Secondary Control of Islanded Microgrids for Fast Convergence Considering Arbitrary Sampling

: This paper proposes a novel distributed secondary control of MGs for fast convergence considering asynchronous sampling. With the employment of the algorithm, optimal power sharing and voltage restoration are implemented simultaneously. First, the hierarchical control objectives concerned with economic operation and voltage quality are introduced. Then, the execution process of the fast convergence algorithm is described for weighted average state estimation, with the illustration of corresponding features and the application in cooperative control. Further, the relevant stability issue is discussed based on large-signal dynamic modeling and a sufﬁcient stability condition is derived based on the Lyapunov–Krasovskii theory. Our approach offers superior reliability, ﬂexibility and robustness because of the loose implementation in terms of its performance concern, which is essential when the distributed consensus protocol is likely to yield toward deviation or even instability under arbitrary sampling and delays. The effectiveness of the proposed methodology is veriﬁed via simulations.


Introduction
With the increasing penetration of renewable energy resources, MGs have been manifested as a promising integration of DGs, storages and loads within identifiable electrical boundaries [1][2][3]. The appealing advantage of MG is that it can enter islanding mode in case of spontaneous events, which facilities uninterrupted power supplies [4,5]. Endowed with the superiorities of low complexity and high efficiency, the DC microgrid has a promising future for extensive application, where a hierarchical control structure with droop control installed in the primary control layer is commonly employed to ensure the stable and efficient operation [6][7][8]. However, voltage deviations and inappropriate current sharing caused by distinct output impedance are generally unavoidable in droop control.
Therefore, secondary control with the objective of system coordination is introduced to compensate for the drawbacks in primary control. The centralized architecture was adopted as an implementation pattern for cooperative control, where the collection and calculation of global information is implemented utilizing a MGCC. However, the requirement of a complex communication network reduces its reliability [9,10]. Inspired by the cooperative multi-agent technology, distributed secondary control has gained increasing popularity, where each DG receives information from its neighbors via a sparse communication network [11]. The consensus protocol provides an effective approach for distributed cooperative control based on the estimation and elimination of the disagreement across the system, and has become a mainstream algorithm in current research [12][13][14][15][16]. Inspired by the robust and optimal control theory, a new distributed control architecture was proposed in [14], where cooperative control objectives can be realized in the presence of system uncertainties. Based on the feedback linearization and Artstein transformation methods, a novel distributed finite-time control scheme which is resilient to communication failures (1) The objectives of optimal load sharing and voltage restoration can be achieved simultaneously with a fast convergence rate due to indirect information acquisition from nodes that are multiple hops away and the exclusion of repetitive information. Further, the relevant stability issue is discussed. (2) Compared with the consensus algorithm, the proposed algorithm shows a superior robustness in cases of asynchronous execution. (3) The optimal load sharing criteria is derived based on simplifying the problem into a constrained extremum search of the objective function utilizing the Lagrange multiplier method.
The reminder of this paper is organized as follows: Section 2 introduces the hierarchical control and formulates the asynchronous problems. A novel secondary control scheme is proposed in Section 3 with stability analysis elaborated in Section 4. Simulations are provided in Section 5; conclusions are drawn in Section 6.

Control Objective
The hierarchical control structure for DC MG is presented in Figure 1, which is equipped with two hierarchies to implement different control objectives.
Generally, droop control is utilized to generate the reference for the voltage and current loops, which is formulated as where i i can be measured using a low-pass filter with a cut-off frequency ω c : .
Processes 2021, 9,971 3 of 17 Processes 2021, 9,   Generally, droop control is utilized to generate the reference for the voltage and current loops, which is formulated as where ii can be measured using a low-pass filter with a cut-off frequency ωc: On account of the diverse output resistances, the current sharing under droop control would be unsatisfactory. Generally, the desired allocation criteria for output currents can be expressed as γ1i1 = γ2i2 =…= γnin, i.e., DGs are supposed to share the load currents inversely proportional to the droop coefficient. However, taking the economic issue as a primary consideration, the optimal load sharing criteria is derived and applied in this paper. The typical cost function of each distributed generation can be modeled as Further, a constrained optimization problem can be formulated as an objective function: with a network constraint on supply and demand balance: The Lagrange multiplier method simplifies the aforementioned problem to the extremum search of the following Lagrange function by virtue of introducing a multiplier η: Employing the first-order derivative approach, the equation follows that Accordingly, the incremental cost can be defined as ηi(ii) = 2αiii + βi and the economic optimization condition (6) yields from (5): On account of the diverse output resistances, the current sharing under droop control would be unsatisfactory. Generally, the desired allocation criteria for output currents can be expressed as γ 1 i 1 = γ 2 i 2 = . . . = γ n i n , i.e., DGs are supposed to share the load currents inversely proportional to the droop coefficient. However, taking the economic issue as a primary consideration, the optimal load sharing criteria is derived and applied in this paper. The typical cost function of each distributed generation can be modeled as Further, a constrained optimization problem can be formulated as an objective function: minC(i i ) = ∑ n i=1 C i (i i ) with a network constraint on supply and demand balance: The Lagrange multiplier method simplifies the aforementioned problem to the extremum search of the following Lagrange function by virtue of introducing a multiplier η: Employing the first-order derivative approach, the equation follows that Accordingly, the incremental cost can be defined as η i (i i ) = 2α i i i + β i and the economic optimization condition (6) yields from (5): i.e., the optimal operation can be achieved when the incremental costs of all DGs are equal [25]. Considering the instinct tradeoff between power sharing and individual voltage restoration, the average voltage across MG is expected to be regulated to the reference so that all voltages can be maintained within an acceptable range; hence, an observer is required for average voltage estimation without knowledge of global information.
Processes 2021, 9, 971 4 of 17 The secondary control selects input u i such that η i →η j , and V i →V ref to guarantee economic superiority and voltage restoration simultaneously.

Inconsistent Information Exchange in Consensus-Based Cooperative Control
For the sake of implementation, a discrete form of secondary control scheme is formulated as where a ij = 1 suggests DGi receives information from DGj, otherwise a ij = 0; the associated Laplace matrix L can be defined as The accuracy of the consensus-based control introduced above is greatly dependent on the synchronicity of information interaction. However, as depicted in Figure 2, inherent asynchronization is inevitable in the practical execution of distributed self-triggered systems given the absence of a centralized synchronization clock and independent control pattern of each DG, which is manifested as in consensus protocol. Where ε represents the interaction strength between DGs.

Cooperative Control Using Fast Convergence Algorithm
Recalling the control objectives discussed in Section II, the implementation of system cooperation requires both consistent incremental costs and average voltage restoration. If the unweighted average of incremental costs are defined as η, i.e., On this basis, in view of the challenges of asynchronous sampling and time delays in the distributed consensus control, the fast convergence algorithm [27] is utilized in secondary control. In the algorithm, the dynamic state yi and the weight wi are designated as the inputs, where wi is set to 1 for each DG in the unweighted average estimation required by cooperative control; two intermediate variables i s  , i x  denoting the sum of the scales and scaled states, respectively, are introduced to assist the calculation of the estimated average value ˆi x , which comes to be the output, and two new variables si→j, xi→j are defined to be the scale and scaled average state estimation transmitted from the ith node to the jth node and will be initialized to si→j(0) = ωi and xi→j(0) = yi, respectively. The iterative execution of the algorithm is presented in Algorithm 1.

Algorithm 1:
Fast convergence algorithm Input: yi and wi are designated as the inputs. On account of the dependency of the consensus algorithm on information synchronization, the system would ultimately converge to a consensus equilibrium deviated from the accurate value as ∆cx = (u − u 2 )∆z, (10) or even come to be unstable [26]. Where u 1 , u 2 represents the left eigenvector corresponding to the eigenvalue 1 of the matrix A s (listed in the Appendix A) and the Laplacian matrix, respectively; Essentially, the conventional consensus protocols assist in the achievement of system cooperation based on disagreement estimation and elimination between local and neighboring states and are valid only when the sampling intervals are restricted to be the same.

Cooperative Control Using Fast Convergence Algorithm
Recalling the control objectives discussed in Section 2, the implementation of system cooperation requires both consistent incremental costs and average voltage restoration. If the unweighted average of incremental costs are defined as η, i.e., η = ∑ n i=1 η i /n, then the objective of identical incremental costs can be equivalently converted to achieving the equilibrium where the incremental cost of each DG is equal to the average value η.
On this basis, in view of the challenges of asynchronous sampling and time delays in the distributed consensus control, the fast convergence algorithm [27] is utilized in secondary control. In the algorithm, the dynamic state y i and the weight w i are designated as the inputs, where w i is set to 1 for each DG in the unweighted average estimation required by cooperative control; two intermediate variables s i , x i denoting the sum of the scales and scaled states, respectively, are introduced to assist the calculation of the estimated average valuex i , which comes to be the output, and two new variables s i→j , x i→j are defined to be the scale and scaled average state estimation transmitted from the ith node to the jth node and will be initialized to s i→j (0) = ω i and x i→j (0) = y i , respectively. The iterative execution of the algorithm is presented in Algorithm 1.

Algorithm 1: Fast convergence algorithm
Input: y i and w i are designated as the inputs. Output: The average estimation of y i denoted as x i is the output. For the kth iteration of the ith DG, execute the following calculation: Then, for each neighbor of the ith DG, calculate the following information: and transmit them to the jth DG.
For the ith node, the detailed execution of average incremental cost or voltage estimation conducted per iteration is illustrated as follows, which consists of three steps: Step 1: Calculate the auxiliary variables s i , x i based on information sampled locally and received from neighbors, which represents all information known by the ith node.
Step 2: Estimate the average value of the incremental cost η i or voltage V i using the auxiliary variables derived in Step 1.
Step 3: For any j ∈ N i , calculate the information to be transmitted from the ith DG to the jth DG in preparation for the next iteration, which includes all information known by the ith DG except for those acquired from the jth DG.
As can be deduced from the above steps, in the first iteration of each DGi, average estimation is conducted based on the limited knowledge of the neighboring states and the form of the proposed algorithm is consistent with the consensus algorithm, whereas information of DGs that are one more hop away from DGi will be accessible in each of the following iterations; thus, accurate average estimation can be implemented with states of all DGs acquired within finite iterations. Subsequently, the average incremental cost or voltage estimated by each DG will converge to the accurate unweighted average value gradually, that is:η Accordingly, the optimal current sharing controller can be designed as: Processes 2021, 9, 971 6 of 17 While as for the voltage regulation control, the dynamic consensus problem is involved and the controller can be established based on the deviation between the average and nominal value of output voltage; hence, the integrated control input can be ultimately formulated as: The concrete implementation structure of the cooperative control is depicted in Figure 3.
gradually, that is: Accordingly, the optimal current sharing controller can be designed as: While as for the voltage regulation control, the dynamic consensus problem is involved and the controller can be established based on the deviation between the average and nominal value of output voltage; hence, the integrated control input can be ultimately formulated as: The concrete implementation structure of the cooperative control is depicted in Figure

Remarks of Algorithm
Convergence: Benefiting from the efficient information transmission, the fast convergence rate can be achieved using the proposed algorithm. Provided the ith node is reachable to the jth node through (dij−1) intermediate nodes in the communication topology and the farthest distance between two nodes is d = max(dij), then it can be concluded that global

Remarks of Algorithm
Convergence: Benefiting from the efficient information transmission, the fast convergence rate can be achieved using the proposed algorithm. Provided the ith node is reachable to the jth node through (d ij −1) intermediate nodes in the communication topology and the farthest distance between two nodes is d = max(d ij ), then it can be concluded that global information can be broadcast to each node after d iterations, implying the system will converge asymptotically after at least d iterations. Compared with the limited knowledge of neighboring information by each node in the consensus algorithm, the information interaction efficiency is greatly improved. Additionally, information derived from one node will not be sent back to it again from any neighbors, indicating the avoidance of repetitive data transmission, which further contributes to the fast convergence performance of the proposed algorithm.
Asynchronous implementation: Another attractive feature of the algorithm is its adaptation to asynchronous sampling situations. In this case, initial information y i , w i of each node i will be transmitted to its neighbors at its local control instant, with the time index of the ith DG denoted as k i incremented. Then, local average estimation and calculation of s i→j (k i ), x i→j (k i ) will be conducted once information from all the neighbors of the ith DG has been received. During the process, the correctness of data is guaranteed. In contrast to the strict requirement for the accurate measurement of disagreement between local and neighboring states from the same timeslot in consensus algorithm, the convergence of the proposed algorithm relies on the accessibility of global information, which can be implemented ultimately through the aforementioned asynchronous execution, only taking a longer period.

Stability Analysis
In this section, the sufficient stability condition for the proposed control scheme is derived based on the large-signal dynamic model.
The large-signal dynamic model for the system is first established, which consists of an inverter, network and loads.
Considering that the voltage and current loop possesses much faster dynamics compared with the power loop, the inverter can be modeled as follows, while ignoring the dynamics of the double-loop: where the estimated average states can be obtained based on the control scheme proposed in Section 3. A: Accordingly, η can be calculated by: Supposing n DGs are connected through the network displayed in Figure 4, the output current can be then expressed as: Accordingly, η can be calculated by: Supposing n DGs are connected through the network displayed in Figure 4, the output current can be then expressed as: where Based on the node voltage equation, it yields where ysi = ylinei + yline(i−1) + yi + yloadi with ylinei and yloadi representing the line and load admittances, respectively. Therefore, the relationship between output current and voltage can be established as: where Y′ is the admittance matrix as shown in the Appendix A.
Combing (2) and (16)~(32), the large-signal dynamic model of overall system can be arranged in a compact form Based on the node voltage equation, it yields where y si = y linei + y line(i−1) + y i + y loadi with y linei and y loadi representing the line and load admittances, respectively. Therefore, the relationship between output current and voltage can be established as: where Y is the admittance matrix as shown in the Appendix A.
Combing (2) and (16)~(32), the large-signal dynamic model of overall system can be arranged in a compact form z(k + 1) = f (z(k)), where Then, we will give the sufficient convergence condition for the system in (33). Theorem 1. The system will convergence asymptotically as long as the following inequality holds: where J (z 0 ) = (J(z 0 ) − I)/T with I representing the identity matrix and J representing the Jacobi matrix of f(z), which can be derived as Proof. First, (33) can be rewritten as: Accordingly, its continuous counterpart can be derived as z and the derivative of V(z) can be expressed as According to the derivation rule of complex functions, the following equation holds for the time derivative of f (z): .
Given the condition that J is negative definite, it can be deduced from (40) that . V(z) is negative definite, leading to the conclusion that the system will come to be stable asymptotically.

Simulation Results
Case studies are carried out to verify the effectiveness of the proposed secondary control using the test MG as shown in Figure 5, where five DGs with identical rating and three loads are connected to the voltage bus. The system and control parameters are listed in Table 1. Figure 5. Configuration of the test MG.

Conventional Performance
The performance of the proposed scheme is compared with the consensus protocol under a unified sampling period T = 0.01 s. The MG is initially controlled by droop control, and the cooperative secondary control is activated at t = 3 s, with an additional load (10 kW) attached to the bus at t = 7 s. The corresponding active powers, output voltages and incremental costs are shown in Figure 6b. For comparison with the distributed consensus control, the MG is operated with the same controller gains and communication topology, with the results shown in Figure 6a. As shown in both Figure 6a,b, the inappropriate power sharing and voltage deviation emerge initially due to droop characteristics. After the activation of secondary control, expected load sharing and voltage restoration are achieved and can be maintained in spite of the increase or decrease in the load. Nevertheless, the transient process in Figure 6a takes a much longer time than Figure 6b. With a further enlargement of the sampling period to 0.05 s, the system installed with consensus-based cooperative controller fails to remain stable, due to the growing oscillation in the curves in Figure 7a. By contrast, the active powers and voltages in Figure 7b show a slight fluctuation and comes to be stable rapidly, which indicates that the proposed algorithm has a superior robustness to sampling periods.

Conventional Performance
The performance of the proposed scheme is compared with the consensus protocol under a unified sampling period T = 0.01 s. The MG is initially controlled by droop control, and the cooperative secondary control is activated at t = 3 s, with an additional load (10 kW) attached to the bus at t = 7 s. The corresponding active powers, output voltages and incremental costs are shown in Figure 6b. For comparison with the distributed consensus control, the MG is operated with the same controller gains and communication topology, with the results shown in Figure 6a. As shown in both Figure 6a,b, the inappropriate power sharing and voltage deviation emerge initially due to droop characteristics. After the activation of secondary control, expected load sharing and voltage restoration are achieved and can be maintained in spite of the increase or decrease in the load. Nevertheless, the transient process in Figure 6a takes a much longer time than Figure 6b. With a further enlargement of the sampling period to 0.05 s, the system installed with consensusbased cooperative controller fails to remain stable, due to the growing oscillation in the curves in Figure 7a. By contrast, the active powers and voltages in Figure 7b show a slight fluctuation and comes to be stable rapidly, which indicates that the proposed algorithm has a superior robustness to sampling periods.

Arbitrary Sampling
The asynchronous consensus concern that is inherent in distributed control is investigated in this case. Due to the absence of a centralized clock, uncertainties inevitably occur and are assumed to be within ±10%Ti on the basis of the individual sampling periods T1 = 0.05 s, T2 = 0.01 s, T3 = 0.02 s, T4 = 0.06 s, T5 = 0.07 s, with associated arbitrary sampling intervals shown in Figure 8 [28]. Simulation results in this case are displayed in Figure 9,

Arbitrary Sampling
The asynchronous consensus concern that is inherent in distributed control is investigated in this case. Due to the absence of a centralized clock, uncertainties inevitably occur and are assumed to be within ±10%T i on the basis of the individual sampling periods T 1 = 0.05 s, T 2 = 0.01 s, T 3 = 0.02 s, T 4 = 0.06 s, T 5 = 0.07 s, with associated arbitrary sampling intervals shown in Figure 8 [28]. Simulation results in this case are displayed in Figure 9, from which it can be observed that the system under the consensus protocol is obviously unstable and the system with the proposed control remains stable due to decaying oscillations. Therefore, the proposed control scheme demonstrates excellent robustness to arbitrary sampling compared with the conventional consensus algorithm because of its inherent asynchronous implementation, elaborated in Section 3. On the other hand, considering the maximum allowable sampling interval T = 0.05 s in the synchronous case, the communication cost can be reduced drastically by 28.57% from the perspective of DG5. from which it can be observed that the system under the consensus protocol is obviously unstable and the system with the proposed control remains stable due to decaying oscillations. Therefore, the proposed control scheme demonstrates excellent robustness to arbitrary sampling compared with the conventional consensus algorithm because of its inherent asynchronous implementation, elaborated in Section III. On the other hand, considering the maximum allowable sampling interval T = 0.05 s in the synchronous case, the communication cost can be reduced drastically by 28.57% from the perspective of DG5.

Discussions
As can be deduced from Figure 6, economic load sharing and voltage restoration can be achieved simultaneously with a faster convergence rate using the proposed control strategy, indicating the proposed scheme is effective and has a loose constraint on communication topology for better convergence performance. Moreover, based on comparison between Figure 7a,b, it can obviously be seen that the proposed control strategy enjoys enhanced stability in the case of large sampling intervals, which would contribute to lower controlling cost. Ultimately, the better robustness to arbitrary sampling of the proposed from which it can be observed that the system under the consensus protocol is obviously unstable and the system with the proposed control remains stable due to decaying oscillations. Therefore, the proposed control scheme demonstrates excellent robustness to arbitrary sampling compared with the conventional consensus algorithm because of its inherent asynchronous implementation, elaborated in Section III. On the other hand, considering the maximum allowable sampling interval T = 0.05 s in the synchronous case, the communication cost can be reduced drastically by 28.57% from the perspective of DG5.

Discussions
As can be deduced from Figure 6, economic load sharing and voltage restoration can be achieved simultaneously with a faster convergence rate using the proposed control strategy, indicating the proposed scheme is effective and has a loose constraint on communication topology for better convergence performance. Moreover, based on comparison between Figure 7a,b, it can obviously be seen that the proposed control strategy enjoys enhanced stability in the case of large sampling intervals, which would contribute to lower controlling cost. Ultimately, the better robustness to arbitrary sampling of the proposed

Discussion
As can be deduced from Figure 6, economic load sharing and voltage restoration can be achieved simultaneously with a faster convergence rate using the proposed control strategy, indicating the proposed scheme is effective and has a loose constraint on communication topology for better convergence performance. Moreover, based on comparison between Figure 7a,b, it can obviously be seen that the proposed control strategy enjoys enhanced stability in the case of large sampling intervals, which would contribute to lower controlling cost. Ultimately, the better robustness to arbitrary sampling of the proposed scheme is demonstrated by Figure 9. In conclusion, the secondary control method proposed in this paper shows great superiority compared with conventional consensus-based secondary control, and can be well integrated with the existing hierarchical control strategies [29][30][31].

Conclusions
In this paper, a novel algorithm which addresses the average consensus problem with higher transmission efficiency is introduced and applied in cooperative control of microgrids. Employing the approach of the Lyapunov function, the sufficient stability condition for the algorithm is derived. Comparative study has revealed the superiority of the algorithm over the consensus algorithm in terms of convergence rate and its adaptability to the arbitrary sampling case.
The matrices in Jacobi matrix can be derived as: