Accident diagnosis of a PWR fuel pin during unprotected loss of flow accident with support vector machine

Nuclear Engineering and Design 352 (2019) 110184

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Accident diagnosis of a PWR fuel pin during unprotected loss of flow accident with support vector machine Dongjune Changa, Maolong Liub, Youho Leec,

T

⁎

a

Department of Nuclear Engineering, University of New Mexico, Albuquerque, NM 87131, USA School of Nuclear Science and Engineering, Shanghai Jiao Tong University, Shanghai, China c Department of Nuclear Engineering, Seoul National University, Seoul, Republic of Korea b

ARTICLE INFO

ABSTRACT

Keywords: Loss of flow accident (LOFA) Machine learning Support vector machine

In this study, we conducted various flow rate change simulations for a fuel pin during unprotected LOFA using MARS. The obtained transient outlet temperature profiles were used to establish a relationship with peak fuel temperatures and flow rate changes, using Support Vector Machine (SVM). Unless the number of training data is scarce, the SVM trained with the core outlet temperature gives an accurate prediction (R2 > 0.9) for peak cladding surface temperature, and mass flow rate changes in the early phase of LOFA transience (~0.5 s). It illuminates that key accident characteristics are well reflected in the early response of reactor core behavior (i.e., core outlet temperature). This implies that the possibility of (1) realizing an accident diagnosis framework different from today's practice which relies on the accumulated response of reactor behavior over an extended accident progression, and (2) providing an effective guideline for accident mitigation strategies in the early phase of accident progression. The high predictability (i.e., R2 > 0.9) presented in the early phase of unprotected LOFA indicates core outlet temperature is strongly correlated to both flow rate change, and peak cladding surface temperature during the entire transience. With these strong correlations between different physical parameters, the traditional boundaries of physical locations and physical quantities in detecting accident response and progression may be reduced, allowing the possibility of interdependent detector systems.

1. Introduction A loss of flow accident (LOFA) is a design basis accident that causes a loss of designed reactor coolability due to pump failure during operation. In recent years, an increasing number of studies have been conducted to analyze nuclear accidents with machine-learning methods (de Oliveira and de Almeida, 2013; Fernandez et al., 2017; He and Lee, 2018; Kim et al., 2015; Liu et al., 2013, 2018; Mathew et al., 2018; Na et al., 2008; Wu et al., 2018; Zio et al., 2010). In the field of fault diagnosis of a nuclear power plant, operational support systems with artificial intelligence have been developed to help operators mitigate failures. Specifically, fuzzy logic techniques for predicting failure scenarios in nuclear systems (Hines et al., 1997; Zio et al., 2010) and probabilistic support vector machines (SVMs) for monitoring the state of components of nuclear power plants (Liu et al., 2013) have been studied. Kim et al. (2015) used an SVM to classify break position and size in the case of loss of coolant accidents. Fernandez et al. (2017) used an artificial neural network (ANN) with multiple sensors to evaluate the ability to predict system behavior during various core power inputs and

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a LOFA. Machine learning has also been used to advance the prediction of various two-phase flow phenomena, including flow-regime transitions (Lokanathan and Hibiki, 2016), modeling of Pressurized Water Reactor (PWR) pressurization by ANN (de Oliveira and de Almeida, 2013), and critical heat flux (CHF) prediction (He and Lee, 2018). In addition, some nuclear research reflects the fast progress of machinelearning approaches by selecting ANN to provide predictions about the embrittlement of reactor pressure (Mathew et al., 2018) or performance (Wu et al., 2018), and advanced deep neural networks have also been used to model boiling heat transfer (Liu et al., 2018). From the authors’ perspective, such recent advances in machine learning suggest that it can also be applied to unprotected LOFA analysis and prediction. In this study, SVM was selected to find cases that can help classify the arbitrarily discretized responses of mass flow rate change in several cases that are helpful for predicting accidents. SVM is theoretically attractive for two reasons (Abe, 2005; Auria and Moro, 2008): (1) SVM can convey a unique solution as the loss function is convex after selecting the appropriate kernel; (2) Even though the training set has some biases, it can provide good generalization if the

Corresponding author. E-mail address: [email protected] (Y. Lee).

https://doi.org/10.1016/j.nucengdes.2019.110184 Received 10 December 2018; Received in revised form 30 June 2019; Accepted 30 June 2019 Available online 09 July 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.

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parameters have been properly configured. This is an advantage over Neural Networks, which has multiple local minima-related solutions and may not be robust over various samples. As a first step, this study used an SVM and transient reactor outlet temperature input to investigate the predictability of loss of reactor core flow upon a reactor pump outage. The results, in terms of predictability, address the possibility of establishing a relationship between flow change and readily measurable reactor core outlet temperature. This, in turn, implies the feasibility of advanced accident diagnosis by removing the traditional boundary between different, yet strongly correlated, physical quantities. As a next step, we used an SVM to investigate the correlation between local fuel temperature (i.e., peak cladding and centerline temperature) and reactor outlet temperature during an unprotected LOFA. We then explored the predictability of unmeasurable local temperature using readily measurable temperatures, advancing the predictability of reactor core damage during a LOFA. This study represents an initial attempt to remove the boundaries between different physical quantities (i.e., change in flow rate and core outlet temperature), and between different locations (i.e., local fuel temperature and core outlet temperature) during a LOFA. 1.1. Significance of core outlet temperature, Tout In this study, the reactor core outlet temperature, Tout, is the primary parameter of interest. Obtained transient Tout (t) can be seen as an outcome of continuity, momentum, and energy balance with constitutive models, which are related to the transient flow change, min (t), and local temperature information Tlocal (t), such as the fuel peak temperature, as illustrated in Fig. 1. The relationship between Tout (t) and min (t) is not straightforward; the energy transfer rate from fuel to the coolant during transient flow is affected by the energy deposition rate on the fuel structures. Energy transfer during transience is determined together with continuity and momentum, all of which are coupled to constitutive equations and correlations. Nuclear system codes such as RELAP5-3D and MARS mechanistically solve these coupled equations in a forward direction, as shown in Fig. 2. Accident diagnosis tackles the problem in the opposite direction. It uses a measurable response parameter, Tout, to find the cause of the transience, min (t ) . This reversed process can be best executed by finding an inverse operator of the forward process. In this study, we obtain the inverse operator by training an SVM with change in min (t ) and the Tout (t) obtained from a MARS simulation. Tout (t) is chosen as the primary response parameter of interest because it contains information of transient loss of flow and is a readily measurable parameter during reactor accidents. This study focuses on the machine learning application for accident diagnosis upon the flow rate change. Creating the data from MARS, this study relies on the constitutive equations embedded in MARS, whose validity was confirmed by separate effect tests. In that regard, the quantification of uncertainties of the constitutive equations is an entirely different task that is conducted for the validation of the code. Hence, relying on the sanity of the code prediction, this computational study does not take into account the uncertainties of the constitutive relations. The demonstration of the proposed concept via transient data created by MARS will illuminate the possibility of applying the same concept to the real situation.

Fig. 1. The significance of Tout during a LOFA resulting from min (t) .

2. Methods A PWR nuclear fuel pin is modeled in the MARS code (Lee et al., 1998). Nominal PWR steady-state nuclear fuel is perturbed with a sudden change of inlet mass flow rate to simulate unprotected LOFA transience. Detailed fuel geometry and thermal hydraulics parameters are summarized in Tables 1 and 2, respectively. The axial heating power profile of the fuel rod is approximated by a cosine function of the average linear heat generation of 17.86 kW/m.

Fig. 2. Schematic for a method of typical accident analysis, and a method of accident diagnosis.

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Table 1 Geometry parameters for the reference PWR fuel pin. Heated fuel height Rod to rod pitch Fuel rod outside diameter Fuel pellet diameter Cladding inner diameter Cladding thickness Gap diameter

To minimize

2

w

= wT w, subject to yi (wT x i + b)

1,

i

(1)

This constrained optimization problem is reorganized into:

3.88 m 12.60 mm 9.50 mm 8.19 mm 4.18 mm 11.78 mm 0.57 mm

To minimize

1 2

w 2 , subject to 1

yi (wT xi + b)

0,

i

(2)

After the gradient of its Lagrangian multiplier L is set to zero with respect to w and b, the equation becomes:

L=

Table 2 Operation parameters for PWR pin simulation. Inlet temperature Operation pressure Pin-average linear heat generation Peak linear heat generation Steady-state mass flux Inlet temperature Operation pressure

1 2

293.10 °C 15.51 MPa 17.86 kW/m 28.05 kW/m 3675.40 kg/m2·s 293.10 °C 15.51 MPa

w=

1 T w w+ 2 n i=1

n

yi (wT x i + b))

i (1

(3)

i=1

n i yi x i ,

i yi

= 0,

i

0

(4)

i=1

We can then obtain the dual optimization form for an SVM: n

max. W( ) =

i i=1

1 2

n i = 1, j = 1

i j yi yj x i

Tx

j

subject to C

i

n

0,

A total of 97 independent MARS simulations were conducted to simulate an unprotected LOFA by introducing a sudden flow-rate reduction of up to 3% of the nominal steady-state flow rate (i.e., 100% → 3%). Transient outlet temperatures and local fuel temperatures are reported for training the SVM for each simulated case.

i=1

i yi

= 0.

(5)

where i and j are the Lagrange multipliers and C is a regularization parameter (Gunn et al., 1998; Smola and Schölkopf, 2004). This is a convex quadratic programming (QP) problem for which the global maximum of i can always be found and numerous established solving tools are available. A result for the above QP problem can be obtained efficiently using the optimal values of i and j , which are termed support vectors (Drucker et al., 1997). In addition to linear separability of mapped data, the algorithm can be implemented even if only the inner product can be calculated without explicitly knowing the mapping function. Kernel techniques were first introduced by Vapnik (1992) as the transformation from input data x i into a higher-dimensional space. The algorithm also showed that it can make the problem easier to solve. The kernel function K is given as the inner product K (x i , xj ) = (x i ) (x j ) of the mapping function, which can be transformed from the original input space, x , to the higher-dimensional feature space, (x ) . Various kernel features such as the linear kernel in Fig. 3(a), the homogeneous polynomial kernel, the Gaussian radial basis function kernel (RBF) in Fig. 3(b), and the hyperbolic tangent have been introduced. Using this

2.1. Support vector machine for LOFA Since they were introduced by Cortes and Vapnik (1995), SVMs have become a widely used classification method. SVM algorithms offer advantages for pattern classification and model regression (Vapnik, 1998; Brown et al., 2000; Evgeniou et al., 2000; Furey et al., 2000; Guo et al., 2000; Tuba and Bacanin, 2015). The first SVM algorithm was designed for classifications that determine where the new point belongs. The given data points {x1, , xn} belong to two classes yi {1, 1} where the class label of remark x i is for i = 1, , n respectively. An SVM finds a hyperplane w , also called the decision boundary, that best separates the data points in the training set by class label {1, 1} . The governing equation of an SVM is formulated as follows:

Fig. 3. Support vector machine with kernel function (a) Linear Kernel (b) Gaussian RBF Kernel. 3

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kernel function, the dual-optimization formula of the SVM becomes: n

max. W( ) =

i i=1

1 2

While the outlet flow temperature escalates, the fuel surface experiences change in heat transfer modes due to overheating. An increasing drop in the flow rate eventually leads to appreciable two-phase heat transfer, characterized by a nucleate, transition, and film (postCHF) boiling regime. The higher bulk fluid temperature toward the end of the flow channel, together with the cosine power profile, causes the onset of the film boiling (occurrence CHF) to occur in the upper half of the fuel rod. The changes in heat transfer modes result in heat transfer rate changes. Fig. 5(a)–(f) show heat transfer coefficients along the fuel length with respect to LOFA duration. The onset of film boiling deteriorates heat transfer rates expressed as heat transfer coefficients. Such dynamic heat transfer rate evolutions associated with the boiling mode changes affect the energy transfer rate to the bulk fluid, which is directly related to the rate at which the outlet flow temperature finds the new equilibrium, as shown in Fig. 5. As a result, the rate of increase in outlet flow temperature Tout during a LOFA contains information related to dynamic heat transfer rate changes. The peak fuel and cladding temperatures during LOFA transience are correlated with the reduced flow rate. Fig. 6 shows the peak fuel and cladding temperatures during LOFA transience with respect to the reduced inlet mass flux. The peak temperatures are insensitive to flow reduction until a threshold point, beyond which they increase with reducing flow rates. These thresholds’ reduced flow rates correspond to the cases in which the flow is low enough to cause the onset of film boiling. Any reduction in flow rate beyond this point further deteriorates heat transfer rates, resulting in fuel and cladding temperature escalation. The presented results imply the possibility of predicting the local peak fuel and cladding temperature by determining the reduced flow rate, which can be predicted by the outlet flow rate. In effect, the peak local fuel and cladding temperatures may be predicted using only the measurable outlet flow temperature as an input. The aforementioned relationship between different physical quantities during an unprotected LOFA suggests the possibility of finding an inverse operator for accident diagnosis (Fig. 2). In this study, an SVM trained with MARS simulation results is chosen for the accident diagnosis operator.

n i = 1, j = 1

i j yi yj K

(x j , xj ) subject to C

i

n

0, i=1

i yi

=0

(6)

Following the publication of comparisons of methods for multi-class support vector machines by Hsu and Lin (2002a,b), many SVMs for multi-class classification, such as BSM and LIBSVM, have been widely implemented. We used (L2-regularized) hinge loss as loss function which comes from a maximum likelihood estimate of our model’s parameters. Also, the kernel function was selected as a Gaussian Basis function among non-linear basis functions. How to effectively extend an SVM for multi-class classification is an on-going issue, but it is beyond the scope of the present study. We used an error-correcting output codes (ECOC) model (Escalera et al., 2010) using K (K 1) 2 binary SVM models and a “one-versusone” coding design in which K is the number of unique class labels. In an ECOC strategy, it is possible to create a classifier that targets classifications between different class subsets. That is, these subsets can divide the main classification problem into sub-classification tasks. For predictions using MARS simulated results, we created a training SVM template using an ECOC model and a Gaussian RBF linear kernel. 2.2. MARS simulation results for a nuclear fuel pin during an unprotected LOFA A reduced flow rate causes outlet temperature to increase. Fig. 4 shows outlet temperature transience for tested flow-reduction cases. The outlet temperature finds a new equilibrium after the onset of flow reduction. The rate at which it reaches equilibrium is determined by heat transfer rates coupled to continuity, momentum, and energy balance with constitutive equations. In Fig. 4, mass flux (G) is defined as the mass flow rate divided by flow area, and mass flux ratio (G ratio) is referred as to the ratio of mass flow rate after and before the event. As the mass flux before the events is all the same, the G ratio indicates how the mass flux changes.

Fig. 4. Overall outlet temperature profiles as decrease of the mass flux ratio. Each line represents a more 3% reduction of mass flow rate compared to the next lower line. 4

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Fig. 5. Difference of flow regimes and heat transfer coefficients due to decrease in the G ratio from G100% into (a) (b) G99%; (c) and (d) G23%; and (e) and (f) G3%, respectively.

MSE (y, y ) =

3. SVM-aided accident diagnosis procedures From the previous section, we simulated the change of various parameters according to the sudden change of 97 mass flow rates through MARS simulation. To obtain two different inverse operators for accident diagnosis, the SVM is trained with MARS results to correlate the following physical quantities: (1) Tout(t) and changed inlet mass flux (G) (2) Tout(t) and peak fuel and cladding temperature Above all, the reason why Tout is selected as the main quantities to develop the SVM model is that Tout has the most meaningful consequences from the energy perspective. Since the case of Tin is kept nearly constant during the instant interval, the change in the mass flow rate brings out the heat transfer effect and boiling phenomenon in the pin, the ultimate result of which is the change of Tout. Therefore, when applying these input and output data sets, the SVM algorithm, which creates the relationship between the input and output, includes Tout as the output data set, which leads to several complicated results. Here, Tout(t) is treated as a time-series vector. Because Tout(t) reached a new equilibrium within three seconds (Fig. 4), the data placed in the multi-class SVM input are limited to the three seconds of transience. Even though the SVM is well known as the more correct classifier, to discretize the two-phase flow behavior changes (due to the abrupt mass flow reduction), we made 97 gradual change groups into 1% to 97% of the mass flow rate reduction implicitly through MARS. That is, the use of SVMs in this paper divides this continuous effect into discrete groups implicitly and determines to which groups the corresponding results of the changes will belong. From the authors’ perspective, there must be enough data to develop the relationship between two parameters that are not straightforward such as (1) and (2) and find a predicted solution as a more correct classifier. As in most machine-learning algorithms, SVM requires splitting data into training sets and test sets. We constructed an SVM model using training sets and tested it by quantifying the error for the model with a test set. In this paper, we use two metrics to measure the error, such as mean square error (MSE) and coefficient of determination (R2 ), defined in Eqs. (7) and (8), respectively. n i=1 n i=1

(yi

y )2

(yi

y¯) 2

n

(yi

y )2

i=1

(8)

where n is the number of data, yi is the real value and y and y is the prediction and averaged value. For making an SVM model for (1), the parameter G ratio (defined in Section 4) and its corresponding Tout(t) became the label and the data of the SVM training data set, respectively. In a similar manner, for an SVM model for (2), the peak fuel (or cladding temperature) and its corresponding Tout(t) became the label and the data of the SVM training data set, respectively. For each model, the effects of training data number and LOFA transient time on prediction accuracy are examined. To ensure the training data number sensitivity, 29 evenly distributed MARS simulation results were fixed for the tests while the number of randomly chosen training data progressively increased from 9 to 68, which indicates 10 (10%), 19 (20%), and 29 (30%), 39 (40%), 49 (51%), 58 (60%), and 68 (70%) of these 97 MARS simulating results. In this case, the transient time of MARS simulation is fixed to 3.0 s. For this LOFA transient time sensitivity, the MARS simulation results used for SVM training progressively increased from 0.5 to 3.0 s. In fact, machine learning-based research can involve predicting how the system will behave in the next time step using the current information. For instance, the next time step in the signal system or the control system may be important for predicting the consequences in the next time. One of the most frequently used algorithms such as a linear or non-linear Kalman filter can be used to predict the sensor signal from a linear or non-linear system in a few future time steps. However, our study focuses on diagnosing the present situation through data that has been made up to now rather than predicting what phenomena will occur at the next time step. In this scheme, a portion of the LOFA time was selected in the training and test procedure.

Fig. 6. Difference of local fuel peak temperatures according to changed inlet mass flux (G) from G100% into G a%.

R2 (y , y ) = 1

1 n

3.1. Prediction of the flow rate change min (t ) using coolant outlet temperature Tout(t) The prediction accuracy for the flow rate increases as the amount of training data increases as shown in Fig. 7 and Appendix 1-1(a). The result shows that a training data ratio of more than 20%, which corresponds to 19 data out of total 97 data, gives 99% of the prediction accuracy. Similarly, the MSE approached zero as the training data ratio increased. This indicates that building an SVM relationship between mass flow rate and core outlet temperature does not need a large quantity of training data. Fig. 8 and Appendix 1-1(b) show the LOFA transient time sensitivity for the prediction accuracy of the flow rate change using the core outlet temperature. The R2 and MSE in the cases of the 10, 20, 30, and 70% training data ratios in this graph, mainly for the points where the data indicate changes, are selectively depicted to ensure clarity in the graph changes. The prediction accuracy of the flow rate change increased with the LOFA transient time. In the first 0.5 s of the LOFA transient time, the prediction accuracy reaches at its maximum value for all tested training data ratio. When the number of the training data was less than or near 10% of the entire data set, the saturated accuracy dropped to 0.9 in R2. This demonstrates that as long as training data ratio is higher than 20%, the SVM model trained with the core outlet temperature gives a highly accurate (0.99 in R2) prediction for the flow rate changes in the first 0.5 s of the LOFA transient time. This implies that SVM can instantaneously give an accurate prediction of flow rate change upon the occurrence of transience.

(7)

6

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Fig. 7. Prediction accuracy of flow rate change (G) using core outlet temperature, with respect to the number of training data. 10% Training Data Ratio represents ~9 data out of total 97 data sets. The LOFA transient time is fixed to 3.0 s. (a) R2, (b) Actual vs predicted value.

3.2. Prediction of the peak cladding temperature using coolant outlet temperature Tout(t)

One can pay attention to why SVM does not accurately predict the peak fuel centerline temperature at higher training data ratios (e.g., > 40%). The training data set in this study was set as the combination of a time series of the core outlet temperature and local fuel temperatures. The fact that SVM prediction accuracy decreases with the increasing training data is clear evidence that its predictability cannot be assured in the given problem. Hence, it can be concluded that no working correlation can be established between the core outlet temperature and fuel centerline temperature using the proposed SVM method. The difficulty correlating the bulk fluid temperature (outlet temperature) with fuel centerline temperature is considered to be related to the distance from the coolant side, which is directly relevant to the layers of thermal resistances. As the distance from the coolant increases, the prediction accuracy degrades further, illuminating that the information propagation is hindered with increasing the layers of thermal resistances. Note that the SVM algorithm regarding the fuel centerline temperature does not ensure high accuracy at a 30–60% training data ratio. Generally, when a good data set is acquired, the accuracy increases when the training data ratio increases. In the case of a mass flow reduction as shown in Figs. 7 and 8, we had training data with 97 mass flow reduction sets. These training sets contain the 1-to-1 correspondence that is highly related to the G drop ratio in a specific region.

Prediction accuracy (R2 and MSE) of peak fuel and cladding temperature using coolant outlet temperature is shown in Fig. 9. It is clearly inferable in Fig. 9 that the SVM model can only give accurate predictions of cladding and fuel surface temperatures while the fuel centerline temperature is hard to be correlated to the core outlet temperature transience. This speaks to the possibility of predicting the core damage initiated by cladding degradation, using the core outlet temperature. The effect of the LOFA transient time on the predictability of peak cladding outer surface temperature is examined as shown in Fig. 10. After ~0.5 s, the prediction gives a relatively high accuracy (> 0.8) as long as the number of training data is sufficient. In this case, training data greater than 9 (10% in training data ratio) is shown to be the minimum number, as can be inferred in Fig. 10. The LOFA transient time sensitivity demonstrates that extended transience does not necessarily lead to a more accurate prediction for cases of intermediate numbers of training data sets (i.e., 20, 30 and 40% in training data ratio in Fig. 10). This implies that the changes in the core outlet temperature in the early phase of the unprotected LOFA contain the most relevant information to the peak cladding outer surface temperature of the entire transience.

Fig. 8. Prediction accuracy of flow rate change (G) using core outlet temperature, with respect to LOFA transient time. 10% Training Data Ratio represents ~9 data out of total 97 data sets. (a) R2, (b) Actual vs predicted value. 7

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Fig. 9. Prediction accuracy of peak fuel temperatures using core outlet temperature, with respect to the number of training data. 10% Training Data Ratio represents ~9 data out of total 97 data sets. The LOFA transient time is fixed to 3.0 s. (a) R2, (b) MSE.

However, as shown in Fig. 6, local peak temperatures do not have such uniform monotonic data sets, unlike mass flow reduction according to the changes in mass flow reduction. This inherent limitation of machine learning due to the non-monotonic training data set seems to affect the prediction accuracy. Adding the fact that the more distant the coolant, the more likely this tendency will be on the limitation, the predicted value of the peak fuel centerline does not show good results even though the training data ratio had increased.

unprotected LOFA. This may imply that SVM-aided accident diagnosis, when combined with the current practice, can lead to comprehensive accident diagnoses and reactor safety analyses leveraging rich information covering from the beginning to the extended accident progression. Yet, accurately detecting the instantaneous response of reactor core will necessitate advanced core instrumentation technology.

4. Discussion: implications for SVM-aided LOFA diagnosis

It is important to note that the presented cases are limited to unprotected LOFA. The demonstrated capability of instantaneously predicting the resulting peak fuel temperature, hence core damage, can provide an effective guideline for accident mitigation strategies. That is, SVM can be utilized to inform reactor operators of potential consequences of accidents in the early phase of the accident, which may help them decide accident coping strategies.

4.2. Guidance for accident mitigation strategies

4.1. Instantaneous accident diagnosis It is remarkable that all of the presented results and analyses demonstrate high predictability of flow change as well as peak fuel temperatures in the early phase of unprotected LOFA (i.e., first 0.5 s of the LOFA transience). This means that key accident characteristics are wellreflected in the early response of the core parameter changes (i.e., core outlet temperature). This speaks to the possibility of an accident diagnosis framework different from today’s practice which relies on the accumulated response of reactor behavior over an extended accident progression. However, with SVM, the early or almost instantaneous response of reactor parameters can tell the cause and consequences of

4.3. Strongly correlated response It is noteworthy that the presented SVM models can achieve exceptionally high prediction accuracy (i.e., R2 > 0.9) even with a limited number of training data. This clearly indicates that many reactor response parameters, including ones presented in this study such as Tout

Fig. 10. Prediction accuracy of peak cladding outer surface temperature using core outlet temperature, with respect to LOFA transient time. 10% Training Data Ratio represents ~9 data out of total 97 data sets. (a) R2, (b) MSE. 8

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profiles, some of which were replaced by adding noise. To investigate the noise effect, the test procedure was accomplished to predict the change in the flow rate after establishing the SVM model based on the modified training data as in Section 3.1. Fig. 12 shows the individual predicted results depending on the degree of noise. Here, four different situations were tested: (1) noise factors applied (“Without Noise”), (2) only one training set applied “With Noise (One training set)”, (3) all training sets applied “With Noise (All training sets),” (4) all core outlet temperature data sets applied “With Noise (All data sets).” This result implies that differences obviously exist, but it can be judged that the reduction of R2 or increase of MSE due to noise addition is insufficient by as much as the tendency of the graph even though there is added noise. Practically, temperature sensing is typically done through the Data Acquisition DAQ board by converting the analog signal into a digital signal. Since the time constant for the temperature change is significantly slower than for other electrical systems, the temperature change will be insignificant if the sampling period is fast enough. To fundamentally prevent noise from the temperature sensor, it is typical to design a low-pass filter (or a band-pass filter for suppressing a specific frequency) that can remove a high-frequency component. By accepting the results of Fig. 12, MARS data used in this paper should be regarded as filtered data through a moving average method or such filters.

Fig. 11. Two cases of the core outlet temperature profiles with gaussian random noise as the mass flux (G) decreases from 100% to 99% and from 100% to 3%.

5. Conclusion

vs G , and Tout vs Tcladding, peak, are strongly correlated. Leveraging such strong correlations, one may be able to have access to information that is not directly measurable, such as core damage initiation. Therefore, the traditional boundaries of physical locations and physical quantities in detecting the accident response and progression may be reduced, allowing the possibility of interdependent detector systems.

We conducted various flow rate change simulations for a fuel pin during an unprotected LOFA using MARS. The obtained outlet temperatures were used to establish a relationship with peak fuel temperatures and flow rate changes, using SVM. Unless the number of training data is scarce, the SVM gives an accurate prediction for peak cladding temperature, and mass flow rate changes in the early phase of LOFA transience (~0.5 s). It can be said that key accident characteristics are well-reflected in the early changes of core parameter (i.e., core outlet temperature). This implies the possibility of (1) realizing an accident diagnosis framework different from today’s practice which relies on the accumulated response of reactor behavior over an extended accident progression, and (2) providing an effective guideline for accident mitigation strategies in the early phase of accident progression. The exceptionally high prediction accuracy (i.e., R2 > 0.9) presented in this study indicates many reactor response parameters, including ones presented in this study such as Tout vs G , and Tout vs Tcladding, peak, are strongly correlated during an unprotected LOFA. Therefore, the traditional boundaries of physical locations and physical quantities in detecting accident response and progression may be reduced, allowing the possibility of interdependent detector systems.

4.4. Random noise effects from synthesis data The core outlet temperature profiles in Section 2.2 were generated through the system code, MARS. However, one might usually gather such temperature data using a temperature sensor such as a thermocouple in a laboratory, which means that the data can inherently include random noise. This section explored how synthesizing data with random noise affects the machine learning method to identify the uncertainties from the inherent random noise. Fig. 11 shows two samples of the core outlet temperature profiles with random noise when the mass flux (G) decreased from 100% to 99% and from 100% to 3%. The added noise is assumed as the Gaussian noise whose sigma and mean value are set to 0.2 and the value of each existing core outlet temperature, respectively. The training data is set to the modified core outlet temperature

Fig. 12. Noise effect on prediction accuracy of flow rate change (G) using core outlet temperature, with respect to the number of training data. 10% Training Data Ratio represents ~9 data out of total 97 data sets. The LOFA transient time is fixed to 3.0 s. Each line indicates degree of noise effects: no noise factors applied (‘Without Noise’), only one of the training set applied ‘With Noise (One train set)’, all of the training set applied ‘With Noise (All train sets)’, all core outlet temperature data set applied ‘With Noise (All data sets)’ (a) R2, (b) MSE.

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Appendix See Appendix 1-1.

Appendix 1-1. Prediction accuracy of flow rate change (G) using core outlet temperature: MSE (a) with respect to the number of training data (corresponding to Fig. 7(a)) (b) with respect to LOFA transient time (corresponding to Fig. 8(a)).

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