关键词: 可靠性, 维修, 马尔可夫模型, 组件重要度

Abstract: With the rapid development of economy, the scale of complex systems has become larger and larger, and system riskiness has increased. When a disaster causes a component of a complex system to fail, it can cause the entire complex system to be hit hard, resulting in huge economic losses. The blockage of the Suez Canal in March 2021 caused the “Butterfly Effect”. According to the world’s insurance giant, Allianz Insurance Group, the Suez Canal blockage is estimated to cause losses to international trade in the range of $6 billion to $10 billion per week. As disasters continue to occur, people gradually focus on pre-disaster prevention and post-disaster repair of the system, with a view to reducing the damage caused by disasters. Since it is difficult to prevent disaster and the damage caused by disasters is huge, it is necessary to focus on the reliability and post-disaster repair analysis of complex systems. Disasters in different severity cause different economic losses to complex systems, for example, the stronger the earthquake, the greater the harm caused by earthquakes. Considering the impact of the disaster level on the performance of complex systems can help to improve the precision management capabilities of post-disaster repairs.
Based on the above background, the paper studies the repair strategies for complex systems under multi-level disasters. Firstly, disasters can be classified into different levels, and the disaster levels are studied based on the Markov model. Then, the component state is abstracted into multiple discrete states, and the performance analysis is carried out for multi-state complex systems based on the Markov model. Based on the performance changes of the complex system before and after the disaster, system losses are analyzed under multiple levels of disasters. System losses include direct and indirect losses; direct losses refer to the damage caused by the disaster to the infrastructure and are related only to the post-disaster state of the failed component; indirect losses refer to the losses of the system due to the degradation of the system performance. A loss optimization model is established, i.e., the repair set of failed components is determined with the objective of minimizing the total system loss. Because of resource constraints, only one failed line can be repaired at a time, and because different lines have different impacts on system losses, the order in which failed lines are repaired must be determined. Finally, the integrated importance measure of the complex system is established to characterize the degree of impact of repairing the failed line on the total loss of the system. The larger the value of the integrated importance measure, the higher the degree of reduction of the total loss of the system caused by repairing the failed line, and the higher the priority of repairing the failed line, thus the repair order of each component in the repair set is determined.
In the paper, the feasibility of the model is demonstrated using the example of the IEEE 18-node standard distribution system. The IEEE 18-node standard distribution system contains 18 bus nodes and 17 lines. Assuming there are 3 states for the disaster level and 5 states for the line, the state transfer rate matrices for the disaster and the line are given respectively. The maximum repair time is given as T=16days. The set of failed lines is given as F={l1,l3,l7,l8,l10,l12,l13,l14,l17} and the failed states of the lines under different levels of disaster are given. Considering the limitation of repair time, firstly, we find that there are 36 possibilities to meet the repair time limitation of the repair set. Then, we calculate the total loss under the 36 repair sets, and when the repair set is R12, i.e., when emergency repair is carried out on the failedlines in R12, the total loss of the system is the smallest, which is 222,500 yuan. Therefore, failed lines l1, l7, l8, l10, l13, l14 and l17 are selected for emergency repair. The integrated importance measure value of each line in the set of repair lines is calculated based on the integrated importance measure formula. The integrated importance measure of lines l1, l7, l8, l10, l13, l14 and l17 are 0.9, 2.799, 8.72101, 6.1226, 4.7497, 2.3250 and 4.1452, respectively. Thus, the repair priority of the failed line is calculated to be l8,l10,l13,l17,l7,l14, l1 in descending order.
The deficiency of this paper is that only a general classification of the disaster level is made, however, the classification standard of different categories of disasters is not the same in real life, and in the future research, we will further extend the model to conduct a more in-depth study of the disaster level. In addition, this paper only studies the post-disaster repair strategy under the objective of loss minimization model.However, the resilience is also one of the important considerations in real life.In the future, we will introduce the resilience theory to study the post-disaster repair research under the dual objectives of loss and resilience.

Key words: reliability, repair, Markov model, component importance measure