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times. The weight updates are performed in batch at the end of each training sequence of length T. No momentum term nor an adaptive learning rate [6] turned out necessary for increasing the efficiency of the training, in this case.
Ten training runs have been carried out to set the number of delays (orders of the MA and AR parts of the synaptic filters) so as to obtain a satisfactory performance of the LRNN, measured in terms of a small root mean square error (RMSE) on the training set.
As a result of these training runs, the MA and AR orders of the IIR synaptic filters have been set to 12 and 10, respectively, for both the hidden and the output neurons.
3.3. Results
The trained LRNN is first verified with respect to its capability of reproducing the transients employed for the training itself. This capability is a minimum requirement, which however does not guarantee the proper general functioning of the LRNN when new transients, different from those of training, are fed into the network. The evolution of the flux, normalized with respect to the steady state value ®0, corresponding to one sample training transients is shown in Figure 2: as expected, the LRNN estimate of the output (crosses) is in satisfactory agreement with the actual transient (circles).
Notice the ability of the LRNN of dealing with both the short-term dynamics governed by the instantaneous variations of the forcing function (i.e., the reactivity step) and the long-term dynamics governed by Xe oscillations.
Training transient: step forcing function
Figure 2. Comparison of the model-simulated normalized flux (circles) with the LRNN-estimated one (crosses), for two sample transients of the training set
3.3.1. Validation phase: training like dynamics
The procedure for validating the generalization capability of the LRNN to transients different from those of training is based on Nt = 80 transients of T = 2000 minutes each, initiated again by step variations in the forcing function p(t) as in eq. (15), with timing and amplitude randomly sampled in the same ranges as in the training phase.
The results reported in Figure 3 confirm the success of the training since the LRNN estimation errors are still small for these new transients. Furthermore, the computing time is about 5000 times lower than that required by the numerical solution of the model. This makes the LRNN model very attractive for real time applications, e.g. for control or diagnostic purposes, and for applications for which repeated evaluations are required, e.g. for uncertainty and sensitivity analyses.
3.3.2. Test phase
The generalization capabilities of the trained and validated LRNN have been then tested on a new set of transients generated by forcing functions variations quite different from those used in both the training and the validation phases. The test set consists of three
transients batches created by three functional shapes of the forcing function p(t) never seen by the LRNN:
Validation transient step forcing function
Figure 3. Comparison of the model-simulated normalized flux (circles) with the LRNN-estimated one (crosses), for one sample transient of the validation set
> A ramp function:
p(t ) =
'pot < Ts
= <po + (Ap /Tv )t - (Ap /Tv)Ts,Ts < t < Ts + Tv (16)
Po + Ap, t > Ts + Tv
where the steady-state time interval Ts (0 < Ts < 2000 min), the ramp variation time interval Tv (0 < Tv < 2000 min) and the reactivity variation amplitude Ap (-5 10-4 < Ap < +5T0-4) are randomly extracted in their ranges of variation in order to generate the different transients;
> A sine function:
p(t) = Ap • sin(2nft), (17)
where f is the oscillation frequency (1 < f < 2 min -1) and Ap (-510-4 < Ap < +5^10-4) is the reactivity variation amplitude;
> Random reactivity variation amplitude with a uniform probability density function between -5 10-4 and +5T0"4.
A total of Nt = 80 temporal sequences has been simulated for each batch, producing a total of 240 test transients. The temporal length and the sampling time steps of each transient are the same as those of the training and validation sets (2000 and 40 minutes, respectively).
Figures 4, 5 and Figure 6 show a satisfactory agreement of the LRNN estimation with the model simulation, even for cases quite different from the dynamic evolution considered during training.
Test transient: ramp forcing function
Figure 4. Comparison of the model-simulated normalized flux (circles) with the LRNN-estimated one (crosses), for one sample ramp transient of the test set
Test transient: sine forcing function
Time (min.)
Figure 5. Comparison of the model-simulated normalized flux (circles) with the LRNN-estimated one (crosses), for one sample sinusoidal transient of the test set
Test transient: random forcing function
Figure 6. Comparison of the model-simulated normalized flux (circles) with the LRNN-estimated one (crosses), for one sample random transient of the test set
These results are synthesized in Table 1, in terms of the following performance indices: root mean square error (RMSE) and mean absolute error (MAE).
Table 1. Values of the performance indices (RMSE and MAE) calculated over the training, validation and test sets for the LRNN applied to the reactor neutron flux estimation
ERRORS
Set Forcing function n. of sequences RMSE MAE
Training Step 250 0.0037 0.0028
Validation Step 80 0.0098 0.0060
Test Ramp 80 0.0049 0.0039
Sine 80 0.0058 0.0051
Random 80 0.0063 0.0054
4. Conclusion
Dynamic reliability analyses entail the rapid simulation of the system dynamics under the different scenarios and configurations, which occur during the system stochastic life evolution. However, the complexity and nonlinearities of the involved processes are such that analytical modelling becomes burdensome, if at all feasible.
In this paper, the framework of Locally Recurrent Neural Networks (LRNNs) for non-linear dynamic simulation has been presented in detail. The powerful dynamic modelling capabilities of this type of neural networks has been demonstrated on a case study concerning the evolution of the neutron flux in a nuclear reactor as described by a simple model of literature, based on a one group, point kinetics equation with nonlinear power reactivity feedback, coupled with the Xenon and Iodine balance equations.
An Infinite Impulse Response-Locally Recurrent Neural Network (IIR-LRNN) has been successfully designed and trained, with a Recursive Back-Propagation (RBP) algorithm, to the difficult task of estimating the evolution of the neutron flux, only knowing the reactivity evolution, since the other non measurable system state variables, i.e. Xenon and Iodine concentrations, remain hidden.
The findings of the research seem encouraging and confirmatory of the feasibility of using recurrent neural network models for the rapid and reliable system simulations needed in dynamic reliability analysis.
References
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[2] Aldemir, T., Torri, G., Marseguerra, M., Zio, E. & Borkowski, J. A. (2003). Using point reactor models and genetic algorithms for on-line global xenon estimation in nuclear reactors. Nuclear Technology, 143, No. 3, 247-255.
[3] Back, A. D. & Tsoi, A. C. (1993). A simplified gradient algorithm for IIR synapse multi-layer perceptron. Neural Comput. 5: 456-462.
[4] Back, A. D. et al. (1994). A Unifying View of Some Training Algorithms for Multilayer Perceptrons with FIR Filter Synapses. Proc. IEEE Workshop Neural Netw. Signal Process. : 146.
[5] Boroushaki, M. et al. (2003). Identification and control of a nuclear reactor core (VVER) using recurrent neural networks and fuzzy system. IEEE Trans. Nucl. Sci. 50(1): 159-174.
[6] Campolucci, P. et al. (1999). On-Line Learning Algorithms of Locally Recurrent Neural Networks. IEEE Trans. Neural Networks 10: 253-271.
[7] Carlos, S., Ginestar, D., Martorell, S. & Serradell, V. (2003). Parameter estimation in thermalhydraulic models using the multidirectional search method. Annals of Nuclear Energy 30, 133-158.
[8] Chernick, J. (1960). The dynamics of a xenon-controlled reactor. Nuclear Science and Engineering 8: 233-243.
[9] Cojazzi, G., Izquierdo, J.M., Melendez, E. & Sanchez-Perea, M. (1992). The Reliability and Safety Assessment of Protection Systems by the Use of Dynamic Event Trees (DET). The DYLAM-TRETA package. Proc. XVIII annaul meeting Spanish Nuclear Society.
[10] Devooght, J. & Smidts, C. (1992). Probabilistic Reactor Dynamics I. The Theory of Continuous Event Trees, Nucl. Sci. and Eng. 111, 3, pp. 229-240.
[11] Haykin, S. (1994). Neural networks: a comprehensive foundation. New York: IEEE Press.
[12] Hochreiter, S. & Schmidhuber, J. (1997). Long short-term memory. Neural Computation, 9(8): 17351780.
[13] Izquierdo, J.M., Hortal, J., Sanchez-Perea, M. & Melendez, E (1994). Automatic Generation of dynamic Event Trees: A Tool for Integrated Safety Assessment (ISA), Reliability and Safety Assessment of Dynamic Process System NATO-ASI Series F, Vol. 120 Springer-Verlag, Berlin.
[14] Labeau, P. E. & Zio, E. (1998). The Cell-to-Boundary Method in the Frame of Memorization-Based Monte Carlo Algorithms. A New Computational Improvement in Dynamic Reliability, Mathematics and Computers in Simulation, Vol. 47, No. 2-5, 329-347.
[15] Labeau, P.E. (1996). Probabilistic Dynamics: Estimation of Generalized Unreliability Trhough Efficient Monte Carlo Simulation, Annals of Nuclear Energy, Vol. 23, No. 17, 1355-1369.
[16] Marseguerra, M. & Zio, E. (1996). Monte Carlo approach to PSA for dynamic process systems, Reliab. Eng. & System Safety, vol. 52, 227-241.
[17] Narendra, K. S. & Parthasarathy, K. (1990). Identification and control of dynamical systems using neural networks. IEEE Trans. Neural Networks 1: 4-27.
[18] Pearlmutter, B. (1995). Gradient Calculations for Dynamic Recurrent Neural networks: a Survey. IEEE Trans. Neural Networks 6: 1212.
[19] Siegelmann, H. & Sontag, E. (1995). On the Computational Power of Neural Nets. J. Computers and Syst. Sci. 50 (1): 132.
[20] Siu, N. (1994). Risk Assessment for Dynamic Systems: An Overview, Reliab. Eng. & System Safety, vol. 43, 43-74.
Appendix: the Recursive Back-Propagation (RBP) Algorithm for batch training
Consider one training temporal sequence of length T and denote by dr(t), r = 1, 2, ..., N4, the desired output value of the training sequence at time t.
The instantaneous squared error at time t, e2(t), is defined as the sum over all N4 output nodes of the squared deviations of the network outputs xMr(t) from the corresponding desired value in the training temporal sequence, dr(t):
NM
e 2(t) = (t)]2, (1')
r=1
where
er (t) = dr (t) - xM (t). (2')
The training algorithm aims at minimizing the global squared error E2 over the whole training sequence of length T,
T
E2 =X e 2(t), (3')
This is achieved by modifying iteratively the network weights wkji(p), vkji(p-) along the gradient descent,
viz.
Aw
k M dE
2 dWfl(P)
(4')
. k m dE2 Avk - ^
where p is the learning rate.
fl(p) 2 fok z uvji(p)
Introducing the usual backpropagating error and delta quantities with respect to the output, Xj(t), and input, skj(t), of the generic node j of layer k:
ek (t) = , (5')
k ^ 1 dE
1 ^ 2 X (t)
_k/, 1 dE2 1 dE2 dxk (t ) o, (t) = -
1 2 dsk (t) 2 dxk (t) dsk} (t)
(6')
= e, (t)fk\s[ (t)}
the chain rule for the modification (4') of the MA and AR synaptic weights wkll(P), vkjl(j>) can be written as
Awkii(p) =-MI dsk(t) w(p)
2
t=1
^ k dsk (t )
= Y.mô) (t ) j" 1 dwfl(p)
t=1
H ( p )
= -Ey
2 tl dsk (t) dv
dE2 dsk (t)
k
fl (P)
(7')
^ k dsk (t)
=y^k (t)—
^ 1 dvk
t=1
j (p)
Note that the weights updates (7') are performed in batch at the end of the training sequence of length T. From (10),
ds] (t) = dyj (t) ; dsk (t) = cyj (t)
w(p) w(p) ' dvkfl(p) dvkfl(p):
(8')
so that from the differentiation of (11) one obtains
8sk (t) k-1, , % k 8sk (t -T
= x,k-1 (t - p) + Z vk-, (9')
dw n t=1 dw \
ji(p) jKp)
dsk (t) k, ^ Ikkik dsj (t -T)
1 = y j(t - p) + Z vMt)-j-k-. (10')
a. k j r > ^ j (T) _ k
dv ^ t=1 dv ■„ ^
iKp) iKp)
To compute Skj(t) from (6'), we must be able to compute ekj(t). Applying the chain rule to (5'), one has
Nk+1 T 1 dE 2 dsk+1(t)
e] (t) = Z Z - ^-dE--, k < M . (11')
1 q=1 T=1 2 +1(t) cx] (t)
Under the hypothesis of synaptic filter temporal causality (according to which the state of a node at time t influences the network evolution only at successive times and not at previous ones), the summation along the time trajectory can start from t = t. Exploiting the definitions (6') and (8'), changing the variables as t -p ^ t and considering that for the output layer, i.e. k = M, the derivative dE2/dxM](t) can be computed directly from (2'), the back-propagation of the error through the layers can be derived
:(t) =
e1 (t) (eq.2'), k = M
rk+1
T-t N
z z s]+'(t + p)
p=0 q=1
dykq+l(t+p) dxk (t)
(12')
k < M,
e
where from (11)
dyj(t + p) = min(1Z+1,p) k+1 j + p -t) dxk (t) tZ1 Vq1 (T) cXk (t)
(13')
+
\wjP ^ < p < L+1 -1
0, otherwise.
Dourmas N. Georgios, Nikitakos V. Ninitas, Lambrou A. Maria - A METHODOLOGY FOR RATING AND RANKING HAZARDS IN MARITIME FORMAL SAFETY ASSESSMENT USING FUZZY LOGIC
A METHODOLOGY FOR RATING AND RANKING HAZARDS IN MARITIME FORMAL SAFETY ASSESSMENT USING FUZZY
LOGIC
Dourmas N. Georgios, Nikitakos V. Ninitas, Lambrou A. Maria
University of the Aegean, Dept. of Shipping Trade and Transport, Chios, Greece
Keywords
decision making, Formal Safety Assessment, hazard identification, marine safety, fuzzy logic Abstract
Formal safety assessment of ships has attracted great attention over the last few years. This paper, following a brief review of the current status of marine safety assessment is focused on the hazards identification (HAZID) and prioritisation process. A multicriteria decision making framework, which is based on experts' estimation, is then proposed for hazards evaluation. Additionally in this paper many aspects of the evaluation framework are presented including the synthesis of evaluation teams, the assessment of the importance of criteria, the evaluation of the consequences of the alternative hazards and the final ranking of the hazards. The proposed methodology has the innovative feature of embodying techniques of fuzzy logic theory into the classical multicriteria decision analysis. The paper concludes by exploring the potentiality of the above methodology in providing a robust and flexible evaluation framework suitable to the characteristics of a hazard evaluation problem.
1. Introduction
Hazard identification (HAZID) is the first and in many ways the most important step in a risk assessment. This paper, following a brief review of the current status of marine safety assessment is focused on the hazards identification and prioritisation process. Hazard Identification is the process of systematically identifying hazards and associated events that have the potential to result in a significant consequence. The aim of HAZID is first to produce a list of all possible hazards and second to evaluate them in order to prioritise them. In order to support the evaluating procedure we propose as a tool the Multicriteria Decision Analysis (MCDA). The reason is that the final decision depends on criteria, which correlate the potential hazardous scenarios with different consequences.
MCDA deals with the problem of ranking various alternatives in the presence of multiple criteria. Up to now, there are a variety of methods that one can choose from solving a multicriteria decision problem, the most famous being the maximin, the weighted average, the multicriteria utility evaluation and the Analytical Hierarchical Process [13].
All the aforementioned methods assume that the decision maker is able to provide exact assessments on the importance of the importance of evaluation criteria on the impact of alternatives. However, owing to the availability and subjectivity of information, it is very difficult to obtain exact assessment data as concerns the fulfilment of the requirements of the criteria or the relative importance of each criterion. Classical decision-making methodologies are thus criticized for over-simplifying the decision-making process by "forcing" the experts to express their views on pure numeric scales. It is common evidence that assessments made by experts are mostly of subjective and qualitative nature.
Fuzzy sets theory, originally proposed by L. A. Zadeh [22], is an effective means to deal with the "vagueness" of human judgement. This theory offers us tools to handle linguistic terms as the ones
Dourmas N. Georgios, Nikitakos V. Ninitas, Lambrou A. Maria - A METHODOLOGY FOR RATING AND RANKING HAZARDS IN MARITIME FORMAL SAFETY ASSESSMENT USING FUZZY LOGIC
mentioned above by converting them to suitable fuzzy sets and numbers. "Fuzzy" multicriteria decision analysis methods allow us to integrate linguistic assessments and weights in a multicriteria decision analysis setting [11], [14].
After fuzzy sets general methodology presentation this paper proposes an application to evaluate and rank a number hazards. We assume a multi-criteria decision making framework, where sets of general and domain-specific criteria are used to judge the relative impact of evaluating hazards. The proposed methodology has the innovative feature of embodying techniques of fuzzy logic theory into the classical multicriteria decision analysis.
2. Hazard identification
Hazard identification (HAZID) is the first and in many ways the most important step in a risk assessment. An overlooked hazard is likely to introduce more error into the overall risk estimate than an inaccurate consequence model or frequency estimate. The aim of the HAZID is to produce, therefore, a comprehensive list of all hazards. The list should include all foreseeable hazards, but it should also avoid double counting by including the same hazard under more than one heading. In order to distinguish between hazards and consequences, it is advisable to start with defining a "hazard". In formal ship safety assessment, a hazard is defined as "a physical situation with potential for human injury, damage to property, damage to the environment or some combination" [12].
Therefore, ship 'grounding' is considered as a possible consequence of hazards related, for example, to navigation error/failure, and not as a hazard itself. Similarly, 'navigation' 'ship manoeuvring', etc. are considered as hazardous operations because a component failure could lead to a chain of unwanted outcomes.
HAZID is concerned with using "brainstorming" technique involving trained and experienced personnel to determine the hazards. HAZID is, most of the time a qualitative exercise strongly based on expert judgement. Many different methods are available for hazard identification and some of them have become standard for particular applications. Experience proved that there is no need to specify which technique should be used in particular cases. Typically, the system being evaluated is divided into parts and the team leader chooses the methodology, which can be standard technique, a modification of one of these or, usually, a combination of several. In other words, the technique used is not that important since each group can follow a methodology of combined techniques. The most important thing is that the HAZID has to be creative in order to obtain comprehensive coverage of hazards skipping as fewer areas as it could practicably be. Also, it is very important that the conclusions of HAZIDs will be discussed and documented during a final session, so that they represent the views of the group rather than of an individual.
Various scientific safety assessment approaches such as Preliminary Hazards Analysis (PHA), Failure Mode, Effects and Criticality Analysis (FMECA) and Hazard and Operability (HAZOP) study can be applied in this step [21].
3. Hazard analysis
Hazard analysis approach is considered a suitable tool for ship safety assessment. In this approach it is assumed that each specific hazard can be represented by one or several threats that have the potential to lead to an incident or top (initiating) event [18]. A threat can be a specific hazard or a more detailed representation of a specific hazard. Each accidental event may lead to unwanted consequences. If a Hazard is released, the accidental event can escalate to one of the several possible consequences. To prevent escalation, the mitigation measures, emergency preparedness and escalation control measures need to be in place to stop chain of events propagation and/or to minimize the consequences of escalation [19]. At the table 1 are described some general hazards, which are analysed in more detailed hazards.
Table 1. List of hazards
