# Disturbance-Observer-Based Adaptive Fuzzy Control for Islanded Distributed Energy Resource Systems

Authors: Chengshun Yang; Tao Hua; Yuchen Dai; Guofu Liu; Xiaoning Huang; Dongdong Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

With the aim to improve the antidisturbance ability of the islanded distributed energy resource (DER) systems, a disturbance-observer-based adaptive fuzzy sliding mode control (DAFSC) voltage controller is designed based on indirect vector control, which implements the voltage tracking and improves the self-regulation ability of the islanded DER systems. Firstly, the circuit diagram and the mathematical model of the DER system are presented. Then, the second-order sliding mode differentiator is designed to solve the problem of calculation expansion in the backstepping control method. To solve the influence of lumped disturbance on the system, a disturbance-observer is proposed to observe the unknown disturbance and compensate the controller feed-forward. Moreover, fuzzy control is proposed to reduce the dependence of the control effect on model accuracy. Finally, the stability of the controller is verified by Lyapunov stability theory, and the hardware in the loop results is given to verify that the control effect of the proposed DAFC controller has better dynamic performance compared with proportion-integral (PI) and the backstepping control strategy.


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## Body

## 1. Introduction

With the gradual depletion of traditional energy and the increasing awareness of environmental protection, distributed energy resource (DER) is getting more and more attention because of its excellent economy and environmental protection, and has been developing rapidly and widely used in distribution networks [1–6]. On the face of the Earth, there are many remote areas with a lot of solar energy, such as mountains and islands; then, a DER system can be built to supply power to the microgrid by using photovoltaic panels, batteries, and power electronics [7, 8]. There are many types of microgrid, which can be divided into island and grid connected according to whether it is interconnected with the public power grid; according to the current division, it can be divided into DC, AC, and AC\DC hybrid microgrid [9, 10]. Because of regional reasons, some DER systems cannot interconnect with the enormous power grid [11, 12], and then the DER system is operating in the island mode. Due to its small capacity and lack of mutual power transmission with the large power grid, the islanded DER system is challenging to maintain the output voltage and frequency stability when the load changes if the system lacks proper control. Thus, this will reduce the quality of the output voltage. As a result, it can cause damage to the power supply system and the electrical types of equipment [13, 14]. Therefore, this paper mainly focuses on ensuring the output voltage stability of the islanded DER system. In addition, in the isolated island microgrid, due to its small capacity and poor system inertia, a more advanced controller must be selected in the selection of its controller to ensure the stable operation of the system [15].Some scholars have done much research on the control strategies of islanded operation DER system. In reference [16], a fractional-order sliding mode control strategy is utilized in islanded DER system. The proposed control method in this article implements the tracking of the voltage under balanced load, unbalanced load, and nonlinear load, and the output voltage can be kept stable when the current is disorganized. However, the amplitude fluctuation of d-axis voltage is larger as the simulation result shows, and there is no control over the uncertainty part of the model. In reference [17], the droop control strategy is used to control the voltage and reactive power. Meanwhile, the distribution of active power in multiple microgrids is realized. However, the controller lacks consideration of model uncertainties as well, and the frequency recovery of the system is slow when the load is switched. Reference [18] presents an optimal secondary voltage control strategy based on droop and PQ control method for islanded multibus microgrid to tradeoff multibus voltage regulation and reactive power allocation between voltage modulation units, but the nonlinear load is not taken into account in the simulation; therefore, the simulation results only proved the control effect under linear load. Reference [19] utilizes a control strategy based on feed-forward signals in order to eliminate the impact of the load. However, the load is assumed to be balanced in the simulation, as in reference [18], the change in the nonlinear load is not considered. In the actual operation of the power system, there will be a large number of power electronics devices in the load. Thus, the load connected to the system is not limited to the linear load. In [20], a nonlinear model predictive control (NMPC) algorithm is utilized in the DER system to deal with the active power generation on the power generation side and realization of load forecasting. However, the output voltage is still fluctuating. Reference [21] proposes an algorithm for distributed generation (DG) units in the distribution network in a microgrid central controller to choose the best droop settings, and the security margins of islanded microgrid voltage have been improved in the article.In recent years, nonlinear control technology has been developed rapidly [22]. Backstepping control, as one of the nonlinear control methods, was first proposed by American professor Kokotovic et al. in 1991 [23]. Backstepping control is based on the control method of reducing the order of nonlinear systems, and in this case, the high-order nonlinear system is decomposed into several subsystems which do not exceed the highest order of the system; then, the error volume, virtual control volume, and Lyapunov function of each subsystem are defined from the bottom layer to the whole system. Finally, the controller can be designed for the whole system by using the stability condition of the Lyapunov function to ensure the stability of the system [24, 25]. But there is also a shortage of backstepping control methods, the differential swelling [26]. In order to solve this problem, the command filter is applied to the backstepping control method, which aims to solve the problem of the repeated derivation of the virtual controller in the backstepping controller, therefore reducing the amount of calculation in the system [27]. In the modeling of the DER system, the mathematical model is imprecise, and the parameters in the model are uncertain due to the dynamic characteristics of the system [28]. So, the adaptive method is used in this article to estimate the error of the model. Meanwhile, the projection operator is used to guarantee the boundedness of the estimate. An adaptive command-filtered backstepping (ACB) controller is used in reference [29] to solve the problem in induction motor drive systems. The results show the proposed controller has achieved a better control effect in the induction motor drive systems. In reference [30], a neural network is combined with the ACB controller to solve the problem in induction motor drive systems, and the better effect of the proposed control method has been proved compared with dynamic surface control by simulation results. In [31], the ACB controller combined with the observer is used in uncertain nonlinear systems, and two examples are provided to prove the validity of the controller. However, the boundedness of adaptive estimation is not considered in all references [29–31].In this paper, a disturbance-observer-based adaptive fuzzy sliding mode control (DAFSC) method is proposed for the islanded DER system, and the control objective is to stabilize the system’s output voltage without considering the current and the dynamic characteristics of the load. The main innovations of this paper are as follows:(1)
The unmodeled part of the system dynamics model is regarded as lumped disturbance, and a disturbance-observer is designed to observe the lumped disturbance(2)
In order to reduce the dependence of control effect on model accuracy, fuzzy logic system (FLS) is used to approximate the nonlinear part of the dynamics model(3)
A second-order sliding mode differentiator (SOSMD) is designed to estimate the virtual control signal and avoid the direct derivation of the virtual controllerFirstly, according to the control objective of VSC, the voltage mathematical model of the inverter is established by using Kirchhoff’s voltage and current law. Secondly, based on the established mathematical model, a second-order sliding mode differentiator is designed for the energy storage inverter to realize the tracking control of the system output voltage. Since the control effect of the model-based control method greatly depends on the model accuracy, in order to eliminate the influence of the unmodeled part of the system on the system, the unmodeled part is regarded as a centralized disturbance, and a disturbance-observer is designed for estimation, which further improves the effect of the controller. In addition, the controller integrates the integral synovial control. Therefore, the robustness and anti-interference ability of the controller have been greatly improved. Finally, in order to get rid of the influence of model accuracy on the control effect and improve the performance of the controller to the greatest extent, the problems of high-order nonlinearity and parameter uncertainty in the DER system are solved based on fuzzy logic system, so that the controller is more in line with engineering application.The sections of this paper are structured as follows. Section2 gives the circuit structure and dynamic model of the islanded DER system. The DAFSC controller is designed for the islanded DER system in Section 3, and the Lyapunov stability theorem proves the controller’s stability. In Section 4, the control effect of the proposed DAFSC controller has been demonstrated by comparing with proportion-integral (PI) and backstepping control strategy. Further, in Section 5, some conclusions are given.


## 2. Problem Statement and Preliminaries

### 2.1. Structure of the Islanded DER System

The structure of the islanded DER system is shown in Figure1. From Figure 1, we can obtain that the system consists of distributed energy resource udc, DC link capacitor Cdc, voltage source converter (VSC), control unit, three-phase load, and three-phase LC filter, where L and Cf represent the value of capacitance and inductance, and R represents the ohmic loss of the filter and VSC. The three-phase AC variables utabc, itabc, usabc, and isabc signify the AC side voltage and current of VSC and the DER system output voltage and current, respectively. In actual operation, the DC side of the system is connected to photovoltaic arrays in order to offer stable DC voltage. Thus, in order to ensure the stability of power supply, this paper uses a DC voltage source to assume the photovoltaic array offers a stable DC voltage.Figure 1
Structure of the islanded DER system.The designed controller is based on vector control method, that is, three-phase AC variable which the controller needed is transformed todq-frame from abc-frame, next, the signals in dq-frame are delivered into the controller to calculate with the dq-frame reference voltage usdref, usqref, then, the controller produces modulated signals mdq in dq-frame, and the signals in dq-frame are transformed back to abc-frame sent to the pulse width modulation (PWM) generator. In the PWM generator, the six-pulse signal is produced from comparing three-phase modulation wave and triangular wave with an amplitude of 1, and the six-pulse signal is used to control the VSC.In the design of the microgrid controller interconnected with the external power grid, we use the phase-locked loop (PLL) to lock the frequency and phase between the microgrid and the external grid. Considering that the islanded DER system is not interconnected with the external power grid, it is unable to provide a stable frequency reference by the system itself. Therefore, in this paper, the frequency and angular velocity are provided by a voltage-controlled oscillator (VCO) and then transmission to thedq converters [32].


### 2.2. Dynamic Model of the Islanded DER System

By using Kirchhoff’s voltage and current law, the mathematical model of islanded DER system in Figure1 has been established in this section. This paper studies the control of voltage source converter. The hybrid energy storage power supply is simplified to DC power supply. By stabilizing the DC voltage source and controlling the inverter in the system, the AC bus voltage of the islanded DER system is controlled [16, 33]. The relationship between voltage and current in the system is described as follows [34]:(1)Cfdus⟶dt=i⟶−is⟶,(2)Ldi⟶dt=ut⟶−us⟶−Ri⟶,where i⟶, is⟶, ut⟶, and us⟶ represent the space phasor of itabc, isabc, utabc, and usabc, respectively. To eliminate the variable i⟶ from (1) and (2), one obtains(3)d2us⟶dt2=−RLdus⟶dt−us⟶LCf−1Cfdis⟶dt−Ris⟶LCf+ut⟶LCf.The relationship between three-phase PWM modulating signalsm⟶ and output voltage of the VSC ut⟶ can be defined as(4)ut⟶=udc2m⟶.Thus, (3) can be described as(5)d2us⟶dt2=−RLdus⟶dt−us⟶LCf−1Cfdis⟶dt−Ris⟶LCf+udcm⟶2LCf.Transform each space phasor in (5) to the dq-frame, and (5) can be rewritten as(6)dusd+jusqejφdt2=−RLdusd+jusqejφdt,−Risd+jisqejφLCf+udcmd+jmqejφ2LCf,−usd+jusqejφLCf−1Cfdusd+jusqejφdt,where φ represents the rotation angle between α-axis and d-axis in park transformation provided by VCO. Split the left-hand side of the (6) to d-axis component and q-axis component, and (6) can be written as follows with considering the uncertainty of the model:(7)d2usddt2=2ωdusqdt−RLdusddt+ω2−1LCfusd+RωLusq−1Cfdisddt+ωCfisq−RLCfisd+udc2LCfmd+δd,(8)d2usqdt2=−2ωdusddt−RLdusqdt+ω2−1LCfusq−RωLusd−1Cfdisqdt−ωCfisd−RLCfisq+udc2LCfmq+δq,where δd and δq represent the lumped disturbances of d-axis and q-axis model, respectively.


## 2.1. Structure of the Islanded DER System

The structure of the islanded DER system is shown in Figure1. From Figure 1, we can obtain that the system consists of distributed energy resource udc, DC link capacitor Cdc, voltage source converter (VSC), control unit, three-phase load, and three-phase LC filter, where L and Cf represent the value of capacitance and inductance, and R represents the ohmic loss of the filter and VSC. The three-phase AC variables utabc, itabc, usabc, and isabc signify the AC side voltage and current of VSC and the DER system output voltage and current, respectively. In actual operation, the DC side of the system is connected to photovoltaic arrays in order to offer stable DC voltage. Thus, in order to ensure the stability of power supply, this paper uses a DC voltage source to assume the photovoltaic array offers a stable DC voltage.Figure 1
Structure of the islanded DER system.The designed controller is based on vector control method, that is, three-phase AC variable which the controller needed is transformed todq-frame from abc-frame, next, the signals in dq-frame are delivered into the controller to calculate with the dq-frame reference voltage usdref, usqref, then, the controller produces modulated signals mdq in dq-frame, and the signals in dq-frame are transformed back to abc-frame sent to the pulse width modulation (PWM) generator. In the PWM generator, the six-pulse signal is produced from comparing three-phase modulation wave and triangular wave with an amplitude of 1, and the six-pulse signal is used to control the VSC.In the design of the microgrid controller interconnected with the external power grid, we use the phase-locked loop (PLL) to lock the frequency and phase between the microgrid and the external grid. Considering that the islanded DER system is not interconnected with the external power grid, it is unable to provide a stable frequency reference by the system itself. Therefore, in this paper, the frequency and angular velocity are provided by a voltage-controlled oscillator (VCO) and then transmission to thedq converters [32].


## 2.2. Dynamic Model of the Islanded DER System

By using Kirchhoff’s voltage and current law, the mathematical model of islanded DER system in Figure1 has been established in this section. This paper studies the control of voltage source converter. The hybrid energy storage power supply is simplified to DC power supply. By stabilizing the DC voltage source and controlling the inverter in the system, the AC bus voltage of the islanded DER system is controlled [16, 33]. The relationship between voltage and current in the system is described as follows [34]:(1)Cfdus⟶dt=i⟶−is⟶,(2)Ldi⟶dt=ut⟶−us⟶−Ri⟶,where i⟶, is⟶, ut⟶, and us⟶ represent the space phasor of itabc, isabc, utabc, and usabc, respectively. To eliminate the variable i⟶ from (1) and (2), one obtains(3)d2us⟶dt2=−RLdus⟶dt−us⟶LCf−1Cfdis⟶dt−Ris⟶LCf+ut⟶LCf.The relationship between three-phase PWM modulating signalsm⟶ and output voltage of the VSC ut⟶ can be defined as(4)ut⟶=udc2m⟶.Thus, (3) can be described as(5)d2us⟶dt2=−RLdus⟶dt−us⟶LCf−1Cfdis⟶dt−Ris⟶LCf+udcm⟶2LCf.Transform each space phasor in (5) to the dq-frame, and (5) can be rewritten as(6)dusd+jusqejφdt2=−RLdusd+jusqejφdt,−Risd+jisqejφLCf+udcmd+jmqejφ2LCf,−usd+jusqejφLCf−1Cfdusd+jusqejφdt,where φ represents the rotation angle between α-axis and d-axis in park transformation provided by VCO. Split the left-hand side of the (6) to d-axis component and q-axis component, and (6) can be written as follows with considering the uncertainty of the model:(7)d2usddt2=2ωdusqdt−RLdusddt+ω2−1LCfusd+RωLusq−1Cfdisddt+ωCfisq−RLCfisd+udc2LCfmd+δd,(8)d2usqdt2=−2ωdusddt−RLdusqdt+ω2−1LCfusq−RωLusd−1Cfdisqdt−ωCfisd−RLCfisq+udc2LCfmq+δq,where δd and δq represent the lumped disturbances of d-axis and q-axis model, respectively.


## 3. Design of the DAFSC Controller

### 3.1. Design of the Disturbance-Observer

In this paper, the disturbance-observer is designed to observe the lumped disturbances of the system. The models (7) and (8) can be rewritten as(9)x˙1.a=x2.ax˙2.a=fax+gma+δa,where a=1,2, a=1 indicates the model of d-axis, and a=2 indicates the model of q-axis. Moreover, x1.d=usd, x1.q=usq, g=udc/2LCf, and(10)fdusd,usq,isd,isq=2ωdusqdt−RLdusddt+ω2−1LCfusd+RωLusq−1Cfdisddt+ωCfisq−RLCfisd,fqusd,usq,isd,isq=−2ωdusddt−RLdusqdt+ω2−1LCfusq−RωLusd−1Cfdisqdt−ωCfisd−RLCfisq.It is assumed that the lumped disturbance is differentiable and bounded. Taking the lumped disturbance as an additional state variable of the system, we havex3.a=δa, δ˙a=ρa, and ρa≤ρd.a. Thus, (9) can be rewritten as(11)x˙1.a=x2.ax˙2.a=fax+gma+x3.ax˙3.a=δ˙a=ρat..Based on the model (11), the disturbance-observer can be designed as follows:(12)x^˙1.a=x^2.a−λ1.asigb+1/2x˜1.ax^˙2.a=fax+gma+x^3.a−λ2.asigb+1/2x˜1.ax^˙3.a=−λ3.asigb+1/2x˜1.a,.where x^i.a is the estimated value of xi.a, x˜i.a=x^i.a−xi.a are the observation errors, λi>0 is the observer gain, 0<b<1, and sigγx=sgnxxγ.Then, the observer error system can be calculated as(13)x˜˙1.a=x˜2.a−λ1.asigb+1/2x˜1.ax˜˙2.a=x˜3.a−λ2.asigb+1/2x˜1.ax˜˙3.a=−ρa−λ3.asigbx˜1.a..Theorem 1 (see [35]).
Assuming that the lumped disturbanceδa is differentiable and bounded, considering the disturbance-observer (12), there exist the observer gains λ1.a>0, λ2.a>0, λ3.a>0, and 0<b<1, such that the observation error can be converged, that is, x3.a≤ℓa.


### 3.2. Design of the Virtual Controller

Definition ofd-axis tracking errors is as follows:(14)e1=usd−usdref,(15)e2=u˙sd−ydc=yd−ydc,where ydc is the output signal of the virtual controller pass through the SOSMD. Take the derivative of two tracking errors in (14) and (15):(16)e˙1=u˙sd−u˙sdref,e˙2=y˙d−y˙dc.In order to stabilize (15), the Lyapunov function is designed as follows:(17)Vd=12e12.And, the derivative of the Lyapunov function is calculated as(18)V˙d=e1e˙1=e1yd−u˙sdref=−k1e12+e1k1e1+yd−u˙sdref,where k1 is a constant greater than zero. To make V˙d≤0, the virtual controller ydd is chosen as(19)ydd=−k1e1+u˙sdref.When a back stepping control strategy is designed in a high-order nonlinear system, due to repeated guidance to the virtual controller, the amount of calculation is expanded, and the controller’s response speed and control effect are affected. As a result, in view of this problem, the SOSMD is designed to estimate the virtual control signal, which is designed as(20)η˙1.a=τ1.aτ1.a=−β1η1.a−yad1/2signη1.a−yad+η2.aη˙2.a=−β2signη2.a−τ1.a,,where β1 and β2 are positive constants and η1.a and τ1.a are the estimated values of yac and y˙ac, respectively.However, a filter error will be generated during the operation of the SOSMD. To compensate for the filtering error, the tracking errore1 should be redefined as(21)e¯1=e1−ε1,where the ε1 is the compensation signal of filter error defined as(22)ε.1=−k1ε1+ydc−ydd.Definition ofq-axis tracking errors is as follows:(23)e3=usq−usqref,(24)e4=u˙sq−yqc=yq−yqc,Theyqc is the output of the q-axis SOSMD in (24). The derivative of (23) and (24) is calculated as(25)e˙3=u˙sq−u˙sqref,e˙4=u˙q−u˙qc.To stabilize (25), the Lyapunov function is defined as(26)Vq=12e32,The derivative of (26) is calculated as(27)V˙q=e3e˙3=e3yq−u˙sqref=−k3e32+e3k3e3+yq−u˙sqref,where k3 is a constant greater than zero. Also, for the sake of making V˙q≤0, the virtual controller yqd is chosen as(28)yqd=−k3e3+u˙sqref.Same as thed-axis design process, the virtual controller yqd is sent to the SOSMD to get signal yqc and y˙qc. Redefine the tracking error e3 as(29)e¯3=e3−ε2,where the compensation signal of filter error ε2 is designed as(30)ε.2=−k3ε2+yqc−yqd.According to (22), the derivative of error e¯1 can be calculated as(31)e¯˙1=e˙1−ε.1=u˙sd−ydc+k1ε1−k1e1−u˙sdref+u˙sdref=−k1e¯1+e2.Similarly, on the basis of (30), the derivative of e¯3 can be calculated as(32)e¯˙3=e˙3−ε.2u˙sq−yqc+k3ε2−k3e3−u˙sqref+u˙sqref=−k3e¯3+e4.The derivative of errorse2 and e4 is calculated as follows:(33)e˙2=y˙d−y˙dc=fdusd,usq,isd,isq+gmd+δd−y˙dc,(34)e˙4=y˙q−y˙qc=fqusd,usq,isd,isq+gmq+δq−y˙qc.According to the following fuzzy rules to construct the fuzzy system [36–38],(35)IFx1isA1l1and⋯andxnisAnln,THENudisSl1⋯ln,where l1=1,2⋯,mi,i=1,2,…,n.Using product reasoning machine, single-value fuzzer and center-average fuzzer design the following fuzzy controllers:(36)fx|θ=∑i=1NΦi∏j=1nμjixj∑i=1N∏j=1nμjixj=ΦTYx,where Φ=ϕ1,ϕ2,…,ϕnT is the unknown idea parameter vector; μjixj is the memory of fuzzy system; and Υx=γ1x,γ2x,…,γnxT is the fuzzy basis function vector, in which(37)Yx=∏i=1nμjixj∑i=1N∏j=1nμjixj.Lemma 1 (see [39]).
A continuous functionfx is delimited on a compact set Ω. For any positive scalar ι, there exists a FLS ΦTΥx that meets the following relationship:(38)supx∈Ωfx−ΦTΥx≤ι.According to the approximation principle of fuzzy system, ifgx is defined as the function set of the system, for any constant ε>0 there exists a fuzzy system satisfying gx−fx≤ε.Considering the nonlinear parts of the dynamics model is approximated by FLS, which is expressed as(39)fd=ΦdTΥdx+od,fq=ΦqTΥqx+oq.where oa≤ιa, and we can get the unequal relation as(40)e2fd≤Φd22hd2ΥdTΥde22+12hd2+12ιd2+12e22.(41)e4fq≤Φq22hq2ΥqTΥqe42+12hq2+12ιq2+12e42.In order to stabilize the whole system, the Lyapunov function is defined again(42)Vt=12e¯12+e22+e¯32+e42+Ξ˜2Γ,where Γ is the positive scalar, Ξ=maxΦd2,Φq2, Ξ^ is the estimate value of Ξ, Ξ˜ is the estimation error satisfying Ξ˜=Ξ^−Ξ, and ha is the positive scalar.The derivative of Lyapunov function (42) is calculated as(43)V˙t=e¯1e¯˙1+e2e˙2+e¯3e¯˙3+e4e˙4+Ξ˜Ξ^˙Γ.Substituting (32)–(35), (40), and (41) into (43), one obtains(44)V˙t≤−k1e¯12−k2e22−k3e¯32−k4e42+e2e¯1+Ξ^2hd2ΥdTΥde2+gmd+δd−y˙dc+k2e2+e4e¯3+Ξ^2hq2ΥqTΥqe4+gmq+δq−y˙qc+k4e4+Ξ˜ΓΞ^˙−Γ2hd2Υd2Υde22+Γ2hq2Υq2Υqe42+12hd2+12ιd2+12e22+12hq2+12ιq2+12e42,where k2 and k4 are positive scalar.Based on (44), the d-axis and the q-axis controllers are designed as(45)md=1g−k2e2−e¯1−Ξ^2hd2ΥdTΥded−δ^d+y˙dc,(46)mq=1g−k4e4−e¯3−Ξ^2hq2ΥqTΥqeq−δ^q+y˙qc.The parameter updating lawΞ^˙ is constructed as(47)Ξ^˙=Γ2hd2Υd2Υded2+Γ2hq2Υq2Υqeq2−σΞ^.Substituting (45), (46), and (47) into (44), the derivative of Lyapunov function V˙t can be expressed as(48)V˙t≤−k1e¯12−k2e22−k3e¯32−k4e42−σΞ˜Ξ^Γ+e2ℓd+e4ℓq+12hd2+12ιd2+12e22+12hq2+12ιq2+12eq2.Lemma 2 (see [40]).
Young’s inequality: for any non-negative real numberx,y, the following inequality holds:(49)xy≤xpp+yqq,where p>1 and 1/p+1/q=1. If and only if xp=yq, the equal sign holds.Based on Young’s inequality, one obtains(50)e2ℓd≤12e22+12ℓd2,e4ℓq≤12e42+12ℓq2,−Ξ˜Ξ^≤−Ξ˜22+Ξ22.Then, (48) can be rewritten as(51)V˙t≤−k1e¯12−k2−1e22−k3e¯32−k4−1e42−σΞ˜22Γ+12hd2+12ιd2+12hq2+12ιq2+12ℓd2+12ℓq2+σΞ22Γ≤−AVt+ℬ.where A=min2k1,2k2−1,2k3,2k4−1,σ and ℬ=+1/2hd2+1/2ιd2+1/2hq2+1/2ιq2+1/2ℓd2+1/2ℓq2+σΞ2/2Γ. As a result, we can get the conclusion that the whole system is asymptotically stable at the origin.


## 3.1. Design of the Disturbance-Observer

In this paper, the disturbance-observer is designed to observe the lumped disturbances of the system. The models (7) and (8) can be rewritten as(9)x˙1.a=x2.ax˙2.a=fax+gma+δa,where a=1,2, a=1 indicates the model of d-axis, and a=2 indicates the model of q-axis. Moreover, x1.d=usd, x1.q=usq, g=udc/2LCf, and(10)fdusd,usq,isd,isq=2ωdusqdt−RLdusddt+ω2−1LCfusd+RωLusq−1Cfdisddt+ωCfisq−RLCfisd,fqusd,usq,isd,isq=−2ωdusddt−RLdusqdt+ω2−1LCfusq−RωLusd−1Cfdisqdt−ωCfisd−RLCfisq.It is assumed that the lumped disturbance is differentiable and bounded. Taking the lumped disturbance as an additional state variable of the system, we havex3.a=δa, δ˙a=ρa, and ρa≤ρd.a. Thus, (9) can be rewritten as(11)x˙1.a=x2.ax˙2.a=fax+gma+x3.ax˙3.a=δ˙a=ρat..Based on the model (11), the disturbance-observer can be designed as follows:(12)x^˙1.a=x^2.a−λ1.asigb+1/2x˜1.ax^˙2.a=fax+gma+x^3.a−λ2.asigb+1/2x˜1.ax^˙3.a=−λ3.asigb+1/2x˜1.a,.where x^i.a is the estimated value of xi.a, x˜i.a=x^i.a−xi.a are the observation errors, λi>0 is the observer gain, 0<b<1, and sigγx=sgnxxγ.Then, the observer error system can be calculated as(13)x˜˙1.a=x˜2.a−λ1.asigb+1/2x˜1.ax˜˙2.a=x˜3.a−λ2.asigb+1/2x˜1.ax˜˙3.a=−ρa−λ3.asigbx˜1.a..Theorem 1 (see [35]).
Assuming that the lumped disturbanceδa is differentiable and bounded, considering the disturbance-observer (12), there exist the observer gains λ1.a>0, λ2.a>0, λ3.a>0, and 0<b<1, such that the observation error can be converged, that is, x3.a≤ℓa.


## 3.2. Design of the Virtual Controller

Definition ofd-axis tracking errors is as follows:(14)e1=usd−usdref,(15)e2=u˙sd−ydc=yd−ydc,where ydc is the output signal of the virtual controller pass through the SOSMD. Take the derivative of two tracking errors in (14) and (15):(16)e˙1=u˙sd−u˙sdref,e˙2=y˙d−y˙dc.In order to stabilize (15), the Lyapunov function is designed as follows:(17)Vd=12e12.And, the derivative of the Lyapunov function is calculated as(18)V˙d=e1e˙1=e1yd−u˙sdref=−k1e12+e1k1e1+yd−u˙sdref,where k1 is a constant greater than zero. To make V˙d≤0, the virtual controller ydd is chosen as(19)ydd=−k1e1+u˙sdref.When a back stepping control strategy is designed in a high-order nonlinear system, due to repeated guidance to the virtual controller, the amount of calculation is expanded, and the controller’s response speed and control effect are affected. As a result, in view of this problem, the SOSMD is designed to estimate the virtual control signal, which is designed as(20)η˙1.a=τ1.aτ1.a=−β1η1.a−yad1/2signη1.a−yad+η2.aη˙2.a=−β2signη2.a−τ1.a,,where β1 and β2 are positive constants and η1.a and τ1.a are the estimated values of yac and y˙ac, respectively.However, a filter error will be generated during the operation of the SOSMD. To compensate for the filtering error, the tracking errore1 should be redefined as(21)e¯1=e1−ε1,where the ε1 is the compensation signal of filter error defined as(22)ε.1=−k1ε1+ydc−ydd.Definition ofq-axis tracking errors is as follows:(23)e3=usq−usqref,(24)e4=u˙sq−yqc=yq−yqc,Theyqc is the output of the q-axis SOSMD in (24). The derivative of (23) and (24) is calculated as(25)e˙3=u˙sq−u˙sqref,e˙4=u˙q−u˙qc.To stabilize (25), the Lyapunov function is defined as(26)Vq=12e32,The derivative of (26) is calculated as(27)V˙q=e3e˙3=e3yq−u˙sqref=−k3e32+e3k3e3+yq−u˙sqref,where k3 is a constant greater than zero. Also, for the sake of making V˙q≤0, the virtual controller yqd is chosen as(28)yqd=−k3e3+u˙sqref.Same as thed-axis design process, the virtual controller yqd is sent to the SOSMD to get signal yqc and y˙qc. Redefine the tracking error e3 as(29)e¯3=e3−ε2,where the compensation signal of filter error ε2 is designed as(30)ε.2=−k3ε2+yqc−yqd.According to (22), the derivative of error e¯1 can be calculated as(31)e¯˙1=e˙1−ε.1=u˙sd−ydc+k1ε1−k1e1−u˙sdref+u˙sdref=−k1e¯1+e2.Similarly, on the basis of (30), the derivative of e¯3 can be calculated as(32)e¯˙3=e˙3−ε.2u˙sq−yqc+k3ε2−k3e3−u˙sqref+u˙sqref=−k3e¯3+e4.The derivative of errorse2 and e4 is calculated as follows:(33)e˙2=y˙d−y˙dc=fdusd,usq,isd,isq+gmd+δd−y˙dc,(34)e˙4=y˙q−y˙qc=fqusd,usq,isd,isq+gmq+δq−y˙qc.According to the following fuzzy rules to construct the fuzzy system [36–38],(35)IFx1isA1l1and⋯andxnisAnln,THENudisSl1⋯ln,where l1=1,2⋯,mi,i=1,2,…,n.Using product reasoning machine, single-value fuzzer and center-average fuzzer design the following fuzzy controllers:(36)fx|θ=∑i=1NΦi∏j=1nμjixj∑i=1N∏j=1nμjixj=ΦTYx,where Φ=ϕ1,ϕ2,…,ϕnT is the unknown idea parameter vector; μjixj is the memory of fuzzy system; and Υx=γ1x,γ2x,…,γnxT is the fuzzy basis function vector, in which(37)Yx=∏i=1nμjixj∑i=1N∏j=1nμjixj.Lemma 1 (see [39]).
A continuous functionfx is delimited on a compact set Ω. For any positive scalar ι, there exists a FLS ΦTΥx that meets the following relationship:(38)supx∈Ωfx−ΦTΥx≤ι.According to the approximation principle of fuzzy system, ifgx is defined as the function set of the system, for any constant ε>0 there exists a fuzzy system satisfying gx−fx≤ε.Considering the nonlinear parts of the dynamics model is approximated by FLS, which is expressed as(39)fd=ΦdTΥdx+od,fq=ΦqTΥqx+oq.where oa≤ιa, and we can get the unequal relation as(40)e2fd≤Φd22hd2ΥdTΥde22+12hd2+12ιd2+12e22.(41)e4fq≤Φq22hq2ΥqTΥqe42+12hq2+12ιq2+12e42.In order to stabilize the whole system, the Lyapunov function is defined again(42)Vt=12e¯12+e22+e¯32+e42+Ξ˜2Γ,where Γ is the positive scalar, Ξ=maxΦd2,Φq2, Ξ^ is the estimate value of Ξ, Ξ˜ is the estimation error satisfying Ξ˜=Ξ^−Ξ, and ha is the positive scalar.The derivative of Lyapunov function (42) is calculated as(43)V˙t=e¯1e¯˙1+e2e˙2+e¯3e¯˙3+e4e˙4+Ξ˜Ξ^˙Γ.Substituting (32)–(35), (40), and (41) into (43), one obtains(44)V˙t≤−k1e¯12−k2e22−k3e¯32−k4e42+e2e¯1+Ξ^2hd2ΥdTΥde2+gmd+δd−y˙dc+k2e2+e4e¯3+Ξ^2hq2ΥqTΥqe4+gmq+δq−y˙qc+k4e4+Ξ˜ΓΞ^˙−Γ2hd2Υd2Υde22+Γ2hq2Υq2Υqe42+12hd2+12ιd2+12e22+12hq2+12ιq2+12e42,where k2 and k4 are positive scalar.Based on (44), the d-axis and the q-axis controllers are designed as(45)md=1g−k2e2−e¯1−Ξ^2hd2ΥdTΥded−δ^d+y˙dc,(46)mq=1g−k4e4−e¯3−Ξ^2hq2ΥqTΥqeq−δ^q+y˙qc.The parameter updating lawΞ^˙ is constructed as(47)Ξ^˙=Γ2hd2Υd2Υded2+Γ2hq2Υq2Υqeq2−σΞ^.Substituting (45), (46), and (47) into (44), the derivative of Lyapunov function V˙t can be expressed as(48)V˙t≤−k1e¯12−k2e22−k3e¯32−k4e42−σΞ˜Ξ^Γ+e2ℓd+e4ℓq+12hd2+12ιd2+12e22+12hq2+12ιq2+12eq2.Lemma 2 (see [40]).
Young’s inequality: for any non-negative real numberx,y, the following inequality holds:(49)xy≤xpp+yqq,where p>1 and 1/p+1/q=1. If and only if xp=yq, the equal sign holds.Based on Young’s inequality, one obtains(50)e2ℓd≤12e22+12ℓd2,e4ℓq≤12e42+12ℓq2,−Ξ˜Ξ^≤−Ξ˜22+Ξ22.Then, (48) can be rewritten as(51)V˙t≤−k1e¯12−k2−1e22−k3e¯32−k4−1e42−σΞ˜22Γ+12hd2+12ιd2+12hq2+12ιq2+12ℓd2+12ℓq2+σΞ22Γ≤−AVt+ℬ.where A=min2k1,2k2−1,2k3,2k4−1,σ and ℬ=+1/2hd2+1/2ιd2+1/2hq2+1/2ιq2+1/2ℓd2+1/2ℓq2+σΞ2/2Γ. As a result, we can get the conclusion that the whole system is asymptotically stable at the origin.


## 4. Results and Analysis

In order to verify the effectiveness of the designed controller, according to the islanded DER model shown in Figure1, a simulation model is built by using MATLAB/Simulink software in this section. The parameters in the DER system are listed in Table 1. For the sake of testing the dynamic characteristics of the controller, different types of load are used during the simulation process, including balanced load, unbalanced load, and nonlinear load. Table 2 shows the parameters of different types of loads.Table 1
The basic parameters of the islanded DER system.
ParametersValueDescriptionL300μHFilter inductorCf500μFFilter capacitanceR3mΩFilter resistanceudc1800VDC link voltageω100π rad/sNominal angular frequencyTable 2
Different types of load.
Load typeLoad descriptionTransformer ratioBalanced loadThree star-connectedRL series load with R = 190 Ω and L = 327 μH.5kVrms/3.3 kVrmsUnbalanced loadRL series Load with R = 150 Ω and L = 1.24 H in phase-a, RL series load with R = 140 Ω and L = 229 mH in phase-b, phase-c is open.Nonlinear loadUncontrollable three-phase bridge rectifier circuit, seriesRL circuits with R = 630 Ω and L = 1.67 H in DC side.It is important to adjust the controller and adaptive parameters to make sure the controller has a good effect. The adaptive parametersr1 and r2 are set to zero at first; owing to the control process in dq-frame, there is no coupling between the d-axis and q-axis; therefore, the d-axis controller parameters k1, k2 and q-axis controller parameters k3, k4 can be adjusted separately. Then, adjust the adaptive parameters to make the system achieve a satisfactory state. The controller parameters and adaptive parameters are listed in Table 3.Table 3
The basic parameters of the islanded DER system.
ParametersValueDescriptionk1, k210000, 12000Gains ofd-axis controllerk3, k49000, 13000Gains ofq-axis controllerΓ, σ10, 0.1Gains of adaptiveλ1.a3×10000Gains of disturbance-observerλ2.a3×100002Gains of disturbance-observerλ3.a100002Gains of disturbance-observerRemark 1.
The parameter selection steps of the DAFSC controller are as follows: firstly, determine the membership function and set the value of adaptive rateΓ from small to large so that the adaptive fuzzy logic system can achieve the goal of approaching the nonlinear function in the inverter model of the energy storage system. The adaptive learning rate σ is selected between 0.001 ˜ 1. A too high learning rate will cause the instability of FLS, and too small will slow down the adaptive convergence speed. After that, the inner loop parameters k2,k4 are adjusted. Then, the outer loop parameters k1,k3 are adjusted to realize the accurate tracking control of the output voltage of DER. Finally, the damping and bandwidth of the instruction filter and the gain coefficient of the disturbance-observer are adjusted. The performance of the command filter depends on the damping and bandwidth. During the debugging of the command filter, it should be observed whether the output signal can approach the input signal. If so, the damping and bandwidth values at this time are appropriate. To adjust the gain coefficient of the disturbance-observer, it is necessary to observe whether the observed value can approach the actual disturbance value. If so, the gain coefficient at this time is appropriate. The value of the gain coefficient should not be too large. Otherwise, it will increase the influence of the estimated peak.
The whole simulation process lasts 2.5 s. At the start of the simulation, the balanced load is switched on, the reference voltage ofd-axis usdref starts to rise straight from 0 V and reaches 480 V when t = 0.1 s, at t = 2 s, usdref is changed stepwise to 530 V from 480 V, and the q-axis reference voltage usqref is maintained at 0 V in the whole simulation. On the load side, the load of the system is switched into unbalanced loads and nonlinear loads at t = 0.9 s and t = 1.8 s, respectively.
Figures2 and 3 show the voltage tracking effect and tracking error surfaces under proportion-integral (PI) control, respectively. As we can see from Figures 2 and3, when the load is switched from three-phase balanced load to three-phase unbalanced load at t = 0.9 s, the d-axis voltage tracking error rises from 2 V to about 5 V, and q-axis voltage tracking error rises from 4 V to about 6 V; when the load is switched from three-phase unbalance load to nonlinear load at t = 1.8 s, the voltage tracking error is further expanded.
Figures4 and 5 show the voltage tracking effect and tracking error surfaces under backstepping control strategy, respectively. By comparison with Figures 4 and 5 under PI control strategy, it can be seen that the voltage keeps steady tracking when the load changes. However, the voltage tremor of d-axis and q-axis is still too large. Moreover, when usdref is changed stepwise to 510 V from 450 V at t = 2 s, the output voltage of the d-axis produces an overshoot of about 30 V when the step signal occurs.
Figures6 and 7 show the voltage tracking effect and tracking error surfaces under the DAFSC strategy, respectively. When the load type is switched in the system at t = 0.9 s and t = 1.8 s, the tracking error does not produce fluctuations by using the DAFSC controller. In addition, when a step signal is generated by d-axis reference voltage at t = 2 s, the d-axis output voltage resumes tracking quickly and smoothly, and no overshoot and distortion appear. Furthermore, the tracking error of d-axis and q-axis is maintained within 1 V in the whole process of simulation under the DAFSC strategy. Therefore, through the simulation of the three kinds of control methods, it is proved that the DAFSC controller designed in this paper has better performance and robustness than traditional PI and backstepping control.
Figures8 and 9 show the three-phase voltage and current waveforms at the time of load switching under the DAFSC strategy. It can be seen that the three-phase current is distorted when the load is switched to an unbalanced load or nonlinear load. Nonetheless, the voltage remained stable. Figure 10 contrasts the input signal yd and output signal ydc of the SOSMD. The input and output signals are almost identical. Thus, the derivative of yd can be approximately equal to the derivative of ydc. Figure 11 illustrates the estimation of lumped disturbance by disturbance-observer. Figure 12 compares the voltage control performance further, the real-time total harmonic distortion (THD) analysis of DER system output voltage under PI, backstepping, and DAFSC strategy. As can be seen, the proposed method ranks first on the current control performance with the least THD in all cases.Figure 2
Voltage tracking effect under PI control strategy.Figure 3
Tracking error surfaces under PI control strategy.
(a)(b)Figure 4
Voltage tracking effect under backstepping control strategy.Figure 5
Tracking error surfaces under backstepping control strategy.
(a)(b)Figure 6
Voltage tracking effect under DAFSC strategy.Figure 7
Tracking error surfaces under DAFSC strategy.
(a)(b)Figure 8
Waveforms of voltage and current under DAFSC strategy att = 0.85 s–1 s.
(a)(b)Figure 9
Waveforms of voltage and current under DAFSC strategy att = 1.75 s–1.9 s.
(a)(b)Figure 10
Input and output signal of SOSMD.Figure 11
Estimation of lumped disturbance.
(a)(b)Figure 12
THD of DER system output voltage under different controllers.


## 5. Conclusion

This paper proposed a DAFSC method to deal with the voltage control problem for islanded DER systems, realizing the stable tracking of the system output voltage to the reference voltage no matter how the current is distorted. Meanwhile, the SOSMD is designed in the controller to solve the problem of calculation expansion in the backstepping control strategy. Then, the lumped disturbance of the system is observed by a designed disturbance-observer. Moreover, the nonlinear part of the dynamics model is approximated by FLS. The results indicate that the DAFSC strategy has better dynamic performance than the PI controller and backstepping controller, both on-load switching and reference voltage mutation. Moreover, the buffeting of the output voltage is kept within a small range by using the DAFSC controller. In the future work, our main research will be on how DER can effectively achieve better in the context of carbon neutrality and carbon peaks, and full utilization to achieve carbon emission reduction and related research on distributed energy storage systems.



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*Source: 1527705-2022-02-03.xml*