Stability and Distributed Optimization for AC Microgrid Considering Line Losses and Time Delay

Abstract

With the development of distributed renewable generations, a large number of distributed generations (DGs) are connected to the microgrid. Therefore, distributed cooperative algorithms are more suitable for optimal dispatch of the microgrid than centralized algorithms. This paper proposes a novel distributed cooperative control method for optimal dispatch of microgrids, considering line losses and time delay. First, the optimization model of the microgrid considering line losses is established, and optimality conditions are obtained. Second, a novel distributed optimization method considering line losses is proposed, where the primary control is to achieve optimal dispatch, and the secondary control is to achieve frequency regulation. Third, the stability of the system under time delay is analyzed, and the robust stability conditions are obtained. Finally, simulation results verify the the effectiveness of the proposed method.

1. Introduction

Economic dispatch (ED) and frequency regulation are important for the stable and economic operation of the power system [1,2]. Generally, there are three types of optimal dispatch methods, i.e., centralized [3,4,5,6,7,8,9,10,11], decentralized [1,2,12,13,14,15,16,17] and distributed optimization methods [18,19,20,21,22,23,24]. The centralized optimal dispatch usually has two processes (i.e., “decision” and “control”). In the first process, the central processing unit collects global information, solves optimization problems and produces the references about DG’s output voltage and power. The second process is to track the references by the PI controller. The main task of the centralized optimal dispatch problem is developing an algorithm to solve the optimization problems. Here, the optimization problems are usually formulated using global parameters, such as line impedances, flow limits, cost functions, and load parameters [3,4]. The centralized optimizations are usually based on the traditional optimization techniques, such as convex optimization methods [5,6,7,8] and heuristic optimization methods [9,10,11,12,13,14]. Due to the power low constraint, the ED problem is usually non-convex. To overcome this problem, some convex relaxation methods, such as the semi-definite programming [6] and second-order cone programming [8], are proposed. For the ED problem with more non-convex and stringent constraints, the heuristic optimization methods, such as the genetic algorithm [9], particle swarm optimization [10,11], the mine blast algorithm [12], the strawberry plant propagation algorithm [13] and the deep neural network approach [14], are proposed to find the solution of the ED problem. Moreover, two analytical methods are proposed to reduce the active and reactive power losses in [15,16]. For modern power systems with large amounts of renewable energy, an optimized solution for a renewable energy-based generation system is proposed to face a power shortfall and load shedding [17]. An optimal sizing and assessment method for an islanded renewable-energy-based generation system considering conventional dispatch methodologies is presented in [18]. A comparative analysis of peak load shaving strategies for an isolated microgrid using actual data is presented in [19]. More optimization techniques for hybrid microgrid systems can be seen in [20,21,22]. For the ED problem with centralized frameworks, the optimization models are usually detailed and accurate, so it usually provides the global or near-global optimal solution in the actual system. However, it depends highly on the intense communication and usually suffers from the single point of the failure problem, which increases the communication cost and reduces the reliability of the microgrid.

To reduce the communication cost for the ED problem, some decentralized approaches based on droop control are proposed. The biggest advantage of the decentralized approaches is that it just uses the local information and does not need communications [23,24,25,26,27,28]. Several decentralized methods based on the equal cost principle are proposed in [23,24,26], whose core idea is to make the cost of each distributed generation (DG) equal based on P-f droop control. Therefore, the output power of the expensive DG is small, while the cheaper DG produces more power, thus reducing the operation cost of the microgrid. In [25,27], nonlinear decentralized optimization approach is proposed based on polynomial approximation, whose main idea is selecting the optimal coefficients of the polynomial to achieve optimal dispatch. Generally, these methods can reduce the operation cost to a certain extent but cannot achieve global optimality. For this reason, several decentralized methods towards global optimal dispatch are proposed [1,28], whose core is the equal incremental cost principle (EICP). Here, incremental costs (i.e., the derivative of the cost function with respect to the DG’s output power) are subtly embedded into the P-f droop control. When the system achieves frequency synchronization, the incremental cost of each DG will be the same; thus, the optimal dispatch will be achieved according to the ECIP. Since the ECIP is only applied to the convex problem, reference [2] further studies under what conditions the global optimal dispatch can be realized in a decentralized manner for a microgrid. It is proved that if and only if the so-called optimal solution functions are all strictly monotonically increasing, the global optimal dispatch can be achieved by the nonlinear droop control in [2]. Compared with the centralized optimal dispatch, the decentralized method does not need to solve the optimization problems, which need the load’s real-time information. It only needs to make the incremental cost equal based on droop control. Therefore, decentralized optimization approaches are able to reduce the cost without communications and achieve plug-and-play. However, it is only suitable for the simplified optimization models. For example, the resistance of the cable is neglected, and the line loss is neglected. Moreover, decentralized optimization approaches are based on droop control, resulting in frequency deviation in the steady state.

The distributed optimal strategy, which only needs the neighbouring communication, is more suitable for the microgrid than the centralized method. Compared with the decentralized method, the distributed optimal strategy can achieve optimal dispatch and frequency regulation simultaneously. Several distributed consensus optimal methods [29,30,31,32,33,34,35,36,37,38,39,40] are proposed to solve the ED problem, such as the distributed projected gradient method [29], the distributed $\lambda$-consensus algorithm [30,31,32,33,34] and the alternating direction method of multipliers (ADMM) method [35]. However, these methods focus on solving the optimization problems in a distributed manner, so they needs the loads’ real-time information. Due to the presence of the line loss, the load information is not easy to obtain. In [36,37], the distributed secondary control methods, considering economic dispatch, are proposed. By making the incremental cost equal based on the distributed consensus control, the optimal dispatch will be achieved in the steady state. So, it does not need the load’s real-time information. Moreover, frequency regulation is also achieved. To achieve optimal dispatch and frequency regulation in finite time, two methods based on distributed finite time control are proposed in [38,39]. To reduce the communication burden, an economic dispatch method based on distributed event-triggered control is proposed in [40]. However, the works in [36,37,38,39,40] do not consider the effect of line loss. In a low-voltage microgrid, the cable’s resistance cannot be neglected, and the line loss will bring an error in optimal dispatch. Moreover, the cable’s resistance and time delay may lead to system instability, which has not been solved well.

In summary, the centralized methods are able to deal with the detailed and accurate economic dispatch model and provide the global optimal solution. However, it relies on high bandwidth communications, which are too expensive for the microgrid, and it suffers from the single point of the failure problem. The distributed optimization methods only need low bandwidth communications, which are suitable for the microgrid. However, existing distributed optimization methods are more suitable for the simple ED model. For example, the line loss is assumed as zero, and complex power flows are neglected, i.e., the assumptions in the existing distributed optimization methods are too strong. Moreover, the existing distributed optimal dispatch methods may have one or more of the following drawbacks.

(1) It needs the real-time information of the load, which is difficult to obtain.

(2) The line loss is not considered in optimal dispatch, which may cause deviation between the system operation point and the optimal solution. Moreover, the effect of cable resistance on the stability has not been considered.

(3) The optimal dispatch depends on communications, which will not be realized if some communication link fails such that the communication network does not have a spanning tree.

(4) The low bandwidth communication usually has a time delay, which may lead to instability.

This paper proposes a distributed cooperative control method for optimal dispatch considering line losses. The main contributions are the following.

1. A distributed cooperative control method is proposed to achieve optimal dispatch and frequency regulation. First, the optimization model considering line losses is established, and optimality conditions are obtained. Second, cost optimization and frequency regulation are implemented in the primary and secondary control, respectively. Therefore, the optimal dispatch can be achieved without communications, and only frequency recovery depends on communications.

2. The stability of the system considering the cable’s resistance and communication time delay is analyzed, and stability conditions are provided. First, the stability condition of the system with no communication is derived. Second, the stability condition of the system without time delay is obtained based on the singular perturbation theorem. Finally, an analytical stability condition for the system under time delay is provided, which easily to gives guideline for the system design.

Moreover, main innovations are highlighted below.

(1) The optimal dispatch algorithm does not depend on communications. The optimal (or sub-optimal) dispatch can be realized even if there is no communication.

(2) The proposed control scheme does not need the real-time information of the load.

(3) The proposed control scheme can realize optimal dispatch and frequency recovery simultaneously if the communication is normal.

The remainder of this paper is organized as follows: Section 2 introduces the optimization model for optimal dispatch considering line losses and related works. Section 3 describes the proposed distributed cooperative control scheme. Section 4 presents the stability analysis of the system. Case studies are presented in Section 5. Section 6 concludes the paper.

2. Related Works and Problem Formulations

In this section, the optimization problem for the ED considering line losses is introduced. Moreover, some related works and their drawbacks are presented, and the motivations of this paper are provided.

2.1. The Economic Dispatch Problem Considering Line Losses

Consider a microgrid with n dispatchable DGs and other undispatchable DGs; the operation cost function of the dispatchable DGs is usually given by [31]

$$f\left(p\right)=\sum _{i=1}^n{f}_i\left(p_i\right),p_i=a_i{p}_i^2+b_i{p}_i+c_i$$

(1)

where $p_i$ and $f_i\left(p_i\right)$ are the output power and operation cost function of the i-th DG, respectively, and $a_i$, $b_i$ and $c_i$ are coefficients and both positive numbers. The main applications include data centers, ship power supply systems, etc. Then, the optimization model for the ED takes the following form

$$\begin{array}{cc}\hfill & \underset{p_i}{min}f\left(p\right)\hfill \\ \hfill s.t.& \sum _{i=1}^n{p}_i=p_{load}+p_{loss},\hfill \\ \hfill & 0\le p_i\le p_{max,i}\hfill \end{array}$$

(2)

where $p_{max,i}$ is the maximal output power, $p_{loss}$ is the line loss, and $p_{load}$ is the equivalent load power, which is equal to the total load power minus the unschedulable DG’s power. Then, the Lagrangian operator for the optimization model is given by

$$L(p,\lambda )=f\left(p\right)+\lambda (p_{load}+p_{loss}-\sum _{i=1}^n{p}_i)+\sum _{i=1}^{2n}{\mu }_i{\varphi }_i$$

(3)

where $\lambda$ is the Lagrangian multiplier about the equality constraint, ${\varphi }_i$ is the inequality constraint, and ${\mu }_i$ is its Lagrangian multiplier. Then, the optimality conditions (i.e., KKT conditions) are given by

$$\left\{\begin{array}{cc}\hfill & \frac{\partial f_1\left(p_1\right)}{\partial p_1}-\lambda (1-\frac{\partial p_{load}}{\partial p_1}-\frac{\partial p_{loss}}{\partial p_1})=0\hfill \\ \hfill & \frac{\partial f_1\left(p_2\right)}{\partial p_2}-\lambda (1-\frac{\partial p_{load}}{\partial p_2}-\frac{\partial p_{loss}}{\partial p_2})=0\hfill \\ \hfill & \cdots \hfill \\ \hfill & \frac{\partial f_1\left(p_n\right)}{\partial p_n}-\lambda (1-\frac{\partial p_{load}}{\partial p_n}-\frac{\partial p_{loss}}{\partial p_n})=0\hfill \\ \hfill & p_{load}+p_{loss}-\sum _{i=1}^n{p}_i=0\hfill \end{array}\right.$$

(4)

Usually, loads are connected to the grid through power electronics interfaces (i.e., point-of-load (POL) converters). Then, even if the input voltage of the POL converter changes, the output voltage of the POL converter will remain unchanged due to the regulation of the converter. Therefore, the load with the POL converter behaves as an instantaneous constant power load, whose power depends on the output power instead of the input voltage of the POL converter. Here, the load is assumed as a constant power load, i.e., $\frac{\partial p_{load}}{\partial p_i}=0$. It is highlighted that $p_{load}$ may vary with time but not with $p_i$. Since $\frac{\partial p_{load}}{\partial p_i}=0$, the optimality conditions in (4) becomes

$$\left\{\begin{array}{cc}\hfill & \frac{\partial f_1\left(p_1\right)}{\partial p_1}-\lambda (1-\frac{\partial p_{loss}}{\partial p_1})=0\hfill \\ \hfill & \frac{\partial f_2\left(p_2\right)}{\partial p_2}-\lambda (1-\frac{\partial p_{loss}}{\partial p_2})=0\hfill \\ \hfill & \cdots \hfill \\ \hfill & \frac{\partial f_n\left(p_n\right)}{\partial p_n}-\lambda (1-\frac{\partial p_{loss}}{\partial p_n})=0\hfill \\ \hfill & p_{load}+p_{loss}-\sum _{i=1}^n{p}_i=0\hfill \end{array}\right.$$

(5)

Let ${\lambda }_i={(1-\frac{\partial p_{loss}}{\partial p_i})}^{-1}\frac{\partial f_1\left(p_i\right)}{\partial p_i}$; the optimal solution about $p_i$ can be obtained by the following optimality conditions

$$\left\{\begin{array}{cc}\hfill & {\lambda }_1={\lambda }_2=\cdots ={\lambda }_n\hfill \\ \hfill & p_{load}+p_{loss}-\sum _{i=1}^n{p}_i=0\hfill \end{array}\right.$$

(6)

Let $p^{*}$ denote the solution of (5). Clearly, $p^{*}$ is unique.

2.2. The Existing Distributed Optimal Dispatch Approaches

For the ED problem in (2), the core task is to make the system operation point satisfy the optimality conditions in (6) through distributed control. There are mainly two types of methods. The first is to solve (5) using a distributed consensus algorithm and then control the DG’s output power to track $p_i^{*}$ [31,32,33,41]. The second method does not solve (5) directly, and it uses the distributed consensus control to make ${\lambda }_1={\lambda }_2=\cdots ={\lambda }_n$ in the steady state [36]. Because the first constraint in (1) is a physical constraint (i.e., it always holds), the system operation point coincides with $p^{*}$ when the system achieves steady state. Thus, the optimal dispatch of the microgrid is realized.

The core algorithm of the first method is briefly summarized below.

$$\left\{\begin{array}{cc}\hfill & {\lambda }_i(k+1)=\sum _{j=1}^n{a}_{ij}{\lambda }_i\left(k\right)+k_i(\sum _{i=1}^n{p}_i\left(k\right)-p_{load}+p_{loss}\left(k\right))\hfill \\ \hfill & {\lambda }_i\left(k\right)={(1-\frac{\partial p_{loss}}{\partial p_i}\left(k\right))}^{-1}(2a_i{p}_i\left(k\right)+b_i)\hfill \end{array}\right.$$

(7)

where $a_{ij}>0$ if there is a communication link between the i-th and j-th DG and $a_{ij}=0$ otherwise. Moreover, $a_{ij}=a_{ji}$ and ${\sum }_{j=1}^n{a}_{ij}=1$. According to results in [21], $p_i\left(k\right)$ will converge to $p^{*}$. Clearly, the algorithm in (7) needs the load real-time information, which is usually difficult to obtain.

For the second method, it usually assumes that the cable resistance is zero, thus, $\frac{\partial p_{loss}}{\partial p_n}=0$. Then, a distributed $\lambda$-consensus control method is proposed in [36], whose mathematical expression is briefly given by

$$\begin{array}{cc}\hfill & {\dot{\theta }}_i={\omega }_i\hfill \\ \hfill & {\tau }_i{\dot{\omega }}_i={\omega }^{*}-\omega -k_i(p_i-P_i)\hfill \\ \hfill & {\dot{P}}_i=-\kappa (\frac{\partial f_i\left(p_i\right)}{\partial p_i}-{\mu }_i)-\gamma (w-w^{*})\hfill \\ \hfill & {\dot{\mu }}_i=-\kappa (p_i-P_i)-\alpha \sum _{j=1}^n{a}_{ij}({\mu }_i-{\mu }_j)-\beta \int \sum _{j=1}^n{a}_{ij}({\mu }_i-{\mu }_j)dt\hfill \end{array}$$

(8)

When the system achieves the steady state, (9) holds

$$\frac{\partial f_1\left(p_1\right)}{\partial p_1}=\frac{\partial f_2\left(p_2\right)}{\partial p_2}=\cdots =\frac{\partial f_n\left(p_n\right)}{\partial p_n}$$

(9)

Then, the system equilibrium coincides with the optimal solution $p^{*}$, i.e., the optimal dispatch is realized. Here, the $\lambda$-consensus algorithm is embedded into the frequency control scheme.

Remark 1. 

Compared with the method in [31,32,33,34,41], it does not need the real-time information. However, the order of the control algorithm in [36] is high (the system in (8) is a 5-order system), which will increase the difficulty of stability analysis. Moreover, the λ-consensus algorithm depends on communications; the global optimal dispatch will not be realized if part of the communication link fails such that the communication network does not have a spanning tree. (The communication network has a spanning tree if there always exists a communication path connection for any two nodes [42].)

2.3. Motivations of This Paper

This paper investigates the ED problem of the microgrid considering line losses and communication delay. Specifically, we propose a distributed cooperative control method for optimal dispatch which has the following advantages.

(1) The optimal dispatch algorithm does not depend on communications. The optimal (or sub-optimal) dispatch can be realized even if there is no communication.

(2) The proposed control scheme does not need the real-time information of the load.

(3) The proposed control scheme can realize optimal dispatch and frequency recovery simultaneously if the communication is normal.

3. The Proposed Distributed Cooperative Control Method for Optimal Dispatch Considering Line Losses

3.1. The Optimality Conditions of the ED Problem Considering Line Losses

In this paper, the equivalent model of the microgrid is presented in Figure 1. There are n dispatchable DGs; all the loads and undispatchable DGs are equivalent as an instantaneous constant power load [43,44,45]. Moreover, the reactive power of the load is assumed as zero. Let $p_{load}$ denote the active power of the load. Let $V{e}^{j{\theta }_i},V{e}^{j{\theta }_2},\cdots ,V{e}^{j{\theta }_n}$ denote the voltages of the DGs, and $X_1{e}^{j{\alpha }_1},X_2{e}^{j{\alpha }_2},\cdots ,X_n{e}^{j{\alpha }_n}$ denote the cable’s impedance, where V, ${\alpha }_i$ and $X_i$ are positive constant scalars. Let $V_L{e}^{j{\theta }_L}$ denote the load voltage and P be the active power of the load.

The complex power of the i-th DG is given by

$$\begin{array}{cc}\hfill s_i=p_i+j{q}_i& =V{e}^{j{\theta }_i}\left(\frac{V{e}^{-j{\theta }_i}-V_L{e}^{-j{\theta }_L}}{X_i{e}^{-j\alpha }}\right)\hfill \\ \hfill & =V\left(\frac{V{e}^{j\alpha }-V_L{e}^{j({\theta }_i-{\theta }_L+{\alpha }_i)}}{X_i}\right)\hfill \end{array}$$

(10)

From (10), the DG’s active power is obtained as

$$\begin{array}{c}\hfill p_i=\frac{V^2}{X_i}cos{\alpha }_i-\frac{V{V}_L}{X_i}cos({\theta }_i-{\theta }_L+{\alpha }_i)\end{array}$$

(11)

Likewise, the complex power of the load is given by

$$\begin{array}{cc}\hfill s_L=p_{load}+j{q}_{load}& =V_L{e}^{j{\theta }_L}\sum _{i=1}^n\frac{V{e}^{-j{\theta }_i}-V_L{e}^{-j{\theta }_L}}{X_i{e}^{-j{\alpha }_i}}\hfill \\ \hfill & =V_L\sum _{i=1}^n\frac{V{e}^{j({\theta }_L-{\theta }_i+{\alpha }_i)}-V_L{e}^{j{\alpha }_i}}{X_i}\hfill \end{array}$$

(12)

Then, the power balance equation of the load is obtained as

$$\begin{array}{cc}\hfill & \sum _{i=1}^n(\frac{V{V}_L}{X_i}cos({\alpha }_i-({\theta }_i-{\theta }_L))-\frac{V_L^2}{X_i}cos{\alpha }_i)=p_{load}\hfill \\ \hfill & V_L\sum _{i=1}^n(\frac{V_L}{X_i}sin{\alpha }_i-\frac{V}{X_i}sin({\alpha }_i-({\theta }_i-{\theta }_L))=0\hfill \end{array}$$

(13)

Then, the total line loss is obtained as

$$\begin{array}{cc}\hfill & p_{loss}=\sum _{i=1}^n{p}_i-p_{load}\hfill \\ \hfill & =\sum _{i=1}^n(\frac{V^2}{X_i}cos{\alpha }_i-\frac{V{V}_L}{X_i}cos({\theta }_i-{\theta }_L+{\alpha }_i))-\sum _{i=1}^n(\frac{V{V}_L}{X_i}cos({\alpha }_i-({\theta }_i-{\theta }_L))-\frac{V_L^2}{X_i}cos{\alpha }_i)\hfill \\ \hfill & =\sum _{i=1}^n\frac{cos{\alpha }_i}{X_i}(V^2+V_L^2-V{V}_{L}cos({\theta }_i-{\theta }_L))\hfill \end{array}$$

(14)

Usually, $|{\theta }_i-{\theta }_L|$ is very small. Then, we will derive the approximate expression of $\frac{\partial p_{loss}}{\partial p_i}$ by the following assumptions

$$sin({\theta }_i-{\theta }_L)\approx {\theta }_i-{\theta }_L,cos({\theta }_i-{\theta }_L)\approx 1$$

(15)

Under the assumptions in (15), we have

$$\left\{\begin{array}{cc}\hfill & p_i=\frac{V(V-V_L)}{X_i}cos{\alpha }_i+sin{\alpha }_i\frac{V{V}_L}{X_i}({\theta }_i-{\theta }_L)\hfill \\ \hfill & p_{loss}=\sum _{i=1}^n\frac{cos{\alpha }_i}{X_i}{(V-V_L)}^2\hfill \\ \hfill & \sum _{i=1}^n(\frac{V_L}{X_i}sin{\alpha }_i-\frac{V}{X_i}(sin{\alpha }_i-({\theta }_i-{\theta }_L)cos{\alpha }_i))=0\hfill \end{array}\right.$$

(16)

From the first equation in (16), we have

$${\theta }_i-{\theta }_L={\left(\frac{V{V}_L}{X_i}sin{\alpha }_i\right)}^{-1}(p_i-\frac{V(V-V_L)}{X_i}cos{\alpha }_i)$$

(17)

Substituting (17) into the third equation in (16), we obtain

$$\left(\sum _{i=1}^n\frac{sin{\alpha }_i}{X_i}\right)(V_L-V)V_L+\sum _{i=1}^{n}cot{\alpha }_i{p}_i+\left(\sum _{i=1}^n\frac{co{s}^2{\alpha }_i}{X_{i}sin{\alpha }_i}\right)(V_L-V)V=0$$

(18)

From (18), (19) is derived

$$(\sum _{i=1}^n\frac{1}{X_{i}sin{\alpha }_i}\frac{V}{V-V_L}-\sum _{i=1}^n\frac{sin{\alpha }_i}{X_i}){(V_L-V)}^2=\sum _{i=1}^{n}cot{\alpha }_i{p}_i$$

(19)

Then, we obtain

$$p_{loss}\approx \frac{ϵ{\displaystyle \sum _{i=1}^n}\frac{cos{\alpha }_i}{X_i}}{{\displaystyle \sum _{i=1}^n}\frac{1}{X_{i}sin{\alpha }_i}-ϵ{\displaystyle \sum _{i=1}^n}\frac{sin{\alpha }_i}{X_i}}\sum _{i=1}^{n}cot{\alpha }_i{p}_i$$

(20)

where $ϵ=\frac{V-V_{L,min}}{V}$ which is the largest acceptable voltage deviation ratio. Then, the approximate expression of $\frac{\partial p_{loss}}{\partial p_i}$ is given by

$$\frac{\partial p_{loss}}{\partial p_i}=\beta cot{\alpha }_i,\beta =\frac{ϵ{\displaystyle \sum _{i=1}^n}\frac{cos{\alpha }_i}{X_i}}{{\displaystyle \sum _{i=1}^n}\frac{1}{X_{i}sin{\alpha }_i}-ϵ{\displaystyle \sum _{i=1}^n}\frac{sin{\alpha }_i}{X_i}}$$

(21)

Optimality Conditions. Therefore, by invoking the result in (5), the optimality conditions for the economic problem considering line losses are obtained as

$$\frac{1}{1-\beta cot{\alpha }_1}\frac{\partial f_1\left(p_1\right)}{\partial p_1}=\frac{1}{1-\beta cot{\alpha }_2}\frac{\partial f_2\left(p_2\right)}{\partial p_2}=\cdots =\frac{1}{1-\beta cot{\alpha }_n}\frac{\partial f_n\left(p_n\right)}{\partial p_n}$$

(22)

Note that the optimization problem in (2) is convex since (20) holds. Then, if and only if the system operation point satisfies the optimality conditions in (22), the optimal dispatch is achieved.

3.2. The Proposed Distributed Cooperative Control Method

In this section, we will propose a distributed cooperative control method to achieve optimal dispatch and frequency regulation.

The proposed distributed cooperative control consists of two layers, namely the primary control and the secondary control. The control block diagram of the proposed method is in Figure 2. The primary control is designed as

$${\dot{\theta }}_i={\omega }^{*}-m{(1-\beta cot{\alpha }_i)}^{-1}\frac{\partial f_i\left(p_i\right)}{\partial p_i}+x_i$$

(23)

where m is the coefficient to keep the frequency within the acceptable range, $x_i$ is the second frequency recovery item, and $x_i$ is designed as

$${\dot{x}}_i=-d\sum _{i=1}^n{a}_{ij}(x_i-x_j)+d({\omega }^{*}-{\dot{\theta }}_i)$$

(24)

where d is a positive scalar, $a_{ij}=1$ if there is a communication link between DG i and j, and $a_{ij}=0$ otherwise. Then, the compact form of the system dynamics is given by

$$\left\{\begin{array}{cc}\hfill & \dot{\theta }={\omega }^{*}{1}_n-mK\frac{\partial f\left(p\right)}{\partial p}+x\hfill \\ \hfill & \dot{x}=-d(Lx+\dot{\theta }-{\omega }^{*}{1}_n)\hfill \end{array}\right.$$

(25)

where $k_i={(1-\beta cot{\alpha }_i)}^{-1};K=diag{k}_i$; $1_n={\left[\begin{array}{ccc}1& 1& \cdots 1\end{array}\right]}^T$; x and $\theta$ are the vectors of $x_i$ and ${\theta }_i$, respectively; L is the Laplacian matrix of the communication network, whose off-diagonal element is $a_{ij}$; and the i-th diagonal is ${\displaystyle \sum _{j=1}^n}{a}_{ij}$. The communication network has a spanning tree if there always exists a communication path connection for any two nodes.

Since communication failures happen sometimes, which may have a negative effect on the optimal dispatch, next, we analyze the steady state of the system when the communication is normal and fails completely.

Steady-State Analysis under Normal Communication. It is noted that the system can reach the steady state only when the system frequency is synchronized. Then, when the system reaches the steady state, we have

$$\left\{\begin{array}{cc}\hfill & {\dot{\theta }}_1={\dot{\theta }}_2=\cdots ={\dot{\theta }}_n\hfill \\ \hfill & Lx+{\omega }^{*}{1}_n-\dot{\theta }=0_n\hfill \end{array}\right.$$

(26)

If we multiply the left by $1_n^T$, the second equation in (26) becomes

$$n{\omega }^{*}=1_n^T\dot{\theta }$$

(27)

By invoking the results in the first equation in (26), we have

$${\dot{\theta }}_1={\dot{\theta }}_2=\cdots ={\dot{\theta }}_n={\omega }^{*}$$

(28)

Then, the second equation in (26) becomes

$$Lx=0_n$$

(29)

According to graph theory, (30) holds if and only if the communication network has a spanning tree

$$x_1=x_2=\cdots =x_n$$

(30)

Then, the following is obtained

$$K\frac{\partial f\left(p\right)}{\partial p}=\frac{x}{m}$$

(31)

The result in (31) indicates that (22) holds, i.e., the optimal dispatch is realized.

Remark 2. 

The results show that even if some communications fail, optimal dispatch and frequency recovery can be achieved as long as the communication network has a spanning tree.

Steady-State Analysis when the Communication Fails Completely. When the communication fails completely (i.e., the communication does not have a spanning tree), $x_i$ will be reset as zero. When the system reaches the steady state, the following is obtained

$$K\frac{\partial f\left(p\right)}{\partial p}=\frac{{\omega }^{*}{1}_n-\dot{\theta }}{m}$$

(32)

By invoking the first equation in (26), (32) indicates that (22) holds, i.e., optimal dispatch is realized.

3.3. Comparative Analysis

In the proposed method, the optimal dispatch method is implemented in the primary control, which is based on the droop control. When the system frequency is synchronized, the optimal dispatch is achieved. Comparing with the method in [36], the optimal dispatch in the proposed method does not depend on the communication. Moreover, the proposed method is simpler than the method in (8). Comparing with the method in [29,30,31,32,33], the proposed method does not need the real-time load information. Comparing with [37,38], the proposed method has considered the impact of line losses; therefore, it may have better performance.

Remark 3. 

The critical contributions of this paper are the following. First, the line losses are considered in the optimization model, and optimality conditions are derived based on Lagrange’ multiplier method. Second, the cost optimization and frequency regulation are implemented in the primary and secondary control, respectively. Therefore, the optimal dispatch can be achieved without communications, and only frequency recovery depends on communications. Third, the stability of the system considering the cable’s resistance and communication time delay is analyzed, and stability conditions are provided.

4. Stability Analysis of the System under the Proposed Method

If the system is stable, the global optimal dispatch will be realized. However, the time delay and the cable’s resistances may lead to system instability. Next, we will analyze the system when it has no communication, the communication without delay and the communication with delay.

4.1. Stability Analysis of the System without Communications

When the system has no communication, $x=0_n$. Considering that

$$\frac{\partial f\left(p\right)}{\partial p}=2Ap+b$$

(33)

where $A=diag\left\{a_i\right\}$ and $b=\left[\begin{array}{cccc}{b}_1& b_2& \cdots & b_n\end{array}\right]$. Then, the dynamics of the DGs are given by

$$\dot{\theta }={\omega }^{*}{1}_n-mK(2Ap+b)$$

(34)

By neglecting the dynamics of the $v_L$, the small-signal model of (34) is

$$\Delta \dot{\theta }=-2mKA(M_1\Delta \theta -M_2{\theta }_L)$$

(35)

where $M_1=diag\left\{\frac{V{V}_L}{X_i}sin({\alpha }_i+{\theta }_i^{*}-{\theta }_L^{*})\right\}$, $M_2=M_1{1}_n$, and ${\theta }_i^{*}-{\theta }_L^{*}$ is the phase difference at the steady state. The

$$\Delta {\theta }_L=M_3\Delta \theta ,M_3={\left(\sum _{i=1}^n\frac{V{V}_L}{X_i}sin({\alpha }_i-({\theta }_i^{*}-{\theta }_L^{*}))\right)}^{-1}{1}_n^{T}diag\left\{\frac{V{V}_L}{X_i}sin({\alpha }_i-({\theta }_i^{*}-{\theta }_L^{*}))\right\}$$

(36)

Then, (35) becomes

$$\Delta \dot{\theta }=-2mKA{M}_4\Delta \dot{\theta },M_4=M_1-M_2{M}_3$$

(37)

Then, the system in (34) is stable if $KA{M}_4$ has only one zero eigenvalue and the real parts of other eigenvalues are positive. The presence of the simple zero eigenvalue corresponds to the rotational invariance of the system [46]. The main results follow.

Theorem 1. 

The system in (34) is stable if there exists a positive definite matrix P such that $PKA{M}_4+M_4^{T}KAP$ is a positive semi-definite matrix and has a simple zero eigenvalue.

Proof. 

According to Lyapunov’s inertia theorem [47], if there exists a positive definite matrix P such that $PKA{M}_4+M_4^{T}KAP$ is a positive semi-definite matrix and has a simple zero eigenvalue, then the real part of the eigenvalue of $KA{M}_4$ has a non-negative part. Moreover, the Jacobian matrix must have a simple zero eigenvalue due to its rotational invariance. Then, Theorem 1 is proved. □

Usually, $|{\theta }_i-{\theta }_L|$ is very small. Then, we assume ${\theta }_i-{\theta }_L=0$, and $M_4$ becomes

$$M_4=diag\left\{\frac{V{V}_L}{X_i}sin{\alpha }_i\right\}-{\left(\sum _{i=1}^n\frac{V{V}_L}{X_i}sin{\alpha }_i\right)}^{-1}diag\left\{\frac{V{V}_L}{X_i}sin{\alpha }_i\right\}{1}_n{1}_n^{T}diag\left\{\frac{V{V}_L}{X_i}sin{\alpha }_i\right\}$$

(38)

which is a Schur complement of the following positive semi-definite matrix with a simple zero eigenvalue

$$M_5=\left[\begin{array}{cc}diag\left\{\frac{V{V}_L}{X_i}sin{\alpha }_i\right\}& -diag\left\{\frac{V{V}_L}{X_i}sin{\alpha }_i\right\}{1}_n\\ -1_n^{T}diag\left\{\frac{V{V}_L}{X_i}sin{\alpha }_i\right\}& {\displaystyle \sum _{i=1}^n}\frac{V{V}_L}{X_i}sin{\alpha }_i\end{array}\right]$$

(39)

Therefore, the system is stable if the phase difference $|{\theta }_i-{\theta }_L|$ is small enough. In summary, the system without communication is stable if Theorem 1 holds.

4.2. Stability Analysis of the System without Communication Delay

When the communication is normal and has no time delay, the system dynamics are given by

$$\left\{\begin{array}{cc}\hfill & \dot{\theta }={\omega }^{*}{1}_n-mK\frac{\partial f\left(p\right)}{\partial p}+x\hfill \\ \hfill & \dot{x}=-d(L+I)x-dmK\frac{\partial f\left(p\right)}{\partial p}\hfill \end{array}\right.$$

(40)

Then, the small-signal model of the system in (40) is obtained as

$$\left[\begin{array}{c}\Delta \dot{\theta }\\ \Delta \dot{x}\end{array}\right]=\left[\begin{array}{cc}-2mKA{M}_4& I\\ 2dmKA{M}_4& -d(L+I)\end{array}\right]\left[\begin{array}{c}\Delta \theta \\ \Delta x\end{array}\right]$$

(41)

Next, we will derive the stability condition based on the singular perturbation theorem [48]. The main results follow.

Theorem 2. 

Assume the condition in Theorem 1 holds. Then, there is a $d^{*}$ such that the system in (40) is stable when $0<d\le d^{*}$.

Proof. 

Assume d is small enough.Then, the system in (41) takes the following form

$$\left[\begin{array}{c}d\Delta \dot{\theta }\\ \Delta \dot{x}\end{array}\right]=d\left[\begin{array}{cc}-2mKA{M}_4& I\\ 2mKA{M}_4& -(L+I)\end{array}\right]\left[\begin{array}{c}\Delta \theta \\ \Delta x\end{array}\right]$$

(42)

which is a singular perturbation system. According to Theorem 11.4 in [48], the system is stable if the bound-layer system and the reduced-order system are both stable.

In fact, the bound-layer system (i.e., the fast subsystem ) is the same as the system in (37). According to Theorem 1, it is stable if $M_4+M_4^T$ is a positive semi-definite matrix and has a simple zero eigenvalue. The reduced-order system (i.e., the slow system) is given by

$$\Delta \dot{x}=-L\Delta x$$

(43)

It is easy to find that the reduced-order system is stable. Therefore, the system in (40) is stable if the conditions in Theorem 2 hold. The proof is accomplished. □

Remark 4. 

In this part, the stability analysis is carried out based on the singular perturbation theorem. If the bound-layer system and the reduced-order system are both stable, there is $d^{*}$ such that the system is stable if $0<d\le d^{*}$. However, how do we compute the critical value $d^{*}$? In fact, there are mainly two methods. The first method is presented in the proof of Theorem 11.4 in [48]. The second is the numerical method such as the root locus method.

4.3. Stability Analysis of the System with Communication Delay

In a distributed control framework, the low-bandwidth neighbor communication is usually used. Therefore, the time delay usually occurs. Next, we will derive the stability conditions of the system under time delay.

The small-signal model of the system under time delay is given by

$$\left[\begin{array}{c}d\Delta \dot{\theta }\left(t\right)\\ \Delta \dot{x}\left(t\right)\end{array}\right]=d\left[\begin{array}{cc}-2mKA{M}_4& I\\ 2mKA{M}_4& -I\end{array}\right]\left[\begin{array}{c}\Delta \theta \left(t\right)\\ \Delta x\left(t\right)\end{array}\right]-d\left[\begin{array}{cc}O& \\ & L\end{array}\right]x(t-\tau )$$

(44)

where $\tau$ is the time delay. It is noted that only state variables that need to be communicated are delayed. The main results follow.

Theorem 3. 

Assume the condition in Theorem 1 holds, and $0<d\le d^{*}$ ($d^{*}$ is defined in Theorem 2). Then, the system with delay is stable if

$$\tau <\frac{\pi }{2d{\lambda }_{max}\left(L\right)}$$

(45)

Proof. 

Since d is small, the system with delay takes the form in (44), which is also a singular perturbation system. Moreover, the bound-layer system is the same as (37), and the reduced-order system is given by

$$\Delta \dot{x\left(t\right)}=-L\Delta x(t-\tau )$$

(46)

Then, according to the result in [42], the system in (46) is stable if (45) holds. □

5. Simulations

To verify the effectiveness of the proposed distributed cooperative control method, the simulation model of an AC microgrid containing four DGs and one load is established based on Matlab/Simulink. The topology is a star which is shown in Figure 1. DGs are modeled as controlled voltage sources, and the dynamics of the filters of the converters are neglected. Each DG is under the proposed distributed cooperative control method. Cables are resistive–inductive lines, and the load is a constant power load. Specific parameters follow.

Physical Parameters. The voltage amplitude of the DG is designed as $V=220$ V, and all DGs have the same voltage amplitude. Cable impedances are $X_1{e}^{j{\alpha }_1}=5e^{j\frac{\pi }{3}}\Omega$, $X_2{e}^{j{\alpha }_2}=2e^{j\frac{\pi }{6}}\Omega$, $X_3{e}^{j{\alpha }_3}=5e^{j\frac{\pi }{3}}\Omega$ and $X_4{e}^{j{\alpha }_4}=4e^{j\frac{\pi }{6}}\Omega$. The reactive power of the load is 0, and the active power of the load is designed as $p_{load}=2$ kW at $0<t\le 5$ s, $p_{load}=2.5$ kW at $5<t\le 10$ s, $p_{load}=4$ kW at $10<t\le 15$ s and $p_{load}=5.5$ kW at $15<t$ s. The parameters of the cost functions are $a_1=0.01$, $b_1=40$, $a_2=0.02$, $b_2=40$, $a_3=0.01$, $b_3=10$, $a_4=0.04$, $b_4=20$, $c_1=c_2=c_3=c_4=0$.

Control Parameters. The Laplacian matrix of the communication network is designed as

$$L=\left[\begin{array}{cccc}2& -1& 0& -1\\ -1& 2& -1& 0\\ 0& -1& 2& -1\\ -1& 0& -1& 2\end{array}\right]$$

Assume the $ϵ=0.1$. Then, K is obtained as $k_1=1.27,k_2=1.04,k_3=1.27$, $k_4=1.04$, $K=\left[\begin{array}{cccc}1.27& 1.04& 1.27& 1.04\end{array}\right]$. $d=5$ and $m=0.01$.

5.1. Stability Test of the System under the Proposed Method

To test the proposed adaptive protection scheme, the following three cases are designed.

Case 1. The communication is normal and has no time delay.

Case 2. The communication is normal but has a time delay and $\tau =0.06$ s.

Case 3. The communication is normal but has a time delay, and $\tau =0.13$ s at $0<t\le 5$ s. The communication is deprecated when $t>5$ s.

Case 4. The communication is normal and has no time delay. The fourth DG starts to work at $t=5$ s and stops working at $t=15$ s, while the other DGs are normal.

Stability Analysis for Cases 1–3. According to Theorems 1 and 2, if there exists a positive definite matrix P such that $PKA{M}_4+M_4^{T}KAP$ is a positive semi-definite matrix and has a simple zero eigenvalue, the system in Case 1 is stable if d is sufficiently small. By calculations, the second small eigenvalue of $KA{M}_4$ is 198.0, 197.2, 193.2 and 187.8 with the load changes. Therefore, the system with no communication is stable, and the system in Case 1 is stable if d is sufficiently small. By calculations, $d^{*}$ is obtained as $d^{*}=43.5$. Since $d=5<d^{*}=43.5$, therefore, the system in Case 1 will be stable. According to Theorem 3, the system is stable if $\tau <\frac{\pi }{2d{\lambda }_{max}\left(L\right)}=\pi /40=0.0785$. Therefore, the system in Case 2 will be stable. For the system in Case 3, the system may be unstable when $0<t<5$ s since $\tau =0.13>0.0785$. However, when $t>5$ s, the communication is deprecated, and the system will be stable. Meanwhile, the optimal dispatch is realized, and only the frequency recovery cannot be achieved.

Plug-and-play capability analysis in Case 4. In the proposed method, only the control parameter $\beta$ depends on the cable impedances of the whole system. When a DG joins or exits, we only need to update the value of $\beta$. Then, the optimal dispatch and frequency regulation will be realized. In Case 4, the fourth DG joins at $t=5$ s. Then, $\beta$ should change from $1.17$ to $1.22$. Likewise, the fourth DG exists at $t=15$ s, and $\beta$ should change from $1.22$ to $1.17$.

Reviews about Simulation Results. Simulation results of Cases 1, 2 and 3 are presented in Figure 3, Figure 4 and Figure 5. Simulation results about the total line loss (i.e., the difference between the total power demanded and the total power dispatched) and the total cost is shown in Figure 6. Results in Figure 3 and Figure 4 show that the system is stable, which verifies the correctness of the stability results in Theorems 2 and 3. Results in Figure 3c and Figure 4c show that ${\lambda }_i$ is synchronized, and the frequency is regulated to 50 Hz, which verifies that the proposed method can achieve optimal dispatch and frequency recovery. In Case 3, due to the time delay, the system is unstable. When the communication is deprecated, the system is stable. Moreover, results in Figure 5c show that ${\lambda }_i({\lambda }_i=k_i\frac{\partial f_i\left(p_i\right)}{\partial p_i})$ is synchronized after the communication with a heavy delay is deprecated, which verifies that the optimal dispatch in the proposed method does not depend on the communication.

In Case 1 and Case 2, the communication is normal (it may have a slight delay), so the frequency is regulated to 50Hz due to the regulation of the secondary control. In Case 3, communication failure occurs at $t=5$ s, and there is no communication when $t>5$ s. So, frequency recovery is not achieved, and lambda consensus is achieved, i.e., optimal dispatch is realized. In Case 3, the communication is normal but has a time delay (0.13 s) at $0<t<5$ s. When $t=5$ s, communication failure occurs, and the communication is deprecated when $t>5$ s. Simulation results in Figure 4b and Figure 5b show that the system is stable, and optimal dispatch is achieved when $t>5$ s (i.e., the communication is deprecated). Therefore, the simulation results verify that the proposed method can achieve optimal dispatch without communications.

It is highlighted that the active power mismatch (i.e., the difference between the total power demanded and the total power dispatched) is the line loss, whose mathematical expression is ${\displaystyle \sum _{i=1}^n}{I}_i^2{X}_{i}cos{\alpha }_i$, where $I_i$ is the output current of the i-th DG. When the power of the load increases, the output active power and output current will increase. Therefore, the line loss will increase if the load or ${\lambda }_i$ increases. Due to the line loss, the total power dispatched is always larger than the total power demanded, i.e., the active power mismatch always exists. When the line is purely inductive or the line resistance is very small, i.e., ${\alpha }_i$ is equal or close to $\frac{\pi }{2}$, the line loss is small, and the active power mismatch is near 0. This paper mainly considers the optimal dispatch of the microgrid with line losses (i.e., the line resistance is large). In the tested microgrid, the line resistances are large, $X_{1}cos{\alpha }_1=2.5\Omega$, $X_{2}cos{\alpha }_2=1.732\Omega$, $X_{3}cos{\alpha }_3=2.5\Omega$, $X_{4}cos{\alpha }_4=3.464\Omega$, and the output currents are 6.39 A, $4.90$ A, $13.95$ A and $3.59$ A, respectively. So, when $t>15$ s, the line loss in Figure 6a is $674.82$ W. In fact, the total power dispatched (6174.82 W) = the total power demanded (5.5 kW) + the line loss (674.82 W). Therefore, the active load demand is completely fulfilled under the proposed method.

In Case 4, the fourth DG is out of service when $0<t<5$ s and $15<t<20$ s, so the output power $p_4$ is zero during this time, which is shown in Figure 7a. For the value of ${\lambda }_4$ when the fourth DG is out of service, on the one hand, the incremental cost (lambda) is derived by ${\lambda }_4=k_4\frac{\partial f_4\left(p_4\right)}{\partial p_4}=k_4(2a_4{p}_4+b_4)$, so ${\lambda }_4=k_4{b}_4=20.8$ when $p_4=0$, which is shown in Figure 7b. On the other hand, once a DG is out of service, its cost is always 0, so the incremental cost should be 0. In fact, the incremental cost of the DG that stops working can be 20.8, 0 or any other number because once a DG is out of service, it will not communicate with other DGs, and the information of the incremental cost will not be sent to other DGs. That is, the value of this incremental cost does not participate in any calculations, and we only control three other DGs by the proposed method to achieve optimal dispatch. When the fourth DG operates normally, the incremental cost (${\lambda }_4$) should be ${\lambda }_4=k_4\frac{\partial f_4\left(p_4\right)}{\partial p_4}$. Therefore, for the consistency of the expression of ${\lambda }_4$, we take ${\lambda }_4=k_4(2a_4{p}_4+b_4)$ when $p_4=0$. So, when the fourth DG is out of service, its incremental cost is 20.8 instead of 0 in Figure 7b.

In Figure 7d, it is noted that when a DG is out of service, it will not contribute towards the total cost. So, the total cost in Case 4 is ${\sum }_{i=1}^4{f}_i\left(p_i\right)$ when $5\le t\le 15$ s and ${\sum }_{i=1}^3{f}_i\left(p_i\right)$ when $0<t<5$ s and $15<t<20$ s. Case 4 and Case 2 are the same when $5\le t\le 15$ s, and Case 4 lacks a cheaper DG (i.e., the fourth DG) when $0<t<5$ s and $15<t<20$ s. Therefore, the total cost in Case 4 (i.e., ${\sum }_{i=1}^3{f}_i\left(p_i\right)$) is larger than the total cost in Case 2 when $0<t<5$ s and $15<t<20$ s, which is shown in Figure 6b and Figure 7d.

The fourth DG joins at $t=5$ s and exists at $t=15$ s. Figure 7a shows that the system can operate normally. Moreover, the $\lambda -consensus$ is always realized, which shows that the proposed method can achieve plug-and-play.

5.2. Compared with the Existing Method in [36]

Compared with the existing methods, the biggest advantage of the proposed method is that the optimal dispatch does not depend on the communication. Moreover, the existing method in [29,30,31,32,33] needs real-time information, which increases the cost and is difficult to obtain. The method in [36] does not need real-time load information, but it does not consider the effect of the cable resistance, which may bring an error in optimal dispatch. The comparison results between the proposed method and the method in [36] are presented in

Table 1. Comparison results between the proposed method and the method in [36].

	The Costs of Case 1/ Case 4 under the Proposed Method	The Costs of Case 1/Case 4 under the Method in [36]
$p_{load}$ = 2 kW	64,719.2/69,474.5	64,796.8/69,637.7
$p_{load}$ = 2.5 kW	89,269.8/89,269.8	89,540.0/89,540.0
$p_{load}$ = 4 kW	179,104.4/179,104.4	181,067.6/181,067.6
$p_{load}$ = 5.5 kW	296,492.8/329,715.1	302,075.7/336,635.2

, which shows that the operation cost $f\left(p\right)$ of the system under the proposed method is lower than the system under the method in [36]. The operation cost curves of the proposed method and the existing method in [36] are provided in Figure 8. It is shown that the proposed method has a lower operation cost.

5.3. Discussions about the Performance of the Proposed Method under Parameter Uncertainties

One of the advantages of the proposed method does not need the information of the load, which implies that the proposed method is strongly robust under load uncertainties. In Cases 1–3, the load changes over time. However, the results in Figure 6 show that ${\lambda }_i$-consensus is realized (i.e., optimal dispatch is achieved) even if the load is time-varying. On the other hand, the parameter matrix K depends on the cable impedances, especially the ratio $R/X$. If the cable impedances are uncertain, these deviations may have a non-negligible negative impact on optimal dispatch. This is a defect of the proposed method, and we will investigate and try to address it in future work.

6. Conclusions

In this paper, a novel distributed cooperative control method is proposed, which can achieve optimal dispatch and frequency regulation at the same time. Specifically, when communication is normal, optimal dispatch and frequency regulation can be realized simultaneously. Even if all the communication fails, the optimal dispatch is still realized. Moreover, the stability conditions of the system considering the effect of the resistance and time delay are derived. The simulation results verify the effectiveness of the proposed method. In future work, we will investigate the optimal dispatch scheme for the system with a meshed high $R/X$ ratio network and improve the robustness against the uncertainties of cable impedances.