# Equivalent SM Controller for Load-Sharing and Dynamic Performance in a DC Microgrid Application

Authors: Muhammad Rashad; Uzair Raoof
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Load-sharing and voltage regulation are key challenges in low voltage dc microgrid systems. Droop-based decentralized control methods are used to control dc grids which cannot achieve these challenges simultaneously due to unequal voltages and load power variations. Centralized control schemes can achieve these challenges using high bandwidth communication. However, the failure of a single central controller can degrade performance and reliability. In addition, cascaded proportional-integral (PI) type controllers are used to control the voltage and current loop of each power electronic converter. These controllers suffer stability and show slower dynamic responses. Furthermore, control parameters cannot be optimized for different operating conditions. To address these challenges, a distributive control method using an equivalent sliding mode (SM) controller utilizing low bandwidth communication is proposed in this paper. The distributive method improves reliability and achieves load sharing and voltage regulation simultaneously. Furthermore, in SM, both voltage and current loop operate in the parallel configuration which improves the dynamic response and shows stability in all operating conditions. To analyze, a system model is developed, and existence and stability conditions are verified. Detailed simulations are carried out to show the steady-state and transient performance. The presented results show the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

A microgrid is a modern way to satisfy the end-user power demands. It is a grid located close to the end-user and has its power generation based on renewable, nonrenewable, and batteries that can operate in grid-connected and islanded mode [1]. Most of the renewable sources, fuel cells and batteries, are dc sources or friendly to dc. In an ac microgrid, which is the conventional form of power distribution, integration of these sources with an ac line is difficult. In addition, most of the modern domestic and commercial loads require dc power or they are dc friendly. Therefore, to connect these loads with an ac line involves an extra stage of power electronic (PE) converters which adds additional power loss [2, 3]. Thus, dc microgrids are getting popular for the integration of renewable sources and dc or dc friendly loads. DC microgrids for data centers, marine systems, and residential applications are reported in [4–8]. It is expected that system efficiency will be increased by 10–22% in dc microgrids [9–11]. Furthermore, frequency synchronization and power factor correction circuits are no more required which relax the microgrid system. These advantages attract the researchers for dc microgrid applications.A DC microgrid system interconnecting dissimilar sources and load is shown in Figure1. AC sources are integrated with dc line through ac to dc PE converters, whereas dc sources and battery are integrated through dc to dc PE converters. DC microgrids are required to achieve better load sharing and voltage regulation performance. In a parallel configuration, only one source dominates and shares load because the output resistances of sources are of low value. Consequently, a small difference in output resistances will cause the source with output voltage a few millivolts higher to keep sharing all current [12]. In addition, dc bus instability is a key concern when operated in constant power loads (CPLs) applications [13–15].Figure 1
DC microgrid system.To address the challenge, control methods can be divided into centralized droop control and decentralized droop control [16–19]. The centralized control method can achieve good load sharing and voltage regulation performance based on a single central controller utilizing a high bandwidth communication channel [20–24]. In this type, the performance is highly dependent on the central controller. However, if the central controller fails, it will immediately degrade the performance and reliability, whereas decentralized droop-controlled microgrids operate without any communication channel [1, 15, 23–26]. In decentralized droop control, resistance is programmed in the output of each source which linearly decreases the voltage with the increase in output power. The advantage achieved is low cost and relaxed scalability. However, both objectives, load sharing and voltage regulation, cannot be achieved simultaneously. Alternatively, a distributive control based on a multi-agent system is proposed in [27, 28] for flexibility and computational efficiency. In [29], a coordination control based on an agent system to charge the batteries of vehicles through a power distribution network at low rates is presented. Another approach based on a multi-agent consensus algorithm is presented in [30] to resolve the grid impact. However, only single problem voltage regulation or frequency restoration can be resolved through these techniques utilizing high computational devices through a high bandwidth communication link. In addition, microgrids suffer from circulating currents which cannot be eliminated only by maintaining the voltage magnitudes or angles.To address the challenge, a distributive control employing low bandwidth communication is proposed in this paper. For load sharing purposes, only the current or power is to be shared among the paralleled sources. So, a low bandwidth communication channel is required. In [31], a dc bus signaling is reported to communicate among sources. In dc signaling, bus voltages are used to communicate the control decisions, which limits the voltage levels. A current sharing bus is reported for communication in [32]. Since sources in microgrid may exist over a distance, the current sharing bus will introduce a power loss and noise which may degrade performance. In [33], power line signaling (PLS) is reported for communication. PLS is low-speed and requires filters to separate the signaling information from power lines. It requires additional cost and suffers from electromagnetic interference issues with power lines. Hence, distributive control utilizing controller area network (CAN) based communication is proposed in this paper.To control the inner voltage and current loop of PE converters in parallel configuration, traditional linear control techniques (PI, PID, lead, or lag) can be used in cascaded connections. However, the requirement of a linearized model and reduction in bandwidth of closed-loop systems make these techniques impossible to guarantee global stability and transient performance [34–37]. In addition, the parameters of the controller are designed based on the operating conditions of the system. These conditions vary with the line and load applied to the system. Therefore, it is difficult to optimize control parameters for these techniques at different operating conditions [38, 39]. Alternatively, an SMC technique as reported in [40] is proposed for stability in any operating condition. A similar technique based on hysteresis comparator is proposed in [41] to regulate the voltage of bidirectional quadratic boost converter to interconnect different dc sources. In addition, in [42, 43], an SM controller based on hysteresis comparator is proposed to control the dc to dc buck converter for the stability of the dc microgrid. Hysteresis band in switching action creates an average nonzero value which introduces error in the voltages. Furthermore, a hysteresis controller suffers a problem of switching frequency variation. Similarly, a cascaded voltage and current loop are proposed in [44] for dc to dc bidirectional converter. In this case, the outer voltage loop is designed through a conventional PI controller, whereas the inner current loop is designed through a fixed frequency SMC. Similar cascaded control for the battery charger of an electric vehicle is proposed in [45]. In this scenario, the inner inductor current loop is designed through discrete SMC for which the reference current is generated using a PI controller in an outer voltage loop. One more SM controller is proposed in [46] without any compensation of duty cycle variation for the interconnection of parallel hybrid energy storage system through bidirectional dc to dc converter.In this paper, a Pulse Width Modulation (PWM) based equivalent SM controller is proposed for load sharing and transient performance. A sliding surface in a sliding phase is proposed in such a way that both the voltage and the current loops operate in the parallel configuration which improves the dynamic response of the system. The contribution of the study is as follows:(i)
A distributive control utilizing low bandwidth communication is proposed.(ii)
DC microgrid system is modeled subject to load sharing and stability problems.(iii)
The sliding surface of the SM controller is designed to operate both voltage and current loops in a parallel configuration.(iv)
The analysis is performed for the existence of the system under SM operation. In addition, equivalent control is derived which is mapped to PWM function to control the PE converters.(v)
The stability is analyzed through a different approach which involves ideal sliding dynamics at equivalent control. The stability analysis is performed at the equilibrium point for stability conditions.(vi)
Results with constant resistive and power loads are presented to validate the performance of the proposed technique. In addition, the transient performance is investigated for different control coefficients and results are presented.Section2 investigates the droop control in dc microgrid with its limitations. Section 3 presents the distributive control architecture for load sharing purposes. Furthermore, this section presents the system model, existence condition, and equivalent controller through SM control. Moreover, the stability of the SM controller is presented in this section. Simulation results are presented in Section 4 and finally, Section 5 concludes this paper.


## 2. Droop Control

Droop control is a decentralized method used for load sharing among sources in the dc microgrid system. In this method, control input to the converters depends on the measured local quantities as shown in Figure2. Droop control generates reference voltage signal based on the locally sensed current and voltage which is used by inner cascaded voltage and current loop to generate PWM signal for PE converter.Figure 2
DC microgrid with droop control.Droop-controlled dc microgrids cannot achieve load-sharing and voltage regulation performance simultaneously due to reasons investigated in [1, 16, 42, 43].


## 3. Proposed Distributive Control

Distributive control is a substitute for a single central controller which is distributed in the controller of each source in the dc microgrid system. The proposed distributive control is shown in Figure3.Figure 3
Distributive control.Large droop gains achieve current sharing, but voltage regulation among sources deviates from the desired value. To alleviate the deviation created by droop action, the voltage relationship of droop control as defined in [43] is shifted by Δvj as follows:(1)vjref=vi+Δvj−ijRdj,where vjref,vi,ij,andRdj are reference voltages, source voltages, source current, and droop resistance, respectively. And Δvj is voltage shift which depends on the load current. As the load increases, Δvj should increase to compensate for the effect created by ijRdj and hence, making the reference voltage close to the source nominal voltage. To find Δvj, each source communicates its supplied current with the others utilizing a low bandwidth communication CAN as shown in Figure 3. Based on the communicated currents, the controller of each source determines the average supplied current by all sources as(2)ijave=∑m=1nimn,where im is supplied current by source-m. The voltage shift Δvj is determined using average current ijave as(3)Δvj=kjijave,where kj is shift gain of source-j. The value of kj is selected close to the droop gain Rdj such that it generates a voltage shift Δvj which will compensate for the effect ijRdj. Selection criteria of kj is reported in [24].


### 3.1. The Proposed Sliding Mode Controller

The reference signal generated by droop compensation is used by the voltage and current loop to generate PWM signal for the PE converter in each source. Traditionally, PI controllers are used to realize the cascaded voltage and current loop of each converter. Global stability through PI controllers cannot be ensured on the desired equilibrium point [47, 48]. In addition, these controllers show slow dynamic responses. Moreover, the control parameters cannot be optimized for different operating conditions [38, 39]. Therefore, an equivalent SM controller is proposed in this paper. SM is a nonlinear controller used of variable structured systems and ensures stability for all operating conditions [49]. In SM, both voltage and the current loop operate in a parallel configuration which improves the dynamic response of the system.


#### 3.1.1. Modeling

A dc microgrid system interconnecting dissimilar sources with load is shown in Figure1. One source model with dc to dc boost PE converter is shown in Figure 4. The source nominal voltage and current are denoted with vi and is, respectively, whereas the output voltage is represented by vo with associated line resistance by Rline.Figure 4
Model of one source with dc microgrid.The differential equation of inductor current and output voltage of modeled source can be expressed as(4)diLdt=vi−1−uvoL,(5)dvodt=1−uiL−iC,where iL, i, L, and C are inductor current, output source current, inductance, and capacitance, respectively, whereas u is the switching state of the converter switch Q.


#### 3.1.2. Sliding Surface Design

Most of the SM controllers contain one or multiple system states in the sliding surface [50, 51]. This paper deals with load-sharing and voltage regulation. Therefore, the selected sliding surface includes state variables of current and voltage errors. In addition, integral action of current and voltage errors is included to further reduce the steady-state voltage and current errors. The sliding surface Ψ proposed is expressed as follows:(6)Ψ=α1x1+α2x2+α3x3,where α1,α2, andα3 represent the sliding coefficients, whereas state variables x1,x2andx3 represent the current error, voltage error, and integral of current and voltage error. The state variables are stated as follows:(7)x1=iref−iL,x2=vref−βvo,x3=∫x1+x2dt,where β,vrefandiref are represents the voltage sensing ratio, reference voltage, and reference current, respectively. The reference voltage vref is generated through compensated droop action given in equation (1). The inductor reference current iref is generated by the amplified voltage error and is expressed as follows:(8)iref=Kvref−βvo,where K is the amplified gain. Differentiating equation (7) and substituting equations (4) and (5), the system dynamics can be expressed as follows:(9)x1˙=diref−iLdt=−βKCic−vi−1−uvoL,x2˙=dvref−βvodt=−βCic,x3˙=d∫x1+x2dtdt=K+1vref−βvo−iL,where ic is the instantaneous current of the capacitor and u represents the switching state of the converter, respectively.


#### 3.1.3. Reachability Condition

The reachability condition defines that the system dynamics will reach the sliding surface. The control law for reachability condition can be expressed as follows:(10)u=1,whenΨ>0,0,whenΨ<0.The law expressed in equation (10) is the definition of hitting condition which states that trajectories of the system under operation will ultimately reach the sliding surface Ψ which is the basic requirement of the system under SM [49]. For the existence of SM operation, the condition represented in equation (11) is required to be satisfied.(11)Ψ=0,Ψ˙=0.To ensure that equation (11) is satisfied, the existence condition must be satisfied which is examined in Section 3.1.4.


#### 3.1.4. Existence Condition

Existence condition must be guaranteed to ensure that the system trajectories will always converge on sliding surfaceΨ [49]. It is defined as follows:(12)limΨ⟶0Ψ.Ψ˙<0.Differentiating equation (6) and substituting in equation (12) results as follows:(13)α1−βKCic−viL−α2βCic+α3K+1vref−βvo−iL,<0,α1−βKCic−vi−voL−α2βCic+α3K+1vref−βvo−iL,>0.Satisfying equation (13) will guarantee that the operation of SM will exist and system trajectories will be directed towards the sliding surface.


#### 3.1.5. Equivalent Controller

The ideal SM controller switches the PE converter at an infinite frequency which is not possible practically. Practical power switches suffer time delays and switching limitations. Practical SM controllers can be designed through two methods: hysteresis modulation and the PWM control method. In the hysteresis method, the hysteresis band is used to generate switching action. This band produces a non-zero average value which creates an error in output voltages. In addition, the hysteresis band suffers the problem of switching frequency variation. In an equivalent control method, problems faced through the hysteresis band can be avoided. Therefore, a PWM control using an equivalent SM controller is proposed in this paper.According to invariance property [49], equivalent control can be expressed as follows:(14)dΨdt|u=ueq=α1x1˙+α2x2˙+α3x3˙=0,where the range of ueq is given as follows:(15)0<ueq<1.Substituting equation (9) into equation (14) simplifies as follows:(16)ueq=1−K2voic−vivo+K1vovref−βvo−K3voiL,where(17)K1=α3α1LK+1,K2=βLCK+α2α1,K3=α3α1L.The coefficientsK1,K2,andK3 are dependent on sliding coefficients which must be selected upon satisfying the existence condition defined in equation (13). The PE converters operate at fixed frequency through an equivalent controller which is realized through PWM. The control function in equation (16) can be transformed to the duty ratio d of PWM, where 0<d=vc/vramp<1 can be represented as follows:(18)vc=K1vref−βvo−K2ic−K3iL+vo−vi,vramp=v0,where vc and vramp are control and ramp signals which are required to generate PWM signals. The architecture of the equivalent SM controlled PE boost converter is shown in Figure 5.Figure 5
The architecture of the equivalent SM controlled PE boost converter.


#### 3.1.6. Stability Analysis

The proposed sliding surface in equation (6) is composed of both voltage and current errors as state variables that are highly nonlinear. Therefore, stability analysis cannot be performed through linear Ackermann’s formula which automatically satisfies the stability of sliding coefficients [52]. A different approach is required for such systems which involves ideal sliding dynamics at equivalent control and performing analysis at equilibrium point for stability condition [53, 54].(1) Ideal Sliding Dynamics and Equilibrium Point. To represent the system dynamics in ideal form, replace u by ueq in equations (4) and (5) and then substitute the equivalent control ueq from equation (16) into equations (4) and (5) which simplifies systems dynamics as follows:(19)diLdt=viL−K2ic+vi−K1vref−βvo+K3iLL,dvodt=iLK2/voic+vi/vo−K1/vovref−βvo+K3/voiLC−iC.The equation in equation (19) denotes the ideal sliding dynamics of the system under SM operation.For the equilibrium point, assume that a point exists on the sliding surface. Ideal system dynamics ultimately settle on this point. Therefore, at equilibriumdiL/dt=dvo/dt=0. Hence, the steady state of the system can be represented as follows:(20)IL=V02ViRL,where IL, Vi,VoandRL are steady-state inductor current, input voltage, output voltage, and load resistance at the equilibrium point, respectively.(2) Linearizing Ideal Dynamics. Linearization is performed considering that system is operating at a steady-state i.e., vi=Vi,iL=IL,vref−βvo=0andIref−IL=0 and that IL≫iL˜ and Vo≫vo˜. The ideal sliding dynamics represented in equation (19) can be represented as linearized ideal dynamics at equilibrium point as follows:(21)diL˜dt=a11iL˜+a12vo˜,dvo˜dt=a21iL˜+a22vo˜,where(22)a11=K3ViRLK2LVo−LViRL,a12=K1βViRL−2K2Vi+Vi2RL/VoK2LVo−LViRL,a21=K2Vi−K3Vo−Vi2RL/VoK2VoC−CViRL,a22=K2Vo/RL−K1βVoK2VoC−CViRL−1RLC.The characteristic equation of the linearized system in equation (21) can be expressed as follows:(23)s2−a11+a22s+a11a22−a12a21=0.The condition for stability of this system is given as follows:(24)a11+a22<0,a11a22−a12a21>0.Fora11+a22<0, the stability condition can be expressed as follows:(25)K3CViRL−K1LβVo+LViK2Vo−ViRL<0.That is,(26)K3ViRLCLβVo+ViβVo<K1,whenK2>ViRLVo,K3ViRLCLβVo+ViβVo>K1,whenK2<ViRLVo,and for a11a22−a12a21>0, the stability condition can be expressed as follows:(27)2K3V03K2−K1βRL+ViV02K2K1βRL−2K2+ViV02RL3K2−K1βRL−Vi3RL2>0.The condition of existence in equation (13) and the condition for stability in equations (26) and (27) are the requirements of selecting the control variables K1, K2, and K3. Satisfying these conditions will ensure that SM operation will exist and it will be stable.(3) Stability Analysis Through Closed-Loop DynamicsAlternatively, stability analysis can be performed through closed-loop system dynamics. The closed-loop dynamics of the modeled system in the s-domain can be obtained by putting the sliding surface, Ψ=0 in equation (6) as follows:(28)α1x1s+α2x2s+α3sx1s+x2s=0.Simplifying equation (28) results as follows:(29)x1sx2s=irefs−iLsvrefs−βvos=−α3+α2sα3+α1s.The dynamics of the output voltage of a source as modeled in equation (5) is imposed by equation (29). In addition, the SM controller generates switching action u to derive the switch of PE converter and equation (5) must include SM dynamics. To remove u from equation (5), the average switching action over the period T is defined as follows:(30)d=1T∫0Tudt.And consideringd′=1−d, the dynamics of the output voltage of each source in equation (5) can be represented in the s-domain as follows:(31)Csvos=d′iLs−is.Combining equations (29) and (31), the complete closed-loop dynamics of the output voltage of each source can be expressed as follows:(32)vos=−α3+α1sCα1s2+βd′α2+Cα3s+βd′α3is+α3+α2sd′Cα1s2+βd′α2+Cα3s+βd′α3vrefs+α3+α1sd′Cα1s2+βd′α2+Cα3s+βd′α3irefs.Since the reference voltage and current in equation (31) are of constant values, the closed-loop dynamics depend only on the perturbation introduced by the output source current. Therefore, the dynamics in equation (32) can be expressed as follows:(33)vosis=−α3/α1+sCs2+Kp1s+Kp2,where Kp1=βd′α2+Cα3/α1 and Kp2=βα3d′/α1. Finally, equation (32) shows that the coefficients Kp1 and Kp2 must be selected positive values to ensure stable SM dynamics, or else the dynamics will exhibit poles that will lead to unstable behavior.


## 3.1. The Proposed Sliding Mode Controller

The reference signal generated by droop compensation is used by the voltage and current loop to generate PWM signal for the PE converter in each source. Traditionally, PI controllers are used to realize the cascaded voltage and current loop of each converter. Global stability through PI controllers cannot be ensured on the desired equilibrium point [47, 48]. In addition, these controllers show slow dynamic responses. Moreover, the control parameters cannot be optimized for different operating conditions [38, 39]. Therefore, an equivalent SM controller is proposed in this paper. SM is a nonlinear controller used of variable structured systems and ensures stability for all operating conditions [49]. In SM, both voltage and the current loop operate in a parallel configuration which improves the dynamic response of the system.


### 3.1.1. Modeling

A dc microgrid system interconnecting dissimilar sources with load is shown in Figure1. One source model with dc to dc boost PE converter is shown in Figure 4. The source nominal voltage and current are denoted with vi and is, respectively, whereas the output voltage is represented by vo with associated line resistance by Rline.Figure 4
Model of one source with dc microgrid.The differential equation of inductor current and output voltage of modeled source can be expressed as(4)diLdt=vi−1−uvoL,(5)dvodt=1−uiL−iC,where iL, i, L, and C are inductor current, output source current, inductance, and capacitance, respectively, whereas u is the switching state of the converter switch Q.


### 3.1.2. Sliding Surface Design

Most of the SM controllers contain one or multiple system states in the sliding surface [50, 51]. This paper deals with load-sharing and voltage regulation. Therefore, the selected sliding surface includes state variables of current and voltage errors. In addition, integral action of current and voltage errors is included to further reduce the steady-state voltage and current errors. The sliding surface Ψ proposed is expressed as follows:(6)Ψ=α1x1+α2x2+α3x3,where α1,α2, andα3 represent the sliding coefficients, whereas state variables x1,x2andx3 represent the current error, voltage error, and integral of current and voltage error. The state variables are stated as follows:(7)x1=iref−iL,x2=vref−βvo,x3=∫x1+x2dt,where β,vrefandiref are represents the voltage sensing ratio, reference voltage, and reference current, respectively. The reference voltage vref is generated through compensated droop action given in equation (1). The inductor reference current iref is generated by the amplified voltage error and is expressed as follows:(8)iref=Kvref−βvo,where K is the amplified gain. Differentiating equation (7) and substituting equations (4) and (5), the system dynamics can be expressed as follows:(9)x1˙=diref−iLdt=−βKCic−vi−1−uvoL,x2˙=dvref−βvodt=−βCic,x3˙=d∫x1+x2dtdt=K+1vref−βvo−iL,where ic is the instantaneous current of the capacitor and u represents the switching state of the converter, respectively.


### 3.1.3. Reachability Condition

The reachability condition defines that the system dynamics will reach the sliding surface. The control law for reachability condition can be expressed as follows:(10)u=1,whenΨ>0,0,whenΨ<0.The law expressed in equation (10) is the definition of hitting condition which states that trajectories of the system under operation will ultimately reach the sliding surface Ψ which is the basic requirement of the system under SM [49]. For the existence of SM operation, the condition represented in equation (11) is required to be satisfied.(11)Ψ=0,Ψ˙=0.To ensure that equation (11) is satisfied, the existence condition must be satisfied which is examined in Section 3.1.4.


### 3.1.4. Existence Condition

Existence condition must be guaranteed to ensure that the system trajectories will always converge on sliding surfaceΨ [49]. It is defined as follows:(12)limΨ⟶0Ψ.Ψ˙<0.Differentiating equation (6) and substituting in equation (12) results as follows:(13)α1−βKCic−viL−α2βCic+α3K+1vref−βvo−iL,<0,α1−βKCic−vi−voL−α2βCic+α3K+1vref−βvo−iL,>0.Satisfying equation (13) will guarantee that the operation of SM will exist and system trajectories will be directed towards the sliding surface.


### 3.1.5. Equivalent Controller

The ideal SM controller switches the PE converter at an infinite frequency which is not possible practically. Practical power switches suffer time delays and switching limitations. Practical SM controllers can be designed through two methods: hysteresis modulation and the PWM control method. In the hysteresis method, the hysteresis band is used to generate switching action. This band produces a non-zero average value which creates an error in output voltages. In addition, the hysteresis band suffers the problem of switching frequency variation. In an equivalent control method, problems faced through the hysteresis band can be avoided. Therefore, a PWM control using an equivalent SM controller is proposed in this paper.According to invariance property [49], equivalent control can be expressed as follows:(14)dΨdt|u=ueq=α1x1˙+α2x2˙+α3x3˙=0,where the range of ueq is given as follows:(15)0<ueq<1.Substituting equation (9) into equation (14) simplifies as follows:(16)ueq=1−K2voic−vivo+K1vovref−βvo−K3voiL,where(17)K1=α3α1LK+1,K2=βLCK+α2α1,K3=α3α1L.The coefficientsK1,K2,andK3 are dependent on sliding coefficients which must be selected upon satisfying the existence condition defined in equation (13). The PE converters operate at fixed frequency through an equivalent controller which is realized through PWM. The control function in equation (16) can be transformed to the duty ratio d of PWM, where 0<d=vc/vramp<1 can be represented as follows:(18)vc=K1vref−βvo−K2ic−K3iL+vo−vi,vramp=v0,where vc and vramp are control and ramp signals which are required to generate PWM signals. The architecture of the equivalent SM controlled PE boost converter is shown in Figure 5.Figure 5
The architecture of the equivalent SM controlled PE boost converter.


### 3.1.6. Stability Analysis

The proposed sliding surface in equation (6) is composed of both voltage and current errors as state variables that are highly nonlinear. Therefore, stability analysis cannot be performed through linear Ackermann’s formula which automatically satisfies the stability of sliding coefficients [52]. A different approach is required for such systems which involves ideal sliding dynamics at equivalent control and performing analysis at equilibrium point for stability condition [53, 54].(1) Ideal Sliding Dynamics and Equilibrium Point. To represent the system dynamics in ideal form, replace u by ueq in equations (4) and (5) and then substitute the equivalent control ueq from equation (16) into equations (4) and (5) which simplifies systems dynamics as follows:(19)diLdt=viL−K2ic+vi−K1vref−βvo+K3iLL,dvodt=iLK2/voic+vi/vo−K1/vovref−βvo+K3/voiLC−iC.The equation in equation (19) denotes the ideal sliding dynamics of the system under SM operation.For the equilibrium point, assume that a point exists on the sliding surface. Ideal system dynamics ultimately settle on this point. Therefore, at equilibriumdiL/dt=dvo/dt=0. Hence, the steady state of the system can be represented as follows:(20)IL=V02ViRL,where IL, Vi,VoandRL are steady-state inductor current, input voltage, output voltage, and load resistance at the equilibrium point, respectively.(2) Linearizing Ideal Dynamics. Linearization is performed considering that system is operating at a steady-state i.e., vi=Vi,iL=IL,vref−βvo=0andIref−IL=0 and that IL≫iL˜ and Vo≫vo˜. The ideal sliding dynamics represented in equation (19) can be represented as linearized ideal dynamics at equilibrium point as follows:(21)diL˜dt=a11iL˜+a12vo˜,dvo˜dt=a21iL˜+a22vo˜,where(22)a11=K3ViRLK2LVo−LViRL,a12=K1βViRL−2K2Vi+Vi2RL/VoK2LVo−LViRL,a21=K2Vi−K3Vo−Vi2RL/VoK2VoC−CViRL,a22=K2Vo/RL−K1βVoK2VoC−CViRL−1RLC.The characteristic equation of the linearized system in equation (21) can be expressed as follows:(23)s2−a11+a22s+a11a22−a12a21=0.The condition for stability of this system is given as follows:(24)a11+a22<0,a11a22−a12a21>0.Fora11+a22<0, the stability condition can be expressed as follows:(25)K3CViRL−K1LβVo+LViK2Vo−ViRL<0.That is,(26)K3ViRLCLβVo+ViβVo<K1,whenK2>ViRLVo,K3ViRLCLβVo+ViβVo>K1,whenK2<ViRLVo,and for a11a22−a12a21>0, the stability condition can be expressed as follows:(27)2K3V03K2−K1βRL+ViV02K2K1βRL−2K2+ViV02RL3K2−K1βRL−Vi3RL2>0.The condition of existence in equation (13) and the condition for stability in equations (26) and (27) are the requirements of selecting the control variables K1, K2, and K3. Satisfying these conditions will ensure that SM operation will exist and it will be stable.(3) Stability Analysis Through Closed-Loop DynamicsAlternatively, stability analysis can be performed through closed-loop system dynamics. The closed-loop dynamics of the modeled system in the s-domain can be obtained by putting the sliding surface, Ψ=0 in equation (6) as follows:(28)α1x1s+α2x2s+α3sx1s+x2s=0.Simplifying equation (28) results as follows:(29)x1sx2s=irefs−iLsvrefs−βvos=−α3+α2sα3+α1s.The dynamics of the output voltage of a source as modeled in equation (5) is imposed by equation (29). In addition, the SM controller generates switching action u to derive the switch of PE converter and equation (5) must include SM dynamics. To remove u from equation (5), the average switching action over the period T is defined as follows:(30)d=1T∫0Tudt.And consideringd′=1−d, the dynamics of the output voltage of each source in equation (5) can be represented in the s-domain as follows:(31)Csvos=d′iLs−is.Combining equations (29) and (31), the complete closed-loop dynamics of the output voltage of each source can be expressed as follows:(32)vos=−α3+α1sCα1s2+βd′α2+Cα3s+βd′α3is+α3+α2sd′Cα1s2+βd′α2+Cα3s+βd′α3vrefs+α3+α1sd′Cα1s2+βd′α2+Cα3s+βd′α3irefs.Since the reference voltage and current in equation (31) are of constant values, the closed-loop dynamics depend only on the perturbation introduced by the output source current. Therefore, the dynamics in equation (32) can be expressed as follows:(33)vosis=−α3/α1+sCs2+Kp1s+Kp2,where Kp1=βd′α2+Cα3/α1 and Kp2=βα3d′/α1. Finally, equation (32) shows that the coefficients Kp1 and Kp2 must be selected positive values to ensure stable SM dynamics, or else the dynamics will exhibit poles that will lead to unstable behavior.


## 3.1.1. Modeling

A dc microgrid system interconnecting dissimilar sources with load is shown in Figure1. One source model with dc to dc boost PE converter is shown in Figure 4. The source nominal voltage and current are denoted with vi and is, respectively, whereas the output voltage is represented by vo with associated line resistance by Rline.Figure 4
Model of one source with dc microgrid.The differential equation of inductor current and output voltage of modeled source can be expressed as(4)diLdt=vi−1−uvoL,(5)dvodt=1−uiL−iC,where iL, i, L, and C are inductor current, output source current, inductance, and capacitance, respectively, whereas u is the switching state of the converter switch Q.


## 3.1.2. Sliding Surface Design

Most of the SM controllers contain one or multiple system states in the sliding surface [50, 51]. This paper deals with load-sharing and voltage regulation. Therefore, the selected sliding surface includes state variables of current and voltage errors. In addition, integral action of current and voltage errors is included to further reduce the steady-state voltage and current errors. The sliding surface Ψ proposed is expressed as follows:(6)Ψ=α1x1+α2x2+α3x3,where α1,α2, andα3 represent the sliding coefficients, whereas state variables x1,x2andx3 represent the current error, voltage error, and integral of current and voltage error. The state variables are stated as follows:(7)x1=iref−iL,x2=vref−βvo,x3=∫x1+x2dt,where β,vrefandiref are represents the voltage sensing ratio, reference voltage, and reference current, respectively. The reference voltage vref is generated through compensated droop action given in equation (1). The inductor reference current iref is generated by the amplified voltage error and is expressed as follows:(8)iref=Kvref−βvo,where K is the amplified gain. Differentiating equation (7) and substituting equations (4) and (5), the system dynamics can be expressed as follows:(9)x1˙=diref−iLdt=−βKCic−vi−1−uvoL,x2˙=dvref−βvodt=−βCic,x3˙=d∫x1+x2dtdt=K+1vref−βvo−iL,where ic is the instantaneous current of the capacitor and u represents the switching state of the converter, respectively.


## 3.1.3. Reachability Condition

The reachability condition defines that the system dynamics will reach the sliding surface. The control law for reachability condition can be expressed as follows:(10)u=1,whenΨ>0,0,whenΨ<0.The law expressed in equation (10) is the definition of hitting condition which states that trajectories of the system under operation will ultimately reach the sliding surface Ψ which is the basic requirement of the system under SM [49]. For the existence of SM operation, the condition represented in equation (11) is required to be satisfied.(11)Ψ=0,Ψ˙=0.To ensure that equation (11) is satisfied, the existence condition must be satisfied which is examined in Section 3.1.4.


## 3.1.4. Existence Condition

Existence condition must be guaranteed to ensure that the system trajectories will always converge on sliding surfaceΨ [49]. It is defined as follows:(12)limΨ⟶0Ψ.Ψ˙<0.Differentiating equation (6) and substituting in equation (12) results as follows:(13)α1−βKCic−viL−α2βCic+α3K+1vref−βvo−iL,<0,α1−βKCic−vi−voL−α2βCic+α3K+1vref−βvo−iL,>0.Satisfying equation (13) will guarantee that the operation of SM will exist and system trajectories will be directed towards the sliding surface.


## 3.1.5. Equivalent Controller

The ideal SM controller switches the PE converter at an infinite frequency which is not possible practically. Practical power switches suffer time delays and switching limitations. Practical SM controllers can be designed through two methods: hysteresis modulation and the PWM control method. In the hysteresis method, the hysteresis band is used to generate switching action. This band produces a non-zero average value which creates an error in output voltages. In addition, the hysteresis band suffers the problem of switching frequency variation. In an equivalent control method, problems faced through the hysteresis band can be avoided. Therefore, a PWM control using an equivalent SM controller is proposed in this paper.According to invariance property [49], equivalent control can be expressed as follows:(14)dΨdt|u=ueq=α1x1˙+α2x2˙+α3x3˙=0,where the range of ueq is given as follows:(15)0<ueq<1.Substituting equation (9) into equation (14) simplifies as follows:(16)ueq=1−K2voic−vivo+K1vovref−βvo−K3voiL,where(17)K1=α3α1LK+1,K2=βLCK+α2α1,K3=α3α1L.The coefficientsK1,K2,andK3 are dependent on sliding coefficients which must be selected upon satisfying the existence condition defined in equation (13). The PE converters operate at fixed frequency through an equivalent controller which is realized through PWM. The control function in equation (16) can be transformed to the duty ratio d of PWM, where 0<d=vc/vramp<1 can be represented as follows:(18)vc=K1vref−βvo−K2ic−K3iL+vo−vi,vramp=v0,where vc and vramp are control and ramp signals which are required to generate PWM signals. The architecture of the equivalent SM controlled PE boost converter is shown in Figure 5.Figure 5
The architecture of the equivalent SM controlled PE boost converter.


## 3.1.6. Stability Analysis

The proposed sliding surface in equation (6) is composed of both voltage and current errors as state variables that are highly nonlinear. Therefore, stability analysis cannot be performed through linear Ackermann’s formula which automatically satisfies the stability of sliding coefficients [52]. A different approach is required for such systems which involves ideal sliding dynamics at equivalent control and performing analysis at equilibrium point for stability condition [53, 54].(1) Ideal Sliding Dynamics and Equilibrium Point. To represent the system dynamics in ideal form, replace u by ueq in equations (4) and (5) and then substitute the equivalent control ueq from equation (16) into equations (4) and (5) which simplifies systems dynamics as follows:(19)diLdt=viL−K2ic+vi−K1vref−βvo+K3iLL,dvodt=iLK2/voic+vi/vo−K1/vovref−βvo+K3/voiLC−iC.The equation in equation (19) denotes the ideal sliding dynamics of the system under SM operation.For the equilibrium point, assume that a point exists on the sliding surface. Ideal system dynamics ultimately settle on this point. Therefore, at equilibriumdiL/dt=dvo/dt=0. Hence, the steady state of the system can be represented as follows:(20)IL=V02ViRL,where IL, Vi,VoandRL are steady-state inductor current, input voltage, output voltage, and load resistance at the equilibrium point, respectively.(2) Linearizing Ideal Dynamics. Linearization is performed considering that system is operating at a steady-state i.e., vi=Vi,iL=IL,vref−βvo=0andIref−IL=0 and that IL≫iL˜ and Vo≫vo˜. The ideal sliding dynamics represented in equation (19) can be represented as linearized ideal dynamics at equilibrium point as follows:(21)diL˜dt=a11iL˜+a12vo˜,dvo˜dt=a21iL˜+a22vo˜,where(22)a11=K3ViRLK2LVo−LViRL,a12=K1βViRL−2K2Vi+Vi2RL/VoK2LVo−LViRL,a21=K2Vi−K3Vo−Vi2RL/VoK2VoC−CViRL,a22=K2Vo/RL−K1βVoK2VoC−CViRL−1RLC.The characteristic equation of the linearized system in equation (21) can be expressed as follows:(23)s2−a11+a22s+a11a22−a12a21=0.The condition for stability of this system is given as follows:(24)a11+a22<0,a11a22−a12a21>0.Fora11+a22<0, the stability condition can be expressed as follows:(25)K3CViRL−K1LβVo+LViK2Vo−ViRL<0.That is,(26)K3ViRLCLβVo+ViβVo<K1,whenK2>ViRLVo,K3ViRLCLβVo+ViβVo>K1,whenK2<ViRLVo,and for a11a22−a12a21>0, the stability condition can be expressed as follows:(27)2K3V03K2−K1βRL+ViV02K2K1βRL−2K2+ViV02RL3K2−K1βRL−Vi3RL2>0.The condition of existence in equation (13) and the condition for stability in equations (26) and (27) are the requirements of selecting the control variables K1, K2, and K3. Satisfying these conditions will ensure that SM operation will exist and it will be stable.(3) Stability Analysis Through Closed-Loop DynamicsAlternatively, stability analysis can be performed through closed-loop system dynamics. The closed-loop dynamics of the modeled system in the s-domain can be obtained by putting the sliding surface, Ψ=0 in equation (6) as follows:(28)α1x1s+α2x2s+α3sx1s+x2s=0.Simplifying equation (28) results as follows:(29)x1sx2s=irefs−iLsvrefs−βvos=−α3+α2sα3+α1s.The dynamics of the output voltage of a source as modeled in equation (5) is imposed by equation (29). In addition, the SM controller generates switching action u to derive the switch of PE converter and equation (5) must include SM dynamics. To remove u from equation (5), the average switching action over the period T is defined as follows:(30)d=1T∫0Tudt.And consideringd′=1−d, the dynamics of the output voltage of each source in equation (5) can be represented in the s-domain as follows:(31)Csvos=d′iLs−is.Combining equations (29) and (31), the complete closed-loop dynamics of the output voltage of each source can be expressed as follows:(32)vos=−α3+α1sCα1s2+βd′α2+Cα3s+βd′α3is+α3+α2sd′Cα1s2+βd′α2+Cα3s+βd′α3vrefs+α3+α1sd′Cα1s2+βd′α2+Cα3s+βd′α3irefs.Since the reference voltage and current in equation (31) are of constant values, the closed-loop dynamics depend only on the perturbation introduced by the output source current. Therefore, the dynamics in equation (32) can be expressed as follows:(33)vosis=−α3/α1+sCs2+Kp1s+Kp2,where Kp1=βd′α2+Cα3/α1 and Kp2=βα3d′/α1. Finally, equation (32) shows that the coefficients Kp1 and Kp2 must be selected positive values to ensure stable SM dynamics, or else the dynamics will exhibit poles that will lead to unstable behavior.


## 4. Results and Discussion

A dc microgrid system connecting two sources through connecting lines as shown in Figure6 is simulated to analyze the load sharing performance. Each source comprises a dc to dc boost converter which is selected to support a maximum of 50 V and 10 A voltage and current ratings, respectively. Thus, the boost converter is supporting a maximum of 500 W Power. The parameters of the converter, node, and connecting lines are shown in Tables 1–3, respectively.Figure 6
A dc microgrid with two sources.Table 1
Parameters of dc to dc boost converter.
ParametersValueNominal voltage48 VSwitching frequency10 kHzInductor, L100µHCapacitor, C4000µFTable 2
Parameters of nodes.
ParametersNode-1Node-2Nominal voltage48 VSource rated power250 W500 WLoad resistance6ΩVoltage and current regulation≤±5%Table 3
Parameters of connecting lines.
ParametersLine-1Line-2Current rating20 ACable resistance205 mΩ2 mΩThe condition of existence in equation (13) and the condition for stability in equations (26) and (27) are the requirements of selecting the control variables K1, K2, and K3 in terms of the system’s specifications. The control coefficients satisfying these conditions are selected based on the observed performance as given below:(i)
The regulation performance in the steady-state condition is improved with the increment in the value ofK1, but it creates more oscillations in the transient response with higher overshoot. Consequently, the settling time of the transient is prolonged.(ii)
A very small increment in the value ofK2 improves the performance in steady-state and transient conditions with small overshoot and fewer oscillations. Consequently, settling time is shortened. However, the range to change the value of K2 is very small.(iii)
A small increment in the valueK3 improves the oscillations and settling time of transient response. But, regulation performance in the steady-state condition has deteriorated.Thus, according to the above-mentioned observations, an approach to design these coefficients is to first select high values ofK1 and K2 and a random small value of K3 without compromising the stability and existence conditions. After selecting the coefficients, these can be adapted according to the desired steady-state and transient performance.


### 4.1. Steady-State Performance with Constant Resistive Load

To observe and compare the steady-state performance for equal load sharing, the system shown in Figure6 is simulated with PI-based droop control and the results are shown in Figures 7 and 8. For small droop gains Rd1=Rd2=0.2Ω, currents shared by sources 1 and 2 are 2.9 A and 4.9 A, respectively. For equal sharing, the desired current is 4 A at nominal load. The maximum current and voltage deviations observed for droop control are 27.5% and 2.5%, respectively. These results show that the small droop values can ensure voltage regulation performance, but the load sharing among sources violates the acceptable limits and is not desired. For large droop gains Rd1=Rd2=1.9Ω, currents shared by sources 1 and 2 are 3.8 A and 4.8 A, respectively. In this case, the maximum current and voltage deviations observed are 5% and 16%, respectively. These results show that large droop values satisfy current sharing limits, but voltage regulation has crossed the acceptable limits. Therefore, load-sharing and voltage regulation cannot be realized simultaneously with PI-based droop control.Figure 7
The voltages of sources 1 and 2 with PI-based droop and SM control. (a)v1. (b)v2.
(a)(b)Figure 8
The currents of sources 1 and 2 with PI-based droop and distributive SM control. (a)i1. (b) i2.
(a)(b)The same dc microgrid system in Figure6 is simulated with proposed distributive architecture using distributive SM controller and results are shown in Figures 7 and 8. The observed node voltages and currents by sources are 47.8 V, 47.4 V, 3.85 A, and 4.05 A, respectively. In this case, the maximum voltage and current deviations are 1.25% and 3.75%, respectively. This shows that both load-sharing and voltage regulation are within desired constraints and simultaneously achieved which validate the steady-state performance of the proposed control method for resistive load.


### 4.2. Steady-State Performance with Constant Power Load

To compare the performance with CPL, dc microgrid shown in Figure6 is simulated with a CPL of 500 W, and the results are shown in Figure 9. The load voltage and current with PI-based droop control are 33 V, and 15.1 A while with distributive SMC are 46.5 V, and 10.7 A, respectively. In this scenario, both voltage and current deviations with PI-based droop control are 30.25% and 31%, respectively, which are high values and beyond the acceptable limits. Thus, PI-based droop control cannot ensure the voltage and current regulations performance with CPL. However, these deviations with proposed distributive SMC are 3.13%, and 2.61%, respectively. These values show that both voltage regulation and load sharing are achieved simultaneously and within the desired limits which confirm the steady-state performance of the proposed technique with CPL.Figure 9
Load voltage and current with PI-based droop control and SM control. (a) Load voltage. (b) Load current.
(a)(b)In addition, the node voltages and currents by sources with PI-based droop control and distributive SMC are shown in Figures10 and 11. The maximum observed voltage and current deviations with PI-based droop control are 31.25% and 22.8%, respectively, which are crossing the desired limits and not acceptable. However, these deviations with proposed distributive SMC are 3.23% and 4.4%, respectively. These values confirm that both load sharing and voltage regulation at nodes are within desired limits. Hence, this validates the steady-state performance of the proposed technique with CPL.Figure 10
The voltages of sources 1 and 2 with PI-based droop and SM control with CPL. (a)v1. (b)v2.
(a)(b)Figure 11
The currents of sources 1 and 2 with PI-based droop and distributive SM control with CPL. (a)i1. (b) i2.
(a)(b)


### 4.3. Transient Performance

To observe and compare the transient performance, a CPL is switched from 750 W to 500 W at 0.2 s and load voltage is shown in Figure12. The voltage regulations before and after the transient condition with PI-based droop control are 15.66% and 12.9%, respectively. And the observed transient time, peak time, and percentage overshoot are 30 ms, 6 ms, and 4.12%, respectively, whereas the voltage regulations before and after the transient condition with proposed distributive SMC are 4.34%, and 2.67%, respectively. In this case, the transient time, peak time, and percentage overshoot are 6 ms, 3.5 ms, and 1.6%, respectively. These parameters show that the produced transient with the proposed technique will settle down in less time with a small value of overshoot. This validates the transient performance of the SM controller.Figure 12
Transient in load voltage when a CPL is switched from 750 W to 500 W at 02 s.Moreover, the transient response is further investigated with different values of control coefficientsK1,K2,andK3 and results are shown in Figure 13. The values of voltage regulation, transient time, peak time, and percentage overshoot when a resistive load is switched from 6 Ω to 12 Ω at 0.2 s are summarized in Table 4. These results show that the control coefficients can be adapted according to the desired steady-state and transient performance.Figure 13
Transient response with SM controller for different values ofK1,K2,andK3. (a) k1 = 80, k2 = 3, k3 = 3. (b) k1 = 50, k2 = 3, k3 = 3. (c) k1 = 10, k2 = 3, k3 = 3. (d) k2 = 0.5, k1 = 80, k3 = 3. (e) k2 = 3, k1 = 80, k3 = 3. (f) k2 = 6, k1 = 80, k3 = 3. (g) k3 = 2, k1 = 80, k2 = 3. (h) k3 = 5, k1 = 80, k3 = 3. (i) k3 = 8, k1 = 80, k3 = 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 4
Performance parameters for different values ofK1, K2, and K3.
Control variablesTransient time (ms)Peak time (ms)Maximum overshoot (%)Voltage regulation (%)Before transientAfter transientK2 = 3K3 = 3K1 = 801521.791.691.37K1 = 504531.692.71.48K1 = 103550.6335.151.48K1 = 80K3 = 3K2 = 0.5202.51.92.411.48K2 = 3162.21.881.691.39K2 = 6182.31.581.271.05K1 = 80K2 = 3K3 = 21921.981.271.13K3 = 5502.41.232.671.44K3 = 8502.40.551.161.44


### 4.4. Performance with Variation in Connecting Line Resistances

To verify the performance of the proposed controller with the variation in connecting line resistances, the system in Figure6 is simulated and results are summarized in Table 5. Each column and row represents the line resistances Rline1 and Rline2, respectively. There are two entries in Table 5: first is representing the maximum value of voltage regulation at load and the second is representing the maximum value of current deviation by sources. It can be seen that the maximum current deviation is introduced only by the line resistance Rline1 as its value is high. Also, the value of the current deviation increases as Rline1 increases. Table 5 shows that both voltage regulation and load sharing by sources are simultaneously achieved and within tolerable limits which validate the performance of the proposed technique with the variation in line resistances.Table 5
Maximum voltage regulation and current deviation with variation in connecting line resistances.
Rline1=0.5×205mΩRline1=1×205mΩRline1=1×205mΩRline1=0.5×205mΩ1.04%,2.5%1.04%,3.65%1.25%,4.95%Rline1=1×2mΩ1.04%,2.5%1.04%,3.65%1.25%,4.95%Rline1=2×2mΩ1.04%,2.5%1.04%,3.65%1.25%,4.95%


### 4.5. Fail-Safe Performance

A key advantage of distributive control is that it shows high reliability. To prove this, a dc microgrid with three sources shown in Figure14 is simulated with line resistances Rline1=205mΩ,Rline2=2mΩ and Rline3=2mΩ for source failure conditions and results of voltages and currents are shown in Figures 15 and 16. Before source failure, the node and load voltages are 49.5 V, 49 V, 49.3 V, and 49 V, respectively. And the source and load currents are 2.65 A, 2.69 A, 2.71 A, and 8.1 A, respectively. The observed maximum voltage and current deviations are 3.03% and 1.47%, respectively. At 0.2 s, source 3 fails to supply the load as shown in Figure 16(d). After this failure, sources 1 and 2 can share the load. In this case, node voltages and supplied currents are 47.5 V, 46.9 V, and 3.86 A, 4.1 A with a maximum voltage and current deviations of 2.34% and 3.5%, respectively. This shows that both voltage regulation and load sharing before and after source failure conditions are within acceptable limits, which confirms the reliable performance of the proposed distributive SM controller under source failure conditions.Figure 14
A dc microgrid with three sources.Figure 15
Load and node voltages when source-3 fails to supply current. (a) Load voltage. (b)v1. (c)v2. (d)v3.
(a)(b)(c)(d)Figure 16
Load and source currents when source-3 fails to supply current. (a) Load current. (b)i1. (c) i2. (d) i3.
(a)(b)(c)(d)


## 4.1. Steady-State Performance with Constant Resistive Load

To observe and compare the steady-state performance for equal load sharing, the system shown in Figure6 is simulated with PI-based droop control and the results are shown in Figures 7 and 8. For small droop gains Rd1=Rd2=0.2Ω, currents shared by sources 1 and 2 are 2.9 A and 4.9 A, respectively. For equal sharing, the desired current is 4 A at nominal load. The maximum current and voltage deviations observed for droop control are 27.5% and 2.5%, respectively. These results show that the small droop values can ensure voltage regulation performance, but the load sharing among sources violates the acceptable limits and is not desired. For large droop gains Rd1=Rd2=1.9Ω, currents shared by sources 1 and 2 are 3.8 A and 4.8 A, respectively. In this case, the maximum current and voltage deviations observed are 5% and 16%, respectively. These results show that large droop values satisfy current sharing limits, but voltage regulation has crossed the acceptable limits. Therefore, load-sharing and voltage regulation cannot be realized simultaneously with PI-based droop control.Figure 7
The voltages of sources 1 and 2 with PI-based droop and SM control. (a)v1. (b)v2.
(a)(b)Figure 8
The currents of sources 1 and 2 with PI-based droop and distributive SM control. (a)i1. (b) i2.
(a)(b)The same dc microgrid system in Figure6 is simulated with proposed distributive architecture using distributive SM controller and results are shown in Figures 7 and 8. The observed node voltages and currents by sources are 47.8 V, 47.4 V, 3.85 A, and 4.05 A, respectively. In this case, the maximum voltage and current deviations are 1.25% and 3.75%, respectively. This shows that both load-sharing and voltage regulation are within desired constraints and simultaneously achieved which validate the steady-state performance of the proposed control method for resistive load.


## 4.2. Steady-State Performance with Constant Power Load

To compare the performance with CPL, dc microgrid shown in Figure6 is simulated with a CPL of 500 W, and the results are shown in Figure 9. The load voltage and current with PI-based droop control are 33 V, and 15.1 A while with distributive SMC are 46.5 V, and 10.7 A, respectively. In this scenario, both voltage and current deviations with PI-based droop control are 30.25% and 31%, respectively, which are high values and beyond the acceptable limits. Thus, PI-based droop control cannot ensure the voltage and current regulations performance with CPL. However, these deviations with proposed distributive SMC are 3.13%, and 2.61%, respectively. These values show that both voltage regulation and load sharing are achieved simultaneously and within the desired limits which confirm the steady-state performance of the proposed technique with CPL.Figure 9
Load voltage and current with PI-based droop control and SM control. (a) Load voltage. (b) Load current.
(a)(b)In addition, the node voltages and currents by sources with PI-based droop control and distributive SMC are shown in Figures10 and 11. The maximum observed voltage and current deviations with PI-based droop control are 31.25% and 22.8%, respectively, which are crossing the desired limits and not acceptable. However, these deviations with proposed distributive SMC are 3.23% and 4.4%, respectively. These values confirm that both load sharing and voltage regulation at nodes are within desired limits. Hence, this validates the steady-state performance of the proposed technique with CPL.Figure 10
The voltages of sources 1 and 2 with PI-based droop and SM control with CPL. (a)v1. (b)v2.
(a)(b)Figure 11
The currents of sources 1 and 2 with PI-based droop and distributive SM control with CPL. (a)i1. (b) i2.
(a)(b)


## 4.3. Transient Performance

To observe and compare the transient performance, a CPL is switched from 750 W to 500 W at 0.2 s and load voltage is shown in Figure12. The voltage regulations before and after the transient condition with PI-based droop control are 15.66% and 12.9%, respectively. And the observed transient time, peak time, and percentage overshoot are 30 ms, 6 ms, and 4.12%, respectively, whereas the voltage regulations before and after the transient condition with proposed distributive SMC are 4.34%, and 2.67%, respectively. In this case, the transient time, peak time, and percentage overshoot are 6 ms, 3.5 ms, and 1.6%, respectively. These parameters show that the produced transient with the proposed technique will settle down in less time with a small value of overshoot. This validates the transient performance of the SM controller.Figure 12
Transient in load voltage when a CPL is switched from 750 W to 500 W at 02 s.Moreover, the transient response is further investigated with different values of control coefficientsK1,K2,andK3 and results are shown in Figure 13. The values of voltage regulation, transient time, peak time, and percentage overshoot when a resistive load is switched from 6 Ω to 12 Ω at 0.2 s are summarized in Table 4. These results show that the control coefficients can be adapted according to the desired steady-state and transient performance.Figure 13
Transient response with SM controller for different values ofK1,K2,andK3. (a) k1 = 80, k2 = 3, k3 = 3. (b) k1 = 50, k2 = 3, k3 = 3. (c) k1 = 10, k2 = 3, k3 = 3. (d) k2 = 0.5, k1 = 80, k3 = 3. (e) k2 = 3, k1 = 80, k3 = 3. (f) k2 = 6, k1 = 80, k3 = 3. (g) k3 = 2, k1 = 80, k2 = 3. (h) k3 = 5, k1 = 80, k3 = 3. (i) k3 = 8, k1 = 80, k3 = 3.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Table 4
Performance parameters for different values ofK1, K2, and K3.
Control variablesTransient time (ms)Peak time (ms)Maximum overshoot (%)Voltage regulation (%)Before transientAfter transientK2 = 3K3 = 3K1 = 801521.791.691.37K1 = 504531.692.71.48K1 = 103550.6335.151.48K1 = 80K3 = 3K2 = 0.5202.51.92.411.48K2 = 3162.21.881.691.39K2 = 6182.31.581.271.05K1 = 80K2 = 3K3 = 21921.981.271.13K3 = 5502.41.232.671.44K3 = 8502.40.551.161.44


## 4.4. Performance with Variation in Connecting Line Resistances

To verify the performance of the proposed controller with the variation in connecting line resistances, the system in Figure6 is simulated and results are summarized in Table 5. Each column and row represents the line resistances Rline1 and Rline2, respectively. There are two entries in Table 5: first is representing the maximum value of voltage regulation at load and the second is representing the maximum value of current deviation by sources. It can be seen that the maximum current deviation is introduced only by the line resistance Rline1 as its value is high. Also, the value of the current deviation increases as Rline1 increases. Table 5 shows that both voltage regulation and load sharing by sources are simultaneously achieved and within tolerable limits which validate the performance of the proposed technique with the variation in line resistances.Table 5
Maximum voltage regulation and current deviation with variation in connecting line resistances.
Rline1=0.5×205mΩRline1=1×205mΩRline1=1×205mΩRline1=0.5×205mΩ1.04%,2.5%1.04%,3.65%1.25%,4.95%Rline1=1×2mΩ1.04%,2.5%1.04%,3.65%1.25%,4.95%Rline1=2×2mΩ1.04%,2.5%1.04%,3.65%1.25%,4.95%


## 4.5. Fail-Safe Performance

A key advantage of distributive control is that it shows high reliability. To prove this, a dc microgrid with three sources shown in Figure14 is simulated with line resistances Rline1=205mΩ,Rline2=2mΩ and Rline3=2mΩ for source failure conditions and results of voltages and currents are shown in Figures 15 and 16. Before source failure, the node and load voltages are 49.5 V, 49 V, 49.3 V, and 49 V, respectively. And the source and load currents are 2.65 A, 2.69 A, 2.71 A, and 8.1 A, respectively. The observed maximum voltage and current deviations are 3.03% and 1.47%, respectively. At 0.2 s, source 3 fails to supply the load as shown in Figure 16(d). After this failure, sources 1 and 2 can share the load. In this case, node voltages and supplied currents are 47.5 V, 46.9 V, and 3.86 A, 4.1 A with a maximum voltage and current deviations of 2.34% and 3.5%, respectively. This shows that both voltage regulation and load sharing before and after source failure conditions are within acceptable limits, which confirms the reliable performance of the proposed distributive SM controller under source failure conditions.Figure 14
A dc microgrid with three sources.Figure 15
Load and node voltages when source-3 fails to supply current. (a) Load voltage. (b)v1. (c)v2. (d)v3.
(a)(b)(c)(d)Figure 16
Load and source currents when source-3 fails to supply current. (a) Load current. (b)i1. (c) i2. (d) i3.
(a)(b)(c)(d)


## 5. Conclusion

A microgrid is a modern way to satisfy the end-user power demands which can locally generate and consume power and can work in grid-tied mode. DC microgrids are attractive due to their high efficiency in the presence of renewable sources. Load sharing and voltage regulation are key challenges in low voltage dc microgrid systems. Droop-based decentralized control methods need a tradeoff between the value of droop gain and voltage regulation. Centralized control schemes can achieve these challenges using high bandwidth communication. However, the failure of a single central controller can degrade performance and reliability. In addition, PI controllers are used to control the voltage and current loop of each PE converter which suffer stability and present a slower transient response. Furthermore, control parameters cannot be optimized for different operating conditions. To address challenges, a distributive control method using an equivalent SM controller utilizing low bandwidth communication is proposed in this paper. It improves reliability and achieves load sharing and voltage regulation simultaneously. A system model is developed and existence and stability conditions are verified through SM. Presented steady-state and transient results validate the performance of the proposed control method.



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*Source: 9700915-2022-03-23.xml*