# Fast Synergetic Control for Chaotic Oscillation in the Power System Based on Input-Output Feedback Linearization

Authors: Ling Liu; Jiangbin Wang; Chongxin Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper presents a fast synergetic control scheme for chaotic oscillation in a three-bus power system model. First, the coupling dynamic model of a controlled power system with the current source converter-based STATCOM device and energy storage device is established. Then, the input-output linearization process for the controlled power system is derived step by step, the control problem for the complex nonlinear power system model is completely transformed into the control of linear systems, and a fast synergetic control scheme is proposed for these linear systems. Since the designed control inputs contain complex system functions which are very difficult to obtain and reduce the engineering practicability of the designed controllers, the assumption that system functions are bounded is introduced into the controller design process, and the controllers are redesigned. The remarkable advantages of the proposed control method are that it improves the rapidity of traditional synergetic control and avoids complex system functions in control inputs. Finally, the effectiveness and the superiority of the control scheme are verified by simulation results.


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## Body

## 1. Introduction

Power system is a typical high-dimensional, complex, and strong coupling dynamic system. Just as many other nonlinear systems exhibit bifurcation, chaos, and other complex nonlinear dynamic behaviors, so do power systems [1–5]. In recent years, scholars have studied how to effectively control chaos in the power system and obtained rich research results [6–8]. At present, the focus of research is to analyze and control chaotic oscillation in a three-bus power system [8–15]. Different from controlling chaos in some simple chaotic systems, the chaos control problem of the power system has obvious physical background. However, the control methods in [12–14] often fail to consider the physical background of the power system and control chaos by means of full-state regulation, which makes the designed control inputs not practical.From the perspective of the power system, the problem of chaos control in the power system does not belong to a chaos control but a power system control problem, and controlling the power system needs to introduce control devices. Among numerous power system control devices, flexible alternating current transmission system (FACTS) devices have superior performance and can provide many superior options [16, 17]. However, the existing chaos control methods for the power system always ignore the use of the FACTS device to control chaotic oscillation in the power system [9–14]. Among the members of FACTS devices, unified power flow controller (UPFC) device has rich control functions and the most excellent performance, and it can adjust active power and reactive power at the same time. However, the structure of the device is complex, and the control input variables are numerous [18–20]; then, it is not conducive to design controllers. Since parallel devices in FACTS devices are easier to implement than series devices, static VAR compensator (SVC) device and static synchronous compensator (STATCOM) device have been widely used in power system control as two typical parallel devices [21–25]. Compared with the SVC device, the STATCOM device is superior in many technical performances [26, 27]; then, the STATCOM device becomes the best choice among these control devices. At present, STATCOM devices have been widely used for reactive compensation and voltage regulation in the power system [24, 25]. There are two types of STATCOM devices: voltage source converter-based device and current source converter-based device. Although the technology of the former is more mature and it is widely used as pointed out in [28, 29], compared with voltage source converter-based STATCOM, current source converter-based STATCOM has more application potential due to its advantages of less harmonic injection. STATCOM device is mainly used for reactive power control of the power system, and it is often necessary to control active power by introducing the energy storage device [30]. Therefore, it has practical value to design controllers for the chaotic power system by combining the current source converter-based STATCOM with the energy storage device.To control complex dynamic systems in engineering, the input-output feedback linearization method based on differential geometry theory provides quite universal methods for these high-dimensional complex systems [31–34]. In order to make the designed controllers have strong robustness, the input-output feedback linearization method is often combined with sliding mode variable structure control. However, the input-output feedback linearization-based sliding mode control methods in [35–37] often use an exponential reaching law with discontinuous terms; then, the chattering problem still widely exists in these control methods. As another variable structure control method, synergetic control can achieve similar control effect with sliding mode control on the basis of overcoming chattering [38]. Therefore, it is of great significance to propose the input-output feedback linearization-based synergetic control method for suppressing the chaotic oscillation in the complex power system.Based on the above discussion, the dynamic models of the current source converter-based STATCOM device and energy storage device in [28–30] are introduced for the three-bus power system, and a complex controlled power system dynamic model is constructed. In order to control chaotic oscillation in the system, the whole nonlinear system is linearized step by step, and a fast synergetic control is proposed for linear systems. Since the traditional synergetic control method has the defect of slow convergence speed, we improve the traditional method by adding an exponential term to the dynamic equations satisfied by macrovariables, thus effectively improving the speed of the system state reaching the invariant manifold. The main contributions of the paper are as follows: (1) a complex coupling dynamic model composed of the power system model, energy storage device model, and current source converter-based STATCOM model is established for chaos control in a three-bus power system; (2) compared with the widely used input-output feedback linearization-based sliding mode control, the proposed input-output feedback linearization-based synergetic control completely avoids the chattering problem; (3) compared with traditional synergetic control, the proposed fast synergetic control improves the rapidity of control by adding an exponential term to the dynamic equation satisfied by the macrovariables of traditional synergetic control. As the system state is far away from the invariant manifold, the exponential term grows exponentially, which significantly accelerates the speed of the system state reaching the invariant manifold and improves the rapidity of traditional synergetic control.The rest of the paper is organized as follows. In Section2, the coupled dynamic model including the power system model, energy storage device model, and STATCOM device model is constructed, and chaotic dynamic behavior in the power system is analyzed. In Section 3, the design process of the fast synergetic controller based on input-output feedback linearization is presented. In Section 4, the effectiveness and superiority of the proposed method are verified by numerical simulation. Finally, the conclusion is given in Section 5.


## 2. Power System Model and Dynamic Analysis

The power system studied is a three-bus power system. The generator bus of the power system is connected to the energy storage device, and the load bus is connected to the energy storage device and the current source converter-based STATCOM device. The circuit diagram of the controlled power system formed after connecting these control devices is shown in Figure1.Figure 1
Wiring diagram of the controlled power system.The coupling dynamic model of the power system in Figure1 is established as follows:(1)δ˙m=ω=f1,ω˙=1M⋅−dmω+Pm+EmYmVLsinδL−δm−θm+Em2Ymsinθm−Pes1=f2,P˙es1=−1TesPes1+Kes1Tes⋅ues1=f3+Kes1Tes⋅ues1,δ˙L=1Kqw⋅−KqwVL2−KqvVL+E0′Y0′VLcosδL+θ0′+EmYmVLcosδL−δm+θm−Y0′cosθ0′+YmcosθmVL2−Q0−Q1−QST=f4,I˙q=−ωId−RLIq+1LVq=f5,V˙q=−1CIq−ωVd+1C⋅uST1=f6+1C⋅uST1,I˙dc2=−2RdcLdcIdc2−3EdLdc⋅nId=f7,I˙d=−RLId+ωIq−1L⋅Edn+1LVd=f8,V˙d=−1CId+ωVq+1C⋅uST2=f9+1C⋅uST2,V˙L=1TKqwKpv⋅KpwKqv2VL2+KpwKqv−KqwKpvVL+Kqw−E0′Y0′VLsinδL+θ0′−EmYmVLsinδL−δm+θm+Y0′sinθ0′+YmsinθmVL2−P0−P1−Pes2−KpwE0′Y0′VLcosδL+θ0′+EmYmVLcosδL−δm+θm−Y0′cosθ0′+YmcosθmVL2−Q0−Q1−QST=f10,P˙es2=−1TesPes2+Kes2Tes⋅ues2=f11+Kes2Tes⋅ues2,where δm and ω represent the generator power angle and frequency deviation; δL and VL represent the phase angle and amplitude of load bus voltage; Pes1 and Pes2 indicate the active power absorbed by two energy storage devices from the generator bus and load bus; Id and Iq denote the d-axis and q-axis components of the AC side current of the STATCOM; Vd and Vq represent the d-axis and q-axis components of the AC side voltage; Idc2 is the square of its DC side current; and QST=−3EdIq/2 represents the reactive power injected into the load bus by the STATCOM. The eleven variables mentioned above are state variables of the controlled power system. ues1 and ues2 represent control inputs of the two energy storage devices; uST1 and uST2 represent control inputs of the STATCOM. These four variables are input variables of the controlled power system.Values of constant parameters in system (1) are listed as follows: Em=1.05, Ym=5.0, θm=0, E0′=20, Y0′=0.1665, θ0′=0, dm=0.05, M=0.01464, Kpw=0.4, Kqv2=2.1, Kqw=−0.03, Kqv=−2.8, Kpv=0.3, T=8.5, P0=0.6, Q0=1.3, P1=0, Q1=2.9, Pm=1.102, Ed=1, Rdc=0.1, Ldc=0.5, n=1, R=0.01, L=0.1, C=1.5, and ω=1. The physical meanings of parameters in the four-dimensional power system model and STATCOM device model are available in [15, 28]. Most values of power system parameters are taken from [15] except for some important bifurcation parameters Pm and Q1. The initial values of state variables of system (1) are taken as follows: δm0=0.29, ω0=0.2, Pes10=0, δL0=0.23, Iq0=0, Vq0=0, Idc20=0, Id0=0, Vd0=0, VL0=0.8, and Pes20=0. All the aforementioned values are in per unit except for angles, which are in degrees. Then, system (1) becomes a deterministic system with specific parameters and initial values. The first, second, fourth, and tenth equations of system (1) are considered, we let Pes1=0, Pes2=0, and QST=0, and then the dynamic system composed of these four equations is the original four-dimensional power system model. Under the given system parameters and initial values, chaotic oscillation occurs, and the strange attractor in the system is shown in Figure 2.Figure 2
Strange attractor in the power system.
(a)(b)


## 3. Chaos Controller Design

In order to control chaotic oscillation in the power system, it is necessary to design a chaos controller for controlled power system (1). The control objective of system (1) is to restore the power system to the synchronous operation state, control its bus voltage to the rated value, and maintain the DC side current of the STATCOM device to its reference value. The specific control objectives of system (1) are set as follows: δmref=0, δLref=0, Idcref2=1, and VLref=1. The system output functions y1, z1, w1, and m1 (i.e., output variables) are expressed as follows:(2)y1=L1δm,z1=L2δL,w1=L3Idc2−1,m1=L4VL−1,where L1>0, L2>0, L3>0, and L4>0.For convenience of expression, system (1) is written as a standard form of the multi-input multioutput nonlinear system as follows:(3)x˙=fx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,y1=h1x=L1δm,z1=h2x=L2δL,w1=h3x=L3Idc2−1,m1=h4x=L4VL−1,where(4)x=δm,ω,Pes1,δL,Iq,Vq,Idc2,Id,Vd,VL,Pes2T,fx=f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11T,ges1x=0,0,Kes2Tes,0,0,0,0,0,0,0,0T,gST1x=0,0,0,0,0,1C,0,0,0,0,0T,gST2x=0,0,0,0,0,0,0,0,1C,0,0T,ges2x=0,0,0,0,0,0,0,0,0,0,Kes2TesT.


### 3.1. Input-Output Feedback Linearization Process

The core idea of input-output feedback linearization is to transform the control problem of a complex nonlinear system into the control of linear systems through nonlinear coordinate transformation. The core step is to derive the output function step by step until at least one control input appears in the linear system in the form of explicit function [39]. The following are the derivations of the output functions:(1)
The derivatives of the first output functiony1 in (2) are obtained step by step. The first derivative of y1 is(5)y˙1=∂h1x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lfh1x+Lges1h1x⋅ues1+LgST1h1x⋅uST1+LgST2h1x⋅uST2+Lges2h1x⋅ues2.Since(6)Lges1h1x=0,LgST1h1x=0,LgST2h1x=0,Lges2h1x=0,system (5) becomes(7)y˙1=Lfh1x=L1⋅f1=L1⋅ω.There is no control input in system (7); then, continue to calculate the second derivative of y1 as follows:(8)y¨1=∂Lfh1x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf2h1x+Lges1Lfh1x⋅ues1+LgST1Lfh1x⋅uST1+LgST2Lfh1x⋅uST2+Lges2Lfh1x⋅ues2.In (8), since(9)Lges1Lfh1x=0,LgST1Lfh1x=0,LgST2Lfh1x=0,Lges2Lfh1x=0,system (8) becomes(10)y¨1=Lf2h1x=L1⋅f2.There is still no control input in system (10) in the form of the explicit function; we continue to calculate the third derivative of y1 as follows:(11)y…1=∂Lf2h1x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf3h1x+Lges1Lf2h1x⋅ues1+LgST1Lf2h1x⋅uST1+LgST2Lf2h1x⋅uST2+Lges2Lf2h1x⋅ues2.Since(12)Lges1Lf2h1x=−L1Kes1M⋅Tes,LgST1Lf2h1x=0,LgST2Lf2h1x=0,Lges2Lf2h1x=0,system (11) becomes(13)y…1=Lf3h1x+Lges1Lf2h1x⋅ues1.SinceLges1Lf2h1x≠0, the control input ues1 will exist in system (13) in the form of an explicit function, and system (13) is equivalent to a linear system as follows:(14)y˙1=y2,y˙2=y3,y˙3=v1,wherev1=Lf3h1x+Lges1Lf2h1x⋅ues1, representing the virtual control input of system (14).In this way, the linear relationship between the virtual control inputv1 and the output function y1 is established, and the controller design problem of the control input ues1 in nonlinear system (1) is transformed into the design process of the control input v1 in linear system (14). The transformation relationship between ues1 and v1 is(15)ues1=−Lf3h1x+v1Lges1Lf2h1x.(2)
The derivatives of the second output functionz1 in (2) are obtained step by step until at least one control input appears in the system in the form of the explicit function. First, the first derivative of z1 is obtained as follows:(16)z˙1=∂h2x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lfh2x+Lges1h2x⋅ues1+LgST1h2x⋅uST1+LgST2h2x⋅uST2+Lges2h2x⋅ues2.In (16), since(17)Lges1h2x=0,LgST1h2x=0,LgST2h2x=0,Lges2h2x=0,system (16) becomes(18)z˙1=Lfh2x=L2⋅f4.Since there is no control input in system (18) in the form of an explicit function, we continue to calculate the second derivative of z1 as follows:(19)z¨1=∂Lfh2x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2.=Lf2h2x+Lges1Lfh2x⋅ues1+LgST1Lfh2x⋅uST1+LgST2Lfh2x⋅uST2+Lges2Lfh2x⋅ues2.In (19), since(20)Lges1Lfh2x=0,LgST1Lfh2x=0,LgST2Lfh2x=0,Lges2Lfh2x=0,system (19) becomes(21)z¨1=Lf2h2x.Since there is no control input in (21) in the form of an explicit function, the third derivative of z1 is obtained as follows:(22)z…1=∂Lf2h2x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf3h2x+Lges1Lf2h2x⋅ues1+LgST1Lf2h2x⋅uST1+LgST2Lf2h2x⋅uST2+Lges2Lf2h2x⋅ues2.In (22),(23)Lges1Lf2h2x=0,LgST2Lf2h2x=0,Lges2Lf2h2x=0,LgST1Lf2h2x=∂Lf2h2x∂Vq⋅1C=L2⋅1Kqw⋅3Ed2⋅1LC.In this way, system (21) is transformed into(24)z…1=Lf3h2x+LgST1Lf2h2x⋅uST1.Similarly, sinceLgST1Lf2h2x≠0, the control input uST1 in (24) appears in the form of an explicit function, while system (24) is equivalent to a linear system:(25)z˙1=z2,z˙2=z3,z˙3=v2,wherev2=Lf3h2x+LgST1Lf2h2x⋅uST1, representing the virtual control input of system (25).In this way, the controller design ofuST1 for nonlinear system (1) is transformed into the design process of the control input v2 for linear system (25). The transformation relationship between uST1 and v2 is as follows:(26)uST1=−Lf3h2x+v2LgST1Lf2h2x.(3)
The derivatives of the third output functionw1 in (2) are obtained step by step until at least one control input appears in the system in the form of an explicit function, and the first derivative of w1 is obtained as follows:(27)w˙1=∂h3x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2.=Lfh3x+Lges1h3x⋅ues1+LgST1h3x⋅uST1+LgST2h3x⋅uST2+Lges2h3x⋅ues2.In (27), since(28)Lges1h3x=0,LgST1h3x=0,LgST2h3x=0,Lges2h3x=0,the system can be changed into(29)w˙1=Lfh3x=L3⋅f7.Since there is no control input in system (27) in the form of an explicit function, we continue to calculate the second derivative of w1 as follows:(30)w¨1=∂Lfh3x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf2h3x+Lges1Lfh3x⋅ues1+LgST1Lfh3x⋅uST1+LgST2Lfh3x⋅uST2+Lges2Lfh3x⋅ues2.Since(31)Lges1Lfh3x=0,LgST1Lfh3x=0,LgST2Lfh3x=0,Lges2Lfh3x=0,system (30) becomes(32)w¨1=Lf2h3x.Since there is no control input in (32) that appears in the form of an explicit function, we continue to calculate the third derivative of w1 as follows:(33)w…1=∂Lf2h3x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf3h3x+Lges1Lf2h3x⋅ues1+LgST1Lf2h3x⋅uST1+LgST2Lf2h3x⋅uST2+Lges2Lf2h3x⋅ues2.In (33), since(34)Lges1Lf2h3x=0,LgST1Lf2h3x=0,Lges2Lf2h3x=0,LgST2Lf2h3x=∂Lf2h3x∂Vd⋅1C=L3⋅−3EdLdc⋅n⋅1LC,the system becomes(35)w…1=Lf3h3x+LgST2Lf2h3x⋅uST2.In (35), since LgST2Lf2h3x≠0, uST2 appears in system (33) in the form of an explicit function, and system (35) is equivalent to a linear system as follows:(36)w˙1=w2,w˙2=w3,w˙3=v3,wherev3=Lf3h3x+LgST2Lf2h3x⋅uST2, representing the virtual control input of system (36).In this way, the controller design ofuST2 for system (1) is transformed into the design process of control input v3 for linear system (36). That is, as long as uST2 is designed for system (36) to make the whole linear system stable to the origin, the third output of system (1) will be controlled to the target. The transformation relationship between uST2 and v3 is as follows:(37)uST2=−Lf3h3x+v3LgST2Lf2h3x.(4)
The derivatives of the fourth output functionm1 in (2) are obtained step by step until at least one control input appears in the system in the form of an explicit function, and the first derivative of m1 is obtained as follows:(38)m˙1=∂h4x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lfh4x+Lges1h4x⋅ues1+LgST1h4x⋅uST1+LgST2h4x⋅uST2+Lges2h4x⋅ues2.In (36), since(39)Lges1h4x=0,LgST1h4x=0,LgST2h4x=0,Lges2h4x=0,system (38) can be changed into(40)m˙1=Lfh4x=L4⋅f10.Since there is no control input in system (40) in the form of an explicit function, we continue to calculate the second derivative of m1 as follows:(41)m¨1=∂Lfh4x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf2h4x+Lges1Lfh4x⋅ues1+LgST1Lfh4x⋅uST1+LgST2Lfh4x⋅uST2+Lges2Lfh4x⋅ues2.In (41), since(42)Lges1Lfh4x=0,LgST1Lfh4x=0,LgST2Lfh4x=0,Lges2Lfh4x=−L4Kes2T⋅Kpv⋅Tes,the system becomes(43)m¨1=Lf2h4x+Lges2Lfh4x⋅ues2.In (43), since Lges2Lfh4x≠0, ues2 appears in system (43) in the form of an explicit function, and system (43) is equivalent to a linear system:(44)m˙1=m2,m˙2=v4,where v4=Lf2h4x+Lges2Lfh4x⋅ues2, representing the virtual control input of system (44).In this way, the design of controllerues2 for nonlinear system (1) is transformed into the controller design problem for linear system (44). The transformation relationship between ues2 and v4 is as follows:(45)ues2=−Lf2h4x+v4Lges2Lfh4x.By using the input-output feedback linearization method, the controller design problems of four control inputsues1, uST1, uST2, and ues2 in system (1) are transformed into the stabilization of linear systems (14), (25), (36), and (44). If the linear systems are controlled to the origin by designing control inputs ues1, uST1, uST2, and ues2, then the four outputs in (2) can be restored to the origin, and the control objective of system (1) can be realized. After the above derivation, the linearization process of controlled power system (1) is completed.


### 3.2. Fast Synergetic Controller Design

Synergetic control inputs can be designed for linear systems (14), (25), (36), and (44) to control them to the origin. Since four control inputs need to be designed, four macrovariables are defined as follows:(46)ψ1=y3+α2y2+α1y1,ψ2=z3+β2z2+β1z1,ψ3=w3+γ2w2+γ1w1,ψ4=m2+η1m1,where values of parameters α2, α1, β2, β1, γ2, γ1, and η1 should make the roots of characteristic equations s2+α2s+α1=0, s2+β2s+β1=0, s2+γ2s+γ1=0, and s+η1=0 located in the left half plane of the complex plane.To make the system state reach the invariant manifoldsψ1=0, ψ2=0, ψ3=0, and ψ4=0, respectively, let the dynamic equations satisfied by macrovariables be(47)T1ψ˙1+eψ1ψ1=0,T2ψ˙2+eψ2ψ2=0T3ψ˙3+eψ3ψ3=0,T4ψ˙4+eψ4ψ4=0,where T1>0, T2>0, T3>0, and T4>0.Remark 1.
According to the traditional synergetic control [40–45], equations in (47) should be expressed as follows:(48)T1ψ˙1+ψ1=0,T2ψ˙2+ψ2=0,T3ψ˙3+ψ3=0,T4ψ˙4+ψ4=0.
However, sinceeψi≥1i=1,2,3,4 in (47), the larger ψi is, the larger the coefficient eψi is, and it will grow exponentially. That is to say, with the same values of Tii=1,2,3,4, the farther the system state is from the invariant manifold, the faster the coefficient eψi increases exponentially, and it significantly improves the speed of the system state reaching the invariant manifold.
Combining (46) and (47) with the last equation of (14), (25), (36), and (44), the control inputs are obtained as follows:(49)ues1=−1Lges1Lf2h1xLf3h1x+α2y3+α1y2+eψ1⋅ψ1T1,uST1=−1LgST1Lf2h2xLf3h2x+β2z3+β1z2+eψ2⋅ψ2T2,uST2=−1LgST2Lf2h3xLf3h3x+γ2w3+γ1w2+eψ3⋅ψ3T3,ues2=−1Lges2Lfh4xLf2h4x+η1m2+eψ4⋅ψ4T4.
However, due to the limitation of relative degree, control inputs (49) contain complex system functions with derivative terms Lf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x. In particular, Lf3h2x and Lf2h4x are more complex and extremely difficult to obtain, and they also reduce the engineering practicability of controllers (49). Therefore, we assume that Lf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x satisfy assumptions (50) and redesign the controllers.(50)Lf3h1x≤k1tanhψ1ε1,Lf3h2x≤k2tanhψ2ε2,Lf3h3x≤k3tanhψ3ε3,Lf2h4x≤k4tanhψ4ε4,where k1, k2, k3, k4, ε1, ε2, ε3, and ε4 are all positive constants, and the values of constants ε1, ε2, ε3, and ε4 are far less than 1.Remark 2.
The forms of output functionsy1, z1, w1, and m1 can also be set as follows:(51)y1=δm,z1=δL,w1=Idc2−1,m1=VL−1.
That is to say, if we letL1=1, L2=1, L3=1, and L4=1 in outputs (2), we can get outputs (51). However, from the above input-output feedback linearization derivation process, it can be obtained that Lf3h1x∝L1, Lf3h2x∝L2, Lf3h3x∝L3, and Lf2h4x∝L4. Considering assumption (50), the reason why the outputs are set to (2) instead of (51) is that the constant parameters L1, L2, L3, and L4 can be used to limit the amplitude of four system functions Lf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x so that the values of parameters k1, k2, k3, and k4 are not too large. Since parameters L1, L2, L3, and L4 play the role of limiting system function amplitudes, the values of these four parameters should be far less than 1.Theorem 1.
Let the system functionsLf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x satisfy assumptions (50). If the control inputs for system (1) are designed as in (52), the four macrovariables ψ1, ψ2, ψ3, and ψ4 will asymptotically reach the invariant manifolds ψ1=0, ψ2=0, ψ3=0, and ψ4=0.(52)ues1=−1Lges1Lf2h1xk1⋅tanhψ1ε1+α2y3+α1y2+eψ1⋅ψ1T1,uST1=−1LgST1Lf2h2xk2⋅tanhψ2ε2+β2z3+β1z2+eψ2⋅ψ2T2,uST2=−1LgST2Lf2h3xk3⋅tanhψ3ε3+γ2w3+γ1w2+eψ3⋅ψ3T3,ues2=−1Lges2Lfh4xk4⋅tanhψ4ε4+η1m2+eψ4⋅ψ4T4.Proof.
We first prove that macrovariableψ1 can reach the invariant manifold ψ1=0 under the control of ues1 in (52). Construct a Lyapunov function V1=ψ12/2, combine (14) and (46) with the control input ues1 in (52), and consider assumption (50); the derivative of V1 with respect to time can be obtained as follows:(53)V˙1=ψ1⋅ψ˙1=ψ1⋅Lf3h1x+Lges1Lf2h1x⋅ues1+α2y3+α1y2,=ψ1⋅Lf3h1x+ψ1⋅Lges1Lf2h1x⋅ues1+α2y3+α1y2,≤ψ1⋅k1tanhψ1ε1+ψ1⋅Lges1Lf2h1x⋅ues1+α2y3+α1y2,=ψ1⋅k1tanhψ1ε1+Lges1Lf2h1x⋅ues1+α2y3+α1y2,=−eψ1⋅ψ12T1≤0.
It shows from (53) that macrovariable ψ1 reaches the invariant manifold ψ1=0 under the control input ues1 in (52). When the macrovariable reaches the invariant manifold, the first output of system (1) is controlled.
The proof process of macrovariablesψ2, ψ3, and ψ4 reaching the invariant manifold ψ2=0, ψ3=0, and ψ4=0 under control inputs uST1, uST2, and ues2 in (52), respectively, is similar to the derivation process in (53), and there is no further proof here. Obviously, compared with inputs in (49), control inputs in (52) do not contain complex system functions Lf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x; then, controllers in (52) are more practical.


## 3.1. Input-Output Feedback Linearization Process

The core idea of input-output feedback linearization is to transform the control problem of a complex nonlinear system into the control of linear systems through nonlinear coordinate transformation. The core step is to derive the output function step by step until at least one control input appears in the linear system in the form of explicit function [39]. The following are the derivations of the output functions:(1)
The derivatives of the first output functiony1 in (2) are obtained step by step. The first derivative of y1 is(5)y˙1=∂h1x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lfh1x+Lges1h1x⋅ues1+LgST1h1x⋅uST1+LgST2h1x⋅uST2+Lges2h1x⋅ues2.Since(6)Lges1h1x=0,LgST1h1x=0,LgST2h1x=0,Lges2h1x=0,system (5) becomes(7)y˙1=Lfh1x=L1⋅f1=L1⋅ω.There is no control input in system (7); then, continue to calculate the second derivative of y1 as follows:(8)y¨1=∂Lfh1x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf2h1x+Lges1Lfh1x⋅ues1+LgST1Lfh1x⋅uST1+LgST2Lfh1x⋅uST2+Lges2Lfh1x⋅ues2.In (8), since(9)Lges1Lfh1x=0,LgST1Lfh1x=0,LgST2Lfh1x=0,Lges2Lfh1x=0,system (8) becomes(10)y¨1=Lf2h1x=L1⋅f2.There is still no control input in system (10) in the form of the explicit function; we continue to calculate the third derivative of y1 as follows:(11)y…1=∂Lf2h1x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf3h1x+Lges1Lf2h1x⋅ues1+LgST1Lf2h1x⋅uST1+LgST2Lf2h1x⋅uST2+Lges2Lf2h1x⋅ues2.Since(12)Lges1Lf2h1x=−L1Kes1M⋅Tes,LgST1Lf2h1x=0,LgST2Lf2h1x=0,Lges2Lf2h1x=0,system (11) becomes(13)y…1=Lf3h1x+Lges1Lf2h1x⋅ues1.SinceLges1Lf2h1x≠0, the control input ues1 will exist in system (13) in the form of an explicit function, and system (13) is equivalent to a linear system as follows:(14)y˙1=y2,y˙2=y3,y˙3=v1,wherev1=Lf3h1x+Lges1Lf2h1x⋅ues1, representing the virtual control input of system (14).In this way, the linear relationship between the virtual control inputv1 and the output function y1 is established, and the controller design problem of the control input ues1 in nonlinear system (1) is transformed into the design process of the control input v1 in linear system (14). The transformation relationship between ues1 and v1 is(15)ues1=−Lf3h1x+v1Lges1Lf2h1x.(2)
The derivatives of the second output functionz1 in (2) are obtained step by step until at least one control input appears in the system in the form of the explicit function. First, the first derivative of z1 is obtained as follows:(16)z˙1=∂h2x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lfh2x+Lges1h2x⋅ues1+LgST1h2x⋅uST1+LgST2h2x⋅uST2+Lges2h2x⋅ues2.In (16), since(17)Lges1h2x=0,LgST1h2x=0,LgST2h2x=0,Lges2h2x=0,system (16) becomes(18)z˙1=Lfh2x=L2⋅f4.Since there is no control input in system (18) in the form of an explicit function, we continue to calculate the second derivative of z1 as follows:(19)z¨1=∂Lfh2x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2.=Lf2h2x+Lges1Lfh2x⋅ues1+LgST1Lfh2x⋅uST1+LgST2Lfh2x⋅uST2+Lges2Lfh2x⋅ues2.In (19), since(20)Lges1Lfh2x=0,LgST1Lfh2x=0,LgST2Lfh2x=0,Lges2Lfh2x=0,system (19) becomes(21)z¨1=Lf2h2x.Since there is no control input in (21) in the form of an explicit function, the third derivative of z1 is obtained as follows:(22)z…1=∂Lf2h2x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf3h2x+Lges1Lf2h2x⋅ues1+LgST1Lf2h2x⋅uST1+LgST2Lf2h2x⋅uST2+Lges2Lf2h2x⋅ues2.In (22),(23)Lges1Lf2h2x=0,LgST2Lf2h2x=0,Lges2Lf2h2x=0,LgST1Lf2h2x=∂Lf2h2x∂Vq⋅1C=L2⋅1Kqw⋅3Ed2⋅1LC.In this way, system (21) is transformed into(24)z…1=Lf3h2x+LgST1Lf2h2x⋅uST1.Similarly, sinceLgST1Lf2h2x≠0, the control input uST1 in (24) appears in the form of an explicit function, while system (24) is equivalent to a linear system:(25)z˙1=z2,z˙2=z3,z˙3=v2,wherev2=Lf3h2x+LgST1Lf2h2x⋅uST1, representing the virtual control input of system (25).In this way, the controller design ofuST1 for nonlinear system (1) is transformed into the design process of the control input v2 for linear system (25). The transformation relationship between uST1 and v2 is as follows:(26)uST1=−Lf3h2x+v2LgST1Lf2h2x.(3)
The derivatives of the third output functionw1 in (2) are obtained step by step until at least one control input appears in the system in the form of an explicit function, and the first derivative of w1 is obtained as follows:(27)w˙1=∂h3x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2.=Lfh3x+Lges1h3x⋅ues1+LgST1h3x⋅uST1+LgST2h3x⋅uST2+Lges2h3x⋅ues2.In (27), since(28)Lges1h3x=0,LgST1h3x=0,LgST2h3x=0,Lges2h3x=0,the system can be changed into(29)w˙1=Lfh3x=L3⋅f7.Since there is no control input in system (27) in the form of an explicit function, we continue to calculate the second derivative of w1 as follows:(30)w¨1=∂Lfh3x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf2h3x+Lges1Lfh3x⋅ues1+LgST1Lfh3x⋅uST1+LgST2Lfh3x⋅uST2+Lges2Lfh3x⋅ues2.Since(31)Lges1Lfh3x=0,LgST1Lfh3x=0,LgST2Lfh3x=0,Lges2Lfh3x=0,system (30) becomes(32)w¨1=Lf2h3x.Since there is no control input in (32) that appears in the form of an explicit function, we continue to calculate the third derivative of w1 as follows:(33)w…1=∂Lf2h3x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf3h3x+Lges1Lf2h3x⋅ues1+LgST1Lf2h3x⋅uST1+LgST2Lf2h3x⋅uST2+Lges2Lf2h3x⋅ues2.In (33), since(34)Lges1Lf2h3x=0,LgST1Lf2h3x=0,Lges2Lf2h3x=0,LgST2Lf2h3x=∂Lf2h3x∂Vd⋅1C=L3⋅−3EdLdc⋅n⋅1LC,the system becomes(35)w…1=Lf3h3x+LgST2Lf2h3x⋅uST2.In (35), since LgST2Lf2h3x≠0, uST2 appears in system (33) in the form of an explicit function, and system (35) is equivalent to a linear system as follows:(36)w˙1=w2,w˙2=w3,w˙3=v3,wherev3=Lf3h3x+LgST2Lf2h3x⋅uST2, representing the virtual control input of system (36).In this way, the controller design ofuST2 for system (1) is transformed into the design process of control input v3 for linear system (36). That is, as long as uST2 is designed for system (36) to make the whole linear system stable to the origin, the third output of system (1) will be controlled to the target. The transformation relationship between uST2 and v3 is as follows:(37)uST2=−Lf3h3x+v3LgST2Lf2h3x.(4)
The derivatives of the fourth output functionm1 in (2) are obtained step by step until at least one control input appears in the system in the form of an explicit function, and the first derivative of m1 is obtained as follows:(38)m˙1=∂h4x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lfh4x+Lges1h4x⋅ues1+LgST1h4x⋅uST1+LgST2h4x⋅uST2+Lges2h4x⋅ues2.In (36), since(39)Lges1h4x=0,LgST1h4x=0,LgST2h4x=0,Lges2h4x=0,system (38) can be changed into(40)m˙1=Lfh4x=L4⋅f10.Since there is no control input in system (40) in the form of an explicit function, we continue to calculate the second derivative of m1 as follows:(41)m¨1=∂Lfh4x∂xfx+ges1xues1+gST1xuST1+gST2xuST2+ges2xues2,=Lf2h4x+Lges1Lfh4x⋅ues1+LgST1Lfh4x⋅uST1+LgST2Lfh4x⋅uST2+Lges2Lfh4x⋅ues2.In (41), since(42)Lges1Lfh4x=0,LgST1Lfh4x=0,LgST2Lfh4x=0,Lges2Lfh4x=−L4Kes2T⋅Kpv⋅Tes,the system becomes(43)m¨1=Lf2h4x+Lges2Lfh4x⋅ues2.In (43), since Lges2Lfh4x≠0, ues2 appears in system (43) in the form of an explicit function, and system (43) is equivalent to a linear system:(44)m˙1=m2,m˙2=v4,where v4=Lf2h4x+Lges2Lfh4x⋅ues2, representing the virtual control input of system (44).In this way, the design of controllerues2 for nonlinear system (1) is transformed into the controller design problem for linear system (44). The transformation relationship between ues2 and v4 is as follows:(45)ues2=−Lf2h4x+v4Lges2Lfh4x.By using the input-output feedback linearization method, the controller design problems of four control inputsues1, uST1, uST2, and ues2 in system (1) are transformed into the stabilization of linear systems (14), (25), (36), and (44). If the linear systems are controlled to the origin by designing control inputs ues1, uST1, uST2, and ues2, then the four outputs in (2) can be restored to the origin, and the control objective of system (1) can be realized. After the above derivation, the linearization process of controlled power system (1) is completed.


## 3.2. Fast Synergetic Controller Design

Synergetic control inputs can be designed for linear systems (14), (25), (36), and (44) to control them to the origin. Since four control inputs need to be designed, four macrovariables are defined as follows:(46)ψ1=y3+α2y2+α1y1,ψ2=z3+β2z2+β1z1,ψ3=w3+γ2w2+γ1w1,ψ4=m2+η1m1,where values of parameters α2, α1, β2, β1, γ2, γ1, and η1 should make the roots of characteristic equations s2+α2s+α1=0, s2+β2s+β1=0, s2+γ2s+γ1=0, and s+η1=0 located in the left half plane of the complex plane.To make the system state reach the invariant manifoldsψ1=0, ψ2=0, ψ3=0, and ψ4=0, respectively, let the dynamic equations satisfied by macrovariables be(47)T1ψ˙1+eψ1ψ1=0,T2ψ˙2+eψ2ψ2=0T3ψ˙3+eψ3ψ3=0,T4ψ˙4+eψ4ψ4=0,where T1>0, T2>0, T3>0, and T4>0.Remark 1.
According to the traditional synergetic control [40–45], equations in (47) should be expressed as follows:(48)T1ψ˙1+ψ1=0,T2ψ˙2+ψ2=0,T3ψ˙3+ψ3=0,T4ψ˙4+ψ4=0.
However, sinceeψi≥1i=1,2,3,4 in (47), the larger ψi is, the larger the coefficient eψi is, and it will grow exponentially. That is to say, with the same values of Tii=1,2,3,4, the farther the system state is from the invariant manifold, the faster the coefficient eψi increases exponentially, and it significantly improves the speed of the system state reaching the invariant manifold.
Combining (46) and (47) with the last equation of (14), (25), (36), and (44), the control inputs are obtained as follows:(49)ues1=−1Lges1Lf2h1xLf3h1x+α2y3+α1y2+eψ1⋅ψ1T1,uST1=−1LgST1Lf2h2xLf3h2x+β2z3+β1z2+eψ2⋅ψ2T2,uST2=−1LgST2Lf2h3xLf3h3x+γ2w3+γ1w2+eψ3⋅ψ3T3,ues2=−1Lges2Lfh4xLf2h4x+η1m2+eψ4⋅ψ4T4.
However, due to the limitation of relative degree, control inputs (49) contain complex system functions with derivative terms Lf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x. In particular, Lf3h2x and Lf2h4x are more complex and extremely difficult to obtain, and they also reduce the engineering practicability of controllers (49). Therefore, we assume that Lf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x satisfy assumptions (50) and redesign the controllers.(50)Lf3h1x≤k1tanhψ1ε1,Lf3h2x≤k2tanhψ2ε2,Lf3h3x≤k3tanhψ3ε3,Lf2h4x≤k4tanhψ4ε4,where k1, k2, k3, k4, ε1, ε2, ε3, and ε4 are all positive constants, and the values of constants ε1, ε2, ε3, and ε4 are far less than 1.Remark 2.
The forms of output functionsy1, z1, w1, and m1 can also be set as follows:(51)y1=δm,z1=δL,w1=Idc2−1,m1=VL−1.
That is to say, if we letL1=1, L2=1, L3=1, and L4=1 in outputs (2), we can get outputs (51). However, from the above input-output feedback linearization derivation process, it can be obtained that Lf3h1x∝L1, Lf3h2x∝L2, Lf3h3x∝L3, and Lf2h4x∝L4. Considering assumption (50), the reason why the outputs are set to (2) instead of (51) is that the constant parameters L1, L2, L3, and L4 can be used to limit the amplitude of four system functions Lf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x so that the values of parameters k1, k2, k3, and k4 are not too large. Since parameters L1, L2, L3, and L4 play the role of limiting system function amplitudes, the values of these four parameters should be far less than 1.Theorem 1.
Let the system functionsLf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x satisfy assumptions (50). If the control inputs for system (1) are designed as in (52), the four macrovariables ψ1, ψ2, ψ3, and ψ4 will asymptotically reach the invariant manifolds ψ1=0, ψ2=0, ψ3=0, and ψ4=0.(52)ues1=−1Lges1Lf2h1xk1⋅tanhψ1ε1+α2y3+α1y2+eψ1⋅ψ1T1,uST1=−1LgST1Lf2h2xk2⋅tanhψ2ε2+β2z3+β1z2+eψ2⋅ψ2T2,uST2=−1LgST2Lf2h3xk3⋅tanhψ3ε3+γ2w3+γ1w2+eψ3⋅ψ3T3,ues2=−1Lges2Lfh4xk4⋅tanhψ4ε4+η1m2+eψ4⋅ψ4T4.Proof.
We first prove that macrovariableψ1 can reach the invariant manifold ψ1=0 under the control of ues1 in (52). Construct a Lyapunov function V1=ψ12/2, combine (14) and (46) with the control input ues1 in (52), and consider assumption (50); the derivative of V1 with respect to time can be obtained as follows:(53)V˙1=ψ1⋅ψ˙1=ψ1⋅Lf3h1x+Lges1Lf2h1x⋅ues1+α2y3+α1y2,=ψ1⋅Lf3h1x+ψ1⋅Lges1Lf2h1x⋅ues1+α2y3+α1y2,≤ψ1⋅k1tanhψ1ε1+ψ1⋅Lges1Lf2h1x⋅ues1+α2y3+α1y2,=ψ1⋅k1tanhψ1ε1+Lges1Lf2h1x⋅ues1+α2y3+α1y2,=−eψ1⋅ψ12T1≤0.
It shows from (53) that macrovariable ψ1 reaches the invariant manifold ψ1=0 under the control input ues1 in (52). When the macrovariable reaches the invariant manifold, the first output of system (1) is controlled.
The proof process of macrovariablesψ2, ψ3, and ψ4 reaching the invariant manifold ψ2=0, ψ3=0, and ψ4=0 under control inputs uST1, uST2, and ues2 in (52), respectively, is similar to the derivation process in (53), and there is no further proof here. Obviously, compared with inputs in (49), control inputs in (52) do not contain complex system functions Lf3h1x, Lf3h2x, Lf3h3x, and Lf2h4x; then, controllers in (52) are more practical.


## 4. Simulation Results

The effectiveness of the proposed control scheme is verified by numerical simulation using MATLAB software. Values of control parameters are selected as follows:Tes=1, Kes1=1, Kes2=1, α1=16, α2=8, β1=16, β2=8, γ1=16, γ2=8, η1=5, L1=0.2, L2=0.2, L3=0.2, L4=0.2, k1=500, k2=500, k3=500, k4=500, ε1=0.2, ε2=0.2, ε3=0.2, ε4=0.2, T1=0.5, T2=0.5, T3=0.5, and T4=0.5. All values of the aforementioned parameters are in per unit except for angles, which are in degrees. After the four controllers (52) are put into operation, time responses of state variables of the whole controlled power system (1) are given as in Figures 3 and 4. It can be seen from Figure 3 that chaotic oscillation in the four-dimensional power system is effectively controlled, and it can be seen from Figure 4 that the state variables of the STATCOM device and energy storage device are also restored to the equilibrium state, and then the whole controlled power system can be restored to the stable operation state.Figure 3
Time responses of the four-dimensional power system.Figure 4
Time responses of the two control devices.Figure5 displays the state evolution process of the uncontrolled power system and the controlled power system in the phase plane. As shown in Figure 5, the uncontrolled power system gradually forms strange attractors in the δm−ω phase plane and δL−VL phase plane. While the controlled system is different, the state of the whole controlled power system gradually evolves into fixed point B through initial point A in the δm−ω phase plane, and the system state gradually evolves into fixed point D through initial point C in the δL−VL phase plane. All the above simulation results verify the effectiveness of the proposed controllers.Figure 5
Evolution diagram of the system state in the phase plane.
(a)(b)In order to highlight the superiority of the proposed fast synergetic control method (47) over the traditional synergetic control method (48), the time responses of macrovariable (46) under the controllers given by the two methods are shown in Figure 6. In Figure 6, ψ1, ψ2, ψ3, and ψ4, respectively, represent the macrovariables under the fast synergetic controllers (52), while ψtr1, ψtr2, ψtr3, and ψtr4, respectively, represent the macrovariables under the traditional synergetic controller (the expression of the traditional synergetic controller can be obtained by letting eψi=1i=1,2,3,4 in (52)). It can be seen from Figure 6 that the evolution speed of each macrovariable under the proposed fast synergetic controller is obviously faster than that under the traditional synergetic controller.Figure 6
Time responses of macrovariables under the proposed fast synergetic controller and the traditional synergetic controller.


## 5. Conclusions

(1)
Aiming at chaos control of a three-bus power system, a nonlinear dynamic system model of the complex controlled power system is constructed. The nonlinear system is linearized step by step, the control problem of the nonlinear power system is transformed into controlling linear systems, and a fast synergetic control scheme is proposed to suppress the chaotic oscillation in the power system.(2)
Although the power system dynamic model with the current source converter-based STATCOM device and energy storage device is only for the four-dimensional power system, the coupling method of the dynamic system can be extended to other three-bus power systems to build other controlled power system dynamic models and put forward similar control methods. Although only the current source converter-based STATCOM device is studied and its chaos control inputs are designed, the voltage source converter-based STATCOM device can also be used to build a similar dynamic model for controlling chaos in the three-bus power system and propose a similar control method.(3)
Different from the widely used input-output feedback linearization-based sliding mode control, the proposed input-output feedback linearization-based fast synergetic control scheme can provide continuous control laws for the controlled power system, and then the proposed control method has advantages in overcoming the chattering problem. Compared with the traditional synergetic control method, exponential terms are added to the coefficient terms of dynamic equations satisfied by the macrovariables, and the coefficient terms grow exponentially when the system state is far away from the invariant manifolds; then, the proposed fast synergetic control scheme significantly improves the rapidity of traditional synergetic control.The results of the paper show that chaos control in the power system is ultimately attributed to the control of its active power and reactive power, and the unified power flow controller (UPFC) has the ability to regulate active and reactive power at the same time. Therefore, UPFC can be considered to control chaos in the power system in our future work.



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*Source: 8404192-2021-07-20.xml*