the pressure drop due to spacer grids, inlet and exit geometry also classified as form losses. The coefficient ξ2 gives the pressure drop due to friction along the core channel.

C. Power Effects At full power the coupling term between the kinetics and

thermal-hydraulics is given by the reactivity feedback coefficients i.e. the fuel and moderator temperature coefficient. Here only the fuel temperature coefficient has

been introduced since the MTC is negligible

(11))( 0fff TT −= αρ

D. The Whole System The whole system is described by equations (3), (4), (5),

(6), (7), (10) and (11). The state variables are the normalized neutron density (or power density), the normalized precursor concentration, the fuel temperature and the coolant temperature. The control system acts on the reactivity by means of the regulating rod. Thus the equation (11) becomes:

(12))()( 00 hhTT hfff −+−= ααρ

where h0 is the critical height of the regulating rod. Thus the input variables are the core inlet coolant temperature, Tin, and the rod excursion, h-h0. The model was implemented in Simulink® which permits to easily treat the non linearity of the system. In Table I all the physical quantities and parameters of the model are summarized.

TABLE I

PHYSICAL QUANTITIES AND PARAMETERS OF THE MODEL

Symbol Quantity Steady Values (when meaningful)

P0 Nominal reactor power 250kWatt Λ Mean Neutron

generation time 50µsec

β Delayed Neutron Fraction

0.00730

λ Precursor Group Decay constant

0.0768sec-1

K Global heat transfer coefficient from fuel to coolant

1.9752 103W/ºC

τf Fuel Time Constant 5sec Cc Coolant Specific heat

Capacity 4186J/Kg ºC

Γ Coolant Mass Flow Rate

8.7060Kg/sec

Fig.1. Simulink® coupled thermo-hydraulics and neutronics Model of the TRIGA Mark II

Tout Core Outlet Coolant

temperature 26.86 ºC

Tin Core inlet Coolant Temperature

20 ºC

Tf Average Fuel Temperature

150 ºC

ν Coolant Thermal Expansion Coefficient

2.0680·10-4 ºC-1

L Core Height 0.7224m αf

Fuel Temperature Reactivity Coefficient

-11·10-5 ºC-1

αh Regulating Rod Worth coefficient

33.4·10-5cm-1

E. Free Response In order to check the goodness of the model, the free

response of the system to a positive step variation of the core inlet coolant temperature and to a step variation of the reactivity were studied. Here only the free response to a reactivity step of about 10c$ (equal to the reactivity insertion step of the controller) is shown for brevity. An instantaneous increase of the reactivity causes an increase of the system power and consequently of the fuel temperature (Fig.2). The increase of fuel temperature has the effect to change the

reactivity of the system causing a decrease of the power. At the end of the transient the system reaches a new equilibrium at a higher power level (Fig. 4) and at a fuel temperature higher of a quantity given by ∆Tf=∆ρ/αf.

III. COMPARISON WITH EXPERIMENTAL DATA The experimental data were measured with the new ARPC

set to “relay mode”. In this mode the controller reads the measured output which in this case consists in an electric current coming from a compensated ionization chamber placed in proximity of the reactor core (outside the core reflector). The electric current is proportional to the neutron thermal flux. In this way it is possible to indirectly measure the reactor power. When the power exceeds the nominal value P0 the controller acts on the Rod Control System (RCS) and negative reactivity is introduced in the system by inserting the regulating rod of 2.2cm in roughly 5sec. The controller acts exactly in the opposite way when the reactor power drops to a value below the nominal value P0. Thus the change of reactivity at the end of the rod insertion and withdrawing step is respectively -10c$ or + 10c$.

150

151

152

153

154

155

156

157

0 20 40 60 80 100 120

Tc(ºC

)

Time(sec)

Fig. 2.Fuel Temperature Behavior caused by a step variation of the reactivity.

1

1.02

1.04

1.06

1.08

1.1

1.12

0 20 40 60 80 100 120

P/P 0

Time(sec)

Fig. 4.Normalized Power Behavior caused by a step variation of reactivity.

0

0.02

0.04

0.06

0.08

0.1

0.12

0 20 40 60 80 100 120

reac

tivity

($)

Time (sec)

Fig. 3.System Reactivity Behavior caused by a step variation of the reactivity.

5.6350E-12

5.6450E-12

5.6550E-12

5.6650E-12

0 5 10 15 20 25

Cur

rent

I(A

)

Time(sec)

Fig. 5. SPND current measurement at full power (250kW).

During the rod excursion the controller is always enabled. he reactor Power Level registered by the Linear Power Channel Recorder of the I&C System, shows a power oscillation period of about 7.2 sec with an amplitude of about ±3% around the nominal value of 250 kW (P0). These quantities are related to the insertion/withdrawing velocity of the regulating rod. The same control logic was simulated and the results (Fig. 6) show a fair agreement with the experimental data.

In addition (Fig. 5) shows the electric current measured by a rhodium based Self-Powered Neutron Detector (SPND) developed by the Instrumentation and Control Department of the National Atomic Energy Commission of Argentina. A SPND is composed by a material with relatively high cross section to interact with thermal neutrons leading a subsequent beta or gamma decay. Basically the SPND collects the charges generated by the emitter material due to the interaction with thermal neutrons and generates a current of the order of pA that can be measured by an electrometer. The current generated by the SPND is proportional to the rate at which neutrons are captured by the emitter. In our case, this current is measured by a Keithley Eletrometer Mod.6517A that is connected to a computer via RS-232 using an acquisition program based on Matlab. The SPND is positioned far from the reactor core, in the thermal column of the reactor, during operation at 250 kW: again the oscillation period is found to be 7.2 sec while the amplitude is about ±0.3%.

IV. PID OPTIMIZATION The simple relay logic, currently used for the TRIGA

reactor power control, can be not sufficient when an accurate control of the power is needed. A better approach is certainly provided by a PID or PI regulator. Here the continuous version is discussed and some results of the optimization process are shown.

In general the action, C(t), of a PID (Proportional- Integral- Derivative) [7] controller is given by the following relation:

)13()()()()(0∫++=t

idp dekdt

tdektektC ττ

The quantity e is the difference between the set point (ψ0=1) and the actual value of the process variable (ψ). The coefficient kp, ki and kd are the tuning parameters. Preliminarily we have chosen to set the derivative term to zero. Usually in a NPP a feed forward logic is used instead of a derivative in order to correct the output for large variations. In our case the action consists in the regulating rod excursion expressed in cm. Initially the ki was held to zero and different values of kp were tested. (Fig. 7) shows the results of the response to a unitary step variation of the core inlet coolant temperature for kp=1, 5, 10.

Fig.7. Normalized Power Variation Behavior under the action of proportional controller for different gain factors. For higher values of the proportional gain the system shows a faster response to the perturbation. For kp=10 after 50 sec the normalized power is equal to 0.994, whereas in the free response to the same perturbation (Fig.8) the power is 0.992.

0.991

0.992

0.993

0.994

0.995

0.996

0.997

0.998

0.999

1

0 20 40 60 80 100

P/P0

time(sec) Fig. 8. Free response to a unitary step variation of Tin As it can be noticed, the proportional alone is not able to give a steady state error (SSE) equal to zero in fact in order to have a finite output a finite error is needed [7]. On the contrary the integral gives a SSE equal to zero at the end of the transient. The dependence of the controller on the integral gain was

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 5 10 15 20 25

Nor

m. P

ower

var

iatio

nδψ

Time(sec)

Fig. 6.Normalized Power Variation Behavior under the action of the relay controller.

studied keeping the kp set to 10 varying the coefficient ki from 0.5 to 8. Fig. 9 shows the results of the simulations. For ki=0.5 the power, after a sharp drop, rises to a value of 0.995P0 after 20sec. A faster response was obtained for ki=8 where the power rises to a value of 0.999P0

0.994

0.995

0.996

0.997

0.998

0.999

1

0 10 20 30 40 50

Nor

mal

ized

Pow

er

time(sec)

ki=8ki=4

ki=2

ki=1

ki=0.5

Fig. 9. Response to a Tin step variation for different values of the coefficient ki of the PID

V. CONCLUSION A dynamic model of the TRIGA Mark II reactor was developed in order to optimize a preliminary PID logic for the new ARPC recently installed at L.E.N.A.. Experimental results of the controller in relay mode were compared with the simulation results showing the goodness of the model. The analysis showed that a continuous PID offers generally better results than the relay which causes power oscillation with amplitude of 3% of the nominal power. However further investigations and experimental activities are necessary in order to reach the best configuration of the new ARPC.

REFERENCES [1] James J. Duderstadt, Louis J. Hamilton, “Nuclear Reactor Analysis”

John Wiley $ Sons, Inc., 1976. [2] Roman Shaffer, Weidonghe, and Robert M. Edwards, “Design and

Validation of optimized Feedforward with Robust Feedback Control of a Nuclear Reactor “, Nuclear Plant Operations and Control, 2001.

[3] M. S. Ash, Nuclear reactor kinetics, New York: McGraw-Hill.

[4] “SIMULINK, Dynamic System Simulation Language User’s Guide,” The Mathworks, Natick, Massachusetts , 1994

[5] General Atomics, 1967. GA-7882, Kinetic Behavior of TRIGA Reactors, pp. 6–32.

[6] Frank P. Incropera, David P. De Witt, “Fundamentals of Heat and Mass Transfer”, 4th Edition John Wiley &Sons, Inc. 1996.

[7] Karl J. Astrom, T. Hagglund, “PID Controllers: Theory, Design, and Tuning”, 2nd Edition