# Digital Twin Concepts with Uncertainty for Nuclear Power Applications

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## Abstract

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## 1. Introduction

## 2. Brief Overview of Nuclear Power Systems

## 3. Defining Digital Twins for Nuclear Power Systems

#### 3.1. Review of Digital Twin Definitions

- Tuegel et al. [5] define the DT as being “ultrarealistic in geometric detail, including manufacturing anomalies, and in material detail, including the statistical microstructure level, specific to this aircraft tail number”. Here, the DT concept is focused on high-fidelity simulation by the finite element method (FEM) and computational fluid dynamics (CFD) for the prediction and management of the structural life of aircraft. The authors note that another key feature of their concept is the ability to “translate uncertainties in inputs into probabilities of obtaining various structural outcomes”.
- Glaessgen and Stargel [6] have a similar definition centering on ultra-high-fidelity simulation being integrated with a vehicle’s health management system. In this paper, the authors focus more on certification of the vehicles and a reliance on the assumed similitude of data used for certification. They identify this as a shortcoming to be addressed by DTs. Their definition for a DT is “an integrated multiphysics, multiscale, probabilistic simulation of an as-built vehicle or system that uses the best available physical models, sensor updates, fleet history, etc., to mirror the life of its corresponding flying twin”.
- Boschert and Rosen [7] provide a very general definition with “the Digital Twin itself refers to a comprehensive physical and functional description of a component, product or system, which includes more or less all information which could be useful in all—the current and subsequent—life cycle phases”. Here, the authors acknowledge that the DT concept is variable in terms of where it is applied in the product life cycle and the overall fidelity of data and models encompassed.
- Chen [8] similarly broadens the usage of DT and defines it as “a computerized model of a physical device or system that represents all functional features and links with the working elements”.
- Schluse et al. [10] take a slightly different perspective on DTs focusing on the value as an asset for experimentation. Nevertheless, many of the same fundamental requirements arise from their definition of experimental DTs as “a one-to-one replica of a real system incorporating all components and aspects relevant to use simulations for engineering purposes but also inside the real system during real-world operations”.
- Tao et al. [21] describe the DT in a few ways. First, it is a concept “associated with cyber-physical integration.“ Further, DTs create “high-fidelity virtual models of physical objects in virtual space in order to simulate their behaviors in the real world and provide feedback”, and “reflects a bi-directional dynamic mapping process”.
- Rasheed et al. [3] state that a “Digital twin can be defined as a virtual representation of a physical asset enabled through data and simulators for real-time prediction, optimization, monitoring, controlling, and improved decision making”.
- Lin et al. [19] also offer a good definition with: “A DT is a digital representation of a physical asset or system that relies on real-time and history data for inferring complete reactor states, finding available control actions, predicting future transients, and identifying the most preferred actions”.

#### 3.2. Analysis of Key Characteristics

#### 3.3. Proposed Definition

- The digital model may or may not be associated with a physical asset. Thus, it need not integrate with physically measured or sensed quantities. This is how we might think of most of the existing M&S efforts in nuclear engineering. Full simulation models of planned or existing reactors, and their systems and components, capable of simulating the system physics and dynamics comprise the digital model. What distinguishes the digital model from the digital twin is that information generated by the digital model is not automatically integrated with the physical asset.
- The digital shadow extends the digital model by incorporating information from an existing physical asset to update the digital model, but does not utilize any information generated by the digital representation in the physical asset. We note that digital representations of historic facilities that no longer exist can qualify as digital shadows. The inverse of the digital shadow, where information only flows from a digital model to a physical asset, is not a coherent paradigm for useful engineering analysis as there is a physical system operating with essentially no connection to reality. Therefore, this situation is not explicitly defined or discussed further (For the curious reader, this paradigm essentially aligns with Plato’s Allegory of the Cave [22] or Putnam’s more contemporary “Brain in a vat” [23]).
- The digital twin is therefore the “closed loop” model of the physical asset and the digital representation(s). The digital twin exchanges information in real-time with the physical system to update its state and perform predictive calculations that are then used to inform decisions and control actions on the physical asset.

- the initial conceptual design,
- the engineering design,
- procurement and construction,
- the operational phase that undergoes intermediate service, and
- and finally the decommissioning phase.

- Prior to the existence of any information exchange between the digital and physical assets, the digital object is described as a digital model. This often encompasses the conceptual and engineering design phases.
- The digital model may exist alongside the physical twin and indefinitely, if there is no information exchange with the physical asset.
- Following the creation of a physical asset, a digital shadow may be created that incorporates information from the physical asset in either an automated or manual sense, but it does not provide information back to the physical asset.
- The digital shadow may also persist indefinitely.
- The digital twin exists only as long as there is a physical asset.
- The digital twin has real-time, automated, two-way information exchange between the digital representation and physical asset.
- The digital twin may involve a set of models of varying fidelity and complexity.
- A digital twin has a corresponding digital model and digital shadow. The digital model and digital shadow are specific aspects of the twin.

## 4. Historical and Contemporary Digital Representations of Nuclear Power Systems

#### 4.1. Common Nuclear Engineering Simulation Tools

#### 4.2. Relevant Non-Nuclear Commercial Simulation Tools

#### 4.3. Emerging Tools and Capabilities

## 5. Enabling Technologies and Challenges for Digital Twins of Nuclear Power Systems

#### 5.1. Enabling Technologies

#### 5.1.1. System Dynamics Modeling

#### 5.1.2. Model Based Controllers

#### 5.1.3. Automated ROM Construction

#### 5.1.4. Functional Mockup Interfaces

#### 5.2. A Digital Twin Paradigm

#### 5.3. Challenges to Realizing Digital Twins

#### 5.3.1. Security

#### 5.3.2. Integration with Prognostics and Health Management

#### 5.3.3. Data Collection, Curation, Transmission, and Integration

#### 5.3.4. Integration with Risk Assessments

#### 5.4. Computing Infrastructure and Reliability

#### 5.4.1. Standardization

#### 5.4.2. Leverage the Progress in High-Fidelity Advanced Modeling Simulation

#### 5.4.3. Uncertainty Quantification

## 6. Uncertainty Quantification for Digital Twins

#### 6.1. Forward UQ

#### 6.2. Inverse UQ

#### 6.3. Optimization under Uncertainty

#### 6.4. Challenges in UQ for Digital Twins

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Gen-IV advanced reactor designs [20].

**Figure 3.**Relationship of Digital Model, Digital Shadow, Digital Twin, and the Physical Asset using images and models of the Ford Nuclear Reactor as an example. Large arrows represent automated information exchange and small arrows represent manual data exchange.

**Figure 4.**Illustration of physical asset life cycle phases and associated lifetimes and activity of the digital model, twin, and shadow. The width of these lifetimes indicates level of interaction with the digital object.

**Figure 5.**A taxonomy of model order reduction techniques in terms of complexity and knowledge of physics.

**Figure 6.**A depiction of the interface stack operating on the reactor model. Each arrow can be implemented with FMI where each box represents a different software tool. (Source: Touran et al. [68]).

**Table 1.**Number of unknowns by type of equation for a real-time, full-scope plant simulator. The $\mathcal{O}$-notation indicates order of magnitude. For example, “$\mathcal{O}\left(1000\right)$” in the second row, second column means the “Reactor (x,y)” equations need to solve thousands of unknowns from the discretized differential equations.

Component | Differential Eqn. | Algebraic Eqn. | Boolean Eqn. |
---|---|---|---|

Reactor (x,y) | $\mathcal{O}\left(1000\right)$ | $\mathcal{O}\left(1000\right)$ | – |

Reactor Axial | $\mathcal{O}\left(10\right)-\mathcal{O}\left(100\right)$ | $\mathcal{O}\left(10\right)-\mathcal{O}\left(100\right)$ | – |

Steam Generator | $\mathcal{O}\left(1000\right)$ | – | |

Turbine | $\mathcal{O}\left(100\right)$ | $\mathcal{O}\left(100\right)-\mathcal{O}\left(1000\right)$ | – |

Feed Pumps | $\mathcal{O}\left(10\right)$ | $\mathcal{O}\left(10\right)-\mathcal{O}\left(100\right)$ | – |

Control Systems | $\mathcal{O}\left(10\right)-\mathcal{O}\left(100\right)$ | $\mathcal{O}\left(1000\right)$ | – |

Electrical System | $\mathcal{O}\left(10\right)-\mathcal{O}\left(100\right)$ | $\mathcal{O}\left(1000\right)$ | – |

Protection System | – | – | $\mathcal{O}\left(1000\right)$ |

Total | $\mathcal{O}(10,000)$ | $\mathcal{O}\left(1000\right)-\mathcal{O}(10,000)$ | $\mathcal{O}\left(1000\right)$ |

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Kochunas, B.; Huan, X. Digital Twin Concepts with Uncertainty for Nuclear Power Applications. *Energies* **2021**, *14*, 4235.
https://doi.org/10.3390/en14144235

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Kochunas B, Huan X. Digital Twin Concepts with Uncertainty for Nuclear Power Applications. *Energies*. 2021; 14(14):4235.
https://doi.org/10.3390/en14144235

**Chicago/Turabian Style**

Kochunas, Brendan, and Xun Huan. 2021. "Digital Twin Concepts with Uncertainty for Nuclear Power Applications" *Energies* 14, no. 14: 4235.
https://doi.org/10.3390/en14144235