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A resilient bi‐level control strategy for power sharing and voltage balancing in bipolar DC microgrids

A resilient bi‐level control strategy for power sharing and voltage balancing in bipolar DC... INTRODUCTIONBackground and motivationProliferation of DC‐nature renewable energy sources, along with the fast increase of DC loads in the modern electrical grid systems have motivated the idea of using DC power systems [1–3]. The DC electrical grids are categorized into two types: unipolar power systems and bipolar power systems [4]. Compared to the unipolar systems, bipolar systems provide more flexibility by allowing the connection of distributed generators (DGs) and loads into two voltage levels, that is, (1) between neutral point and either positive or negative pole, and (2) between positive pole and negative pole. Besides the voltage‐related advantages mentioned above, the bipolar systems benefit from high reliability due to the independent operation of two poles, which allows the half capacity of the system to remain available in fault situations [5].On the other hand, bipolar DC systems face asymmetrical operations when the power distribution in the two poles is not identical [6]. Such asymmetry puts the voltages of poles into an unbalanced condition, which consequently leads to power quality deterioration, voltage instability, and under a severe situation as the worst case, serious damage to the whole power conversion units [7]. As a result, introducing a solution capable of equalizing the DC‐bus voltages is indispensable in the bipolar DC microgrids. Addressing this challenge from the power control perspective (i.e. without any additional power electronic equipment) is the main motivation of this study.Literature reviewMost of the leading methods of voltage balancing are power electronics solutions, and there are very few works on addressing the problem from the power system perspective. The existing device‐level solutions to mitigate voltage imbalance include: (i) employing a balancing power electronic blocks called voltage balancer (VB), (ii) installation of distributed DC electric springs (DC‐ESs), and (iii) redistribution of unipolar loads.The VBs operate to make the positive pole voltage of the converter outlet equal to the negative pole voltage [8]. Different types of VB topologies and relevant control strategies have been proposed in previous studies [9–13]. It is worth mentioning that VBs are only responsible for voltage‐balancing control tasks regardless of voltage value at the outlet of the converter, that is, they have no responsibility to restore the voltage of the system to the nominal value. Moreover, the VBs cannot function as a DC‐DC converter in applications such as photovoltaic systems and battery storage systems [14]. A DC‐ES‐based method for unbalanced voltage suppression in a bipolar DC systems is introduced in [15]. In this method, DC‐ES is connected in series with non‐critical load in each pole to maintain the voltage across the critical load in the nominal value. In the two aforementioned methods, suppression of unbalanced voltage is accomplished through additional power electronics equipment and only the local voltage quality is guaranteed. Redistribution of unipolar loads in order to mitigate voltage imbalance can be achieved by a centralized control approach [8, 16, 17]. In this approach, a full communication channel is required to collect the required information from each node and sent them to the control center. Then, the switch signal will be generated and sent to the commutation switch. Due to the centralized communication infrastructure and additional switching devices, this approach is not sufficiently cost‐effective [16]. Besides this, it is noteworthy that this method belongs to the energy management level rather than power management, which means the time scale in this method is large, and it cannot be implemented in the real‐time microgrid operation [8, 18].A different approach to voltage balancing is the use of three‐level DC‐DC converters (TLCs) [19]. TLCs have the abilities of both serving as a DC‐DC interface converter for DGs (e.g. photovoltaic systems, fast chargers, and battery storage systems) and voltage balancing control [14]. In contrast to the conventional VB, three‐level converter‐based voltage‐balancing control systems require no additional balancing circuits. Assuming the interface converter of DGs in a bipolar DC microgrid is a three‐level DC‐DC converter‐based VB, some control strategies have been proposed to address power sharing and voltage balancing tasks. In [20], the conventional droop control has been employed in a grid‐connected bipolar DC microgrid; however, it suffers from poor power sharing and voltage imbalance. To cope with the drawbacks of droop strategy, an improved droop control approach has been presented in [21]. Despite satisfactory operation, it uses complex and high bandwidth communication networks which requires information of all DGs, thus it will be vulnerable to a single point of failure. Tavakoli et al. [22] have proposed two control strategy for power sharing and voltage regulation of bipolar DC microgrids by using the controller area network (CAN). In these two networke‐based control strategy, the CAN bus is utilized to carry control signals and realize information sharing among DGs. Although the CAN‐based control strategy eliminates the need for a central control unite, it utilizes complex and all‐to‐all high bandwidth communication networks where any single communication failure renders the whole microgrid inoperable.More recently, a distributed cooperative control scheme for bipolar DC microgrids was proposed, by which the average voltages of the positive and negative poles of the system converge to the nominal value [23]. However, this paper only solves the unbalanced voltage problem while accurate power sharing of DGs is not taken into consideration. It is worth mentioning that the previously proposed communication‐based methods in [21–23] are susceptible to additive noise and time delays in practical communication networks which can impair the whole system functionality while their effects have not been taken into account. To the best of the authors' knowledge, there are no control strategies in the literature that establishes simultaneously accurate power sharing and voltage balancing in the bipolar DC microgrids, while considering practical issues such as line impedances, communication time delays, and noises. This is the main motivation of this study.Key contributions and organizationWithin the context alluded above, the aim of this study is to propose a resilient bi‐level control strategy to balance the voltage of poles, and share the load among DGs in proportion to their capacities. In the primary level, a fully distributed, failure‐resilient, consensus‐based cooperative control method with consideration of both communication time delays and noises is developed to achieve precise power‐sharing among DGs. In this method, only a sparse communication network is required to exchange local information among neighbouring DGs. In the secondary level, a washout filter‐based control strategy is developed to restore the voltage of each pole to the desired value, where it operates without any communication networks and extra control loops. Different from the previous works in the field of three‐level DC‐DC converter‐based bipolar DC microgrid control, the salient contributions of the proposed control scheme are outlined below:A fully distributed, noise‐resilient, robust, consensus‐based control method with consideration of communication delays is developed to achieve an accurate power sharing among DGs in a bipolar DC microgrid. One of the distinct features of this control loop compared to the existing communication‐based techniques in bipolar DC microgrids (e.g. see [21, 22]) is that it uses only neighbour‐to‐neighbour (not all‐to‐all) information over a sparse communication network where no single link failure can impair the normal function of the whole system. It is noteworthy that the proposed method is robust against cyber communication time delays and noises, while in the previous works [21–23], the communication links have been assumed ideal without any noise and time delay. Moreover, as opposed to extensive literature having proposed distributed consensus technique for DC microgrids, the developed strategy in this work requires only the output current state variable instead of two state variables (i.e. output voltage and current output). Therefore, the system communication burden is effectively reduced and the reliability is enhanced.A washout filter in feedback control is developed to restore the voltage of each pole to the desired value. It establishes voltage balancing between positive‐ and negative‐pole to the neutral point. This method operates without any communication links and additional control loops, which makes the controller simple to implement.Cooperation of the proposed consensus‐based power sharing method and washout filter‐based voltage regulator method simultaneously accomplishes proportional power sharing and voltage balancing, while the current control methods of bipolar DC microgrids (see [21–23]) are able to establish only voltage balancing or proportional power sharing, when the impedances of the lines are not negligible.In contrast with the existing communication‐based methods [21, 22], the proposed control strategy does not need prior knowledge of the global parameters such as the number of DGs. Therefore, it offers expandability, more flexibility, and plug‐in and plug‐out feature.BIPOLAR DC MICROGRID STRUCTURE AND CONVENTIONAL DROOP CONTROLA typical configuration of a bipolar DC microgrid is shown in Figure 1, which comprises three main elements: (i) renewable energy source interface converters (RESICs) that interconnect renewable sources to the bipolar DC terminals, (ii) loads , and (iii) TLCs that act as energy storage interface converter and voltage balancing in the system. Since in the island operation of microgrids only the energy storages, as dispatchable DGs of the system, are responsible for voltage regulation, the focus of this work is control of TLCs, whereas the other power supplies in the system including renewable resources are considered as variable current sources.1FIGURETypical structure of an islanded bipolar DC microgridBlock diagram of a bipolar bidirectional DC/DC boost converter which plays the role of voltage balancer and equipped with the conventional droop control method is illustrated in Figure 2. A bipolar DC microgrid with two DGs is illustrated in Figure 3, where each converter is simplified by applying the Thevenin equivalent model. It is noteworthy that in the whole article, subscript ′′m′′=u,l$^{\prime \prime }m^{\prime \prime }={u,l}$ stands for pole label. The expression of the droop control method is as follows:1Vm,i=V∗−Rm,iim,i\begin{equation} {V_{m,i}=V^*-R_{m,i} \; i_{m,i}} \end{equation}where Vm,i$V_{m,i}$ represents the output voltage of ith DG in pole ′′m′′$^{\prime \prime }m^{\prime \prime }$, V∗$V^*$ is the global reference voltage of both poles, im,i$i_{m,i}$ is the output current of ith DG in pole ′′m′′$^{\prime \prime }m^{\prime \prime }$, and Rm,i$R_{m,i}$ is the droop gain of ith DG in pole ′′m′′$^{\prime \prime }m^{\prime \prime }$. Notice that the subscript “i” indicates ith DG. In the following equations, everywhere the subscript of each variable modifies from “i” to “j”, which indicates the jth DG instead of ith DG.2FIGURESchematic of bipolar bidirectional DC/DC boost converter with conventional droop control method3FIGURESimplified model of a two‐DG bipolar DC microgrid in Thevenin formFrom Figure 3, the voltage of load in each pole ′′m′′$^{\prime \prime }m^{\prime \prime }$ can be derived as (2)2Vload,m=Vm,i*−Rm,iim,i−Rlinem,iim,iVload,m=Vm,j*−Rm,jim,j−Rlinem,jim,j,\begin{equation} \left\{ \def\eqcellsep{&}\begin{array}{c} {{V}_{\textit{load},m}={V}_{m,i}^{\ast}-{R}_{m,i}{i}_{m,i}-{R}_{\textit{lin}{e}_{m,i}}{i}_{m,i}}\\[6pt] {{V}_{\textit{load},m}={V}_{m,j}^{\ast}-{R}_{m,j}{i}_{m,j}-{R}_{\textit{lin}{e}_{m,j}}{i}_{m,j}}\end{array} \right., \end{equation}where Rlinem,i$R_{line_{m,i}}$ and Rlinem,j$R_{line_{m,j}}$ are the line resistance. Equations (3) is yielded from (2):3im,iim,j=Rm,j+Rlinem,jRm,i+Rlinem,i.\begin{equation} {\frac{i_{m,i}}{i_{m,j}} = \frac{R_{m,j} + R_{line_{m,j}}}{R_{m,i} + R_{line_{m,i}}}} .\end{equation}It is noteworthy that in a small DC microgrid system, the lines resistances have a low value. Considering this assumption and selecting the droop gains much larger than the line resistances, that is, Rm,i≫Rlinem,i,Rm,j≫Rlinem,j$ {{{R_{m,i} \gg R_{line_{m,i}}}}, {{R_{m,j} \gg R_{line_{m,j}}}}}$, the following equation is inferred from (3), which indicates a proportional power sharing among DGs in the mthe pole (i.e. in the both upper and lower poles):4im,iim,j≈Rm,jRm,i\begin{equation} {\frac{i_{m,i}}{i_{m,j}} \approx \frac{R_{m,j} }{R_{m,i}}} \end{equation}However, the above assumption is only acceptable for small systems while in the large systems due to the large value of lines resistances, (4) cannot be inferred [24]. Therefore, to achieve a proportional power sharing, the droop gains can be increased; however, the larger droop gain leads to more voltage deviation. Thus, the main objective of this study is to address a control strategy that simultaneously ensures accurate power sharing among DGs and restoring the voltage of each pole to the desired value.DESCRIPTION OF PROPOSED CONTROL SCHEMETo overcome the drawbacks of the conventional droop control, a combined control structure is proposed for proportional power sharing and voltage balancing in bipolar DC microgrids. The overall diagram of the proposed control structure is illustrated in Figure 4. Besides the local inner voltage control loop, the proposed control structure consists of two parts:proportional power sharing, andvoltage regulation.4FIGUREThe overall schematic diagram of the proposed control methodThe controller design is a combination of a distributed consensus‐based cooperative method and a washout filter‐based method.Proportional power sharingIn this section, a distributed consensus‐based cooperative control method with consideration of both communication time delays and noises is developed to synchronize the output currents of all DGs (in per‐unit) to a common value.1)Preliminaries of graph theoryIn order to model the communication network among DGs, a directed graph G=(V,E)$\mathcal {G=(V,E)}$ with vertices V$\mathcal {V}$={1,2,…,N}$=\lbrace 1, 2, \ldots, N\rbrace$ and edges E⊆V×V$\mathcal {E} \subseteq \mathcal {V} \times \mathcal {V}$ is introduced here. Each vertex denotes a DG. The edge (i,j)$(i,j)$ (pointing from j to i) indicates that vertex i can receive information from vertex j, with a weighting factor aij$a_{ij}$. The neighbours of vertex i are denoted by Ni$N_{i}$, which contains all the vertices which send information to that specific vertex, that is, j∈Ni$j\in N_{i}$ if (j,i)∈$(j,i)\in$E$\mathcal {E}$. The communication structure between the vertexes is explained by the adjacency matrix A$\mathcal {A}$=[aij]∈RN×N$=[a_{ij}]\in {R}^{N \times N}$, where aij=1$a_{ij}=1$ if there is an edge from vertex j to vertex i, and aij=0$a_{ij}=0$ otherwise. The graph Laplacian matrix is defined as L=D−A$\mathcal {L=D-A}$, where D$\mathcal {D}$ is called in‐degree matrix that is a diagonal matrix, that is, D$\mathcal {D}$=diag(di)⊆RN×N$=diag(d_{i})\subseteq {R}^{N \times N}$ with di=∑j∈Niaij$d_{i}=\sum _{j\in N_{i}} {a_{ij}}$.2)Basics of distributed cooperative controlThe scalar information state, xi$x_{i}$, is dedicated to each communication vertex, i. Each vertex requires to receive its neighbours information, j(j∈Ni)$j \: (j\in N_{i})$, via sparse communication network to update its state, xi$x_{i}$. The update rule is based on the consensus protocol for regulator synchronization problem as:5ẋi(t)=∑i=1naijxj(t)−xi(t)\begin{equation} {\dot{x}_{i}(t)=\sum _{i=1}^{n} {{a}_{ij}\Bigl (x_{j}(t)-x_{i}(t)\Bigr )}} \end{equation}where aij${a}_{ij}$ represents the elements of the communication digraph adjacency matrix.3)Design of distributed cooperative controller with communication time delays and noises for power sharingIn this part, based on the regulator synchronization problem protocol given in (5), a distributed cooperative control strategy with time delays and multiplicative noises in the communication layer is developed to achieve a proportional power sharing in the bipolar DC microgrid, as seen in Figure 4. Since the measured current signals are corrupted with a significant amount of high‐frequency noise, a distributed resilient control strategy in mean square consensus protocol is developed here to ensure the stability of the system. Therefore, to control the current of each pole of ith DG , that is, iu,i$i_{u,i}$ and il,i$i_{l,i}$, the following current sharing mismatch in each pole is calculated6um,i=K∑j∈Niem,ji(t),\begin{align} {u_{m,i}} = K \sum _{j \in N_{i}} e_{m,ji} (t), \end{align}notice that subscript ′′m′′$^{\prime \prime }m^{\prime \prime }$={u,l}$=\lbrace u, l\rbrace$ stands for the pole of the system, and the subscripts ′′i′′$^{\prime \prime }i^{\prime \prime }$ and ′′j′′$^{\prime \prime }j^{\prime \prime }$ represent the ith and jth DG, respectively. K represents a positive matrix to be designed. em,ji(t)$e_{m,ji}(t)$ is the state measurement of ith DG from its neighbour DG j7em,ji(t)=im,jpu(t−τ1)−im,ipu(t−τ1)+fji(im,jpu(t−τ2)−im,ipu(t−τ2))γji(t),j∈Ni,\begin{eqnarray} e_{m,ji} (t) &=& i_{m,j}^{pu} (t-\tau _1)-i_{m,i}^{pu} (t-\tau _1) \nonumber \\ && +\ f_{ji} (i_{m,j}^{pu} (t-\tau _2)-i_{m,i}^{pu} (t-\tau _2)) \gamma _{ji}(t), \ j \in N_i ,\qquad \end{eqnarray}where γji(t)$\gamma _{ji} (t)$ ∈R$\in R$ indicates the measurement noise; τ1,τ2$\tau _1, \tau _2$ are the time delays, and fji(.)$f_{ji}(.)$ represents the noise variance. As is seen em,ji(t)$e_{m,ji} (t)$ is composed of two segments. The first segment of protocol im,jpu(t−τ1)−im,ipu(t−τ1)$i_{m,j}^{pu} (t-\tau _1)-i_{m,i}^{pu} (t-\tau _1)$ indicates the deterministic term, and the second segment fji(im,jpu(t−τ2)−im,ipu(t−τ2))γji(t)$ f_{ji} (i_{m,j}^{pu} (t-\tau _2)-i_{m,i}^{pu} (t-\tau _2)) \gamma _{ji}(t)$ indicates the stochastic term. We assume that the noises are independent Gaussian white noises, and assure the following assumption.1AssumptionThe process noise γji(t)∈R$\gamma _{ji} (t) \in R$ assures ∫0tγji(t)(s)ds=ωji(t)$\int _{0}^{t} \gamma _{ji} (t)(s)ds = \omega _{ji} (t)$, t≥0$t \ge 0$, i=1,2,…,N$i=1,2,\ldots,N$, j∈Ni$j \in N_i$, where {ωji(t),i=1,2,…,N}$\lbrace \omega _{ji}(t), i=1,2,\ldots,N \rbrace$ points independent two‐dimensional Brownian motions defined on the complete probability space (Ω,F,P)$(\Omega , F, P)$ with a filtration Ft(t≥0)$F_t (t \ge 0)$ satisfying the usual conditions, namely, it is right continuous and increasing while F0 contains all P‐null sets.1Remark{λiu(A)}i$\lbrace \lambda _i^{u} (\mathcal {A} ) \rbrace _i$ indicates the unstable eigenvalues of A$\mathcal {A}$, that is, Re(λiu(A))≥0$ Re(\lambda _i^{u} (\mathcal {A} )) \ge 0$. We define λ0u=∑iRe(λiu(A))$\lambda _0^u = \sum _{i} Re(\lambda _i^{u} (\mathcal {A} ))$.The results of this part are summarized as below:1TheoremThe distributed cooperative control protocol8um,i=K∑j∈Niem,ji(t)em,ji(t)=im,jpu(t−τ1)−im,ipu(t−τ1)+fji(im,jpu(t−τ2)−im,ipu(t−τ2))γji(t),j∈Ni\begin{eqnarray} {u_{m,i}} &=& K \sum _{j \in N_{i}} e_{m,ji} (t) \nonumber \\ e_{m,ji} (t) &=& i_{m,j}^{pu} (t-\tau _1)-i_{m,i}^{pu} (t-\tau _1) \nonumber \\ && +\ f_{ji} (i_{m,j}^{pu} (t-\tau _2)-i_{m,i}^{pu} (t-\tau _2)) \gamma _{ji}(t), \quad j \in N_i \qquad \end{eqnarray}can ensure that the current is shared proportionally among DGs without regard to time‐delay and noise in communication links.Notice that the controller exists if Assumption 1 is valid, fji$f_{ji}$ = σji(i)≥0$\sigma _{ji} (i) \ge 0$, max(Re(λ(A)))>0$max (Re (\lambda (\mathcal {A}))) &gt; 0$, and 4λ0uσ¯2≤λ1$4 \lambda _0^{u} \bar{\sigma }^2 \le \lambda _{1}$.The upper bound of delays in the communication links can be obtained as9τ1*=min(12P,λ1−4σ¯(λ0u+ε)6(λ0u+ε)λN).\begin{align} \tau _1 ^{\ast } = min (\dfrac{1}{2 P }, \dfrac{\lambda _{1}-4 \bar{\sigma } (\lambda _0^{u}+\epsilon ) }{6(\lambda _0^{u}+\epsilon )\lambda _{N}}). \end{align}Notice that P>0$P&gt;0$ is designed by solving the Ricatti equation as:102P−2αP21+P+1=0,\begin{align} 2P- \dfrac{2 \alpha P^2}{1+P} +1=0, \end{align}with α∈(λ0u,λ0u+ε)$\alpha \in (\lambda _0^{u}, \lambda _0^{u}+\epsilon )$, ε∈(0,λ1−4σ¯2λ0u4σ¯2)$\epsilon \in (0, \dfrac{ \lambda _{1}-4 \bar{\sigma }^2 \lambda _0^{u}}{4\bar{\sigma }^2})$, and σ¯2=maxi=1,j=0Nσij2$\bar{\sigma }^2 = max_{i=1,j=0}^{N} \sigma _{ij}^2$.Eventually, the K in (8) can be obtained as11K=kP1+P,\begin{align} K = \dfrac{kP}{1+P} , \end{align}where k∈(k̲,k¯)$k \in (\underline{k}, \bar{k})$, k̲=[λ1−λ12−2αρ]ρ$\underline{k} = \dfrac{[\lambda _{1}- \sqrt {\lambda _{1} ^2-2 \alpha \rho ]}}{\rho }$, k¯=[λ1+λ12−2αρ]ρ$\bar{k} = \dfrac{[\lambda _{1}+\sqrt {\lambda _{1} ^2-2 \alpha \rho ]}}{\rho }$, and ρ=(2σ¯2+3λNτ1)λ1$\rho = (2 \bar{\sigma }^2+ 3\lambda _{N} \tau _1)\lambda _{1}$.The proof is given in Appendix.Then, the calculated current sharing mismatches, em,i${e}_{m,i}$, in each pole passes through a PI controller to generate the current sharing correction terms, δm,iv$\delta _{m,i}^v$ in each pole, as shown in Figure 4. This terms in each pole is added into the droop function (1) to eliminate the discrepancy between per‐unit current of converter i and per‐unit current of neighbour converters in the steady state, that is, limt→+∞em,i=0$ \lim _{t\rightarrow +\infty }e_{m,i}=0$, where i≠j$i\ne j$. Therefore, the conventional droop equation given in (1) is revised as:12Vm,i=Vm∗−Rm,iim,i+δm,iv.\begin{equation} {V_{m,i}=V_{m}^*-R_{m,i} \; i_{m,i}+\delta _{m,i}^v} .\end{equation}This adjustment reduces current mismatch among neighbour DGs, and ultimately among all DGs of the microgrid. Under normal operation, the per‐unit currents of all converters reach to a common value at the steady‐state, and consequently, the mismatch terms, em,i${e}_{m,i}$, become zero. However, voltage deviation due to the droop controller function is maintained.Voltage regulationInspired by previous applications of washout filters in addressing former challenges [25–27], a washout filter‐based control strategy for voltage balancing in bipolar DC microgrids is developed in this section. The proposed strategy works without any communication links and without any extra control loops.1) Design of washout filter‐based controller for voltage regulationTo eliminate the voltage deviation created by the droop resistor and restore the voltage amplitude to the rated value in each pole, a strategy based on the utilization of a washout filter in the feedback path is designed here. To this end, the equation (12) is written in the following form:13Vm,i−(Vm∗+δm,iv)+Rm,iim,i=0.\begin{equation} {V_{m,i}-(V_{m}^*+\delta _{m,i}^v)+R_{m,i} \; i_{m,i}=0} .\end{equation}By considering ΔVm,i=Vm,i−(Vm∗+δm,iv)$\Delta V_{m,i}=V_{m,i}-(V_{m}^*+\delta _{m,i}^v)$ and taking time derivative from (13), the following equation is achieved:14ddtΔVm,i+Rm,iddtim,i=0.\begin{equation} {\frac{d}{dt} \Delta V_{m,i} + R_{m,i} \frac{d}{dt} i_{m,i}=0} .\end{equation}Through Equation (14), proportional power sharing is ensured in the steady state. However, voltage regulation due to droop resistor can not be compensated. This can be accomplished by adding a factor of ΔVm,i$\Delta V_{m,i}$ to (14) as:15ddtΔVm,i+Rm,iddtim,i+kmΔVm,i=0.\begin{equation} {\frac{d}{dt} \Delta V_{m,i} + R_{m,i} \frac{d}{dt} i_{m,i}+k_{m} \Delta V_{m,i}=0} .\end{equation}Since the derivative terms of (15) are zero in the steady state (i.e. ddtΔVm,i=0$\frac{d}{dt} \Delta V_{m,i}=0$ and ddtim,i=0$\frac{d}{dt} i_{m,i}=0$), the ΔVm,i$\Delta V_{m,i}$ is forced to be zero in (15). Therefore, the voltage in pole ′′m′′$^{\prime \prime }m^{\prime \prime }$ is restored to its nominal values. Applying Laplace transform to (15), it is expressed as16sΔVm,i(s)+Rm,isim,i(s)+kmΔVm,i(s)=0,\begin{equation} {s \; \Delta V_{m,i}(s) + R_{m,i} \; s \; i_{m,i}(s)+k_{m} \Delta V_{m,i}(s)=0} , \end{equation}which can be rewritten as a transfer function as below17ΔVm,i(s)im,i(s)=−Rm,iss+km.\begin{equation} { \frac{\Delta V_{m,i}(s)}{ i_{m,i}(s)} =-\frac{ R_{m,i} \; s}{s+k_{m}}} . \end{equation}This equation can be stated in the more common form as18Vm,i=Vm∗+δm,iv−Rm,iss+kmim,i.\begin{equation} {V_{m,i}=V_{m}^*+\;\delta _{m,i}^v-\; \frac{ R_{m,i} \; s}{s+k_{m}} \; i_{m,i}} .\end{equation}Finally, as mentioned before, the subscript ′′m′′$^{\prime \prime }m^{\prime \prime }$ stands for the pole label, hence, the equation (18) can be written for each pole as:19Vu,i=Vu*+δu,iv−Russ+kuiu,iVl,i=Vl*+δl,iv−Rlss+klil,i.\begin{equation} \left\{ \def\eqcellsep{&}\begin{array}{c} {{V}_{u,i}={V}_{u}^{\ast}+{\delta}_{u,i}^{v}-\dfrac{{R}_{u}s}{s+{k}_{u}}{i}_{u,i}}\\[9pt] {{V}_{l,i}={V}_{l}^{\ast}+{\delta}_{l,i}^{v}-\dfrac{{R}_{l}s}{s+{k}_{l}}{i}_{l,i}}\end{array} \right.. \end{equation}As it can be seen in (18), the proposed method uses a washout filter in the feedback loop. A washout filter is a first‐order high‐pass filter (HPF), HHPF(s)=ss+km$ H_{HPF} (s)= \frac{s}{s+k_{m}}$, that washes out (rejects) steady state component of signal, while passing transient component of the signal [25]. Therefore, the large voltage deviation on the droop resistor can be compensated by using a washout filter in the feedback path at the steady state20Vu,i=Vu∗+δu,ivVl,i=Vl∗+δl,iv.\begin{equation} {\begin{cases} V_{u,i}=V_{u}^*+\;\delta _{u,i}^v\\ [8pt]{V_{l,i}=V_{l}^*+\;\delta _{l,i}^v} \end{cases}} .\end{equation}It is noteworthy that the terms δu,iv$\delta _{u,i}^v$ and δl,iv$\delta _{l,i}^v$ in (20) are very small which have little effect on the voltage deviation from the desired value.Notice that in order to guarantee the dynamic stability of the whole system and to keep the droop control effective, the restrictive conditions should be considered in designing the parameters of washout‐based control. According to Figure 4, we have:21im,i=HLPF(s)Im,i=ωcs+ωcIm,i,\begin{equation} {i_{m,i}=\; H_{LPF} (s)\; I_{m,i}= \; \frac{ \omega _{c}}{s+\omega _{c}} \; I_{m,i}} ,\end{equation}where Im,i$I_{m,i}$ is the instantaneous current, im,i$i_{m,i}$ is the filtered current, HLPF(s)$ H_{LPF} (s)$ is the low‐pass filter (LPF) transfer function, and ωc$ \omega _{c}$ is the cut‐off frequency of the low‐pass filter. Considering (21), we can rewrite (18) as22Vm,i=Vm∗+δm,iv−Rm,iss+kmωcs+ωcIm,i.\begin{equation} {V_{m,i}=V_{m}^*+\;\delta _{m,i}^v-\;R_{m,i} \; \frac{s}{s+k_{m}} \; \left(\frac{ \omega _{c}}{s+\omega _{c}} \; I_{m,i}\right)} . \end{equation}As we have assumed HHPF(s)=ss+km$ H_{HPF} (s)= \frac{s}{s+k_{m}}$ and HLPF(s)=ωcs+ωc$H_{LPF} (s) = \frac{ \omega _{c}}{s+\omega _{c}}$, (22) can be written as23Vm,i=Vm∗+δm,iv−Rm,iHHPF(s)HLPF(s)Im,i.\begin{equation} {V_{m,i}=V_{m}^*+\;\delta _{m,i}^v-\;R_{m,i} \; H_{HPF} (s) \; H_{LPF} (s) \; I_{m,i}} .\end{equation}From (23), it is seen that an HPF and an LPF have been cascaded and formed a band‐pass filter (BPF). Thus, the parameters of the washout filter‐based controller are influenced by the cut‐off frequency of BPF. In this way, as discussed in [27], stable and appropriate operation of the developed washout filter‐based controller in the secondary level is achieved, when the cut‐off frequency of the HPF satisfies the following condition:24km<ωc.\begin{equation} {k_{m}&lt;\omega _{c}} .\end{equation}Equation (23) determines a restrictive condition in the parameter designing procedure of washout filter‐based controller. If the cut‐off frequency restraint condition in (23) does not consider, the output current signals Im,i$I_{m,i}$ which pass through ill‐conditioned BPF will be oscillating; consequently, the system becomes unstable.SIMULATION RESULTSTo illustrate the effectiveness of the proposed control structure shown in Figure 4, the bipolar DC microgrid depicted in Figure 5 is simulated in five case studies. The system electrical and control parameters are given in Table 1. A ring‐shape communication topology in the cyber‐layer is designed, as seen in Figure 5, which keeps its own connectivity with any communication failure or converter failure. The rated voltage level of the test system is set as ±400 V. The communication time delay quantity is set as τ=20ms$\tau = 20 ms$. Moreover, the additive noise with σ2=0.1$\sigma ^{2}=0.1$ is considered for communication links.5FIGUREThe structure of the simulation test system with implemented communication networks1TABLEParameters of the systemItemsParameterValuePVPmax$_{max}$20 kWPower ratings of DGsPDG#1$P_{DG \#1}$ and PDG#2$P_{DG \#2}$7 kWPDG#3$P_{DG \#3}$ and PDG#4$P_{DG \#4}$14 kWVin$_{in}$200 VVout$_{out}$2× 400 VConverter parametersLm , Ls134 μH, 100 μHrL$r_{L}$ , rc$r_{c}$0.2 Ω, 0.001 ΩC1,C2$C_{1}, C_{2}$200 μFfsw$_{sw}$20 kHzLine#1$\#1$R=0.1 Ω, L=70 μHLine#2$\#2$R=0.15 Ω, L=100 μHTransmission linesLine#3$\#3$R=0.1 Ω, L=70 μHLine#4$\#4$R=0.12 Ω, L=90 μHLine#5$\#5$R=0.05 Ω, L=50 μHLoad#1$\#1$R=40 ΩLoadsLoad#2$\#2$R=30 ΩLoad#3$\#3$R=200 ΩLoad#4$\#4$R=100 ΩRu,Rl$R_{u}, R_{l}$ (DG#1$\#1$ & DG#2$DG\#2$)4 V/A, 4 V/AControl parametersRu,Rl$R_{u}, R_{l}$ (DG#3$\#3$ & DG#4$DG\#4$)2 V/A, 2 V/Aku,kl$k_{u}, k_{l}$4K11.52ωc$\omega _{c}$50πPerformance assessmentIn this case, the performance of the proposed controller is studied in comparison with the conventional droop control. The test system undergoes the following three stages:Stage 1 (0‐3 s): At t=0s$t=0 s$, the system is controlled using the conventional droop controller, and the PV system does not inject any power to the system, that is, PPV=0kW$P_{PV} =0 kW$.Stage 2 (3–6 s): At t=3s$t=3 s$, the PV system begins to inject 16kW$16 kW$ to the upper pole.Stage 3 (6–9 s): At t=6s$t=6 s$, the proposed controller is engaged.The simulation results are illustrated in Figure 6. During stage 1, when only the conventional droop control is applied, the voltage of DGs in both positive and negative pole has deviated severely from the desired value ±400V$\pm 400 V$, as seen in Figure 6e–h. The voltage of the main busbar is also shown in Figure 6d, where the voltage deviation due to the line resistances is greater than the output voltage of DGs. Moreover, although the droop gains have been designed in proportion to the DGs' capacity ratio, the effect of the lines' impedances has incapacitated the droop mechanism which results in an imprecise power sharing in both positive and negative poles, respectively, as shown in Figures 6b and 6c.6FIGUREComparative studies of the conventional droop control and the proposed controller. (a) The output power of PV. (b,c) The output currents of DGs in the upper and lower poles, respectively. (d) The voltages of main busbar. (e–h) The output voltages of DG#1, DG#2, DG#3 and DG#4, respectivelyThen, the PV system begins to inject 16 kW to the network in Stage 2, as seen in Figure 6a. In this stage, the generated power by the PV system is higher than the load demand in the upper pole. Hence, the surplus power is transferred to the lower pole by means of the bipolar boost converters. However, a precise load‐sharing among DGs is not met owing to the negative impacts of line impedances in the upper and lower poles as seen in Figures 6b and 6c, respectively. Not only that, but the impacts of droop controllers lead to an unacceptable voltage discrepancy between voltages of the upper and lower poles of DGs (see Figure 6e–h). Due to the voltage drop over the line impedances, the voltage discrepancy between the voltage of the upper and lower poles in the main busbar, that is, 29 V, is more pronounced (see Figure 6d). Such a significant voltage discrepancy between the upper and lower poles is unacceptable and should be compensated.The proposed controller is activated in stage 3. Consequently, the voltage of each pole is restored extremely close to the desired reference value so that the voltage discrepancy between two poles at the output of all DGs almost reaches to zero (see Figure 6e–h). The voltage difference in the main busbar voltages is decreased to 3V, which is very satisfactory in comparison with 29V in the conventional droop method, as seen in Figure 6d. It is worth mentioning that this voltage discrepancy in the main busbar is caused due to the insignificant voltage drop on the DC distribution line. This is because the proposed method is a source supply‐side method and unlike the load‐side methods which use additional devices [15], does not have direct access to control the main busbar. Furthermore, Figures 6b and 6c show the output current of the upper and lower terminals of sources, respectively. As it is observed, the proposed controller can ensure proper current sharing among sources in both poles, where DG#3 and DG#4 carry twice the current as DG#1 and DG#2. The zoom‐in simulation results of Figure 6b,c clearly illustrate the accuracy of the proposed controller in comparison with the conventional droop controller.Robustness against load variations and a link‐failureFrom the perspective of robustness, two criteria can be evaluated, namely, robustness against load changes and resiliency to a single link failure. To assess this feature of the proposed control strategy, the system goes through the following 2 stages as follows:Stage 1 (0–5 s): At t=0s$t=0 s$, the proposed controller is activated with the original ring‐shape communication topology, and the PV system injects 16 kW to the upper pole. At t=2s$t=2 s$, Load#2 is changed from 30 to 40Ω between both positive‐ and negative‐pole to neutral point. Besides this, at t=4s$t=4 s$, Load#4 is connected between the positive and negative poles of the system.Stage 2 (5–10 s): At t=5s$t=5 s$, the communication link between DG#2 and DG#3 is failed. Then, at t=4s$t=4 s$, Load#2 is returned to its initial value, 20 Ω. Finally, Load#4 is disconnected from the system at t=8s$t=8 s$.The results for this study are illustrated in Figure 7. It is observed from Figure 7a,b, during different load conditions detailed in stage 1, the proposed controller can share the load among DGs in both poles in proportion to DGs' capacity. Besides this, the voltages at the DGs output as well as the main busbar have been satisfactorily kept balanced between two poles independent of load changes (see Figures 7c and 7d, respectively). As a result, the proposed scheme remains unaffected under different load changes with the ring‐shaped communication network, and guarantees both proportional power sharing and voltage balancing in the steady‐state.7FIGUREPerformance of the system in case of load changes and link‐failure. (a,b) The output currents of DGs in the upper and lower poles, respectively. (c) The output voltages of DGs. (d) The voltages of main busbarThen, a single link failure occurs in stage 2, that is, the communication channel between DG#2 and DG#3 is disconnected at t=5s$t=5 s$, which results in the obstruction of information exchange between these two DGs. As seen from the simulation results at this point, proportional power sharing and voltage balancing are maintained unaffected, although the link failure has occurred (see Figure 7). To demonstrate the performance of the proposed strategy after a link failure, Load#2 and Load#4 are consecutively returned to their initial condition at t=6s$t=6 s$ and t=8s$t=8 s$. As seen in Figure 7, the performance of the proposed control strategy is not disturbed, and proportional power sharing and voltage balancing are achieved in all conditions, which clearly proves the system resiliency against link failure. However, the transient response in stage 2 is slightly slow in comparison with stage 1. This is because any link failure restricts information transmission. As a result, the proposed strategy is robust against the load changes and communication link failure.Plug‐in and plug‐out featureIn this study, the plug‐in and plug‐out feature of the proposed controller is investigated. The test system undergoes the following three stages:Stage 1 (0–3 s): At t=0s$t=0 s$, the proposed controller is activated, and the PV system injects 16 kW to the upper pole.Stage 2 (3–6 s): At t=3s$t=3 s$, DG#3 is plugged out.Stage 3 (6–9 s): At t=6s$t=6 s$, DG#3 is plugged in.The simulation results are shown in Figure 8. From Figure 8a,b, it is observed that when DG#3 is plugged out from the system in stage 2, the output currents of other DGs change to satisfy demand‐generation equality. Then the proposed strategy operates to share the load among the remaining three DGs in proportion to their capacity in both poles as seen in Figures 8a and 8b, respectively. Then, when DG#3 is plugged back in stage 3, the load is proportionally shared again among all the four DGs in both upper and lower poles, as seen in Figures 8a and 8b, respectively. Besides establishing proportional power sharing among DGs when DG#3 is plugged out from the system in stage 2 and returned in stage 3, the voltages are desirably maintained balanced in the main busbar as seen in Figure 8d. Due to page limitation, only the output voltage of DG#3 is shown in Figure 8c. This study clearly demonstrates the plug‐in and plug‐out feature of the proposed controller.8FIGUREPlug‐in and plug‐out capability. (a,b) The output currents of DGs in the upper and lower poles, respectively. (c) The output voltages of DG#3. (d) The voltages of main busbarPerformance evaluation with larger additive noiseIn this case study, the resiliency of the proposed control strategy against large additive noise is testified. It is assumed that the test system undergoes the following stages:Stage 1 (0–2 s): At t=0s$t=0 s$, the proposed controller is activated, and the PV system injects 16 kW to the upper pole.Stage 2 (2–4 s): At t=2s$t=2 s$, the variance of the additive noise is changed from σ2=0.1$\sigma ^2= 0.1$ to σ2=0.5$\sigma ^2= 0.5$.Stage 3 (4–6 s): At t=4s$t=4 s$, Load#2 is changed from 30Ω to 40Ω between both positive‐ and negative‐pole to neutral point.Stage 4 (6–8 s): At t=6s$t=6 s$, Load#4 is connected between the positive and negative poles of the system.From Figure 9a–c, it is seen that even if the additive noise interference is multiplied, the voltage of poles becomes stably balanced, and proportional current sharing among sources in both poles is achieved. Figure 10a,b shows the measured current signals, which are corrupted with high‐frequency white noise.9FIGUREPerformance of the system in case of change of additive noise parameters. (a,b) The output currents of DGs in the upper and lower poles, respectively. (c) The output voltages of DGs10FIGUREDepiction of measured currents affected by high‐frequency white noise. (a) The measured current signals in the upper poles. (b) The measured current signals in the lower polesComparison with an existing method in [22]In this case, the proposed control strategy is compared with the most significant available method in the literature [22], that is, equivalent droop using the communication network. The simulation stage proceeds as follows:Stage 1 (0–2 s): At t=0s$t=0 s$, the system is controlled using the conventional droop controller, and the PV system does not inject any power to the system, that is, PPV=0$P_{PV} =0$ kW.Stage 2 (2–4 s): At t=2s$t=2 s$, the PV system begins to inject 16kW$16 kW$ to the upper pole.Stage 3 (4–6 s): At t=4s$t=4 s$, the proposed controller in [22] is engaged.Stage 4 (6–8 s): At t=6s$t=6 s$, a 20 ms$ms$ communication delay and additive noise with σ2=0.1$\sigma ^2= 0.1$ are added in the communication links.The results for this case study are shown in Figure 11. The performance of the system in stage 1 and 2 are like the performance of the system in stage 1 and 2 of Case A. When the proposed controller is activated in stage 3, as seen from Figure 11c, the voltages of DGs are restored to the nominal value which results in a balanced voltage condition. Consequently, voltage at the main busbar is also restored to the desirable value as seen in Figure 11d. However, this strategy is unable to share the power precisely among the DGs in proportion to the capacity of DGs in each pole as seen in Figure 11a,b. In contrast, as discussed in Section 4.1, our proposed strategy can simultaneously balance the voltage and share the power accurately among the DGs. Besides this, when the communication delay and additive noise are added in stage 4, from the results shown in Figure 11, it is clear that the control scheme in [22] cannot continue to operate with time delay and additive noise and is in fact unstable. In contrast, as seen in the previous simulation studies, our proposed strategy has a desirable performance with the existence of communication time delay and additive noise. Moreover, since this strategy needs a knowledge update for the system's information such as the number of DGs, it can not support plug‐in and play‐out feature, unlike our proposed strategy. This obviously demonstrates the advantages of our proposed strategy compared with the previously‐published ones in [22].11FIGUREPerformance assessment of the proposed equivalent droop using the communication network [22]. (a,b) The output currents of DGs in the upper and lower poles, respectively. (c) The output voltages of DGs. (d) The voltages of main busbarCONCLUSIONHere, a new combined control structure has been presented for a bipolar DC microgrid, which is a combination of a distributed consensus‐based cooperative method and a washout filter‐based method. It not only precisely guarantees proportional power sharing, but also satisfactorily balances the voltage of poles. To achieve an accurate power sharing, a distributed consensus‐based cooperative control strategy with consideration of both communication time delays and noises is utilized in the first level, under which each DG participates in the power sharing in proportion to its own capacity ratio. The voltage balancing problem is solved by utilizing the dynamic feedback incorporating washout filters in the second level. This level of proposed controller eliminates the steady‐state deviation in the output voltage amplitude without using any communication links and additional control loops. Along with the two main objectives mentioned above, supporting the plug‐and‐play feature of microgrid and also, benefiting from robustness, resiliency, expandability, and flexibility are the key features of the presented algorithm in comparison to the existing control methods. Finally, the simulation studies validate the effectiveness of the proposed control strategy, in terms of providing accurate power sharing and voltage balancing with sparse communication traffic, robustness to load changes, resilience to communication link failure, and feature of plug‐in and plug‐out of DGs.CONFLICT OF INTERESTThe authors have declared no conflict of interest.REFERENCESXing, L., Guo, F., Liu, X., Wen, C., Mishra, Y., Tian, Y.‐C.: Voltage restoration and adjustable current sharing for DC microgrid with time delay via distributed secondary control. IEEE Trans. Sustainable Energy 12(2), 1068‐1077 (2021)Yuan, Q.‐F., Wang, Y.‐W., Liu, X.‐K., Lei, Y.: Distributed fixed‐time secondary control for dc microgrid via dynamic average consensus. IEEE Trans. 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Power Syst. 34(5), 3573–3581 (2019)APPENDIXProofThe theorem 3.3 is proved as following. Assume that δi=ii−i0$\delta _i = i_i- i_0$, i=1,2,…,N$i=1,2,\ldots,N$. Vector y defines as [i1T,…,iNT]T$[i_1^T,\ldots,i_N^T]^T$, and vector δ defines as [δ1T,…,δNT]T$[\delta _1^T,\ldots,\delta _N^T]^T$. The closed‐loop network dynamics can be achieved with the equations (8) and as follows:25di(t)=(IN⊗A)i(t)dt−(L⊗K)(i(t−τ1)dt+dM1),\begin{align} di(t)= (I_N \otimes A) i(t) dt - (L \otimes K) (i(t-\tau _1)dt + dM_1), \end{align}where L is graph Laplacian matrix and K represents a positive matrix to be designed. τ1,τ2$\tau _1, \tau _2$ are the time delays. Notice that ⊗ shows the Kronecker product, A=1$A=1$, and26M1=∑i,j=1Naijσji∫0t[Si,j⊗K]δ(s−τ2)dwji(s),\begin{align} M_1 = \sum _{i,j=1}^{N} a_{ij} \sigma _{ji} \int _0^{t} [S_{i,j} \otimes K] \delta (s- \tau _2) dw_{ji}(s), \end{align}Si,j=[sk]N×N$S_{i,j} = [s_{k}]_{N \times N}$ indicates N×N$N \times N$ matrix in which sii=−aij,sij=aij$s_{ii} = -a_{ij}, s_{ij} = a_{ij}$ and all other elements being zero. aij${a}_{ij}$ represents the elements of the communication digraph adjacency matrix. By the definition of δ(t)$\delta (t)$, next27dδ(t)=(IN⊗A)δ(t)dt−(L⊗K)(δ(t−τ1)dt+dM2),\begin{align} d \delta (t)= (I_N \otimes A) \delta (t) dt - (L \otimes K) (\delta (t-\tau _1)dt + dM_2), \end{align}where28M2=−∑i=1Naijσ0i∫0t[S¯i⊗K]δ(s−τ2)dw0i(s),\begin{align} M_2 = - \sum _{i =1}^{N} a_{ij} \sigma _{0i} \int _0^{t} [\bar{S}_i \otimes K] \delta (s- \tau _2) dw_{0i}(s), \end{align}S¯i=[sk]N×N$\bar{S}_i = [s_{k}]_{N \times N}$ is an N×N$N \times N$ matrix with nii=bi$n_{ii} = b_i$ and the whole other elements are zero. Let δ¯(t)=(ϕ−1⊗In)δ(t)$\bar{\delta }(t) = (\phi ^{-1} \otimes I_n) \delta (t)$, where ϕ is a positive definite matrix. Then we have29dδ¯(t)=A0δ¯(t)dt+A1δ¯(t−τ1)dt+dM3,\begin{align} d \bar{\delta } (t) = A_0 \bar{\delta } (t) dt + A_1 \bar{\delta } (t- \tau _1) dt + d M_3, \end{align}where A0=IN⊗A$A_0 = I_N \otimes A$, A1=−Γ0⊗K$A_1 = - \Gamma _0 \otimes K$, where Γ0 is a positive gain,30M3=∑i,j=1Nσji∫0t[(ϕTSi,jϕ)⊗K]δ¯(s−τ2)dwji(s)−∑i=1Nσ0i∫0t[(ϕTSi¯ϕ)⊗K]δ¯(s−τ2)dw0i(s).\begin{align} M_3 &= \sum _{i,j=1}^{N} \sigma _{ji} \int _0^{t} [(\phi ^T S_{i,j} \phi ) \otimes K] \bar{\delta } (s- \tau _2) dw_{ji}(s) \nonumber \\ &\quad - \sum _{i =1}^{N} \sigma _{0i} \int _0^{t} [ (\phi ^T \bar{S_i} \phi ) \otimes K] \bar{\delta } (s- \tau _2) dw_{0i}(s). \end{align}Let us define P¯=IN⊗P$\bar{P} = I_N \otimes P$, also, we have31⟨M3,PM3⟩(t)=∑i,j=1Nσji2∫0t[δ¯T(s−τ2)((ϕTSi,jϕ)T(ϕTSi,jϕ))⊗(K)TPK]δ¯(s−τ2)ds+∑i=1Nσ0i∫0tδ¯T(s−τ2)[(ϕTS¯iϕ)⊗(K)TPK]δ¯(s−τ2)ds.\begin{align} & \langle M_3, P M_3 \rangle (t) = \sum _{i,j=1}^{N} \sigma _{ji}^2 \int _0^{t} [\bar{\delta } ^T (s- \tau _2) ((\phi ^T S_{i,j} \phi )^T \nonumber \\ & (\phi ^T S_{i,j} \phi )) \otimes (K)^T PK ] \bar{\delta } (s- \tau _2) ds + \sum _{i =1}^{N} \sigma _{0i} \nonumber \\ & \int _0^{t} \bar{\delta } ^T (s- \tau _2) [(\phi ^T \bar{S}_i \phi ) \otimes (K)^T PK ] \bar{\delta } (s- \tau _2) ds. \end{align}It is noteworthy that, ∑i,j=1N=Si,jTSi,j=2L$\sum _{i,j=1}^{N} = S_{i,j}^T S_{i,j} = 2L$ and ∑i=1NS¯iTS¯i=b$ \sum _{i =1}^{N} \bar{S}_i ^T \bar{S}_i = b$. Accordingly, ∑i,j=1N(ϕTSi,jϕ)T(ϕTSi,jϕ)=2ϕTLϕ$\sum _{i,j=1}^{N} (\phi ^T S_{i,j} \phi )^T (\phi ^T S_{i,j} \phi ) = 2 \phi ^T L \phi$ and ∑i=1N(ϕTS¯iϕ)T(ϕTS¯iϕ)=ϕTD0ϕ$ \sum _{i =1}^{N} (\phi ^T \bar{S}_i \phi )^T (\phi ^T \bar{S}_i \phi ) = \phi ^T D_0 \phi$ are obtained. Hence, we have32⟨M3,PM3⟩(t)≤2σ¯2δ¯T(s−τ2)Dδ¯(s−τ2)dt,\begin{align} \langle M_3, P M_3 \rangle (t) \le 2 \bar{\sigma }^2 \bar{\delta }^T (s- \tau _2) D \bar{\delta } (s- \tau _2) dt, \end{align}where D = Γ0⊗(KTPK)$ \Gamma _0 \otimes (K^T PK)$. By using conditions of Theorem 1, under K=kPIm+TP$K = \dfrac{kP}{I_m+ TP}$, A¯TP¯+P¯A¯+(A¯TP¯A¯+A1TP¯A1)τ1+σ¯2D<0$\bar{A}^T \bar{P} + \bar{P} \bar{A} + (\bar{A}^T \bar{P} \bar{A}+ A_1^T \bar{P} A_1) \tau _1 + \bar{\sigma }^2 D &lt; 0$ is obtained, in which A¯=A0+A1$\bar{A} = A_0 + A_1$. We note that it can be guaranteed by33WiTP+PWi+WiTPWiτ1+(λi2τ1+2λiN−1Nσ¯2)KTPK<0,\begin{align} &W_i^T P+ P W_i + W_i ^T P W_i \tau _1 + (\lambda _i ^2 \tau _1 + 2 \lambda _i \dfrac{N-1}{N} \bar{\sigma }^2) \nonumber \\ &K^T PK &lt; 0, \end{align}where Wi=A−λiK$W_i = A- \lambda _i K$. By using the following inequality34(z+h)TQ(z+h)≤2zTQz+2hTQh,z,h∈Rn,Q>0\begin{align} (z+h)^TQ(z+h) \le 2 z^T Q z + 2h^T Q h,\nobreakspace \nobreakspace z,h \in R^n,\nobreakspace \nobreakspace Q &gt; 0 \end{align}hence,35WiTPWi≤2ATPA+2λi2KTBTPBK.\begin{align} W_i ^T P W_i \le 2 A^T PA + 2 \lambda _i ^2 K^T B^T PBK. \end{align}Inserting K=k(I+P)−1P$K = k(I+ P)^{-1}P$ and (35) into (33), (33) is assured by36Γi=ATP+P+2τ1P−ζiP(Im+P)−1P\begin{align} \Gamma _i &= A^TP+P+ 2 \tau _1 P - \zeta _i P (I_m+ P)^{-1} P \end{align}where ζi=2kλi−(2N−1Nσ¯2+3λNτ1)λik2$\zeta _i = 2k \lambda _i - (2 \dfrac{N-1}{N} \bar{\sigma }^2 + 3 \lambda _N \tau _1) \lambda _i k^2$ and P is the solution to (10). According to the equation (23) in [29], we have37Γi<(2α−ζi)P2(I+P),\begin{align} \Gamma _i &lt; (2 \alpha - \zeta _i) \frac{ P^{2}}{(I+ P)} , \end{align}where for k∈(k̲,k¯)$k \in (\underline{k}, \bar{k})$, (2α−ζi)<0$(2 \alpha - \zeta _i) &lt;0$. Hence, from (37), Γi<0$\Gamma _i &lt; 0$ is concluded. Hence, according to the Theorem 2.2 in [29], E∥δ(t)¯∥2≤C0e−γ0t$E \parallel \bar{\delta (t)} \parallel ^2 \le C_0 e^{-\gamma _0 t}$, limsupt→∞1tlog∥δ(t)¯∥≤−γ02$lim sup_{t \rightarrow \infty } \dfrac{1}{t} log \parallel \bar{\delta (t)} \parallel \le - \dfrac{\gamma _0}{2}$, a.s., where C0 and γ0 are positive constants. Consequently, the proposed strategy indicates the mean square and approximately sure consensus.□$\Box$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "IET Generation, Transmission & Distribution" Wiley

A resilient bi‐level control strategy for power sharing and voltage balancing in bipolar DC microgrids

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Wiley
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© 2022 The Institution of Engineering and Technology.
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1751-8695
DOI
10.1049/gtd2.12530
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Abstract

INTRODUCTIONBackground and motivationProliferation of DC‐nature renewable energy sources, along with the fast increase of DC loads in the modern electrical grid systems have motivated the idea of using DC power systems [1–3]. The DC electrical grids are categorized into two types: unipolar power systems and bipolar power systems [4]. Compared to the unipolar systems, bipolar systems provide more flexibility by allowing the connection of distributed generators (DGs) and loads into two voltage levels, that is, (1) between neutral point and either positive or negative pole, and (2) between positive pole and negative pole. Besides the voltage‐related advantages mentioned above, the bipolar systems benefit from high reliability due to the independent operation of two poles, which allows the half capacity of the system to remain available in fault situations [5].On the other hand, bipolar DC systems face asymmetrical operations when the power distribution in the two poles is not identical [6]. Such asymmetry puts the voltages of poles into an unbalanced condition, which consequently leads to power quality deterioration, voltage instability, and under a severe situation as the worst case, serious damage to the whole power conversion units [7]. As a result, introducing a solution capable of equalizing the DC‐bus voltages is indispensable in the bipolar DC microgrids. Addressing this challenge from the power control perspective (i.e. without any additional power electronic equipment) is the main motivation of this study.Literature reviewMost of the leading methods of voltage balancing are power electronics solutions, and there are very few works on addressing the problem from the power system perspective. The existing device‐level solutions to mitigate voltage imbalance include: (i) employing a balancing power electronic blocks called voltage balancer (VB), (ii) installation of distributed DC electric springs (DC‐ESs), and (iii) redistribution of unipolar loads.The VBs operate to make the positive pole voltage of the converter outlet equal to the negative pole voltage [8]. Different types of VB topologies and relevant control strategies have been proposed in previous studies [9–13]. It is worth mentioning that VBs are only responsible for voltage‐balancing control tasks regardless of voltage value at the outlet of the converter, that is, they have no responsibility to restore the voltage of the system to the nominal value. Moreover, the VBs cannot function as a DC‐DC converter in applications such as photovoltaic systems and battery storage systems [14]. A DC‐ES‐based method for unbalanced voltage suppression in a bipolar DC systems is introduced in [15]. In this method, DC‐ES is connected in series with non‐critical load in each pole to maintain the voltage across the critical load in the nominal value. In the two aforementioned methods, suppression of unbalanced voltage is accomplished through additional power electronics equipment and only the local voltage quality is guaranteed. Redistribution of unipolar loads in order to mitigate voltage imbalance can be achieved by a centralized control approach [8, 16, 17]. In this approach, a full communication channel is required to collect the required information from each node and sent them to the control center. Then, the switch signal will be generated and sent to the commutation switch. Due to the centralized communication infrastructure and additional switching devices, this approach is not sufficiently cost‐effective [16]. Besides this, it is noteworthy that this method belongs to the energy management level rather than power management, which means the time scale in this method is large, and it cannot be implemented in the real‐time microgrid operation [8, 18].A different approach to voltage balancing is the use of three‐level DC‐DC converters (TLCs) [19]. TLCs have the abilities of both serving as a DC‐DC interface converter for DGs (e.g. photovoltaic systems, fast chargers, and battery storage systems) and voltage balancing control [14]. In contrast to the conventional VB, three‐level converter‐based voltage‐balancing control systems require no additional balancing circuits. Assuming the interface converter of DGs in a bipolar DC microgrid is a three‐level DC‐DC converter‐based VB, some control strategies have been proposed to address power sharing and voltage balancing tasks. In [20], the conventional droop control has been employed in a grid‐connected bipolar DC microgrid; however, it suffers from poor power sharing and voltage imbalance. To cope with the drawbacks of droop strategy, an improved droop control approach has been presented in [21]. Despite satisfactory operation, it uses complex and high bandwidth communication networks which requires information of all DGs, thus it will be vulnerable to a single point of failure. Tavakoli et al. [22] have proposed two control strategy for power sharing and voltage regulation of bipolar DC microgrids by using the controller area network (CAN). In these two networke‐based control strategy, the CAN bus is utilized to carry control signals and realize information sharing among DGs. Although the CAN‐based control strategy eliminates the need for a central control unite, it utilizes complex and all‐to‐all high bandwidth communication networks where any single communication failure renders the whole microgrid inoperable.More recently, a distributed cooperative control scheme for bipolar DC microgrids was proposed, by which the average voltages of the positive and negative poles of the system converge to the nominal value [23]. However, this paper only solves the unbalanced voltage problem while accurate power sharing of DGs is not taken into consideration. It is worth mentioning that the previously proposed communication‐based methods in [21–23] are susceptible to additive noise and time delays in practical communication networks which can impair the whole system functionality while their effects have not been taken into account. To the best of the authors' knowledge, there are no control strategies in the literature that establishes simultaneously accurate power sharing and voltage balancing in the bipolar DC microgrids, while considering practical issues such as line impedances, communication time delays, and noises. This is the main motivation of this study.Key contributions and organizationWithin the context alluded above, the aim of this study is to propose a resilient bi‐level control strategy to balance the voltage of poles, and share the load among DGs in proportion to their capacities. In the primary level, a fully distributed, failure‐resilient, consensus‐based cooperative control method with consideration of both communication time delays and noises is developed to achieve precise power‐sharing among DGs. In this method, only a sparse communication network is required to exchange local information among neighbouring DGs. In the secondary level, a washout filter‐based control strategy is developed to restore the voltage of each pole to the desired value, where it operates without any communication networks and extra control loops. Different from the previous works in the field of three‐level DC‐DC converter‐based bipolar DC microgrid control, the salient contributions of the proposed control scheme are outlined below:A fully distributed, noise‐resilient, robust, consensus‐based control method with consideration of communication delays is developed to achieve an accurate power sharing among DGs in a bipolar DC microgrid. One of the distinct features of this control loop compared to the existing communication‐based techniques in bipolar DC microgrids (e.g. see [21, 22]) is that it uses only neighbour‐to‐neighbour (not all‐to‐all) information over a sparse communication network where no single link failure can impair the normal function of the whole system. It is noteworthy that the proposed method is robust against cyber communication time delays and noises, while in the previous works [21–23], the communication links have been assumed ideal without any noise and time delay. Moreover, as opposed to extensive literature having proposed distributed consensus technique for DC microgrids, the developed strategy in this work requires only the output current state variable instead of two state variables (i.e. output voltage and current output). Therefore, the system communication burden is effectively reduced and the reliability is enhanced.A washout filter in feedback control is developed to restore the voltage of each pole to the desired value. It establishes voltage balancing between positive‐ and negative‐pole to the neutral point. This method operates without any communication links and additional control loops, which makes the controller simple to implement.Cooperation of the proposed consensus‐based power sharing method and washout filter‐based voltage regulator method simultaneously accomplishes proportional power sharing and voltage balancing, while the current control methods of bipolar DC microgrids (see [21–23]) are able to establish only voltage balancing or proportional power sharing, when the impedances of the lines are not negligible.In contrast with the existing communication‐based methods [21, 22], the proposed control strategy does not need prior knowledge of the global parameters such as the number of DGs. Therefore, it offers expandability, more flexibility, and plug‐in and plug‐out feature.BIPOLAR DC MICROGRID STRUCTURE AND CONVENTIONAL DROOP CONTROLA typical configuration of a bipolar DC microgrid is shown in Figure 1, which comprises three main elements: (i) renewable energy source interface converters (RESICs) that interconnect renewable sources to the bipolar DC terminals, (ii) loads , and (iii) TLCs that act as energy storage interface converter and voltage balancing in the system. Since in the island operation of microgrids only the energy storages, as dispatchable DGs of the system, are responsible for voltage regulation, the focus of this work is control of TLCs, whereas the other power supplies in the system including renewable resources are considered as variable current sources.1FIGURETypical structure of an islanded bipolar DC microgridBlock diagram of a bipolar bidirectional DC/DC boost converter which plays the role of voltage balancer and equipped with the conventional droop control method is illustrated in Figure 2. A bipolar DC microgrid with two DGs is illustrated in Figure 3, where each converter is simplified by applying the Thevenin equivalent model. It is noteworthy that in the whole article, subscript ′′m′′=u,l$^{\prime \prime }m^{\prime \prime }={u,l}$ stands for pole label. The expression of the droop control method is as follows:1Vm,i=V∗−Rm,iim,i\begin{equation} {V_{m,i}=V^*-R_{m,i} \; i_{m,i}} \end{equation}where Vm,i$V_{m,i}$ represents the output voltage of ith DG in pole ′′m′′$^{\prime \prime }m^{\prime \prime }$, V∗$V^*$ is the global reference voltage of both poles, im,i$i_{m,i}$ is the output current of ith DG in pole ′′m′′$^{\prime \prime }m^{\prime \prime }$, and Rm,i$R_{m,i}$ is the droop gain of ith DG in pole ′′m′′$^{\prime \prime }m^{\prime \prime }$. Notice that the subscript “i” indicates ith DG. In the following equations, everywhere the subscript of each variable modifies from “i” to “j”, which indicates the jth DG instead of ith DG.2FIGURESchematic of bipolar bidirectional DC/DC boost converter with conventional droop control method3FIGURESimplified model of a two‐DG bipolar DC microgrid in Thevenin formFrom Figure 3, the voltage of load in each pole ′′m′′$^{\prime \prime }m^{\prime \prime }$ can be derived as (2)2Vload,m=Vm,i*−Rm,iim,i−Rlinem,iim,iVload,m=Vm,j*−Rm,jim,j−Rlinem,jim,j,\begin{equation} \left\{ \def\eqcellsep{&}\begin{array}{c} {{V}_{\textit{load},m}={V}_{m,i}^{\ast}-{R}_{m,i}{i}_{m,i}-{R}_{\textit{lin}{e}_{m,i}}{i}_{m,i}}\\[6pt] {{V}_{\textit{load},m}={V}_{m,j}^{\ast}-{R}_{m,j}{i}_{m,j}-{R}_{\textit{lin}{e}_{m,j}}{i}_{m,j}}\end{array} \right., \end{equation}where Rlinem,i$R_{line_{m,i}}$ and Rlinem,j$R_{line_{m,j}}$ are the line resistance. Equations (3) is yielded from (2):3im,iim,j=Rm,j+Rlinem,jRm,i+Rlinem,i.\begin{equation} {\frac{i_{m,i}}{i_{m,j}} = \frac{R_{m,j} + R_{line_{m,j}}}{R_{m,i} + R_{line_{m,i}}}} .\end{equation}It is noteworthy that in a small DC microgrid system, the lines resistances have a low value. Considering this assumption and selecting the droop gains much larger than the line resistances, that is, Rm,i≫Rlinem,i,Rm,j≫Rlinem,j$ {{{R_{m,i} \gg R_{line_{m,i}}}}, {{R_{m,j} \gg R_{line_{m,j}}}}}$, the following equation is inferred from (3), which indicates a proportional power sharing among DGs in the mthe pole (i.e. in the both upper and lower poles):4im,iim,j≈Rm,jRm,i\begin{equation} {\frac{i_{m,i}}{i_{m,j}} \approx \frac{R_{m,j} }{R_{m,i}}} \end{equation}However, the above assumption is only acceptable for small systems while in the large systems due to the large value of lines resistances, (4) cannot be inferred [24]. Therefore, to achieve a proportional power sharing, the droop gains can be increased; however, the larger droop gain leads to more voltage deviation. Thus, the main objective of this study is to address a control strategy that simultaneously ensures accurate power sharing among DGs and restoring the voltage of each pole to the desired value.DESCRIPTION OF PROPOSED CONTROL SCHEMETo overcome the drawbacks of the conventional droop control, a combined control structure is proposed for proportional power sharing and voltage balancing in bipolar DC microgrids. The overall diagram of the proposed control structure is illustrated in Figure 4. Besides the local inner voltage control loop, the proposed control structure consists of two parts:proportional power sharing, andvoltage regulation.4FIGUREThe overall schematic diagram of the proposed control methodThe controller design is a combination of a distributed consensus‐based cooperative method and a washout filter‐based method.Proportional power sharingIn this section, a distributed consensus‐based cooperative control method with consideration of both communication time delays and noises is developed to synchronize the output currents of all DGs (in per‐unit) to a common value.1)Preliminaries of graph theoryIn order to model the communication network among DGs, a directed graph G=(V,E)$\mathcal {G=(V,E)}$ with vertices V$\mathcal {V}$={1,2,…,N}$=\lbrace 1, 2, \ldots, N\rbrace$ and edges E⊆V×V$\mathcal {E} \subseteq \mathcal {V} \times \mathcal {V}$ is introduced here. Each vertex denotes a DG. The edge (i,j)$(i,j)$ (pointing from j to i) indicates that vertex i can receive information from vertex j, with a weighting factor aij$a_{ij}$. The neighbours of vertex i are denoted by Ni$N_{i}$, which contains all the vertices which send information to that specific vertex, that is, j∈Ni$j\in N_{i}$ if (j,i)∈$(j,i)\in$E$\mathcal {E}$. The communication structure between the vertexes is explained by the adjacency matrix A$\mathcal {A}$=[aij]∈RN×N$=[a_{ij}]\in {R}^{N \times N}$, where aij=1$a_{ij}=1$ if there is an edge from vertex j to vertex i, and aij=0$a_{ij}=0$ otherwise. The graph Laplacian matrix is defined as L=D−A$\mathcal {L=D-A}$, where D$\mathcal {D}$ is called in‐degree matrix that is a diagonal matrix, that is, D$\mathcal {D}$=diag(di)⊆RN×N$=diag(d_{i})\subseteq {R}^{N \times N}$ with di=∑j∈Niaij$d_{i}=\sum _{j\in N_{i}} {a_{ij}}$.2)Basics of distributed cooperative controlThe scalar information state, xi$x_{i}$, is dedicated to each communication vertex, i. Each vertex requires to receive its neighbours information, j(j∈Ni)$j \: (j\in N_{i})$, via sparse communication network to update its state, xi$x_{i}$. The update rule is based on the consensus protocol for regulator synchronization problem as:5ẋi(t)=∑i=1naijxj(t)−xi(t)\begin{equation} {\dot{x}_{i}(t)=\sum _{i=1}^{n} {{a}_{ij}\Bigl (x_{j}(t)-x_{i}(t)\Bigr )}} \end{equation}where aij${a}_{ij}$ represents the elements of the communication digraph adjacency matrix.3)Design of distributed cooperative controller with communication time delays and noises for power sharingIn this part, based on the regulator synchronization problem protocol given in (5), a distributed cooperative control strategy with time delays and multiplicative noises in the communication layer is developed to achieve a proportional power sharing in the bipolar DC microgrid, as seen in Figure 4. Since the measured current signals are corrupted with a significant amount of high‐frequency noise, a distributed resilient control strategy in mean square consensus protocol is developed here to ensure the stability of the system. Therefore, to control the current of each pole of ith DG , that is, iu,i$i_{u,i}$ and il,i$i_{l,i}$, the following current sharing mismatch in each pole is calculated6um,i=K∑j∈Niem,ji(t),\begin{align} {u_{m,i}} = K \sum _{j \in N_{i}} e_{m,ji} (t), \end{align}notice that subscript ′′m′′$^{\prime \prime }m^{\prime \prime }$={u,l}$=\lbrace u, l\rbrace$ stands for the pole of the system, and the subscripts ′′i′′$^{\prime \prime }i^{\prime \prime }$ and ′′j′′$^{\prime \prime }j^{\prime \prime }$ represent the ith and jth DG, respectively. K represents a positive matrix to be designed. em,ji(t)$e_{m,ji}(t)$ is the state measurement of ith DG from its neighbour DG j7em,ji(t)=im,jpu(t−τ1)−im,ipu(t−τ1)+fji(im,jpu(t−τ2)−im,ipu(t−τ2))γji(t),j∈Ni,\begin{eqnarray} e_{m,ji} (t) &=& i_{m,j}^{pu} (t-\tau _1)-i_{m,i}^{pu} (t-\tau _1) \nonumber \\ && +\ f_{ji} (i_{m,j}^{pu} (t-\tau _2)-i_{m,i}^{pu} (t-\tau _2)) \gamma _{ji}(t), \ j \in N_i ,\qquad \end{eqnarray}where γji(t)$\gamma _{ji} (t)$ ∈R$\in R$ indicates the measurement noise; τ1,τ2$\tau _1, \tau _2$ are the time delays, and fji(.)$f_{ji}(.)$ represents the noise variance. As is seen em,ji(t)$e_{m,ji} (t)$ is composed of two segments. The first segment of protocol im,jpu(t−τ1)−im,ipu(t−τ1)$i_{m,j}^{pu} (t-\tau _1)-i_{m,i}^{pu} (t-\tau _1)$ indicates the deterministic term, and the second segment fji(im,jpu(t−τ2)−im,ipu(t−τ2))γji(t)$ f_{ji} (i_{m,j}^{pu} (t-\tau _2)-i_{m,i}^{pu} (t-\tau _2)) \gamma _{ji}(t)$ indicates the stochastic term. We assume that the noises are independent Gaussian white noises, and assure the following assumption.1AssumptionThe process noise γji(t)∈R$\gamma _{ji} (t) \in R$ assures ∫0tγji(t)(s)ds=ωji(t)$\int _{0}^{t} \gamma _{ji} (t)(s)ds = \omega _{ji} (t)$, t≥0$t \ge 0$, i=1,2,…,N$i=1,2,\ldots,N$, j∈Ni$j \in N_i$, where {ωji(t),i=1,2,…,N}$\lbrace \omega _{ji}(t), i=1,2,\ldots,N \rbrace$ points independent two‐dimensional Brownian motions defined on the complete probability space (Ω,F,P)$(\Omega , F, P)$ with a filtration Ft(t≥0)$F_t (t \ge 0)$ satisfying the usual conditions, namely, it is right continuous and increasing while F0 contains all P‐null sets.1Remark{λiu(A)}i$\lbrace \lambda _i^{u} (\mathcal {A} ) \rbrace _i$ indicates the unstable eigenvalues of A$\mathcal {A}$, that is, Re(λiu(A))≥0$ Re(\lambda _i^{u} (\mathcal {A} )) \ge 0$. We define λ0u=∑iRe(λiu(A))$\lambda _0^u = \sum _{i} Re(\lambda _i^{u} (\mathcal {A} ))$.The results of this part are summarized as below:1TheoremThe distributed cooperative control protocol8um,i=K∑j∈Niem,ji(t)em,ji(t)=im,jpu(t−τ1)−im,ipu(t−τ1)+fji(im,jpu(t−τ2)−im,ipu(t−τ2))γji(t),j∈Ni\begin{eqnarray} {u_{m,i}} &=& K \sum _{j \in N_{i}} e_{m,ji} (t) \nonumber \\ e_{m,ji} (t) &=& i_{m,j}^{pu} (t-\tau _1)-i_{m,i}^{pu} (t-\tau _1) \nonumber \\ && +\ f_{ji} (i_{m,j}^{pu} (t-\tau _2)-i_{m,i}^{pu} (t-\tau _2)) \gamma _{ji}(t), \quad j \in N_i \qquad \end{eqnarray}can ensure that the current is shared proportionally among DGs without regard to time‐delay and noise in communication links.Notice that the controller exists if Assumption 1 is valid, fji$f_{ji}$ = σji(i)≥0$\sigma _{ji} (i) \ge 0$, max(Re(λ(A)))>0$max (Re (\lambda (\mathcal {A}))) &gt; 0$, and 4λ0uσ¯2≤λ1$4 \lambda _0^{u} \bar{\sigma }^2 \le \lambda _{1}$.The upper bound of delays in the communication links can be obtained as9τ1*=min(12P,λ1−4σ¯(λ0u+ε)6(λ0u+ε)λN).\begin{align} \tau _1 ^{\ast } = min (\dfrac{1}{2 P }, \dfrac{\lambda _{1}-4 \bar{\sigma } (\lambda _0^{u}+\epsilon ) }{6(\lambda _0^{u}+\epsilon )\lambda _{N}}). \end{align}Notice that P>0$P&gt;0$ is designed by solving the Ricatti equation as:102P−2αP21+P+1=0,\begin{align} 2P- \dfrac{2 \alpha P^2}{1+P} +1=0, \end{align}with α∈(λ0u,λ0u+ε)$\alpha \in (\lambda _0^{u}, \lambda _0^{u}+\epsilon )$, ε∈(0,λ1−4σ¯2λ0u4σ¯2)$\epsilon \in (0, \dfrac{ \lambda _{1}-4 \bar{\sigma }^2 \lambda _0^{u}}{4\bar{\sigma }^2})$, and σ¯2=maxi=1,j=0Nσij2$\bar{\sigma }^2 = max_{i=1,j=0}^{N} \sigma _{ij}^2$.Eventually, the K in (8) can be obtained as11K=kP1+P,\begin{align} K = \dfrac{kP}{1+P} , \end{align}where k∈(k̲,k¯)$k \in (\underline{k}, \bar{k})$, k̲=[λ1−λ12−2αρ]ρ$\underline{k} = \dfrac{[\lambda _{1}- \sqrt {\lambda _{1} ^2-2 \alpha \rho ]}}{\rho }$, k¯=[λ1+λ12−2αρ]ρ$\bar{k} = \dfrac{[\lambda _{1}+\sqrt {\lambda _{1} ^2-2 \alpha \rho ]}}{\rho }$, and ρ=(2σ¯2+3λNτ1)λ1$\rho = (2 \bar{\sigma }^2+ 3\lambda _{N} \tau _1)\lambda _{1}$.The proof is given in Appendix.Then, the calculated current sharing mismatches, em,i${e}_{m,i}$, in each pole passes through a PI controller to generate the current sharing correction terms, δm,iv$\delta _{m,i}^v$ in each pole, as shown in Figure 4. This terms in each pole is added into the droop function (1) to eliminate the discrepancy between per‐unit current of converter i and per‐unit current of neighbour converters in the steady state, that is, limt→+∞em,i=0$ \lim _{t\rightarrow +\infty }e_{m,i}=0$, where i≠j$i\ne j$. Therefore, the conventional droop equation given in (1) is revised as:12Vm,i=Vm∗−Rm,iim,i+δm,iv.\begin{equation} {V_{m,i}=V_{m}^*-R_{m,i} \; i_{m,i}+\delta _{m,i}^v} .\end{equation}This adjustment reduces current mismatch among neighbour DGs, and ultimately among all DGs of the microgrid. Under normal operation, the per‐unit currents of all converters reach to a common value at the steady‐state, and consequently, the mismatch terms, em,i${e}_{m,i}$, become zero. However, voltage deviation due to the droop controller function is maintained.Voltage regulationInspired by previous applications of washout filters in addressing former challenges [25–27], a washout filter‐based control strategy for voltage balancing in bipolar DC microgrids is developed in this section. The proposed strategy works without any communication links and without any extra control loops.1) Design of washout filter‐based controller for voltage regulationTo eliminate the voltage deviation created by the droop resistor and restore the voltage amplitude to the rated value in each pole, a strategy based on the utilization of a washout filter in the feedback path is designed here. To this end, the equation (12) is written in the following form:13Vm,i−(Vm∗+δm,iv)+Rm,iim,i=0.\begin{equation} {V_{m,i}-(V_{m}^*+\delta _{m,i}^v)+R_{m,i} \; i_{m,i}=0} .\end{equation}By considering ΔVm,i=Vm,i−(Vm∗+δm,iv)$\Delta V_{m,i}=V_{m,i}-(V_{m}^*+\delta _{m,i}^v)$ and taking time derivative from (13), the following equation is achieved:14ddtΔVm,i+Rm,iddtim,i=0.\begin{equation} {\frac{d}{dt} \Delta V_{m,i} + R_{m,i} \frac{d}{dt} i_{m,i}=0} .\end{equation}Through Equation (14), proportional power sharing is ensured in the steady state. However, voltage regulation due to droop resistor can not be compensated. This can be accomplished by adding a factor of ΔVm,i$\Delta V_{m,i}$ to (14) as:15ddtΔVm,i+Rm,iddtim,i+kmΔVm,i=0.\begin{equation} {\frac{d}{dt} \Delta V_{m,i} + R_{m,i} \frac{d}{dt} i_{m,i}+k_{m} \Delta V_{m,i}=0} .\end{equation}Since the derivative terms of (15) are zero in the steady state (i.e. ddtΔVm,i=0$\frac{d}{dt} \Delta V_{m,i}=0$ and ddtim,i=0$\frac{d}{dt} i_{m,i}=0$), the ΔVm,i$\Delta V_{m,i}$ is forced to be zero in (15). Therefore, the voltage in pole ′′m′′$^{\prime \prime }m^{\prime \prime }$ is restored to its nominal values. Applying Laplace transform to (15), it is expressed as16sΔVm,i(s)+Rm,isim,i(s)+kmΔVm,i(s)=0,\begin{equation} {s \; \Delta V_{m,i}(s) + R_{m,i} \; s \; i_{m,i}(s)+k_{m} \Delta V_{m,i}(s)=0} , \end{equation}which can be rewritten as a transfer function as below17ΔVm,i(s)im,i(s)=−Rm,iss+km.\begin{equation} { \frac{\Delta V_{m,i}(s)}{ i_{m,i}(s)} =-\frac{ R_{m,i} \; s}{s+k_{m}}} . \end{equation}This equation can be stated in the more common form as18Vm,i=Vm∗+δm,iv−Rm,iss+kmim,i.\begin{equation} {V_{m,i}=V_{m}^*+\;\delta _{m,i}^v-\; \frac{ R_{m,i} \; s}{s+k_{m}} \; i_{m,i}} .\end{equation}Finally, as mentioned before, the subscript ′′m′′$^{\prime \prime }m^{\prime \prime }$ stands for the pole label, hence, the equation (18) can be written for each pole as:19Vu,i=Vu*+δu,iv−Russ+kuiu,iVl,i=Vl*+δl,iv−Rlss+klil,i.\begin{equation} \left\{ \def\eqcellsep{&}\begin{array}{c} {{V}_{u,i}={V}_{u}^{\ast}+{\delta}_{u,i}^{v}-\dfrac{{R}_{u}s}{s+{k}_{u}}{i}_{u,i}}\\[9pt] {{V}_{l,i}={V}_{l}^{\ast}+{\delta}_{l,i}^{v}-\dfrac{{R}_{l}s}{s+{k}_{l}}{i}_{l,i}}\end{array} \right.. \end{equation}As it can be seen in (18), the proposed method uses a washout filter in the feedback loop. A washout filter is a first‐order high‐pass filter (HPF), HHPF(s)=ss+km$ H_{HPF} (s)= \frac{s}{s+k_{m}}$, that washes out (rejects) steady state component of signal, while passing transient component of the signal [25]. Therefore, the large voltage deviation on the droop resistor can be compensated by using a washout filter in the feedback path at the steady state20Vu,i=Vu∗+δu,ivVl,i=Vl∗+δl,iv.\begin{equation} {\begin{cases} V_{u,i}=V_{u}^*+\;\delta _{u,i}^v\\ [8pt]{V_{l,i}=V_{l}^*+\;\delta _{l,i}^v} \end{cases}} .\end{equation}It is noteworthy that the terms δu,iv$\delta _{u,i}^v$ and δl,iv$\delta _{l,i}^v$ in (20) are very small which have little effect on the voltage deviation from the desired value.Notice that in order to guarantee the dynamic stability of the whole system and to keep the droop control effective, the restrictive conditions should be considered in designing the parameters of washout‐based control. According to Figure 4, we have:21im,i=HLPF(s)Im,i=ωcs+ωcIm,i,\begin{equation} {i_{m,i}=\; H_{LPF} (s)\; I_{m,i}= \; \frac{ \omega _{c}}{s+\omega _{c}} \; I_{m,i}} ,\end{equation}where Im,i$I_{m,i}$ is the instantaneous current, im,i$i_{m,i}$ is the filtered current, HLPF(s)$ H_{LPF} (s)$ is the low‐pass filter (LPF) transfer function, and ωc$ \omega _{c}$ is the cut‐off frequency of the low‐pass filter. Considering (21), we can rewrite (18) as22Vm,i=Vm∗+δm,iv−Rm,iss+kmωcs+ωcIm,i.\begin{equation} {V_{m,i}=V_{m}^*+\;\delta _{m,i}^v-\;R_{m,i} \; \frac{s}{s+k_{m}} \; \left(\frac{ \omega _{c}}{s+\omega _{c}} \; I_{m,i}\right)} . \end{equation}As we have assumed HHPF(s)=ss+km$ H_{HPF} (s)= \frac{s}{s+k_{m}}$ and HLPF(s)=ωcs+ωc$H_{LPF} (s) = \frac{ \omega _{c}}{s+\omega _{c}}$, (22) can be written as23Vm,i=Vm∗+δm,iv−Rm,iHHPF(s)HLPF(s)Im,i.\begin{equation} {V_{m,i}=V_{m}^*+\;\delta _{m,i}^v-\;R_{m,i} \; H_{HPF} (s) \; H_{LPF} (s) \; I_{m,i}} .\end{equation}From (23), it is seen that an HPF and an LPF have been cascaded and formed a band‐pass filter (BPF). Thus, the parameters of the washout filter‐based controller are influenced by the cut‐off frequency of BPF. In this way, as discussed in [27], stable and appropriate operation of the developed washout filter‐based controller in the secondary level is achieved, when the cut‐off frequency of the HPF satisfies the following condition:24km<ωc.\begin{equation} {k_{m}&lt;\omega _{c}} .\end{equation}Equation (23) determines a restrictive condition in the parameter designing procedure of washout filter‐based controller. If the cut‐off frequency restraint condition in (23) does not consider, the output current signals Im,i$I_{m,i}$ which pass through ill‐conditioned BPF will be oscillating; consequently, the system becomes unstable.SIMULATION RESULTSTo illustrate the effectiveness of the proposed control structure shown in Figure 4, the bipolar DC microgrid depicted in Figure 5 is simulated in five case studies. The system electrical and control parameters are given in Table 1. A ring‐shape communication topology in the cyber‐layer is designed, as seen in Figure 5, which keeps its own connectivity with any communication failure or converter failure. The rated voltage level of the test system is set as ±400 V. The communication time delay quantity is set as τ=20ms$\tau = 20 ms$. Moreover, the additive noise with σ2=0.1$\sigma ^{2}=0.1$ is considered for communication links.5FIGUREThe structure of the simulation test system with implemented communication networks1TABLEParameters of the systemItemsParameterValuePVPmax$_{max}$20 kWPower ratings of DGsPDG#1$P_{DG \#1}$ and PDG#2$P_{DG \#2}$7 kWPDG#3$P_{DG \#3}$ and PDG#4$P_{DG \#4}$14 kWVin$_{in}$200 VVout$_{out}$2× 400 VConverter parametersLm , Ls134 μH, 100 μHrL$r_{L}$ , rc$r_{c}$0.2 Ω, 0.001 ΩC1,C2$C_{1}, C_{2}$200 μFfsw$_{sw}$20 kHzLine#1$\#1$R=0.1 Ω, L=70 μHLine#2$\#2$R=0.15 Ω, L=100 μHTransmission linesLine#3$\#3$R=0.1 Ω, L=70 μHLine#4$\#4$R=0.12 Ω, L=90 μHLine#5$\#5$R=0.05 Ω, L=50 μHLoad#1$\#1$R=40 ΩLoadsLoad#2$\#2$R=30 ΩLoad#3$\#3$R=200 ΩLoad#4$\#4$R=100 ΩRu,Rl$R_{u}, R_{l}$ (DG#1$\#1$ & DG#2$DG\#2$)4 V/A, 4 V/AControl parametersRu,Rl$R_{u}, R_{l}$ (DG#3$\#3$ & DG#4$DG\#4$)2 V/A, 2 V/Aku,kl$k_{u}, k_{l}$4K11.52ωc$\omega _{c}$50πPerformance assessmentIn this case, the performance of the proposed controller is studied in comparison with the conventional droop control. The test system undergoes the following three stages:Stage 1 (0‐3 s): At t=0s$t=0 s$, the system is controlled using the conventional droop controller, and the PV system does not inject any power to the system, that is, PPV=0kW$P_{PV} =0 kW$.Stage 2 (3–6 s): At t=3s$t=3 s$, the PV system begins to inject 16kW$16 kW$ to the upper pole.Stage 3 (6–9 s): At t=6s$t=6 s$, the proposed controller is engaged.The simulation results are illustrated in Figure 6. During stage 1, when only the conventional droop control is applied, the voltage of DGs in both positive and negative pole has deviated severely from the desired value ±400V$\pm 400 V$, as seen in Figure 6e–h. The voltage of the main busbar is also shown in Figure 6d, where the voltage deviation due to the line resistances is greater than the output voltage of DGs. Moreover, although the droop gains have been designed in proportion to the DGs' capacity ratio, the effect of the lines' impedances has incapacitated the droop mechanism which results in an imprecise power sharing in both positive and negative poles, respectively, as shown in Figures 6b and 6c.6FIGUREComparative studies of the conventional droop control and the proposed controller. (a) The output power of PV. (b,c) The output currents of DGs in the upper and lower poles, respectively. (d) The voltages of main busbar. (e–h) The output voltages of DG#1, DG#2, DG#3 and DG#4, respectivelyThen, the PV system begins to inject 16 kW to the network in Stage 2, as seen in Figure 6a. In this stage, the generated power by the PV system is higher than the load demand in the upper pole. Hence, the surplus power is transferred to the lower pole by means of the bipolar boost converters. However, a precise load‐sharing among DGs is not met owing to the negative impacts of line impedances in the upper and lower poles as seen in Figures 6b and 6c, respectively. Not only that, but the impacts of droop controllers lead to an unacceptable voltage discrepancy between voltages of the upper and lower poles of DGs (see Figure 6e–h). Due to the voltage drop over the line impedances, the voltage discrepancy between the voltage of the upper and lower poles in the main busbar, that is, 29 V, is more pronounced (see Figure 6d). Such a significant voltage discrepancy between the upper and lower poles is unacceptable and should be compensated.The proposed controller is activated in stage 3. Consequently, the voltage of each pole is restored extremely close to the desired reference value so that the voltage discrepancy between two poles at the output of all DGs almost reaches to zero (see Figure 6e–h). The voltage difference in the main busbar voltages is decreased to 3V, which is very satisfactory in comparison with 29V in the conventional droop method, as seen in Figure 6d. It is worth mentioning that this voltage discrepancy in the main busbar is caused due to the insignificant voltage drop on the DC distribution line. This is because the proposed method is a source supply‐side method and unlike the load‐side methods which use additional devices [15], does not have direct access to control the main busbar. Furthermore, Figures 6b and 6c show the output current of the upper and lower terminals of sources, respectively. As it is observed, the proposed controller can ensure proper current sharing among sources in both poles, where DG#3 and DG#4 carry twice the current as DG#1 and DG#2. The zoom‐in simulation results of Figure 6b,c clearly illustrate the accuracy of the proposed controller in comparison with the conventional droop controller.Robustness against load variations and a link‐failureFrom the perspective of robustness, two criteria can be evaluated, namely, robustness against load changes and resiliency to a single link failure. To assess this feature of the proposed control strategy, the system goes through the following 2 stages as follows:Stage 1 (0–5 s): At t=0s$t=0 s$, the proposed controller is activated with the original ring‐shape communication topology, and the PV system injects 16 kW to the upper pole. At t=2s$t=2 s$, Load#2 is changed from 30 to 40Ω between both positive‐ and negative‐pole to neutral point. Besides this, at t=4s$t=4 s$, Load#4 is connected between the positive and negative poles of the system.Stage 2 (5–10 s): At t=5s$t=5 s$, the communication link between DG#2 and DG#3 is failed. Then, at t=4s$t=4 s$, Load#2 is returned to its initial value, 20 Ω. Finally, Load#4 is disconnected from the system at t=8s$t=8 s$.The results for this study are illustrated in Figure 7. It is observed from Figure 7a,b, during different load conditions detailed in stage 1, the proposed controller can share the load among DGs in both poles in proportion to DGs' capacity. Besides this, the voltages at the DGs output as well as the main busbar have been satisfactorily kept balanced between two poles independent of load changes (see Figures 7c and 7d, respectively). As a result, the proposed scheme remains unaffected under different load changes with the ring‐shaped communication network, and guarantees both proportional power sharing and voltage balancing in the steady‐state.7FIGUREPerformance of the system in case of load changes and link‐failure. (a,b) The output currents of DGs in the upper and lower poles, respectively. (c) The output voltages of DGs. (d) The voltages of main busbarThen, a single link failure occurs in stage 2, that is, the communication channel between DG#2 and DG#3 is disconnected at t=5s$t=5 s$, which results in the obstruction of information exchange between these two DGs. As seen from the simulation results at this point, proportional power sharing and voltage balancing are maintained unaffected, although the link failure has occurred (see Figure 7). To demonstrate the performance of the proposed strategy after a link failure, Load#2 and Load#4 are consecutively returned to their initial condition at t=6s$t=6 s$ and t=8s$t=8 s$. As seen in Figure 7, the performance of the proposed control strategy is not disturbed, and proportional power sharing and voltage balancing are achieved in all conditions, which clearly proves the system resiliency against link failure. However, the transient response in stage 2 is slightly slow in comparison with stage 1. This is because any link failure restricts information transmission. As a result, the proposed strategy is robust against the load changes and communication link failure.Plug‐in and plug‐out featureIn this study, the plug‐in and plug‐out feature of the proposed controller is investigated. The test system undergoes the following three stages:Stage 1 (0–3 s): At t=0s$t=0 s$, the proposed controller is activated, and the PV system injects 16 kW to the upper pole.Stage 2 (3–6 s): At t=3s$t=3 s$, DG#3 is plugged out.Stage 3 (6–9 s): At t=6s$t=6 s$, DG#3 is plugged in.The simulation results are shown in Figure 8. From Figure 8a,b, it is observed that when DG#3 is plugged out from the system in stage 2, the output currents of other DGs change to satisfy demand‐generation equality. Then the proposed strategy operates to share the load among the remaining three DGs in proportion to their capacity in both poles as seen in Figures 8a and 8b, respectively. Then, when DG#3 is plugged back in stage 3, the load is proportionally shared again among all the four DGs in both upper and lower poles, as seen in Figures 8a and 8b, respectively. Besides establishing proportional power sharing among DGs when DG#3 is plugged out from the system in stage 2 and returned in stage 3, the voltages are desirably maintained balanced in the main busbar as seen in Figure 8d. Due to page limitation, only the output voltage of DG#3 is shown in Figure 8c. This study clearly demonstrates the plug‐in and plug‐out feature of the proposed controller.8FIGUREPlug‐in and plug‐out capability. (a,b) The output currents of DGs in the upper and lower poles, respectively. (c) The output voltages of DG#3. (d) The voltages of main busbarPerformance evaluation with larger additive noiseIn this case study, the resiliency of the proposed control strategy against large additive noise is testified. It is assumed that the test system undergoes the following stages:Stage 1 (0–2 s): At t=0s$t=0 s$, the proposed controller is activated, and the PV system injects 16 kW to the upper pole.Stage 2 (2–4 s): At t=2s$t=2 s$, the variance of the additive noise is changed from σ2=0.1$\sigma ^2= 0.1$ to σ2=0.5$\sigma ^2= 0.5$.Stage 3 (4–6 s): At t=4s$t=4 s$, Load#2 is changed from 30Ω to 40Ω between both positive‐ and negative‐pole to neutral point.Stage 4 (6–8 s): At t=6s$t=6 s$, Load#4 is connected between the positive and negative poles of the system.From Figure 9a–c, it is seen that even if the additive noise interference is multiplied, the voltage of poles becomes stably balanced, and proportional current sharing among sources in both poles is achieved. Figure 10a,b shows the measured current signals, which are corrupted with high‐frequency white noise.9FIGUREPerformance of the system in case of change of additive noise parameters. (a,b) The output currents of DGs in the upper and lower poles, respectively. (c) The output voltages of DGs10FIGUREDepiction of measured currents affected by high‐frequency white noise. (a) The measured current signals in the upper poles. (b) The measured current signals in the lower polesComparison with an existing method in [22]In this case, the proposed control strategy is compared with the most significant available method in the literature [22], that is, equivalent droop using the communication network. The simulation stage proceeds as follows:Stage 1 (0–2 s): At t=0s$t=0 s$, the system is controlled using the conventional droop controller, and the PV system does not inject any power to the system, that is, PPV=0$P_{PV} =0$ kW.Stage 2 (2–4 s): At t=2s$t=2 s$, the PV system begins to inject 16kW$16 kW$ to the upper pole.Stage 3 (4–6 s): At t=4s$t=4 s$, the proposed controller in [22] is engaged.Stage 4 (6–8 s): At t=6s$t=6 s$, a 20 ms$ms$ communication delay and additive noise with σ2=0.1$\sigma ^2= 0.1$ are added in the communication links.The results for this case study are shown in Figure 11. The performance of the system in stage 1 and 2 are like the performance of the system in stage 1 and 2 of Case A. When the proposed controller is activated in stage 3, as seen from Figure 11c, the voltages of DGs are restored to the nominal value which results in a balanced voltage condition. Consequently, voltage at the main busbar is also restored to the desirable value as seen in Figure 11d. However, this strategy is unable to share the power precisely among the DGs in proportion to the capacity of DGs in each pole as seen in Figure 11a,b. In contrast, as discussed in Section 4.1, our proposed strategy can simultaneously balance the voltage and share the power accurately among the DGs. Besides this, when the communication delay and additive noise are added in stage 4, from the results shown in Figure 11, it is clear that the control scheme in [22] cannot continue to operate with time delay and additive noise and is in fact unstable. In contrast, as seen in the previous simulation studies, our proposed strategy has a desirable performance with the existence of communication time delay and additive noise. Moreover, since this strategy needs a knowledge update for the system's information such as the number of DGs, it can not support plug‐in and play‐out feature, unlike our proposed strategy. This obviously demonstrates the advantages of our proposed strategy compared with the previously‐published ones in [22].11FIGUREPerformance assessment of the proposed equivalent droop using the communication network [22]. (a,b) The output currents of DGs in the upper and lower poles, respectively. (c) The output voltages of DGs. (d) The voltages of main busbarCONCLUSIONHere, a new combined control structure has been presented for a bipolar DC microgrid, which is a combination of a distributed consensus‐based cooperative method and a washout filter‐based method. It not only precisely guarantees proportional power sharing, but also satisfactorily balances the voltage of poles. To achieve an accurate power sharing, a distributed consensus‐based cooperative control strategy with consideration of both communication time delays and noises is utilized in the first level, under which each DG participates in the power sharing in proportion to its own capacity ratio. The voltage balancing problem is solved by utilizing the dynamic feedback incorporating washout filters in the second level. This level of proposed controller eliminates the steady‐state deviation in the output voltage amplitude without using any communication links and additional control loops. Along with the two main objectives mentioned above, supporting the plug‐and‐play feature of microgrid and also, benefiting from robustness, resiliency, expandability, and flexibility are the key features of the presented algorithm in comparison to the existing control methods. Finally, the simulation studies validate the effectiveness of the proposed control strategy, in terms of providing accurate power sharing and voltage balancing with sparse communication traffic, robustness to load changes, resilience to communication link failure, and feature of plug‐in and plug‐out of DGs.CONFLICT OF INTERESTThe authors have declared no conflict of interest.REFERENCESXing, L., Guo, F., Liu, X., Wen, C., Mishra, Y., Tian, Y.‐C.: Voltage restoration and adjustable current sharing for DC microgrid with time delay via distributed secondary control. IEEE Trans. Sustainable Energy 12(2), 1068‐1077 (2021)Yuan, Q.‐F., Wang, Y.‐W., Liu, X.‐K., Lei, Y.: Distributed fixed‐time secondary control for dc microgrid via dynamic average consensus. IEEE Trans. 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Power Syst. 34(5), 3573–3581 (2019)APPENDIXProofThe theorem 3.3 is proved as following. Assume that δi=ii−i0$\delta _i = i_i- i_0$, i=1,2,…,N$i=1,2,\ldots,N$. Vector y defines as [i1T,…,iNT]T$[i_1^T,\ldots,i_N^T]^T$, and vector δ defines as [δ1T,…,δNT]T$[\delta _1^T,\ldots,\delta _N^T]^T$. The closed‐loop network dynamics can be achieved with the equations (8) and as follows:25di(t)=(IN⊗A)i(t)dt−(L⊗K)(i(t−τ1)dt+dM1),\begin{align} di(t)= (I_N \otimes A) i(t) dt - (L \otimes K) (i(t-\tau _1)dt + dM_1), \end{align}where L is graph Laplacian matrix and K represents a positive matrix to be designed. τ1,τ2$\tau _1, \tau _2$ are the time delays. Notice that ⊗ shows the Kronecker product, A=1$A=1$, and26M1=∑i,j=1Naijσji∫0t[Si,j⊗K]δ(s−τ2)dwji(s),\begin{align} M_1 = \sum _{i,j=1}^{N} a_{ij} \sigma _{ji} \int _0^{t} [S_{i,j} \otimes K] \delta (s- \tau _2) dw_{ji}(s), \end{align}Si,j=[sk]N×N$S_{i,j} = [s_{k}]_{N \times N}$ indicates N×N$N \times N$ matrix in which sii=−aij,sij=aij$s_{ii} = -a_{ij}, s_{ij} = a_{ij}$ and all other elements being zero. aij${a}_{ij}$ represents the elements of the communication digraph adjacency matrix. By the definition of δ(t)$\delta (t)$, next27dδ(t)=(IN⊗A)δ(t)dt−(L⊗K)(δ(t−τ1)dt+dM2),\begin{align} d \delta (t)= (I_N \otimes A) \delta (t) dt - (L \otimes K) (\delta (t-\tau _1)dt + dM_2), \end{align}where28M2=−∑i=1Naijσ0i∫0t[S¯i⊗K]δ(s−τ2)dw0i(s),\begin{align} M_2 = - \sum _{i =1}^{N} a_{ij} \sigma _{0i} \int _0^{t} [\bar{S}_i \otimes K] \delta (s- \tau _2) dw_{0i}(s), \end{align}S¯i=[sk]N×N$\bar{S}_i = [s_{k}]_{N \times N}$ is an N×N$N \times N$ matrix with nii=bi$n_{ii} = b_i$ and the whole other elements are zero. Let δ¯(t)=(ϕ−1⊗In)δ(t)$\bar{\delta }(t) = (\phi ^{-1} \otimes I_n) \delta (t)$, where ϕ is a positive definite matrix. Then we have29dδ¯(t)=A0δ¯(t)dt+A1δ¯(t−τ1)dt+dM3,\begin{align} d \bar{\delta } (t) = A_0 \bar{\delta } (t) dt + A_1 \bar{\delta } (t- \tau _1) dt + d M_3, \end{align}where A0=IN⊗A$A_0 = I_N \otimes A$, A1=−Γ0⊗K$A_1 = - \Gamma _0 \otimes K$, where Γ0 is a positive gain,30M3=∑i,j=1Nσji∫0t[(ϕTSi,jϕ)⊗K]δ¯(s−τ2)dwji(s)−∑i=1Nσ0i∫0t[(ϕTSi¯ϕ)⊗K]δ¯(s−τ2)dw0i(s).\begin{align} M_3 &= \sum _{i,j=1}^{N} \sigma _{ji} \int _0^{t} [(\phi ^T S_{i,j} \phi ) \otimes K] \bar{\delta } (s- \tau _2) dw_{ji}(s) \nonumber \\ &\quad - \sum _{i =1}^{N} \sigma _{0i} \int _0^{t} [ (\phi ^T \bar{S_i} \phi ) \otimes K] \bar{\delta } (s- \tau _2) dw_{0i}(s). \end{align}Let us define P¯=IN⊗P$\bar{P} = I_N \otimes P$, also, we have31⟨M3,PM3⟩(t)=∑i,j=1Nσji2∫0t[δ¯T(s−τ2)((ϕTSi,jϕ)T(ϕTSi,jϕ))⊗(K)TPK]δ¯(s−τ2)ds+∑i=1Nσ0i∫0tδ¯T(s−τ2)[(ϕTS¯iϕ)⊗(K)TPK]δ¯(s−τ2)ds.\begin{align} & \langle M_3, P M_3 \rangle (t) = \sum _{i,j=1}^{N} \sigma _{ji}^2 \int _0^{t} [\bar{\delta } ^T (s- \tau _2) ((\phi ^T S_{i,j} \phi )^T \nonumber \\ & (\phi ^T S_{i,j} \phi )) \otimes (K)^T PK ] \bar{\delta } (s- \tau _2) ds + \sum _{i =1}^{N} \sigma _{0i} \nonumber \\ & \int _0^{t} \bar{\delta } ^T (s- \tau _2) [(\phi ^T \bar{S}_i \phi ) \otimes (K)^T PK ] \bar{\delta } (s- \tau _2) ds. \end{align}It is noteworthy that, ∑i,j=1N=Si,jTSi,j=2L$\sum _{i,j=1}^{N} = S_{i,j}^T S_{i,j} = 2L$ and ∑i=1NS¯iTS¯i=b$ \sum _{i =1}^{N} \bar{S}_i ^T \bar{S}_i = b$. Accordingly, ∑i,j=1N(ϕTSi,jϕ)T(ϕTSi,jϕ)=2ϕTLϕ$\sum _{i,j=1}^{N} (\phi ^T S_{i,j} \phi )^T (\phi ^T S_{i,j} \phi ) = 2 \phi ^T L \phi$ and ∑i=1N(ϕTS¯iϕ)T(ϕTS¯iϕ)=ϕTD0ϕ$ \sum _{i =1}^{N} (\phi ^T \bar{S}_i \phi )^T (\phi ^T \bar{S}_i \phi ) = \phi ^T D_0 \phi$ are obtained. Hence, we have32⟨M3,PM3⟩(t)≤2σ¯2δ¯T(s−τ2)Dδ¯(s−τ2)dt,\begin{align} \langle M_3, P M_3 \rangle (t) \le 2 \bar{\sigma }^2 \bar{\delta }^T (s- \tau _2) D \bar{\delta } (s- \tau _2) dt, \end{align}where D = Γ0⊗(KTPK)$ \Gamma _0 \otimes (K^T PK)$. By using conditions of Theorem 1, under K=kPIm+TP$K = \dfrac{kP}{I_m+ TP}$, A¯TP¯+P¯A¯+(A¯TP¯A¯+A1TP¯A1)τ1+σ¯2D<0$\bar{A}^T \bar{P} + \bar{P} \bar{A} + (\bar{A}^T \bar{P} \bar{A}+ A_1^T \bar{P} A_1) \tau _1 + \bar{\sigma }^2 D &lt; 0$ is obtained, in which A¯=A0+A1$\bar{A} = A_0 + A_1$. We note that it can be guaranteed by33WiTP+PWi+WiTPWiτ1+(λi2τ1+2λiN−1Nσ¯2)KTPK<0,\begin{align} &W_i^T P+ P W_i + W_i ^T P W_i \tau _1 + (\lambda _i ^2 \tau _1 + 2 \lambda _i \dfrac{N-1}{N} \bar{\sigma }^2) \nonumber \\ &K^T PK &lt; 0, \end{align}where Wi=A−λiK$W_i = A- \lambda _i K$. By using the following inequality34(z+h)TQ(z+h)≤2zTQz+2hTQh,z,h∈Rn,Q>0\begin{align} (z+h)^TQ(z+h) \le 2 z^T Q z + 2h^T Q h,\nobreakspace \nobreakspace z,h \in R^n,\nobreakspace \nobreakspace Q &gt; 0 \end{align}hence,35WiTPWi≤2ATPA+2λi2KTBTPBK.\begin{align} W_i ^T P W_i \le 2 A^T PA + 2 \lambda _i ^2 K^T B^T PBK. \end{align}Inserting K=k(I+P)−1P$K = k(I+ P)^{-1}P$ and (35) into (33), (33) is assured by36Γi=ATP+P+2τ1P−ζiP(Im+P)−1P\begin{align} \Gamma _i &= A^TP+P+ 2 \tau _1 P - \zeta _i P (I_m+ P)^{-1} P \end{align}where ζi=2kλi−(2N−1Nσ¯2+3λNτ1)λik2$\zeta _i = 2k \lambda _i - (2 \dfrac{N-1}{N} \bar{\sigma }^2 + 3 \lambda _N \tau _1) \lambda _i k^2$ and P is the solution to (10). According to the equation (23) in [29], we have37Γi<(2α−ζi)P2(I+P),\begin{align} \Gamma _i &lt; (2 \alpha - \zeta _i) \frac{ P^{2}}{(I+ P)} , \end{align}where for k∈(k̲,k¯)$k \in (\underline{k}, \bar{k})$, (2α−ζi)<0$(2 \alpha - \zeta _i) &lt;0$. Hence, from (37), Γi<0$\Gamma _i &lt; 0$ is concluded. Hence, according to the Theorem 2.2 in [29], E∥δ(t)¯∥2≤C0e−γ0t$E \parallel \bar{\delta (t)} \parallel ^2 \le C_0 e^{-\gamma _0 t}$, limsupt→∞1tlog∥δ(t)¯∥≤−γ02$lim sup_{t \rightarrow \infty } \dfrac{1}{t} log \parallel \bar{\delta (t)} \parallel \le - \dfrac{\gamma _0}{2}$, a.s., where C0 and γ0 are positive constants. Consequently, the proposed strategy indicates the mean square and approximately sure consensus.□$\Box$

Journal

"IET Generation, Transmission & Distribution"Wiley

Published: Sep 1, 2022

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