A modeling and simulation framework for the reliability/availability assessment of a power transmission grid subject to cascading failures under extreme weather conditions Francesco Cadini, Gian Agliardi, Enrico Zio
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Francesco Cadini, Gian Agliardi, Enrico Zio. A modeling and simulation framework for the reliability/availability assessment of a power transmission grid subject to cascading failures un- der extreme weather conditions. Applied Energy, Elsevier, 2017, 185, Part 1, pp.267-279. 10.1016/j.apenergy.2016.10.086. hal-01652288
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A modeling and simulation framework for the reliability/availability assessment of a power transmission grid subject to cascading failures under extreme weather conditions ⇑ Francesco Cadini a, , Gian Luca Agliardi a, Enrico Zio a,b a Dipartimento di Energia, Politecnico di Milano, Via La Masa 34, 20156, Italy b Chair System Science and the Energy Challenge, Fondation Electricite’ de France (EDF), CentraleSupélec, Université Paris-Saclay, Grande Voie des Vignes, 92290 Chatenay-Malabry, France highlights
We combine stochastic extreme weather model and realistic power grid fault dynamics. We introduce a novel restoration model accounting for the weather conditions. The complete model is quantified by means of a customized sequential Monte Carlo. Reliability/availability indicators show importance of rare extreme weather events. article info abstract
Article history: Electrical power transmission networks of many developed countries are undergoing deep transforma- Received 20 January 2016 tions aimed at (i) facing the challenge offered by the stochastically fluctuating power contributions Received in revised form 15 September due to the continuously growing connections of renewable power generating units and (ii) decreasing 2016 their vulnerability to failures or malicious attacks and improving their resilience, in order to provide more Accepted 24 October 2016 reliable services, thus increasing both safety and profits. In this complex context, one of the major con- Available online 7 November 2016 cerns is that related to the potentially catastrophic consequences of cascading failures triggered by rare and difficult to predict extreme weather events. In this work, we originally propose to combine an Keywords: extreme weather stochastic model of literature to a realistic cascading failure simulator based on a direct Power grids Cascading failures current (DC) power flow approximation and a proportional re-dispatch strategy. The description of the Extreme weather dynamics of the network is completed by the introduction of a novel restoration model accounting for Wind storm the operating conditions that a repair crew may encounter during an extreme weather event. The result- Lightning ing model is solved by a customized sequential Monte Carlo scheme in order to quantify the impact of Monte Carlo simulation extreme weather events on the reliability/availability performances of the power grid. The approach is Reliability demonstrated with reference to the test case of the IEEE14 power transmission network. Availability Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction and the study of strategies for the mitigation of the consequences have gained increased attention. However, these tasks are difficult Electric energy has an impact on all modern life activities. because of the inherent complexities of the power transmission Consequently, the public expectation for reliable electric energy networks, the complex physical laws governing the power flow delivery has significantly grown in the past decades. For this, the dynamics and the uncertainties affecting the boundary conditions so-called power grids, i.e., the interconnected systems that trans- (operational and environmental) in which they operate. Additional mit electric power from where it is generated to where it is con- difficulties come from the evolution of society and energy markets, sumed, play a fundamental role. Thus, it comes with no surprise that demand changes and adaptations of the large-scale power that the assessment of the risk of failures of these infrastructures transmission infrastructures for continuing to satisfy the require- ments of reliability/availability under the continuous increase in ⇑ Corresponding author. electric power demand, avoiding congestions and overloads, so as E-mail address: [email protected] (F. Cadini). to improve safety and increase economic margins. In addition, http://dx.doi.org/10.1016/j.apenergy.2016.10.086 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved. 268 F. Cadini et al. / Applied Energy 185 (2017) 267–279 power transmission systems can no longer be considered solely as network performances in presence of failures triggered by the a means for energy delivery; rather, they are operated as an elec- extreme weather events. tricity market trading platform for shifting power volumes across With regards to extreme weather event modeling for power different regions or even different countries, for economic reasons. grids reliability/availability estimation, the common approach is Furthermore, in recent years we have witnessed an exponential that of resorting to the ‘‘two-state” model [9], or to its improved, growth in the connection of renewable power generating units ‘‘three-state” version [10], where the weather conditions are (photovoltaic power plants, wind farms, biomass plants, etc.) and divided into two or three classes, normal, adverse and, in case of poly-generation systems (electrical power, heat and cooling), three state models, major disaster, and the stochastic transitions which add even more complexity to the system. In fact, such con- among the different classes are modeled as homogeneous Poisson nections were obviously not planned in the initial design of the processes with constant transition rates, to be estimated from power networks, which now need to adapt to accommodate their weather data. fluctuating power contributions, difficult to predict and geograph- The major drawbacks of these approaches are that (i) they are ically distributed [1]. not flexible enough to represent the wide range of possible In this new setting, concerns come for the protection from fail- weather events types and intensity levels and (ii) their rates of ures and attacks, and their possible cascading effects, i.e., the prop- occurrence are, in general, dependent on time, due to their typi- agation of disconnections of components from the grid due to the cally seasonal behaviors. Significant advances towards a more real- power flows redistribution (manually or automatically) following istic modeling of the impact of weather events on power networks the initial failure or attack. Cascading failures may lead to catas- reliability/availability is proposed in [11] and further extended in trophic disconnections of wide portions of the power transmission [16]. In those works, non-homogeneous Poisson distributions of networks, as epitomized by several large-scale blackouts occurred the occurrence times and suitable distributions of the events inten- in developed countries in the recent years, such as for example the sities are derived on the basis of available weather statistics; fur- blackout in Italy on September, 28 2003 [2], that in the United thermore, the failure rates of the networks’ lines are States and Canada on August 14, 2003 [3] or that in Europe on mathematically related to the times of occurrence and the intensi- November, 6 2006 [4]. ties of the weather events considered, following the approaches Normally, a power transmission network is designed and oper- proposed in previous works of literature. In fact, it is recognized ated so that a single component disconnection does not give rise to that the failure rates of the grid’s components, typically taken con- any load shedding, so that cascading failures do not occur (N 1 stant, are actually significantly increased by the occurrence of sev- criterion [2]); however, rare combinations of circumstances, ere weather events. For example, quadratic and exponential uncommon events or inadequate countermeasures may result in models are introduced in [12] and [13], respectively, to describe further line disconnections, eventually leading to disruptive conse- the overhead line failure rates as functions of the wind speed; a lin- quences. In fact, the operative conditions at a given time are deter- ear model is presented in [14] to model the relationship between mined by many, usually correlated factors, such as power demand, the number of line interruptions and that of lightning flashes; generating capacity, economic optimizations, technical limits and the combined effect of high winds and lightning, possibly occurring environmental conditions. Some of these factors are defined as during severe storms, on the grid components failure rates are ana- operative limits or determined by technical-economical optimiza- lyzed in [15]. The resulting model is then solved by means of a tions (e.g., the power generation) and, as such, they are more pre- sequential Monte Carlo scheme simulating the annual ‘‘lives” of dictable. Others, such as the power demand and the environmental the power grid subject to the weather events. conditions, are uncertain by nature and many efforts are, thus, The approaches illustrated above, however, have been only devoted to quantify the effects of these uncertainties on the power applied either to very simple, representative power grids with production and transportation capabilities, like, for example in [5] few components [10] or to power distribution grids (e.g., a the and [6], where the fluctuations of the power demand and of a Swedish rural reliability test system in [16] or an actual distribu- specific environmental feature, i.e. the wind speed, are taken into tion system in the Northeast US in [17]). Limited research efforts account, respectively. have been made in developing ways for performing weather- Furthermore, human errors, intentional attacks or extremely related reliability/availability analyses on power transmission net- severe natural contingencies, such as earthquakes, volcanic erup- work models including realistic representations of cascading fail- tions and weather related events (floodings, wind storms, light- ure phenomena. For example, in [18] simple stochastic models ning, icing, heat and cold waves, etc.), may even directly fail the representing the occurrences of extreme weather events are cou- components of the network [7]. pled to grid cascading failure models, where the network reliability In this context, we address the problem of quantifying the reli- is abstractly quantified with respect to the existence of connecting ability/availability of a transmission grid specifically with respect paths between grid nodes; however, the realistic electrical behav- to the occurrence of extreme weather events, which have been ior of the networks are not taken into account. identified as major threats to the operation of power networks In this work, we originally propose to resort to the stochastic [8]. Indeed, the decision making process of the grid designers model for representing the uncertainties related to extreme and operators should be driven by the results of cost-benefit anal- weather events introduced in [16] and further developed in [11], yses, accounting for the reliability/availability of the system and and to combine it to a realistic cascading failure simulation model the costs required to design, build and operate it. Proper assess- based on a MATLAB-based DC power flow approximation of a ment of the transmission grid reliability/availability allows power transmission network [19] and a proportional re-dispatch informed decisions on grid design and operation by the players strategy [20]. Furthermore, in order to offer an even more realistic involved in the utility management upon. Also, reliability/avail- description of the power grid dynamics in response to extreme ability analysis is a key factor for supporting the power quality reg- weather events, we develop an original restoration model based ulation of transmission grids in many developed countries. on the definition of suitable repair ‘‘velocities” correlated to the The problem addressed is twofold: first, a realistic, possibly occurrence of the different events. stochastic, model of the extreme weather event occurrences must The complexity of the resulting model is such that its resolution be developed and its parameters properly identified on the basis of for the estimation of the system reliability/availability indices is available weather statistics; then, this model is coupled with real- not easy. We, then, develop a customized, sequential Monte Carlo istic transmission grid dynamic models for the computation of the simulation scheme, where several annual histories of power F. Cadini et al. / Applied Energy 185 (2017) 267–279 269
transmission network lives, subject to both normal and extreme senting both the occurrence time ti and the network line affected weather events-related failures, are simulated and the conse- by the failure li; i ¼ 1; ...; NL, with NL being the total number of quences are evaluated. The proposed modeling and simulation lines of the transmission network under consideration. The framework is applied for quantifying the impact of extreme sequence of failure occurrences is determined, in turn, by the com- weather events on a power transmission network of reference, bination of a ‘‘normal” line failure model, representing, for exam- the IEEE14 [21]. ple, failures due to aging, malfunctioning, human errors and The rest of the paper is structured as follows. Section 2 attacks in general, and the ‘‘extreme weather” failure stochastic describes the proposed framework for coupling the extreme model, where the line failures are related to the random occur- weather event model to the cascading failure simulation model. rence of extreme weather events, represented by the sequence
In particular, Section 2.1 describes the line failure mechanism E ¼fe1; e2; ...; emg, where m is the total number of events over ; ; ; ; ...; and its stochastic representation. s 2.2 and 2.3 illustrate the the given time horizon (one year) and ej ¼ftj pj Ijg j ¼ 1 m cascade model and the restoration strategy, respectively, while is a sequence of elements representing the occurrence time tj, the Sequential Monte Carlo procedure is presented in Section 2.4. the type of weather event pj and its intensity Ij. Similarly to [16] The IEEE14 test network specifications and the results are and [29], the severe weather events considered in this work are described in Section 3. In Section 4, some conclusions are drawn. exceedingly high winds due to storms and lightning, under the assumption that the entire network experiences the same weather conditions. 2. The model An original deterministic restoration model is, then, coupled to the cascading failure model, so that, when a cascading failure In this Section, we illustrate the framework for stochastic extinguishes, the line initially failed is repaired and the network modeling of the cascading failure behavior of a transmission power is assumed to re-start from ‘‘as good as new” conditions. As it will network under uncertain weather conditions. In cascading failure be shown in Section 2.3, the restoration process is also affected by analyses, the power transmission network is typically modeled as the stochastic weather conditions. a graph G ¼ðV; EÞ, where V represents the set of vertices and E Note that, since the input F is a stochastic, multivariate vari- the set of edges. Usually, it is assumed that only elements belong- y able described by its probability density function (pdf) dðF Þ, the ing to one of the sets V or E are subjected to failure and, corre- y outputs of the deterministic cascading failure model, i.e., the relia- spondingly, the analyses are carried on under the assumptions of, bility/availability indicators EENS, ALS, ART and AFF, are also respectively, node removals or edge removals (examples of these two different approaches can be found in [22–24] and [20,25– uncertain and described by pdfs dðEENSÞ, dðALSÞ, dðARTÞ and 27], respectively). In this work, we restrict our attention only to dðAFFÞ, respectively. edge removals, since power transmission line failures are more common than bus failures [28]. 2.1. ‘‘Normal” and extreme-weather stochastic models of failure The proposed framework aims at modeling the effects of ran- dom line failures due to both normal operation and extreme The random occurrence of ‘‘normal” failures is modeled as a weather conditions in a power transmission network, accounting homogeneous Poisson process, as proposed in [16]: also for the subsequent restoration processes. The uncertainty in k ½knorm t k the occurrence of the failure events affects the network system PðN ðtÞ¼kÞ¼ e norm t; k ¼ 0; 1; 2 ... ð1Þ f k! reliability/availability in terms of the expected energy not served
(EENS [GWh/y]), the average load shed due to failure events where Nf ðtÞ [occ/km] is the number of failures per kilometer of grid
(ALS [MW/occ]), the average frequency of the power interruptions line occurring in the time period (0,t], knorm [occ/km/h], assumed to (AFF [occ/y]) and their average duration (ART [h/y or h/occ]), which be constant throughout one year, is the failure rate under normal will be defined in Section 3. weather conditions per kilometer of grid line and PðNf ðtÞ¼kÞ is The scheme of the conceptual model adopted is shown in Fig. 1. the probability that k failures occur in the period (0,t], per kilometer The deterministic cascading failure model allows the computation of grid line. of the consequences of the failure of an individual line of a given On the other hand, the random occurrence of extreme weather network. The input to this model is the multivariate distribution event-related failures [16] is modeled as a non-homogeneous Pois- of the sequence of initial line failure occurrences, son process with a parameter V eðtÞ dependent on the stochastic F ; ; ...; y ¼ff 1 f 2 f ng, where n is the total number of line failure weather conditions during the year. The weather conditions are events over a given time horizon (assumed to be one year in this determined by the stochastic sequence of weather events ; ; ; ...; E ; ; ...; work), with each sequence element f i ¼fti lig i ¼ 1 n, repre- y ¼fe1 e2 emg. Similarly to [16], we model the occurrence of
Fig. 1. Conceptual scheme of the proposed stochastic framework for the estimation of power grid reliability/availability indices. 270 F. Cadini et al. / Applied Energy 185 (2017) 267–279
two types of events, i.e. exceedingly high winds and lightning, the random variable wwind [m/s], i.e. the wind intensity conditional assuming they are independent. Wind storms are assumed to give on the occurrence of a wind storm event, can be written as: rise to the falling of trees not properly trimmed and to damage w ¼ w þ Dw ð4Þ structures like cables and poles, whereas lightning may directly wind crit damage the electrical equipment. where wcrit is fixed and Dw is the random variable assumed to be The occurrence of the weather events is modeled by a non- described by the Weibull distribution with the parameters reported homogeneous Poisson Process (NHPP) [1]: in Table 1. The durations of the wind storms (Dwind) events are k assumed to follow a Weibull distribution with the parameters ½V ðtÞ PðN ðtÞ¼kÞ¼ p e VðtÞ; k ¼ 0; 1; 2 ... ð2Þ reported in Table 1. p k! On the other hand, the lightning intensity level is quantified where NpðtÞ [occ] is the number of events of type p occurring in the through the number of flashes that hit the ground per square- 2 time period (0,t], PðNpðtÞ¼kÞ is the probability that k events of type kilometer and hour Ng ðtÞ [occ/h/km ], which is assumed to be a p occur in the period (0,t] and V pðtÞ [occ] is the average number of random variable distributed according to the Weibull distribution occurrences of event p over the time period (0,t]. The time- with the parameters reported in Table 1 [16]. As opposed to the dependent parameter V pðtÞ can be expressed as: high wind events, lightning gives rise to line failures regardless Z t its ground flash density, so that no critical threshold is considered. V pðtÞ¼ vpðt Þdt ð3Þ The durations of the lightning events (Dlight) are assumed to follow 0 a lognormal distribution with the parameters reported in Table 1. where v pðtÞ [occ/h] is the rate of occurrence of event p (wind storms The number of line failure occurrences due to each type of event or lightning), taken to be stepwise constant on a monthly basis in is assumed to be distributed according to a NHPP: order to account for the seasonality of the weather events. Fig. 2, R k t R k t taken from [16], shows the value of the rates of occurrence of wind ð 0 f ðt Þdt Þ k ðt Þdt PðN ðtÞ¼kÞ¼ e 0 f ; k ¼ 0; 1; 2 ... ð5Þ storms and lightning as a function of time, used in this work. f k! According to [16], the identification of the parameters used in the where N ðtÞ is the number of line failures per kilometer occurring in stochastic models described above is based on the weather observa- f the time period (0,t], k ðtÞ [occ/km/h] is the failure rate per kilome- tions collected in the Swedenergy [30] and the SMHI (i.e. the mete- f ter at time t and PðN ðtÞ¼kÞ is the probability that k failures occur orological agency of Sweden) databases [31] and referring to two f in the period (0,t] per kilometer. locations in the Swedish East and West coasts. Failure and weather According to [16], since the failure processes are due to inde- statistics were gathered separately and, then, connected into a data- pendent, different physical phenomena, then the total failure rate base, obtaining the weather conditions to which the grid was kðtÞ can be written as: exposed when a line failed.
The occurrences of each type of weather event are, then, charac- kðtÞ¼knorm þ kwindðwwindðtÞÞ þ klightðNgðtÞÞ ð6Þ terized by stochastic intensity levels and durations [16]. To repre- k sent the intensity level of a wind event, the maximum mean wind where windðwwindðtÞÞ represents the line failure rate per kilometer k speed over a 10-min period measured during a three-hour period is due to wind storms at time t and lightðNg ðtÞÞ is the line failure rate used. The lightning intensity is quantified through the number of cloud-to-ground flashes per kilometer and hour that affects the Table 1 service areas. Parameters of the pdfs characterizing the extreme weather model [16]. According to [16], a wind storm event is considered to occur each time the wind speed (intensity) overcomes a critical threshold Distribution Scale, a Shape, b Wind storm duration (D Þ Weibull 9.89 1.17 wcrit : in fact, high winds give rise to failures only when the friction wind D force acting on trees and poles overcome a critical value. Here, sim- Wind storm intensity ( wÞ Weibull 1.23 1.05 Lightning duration (D Þ Weibull 0.96 0.85 ilarly to [16], we take w ¼ 8 [m/s], since it is observed that, light crit l r above this value, the number of failures due to wind rapidly Lightning intensity (N ðtÞÞ Lognormal 5.34 1.07 increases with wind speed. Thus, operatively, it is assumed that g
Fig. 2. Rates of occurrence for wind storms and lightning events estimated from weather data [16]. F. Cadini et al. / Applied Energy 185 (2017) 267–279 271
per kilometer due to lightning at time t. The failure rates kwind and 1.0 per unit in magnitude, (ii) transmission line resistance is klight depend on the occurrence of the weather events as follows. neglected and voltage angles differences are assumed to be small such that [34]: A. Wind storm failure rate: The effects of wind speed are (i) increased pressure over trees, possibly falling on overhead 8k; j 2 V sinðhk hjÞ hk hj ð10Þ lines and/or (ii) direct friction on poles and lines of the sys- where h is the voltage angle at bus k. Let b be the n-dimensional tem, possibly making them fall or touch external bodies. k vector of voltage angles, where a component is chosen to be equal According to [16] and to the previous work in [12], a model proportional to the square of wind velocity is used, resem- to 0 (reference bus), and bkj is the susceptance of the line joining Br bling the equation for the drag force acting on an object in node k to node j. The n n matrix at the r-th iteration is defined a fluid: by: ! X 2 r r wwindðtÞ B ¼ b k ðw ðtÞÞ ¼ a 1 k ð7Þ kk bus j connected to bus i kj wind wind 2 norm ð11Þ wcrit r Bkj ¼ bkj where a is a positive scaling parameter. When either there are The DC load flow equations are: no wind events or the wind speed w is below wcrit, then r r r kwind ¼ 0. W ¼ B h ð12Þ B. Lightning failure rate: Lightning directly hits the subcompo- Wr nents of the network, causing their failures. Similarly to where is the vector of the node power outputs, which sum to [16] and to the previous work in [14], a linear relationship zero due to the power balance constraint between power genera- Br is used between the failure rate due to lightning tion and demand within the network. The matrix has rank N 1; removing the row corresponding to the reference bus and klightðNg ðtÞÞ and the ground flash density Ng ðtÞ: taking the inverse, one obtains: klightðNg ðtÞÞ ¼ ðbNgðtÞÞknorm ð8Þ hr ¼ XrWr ð13Þ where b is a positive scaling parameter. When there are no
lightning events, then klight ¼ 0: Finally, the power flow in line li connecting bus k and j is com- Note that a rare, but possible, event is the simultaneous occur- puted as: rence of a light storm and a high wind event. According to (6) the r r r r P ¼ b ðh h Þð14Þ total failure rate kðtÞ is given by the summation of all the three li kj k j contributions. For further details on the DC power flow approximation, the The specific failure rate of the generic i-th line of length interested reader may refer to [34]. li; i ¼ 1; ...; NL is, then: During the propagation of the cascading failure, the grid could become separated in multiple connected components, also called kiðtÞ¼kðtÞ li ð9Þ islands [33]. The newly formed islands may not satisfy the power balance, thus not allowing further DC computation. A simple re- 2.2. Cascading failure model dispatch routine is embedded in the model to overcome this issue,
as follows. Considering the island I G, where ðGenI; LoadIÞ are the In this paper, we resort to the cascading failure model devel- sets of generator and load buses in I, at each iteration r three differ- oped in [33] and applied also in [32]. The model simulates the evo- ent unbalanced situations could occur: lution of cascading failures, triggered by an initial set of disconnected lines Ldisc, relying on the direct current (DC) power 1. The power supply does not meet the demand but the generators flow approximation [20]. Operatively, the disconnected set Ldisc have enough reserve to accommodate the surplus: includes both the line damaged by the failure f i and, possibly, other X X X X r 1 0 0 unavailable lines at time ti due to restoration. Starting from the ini- < < PGk PDk and PDk PMAXk ð15Þ 0 tial grid G ¼ G, the general cascade algorithm reads [33]: k2GenI k2LoadI k2LoadI k2GenI Then, the power production of each generator belonging to I is 1 ; P1 1 1. G ¼ðV M fLdiscgÞ. Compute vector of power flow in G . increased proportionally: 2. If required, adjust load and generation (re-dispatch). r r : r r 1 3. Stage r. Compute P power flow vector in G . 8k 2 GenI PGk ¼ PGk 4. Set Or, i.e., the set of lines overloaded (and disconnected) at r 1 X X PMAXk PGk 0 r 1 þ P P PDk PGk stage r. r 1 LoadI LoadI k2Gen PMAXk k2Gen PGk 5. If jOrj P 1, set Grþ1 ¼ðV; Mr OrÞ and go to step 2, otherwise, I I 16 end the cascade. ð Þ
: r 0 The stage r corresponds to a line disconnection event in the 8j 2 LoadI PDj ¼ PDj ð17Þ cascading failure simulation, which can result in lines disconnec- 2. The power supply does not meet the demand and does not have tion jOrj P 1 or not jOrj¼0. A re-dispatch action (step 3) is a enough reserve to meet the surplus: consequence of the islanding process, which occurs when an X X X X entire portion of the network G becomes disconnected from the r 1 < 0 0 > PGk PDk and PDk PMAXk ð18Þ grid, thus experiencing an excess either in the power supply or k2GenI k2LoadI k2LoadI k2GenI in the power demand. The vector Pr is computed throughout the cascade by means of the DC power flow model, a linear Then, the power demand of each load belonging to I is approximation of the AC power flow model, which relies on the decreased proportionally and the power supply is fixed to the max- following assumptions: (i) all bus voltage phasors are assumed imum limit: 272 F. Cadini et al. / Applied Energy 185 (2017) 267–279 P PMAXk (iii) the duration of the cascade can be neglected with respect to 8k 2 Load : PDr ¼ PD0 Pk2GenI ð19Þ I k k 0 the times of repairs. k2Load PDk I The variability of the weather conditions introduces uncertain- ties in the time required for the repair, as adverse environmental 8j 2 Gen : PGr ¼ PMAX ð20Þ I j j conditions usually slow down or even prevent the crew from inter- Meanwhile, if in the island I the power supply exceeds the vening. The restoration time is the overall time needed to reacti- demand, i.e.: vate the line, i.e. the total time required for the arrival of the X X crew, for the repair itself, and for switching back and reactivating r 1 > 0 PGk PDk ð21Þ all the disconnected components until the initial steady state is k2GenI k2LoadI reached again. In case of multiple failures or sequential attacks to the networks, it is assumed that multiple crews can be dispatched then, the power production of each generator belonging to I is to repair the failed lines. decreased proportionally: The dependence of the repair times on the weather conditions is P v PD0 modeled by means of a restoration velocity rest, defined as the r r 1 k2LoadI k 8k 2 GenI : PG ¼ PG P ð22Þ ‘‘portion” of the total required repair effort carried on in one hour. k k PGr 1 k2GenI k For simplicity, but with no loss of generality, we assume that the failed lines always require the same amount of repair effort to be : r 0 8j 2 LoadI PDj ¼ PDj ð23Þ fully restored, regardless their type or length. The restoration velocity, on the other hand, depends on the severity of the natural 0 In all the above equations, PDj denotes the power demand of disturbances under which the repair is carried on. Under normal node j before the initial failure. circumstances, the restoration velocity for overhead-uninsulated
The major drawback of this strategy is related to the fact that it lines is assumed to be fixed, and equal to vnorm ¼ 20 [%/h] or, in does not take into account the capacities of the surviving lines in other words, the undisturbed repair time is always trep;norm ¼ 5 the system, in order to more effectively re-dispatch the power [h], in line with what suggested in [37]. Given a 1-year sequence flows. For example, in [20] the power supply and demand is of weather events and line failures, if the repairs take place during adjusted by means of a linear optimization, whereas in [35] a con- a severe weather occurrence, then, a new, lower restoration veloc- troller proportionally sheds load or increases generation depend- ity vrest is assumed for the repair of the failed line: ing on the situation. However, with this strategy, the simulation 8 v norm ; > of cascading failures are much faster, thus allowing to demonstrate > e for wðtÞ wcrit and Ng ¼ 0 <1þ ðwwindðtÞ wcrit Þ v the feasibility of the statistical analyses and the optimizations pre- v norm ; restðtÞ¼ 1þuN for wðtÞhwcrit and Ng i0 sented in this paper at reasonable computational efforts. > g : vnorm e u ; for wðtÞ > wcrit and Ng > 0 At step 4 of the procedure for simulating cascading failures, an ½1þ ðwwindðtÞ wcrit Þ þ½1þ Ng outage model is required to identify the transmission lines which ð26Þ fail during iteration r. To this aim, we exploit the concept of effec- e u tive power flow [33,36], where the transients of the power flows xli where and are parameters with positive values. As the intensity in the network lines after a disconnection event are modeled as: of the severe weather events grows, the restoration velocity decreases, increasing the duration of the outage and the interrup- r r xl ðt þ 1Þ¼ð1 aÞxl ðtÞþaP ; l 2 E ð24Þ i i li tion in the power delivery. During a given realization of a sequence of weather events and failures, when the weather conditions go r 1 v x ð0Þ¼H ð25Þ back to normality, the restoration velocity is again set to norm. li li For example, if a line failure occurs during a high wind event and where the parameter a 2ð0; 1 is called thermal inertia of the grid, the repair is not finished before the weather event ends, then the r r P is the i-th component of the power flow vector P , t is a discrete repair action continues with the normal restoration velocity vnorm; li r 1 on the contrary, if an outage occurs under normal circumstances time index and H is the power flowing through line li at the r 1 li and a natural disturbance takes place before the restoration is com- -th failure event. The grid inertia a is a parameter introduced in pleted, the restoration of the line continues with a lower velocity. order to model the ‘‘reactiveness” or ‘‘memory” of the grid with In the proposed model, we further assume that, while a line of respect to any changes: if a ¼ 1, the system is memoryless and r the grid is under repair, the other components of the network are xl ðt þ 1Þ¼P , thus implying an instantaneous power flow change i li still subjected to failure, in agreement with the ‘‘failure bunching a < after the r-th failure event, whereas 1 implies a transient phase due to adverse weather” concept illustrated in [10]. In case this which becomes longer as a approaches 0 [33].Ifx ðtÞ is larger than li happens, the network is in a weaker configuration and the failure line capacity, then line li fails and the propagation of the failure cas- of another component has, in general, even more severe conse- cade continues. If multiple lines are overloaded, the line undergoing quences, also depending on whether it affects the disconnected the fastest transient is disconnected; otherwise, if no line is over- portion of the grid or the still operating part. In the first case, no loaded, the cascade stops. further damages are considered, since no power is dispatched In this paper, for simplicity, the cascade is assumed instanta- and no further load can be shed. In the second case, the failure of neous, i.e. its duration can be neglected with respect to the restora- another line and the possible, consequent cascade likely lead to tion times (see next Section). more catastrophic consequences, as a smaller subset of lines is available for the re-dispatching of the power supply. The repairs 2.3. Restoration model of all the failed lines are assumed to be simultaneously carried on at the same velocities. In conclusion, the occurrence of severe Each time the network suffers a damage, the initially failed line weather events causes, on average, the restorations to last longer, is repaired, so that the network is restored to its initial conditions. due to the combined effects of a reduced vrest and the possible We assume that: (i) to perform the reconnection, a crew of repair- occurrence of multiple failures. men is dispatched as soon as the cascade ends; (ii) the network For the following case study of Section 3 below, the parameters becomes fully functional as soon as the initial line is repaired; defining the restoration velocity in disturbed conditions (28) are F. Cadini et al. / Applied Energy 185 (2017) 267–279 273 arbitrary set to e ¼ 0:4 and u ¼ 40, on the basis of engineering customized sequential time Monte Carlo scheme for generating common sense. repeated realizations of several histories of the life of a power transmission network. In what follows, we briefly describe the 2.4. Sequential time Monte Carlo solution method, whose flowchart is shown in Fig. 3. For further details on the procedure, the interested reader may refer to [16] The complete model, i.e., the stochastic model of line failures and [17]. driven by ‘‘normal” and extreme weather events combined with The sequential Monte Carlo scheme proceeds as follows. The the DC power flow-based cascading failure model, lends itself to time horizon of the simulation is 1 year, i.e., 8760 h. At the begin- a simulation-based numerical resolution scheme for estimating ning of each year, the grid is assumed to be fully operational and the expected values of the reliability/availability indices of interest. the weather conditions to be normal. First, the times of occur- To this aim, similarly to [16] and [17], we propose to resort to a rences of the ‘‘normal” failures throughout the year are generated
Fig. 3. Flowchart of the Monte Carlo simulation algorithm. 274 F. Cadini et al. / Applied Energy 185 (2017) 267–279
by sampling from the exponential distribution associated to the and NL links, and its complete configuration, i.e. the network topol- homogeneous Poisson distribution (1) with total parameter ogy, loads and generators. Although the size of the system is small, P k NL k k its composition is regarded complete with respect to the character- normTOT ¼ i ð normi Þ, where normi is the failure rate of line i due istics of interest for the purpose of the modeling and simulation to ‘‘normal” causes. The failed line i of length li is sampled from a k l framework development proposed in the present work. In order probability mass distribution with values P norm;i i , i 1; ...; N . NL ¼ L ðk ; l Þ i norm i i to compute the failure probabilities, the length of the lines is Then, the occurrence times of the two types of severe weather needed. It is possible to infer the line length from the available data events are independently sampled from the distributions associ- in [21], using the total impedance Z reported in both [38] and [39]: ated to the non-homogeneous Poisson distributions (2) with Zi ¼ðri þ jxiÞli ð27Þ parameters vwindðtÞ and vlightðtÞ, respectively. The intensities and durations of the weather events in the annual sequence are sam- where Zi is the impedance of the line i, r and x are the resistance and pled from the corresponding distributions (Table 1). The times of reactance per unit of length, respectively. Assuming as power base the failures possibly occurring during each weather event are sam- Sbase ¼ 100 [MVA] [21] and using the nominal voltage V base of each pled from the Poisson distribution with total parameter P line, the base impedance Zbase can be obtained: k NL k pTOT ¼ i ð pðtÞiÞ, where p refers to the extreme weather event 2 type. If the sampled failure time falls within the duration of the V base Zbase ¼ ð28Þ extreme weather event under consideration, then, the sample is Sbase accepted; otherwise, it is rejected. The line of the grid actually Multiplying Z for the specific reactance of each line and using undergoing the failure is, then, sampled from the probability mass base a conversion factor of 0.7 [O/mile], the length of each line can be distribution built similarly to the case of the ‘‘normal” failures inferred, as proposed in [40] and already applied to the IEEE14 net- described above. work in [39]. Lines containing transformers are considered to have Finally, the three sequences of events are superimposed. In cor- null length. respondence of each failure of the sequence, the cascading failure Table 3 shows the main network data, i.e. the lines types, their model is run until a new, stable configuration is found. At the operating voltages and lengths, the corresponding ‘‘normal” failure end of the cascading failure event, the repair of the initial line rates knorm (assumed to be related only to the line voltage level, as starts with a restoration velocity vrestðtÞ computed according to proposed in [41] and also used in [39]) and the resulting annual the model described above (28). The possible occurrence of multi- failure probabilities. The failure rates parameters in (7) and (8) ple failures within the same weather event is treated as described a : 2 b : 3 at the end of Section 2.3. are chosen to be ¼ 1 13 10 and ¼ 3 10 10 , on the basis of engineering common sense. A total of NS ¼ 4000 histories of one year life of the IEEE14 network are simulated, with the proposed 3. Application Monte Carlo simulation scheme. Fig. 4 shows the outcome of a realization of weather events and The proposed modeling and simulation framework is applied to line failures occurrences during a portion of one year. Each squared the IEEE14 test case [21]. The grid model is composed by two kinds wave represents the occurrence of an event, its starting time and of components, nodes and lines, which are representative of all the duration. The solid line is referred to the occurrence of a wind subcomponents. The test network represents a power transmission storm, the circles indicate the weather-related failure times and system with 20 links (NL ¼ 20Þ, mainly overhead, uninsulated the dashed line is associated to the consequent restoration pro- lines. The system operates with two different voltage levels, 132 cesses. In this particular realization, the wind storm event gives and 230 kV: to allow this, transformers are present. Each node rep- rise to multiple line failures. The event is characterized by a wind resents a generating unit, a load point or a transmission/switch speed of wwind ¼ 9:42 [m/s], lasting for Dwind ¼ 31:6 [h] and causing bus. The specific topology and data regarding generating capacity four lines failures. The corresponding restoration velocity, com- can be found in [21]. In the following, we shall use G to denote puted by (28), is vrestðtÞ¼0:638, which would imply a repair dura- the topology ðV; NLÞ of the power network, with jVj¼14 nodes tion trepðtÞ¼7:84 [h]. However, after the occurrence of the initial
Fig. 4. Example of a sequence of weather-related and failure events: the start, end and duration of a wind storm event are indicated by the solid line; the failures due to the wind storms are represented by the circles; the start, the end and the duration of the repairs are indicated by the dashed line. F. Cadini et al. / Applied Energy 185 (2017) 267–279 275
Table 2 Details of the sequence of Fig. 4.
Failure Time [h] Line Targeted li (unavailable lines) Cascade lines li disconnection Load shed [MW] 1 4479.7 12 – 0 2 4480.2 3 (12) 6 9.54 3 4489.0 13 (3, 6, 12) 19, 18, 17, 16 136.47 4 4498.7 17 20 16.11
Table 3 Data for the IEEE14 transmission network ([39] and [41]).
Node i Node j Line type and voltage Length [km] Failure rate knorm [1/km/y] Annual failure probability (‘‘normal”) 1 2 132 kV line 22 1.086 10 2 0.216 1 5 132 kV line 81 1.086 10 2 0.585 2 3 132 kV line 72 1.086 10 2 0.542 2 4 132 kV line 64 1.086 10 2 0.501 2 5 132 kV line 63 1.086 10 2 0.495 3 4 132 kV line 62 1.086 10 2 0.490 4 5 132 kV line 15 1.086 10 2 0.150 4 7 132/230 kV transformer – – – 4 9 132/230 kV transformer – – – 5 6 132/230 kV transformer – – – 6 11 230 kV line 219 5.429 10 2 0.696 6 12 230 kV line 282 5.429 10 2 0.784 6 13 230 kV line 144 5.429 10 2 0.542 7 8 132/230 kV transformer – – – 7 9 132/230 kV transformer – – – 9 10 230 kV line 93 5.429 10 2 0.396 9 14 230 kV line 298 5.429 10 2 0.802 10 11 230 kV line 212 5.429 10 2 0.684 12 13 230 kV line 220 5.429 10 2 0.697 13 14 230 kV line 384 5.429 10 2 0.876
failure at t ¼ 4479:7 [h], a second and third failures occur before tion time associated to the cascade failure event i caused by the the conclusion of the repair of the first failure event. Since the event e: Note that the index ARTy bears some similarities with the reconnection of all the disconnected lines is assumed to be per- SAIDI indicator typically used for quantifying the reliability of formed at the same time (Section 2.3), any further line failure, power distribution networks (see, for example, [16] and [17]). The occurring within the restoration period of a previous damage, pre- index ARTi [h/occ] is the average duration of a single outage, here vents the repairs to be finished on schedule and a new restoration estimated as: period begins. In the presented case, the three successive failure P Ne;tot trep;e events result in a restoration duration of trepðtÞ¼17:2 [h]. Table 2 d i i ARTi ¼ ð30Þ reports the storyline corresponding to the failure sequence shown Ne;tot in Fig. 4, giving the failures occurrence times (column 1), the grid where Ne;tot is defined as the overall number of failure occurrences line damaged by the severe weather event (column 2), the lines dis- due to the event of type e in all the Ns simulations: connected after the cascading failure events (column 3) and the load shed after each cascading failure. It can be seen that the size of the XNS N ; ¼ N ð31Þ subset of disconnected lines increases from the first to the third fail- e tot ej ure and, accordingly, also the undelivered power does. The fourth j¼1 and last failure, instead, occurs after the repairs are completed, so The AFF [occ/y] index quantifies the average number of failure that a larger subset of lines is again available for re-dispatching occurrences in the grid in one year, here estimated as: the power and, consequently, the final power shed is lower. P NS d Ne As anticipated in Section 2, we quantify the reliability/availabil- AFF ¼ i¼1 i ð32Þ ity of the power transmission network by means of the following NS indices: (i) the System Average Restoration Time Index (ART), (ii) The EENS [GW h/y] indicator represents the annual average the System Average Failure Frequency Index (AFF) and (iii) the amount of energy not delivered by the transmission network due Expected Energy Not Supplied (EENS). to the disruptive events, here estimated as: The ART indicator is estimated in two different versions: ARTy PNe XNS j [h/y] and ARTi [h/occ]. The index ARTy [h/y] is the average duration d LSe trep;e EENS ¼ i i i ð33Þ of the outages over one year: N j¼1 S PNe XNS j ; d trep ei Finally, since we are interested also in quantifying the average ART ¼ i ð29Þ y N j¼1 S impact of cascading failures on the network reliability/availability, we further introduce the Average Load Shed index (ALS [MW/occ]) where Nej is the number of failure occurrences due to the event e as the average power not served at the end of a failure and, possi- ; (‘‘normal” or weather-related) in the year j and trep ei is the restora- bly, subsequent cascade event, here estimated as: 276 F. Cadini et al. / Applied Energy 185 (2017) 267–279 P Ne;tot ning or their simultaneous occurrence (lightning during high d i¼1 LSei ALS ¼ ð34Þ winds). Ne;tot d The average number of failures in one year (AFF, column 3, where LS [MW/occ] is the load shedding after a cascade event ei Table 4) is larger in case 2, demonstrating that the extreme weather caused by the event e: events have a significant impact on the reliability/availability per- Note that the EENS indicator is widely used in analyses of both formances of the network, even if their total average duration rep- transmission and distribution grids and the ALS is a common mea- resents only a small fraction of one year, as shown above. sure of the impact of cascading failures in transmission grids. Other significant consequences of the severe weather events are Following the approach proposed in [16], in order to measure the increases in the average durations of the single interruption of the contribution to the reliability/availability indices due to the d supply (ARTi ) and in the annual average duration of the restoration extreme weather events, the simulations are carried on in two dif- d d ferent settings: in the first case (case 1), we assume that the net- processes (ARTy ). It can be seen that the relative increase in ARTi is d work is subjected only to ‘‘normal” failures, i.e. the cascades are significantly smaller than that in ARTy , being due only to the triggered by line failures only due to normal operating conditions; effects of the restoration model. In particular, the increase is due in the second case (case 2), the occurrences of wind storms and to the simultaneous effects of (i) the lower restoration velocity lightning are included in the simulation. vrestðtÞ during the weather event, directly correlated to its intensity b and (ii) the occurrence of multiple line failures within the same With regards to case 2, the estimates of the mean duration, De, ^ extreme weather event, since more lines have to be repaired in and of the average number of occurrences of event e, ne, are com- d puted as: order for the grid to be restored. On the other hand, the ARTy is also P affected by the average annual number of interruptions. NS ^ i¼1ne;i As said above, each time the system is subjected to multiple ne ¼ ð35Þ NS failures within the same extreme weather event, it also experi- P ences larger load shedding magnitudes at the end of the cascades, ne;tot due to the progressive reduction of the available lines, as shown b i¼1 De;i De ¼ ð36Þ earlier. This is shown by the estimates of the load shed index ne;tot d ALS, as shown in column 4, Table 5, where the separate contribu- where ne;tot is the total number of events e among all the NS tions due to the different weather event types are reported. In fact, simulations. The resulting average occurrences of high wind and as the failure rate increases, the probability of multiple failures ^ : lightning events are estimated to be nwind ¼ 53 2 [occ/y] and grows and the magnitude of the final outages increases. For exam- b Nlight ¼ 48:5 [occ/y], respectively, while their mean durations are ple, the maximum value of the failure rate occurs when a wind b b storm and a lightning event occur simultaneously, and, coherently Dwind ¼ 9:78 [h] and Dlight ¼ 1:21 [h], respectively. The portion of the year interested by extreme weather events is: the maximum ALS is observed in correspondence of this kind of event. Note also that the duration of the weather event has a sig- b b ^ d T wind ¼ Dwind nwind ¼ 520 ½h nificant effect on the value of the index ALS. Comparing the ALS b b ^ index due to high wind, lightning (column 4, Table 5) and the T light ¼ Dlight nlight ¼ 58 ½h one calculated during ‘‘normal” conditions (column 4, Table 4)it b b T wind þ T light can be noticed that a sustained extreme weather event (as wind ¼ 6:6% 8760h storms) can imply severe consequences, while a short one (such Table 4 shows the results obtained in the two cases, where the as lightning) causes an average load shed similar to the one under symbol l indicates the sample means of the reliability/availability normal circumstances. indices and r their sample standard deviations. Another important consequence of the larger number of failures Table 5 shows the individual contributions to the reliability/ during extreme weather and of the associated larger restoration availability indices due to the occurrence of each type of extreme times, is a larger value of the index EENS, as it can be seen from weather event considered, including their possible simultaneous Table 4. Table 5 also shows that the wind storm events provide occurrence. In other words, the Table shows the values of the indi- the largest contribution to EENS, mostly due to their longer dura- cators computed by taking into account only the contribution of a tions, which significantly affect the related restoration velocities b single weather event typology, i.e., either only high winds, light- (Dwind ¼ 9:36 [h]).
Table 4 Results under normal failures. Average of 4000 repetitions of the simulations.
IEEE14 d d d d d ARTi [h/occ] ARTy [h/y] AFF [occ/y] ALS [MW/occ] EENS [GW h] l r l r l r l r l r
Case 1 – No Extreme Weather 5.06 1.01 71.68 19.61 13.61 3.79 84.90 107.51 5.80 1.96 Case 2 – With Extreme Weather 5.98 0.89 175.0 31.62 29.23 5.06 92.30 18.97 16.60 4.42
Table 5 Results for weather-related failures. Average of 4000 repetitions of the simulations.
IEEE14 d d d d d ARTi [h/occ] ARTy [h/y] AFF [occ/y] ALS [MW/occ] EENS [GW h] l r l r l r l r l r
High wind 6.67 1.90 99.15 25.93 14.86 3.79 103.10 95.50 10.43 3.86 Lighting 5.32 1.77 3.89 4.43 0.73 0.70 85.60 88.54 0.33 0.57 High wind + lightning 6.83 2.02 0.22 1.26 0.031 0.19 110.50 94.23 0.03 0.13 F. Cadini et al. / Applied Energy 185 (2017) 267–279 277
Interestingly, in our application we observe a general, signifi- flexible restoration model which allows to capture the uncertain- cant increase in the estimates of the average reliability/availability ties related to the repair processes. indicators when severe weather events are included in the model, The framework has been developed by originally combining as opposed to the results obtained in [16], where a distribution stochastic models for describing uncertain weather conditions network is analyzed. In fact, even though the duration of extreme with a cascading failure model based on a DC approximation of weather events is only a small fraction of the grid operating time, the power flows and a proportional re-dispatch strategy. The novel the consequences are harsher than in a distribution system, due to restoration model accounts for the harsher conditions offered by the multiplying effect of cascading failures. This behavior is less extreme weather events, under which a repair crew may be forced d to operate, thus realistically representing the downtime of a power evident for the indicator ARTi because, as stated above, it is related to the cascading failure consequences only indirectly, through the grid upon a cascading failure event. restoration model. The standard deviations of the reliability/avail- The model evaluation has been performed by a customized ability indicators, which are important measures of the operator’s sequential time Monte Carlo simulation scheme of resolution. exposure to risk, are also observed to be higher in case extreme The main assumptions of the proposed framework, introduced weather events are included in the framework, in agreement with in the paper, are here critically revisited: the results obtained in [16]. Note that the proposed framework should be used by the grid (a) Only failures of the grid’s lines are taken into account. The operators to (i) check the current grid reliability/availability perfor- failures of other grid’s components, such as buses, substa- mances with respect to any thresholds set by the regulatory author- tions, and transformers are not included in the analysis. This ities and (ii) to compare alternative actions to be undertaken, in is a common approach followed in many works of literature support of a decision making process involving grid improvements (since power transmission line failures are more common and operation and maintenance strategies. In particular, the differ- than bus failures [28]) which may lead to an overestimation ent options could be ranked according to the values of the indica- of the reliability/availability of the grid. tors reported in Table 4, possibly assigning different weights to (b) The cascading failure model for the estimation of the line the various indexes according to the final objective of the decision failures’ consequences is based on a DC approximation of making process. Then, in order to further inform the decision mak- the power flows. This is also a very popular strategy adopted ing process, the results of Table 5 could be used to identify the in both research works and actual applications, especially typology of weather event to which the grid is mostly exposed to when the network analyzed is very large, making the AC from a reliability/availability point of view, so as to possibly steer power flow computations almost infeasible in many practi- the investments towards more tailored interventions. cal applications. As recognized in (Kenneth Van den Bergh, The computational time required by the simulations is mainly Erik Delarue and William D’Haeseleer, DC power flow in unit driven by the number of calls to the cascading failure model. The commitment models, TME Working Paper - Energy and Envi- computational time is approximately 4.5 h for Case 1 and 10 h ronment (2014)): ‘‘In general, one can conclude that for high for Case 2, on a 2.26 GHz Intel Core 2 Duo processor with 2 GB of voltage grids - which are mostly the ones looked at in unit RAM. In Case 2, the occurrence of extreme weather events leads commitment models - the accuracy of DC power flows is to a significant increase in the number of line failures, each of around 5%, compared to AC power flow and averaged over which requires a cascading failure simulation. The computational all lines. The deviation on single line flows can be much lar- time is expected to increase more than linearly with the size of ger. Hence, the accuracy of DC power flow is acceptable in the grid, as a consequence of the combined effects of (i) the large the scope of unit commitment models, but one should annual average number of failures, simply due to the long total always keep the limitations of DC power flow in mind and extension of the lines and (ii) the large computational resources be careful with drawing conclusions about single lines”. required by a single cascading failure simulation on a more com- Since the purpose of our work is that of analyzing average plex network. Rough estimates carried out by running a few cas- behaviors and not single line’s performances, we consider cading failure simulations for the IEEE118 reference network the DC approximation as the best option, especially in view suggest that the proposed analysis would require computational of a possible application of our framework in optimization times of the order of days/one week on the same machine. This or sensitivity problems requiring multiple runs. means that an analysis of a more realistic grid is probably still fea- Another simplification made in the proposed cascading sible, although more complex analyses, e.g. including uncertainty failure model is that of using a simple, proportional power and sensitivity analyses or optimizations, would require either to flow re-dispatch strategy to balance the islands forming in resort to processor clusters, or to identify strategies for accelerat- the grid when many of the grid’s line are disconnected as ing the computations, possibly based on the use of surrogate, the cascading failure proceeds. In general, optimizations approximating models. techniques are used to re-balance the islands in order to guarantee that the loads are not shed while minimizing the losses. These methods can be easily embedded in the 4. Conclusions proposed framework, but they require much longer com- putational times. We chose not to do that in order to be In this work, we have developed a modeling and simulation able to perform our research in acceptable times according framework for the quantification of reliability/availability indexes to the computational power available. of power transmission grids, which represents a fundamental task (c) The distributions of the times of occurrence of the severe for the cost-effective development, operation, maintenance and weather events and of the line failures, and the distributions improvement of these complex infrastructures. of the weather event intensities are taken from the works of More specifically, the purpose of the proposed framework is literature cited in the Introduction. The parameters of the that of allowing for a more realistic reliability/availability evalua- distributions must indeed be identified on the basis of tion, by including a representation of the cascading failure dynam- weather statistics collected in the geographical region of ics triggered by extreme weather events and by introducing a new, interest. 278 F. Cadini et al. / Applied Energy 185 (2017) 267–279
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