- Email: [email protected]
Modeling nuclear reactor core dynamics with recurrent neural networks
NEUROCOMPUTING
ELSEVIER
Neurocomputing
15 (1997) 363-381
Modeling nuclear reactor core dynamics with recurrent neural networks ’ Tiilay Ada11 al*, Bora Bakal a, M. Kemal Siimnez b, Reza Fakory ‘, C. Oliver Tsaoi” a Information Technology Laboratory, Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, Baltimore MD 21250, USA b Institute for Systems Research, University of Maryland, College Park, MD 20742, USA ’ Simulation, Systems & Services Technologies Company (S3 Technologies), Columbia MD 21045, USA
Abstract
A recurrent multilayer perceptron (RMLP) model is designed and developed for simulation of core neutronic phenomena in a nuclear power plant, which constitute a non-linear, complex dynamic system characterized by a large number of state variables. Training and testing data are generated by REMARK, a first principles neutronic core model [ 161. A modified backpropagation learning algorithm with an adaptive steepness factor is employed to speed up the training of the RMLP. The test results presented exhibit the capability of the recurrent neural network model to capture the complex dynamics of the system, yielding accurate predictions of the system response. The performance of the network is also demonstrated for interpolation, extrapolation, fault tolerance due to incomplete data, and for operation in the presence of noise. Keywords:
Nuclear reactor core dynamics; Recurrent neural networks
1. introduction In recent years, there has been considerable interest in modeling dynamic systems with neural networks. The basic motivation is the ability of neural networks to create
’ This work is supported by Maryland Industrial Partnerships 1214.12 and MIPS-1214.24. * Corresponding author. Email: [email protected] 0925-23 12/97/$17.00 Copyright PZISO925-2312(97)00018-O
and S3 Technologies
0 1997 Elsevier Science B.V. All rights reserved
under grants no. MIPS-
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data-driven representations of the underlying dynamics with less reliance on accurate mathematical/physical modeling. There exist many problems for which such data-driven representations offer advantages over more traditional modeling techniques, such as availability of fast hardware implementations, ability to cope with noisy/incomplete data, and avoidance of complicated internal state representations. In this work, we make a system identification of core neutronic phenomena in a nuclear reactor using a Recurrent Multilayer Perceptron (RMLP). The training data is obtained from REMARK, a first principles neutronic core model which can model both normal operation and fast and localized transients. REMARK in its plant specific versions is currently being used to train personnel who will be working in nuclear power plants and is tested with a number of standard real plant data to verify its approximation capability. We can simulate various scenarios in nuclear plant operation such as reactor and pump trips and generate training and test data sets which obviously cannot easily be obtained in a real nuclear plant. We introduce a novel modified backpropagation scheme for training the RMLP to speed up learning. The proposed scheme keeps the network weights active in early stages of learning by adapting steepness of the activation function and using an annealing schedule to ensure convergence. It is demonstrated that, by the procedure, learning speed is improved considerably, a feat especially important in a modeling task of this size which consists of some 146 inputs and 52 outputs. The performance of RMLP model is evaluated in terms of its prediction performance on unseen test sets for interpolation and extrapolation, fault tolerance in cases of incomplete data, and for operation in the presence of additive white gaussian noise. Accurate predictions provided by RMLP demonstrate its success in capturing the underlying complex dynamics of the neutronic core. The specific advantages of RMLP for core neutronic modeling are: (i) Providing speed-ups in system prediction by using dedicated hardware which provide much needed faster than real-time prediction power; (ii) Better robustness to noisy data, ability to provide estimates of the system response in the presence of missing measurements (possibly due to malfunctioning equipment) which are harder to deal with in a model based implementation; (iii) The capability to start predictions from arbitrary initial conditions which allow recovery of transients without the need to set internal parameters necessary in model based implementations. The next section, Section 2, introduces the notation for the recurrent neural network structure employed in this application and for backpropagation learning. In the third section, the modified delta learning rule is introduced for a globally recurrent MLP, and its operation is discussed. The fourth section provides data acquisition details and finally the fifth section presents results that demonstrate advantages of the neural network approach over conventional approaches in terms of properties such as robustness in cases of noisy/incomplete data and extrapolation and interpolation capabilities. The appendix includes an overview of the dynamics of the neutronic core model REMARK and describes data acquisition using REMARK.
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2. The RMLP model We employ a recurrent network structure to capture the dynamics of the nuclear reactor core as represented by transients under different operating conditions. We adopt a global RMLP which incorporates a unit delayed connection between the output layer and the input layer. The structure is illustrated in Fig. 1. Ref. [8] presents an experimental comparative study of various recurrent architectures on different classes of problems and reports that the RMLP and Narendra and Parthasarathy’s [lo] model outperform other recurrent architectures that they consider for a nonlinear system identification problem. In [l], we show that the RMLP structure provides a powerful representation of the core dynamics outperforming a time delay neural network (TDNN) structure in that it provides a much better approximation of the input dynamics with a network size much smaller. We introduce the notation for the globally recurrent MLP as follows: Let x(n) denote the N x 1 external input vector applied to the network at time n, and let y(n) denote the corresponding A4 x 1 vector of outputs at time n. The external input vector x(n) and one step delayed output vector y(n - 1) are concatenated to form the input layer
xl(n)
Fig. 1. RMLP structure.
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to the network. The network is fully interconnected with V(n) and W(n) as its weight matrices. Let Uij(n) denote the connection weight between ith input node and jth hidden neuron, and wV(n) denote the connection weight between ith hidden neuron and jth output neuron. The output of kth output neuron at time n is given by . ~N+hf _
R‘t!
f v,w *
(1)
and we define the kth neuron output as Y/C(~)=
[fV,W(x(n),y(n
with
-
l))lk
=
“(Sk@))
(2)
J Sk(n)
= c
(3)
wjk(n)zj(n)
j=l and zk(n)
= Q
5
&k(n)%(n)
+ 5
i=l
~(N+m)k(n)ynt(n
-
1)
(4)
3
in=1
where J is the number of neurons in the hidden layer, a(.) is the node activation function, and k = 1,. . . ,M. With the notation introduced above, the weights and v+ are adapted by the steepest descent rule wjk
Wjk(n + 1) = wjk(n)
-
Wn)
CL-
(5)
awjk(n)
(similarly for Vij) such that the instantaneous squared error
E(n) = ;
k@k(n)
-
(6)
yk(n)>2
k=l
between the desired dk(n) and the network outputs yk(n) is minimized over all outputs. The instantaneous gradient terms for the weights are given by (7) for the hidden to output layer weights, and 41
- $(n))xi(n)
gwjkcn)(dk(n)
-
yk(n))(l
-
YR2Cn)h
k=l
i <
aE(n)=
N,
a&j(n) 41
- zj(n)>yi-dn
- 112
wjk(n)(dk(n)
-
YkCnIMl
-
d(n)),
k=l
N
(8)
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for the update of input to hidden signal.
layer weights through backpropagation
of the error
3. The modified delta learning rule In this section, we introduce a modified backpropagation scheme for training the RMLP. The algorithm is a gradient descent technique with a dynamic steepness factor in the activation functions of the nodes in the hidden layer. We define the activation functions as ~(a~) = tanh($),
(9)
where 2 is the steepness that -
N
factor, and Uj is the total input of the jth hidden neuron
-
such
M
@j = >, Ug(n)Xi(n) + >, U(N+m)j(n)Ym(n - l), Convergence of backpropagation-type ploying sigmoid-like activation functions
i = 1,. . . , J.
learning algorithms in neural networks emsuffer from premature increases in some node
weights which force node outputs to the saturated region of the activation function in early stages of learning. The learning rate of the nodes is thus diminished due to vanishingly small gradients. There have been several proposed fixes for this problem in the literature, such as independent learning rates for each weight, adaptation of learning rates during learning [5,9,11], scaling [13] or adding a constant [6] to the the gradient term backpropagated to the previous layers. We propose the following technique for adaptation of the steepness factor L of the activation functions in the hidden layer depending on the magnitude of the inputs to the hidden layer node. The goal is to keep the weights active during the early stages of learning by adjusting steepness according to a representative node selected by the maximum magnitude of the input to hidden node. To ensure convergence, an annealing schedule which, as training progresses, decreases the steepness adjustment is also proposed. The resulting algorithm takes aggressive steps early due to the large gradients and converges as the steepness adjustment is cooled down. We can write the adaptive procedure for the steepness of the activation function as A(n) =
A0 max{4,a2,...,crJ}
P(n)+ A(1 - p(n)),
(11)
where p(n)
is the annealing parameter which can be chosen such that p( 1) = 1 and as it -+ CCLA typical choice for p(n) is a sequence of order O(nP) for some tl > 0. We demonstrate the effect of annealing in the update with a simple example. Consider the nonlinear system identification problem where the system to be identified is described by the relationship p(n)
--+ 0
d@)
=
x(n)(l + 0 - 1)) 1 +&(n
- 1)
(12)
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Fig. 2. Effect of steepness adaptation Solid line: standard backpropagation, with the annealing procedure.
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and annealing procedure for the simple system identification problem. dotted line: steepness adaptation, dash-dot line: steepness adaptation
and the input is chosen as a monotonously decreasing ramp in the range [-1 J]. A 2-3-l globally RMLP is trained with 80 training samples. In Fig. 2 we show the initial speed up in the learning with the steepness update with (p(n) = l/n) and without annealing (p(n) = 1) for 50 epochs. The adaptive steepness learning curves for both cases (with and without annealing) show a considerable speed up in convergence. Also, with annealing, the network achieves a much lower final training error. Fig. 3 indicates a similar trade-off for the same system identification problem (12) where this time lo is let to change with no annealing. In the nuclear reactor core modeling problem, the modification of backpropagation algorithm results in considerable improvement in terms of both the learning characteristics and accuracy. This improvement is illustrated in Fig. 4 by comparing the two learning curves, that are obtained with (dotted line) and without (solid line) the adaptation of the activation function. Since there is a natural convergence in the value of n(n), we do not employ annealing for this problem, hence p(n) is set to 1. The value of & is chosen as the steepness value J.(n) naturally converges to, which is 6 in our implementation. The behavior of the adaptive steepness factor n(n) for our reactor core problem is shown in Fig. 5. With the adaptation, the number of iterations required to reach the same training error is much smaller than the number of iterations required without the adaptation. The final value of the training error is also reduced considerably.
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Fig. 3. Effect of As in steepness adaptation Curve 2: 1s = 1.0, Curve 3: J&j= 2.0.
0
200
for the simple system identification
400
600
(a) Number of epochs
0
200
problem.
400
Curve 1: I.0 = 0.5,
600
600
(b) Number of epochs
Fig. 4. Effect of steepness adaptation for reactor core modejjng. (a) Learning curves for single pump trip scenario, (b) learning curves for two pump trip scenario. Curves are plotted with (dotted) and without (solid) the adaptation of the steepness factor.
4. Data acquisition and implementation The version of REMARK we use has 146 external inputs and 52 external outputs that characterize the reactor core physics, and internal state variables as discussed in the appendix. To generate training and test data sets, REMARK is run with various scenarios which describe different operation conditions of the nuclear plant. All of the scenarios used in our experiments assume a disturbance of the system from its steady state operating condition, i.e., modeling of various transient responses of the system. These transients are achieved by setting the internal parameters of REMARK. The steady state response of the reactor core as we show in [3], can be achieved by a simple feedforward structure. In [3], we present a full model of the system (the steady state and the
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Fig. 5. Adaptive steepness l(n) behavior during training for reactor core modeling
for two pump trip scenario.
three types of transient responses considered) using a switch mechanism between three RMLPs and a simple feedforward MLP. The three transient scenarios are core trip, single pump trip, and double pump trips. In our implementation, the switch uses REMARK internal variables for control of the switching mechanism. In a real plant scenario, various plant parameters can be used for this purpose, such as using the rod positions for checking for reactor trip [3]. The structure of the switch mechanism is excluded in this paper because of the additional introduction it necessitates to the structure of REMARK. Due to the configuration of the circulation pumps around the reactor core (Fig. 14), tripping different combination of circulation pumps (for the double pump trip scenario) and tripping a different pump (for the single pump trip scenario) result in different transient characteristics. When the pump trips are coupled with a reactor shut down, rapid transients occur with reactor shut down having the dominant effect. For pump trip(s) without reactor shut down, we observe slower transient characteristics. For single pump trips, the power distribution in the reactor core varies depending on the location of the pump tripped. Therefore, to evaluate the generalization performance of the network we use data generated by tripping a different pump than the one used in training. For double pump trips, we consider tripping two pumps on the opposing sides of the cooling system (see Fig. 14) since this scenario does not cause system unbalance due to reverse leakage of coolant fluid as the parallel pump trips do. Again for testing we use data collected by tripping a different two-pump combination than the one used in training. We collect samples generated by REMARK with 4Hz sampling frequency. The network is trained by using the first 1000-1500 samples (samples l/4 s apart) and the generalization performance is tested on a different data set (obtained by tripping a different pump or combination of pumps). We also test for samples after the initial n samples used in training to see the prediction capability of the network
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into the future. Also, of particular interest is starting the network from an arbitrary (in our case all zeros) initial condition instead of its steady-state value to see the ability of the network to generalize to this case. The performance of the network in this test (start-up from an arbitrary initial condition) presents a particular advantage of the neural network modeling with respect to first principles modeling as will be explained later. The inputs to REMARK are: liquid temperature (9 units), liquid density (9 units), gas density (9 units), boron concentration (9 units), fuel rod temperature (36 units), cladding temperature (inner surface (36 units) and outer wall (36 units)), vessel saturation temperature (1 unit) (thermal-hydraulic conditions supplied to REMARK by RETACT [ 161) and the control rod positions (1 unit). The outputs are: power (1 unit), prompt fission power (1 unit), decay heat (1 unit), Xenon number density (1 unit), effective multiplication factor (1 unit), core reactivity (1 unit), normalized heat slab power distribution (36 units), power range monitor reading (8 units), and source range monitor reading (2 units). After data collection from REMARK, a normalization scheme is applied to transform the ranges of variables into the interval [ - l,l] providing a common dynamic range for each variable for presentation to the neural network. The RMLP structure employed to capture the system dynamics has an input vector of size 198 (146 external inputs and unit delayed 52 network output variables), 52 output nodes, and 60 hidden nodes which is determined empirically by scanning a range 3&120 hidden nodes in increments of 10. The value of 20 used in the implementation is 6 which is chosen as the steepness value n(n) naturally converges to with no annealing, i.e. with p(n) = 1 in (11).
5. Results As explained in Section 4, we consider three cases that result in three major types of transient responses in nuclear plants which are the single and double pump trips and the total reactor shut down. Within each type, different transients occur depending on the location of the pump(s) tripped. Since the three types of transients are distinctively different, for achieving a good tradeoff between the required accuracy of prediction performance and the network generalization, three separate RMLPs are trained for each of the three scenarios (single, double, and total reactor trips). The control mechanism selecting between these RMLPs is discussed in the previous section and in detail in [3]. The results show that the RMLP can capture the complex dynamics of the system, yielding satisfactory predictions of the expected transients for all the cases considered. In Fig. 6, we show RMLP prediction curves for four output variables for fast transient prediction, i.e., with reactor shut down. As seen in the figure the RMLP yields a close prediction of the system output for this fast transient which is particularly important in a nuclear power plant. To study the capabilities of RMLP for modeling core dynamics and to highlight its advantages over the conventional approaches we consider three specific problems and investigate extrapolation and interpolation capabilities of the network, its robustness with respect to noisy/incomplete data, and its transient recovery capabilities.
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Decay Heat lr
-1’
0
Xenon Number Density 1
500
1000
1 1500
Effective Multiplication Factor
_ITz 0
500
1000
Fig. 6. RMLP test results: Neural network time ( f S) (reactor trip scenario).
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lr
-1’
0
,
500
1000
I 1500
Normalized Heat Slab Power Distribution
1500
_LcTI 0
(dotted line) vs. REMARK
500
1000
1500
(solid line) outputs as a function
of
An important issue in a system modeling problem is the long term prediction capabilities of the network. This becomes an even more critical issue in a problem of this size which would imply training for large amounts of data for long-term modeling. As explained in the previous section the data used in training are the first 1000-1500 samples taken with 4Hz frequency. Naturally, an important performance consideration is the long term prediction capability of the network, i.e. when it is trained on the first n samples, looking at the prediction performance of the network beyond these n samples. In a normal operation scenario, everything except the outputs will be supplied for prediction performance. We test the long term prediction, i.e. the extrapolation capabilities, of the network by training the network with the first 1000 samples in the sequence (taken every 114 s) and then performing the test with a complete sequence of 4000 samples (including the first 1000 used for training). These results are shown in Fig. 7 for a single pump shut-down scenario. As seen in the figure, the network does a very good job of predicting the future system response. Also, the interpolation capabilities are tested by training the network with a data set down-sampled 10 : 1. That is, only one sample in every 10 samples is used for training (400 out of a total of 4000). This could be of great advantage in cases where the size of the training data is very large. These results are shown in Fig. 8. Note that the point tested here is that if because of storage or memory capabilities the data used for training is subsampled, how well the network can do the interpolation among the samples used for training for the samples that are missing in training. The network is also tested for fault tolerance, i.e., with incomplete data which might be because of a malfunctioning equipment; a harder problem to deal with in a model based approach. For this purpose, two different sets of four randomly selected input variables are set to zero to represent a faulty condition, and the experiment is repeated twice for each set. The test is performed with the reactor shut down scenario. In the
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Normalized Core Thermal Power _iI‘--i-i
0
Normalized Heat Slab Power Di.Mbution _py=yy
1000
3000
2000
4000
0
0
2000
3000
4000
Source Range Monitor Reading
Normalized Heat Slab Power Distribution
!1_1---
1000
_!=I
1000
3000
2000
4000
0
1000
2000
3000
4000
Fig. 7. Test results: evaluation of RMLP extrapolation capabilities. The network is trained only with the first 1000 samples. Neural network (dotted line) vs. REMARK (solid line) outputs as a function of time (is) (single pump trip scenario).
Normalized Heat Slab Power Distribution
Decay Heat
[\-I
0
_[ri
1000
2000
3000
4000
0
Normalized Heat Slab Power Distribution
_;vi
0
1000
2000
3000
4000
Power Range Monitor Reading
;=
1000
2000
3000
4000
0
1000
2000
3000
4000
Fig. 8. Test results: evaluation of RMLP extrapolation capabilities. The network is trained with a 10: 1 down-sampled data. Neural network (dotted line) vs. REMARK (solid line) outputs as a function of time ($ s) (single pump trip scenario).
first realization, where liquid density ( 1 ), gas density (1 ), and fuel rod temperatures (2) are set to zero, RMLP generated the same prediction accuracy as in the normal operation (similar to Fig. 6). However, for the second realization where liquid temperatures (2) inner surface cladding temperature (1 ), and outer wall cladding temperature (1) are set to zero, 3 outputs (Xenon m.unber density, effective multiplication factor, and core reactivity) out of 52 showed a maximum of 15% deviation from the original
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314
Decay Heat
Core Reactivity
‘1
I
-I’
500
1000
Normalized Heat Slab Power Distribution 11 1
j/f----
-A-
500
1500
1000
Normalizad Heat Slab Power Distribution
pL.___. I
-1’
500
1000
1500
500
1500
1000
1500
Core Readvity
Decay Heat 0.5 ‘I
4
2
6
Normalized Heat Slab Power Distribution
-0.75
4
6
._ ‘y.-2
...
8
Normalized Heat Slab Power Distribution 0.6 .‘..,
-::r] -0.7’
2
6
I
,_._,-..- .._-._._--
4
6
8
0’
2
4
6
I 8
Fig. 9. First two rows: the prediction curves (dash-dot: from known initial conditions, dotted: from unknown initial conditions) vs. REMARK outputs (solid) as a function of time (is). Bottom two rows: the first 8 data samples for the observation of recovery from unknown initial conditions.
prediction curves only at the steady-state. The scenario that is used for this test is the reactor shut down. Model based implementations, in general, require proper setting of some internal variables which might create problems in situations where, for example in a real plant, the predictions are to be made from an arbitrary initial condition. We tested the RMLP prediction by starting from an arbitrary point on the transient curve (i.e. not from its steady-state value) and by setting the initial outputs to zero. As shown in Fig. 9 the recurrent structure converges to the desired response in about 5 data samples, demonstrating a very important advantage, recovery from arbitrary initial conditions, over the conventional techniques. The results shown are for a double pump trip scenario.
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Liquid Temperature
Liquid Temperature 21
1
_;\--i 0
500
1000
1500
a~ 21
0
500
Fig. 10. Input sequence without noise (left), input sequence with an additive gaussian (right) both as a function of time (is).
Normalized Core Thermal Power
1000
1500
noise of variance
0.01
Normalized Heat Slab Power Distribution 1
‘l 0.5 0 -0.5 r. -1’
500
0
1000
I 1500 Power Range Monitor Reading
Normalized Heat Slab Power Distribution ‘I ‘I 0.5.
0.5
0.
0 -0.5
-0.5. -1;
I, 0
‘1...._ 500
1000
1500
Fig. 11. Predictions with the noisy input as a function line: network output).
-1’
0
of time (is).
’ 500
1000
(Solid line: REMARK
1 00 output, dotted
Finally, the performance of the network is tested in the presence of white gaussian noise. An additive noise of variance of 0.01 is introduced at the input of the network to the variables normalized to [-l,l]. Samples of noisy predictions are given in Fig. 11. Fig. 10 is added to illustrate a comparison between the input and output noise levels. Note that, the noise effect is almost negligible when the system is in steady state. However, during transitions noise effects are more emphasized. We have shown that by a globally recurrent MLP structure, where we have employed a modified backpropagation scheme to speed up the training, we can effectively capture the dynamics of the reactor core. The neural network model offers the following specific
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advantages over conventional methods: It provides considerable speed-ups for faster than real-time prediction of the system output by using dedicated hardware under a variety of conditions; is robust with respect to noisy/incomplete data; has the capability to recover when started from arbitrary initial conditions. Also, the network is capable of providing good extrapolation/interpolation performance.
6. Appendix
The first-principles engineering models REal Time Advanced Core Thermo-hydraulics (RETACT) and REal-time Multi-group Advanced Reactor Kinetics (REMARK) [16] are designed and applied to simulate the thermal hydraulic and neutronic phenomena of nuclear reactors. REMARK is a two energy group, three dimensional, finite-difference, diffusion theory core model developed for real time simulation of reactor core physics. It can simulate the core neutron physics during normal operation (startup, shutdown, and power manipulation), anticipated transient occurrence, and accidental transient events. In addition, the REMARK core model can accurately reproduce the core thermal power behavior in fast and localized transients. In other words, it provides higher fidelity in the core thermal power feedbacks to the thermal-hydraulic model and the nuclear instrumentation model. There are three systems interfaced with REMARK, the thermal-hydraulic (TH) system, the control rod drive (RD) system, and the neutron instrumentation (NI) system. The thermal-hydraulic conditions of the reactor core, including fuel rod temperature, fuel cladding temperature, moderator temperature/density, coolant liquid phase temperature/ density, coolant gas phase temperature/density, coolant void fraction, and boron concentration in coolant, are calculated in the TH system and they constitute the inputs to REMARK. In addition, the control rods provide a mechanism to control reactor thermal power by operators. Due to the fact that the TH system has a coarser nodalization in the active core region compared to the REMARK model, the values calculated in the TH system are mapped into the fine meshes of REMARK based on power distribution. Based on the mapped TH conditions in each mesh, REMARK calculates the thermal power and power distribution, and then feeds these values back to the TH system. RD system provides the control rod positions and the NI system provides the locations of neutron detectors to REMARK, and obtains the flux readings from REMARK [16]. Also note that NI is not included as an input to the neural network, since it is an internal parameter that is used by the software structure of REMARK. These interactions are illustrated in Fig. 12 together with the neural network training set-up. The core neutronic model REMARK is governed by two differential equations, the system diffusion equation and the delayed neutron precursor equations. The system diffusion equation is given by 1 &&r,t) - = VDV@(r, t) - C&&r, t) + @f@(r, v at
t),
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,
1
/
rod
311
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CR
REMARK
positions (RI3
, t
TH
A -7
j+ NN
Fig. 12. Neural network training
set-up.
where @(r, t) is the neutron flux and D is the neutron cross-section depends on the water and fuel temperatures and the neutron poison Nboron,
Nwlon
and
f%kx-mium~
The delayed neutron
precursor
1 @l(r, t) = VD1Vd+(r,t) v, at
-~
1 a@2@,t)
-~
v2
at
Xi - -X z-
density which concentrations
equations
are of the form:
- &l@l(r,t)
+ C1_Z@l(r,t)
+(l - PI (VCfl@l(r,t) + vCf2@2(r,t))
+C
= V&V@2(r, t) - &@2(r, t) + Ci-.2@i(r,
t),
+ Pi(vCf~@l(r,t) + vCf2@2(r,t))
p(t).
ACi
+
Lt,
(14)
(15)
(16)
In the neutron precursor equations (14)-(16), Eq. (14) is used to solve @l(r, t) using Ci in (16), and Eq. (15) is used to solve c&(r, t) using &(r, t) in Eq. (14). The flux distribution is then used to compute the reactor thermal power variables and the
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Fig. 13. Nodalilzation
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of the Reactor Core in REMARK.
Fig. 14. Configuration
of the coolant pumps.
other related quantities. These include reactor thermal power (nuclear fission heat and decay heat), number densities of fission products (xenon and samarium), core reactivity, reactor period, neutron flux instrumentation readings, and 3-dimensional power distribution in the core (Fig. 13). These outputs either provide heat to the coolant circulation and steam generation systems, or give indications for the purpose of reactor control. The version of REMARK we use has 146 external inputs and 52 external outputs that characterize the reactor core physics, and internal state variables which are
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used for the internal computations in the core nodalization shown in Fig. 13. Through proper setting of the internal variables, REMARK can generate real-time data corresponding to various scenarios which describe different operation conditions of the nuclear plant. Due to the configuration of the circulation pumps around the reactor core (Fig. 14), tripping different combination of circulation pumps with or without a total reactor shut down result in different transient characteristics. The training and testing data for the RMLP are obtained through various combinations of such operating scenarios.
References [l] T. Adali, B. Bakal, R. Fakory, D. Komo, M.K. Sonmez, C.O. Tsaoi, Simulation of nuclear reactor core dynamics by recurrent neural networks, in: Computer-Based Human Support Systems: Technology, Methods and Future, American Nuclear Society, Philadelphia, PA, June 1995, pp. 475-479. [2] B. Bakal, Temporal processing with neural networks: nuclear reactor core simulation and NOX prediction in fossil fuel plants, MS. Thesis, University of Maryland Graduate School, Baltimore, MD, 1996. [3] B. Bakal, T. Adali, D. Komo, Symnet: Technical Manual, S3 Technologies, Columbia, MD, USA. [4] B. Bakal, T. Adali, R. Fakory, D. Komo, M.K. Sonmez, 0. Tsaoi, Modeling core neutronics by recurrent neural networks, in: Proc. World Congress on Neural Networks, Washington, DC, July 1995, vol. 2, pp. 504-508. [5] M. Brown, P.C. An, C.J. Harris, H. Wang, How biased is your multi-layer perceptron?, World Congress on Neural Networks 3, Portland, OR, 1993, pp. 507-511. [6] S.E. Fahlman, An empirical study of learning speed in back-propagation networks, Report CMU-CS90-100, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 1988. [7] J. Hertz, A. Krogh, R.G. Palmer, Introduction to the Theory of Neural Computation, Addison Wesley, Reading, MA, 1991. [8] B.G. Home, C.L. Giles, An experimental comparison of recurrent neural networks, in: Neural Information Processing Systems 7. [9] R.A. Jacobs, Increased rates of convergence through learning rate adaptation, Neural Networks 1 (1988) 295-307. [IO] K.S. Narendra, K. Parthasarathy, Identification and control of dynamical systems using neural networks, IEEE Trans. Neural Networks l(1) (1990). [I I] C. Pal, I. Hagiwara, N. Kayaba, S. Morishita, Dynamic system identification by neural network a new, fast learning method based on error back propagation, J. Intelligent Mat. Syst. Structures 5 (1994) 127-135. [12] A.G. Parlos, K.T. Chong, A.F. Atiya, Application of the recurrent multilayer perceptron in modeling complex dynamics, IEEE Trans. Neural Networks 5(2) (1994) 255-266. [13] A.K. Rigler, J.M. Irvine, T.P. Vogl, Resealing of variables in back propagation learning, Neural Networks 4 (1991) 225-229. [14] P.S. Sastry, G. Santharam, K.P. Unnikrishnan, Memory neuron networks for identification and control of dynamical systems, IEEE Trans. Neural Networks 5(2) (1994) 306319. [15] G. Thimm, P. Moerland, E. Fiesler, The interchangeability of learning rate and gain in backpropagation neural networks, Neural Computation, submitted. [16] C.O. Tsaoi, REMARK: REal Time Multigroup Advanced Reactor Kinetics, testing results for PWR, 1993 SCS Simulation Multiconference, Arlington, VA, 1993. [17] R.E. Uhrig, Use of neural networks in nuclear power plants, ISA Trans. 32 (1993) 139-145.
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T&y AdaIr received the B.S. degreee from the Middle East Technical University, Ankara, Turkey in 1987 and the M.S. and Ph.D. degrees from the North Carolina State University, Raleigh in 1988 and 1992, respectively, all in electrical enginering. In 1992, she joined the Department of Computer Science and Electrical Engineering at the University of Maryland, Baltimore Country as an assistant professor. She has served on the program committees of a number of international conferences including the European Signal Processing Conference (EUSIPCO), IAESTED International Conference on Signal and Image Processing, and the IEEE Intemational Workshop on Neural Networks for Signal Processing (NNSP). She is the guest editor of a special issue on Neural Networks for Biomedical Imaging/Image Processing for the Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology r on Neural Networks. Her research interests are in statistical signal processing, neural computation, adaptive signal processing and their applications in channel equalization, system identification, time-series prediction, and image analysis. Dr. Adah is the recipient of a 1997 National Science Foundation CAREER award.
Bora Bakal received his B.S. degree in Electrical and Electronics Engineering from Middle East Techinical University, Ankara, Turkey, in 1993. He received his M.S. degree in Electrical Engineering from the University of Maryland, Baltimore Country, Baltimore, MD, in 1995. His interest area is signal processing with neural networks. He is currently employed as a software engineer by Faraday National Corporation, Chantilly, VA.
M. KemaI Siinmez received his B.S. degree in Electrical and Electronics Engineering from Middle East Technical Universitv, Ankara, Turkev in 1989, and his MS. degree from North Carolina State Univemity, Raleigh, NC in 1991. He is currently pursuing his Ph.D. degree in Electrical Engineering. He is with Speech Technology and Research Laboratory of SRI International since 1996. University of Maryland. College Park, MD. His research areas include statistical signal processing and artifical neural networks.
Reza Fakery received his M.S. degrees in Nuclear Engineering and Mechanical Engineering. from Massachusetts Institute of Technoloav in 1977. and his Ph.D. decree & Nucl&r Engineering from Rensselaer Polyte&% Institute, USA in 1983. &rrently, he is the principle investigator at S3 Technologies Company, Columbia, MD, USA in the research and development department in the areas of thermal-hydraulic and RETACTR enhancement, simulation of severe accident in PWRs/BWRs, and RETACT code standardization. He also had worked as staff scientist at Computer Simulation Company where he developed advanced models for simulation of NSSS and reactor core systems on the Arkansas Nuclear I simulator (ANO), and as a staff scientist at Simulation Associates, Inc., USA, where he was responsible for model development
on NSSS for Surry, North Anna and Grafen Reinfield (Germany)
simulators.
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C. Oliver Tsaoi received his B.S. from National Tsing-Hua University in 1979, his MS. and Ph.D. degrees from Purdue University in 1987 and 1990, all in Nuclear Engineering. He is currently the Engineering Manager responsible for the development and application of the core neutronics model, REMARK, in S3 Technologies Company, Columbia, MD, USA.
ELSEVIER
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Modeling nuclear reactor core dynamics with recurrent neural networks ’ Tiilay Ada11 al*, Bora Bakal a, M. Kemal Siimnez b, Reza Fakory ‘, C. Oliver Tsaoi” a Information Technology Laboratory, Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, Baltimore MD 21250, USA b Institute for Systems Research, University of Maryland, College Park, MD 20742, USA ’ Simulation, Systems & Services Technologies Company (S3 Technologies), Columbia MD 21045, USA
Abstract
A recurrent multilayer perceptron (RMLP) model is designed and developed for simulation of core neutronic phenomena in a nuclear power plant, which constitute a non-linear, complex dynamic system characterized by a large number of state variables. Training and testing data are generated by REMARK, a first principles neutronic core model [ 161. A modified backpropagation learning algorithm with an adaptive steepness factor is employed to speed up the training of the RMLP. The test results presented exhibit the capability of the recurrent neural network model to capture the complex dynamics of the system, yielding accurate predictions of the system response. The performance of the network is also demonstrated for interpolation, extrapolation, fault tolerance due to incomplete data, and for operation in the presence of noise. Keywords:
Nuclear reactor core dynamics; Recurrent neural networks
1. introduction In recent years, there has been considerable interest in modeling dynamic systems with neural networks. The basic motivation is the ability of neural networks to create
’ This work is supported by Maryland Industrial Partnerships 1214.12 and MIPS-1214.24. * Corresponding author. Email: [email protected] 0925-23 12/97/$17.00 Copyright PZISO925-2312(97)00018-O
and S3 Technologies
0 1997 Elsevier Science B.V. All rights reserved
under grants no. MIPS-
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data-driven representations of the underlying dynamics with less reliance on accurate mathematical/physical modeling. There exist many problems for which such data-driven representations offer advantages over more traditional modeling techniques, such as availability of fast hardware implementations, ability to cope with noisy/incomplete data, and avoidance of complicated internal state representations. In this work, we make a system identification of core neutronic phenomena in a nuclear reactor using a Recurrent Multilayer Perceptron (RMLP). The training data is obtained from REMARK, a first principles neutronic core model which can model both normal operation and fast and localized transients. REMARK in its plant specific versions is currently being used to train personnel who will be working in nuclear power plants and is tested with a number of standard real plant data to verify its approximation capability. We can simulate various scenarios in nuclear plant operation such as reactor and pump trips and generate training and test data sets which obviously cannot easily be obtained in a real nuclear plant. We introduce a novel modified backpropagation scheme for training the RMLP to speed up learning. The proposed scheme keeps the network weights active in early stages of learning by adapting steepness of the activation function and using an annealing schedule to ensure convergence. It is demonstrated that, by the procedure, learning speed is improved considerably, a feat especially important in a modeling task of this size which consists of some 146 inputs and 52 outputs. The performance of RMLP model is evaluated in terms of its prediction performance on unseen test sets for interpolation and extrapolation, fault tolerance in cases of incomplete data, and for operation in the presence of additive white gaussian noise. Accurate predictions provided by RMLP demonstrate its success in capturing the underlying complex dynamics of the neutronic core. The specific advantages of RMLP for core neutronic modeling are: (i) Providing speed-ups in system prediction by using dedicated hardware which provide much needed faster than real-time prediction power; (ii) Better robustness to noisy data, ability to provide estimates of the system response in the presence of missing measurements (possibly due to malfunctioning equipment) which are harder to deal with in a model based implementation; (iii) The capability to start predictions from arbitrary initial conditions which allow recovery of transients without the need to set internal parameters necessary in model based implementations. The next section, Section 2, introduces the notation for the recurrent neural network structure employed in this application and for backpropagation learning. In the third section, the modified delta learning rule is introduced for a globally recurrent MLP, and its operation is discussed. The fourth section provides data acquisition details and finally the fifth section presents results that demonstrate advantages of the neural network approach over conventional approaches in terms of properties such as robustness in cases of noisy/incomplete data and extrapolation and interpolation capabilities. The appendix includes an overview of the dynamics of the neutronic core model REMARK and describes data acquisition using REMARK.
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2. The RMLP model We employ a recurrent network structure to capture the dynamics of the nuclear reactor core as represented by transients under different operating conditions. We adopt a global RMLP which incorporates a unit delayed connection between the output layer and the input layer. The structure is illustrated in Fig. 1. Ref. [8] presents an experimental comparative study of various recurrent architectures on different classes of problems and reports that the RMLP and Narendra and Parthasarathy’s [lo] model outperform other recurrent architectures that they consider for a nonlinear system identification problem. In [l], we show that the RMLP structure provides a powerful representation of the core dynamics outperforming a time delay neural network (TDNN) structure in that it provides a much better approximation of the input dynamics with a network size much smaller. We introduce the notation for the globally recurrent MLP as follows: Let x(n) denote the N x 1 external input vector applied to the network at time n, and let y(n) denote the corresponding A4 x 1 vector of outputs at time n. The external input vector x(n) and one step delayed output vector y(n - 1) are concatenated to form the input layer
xl(n)
Fig. 1. RMLP structure.
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to the network. The network is fully interconnected with V(n) and W(n) as its weight matrices. Let Uij(n) denote the connection weight between ith input node and jth hidden neuron, and wV(n) denote the connection weight between ith hidden neuron and jth output neuron. The output of kth output neuron at time n is given by . ~N+hf _
R‘t!
f v,w *
(1)
and we define the kth neuron output as Y/C(~)=
[fV,W(x(n),y(n
with
-
l))lk
=
“(Sk@))
(2)
J Sk(n)
= c
(3)
wjk(n)zj(n)
j=l and zk(n)
= Q
5
&k(n)%(n)
+ 5
i=l
~(N+m)k(n)ynt(n
-
1)
(4)
3
in=1
where J is the number of neurons in the hidden layer, a(.) is the node activation function, and k = 1,. . . ,M. With the notation introduced above, the weights and v+ are adapted by the steepest descent rule wjk
Wjk(n + 1) = wjk(n)
-
Wn)
CL-
(5)
awjk(n)
(similarly for Vij) such that the instantaneous squared error
E(n) = ;
k@k(n)
-
(6)
yk(n)>2
k=l
between the desired dk(n) and the network outputs yk(n) is minimized over all outputs. The instantaneous gradient terms for the weights are given by (7) for the hidden to output layer weights, and 41
- $(n))xi(n)
gwjkcn)(dk(n)
-
yk(n))(l
-
YR2Cn)h
k=l
i <
aE(n)=
N,
a&j(n) 41
- zj(n)>yi-dn
- 112
wjk(n)(dk(n)
-
YkCnIMl
-
d(n)),
k=l
N
(8)
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for the update of input to hidden signal.
layer weights through backpropagation
of the error
3. The modified delta learning rule In this section, we introduce a modified backpropagation scheme for training the RMLP. The algorithm is a gradient descent technique with a dynamic steepness factor in the activation functions of the nodes in the hidden layer. We define the activation functions as ~(a~) = tanh($),
(9)
where 2 is the steepness that -
N
factor, and Uj is the total input of the jth hidden neuron
-
such
M
@j = >, Ug(n)Xi(n) + >, U(N+m)j(n)Ym(n - l), Convergence of backpropagation-type ploying sigmoid-like activation functions
i = 1,. . . , J.
learning algorithms in neural networks emsuffer from premature increases in some node
weights which force node outputs to the saturated region of the activation function in early stages of learning. The learning rate of the nodes is thus diminished due to vanishingly small gradients. There have been several proposed fixes for this problem in the literature, such as independent learning rates for each weight, adaptation of learning rates during learning [5,9,11], scaling [13] or adding a constant [6] to the the gradient term backpropagated to the previous layers. We propose the following technique for adaptation of the steepness factor L of the activation functions in the hidden layer depending on the magnitude of the inputs to the hidden layer node. The goal is to keep the weights active during the early stages of learning by adjusting steepness according to a representative node selected by the maximum magnitude of the input to hidden node. To ensure convergence, an annealing schedule which, as training progresses, decreases the steepness adjustment is also proposed. The resulting algorithm takes aggressive steps early due to the large gradients and converges as the steepness adjustment is cooled down. We can write the adaptive procedure for the steepness of the activation function as A(n) =
A0 max{4,a2,...,crJ}
P(n)+ A(1 - p(n)),
(11)
where p(n)
is the annealing parameter which can be chosen such that p( 1) = 1 and as it -+ CCLA typical choice for p(n) is a sequence of order O(nP) for some tl > 0. We demonstrate the effect of annealing in the update with a simple example. Consider the nonlinear system identification problem where the system to be identified is described by the relationship p(n)
--+ 0
d@)
=
x(n)(l + 0 - 1)) 1 +&(n
- 1)
(12)
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Fig. 2. Effect of steepness adaptation Solid line: standard backpropagation, with the annealing procedure.
IS (1997)
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and annealing procedure for the simple system identification problem. dotted line: steepness adaptation, dash-dot line: steepness adaptation
and the input is chosen as a monotonously decreasing ramp in the range [-1 J]. A 2-3-l globally RMLP is trained with 80 training samples. In Fig. 2 we show the initial speed up in the learning with the steepness update with (p(n) = l/n) and without annealing (p(n) = 1) for 50 epochs. The adaptive steepness learning curves for both cases (with and without annealing) show a considerable speed up in convergence. Also, with annealing, the network achieves a much lower final training error. Fig. 3 indicates a similar trade-off for the same system identification problem (12) where this time lo is let to change with no annealing. In the nuclear reactor core modeling problem, the modification of backpropagation algorithm results in considerable improvement in terms of both the learning characteristics and accuracy. This improvement is illustrated in Fig. 4 by comparing the two learning curves, that are obtained with (dotted line) and without (solid line) the adaptation of the activation function. Since there is a natural convergence in the value of n(n), we do not employ annealing for this problem, hence p(n) is set to 1. The value of & is chosen as the steepness value J.(n) naturally converges to, which is 6 in our implementation. The behavior of the adaptive steepness factor n(n) for our reactor core problem is shown in Fig. 5. With the adaptation, the number of iterations required to reach the same training error is much smaller than the number of iterations required without the adaptation. The final value of the training error is also reduced considerably.
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Fig. 3. Effect of As in steepness adaptation Curve 2: 1s = 1.0, Curve 3: J&j= 2.0.
0
200
for the simple system identification
400
600
(a) Number of epochs
0
200
problem.
400
Curve 1: I.0 = 0.5,
600
600
(b) Number of epochs
Fig. 4. Effect of steepness adaptation for reactor core modejjng. (a) Learning curves for single pump trip scenario, (b) learning curves for two pump trip scenario. Curves are plotted with (dotted) and without (solid) the adaptation of the steepness factor.
4. Data acquisition and implementation The version of REMARK we use has 146 external inputs and 52 external outputs that characterize the reactor core physics, and internal state variables as discussed in the appendix. To generate training and test data sets, REMARK is run with various scenarios which describe different operation conditions of the nuclear plant. All of the scenarios used in our experiments assume a disturbance of the system from its steady state operating condition, i.e., modeling of various transient responses of the system. These transients are achieved by setting the internal parameters of REMARK. The steady state response of the reactor core as we show in [3], can be achieved by a simple feedforward structure. In [3], we present a full model of the system (the steady state and the
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Fig. 5. Adaptive steepness l(n) behavior during training for reactor core modeling
for two pump trip scenario.
three types of transient responses considered) using a switch mechanism between three RMLPs and a simple feedforward MLP. The three transient scenarios are core trip, single pump trip, and double pump trips. In our implementation, the switch uses REMARK internal variables for control of the switching mechanism. In a real plant scenario, various plant parameters can be used for this purpose, such as using the rod positions for checking for reactor trip [3]. The structure of the switch mechanism is excluded in this paper because of the additional introduction it necessitates to the structure of REMARK. Due to the configuration of the circulation pumps around the reactor core (Fig. 14), tripping different combination of circulation pumps (for the double pump trip scenario) and tripping a different pump (for the single pump trip scenario) result in different transient characteristics. When the pump trips are coupled with a reactor shut down, rapid transients occur with reactor shut down having the dominant effect. For pump trip(s) without reactor shut down, we observe slower transient characteristics. For single pump trips, the power distribution in the reactor core varies depending on the location of the pump tripped. Therefore, to evaluate the generalization performance of the network we use data generated by tripping a different pump than the one used in training. For double pump trips, we consider tripping two pumps on the opposing sides of the cooling system (see Fig. 14) since this scenario does not cause system unbalance due to reverse leakage of coolant fluid as the parallel pump trips do. Again for testing we use data collected by tripping a different two-pump combination than the one used in training. We collect samples generated by REMARK with 4Hz sampling frequency. The network is trained by using the first 1000-1500 samples (samples l/4 s apart) and the generalization performance is tested on a different data set (obtained by tripping a different pump or combination of pumps). We also test for samples after the initial n samples used in training to see the prediction capability of the network
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into the future. Also, of particular interest is starting the network from an arbitrary (in our case all zeros) initial condition instead of its steady-state value to see the ability of the network to generalize to this case. The performance of the network in this test (start-up from an arbitrary initial condition) presents a particular advantage of the neural network modeling with respect to first principles modeling as will be explained later. The inputs to REMARK are: liquid temperature (9 units), liquid density (9 units), gas density (9 units), boron concentration (9 units), fuel rod temperature (36 units), cladding temperature (inner surface (36 units) and outer wall (36 units)), vessel saturation temperature (1 unit) (thermal-hydraulic conditions supplied to REMARK by RETACT [ 161) and the control rod positions (1 unit). The outputs are: power (1 unit), prompt fission power (1 unit), decay heat (1 unit), Xenon number density (1 unit), effective multiplication factor (1 unit), core reactivity (1 unit), normalized heat slab power distribution (36 units), power range monitor reading (8 units), and source range monitor reading (2 units). After data collection from REMARK, a normalization scheme is applied to transform the ranges of variables into the interval [ - l,l] providing a common dynamic range for each variable for presentation to the neural network. The RMLP structure employed to capture the system dynamics has an input vector of size 198 (146 external inputs and unit delayed 52 network output variables), 52 output nodes, and 60 hidden nodes which is determined empirically by scanning a range 3&120 hidden nodes in increments of 10. The value of 20 used in the implementation is 6 which is chosen as the steepness value n(n) naturally converges to with no annealing, i.e. with p(n) = 1 in (11).
5. Results As explained in Section 4, we consider three cases that result in three major types of transient responses in nuclear plants which are the single and double pump trips and the total reactor shut down. Within each type, different transients occur depending on the location of the pump(s) tripped. Since the three types of transients are distinctively different, for achieving a good tradeoff between the required accuracy of prediction performance and the network generalization, three separate RMLPs are trained for each of the three scenarios (single, double, and total reactor trips). The control mechanism selecting between these RMLPs is discussed in the previous section and in detail in [3]. The results show that the RMLP can capture the complex dynamics of the system, yielding satisfactory predictions of the expected transients for all the cases considered. In Fig. 6, we show RMLP prediction curves for four output variables for fast transient prediction, i.e., with reactor shut down. As seen in the figure the RMLP yields a close prediction of the system output for this fast transient which is particularly important in a nuclear power plant. To study the capabilities of RMLP for modeling core dynamics and to highlight its advantages over the conventional approaches we consider three specific problems and investigate extrapolation and interpolation capabilities of the network, its robustness with respect to noisy/incomplete data, and its transient recovery capabilities.
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Decay Heat lr
-1’
0
Xenon Number Density 1
500
1000
1 1500
Effective Multiplication Factor
_ITz 0
500
1000
Fig. 6. RMLP test results: Neural network time ( f S) (reactor trip scenario).
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lr
-1’
0
,
500
1000
I 1500
Normalized Heat Slab Power Distribution
1500
_LcTI 0
(dotted line) vs. REMARK
500
1000
1500
(solid line) outputs as a function
of
An important issue in a system modeling problem is the long term prediction capabilities of the network. This becomes an even more critical issue in a problem of this size which would imply training for large amounts of data for long-term modeling. As explained in the previous section the data used in training are the first 1000-1500 samples taken with 4Hz frequency. Naturally, an important performance consideration is the long term prediction capability of the network, i.e. when it is trained on the first n samples, looking at the prediction performance of the network beyond these n samples. In a normal operation scenario, everything except the outputs will be supplied for prediction performance. We test the long term prediction, i.e. the extrapolation capabilities, of the network by training the network with the first 1000 samples in the sequence (taken every 114 s) and then performing the test with a complete sequence of 4000 samples (including the first 1000 used for training). These results are shown in Fig. 7 for a single pump shut-down scenario. As seen in the figure, the network does a very good job of predicting the future system response. Also, the interpolation capabilities are tested by training the network with a data set down-sampled 10 : 1. That is, only one sample in every 10 samples is used for training (400 out of a total of 4000). This could be of great advantage in cases where the size of the training data is very large. These results are shown in Fig. 8. Note that the point tested here is that if because of storage or memory capabilities the data used for training is subsampled, how well the network can do the interpolation among the samples used for training for the samples that are missing in training. The network is also tested for fault tolerance, i.e., with incomplete data which might be because of a malfunctioning equipment; a harder problem to deal with in a model based approach. For this purpose, two different sets of four randomly selected input variables are set to zero to represent a faulty condition, and the experiment is repeated twice for each set. The test is performed with the reactor shut down scenario. In the
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Normalized Core Thermal Power _iI‘--i-i
0
Normalized Heat Slab Power Di.Mbution _py=yy
1000
3000
2000
4000
0
0
2000
3000
4000
Source Range Monitor Reading
Normalized Heat Slab Power Distribution
!1_1---
1000
_!=I
1000
3000
2000
4000
0
1000
2000
3000
4000
Fig. 7. Test results: evaluation of RMLP extrapolation capabilities. The network is trained only with the first 1000 samples. Neural network (dotted line) vs. REMARK (solid line) outputs as a function of time (is) (single pump trip scenario).
Normalized Heat Slab Power Distribution
Decay Heat
[\-I
0
_[ri
1000
2000
3000
4000
0
Normalized Heat Slab Power Distribution
_;vi
0
1000
2000
3000
4000
Power Range Monitor Reading
;=
1000
2000
3000
4000
0
1000
2000
3000
4000
Fig. 8. Test results: evaluation of RMLP extrapolation capabilities. The network is trained with a 10: 1 down-sampled data. Neural network (dotted line) vs. REMARK (solid line) outputs as a function of time ($ s) (single pump trip scenario).
first realization, where liquid density ( 1 ), gas density (1 ), and fuel rod temperatures (2) are set to zero, RMLP generated the same prediction accuracy as in the normal operation (similar to Fig. 6). However, for the second realization where liquid temperatures (2) inner surface cladding temperature (1 ), and outer wall cladding temperature (1) are set to zero, 3 outputs (Xenon m.unber density, effective multiplication factor, and core reactivity) out of 52 showed a maximum of 15% deviation from the original
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314
Decay Heat
Core Reactivity
‘1
I
-I’
500
1000
Normalized Heat Slab Power Distribution 11 1
j/f----
-A-
500
1500
1000
Normalizad Heat Slab Power Distribution
pL.___. I
-1’
500
1000
1500
500
1500
1000
1500
Core Readvity
Decay Heat 0.5 ‘I
4
2
6
Normalized Heat Slab Power Distribution
-0.75
4
6
._ ‘y.-2
...
8
Normalized Heat Slab Power Distribution 0.6 .‘..,
-::r] -0.7’
2
6
I
,_._,-..- .._-._._--
4
6
8
0’
2
4
6
I 8
Fig. 9. First two rows: the prediction curves (dash-dot: from known initial conditions, dotted: from unknown initial conditions) vs. REMARK outputs (solid) as a function of time (is). Bottom two rows: the first 8 data samples for the observation of recovery from unknown initial conditions.
prediction curves only at the steady-state. The scenario that is used for this test is the reactor shut down. Model based implementations, in general, require proper setting of some internal variables which might create problems in situations where, for example in a real plant, the predictions are to be made from an arbitrary initial condition. We tested the RMLP prediction by starting from an arbitrary point on the transient curve (i.e. not from its steady-state value) and by setting the initial outputs to zero. As shown in Fig. 9 the recurrent structure converges to the desired response in about 5 data samples, demonstrating a very important advantage, recovery from arbitrary initial conditions, over the conventional techniques. The results shown are for a double pump trip scenario.
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Liquid Temperature
Liquid Temperature 21
1
_;\--i 0
500
1000
1500
a~ 21
0
500
Fig. 10. Input sequence without noise (left), input sequence with an additive gaussian (right) both as a function of time (is).
Normalized Core Thermal Power
1000
1500
noise of variance
0.01
Normalized Heat Slab Power Distribution 1
‘l 0.5 0 -0.5 r. -1’
500
0
1000
I 1500 Power Range Monitor Reading
Normalized Heat Slab Power Distribution ‘I ‘I 0.5.
0.5
0.
0 -0.5
-0.5. -1;
I, 0
‘1...._ 500
1000
1500
Fig. 11. Predictions with the noisy input as a function line: network output).
-1’
0
of time (is).
’ 500
1000
(Solid line: REMARK
1 00 output, dotted
Finally, the performance of the network is tested in the presence of white gaussian noise. An additive noise of variance of 0.01 is introduced at the input of the network to the variables normalized to [-l,l]. Samples of noisy predictions are given in Fig. 11. Fig. 10 is added to illustrate a comparison between the input and output noise levels. Note that, the noise effect is almost negligible when the system is in steady state. However, during transitions noise effects are more emphasized. We have shown that by a globally recurrent MLP structure, where we have employed a modified backpropagation scheme to speed up the training, we can effectively capture the dynamics of the reactor core. The neural network model offers the following specific
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advantages over conventional methods: It provides considerable speed-ups for faster than real-time prediction of the system output by using dedicated hardware under a variety of conditions; is robust with respect to noisy/incomplete data; has the capability to recover when started from arbitrary initial conditions. Also, the network is capable of providing good extrapolation/interpolation performance.
6. Appendix
The first-principles engineering models REal Time Advanced Core Thermo-hydraulics (RETACT) and REal-time Multi-group Advanced Reactor Kinetics (REMARK) [16] are designed and applied to simulate the thermal hydraulic and neutronic phenomena of nuclear reactors. REMARK is a two energy group, three dimensional, finite-difference, diffusion theory core model developed for real time simulation of reactor core physics. It can simulate the core neutron physics during normal operation (startup, shutdown, and power manipulation), anticipated transient occurrence, and accidental transient events. In addition, the REMARK core model can accurately reproduce the core thermal power behavior in fast and localized transients. In other words, it provides higher fidelity in the core thermal power feedbacks to the thermal-hydraulic model and the nuclear instrumentation model. There are three systems interfaced with REMARK, the thermal-hydraulic (TH) system, the control rod drive (RD) system, and the neutron instrumentation (NI) system. The thermal-hydraulic conditions of the reactor core, including fuel rod temperature, fuel cladding temperature, moderator temperature/density, coolant liquid phase temperature/ density, coolant gas phase temperature/density, coolant void fraction, and boron concentration in coolant, are calculated in the TH system and they constitute the inputs to REMARK. In addition, the control rods provide a mechanism to control reactor thermal power by operators. Due to the fact that the TH system has a coarser nodalization in the active core region compared to the REMARK model, the values calculated in the TH system are mapped into the fine meshes of REMARK based on power distribution. Based on the mapped TH conditions in each mesh, REMARK calculates the thermal power and power distribution, and then feeds these values back to the TH system. RD system provides the control rod positions and the NI system provides the locations of neutron detectors to REMARK, and obtains the flux readings from REMARK [16]. Also note that NI is not included as an input to the neural network, since it is an internal parameter that is used by the software structure of REMARK. These interactions are illustrated in Fig. 12 together with the neural network training set-up. The core neutronic model REMARK is governed by two differential equations, the system diffusion equation and the delayed neutron precursor equations. The system diffusion equation is given by 1 &&r,t) - = VDV@(r, t) - C&&r, t) + @f@(r, v at
t),
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1
/
rod
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CR
REMARK
positions (RI3
, t
TH
A -7
j+ NN
Fig. 12. Neural network training
set-up.
where @(r, t) is the neutron flux and D is the neutron cross-section depends on the water and fuel temperatures and the neutron poison Nboron,
Nwlon
and
f%kx-mium~
The delayed neutron
precursor
1 @l(r, t) = VD1Vd+(r,t) v, at
-~
1 a@2@,t)
-~
v2
at
Xi - -X z-
density which concentrations
equations
are of the form:
- &l@l(r,t)
+ C1_Z@l(r,t)
+(l - PI (VCfl@l(r,t) + vCf2@2(r,t))
+C
= V&V@2(r, t) - &@2(r, t) + Ci-.2@i(r,
t),
+ Pi(vCf~@l(r,t) + vCf2@2(r,t))
p(t).
ACi
+
Lt,
(14)
(15)
(16)
In the neutron precursor equations (14)-(16), Eq. (14) is used to solve @l(r, t) using Ci in (16), and Eq. (15) is used to solve c&(r, t) using &(r, t) in Eq. (14). The flux distribution is then used to compute the reactor thermal power variables and the
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Fig. 13. Nodalilzation
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of the Reactor Core in REMARK.
Fig. 14. Configuration
of the coolant pumps.
other related quantities. These include reactor thermal power (nuclear fission heat and decay heat), number densities of fission products (xenon and samarium), core reactivity, reactor period, neutron flux instrumentation readings, and 3-dimensional power distribution in the core (Fig. 13). These outputs either provide heat to the coolant circulation and steam generation systems, or give indications for the purpose of reactor control. The version of REMARK we use has 146 external inputs and 52 external outputs that characterize the reactor core physics, and internal state variables which are
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used for the internal computations in the core nodalization shown in Fig. 13. Through proper setting of the internal variables, REMARK can generate real-time data corresponding to various scenarios which describe different operation conditions of the nuclear plant. Due to the configuration of the circulation pumps around the reactor core (Fig. 14), tripping different combination of circulation pumps with or without a total reactor shut down result in different transient characteristics. The training and testing data for the RMLP are obtained through various combinations of such operating scenarios.
References [l] T. Adali, B. Bakal, R. Fakory, D. Komo, M.K. Sonmez, C.O. Tsaoi, Simulation of nuclear reactor core dynamics by recurrent neural networks, in: Computer-Based Human Support Systems: Technology, Methods and Future, American Nuclear Society, Philadelphia, PA, June 1995, pp. 475-479. [2] B. Bakal, Temporal processing with neural networks: nuclear reactor core simulation and NOX prediction in fossil fuel plants, MS. Thesis, University of Maryland Graduate School, Baltimore, MD, 1996. [3] B. Bakal, T. Adali, D. Komo, Symnet: Technical Manual, S3 Technologies, Columbia, MD, USA. [4] B. Bakal, T. Adali, R. Fakory, D. Komo, M.K. Sonmez, 0. Tsaoi, Modeling core neutronics by recurrent neural networks, in: Proc. World Congress on Neural Networks, Washington, DC, July 1995, vol. 2, pp. 504-508. [5] M. Brown, P.C. An, C.J. Harris, H. Wang, How biased is your multi-layer perceptron?, World Congress on Neural Networks 3, Portland, OR, 1993, pp. 507-511. [6] S.E. Fahlman, An empirical study of learning speed in back-propagation networks, Report CMU-CS90-100, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, 1988. [7] J. Hertz, A. Krogh, R.G. Palmer, Introduction to the Theory of Neural Computation, Addison Wesley, Reading, MA, 1991. [8] B.G. Home, C.L. Giles, An experimental comparison of recurrent neural networks, in: Neural Information Processing Systems 7. [9] R.A. Jacobs, Increased rates of convergence through learning rate adaptation, Neural Networks 1 (1988) 295-307. [IO] K.S. Narendra, K. Parthasarathy, Identification and control of dynamical systems using neural networks, IEEE Trans. Neural Networks l(1) (1990). [I I] C. Pal, I. Hagiwara, N. Kayaba, S. Morishita, Dynamic system identification by neural network a new, fast learning method based on error back propagation, J. Intelligent Mat. Syst. Structures 5 (1994) 127-135. [12] A.G. Parlos, K.T. Chong, A.F. Atiya, Application of the recurrent multilayer perceptron in modeling complex dynamics, IEEE Trans. Neural Networks 5(2) (1994) 255-266. [13] A.K. Rigler, J.M. Irvine, T.P. Vogl, Resealing of variables in back propagation learning, Neural Networks 4 (1991) 225-229. [14] P.S. Sastry, G. Santharam, K.P. Unnikrishnan, Memory neuron networks for identification and control of dynamical systems, IEEE Trans. Neural Networks 5(2) (1994) 306319. [15] G. Thimm, P. Moerland, E. Fiesler, The interchangeability of learning rate and gain in backpropagation neural networks, Neural Computation, submitted. [16] C.O. Tsaoi, REMARK: REal Time Multigroup Advanced Reactor Kinetics, testing results for PWR, 1993 SCS Simulation Multiconference, Arlington, VA, 1993. [17] R.E. Uhrig, Use of neural networks in nuclear power plants, ISA Trans. 32 (1993) 139-145.
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T&y AdaIr received the B.S. degreee from the Middle East Technical University, Ankara, Turkey in 1987 and the M.S. and Ph.D. degrees from the North Carolina State University, Raleigh in 1988 and 1992, respectively, all in electrical enginering. In 1992, she joined the Department of Computer Science and Electrical Engineering at the University of Maryland, Baltimore Country as an assistant professor. She has served on the program committees of a number of international conferences including the European Signal Processing Conference (EUSIPCO), IAESTED International Conference on Signal and Image Processing, and the IEEE Intemational Workshop on Neural Networks for Signal Processing (NNSP). She is the guest editor of a special issue on Neural Networks for Biomedical Imaging/Image Processing for the Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology r on Neural Networks. Her research interests are in statistical signal processing, neural computation, adaptive signal processing and their applications in channel equalization, system identification, time-series prediction, and image analysis. Dr. Adah is the recipient of a 1997 National Science Foundation CAREER award.
Bora Bakal received his B.S. degree in Electrical and Electronics Engineering from Middle East Techinical University, Ankara, Turkey, in 1993. He received his M.S. degree in Electrical Engineering from the University of Maryland, Baltimore Country, Baltimore, MD, in 1995. His interest area is signal processing with neural networks. He is currently employed as a software engineer by Faraday National Corporation, Chantilly, VA.
M. KemaI Siinmez received his B.S. degree in Electrical and Electronics Engineering from Middle East Technical Universitv, Ankara, Turkev in 1989, and his MS. degree from North Carolina State Univemity, Raleigh, NC in 1991. He is currently pursuing his Ph.D. degree in Electrical Engineering. He is with Speech Technology and Research Laboratory of SRI International since 1996. University of Maryland. College Park, MD. His research areas include statistical signal processing and artifical neural networks.
Reza Fakery received his M.S. degrees in Nuclear Engineering and Mechanical Engineering. from Massachusetts Institute of Technoloav in 1977. and his Ph.D. decree & Nucl&r Engineering from Rensselaer Polyte&% Institute, USA in 1983. &rrently, he is the principle investigator at S3 Technologies Company, Columbia, MD, USA in the research and development department in the areas of thermal-hydraulic and RETACTR enhancement, simulation of severe accident in PWRs/BWRs, and RETACT code standardization. He also had worked as staff scientist at Computer Simulation Company where he developed advanced models for simulation of NSSS and reactor core systems on the Arkansas Nuclear I simulator (ANO), and as a staff scientist at Simulation Associates, Inc., USA, where he was responsible for model development
on NSSS for Surry, North Anna and Grafen Reinfield (Germany)
simulators.
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C. Oliver Tsaoi received his B.S. from National Tsing-Hua University in 1979, his MS. and Ph.D. degrees from Purdue University in 1987 and 1990, all in Nuclear Engineering. He is currently the Engineering Manager responsible for the development and application of the core neutronics model, REMARK, in S3 Technologies Company, Columbia, MD, USA.