A distributed algorithm for aggregative games of disturbed Euler-Lagrange systems

Abstract

An aggregative game of disturbed Euler-Lagrange systems is studied in this paper. The cost function of each player depends on its own decision and the aggregate of all decisions. Different from the well-known aggregative games, the second-order nonlinear dynamic of every player is considered in our problem, and every player is influenced by exogenous disturbances. To seek the Nash equilibrium, a distributed algorithm is developed via state feedback, gradient descent, and internal model. The convergence of the algorithm is analyzed with the help of variational analysis and Lyapunov stability theory. It shows that the Euler-Lagrange players with the proposed algorithm, can asymptotically converge to the Nash equilibrium, even though exogenous disturbances have impact on the behaviors of the players. Finally, a numerical simulation is given to illustrate the effectiveness of our algorithm.

Introduction

Game problems have been widely investigated in economics, ¹ environmental studies, ² communication networks, ³ smart grids, ⁴ and military defense. ⁵ In recent years, Nash equilibrium (NE) seeking problems have attracted considerable attention, and some distributed algorithms for game problem have been developed (see Maojiao and Guoqiang, ⁶ Yi and Pavel, ⁷ Deng, ⁸ and Ma et al.^9,10).

In aggregative games, each player has its cost function which depends on its decision and the aggregate of all players’ decisions. Our objective is to seek the NE of the aggregative game. To this end, many distributed algorithms have been investigated. For example, the authors proposed distributed algorithms for price-based demand response games based on consensus techniques in Maojiao and Guoqiang ⁶ and Deng, ⁸ studied the generalized Nash equilibrium seeking algorithm for nonsmooth aggregative games. In Koshal et al., ¹¹ distributed algorithms for aggregative games were exploited where the players’ objectives are coupled through a more general form of the aggregate function. Various constraints from environment are omnipresent in many applications (see Paccagnan et al. ¹² and Grammatico ¹³ ). For this reason, aggregative games with different constraints attracted much attention. Paccagnan et al. ¹⁴ presented a distributed algorithm for game problems with linear inequality coupling constraints. Liang et al. ¹⁵ exploited distributed algorithms for aggregative games with coupled constraints and nonlinear aggregates. Besides, Wang et al. ¹⁶ studied the energy-optimized games for wireless sensor networks. In practice, the players in the game may have their own physical dynamics such as electricity market ¹⁷ and energy resources. ¹⁸ However, the aforementioned works do not involve physical systems.

With the development of cyber-physical system, distributed strategies which consider the dynamics of physical systems have been extensively studied. Deng ¹⁹ investigated distributed algorithms for second-order multi-agent systems to solve resource allocation problems, and Wang et al. ²⁰ studied nonsmooth convex optimization problems of second-order multiagent systems. Deng and Hong ²¹ and Zhang et al. ²² studied distributed optimization problems of EL systems. However, few results about aggregative games for disturbed EL systems have been investigated. How to design distributed algorithms for seeking NE with EL systems is more attractive and realistic, because the EL systems can accommodate many multi-agent systems such as optimal network performance games in unmanned aerial vehicle (UAV) networks. ²³ Most existing distributed algorithms for seeking Nash equilibrium, such as Maojiao and Guoqiang, ⁶ Deng, ⁸ and Liang et al. ¹⁵ cannot be used to solve the problem directly, due to the EL dynamics and exogenous disturbances.

In practice, different kind of disturbances, from environment, measurement, or communication (see Lin and Ren ²⁴ and Wang et al. ²⁵ ) may affect the performance of the physical systems. Distributed optimization for a class of nonlinear multi-agent systems with disturbance rejection was studied in Wang et al. ²⁵ and Deng et al. ²⁶ investigated event-triggered distributed optimization for disturbed multi-agent systems. Deng and Nian ²⁷ studied aggregative games with disturbed systems over digraphs, and Zou et al. ²⁸ investigated the higher-order dynamics with stochastic disturbances. On the other hand, Internal Model is an effective technique to solve different type of disturbances (refer to Wang et al., ²⁵ Davison, ²⁹ and Huang ³⁰ for details).

In this paper, we investigate the aggregative game problem of nonlinear EL systems with exogenous disturbances. A distributed algorithm is designed for the EL systems to seek the Nash equilibrium. The contributions are listed as follows:

• (i) This paper studies the aggregative games of disturbed EL systems and investigates how the EL players autonomously seek the NE, even though they are influenced by exogenous disturbances. Our problem is an extension of the NE seeking problem in Maojiao and Guoqiang, ⁶ Deng and Hong, ²¹ Zhang et al., ²² and Deng and Nian ³¹ by considering EL systems and/or exogenous disturbances. Owing to the dynamics of EL systems and exogenous disturbances, most existing distributed algorithms for seeking Nash equilibrium of aggregative games, such as Maojiao and Guoqiang, ⁶ Deng, ⁸ and Liang et al. ¹⁵ cannot be applied to our problem directly.

• (ii) Based on state feedback and gradient descent, we design a distributed Nash equilibrium seeking algorithm, in which internal model principle is utilized to reject exogenous disturbances. The convergence of our algorithm is analyzed by virtue of Lyapunov stability theory and variational analysis. We prove that EL players with the proposed algorithm can asymptotically converge to the Nash equilibrium of the aggregative game, though the EL systems are influenced by exogenous disturbances. Moreover, by the proposed algorithm, all players do not exchange their decision variables and local cost functions, which protects the privacy of every player.

The remainder of this paper is as follows. The preliminaries and mathematical formulation are given. Then, we designed a distributed algorithm for the EL systems to seek the Nash equilibrium of the aggregative game and provided the convergence analysis. Finally, an examples of smart grids is provided to verify the effectiveness of the proposed algorithm, and the conclusion is given.

Notation: ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space. $\mathbb{R}$ is the set of all positive real numbers. ⊗ represents the Kronecker product. $x_i$ is the $i$ th element of vector $x$ , and $x^T$ is the transpose of $x$ . $I_n$ is $n$ -dimensional identity matrix.

Preliminaries and formulation

In this section, we introduce some concepts on graph theory and convex analysis. Then, the problem of this paper is formulated.

Preliminaries

We briefly introduce some concepts on graph theory (referring to Mesbahi and Egerstedt ³² ). A weighted undirected graph $\mathcal{G}:=\{\mathcal{V},\mathcal{E},\mathcal{A}\}$ is considered in our problem. $\mathcal{V}=\{1,...,N\}$ is the node set, and $\mathcal{E}\in \mathcal{V}\times \mathcal{V}$ is the edge set, respectively. $\mathcal{A}:=[a_{\mathrm{ij}}{]}_{N\times N}$ is the adjacency matrix where $a_{\mathrm{ij}}$ is the weighting of $(i,j)$ . If $j$ is a neighbor of $i$ , nodes $(i,j)\in \mathcal{E}$ is an edge of $\mathcal{G}$ . $a_{\mathrm{ij}}>0$ if $(i,j)\in \mathcal{E}$ , otherwise $a_{\mathrm{ij}}=0$ . Moreover $a_{\mathrm{ii}}=0$ for all $i\in \mathcal{V}$ which means player $i$ does not connect with itself. The degree of node $i$ is $\mathrm{de}{g}_i=\sum _{j=1}^N{a}_{\mathrm{ij}}$ . The Laplacian matrix of $\mathcal{G}$ is $L=\mathcal{D}-\mathcal{A}$ with $\mathcal{D}=\mathrm{diag}\{\mathrm{de}{g}_1,...,\mathrm{de}{g}_N\}$ . if $(i,j)\in \mathcal{E}$ , and $(j,i)\in \mathcal{E}$ for every pair of $(i,j)$ , $\mathcal{G}$ is an undirected graph. The eigenvalues of $L$ are denoted by ${\lambda }_1,\cdots ,{\lambda }_N$ , and ${\lambda }_i\le {\lambda }_j,(\forall i\le j)$ . If there exists a path for every pair of vertices in $\mathcal{V}$ , graph $\mathcal{G}$ is connected.

The following definitions are given in Rockafellar and Wets ³³ which are well-known in convex analysis.

$$\mathrm{A}\mathrm{function}f:{\mathbb{R}}^m\to \mathbb{R}\mathrm{is}\mathrm{convex}\mathrm{on}{\mathbb{R}}^{\mathrm{m}}\mathrm{if}$$

$$\begin{array}{c}f(\alpha x+(1-\alpha )y)\le \alpha f(x)+(1-\alpha )f(y),\hfill \\ \multicolumn{1}{c}{\forall x,y\in {\mathbb{R}}^m,\forall \alpha \in [0,1].}\end{array}$$

(1)

A function $f:{\mathbb{R}}^m\to {\mathbb{R}}^m$ is $\omega$ -strongly monotone ( $\omega >0$ ) on ${\mathbb{R}}^m$ if

$$(x-y{)}^T(f(x)-f(y))\ge \omega \parallel x-y{\parallel }^2,\forall x,y\in {\mathbb{R}}^m.$$

(2)

A function $f:{\mathbb{R}}^m\to {\mathbb{R}}^m$ is $\theta$ -Lipschitz $(\theta >0)$ on ${\mathbb{R}}^m$ if

$$\parallel f(x)-f(y)\parallel \le \theta \parallel x-y\parallel ,\forall x,y\in {\mathbb{R}}^m.$$

(3)

Problem formulation

In this paper, an aggregative game with $N$ players over undirected graph $\mathcal{G}$ is considered. Player $i\in \mathcal{V}$ has a cost function $J_i(q_i,q_{-i}):{\mathbb{R}}^{\mathrm{Nm}}\to \mathbb{R}$ , where $q_i\in {\mathbb{R}}^m$ is the decision variable of player $i$ and $q_{-i}=\mathrm{col}(q_1^T,...,q_{i-1}^T,q_{i+1}^T,...,q_N^T{)}^T$ . The cost function can be rewritten as ${\vartheta }_i(q_i,\sigma (q))$ , $\forall i\in \mathcal{V}$ , where the aggregate $\sigma (q):=\frac{1}{N}\sum _{i=1}^N{\phi }_i(q_i)$ , and ${\phi }_i(q_i):{\mathbb{R}}^m\to {\mathbb{R}}^n$ is a linear function. The objective of player $i$ is to minimize its cost function $J_i(q_i,q_{-i})$ by adjusting its decision variable $q_i$ . Strictly speaking, player $i$ is faced with the problem:

$$\underset{q_i\in {\mathbb{R}}^m}{min}{J}_i(q_i,q_{-i}).$$

(4)

By the following definition, the Nash equilibrium of game (4) is illustrated (see Maojiao and Guoqiang, ⁶ Paccagnan et al., ¹⁴ and Liang et al. ¹⁵ ).

• Definition 1. $q^{*}:=\mathrm{col}(q_1^{*T},...,q_N^{*T}{)}^T$ is a Nash equilibrium of the aggregative game (4) if $J_i(q_i^{*},q_{-i}^{*})\le J_i(q_i,q_{-i}^{*}),\forall q_i\in {\mathbb{R}}^m,i\in \mathcal{V}$ .

The above definition implies that player $i$ cannot reduce its objective function by changing its own decision unilaterally when they converge to Nash equilibrium.

Define the following mapping for subsequent analysis:

$$F_i(q_i,\sigma (q)):={\nabla }_{q_i}{J}_i(q_i,q_{-i}),$$

(5a)

$$G_i(q_i,y_i):={F_i(q_i,\sigma (q))|}_{\sigma (q)=y_i},$$

(5b)

$${\mathrm{\Phi }}_{\epsilon }(q,y):=\left[\begin{array}{c}G(q,y)\hfill \\ \multicolumn{1}{c}{\epsilon (y-\phi (q))}\end{array}\right],$$

(5c)

where $G(q,y)=\mathrm{col}(G_1^T(q_1,y_1),\cdots ,G_N^T(q_N,y_N){)}^T$ , $y=\mathrm{col}(y_1^T,\cdots ,y_N^T{)}^T\in {\mathbb{R}}^{\mathrm{Nn}}$ , $\phi (q)=\mathrm{col}({\phi }_1^T(q_1),\cdots ,{\phi }_N^T(q_N){)}^T$ , and $\epsilon >0$ . Here $G_i(q_i,y_i)$ is the estimation of the gradient of $J_i(q_i,q_{-i})$ in $q_i$ , since $y_i$ is the estimation of the aggregate $\sigma (q)$ . $G_i(q_i,y_i)=F_i(q_i,\sigma (q))$ when $y_i=\sigma (q)$ .

We make the following assumptions, which were widely used, such as Maojiao and Guoqiang, ⁶ Koshal et al., ¹¹ Paccagnan et al., ¹⁴ and Liang et al. ¹⁵

• Assumption 1. The graph $\mathcal{G}$ is undirected and connected.

• Assumption 2. The cost function $J_i(q_i,q_{-i})$ is continuously differentiable in $q$ and convex in $q_i$ for any fixed $q_{-i}$ , $i\in \mathcal{V}$ .

• Assumption 3. The map ${\mathrm{\Phi }}_{\epsilon }$ is $\omega$ -strongly monotone and $\theta$ -Lipchitz for some $\epsilon >0$ on ${\mathbb{R}}^{N(m+n)}$ .

${\mathrm{\Phi }}_{\epsilon }$ is strongly monotone, if and only if the Jacobian matrix $\mathcal{J}{\mathrm{\Phi }}_{\epsilon }$ is uniformly positive definite (referring to Facchinei and Pang ³⁴ ).

The NE of the aggregative game (4) is unique under Assumptions 2 and 3 (referring to Theorem 2.2.3 of Facchinei and Pang ³⁴ ). In this paper, we only consider the case of finite decision variables. In other words, $0\le \parallel q^{*}\parallel <+\infty$ .

Define:

$$F(q):=\mathrm{col}{({\nabla }_{q_1}{J}_1(q_1,q_{-1}),\dots ,{\nabla }_{q_N}{J}_N(q_N,q_{-N}))}^T.$$

$F(q)=G(q,y)$ , when $y_i=\sigma (q),\forall i\in \mathcal{V}$ . Besides, $F(q)$ is strongly monotonic and Lipschitz continuous in $q$ together with the strong monotonicity and Lipschitz continuity of ${\mathrm{\Phi }}_{\epsilon }$ , respectively.

The Nash equilibrium of the aggregative game (4) is clarified by the following lemma:

• Lemma 1. Based on Assumption 2, $q^{*}=\mathrm{col}(q_1^{*T},...q_N^{*T}{)}^T$ is the Nash equilibrium of the aggregative game (4) if

$${\nabla }_{q_i}{J}_i(q_i^{*},q_{-i}^{*})=0_m,i\in \mathcal{V}.$$

(6)

Proof. The solution to $\mathrm{VI}({\mathbb{R}}^{N\times m},F)$ satisfying (6), is the Nash equilibrium of the aggregative game (4) (referring to Theorem 3.9 and Theorem 4.8 in Facchinei and Kanzow ³⁵ ).

On the other hand, EL dynamics can accommodate to many multi-agent systems, like UAV networks and multi-robot systems (see Li et al. ²³ and Mishra et al. ³⁶ ). In our work, the player has the following EL dynamics:

$$M_i\left(q_i\right){\stackrel{··}{q}}_i+C_i(q_i,{\stackrel{·}{q}}_i){\stackrel{·}{q}}_i+g_i\left(q_i\right)={\tau }_i+d_i,$$

(7)

where $M_i(q_i)\in {\mathbb{R}}^{m\times m}$ is the positive definite inertia matrix, $C_i(q_i,{\stackrel{·}{q}}_i){\stackrel{·}{q}}_i\in {\mathbb{R}}^m$ is the Coriolis and centripetal force vector, $g_i(q_i)\in {\mathbb{R}}^m$ represents the gravitational effect, ${\tau }_i\in {\mathbb{R}}^m$ is the control input. $d_i\in {\mathbb{R}}^m$ is the exogenous disturbance which is generated by

$$d_i=C{w}_i,{\stackrel{·}{w}}_i=S{w}_i,$$

(8)

where $S\in {\mathbb{R}}^{n\times p}$ and $C\in {\mathbb{R}}^{m\times p}$ are transition matrix and observation matrix, respectively. $w_i\in {\mathbb{R}}^p$ . In this paper, we assume that all the eigenvalues of $S$ are distinctly lying on the imaginary axis, which implies the disturbance is bounded.

The EL systems (7) have the following properties (referring to Spong ³⁷ ):

1. $M_i(q_i)$ is positive definite and is bounded for any $q_i$ , and ${\stackrel{·}{M}}_i(q_i)$ is linearly bounded with respect to ${\stackrel{·}{q}}_i$ . More specifically, there exist positive constants $k_{\underset{¯}{m}i},k_{\overline{m}i}$ and $k_{\stackrel{·}{m}i}$ such that $k_{\underset{¯}{m}i}{I}_m\le M_i(q_i)\le k_{\overline{m}i}{I}_m$ and $\parallel {\stackrel{·}{M}}_i(q_i)\parallel \le k_{\stackrel{·}{m}i}\parallel {\stackrel{·}{q}}_i\parallel$ .

Our purpose is to design a distributed algorithm for the aggregative game (4) of EL system (7) with exogenous disturbance (8) such that the players can seek the Nash equilibrium of the game (4).

• Remark 1. Different from the well-studied aggregative games,^6,8,11,14,15 and we consider the EL dynamics for each player which is influenced by exogenous disturbances. Most existing algorithms such as Maojiao and Guoqiang, ⁶ Deng, ⁸ Koshal et al., ¹¹ Paccagnan et al., ¹⁴ and Liang et al. ¹⁵ can not be utilized to our problem directly due to the dynamics of EL systems and exogenous disturbances.

Main results

In this section, a distributed Nash equilibrium seeking algorithm is proposed. By the proposed algorithm, the EL system (7) converges to the Nash equilibrium of the aggregative game (4). Then, we analyze the convergence of our algorithm by Lyapunov stability theory and variational analysis.

Distributed algorithm design

We give the following lemma about the exogenous disturbances, which is introduced in Wang et al., ²⁵ Deng et al., ²⁶ Deng and Nian, ²⁷ and Huang. ³⁰

• Lemma 2. Take $p(\lambda )={\lambda }^s+p_1{\lambda }^{s-1}+\cdots +p_s$ as the minimal polynomial of $S$ with some real numbers $p_1,...,p_s$ ( $p_i,\forall i\in \{1,2,\cdots ,s\}$ is real number, since the disturbance is supposed to be bounded) and the disturbance (8) can be described as

$$d_i=(I_n\otimes \psi ){\mu }_i,{\stackrel{·}{\mu }}_i=(I_n\otimes \mathrm{\Phi }){\mu }_i,$$

(9)

where ${\mu }_i=\mathrm{col}({\mu }_{i1}^T,\cdots ,{\mu }_{\mathrm{in}}^T{)}^T$ and ${\mu }_{ij}=\mathrm{col}(d_{ij}(t),\frac{d{d}_{ij}(t)}{dt},\cdots ,\frac{d^{s-1}{d}_{ij}(t)}{d{t}^{s-1}}{)}^T$ with $j\in \{1,\cdots ,n\}$ , $\psi =\left[{1|0}_{1\times (s-1)}\right]$ , and $\mathrm{\Phi }=\left[\begin{array}{cc}0\hfill & I_{s-1}\hfill \\ \multicolumn{1}{c}{-p_s}& -p_{s-1}\cdots -p_1\hfill \end{array}\right]$ .

Proof. Referring to Lemma 3 in Deng and Nian. ²⁷

• Remark 2. By Lemma 2, we have a vector $\xi \in {\mathbb{R}}^s$ such that $H=\mathrm{\Phi }+\xi \psi$ , and $H$ is Hurwitz due to the observability of the pair $(\psi ,\mathrm{\Phi })$ . Consequently, there exists a positive definite symmetric matrix $P$ such that $H^{T}P+\mathrm{PH}=-2I_s$ is satisfied.

Based on internal model principle, ${\eta }_i$ is designed for player $i$ to reject the exogenous disturbances:

$$\begin{array}{c}{\stackrel{·}{\eta }}_i=(I_n\otimes H){\eta }_i+(I_n\otimes \xi )(-k{M}_i(q_i){\stackrel{·}{q}}_i\hfill \\ -M_i(q_i)G_i(q_i,y_i)-(I_n\otimes \psi ){\eta }_i),i\in \mathcal{V}.\hfill \end{array}$$

(10)

Based on (9), (10), state feedback and gradient descent, the distributed algorithm is designed for the aggregative game (4):

$$\begin{array}{c}{\tau }_i=g_i\left(q_i\right)+C_i(q_i,{\stackrel{·}{q}}_i){\stackrel{·}{q}}_i-k{M}_i\left(q_i\right){\stackrel{·}{q}}_i\hfill \\ -M_i\left(q_i\right)G_i(q_i,y_i)-I_n\otimes \psi {\eta }_i,\hfill \end{array}$$

(11a)

$$\begin{array}{cc}{\stackrel{·}{y}}_i=\hfill & {\upsilon }_{\mathrm{yi}},\hfill \end{array}$$

(11b)

$$\begin{array}{c}{\stackrel{·}{\upsilon }}_{\mathrm{yi}}=-k{\upsilon }_{\mathrm{yi}}-\epsilon (y_i-{\phi }_i\left(q_i\right))-\sum _{j=1}^N{a}_{\mathrm{ij}}(y_i-y_j)\hfill \\ -\sum _{j=1}^N{a}_{\mathrm{ij}}(z_i-z_j),\hfill \end{array}$$

(11c)

$$\begin{array}{cc}{\stackrel{·}{z}}_i=\hfill & \sum _{j=1}^N{a}_{\mathrm{ij}}(y_i-y_j)+\sum _{j=1}^N{a}_{\mathrm{ij}}({\upsilon }_{\mathrm{yi}}-{\upsilon }_{\mathrm{yj}}),\hfill \end{array}$$

(11d)

where $k$ is the control gain to be determined latter.

The algorithm (11) can be decomposed into two part: game part (11a) and gradient estimation part (11b) − (11d). In more details, $g_i\left(q_i\right)+C_i(q_i,{\stackrel{·}{q}}_i){\stackrel{·}{q}}_i$ denotes feedback item for nonlinear elimination, and we refer to $k{M}_i\left(q_i\right){\stackrel{·}{q}}_i$ as damping. $M_i\left(q_i\right)G_i(q_i,y_i)$ is a game item for seeking the Nash equilibrium, and $\psi {\eta }_i$ is for rejecting disturbances. In the gradient estimation part (11b)−(11d), $y_i$ is the estimation of the aggregate $\sigma (q)$ , $G_i(q_i,y_i)$ is the estimation of the gradient of $J_i(q_i,q_{-i})$ with respect to $q_i$ , and ${\upsilon }_{\mathrm{yi}},z_i$ are auxiliary variables. Obviously, the algorithm protects the privacy of each player, since every player do not share its decision and cost function with its neighbors. The illustration of (11) is showed in Figure 1.

Figure 1.

The illustration of the proposed algorithm (11).

Convergence analysis

The convergence is analyzed in this section.

Define ${\tilde{\eta }}_i={\eta }_i-{\mu }_i$ . Combining (9) and (10), we have

$$\begin{array}{c}{\stackrel{·}{\stackrel{~}{\eta }}}_i=(I_n\otimes H){\tilde{\eta }}_i+(I_n\otimes \xi )(-k{M}_i(q_i){\stackrel{·}{q}}_i\hfill \\ -M_i(q_i)G_i(q_i,y_i)-(I_n\otimes \psi ){\tilde{\eta }}_i),\hfill \end{array}$$

(12)

which can be regarded as the compensating error. It should be noticed that the exogenous disturbance is rejected as ${\tilde{\eta }}_i$ goes to 0.

Let $\tilde{x}=\mathrm{col}(q^T,y^T{)}^T$ , $\tilde{\upsilon }=\mathrm{col}{({\stackrel{·}{q}}^T,{\upsilon }_y^T)}^T,z=\mathrm{col}{(z_1^T,\dots ,z_N^T)}^T$ , $\stackrel{.}{q}=col{({\stackrel{.}{q}}_1^T,\dots ,{\stackrel{.}{q}}_N^T)}^T,{\upsilon }_y=col{({\upsilon }_{y1}^T,\dots ,{\upsilon }_{yN}^T)}^T$ and $\tilde{\eta }=\mathrm{col}({\tilde{\eta }}_1^T,\dots ,{\tilde{\eta }}_N^T{)}^T$ , $M(q)=\mathrm{diag}(\frac{1}{M_1(q_1)},\dots ,\frac{1}{M_N(q_N)})$ . Then, by combining (7), (11), and (12), we have

$$\stackrel{·}{\stackrel{~}{x}}=\tilde{\upsilon },$$

(13a)

$$\stackrel{·}{\stackrel{~}{\upsilon }}=-k\tilde{\upsilon }-{\varphi }_{\epsilon }(\tilde{x})-H_1\tilde{x}-H_{2}z-H_{3}M(q)I_n\otimes \psi \tilde{\eta },$$

(13b)

$$\stackrel{·}{z}=H_4(\tilde{x}+\tilde{\upsilon }),$$

(13c)

where $H_1=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& L\otimes I_n\hfill \end{array}\right]$ , $H_2=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{L\otimes I_n}\end{array}\right]$ , $H_3=\left[\begin{array}{c}{I}_{N\times m}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$ , and $H_4=\left[\begin{array}{cc}0\hfill & L\otimes I_n\hfill \end{array}\right]$ .

With the help of Lemma 1, we have the following conclusion about the equilibrium point of the system (13).

• Theorem 1. Based on Assumptions 1–2, the aggregative game (4) is considered. If $({\tilde{x}}^{*},{\tilde{\upsilon }}^{*},z^{*})$ is the equilibrium point of the system (13), then $q^{*}$ is a Nash equilibrium of the aggregative game (4), where ${\tilde{x}}^{*}=\mathrm{col}(q^{*T},y^{*T}{)}^T$ . Conversely, if $q^{*}$ is a Nash equilibrium of the aggregative game (4), there exist $y^{*}\in {\mathbb{R}}^{\mathrm{Nn}},{\tilde{\upsilon }}^{*}\in {\mathbb{R}}^{\mathrm{Nn}},z^{*}\in {\mathbb{R}}^{\mathrm{Nn}}$ such that $({\tilde{x}}^{*},{\tilde{\upsilon }}^{*},z^{*})$ is an equilibrium point of the system (13).

Proof. When the compensating error term (12) and the system (13) are at equilibrium, we have

$${\tilde{\upsilon }}^{*}=0_{N(m+n)},$$

(14a)

$$\begin{array}{c}-H_{3}M(q)I_n\otimes \psi \tilde{\eta }-k{\tilde{\upsilon }}^{*}\hfill \\ -{\varphi }_{\epsilon }\left({\tilde{x}}^{*}\right)-H_1{\tilde{x}}^{*}-H_2{z}^{*}=0_{N(m+n)},\hfill \end{array}$$

(14b)

$$\begin{array}{cc}{H}_4({\tilde{x}}^{*}+{\tilde{\upsilon }}^{*})\hfill & {=0}_{N(m+n)},\hfill \end{array}$$

(14c)

$$\begin{array}{c}(I_n\otimes H){\tilde{\eta }}_i^{*}+(I_n\otimes \xi )(-k{M}_i(q_i){\stackrel{·}{q}}_i^{*}\hfill \\ -M_i(q_i)G_i(q_i^{*},y_i^{*})-(I_n\otimes \psi ){\tilde{\eta }}_i^{*})=0_{\mathrm{ns}}.\hfill \end{array}$$

(14d)

From (14a) and (14b), we have $-k{M}_i(q_i){\stackrel{·}{q}}_i-M_i(q_i)G_i(q_i,y_i)-(I_n\otimes \psi ){\tilde{\eta }}_i=0$ . Combining (14d), we have $(I_n\otimes H){\tilde{\eta }}^{*}=0$ , which means the exogenous disturbance is rejected. Combining (14a), (14c), and $(I_n\otimes H){\tilde{\eta }}^{*}=0$ , we have $G_i(q_i^{*},y_i^{*})=0$ . Moreover, (14a) and (14c) yield $(L\otimes I_m)y^{*}=0_{\mathrm{Nn}}$ . we have $y_1^{*}=\cdots =y_N^{*}$ from $(L\otimes I_m)y^{*}=0_{\mathrm{Nn}}$ . Based on (14a), (14b), and $1_N^{T}L=0_N^T$ , we have $y_1^{*}+\cdots +y_N^{*}={\phi }_1(q_1^{*})+{\phi }_2(q_2^{*})+\cdots +{\phi }_N(q_N^{*})$ such that $y_i^{*}=\frac{1}{N}\sum _{j=1}^N{\phi }_j(q_j^{*})$ . Finally, $y_i^{*}=\frac{1}{N}\sum _{j=1}^N{\phi }_j(q_j^{*})$ which means $F_i(q_i^{*},\sigma (q))=G_i(q_i^{*},y_i^{*})=0_m,\forall i\in \mathcal{V}$ . In the aggregative game (4), $q^{*}$ is the Nash equilibrium according to Lemma 1.

Theorem 1 reveals that if (13) is stable, the system (7) converges to the Nash equilibrium of the non-cooperative aggregative game (4) with the help of the algorithm (11). The following theorem shows the stability of (13).

• Theorem 2. Based on Assumptions 1–3, the aggregative game (4) of EL system (7) with exogenous disturbance (9) is investigated. The decision of the EL system (7) with algorithm (11) asymptotically converges to the Nash equilibrium of the aggregative game (4).

Proof. First, we make the following coordinate transformation,

$$\{\begin{array}{cc}\overline{x}\hfill & =[{\overline{x}}_q^T,{\overline{x}}_y^T{]}^T=\tilde{x}-{\tilde{x}}^{*},\hfill \\ \multicolumn{1}{c}{\overline{\upsilon }}& =[{\overline{v}}_q^T,{\overline{v}}_y^T{]}^T=\tilde{\upsilon }-{\tilde{\upsilon }}^{*},\hfill \\ \multicolumn{1}{c}{\overline{z}}& =z-z^{*},\hfill \\ \multicolumn{1}{c}{\overline{\eta }}& =\tilde{\eta }-(I_N\otimes \xi ){\overline{\upsilon }}_q,\hfill \end{array}$$

(15)

where ${\overline{x}}_q,{\overline{\upsilon }}_q\in {\mathbb{R}}^{\mathrm{Nm}}$ , and ${\overline{x}}_y,{\overline{\upsilon }}_y\in {\mathbb{R}}^{\mathrm{Nn}}$ .

Based on (13), (14a), (14b), and (14c), we have

$$\{\begin{array}{c}\stackrel{·}{\overline{x}}=\overline{\upsilon },\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\overline{\upsilon }}=-k\overline{\upsilon }-h-H_1\overline{x}-H_2\overline{z}-H_{3}M(q)I_n\otimes \psi \overline{\eta },}\\ \multicolumn{1}{c}{\stackrel{·}{\overline{z}}=H_4(\overline{x}+\overline{\upsilon }),}\\ \multicolumn{1}{c}{\stackrel{·}{\overline{\eta }}=I_N\otimes H\overline{\eta }+I_N\otimes H\xi {\overline{\upsilon }}_q,}\end{array}$$

(16)

where $h={\mathrm{\Phi }}_{\epsilon }(\tilde{x})-{\mathrm{\Phi }}_{\epsilon }({\tilde{x}}^{*})$ .

By (15), $q^{*}$ converges to the Nash equilibrium of the aggregative game (4) as $\overline{x}$ approaches $0$ . Thus, our problem is converted to analyze the convergence of $\overline{x}$ .

Make the following orthogonal transformation,

$$\begin{array}{c}{\chi }_q=\mathrm{col}{({\chi }_{q1}^T,{\chi }_{q2}^T)}^T=({[rR]}^T\otimes I_m){\overline{x}}_q,\hfill \\ {\chi }_y=\mathrm{col}{({\chi }_{y1}^T,{\chi }_{y2}^T)}^T=({[rR]}^T\otimes I_n){\overline{x}}_y,\hfill \\ {\upsilon }_q=\mathrm{col}{({\upsilon }_{q1}^T,{\upsilon }_{q2}^T)}^T=({[rR]}^T\otimes I_m){\overline{\upsilon }}_q,\hfill \\ {\upsilon }_y=\mathrm{col}{({\upsilon }_{y1}^T,{\upsilon }_{y2}^T)}^T=({[rR]}^T\otimes I_n){\overline{\upsilon }}_y,\hfill \\ \delta =\mathrm{col}{({\delta }_1^T,{\delta }_2^T)}^T=({[rR]}^T\otimes I_n)\overline{z},\hfill \end{array}$$

(17)

where ${\chi }_{x1},{\nu }_{q1}\in {\mathbb{R}}^m$ , ${\chi }_{x2},{\nu }_{q2}\in {\mathbb{R}}^{(N-1)m}$ , ${\chi }_{y1},{\nu }_{y1},{\delta }_1\in {\mathbb{R}}^n,{\chi }_{y2},{\nu }_{y2},{\delta }_2\in {\mathbb{R}}^{(N-1)n},r=\frac{1}{\sqrt{N}}{1}_N,r^{T}R=0_{N-1}^T,R^{T}R=I_{N-1}$ and $R{R}^T=I_N-\frac{1}{N}{1}_N{1}_N^T$ .

Let ${\chi }_1=\mathrm{col}{({\chi }_{q1}^T,{\chi }_{y1}^T)}^T,{\chi }_2=\mathrm{col}{({\chi }_{q2}^T,{\chi }_{y2}^T)}^T,{\nu }_1=\mathrm{col}{({\nu }_{q1}^T,{\nu }_{y1}^T)}^T$ , and ${\nu }_2=\mathrm{col}{({\nu }_{q2}^T,{\nu }_{y2}^T)}^T$ . We rewrite (16) into two subsystems as follows to simplify convergence analysis,

$$\{\begin{array}{c}{\stackrel{·}{\chi }}_1={\nu }_1,\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\nu }}_1=-k{\nu }_1-{\hat{r}}^{T}h-{\hat{r}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta },}\\ \multicolumn{1}{c}{{\stackrel{·}{\delta }}_1{=0}_{N-1},}\end{array}$$

(18)

$$\{\begin{array}{c}{\stackrel{·}{\chi }}_2={\nu }_2,\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\nu }}_2=-k{\nu }_2-{\hat{R}}^{T}h-\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{R^T\mathrm{LR}\otimes I_n}\end{array}\right]{\chi }_{y2}}\\ \multicolumn{1}{c}{-\left[\begin{array}{c}0,\hfill \\ \multicolumn{1}{c}{R^T\mathrm{LR}\otimes I_n}\end{array}\right]{\delta }_2-{\hat{R}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta },}\\ \multicolumn{1}{c}{{\stackrel{·}{\delta }}_2=\left[\begin{array}{cc}0\hfill & R^T\mathrm{LR}\otimes I_n\hfill \end{array}\right]{\chi }_2}\\ \multicolumn{1}{c}{+\left[\begin{array}{cc}0\hfill & R^T\mathrm{LR}\otimes I_n\hfill \end{array}\right]{\nu }_2,}\end{array}$$

(19)

where $\hat{r}=\left[\begin{array}{cc}r\otimes I_m\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& r\otimes I_n\hfill \end{array}\right],\hat{R}=\left[\begin{array}{cc}R\otimes I_m\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& R\otimes I_n\hfill \end{array}\right]$ .

Take the candidate Lyapunov function as follows:

$$\begin{array}{c}{V}_1=\frac{1}{2}\parallel {\chi }_1+{\nu }_1{\parallel }^2+\frac{1}{2}(k-1)\parallel {\chi }_1{\parallel }^2+\frac{1}{2}\parallel {\chi }_2+{\nu }_2{\parallel }^2\hfill \\ +\frac{1}{2}(k-1)\parallel {\chi }_2{\parallel }^2+\frac{1}{2}\parallel {\delta }_2{\parallel }^2+\frac{1}{2}\gamma \parallel {\nu }_{y2}+{\delta }_2{\parallel }^2\hfill \\ +\frac{\parallel \psi {\parallel }^2}{k_{\min }^2\beta }\overline{\eta }(I_N\otimes P)\overline{\eta },\hfill \end{array}$$

(20)

where $k_{\min }=\min (k_{\underset{¯}{m}1},k_{\underset{¯}{m}2},\cdots ,k_{\underset{¯}{m}N})$ .

The derivative of $V_1$ along (18) and (19) is,

$$\begin{array}{c}{\stackrel{·}{V}}_1=-{\chi }_1^T{\hat{r}}^{T}h-{\chi }_2^T{\hat{R}}^{T}h-(k-1)\parallel {\upsilon }_1{\parallel }^2-(k-1)\parallel {\upsilon }_2{\parallel }^2\hfill \\ -{\upsilon }_1^T{\hat{r}}^{T}h-{\upsilon }_2^T{\hat{R}}^{T}h-{\chi }_{y2}^T(R^T\mathrm{LR}\otimes I_n){\chi }_{\upsilon 2}\hfill \\ -{\upsilon }_{y_2}^T(R^T\mathrm{LR}\otimes I_n){\chi }_{y2}+\gamma (-k{\delta }_2^T{\upsilon }_{y2}-{\delta }_2^T[0R^T]h\hfill \\ -{\delta }_2^T(R^T\mathrm{LR}\otimes I_n){\delta }_2+{\upsilon }_{y2}^T(R^I\mathrm{LR}\otimes I_n){\upsilon }_{y2}-k\parallel {\upsilon }_{y2}{\parallel }^2\hfill \\ -{\upsilon }_{y2}^T\left[\begin{array}{cc}0\hfill & R^T\hfill \end{array}\right]h)-{\chi }_1^T{\hat{r}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta }\hfill \\ -{\upsilon }_1^T{\hat{r}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta }-{\chi }_2^T{\hat{R}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta }\hfill \\ -{\upsilon }_2^T{\hat{R}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta }+\frac{\parallel \psi {\parallel }^2}{k_{\min }^2\beta }(-2\parallel \overline{\eta }{\parallel }^2\hfill \\ +2{\overline{\eta }}^T(I_N\otimes \mathrm{PH}\xi ){\overline{\upsilon }}_q).\hfill \end{array}$$

Then ${\stackrel{·}{V}}_1$ is analyzed item by item. Base on Assumption 3, the following inequalities hold,

$${\chi }_1^T{\hat{r}}^{T}h+{\chi }_2^T{\hat{R}}^{T}h\ge \omega ({\Vert {\chi }_1\Vert }^2+{\Vert {\chi }_2\Vert }^2),$$

(21)

$$\begin{array}{c}{\upsilon }_1^T{\hat{r}}^{T}h+{\upsilon }_2^T{\hat{R}}^{T}h\le \frac{1}{2}(\frac{{\theta }^2}{\omega }({\Vert {\upsilon }_1\Vert }^2+{\Vert {\upsilon }_2\Vert }^2)\hfill \\ +\omega ({\Vert {\chi }_1\Vert }^2+{\Vert {\chi }_2\Vert }^2)).\hfill \end{array}$$

(22)

According to Schur Complement Lemma,

$$F:=\left[\begin{array}{cc}{R}^T\mathrm{LR}\otimes I_n\hfill & \frac{1}{2}{R}^T\mathrm{LR}\otimes I_n\hfill \\ \multicolumn{1}{c}{\frac{1}{2}{R}^T\mathrm{LR}\otimes I_n}& \frac{1}{4}{\lambda }_{N}I\hfill \end{array}\right]\ge 0$$

which implies that

$$\begin{array}{c}{\chi }_{y2}^T(R^T\mathrm{LR}\otimes I_n){\chi }_{y2}+{\upsilon }_{y2}^T(R^T\mathrm{LR}\otimes I_n){\chi }_{y2}\hfill \\ \multicolumn{1}{c}{=\left[\begin{array}{cc}{\chi }_{y2}^T\hfill & {\upsilon }_{y2}^T\hfill \end{array}\right]F\left[\begin{array}{c}{\chi }_{y2}\hfill \\ \multicolumn{1}{c}{{\upsilon }_{y2}}\end{array}\right]-\frac{1}{4}{\lambda }_N{\Vert {\upsilon }_{y2}\Vert }^2}\\ \multicolumn{1}{c}{\ge -\frac{1}{4}{\lambda }_N{\Vert {\upsilon }_{y2}\Vert }^2.}\end{array}$$

(23)

Besides, it results from $\mathrm{ab}\le \frac{c}{2}{a}^2+\frac{1}{2c}{b}^2,\forall c>0$ that

$$-k{\delta }_2^T{\upsilon }_{y2}\le \frac{1}{2}k(\frac{{\lambda }_2}{k+1}{\Vert {\delta }_2\Vert }^2+\frac{k+1}{{\lambda }_2}{\Vert {\upsilon }_{y2}\Vert }^2),$$

(24)

$$-{\delta }_2^T\left[0R^T\right]h\le \frac{1}{2}({\lambda }_2\Vert {\delta }_2{\Vert }^2+\frac{1}{{\lambda }_2}({\Vert {\chi }_1\Vert }^2+{\Vert {\chi }_2\Vert }^2)),$$

(25)

$$-{\upsilon }_{y2}^T\left[0R^T\right]h\le \frac{1}{2}(2k{\Vert {\upsilon }_{y2}\Vert }^2+\frac{1}{2k}({\Vert {\chi }_1\Vert }^2+{\Vert {\chi }_2\Vert }^2)).$$

(26)

Also, we have

$$\begin{array}{c}\multicolumn{1}{c}{-{\chi }_1^T{\hat{r}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta }-{\chi }_2^T{\hat{R}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta }}\\ \multicolumn{1}{c}{-{\nu }_1^T{\hat{r}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta }-{\nu }_2^T{\hat{R}}^T{H}_{3}M(q)I_n\otimes \psi \overline{\eta }}\\ \multicolumn{1}{c}{=-({\chi }_1^T{\hat{r}}^T+{\chi }_2^T{\hat{T}}^T)H_{3}M(q)\otimes \psi \overline{\eta }}\\ \multicolumn{1}{c}{-({\nu }_1^T{\hat{r}}^T+{\nu }_2^T{\hat{T}}^T)H_{3}M(q)\otimes \psi \overline{\eta },}\end{array}$$

(27)

and due to the orthogonal transformation, we have

$$\begin{array}{c}-({\chi }_1^T{\hat{r}}^T+{\chi }_2^T{\hat{R}}^T)H_{3}M(q)I_n\otimes \psi \overline{\eta }\hfill \\ -({\nu }_1^T{\hat{r}}^T+{\nu }_2^T{\hat{R}}^T)H_{3}M(q)I_n\otimes \psi \overline{\eta }\hfill \\ =-({\overline{x}}_q{)}^{T}M(q)I_n\otimes \psi \overline{\eta }-({\overline{\upsilon }}_q{)}^{T}M(q)I_n\otimes \psi \overline{\eta }.\hfill \end{array}$$

(28)

Similarly, by the properties of EL systems $k_{\mathrm{mi}}{I}_m\le M_i(q_i)\le k_{(\overline{m}i)}{I}_m$ ( $M(q)=\mathrm{diag}(\frac{1}{M_1(q_1)},\dots ,\frac{1}{M_N(q_N)})$ ) and $\mathrm{ab}\le \frac{c}{2}{a}^2+\frac{1}{2c}{b}^2,\forall c>0$ , it’s easy to show that

$$\begin{array}{c}-({\overline{x}}_q{)}^{T}M(q)I_n\otimes \psi \overline{\eta }-({\overline{\upsilon }}_q{)}^{T}M(q)I_n\otimes \psi \overline{\eta }\hfill \\ \le \frac{\beta }{2}\parallel {\overline{x}}_q{\parallel }^2+\frac{\beta }{2}\parallel {\overline{\upsilon }}_q{\parallel }^2+\frac{1}{\beta }\parallel M(q)I_n\otimes \psi \overline{\eta }{\parallel }^2\hfill \\ \le \frac{\beta }{2}\parallel {\overline{x}}_q{\parallel }^2+\frac{\beta }{2}\parallel {\overline{\upsilon }}_q{\parallel }^2+\frac{\parallel \psi {\parallel }^2}{k_{\min }^2\beta }\parallel \overline{\eta }{\parallel }^2.\hfill \end{array}$$

(29)

In the internal model item, there is

$$\begin{array}{c}-2\parallel \overline{\eta }{\parallel }^2+2{\overline{\eta }}^T(I_N\otimes \mathrm{PH}\xi ){\overline{\upsilon }}_q\hfill \\ \le -\parallel \overline{\eta }{\parallel }^2+{\xi }^T{H}^T{P}^{2}H\xi \parallel {\overline{\upsilon }}_q{\parallel }^2.\hfill \end{array}$$

(30)

From the orthogonal transformation, obviously we have

$$\begin{array}{c}\parallel {\chi }_1{\parallel }^2+\parallel {\chi }_2{\parallel }^2\ge \parallel {\overline{x}}_q{\parallel }^2,\hfill \\ \multicolumn{1}{c}{\parallel {\nu }_1{\parallel }^2+\parallel {\nu }_2{\parallel }^2\ge \parallel {\overline{\nu }}_q{\parallel }^2.}\end{array}$$

(31)

With (21)−(31), we have

$$\begin{array}{c}{\stackrel{·}{V}}_1\le -(\frac{1}{2}\omega -\gamma (\frac{1}{2{\lambda }_2}+\frac{1}{4k})-\frac{\beta }{2})({\Vert {\chi }_1\Vert }^2+{\Vert {\chi }_2\Vert }^2)\hfill \\ -(k-\frac{{\theta }^2}{\omega }-1-\frac{\beta }{2}-\frac{\parallel \psi {\parallel }^2}{k_{\min }^2\beta }{\xi }^T{H}^T{P}^{2}H\xi ))\Vert {\upsilon }_1{\Vert }^2\hfill \\ -(k-\frac{{\theta }^2}{\omega }-\frac{1}{4}{\lambda }_N-\gamma ({\lambda }_N+\frac{k(k+1)}{2{\lambda }_2})\hfill \\ -\frac{\beta }{2}-\frac{\parallel \psi {\parallel }^2}{k_{\min }^2\beta }{\xi }^T{H}^T{P}^{2}H\xi )\parallel {\upsilon }_2{\parallel }^2,\hfill \end{array}$$

(32)

where

$$\begin{array}{c}0<\gamma <\min \{{\gamma }_1,{\gamma }_2\},\hfill \\ {\gamma }_1=\frac{2\omega {\lambda }_{2}k}{2k+{\lambda }_2-2\beta {\lambda }_{2}k},\hfill \\ {\gamma }_2=\frac{2{\lambda }_2(k-\frac{{\theta }^2}{\omega }-\frac{1}{4}{\lambda }_N-\frac{\beta }{2}-\frac{\parallel \psi {\parallel }^2}{k_{\min }^2\beta }{\xi }^T{H}^T{P}^{2}H\xi )}{k(k+1)+2{\lambda }_2{\lambda }_N}\hfill \end{array}$$

and $k>\frac{{\theta }^2}{\omega }+\frac{1}{4}{\lambda }_N+1+\frac{\beta }{2}+\frac{\parallel \psi {\parallel }^2}{k_{\min }^2\beta }{\xi }^T{H}^T{P}^{2}H\xi$ . It results from (32) that $q$ asymptotically converges to the Nash equilibrium $q^{*}$ .

• Remark 3. Internal model principle is used in the algorithm (11), and exogenous disturbances are rejected. Sequentially, revealed by theorem 2, the algorithm (11) achieves asymptotical convergence.

Simulations

The competition between multi-agent energy resources in the electricity market can be viewed as a kind of Nash-Cournot game (referring to Hobbs and Pang ¹⁸ and Liu et al. ³⁸ ). In this case, we consider six generation systems for the Nash-Cournot game to verify our algorithm. The communication topology is formulated as an undirected graph, which is shown in Figure 2. The cost function of player $i\in \{1,...,6\}$ is

$$J_i(P_i,P_{-i})=c_i(P_i)-p(\sigma (P))P_i,$$

(33)

Figure 2.

The communication topology.

where $P_i$ is the generation power of player $i$ , $P_{-i}=[P_1^T,\cdots ,P_{i-1}^T,P_{i+1}^T,\cdots ,P_N^T{]}^T$ , and $P=[P_1^T,\cdots ,P_N^T{]}^T$ . $c_i(P_i)$ is the cost of generation system. $p(\sigma (P))$ is the electricity price. $c_i(x_i)={\alpha }_i+{\beta }_i{P}_i+{\gamma }_i{P}_i^2$ , where ${\alpha }_i,{\beta }_i,{\gamma }_i$ are the parameters of the electric power system $i$ (referring to Binetti et al. ³⁹ for more details about the generators). Besides, the aggregate function is $\sigma (P)=\frac{1}{N}\sum _{i=1}^N{P}_i$ and $p(\sigma (P))=p_0-\mathrm{aN}\sigma (P)$ , where $p_0$ and $a$ are constants. The electric power system $i\in \mathcal{V}$ is impacted by the following disturbance which has been widely used in the literatures (see other works^24,26,40,41),

$$d_i(t)=a_i\sin (w_{i}t+c_i),$$

(34)

where $a_i,w_i,c_i$ are the parameters of the disturbance. Furthermore, convert (34) into the form of (9) with $C=\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right]$ , $S=\left[\begin{array}{cc}0\hfill & w_i\hfill \\ \multicolumn{1}{c}{-w_i}& 0\hfill \end{array}\right]$ , $w_i(0)=\left[\begin{array}{c}{a}_i\sin c_i\hfill \\ \multicolumn{1}{c}{a_i\cos c_i}\end{array}\right]$ .

When the mechanical and electromagnetic losses are neglected, the dynamics of the generator can be simplified to the following form (referring to Guo et al. ¹⁷ ).

$$\{\begin{array}{c}{\stackrel{·}{P}}_i=-\frac{1}{T_{\mathrm{mi}}}{P}_i+\frac{K_{\mathrm{mi}}}{T_{\mathrm{mi}}}{X}_{\mathrm{ei}},\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{X}}_{\mathrm{ei}}=-\frac{K_{\mathrm{ei}}}{T_{\mathrm{ei}}{R}_i{\omega }_0}{\omega }_i-\frac{1}{T_{\mathrm{ei}}}{X}_{\mathrm{ei}}+\frac{1}{T_{\mathrm{ei}}}{u}_i+d_i,}\\ \multicolumn{1}{c}{{\stackrel{·}{\omega }}_i=-\frac{D_i}{2H_i}{\omega }_i,i\in \mathcal{V},}\end{array}$$

(35)

where $X_{\mathrm{ei}}\in \mathbb{R}$ is the valve opening, and ${\omega }_i\in \mathbb{R}$ is the relative speed of the $i$ th turbine-generator system. $K_{\mathrm{mi}}$ is the gain of the $i$ th machine’s turbine. $T_{\mathrm{mi}}$ and $T_{\mathrm{ei}}$ are the time constants. $R_i$ is the regulation constant, and $D_i$ is the per unit damping constant. $H_i$ is the inertia constant. ${\omega }_0$ is the synchronous machine speed, and $u_i\in \mathbb{R}$ is the control.

The parameters of the turbine-generator systems are showed in Table 1. In this case, we let ${\omega }_0=315.159,p_0=200$ , and $a=0.1$ . ${\mathrm{\Phi }}_{\epsilon }$ is $\omega$ -strongly monotone and $\theta$ -Lipschitz continuous by letting $\epsilon =0.6$ , where $\omega =1.2$ and $\theta =5.6$ . Let $k=40$ to keep the convergence of the multi-agent system (13). Besides, $y_i(0),{\upsilon }_{\mathrm{yi}}(0)$ and $z_i(0)$ are zeros.

Table 1.

The parameters of generators.

	$T_{\mathrm{mi}}$	$T_{\mathrm{ei}}$	$K_{\mathrm{mi}}$	$K_{\mathrm{ei}}$	$D_i$	$H_i$	$R_i$	${\alpha }_i$	${\beta }_i$	${\gamma }_i$	$P_i(0)$	$X_{\mathrm{ei}}(0)$	${\omega }_i(0)$
Generator $\#1$	0.35	0.10	1.0	1.0	5.0	4.0	0.05	5	12	1.0	30	6	4.3
Generator $\#2$	0.30	0.12	1.1	1.1	4.0	3.5	0.04	8	10	0.5	25	5	3.5
Generator $\#3$	0.28	0.08	0.9	0.9	3.0	2.8	0.03	6	11	0.8	20	4	3.0
Generator $\#4$	0.40	0.11	1.2	1.2	4.5	4.2	0.06	9	11	0.7	35	7	4.8
Generator $\#5$	0.43	0.90	0.8	0.8	3.5	3.0	0.04	7	13	1.1	28	5	4.0
Generator $\#6$	0.35	0.10	1.0	1.0	5.0	4.0	0.05	8	14	0.6	37	8	5.0

The parameters of the disturbances are $(a_1,a_2,a_3,a_4,a_5,a_6)=\sqrt{2}(1,3,2,5,4,6)$ , $w_i=1$ , and $c_i=\frac{\pi }{4},\forall i\in \mathcal{V}$ . By Lemma 2, we have $\mathrm{\Phi }=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill \end{array}\right]$ and $\psi =\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right]$ . Let $\xi ={\left[\begin{array}{cc}-1\hfill & 0.8\hfill \end{array}\right]}^T$ such that $H=\left[\begin{array}{cc}-1\hfill & 1\hfill \\ \multicolumn{1}{c}{-0.2}& 0\hfill \end{array}\right]$ is Hurwitz.

The output powers of the turbine-generator systems and convergence of the cost functions are shown in the Figures 3 and 4, respectively. $P_i^{*}$ is the Nash equilibrium, and the output powers of the $i$ th generation can converge to $P_i^{*}$ . The exogenous disturbances are rejected under the algorithm (11), which verifies the effectiveness of our algorithm.

Figure 3.

The evolutions of output powers of turbine-generator systems.

Figure 4.

The evolutions of the cost functions of turbine-generator systems.

Conclusion

This paper has studied the aggregative games of second-order nonlinear multi-agent systems, where the dynamics of the players are described by EL equations and are influenced by exogenous disturbances. We have designed a distributed algorithm to seek the NE of the game, where internal model principle is used to deal with the exogenous disturbances. Moreover, we have analyzed the convergence of the algorithm. By the proposed method, the EL players can not only reject the exogenous disturbances but also asymptotically converge to the NE of the game. An example of smart grids has verified the effectiveness of our algorithm.