# Reliability Analysis of Nuclear Power Plant Electrical System Considering Common Cause Failure Based on GO-FLOW

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## Abstract

**:**

## 1. Introduction

## 2. Reliability Modeling of Nuclear Power Plant Electrical Systems Based on the GO-FLOW Method

- (1)
- Failure analysis of the external power supply system: In the GO-FLOW diagram of the external power supply system, signal flow 54 represents the entire external power supply system, and signal flows 34 and 53 represent the outputs of the two-way power supply of the external power supply system, respectively. The logical relationship of the two-way power supply outputs is indicated by OR gate operator 54, and when there is no power supply in the two-way power supply because of a loss of the external power supply system, this is called a LOOP event.
- (2)
- Analysis of the failure of the electrical system and plant-wide power failure: When the external power system of a nuclear power plant is lost, the external power system and the diesel generator relate to the contingency gate operators 55 and 59 in the diagram. Signal flow 58 and 62, respectively, represent the backup system two-way power supply output, with OR gate operator 63 representing the two-way power supply output of the logical relationship. The output signal 63 failure means that the two-way power supplies have no power supply output because of the nuclear power plant’s plant-wide power outage. This is called an SBO event.

## 3. A Common Fault Failure Analysis Method for Repairable Systems Based on the GO-FLOW Approach

#### 3.1. Theoretical Basis of GO-FLOW Method to Deal with Common Fault Failure

_{0}is the probability of system failure without considering the common fault failure, F(0,0) is the probability of system failure when the probability of failure of unit A and unit B are both taken as 0, F(1,1) is the probability of system failure when the probability of failure of unit A and unit B are both taken as 1, and the probability of common fault failure of unit A and unit B is C

_{A,B}.

_{k}is as follows:

_{0}is the probability of system failure without considering common fault failure, C

_{j}is the probability of common fault failure for group j, $F(1,1,\cdots )$ is the probability of system failure when all common cause unit failure probabilities are taken as 1, and $F\left(0,0,\cdots \right)$ is the probability of system failure when all common cause unit failure probabilities are taken as 0.

_{0}, and ② system unreliability caused by various common fault failure events.

#### 3.2. Model for Calculating the Probability of Co-Causal Failure of Repairable Systems

- State 0: Parts A and B are intact,
- State 1: Component A has failed, and component B is intact,
- State 2: Component A is intact, and component B has failed,
- State 3: Non-co-causal simultaneous failure of components A and B,
- State 4: Co-causal failure of components A and B.

## 4. Reliability Analysis Process of Nuclear Power Plant Electrical Systems Based on GO-FLOW Method Considering Common Fault Failure

_{0}. If the system does not have a common fault failure unit, then the system failure rate is F

_{0}; if the system has a common fault failure unit, it is necessary to determine the number of unit groups in the system that have a common fault failure, k, the system failure rate of the common fault failure F

_{1}, the number of units in the common fault failure units of the unit group j and s, and the common fault failure of the parameter model. If the common fault failure unit can be repaired, the common fault failure rate C

_{j}of the repairable common fault failure unit is calculated according to Equation (4); if it cannot be repaired, the common fault failure probability C

_{j}of the jth group of the common fault failure units is calculated. The failure probability of the common failure unit is set to 1 and 0, and the failure probability of the common fault failure unit is calculated at $F(1,1,\cdots )$ and $F\left(0,0,\cdots \right)$, respectively, and at this time, the common fault failure rates of the s units in the jth group of common fault failure units can be calculated by Equation (2); each loop increases j by 1; when j is greater than k + 1, the output system failure rate is F; otherwise, the calculation operation is repeated.

## 5. Example Analysis

#### 5.1. Calculations Introduction

^{−3}times/machine·years, and the average repair time is 100 h. The failure frequency and average repair time of some repairable components are shown in Table 1, and the overhaul frequency and overhaul time are shown in Table 2. Among them, the isolation switches in the system are considered to be non-failure parts, and the success probability is always considered to be 1. The equivalent failure rate, equivalent repair rate, average success probability, and average failure probability of each part can be found out through the comprehensive processing, respectively (Table 3).

#### 5.2. Reliability Analysis of Nuclear Power Plants’ External Power Supply System and Electrical System

- (1)
- Scene 1: The external power supply system in the case of shared signals.
- (2)
- Scene 2: The external power system when shared signals are not considered.
- (3)
- Scene 3: The electrical system not considering shared signals.
- (4)
- Scene 4: The electrical system considering shared signals.

- (1)
- For a nuclear power plant’s external power supply system, the system failure probability without considering the shared signals grows rapidly from 1.777503 × 10
^{−5}at time point 0 to 5.760142 × 10^{−5}at time point 9, and is finally maintained at a relatively smooth state. The system failure rate considering the shared signal grows rapidly from 1.966868 × 10^{−5}at time point 0 to 6.048316 × 10^{−5}at time point 16, and is finally maintained at a relatively smooth state. For the nuclear power plant power supply system, the system failure probability without considering the shared signals is basically maintained near 0. The system failure rate considering the shared signals grows rapidly from 1.456204 × 10^{−5}at time point 0 to 4.721015 × 10^{−5}at time point 14, and is finally maintained at a relatively smooth state. In the early stage of the operation of the nuclear power plant, the failure probability of the external power system and the electrical system increases significantly with time, which is mainly caused by the increase in the failure probability of the equipment and components in the nuclear power plant as well as the connection between the nuclear power plant and the external power grid over time. However, after the 6th year of operation, the probability of failure of the external power system tends to level off, which is mainly caused by the high maintenance rate and regular overhaul of the equipment and components in the nuclear power plant as well as the connection between the nuclear power plant and the external power grid. Overall, the reliability of the external power supply system and the electrical system of the nuclear power plant is quite high. - (2)
- Compared with the external power supply system, the electrical system has also added two emergency diesel generators to supply power to the emergency bus, resulting in the reliability of the electrical system of the nuclear power plant being two orders of magnitude higher than that of the external power supply system, and the data are in line with the design logic of gradual enhancement of the mitigation measures from the LOOP event to the SBO event.
- (3)
- The results after considering the shared signals are very different from the results without considering the shared signals. Therefore, it is necessary to consider the effect of shared signals for external power systems and electrical systems in nuclear power plants, as well as in redundant systems where special treatment of shared signals is necessary.

#### 5.3. Analysis of the Impact of Single Power Supply Failure on the Reliability of Electrical Systems

^{−5}at time point 0 to 0.423153 × 10

^{−5}at time point 10, and is finally maintained at a relatively stable state. The failure probability of emergency diesel generator failure grows rapidly from 1.777507 × 10

^{−5}at time point 0 to 5.760145 × 10

^{−5}at time point 9, and finally remains relatively flat. And it can be seen from Figure 6 that when the emergency diesel generator set fails, the probability of system failure is the largest; the system failure rate due to the failure of the auxiliary external power supply, the main external power supply, and the main turbine generator decreases in order. This order is exactly opposite to the order of power supply priority of the electrical system power supply of the nuclear power plant, i.e., the higher the power supply priority, the lower the probability of its failure leading to the failure of the electrical system.

#### 5.4. Analysis of the Impact of Common Cause Failure Factors on the Reliability of Electrical Systems

- (1)
- Assuming that no common fault failure occurs at startup of auxiliary transformers 1 and 2, the runtime common cause failure is modeled using the β-factor model β = 0.1. Then, the common fault failure rate at operation is $c=\lambda {\beta}_{1}=0.2776\times {10}^{-3}$, and the common fault failure probability of the auxiliary transformer is calculated by applying Equation (4) as ${C}_{1}\left(t\right)=2.664936\times {10}^{-0.5}-2.66493936\times {10}^{-0.5}\times {e}^{-85.465477}$
- (2)
- Nuclear power units are equipped with emergency diesel generators, which are called EMP and EMQ. Assuming that the probability of startup failure of EMP and EMQ is $\gamma =0.0236$ and the common fault failure is modeled using a β-factor model with B = 0.05 at startup and A = 0.1 at operation, then the probability of initial success of the standby diesel generator EMP and EMQ after considering startup failure is $1-\gamma =0.9764$ Then, their initial common fault failure probability is ${\gamma}_{c}=\gamma {\beta}_{0}=0.00118$, the common fault failure rate at operation is $c={\lambda}_{LGP,LGQ}{\beta}_{1}=17.4324$, and the common fault failure probability of the diesel generator is calculated by applying Equation (4) as$${C}_{2}(t)=0.00495-0.00377{e}^{-3521.4324t}$$

- (3)
- Applying the GO-FLOW method to first calculate the failure probability of the system that does not contain the common fault failure, and then calculate the failure probability of the system that contains the common fault failure according to Equation (4), the total unavailability change curve of the system is obtained, as shown in Figure 7.

^{−7}at time point 0 to 4.720932 × 10

^{−7}at time point 13, and is finally maintained a relatively smooth state. The outage probability considering common fault failure increases from 1.669504 × 10

^{−7}at time point 0 to 7.575017 × 10

^{−7}at time point 14, and is finally maintained at a relatively smooth state. The group of common fault failure components considered has a greater impact on the overall reliability analysis of the electrical system of a nuclear power plant. Therefore, in practical engineering applications, especially for redundant systems, the common fault failure factors should be fully considered.

#### 5.5. Analysis of the Impact of Additional Standby Units on the Reliability of the Electrical System

^{−3}at time point 4, and then slowly decreases to 0 at time point 24. The power failure probability with the addition of the fifth diesel generator first increases rapidly from 0 at time point 0 to 5.96036 × 10

^{−4}at time point 2, steeply decreases to 0.140665 × 10

^{−4}, slowly increases to 0.468085 × 10

^{−4}at time point 6, and finally, slowly decreases to 0 at time point 24. And from Figure 8, if there is no additional diesel generator and only EMP and EMQ are on standby, the probability of plant-wide power outage of the nuclear power plant reaches its maximum at about 4 h due to the failure of the generator itself, and with the increase in the operation time, the probability of power outage decreases due to the repair of the external power source. With the increase in operation time, the probability of power failure decreases due to the repair of the external power source. The commissioning of additional diesel generators can greatly reduce the probability of power failure. Before the normal operation of the additional diesel generator, the probability of plant-wide power failure reaches the maximum in 2 h, while the probability of power failure decreases abruptly after 2 h.

## 6. Conclusions

- (1)
- The constructed GO-FLOW model of the external power supply system, auxiliary power supply system, and standby power supply system of a nuclear power plant considers the multi-mode repairable component reliability parameter equivalence model and the improved quantitative calculation method of the common signaling system, which improves the accuracy of the reliability analysis of the electrical system of a nuclear power plant.
- (2)
- The group of common fault failure components has a greater impact on the overall reliability analysis of a nuclear power plant’s electrical system, and for redundant systems, common fault failure factors should be fully considered.
- (3)
- The addition of standby diesel generators can greatly reduce the probability of power outages and can effectively improve the reliability of the electrical system in nuclear power plants.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**GO-FLOW diagram of the electrical system of a nuclear power plant. (

**a**) Main power system. (

**b**) Auxiliary power systems. (

**c**) Backup power systems.

**Figure 4.**Reliability analysis flow of nuclear power plant electrical system based on GO-FLOW method considering common fault failure.

Unit Name | Malfunctions Number of Modes | Failure Frequency (Times/Year) | Average Repair Time (Hours) |
---|---|---|---|

main steam turbine generator | 1 | 1.500 | 7.300 × 10 |

mains external power supply | 1 | 2.300 × 10^{−3} | 1.000 × 10^{2} |

auxiliary external power supply | 1 | 8.468 | 3.351 |

emergency diesel generator | 1 | 1.743 × 10^{2} | 5.000 |

main transformer | 2 | 2.015 × 10^{−3} | 1.000 × 10 |

8.760 × 10^{−3} | 7.500 × 10^{2} | ||

auxiliary transformer | 2 | 5.694 × 10^{−3} | 1.000 × 10 |

8.760 × 10^{−3} | 4.000 × 10^{2} | ||

auxiliary transformers | 2 | 1.139 × 10^{−2} | 1.000 × 10 |

1.139 × 10^{−2} | 4.000 × 10^{2} | ||

main electrical connection | 1 | 2.059 × 10^{−2} | 1.060 × 10^{2} |

busbar | 2 | 3.679 × 10^{−3} | 5.000 × 10 |

4.643 × 10^{−4} | 7.200 × 10 | ||

generator export circuit-breaker | 1 | 4.890 × 10^{−2} | 1.116 × 10^{2} |

circuit-breaker | 1 | 3.030 × 10^{−2} | 5.750 × 10 |

isolating switch | 1 | 0 | 0 |

Unit Name | Unit State | Frequency of Overhaul (Times/Year) | Overhaul Time (Hours) |
---|---|---|---|

main steam turbine generator | examine and fix (a motor) | 1.000 | 2.800 × 10^{2} |

main transformer | examine and fix (a motor) | 2.000 × 10^{−1} | 1.600 × 10^{2} |

main electrical connection | examine and fix (a motor) | 5.000 × 10^{−1} | 2.400 × 10 |

circuit-breaker | examine and fix (a motor) | 7.872 × 10^{−1} | 8.100 × 10 |

Unit Name | Equivalent Failure Rate | Equivalent Maintenance Rate | Average Probability of Success | Mean Failure Probability |
---|---|---|---|---|

main steam turbine generator | 2.50000 | 5.62260 × 10 | 0.957429 | 4.2571 × 10^{−3} |

circuit-breaker 1 | 4.89000 × 10^{−2} | 7.84946 × 10 | 0.999377 | 6.2258 × 10^{−4} |

auxiliary transformer | 1.44540 × 10^{−2} | 3.55572 × 10 | 0.999594 | 4.0633 × 10^{−4} |

circuit-breaker 2 | 8.17500 × 10^{−1} | 1.09430 × 10^{2} | 0.992585 | 7.4151 × 10^{−3} |

busbar LGA | 4.14348 × 10^{−3} | 7.00386 × 10^{2} | 0.999994 | 5.9160 × 10^{−6} |

circuit-breaker 3 | 8.17500 × 10^{−1} | 1.09430 × 10^{2} | 0.992585 | 7.4151 × 10^{−3} |

busbar LGB | 4.14348 × 10^{−3} | 7.00386 × 10^{2} | 0.999994 | 5.9160 × 10^{−6} |

circuit-breaker 4 | 8.17500 × 10^{−1} | 1.09430 × 10^{2} | 0.992585 | 7.4151 × 10^{−3} |

circuit-breaker 5 | 8.17500 × 10^{−1} | 1.09430 × 10^{2} | 0.992585 | 7.4151 × 10^{−3} |

busbar LGD | 4.14348 × 10^{−3} | 7.00386 × 10^{2} | 0.999994 | 5.9160 × 10^{−6} |

circuit-breaker 6 | 8.17500 × 10^{−1} | 1.09430 × 10^{2} | 0.992585 | 7.4151 × 10^{−3} |

busbar LGC | 4.14348 × 10^{−3} | 7.00386 × 10^{2} | 0.999994 | 5.9160 × 10^{−6} |

circuit-breaker 7 | 8.17500 × 10^{−1} | 1.09430 × 10^{2} | 0.992585 | 7.4151 × 10^{−3} |

busbar LHA | 4.14348 × 10^{−3} | 7.00386 × 10^{2} | 0.999994 | 5.9160 × 10^{−6} |

busbar LHB | 4.14348 × 10^{−3} | 7.00386 × 10^{2} | 0.999994 | 5.9160 × 10^{−6} |

mains external power supply | 2.30000 × 10^{−3} | 8.76000 × 10 | 0.999974 | 2.6255 × 10^{−5} |

main electrical connection | 7.50000 × 10^{−1} | 4.10625 × 10^{2} | 0.998177 | 1.8232 × 10^{−3} |

main transformer | 2.10775 × 10^{−1} | 4.78461 × 10 | 0.995614 | 4.3860 × 10^{−3} |

auxiliary transformer | 8.46800 | 2.61414 × 10^{3} | 0.996771 | 3.2288 × 10^{−3} |

circuit-breaker 8 | 8.17500 × 10^{−1} | 1.09430 × 10^{2} | 0.992585 | 7.4151 × 10^{−3} |

isolating switch 1 | 0 | $+\infty $ | 1 | 0 |

auxiliary transformers 1 | 2.27760 × 10^{−2} | 4.27316 × 10 | 0.999467 | 5.3281 × 10^{−4} |

9LGM | 4.14348 × 10^{−3} | 7.00386 × 10^{2} | 0.999994 | 5.9160 × 10^{−6} |

isolating switch 2 | 0 | $+\infty $ | 1 | 0 |

circuit-breaker 9 | 8.17500 × 10^{−1} | 1.09430 × 10^{2} | 0.992585 | 7.4151 × 10^{−3} |

isolating switch 3 | 0 | $+\infty $ | 1 | 0 |

auxiliary transformers 2 | 2.27760 × 10^{−2} | 4.27316 × 10 | 0.999467 | 5.3281 × 10^{−4} |

9LGE | 4.14348 × 10^{−3} | 7.00386 × 10^{2} | 0.999994 | 5.9160 × 10^{−6} |

isolating switch 4 | 0 | $+\infty $ | 1 | 0 |

emergency diesel generator LGP | 1.74324 × 10^{2} | 1.75200 × 10^{3} | 0.909504 | 9.0496 × 10^{−2} |

isolating switch 5 | 0 | $+\infty $ | 1 | 0 |

emergency diesel generator LGQ | 1.74324 × 10^{2} | 1.75200 × 10^{3} | 0.909504 | 9.0496 × 10^{−2} |

isolating switch 6 | 0 | $+\infty $ | 1 | 0 |

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**MDPI and ACS Style**

Wang, Z.; Sun, Y.; Zhao, J.; Dong, X.; Chen, C.; Wang, B.; Wu, H.
Reliability Analysis of Nuclear Power Plant Electrical System Considering Common Cause Failure Based on GO-FLOW. *Sustainability* **2023**, *15*, 14071.
https://doi.org/10.3390/su151914071

**AMA Style**

Wang Z, Sun Y, Zhao J, Dong X, Chen C, Wang B, Wu H.
Reliability Analysis of Nuclear Power Plant Electrical System Considering Common Cause Failure Based on GO-FLOW. *Sustainability*. 2023; 15(19):14071.
https://doi.org/10.3390/su151914071

**Chicago/Turabian Style**

Wang, Zhijian, Yao Sun, Jie Zhao, Xuzhu Dong, Chen Chen, Bo Wang, and Haocheng Wu.
2023. "Reliability Analysis of Nuclear Power Plant Electrical System Considering Common Cause Failure Based on GO-FLOW" *Sustainability* 15, no. 19: 14071.
https://doi.org/10.3390/su151914071