# Adaptive Sliding-Mode Control in Bus Voltage for an Islanded DC Microgrid

Authors: Dan Zhang; Jie Wang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

The control of bus voltage is a crucial task for the stable operation of islanded DC microgrids. To improve the DC bus voltage control dynamics and stability, this paper proposes an adaptive sliding-mode control method based on large-signal model. The sliding-mode control, adaptive observation, and fix-frequency pulse width modulation technology are adopted and combined efficiently, which guarantee stable bus voltage and the constant switching frequency of closed-loop system, regardless of how the parameters vary with the variable constant-power loads and uncertainties. In addition, the reference values can be quickly tracked by the state variables using the proposed method without any additional sensors/hardware circuits. Therefore, this method is beneficial for the scalability and plug-play of the distributed generators and loads within the DC microgrids. The performance of the proposed control method has been successfully verified in simulation.


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## Body

## 1. Introduction

The microgrid (MG) has been widely applied in a wide variety of practical fields, such as islands [1, 2], remote areas [3], industrial power systems [4], commercial buildings [5], residential homes [6], and university campuses [7]. This kind of technology can yield a high-reliable regional power system with favourable economy and flexible power supply capacity. As a crucial technology of smart grid, the MG can make up the disadvantages existing in centralized power supply grid and alleviate the influence of distributed generators (DGs) fluctuation on utility grid [8, 9]. Owning to the aforesaid superiorities, the MG has become an excellent candidate for accepting high permeability DGs into the power grid. Compared with the AC MGs, DC MGs can achieve lower expenditure and higher efficiency by eliminating the extra power electronic conversion stages, since DC power is directly utilized by many renewable DGs, energy storage systems, and loads [10]. Additionally, high controllability is offered by DC system, since it does not have issues associated with synchronization between the DGs, reactive power flows, harmonic currents, and DC/AC conversion losses, which are intrinsic to the AC MG [11]. Moreover, the DC MG can run stand-alone and provide reliable power supply to the loads, although it fully decouples from the utility grid. For the aforementioned factors, the DC MG is gaining attention incrementally.In island mode, the stability of bus voltage is the critical criterion to assess the stability of a DC MG. When an islanded DC MG encounters abrupt changes from loads and DC sources or is required to supply power for constant-power loads (CPLs), Emadi et al. [12, 13] shows that the stability of bus voltage will meet greater challenges. Several control strategies have been proposed based on improved PI controllers in [14, 15] to improve the stability of bus voltages. In [16], they propose superior linear controllers that perform well in DC distributed networks with CPLs and improve the performance and stability of the system. The aforesaid approaches are linearization control methods with limitations to control accuracy, sensitivity, and robustness to parameter variations. Therefore, the nonlinear control strategies have been increasingly studied in MGs.In [17, 18], the droop and sliding-mode control (SMC) approaches are employed to design controllers, respectively. However, the adopted small-signal models are unable to meet global stability requirements when operating points are allowed to work on voltage levels away from the linearization point. In [19], a feedback linearization method is derived on the basis of the Lyapunov stability theory to overcome the impact of CPLs on global stability in DC MG, without considering the uncertainties in the real system. Thus, the robustness of this control system is restricted. Higher order sliding model control (HOSMC) is studied deeply and applied to MGs by Michele [20] and Satish et al. [21]. This method can alleviate chattering and exert the ability of variable structure control for large-signal processing. However, it is difficult to obtain the uncertain boundary and abate the dependence of the control system on the model, which limit the application of this approach in engineering. In order to ensure robustness of control systems under loads and sources variations, a sliding-mode control method based on a washout filter with hysteresis band (HB) is proposed [22, 23]. The disadvantages of this technique are that there exist variable switching frequency and high sensitivity to noise. Su et al. [24] employ an adaptive total sliding-mode control (ATSMC) to ensure the stability and dynamic performances of the MG. The advantages of SMC in design and its robustness for parameter uncertainties and external disturbance are shown obviously. In [25], an adaptive nonlinear observation method is proposed for a boost power factor correction (PFC) converter to estimate the load value without load information. This method offers a reliable and cost-effective solution with minimized sensors. In [26], by employing PWM technique in SMC, a fixed-frequency pulse width modulation- (PWM-) based SM controller can be obtained. The frequency of switching signal is constant, regardless of how the duty cycle varies with control signal. The antecedent control methods used in MGs have been well studied and achieved positive control effects. However, single method is incapable of solving hybrid issues, such as variable structure characteristics existing in converters, frequency fluctuation, system parameter variation, and disturbance caused by CPLs. Therefore, it is necessary to associate with several control methods to iron out complex problems.The main contributions of this paper are as follows. The first one is that an advanced bus voltage controller based on adaptive sliding-mode control (ASMC) method combined with fixed-frequency PWM technology using large-signal modelling theory is proposed. Second contribution is to apply the proposed control method to an islanded DC MG with CPLs. Under uncertain external disturbances, the adaptive observer can accurately estimates the values of loads and battery voltage without any additional sensors/hardware circuits. Therefore, it is beneficial for the scalability and plug-play of the DGs and loads within the DC MG. The global SMC method can force the state variables to track their reference values quickly and stably in the presence of CPLs by controlling bidirectional buck/boost converter. Also, fix-frequency PWM is adopted to make the switching frequency constant, which greatly simplifies the design complication of the filter compared with HB method. The third contribution is that the performance of the proposed controller has been validated through theoretical proof and simulation studies in MATLAB. The proposed composite control method offers a very reliable and cost-effective solution for various applications such as MGs, telecommunications, plug-in electric vehicle, aerospace, and so on.The paper is organized as follows. In Section2, the configuration and large-signal model of the DC MG under isolated operation mode is described. Section 3 presents the theoretical derivations and design of a fixed-frequency PWM-based ASMC for stand-alone DC MG. Also, a detailed analysis of the stability problem is provided in this section. Section 4 demonstrates the performances and comparison of two closed-loop control systems adopted different control methods which are HB-based SMC and the proposed control scheme. Finally, Section 5 concludes the article.


## 2. DC Microgrid System Description

### 2.1. Microgrid Configuration

The structure of an isolated DC MG is depicted in Figure1, where all DGs, active loads (CPLs), resistive load, and energy storage device are connected to the DC bus. In such power electronic converter dominated systems, tightly regulated point-of-load converters (POLs) paired with motors, actuators, and even simple resistive loads behave as CPLs. The total power of CPLs in MG is PL. This means that the power absorbed from sources by CPLs remains constant during system operation. Moreover, DGs (photovoltaic power and fuel cell) and connected DC-DC boost converters are together seen as constant-power sources (CPSs). The total power of CPSs is PS. This means that, despite the variation on the bus voltage, the current provided by the power converter adapts to keep injecting a constant power into MG. As CPLs and CPSs behave in the same way, they can be modelled as a lumped CPL, calculated as P=PL+PS [27].Figure 1
Structure of isolated DC microgrid.Figure2 shows the energy flow of DC MG. It can be seen from Figure 2 that the directions of power flow through CPSs, CPLs, and resistive load are unidirectional. The directions of power flow through CPSs are from power sources to the DC bus; inversely, the directions of CPLs and resistive load are from DC bus to loads. However, the power flow direction in bidirectional buck/boost converter is bidirectional. When the direction of the power flow is from battery to the DC bus, it is obvious that the power provided by sources is insufficient supply to meet the load demand. Therefore, the battery is in discharge state, P≥0. On the other hand, when the power offered by sources is more than load demands, the battery is in charge state, P≤0. At this time the direction of power flow is from DC bus to battery and the power sources can supply electric power for entire MG system.Figure 2
Power flow of DC microgrid.


### 2.2. Microgrid Modelling

With purpose of theoretical analysis, an equivalent topology of DC MG is applied in Figure3. The bus current is described as follows: (1)ibus=iR+iP,iR=vbusR,iP=Pvbus,where R and P denote the resistance value of resistive load and the power of aforementioned lumped CPL; iR and iP are the currents flowing through resistive load and lumped CPL, respectively. Both R and P are time-variable [28].Figure 3
Equivalent topology of DC microgrid.Only the case of current continuous conduction mode (CCM) is considered in this paper; hence, during this mode, the dynamic model of MG system is given by(2)LdiLdt=vS-uvbus,Cdvbusdt=uiL-ibus,where iL, vbus, and vs represent inductive current, bus voltage, and battery voltage, respectively. u denotes control input of MG system. S1 and S2 switch on complementarily. When S1 is in conducting state, u is equal to 1; conversely, u is equal to 0. Therefore, the switching control law is adopted as(3)u=u+=1,u-=0.For the simplified analysis, we replaceiL and vbus by x1 and x2, then (2) would be shown as(4)x˙1=vSL-ux2L,x˙2=ux1C-Mx2C-NCx2,where M=1/R∈R+, N=P∈R, and vS∈R+.


## 2.1. Microgrid Configuration

The structure of an isolated DC MG is depicted in Figure1, where all DGs, active loads (CPLs), resistive load, and energy storage device are connected to the DC bus. In such power electronic converter dominated systems, tightly regulated point-of-load converters (POLs) paired with motors, actuators, and even simple resistive loads behave as CPLs. The total power of CPLs in MG is PL. This means that the power absorbed from sources by CPLs remains constant during system operation. Moreover, DGs (photovoltaic power and fuel cell) and connected DC-DC boost converters are together seen as constant-power sources (CPSs). The total power of CPSs is PS. This means that, despite the variation on the bus voltage, the current provided by the power converter adapts to keep injecting a constant power into MG. As CPLs and CPSs behave in the same way, they can be modelled as a lumped CPL, calculated as P=PL+PS [27].Figure 1
Structure of isolated DC microgrid.Figure2 shows the energy flow of DC MG. It can be seen from Figure 2 that the directions of power flow through CPSs, CPLs, and resistive load are unidirectional. The directions of power flow through CPSs are from power sources to the DC bus; inversely, the directions of CPLs and resistive load are from DC bus to loads. However, the power flow direction in bidirectional buck/boost converter is bidirectional. When the direction of the power flow is from battery to the DC bus, it is obvious that the power provided by sources is insufficient supply to meet the load demand. Therefore, the battery is in discharge state, P≥0. On the other hand, when the power offered by sources is more than load demands, the battery is in charge state, P≤0. At this time the direction of power flow is from DC bus to battery and the power sources can supply electric power for entire MG system.Figure 2
Power flow of DC microgrid.


## 2.2. Microgrid Modelling

With purpose of theoretical analysis, an equivalent topology of DC MG is applied in Figure3. The bus current is described as follows: (1)ibus=iR+iP,iR=vbusR,iP=Pvbus,where R and P denote the resistance value of resistive load and the power of aforementioned lumped CPL; iR and iP are the currents flowing through resistive load and lumped CPL, respectively. Both R and P are time-variable [28].Figure 3
Equivalent topology of DC microgrid.Only the case of current continuous conduction mode (CCM) is considered in this paper; hence, during this mode, the dynamic model of MG system is given by(2)LdiLdt=vS-uvbus,Cdvbusdt=uiL-ibus,where iL, vbus, and vs represent inductive current, bus voltage, and battery voltage, respectively. u denotes control input of MG system. S1 and S2 switch on complementarily. When S1 is in conducting state, u is equal to 1; conversely, u is equal to 0. Therefore, the switching control law is adopted as(3)u=u+=1,u-=0.For the simplified analysis, we replaceiL and vbus by x1 and x2, then (2) would be shown as(4)x˙1=vSL-ux2L,x˙2=ux1C-Mx2C-NCx2,where M=1/R∈R+, N=P∈R, and vS∈R+.


## 3. Control Scheme

### 3.1. Adaptive Observer

The adaptive observer is gaining increasing attention due to its low requirement to the accuracy of system mathematical model. In addition, this observer has an excellent robustness to variable parameters and external disturbances. In view of the above advantages existing in adaptive observer, based on the theory of parameter estimation, this paper designs a following adaptive observer according to the MG model and its feature:(5)x^˙1=v^SL-ux^2L-K1x1-x^1,x^˙2=ux^1C-M^x2C-N^Cx2+K2x2-x^2,where observer gains are K1>0 and K2>0; x^1, x^2, M^, N^, and v^S denote the estimates of x1, x2, M, N, and vS. x1 and x2 are observable state variables. Let x~1=x1-x^1, x~2=x2-x^2, M~=M-M^, N~=N-N^, and v~S=vS-v^S, then the error equations of adaptive observer can be written as follows:(6)x~˙1=v~SL-ux~2L-K1x~1,x~˙2=ux~1C-M~x2C-N~Cx2-K2x~2.Select a positive definite Lyapunov function as the following form:(7)V=12Lx~12+12Cx~22+12γ1M~2+12γ2N~2+12γ3v~S2,where the adaptive gains satisfy γ1>0, γ2>0 and γ3>0. The derivative of Lyapunov function is calculated by(8)V˙=Lx~1x~˙1+Cx~2x~˙2+1γ1M~M~˙+1γ2N~N~˙+1γ3v~Sv~˙S.Using (6), the derivative of the Lyapunov function can be derived as(9)V˙=-K1Lx~12-K2Cx~22-M~x2x~2+M^˙γ1-N~x~2x2+N^˙γ2+v~Sx~1-1γ3v^˙S.To eliminate the influence of uncertainties in the right side of (9) on state convergence, adaptive rates are structured. These adaptive rates are utilized to eliminate the influence of unknown bounds on the convergence of state error estimation; thus, they are selected as follows:(10)M^˙=-γ1x2x~2,N^˙=-γ2x~2x2,v^˙S=γ3x~1.We can getV˙=-K1Lx~12-K2Cx~22≤0, which means V˙ is negative semidefinite. Further, based on the Lasalle invariant theorem, x~1 and x~2 converge to 0 asymptotically.


### 3.2. ASMC

For the state spacex={x1,x2}∈R2, there exists a discontinuity manifold ∑:{x∈R2:Sx=0}, which divides state space into two parts by using the time-varying scalar sliding-mode switching function described as(11)Sx=S=x^1-x1_ref,where x1_ref denotes reference value of x1.In this paper, the current estimation error is selected as a sliding-mode switching function. Since the steady-state error of bus voltage is very small, and there is no need to add a voltage error integration term to the sliding surface. This switching function is a time-variant sliding surface, which is different from the traditional fixed-length sliding surface. It would utilize characteristics of adaptive sliding observer to achieve the function of sliding control. On the other hand, since the parameters of switching function are adaptive time-variant, the system would be in the sliding mode att=0 and keep sliding to the expected state on sliding surface with initial conditions selected appropriately. All above facts can realize the quickness and accuracy of control performance.When system reaches its steady state, based on conservation of energy, the reference value of inductive current can be calculated as follows:(12)IL_ref=V2bus_refM+NvS,where IL_ref and Vbus_ref denote reference values of inductive current and bus voltage at steady state, respectively. And x1_ref can be obtained as(13)x1_ref=V2bus_refM^v^S+N^v^S.Therefore, the derivative ofS can be described by(14)S˙=v^SL-ux^2L+K1x~1-N^˙v^S-N^v^˙Sv^S2-M^˙v^SV2bus_ref-M^v^˙SV2bus_refv^S2.By utilizing invariant condition, the equivalent control can be written in the following form:(15)ueq=v^S+K1Lx~1x^2+Lγ2x~2v^S+Lγ3x~1x2N^v^S2x2x^2+LV2bus_refγ1x2x~2v^S+γ3x~1M^v^S2x^2,where ueq is smooth function of discrete input function u. Based on both concept of equivalent control and the necessary and sufficient condition of sliding mode, there exists a following relationship:(16)0<ueq<1.Therefore, we can obtain(17)0<v^S+K1Lx~1+LN^˙v^S-LN^v^˙Sv^S2-LV2bus_refM^˙v^S-M^v^˙Sv^S2<x^2.Moreover, initial conditions should meet (18) to guarantee the occurrence of sidling mode at t=0; that is, the system state variables are sliding on sliding surface initially. For control system, there has been S˙x=0.(18)x^20>0,v^S0>0,M^0>0,S0=x^10-x1_ref0=0.


### 3.3. Composite Control System Structure

The control system structure of microgrid is designed in Figure4. The system includes four components: model of microgrid, adaptive observer, sliding adaptive controller, and PWM generator. Each part has the corresponding expression.Figure 4
Block diagram of control system.SMC is applied into this paper, since the feature of itself variable structure agrees with the characteristic of converter. However, two important characteristics existing in SMC are the switching frequency variation and chattering phenomenon. The chattering problem can be effectively improved by using the ASMC. There are two methods to overcome the defects of variable switching frequency as well as take advantage of SMC simultaneously. One is to add additional hardware circuit to fix switching frequency. It is obviously that this method will increase the cost of system and is adverse to the scalability and plug-play of the distributed generators and loads in DC MG. The other scheme is to combine SMC with other control methods. Since the model used in this paper is large-signal model, the common PWM voltage/current mode control is unsuitable to large-signal situation. Moreover, PWM controller can not show an absolute robustness to input and output variation. Consequently, fixed-frequency PWM-based ASMC, a combination of SMC, PWM method, and adaptive control method, is adopted in this paper. Not only is the influence of power supplies and loads fluctuation on the stability of the system considered, but also the problem that the controller cannot be realized when the input and output are variable and unknown is conquered. In addition, this controller achieves its optimal performance at the possible conditions [29, 30].Fixed-frequency PWM-based ASMC is equivalent to a voltage and current dual-loop control using PWM modulation. Particularly, it is unnecessary for this control method to establish small-signal model and to design the control parameters by utilizing the magnitude-frequency characteristic in classical control theory, so that the design process is relatively simple. Moreover, the proposed control method is derived from modelling of system large signal, which is suitable to wide range of the signal disturbance. Fixed-frequency PWM-based ASMC method has similar fundamental principles with classical PWM methods; the frequency of switching signals is fixed by fastening the frequency of triangle-wave signal. In order to apply the proposed control strategy in DC MG system, it is necessary to convert smoothing function(ueq) to the instantaneous duty cycle of the converter represented by D; then D is used in the bidirectional buck/boost converter to realize the proposed control scheme [31].


## 3.1. Adaptive Observer

The adaptive observer is gaining increasing attention due to its low requirement to the accuracy of system mathematical model. In addition, this observer has an excellent robustness to variable parameters and external disturbances. In view of the above advantages existing in adaptive observer, based on the theory of parameter estimation, this paper designs a following adaptive observer according to the MG model and its feature:(5)x^˙1=v^SL-ux^2L-K1x1-x^1,x^˙2=ux^1C-M^x2C-N^Cx2+K2x2-x^2,where observer gains are K1>0 and K2>0; x^1, x^2, M^, N^, and v^S denote the estimates of x1, x2, M, N, and vS. x1 and x2 are observable state variables. Let x~1=x1-x^1, x~2=x2-x^2, M~=M-M^, N~=N-N^, and v~S=vS-v^S, then the error equations of adaptive observer can be written as follows:(6)x~˙1=v~SL-ux~2L-K1x~1,x~˙2=ux~1C-M~x2C-N~Cx2-K2x~2.Select a positive definite Lyapunov function as the following form:(7)V=12Lx~12+12Cx~22+12γ1M~2+12γ2N~2+12γ3v~S2,where the adaptive gains satisfy γ1>0, γ2>0 and γ3>0. The derivative of Lyapunov function is calculated by(8)V˙=Lx~1x~˙1+Cx~2x~˙2+1γ1M~M~˙+1γ2N~N~˙+1γ3v~Sv~˙S.Using (6), the derivative of the Lyapunov function can be derived as(9)V˙=-K1Lx~12-K2Cx~22-M~x2x~2+M^˙γ1-N~x~2x2+N^˙γ2+v~Sx~1-1γ3v^˙S.To eliminate the influence of uncertainties in the right side of (9) on state convergence, adaptive rates are structured. These adaptive rates are utilized to eliminate the influence of unknown bounds on the convergence of state error estimation; thus, they are selected as follows:(10)M^˙=-γ1x2x~2,N^˙=-γ2x~2x2,v^˙S=γ3x~1.We can getV˙=-K1Lx~12-K2Cx~22≤0, which means V˙ is negative semidefinite. Further, based on the Lasalle invariant theorem, x~1 and x~2 converge to 0 asymptotically.


## 3.2. ASMC

For the state spacex={x1,x2}∈R2, there exists a discontinuity manifold ∑:{x∈R2:Sx=0}, which divides state space into two parts by using the time-varying scalar sliding-mode switching function described as(11)Sx=S=x^1-x1_ref,where x1_ref denotes reference value of x1.In this paper, the current estimation error is selected as a sliding-mode switching function. Since the steady-state error of bus voltage is very small, and there is no need to add a voltage error integration term to the sliding surface. This switching function is a time-variant sliding surface, which is different from the traditional fixed-length sliding surface. It would utilize characteristics of adaptive sliding observer to achieve the function of sliding control. On the other hand, since the parameters of switching function are adaptive time-variant, the system would be in the sliding mode att=0 and keep sliding to the expected state on sliding surface with initial conditions selected appropriately. All above facts can realize the quickness and accuracy of control performance.When system reaches its steady state, based on conservation of energy, the reference value of inductive current can be calculated as follows:(12)IL_ref=V2bus_refM+NvS,where IL_ref and Vbus_ref denote reference values of inductive current and bus voltage at steady state, respectively. And x1_ref can be obtained as(13)x1_ref=V2bus_refM^v^S+N^v^S.Therefore, the derivative ofS can be described by(14)S˙=v^SL-ux^2L+K1x~1-N^˙v^S-N^v^˙Sv^S2-M^˙v^SV2bus_ref-M^v^˙SV2bus_refv^S2.By utilizing invariant condition, the equivalent control can be written in the following form:(15)ueq=v^S+K1Lx~1x^2+Lγ2x~2v^S+Lγ3x~1x2N^v^S2x2x^2+LV2bus_refγ1x2x~2v^S+γ3x~1M^v^S2x^2,where ueq is smooth function of discrete input function u. Based on both concept of equivalent control and the necessary and sufficient condition of sliding mode, there exists a following relationship:(16)0<ueq<1.Therefore, we can obtain(17)0<v^S+K1Lx~1+LN^˙v^S-LN^v^˙Sv^S2-LV2bus_refM^˙v^S-M^v^˙Sv^S2<x^2.Moreover, initial conditions should meet (18) to guarantee the occurrence of sidling mode at t=0; that is, the system state variables are sliding on sliding surface initially. For control system, there has been S˙x=0.(18)x^20>0,v^S0>0,M^0>0,S0=x^10-x1_ref0=0.


## 3.3. Composite Control System Structure

The control system structure of microgrid is designed in Figure4. The system includes four components: model of microgrid, adaptive observer, sliding adaptive controller, and PWM generator. Each part has the corresponding expression.Figure 4
Block diagram of control system.SMC is applied into this paper, since the feature of itself variable structure agrees with the characteristic of converter. However, two important characteristics existing in SMC are the switching frequency variation and chattering phenomenon. The chattering problem can be effectively improved by using the ASMC. There are two methods to overcome the defects of variable switching frequency as well as take advantage of SMC simultaneously. One is to add additional hardware circuit to fix switching frequency. It is obviously that this method will increase the cost of system and is adverse to the scalability and plug-play of the distributed generators and loads in DC MG. The other scheme is to combine SMC with other control methods. Since the model used in this paper is large-signal model, the common PWM voltage/current mode control is unsuitable to large-signal situation. Moreover, PWM controller can not show an absolute robustness to input and output variation. Consequently, fixed-frequency PWM-based ASMC, a combination of SMC, PWM method, and adaptive control method, is adopted in this paper. Not only is the influence of power supplies and loads fluctuation on the stability of the system considered, but also the problem that the controller cannot be realized when the input and output are variable and unknown is conquered. In addition, this controller achieves its optimal performance at the possible conditions [29, 30].Fixed-frequency PWM-based ASMC is equivalent to a voltage and current dual-loop control using PWM modulation. Particularly, it is unnecessary for this control method to establish small-signal model and to design the control parameters by utilizing the magnitude-frequency characteristic in classical control theory, so that the design process is relatively simple. Moreover, the proposed control method is derived from modelling of system large signal, which is suitable to wide range of the signal disturbance. Fixed-frequency PWM-based ASMC method has similar fundamental principles with classical PWM methods; the frequency of switching signals is fixed by fastening the frequency of triangle-wave signal. In order to apply the proposed control strategy in DC MG system, it is necessary to convert smoothing function(ueq) to the instantaneous duty cycle of the converter represented by D; then D is used in the bidirectional buck/boost converter to realize the proposed control scheme [31].


## 4. Stability Analysis

Proposition 1.
In sliding mode, the closed-loop MG control system consisting of (4)–(6), (10), and (15) is asymptotically stable, and state variables (x1,x2) converge to the desired equilibrium (IL_ref,Vbus_ref).Proof.
By substitutingx1=x~1+x^1, x2=x~2+x^2, (10), (15), and S˙x=0 into (5), we obtain (19)x^˙2=1Cx^2V2bus_refM^+V2bus_refK1Lx~1M^v^S+K1Lx~1N^v^S+LV2bus_refγ3x~1M^N^v^S3+LV2bus_refγ2x~2M^v^S2x2+Lγ3x~1N^2v^S3+LV2bus_refN^γ3x~1M^v^S3+LV2bus_refγ1x2x~2N^v^S2+Lγ2N^x~2v^S2x2+LV4bus_refγ1x2x~2M^v^S2+LV4bus_refγ3x~1M^2v^S3+N^-M^x~2+x^2C-N^Cx~2+x^2+K2x~2.
Therefore, we yield the following equation:(20)x^2x^˙2=M^CV2bus_ref-x^22+1CV2bus_refK1LM^v^S+K1LN^v^S+LV4bus_refγ3M^2v^S3+LV2bus_refN^γ3M^v^S3+LV2bus_refγ3M^N^v^S3+Lγ3N^2v^S3x~1+1CLV4bus_refγ1x2M^v^S2+Lγ2N^v^S2+LV2bus_refγ2M^x2v^S2+LV2bus_refγ1x2N^v^S2-M^x^2+K2Cx^2x~2+1CN^-N^x^2x~2+x^2.
After rearranging terms, (20) is rewritten as(21)dx^22dt=2M^CV2bus_ref-x^22+ψ+h,where h and ψ are determined by(22)h=2CN^-x^2N^x~2+x^2,ψ=2CV^SV2bus_refK1LM^+LV4bus_refγ3M^2v^S2+LV2bus_refγ3M^N^v^S2+K1LN^+LV2bus_refN^γ3M^v^S2+Lγ3N^2v^S2x~1+2Cv^S2Lγ2N^+LV4bus_refγ1x2M^+LV2bus_refγ2M^x2-x^2v^S2M^+K2Cx^2v^S2+LV2bus_refγ1x2N^Sx~2.
Sincex~1 and x~2 converge to 0, ψ and h converge to 0; thus, x^2 converges to ±Vbus_ref. In addition, according to (17), x^2>0; therefore, it is obvious that x^2 and x2 converge asymptotically to Vbus_ref. By substituting (15) into (4), we can obtain xL_ref converges to IL_ref; thus x^1 and x1 converge to IL_ref.Remark 2.
In order to realize the fixed-frequency PWM-based SMC in bidirectional buck/boost converter, it is necessary to transform the smooth function to the demand instantaneous duty ratio of converter [32].


## 5. Simulation

To validate the feasibility of the proposed adaptive sliding-mode controller in islanded DC MG, a real-time simulation circuit is built in MATLAB. The simulation parameters are shown in Table1.Table 1
Simulation parameters.
Name
Parameter
Value
Bus reference voltage
V
b
u
s
_
r
e
f
380 V
Battery nominal voltage
v
S
300 V
Capacitance
C
1000μF
Inductance
L
5 mH
Switching frequency
f
S
20 kHz
Resistive loads
R
100 ΩThe observer gains areK1=K2=685 with the adaptive gains γ1=3, γ2=150, and γ3=150. The initial conditions are selected as x^20=365V, v^S0=303V, M^0=1×10-2, N^0=-4000W, and x^10=-8.44A. The sliding mode occurs at t=0, so that S(0)=0.DGs and CPLs are the primary factors that affect the stability of MG, so simulation has been conducted under the following operating conditions. Four-step changes were applied in the powerP, shown in Figure 5.1.
t=0s, P=-4000W;2.
t=0.1s, P=-4000W→2000W;3.
t=0.2s, P=2000W→-2000W;4.
t=0.4s, P=-2000W→1000W.Figure 5
Change of lumped power.


### 5.1. Hysteresis Band-Based SMC

In this scenario, SMC is adopted in DC MG system and HB method is applied to limit the switching frequency. The real-time simulation response of the system corresponding to this control method is shown in Figures6–8. It can be seen from Figure 6, under the impact of CPLs, which the bus voltage reaches to its reference value in 20 ms with larger transient values and chattering amplitude, voltage fluctuations in the range of −1.2%~+1.6%. The inductive current tracks its reference value perfectly but has high startup value, chattering amplitude, and step values at the turning points, showed in Figure 7. The controller ensures DC bus voltage regulation within permissible limits in the case of the power P mutation. However, the adopted frequency limit method can solve high switching frequency problem; it can do nothing to make the frequency constant. Thus, this approach makes it difficult to design the input and output filter. Figure 8 shows the charge and discharge status of battery. When the power flows from battery to the DC bus (P≥0), the battery is in discharge state. On the contrary, when the power flows from DC bus to battery (P≤0), the battery is in charge state.Figure 6
DC bus voltage under hysteresis band-based SMC.Figure 7
Inductive current under hysteresis band-based SMC.Figure 8
Battery status under hysteresis band-based SMC.


### 5.2. Fix-Frequency PWM-Based ASMC

In this scenario, response waveforms of MG system are shown in Figures9–11, which describe DC bus voltage, inductive current, and state of charge, respectively. It can be seen from Figure 9, under the 380 V bus reference voltage, which voltage state variable can trace the reference voltage within 8 ms. Moreover, fluctuation range of bus voltage is −1%~1.3% in the case of CPLs mutation. That is, the adaptive sliding-mode controller ensures both the bus voltage fluctuating at a permissible range and converging to reference value at a fast rate. Compared with the results in Figure 6, the maximum fluctuation range of dc bus voltage has been reduced 0.3% with less than 5% allowable range of voltage fluctuation. The voltage regulation time has also been shortened 12 ms. In addition, it can be seen from the smooth output waveform that the chattering phenomenon has been effectively improved. Figure 10 shows that inductive current can trace the reference current perfectly starting from the initial state. Although there is a slight change in the inductor current in the case of CPLs mutation, it is quickly adjusted to the reference value. Compared with the results in Figure 7, the maximum fluctuation range and regulation time of inductive current have been substantially reduced. It can be seen from the smooth output current waveform that the chattering phenomenon has been effectively improved. Accordingly, the system state variables have been well maintained on the sliding surface and the dynamic response and transient values of inductive current have been improved in the whole process. Figure 11 shows the charge and discharge status of battery. The overall trend is similar to that in Figure 8. The change of battery status agrees with the trend of P variation. When P is less than 0, the battery is in charge status; otherwise, battery is in discharge status. In addition, the PWM is used in this control method to make the frequency constant; thus, the switching frequency of the system is the same as that of the triangular wave.Figure 9
DC bus voltage under fix-frequency PWM-based ASMC.Figure 10
Inductive current under fix-frequency PWM-based ASMC.Figure 11
Battery status under fix-frequency PWM-based ASMC.Overall, it can be seen that the designed control system has a good robustness to deal with the parameters uncertainty and CPLs of DC system, and there is a good match among the theoretical analysis and simulation.


## 5.1. Hysteresis Band-Based SMC

In this scenario, SMC is adopted in DC MG system and HB method is applied to limit the switching frequency. The real-time simulation response of the system corresponding to this control method is shown in Figures6–8. It can be seen from Figure 6, under the impact of CPLs, which the bus voltage reaches to its reference value in 20 ms with larger transient values and chattering amplitude, voltage fluctuations in the range of −1.2%~+1.6%. The inductive current tracks its reference value perfectly but has high startup value, chattering amplitude, and step values at the turning points, showed in Figure 7. The controller ensures DC bus voltage regulation within permissible limits in the case of the power P mutation. However, the adopted frequency limit method can solve high switching frequency problem; it can do nothing to make the frequency constant. Thus, this approach makes it difficult to design the input and output filter. Figure 8 shows the charge and discharge status of battery. When the power flows from battery to the DC bus (P≥0), the battery is in discharge state. On the contrary, when the power flows from DC bus to battery (P≤0), the battery is in charge state.Figure 6
DC bus voltage under hysteresis band-based SMC.Figure 7
Inductive current under hysteresis band-based SMC.Figure 8
Battery status under hysteresis band-based SMC.


## 5.2. Fix-Frequency PWM-Based ASMC

In this scenario, response waveforms of MG system are shown in Figures9–11, which describe DC bus voltage, inductive current, and state of charge, respectively. It can be seen from Figure 9, under the 380 V bus reference voltage, which voltage state variable can trace the reference voltage within 8 ms. Moreover, fluctuation range of bus voltage is −1%~1.3% in the case of CPLs mutation. That is, the adaptive sliding-mode controller ensures both the bus voltage fluctuating at a permissible range and converging to reference value at a fast rate. Compared with the results in Figure 6, the maximum fluctuation range of dc bus voltage has been reduced 0.3% with less than 5% allowable range of voltage fluctuation. The voltage regulation time has also been shortened 12 ms. In addition, it can be seen from the smooth output waveform that the chattering phenomenon has been effectively improved. Figure 10 shows that inductive current can trace the reference current perfectly starting from the initial state. Although there is a slight change in the inductor current in the case of CPLs mutation, it is quickly adjusted to the reference value. Compared with the results in Figure 7, the maximum fluctuation range and regulation time of inductive current have been substantially reduced. It can be seen from the smooth output current waveform that the chattering phenomenon has been effectively improved. Accordingly, the system state variables have been well maintained on the sliding surface and the dynamic response and transient values of inductive current have been improved in the whole process. Figure 11 shows the charge and discharge status of battery. The overall trend is similar to that in Figure 8. The change of battery status agrees with the trend of P variation. When P is less than 0, the battery is in charge status; otherwise, battery is in discharge status. In addition, the PWM is used in this control method to make the frequency constant; thus, the switching frequency of the system is the same as that of the triangular wave.Figure 9
DC bus voltage under fix-frequency PWM-based ASMC.Figure 10
Inductive current under fix-frequency PWM-based ASMC.Figure 11
Battery status under fix-frequency PWM-based ASMC.Overall, it can be seen that the designed control system has a good robustness to deal with the parameters uncertainty and CPLs of DC system, and there is a good match among the theoretical analysis and simulation.


## 6. Conclusions

An ASMC-based bus voltage control strategy was proposed in this paper, which is especially suitable for an isolated DC MG. With the proposed control strategy, bus voltage can be fast and accurately tracked without additional sensors/hardware circuits and high bandwidth communications, which is key for the system scalability and maintaining the plug-and-play feature of the DGs. By using the ASMC method to estimate the uncertainty and external disturbance (CPL et al.), the system parameters and switching function can be adjusted in time to weaken the chattering. Based on the ASMC, the fix-frequency PWM-based ASMC can make the frequency constant, which greatly reduces the difficulty of the design of the filter. The simulation results show that the designed controller has the robustness to large disturbance. Moreover, the proposed MG control method is of significance to simplify the design of MG control system, to reduce the costs of MG projects and to provide stable power supply for island, remote areas without electricity, and other demand areas.



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*Source: 8962086-2017-12-24.xml*