Consensus active power sharing for islanded microgrids based on distributed angle droop control

Funding information National Key R&D Program of China, Grant/Award Numbers: 2017YFB0902900, 2017YFB0902904; National Nature Science Foundation of China, Grant/Award Numbers: 51777134, 52061635103 Abstract The implementation of synchronous phasor measurement units (PMUs) in distribution networks provide accurate and GPS-based synchronous data which support the grid flexible reconfiguration, such as island partition and island reconnection. Utilizing the high resolution PMU data, the phase angle droop control can be implemented on islanded microgrids (MGs) without causing the deviation on frequency brought from traditional frequency droop control, consequently support the synchronous and seamless transition. In this paper, a phase angle droop control method considering the incremental cost of distributed generation (DG) is proposed utilizing distributed finite-time protocol to realize cost minimization and consensus active power sharing of DG units. In addition, an observer-based voltage controller with finite-time convergence velocity is implemented to meet the target of DG units’ voltage recovery and the accuracy of reactive power sharing. The proposed method is implemented through a sparse communication topology and exchange information with their own neighbours only. The effectiveness, robustness and scalability of the proposed distributed control method are verified through case studies.


INTRODUCTION
The randomness and volatility brought by the widely spread integration of distributed generation (DG) have challenged the operation and control of traditional power grid [1,2]. Microgrids (MGs) that manage the local DG, energy storage systems and loads to mitigate the negative impacts of renewables with or without the support from the main grid have gathered significant attention worldwide [3]. Generally, when the MGs operate in grid-tied mode, DG units are mainly controlled by P-Q controller to manage the power output to the main grid while maximizing the utilization of renewable resources. When the MGs are in islanded mode, DG units are not only responsible for maintaining power balance within the island and avoid unnecessary load shedding, but in charge of providing necessary voltage and frequency reference for stable operation.
For islanded MGs, the basic control and management method is hierarchical control which consists of primary control, secondary control, and tertiary control [4,5]. The primary This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2021 The Authors. IET Renewable Power Generation published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology control is proposed to provide voltage and frequency for the MG, droop control is commonly adopted. Droop control simulates the primary frequency modulation characteristics of synchronous generator and provides voltage amplitude and frequency references of isolated MG through droop characteristics. However, due to the nature of droop control itself, the voltage magnitude and frequency at steady state are obviously deviated from the rated value. Therefore, secondary control of MG is considered to eliminate the voltage amplitude and frequency deviation caused by primary control. In addition, DG units' active and reactive power output can be allocated proportionally. The tertiary control of MG, known as MG energy management, aims to manage the MG from system level which mainly concentrates on the optimal economic operation. In addition, the optimal management of MG clusters are involved in tertiary control layer [6].
With the recent development of distribution phasor measurement unit (DPMU) technology, voltage phase angle can be used directly for load sharing in MGs. Compared with the conventional frequency droop control, the angle droop control enables the DG to meet appropriate power distribution without steady state frequency deviation and enhances the stability and transient performance in MGs with high penetration of DG [7,8]. But the main drawback of angle droop control is the relative poor power sharing capability comparing with frequency droop control, and the feeder impedance has significant impact on power sharing. To alleviate the effect of line impedance on power sharing, the measured voltage phase angle of the point of common coupling (PCC) by DPMU is utilized as a feedback signal to improve the performance of the traditional angle droop control [9,10]. Besides, the concepts of droop-free strategy are introduced to realize the restoration of the average voltage amplitude to the rated value under the distributed consistency control protocol and achieve the proportional distribution of active and reactive power among DG units [11,12].
The idea of distributed consistency based on multi-agent system (MAS) has been widely explored in the secondary control of isolated MGs [13][14][15][16][17][18][19][20][21][22][23][24]. All the DG units in the MG and the associate communication system can be represented by a balanced connection diagram with weighted value. Peer-topeer communication protocol is used to realize the information exchange between adjacent agents, and each agent can reach the desired value under the corresponding consistency control protocol. A frequency and voltage control strategy was designed with linear consensus feedback controller in [14], the proposed method can converge within asymptotical time. A distributed secondary control of MGs considering the communication time delay was proposed in [15] to handle clustered plug-and-play operation. Considering the contradiction between accurate reactive power sharing and voltage restoration, distributed average voltage control algorithms have been proposed in [16,17] and a distributed-averaging proportional-integral (DAPI) controller has been proposed in [18] to achieve precise frequency regulation and active power sharing.
Recently, distributed consensus algorithms with finite-time convergence control protocol have been studied in-depth for secondary control. In [19], a distributed, bounded, and convergent consistency control protocol has been proposed for the precise distribution of active power and restoration of frequency in finite time. In [20], a bounded finite time convergence protocol was proposed for frequency, voltage recovery and accurate active power sharing. Through input-output feedback linearization, the voltage restoration has been converted to a first order tracking problem, the frequency restoration and accurate active power sharing can be achieved through a finite time convergence protocol in [21] and [22]. In [23] and [24], two control protocols with different convergence rates have been developed, which can restore the frequency and the mean value of the voltage amplitude of all DG units to the expected values, and manage the active and reactive power proportionally. A tracker consensus approach based on model predictive control (MPC) for voltage regulation has been presented, and a distributed proportional integral frequency control method combined with a finite time observer to estimate the global reference has also been developed in [25].
The optimal active power sharing has been usually considered in tertiary control. In recent literature, research has been conducted to bridge the time scale gap between frequency restoration and economic operation in [26][27][28][29]. The cost-based droop scheme by embedding the incremental costs into the droop control has been developed in [26], which can guarantee the total active power generation cost minimization in MGs via droop control approach. In [27], a distributed consensus protocol with communication time delay is considered for frequency regulation, optimal active power sharing, voltage regulation and accurate reactive power sharing. In [28], an event-triggered optimal control strategy considering time varying demand and communication topologies has been proposed to achieve frequency regulation and economic operation. The economic dispatch issues have also been studied for a number of smart grids in [29] subject to unknown communication uncertainties by employing adaptive consensus based dispatch algorithms. In [30], a simple event triggered condition is designed to ensure economic dispatch and frequency restoration, which can bridge the time scale gap between secondary and tertiary control while reducing communication burden.
However, the above papers resolve the voltage regulation, frequency recovery and power distribution issues based on frequency droop control methods that solve the issues iteratively utilizing the feedback signal. Consequently, in order to achieve stable operation of islanded MG without additional frequency modulation, the proposed algorithm that combine the optimal economic power allocation method with angle droop control based on distributed consensus approach is proposed in this paper. First, a consistency control protocol with finite time convergence is proposed, which can realize the active power economic allocation of DG units within finite time. Second, for the voltage amplitude controller, the consistency control protocol of asymptotic convergence speed is adopted to achieve consistency of voltage of all DG units and the proportional distribution of reactive power. The two control approaches are designed with different convergence rate, which can realize phase and voltage control decoupled.
The main contributions of this paper can be summarised as follows: 1. A distributed secondary control protocol is proposed for voltage control, optimal active power and reactive power sharing in an islanded MG utilizing the information from neighbours only through a sparse communication network. 2. A secondary finite time convergence protocol is utilised in angle droop control that can regulate frequency to the set point without error, and the optimal active power sharing can be achieved in finite time. 3. An observer-based voltage controller and distributed control protocol with asymptotical convergence is developed to realize the voltage recovery and accurate reactive power sharing within an islanded MG.
The rest of the paper is organized as follows. In Section 1, preliminaries about graph theory and phase angle droop control are introduced. The proposed distributed consensus control protocol is explained in Section 2. In Section 4, the proposed control protocol is validated via case studies. Concluding remarks are given in Section 5.

Graph theory
All the DG units in an MG can be managed via an MAS, in which each DG is considered as an individual agent, and the communication connection between the DG represents the link between agents. Therefore, an undirected graph G = (v, ∈, A) can be used to represent the relationship among DG units.
The remaining elements of matrix A are 0s. The neighbour nodes of the ith DG is defined as: In the communication diagram, it is assumed that each node does not exist a self-loop, namely a ii = 0. Hence, the in-degree, deg in , and outdegree, deg out , of each node can be defined as: From the definition of in-degree and out-degree, it can be observed that the degree matrix of graph G is a diagonal matrix, According to the adjacency matrix A and degree matrix of the graph Δ, the Laplace matrix L of the graph can be defined as: For a directed graph G, if any node in G satisfies ∑ j ≠i a i j = ∑ j ≠i a ji , then, G is an equilibrium graph. If a graph with bidirectional communication, it is an equilibrium graph which is equivalent to an undirected graph. Lemma 1. [23]: For an undirected graph, the following properties are valid: (3) where 2 (L) is the second smallest eigenvalue of L, which is also defined as algebraic connectivity.

Lemma 2. [31]
: For a scalar system: where m, n, p, q are all positive odd integers, at the same time m > n, p < q, , > 0. Then, the equilibrium of Equation (6) is globally finite time stable and the setting time is bounded as:

Angle droop control
As shown in Figure 1, the complex power delivered from the ith DG to the AC bus is: where the DG's output voltage is v mag,i ∠ i , the AC bus voltage is v bus,i ∠0, and the combined LCL filter denoted as Z i ∠ i . From Equation (8), the active and reactive power from the ith DG to AC bus can be obtained as: Assuming the line impedance is pure inductive, i ≈ 90 • ; and the phase difference between the ith DG and the AC bus, i , is small enough, Equation (9) can be reformulated as: Hence, it can be seen from Equation (10) that the active power output of DG is directly proportional to the voltage phase angle if the voltage phase angle variation is small enough. In other words, the active and reactive power of DG can be managed by controlling the voltage phase angle and magnitude.
In traditional P-f and Q-V droop control, the voltage phase angle is indirectly changed by changing the frequency. However, the Pand Q-V droop control approach can directly control the phase angle and magnitude of the DG voltage output, it can be expressed as: where ni and V ni are the nominal voltage angle and magnitude, respectively; i and V i are the voltage angle and magnitude of DG i respectively; m Pi and n Qi are the droop control parameters.

Optimal active power sharing
The types of DG within MGs generally consist of gas turbine, fuel cells and other renewable energy resources with energy storage to provide stable power output. The operating costs include fuel prices of distributed power supply, battery maintenance and replacement costs, processing fees and pollutant discharge, penalties etc. According to the existing literature, the operating costs of different distributed power sources can be represented as the fitting quadratic function [26][27][28]: where i , i , i are the corresponding parameters of the cost function, respectively; P i is the active power output of the ith DG. In order to minimize the total generation operational cost under the condition of maintaining the load balance in the isolated MG, it is necessary to keep the equivalent incremental cost of all DG in the MG consistent. The incremental cost of DG is expressed as follows: Therefore, when the optimal active power cost is considered, the angle droop control form can be changed to: where k is a positive constant coefficient. By replacing the angle expression part of angle droop control with the above equation, a new expression can be obtained as follows: It is worth noting that the increment cost should be equal when meeting the steady state, which means the optimal active power sharing is obtained [28].

DISTRIBUTED SECONDARY CONTROL
In this paper, the control objectives of the distributed consensus control protocol are determined as follows: 1. To achieve the optimal active power sharing, where T f is optimal active power sharing restoration time. 2. To restore the average voltages of all DG units to the reference values, 3. To realize the accurate reactive power sharing,

Finite time optimal active power sharing
In angle droop control of isolated MGs, the voltage phase angle is a local variable which is different from frequency droop control that use global variable, and finally become a fixed value in the steady state. Therefore, the voltage phase angle is not considered to be constrained and restored in angle droop control. At the same time, the voltage phase angle difference between each node is very small in actual MG system, and the variation fluctuates within a very small range. Therefore, the first derivative value of the voltage phase angle is assumed to be 0. In addition, under the inductive environment of the equivalent output impedance of the inverter, the active power output of the inverter is proportional to the voltage phase angle, so the active power output of the inverter can be controlled by altering the voltage phase angle. Therefore, taking differentiation on both sides of angle droop control in Equation (11), then: Therefore, the reference value of voltage phase angle can be expressed as: Different from the consistent algorithm of asymptotic convergence speed proposed in [13] and [14] to achieve the precise distribution target of DG active power output, the distributed cooperative control protocol with finite time convergence is utilized in this paper, the control input u Pi can be represented with the following form: where , > 0, (.) m∕n = sign(.)|.| m∕n , m > n > 0, 0 < p < q, the final global consistency variable is selected as the incremental cost of DG, shown as i (P i ). The advantage of the consistent control mode of finite time convergence for active power is the inconsistent convergence speed between reactive power and voltage amplitude. Therefore, the coupling effect of active and reactive power regulation can be alleviated to some extent. (21), the optimal active power sharing in Equation (16) can be achieved when the system converges to the steady state, with the upper time limit T f given in Equation (29).

Theorem 1. Let the undirected communication graph G among DG units connected and balanced, by using the distributed finite time control protocol shown in Equation
Proof: According to the previous analysis results, to achieve the final optimal active power sharing, the necessary and sufficient condition is to make all the incremental cost of DG to be equal, this can be written as: where Ψ is a constant value, namely the marginal cost.
Taking e Pi (t ) = i (P i ) − Ψ, then, the following expression can be obtained: Selecting the following Lyapunov candidate Equation V 1 : The first derivative of Equation (24) is expressed as: (25) where N represents the number of DG units. . Two new Laplace matrices L B and L C are considered, making G B = 2e T P L B e P and G C = 2e T P L C e P . When the communication topology is connected, then 1 T N e P = 0. So, the following equation can be proved by Lemma 1.
where = min{ 2 (L B ), 2 (L C )} > 0. And when V1 ≠ 0, let y = √ 4 V 1 then: According to Lemma 2: lim t →T e P (t ) = 0. And the setting time is bounded by: Therefore, the distributed cooperative control protocol with above equation can realize optimal distribution of all DG units' active power output in the steady state. This completes the proof of Theorem 1.

Voltage regulation and accurate reactive power sharing
Due to the influence of line impedance in MGs, the restoration of voltage amplitude and the precise distribution of reactive power among different DG units cannot be realized simultaneously [14,16]. In order to make a balance between the two targets, namely the precise power distribution and voltage recovery, a distributed cooperative control with voltage observer is designed in this paper. Therefore, the average value of the voltage output of all DG units is consistent with the given reference and satisfy the precise distribution of reactive power among DG units at the same time. The voltage observer can be represented as follows:̇i where the average value of voltage amplitude converge asymptotically. Then, the consistency algorithm with a virtual leader node which receives the reference value is used to restore the average voltage to the reference, the voltage observer is required to realize the consistency of the average voltage as soon as possible. In this paper, the voltage observer is constructed by using the corresponding distributed consistency control protocol with finite time convergence. A distributed voltage observer is designed as: where k 1 , k 2 > 0, 0 < 1 < 1, 2 > 1. In this way, it is assumed that each DG can exchange its own voltage observation val-ues with its adjacent DG nodes and update its own observation by processing the neighbours' estimated valuesv j and the local voltage measurement v i , and finally achieve the consistency of all voltage observations based on the consistency algorithm of finite time convergence.
To analyse the stability of Equation (31), differentiating it yields:̄v The global observer dynamic can be written as: Considering the candidate Lyapunov function: The communication topology is undirected and connected, 0 is the smallest eigenvalue of L and V 2 = 0 if and only ifv ∈ span{1 n }. Let z i = ∑ j ∈Ni a i j (v j −v i ), then differential expression of Equation (31) can be written as: According to the results of [32], if the communication links among DG units are connected, the voltage observer shown in Equation (32) can make each DG's voltage observation converge to the average value of all DG units' actual voltage magnitudes within finite time.
After all DG units' output estimated voltages tend to the average value, the distributed consistency control with leading node is used to achieve the consistency between the average and the rated value. Meanwhile, the accurate distribution of reactive power among DG units is realized. Hence, after considering the estimated average voltage and the dynamic characteristics of reactive power, the corresponding state space equation can be obtained as: Hence, the distributed consensuses control rule for estimated average voltage and reactive power sharing can be designed as: (37) where g i is the pinning gain which is non-zero for the DG that has direct access to the reference voltage value, k v and k Q are positive gains.
Through the distributed consensus control method mentioned above, the average voltage amplitude of all DG units can converge to the expected value in accordance with the speed of asymptotical convergence, and reactive power can also achieve the goal of proportional distribution. Comparing to the incremental cost convergence achieved with finite time, the traditional asymptotic convergence controller enables the voltage regulation and reactive power sharing in a slower speed. Consequently, the coupling of active and reactive power control can be alleviated in some extent.

CASE STUDIES
In order to verify the effectiveness of the proposed control method shown in Figure 2, the testing model of isolated MG has been built in MATLAB/Simulink environment. The MG contains 5 DG units. The topology of the MG and associate  Tables A1, A2 and A3 respectively in Appendix.

Case 1: Load variations
As shown in Figure 4(a-d), only the angle droop control method is applied to the testing islanded MG initially. At t = 1 s, load2 is connected and at t = 2 s, the proposed control strategy is activated, at t = 3 s, an additional load6 of 50 kW + 10 kVar is connected to the MG at node5 and then removed at t = 4 s. Before using the proposed distributed consensus secondary control, the active and reactive power are not rational distributed. At the same time, the estimated average voltage deviate from the rated value. In addition, the DG's incremental cost consensus value is not achieved, which means the target of the overall operating cost minimization of DG cannot be guaranteed. On the contrary, after applying the proposed secondary control strategy, the DG units' active and reactive power output are distributed managed with their own ratio. And all incremental costs of DG units are consistent after adding the proposed control strategy. By using the finite time convergence protocol, the incremental costs can converge to a new steady state effectively and the active power can be reassigned during the load variations.
According to the results shown in Figure 4(c,d), it can be found that the control of reactive power adopts the consistent control protocol with asymptotical convergence rate is relatively slow compared with the finite time convergence speed of optimal active power sharing.
It can be seen from Figure 5, a voltage observer with finite time convergence is first used to obtain the average value of the voltage amplitude for all the DG voltage, then the distributed control protocol with asymptotically convergent speed is used to restore the average value of the voltage to the given reference. Therefore, the voltage response speed of DG is obviously faster than the speed of convergence to the reference value of the average voltage. This study demonstrates the performance of consistency control protocol with different convergence rates.

Case 2: The plug-and-play capability
This case study aims to test the influences of DG units' plugand-play operation. At t = 1 s, the proposed distributed consensus control method is added to the controller. At t = 2 s, DG 5 After removing DG 5 from the MG, the rest DG units share the load proportional. The active and reactive power outputs of DG 5 is gradually decrease to 0 due to the LCL filter at t = 2 s. However, when DG 5 is reconnected to the MG, a large power fluctuation occurred instantly due to the lack of synchronous connection.

Case 3: Different communication topologies
When the communication topology is balanced, the corresponding graph is connected and balanced. Therefore, the Laplace matrix L is non-singular, and the n eigenvalues of L are ordered as 0 = 1 (L) < 2 (L) ≤ ⋯ ≤ n (L) ≤ 2Δ, where Δ = max i deg out ( i ). Here, 2 (L) as the dominant pole of L, is defined as the algebraic connectivity, which has a sig- nificant influence on the system dynamic [15]. From Equation (26), it can be found that the second minimum eigenvalue of the Laplacian matrix of communication topology has obvious influence on the upper limit of the final convergence time. Hence, this part is mainly designed to test the speed of convergence under different communication topologies shown in Figure 8, the corresponding algebraic connectivity and convergence time with different topologies are shown in Table 1.
At 2-4 s of the simulation, the communication topology is in line type, and 4-7 s, the communication topology is in ring  type; and at t = 7 s, the communication topology changes to mesh. In addition, load 6 is connected to the MG at t = 2, 5 and 8 s, and disconnect it from the MG at t = 3, 6, 9 s, respectively. It can be seen from the results shown in Figure 9(a) that the communication connection degree between the DG varies under different communication topologies, but as long as each DG can exchange information through communication, all DG in the islanded MG can timely respond to load variations. From the incremental cost results shown in Figure 9(b), as long as all DG can exchange information through communication topology, it can also ensure that all DG units' incremental costs are consistent with the altered communication topologies. In addition, by combining the local amplification results and the upper limit of convergence time calculated theoretically, more communication connections between DG units are utilized, faster the convergence rate of the incremental cost. The incremental cost can converge to a constant value within 0.4 s under line topology and converge within 0.2 s with mesh topology. However, comparing the convergence time of the ring topology and mesh topology, it can be found that the increment of convergence time is very small with the communication link increases. Therefore, it is not merit to improve the convergence speed slightly by reinforcing the communication link.

Case 4: Communication delay
This case study aims to test the performance of the proposed distributed consensus control method under different communication delays. It can be observed from Figures 10 and 11 that the performance of the system become more oscillating as the communication delay increase. After the communication delay exceeds 20 ms, the system can be unstable. This is because the consensus control protocols are designed based on Lyapunov method, and the stability margin decreases with the increase of communication delays. It worth noting that the communication delays are usually on the time scale of milliseconds or tens of milliseconds [33]. Thus, this method can be used in the real systems.

Case 5: Scalability test
This case study is designed to test the proposed distributed consensus control on a modified IEEE 33 bus distribution system [34], as shown in Figure 12. The results of incremental costs of DG, reactive power sharing ratio and average voltage of DG are shown in Figure 13. The incremental costs can reach consensus within 0.5 s; the reactive power sharing ratio and average voltage of DG can reach consensus within 1.5 s, which verify the effectiveness of the proposed method for large system.
Then, additional DG units have been added at node 2, node 14, and node 26 respectively to verify the performance of the proposed method under heavy DG integration. The results of incremental costs of DG, reactive power sharing ratio and average voltage of DG are shown in Figure 14 respectively.
As shown in Table 2, with the increasing level of DG integration, the upper limit of convergence time of the consistent algorithm is also growing, in particular the voltage controller with asymptotic convergence rates. This is because the increase of DG will lead to more complex communication and calculation of the consistency algorithm. Besides, it can be noted that the convergence time and the maximum tolerable communication time delay rely on the eigenvalues of the communication topology. Therefore, for large power system with many DG units, the maximum tolerable communication delay may become small, which means that the scalability of the distributed consensus control is constrained by the communication delay. Therefore, a  proper management scheme such as the deferrable load aggregation method [35] can be considered in the control of large scale DG systems. In order to decrease the communication and the corresponding computation burden, the DG units can be aggregated with clustering algorithm according to its capacity, type, and distance etc.

CONCLUSION
In this paper, a phase angle droop control method considering the incremental cost of DG is proposed utilizing distributed finite-time protocol to realize cost minimization and consensus active power sharing within an islanded MG. By utilizing the information of neighbours only, the distributed protocol enables efficient distributed data processing, improves the anti-interference ability of communication and demonstrates extensive scalabilities. The proposed protocol can realize optimal active power sharing within finite-time, average voltage regulation and reactive power sharing accurately. In addition, an observer-based voltage controller and distributed control protocol with asymptotical convergence which facilitate the decoupled design for voltage controller are beneficial for voltage recovery and accurate reactive power sharing. The performance of load variations, plug-and-play capability, communication robustness and system scalable expansion have been fully verified via various simulations in this paper.