Distributed event‐triggered secondary control for microgrids applicable to directed communication graph

Recently, distributed cooperative control has been popularly used in the secondary control of microgrids due to its reliability and flexibility. However, traditional distributed control schemes based on fixed‐cycle communication usually result in a waste of communication resources. Thus, this paper proposes an event‐triggered distributed control strategy that can significantly reduce the requirements of communication. In addition, compared with existing schemes, the proposed algorithm can achieve the secondary control goal in a directed graph, which can be applied to more scenarios, especially for weaker communication conditions. The stability of the strategy is investigated with the Lyapunov method. The simulation results verify the validity and effectiveness of the proposed scheme.


Primary Control
Centralized control collects the variables of each DG through the microgrid central controller (MGCC).In Reference 8, MGCC sends the power command according to the power allocation factor by collecting the frequency difference of each DG to achieve global optimum.Reference 9 studied the MG adaptive droop control strategy and proposed the stability range of the droop coefficient for different working conditions.However, once the MGCC fails or the communication network fails, it is hard for the MG to maintain stability. 10istributed control improves the reliability of the system by distributing the global objectives to local controllers of each DG without the supervision of MGCC. 11Reference 12 introduces a distributed consistency control method based on a multi-agent system (MAS).Within the MAS framework, MG's communication topology can be abstracted as a graph.The MG is regarded as a set of agents that only communicate with its neighbors, and the whole system will gradually reach a balanced state.This can not only effectively reduce the communication cost, but also get rid of the dependence on MGCC which significantly improve the reliability of MGs. 13 However, traditional distributed control usually employs a fixed control cycle, which means that even in the steady process, the bandwidth of the communication network is also challenged. 14Considering the uncertainty and intermittency of renewable resources, the control mechanism can be further optimized. 15The event-triggered control mechanism can effectively alleviate the pressure of MG communication.In, 16 an event-triggered control algorithm is designed, and its stability is proved by the Lyapunov method.Reference 17 proposes a finite/fixed event algorithm with an event-triggered mechanism, which has strong anti-interference ability and improves the robustness of the control scheme.Based on event-triggered control (ETC), Reference 18 designs a voltage observer to further reduce the communication requirements.Reference 19 proposes a novel distributed control strategy to improve synchronization speed.Reference 20 constructs an event-triggered distributed control method to reduce communication pressure and improve system stability.
However, all the event-triggered schemes mentioned above are based on undirected communication graphs, which require higher communication bandwidth compared to the algorithms based on directed communication graphs.On one hand, IEC61850, IEC104, and other protocols commonly used in the power system need higher bandwidth to achieve two-way communication, but fewer resources to achieve one-way communication.On the other hand, new communication methods such as power line communication (PLC) provide more choices for the construction of MG communication networks, and one-way communication algorithms can greatly reduce the production cost of such equipment.In addition, with the expansion of MG scales, the challenges of the communication network are gradually increasing, and the directed-graph-based topology can further reduce the demand for communication bandwidth.However, there is little work on distributed ETC for directed graph topology.Therefore, the study is still insufficient and needs further research.
And, there is little work on distributed ETC for directed graph topology.Therefore, the study is still insufficient and needs further research.Motivated by it, this paper proposes an ETC-based distributed secondary control scheme for a directed graph, which could significantly reduce the communication rate, as well as ensure the stability of the whole system.
The main contributions of this paper are concluded as follows: 1.A fully ETC-based distributed control for AC microgrid is proposed, which can realize frequency/voltage recovery and accurate power sharing.While achieving control objectives, it can reduce communication burden; 2. The control algorithm is designed on the directed graph, which can further reduce the communication rate compared with the traditional undirected graph methods; 3. The convergence of the proposed controller is investigated by the Lyapunov method.
The remaining chapters of this paper are arranged as follows: Section 2 gives the preliminary.The secondary controller based on ETC for the directed graph is proposed and analyzed in Section 3. In Section 4, the effectiveness of the proposed scheme is verified by simulation with MATLAB/Simulink.Section 5 concludes the paper.

Graph theory
In this paper, G = (v, , W) represents a graph, consisting of a nonempty finite set of N vertexes, where N×N represents the weight matrix.For arbitrary two nodes in the weighted directed network v i and v j , if node v i can receive information directly from v j , then the value of  ij is 1, otherwise, this value is zero.
The Laplace matrix

Primary control
By primary control, each DG can regulate its output voltage stably, whose equivalent circuit is shown in Figure 2.Where i l represents the inverter output current, while U out and i out , respectively, represent the output voltage and current of the inverter after passing through the LC filter.The controller is composed of four parts: power calculation, droop controller, inner dual-loop regulator, and sinusoidal pulse width modulator. 21he droop control algorithm can not only realize the power-sharing of MG, but also avoid communication between devices. 22The mechanism of droop control can be inferred as follows: where  i and V i represent the frequency and voltage of DG i , respectively, P i and Q i represent the active power and reactive power, respectively. ni and V ni represent the rated frequency and rated voltage, respectively.active power droop coefficient and reactive power droop coefficient, respectively, which are usually set by their rated capacity. 23But for droop characteristics, the frequency and voltage of the MG deviate from their rated values. 24

Distributed secondary control
To solve the aforementioned issues caused by the droop mechanism, it is necessary to execute frequency/voltage recovery and accurate power sharing.Therefore, the objective of secondary control can be written as: Based on the distributed control strategy in graph theory, 25 the controller can be designed as: W ij is the weight value between nodes i, j.If node i can accept the message from node j, W ij = 1, otherwise W ij = 0.If node i can accept the reference value, d i = 1.Then, the above controller could be rewritten discretely as: x i (k + 1) and x i (k) represent the information transferred by DG i at kth and (k + 1)th iterations, respectively.The global information interaction process of the MG can be expressed as follows: where X is the information exchange matrix and E is the information update matrix.After a certain iteration, the state variable of each DG can converge to the given value, and the distributed secondary control is realized.

Event-triggered controller design
The secondary control of MGs aims to compensate for voltage and frequency deviations caused by primary control. 26The distributed control laws for frequency and voltage restoration are similar.Taking frequency restoration as an example, during the process of frequency restoration, in order to achieve the objective of distributed secondary control, it is necessary to provide the parameter  ni in Equation ( 1) to synchronize the frequency and reference values by utilizing information from local and neighboring entities.Therefore, the nominal set point of droop control  ni can be obtained from: Distributed secondary frequency coordination control is usually designed as follows: The controller of each distributed generator relies on continuous state feedback, which implies lower communication efficiency.In practice, distributed controllers do not necessarily require periodic and frequent operations.Therefore, this paper proposes a method of event-triggering.By designing an event-triggering function, communication is only activated when the triggering condition is satisfied.Thus, the conventional distributed controller Equation ( 4) can be reformulated as follows: where f i (t) represents the event-triggering function, and Th represent the threshold of the event-triggering function, T represents the set of points that DGs can communicate, T represents the set of times when communication is not activated, and t i k+1 represents the time of the (k + 1)th event trigger.The neighbor node information serves as local control parameters to adjust the state variables of the local DG to their desired values, thereby accomplishing the objectives of secondary control.According to the Equation ( 9), the distributed ETC of frequency for DG i is designed as: where x w i (t) and v w i (t) represent the frequency value and frequency variation value of node i at time t, while T w and T w correspond to the sets of time points triggering communication and prohibiting communication in the frequency controller, respectively.
Given that the allocation of active power in the secondary control is dependent on the installed capacity of distributed generators, the distributed ETC of active power for DG i is designed as: where x P i (t) and v P i (t)∕D pi represent the active power value and active power variation value of node i at time t, while T P and T P correspond to the sets of time points triggering communication and prohibiting communication in the active power controller, respectively.
During the active power sharing process, each DG receives the active power and corresponding droop coefficients from neighboring nodes.Based on this information, the local reference power value is computed.This reference value is then utilized as a control parameter for the local inverter, ultimately achieving the equal distribution of active power.
The controller for voltage restoration and reactive power sharing exhibits similarities with the frequency and active power controllers described earlier.The overall event-triggered control diagram for the secondary control of the system is depicted in Figure 3: The key of the event-triggered mechanism is to design the time point when to update the state variables. 16Therefore, two kinds of errors are defined, the local error e i (t) and the global error  i (t): The local error represents the difference between the real-time state value and the last broadcast state value, while the global error represents the difference between the real-time state value and the reference value.On this basis, the trigger time and trigger function are defined as: where a positive number Th represents the trigger threshold and a constant k represents the trigger gain.For frequency and voltage restoration, x ref in Equation ( 14) represents the reference values for frequency and voltage, respectively.And, the reference values for active power and reactive power could be computed in Equation (3).

Stability analysis
For the Laplace matrix of a strongly connected digraph, its algebraic connectivity is defined as follows: where For the system of Equation ( 9), a Lyapunov function can be designed as: where ) T , ⊗ means the Kronecker product and c is a non-zero constant.
Substitute P into Equation ( 17): Obviously, V(t) ≥ 0 and if x(t) = 0 N ⊗ I N , then V(t) = 0.For t ∈ T, ̇x(t) = 0, V(t) = 0, where ̇x(t) and V(t) represents the derivative of x(t) and V(t), respectively.For t ∈ T, let B = −cL, then: where  2 (L) = min Obviously, if a  (L) > 0, then V(t) ≤ 0, which implies that Equation ( 9) is gradually stable.If the system is a strongly connected graph, then a  (L) > 0, and the proof is given below.Lemma 1. Assuming that the Laplace matrix is irreducible, there must be a positive vector x, so that L T x = 0.
Proof.Select a positive integer l to satisfy l −  N (L) > 0 and l − L ii > 0. Then the matrix lI N − L is positive definite,  (lI Assuming that the Laplace matrix L is irreducible, then there must be a positive definite diagonal matrix Proof.L is symmetric.According to Lemma 1, there is a  = ( 1 ,  2 , • • • ,  n ) T that satisfies  T L = 0.  is the eigenvector with zero eigenvalue of Laplace matrix L. Because  = Ξ1 N , then L T 1 N = 0. Therefore, L T Ξ is a matrix whose sum of each row zero.And because of ∑ N j=1 L ij = 0, ΞL1 N = 0.The sum of each line of L is zero.And L is symmetric, so the sum of each column of L is zero.The certification is complete.▪ Lemma 3. Suppose that the matrix L is symmetrically irreducible and satisfies Proof.Let Λ be the diagonal incidence matrix of L, then there is a matrix P = (P 1 , P 2 , • • • , P N ) satisfying L = PΛP T , y = P T x.Obviously: Similarly: The proof is completed.▪ Corollary.If the Laplace matrix L is irreducible, by selecting the positive vector ,  T L = 0, then, a  (L) > 0 is made.
Proof.According to Lemma 2, there is a positive vector It is obvious that 1 is the left eigenvector with the eigenvalue of zero, and The certification is complete.To sum up, when the system is a strongly connected graph, the consistency strategy represented by Equation ( 9) can be gradually stable.

CASE STUDY
To verify the effectiveness of the distributed event-triggered secondary control strategy proposed in this paper, a simulation model of 8DGs islanded MG is constructed.Its electrical connection and communication network are shown in Figure 4.The detailed parameters of the test system are shown in Table 1.The parameter settings of the dual-loop PI controller are presented in Table 2, which have a direct influence on the controller's performance and stability.The time step of the test system is set to 1 × 10 −5 s.The reference frequency is set to  ref = 2f, f = 50 Hz, and the reference voltage is set to v ref = 1 p.u.And only DG1 can access the reference value in the test system.

Controller performance
In this section, the proposed control strategy was tested to validate the frequency/voltage restoration and active/reactive power sharing.Initially, Load 1 -Load 8 in Figure 4 were all connected to the test network while Load 9 was disconnected, and the system only implemented droop control.At t = 1 s, Load 9 was connected to the test network, and the system continued to operate with only droop control.At t = 1.5 s, the proposed distributed controller began to operate.At t = 3 s, Load 9 was disconnected from the test network, and the system continued to operate under the joint action of droop control and distributed control.The test ended at t = 4 s. Figure 5 illustrates the testing results of the proposed control strategy for frequency restoration and active power sharing.It is evident that during the system initialization phase, each DG distributes active power according to its respective droop coefficients, which stabilizes the system frequency at 50 Hz.At t = 1 s, Load 9 is connected to the network in Figure 4, and each DG distributes active power based on their droop coefficients, resulting in a frequency drop to 49.95 Hz.At t = 1.5 s, the proposed ETC begins to function, and from t = 1.5 to t = 2.5 s, under the action of the distributed controller, the system gradually recovers the frequency to the reference value of 50 Hz while accurately distributing the active power among each DG based on their droop coefficients.At t = 3 s, the disconnection of Load 9 causes the system frequency to fluctuate.However, with the joint action of droop control and the distributed controller, the system restores the frequency to the reference value and allocates the active power before t = 4 s.The simulation results verify the effectiveness of the proposed event-triggered mechanism.
Similar to Figure 5, Figure 6 demonstrates the performance of the proposed voltage restoration and reactive power allocation control strategy.It can be observed that, following the system load increase at t = 1 s, the terminal voltage of each DG drops due to the non-negligible line impedance in the system, leading to unequal distribution of reactive power among the DGs.At t = 1.5 s, the ETC is activated, restoring the system voltage and enabling accurate allocation of reactive power based on the droop coefficients.After t = 3 s, the ETC restores the voltage deviation caused by the disconnection of Load 9 and maintains the proportional distribution of reactive power among DGs, validating the effectiveness of the proposed controller for voltage restoration and reactive power allocation.
Then the parameters' influence on the ETC rate is compared by changing the threshold value in the trigger function.Lower threshold implies that the trigger function is more sensitive to the system state changes and the communication is triggered more frequently.In contrast, higher thresholds require a greater deviation from the equilibrium state before triggering communication.
Figure 7 shows the comparison of the ETC rates for different threshold values.It can be seen that when the threshold value is small, the communication frequency is high, resulting in a smoother waveform during t = 1.5-2 s.On the other hand, a larger threshold value leads to a lower communication frequency and more significant step changes in the waveform during the same period.Moreover, it is worth noting that although a lower threshold value leads to a smoother waveform, it also results in more frequent communication, which may increase the network traffic and communication overhead.Therefore, a trade-off should be made between communication frequency and system performance when selecting the threshold value for the trigger function.

Comparison with the traditional way
This part is to the advantage of the proposed distributed ETC compared with the traditional way.
Although the communication overheads differ significantly between the traditional cycle-triggered strategy and the proposed ETC strategy, they have comparable control performances in terms of frequency recovery.As shown in Figure 8, the cycle-triggered strategy has a fixed communication frequency, which results in a certain degree of communication redundancy though the system state does not require updates.In contrast, the ETC mechanism adapts to the system state and only triggers communication when necessary, resulting in a lower overall communication frequency.This not only reduces communication overhead but also improves the utilization of communication resources.Therefore, the proposed ETC mechanism has potential benefits in terms of communication efficiency and resource allocation.
Figure 9 displays the communication frequency under the ETC mechanism, which is an asynchronous triggering method compared to the fixed-time trigger mechanism with a communication period of 0.1 ms.It can be observed that the communication frequency is higher during the transient response period when the system experiences rapid changes and lower during the steady-state period.This indicates that the ETC mechanism can adjust the communication frequency according to the system dynamics, which reduces the communication burden while ensuring the effectiveness of communication.The asynchronous triggering mechanism also allows for more flexibility in communication, which can improve the system's adaptability to different operating conditions.

Plug-and-play capability
This part analyzes the plug-and-play capability of the proposed distributed ETC.At t = 0 s, the test system consists of DGs 1-8 and loads 1-8.Assuming DG 7 and Load 7 are plugged out at t = 1.5 s, and DG 9 (with the same parameters as DG 5 ) is plugged in at t = 2.5 s, with the relevant line impedance set to 0.21 + 0.31j.After DG 7 is plugged out, the communication topology of the test system changes, and DG 8 will no longer receive information from DG 7 .Instead, it will start receiving information from DG 3 .Similarly, after DG 9 is plugged in, DG 9 will receive information from DG 3 and transmit local information to DG 8 based on the proposed event-triggering mechanism in this paper.During this process, the disconnection and connection of Load 9 will occur at the same previously described timing as mentioned in the previous context, without any changes.
As shown in Figures 10 and 11, after the removal of DG 7 in the test network, the remaining seven DGs are able to restore the system frequency and voltage to the reference values between t = 1.5 s and t = 2.5 s.Simultaneously, active and reactive power are redistributed according to the droop coefficients of each DG.At t = 2.5 s, DG 9 is plugged in the test network, and the system remains stable while redistributing power.Despite the unavailability of the original communication link due to the removal of DG 7 , the altered communication topology remains strongly connected, ensuring overall system controllability.The experimental results are consistent with the theoretical analysis, validating the plug-and-play capability of the proposed method.

CONCLUSION
In this paper, an ETC-based distributed secondary control scheme for AC MG under directed communication is proposed.
Compared with the cycle-triggered strategy, the proposed method could greatly reduce the communication frequency while ensuring the system is stable.The stability of the proposed approach is proved through the Lyapunov function.Furthermore, the proposed method is applied in the test system.The simulation results are consistent with the theoretical analysis of the proposed control method.However, the crucial factor of time delay was not extensively examined in this study.Therefore, future research can focus on investigating the dynamic topology aspect while considering the effect of time delay on the controller, and subsequently, developing more appropriate control algorithms.

1
The schematic diagram of hierarchical control.

F I G U R E 2
D pi and D qi represent the Droop control model of distributed generation.

F I G U R E 5
Frequency and active power of each DG.F I G U R E 6Voltage and reactive power of each DG.

F I G U R E 7
Influence of parameters on event-triggered control.

F I G U R E 8
System frequency change in two modes.

9
Event instants for the proposed distributed secondary control scheme.

F I G U R E 10
Frequency and active power response with plug-and-play operation.FI G U R E 11Voltage and reactive power response with plug-and-play operation.
F I G U R E 4 Distributed event-triggered control test network.TA B L E 1