A comprehensive resilience evaluation method for power distribution systems based on weighted Page-Rank algorithm

Resilience evaluation plays a critical role in the decarbonized development of modern power distribution systems since it objectively assesses how the distribution system performs in the face of stochastic renewable generation, increasing load demand, and promising to cut carbon emissions. This study proposed a comprehensive resilience evaluation method for regional distribution systems. The proposed resilience evaluation system comprises three sub-groups: performance, economy, and connections. To improve the objectivity of the proposed evaluation method, the weighting factors are calculated based on the Analytic Hierarchy Process (AHP) and weighted Page-ranking algorithm, respectively. Specifically, AHP is adopted to calculate the weighting factors for different metrics. At the same time, the original Page-ranking algorithm is improved according to the importance of the buses and then used to calculate the weighting factors for different buses within the same metric. The proposed method is applied to a real-world distribution network in China. Numerical results show that the proposed method can objectively evaluate the overall resilience level of the distribution network, including the source side, the network, and the load side. The evaluation results can help the decision-makers to make resilience enhancement in an economical and targeted way.


Introduction
In recent years, frequent extreme disasters have posed great challenges to the operation of urban power grids, seriously endangering urban energy security and social development.On the other hand, with the background of energy structure transformation and low-carbon development, the large-scale integration of stochastic distributed renewable energy has brought new challenges to the operation of urban power distribution systems [1].Enhancing the performance of urban power distribution networks with an increasing proportion of new energy in the face of uncertainty, faults, disturbances, and sudden changes in load has become an urgent task.
In this context, the concept of resilience has been introduced into the power system.Resilience can be defined as the ability of the power system to withstand and recover from "low-frequency, high impact" events such as deliberate attacks, accidents, or naturally occurring incidents while minimizing the negative impacts, both short-term and long-term, of these events on the power system [2]- [3].For an electric power distribution system, since it is directly connected to the consumers and is pivotal to the continuous power supply, its performance in the face of severe disturbance significantly impacts the reliability of the distribution power system.The resilience of distribution systems [4] does not guarantee a consistent power supply in extreme conditions.Still, it enables flexibility, an active response to a major disturbance, and a fast restoration to a stable state with affordable damage.Therefore, resilience evaluation plays a critical role in the decarbonized development of modern electric distribution systems since it objectively assesses how the distribution network performs in the face of stochastic renewable generation, increasing load demand, and the promise to cut carbon emissions.In [5], researchers study the resilience evaluation of active distribution systems based on situational awareness estimation.Prevalent reliability indices like the System Average Interruption Frequency Index (SAIFI) and System Average Interruption Duration Index (SAIDI) [6] can also contribute to assessing the resilience of the power system.Still, these metrics are designed for postcontingency stage assessment and usually fail to identify the withstand capability of the distribution systems.To evaluate the system's ability to withstand disturbance, Phillips et al. [7] proposes an adaptive-capacity-based index.In [8], a set of resilience metrics includes modified traditional reliability indices, a new metric for network structure assessment, and a metric for recovery difficulty.These metrics might be sufficient for their research goals but are not comprehensive.
On the other hand, most studies often need to consider the importance of different buses when calculating the resilience metrics.Although it is straightforward to assign weighting factors to different buses, the subjective weighting factors make the evaluation results less practical.Recently, researchers have demonstrated that complex network theory [9] is a promising candidate for modelling the system and identifying critical buses and lines [10].
To address the limitations of previous studies, this paper proposes a comprehensive resilience evaluation method for regional distribution systems.The proposed resilience evaluation system comprises three sub-groups: Performance, economy, and connections.Furthermore, AHP and an improved Page-ranking algorithm are used to alleviate the subjectivity of the assessment.A real-world application is presented as a case study to show the effectiveness of the proposed assessment methodology.

The proposed resilience evaluation framework
The proposed resilience evaluation system comprises three sub-groups: Performance, economy, and connections.A general structure of the evaluation framework is established and illustrated in Figure 1.More specific details are expanded in the following Sub-sections 2.1-2.3.

Metrics for performance evaluation
Performance evaluation is designed to assess the resilience performance of a distribution system from the following aspects: 1) Topological structure and automation coverage to withstand a disturbance; 2) Flexible sources to support the system during a disturbance; 3) Reliability assessment.
2.1.1.Flexibility metric.Flexibility metric refers to the percentage of flexible sources in all sources.The operators of the grid should determine the definition of flexible sources.Since the flexible sources are calculated based on buses, the algorithm introduced in Section 2.4 further improves the metric.

Demand response metric.
The flexibility metric refers to the percentage of loads participating in demand response programs.It should be noted that the electric vehicle is excluded from the loads since there is a dedicated metric for EVs in the evaluation framework.Since the loads participated in demand response programs are calculated based on buses, the metric is further improved by the algorithm introduced in Section 2.4.

Electric vehicle metric.
Like the demand response metric, the electric vehicle metric is defined as the percentage of EVs participating in demand response programs (in all EVs).Since EVs that participated in demand response programs are calculated based on buses, the metric is further improved by the algorithm introduced in Section 2.4.

Topology metric.
Topology metric refers to the percentage of the distribution network that satisfies the specific grid codes.A satisfied network can support power flow during a disturbance, enabling a prolonged power supply to customers even when most local power sources are unavailable.

Automation metric.
The automation metric is the percentage of automatic distribution systems in a regional distribution network.Distribution automation that offers rapid detection and actions is critical during a disturbance.[6] is used to assess the reliability of the distribution systems.

Reliability metric (SAIFI).
Another prevalent reliability metric, the System Average Interruption Frequency Index (SAIFI) [6], is also adopted to assess the reliability of the distribution systems.It should be noted that a reliability metric alone cannot evaluate the resilience of a distribution system when many components are damaged or a prolonged outage occurs.

Metrics for economy evaluation
This sub-group is designed to evaluate the monetary performance of the distribution system in terms of resilience.The cost evaluation is important since redundancy can improve the resilience performance of the distribution system, but unnecessary investment and low-utility rate should be avoided.

Unit cost for increased electricity.
It is defined as the monetary investment cost for every 1 kW/h in a benchmark year over the current monetary investment cost for every 1 kW/h.

Peak and valley metric.
It refers to the total load over the average difference between peak load and valley load.A smaller difference is favoured for the resilience performance of the distribution system.Since the peak and valley load statistics are collected based on buses, the algorithm introduced in Section 2.4 further improves the metric.

Distribution line loss metric.
This metric adopts a widely used distribution line loss rate, defined as the percentage of the lost electricity in the overall electricity.The metric is a reciprocal of the distribution line loss rate.

Utility metric.
It is defined as the reciprocal value of the utility rate of equipment.This metric is designed to avoid unnecessary and redundant equipment investment.Although backup power sources or distribution lines are helpful during a contingency, overestimated and conservative asset investment would lead to a low utility rate and a prolonged investment return.

Metrics for connection evaluation
This sub-group is designed to assess the connection of the distribution system in terms of resilience.

DG penetration metric.
It is defined as the penetration level (presented as a percentage) of the distributed generators.

Electricity metric.
This metric is defined as the percentage of electricity in all energy consumption in a distribution system.Compared with gas, heat, and other energy sources, electricity is more flexible than its counterparts and can be used as an alternative for gas and heat on many occasions.

Energy storage metric.
This metric is defined as the ratio of energy storage capacity over the total generation capacity of the distributed generators.A higher energy storage installation in a distribution system is beneficial since it can provide an emergency power supply to users.Energy storage devices are installed on buses.The metric is further improved by the algorithm introduced in Section 2.4.

IoT metric.
This metric is defined as the IoT (Internet of Things) ratio of the distribution system, describing the extent of connectivity in terms of IoT.IoT is installed at each stage of the distribution system for monitoring and communication, enabling a more rapid response to major disturbances and more efficient utilization of flexible and conventional power sources.

Weighted Page-Ranking method
Some metrics introduced in Section 2.1-2.3 are calculated using different buses.It should be noted that there are critical buses and less important buses.However, the weighting factors of different buses are conventionally considered universal or determined subjectively.This section introduces a weighted Page-ranking method to improve the objectivity of the following metrics: Flexibility, Demand response, EV, Peak and valley, and Energy storage.
The original PageRank algorithm was originally proposed by the founders of Google [11].The hyperlink structure is adopted to measure the importance of web pages.Specifically, a webpage is important if other important pages in the PageRank algorithm point to it.A higher PageRank score (PR score) indicates a higher importance.PageRank algorithm is widely used to identify critical nodes in a complex network in many engineering problems.However, the original PageRank algorithm solely depends on the network structure and ignores additional information.Therefore, there are more practical methods for the distribution of power systems.On the other hand, assigning weighting factors to different buses is arbitrary since there is no universal standard for a more important bus.In this section, the original PageRank algorithm is integrated with the proposed metrics, and the advantages of this method are as follows: 1) Weighted PageRank not only considered the topological importance of the buses but also the other technical and economic related information; 2) The weighting factors are objectively determined, instead of subjective ones.
For any bus in a distribution system, its weighted PR score for a specific metric can be described as: represents the bus and the   is the set of buses that with inward power flow to   .|  | is the outward power flow from |  |.   is one of the metrics considered in this section for bus I, including flexibility metric, demand response metric, EV metric, peak and valley metric, and energy storage metric.
Assuming a distribution system has b buses, the PR of a bus can be calculated iteratively according to Equation (1).Alternatively, Equation (1) can be written as Equation ( 2) compactly: where k is the iteration number and  () is the PR vector. is a parameter ranging from 0 to 1.  is a unit column vector and  is a row-normalized matrix with ℎ  .ℎ  =0 if there is no connection from bus i to j and ℎ  =1/|  | if there is such a connection. is a binary vector, where   =1 if   is dangling node in the network and   =0 otherwise.
To comply with other metrics without PR score, the flexibility metric, demand response metric, EV metric, peak and valley metric, and energy storage metric should be further processed.The normalization can be described as Equation (5).

Application in a distribution system with AHP
In this section, the proposed method is implemented in a distribution system in Zhejiang Province, China, with 86 distribution lines and 5 substations, all at 10 kV level.To avoid arbitrary weighting factors among different metrics, AHP (Analytic hierarchy process) [12] is adopted in the case study to alleviate the subjectivity.As shown in Table 1, the calculated weighting factors for performance, economy, and connection metrics are 47.21%, 29.62%, and 23.17%, respectively.According to Section 2.4, the weighted PageRank scores are calculated, and the weighted PageRank score for Flexibility is presented as an example as shown in Figure 2.With weighting factors for metrics and the proposed metrics in Section 2, the final resilience evaluation can be calculated, and the results are listed in Table 2.For metrics with weighted PR Scores, it can be concluded from the table below that, after modifying the evaluation results with the weighted PR algorithm, the evaluation results are more aligned with the electrical engineering practice since more importance buses are assigned with higher and objective weighting factors.

Figure 2 .
Figure 2. Weighted PR score for Flexibility (to be normalized).

Table 1 .
Weighting factors for all metrics.