IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 63, NO. 5, OCTOBER 2016
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Reactivity Estimation Based on an Extended State
Observer of Neutron Kinetics
Zhe Dong, Xiaojin Huang, Duo Li, and Zuoyi Zhang
Abstract— Reactivity is a key physical parameter that directly
reflects the balance between neutron generation and consumption
in every nuclear fission reactor. In this paper, by regarding
the reactivity as an extended state-variable of point kinetics,
a novel extended state-observer (ESO) is proposed. Theoretic
analysis shows that this newly-built ESO provides globally
asymptotically bounded estimation for the reactor neutron flux,
concentrations of delayed neutron precursors and reactivity.
Numerical simulation results illustrate the influence of the
observer parameter to its performance, and show that this ESO
can provide better reactivity estimation than the classical inverse
point kinetics (IPK) method. The ESO has a simple form, and
has only one parameter to be tuned online, which can induce an
easy engineering implementation.
Index Terms— Extended state-observer, reactivity estimation.
I. I NTRODUCTION
R
EACTIVITY is a key physical parameter of every
nuclear fission reactor which directly reflects the balance between neutron generation and consumption. Reactivity estimation is crucial for the reactor monitoring and
measurement of control rod worth. There are mainly three
methods for reactivity estimation, i.e. the period, kinetics and
observer based methods. For the period method, the reactivity
is calculated from the in-hour equation by the use of the
measurement of stable or asymptotic reactor period, which
only works for positive periods [1]. For the kinetics method,
reactivity estimation can be made continuously by solving the
inverse point kinetics (IPK) equation, i.e. a differential-integral
equation containing both differentiation of the neutron flux and
integration of the concentrations of the delayed neutron precursors. Some analog [2] and digital [3]–[7] reactivity meters
based upon the IPK method were developed and implemented
practically. In order to improve the precision of numerical
differentiation and integration in solving the IPK equation,
the Euler-Maclaurin formula [8] and Lagrange method [9] are
introduced for higher performance of numerical differentiation
and integration respectively. Actually, the discretization of the
IPK equation can be realized in not only the time domain
Manuscript received June 28, 2016; revised August 4, 2016; accepted
August 8, 2016. Date of publication August 16, 2016; date of current version
October 11, 2016. This work was supported in part by the National S&T
Major Project, and Natural Science Foundation of China (NSFC) under
Grant 61374045.
The authors are with the Institute of Nuclear and New Energy Technology, Collaborative Innovation Centre of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of
Ministry of Education, Tsinghua University, Beijing 100084, China (e-mail:
dongzhe@mail.tsinghua.edu.cn).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNS.2016.2600648
but also the frequency domain. Based on the relationship
between Laplace transform and Z -transform, a finite impulse
response (FIR) filter was proposed for solving IPK equation [10], and the parameters of this FIR filter can be optimized
by the least square (LS) method for the reduction of nuclear
power fluctuation [11]. Because differentiation may enlarge
the negative influence of measurement noise, Shimazu gave a
simple reactivity estimation approach through omitting the differentiation part of IPK equation, which induces a good noise
filtering feature [12]. Some nonlinear techniques such as the
sliding mode differentiator [13] were applied to improve the
differentiation estimation in the IPK equation [14]. Observerbased method is also an important reactivity estimation method
developed since 1990s. Through regarding the reactivity as a
constant parameter, Raćz transferred the reactivity measurement problem into a typical parameter estimation problem, and
then realized the estimation based upon Kalman filter [15].
By considering the nonlinearity in the point kinetics (PK)
equation, the extended Kalman filter (EKF) is then applied for
reactivity estimation [16], [17]. The qualitative performance
comparison between the EKF and IPK methods [18] shows
that the former approach is better than the latter one. Some
other observation techniques were also given for reactivity
estimation, e.g. Wang, Aldemir and Utkin gave a sliding mode
observer based method [19].
The extended state-observer (ESO) proposed by Han [20]
is a crucial technique for estimating system uncertainty and
realizing active disturbance rejection control (ADRC), which
has been widely utilized in practical engineering [21]. Reactivity is affected by many properties of a reactor core such as
the composition, geometry, temperature, pressure and ability
of producing fission neutrons, which is too difficult to be
modelled and can be seen as an uncertainty. Based on the
idea of ESO, it is reasonable to regard the reactivity as an
extended state variable, and design a proper ESO to estimate
it. However, the classical ESO [20], [21] is only suitable for
the dynamic systems in Brunovsky form, and is not applicable
for reactivity estimation due to the form of PK equation is not
canonical. Motivated by this, a novel extended state-observer is
proposed for point neutron kinetics, which is then applied for
the reactivity estimation. This ESO guarantees globally asymptotically bounded estimation for the neutron flux, precursor
concentrations and reactivity. This ESO also has a simple
form which leads to easy practical implementation. Finally,
numerical simulation results not only illustrate the relationship
between observer parameter and its performance but also show
that this newly-built ESO is stronger than classical IPK method
in the noisy environment.
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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 63, NO. 5, OCTOBER 2016
II. E XTENDED S TATE -O BSERVER OF N EUTRON K INETICS
In this section, a new extended state-observer of neutron
kinetics is proposed for estimating the total reactivity, and
theoretical performance analysis is also given.
A. Extended State-Space Model of Neutron Kinetics
The point reactor kinetics (PRK) model gives a satisfactory
description for the time-domain behavior of a nuclear reactor,
which can be expressed as [22], [23]:
⎧
6
⎪
⎪
βi
ρ−β
⎨
ṅr =
nr +
Cr,i ,
(1)
i=1
⎪
⎪
⎩Ċ = λ n − C ,
i = 1, · · · , 6,
r,i
i
r
r,i
where n r is the relative nuclear power, cr,i , βi and λi are
respectively the relative concentration, fraction and decay
constant of the i th group delayed neutron precursors, β =
β1 + β2 + . . . + β6 is the total fraction of delayed neutron
precursors, is the effective prompt neutron lifetime, ρ is the
reactivity. Here, the relative values means normalized values.
For an instance, n r = n/n 0 , where n is the neutron flux, and
n 0 is the neutron flux at full power-level.
It is not loss of generality to assume that
ρ̇ = G ρ (t) ,
(2)
where
ei = x̂ i − x i , i = 1, · · · , 8,
positive constants ri (i = 1, 2) are the observer gains,
and scalar ε is called the perturbation constant satisfying 0 < ε < 1. It is assume here that x 8 and ei are all bounded.
Then, the observation error e =[e1 , . . . , e8 ]T converges globally asymptotically to a bounded set if
r 1 > ε + c1 ,
and
r2 > max
x = x1
···
x8
T
= nr
Cr,1
···
Cr,6
ρ
T
where only x 1 can be obtained through measurement directly,
i.e. system output y of (5) is given by
y = x1 ,
V (e) =
(6)
and it is clear that y > 0.
B. Extended State Observer Design
The following theorem, which is the main result of this
paper, gives a globally bounded state-observer for system (5).
Theorem. Consider system (5) with its state-observer taking
the form as
⎧
6
⎪
y x̂ 8
r1
βi ⎪
⎪
˙1 = −
⎪
x̂
x̂ 1 − x̂ i+1 +
−
ye1 ,
⎪
⎨
ε
i=1
(7)
⎪
x̂˙i+1 = λi x̂ 1 − x̂ i+1 − 2λi e1 , i = 1, · · · , 6,
⎪
⎪
⎪˙
r
⎪
⎩x̂ 8 = − 2 ye1 ,
ε
(10)
6
r 2 − r 1 βi 2
1 T
ē P ē +
e ,
2
2ε
λi i+1
(12)
i=1
where
T
ē = e1 e8 ,
−1
ε (r2 − r1 ) −1
P=
.
−1
1
, (4)
the corresponding extended state-space model given
by (1) and (2) can be rewritten as
⎧
6
⎪
βi
x1 x8
⎪
⎪
⎪
ẋ
,
=
−
(x 1 − x i+1 ) +
1
⎨
i=1
(5)
⎪
ẋ i+1 = λi (x 1 − x i+1 ) ,
i = 1, · · · , 6,
⎪
⎪
⎪
⎩
ẋ 8 = G ρ ,
r1 + 1 ,
Define function V of observation error e as
(3)
The meaning of equation (2) is that total reactivity ρ can
be viewed as an extended state-variable.
Then, by defining
c2 ε + r12
r1 − ε
(9)
where ci (i = 1, 2) are arbitrarily given constants.
Proof: From system model (5) and extended stateobserver (7), the dynamics of state-observation error can be
written as
⎧
6
⎪
r1
βi
ye8
⎪
⎪
⎪
ė1 = −
−
ye1 ,
(e1 − ei+1 ) +
⎪
⎨
ε
i=1
(11)
⎪
ė1+i = −λi (e1 + ei+1 ) , i = 1, · · · , 6,
⎪
⎪
⎪
r
⎪
⎩ė8 = − 2 ye1 − G ρ .
ε
where function G 1 is norm-bounded, i.e.
Gρ ≤ hρ .
(8)
(13)
(14)
From inequality (10) and the property of 0 < ε < 1,
matrix P is strict positive definite, which means that function V given by equation (12) is a proper Lyapunov function
for observation error dynamics (11).
Differentiate function V along the trajectory given by (11),
1 V̇ (e) = ēT AT P + PA ē + ēT Pϕ e, G ρ
2
6
r2 − r1 βi ei+1 (e1 + ei+1 ),
−
(15)
ε
i=1
where
y −ε−1r1 1
,
−ε−1r2 0
T
6
βi
ϕ e, G ρ = −
.
(e1 − ei+1 ) G ρ
A=
(16)
(17)
i=1
Based on equations (14) and (16) as well as inequality (10),
it can be computed that
2
y (r1 −ε)r2 2 −r1 0
y cε2 0
1 T
ε
A P + PA = −
,
<−
2
0 1
0
1
(18)
DONG et al.: REACTIVITY ESTIMATION BASED ON AN EXTENDED STATE OBSERVER OF NEUTRON KINETICS
Fig. 1.
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Simulation results in case A1.
TABLE I
PARAMETERS OF THE P OINT K INETICS
and
ēT Pϕ = −
+
r2 − r1
ε
6
βi e1 (e1 − ei+1 )
i=1
6
e8 βi (e1 −ei+1 ) + G ρ (e1 − e8 ) .
(19)
i=1
Substitute (18) and (19) to equation (15),
6
y
r2 − r1 c2 y + (r2 − r1 ) β 2
2
e1 − e82 −
βi ei+1
ε
ε
i=1
+ θ e, G ρ ,
(20)
V̇ (e) < −
where
6
e8 βi (e1 − ei+1 ).
θ e, G ρ = G ρ (e1 − e8 ) +
(21)
i=1
From (3), it is reasonable to suppose that there exists a large
enough positive constant M satisfying inequality
(22)
θ e, G ρ ≤ M.
Remark 1. It is easy to see from (23) that when the
perturbation constant ε becomes smaller, the final observation
error is smaller. However, from observer expression (7), a
smaller ε induces a higher sensitivity to the noises. On the
Then, based upon (20) and (22), observation error e globally other hand, although a larger ε induces higher observation
asymptotically converges to the bounded set given by
error, it reduces the sensitivity to the noises. Thus, εshould be
carefully
tuned practically.
6
βi 2
.
= e ∈ R8
e
≤
Mε
c2 y + 1 e12 + εye82 +
Remark
2. From (7), there is no online iterative calculation
i+1
i=1
of the estimation error covariance matrices as that done in the
(23) EKF method, and there is no need to know the statistical characteristics of the noise. Since observer (7) can be implemented
This completes the proof of this theorem.
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Fig. 2.
IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 63, NO. 5, OCTOBER 2016
Simulation results in case A2.
on the widely used digital platforms conveniently, and since
the only one parameter to be tuned online is perturbation constant ε. Thus, the ESO-based reactivity estimation approach is
suitable for practical engineering implementations.
III. S IMULATION S TUDY W ITH D ISCUSSIONS
The ESO-based reactivity estimation approach proposed
in section II is verified in this section through numerical
simulation. The influence of perturbation constant ε to the
estimation performance is first shown, and then the ESO-based
method is compared to the classical IPK method in a noisy
environment.
A. Numerical Simulation Results
This numerical simulation is performed in Matlab/Simulink
platform. The point kinetics with six groups of delayed neutron precursors is adopted, whose parameters are all given
in Table I. These values come from the nuclear heating
reactor (NHR) [24], [25]. In this simulation, the influence of
perturbation constant ε is studied, and then the performance
of the ESO-based method is compared to that of the classical
IPK method. In the simulation, observer gains ri (i = 1, 2)
are chosen to be r1 = 1.0 and r2 = 5.0 so that (9) and (10)
are satisfied.
Case A (Reactivity Step without Noise):
The case study of reactivity step increase and decrease in
2% full power (FP) without measurement noise are performed.
Fig. 3.
Measurement noise in case B.
A1: The neutron kinetics operates at 2% FP steadily and
in-itially, and a reactivity step increase of 0.1
(starts at 1000s. The responses of the estimation errors of the
relative nuclear power, concentrations of the second and sixth
groups of precursors and the estimation of reactivity variation
with different ε are shown in Fig. 1.
A2: A Reactivity step decrease of 0.5$ starts at 1000s. The
responses corresponding to different εare given in Fig. 2.
Case B (Reactivity Step with Noise):
DONG et al.: REACTIVITY ESTIMATION BASED ON AN EXTENDED STATE OBSERVER OF NEUTRON KINETICS
Fig. 4.
2695
Simulation results in case B1.
The simulation study in the case of reactivity step increase
and decrease at 2% FP in a noisy environment are performed.
The noise added to the simulation is shown in Fig. 3, which
is rational to verify the observer performance in a noisy
environment and is generated by Simulink block of bandlimited white noise with the sample time of 0.1s and power
spectral density (PSD) of 1e-8. Moreover, different values of
constant ε are adopted.
B1: The neutron kinetics runs at 2% FP steadily and
initially, and a reactivity step increase of 0.1$ (starts at 1000s.
The responses of estimation errors of relative nuclear power
and concentrations of the second and sixth precursors and the
estimated reactivity based on the ESO-based and classical IPK
methods and simulated reactivity are shown in Fig. 4.
B2: A reactivity step decrease of 0.5$ starts at 1000s. The
responses of the main state-variables are illustrated in Fig. 5.
Case C (Reactivity Ramp with Noise):
The neutron kinetics runs at 2% FP steadily and initially,
and then a reactivity ramp increase of 0.2$ (with the speed
of 0.1$/min starts at 1000s. The responses of the estimation
errors of the relative nuclear power and concentrations of the
second and sixth precursors as well as the estimated values of
reactivity based upon both the ESO-based and classical IPK
methods and simulated reactivity are shown in Fig. 6.
B. Discussions
From Figs. 1 and 2, the transition periods and overshoots
of the estimation errors of the relative nuclear power and
precursor concentrations as well as the estimated reactivity
become smaller as perturbation constant ε is chosen to be
smaller. If ε is large enough, then there are steady estimation
errors. Actually, from (20) and (23), estimation error vector
e converges to a smaller bounded set if ε becomes smaller,
which explains the numerical phenomena from the theoretic
point of view.
From the simulation results in Fig. 4 which corresponds
to the performance comparison between the ESO and IPK
based methods in the case of reactivity step increase at 2%FP,
the ESO-based method can be utilized instead of the IPK
method in the power operation range. However, in the case
of reactivity step and ramp decrease with the amplitude of
0.5$ at 2%FP, due to the neutron kinetics quickly becomes
subcritical, the performance of the IPK method deteriorates
quickly, but from Fig. 5, ESO based method can still give
a satisfactory estimation of neutron flux, precursor concentrations and reactivity. Actually, from the expression of IPK
equation
dnr
nr dt
t
6
βi
Cri (0) e−λi t + λi
nr (τ ) e−λi (t −τ ) dτ ,
−
nr
0
ρ=β+
i=1
(24)
as relative neutron flux n r converges to zero, the reactivity
estimation becomes more and more sensitive to the fluctuation
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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 63, NO. 5, OCTOBER 2016
Fig. 5.
Simulation results in case B2.
Fig. 6.
Simulation results in case C.
DONG et al.: REACTIVITY ESTIMATION BASED ON AN EXTENDED STATE OBSERVER OF NEUTRON KINETICS
of the subcritical state since n r is in the denominator. Due to
the reactivity estimation given by ESO (7) does not contain any
division operation of the measured relative neutron flux, the
estimation performance of the ESO-based method can be still
guaranteed in a very low power-level or even in the subcritical
state. From Fig. 5, the estimation is sensitive to the noise
if perturbation constant ε is small enough, as we can see
that there exists a delayed offset. Actually, from the first and
third equations of ESO (7), when ε is smaller, the sensitivity
to noises is higher, which induces the offset. Moreover, as
we have discussed in the case of no noise, if ε is larger,
from (20) and (23), estimation error e converges to a larger
bounded set, which induces a steady estimation error. Thus,
in the practical engineering, perturbation constant ε should be
carefully chosen so that there is a satisfactory tradeoff between
the estimation precision and noise sensitivity.
From Fig. 6, the observation errors of the state-variables
in the case of reactivity ramp are all smaller than those in
the case of reactivity step, which is caused by the lack of the
prompt reactivity change in the ramp case. Although, there
is a steady estimation error for the reactivity, the estimation
process is converged, and the steady error is acceptable.
In a summary, ESO (7) can be applied for reactivity estimation instead of classical IPK method in the power range, and is
much stronger than IPK method in the subcritical state. Due
to there is no computation about the estimation covariance
matrices, the computing load of applying the ESO-based
method is much smaller than that of the EKF-based method.
Thus, ESO (7) can give effective reactivity estimation with
easy engineering implementation.
IV. C ONCLUSIONS
Reactivity estimation of a nuclear reactor is very important
for monitoring shutdown margins, calibration of control rods,
qualification of the worth of fuel elements, and fault detection.
Reactivity estimation method is the basis of building reactivity
meters for facilitating continuous reactivity surveillance in
the cases of shutdown, start up and power-range operations.
In this paper, a new simple ESO of the point neutron kinetics
is proposed for reactivity estimation, which guarantees globally asymptotically bounded estimation for the neutron flux,
precursor concentrations and reactivity. Numerical simulation
results show that this ESO can be utilized for reactivity
estimation instead of the classical IPK method in the power
range and in the subcritical case. Only one parameter of the
ESO, i.e. perturbation constant ε needs to be tuned online
for a satisfactory tradeoff between the estimation precision
and noise sensitivity. One of the future work is implementing
the ESO in a digital measurement platform for engineering
verification.
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