Reliability assessment of integrated energy systems based on complex network theory

As integrated energy systems (IES) continue to undergo development and advancement, the degree of coupling between subsystems and their complexity are also on the rise, posing a challenge for the reliability assessment of such systems. The conventional method is unable to explain the interaction between subsystems as well as system‐level phenomena. The complex network theory is needed to provide a deeper analysis. Firstly, this paper introduces the basic concept and analysis tools of complex network theory. Secondly, the topological network model and the identification indexes of weak links for the IES are summarized as the basis for reliability assessment. Then, the reliability assessment indexes and assessment methods are sorted out for the IES based on the complex network theory; and further, the measures to optimize the reliability of the IES are put forward. Finally, from the perspective of complex network theory, directions for future research on the reliability assessment of IES are suggested.

sources throughout the whole process of planning, design, construction, and operation to make full use of renewable energy. 1 Like the existing power supply, gas supply, and cooling/heating, safety and reliability are the most basic requirements for IES and the foundation for ensuring the stability of regional energy consumption. The necessary reliability assessment can be qualitatively or quantitatively represented for the risk level of system's energy supply interruption to guide the production practice activities such as system planning and operation. The reliability of an IES includes adequacy and safety. The adequacy means that the system can meet users' peak load requirements at any time. Safety means that all equipment in the system must operate under the conditions of not exceeding the permissible pressure and flow, irrespective of during normal operation or in the case of an accident, representing the ability to maintain consecutive energy supply for a short period of time and avoid large-scale energy supply interruption if an accident occurs in the system. Compared with traditional single energy supply systems such as electric power system and natural gas system, IES has their characteristics and a more complex network structure based on them. With continuous expansion of the project scale of the IES, it will have more and more internal nodes. Together with the dynamic characteristics of flexible loads, energy storage and units, etc., the structure of the system will gradually transit to complex networks from traditional bus-integrated network and different energy systems are increasingly coupled tightly, promoting coordination, and complementation of energy sources. However, the coupled two-way energy flow promotes the two-way transmission of cascading faults. 2 A single faulty energy subsystem will result in a load redistribution in the whole system, and the resulted problems including network congestion may give rise to new risks or faulty equipment. As a result, the energy demands of a large number of users are difficult to be met, thus resulting in cascading fault in IES. Therefore, how to identify the energy supply reliability of the IES is an urgent priority. At present, there is a certain research foundation for reliability evaluation of IES. In Reference 3, the factors affecting the power supply reliability of an electricity-gas coupled IES are analyzed comprehensively, and an analytic algorithm for power supply reliability evaluation of the electricity-gas coupled IES is proposed. Reference 4 uses a sequential Monte Carlo simulation to evaluate the reliability of integrated power-gas systems, and gives new reliability indices. Reference 5 considers the multi-time scale of IES, develops an operating strategy that takes into account the inertia of the thermodynamic system and the interaction of multiple energy sources, and proposes how to evaluate the reliability of IES.
The above research results have laid a good foundation for the research on the reliability of IES. However, the following problems still exist for the current research: (1) The current research on the reliability of IES energy supply mainly focuses on the scale of Micro-Grid or the correlation between equipment, without analyzing the system topology, thus being unable to restore the complex relationship among the subsystems and their dynamic laws. And, as the energy coupling technology advances, the scale of IES will also be gradually expanded from the users' Micro-Grid level to the regional level; (2) With the continuous popularization and penetration of energy conversion equipment, diversified energy storage and electric vehicles, more and more two-way dynamic nodes will be designed in the IES, dynamic changes may occur on the system structure. This means that a faulty node is probably to cause cascading faults. However, the conventional method cannot represent the energy supply reliability and dynamic transmission characteristics in the case of an accident of the system from the perspective of topology; (3) The current method does not combine the topology of the system with the physical parameters and operating characteristics of the studied systems to identify the weak links in the system and assess the system reliability from the topological structure and operating state. With continuous expansion of the IES scale, deepening energy coupling technology, and the uninterrupted increase of two-way dynamic nodes, the IES is designed with a larger node scale, a more complex network topology and a more far-reaching and lasting fault transmission mechanism, which are all representing the typical characteristics of complex networks. Therefore, complex network theory is applicable to IES. With application of complex network theory, complex systems can be studied from the perspective of the system as a whole to reveal the stability of complex network topology and the dynamic transmission characteristics if attacked, analyze cascading fault models and internal transmission mechanism. This method overcomes the shortcomings of the conventional methods, for example, ignoring the interaction among subsystems, and failure to explain the emergent properties of complex systems on the system level. 6 A method based on the complex network theory can structurally represent the statistical characteristics of the IES, understand, and explain the operation mode of the network, so as to account for the progression trend of faults. With reliability assessment, the weak links that have a greater impact on the system can be identified and vulnerable lines and nodes are found. And, targeted protection measures are taken to improve the interference immunity and the system reliability. Applying complex network theory to the reliability assessment of an IES can provide a means for ensuring safe and reliable operation of the IES in the future. 7 Firstly, this paper introduces the basic concept and analysis tools of complex network theory. Secondly, the topological network models of the IES are summarized, and the identification indexes of weak links are summarized as the basis for reliability assessment after comprehensively considering the system topology and operating state. Then, the reliability assessment indexes and assessment methods are sorted out for the IES based on the complex network theory; and further, the measures to optimize the reliability of the IES are put forward. Finally, from the perspective of complex network theory, directions for future research on the reliability assessment of IES are suggested.

Basic concept of complex network theory
For the basic method of complex network theory, a complex system is abstracted as the connecting relationships between nodes and edges, which are statistically summarized. It is formally similar to the topological analysis, that is, a as a tool describing a network system. A complex network usually consists of nodes and edges, where the former is the basic elements in the network, representing different individuals in the real system, while the latter connects the nodes in the system, representing the relationships between individuals. Thus, a complex network can be simply expressed as: Where: G n is the complex network chart; N and E respectively represent the set of all the nodes and sides; n i is node i; and e ij is the side connecting the node i and the node j.
In 1998, the concept of the small-world network was proposed by Watts and Strogatz, 8 which means that the average distance between nodes is kept at a small value, regardless of the size of the network. The small-world property refers to the feature that a network has a large clustering coefficient and small average distance. Watts defined the following topographic characteristic of a network as the small-world property: Where: C is the clustering coefficient; D is the minimum average distance of the network; and the subscript random represents the stochastic network. In 1999, Barabasi and Albert brought forward a scale-free network model. 9 The scale-free property of a network means that there are a tiny number of nodes with a large degree in the network and the degree values of most nodes are mainly near several smaller values and the node degrees of the network are distributed in a form of power law. The BA scale-free network 9 is the most common scale-free network model, which is constructed according to the growth characteristics of the network itself and the priority connection between nodes; that is, if it is necessary to add a new node in the network, the new node will have a higher possibility of being connected to an old node with a higher priority number.

Analysis tool of complex network
The IES network may be modeled and visualized and the data are computed and analyzed with Python language based NetworkX. The NetworkX is a Python based software package, which is used to create and operate the complex network and to learn the structure, dynamics, and functions of the complex network. With NetworkX, many kinds of complex network model may be created, the models are visualized, all kinds of complex network algorithms are accessed to compute relevant parameters of the network and analyze its structure and characteristics. The core functions of the Net-workX complex network analysis software include: (1) Constructing complex network: A network can be constructed with its built-in model or special types of network may be built by writing functions; (2) Network visualization: The integrated complex network is visualized to more intuitively represent the elements and interactive relationship of the complex network; (3) Computing complicated network characteristic parameter: Accessing the algorithm can computer all kinds of network characteristic parameters; (4) Statistical characteristic analysis: NetworkX features strong capability to analyze complex network characteristics and may perform statistical analysis of the built network with the existing functions.

TOPOLOGICAL NETWORK MODEL OF IES
Integrated energy systems (IES) are composed of coupled heterogeneous systems. On the one hand, this structure features complex system components, obvious model differences, and diverse operation modes; on the other hand, there is interaction between coupled systems, so that any fault will spread among different energy subsystems, which makes the system and its characteristics quite complex. Therefore, establishing a reliability assessment model for IES coupled with multi-energy flow is essential for accurate assessment of the reliability of IES. It also matters for accurately describing the characteristics of different energy forms and the complex coupling relationships between them. 10

Model of coupling link
In contrast to conventional power systems, the change in reliability of IES is mainly caused by the coupling among its subsystems. Therefore, it is necessary to establish an appropriate coupling model to evaluate the reliability of the IES. References 11, 12 introduce an energy hub to describe the coupling relationship of various energy resources such as electricity, gas, and heat of the IES to implement the conversion, storage and distribution among the energy resources. According to the conversion equation L = CP, the energy hub model describes the coupling processes of multi-energy flow, of which relationship between input and output follows the Equation (4): Where: L , L and L are respectively the electric power, natural gas and thermodynamic outputs; P , P and P represent respectively the electric power, natural gas and thermodynamic inputs; the matrix C is a coupling matrix, of which each coupling coefficient corresponds to a coupling relationship between a specific input and a specific output.
The energy hub shown in Figure 1 can be taken as an example. 13,14 This energy hub consists of a transformer (T), gas turbine (GT), gas boiler (GB) and heat exchanger (H), of which the conversion formula of the input and output is shown according to Equation (5): Where: is the distribution coefficient; T is the efficiency of transformer; GT and GT are respectively the efficiency of the gas turbine converted to electrical energy and thermal energy; GB is the efficiency of the gas boiler; and H is the efficiency of the thermal energy exchanger. The energy hub model can comprehensively consider multiple energy coupling forms from the whole system. 15 However, in view of the built-in power balance and energy conversion relationship for the coupling matrix, the difficulty F I G U R E 1 Sketch map of energy hub model for identification and acquisition of key parameters will be increased with increased structural complexity of the energy hub model; thus, the coupling matrix will be difficult to obtain directly. Furthermore, the above models all have some shortcomings in scalability; specifically, the above models require prior knowledge or a preset configuration of the internal structure of the system and the specific energy conversion equipment when modeling. If the structure or equipment changes, the model will fail and needs to be rebuilt from scratch.

Model of system topology
At present, the analysis of the network characteristics of an electric power system by some researchers have demonstrated that an electric power system has small world, free scale and other complex network characteristics. 8,16,17 They also explored the applications of the complex network theory in identifying weak links of an electric power system, risk assessment and other fields. [18][19][20] Based on this, some researchers have further analyzed the complex network characteristics of the IES and demonstrated the applicability of the complex network theory in the study of IES. 7,21 Therefore, IES can be studied from the perspective of complex networks based on the above theoretical basis, and the network model of IES can be simplified as a complex network diagram. The principle of its simplification can be simply described as follows: (1) The components, elements, etc. included in the generator, the heat source, gas source, load, gas storage tank, and energy hub are simplified as nodes; (2) The branches such as transmission lines, gas pipelines, and thermodynamic network water supply/return pipelines are simplified as sides. The problem that different parameters are caused by different physical structures of the transmission lines and pipelines are ignored, which are considered as promiscuous networks. And, if one of the water supply or return pipeline in the thermodynamic network pipeline is faulty, the water supply and return lines will be failed. The water supply and return pipelines are simplified as one side. If one of the water supplies or return pipelines in the heat network pipeline is faulty, the water supply and return lines will fail. The water supply and return pipelines are simplified as one edge. Programming is made in the Python language based NetworkX software package to abstract the integrated energy network model diagram as a complex network model diagram. Taking the energy hub based topological graph of the electricity-gas-heat IES shown in Figure 2 as an example, it is simplified as the complex network model diagram shown in Figure 3 with NetworkX. The topology model of an IES based on complex network theory usually only considers the system structure and the relationships between its parts. The equipment or links in the system are abstracted as nodes and edges to establish the network topology. However, due to its treating all nodes and branches equally and normalizing their weights, the static model is prone to error. Therefore, it is necessary to establish a weighted topology model of IES. There are two common methods of edge weighting: 1. The impedance of the edge is used as the weight with which to evaluate the electrical distance between nodes. 22 References 23, 24 introduce the impedance value of the line to the topology model of the power grid. The shortest path is defined as the path with the minimum sum of the impedance value of the line between two nodes, and the path with the lowest network loss is searched by the Dijkstra algorithm. On the basis of the above definition, Reference 25 introduces the concept of electrical distance to IES, which is improved according to the features of the gas network and the heat network. The shortest path between nodes is searched by computing the sum of the equivalent impedances of the lines.

Natural gas station
Electric grid network Thermodynamic network Natural gas network

F I G U R E 4
General branches of gas network and heat network 2. The output of energy supply equipment and the state of power flow are used as the weight of the edge for evaluating the importance of the branch and its influence on efficiency. In Reference 25, the weight of a side is defined as the ratio of the branch transmitted energy S ij to the system rated capacity S R . And, with consideration of the directivity of the energy flow, the side weighting coefficient ij is defined as follow: And, the ratio of node transmitted total energy to the system rated capacity is considered as the node weight s i , which reflects the importance of the node to ensure the successful energy transmission. The node-weighting coefficient i is defined as follow: Where: S i = ∑ S ij . The above-mentioned two methods of weight are not contradictory. The equivalent impedance of the line is used as the distance weight for searching the shortest path, while the branch power flow is used as the edge weight for assessing the efficiency of energy supply. In addition, Reference 20 establishes a weighted topology model based on component reliability parameters, which improves the identification accuracy for units with the same structural characteristics in the network.
Integrated energy systems (IES) are coupled with multi-energy flow and represent complex multi-input and multi-output systems that involve the mutual conversion of multiple energy forms. Different energy systems have different dynamic characteristics and time scales 26,27 ; therefore, it is necessary to build a unified model of multi-energy flow coupling systems that account for time scale characteristics. 28,29 In order to realize the formal unification of the heterogeneous energy network model, in Reference 30 the heterogeneous energy flow is calculated. For this purpose, the gas network and thermodynamic network are represented as one energy circuit diagram, which is connected topologically by several branches containing such energy circuit components as gas resistors, thermal resistors based on the united energy circuit theory proposed in References 31, 32. It is shown in Figure 4.
Where: G b is the pipeline flow; s , m is the pressure at the head and end of the branch; z b are the air resistance and thermal resistance; E b is the pressure source corresponding to the turbocharger; k b is the component parameter of the controlled source.
In Reference 33, the generalized electrical circuit model is proposed for the branch layer of multiple-energy-source network to reveal the generality of the energy flow of the multiple-energy-source network based on the Laplace transform. Based on Reference 33, in Reference 34, the generalized circuit analysis theory of the network layer is brought forward to address the difficult problem of joint analysis of the multiple-energy-source network. In order to solve the problem of multi-time scale in reliability modeling, Reference 1 introduces time-varying characteristics in the energy output link using a Monte Carlo timing simulation. When constructing the assessment index for identifying weak nodes, Reference 30 considers the delayed response characteristics of the gas and thermodynamic networks. Reference 4 considers the multi-time-scale characteristics of IES and develops an operating strategy that takes into account the inertia of the thermodynamic system and the interaction of multiple energy sources.

Vulnerability indexes of topological structure
Generally speaking, indexes such as degree and betweenness are selected to measure the importance of components and lines in power system network, natural gas network, and thermodynamic network. The vulnerability indexes of topological structure usually include the following classifications:

Degree of node
This refers to the number of nodes adjacent to a certain node. The larger the degree, the stronger the influence of such node on nearby ones, and the higher the importance of such node in the network. However, the degree of conventional nodes does not account for the physical characteristics of the network, and cannot distinguish the difficulty level of cascading faults caused by different pipes and lines. Therefore, Reference 30 introduces the concept of M-order neighbor number with consideration of the physical properties of IES for the purpose of improving the degree of node. The M-order neighbor number of Node i means the total number of the node of such node within a certain impedance range, which is defined as follows: Where: |Z| ij is the equivalent impedance of the shortest path between Node i and Node j; |Z| is the threshold of the given equivalent impedance; I(⋅) is the index function. I(⋅) is taken as 1 if the impedance of the shortest path between Node i and Node j is less than the threshold; or, it is taken as 0.

Betweenness
The betweenness of a node refers to the ratio of the times that all the shortest paths in a network pass through Node i to the total number of the shortest paths. The betweenness of a line refers to the ratio of the times that all the shortest paths pass through the line (i, j) to the total number of the shortest paths. However, the betweenness only considers the topology structure of the network but not the influence of the distribution and output of the energy supply equipment on the system and the power transmission characteristics of the network. To address the above problems, Reference 35 introduces the concepts of weighted line betweenness and weighted node betweenness, and modifies them according to the characteristics of IES. The weighted line betweenness B ij the line (i, j) is defined as the sum of the loads passed through and borne by the shortest paths between all the energy-supply equipment in the system and the loads, as shown in Equation (12): Where: P m is the output of the energy-supply equipment m; m is the set for the energy-supply equipment of the shortest path to the load via the line (i, j).
The weighted line betweenness b i of Node i is defined as the sum of the weighted line betweennesses of all the sides connected with the node as shown in Equation (13): Where: i is the set of all the nodes connected with Node i.

Line capacity limit
For the transmission energy line, the limiting factor of total transmission capability of the line needs to be considered. In existing research, the total transmission capability is taken as the weight in the existing study, which cannot accurately reflect the influence of power flow limit. To remedy this deficiency, Reference 22 defines the capacity limit index of the line as the difference between the transmitted line energy and the rated line capacity.

Vulnerability indexes of operating state
To identify the weak links, not only the influence of the nodes and lines on the network topology, but also their influence on the system operating state will be taken into consideration. From the perspective of operating state, to identify the weak links of a power system, the Reference 36 puts forward the load power flow entropy index, which addresses the defect that the load power flow entropy index cannot distinguish heavy-load branches from light-load branches. In Reference 37, a Theil's entropy is used to build assessment indexes for the degree of balance for the power flow growth rate of a power grid branch considering the voltage level. Based on the above research and according to the unified energy path theory, Reference 30 firstly expresses the gas network and heat network as an energy path diagram connected by branches containing impedance like the power grid and then introduces the entropy theory into the research of the IES. According to the Theil's entropy principle, the heterogeneous energy sources are divided into different zones. The improved Theil's entropy index is built to represent the degree of balance of power flow and pressure surge according to the degree of change in power flow and pressure in each zone and the tolerance margin of the system.

4.2.1
Theil's entropy index of node pressure growth rate During the operation of the IES, the node pressure in the system will be rapidly changed with the state changes or the occurrence of fault disturbance. The Theil's entropy index T i of node pressure growth rate is defined to describe the deviation trend of node pressure when the load fluctuates. The smaller the Theil's entropy T i of the node pressure growth rate is, the more the offset system pressure is evenly distributed to each node after the node i load fluctuates. When the node i load in the network fluctuates, it will affect the pressure at the node j, and the tolerance of the node j to the pressure fluctuation is expressed by j .
Where: ji is the pressure amplitude at the node i after the load fluctuation at the node j; j0 is the pressure amplitude at the node j in the initial state; jN is the rated pressure at the node j.
The IES is divided into three zones including power grid, gas network, and heat network. And, the coupling elements are regarded as both the source node and the load node of their respective zones according to their energy flow conversion characteristics, which will be considered repeatedly during calculation. Given that an IES is designed with N n nodes, N l branches, and S zones, and the S th zone is provided with N S n nodes and N S l branches, Theil's entropy index of node pressure growth rate in the S th zone is (15) Where: S j is the pressure growth rate at the node j in the S th zone; S is the sum of the node pressure growth rates in the S th zone.
The Theil's entropy of the pressure growth rate of the whole system can be defined as follows: Where: T k is the degree of imbalance of the pressure growth rate within the zone; T b is the degree of imbalance of the pressure growth rate between zones; is the sum of the overall pressure growth rate of the system.

4.2.2
Theil's entropy index of branch power flow surge rate Overloading line transmission energy, faulty line, and power flow transfer will all change the operating state of the system. In this study, the degree of balance of line power flow surge rate is used to measure the distribution law of power flow surge in the system after line disturbance. The smaller the line power flow surge rate T Fi is, the more balanced the power flow surge is distributed to each line after the line i in the system is disconnected. Given that the system is under normal operating state, when the line i is disturbed, the power flow surge rate generated by the line j is: Where: F ji is the actual power flow of the line j after the line i is disturbed; F j0 is the initial power flow value of the line j; F jN is the rated transmission capacity of the line j; The Theil's entropy index of the defined line power flow surge rate in the S th zone: Where: S j is the power flow surge rate of the line j in the S th zone; S is the sum of line power flow surge rates in of the S th zone.
The Theil's entropy of the power flow surge rate of the whole system can be defined as follows: Where: T kF is the degree of imbalance of the power flow surge rate within the zone; T bF is the degree of imbalance of the power flow surge rate between zones; is the sum of the overall power flow surge rate of the system.

Integrated assessment indexes of weak links
The above studies mostly use a single index to identify the weak links of the network, but using only one index will result in a somewhat one-sided description of uncertain information. In the light of this shortcoming, multiple indexes can be selected to comprehensively assess the importance of nodes and lines. Reference 38 uses the analytic hierarchy process to determine the weight of each index, and proposes comprehensive assessment indexes for nodes and lines, respectively. However, the classic multi-attribute assessment methods seldom consider the influence of decision makers on the results of the assessment when weighting each index. Therefore, References 17, 39 make comprehensive use of the Entropy method and analytic hierarchy process to weight each index objectively and subjectively. This approach cannot only reflect the objective risks existing in the system structure, but can also take into account the judgment ability of experts' experience.
To comprehensively identify the weak nodes and lines in the system, the importance indexes of weighted node betweenness and weighted line betweenness are considered from the perspective of topology. And, from the perspective of operating state, the Theil's entropy of node pressure growth rate and the Theil's entropy of branch power flow surge rate are considered to build a comprehensive vulnerability assessment index.
Where: C n,i is the comprehensive vulnerability assessment index of the node; C l,i is the comprehensive vulnerability assessment index of the line.

Reliability assessment indexes
In case of the fault of the IES itself or external attack, the IES lines, pipelines or equipment may cease operating, and the resulting impairment of the network transmission performance in a running network is to be assessed with certain indexes. 40,41 In consideration of the physical properties of IES, at present, the indexes measuring the reliability of IES include:

Breakdown threshold
In the connectivity-based cascading fault model, it is considered that only the nodes in the maximum connected subgraph can work normally, and the other subgraphs and nodes will fail. Therefore, in Reference 7, the system reliability index is represented as the ratio of the number of nodes of the maximum connected subgraph in the network after an attack to the total number of nodes of the original network. However, when renewable energy sources are designed in the power grid or gas storage tanks are equipped in the natural gas network, the nodes in the non-maximum connected subgraph may also operate normally. Therefore, it is limited that the system reliability is quantified by assessing the scale of the maximum connected subgraphs after the network is attacked. In Reference 42, the system reliability is more practically quantified by studying the minimum number of removed nodes that cause the entire system to break down.

Network effectiveness indexes
The network effectiveness index represents the strength of the connectivity between nodes and measures the global energy transmission capability of IES. The higher the network effectiveness index is, the higher the efficiency of energy transmission and conversion between nodes is, which is conducive to ensuring the reliability of system energy supply. In Reference 25, the concept of distance in traditional graph theory is replaced with the weighted distance of the electric power/natural gas/thermodynamic system, and the network effectiveness g is defined as the average value of the reciprocal of the shortest path length between the energy supply node and the energy demand node in the system, namely: Where: S G , S W , S S , S R , S D are respectively the set of generator, gas source, gas storage, heat source, and load nodes; N G , N W , N S , N R are respectively the number of generator, gas source, gas storage, and heat source nodes; d ij is the weighted distance of power/natural gas/thermal power system that may exist between the node i and the node j; Z , Z , Z are respectively the equivalent impedances of power network transmission and distribution lines, natural gas network gas pipelines, and thermodynamic network heat-supply pipelines.

Index of load loss rate
Because the major task of the energy system is to provide reliable load supply, the load loss shall also be considered for the reliability assessment. In Reference 43, after the system is attacked, the energy-supply proportion attained from locally balanced power of the areas is defined as the index of maximum power supply range F max , as shown in Equation (23): After the system is attacked, the load loss rate is ,as shown in Equation (24): Where: P ,j , P ,j , P ,j are respectively the output of the energy-supply equipment j of the electric power, natural gas and thermodynamic systems in the Zone I after being attacked; L ,k , L ,k , L ,k are respectively the load of the load node k of the electric power, natural gas, and thermodynamic systems in the Zone I after being attacked; K is the total number of the zone; F is the maximum power supply range of the system before being attacked. Another issue needing attention is that the economic loss caused by the unit load loss of different nodes in the same network may be different due to the scale of complex networks. For this reason, Reference 20 introduces an economic index to further distinguish the degree of influence of nodes in the network through economic post-assessment. Meanwhile, in the context of the rapid development of renewable energy such as wind power, some papers proposed risk indexes for coupling systems to help account for the uncertainty of intermittent energy sources. 44,45 At present, the major problems for the indexes of the reliability assessment include the following: Each energy flow system in a given IES has its own models and control methods, which means that the transmission and conversion characteristics are also different for each energy flow system. For this reason, it is difficult to use a unified standard to measure the influence of each energy flow system on network reliability. In response to this problem, Reference 1 proposes that research should be done on a reliability assessment index system for the characteristics of different energy supply forms, which would solve the difficulty of comprehensive measurement of the reliability level of multiple energy supply forms. In addition, it will also be necessary to study the influence index of coupling relationships between various energy flow systems on the overall reliability of IES.

Assessment method
The procedure for the reliability assessment of IES is described as follows: 1. The above-mentioned calculated indexes are sorted by magnitude to identify the relative importance of the nodes and lines in the complex network model and identify the key links of the network. The topological structure vulnerability index, operating state vulnerability index and the comprehensive assessment index of weak links mentioned in Section 4 are calculated according to the formula. And, the above-mentioned calculated indexes are sorted by magnitude to identify the importance of the nodes and lines in the network and identify the key links of the network. 2. The different attack strategies are customized according to the demands of reliability analysis to simulate the external attacks on the system. And, the nodes of attack or the size of lines, that is, the number of the nodes or lines to be removed, are selected as needed. 3. The reliability assessment indexes, such as network effectiveness and load loss rate of the system, are calculated for each attack.
4. The fault impact degree under different attack strategies and the fault scale that the system can withstand can then be assessed through comparative analysis of survivability simulation results. 35 See Figure 5 for the flow diagram of the reliability assessment. Conventional network attack strategies can be categorized in terms of objects destroyed (destroying nodes or destroying lines) and in terms of intentionality (random attacks and deliberate attacks). And, for the random attack, a destroy from the natural world (e.g., a natural disaster) is simulated, of probability of occurrence is random. Therefore, the order of the destroy is not controllable; for the deliberate attack, the destroy from the enemies of the system or terrorists is simulated, which is deliberate. For a deliberate attack, the attacker usually starts with the most important components or lines and destroys the system components in descending order of importance. The exact sequence of destruction varies according to the attacker's different determination of the importance for the nodes and edges. Reference 7 compares the damage degree of the system under three different attack strategies, that is, preferentially destroying nodes with high betweenness, nodes with high degree, and nodes with high importance.
A deliberate attack may be static or dynamic. For a static attack, the first ranked nodes and lines are deleted according to the importance of the nodes and lines computed for the whole network without considering the network structural changes that occur during their removal; for a dynamic attack, the importance of the nodes and lines are re-computed in consideration of network structural changes to delete critical nodes and lines in a targeted way for the purpose of quickly destroying the network. 46 Reference 22 further describes the importance of the key links in the network and establishes five fault attack modes, namely random attack, random matrix theory and entropy theory-based static deliberate attack, electric betweenness-based static deliberate attack, line influence comprehensive index-based static deliberate attack and line influence comprehensive index-based dynamic deliberate attack. Normally, the attacker may develop different attack strategies depending on what system information he has attained. 47 An organizational attacker who has background knowledge may implement selective and precise attacks on the targets. Therefore, from the view of the attackers, various attack modes are built based on different information. Working from the attacker's perspective, Reference 48 proposes four kinds of attack scenarios, namely zero information-based random attack, topology information-based attack, topology and coupling information-based attack, and comprehensive information-based attack, and compares the damage degree on system function under different attack scenarios; the study demonstrated that strengthening information confidentiality can improve system reliability.
In the present literature, researchers have elucidated the reliability of a system in the case that the nodes or lines are simultaneously attacked. However, in practice, such attacks are probably carried out in a particular sequence. Therefore, in Reference 49, the order of attack is further considered and a new attack strategy, that is, sequential attack, is proposed. The sequential attack means to remove a certain number of nodes or lines from the system in turn, one for each; the synchronous attack means to attack simultaneously a certain number of nodes or lines of the system. In Reference 50, the cases show that the system reliability under a sequential attack is worse than that under a synchronous attack. However, the implementation of a sequential attack is much easier than for a synchronous attack, and the former can reflect the load loss of a system at different moments in the case that cascading faults occur in the system. In practice, both means of attack may be of reference value.

Case study
To verify the effectiveness of the above-mentioned reliability assessment method, the simulation analysis is carried out by taking a real electricity-gas-thermal IES shown in Figure 2 as an example. The relevant parameters of the system can be found in the Reference 51. The equivalent topology is shown in Figure 3, which includes 35 nodes and 37 edges. It is expressed as a unified energy path diagram according to Formula (8). Then, the weighted node betweenness, weighted line betweenness, Theil's entropy of node pressure growth rate and Theil's entropy of branch power flow surge rate of all nodes and lines of the system are calculated according to Formulas (12), (13), (16) and (19). And, the comprehensive vulnerability assessment indexes of nodes and lines are obtained according to Formulas (20) and (21). And, the weighted node betweenness, the weighted line betweenness and the comprehensive vulnerability assessment index are sorted in a descending order to obtain the top 30% of nodes and lines (11 nodes and 12 lines), as shown in Figure 6. By removing these key nodes and lines in turn, random attacks and calculated attacks are simulated to compute the network effectiveness and load loss rate of IES at different faults. With the degree of destroy (i.e., the ratio of the number of attacked nodes or lines to the total number of nodes or lines) as the abscissa, the network effectiveness function curve and the load loss rate function curve of the IES network in 6 different failure modes are drawn, as shown in Figures 7 and 8.
It may be concluded as follows through comparative analysis of Figures 6 to 8: 1. Some nodes and lines in the IES are in many optimized energy-supply paths with higher indexes of weighted node betweenness and weighted line betweenness and its failure may cause increased energy-supply loss and reduced efficiency and even affect the smooth energy supply. Therefore, it is necessary to perform maintenance work. For example, Node 5 and Lines (12,34) are the key links of energy transmission, and routed by many energy supply paths. A fault (if occurs) will cause serious energy supply shortage. 2. According to comprehensive vulnerability assessment index of nodes and lines, the comprehensive vulnerability of nodes and lines adjacent to power sources, gas sources, heat sources and energy conversion equipment is higher because these nodes and lines act as relays. If disconnected, they will have a greater impact on the upstream and downstream lines. For example, Node 12 and Lines (26,34) are adjacent to the energy conversion equipment; the comprehensive vulnerability is relatively high. A fault (if occurs) will have a greater impact on the power supply and heating. In an IES, if the line is attacked, the system shows stronger reliability compared with the attack on the energy components or load nodes. Figures 6 and 7 show that, if the node is attacked, the curve of each index drops faster compared with the attacked line, indicating that component or load fault has a greater impact on the system performance. The reason is offered as follows: The connected lines will inevitably be affected if a node in the network is at fault, while the nodes may not be affected if a line in the network is at fault. 4. When the IES is under random attack, it demonstrates stronger reliability than that under a calculated attack. Figures 6  and 7 show intuitively that the index curve when nodes and lines are under random attack drops more slowly than the index curves under other strategies. The reason is offered as follows: The degree of destroy cannot cause the system breakdown immediately and new energy islands or connected subgraphs will not be generated at once when the system is subjected to random attacks, thus keeping the network effectiveness and load loss rate unchanged within a certain range.

MEASURES FOR OPTIMIZATION OF IES RELIABILITY
Based on the weak nodes and lines identified in the system, corresponding measures can be proposed to improve the system reliability from the perspective of complex network.

Protect key links and add autonomous nodes
The network is quickly collapsed or split by finding a small number of nodes or connecting edges, and so forth. Conversely, if priority is given to protecting these key links, it is possible to alleviate or suppress network breakdown accident. In Reference 52, the system robustness is improved by strengthening the protection and defense of nodes with a high degree of internal connectivity. And, in some studies, the impact from a fault is buffered by setting up autonomous nodes. The cause of a fault is found and the transmission path of the fault in the network is disconnected to avoid the sudden collapse and fragmentation of the network if attacked. In Reference 53, a scheme of adding autonomous nodes is proposed to reduce the network degree of coupling. The autonomous power station is designed with a backup communication system, which can still run if the server fails, thus improving the robustness of the system.

Network reconstruction
The concept of network reconstruction appeared first in the study of distribution network, which refers to taking certain measures to maintain the network topology and restore the network connection if the network structure is broken or its performance is deteriorated. It can reduce the network energy consumption and balance the network loads if the network can operate normally, and restore quickly the power supply to non-faulty zones if the network is at fault. By adding corresponding backup links and nodes in the network or taking other repair measures, the general reconstruction strategy can reduce the shortest path between network nodes to improve the connectivity of the network. Therefore, the FESTA algorithm, 54 the Spider-Web Heuristic algorithm 55 and CORP algorithm 56 and so forth. have been proposed and applied for network topology repair, which may eventually reconstruct a robust topology with minimum relaying nodes and maintain network connectivity, with different locations and methods of placing relay nodes. However, for the repaired network, it is probably to fail again due to the failure of one of the relay nodes deployed in it. Therefore, more and more people focus on the improvement of fault tolerance and robustness of the repaired network. Combined the study on fractal mechanism, it is found in Reference 57 that the direct connection of hub nodes will make the network more fragile. Accordingly, a topology improvement strategy for complex network is designed based on the maximum edge betweenness spanning tree, which can improve the fractal characteristics of the network, reduce the small-world characteristics, and therefore can significantly improve the robustness of the network, and address low economy of conventional methods.

Optimization of fault recovery measures
The above-mentioned two measures are pre-accident preventive measures. In Reference 58, a post-accident optimized recovery measure is put forward. After the occurrence of many faulty nodes, priority is given to recovering those nodes with high number of degree, high betweenness or high PageRank in the network, which can reduce losses and the negative impact of vulnerability. In Reference 59, the results of the system returning to a normal state are compared under the three mechanisms of random recovery, target recovery and dependency recovery, and the study shows that the system reliability is the best under the dependency recovery mechanism. In addition to considering network static structure and node repair ability, the network evolution characteristics are also taken into consideration in some references. In Reference 60, an edge-connectivity compensation repair method by adding limit parameters is proposed to address the situation that the continuous attack evolves with the network and the node cannot be repaired. Simulation experiment results verified that this repair strategy can ensure that more than 85% of the nodes can remain connected during the evolution of the network without changing the scale-free structural characteristics of the network.

CONCLUSION
In conclusion, it can be seen that progress and achievements have been made for complex network theory in the reliability assessment of IES. However, the limitations of current research have also been exposed. In the future, research will mainly focus on the following issues: 1. IES is a complex system with multi-energy flow coupling. Because different energy flows have also different characteristics, time scales and coupling modes-in addition to many uncertain factors-it is necessary to build a unified model of multi-energy flow coupling systems that takes time scale characteristics into account. The complex coupling relationships among various energy forms will lead to cascading faults. Therefore, it is necessary to establish coupling models to quantitatively describe coupling relationships in reliability assessment. However, the issue of accounting for time scales and coupling relationships in reliability modeling is very complex and is rarely addressed by the existing reliability assessment models. This presents an urgent problem in reliability assessment modeling. 2. At present, no unified reliability assessment index system is available for IES. In order to achieve comprehensive measurement of the reliability level of multiple energy supply forms, a reliability assessment index system needs to be studied to account for the characteristics of different energy supply forms, including electricity, gas, cooling and heating. In addition, it is also necessary to study the influence index of coupling relationships among various energy flow systems on the overall reliability of IES, which is an urgent problem in reliability assessment indexes. Along with the targeted indexes of different energy forms, the importance index of the key components or links of IES reflect the weak links of each energy system. This should be one of the future topics of research on reliability assessment indexes.
3. At present, complex network theory-based reliability assessment of IES mostly focuses on static topology analysis and rarely combines complex network theory with the physical parameters and operational characteristics of the studied system. Furthermore, the application demands of system reliability assessment include accuracy and rapidity. However, the fact is that different energy forms of IES are coupled with each other and that energy can flow between the energy subsystems. This entails a significant increase in the complexity of modeling and computing for reliability assessment. Therefore, it is necessary to explore more practical, accurate, and efficient methods of reliability assessment.
The reliability of IES is the basis for ensuring the stability of regional energy use. System reliability assessment can better ensure the safe and reliable operation of IES. The conventional method cannot explain system-level phenomena and ignores the interactions between subsystems. Therefore, the reliability of IES is analyzed from the perspective of complex networks. In this paper, the topological network model and the identification indexes of weak links for the IES are summarized. The research progress and applications for reliability assessment indexes and assessment methods are also sorted out for IES based on complex network theory. On this basis, measures for optimizing the reliability of IES are put forward. Finally, the deficiencies of existing studies are described and future topics of research for the reliability assessment of IES are summarized.