# Adaptive Dynamic Surface Control for Generator Excitation Control System

Authors: Zhang Xiu-yu; Liu Cui-ping; Wang Jian-guo; Lin Yan
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

For the generator excitation control system which is equipped with static var compensator (SVC) and unknown parameters, a novel adaptive dynamic surface control scheme is proposed based on neural network and tracking error transformed function with the following features: (1) the transformation of the excitation generator model to the linear systems is omitted; (2) the prespecified performance of the tracking error can be guaranteed by combining with the tracking error transformed function; (3) the computational burden is greatly reduced by estimating the norm of the weighted vector of neural network instead of the weighted vector itself; therefore, it is more suitable for the real time control; and (4) the explosion of complicity problem inherent in the backstepping control can be eliminated. It is proved that the new scheme can make the system semiglobally uniformly ultimately bounded. Simulation results show the effectiveness of this control scheme.


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## Body

## 1. Introduction

With the development of power system, the requirements of the load are increasing. However, the ability of the transmission network is limited and the electric power system has gradually reached its operating limit; the world power systems tend to extend power grid scale and develop larger power system. However, the power systems are more likely to encounter some problems such as oscillations and complex nonlinear phenomena due to the extension of power grid scale and some emergencies. Therefore, ways of maintaining the reliability and stability of power system are attracting more and more research interest [1–4]. Moreover, the promotion of the control ability of the generator excitation system can help us to monitor the states of system and predict system performance; it is the most effective way to improve the performance of the whole excitation control system and it has been widely of concern in recent years [5–7].The most common method for designing generator excitation system controller is the direct-feedback-linearization, which transforms the nonlinear model into a linear one by some mathematical ways. However, it is only effective to the system when special conditions are satisfied [8, 9]. Now, more and more nonlinear control techniques are being applied to the generator excitation systems. In [10], an output feedback controller combined with high-order sliding-mode techniques is proposed to guarantee the transient stability of the power systems. In [11], for the nonlinear characteristics of the transient process in power system whose parameters are known, Colbia-Vega and his collaborators designed a controller for steam-valving and excitation system, which can stabilize the voltage and improve power quality. In [12], an adaptive fuzzy zero dynamic and sliding model control is proposed to cope with the unknown parameters, and the fuzzy logic rules are used to approximate the unknown nonlinear functions. For the time-varying and uncertainties of the reactance and voltage in one-machine infinite-bus power system, Zhang and Zhu in [13] designed an adaptive excitation controller by using strong tracking filter in the design procedure of excitation controller of generators, and the stability of the control system is improved when it is accompanied with some disturbances; however, it has the explosion of complicity problem inherent in the backstepping control. In [14], an adaptive filter control for generator excitation system was proposed based on IEEE 14-bus.Motivated by the previous work ([12, 13], etc.) this paper has successfully fused adaptive dynamic surface control with neural network and tracking error transformed function to design a robust adaptive controller for the nonlinear power system. The features of our control scheme are as follows:(i)
the tracking error of the power angle and voltage at the SVC can be transformed to an arbitrarily prespecified performance by an error transformed function;(ii)
the neural network based adaptive dynamic surface controller is designed without doing any linearization to the excitation generator model. Therefore, the design procedures and the final control law are greatly simplified and the control accuracy is improved;(iii)
by using the neural network, the structure and the parameters of the control system can be totally unknown;(iv)
by estimating the norm of the weighted vector of neural network instead of the weighted vector itself, the computational burden is greatly reduced and it is more suitable for the real time control.This paper is organized as follows. In Section2, the mathematical model of excitation system of power generator is introduced and the control purpose is formulated. In Section 3, the design procedure of the neural network and the tracking error transformed function based adaptive DSC scheme is presented. Section 4 gives the stability analysis for the proposed scheme. Finally, simulation examples are given to demonstrate the effectiveness of the proposed design method.


## 2. The Excitation System of Power Generator

### 2.1. Mathematical Model

Assume that the SVC device is installed in a single-machine infinite-bus system; the structure and the equivalent circuit diagram are shown in Figure1 [15].Figure 1
Single machine infinite bus system with SVC.To proceed, the following assumptions are made.Assumption 1.
Ignoring the dynamic process of rapid excitation equipment means the control voltageuf is equal to the generator excitation voltage Ef, Ef=uf.Assumption 2.
The mechanical power of generators remains unchanged in transient stability process; that is,ΔPm=0.Then, the mathematical model of the generator excitation control system equipped with SVC can be described as follows [16–18]:
(1)δ˙=ω-ω0,ω˙=ω0HPm0-DH(ω-ω0)-ω0HEq′VsXdΣ′sinδ+d1,Eq′˙=-XdΣTd0XdΣ′Eq′+1Td0Xd-Xd′XdΣ′Vscos⁡δ+1Td0uf+d2,B˙L=1TC(-BL+BC+uB),
where δ is the power angle of the generator running rotor; ω is the rotor speed of the generator (rad/s); D is the damping constant; H is the inertia constant; ω0=2πf0 is the synchronous machine speed (rad/s); f0 is the rated frequency (Hz); Pm is the mechanical power of motor output; Td0 is the direct axis transient short circuit time constant (s); TC is the time constant of adjusting system and SVC (s); Eq′ is the transient electromotive force (EMF) of the generator; Vs=1 is the voltage of the system; uf is the control voltage of excitation equipment; uB is the input of SVC; BL is the adjustable equivalent susceptance in SVC; BC is the initial value of adjustable susceptance; d1 is position bounded torque interference of the rotor; and d2 is electromagnetic transient interference of the stator:
(2)X1=Xd′+XT,X3=Xd+XT,XdΣ=X3+X2+X3X2(BL-BC),XdΣ′=X1+X2+X1X2(BL-BC),
where X2 is the transmission line reactance; Xd is the generator direct axis reactance; Xd′ is the direct axis transient reactance of the generator; and XT is the transformer reactance.When the system encounters the external disturbances, the stability of the power angle and SVC access point voltage can be effectively improved by combining the regulation of SVC with generator-excitation adjustment. In this paper, we take increment of the power angle and accessing point voltage as the output value of the system to design our adaptive dynamic surface controller; that is,(3)y1=δ-δ0,y2=Vm-Vref,
where Vm is the accessing point voltage and Vref is the reference voltage. According to the circuit, we have
(4)Vm=(X2Eq′)2+(X1Vs)2+2X1X2Eq′Vscos⁡δXdΣ′.Based on the above descriptions, letx1=δ-δ0, x2=ω-ω0, x3=Eq′, and x4=Vm-Vref; then, (1) can be expressed as the following form:
(5)x˙1=x2,x˙2=-ω0HVssin⁡(x1+δ0)XdΣ′x3-DHx2+ω0HPm0+d1,x˙3=1Td0uf-XdΣTd0XdΣ′x3+1Td0Xd-Xd′XdΣ′Vscos⁡δ+d2,x˙4=-VssinδX1X2Vm(XdΣ′)2x2x3-X1X2VmXdΣ′·1Tc(-BL+BC)+X22x3+X1X2Vscos⁡δVm(XdΣ′)2·(-XdΣTd0XdΣ′x3+1Td0Xd-Xd′XdΣ′·Vscos⁡δ)+ufTd0Vm(XdΣ′)2·(X22x3+X1X2Vscos⁡δ)+X1X2TCXdΣ′(-x4uB).
Let
(6)g2=-ω0HVssin⁡(x1+δ0)XdΣ′,g3=1Td0,g4=X1X2TCXdΣ′,f2(x-2)=-DHx2+ω0HPm0,uB′=-x4uB,f4(x-4)=-VssinδX1X2Vm(XdΣ′)2x2x3-X1X2VmXdΣ′·1Tc(-BL+BC)+X22x3+X1X2Vscos⁡δVm(XdΣ′)2·(-XdΣTd0XdΣ′x3+1Td0·Xd-Xd′XdΣ′Vscos⁡δ)+X22x3+X1X2Vscos⁡δTd0Vm(XdΣ′)2uf,f3(x-3)=-XdΣTd0XdΣ′x3+1Td0Xd-Xd′XdΣ′Vscos⁡δ,
where x-i:=[x1,…,xi]T∈Ri, i=1,…,4, are state vectors. Then, the system (5) can be converted into the following two independent subsystems (7) and (8), whose relative degree is 3 and 1, respectively:
(7)x˙1=x2,x˙2=g2x3+f2(x-2)+d1,x˙3=g3uf+f3(x-3)+d2,y1=x1,(8)x˙4=g4uB′+f4(x-4),y2=x4.Now, we can design our adaptive dynamic surface control law of the power system based on the following assumptions.Assumption 3.
The value ofg2 is unknown but the sign of g2 is known; it is assumed that there exist two constants gm and g2max⁡ satisfying
(9)0<gm≤|g2|≤g2max⁡.Assumption 4.
The reference signalyr is smooth and bounded; [yr,y˙r,y¨r]T belong to a compact set for all t≥0.Remark 5.
Assumptions1 and 2 which are the same as those in [15] are used to construct the model of the excitation system of power generator. Assumption 3 is used in Section 4, such as in the proof of (59), (65), and (67). We also emphasize that gm and g2max⁡ are not required in the implementation of our control law but are required for analysis only. Assumption 4 is common in all dynamic surface control schemes.


### 2.2. RBF Neural Network

In general, the neural network is a multi-input single-output system [19]; the mathematical expression is
(10)Y=ϑTψ(ξ),
where ξ∈Rn is the input of RBF neural network; Y∈R is the output of RBFNN, ϑ∈RN is an N-dimensional parameter vector, and here N is the number of neurons; ψ(ξ):Rn→RN is nonlinear vector function and ψ(ξ)=[ψ1(ξ),…,ψN(ξ)]T and ψk(ξ) is the Gauss function.Lemma 6.
RBFNN is a universal approximator in the sense that, given any real continuous functionF(·):Ωξ→R with Ωξ⊆Rn a compact set and any δm>0, by appropriately choosing σk and ζk, k=1,…,N, for some sufficiently large integer N, there exists an RBF network [20], ∀ξ∈Ωξ⊆Rn, |δ|≤δm,
(11)F(ξ)=ϑ*Tψ(ξ)+δ(ξ),
where ϑ* is the optimal weight vector of ϑ and δ(ξ) is approximation error and can be defined as
(12)ϑ*=argmin⁡ϑ∈Rn{sup⁡ξ∈Ωξ|Y(ξ)-F(ξ)|},δ(ξ)=F(ξ)-ϑ*Tψ(ξ).


### 2.3. Error Transformed Function

We define the tracking error:(13)e∶=y-yr,
where yr is the desired trajectory. According to [21, 22], the performance function and error transformed function are defined as follows: a performance function ϖ(t):R+→R--{0} is defined as a positive smooth decreasing function, such that, for all t≥0,
(14)-σϖ(t)<e(t)<ϖ(t),ife(0)>0,-ϖ(t)<e(t)<σϖ(t),ife(0)<0,
where 0<σ<1 and limt→∞=ϖ∞>0 and ϖ∞ is the maximum allowable tracking error of steady-state. To convert (14) into an equivalent unconstrained one, define the error transformed function
(15)e(t)∶=ϖ(t)Φ(S1),
where S1 is the transformed error and Φ(S1) is a smooth and strictly monotone increasing function; the inverse function has the following properties:
(16)-σ<Φ(S1)<1,ife(0)>0,-1<Φ(S1)<σ,ife(0)<0,
where
(17)lim⁡S1→-∞Φ(S1)=-σ,lim⁡S1→∞Φ(S1)=1,ife(0)>0,lim⁡S1→-∞Φ(S1)=-1,lim⁡S1→∞Φ(S1)=σ,ife(0)<0.As (17) shows, if S1∈L∞, (16) holds. Take ϖ(t)>0 and (15) into consideration, and we can have -σϖ(t)<ϖ(t)Φ(S1)<1 (if e(0)>0) or -ϖ(t)<ϖ(t)Φ(S1)<σϖ(t) (if e(0)<0); that is, (14) holds. Hence, from the above analysis, to achieve the prespecified tracking performance, we just only need to show that S1∈L∞, where the strictly increasing property of Φ(S1) guarantees that we can obtain
(18)S1=Φ-1(e(t)ϖ(t)).Note that the casee(0)=0 can be incorporated into e(0)>0 or e(0)<0. At the same time, σ cannot be chosen to be zero due to S1(0) being infinite.


## 2.1. Mathematical Model

Assume that the SVC device is installed in a single-machine infinite-bus system; the structure and the equivalent circuit diagram are shown in Figure1 [15].Figure 1
Single machine infinite bus system with SVC.To proceed, the following assumptions are made.Assumption 1.
Ignoring the dynamic process of rapid excitation equipment means the control voltageuf is equal to the generator excitation voltage Ef, Ef=uf.Assumption 2.
The mechanical power of generators remains unchanged in transient stability process; that is,ΔPm=0.Then, the mathematical model of the generator excitation control system equipped with SVC can be described as follows [16–18]:
(1)δ˙=ω-ω0,ω˙=ω0HPm0-DH(ω-ω0)-ω0HEq′VsXdΣ′sinδ+d1,Eq′˙=-XdΣTd0XdΣ′Eq′+1Td0Xd-Xd′XdΣ′Vscos⁡δ+1Td0uf+d2,B˙L=1TC(-BL+BC+uB),
where δ is the power angle of the generator running rotor; ω is the rotor speed of the generator (rad/s); D is the damping constant; H is the inertia constant; ω0=2πf0 is the synchronous machine speed (rad/s); f0 is the rated frequency (Hz); Pm is the mechanical power of motor output; Td0 is the direct axis transient short circuit time constant (s); TC is the time constant of adjusting system and SVC (s); Eq′ is the transient electromotive force (EMF) of the generator; Vs=1 is the voltage of the system; uf is the control voltage of excitation equipment; uB is the input of SVC; BL is the adjustable equivalent susceptance in SVC; BC is the initial value of adjustable susceptance; d1 is position bounded torque interference of the rotor; and d2 is electromagnetic transient interference of the stator:
(2)X1=Xd′+XT,X3=Xd+XT,XdΣ=X3+X2+X3X2(BL-BC),XdΣ′=X1+X2+X1X2(BL-BC),
where X2 is the transmission line reactance; Xd is the generator direct axis reactance; Xd′ is the direct axis transient reactance of the generator; and XT is the transformer reactance.When the system encounters the external disturbances, the stability of the power angle and SVC access point voltage can be effectively improved by combining the regulation of SVC with generator-excitation adjustment. In this paper, we take increment of the power angle and accessing point voltage as the output value of the system to design our adaptive dynamic surface controller; that is,(3)y1=δ-δ0,y2=Vm-Vref,
where Vm is the accessing point voltage and Vref is the reference voltage. According to the circuit, we have
(4)Vm=(X2Eq′)2+(X1Vs)2+2X1X2Eq′Vscos⁡δXdΣ′.Based on the above descriptions, letx1=δ-δ0, x2=ω-ω0, x3=Eq′, and x4=Vm-Vref; then, (1) can be expressed as the following form:
(5)x˙1=x2,x˙2=-ω0HVssin⁡(x1+δ0)XdΣ′x3-DHx2+ω0HPm0+d1,x˙3=1Td0uf-XdΣTd0XdΣ′x3+1Td0Xd-Xd′XdΣ′Vscos⁡δ+d2,x˙4=-VssinδX1X2Vm(XdΣ′)2x2x3-X1X2VmXdΣ′·1Tc(-BL+BC)+X22x3+X1X2Vscos⁡δVm(XdΣ′)2·(-XdΣTd0XdΣ′x3+1Td0Xd-Xd′XdΣ′·Vscos⁡δ)+ufTd0Vm(XdΣ′)2·(X22x3+X1X2Vscos⁡δ)+X1X2TCXdΣ′(-x4uB).
Let
(6)g2=-ω0HVssin⁡(x1+δ0)XdΣ′,g3=1Td0,g4=X1X2TCXdΣ′,f2(x-2)=-DHx2+ω0HPm0,uB′=-x4uB,f4(x-4)=-VssinδX1X2Vm(XdΣ′)2x2x3-X1X2VmXdΣ′·1Tc(-BL+BC)+X22x3+X1X2Vscos⁡δVm(XdΣ′)2·(-XdΣTd0XdΣ′x3+1Td0·Xd-Xd′XdΣ′Vscos⁡δ)+X22x3+X1X2Vscos⁡δTd0Vm(XdΣ′)2uf,f3(x-3)=-XdΣTd0XdΣ′x3+1Td0Xd-Xd′XdΣ′Vscos⁡δ,
where x-i:=[x1,…,xi]T∈Ri, i=1,…,4, are state vectors. Then, the system (5) can be converted into the following two independent subsystems (7) and (8), whose relative degree is 3 and 1, respectively:
(7)x˙1=x2,x˙2=g2x3+f2(x-2)+d1,x˙3=g3uf+f3(x-3)+d2,y1=x1,(8)x˙4=g4uB′+f4(x-4),y2=x4.Now, we can design our adaptive dynamic surface control law of the power system based on the following assumptions.Assumption 3.
The value ofg2 is unknown but the sign of g2 is known; it is assumed that there exist two constants gm and g2max⁡ satisfying
(9)0<gm≤|g2|≤g2max⁡.Assumption 4.
The reference signalyr is smooth and bounded; [yr,y˙r,y¨r]T belong to a compact set for all t≥0.Remark 5.
Assumptions1 and 2 which are the same as those in [15] are used to construct the model of the excitation system of power generator. Assumption 3 is used in Section 4, such as in the proof of (59), (65), and (67). We also emphasize that gm and g2max⁡ are not required in the implementation of our control law but are required for analysis only. Assumption 4 is common in all dynamic surface control schemes.


## 2.2. RBF Neural Network

In general, the neural network is a multi-input single-output system [19]; the mathematical expression is
(10)Y=ϑTψ(ξ),
where ξ∈Rn is the input of RBF neural network; Y∈R is the output of RBFNN, ϑ∈RN is an N-dimensional parameter vector, and here N is the number of neurons; ψ(ξ):Rn→RN is nonlinear vector function and ψ(ξ)=[ψ1(ξ),…,ψN(ξ)]T and ψk(ξ) is the Gauss function.Lemma 6.
RBFNN is a universal approximator in the sense that, given any real continuous functionF(·):Ωξ→R with Ωξ⊆Rn a compact set and any δm>0, by appropriately choosing σk and ζk, k=1,…,N, for some sufficiently large integer N, there exists an RBF network [20], ∀ξ∈Ωξ⊆Rn, |δ|≤δm,
(11)F(ξ)=ϑ*Tψ(ξ)+δ(ξ),
where ϑ* is the optimal weight vector of ϑ and δ(ξ) is approximation error and can be defined as
(12)ϑ*=argmin⁡ϑ∈Rn{sup⁡ξ∈Ωξ|Y(ξ)-F(ξ)|},δ(ξ)=F(ξ)-ϑ*Tψ(ξ).


## 2.3. Error Transformed Function

We define the tracking error:(13)e∶=y-yr,
where yr is the desired trajectory. According to [21, 22], the performance function and error transformed function are defined as follows: a performance function ϖ(t):R+→R--{0} is defined as a positive smooth decreasing function, such that, for all t≥0,
(14)-σϖ(t)<e(t)<ϖ(t),ife(0)>0,-ϖ(t)<e(t)<σϖ(t),ife(0)<0,
where 0<σ<1 and limt→∞=ϖ∞>0 and ϖ∞ is the maximum allowable tracking error of steady-state. To convert (14) into an equivalent unconstrained one, define the error transformed function
(15)e(t)∶=ϖ(t)Φ(S1),
where S1 is the transformed error and Φ(S1) is a smooth and strictly monotone increasing function; the inverse function has the following properties:
(16)-σ<Φ(S1)<1,ife(0)>0,-1<Φ(S1)<σ,ife(0)<0,
where
(17)lim⁡S1→-∞Φ(S1)=-σ,lim⁡S1→∞Φ(S1)=1,ife(0)>0,lim⁡S1→-∞Φ(S1)=-1,lim⁡S1→∞Φ(S1)=σ,ife(0)<0.As (17) shows, if S1∈L∞, (16) holds. Take ϖ(t)>0 and (15) into consideration, and we can have -σϖ(t)<ϖ(t)Φ(S1)<1 (if e(0)>0) or -ϖ(t)<ϖ(t)Φ(S1)<σϖ(t) (if e(0)<0); that is, (14) holds. Hence, from the above analysis, to achieve the prespecified tracking performance, we just only need to show that S1∈L∞, where the strictly increasing property of Φ(S1) guarantees that we can obtain
(18)S1=Φ-1(e(t)ϖ(t)).Note that the casee(0)=0 can be incorporated into e(0)>0 or e(0)<0. At the same time, σ cannot be chosen to be zero due to S1(0) being infinite.


## 3. Adaptive Dynamic Surface Control of Generator Excitation System

According to the mathematical model of the generator excitation system that is shown in (7), we design adaptive dynamic surface controller about power angle subsystem as follows.Step 1.
Let the first surface error and the tracking error as (18) be defined:
(19)S˙1=Ψ[-ϖ˙ϖe+x2-y˙r],Ψ:=1ϖ∂Φ-1∂(e/ϖ),
where yr is an ideal output of the power angle; we suppose that yr is a smooth and continuous function, e:=y1-yr=x1-yr.
Consider the following quadratic function:(20)V1=12S12.
We can obtain that
(21)V˙1=S1[Ψ(-ϖ˙ϖe-y˙r)+Ψ(x2-x2d)+Ψx2d],
where x2d is the virtual control signal that will be designed; choose the virtual control law as follows to stabilize (21):
(22)x2d=[-k1S1-Ψ(-(ϖ˙/ϖ)e+ΨS1-y˙r)]Ψ.
In order to avoid “differential explosion,” the new variablez2 is obtained by first-order low pass filter:
(23)τ2z˙2+z2=x2d,
where τ2 is the filter time constant.Step 2.
Define the second surface error:(24)S2=x2-z2.
Consider the following quadratic function:(25)V2=12(S22+gmγv2v~22).
The time derivative of V2 is
(26)V˙2=S2[g2x3+f2(x-2)+d1-z˙2]+gmγv2v~2v^˙2.
We need to use neural network to approximate the unknown terms on a compact set Ωξ2:
(27)1g2[f2(x-2)-z˙2+d1+32g22S2]=θδ2*Tψ2(ξ2)+δ2*(ξ2),
where ξ2:=(x1,x2,x2d,z2)∈Ωξ2⊂R4; define υ~2=υ^2-υ2*, and γυ2 is a positive design parameter. Let the estimated value of υ2*=(1/gm)∥θδ2*T∥2 be υ^2, according to inequality
(28)g2S2θδ2*Tψ2≤gmα22S22υ2*ψ2Tψ22+g2max⁡22α22,(29)g2S2δ2≤12g22S22+12δ2*2(ξ2).
From (27) and (28), we have
(30)V˙2≤S2[g2(x3-x3d)+g2x3d+gmα22S2υ2*ψ2Tψ22]-g22S22+12δ2*2+g2max⁡22α22+gmγv2v~2v^˙2,
where x3d is the virtual control signal to be designed; let the power angle actual control law and parameter adjustment law, respectively, be as follows:
(31)x3d=sgn⁡(g2)(-k2S2-α22S2υ^2ψ2Tψ22),(32)v^˙2=γv2[α22S22ψ2Tψ22-σv2v^2].
The new variable z3 is obtained by first-order low pass filter:
(33)τ3z˙3+z3=x3d,
where τ3 is the filter time constant. In order to avoid “differential explosion” problem brought by backstepping control, we use z3 instead of x3d in the following.Step 3.
Define the third surface error:(34)S3=x3-z3.
Consider the following quadratic function:
(35)V3=12(1g3S32+1γv3v~32).
The time derivative of V3 is
(36)V˙3=S3[uf+f3(x-3)g3+d2g3-z˙3g3]+1γv3v~3v^˙3.
Similar to second step, we use neural network to approximate the unknown term in a compact set:
(37)1g3[f3(x-3)-z˙3+d2]=θδ3*Tψ3(ξ3)+δ2*(ξ3),
where ξ3:=(x1,x2,x3,x3d,z3)∈Ωξ3⊂R5; let υ~3=υ^3-υ3*, γυ3 is positive design parameter, and υ^3 is the estimate value of υ3*=∥θδ3*T∥2. Consider the hypothesis; we have
(38)V˙3≤S3[uf+α32S3υ3*ψ3Tψ32+S32]+12δ3*2(ξ3)+12α32+1γv3v~3v^˙3.
Let the power angle actual control law and parameter adjustment law be, respectively, as follows:
(39)uf=-k3S3-α32S3υ^3ψ3Tψ32-S32,(40)v^˙3=γv3[α32S32ψ3Tψ32-σv3v^3].
Similarly, we design controller for the voltage subsystem (8); surface error is defined in (18):
(41)S˙4=Ψ2[-ϖ˙2ϖ2e2+g4uB′+f4(x-4)-y˙r2],Ψ2∶=1ϖ2∂Φ-1∂(e2/ϖ2),
where e2∶=y2-yr2=x4-yr2, yr2 is the ideal voltage reference signal; consider the following quadratic function:
(42)V4=12(1g4S42+1γv4v~42).
The time derivative of V4 is
(43)V˙4=S4[Ψ2g4(-ϖ˙2ϖ2e2-y˙r2)+Ψ2f4(x-4)g4+Ψ2uB′]+1γv4v~4v^˙4.
We use neural network to approximate f4(x-4):
(44)1g4Ψ2[f4(x-4)-ϖ˙2ϖ2e2-y˙r2]=θδ4*Tψ4(ξ4)+δ4*(ξ4),
where ξ4∶=(x4,ϖ2,ϖ˙2,e2,y˙r2)∈Ωξ4⊂R5; let υ~4=υ^4-υ4*, γυ3 is a positive design parameter, and υ^4 is an estimate value of υ4*=∥θδ4*T∥2:
(45)V˙4≤S4[α42S4υ4*ψ4Tψ42+Ψ2uB′+12S4]+12δ4*2(ξ4)+12α42+1γv4v~4v^˙4.
Design voltage control law and parameter adjustment law are, respectively, as follows:
(46)uB′=1Ψ2[-k4S4-12S4-α42S4υ^4ψ4Tψ42],v^˙4=γv4[α42S42ψ4Tψ42-σv4v^4].Remark 7.
We emphasize our dynamic surface control scheme which, combined with tracking error transformed function (15), is also suitable for the half-car active suspension system that is described in [23, 24].


## 4. Stability Analysis

This section will conduct stability analysis for dynamic surface control scheme on the generator excitation system. Although the control law is simple, the stability analysis is relatively complex due to the introduction of first-order low-pass filters. For the first subsystem in (7), to begin with, we define
(47)ye2=z2-x2d=z2+[k1S1+Ψ(-(ϖ˙/ϖ)e+ΨS1-y˙r)]Ψ,(48)ye3=z3-x3d=z3±(k2S2+α22S2υ^2ψ2Tψ22),
where x2d and x3d are designed in (22) and (31), respectively. From (23) and (33), we have
(49)z˙2=(x2d-z2)τ2=-ye2τ2,z˙3=(x3d-z3)τ3=-ye3τ3.
Then, the time derivative of (47) and (48) is
(50)y˙e2=-ye2τ2+k1S˙1Ψ+k1S1Ψ˙Ψ2+(ϖ¨ϖe-ϖ˙2ϖ2e+ϖ˙ϖe˙)+(Ψ˙S1+ΨS˙1)-y¨r=-ye2τ2+B2(S1,S2,y2,ϖ,ϖ˙,ϖ¨,yr,y˙r,y¨r),y˙e3=-ye3τ3±[(∑j=12∂ψ2∂xjx˙j+∂ψ2∂S2)k2S˙2+α22S˙2υ^2ψ2Tψ22+α22S2υ^˙2ψ2Tψ22iimmmm+α22S2υ^˙2ψ2×(∑j=12∂ψ2∂xjx˙j+∂ψ2∂S2)]=-ye3τ3+B3(S1,S2,S3,y2,y3,υ^2,ϖ,ϖ˙,ϖ¨,yr,y˙r,y¨r),
where B2 and B3 are continuous functions.Theorem 8.
Consider the closed-loop control system composed by object (7), first-order low-pass filter (23), (33), the actual control law (39), and parameter adjustment law (40):
(51)V=V1+V2+V3+12ye22+12ye33,
where V1, V2, and V3, respectively, are defined by (20), (25), and (35). We suppose that for the given positive constant δi*, (27), (37), and (44) meet |δi(ξi)|≤δi*, i=2,3,4, in a compact set. For any given positive parameter p, if
(52)V(0)≤p,
then through the proper selection of the design parameters k1, k2, k3,γυ2,γυ3, συ2, and συ3 all the variables in the closed-loop system are uniformly ultimately bounded. The tracking performance of the system can be specified in advance.Proof.
The derivative ofV by considering (51) is
(53)V˙=V˙1+V˙2+V˙3+ye2y˙e2+ye3y˙e3.
From (24) and (47), we have
(54)x2=S2+ye2+x2d.
And consider the inequality
(55)ΨS1S2≤12Ψ2S12+12S22,ΨS1ye2≤12Ψ2S12+12ye22.
Substituting (54) and (55) and virtual control law (22) into (21), we have
(56)V˙1≤-k1S12+12S22+12ye22.
Using (34) and (48),
(57)x3=S3+ye3+x3d.
At the same time,
(58)g2S2S3≤12g22S22+12S32,g2S2ye3≤12g22S22+12ye32.
Substitute (57) and (58), virtual control law (31), and parameter adjustment law (32) into (30):
(59)V˙2≤-k2|g2|S22+12S32+12ye32+12δ2*2+g2max⁡22α22-gmσv2v^2v~2.
From (39) and (40), (38) can be transformed as
(60)V˙3≤-k3S32+12δ3*2+12α32-σv3v^3v~3.
According to Assumption 4,
(61)Π∶={(yr,y˙r,y¨r):yr2+y˙r2+y¨r≤B0}
is a compact set in R3, and B0>0. M2 is the maximum of |B2| in Π; M3 is the maximum of |B3| at Π. For any p>0, using inequality, we can obtain that
(62)|ye2B2|≤ye22B222μ+μ2≤ye22M222μ+μ2,(63)|ye3B3|≤ye32B322μ+μ2≤ye32M322μ+μ2,
where μ is a positive constant parameter
(64)1τ2=12+M222μ+α0,1τ3=12+M322μ+α0.
Furthermore,
(65)-gmσv2v^2v~2≤-σv2gm2v~22+σv2gm2v2*2,(66)-σv3v^3v~3≤-σv32v~32+σv32v3*2.
Taking (56), (59), (60), and (62)–(66) into consideration, (53) can be transformed into the following equation:
(67)V˙≤-k1S12-(k2gm-12)S22-(k3-12)S32-α0ye22-α0ye32-σv2gm2v~22-σv32v~32+C,
where
(68)C=σv2gm2v2*2+σv32v3*2+μ+12δ2*2+12δ3*2+g2max⁡22α22+12α32.
And α0 is a positive design parameter:
(69)α0≤min⁡{k1,k2gm-12,k3-12,γv2σv22,γv3σv32}.
From (67), we have
(70)V˙≤-2α0V+C.
Let
(71)α0>C2p.
We can see that when V=p, V˙≤0. So, V≤p is an invariant set. If V(0)≤p, then, for all t≥0, V(t)≤p. From (70), ∀t≥0,
(72)0≤V(t)≤C2α0+{V(0)-C2α0}e-2α0t.
Then,
(73)lim⁡t→∞V(t)=C2α0.
Similar to the second subsystems, we substitute (46) into (45):
(74)V˙4≤-k4S42-συ42υ~42+12δ4*2+12α42+συ42υ4*2.
Let
(75)k4≥ag4,(76)σv4≥2aγv4,(77)C2=σv42v4*2+12δ4*2+12α42.
We have
(78)V˙4≤-2aV+C2,(79)lim⁡t→∞⁡V4(t)=C2a.
As a result, all the signals such asS1, S2, S3, S4, k1, k2, k3, k4, σv2, σv3, σv4, α0, and a in the closed-loop system are uniformly bounded. In addition, (69), (75), and (76) show that, by selecting appropriate design parameters, α0, a can be infinite; (73) and (79) show that the tracking error can converge to an arbitrary small value.


## 5. Simulation Analysis

The single machine infinite bus power system with SVC is shown in Figures1 and 2, assuming that the amplitude and frequency of receiving end bus voltage are not changed.Figure 2
The equivalent circuit diagram.System parameters areXd′=0.344, Xd=0.982, XT=0.1, D=2, XdΣ′=0.6967, BL=1.55, BC=0.5, Td0=5, TC=2, H=13, Eq′=1.059, δ0=40, ω0=100π rad/s, Pm0=1, Vs=1, Vref=1, X2=0.35, and XdΣ=1.4.In the simulation the initial values of the states arex1(0)=40, x2(0)=100π, x3(0)=1.05, and x3(0)=1. For NN Gaussian function ψj(ξi)=exp⁡[-(ξi-ζj)T(ξi-ζj)/ηj2], i=2,3,4, we choose 21 nodes with the centers of the basis functions ζj evenly spaced in [-40,+40] × [-314,+3144] × [-2,+2] × [-1,+1] and width ηj=1,forj=1,…,21. Also, we choose a smooth and strictly monotone increasing function, Φ2(S4)=(2/π)arctan(S4), ϖ(t)=0.65e-8t+0.002 and S4=tan((π/2)·(e2(t)/w-2(t))), which means that Ψ2=(π/2)·(1/ϖ2)·cos⁡-2((π/2)·(e2/ϖ2)).We suppose that three-phase short-circuit occurred on the terminal of the line att=8 s and eliminate the fault at t=9 s; the simulation results are shown in Figures 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13. Figure 3 shows that, by introducing the error transformed function, the system can be stabilized promptly when it encountered some emergencies such as three-phase short-circuit which means that our control scheme makes the generator excitation control system have strong robustness. The tracking error of power angle can be prespecified by using the performance function ϖ(t)=0.65e-8t+0.002. Figures 4 and 5 are actual control of power angle and actual voltage control law of the access point voltage, respectively. From Figure 6–Figure 8 we can know that all the states in the control system recover to their desired values in a short time by using our dynamic surface control scheme. Figure 9 to Figure 11 show the estimations of weight vector norm of the neural network. Figure 12 is the access point voltage increment Vm-Vref, whose tracking process can also be arbitrarily prespecified by using performance function as that in Figure 3.Figure 3
Tracking error and the performance function.Figure 4
The actual control lawuf of power angle.Figure 5
The actual voltage control lawuB′ of access point voltage.Figure 6
The rotor angleδ of generator.Figure 7
The rotor speedω of generator.Figure 8
The transient EMFEq′ of the generator.Figure 9
The estimated norm of the weighted vectorυ^2, with υ2*=∥θδ2*T∥2.Figure 10
The estimated norm of the weighted vectorυ^3, with υ3*=∥θδ3*T∥2.Figure 11
The estimated norm of the weighted vectorυ^4, with υ4*=∥θδ4*T∥2.Figure 12
The access point voltage incrementVm-Vref.Figure 13
The comparison of our NN based dynamic surface control scheme and backstepping.Remark 9.
The controllers that have been proposed in [12, 13] are based on backstepping technique, which can cause the explosion of complicity problem that means the controller will become complex when the relative degree of the control system is high. Also, our novel neural network based adaptive dynamic surface controller is designed without doing any linearization to the excitation generator model. Therefore, the design procedures and the final control law are greatly simplified and the control accuracy is improved. Figure 13 has shown the comparison of the tracking error between our control scheme and backstepping control scheme.


## 6. Conclusion

This paper proposed a novel neural network and tracking error transformed function based on adaptive dynamic surface control scheme for the generator excitation control system which is equipped with SVC. In this scheme, both the designing procedures and the control law are simplified because it does not need to make any transformation to the excitation generator model. The prespecified tracking performance can be guaranteed by using the tracking error transformed function. The norm of the weighted vector of neural network is estimated which replaces the estimation of the weighted vector itself; therefore, the computational burden is greatly reduced. Also, the explosion of complicity problem inherent in the backstepping control can be eliminated. It is proved that the new scheme can make the system semiglobally uniformly ultimately bounded.



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*Source: 481936-2014-07-07.xml*