Qinxue LI,Bugong XU ,Shanbin LI,Yonggui LIU ,Delong CUI

1.School of Automation Science and Engineering,South China University of Technology,Guangzhou Guang dong 510640,China;

2.College of Computer and Electronic Inform ation,Guangdong University of Petrochemical Technology,Maoming Guangdong 525000,China

1 Introduction

The design of estimation and control algorithm s or strategies for cyber physical system s(CPS),which is resilient against malicious attacks is a new and hard work in recent years.Many researchers focus on designing the more resilient estimators and controllers[1–4]or attack-detection and identification technology[5,6]to reduce dam age from the attacks,but few researchers pay attention to reconstruction of the transmitted data from the sensors to estimators or from controllers to actuators,which is incomplete data after attacks.

In general,the measurements from sensors are very important for state estimator and vulnerable to cyber attacks,which can be processed in two ways.The first w ay is that polluted(or corrupted)measurements are all received by state estimator,then anti-attack state estimators are designed elaborately.For example,state recovery technology is achieved by l1/lrdecoder[2],attack reconstruction[7,8]or state reconstruction[9,10],which is similar to the error correction and thus relevant techniques,such as signal separation,fault-tolerant or noise-tolerant state estimator,but all of these researches mentioned above must need the known model of state equation.Inspired by the idea of data reconstruction,we pay attention to the measurement reconstruction under the unknown model of state equation after discarding the polluted measurements.Therefore,discarding the polluted data is the second processing way for measurements,which is similar to remove outliers of data and lends to less and less clean measurements used by state estimator.With the reduction of clean measurements,large estimating errors or failures of the general anti-noise state estimator(such as Kalman,weighted least squares)will appear,hence the measurement reconstruction is necessary to be considered.

In the case of unknown model of state equation,we focus on the reconstruction of measurements under cyber attacks,which is different from the idea of state reconstruction or resilient state estimator before.If the exchanged or transmitted measurements are reconstructed accurately under the cyber attacks,the design of secure state estimator is not the most important,which is a new idea to solve the security problem under malicious attacks for CPS(for instance,modern power system,namely smart grid[11]).In other words,we do not design the anti-attack state estimator but focus on reconstructing the enough measurements from the incomplete(or residual)clean measurements,including the triggering time of the reconstruction and high precision reconstruction method.Inspired by[12],the indirect and direct methods of state estimation are developed based on compressive sensing(CS),which reduce the burden of communication and data processing.Instead,the CS is draw n to restore the measurements from in complete but clean measurements under the cyber attacks scenario.

It is very important to design the dictionary matrix for optimal sparse representation for measurements and the measurement matrix of CS.There are three main ways to obtain the dictionary matrix in CS[12–14].The first way is the traditional transformation basis,such as discrete wavelet trans form(DWT)basis[12],which is simple and fast but depends on whether or not the characteristics of the signal are consistent with the characteristics of the basis vectors.The second way is joint dictionary,in which vectors come from different classes of basis[13],whose effect is better than the first one but still not the best sparse representation.Finally,improved sparsity can be achieved by choosing the dictionary for the signals adaptively,that is,dictionary learning[14].For the design of measurement matrix,the Gauss or Bernoulli random matrices are usually proposed[12,15],which have a good effect in the field of image reconstruction before.In this paper several improvements have been done for the CS under the cyber attacks to CPS.

The approaches and contributions of the paper are summarized as follows:

?For the polluted measurements under malicious attacks,the observability analysis is used to decide the time to trigger the reconstruction and the damage level from attacks.The reconstruction of measurements can be triggered as long as the damage level of the attacks makes the system losing observability,which can reduce computational burden.

?With unknown model of state equation,the improved CS is applied in reconstruction of measurements against deception attacks for CPS.In particular,the dictionary learning by K-singular value decomposition(KSVD)is proposed to form the over-completed dictionary matrix for the power measurements of power system,which never been adopted before in this field.

?Since the irregularity of residual measurements,instead of the traditional Gauss or Bernoulli random matrix,a kind of sampling matrix is designed as the measurement matrix of CS for specific CPS,such as power system,which improves the effect of reconstruction.

?A new state estimation strategy against the cyber attacks based on linear model is introduced in this paper,which can improve the resilience of system and is meaningful and different idea from the design of the resilient estimator with the known model of the state equation.

As the typical CPS,security of the smart grid under malicious attacks is widely concerned.Therefore,we apply our proposed reconstruction method on smart grid(modern power system).The rest of this paper is organized as follow s.In Section 2,the general state estimation Model of power system(Section 2.1)and Gaussian Markov Random Field associated with attack detection and identification(Section 2.2)are introduced,and stealthy deception attacks strategy is described in Section 2.3.The attack detection and identification technology is presented in Section 3.1,which not only can detect the stealthy deception attacks but also identify the attacked nodes of system.Section 3.2 introduce observability analysis,which can decide the time to do the reconstruction and the dam age level from attacks.The reconstruction process of measurements is proposed in Section 3.3,including the design of over-completed dictionary and measurement matrix,in addition,recovery strategy is also chosen.The whole state estimation strategy is summarized in Section 3.4.Finally,in Section 4,qualitative and quantitative analyses of simulation experiments are completed on 6-bus power system to show the effect of our proposed method,and conclusion is seen in Section 5.

Notation0nmeans a null vector of dimension n,and Indenotes that I is the identity matrix and n is the state dimension.b1∝b2denotes that b1is proportional to b2.For a vector ·,‖·‖0denotes the number of nonzero elements for this vector.‖·‖1and ‖·‖2denotes the 1-norm and 2-norm of the vector ·,respectively.∩ is used to denote the union set.Finally,diag{b3}denotes that the vector b3are diagonal elements of a matrix.

2 Preliminaries and problem formulation

The proposed overall framework in this paper is displayed in Fig.1.The parts in the dashed-line box of Fig.1 are our main works,which are also the proposed state estimation strategy and have a great contribution to improve the resilience of the strategy under cyber attacks.

The state estimation strategy is applied to the power system in this paper,which is a major component of energy management systems(EMS)and relies on the supervisory control and data acquisition(SCADA).This is a process about optimal estimation for the power system state using noisy data from power meters and system para meters on the state estimation model.However,the measurements are collected by SCADA via remote terminal units(RTU),which is prone to the cyber attacks.We consider the stealthy deception attacks that consist of a set of com promised power meters whose readings are altered by the intelligent attacker,which lead to the polluted power measurements of nodes.The general state estimation model of the system and stealthy deception attack strategy is elaborated as follow s,respectively.In addition,the know ledge of a Gaussian Markov random field is introduced,which is the foundation for attack detection and identification method.

Fig.1 The proposed overall framework.

2.1 General system model

The nonlinear measurement model of power system for state estimation(SE)is show n in(1),whose relationship between measurements and states in detail can be found in[16].

where the state x=(θT,VT)Tdenote the phase angles and magnitudes of the bus voltage,the measurements z∈Rm,and e～N(0,R)is the measurement noise.

To test the method proposed in this paper,a DC approximation model for the measurement equations(1)is adopted.Specifically,the DC model is obtained by assuming that all branch resistances and shunt elements are neglected.In addition,the bus voltage magnitudes are already known and all equal to 1 p.u.[16,17].With linear correlation between the measurements z and the state x(voltage phase angles θ in this paper),the DC model is given by

1)If zk=Pij,then

2)If zk=Pi,then

w here the voltage phase angles x=θ∈Rn,H is the m easurem ent Jacobian m atrix.Pijis real power flow on line i?j.Piis real power injection at bus i.Niis the set of buses connected to bus i,and Xijdenote that the reactance of branch between nodes i and j.

2.2 Gaussian Markov random field

A Gaussian Markov random field(GMRF)[18]is a family whose factor is a finite-dimensional random vector following a Gaussian distribution according to a given graph G=(V,E),with V={1,2,...,n}.The vector of Gaussian random variables X=[x1,x2,...,xn]is considered,where each node i∈V is associated with a scalar Gaussian random variable xi.A probability density function of a GMRF can be parameterized as follows on G:

w here J is the potential or information matrix,which is a positive-definite symmetric matrix.The sparsity pattern of J corresponds to that of the graph G.More precisely,

and the vector h is a vertex potential vector.In general,graph G=(V,E)is known as the Markov graph(graphical model)underlying the joint probability distribution fX(x),where the node set V represents random variable set{xi}(voltage phase angle θiin this paper)and the edge set E is defined in order to satisfy the local Markov property.Σ represents covariance matrix for voltage phase angles.

2.3 Stealthy deception attacks

It assumes that the attacker can obtain the know ledge of the bus-branch model,and the goal of the attacker is to deceive the traditional bad data detection(BDD)(such as the largest normalized residual,LNR)and state estimator(SE)(such as the weighted least square)[19].The measurements are polluted as follows:

where za∈Rmis the polluted measurements and a∈Rmis the attack vector.(7)assumes that a is independent of the measurement z in the network.However,(7)is not enough,the cost of attacks is always hoped to be lower.Therefore,the stealthy deception attack a with the minimum cost can be formulated by solving the problem

Rem ark 1It is noted that the stealth of the(7)in power system have been proved in[20].Based on the(7)and the(8),the attacks should have some functions.First,the attack strategy is designed to ensure that the SE algorithm still converges under attacks.Second,the attacker hopes the estimated measurementis equal to the actual polluted measurementsuch that,which is more stringent for the practical nonlinear power system s,since the stealthy deception attacks is built on DC approximation model.Therefore,the limitations of this linear attack strategy mentioned in[19]should be considered,including varying model matrix H that is acquired by the attacker and saturation limits,which make sure the attack strategy(7)is optimal and stealth under the traditional BDD(LNR)and SE(WLS)for a practical nonlinear system.Therefore,the stealthy deception attacks are coordinated attacks,which need corrupt a set of measurements but not just the target measurements[19].

Since the attacker is intelligent,we need m ore effective attack detection and identification technologies and state estimation strategies to resist the stealthy deception attacks,which is important for the subsequent supervisory control.

3 Methodology

3.1 Attack detection and identification

The voltage phase angles(θ)in DC model can form a GMRF and argue that their Markov graph is dictated by the structure of pow er grid[18].Therefore,the conditional covariance test CM IT is introduced to learn the structure of the power grid.It should be noted that since the CMIT for structure learning of phase angle data is robust against measurement noise(see[18],Section IIIA),the measurement noise e can be ignored.From(2)and(5),we get

where xais the vector of phase angles when the system is under attacks.H?1denotes the pseudo-inverse of matrix H.since a is independent on the measurement z,we have the covariance matrix as follows:

When the attack occurs,the attacker tends to insert non-constant vectors a during some samples,since constant attack vectors could not follow the measurements in change during some samples to satisfy(7),which can be easily detected.Thus Σ(a,a)≠ 0 and

It is showed that the Markov graph of the voltage angles θ is consistent with the power grid graph under no attack circum stances,but changes once the power grid has been attacked on(11)and(12).Hencea discrepancy between the calculated Markov graph under attacks and learned Markov graph under no attacks should trigger the attack alarm.The attack detection method can detect the stealthy deception attacks in Section 2.3,which is elaborated in[18].

Moreover,the correlation anomaly metric[21]is used to identify the attacked nodes.As soon as the attack is detected,the anomaly score for each node is computed through differences between the attacked information matrix and the information matrix corresponding to the current topology of the power grid.The nodes with highest anomaly scores are announced as the nodes under attacks.This attack identification technology has been proved to be valid for stealthy deception attacks in Section 2.3[18].

3.2 Observability analysis

From the(8),it is easy to see that the attack vector is always sparse.Therefore,discarding few polluted measurements and isolating the attacked parts after attack detection and identification are feasible.However,less and less clean measurements could lend to large estimating errors or failures of the general anti-noise state estimator,which is the time to trigger the reconstruction of measurements.

This section,the observability analysis of the CPS is used to decide the time to trigger the reconstruction and the damage level from attacks,which is based on orthogonal transformation algorithm,including checking observability,identifying critical measurements,determining observable/unobservable islands.Variable descriptions of observability analysis are shown in Table1.

Table 1 Variable description about observability analysis.

The algorithm[22]is presented as follow s:

Step 1(Initialization) Set matrix W=In,vector U=0n.

Fori=1:m(m is the dimension of the measurements).

Step 2Calculate the dot productsj=1,...,n,namely,the dot products of the i throw of matrix H by the j th column of matrix W.

Step 3If the first no n null valuecorresponding to a null element in position p on vector U is found,which indicate that the column p of Wis pivot,and then the p th component of vector U should be replaced by i.Else there is no such colum n,append i to vector V and vector Tito matrix S and go to Step 5.

Step 4Update the matrix W.If j=p,W(:,p)=W(:,k=1,...,n.

Step 5If i=m,end.Otherwise,i=i+1 and jumps to Step 2.

End

Step 6If Ucontain sonly one zero element,the state is observable;otherwise,it is not.

3.3 Reconstruction of measurements

Reconstruction of measurements is the foundation for state estimation,which can be achieved by CS in the paper.Through the learning of the measurement data under no attacks,the over-completed dictionary is formed to provide prior information for possible triggered reconstruction at any time.There are three steps in the reconstruction process,including design of the over completed dictionary(dictionary learning is proposed),compressive observation(how to design the measurement matrix in CS)and recovery of in completed data(power flow measurements z),where the first two steps are coding procedure and the last step is decoding procedure.The problem can be formulated as follows:recover s from know ledge of

w here y∈Rm1(m1?m)are residual measurements data.Ψ is m1× m measurement matrix,and Φ is sparse transformation basis or dictionary matrix.As significant coefficient,s contains the representation sparse coefficients of the signal z(z=Φs)that must be a K-sparse vector(K＜m1?m).A=ΨΦ is a sensing matrix.It is quite intuitive to recover s from know ledge of y by solving

However,the fact is that there is certain error in the solution to(14),hence the optimization problem can be rep laced by a simple approximate solution:

whereεis a very small constant.It should be pointed out that the algorithm s(14)and(15)are NP-hard,therefore,the l0norm is often substituted by the closest convex norm,namely the l1norm[23].Here the problem is equivalent to a much simpler linear programming problem.The measurement signals can be reconstructed by solving the following problem:

In fact,(16)is equivalent to basis pursuit de-noising(BPDN)[24].In addition,regularized orthogonal matching pursuit(ROMP)[25],sparsity adaptive matching pursuit(SAMP)[26],basis pursuit(BP)and iterative thresholding(IS)[27]are adopted by researchers to completed the reconstruction.

3.3.1 Design of over-completed dictionary

From the described above,it is easy to find that the Ψ and Φ have key roles in the whole reconstruction process,especially Φ.Designing or choosing the Φ for optimal sparse representation for data,which is sim ilar to(14)and(15).In other words,the data must be sparse in some transform basis vectors.

Rem ark 2The design of the dictionary for optimal sparse representation for data is extremely critical.The traditional transformation basis depends on whether or not the characteristics of the signal are consistent with the characteristics of the single basis vector,hence it has great limitation,such as discrete wavelet transform(DWT)basis[12],discrete cosine transform(DCT)[28]and so on.The joint dictionary includes different classes of basis[13],whose effect is better than the single basis but still has the similar problem with the traditional transformation basis.Therefore,im proved sparse representation in dictionary learning is formed adaptively according to the specific characteristics of signal,which has no limitations and is more suitable.

In this paper,the dictionary learning on K-SVD algorithm is used to obtain the over-completed dictionary,which is generalization of the K-means clustering pro-cess.To better fit the data,the method alternates between sparse coding of the examples based on the current dictionary and a process of updating the dictionary atom s,completing not only the update of the dictionary but the update of the sparse representations,which render convergence[14].The K-SVD algorithm is adaptive and can work with any pursuit methods(e.g.,matching pursuit or basis pursuit).

Assume that Φ ∈ Rm×Kdenote over-completed dictionary after training.z∈Rmand s∈RKis training samples and corresponding sparse representation coefficient vector.are M sets of training data samples.are M sets of coefficient vector.Thus the dictionary learning process can be represented as an optimization problem:

where Tmaxis the maximum value of the number of nonzero entries in si.The dictionary learning process is described as follows:

1)Initialization:there are two ways to initialization dictionary.First one is initialized by giving a traditional transformation basis dictionary(such as the DWT dictionary).The other one is initialized by the training sam p les them selves.

2)Sparse coding:use any pursuit algorithm(such as BP)to compute the representation vectorsifor each ziby approximating the solution of

3)Dictionary update:update dictionary Φ after fixed vector si.Denote vector ?kas k th column of Φ.The decomposition form of the sample sets can be expressed as

Note that ωkis pointing to examples{zi}that use the atom ?k.Define Ωkas M × |ωk|matrix with ones on the(ωk(i),i)th entries and zeros elsewhere.Then there have the definitions as follows:

where Ekandis shrinkage result by discarding of the zero entries,respectively.indicate a selection of error columns that correspond to examples who use the atom ?k.Thenis decomposed by SVD render thatwhere U and V are two mutually orthogonal matrices and Δ is diagonal matrix:

Note that σi(i=1,2,...,r)is all nonzero singular value of,and r is the rank of the matrix.

If Δ(1,1)is the maximum singular value of Δ,and then the atom ?kshould be rep laced by the first column of U.The coefficient vectoris updated by the first column of V multiplied by Δ(1,1),until the update of the atom ?kis finished.If the convergence condition is satisfied or to the maximum number of iterations that indicate a new over-completed dictionary will been done,otherwise go to Step 2).

3.3.2 Designing measurement matrix and recovery strategy

Due to the irregularity of in complete measurements or residual measurements,we can not select the traditional Gauss or Bernoulli random matrix as measurement matrix,which are widely used in image processing.In this paper,the sampling matrix Ψ(m1× m)is used as the measurement matrix,which is an under determined matrix.m1?m implies residual character of the measurements.We define that Z is the set of the residual measurements,C is the set of the complete measurements,which is our reconstruction target.i denote that the number of the residual measurements in Z,and j is the position number of the residual measurements in C.We have the sampling matrix Ψ as follows:

To restore or reconstruct the measurements,the matrix A should meet the following restricted isometry property(RIP)[23]:

Definition 1(Restricted isometry constant(RIC))For any K-sparse vector s and δk∈ (0,1),A has the RIP of order k such that

It is noted that the RIC δ is always adopted in stead of the RIP,which is the sm allest value of the δk.

In order to test the effect of our over-completed dictionary,recovery strategy should not contains only one recovery algorithm.As the last step of the reconstruction,the recovery algorithm in the paper are adopted including several algorithm s,such as BP,BPDN,ROMP,SAMP and iterative soft thresholding(IST),which can prove the universality of the method proposed.Note that the BP,BPDN,ROMP,SAMP and IST have been widely used in the field of image processing,which can be seen in[24–27,29].

3.4 State estimation strategy

The w hole state estimation strategy around in the dashed-line box of Fig.1 is presented as follow s:

Algorithm flow(whole state estimation strategy)

1)Receive the measurements and attack information(whether or not the nodes are attacked and the set of attacked nodes)from the attack detector(attack detection and identification).

2)The observability analysis is run according to the measurements.

3)If the system is observable,the Gauss-New ton’s method[16]is adopted in state estimator,otherwise,the measurements should be reconstructed by our proposed in Section 3.3:First,the over-completed dictionary can be formed by dictionary learning on K-SVD,namely,matrix Φ is obtained.Second,the measurement matrix Ψ is designed according to the actual situation,for instance,the information of attacked nodes.Third,the optimal s?is obtained by solving the problem(16)by the IST,since the effect of IST on data recovery is verified to be the best among the five recovery algorithms in simulation experiments.Finally,the z?can be calculated by(17).

It should be noted that the whole state estimation strategy is a general solution,although our proposed state estimation strategy is based on linear model.

4 Simulation experiments

The IEEE 6-bus power system[30]is considered to investigate the effect of the proposed method,which includes 3 generators,6 buses,8 transmission lines.The measurement configuration of a6-bus powersy stem are shown in Fig.2.If there is no attacks,7 power flow measurements({P12,P14,P24,P25,P36,P45,P56})are transmitted to state estimator.However,owing to the stealthy deception attacks,some of measurements could be discarded after attack detection and identification,the residual measurements are called incomplete measurements,which need be restored to 7 original power flow measurements to ensure precision of SE.Sinceat tack detection and identification in Section 3.1 has been proved to detect the attacks in Section2.3 and identify the attacked nodes successfully in[18],we pay main attention to the reconstruction method of measurements in this paper.

Fig.2 Measurement configuration of a 6-bus power system.

Two cases in simulation experiments are considered,which are shown in Table2.Define η =m1/m×100%,which indicate the damage level from attacks,where m1denotes the number of available clean measurements(some measurements could have been discarded as soon as possible once corresponding nod es are attacked nodes that should be isolated),m is 7 that denotes maxim um number of measurements(assume that there is no packet loss and delay).In case 1,there are 4 available clean measurements(U={P14,P36,P45,P56}),which lead the system to be unobservable,where the observability analysis method in Section3.2 is em ployed.With less available measurements,the system also loses observability in case 2.Hence the proposed reconstruction method in Section 3.3 should be used to restore the measurements for cases 1 and 2.

Table 2 Available measurements and observability in detail for cases.

4.1 Simulation experiment analysis in a highly coherent dictionary

As mentioned in Section 3.3.1,the over-completed dictionary(OcD)can be initialized by giving a traditional transformation basis dictionary(such as the DST or the joint dictionary)or the training samples themselves.Through experiments,we find that different initialization methods influence the coherence of the atoms of the dictionary and the effects of reconstruction directly.At first,the training samples are adopted as the initial dictionary,which is better theoretically but lend to a highly coherent dictionary in this application scenario.

4.1.1 Simulation experiment analysis for over-completed dictionary

In the reconstruction process,the effect of OcD proposed in this paper and the joint dictionary(JD)are com pared.In the JD,many sparse bases(discrete cosine transform(DCT),discrete sine transform(DST),discrete Fourier transform(DFT),discrete wavelet transform(DWT),discrete Hartley transform(DHT),discrete Wtrans form(DWT))are em braced as far as possible to improve sparse representation,where the orthogonal matching pursuit(OMP)is em ployed to select optimal sparse basis to form the final JD.Fur therm ore,the BP and BPDN,ROMP,SAMP,IST are all introduced to display the reconstruction results at last sampling time in Fig.3 when m1=2(case 2),where “Original”are the 7 original power flow measurements that the reconstruction process wants to fit.

From Fig.3,we observe that reconstruction effect under OcD proposed in this paper(Fig.3(b))is obviously better than the effect under the JD(Fig.3(a)),whatever how many sparse bases are embraced in the JD.That is because the dictionary learning proposed in this paper is flexible and adaptive in essence and is not formed by the fixed property sparse bases,which bring the optimal performance.Especially in the case 2,only 29%available clean measurements are left,the five recovery algorithm s still restore the measurements well(Fig.3(b)).

where zjandis reconstruction targeted values and reconstruction values of measurements,respectively.

Fig.3(a)The reconstruction of measurement values on JD;and(b)The reconstruction of measurement values on OcD proposed in this paper,when m1=2(number of the remaining measurements).

In order to give a full exhibition for the superiority of OcD,the integrated normalized absolute error(INAE)is introduced for the quantitative analysis.

Only one sampling time is not enough,hence Fig.4 disp lay the INAE of reconstruction on JD(Fig.4(a))and OcD(Fig.4(b))at 10 sampling times respectively,when m1=2.The maximum of INAE of reconstruction on JD is about 1.37 in Fig.4(a)for ROMP and SAMP at 10th sampling time,while the maximum of INAE of reconstruction on OcD is about 0.15 in Fig.4(b)for BPDN at 10th sampling time.The maximum in Fig.4(a))is more than 9 times as much as the maxim um in Fig.4(b).

In addition,the exact quantitative analysis(mean value of INAE for 10 sampling times)for reconstruction process is shown in Table3 for both cases1 and 2.Either in case 1 or case 2,the mean values of INAE on OcD(Mean-INAE-OcD)are all more Lower than the mean values of INAE on JD(Mean-INAE-JD)by five recovery algorithm s,where the mean values of INAE on OcD by IST are lowest among the five recovery algorithm s both in cases1 and 2.In other words,the effect of IST on OcD for data recovery can be verified to be the best among the five recovery algorithm s in Table3.Therefore,from Fig.4 and Table3,the superiority of OcD proposed in this paper is reflected further.

Fig.4(a)The INAE of reconstruction on JD at 10 sampling times;and(b)The INAE of reconstruction on OcD at 10 sampling times,when m1=2(number of the available clean measurements).

Table 3 Quantitative analysis(mean value of INAE for 10 sampling times)for reconstruction process.

4.1.2 Simulation experiment analysis for measurement matrix

In order to choose the most suitable measurement matrix for the reconstruction process in this paper and verify our theoretical analysis results,the influence of the chosen measurement matrix on the reconstruction of measurements by IST under the OcD is shown in Fig.5.Three kinds of measurement matrices are compared,including the traditional Gauss random matrix,Bernoulli random matrix and the sampling matrix proposed in this paper.It is noted that the first two kinds of measurement matrix are widely applied to image processing,but under this reconstruction scenario,the proposed sampling matrix can fit reconstruction targeted values perfectly,which is m ore suitable than the two traditional measurement matrixes in Fig.5.

Fig.5 The influence of the chosen measurement matrix on the reconstruction of measurements by IST under the OcD.

Finally,significance of reconstruction for measurements is displayed in Fig.6 through the results of the error of state estimator for voltage phase angle(Va).The maximal error of state estimator for Va(?)under no reconstruction is about 11.78 at the 6th sampling time,while the corresponding maximal error under reconstruction by our method in this paper is only 0.5848 at the 7th sampling time.In summary,the proposed reconstruction method for measurements and state estimation strategy improve the accuracy of state estimation under the malicious intelligence attacks extremely.

Fig.6(a)Error of state estimator for Va(?)under no reconstruction;and(b)Error of state estimator for Va(?)under reconstruction by our proposed method in this paper,when m1=2.

Rem ark 3From the analysis results above,the measurements are reconstructed well by the method proposed,especially by the IST.In other words,state estimation strategy on reconstruction of the measurements is more resilient to the intelligence attacks,which has certain self-recovery capability when received measurements are polluted by malicious attackers.However,the RIC value of the IST is maximum value among the five recovery algorithm s,whose performance is the best one.What is the reason for this result?From[31]and[32],there are two kinds of recovery algorithms,that are the greedy algorithm s(such as OMP and BP)and the convex optimization algorithms(such as IST),where the effects of the greedy algorithm s largely depend on the lowly coherent atoms of the dictionary,while the convex optimization algorithms maybe not,instead,the convex optimization algorithm s maybe have perfect performance on the highly coherent dictionary,like the IST in this paper,which can reconstruct the power flow measurements beyond the RIP constraint.In the next section,the simply simulation experiments analysis on lower coherent dictionary are completed for comparison.

4.2 Simulation experiment analysis in a lower coherent dictionary

In this section,the OcD is initialized by giving a traditional transformation basis(such as the DST mentioned before),the RIC value of BP,BPDN,ROMP,SAMP and IST are 0.1297,0.5899,?0.6043,?0.6043 and 0.1297,which show the success rate of reconstruction of BP and IST are the possible highest.The INAE of reconstruction on JD and OcD(initialized by the DST)at 10 sampling times are shown in Fig.7.

The good perform an ce of the proposed method is also proved from Fig.7,although there are some different effects for the five recovery algorithms.

Fig.7 (a)The INAE of reconstruction on JD at 10 sampling times;and(b)The INAE of reconstruction on OcD at 10 sampling times,when m1=2(number of the available clean measurements).

5 Conclusions

In this paper,the resilience of state estimation strategy in cyber physical systems under intelligent stealthy deception attacks is studied.We do not focus on design of the state estimator but on reconstruction of the residual measurements with the unknown model of state equation.The dictionary learning by K-SVD is introduced to form the over-completed dictionary for the measurements.Furthermore,different from CS in image processing,a special sampling matrix is designed as the measurement matrix to improve the performance of reconstruction.Simulation experiments of two cases for 6-bus power system show that not only the proposed over-completed dictionary(OcD)is more flexible and adaptive than the joint dictionary but also effect of the designed measurement matrix on the reconstruction of measurements by IST under the OcD is much better than that of traditional Gauss and Bernoulli random matrix.In addition,the results of qualitative and quantitative analyses indicate that the measurements are re-stored very well on our proposed method by five general recovery algorithm s,even when 29%available measurements are left.The proposed method in this paper p lay a key role in the whole state estimation strategy.In a word,from a new perspective,the resilience of state estimation strategy is improved.

In the future,the distributed reconstruction method of state estimation strategy for large scale CPSs will be focused on.In addition,there are some unsolved problem in the paper will be studied,such as reconstruction conditions for IST on highly coherent dictionary except for RIP,which indicates that there are more or less differences between theory and application.

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