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Security assessment of cross-border electricity interconnections

Cross-border electricity interconnections are important for ensuring energy exchange and addressing undesirable events such as power outages and blackouts. This paper assesses the performance of interconnection lines by measuring their impacts on the main reliability and vulnerability indicators of interconnected power systems. The reliability study is performed using the sequential Monte Carlo simulation technique, while the vulnerability assessment is carried out by proposing a cascading failures methodology. The conclusions obtained show that highly connected infrastructures have simultaneously high reliability and limited robustness, which suggests that both approaches show different operational characteristics of the power system. Nevertheless, an appropriate increase in the number and capacity of the interconnections can help to improve both security parameters of the power supply. Seven case studies are performed based on the IEEE RTS-96 test system. The results can be used to help transmission system operators better understand the behaviour and performance of electrical networks.


Introduction
Power systems are increasingly large and complex due to the rapid development of modern societies, which have created innumerable challenges of analysis for infrastructure operators. This prolific growth has been accompanied by the construction of new plants and assets to maintain the levels of reliability and security of the electricity grid. More recently, in the member countries of the European Union, this progress has also included improvements in cross-border electricity interconnections to increase the electricity supply and integrate more renewable energy sources into the energy markets [1]. The merging of two or more electricity infrastructures has created numerous benefits for the economy and the population, since a country can rely on neighbours for importing the necessary electricity and becoming better equipped to manage undesirable events, such as weather events, failures and blackouts. Additionally, the need to build new power plants is reduced, and the optimal management of renewable energies is facilitated, among many other factors [1][2][3]. Therefore, connecting isolated power systems is essential for supply security.
In the case of coupled electrical infrastructures, a failure within one of the systems has the potential to propagate throughout the network. For example, the European blackout in 2006 was due to a routine disconnection of a single power line in northwest Germany, which caused more than 15 million customers in five countries of the Union for the Coordination of the Transmission of Electricity (UCTE) to be without electricity for approximately two hours [4]. These critical infrastructures must operate safely and reliably.
These aspects highlight the importance of managing these risks and vulnerabilities. Traditionally, the risk management approach that ensures proper operating conditions of electrical infrastructure has prevailed since it identifies the most potential threats and hazards. However, in risk management, two complementary approaches can be considered -high-probability but low-impact events or low-probability but high-impact events [5][6][7]. The first class of events corresponds to the reliability approach, since it considers n-1 and n-2 contingencies, while the second class of events corresponds to the vulnerability approach since it considers n-k contingencies [8].
This article considers that both perspectives of risk analysis should be integrated within a unified decision framework to compensate for the limitations of a single approach in the study of cross-border interconnections. The reliability analysis, as the focus of risk management in power networks, should be complemented with a vulnerability analysis to improve the planning of the system's expansion and better understand the structural performance of the networks. Feeder links can improve the levels of reliability but fail to achieve improvements in the levels of vulnerability, or they can improve the vulnerability but simultaneously worsen the reliability. In these situations, the construction of new connecting lines must consider both approaches to improve the performance and behaviours of the interconnected systems.
Reliability can be defined as the capacity of the power system to continuously satisfy demand with an acceptable level of quality or perform a required function in given environmental and operational conditions for a given period [9][10][11][12]. Other similar definitions are provided in [13,14]. Several approaches can be employed to conduct https://doi.org/10.1016/j.ress.2020.106950 Received 20 September 2019; Received in revised form 19 February 2020; Accepted 23 March 2020 reliability studies, from analytical models to Monte Carlo probabilistic models. The latter is a more flexible methodology than analytical models but requires more calculation time, especially when operating conditions and complex system states are considered. The reliability of electrical infrastructure is measured in terms of the indices presented in Section 3.
Reliability assessments are based on the probabilistic estimation of the failures and the negative quantification of these events or incidents [15][16][17]. These estimates are purely quantitative since they are constructed considering the estimated probabilities, which can generate some uncertainty in the final results [18,19]. Some authors argue that risk management should also incorporate concepts such as robustness and resilience to further improve the security of power networks [15,18]. In this sense, the vulnerability approach is the methodology with the best characteristics to address these concepts.
Vulnerability is the degree with which an electrical network degrades due to a loss of functioning of its elements after an attack or failure [20,21]. Vulnerability can also be defined as the inability of an infrastructure to withstand variations in voltages and the effects of failures [22]. The term robustness, complementary to the term vulnerability, is used to define the ability of a system to maintain, at least partially, the electrical supply in the case of unforeseen events or incidents [23]. For example, the single interruption of a transmission line can cause the overload of other lines, which increases the likelihood that other assets will fail and cause a system-wide failure [24]. Unlike the reliability approach, the vulnerability approach does not consider the probabilities of failures or threats but rather quantifies the effect or impact of the disturbances on the structural performance of the network to identify potential threats. Vulnerability is an inherent characteristic of power systems.
Few documents discuss the two approaches, and the results are contrasted in parallel [25][26][27], that is, systematic comparisons to identify how the two approaches can be applied in a complementary manner are lacking.
This article addresses the lack of comparative studies analysing the impact of cross-border interconnections with reliability and vulnerability assessment approaches. The objective is to show, using empirical evidence, that feeder links have an important role in supply security but simultaneously have the potential to cause disturbances in a joint network since they can propagate failures or disruptions [2]. The network selected to carry out this study is based on the well-known IEEE RTS-96 test system, from which seven different case studies are extracted [28]. This system is chosen as it is a sufficiently small network for performing a large number of studies with a reasonable solution time but sufficiently detailed to reflect the actual complexities involved in reliability and vulnerability approaches. This system includes main generation and transmission facilities in a simple but representative model of real interconnected infrastructures. The premise is to illustrate, as much as possible, different technologies and configurations in power systems [28].
Two studies are executed using the procedures and provisions identified in their respective fields of research. The reliability approach is performed using the sequential Monte Carlo simulation technique, and the vulnerability approach is executed using a cascading failure procedure [29][30][31]. The discussion seeks to enrich the scientific debate by demonstrating that results from the reliability and vulnerability analyses can be combined and discussed in parallel and show how these approaches can provide complementary information about the interconnected infrastructure.
The remainder of this document is organised as follows: Section 2 provides a review of the most representative documents of the reliability and vulnerability approaches. Section 3 describes the procedures for evaluating the reliability and vulnerability of interconnected power systems. Section 4 presents the case studies based on the IEEE RTS-96 test network and separately shows the numerical results of reliability and vulnerability. Section 5 discusses the findings jointly, and Section 6 summarises the main conclusions of this work.

Reliability and vulnerability approaches
This section presents some important characteristics of the reliability and vulnerability studies of power systems. These characteristics, although not exhaustive, describe the fundamental principles involved with each technique.

Evaluation of reliability
Reliability is the ability of a power grid to operate at the required level of quality over a long period. The most analysed contingency events are those that correspond to n-1 and n-2, which represent the loss of one or two infrastructure assets, respectively. Reliability is broken down into system adequacy and system security [14,32]. System adequacy evaluates whether the capacity of energy generation is adjusted to the demand and the limitations of the network in stable conditions without contingencies. System security analyses the behaviour and performance of a network against the nontrivial loss of a generator or transmission line. The purpose of the reliability analysis is to describe the behaviour of the network and calculate indices that describe intrinsic characteristics, such as energy not supplied, demand not supplied, unmet demand, among others [13,15]. The study performed in this article focuses on the security of the system as related to the performance of the interconnected network.
This type of evaluation can be performed from an analytical or simulation perspective. The analytical technique represents the infrastructure via analytical models and evaluates the indices using mathematical solutions. This approach requires assumptions to simplify the problem and produce an analytical model of the network so that the resulting analysis may lose some or much of its meaning. The solution using this technique is obtained in a relatively short time [14,33,34]. The Monte Carlo simulation technique estimates the reliability indicators that simulate the process and random behaviour of the network via experiments. Theoretically, this approach can take into account all of the aspects and contingencies inherent to the planning, design and operation of the infrastructure. These aspects include random events, such as power outages and repairs of components constructed from general probability distributions, component behaviours and power variations [33,34]. In general, the analytical technique is efficient when the operating conditions are not complex, or the probability that assets will fail are quite slim, and the Monte Carlo technique is preferable for complex operating conditions or a large number of events. This latter approach is employed in this study for the reliability analysis of the proposed case studies [33].
A reliability analysis based on the Monte Carlo simulation approach generally employs two techniques: sequential and non-sequential. On the one hand, the sequential technique simulates the real chronological process and the random behaviour of the electrical infrastructure [33,34]. This technique is performed by dividing the simulations into periods and considering the contingencies produced in each case. The sets of events are directly related to the previous simulation conditions. The indicators are quantified when interruptions over one year are considered [15,20]. On the other hand, the non-sequential technique samples the system in such a way that each time step, system state or failure event is considered independently, which produces a nonchronological network state, i.e., less realistic state [33,34]. This research applies the sequential Monte Carlo simulation technique since it is a more flexible and precise approach than the non-sequential technique in addition to representing a more realistic behaviour of the coupled infrastructure.

Evaluation of vulnerability
The vulnerability of the power system has been a well-studied area of research in the last decade. Despite being a common term, more than twenty definitions of this concept are provided in the scientific literature [35]. Some authors suggest that vulnerability can involve social, organisational, economic, environmental, territorial, physical and systematic aspects [36]. However, the majority of studies focus on systematic and physical vulnerabilities [36]. For this article, vulnerability is an internal characteristic of electrical systems that is related to the inability to withstand the effects of failures [37].
Vulnerability can be studied in two different ways: functional vulnerability and structural vulnerability. Functional vulnerability is related to the operational aspects and conditions of infrastructure, while structural vulnerability is related to a decrease in the performance of a network after a failure or event. For more information, consult [38]. The study of the impact of cross-border interconnections in interconnected power systems is carried out from the perspective of structural vulnerability, as it evaluates the combined performance of networks when they are subject to multiple n-k contingencies, i.e., cascading failures. The initial hypothesis of this study suggests that cross-border connections can cause a decrease or drop in total network performance as disturbances can propagate through these lines.
The assessment of vulnerability has many objectives according to the uses of the researchers, such as identifying the critical components that require protection, determining possible undesirable events, classifying the components according to their consequences, identifying potential vulnerabilities, identifying existing countermeasures and estimating the degree of vulnerability of each component [14,36,39,40]. This type of study, unlike the study of reliability, does not consider the probabilities of failure of the elements; instead, it exhaustively quantifies the impact caused in the topology by the systematic removal of the assets. Contingency studies are among the criteria most employed to evaluate the degree of impact of an event [41].

Discussion
Reliability and vulnerability studies of power networks are welldocumented approaches in the scientific literature. However, the security evaluation of cross-border interconnections in interconnected electrical systems is a recent area of research with a lack of studies that address both concepts in parallel. As a result, this study seeks to contribute to bridging the gap to the apparent lack of related studies in this field of research.
Interconnected electrical infrastructures are operated by different independent Transmission System Operators (TSOs), who may unintentionally perform actions that could damage network operation. Therefore, the exchange of information should be considered when analysing the joint vulnerability and reliability of networks [42].
The coupling of systems with different operational or topological characteristics can drastically increase the joint vulnerability of the networks, although from the point of view of reliability, improvements are possible [43,44]. For example, when two infrastructures of different sizes are coupled, larger networks may be less vulnerable than small networks when few interconnections exist due to less propagation of the disturbances of the other joint systems. Meanwhile, small networks can be more vulnerable than large networks when many interconnections exist due to the substantial difference in the scale of the systems. Conversely, the interconnected systems would be much more reliable from a reliability approach since the infrastructures would be less sensitive to power outages caused by a malfunction. Both situations shown here can compromise the coupling of power networks.
Vulnerability studies with the potential to identify and classify the most severe failures or impacts can serve as a technique to determine and optimise the best feeder links, improve the coupling of infrastructure and mitigate undesirable events [45,46].
Another essential aspect of recent consideration in the security of interconnected systems is the constant occurrence of cyberattacks against control systems. Vulnerability studies identify assets whose destruction or weakening present an extremely high risk to the coupled system [47,48]. This approach also helps measure the evolution of interconnected networks both before and after contingency events [49,50]. In scientific literature, several traditional techniques for carrying out these tasks are identified [51][52][53].
The combined analysis of the results of the reliability and vulnerability studies can offer a new perspective for evaluating the security of cross-border interconnections since the information generated by both approaches enables us to improve the understanding of these links. This article provides new knowledge in this area of research.

Procedure to quantify the impacts of interconnections on reliability and vulnerability
This section describes the methodology and statistical indicators used to study the impact of cross-border interconnections on the reliability and vulnerability of coupled systems.

Methodology for the reliability calculation
The study of reliability requires the calculation of statistical indicators to measure the efficiency of the electrical infrastructure. The computation of these indicators is based on the frequency, duration, and magnitude (or the probability) of adverse effects on the electric supply. Regarding the security dimension of reliability, n-1 security principle is usually employed [54]. This analysis aims to prevent emergency conditions in the power grid, including the propagation of incidents from one system to another [54]. Failure events are statistically independent, as asset disruptions are not related to other contingencies occurring in another location in the network. This simulation process realistically takes into account all aspects inherent in the design and operation of the infrastructure. In this regard, TSOs have a variety of indexes that allow them to measure the severity of a disruptive event and to establish effective mitigation measures.
To calculate the security indicators, the sequential Monte Carlo technique has an orderly procedure of steps that can be summarised as follows [33,34]: Step 1. Determine the status of components susceptible to failures. The two possible states of the assets of an electrical network are normal or failure. Initially, all assets are assumed to be in a normal state.
Step 2. Calculate the time the components spend in each state. The time to failure (TTF) and time to repair (TTR) are calculated sequentially using the failure rates (λ), uniformly distributed random numbers between [0,1] (r) and mean time to repair (MTTR) using equations (1) and (2), respectively. The r-values are calculated using congruential generators [33].
This step is repeated for a specific amount of time, usually one year.
Step 3. Provide the overlap times of the failures for an annual horizon with an hourly resolution (8760 steps).
Step 4. Run optimal power flow study after a failure in the assets.
Step 5. Measure the reliability indicators in equations (3) where E j, i is the energy not supplied (MW) to the electrical network in the j-th power interruption in year i, N y is the total number of years simulated and N i is the total number of interruptions in year i.
where D j, i is the duration of the j-th power interruption in year I (hours).

■ Loss of load probability [%]
= LOLE LOLP 8760 Step 6. Repeat the previous steps until the EENS covariance index is less than a predefined tolerance.
It should be noted that it is possible to take into account statistical uncertainty in the input data when calculating the above indicators. For this purpose, different probability functions associated with component failure and restoration activities can be used. The objective is not only to calculate reliability indices in the form of expected values of random variables, but also the distributions of these indices. The most commonly used probability distributions are Weibull, normal and lognormal [55]. This modified approach assumes that the components have the same adjacent distribution and that the failure and repair processes of the assets follow exponential distributions [56].

Plexos software
The Plexos software was used to develop the reliability studies in the proposed cases, quantify the statistical indexes presented in the previous point, and measure the impact of cross-border interconnections on the reliability of electrical networks. Plexos is a tool used to model and plan electricity and gas markets that is widely used by the Energy Regulatory Commission (ERC) of Ireland and the California Independent System Operator (CAISO), among many other public and private organisations [29,57,58]. This software can calculate the reliability indices of LOLP, LOLE, EDNS and EENS from the PASA simulation phase using convolution. However, it can also use the detailed chronological simulation of ST Schedule to produce the same indicators using Monte Carlo [59]. This latter approach is used in this article. Although Plexos is a well-known software for conducting reliability studies, this subsection illustrates the applicability of the program as an accurate and efficient modelling tool.
For the case study, the well-known IEEE RTS-96 test network was selected; its technical and operational characteristics are described in detail in Appendix A [28], and the software was configured to perform 500 iterations using the sequential Monte Carlo technique with a preset tolerance of 1.5%. The simulation data for each area can be found in [15]. The results were compared with those reported in references [25] and [32]. The programme was run on a personal computer with an Intel® Core™ i5, 1.80 GHz CPU and 6 GB RAM. The simulation time was 8.43 hours. Table 1 contains the results of the reliability indicators obtained in this article in comparison with the results shown in references [6] and [26]. As can be seen, the results calculated with the Plexos program compare well with those reported in the references, indicating that the model used provides results similar to those calculated with the traditional methods already known. This shows that the functional model can be applied to the case studies presented in the following section.

Methodology for the vulnerability calculation
Structural vulnerability evaluates the physical characteristics of the topology of the infrastructure when subjected to various n-k contingencies. The aim is to prove whether cross-border interconnections cause a decrease in network performance when disruptions propagate through these links to the other interlinked systems.

Vulnerability metric
The appraisal of the performance of the interconnected infrastructures is done through an index that quantifies the power supply during the disintegration process. Similar to other studies [60][61][62], the load shedding (LS) indicator is used to determine the effect of cascading failures on the coupled network. The contingencies considered correspond to random events such as component failures, protective devices malfunctions, and human errors [63]. In this paper, cascading failures are successive events of faults in the system's elements, even if the failed assets are not adjacent.
The LS index is calculated as follows: where P Di LC is the power load that remains electrically connected, following a node i disruption (MW), and P Di BC is the power load under the base case (MW). The LS metric varies between 1 and 0. Therefore, as the LS index decreases or approaches zero, the impact on the entire infrastructure increases. To calculate LS, a DC optimal power flow (DCOPF) study was used, which considered the active power as the most important parameter [64].

Structural vulnerability algorithm
Algorithm 1 quantifies the impact of interconnections on interconnected power systems subject to multiple correlated n-k contingencies. The procedure incorporates equation (9) depending on the assets or elements eliminated. The algorithm begins by running a DCOPF study to determine the total active power under the base case In addition, the bus removed is never the slack bus. This bus is not removed because it is responsible for providing energy balances in the flow equations. Although other buses can be chosen as references, this is not done because the objective is to have a single measurement network. The reason for this is that islands are generated during the process of disintegrating the grid; therefore, it is necessary only to select the subnet containing the slack bus. Algorithm 1 uses the Depth First Search (DFS) algorithm to determine the islands in the network [66]. Then, DCOPF is only executed on the selected subnet to determine the active power load that remains electrically connected, following the removal of node i, i.e., P Di LC . The algorithm ends when no more assets can be removed or if the sample threshold is reached.

Case studies
This section presents the case studies constructed from the IEEE RTS-96 test network, from which seven different interconnected networks were extracted [28]. The reliability and vulnerability results for each system studied are also reported.

Description of the case studies
The IEEE RTS-96 system is a good case study because it represents, in real terms, a coupled infrastructure consisting of three identical power systems [28]. Each electrical network consists of 24 buses, 32 generators and 38 power lines and transformers. This study considers a failure rate of 0.001/year, a 24-hour MTTR and an annualised peak power demand of 2850 MW in each infrastructure [15]. The technical data of the network can be consulted in Appendix A [28].
This system has three equal areas that are connected as follows: Area 1 is connected to Area 2 by three power transmission lines, Area 2 is connected to Area 3 by a transmission line, and Area 3 is connected to Area 1 through an additional bus, an electric transformer and a transmission line.
For simulation purposes, only Areas 1 and 2 (herein called Areas A and B) with their three respective interconnecting links were considered, as shown in Fig. 1. By alternately combining the three interconnectors, seven different networks were extracted for analysis.
In isolated power systems, load shedding can occur when generation does not meet the load demand. In interconnected power systems, this deficiency can be met by exchanging power from other systems. This support depends on the available capacity, the operational reserve, the interconnection constraints and the type of agreement between the infrastructures. That is, the capacity of the tie lines imposes limits on assistance from one system to another -the above factor impact on the reliability levels of the interconnected systems. Table 2 shows the systems studied, the interconnection lines considered, and the capacities of the interconnection lines.
It should be noted that Network 1 consists of Areas A and B connected with transmission lines , [13][14][15] and [07-03]. Network 2 also consists of Areas A and B, but they are connected with links  and [13][14][15]. Networks 3 and 4 also connect the two areas with two of the possible lines, while Networks 5 to 7 each use a single interconnector. Table 3 shows the reliability results of the different networks studied using the Plexos software. Five hundred iterations are considered for each system and covariance of less than 2%. Figure 2 shows the convergence results of the EENS index when the sequential Monte Carlo simulation technique is applied.

Effect of interconnections on reliability
The results in Table 3 show that reliability improves as the number of links between the systems increases. The best-case corresponds to the interconnection of the two areas with three power lines (Network 1), where the reliability improves by 71% compared to the isolated grid.
The values of the EENS metric (%) are better in the case of the interconnected areas of the first six case studies, where small improvements are seen between the cases of Networks 1 to 6 as the number and capacity of the interconnections increase. Thus, the minimum EENS value is 0.151% in Network 1 compared to 0.164% in Network 6.
However, the EENS is more than two times higher in the case of the grid-connected with the line with the lower capacity [07-03] (Network 7) than in the other cases. Nevertheless, when the results of the system for case 7 are compared with those shown in Table 1, which correspond to an isolated grid, EENS decreases from 0.52% to 0.35%, which indicates that a single interconnection line has the potential to improve the reliability. The LOLP and LOLE indices have similar results.
Interconnections have an important role because they serve as support elements between the different areas that make up an interconnected system. The efficient use of these connections is related to the power that they can transmit to the joint systems. It is therefore proposed to study this feature using the indicators of exchanged energy where N y is the total number of years simulated, L is the total number of interconnections lines between areas, and F is the active power.  Table 4 shows the line loading of the interconnections for the seven grids studied. It should be noted that Network 1 has the highest annual PEE indicator, which implies that the use of the interconnections between highly connected grids is high. This means that the higher the interconnection capacity is, the greater the power exchange between areas is. This benefits the systems because it facilitates mutual support in case of contingencies.
Additionally, the EE results in Table 4 show that Area B strongly supports Area A in all of the systems studied, which could be due to the technical characteristics of the assets that work as an interconnection. Furthermore, the MPLC results indicate that the capacity of the interconnection lines is sufficient: the busiest line corresponds to link  in the case study of Network 7, which has an annual maximum utilisation of 47.89% of its capacity, and the average of the remaining lines is approximately 25%.
In short, interconnections increase the reliability of interconnected systems; however, the capacity of the links in the energy exchange and the buses or assets selected as interconnecting elements should always be considered.

Effect of interconnections on vulnerability
According to the methodology described in Section 3.2, the disintegration of the systems was carried out by introducing random failures and calculating the LS statistical indicator during the disintegration process. The results were obtained by averaging 1000 simulation samples for each infrastructure studied. When all of the buses are initially connected, the LS index is equal to 1. Then, as the grid disintegrates due to the propagation of cascading failures, the LS index decreases to 0, which means that the power supplies to all loads on the grid have been interrupted. Figure 3 shows the performances of the seven interconnected networks under cascading failures and the vulnerability curve of a separated Area A.
The results show that Area A initially collapses with the removal of approximately 80% of the buses. In the remaining cases, the grids disintegrate more quickly because the removal of 60-70% of the buses is more than sufficient to cause a widespread blackout. Furthermore,    when the cases linked by a single cross-border link (e.g., Network 5) are considered, the vulnerability worsens significantly compared to the separate case (Area A). However, by connecting the grids with two links, such as Networks 2, 3 and 4, the infrastructure's vulnerability begins to decrease because the disintegration process is slower. If a third connecting link is added, the results do not improve significantly because the structural performance never reaches the vulnerability levels of the isolated power system. A comparison of Figs. 3(a) and (b) shows that the network with interconnections  and [13][14][15] provides the entire energy system better efficiency compared to the network with interconnections  and [07-03]. These empirical findings suggest that certain links provide a more robust topological structure against the propagation of cascading failures. For example, connecting two grids using links , [13][14][15] and [07-03] does not significantly increase the vulnerability with respect to the separate system; however, if only the case with link [07-03] is considered, the vulnerability increases dramatically compared to the case described above.
In summary, the results show that interconnections have a substantial effect on the topology because they increase the propagation of cascading events in interconnected systems. However, the number and capacity of the interconnections are relevant to minimise the impact of the propagation of grid disruptions. The findings obtained here suggest similar conclusions to those shown in the papers [43] and [44].

Discussion of results
This study shows that reliability and vulnerability assessments are useful tools for TSOs to understand the performance and limitations of their critical infrastructure better. Therefore, this section focuses on the comparison and discussion of both approaches.
Until now, the two techniques have been used to analyse and examine different aspects of networks; however, this article considers that both approaches can be used in combination to provide a broader view of the operational performance of power systems.
The reliability results were obtained through statistical indicators that describe the operating conditions of the grid after an n-1 contingency, and the vulnerability results were plotted as decay curves that represent the topological behaviour of the studied systems against n-k contingencies. Therefore, it is proposed to determine damage areas (DAs) to accurately assess the performance of the decay curves of each result in Fig. 3. DA is defined as the region under the cascading failures curve [67]. This metric is calculated as follows: 1 Determine the equations f(x) of the decay curves. 2 Calculate the integral of the equations using the fraction of removed buses as the minimum and maximum limits.
Curves near the abscissa represent greater damage to the infrastructure; thus, a small DA represents a case of severe damage, which coincides with a smaller area under the curve. Conversely, higher DA values represent less severe scenarios for the system. The use of this metric allows vulnerability indicator to be compared with reliability indicators. Figure 4 compares the EENS (%) reliability indicators from Table 3 and the vulnerability values obtained from the plots in Fig. 3. First, the vulnerability results show that isolated systems (Area A) are more robust than interconnected grids. The isolated system (Area A) has a DA value of 35, which is significantly better than the vulnerability indicators for the other case studies corresponding to the interconnection of the two areas, which have DA values between 24 and 31. Specifically, cross-border interconnections can increase the impact of a disturbance on the vulnerability of the entire power system. Of all the cases analysed, the smallest loss of robustness is achieved in the case study of Network 1, i.e., when there is greater interconnection capacity between the two areas (through lines , [13][14][15] and [07-03]).
Therefore, the reliability results show that interconnected systems are more reliable than isolated systems. The information in Table 2 shows that the most interconnected case (Network 1) is less sensitive to power outages caused by a malfunction; it is more reliable. In addition, the least connected infrastructure, which is Network 7, has a higher Expected Energy Not Supplied indicator (EENS); however, this case still has better reliability than the case of the isolated grid of Area A (see EENS results in Table 1).
A joint analysis of the vulnerability and reliability indicators shown in Fig. 4 shows that Networks 1, 2, 3, 4 and 6 have small differences in both vulnerability and reliability. These cases represent a compromise solution for the design of interconnections between power systems, so the final design will only depend on the requirements of energy exchange to improve the system adequacy of the joint system (Network 1 offers an interconnection capacity of 1424 MW, Network 2 of 1216 MW, Network 3 and 4 of 816 MW, Network 6 of 608 MW). However, Networks 5 and 7 are characterised by lower robustness against cascading failures, and Network 7 (which corresponds to the interconnection of the two areas with a single low-capacity line [07-03]) has the worst reliability and the worst vulnerability of all the cases studied.
In contrast, the best results for both indicators are obtained in Network 1, which indicates that when it is necessary to connect electrical systems with cross-border lines to improve the security of supply, a greater interconnection capacity helps to improve the reliability and also provides the smallest loss of robustness in the interconnected power system.

Conclusions
In this article, the impacts of cross-border electricity interconnections on the reliability and vulnerability of interconnected power systems were studied. The reliability assessment was carried out by quantifying the main traditional indicators, while the vulnerability assessment was performed by considering a process of cascading failures and measuring the energy not supplied during the grid disintegration process. Both approaches were used in a comprehensive study framework that considered the fundamental specifications of their respective research fields. The simulation case studies corresponded to seven different systems obtained from the IEEE RTS-96 network.
The results showed that interconnections increase the reliability but also decrease the robustness of the entire system. This means that connection lines improve the energy exchange between the different areas that make up an interconnected infrastructure and simultaneously increase the probability that disruptions in one of the systems propagate to the others. In that case, having greater interconnection capacity is the best compromise solution for the design of transmission network topologies under reliability and robustness criteria. Thus, interconnected energy systems behave differently depending on how they are studied; however, the results obtained here could help TSOs to study the interconnection of two or more grids from a more comprehensive point of view.

Acknowledgements
This study was supported by the Ministry of Economy and Competitiveness, Spain, under the project ENE2016-77172-R.