Distributed Economic Control Method for Energy Storage Unit with Random Time Delay

Abstract

The operation cost waste in the charge and discharge process cannot be ignored for islanded microgrids with energy storage units. Different from the economic dispatch methods focused on the power pricing and bid coefficient in the tertiary control layer, this paper designed a power management algorithm for the economic operation of energy storage units in the secondary control layer. The strategy is devoted to minimizing the total charge power cost of energy storage units while satisfying the constraint of power balance. Meanwhile, the maximized allowable delay bounds were derived to guarantee the communication network reliability when against random time delays. The system stability was analyzed by the Lyapunov functions. Finally, the numerical simulations on the modified IEEE 30 feeder model verify the superiority of the provided algorithm.

1. Introduction

Unlike AC microgrids influenced by reactive power regulation and frequency fluctuations, DC microgrids are easier to control and attract academic interest. The hierarchical structure is a conventional method used to control islanded DC microgrids. Based on different time scales, the hierarchical structure can be operated with separated control layers, including primary, secondary, and tertiary control layers [1,2]. In the primary control layer, the voltage-based droop curve can adjust the active power output to maintain voltage stability after disturbance and generate steady-state voltage deviation. In the secondary control layer, voltage deviation can be eliminated with the distributed controller. In the tertiary control layer with the longest time-scale, the output of the power system can be dispatched in the economical operation [3,4,5].

Most previous studies focused on power scheduling to realize economic dispatch at the tertiary control layer. Specifically, multiple kinds of neural network algorithms have been applied in the tertiary control layer to regulate power allocation for economic dispatch. In [6], an improved hybrid bat method is proposed to achieve balanced power distribution. This method considers the reactive power dispatch and optimizes the combined scheme. Ref. [7] proposed a squirrel search method to address the environmental and economic active power sharing issue by ensuring the diversity of Pareto-optimal solutions. Some extreme scenarios were considered in [8] to guarantee the day-ahead economic dispatch of islanded microgrids with wind power. Based on a quadratic programming method, the proposed economic dispatch could solve wind power limitation and load-shedding problems. In order to address the economic dispatch issue with different scenarios, ref. [9] combined the Lagrangian relaxation method and the proximal method for faster convergence. Moreover, power pricing is an essential factor in addressing the economic dispatch issue of multi-microgrid systems. To realize power dispatch following demand forecasts, a temporal locational marginal pricing method is provided in [10]. However, neural network algorithms, scenario-based solutions, and pricing strategies focused on the day-ahead power dispatch at the tertiary control layer but ignored the economic dispatch at the secondary control layer.

In the secondary control layer, the economic dispatching issue can be divided into two aspects: the generation cost minimization of the generator unit and the charging/discharging power cost minimization of the energy storage unit. For generator units, ref. [11] presented an automatic generation control method to establish the necessary and sufficient conditions for maintaining the global minimum of steady-state synchronous generator outputs. Considering the constraints of local generators, ref. [12] designed a distributed optimization method to achieve the optimal method of the economic dispatch issue. Ref. [13] addressed the network-overloading problem using particle swarm optimization. In particular, the excessive loads could be balanced with the operating generators. Local constraints could be satisfied, and the load could be dispatched to each distributed generation (DG) using a decentralized optimization method with a penalty function in [14]. Compared with power generation units, there are few research studies on the economic dispatch of energy storage units. By employing a mixed-integer linear programming strategy, ref. [15] calculated the lowest levelized cost of energy storage units (ESUs) and evaluated the generation cost, hence determining the Pareto optimal storage configuration. Considering the microgrids with multisource, ref. [16] designed a multi-objective optimal method to achieve a profitable comprehensive income and eliminate factors that cause system instability. In order to expand the economic application of energy storage units, this paper cares about the solutions used to minimize the operation cost of batteries during the charging/discharging process.

The communication network plays a key role in the control process at the secondary control layer. In order to maintain the robustness of communication links, ref. [17] presented a consensus-based distributed method used to realize the optimal dispatch of active power. Meanwhile, the delay bound was calculated by the generalized Nyquist criterion in [17]. Ref. [18] focused on the coexistence delay of the communicate process and transmit process and provided a novel combined-delay-based augmented Lyapunov functional to describe the sampling pattern. Based on the linear matrix inequalities, the system stability could be guaranteed. By employing an aperiodic communication method, ref. [19] relaxed the periodicity of the intermittent data transmission and saved the communication resource. Furthermore, the possibility of time delay is not fixed in the communication network. Hence, the influence of random delay cannot be ignored. To solve the random communication delay problem, ref. [20] proposed a random delay model with conditional probabilities and provided a novel state estimator with a random parameter matrix Kalman filter. To satisfy the constraint of supply/demand power, ref. [21] calculated the maximized allowable bound to maintain the system stability with an economical operation. However, the impact of random delay on the information transmit process of ESUs has received little attention.

Summarizing the above analysis, the current distributed economic power dispatch method did not address the operating waste issue of energy storage units in the secondary control layer well. Meanwhile, the power balance constraint of batteries should be considered in the control process. To maintain the communication network stability in the secondary control layer, it is essential to further research how to deal with randomly occurring time delays in communication links.

In order to address the above shortcomings, this paper presents a distributed cooperative economic dispatch algorithm for energy storage units. The salient novel contributions of this paper are:

1. Different to economic dispatch methods for generator units, we provided a distributed cooperative method to minimize the total charge power cost of energy storage units. A second-order polynomial function was used to describe the operating cost of batteries, and Lagrange optimization methods were used to coordinate incremental costs.

2. In order to meet the constraint of power balance, we added a power mismatch auxiliary variable to the controller. This variable can adjust the injected/output power of each node and ensure the balance constraint during the control process.

3. To work against the random time delay in the communication network, we derived the maximum allowable boundary of delay based on the probability of delay occurrence. The closed-loop system stability analysis and simulation show the robustness of the presented solution.

This paper is organized as follows. Preliminaries are provided in Section 2. An economic distributed algorithm for energy storage units with random time delay is developed in Section 3. In Section 4, different cases are proposed to verify illustrative results. Finally, Section 5 concludes the paper.

2. Preliminaries

The islanded DC microgrid consists of a physical network and communication network. The physical network is composed of six DG nodes, as shown in the red box in Figure 1. Each DG is connected to the DC bus by the corresponding DCDC converter, which contains the local primary controller and the distributed secondary controller. Through the local primary controller, each node can eliminate voltage deviation and power fluctuation through droop control. Based on the communication network (black dotted line) in Figure 1, the distributed secondary controller can exchange state information and realize the control objectives.

2.1. Algebraic Graph Theory

Complex network system G(N,E) can describe the islanded DC power system with an energy storage unit. N is the node set, and each DG node contains a PV unit and a battery unit. Edge set E is E ∈ N × N. Moreover, the communication network for the complex network system can be described as adjacency matrix A = [a_ij] and Laplace matrix L = [l_ij], where a_ij describes the coupling relationship between node i and j, and l_ii = ∑_i≠j a_ij, l_ij = −a_ij. Note that the Laplace matrix satisfies ∑l_ij = 0. The nominal value is given by the virtual leader, while gain matrix B = diag{g₁, …, g_N} represents the communication links with the virtual leader node. Specifically, V* is set as the nominal voltage, and the total load P_L is a constant.

2.2. Economic Dispatch

In order to realize the economic operation of energy storage units, the total operation cost of batteries should be minimized; hence, the optimization problem can be defined as:

$$\begin{array}{l}\min f(P_{B,i})=\min ({\displaystyle \sum _{i=1}^N{\nu }_i{P}_{B,i}})=\min ({\displaystyle \sum _{i=1}^N-{\alpha }_i{P}_{B,i}^2+{\beta }_i{P}_{B,i}})\hfill \\ s.t.{\displaystyle \sum _{i=1}^N{P}_{B,i}}={\displaystyle \sum _{i=1}^N{P}_{G,i}}-{\displaystyle \sum _{i=1}^N{P}_{L,i}}\hfill \end{array}$$

(1)

The control objective in (1) is to minimize the total charge cost and battery consumption power. The charge efficiency defines the actual charge power and charge power, which can be described as ν_i = −α_iP_B,i + β_i. Inspired by [22], the charge efficiency function is a fitted polynomial curve, where α_i is the first-order fitting parameter, and β_i is the constant fitting parameter. Moreover, the power balance constraint should be satisfied with the total generation power ∑P_G,i and the total load demand ∑P_L,i. Then, the increment cost variable ${\varpi }_i$ is used to solve the optimization problem by the Lagrange function $Lag={\displaystyle \sum _{i=1}^n{f}_i(P_{B,i})}+{\displaystyle \sum _{i=1}^n{\varpi }_i(P_{G,i}-P_{L,i}-P_{B,i})}$.

2.3. Control Structure

In the islanded DC power system, the node voltage and active power output are the most important control objectives. Based on the voltage droop control rule [23], the voltage V_i changes with active power output P_i in the ith converter through

$$V_i-V_i^{*}=D_i(P_i^{*}-P_i),i\in v_I$$

(2)

where P_i*, D_i, and V* are the nominal power, droop parameter, and nominal voltage, respectively. It can be seen that, once the local load P_L,i fluctuates with ∆P_L,i, the voltage deviation ${\widehat{V}}_i=V_i-V^{*}$ becomes nonzero. The secondary controller in the DC/DC converter then adjusts P_i* for the power compensation ∆P_L,i. Consider the injected/output power balance. There exists P_i = P_G,i − P_B,i in each node, where P_G,i and P_B,i are the power generation and the charging power for the battery unit, respectively. Hence, to force ${\widehat{V}}_i=0$ for all agents, the secondary controller forces the nominal power to track the reference P_i* = P^*_i,ref = P_G,i − P_B,i^ref. The control objective of the primary droop controller and the secondary controller can be defined as:

$$\begin{array}{l}\underset{t\to \infty }{\lim }{P}_i^{*}=P_{i,ref}^{*}=P_{G,i}-P_{B,i}^{ref},i\in v_I\hfill \\ \underset{t\to \infty }{\lim }{\widehat{V}}_i=0,i\in v_I\hfill \end{array}$$

(3)

3. Proposed Control Scheme

Inspired by [21,22], this paper constructed an islanded power system that consists of a physical power system and communication network. The physical power system is composed of multiple distributed generation (DG) nodes, each containing a renewable energy source (RES) generation unit and an energy storage unit. Meanwhile, the communication links between the DC/DC converters constitute the communication network. In order to reduce the charging power loss in the physical power network and avoid the random delay in the communication system, we provided a distributed cooperative control algorithm for the economic operation of a battery with random delay effects.

3.1. Distributed Cooperative Control for Economic Operation

3.1.1. Voltage Restoration

To eliminate voltage deviations and force voltages to follow the reference values, we propose a leader–follower control method as follows:

$${\dot{V}}_i^B(t)={\displaystyle {\sum }_{\{i,j\in N_c\}}{a}_{ij}(V_j^B(t)-V_i^B(t))}+g_i(V_i^{*}-V_i^B(t))$$

(4)

When the node can connect to the virtual leader node, the connection parameter g_i ≠ 0; otherwise, g_i = 0. Considering all nodes, Equation (4) can be combined into matrix form ${\dot{V}}^B=-(L+G)V^B$, where G is the communication connection matrix.

3.1.2. Power Balance

The operation cost of the power system can be minimized when increment costs of different batteries are controlled to be synergized. Hence, a distributed consensus algorithm with second-order dynamics was designed to realize the control objective. Meanwhile, the auxiliary power mismatch variable ρ_i was added into the control process to meet the constraint of power balance:

$$\begin{array}{l}{\dot{P}}_{B,i}(t)=-2{\alpha }_i{\dot{\varpi }}_i^B(t),\hfill \\ {\dot{\varpi }}_i^B(t)={\displaystyle {\sum }_{j\in Nc}{a}_{ij}(({\varpi }_j^B(t)-{\varpi }_i^B(t)))}+{\rho }_i^B(t)\hfill \\ {\dot{\rho }}_i^B(t)={\displaystyle {\sum }_{j\in Nc}{a}_{ij}(({\rho }_j^B(t)-{\rho }_i^B(t))}-{\dot{P}}_i^B(t)\hfill \end{array}$$

(5)

Based on the relationship between P_B,i and ${\varpi }_i$, we can obtain the corresponding charging power when incremental costs are consistent. Furthermore, the power mismatch variable ρ_i ensures the balance of injected/released power in the control process.

Similarly, Formula (5) can be expressed as matrix form (6) when combining all nodes:

$$\begin{array}{l}{\dot{P}}_B(t)=-2\alpha {\dot{\varpi }}^B(t),\hfill \\ {\dot{\varpi }}^B(t)=-L{\varpi }^B(t)+{\rho }^B(t)\hfill \\ {\dot{\rho }}^B(t)=-L{\rho }^B(t)+2\alpha {\dot{\varpi }}^B(t)\hfill \end{array}$$

(6)

Through Formulas (4) and (5), we can achieve the control objectives of voltage restoration and economic dispatch while maintaining power balance, which is shown as:

$$\left\{\begin{array}{l}\underset{t\to \infty }{\lim }{V}_i^B(t)=V_i^{*}\hfill \\ \underset{t\to \infty }{\lim }{\varpi }_i^B(t)={\varpi }_j^B(t),\forall i,j\in N\hfill \end{array}\right.$$

(7)

Continuous information exchange is necessary for realizing the control objectives, so it is essential to ensure communication reliability. To avoid the effect of random delay on the control process, we contributed to proposing practical constraints.

3.2. Distributed Cooperative Control for Economic Dispatch with Random Delay Effect

The states of converters are transmitted via the communication network, which is formed as the data packet. Each local controller updates its own state when receiving the latest data packet. Consider the randomly occurring packet dropouts. The variables ε(t) are proposed to describe the transmission status. ε(t) are independent Bernoulli-distributed white sequences with

$$\left\{\begin{array}{l}P(\epsilon (t)=1)=E\{\epsilon (t)\}={\epsilon }^{\prime }\hfill \\ P(\epsilon (t)=0)=1-{\epsilon }^{\prime }\hfill \end{array}\right.$$

(8)

where ε’ ∈ [0, 1] is defined as the random dropout rates in the communication process.

Subject to random packet loss, the control input should be modified with the probabilistic random variable ε(t) in the signal transmission:

$$\begin{array}{l}{\dot{V}}_i^B(t)={\displaystyle {\sum }_{\{j\in N_c\}}\epsilon (t)\left(a_{ij}(V_j^B(t)-V_i^B(t))+g_i(V_i^{*}-V_i^B(t))\right)+}\hfill \\ (1-\epsilon (t))\left(a_{ij}(V_j^B(t-\tau )-V_i^B(t-\tau )+g_i(V_i^{*}-V_i^B(t))\right)\hfill \\ {\dot{\varpi }}_i^B(t)={\displaystyle {\sum }_{\{j\in N_c\}}\epsilon (t)\left(a_{ij}({\varpi }_j^B(t)-{\varpi }_i^B(t))+{\rho }_i^B(t)\right)+}\hfill \\ (1-\epsilon (t))\left(a_{ij}({\varpi }_j^B(t-\tau )-{\varpi }_i^B(t-\tau ))+{\rho }_i^B(t-\tau )\right)\hfill \\ {\dot{\rho }}_i(t)={\displaystyle {\sum }_{\{j\in N_c\}}\epsilon (t)\left(a_{ij}({\rho }_j^B(t)-{\rho }_i^B(t))-{\dot{P}}_i^B(t)\right)+}\hfill \\ (1-\epsilon (t))\left(a_{ij}({\rho }_j^B(t-\tau )-{\rho }_i^B(t-\tau ))-{\dot{P}}_i^B(t-\tau )\right)\hfill \end{array}$$

(9)

Obviously, economic dispatch variable $\varpi$ and power imbalance variable ρ are applied in Equation (9) to design the control input. Suppose that different variables communicate with various information channels. It will be easier for the stabilization of systems to become broken. Therefore, we transformed Equation (9) into the matrix form, which aims to package these two variables into a single packet to reduce the risk of communication.

$$\begin{array}{l}\left[\begin{array}{l}{\dot{\varpi }}^B(t)\hfill \\ {\dot{\rho }}^B(t)\hfill \end{array}\right]=\epsilon (t)\left[\begin{array}{cc}-(L^B)& I_N\\ 2{\alpha }^B(L^B)& -L^B-2{\alpha }^B\end{array}\right]\left[\begin{array}{l}{\varpi }^B(t)\hfill \\ {\rho }^B(t)\hfill \end{array}\right]+\\ (1-\epsilon (t))\left[\begin{array}{cc}-(L^B)& I_N\\ 2{\alpha }^B(L^B)& -L^B-2{\alpha }^B\end{array}\right]\left[\begin{array}{l}{\varpi }^B(t-\tau )\hfill \\ {\rho }^B(t-\tau )\hfill \end{array}\right]\end{array}$$

(10)

For the sake of simplicity, we converted Equation (10) to:

$$\begin{array}{l}{\dot{\gamma }}_B(t)=\epsilon (t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)+\\ (1-\epsilon (t))(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t-\tau )\end{array}$$

(11)

where ${\gamma }_B(t)=[{\varpi }_1^B(t),{\rho }_1^B(t),\cdots ,{\varpi }_N^B(t),{\rho }_N^B(t)]$, $S_1=\left[\begin{array}{cc}-1& 0\\ 2{\alpha }^B& -1\end{array}\right]$, $S_2=\left[\begin{array}{cc}0& 1\\ 0& -2{\alpha }^B\end{array}\right]$.

In the communication network of the closed-loop power system, each agent has an information sender and receiver. The sender can package its own status information to neighbors, as shown by the blue line, whereas the receiver can obtain the status information of neighbors, as shown by the green line. It is evident that the data transmission process is interfered with by random delay, whose occurrence probability is ε’.

Note that the random time delays in communication links can be constructed as stochastic variable functions satisfying the Bernoulli-distributed white sequence.

Definition 1.

The model (11) is exponentially mean-square stable by the positive constants b₁ and b₂, so there is $E\left\{{‖x(t)‖}^2\right\}\le b_1{e}^{-b_{2}t}\underset{k\in [-\tau ,0]}{\sup }E\left\{{‖x(k)‖}^2\right\}$, where $x(\cdot )$ can represent $V(\cdot ),\varpi (\cdot ),\rho (\cdot ),\mu (\cdot ),\gamma (\cdot )$ in this paper.

Based on Equations (2) and (9), a primary/secondary control framework can be designed to realize the control objectives (7), which is shown in Figure 2. In the secondary control layer, each agent exchanges state information with its neighbors in the communication network. Some information packets can be transmitted successfully, which are marked by red lines. Furthermore, the other information packets are affected by random delays, which are marked by green lines. Then, the secondary controller on each converter calculates the corresponding reference voltage and the reference charging power through the collected state data affected and not affected by the delay and sends the corresponding reference value to the primary controller. The primary controller then calculates the deviation between the actual voltage and the reference voltage and the deviation between the actual charging power and the reference charging power, and finally adjusts the output of the DC/DC converter through the PWM wave.

3.3. Stability Analysis

3.3.1. Voltage Restoration

Inspired by [21], the error system $\mu (t)={[{\mu }_1^V(t),\cdots {\mu }_i^V(t),\cdots {\mu }_N^V(t)]}^T$ should be defined to ensure the stability of the voltage restoration process, where ${\mu }_i^V(t)=V_i^{*}-V_i(t)$. Hence:

$$\dot{\mu }(t)=-\epsilon (t)(L_V+G_V)\mu (t)-(1-\epsilon (t))(L_V+G_V)\mu (t-\tau (t))$$

(12)

where L_V is the Laplace matrix and G_V is the gain matrix.

Theorem 1.

Given communication parameter ${\psi }_V\in [0,{\lambda }_1(L_V+G_V)]$, the close-loop system (12) has exponential mean-square stability if there exists a delay bound ${\tau }_{max}^V=\min \{\frac{{\lambda }_1((L_V+G_V))-{\psi }_V}{{\lambda }_{\max }({(L_V+G_V)}^T(L_V+G_V))},\frac{{\psi }_V}{(1-{\epsilon }^{\prime }){\lambda }_{\max }({(L_V+G_V)}^T(L_V+G_V))}\}$ for the communication delay τ^V. Then, the voltage can be forced to the reference in the control process.

Proof of Theorem 1.

Consider the Lyapunov function candidate:

$$\begin{array}{l}{V}_V(t)=V_{V,1}(t)+V_{V,2}(t)+V_{V,3}(t),\hfill \\ V_{V,1}(t)=\frac{1}{2}{\mu }^T(t)\mu (t),\hfill \\ V_{V,2}(t)={\psi }_V{\displaystyle {\int }_{t-\tau }^t{\mu }^T(s)\mu (s)ds},\hfill \\ V_{V,3}(t)={\displaystyle {\int }_{-\tau }^0{\displaystyle {\int }_{t+\vartheta }^t{\dot{\mu }}^T(s)\dot{\mu }(s)dsd\vartheta }}.\hfill \end{array}$$

(13)

Hence, we have:

$$\begin{array}{l}{\dot{V}}_{V,1}(t)={\mu }^T(t)\dot{\mu }(t)={\mu }^T(t)[-\epsilon (t)(L_V+G_V)\mu (t)-(1-\epsilon (t))(L_V+G_V)\mu (t-\tau (t))]\hfill \\ =-\epsilon (t){\mu }^T(t)(L_V+G_V)\mu (t)-(1-\epsilon (t)){\mu }^T(t)(L_V+G_V)\mu (t-\tau (t)),\hfill \\ {\dot{V}}_{V,2}(t)={\psi }_V{\mu }^T(t)\mu (t)-{\psi }_V{\mu }^T(t-\tau )\mu (t-\tau ),\hfill \\ {\dot{V}}_{V,3}(t)=\tau {\dot{\mu }}^T(t)\dot{\mu }(t)-{\displaystyle {\int }_{t-\tau }^t{\dot{\mu }}^T(s)\dot{\mu }(s)ds}\hfill \\ =\tau {[-\epsilon (t)(L_V+G_V)\mu (t)-(1-\epsilon (t))(L_V+G_V)\mu (t-\tau (t))]}^T\hfill \\ \ast [-\epsilon (t)(L_V+G_V)\mu (t)-(1-\epsilon (t))(L_V+G_V)\mu (t-\tau (t))]-{\displaystyle {\int }_{t-\tau }^t{\dot{\mu }}^T(s)\dot{\mu }(s)ds}\hfill \\ =\tau {\epsilon }^2(t){\mu }^T(t){(L_V+G_V)}^T(L_V+G_V)\mu (t)+\tau {(1-\epsilon (t))}^2{\mu }^T(t-\tau ){(L_V+G_V)}^T(L_V+G_V)\mu (t-\tau )\hfill \\ +2\tau \epsilon (t)(1-\epsilon (t)){\mu }^T(t){(L_V+G_V)}^T(L_V+G_V)\mu (t-\tau )-{\displaystyle {\int }_{t-\tau }^t{\dot{\mu }}^T(s)\dot{\mu }(s)ds}.\hfill \end{array}$$

(14)

In ${\dot{V}}_{V,1}(t)$, the last term follows:

$$\begin{array}{l}-(1-\epsilon (t)){\mu }^T(t)(L_V+G_V)\mu (t-\tau )\hfill \\ =-(1-\epsilon (t)){\mu }^T(t)(L_V+G_V)\mu (t)+{\displaystyle {\int }_{t-\tau }^t{[(1-\epsilon (t))(L_V+G_V)\mu (t)]}^T\dot{\mu }(s)ds}\hfill \\ \begin{array}{l}\le -(1-\epsilon (t)){\mu }^T(t)(L_V+G_V)\mu (t)+\tau {(1-\epsilon (t))}^2{\mu }^T(t){(L_V+G_V)}^T(L_V+G_V){\mu }^T(t)\\ +{\displaystyle {\int }_{t-\tau }^t{\dot{\mu }}^T(s)\dot{\mu }(s)ds}\end{array}\hfill \end{array}$$

(15)

Note that ε(t) (1 − ε(t)) = 0:

$$\begin{array}{l}{\dot{V}}_V={\dot{V}}_{V,1}+{\dot{V}}_{V,2}+{\dot{V}}_{V,3}\le -\epsilon (t){\mu }^T(t)(L_V+G_V)\mu (t)\hfill \\ +{\psi }_V{\mu }^T(t)\mu (t)-{\psi }_V{\mu }^T(t-\tau )\mu (t-\tau )-(1-\epsilon (t)){\mu }^T(t)(L_V+G_V)\mu (t)\hfill \\ \begin{array}{l}+\tau {(1-\epsilon (t))}^2{\mu }^T(t){(L_V+G_V)}^T(L_V+G_V){\mu }^T(t)\\ +\tau {\epsilon }^2(t){\mu }^T(t){(L_V+G_V)}^T(L_V+G_V)\mu (t)\end{array}\hfill \\ +\tau {(1-\epsilon (t))}^2{\mu }^T(t-\tau ){(L_V+G_V)}^T(L_V+G_V)\mu (t-\tau )\hfill \end{array}$$

(16)

Then, it can be obtained that:

$${\dot{V}}_V\le [{\mu }^T(t){\mu }^T(t-\tau )]M_V{[{\mu }^T(t){\mu }^T(t-\tau )]}^T$$

(17)

where

$$\begin{array}{l}{M}_V=\left[\begin{array}{cc}-(L_V+G_V)+{\psi }_V{I}_N+\tau {(1-\epsilon (t))}^2{L}_{GV}+\tau {\epsilon }^2(t)L_{GV}& 0\\ 0& -{\psi }_V{I}_N+\tau {(1-\epsilon (t))}^2{L}_{GV}\end{array}\right],\hfill \\ L_{GV}={(L_V+G_V)}^T(L_V+G_V).\hfill \end{array}$$

(18)

By $E\left\{{(1-\epsilon (t))}^2\right\}=E\{1-2\epsilon (t)+{\epsilon }^2(t))\}=1-{\epsilon }^{\prime }$, we have:

$$E\left\{{\dot{V}}_V\right\}\le [{\mu }^T(t){\mu }^T(t-\tau )]M_V^E{[{\mu }^T(t){\mu }^T(t-\tau )]}^T$$

(19)

where

$$M_V^E=\left[\begin{array}{cc}-(L_V+G_V)+{\psi }_V{I}_N+\tau L_{GV}& 0\\ 0& -{\psi }_V{I}_N+\tau (1-{\epsilon }^{\prime })L_{GV}\end{array}\right]$$

(20)

In order to satisfy $E\left\{\dot{V}\right\}\le 0$, there exist the following inequalities:

$$\left\{\begin{array}{l}-(L_V+G_V)+{\psi }_V{I}_N+\tau L_{GV}<0\hfill \\ -{\psi }_V{I}_N+\tau (1-{\epsilon }^{\prime })L_{GV}<0\hfill \end{array}\right.$$

(21)

which can be expressed by:

$$\left\{\begin{array}{l}{\tau }^V<\frac{{\lambda }_1(L_V+G_V)-{\psi }_V}{{\lambda }_{\max }({(L_V+G_V)}^T(L_V+G_V))}\hfill \\ {\tau }^V<\frac{{\psi }_V}{(1-{\epsilon }^{\prime }){\lambda }_{\max }({(L_V+G_V)}^T(L_V+G_V))}\hfill \end{array}\right.$$

(22)

Therefore, the allowable bound of delays are:

$${\tau }^V<{\tau }_{\max }^V=\min \left\{\frac{{\lambda }_1(L_V+G_V)-{\psi }_V}{{\lambda }_{\max }({(L_V+G_V)}^T(L_V+G_V))}\frac{{\psi }_V}{(1-{\epsilon }^{\prime }){\lambda }_{\max }({(L_V+G_V)}^T(L_V+G_V))}\right\}$$

(23)

. □

3.3.2. Economic Power Sharing

In order to guarantee economic power sharing among energy storage units, the stability of the communication network must be verified. Error vector ${\overline{\gamma }}_B(t)=[{\overline{\varpi }}_1^B(t),{\overline{\rho }}_1^B(t),\cdots ,{\overline{\varpi }}_i^B(t),{\overline{\rho }}_i^B(t),\cdots ,{\overline{\varpi }}_N^B(t),{\overline{\rho }}_N^B(t)]$ is applied to prove the system stability, where

$$\begin{array}{l}{\overline{\varpi }}_i^B(t)={\varpi }_i^B(t)-\frac{1}{N}{\displaystyle \sum _{i=1}^N{\varpi }_i^B(0)}\hfill \\ {\overline{\rho }}_i^B(t)={\rho }_i^B(t)-\frac{1}{N}{\displaystyle \sum _{i=1}^N{\rho }_i^B(0)}\hfill \end{array}$$

(24)

In addition, the error system of the controller can be described as:

$${\dot{\overline{\gamma }}}_B(t)=\epsilon (t)(L_B\otimes S_1+I_N\otimes S_2){\overline{\gamma }}_B(t)+(1-\epsilon (t))(L_B\otimes S_1+I_N\otimes S_2){\overline{\gamma }}_B(t-\tau )$$

(25)

Theorem 2.

Given communication parameter ${\psi }_B\in [0,{\lambda }_1(L_B\otimes S_1+I_N\otimes S_2)]$, the close-loop system (25) has exponential mean-square stability if there exists a delay bound ${\tau }_{max}^B=\min \{\frac{{\lambda }_1(L_B\otimes S_1+I_N\otimes S_2)-{\psi }_B}{{\lambda }_{\max }({(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2))},\frac{{\psi }_B}{(1-\epsilon \prime ){\lambda }_{\max }({(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2))}\}$. Then, the incremental cost $\varpi$ can be synergized, and the economic power sharing can be realized with the power balance constraint.

Proof of Theorem 2.

Construct a Lyapunov functional as:

$$\begin{array}{l}{V}_B(t)=V_{B,1}(t)+V_{B,2}(t)+V_{B,3}(t)\hfill \\ V_{B,1}(t)=\frac{1}{2}{\gamma }_B{}^T(t){\gamma }_B(t),\hfill \\ V_{B,2}(t)={\psi }_B{\displaystyle {\int }_{t-\tau }^t{\gamma }_B{}^T(x){\gamma }_B(x)dx},\hfill \\ V_{B,3}(t)={\displaystyle {\int }_{-\tau }^0{\displaystyle {\int }_{t+y}^t{\dot{\gamma }}_B{}^T(x){\dot{\gamma }}_B(x)dxdy}}\hfill \end{array}$$

(26)

By using the time derivative of (26), we can obtain:

$$\begin{array}{l}{\dot{V}}_{B,1}(t)=-\epsilon (t){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)-(1-\epsilon (t)){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t-\tau (t))\hfill \\ {\dot{V}}_{B,2}(t)={\psi }_B{\gamma }_B{}^T(t)\gamma (t)-{\psi }_B{\gamma }_B{}^T(t-\tau ){\gamma }_B(t-\tau )\hfill \\ \begin{array}{l}{\dot{V}}_{B,3}(t)=\tau {\epsilon }^2(t){\gamma }_B{}^T(t){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)\hfill \\ +\tau {(1-\epsilon (t))}^2{\gamma }_B{}^T(t-\tau ){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t-\tau )\hfill \end{array}\hfill \\ +2\tau \epsilon (t)(1-\epsilon (t)){\gamma }_B{}^T(t){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t-\tau )-{\displaystyle {\int }_{t-\tau }^t{\dot{\gamma }}_B{}^T(x){\dot{\gamma }}_B(x)dx}\hfill \end{array}$$

(27)

Note that ${\dot{V}}_{B,1}(t)$ follows the inequality:

$$\begin{array}{l}{\dot{V}}_{B,1}(t)=-\epsilon (t){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)-(1-\epsilon (t)){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t-\tau (t))\hfill \\ =-\epsilon (t){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)-(1-\epsilon (t)){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)\hfill \\ +{\displaystyle {\int }_{t-\tau }^t{[(1-\epsilon (t))(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)]}^T{\dot{\gamma }}_B(x)dx}\hfill \\ \le -\epsilon (t){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)\hfill \\ -(1-\epsilon (t)){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)+{\displaystyle {\int }_{t-\tau }^t{\dot{\gamma }}_B{}^T(x){\dot{\gamma }}_B(x)dx}\hfill \\ +\tau {(1-\epsilon (t))}^2{\gamma }_B{}^T(t){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B{}^T(t)\hfill \end{array}$$

(28)

Then,

$$\begin{array}{l}{\dot{V}}_B\le -\epsilon (t){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)+{\psi }_B{\gamma }_B{}^T(t){\gamma }_B(t)-{\psi }_B{\gamma }_B{}^T(t-\tau ){\gamma }_B(t-\tau )\hfill \\ -(1-\epsilon (t)){\gamma }_B{}^T(t)(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)\hfill \\ \begin{array}{l}+\tau {(1-\epsilon (t))}^2{\gamma }_B{}^T(t){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B{}^T(t)\hfill \\ +\tau {\epsilon }^2(t){\gamma }_B{}^T(t){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t)\hfill \end{array}\hfill \\ +\tau {(1-\epsilon (t))}^2{\gamma }_B{}^T(t-\tau ){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2){\gamma }_B(t-\tau )\hfill \end{array}$$

(29)

It can be shown that:

$${\dot{V}}_B\le [{\gamma }_B{}^T(t){\gamma }_B{}^T(t-\tau )]M_B{[{\gamma }_B{}^T(t){\gamma }_B{}^T(t-\tau )]}^T$$

(30)

where

$$\begin{array}{l}{M}_B=diag\{-(L_B\otimes S_1+I_N\otimes S_2)+{\psi }_B{I}_N+\\ \tau {(1-\xi (t))}^2{(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2)+\\ \tau {\xi }^2(t){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2),\\ \tau {(1-\xi (t))}^2{(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2)-{\psi }_B{I}_N\}\end{array}$$

(31)

Apply mathematical expectations of ${\dot{V}}_B$, such that:

$$E\left\{{\dot{V}}_B\right\}\le [{\gamma }^T(t){\gamma }^T(t-\tau )]M_B^E{[{\gamma }^T(t){\gamma }^T(t-\tau )]}^T$$

(32)

where

$$\begin{array}{l}{M}_B^E=diag\{-(L_B\otimes S_1+I_N\otimes S_2)+{\psi }_B{I}_N+\\ \tau {(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2),\\ -{\psi }_B{I}_N+\tau (1-{\epsilon }^{\prime }){(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2)\}\end{array}$$

(33)

Then, we can obtain the allowable delay bound:

$$\left\{\begin{array}{l}{\tau }^B<\frac{{\lambda }_1(L_B\otimes S_1+I_N\otimes S_2)-{\psi }_B}{{\lambda }_{\max }({(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2))}\hfill \\ {\tau }^B<\frac{{\psi }_B}{(1-{\epsilon }^{\prime }){\lambda }_{\max }({(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2))}\hfill \end{array}\right.$$

(34)

In addition, the upper bound of τ^B can be expressed as:

$$\begin{array}{l}{\tau }^B<{\tau }_{{\max }^B}=\min \{\frac{{\lambda }_1(L_B\otimes S_1+I_N\otimes S_2)-{\psi }_B}{{\lambda }_{\max }({(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2))},\\ \frac{{\psi }_B}{(1-{\epsilon }^{\prime }){\lambda }_{\max }({(L_B\otimes S_1+I_N\otimes S_2)}^T(L_B\otimes S_1+I_N\otimes S_2))}\}\end{array}$$

(35)

. □

Remark 1.

Note that there are two allowable delay upper bounds τ^V_max and τ^B_max in the process of voltage recovery control and SOC balance economy control, respectively. Therefore, the upper limit of the allowable delay of the entire system should take the minimum value between the two; that is, τ_max = min{τ^V_max, τ^B_max}.

4. Simulation

By employing an islanded microgrid modified from the IEEE 30 bus test system, the algorithm validity and model rationality can be verified when against random communication delay. The system includes 6 DGs, 4 transformers, and 37 lines. A 50 kWh battery was chosen as the energy storage unit in DG node. Each node has a 0.6 kΩ local load, and the line impedances are converted into the local load. Moreover, a 0.4 kW load connected to node 7 can be cut out or reconnected by the breaker, which is shown in Figure 3. The control coefficient, electrical coefficient, and reference value are shown in

Table 1. Coefficients and related parameters of the power system.

Parameter	Symbol	Value
Nominal voltage	V*	220 V
DG power	PG,i	{12, 10, 11, 11, 12, 11} kW
Local load	PL,i	{6, 7.5, 8, 8.5, 9.5, 6} kW
Total demand	PL	46 kW
Droop coefficient	Di	{10, 5, 4, 5, 2, 8}
Charging coefficient	αi	−0.038
Charging coefficient	βi	0.118
Time delay	τ	[0, 0.16] s
Voltage restoration delay bound	τVmax	0.22 s
Economic control delay bound	τBmax	0.18 s
Maximum allowable delay bound	τmax	0.18 s

. By employing the stability analysis, the maximum delay boundary of the voltage recovery control algorithm is 0.22 s, whereas that of the economic control algorithm is 0.18 s. The whole system can be stable when the delay range is 0–0.16 s.

4.1. Case 1: Normal Operation

The real-time simulation results with random communication delay are demonstrated in Figure 4. In the control period t = 0–4 s, the increment cost of different agents can be synergized to −0.62 with the impact of random delay. The power mismatch variable can be forced to zero in the control process, which means that the power mismatch between different agents is eliminated with the distributed secondary controllers. Based on the increment cost variables and power mismatch variables, the references of charge power are calculated in the corresponding converters. Then, the actual charge power can follow the reference with the primary controllers. Hence, the total operation cost can be minimized with the power balance constraint, and the voltages can be controlled to the nominal values.

4.2. Case 2: Load Variation

The effectiveness of the presented algorithm in the situation of random delay and load fluctuation is verified in this subsection.Between t = 3 s and t = 6 s, the added load connects to the DG5. Moreover, an extra 0.5 kW load is added in DG4 at t = 4 s and disconnected at t = 5 s. It can be seen that, with the connection and disconnection of the load, the power mismatch variable ρ will produce large fluctuations, thus destroying the stability of the incremental cost variable. However, the designed controller can re-calculate the corresponding charging power reference value according to the balance constraint in a relatively short time. With the actual charging power following the above reference value, the system can be restored to stability, and the voltage fluctuation caused by load variations can be eliminated, as shown in Figure 5.

4.3. Case 3: Communication Failure

The stability of communication links plays a vital role in the control process. In order to verify the robustness of the designed communication model, case 3 simulates a single point fault that occurs on the communication link between DG4 and DG1 at t = 0.1 s and t = 4 s, respectively, and disappears at t = 0.8 s and t = 4.5 s. When t = 0.1 s, the synchronization rates of the incremental cost variable ${\varpi }_4$, the power imbalance variable ρ₄, and the voltage V₄ in DG4 are significantly lower than those of the normal operation. However, these three variables can still quickly enter the cooperative state when the communication link is restored at t = 0.8 s. After t = 4 s, the system enters a stable state; hence, the single point fault on the communication link has little impact on the system. The incremental cost variable, power mismatch variable, and voltage maintain stability with communication failure, as shown in Figure 6.

4.4. Case 4: Comparison

Moreover, the effectiveness of the presented algorithm against random communication delay can be verified by comparing it with the traditional distributed secondary control algorithm. Due to data loss and information errors when receiving neighbor information, each agent generates an incorrect reference value when updating its own state. Therefore, the traditional secondary control algorithm produces drastic fluctuations, and the whole system cannot be synergized. The stability of the network is destroyed, and the voltage and charging power cannot track the reference value, which is shown in Figure 7.

5. Conclusions

In order to realize voltage restoration and economic dispatch with a power balance constraint, a novel distributed cooperative algorithm was proposed in this paper. In order to further work against the random delay in communication links, an independent Bernoulli-distributed white sequence was applied to ensure the system stability. Based on an IEEE 30-node feeder model, the simulation results show the effectiveness and robustness of the proposed algorithm.

Our future work is to expand the application of the presented method, including charge/discharge mode switching, SOC balance, and AC/DC hybrid microgrid development.