Impact of inter-network assortativity on robustness against cascading failures in cyber–physical power systems
Graphical abstract
Introduction
Catastrophic cascading failures triggered by small-scale loss of facilities in power systems have caused huge economic losses and seriously negative social impacts, e.g., the 2003 American blackout [1], the 2003 Italian blackout [2], the 2006 European blackout [3], the 2011 Brazilian blackout [4], the 2012 Indian blackout [5] and the 2015 Ukrainian blackout [6]. Understanding how cascading failures spread in power grids attracts a lot of interest [7]. It helps design more robust interdependent power systems against cascading failures.
In the past decade, the study of cascading failures in power system has focused on the understanding of the propagation behavior of the failures in single-layer power grid. For example, the overload models, such as the Bak–Tang–Wiesenfeld sandpile model [8], the CASCADE model [9], the OPA model [10] at a fast time scale, the betweenness-based overload model [11], the capacity model [12], etc., are proposed to identify the overloaded electrical components and remove them from the network to push the cascade process. To capture the propagation behaviors of cascading failures statistically, the stochastic models, such as the OPA model [10] at a slow time scale, the hidden failure model [13], the percolation model [14], the influence model [15] and the branching process model [16], are proposed based on stochastic theory. Furthermore, recently, a large-scale simulation model [17] with the realistic structural and dynamical parameters of power grids is used to find out the vulnerable structures.
However, the modern power system is monitored and controlled by a cyber network, such as Supervisory Control and Data Acquisition (SCADA) system with devices including Automatic Generation Controls (AGCs) and Phasor Measurement Units (PMUs) [18]. Hence, the modern power system is often modeled as the cyber–physical power system (CPPS). In CPPS, the failures that occur in the physical power grid layer (PPGL) can cause the failures in the cyber network layer (CNL), e.g., the 2003 Italian blackout [2] triggered by the shutdown of a power station. Similarly, the failures that occur in CNL can cause the failures in PPGL, e.g., the Ukrainian blackout [6] triggered by the false data injection attack. These initial failures lead to the cascading failures spreading within the PPGL and CNL, and through their inter-network structures.
To study the cascading failure dynamics in CPPS, we need a proper model to describe the failure spreading based on different failure mechanisms. Currently, the cascading failure models can be categorized into two groups. One is the percolation model and the other includes different realistic models considering real power-grid and cyber failures. Besides, we need to build different inter-network structures to systematically study the impact of the interdependence on cascading failures.
The percolation model [19] is widely used to study cascading failures in complex networks. The cascade process evolves with removing nodes and edges from giant components. It is found that the robustness of the system exhibits a first-order percolation transition under random failures. The multi-layer [20], [21] networks make the system more vulnerable than the single-layer networks. The percolation model is applied to the CPPSs with one-to-one or k-to-n interdependence [22], [23], [24]. Unfortunately, although the percolation model is mathematically elegant, they cannot accurately describe the propagation of cascading failures in real CPPS. Therefore, different models have been proposed considering realistic failure mechanisms of power grids and cyber networks, e.g., the dynamic power flow model and the data exchanging model [25] are adopted to analyze the interdependence impacts on cascading failures and the realistic network operational settings [26] is also used to study the assortative and disassortative interdependence for densely or sparsely coupled networks. The HINT model [18] is proposed to train machine learning methods based on novel features for predicting the effects of the cascading failures. The minimum load reduction model [27] is used to simulate the dispatching process based on the existing of community structures in CPPS when the cascading failure happens. The probabilistic load flow model [28] is employed to examine the impacts of uncertainties on the vulnerability of the CPPS.
The assortative or disassortative interdependence is critically important to the robustness against cascading failures in CPPS. There are various metrics to estimate the ‘importance’ of nodes, such as degree [29], [30], betweenness [31], harmonic closeness [32] and node destructiveness [26]. These metrics of importance are used to generate assortative, disassortative and random interdependence. The simple coupling scheme is conducted to build the assortative or disassortative interdependence in a simple way, such as the max–max or max–min linkage pattern between layers according to the metrics of ‘importance’. The assortative coupling scheme ranks the nodes in PPGL and CNL in the order of the ‘importance’ metrics and builds the connection of the two nodes with similar importance from two layers while the disassortative coupling scheme links a more important node in PPGL to a less important node in CNL. The random coupling scheme randomly links two nodes in PPGL and CNL. The simple coupling scheme can only generate the extreme case of the max–max links for the assortative interdependence or the max–min disassortative interdependence. The inter-network assortativity can obviously influence the failure propagation. For example, the assortative interdependence based on betweenness can better resist cascading failures compared to disassortative or random interdependence under the intentional attack [33]. Also, the assortative interdependence based on degree is generally more robust to random failure [29]. More specifically, it is found that assortative interdependence is more robust for dense coupling while disassortative interdependence is more robust for sparse coupling [26]. Assortative interdependence from PPGL to CNL or disassortative interdependence from CNL to PPGL can enhance the robustness against the cascades under targeted attack [34]. But in fact, the both layers in a real CPPS are neither complete assortative nor complete disassortative interdependent. It has been found that the optimal interdependence has a moderate assortative value [35]. So, the robustness against cascading failures in some intermediate states among the complete assortative, disassortative and random inter-network coupling is quite worthy of investigation. The stochastic structural algorithm [36] can generate the CPPSs with adjustable interdependence.
This paper proposes a new model, called POD, to analyze cascading failures in CPPS. Unlike previous models, POD adopts an optimized load shedding policy [37], [38] to describe the cascade process considering three types of non-avoidable failures, including the Power-loss failure, the Out-of-control failure, and the Data-blocking failure. Although these non-avoidable failures are studied extensively, the previous studies neglect the impact of mitigation control policies on the propagation of failures. The POD model is proposed to study cascading failures in consideration of the influence of mitigation control policies in power systems.
The propagation of failures in real power systems is influenced by the mitigation control policies. According to the official reports on the blackouts [1], many mitigation control policies are used to prevent power grids from blackouts triggered by cascading failures. Real power systems, such as the western US grid, have over a hundred mitigation control policies [39]. More complicated mitigation control strategies [40], [41] are investigated and applied in the modern power system. But, most previous cascading failure models neglect the impact of these mitigation control policies [25], [26], [27].
Mitigation control policies [37], [38], [42], [43], [44] are designed to mitigate the cascading failures based on the fundamental idea of reducing the range of failure propagation, e.g., cutting off the propagation path of cascading failures. Therefore, the cascade process will be strongly influenced by the mitigation control policy. To better model the cascade process in real power systems, we must adopt the important mitigation control policies in the model. For simplicity and without loss of generality, we adopt the optimized load shedding policy [37], [38] in the POD model. According to the optimized load shedding policy, the failures are limited within non-avoidable failed components. The POD model is suitable to describe the cascade process in a real scenario when the cascading failure propagation is adequately and immediately contained by the mitigation control policies. Adopting the optimized load shedding policy in the POD model makes the POD model close to the realistic situations for the cascading failures in CPPS.
The POD model adopts the islanding mode. The islanding mode is vital in the modern power system [45] to prevent cascading blackouts. In previous studies, the control center is often located in the hub node with the highest degree or betweenness [25], [27], [46] for the centralized control. However, as the modern large power systems develop, the controls can be available from the distributed control centers [47]. So, in the POD model, each CNL node is considered as a distributed control center [48] for the islanding operation in CPPS.
It is not easy to control the assortativity of interdependence for the generated CPPSs. The RAndom Interacting Network (RAIN) model [49] is a general framework to generate the multi-to-multi interdependent networks. To understand the impact of inter-network structures on the cascade process, we extend the RAIN model based on the probability matrix to generate one-to-one assortative interdependence for a range of assortativity coefficients in terms of ‘importance’ metrics.
We exploit the POD model to systemically investigate the impact of assortative interdependence on robustness against cascading failures for the distributed and centralized power grids. Structurally, we change the assortative interdependence based on the extended RAIN model by fixing the topologies of PPGL and CNL. Our experimental results show that the impact of inter-network assortativity is really complicated. The increasing inter-network assortativity can neither always enhance the CPPS robustness under random attacks [29] nor always reduce the CPPS robustness under targeted attacks [30] as the previous studies said.
To understand the complicated relationship between the assortative interdependence and robustness, the network fragmentation and layer compatibility are introduced to explain our simulation results. Given different assortative interdependence, the interplay of network fragmentation and layer compatibility leads to varieties of evolution patterns of cascading failures in CPPS. Slighter fragmentation and better compatibility make the CPPSs more robust against cascading failures.
Our main contributions in this article are:
- (1)
The POD model is proposed to simulate cascading failures considering the optimized load shedding policy, the three types of non-avoidable failure mechanisms and the islanding scheme.
- (2)
The extended RAIN model can generate CPPSs with different one-to-one interdependence according to a new connection strategy based on the interdependent network adjacency matrix.
- (3)
We find that the assortative interdependence can mitigate, aggravate or have little impact on cascading failures considering the centralized or distributed PPGL, scale-free or small-world CNL and degree-based or betweenness-based interdependence.
- (4)
Fragmentation and compatibility are introduced to summarize and explain the complicated impact of inter-network assortativity on robustness against cascading failures in CPPS.
Section snippets
The POD model
The POD model is proposed to simulate cascading failures considering the optimized load shedding policy, the islanding scheme and three types of non-avoidable failure mechanisms, including the power-loss failure mechanism, the out-of-control failure mechanism, and the data-blocking failure mechanism.
The power-loss failure mechanism [50] refers to failed load nodes cut off from generators. The out-of-control failure mechanism [50] refers to the failed nodes disconnected from the CNL nodes or
Islanding mode in the POD model
The CPPS in Fig. 4 has the European high voltage power grid [67] as the PPGL with 1354 nodes and 1710 lines in which 100 nodes are randomly chosen to be generator nodes (in yellow) with power injection and the remaining are load nodes (in blue) with in Fig. 4(a). The CNL is a scale-free network with 1354 nodes and 1993 lines generated with scale-free preferential attachment scheme in Section 2.2.2. In order to make cascading failures take place, 135 load nodes (about 10%
Discussion
The robustness against cascading failures is influenced by the combination of the three structural properties of CPPS, the centralized/ distributed PPGL, the scale-free/small-world CNL, and the betweenness-based/degree-based interdependence in Fig. 5. To reveal how the combined impacts on the cascade process, network fragmentation and layer compatibility are introduced. It is noted that the two metrics of fragmentation and compatibility in two-layer CPPS are the extension of the fragmentation
CRediT authorship contribution statement
Hao Liu: Development of methodology; Creation of models; Original draft preparation. Xin Chen: Ideas; Designing the project; Development of methodology and creation of models; Performing numerical simulations; Supervising the findings of this work; Original draft preparation; Writing the manuscript; Acquisition of financial support for the project leading to this publication. Long Huo: Development of methodology; Creation of models. Yadong Zhang: Implementation of the computer code and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
Funding
This work was supported by the National Natural Science Foundation of China [grant numbers 21773182 (B030103) and 51201175]
References (71)
- et al.
Impact of degree heterogeneity on attack vulnerability of interdependent networks
Sci Rep
(2016) - et al.
Cascade-robustness optimization of coupling preference in interconnected networks
Chaos Solitons Fractals
(2016) - et al.
Renormalization group analysis of the small-world network model
Phys Lett A
(1999) - et al.
Vulnerability and cosusceptibility determine the size of network cascades
Phys Rev Lett
(2017) Final report on the August 14, 2003 blackout in the United States and Canada: Causes and recommendationsTech. Rep.
(2004)The Italian 2003 blackout
- et al.
Analysis, control, and economic impact assessment of major blackout events
Eur Trans Electr Power
(2008) - et al.
Monitoring of a large scale event in the Brazilian power system by WAMS
- et al.
Investigation on 2012 Indian blackout
Analysis of the cyber attack on the Ukrainian power grid, Vol. 388
(2016)
Curtailing cascading failures
Science
Sandpile on scale-free networks
Phys Rev Lett
A loading-dependent model of probabilistic cascading failure
Probab Engrg Inform Sci
An initial model fo complex dynamics in electric power system blackouts
Edge overload breakdown in evolving networks
Phys Rev E
Cascade-based attacks on complex networks
Phys Rev E
Expose hidden failures to prevent cascading outages
IEEE Comput Appl Power
Scaling and percolation in the small-world network model
Phys Rev E
The influence model
IEEE Control Syst Mag
Estimating the propagation and extent of cascading line outages from utility data with a branching process
IEEE Trans Power Syst
Small vulnerable sets determine large network cascades in power grids
Science
A realistic model for failure propagation in interdependent cyber-physical systems
IEEE Trans Netw Sci Eng
Catastrophic cascade of failures in interdependent networks
Nature
The extreme vulnerability of interdependent spatially embedded networks
Nat Phys
Detecting critical nodes in interdependent power networks for vulnerability assessment
IEEE Trans Smart Grid
Robustness of interdependent cyber-physical systems against cascading failures
IEEE Trans Automat Control
Balancing system survivability and cost of smart grid via modeling cascading failures
IEEE Trans Emerg Top Comput
Small cluster in cyber physical systems: Network topology, interdependence and cascading failures
IEEE Trans Parallel Distrib Syst
Cascading failure analysis considering interaction between power grids and communication networks
IEEE Trans Smart Grid
Robustness of interdependent power grids and communication networks: A complex network perspective
IEEE Trans Circuits Syst II
Modeling and vulnerability analysis of cyber-physical power systems based on community theory
IEEE Syst J
A stochastic model of cascading failure dynamics in cyber-physical power systems
IEEE Syst J
Inter-similarity between coupled networks
Epl
Cascading-failure-resilient interconnection for interdependent power grid - optical networks
Cascading failures in interdependent scale-free networks of different coupling preferences with the harmonic closeness
Europhys Lett
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Hao Liu and Xin Chen contributed equally to this work.