Distributed Finite-Time Cooperative Economic Dispatch Strategy for Smart Grid under DOS Attack

Abstract

This paper proposes an energy management strategy that can resist DOS attacks for solving the Economic Dispatch Problem (EDP) of the smart grid. We use the concept of energy agent, which acts as a hub for the smart grid, and each EA is an integrated energy unit that converts, stores, and utilizes its local energy resources. This approach takes into account the coupling relationship between energy agents (EA) and utilizes the Lyapunov function technique to achieve finite-time solutions for optimization problems. We incorporate strategies to resist DOS attacks when analyzing finite-time convergence using the Lyapunov technique. Based on this, a finite convergence time related to DOS attack time is derived. The integral sliding mode control strategy is adopted and the Lyapunov method is used to analyze it, so that the algorithm can resist DOS attacks and resist external disturbances. Through theoretical analysis, it is shown that the strategy is capable of converging to the global optimal solution in finite time even if it is attacked by DOS. We conducted case studies of six-EA and ten-EA systems to verify the effectiveness of this strategy. The proposed strategy has potential for deployment in distributed energy management systems that require resilience against DOS attacks.

1. Introduction

In recent years, much attention has been devoted to the smart grid as a potential solution to the world’s environmental issues and resource shortages. By integrating advanced communication technology with power networks, smart grids enable the integrated and efficient use of energy resources, absorb significant proportions of clean energy, and optimize energy utilization efficiency, ultimately realizing intelligent and stable power network operations and reduced carbon emissions [1,2,3,4]. Smart grid economic dispatch problems (EDP) have become a hot topic, as they provide a means to reduce total costs through coordinated management of different-capacity equipment and loads, thus ensuring efficient and smooth operation of the smart grid.

Currently, the prevailing approach to solving EDP is centralized. However, this method has significant challenges such as high sensitivity to single points of failure and privacy constraints [5]. To improve the flexibility, stability, and privacy of the system, distributed solutions are being researched and developed as the best alternative to solve EDP. Distributed methods, such as Alternating Direction Method of Multipliers (ADMM), gradient descent, and consensus-based methods, are progressively replacing centralized methods. ADMM algorithms are highly efficient, and the integration of consensus-based methods with ADMM algorithms has maturity greatly. For instance, Attarha et al. [6] proposes a distributed consensus ADMM algorithm to optimize energy generation and consumption for each participant, while considering system robustness. The consensus-based solutions are also becoming more sophisticated, and in [7], Li et al. proposed a distributed double-consensus approach to facilitate the cooperative energy management of multiple WEs, thereby introducing the concept of WEs. Recently, distributed solutions have evolved based on specific application needs, such as distributed Newtonian descent [8,9,10], and distributed methods based on neurodynamics [11,12].

However, the existing methods such as ADMM, gradient descent, and consensus-based algorithms used for solving the local cost function of EDP are independent, which means the coupling relationship among energy agents is not considered. Hence, these algorithms cannot solve the coupling optimization problem where energy agents interact with each other. Currently, there are few studies in this area. In [13], Knudsen et al. studied the wake effect of wind turbines regarding the influence of capacity equipment. Still, this research solely focuses on the modeling of wind turbines; the problem of distributed coupling optimization for smart grids needs further exploration. In [14], an optimization algorithm was mentioned to solve the distributed coupling optimization problem for the smart grid. However, this approach relies on a stable communication network and does not account for the huge impact of DOS attacks on the communication network of the smart grid.

Distributed methods have significantly contributed to solving the EDP of smart grids. However, with the advancement of industrial technology, convergence speed has emerged as a critical issue for the development of smart grids. Although there are presently numerous convergence methods for EDP [15,16,17], finite time convergence has become more critical. Dai et al. [18] popularized finite-time stability theory in smart grids. In [19] Majid and Amin proposed a finite-time optimization approach to power systems using the Lyapunov function technique. However, the dependable operation of these approaches is based on network stability, and there is no way to handle network attacks. DOS attack, for instance, interferes with the reliable operation of distributed communication structures by consuming system resources. Communication network disruption leads to extensive economic losses due to a lack of prompt exchange of accurate real-time data between the agents involved. Academic research has focused on cyber-attacks on smart grids [20,21]. For example, in [22], Zhang et al. mentioned a robust method based on mixed integer linear programming to mitigate DOS attack impacts. Distributed anomaly detection technology has equally proven effective in solving problems of DOS attacks. In [23], Çakır et al. proposed a machine learning-based model for detecting DOS attacks on smart grids. In [24], Huang et al. also suggested an energy dispatch strategy that isolates the detected fault points. However, more effective methods are required to resist the impacts of DOS attacks, especially those that can solve the EDP of smart grids in a finite time after a network attack. The latest method, proposed by Li et al. [25], can effectively solve the problem of economic dispatch under DOS attacks and the double-gradient descent elastic energy management approach; however, unfortunately, in cannot solve the optimization problem in a finite time. The obtained conclusions about state-of-the-art-works are shown in

Table 1. The obtained conclusions about state-of-the-art-works.

Reference	Finite-Time Convergence or Not	Resistant to DOS Attack or Not	Coupled Cost Function Is Considered	Features of Paper
[14]	Yes	No	Yes	Proposed a finite time convergence algorithm considering coupled agents
[18]	Yes	No	No	Popularized finite-time stability theory analyzed by Lyapunov function technique
[25]	No	Yes	No	Proposed a strategy that can resist DOS attacks

.

This paper presents a new finite-time elastic optimization strategy capable of solving EDP with coupled local cost functions in the presence of Distributed Denial of Service (DOS) attacks. The main contributions are:

1.

A cost function model taking coupling relationships into account, which is more complex than previous models that did not consider these relationships. The capacity or consumption of EAs with coupling relationships is included in the intersection terms of local cost functions. Furthermore, we propose an elastic optimization approach that resists the destruction of the smart grid’s distributed coupling optimization during DOS attacks;

2.

We present a finite-time elasticity optimization strategy that adopts sliding mode control and eliminates the information loss incurred by DOS attacks. We analyze the convergence characteristics of the proposed algorithm by using the Lyapunov method and design a finite-time convergence algorithm that ensures the EDP solution can be reached, even after a DOS attack occurs.

The remainder of this paper is organized as follows. Section 2 introduces the system structure, EA model, communication topology, and formulates the EDP that needs to be solved. In Section 3, we provide a brief introduction on DOS attacks, describe the lemma used, propose the elastic optimization algorithm, and analyze its convergence. In Section 4, we verify the results of our proposed algorithm through simulation experiments. Finally, we conclude our findings in Section 5.

2. System Structure and Model

Smart grid refers to the utilization of advanced power technology and equipment information and communication technology to achieve intelligent monitoring, analysis, and decision control of the grid in a systematic manner. The smart grid is designed to be safe, reliable, efficient, highly automated, and equipped with interactive functions for self-healing. In order to establish an effective communication network topology, we introduce the Energy Agent (EA) as the core component of each smart grid in the form of intelligent communication devices. The EA is used for analyzing the energy flow within the large smart grid, which is composed of multiple interconnected smart grids. As the hub of the smart grid, each EA is an integrated energy unit that can convert, store, and utilize its local energy resources, including distributed renewable energy generators (DRGS), distributed fuel generators (DFGS), distributed power storage (DES), and energy loads. Additionally, multiple smart grids are connected through the EA to form a large smart grid, which enables the partial and overall optimal management of energy.

2.1. EA Model

The supply and demand balance constraints for energy conversion, storage, and utilization (i.e., input and output) of EA at time t are given by

$$p_{i,t}^{drg}+p_{i,t}^{dfg}+p_{i,t}^{es}-l_{i,t}^p=p_{i,t}^{out},$$

(1)

where $l_{i,t}^p$ is the overall power load at time t and equal to the sum of must-run power loads and controllable power loads, expressed by the equation $l_{i,t}^p=l_{i,t}^{p,mr}+l_{i,t}^{p,ctrl}$. $l_{i,t}^{p,mr}$ indicates that the load must be run and $l_{i,t}^{p,ctrl}$ represents controllable load. The notation meanings are shown in

Table 2. Frequency used notation.

Notations	Description
$i,t$	Index for EA and time
$p$	Electricity power
$drg,dfg$	Index for DRG, DFG
$es$	Index for DES

At time t, the cost function of the equipment included in the EA is shown below.

(1)

The cost function of DRG is

$$C\left(p_{i,t}^{drg}\right)=a_i^{p,r}{p}_{i,t}^{drg}+b_i^{p,r}\exp \left({\eta }_i^{p,r}\frac{p_{i,t}^{drg,\max }-p_{i,t}^{drg}}{p_{i,t}^{drg,\max }-p_{i,t}^{drg,\min }}\right),$$

(2)

where $r$ is the index for renewable devices; $a_i^{p,r}$ and $b_i^{p,r}$ are non-negative cost coefficients; and ${\eta }_i^{p,r}$ is penalty coefficient.

(2)

The cost function of DFG is

$$C\left(p_{i,t}^{dfg}\right)=a_i^{p,f}{\left(p_{i,t}^{dfg}\right)}^2+b_i^{p,f}{p}_{i,t}^{dfg}+c_i^{p,f},$$

(3)

where $a_i^{p,f}$, $b_i^{p,f}$, $c_i^{p,f}$ represent non-negative cost coefficients; the symbol $f$ denotes fuel devices.

(3)

The cost function of ES is

$$C\left(p_{i,t}^{es}\right)=a_i^{p,es}{\left(p_{i,t}^{es}+b_i^{p,es}\right)}^2,$$

(4)

where $a_i^{p,es}$ and $b_i^{p,es}$ are non-negative cost coefficients, and symbol.

(4)

The cost function of energy loads

We believe that the benefit function of the energy load is consistent with the cost function form, and when $p_{i,t}$ is less than zero, it is considered that this energy form of EA is converted into a load state. When ${\displaystyle {\sum }_{i=1}^n{p}_{i,t}=0}$ is satisfied, it is considered that the balance between supply and demand is met between all EAs.

2.2. Problem Formulation

2.2.1. Graph Theory

There are multiple EAs, and each EA can interact with its neighbor EAs to achieve internal energy demand and global optimal energy distribution by selling surplus energy or purchasing deficient energy. Here, we introduce graph theory to model the energy flow process.

The topology graph between agents is donated as $G=\left(V,\xi ,W\right)$, where $V=\left\{v_1,v_2,\dots ,v_n\right\}$ is a finite set of vertices representing all EAs, $\xi \subset V\times V$ is the edge set representing the communication links, $W=\left[w_{ij}\right]\in {ℝ}^{n\times n}$ is the vertex–edge incidence matrix. The Laplacian matrix of the graph $G$ is defined as follows

$$L_{ij}=\left\{\begin{array}{l}d\left(v_i\right) \mathrm{if} i=j,\\ -W_{ij} \mathrm{if} i\ne j \mathrm{and} G \mathrm{is} \mathrm{connected},\\ 0 \mathrm{else},\end{array}\right.$$

(5)

$d\left(v_i\right)={\displaystyle {\sum }_{i=1,i\ne j}^n{W}_{ij}}$ denotes the degree of vertex $v_i$. The pairs of vertices in $\xi$ are ordered; that is, every edge in $\xi$ is oriented, then $G$ is a digraph. When $G$ is undirected, the pairs of vertices are unordered with $w_{ij}=w_{ji}, \forall i\ne j$. The second smallest eigenvalue of the Laplacian matrix of a graph, ${\lambda }_2$, is defined as algebraic connectivity. If the graph is connected, then the algebraic connectedness is greater than zero, and the graph is divided into two parts when the value is zero. In this paper, we use the undirected connected graph.

2.2.2. Energy Management of EAs

In order to achieve the optimal energy production and utilization of each EA under the balance of supply and demand, we can describe the Economic Dispatch Problem of n agents as follows:

$$\begin{array}{l}\min \mathrm{C}\left(x\right)={\displaystyle {\sum }_{i=1}^n\left(C_i\left(x\right)+{\displaystyle {\sum }_{j\in N_i^C}{C}_i^j}\left(x\right)\right)},\\ \mathrm{s}.\mathrm{t}. {\displaystyle {\sum }_{i=1}^n{x}_i\left(t\right)=0, \forall t\ge 0} ,\end{array}$$

(6)

where $C\left(x\right)$ is the objective function.

$x$ is the optimization variable. We assume the optimization variables $x_i\left(t\right)$ is

$$\stackrel{·}{x_i\left(t\right)}=u_{i1}\left(t\right)+u_{i2}\left(t\right), i=1,2,\dots ,n,$$

(7)

where the $u_i\left(t\right)$ is the control algorithm of agent $i$. When there is no DOS attack, we achieve finite-time convergence by designing $u_{i1}\left(t\right)$ and $u_{i2}\left(t\right)$.

Considering the local interdependence between EAs, the cost ${\displaystyle {\sum }_{j\in N_i^C}{C}_i^j}\left(x\right)$ imposed by the $jth$ agent on the $ith$ agent is added on the basis of the $ith$ EA in addition to its own constant power generation cost $C_i(x)$. $C_i^j(x)={\displaystyle {\sum }_{j=1}^n{\gamma }_j^i{x}_j}$. Meanwhile, according to the model mentioned above, we can express $C_i\left(x\right)$ as

$$C_i\left(x\right)=C\left(p_{i,t}^{drg}\right)+C\left(p_{i,t}^{dfg}\right)+C\left(p_{i,t}^{es}\right)+C\left(l_{i,t}\right).$$

(8)

Assumption 1.

The team objective function  $C\left(x\right)$ is twice continuously differentiable, and strongly convex, satisfying the condition that ${\nabla }^{2}C(x)\ge \theta I_n>0$ for real value $\theta >0$.

Definition 1.

In a network with n agents,$N_i^C\subset V$ is the set of critical neighbors of agent i, which consists of the agent i neighbors where their decision variables directly appear in the local cost  $C_i(x)$. Information exchanging with these is required for agent i to compute  $C_i(x)$.

Remark 1.

According to the linear equation constraint of the convex function [26], $x^{*}$ is the optimal solution to problem (9) if and only if $\nabla C(x^{*})=c{1}_n$ for $c\in ℝ$.

3. Main Algorithm

3.1. DOS Attack

A DOS attack can cause a broken communication link. The core content of our research is to use distributed methods to solve the problem of integrated energy systems, which require a distributed network structure to implement distributed algorithms, but this is vulnerable to DOS attacks. When it is attacked by a network, it will lead to the continuous loss of information shared between nodes, and there is a certain probability that the balance of supply and demand between all nodes will be affected. We believe that a DOS attack cannot destroy the entire communication link, even if only one communication line is left in the worst case. Starting from $t_0$, the total time $\left[t_0,\left.t\right]\right.$ can be divided into attacked periods and normal periods. After each attack, there is a recovery period $t_r$. This period counts towards normal periods. We define $\left(t_k,t_k+{\tau }_k\right]$ as the period during which the kth DOS attack was received. $t_k$ and ${\tau }_k$ are the corresponding launched attack instant and the attack dwell time, respectively. The total dwell time of the DOS attack is $\Im ={\displaystyle {\sum }_{\forall k}{\tau }_k}$, whose proportion of the total is $\phi =\Im /(t-t_0)$. DOS attack periods and normal periods are defined as ${\varphi }_{ap}={\displaystyle \sum _{\forall k}\left(t_k,t_k+{\tau }_k\right]\cap }\left[t_0,t\right]$ and ${\varphi }_{np}={\displaystyle \sum _{\forall k}\left(t_k+{\tau }_k,t_{k+1}\right]\cap }\left[t_0,t\right]$, respectively. If a communication link is subjected to a DOS attack, both of its end nodes are considered to be under a network attack.

3.2. Notation

We denote $1_n={\left[1,1,1,\dots ,1\right]}^T\in {ℝ}^n$, and $I_n$ as the $n\times n$ identity matrix. For $n\times n$, and $\alpha \in ℝ$, let $\mathrm{sgn}{(x)}^{\alpha }=\left[\mathrm{sgn}{(x_i)}^{\alpha }\right]\in {ℝ}^n$, where $\mathrm{sgn}{(x)}^{\alpha }=sig(x_i){\left|x_i\right|}^{\alpha }$, $sig\left(\cdot \right)$ represents the signum function, and $\left|x_i\right|$ denotes the absolute value of $x_i$. For a matrix $A\in {ℝ}^{n\times k}$, $A^T$ denotes the transpose of $A$. For a symmetric matrix $A=A^T\in {ℝ}^{n\times n}$, its eigenvalues are denoted as ${\lambda }_1(A)\le {\lambda }_2(A)\le ,\dots \le {\lambda }_n(A)$ in a non-decreasing order. $\nabla f(s)$ and ${\nabla }^{2}f(s)$, respectively, denote the gradient and the Hessian matrix of the function $f(s)$.

3.3. Useful Lemmas

All of the lemmas mentioned below are used for mathematical derivation in Section 3.4. When these lemmas are used, where they are used is mentioned.

Lemma 1

([27]). Let  ${\xi }_1,{\xi }_2,{\xi }_3,\mathrm{\dots }{\xi }_n\ge 0$. For all real values  $0<p\le 1$ we have ${{\displaystyle {\sum }_{i=1}^n{\xi }_i^p\ge \left({\displaystyle {\sum }_{i=1}^n{\xi }_i}\right)}}^p$. If $1<p<\infty$, then, ${\displaystyle {\sum }_{i=1}^n{\xi }_i^p}\ge n^{1-p}{({\displaystyle {\sum }_{i=1}^n{\xi }_i})}^p$. This lemma is used in Formulae (31), (34), and (39) to transform mathematical formulae to prove finite time convergence.

Lemma 2

([28]). Let G be an undirected and connected graph with  $L(A)=\left[l_{ij}\right]\in {ℝ}^{n\times n}$. Then, for all$\xi =\left[{\xi }_i\right]\in {ℝ}^n$, we have${\xi }^{T}L(A)\xi =\frac{1}{2}{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}{({\xi }_j-{\xi }_i)}^2}$. This lemma is used in Formula (40) to construct mathematical expressions for the use of lemma 3.

Lemma 3

([18]). Consider the system  $\dot{x}=f\left(t,x\right)$, where  $x\in {ℝ}^n$ and $f:ℝ\times {ℝ}^n\to {ℝ}^n$. Let the origin be an equilibrium point for the system. Let there exist a continuous radially unbounded function $V(x)\ge 0$ with  $V(x)=0\iff x=0$. If $\dot{V}(x)\le -{\beta }_{0}V(x)-{\beta }_1{V}^{q_1}(x)-{\beta }_2{V}^{q_2}(x)$, where  ${\beta }_0,{\beta }_1,{\beta }_2>0$, $0<q_1<1$, and $q_2>1$, the origin is a globally finite-time stable equilibrium with the settling time $T_S\le \frac{\ln (1+{\beta }_0/{\beta }_1)}{{\beta }_0(1-q_1)}+\frac{\ln (1+{\beta }_0/{\beta }_2)}{{\beta }_0(q_2-1)}$. This lemma is used in the Formulae (32), (35), and (60) to calculate the upper limit of time for convergence.

Lemma 4

([19]). Consider $x,y,p\in ℝ$, and $p\ge 1$. Then,  $\left|\mathrm{sgn}{(x)}^p-\mathrm{sgn}{(y)}^p\right|\ge 2^{1-p}{\left|x-y\right|}^p$. This lemma is used in Formula (38) to transform mathematical formulae to prove finite time convergence.

Lemma 5

([19]). Let$A\in {ℝ}^{n\times n}$  be a symmetric positive semi-definite matrix, with $0={\lambda }_1(A)=\cdots ={\lambda }_{k-1}(A)<{\lambda }_k(A)\le \cdots \le {\lambda }_n(A)$ as the eigenvalues. Then,  $x^{T}AAx\ge$${\lambda }_k(A)x^{T}Ax$$, \forall x\in {ℝ}^n$. This lemma is used in Formula (48) to calculate the form of the Lyapunov function under DOS attacks.

Lemma 6

([19]). Let $A\in {ℝ}^{n\times n}$ be a given positive semi-definite matrix such that ${\lambda }_1(A)=0$ is a simple eigenvalue of $A$ with the corresponding eigenvector $1_n$. Let $C(x)$ be a θ-strongly convex function with respect to $x\in {ℝ}^n$, i.e.,${\nabla }^{2}C(x)\ge \theta I_n>0$ for some positive values θ. Suppose, $x^{*}$ is a solution of the following convex optimization problem,

$$\left\{\begin{array}{c}{\min }_{x}C(x),\\ s.t:{\displaystyle {\sum }_{i=1}^n{x}_i=0},\end{array}\right.$$

(9)

Then, the following inequality holds,

$$\frac{1}{2}\theta {\lambda }_2(A)\left(C(x)-C(x^{*})\right)\le \nabla C{(x)}^{T}A\nabla C(x).$$

(10)

This lemma is used in Formulae (41) and (48) to calculate the form of the Lyapunov function of the algorithm without DOS attack and under DOS attack, respectively.

3.4. Distributed Optimization Strategies to Resist DOS Attacks

Inspired by the sliding-mode control and the energy management strategy based on distributed elastic double-gradient descent, we propose a distributed finite-time convergence algorithm to solve the Energy Distribution Problem (EDP) with energy agents. The algorithm is designed to resist DOS attacks, and its control variables will switch to different states depending on the communication environment. The algorithm can mainly be categorized into three states. Specifically, when the algorithm is not under DOS attack, its mathematical expression is given as follows:

$$\dot{x}=u_{i1}(t)+u_{i2}(t),$$

(11)

$$u_{i1}=k_0{\displaystyle \sum _{j\in N_i}{a}_{ij}({\eta }_j-{\eta }_i)+k_1}{\displaystyle \sum _{j\in N_i}{a}_{ij}sig(}{\eta }_j-{\eta }_i)+k_2{\displaystyle \sum _{j\in N_i}{a}_{ij}(\mathrm{sgn}(}{\eta }_j{)}^p-\mathrm{sgn}{({\eta }_i)}^p),$$

(12)

$${\eta }_i=\frac{\partial }{\partial x_i}{C}_i(x)+{\displaystyle \sum _{j\in N_i^c}\frac{\partial }{\partial x_i}}{C}_j(x),$$

(13)

$$u_{i2}=-c_0{s}_i-c_{1}sig(s_i)-c_2\mathrm{sgn}{(s_i)}^{\alpha },$$

(14)

$$s_i(t)=x_i(t)-{\displaystyle {\int }_{t_0}^t{u}_{i1}(\tau )d\tau }.$$

(15)

where $k_0$, $k_1$, $k_2$, $c_0$, $c_1$, $c_2$, $p$, $\alpha$ are adjustable constants. $u_{i1}$ and $u_{i2}$ are part of the algorithm, ${\eta }_i$ is the ith element of $\nabla C(x)$, and $s_i(t)$ is auxiliary variable. Their roles are described in detail in this section. When subjected to a DOS attack, the state switches to the following form:

$$\dot{x}=u_{i1}^{\prime }(t)+u_{i2}^{\prime }(t),$$

(16)

$$u_{i1}^{\prime }=k_0{\displaystyle \sum _{j\in N_i}{a}_{ij}({\eta }_j(t_k)-{\eta }_i)+k_1}{\displaystyle \sum _{j\in N_i}{a}_{ij}sig(}{\eta }_j(t_k)-{\eta }_i),$$

(17)

$${\eta }_i=\frac{\partial }{\partial x_i}{C}_i(x)+{\displaystyle \sum _{j\in N_i^c}\frac{\partial }{\partial x_i}}{C}_j(x),$$

(18)

$$u_{i2}^{\prime }=-c_0{s}_i-c_{1}sig(s_i)-c_2\mathrm{sgn}{(s_i)}^{\alpha },$$

(19)

$$s_i(t)=x_i(t)-{\displaystyle \sum _{\forall k}{\displaystyle {\int }_{t_{k-1}}^{t_k}{u}_{i1}(\tau )d\tau }}-{\displaystyle \sum _{\forall k}{\displaystyle {\int }_{t_k}^{t_k+{\tau }_k}{u}_{i1}^{\prime }(\tau )d\tau }},$$

(20)

$$w_i(t)=x_i(t)-{\displaystyle {\int }_{t_k}^{t_k+{\tau }_k}{u}_{i1}(\tau )d\tau }.$$

(21)

In the aforementioned equation, $u_{i1}^{\prime }$ and $u_{i2}^{\prime }$ represent the $u_{i1}$ and $u_{i2}$ at the time of receiving DOS attack. They may not be the same in form, but they function the same, the details of which are explained later in this section. $w_i(t)$ denotes the auxiliary variable, which is utilized to restore the system to its original state after a Denial-of-Service (DOS) attack. Once the incident terminates, the algorithm switches its state once again to initiate adaptive recovery for any information loss, supply, and demand imbalances, or other negative impacts caused by the DOS attack.

$$\dot{x}=u_{i1}(t)+u_{i2}(t)+v_i(t),$$

(22)

$$u_{i1}=k_0{\displaystyle \sum _{j\in N_i}{a}_{ij}({\eta }_j-{\eta }_i)+k_1}{\displaystyle \sum _{j\in N_i}{a}_{ij}sig(}{\eta }_j-{\eta }_i)+k_2{\displaystyle \sum _{j\in N_i}{a}_{ij}(\mathrm{sgn}(}{\eta }_j{)}^p-\mathrm{sgn}{({\eta }_i)}^p),$$

(23)

$${\eta }_i=\frac{\partial }{\partial x_i}{C}_i(x)+{\displaystyle \sum _{j\in N_i^c}\frac{\partial }{\partial x_i}}{C}_j(x),$$

(24)

$$u_{i2}=-c_0{s}_i-c_{1}sig(s_i)-c_2\mathrm{sgn}{(s_i)}^{\alpha },$$

(25)

$$s_i(t)=x_i(t)-{\displaystyle {\int }_{t_0}^t{u}_{i1}(\tau )d\tau },$$

(26)

$$v_i(t)=-e_0{r}_i-e_{1}sig(r_i)-e_2\mathrm{sgn}{(r_i)}^{\beta },$$

(27)

$$r_i(t)=x_i(t)-w_i(t_k+{\tau }_k)-{\displaystyle {\int }_{t_0}^t{u}_i(t)}.$$

(28)

where $e_0$, $e_1$, $e_2$, $\beta$ is an adjustable constant, $v_i(t)$ is a newly added part of the algorithm in the third part, and $r_i(t)$ is a newly added auxiliary variable in this part. The algorithm is divided into three states.

When not attacked by the network, the first state of the algorithm is executed, and the state is divided into two parts. One part is $u_{i1}$, based on the KKT condition, so that the local gradient reaches consistency in a finite time, so that the objective function can converge to the optimal solution in a finite time, and the sum of $u_{i1}$ and $u_{i2}$ of the node is zero has always been established; that is: ${\displaystyle {\sum }_{\forall i}{u}_{i1}+u_{i2}}=0$. The derivatives of $x_i$ of all nodes are added to zero, so that the initial value meets the balance of supply and demand, which can ensure the global balance of supply and demand. The other part is $u_{i2}$; the purpose of $u_{i2}$ is to cancel out the fluctuations caused by too much inertia caused by the rapid convergence of $u_{i1}$, and also to eliminate the disturbance caused by the external environment. $u_{i2}$ can also achieve finite-time convergence and assist the overall algorithm to achieve finite time smooth convergence. When the network is not subject to DOS attacks, or the network is attacked by DOS, but the energy agent is not a node that is attacked by DOS, we use the information received in real time to calculate and execute the algorithm of the first state.

The second state of the algorithm is switched when attacked by DOS, and its core is also two parts: $u_{i1}^{\prime }$ and $u_{i2}^{\prime }$, which act the same as the first state of the algorithm, except that the $u_{i1}^{\prime }$ of the second state is less than the last item $k_2{\displaystyle {\sum }_{j\in N_i}{a}_{ij}(\mathrm{sgn}{({\eta }_j)}^p-\mathrm{sgn}{({\eta }_i)}^p})$ of the first state $u_{i1}$. Although $u_{i2}^{\prime }$ and $u_{i2}$ may seem different in form, the core methods of both are the same and have the same function. $w_i(t)$ is an auxiliary variable designed for the third state, and its role can be understood as a sliding surface designed to maintain the balance of supply and demand. When subjected to a DOS attack, and the energy agent is a node attacked by DOS, the latest received information is used to calculate ${\eta }_j(t_k)$ and update the control variables.

When the DOS attack ends, the algorithm transitions to a third state, and the third state is the recovery algorithm, in which the control variables are updated with the information received in real time. This part of the algorithm consists of $u_{i1}$, $u_{i2}$, and $v_i(t)$, where $u_{i1}$ and $u_{i2}$ are the same as the first state in action and form. The design of $v_i(t)$ is inspired by sliding film control, using the second part of $w_i(t)$, forming a sliding surface of supply and demand balance, so that the results of the algorithm restore the balance between supply and demand, and the algorithm in the recovery state can achieve finite-time convergence, which can be adaptively adjusted to the first state without DOS attack after the convergence is completed.

By implementing the above algorithm, all energy agents can converge to the optimal state in a finite time.

3.5. Convergence Analysis

In order to test the performance of the algorithm and analyze its convergence. We first introduce the second part of the algorithm $u_{i2}$. $u_{i2}$ can cancel the fluctuation caused by the $u_{i1}$ part and can also eliminate environmental interference and improve the robustness of the system. Fluctuations can be suppressed for a finite time by $u_{i2}$. Using the derivative of (21):

$${\dot{s}}_i(t)={\dot{x}}_i(t)-u_{i1}(t)=u_{i2}=-c_0{s}_i-c_{1}sig(s_i)-c_2\mathrm{sgn}{(s_i)}^{\alpha }.$$

(29)

Construct $V_d(t)={\displaystyle {\sum }_{i=1}^n\frac{1}{2}}{s}_i{}^2$ and derive $V_d(t)$ to obtain

$${\dot{V}}_d(t)={\displaystyle {\sum }_{i=1}^n{s}_i}{\dot{s}}_i=-c_0{\displaystyle {\sum }_{i=1}^n{s}_i{}^2-c_1{\displaystyle {\sum }_{i=1}^n\left|s_i\right|}}-c_2{{\displaystyle {\sum }_{i=1}^n\left|s_i\right|}}^{\alpha +1}.$$

(30)

According to Lemma 1:

$$\begin{gathered} {\dot{V}}_d(t)\le -c_0{{\displaystyle {\sum }_{i=1}^n{s}_i{}^2-c_1\left({\displaystyle {\sum }_{i=1}^n{s}_i{}^2}\right)}}^{\frac{1}{2}}-c_2{n}^{\frac{1-\alpha }{2}}{\left({\displaystyle {\sum }_{i=1}^n{s}_i{}^2}\right)}^{\frac{\alpha +1}{2}}, \\ \le -c_0{V}_d(t)-c_1{V}_d{(t)}^{\frac{1}{2}}-c_2{n}^{\frac{1-\alpha }{2}}{V}_d{(t)}^{\frac{\alpha +1}{2}}. \end{gathered}$$

(31)

Thus, from Lemma 3, the following can be derived: $T_{S1}\le \frac{2\ln (1+c_0/c_1)}{c_0}+\frac{2\ln (1+c_0/c_2{n}^{\frac{\alpha +1}{2}})}{c_0(\alpha -1)}$.

The proof is thus completed. For all nodes $i=1,2,\dots ,n$, $s_i(t)$ will converge to zero by $T_{S1}$.When the node is attacked by DOS attack, the algorithm switches to the second state, but $s_i(t)$ does not participate in communication, and convergence is the same as when it is not attacked by DOS, for all three states of the algorithm, $s_i(t)$ is the same, and the function of $u_{i2}$ is the same.

The role of $v_i(t)$ in the third state of the algorithm is to recover from the impact of DOS attacks, using the derivative of (28).

$$\begin{gathered} {\dot{r}}_i(t)={\dot{x}}_i(t)-u_{i1}(t)-u_{i2}(t)=v_i(t), \\ =-e_0{r}_i-e_{1}sig(r_i)-e_2\mathrm{sgn}{(r_i)}^{\beta }. \end{gathered}$$

(32)

Construct $V_b(t)={\displaystyle {\sum }_{i=1}^n\frac{1}{2}}{r}_i{}^2$ and derive $V_b(t)$ to obtain:

$$\begin{array}{l}{\dot{V}}_b(t)={\displaystyle {\sum }_{i=1}^n{r}_i}{\dot{r}}_i,\\ \hspace{1em}\hspace{1em} =-e_0{\displaystyle {\sum }_{i=1}^n{r}_i{}^2-e_1{\displaystyle {\sum }_{i=1}^n\left|r_i\right|}}-e_2{{\displaystyle {\sum }_{i=1}^n\left|r_i\right|}}^{\beta +1}.\end{array}$$

(33)

According to Lemma 1:

$$\begin{gathered} {\dot{V}}_b(t)\le -e_0{{\displaystyle {\sum }_{i=1}^n{r}_i{}^2-e_1\left({\displaystyle {\sum }_{i=1}^n{r}_i{}^2}\right)}}^{\frac{1}{2}}-e_2{n}^{\frac{1-\beta }{2}}{\left({\displaystyle {\sum }_{i=1}^n{r}_i{}^2}\right)}^{\frac{\beta +1}{2}}, \\ \le -e_0{V}_b(t)-e_1{V}_b{(t)}^{\frac{1}{2}}-e_2{n}^{\frac{1-\beta }{2}}{V}_b{(t)}^{\frac{\beta +1}{2}}. \end{gathered}$$

(34)

Thus, from Lemma 3, the following can be derived: $T_{S2}\le \frac{2\ln (1+e_0/e_1)}{e_0}+\frac{2\ln (1+e_0/(e_2{n}^{\frac{\beta +1}{2}}))}{e_0(\beta -1)}$.

$u_{i2}$ has counteracted the fluctuations caused by part $u_{i1}$, and interference can also be eliminated. The third state of the algorithm $v_i(t)$ recovers from the impact of DOS attacks in a finite time. During the period when not under DOS attack:

$${\dot{x}}_i(t)=k_0{\displaystyle {\sum }_{j\in N_i}{a}_{ij}({\eta }_j-{\eta }_i)+k_1{\displaystyle {\sum }_{j\in N_i}{a}_{ij}sig({\eta }_j-{\eta }_i)+k_2{\displaystyle {\sum }_{j\in N_i}{a}_{ij}\left(\mathrm{sgn}{({\eta }_j)}^p-\mathrm{sgn}{({\eta }_i)}^p\right)}}}.$$

(35)

Let $x^{*}={\left[x_1^{*},x_x^{*},\dots ,x_n^{*}\right]}^T$ be an optimal solution of the convex optimization problem (6) for which $\nabla C(x^{*})=c{1}_n$ for $c\in ℝ$ (see Remark 1). Construct $V_a(t)=C(x)-C(x^{*})$.

From Assumption 1, $V_a(t)$ is positive semi-definite and $V_a(t)$ = 0 if and only if $C(x)=C(x^{*})$. From (13), ${\eta }_i$ is the ith element of $\nabla C(x)$. Therefore, taking the time derivative of $V_a(t)$ yields ${\dot{V}}_a(t)={\displaystyle {\sum }_{i=1}^n{\eta }_i{\dot{x}}_i}$.

During any normal period $\left(t_k+{\tau }_k,t_{k+1}\right]$:

$$\begin{gathered} {\dot{V}}_a(t)=k_0{\displaystyle \sum _{i=1}^n{\eta }_i}{\displaystyle \sum _{j\in N_i}{a}_{ij}({\eta }_j-{\eta }_i)+k_1}{\displaystyle \sum _{i=1}^n{\eta }_i}{\displaystyle \sum _{j\in N_i}{a}_{ij}sig(}{\eta }_j-{\eta }_i) \\ +k_2{\displaystyle \sum _{i=1}^n{\eta }_i}{\displaystyle \sum _{j\in N_i}{a}_{ij}(\mathrm{sgn}(}{\eta }_j{)}^p-\mathrm{sgn}{({\eta }_j)}^p). \end{gathered}$$

(36)

From (37), it follows that:

$$\begin{gathered} {\dot{V}}_a(t)=-\frac{1}{2}{k}_0{\displaystyle \sum _{i,j=1}^n{a}_{ij}{({\eta }_j-{\eta }_i)}^2-\frac{1}{2}}{k}_1{\displaystyle \sum _{i,j=1}^n{a}_{ij}\left|{\eta }_j-{\eta }_i\right|} \\ -\frac{1}{2}{k}_2{\displaystyle \sum _{i,j=1}^n{a}_{ij}(}{\eta }_j-{\eta }_i)(\mathrm{sgn}{({\eta }_j)}^p-\mathrm{sgn}{({\eta }_i)}^p). \end{gathered}$$

(37)

Since $sig({\eta }_j-{\eta }_i)=sig(\mathrm{sgn}{({\eta }_j)}^p-\mathrm{sgn}{({\eta }_i)}^p)$, we can have $({\eta }_j-{\eta }_i)(\mathrm{sgn}{({\eta }_j)}^p-\mathrm{sgn}{({\eta }_i)}^p)$ $=\left|{\eta }_j-{\eta }_i\right|\left|\mathrm{sgn}{({\eta }_j)}^p-\mathrm{sgn}{({\eta }_i)}^p\right|$. Then, by simple calculations, from Lemma 4:

$$\begin{gathered} {\dot{V}}_a(t)\le -\frac{1}{2}{k}_0{\displaystyle \sum _{i,j=1}^n{a}_{ij}{({\eta }_j-{\eta }_i)}^2-\frac{1}{2}}{k}_1{\displaystyle \sum _{i,j=1}^n{a}_{ij}\left|{\eta }_j-{\eta }_i\right|}-2^{-p}{k}_2{{\displaystyle \sum _{i,j=1}^n{a}_{ij}\left|{\eta }_j-{\eta }_i\right|}}^{p+1}, \\ \le -\frac{1}{2}{k}_0{\displaystyle \sum _{i,j=1}^n{a}_{ij}{({\eta }_j-{\eta }_i)}^2-\frac{1}{2}}{k}_1{\displaystyle \sum _{i,j=1}^n(a_{ij}^2}{({\eta }_j-{\eta }_i)}^2{)}^{\frac{1}{2}}-2^{-p}{k}_2{\displaystyle \sum _{i,j=1}^n(a_{ij}^{\frac{2}{p+1}}}{({\eta }_j-{\eta }_i)}^2{)}^{\frac{p+1}{2}}. \end{gathered}$$

(38)

According to Lemma 1,

$$\begin{gathered} {\dot{V}}_a(t)\le -\frac{1}{2}{k}_0{\displaystyle \sum _{i,j=1}^n{a}_{ij}{({\eta }_j-{\eta }_i)}^2}-\frac{1}{2}{k}_1{\left({\displaystyle \sum _{i,j=1}^n{a}_{ij}^2{({\eta }_j-{\eta }_i)}^2}\right)}^{\frac{1}{2}}-2^{-p}{k}_2{n}^{\frac{1-p}{2}}{\left({\displaystyle \sum _{i,j=1}^n{a}_{ij}^{\frac{2}{p+1}}{({\eta }_j-{\eta }_i)}^2}\right)}^{\frac{p+1}{2}}, \\ {\dot{V}}_a(t)\le -\frac{1}{2}{k}_0{\displaystyle \sum _{i,j=1}^n{a}_{ij}{({\eta }_j-{\eta }_i)}^2}-\frac{1}{2}{k}_1{\left({\displaystyle \sum _{i,j=1}^n{a}_{ij}^2{({\eta }_j-{\eta }_i)}^2}\right)}^{\frac{1}{2}}-2^{-p}{k}_2{n}^{\frac{1-p}{2}}{\left({\displaystyle \sum _{i,j=1}^n{a}_{ij}^{\frac{2}{p+1}}{({\eta }_j-{\eta }_i)}^2}\right)}^{\frac{p+1}{2}}, \end{gathered}$$

(39)

let $A_0=\left[a_{ij}\right]$, $A_1=\left[a_{ij}^2\right]$, and $A_2=\left[a_{ij}^{\frac{2}{p+1}}\right]$; therefore, from Lemma 2:

$${\dot{V}}_a(t)\le -\frac{1}{2}{k}_0(2{\eta }^{T}L(A_0)\eta )-\frac{1}{2}{k}_1{(2{\eta }^{T}L(A_1)\eta )}^{\frac{1}{2}}-2^{-p}{k}_2{n}^{\frac{1-p}{2}}{(2{\eta }^{T}L(A_2)\eta )}^{\frac{p+1}{2}},$$

(40)

where $L(A_0)$, $L(A_1)$, and $L(A_2)$ are the Laplacian matrices of undirected graphs $G(A_0)$, $G(A_1)$, and $G(A_2)$, respectively. From Lemma 6:

$$\left\{\begin{array}{c}\frac{1}{2}\theta {\lambda }_2(L(A_0))(C(x)-C(x^{*}))\le {\nabla }^{T}C(x)L(A_0)\nabla C(x),\\ \frac{1}{2}\theta {\lambda }_2(L(A_1))(C(x)-C(x^{*}))\le {\nabla }^{T}C(x)L(A_1)\nabla C(x),\\ \frac{1}{2}\theta {\lambda }_2(L(A_2))(C(x)-C(x^{*}))\le {\nabla }^{T}C(x)L(A_2)\nabla C(x),\end{array}\right.$$

(41)

Since $\eta =\nabla C(x)$, we obtain

$$\begin{gathered} {\dot{V}}_a(t)\le -\frac{1}{2}{k}_0\theta {\lambda }_2(L(A_0))V_a(t)-\frac{1}{2}{k}_1{\theta }^{\frac{1}{2}}{\lambda }_2{(L(A_1))}^{\frac{1}{2}}{V}_c{(t)}^{\frac{1}{2}} \\ -2^{-p}{k}_2{\theta }^{\frac{p+1}{2}}{\lambda }_2{(L(A_2))}^{\frac{p+1}{2}}{n}^{\frac{1-p}{2}}{V}_a{(t)}^{\frac{1+p}{2}}. \end{gathered}$$

(42)

During any attack period $\left(t_k,t_k+{\tau }_{\mathrm{k}}\right]$,

$$V_a^{\prime }(t)=k_0{\displaystyle {\sum }_{i=1}^n{\eta }_i{\displaystyle {\sum }_{j\in N_i}{a}_{ij}({\eta }_j(t_k)-{\eta }_i)}}+k_1{\displaystyle {\sum }_{i=1}^n{\eta }_i{\displaystyle {\sum }_{j\in N_i}{a}_{ij}sig({\eta }_j}(t_k)-{\eta }_i)}.$$

(43)

In period $\left(t_k,t_k+{\tau }_{\mathrm{k}}\right]$, if it is not subject to a DOS attack, the normal communication value under normal conditions is ${\eta }_j$. However, during the DOS attack period, we can only update the control variable with the latest received data ${\eta }_j(t_k)$; hence, we let the difference between the two be $u_j$.

$${\eta }_j(t_k)={\eta }_j+u_j,$$

(44)

from which we can have,

$$\begin{gathered} V_a^{\prime }=k_0{\displaystyle {\sum }_{i=1}^n{\eta }_i{\displaystyle {\sum }_{j\in N_i}{a}_{ij}({\eta }_j-{\eta }_i)+k_0}}{\displaystyle {\sum }_{i=1}^n{\eta }_i}{\displaystyle {\sum }_{j\in N_i}{a}_{ij}{u}_j} \\ +k_1{\displaystyle {\sum }_{i=1}^n{\eta }_i{\displaystyle {\sum }_{j\in N_i}{a}_{ij}sig({\eta }_j+u_j-{\eta }_i)}}, \\ =-\frac{1}{2}{k}_0{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}{({\eta }_j-{\eta }_i)}^2-\frac{1}{2}}{k}_0{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}({\eta }_j-{\eta }_i)u_j} \\ -\frac{1}{2}{k}_1{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}({\eta }_j-{\eta }_i)sig(}{\eta }_j+u_j-{\eta }_i). \end{gathered}$$

(45)

$u_j$ is a function that changes over time, and in the worst case, $u_j$ is an incrementing function and the maximum value of the slope is $k_{\max }$ within the attack time ${\tau }_k$,

$$\begin{gathered} V_a^{\prime }=-\frac{1}{2}{k}_0{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}{({\eta }_j-{\eta }_i)}^2}-\frac{1}{2}{k}_0{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}({\eta }_j-{\eta }_i)}{k}_{\max }\tau \\ -\frac{1}{2}{k}_1{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}({\eta }_j-{\eta }_i)sig({\eta }_j+u_j-{\eta }_i)}, \end{gathered}$$

(46)

in the worst case, $k_{\max }$ is the opposite of ${\eta }_j-{\eta }_i$, and $sig({\eta }_j+u_j-{\eta }_i)$ is the opposite of ${\eta }_j-{\eta }_i$,

$$V_a^{\prime }\le -\frac{1}{2}{k}_0{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}{({\eta }_j-{\eta }_i)}^2}+\frac{1}{2}{k}_0\left|k_{\max }\right|\tau {\displaystyle {\sum }_{i,j=1}^n{a}_{ij}\left|{\eta }_j-{\eta }_i\right|}+\frac{1}{2}{k}_1{\displaystyle {\sum }_{i,j=1}^n{a}_{ij}}\left|{\eta }_j-{\eta }_i\right|.$$

(47)

From Lemma 5 and Lemma 6, we add a limiting factor $m$:

$$V_a^{\prime }\le -\frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)V_a(t)+\frac{1}{2}m\left(k_0\left|k_{\max }\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n{\left(L(A_1)\right)}^{\frac{1}{2}}{V}_a{}^{\frac{1}{2}}(t).$$

(48)

During DOS attacks periods, algorithms cannot converge. Based on the above analysis, we analyze the convergence of the whole.

$$V\left(t\right)=C\left(x\right)-C\left(x^{*}\right).$$

(49)

Let us start with the case at $V\left(t\right)\ge 1$:

$$\begin{gathered} \stackrel{·}{V_a}(t)\le -\frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)V_a\left(t\right)-\frac{1}{2}k{\theta }^{\frac{1}{2}}{\lambda }_2^{\frac{1}{2}}\left(L(A_1)\right)V_a^{\frac{1}{2}}\left(t\right) \\ -2^{-P}{k}_2{\theta }^{\frac{P+1}{2}}{\lambda }_2^{\frac{P+1}{2}}\left(L(A_2)\right)n^{\frac{1-P}{2}}{V}_a^{\frac{P+1}{2}}\left(t\right), \end{gathered}$$

(50)

$$\stackrel{·}{V_a}(t)\le -\frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)V_a\left(t\right),$$

(51)

$$V_a\left(t\right)\le e^{-\frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)}{V}_a\left(t_0\right),$$

(52)

$$\stackrel{·}{V_a^{\prime }}(t)\le \frac{1}{2}m\left(k_0\left|k_{\max }\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L(A_1)\right)V_a^{\frac{1}{2}}\left(t\right),$$

(53)

$$V_a^{\prime }\left(t\right)\le e^{\frac{1}{2}m\left(k_0{k}_{\max }\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L(A_1)\right)t}{V}_a\left(t_0\right),$$

(54)

$$\phi =\frac{{\displaystyle \sum _{k=1}^n{\tau }_k}}{t}.$$

(55)

We combine situations that are subject to DOS attacks with situations that are not subject to DOS attacks:

$$V\left(t\right)\le V_a\left(\left(1-\phi \right)t\right){\mathrm{e}}^{\frac{1}{2}m\left(k_0{k}_{\max }\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_2^{\frac{1}{2}}\left(L(A_0)\right)\phi t}\le V_a\left(\left(1-\phi \right)t\right){\left({\mathrm{e}}^{\frac{1}{2}m\left(k_0{k}_{\max }\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_2^{\frac{1}{2}}\left(L(A_1)\right)\phi t}\right)}^{\frac{\mathrm{P}+1}{2}},$$

(56)

$$\begin{array}{cc}\hfill & \stackrel{\cdot }{V}\left(t\right)\le \left[\left(1-\phi \right)\left(-\frac{1}{2}{k}_0\theta {\lambda }_2\left(L\left(A_0\right)\right)\right)V_a\left(\left(1-\phi \right)t\right)-\frac{1}{2}{k}_1{\theta }^{\frac{1}{2}}{\lambda }_2^{\frac{1}{2}}\left(L\left(A_1\right)\right){V_a}^{\frac{1}{2}}\left(\left(1-\phi \right)t\right)\right.\hfill \\ \hfill & \left.-2^{-P}{k}_2{\theta }^{\frac{P+1}{2}}{\lambda }_2^{\frac{P+1}{2}}\left(L\left(A_2\right)\right)n^{\frac{1-P}{2}}{V_a}^{\frac{P+1}{2}}\left(\left(1-\phi \right)t\right)\right]{\left({\mathrm{e}}^{\frac{1}{2}m\left(k_0{k}_{max}\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)\phi t}\right)}^{\frac{\mathrm{P}+1}{2}}\hfill \\ \hfill & +\frac{P+1}{4}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)\phi V_a\left(\left(1-\phi \right)t\right){\left({\mathrm{e}}^{\frac{1}{2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)\phi t}\right)}^{\frac{\mathrm{P}+1}{2}}.\hfill \end{array}$$

(57)

Let $\frac{p+1}{4}m\left(k_0\left|k_{\max }\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L(A_1)\right)\phi \le \alpha \frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)$, $\alpha <\left(1-\phi \right)$,

$$\begin{array}{cc}\hfill & \stackrel{\cdot }{V}\left(t\right)\le \left(1-\phi -\alpha \right)\left(-\frac{1}{2}{k}_0\theta {\lambda }_2\left(L\left(A_0\right)\right)\right)V_a\left(\left(1-\phi \right)t\right){\left({\mathrm{e}}^{\frac{1}{2}m\left(k_0{k}_{max}\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)\phi t}\right)}^{\frac{\mathrm{P}+1}{2}}\hfill \\ \hfill & -\frac{1}{2}{k}_1{\theta }^{\frac{1}{2}}{\lambda }_{{}_2}^{\frac{1}{2}}\left(L\left(A_2\right)\right){V_a}^{\frac{1}{2}}\left(\left(1-\phi \right)t\right){\left({\mathrm{e}}^{\frac{1}{2}m\left(k_0{k}_{max}\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)\phi t}\right)}^{\frac{\mathrm{P}+1}{2}}\hfill \\ \hfill & -2^{-P}{k}_2{\theta }^{\frac{P+1}{2}}{\lambda }_2^{\frac{P+1}{2}}\left(L\left(A_2\right)\right)n^{\frac{1-P}{2}}{V_a}^{\frac{P+1}{2}}\left(\left(1-\phi \right)t\right){\left({\mathrm{e}}^{\frac{1}{2}m\left(k_0{k}_{max}\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)\phi t}\right)}^{\frac{\mathrm{P}+1}{2}},\hfill \end{array}$$

(58)

$$\begin{array}{l}\dot{V}(t)\le \left(1-\phi -\alpha \right)\left(-\frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)\right)V\left(t\right)\\ -\frac{1}{2}{k}_1{\theta }^{\frac{1}{2}}{\lambda }_{{}_2}^{\frac{1}{2}}\left(L(A_1)\right)V^{\frac{1}{2}}\left(t\right){\left({\mathrm{e}}^{\frac{1}{2}m\left(k_0{k}_{\max }\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L(A_1)\right)\phi t}\right)}^{\frac{\mathrm{P}}{2}}\\ -{\left(\frac{1}{2}\right)}^P{k}_2{e}^{\frac{P+1}{2}}{\lambda }_2^{\frac{p+1}{2}}\left(L\left(A_2\right)\right)n^{\frac{1-P}{2}}{V}^{\frac{P+1}{2}}\left(t\right).\end{array}$$

(59)

Because ${\mathrm{e}}^{\frac{1}{2}m\left(k_0{k}_{\max }\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L(A_1)\right)\phi t}>1$, we can obtain

$$\begin{array}{l}\stackrel{·}{V}(t)\le \left(1-\phi -\alpha \right)\left(-\frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)\right)V\left(t\right)-\frac{1}{2}{k}_1{\theta }^{\frac{1}{2}}{\lambda }_{{}_2}^{\frac{1}{2}}\left(L(A_1)\right)V^{\frac{1}{2}}\left(t\right)\\ -2^{-P}{k}_2{\theta }^{\frac{P+1}{2}}{\lambda }_2^{\frac{P+1}{2}}(L(A_2))n^{\frac{1-P}{2}}{V}^{\frac{P+1}{2}}\left(t\right).\end{array}$$

(60)

This form satisfies the basic form of Lemma 3, from which it can be concluded that when $V(t)>1$, $V(t)$ converges to $V(t)=1$ by ${\widehat{t}}_1$.

$${\widehat{t}}_1\le \frac{2\ln (1+I_1/I_2)}{I_1(p-1)}.$$

(61)

where $I_1=\frac{(1-\phi -\alpha )k_0\theta {\lambda }_2\left(L\left(A_0\right)\right)}{2}$, $I_2=2^{-p}{k}_2{\theta }^{\frac{p+1}{2}}{\lambda }_2^{\frac{p+1}{2}}\left(L\left(A_2\right)\right)n^{\frac{1-p}{2}}$.

When $0\le V(t)\le 1$, under the situation as that without a DOS attack,

$$\stackrel{·}{V_a}(t)\le -\frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)V_a\left(t\right)-\frac{1}{2}{k}_1{\theta }^{\frac{1}{2}}{\lambda }_2^{\frac{1}{2}}\left(L(A_1)\right)V_a^{\frac{1}{2}}\left(t\right).$$

(62)

We use $a$ for $\frac{1}{2}{k}_0\theta {\lambda }_2\left(L(A_0)\right)$ and $b$ for $\frac{1}{2}{k}_1{\theta }^{\frac{1}{2}}{\lambda }_2^{\frac{1}{2}}\left(L(A_1)\right)$. Next, we use the reciprocal to find the original function. The condition $0<V_a\left(t\right)\le 1$ must be met,

$$\stackrel{·}{V_a}(t)\le -a{V}_a\left(t\right)-b{V}_a{}^{\frac{1}{2}}\left(t\right),$$

(63)

$$\frac{d{V}_a}{dt}<-a{V}_a-b{V}_a{}^{\frac{1}{2}},$$

(64)

$$\frac{dt}{d{V}_a}>\frac{-1}{a{V}_a+b{V}_a{}^{{}^{\frac{1}{2}}}},$$

(65)

$$dt\le \frac{-d{V}_a}{a{V}_a+b{V}_a{}^{{}^{\frac{1}{2}}}},$$

(66)

then, we let $h=\sqrt{V_a}$, and $V_a=h^2$ is the same,

$$d{V}_a=2h\cdot dh,$$

(67)

$$dt\le \frac{-2hdh}{a{h}^2+bh},$$

(68)

$$dt\le \frac{-2}{ah+b}dh,$$

(69)

$${\displaystyle \int dt\le {\displaystyle \int \frac{-2}{ah+b}}}dh,$$

(70)

$$t+c\le \frac{-2}{a}\ln \left(ah+b\right),$$

(71)

$$-\frac{a}{2}\left(t+c\right)\ge \ln \left(ah+b\right),$$

(72)

$$ah+b\le e^{-\frac{a}{2}\left(t+c\right)},$$

(73)

$$ah\le e^{-\frac{a}{2}\left(t+c\right)}-b,$$

(74)

$$h\le \frac{1}{a}{e}^{-\frac{a}{2}\left(t+c\right)}-\frac{b}{a},0\le h<1,$$

(75)

where $c$ is constant. If $t\to +\infty$, $h\to -b/a$. Moreover, $h$ is a monotonically decreasing function, so there must be one and only one value of $t$ which can satisfy the function with a value of zero, and the meaningful range is also $0\le h\le 1$,

$$V_a(t)\le h^2=\frac{1}{a^2}{e}^{-at}V\left(t_0\right)-\frac{2b}{a^2}{e}^{\frac{-a}{2}t}{V}^{\frac{1}{2}}\left(t_0\right)+\frac{b^2}{a^2},0\le V_a(t)\le 1.$$

(76)

Combine the period of attack and the normal period to form the following formula:

$$V\left(t\right)\le \frac{1}{a^2}{V}_a^{\prime }\left(\phi t\right)e^{-a\left(1-\phi \right)t}-\frac{2b}{a^2}{V}_a{{}^{\prime }}^{\frac{1}{2}}\left(\phi t\right)e^{\frac{{}^{-a\left(1-\phi \right)t}}{2}}+\frac{b^2}{a^2},$$

(77)

Because of $0\le V_a^{\prime }\le 1$, we obtain $V_a{{}^{\prime }}^{\frac{1}{2}}(\phi t)\ge V_a^{\prime }(\phi t)$,

$$V\left(t\right)\le \frac{1}{a^2}{V}_a^{\prime }\left(\phi t\right)e^{-a\left(1-\phi \right)t}-\frac{2b}{a^2}{V}_a^{\prime }\left(\phi t\right)e^{\frac{{}^{-a\left(1-\phi \right)t}}{2}}+\frac{b^2}{a^2},$$

(78)

$$\begin{array}{cc}\hfill & \stackrel{\cdot }{V}\left(t\right)<\frac{1}{a^2}\phi \left(-\frac{1}{2}{k}_0\theta {\lambda }_2\left(L\left(A_0\right)\right){V_a}^{\prime }\left(\phi t\right)+\frac{1}{2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)V_a{{}^{\prime }}^{\frac{1}{2}}\left(t\right)\right)e^{-a\left(1-\phi \right)t}\hfill \\ \hfill & -\frac{\left(1-\phi \right)}{a}{V_a}^{\prime }\left(\phi t\right)e^{-a\left(1-\phi \right)t}+\left(1-\phi \right)\frac{b}{a}{e}^{\frac{{}^{-at\left(1-\phi \right)}}{2}}{V_a}^{\prime }\left(\phi t\right)\hfill \\ \hfill & -\phi \frac{2b}{a^2}{e}^{-\frac{at}{2}}\left(-\frac{1}{2}{k}_0\theta {\lambda }_2\left(L\left(A_0\right)\right){V_a}^{\prime }\left(\phi t\right)+\frac{1}{2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)V_a{{}^{\prime }}^{\frac{1}{2}}\left(t\right)\right),\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill & \stackrel{\cdot }{V}\left(t\right)\le -\frac{\phi k_0\theta {\lambda }_2\left(L\left(A_1\right)\right)}{2a^2}{V_a}^{\prime }\left(\phi t\right)e^{-a\left(1-\phi \right)t}+\frac{\phi }{2a^2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)V_a{{}^{\prime }}^{\frac{1}{2}}\left(t\right)e^{-a\left(1-\phi \right)t}\hfill \\ \hfill & -\frac{\left(1-\phi \right)}{a}{V_a}^{\prime }\left(\phi t\right)e^{-a\left(1-\phi \right)t}+\frac{\left(1-\phi \right)b}{a}{e}^{-\frac{a\left(1-\phi \right)t}{2}}{V_a}^{\prime }\left(\phi t\right)+\frac{b\phi }{a^2}{k}_0\theta {\lambda }_2\left(L\left(A_0\right)\right)e^{-\frac{a\left(1-\phi \right)t}{2}}{V_a}^{\prime }\left(\phi t\right)\hfill \\ \hfill & -\frac{b\phi }{a^2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)V_a{{}^{\prime }}^{\frac{1}{2}}\left(t\right)e^{-\frac{a\left(1-\phi \right)t}{2}},\hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill & \stackrel{\cdot }{V}\left(t\right)\le -\left(\frac{\phi k_0\theta {\lambda }_2\left(L\left(A_0\right)\right)}{2a^2}+\frac{\left(1-\phi \right)}{a}\right){V_a}^{\prime }\left(\phi t\right)e^{-a\left(1-\phi \right)t}\hfill \\ \hfill & +\frac{\phi }{2a^2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)V_a{{}^{\prime }}^{\frac{1}{2}}\left(t\right)e^{-a\left(1-\phi \right)t}\hfill \\ \hfill & +\frac{\left(1-\phi \right)b}{a}{e}^{-\frac{a\left(1-\phi \right)t}{2}}{V_a}^{\prime }\left(\phi t\right)+\frac{b\phi }{a^2}{k}_0\theta {\lambda }_2\left(L\left(A_1\right)\right)e^{-\frac{a\left(1-\phi \right)t}{2}}{V_a}^{\prime }\left(\phi t\right)\hfill \\ \hfill & -\frac{b\phi }{a^2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)V_a{{}^{\prime }}^{\frac{1}{2}}{\left(t\right)}^{-\frac{a\left(1-\phi \right)t}{2}},\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill & \stackrel{\cdot }{V}\left(t\right)\le -\left(\frac{\phi k_0\theta {\lambda }_2\left(L\left(A_0\right)\right)}{2a^2}+\frac{\left(1-\phi \right)}{a}\right){V_a}^{\prime }\left(\phi t\right)e^{-at}\hfill \\ \hfill & +\frac{\phi }{2a^2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)V_a{{}^{\prime }}^{\frac{1}{2}}\left(t\right)e^{-\frac{at}{2}}+\frac{\left(1-\phi \right)b}{a}{e}^{-\frac{at}{2}}{V}_a{{}^{\prime }}^{\frac{1}{2}}\left(\phi t\right)\hfill \\ \hfill & +\frac{b\phi }{a^2}{k}_0\theta {\lambda }_2\left(L\left(A_0\right)\right)e^{-\frac{at}{2}}{V}_a{{}^{\prime }}^{\frac{1}{2}}\left(\phi t\right)-\frac{b\phi }{a^2}m\left(k_0\left|k_{max}\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)V_a{{}^{\prime }}^{\frac{1}{2}}{\left(t\right)}^{-\frac{a\left(1-\phi \right)t}{2}}.\hfill \end{array}$$

(82)

As for $V\left(t\right)=\frac{1}{a^2}{V}_a^{\prime }\left(\phi t\right)e^{-a(1-\phi )t}-\frac{2b}{a^2}{V}_a{{}^{\prime }}^{\frac{1}{2}}\left(\phi t\right)e^{\frac{-a(1-\phi )t}{2}}+\frac{b^2}{a^2}$, in this situation, we consider the worst-case scenario: $V_a^{\prime }\left(\phi t\right)=1$. When $t=-\frac{1}{a(1-\phi )}\ln b^2$, $V\left(t\right)=0$, and $-\frac{2b}{a^2}{e}^{\frac{-a}{2}t}+\frac{b^2}{a^2}$ is monotonically increasing with time; so, within the feasible range, $-\frac{2b}{a^2}{e}^{\frac{-a}{2}t}{V}_a^{\prime }\left(\phi \mathrm{t}\right)+\frac{b^2}{a^2}<0$ is always satisfied. Then, we have $\frac{V_a^{\prime }\left(\phi t\right)}{a^2}{e}^{-at}>V\left(t\right)$, and $\frac{1}{a}{e}^{-\frac{a}{2}t}{V}_a{{}^{\prime }}^{\frac{1}{2}}\left(\phi t\right)e^{-\frac{a}{2}t}>V^{\frac{1}{2}}\left(t\right)$.

From this, we obtain,

$$\begin{array}{l}\stackrel{·}{V}(t)\le -\left(\frac{\phi k_0\theta {\lambda }_2\left(L(A_0)\right)}{2a^2}+\frac{\left(1-\phi \right)}{a}\right)a^{2}V\left(t\right)\\ -\left(\frac{b\phi m\left(k_0\left|k_{\max }\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)}{a^2}-\frac{b\phi k_0\theta {\lambda }_2\left(L(A_0)\right)}{a^2}-\frac{b\left(1-\phi \right)}{a}\right.\\ \left.-\frac{\phi m\left(k_0\left|k_{\max }\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)}{2a^2}\right)V_a{{}^{\prime }}^{\frac{1}{2}}\left(\phi t\right)e^{\frac{{}^{-at}}{2}}.\end{array}$$

(83)

Further conversion and merging:

$$\begin{array}{l}\stackrel{·}{V}(t)\le -\left(\frac{\phi k_0\theta {\lambda }_2\left(L(A_0)\right)}{2}+a\left(1-\phi \right)\right)V\left(t\right)\\ -\left(\frac{b\phi m\left(k_0\left|k_{\max }\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)}{2a}-\frac{b\phi k_0\theta {\lambda }_2\left(L\left(A_0\right)\right)}{a}-b\left(1-\phi \right)\right)V^{\frac{1}{2}}\left(t\right).\end{array}$$

(84)

We let

$$\frac{\phi k_0\theta {\lambda }_2\left(L(A_0)\right)}{2a^2}+\frac{\left(1-\phi \right)}{a}=I_3,$$

(85)

$$\frac{b\phi m\left(k_0\left|k_{\max }\right|\tau +k_1\right){\theta }^{\frac{1}{2}}{\lambda }_n^{\frac{1}{2}}\left(L\left(A_1\right)\right)}{2a}-\frac{b\phi k_0\theta {\lambda }_2\left(L\left(A_0\right)\right)}{a}-b\left(1-\phi \right)=I_4.$$

(86)

Both $I_3$ and $I_4$ are greater than zero. We can convert the final form to

$$\stackrel{·}{V}(t)\le -I_{3}V(t)-I_4{V}^{\frac{1}{2}}(t).$$

(87)

From [13], when $0\le V(t)\le 1$, $u_{i1}$ can converge to zero within ${\widehat{t}}_2\le \frac{2\ln (1+I_3/I_4)}{I_3}$.

Theorem 1.

Obtained from the above proof. When subjected to a DOS attack, using the energy management strategy proposed in this paper can achieve finite-time convergence. The  $u_{i1}$ can complete convergence by  $T_{S3}\le {\widehat{t}}_1+{\widehat{t}}_2$. Combined with (33) and (36), when subjected to DOS attacks, the distributed optimization strategy can be optimized within $T\le T_{S1}+T_{S2}+T_{S3}$.

4. Simulation Test

This section presents two cases to confirm the efficacy of the strategy. Firstly, the ability of the six-EA system to resolve the issue within a finite time frame is examined under DOS attacks. The six-EA system is a smart grid consisting of six EAs. EAs can exchange information between one another through communication lines and can also exchange energy through power lines. Each EA can perform distributed calculations, execute the algorithm proposed in this paper according to the given parameters set in advance, and then calculate the optimal economic dispatch strategy by constantly exchanging real-time information. When subjected to a DOS attack, the DOS-attacked EA executes the second part of the algorithm, and after the DOS attack ends, the EA executes the third part of the algorithm and then adaptively adjusts to the first part of the algorithm. An EA that is not attacked by DOS always executes the first part of the algorithm. Figure 1 displays the physical topology and communication topology of the six-EA system. In Figure 1, the green dotted line represents the communication line, the blue implementation represents the power line, and each black dotted box represents an EA, which considers distributed power generator, load, and distributed energy storage in this case study. Each EA includes different devices. Under normal conditions, the connection relationship of the six nodes can be observed in Figure 2a. When a DOS attack occurs, the communication link would be impacted, and thus, it is presumed that the attacked link cannot communicate. The DOS attack time sequence is shown in Figure 3. During a DOS attack, the communication link is illustrated in Figure 2b.

The initial values for these six EAs are set to [4.2; −3.1; 2.5; −3.5; 1.6; −1.7]; we set $[k_0,k_1,k_2,p,c_0,c_1,c_2,\alpha ,e_0,e_1,e_2,\beta ]$ = $[5,5,0.1,1.5,2,1,1,1.5,1.1,2,0.4,1.5]$. The coupling coefficients between the device for each EA and the parameters of the cost function are shown in

Table 3. The parameters of the cost function of each EA.

	${\mathit{a}}_{\mathit{i}}$	${\mathit{b}}_{\mathit{i}}$	${\mathit{c}}_{\mathit{i}}$	${\mathit{\gamma }}_{\mathbf{1}}^{\mathit{i}}$	${\mathit{\gamma }}_{\mathbf{2}}^{\mathit{i}}$	${\mathit{\gamma }}_{\mathbf{3}}^{\mathit{i}}$	${\mathit{\gamma }}_{\mathbf{4}}^{\mathit{i}}$	${\mathit{\gamma }}_{\mathbf{5}}^{\mathit{i}}$	${\mathit{\gamma }}_{\mathbf{6}}^{\mathit{i}}$
$E{A}_1$	0.041	7.88	170	0	0.09	0.08	0.12	0	0
$E{A}_2$	0.0431	7.85	120	0.2	0	0.19	0.12	0.21	0
$E{A}_3$	0.052	7.82	160	0.08	0.23	0	0.31	0	0.13
$E{A}_4$	0.0421	7.8	130	0.02	0.11	0.15	0	0.24	0.17
$E{A}_5$	0.055	7.62	90	0	0.03	0	0.07	0	0.13
$E{A}_6$	0.049	7.82	110	0	0	0.13	0.17	0.21	0

.

4.1. Convergence and Optimality Analysis

Based on the above information, it can be calculated that within time $t\le 2$ s, the algorithm will complete convergence. As shown in Figure 4a, $x_i$ represents the generation or consumption of electrical energy for each energy agent. As can be seen from the Figure 4a, $x_i$ has converged after a short fluctuation, and the convergence time is less than 2 s. We change the initial value 10 times, and the final result converges to the same result when $t\le 2$ s. The $x_i$ of the ten-EA system is shown in Figure 4b, from which it can be seen that at time $t\le 3$ s, the strategy successfully resists DOS attacks and completes convergence.

We use a centralized method to solve this problem for EDP under DOS-free attacks. The centralized method calculated that the final minimum cost of the six-EA system was 779.4. If the strategy proposed in this paper is used to solve EDP under DOS attack, the lowest cost is calculated to be 782, and the gap is less than 0.4%, which is within the allowable range. We use the converged $x^{*}$ to calculate $\nabla C(x)$, and finally we obtain $\nabla C(x^{*})=\left[8.3830;8.3830;8.3830;8.3830;8.3830;8.3830\right]$, which satisfies Remark 1. The optimality of the proposed strategy is verified.

4.2. Comparative Analysis

We compare the use of this strategy and the absence of this strategy under DOS attacks. Without this strategy, we believe that only the first part of this algorithm is executed, and the protocol cannot be changed when subjected to DOS attacks. The results of the six-EA system and the ten-EA system without this strategy are shown in Figure 5a,b, and the lowest cost shown in

Table 4. Lowest cost for different strategies.

6-EA System	Lowest Cost	10-EA System	Lowest Cost
The centralized strategy	779.4	The centralized strategy	1325.5
The distributed strategy proposed in this paper under DOS attack	782	The distributed strategy proposed in this paper under DOS attack	1329.4
Normal distributed strategy under DOS attacks	803.3	Normal distributed strategy under DOS attacks	1382.7

.

As can be seen from Figure 5a, if the strategy is not used, the six-EA system becomes unstable when the EAs are attacked by DOS. After the DOS attack ends, $x_i$ will experience fluctuations for a period of time to gradually stabilize. By comparing Table 4, it can be seen that the final convergence value of $x_i$ is not the optimal solution. After a DOS attack, the cost nadir shifted, which is caused by information loss and supply and demand imbalance caused by DOS attacks.

It can also be seen from Figure 5b that if the algorithm proposed in this paper is not implemented, the ten-EA system will be very unstable during DOS attacks, and the value of $x_i$ will produce a large runout, which will seriously affect the stability of the system and the balance of supply and demand. After the DOS attack is over, the value of $x_i$ cannot tend to be stable, which seriously affects the convergence of the algorithm and the optimality of the result.

5. Conclusions

This paper investigates the distributed economic dispatch problem for smart grids in the presence of Denial-of-Service (DOS) attacks. A novel energy management strategy is proposed, which not only resists DOS attacks but also recovers lost information when there is an interruption in the communication link. This approach exhibits strong robustness and adaptability against DOS attacks and deploys the Lyapunov method for achieving optimal energy management within a finite time. The integral sliding mode technique is applied to eliminate disturbances caused by external factors. The effectiveness of the proposed method is verified by Lyapunov analysis, and the convergence time of the algorithm under DOS attack is analyzed—which is related with the length of DOS attack period—and Theorem 1 is obtained. Lastly, simulation results are used to validate the effectiveness and efficiency of the new approach. In the future, we hope to extend this approach to integrated energy systems to solve more complex energy management problems.