mitigating the risk of cascading blackouts: a data...

Received May 7, 2018, accepted July 6, 2018, date of publication July 16, 2018, date of current version August 7, 2018.

Digital Object Identifier 10.1109/ACCESS.2018.2855153

Mitigating the Risk of Cascading Blackouts:A Data Inference Based Maintenance MethodFENG LIU 1, (Member, IEEE), JINPENG GUO 1, (Student Member, IEEE),XUEMIN ZHANG1, (Member, IEEE), YUNHE HOU 2, (Senior Member, IEEE),AND SHENGWEI MEI 1, (Fellow, IEEE)1State Key Laboratory of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing 100084, China2Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong

Corresponding author: Feng Liu ([email protected])

This work was supported by the National Natural Science Foundation of China under Grant 51621065.

ABSTRACT The risk of cascading blackouts (RCB) is of great significance in practice because cascadingoutages can have catastrophic consequences. As there is a positive relationship between the probability ofcascading blackouts and that of component failures, an effective way to mitigate the RCB is to performmaintenance. However, this approach is of limited value when considering extremely complicated cascadingoutages, such as those in particularly large systems. In this paper, we propose a methodology to efficientlyidentify the most influential component(s) for mitigating the RCB in a large-power system based oninference from the simulation data. First, we establish a data-based analytic relationship between the adoptedmaintenance strategies and the estimated RCB. Then, we formulate the component maintenance decision-making problem as a nonlinear 0–1 programming problem. We then quantify the credibility of the estimatedRCB and develop an adaptive method to determine the minimum required number of simulations, which is acrucial parameter in the optimizationmodel. Finally, we devise two heuristic algorithms to efficiently identifyapproximately optimal solutions. The proposed method is then validated by way of numerical experimentsbased on IEEE standard systems and an actual provincial system.

INDEX TERMS Data inference, maintenance strategy, risk of cascading blackouts.

I. INTRODUCTIONUnder certain conditions, system disturbances or componentoutages in a power system can trigger a sequence of com-ponent failures, or cascading outages, that can have seri-ous consequences, even catastrophic blackouts. Althoughthe probability of such catastrophic blackouts is very small,the results of theoretical research and practical experienceindicate that the risk of cascading blackouts (RCB) shouldnot be ignored [1]–[4].

Intuitively, component maintenance is seen as an effec-tive means to mitigate RCB since it directly reduces theprobability of component failures [5], [6]. In practice, of thelarge number of components in a power system, a smallsubset exerts a disproportionately large influence on theRCB [7]–[9]. Therefore, by preferentially maintaining thesubset of the most influential components, the RCB can beminimized with limited resources. It should be noted thatthis is not a new idea, but has been extensively deployedin conventional reliability-centered maintenance or risk-based maintenance methods. In [10], a risk-based resource

optimization model for transmission system maintenance isproposed where both the maintenance strategies and corre-sponding risk reductions serve as input data. From a dif-ferent perspective, risk can be defined and calculated viascenario enumeration in which components associated withhigh-risk scenarios are selected for maintenance [11]. In amore rigorous approach, the researchers in [12] leverage0-1 integer programming to optimize the system risk usinglimited resources in power systems, where the system risk isdefined as the sum of the component risks and is calculatedby enumeration. A similar optimization model is presentedin [6]. In all of the above methods, maintenance strategiesare selected based on evaluating the risk with respect to themaintenance strategies under consideration.

While the aforementioned methods have been shown towork well in both reliability-centered maintenance or risk-based maintenance methods, this is not the case when cas-cading outages are considered. In fact, few practical methodsare available for analyzing cascading outages in large systemsdue to the complexity of the required computations.

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F. LIU et al.: Mitigating the RCB: Data Inference-Based Maintenance Method

As is well known, the propagation of a cascading outageis a complicated dynamic process involving various ran-dom factors, which makes it impossible to analytically cal-culate the RCB. Hence, statistical methods like the MonteCarlo (MC) method are often employed to indirectly estimatethe RCB. However, to realize a credible estimation, the MCmethod must generate a large number of samples via cas-cading blackout simulations [13]–[15], which is extremelytime consuming [16] for large systems in particular, andmaintenance only exacerbates this problem. Since there is noanalytic relationship to bridge the estimated RCB with theprobability of component failure that vary with maintenancestrategies, samples used to estimate the RCB with respect toa particular maintenance strategy cannot be used for another.This implies that whenever themaintenance strategy changes,all blackout samples must be completely regenerated by con-ducting blackout simulations. Considering the large scale ofreal power systems and the immense number of possiblemaintenance strategies, both sample generation and estima-tion of RCB are extremely time-consuming. This furtherincreases the intractability of the corresponding maintenanceoptimization problem.

In addition to the MC method, importancesampling [17]–[19] and Splitting [20]–[22] are two otherefficientmethods to simulate cascading outages. In the impor-tance sampling method, the probability of severe cascadingoutages is ‘amplified’ so that more rare cascading blackoutscan be captured. As for the Splitting method, the basic ideais to divide the simulation process into sub-levels and thencopy samples in the beginning of each sub-level, so that morerare cascading blackouts can be obtained. It is worthy ofnoting that despite the important sampling method and theSplitting method can remarkably improve the simulation effi-ciency, they cannot explicitly reveal the relationship betweencomponent failure probabilities and the corresponding RCB.Therefore, they face with similar difficulties to the MCmethod in the maintenance optimization problem.

Guo et al. [23] exploit the information pertaining to cas-cading blackouts that is buried in the data of cascadingblackout simulations in their development of a semi-analyticmethod to characterize the relationship between the unbiasedestimation of the RCB and component failure probabilities.With this approach, it is possible to directly estimate theRCB under varying component failure probabilities with noneed to regenerate any samples. In other words, this poten-tially suggests an efficient RCB evaluation approach thatcan address varying maintenance strategies. In this paper,we extend that approach to address the optimal componentmaintenance problem while considering the RCB. The majorcontributions of our work are threefold:

1) An analytic relationship between the estimated RCBand maintenance strategy is established based on infer-ence from the cascading blackout simulation data.On this basis, the optimal component maintenanceproblem is formulated as a nonlinear 0-1 programmingproblem. In the model, the evaluation of the RCB with

respect to varying maintenance strategies is explicitlyexpressed based on the initial simulation data.

2) To guarantee the validity of the RCB evaluationresult when solving the proposed optimization model,we propose an adaptive method to determine an appro-priate sample size via credibility analyses based oninference from the simulation data.

3) The proposed optimizationmodel is a high-dimensionalnonlinear 0-1 programming problem. However, forsimplicity, we propose two simple heuristic algorithmsto search for nearly-optimal solutions with very highefficiency. This guarantees the proposed methodologycan be employed in real large power systems.

The rest of this paper is organized as follows. Section IIprovides an overview of the basic definitions and notations,based on which the optimization problem is formulated.In Section III, based on credibility analyses of the estimatedRCB, the sample size, which is the critical parameter, is deter-mined. The two high-efficiency heuristic algorithms andaccompanying holistic procedure are presented in Section IV,while several case studies are described in Section V. Finally,our conclusions are presented in Section VI.

II. PROBLEM STATEMENT AND FORMULATIONIn this work, we develop a method to mitigate the RCBby maintaining a few key components in large systems.In essence, the influence of maintenance on the componentfailure probabilities can be estimated based on historical dataand experience. Therefore, the critical issue lies in char-acterizing the relationship between the component failureprobabilities and RCB (or more precisely, the estimationof the RCB). As long as the relationship is obtained, it isstraightforward to determine suitable maintenance strategiesby enumeration or optimization. To this end, an analyticrelationship is preferable for optimizing the maintenancestrategy. To realize this, we employ the method describedin [23] to analytically characterize this relationship throughinference from the simulation data, and then to explicitlyformulate the optimization problem.

A. DEFINITIONSWebegin by providing basic definitions of cascading outages,the component failure probability function, and the RCB.

Invoking the description in [17], in a broad class of steady-state simulation models, an n-stage cascading outage can bedefined as a Markov sequence denoted by

Z := {X0,X1, . . . ,Xj, . . . ,Xn,Xj ∈ X ,∀j ∈ N}, (1)

with respect to probability series g(Z ). Here, N :=

{1, 2, · · · , n} is the set of cascading stages; j is the stagelabel; and Xj represents the state variables of the sys-tem at stage j, which can be ON/OFF states of compo-nents, power injections at each bus, etc. In particular, X0is the initial system state of the cascading outage, which isassumed to be deterministic. The state space X is assumedto be finite. With regard to a specific cascading outage,

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z = {x0, . . . , xj, . . . , xn} and its probability (series) is denotedby g(z) := g(x0, . . . , xj, . . . , xn). At stage j, the failure prob-ability of component k is denoted by

ϕk (xj) := Pr(component k fails at xj ), (2)

where ϕk is referred to as the component failure probabilityfunction, which is determined by the inherent characteristicsof the component, e.g., the component type, operating condi-tion, etc. In the literature, there are various forms of ϕk [3],[13]–[15], [24]. On the other hand, if a certain componentk is maintained, the failure probability function will changeaccordingly. We denote the failure probability function ofcomponent k after maintenance by ϕ̄k . It is worthy of notingthat it is difficult to estimate the impact of maintenance onthe component failure probability and choose appropriate ϕ̄kin practice. Because there are many maintenance methodsand various kinds of components with different ages, whilethe accurate data related to the specific component and main-tenance method is far from sufficient. However, some exist-ing methods can facilitate dealing with this issue [25]–[27].In this work, we focus on the influence of component failureprobabilities on RCB and the maintenance-based risk man-agementmethod. Therefore, we simply employ generic formsof ϕk and ϕ̄k , and do not discuss in detail the specific methodsto obtain ϕk and ϕ̄k in this paper.

Note that a cascading outage can involve load shedding.The general definition of the RCB associated with loadshedding is as in [17]. Specifically, the load shedding ofa cascading outage is denoted Y , which can be consideredas a function of the corresponding cascading outage, i.e.,Y = h(Z ). Then, the RCB with respect to g(Z ) and a givenload shedding level Y0 is

Rg(Y0) := E(Y · δ{Y≥Y0}) =∑z∈Z

g(z)h(z)δ{h(z)≥Y0}, (3)

where Z is the set of all possible cascading outages andδ{Y≥Y0} is an indicator function of the events {Y ≥ Y0}, givenby

δ{Y≥Y0} :=

{1 if Y ≥ Y0;0 otherwise.

(4)

The RCB defined in (3) is the expectation of load sheddingbeyond the given level Y0. When Y0 = 0, this is consistentwith the traditional definition of blackout risk. When Y0 > 0,this is consistent with the risk of those events with seriousconsequences, i.e., those with a load shedding greater than Y0.

B. ANALYTIC RELATIONSHIP INFERENCE FROM THESIMULATION DATAInvoking the Markov property and conditional probabilityformula, g(z) can be rewritten as

g(z) = g(xn, · · · , x1, x0) =n−1∏j=0

gj+1(xj+1|xj), (5)

where gj+1(xj+1|xj) represents the transition probability fromstate xj to state xj+1. Considering the failure components atstage j, (5) is equivalent to

g(z) =n−1∏j=0

∏k∈Fj

ϕk (xj) ·∏k∈F̄j

(1− ϕk (xj))

, (6)

where Fj is the component set consisting of the componentsthat are defective at xj+1 but operate normally at xj, while F̄jconsists of the components that function normally at xj+1.Since maintenance only influences some of the compo-

nents in the system, we consider the items related to a specificcomponent k ∈ K in (6) and define

0(ϕk , z) :=

n−1∏j=0

(1− ϕk (xj)) : if nk = n

ϕk (xnk )nk−1∏j=0

(1− ϕk (xj)) : otherwise

(7)

where K is the set that includes all components; and nk is thestage at which component k fails. In particular, if componentk does not fail in the whole cascading process, let nk := n.Then

g(z) =∏k∈K

0(ϕk , z). (8)

Substituting (8) into (3) yields

Rg(Y0) =∑z∈Z

(h(z)δ{h(z)≥Y0} ·

∏k∈K

0(ϕk , z)

). (9)

The inherent relationship between Rg(Y0) and ϕk is indi-cated in (9). However, since |Z| and |K | 1 may be verylarge, it cannot be directly used in optimization. Alternatively,we use an approximation of Rg(Y0) in terms of a set ofsamples, which is given by

R̂g(Y0) =1N

N∑i=1

h(zi)δ{h(zi)≥Y0}. (10)

In (10),N is the number of independent identically distributed(i.i.d.) samples generated using specific blackout simulationmodels with respect to g(Z ) (or more precisely, ϕk , k ∈ K );zi = {x i0, . . . , x

ij , . . . , x

ini} is the i-th sample, where ni is the

number of stages in the i-th sample. In particular, we defineZg := {zi, i = 1, · · · ,N }, which represents the sample setgenerated with respect to g(Z ).When some of the ϕk values change due to maintenance,

g(Z ) will be converted into another probability series that isdenoted f (Z ). With regard to Zg, Guo et al. [23] state that theunbiased estimation of Rf (Y0) is given by

R̂f (Y0) =1N

N∑i=1

w(zi)h(zi)δ{h(zi)≥Y0}, (11)

1| · | represents the cardinality of a set.

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where w(zi) is the sample weight of zi and is defined as

w(zi) :=f (zi)g(zi)

, (∀zi ∈ Z). (12)

According to (11), it is interesting to note that R̂f (Y0) onlyrequires the information of zi that is generated in the blackoutsimulations with respect to g(Z ). This implies that when g(Z )changes to f (Z ), the RCB can be directly obtained usingthe information of Zg, rather than by regenerating samplesvia blackout simulations. Therefore, this approach provides ameans to explicitly express the RCB with respect to mainte-nance strategies, which will greatly facilitate the formulationof optimal maintenance decision-making problems.

C. FORMULATION OF MAINTENANCESTRATEGY OPTIMIZATIONBinary variablemk represents the maintenance status of com-ponent k . If component k is maintained, mk = 1; otherwise,mk = 0. Vector M := {mk , k ∈ K } represents the mainte-nance strategy. For a specific sample zi, we have

f (zi) =∏k∈K

[mk0(ϕ̄k , zi)+ (1− mk )0(ϕk , zi)

]. (13)

Substituting (13) into (11) yields

R̂f (Y0)

=1N

N∑i=1

∏k∈K

[mk0(ϕ̄k , zi)+(1−mk )0(ϕk , zi)

]∏k∈K

0(ϕk , zi)h(zi)δ{h(zi)≥Y0}

=1N

N∑i=1

[∏k∈K∗

(1+mk

(0(ϕ̄k , zi)0(ϕk , zi)

− 1))

h(zi)δ{h(zi)≥Y0}

],

(14)

where K∗ is the set that includes all components available formaintenance. Thus, K∗ ⊆ K and mk = 0,∀k /∈ K∗.

The relationship between the estimated RCB and main-tenance strategies is detailed in (14), which is an explicitexpression that provides an estimate of the RCB. Tominimizethe RCB with the limited number of components consideredto be in maintenance, the following optimization problem canbe formulated

minmk

1N

N∑i=1

[∏k∈K∗

(1+mk

(0(ϕ̄k , zi)0(ϕk , zi)

−1))h(zi)δ{h(zi)≥Y0}

]s.t.

∑k∈K∗

mk ≤ Mmax , (15)

where mk , k ∈ K∗ are the decision variables, Mmax is apredefined parameter that stands for the maximum num-ber of components considered in maintenance, and 0(ϕ, zi),0(ϕ̄k , zi), h(zi), and δ{h(zi)≥Y0} are variables that rely on thesamples. For simplicity, we define an N -dimension vector Cand two |K∗| × N matrices, P and Q, as follows

Ci := h(zi)δ{h(zi)≥Y0} ∀i

Pki := 0(ϕk , zi) ∀i, k

Qki := 0(ϕ̄k , zi) ∀i, k.

Remark 1: Note that in practice, cascading outagesalways occur in power systems that are heavily loaded.Therefore, we assume the initial state is deterministicwhen formulating cascading outages. Similarly, in practice,the proposed optimization model (15) is employed in typicalheavily loaded system states. On the other hand, it is worthnoting that these formulations and the proposed methodologyserve as the first step in related analyses. When multipleinitial states are considered, the methodology can be sepa-rately employed in each initial state, and the final RCB canbe calculated by the average of the RCBs in multiple initialstates. The proof of this will not be provided here as it isbeyond the scope of this paper.

In large power systems that include thousands of compo-nents, (15) is a high-dimensional 0-1 programming problem,the solution for which involves the following two issues.

1) DETERMINING APPROPRIATE SAMPLE SIZE NIn the optimization model described by (15), the sample sizeN is a crucial parameter because the objective function isan unbiased estimation of the RCB and the estimation errorrelies on N . To guarantee the credibility of the estimation,a sufficiently largeN is required. On the other hand, anN thatis too largewill drastically increase the computational burden.Unfortunately, it is not trivial to determine an appropriatesample size. In this regard, we analyzed the variance of theestimated RCB based on inferences from the simulation data,which enabled us to develop an adaptive method to determinea suitable sample size that achieves a good trade-off betweenthe estimation accuracy and computational complexity. Thedetails of this method are provided in Sections III and IV-B.

2) REDUCING COMPUTATIONAL COMPLEXITYEven after an appropriate sample sizeN has been determined,it is still challenging to solve the optimization problem in(15). The number of decision variables in (15) equals |K∗|,and there are 2|K

∗|− 1 product terms of decision variables in

the objective function. Therefore, the optimization problemin (15) for a real large system may have thousands of deci-sion variables and an astronomical number of product terms.For this type of high-dimensional 0-1 integer programmingproblem, enumeration and conventional optimization algo-rithms, such as branch and bound, are demonstrably inef-fective. To address these limitations, we devised two simpleyet efficient algorithms to identify nearly-optimal solutionsof (15) in a heuristic manner, which is a suitable approach inpractice for large systems. The details of these algorithms areprovided in Section IV.

III. DETERMINING N BASED ON CREDIBILITY ANALYSESDue to the inherent uncertainty of the samples in Zg, the esti-mated RCB, R̂f (Y0), given in (11) always includes a certainamount of estimation error. In general, increasing the samplesize reduces the estimation error. However, to determine an

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appropriate sample size of Zg, i.e., N , the influence of Non the estimation error must be characterized via a varianceanalysis based on inference from the simulation data.

A. VARIANCE-BASED CREDIBILITY ANALYSESWe first consider the relative error bound of (11) with respectto a specific Zg. Specifically, we denote ε as the relative errorbound of R̂f (Y0) with a confidence level β. Then, ε shouldbound the probability of Rf (Y0) within I := [R̂f (Y0)/(1 +ε), R̂f (Y0)/(1− ε)] with a probability β, i.e.

Pr

(|Rf (Y0)− R̂f (Y0)|

Rf (Y0)≤ ε

)= β, (16)

where 2ε/(1−ε2) is defined as the relative length of the errorbound used to quantify the credibility of the estimated RCB.Note that (16) is equivalent to

Pr

(−εRf (Y0)√Df (Y0)

≤Rf (Y0)− R̂f (Y0)√

Df (Y0)≤εRf (Y0)√Df (Y0)

)= β,

(17)

where Df (Y0) is the estimation variance of R̂f (Y0) givenby (11). Invoking the Central Limit Theorem, when N issufficiently large, we have:

Rf (Y0)− R̂f (Y0)√Df (Y0)

∼ Norm(0, 1).

In this context, ε can be determined by solving (17), yielding

ε =8−1(1/2+ β/2)

√Df (Y0)

Rf (Y0), (18)

where 8 is the cumulative density function of the standardnormal distribution, and 8−1 is the inverse function of 8.According to (18), a small enough Df (Y0) is required

to ensure the credibility of the blackout risk estimation.However, as the distribution of R̂f (Y0) is unknown, Df (Y0)cannot be calculated accurately in practice. To address this,we propose a proposition for estimating Df (Y0) based on aspecific Zg.Proposition 1: Given Zg, g(Z ), and f (Z ), an unbiased esti-

mation of Df (Y0) is given by

D̂f (Y0) =1

(N − 1)N

N∑i=1

[w(zi)h(zi)δ{h(zi)≥Y0} − R̂f (Y0)

]2.

(19)Proof: We first define random variable Li(Y0) as

Li(Y0) :=f (zi)g(zi)h(z

i)δ{h(zi)≥Y0}, i = 1, · · · ,N . Since zi isi.i.d., Li(Y0) is also i.i.d. Therefore, the variance of randomvariables Li(Y0) are the same. Specifically, we denote thevariance of Li by df (Y0). We then have

Df (Y0) = D

(1N

N∑i=1

Li(Y0)

)=

1ND(Li(Y0)) =

1Ndf (Y0),

(20)

whereD is an operator of the variance calculation. The secondequation holds by noting that Li(Y0), i = 1, · · · ,N are i.i.d.According to the definition of the sample variance in [28],

the unbiased estimation of df (Y0) based on Zg is given by

d̂f (Y0) =1

N−1

N∑i=1

[w(zi)h(zi)δ{h(zi)≥Y0}−R̂f (Y0)

]2. (21)

Therefore, by substituting (21) into (20), the unbiased esti-mation of Df (Y0) can be determined using (19).

Combining (11), (18), and (19), ε can be estimated by

ε̂ =8−1(1/2+ β/2)

√D̂f (Y0)

R̂f (Y0). (22)

On the other hand, according to (18) and (20), we have

ε ∝

√df (Y0)

√NRf (Y0)

. (23)

Since df (Y0) is deterministic for specific Zg and f (Z ), whenthe error bound of the estimated RCB in (11) does not satisfythe predefined requirement, the sample size N should beenlarged and Zg should be updated, as explained next.

B. DETERMINING THE SAMPLE SIZE NIn this subsection, the previous credibility analyses areemployed to determine an appropriate N based on inferencefrom the simulation data in Zg. In essence, if the estimationerror bound is determined (according to the requirement ofpractical utilization), i.e.,ε̄ = 10%, then (22) gives

D̄f (Y0) =

(ε̄R̂f (Y0)

8−1(1/2+ β/2)

)2

. (24)

In (24), D̄f (Y0) represents the maximal estimation variancerequired so that the relative error is within a given error boundε̄ with a confidence level of β.

To ensure that Df (Y0) is less than D̄f (Y0), the requiredsample size N̄ can be obtained using (20) and (21) as

N̄ =d̂f (Y0)

D̄f (Y0)=

d̂f (Y0)

R̂2f (Y0)

(8−1(1/2+ β/2)

ε̄

)2

. (25)

According to (25), when N ≤ N̄ , the sample size is notlarge enough to guarantee the credibility of the blackoutrisk estimation and more samples are required to reduce theestimation variance. This provides a systematic approach tothe optimization problem (15) when determining an appropri-ate sample size. The corresponding algorithm can be easilyimplemented in practice, as described in Section IV-B.

It is noteworthy that both d̂f (Y0) and R̂f (Y0) in (25) arecalculated based only on the sample set Zg. Hence, N̄ isessentially inferred from the simulation data in Zg and thereis no need to regenerate any additional samples as long as thesample size condition N > N̄ holds.

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IV. SOLUTION ALGORITHMSA. HEURISTIC ALGORITHMSAs mentioned previously, the optimization model (15) is ahigh-dimensional nonlinear 0-1 programming problem that isconsidered to be NP-hard. Many nearly-optimal solutions areable to mitigate the RCB in practice, as will be demonstratedin the Case Studies. We now devise two heuristic algorithmsto identify nearly-optimal solutions to (15) on a fixed Zg.Since one significant obstacle is the large number of deci-

sion variables, a simplistic approach is to reduce the numberof components considered for maintenance, i.e., |K∗|. Withthis in mind, we employ a sensitivity-based approach in thedesign of the following algorithm.

Algorithm 1• Step 1: Sensitivity analysis. Construct |K∗| scenarios inwhich the ϕk of a single component k ∈ K changes to ϕ̄k .Then, estimate the RCB using (11).• Step 2: Scenario reduction. Choose theMk componentswith larger reductions in the RCB to make up the Km.Here,Mk is a predefined parameter based onMmax and theexperience. In particular, Mmax < Mk = |Km| < |K∗|.• Step 3: Solving the optimization problem. Substitute K∗

in (15) with Km and solve (15) by enumeration.

Since (11) only requires a few algebraic calculations,it is efficient to estimate the RCB in each scenario.Suppose the average calculation time of a single sce-nario is ts, then the total calculation time is (|K∗| +C(Mk ,Mmax))ts.Here,C(Mk ,Mmax) is the number ofMmax−combinations from Mk elements. As the maximalnumber of components that are considered for maintenanceMmax is often small in practice, Mk can be chosen to be amoderate number. Then, the number of scenarios and thecomputation time by enumeration will be acceptable.

Nevertheless, as Mk and Mmax grow larger, such as inan actual large power system, the number of scenarios(|K∗|+C(Mk ,Mmax)) and computation time increase dramat-ically. To address this, we propose the following additionalalgorithm.

Algorithm 2• Step 1: Initialization. Let Km = ∅. Then its complemen-tary set, which is denoted by K̄m, equals K∗.• Step 2: Estimation of RCB. Construct |K̄m| scenarios.In each scenario, for all components in Km and a singlecomponent in K̄m, change their ϕk to ϕ̄k and estimate theRCB with (11).• Step 3: Iteration. Choose the scenario with the lowestRCB. Move the corresponding component from K̄m to Km.If |Km| = Mmax , the procedure ends; otherwise, go backto Step 2.

In Algorithm II, the components that must be maintainedare determined successively. In each round, the component

FIGURE 1. Flowchart of the methodology.

that can most effectively reduce the RCB is selected. There-fore, Algorithm II actually belongs to a family of so calledgreedy algorithms. The total number of scenarios is (2|K∗|−Mmax + 1)Mmax/2. Since the number of scenarios increaseslinearly with |K∗|, the optimization process remains efficienteven in a large system.

B. PROCEDUREThe methodology (see Fig. 1) can be summarized as follows.

Procedure• Step 1: Initialization. Initialize the system. In particular,determine ϕk and ϕ̄k for each component in the system. Theinitial sample size is N = N0.• Step 2: Sample generation. Generate N samples basedon the specific blackout model and ϕk , k ∈ K . The sampleset is Zg. Based on Zg, ϕk , and ϕ̄k , calculate matrices C , P,and Q.• Step 3: Maintenance optimization strategy. Solve theoptimization problem (15) using Algorithm I or II.• Step 4: Credibility evaluation. For the optimal mainte-nance strategy, calculate the necessary sample size N̄ with(25). If N̄ ≥ N , generate another N̄ − N samples and addthem into Zg. Then, update C , P, Q and go back to Step 3;Otherwise, directly choose the optimal one according tothe results in Step 3.

The proposed procedure incorporates an adaptive selectionof the sample size based on a credibility evaluation, as shownin Step 4. In this way, the sample size can be minimizedwhile the estimation error is within a predefined limit withrespect to a given confidence level. It is noteworthy that tobenefit from the inference from blackout simulation data, allsamples are generated with respect to g(Z ) only and not for

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different f (Z ). This results in a significant reduction in thesampling time and simplifies the implementation.

V. CASE STUDIESA. SETTINGSIn this section, numerical experiments are carried out ona IEEE 57-bus system, IEEE 300-bus system, and anactual provincial power system in China with a simplifiedORNL-PSerc-Alaska (OPA) model that omits slow dynam-ics [13]. In the simulations, a traditional MC method isemployed to consider the random failures of both the trans-mission lines and power transformers. The failure probabilityfunctions of the transmission lines and power transformersare the same as those in [14] and [24], respectively. In allcases in this work, typical parameters are applied, and forsimplicity, only the maintenance of power transformers isconsidered.

The simulation model with specific component failureprobability functions mentioned above is only employed todemonstrate the proposed method, and it is expected thatmore realistic models and settings will be adopted in actualsituations.

B. IEEE 57-BUS SYSTEMIn this simulation, we employed a small system that included53 transmission lines and 17 power transformers to demon-strate the difficulties encountered in mitigating the RCB viacomponent maintenance and to highlight some salient fea-tures of the proposed method.

1) RCB ESTIMATIONHere, we demonstrate the unbiasedness of (11) and the inher-ent estimation error caused by the randomness of the samples.First, we constructed a series of Zg with increasing samplesizes and randomly changed the ϕk of four components. Then,we calculated the RCB with Y0 = 200 using (11). Next,we estimated ε with respect to β = 90% using (22) andcalculated the corresponding error bounds. For comparison,we directly regenerated the samples with respect to the newϕ̄k and estimated the RCB using (9). The results are presentedin Fig. 2.

Fig. 2 shows that the estimation errors between the cal-culations and direct sampling become smaller as the samplesize increases until they are almost the same when the sam-ple size is large enough. This indicates the effectiveness ofthe RCB estimation given by (11). This can be illustratedmore rigorously by noting the relative length of the errorbounds (see Fig. 3) also becomes smaller as the sample sizeincreases.

At the same time, it is worth noting that Fig. 3 also indicatesthat some amount of estimation error always exists. Moreimportantly, when the sample size is small, the estimationerror can be very large. Intuitively, optimized maintenancestrategies based on a small Zg may have a large devia-tion in their performance of mitigating the RCB. Therefore,

FIGURE 2. RCB estimation with sampling and calculation.

FIGURE 3. Relative length of the error bounds.

TABLE 1. RCB after component maintenance.

considering the calculation efficiency and possible estimationerror, it is important that the sample size of Zg, i.e., N , shouldbe selected appropriately.

2) INFLUENCE OF MAINTENANCE ON THE RCBWe constructed a Zg that included 100, 000 samples, basedon which we considered the simultaneous maintenanceof components. Since there were only 17 transformers inthis small system, we directly enumerated all the possi-ble strategies and compared the respective risk reductionperformance. We first considered the maintenance of asingle component. The RCBs with respect to Y0 = 0and Y0 = 200 after maintenance are given in Table 1.In particular, the original RCBs were R̂g(0) = 6.74 andR̂g(200) = 5.14.The results in Table 1 intuitively indicate that component

maintenance is an effective way to mitigate the RCB. More-over, a few critical components in the system have muchgreater influence on the RCB than the others. For example,

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TABLE 2. Optimal maintenance strategies (Y0 = 0).

TABLE 3. Optimal maintenance strategies (Y0 = 200).

the maintenance of No. 7 transformer can result in a 2.8%reduction in the RCB with respect to Y0 = 0, while theaverage reduction is only 0.9%. This result confirms theefficacy of optimizing the maintenance strategy in mitigatingthe RCB.

Next, we considered maintaining four transformers simul-taneously and then estimated the RCBs with all possiblemaintenance strategies. The strategies with the largest reduc-tions in the RCBs with respect to the two Y0 are pre-sented in Tables 2 and 3, respectively. The results illustratethe complicated relationship between maintenance strategies(or more precisely, component failure probabilities) and theRCB. Note that the degree of risk reduction associated withthe four components are much smaller than the sum of thefour individual components. Moreover, the optimal compo-nents to be maintained (see Table 2 and Table 3) are notsimply combinations of components with smaller RCBs (seeTable 1). This indicates that the influence of component fail-ure probabilities on the RCB is essentially nonlinear, whichimplies that one cannot quantify the influence of maintainingmultiple components solely based on the influence of theindividual components.

Moreover, Tables 2 and 3 show that there are many nearly-optimal maintenance strategies whose risk reduction ratiosare very close to the optimal one, even though the result-ing components to be maintained are not identical. Con-sidering the possible estimation error of the blackout riskestimation, it is reasonable to also deploy these nearly-optimal maintenance strategies. In other words, there is noneed to rigorously solve the optimization problem (15), par-ticularly when finding a solution is very time consuming.With this in mind, we employed two heuristic algo-rithms to efficiently find (nearly-)optimal solutions in thiswork.

TABLE 4. Optimal maintenance strategies with different algorithms(Y0 = 200).

TABLE 5. Optimal maintenance strategies with different algorithms(Y0 = 100).

3) HEURISTIC ALGORITHMSTo compare the two heuristic algorithms, we selected differ-ent parameters, |Zg|, Y0,Mmax , andMk , and compared the cor-responding optimal maintenance strategies. Here, we presenttwo typical cases. Specifically, |Zg| = 35000 and Mmax = 4.We employed Algorithm I withMk = 8, 12 and Algorithm IIto determine the optimal maintenance strategies with respectto Rf (100) and Rf (200), respectively, and the results areshown in Tables 4 and 5.

As shown in Table 4, both algorithms produced the samemaintenance strategy and achieved the global optimum,2

while in Table 5, Algorithm I with Mk = 8 and Algorithm IIprovided sub-optimal solutions. The reason is due to theinherent nonlinearity between the component failure proba-bilities and the RCB, as well as the inevitable uncertainty inthe samples. However, it is worth noting that, based on ourexperience, Algorithm I with a moderateMk and Algorithm IIproduce effective maintenance strategies in most cases, andare therefore suitable for large systems.

C. IEEE 300-BUS SYSTEMIn this case, the complete proposed method is tested usingdata from the IEEE 300-bus system with 304 lines and107 transformers. Specifically, we set ε = 10%, β = 95%,Mmax = 8, Mk = 30, N0 = 5000, Y0 = 100.

1) SOLVING PROCESSFirst, the solving process with Algorithm I is presentedin Table 6.

As shown in Table 6, we began with a Zg that included5,000 samples, with which we obtained an optimal main-tenance strategy. However, the relative error requirement εcould not be satisfied. Therefore, 8, 0000 samples were added

2Note that since |Zg| is different in Tables 3 and 4, the largest reductionsin the RCB obtained by enumeration are different, which indicates thatdetermining the sample size based on credibility analyses has practicalsignificance.

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TABLE 6. Solving process with Algorithm I.

TABLE 7. Solving process with Algorithm II.

FIGURE 4. Computation time with two algorithms.

into Zg according to (25) in step 2. Then, the optimal main-tenance strategy and ε̂ were recalculated. This process wasrepeated in step 3 to determine the final optimal maintenancestrategy. The level of reduction in the RCB with respect toY0 = 100 was 21.5%.

The solution process with Algorithm II is summarizedin Table 7. In this case, the optimal strategies in each stepare actually the same as the ones of Algorithm I. Note that thecomponents of each strategy in Table 7 are ordered accordingto the decision process. Even though the results provided bythe two algorithms are similar in steps 2 and 3, the decisionprocesses are different. This observation again demonstratesthe complicated relationship between the component failureprobabilities and the RCB.

2) COMPUTATION TIMEWeconducted all of the tests on a computer with an Intel XeonE5-2670 processor running at 2.6 GHz and 64GB ofmemory.The computation times of the two algorithms are presentedin Fig. 4. Note that the computation time for sampling andcalculating C,P,Q in each step depends on the number ofadditional samples, while the optimization time depends onthe number of total samples. From the figure, it can be seenthat the computation time for sampling was much larger thanthat for the other times. In our cases, 107min were required tocomplete sampling. The requirement for repeated sampling iswhy traditional methods are extremely inefficient at directlyestimating the RCB for all maintenance strategies. In contrast,our methodology enables a very efficient estimation of the

FIGURE 5. Optimal maintenance strategy in the provincial system.

RCB under varying maintenance strategies without requiringthe regeneration of any samples. In addition, the optimizationtime of Algorithm II is smaller than that of Algorithm I asit involves fewer maintenance scenarios. However, as Algo-rithm II may result in sub-optimal solutions in some cases,e.g., Table 4, it is preferable for large-scale systems withmany candidate components considered for maintenance.

D. AN ACTUAL PROVINCIAL SYSTEMIn this case, an actual provincial power system in China isemployed to show the advantages of the proposed method.In particular, this system includes 1,122 buses, 1,230 trans-mission lines, and 562 transformers. Considering the largescale of the system, the chosen parameters were Y0 = 1500andMmax = 20. Under these conditions, even if a ‘small’Mkis chosen in Algorithm I, the number of possible maintenancescenarios is astronomical, e.g., if Mk = 40, the numberof scenarios is greater than 100 billion. Therefore, we onlyemployed Algorithm II when determining the maintenancestrategy.

Algorithm II realizes the final optimal maintenance strat-egy shown in Fig. 5. Specifically, the transformers on thehorizontal axis are ordered by the decision process in Algo-rithm II, and the reduction ratio of the RCB after each trans-former is maintained is shown on the vertical axis. Sincethe most effective transformer is maintained in each step ofAlgorithm II, the increase in the reduction ratio of the RCBbecomes slower along with the decision process. The finalreduction ratio is 23.1%. The computational time requiredfor the entire procedure was 3,134 min, and Fig. 6 shows thespecific proportions of different parts. From the figure, it canbe seen that the sampling required almost all of the time inthis large system. In contrast, Algorithm II is extremely effi-cient. This not only verifies the applicability of the proposedalgorithm in actual large systems, but also demonstrates thathigh-efficiency samplingmethods [17], [20] can be employedto further improve the scalability and efficiency of themethodology.

E. SAMPLING TIMEIn this case, we further compare the computational time takenfor sampling when using the traditional MC method. In three

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FIGURE 6. Computation time in the provincial system.

TABLE 8. Average time to generate one sample.

test systems, the average times to generate one sample arelisted in Table 8.

According to Table 8, as the scale of the system grows up,the average time to generate one sample increases obviously.The main reason can be attributed to two aspects:

1) As the system scale increases, the calculation of indi-vidual power flows turns to be much more time-consuming.

2) As the number of components in the system increases,the average path length of individual cascading outagesbecomes longer. That further leads to higher intensityof computation.

Additionally, noting that both the state space of cascad-ing outages and the number of possible maintenance strate-gies are fairly large, the sample size required for cascadingoutage simulations considering impacts of maintenance isinevitably huge. It indicates that, in a maintenance optimiza-tion problem, the total compuational burden for generatingsamples are very heavy, particularly when a large scale sys-tem is considered. In such a circumstance, traditional meth-ods relying on conducting additional simulations appear to beintractable.

VI. CONCLUSION WITH REMARKSIn this paper, we propose an efficient methodology tooptimize component maintenance strategies for effectivelymitigating the RCB. In the proposed method, an analyticrelationship between the estimated RCB and maintenancestrategies is developed through inference from simulationdata. The results of theoretical analyses and numerical exper-iments confirm that:

1) A small set of components in power systems exert greatinfluence on the propagation of cascading outages, andreducing their failure probabilities through componentmaintenance can significantly mitigate the RCB.

2) Variance-based analyses are employed to further eluci-date the credibility of the estimated RCB while consid-ering different maintenance strategies.

3) The proposed heuristic algorithms are found to effi-ciently optimize the component maintenance strategiesin large power systems.

The proposed method also can be applied in otherfileds. For example, it can be employed to identify criti-cal buses or branches in complex networks. Moreover, notethat as the scale of the system increases, the sampling pro-cess will take more time. We intend to incorporate somerecently developed high-efficiency sampling methods, suchas Sequential Importance Sampling [17] and Splitting [20],to further improve the performance of this method in futurework. Another area of ongoing work is to include a morepractical formulation of maintenance strategy optimizationand the corresponding solving algorithms.

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FENG LIU (M’12) received the B.Sc. andPh.D. degrees in electrical engineering fromTsinghua University, Beijing, China, in 1999 and2004, respectively. From 2015 to 2016, he wasa Visiting Associate with the California Insti-tute of Technology, CA, USA. He is currently anAssociate Professor with Tsinghua University. Hehas authored/co-authored over 100 peer-reviewedtechnical papers and two books, and holds over20 issued/pending patents. His research interests

include power system stability analysis, optimal control and robust dispatch,game theory and learning theory, and their applications to smart grids.He was a Guest Editor of the IEEE TRANSACTIONS ON ENERGY CONVERSION.

JINPENG GUO (S’17) received the B.Sc. degreein electrical engineering from Tsinghua Univer-sity, Beijing, China, in 2013, where he is currentlypursuing the Ph.D. degree in electrical engineer-ing. His research interests include the modelingand simulation of blackouts, risk assessment, andmanagement.

XUEMIN ZHANG (M’06) received the B.S.and Ph.D. degrees in electrical engineering fromTsinghua University, Beijing, China, in 2001 and2006, respectively. She is currently an AssociateProfessor with the Department of Electrical Engi-neering, Tsinghua University. Her research inter-ests include power system analysis and control.

YUNHE HOU (M’08–SM’15) received the B.E.and Ph.D. degrees from the Huazhong Univer-sity of Science and Technology, Wuhan, China,in 1999 and 2005, respectively. He was a Post-Doctoral Research Fellow at Tsinghua University,Beijing, China, from 2005 to 2007. He was a Vis-iting Scholar with Iowa State University, Ames,IA, USA, and a Research Scientist with Univer-sity College Dublin, Dublin, Ireland, from 2008 to2009. He was also a Visiting Scientist with the

Laboratory for Information and Decision Systems, Massachusetts Instituteof Technology, Cambridge, MA, USA, in 2010. He is currently with TheUniversity of Hong Kong as a Research Assistant Professor. His researchinterests include electricity market theory, power system resource planning,and self-healing smart grid construction.

SHENGWEI MEI (SM’06–F’15) received theB.Sc. degree in mathematics from Xinjiang Uni-versity, Ürümqi, China, the M.Sc. degree in opera-tions research from Tsinghua University, Beijing,China, and the Ph.D. degree in automatic controlfrom the Chinese Academy of Sciences, Beijing,in 1984, 1989, and 1996, respectively. He is cur-rently a Professor with Tsinghua University. Hisresearch interests include power system analysisand control, game theory, and its application inpower systems.

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