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Open AccessTechnical NotePost Publication Peer ReviewVersion 2, Approved

On Singular Perturbation of Neutron Point Kinetics in the Dynamic Model of a PWR Nuclear Power Plant (Version 2, Approved)

by Xiangyi Chen 1 and Asok Ray 1,2,*
1
Department of Nuclear Engineering, The Pennsylvania State University, University Park, PA 16802, USA
2
Department of Mechanical Engineering, The Pennsylvania State University, University Park, PA 16802, USA
*
Author to whom correspondence should be addressed.
Received: 13 March 2020 / Accepted: 11 April 2020 / Published: 27 May 2020
(This article belongs to the Section Thermal Engineering and Sciences)
Peer review status: 2nd round review Read review reports

Reviewer 1 Seddon Atkinson Department of Materials Science and Engineering, Sir Robert Hadfield Building, The University of Sheffield, Sheffield S10 2TN, UK Reviewer 2 Abdalla Abou Jaoude Idaho National Laboratory
Version 1
Original
Approved with revisions
Authors' response
Approved with revisions
Authors' response
Version 2
Approved
Approved Approved
Version 2, Approved
Published: 27 May 2020
DOI: 10.3390/sci2020036
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Version 1, Original
Published: 26 April 2020
DOI: 10.3390/sci2020030
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This short communication makes use of the principle of singular perturbation to approximate the ordinary differential equation (ODE) of prompt neutron (in the point kinetics model) as an algebraic equation. This approximation is shown to yield a large gain in computational efficiency without compromising any significant accuracy in the numerical simulation of primary coolant system dynamics in a PWR nuclear power plant. The approximate (i.e., singularly perturbed) model has been validated with a numerical solution of the original set of neutron point-kinetic and thermal–hydraulic equations. Both models use variable-step Runge–Kutta numerical integration. View Full-Text
Keywords: PWR nuclear power plants; point kinetics; singular perturbation PWR nuclear power plants; point kinetics; singular perturbation
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MDPI and ACS Style

Chen, X.; Ray, A. On Singular Perturbation of Neutron Point Kinetics in the Dynamic Model of a PWR Nuclear Power Plant. Sci 2020, 2, 36.

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1

Reviewer 1

Sent on 30 Apr 2020 by Seddon Atkinson | Approved with revisions
Department of Materials Science and Engineering, Sir Robert Hadfield Building, The University of Sheffield, Sheffield S10 2TN, UK

Overall a really good article which is well presented. 

I feel it might add a bit more depth to the article if the author included a bit more information with regards to the systems codes which use these methods and how this improved method could be implemented. 

The time saving of an order of magnitude is a useful conclusion, however, did the author identify any cases where the time required for these transients were excessive using the standard methods?

Please increase the size of the omega in Figure 2. 

 

Response to Reviewer 1

Sent on 12 Jul 2020 by Xiangyi Chen, Asok Ray

The code of this work is uploaded to github. One can find the code here: https://github.com/chenxiangyi10/MDPI-sci-On-Singular-Perturbation-of-Neutron-Point-Kinetics-in-the-Dynamic-Model-of-a-PWR. In last version, we reported “The computation time and the number of integration steps for the simulation using Equation (4) are 412 milliseconds and 1661, respectively. In contrast, the simulation using Equation (22) takes computation time of ~50 milliseconds and the number of integration steps is 197.” In this new implementation that are uploaded in the github, “The computation time and the number of integration steps for the simulation using Equation (4) are 564 milliseconds and 1661, respectively. In contrast, the simulation using Equation (22) takes computation time of ~67 milliseconds and the number of integration steps is 197.” You can find the ratios of the running time are similar: 8.4 for the old implementation and 8.2 for the new implementation. The manuscript is revised accordingly.

In the introduction section, it has been addressed that “a challenge in the transient analysis of nuclear plants for design of (real-time) monitoring and (active) control systems is to construct a dynamic model that would be computationally efficient and yet serve the purpose at hand.” For example, in the field of expert system of the nuclear power station, the simulation time reduction by one order makes fundamental difference. It could help the operators/systems identify the type of transient in time. Considering the artificial intelligence are gradually incorporated in the expert system, the system development is unlikely to be carried out by using complied language but by interpreted language like python and matlab because the availability of the supporting packages. However the interpreted language has the issue of low implementation speed which could prevent the efficiency of the expert system. Though the hardware and software are changing time by time that efficiency issues come in and be fixed out. The authors consider the efficiency of the dynamical system alone will find its outlets on its own merit and will not envision the application circumstances in detail in the manuscript. However, it is important to keep in mind about the valid domain that the method can be used. The valid domain of the method requires the small positive ϵ. In the case of point kinetics, ϵ(t)=-Λ/(ρ(t)-β). The accident of “control rod ejection”, for example, causes large reactivity insertion obviously is not under the scope. A small positive ϵ requires ρ(t) is smaller than β and not too close to β that their difference is in the order of Λ. One can monitoring whether the scope is valid when this method is implemented. The example in 3.3 has nothing to do with the load following. We mentioned the load following for explaining the thermal-hydraulic process is slow dynamic and neutron kinetic process is fast dynamic.

The comment is unclear. The authors did not include omega in Figure 2.

Reviewer 2

Sent on 22 May 2020 by Abdalla Abou Jaoude | Approved with revisions
Idaho National Laboratory

The paper is very well written and insightful. It is very beneficial to the community at large.

I have a few minor suggestions that may help readers better understand the potential for this work:

I would encourage the authors to include a discussion about the scope of applicability of their proposed approach. It would be helpful to perhaps list a few example transient scenarios and distinguishing between those where the assumptions hold and ones where it does not (perhaps in a table?). The example regarding prompt reactivity insertion is very helpful, but what about cases beyond this? It is hard to imagine other accident scenarios that fit the 'abrupt and singular perturbation' requirement. The authors alluded to load following in the introduction, these feedbacks are slower than the control rod ejection case considered. The example in 3.3 then adds to the confusion as it is almost alluded that fluctuations in loads cause fluctuation in the control rods, which should be an independent system. Perhaps some minor rephrasing here could help.

On a similar note, it is unclear what is meant by the term 'reduction in the dimension of the state space' in page 5. Is this a sudden change in the core geometry somehow?

Lastly, some thoughts on how this approach would apply to reactors beyond PWRs could be helpful. For instance, a fast reactor would have a smaller beta-eff, would that impact the applicability of the novel approach?

Response to Reviewer 2

Sent on 12 Jul 2020 by Xiangyi Chen, Asok Ray

The theoretical valid domain of the singular perturbation of in the point kinetics is discussed in the section 3. A full scope nonlinear dynamical simulation code of PWR is under development. A plenty of scenarios will be implemented with and without singular perturbation in the future work.

The valid domain of the method requires the small positive ϵ. In the case of point kinetics, ϵ(t)=-Λ/(ρ(t)-β). The accident of “control rod ejection”, for example, causes large reactivity insertion obviously is not under the scope. A small positive ϵ requires ρ(t) is smaller than β and not too close to β that their difference is in the order of Λ. One can monitoring whether the scope is valid when this method is implemented. The example in 3.3 has nothing to do with the load following. We mentioned the load following for explaining the thermal-hydraulic process is slow dynamic and neutron kinetic process is fast dynamic.

The dimension of the state space means the number of state variables in the ODEs.

Yes, it could impact the applicability. See the response to the comment 2.

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