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Periodic Output Feedback Control of a Large Nuclear Reactor
Copyright @ IFAC Periodic Control Systems. Cemobbio-Como, Italy, 2001
PERIODIC OUTPUT FEEDBACK CONTROL OF A LARGE NUCLEAR REACTOR Chandan Nene' B. Bandyopadhyay' A. P. Tiwari ••
• Systems and Control Engineering Group, Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400 076, INDIA . •• Reactor Control Division, Bhabha Atomic Research Centre, Trombay, Mumbai - 400 085, INDIA.
Abstract: The paper presents the design of piece-wise constant periodic output feedback control for a discrete three time-scale system resulting from the discretization of a continuous-time standard three time-scale system. By a suitable linear transformation of state variables the given continuous-time system model is converted into a block diagonal form in which the three time scale system is decoupled into a slow , a fast and a faster subsystem, respectively. The periodic output feedback gain for the slow subsystem is then calculated from the output injection gain computed from the slow subsystem and the same for the fast subsystems are set equal to zero. Finally the periodic output feedback gain for the composite system is obtained using all the above mentioned periodic output feedback gains . The method has been applied to a large Pressurized Heavy Water Reactor (PHWR) for control of Xenon-induced spatial oscillations and the efficacy of control has been demonstrated by simulation of transient behaviour of the nonlinear model of the PHWR. Copyright ©2001 IFAC. Keywords: Nuclear Reactor , Space-time Kinetics, Singular perturbations and time-scale methods, Optimal Control.
1. INTRODUCTION
feedback concept. However, since late 80s new design trends, which would render better closed loop response characteristics, have emerged. Hence, there is a renewed interest in looking at the problem of spatial control.
In a large Pressurized Heavy Water Reactor, such as in the 500 MWe PHWR, the neutronic coupling among different core-regions is so week that any deliberate attempt to operate the reactor with flattened axial and radial flux distribution is also accompanied with the xenon-induced spatial oscillations, which require control. If the oscillations in the power distribution are left uncontrolled, the power density and the time rate of change of power at some locations in the core may exceed the respective limits which may cause fuel failure. In several heavy water reactors, operational in different parts of the world, means for automatic control of power distribution have been devised. Apparently, these control systems have been designed utilizing the usual static output
The design of automatic power distribution control systems for nuclear reactors is a complicated task because the representation of the spatial effects in a nuclear reactor requires a large number of equations and the model is characterised by the simultaneous presence of both the slow and fast varying modes. While applying the design methods, special care must be taken to overcome the stiffness and ill-£onditioning problems. In this paper, the singularly perturbed structure of the nuclear reactor model is exploited to decompose it into a slow subsystem, which is unstable, and 43
two fast subsystems which are stable. An output injection gain is then designed for the slow subsystem only and the output injection gains for the fast subsystems are set equal to zero. An ingenious method is suggested whereby the periodic output feedback gains for the original system are determined without any significant difficulty. It is shown that the periodic output feedback control yields a satisfactory closed loop response in the case of the 500 MWe PHWR. The proposed design method eliminates the problems of stiffness and ill-conditioning.
(4)
dX dt'
= /x
[~/lirPIi -
[AX
+ ~/2irP2iJ + A/li
+ (7axrP2d Xi ,
dHi
Tt = -mqi,
Pi =
Eell V; [~/lirPli
1
2. MODEL OF THE PHWR
lli = - - - ,
l2i
=
VliI:ali
h
dt
[1 +
+ (1 -
Wlii
+ ~1-2i]
~
. ah
(3) [v~ JlirPli
N
A. .
'l'h
+~
Wlij
A.
.
~ ~ . '1'1) j=l ah
+ v~ 12i4>2iJ
~ali
1
+~ACi'
(1)
L..ali
-
[
1+
(7ax --Xi
~a2i
i] + -K -Hi
~a2i
rP2i,
+ ~/2irP2iJ,
(7)
1
---, V2i~a2i
where N denotes the number of zones in the reactor, rPli, rP2i, C i , fi' Xi and P; denote respectively the fast group flux, thermal group flux , effective one group delayed neutron precursor concentration, iodine concentration, xenon concentration and power level in the zone i; V; denotes the volume of zone i; Eel I denotes the average energy liberated in a fission reaction; A, A/ and AX denote the decay constants respectively for the effective one group delayed neutron precursor, iodine and xenon ; v denotes the average neutron yield in thermal fission, (3 the delayed neutron fraction and // and / x the fission yields of iodine and xenon respectively; Vii and V2i denote respectively the average speed of fast and thermal neutrons in the reactor; Wlii, Wlij, W2 ii and W2ij are nodal coupling coefficients characterizing the leakages of fast and thermal neutron fluxes among zones; ~ali and ~a2i denote respectively the fast and thermal absorption crosssection, ~ lli and ~ 12i the fission crosssections, and ~1-2i the slowing down crosssection for fast neutrons; (7ax denotes the microscopic thermal neutron absorption crosssection of xenon; Hi denotes the water level in the zone controller, qi the signal to the inflow control valve of the ith zone, m the rate of change of water level in the zone controllers for unit change in the input signal to the control valve and Ki denotes the change in thermal group neutron absorption crossection per unit change in the water level(Tiwari et al., 1996; Tiwari et al., 2000).
Employing the nodal approximation of the two group diffusion equations, the following dynamic equations can be derived for the 500 MWe PHWR(Nene, 2001):
=_
(6)
(i=1 , 2, ... ,N)
The space-time kinetic behaviour in a nuclear reactor is usually described by multigroup diffusion equations(Duderstadt and Hamilton, 1983; Henry, 1976; Stammler and Abbate, 1983). However, in the case of control design problems, the state equations are obtained by suitable approximation of the neutron diffusion equations and associated equations describing the dynamics of delayed neutron precursors, for the treatment of the fast transients. In spatial control problems involving slow transients, equations describing the dynamics of iodine and xenon are also needed. For a realistic modeling, the thermal reactivity feedback should also be considered. However, this effect is very small in PHWRs and it may be ignored while dealing with the spatial control problem. Also a two group formulation of the diffusion equations with one effective group of delayed neutrons is usually sufficient.
l drPli
(5)
Accuracy of the model shall be in general better for large N. However, in case of the 500 MWe PHWR, satisfactory accuracy is obtained for N = 14. Thus there are 84 differential equations needed to describe the space-time behaviour of the reactor.
(2)
The above equations can be linearized around the steady state operating point to obtain the following standard state space representation: 44
Y=[MSMfMfll[~;Zjj ].
where All ... A33, B l of suitable dimensions,
,
lvh and
(11)
Note that the original state equation having three widely separated clusters of eigenvalues is decoupled into a slow subsystem (As> Bs) of order 56, having eigenvalues of very small magnitude, a fast subsystem (Aj, B j ) of order 14 having eigenvalues of large magnitude and a faster subsystem (Afj, BfI) of order 14 having eigenvalues of very large magnitude. Moreover, the fast and faster subsystems are stable and have eigenvalues with large negative real parts.
M3 are matrices
3. CONTROL OF THE PHWR
Z3 = [
Y oQ
=
OifJll ifJ1l0
[OPl PlO'"
= [Oql
For control of the 500 MWe PHWR, an observer based state feedback can be suggested. However, the observer based controller has generally very complex structure(Tiwari and Bandyopadhyay, 1998) . Moreover, even a small variation in the model parameters and the reactor parameters, may result into significant degradation of the closed loop performance. In contrast to this, the control of the reactor based on the periodic output feedback is simpler and the variation of the parameters does not pose much problem as only the feedback of outputs is required . A brief description of the technique and its extension to multi- time-scale systems is given in the following.
oifJlN ] T ifJlNo OPN]T PNO
. . . oq N IT ,
and ifJliO, ifJ2iO, CiQ, liO, X iO and PiO denote the steady state values of the respective quantities. It is worth to note that the states Zl consists of quantities varying slowly with time, Z2 consists of quantities varying rapidly and Z3 of those varying very rapidly. The state space model corresponding to this particular grouping of the states makes the application of time-scale methods, convenient(Kokotovic et ai., 1986). The eigenvalues of the system matrix in the state equation (8), are seen to lie in three widely spaced clusters indicating that the PHWR model possesses the simultaneous presence of both the fast and slow phenomena, which is known to cause illconditioning and severe computational problems in control design and simulation. The singular perturbation approach is particularly suited in such a case. A particular property of the standard singularly perturbed model of special interest is that the model given by (8) and (9), can be transformed into the following block diagonal form(Kokotovic et ai., 1986):
3. 1 Periodic Output Feedback It has been established by Chammas and Leondes (1979) that the poles of a controllable and observable system discretized at the output sampling rate can be arbitrarily assigned by a piecewise constant periodic output feedback , provided the number of changes in the gain during one output sampling interval is greater than or equal to the controllability index of the system. The approach can be briefly described as follows (Werner and Furuta, 1995).
Consider the linear time invariant model i= Az
y=Mz .
[~; lOQ,
(12) (13)
The discrete model corresponding to the above state equation, for sampling at the rate "*' called as the output sampling rate, is
is 1 [As 0 00 1 [ififl = 00 Af0 AfI +
+ Bu,
z[(k (10) where
BfI 45
+ 1)7] = ATz[kT] + B'Tu[kT], y[kT] = M z[kT],
(14) (15)
J
B.,. =
causes excessive variation of input during an output sampling interval though it meets the performance specifications at the output sampling instants. Hence , it is necessary to improve the closed loop behaviour of the system in some way. For this, the approach of Werner and Furuta (1995) is used. One begins by selecting a performance index which consists of a quadratic term in state, z , and a quadratic term in input, u , of the model given by (18) and a term that relaxes, to some extent , the condition that K L achieves the same right hand side as in (26). The consideration leads to a two point boundary value problem which is solved to obtain K L , that optimizes the closed loop system behaviour, during the output sampling intervals.
( 16)
.,.
eAt Bdt.
(17)
o Now , let the output sampling interval , T , be divided into Ne subintervals, 06. = !'c' Then , the model corresponding to the sampling at the rate will be
i
z[(l
+ l)o6.J = A~z[lo6.J + B~u [lo6.J, y[lo6.J = M z[lo6.],
(18) (19)
where A~ = eA~ ,
(20)
J
3.2 Periodic Output Feedback Control for Three Time Scale Systems
~
B~=
eAtBdt .
(21)
o
As the 500 MWe PHWR is a three time scale system , the attention is now turned to the determination of periodic output feedback gain sequence, K L , for system as one described by equations (10) and (11). Here we have to consider the state z , in (12) as composite state vector of slow, fast and
Further , from (16) and (20) A.,. A~c. Now let G be an output injection gain that yields the required performance for the system given by (14) and (15) , i.e. the eigenvalues of (A.,. + GM) are at the desired locations in the circle of unit radius. Then there exists a sequence of gains
JT
. T faster states , t.e., Z = [ ZsT ZjT zff . The task of finding KL from (26) for a G may seem trivial albeit the problems of ill- conditioning, usually encountered in case of multi-time scale systems. To overcome this difficulty, the approach in which all the slow, fast and faster phenomenon are separately considered is required. To begin with , consider the following discrete model for the system described by (10) corresponding to the sampling at the rate
(22) such that when the control input is applied according to the rule,
u(t)
= KLY[kTJ
(23)
i:
where,
kT + lo6. :::; t < kT + (I + 1)06. K/+N=K/ ,
and ,
ZS[(l+I)o6. J zj[(1 + 1)06.] [ zff[(l + l)o6.J
I=O,1 , .. . N e -1 ,
the closed loop system corresponding to (14) and (23) given by,
1
[l At, A~J
where
+
[~~~ 1 B~ff
U
(27)
and the output equation corresponding to the sampling at the rate ~
is stable. Assuming that the system given by (14) and (15) is observable , it is possible to determine the output injection gain, G , such that (A.,. +GM) has desired performance characteristics. Hence, a periodic output feedback gain sequence K L, determined from the relation
(28)
(26)
As all the modes are decoupled, it would be appropriate to consider the output injection matrix as,
would realize the output injection gain G. However, it is found that the time-varying gain J( L
(29) 46
where, G s , G I and Gft denote the output injection matrices for the slow , fast and faster subsystems, respectively. Applying this G to the discrete system corresponding to the sampling at the rate ~, the closed loop system matrix is obtained as
A.,.+GM=A~C +GM
1 .03
Global power vs. time in sec.
r-----'---~----------___,
1.02 1 .01
0.99
(30)
0 . 98
Since both the fast and faster subsystems are stable, the output injection gains for both of these subsystems may be set equal to zero i.e., G I = 0 and G I I = O. This consideration simplifies (30) to
0 . 970~-----1~0~0~----2~0-0~----~300
Fig. 1. Variation of normalized global power. Zone1 pO\Ner vs. time in sec .
, .05
Using (22) and (25), the equation (26) can also be written as
0 . 95~----~~----~2~0~0~---~3 ~00
Fig. 2. Variation of normalized power in zone-I. the system given by (18) exhibits the three timescale behaviour while N e is larger than or equal to the controllability index of the system . Also the value of T so selected should not cause loss of controllability and observability, which might happen due to sampling(Kuo, 1992) . The fastest variable amongst the slow variables is the delayed neutron precursor concentration. The numerically largest eigenvalue of As was found to be 8.2138 x 10- 2 rad/ sec. In accordance with the considerations discussed above, the output sampling interval , T , is chosen as 5 s. Further, let Ne = 5 which yields ~ = T / Ne = 1 s for which the discrete system (18) also possesses 3 time scale behaviour. The output injection gain matrix Go which stabilizes the slow subsystem was obtained by the optimal control technique and the corresponding periodic output feedback gains, K L , by the optimization method of Werner and Furuta (1995). It is seen that the periodic output feedback gains K L determined using the proposed approximations also stabilizes the composite system .
(32)
If sampling period, T, and the number of gain changes, Ne, are chosen appropriately, the discrete system (18) would also possess three time scale property(Naidu and Rao, 1985) , i.e., the eigenvalues of A Cl , and A Cl " will be very small. Hence , dA A2 ANc-l A Cl" A 2Cl,," ', ANc-l Cl, an Cl", Cl" , "' , Cl" will be so small that they can be neglected in (32) to obtain the simplified equation
(33)
which can be solved for Ko, K 1 , . .. , K Nc- 2 and set KNc-l = O. It should be observed that (33) involves only the slow subsystem mat ices.
To demonstrate the efficacy of the controller when placed in closed loop with the reactor, the behaviour of the system may be studied from simulation of nonlinear equations (1) - (6), for transient variation of reactivity which was simulated by perturbing the thermal neutron absorption cross section in zone 1, L a 21' The perturbation consisted of increasing La21 linearly by 1.4% in 2.5 s and then reducing it back to original value in the next 2.5 s. The simulation was carried out using SIMULINK and the ode15s solver (for stiff equations) was used for computations. The variation of global
4. RESULTS The proposed method is applied for designing the piecewise constant periodic output feedback control for the 500 MWe PHWR. The output sampling period , T, and the input sampling period, ~, should be chosen in such a way that
47
sampling period, T is also large i.e. sampling rate is not required to be very high. This renders a considerable reduction in the cost of associated processing hardware.
power and power level , xenon concentration and iodine concentration in zone-I, is shown in Fig. 1 to 4. It is seen that the global power settles quickly following the transient. The zonal power levels also settle to their respective equilibrium values in about 200 s. The changes in xenon and iodine concentrations is also small although it takes very large time for these variables to settle at the respective equilibrium values . Such a response is considered satisfactory for the operation of the 500 MWe PHWR.
Here, the attention is focussed on controlling the xenon-induced spatial oscillations in a large PHWR though the method can be directly applied to other types of thermal reactors which are described by the nodal model. The method for computing the periodic output feedback gains can be applied directly to other large scale systems.
Zone1 xenon vs. time in sec.
1 .0001
6. REFERENCES
10~--------~5~----------1~0----------~ 15 X
10
4
Fig. 3. Variation of normalized xenon concentration in zone-I .
;-----~ . -.--;-----~-----;;;-----!" .1 0 '
Fig. 4. Variation of normalized iodine concentration in zone-I.
5. CONCLUSION It is well known that models of large scale physical systems are characterised by the presence of fast-decaying dynamic modes which generally do not require control. This usual property is exploited here and an approach for designing piecewise constant periodic output feedback control for multi-time-scale continuous-time systems is formulated . The method can also be applied to systems which are directly represented by standard singularly perturbed discrete-time model. It would be only necessary to transform the given model into the block diagonal form , in which fast subsystems are decoupled. Because ~ is permitted to be several times large compared to the largest time constant of the fast subsystem, speed of actuators would generally not be a limit in achieving the desired performance. Moreover, the output 48
Chammas, A . B . and C . T . Leondes (1979) . Pole assignment by piecewise constant output feedback. International Journal of Control 29, 31- 38 . Duderstadt , J a mes J . and Louis J . Hamilton (1983). Nuclear Reactor Analysis. John Wiley & Sons Inc .. New York. Henry, A . F . (1976) . Nuclear Reactor Analysis. The MIT Press. Cambridge . Kokotovic , P. V., H. K. Khalil and John O 'Reilly (1986) . Singular P erturbation Methods in Control A nalysis and Design. Academic Press. New York . Kuo, B . C . (1992) . Digital Control Systems. Saunders College Pub!.. Ft. Worth . Naidu , D. S. and A. K. Rao (1985) . Lecture Notes in Mathematics: Singular Perturbation Analysis of Discrete Control Systems. pp. 5062. Vol. 1154. Springer- Verlag. Berlin Heidelberg. Nene, Chandan R. (2001). Modeling and control of a la rge nuclear reactor. M. Tech Dissertation . Indian Institute of Technology Bombay, India. Stammler, R. J. J . and M. J . Abbate (1983) . Methods of Steady State Reactor Physics. Academic Press. Tiwari, A . P . and B. Bandyopadhyay (1998) . Control of xenon induced spatial oscillations in a large PHWR. In: Proceedings of the IEEE Region 10 Conference on Global Connectivity in Energy, Computer Communication and Control (TENCON- 98). New Delhi, India. Tiwari, A. P., B . Bandyopadhyay and G. Govindarajan (1996) . Spatial control of a large pressurized heavy water reactor. IEEE Transactions on Nuclear Science 43, 2440-2453. Tiwari, A . P. , B . Bandyopadhyay and H. Werner (2000). Spatial control of a large PHWR by piecewise constant periodic output feedback. IEEE Transactions on Nuclear Science 47, 389-402. Werner, H. and K. Furuta (1995). Simultaneous stabilization based on output measurement. Kybernetika 31(4),395-411.
PERIODIC OUTPUT FEEDBACK CONTROL OF A LARGE NUCLEAR REACTOR Chandan Nene' B. Bandyopadhyay' A. P. Tiwari ••
• Systems and Control Engineering Group, Department of Electrical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400 076, INDIA . •• Reactor Control Division, Bhabha Atomic Research Centre, Trombay, Mumbai - 400 085, INDIA.
Abstract: The paper presents the design of piece-wise constant periodic output feedback control for a discrete three time-scale system resulting from the discretization of a continuous-time standard three time-scale system. By a suitable linear transformation of state variables the given continuous-time system model is converted into a block diagonal form in which the three time scale system is decoupled into a slow , a fast and a faster subsystem, respectively. The periodic output feedback gain for the slow subsystem is then calculated from the output injection gain computed from the slow subsystem and the same for the fast subsystems are set equal to zero. Finally the periodic output feedback gain for the composite system is obtained using all the above mentioned periodic output feedback gains . The method has been applied to a large Pressurized Heavy Water Reactor (PHWR) for control of Xenon-induced spatial oscillations and the efficacy of control has been demonstrated by simulation of transient behaviour of the nonlinear model of the PHWR. Copyright ©2001 IFAC. Keywords: Nuclear Reactor , Space-time Kinetics, Singular perturbations and time-scale methods, Optimal Control.
1. INTRODUCTION
feedback concept. However, since late 80s new design trends, which would render better closed loop response characteristics, have emerged. Hence, there is a renewed interest in looking at the problem of spatial control.
In a large Pressurized Heavy Water Reactor, such as in the 500 MWe PHWR, the neutronic coupling among different core-regions is so week that any deliberate attempt to operate the reactor with flattened axial and radial flux distribution is also accompanied with the xenon-induced spatial oscillations, which require control. If the oscillations in the power distribution are left uncontrolled, the power density and the time rate of change of power at some locations in the core may exceed the respective limits which may cause fuel failure. In several heavy water reactors, operational in different parts of the world, means for automatic control of power distribution have been devised. Apparently, these control systems have been designed utilizing the usual static output
The design of automatic power distribution control systems for nuclear reactors is a complicated task because the representation of the spatial effects in a nuclear reactor requires a large number of equations and the model is characterised by the simultaneous presence of both the slow and fast varying modes. While applying the design methods, special care must be taken to overcome the stiffness and ill-£onditioning problems. In this paper, the singularly perturbed structure of the nuclear reactor model is exploited to decompose it into a slow subsystem, which is unstable, and 43
two fast subsystems which are stable. An output injection gain is then designed for the slow subsystem only and the output injection gains for the fast subsystems are set equal to zero. An ingenious method is suggested whereby the periodic output feedback gains for the original system are determined without any significant difficulty. It is shown that the periodic output feedback control yields a satisfactory closed loop response in the case of the 500 MWe PHWR. The proposed design method eliminates the problems of stiffness and ill-conditioning.
(4)
dX dt'
= /x
[~/lirPIi -
[AX
+ ~/2irP2iJ + A/li
+ (7axrP2d Xi ,
dHi
Tt = -mqi,
Pi =
Eell V; [~/lirPli
1
2. MODEL OF THE PHWR
lli = - - - ,
l2i
=
VliI:ali
h
dt
[1 +
+ (1 -
Wlii
+ ~1-2i]
~
. ah
(3) [v~ JlirPli
N
A. .
'l'h
+~
Wlij
A.
.
~ ~ . '1'1) j=l ah
+ v~ 12i4>2iJ
~ali
1
+~ACi'
(1)
L..ali
-
[
1+
(7ax --Xi
~a2i
i] + -K -Hi
~a2i
rP2i,
+ ~/2irP2iJ,
(7)
1
---, V2i~a2i
where N denotes the number of zones in the reactor, rPli, rP2i, C i , fi' Xi and P; denote respectively the fast group flux, thermal group flux , effective one group delayed neutron precursor concentration, iodine concentration, xenon concentration and power level in the zone i; V; denotes the volume of zone i; Eel I denotes the average energy liberated in a fission reaction; A, A/ and AX denote the decay constants respectively for the effective one group delayed neutron precursor, iodine and xenon ; v denotes the average neutron yield in thermal fission, (3 the delayed neutron fraction and // and / x the fission yields of iodine and xenon respectively; Vii and V2i denote respectively the average speed of fast and thermal neutrons in the reactor; Wlii, Wlij, W2 ii and W2ij are nodal coupling coefficients characterizing the leakages of fast and thermal neutron fluxes among zones; ~ali and ~a2i denote respectively the fast and thermal absorption crosssection, ~ lli and ~ 12i the fission crosssections, and ~1-2i the slowing down crosssection for fast neutrons; (7ax denotes the microscopic thermal neutron absorption crosssection of xenon; Hi denotes the water level in the zone controller, qi the signal to the inflow control valve of the ith zone, m the rate of change of water level in the zone controllers for unit change in the input signal to the control valve and Ki denotes the change in thermal group neutron absorption crossection per unit change in the water level(Tiwari et al., 1996; Tiwari et al., 2000).
Employing the nodal approximation of the two group diffusion equations, the following dynamic equations can be derived for the 500 MWe PHWR(Nene, 2001):
=_
(6)
(i=1 , 2, ... ,N)
The space-time kinetic behaviour in a nuclear reactor is usually described by multigroup diffusion equations(Duderstadt and Hamilton, 1983; Henry, 1976; Stammler and Abbate, 1983). However, in the case of control design problems, the state equations are obtained by suitable approximation of the neutron diffusion equations and associated equations describing the dynamics of delayed neutron precursors, for the treatment of the fast transients. In spatial control problems involving slow transients, equations describing the dynamics of iodine and xenon are also needed. For a realistic modeling, the thermal reactivity feedback should also be considered. However, this effect is very small in PHWRs and it may be ignored while dealing with the spatial control problem. Also a two group formulation of the diffusion equations with one effective group of delayed neutrons is usually sufficient.
l drPli
(5)
Accuracy of the model shall be in general better for large N. However, in case of the 500 MWe PHWR, satisfactory accuracy is obtained for N = 14. Thus there are 84 differential equations needed to describe the space-time behaviour of the reactor.
(2)
The above equations can be linearized around the steady state operating point to obtain the following standard state space representation: 44
Y=[MSMfMfll[~;Zjj ].
where All ... A33, B l of suitable dimensions,
,
lvh and
(11)
Note that the original state equation having three widely separated clusters of eigenvalues is decoupled into a slow subsystem (As> Bs) of order 56, having eigenvalues of very small magnitude, a fast subsystem (Aj, B j ) of order 14 having eigenvalues of large magnitude and a faster subsystem (Afj, BfI) of order 14 having eigenvalues of very large magnitude. Moreover, the fast and faster subsystems are stable and have eigenvalues with large negative real parts.
M3 are matrices
3. CONTROL OF THE PHWR
Z3 = [
Y oQ
=
OifJll ifJ1l0
[OPl PlO'"
= [Oql
For control of the 500 MWe PHWR, an observer based state feedback can be suggested. However, the observer based controller has generally very complex structure(Tiwari and Bandyopadhyay, 1998) . Moreover, even a small variation in the model parameters and the reactor parameters, may result into significant degradation of the closed loop performance. In contrast to this, the control of the reactor based on the periodic output feedback is simpler and the variation of the parameters does not pose much problem as only the feedback of outputs is required . A brief description of the technique and its extension to multi- time-scale systems is given in the following.
oifJlN ] T ifJlNo OPN]T PNO
. . . oq N IT ,
and ifJliO, ifJ2iO, CiQ, liO, X iO and PiO denote the steady state values of the respective quantities. It is worth to note that the states Zl consists of quantities varying slowly with time, Z2 consists of quantities varying rapidly and Z3 of those varying very rapidly. The state space model corresponding to this particular grouping of the states makes the application of time-scale methods, convenient(Kokotovic et ai., 1986). The eigenvalues of the system matrix in the state equation (8), are seen to lie in three widely spaced clusters indicating that the PHWR model possesses the simultaneous presence of both the fast and slow phenomena, which is known to cause illconditioning and severe computational problems in control design and simulation. The singular perturbation approach is particularly suited in such a case. A particular property of the standard singularly perturbed model of special interest is that the model given by (8) and (9), can be transformed into the following block diagonal form(Kokotovic et ai., 1986):
3. 1 Periodic Output Feedback It has been established by Chammas and Leondes (1979) that the poles of a controllable and observable system discretized at the output sampling rate can be arbitrarily assigned by a piecewise constant periodic output feedback , provided the number of changes in the gain during one output sampling interval is greater than or equal to the controllability index of the system. The approach can be briefly described as follows (Werner and Furuta, 1995).
Consider the linear time invariant model i= Az
y=Mz .
[~; lOQ,
(12) (13)
The discrete model corresponding to the above state equation, for sampling at the rate "*' called as the output sampling rate, is
is 1 [As 0 00 1 [ififl = 00 Af0 AfI +
+ Bu,
z[(k (10) where
BfI 45
+ 1)7] = ATz[kT] + B'Tu[kT], y[kT] = M z[kT],
(14) (15)
J
B.,. =
causes excessive variation of input during an output sampling interval though it meets the performance specifications at the output sampling instants. Hence , it is necessary to improve the closed loop behaviour of the system in some way. For this, the approach of Werner and Furuta (1995) is used. One begins by selecting a performance index which consists of a quadratic term in state, z , and a quadratic term in input, u , of the model given by (18) and a term that relaxes, to some extent , the condition that K L achieves the same right hand side as in (26). The consideration leads to a two point boundary value problem which is solved to obtain K L , that optimizes the closed loop system behaviour, during the output sampling intervals.
( 16)
.,.
eAt Bdt.
(17)
o Now , let the output sampling interval , T , be divided into Ne subintervals, 06. = !'c' Then , the model corresponding to the sampling at the rate will be
i
z[(l
+ l)o6.J = A~z[lo6.J + B~u [lo6.J, y[lo6.J = M z[lo6.],
(18) (19)
where A~ = eA~ ,
(20)
J
3.2 Periodic Output Feedback Control for Three Time Scale Systems
~
B~=
eAtBdt .
(21)
o
As the 500 MWe PHWR is a three time scale system , the attention is now turned to the determination of periodic output feedback gain sequence, K L , for system as one described by equations (10) and (11). Here we have to consider the state z , in (12) as composite state vector of slow, fast and
Further , from (16) and (20) A.,. A~c. Now let G be an output injection gain that yields the required performance for the system given by (14) and (15) , i.e. the eigenvalues of (A.,. + GM) are at the desired locations in the circle of unit radius. Then there exists a sequence of gains
JT
. T faster states , t.e., Z = [ ZsT ZjT zff . The task of finding KL from (26) for a G may seem trivial albeit the problems of ill- conditioning, usually encountered in case of multi-time scale systems. To overcome this difficulty, the approach in which all the slow, fast and faster phenomenon are separately considered is required. To begin with , consider the following discrete model for the system described by (10) corresponding to the sampling at the rate
(22) such that when the control input is applied according to the rule,
u(t)
= KLY[kTJ
(23)
i:
where,
kT + lo6. :::; t < kT + (I + 1)06. K/+N=K/ ,
and ,
ZS[(l+I)o6. J zj[(1 + 1)06.] [ zff[(l + l)o6.J
I=O,1 , .. . N e -1 ,
the closed loop system corresponding to (14) and (23) given by,
1
[l At, A~J
where
+
[~~~ 1 B~ff
U
(27)
and the output equation corresponding to the sampling at the rate ~
is stable. Assuming that the system given by (14) and (15) is observable , it is possible to determine the output injection gain, G , such that (A.,. +GM) has desired performance characteristics. Hence, a periodic output feedback gain sequence K L, determined from the relation
(28)
(26)
As all the modes are decoupled, it would be appropriate to consider the output injection matrix as,
would realize the output injection gain G. However, it is found that the time-varying gain J( L
(29) 46
where, G s , G I and Gft denote the output injection matrices for the slow , fast and faster subsystems, respectively. Applying this G to the discrete system corresponding to the sampling at the rate ~, the closed loop system matrix is obtained as
A.,.+GM=A~C +GM
1 .03
Global power vs. time in sec.
r-----'---~----------___,
1.02 1 .01
0.99
(30)
0 . 98
Since both the fast and faster subsystems are stable, the output injection gains for both of these subsystems may be set equal to zero i.e., G I = 0 and G I I = O. This consideration simplifies (30) to
0 . 970~-----1~0~0~----2~0-0~----~300
Fig. 1. Variation of normalized global power. Zone1 pO\Ner vs. time in sec .
, .05
Using (22) and (25), the equation (26) can also be written as
0 . 95~----~~----~2~0~0~---~3 ~00
Fig. 2. Variation of normalized power in zone-I. the system given by (18) exhibits the three timescale behaviour while N e is larger than or equal to the controllability index of the system . Also the value of T so selected should not cause loss of controllability and observability, which might happen due to sampling(Kuo, 1992) . The fastest variable amongst the slow variables is the delayed neutron precursor concentration. The numerically largest eigenvalue of As was found to be 8.2138 x 10- 2 rad/ sec. In accordance with the considerations discussed above, the output sampling interval , T , is chosen as 5 s. Further, let Ne = 5 which yields ~ = T / Ne = 1 s for which the discrete system (18) also possesses 3 time scale behaviour. The output injection gain matrix Go which stabilizes the slow subsystem was obtained by the optimal control technique and the corresponding periodic output feedback gains, K L , by the optimization method of Werner and Furuta (1995). It is seen that the periodic output feedback gains K L determined using the proposed approximations also stabilizes the composite system .
(32)
If sampling period, T, and the number of gain changes, Ne, are chosen appropriately, the discrete system (18) would also possess three time scale property(Naidu and Rao, 1985) , i.e., the eigenvalues of A Cl , and A Cl " will be very small. Hence , dA A2 ANc-l A Cl" A 2Cl,," ', ANc-l Cl, an Cl", Cl" , "' , Cl" will be so small that they can be neglected in (32) to obtain the simplified equation
(33)
which can be solved for Ko, K 1 , . .. , K Nc- 2 and set KNc-l = O. It should be observed that (33) involves only the slow subsystem mat ices.
To demonstrate the efficacy of the controller when placed in closed loop with the reactor, the behaviour of the system may be studied from simulation of nonlinear equations (1) - (6), for transient variation of reactivity which was simulated by perturbing the thermal neutron absorption cross section in zone 1, L a 21' The perturbation consisted of increasing La21 linearly by 1.4% in 2.5 s and then reducing it back to original value in the next 2.5 s. The simulation was carried out using SIMULINK and the ode15s solver (for stiff equations) was used for computations. The variation of global
4. RESULTS The proposed method is applied for designing the piecewise constant periodic output feedback control for the 500 MWe PHWR. The output sampling period , T, and the input sampling period, ~, should be chosen in such a way that
47
sampling period, T is also large i.e. sampling rate is not required to be very high. This renders a considerable reduction in the cost of associated processing hardware.
power and power level , xenon concentration and iodine concentration in zone-I, is shown in Fig. 1 to 4. It is seen that the global power settles quickly following the transient. The zonal power levels also settle to their respective equilibrium values in about 200 s. The changes in xenon and iodine concentrations is also small although it takes very large time for these variables to settle at the respective equilibrium values . Such a response is considered satisfactory for the operation of the 500 MWe PHWR.
Here, the attention is focussed on controlling the xenon-induced spatial oscillations in a large PHWR though the method can be directly applied to other types of thermal reactors which are described by the nodal model. The method for computing the periodic output feedback gains can be applied directly to other large scale systems.
Zone1 xenon vs. time in sec.
1 .0001
6. REFERENCES
10~--------~5~----------1~0----------~ 15 X
10
4
Fig. 3. Variation of normalized xenon concentration in zone-I .
;-----~ . -.--;-----~-----;;;-----!" .1 0 '
Fig. 4. Variation of normalized iodine concentration in zone-I.
5. CONCLUSION It is well known that models of large scale physical systems are characterised by the presence of fast-decaying dynamic modes which generally do not require control. This usual property is exploited here and an approach for designing piecewise constant periodic output feedback control for multi-time-scale continuous-time systems is formulated . The method can also be applied to systems which are directly represented by standard singularly perturbed discrete-time model. It would be only necessary to transform the given model into the block diagonal form , in which fast subsystems are decoupled. Because ~ is permitted to be several times large compared to the largest time constant of the fast subsystem, speed of actuators would generally not be a limit in achieving the desired performance. Moreover, the output 48
Chammas, A . B . and C . T . Leondes (1979) . Pole assignment by piecewise constant output feedback. International Journal of Control 29, 31- 38 . Duderstadt , J a mes J . and Louis J . Hamilton (1983). Nuclear Reactor Analysis. John Wiley & Sons Inc .. New York. Henry, A . F . (1976) . Nuclear Reactor Analysis. The MIT Press. Cambridge . Kokotovic , P. V., H. K. Khalil and John O 'Reilly (1986) . Singular P erturbation Methods in Control A nalysis and Design. Academic Press. New York . Kuo, B . C . (1992) . Digital Control Systems. Saunders College Pub!.. Ft. Worth . Naidu , D. S. and A. K. Rao (1985) . Lecture Notes in Mathematics: Singular Perturbation Analysis of Discrete Control Systems. pp. 5062. Vol. 1154. Springer- Verlag. Berlin Heidelberg. Nene, Chandan R. (2001). Modeling and control of a la rge nuclear reactor. M. Tech Dissertation . Indian Institute of Technology Bombay, India. Stammler, R. J. J . and M. J . Abbate (1983) . Methods of Steady State Reactor Physics. Academic Press. Tiwari, A . P . and B. Bandyopadhyay (1998) . Control of xenon induced spatial oscillations in a large PHWR. In: Proceedings of the IEEE Region 10 Conference on Global Connectivity in Energy, Computer Communication and Control (TENCON- 98). New Delhi, India. Tiwari, A. P., B . Bandyopadhyay and G. Govindarajan (1996) . Spatial control of a large pressurized heavy water reactor. IEEE Transactions on Nuclear Science 43, 2440-2453. Tiwari, A . P. , B . Bandyopadhyay and H. Werner (2000). Spatial control of a large PHWR by piecewise constant periodic output feedback. IEEE Transactions on Nuclear Science 47, 389-402. Werner, H. and K. Furuta (1995). Simultaneous stabilization based on output measurement. Kybernetika 31(4),395-411.