Don’t go chasing artificial waterfalls: Artificial line limits and cascading failures in power grids

Research on cascading failures in power-transmission networks requires detailed data on the capacity of individual transmission lines. However, these data are often unavailable to researchers. Consequently, line limits are often modeled by assuming that they are proportional to some average load. However, there is scarce research to support this assumption as being realistic. In this paper, we analyze the proportional loading (PL) approach and compare it to two linear models that use voltage and initial power ow as variables and are trained on the line limits of a real power network that we have access to. We compare these arti cial line-limit methods using four tests: the ability to model true line limits, the damage done during an attack, the order in which edges are lost, and accuracy ranking the relative performance of di erent attack strategies. We nd that the linear models are the top-performing method or are close to the top in all the tests we perform. In comparison, the tolerance value that produces the best PL limits changes depending on the test. The PL approach was a particularly poor t when the line tolerance was less than two, which is the most commonly used value range in cascading failure research. We also nd indications that the accuracy of modeling line limits does not indicate how well a model will represent grid collapse. The ndings of this paper provide an understanding of the weaknesses of the PL approach and o er an alternative method of line-limit modeling. © 2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5115493 The networked structure of power-transmission gridsmakes them susceptible to cascading failures. A cascading failure occurs when a single failure or a small number of failures propagate through a system, wreaking havoc. While major blackouts caused by cascading failures are rare, they a ect a very large number of people and have signi cant nancial consequences. This paper considers the resilience of the power grid from the perspective of network science. In this context, cascading failures are not simulated using random failure but attacks that continue until the complete collapse of the network. The potential for such threats comes from either physical or cyber-physical attacks. The rst known cyber-physical attack was in Ukraine which caused power loss to 200 000 people. Since then the perceived threat of cyberphysical attacks has grown.This paper explores a commonly used method to simulate cascading failures in the power grid, known as proportional loading, and nds that inaccurate estimation of the system tolerance produces modeling results that deviate substantially from the behavior of the grid under attack when using the real line limits.


I. INTRODUCTION
Power networks are an essential part of modern civilization. Power outages caused by either random failure or targeted attack can cascade through the power system and cause massive damage. In the last two decades, there have been several examples of cascading failures causing a loss of power to tens, sometimes hundreds of millions of people. In 2003, a major blackout in the northeast portion of the USA was estimated to have cost approximately $6 billion. 1 In 2012, a cascading failure in the Indian power grid caused a loss of power to 600 × 10 6 people. 2 More recently, in 2019, 3 a failure in the interconnector between Argentina and Uruguay caused a blackout that a ected nearly the entirety of both countries, which was close to 50 × 10 6 people. In addition to random failures, targeted cyberphysical attacks have become a growing threat to the power grid. 5,7 Cyber-physical attacks are so-called because they use cyber attacks to cause physical e ects. The most successful cyber-physical attacks were against the Ukrainian power grid in 2015 and 2016 causing a loss of power to 200 000 people. 6,8 Given the potential magnitude Chaos ARTICLE scitation.org/journal/cha of cascading failures and the threat posed by a deliberate attack, it is not surprising that researchers are looking for ways to reduce the impact and frequency of such failures. One method of understanding cascading failures is through network science, where cascading failures are stimulated using targeted attacks on network nodes or edges. Substantial work in this regard has focused on developing "vulnerability metrics" or "attack strategies" that identify the order in which nodes should be attacked to cause maximum damage to the power grid. When researchers rst began analyzing power grids using network science, the techniques applied used purely topological information about the power-grid structure. 9 As research developed, the power-grid's electrical properties were incorporated into the analysis, thereby creating the "extended topology." 10 Speci cally, the extended topology integrates the power owing in the network with the topological features. In most cases, electricity is transmitted through the power grid using alternating current (AC) at high voltage to reduce power loss, although direct current (DC) can be used point-to-point over long distances or underwater (e.g., the UK interconnectors to France, Ireland, and The Netherlands). However, as it is challenging to solve the AC power-ow equations, researchers often use DC power ow equations which can be solved using linear algebra. A recent literature review revealed that 81% of studies that involved power ow used DC power ow. 9 We use the DC power ow calculations, f = CA(A T CA) −1 p, described by Pepyne 11 and Arianos et al. 12 In this equation, A is the adjacency matrix with the slack bus removed to make the system invertible, C is a diagonal matrix of the line susceptance, and p is the power injected at each node. The slack bus absorbs or supplies additional power as demanded in order to ensure that the system is balanced.
Numerous studies create synthetic networks [11][12][13][14] or use the topological structure of a real power grid without the line limits. 13,[15][16][17][18] It is necessary to know these limits to detect when a line has tripped in a simulation. Beyond cascading failure research, equivalencing is used to reduce large complex power networks to smaller sub grids in order to study local phenomena. In such cases, estimating the line limits of the equivalenced networks is important so as to be able to perform accurate analyses. [19][20][21] Although a few open-data solutions are being developed, 22,23 the lack of datasets with line limits remains a problem. Little modeling or simulation work has been done using real line limits 24 for this reason. If these limits are not available in the data, an estimate must be made. A few studies 25,26 have used statistical techniques to provide a distribution of possible line limits. However, a more common technique for this is to use proportional loading (PL). When using the PL approach, the line limits are set at a xed proportion of the amount of power owing in each line at initiation. 11,15,17,[27][28][29][30] The PL of networks is usually de ned as where f c i is the absolute power ow for line i under initial conditions, f max i is the line limit, and α is the tolerance factor. The PL assumes that the line limits of the network are proportional to the power ow; however, there is little evidence supporting this.
As there is no direct comparison between PL and real line limits, it is di cult to know how accurately PL and the simulations based on it re ect real-grid behavior or whether a more realistic method can be created. In this paper, we address this gap. We have access to a dataset that includes the generation and load nodes with capacities in megawatts, as well as the line limits in megawatts for a single base load pro le of the power grid. The dataset is a simpli ed version of the UK power grid and is based on the Electrical Ten Year Statement (ETYS) produced by the national grid.
Using the base load pro le, we compare the real line limits of the network against the PL values of α between 1 and 50, the results produced by two linear models, and topological analysis. We simulate random attacks on the power grid and analyze how well the articial line limits model the damage caused by the attack as well as the order in which the edges are lost due to cascading. We also measure how well each line-limit method ranks the relative e ectiveness of di erent grid-attack strategies. The ndings of this paper provide an understanding of the weaknesses of the PL approach and provide an alternative method of line-limit modeling using linear models.

II. METHOD
1. We examine the network's real line-limit distribution. We then compare the accuracy of PL (proportional loading) and the modeled limits (using two di erent linear models) against the real line limits. 2. We simulate random attacks on the grid using the DC power ow model. We attack the grid until it collapses completely, repeating the process 100 times. We compare the mean damage and standard deviation of the arti cial line limits and the real line limits at each stage of the attacks. 3. We compare the rank order in which edges are lost due to cascades between the arti cial and real line limits. Then, we calculate the correlation coe cient, which shows us which articial line-limit method most accurately represents the behavior of grid collapse. 4. In order to test whether the arti cial line limits can compare vulnerability metrics, we attack the grid using ve di erent strategies. We measure the ability of the arti cial line limits to accurately represent the true ranking of each strategy using the real line limits.
The dataset we use in this paper describes the physical and electrical structures of a simpli ed version of the UK national grid. The dataset contains 512 nodes representing substations around the UK. The nodes are connected with 698 transmission lines that are rated 132 kV, 275 kV, and 400 kV. The network has a mean degree of 2.73, the average unweighted nodal distance is 11.7, and the clustering and mean normalized centrality are at 0.1. The low clustering coe cient, close to zero, suggests that the network tends to not form connected cliques, thereby making the network more vulnerable to disruption when individual nodes are lost. The dataset includes all the information required to perform the DC power ow calculations. This information includes line-node connections, line-reactance, base load node load/demand, and node generation. Unusually for a power-grid dataset, it also provides line limits, thereby making it possible to test the PL approach. This paper de nes the simulation using ve parameters: physics model, element, attack type, removal method, and load pro le. The physics model used in this analysis is DC ow or topological-that is, the power model uses DC power ow or is simply a topological analysis. The elements attacked are the nodes. The attack type is " xed," meaning that the order in which the nodes will be removed Chaos ARTICLE scitation.org/journal/cha

Physics
Cascading DC, topological Element Node Attack type Fixed Removal Sequential Load pro le Single is generated before the attack begins. The order in which nodes are removed is sequential. This means that only a single node is targeted for removal each round. However, other nodes may be lost due to a cascading failure. A summary of the parameters is shown in Table I. These simulation parameters form the acronym PEARL. (The PEARL framework is discussed in greater depth in supplementary material S1.) The last parameter of the PEARL settings is the load prole-although ideally we would use multiple demand load pro les for our analysis to explore the e ects of time of day and seasonality, such data are unavailable to us. However, we do have a single load pro le provided with the dataset, and we use this. This load pro le is based on the year-round base load. The base-load data are provided by the UK power suppliers in their seven-year statements. In general, it is advisable to use multiple load pro les when performing cascading failure analysis on the power-grid, as cascade behavior can di er depending on line loading. 31 In this paper, we only have access to a single load pro le; however, the goal of this paper is not to demonstrate any weaknesses of the UK power grid itself but to highlight the weaknesses of proportional loading. For such a goal, only a single line limit is necessary.
In this analysis, there are no restrictions on the generators. This implies that a generator of 100 MW in a subgraph with a load node of 1 MW will reduce its power output to balance the demands of the subgraph. This is a simpli cation and, in reality, there would be constraints on the output of the power station; however, that is beyond the scope of this paper.

A. Artificial line limits and real line limits
In order to understand how arti cial line limits di er from real line limits, we use 13 α values for PL, the topological approach, and the line limits generated from two linear models. We use α values of 1.05, 1.1, 1.2, 1.3, 1.5, 2, 3, 5, 7, 10, 15, 20, and 50. The α values were selected on the basis of values used in other papers, typically 1 to 5, and extended until 50. The topological analysis is identical to an α level of in nity. With low values of α, node removal tends to cause larger cascading failures as the system has less spare carrying capacity; larger values of α tend to cause smaller cascades.
Line limits are caused by three main factors: thermal-limits, voltage drop, and steady state stability of frequency. In addition, certain line limits are imposed to prevent dynamic instability at a system level. These factors are all related to the line length and its voltage. 32 To keep the models similar to PL, we create a model that uses only the initial power ow on the line and a second model that uses initial power ow and the voltage, as voltage data are available in several open datasets. [33][34][35] The voltage power ow (volt PF) model has the form y i = β 0 1 + β f x if + β v x iv , where y i is the line limit of the ith power line to the log base 10, x if is the initial power ow of the ith power line in megawatts, and x iv represents the voltage level of line i. The coe cients are β f and β v , while β 0 is the model bias. The power-ow-only model (PF model) has the form y i = β 0 1 + β f x if . The models are trained using 10-fold cross-validation, using ordinary least squares regression. Tenfold cross-validation is used as we only have access to the real line limits of a single power network. We train the model using 90% of the data and predict for the out of sample 10%; this will be repeated to predict for all parts of the network. The coe cients shown in this paper are the average across the ten folds.
(For code details on model generation, see the section "Modeling Line Limits" of the GitHub code. 36 ) These models will be compared to the PL approach throughout the paper. Ideally, we could use the models to predict the line limits of a separate network. However, we only have access to the real line limits of a single network. Since this is the case, we will use the line limits predicted in each of the validation sets as the predicted line limits for the attack simulation. The articial line limits are compared to the real line limits using R 2 , root mean square error (RMSE), and mean average percent error (MAPE).
The model we use is deterministic and based on the frequentest approach to statistics; this di ers from the work of Kim and Motter 25 and Elyas and Wang 26 whose generative models use the Bayesian framework. The Bayesian approach to modeling can provide a better understanding of the range of possible results; however, it comes at the cost of greatly increased computation. In this paper, we do not seek to nd an optimum line limit simulation model simply to create a statistical model to compare against PL. We use a simple linear model because it provides line limits that are based on a real system and are easily interpretable, with a low computational overhead.

B. Comparing attack damage between the PL approach and real line limits
We simulate the grid under attack by assigning a random-node attack rank to each network node. We then remove the node with Rank 1. Since node removal can cause overloading, we recalculate the power ow and remove any lines that exceed their maximum line limit. We then rebalance the power and load in the network. Line removal can cause the network to break into subcomponents; so, we remove any nodes that are in a subcomponent that has no power. We then recalculate power ow. This process continues until no further removals occur. We then nd the node with the next lowest attack rank and remove it, repeating the process until no nodes remain in the network. The process is described in Algorithm 1 for graph G, the set of n nodes/vertices V, and the set of m edges E. A schematic of the process is shown in Fig. 1.
During the attack simulation, two di erent damage metrics are used to analyze attack progression giant-component size, measured as the largest connected component, and blackout size, measured as total MW lost. These metrics are compared to the graph's original state using P x = P 1 −Px P 1 , where P 1 is the complete graph and P x is the graph after attack x. This method of measuring damage returns a percentage between 0 (no damage) and 100 (complete grid collapse).

ARTICLE
scitation.org/journal/cha Algorithm 1 Attack the grid 1: procedure A T G (G) 2: 3: E ← E(G) 4: while V = ∅ do 5: remove min i (v i ) i is the order in which nodes will be attacked 6: repeat 7: calculate power ow in G 8: for e ∈ E do 9: remove e if f e > f max e Power ow exceeds line limit 10: end for 11: rebalance generation and supply 12: for v ∈ V do 13: remove v if subgraph has no power 14: end for 15: until f e < f max e for all e 16: end while 17: end procedure Note that using the largest connected component as a measure might not be representative of the physical processes that occur on the power grid. For example, say we have a network that has 100 nodes but only two generators, where the rst generator produces 99% of the electricity and the other generates the remaining 1%. If the big generator fails, the giant-component size is still 99%, thereby indicating that the network is almost una ected even though there is only 1% of the necessary power. Despite this drawback, the metric's simplicity has made it a popular choice, 27,37-39 and, thus, it will be included here. Blackout size is a metric from the extended topology. It measures the loss of system power in megawatts. This metric is popular and has several di erent names with slightly di erent implementations that produce similar results. It has been called "loss of load," 40 "blackout size," 24 "load shed," 41 "power supply," 30 and "total loss of power," 42 among other names. This metric does not su er the problem described above regarding the size of the giant component.
C. Comparing the order in which edges are lost due to cascade Accurately modeling the damage done in an attack provides substantial insight into a grid's vulnerability. Sometimes, however, it is also useful to know the order in which nodes or edges were lost, as this information is used in certain vulnerability metrics. 28,31,43 One caveat here, though, is that if the order of the nodes being lost di ers depending on the line limits, the results of such analyses will not be reliable. In order to explore the robustness of loss order in relation to line limits, we correlate the order in which nodes are lost during an attack for arti cial line limits with real line limits.
Tables II and III provide a toy example of how we compare nodeloss order. First, k node-removal orders are generated. Then, all k simulations are run using each line-limit type. Table III shows Simulation 1 for the real line limits and a line limit of α = 3. In the table, the bold numbers indicate a node that has been targeted for removal, while the other nodes are removed by the cascade. We nd the similarity of network collapse by correlating the round lost of nodes that were lost to cascade. This implies we exclude nodes that were targeted for removal. As a result, in the example, only nodes F, G, and H can be compared.
For each of the k simulations, we use Spearman's correlation where rg x,k and rg y,k are the rank of x k and y k , respectively, for the n nodes in the network, σ is the variance of the rank of the nodes, and cov() is the covariance of the rank of the nodes. In this case, x k is the vector of the node-removal rounds of  the arti cial line limit for Simulation k and y k is the vector of the node-removal round for the real line limits for Simulation k. In the example, this means x 1 = 2, 4, 4 and y 1 = 1, 3, 2, thereby yielding a correlation of 0.866. For the real experiment, we generate 100 attack orders for the 512 nodes, producing 100 correlation scores per line limit method. This reveals the extent of similarity between the collapse order of the arti cial line limits and that of the real line limits

D. Comparing vulnerability ranking accuracy
The choice of the attack strategy can have a substantial impact on results. Di erent strategies damage the network at di erent rates throughout the attack. When comparing attack strategies, it may be that only the relative performance is important. In this analysis, we compare the ability of di erent line-limit methods to accurately rank attack strategies. We compare ve di erent attack strategies across TABLE III. The numbers indicate the round in which the node was lost in Simulation 1 for the real limits and α = 3. Nodes that are shown bold were targeted for removal and so are not included when calculating removal similarity.

Node ID Real limits
eight α values, the modeled limits, and the topological limits. The α values used in this analysis are 1.05, 1.1,1.5, 2, 5, 10, 20, and 50. The attack strategies we use in this analysis are a mixture of topological methods and extended topology. The strategies are entropic degree 10 using line limit, entropic degree using initial power ow, degree, centrality, and electrical centrality. 40 The centrality and degree attack strategies prioritize the most central nodes. Entropic degree using power ow and electrical centrality generally prioritize attacking the power sinks and the southern part of the UK. Entropic degree line-limit lies between the centrality and electrical centrality methods.
Using each attack strategy to de ne the node-removal order, we simulate an attack until we achieve complete grid collapse. After each attack round (node removed), we rank the strategies by total blackout size. The strategy that caused the most damage to the network is given  Rank 1 and the strategy that caused the least damage is given Rank 5.
We then compare the rankings obtained by each arti cial limit with those obtained by the real line limits. Using RMSE, we evaluate which line-limit method has the lowest error relative to the real line limits.

III. RESULTS
The results are divided into two sections. The rst section describes the results of the linear model used to generate line limits, and the second section compares the performance of di erent arti cial line limits to the real line limits.

A. Modeling line limits
We rst create a linear model that predicts line limits. The variables used are voltage and initial power ow. These variables create two models, the PF model (power ow model) and volt PF model (voltage power ow model). The network is represented in Fig. 2 and uses a UK geographical location and a graph space representation using force expansion. As we only have a single network, we use tenfold cross-validation to train on 90% of the data and then predict using the remaining 10%. The model coe cients are all quite stable, with a proportionally small standard deviation. The coe cients were all positive across all folds. The volt PF model shows that as voltage increases so does the line limit. This is intuitive, as high-voltage cables are often used for bulk-power transmission (see Table IV).

B. Comparing the performance of different line limits
An inspection of the power-grid tolerance distribution under initial conditions shows that the system is not proportionally loaded. Figure 3 depicts the α distribution of the power grid under initial loading. The power grid has a mean of α = 5.12 and a median of α = 6.12. These tolerances are much higher than those used in the literature, which are typically less than two. We also compared the line capacity against line load and found a load capacity relationship similar to that found in Kim and Motter. 25 That pattern is C − L ∼ L τ , where C is the capacity, L is the line load, and τ is the network exponential. This pattern persists when the data are broken out by voltage; further details can be found in supplementary material S2.
Correlating PL (proportional loading) limits with real limits yields an R 2 of 0.5. Using a range of α values, we nd that the minimum MAPE (mean average percentage error) and RMSE (root mean square error) are 0.52 and 1380 with α values of 3.6 and 3.5, respectively. The volt PF model greatly outperforms all PL models and the PF model (see Table V). The PF model has very poor performance across all three metrics, which shows that it is not an e ective method to accurately represent line limits.
We explore the damage done during the random attack across di erent values of α. It is evident that α has a logarithmic relationship with the damage for a given number of attacks. Figure 4 illustrates that, for both damage metrics, the α levels only converge at grid collapse and do not cross each other before that.  Figure 5 shows that, despite its poor ability to accurately measure line limits, the PF model is the most accurate method of measuring blackout size, with an RSME of 0.077. The volt PF model performs slightly worse than the α = 5 (the system mean) in terms of blackout size. These models have RMSE scores of 0.085 and 0.082, respectively.
For the giant component, the volt PF model is the best performer, with an RMSE error of 0.088. The volt PF model is followed by α = 7 and α = 10, which have RMSE scores of 0.103 and 0.104, respectively. The PF model comes fourth with an RMSE score of 0.117. The error relationship between blackout size and optimum α is intuitive, as it matches the average α of the real line limits. The reason for the giant-component optimum α is less apparent and may be linked to the topological structure. The linear models perform well in both cases.
Next, we compare the standard deviation of damage across all 100 simulations. The volt PF and the PF only models have very similar results with RMSE scores of 0.022 and 0.023, respectively. The accuracy of the linear models comes from the ability to represent the hump of the real limits shown in Fig. 6, something no value of α can replicate.
When examining the RMSE of the standard deviation of the damage of the giant component, we see that the volt PF model, the PF only model, and α = 7 have similar scores, 0.018, 0.019 and 0.14, respectively.
We compare the correlation of the order of node loss between each arti cial line limit and the real line limits. The correlation across all 100 simulations is shown in Fig. 7. The gure shows that, as α increases, the node-loss order similarity increases following a logarithmic growth curve with the topological analysis most similar to the real line limits. The topological analysis has a mean correlation of 0.853, more than three times as high as α = 1.05, which has a mean of 0.277. The α = 50 model narrowly outperforms the volt PF model, which have correlation scores of 0.847 and 0.842, respectively. The PF model has a signi cantly lower correlation score of 0.730. Figure 8 shows plots of the damage caused by the ve attack strategies in four of the ten line-limit scenarios. As can be seen, the rankings of the attack strategies change as the number of nodes removed increases. We analyze how well each line-limit method re ects the relative performance of the attack strategies. We nd that the volt PF model (RMSE 0.638) comes out at the top, outperforming the α = 5 (0.776), while the PF model (1.02) comes fth, narrowly beaten by α = 2 (0.971) and α = 10 (0.987) (see Fig. 9). The values of α of 1.5 or less have a considerably greater error in ranking di erent attack strategies.
The fact that the low α values are much worse at ranking the performance of attack strategies is important, as the majority of PL papers use α values of less than ve. One reason for this may be that the cascade size is bigger with lower α values. An interesting nding is that in this speci c case the change in α as the attack develops is minimized when the α is equal to the base case α of the system using the real line limits. Figure 10 shows a large drop in α during an attack, when the initial α of the system is much higher than the true system mean α of 5.1. In contrast, the load level ( 1 α ) shows a large drop, during an attack, when the initial α of the system is much lower than the true system α. When the initial α is set close to the true system α, the sum of the change in α and load level, Total = α + 1 α , from initialization is minimized (see Fig. 11). This indicates that there may be a relationship between the position of an edge in the network topology and its line-limit, an idea related to Elyas and Wang. 26 Although no conclusions can be drawn from this single result, it is included as it may be of value to other researchers.

IV. DISCUSSION
This study found that using linear models to estimate powergrid line limits gives consistently high performance. The volt PF model (voltage and power ow model) performs very well across all tests, while the PF model (power ow only model) performs better than most α values but is not outstanding. We nd that the optimum tolerance α is not consistent and changes depending on what is being measured. Although α = 5, the value closest to the system mean of α = 5.12 performed well; sometimes other α values, such as 7 or topological, provided the best performance in speci c analyses. Results based on lower values of α may imply that the risk of cascading blackouts in some networks is greater than it actually is and provide a false indication of the robustness of power networks to cascading failure.
An interesting nding was that the ability of the linear model to measure line limits does not re ect the ability of the linear model to accurately represent power-grid collapse behavior. Although the volt PF model is much more accurate at estimating line limits, the PF is more accurate at modeling blackout size during grid collapse. This may be because the PF model makes large errors on edges that do not overload and smaller errors on edges that do. If a line is unlikely to overload, the limit is irrelevant. A complicating factor is that the PF model has relatively poor correlation with the order in which the nodes are lost. This result is probably related to the overloaded lines, of which the PF model has more on average than the volt PF model or the real line limits. Further research is needed to explore the relationship between modeling line limits and modeling collapse damage, particularly with regard to the importance of accurately identifying lines that may overload. Further details on model performance may be found in supplementary material S2.
A limitation of this paper is that it uses a static analysis of cascading failures. Work that uses transient analysis of cascading failures 15,17,18 can show networks to be more vulnerable than static analysis, as failures can occur due to synchronization issues. However, evidence suggests 18 that if this analysis was run using transient modeling, low values of α would be more a ected. As such, we believe that the static nature of this analysis does not have a substantial outcome on the results.
In this paper, we explored how line limits a ect the collapse behavior of the power grid using a single load pro le which was the base case load pro le. Ideally cascading failure research uses a range of load pro les, such as di erent time of day or season. Using di erent load pro les enables researchers to draw more generally applicable conclusions. However, this paper does not seek to nd a solution to any possible weaknesses in the UK power grid topology Chaos ARTICLE scitation.org/journal/cha FIG. 10. When the PL value is much higher than the true system mean α, there will be a large drop in the observed α value as the attack develops. If the PL value is much lower than α, then there will be a drop in the 1 α value as the attack develops. This indicates that there may be a link between the topology of the network and the edge line limit. but instead to highlight the weaknesses of using proportional loading in power grid analysis. Consequently, the availability of only a single load pro le is of less importance than it otherwise might have been, as we can demonstrate the weaknesses of α with only a single load pro le.
Overall, we nd that the line limits follow a simple linear model using voltage and initial power ow. Although the linear model has only been trained using a single load pro le and only a single power network, it provides value as it is at a much more realistic system tolerance than arbitrarily selecting α because it uses the base load case FIG. 11. By summing the absolute maximum difference in values for α and load level, we find that the minimum value is α = 5, which is also the closest simulated value to the true mean α of 5.12.

ARTICLE
scitation.org/journal/cha for the UK grid. Moreover, the volt PF model uses the fact that higher voltage lines generally have more capacity to obtain information that is independent of the initial power ow state.

V. CONCLUSION
The title of this paper suggests that it is unwise to use articial line limits when studying the collapse behavior of power grids under attack. In fact, given the importance of arti cial line limits in this eld, we merely propose that researchers must carefully consider how they set these arti cial line limits. This is because, as we have shown, arti cial line limits far from reality can produce results which themselves are far from reality and do not represent real-grid behavior. This study demonstrates the importance of using values based on real engineered systems rather than arbitrary representations when researching cascading failures on the power grid. In cases where researchers have no knowledge of the mean loading of the network and α will be selected arbitrarily, we recommend using the voltage power ow model developed in this study. Use of this linear model in conjunction with the network's base load pro le will reduce the overall size of cascades (compared to a small value of α); however, the evidence presented in this paper suggests that the results of analyses using the model will be much more closely related to the real behavior of the grid.

SUPPLEMENTARY MATERIAL
This paper includes two additional sections in the form of supplementary materials S1 and S2. S1 provides more information on the PEARL analysis framework and acts as partial documentation for the associated R package. S2 provides a more in depth look at the performance of the models for researchers who are considering using them.