Optimal Cooperative Secondary Control for Islanded DC Microgrids via a Fully Actuated Approach

Yi Yu ,,, Guo-Ping Liu ,,, Yi Huang , and Peng Shi ,,

Abstract—DC-DC converter-based multi-bus DC microgrids(MGs) in series have received much attention, where the conflict between voltage recovery and current balancing has been a hot topic.The lack of models that accurately portray the electrical characteristics of actual MGs while is controller design-friendly has kept the issue active.To this end, this paper establishes a large-signal model containing the comprehensive dynamical behavior of the DC MGs based on the theory of high-order fully actuated systems, and proposes distributed optimal control based on this.The proposed secondary control method can achieve the two goals of voltage recovery and current sharing for multi-bus DC MGs.Additionally, the simple structure of the proposed approach is similar to one based on droop control, which allows this control technique to be easily implemented in a variety of modern microgrids with different configurations.In contrast to existing studies, the process of controller design in this paper is closely tied to the actual dynamics of the MGs.It is a prominent feature that enables engineers to customize the performance metrics of the system.In addition, the analysis of the stability of the closed-loop DC microgrid system, as well as the optimality and consensus of current sharing are given.Finally, a scaled-down solar and battery-based microgrid prototype with maximum power point tracking controller is developed in the laboratory to experimentally test the efficacy of the proposed control method.

I.INTRODUCTION

DUE to the growing demand for energy in recent years,microgrids (MGs) have received increased attention and have been the subject of extensive research due to their flexibility to integrate clean power sources such as energy storage systems (ESSs) and renewable energy sources (RESs) [1], [2].On the other hand, recent frequent blackouts as a result of natural disasters indicate that we should not rely on the grid alone, but protect and increase the resilience of the power and energy system through the use of distributed energy resources(DERs) [3].RESs-based MGs have the ability to work both independently and in parallel with the main grid, making them a technically viable option for improving power supply reliability and energy efficiency.This has also led to their gradual development as an important source of supply consistency [4].Therefore, MGs are not only a potential alternative for green operation of power generation systems, but also considered as an effective technological architecture to improve the performance of future grids [5].

DC MGs typically consist of DC-DC power converters, possibly with a cascaded distributed power architecture, feeding loads through a common DC bus.DC MGs can effectively integrate local distributed generation (DG) units and ESSs directly into the DC load, making it a valuable architecture for combining multiple RESs [6], [7].Compared to inverter-based AC MGs, it has higher efficiency, fewer conversion stages and more reliable operation due to fewer power electronic components [8], [9].In addition, fewer conversion phases make DC MGs attractive, such as no reactive power, economical and mostly autonomous operation, and less complex control and management [10].Further, the rapid improvement in ESSs performance over the last decade has made DC MGs an economical alternative to AC MGs, which also helps address the issue of energy savings.However, there are still some challenges in DC MGs, such as efficient power management and bus voltage control under certain conditions [11], as well as abnormal disturbances caused by communication networks[12]-[14].Moreover, the stability of these systems is a major concern.

Hierarchical control architecture is commonly used in MGs at present, which corresponds to primary, secondary and tertiary control.Droop control has been the most popular primary control method for regulating the voltage and current sharing in shunt converters.To compensate for unacceptable voltage dips and current/power mismatches caused by droop control, secondary control comes into being.Existing secondary control techniques are typically categorized into centralized and distributed control, both based on the communication infrastructure to provide the desired performance [15].In a centralized scheme, a master controller exists for the monitoring and upper level regulation tasks of the entire system.However, global communication is required to bring in the necessary information for corrections, and the reliability of the master controller has a significant impact on the operation of the entire MGs.Unlike a centralized structure, a distributed control approach offers more scalability and higher reliability,while non-centralized data interaction makes it robust to single points of failure [16].The distributed paradigm is becoming dominant due to the aforementioned advantages as well as the large number of DG units and their geographical distance from each other [17], [18].For example, in [19], a delay-compensated distributed secondary control strategy adapted to switching topologies was proposed, which allows for the recovery of the voltage and frequency of the DGs and accomplishes active power sharing in finite time.

Typical DC MGs are available as single bus systems and multiple bus systems.In a small-scale DC MG, each RESsbased generation unit is connected in parallel to a single DC bus via local feeder.In recent research, the tasks of current sharing and voltage regulation are mitigated in a single bus DC MG system [20].For example, the authors in [21] realized bus voltage recovery and current sharing for a single-bus DC microgrid in the presence of heterogeneous communication delays under the sampling control framework.In a largescale DC MG spreading over a wide area, each DER itself has a local DC bus.These buses converge at a point of common coupling (PCC), and interconnect through transmission lines to form a multi-bus framework.Despite the improved reliability in multi-bus DC MGs, the impedance of the coupling lines poses a conflict between power sharing and voltage regulation.

The approaches in the existing literature for the above-mentioned regulation contradictions have focused on droop control with correction terms.For example, a method combining the consensus-based average voltage observer and the current cooperative control was proposed in [22] to achieve accurate proportional current-sharing amongst DG units.Although the voltage recovery and current equalization problems in multibus DC MGs are somewhat addressed in the aforementioned works, these efforts fail to allow the design of model-based controller approaches such as linear quadratic (LQ) control,H∞control, or model predictive control (MPC) [23], which compromise the customizability of microgrid system performance.

For the modeling of DC MGs, two main approaches are of interest.On the one hand, there is the black-box modeling method based on measurement data, which relies on the actual operating state of the power system [20].For example, in [24],the control design of a DC microgrid was accomplished in a purely numerical way based on experimental data.This method bypasses system modeling, and maintains the stability of the microgrid well.On the other hand, there is the principle-based white-box modeling approach, which depends on a priori knowledge of the physical characteristics of the MG system [25].For example, the effort of [26] lay in proposing a consensus-based secondary control strategy for achieving current sharing and voltage balancing in a multi-bus DC microgrid, which relies on a principle-based model of the physical microgrid.Regrettably, for multi-bus DC microgrids, models that can facilitate secondary controller design and corresponding model-based solutions for voltage recovery and precise current sharing are still missing.

To fill this gap, this paper systematically models a multi-bus DC MG and proposes a novel accurate and fast cooperative optimal controller based on the high-order fully actuated(HOFA) system theory for voltage recovery and proportional current sharing as well as economic distribution of currents.Detailed experiments verify the effectiveness of the proposed fully actuated optimal cooperative (FAOC) secondary control in terms of robustness performance under load variation and failure of the DG unit.Moreover, a comparative case study with a recent secondary control protocol shows that the proposed FAOC strategy provides satisfactory performance with respect to the voltage and current traces of the DG units.In particular, this paper adds to the existing studies with the following contributions.

1) In this paper, primary control and underlying device-level control of DC-DC converter-based MG is considered as a whole, and the studied multi-bus MG with series connection is extensively modeled.To facilitate controller design, the obtained comprehensive system model is converted to a fully actuated linear form, which is simple in form and reflects the relationship between voltage and current as well as other electrical characteristics of the actual power system.

2) Based on the derived model, a quadratic cost function is designed for the DC MG, and a FAOC secondary control strategy is further proposed, which not only achieves the basic voltage regulation and current distribution tasks, but also enables the DC MG to obtain a fast dynamic response and high quality steady-state performance.

3) In addition to focusing on the precise proportional current sharing, tertiary economic scheduling and secondary control are integrated, i.e., the optimal power exchange between DG units is accomplished only by the proposed distributed secondary control.

4) Unlike common numerical simulations and circuit simulations in SIMULINK or PSCAD software, this paper develops a prototype of a solar-based low-voltage DC MG with energy storage devices to verify the correctness of the developed DC microgrid model.

Notations: Throughout the paper, the superscriptTstands for the matrix transpose.Inand 0 indicate unit and zero matrices with compatible dimensions.Additionally, if the dimensions of the matrices are not explicitly stated, they are assumed to be compatible with algebraic operations.In symmetric matrix expressions, an asterisk (∗) is used to denote the entries resulting from symmetry.The notationP＞0 is used to denote its positive definiteness.⊗ indicates the Kronecker product operator.

II.CONTROL FRAMEWORK FOR A SERIES-CONNECTED MULTI-BUS DC MGS IN ISLANDED OPERATION

In this section, a basic description of a DC microgrid with multiple buses connected in series is first presented.On this basis, the DC microgrid is modeled.Thereafter, an integrated control framework for solving the conflict between voltage restoration and current regulation faced by a series-connected multi-bus DC microgrid is provided.

A. Series-Connected Multi-Bus DC MGs

DERs are connected in series through power lines to form an islanded low-voltage DC microgrid, and its structure is shown in Fig.1.The distributed energy-based generation unit consists of RESs (e.g., wind turbines, photovoltaics, and batteries), DC-DC converters, RLC filters, and local loads, as detailed in Fig.2.Due to the presence of line impedancethe regulation task of the series-connected multi-bus DC microgrid presented in Fig.1 is more diverse and challenging than that of a parallel single-bus DC microgrid.

Fig.1.Series-connected DC MGs formed by DGs interconnected through power lines.

Fig.2.Electrical structure of ith DG with resistive load.

It is well known that the intermittent (e.g., solar power) and stochastic (e.g., wind power) nature of RESs brings volatility to distributed generation units, having been an important factor affecting the reliability of microgrid systems.These fluctuations are mostly compensated for in practical applications with the help of energy storage solutions, and here is no exception.Namely, it is assumed that the volatility of RESs can be well resolved.Furthermore, similar to [25], suppose that the DC MG under study satisfies the condition for the use of quasi stationary line (QSL) estimation, i.e., the inductance on the tie line is sufficiently small.Then, based on Kirchhoff’s voltage and current laws, the dynamics equations of the DC microgrid system can be given as follows:

whereVo,j∈Nip, is the voltage at the port linked with theith DER, and Nipis the set of physically neighboring DER units connected toith DER through the power line.Note that the variableiappearing in the description of the system that follows represents the index of theith DER unit, wherei=1,2,...,NandNis the number of DER units.The definitions of other symbols in (1) and Fig.2 are provided in Table I.

TABLE I SYMBOLS PRESENT IN (1)

B. DC MGs Modeling With HOFA System Approaches

Equation (1) fails to thoroughly characterize the full dynamics of a series-connected DC microgrid, under the commonly used hierarchical control structure.To best describe the actual microgrid system, a comprehensive modeling of its underlying circuit model, the inner control loops at the device level,and the primary controls in the hierarchical control framework is required.

First, the dual closed-loop control at the device level is indispensable for DC microgrids.Specifically, for theith DER, the dynamics of the voltage controller in the outer loop is given as follows:

and the current controller the inner loop is described as

whereAiiandAi jare presented in (7), shown at the bottom of the page,Bii=[0,Vin,iKpciKpvi/Lfi,1,Kpvi]T.

For the large-signal model above, let the inductance current be the system output, i.e.,yi(t)=hi(xi(t)) , wherehi(xi(t))=Cixi(t),Ci=[0,1,0,0].It can be seen that the DC microgrid system expressed by (6) is under-actuated, since there is only one control input while there are two states to be controlled.Therefore, differentiating the inductance current ofith DG unit yields

wherebc,i=CiBii=Vin,iKpciKpvi/Lfi.

Consideringbc,i≠0 and inspired by the HOFA method[27], the following fully actuated secondary controller can be designed:

Assuming that the states in the DC MG system (6) are measurable, the auxiliary controller is determined as follows:

Further, with (9) and (10) substituted into the large signal model (6), the linear fully actuated multi-agent system (FAMAS) model is obtained as follows:

It should be noted that the one shown in Fig.1 is a commonly used topology.However, there are many variants of multi-bus DC microgrids based on this architecture, such as multiple DERs on one bus.The modeling of these microgrids can benefit from the methodology proposed in this paper, but requires specific analysis and further development.

Remark 1: In general, the voltage and current dynamics of microgrids are not appropriate to be considered completely aside from their physical constraints, but should on the basis of mechanistic knowledge such as Kirchhoff’s voltage and current laws.At the same time, it is necessary to build a bridge between voltage and current dynamics, so that they can be decoupled from a controller design perspective.In the control framework of the algebraic equation-based DC microgrid model, the intrinsic physical properties of the microgrid are usually neglected in controller design, but are only concerned in the part of the system stability analysis, which is also similar to the above argument.Moreover, the stability analysis often relies on global circuit and topology information, which is typically difficult to acquire during the controller design phase.The developed FA-MAS model (11) takes into account the electrical characteristics of the actual microgrid in a distributed fashion at the controller design stage, thus overcoming the above two problems.

C. Control Framework for Conflict Between Voltage and Current Regulation in Series-Connected DC MGs

As pointed out by Nasirianet al.in [22], for a multi-bus DC microgrid connected by power lines in series, there is an inherent contradiction in accurate voltage recovery and current management due to the line impedance between the buses.This conflict can be discussed in two aspects.On the one hand, according to Fig.2, if the voltages of all DERs at PCCs are restored to a certain desired value, no current will flow on the transmission line, i.e.,Ic,ij=0.With consistent voltage, the microgrid will degenerate into a multi-agent system consisting of several mutually isolated unit circuits.Therefore, restoring the voltage at each PCC to the desired value would not be appealing, except for the voltage at the bus where some critical loads are located.On the other hand, if the effort is devoted only to the economic distribution of the output current, then it may result in the bus voltages of the microgrid not being in a reasonable range, which is also undesirable.In addition, a previous study [28] theoretically demonstrated that consistent recovery of voltage and accurate sharing of current can not be guaranteed at the same time.

Inspired by the above discussion, this paper introduces a secondary control framework consisting of voltage regulator with limiter and cooperative current controller to resolve the contradiction.It is worth mentioning that the feasibility of this solution is due to the fact that the current can be shared with a small voltage difference.In the absence of secondary control,the reference valuereceivedby the innercontrolloopsis,and the one by thedroopcontrol interfaceis,as givenby(2) and (5).When the secondary control is activated, the voltage regulation with limiter takes the first action, and the dual closed-loop control receives the actual reference valueas

Remark 2: In previous work, scholars have also performed some studies on the design of secondary controllers under the coupled large-signal models and the independent first-order multi-agent models of DC microgrids, respectively.For example, based on a coupled large-signal model, a secondary control method was proposed in [25] to actively compensate for the communication delays.However, the use of coupled multi-agent model leads to the requirement for a fully connected communication network.In another result [9], a secondary security control strategy was presented based on a first-order multi-agent model of DC microgrids.However, we found that the first-order model itself does not consider the relationship between voltage and current.In this way, it seems that both accurate voltage and current regulation tasks can be achieved, which is contradictory to the actual situation.So this independent first-order model can easily lead to misoperation in controller design.Only by analyzing the coupling relations inherent in the circuit in advance can a compromise be made between the two control objectives [22].Afterwards, based on these problems encountered during previous studies, it took a lot of efforts in this paper to build the FA-MAS model for DC microgrids, which provides an appropriate and accurate description for system dynamics, while facilitating the secondary controller design.

III.DESIGN OF FAOC SECONDARY CURRENT CONTROL

The main task of this section is to design distributed optimal cooperative current control strategies for the series-connected DC microgrid, including an optimal proportional current sharing controller and an optimal economic load sharing controller based on generation cost, which will be presented in Sections III-B and III-C, respectively.To illustrate the generality and feasibility of the proposed secondary controller, the stability, consensus and optimality analysis of the DC microgrid under the proposed controller is also provided.

A. Setting of Communication Network Based on Graph Theory

In the fashion of distributed control, an undirected graph is used to represent the communication connections between the

B. Optimal Cooperative Current Sharing for Multi-Bus DC MGs

In order to improve the lifetime of the devices in DG units,the current needs to be equalized over the whole microgrid according to a capacity-dependent distribution factor [21], i.e.,it needs to be ensured that

whereIso,i(t)=Iti(t)/ηi, ηiis the given distribution factor.According to (11), the dynamics of the weighted output current can be obtained as

To ensure accurate sharing of output currents, the following distributed controller is designed for each DER-based generation unit:

wherecsis the coupling gain to be designed andFl,sis the local feedback gain of each generation unit to be determined.For (15), the overall current sharing error can be written as

wherees(t)=[es,1(t),es,2(t),...,es,N(t)]T,Iso(t)=[Iso,1(t),Iso,2(t),...,Iso,N(t)]T.Further, the global form of the controller of the DC microgrid can be given as

whereup(t)=[up,1(t),up,2(t),...,up,N(t)]T.As a result, the global closed-loop dynamics of the weighted output currents

can be presented as

Before giving specific results about the above cooperative current controller, the following Lemma is first introduced.

Lemma 1: For the linear systemx˙(t)=Ax(t)+Bu(t), as (14),assumeu(t) is a stabilizing control input.If there exists a positive definite matrixP, and a positive semi-definite matrixQsatisfying the following conditions:

thenu(t)=ψ(x(t))=Fx(t) is optimal regarding the following performance criterion:

whereG(x(τ),u(τ))=xT(τ)Qx(τ)+uT(τ)Ru(τ).Further, the optimal value of this performance index can be determined asJ(x(0),u(t))=V(x(0)).

We will elaborate on how to design the control gain of the distributed control protocol (16) so that the considered DC microgrid accomplishes accurate current sharing in a globally optimal manner and maintains the stability of the closed-loop microgrid system.

The feedback gain of the current sharing controller is designed as

Then, when the graphGis undirected and connected, and when the coupling gain satisfiescs＞0, the DC microgrid can be stabilized to the null space ofand achieve accurate proportional current sharing, the voltage can be restored to the allowed range, and the FAOC controller (16) is optimal with respect to the following performance index:

Proof: First, the proof of stability and consensus of the closed-loop FA-MAS is given.Take the Lyapunov candidate function aswhereTherefore, it follows from the positive semi-definiteness ofL that the Lyapunov function only has a value of 0 on the the null space of L.

SincePa,s=csRa,sL , whenFl,s=RBT Pl,s, one obtains

which impliestheasymptoticstabilityofIso(t)inthenull space ofQgs.As shown by thestabilityof theweightedoutput currents, the dynamics of the underlying PI control is also stabilized, so the voltage is restored to the desired range by the limiter.Furthermore, when the communication graphGof a DC microgrid is undirected and connected, 0 is the simple eigenvalue of the Laplacian matrix L of that graph.Therefore,1 is the kernel of the non-frequent zero space of L and the consensus of the weighted output currents can be reached, i.e.,(13) can be achieved.

Next, the proof of the optimality of the proposed secondary control method is provided.Similar to the definition of, letRg,s=Ra,s⊗Rl,s.Therefore, from=Pa,s⊗Pl,s, one has

which is the algebraic Riccati equation for the global FAMAS.RegardingA=0 in (14), the above equation is equivalent toConsideringIN⊗Aas a whole, it can be seen from the above equation that the DC microgrid system in global form meets the algebraic Riccati equation, i.e., it satisfies the third condition in Lemma 1.

In addition, by substituting the abovePa,sandFl,sinto the designed FAOC current controller (18), there is

According toLemma 1,specifyingQandRin (21),i.e.,is obtained that the designed FAOC current controller is optimal with respect to (24).■

C. Optimal Cooperative Power Management for Multi-Bus DC MGs

Unlike the task in Section III-B, this subsection aims to achieve optimal load sharing among DERs in the secondary control layer for the economic operation of the microgrid considering the generation revenue.For this purpose, we first describe the generation cost of each DER, then the dynamics of incremental costs obtained from the optimization of the generation cost and the system (14) are modeled as a whole,and finally a distributed optimal load sharing controller is designed.

The quadratic cost function [30] is used here to characterize the generation cost of DERi

with cost coefficients αi＞0, βi＞0 and γi＞0.The load allocation for economic operation aims to find the optimalIti(t) to minimize the total generation cost under the constraint of supply-demand balance.Therefore, the economic operation of the microgrid is equivalent to the following optimization problem:

whereIDis the load demand.To solve the above optimization problem, the relevant Lagrangian function is given as

where λtiis the Lagrangian multiplier.According to the firstorder optimality condition, one has

which implies

According to the results of [31], a sufficient condition for minimizing the generation costs of DERs while maintaining a supply-demand balance is that the incremental costs of each DER must be equal, i.e.,

To ensure the consensus of the incremental cost, the following distributed feedback controller is designed for each distributed energy-based generation unit:

wherecλis the coupling gain to be designed andFl,λis the local feedback gain for each generation unit.The best choice of the aforementioned control gains is detailed below.

For (37), the tight form of the local incremental cost error can be written as

whereud=[ud1,ud2,...,udN]T.Up to this point, the global closed-loop dynamics of the incremental cost can be given as

Similar to Conclusion 1, we will give results on the stability of the closed-loop DC microgrid system and the global optimality of the load distribution scheme.

The feedback gain of the current sharing controller is designed as

Then, if the graphGis undirected and connected, and if the coupling gain meetscλ＞0, the DC microgrid can achieve precise economic current sharing, the voltage can be restored to the allowed range, and the closed-loop DC microgrid system can be stabilizing to the null space ofQλggiven in (45)and the FAOC controller (38) is optimal with respect to the following performance index:

whereRg,λ=Ra,λ⊗Rl,λ,

Proof: The procedure follows that of Conclusion 1 and is not repeated here.■

Up to this point, we can summarize the proposed control strategy in Fig.3, in which the data flow can be clearly seen.

Remark 3: It is worth mentioning that most of the existing distributed cooperative algorithms achieve optimal power sharing or voltage control based only on droop control with correction terms.In contrast, this paper establishes a comprehensive large-signal model of the multi-bus DC microgrid in series and designs a model-based secondary controller over the methodology of a high-order fully actuated system.This novel perspective on system dynamics is rarely mentioned in existing efforts with a droop control interface or a standalone first-order integrator model.These practices isolate the underlying physical characteristics from the upper control levels,and focus only on the dynamic behaviors of the designed controller.In addition, unlike previous work, this paper implements voltage recovery and optimal load sharing (economic operation of DG units) in the secondary control level rather than the tertiary control level, which also reflects the benefits of establishing the comprehensive microgrid model.

IV.EXPERIMENTAL VERIFICATION

In this section, the effectiveness of the FAOC secondary control scheme proposed in this paper will be verified by experimental case studies on a series-connected multi-bus DC MG.

A. Hardware Setup and Experimental Procedure

Fig.3.Structure of the optimal cooperative secondary control strategies based on the FA-MAS model: and , respectively denote proportional and economic current sharing.

Fig.4.Layout of the islanded DC MG prototype: a) Monitor and control desk; b) Oscilloscope (voltage); c) Oscilloscope (current); d) Switch for load change; e) Resistance for load variation; f) Tie lines; g) Resistive loads;h) Switch for plug-and-play; i) Current probes; j) PV panels with MPPT controller; k) Storage batteries; l) DC-DC converters (4 sets); m) OPAL-RT;n) DC sources.

The scaled-down islanded microgrid testbed is given in Fig.4.It consists of four generation units, power lines connecting each unit, and a hardware controller.The power supply, the DC-DC buck converter, the LC filter, and the local resistive load are the specific components of each power generation unit.In order to reproduce as much as possible in the laboratory a power generation system based on real RESs, a set of photovoltaic (PV) panels with batteries acts as a DER, while the other three are substituted with DC supplies.The OPALRT (OP5700) works as a hardware controller rather than a target machine, collecting analog signals from each DER’s voltage and current sensors via cables.The OP5700 executes the designed FAOC control algorithm and generates the signal to drive the IGBT through the built-in FPGA-based module.

Fig.5.Architecture of the test bench composed of physical converters and local controllers.

Fig.5 illustrates the topology of the basic physical connections and the control logic of the test system.The communication topology utilized for the DC microgrid prototype is the same as the physical connection graph, namely, a strongly connected ring structure with the adjacency matrix of AG=[0,1,0,1;1,0,1,0;0,1,0,1;1,0,1,0] andVG={1,2,3,4}.If not specified, this topology will be adopted for all subsequent cases.The nominal voltage is set to=48Vfori∈VGby the tertiary control layer, which is also a typical value in low-voltage DC distribution networks.The permissible percentage of output voltage deviation from the nominal voltage setpoint is set to δ=3%.That is ϵ ≈1.5 V.The input voltage levels of the four DERs are 80 V, 80 V, 90 V and 100 V.The coefficient of proportional current sharing is set to η=[1.5,2,2,1]and the generation cost coefficients of the DC microgrid are taken as α=[0.375,0.25,0.25,0.375], β=[1.349,0.9,0.9,1.349], γ=[0.86,0.86,0.86,0.86].In addition,the samplingperiodintheOP5700 isconfigured as2×10-5s.Otherspecificationsofthe DERs instandalone mode,transmission lines and loads are summarized in Table II.

TABLE II SETTINGS OF SCALED-DOWN MG AND PRIMARY CONTROL

Subsequent case studies are conducted covering three scenarios: 1) basic performance testing and comparison; 2) performance testing under load variations; and 3) evaluation of plug-and-play capability.In order to completely demonstrate the results of above cases, the whole experiment needs to go through four stages as follows.

Phase 1: activating the local droop control and enabling the grid integration of all DERs att1.

Phase 2: executing the proposed FAOC secondary control scheme for the DC MG att2.

Phase 3: adding a constant loadRL,cutto the local loadRL1att3and removing it from the bus att4.

Phase 4: removing DER4from the DC MG prototype att5and reconnecting it to the MG att7.It should be pointed out that to implement the grid connection, the voltage of DER4is raised to near 48 V using local droop control during [t6,t7).

B. Comparison and Fundamental Performance Assessment

In this subsection, the FAOC method proposed in this paper is compared with the one in [26] to demonstrate its basic performance in DC microgrids.To this end, the experimental results under the following case studies are presented 1) voltage balancing and proportional current sharing under the existing consensus-based secondary control method, 2) voltage balancing and proportional current sharing under the proposed method, and 3) voltage balancing and power management under the proposed method.

Case I:Results of the Existing Observer-Based Methods

For a multi-bus DC microgrid in series, Tucciet al.[26]proposed the following consensus-based secondary scheme to adjust the reference value of each primary voltage regulator

Therefore, in the case of testing the existing consensusbased scheme, the PnP voltage regulator works alone in the first stage and activates the above-mentioned secondary control (46) only after the system is stabilized.As can be seen from the waveforms shown in Fig.6, the above-mentioned secondary control strategy allows for voltage balancing and precise proportional sharing of current in the microgrid.

Case II:Results Under the Proposed FAOC Control

1)Waveforms of the DC MG Under Proportional Current-Sharing Scheme

In this case, by choosingQl,s=1 andRl,s=0.1, thePl,scan be calculated by the algebraic Riccati equation asPl,s=0.3162.From (23), we have the feedback gain matrixFl,s=3.126, and then there areQg,s=[6,-4,2,-4;∗,6,-4,2;∗,∗,6,-4;∗,∗,∗,6]andRg,s=diag{0.1,0.1,0.1,0.1} in the cost function (24).The gain factor in the FAOC secondary controller (16) is set tocs=1.Fig.7 plots the output responses of the DC microgrid in this case.The experimental results show that the voltage at each PCC is quickly stabilized and restored to an acceptable range and the currents are quickly and accurately shared among the DG units due to the activation of the proposed FAOC secondary control att2.

2)Waveforms of the DC MG Under Economic OperationScheme

Fig.6.Evolution of (a) measured voltages at PCCs and (b) output currents under the secondary control proposed in [26].

Fig.7.Evolution of (a) measured voltages at PCCs and (b) output currents under the FAOC control scheme for accurate proportional current sharing.

In addition to achieving the proportional current sharing in[26], this paper further considers the economic operation of the multi-bus DC microgrid and designs a cooperative power management strategy.The parameters in LQR are the same as those in the previous case.As can be seen from the waveforms shown in Fig.8, the proposed FAOC secondary control features fast regulation profile and satisfactory steady-state performance and is able to recover the voltage at each PCC of the islanded DC microgrid to within the given range.Meanwhile, the output currents of the four DERs in Fig.8 achieve consensus quickly, and are divided into two categories as a result of the fact that DERs #1, #4 as well as DERs #2 and #3 have the same cost factors.The waveforms of incremental costs in Fig.9 also suggest that the proposed FAOC scheme maintains optimal load sharing among the DERs.

Fig.8.Evolution of (a) measured voltages at PCCs and (b) output currents under the FAOC control scheme for economic operation of MGs.

Fig.9.Evolution of incremental costs under the FAOC control scheme for economic operation of MGs.

Based on the above demonstration and presentation of the basic results, the following observations and discussions are made.

i) Compared to the waveforms obtained in Case I, the proposed FAOC method achieves better regulation performance,such as smaller overshoot, and smoother output voltage and current response.In particular, the completion of voltage recovery and current distribution under both tests in Case II is significantly faster than that of Case I.Notice that the time interval for each division in Fig.6 is 4 s, while that in Figs.7 and 8 is 2 s.

ii) On the other hand, the performance metricsJunder Case I and the proportional current-sharing case study with the proposed method in Case II areJ= 27.34 andJ= 15.67, respectively.It should be noted that the cumulative cost functionJ,which is actually the accumulation of instantaneous costs over an infinite period of time, is calculated here for the constructed DC microgrid operating for quite a long time after the secondary control is activated.Namely,J(Iso(t2),u(t))=whereT0∈[t2,t3).It can be seen that the existing consensus-based control strategy is only concerned with proportional current sharing, while the FAOC controller makes the DC microgrid system not only exhibit fast and smooth performance, but also operate in an optimal manner with respect to the performance index function (24).

iii) In addition to the above waveform results, the generation costs under the two tests of Case II can be calculated asC=57.6180 for the first scene andC= 48.2834 for the second scene in the steady-state, respectively.This indicates that the power management strategy proposed in Section III-C can ensure the economic operation of the DC microgrid.

C. Evaluation for Load Changes

This subsection tests the performance of the proposed FAOC control strategy in microgrids facing load variations,i.e., the experiment corresponding to Phase 3 is performed.It is worth mentioning that only the FAOC secondary strategy proposed in Section III-B is examined here in order to avoid overly redundant experimental cases.

The voltage response profiles of the DC microgrid under load variations are given in Fig.10(a).At the momentst3andt4, the output voltages of the DC microgrid both rapidly recover to the given range and are stabilized after a few seconds of fluctuation due to the change of local load near DER1.Concurrently, the current response curves in this case are depicted in Fig.10(b).It can be seen that, the output current of each DG unit varies with the loads in the microgrid, but the total load current can be quickly and precisely shared among DERs according to their respective sharing ratio.The output currents under load changes are still divided into three sets of curves at steady state, which is consistent with the outcomes under the basic performance test.Further, to clearly demonstrate the responses of the DC microgrid system at the moments of load changing, a zoomed-in view of the voltage and current is given in Fig.11.The four subgraphs reflect that both the voltage and current jitter at the two instants,t3andt4,but they can quickly recover to the desired steady state.From the above results and discussions, it is seen that the proposed FAOC control scheme has excellent adaptability to load changes.

D. Plug-and-Play Feature Test

Accounting for the characteristics of DERs, such as intermittency, this subsection examines whether the proposed FAOC controller can still achieve the expected control performance when a DG unit is plugged in or out of the microgrid.It is worth mentioning that DER4powered by the PV panels is physically plugged out from the DC microgrid att5and then plugged back into the microgrid system att7, where an air switch acts as the actuator for these operations.As shown in Fig.12, when DER4is disconnected from the microgrid, it is automatically disconnected from its neighbors at the information layer, since the failure of the DG unit also implies the loss of all communication links connected to it.

Fig.10.Evolution of (a) measured voltages at PCCs and (b) output currents under the FAOC control scheme in case of load variations.

Fig.11.Responses of voltages in (a) load increase; (b) load decrease instant, and responses of currents in (c) load increase; (d) load decrease instant.

As shown in Fig.13(a), the detachment of DER4att5causes some fluctuations in the output voltages.In particular, the voltage of DER4at the PCC slowly decays to 0 due to the shunt capacitor,whilethe output voltagesof the remaining threeDERsquicklystabilizeinthe desiredinterval[(1-ε),(1+ε)]afterafew secondsoftransients.As thesupply capacity of the generation unit is resumed att6and DER4is grid-connected to the microgrid att7, all output voltages can be quickly returned to their pre-failure levels.As shown in Fig.13(b), similar to the load variations in Fig.10(b), the output currents of DER1, DER2and DER3stabilize quickly after brief fluctuations due to the failure of DER4and still follow the desired current sharing in the microgrid formed by the remaining DERs.Similarly, with the return of DER4att7, the microgrid is restored to its pre-fault operating state and maintains the accurate proportional current sharing.

Fig.12.Communication topology changes when DER4 plug-in and plug-out.

Fig.13.Evolution of (a) measured voltages at PCCs and (b) output currents under the FAOC control scheme in case of plug-and-play.

The above experimental results and discussions suggest that the microgrid exhibits resilience towards load variations and satisfactory plug-and-play properties under the proposed secondary control strategy, while achieving optimal regulation with respect to a performance metric function.

V.CONCLUSION

Considering the multi-bus DC MGs, the FAOC secondary control method is proposed in this paper to ensure voltage restoration and accurate current sharing as well as economic current distribution.A comprehensive model of the DC MGs is performed prior to the design of the secondary control protocol, which greatly portrays the various electrical characteristics and physical constraints of the actual MG system.At the same time, the developed model has a simple and control design-oriented structure, which is significantly different from the traditional modeling approach.The FAOC secondary controls (16) and (38) are proposed based on the developed model.The proposed controllers feature the same simple design pattern as the widely used droop interface-based methodology under the decoupling effect of high-order fully actuated system theory.With the proposed FAOC control strategies,the optimal power allocation in a multi-bus DC MG can be achieved by secondary control only, i.e., the two goals of economical current distribution and current sharing amongst DG units can be realized with the help of the suggested virtual control.The stability, consensus and optimality analysis of the closed-loop DC MG system under FAOC scheme is presented, which theoretically ensures the feasibility and generality of the developed methods.Extensive case tests are executed on the laboratory-scale MG system, and the experimental results confirm the correctness of the developed model, the effectiveness of the designed strategies, and the typical characteristics of the MG such as plug-and-play capability.

Future research will be devoted to the design of distributed optimal robust secondary control that can tolerate process and measurement noises.In addition, investigating how to refine the model-based distributed optimal secondary control from the category of asymptotic convergence to that of finite time or fixed time convergence [33] is part of future development.