# Modeling and Dynamical Analysis of Multi-Area Network with a Third-Order Chaotic Power System

Authors: Lixin Yang; Jie Gao; Jie Ma
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

The synchronization of states is important to sustain the energy of consumers at any given time for power networks. This paper focuses on the multi-area power network model and then analyzes the cluster synchronization of this kind of network comprised of a third-order chaotic power system. Specifically, we investigate the rich dynamic properties of the single third-order power system. Furthermore, the multi-area network model with the chaotic power system is proposed and the adaptive controller is designed to achieve cluster synchronization. Combining analytical considerations with numerical simulations on a small-scale network, we address the cluster synchronous performance in the multi-area power network. Therefore, our results can provide a basic physical picture for power system dynamics and enable us to further understand the complex dynamical behavior in the multi-area power network.


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## Body

## 1. Introduction

As we know, modern power networks play a crucial role in our modern society today. Power networks refer to complex interconnected systems for routing power via transmission lines [1–3].Generally, they consist of a large number of heterogeneous elements that operate interlinked, creating a multiplex network. In this multiplex network, the interlinked nodes either play the role of an energy generator or an energy consumer. Furthermore, it operates normally only if the total demand power matches the total supply from all the generators [4]. Meanwhile, its dynamical behavior is extremely important for stable operation of the power network. One of the most common and useful models for studying power systems is the canonical single machine infinite bus system, which takes into account its electro-behavior [5]. Furthermore, the single machine infinite bus system refers to a power network that corresponds to an oscillatory power-grid node and a connected system as an environment. In addition, the Kuramoto-like model is a standard mathematical model to investigate the dynamics of the power network [6–9]. However, the Kuramot-like model does not take into account the physical characteristics of the power grid. Therefore, most of the existing works have missed a detailed analysis of the nonlinear dynamic behaviors for the third-order power system.Due to the increasing of renewable energy sources on power production, questions concerning the limits, quantification, and control of power grid stability face new challenges. Therefore, the power system represents a distributed network carrying many small units of energy to the consumers instead of large units of energy coming from a few power plants. This implies that the power system will undergo a lot of challenges concerning grid topology [10–12]. As a consequence, the power network will be divided into small areas. As a matter of fact, the real power network usually has a multi-area structure. However, in many recent studies on power system dynamics, this special topology structure is neglected.Based on the above discussion, in the present paper, we focus on a more realistic third-order system model to describe the multi-area power network. The investigation of power systems has been recently addressed from a nonlinear dynamics point of view. Meanwhile, synchronization is vital for the stable operation and control of the power network, and it is necessary to ensure appropriate operation of power generation [13–16]. From the dynamics viewpoint, synchronization is essential for the proper functioning of the power network. Its loss can lead to cascading failure [17–19]. In virtue of multi-area structures, new patterns of synchronization are relevant [15]. In particular, cluster synchronization plays a crucial role on keeping the power balance [20–24]. This synchronous pattern implies that nodes in different areas can achieve different synchronous states to cope with energy imbalance. Thus, we are going to address the cluster synchronization in the multi-area network.This paper is organized as follows. First, we present a third-order power system model in Section2. Afterwards, the characteristics of dynamic behavior for a single power system are investigated in Section 3. We propose the multi-area power network model and analyze the cluster synchronization in Section 4. Finally, we present our results, followed by a summary in Section 5.


## 2. Analysis of Dynamics of the Third-Order Power System

### 2.1. The Third-Order Power System Model

The simple third-order power model is illustrated in Figure1, which can be viewed as a generalized case. In terms of the power system, Vt denotes generator, xt represents transformer reactance, xL is the transmission line reactance, and Vs and Vt denote infinite bus voltage and synchronous generator terminal voltage, respectively.Figure 1
Schematic diagram of the single machine infinite bus system.In what follows, we will focus on the third-order single machine infinite bus system with the excitation mechanism model, which is given by(1)δ˙=ω0ω,ω˙=1HPm−Dω−EVsxdΣ′sinδ,E˙=1Td0′−xdΣxdΣ′E+xd−xd′xdΣ′Vscosδ+Ef,where δ denotes the generator rotor angle, ω represents the deviation of rotor angular velocity from synchronous angular velocity, E is the transient voltage, ω0, D, H denote synchronous angular velocity, damping coefficient, and inertia constant, respectively, Ef is the excitation voltage, and Pm represents the mechanical power of generator.Without loss of generality, the third-order single machine infinite bus system can be cast into a simplified model, namely,(2)δ˙=ω0ω,Hω˙=−γω+Pm−BVsEsinδ,αE˙=−1+XBE+XBVscosδ+Ef.For convenience, we setB=1/xdΣ′,X=xd−xd′,D=γ,Td0′=α,xdΣ/xdΣ′=1+XB.Moreover, we select the parametersB=1,X=1,α=2,Ef=1,Vs=1. Thus, we can obtain the third-order single machine infinite bus system from equation (2):(3)x˙=dy,y˙=−ay+b−czsinx,z˙=−z+0.5cosx+0.5,where x denotes the generator rotor angle, y represents the transient voltage, and z is the relative speed. In what follows, we aim to study the rich dynamic behavior of system (3).


### 2.2. Basic Properties of the System

In this section, the chaotic dynamics of the third-order power system are analyzed. According to the Wolff algorithm, the largest Lyapunov exponents of the system areλ1=0.1599,λ2=−0.0264,λ3=−0.9935, and the system displays chaotic state. We consider the vector field divergence of the system, and one can obtain(4)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z=0−a−1=−a−1.Thus, we can get(5)∑i=13λi=TrJ=divV.Then,(6)divV=∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z=TrJ=−a−1=λ1+λ2+λ3=−0.86.In particular,dv/dt=e−a+1=e−0.86. If a+1>0, system (3) will be a dissipative system.Furthermore, we can calculate the Kaplan–Yorke dimension for chaotic system (3):(7)D=j+1λj+1∑i=1jλj=2+0.1599−0.02640.9935=2.134.That is, system (3) displays chaotic behavior.In order to better understand the dynamic behavior of the third-order power system, we show the phase diagrams and the evolution of variables of the system under different parameters.Firstly, we take the parametera,b,c,d=0.5,−0.08,1,1 and the initial values x,y,z=1,0.1,1, and the system is in periodic state, as shown in Figures 2(a) and 2(b).Figure 2
(a) Phase diagram of system (3). (b) The evolution of the state variables with time.
(a)(b)In what follows, we select the parametera,b,c,d=0.5,0.12,1,1 and the initial value x,y,z=1,0.1,1; as can be seen from Figures 3(a) and 3(b), one can find that the system converges to a fixed point. That is, the values of parameters play an important role on the dynamics for the power system.Figure 3
(a) Phase diagram of system (3). (b) The evolution of the state variables with time.
(a)(b)


### 2.3. Analysis of Bifurcation Condition for the Power System

In this section, we focus on investigating the bifurcation behavior of the third-order chaotic power system. Firstly, according to equation.(3), the Jacobian matrix can be calculated as follows:(8)J=0ω00−BVsE0cosδ0H−γH−BVssinδ0H−XBVssinδ0α0−1+XBα,where δ0,0,E0 denotes the equilibrium point of system (3), and from equation (8), we get(9)λE−J=λ−ω00−BVsE0cosδ0Hλ+γHBVssinδ0HXBVssinδ0α0λ+1+XBα=0.Based on the above algebraic equation, we can obtain that(10)λ3+H1+XB+γαHαλ2+γ1+XB+αω0BVsE0cosδ0Hαλ+ω0BVs1+XBE0cosδ0−XBVssin2δ0Hα=0.In the following, we suppose that(11)λ3+H1+XB+γαHαλ2+γ1+XB+αω0BVsE0cosδ0Hαλ+ω0BVs1+XBE0cosδ0−XBVssin2δ0Hα=λ3+pλ2+q2λ+pq2,where(12)p=H1+XB+γαHα,q2=γ1+XB+αω0BVsE0cosδ0Hα,pq2=ω0BVs1+XB+γαHα⋅γ1+XB+αω0BVsE0cosδ0Hα=ω0BVs1+XBE0cosδ0−XBVssin2δ0Hα1+XBHγ2+αHω0BVsE0cosδ0+H21+XB2αγ+ω0XB2Vs2sin2δ0=0.Thus, the equation of bifurcation curve can be described by the following equations:(13)Pm=BVsE0sinδ0,1+XBE0=Ef+XBVscosδ0,1+XBHγ2+αHω0BVsE0cosδ0+H21+XB2αγ+ω0XB2Vs2sin2δ0=0.By means of bifurcation diagram, dynamic behaviors with varying system parametera are investigated. Let the system parameter a vary from −0.2 to 1 with the step size of 0.001, and the other parameters are taken as b=0.5,c=1,d=1; the bifurcation diagram can provide an overall perspective of the dynamics of the system, which is depicted in Figure 4.Figure 4
Dynamic behaviors of the chaotic third-order power system witha.From the bifurcation diagram, it can be observed that with the increase ofa from −0.2, system (3) is chaotic over most of the scope a∈−0.2,−0.1, and when a∈0.2,1, the system converges to a fixed point. With the value of a increasing from −0.1, system (3) presents Hopf bifurcation.


## 2.1. The Third-Order Power System Model

The simple third-order power model is illustrated in Figure1, which can be viewed as a generalized case. In terms of the power system, Vt denotes generator, xt represents transformer reactance, xL is the transmission line reactance, and Vs and Vt denote infinite bus voltage and synchronous generator terminal voltage, respectively.Figure 1
Schematic diagram of the single machine infinite bus system.In what follows, we will focus on the third-order single machine infinite bus system with the excitation mechanism model, which is given by(1)δ˙=ω0ω,ω˙=1HPm−Dω−EVsxdΣ′sinδ,E˙=1Td0′−xdΣxdΣ′E+xd−xd′xdΣ′Vscosδ+Ef,where δ denotes the generator rotor angle, ω represents the deviation of rotor angular velocity from synchronous angular velocity, E is the transient voltage, ω0, D, H denote synchronous angular velocity, damping coefficient, and inertia constant, respectively, Ef is the excitation voltage, and Pm represents the mechanical power of generator.Without loss of generality, the third-order single machine infinite bus system can be cast into a simplified model, namely,(2)δ˙=ω0ω,Hω˙=−γω+Pm−BVsEsinδ,αE˙=−1+XBE+XBVscosδ+Ef.For convenience, we setB=1/xdΣ′,X=xd−xd′,D=γ,Td0′=α,xdΣ/xdΣ′=1+XB.Moreover, we select the parametersB=1,X=1,α=2,Ef=1,Vs=1. Thus, we can obtain the third-order single machine infinite bus system from equation (2):(3)x˙=dy,y˙=−ay+b−czsinx,z˙=−z+0.5cosx+0.5,where x denotes the generator rotor angle, y represents the transient voltage, and z is the relative speed. In what follows, we aim to study the rich dynamic behavior of system (3).


## 2.2. Basic Properties of the System

In this section, the chaotic dynamics of the third-order power system are analyzed. According to the Wolff algorithm, the largest Lyapunov exponents of the system areλ1=0.1599,λ2=−0.0264,λ3=−0.9935, and the system displays chaotic state. We consider the vector field divergence of the system, and one can obtain(4)∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z=0−a−1=−a−1.Thus, we can get(5)∑i=13λi=TrJ=divV.Then,(6)divV=∇V=∂x˙∂x+∂y˙∂y+∂z˙∂z=TrJ=−a−1=λ1+λ2+λ3=−0.86.In particular,dv/dt=e−a+1=e−0.86. If a+1>0, system (3) will be a dissipative system.Furthermore, we can calculate the Kaplan–Yorke dimension for chaotic system (3):(7)D=j+1λj+1∑i=1jλj=2+0.1599−0.02640.9935=2.134.That is, system (3) displays chaotic behavior.In order to better understand the dynamic behavior of the third-order power system, we show the phase diagrams and the evolution of variables of the system under different parameters.Firstly, we take the parametera,b,c,d=0.5,−0.08,1,1 and the initial values x,y,z=1,0.1,1, and the system is in periodic state, as shown in Figures 2(a) and 2(b).Figure 2
(a) Phase diagram of system (3). (b) The evolution of the state variables with time.
(a)(b)In what follows, we select the parametera,b,c,d=0.5,0.12,1,1 and the initial value x,y,z=1,0.1,1; as can be seen from Figures 3(a) and 3(b), one can find that the system converges to a fixed point. That is, the values of parameters play an important role on the dynamics for the power system.Figure 3
(a) Phase diagram of system (3). (b) The evolution of the state variables with time.
(a)(b)


## 2.3. Analysis of Bifurcation Condition for the Power System

In this section, we focus on investigating the bifurcation behavior of the third-order chaotic power system. Firstly, according to equation.(3), the Jacobian matrix can be calculated as follows:(8)J=0ω00−BVsE0cosδ0H−γH−BVssinδ0H−XBVssinδ0α0−1+XBα,where δ0,0,E0 denotes the equilibrium point of system (3), and from equation (8), we get(9)λE−J=λ−ω00−BVsE0cosδ0Hλ+γHBVssinδ0HXBVssinδ0α0λ+1+XBα=0.Based on the above algebraic equation, we can obtain that(10)λ3+H1+XB+γαHαλ2+γ1+XB+αω0BVsE0cosδ0Hαλ+ω0BVs1+XBE0cosδ0−XBVssin2δ0Hα=0.In the following, we suppose that(11)λ3+H1+XB+γαHαλ2+γ1+XB+αω0BVsE0cosδ0Hαλ+ω0BVs1+XBE0cosδ0−XBVssin2δ0Hα=λ3+pλ2+q2λ+pq2,where(12)p=H1+XB+γαHα,q2=γ1+XB+αω0BVsE0cosδ0Hα,pq2=ω0BVs1+XB+γαHα⋅γ1+XB+αω0BVsE0cosδ0Hα=ω0BVs1+XBE0cosδ0−XBVssin2δ0Hα1+XBHγ2+αHω0BVsE0cosδ0+H21+XB2αγ+ω0XB2Vs2sin2δ0=0.Thus, the equation of bifurcation curve can be described by the following equations:(13)Pm=BVsE0sinδ0,1+XBE0=Ef+XBVscosδ0,1+XBHγ2+αHω0BVsE0cosδ0+H21+XB2αγ+ω0XB2Vs2sin2δ0=0.By means of bifurcation diagram, dynamic behaviors with varying system parametera are investigated. Let the system parameter a vary from −0.2 to 1 with the step size of 0.001, and the other parameters are taken as b=0.5,c=1,d=1; the bifurcation diagram can provide an overall perspective of the dynamics of the system, which is depicted in Figure 4.Figure 4
Dynamic behaviors of the chaotic third-order power system witha.From the bifurcation diagram, it can be observed that with the increase ofa from −0.2, system (3) is chaotic over most of the scope a∈−0.2,−0.1, and when a∈0.2,1, the system converges to a fixed point. With the value of a increasing from −0.1, system (3) presents Hopf bifurcation.


## 3. Analysis of Multi-Area Network Coupled with Chaotic Power System

We are witnessing a time of drastic changes in the operation of power grids caused by the necessity to reduce global warming caused by large emission of carbon dioxide gases. Thus, more and more decentralized renewable energy sources are replacing centralized power generation. The strategy can change the effective grid structure from a fully connected to a locally connected one. Hence, large-scale real power networks will be divided into many areas. This implies that the power network usually has multi-area structure. Generally, the interactions between nodes (generators and consumers) in the same area are identical and those in different areas are nonidentical. Hence, there exist diverse coupling forms in the multi-area power network. For better describing this kind of phenomenon, the multi-area power network model is presented. Also, the schematic diagram of the multi-area power network is shown in Figure5. Here, we suppose that the node’s dynamics in different areas are identical, which is described as system (3).Figure 5
The multi-area power network topology structure.Furthermore, alternating voltage of the power plants is required to be synchronized around a certain specific frequency; otherwise, severe problems like large blackouts may be occur in a large area. Thus, cluster synchronization is essential for the proper functioning of the multi-area power network. In the following, we will investigate the multi-area power network model and its cluster synchronization.


### 3.1. The Multi-Area Power Network Model

In this section, we consider a complex multi-area power network withM areas and each node is a third-order power system. Suppose that the kth area is composed of Nk nodes. Then, the general multi-area power network can be described as(14)x˙it=Fkx˙it+∑j=1,j≠iNkaijcijkkxjt−xit+∑l=1,l≠kM∑j=1Nlaijtcijklxjt−cijlxiti=1,2,…,Nk,k=1,2,…,M,where xik=xi1k,xi2k,…,xinkkT∈Rnk denotes the state vector of the ith node in the kth area. Fk describes the node’s dynamics of the k th area. The matrix Ckk=diagckk1,ckk2,…,ckknk is the inner coupling matrix which denotes the internal connection in k th area, and Ckl=cklrs∈Rnk×nl and Ckkl=diagckkl1,ckkl2,…,ckklnk are the inner coupling matrices between the k th and l th areas; if the r th element of xit in the k th area is influenced by the l th element of xjt in the l th, then ckklr≠0,cklrs≠0; otherwise, ckklr=0,cklrs=0. Also, matrix A=aijN×N denotes the topology structure of the entire multi-area power network, which is described as follows: if there exists a connection between node i and j, then aij=aji>0i≠j; otherwise, aij=aji=0i≠j.Suppose that the sets of subscripts of these areas areG1=1,2,…,N1,G2=N1+1,N1+2,…,N1+N2,…,GM=N1+⋯+NM−1+1,⋯,N, where N1+N2+⋯+NM=N. The coupling matrix A can be written in the following form:(15)A=A11A12⋯A1MA21A22⋯A2M⋮⋱⋮AM1AMM,where App∈Rrp×rpp=1,2,…,M show the coupling pattern in the same area and Apq∈Rrp×rqp=1,2,…,M,q=1,2,…,M,p≠q are the coupling schemes among different areas.In order to achieve cluster synchronization of the multi-area power network, the control inputsuit∈Rn are added and the controlled multi-area power network can be characterized by the following equation:(16)x˙tt=Fkx˙tt+∑j=1,j≠iNkaijtcijkkxjt−xit+∑l=1,l≠kM∑j=1Nlaijtcijklxjt−cijklxjt−cijlxit+uiti=1,2,…,Nk,k=1,2,…,M.Define the synchronous errors(17)eit=xit−skti=1,2,…,Nk,where skt is a solution of an isolated node in the kthand satisfies s˙kt=Fkskt. The network can realize the cluster synchronization, if limt⟶∞eit=0,i=1,2,…,N.


### 3.2. Cluster Synchronization of Multi-Area Power Network

For simplicity, we suppose that all the inner coupling patterns between different areas are identical, and equation (14) can be derived as(18)x˙it=Fkxit+∑j=1,i≠jNkaijtcijkkxjt−xit+uiti=1,2,…,Nk,k=1,2,…,M.According to the above definition of the error variables, we can get the error dynamical system as follows:(19)eit=Fkxit−Fksit+∑j=1Nkaijcijkkejt+∑j=1Naijcijkksjt+uit.Since the above coupling condition holds, one can obtain(20)∑j=1Naijcijkksjt=0,i∈Gϕi−G¯ϕi,where Gϕi represents all the nodes in the area and G¯ϕi represents the nodes in other areas.Subsequently, the control scheme is given via adaptive pinning control idea. The adaptive controller is designed as follows:(21)uit=−∑j=1Naijcijkksjt−qitcijkkxit−skt,i∈ϕ˜k,k=1,2.…,M0,otherwise,where the coupling strength aijt=aij1t,aij2t,…,aijntT between the nodes and the feedback gains adopts the following adaptive strategy:(22)a˙ijt=βxjt−sjt−xit−sit,i∈ϕ˜k,k=1,2,…,M,q˙i=αeit22,where α>0,β>0 are the adaptive gains.Theorem 1.
For multi-area power network (18), cluster synchronization can be achieved under designed controllers (21) and (22).Proof.
Construct a Lyapunov function as(23)Vt=12∑i=1NeitTeit+12α∑i=1Nqit−qi∗2+12β∑εNbij−aijTbij−aij.
Then, the derivative ofVt can be calculated as follows:(24)V˙=∑i=1NeitTe˙it+1α∑i=1Nqit−qi∗q˙it+1β∑εNbij−aijTa˙ij,=∑k=1M∑i=1NkeiTtFkxit−Fksit+∑j=1Naijcijkkejt−qicijkkxi−skt+∑i=1Nqit−qi∗eiTteit−∑ξbij−aijTxjt−sjt−xit−sit≤ωk∑k=1M∑i=1NkeiTteit+∑k=1M∑i=1NkeiTt∑j=1NaijCkkejt−qi∗Ckkeit−∑ξbij−aijTxjt−sjt−xit−sit=∑k=1M∑i=1NkwkeiTteit+∑j=1nγje˜jTtPe˜jt−∑j=1nγje˜jTtQe˜jt−∑ξbij−aijTxjt−sjt−xit−sit≤∑i=1ne˜jTtwiIN+γjP−Qe˜jt−∑ξbij−aijTxjt−sjt−xit−sit,where P=A⊗Ckk,Q=qi∗Ckk.
Thus, there exists a positive constantbij larger than the corresponding coupling strength aij, i.e.,(25)∑ξbij−aijTejt−eit≥0.
Therefore, one can take sufficiently large positive constantsqi∗, i.e., V˙<0. That is, cluster synchronization can be achieved via designed adaptive controllers (21) and (22).


## 3.1. The Multi-Area Power Network Model

In this section, we consider a complex multi-area power network withM areas and each node is a third-order power system. Suppose that the kth area is composed of Nk nodes. Then, the general multi-area power network can be described as(14)x˙it=Fkx˙it+∑j=1,j≠iNkaijcijkkxjt−xit+∑l=1,l≠kM∑j=1Nlaijtcijklxjt−cijlxiti=1,2,…,Nk,k=1,2,…,M,where xik=xi1k,xi2k,…,xinkkT∈Rnk denotes the state vector of the ith node in the kth area. Fk describes the node’s dynamics of the k th area. The matrix Ckk=diagckk1,ckk2,…,ckknk is the inner coupling matrix which denotes the internal connection in k th area, and Ckl=cklrs∈Rnk×nl and Ckkl=diagckkl1,ckkl2,…,ckklnk are the inner coupling matrices between the k th and l th areas; if the r th element of xit in the k th area is influenced by the l th element of xjt in the l th, then ckklr≠0,cklrs≠0; otherwise, ckklr=0,cklrs=0. Also, matrix A=aijN×N denotes the topology structure of the entire multi-area power network, which is described as follows: if there exists a connection between node i and j, then aij=aji>0i≠j; otherwise, aij=aji=0i≠j.Suppose that the sets of subscripts of these areas areG1=1,2,…,N1,G2=N1+1,N1+2,…,N1+N2,…,GM=N1+⋯+NM−1+1,⋯,N, where N1+N2+⋯+NM=N. The coupling matrix A can be written in the following form:(15)A=A11A12⋯A1MA21A22⋯A2M⋮⋱⋮AM1AMM,where App∈Rrp×rpp=1,2,…,M show the coupling pattern in the same area and Apq∈Rrp×rqp=1,2,…,M,q=1,2,…,M,p≠q are the coupling schemes among different areas.In order to achieve cluster synchronization of the multi-area power network, the control inputsuit∈Rn are added and the controlled multi-area power network can be characterized by the following equation:(16)x˙tt=Fkx˙tt+∑j=1,j≠iNkaijtcijkkxjt−xit+∑l=1,l≠kM∑j=1Nlaijtcijklxjt−cijklxjt−cijlxit+uiti=1,2,…,Nk,k=1,2,…,M.Define the synchronous errors(17)eit=xit−skti=1,2,…,Nk,where skt is a solution of an isolated node in the kthand satisfies s˙kt=Fkskt. The network can realize the cluster synchronization, if limt⟶∞eit=0,i=1,2,…,N.


## 3.2. Cluster Synchronization of Multi-Area Power Network

For simplicity, we suppose that all the inner coupling patterns between different areas are identical, and equation (14) can be derived as(18)x˙it=Fkxit+∑j=1,i≠jNkaijtcijkkxjt−xit+uiti=1,2,…,Nk,k=1,2,…,M.According to the above definition of the error variables, we can get the error dynamical system as follows:(19)eit=Fkxit−Fksit+∑j=1Nkaijcijkkejt+∑j=1Naijcijkksjt+uit.Since the above coupling condition holds, one can obtain(20)∑j=1Naijcijkksjt=0,i∈Gϕi−G¯ϕi,where Gϕi represents all the nodes in the area and G¯ϕi represents the nodes in other areas.Subsequently, the control scheme is given via adaptive pinning control idea. The adaptive controller is designed as follows:(21)uit=−∑j=1Naijcijkksjt−qitcijkkxit−skt,i∈ϕ˜k,k=1,2.…,M0,otherwise,where the coupling strength aijt=aij1t,aij2t,…,aijntT between the nodes and the feedback gains adopts the following adaptive strategy:(22)a˙ijt=βxjt−sjt−xit−sit,i∈ϕ˜k,k=1,2,…,M,q˙i=αeit22,where α>0,β>0 are the adaptive gains.Theorem 1.
For multi-area power network (18), cluster synchronization can be achieved under designed controllers (21) and (22).Proof.
Construct a Lyapunov function as(23)Vt=12∑i=1NeitTeit+12α∑i=1Nqit−qi∗2+12β∑εNbij−aijTbij−aij.
Then, the derivative ofVt can be calculated as follows:(24)V˙=∑i=1NeitTe˙it+1α∑i=1Nqit−qi∗q˙it+1β∑εNbij−aijTa˙ij,=∑k=1M∑i=1NkeiTtFkxit−Fksit+∑j=1Naijcijkkejt−qicijkkxi−skt+∑i=1Nqit−qi∗eiTteit−∑ξbij−aijTxjt−sjt−xit−sit≤ωk∑k=1M∑i=1NkeiTteit+∑k=1M∑i=1NkeiTt∑j=1NaijCkkejt−qi∗Ckkeit−∑ξbij−aijTxjt−sjt−xit−sit=∑k=1M∑i=1NkwkeiTteit+∑j=1nγje˜jTtPe˜jt−∑j=1nγje˜jTtQe˜jt−∑ξbij−aijTxjt−sjt−xit−sit≤∑i=1ne˜jTtwiIN+γjP−Qe˜jt−∑ξbij−aijTxjt−sjt−xit−sit,where P=A⊗Ckk,Q=qi∗Ckk.
Thus, there exists a positive constantbij larger than the corresponding coupling strength aij, i.e.,(25)∑ξbij−aijTejt−eit≥0.
Therefore, one can take sufficiently large positive constantsqi∗, i.e., V˙<0. That is, cluster synchronization can be achieved via designed adaptive controllers (21) and (22).


## 4. Numerical Simulations

In this section, several numerical examples are presented to verify the theoretical results about cluster synchronization in the multi-area power network. In the following simulations, we take the network topology structure shown in Figure1 as example. Moreover, we select the node dynamics of the different area as the third-order power system:(26)x˙i1=dxi2,x˙i2=−axi2+b−czsinxi1,x˙i3=−xi3+0.5cosxi1+0.5,with i=1,2,…,18.Figures6–8 show the evolution of state variables and the error time of each area; from these figures, one can observe that the different areas synchronize to different chaotic orbits and errors evolve toward zero. An intuitive representation of the evolution of the components of the state variables of the multi-area network can be found in Figures 6(a)–8(a). In these figures, the top subfigures show the amplitudes of state variables in different areas and the bottom subfigures represent the corresponding evolution of state variables. In addition, Figures 6–8 show the state trajectories of the nodes in different areas with different initial conditions. As can be seen from these figures, state variablies evolve in the same direction and get closer with time and finally coincide under the designed controller. It implies that cluster synchronization can be realized for the multi-area power network.Figure 6
The evolution of state variables in the first area and the synchronization error.
(a)(b)Figure 7
The evolution of state variables and the synchronization error in the second area.
(a)(b)Figure 8
The evolution of state variables and the synchronization error in the third area.
(a)(b)


## 5. Conclusions

As we know, real power networks have complex structures. This paper investigated the multi-area power network to model the network topology underlying high-voltage transmission grids, while the single node dynamics are described by the third-order chaotic power system. We studied the rich dynamics of the single third-order system via theoretical analysis and numerical simulations. The dynamic behaviors of the system are depicted by means of the phase portrait and Hopf bifurcation analysis. In addition, it is noted that real power networks usually have multi-area structure. Therefore, we analyze the cluster synchronization behavior of the multi-area power network. Furthermore, we presented an adaptive feedback control scheme for achieving cluster synchronization.Even though we investigated these phenomena in a small-scale multi-area power network, we would like to emphasize that small-scale power network structure is indeed in reality.



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*Source: 8421754-2021-10-28.xml*