Journal of Modern Power Systems and Clean Energy

ISSN 2196-5625 CN 32-1884/TK

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Fixed-time Distributed Voltage and Reactive Power Compensation of Islanded Microgrids Using Sliding-mode and Multi-agent Consensus Design  PDF

  • Mohamed Ghazzali
  • Mohamed Haloua
  • Fouad Giri
Electrical Engineering Department, Mohammadia School of Engineers, University Mohammed V, Rabat, Morocco; Caen Automation Laboratory (LAC), Caen Normandie University (UNICAEN), Caen, France

Updated:2022-01-21

DOI:10.35833/MPCE.2019.000308

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Abstract

This paper investigates a fixed-time distributed voltage and reactive power compensation of islanded microgrids using sliding-mode and multi-agent consensus design. A distributed sliding-mode control protocol is proposed to ensure voltage regulation and reference tracking before the desired preset fixed-time despite the unknown disturbances. Accurate reactive power sharings among distributed generators are maintained. The secondary controller is synthesized without the knowledge of any parameter of the microgrid. It is implemented using a sparse one-way communication network modeled as a directed graph. A comparative simulation study is conducted to highlight the performance of the proposed control strategy in comparison with finite-time and asymptotic control systems with load power variations.

I. Introduction

MICROGRIDS are small-scale electrical distribution networks, consisting of distributed power sources, loads and energy storage systems. Primary control of microgrids maintains the stability of frequency and voltage, which causes the magnitudes of frequency and voltage to deviate from their nominal values. Emerged as a natural control system for microgrids, distributed secondary control restores the frequency and the voltage to their nominal operation points and achieves accurate power sharing in both grid-connected and islanded operation modes.

Distributed secondary control has been widely discussed in literature [

1]-[11]. Several techniques including proportional-integral (PI) control [9], feedback control [3]-[6], [10]-[12], adaptive control [6], [13] and model predictive control (MPC) [14], [15] among others are developed to address secondary control objectives of microgrids, which are voltage control, frequency control and power sharing. However, many shortcomings are to be addressed. In [9], a distributed PI controller is proposed for secondary control of voltage and frequency. Only small signal stability is guaranteed which means that large external disturbances, i.e., load variations during peak hours and the connection or disconnection of a power source, can destabilize the power system and cause voltage or frequency to swell or sag. In [5], [6], [10]-[12], feedback linearization and variants of feedback control are used to achieve secondary control of microgrids. However, a nonlinear model of AC microgrids is used in the design of the distributed control protocols that require detailed parameters of the microgrid. Practically, load parameters and transmission line impedances among other parameters are unknown and time-variant, which is challenging to apply the controllers in real applications. Adaptive control is suggested in [6], [13] for secondary control of AC microgrids. A major drawback of this approach is high computation complexity of the control law. In fact, in the adaptation technique, the control gains are updated in real time via a complicated and time-consuming mathematical approach as linear matrix inequalities (LMIs) or differential equation systems in [6] and neural networks in [13]. Thus, a control hardware with high computation capacity is required, which increases the implementation cost of the control system. In the case of neural network based adaptive control, a prior learning phase of the neurons based on experimental data is required for the design of the controller, which complicates the design, tuning and implementation of the controller. In [14], [15], the model of predictive secondary controllers is proposed for AC microgrids. The MPC approach is based on the model, which indicates that a model of the microgrid is required to calculate the output of the controller. Consequently, its robustness will depend on the accuracy of the model, which makes the controller vulnerable to modeling uncertainties. In addition, a fully detailed model of the microgrid can not be used in the design procedure as some parameters are practically unknown or difficult to obtain, which means that the unmodeled dynamics of the microgrid will always hinder the performance of the controller.

Furthermore, the distributed secondary controllers can be categorized into asymptotic controllers [

2], [5]-[10] and finite-time controllers [1], [3], [4]. In asymptotic control, the reference is only a limit of the terminal voltage or frequency of DG when time goes to infinity. Practically, reference tracking and disturbance rejection cannot be achieved in a finite time as the tracking error cannot be canceled using asymptotic control protocols. In finite-time control, the reference is reached in a finite time. The settling-time depends on the uncertain initial conditions of the system. An explicit relation between the parameters of the controller and the settling-time is not always available.

Sliding-mode control (SMC) is widely used to design the control systems and has been applied in the areas including robotics [

16]-[19], power converter control [20]-[23], motor control [24]-[27], and power systems [28]-[32]. SMC is designed to achieve reference tracking and regulation despite uncertainties of the parameter and external disturbance, which makes it suitable for applications with unknown dynamics and under uncertain conditions. In power systems, parameter uncertainties and external disturbances are constant issues [33]-[37]. In fact, most of the parameters are time-varying, which makes exact measurement of these parameters a challenging task. The fluctuations of voltage and frequency are caused by the nonlinear nature of loads and their unpredictable power consumption variations. As a result, SMC is suitable for microgrid control as it is robust to the unmodeled dynamics, variated parameters, and unknown disturbances. In addition, SMC has a straightforward design procedure and several variants of SMC have been designed with continuous and discontinuous outputs for several classes of linear and nonlinear systems. Therefore, the sliding-mode approach can be easily applied.

Considering the advantages of SMC and obviating the shortcomings of the secondary microgrid controllers, a fixed-time distributed SMC-based control approach is proposed for the secondary control of AC microgrid. Hence, the voltage regulation and reference tracking before the desired preset fixed-time can be ensured. And the accuracy of reactive power sharing is ensured among distributed generators (DGs) at its nominal levels. These objectives are guaranteed despite the uncertain microgrid parameters and the unknown disturbances. Moreover, the design procedure is model-free since no prior knowledge of load power demand, transmission line impedance or the microgrid topology is required.

The rest of the paper is organized as follows. In Section II, a large-signal dynamic model of islanded microgrids is presented. Section III presents the design of the fixed-time distributed secondary controller with voltage and reactive power sharing. A comparative simulation study on finite-time asymptotic secondary controllers and conventional power sharing protocol with load power variations is presented in Section IV. Finally, Section V concludes the paper.

II. Large-signal Dynamical Model of Islanded Microgrids

An islanded microgrid is adopted with n DGs, where every DG i, i{1,2,...,n} contains a primary energy source connected to a voltage-source converter (VSC), an RL series filter, a step-up transformer (Y-Δ) with transformation ratio mi, and a shunt capacitor Ct,i attenuating the impact of high-frequency voltage harmonics of the local load. Each DG i is connected to a set Ni{1,2,...,n} of neighboring DGs at the corresponding point of common coupling (PCC) through transmission lines modeled as a RL series circuits. A schematic of two connected DGs is depicted in Fig. 1, where Vi,d* and Vi,q* are the references of the direct and quadratic components of the output voltage of DG i, respectively; Vt,i, It,i, Rt,i, Lt,i are the voltage, current, resistance, and impetance of VSC terminal, respectively; and the subscript i+1 denotes DG i+1.

Fig. 1 Schematic of two connected DGs.

DG i can be modeled in the d-q framework by the follwoing large-signal dynamical model [

38]:

Ii,d=miIt,i,d-Ct,iddtVi,d+Ct,iωnVi,qIi,q=miIt,i,q-Ct,iddtVi,q-Ct,iωnVi,dddtxi=Aixi+jNiAijxj+Biui+DiwiVt,i,d=Vt,i,d*Vt,i,q=Vt,i,q*Vt,i,d*=-ωnLt,iIt,i,q+Kpci(It,i,d*-It,i,d)+Kici(It,i,d*-It,i,d)dVt,i,q*=ωnLt,iIt,i,d+Kpci(It,i,q*-It,i,q)+Kici(It,i,q*-It,i,q)dIt,i,d*=FiIi,d-ωnCt,iVi,q+Kpvi(Vi,d*-Vi,d)+Kivi(Vi,d*-Vi,d)dIt,i,q*=FiIi,q-ωnCt,iVi,d+Kpvi(Vi,q*-Vi,q)+Kivi(Vi,q*-Vi,q)d (1)

where Ii,d and Ii,q are the direct and quadratic components of the output current of DG i, respectively; Vi,d and Vi,q are the direct and quadratic components of the output voltage of DG i, respectively, and the output voltage of DG i is also the voltage of PCC i; Fi is the feed-forward coefficent; Vt,i,d* and Vt,i,q* are the references of the direct and quadratic components of the terminal voltage of VSC, respectively, which are calculated by the local voltage and current of VSC controller [

39]; Vt,i,d and Vt,i,q are the direct and quadratic components of VSC terminal voltage, respectively; It,i,d* and It,i,q* are the references of the direct and quadratic components of the VSC terminal current, respectively; It,i,d and It,i,q are the direct and quadratic components of the VSC terminal current, respectively; IL,i,d and IL,i,q are the direct and quadratic components of the current of load i; ωn=2πfn is the nominal pulsation with fn=50 Hz nominal frequency; Ai, Aij, Bi, and Di are the state-space matrices [40]; xi=[Vi,d,Vi,q,It,i,d,It,i,q]T is the state vector; ui=[Vt,i,d,Vt,i,q] is the input; Kpvi, Kivi, Kpci, and Kici are the control gains of the PI controllers in the internal voltage and current control loops designed for high-frequency disturbance rejection and the filter output damping to avoid any resonance with the external network, respectively [39], which are tuned using the symmetrical optimum tool [41]; and wi=[IL,i,d,IL,i,q] is the load current consumption considered as a known disturbance and an exogenous input to the system for i{1,2,...,n}. With the relatively high switching frequency of VSC, it is safe to neglect the switching artifact via average-value modeling. Since the dynamics of the DC-bus can be safely neglected assuming an ideal source from the DG side [39], we can obtain Vt,i,d=Vt,i,d* and Vt,i,q=Vt,i,q*.

The state-space matrices Ai, Aij, Bi, and Di are defined as [

40]:

Ai=-1Ct,ijNiRijZij2ωi-1Ct,ijNiXijZij2miCt,i0-ωi+1Ct,ijNiXijZij2-1Ct,ijNiRijZij20miCt,i-Rt,iLt,i0-miLt,iωi0-miLt,i-ωi-Rt,iLt,i (2)
Aij=1Ct,iRijZij2XijZij200-XijZij2RijZij20000000000Bi=00001Lt,i001Lt,i (3)
Di=-1Ct,i00-1Ct,i0000 (4)

where ωi=2πfi is the output pulsation of DG i and fi is its output frequency; Zij=Rij2+ωi2Lij2 is the transmission line impedance; and Xij=ωiLij is the transmission line reactance.

The output active and reactive power Pi and Qi can be calculated from the output voltage and current in the d-q frame using:

ddtPi=-fciPi+fci(Vi,dIi,d+Vi,qIi,q)ddtQi=-fciQi+fci(Vi,qIi,d-Vi,dIi,q) (5)

where fci is the cut-off frequency of the low-pass filters used to extract the fundamental component of Pi and Qi.

The communication network of the microgrid is described using graphs. Consider a n-order weighted directed graph (digraph) G=(V,E) with V=(v1,v2,...,vn) set of nodes and EV×V set of directed edges. A weight aij0 is associated with every edge. aij>0 if there is an edge from node j to node i. If agent j communicates its state information to agent i, then aij=0. A=(aij)n×n is called the adjacency matrix of the graph G. The graph Laplacian matrix of G, L(G)=L, is defined as L=D-A, where D=diag{d1,d2,...,dn} and di=j=1naij.

Unlike the existing models of microgrids, the interactions between DGs are considered in the proposed model. It also includes the nonlinearties introduced by the filter, the shunt capacitance, the loads, and the step-up transformers.

III. Fixed-time Distributed Secondary Voltage and Reactive Power Control

The control system developed in this paper is a secondary control strategy with voltage and reactive power sharing in hierarchical control framework. The primary control is given as:

Vi,d*=Vind-DqQiVi,q*=0 (6)

where Vind and Dq are the secondary control output voltage and reactive power droop coefficient, respectively. DG i generates the desired voltage reference, thus Vi,d*=Vi,d and Vi,q*=Vi,q. Then, (6) can be written as:

Vi,d=Vind-DqQiVi,q=0 (7)

Since Vqi=0, the output voltage magnitude of DG i Vi,mag satisfies Vi,mag=Vi,d2+Vi,q2. Thus, controlling the output voltage magnitude is the same as controlling its direct component.

The primary voltage and reactive power control aligns the output voltage magnitude to the d-axis of the voltage reference.

Secondary voltage control is designed to ensure a good trade-off between the conflicting objectives of voltage regulation and maintain the reactive power sharing accuracy in the same pattern as in the primary control:

Dq1Q1=Dq2Q2=...=DqnQn=ΔVmax (8)

where ΔVmax is the maximum allowable voltage deviations.

Differentiating the voltage droop characteristics twice will yield:

d2dt2Vind=d2dt2Vi,d+Dqd2dt2Qi=uvQi (9)

where uvQi is the auxiliary control input. According to (9), the secondary voltage and reactive power control of islanded microgrids can be transformed to a leader-follower second-order consensus problem for the following linear second-order multi-agent system:

d2dt2Vind=uvQii{1,2,...,n} (10)

Definition: the leader-follower second-order consensus in multi-agent systems is presented as:

d2dt2zi(t)=vi(t)i{1,2,...,n} (11)

where zi is the position of the ith agent; and vi is the ith agent control input.

Leader-follower second-order consensus in multi-agent system (11) is achieved under any initial conditions and i{1,2,...,n}. We can obtain:

limt+||zi(t)-z0(t)||=0limt+ddtzi(t)-ddtz0(t)=0 (12)

We propose the non-singular terminal sliding-mode auxiliary control law uvQi based on the voltage magnitude and the reactive power information from the neighbors of DG i:

uvQi=uvi+uQi (13)
uvi=jNiaij+bi-1-γvisign(si(evip,eviv))+jNiaijuj-pi-2qipiπpi4Ticsqipi/(pi-2qi)eviv1-[2qi/(pi-2qi)]Fpi,qi(evip)+2ξv2ηvTcrμτ((eviv)2qi/(pi-2qi))(eviv)-[2qi/(pi-2qi)]((si(evip,eviv))1-(2ηv/ξv)+n2ηv/ξv((si(evip,eviv))2)2ηv/ξv(si(evip,eviv))1-2ηv/ξv) (14)
uQi=jNiaij+bi-1-γQisign(si(eQip,eQiv))+jNiaijuj-pi-2qipiπpi4Ticsqipi/(pi-2qi)eQiv1-[2qi/(pi-2qi)]Fpi,qi(eQip)+2ξq2ηqTcrμτ((eQiv)2qi/(pi-2qi))(eQiv)-2qi/(pi-2qi))(si(eQip,eQiv))1-2ηq/ξq)+n2η/ξ((si(eQip,eQiv))2)2ηq/ξq(si(eQip,eQiv))1-2ηq/ξq) (15)

where eip=jNiaij(vi-vj)+bi(vi-v0) and eiv=jNiaij(v˙i-v˙j)+bi(v˙i-v˙0) are the voltage tracking errors and the derivative, respectively; eQip=jNiaij(DQiQi-DQjQj) and eQiv=jNiaij(DQiQ˙i-DQjQ˙j) are the reactive power sharing errors and the derivative, respectively, and DQi and DQj are the reactive power droop coefficients; Tcr is a time constant; Tics is the lowest upper-bound of the sliding-time; bi>0 is the pinning gain and it is non-zero only for the agents that have access to the reference voltage amplitude v0; and pi, qi, ηv, ξv, ηq, ξq, γvi, and γQi are the control parameters that verifies the following conditions:

0<qi/pi<0.250<ηv/ξv<0.50<ηq/ξq<0.5γvi>0γQi>0 (16)

Denote B=diag{bi}i{1,2,...,n}, and DQi is the reactive power droop coefficient. Fpi,qi() and μτ() with τ>0 are defined as:

Fpi,qi(x)=(cos h((x2)qi/pi))pi/(pi-2qi)+2qipipi(pi-2qi)·(eip)2qi/pisin h((x2)qi/pi)(cos h((x2)qi/pi))2qi/(pi-2qi) (17)
μτ(x)=sinπx22τ2|x|τ1otherwise (18)

si(x,y) is the sliding surface defined as:

si(x,y)=ypi/(pi-2qi)+πpi4Ticsqix(cos h((x2)qi/pi))pi/(pi-2qi) (19)

Figure 2 shows the block diagram of the distributed fixed-time voltage and reactive power controller.

Fig. 2 Block diagram of fixed-time distributed voltage and reactive power controller.

Tcr+2τ(pi-2qi)/(2qi)/γi is an upper bound of the reaching time of the secondary SMC of DG i, i{1,2,...,n}.

The desired prefixed upper bound of the settling time is expressed as:

T=Tcr+maxi(Tics)+maxi2τ(pi-2qi)/(2qi)γi (20)

Theorem: if the control protocol in (15), (16), (17), and (18) is applied, then the fixed-time leader-follower consensus tracking is achieved before the prefixed settling time upper bound (20). Thus, voltage regulation and reference tracking are met in the fixed-time Tsettling without hindering reactive power sharing, i.e., TsettlingT is verified.

Proof: the result in the Theorem is valid for uvi and uQi, separately [

42]. However, to prove that the result is valid for the superposition of the two nonlinear control laws, i.e., uvQi=uvi+uQi, we consider the Lyapunov function V=i=1n(si2(evip,eviv)+si2(eQip,eQiv)) . Using μτ(x)+μτ(y)μτ(x+y)/2 with μτ(x+y)/2=μτ/2(x+y) and xa+ya(x+y)a, x,y,aR+*, and following similar demonstration steps as in [42], it is concluded that limttrV=0. Then, limttrsi(x,y)=0, where tr is the reaching time of the sliding surface si(x,y)=0. Then, tr<Tcr+maxi2τ(pi-2qi)/(2qi)/γi is verified. On the sliding surface, i.e., when si=0, if we denote ei {evip,eQip,eviv,eQiv}, the sliding motion dynamics can be described by e˙i=-πpi/4qiTics(eip)1-(2q/p)cos h(((eip)2)q/p). Thus we can obtain 2eie˙i=-πpi/2qiTics(eip)2(1-(q/p))cos h(((eip)2)q/p). By denoting Ve=ei2, the aforementioned equation can be written as qpVe(q/p)-1dVecos h(Veq/p)=-π2Ticsdt, which is equivalent to dVeq/pcos h(Veq/p)=π2Ticsdt. Integrating this equation from tr to t yields arcsin(tan h((ei2(t))q/p))=-π(t-tr)/2Tics+arcsin(tan h((ei2(tr))q/p)).At the settling-time ts of the sliding motion, ei=0. Thus, ts=(2Tics/π)arcsin(tan h((ei2(tr))q/p))Tics. The tracking errors evip, eQip, eviv, eQiv, i{1,2,...,n} converge to zero at Tsettling. Then TsettlingTcr+maxi(Tics)+maxi2τ(pi-2qi)/(2qi)/γi is verified. Since vi-v0=(L+B)evip, the reference voltage magnitude is reached at Tsettling.

Remark 1: the term 2τ(pi-2qi)/(2qi)/γi can be neglected for sufficient high values of γi and low values of τ, and T=Tcr+maxi(Tics) is verified by the upper bound of the prefixed settling time. However, for very small values of Tcr+maxi(Tics) , the effect of the term 2τ(pi-2qi)/(2qi)/γi can be reduced but can not be negligible without hindering the controller performance as high values of γi increase the chattering effect.

Remark 2: an important feature of the proposed control approach is that the design procedure is straightforward. No knowledge of the microgrid parameter is required. The desired settling-time upper bounds are specified directly in the control law, which makes the tuning process simple. And the only step left is to choose γi and τ as explained in Remark 2. In addition, the convergence at the desired settling-time is mathematically guaranteed despite the unknown disturbances.

IV. Comparative Simulation Study

The performance of the proposed fixed-time secondary voltage control is verified with load power variations in comparison with finite-time and asymptotic secondary voltage control. The performance of the proposed protocol of fixed-time reactive power sharing in maintaining power sharing accuracy is compared with the conventional one in [

3], [4] and others representing the benchmark for power sharing accuracy, i.e, the nominal level to be maintained within the microgrid. The test system used in the simulations is a low-voltage islanded AC microgrid (nominal frequency f0=ω0/(2π)=50 Hz and nominal voltage magnitude V0=380 V) containing four DGs and four loads connected via the communication network. And the network is modeled by the directed graph depicted in Fig. 3. The arrows in the graph indicates the voltage and reactive power data flow direction. DG1 is the leader node, thus the only DG gets access to the reference voltage magnitude with the pinning gain b1=1. For the other DGs, bi=0, i{2,3,4}.

Fig. 3 Topology of microgrid.

Except for the internal voltage and current loop parameters, the DG specifications are adopted from [

40]. The filter parameters and the droop coefficients are summarized in Table I. The droop coefficients are assumed so that the voltage deviations are less than the maximum allowable voltage deviation below 10% [43]. Therefore, to achieve high control performance, the maximum allowable voltage deviation is ΔVmax=2%. The shunt capacitance Ct=62.86  μH and the transformer ratio k=0.6/13.8 are used for the four DGs. Active and reactive power variations of the loads are presented in Figs. 4 and 5. The control gains of the internal voltage and current loops are tuned as follows: Kpv=Kpvi=150, Kiv=Kivi=250, Kpc=Kpci=100, and Kic=Kici=10, i{1,2,3,4}.

Table I Microgrid Parameters
DGFilter parameterDroop coefficient
Rt  (mΩ)Lt  (μH)DpDq
1 1.2 93.7 2.08×10-5 1.210×10-3
2 1.6 94.8 2.10×10-5 1.214×10-3
3 1.5 107.7 2.11×10-5 1.217×10-3
4 1.5 90.6 2.12×10-5 1.218×10-3

Fig. 4 Active power variation of load.

Fig. 5 Reactive power variation of load.

The parameters of voltage and reactive power controllers are set as: qi/pi=3/37, ηv/ξv=ηq/ξq=0.1, Tcr=5  ms and Tics=5  ms, i{1,2,3,4}. As Tcr+Tics is very small, several values of τ and γi are tested to reduce the term 2τ(pi-2qi)/(2qi) without increasing the chattering effect. τ=1×10-3, γvi=2.4×106, and γQi=10 achieve the desired effect and will be used in this paper.

To highlight the efficiency of the proposed fixed-time control, the finite-time and the asymptotic secondary controllers in [

4] and [10] have the same settling-time as those of the proposed controller.

The simulations are conducted considering the following scenario.

1) At t=0: the simulation is initialized and the primary control is activated.

2) At t=0.8 s: the proposed secondary control system is applied.

3) At t=1 s: active power and reactive power of load 4 are increased.

4) At t=1.2 s: active power and reactive power of load 2 are decreased.

5) At t=1.4 s: load 1 is disconnected from the microgrid.

The results of the simulations are shown in Figs. 6-8. As shown in Fig. 6, the droop-based primary control maintains the microgrid voltage stability. Accurate power sharing can be shown in Fig. 8, which does not cancel the reference tracking error and maintain the microgrid back to the nominal operation conditions. Thus, the primary control results in a deviation in the voltage amplitude of DGs. As a result, the secondary control is required for voltage restoration.

Fig. 6 DG1 output voltage using proposed fixed-time control, finite-time control and asymptotic control. (a) Output voltage. (b) Zoom on time lapse [0.79 s, 0.815 s]. (c) Zoom on time lapse [0.99 s, 1.02 s]. (d) Zoom on time lapse [1.19 s, 1.22 s]. (e) Zoom on time lapse [1.39 s, 1.42 s].

Fig. 7 Control signal Vind of proposed fixed-time controller, finite-time control and asymptotic control. (a) Control signal Vind. (b) Zoom on time lapse [0.8 s, 2 s].

Fig. 8 Error of reactive power sharing using proposed and conventional approaches. (a) Dq2Q2-Dq1Q1. (b) Dq3Q3-Dq1Q1. (c) Dq4Q2-Dq4Q1.

At t=0.8 s, the secondary control is activated. Figure 6 shows that the fixed-time secondary control restores the voltage magnitude to its reference before the upper bound of prefixed settling-time of 10 ms with less than 2 V overshoot while maintaining smooth output voltage. However, the finite-time controller provides a fluctuating output voltage with over 12 consecutive overshoots before reaching the reference, which means that the proposed approach enhances the tracking dynamics by 90% compared with that of finite-time control. Unlike the designed fixed-time approach, the asymptotic control also leads to voltage fluctuation during its convergence to the reference. And the reference tracking can not be achieved in a finite time. In the asymptotic control, the reference is only a limit of DG terminal voltage when time goes to infinity. In practical, this means that the reference tracking error can not be canceled using asymptotic control protocols, while the proposed fixed-time control ensures voltage reference tracking before a prefixed maximum settling time despite the external disturbances. For example, the reaching time of the reference is guaranteed to be finite and fixed in advance. In this paper, the fixed-time approach provides 100% better performance than asymptotic approaches on both transient dynamics and steady-state basis.

The primary reactive power control represents the benchmark for power sharing accuracy, i.e., the nominal level to be maintained within the microgrid. The proposed fixed-time secondary control achieves efficient voltage reference tracking while maintaining 100% of power sharing accuracy.

Load power variations begin at t=1 s as shown in Figs. 4 and 5. Figure 6 shows that by using fixed-time, finite-time and asymptotic controllers, similar voltage fluctuations occur. Unlike other approaches, the proposed controller rejects the voltage fluctuations before the upper bound of prefixed settling-time of 10 ms, which achieves accurate reactive power sharing.

The control signals displayed in Fig. 7 show that the proposed control system provides better performance with smaller control signal than finite-time and asymptotic approaches. The proposed system has faster dynamics and a smaller steady-state value, which justifies the voltage regulation and reference tracking speed of the proposed approach.

Figure 8 shows that the proposed reactive power controller maintains power sharing accuracy at its benchmark level in presence of load power variations. The voltage control has not been conducted at the expense of reactive power control. Therefore, the proposed power controller have successfully provided better voltage regulation and reference tracking performance while maintaining reactive power sharing accuracy at its nominal level.

V. Conclusion

In this paper, a fixed-time distributed voltage and reactive power secondary control approach for islanded AC microgrids has been designed. The proposed distributed sliding-mode controller ensures voltage regulation and reference tracking before the upper bound of prefixed settling-time despite the unknown disturbances. And accurate reactive power sharing among DGs is maintained. The comparative simulation conducted with load power variations confirms the performance of the controller in voltage regulation and reference tracking before the desired fixed-time, and the accuracy of reactive power sharing is maintained. Simulation results show that the proposed fixed-time control provides better performance in term of voltage regulation and reference tracking than finite-time and asymptotic approaches, which can achieve fast regulation reference tracking at the expense of the system stability with severe voltage fluctuations.

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