# An Improved Adaptive Passive Coordinated Control for Generator Excitation and STATCOM

Authors: Chaofan Du; Lei Zhang; Yuanhui Tang; Jiahao Zhu; Kaifeng Wang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

An improved adaptive backstepping passive coordinated control strategy is proposed to deal with the uncertainty of damping coefficient and mechanical power and external disturbance on the system stability. A nonlinear coordinated controller of generator excitation and STATCOM (static synchronous compensator) is designed to improve its adaptability. The adaptive estimation laws of damping coefficient and mechanical power are designed by using I&I (immersion and invariance) adaptive control to improve the adaptive ability of the controller. Based on the backstepping recursive method, the control law of STATCOM is designed and combines the passivity theory, thus acquiring the control law of generator excitation. In the backstepping design, the derivative of the virtual control input and external disturbance are considered to be uncertain, and the nonlinear damping algorithm is introduced to reduce the “calculation explosion.” Simulation results show that the designed controller improves the stability of the power system and shows strong adaptability and robustness.


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## Body

## 1. Introduction

At present, the scale and structure of power grid is more and more large and complex, and the transmission system has developed into a large capacity, long-distance, and ultra-high voltage transmission form [1, 2]. The following problem is the stability of the power system. Flexible AC Transmission System (FACTS) can solve the stability problem of the power system economically and effectively [3, 4]. The static synchronous compensator (STATCOM) is a good FACTS device to improve the voltage quality of the control installation point, adjust the reactive power of the power grid, increase the transmission capacity of the line, and improve the transient stability of the power system [5, 6]. The generator excitation system has always played an important role in solving the stability problem of the power system [7]. In the transmission system equipped with STATCOM, the interaction of the different controllers is the kernel that impacts the system dynamics. Therefore, it is extremely valuable to research on how to design the coordinated controller of generator excitation and STATCOM.Many uncertain parameters in the power system are the main factors affecting the stability of the transmission system [8]. When establishing the mathematical model of the transmission system equipped with STACOM, some approximate processing is usually done or the parameters are assumed to be known, leading to the existence of uncertain parameters [9]. However, when the system suffers from disturbances such as generator fault or fault load removal during operation, the current network structure of the system will be different from that of modeling. In the early stage, the linear control method [10] ignores the uncertainty in system modeling, and its design of parameters are not robust, when system changes. To solve the shortcomings of the linear control method, adopting nonlinear control methods to design the system has become a popular research subject in recent years, such as the target holographic feedback method, fuzzy control theory [11], sliding mode variable control method [12], passive control method [13], Hamiltonian theory [14], and adaptive backstepping method [15]. Among the above control methods, the adaptive backstepping method, as a simple and effective nonlinear control method, has been widely used in designing the coordinated controller of excitation and STACOM, achieving good results [15]. The designed controller based on the adaptive backstepping method [15] can only estimate a single parameter, and this algorithm has poor estimation ability for a single parameter. Inspired by [16], I&I adaptive control algorithm can design parameter adaptive laws for multiple uncertain parameters in the system model at the same time, estimating multiple parameters and improving the estimation ability of parameters.In addition, there is a problem of “calculation explosion” when using the backstepping method in higher-order systems. The existing literature mainly adopts dynamic surface control [17] and nonlinear tracking differentiator algorithm [18] to solve this problem. However, dynamic surface control can make the tracking error of the system converge exponentially to the bounded set, which is not globally asymptotically stable; the design process of nonlinear tracking differentiator algorithm is complex and difficult to understand. Pan et al. [19] regard the derivative of the backstepping virtual control as an uncertain term, introduce a robust term to solve the problem of “calculation explosion,” and the system satisfies the global asymptotic stability. Therefore, this paper adopts the idea of [19] and introduces the nonlinear damping algorithm to deal with uncertain items. This algorithm has a simple design process, overcomes the defects of the above two control methods, and has strong robustness.Contributions of this paper: considering the influence of the uncertain two parameters on the stability of generator excitation and STATCOM coordinated control system, based on I&I adaptive control, the adaptive estimation laws are designed, which breaks through the certainty equivalence principle, solves the bottleneck problem of multiparameters’ identification in the system, and enhances the adaptive ability of the controller. Moreover, the virtual control input and unknown disturbance in the system are regarded as uncertain items, and the nonlinear damping algorithm is used to compensate the uncertain items to solve the inverse problem.


## 2. Passivity and Design Method

Definition 1.
Nonlinear systems as follows:(1)x˙=fx+gxu,y=hx,where x∈Rn, u∈Rm, and f, g, and h are all fairly smooth function vectors.
If there is a semipositive definite functionVx (V: x∈R, V0=0), such that the following inequality,(2)Vxt−Vx0≤∫0tyTτuτdτ,holds for ∀t≥0 and u∈Rm, the nonlinear system (equation (1)) is called a passive system, where Vx is the energy storage function and equation (2) is called dissipation inequality.Definition 2.
For the passive system (1) and the energy storage function Vx, there is a positive definite function Qx, such that(3)Vxt−Vx0≤∫0tyTτuτdτ−∫0tQxdτ.
If∀t≥0, u∈Rm, and equation (3) holds, then the resultant system (equation (1)) is strictly passive.
According to equations (2) and (3), we can know that the zero output of the system is Lyapunov stability. If the system is strictly passive, then the zero output of the system is Lyapunov asymptotic stability [20].Theorem 1.
For a two-input system, an input-output pairu,y is selected for which the relative degree is one order, and the system (equation (1)) can be converted into the following form:(4)z˙=qz,y+pz,yu2,y˙=αz,y+β1z,yu1+β2z,yu2.The control law of (4) is(5)u1=β1−1z,y−β2z,yu2z−αz,y−∂W∂zp˜z,y+v,u2=γz,where p˜z,y=qz,0+pz,0γz. Then, the Lyapunov function Wz such that(6)W˙=∂W∂zqz,0+pz,0γz<−αz,where α is a k-class function.Take the energy storage functionVz as(7)V=Wz+12y2,which satisfy V≥0 and V˙≤0. Therefore, the system (equation (4)) can be guaranteed to be asymptotically stable.It can be easily verified that the storage function satisfies for system (4), which proves the passivity of the pair [20].


## 3. Power System Model

The research object of this paper is the single machine infinite bus (SMIB) power system structure with STATCOM considering generator excitation control, as shown in Figure1. The generator excitation control equipment is connected to the power transmission line with the circuit breaker through the transformer, and STATCOM is installed at the midpoint of the line. Research shows that STATCOM can effectively improve the transmission capacity of the system, extend the transmission distance, and give full play to STATCOM’s role in power system stability [14, 21].Figure 1
Configuration of generator excitation and STATCOM coordinated control system.Assumption 1.
Not considering the effects of the governor.Assumption 2.
Not considering the stator loop resistance and the rotor damping winding.
In case of Assumptions1 and 2, the generator excitation system studied in this paper adopts the third-order mathematical model. The topology of STATCOM is voltage type shown in Figure 2, and the first-order controllable current source model is adopted. Therefore, the state equation of the generator excitation and STATCOM coordination control system shown in Figure 1 is as follows:(8)δ˙=ω−ω0,ω˙=−DHω−ω0+ω0HPm−Pe+w1,E˙q′=−xdΣxdΣ′Td0Eq′+xd−xd′xdΣ′Td0Vscosδ+1Td0uf+w2,I˙q=1Tq−Iq+Iq0+us+w3,where δ is the generator rotor angle, ω is the generator rotor angular speed, H is the inertia constant, Pm is the mechanical power, D is the unit damping coefficient; Pe=Eq′Vssinδ/xdΣ′ is the delivered electrical power, Eq′ is the generator q-axis transient electromotive force, Vs is the bus voltage of the infinite bus system, set to 1 p.u, xdΣ=xd+XL+XT, and xdΣ′=xd′+XL+XT.
xdΣ and xdΣ′ are the equivalent steady-state reactance and transient reactance of transmission line, respectively. Among, xd and xd′ are the generator d-axis steady-state reactance and transient reactance, respectively. XT is the reactance of the transformer; XL=XL1+XL1, XL1 and XL2 are the equivalent transmission line reactance, respectively; Td0 is the open-circuit transient time constant of d-axis; Tq is the inertial time constant of STATCOM; Iq is the output current of the equivalent controllable power supply of STATCOM; uf and us are the control input of the generator excitation and STATCOM, respectively. w=w1w2w3T is the uncertainty disturbance applied to the generator rotor, admittance, and STATCOM controller, and wit≤hi holds, and hi is a positive number.
In the coordinated control system, select the state variables of the system asx1x2x3x4T=δ−δ0ω−ω0Iq−Iq0Eq′−Eq0′T, where the initial values correspond to the variables of δ0, ω0, Iq0, and Eq0'. Assume that the input is us and uf, the output y=ΔEq′, θ1=−D/H and θ2=Pm, and θ1 and θ2 is the uncertain parameter. The system model (8) can be manipulated as(9)x˙1=x2,x˙2=θ1x2+a1θ2−a1ΔPe+w1,x˙3=a2x3+a3us+w3,y˙=a4y+a5cosx1+δ0+a6uf+w2,where a1=ω0/H, a2=−1/Tq, a3=1/Tqa4=−xdΣ/Td0xdΣ′, a5=xdΣ−xdΣ′Vs/Td0xdΣ′, a6=1/Td0, b=−xdΣEq0′/Td0xdΣ′, ΔPe=y+Eq0′Vssinx1+δ0/xdΣ′+Δx−Eq0′Vssinδ0/xdΣ′, and Δx=XL1XL2ΔIq.
To facilitate the design of the controller, (9) can be written as follows:(10)x˙1x˙2x˙3=x2θ1x2+a1θ2+a1VSEq0′sinx1+δ0xdΣ′+Δx−sinδ0xdΣ′+w1a2x3+w3+0a1Vssinx1+δ0xdΣ′+Δx0y+00a3u1=f1x+f2xy+gxus,y˙=a4y+a5cosx1+δ0+a6uf+b+w2.
Control objective is as follows. An I&I adaptive robust coordinated control strategy is designed for system (8) with multiple parameter uncertainties and unknown disturbances. Design the adaptive laws to recognize the uncertain parameters. Design the coordinated control laws for the generator excitation and the STATCOM so that the control system is robust for the uncertain disturbances and the error system has a globally asymptotical stability at the equilibrium point which indicates that the state error variables converge to zero.Figure 2
Topology structure of voltage-type STATCOM.


## 4. Controller Design

The design process of generator excitation and STATCOM-coordinated controller is mainly as follows: under the condition of no excitation control, aiming at the coordinated control system (2) with uncertain parameters, damping coefficient, and mechanical power. Firstly, the I&I control method is adopted to design the uncertain parameter adaptive law; then, the STATCOM control law is designed by using the backstepping method and nonlinear damping algorithm. Applying passivity theory, the excitation control law is designed.According to the idea of passivity (9), the state equation without excitation control is(11)x˙1=x2,x˙2=θ1x2+a1θ2−a1ΔPe1+w1,x˙3=a2x3+a3us+w3,where ΔPe1=y+Eq0′Vssinx1+δ0/xdΣ′+Δx−Eq0′Vssinδ0/xdΣ′.


### 4.1. Parameter Adaptive Law Design

According to the I&I adaptive control method, an adaptive law is designed for two uncertain parameters at the same time. The process is as follows.Firstly, the parameter estimation errors of two uncertain parameters are defined as follows:(12)z1=θ^1−θ1+β1x1,x2,z2=θ^2−θ2+β2x1,x2,where θ^1 and θ^2 are estimates of θ1 and θ2, respectively, and β1x1,x2 and β2x1,x2 are the functions to be designed, respectively.Then, by substituting (11) into (12), therefore, (12) is obtained as(13)z˙1=θ^˙1+∑k=12∂β1∂xk×x˙k=θ^˙1+∂β1∂x1x˙1+∂β1∂x2θ1x2+a1θ2−a1ΔPe1+w1,z˙2=θ^˙2+∑k=12∂β2∂xk×x˙k=θ^˙2+∂β2∂x1x˙1+∂β2∂x2θ1x2+a1θ2−a1ΔPe1+w1.The adaptive laws for designing two uncertain parameters are as follows:(14)θ^˙1=−∂β1∂x1x2−∂β1∂x2θ^1+β1x2+a1θ^2+β2−a1ΔPe1−h1,θ^˙2=−∂β2∂x1x2−∂β2∂x2θ^1+β1x2+a1θ^2+β2−a1ΔPe1−h1.Substituting (14) into (13), we can obtain(15)z˙1=−∂β1∂x2z1x2+z2a1−h1−w1,z˙2=−∂β2∂x2z1x2+z2a1−h1−w1.Take the adjustable functions as follows:(16)β1x1,x2=ρx22,β2x1,x2=ρa1x2,where ρ>0.Construct the parameter estimation error system (15) with the alternative Lyapunov function (CLF) as(17)Vz1,z2=12z12+12z22.The derivative ofVz1,z2 is(18)V˙z1,z2=z1z˙1+z2z˙2=−∂β1∂x2z1x2+z2a1−h1−w1z1−∂β2∂x2z1x2+z2a1−h1−w1z2≤−ρz1x2+z2k1−h1−w122≤0.According to (18), Vz1,z2 is negative under the parameter adaptive law (14). Therefore, according to LaSalle’s theorem, the designed parameter adaptive law can ensure the asymptotic stability of the dynamics of the parameter estimation error.


### 4.2. Design Control Law of STATCOMu1

Step 1
: the state error function of the system is defined as(19)e1=x1,e2=x2−x2∗,where x2∗ is “virtual control quantity” and its control law will be designed in the following backstepping method.Step 2
: firstly, the derivation ofe1 is given by(20)e˙2=θx2+k1Pm−k2x3+Eq0′xdΣ′sinx1+δ0+w1+c1x2.According to the idea of document [19], e2 is regarded as an uncertain term and Assumption 1: e2≤χ1, where χ1 is the normal number to be designed.Applying the nonlinear damping algorithm,x2∗ is designed as follows:(21)x2∗=−k1e1−χ12e12λ1,where k1>0, λ1>0, and χ12e1/2λ1 is the nonlinear damping term to compensate for uncertainties [19].Take the Lyapunov function of (11) as(22)V1=12e12.Then, the derivation ofV1 can be described as(23)V˙1=e1e˙1=e1e2+x2∗=e1e2−k1e12−χ12e122λ1=−k1e12−χ12e122λ1+e1e2≤−k1e12+χ1e1−χ12e122λ1.Whene1≥2λ1/χ1 is satisfied, we have(24)V˙1≤−k1e12≤0.Step 3
: the derivation ofe2 is given by(25)e˙2=x˙2−x˙2∗=θ1x2+a1θ2−a1ΔPe1+w1−x˙2∗=θ1x2+a1θ2−a1ΔPed+a1e3+w1−x˙2∗.Similarly, according to the idea of [19], a1e3−x˙2∗ is regarded as an uncertain term and Assumption 2: a1e3+w1−x˙2∗≤χ2, where χ2 is the normal number to be designed.Applying the nonlinear damping algorithm,ΔPe1d is designed as follows:(26)ΔPe1d=1a1θ^1+12x22x2+θ^2+a1x2+k2e2+χ22e22λ2,where k2>0, λ2>0, and χ22e2/2λ2 is the nonlinear damping term to compensate for uncertainties [19].The extended Lyapunov function is(27)V2=V1+12e22.Then, the derivation ofV2 can be described as(28)V˙2=V˙1+e2e˙2=−k1e12+e2θ1x2+a1θ2−a1ΔPe1d+a1e3+w1−x˙2∗=−k1e12−k2e22−e2z1x2+z2a1+e2a1e3+w1−x˙2∗−χ22e222λ2≤−k1e12−k2e22−e2z1x2+z2a1+χ2e2−χ22e222λ2.According to the abovementioned I&I parameter estimation error, derivative dynamics is asymptotically stable; then,limzi=1,2⟶∞e2z1x2+z2a1=0 holds.Whene2≥2ε2/χ2 is satisfied, we have(29)V˙2≤−k1e12−k2e22−e2z1x2+z2a1≤0.Step 4
: definee3=ΔPed−ΔPe1d, and set ΔPe1d as the virtual control quantity.The derivation ofe3 is given by(30)e˙3=ΔP˙e−ΔP˙ed=c1cosx1+δ0x2−c2sinx1+δ0a2x3+a3us+w3−ΔP˙e1d=c1cosx1+δ0x2−c2sinx1+δ0a2x3+a3us−ΔP˙e1d+c2sinx1+δ0w3,where c1=Eq0′Vs/xdΣ′ and c2=Eq0′VsxL1xL2/xdΣ′2.Similarly,−ΔP˙e1d−c2sinx1+δ0w3 is regarded as an uncertain term, and according to idea of [19], −ΔP˙e1d−c2sinx1+δ0w3≤χ3, where χ3 is the normal number to be designed.Applying the nonlinear damping algorithm, whensinx1+δ0≠0, us is designed as follows:(31)us=1a3c2sinx1+δ0c1x2cosx1+δ0−a2c2sinx1+δ0+k3e3+χ32e32λ3.Take the global Lyapunov function of (11) as follows:(32)V3=V2+12e32.Then, the derivation ofV3 can be described as(33)V˙3=V˙2+e3e˙3=V˙2+e3c1cosx1+δ0x2−c2sinx1+δ0a2x3+a3us+w3−ΔP˙e1d=−k1e12−k2e22−e2z1x2+z2a1−k3e32+e3−ΔP˙e1d−c2sinx1+δ0w3−χ32e322λ3≤−k1e12−k2e22−e2z1x2+z2a1−k3e32+χ3e2−χ32e322λ3.Whene3≥2λ3/χ3 is satisfied, we have(34)V˙3≤−k1e12−k2e22−k3e32−e2z1x2+z2a1.


### 4.3. Design the Control Law of Generator Excitationu2

LetWe=V3, and take the storage function of the whole system as(35)V=We+12y2.According to equation (5), u2 is designed as(36)uf=1a6e2ω0VsHxdΣ′sinx1+δ0−a5cosx1+δ0−b+v.Then, the derivation ofV can be described as(37)V˙=∂We∂ee˙|y=0+yy˙=∂We∂ee˙|y=0+a4y2+vy≤a4y2+vy≤vy.Due toV˙≤vy , the passivity condition is satisfied. In addition, take v=−k4y, where k4>0. According to Theorem 1, system (8) can be guaranteed to be asymptotically stable at the equilibrium.To sum up, the control law of the generator and STATCOM coordination controller is as followed:(38)us=1a3c2sinx1+δ0c1x2cosx1+δ0−a2c2sinx1+δ0+k3e3+χ32e32λ3,uf=1a6e2ω0VsHxdΣ′sinx1+δ0−a5cosx1+δ0−b+v.Therefore, the closed-loop error system of the system (equation (8)) is(39)e˙1=e2−k1e1−χ12e12λ1,e˙2=−ρz1x2+z2a1−k2e2−χ22e22λ2,e˙3=−k3e3−χ32e32λ3,y˙=a4y−e2a1Vssinx1+δ0xdΣ′+v.According to formula (37), the state error variables of the closed-loop system (39) converge to zero. Therefore, the state variables of system (8) are stable at the equilibrium point.Compared with the adaptive backstepping coordinated control method adopted in [15], the advantages are as follows. (1) Based on the I&I adaptive control method, the adaptive laws of two uncertain parameters in the system are designed, and two tunable functions are added to the adaptive laws, which improves the flexibility and control ability of parameter estimation. (2) The design of adaptive laws is separated from the design of control laws, and the nonlinear damping term is introduced in the backstepping method to compensate the uncertain term, which solves the problem of “calculation expansion” to a certain extent.


## 4.1. Parameter Adaptive Law Design

According to the I&I adaptive control method, an adaptive law is designed for two uncertain parameters at the same time. The process is as follows.Firstly, the parameter estimation errors of two uncertain parameters are defined as follows:(12)z1=θ^1−θ1+β1x1,x2,z2=θ^2−θ2+β2x1,x2,where θ^1 and θ^2 are estimates of θ1 and θ2, respectively, and β1x1,x2 and β2x1,x2 are the functions to be designed, respectively.Then, by substituting (11) into (12), therefore, (12) is obtained as(13)z˙1=θ^˙1+∑k=12∂β1∂xk×x˙k=θ^˙1+∂β1∂x1x˙1+∂β1∂x2θ1x2+a1θ2−a1ΔPe1+w1,z˙2=θ^˙2+∑k=12∂β2∂xk×x˙k=θ^˙2+∂β2∂x1x˙1+∂β2∂x2θ1x2+a1θ2−a1ΔPe1+w1.The adaptive laws for designing two uncertain parameters are as follows:(14)θ^˙1=−∂β1∂x1x2−∂β1∂x2θ^1+β1x2+a1θ^2+β2−a1ΔPe1−h1,θ^˙2=−∂β2∂x1x2−∂β2∂x2θ^1+β1x2+a1θ^2+β2−a1ΔPe1−h1.Substituting (14) into (13), we can obtain(15)z˙1=−∂β1∂x2z1x2+z2a1−h1−w1,z˙2=−∂β2∂x2z1x2+z2a1−h1−w1.Take the adjustable functions as follows:(16)β1x1,x2=ρx22,β2x1,x2=ρa1x2,where ρ>0.Construct the parameter estimation error system (15) with the alternative Lyapunov function (CLF) as(17)Vz1,z2=12z12+12z22.The derivative ofVz1,z2 is(18)V˙z1,z2=z1z˙1+z2z˙2=−∂β1∂x2z1x2+z2a1−h1−w1z1−∂β2∂x2z1x2+z2a1−h1−w1z2≤−ρz1x2+z2k1−h1−w122≤0.According to (18), Vz1,z2 is negative under the parameter adaptive law (14). Therefore, according to LaSalle’s theorem, the designed parameter adaptive law can ensure the asymptotic stability of the dynamics of the parameter estimation error.


## 4.2. Design Control Law of STATCOMu1

Step 1
: the state error function of the system is defined as(19)e1=x1,e2=x2−x2∗,where x2∗ is “virtual control quantity” and its control law will be designed in the following backstepping method.Step 2
: firstly, the derivation ofe1 is given by(20)e˙2=θx2+k1Pm−k2x3+Eq0′xdΣ′sinx1+δ0+w1+c1x2.According to the idea of document [19], e2 is regarded as an uncertain term and Assumption 1: e2≤χ1, where χ1 is the normal number to be designed.Applying the nonlinear damping algorithm,x2∗ is designed as follows:(21)x2∗=−k1e1−χ12e12λ1,where k1>0, λ1>0, and χ12e1/2λ1 is the nonlinear damping term to compensate for uncertainties [19].Take the Lyapunov function of (11) as(22)V1=12e12.Then, the derivation ofV1 can be described as(23)V˙1=e1e˙1=e1e2+x2∗=e1e2−k1e12−χ12e122λ1=−k1e12−χ12e122λ1+e1e2≤−k1e12+χ1e1−χ12e122λ1.Whene1≥2λ1/χ1 is satisfied, we have(24)V˙1≤−k1e12≤0.Step 3
: the derivation ofe2 is given by(25)e˙2=x˙2−x˙2∗=θ1x2+a1θ2−a1ΔPe1+w1−x˙2∗=θ1x2+a1θ2−a1ΔPed+a1e3+w1−x˙2∗.Similarly, according to the idea of [19], a1e3−x˙2∗ is regarded as an uncertain term and Assumption 2: a1e3+w1−x˙2∗≤χ2, where χ2 is the normal number to be designed.Applying the nonlinear damping algorithm,ΔPe1d is designed as follows:(26)ΔPe1d=1a1θ^1+12x22x2+θ^2+a1x2+k2e2+χ22e22λ2,where k2>0, λ2>0, and χ22e2/2λ2 is the nonlinear damping term to compensate for uncertainties [19].The extended Lyapunov function is(27)V2=V1+12e22.Then, the derivation ofV2 can be described as(28)V˙2=V˙1+e2e˙2=−k1e12+e2θ1x2+a1θ2−a1ΔPe1d+a1e3+w1−x˙2∗=−k1e12−k2e22−e2z1x2+z2a1+e2a1e3+w1−x˙2∗−χ22e222λ2≤−k1e12−k2e22−e2z1x2+z2a1+χ2e2−χ22e222λ2.According to the abovementioned I&I parameter estimation error, derivative dynamics is asymptotically stable; then,limzi=1,2⟶∞e2z1x2+z2a1=0 holds.Whene2≥2ε2/χ2 is satisfied, we have(29)V˙2≤−k1e12−k2e22−e2z1x2+z2a1≤0.Step 4
: definee3=ΔPed−ΔPe1d, and set ΔPe1d as the virtual control quantity.The derivation ofe3 is given by(30)e˙3=ΔP˙e−ΔP˙ed=c1cosx1+δ0x2−c2sinx1+δ0a2x3+a3us+w3−ΔP˙e1d=c1cosx1+δ0x2−c2sinx1+δ0a2x3+a3us−ΔP˙e1d+c2sinx1+δ0w3,where c1=Eq0′Vs/xdΣ′ and c2=Eq0′VsxL1xL2/xdΣ′2.Similarly,−ΔP˙e1d−c2sinx1+δ0w3 is regarded as an uncertain term, and according to idea of [19], −ΔP˙e1d−c2sinx1+δ0w3≤χ3, where χ3 is the normal number to be designed.Applying the nonlinear damping algorithm, whensinx1+δ0≠0, us is designed as follows:(31)us=1a3c2sinx1+δ0c1x2cosx1+δ0−a2c2sinx1+δ0+k3e3+χ32e32λ3.Take the global Lyapunov function of (11) as follows:(32)V3=V2+12e32.Then, the derivation ofV3 can be described as(33)V˙3=V˙2+e3e˙3=V˙2+e3c1cosx1+δ0x2−c2sinx1+δ0a2x3+a3us+w3−ΔP˙e1d=−k1e12−k2e22−e2z1x2+z2a1−k3e32+e3−ΔP˙e1d−c2sinx1+δ0w3−χ32e322λ3≤−k1e12−k2e22−e2z1x2+z2a1−k3e32+χ3e2−χ32e322λ3.Whene3≥2λ3/χ3 is satisfied, we have(34)V˙3≤−k1e12−k2e22−k3e32−e2z1x2+z2a1.


## 4.3. Design the Control Law of Generator Excitationu2

LetWe=V3, and take the storage function of the whole system as(35)V=We+12y2.According to equation (5), u2 is designed as(36)uf=1a6e2ω0VsHxdΣ′sinx1+δ0−a5cosx1+δ0−b+v.Then, the derivation ofV can be described as(37)V˙=∂We∂ee˙|y=0+yy˙=∂We∂ee˙|y=0+a4y2+vy≤a4y2+vy≤vy.Due toV˙≤vy , the passivity condition is satisfied. In addition, take v=−k4y, where k4>0. According to Theorem 1, system (8) can be guaranteed to be asymptotically stable at the equilibrium.To sum up, the control law of the generator and STATCOM coordination controller is as followed:(38)us=1a3c2sinx1+δ0c1x2cosx1+δ0−a2c2sinx1+δ0+k3e3+χ32e32λ3,uf=1a6e2ω0VsHxdΣ′sinx1+δ0−a5cosx1+δ0−b+v.Therefore, the closed-loop error system of the system (equation (8)) is(39)e˙1=e2−k1e1−χ12e12λ1,e˙2=−ρz1x2+z2a1−k2e2−χ22e22λ2,e˙3=−k3e3−χ32e32λ3,y˙=a4y−e2a1Vssinx1+δ0xdΣ′+v.According to formula (37), the state error variables of the closed-loop system (39) converge to zero. Therefore, the state variables of system (8) are stable at the equilibrium point.Compared with the adaptive backstepping coordinated control method adopted in [15], the advantages are as follows. (1) Based on the I&I adaptive control method, the adaptive laws of two uncertain parameters in the system are designed, and two tunable functions are added to the adaptive laws, which improves the flexibility and control ability of parameter estimation. (2) The design of adaptive laws is separated from the design of control laws, and the nonlinear damping term is introduced in the backstepping method to compensate the uncertain term, which solves the problem of “calculation expansion” to a certain extent.


## 5. Simulation Results and Analysis

MATLAB is adopted for simulation analysis. We construct the simulation system model represented by formula (8) with S function. And, the system simulation parameters are as follows [14, 17]: H=5.9, Eq′=1.0, ω0=314.15, δ0=50∘, Tq=0.02, Td0=0.59, xd=1.5, xd′=0.5, XL1=0.2, XL=2.2XL2=0.1, and XT=0.7. The virtual control variable parameters are as follows: k1=200, k2=10, k3=20, k4=450, χ1=6, χ2=20, χ3=0.1, λ1=0.55, λ2=20, and λ3=0.02. Under the same conditions, the design in this paper and the adaptive backstepping method in [15] are divided into two cases for simulation analysis. Among them, the simulation diagrams of STATCOM control input us and excitation control input uf within the physical change range are shown in Figures 3 and 4.Figure 3
Control input ofus.Figure 4
Control input ofuf.


### 5.1. Simulation Analysis under Small Interference

Whent=0, the power angle of the generator deviates from the equilibrium point, which indicates that the system has suffered small interference. Figures 5–8 show the simulation results of generator power angle, generator angular velocity, generator potential, and equivalent current of STATCOM. In the figure, the solid line is the simulation result under the control of the improved adaptive backstepping passive coordinated control (IABPC) proposed in this paper, and the dashed line is the adaptive backstepping coordinated control (ABCC) used in the literature’s [15] simulation result.Figure 5
Transient response curve of state variablex1.Figure 6
Transient response curve of state variablex2.Figure 7
Transient response curve of state variablex3.Figure 8
Transient response curve of state variablex4.It can be seen from Figures5–8 that when the system suffers from small interference, compared with the ABCC control method, the IABPC control method can quickly quell the power oscillation, and the time for restoring stability is shortened, which proves that IABPC controller has better static performance.


### 5.2. Simulation Analysis under the Condition of Large Interference

Whent=0.1, three-phase short circuit occurs at the outlet of the bus at the high voltage side of the transformer, and the fault is removed in 0.15 s. The simulation results are shown in Figures 9–12. It can be seen from Figures 9–12 that when the system is greatly disturbed, the IABPC control method can quickly stabilize the disturbed system and make it return to a stable state with fast convergence speed and small oscillation range and amplitude, which improves the transient stability of the system. Its effect is better than that of the ABCC control method.Figure 9
Transient response curve of state variablex1.Figure 10
Transient response curve of state variablex2.Figure 11
Transient response curve of state variablex3.Figure 12
Transient response curve of state variablex4.To verify the strong adaptive ability of the controller designed in this paper, the IABPC method and the ABCC method are compared and analyzed, and the simulation results are shown in Figures13–14. In the simulation, if the damping coefficient D = 1, the parameter = −0.143; if the damping coefficient Pm = 1, the parameter = 1. From Figures 13 and 14, it can be seen that compared with the ABCC method, the adaptive estimation law designed by the IABPC method is closer to the set value and has stronger adaptive ability.Figure 13
Response of uncertain parameter estimationθ1.Figure 14
Response of uncertain parameter estimationθ2.


## 5.1. Simulation Analysis under Small Interference

Whent=0, the power angle of the generator deviates from the equilibrium point, which indicates that the system has suffered small interference. Figures 5–8 show the simulation results of generator power angle, generator angular velocity, generator potential, and equivalent current of STATCOM. In the figure, the solid line is the simulation result under the control of the improved adaptive backstepping passive coordinated control (IABPC) proposed in this paper, and the dashed line is the adaptive backstepping coordinated control (ABCC) used in the literature’s [15] simulation result.Figure 5
Transient response curve of state variablex1.Figure 6
Transient response curve of state variablex2.Figure 7
Transient response curve of state variablex3.Figure 8
Transient response curve of state variablex4.It can be seen from Figures5–8 that when the system suffers from small interference, compared with the ABCC control method, the IABPC control method can quickly quell the power oscillation, and the time for restoring stability is shortened, which proves that IABPC controller has better static performance.


## 5.2. Simulation Analysis under the Condition of Large Interference

Whent=0.1, three-phase short circuit occurs at the outlet of the bus at the high voltage side of the transformer, and the fault is removed in 0.15 s. The simulation results are shown in Figures 9–12. It can be seen from Figures 9–12 that when the system is greatly disturbed, the IABPC control method can quickly stabilize the disturbed system and make it return to a stable state with fast convergence speed and small oscillation range and amplitude, which improves the transient stability of the system. Its effect is better than that of the ABCC control method.Figure 9
Transient response curve of state variablex1.Figure 10
Transient response curve of state variablex2.Figure 11
Transient response curve of state variablex3.Figure 12
Transient response curve of state variablex4.To verify the strong adaptive ability of the controller designed in this paper, the IABPC method and the ABCC method are compared and analyzed, and the simulation results are shown in Figures13–14. In the simulation, if the damping coefficient D = 1, the parameter = −0.143; if the damping coefficient Pm = 1, the parameter = 1. From Figures 13 and 14, it can be seen that compared with the ABCC method, the adaptive estimation law designed by the IABPC method is closer to the set value and has stronger adaptive ability.Figure 13
Response of uncertain parameter estimationθ1.Figure 14
Response of uncertain parameter estimationθ2.


## 6. Conclusion and Future Work

In this paper, an improved adaptive backstepping passive coordinated control strategy for generator excitation and STATCOM is designed by adopting I&I adaptive control, backstepping method, and passivity theory. The uncertainties of parameters and disturbances are taken into account in the coordinated control system. The derivative of the virtual control input is treated as an uncertainty and nonlinear damping algorithm is applied to solve the “calculation explosion” problem inherent in the backstepping method. The proposed coordinated control strategy in this paper only considers the influence of two uncertain parameters on system stability, while the system model also contains other uncertain parameters, such as line reactance, transformer reactance, and electromagnetic power. In the later research, the influences of multiple uncertain parameters in the system will be considered to ensure the stability of the control system. In addition, the situation in practical engineering application is often complicated, and the multimachine power system is usually more likely to work for real. Therefore, this method will be extended in the multimachine power system.



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*Source: 6674542-2021-08-03.xml*