# Prescribed Performance-Based Adaptive Terminal Sliding Mode Control for Virtual Synchronous Generators

Authors: Zhenlan Dou; Lianqing Tang; Yongbao Sun; Chunyan Zhang; Weilin Yang; Dezhi Xu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Due to the lack of inertia and damping in the distributed power system with power electronic devices as interface, the stability of the power system will be adversely affected when the distributed power system is connected to the grid. In order to solve the problem of low inertia and improve stability, an adaptive terminal sliding mode control method based on the electromagnetic transient characteristics of virtual synchronous generators is proposed. During the design process, projection operators and parameter adaptive law control methods are introduced to compensate for parameter errors caused by environmental interference. Furthermore, by utilizing a prescribed performance function, the terminal sliding surface of the system will have the desired dynamic response, and variations of the system tracking error can be restricted. Simulation results show that the proposed controller has improved control accuracy and response speed, frequency stability, and power response.


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## Body

## 1. Introduction

The development of grid-connected distributed energy systems with grid-connected converters has been expanding due to increasing energy demand and environmental concerns [1]. The stability of grid frequency could seriously threaten due to the unpredictable power supplied by distributed generation (DG) units. However, when DG is combined with supply side energy storage, the power and frequency fluctuations of the distributed system can be mitigated [2–4]. These systems still suffer from low inertia such that they cannot implicitly provide the necessary inertial support to compensate for microgrids with poor stability problem of the system which has always existed; that is, when quality and reliability of the power supply are disturbed, the grid-connected converter cannot provide the necessary inertial support for the poor stable microgrid [5].In order to solve the system instability problem described above, the virtual synchronous generators (VSGs) have been proposed [6]. This is achieved by introducing the mechanical and electromagnetic equations of the synchronous generator (SG) and simulating its external characteristics. The virtual inertia and primary frequency can be adjusted to increase the virtual inertial of the inverter and improve the quality of the system output response [7, 8]. In general, two different control mode structures have been studied that treat the VSG as a voltage-source or current-source. The current-source-mode converter has been shown to inadequately simulate the rotor motion characteristics of SG, thereby making voltage-source-mode the preferred method [9, 10]. In [11], a detailed VSG model is presented and the system stability is determined from an Eigenvalues analysis. The main purpose of this method is to adjust the inverter’s power control loop, but in order to respond to changes in grid frequency, power grid frequency detection and control should also be taken into account. In [12, 13], a VSG regulation scheme for active and reactive power is proposed, to achieve seamless switching function that can be connected and disconnected from the grid. The system virtual inertia and damping characteristics and its tuning method are analyzed in detail. It should be noted that although the above method guarantees a simple control system architecture, the implementation of grid failure and transient protections systems is complicated [14].The VSG control strategy is mainly based on the classic second-order SG model. When controlled as a voltage-source, it has similar characteristics to the droop control, load changes result in mismatched the grid angle, and frequency when operating in island mode [10]. Therefore, when connected to the grid, a phase-locked-loop (PLL) is required to obtain the frequency and initial angle to prevent deviation. This technique relies upon grid-DG communication that becomes more susceptible to interruptions as the number of DG systems increases [15, 16]. Power synchronization technology is proposed in [17] that eliminates the communication dependence of a PLL to keep the output voltage consistent with the grid voltage. However, this system requires a presynchronization process that can prevent plug-and-play converter deployment and is susceptible to instability caused by short-term voltage out-of-sync events. In addition, the method based on frequency derivative can effectively enhance the virtual inertia of the system and improve the response speed of the system [18, 19]. In [20], a comprehensive analysis of the effects of virtual inertia is performed to obtain the load disturbance frequency in grid-connected mode quickly and stably. In [21], a VSG control strategy based on RBF neural network is proposed and has good performance in damping of oscillation. In [22], the VSG parameters are adjusted by using quantitative feedback theory (QFT) to ensure the robust stability and performance of the system. To the best of the authors’ knowledge, the above research work does not use the VSG adaptive terminal sliding mode control method based on preset performance to solve the transient events of grid-connected and off-grid.Backstepping control (BSC) can simplify the system structure for nonlinear systems and can recursively prove system stability when combined with the Lyapunov function. With the introduction of adaptive methods and projection operators, uncertain parameters in the system are allowed to be effectively estimated, and the estimated values are then constrained to prevent exceeding predetermined limits [23–25]. In addition, sliding mode control (SMC) has been proven to handle complex systems due to its advantages of reliability, anti-interference, and simple design [26]. However, chattering is one of the main factors affecting the accuracy of sliding mode control [27]. Terminal sliding mode control is an effective control method to deal with system uncertainty, especially for nonlinear systems [28]. Compared with traditional sliding mode control, the terminal sliding mode control can make the tracking error reach to zero in finite time and has the advantages of eliminating chattering and robustness against external disturbance and parameter perturbation [29, 30]. All these advantages greatly improve the performance of the controller. In [31], a backstepping controller that combines command-filter and integral sliding mode control is used to investigate microgrid dynamics. The prescribed performance control (PPC) method uses a performance function and error transformation to achieve predetermined convergence speed, overshoot, and track error performance [32].In this paper, a prescribed performance-based adaptive terminal sliding mode backstepping controller for the nonlinear VSG system error model is proposed. The projection operator is used to constrain the system parameter error and limits the sliding surface to predetermined values when combined with the prescribed performance function. The contributions of this paper are as follows:(i)
The projection operator adaptive law is used to estimate the virtual inertia parameters and damping parameters of the system, so as to improve the accuracy of the system model(ii)
The parameter error of the system is constrained by the projection operator, and the sliding mode surface is limited to the predetermined range combined with the prescribed performance function(iii)
The controller is implemented in state space, and the Lyapunov function is used to verify the system stability(iv)
The designed controller can operate stably in a wide range of operation scenarios, including island, grid-connected, and transient eventsThe paper is organized as follows. In section2, the systems mathematical model is established. A state space equation and an adaptive terminal SMC strategy based on prescribed performance are presented in section 3. Simulation results are presented in section 4 and compared with other control methods to demonstrate its improved performance. Conclusions are given in section 5.


## 2. Overview of the Model

In Figure1, the primary model of VSG inverter in MG is introduced in detail, which plays the role of DC-AC conversion and power transmission [7, 33]. The resistance Rs and inductance Ls represent the stator impedance in an ideal synchronous motor as shown in Figure 2. Among them, i˜=iaibic represents the current of the stator winding, and the voltage of the capacitor terminal u˜=uaubuc is the terminal voltage of SG. Rf and Lf represent the impedance of the rotor. The voltage generation principle is similar to the back electromotive force (EMF) in SG:(1)e˜=ωψfsinθ˜.Figure 1
Power part of a synchronverter.Figure 2
Structure of an idealised three-phase round-rotor SG.The mathematical model of the VSG can be formulated as shown in Figure3, Tref and Te are, respectively, the virtual input mechanical torque and the virtual electromagnetic torque of the rotor, J is the virtual inertia, Dp is the virtual mechanical friction coefficient, which is also the drooping coefficient of the frequency droop control loop, and θ and θ˙ are the rotor angle and the virtual angular speed.Figure 3
Rotor mathematical model of the synchronverter.The equation of mechanical dynamic can be expressed as(2)Jθ¨=Tref−Te−Dpθ˙.The output power can be obtained by the inner product of the three-phase voltage and current:(3)Pe=i˜,e˜=ωψfi˜,sinθ˜,where i˜=icosφicosφ−2π/3icosφ+2π/3andθ˜=θθ−2π/3θ+2π/3.Formula (3) can be transformed as follows:(4)Pe=32ωψficosθ−φ=32ωψficosδ,where δ=θ−φ is the power angle.Then, the virtual electromagnetic torque can be expressed as follows:(5)Te=Peω=32ψficosδ.Correspondingly, the solution of reactive power is similar to active power, which can be obtained as follows:(6)Q=32wψfisinδ.The virtual excitation module is similar to the excitation control system of SG, and the terminal voltage and reactive power are regulated by controlling the virtual potential of VSG [34, 35]. The controlling process is shown in Figure 4. The reference voltage Vref can be obtained from the formula as follows:(7)Vref=Vs+mQref−Qe,where Vs and Vm are the no-load voltage and actual voltage amplitude, Qe is the actual reactive power, Qref is the reference reactive power, and m is the droop factor. A PI controller is used to track the reference voltage error to generate a control signal ψf. In order to make the VSG acts as a real SG, a low-pass filter is being used to simulate the flux decay.Figure 4
Flux control loop.Equation due to the influence of the rotor wingding inductance [8] is given as follows:(8)Gfluxs=kaτas+c,where τa is the flux loop time constant which is used to appropriate voltage transient response.


## 3. Proposed Control Topology and Design Process

### 3.1. Prescribed Performance Control (PPC)

The prescribed performance control has the following definition [32, 36].Definition 1.
A smooth functionρ:ℜ+⟶ℜ+ will be called a performance function if(1)
ρt is positive and decreasing(2)
limt⟶∞ρt=ρ∞>0
This paper selects the following function as the default performance function:(9)ρt=ρ0−ρ∞exp−ht+ρ∞,where ρ0 is the initial value of the performance function, ρ∞ is the allowable range of the system steady-state error, and h is the convergence rate of the performance function.
In this paper, the target of control is the terminal sliding surfaces2; apparently, the s2 boundary is guaranteed to have the following equivalent:(10)−δρt<s2<ρtifs2≥0,−ρt<s2<δρtifs2<0,for all t≥0, where 0≤δ≤1.


### 3.2. Error Transformation

In order to make the control objects2 easy to implement, the inequality constraint shown in (10) can be transformed into the form of the equality constraint, and the following form of error transformation is introduced:(11)s2t=ρtϑεt,where εt is the transformed error and ϑεt is a function possessing the following properties:(1)
ϑεt is smooth and strictly increasing. Therefore, the conversion error εt can be obtained by inverse transformation of (11):(12)εt=ϑ−1s2tρt.(2)
The variables are constrained as follows.−δ<ϑεt<1s2t≥0−1<ϑεt<δs2t<0.Then, we can choose the typical form ofϑεt as follows:(13)ϑεt=expε−δexp−εexpε+exp−εifs2t≥0,δexpε−exp−εexpε+expεifs2t<0.Equation (13) can be expressed as follows:(14)εt=12lnz+δ1−zifs2t≥0,12lnz+1δ−zifs2t<0.The derivative of Equation (14) is as follows:(15)ε˙t=∂ϑ−1∂z∂z∂t=gρt,s2t1ρts˙2t−ρ˙s2tρt,where z=s2t/ρt.Remark 1.
With the monotonously decreasing property of performance function and error transformation property, the system error will converge to a predetermined range in finite time by selecting appropriate parametersρ0, ρ∞, and h. Notice that ρ0 shall be selected relatively large considering the maximum overshoot of s2 and ρ∞ shall be selected relatively small to achieve tracking performance. Besides, considering the transient process of grid-connected and off-grid, h should be taken appropriately to avoid singularity problems.


### 3.3. Voltage Control Scheme

The overall structure is presented in Figure5. In order to obtain the dynamics of δ, it is assumed that dynamics of voltage angle are faster than dynamics of current angle [37], which means that the rate of δ change of is proportional to voltage frequency deviation from its rated value. Then, comparing the load angle δ, frequency ω, and electromagnetic torque Te with their equilibrium points, the state variables load angle error, frequency error, and electromagnetic torque error are obtained, as shown in Figure 5.(16)Δδ˙=ΔwΔw˙=−DJΔw−1JΔTe.Figure 5
The structure of the proposed controller for the VSG system.Derivation of (5) to obtain the dynamic torque T˙e is as follows:(17)T˙e=32ψ˙ficosδ+32ψfi˙cosδ−32ψfiΔwsinδ.According to the nature of the low-pass filter circuit, whereψ˙f is obtained from (8), (18)ψ˙f=kaτau−cτaψf.Substituting (18) into (17),(19)T˙e=32kaτauicosδ−cτaTe+32ψfi˙cosδ−32ψfiwsinδ.The transient form of (19) using the similar method in [34] is as follows:(20)ΔT˙e=32kaτauicosΔδ−cτaΔTe+32ψfi˙cosΔδ−32ψfiΔwsinΔδ.The overall system state space model is as follows:(21)Δδ˙=Δw,Δw˙=σ1Δw+σ2ΔTe,ΔT˙e=f1i,Δδu+f2i,ψf,Δδ,Δw,ΔTe.Here,σ1=−D/J, σ2=−1/J, and f1i,Δδu=3/2ka/τauicosδf2i,ψf,Δδ,Δw,ΔTe=3/2ψfi˙cosΔδ−3/2ψfiΔwsinΔδ−c/τaΔTe.


### 3.4. Adaptive Terminal Sliding Mode Control Strategy with PPC

The state space model has been described in the above section and in this section. The detailed controller design procedure proposed for VSG is described as follows. Defining the tracking error variablese1, e2, and e3,(22)e1=Δδ−Δδref,e2=Δw−Δwrefs,e3=ΔTe−ΔTrefs,where Δδref is the desired reference signal. Δwrefs and ΔTrefs are the frequency and torque reference. The deviation of (22) can be calculated as follows:(23)e˙1=Δw−Δδ˙ref,e˙2=σ1Δw+σ2ΔTe−Δw˙refs,e˙3=f1i,Δδu+f2i,ψf,Δδ,Δw,ΔTe−ΔT˙refs.Defining the terminal sliding mode surface,(24)s1=e2+∂1∫0−te2dtp1/q1,s2=e3+∂2∫0−te3dtp2/q2,where ∂1>0 and ∂2>0 represent the designed constants of sliding mode surface and p1, p2, q1, and q2 are the positive odd numbers, where 1<p1/q1<2 and 1<p2/q2<2.The derivative of (24) can be calculated as follows:(25)s˙1=e˙2+∂1p1q1e2∫0−te2dtp1/q1−1=σ1Δw+σ2ΔTe−Δw˙ref+∂1p1q1e2∫0−te2dtp1/q1−1,s˙2=e˙3+∂2p2q2e3∫0−te3dtp2/q2−1=f1i,Δδu+f2i,ψf,Δδ,Δw,ΔTe−ΔT˙refs+∂2p2q2e3∫0−te3dtp2/q2−1.Using the backstepping method to stabilize the load angleδ, the Lyapunov function is selected as follows:(26)V1=12e12.The derivation of (26) is(27)V˙1=e˙1e1=e1Δw−Δδ˙ref.According to the requirements of the Lyapunov stability condition, the equationV˙1<0 needs to be satisfied; therefore, the optimal choice of the virtual controller Δwref is as follows:(28)Δwref=−k1e1+Δδ˙ref,where k1>0,(29)V˙1=−k1e12.Then, the Lyapunov functionV2 is selected to stabilize the frequency error Δw as follows:(30)V2=12e12+s12.The derivation of (30) is as follows:(31)V˙2=−k1e12+s˙1s1.Similarity, in order to meet the equationV˙2<0, the constraint conditions of controller s˙1 can be designed as follows:(32)−λ1sgns1=σ1Δw+σ2ΔTe−Δw˙ref+∂1p1q1e2∫0−te2dτp1/q1−1,where λ1>0 and sgn⋅ means(33)sgnx=xxifx≠0,0ifx=0.Substituting (32) into (31), obtain V˙2 as follows:(34)V˙2=−k1e12−λ1s1.Therefore, the virtual controllerΔTrefs is as follows:(35)ΔTrefs=1σ2−λ1sgns1−σ1Δw+Δw˙ref−∂1p1q1e2∫0−te2dτp1/q1−1.Considering the parameter error at the same time, the form of the virtual error variable is obtained as follows:(36)ΔT^refs=1σ^2−λ1sgns1−σ^1Δw+Δw˙ref−∂1p1q1e2∫0−te2dτp1/q1−1,where σ^1 and σ^2 are the parameter estimation values.Then, using the above mentioned prescribed performance control method, the terminal sliding mode surfaces2 can be converted to ε through error conversion. Therefore, the new Lyapunov equation can be expressed as follows:(37)V3=12e12+s12+ε2+σ˜12r1+σ˜22r2,where r1andr2 are the gains of adaptive law, σ˜1=σ^1−σ1, σ˜2=σ^2−σ2, and the derivation of (37) is as follows:(38)V˙3=V˙2−k2ε2+εgρt,s2t1ρts˙2t−ρ˙s2tρt+k2ε+σ˜1r1σ^˙1−r1s1Δw+σ˜2r2σ^˙2−r2s1ΔTe,where k2>0, and the controller u can be designed as follows:(39)u=1f1i,Δδ−ρk2ε/gρt,s2+ρ˙s2ρ−f2i,ψf,Δδ,Δw,ΔTe+ΔT^refs−∂2p2q2e3∫0−te3dtp2/q2−1.The projection operator adaptive law can be designed as follows:(40)σ^˙1=r1Projσ^1,s1Δw,σ^˙2=r2Projσ^2,s1ΔTe,where function Proj⋅,⋅ is the projection operator adaptive law [38] valid for a robust adaptive controller which needs multiple differentiation, designed as follows:(41)Projξ^,ς=0,ifξ^=ξmaxandς>0,0,ifξ^=ξminandς<0,ς,otherwise,and Proj⋅,⋅ has the conclusion as follows:(42)Property1ξ^∈Ωξ≜ξ^:ξmin≤ξ^≤ξmax,Property2ξ˜Projξ^,ς−ς≤0,∀ς.Taking (39) and (40) into (38), the Lyapunov funtion can be expressed as follows:(43)V˙3=−k1e12−λ1s1−k2ε2≤0.Therefore, according to the above derivation process, equationV˙3≤0 can prove that the designed controller meets the Lyapunov stability condition.


## 3.1. Prescribed Performance Control (PPC)

The prescribed performance control has the following definition [32, 36].Definition 1.
A smooth functionρ:ℜ+⟶ℜ+ will be called a performance function if(1)
ρt is positive and decreasing(2)
limt⟶∞ρt=ρ∞>0
This paper selects the following function as the default performance function:(9)ρt=ρ0−ρ∞exp−ht+ρ∞,where ρ0 is the initial value of the performance function, ρ∞ is the allowable range of the system steady-state error, and h is the convergence rate of the performance function.
In this paper, the target of control is the terminal sliding surfaces2; apparently, the s2 boundary is guaranteed to have the following equivalent:(10)−δρt<s2<ρtifs2≥0,−ρt<s2<δρtifs2<0,for all t≥0, where 0≤δ≤1.


## 3.2. Error Transformation

In order to make the control objects2 easy to implement, the inequality constraint shown in (10) can be transformed into the form of the equality constraint, and the following form of error transformation is introduced:(11)s2t=ρtϑεt,where εt is the transformed error and ϑεt is a function possessing the following properties:(1)
ϑεt is smooth and strictly increasing. Therefore, the conversion error εt can be obtained by inverse transformation of (11):(12)εt=ϑ−1s2tρt.(2)
The variables are constrained as follows.−δ<ϑεt<1s2t≥0−1<ϑεt<δs2t<0.Then, we can choose the typical form ofϑεt as follows:(13)ϑεt=expε−δexp−εexpε+exp−εifs2t≥0,δexpε−exp−εexpε+expεifs2t<0.Equation (13) can be expressed as follows:(14)εt=12lnz+δ1−zifs2t≥0,12lnz+1δ−zifs2t<0.The derivative of Equation (14) is as follows:(15)ε˙t=∂ϑ−1∂z∂z∂t=gρt,s2t1ρts˙2t−ρ˙s2tρt,where z=s2t/ρt.Remark 1.
With the monotonously decreasing property of performance function and error transformation property, the system error will converge to a predetermined range in finite time by selecting appropriate parametersρ0, ρ∞, and h. Notice that ρ0 shall be selected relatively large considering the maximum overshoot of s2 and ρ∞ shall be selected relatively small to achieve tracking performance. Besides, considering the transient process of grid-connected and off-grid, h should be taken appropriately to avoid singularity problems.


## 3.3. Voltage Control Scheme

The overall structure is presented in Figure5. In order to obtain the dynamics of δ, it is assumed that dynamics of voltage angle are faster than dynamics of current angle [37], which means that the rate of δ change of is proportional to voltage frequency deviation from its rated value. Then, comparing the load angle δ, frequency ω, and electromagnetic torque Te with their equilibrium points, the state variables load angle error, frequency error, and electromagnetic torque error are obtained, as shown in Figure 5.(16)Δδ˙=ΔwΔw˙=−DJΔw−1JΔTe.Figure 5
The structure of the proposed controller for the VSG system.Derivation of (5) to obtain the dynamic torque T˙e is as follows:(17)T˙e=32ψ˙ficosδ+32ψfi˙cosδ−32ψfiΔwsinδ.According to the nature of the low-pass filter circuit, whereψ˙f is obtained from (8), (18)ψ˙f=kaτau−cτaψf.Substituting (18) into (17),(19)T˙e=32kaτauicosδ−cτaTe+32ψfi˙cosδ−32ψfiwsinδ.The transient form of (19) using the similar method in [34] is as follows:(20)ΔT˙e=32kaτauicosΔδ−cτaΔTe+32ψfi˙cosΔδ−32ψfiΔwsinΔδ.The overall system state space model is as follows:(21)Δδ˙=Δw,Δw˙=σ1Δw+σ2ΔTe,ΔT˙e=f1i,Δδu+f2i,ψf,Δδ,Δw,ΔTe.Here,σ1=−D/J, σ2=−1/J, and f1i,Δδu=3/2ka/τauicosδf2i,ψf,Δδ,Δw,ΔTe=3/2ψfi˙cosΔδ−3/2ψfiΔwsinΔδ−c/τaΔTe.


## 3.4. Adaptive Terminal Sliding Mode Control Strategy with PPC

The state space model has been described in the above section and in this section. The detailed controller design procedure proposed for VSG is described as follows. Defining the tracking error variablese1, e2, and e3,(22)e1=Δδ−Δδref,e2=Δw−Δwrefs,e3=ΔTe−ΔTrefs,where Δδref is the desired reference signal. Δwrefs and ΔTrefs are the frequency and torque reference. The deviation of (22) can be calculated as follows:(23)e˙1=Δw−Δδ˙ref,e˙2=σ1Δw+σ2ΔTe−Δw˙refs,e˙3=f1i,Δδu+f2i,ψf,Δδ,Δw,ΔTe−ΔT˙refs.Defining the terminal sliding mode surface,(24)s1=e2+∂1∫0−te2dtp1/q1,s2=e3+∂2∫0−te3dtp2/q2,where ∂1>0 and ∂2>0 represent the designed constants of sliding mode surface and p1, p2, q1, and q2 are the positive odd numbers, where 1<p1/q1<2 and 1<p2/q2<2.The derivative of (24) can be calculated as follows:(25)s˙1=e˙2+∂1p1q1e2∫0−te2dtp1/q1−1=σ1Δw+σ2ΔTe−Δw˙ref+∂1p1q1e2∫0−te2dtp1/q1−1,s˙2=e˙3+∂2p2q2e3∫0−te3dtp2/q2−1=f1i,Δδu+f2i,ψf,Δδ,Δw,ΔTe−ΔT˙refs+∂2p2q2e3∫0−te3dtp2/q2−1.Using the backstepping method to stabilize the load angleδ, the Lyapunov function is selected as follows:(26)V1=12e12.The derivation of (26) is(27)V˙1=e˙1e1=e1Δw−Δδ˙ref.According to the requirements of the Lyapunov stability condition, the equationV˙1<0 needs to be satisfied; therefore, the optimal choice of the virtual controller Δwref is as follows:(28)Δwref=−k1e1+Δδ˙ref,where k1>0,(29)V˙1=−k1e12.Then, the Lyapunov functionV2 is selected to stabilize the frequency error Δw as follows:(30)V2=12e12+s12.The derivation of (30) is as follows:(31)V˙2=−k1e12+s˙1s1.Similarity, in order to meet the equationV˙2<0, the constraint conditions of controller s˙1 can be designed as follows:(32)−λ1sgns1=σ1Δw+σ2ΔTe−Δw˙ref+∂1p1q1e2∫0−te2dτp1/q1−1,where λ1>0 and sgn⋅ means(33)sgnx=xxifx≠0,0ifx=0.Substituting (32) into (31), obtain V˙2 as follows:(34)V˙2=−k1e12−λ1s1.Therefore, the virtual controllerΔTrefs is as follows:(35)ΔTrefs=1σ2−λ1sgns1−σ1Δw+Δw˙ref−∂1p1q1e2∫0−te2dτp1/q1−1.Considering the parameter error at the same time, the form of the virtual error variable is obtained as follows:(36)ΔT^refs=1σ^2−λ1sgns1−σ^1Δw+Δw˙ref−∂1p1q1e2∫0−te2dτp1/q1−1,where σ^1 and σ^2 are the parameter estimation values.Then, using the above mentioned prescribed performance control method, the terminal sliding mode surfaces2 can be converted to ε through error conversion. Therefore, the new Lyapunov equation can be expressed as follows:(37)V3=12e12+s12+ε2+σ˜12r1+σ˜22r2,where r1andr2 are the gains of adaptive law, σ˜1=σ^1−σ1, σ˜2=σ^2−σ2, and the derivation of (37) is as follows:(38)V˙3=V˙2−k2ε2+εgρt,s2t1ρts˙2t−ρ˙s2tρt+k2ε+σ˜1r1σ^˙1−r1s1Δw+σ˜2r2σ^˙2−r2s1ΔTe,where k2>0, and the controller u can be designed as follows:(39)u=1f1i,Δδ−ρk2ε/gρt,s2+ρ˙s2ρ−f2i,ψf,Δδ,Δw,ΔTe+ΔT^refs−∂2p2q2e3∫0−te3dtp2/q2−1.The projection operator adaptive law can be designed as follows:(40)σ^˙1=r1Projσ^1,s1Δw,σ^˙2=r2Projσ^2,s1ΔTe,where function Proj⋅,⋅ is the projection operator adaptive law [38] valid for a robust adaptive controller which needs multiple differentiation, designed as follows:(41)Projξ^,ς=0,ifξ^=ξmaxandς>0,0,ifξ^=ξminandς<0,ς,otherwise,and Proj⋅,⋅ has the conclusion as follows:(42)Property1ξ^∈Ωξ≜ξ^:ξmin≤ξ^≤ξmax,Property2ξ˜Projξ^,ς−ς≤0,∀ς.Taking (39) and (40) into (38), the Lyapunov funtion can be expressed as follows:(43)V˙3=−k1e12−λ1s1−k2ε2≤0.Therefore, according to the above derivation process, equationV˙3≤0 can prove that the designed controller meets the Lyapunov stability condition.


## 4. Simulation Results and Analysis

In this section, a simulation model demonstrates the control performance of the proposed controller for the VSG system. Figure5 summarizes the detailed control method of this article, and the configuration of the VSG system and adjustable parameters system is shown in Table 1.Table 1
Parameters of VSG mode and controller parameter values.
ParametersValuesw314rads,Pref10kwUdc1100v,Uac510vJ0.4kg⋅m2,Dp28.2611Modelm0.0018,kp50,ki200ka0.0005,c1,τa0.05Parametersλ10.1∂1,∂20.9,1.5p1,p25,5Gains of SMCq1,q23,3Controller∂1,∂20.9,1.5p1,p25,5q1,q23,3k1,k210,0.9Gains of adaptive lawr1,r2100,100PP controllerh=10,ρ0=32,ρ∞=0.6The simulation results verify the control performance of the proposed controller under grid-connected and off-grid conditions, and the rated load is set at 10kW. The simulation results are shown as follows. Figure6 shows variation of active power when the system is connected to the grid at 0.4s and then disconnected again to the island state at 1.8s. The simulation results show that the controller designed in this paper can make the active power output smoother in the switching process of grid-connected and island, the oscillation range is smaller than the command-filtered backstepping controller (CBC) and terminal sliding mode controller (SMC) without adaptive law, and the control performance is better. Figure 7 and Figure 8 also show the frequency, voltage, and current changes of the system in the transient process. It is seen that the frequency changes tend to the rating. The waveforms of voltage and current have no obvious change when the system is connected to the grid and disconnected to the island state.Figure 6
Active power response in transient conditions.Figure 7
Frequency response in transient conditions.Figure 8
Current (a) and voltage (b) response in transient conditions.
(a)(b)Figure9 shows the comparison of tracking errors e1, e2, and e3 under the three control methods. It can be seen that the tracking errors e1 and e2 of the virtual control quantity Δδ and the virtual control quantity Δw are significantly smaller than the strategy of CBC, indicating that the proposed control strategy can better achieve error control and ensure the stability of the system. In addition, the tracking error error e2 of the adaptive terminal sliding mode control with prescribed performance is smoother and more stable in the transient adjustment process. Similarly, e3 can more accurately follow the system’s adjustments to achieve state stability.Figure 9
e1 (a), e2b, and e3c comparison of three methods.
(a)(b)(c)From Figure10, the terminal sliding mode surface s1 and the terminal sliding mode surface s2 with the prescribed performance are shown, and we can observe that s1 and s2 satisfy robust convergence. In addition, s2 also meets requirements for performance functions and is limited to a predetermined range. As can be seen from Figure 11, compared with the method proposed in [39], the control method adopted in this paper can make the output active power and frequency smoother in the process of grid-connected and off-grid and can reach the rated value more quickly after grid-connected and off-grid with less fluctuation.Figure 10
The responses of sliding mode surfaces1 (a) and prescribed performance function ρ and sliding mode surface s2 (b).
(a)(b)Figure 11
Frequency response in transient conditions (a) and active power response in transient conditions (b).
(a)(b)


## 5. Conclusion

An adaptive terminal sliding mode control strategy based on a prescribed performance function is proposed that improves VSG inverter stability in isolated islands, grid-connected, and transient operations. Compared with the traditional voltage-source inverter, the improved stability is realized by simulating a SG to introduce virtual damping and inertia. A projection operator and the parameter adaptive law are used to effectively estimate and limit the parameter values. The introduction of terminal sliding mode control method improves the system robustness of the system under transient situations and effectively limits the fluctuation range of the sliding mode surface through the prescribed performance. Simulation results show that the system can be better achieved including the fast response times and improved robustness and the adaptive estimation of uncertain parameter based on the linear virtual damping and inertia control and the nonlinear backstepping sliding mode control strategy under different conditions.In the future work, we will study the cooperative control of multiple VSGs and introduce multiagent information exchange network on this basis.



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*Source: 5742759-2022-04-26.xml*