# An Optimal DoS Attack Strategy Disturbing the Distributed Economic Dispatch of Microgrid

Authors: Yihe Wang; Mingli Zhang; Kun Song; Tie Li; Na Zhang
Journal: Complexity
 (2021)

---

## Abstract

As a promising method with excellent characteristics in terms of resilience and dependability, distributed methods are gradually used in the field of energy management of microgrid. However, these methods have more stringent requirements on the working conditions, which will make the system more sensitive to communication failures and cyberattacks. As a result, it is both theoretical merits and practical values to investigate the malicious effect of cyber attacks on microgrid. This paper studies the distributed economic dispatch problem (EDP) under denial-of-service (DoS) attacks for the microgrid, in which each generator can communicate with its neighbors and has the computational capability to implement local operation. Firstly, a DoS attack model is proposed, in which the DoS attacker intentionally jams the communication channel to deteriorate the performance of the microgrid. Then, the evolution mechanism of the dispatch system of the microgrid under different attack scenarios is adequately discussed. On this basis, an optimal attack strategy based on enumerating-search algorithm is presented to allocate the limited attack resources reasonably, so as to maximize the effect of DoS attacks. Finally, the validity of the theoretical studies about the attack effect under different scenarios and the effectiveness of the proposed enumerating-search-based optimal attack strategy are illustrated through the simulation examples on the IEEE 57-bus system and IEEE 39-bus system, respectively.


---

## Body

## 1. Introduction

The intelligent microgrid integrates various renewable generators and energy storage systems within a distributed system to achieve the goal of energy-saving and efficient operation and has been widely recognized as the future of power system. As a multidimensional complex system, the microgrid includes widely distributed sensors, capacity of control and optimization computation, and well-developed communication networks [1]. Now, it has been widely recognized as a typical cyber-physical system (CPS), and various fundamental problems of the microgrid have been reinvestigated in the perspective of CPS in the last few years, such as droop control [2], economic dispatch [3], and zero-sequence current suppression control [4].Like the traditional power system, economic dispatch is one of the most concerned issues in the operation of the microgrid. Economic dispatch of the microgrid with the goal of minimizing operating costs can realize the collaborative optimization and energy distribution among units. Generally speaking, the methods to solve the economic dispatch problem (EDP) of the microgrid can be classified into centralized methods and distributed methods. Although the centralized methods are efficient, they need central authorities to collect global information of all the controlled units. These methods are sensitive to single-point failure and increase the communication burden of system [5]. On the opposite, the distributed methods without central authorities can carry out the economic dispatch with less communication burden and better scalability and privacy [6]. As a consequence, scholars have investigated the EDP for the microgrid based on various distributed methods in the past few years [7–10]. For example, a fully distributed algorithm proposed in [7] can fit well with arbitrary initializations. Moreover, it enabled generators to collectively obtain the mismatch between demand and total power output as a feedback mechanism to adjust actual power generation. In order to maximize the social welfare, a distributed consensus algorithm based on augmented Lagrangian and projection penalty function was proposed in [8]. Literature [9] presented a distributed double-Newton descent algorithm for cooperative energy management with faster convergence speed in a fully distributed fashion. In [10], a novel distributed primal-dual augmented subgradient algorithm was investigated, which can solve a class of distributed constrained optimization problems over a general unbalanced directed communication network. However, the above literature assumes that these algorithms are executed under ideal communication conditions and the possible cyberattacks in the communication network are not considered. In practice, once the malicious cyberattacks occur in the communication network of the microgrid, the data transmission will be disturbed, and the execution of those algorithms will be affected. Consequently, the optimal operation points of the system cannot be obtained, and even the stability is destroyed [11]. Therefore, it is of theoretical and practical significance to investigate the EDP under various cyberattacks.Until now, two types of cyberattacks have received the most attention: denial-of-service (DoS) attacks and deception attacks [12]. As a common type of cyberattacks in CPS, DoS attacks attempt to make the network resource unavailable for its intended users. This kind of attack blocks real traffic by sending a mass of invalid information that takes up all the communication bandwidth. Currently, research on DoS attacks has two main perspectives as follows. (1) From the perspective of system defense, in order to mitigate the vulnerability of system under attacks, many robust methods have been investigated in recent years [13–15], such as event/self-triggered control [13], the confidence-level-based resilient strategy [14], and intensity-dependent consensus protocol [15]. (2) From the perspective of DoS attacker, in order to increase the probability of successful attacks and enhance the effect of attacks, the attacker tends to formulate various optimal attack strategies according to their working conditions. Recently, there are some studies on formulating optimal attack strategies. For example, literature [16] proposed an optimal attack strategy under limited attack resources, and this strategy can damage the estimation performance of the system by deciding which sensor to attack. And, on this basis, the optimal attack time and attack mode are determined in [17] to maximum the impact of DoS attacks on the remote state estimation of the power system. However, the above research on the optimal attack strategies only discusses the DoS attacks against the state estimation, and the goal of it is to undermine the security and stability of systems. Until now, there are few literatures on designing the optimal DoS attacks strategies to damage the economic performance of the power system. However, the degradation caused by DoS attacks to the economic performance of the power system cannot be ignored, which may have a great impact on the national economy. As a result, it is necessary to study under what strategy the attack will cause the maximum degree of economic performance degradation, i.e., the optimal attack strategy disturbing the distributed economic dispatch of the microgrid.Based on the above analysis, the motivation of the current work is in the following three aspects:(1)
As a common type of cyberattacks, DoS attacks are highly destructive, which may cause severe consequences for the power system.(2)
As far as we know, there are few literatures about the impact of DoS attacks on the economic dispatch of the power system. However, it is essential to analyze potential consequences of these attacks on the economic performance of the system.(3)
Research on optimal attack strategies can guide the formulation of defense measures for energy systems, and one needs to understand what the worst effect of attacks might be.In this paper, we investigate the economic dispatch problem of the microgrid under DoS attacks. Firstly, the model of DoS attacks is established according to their working mechanism. Then, the impact of DoS attacks on the economic dispatch of the microgrid under different scenarios is investigated. On this basis, an optimal attack strategy that can maximize the effect of attacks with limited attack resources is proposed. The main contributions of this paper are summarized as follows:(1)
In this paper, the EDP of the microgrid under DoS attacks is investigated, which further enriches the theoretical research on DoS attacks in the power system.(2)
The time-varying communication network caused by DoS attacks can be divided into two types, i.e., connected and disconnected. How the two scenarios of attacks affect economic dispatch is analyzed in this paper (Theorems1 and 2).(3)
Different from the existing literatures, where optimal attack strategies were always put forward to increase the state estimation error of the power system, the optimal attack strategy proposed in this paper aims at damaging economic performance of the microgrid.(4)
An enumerating-search algorithm is proposed to find out the optimal attack strategy for DoS attacker (Algorithm1).Algorithm 1: Solution of optimal attack strategy.
Input:PD;αi;βi;Pimax;Pimin;Pcut=0;δmax=0;ϕ;ΛComputeλ∗=PD+∑i=1Nβi/2αi/∑i=1N1/2αiEnumerate the set of all attack schemesX=X1,X2,… according to constraints (31) and (33), and each attack strategy Xj can be further represented as Xj=i|∃i∈V, which represents the units’ isolated set caused by an attack actionDefinem=cardXforj = 1:m dofori = 1:N doifi∈Xj thenCompute total output lower limit of isolated generatorsPcut=Pcut+Pimin;elseComputeβ/2α=β/2α+βi/2αi;Compute1/2α=1/2α+1/2αi;end ifend forComputeδ=PD−Pcut+β/2α/1/2α−λ∗;ifδ>δmax thenUpdateδmax=δ;UpdateX∗ according to Xj and ϕ;end ifend forOutput:δmax; X∗.The rest of the paper is organized as follows. Section2 revisits some preliminaries, and the problems of interest are summarized. Section 3 illustrates the degradation of system economic performance under the different scenarios of attacks through complete theoretical analysis. Section 4 derives the optimization model of attack strategy and its corresponding algorithm to find out the solution. The simulation examples and conclusion are appeared in Sections 5 and 6, respectively.


## 2. Preliminaries

### 2.1. Graph Theory

A graphG is always used to model the network topology of a system. A graph can be denoted as G=V,E,A, where V=1,2,3… is a finite set of points, and E⊆V×V is an edge set of pairs of nodes. In a directed graph, the edge from i to j is denoted by an ordered pairs i,j∈E, which means information can be transferred from i to j. A=aij is a matrix named adjacency matrix, aij denotes the weight of the edge i,j. If the agent i and j can communicate with each other, they will be defined as neighbors. The in-neighbors of the ith node are denoted by the set Ni+=j|j,i∈E. Similarly, the out-neighbors of the ith node are denoted by the set Ni−=j|i,j∈E. Especially, each vertex belongs to both its in-neighbor and out-neighbor, i.e., i∈Ni+ as well as i∈Ni−. The in-degree and out-degree of the ith vertex are defined as di+=Ni+ and di−=Ni−, respectively. Strongly connected graph can be defined if there are paths between any pair of two vertices in a directed graph, i.e., each vertex is reachable for all the other vertices.


### 2.2. Stochastic Matrix

A nonnegative square matrixM is defined as row stochastic matrix if M1=1, where 1 is a column vector with all the elements equal to 1. Similarly, a nonnegative square matrix N is defined as column stochastic matrix if 1TN=1T, and a nonnegative square matrix H is double stochastic matrix if it satisfies both H1=1 and 1TH=1T.Moreover, based on the Perron–Frobenius theorem [18], for a row stochastic matrix M, there exists such an identity column vector μ satisfying μTM=μT. Similarly, for a column stochastic matrix N, there also exists an identity column vector ξ satisfying Nξ=ξ.


### 2.3. Analytic Solution to EDP

The microgrid is essentially a typical multiagent system, in which each generator can be regarded as an agent. The operation cost function of theith generation is given by the following quadratic form [19]:(1)CiPi=αiPi2+βiPi+γi,where Pi is the power generated by generator i and βi, γi, and αi>0 are the operation cost parameters of generator i.Assume there is a microgrid system withn generators, the objective of the economic dispatch of the microgrid is to minimize the total operation cost, which is given by(2)min∑i=1nCiPi.The EDP of the microgrid needs to be solved under the power balance constraint and power generation constraint as follows:(3)∑i=1nPi=PD,(4)Pimin≤Pi≤Pimax,fori=1,2,3…n,where Pimin and Pimax are the lower bound and upper bound of the output power associated with the ith generation, respectively. PD denotes the total power demand satisfying ∑i=1nPimin≤PD≤∑i=1nPimax, i.e., the problem is solvable.The above optimization problem can be solved by the Lagrange multiplier method [20]. The Lagrange multiplier can be expressed as follows:(5)λi=tialCiPitialPi=2αiPi+βi.Because the Lagrange multiplierλi is equal to the incremental cost tialCiPi/tialPi, we directly use λi to represent the incremental cost of units in the following paragraphs.With the consideration of generation constraint in (4), the power generated by generator i is given by(6)2αiPi+βi=λ∗,forPimin≤Pi≤Pimax,2αiPi+βi<λ∗,forPi=Pimax,2αiPi+βi>λ∗,forPi=Pimin,where λ∗ is the incremental cost corresponding to the optimal solution of the EDP.


### 2.4. Problems of Interest

Based on a well-developed distributed algorithm to solve the EDP of the smart grid, the main problems we are interested in consist of the following:(1)
What is the possible effect of DoS attacks on EDP when the attacker can be detected hardly by the monitor in the smart grid?(2)
How to quantify the degradation of economic performance caused by DoS attacks?(3)
Does there exists an optimal attack strategy that renders maximum degradation of economic performance caused by the attacker?


## 2.1. Graph Theory

A graphG is always used to model the network topology of a system. A graph can be denoted as G=V,E,A, where V=1,2,3… is a finite set of points, and E⊆V×V is an edge set of pairs of nodes. In a directed graph, the edge from i to j is denoted by an ordered pairs i,j∈E, which means information can be transferred from i to j. A=aij is a matrix named adjacency matrix, aij denotes the weight of the edge i,j. If the agent i and j can communicate with each other, they will be defined as neighbors. The in-neighbors of the ith node are denoted by the set Ni+=j|j,i∈E. Similarly, the out-neighbors of the ith node are denoted by the set Ni−=j|i,j∈E. Especially, each vertex belongs to both its in-neighbor and out-neighbor, i.e., i∈Ni+ as well as i∈Ni−. The in-degree and out-degree of the ith vertex are defined as di+=Ni+ and di−=Ni−, respectively. Strongly connected graph can be defined if there are paths between any pair of two vertices in a directed graph, i.e., each vertex is reachable for all the other vertices.


## 2.2. Stochastic Matrix

A nonnegative square matrixM is defined as row stochastic matrix if M1=1, where 1 is a column vector with all the elements equal to 1. Similarly, a nonnegative square matrix N is defined as column stochastic matrix if 1TN=1T, and a nonnegative square matrix H is double stochastic matrix if it satisfies both H1=1 and 1TH=1T.Moreover, based on the Perron–Frobenius theorem [18], for a row stochastic matrix M, there exists such an identity column vector μ satisfying μTM=μT. Similarly, for a column stochastic matrix N, there also exists an identity column vector ξ satisfying Nξ=ξ.


## 2.3. Analytic Solution to EDP

The microgrid is essentially a typical multiagent system, in which each generator can be regarded as an agent. The operation cost function of theith generation is given by the following quadratic form [19]:(1)CiPi=αiPi2+βiPi+γi,where Pi is the power generated by generator i and βi, γi, and αi>0 are the operation cost parameters of generator i.Assume there is a microgrid system withn generators, the objective of the economic dispatch of the microgrid is to minimize the total operation cost, which is given by(2)min∑i=1nCiPi.The EDP of the microgrid needs to be solved under the power balance constraint and power generation constraint as follows:(3)∑i=1nPi=PD,(4)Pimin≤Pi≤Pimax,fori=1,2,3…n,where Pimin and Pimax are the lower bound and upper bound of the output power associated with the ith generation, respectively. PD denotes the total power demand satisfying ∑i=1nPimin≤PD≤∑i=1nPimax, i.e., the problem is solvable.The above optimization problem can be solved by the Lagrange multiplier method [20]. The Lagrange multiplier can be expressed as follows:(5)λi=tialCiPitialPi=2αiPi+βi.Because the Lagrange multiplierλi is equal to the incremental cost tialCiPi/tialPi, we directly use λi to represent the incremental cost of units in the following paragraphs.With the consideration of generation constraint in (4), the power generated by generator i is given by(6)2αiPi+βi=λ∗,forPimin≤Pi≤Pimax,2αiPi+βi<λ∗,forPi=Pimax,2αiPi+βi>λ∗,forPi=Pimin,where λ∗ is the incremental cost corresponding to the optimal solution of the EDP.


## 2.4. Problems of Interest

Based on a well-developed distributed algorithm to solve the EDP of the smart grid, the main problems we are interested in consist of the following:(1)
What is the possible effect of DoS attacks on EDP when the attacker can be detected hardly by the monitor in the smart grid?(2)
How to quantify the degradation of economic performance caused by DoS attacks?(3)
Does there exists an optimal attack strategy that renders maximum degradation of economic performance caused by the attacker?


## 3. Performance Analysis of Dispatch System under DoS Attacks

### 3.1. DoS Attacks’ Model

Assume the DoS attacker is a single source of attacks in the communication network environment of the microgrid, and the DoS attacker sends the invalid requests to occupy the network resources. The DoS attacks target various components including smart meters and communication links, resulting in limited availability of these components. In this paper, the objectives of DoS attacks are to hinder the transmission of information on the communication links, thereby degrading the economic performance of the microgrid.Stochastic DoS attacks can occur at any communication moment and any communication link. To clearly describe the effect of DoS attacks on the economic dispatch of the microgrid, the model of DoS attacks needs to be established. Define an edge setUek⊆E to represent the communication links attacked at iteration k, k∈0,1,…. The information transmission over the compromised communication links is impeded.The effect of DoS attacks on the communication network is equal to packet dropout, which fails the information transmission among agents. Actually, the communication pathi,j∈E can be regarded as unconnected if the information transmission on the edge is failed. It is reasonable to assume that the network topology could be time-varying and possibly disconnected when the communication network is attacked by the DoS attacker. In this paper, EDP over unreliable network with DoS attacks is regarded as the EDP under time-varying communication topology. Moreover, we assume that once DoS attacks disappear, any disconnected links caused by attacks will be recovered into the connected state by communication restoration mechanism.Ak=aijk∈ℕn×n is the adjacency matrix corresponding to the communication network topology under DoS attacks at iteration k. It can be defined as the following form:(7)aijk+1=0,i,j∈Uek+1,1,i,j∉Uek+1.The updating matricesMk=mijk,Nk=nijk∈ℝn×n can be defined according to Ak as follows:(8)mijk=1di+k,j∈Ni+,0,otherwise∀i,j∈V,nijk=1dj−k,i∈Nj−,0,otherwise,∀i,j∈V.It can be deduced thatMk is a row stochastic matrix and Nk is a column stochastic matrix.Lemma 1.
Based on the definition, row stochastic matrixM and column stochastic matrix N both have the eigenvalues 1, and the rest eigenvalues are within the unit circle on the complex plane.In this paper, a fully distributed algorithm proposed in [20] to solve the EDP of power system can be transformed into the following form with the consideration of DoS attacks:(9)λik+1=∑j=1nmijkλjk+εyik,(10)Pik+1=Pimax,λ∗>λimax,λik+1−βi2αi,λimin≤λ∗≤λimax,Pimin,λ∗<λimin,(11)yik+1=∑j=1nnijkyjk−Pik+1−Pik,where yi is an auxiliary variable which denotes the estimated local power mismatch and ε is a sufficiently small positive constant named learning gain. λimax=Pimax−βi/2αi and λimin=Pimin−βi/2αi. The distributed algorithms (9)–(11) can solve the economic dispatch problem (2) under constraints (3)-(4). Moreover, the following formula can be derived from (11):(12)∑i=1nyik+1=∑i=1n∑j=1nnijkyik−Pik+1−Pik⟹∑i=1nyik+1+Pik+1=∑i=1nyik+Pik⟹∑i=1nyik+1+Pik+1=∑i=1nyi0+Pi0.It is easy to deduce that∑i=1nPik+1+yik+1=PD will be guaranteed if ∑i=1nPi0+yi0=PD. In other words, the equilibrium between power supply and demand can be guaranteed when the algorithm converges only if the selection of initial value satisfies certain conditions.Most of the existing literature analyzes the impact of attacks based on the assumption of known attack probabilities [17]. In fact, the probability of an attack is generally not fully grasped by the system [21]. In this paper, we explore the effect of DoS attacks without the knowledge of probability distribution information. To this end, a communication network attacked by the DoS attacker is classified into two types:(1)
The communication network topologies keep strongly connected(2)
There exist communication network topologies disconnected, i.e., some units are isolated from the communication network


### 3.2. Strongly Connected Scenario under DoS Attacks

This section investigates the performance analysis of the dispatch system in the scenario which communication topology graphs of the microgrid keep strongly connected under DoS attacks.Under this attacks, the communication topology graphs of the microgrid are time-varying, as shown in Figure1(a), where the solid line represents the normal connected communication links and the dotted line represents compromised communication links caused by DoS attacks. Although the communication network topology no longer maintains a fixed form under DoS attacks, it still keeps strongly connected. In this case, the distributed economic dispatch algorithm (9)–(11) can be carried out to solve the EDP of the microgrid.Figure 1
Two scenarios caused by DoS attacks. (a) Communication network topologies are still strongly connected. (b) Communication network topologies are disconnected.
(a)(b)Theorem 1.
In the scenario of attacks that the communication network topologies keep strongly connected, if the positive constantε in algorithm (9)–(11) is sufficiently small and the variables satisfy the initial condition ∑i=1nPi0+yi0=PD, then the distributed algorithm is stable, and all the variables converge to the solution of the EDP, i.e.,(13)λik⟶λ∗,Pik⟶P∗,yik⟶0,ask⟶∞,∀i∈V.Proof.
Use the eigenvalue perturbation approach [18] to analyze the convergence performance of algorithms (9)–(11). Substituting (9) and (10) into (11), we can obtain(14)λk+1=Mkλk+εyk,(15)yk+1=Nk−εByk+BI−Mkλk,where λ and y are the column stack vectors of λi and yi, respectively, I is an identity matrix, and B=diag1/2α11/2α2…1/2αN. Algorithms (14)–(15) can be written in the following form:(16)λk+1yk+1=MkεIBI−MkNk−εBλkyk.
DefineCk=Mk0BI−MkNk, Δ=0I0−B, and updating matrix Dk=MkεIBI−MkNk−εB=Ck+εΔ.
The updating matrixDk in (16) can be regarded as Ck perturbed by εΔ. It is easy to deduce that Ck is a lower triangular block matrix with the matrices Mk and Nk in its diagonal line. As a result, the eigenvalues of Ck include that of matrices Mk and Nk. According to Lemma 1, it can be obtained that Ck has two eigenvalues ν1k=ν2k=1 at any iteration k.
Assumezik and xik are left and right eigenvectors of matrix Ck with respect to eigenvalues νi, and they satisfy the following conditions:(17)ziTCk=ziTvik,Ckxik=vikxik,ziTxik=1,ziTxjk=0.
According to formula (17), it is not hard to find that z1Tk=1TB1T and z2Tk=uTk0 both are the left eigenvectors of Ck; x1k=0TξkTT and x2k=1T−ηξkTT both are the right eigenvectors of Ck associated with eigenvalues ν1k=ν2k=1, where η=∑i=1N1/2αi is defined. If ε≠0, the eigenvalues of Ck is perturbed by εΔ. According to the eigenvalue derivation theory, the partial derivative of νi corresponding to ε can be expressed as follows:(18)∂νik∂ε=ziTk∂Dk∂εxik.
Thus, it can be deduced that(19)∂ν1k∂ε=z1Tk∂Dk∂εx1k=1TB1T0I0−Bk0ξk=0,∂ν2k∂ε=z2Tk∂Dk∂εx2k=μTk00I0−Bk1−ηξk=−ημTkξk<0.
That meansν1k=1 does not change with ε, and as ε goes up, ν2k goes down from 1. Therefore, there is a small constant σ1 satisfying 0<ε<σ1, and ν2k is restricted to less than 1. Based on the Bauer–Fike theorem [22], there exists an upper bound σ2 such that 0<ε<σ2, then νik<1, i=3,4…2n, i.e., other eigenvalues of Ck lie in the open unit disk. Because the eigenvalues of Dk=Ck+εΔ depend on ε, given a constant σ=minσ1,σ2, if 0<ε<σ, all the eigenvalues of Dk are less than 1, except for one eigenvalue equal to 1.
It is not hard to deduce that1T0TT∈ℕ2n is the right eigenvector of Dk corresponding to ν1k=1 at any time k, i.e., Dk10=10. The rest of the eigenvalues are within the open unit disk, and matrix Dk has 2n independent eigenvectors.
It is reasonable to assume that the initial value can be expressed as a linear combination of all the eigenvectors as shown in the following:(20)λ0y0=c1θ1+c2θ2+⋯+c2nθ2n,where c1,c2,…c2n is a set of constant and θ1,θ2,…θ2n is a set of eigenvectors of Dk. According to formula (16), we can obtain(21)λ1y1=D0λ0y0=ν1c1θ1+ν2c2θ2+⋯+ν2nc2nθ2n.
Thus, it can be deduced that(22)λkyk=∏p=1kDk−pλ0y0.
According to the above analysis, because all the eigenvalues ofDk are less than 1, except for one eigenvalue equal to 1, and the vector 1T0TT corresponds to the eigenvalue 1. It can be conducted that(23)limk⟶+∞λkyk=span10.
Ask⟶∞ and yik⟶0, ∑i=1nPik+yik=∑i=1nPi0+yi0 is always satisfied. Because the initial condition ∑i=1nPi0+yi0=PD is guaranteed, it can be derived that the power supply-demand balance of the system is satisfied ultimately. Thus, Theorem 1 can be obtained.
It worth noting that the research in this paper is aimed at day-ahead economic dispatch of the smart grid, and our concern is whether the algorithm can converge to the optimal solution of the EDP rather than the convergence time. This is also the focus of another attack scenario we discuss in Section3.3. According to Theorem 1, it is not very difficult to obtain that the distributed algorithm executed in the communication environment with DoS attacks still converges to the point which corresponds to the optimal operation point of the microgrid. Through the above analysis, we can get the following inference that, in this scenario of attack that keeps the communication topologies still strongly connected, the DoS attack shows little benefit to the degradation of economic performance for the microgrid. It is a situation that is unwilling to accept by the DoS attacker, whose intent is to disrupt the economic operation of the microgrid as much as possible.


### 3.3. Disconnected Scenario under DoS Attacks

This section investigates the performance analysis of the dispatch system in the scenario that communication topology graphs of the microgrid are disconnected under DoS attacks.If the DoS attacker concentrates on disconnecting all the communication links of a generator, it will briefly prevent the generator from communicating with its neighbors. Under these attacks, at least one generator is isolated from the communication network. It means the time-varying matrixAk has the rows and columns in which all the elements are zero, i.e., the in-degree and out-degree of some vertexes in the communication topology graph are zero. For the convenience of analysis, it is necessary to reduce the dimension of Ak to obtain A′k by removing the rows and columns whose elements are all equal to 0. The corresponding matrices Mk and Nk are transformed to M′k and N′k, respectively.In this scenario, all the generators can be divided into two types, misbehaving generators and well-behaving generators, respectively. DefineAnk⊆V as the set of misbehaving generators at iteration k, i.e., the units isolated from communication network, and Nnk=Ank is the number of misbehaving generators. It is worth noting that Ank presents isolated generators only due to DoS attacks in unreliable communication network, and isolation of units caused by other abnormal factors is beyond the scope of this paper. If i∉Ank, the generator i is involved in economic dispatch as a well-behaving generator, which updates its output power and incremental cost normally; otherwise, the generator i is a misbehaving generator, which maintains its output power and incremental cost until it is reconnected to the communication network after the disappearance of DoS attacks on it.Under these attacks, algorithms (9)–(11) can be rewritten as the following dynamics (24)–(26):(24)λik+1=∑j∉Ankmij′kλjk+εyik,i∉Ank,λik,i∈Ank,(25)Pik+1=Pimax,λi>λimax,λik+1−βi2αi,λimin≤λi≤λimax,Pimin,λi<λimin,(26)yik+1=∑j∉Anknij′kyjk−Pik+1−Pik,i∉Ank,0,i∈Ank,(27)∑j∉AnkPimin≤PD−∑j∈AnkPik≤∑j∉AnkPimax.Theorem 2.
In the scenario of attacks where there exist the communication network topologies disconnected, if the variables satisfy the initial condition∑i=1nPi0+yi0=PD, the learning gain ε in algorithms (24)–(26) is sufficiently small, and there exist enough output power satisfying total power demand, as shown in formula (27); then, the algorithm is stable and the variables of well-behaving generators converge to the suboptimal solution of the EDP, i.e.,(28)λik⟶λ′,i∉Ank,λik⟶λi,i∈Ank,Pik⟶P′,i∉Ank,Pik⟶Pi,i∈Ank,yik⟶0,ask⟶∞,∀i∈V,where λ′≠λ∗ and P′≠P∗ denote the suboptimal incremental cost vector and output power vector of well-behaving generators, respectively, and λi and Pi denote the incremental cost and output power of the misbehaving generator i, respectively. Note that convergence results shown in (28) depend on the attack strategies chosen by the DoS attacker.Proof.
The proof is similar with Theorem1. The process is omitted here for convenience.Remark 1.
The attacker deliberately causes algorithms (24)–(26) to be executed on the premise that formula (27) is satisfied in the microgrid. As a result, the algorithm converges to a stable but not optimal point as shown in Theorem 2, the attacks will be considered online stealthily [23]. The attacks are especially hard to detect by the system, and they can keep stealthy and damage the performance of the microgrid.
It worth noting that misbehaving generators isolated from the communication network will continue to supply electricity to local demand because the isolation action caused by DoS attacks cannot directly disconnect the transmission lines of the misbehaving generators in the physical network. Under this scenario of attack, the distributed algorithm cannot be executed consistently due to the absence of the real updating information of the misbehaving units. According to Theorem2, it is not very difficult to obtain that the distributed algorithm converges to the other convergence point, which is not equal to the optimal operation point of the microgrid. The conclusion can be drawn that, in this attack scenario which make the communication topology graphs disconnected, DoS attacks can interfere with the economic dispatch process of the microgrid to degrade the economic performance of the system. Through the above analysis, this scenario of attacks is a benefit to the DoS attacker. Only if DoS attacks isolate some generators from the communication network can such attacks will be effective for the destruction of economic operation of the microgrid.
Based on the analysis of the effect of attacks on the EDP, the problem that the authors are interested in is to find the largest degradation of economic performance of the microgrid under DoS attacks. In order to quantify the effect of attacks, the change of incremental cost in the attacks’ situation compared to the ideal situation is exploited.


## 3.1. DoS Attacks’ Model

Assume the DoS attacker is a single source of attacks in the communication network environment of the microgrid, and the DoS attacker sends the invalid requests to occupy the network resources. The DoS attacks target various components including smart meters and communication links, resulting in limited availability of these components. In this paper, the objectives of DoS attacks are to hinder the transmission of information on the communication links, thereby degrading the economic performance of the microgrid.Stochastic DoS attacks can occur at any communication moment and any communication link. To clearly describe the effect of DoS attacks on the economic dispatch of the microgrid, the model of DoS attacks needs to be established. Define an edge setUek⊆E to represent the communication links attacked at iteration k, k∈0,1,…. The information transmission over the compromised communication links is impeded.The effect of DoS attacks on the communication network is equal to packet dropout, which fails the information transmission among agents. Actually, the communication pathi,j∈E can be regarded as unconnected if the information transmission on the edge is failed. It is reasonable to assume that the network topology could be time-varying and possibly disconnected when the communication network is attacked by the DoS attacker. In this paper, EDP over unreliable network with DoS attacks is regarded as the EDP under time-varying communication topology. Moreover, we assume that once DoS attacks disappear, any disconnected links caused by attacks will be recovered into the connected state by communication restoration mechanism.Ak=aijk∈ℕn×n is the adjacency matrix corresponding to the communication network topology under DoS attacks at iteration k. It can be defined as the following form:(7)aijk+1=0,i,j∈Uek+1,1,i,j∉Uek+1.The updating matricesMk=mijk,Nk=nijk∈ℝn×n can be defined according to Ak as follows:(8)mijk=1di+k,j∈Ni+,0,otherwise∀i,j∈V,nijk=1dj−k,i∈Nj−,0,otherwise,∀i,j∈V.It can be deduced thatMk is a row stochastic matrix and Nk is a column stochastic matrix.Lemma 1.
Based on the definition, row stochastic matrixM and column stochastic matrix N both have the eigenvalues 1, and the rest eigenvalues are within the unit circle on the complex plane.In this paper, a fully distributed algorithm proposed in [20] to solve the EDP of power system can be transformed into the following form with the consideration of DoS attacks:(9)λik+1=∑j=1nmijkλjk+εyik,(10)Pik+1=Pimax,λ∗>λimax,λik+1−βi2αi,λimin≤λ∗≤λimax,Pimin,λ∗<λimin,(11)yik+1=∑j=1nnijkyjk−Pik+1−Pik,where yi is an auxiliary variable which denotes the estimated local power mismatch and ε is a sufficiently small positive constant named learning gain. λimax=Pimax−βi/2αi and λimin=Pimin−βi/2αi. The distributed algorithms (9)–(11) can solve the economic dispatch problem (2) under constraints (3)-(4). Moreover, the following formula can be derived from (11):(12)∑i=1nyik+1=∑i=1n∑j=1nnijkyik−Pik+1−Pik⟹∑i=1nyik+1+Pik+1=∑i=1nyik+Pik⟹∑i=1nyik+1+Pik+1=∑i=1nyi0+Pi0.It is easy to deduce that∑i=1nPik+1+yik+1=PD will be guaranteed if ∑i=1nPi0+yi0=PD. In other words, the equilibrium between power supply and demand can be guaranteed when the algorithm converges only if the selection of initial value satisfies certain conditions.Most of the existing literature analyzes the impact of attacks based on the assumption of known attack probabilities [17]. In fact, the probability of an attack is generally not fully grasped by the system [21]. In this paper, we explore the effect of DoS attacks without the knowledge of probability distribution information. To this end, a communication network attacked by the DoS attacker is classified into two types:(1)
The communication network topologies keep strongly connected(2)
There exist communication network topologies disconnected, i.e., some units are isolated from the communication network


## 3.2. Strongly Connected Scenario under DoS Attacks

This section investigates the performance analysis of the dispatch system in the scenario which communication topology graphs of the microgrid keep strongly connected under DoS attacks.Under this attacks, the communication topology graphs of the microgrid are time-varying, as shown in Figure1(a), where the solid line represents the normal connected communication links and the dotted line represents compromised communication links caused by DoS attacks. Although the communication network topology no longer maintains a fixed form under DoS attacks, it still keeps strongly connected. In this case, the distributed economic dispatch algorithm (9)–(11) can be carried out to solve the EDP of the microgrid.Figure 1
Two scenarios caused by DoS attacks. (a) Communication network topologies are still strongly connected. (b) Communication network topologies are disconnected.
(a)(b)Theorem 1.
In the scenario of attacks that the communication network topologies keep strongly connected, if the positive constantε in algorithm (9)–(11) is sufficiently small and the variables satisfy the initial condition ∑i=1nPi0+yi0=PD, then the distributed algorithm is stable, and all the variables converge to the solution of the EDP, i.e.,(13)λik⟶λ∗,Pik⟶P∗,yik⟶0,ask⟶∞,∀i∈V.Proof.
Use the eigenvalue perturbation approach [18] to analyze the convergence performance of algorithms (9)–(11). Substituting (9) and (10) into (11), we can obtain(14)λk+1=Mkλk+εyk,(15)yk+1=Nk−εByk+BI−Mkλk,where λ and y are the column stack vectors of λi and yi, respectively, I is an identity matrix, and B=diag1/2α11/2α2…1/2αN. Algorithms (14)–(15) can be written in the following form:(16)λk+1yk+1=MkεIBI−MkNk−εBλkyk.
DefineCk=Mk0BI−MkNk, Δ=0I0−B, and updating matrix Dk=MkεIBI−MkNk−εB=Ck+εΔ.
The updating matrixDk in (16) can be regarded as Ck perturbed by εΔ. It is easy to deduce that Ck is a lower triangular block matrix with the matrices Mk and Nk in its diagonal line. As a result, the eigenvalues of Ck include that of matrices Mk and Nk. According to Lemma 1, it can be obtained that Ck has two eigenvalues ν1k=ν2k=1 at any iteration k.
Assumezik and xik are left and right eigenvectors of matrix Ck with respect to eigenvalues νi, and they satisfy the following conditions:(17)ziTCk=ziTvik,Ckxik=vikxik,ziTxik=1,ziTxjk=0.
According to formula (17), it is not hard to find that z1Tk=1TB1T and z2Tk=uTk0 both are the left eigenvectors of Ck; x1k=0TξkTT and x2k=1T−ηξkTT both are the right eigenvectors of Ck associated with eigenvalues ν1k=ν2k=1, where η=∑i=1N1/2αi is defined. If ε≠0, the eigenvalues of Ck is perturbed by εΔ. According to the eigenvalue derivation theory, the partial derivative of νi corresponding to ε can be expressed as follows:(18)∂νik∂ε=ziTk∂Dk∂εxik.
Thus, it can be deduced that(19)∂ν1k∂ε=z1Tk∂Dk∂εx1k=1TB1T0I0−Bk0ξk=0,∂ν2k∂ε=z2Tk∂Dk∂εx2k=μTk00I0−Bk1−ηξk=−ημTkξk<0.
That meansν1k=1 does not change with ε, and as ε goes up, ν2k goes down from 1. Therefore, there is a small constant σ1 satisfying 0<ε<σ1, and ν2k is restricted to less than 1. Based on the Bauer–Fike theorem [22], there exists an upper bound σ2 such that 0<ε<σ2, then νik<1, i=3,4…2n, i.e., other eigenvalues of Ck lie in the open unit disk. Because the eigenvalues of Dk=Ck+εΔ depend on ε, given a constant σ=minσ1,σ2, if 0<ε<σ, all the eigenvalues of Dk are less than 1, except for one eigenvalue equal to 1.
It is not hard to deduce that1T0TT∈ℕ2n is the right eigenvector of Dk corresponding to ν1k=1 at any time k, i.e., Dk10=10. The rest of the eigenvalues are within the open unit disk, and matrix Dk has 2n independent eigenvectors.
It is reasonable to assume that the initial value can be expressed as a linear combination of all the eigenvectors as shown in the following:(20)λ0y0=c1θ1+c2θ2+⋯+c2nθ2n,where c1,c2,…c2n is a set of constant and θ1,θ2,…θ2n is a set of eigenvectors of Dk. According to formula (16), we can obtain(21)λ1y1=D0λ0y0=ν1c1θ1+ν2c2θ2+⋯+ν2nc2nθ2n.
Thus, it can be deduced that(22)λkyk=∏p=1kDk−pλ0y0.
According to the above analysis, because all the eigenvalues ofDk are less than 1, except for one eigenvalue equal to 1, and the vector 1T0TT corresponds to the eigenvalue 1. It can be conducted that(23)limk⟶+∞λkyk=span10.
Ask⟶∞ and yik⟶0, ∑i=1nPik+yik=∑i=1nPi0+yi0 is always satisfied. Because the initial condition ∑i=1nPi0+yi0=PD is guaranteed, it can be derived that the power supply-demand balance of the system is satisfied ultimately. Thus, Theorem 1 can be obtained.
It worth noting that the research in this paper is aimed at day-ahead economic dispatch of the smart grid, and our concern is whether the algorithm can converge to the optimal solution of the EDP rather than the convergence time. This is also the focus of another attack scenario we discuss in Section3.3. According to Theorem 1, it is not very difficult to obtain that the distributed algorithm executed in the communication environment with DoS attacks still converges to the point which corresponds to the optimal operation point of the microgrid. Through the above analysis, we can get the following inference that, in this scenario of attack that keeps the communication topologies still strongly connected, the DoS attack shows little benefit to the degradation of economic performance for the microgrid. It is a situation that is unwilling to accept by the DoS attacker, whose intent is to disrupt the economic operation of the microgrid as much as possible.


## 3.3. Disconnected Scenario under DoS Attacks

This section investigates the performance analysis of the dispatch system in the scenario that communication topology graphs of the microgrid are disconnected under DoS attacks.If the DoS attacker concentrates on disconnecting all the communication links of a generator, it will briefly prevent the generator from communicating with its neighbors. Under these attacks, at least one generator is isolated from the communication network. It means the time-varying matrixAk has the rows and columns in which all the elements are zero, i.e., the in-degree and out-degree of some vertexes in the communication topology graph are zero. For the convenience of analysis, it is necessary to reduce the dimension of Ak to obtain A′k by removing the rows and columns whose elements are all equal to 0. The corresponding matrices Mk and Nk are transformed to M′k and N′k, respectively.In this scenario, all the generators can be divided into two types, misbehaving generators and well-behaving generators, respectively. DefineAnk⊆V as the set of misbehaving generators at iteration k, i.e., the units isolated from communication network, and Nnk=Ank is the number of misbehaving generators. It is worth noting that Ank presents isolated generators only due to DoS attacks in unreliable communication network, and isolation of units caused by other abnormal factors is beyond the scope of this paper. If i∉Ank, the generator i is involved in economic dispatch as a well-behaving generator, which updates its output power and incremental cost normally; otherwise, the generator i is a misbehaving generator, which maintains its output power and incremental cost until it is reconnected to the communication network after the disappearance of DoS attacks on it.Under these attacks, algorithms (9)–(11) can be rewritten as the following dynamics (24)–(26):(24)λik+1=∑j∉Ankmij′kλjk+εyik,i∉Ank,λik,i∈Ank,(25)Pik+1=Pimax,λi>λimax,λik+1−βi2αi,λimin≤λi≤λimax,Pimin,λi<λimin,(26)yik+1=∑j∉Anknij′kyjk−Pik+1−Pik,i∉Ank,0,i∈Ank,(27)∑j∉AnkPimin≤PD−∑j∈AnkPik≤∑j∉AnkPimax.Theorem 2.
In the scenario of attacks where there exist the communication network topologies disconnected, if the variables satisfy the initial condition∑i=1nPi0+yi0=PD, the learning gain ε in algorithms (24)–(26) is sufficiently small, and there exist enough output power satisfying total power demand, as shown in formula (27); then, the algorithm is stable and the variables of well-behaving generators converge to the suboptimal solution of the EDP, i.e.,(28)λik⟶λ′,i∉Ank,λik⟶λi,i∈Ank,Pik⟶P′,i∉Ank,Pik⟶Pi,i∈Ank,yik⟶0,ask⟶∞,∀i∈V,where λ′≠λ∗ and P′≠P∗ denote the suboptimal incremental cost vector and output power vector of well-behaving generators, respectively, and λi and Pi denote the incremental cost and output power of the misbehaving generator i, respectively. Note that convergence results shown in (28) depend on the attack strategies chosen by the DoS attacker.Proof.
The proof is similar with Theorem1. The process is omitted here for convenience.Remark 1.
The attacker deliberately causes algorithms (24)–(26) to be executed on the premise that formula (27) is satisfied in the microgrid. As a result, the algorithm converges to a stable but not optimal point as shown in Theorem 2, the attacks will be considered online stealthily [23]. The attacks are especially hard to detect by the system, and they can keep stealthy and damage the performance of the microgrid.
It worth noting that misbehaving generators isolated from the communication network will continue to supply electricity to local demand because the isolation action caused by DoS attacks cannot directly disconnect the transmission lines of the misbehaving generators in the physical network. Under this scenario of attack, the distributed algorithm cannot be executed consistently due to the absence of the real updating information of the misbehaving units. According to Theorem2, it is not very difficult to obtain that the distributed algorithm converges to the other convergence point, which is not equal to the optimal operation point of the microgrid. The conclusion can be drawn that, in this attack scenario which make the communication topology graphs disconnected, DoS attacks can interfere with the economic dispatch process of the microgrid to degrade the economic performance of the system. Through the above analysis, this scenario of attacks is a benefit to the DoS attacker. Only if DoS attacks isolate some generators from the communication network can such attacks will be effective for the destruction of economic operation of the microgrid.
Based on the analysis of the effect of attacks on the EDP, the problem that the authors are interested in is to find the largest degradation of economic performance of the microgrid under DoS attacks. In order to quantify the effect of attacks, the change of incremental cost in the attacks’ situation compared to the ideal situation is exploited.


## 4. Attack Strategy Formulation

### 4.1. Attack Capacity Formulation

According to the analysis in Section3.1, DoS attacks can occur at any communication moment and communication link. In practice, the launch of DoS attacks cannot be unrestrained, which is reflected in the constrained attack frequency and attack duration [24]. On the one hand, it is due to the inherent characteristics of DoS attacks, and on the other hand, several technologies of the power system can effectively resist jamming attacks [25]. As a consequence, the DoS attacker needs to formulate an optimal attack strategy on the premise of satisfying its own resource constraints to maximize the attack efficiency and attack effect.We define the total attack resource asΛ, and δnk is the attack level at iteration k, which represents the proportion of isolated units to the total number of units, i.e., δnk=Nnk/n, 0≤δnk≤1. It implies that the more generators that are isolated due to DoS attacks, the higher the attack level the attacks have. Define ϕ=ϕk, k∈0,1,… as the attack decision vector in which ϕk is the decision of the attacker at iteration k, and ϕk=1 indicates that the attacker decides to jam the wireless channel; otherwise, ϕk=0. Define ρkϕk,δkak as the indicator function to represent whether the attacker implements DoS attacks successfully, and the indicator function can be formulated as follows:(29)ρkϕk,δnk=1,ϕk=1,δnk≠0,0,otherwise.


### 4.2. Optimal Attack Strategy

The definition of the incremental cost in (5) implies that incremental cost is the value of the operation cost changing with the generated power. The incremental cost can be regarded as an indicator of economic performance for the power system:(30)argmaxAnk∑k=k1k2ρkλ′−λ∗=argmaxAnk∑k=k1k2ρkPD−∑i∈AnkPik+∑i∉Ankβi/2αi∑i∉Ank1/2αi−λ∗.Thus, the change of incremental cost in the attacks’ situation compared to the ideal situation can be used for a metric to quantify the effect of attacks. On this basis, the problem of designing an optimal attack strategy can be formulated as an optimization problem, for which the optimal solution can be obtained mathematically. This optimization problem aims to maximize the incremental cost deviation. At the same time, in order to ensure the DoS attacks are online stealthy, the attacker should ensure that all the units can meet the power balance requirement after some units are isolated. Furthermore, the DoS attacker needs to formulate an optimal attack strategy on the premise of satisfying its own resource constraints. In the iteration intervalk∈k1,k2, the formulation of an optimal DoS attacks’ strategy can be transformed to the optimization problem as follows:(31)s.t.∑k=k1k2ϕkδnk≤Λ,and27.The design of optimal strategy is completely reasonable because the purpose of the DoS attacks against economic dispatch process is to violate the economy principle and result in the largest degradation of the economic performance of the microgrid. As shown in previous analysis,λ∗ corresponds to the optimal economic dispatch results without DoS attacks and λ′ reflects the solution of the EDP under DoS attacks, and the value of λ′ is related to the attack strategies chosen by the attacker. Therefore, the more λ′ deviates from λ∗, the worse economic performance of the system can be obtained, and the objective function can be formulated as (30). Moreover, if the attack strategy selected violates the power supply-demand principle in (27), the attacks will be detected easily by the monitoring mechanism in the system, which goes against the intention of the attacker to hide its behaviors. Formula (31) considers the constraint of the attack resources from the practical point of view, and the formulation of attack strategy must be limited by it.In this paper, the static attack problem is investigated, and the optimal attack strategy is decided before the attack actions begin. However, the solution of the above objective function is inaccessible because the output power of isolated generators is unknown at that time. There are only some static parameters available for the attacker to formulate the attack strategies; thus, we put forward the following assumptions about DoS attacks and reformulate the optimization model according to them.Assumption 1.
Attacker can collect pivotal information of the microgrid such as operation cost parameters of generators, power upper and lower limits, and the total power demand. The information can be obtained by monitoring normal process of economic dispatch and collecting the relevant information published by authoritative organizations or manufacturers.Assumption 2.
For the objective to maximize the change of incremental cost in the attacks’ situation compared to the ideal situation, the attacker tends to attack the unit when its output power is the lower limit, which increases the burden on well-behaving generators and their incremental cost value.
Based on the assumption of attack pattern and objective shown above, the attacker pursues an expression of the optimal attack strategyX∗. Reformulate (27) and (30), ∑i∈AnkPi can be replaced by ∑i∈AnkPimin, and the optimization problem can be reformulated as follows:(32)X∗=argmaxAnk∑k=k1k2ρkλ′−λ∗=argmaxAnk∑k=k1k2ρkPD−∑i∈AnkPimin+∑i∉Ankβi/2αi∑i∉Ank1/2αi−λ∗,(33)s.t.∑i∉AnkPimin≤PD−∑i∈AnkPimin≤∑i∉AnkPimax,and31,where the parameters’ information of (31)–(33) is available in advance, and it means the optimal attack strategy can be decided before the attack actions begin.Remark 2.
According to the above analysis, in the nonideal communication environment with DoS attacks, the incremental costλ′ obtained by the calculation is related to the selected attack strategy. Because the EDP of the microgrid (2)–(4) is a convex optimization problem, the incremental cost is positively correlated with its corresponding output of the generator, which means that the greater the deviation of λ′ and λ∗ obtained under different attack strategies, the higher the operation cost of the system is. The above optimal attack strategy models (31)–(33) are essential to increase the operating cost of the system, thereby destroying the economic performance of the microgrid.
Based on the analysis of the optimal attack strategy formulation, an algorithm used to obtain this optimal strategy needs to be developed.


### 4.3. Algorithm to Obtain the Optimal Attack Strategy

The above optimization problem is not a simple convex optimization problem, and it cannot be directly solved by analytical methods or CVX toolbox in MATLAB [26]. As a result, this paper proposes an enumerating-search algorithm to solve the above optimization problem, which is described in Algorithm 1.The essence of Algorithm1 is to use the recursive process to exhaustively search for the optimal solution, i.e., the optimal attack strategy which damages the economic performance of the microgrid to the greatest extent. Firstly, enumerate all the possible attack schemes satisfying constraints (31) and (33), where the attack schemes are represented by the sequence number of the isolated units caused by each attack action. Then, we compute the deviation value of incremental cost corresponding to each attack scheme. By comparing all the deviation value, the attack scheme corresponding to the maximum deviation value is pursued by the DoS attacker. Finally, combine this optimal attack scheme with the attack decision vector; the attacker can decide the optimal attack strategy. It is worth noting that, for developing the optimal attack strategy before launching attacks so that the DoS attacks achieve the best effect of attacks, the attacker needs to further decide the attack moments, which can be represented by the attack decision vector ϕ [17]. This part is not the focus of the EDP for the microgrid, so it will not be detailed here.


## 4.1. Attack Capacity Formulation

According to the analysis in Section3.1, DoS attacks can occur at any communication moment and communication link. In practice, the launch of DoS attacks cannot be unrestrained, which is reflected in the constrained attack frequency and attack duration [24]. On the one hand, it is due to the inherent characteristics of DoS attacks, and on the other hand, several technologies of the power system can effectively resist jamming attacks [25]. As a consequence, the DoS attacker needs to formulate an optimal attack strategy on the premise of satisfying its own resource constraints to maximize the attack efficiency and attack effect.We define the total attack resource asΛ, and δnk is the attack level at iteration k, which represents the proportion of isolated units to the total number of units, i.e., δnk=Nnk/n, 0≤δnk≤1. It implies that the more generators that are isolated due to DoS attacks, the higher the attack level the attacks have. Define ϕ=ϕk, k∈0,1,… as the attack decision vector in which ϕk is the decision of the attacker at iteration k, and ϕk=1 indicates that the attacker decides to jam the wireless channel; otherwise, ϕk=0. Define ρkϕk,δkak as the indicator function to represent whether the attacker implements DoS attacks successfully, and the indicator function can be formulated as follows:(29)ρkϕk,δnk=1,ϕk=1,δnk≠0,0,otherwise.


## 4.2. Optimal Attack Strategy

The definition of the incremental cost in (5) implies that incremental cost is the value of the operation cost changing with the generated power. The incremental cost can be regarded as an indicator of economic performance for the power system:(30)argmaxAnk∑k=k1k2ρkλ′−λ∗=argmaxAnk∑k=k1k2ρkPD−∑i∈AnkPik+∑i∉Ankβi/2αi∑i∉Ank1/2αi−λ∗.Thus, the change of incremental cost in the attacks’ situation compared to the ideal situation can be used for a metric to quantify the effect of attacks. On this basis, the problem of designing an optimal attack strategy can be formulated as an optimization problem, for which the optimal solution can be obtained mathematically. This optimization problem aims to maximize the incremental cost deviation. At the same time, in order to ensure the DoS attacks are online stealthy, the attacker should ensure that all the units can meet the power balance requirement after some units are isolated. Furthermore, the DoS attacker needs to formulate an optimal attack strategy on the premise of satisfying its own resource constraints. In the iteration intervalk∈k1,k2, the formulation of an optimal DoS attacks’ strategy can be transformed to the optimization problem as follows:(31)s.t.∑k=k1k2ϕkδnk≤Λ,and27.The design of optimal strategy is completely reasonable because the purpose of the DoS attacks against economic dispatch process is to violate the economy principle and result in the largest degradation of the economic performance of the microgrid. As shown in previous analysis,λ∗ corresponds to the optimal economic dispatch results without DoS attacks and λ′ reflects the solution of the EDP under DoS attacks, and the value of λ′ is related to the attack strategies chosen by the attacker. Therefore, the more λ′ deviates from λ∗, the worse economic performance of the system can be obtained, and the objective function can be formulated as (30). Moreover, if the attack strategy selected violates the power supply-demand principle in (27), the attacks will be detected easily by the monitoring mechanism in the system, which goes against the intention of the attacker to hide its behaviors. Formula (31) considers the constraint of the attack resources from the practical point of view, and the formulation of attack strategy must be limited by it.In this paper, the static attack problem is investigated, and the optimal attack strategy is decided before the attack actions begin. However, the solution of the above objective function is inaccessible because the output power of isolated generators is unknown at that time. There are only some static parameters available for the attacker to formulate the attack strategies; thus, we put forward the following assumptions about DoS attacks and reformulate the optimization model according to them.Assumption 1.
Attacker can collect pivotal information of the microgrid such as operation cost parameters of generators, power upper and lower limits, and the total power demand. The information can be obtained by monitoring normal process of economic dispatch and collecting the relevant information published by authoritative organizations or manufacturers.Assumption 2.
For the objective to maximize the change of incremental cost in the attacks’ situation compared to the ideal situation, the attacker tends to attack the unit when its output power is the lower limit, which increases the burden on well-behaving generators and their incremental cost value.
Based on the assumption of attack pattern and objective shown above, the attacker pursues an expression of the optimal attack strategyX∗. Reformulate (27) and (30), ∑i∈AnkPi can be replaced by ∑i∈AnkPimin, and the optimization problem can be reformulated as follows:(32)X∗=argmaxAnk∑k=k1k2ρkλ′−λ∗=argmaxAnk∑k=k1k2ρkPD−∑i∈AnkPimin+∑i∉Ankβi/2αi∑i∉Ank1/2αi−λ∗,(33)s.t.∑i∉AnkPimin≤PD−∑i∈AnkPimin≤∑i∉AnkPimax,and31,where the parameters’ information of (31)–(33) is available in advance, and it means the optimal attack strategy can be decided before the attack actions begin.Remark 2.
According to the above analysis, in the nonideal communication environment with DoS attacks, the incremental costλ′ obtained by the calculation is related to the selected attack strategy. Because the EDP of the microgrid (2)–(4) is a convex optimization problem, the incremental cost is positively correlated with its corresponding output of the generator, which means that the greater the deviation of λ′ and λ∗ obtained under different attack strategies, the higher the operation cost of the system is. The above optimal attack strategy models (31)–(33) are essential to increase the operating cost of the system, thereby destroying the economic performance of the microgrid.
Based on the analysis of the optimal attack strategy formulation, an algorithm used to obtain this optimal strategy needs to be developed.


## 4.3. Algorithm to Obtain the Optimal Attack Strategy

The above optimization problem is not a simple convex optimization problem, and it cannot be directly solved by analytical methods or CVX toolbox in MATLAB [26]. As a result, this paper proposes an enumerating-search algorithm to solve the above optimization problem, which is described in Algorithm 1.The essence of Algorithm1 is to use the recursive process to exhaustively search for the optimal solution, i.e., the optimal attack strategy which damages the economic performance of the microgrid to the greatest extent. Firstly, enumerate all the possible attack schemes satisfying constraints (31) and (33), where the attack schemes are represented by the sequence number of the isolated units caused by each attack action. Then, we compute the deviation value of incremental cost corresponding to each attack scheme. By comparing all the deviation value, the attack scheme corresponding to the maximum deviation value is pursued by the DoS attacker. Finally, combine this optimal attack scheme with the attack decision vector; the attacker can decide the optimal attack strategy. It is worth noting that, for developing the optimal attack strategy before launching attacks so that the DoS attacks achieve the best effect of attacks, the attacker needs to further decide the attack moments, which can be represented by the attack decision vector ϕ [17]. This part is not the focus of the EDP for the microgrid, so it will not be detailed here.


## 5. Simulation Examples

Numerical simulation is a common method to verify theoretical analysis results as well as the effectiveness of the algorithm proposed. In this section, we firstly chose the IEEE 57-bus system to simulate the microgrid to compare the EDP under the different communication situations. Then, on the IEEE 39-bus system, we discuss the effect of DoS attacks adopting various attack strategies on economic performance of the microgrid.


### 5.1. Case 1: The EDP under Ideal Communication Conditions

There is an IEEE 57-bus system for simulation, and the communication network corresponding to the physical power network is shown in Figure2. This test system includes five generators, whose parameters and initial state values are listed in Tables 1 and 2, respectively. The total power demand of the system is set to 1000 MW.Figure 2
IEEE 57-bus system.Table 1
Parameters of generators.
αiβiγiPiminMWPimaxMW0.0015627.925611506000.001947.853101004000.004827.878502000.002347.853001003000.003127.8810050300Table 2
Initialization.
λi0$/MWhPi0MWyi0MW8.85723001008.82250−1008.764100507.85468200−507.8862410050In this case, under the ideal communication conditions without DoS attacks, the EDP can be solved with the algorithm proposed in [20]. In the iteration interval k∈0,350, the evolution of the output power Pi, incremental cost λi, the total power generated ∑P, and power demand PD is shown in Figure 3. Ultimately, the incremental cost λi, i=1,…,5, converges to a common value λ∗=8.8183$/MWh; meanwhile, the output power P1∗=287.5440MW, P2∗=249.5587MW, P3∗=105.6315MW, P4∗=206.8991MW, and P5∗=150.3666MW, all the generators are operating within their generation constraints, and also the total power demand is satisfied by these power output. The convergence results of the distributed algorithm are the same as that calculated by the centralized algorithm in [27]. The total operation cost is 9692.8$/h, which is the minimum operation cost of the system.Figure 3
The convergence results of the dispatch system under ideal communication conditions. (a) The power output of each generator. (b) The incremental cost of each generator. (c) Auxiliary variabley of each generator. (d) Total power output and load demand.
(a)(b)(c)(d)According to the above analysis, the optimal solution of the EDP can be solved through the distributed economic dispatch algorithm under ideal communication conditions. This case can be used as a comparison of the subsequent simulation cases.


### 5.2. Case 2: The EDP with Strongly Connected Communication Topology under DoS Attacks

Simulation case is performed with the same system model and parameters as those in Section5.1. In this case, we consider the scenario that communication topology graphs keep strongly connected under DoS attacks. Assuming the communication topology changes two times resulted by DoS attacks, the attacker launches DoS attacks firstly in the communication link (1,3) at iteration k=50 until k=100; then, the attacker launches DoS attacks in the communication link (1,4) at iteration k=100 until k=350.In this case, the EDP can be solved with algorithms (9)–(11). In the iteration interval k∈0,350, the evolution of the output power Pi, incremental cost λi, the total power generated ∑P, and power demand PD is shown in Figure 4. Ultimately, the incremental cost λi, i=1,…,5, converges to a common value λ∗=8.8183$/MWh and the output power P1∗=287.5440MW, P2∗=249.5587MW, P3∗=105.6315MW, P4∗=206.8991MW, and P5∗=150.3666MW. It can be concluded that the final convergence results in this case are the same as the results of Case 1.Figure 4
EDP with strongly connected topology under DoS attacks. (a) The power output of each generator. (b) The incremental cost of each generator. (c) Auxiliary variabley of each generator. (d) Total power output and load demand.
(a)(b)(c)(d)According to Figure4, it can be observed that the distributed algorithms (9)–(11) can finally converge to the optimal solution of EDP, although the existences of DoS attacks interfere with the convergence process of the algorithm. As a result, the correctness of the conclusion in Section 3.2 is verified, i.e., this attack scenario where the communication network still keeps strongly connected shows little benefit to the degradation of economic performance of the microgrid.


### 5.3. Case 3: The EDP with Disconnected Communication Topology under DoS Attacks

Simulation is performed with the same system model and parameters as those in Section5.1. In this case, we consider the scenario that there exist communication topology graphs disconnected under DoS attacks. If all the communication paths of a generator are attacked for a certain period of time, the generator is briefly isolated from the communication network. It is important to note that the generator will reconnect to the communication network when the attacks disappear.Assuming the communication topology changes two times because of DoS attacks, the attacker launches DoS attacks on generator 5, firstly, at iterationk=50 until k=100; then, the attacker launches DoS attacks on generator 4 and 5 at iteration k=100 until k=350. In this scenario, these two generators are misbehaving generators because of the DoS attacks on them, which lead to the disconnected communication topology graphs.In this case, algorithms (24)–(26) are carried out to solve the EDP. The evolution of the output power Pi, incremental cost λi, the total power generated ∑P, and power demand PD is shown in Figure 5. Under these DoS attacks, misbehaving generator 5 is isolated from the communication network at the iteration interval k∈50,350, and misbehaving generator 4 is isolated at the iteration interval k∈100,350, while the output power of the well-behaving generators can still satisfy the power demand remained. The incremental cost λi,i=1,2,3, converge to a common value λ′=8.8562$/MWh, λ4 converges to λ4=8.7934$/MWh, and λ5 converges to λ5=8.6893$/MW. The ultimate generators’ output P1′=299.6912MW, P2′=259.3390MW, P3′=109.5680MW, P4′=201.5769MW, and P5′=129.6879MW. All the generations are operating within their generation constraints, and the total load requirement is satisfied ultimately by generators participating in economic dispatch and the generators isolated. The total operation cost of the system is 9693.5$/h, which is higher than that of Case 1, the economic performance of the system declines.Figure 5
EDP with disconnected topology under DoS attacks. (a) The power output of each generator. (b) The incremental cost of each generator. (c) Auxiliary variabley of each generator. (d) Total power output and load demand.
(a)(b)(c)(d)According to Figure5 and the total operation cost, it can be observed that the distributed algorithms (24)–(26) cannot finally converge to the optimal solution of the EDP, and the economic dispatch process is disrupted by malicious attacks. It implies the correctness of conclusion in Section 3.3 is verified, i.e., the attack scenario that makes the communication network disconnected leads to the suboptimal solution of the EDP, which degrades the economic performance of the microgrid.


### 5.4. Case 4: The Optimal Attack Strategy

In this case, we discuss the different attack strategies, which can cause varying degrees of damage to economic performance of the microgrid. We chose the IEEE 39-bus system to simulate the microgrid, and its communication network corresponding to the physical power network is shown in Figure6. This system includes four generators, whose parameters can be chosen in Table 1. The total power load in the microgrid is set to 500MW. Assuming the total attack energy Λ=1 and attack decision vector ϕ=0…010…010…0, it is easy to deduce that δnk must be smaller than 1/2, which means the number of generators isolated caused by each attack action cannot exceed two for the communication network in Figure 6. Then, the set of attack strategies can be obtained according to constraints (31) and (33).Figure 6
IEEE 39-bus system.For the communication network in this case, we can enumerate the set of all attack schemesX=X1,X2,…X10, which include X1=1, X2=2, X3=3, X4=4, X5=1,2, X6=1,3, X7=1,4, X8=2,3, X9=2,4, and X10=3,4; an attack scheme is represented by the serial numbers of the misbehaving generators caused by each attack action. Establishing the objective function and constraints as (31)–(33), the optimization problem can be solved by Algorithm 1, and the calculation results are shown in Figure 7(a). According to Figure 7(a), if X5=1,2 is adopted, the deviation of incremental cost will be maximized. It means the execution of the optimal attack strategy X∗ is to repeat the attack scheme X5=1,2 at each attack moment. Combining the attack decision vector ϕ and attack schemes, different attack strategies can be obtained. The comparison of different attack strategies is shown in Figure 7(b), from which we can easily sum up that optimal attack strategy obtained by using Algorithm 1 indeed causes the most degradation of economic performance compared with other attack strategies. As a result, it is feasible to carry out the proposed enumerating-search algorithm to obtain the optimal attack strategy.Figure 7
The comparison for different attack strategies. (a) Deviation of incremental cost in the attacks’ situation compared to the ideal situation. (b) The effect of attacks under different attack strategies.
(a)(b)


## 5.1. Case 1: The EDP under Ideal Communication Conditions

There is an IEEE 57-bus system for simulation, and the communication network corresponding to the physical power network is shown in Figure2. This test system includes five generators, whose parameters and initial state values are listed in Tables 1 and 2, respectively. The total power demand of the system is set to 1000 MW.Figure 2
IEEE 57-bus system.Table 1
Parameters of generators.
αiβiγiPiminMWPimaxMW0.0015627.925611506000.001947.853101004000.004827.878502000.002347.853001003000.003127.8810050300Table 2
Initialization.
λi0$/MWhPi0MWyi0MW8.85723001008.82250−1008.764100507.85468200−507.8862410050In this case, under the ideal communication conditions without DoS attacks, the EDP can be solved with the algorithm proposed in [20]. In the iteration interval k∈0,350, the evolution of the output power Pi, incremental cost λi, the total power generated ∑P, and power demand PD is shown in Figure 3. Ultimately, the incremental cost λi, i=1,…,5, converges to a common value λ∗=8.8183$/MWh; meanwhile, the output power P1∗=287.5440MW, P2∗=249.5587MW, P3∗=105.6315MW, P4∗=206.8991MW, and P5∗=150.3666MW, all the generators are operating within their generation constraints, and also the total power demand is satisfied by these power output. The convergence results of the distributed algorithm are the same as that calculated by the centralized algorithm in [27]. The total operation cost is 9692.8$/h, which is the minimum operation cost of the system.Figure 3
The convergence results of the dispatch system under ideal communication conditions. (a) The power output of each generator. (b) The incremental cost of each generator. (c) Auxiliary variabley of each generator. (d) Total power output and load demand.
(a)(b)(c)(d)According to the above analysis, the optimal solution of the EDP can be solved through the distributed economic dispatch algorithm under ideal communication conditions. This case can be used as a comparison of the subsequent simulation cases.


## 5.2. Case 2: The EDP with Strongly Connected Communication Topology under DoS Attacks

Simulation case is performed with the same system model and parameters as those in Section5.1. In this case, we consider the scenario that communication topology graphs keep strongly connected under DoS attacks. Assuming the communication topology changes two times resulted by DoS attacks, the attacker launches DoS attacks firstly in the communication link (1,3) at iteration k=50 until k=100; then, the attacker launches DoS attacks in the communication link (1,4) at iteration k=100 until k=350.In this case, the EDP can be solved with algorithms (9)–(11). In the iteration interval k∈0,350, the evolution of the output power Pi, incremental cost λi, the total power generated ∑P, and power demand PD is shown in Figure 4. Ultimately, the incremental cost λi, i=1,…,5, converges to a common value λ∗=8.8183$/MWh and the output power P1∗=287.5440MW, P2∗=249.5587MW, P3∗=105.6315MW, P4∗=206.8991MW, and P5∗=150.3666MW. It can be concluded that the final convergence results in this case are the same as the results of Case 1.Figure 4
EDP with strongly connected topology under DoS attacks. (a) The power output of each generator. (b) The incremental cost of each generator. (c) Auxiliary variabley of each generator. (d) Total power output and load demand.
(a)(b)(c)(d)According to Figure4, it can be observed that the distributed algorithms (9)–(11) can finally converge to the optimal solution of EDP, although the existences of DoS attacks interfere with the convergence process of the algorithm. As a result, the correctness of the conclusion in Section 3.2 is verified, i.e., this attack scenario where the communication network still keeps strongly connected shows little benefit to the degradation of economic performance of the microgrid.


## 5.3. Case 3: The EDP with Disconnected Communication Topology under DoS Attacks

Simulation is performed with the same system model and parameters as those in Section5.1. In this case, we consider the scenario that there exist communication topology graphs disconnected under DoS attacks. If all the communication paths of a generator are attacked for a certain period of time, the generator is briefly isolated from the communication network. It is important to note that the generator will reconnect to the communication network when the attacks disappear.Assuming the communication topology changes two times because of DoS attacks, the attacker launches DoS attacks on generator 5, firstly, at iterationk=50 until k=100; then, the attacker launches DoS attacks on generator 4 and 5 at iteration k=100 until k=350. In this scenario, these two generators are misbehaving generators because of the DoS attacks on them, which lead to the disconnected communication topology graphs.In this case, algorithms (24)–(26) are carried out to solve the EDP. The evolution of the output power Pi, incremental cost λi, the total power generated ∑P, and power demand PD is shown in Figure 5. Under these DoS attacks, misbehaving generator 5 is isolated from the communication network at the iteration interval k∈50,350, and misbehaving generator 4 is isolated at the iteration interval k∈100,350, while the output power of the well-behaving generators can still satisfy the power demand remained. The incremental cost λi,i=1,2,3, converge to a common value λ′=8.8562$/MWh, λ4 converges to λ4=8.7934$/MWh, and λ5 converges to λ5=8.6893$/MW. The ultimate generators’ output P1′=299.6912MW, P2′=259.3390MW, P3′=109.5680MW, P4′=201.5769MW, and P5′=129.6879MW. All the generations are operating within their generation constraints, and the total load requirement is satisfied ultimately by generators participating in economic dispatch and the generators isolated. The total operation cost of the system is 9693.5$/h, which is higher than that of Case 1, the economic performance of the system declines.Figure 5
EDP with disconnected topology under DoS attacks. (a) The power output of each generator. (b) The incremental cost of each generator. (c) Auxiliary variabley of each generator. (d) Total power output and load demand.
(a)(b)(c)(d)According to Figure5 and the total operation cost, it can be observed that the distributed algorithms (24)–(26) cannot finally converge to the optimal solution of the EDP, and the economic dispatch process is disrupted by malicious attacks. It implies the correctness of conclusion in Section 3.3 is verified, i.e., the attack scenario that makes the communication network disconnected leads to the suboptimal solution of the EDP, which degrades the economic performance of the microgrid.


## 5.4. Case 4: The Optimal Attack Strategy

In this case, we discuss the different attack strategies, which can cause varying degrees of damage to economic performance of the microgrid. We chose the IEEE 39-bus system to simulate the microgrid, and its communication network corresponding to the physical power network is shown in Figure6. This system includes four generators, whose parameters can be chosen in Table 1. The total power load in the microgrid is set to 500MW. Assuming the total attack energy Λ=1 and attack decision vector ϕ=0…010…010…0, it is easy to deduce that δnk must be smaller than 1/2, which means the number of generators isolated caused by each attack action cannot exceed two for the communication network in Figure 6. Then, the set of attack strategies can be obtained according to constraints (31) and (33).Figure 6
IEEE 39-bus system.For the communication network in this case, we can enumerate the set of all attack schemesX=X1,X2,…X10, which include X1=1, X2=2, X3=3, X4=4, X5=1,2, X6=1,3, X7=1,4, X8=2,3, X9=2,4, and X10=3,4; an attack scheme is represented by the serial numbers of the misbehaving generators caused by each attack action. Establishing the objective function and constraints as (31)–(33), the optimization problem can be solved by Algorithm 1, and the calculation results are shown in Figure 7(a). According to Figure 7(a), if X5=1,2 is adopted, the deviation of incremental cost will be maximized. It means the execution of the optimal attack strategy X∗ is to repeat the attack scheme X5=1,2 at each attack moment. Combining the attack decision vector ϕ and attack schemes, different attack strategies can be obtained. The comparison of different attack strategies is shown in Figure 7(b), from which we can easily sum up that optimal attack strategy obtained by using Algorithm 1 indeed causes the most degradation of economic performance compared with other attack strategies. As a result, it is feasible to carry out the proposed enumerating-search algorithm to obtain the optimal attack strategy.Figure 7
The comparison for different attack strategies. (a) Deviation of incremental cost in the attacks’ situation compared to the ideal situation. (b) The effect of attacks under different attack strategies.
(a)(b)


## 6. Conclusion

This paper considers an optimal attack strategy against the economic dispatch for the microgrid. Firstly, according to the operating principle of DoS attacks, the unreliable communication network under DoS attacks can be classified into two scenarios, i.e., disconnected and strongly connected scenarios. The theoretical analysis results show that economy performance of the microgrid is damaged in the former case. On this basis, the optimal DoS attack strategy considering the resource constraints of the attacker is formulated to maximize the effect of attacks, which can be obtained by using the enumerating-search algorithm. Finally, the correctness of the theoretical analysis results with respect to the effect of attacks is verified by the simulations on the IEEE 57-bus system, and the effectiveness of the proposed enumerating-search-based optimal attack strategy is illustrated through the simulations on the IEEE 39-bus system. This paper delivers the message that, besides the efforts of designing novel distributed economic dispatch algorithms to guarantee the economic operation for the microgrid, it is equally important to protect the economic dispatch process from malicious attacks and evade potential economic losses.For the future work, how to build the rational model of DoS attacks is the key to solving the EDP of the energy system under DoS attacks. Furthermore, we will synthetically consider the impact of other attacks on the EDP, such as false data injection attacks and replay attacks. Moreover, research on the dynamic optimal attack strategy disrupting the economic dispatch of the energy system is also the focus of our future work.



---

*Source: 5539829-2021-04-02.xml*