Nuclear Engineering and Design 241 (2011) 1469–1477
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Nuclear Engineering and Design
journal homepage: www.elsevier.com/locate/nucengdes
Assessment of the Prony’s method for BWR stability analysis
Javier Ortiz-Villafuerte ∗ , Rogelio Castillo-Durán, Javier C. Palacios-Hernández
Gerencia de Ciencias Aplicadas, Instituto Nacional de Investigaciones Nucleares, Carr. México-Toluca S/N, La Marquesa, Ocoyoacac, Edo. México 52750, Mexico
a r t i c l e
i n f o
Article history:
Received 21 September 2010
Received in revised form 26 January 2011
Accepted 18 February 2011
a b s t r a c t
It is known that Boiling Water Reactors are susceptible to present power oscillations in regions of high
power and low coolant flow, in the power-flow operational map. It is possible to fall in one of such
instability regions during reactor startup, since both power and coolant flow are being increased but not
proportionally. One other possibility for falling into those areas is the occurrence of a trip of recirculation pumps. Stability monitoring in such cases can be difficult, because the amount or quality of power
signal data required for calculation of the stability key parameters may not be enough to provide reliable
results in an adequate time range. In this work, the Prony’s Method is presented as one complementary
alternative to determine the degree of stability of a BWR, through time series data. This analysis method
can provide information about decay ratio and oscillation frequency from power signals obtained during
transient events. However, so far not many applications in Boiling Water Reactors operation have been
reported and supported to establish the scope of using such analysis for actual transient events. This work
presents first a comparison of decay ratio and frequency oscillation results obtained by Prony’s method
and those results obtained by the participants of the Forsmark 1 & 2 Boiling Water Reactor Stability
Benchmark using diverse techniques. Then, a comparison of decay ratio and frequency oscillation results
is performed for four real BWR transient event data, using Prony’s method and two other techniques
based on an autoregressive modeling. The four different transient signals correspond to BWR conditions
from quasi-steady to power oscillations. Power signals from such transients present a challenge for stability analysis, either because of the low number of data points or need of much iteration, and thus reducing
their capability for real time analysis. The results show that Prony’s method can be a complementary
reliable tool in determining BWR’s stability degree.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
Boiling Water Reactors (BWRs) are still susceptible to present
power oscillations in regions of high power and low coolant flow, in
their power-flow operational map. In BWR operation, those instability regions can be reached during reactor startup, since both
power and coolant flow are being increased but not proportionally.
Moreover, operational maneuvering during startup is not in general
independent of performance of fuel type loaded in core, and since
some new fuel assembly designs have smaller coolant flow area
and have shown faster response to certain neutronic perturbations,
startup trajectories in a power-flow map and/or operator maneuvering may need to be adapted to avoid falling in unstable areas.
This issue is of particular interest to power uprates programs, which
are becoming a quite viable option for improved performance of
nuclear power plants, since part of their success is to use new
∗ Corresponding author. Tel.: +52 55 53297200x2463; fax: +52 55 53297340.
E-mail addresses: javier.ortiz@inin.gob.mx, javier.ortizvillafuerte@gmail.com
(J. Ortiz-Villafuerte), rogelio.castillo@inin.gob.mx (R. Castillo-Durán),
javier.palacios@inin.gob.mx (J.C. Palacios-Hernández).
0029-5493/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.nucengdes.2011.02.018
fuel assembly designs. However, it has been inferred (through core
specific safety analyses) that certain operational areas, within the
before-to-up rate power-flow map, could be reached when using
new fuel designs (or a mixed core) at the expected operational conditions, especially during startup. One other possibility for falling
into those areas is because of recirculation pumps trips. In this case,
a reactor can be originally operating at rated conditions, but the
sudden decrease of coolant flow at still high power conditions can
lead to fall into instability zones. Determining the degree of BWR
stability in those unstable zones is necessity, in order to perform
appropriate stabilizing operations.
Continuous monitoring of key parameters related to stability is currently common practice in the nuclear power industry,
since, either during normal operation or during transient events,
monitoring devices become of great help to determine reactor stability degree. In particular, since power oscillations could occur in
BWRs, monitoring equipment is normally required to determine
the figures of merit in stability analysis. Power signal data coming from local power range monitors (LPRMs) or average power
range monitors (APRMs) are mostly the basis for calculating key
stability parameters and therefore determining reactor stability in
a particular situation. However, reliability of those key stability
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J. Ortiz-Villafuerte et al. / Nuclear Engineering and Design 241 (2011) 1469–1477
parameters depends on amount and quality of data used in calculations. Although occurrence of some power oscillation is normally
of not of much safety concern, since those important parameters
associated to stability are continuously tracked by power stability
monitors and thus oscillation suppression can be readily accomplished, it still can occur high amplitude power oscillations during
fast evolving transients, in which the amount and/or quality of signal data required by stability monitoring devices may not be the
minimum enough to provide with reliable estimation of the stability figures of merit, and therefore a instability range can be difficult
to determine in adequate time range. Even in case of stable and
quasi-stable operating conditions, it is still of interests to know the
degree of stability.
In the power industry, normally the figures of merit in BWR stability analysis are decay ratio and oscillation frequency. There exist
several methodologies to determine such parameters, as autoregressive methods, autoregressive and moving average techniques,
autocorrelations, impulse response method, determination of Lyapunov exponents, etc. Some of these methodologies are general,
and thus, variations or combinations of them are used when analyzing power noise signals. Fundaments of such techniques have been
provided by their authors and users and normally published in scientific journals. As an effort to estimate advantages and weaknesses
of the most commonly employed methodologies, benchmark tests
based on their application to actual power signals have been already
performed (NEA/NSC, 2001). Among diverse signal analysis techniques, the Prony’s method has not been applied to BWR stability
analysis, although it has been applied, without explicitly mention
it, as a fitting function to an impulse response. This technique is
widely used in power and energy transport systems problems for
steady-state and transient analyses, providing information about
wave’s amplitude, damping factor, oscillation frequency and phase.
In the next section, a short discussion of the use of decay ratio as
figure of merit in BWR stability is presented. Then, Prony’s method
is described in the way it was applied in this work. Then, in Section
6, it is presented first a comparison of decay ratio and frequency
oscillation results obtained by Prony’s method and those results
obtained by the participants of the Forsmark 1 & 2 Boiling Water
Reactor Stability Benchmark using diverse techniques. Then, a comparison of decay ratio and frequency oscillation results is performed
for four real BWR transient event data, using Prony’s method and
two other methods base on the autoregressive technique. One of
these methods is based on the original computer program developed by March-Leuba (1984), which was adapted to perform the
analyses. The other method computes the impulse response and
then it uses a functional fitting technique to calculate the oscillation frequency and decay ratio. This method was too adapted from
that developed by Tomokai Sazudo from the Japan Atomic Energy
Research Institute (NEA/NSC, 2001). The four different signals correspond to BWR transient events, in which zones of coolant flow
below 40% and power from 30% to about 60% were reached. Concluding remarks, regarding advantages and disadvantages of using
the Prony’s method as a complementary tool in determining the
degree of stability of a BWR are also discussed.
system, during a limit cycle the decay ratio value is considered to
be 1.0, so the oscillation amplitude becomes the figure of merit.
Note, however, that even when a BWR is still in stable condition,
the parameter decay ratio has been questioned as a correct measure of degree of stability, since some experiments have shown that
low values of decay ratio may be misleading about the true degree
of stability (Van der Hagen, 2000). Other argument against using
the decay ratio as the only key indicator of stability is the fact that
it is a measure of linear stability and a clear definition only can
be given for a second order system (NEA/NSC, 2001). Nonetheless,
decay ratio is still recognized as a figure of merit for surveillance
during reactor operation, and it is requested by regulatory authorities as part of fuel reloads safety analyses. Even for next generation
reactors, calculation of decay ratio is given outmost importance for
stability evaluation (Rohde et al., 2010). To calculate decay ratio and
oscillation frequency from power signal data, the most common
model used to represent the dynamics embedded in a power signal
has been the damped harmonic oscillator equation with constant
damping coefficient k and angular frequency ω, equation which is a
linear second order differential equation with constant coefficients:
ẍ + kẋ + ω2 x = Z(t),
where Z(t) is a noise source, assumed as Gaussian white noise with
null average. Solutions to Eq. (1) have also been presented for the
case when fluctuating parts of both k and ω have been introduced
and having some correlation through time, that is, some colorized
noise, (Konno and Kanemoto, 1998). The main case of interest when
solving the homogenous case of Eq. (1) is that leading to oscillatory
motion, which is when the discriminant of the characteristic equation is negative. In this case, the general solution can be casted
as
x(t) = A e˛t sin(ˇt + ),
BWR dynamics is a complex non-linear process, but its inherent
stable behavior in normal operation conditions allows for the use
of linear dynamics models, introducing linearized system of equations when needed. However, if a perturbation is introduced in the
system, in such a way that nonlinearities lead to pass the stable regimen, only nonlinear dynamics models should be used to determine
reactor behavior. For a linear system, its stability can be determined by calculating the decay ratio parameter. For a non-linear
(2)
where A is the amplitude of the motion, is the phase angle, and ˛
and ˇ form the two complex conjugate roots, and according to Eq.
(1) they are given by
−k
˛=
2
ˇ=
and
4ω2 − k2
.
2
(3)
Eq. (2) is also, practically, the same impulse response function
et sin(ωd t) that determines the stability of such a system (MarchLeuba and Smith, 1985), so ˛ and ˇ are the real and imaginary parts
of the pair of complex conjugate poles of that system. In noise analysis of power signals, decay ratio (DR) and fundamental oscillation
frequency (f) are calculated from
DR = exp
2 Re() |Im()|
and
f = |Im()|,
(4)
where Re() and |Im()| mean, respectively, the real and absolute
value of the imaginary parts of the pair of complex conjugate poles
and *, which are associated to the main oscillation in the signal.
By using Eqs. (3) and (4), the decay ratio value is
DR = exp
2. Decay ratio as figure of merit in BWR stability analysis
(1)
−2k
4ω2 − k2
.
(5)
However, the decay ratio obtained from Eq. (5) has been considered
only a rough indication of the stability (NEA/CSNI, 1997). Also, even
if Eq. (5) could be directly used, it would be necessary first to accurately characterize a BWR to determine both k and ω, which would,
further, depend on particular types of fuel loading, reactor geometry (friction distribution), and operating conditions in a power-flow
map. Thus, in practical applications, and * are computed instead,
that is, it is preferred to calculate directly both ˛ and ˇ, so both DR
and f are also determined.
J. Ortiz-Villafuerte et al. / Nuclear Engineering and Design 241 (2011) 1469–1477
3. Prony’s method
Although Prony’s method is quite well known in diverse areas
of the engineering, and therefore it can be found in many textbooks
in much detail (Marple, Jr., 1987), inhere it is briefly introduced for
completeness. This analysis technique models sampled data as a
linear combination of exponential functions. One disadvantage of
this method is that it is known to be sensible to noise content in the
signal being analyzed, but several computational techniques have
been developed to deal with cases with noisy data, depending on
the problem being attacked (Kumaresan et al., 1984; Trudnowski
et al., 1999; Qi et al., 2007; Peng and Nair, 2009). The main difference
between using Prony’s method and autoregressive (AR) and ARmoving average (ARMA) techniques is that the former seeks to fit a
deterministic exponential model to the data, while AR and ARMA
techniques seek to fit a random model to the second order data
statistics.
Assume first that there are N sampled data (x(1), . . . , x(N)) available for analysis, which are equally spaced in the time period of
sampling. Prony’s method estimates a fitting function x̂(n) of order
p to signal data as follows:
p
x̂(n) =
Ak exp
˛k (n − 1) + (j2fk )(n − 1) T + jk ,
k=1
(6)
√
where 1 ≤ n ≤ N, j = −1, T is the frequency of data sampling
[s]; and each term in Eq. (6) has the four elements: amplitudes
Ak ; damping factors ˛k [radians/s]; sinusoidal frequencies fk [Hz];
and phases k [radians]. These all four elements form the k different mode components of the signal being analyzed. The terms in
the exponential function k = (˛k + j2fk ) are the eigenvalues of the
system. Eq. (6), by using Euler’s formula, can be also be casted as
x̂(n) =
p
Ak exp [˛k (n − 1)T ] cos (2fk )(n − 1)T + k ,
(7)
k=1
which could be considered as a discretized equivalent form of the
impulse response function of a second order system given by Eq.
(1), since x(t) can be reconstructed as superposition of oscillation
modes with different amplitude and frequency, given by Eq. (7).
Note, however, that x̂(n) is just a fitting function, that is, Prony’s
method only identifies a model for arbitrary data being analyzed,
but it does not, in general, identify parameters of a system or transfer function.
If a time discrete function of the p-exponential functions in Eq.
(6) is expressed as
x̂(n) =
p
hk zkn−1 ,
(8)
k=1
where the complex constants hk and zk are defined as
hk = Ak exp(jk )
and zk = exp(k T ),
(9)
these represent the model’s discrete residues and poles, respectively. Once zk and hk have been computed, damping coefficients,
frequencies, amplitudes, and phases, in the signal can be obtained:
˛k =
ln|zk |
,
T
(10)
fk =
tan−1 Im(zk )/Re(zk )
[2T ]
Ak = |hk |,
k = tan−1
,
(11)
(12)
Im(hk )
.
Re(hk )
(13)
1471
Thus, Prony’s method directly provides the oscillation frequencies,
Eq. (11), and the decay ratio associated to each oscillation mode is
simply
DRk =
˛k
.
fk
(14)
Trying to determine simultaneously hk and zk , and the optimum
order p, i.e., the value of p which in conjunction with hk and zk
minimizes the total squared error between data and model,
ε=
N
x(n) − x̂(n)
2
(15)
n=1
is a nonlinear problem difficult to solve, so sequential steps have
been identified to solve a similar, but not exactly the same, problem.
The first step is to construct a linear prediction model of the (matrix)
form:
⎡
x(p)
⎤ ⎡ x(p − 1)
⎢ x(p + 1) ⎥ ⎢ x(p)
⎢ x(p + 2) ⎥ ⎢ x(p + 1)
⎢
⎥ ⎢
⎢ . ⎥=⎢ .
⎣ .. ⎦ ⎣ ..
x(N − 1)
x(N − 2)
x(p − 2)
...
x(1)
x(0)
x(p − 1)
...
x(2)
x(1)
x(p)
...
x(3)
x(2)
.
.
.
...
.
.
.
.
.
.
x(N − 3)
...
...
x(N − p − 1)
⎤⎡ a ⎤
1
⎥ ⎢ a2 ⎥
⎥ ⎢ a3 ⎥
⎥⎢ ⎥
⎥⎢ . ⎥
⎦ ⎣ .. ⎦
(16)
ap
to solve for the unknowns ai . Note that an order p has been assumed
and that the ai can be identified as the AR parameters of this lineal
prediction model. In the second step, the impulse response can be
expressed in terms of the roots (poles) zk of the following polynomial:
z p + a1 z p−1 + a2 z p−2 + · · · + ap−1 z + ap = 0.
(17)
Finally, the third step is to substitute the poles in Eq. (8) to solve
for the residues hk , using the least squares technique.
Steps 1 and 2 of the computation process just described can be
considered as a method to solve for poles in an AR procedure. In the
AR methodology, when a system is stable the poles of the system
must fall within the unit circle, in the z plane. In Prony’s analysis,
when the damping coefficients are negative (˛k < 0), the roots of
the system fall also within the unit circle, in the z plane. Thus, as
in the case of the dominant Lyapunov exponent, the magnitude
and sign of the damping coefficient are sufficient to determine if
the system under surveillance is stable or not. However, the degree
of stability, measured through decay ratio, requires knowing the
value of the main oscillation frequency (associated with the most
unstable pole), which is also an output in Prony’s method.
4. Autoregressive-based methods used for calculation of
transient event’s DR and f
Autoregressive (AR) models have been widely used in power
signal analysis. This type of models depends on a white noise drive
mechanism and yields parametric descriptions of second order
statistics (mean square differences, variances, correlations, etc.) of
a random process. For their application to BWR stability analysis, it
is considered that reactivity fluctuations are the white noise driving
source. The following two AR methods were used only for analysis
of the transient events signal data.
The first computer program used for determining DR and f is an
adaptation of the original code STABIL, which is described in much
detail by its own author (March-Leuba, 1984). STABIL provides also
a confidence level associated to the DR value calculated by three
different ways. Briefly, the algorithms employed in STABIL to estimate the DR from power noise measurements can be summarized
in the following steps:
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J. Ortiz-Villafuerte et al. / Nuclear Engineering and Design 241 (2011) 1469–1477
(1) Compute autocorrelation function and estimate DR,
(2) Compute AR model of optimal order, using the Yule-Walker’s
equations and Akaike’s Information Criterion (Akaike, 1974),
(a) AR model impulse response.
(b) AR model frequency-domain pole search.
(3) Estimate the asymptotic DR from:
(4) Validate DR estimates from steps 2, 3a and 3b using heuristic
rules,
(5) Select the largest valid DR as the result, which is a conservative
estimate of reactor’s DR.
The second method was adapted from that presented by
Tomokai Sazudo from the Japan Atomic Energy Research Institute
(NEA/NCS, 2001) in the Forsmark Stability benchmark report. A
computer program was developed to perform all necessary computations. The code has been called MODAR. This method computes
first the impulse response and then, to fit the output of the impulse
response, it uses the following equation:
y(t) = 1 e−2 t cos(3 t + 4 ),
(18)
where y(t) represents the impulse response al time t, and 1 , 2 ,
3 , and 4 are the fitting parameters, from which DR and f can be
determined, as:
DR = e−(22 /3 ) ,
f =
2
.
3
(19)
It can be noted that Eq. (18) is representing the solution of a damped
harmonic oscillator equation, that is, Eq. (2). Also, Eq. (18) is practically the same as Eq. (7), so Prony’s method has been used for
calculating only the main mode, out of all p possible modes, and
from it DR and f are determined. In MODAR, the AR parameters
are computed via the Yule-Walker’s equations and a normalized
Akaike’s Information Criterion. The normalization factor used here
is the value of the variance obtained from a second order model. In
this study, after dismissing data corresponding to the first second,
only the next 10 s of data are used for the fitting process, using a
sequential quadratic programming method, which is a technique
commonly used in optimization problems, and it was coded following the algorithm developed by Schittowski (1985). A short
description of the data used for the comparison tests is presented
next.
5. Description of benchmark and transient event data used
for analysis
A first comparison exercise was carried out between the results,
from power signals corresponding to the Case 1, presented by
the participants of the Forsmark 1 & 2 BWR Stability Benchmark
(NEA/NSC, 2001) and those inhere obtained by applying the Prony
methodology described above. Case 1 was chosen because its objective was to compare performance of diverse methods applied to
determine main stability parameters, since those signals were considered easy to evaluate. Further, in Case 1, the DR ranged, taking
the average values, from 0.458 to 0.715, so the reactor conditions
ranged from stable to quasi-stable.
Four power signals obtained during transient events in which
the BWR/5 units at the Laguna Verde Nuclear Power Plant reached
quasi-stable or unstable zones (Bravo-Sánchez et al., 2002) were
considered next, to test of Prony’s method performance under such
reactor operating conditions. The events are related to problems
with recirculation pumps and startup maneuvers, and the operating conditions correspond to the Zones II and III of the power-flow
map. In general, within Zone II power ranges from 35% to 50% and
core flow ranges from 25% to 45%. Regulations require maneuvers
to leave this zone immediately. In the case of Zone III, power ranges
from 25% to 35% and core flow ranges from 25% to 40%. In this zone
Fig. 1. Raw signal of power evolution during the power oscillation event.
the entrance is controlled, requiring continuous surveillance. The
four events are described briefly next, along with figures showing
the evolution of the power during the transients.
Event 1: Power Oscillation. The first event analyzed is a power
oscillation occurred during startup maneuvers. Although the
maneuvers were followed as approved and power and coolant flow
conditions were such that the reactor was not in an unstable zone,
this event showed that the limits of the exclusion zone were not as
sharp as originally thought. Fig. 1 shows the behavior of the power
during the transient. Signal data were obtained from an APRM at a
sampling rate of 5 Hz. Diverse analyses have determined that the
DR value was above 1.0 and the oscillation frequency (f) was in a
range from 0.50 to 0.54 Hz (NEA/CSNI, 1997; Castillo-Durán et al.,
2008). In this study, the DR and f are calculated for the slightly more
than 200 s period limited by the two vertical lines in Fig. 1. In this
interval, the coolant flow varied from 34% to 32% of the nominal
flow. Clearly, the signal is not stationary.
Event 2: Recirculation Pumps Trip. During some programmed
tests to measure vibration on recirculation system valves, the system logic failed to transfer pump velocity from high to low, leading
to tripping the pumps on both recirculation loops. The reactor operator verified that reactor was stable during the whole event. Fig. 2
shows the power evolution after pumps trip, as measured by an
APRM at a sampling rate of 5 Hz. The power decrease shown in
the figure is due to both the decrease in coolant flow through core
and insertion of control rods. The transient was over when reactor
operator started over the recirculation pumps, first in loop A and
later on loop B. The signal data analyzed are within the limits set
by the vertical lines in Fig. 2. The coolant flow at that 500 s period
was about 28%.
Fig. 2. Raw signal of power evolution during the recirculation pumps trip event.
J. Ortiz-Villafuerte et al. / Nuclear Engineering and Design 241 (2011) 1469–1477
Fig. 3. Raw signal of power evolution during recirculation pumps sudden transfer
to low velocity.
Event 3: Recirculation Pumps Sudden Transfer to Low Velocity.
While operating at practically rated conditions, a spurious signal
of the differential temperature change between the dome steam
temperature and liquid temperature at exit of the recirculation
loop, lead to automatic transfer of pump velocity from high to
low. The reactor went into the Zone II of the power-flow map,
but then a decrease of feedwater temperature occurred, leading
to the power behavior shown in Fig. 3. Coolant flow was about
36%. While taking the reactor out of those unstable zones, reactor operator verified that the reactor was stable during the whole
event. The power decrease shown in the figure is due to insertion of
control rods. The signal comes from an APRM taking data at a sampling rate of 5 Hz, and signal data analyzed are within the limits set
by the vertical lines in Fig. 3, that is, within the first 1000 s of the
data.
Event 4: Entrance in Region III. During startup maneuvers, the
operator took the reactor to a point where a transfer of pump velocity from low to high should be initiated. After extracting control
rods, the reactor went into the Zone III of the power-flow map,
for about 2 min, and then went out. This 2-min transient finished
because of normal feedwater temperature increase and the amount
of Xenon present at that time. There was no need of any special
operation and the reactor was always stable. Fig. 4 shows the power
evolution for the time period before reactor reached, during, and
after leaving the Zone III of the power-flow map. The signal data for
analysis was taken from the interval limited by the vertical lines in
Fig. 4, and those data correspond to the time when the reactor was
in the Zone III and reached the highest power during this event. The
signal comes from an APRM taking data at a sampling rate of 10 Hz.
The coolant flow was about 30%.
1473
Fig. 4. Raw signal of power evolution during the event in which Zone III was reached.
6. Results
Prony’s method is directly applied to preprocessed power noise
signals. Raw signal processing consisted of applying a recursive algorithm to extract DC and fluctuation (noise) components
(Ceceñas-Falcón, 2001), so signals analyzed have zero mean value.
This type of preprocessing was carried out for all signals.
6.1. Comparison against Forsmark benchmark data
The original raw signals were sampled at 25 Hz, but the time
series was decimated to 12.5 Hz, ending with about 4000 data
points. In this study, two tests with the 14 signals were carried out.
In the first test only the first 2024 of the whole time series were used
in the Prony’s method; while the whole 4041 data points was used
in the second test. In both cases, the order was set to 30 (p = 30).
Table 1 presents the results obtained for the main oscillation
frequency in each signal. As it can be noted, the maximum of
all the percent relative differences (RD), respect the benchmark
average (BA) values, decreased in absolute value, from 6.6 to 4.3,
and although for some signals the percent RD increased when
using more data, the root mean square error (rmse) for all signals
decreased 15% when using the whole data series. Clearly, the rmse
is not a figure of merit here, but it shows that Prony’s method performs better when using more data for the functional fitting. Table 1
also shows that Prony’s method results are within one standard
deviation (SD) of the benchmark results, so frequencies obtained by
this method are acceptable too when using half of the data points.
Table 2 shows DR results as in Table 1. As in the case of oscillation
frequency, Prony’s method performs better, in general, when using
Table 1
Comparison of the main oscillation frequency results, between the Prony’s method and values reported in the Forsmark benchmark.
Signals C1
BA f [Hz]
BSD
P-2024 f [Hz]
RD 2024 [%]
P-4041 f [Hz]
RD 4041 [%]
aprm.1
aprm.2
aprm.3
aprm.4
aprm.5
aprm.6
aprm.7
aprm.8
aprm.9
aprm.10
aprm.11
aprm.12
aprm.13
aprm.14
0.454
0.455
0.467
0.474
0.487
0.472
0.509
0.495
0.408
0.437
0.449
0.446
0.401
0.473
0.031
0.036
0.056
0.055
0.034
0.043
0.062
0.071
0.041
0.032
0.052
0.044
0.041
0.054
0.460
0.460
0.498
0.486
0.492
0.471
0.527
0.513
0.418
0.440
0.447
0.460
0.385
0.495
−1.2
−1.1
−6.6
−2.6
−1.1
0.2
−3.5
−3.7
−2.5
−0.7
0.4
−2.9
3.9
−4.5
0.461
0.467
0.487
0.491
0.496
0.475
0.521
0.515
0.406
0.440
0.458
0.461
0.401
0.487
−1.3
−2.5
−4.3
−3.7
−2.0
−0.5
−2.3
−4.0
0.5
−0.6
−2.1
−3.1
0.0
−3.0
BA, benchmark average; BSD, benchmark standard deviation; RD, relative difference; P-2024, Prony’s method results using 2024 data; P-4041, Prony’s method results using
4041 data.
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Table 2
Comparison of decay ratio (associated to the main oscillation frequencies in Table 1 results, between Prony’s method and values reported in Forsmark benchmark.
Signals C1
BA DR
BSD
P-2024 DR
aprm.1
aprm.2
aprm.3
aprm.4
aprm.5
aprm.6
aprm.7
aprm.8
aprm.9
aprm.10
aprm.11
aprm.12
aprm.13
aprm.14
0.487
0.572
0.508
0.497
0.515
0.520
0.591
0.458
0.508
0.519
0.479
0.715
0.498
0.601
0.085
0.110
0.131
0.135
0.121
0.142
0.150
0.103
0.070
0.120
0.165
0.116
0.121
0.189
0.442
0.533
0.481
0.573
0.530
0.482
0.698
0.489
0.576
0.589
0.557
0.751
0.522
0.694
RD 2024 [%]
9.2
6.8
5.2
−15.1
−2.9
7.4
−18.1
−6.7
−13.4
−13.5
−16.4
−5.2
−4.9
−15.5
P-4041 DR
0.503
0.600
0.476
0.504
0.524
0.488
0.642
0.452
0.566
0.522
0.471
0.776
0.507
0.661
RD 4041 [%]
−3.4
−5.0
6.2
−1.3
−1.7
6.3
−8.7
1.4
−11.4
−0.6
1.7
−8.6
−1.8
−10.0
Columns’ titles are the same as in Table 1.
more data points. However, for DR the improvement is noticeable,
since the maximum of all the percent RDs decreased, in absolute
value, from 18.1 to 11.4, although such a change does not correspond to the same signal. Also, except for two signals, when using
more data DR values calculated with the Prony’s method got closer
to the BA values, and the improvement in some cases was more than
10%. It also should be noted that when using 2024 data, Prony’s
results are again still within one standard deviation. Finally, the
rmse for DR calculations for all signals decreased 42% when using
the whole data series.
One feature that can be noted in both Tables is that Prony’s
method in most cases results on higher values than those average
values reported in the benchmark problem, for both DR and f. Thus,
in general, Prony’s method predicts a more unstable system. Therefore, considering that the DR value calculated via Prony’s method
deviated more than 10%, respect the average value reported, for
only one of the signals, and that the RDs are at most 6.6% for the frequency values, performance of Prony’s method compares quite well
for steady and quasi-steady conditions against the performance of
more traditional power noise signal analysis techniques, as the AR,
ARMA, impulse response, etc. techniques, which were employed in
the Forsmark benchmark tests.
6.2. Comparison against transient event signal data
Generally, techniques employed for stability analysis perform
quite well when using stationary or quasi-stationary signals, so
results can be reliable. However, during or after a transient, for
example pumps trips, neutron power signals may lose the stationary feature, and thus the level of uncertainty in decay ratio
calculation increases noticeably. Prony’s method has been used for
transient signals and cases of noticeable distorted wave’s amplitude
and frequency in many areas of engineering dealing with signal
analysis, so it can be considered as an alternative to compute the
DR for such cases.
Power Spectral Density. In order to perform the comparison
between the DR and f of the events described, the Power Spectral
Density (PSD) of each event’s power signal was firstly calculated,
so the dominant frequencies could be determined. Figs. 5–8 show
the PSD corresponding to the signals shown in Figs. 1–4, respectively. As it can be noted, except for the Event 2, the dominant
frequencies are in the range corresponding to the void reactivity
feedback mechanism. In the case of transients of the type of Event
2, experience has shown that shifting pump speeds and control
rod moves can lead to record signals having high noise amplitudes
(Jones and Humpheys, 1989), so adjustments are necessary in stability monitors, to avoid performing analyses yielding misleading
results.
Fig. 5. PSD of signal from Event 1.
Table 3 presents the results obtained by codes STABIL and
MODAR for each of the four events. When using STABIL, the
amounts of data used in the calculations were 1024, 2500, 5000,
and 1000, for the Events 1, 2, 3, and 4, respectively. In all four cases
the sampling rate used was 5 Hz and the number of correlation lags
was 32. In the case of using MODAR, the amounts of data used in
Fig. 6. PSD of signal from Event 2.
J. Ortiz-Villafuerte et al. / Nuclear Engineering and Design 241 (2011) 1469–1477
1475
Table 3
DR and f results obtained by STABIL and MODAR.
Event
STABIL
MODAR
DR
1
2
3
4
DR range [min–max]
0.999
0.249
0.678
0.503
0.997–1.000
0.208–0.286
0.643–0.699
0.209–0.607
f [Hz]
f range [±]
DR
f [Hz]
0.530
0.261
0.511
0.560
0.018
0.010
0.010
0.031
0.980
0.220
0.680
0.560
0.540
0.090
0.520
0.550
Table 4
Prony’s method results from Event 1 signal analysis, using N = 128 and p = 10.
Fig. 7. PSD of signal from Event 3.
Fig. 8. PSD of signal from Event 4.
the calculations were 500 for the three first events, and 1000 for
the Event 4.
The DR and f values from STABIL agree well with description
of the events given above, that is, reactor condition (stable or not)
and the PSD corresponding to each event, see Figs. 5–8. Note in
fk [Hz]
˛k
Ak
k [radians]
0.174
0.541
1.123
1.729
2.266
−0.0842
−0.0359
−0.8035
−0.6509
−0.7803
0.3733
0.1499
0.1359
0.1025
0.1336
1.6584
2.9246
−1.7786
−1.1935
−0.7535
Table 3 that MODAR found as dominant frequency for the recirculation pumps transient that corresponding to the first peak, see
Fig. 6, which is within the frequency range related to controllers, so
it is not used in BWR stability analysis. All other values of DR and
f fall within the intervals provided by STABIL. By taking as reference the results of the best estimate values reported by STABIL, the
maximum relative difference, in absolute value, for the DR results
between these two methods based on AR-techniques, is about 12%,
and for the f results, except for the frequency value from the Event
2, the maximum difference is less than 2%.
Table 4 shows that Prony’s method can be used to identify the
most relevant frequencies embedded in the signal under study, that
is, this method is also a complementary tool for spectral analysis.
The results presented in Table 4 correspond to an analysis of Event
1 signal, and they can be compared to Fig. 5. For this analysis, only
128 data points were used and the order was set to 10. Although
this particular combination of p and N did lead to incorrectly predict that the reactor was still stable, since all ˛k are negative, the
value of the largest ˛k indicates that reactor was close to unstable
conditions. A DR value of 0.936 results from the power oscillation
frequency (0.541 Hz) and the largest ˛k . Only the positive values of
the frequencies calculated are shown in Table 4. It is shown later
that Prony’s method correctly predicted that the reactor reached
unstable conditions by increasing both p and N.
Tables 5–8 show the results obtained for all four events. Examination of ˛k for each p and N corroborated that the actual conditions
of the reactor (stable or unstable) were correctly predicted. Therefore, only those results corresponding to the relevant frequency of
each particular event are shown in Tables 5–8.
Tables 5 and 7 show that results by Prony’s method for Events
1 and 3 agree quite well with those results obtained by STABIL
and MODAR. The maximum RD between the average of the results
from those two codes (see Table 3) and Prony’s method results is
about 12% for DR and 4% for f results. Thus, even when using a low
order (p = 20) and relatively small amount of data (N = 512), Prony’s
method yielded reliable results. As it was the case when comparing against results from the Forsmark benchmark, using more data
reduced the RD between the AR-based and Prony’s methods. For
Table 5
Results obtained by Prony’s method from Event 1 power signal, using the order p and N data values shown.
p/N
˛k
Ak
k [radians]
f [Hz]
DR
20/512
20/1024
30/512
30/1024
0.0056
0.0039
0.0097
0.0039
0.2139
0.4736
0.1690
0.4846
1.8506
−2.6165
−1.5950
−2.7657
0.541
0.539
0.542
0.540
1.010
1.007
1.018
1.007
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J. Ortiz-Villafuerte et al. / Nuclear Engineering and Design 241 (2011) 1469–1477
Table 6
Results obtained by Prony’s method from Event 2 power signal, using the order p and N data values shown.
p/N
˛k
Ak
k [radians]
f [Hz]
DR
20/512
20/1024
30/512
30/1024
30/2500
−0.7170
−0.5368
−0.2600
−0.2495
−0.3300
0.8118
0.4797
0.0781
0.0683
0.1256
−2.3310
−2.6214
−2.2160
2.6285
−2.8544
0.299
0.308
0.256
0.260
0.266
0.091
0.175
0.361
0.383
0.290
Table 7
Results obtained by Prony’s method from Event 3 power signal, using the order p and N data values shown.
p/N
˛k
Ak
k [radians]
f [Hz]
DR
20/512
20/1024
30/512
30/1024
30/5000
−0.2526
−0.2537
−0.2198
−0.2479
−0.2200
0.9849
0.9653
0.8485
0.7496
0.3854
0.6759
−1.0603
−1.0190
−1.8638
−2.1389
0.495
0.506
0.495
0.516
0.522
0.600
0.606
0.641
0.618
0.656
Table 8
Results obtained by Prony’s method from Event 4 power signal, using the order p and N data values shown.
p/N
˛k
Ak
k [radians]
f [Hz]
DR
20/512
20/1024
30/512
30/1024
30/5000
−0.4269
−0.3803
−0.3831
−0.3067
−0.2798
0.6147
0.5894
0.6366
0.6313
0.5714
−2.8945
−3.0573
1.5241
−2.5742
2.2324
0.545
0.553
0.622
0.579
0.591
0.457
0.503
0.540
0.589
0.623
the Event 4 (Table 8), however, when using the whole data set and
highest order (p = 30) it did not reduced the relative differences, and
the minimum differences occurred when using 1024 data. For this
event, the maximum RD is 17% for DR and 12% for the f results. Thus,
the maximum RD for Events 1, 3, and 4, in any case, are under 20%,
which is considered acceptable. Also, the impact of using an order
or 30 did not improve much the results, as relative to the AR-based
techniques, although the order difference is only 10.
The results from Event 2’s power signal (Table 6) are contrary
to the previous results. In this case, the smallest differences with
the AR methods are not less than 25% for the DR. However, it is also
important to note that frequencies outside the range from around
0.4 to 0.8 are not considered for BWR stability analysis. It is important to mention that, for this particular case, STABIL reported an ill
conditioned signal, so it could not find the actual frequency related
to the transient. Thus, both AR methods would require limiting the
analysis to the interval mentioned before a new analysis process
was performed to find the relevant frequencies to stability analysis. Through Prony’s method, instead, a filtering our process can
be performed afterwards, since all frequencies in the signal are
found, according to the order chosen for the computational process. Table 9 shows those frequencies found in the range from 0.4
to 0.8 by this method, and their corresponding ˛k , Ak , k , and DRk .
Table 9
Prony’s method fk , ˛k , Ak , k , and DRk results from Event 2 signal analysis, in the
BWR stability analysis range.
f [Hz]
p = 30; N = 512
0.424
0.592
0.784
p = 30; N = 1024
0.423
0.580
0.788
p = 30; N = 2500
0.427
0.600
0.792
DR
˛k
Ak
k [radians]
0.505
0.458
0.624
−0.2896
−0.4626
−0.3700
0.0349
0.0282
0.0195
−1.8543
−0.4266
2.0087
0.462
0.472
0.671
−0.3268
−0.4350
−0.3144
0.0159
0.0271
0.0274
−2.2805
0.9454
1.9612
0.425
0.377
0.548
−0.3658
−0.5856
−0.4769
0.0323
0.0528
0.0507
−2.0668
0.2403
1.5839
Table 9 shows that the most unstable pole corresponds to
the frequency about 0.6 Hz (see Fig. 6). The other two frequencies are close to the limits commonly corresponding to the BWR
stability range, so, to the author’s view, the actual dominant frequency during the recirculation pumps trip, besides that related
to controllers, was that at around 0.6 Hz. Further, the results corresponding to p = 30 and N = 1024 are considered closer to actual
conditions. Finally, regarding computation speed, Prony’s method
is practically twice faster than STABIL and MODAR, since it requires
practically only one half of the computations in the whole process
of determining both DR and f.
7. Conclusions
Prony’s method has been proved to be a complementary reliable and useful technique for BWR stability analysis. This technique
performed quite well for steady and quasi-steady power signal
analysis, when compared against traditional autoregressive-based
methods. For the case of non-stationary signals, Prony’s method
also showed a robust performance. This method, further, has an
advantage of being a spectral analysis technique. Drawbacks of
this method include that it does not minimize the error between
the model generated and the actual signal data, but it only minimizes the error in the linear predictor stage. This fact has a direct
impact on the computation of ˛ and f, parameters that determine
the degree of reactor stability, via the calculation of the DR. Moreover, for BWR stability analysis, the results presented showed that
it is reasonable to use orders of 20–30, for Prony’s method computations, although trying higher orders is still necessary to state
definite conclusions. It was also shown that the amount of data
available is a key parameter to obtain reliable results.
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