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Machine Learning Methods for Attack Detection inthe Smart Grid

Mete Ozay, Member, IEEE,Inaki Esnaola, Member, IEEE,Fatos T. Yarman Vural, Senior Member, IEEE,Sanjeev R. Kulkarni, Fellow, IEEE,and H. Vincent Poor, Fellow, IEEE

Abstract—Attack detection problems in the smart grid areposed as statistical learning problems for different attack sce-narios in which the measurements are observed in batch oronline settings. In this approach, machine learning algorithmsare used to classify measurements as being either secure orattacked. An attack detection framework is provided to exploitany available prior knowledge about the system and surmountconstraints arising from the sparse structure of the problem inthe proposed approach. Well-known batch and online learningalgorithms (supervised and semi-supervised) are employedwithdecision and feature level fusion to model the attack detectionproblem. The relationships between statistical and geometricproperties of attack vectors employed in the attack scenarios andlearning algorithms are analyzed to detectunobservable attacksusing statistical learning methods. The proposed algorithms areexamined on various IEEE test systems. Experimental analysesshow that machine learning algorithms can detect attacks withperformances higher than the attack detection algorithms whichemploy state vector estimation methods in the proposed attackdetection framework.

Index Terms—Smart grid security, sparse optimization, classi-fication, attack detection, phase transition.

I. I NTRODUCTION

Machine learning methods have been widely proposedin the smart grid literature for monitoring and control ofpower systems [1], [2], [3], [4]. Rudin et al. [1] suggest anintelligent framework for system design in which machinelearning algorithms are employed to predict the failures ofsystem components. Anderson et al. [2] employ machinelearning algorithms for the energy management of loads andsources in smart grid networks. Malicious activity predictionand intrusion detection problems have been analyzed usingmachine learning techniques at the network layer of smartgrid communication systems [3], [4].

In this paper, we focus on the false data injection attackdetection problem in the smart grid at the physical layer.We use the Distributed Sparse Attacks model proposed byOzay et al. [5], where the attacks are directed by injectingfalse data into the local measurements observed by eitherlocal network operators or smart Phasor Measurement Units

M. Ozay is with the School of Computer Science, University ofBirm-ingham, B15 2TT, UK (e-mail: [email protected]). I. Esnaola, S. R.Kulkarni and H. V. Poor are with the Department of ElectricalEngineering,Princeton University, Princeton, NJ 08544, USA (e-mail:jesnaola, kulkarni,[email protected]). I. Esnaola is also with the Department of AutomaticControl and Systems Engineering, University of Sheffield, S1 3JD, UK. F. T.Yarman Vural is with the Department of Computer Engineering, Middle EastTechnical University, Ankara, Turkey (e-mail: [email protected]).

This research was supported in part by the U. S. National ScienceFoundation under Grant CMMI-1435778.

(PMUs) in a network with a hierarchical structure, i.e. themeasurements are grouped into clusters. In addition, networkoperators who employ statistical learning algorithms for attackdetection know the topology of the network, measurementsobserved in the clusters and the measurement matrix [5].

In attack detection methods that employ state vector es-timation, first the state of the system is estimated fromthe observed measurements. Then, the residual between theobserved and the estimated measurements is computed. If theresidual is greater than a given threshold, a data injectionattack is declared [5], [6], [7], [8]. However, exact recovery ofstate vectors is a challenge for state vector estimation basedmethods in sparse networks [5], [9], [10], where the Jacobianmeasurement matrix is sparse. Sparse reconstruction methodscan be employed to solve the problem, but the performanceof this approach is limited by the sparsity of the state vectors[5], [11], [12]. In addition, if false data injected vectorsresidein the column space of the Jacobian measurement matrix andsatisfy some sparsity conditions (e.g., the number of nonzeroelements is at mostκ∗, which is bounded by the size ofthe Jacobian matrix), then false data injection attacks, calledunobservableattacks, cannot be detected [7], [8].

The contributions of this paper are as follows:1) We conduct a detailed analysis of the techniques pro-

posed by Ozay et al. [13] who employ supervised learn-ing algorithms to predict false data injection attacks.In addition, we discuss the validity of the fundamentalassumptions of statistical learning theory in the smartgrid. Then, we propose semi-supervised, online learning,decision and feature level fusion algorithms in a genericattack construction framework, which can be employedin hierarchical and topological networks for differentattack scenarios.

2) We analyze the geometric structure of the measurementspace defined by measurement vectors, and the effectof false data injection attacks on the distance functionof the vectors. This leads to algorithms forlearningthe distance functions,detectingunobservable attacks,estimating the attack strategies andpredicting futureattacks using a set of observations.

3) We empirically show that the statistical learning algo-rithms are capable of detecting both observable andunobservable attacks with performance better than theattack detection algorithms that employ state vectorestimation methods. In addition, phase transitions can beobserved in the performance of Support Vector Machines(SVM) at a value ofκ∗ [14].

http://arxiv.org/abs/1503.06468v1

2

In the next section, the attack detection problem is formu-lated as a statistical classification problem in a network accord-ing to the model proposed by Ozay et al. [5]. In Section II, weestablish the relationship between statistical learning methodsand attack detection problems in the smart grid. Supervised,semi-supervised, decision and feature level fusion, and onlinelearning algorithms are used to solve the classification problemin Section III. In Section IV, our approach is numericallyevaluated on IEEE test systems. A summary of the resultsand discussion on future work are given in Section V.

II. PROBLEM FORMULATION

In this section, the attack detection problem is formalizedas a machine learning problem.

A. False Data Injection Attacks

False Data Injection Attacks are defined in the followingmodel:

z =Hx + n, (1)

wherex ∈ RD contains the voltage phase angles at the buses,z ∈ R

N is the vector of measurements,H ∈ RN×D is the

measurement Jacobian matrix andn ∈ RN is the measurementnoise, which is assumed to have independent components [7].The attack detection problem is defined as that of decidingwhether or not there is an attack on the measurements. Ifthe noise is distributed normally with zero mean, then aState Vector Estimation (SVE) method can be employed bycomputing

x = (HTΛH)−1HT

Λz, (2)

where Λ is a diagonal matrix whose diagonal elementsare given byΛii = ν−2i , and ν2i is the variance ofni,∀i = 1,2, . . . ,N [7], [13]. The goal of the attacker is to inject afalse data vectora ∈ RN into the measurements without beingdetected by the operator. The resulting observation model is

z =Hx + a + n. (3)

The false data injection vector,a, is a nonzero vector, suchthat ai ≠ 0, ∀i ∈ A, where A is the set of indices ofthe measurement variables that will be attacked. The securevariables satisfy the constraintai = 0, ∀i ∈ A, whereA is theset complement ofA [13].

In order to detect an attack, themeasurement residual[7],[13] is examined inℓ2-norm ρ = ∥z −Hx∥22, wherex ∈ RD

is the state vector estimate. Ifρ > τ , where τ ∈ R is anarbitrary threshold which determines the trade-off betweenthe detection and false alarm probabilities, then the networkoperator declares that the measurements are attacked.

One of the challenging problems of this approach is thatthe Jacobian measurement matrices of power systems in thesmart grid are sparse under the DC power flow model [13],[15]. Therefore, the sparsity of the systems determines theperformance of sparse state vector estimation methods [11],[12]. In addition, unobservable attacks can be constructedevenif the network operator can estimate the state vector correctly.For instance, ifa =Hc, wherec ∈ RD is an attack vector, thenthe attack isunobservableby using the measurement residual

ρ [7], [8]. In this work, we show that statistical learningmethods can be used to detect the unobservable attacks withperformance higher than the attack detection algorithms thatemploy a state vector estimation approach. Following themotivation mentioned above, a new approach is proposedusing statistical learning methods.

B. Attack Detection using Statistical Learning Methods

Given a set of samplesS = siMi=1 and a set of labelsY = yi

Mi=1, where (si, yi) ∈ S × Y are independent and

identically distributed (i.i.d.) with joint distributionP , thestatistical learning problem can be defined as constructingahypothesis functionf ∶ S → Y, that captures the relationshipbetween the samples and labels [16]. Then, the attack detectionproblem is defined as a binary classification problem, where

yi =⎧⎪⎪⎨⎪⎪⎩

1, if ai ≠ 0

−1, if ai = 0. (4)

In other words,yi = 1, if the i-th measurement is attacked,andyi = −1 when there is no attack.

In this paper, the model proposed by Ozay et al. [5] isemployed for attack construction where the measurements areobserved in clusters in the network. Measurement matrices,and observation and attack vectors are partitioned intoG

blocks, denoted byGg with ∣Gg ∣ = Ng for g = 1,2, . . . ,G.Therefore, the observation model is defined as

⎡⎢⎢⎢⎢⎢⎣

z1

⋮zG

⎤⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎣

H1

⋮HG

⎤⎥⎥⎥⎥⎥⎦

x +

⎡⎢⎢⎢⎢⎢⎣

a1

⋮aG

⎤⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎢⎣

n1

⋮nG

⎤⎥⎥⎥⎥⎥⎦

, (5)

where zg ∈ RNg is the measurement observed in theg-th

cluster of nodes through measurement matrixHg ∈ RNg×D

and noiseng ∈ RNg , and which is under attackag ∈ R

Ng

with g = 1,2, . . . ,G [5]. Within this framework, each observedmeasurement vector is considered as a sample, i.e.,si ≜ zg,wherezg ∈ RNg1. Taking this into account, the measurementsare classified in two groups,secureandattacked, by computingf(si),∀i = 1,2, . . . ,M .

The crucial part of the traditional attack detection algorithm,which we call State Vector Estimation (SVE), is the estimationof x. If the attack vectors,a, are constructed in the columnspace ofH, then they are annihilated in the computation ofthe residual [7]. Therefore, SVE cannot detect the attacks andthese attacks are calledunobservable. On the other hand, weobserve that the distance between the attacked and the securemeasurement vectors is defined by the attack vector inS. Ifthe attacks are unobservable, i.e.ai = Hci and aj = Hcj ,whereci ∈ RD and cj ∈ RD are the attack vectors, then thedistance betweenzi = zi + ai and zj = zj + aj is computed as

∥zi − zj∥2 =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∥zi − zj∥2 + ∥ai − aj∥2, if i, j ∈ A

∥zi − zj∥2 + ∥ai∥2, if i ∈ A, j ∈ A

∥zi − zj∥2, if i, j ∈ A

,

(6)

1For simplicity of notation, we usei as the index of measurementszi, zi,and attack vectorsai, ∀i = 1,2, . . . ,M .

3

where zi ∈ S and zj ∈ S. In (6), we can extract informationon the attack vectors by observing the measurements. Sincethe distances between secure and attacked measurements arediscriminated by the attack vectors, the attacks can be recog-nized by the learning algorithms which use the information ofthese distances, even if the attacks areunobservable.

Two main assumptions from statistical learning theory needto be taken into account to classify measurements whichsatisfy (6):

1) We assume that(si, yi) ∈ S×Y are distributed accordingto a joint distributionP [17]. In a smart grid setting, thisdistribution assumptionis satisfied for the attack modelsin which the measurementsz are functions ofa, and wecan extract statistical information about both the attackedand secure measurements from the observations.

2) We assume that(si, yi), ∀i, are sampled fromP ,independently and identically. This assumption is alsosatisfied in the smart grid if the entries ofn anda arei.i.d. random variables [16].

In order to explain the significance of the above assumptionsin the smart grid, we consider the following example. Assumethat measurements1,2 ∈ A and 3,4 ∈ A, are given such thaty1, y2 = 1 andy3, y4 = −1. Furthermore, assume thatz1 = 3 ⋅I,z2 = 5 ⋅ I, z3 = 2 ⋅ I and z4 = 4 ⋅ I, whereI = (1,1)T . If theattack vectors areidenticalbut not independent, then the attackvectors can be constructed asa1 = a2 = −1 ⋅ I. As a result, weobserve thatz1 = z3 = 2 ⋅ I and z2 = z4 = 4 ⋅ I. Therefore,our assumption about the existence of a joint distributionP

is not satisfied and we cannot classify the measurements withthe aforementioned approach.

III. A TTACK DETECTION USINGMACHINE LEARNING

METHODS

In this section, the attack detection problem is modeledby statistical classification of measurements using machinelearning methods.

A. Supervised Learning Methods

In the following, the classification functionf is computed ina supervised learningframework by a network operator usinga set of training dataTr = (si, yi)M

Tr

i=1 . The class label,y′i,of a new observation,s′i, is predicted usingy′i = f(s′i). Weemploy four learning algorithms for attack detection.

1) Perceptron:Given a samplesi, a perceptron predictsyiusing the classification functionf(si) = sign(w ⋅ si), wherew ∈ RNi is a weight vector andsign(w ⋅ si) is defined as [17]

sign(w ⋅ si) =⎧⎪⎪⎨⎪⎪⎩

−1, if w ⋅ si < 01, otherwise.

(7)

In the training phase, the weights are adjusted at eachiteration t = 1,2, . . . , T of the algorithm for each trainingsample using

w(t + 1) ∶=w(t) +∆w, (8)

where∆w = γ(yi − f(si))si andγ is the learning rate. Thealgorithm is iterated until a stopping criterion, such as thenumber of algorithm steps, or an error threshold, is achieved.

In the testing phase, the label of a new test sample is predictedby f(s′i) = sign(w(T ) ⋅ s′i).

Despite its success in various machine learning applications,the convergence of the algorithm is assured only when thesamples are linearly separable [17]. For that reason, theperceptron can be successfully used for the detection of theattacks only if the measurements can be separated by ahyperplane. In the following sections, we give examples ofclassification algorithms which overcome this limitation byemploying non-linear classification rules or feature extractionmethods.

2) k-Nearest Neighbor (k-NN): This algorithm labels anunlabeled samples′i according to the labels of itsk-nearestneighborhood in the feature space [17]. Specifically, the ob-served measurementssi ∈ S, ∀i = 1,2, . . . ,M , are takenas feature vectors. The set ofk-nearest neighbors ofs′i,ℵ(s′i) = si(1), si(2), . . . , si(k), is constructed by computingthe Euclidean distances between the samples [18], wherei(1), i(2), . . . , i(M) are defined as

∥s′i − si(1)∥2 ≤ ∥s′i − si(2)∥2 ≤ . . . ≤ ∥s

′i − si(M)∥2. (9)

Then, the most frequently observed class label is computedusing majority voting among the class labels of the samplesin the neighborhood, and assigned as the class label ofs

′i [19].

One of the challenges ofk-NN is thecurse of dimensionality,which is the difficulty of the learning problem when thesample size is small compared to the dimension of the featurevector [17], [19], [20]. In attack detection, this problem canbe handled using the following approaches:

Feature selection algorithms can be used to reduce thedimension of the feature vectors [19], [20]. Developmentof feature selection algorithms may be a promising di-rection for smart grid security, and is an interesting topicfor future work.

Kernel machines, such as SVMs, can be used to map thefeature vectors inS to Hilbert spaces, where the featurevectors are processed implicitly in the mappings and thecomputation of the learning models. We give the details ofthe kernel machines and SVMs in the following sections.

The samples can be processed in small sizes, e.g. byselecting a single measurement vector as a sample,which leads to one-dimensional samples. We employthis approach in Section IV. If the sample size is large,distributed learning and optimization methods can be used[5], [15].

3) Support Vector Machines:We seek a hyperplane thatlinearly separates attacked and secure measurements into twohalf spaces using hyperplanes in aD′ dimensional featurespace,F , which is constructed by a non-linear mappingΨ ∶ S → F [13], [21]. A hyperplane is represented by a weightvectorwΨ ∈ R

D′ and a bias variableb ∈ R, which results in

wΨ ⋅Ψ(s)+ b = 0, (10)

whereΨ(s) is the feature vector of the sample that lies on thehyperplane inF as shown in Fig. 1. We choose the hyperplanethat is at the largest distance from the closest positive and

4

Class of attacked

measurements

Class of secure

measurements

ked

Class of secure

measurements

(a) Attack detection using the linearly separable dataset.

Class of attacked

measurements

Class of secure

measurements

Class of attacked

Class of secure

measurements

(b) Attack detection using the linearly non-separable dataset.

Fig. 1: Classification using SVM. Positive and negative sam-ples which belong to the class of attacked and secure mea-surements are depicted by disk and star markers, respectively.Support vectors and misclassified samples are depicted bydashed circles and hexagonal markers, respectively.

negative samples. This constraint can be formulated as

yi(wΨ ⋅Ψ(s)+ b) − 1 ≥ 0, ∀i = 1,2, . . . ,MTr. (11)

Since d+ = d− =1

∥wΨ∥2, whered+ and d− are the shortest

distances from the hyperplane to the closest positive andnegative samples respectively, a maximum margin hyperplanecan be computed by minimizing∥wΨ∥2.

If the training examples in the transformed space are notlinearly separable (see Fig. 1.b), then the optimization prob-lem can be modified by introducing slack variablesξi ≥ 0,∀i = 1,2, . . . ,MTr, in (11) which yields

yi(wΨ ⋅Ψ(si) + b) − 1 + ξi ≥ 0 ,∀i = 1,2, . . . ,MTr. (12)

The hyperplanewΨ is computed by solving the followingoptimization problem in primal or dual form [21], [22], [23]

minimize ∥wΨ∥2

2+C

MTr

∑i=1

ξi

subject to yi(wΨ ⋅Ψ(si) + b) − 1 + ξi ≥ 0ξi ≥ 0, ∀i = 1,2, . . . ,MTr

(13)

whereC is a constant that penalizes (an upper bound on) thetraining error of the soft margin SVM.

4) Sparse Logistic Regression:In utilizing this approach forattack detection, we solve the classification problem usingtheAlternating Direction Method of Multipliers (ADMM) [24]considering the sparse state vector estimation approach ofOzay et al. [5]. Note that, the hyperplanes defined in (10) canbe computed by employing the generalized logistic regressionmodels presented in [19], which provide the distributions

P (yi∣si) =1

1 + exp(−yi(w ⋅ si + b)), (14)

P (yi∣Ψ(si)) =1

1 + exp(−yi(wΨ ⋅Ψ(si) + b)), (15)

in S andF , respectively. For this purpose, we minimize thelogistic lossfunctions

L(si, yi) = log (1 + exp (−yi(w ⋅ si + b))) , (16)

L(Ψ(si), yi) = log (1 + exp(−yi(wΨ ⋅Ψ(si) + b))) . (17)

Defining a feature matrixS = (sT1 , sT2 , . . . , s

TMtr)T and a label

vector Y = (y1, y2, . . . , yMtr)T , the ADMM optimization

problem [24] is constructed as

minimize L(S,Y) + µ(r)subject to w − r = 0 (18)

where w is a weight vector,r is a vector of optimizationvariables,µ(r) = λ∥r∥1 is a regularization function, andλ isa regularization parameter which is introduced to control thesparsity of the solution [24].

B. Semi-supervised Learning Methods

In semi-supervised learning methods, the information ob-tained from the unlabeled test samples is used during thecomputation of the learning models [25].

In this section, a semi-supervised Support Vector Machinealgorithm, called Semi-supervised SVM (S3VM) [26], [27]is employed to establish the analytical relationship betweensupervised and semi-supervised learning algorithms. In thissetting, the unlabeled samples are incorporated into cost func-tion of the optimization problem (13) as

minimize ∥w∥22 +C1

MTr

∑i=1

LTr(si, yi) +C2

MTe

∑i=1

LTe(s′i), (19)

whereC1 andC2 are confidence parameters, andLTr(si, yi) =max(0,1−yi(wsi+ b)) andLTe(s′i) =max(0,1−∥s′i∥1) arethe loss functions of the training and test samples, respectively.

The main assumption of the S3VM is that the samples inthe same cluster have the same labels and the number of sub-clusters is not large [27]. In other words, attacked and securemeasurement vectors should be clustered in distinct regionsin the feature spaces. Moreover, the difference between thenumber of attacked and secure measurements should not belarge in order to avoid the formation of sub-clusters.

This requirement can be validated by analyzing the featurespace. Following (6), if∥zi−zj∥2+∥ai−aj∥2 ≤ ∥ai∥2+∥aj∥2,and∥zk − zl∥2 ≤ ∥ai∥2 + ∥aj∥2, ∀i, j ∈ A and∀k, l ∈ A, then

5

the samples belonging to different classes are well-separatedin different classes. Moreover, this requirement is satisfied in(19) by adjustingC2 [27]. A survey of the methods which areused to provideoptimalC2 and solve (19) is given in [27].

C. Decision and Feature Level Fusion Methods

One of the challenges of statistical learning theory is to finda classification rule that performs better than a set of rulesofindividual classifiers, or to find a feature set that represents thesamples better than a set of individual features. One approachto solve this problem is to combine a collection of classifiersor a set of features to boost the performance of the individualclassifiers. The former approach is called decision level fusionor ensemble learning, and the latter approach is called featurelevel fusion. In this section, we consider Adaboost [28] andMultiple Kernel Learning (MKL) [29] for ensemble learningand feature level fusion.

1) Ensemble Learning for Decision Level Fusion:Variousmethods such as bagging, boosting and stacking have beendeveloped to combine classifiers in ensemble learning situa-tions [17], [30]. In the following, Adaboost is explained asan ensemble learning approach, in which a collection ofweakclassifiers are generated and combined using a combinationrule to construct astronger classifier which performs betterthan theweakclassifiers [17], [28], [31].

At each iterationt = 1,2, . . . , T of the algorithm, a decisionor hypothesisft(⋅) of the weak classifier is computed withrespect to the distribution on the training samplesDt(⋅) at t by

minimizing the weighted errorǫt =MTr

∑i=1

Dt(i)I(ft(si) ≠ yi),

where I(⋅) is the indicator function. The distribution is ini-tialized uniformlyD1(i) =

1

MTr at t = 1, and is updated by aparameterαt =

1

2log( 1−ǫt

ǫt) as follows [31]

Dt+1(i) =Dt(i) exp−αtyift(si)

Zt

, (20)

whereZt is a normalization parameter, called thepartitionfunction. At the output of the algorithm, a strong classi-fier H(⋅) is constructed for a samples′ using H(s′) =

sign(T

∑t=1

αtft(s′)).

2) Multiple Kernel Learning for Feature Level Fusion:Fea-ture level fusion methods combine the feature spaces insteadof the decisions of the classifiers. One of the feature levelfusion methods is MKL in which different feature mappingsare represented by kernels that are combined to produce anew kernel which represents the samples better than the otherkernels [29]. Therefore, MKL provides an approach to solvethe feature mapping selection problem of SVM. In order tosee this relationship, we first give the dual form of (13)

maximizeMTr

∑i=1

βi − 1

2

MTr

∑i=1

MTr

∑j=1

βiβjyiyjk(si, sj)

subject toMTr

∑i=1

βiyi = 0

0 ≤ βi ≤ C, ∀i = 1,2, . . . ,MTr,

(21)

whereβi is the dual variable andk(si, sj) = Ψ(si) ⋅Ψ(sj) isthe kernel function. Therefore, (21) is a single kernel learning

algorithm which employs a single kernel matrixK ∈ RMTr×MTr

with elementsK(i, j) = k(si, sj). If we define the weighted

combination ofU kernels asK =U

∑u=1

duKu, wheredu ≥ 0 are

the normalized weights such thatU

∑u=1

du = 1, then we obtain

the following optimization problem of the MKL [32]:

maximizeMTr

∑i=1

βi − 1

2

MTr

∑i=1

MTr

∑j=1

βiβjyiyjU

∑u=1

duKu(si, sj)

subject toMTr

∑i=1

βiyi = 0

0 ≤ βi ≤ C, ∀i = 1,2, . . . ,MTr.

(22)

In (22), the kernels withdu = 0 are eliminated, and thereforeMKL can be considered as a kernel selection method. In theexperiments, SVM algorithms are implemented with differentkernels and these kernels are combined under MKL.

D. Online Learning Methods for Real-time Attack Detection

In the smart grid, the measurements are observed in real-time where the samples are collected sequentially in time.In this scenario, we relax the distribution assumption ofSection II.B, since the samples are observed in an arbitrarysequence [33]. Moreover,smartPMUs which employ learningalgorithms, may be required to detect the attacks when themeasurements are observed without processing the whole setof training samples. In order to solve these challenging prob-lems, we may use online versions of the learning algorithmsgiven in the previous sections.

In a general online learning setting, a sequence of trainingsamples (or a single sample) is given to the learning algorithmat each observation or algorithm processing time. Then, thealgorithm computes the learning model using only the givensamples and predicts the labels. The learning model is updatedwith respect to the error of the algorithm which is computedusing a loss function on the given samples. Therefore, theperceptron and Adaboost are convenient for online learninginthis setting. For instance, an online perceptron is implementedby predicting the labelyi of a single samplesi at eachtime t, and updating the weight vectorw using∆w for themisclassified samples withyi ≠ sign(f(si)) [34]. This simpleapproach is applied for the development of online MKL [34]and regression algorithms [35].

E. Performance Analysis

In smart grid networks, the major concern is not just thedetection of attacked variables, but also that of the securevariables with high performance. In other words, we requirethe algorithms to predict the samples with high precision andrecall performance in order to avoid false alarms. Therefore,we measure the true positives (tp), the true negatives (tn), thefalse positives (fp), and the false negatives (fn), which aredefined in Table I.

In addition, the learning abilities and memorization proper-ties of the algorithms are measured by Precision (Prec), Recall(Rec) and Accuracy (Acc) values which are defined as [13]

Prec = tp

tp+fp, Rec = tp

tp+fn, Acc = tp+tn

tp+tn+fp+fn. (23)

6

TABLE I: Definitions of performance measures

Attacked SecureClassified as Attacked tp fp

Classified as Secure fn tn

Precision values give information about the prediction per-formance of the algorithms. On the other hand, Recall valuesmeasure the degree ofattack retrieval. Finally, the totalclassification performance of the algorithms is measured byAccuracy. For instance, ifPrec = 1, then none of the securemeasurements is misclassified as attacked. IfRec = 1, thennone of the attacked measurements is misclassified as secure.If Acc = 1, then each measurement classified as attacked isactually exposed to an attack, and each measurement classifiedas secure is actually a secure measurement.

IV. EXPERIMENTS

The classification algorithms are analyzed in IEEE 9-bus,57-bus and 118-bus test systems in the experiments. Themeasurement matricesH of the systems are obtained fromthe MATPOWER toolbox [36]. The operating points of the testsystems provided in the MATPOWER case files are used ingeneratingz. Training and test data are generated by repeatingthis process50 times for each simulated point and dataset.In the experiments, we assume that the attacker has accessto κ measurements which are randomly chosen to generatea κ-sparse attack vectora with Gaussian distributed nonzeroelements with the same mean and variance as the entries ofz

[5], [13], [15]. We assume that concept drift [37] and datasetshift [38] do not occur. Therefore, we useG = N in thesimulations following the results of Ozay et al. [5].

We analyze the behavior of each algorithm on each systemfor both observable and unobservable attacks by generatingattack vectors with different values ofκ

N∈ [0,1]. More

precisely, if κ ≥N −D + 1, then attack vectors that are notobservable by SVE, i.e.unobservableattacks, are generated[5]. Otherwise, the generated attacks areobservable.

The LIBSVM [39] implementation is used for the SVM,and the ADMM [24] implementation is used for SparseLogistic Regression (SLR).k values of thek-NN algorithmare optimized by searchingk ∈ 1,2, . . . ,

√MTr using leave-

one-out cross-validation, whereMTr is the number of trainingsamples. Both the linear and Gaussian kernels are used for theimplementation of SVM. A grid search method [39], [40], [41]is employed to search the parameters of the SVM in an intervalI = [Imin,Imax], where Imin and Imax are user definedvalues. In order to follow linear paths in the search space,logvalues of parameters are considered in the grid search method[41]. Keerthi and Lin [41] analyzed the asymptotic propertiesof the SVM forI = [0,∞). In the experiments,Imin = −10 ischosen to compute a lower limit2−10 of the parameter valuesfollowing the theoretical results given in [39] and [41]. Sincethe classification performance of the SVM does not changefor parameter values that are greater than a threshold [41],weusedImax = 10 as employed in the experimental analyses in[41]. Therefore, the kernel width parameterσ of a Gaussian

kernel is searched in the intervallog(σ) ∈ [−10,10] and thecost penalization parameterC of the SVM is searched in theinterval log(C) ∈ [−10,10]. The regularization parameter ofthe SLR is computed as

λ = Ωλmax, (24)

whereλmax = ∥Hz∥∞ determines the critical value ofλ abovewhich the solution of the ADMM problem is0 and Ω issearched for in the intervalΩ ∈ [10−3,1] [5], [24], [42]. Anoptimal λ is computed by analyzing the solution (or regu-larization) path of the LASSO type optimization algorithmsusing a given training dataset. As the sparsity of the systemsthat generate datasets increases, lower values are calculated forΩ [5], [24], [42]. The absolute and relative tolerances, whichdetermine values of upper bounds on the Euclidean normsof primal and dual residuals, are chosen as10−4 and 10−2,respectively [24]. The penalty parameter is initially selected as1 and dynamically updated at each iteration of the algorithm[24]. The maximum number of iterations is chosen as104 [5],[24].

In the experiments, we observe that the selection of toler-ance parameters does not affect the convergence rates if theirrelative values do not change. In addition, selection of theinitial value of the penalty parameter also does not affect theconvergence rate if relative values of tolerance parameters arefixed [24]. For instance, similar convergence rates are observedwhen we chose10−4 and10−2, or 10−6 and10−4, as toleranceparameters.∥z −Hxb∥ ≤ τ is computed in order to decidewhether there is an attack using the SVE and assuming a chi-square test with95% confidence in the computation ofτ [6],[13].

A. Results for Supervised Learning Algorithms

The performance of different algorithms is compared forthe IEEE 57-bus system in Fig. 2. Accuracy values of theSVE and perceptron increase asκ

Nincreases in Fig. 2.a and

Fig. 2.b. Additionally, Recall values of the SVE increaselinearly as κ

Nincreases. Precision values of the perceptron are

high and do not decrease, and Accuracy and Recall valuesincrease, sincefn values decrease andtn values increase.In Fig. 2.c, a phase transition aroundκ∗ = N − D + 1, isobserved for the performance of the SVM. Since the distancebetween measurement vectors of attacked and secure variablesincreases asκ

Nincreases following (6), we observe that the

Accuracy, Precision and Recall values of thek-NN increasein Fig. 2.d. Accuracy and Recall values of thek-NN and SLRare above0.9 and do not change asκ

Nincreases in Fig. 2.e.

The class-based performance values of the algorithms aremeasured using class-wise performance indices, where Class-1and Class-2 denotes the class of attacked and secure variables,respectively. The class-wise performance indices are definedas follows:

Class − 1 ∶ Prec − 1 = tp

tp+fp, Rec − 1 =

tp

tp + fn, (25)

Class − 2 ∶ Prec − 2 = tntn+fn

, Rec − 2 =tn

fp + tn. (26)

7

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

SVE

AccuracyPrecisionRecall

(a) SVE.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / NP

erf

orm

an

ce

Perceptron


(b) Perceptron.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Linear SVM


(c) SVM with linear kernel.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

k-NN


(d) k-NN.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

SLR


(e) SLR.

Fig. 2: Results for the IEEE 57-bus system. Accuracy values ofthe SVE and perceptron increase while Precision values of thek-NN and SLR increase asκ

Nincreases. Both Accuracy and

Precision values of the SVM increase and phase transitionsoccur.

In Fig. 3.a, we observe that the Precision, Recall andAccuracy values of the SVE increase asκ

Nincreases for Class-

1. Note that the first value of Acc-1 is observed at0.008.In Fig. 3.b, Precision values for Class-2 decrease with thepercentage of attacked variables, i.e. the number of securevariables that are incorrectly classified by the SVE increases asthe number of attacked variables increases. Although the SVEmay correctly detect the attacked variables asκ

Nincreases, the

secure variables are incorrectly labelled as attacked variables,and therefore, the SVE gives more false alarms than the otheralgorithms.

Performance values for the perceptron are given in Fig.4. We observe that Precision values for Class-1 increase andRecall values do not change drastically for both of the classesas κ

Nincreases. Moreover, we do not observe any performance

increase for the Recall values of the secure class in theperceptron.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Prec−1Rec−1Acc−1

(a) Performance values for Class-1.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Prec−2Rec−2Acc−2

(b) Performance values for Class-2.

Fig. 3: Experiments using the SVE for the IEEE 57-bus testsystem. Note thatfp values increase asκ

Nincreases.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Perceptron

Prec−1Prec−2

(a) Results for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Perceptron

Rec−1Rec−2

(b) Results for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Perceptron

Prec−1Prec−2

(c) Results for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Perceptron

Rec−1Rec−2

(d) Results for the IEEE 118-bus.

Fig. 4: Performance analysis of the perceptron.

In Fig. 5, the results fork-NN are shown. We observethat performance values for Class-1 increase and the valuesfor Class-2 decrease asκ

Nincreases sincek-NN is sensitive

to class-balance and sparsity of the data [43]. In addition,classification hypotheses are computed by forming neigh-borhoods in Euclidean spaces, and theℓ2 norm of vectorsof attacked measurements increases asκ

Nincreases in (6);

therefore, decision boundaries of the hypotheses are biasedtowards Class-1.

Fig. 6 depicts the results for the SLR, where the per-formance values for Class-2 (secure variables) increase asthe system size increases. Moreover, we observe that theperformance values for Class-2 do not decrease rapidly asκN

increases, compared to the other supervised algorithms. Inaddition, the performance values for Class-1 are higher thanthe values of the other algorithms, especially for lowerκ

N

values. The reason is that the SLR can handle the variety in thesparsity of the data asκ

Nchanges. This task is accomplished

8

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

κ / N

Per

form

ance

k-NN

Prec−1Prec−2

(a) Prec. values for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

κ / NP

erfo

rman

ce

k-NN

Rec−1Rec−2

(b) Rec. values for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10.75

0.8

0.85

0.9

0.95

1

κ / N

Per

form

ance

k-NN

Prec−1Prec−2

(c) Prec. values for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10.75

0.8

0.85

0.9

0.95

1

κ / N

Per

form

ance

k-NN

Rec−1Rec−2

(d) Rec. values for the IEEE 118-bus.

Fig. 5: Since thek-NN is sensitive to class-balance andsparsity of the data, performance values for Class-1 increaseand the values for Class-2 decrease asκ

Nincreases. Note that

the performance curves intersect at the critical valuesκ∗.

by controlling and learning the sparsity of the solution in (18)using the training data in order to learn the sparse structure ofthe measurements defined in the observation model (5).

The results of the experiments for the SVM are shown inFig. 7, where a phase transition for the performance values isobserved. It is worth noting that the values ofκ at which thephase transition occurs correspond to the minimum numberof measurement variables,κ∗, that the attacker needs tocompromise in order to construct unobservable attacks [7].κ∗

is depicted as a vertical dotted line in Fig. 7. For instance,κ∗ = 10 and κ∗

N= 0.56 for the IEEE 9-bus test system.

The transitions are observed before the critical points whenthe linear kernel SVM is employed in the experiments forIEEE 57-bus and 118-bus systems. In addition, the phasetransitions of performance values occur at the critical pointswhen Gaussian kernels are used.

B. Results for Semi-supervised Learning Algorithms

We use the S3VM with default parameters as suggestedin [44]. The results of the semi-supervised SVM are shownin Fig. 8. We do not observe sharp phase transitions in thesemi-supervised SVM unlike the supervised SVM, since theinformation obtained from unlabeled data contributes to theperformance values in the computation of the learning models.For instance, Precision values of Class-2 decrease sharplynear the critical point for the supervised SVM in Fig. 7.However, the semi-supervised SVM employs the unlabeledsamples during the computation of the learning model in (19),and partially solves this problem.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Prec−1Prec−2


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Rec−1Rec−2


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Prec−1Prec−2


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Rec−1Rec−2


Fig. 6: Experiments using the SLR. Note that the SLR canhandle the variety in the sparsity of the data asκ

Nchanges.

C. Results for Decision and Feature Level Fusion Algorithms

In this section, we analyze Adaboost and MKL.Decisionstumpsare used as weak classifiers in Adaboost [31]. Eachdecision stump is a single-level two-leaf binary decision treewhich is used to construct a set of dichotomies consistingof binary labelings of samples [31]. The number of weakclassifiers is selected using leave-one-out cross-validation inthe training set. We use MKL with a linear and a Gaussiankernel with the default parameters suggested in the SimpleMKL implementation [32]. The results given in Fig. 9 showthat Recall values of MKL for Class-1 are less than thevalues of Adaboost. In addition, Precision values of MKLdecrease faster than the values of Adaboost asκ

Nincreases

for Class-2. Therefore, thefn values of MKL are greater thanthe values of Adaboost, or in other words, the number ofattacked measurements misclassified as secure by MKL isgreater than that of Adaboost. This phenomenon is observed inthe results for semi-supervised and supervised SVM given inthe previous sections. However, there are no phase transitionsof the performance values of MKL compared to the supervisedSVM.

D. Results for Online Learning Algorithms

We consider four online learning algorithms, namely OnlinePerceptron (OP), Online Perceptron with Weighted Models(OPWM), Online SVM and Online SLR. Note that these algo-rithms are the online versions of the batch learning algorithmsgiven in Section III-A and developed considering the onlinealgorithm design approach given in Section III-D. The detailsof the implementations of the OP, OPWM, Online SVM andSLR are given in [34], [35] and [45].

9

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Linear SVM for the IEEE 9−bus

Prec−1Prec−2Critical Point

(a) Linear SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Rec−1 and Rec−2Critical Point

(b) Linear SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Gaussian SVM for the IEEE 9−bus


(c) Gaussian SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(d) Gaussian SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(e) Linear SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(f) Linear SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(g) Gaussian SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(h) Gaussian SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(i) Linear SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(j) Linear SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(k) Gaussian SVM.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce



(l) Gaussian SVM.

Fig. 7: Experiments using the SVM with linear and Gaussian kernels. Phase transitions of performance values occur at thecritical pointκ∗. See the text for more detailed explanation.

When the OP is used, only the modelw(t) computed usingthe last observed measurement at timet is considered for theclassification of the test samples. On the other hand, we con-

sider an average of the modelswave(t) = 1

T

T

∑t=1

w(t) which is

computed by minimizing margin errors in the OPWM. Resultsare given for the OP in Fig. 10. In the weighted models, weobserve phase transitions of the performance values for Class-2 in Fig. 10.e-Fig. 10.h. However, the phase transitions occurbefore the critical values, and the values of the phase transitionpoints decrease as the system size increases. Additionally, wedo not observe sharp phase transitions in the OP.

In the OP, if the label of a measurements is not correctlylabeled, then the measurement vector is added to a set of sup-porting measurementsS that are used to update the hypothesesin the training process. However, the hypotheses are updatedin the OPWM if a measurements′ is not correctly labeled, andthe vectors ofs′ ands ∈ S are linearly independent. Since thesmallest number of linearly dependent measurements increasesas κ

Nincreases [5], [46], the size ofS decreases and the bias

is decreased towards Class-1. Therefore, false negative (fn)values decrease and false positive (fp) values increase [47]. As

a result, we observe that Recall values of the OP are less thanthat of the OPWM for Class-1. The results of the Online SVMand Online SLR are provided in Fig. 10 for different IEEE testsystems. We observe phase transitions of performance valuesin the Online SVM similar to the batch supervised SVM.

Learning curves of online learning algorithms are given inFig. 11 for both observable attacks generated withκ

N= 0.33

and unobservable attacks generated withκN= 0.66. Since the

cost function of each online learning algorithm is different,the learning performance is measured and depicted usingaccuracy (Acc) defined in (23). In the results, performancevalues of the Online SVM and OPWM increase as the numberof samples increases, since the algorithms employ marginlearning approaches which provide better learning rates asthenumber of training samples increases [34], [45].

Briefly, we suggest using Online SLR for the scenarios inwhich the precision of the classification of secure variablesis important to avoid false alarms. On the other hand, if theclassification of attacked variables with high Precision andRecall values is an important task, we suggest using the OnlinePerceptron.

10

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Semi−supervised SVM with Linear Kernel

Prec−1Prec−2


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / NP

erf

orm

an

ce


Rec−1Rec−2


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Prec−1Prec−2


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Rec−1Rec−2


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Semi−supervised SVM with Gaussian Kernel

Prec−1Prec−2

(e) Results for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Rec−1Rec−2

(f) Results for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Prec−1Prec−2

(g) Results for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Rec−1Rec−2

(h) Results for the IEEE 118-bus.

Fig. 8: Sharp phase transitions are not observed in the semi-supervised SVM unlike the supervised SVM, since the informationobtained from unlabeled data contributes to the performance values in the computation of the learning models.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Adaboost

Prec−1Prec−2

(a) Adaboost for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Adaboost

Rec−1Rec−2

(b) Adaboost for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Adaboost

Prec−1Prec−2

(c) Adaboost for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Adaboost

Rec−1Rec−2

(d) Adaboost for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

MKL

Prec−1Prec−2

(e) MKL for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

MKL

Rec−1Rec−2

(f) MKL for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

MKL

Prec−1Prec−2

(g) MKL for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

MKL

Rec−1Rec−2

(h) MKL for the IEEE 118-bus.

Fig. 9: Experiments using Adaboost and MKL. Note that thefn values of MKL are greater than the values of Adaboost, andthere are no phase transitions of the performance values of MKL compared to the supervised SVM.

11

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Online Perceptron

Prec−1Prec−2

(a) OP for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Online Perceptron

Rec−1Rec−2

(b) OP for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Online Perceptron

Prec−1Prec−2

(c) OP for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Online Perceptron

Rec−1Rec−2

(d) OP for the IEEE 118-bus.

0 0.5 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Online Perceptron with Weighted Models

Prec−1Prec−2

(e) OPWM for the IEEE 57-bus.

0 0.5 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Rec−1Rec−2

(f) OPWM for the IEEE 57-bus.

0 0.5 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Prec−1Prec−2

(g) OPWM for the IEEE 118-bus.

0 0.5 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce


Rec−1Rec−2

(h) OPWM for the IEEE 118-bus.

0 0.5 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Online SVM

Prec−1Prec−2

(i) Online SVM for the IEEE 57-bus.

0 0.5 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Online SVM

Rec−1Rec−2

(j) Online SVM for the IEEE 57-bus.

0 0.5 10

0.2

0.4

0.6

0.8

1

κ / N

Pe

rfo

rma

nce

Online SVM

Prec−1Prec−2

(k) Online SVM for the IEEE 118-bus.

0 0.5 10

0.2

0.4

0.6

0.8

1

κ / NP

erf

orm

an

ce

Online SVM

Rec−1Rec−2

(l) Online SVM for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance

Online Sparse Logistic Regression

Prec−1Prec−2

(m) Online SLR for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance


Rec−1Rec−2

(n) Online SLR for the IEEE 57-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance


Prec−1Prec−2

(o) Online SLR for the IEEE 118-bus.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

κ / N

Per

form

ance


Rec−1Rec−2

(p) Online SLR for the IEEE 118-bus.

Fig. 10: Experiments using the Online Perceptron (OP), Online Perceptron with Weighted Models (OPWM), Online SVM andSLR. Recall values of the OP are less than that of the OPWM for Class-1. Multiple phase transitions of performance valuesof the Online SVM are observed in the IEEE 118-bus system.

V. SUMMARY AND CONCLUSION

The attack detection problem has been reformulated as amachine learning problem and the performance of supervised,semi-supervised, classifier and feature space fusion and onlinelearning algorithms have been analyzed for different attackscenarios.

In a supervised binary classification problem, the attackedand secure measurements are labeled in two separate classes.

In the experiments, we have observed that state of the artmachine learning algorithms perform better than the well-known attack detection algorithms which employ a state vectorestimation approach for the detection of both observable andunobservable attacks.

We have observed that the perceptron is less sensitive andthe k-NN is more sensitive to the system size than the otheralgorithms. In addition, the imbalanced data problem affects

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Pe

rfo

rma

nce

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(a) Observable attacks.

0 1000 2000 3000 4000 5000 60000.5

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Number of SamplesP

erf

orm

an

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(b) Unobservable attacks.

Fig. 11: Learning curves of online learning algorithms.

the performance of thek-NN. Therefore,k-NN may performbetter in small sized systems and worse in large sized systemswhen compared to other algorithms. The SVM performsbetter than the other algorithms in large-scale systems. Intheperformance tests of the SVM, we observe a phase transitionatκ∗, which is the minimum number of measurements that arerequired to be accessible by the attackers in order to constructunobservable attacks. Moreover, a large value ofκ does notnecessary imply high impact of data injection attacks. Forexample, if the attack vectora has small values in all elements,then the impact ofa may still be limited. More important, ifa is a vector with small values compared to the noise, theneven machine learning-based approaches may fail.

We observe two challenges of SVMs in their application toattack detection problems in smart grid. First, the performanceof the SVM is affected by the selection of kernel types. Forinstance, we observe that the linear and Gaussian kernel SVMperform similarly in the IEEE 9-bus system. However, for theIEEE 57-bus system the Gaussian kernel SVM outperformsits linear counterparts. Moreover, the values of the phasetransition points of the performance of the Gaussian kernelSVM coincide with the theoretically computedκ∗ values. Thisimplies that the feature vectors inF , which are computedusing Gaussian kernels, are linearly separable for higher valuesof κ. Interestingly, the transition points missκ∗ in the IEEE118-bus system, which means that alternative kernels areneeded for this system. Second, the SVM is sensitive tothe sparsity of the systems. In order to solve this problem,sparse SVM [48] and kernel machines [49] can be employed.In this paper, we approached this problem using the SLR.However, obtaining anoptimal regularization parameter,λ, iscomputationally challenging [24].

In order to use information extracted from test data in thecomputation of the learning models, semi-supervised methodshave been employed in the proposed approach. In semi-supervised learning algorithms, we have used test data to-gether with training data in an optimization algorithm usedto compute the learning model. The numerical results showthat the semi-supervised learning methods are more robust tothe degree of sparsity of the data than the supervised learningmethods.

We have employed Adaboost and MKL as decision andfeature level fusion algorithms. Experimental results show that

fusion methods provide learning models that are more robustto changes in the system size and data sparsity than the othermethods. On the other hand, computational complexities ofmost of the classifier and feature fusion methods are higherthan that of the single classifier and feature extraction methods.

Finally, we have analyzed online learning methods for real-time attack detection problems. Since a sequence of trainingsamples or just a single sample is processed at each time, thecomputational complexity of most of the online algorithms isless than the batch learning algorithms. In the experiments, wehave observed that classification performance of online learn-ing algorithms are comparable to that of the batch algorithms.

In future work, we plan to first apply the proposed approachand the methods to an attack classification problem for decid-ing which of several possible attack types have occurred giventhat an attack have been detected. Then, we plan to considerthe relationship between measurement noise and bias-varianceproperties of learning models for the development of attackdetection and classification algorithms. Additionally, weplanto expand our analyses for varying number of clustersG andcluster sizesNg, ∀g = 1,2, . . . ,G, by relaxing the assumptionsmade in this work for attack detection in smart grid systems,e.g. when the samples are not independent and identicallydistributed and obtained from non-stationary distributions, inother words, concept drift [37] and dataset shift [38] occur.

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