Case beyond historical severity : Winds, faults, outages, and costs for electric grid

extreme windstorm costs as well as a realistic case for broader emergency preparedness exercises. The former application is illustrated by the preliminary cost-benefit assessment for cabling in the case of an unprecedented windstorm. Finally, the reevaluation of currently used cost rates calls for an account of time dependency, critical services, and impacts on smaller economic and population segments.


Introduction
Disruption from a known and previously manifested threat can, at unprecedented severity, cause extreme impacts that were not considered possible and thus not prepared for.Examples of such disruptions include some of the largest disruptions of energy systems in recent years.In 2022, Russia cut over half of its natural gas supply to the European Union [1] and entirely to five EU member states [2].The threat of such a cut was well known, given multiple supply interruptions to central and eastern European countries during political tensions [3].However, larger cuts against economically more significant countries in 2022 exposed not only the lack of infrastructure to enable alternative supplies [4] but also raised the prices of global liquefied natural gas manifolds [1] and diverted supplies from other regions [5].In 2020, worldwide lockdowns for pandemic management caused an energy consumption collapse that was the largest since the second world war [6].Again, pandemics are a known threat and larger pandemics have occurred.Furthermore, some aspects of the modern world make the occurrence and spread of infectious diseases easier [7].In 2011, Japan experienced one of the largest ever recorded (magnitude 9.0) earthquake and tsunami (some waves more than ten meters high) that devastated coastal areas and led to the meltdown of three reactors in the Fukushima Daiichi nuclear plant [8].The Fukushima accident was a major blow to public trust in the safety of nuclear energy, hampering its development worldwide [9].However, the accident occurred in a country with extensive experience in mitigation and infrastructure adaptation to seismic hazards [10,11].All three cases present impacts of unprecedented severity disruptions that differ fundamentally in costs and nature from impacts of previous disruptions of the same threat type.These cases also indicate the fact that the potential impacts and costs can cascade far beyond the energy sector due to the high dependence of modern societies on interrupted energy, especially electrical energy.
Novel impacts can easily be missed without dedicated assessments, making unprecedented disruption look like a black swan.Black swan is the metaphor referring to an event that is rare, extremely impactful, and predictable in retrospect but not in advance [12].In practice, the unpredictability of black swan events may stem not only from the fundamental nature of the threat but also due to inadequate analysis.Naqvi et al. refer to the problem of suboptimal long-term capital allocation due to a short-term focus in financial analysis with the expression "all swans are black in the dark" [13].Analogously, the opening statement of this paper could be expressed by saying "fat swans are black looking from their shadows" with the weight of the swan referring to the severity of the disruption.Measures to minimize potentially huge costs from fat swans depend on knowledge about such swans.However, shedding light on a fat swan is complicated by structural dependence of a swan's weight -types of the most significant cost factors depend on potential disruption severity.The dependence on weight suggests that the most promising first step in the fat swan study is its weighting to narrow down the study problem.
The large number and variety of potential threats to energy systems [14] necessitate the study of unprecedented disruption to be threat-type specific.Arguably, the most interesting threat type is the one that manifests in the most likely to occur or potentially most impactful future disruption.However, even preliminary estimates for potential impacts of future disruptions across multiple types are a significant research challenge.Therefore, for this study, windstorms are chosen as a historically major cause of electricity interruptions globally [15].Windstorms refer to intense extratropical cyclones that are among the most significant meteorological phenomena affecting the weather of the Northern Hemisphere and are especially damaging for highly forested countries like Finland.In addition to a high forestry rate, Finland is sparsely populated with a high share of electrified heating, which is a critical service in a cold Nordic climate.The mentioned factors make Finland more vulnerable to windstorms even though electricity interruptions are rare and made rarer by recent grid strengthening.That said, a handful of the strongest Finnish windstorms alone account for the majority of all interruptions and likely an even higher share of associated costs [16].While most of the Finnish electricity grid has existed for seventy years [17], it seems reasonable to expect that this period does not capture the strongest windstorm that has occurred or could occur.Analogously to the major disruptions in global energy systems mentioned before, significantly stronger windstorms could differ greatly from historical cases in dominant cost factors and most relevant countermeasures.Costs of hours-long electricity interruptions are structurally different from days-long interruptions that disrupt critical services like water and heat supply [18].Even the technical problem of minimizing overall lost load (LL) after a significantly more damaging windstorm may depend more on repair capacity (number of repair crews and stock of spare parts) than its speed (current Finnish regulation specifies quality requirements for distribution system development in terms of maximum disruption duration [19]).
This work aims to provide the basis for identifying the most relevant cost factors and realistic shock test scenario for energy system development strategies and emergency preparedness exercises (e.g., [20]) in case of historically unprecedented windstorms.The unprecedented windstorm case in this study refers to a windstorm that creates the costliest impact and is costlier than any historical windstorm since the original construction of the electricity grid while still being meteorologically plausible.The aim to go beyond the historical severity range calls for a reexamination of relevant impact factors and suitable impact evaluation methods throughout the impact chain.However, resilience literature that discusses high-impact-low-probability events tends to be abstract and qualitative (e.g., [21]).In contrast, arguably, the spatiotemporal LL profile is the most objective basis for identifying and evaluating relevant cost factors.Generating a realistic LL profile requires a sophisticated account of meteorological and electricity systems.However, few electricity system impact studies that include cases beyond historical severity [22][23][24] have very simplistic wind field representation.At the same time, the absence of statistical methods to generate spatiotemporal extremes and unprecedented windstorm value fields from climate change studies indicate that existing electricity impact models are untested for extremes of interest in this work.Furthermore, extratropical cyclones differ from tropical cyclones by their asymmetry, complicating their representation.Considering electricity system, for countries like Finland, LL profile should be obtained by modeling faults in distribution grids, as practically all windstormrelated interruptions in Finland occur because of them.Additionally, such a profile should be obtained for a large area to capture broader implications (e.g., for national regulation).Despite many studies on windstorms and their impacts on the grid, no study was found to have such a combination of scale and detail.Therefore, the authors specifically developed the impact model covering the whole of Finland at medium voltage (MV) grid detail (documented in [16]) and investigated its suitability for most severe windstorms (documented in [25]).This work goes beyond historical windstorms, producing unprecedented one and assessing its impacts.The specific objectives of the paper are the following: 1. Generate spatiotemporal wind gust speed field for the strongest meteorologically plausible windstorm in Finland.2. Obtain spatiotemporal LL profile for this wind gust field to the present Finnish electricity system.

Illustrate the relevance of an unprecedented case by assessing costs
for obtained LL profile and potential implications for electricity sector development.
The structure of the paper is the following.Section 2 documents a broad exploration of windstorm impact driving factors and their potential scaling methods and plausible magnitude.This exploration includes a review of meteorological literature and reasoning from the first principles.The most relevant factors, methods, and severity limitations are combined to generate a spatiotemporal wind gust speed field for a historically unprecedented windstorm.A notable original addition is the proposed link between the spatial set of return period maximum values and a spatiotemporal value field of an event.Section 3 presents a brief literature review on modeling windstorm impact on the electricity grid and an overview of the impact modeling framework used here.Section 4 presents the modeled impacts of the largest historical and unprecedented windstorm's impacts in terms of spatiotemporal powerline fault and lost load profiles.One potential use of such profiles is illustrated by preliminary cost-benefit reassessment of major grid hardening efforts in case of unprecedented disruption, providing a supplementary perspective to cost-benefit assessment for average annual disruptions.Quantitative reassessment is followed by a qualitative discussion of socioeconomic cost factors that may change significantly with disruption severity and thus be relevant for more sophisticated assessment.Section 5 presents a sensitivity analysis for LL against the most significant impact modeling uncertainties and a list of considerations on study limitations, usability of results, and applicability of methods for other countries and other weather extremes.Section 6 concludes the paper.

Unprecedented windstorm generation
The aim of unprecedented windstorm generation is to develop a meteorologically plausible spatiotemporal field of weather parameters that would cause the costliest disruption in the Finnish electricity system.To the authors' knowledge, no studies have attempted to generate the costliest windstorm.Therefore, an attempt to do so in this work includes the assessment of meteorological plausibility, relevant factors and methods, and filling the remaining methodological gap between a return period values and a windstorm.

Future projections for changing climate
The assessment of meteorological plausibility includes a literature review on expected future windstorms.This literature provides no consistent evidence for change in total windiness despite projected changes in the overall climate.However, there is evidence that extreme wind speeds of extratropical cyclones may get more potent in a changing climate.Variations in projected changes are influenced by factors like the choice of climate model, the variable examined, or the geographic area studied [26].
Climate change affects extratropical cyclones and their associated weather patterns, altering cyclones' wind speed, rainfall, intensity, and frequency.The initial cause of extratropical cyclones, a meridional temperature gradient, weakens as polar regions warm faster than lower latitudes [27] (warming in the Arctic has been up to four times faster than in the rest of the world [28]).While warming can lead to weaker extratropical cyclones and winds, it represents only one of multiple climate-related changes in the atmospheric processes.These changes can influence extratropical cyclone development in opposing ways, leading to high uncertainties about climate change's overall impact.
The studies in the field agree that the total number of extratropical cyclones decreases [29][30][31][32][33], but the intensity of the most extreme cyclones may increase [32,34].Several studies have shown that changes in extratropical cyclones are regionally dependent, with projected increases in intensity in the Southern Hemisphere [35,36] and likely decreases in the Northern Hemisphere [31,36] (the median decrease in identified cyclones is up 5.4% in winter and 6.8% in summer respectively [33]).The changes in intensity are also dependent on the variable [26,37].The projected changes in wind speeds by the end of the century vary depending on the region and are uncertain [38].According to Priestley and Catto (2022), the area impacted by intense wind speeds in extreme windstorms is projected to increase by up to 40% by the end of the century [33].Sinclair et al. additionally found evidence for an increase in the strength of low-level cyclone wind speeds in the vicinity of the warm sector by up to 3.5 m/s [32].The evidence hints that extreme extratropical cyclones may become, in several ways, more extreme in the future.
Regarding the area of interest of this paper, Finland (and northern Europe), the climate projections for mean and extreme wind speeds do not show significant changes except a slight increase (up to 2.5% in extreme wind speeds) in autumn at the end of the 2100 s [39,40].Tropical cyclones are expected to intensify due to climate change, with the tropical cyclone development region expanding eastward [41].This shift increases the risk of transitioned cyclones in northern Europe, which tend to be stronger than regular windstorms and contribute to about 10% of severe windstorms in Europe [41,42].Additionally, there is a projected increase in the number and intensity of cyclones reaching Europe [41], potentially leading to more windstorms like windstorm Mauri occurring in Finland [41,43].Haarsma et al. (2013) predict a wind speed change ranging from 0 to 16 m/s during autumn (August to October) between present and future climate conditions in the North Sea [44].When considering Norway, the North Sea, and the Gulf of Biscay collectively, the count of hurricane-force winds (> 32.6 m/s) during autumn is projected to rise from 2 to 13 throughout the 21st century.The possibility of an increasing number of extratropical cyclones in Northern Europe and Finland may be also influenced by the predicted poleward shift of the storm track locations [45].
Even though the wind and windstorm climate in northern Europe and Finland would not massively change in the future, the risk of windstorm damage to the electrical grid is likely to increase.The likely reasons for this increase are changes in other weather-related parameters that influence the uprooting and breaking of the trees and their falling on the overhead powerlines.Warming temperatures increase the atmospheric moisture content, which can intensify extratropical cyclones by leading to heavier precipitation [46].The climate projections also show a particular increase in winter rainfall amounts in Finland [47].The trees uproot more easily when the soil is wet.Thus, compound hazards, like strong wind and heavy precipitation, may co-occur [48] and affect the power grid more frequently in the future.Typically, soil frost is present during winter in high-latitude countries such as Finland.Soil frost anchors the trees to the ground, making them more resistant to windstorms [49][50][51].However, in a warmer climate, the period of soil frost shortens [52], and the depth of soil frost decreases [53].According to Venäläinen et al. (2020) and Lehtonen et al. (2019) [51,52], this increases the risk of wind damage, especially in the southern and middle parts of Finland [54].Climate change also has other indirect impacts, which weaken the durability of forests and thus may also increase the risk of wind damage to overhead powerlines.The warmer climate provides, for example, more suitable living conditions for insect pests (e.g., spruce bark beetle) and pathogens that attack the forests and weaken the trees' resistance against the strong wind [51,55].
Presented literature suggests that the impacts of climate change on major windstorms in Finland are uncertain but may cause an increase in severity.Such an increase seems unlikely to be significant enough to cause a large change in dominant cost factors for subsequent electricity interruptions.That said, significantly stronger windstorms, while unlikely and uncertain in current and projected climate scenarios, are nonetheless possible and relevant.

Severity determining factors
The severity of windstorm damage to the grid depends on multiple windstorm and surrounding environmental factors.Major windstorm impact severity factors include speed and direction of wind, duration and area covered by grid-damaging winds.The speed of wind gusts is by far the most important factor, as mechanical power created by the wind is proportional to the speed's cube.The relationship between wind gust speed and powerline failure probability is even steeper, potentially reaching power function form with a power of ten as shown by calibration of the model for the most impactful historical Finnish windstorm case [16] and by a study that investigated the impact of wind speed and soil-frost to the powerlines [56].A similar relationship has also been detected in the assessment of the volume of forest damage during the strongest windstorms in Finland [57] and in powerline fragility function derivation from disruption data for the United Kingdom [58].The direction of strong winds in Finland is predominantly west to east, with most of the largest windstorms coming from the North Atlantic [57].Finnish forests have adapted to this condition, with trees grown to be more robust to winds in such a wind direction.The wind-direction-dependent tree strength is illustrated by the 2012 windstorm Antti, which came from the east and caused significantly more extensive damage than its wind speeds would suggest [57].Duration and area with powerline-breaking winds hold a linear relationship with grid damage, given that all other factors remain the same.Additionally, an increase in windstorm duration increases an effective repair time by delaying the start of repair work.An increase in windstorm area increases damages depending on the relative positioning of strong winds to vulnerable powerlines, downstream consumption, and economic activity.Environmental factors can further be broken into those that influence the level of grid damage and those that influence the socioeconomic cost of given grid damage.Grid damage determinants include soil type, its frost and moisture, tree cover with snow or leaves, positioning, and composition of the forest in relation to the grid.Arguably, the most significant cost determinants in the cold Finnish climate are the outside temperature and the level of electrification in space heating.The number of mentioned factors and their interdependencies necessitates the selection of a smaller subset of the most critical factors that could be modified for practical impact quantification.
Reducing the number of modifiable factors in the study does reduce the potential impact of the storm.However, it dramatically simplifies the estimation of meteorologically feasible limits and the modeling of grid impacts.Assessing the potential increase of windstorm impact by modifying a single variable requires knowledge of grid sensitivity to a change of this variable and scaling potential limited by meteorological feasibility.Assessing potential impact increase from modification of multiple variables requires additional assessment of the interaction between these factors.Some factors can coincide with multiplying damage, while others are mutually exclusive.For example, the absence of soil frost weakens the anchoring of trees but occurs only during moderate temperatures when heating demand is low.Meteorologically feasible extremes of multiple variables may not necessarily be feasible at the same time and place.Determining the extent of possible overlap requires complex modeling of the meteorological system itself, which is outside the scope of this paper.Regarding grid impact, apart from windstorm factors, the current version of the model accounts only for temperature-dependent electricity consumption.Considering the complexities involved, a modification of only one factor is chosen for the generation of extreme windstorm representation.
Wind gust speed is the obvious choice for a severity factor to be scaled up to generate extreme windstorm representation, given the rapid increase of fault probability with increasing speed.A high rate of increase means that a meteorologically plausible speed limit needs to be only slightly higher than historical records to produce significantly more destructive windstorms.The level of destructiveness is further constrained by fault saturation, i.e., fault probability no longer matters once the majority of vulnerable segments have already failed.While stronger winds could damage harder grid components (namely high voltage transmission lines and substations, currently considered immune to wind), this requires significantly stronger winds.Even countries with a harsher wind climate than Finland have limited number of faults (NoF) in transmission lines from wind [59].Furthermore, a single fault in a transmission line would have a limited effect on supply as looped grid topology provides alternative routes for every grid point.However, in the case of a few faults in the transmission grid, the impact could be comparable to that of a few thousand faults in the distribution grid.Accounting for such a possibility requires an additional account of the transmission grid, which is outside the scope of the present study.It is worth noting that scaling wind gust speeds above historical records increases NoF not only in historically most affected areas.Scaling wind speeds over the whole country for the whole study period effectively increases the area where grid-damaging gusts occur and prolongs periods during which repairs are not possible.

Main options
Options for generating a spatiotemporal wind gust speed field beyond historical windstorm severity levels include importing foreign, scaling historical, and building a new windstorm value field.
An example of storm "import" is a study of hurricane impacts in the United States that includes a case representing a typhoon from the Philippines [24].The proxy of "import" suitability is expected to be similar to wind climate.Obviously, no other area can provide the perfect match, but neighboring areas may be reasonably close.It is also worth noting that extratropical cyclones are "harder to import" than tropical cyclones because of the significant variations in the frontal structure and strong wind occurrence regions of extratropical cyclones.
Scaling historical windstorms within the country requires a scaling constant for multiplication or addition: where: wg− wind gust, x and y− spatial coordinates, t− time, A− country area.Such a constant could be assumed or based on evidence.Possible sources of evidence for scaling constant include statistical analysis of historical wind extremes or general wind conditions [60], expert knowledge, studies on future wind climate [61], and windstorms in neighboring countries.
A new field of wind gust speeds can be built using models with a wide range of sophistication.The simplest models contain only a few formulas or some random number distribution.Models with few deterministic formulas capture tropical cyclones' geometry and other significant features (e.g., [22]).However, as mentioned before, such a basic approach is challenging for extratropical windstorms as they are not symmetrical.The closest example found for extratropical cyclone generation is a Finnish study that models synthetic pressure and wind fields by assuming, among other things, to have a Gaussian-shaped pressure field that moves at a constant speed [62,63].This model, however, focuses on investigating the effects of wind and pressure fields on sea areas along the Finnish coast and thus is not directly applicable to this study focusing on inland winds.Additionally, the mentioned study uses a numerical weather prediction (i.e., a highly complex) model.Models with random number sampling from predefined distributions represent wind variation better but do not provide means to distinguish spatial windstorm distribution.Random wind speed sampling distributions typically have a Weibull distribution form (e.g., [64]).It is widely used in wind-related engineering applications because of its simple parameter estimation [65], good fit, and suitability for wind speed observation data in many regions of the world [65][66][67].More sophisticated models include a representation of atmospheric physics aspects, which significantly increase model complexity.
This study's selected method for generating wind gust speed field is scaling the historical windstorm field due to its relative simplicity.The scaling is done for the wind gust speed field of the last week of 2011 when two major windstorms, Tapani and Hannu, hit the Finnish power grid one after another.Taken together, Tapani-Hannu windstorms caused the most extensive wind-related electricity disruption on record, with power supply interruption for 570 k consumers [68,69] out of 3.3 mln.Electricity consumers in the country [70].For simplicity, hereafter, Tapani-Hannu is referred to as a single windstorm.The scaling is done with the constant based on statistical analysis, described in the following subsections.

Statistical scaling of historical events
Statistic estimation of the possible extreme event magnitude beyond the historical record range is a common problem solvable with wellestablished extreme value theory (EVT).However, no studies were found to scale spatiotemporal events that contain three-dimensional parameter fields.
The closest studies seem to be scaling of spatial parameter surface for meteorological (mostly precipitation) extremes (e.g., [71][72][73][74][75]).These studies rely on location-specific marginal distributions and spatial correlation of meteorological parameters.Both are obtainable by fitting historical data or its subset limited to extreme events.Besides the omission of temporal dimension, the major challenge with such methods is the arbitrary nature of the definition of extreme spatial event.In this work, the extreme spatial event of the highest interest is the wind gust field that results in the largest LL.However, the link between wind and LL is complicated, requiring the model presented later in subsection 3.2 to quantify.LL maximization appears to be equivalent to the maximization of NoF, which is still a highly complex problem.Statistical modeling for extreme wind surface maximizing NoF may even be more complex than meteorological windstorm modeling.At the same time, further problem simplification to maximization of wind field within specific distribution shapes seems unlikely to lead to the largest LL.Despite the mentioned insufficiencies, EVT provides partial means to inform windstorm scaling and, by extension, a basis for developing a complete method for such scaling.
Windstorm severity scaling with EVT is arguably the most straightforward by using a single macro parameter, such as the storm severity index (SSI).Multiple SSI versions exist, accounting for some variation of the storm's wind speed field, affected area and duration, and potentially other meteorological and non-meteorological variables [76].Representation of wind speed includes wind gust speed at a certain height (e. g., 10 m) and wind speed at a particular atmospheric pressure level (e.g., 925 hPa) over certain absolute (e.g., 25 m/s) or relative (e.g., 98th percentile) threshold [76].Damaging wind speeds around a storm track can determine the affected area through which the storm center travels.A storm track can be identified with other meteorological parameters like relative vorticity [76], but it is not a trivial exercise.More critically, the derivation of SSI, as with any other parametrization, inevitably reduces data dimensionality and results in partial data loss.It is not apparent what type of data dimensionality reduction would be optimal.However, superior compression is expected to be after an EVT use with the most extensive available data set.
EVT provides methods for obtaining probability distributions for extreme value occurrences.The two major methods of relevance here are fixed box and maximum threshold methods that include collecting a subset of historical values and fitting them to predefined form probability density distributions.These distributions can be extrapolated to obtain probabilities for historically unprecedented values.The inverse of these probabilities also shows so-called return periods, during which a maximum value (return period value) is expected to occur once.The value subset in the fixed box method consists of maximums within historical data portions of a single chosen size, e.g., hourly data maximums within each year.The value subset in the maximum threshold method consists of all values above the chosen level.The probability density of values within the fixed box subset is fitted with the Generalized Extreme Value distribution function, which, depending on fitting parameters, takes the Gumbel, Ferchet, or Weibull distribution function form.The probability density of values within the maximum threshold subset is fitted with the Generalized Pareto distribution, which, depending on fitting parameters, takes Exponential, Pareto, or Beta distribution function form.In this work, the fixed box method is chosen as the box size selection seems more straightforward than the threshold level selection.
The extreme value analysis presented here utilizes Finland's available wind gust data in the ERA5 reanalysis [77,78].ERA5 data includes forty-three years (1979-2021) of hourly values for 43 × 51 grid cells of approximately 31 × 31 km 2 size that cover rectangular area projection enclosing Finland, 826.6 mln.Values in total.These values are transformed and aggregated via the following operations.The first operation takes annual maximums for each grid cell covered, reducing the number of values for remaining operations to just 94 k.The second operation checks for the presence of trends throughout the years.The check is done visually for a temporal plot where one-year values are distributed in a range between two years, making individual points easier to distinguish.The resulting Fig. 1 shows no clear trends that should be accounted for return period calculations.The third operation computes each grid cell's fifty-and hundred-year return period values.Fig. 2 shows the most likely and the upper limit of 95% confidence interval values of respective return periods.It is worth noting that differences in Fig. 2 between the two periods are significantly lower than differences between the most likely values and values of the upper limit of 95% confidence interval.It indicates that longer return periods, while mathematically trivial to obtain, produce highly uncertain results.

Linking windstorms to return periods
The linking between the strongest windstorms expected and return periods would allow to fill the remaining methodological gap for statistical windstorm scaling.However, this linking must address different data dimensionality.A windstorm description is a spatiotemporal wind gust speed field for a short period (several hours or days).The return periods compose a spatial surface of values expected to occur once in a long period (here, fifty or hundred years).The projection towards higher dimensionality data, i.e., return periods onto windstorm, requires additional information that EVT methods do not provide.On the other hand, location-specific maximum historical values are equivalent to the location-specific return period values.Wind gust maximums for the forty-three years of historical data available and the week of the Tapani-Hannu windstorm is shown in Fig. 3.The map of forty-three-year maximums closely resembles the map of fifty-year return period most likely values, illustrating a similar nature of the two data sets.However, only a portion of the forty-three-year maximums occur during the strongest Tapani-Hannu windstorm week.This demonstrates that the map of the long-period maximums is highly asynchronous and comprises values from many windstorms and times.
The relationship between the spatial maps of return period values and historical maximums falls short of the relationship between the former map and the spatiotemporal field of windstorm values.By definition, a windstorm with the highest grid damaging potential in a given period should have maximum wind gust speeds at some locations close to or coinciding with the maximum values of those locations throughout the entire period.During short periods, a country-wide mean of maximum values is very sensitive to windstorm location.E.g., a period containing two windstorms that are equal in magnitude but affect opposite sides of the country can give up to two times higher national mean than a period with just one such storm.However, increases in both the largest windstorm values and maximum values of the entire period are expected to be moderate with a duration increase of a multi-decade-Fig.1. Evolution of maximum wind gusts in Finland (1979Finland ( -2021)).long period.Therefore, as the first approximation, the relationship between the largest windstorm values and entire period maximum values is assumed to be linear.More specifically, the remaining methodological gap is filled by assuming that the ratio between the unprecedented and the largest historical windstorm values equals the national mean of ratios between return period values and historical maximums.The national mean of ratios refers to the mean of wind gust speed ratios for grid cells entirely within Finnish territory, i.e., grid cells with black edges in Figs.2-4.The mathematical expression of the assumed link is: x,y wg return period ( x, y ) where: N− number of spatial tiles country covers.Such spatially uniform scaling neglects geographical variations expected to be small for Finland but should be evaluated for larger and geographically more diverse countries.Fig. 4 shows maps of ratios between return period values and forty-three-year-long historical maximums.The longer return period corresponds with only marginally higher most likely values but significantly higher 95% upper confidence interval values.This difference indicates a large level of uncertainties with increased maximum wind speeds and, by extension, the strongest windstorm.Furthermore, the statistical scaling method necessitates a somewhat subjective choice of return period and confidence interval level, though it should be viewed as part of the scenario definition.Here, hundred-year and 95% upper confidence interval values are considered to provide the largest values with sufficient statistical reliability for further analysis.This level corresponds with the national mean of 1.24, chosen as the scaling constant for the Tapani-Hannu windstorm field of wind gust speeds to produce the unprecedented windstorm value field.The maximum wind gust speed in the resulting field on land is 40.0 m/s, which is comparable to the largest wind gust speed on record in the Finnish waters of the Baltic Sea (41.6 m/s, Aapeli windstorm (2019)) [79].The resulting value field also seems plausible based on the authors' meteorological experience and familiarity with wind climate in Finland and neighboring countries.The approach taken in this section for generating an unprecedented but meteorologically plausible windstorm value field utilizes 826 mln.Historical wind gust speed values to derive one scaling constant.While most of the analysis is done with annual maximums (94 k values) and spatial value surfaces (2193 values), the level of contraction represents significant simplification.Despite this level of simplification, no superior approaches were found for generating unprecedented windstorms short of complex physical models for meteorological phenomena.

Linking windstorms to their impacts
As described in the introduction, the objectives of this paper require quantifying windstorm impact on an electricity grid to capture multiple aspects, namely: unprecedented severity, meteorological realism, temporal dimension, electricity load, large scale, and distribution grid level of detail.The first part of this section presents a literature review on modeling windstorms' impacts on an electricity system in search of the most relevant studies for this work.The review indicates that aspects of   interest here are present only in some studies, and not a single study contains the complete combination.The second part of this section presents the modeling framework specifically developed with the complete combination of aspects needed for this study.

Review of windstorm impact studies
Electricity grid modeling studies include various assessments of grid vulnerability to wind hazard and grid impacts in different storm scenarios.Vulnerability assessments for electricity systems cover both network and component aspects.Bases for network vulnerability assessments include network topology (and its relationship to reliability [80]) and graph theory concepts (e.g., betweenness centrality, graph diameter, average path length, and clustering coefficient [81]).Examples of bases for component vulnerability assessments include an overhead powerline segment's fragility (fault probability) as a function of wind gust speed [58] and susceptibility to wind damage considering vegetation present near the powerline [82].While the above-mentioned vulnerability assessments are relatively simple, they do not provide information for the expected share and duration of damage in a vulnerable part of the system in case of extreme windstorms.More sophisticated models link wind gust speed field with grid impacts in the form of damaged components or interrupted supply using statistical and analytical methods [83,84].Statistical models use a large number of predictive variables and methods ranging from generalized linear models to various machine-learning techniques [83].Analytical, or fragility-based, models use electricity system representation composed of elements with known fragility functions (e.g., [64]) and, when recovery is represented, repair times (e.g., [85]).
Several grid impact assessment studies can be found with extreme wind gust speed scenarios when the search only concerns the severity of wind hazard, i.e., neglecting the absence of other modeling aspects relevant to the study of novel cost factors for unprecedented windstorms.Most such studies appear to take the wind gust speed field of major historical storms, often considering tropical cyclones (e.g., hurricanes Harvey [86] and Sandy [84]), or scale it from the historical wind gust field of non-extreme days up to historical records values (e.g., [59]).Given the rarity of major storms, most historical impact case studies use data from three or fewer storms, which is problematic as storms can differ in many impact-driving aspects [87].Storm severity itself varies within a large range that is especially difficult to capture with statistical impact models [87].The impact dependence of storm severity indicates the difficulty of extrapolating statistical models beyond the historical range where they can be fitted.Fragility-based models, in principle, should be easier to apply for unprecedented severity storms, but their suitability is not apparent, and thus, obtaining strong evidence for suitability seems critical.While without such evidence few studies have been found to go beyond the historical record.Bao et al. apply wind speeds for three regions of a small electricity and natural gas grid for twelve hours fixed to a chosen storm class [23].Previously mentioned Guikema et al. study, aiming "to examine the potential outages from a substantially stronger storm", impose track and intensity characteristics of the typhoon from the Philippines on tracks of hurricanes in the United States [24].Salman models grid impacts for hurricane wind field, generated with an analytical formula [22].Parameters of that formula are determined from historical hurricane data and varied for potential climate change impact on hurricane severity and frequency.It should be noted that climate scenarios in Salman study are assessed probabilistically with Monte Carlo simulation that differs from single case studies considered in this work.Also, Guikema et al. [24] and Salman [22] works' present examples of tropical cyclone studies that are common among severe wind impact studies but are nonetheless meteorologically different from less symmetrical windstorms that occur in higher latitudes (extratropical cyclones) [88].Overall, reviewed grid impact studies that modify wind gust speed field to reach historical records or go beyond it appear to use simpler methods than those present in meteorological studies, which complicates the aim of exploring meteorological feasibility limits.

Modeling framework
In the absence of an existing model suited to this paper's objectives, a new modeling framework was developed.Details of this framework are documented in [16], and the framework's three-part structure is shown in Fig. 5.The only model changes since [16] are lognormal form of fragility functions and fixing time distributions fitted for larger dataset as documented in [25].
The first part generates a synthetic (i.e., real-like) electricity grid and electricity consumption profiles.The methodology of this generation utilizes the fact that Finland has many relatively small distribution system operators (DSOs, 77 as of 2020).The presence of many small DSOs effectively makes operator-specific data low-level spatial data on distribution grid placement in the country.Low and medium voltage transformer numbers and consumption are mapped onto each municipality proportional to the population share of each DSO service area that falls within a given municipality.Line lengths are similarly mapped for each municipality, assuming proportionality to an arithmetic mean of DSO service population and area shares within a given municipality.MV line segments between two medium-to-low voltage transformers (136 k over the whole of Finland) are combined into municipal grids assuming typical big-trunk feeder topology.Mapped consumption is combined with available consumption breakdown by consumer sector and consumer type-specific temporal reference profiles.Resulting consumption profiles distinguish dependency on time (hourly resolution), space (municipal resolution), voltage level (medium and low), and consumer sectors (industrial, commercial, and residential).
The second part combines the generated system with fragility functions and fixing time distributions.Fragility function determines the failure probability of MV overhead powerline segment for a given wind gust speed via a commonly used lognormal function form [80]. Shape and scale parameters for this 1 km powerline segment function are obtained from a calibration of the most impactful windstorm case in Fig. 5.The structure of the modeling framework developed in [16].The figure is reproduced from the source paper.
Finnish electricity system history.Calibration refers to running the model multiple times with different fragility function parameters and selecting a pair with the recreated lost load profiles that match the historical profile the most.Specific segment fragility further accounts for segment length and forested area share in a given municipality via the following relationships: where f = segment failure probability, v = wind gust speed, l = length of the reference subsection (here 1 km), L = length of the actual line section, A = area of a municipality.Fixing times are obtained via two-level fitting of interruption times of all wind-related faults in Finland during the 2005-2014 decade.The first level fits duration distributions of interruptions (assuming Weibull distribution) within each storm, which is considered an independent event.The second level fits the obtained distribution shape and scale parameters as a linear function (with a floor for shape parameter) of the peak NoF, resulting in a storm severitydependent distribution.At each time step, the fault-fix algorithm assesses potential faults for all vulnerable components, i.e., undamaged overhead powerline sections, and repair done for all failed segments.A fault occurs when failure probability, i.e., fragility function value for wind gust value at a given time step and municipality, is larger than the randomly generated number from a uniform distribution between zero and one.Failed components are assigned fixing time from the previously obtained severity-dependent distribution with severity represented by a total NoF at a given time step.Repair is represented by lowering the remaining repair time by time step duration in the absence of strong winds (representing a continuing windstorm) and night (representing a need to rest for repair crews).A remaining repair time of zero indicates a change of segment status from failed to fixed.Given the radial topology of modeled grids, LL is simply a sum of consumption downstream of all failed components.The final part of the framework is an application of the model in which the input is the spatiotemporal wind gust speed field, and the output is the spatiotemporal LL profile.The input data with its resolution, sources, and use part of the framework is summarized in Table 1.Annual data depends on the year, while hourly data additionally depends on the time of the year of the study case.In all cases, the same data sources are used.
The ability of the presented modeling framework to generate realistic LL profiles for the highest severity windstorms was explored across multiple indirect lines of evidence in [25].The most crucial evidence of model generality included a recreation of LL profiles for Finland's three most grid-damaging and recent windstorms.The second line of evidence concerned interruption data quality and goodness of fits for fixing time distributions.Both lines of evidence indicated the model's ability to recreate the national LL profile for historical windstorms with uncertainty ranges of around 20%.The third line included a review of the impactfulness of extreme winds in Finland.It showed that despite many windstorm impact driving factors, most severe windstorms appear to have similarities (most notably, lack of soil frost that anchors trees in the windiest winter season), indicating the model's ability to recreate impact profiles for the most impactful future windstorms.
Arguably, the most distinct aspect of the presented framework is the combination of national scale and MV grid detail, which enables the investigation of strategic considerations for typically local fault phenomena.Such scale and detail combination was not seen before in fragility-based electricity system impact modeling literature.In addition to electricity system detail, municipal detail for wind gusts allows a relatively detailed representation of meteorological parameter, which is shaped significantly by local conditions.Another major unique framework's aspect is severity-dependent fixing time distributions.Such distributions combined with model validation for the most impactful and recent windstorms make this impact modeling framework uniquely positioned to study even more severe windstorm impacts.

Historical and unprecedented windstorms' impacts
To quantify the implications of windstorm scaling and grid development, the largest historical and unprecedented windstorms' impacts are modeled for Finnish electricity grids of 2020 and 2011 with 2020 consumption.These grids represent the current electricity system in the country as is and as it would be in the absence of grid developments during the previous decade.The most significant among those grid developments has been the underground cabling of powerlines, which makes them practically immune to strong winds.The cabled MV line share grew from 12.3% in 2011 to 39.0% in 2020 [25].The following assessment of corresponding economic impacts consists of qualitative and quantitative parts.The quantitative part covers a comparison of estimated costs and impact cost savings due to cabling done in the decade of 2011-2020.The qualitative part provides a preliminary exploration of social and economic impacts that are insignificant for moderate interruptions but could become major cost factors for interruption with the magnitude of the unprecedented windstorm case.

Table 1
Summary of the model input data.LV/MV/HV stand for low/medium/high voltage.Interruption record contains data on its occurrence, duration, affected consumption and number of consumers.In 2020, Finland had 309 municipalities, and interruption data is available for DSO service areas aggregated into five regions.The table is reproduced from [25].

Technical impacts
Fig. 6 shows modeling results for four cases (impacts of two windstorms onto two grids) in terms of LL and NoF.The most important aspects of obtained impacts are visible by comparing grid and windstorm cases in relation to each other and the vulnerable (i.e., consisting of overhead lines) grid part.While the two variables give more information about the system, all major trends are the same for both LL and NoF.The comparison of grid cases presents the most prominent aspect of decreasing impact with decreasing system vulnerability.The grid development in the face of both windstorm cases reduces total NoF by around 1.5 times and LL threefold.The impact reduction is expected as grid development has reduced vulnerable consumption from 3.8 to 2.3 GW and the number of vulnerable sections from 111.2 to 73.3 thousands.At the same time, the comparison of windstorm cases shows an even larger difference -24% larger wind gusts in the 2020 grid increase total NoF fivefold and LL tenfold.In the unprecedented windstorm, both grids suffer up to 30% of overhead lines failing and close to 45% of the peak share of vulnerable consumption disconnected.Note that peak LLs represent more than half of the vulnerable consumption values in the figure, as these values show the weekly mean rather than the maximum of vulnerable consumption.The impacts in the northern part of the country remain small, leading to even higher disruption shares for the remaining four regions.The impacts in the east and southeast of the country show a higher increase than impacts in most affected west and southwest parts.The unequal increase indicates partial saturation of damage in the west and southwest and increasing importance of changes in exposure to over changes in severity of wind hazard.
Another major aspect of the results concerns the limited study duration that does not fully capture the impacts of the unprecedented windstorm.The remaining LL level at the end of the week is large, which may be even more important than peak impact levels.The remaining LL in the 2020 grid is 147 MW, which is only 1.5 and 3.5 times less than the peak LL from the historical windstorm in the 2020 and 2011 grids, respectively.The remaining LL for the unprecedented windstorm in the 2011 grid of 660 MW is even larger than the historical LL peak.Each MW of disconnected consumption corresponds roughly to 1 k of disconnected Finns.A non-insignificant portion of the population without power for a week may need to temporarily leave their homes or seek external help.However, seeking such help is simplified by the distributed nature of disruption as practically all municipalities now have significant powerline shares cabled [25].In other words, many households may need external help but could technically get it nearby as some of their neighbors would still have a power supply.
Finally, the large NoF still occurring in the 2020 grid highlights the importance of retaining grid repair capacity.Keeping repair capacity may require new arrangements as traditional incentives for keeping large repair capacity are likely to decline with higher reliability from cabling during moderate windstorms.Note that the approach chosen to derive fixing time distribution parameters for this model did not allow to account for changes in repair capacity that may have already occurred since 2011.

Quantitative economic impacts
Assessment of quantitate economic impact is needed to make decisions for supply security improving measures as both interruptions and measures to reduce them are costly.Economic system development aims to minimize the sum of the two, which mathematically corresponds to the condition when the marginal costs of interruption are equal to the marginal costs of interruption reduction measures [100][101][102].
The standard measure of marginal interruption costs is the value of LL (VoLL), expressed as monetary unit per energy unit of unsupplied demand, e.g., €/MWh [102].VoLL depends on a multitude of factors like customer affected, disruption duration, timing (time of day, day of week, season), presence of prior warning, etc. [102].While a large body of literature exists for VoLL, most estimates are made for interruptions up to eight hours long and almost no for interruptions longer than a day [103].Also, dependency on interruption duration in VoLL estimates is mostly absent [104] and when presentis derived by extrapolating from a few points [103] to replace currently used values by the Finnish energy market regulator [106] based on the study from 2006 [107].AFRY Oy Ltd. provides values for all sectors represented: residential -9.99 €/MWh, commercial -11.26 €/MWh, industrial -14.53 €/MWh.
A range of measures exists to increase distribution grid resilience against windstorms, e.g., improved management of vegetation near powerlines, placement of powerlines along the roads instead of straight through forests, and improved preparation and placement of repair crews [108].However, the most significant measure in terms of both implementation cost and interruption reduction is the underground cabling of powerlines.In the wake of major windstorms during the early 2010s, the reliability standards of the electricity system in Finland were revised with the passing of the Electricity Market Act 588/2013 [19].The regulation requires distribution grids to be designed, constructed, and maintained in a way that outages from windstorms would take no longer than 36 h for rural and no longer than 6 h for urban consumers.To satisfy these reliability standards, DSOs have started major underground cabling efforts, whose economic justification was later questioned by Nurmi et al. [101].
Nurmi et al. compared the costs for cabling needed to satisfy the new reliability requirements with the direct benefits of the corresponding interruption reductions [101].They compared the cost-benefit for the subset of Finnish DSOs for expected annual interruptions in net present value terms.The base number and duration of interruptions are taken as an annual average for 2005-2014 rates with the account of expected changes for tree uprooting risk in future climate scenarios.The number and duration of interruptions due to the implementation of the new reliability standards are assumed to be none in urban areas and half of the baseline rates for rural areas.Finally, the avoided cost computation combines these interruption levels with the momentary and constant ongoing costs for the interrupted household consumer time (€ and €/h) and industry consumer energy (€/kW and €/kWh).The described costbenefit analysis concludes that new reliability standards are a net benefit for urban consumers but a net cost for the whole system.
However, cost-benefit analysis for annual interruption averages does not account for the fact that interruptions from a single major windstorm can be larger.Previously obtained LL profiles provide the means to compute costs for such windstorm cases.Note that the comparison made here between 2011 and 2020 grids also differs from Nurmi et al. comparison of grids with and without the implementation of new reliability standards, as the cabling levels of 2020 are not sufficient for these standards.Both interruption and cabling costs are in 2020 value terms.The account for value changes of money in time (i.e., discounting) is considered unnecessary without knowledge of the timing of the extreme windstorm.Thus, the following assessment presents the simplified economic comparison between the cost of overnight construction and the cost of interruption for one-week-long cases.In studied cases, the momentary interruption (i.e., shut down) costs are considered insignificant for long interruption durations.The costs for infrastructure restoration are also excluded for simplicity but are expected to be lower than the costs of disrupted supply.
The cabling costs between 2011 and 2020 are obtained by multiplying the increased length of MV cabled lines with cost per km.The cost of 59,775 €/km is taken as an average of reported values in multiple Finnish reports based on prices of cabled powerline components reported by the Energy Authority (see details in Supplementary Material S2).No account is made for the cost of maintenance and renewal for overhead lines that the buildup of cabled lines replaces.Nurmi et al. assume the need for renewal in half of the overhead lines is replaced by cabling [101], which, in addition to overhead lines being half as expensive, results in a quarter of the cabling costs being avoided.The computed total without such discount is 2.7 billion €.Fig. 7 shows the comparison of these cabling costs with interruption costs for windstorm cases introduced in the first subsection.In the historical case, the cabling saved 200 million €.For cabling to pay off, 14 such windstorms would need to occur during the cabled line lifetime of 35-50 years [109].In the unprecedented case, interruption costs for the 2011 grid are 82% of cabling costs, which reduces this share by 33%, i. e., single event savings equal half of cabling costs.Significant costs inflicted on both grids are conservative estimates due to the limitation of the one-week study period.

Qualitative economic and social impacts
While all interruption costs can be expressed in monetary terms, analysis in monetary terms does not show changes in the nature of the most important cost factors with growing interruption duration and spread.The information on such changes would provide an important check of VoLL (measurement method) applicability ranges and indicate major challenges facing consumers and emergency response organizations.Intuitively, the potential water and heat supply loss during multiple-day-long interruption seems significantly more important than a relatively brief loss of leisure time or interrupted economic activity for which VoLL assessments may be made.However, determining the most critical cost factors is difficult due to the large number and variety of potentially relevant factors.A deeper investigation of unsupplied consumption requires its breakdown.The most apparent bases for such breakdown seem to be the infrastructure and electricity consumption sectors.
Disrupted operations in critical infrastructures are likely to cause the costliest downstream implications of electricity interruptions.However, no broadly accepted definition exists for critical infrastructures.Generally, the list of such infrastructures tends to be longer in more developed countries [108].In Finland, the Security Committee (an organization consulting the Finnish government on comprehensive security questions) lists basic functions of society considered critical in the country.The basic functions include food, water, finance, transportation, telecommunication, heating, and certain public facilities [110].The critical public facilities include hospitals, emergency response centers, police departments, and fire stations.Schools and daycare centers are also considered of high importance as they enable parents to go to their work, including work to maintain critical infrastructures.Modeling the impacts of electricity interruptions in these sectors is extremely difficult.However, essential facilities and infrastructure components are expected to be supplied via cabled lines (39% of MV lines in 2020 were cabled), and thus largely unaffected even in the studied windstorm cases.This does not include electricity-powered heating and water supply, which are present in a significant share of individual homes.Such homes should have alternative means to obtain heating and water without electricity to avoid significant challenges.Further investigation of the most critical cost factors requires a more detailed sectoral consumption breakdown than a breakdown into industry, commerce, and residence sectors done so far.The most promising breakdown basis for non-critical industry and commerce subsectors appears to be their direct and indirect economic significance.Direct economic significance should be relatively easy to quantify using indicators like added value and employment.The quantification of indirect significance is complicated by the need to account for the importance of a subsector to other subsectors in the economy or population at large.The subsectors of critical importance in industry and commerce are likely to be only those directly related to the provision of critical societal functions discussed earlier, most notably food distribution.The primary value of the residential sector is the well-being of the population.This suggests that a breakdown of the population is the most suitable basis for investigating interruption costs in the residential sector.
While the population can be broken down based on a multitude of socioeconomic factors, the meaningful breakdown and associated study findings are expected to be country-specific.The most relevant study of electricity interruption impacts on Finnish consumers is done by Nikkanen et al. [111].They compared preparedness to and impacts of the 2019 winter windstorm Aapeli among different groups of people.These groups were distinguished by factors like the type of home location (urban versus rural) and building (flat versus detached house), presence of prior interruption experiences, knowledge of the Finnish language, and age.Study results show that socioeconomic factors have only a marginal influence on preparedness for and impacts from the windstorm, potentially due to Finland's relatively uniform distribution of well-being.However, such a comparison of historical windstorm impacts between social groups may not be sufficient for identifying the most vulnerable populations and their main challenges in the face of unprecedented windstorm interruptions.For example, stock of certain preparedness items in stores may run out if everyone goes to them after weather forecast-based warnings.At the same time, the capacity of authorities to respond, perceived by the Nordic public to play a prominent role [111], is likely to be significantly more limited than in typical disruptions.Preparation and later impact mitigation of communities rather than individuals could be much more significant.I.e., some of the most vulnerable individuals may have family or friends to help while even relatively well-off individuals without social capital could fall into more complicated situations.

Study limitations and applicability
The assessment of the limitations and applicability of this study is complicated by the absence of result validation and the large scope of the study.Direct validation of results is impossible for disruption cases beyond the historical severity level.Therefore, the first part of this section provides an indirect reliability check of study methods and results via sensitivity analysis for resulting LL against primary sources of impact modeling uncertainties.To address the wide study scope, the second part of this section presents the list of major considerations for study limitations, the usability of results, and the applicability of methods to other countries and other types of weather extremes across three domains roughly corresponding to the main sections of this paper.

Sensitivity of technical impacts to major uncertainties
The main uncertainties of the presented LL modeling are wind gust speed, component fragility, and component fixing time.Result sensitivity for each uncertain quantity is quantified by running the model 4 times with corresponding parameter values ±10% and ±20% around corresponding values of unprecedented windstorm in the 2020 grid case.20% deviation indicates the expected level of uncertainties (based on model generality analysis in the paper that studied the generality of the model used here [25]).10% deviation provides mid-points, allowing to check for acceleration or deceleration of LL changes from the base case.The same 20% deviation is accounted differently for each quantity depending on its relation to the parameters describing them.Table 2 presents the summary of uncertainty accounts based on the following lines of evidence for uncertainty ranges.
The fixing time uncertainty range is based on two-level fitting where most storm periods fall within ∓20% range of the second fit lines for dependency on storm severity [25].Fixing time is fitted for Weibull distribution, which has scale and shape parameters.The scale parameter closely follows average fixing time distribution values.In contrast, the increase of shape parameter decreases the fixing time distribution average in the value range studied.Therefore, to obtain maximum deviations, the multiplication of the shape parameter for sensitivity cases is done in reverse order of the multiplication of the scale parameter.
The uncertainty range for fragility is based on the level of fit between modeled and historical lost load profiles for the three most impactful and recent windstorms [25].However, the evidence of uncertainty for fragility does not lead to direct evidence of uncertainty for fragility function parameters.To maintain the chosen fragility function form, fragility function parameters were varied by trial and error until fragility deviation from the unprecedented case was reached.As the fragility function is not linear, a 10-20% deviation of interest is limited to the range of 20-35 m/s, where most faults are expected to occur.The chosen parameters and ratio of resulting fragilities with baseline fragilities are shown in Fig. 8.
Wind uncertainty for the field of unprecedented case wind gust speeds is defined by the scaling factor uncertainty, which is defined by the uncertainty of the method chosen for linking return period values with the strongest windstorm value field.However, no evidence is known to quantify this uncertainty.At the same time, a confidence interval of 95% for return period values is considered part of the scenario and does not need to be accounted for again.Thus, wind uncertainty is assumed to be the same as for other quantities, i.e., 20%.20% variation is applied to increase the portion of the scaling constant (i.e., 0.24), as 20% of the whole scaling constant would include a range with the historical windstorm itself.Such a range is considered too wide even  without direct evidence for the uncertainty of unprecedented windstorm value field generation.Fig. 9 shows sensitivity analysis results for established ranges.Across quantities, uncertainties increase, going from fixing time to fragility to the wind.Only wind uncertainty produces relatively high, i.e., >10% or 20%, deviation for LL.This means that improving the accuracy and reliability of the wind gust speed field is the most consequential area for improvement of impact assessment.The distribution of deviations shows no significant surprises except for the fixing time increase, primarily due to the limited study period.Most of the changes occur in the east and southeast of the country, indicating relatively higher sensitivity due to changes in an area exposed to fault-inducing winds rather than severity changes in the most impacted region.Sensitivities are largely decelerating, i.e., 20% deviations are less than twice as large as 10% deviations.The exception here is increasing fragility, which could be explained by the asymmetry in converting expected uncertainty for fragility into expected uncertainty for fragility function parameters, i.e., the top curve in Fig. 8 being more bent and mainly above the ratio of 1.2.Sensitivities are also largely symmetrical except for a larger increase in the western region for all parameters and a larger decrease in all other regions for fixing time.Related fixing time deviations from trends include the slight difference between +20% and + 10% cases and the significantly smaller share of the increase in the east compared to the share of increase for other quantities.All these fixing time deviations could be explained mainly by the increase in fixing times for faults that were already longer than a week and are not captured in week-long model runs.Moreover, duration limitations are concentrated in the longest faults.Fault duration differences between cases are self-evident, while duration differences between regions are determined by the marginal fixing time generation algorithm.According to this algorithm, fixing times are determined by NoF at a given hour.This results in the growth of fixing times with growing NoF when Tapani-Hannu-based unprecedented windstorm travels from west to east.

Limitations and applicability considerations
Table 3 presents a list of the main considerations about study limitations, results' usability, and methods' applicability for other countries and other types of weather extremes.
The main limitations in windstorm scaling are unaccounted aspects known to affect results.Windstorm hazard is represented with only one parameter of wind gust speed, omitting even its duration.The statistical scaling of the wind gust field omits potentially relevant meteorological phenomena.Moreover, the scaling is constant in space and time, with scaling extent based on the chosen acceptable scaling confidence level.Nevertheless, the obtained maximum speeds seem reasonable and are close to the maximum speeds recorded in the Finnish territorial waters of the Baltic Sea.
The applicability of the windstorm scaling method is expected to be high for other countries but limited for other types of weather extremes.Wind gust speed and other meteorological data are available in global databases, even though the quality of its long-term records varies across locations.The spatially constant scaling would require some regional splitting for larger or geographically more diverse countries than Finland.The spatial scaling, even with potential regionalization, is agnostic to the geometry of the windstorm itself.This agnostic nature suggests scaling to be suitable for tropical cyclones.However, the symmetry of tropical cyclones may allow a more sophisticated yet reasonable-effort account of the area with grid-damaging winds.Many other weather extremes are also expected to be dominated by single (or a small number of) factors and be statistically derivable (i.e., no need for modeling meteorological phenomena), given sufficient historical data.On the other hand, non-wind meteorological extremes do not seem to necessitate an account of temporal dimension, which is the main methodological contribution of this work.That is, no need for method to generate field of extreme values over existing methods for producing spatial extreme profiles.
The impact modeling framework is primarily limited by the account of the natural environment, available grid data, and reliance on historical impact data.The modeling representation of the natural environment is limited by forest and lake area shares, omitting other topographic, soil, and vegetation aspects.The account of physical environment aspects is often limited among fragility-based impact models.Thus, an environment account is likely to benefit from knowledge obtained from or complementary use of statistical machinelearning-based models that include hundreds of environmental variables (e.g., [87]).The impact model is limited to DG as it is responsible for the most wind-related disruptions, though this may need reevaluation for unprecedented windstorms.To circumvent the absence of data for DG, country-wide synthetic DGs are generated utilizing DSO-specific data of numerous relatively small DSOs in Finland.Similarly, available grid impact data is necessary for deriving fragility functions and severity-dependent fix time distributions.The statistical methods used in these derivations inevitably omit mechanics on how faults and repair work occur.Fix time allocation algorithm itself depends on a nationwide number of faults at the given hour, which introduces slight windstorm direction bias.The following LL evaluation also requires extending the study period (doable within the current framework) as the currently used duration at its end leaves significant LL in unprecedented disruption.As mentioned, LL profiles were reproduced for the three most impactful and recent windstorms with errors of around 20% in the previous publication [25].Also, the sensitivity analysis shown in the previous subsection indicates uncertainties for the unprecedented windstorm to be of similar magnitude.Both historical and unprecedented windstorm modeling results were analyzed distinguishing five regions in the country according to the interruption data available.The analyzed performance indicates that the model can generate relatively reliable results for considerations of strategic national scale questions.The model has higher granularity (309 Finnish municipalities), but local variations are likely to produce additional uncertainties that cancel out in higher aggregation levels.The additional assessment is required for impact model reliability at lower aggregation levels, which would be indicative of model suitability for more local grid development considerations.
The grid impact modeling framework is also highly applicable to other countries but not so much to other types of weather extremes.The main limitation of the impact model use in other countries is a grid and its impact data of the unconsolidated DSO sector, which is present in many but not all European countries [112].For countries with such data, one should also reassess the account of inaccessible areas (e.g., lakes and mountains) and forests, especially if falling trees onto overhead powerlines is not a dominant fault mechanism.More detailed considerations for applicability to other countries are discussed in the dedicated section of the original paper presenting the modeling framework [16].While wind impacts grid in all countries, other weather hazards impact different parts of the energy system and its surrounding environment.E.g., drought and flood modeling concern more powerplants than powerlines.Furthermore, fragility functions are used to model only mechanical damage.Severity-dependent fixing time representation is expected to be more general.That said, obtaining a fixing time distribution requires significantly more historical data if severity dependence is to be obtained.
The qualitative impact cost assessment is limited by timewise constant VoLLs, available only for a few sectors and obtained considering moderate disruptions.In other words, there is no account of VoLL dependence on disruption severity, i.e., the initial claim motivating the study.The illustrative comparison with cabling measures taken is further limited by simplistic calculation of cabling costs even though it is separate from the focus of this work, i.e., windstorm impact assessment.Despite mentioned limitations, the cost assessment already provides a supplementary perspective for optimal security investments, which are typical assessed considering average annual disruption levels.
The VoLLs and underlying vulnerabilities in cost assessment are country-specific, but the bases for their breakdown are neither country nor disruption-type specific.Finland has aspects (e.g., relatively uniform wealth distribution) that are unlike to apply elsewhere.At the same time, there seems to be a universal need for further breakdown by economic sectors and population segments and the need for VoLL time dependency over a longer duration.It should also be noted that disruption type may affect costs if it simultaneously disrupts other critical infrastructures (e.g., roads) and services (e.g., telecommunication).
More generally, assessment of any type of disruption is expected to require focus on the dominant fault mechanism and reassessment of relevant factors and suitable methods along impact chain steps as done in this work for windstorms.

Conclusions
The key results of this work are given in the following: -The wind gust speed field for the unprecedented but plausible windstorm, which is the scaled-up field of the historically most impactful windstorm by a factor of 1.24.The unprecedented windstorm represents the strongest windstorm expected to occur once in 100 years at the upper limit of 95% confidence interval in extreme-valuetheory (EVT) -based extrapolation of forty-three years of historical data; -The spatiotemporal lost load (LL) profile resulting from the unprecedented windstorm impact on the Finnish electricity grid of 2020.Profile peaks at 45% of vulnerable consumption (roughly a quarter of the national total) and decreases to 32% of the historical LL peak (reached in the 2011 windstorm) at the end of the week.Cumulative LL is ten times higher than the equivalent LL from the historical windstorm hitting the same grid; -The preliminary costs of electricity supply interruption from unprecedented windstorm.The decline of interruption costs due to major cabling done between 2011 and 2020 covers at least half of associated underground powerline cabling costs.
These results present the unprecedented windstorm as an order of magnitude larger disruption to the Finnish electricity system than any historical disruption.Such severity makes the obtained spatiotemporal LL profile an interesting basis for reevaluation of dominant cost factors or emergency preparedness exercises.This remains true even after accounting for a large share of cabled medium voltage (MV) lines in the country's most densely populated southwest parts that reduce windstorm impact multiple times.Additionally, the evidence for the windstorm hazards in a changing climate indicates the potential increase of lower magnitude than in the studied case.In case even a single unprecedented windstorm storm does occur, the benefits of underground cabling undertaken in Finland are large enough to justify a significant part of the cabling costs.Even an unprecedented windstorm that is less severe than the studied one could change the cost-benefit calculus for cabling and thus appear to be important to consider in addition to traditional assessments based on historical and expected supply reliability levels.The discussed bases for cost breakdown by critical societal functions, economic subsectors, and population groups present further support for studying the most crucial cost factors and countermeasures.Arguably, some educated guesses can already be made on the centrality of repair capacity and mechanisms to enable help from neighboring household electricity consumers.That said, additional work is needed to identify and quantify such factors and their implications for the development of the electricity sector, critical services reliant on uninterrupted electricity supply, and broader emergency response capabilities.
The two main directions of future work are to improve the reliability of obtained results and to expand result applicability with a more detailed study of cost factors.The reliability of the results for the main factors accounted for in LL profile modeling appears relatively high as uncertainties from sensitivity analysis are significantly lower than the obtained LL increase.Among the analyzed aspects, wind presents the highest level of uncertainty, which is further exaggerated by the scenario defining the basis of the unprecedented windstorm as the upper limit of a 95% confidence interval in the EVT-based scaling method.Thus, the reliability of the wind gust speed field would significantly benefit from generation by the meteorologically based model over the current statistical method.A significantly simpler but potentially interesting addition for the wind variable would be to account for climate change with the current windstorm scaling method by taking data from climate scenarios as input.The results of grid damage may also significantly depend on and thus be worth considering the vulnerability of harder grid components, namely high voltage overhead powerlines and high to medium voltage substations.Major improvements for the representation of grid repairs include a longer study period, an analytical instead of statistical account method for dependence on windstorm severity, and an account of repair capacity available (namely, the available number of repair crews and stocks of spare parts).The duration of disruption is important given its expected high influence on value of LL (VoLL), which is also yet to be accounted for.The reliable and duration-dependent estimate of disruption cost for the unprecedented windstorm can already provide significant value in electricity sector development such as a different perspective for assessing the optimal level of investment in security measures.However, the applicability of results would greatly benefit from the individual reevaluation of: impacts for each critical societal service, major economic subsector and population group; existing system development and broader emergency response plans; the role of government and civil actors at a time of disruption that is beyond the capacity of public emergency response organizations.
The experience from this work seems to indicate major challenges for the mentioned future work options and studies of unprecedented disruption for different types of threats to energy systems.The assessment of costs includes a long chain of impacts across different systems, which requires the accounting of different fields of knowledge.Moreover, such cost assessment must overcome the challenge of data and existing methods being available primarily or exclusively for historical disruption severity range.In other words, an unprecedented disruption study requires a rethinking of the problem scope and suitable methods at each impact assessment step and, at some steps, pursuing indirect lines of evidence or simplifications.Without rethinking the assessment of unprecedented disruption impacts is unlikely to show implications of most relevance.This paper's extensive documentation of windstorm destructiveness factors, their scaling methods, and plausible limits represent a significant part of such rethinking.Despite inevitably limited scaling scope, the extensive documentation aids the replicability of the reasoning behind scaling.Extensive documentation also supports subsequent work with alternative approaches (e.g., scaling windstorm by storm severity index) or potential expansions (e.g., the account of second storm impact driving factor).

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 2 .
Fig. 2. Wind gust speeds of fifty-and hundred-year return periods in Finland.Upper limit values are for 95% confidence interval.

Fig. 4 .
Fig. 4. Ratios of fifty-and hundred-year return periods and 1979-2021 historical maximums of wind gust speeds in Finland.Upper limit values are for the 95% confidence interval.Values below the subtitles show national averages.

Fig. 6 .
Fig. 6.Spatiotemporal disruption profiles for the largest historical and historically unprecedented windstorms in the Finnish grid of 2011 and 2020.

Fig. 9 .
Fig. 9. Sensitivity analysis of primary sources of uncertainty for the LL profile derivation.Values show deviation from the base case of the historically unprecedented windstorm.

Table 3
List of considerations about study limitations and applicability.