Optimal decision-making method for equipment maintenance to enhance the resilience of power digital twin system under extreme disaster

0 Introduction

In 2003, Prof.Graef of the University of Michigan in the United States introduced a new concept for product lifecycle management, which was later defined as “Digital Twin” by the United States National Aeronautics and Space Administration (NASA) [1].Over the subsequent two decades, Digital Twin technology has gradually expanded into various industries.In the context of power systems,digital twin technology is primarily used for data monitoring,intelligent communication, real-time control, and information mining [2-4].When a power digital twin system operates smoothly, it provides system administrators with real-time data analysis and support.However, when a power system fails, recovery of the digital twin system becomes another critical problem in disaster recovery research [5-6].Postdisaster recovery in power systems is a crucial task, with related research dating back to the 1940s [7].Extensive research has been conducted on the significance and methods of the physical recovery of power systems [8-10].However,for digital twin systems in the power sector, network infrastructure can fail and become dysfunctional [11-13].

Based on the existing research findings, attacks on communication equipment primarily stem from device damage and cyber attack [14-17].Reference [18] developed optimal attack strategies that cause maximum damage to the system.Reference [15] quantified and validated the adverse effects of different attack models on the system operation.Methods for preventing or detecting the occurrence and propagation of attacks have been proposed to reinforce system resilience [17-19]. [20-21] modeled the problem of false data injection in power systems and analyzed it using data-driven approaches.Although extensive research has been conducted on modeling and defense measures for cyberattacks that occur before and during extreme disasters, there is limited research on how to recover the functionality of the network infrastructure after an attack, making it challenging to rapidly improve system resilience [7-8].Reference [22] proposes a hybrid fast-path recovery algorithm for recovering single communication link failures in information-physical systems.However,it remains limited to minor disruptions and unsuitable for severe disaster scenarios.Reference [23] presents an information-physical coordinated recovery strategy considering communication system failures to address information-physical composite failures in transmission systems.In [24-25], a software adjustment scheme was developed to recover the monitoring capabilities of power digital twin systems.However, this process considers only damaged data processors and does not analyze other types of component failures following a severe disaster.Some studies have partitioned power grids after natural disasters and proposed active branch adjustment methods to reduce the risk of branch power exceeding limits and system instability caused by line disconnections during the disaster recovery phase, ultimately enhancing system resilience [26].

In summary, previous research has mainly approached the problem from a single perspective, focusing on either communication networks or power systems, without adequately addressing the specific needs of monitoring and controlling power physical entity systems.This paper introduces a new approach for post-disaster monitoring and rapid data transmission in power digital twin systems.Extensive research has been conducted on scenarios in which communication equipment stops operating after a disaster, primarily focusing on network recovery in a power digital twin system to analyze the information transmission capability of network infrastructure and monitoring ability after a power outage.

This study first analyzes the functions of power physical entity systems and digital twin systems to address the abovementioned problems and establishes an energyinformation operational model based on this analysis.Subsequently, we analyze the operation of the central data processing unit (CDP), communication router (CR), and phasor measurement unit (PMU), and establishes postdisaster recovery models for these devices.Subsequently, a mathematical optimization model is developed to enhance the monitoring capability of the digital twin systems,and branch-and-bound methods are employed for the computation.Finally, the effectiveness of the proposed method is validated by analyzing the IEEE-14 system.

1 Composition of power physical-digital Twin systems

A power digital twin system comprises a physical entity system and a digital twin system, as illustrated in Fig.1.The physical entity and digital twin system operate in real-time synchronization and data exchange.

Fig.1 Components of a power digital twin system

1.1 Power physical entity system

A power physical entity system includes generators,transformers, transmission lines, and end users, as shown in Fig.1.The corresponding mathematical model is an alternating-current (AC) power-flow model.The modeling process utilizes methods such as Kirchhoff’s

laws, impedance modeling, and system dynamics to model the individual power equipment units.Subsequently, a complete computational model of the entire power system is developed by integrating the physical operating models of all subunits and their interaction relationships [27].The operating model is represented in (1):

where Pij,t and Qij,t represent the active and reactive power inflows at the first end of branch ij at time t, PG,j,t and QG,j,t denote the active and reactive power injection at node j connected to a power source at time t; PD, j,t and QD, j,t represent the active and reactive loads at node j connected to loads at time t; Iij,t represents the current magnitude in branch ij at time t; Vi,t represents the voltage magnitude at node i at time t; rij and xij are the resistance and reactance values of branch ij.

Research on power physical entity systems is relatively advanced.In contrast, the interaction between the digital twin system and the physical entity system primarily relies on the mapping relationship between the power nodes and network nodes in the digital twin monitoring module.This enables real-time interaction between systems through five processes: information collection, information sharing,information upload, information processing, decisionmaking, and command execution.The following analysis focuses on the digital twin system.

1.2 Power digital twin system

The primary application of digital twin technology in power systems involves the interaction and correction between physical and digital models, as shown in Fig.1.With the application of digital twin technology in power systems, the modeling of power digital twin systems has gradually evolved from traditional physical modeling to physical data-driven modeling.Relevant research has addressed the limitations of traditional mechanical modeling methods in calculating network-wide power flows under operating conditions with insufficient parameters.New modeling methods can adaptively adjust system models based on historical data, primarily using heuristic algorithms and artificial intelligence methods for parameter identification [28].Digital twin systems rely on various monitoring devices and rapidly responsive data transmission units.The online correction process involves collecting real-time data from different components, lines, nodes,transformers, loads, and batteries in the power system,and conducting advanced applications.Furthermore, realtime data collection of environmental parameters such as temperature, solar radiation, humidity, wind speed, and wind direction in a physical entity system are required to enhance the monitoring capability of the digital twin system.Data with different granularities can be used to update system models and predict the status of new energy sources.

Among the many monitoring communication devices in a digital twin system, the critical monitoring components include the CDP, CR, PMU, and the communication network they form.The PMU is the monitoring terminal of the digital twin system, connected to the power network nodes in the physical entity system, which collects critical information, such as voltage, frequency, and phase angle,from the nodes in real time.This information is then sent to the CR, which, through data-sharing capabilities, transmits critical information to other CRs or CDPs, ultimately storing it internally for data mining, state recognition, and other advanced applications within the digital twin system.CDPs are deployed in the central areas near the digital twin system, while PMUs are directly connected to the nodes in the power physical entity system.To enhance data connectivity across the entire system, multiple CRs are installed between them to facilitate data forwarding and meet the various requirements of the digital twin system.

As this study primarily analyzes the operation of the digital twin system’s communication module in extreme disaster scenarios and considers potential problems in the working environment, the constructed information network operation model can be represented as follows [29]:

where Ia,t represents the collected signal at time t; ΔTattact denotes the communication delay situation; Lc0,t represents the transmission capacity of the system’s information element at time t; IB,t indicates the portion of information loss or offset at time t; Kt+ΔTattact represents the signal distortion rate accounting for communication latency effects.

Post-disaster recovery of the power physical entity system following major disruptions, such as natural disasters is already a mature research problem [30-31].However,components in digital twin systems, such as the CDP, CR, and PMU, are also risk being affected by disasters, leading to their shutdown or the transmission of incorrect data.Therefore,it is essential to rapidly repair the digital twin system monitoring modules and restore the power system monitoring capability.This research focuses on post-disaster recovery methods for physical power entities and digital twin systems.

2 Post-disaster recovery of power physical entity system and digital twin system

After extreme natural disasters strike a power system,its operational level decreases significantly, and human intervention measures must be taken to restore it to normal levels.This process includes the recovery of both the physical entity system and the maintenance of equipment in the digital twin system, as illustrated in Fig.2.

Fig.2 Recovery processes in electrophysical-digital twin systems

As shown in Fig.2, after a power transmission network outage, the primary task is to restore the operation of the generators in the system [32-35].Subsequently, the energy transmission pathways between the generators and loads are repaired to restore the power supply as quickly as possible[36].Considering the area under the curve, the level of recovery can be calculated when the resilience curve begins to exhibit an upward trend until the system returns to a normal load state, considering the area under the curve [37-38].The calculation method for the digital twin system is similar to that for the digital twin system, as shown in Fig.3.The difference is that the process of restoring the information communication inolves a discrete function.

Fig.3 Resilience analysis of the power digital twin system

The process of rapidly restoring the monitoring capability of the digital twin system with enhanced information system resilience is represented as

where fre(ξt) represents the resilience curve; I represents the communication system recovery function; st refers to the system’s repair operations.

2.1 System recovery model after extreme disasters

This study quantifies the recovery of a digital twin system by considering the recovery status of various component types, connectivity between normally operating components, and connectivity status after the recovery of individual monitoring links.This comprehensive representation reflects the post-disaster recovery of an information system, and can be expressed as follows:

where, RI represents the post-disaster recovery status indicator of the information system, ωc, ωl, ωn, ωpmu, ωcdp,and ωcr represent the weight coefficients for the recovery capacity of components, the weight coefficients for connectivity between normally operating components, the weight for bus state monitoring, the weight for repairing PMU, the weight for repairing CDP, and the weight for repairing CR, respectively.Φpmu_x and Φpmu_f represent the number of PMUs already recovered and the total number of PMUs affected by disaster faults, respectively.Φcdp_x and Φcdp_f represent the number of recovered CDPs and the total number affected by disaster faults, respectively.Φcr_x and Φcr_ f represent the number of CRs already recovered and the total number of CRs affected by disaster faults,respectively.χ represents the connectivity between typically operating components, n indicates the number of nodes monitored during system recovery, and N indicates the total number of nodes in the system that lost monitoring capability owing to disasters.

After information disruption in the digital twin system,the primary task for repair personnel is to restore the CDP and repair the PMUs.Subsequently, the data transmission channels are rapidly restored through CR repair.When a digital twin system can monitor all components of the physical entity system or obtain the operating status through advanced computing methods, the system restores its monitoring capability.

The recovery of the physical entity and digital twin systems followed the area integration method for resilience states.However, there are specific differences in the recovery objects.The measurement metric for the recovery of the physical entity system is the electrical transmission capacity of the system, which corresponds to the physical concept of restored available transmission capacity or the supply of loads.For the recovery of the digital twin system,the measurement metric is the information transmission capacity, which corresponds to the degree of transmission of information collected by the PMUs.

2.2 Modeling of the constituent units of the monitoring module of the digital twin system

Owing to independent and unconnected PMUs, CDPs and CRs cannot form a system monitoring path, and the large number of devices in the actual system leads to a large-scale combination of their methods to achieve effective and efficient recovery of system monitoring,making it is necessary to search for and construct secure and functional data transmission paths.Therefore, this study begins by recovering core components as a priority and searches for efficient equipment combinations as a recovery sequence.For analysis convenience, this study represents the operational states of PMUs, CRs, and CDPs using matrices ξ t, τt, and υt, respectively.In the modeling process of this study, it is assumed that once a device has recovered from a faulty state to regular operation through recovery operations, it no longer experiences faults.External disturbances and special perturbations are neglected, and the equipment recovery process is assumed to be irreversible, which is represented as

In this equation, if a device typically operates, its corresponding element in the device status matrix is assigned a value of one; otherwise, it is assigned a value of zero.

In the actual operation process, if the PMU’s collected data are considered usable, it requires the PMU to have the capability for safe operation and establish a secure and stable connection with the corresponding buses of the physical entity system.The obtained data should be recognized by the operational CRs and transmitted to the CDP for data processing and analysis.Similarly, CRs and CDPs must focus on their physical and functional capabilities while considering the degree of work they undertake in the system.

Throughout the recovery process of the digital twin system after extreme disasters, various combinations of PMUs, CRs, and CDPs should be considered.Efficient equipment combinations can rapidly monitor the node data in power physical entity systems.In the same system,multiple paths should satisfy the operational conditions to monitor the operational data of the same node.The most essential aspect of this process involves formulating the optimal network recovery planning problem, observing the system’s operation patterns from a system-wide perspective,and determining the optimal recovery strategies for the monitoring units of the digital twin system.

3 Modeling the optimal recovery problem for digital twin system monitoring units

After extreme natural disasters, multiple network components of the physical entity and digital twin systems cannot operate normally.In such cases, the only method to restore the monitoring capability of the digital twin system involves repairing the damaged components and converting them to normal operating conditions.Repairing damaged parts can take several tens of minutes to several hours [38-39], and a systematic optimization method should be developed to determine the priority of recovery tasks owing to limited resources and personnel constraints for conducting repair tasks simultaneously.

3.1 Optimization model

Since the actual recovery is performed in stages,the maximum resilience recovery of the system can be reformulated as

As mentioned earlier, it is necessary to restore at least one CDP to maintain its normal operating state in the initial stage to complete the system-monitoring tasks.Therefore, if all the CDPs are faulty, the first step is to restore the CDP,while the remaining CDPs can be selectively repaired during recovery.For time t, the recovery plan for the CDP and its associated CR are subject to the following constraints:

Assuming that the communication network formed by the recovered CDPs and CRs was established in the initial recovery steps, with the CRs located in the equipment layer as an intermediate layer, the recovery constraints for the upper-layer CDP devices adjacent to the recovered CRs and lower-layer PMU devices at time t are as follows:

where A is the matrix that represents the association between the CRs and PMUs, Dup,t and Ddown,t represent the operational status of the upper-layer CDP device adjacent to the recovered CR and lower-layer PMU device,respectively.That is, when the upper and lower devices adjacent to the recovered CR have not completed their recovery, they should be repaired sequentially, and only one device can be repaired at a given time.Equation (9) can no longer be applied once adjacent devices have completed their recovery.

Upon completing the recovery of the upper and lower devices adjacent to the already restored CR, the subsequent step is to prioritize the restoration of the other functioning CRs connected to the CDPs during regular operation.Newly restored CRs should be directly or indirectly related to the PMUs at the crucial nodes that must be repaired.It is assumed that the CDPs and CRs have already been restored in the previous steps.

Since the total resources for the recovery of the digital twin system monitoring unit devices are limited, constraints are required to represent the number of PMUs, CRs, and CDPs to be restored at each step, as follows:

where fipmu, ficr, and ficdp represent the required recovery resources, and cpmu, ccr, and ccdp, denote the maximum number of PMUs, CRs, and CDPs, respectively, to be restored in each step.

In an actual repair process, the time required for each type of resource recovery is similar.Equation (10) can be rewritten as

The PMU data communication capabilities must satisfy the data transmission requirements of the PMUs, CRs, and CDPs.Therefore, the data transmission conditions for the PMUs can be represented as

3.2 Computational steps

The objective function to be calculated in this study is given by (6), with the constraints specified in Equations (3-4),(7-9), and (11-12).Owing to many binary variables in the calculation, the branch-and-bound method was used to solve the model proposed in this study.

4 Case study

4.1 Case introduction

The recovery problem for the digital twin system in this study is based on the IEEE-14 bus system.As shown in Fig.4, in the IEEE-14 bus system, a monitoring module for a digital twin system is constructed with nine PMUs, 14 CRs, and three CDPs.Since the monitoring of the generators can be directly consulted with the operator, all nine PMUs are installed on the non-generator buses.The PMUs collect and transmit data to the CRs, facilitating data sharing and information transmission through CRs.The repair time for each component is assumed to be the same, enabling the entire system repair process to be performed in steps.In the scenario considered in this study, it is assumed that all the PMUs, CRs, and CDPs are damaged.The proposed optimization model was computed using MATLAB 2022b.

4.2 Comparison and analysis of different computational methods

By setting the number of recovery resources per step to three, the results of the proposed algorithm are compared and analyzed using path optimization and random algorithms, as shown in Fig.5.

Fig.4 IEEE-14 Power digital twin system topology

Fig.5 System recovery results under different algorithms

As shown in Fig.5, the results of the calculations are compared with [9] and [10] when the number of recovery resources is set to three.The algorithm proposed in this study can restore the monitoring capability of the entire power digital twin system in the 7th step.By contrast,random and path optimization algorithms require nine steps to correct the monitoring capabilities of the system.The main reason for this is that the proposed algorithm can automatically prioritize the critical equipment R9 for data transmission based on the connectivity between power system nodes and PMUs in the digital twin system and the layout of communication routers.Other algorithms are required to help identify critical equipment for information recovery.In Fig.5, the algorithm proposed in this study outperforms different algorithms in terms of the area enclosed by the x-axis (i.e., system resilience),demonstrating the effectiveness of the proposed algorithm.

4.3 Comparison and analysis of recovery under different initial conditions

Further analyses were conducted under different initial recovery conditions.The operation was examined for the other initial recovery nodes, C4 and C9, as indicated in Table.1.

In Table.1, when the initial recovery node is set to C4, no node recovers the monitoring data in Steps 2 and 3.However, in Step 4, the monitoring capability of the system suddenly increases because the power nodes B7, B10, and B11 upload data through their respective CRs, R9 and C9.When the initial recovery node was set to C9, the nodes recovered their monitoring statuses at every step.Since R6 and C6 serve only as information transmission nodes, they are not necessary for recovery.Therefore, the impact of restoring R6 on the repair results is analyzed further.

Fig.6 Calculation results for different recovery methods

As shown in Fig.6, the initial recovery speed of the system significantly increases when restoring C6 and R6.Therefore, an arrangement can be made during the repair process based on actual needs.

Table 1 Repair strategies under different initial value

Recovery procedure Resetter Observable bus Recovery procedure Resetter Observable bus 1 C9 R9 P9 9 1 C4 P4 R4 4 2 R5 R11 P11 11 2 R10 P10 P11 /3 P5 R10 P10 10, 5 3 R7 R11 P7 /4 C4 R4 P4 4 4 C9 P9 R9 9, 7, 10, 11 5 P7 P13 R13 13 5 R13 R14 P14 14 6 R12 R7 P12 12, 7 6 R5 P5 P13 13, 5 7 C6 R6 R14 / 7 R12 P12 12 8 P14 R8 14 8 / /9 R1 R2 R3 / 9 / /

4.4 Comparison and analysis of different recovery resources

The recovery resources are set to increase from two to four per step, while all other conditions remain unchanged,and different scenarios are analyzed.Fig.7 illustrates the three detectability recovery curves for the different scenarios.When the number of recovery resources increases to four, the monitoring capability of the entire system can be restored in only five steps.However,when the number of recovery resources is reduced to two,11 steps are required to restore the monitoring capability of the system.Doubling the number of recovery resources reduces the total number of recovery steps by more than half, from 11 to five.Thus, the proposed algorithm can be used to calculate the optimal allocation of system recovery resources in different working states.Further research should investigate cross-sectional recovery during the recovery process, that is, specific results at certain recovery steps.

Fig.7 Comparison of recovery steps with different recovery resources

Fig.8 shows the recovery process for different recovery resources.The proposed algorithm can determine the optimal recovery path and reduce the time required to rebuild the information network.

Overall, the algorithm proposed in this study outperforms the random and path optimization algorithms,enabling it to find the optimal recovery strategy to quickly restore the monitoring capability of the power digital twin system in the event of an information collapse.

Fig.8 Power system detectability at step 5

5 Conclusion

This paper introduced the concept of post-disaster recovery for power digital twin systems with the intention of studying rational approaches to enhance the postdisaster monitoring capability of such systems after significant disasters.An overall approach for optimal recovery planning was proposed and an optimal linear programming method was developed to rapidly enhance system monitoring capability after extreme disasters.The simulation results were validated using the IEEE-14 node system, demonstrating that the optimization method effectively restored the monitoring capability of power digital twin systems more quickly than traditional algorithms.Comparative studies revealed that the initial repair nodes during system recovery affect the overall monitoring performance.

Nomenclature

NASA National Aeronautics and Space Administration CR Communication Router CDP Central Data Processing Unit PMU Phasor Measurement Unit AC alternating current

Acknowledgments

This work was supported by the State Grid Jilin Province Electric Power Co, Ltd - Research and Application of Power Grid Resilience Assessment and Coordinated Emergency Technology of Supply and Network for the Development of New Power System in Alpine Region(Project Number is B32342210001).

Declaration of Competing Interest

We declare no conflicts of interests.