Review on Energy Resilience

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Applied Energy

journal homepage: www.elsevier.com/locate/apenergy

A power-flow emulator approach for resilience assessment of repairable power grids subject to weather-induced failures and data deficiency

Roberto Rocchettaa, Enrico Ziob,c, Edoardo Patellia,⁎

a Insititue for Risk and Uncertainty, Liverpool University, Liverpool, UK b Department of Energy, Politecnico di Milano, Milan, Italy c Chair System Science and the Energy Challenge Fondation Électricité de France (EDF) CentraleSupélec, Université Paris-Saclay, Grande Voie des Vignes, Chatenay- Malabry, France

H I G H L I G H T S

• Weather extreme conditions affect power grid failures/repairs and data is scarce. • A stochastic resilience framework is proposed for assess the weather-grid interaction. • A power-flow emulator is constructed to reduce the computational cost. • Imprecise probabilistic methodology is used to tackle lack of data issues. • Most relevant weather-grid factors are identified by the global sensitivity analysis.

A R T I C L E I N F O

Keywords: Load curtailing Severe weather Power grids Resilience Global sensitivity Artificial neural network Credal sets

A B S T R A C T

A generalised uncertainty quantification framework for resilience assessment of weather-coupled, repairable power grids is presented. The framework can be used to efficiently quantify both epistemic and aleatory un- certainty affecting grid-related and weather-related factors. The power grid simulator has been specifically designed to model interactions between severe weather conditions and grid dynamic states and behaviours, such as weather-induced failures or delays in components replacements. A resilience index is computed by adopting a novel algorithm which exploits a vectorised emulator of the power-flow solver to reduce the computational efforts. The resilience stochastic modelling framework is embedded into a non-intrusive generalised stochastic framework, which enables the analyst to quantify the effect of parameters imprecision. A modified version of the IEEE 24 nodes reliability test system has been used as representative case study. The surrogate-based model and the Power-Flow-based model are compared, and the results show similar accuracy but enhanced efficiency of the former. Global sensitivity of the resilience index to increasing imprecision in parameters of the probabilistic model has been analysed. The relevance of specific weather/grid uncertain factors is highlighted by global sensitivity analysis and the importance of dealing with imprecision in the information clearly emerges.

1. Introduction

The power grid is the largest man-made critical infrastructure and is extremely complex in both its operations and structure. The weather conditions drifting towards extremes and the increasing use of renew- able energy sources are tightening the interactions between power network states and the external environment. Reliability/availability analysis frameworks have, then, to incorporate weather models and consider interactions between grid states and environmental states, accounting for relevant sources of randomness (i.e. aleatory un- certainty) but also for parameters values imprecision (i.e. epistemic

uncertainty). Power network reliability is a well-defined mathematical concept

[1]. Many frameworks for reliability assessment have been proposed in the past, which generally focus on known threats such as −N 1 or −N 2 failures paradigms [2] or on a predefined contingency set [3,4]. System resilience broadens the reliability concept by accounting for low- probability-high-consequence events (such as severe weather condi- tions [5]) and recovery process of the system. A generally accepted definition of resilience still has to be formulated, an example being ‘the network ability to withstand high impact low probability events, rapidly recovering and improving operations and structures to mitigate the impact of

https://doi.org/10.1016/j.apenergy.2017.10.126 Received 7 July 2017; Received in revised form 7 September 2017; Accepted 31 October 2017

⁎ Corresponding author. E-mail address: [email protected] (E. Patelli).

Applied Energy 210 (2018) 339–350

Available online 09 November 2017 0306-2619/ © 2017 Elsevier Ltd. All rights reserved.

T

similar events in the future’ [6,7]. It can be argued that a main difference between a reliable power grid and a resilient power grid is that, in the latter, low-probability-high-consequence events (e.g. extreme weather events) are specifically considered and handled, with the ability to learn from past occurrences. To achieve this, a comprehensive analysis of the relevant sources of uncertainty should be performed. In particular, lack of data is generally affecting low probability events. To improve overall robustness of the analysis, it is uttermost important to develop and improve frameworks capable of tackling (effectively and efficiently) data deficiency issues. A rigorous quantification of the lack of data af- fecting extreme low-probability-high-consequence events is necessary.

In the last years, many studies have focused on analysing the effect of extreme weather events on the power grid risk and reliability. Some research was carried out with the support of international organizations [8]; other focused on different extreme events such as floods, ice storms, strong wind gusts and more [6,8–13]. More recently, Cadini, Zio and Agliardi [9] proposed a probabilistic reliability/availability assessment framework extended from Ref.[12]. The framework in- corporates a sampler of severe weather conditions and models weather- induced effects on the grid’s components failures and replacements. One of the challenges for the application of the framework is the high computational cost. This is mainly attributable to the number of calls to the cascading failure model (i.e. the power-flow solver): ‘…analysis of a more realistic grid is probably still feasible, although more complex analyses, e.g. including uncertainty and sensitivity analyses or optimizations, would require either to resort to processor clusters, or to identify strategies for accelerating the computations, possibly based on the use of surrogate, ap- proximating models’ [9].

In general, the time needed to compute the load curtailed can be quite small (e.g. that of a single optimal power flow evaluation); this is especially true if the power network size is modest. Unfortunately, optimisation problems cannot be vectorised and power flows have to be solved one-at-a-time. Consequently, any analysis for which a large number of power flow evaluations are needed may result computa- tionally untreatable. Examples of such costly analysis are global sensi- tivity analysis [14], cascading failures analysis [9], or imprecise (gen- eralised) uncertainty quantification analysis [15]. To perform such computationally demanding analysis, a significant reduction in the computational complexity (while reducing marginal accuracy) is needed and emulators can be adequate for this aim. A surrogate model, also known as emulator or meta-model, is a numerically cheap math- ematical approximation of a computationally expensive realistic model [16]. Some examples of popular meta-models are Artificial Neural Networks [17,18], Poly-Harmonic Splines [19] and Kriging models [20]. Surrogates have been extensively applied to reduce time expenses of numerically burdensome models and few works have attempted to use meta-models to analyse power grids, see for instance [21–26]. Amjady et al. [23] proposed an emulator to assess the power system reliability providing as input forced outage rates. Silva et al. employed artificial neural networks to monitor voltage magnitudes [24]. Chen et al. [25] adopted a surrogate-based strategy for optimal power flow inequality constraints aggregation. To the Authors knowledge, none of the reviewed papers attempted to mimic the relationship between the grid components state vector, load profile and power load curtailed within a resiliency assessment framework.

Probabilistic reliability assessments of power grids are traditionally carried out with reference to a well-defined probabilistic characterisa- tion of the uncertain output, whose calculation generally requires a large body of empirical information. A sufficient amount of samples are necessary to properly estimate the underlying probability distributions parameters and often, due to technological limits or time/cost con- straints, the available data is not sufficient for accurately estimating all relevant parameters [27]. In those situations, expert assumptions are made on the probabilistic model, which can lead to erroneous conclu- sions, overestimation of the system performance and a false sense of confidence [28]. Data scarcity often affects the analysis of power grid

resilience and safety [29]. In fact, consider the highly reliable compo- nents (e.g. transformers, underground cables, etc.) of which power grids are made. Those components will likely fail only a few times during their life span (or possibly even never). The lack of statistical failure data makes it difficult to characterise the failure behaviour of these components with confidence. In this situation, a common practice is to estimate the failure rates of the components by considering the few available failure occurrences in similar components. This procedure, so- called “data pooling” [27], assumes similar elements behave as de- scribed by the same probabilistic model. This is a rational assumption, but when (similar) components operate differently (e.g. close/far from their thermal limits or in harsh/mild environments) or undergo dif- ferent maintenance/repairing policies, such assumption is rarely true. In practice, different factors influence the components, leading to dif- ferent failure behaviours even for identical components. In those si- tuations, it is advisable to relax the assumption of a precise probabilistic model, for instance, by accounting for imprecision in the distribution parameters (e.g. in the estimation of components failure rates and events occurrence rates) [28,31]. Generally speaking, a set of plausible distribution families can be considered for describing imprecision (for instance, an envelope of Weibull, Exponential, Normal, etc. CDFs modelled using a non-parametric P-box). However, dealing with several distribution families was not the aim of this work. In this research, the parameters of the probability distribution families (e.g. used to sample the high wind event duration and intensity) and of the components of the grid-weather model are assumed affected by an increasing level of imprecision.

In this paper, a generalised framework is proposed for (imprecise) probabilistic resilience assessment of power networks. The framework has been designed to capture complex coupling between weather con- ditions and power grid operations, by incorporating weather-influenced failures and repairs of the grid’s components. An Artificial Neural Network (ANN) is trained to emulate the total load curtailed given specific lines failures and the load profile, and has been embedded within the framework to increase computational efficiency. Comparison between the novel surrogate-based framework and the original solver shows significant improvement in efficiency at the expense of a small reduction in accuracy. Aleatory uncertainty is accounted for and epis- temic uncertainty is associated with imprecision and lack of knowledge. Both types of uncertainty are propagated by generalised probabilistic methods based on Credal sets and Fuzzy sets. The sensitivity of the resilience index to parameters imprecision is quantified. Aleatory un- certainty propagation and generalised uncertainty propagation (i.e. accounting also for imprecision) are performed and the ANN cap- abilities tested against the full power-flow. The results again show that the use of the ANN meta-model allows advanced sensitivity analysis to be performed on the parameters of the probabilistic model at the ex- pense of a small reduction in accuracy but with a significant gain in computational time.

The rest of the paper is organised as follows. Section 2 introduces the probabilistic model for coupling weather conditions and grid states. The emulator is presented in Section 3. Section 4 presents the overall modelling and computational framework. In Section 5 the generalised probabilistic framework based on Credal sets is described. The case study and results are presented in Section 6. Section 7 presents a dis- cussion of the findings from an applicative perspective and Section 8 closes the paper.

2. A probabilistic model for weather-grid coupling

A power grid topology can be represented by a graph ( , )G N E , where i denotes a node within the node set N and i j( , ) the link between node i and j in the line set E [31–34]. Denote with NL the number of loads, with Nl the number of lines and with Ng the number of generators in G .

Optimal-Power-Flow (OPF) methods can used to solve the network

R. Rocchetta et al. Applied Energy 210 (2018) 339–350

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power dispatch problem [35,36]. In the adopted formulation, loads can be curtailed if necessary. This has indeed very high cost for the grid and will occur only if the cost minimization problem cannot be solved otherwise, for instance, if load demand exceeds power capacity or to avoid line overloads. Mathematically, the problem is defined as follows:

f P Lmin ( , ) P L

g cut ,g cut (1)

where the cost function depends on the power generated and the load curtailed (Lcut).

2.1. Power grid resilience index

The Expectation of Energy-not-Supplied, named ENS[ ]� , has been commonly used as a reliability index in a number of studies. Although initially conceived as a reliability indicator, it is also suitable to the resilience concept [37]. The ENS[ ]� can be obtained by averaging contributions of Ns independent simulations:

= ∑ =ENS

ENS N

[ ] i N

i

s

1 s

� (2)

where the Energy-not-Supplied, ENS, for a given simulation period Tsim, here considered to be 1 year, is obtained as follows:

∑ ∑= = ∈

ENS L t· t

T

i cut i t

1 , ,

sim

N (3)

where Lcut i t, , is the load curtailed at each time t and each node i, ob- tained solving the minimisation problem in Eq. (1). The ENS due to a failure event f can be obtained as ∑ ∑= ∈ L t·t

T i cut i t1 , ,

f N

, where Tf is the duration of the failure event.

2.2. Weather-dependent failures

In this work, weather events can trigger failures in the network. The network state is therefore identified by the combination of a ‘normal weather’ failure mode and the ‘severe weather’ failure mode. A normal weather model represents weather-independent effects such as ageing, malicious attacks or manufacturing errors in general. A severe weather model describes failures which are triggered by extreme weather con- ditions, for instance, lightning-induced dielectric breakdowns or wind- driven structural failures. High winds storms and lightning storms are here considered and, for simplicity but without loss of generality, the entire network is assumed facing the same weather conditions. The random occurrence of failures in ‘normal’ weather conditions is mod- elled as a Homogeneous Poisson process (HPP) [12]:

= = = …−P N t k λ t

k e k N( ( ) )

[ · ] !

0,1, ,f n

k λ t·n

(4)

where λn ⎡⎣ ⎤⎦ occ

h km· represents the line failure rate in normal weather

conditions, =P N t k( ( ) )f is the probability that k failures occur in the

network in the period t(0, ] and N t( )f is the number of failures per km of grid line occurring in the period t(0, ] and measured in ⎡⎣ ⎤⎦

occ km

. The severe weather events (e.g. high wind and lightning storms) are

affected by uncertainty due to climate changes and inherent variability. In general, events are more likely to occur during specific periods of the year. The occurrence of severe weather events is modelled by a Non- Homogeneous Poisson Process (NHPP) [12]:

= = = …−P N t k V t

k e k N( ( ) )

[ ( ) ] !

0,1, ,e e

k V t( )e

(5)

where V t( )e represents the time dependent occurrence rate of the severe weather event =e P N t k, ( ( ) )e is the probability that k events occur in the period t(0, ] and N t( )e is the number of events of type e occurring in the period t(0, ]. The quantity V t( )e can be obtained as:

∫= ′ ′V t v t dt( ) ( )e t e 0 (6)

where ′v t( )e is the time-varying occurrence rate of the event e. In this model, lightning storms and high wind speeds are considered as threatening environmental conditions ∈e lg w{ , }. In Ref. [12], the ratios v t( )w and v t( )lg of occurrence of these conditions are evaluated on a monthly basis and assumed to be stepwise constants, as depicted in Fig. 1.

Once a severe weather scenario occurs (i.e high wind and/or lightning storm), its intensity and duration are described by char- acteristic historically fitted, probability distribution functions. Specifically, the wind storm intensity is obtained as follows [12]:

= +W t W t( ) Δ ( )w crt w (7)

where W t( )w is the wind speed intensity at time t for the wind event w W, crt is the ’critical’ wind speed assumed to be 8⎡⎣ ⎤⎦

m s

and Δw is a random surplus of the critical wind speed threshold. The intensity of a lightning storm is quantified by its lightning ground strike density N t( )g , measured as the number of ground-flashes (or ground-strikes) per unit of time and area ⎡⎣ ⎤⎦

occ h·km2

. The variability of the ground flash density is assumed log-normally distributed, with parameters fitted based on historical records. The probabilistic model for wind storm duration (Dw), lightning storm duration (Dlg) and the respective intensities are summarised in Table 1.

Gen Feb Mar Apr May Jun Jul Aug Sep Opt Nov Dec 0

0.005

0.01

0.015

0.02

0.025

O cc ur an ce Ra te

Fig. 1. The variable occurrence rate of wind storm events, solid line, and lightning events, dashed line (data taken from [12]).

Table 1 Probability distributions for intensity and duration of severe weather events [12].

Distribution Scale (a) Shape (b)

Dw Weibull 9.89 1.17 Dlg Weibull 0.96 0.85

tΔ ( )w Weibull 1.23 1.05 Mean (μ Ng) Standard deviation (σNg)

N t( )g Log-normal −5.34 1.07

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341

High wind speeds can directly or indirectly damage the line struc- ture, e.g. by friction-induced fatiguing of structure’s joints or by moving/breaking trees branches in the proximity of the line. Lightning strikes can damage lines, for instance triggering insulator dielectric breakdown. By considering an overhead line as made of subcomponents in series (the insulation and the mechanical structure) and assuming that the individual subcomponent failure depends on different physical phenomena, the total failure rate can be obtained as follows [12]:

= + +λ t λ λ W t λ N t( ) ( ( ) ) ( ( ) )n w w lg g (8)

were λ w is the contribution to the total line failure rate per km due to high wind speed at time t and λlg the lightning storms contribution. Note that it is possible, although very unlikely, to face the simultaneous occurrence of a lightning storm and a high wind event.

The contribution to the line failure rate due to high wind is ex- pressed as follows [12]:

⎜ ⎟= ⎛ ⎝

− ⎞ ⎠

λ W t λ W t

W α( ( ) )

( ) 1w w n

w

crt w

2

2 (9)

were αw is a regression parameter obtained from failure data. The failure rate due to a wind event has a strong relation with the wind intensity, following a quadratic law. It can be observed that for wind speed less or equal to the critical wind speed ( ⩽W t W( )w crt) the wind contribution to the failure rate is null.

The contribution to the line failure rate due to lightning is obtained as follows [12]:

=λ N t λ β N t( ( ) ) ( )lg g n lg g (10)

βlg is a regression coefficient fitted on historical data and the line failure rate is linearly related to the lightning event intensity. According to [9], the failure rate of the generic line λ t( )i expressed in ⎡⎣ ⎤⎦

occ h

can be ob- tained from the total failure rate by multiplying it by the line length (li).

2.3. Weather-dependent repairs

Delays in the repair of power grid components can be caused by, for instance, ineffective communication between all of the parties involved (e.g. non-cooperative landowner) or harsh weather which slows down the identification of the trouble location (e.g. helicopters cannot be sent to find the fault location and report back). Recently a weather-depen- dent repair model has been proposed, reflecting the realistic sense that the efficiency of repairing crews is strongly affected by the external weather conditions [9]. The model assumes that: (1) a crew of re- pairmen is dispatched with no delay as soon as a failure in one line occurs, (2) the network becomes fully functional as soon as failed lines are replaced, (3) the time of the failure transient is negligible with re- spect to the time to repair. It is clear that the time needed to fully re- place an overhead line increases if the crew operates in harsh weather conditions. In particular if wind and/or lightning events occur, the normal average repair speed (νnorm) is assumed to decrease accordingly to the intensity of the severe weather condition. The repair speed can be defined as follows [9]:

=

⎧

⎨

⎪

⎩ ⎪

⩾ =

< >

⩾ >

+ −

+

+ − + +

ν

W t W N

W t W N

W t W N

, if ( ) & 0

, if ( ) & 0

, if ( ) & 0 repair

ν η W t W w crt g

ν ψ N w crt g

ν η W t W ψ N w crt g

1 ·( ( ) )

1 ·

[1 ·( ( ) ) ] [1 · ]

norm w crt

norm g

norm w crt g (11)

where ψ and η are positive parameters. The normal speed νnorm is set equal to 20 ⎡⎣ ⎤⎦h

% (i.e. 5 [h] are needed to replace a line in normal weather conditions), whilst ψ and η are set to 40 and 0.4 following engineering judgement [9]. Also in this work, a line under repair is not subject to further failures (i.e. its time-to-replacement monotonically decreases) whilst repaired/working lines are subject to failures.

2.4. Probabilistic load demand

Load demands generally display time and space correlations, i.e. the variability of the different power demands depends on the time of the day and their relative position within the network. Stochastic models for load demand have been designed to account for time correlation, e.g. [38]. The model employed here considers daily variability of the average load demand and neglects seasonal and holiday effects. The aggregated load connected to a node i at a time t (L t( )i ) can be de- scribed by a Normal distribution with parameters fitted on historical data [11]:

= −

−

( ) ( ) f L t

π σ t e( ( ) )

1 2

i L

L t μ t

σ t

( ) ( )

2 ( )

i

i Li Li

2

(12)

where L t( )i is the load demand at node i at hour of the day t μ t, ( )Li is the load mean value and σ t( )Li is the standard deviation at node ∈i N and time t.

3. Artificial neural networks: OPF load curtailed emulator

A reliable estimation of the Energy-Not-Supplied requires a high number of simulations to be performed, i.e. many ENS have to be computed over the simulation time Tsim and even more OPF solved (one for each failure). In general, the time needed to solve a OPF is not too large, which is especially true if the size of the network is modest. If necessary, the solution can be fastened by employing parallel com- puting strategies, i.e. distributing the OPF to several cores in a com- puter cluster. Unfortunately, the optimisation problem cannot be vec- torised and the OPF has to be solved one-at-a-time. This can be computationally time–costly if a large number of model evaluations are necessary.

To address the computational challenge, an Artificial Neural Network (ANN) is trained to emulate the power flow solution. An ANN is a mathematical model defining a function →K I Y: where K g( ) is a composition (e.g. non-linear weighted sum) of other weighted functions g x( )i . The basic architecture for a feed forward ANN consists of one input layer, one or more hidden layers and one output layer [39,40]. Each layer employs several artificial neurons, also known as nodes, which are connected to the neurons of the adjacent layers by weighted links. In each neuron, the inputs are first weighted and, then, summed as follows:

∑= + =

g x ω g x b( ) · ( ) i

n

i i 1

where ωi are the weights, g x( )i is the output of the node i in the previous layer and b is the bias, which is generally introduced in the hidden and output layers and acts as a threshold for the argument of the activation function. The sum g x( ) is processed by an activation function K to produce the neuron’s output. A. The employed activation function is the widely used sigmoidal function, defined as follow:

= + −

K g e

( ) 1

(1 )g

An illustrative example of neural network architecture and node functionality is depicted in Fig. 2.

The jth ANN input vector assembles the loads demanded in the nodes and the state vector of the lines connecting the nodes =I L X[ , ]j j. The load vector for the failure state f is = …[ ]L LL , ,f N f1 L , whilst the lines states vector is = …X XX [ , , ]f f1 | |E , where ∈ +L1 � and ∈X {0,1}i and the cardinality of the link set is the number of lines in the grid =N | |l E . The jth input vector Ij is associated to a positive real valued output, the total load curtailed = ∑ ∈Y Lj i cut i j, ,N , obtained solving the minimisation problem in Eq. (1).

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4. The proposed efficient framework

Fig. 3 presents the flow chart of the proposed framework, where the Artificial Neural Network is employed to speed up the calculations. The algorithm starts by inputting parameters of the stochastic model (e.g. shape and scale parameters, critical wind speed, failure rates, etc.) and two main blocks can be seen. The first block is, in essence, an event sampler whilst the second uses the emulator to calculate the load cur- tailed for each sampled failure event f. Then, the ENS is obtained by Eq. (3).

First, ‘normal’ time to failures and occurrence time of severe weather events are sampled using HPP and NHPP, respectively. Then, a vector of Times-to-Events (TTEs) is obtained by sorting the occurrence times and recording the type of event (i.e. normal failure, lighting and/ or strong wind). Then, a sequential Monte Carlo (S-MC) starts, see as example Refs. [9,41]. This iterative procedure terminates if the max- imum simulation time is reached ( >t 8760 [h]) or all the sampled events (normal failure and severe weather) have been analysed. The S- MC procedure is summarised as follows:

1. Set t equal to the occurrence time of first event, TTE( =e 1) and failure index =f 0. If e is a ‘normal’ failure, go to point 2 otherwise go to point 3.

2. Set = +f f 1 and sample a line i from the probability mass dis- tribution with values

∑ =

λ l X

λ l X

· ·

· ·

n i f i

l Nl n i f i

,

1 , and =l N1,.., l. Set =X 0f i, , sample a

load profile (Lf ) accordingly to t. Set the failed line replacement to 100 % and save Xf and Lf for f. Update % from full replacement using TTE( +e 1)-TTE(e) and normal replacement speed. Then, go to point 5.

3. Sample the severe weather duration (Te) and intensity (Ng,Δw), compute the increased total failure rate λ t( ) using Eq. (8). Sample time-to-failure using the HPP, and by inputting λ t( ) and the interval [t +t Te]. If at least one failure event is sampled, set t equal to the next failure occurrence and proceed to step 4, otherwise go to step 5.

4. Set = +f f 1, sample one failed line using the probability mass function

∑ =

λ t l X

λ t l X

( )· ·

( )· ·

i f i

l Nl i f i

,

1 , and =l N1,.., l. Sample a load profile (Lf ) ac-

cordingly to t. Set the failed line replacement to 100 % and save the lines states vector Xf and load profile Lf for failure f. Update lines % from full replacement, using the reduced repairing speed computed as in Eq. (11). If the severe weather failures are all evaluated go to step 5, otherwise set t equal to the occurrence time of the next failure and repeat point 4.

5. If >t Tsim or e is the last event in the TTE list, stop simulation.

Otherwise, set = +e e 1 and =t TTE e( ). If it is a ‘normal’ failure, go to point 2 otherwise go to point 3.

The first unit is used to produce a set of Xf and Lf to be used as vectorised input to a previously trained ANN. The ANN input is an

+ ×N N F( )l L matrix of load profiles and state vectors, where F is the total number of failures faced by the grid in 1 year. A total of Ns in- dependent histories (Ns S-MC) are simulated until convergence of the

ENS[ ]� is obtained. Fig. 4 displays a simplified version of the methods used within the

framework. The procedure used to simulate Ns independent grid years is displayed in the top panel on the left-hand side. The method run by first selecting the OPF solver or the efficient method based on the ANN emulator (i.e. presented in Fig. 3). If the (time costly) S-MC optimal power flow solver is selected, the procedure displayed in the top panel on the right-hand side run Ns times (the diagram has been adapted from [9]). If the efficient method is selected, the procedure displayed in Fig. 3 run Ns times.

The bottom panel in Fig. 4 summarises the overall work flow for the analysis. This is used to present, from an intuitive point of view, how the efficient resilience assessment method is embedded within an ad- vanced uncertainty quantification framework to assess effects of im- precision (i.e. introduced in Section 5). The analysis starts by simulating Ns grid years using the S-MC OPF solver. Then, the ANN is trained as explained in Section 3 and its results are validated against the original model (S-M OPF). Once the emulator is validated, the imprecision af- fecting the poorly known parameters of the grid and weather models is characterised using Credal sets (please see Section 5.1). To conclude, the effect of imprecision on the grid resilience is quantified using ad- vanced uncertainty propagation methods and global sensitivity ana- lysis.

5. A generalised framework for uncertainty quantification

The assessment of power grid reliability indices is traditionally based on well-defined probabilistic models. The definition of these models may require a large body of empirical information to estimate the parameters of the underlying probability distributions. This is not always available in practice and imprecision in the parameters of the deterministic and probabilistic models must be accounted for [28]. Bayesian methods and set-theoretical methods are two of the most widely applied paradigms to deal with epistemic uncertainty. The for- mers are entirely based on probability theory and use probability dis- tributions to describe lack of knowledge (e.g. uniform probability dis- tributions). The set-theoretical models [42] use set-valued descriptors (e.g. intervals) to model epistemic uncertainty, e.g. intervals [43], random sets [44] or fuzzy sets [45]. Intervals are used when variables are only known to be bounded within lower and upper limits whereas Fuzzy Sets can be used to simultaneously analyse different bounded sets. This is particularly helpful if the bounds are not precisely known [30,15]. Credal sets theory [46] provides strong mathematical foun- dation to express sets of probability distributions and is for this reason employed in this work to explore different levels of imprecision, which can affect the parameters of the probabilistic model.

5.1. Credal sets

A Credal set (C) is defined as a set of probability distribution functions. The hyper-parameters p defining the joint probability dis- tribution (HD) can be given as intervals (i.e. an n-orthotope also called hyper-rectangle). In this work, without loss of generality, Credal sets for which distribution families are specified and independent, and interval- described parameters provided are considered. The Credal set is defined as follows:

= < <H η p p p p{ ( ; ) | }DC (13)

Fig. 2. Conceptual scheme of an Artificial Neural Network architecture and the function of an artificial neuron.

R. Rocchetta et al. Applied Energy 210 (2018) 339–350

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where η is the random variable and p and p are hyper-parameters re- presenting lower and upper bounds, respectively.

5.2. Expectations in generalised uncertainty models

When the probabilistic model input is precisely defined, i.e. = =p p p, its output will also be precisely specified, i.e. a crisp prob-

ability distribution and expectation. The precise expectation of the probabilistic model H η p( ; )D is defined as:

∫=H η H η dηp p[ ( ; ) ] ( ; )D D Ω

� (14)

When imprecision is affecting the problem, the expectation becomes imprecise and can be obtained as follows:

∫⎡ ⎣⎢

⎤ ⎦⎥

= ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

< <

< <

H η H η

H η dη p p

p [ ( ; ) ] [ ( ; ) ]

sup

inf ( ; )D

D D

p p p

p p p pΘ ( )

�

� (15)

where Ω is the probability space, Θ is the imprecise probability space dependent on the vector p,� and � are the upper and lower expecta- tions, respectively. In order to compute bounds for the expectation, a minimisation (inf ) maximisation (sup) problem constrained by the

hyper-parameter space bounds p p, has to be solved [47]. For the so- lution, different approaches can be used, for instance vertex methods [48], global optimisation techniques [49,50], sampling or interval ar- ithmetic methods [51]. In this work, the parameters of the coupled grid- weather model are considered affected by imprecision and the ENS[ ]� bounds are approximated by searching its minimum and maximum values within the hyper-parameter space.

6. A case study

The framework has been applied on a modified version of the IEEE- RTS 24 nodes power grid [52], which counts 24 nodes, 17 loads, 32 generating units, 33 transmission lines, 5 transformer links and a total installed capacity of 3.405 GW. The adopted line failure rates in normal weather conditions and lengths are presented in Table 2, the transfor- mers branches have been assumed fully reliable, i.e. =λ 0n l, where lines l are 10–11, 10–12, 9–11, 9–12 and 3–24. The mean load per hour of the day and node is presented in Fig. 5, the standard deviation σ t( )Li is assumed to be 10% of the mean for each t and node.

Fig. 3. A simplified flow-chart for the resilience analysis by sequential Monte Carlo simulation. The probabilistic model is used to sample failures and repairs whilst the Artificial Neural Network is used to compute load cur- tailments.

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6.1. Results: OPF sequential Monte Carlo

Depending on the number of failures sampled within each simulated year, sequential Monte Carlo analysis requires between 4 s and 10 s on a standard desktop machine (8.0 GB RAM and 2 GHz processors) using the full OPF model. Fig. 6 shows 9 failure events randomly occurred between hours 27 and 293. The first 3 lines failures (displayed by thicker lines) occur independently and in either normal or severe weather conditions. The time needed to replace the line increases due to high wind, 5.3 [h] for the first event and 6.2 [h] for the second, without additional failures. Conversely, the failures of lines 15–24, 12–13, 8–9 and 15–16 are driven by a common generating event (high wind), which occurs at hour 177 and with random duration of 3 h. It can be observed that the repair crew is unable to restore the lines in such a small time window.

6.2. Results: traditional UQ and artificial neural network performance

A parallel computing strategy has been used to solve 15,000 in- dependent years; the computational time required for its completion is about 1 h and 15 min on a 20 cores machine cluster with 8.00 Gb ram and a 2.00 GHz Intel® Core™ i5-4590T processor. The average number of failure events per each =T 8760sim [h] (1 year) is estimated to be 311,

Fig. 4. A diagram for the overall work flow of the analysis (in the bottom panel). The procedure to compute the

ENS[ ]� using the S-MC (in top panel on the left-hand side) and the (computationally demanding) algorithm adapted from [9] (in the top panel on the right-hand side).

Table 2 The line failure rates in normal weather conditions and line lengths. The transformers links are assumed perfectly reliable and not reported within the table.

Line i li [km] λn i, ⎡ ⎣

⎤ ⎦

occ km year

Line i li [km] λn i, ⎡ ⎣

⎤ ⎦

occ km year

1-2 4.8 −1.24·10 3 11-13 53.1 −0.19·10 2

1-3 88.5 −0.144·10 2 11-14 53.1 −0.19·10 3

1-5 35.4 −0.233·10 3 12-13 53.1 −0.19·10 3

2-4 53.1 −0.184·10 3 12-23 107.8 −0.121·10 3

2-6 80.4 −0.149·10 3 13-23 96.5 −0.126·10 3

3-9 49.8 −0.19·10 3 14-16 43.4 −0.218·10 3

4-9 43.4 −0.2·10 3 15-16 19.3 −0.42·10 3

5-10 37.0 −0.23·10 3 15-21 54.7 −0.19·10 3

6-10 25.7 −0.32·10 3 15-21 54.7 −0.19·10 3

7-8 25.7 −0.29·10 3 15-24 57.9 −0.177·10 3

8-9 69.2 −0.16·10 3 16-17 28.9 −0.3·10 3

8-10 69.2 −0.16·10 3 16-19 25.7 −0.33·10 3

21-22 75.6 −0.15·10 3 17-18 16.0 −0.5·10 3

20-23 24.1 −0.35·10 3 17-22 117.4 −0.115·10 3

20-23 24.1 −0.35·10 3 18-21 28.9 −0.3·10 3

19-20 44.2 −0.215·10 3 18-21 28.9 −0.3·10 3

19-20 44.2 −0.215·10 3

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of which 73 normal failures 223 wind-induced failures and 15 light- ning-induced failures. The expected load curtailed for each failure event is estimated to be 0.23 [W/failure]. The ENS[ ]� is 147.5 [MWh/yr] with coefficient of variation (CoV) 0.175, and slowly converges after about 2000 simulations.

A total of 100 years worth of failures (about 31,000 events) are randomly selected. For each failure event f, the lines state vector (Xf ) and load samples (L t( )i f, ) are assembled and 70% are used as training set for an ANN. The remaining samples are used for validation (15%) and testing (15%). The network architecture consist of 1 input layer counting 50 nodes, 2 hidden layers (45 and 35 nodes) and 1 single node output layer. The architectural selection was done by trial and error of hidden layers and number of neurons per layer. The topology with lowest mean squared error between the emulator output and the ori- ginal model target was selected. The regression plot is presented in the top panel of Fig. 7 and the regression coefficient is about 0.98–0.99.

This is considered a satisfactory result and no further optimization of the ANN architecture was performed. In the bottom panel of Fig. 7, a comparison between the total load curtailed ∑ ∈ Li cut i t, ,N computed using the OPF and the ANN (diamonds marked line) is reported.

The algorithm displayed by Fig. 3 has been run 15,000 times (i.e. 15,000 simulated years). The ANN has been used to calculate the load curtailed for each failure event and the ENS is obtained. In Fig. 8, the

ENS[ ]� computed is displayed by the dotted line and converges to 145.5 [MWh/yr] with CoV 0.185. Compared to the OPF result (solid line) the error of the ANN is just 0.5% but the reduction in computa- tional time is remarkable 98.8%: in fact, just 58.4 s were needed to solve the 15,000 independent S-MC histories (on the 20 workers cluster).

Fig. 5. The mean of the load value per node and hour of the day, μ t( )Li , for the modified IEEE-RTS [52].

Multiple Failures due to Strong Wind Occurance

Fig. 6. An example of 9 sequential failure events ex- tracted from a simulated year for the grid. Strong Wind occurrence (from hour 177 to hour 180) increase line failure rates and decrease the repair speed, hence, leading to 4 common cause outages.

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6.3. Results of the generalised uncertainty quantification

The probabilistic model describing the uncertainty affecting the grid and weather is extended to include the Credal set

= ∈ < <H η p p p p p{ ( ; ) | }D 45C � (16)

The imprecise parameters p accounted in the analysis are:

= β α W a b a b a b ν μ σ λp [ , , , , , , , , , , , , ]lg w crt D D D D norm N N n iΔ Δ ,w w lg lg w w g g

The parameters uncertainty (intervals) are characterised using the quantity ι which quantifies the extent of the imprecision. Defining the estimated values of the parameters as pest, lower and upper bounds are defined as = −ιp p (1 )est and = + ιp p (1 )est , respectively. The levels of imprecision considered in this analysis are

=ι {0,0.01,0.02,0.03,0.04,0.05,0.06}. A value of ι=0 means that the

probabilistic analysis is performed assuming a perfect estimation of the parameters and equal to pm. If ι = 0.06, the true value of the parameters is ±6% imprecise. Based on previous analysis, the number of samples used to compute ENS[ ]� is set equal to 5000 and each Credal set is propagated by minimising (inf ) and maximising (sup) the ENS[ ]� ex- pectation within the bounds ∈p p p[ , ]. In this work, ENSinf [ ]� and

ENSsup [ ]� are obtained using a derivative-free stochastic optimiser, which constrains p within the hypercube p p[ , ] [49,15]. Fig. 9 shows the nested bounds for the ENS[ ]� displayed as a Fuzzy set and the results are summarised in Table 3. The model based on power-flow (dashed line) and the surrogate model (solid line) are compared. The ANN was effective in capturing the imprecision trend expected from the original model. It is clear that the larger is the imprecision in the parameters, the wider the bounds of the resilience index are. The analysis performed using the original OPF model were very time consuming, thus, for

Fig. 7. The regression plots for the ANN (top panels) and an example of load curtailed computed using the OPF compared to the ANN result (bottom panel).

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comparison purposes just two =ι {0,0.06} were considered. It is also interesting to notice that, although the level of imprecision for each input was relatively modest (maximum of 6%), it produces a severe imprecision on the model expectation (21–29%). This is partially con- firming earlier results in the structural reliability analysis context [15], where sensitivity of the system reliability to the imprecision in the probabilistic model parameters has been pointed out. This issue has proven to be particularly relevant if many imprecise variables are in- volved. In those cases, a solution based on double loop Monte Carlo would have been infeasible, or extremely time consuming. The pro- posed approach and obtained results are indeed useful from a decision maker perspective, for instance, to identify the maximum level of input imprecision (e.g. tolerances) in order to assure a minimum power grid resilience standard (i.e. ENS[ ]� less than a predefined threshold level). Considering for example a minimum allowed resilience level, set to

be ⩽E ENS[ ] 1.6105 ⎡ ⎣

⎤ ⎦

Wh year

, then, the maximum level of tolerance im- precision for the considered parameters has to be limited to just 2%.

6.4. Variance-based global sensitivity

Variance-based sensitivity aims at quantifying the importance of a model input by assessing the expected reduction in model output var- iance induced by a reduction in input variances (e.g. knowing the value of the model input with certainty) [53]. The total effect sensitivity index, i.e. the first order (additive) and higher order (interactions) ef- fects of factor i, can be expressed as − =1 Var E Y X x

Var Y [ [ | ] ]

[ ] i i , where the var-

iance of the expectation of the output Y when fixing the input Xi to the value xi is divided by the total variance of the output. Similarly, the first order sensitivity index can be obtained as =Var E Y X x

Var Y [ [ | ] ]

[ ] i i and it quanti-

fies the additive effects in the model [14].

Global sensitivity analysis has been performed using =Y ENS[ ]� and assuming uniform distributions for the imprecise parameters in the range − +p p[ (1 0.1), (1 0.1) ]est est . For simplicity, the normal failure rates of the lines are set constant to the nominal value parameters. The analysis is performed using the method proposed in Ref. [14] and employing the surrogate-based Sequential Monte Carlo (see flow chart in Fig. 3). The results are presented in Table 3 and in Fig. 10. It can be observed that the model is more sensitive to changes in the replacement speed in normal weather conditions and to changes in the wind dura- tion scale parameter (a Dw), and to wind events over lightnings events in general.

7. Discussion on the use of the framework in a practical context

In real situations, power grid analysts have to complete the needed

Fig. 8. Comparison between ENS[ ]� computed using ANN and the OPF solver.

Fig. 9. Comparison between Fuzzy ENS[ ]� computed using the original OPF model and the ANN surrogate.

Table 3 Comparison between upper and lower ENS[ ]� bounds obtained using the ANN (6 Credal sets) and the OPF (2 Credal sets). The levels of imprecision in the input model parameters and corresponding imprecision in the expectation upper and lower bounds is also pre- sented.

Input OPF (MC) ANN Output (ANN)

ι [%] ENS[ ]� ENS[ ]� ENS[ ]� ENS[ ]� ι [%] � ι [%] �

0 146.9 146.9 145.5 145.5 0 0 1 n.a. n.a. 139.9 151.7 3.8 4.3 2 n.a. n.a. 135.0 158.5 7.2 8.9 3 n.a. n.a. 129.8 163.4 10.7 12.4 4 n.a. n.a. 123.9 172.3 14.8 18.4 5 n.a. n.a. 119.8 177.2 17.6 21.8 6 116.7 185.3 114.5 187.3 21.3 28.8

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assessments quickly, with a limited amount of time. Unavoidably, the information available for the assessment will be limited and in some cases, when the data does not suffice, expert judgment elicitation is the only viable way for carrying out the assessment.

Both problems of computational efficiency and limited data avail- ability have been discussed and tackled in this work. To overcome lack of data issues, a generalised uncertainty quantification framework has been adopted. Computational efficiency has been greatly improved by using a novel emulator of the optimal power flow and a simulation tool allows the impact of extreme weather on the power grid to be assessed.

When the information is poor, inconsistent or limited, it is common in practice to proceed with some (strong or weak) assumptions based on expert judgement. In the power grid reliability assessment context, a typical situation is to have just a few available failure samples for a given (reliable) component type. Similarly, in a weather modelling context, few samples of a (rare) weather event will be available and it will be difficult to assess with confidence its occurrence rates. In those cases, the lack of data is probably a most relevant source of uncertainty. Nevertheless, it is common to assume a well-defined probability model to describe the component failure behaviour (e.g. precise point-valued failure rates) or the weather occurrence (e.g. precise events occurrence rates). These assumptions will likely lead to an underestimation of the effect of the uncertainty. To improve the overall robustness of the analysis, it is necessary to rigorously assess the effect of lack of data on the results. Within an imprecise-information scenario (e.g. few samples, probability distribution not specified or unknown, or known but with vague parameters, conflicting and limited knowledge, linguistic in- comprehension, single or multiple intervals, etc.), generalised (im- precise) probabilistic approaches can be employed to model lack of information with fewer constraining assumptions. Once the uncertainty is propagated through the model, the result will discriminate between the non-reducible uncertainty (aleatory components, randomness, variability) and reducible uncertainty (epistemic component, lack of information). This can trigger a discussion on whether it is necessary to collect more data (which can be costly) to guarantee a probabilistic standard, for instance, the 1-day-in-10-years loss of load probability standard or an expected energy not supplied less than a predefined threshold level.

The proposed framework can be used to tackle those types of issues and within reasonable time thanks to the improvement in computa- tional efficiency by the emulator. As an additional example, consider the sensitivity results. The results show that the lack of data on the repairing crew speed in normal weather conditions and for the prob- abilistic model of the wind is affecting the most the grid resilience precision. Consequently, a decision-maker which recommends an im- precision reduction strategy (i.e. reduce the tolerance intervals on the parameters) would suggest starting the data collection to refine the

wind model and the repairing crews model first. In this way, the as- sessment would profit the most in term of resilience index imprecision reduction and, if the score is not satisfactorily low, a strategy to en- hance the network resilience can be discussed (e.g. by reducing the average repairing crew intervention time).

8. Conclusion

A generalised uncertainty quantification framework for power grid resilience assessment has been presented. A power grid stochastic model accounts for interactions between severe weather conditions, transmission line failures and repairing crew working efficiency. The stochastic model needs accurate estimation of its parameters and data is often insufficient or limited. Thus, the framework has been extended adopting a generalised, non-intrusive uncertainty quantification method. An efficient solution has been proposed employing a vectori- sable Artificial Neural Network to emulate the relation between load curtailments, lines states vectors and load profiles. The proposed sur- rogate reduces the computational time of about 99% although a small error is inevitably introduced in the results. The surrogate accuracy was tested against the original model based on OPF; both classical and generalised stochastic framework were tested and pointed out the goodness of the emulator. The effect of imprecision was tested by propagating nested Credal sets to the Expected-Energy-Not-Supplied. Results pointed out that the precision of the estimator decreases rapidly for increasing imprecision in the model parameters. To conclude, a global sensitivity analysis pointed out which among the wind, lightning and replacement related parameters are key drivers for the uncertainty in the proposed power grid resilience index.

The main contributions of this work can be summarised as follow:

1. A novel simulation method based on a computationally cheap emulator of the optimal power flow is presented and used to speed up the computation of the expected energy not supplied by the network.

2. The method greatly reduces the computational cost of the time-de- manding analysis (up to a 99% reduction).

3. Problems of lack of data are discussed and the efficient simulator, embedded within a generalised uncertainty quantification frame- work, allows the effect of lack of data to be quantified.

4. Sensitivity analysis has allowed to point out which among the im- precise parameters have to be prioritised if further data were col- lected (for highest reduction of the imprecision in the resilience index).

Fig. 10. Total sensitivity indices.

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Acknowledgements

The authors would like to acknowledge the gracious support of this work through the EPSRC and ESRC Centre for Doctoral Training on Quantification and Management of Risk & Uncertainty in Complex Systems & Environments Grant No. (EP/L015927/1)

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