Distributed secondary frequency control and state of charge (SoC) balance for droop-controlled battery energy storage systems (BESSs) with a unified SoC relative variation rate

Abstract

The state of charge (SoC) balance, power sharing, and frequency restoration are common control objectives of battery energy storage systems. However, the SoC balance scheme induced by the power allocation through existing droop controllers can cause the capacity parameters of battery cells to be unequal to the droop coefficient, which is the result of battery capacity degradation. Under this limitation, previous capacity-based droop controllers and the secondary controllers are no longer suitable to address the imprecise power sharing and frequency restoration caused by this problem. Therefore, a power allocation scheme based on the current SoC level ratio is designed to induce a new droop controller and ensure that the SoC simultaneously drops to 0. In order to restore frequency in a distributed manner and obtain the SoC level ratio, a distributed nominal frequency controller, SoC average estimator, and power-sharing controller are designed based on multi-agent systems in both asymptotic and finite-time manners. In the asymptotic scheme, the steady-state performance of the SoC estimator is adjustable, and power sharing and frequency restoration are zero errors. In the finite-time scheme, the influence of parameters on convergence time is well analyzed, and some conservative calculation methods are provided. Several time-domain simulation examples are designed on an improved IEEE-57 bus system to verify the distribution of asymptotic and finite-time schemes.

Keywords

Battery energy storage system multi-agent system distributed control droop control SoC balance

Introduction

A battery energy storage system (BESS) is integrated into the smart grid to suppress the peak and valley supply gap and suppress the randomness and intermittency of renewable energy generation (Lin and Zamora, 2022; Pei et al., 2022), thereby improving power quality and providing reliable uninterrupted power supply (Pei et al., 2022). However, the management of multiple parallel BESSs also faces challenges (Calero et al., 2022; Kang et al., 2022; Rouholamini et al., 2022). The imbalance of state of charge (SoC), serving as an important indicator to measure the level of a BESS, leads to the waste of battery capacity (Calero et al., 2022; Kang et al., 2022; Kebede et al., 2022; Qays et al., 2022; Xu et al., 2022), which increases management costs. The differential restoration of frequency leads to power harmonics, thereby reducing power quality (Hossain Lipu et al., 2022; Kang et al., 2022; Pei et al., 2022; Rouholamini et al., 2022; Xu et al., 2022). Hence, cooperative management/control of multiple parallel BESSs becomes a problem worthy of in-depth study.

Thus, a large number of excellent literature has been committed to solving the above problems. Both distributed and centralized manners have been investigated. For example, the SoC of each battery was monitored by a centralized controller in Gonzalez-Garrido et al. (2020), while an external balancing circuit was applied to realize power sharing. However, the external balancing circuit was going to consume some energy. In practice, the centralized controller is more expensive, and the single-point failures are unavoidable (Calero et al., 2022; Forero-Quintero et al., 2022; Rouholamini et al., 2022). Instead, distributed control represented by multi-agent systems (MASs) can enhance the robustness and scalability of the system, making it more favored for managing BESSs.

For multiple BESSs, many distributed algorithms based on MAS for BESS focus on asymptotic control (Khazaei and Nguyen, 2019; Nguyen and Khazaei, 2021; Raeispour et al., 2020; Zeng et al., 2022), finite-time control (Ding et al., 2020; Zhang et al., 2023), and disturbance rejection/resilience control (Ding et al., 2020; Raeispour et al., 2020), etc., of SoC balancing, power sharing, and frequency restoration. Therefore, in terms of convergence performance (Ding et al., 2020; Khazaei and Nguyen, 2019; Nguyen and Khazaei, 2021; Raeispour et al., 2020; Zeng et al., 2022; Zhang et al., 2023), robustness performance (Ding et al., 2020; Raeispour et al., 2020), etc., these results are exhibited extremely well. The model-free method is also used to promote SoC consensus and frequency regulation based on droop control in Chen et al. (2022). The author improved the power allocation scheme based on capacity using an adaptive term based on SoC in Yang et al. (2021) and achieved SoC consensus. In Yu et al. (2021), the author allocates power based on marginal cost to achieve optimal economic cost operation. These distributed algorithms for BESS are hierarchical and have some commonalities. Therein, power needs to be proportionally shared according to the capacity of each BESS, resulting in SoC consensus and only power participation in frequency modulation based on droop control.

However, the consensus method of SoC based on power exchange has changed the working modes of some BESSs, that is, from discharge mode to charging mode. This not only leads to an increase in losses on the power feeder line, but also causes some existing battery parameters to change due to early entry into the next cycle, thus requiring a re-estimation of the parameters. In other words, the current methods have caused a waste of power and capacity, as well as modeling errors.

Recently, some new developments regarding SoC balance have emerged in (Meng et al., 2021, 2022; Wu et al., 2022; Xing et al., 2019). The results of these studies are all aimed at verifying that when BESSs have a uniform discharge rate, no battery will exit prematurely due to power depletion, in other words, all batteries will run out of energy at the same time. There is a power allocation scheme implied here. Unlike previous balance schemes based on SoC consensus, these results seem to indicate that as long as power is shared according to the current SoC, the goal of achieving time consensus for SoC depletion to 0 can be achieved. Based on this, the previous droop control as a primary control strategy cannot achieve accurate frequency restoration. Thus, inspired by the above-related work, considering the novel power allocation scheme, we replace the original static value with a dynamic droop coefficient based on SoC. This dynamic value includes the rated capacity of the battery, real-time SoC, and its average value. In order to obtain this average value and achieve secondary frequency control in distributed manners, we are going to design two types of distributed estimators for the SoC balance and two active power-sharing controllers, which are later referred to as power-sharing estimators, for multi-BESS networks with the heterogeneous dynamics in this paper, respectively. That is, asymptotic and finite-time schemes are simultaneously designed here. Concisely, the main contributions in this paper are listed as follows:

(i) This article introduces a discharge rate constraint for each battery, expecting each BESS to have the same relative discharge rate. The constraint of the discharge mode introduced in this article ensures that all batteries are depleted simultaneously and that the operating mode of each one will be unchanged before depletion. Compared to existing schemes in Ding et al. (2020), Khazaei and Nguyen (2019), Lu et al. (2023), Nguyen and Khazaei (2021), Raeispour et al. (2020), Zeng et al. (2022), and Zhang et al. (2023), our designed scheme does not require power exchange, thereby reducing losses on power feeders.

(ii) A new frequency droop controller is proposed. Active power is no longer allocated based solely on the rated capacity of each battery, but rather on the rated capacity and the current SoC level of each BESS and the average SoC and output power states in this network. That is to say, the frequency droop coefficient is dynamic and nonconstant, which is different from before (Ding et al., 2020; Khazaei and Nguyen, 2019; Nguyen and Khazaei, 2021; Raeispour et al., 2020; Zeng et al., 2022; Zhang et al., 2023). The advantage of doing so is that the power allocated by this droop control can ensure that all BESSs are in discharge mode at all times and depleted simultaneously.

(iii) In order to obtain the average SoC and output power states and achieve secondary frequency control and SoC balance, two distributed schemes are proposed based on an improved droop control strategy, one is asymptotic and the other is finite time.

Section “Introduction” gives a general description of graph theory. Section “The description of a BESS and : problem statement” introduces a common model of BESS with droop control and common control objectives. Section “Distributed secondary control schemes design” presents asymptotic and finite-time schemes, respectively. Section “Some simulation cases” designs several cases to test the proposed estimators. Section “Conclusions” summarizes this paper.

The description of a BESS and problem statement

A frequency droop-controlled BESS

A BESS usually adopts a hierarchical control scheme. In this paper, we adopt a multi-BESS network, and the simplified dynamics of each one can be described in the dq frame as (1a) to (1d), refer to Ding et al. (2020), Khazaei and Nguyen (2019), Nguyen and Khazaei (2021), Raeispour et al. (2020), and Zhang et al. (2023) for details.

{\overset{\cdot}{E}}_{i} = \frac{- K_{i}^{E}}{3600} {\tilde{P}}_{i} + u_{i}^{E},

(1a)

{\overset{\cdot}{\tilde{P}}}_{i} = u_{i}^{P},

(1b)

ω_{i} = ω_{i}^{*} - K_{i}^{P} {\tilde{P}}_{i},

(1c)

{\overset{\cdot}{ω}}_{i}^{*} = u_{i}^{ω},

(1d)

where $E_{i}$ is the SoC of battery $i$ , $P_{i}$ and $ω_{i}$ are the output active power and the frequency, respectively, of inverter $i$ , $ω_{i}^{*}$ is represented for the nominal value of the frequency, (1c) is droop control, namely the primary control for frequency, $K_{i}^{P}$ is a droop coefficient. Therefore, BESS models conforming to these dynamics are heterogeneous. Previous researchers have favored using $K_{i}^{P} = K_{i}^{E}$ and denoting $P_{i} =^{Δ} K_{i}^{P} {\tilde{P}}_{i}$ , which means that active power is allocated according to the capacity of battery $i$ .

In previous SoC balance schemes, both underactuated and fully actuated schemes are actually induced by output power to achieve SoC consensus. Driven by these schemes, some batteries at low SoC levels are in charging mode for a period of time, while others are in discharge mode. The inconsistency of working modes has led to batteries in charging mode entering the next cycle ahead of schedule. However, we note that the capacity of the battery decreases with an increase in the number of cycles. Specifically, for BESS $i$ in the $f_{i}$ th cycle, its current actual capacity is

K_{i}^{E} = K_{i, m}^{E} η_{i}^{f_{i}},

where $K_{i, m}^{E}$ and $η_{i}$ are the rated capacity and the Coulomb efficiency of BESS $i$ , respectively. Therefore, treating $K_{i}^{E}$ as a constant may cause significant modeling errors.

A novel power allocation strategy and its induced droop controller

The significance of SoC balance is to ensure that the battery cells within the battery pack are simultaneously depleted, thereby fully utilizing the capacity of the battery pack. The previous schemes of directly aligning SoC and proportionally aligning output power both require power exchange between battery packs, resulting in unnecessary waste on power feeders. To alleviate this situation, for a BESS network containing $n$ batteries, the output power of each battery often needs to be determined based on its SoC. Mathematically, batteries have the same relative discharge rate, that isi.e.,

\frac{P_{i}}{E_{i}} = \frac{P_{j}}{E_{j}}, \forall i, j \in {1, 2, \dots, n} .

(2)

Combining (1a) and (2), it can be concluded that the output power of each BESS can be counted as

P_{i} = \frac{E_{i}}{E_{a}} P_{a},

(3)

where $E_{a}$ and $P_{a}$ are the average value of SoCs and the total load power, respectively. As long as the power is allocated according to this scheme, the SoC balance can be guaranteed without the need for additional control inputs.

Based on (5), we can conclude that active power needs to be allocated according to the SoC level. If $K_{i}^{P} = K_{i}^{E} (E_{a} / E_{i})$ , then $P_{a} = K_{i}^{P} {\tilde{P}}_{i}$ . In this way, as a primary controller, the droop control can be modified as follows:

ω_{i} = ω_{i}^{*} - P_{a} .

(4)

Remark 1. For SoC balance schemes that rely on a common frequency droop control shown in (1c) to allocate power, $K_{i}^{P} = K_{i}^{E}$ is constant and a widely used condition, which is very unreasonable from the above analysis. And our redesigned droop control (4) adopts a dynamic droop control gain, to allocate power according to the current SoC level to ensure that all BESSs are in discharge mode and depleted simultaneously.

Control objectives

Frequency restoration helps to provide high-quality electricity. Before that, SoC balance and power-sharing states should be well distributed and estimated. Hence, each BESS is subjected to the following dynamics:

{\overset{\cdot}{E}}_{i} = \frac{- 1}{3600} P_{i},

(5a)

{\overset{\cdot}{\hat{E}}}_{a, i} = u_{i}^{E},

(5b)

P_{i} = \frac{E_{i}}{{\hat{E}}_{a, i}} {\hat{P}}_{a, i},

(5c)

{\overset{\cdot}{\hat{P}}}_{a, i} = u_{i}^{P},

(5d)

ω_{i} = ω_{i}^{*} - {\hat{P}}_{a, i},

(5e)

{\overset{\cdot}{ω}}_{i}^{*} = u_{i}^{ω},

(5f)

where (5c) and (5e) are a power allocation scheme and a modified droop controller, respectively, ${\hat{E}}_{a, i}$ and ${\hat{P}}_{a, i}$ are the estimation of $E_{a}$ and $P_{a}$ , respectively. The detailed objectives are described in Problem 1,

Problem 1. Consider a multi-BESS network including $N$ BESSs with droop control and subjected to dynamics (1a)- to (1d). We intend to design some distributed controllers for each BESS to solve the following problems,:

(1) The real-time active power-sharing state of each battery can be well estimated, and reactive sharing can be achieved, that is,

lim_{t \to \infty} | {\hat{P}}_{a, i} - P_{a} | = 0, \forall i \in {1, 2, \dots, N} .

(2) On the premise that ${\hat{P}}_{a, i}$ is well distributed estimated, a nominal frequency controller is designed to achieve secondary control, that is,

lim_{t \to \infty} | ω_{i} - ω^{r} | = 0, \forall i \in {1, 2, \dots, N},

where $ω^{r}$ denotes the reference frequency.

(3) Thus, to achieve SoC balance, the real-time SoC balance state should be well estimated, that is,

lim_{t \to \infty} | {\hat{E}}_{a, i} - E_{a} | \leq ε, \forall i \in {1, 2, \dots, N},

where $ϵ$ is a prescribed positive constant.

Distributed secondary control schemes design

Graph theory

In general, agents communicate with each other on a topology $G = (V, E)$ . $V$ stands for the set embracing $n$ nodes, each of which represents an agent. Meanwhile, $E$ is defined as the set of edges, if there exists an edge from nodes $j$ to $i$ , $a_{ij} = 1$ , which means the agent $i$ has access to the agent $j$ ’s information; if not, $a_{ij} = 0$ . Besides, self-edge is not considered in this paper, that is $a_{ii} = 0$ for all nodes. Such a graph is called the directed graph, which contains a spanning tree and a root node that has no parent. According to the above statement, we define $A = [a_{ij}]$ as the adjacency matrix, and $D = diag (d_{1}, d_{2}, \dots, d_{n})$ as the in-degree matrix with $d_{i} = \sum_{j \neq i} a_{ij}$ . Therefore, the Laplacian matrix corresponding $G$ is $L = D - A$ .

The leader–follower model is mainly applied in this paper. Let $B = diag {b_{11}, b_{22}, \dots, b_{nn}}$ be an adjacency matrix between the leader and all followers. If the information of the leader can be acquired by the follower $i$ , $b_{ii} = 1$ ; otherwise, $b_{ii} = 0$ . Denote $H = L + B$ .

Assumption 1. Assume that $G$ is a connected undirected graph, and at least one follower can access the leader.

Lemma 1. Mesbahi and Egerstedt (2010) under Assumption 1, $L$ is positively semidefinite, and $H = L + B$ is positively definite.

Under Assumption 1, $L$ has a zero eigenvalue and $n - 1$ positive eigenvalues, and $H$ has $n$ positive eigenvalues. Let $0 = λ_{1} (L) < λ_{2} (L) \leq \dots \leq λ_{n} (L)$ and $λ_{m} = λ_{1} (H) \leq λ_{2} (H) \leq \dots \leq λ_{n} (H) = λ_{M}$ be the non-decreasing eigenvalue sequences of $L$ and $H$ , respectively.

Distributed asymptotical control scheme

Inspired by George and Freeman (2019), Meng et al. (2021), and Wu et al. (2022), the distributed SoC balance and powersharing estimators can be designed for each battery unit here, as shown in (6a) to (6c),

{\hat{E}}_{a, i} = q_{i} + E_{i},

(6a)

{\overset{\cdot}{q}}_{i} = - α q_{i} - β \sum_{j \in N_{i}} a_{ij} ({\hat{E}}_{a, i} - {\hat{E}}_{a, j}) - v_{i},

(6b)

{\overset{\cdot}{v}}_{i} = α β \sum_{j \in N_{i}} a_{ij} ({\hat{E}}_{a, i} - {\hat{E}}_{a, j}),

(6c)

{\overset{\cdot}{\hat{P}}}_{a, i} = - κ \sum_{j \in N_{i}} a_{ij} ({\hat{P}}_{a, i} - {\hat{P}}_{a, j}),

(6d)

where $v_{i}$ and $q_{i}$ are the designed intermediate states. Here, let initial condition ${\hat{E}}_{a, i} = E_{i}$ , $v_{i} = 0, q_{i} = 0$ and ${\hat{P}}_{a, i} = P_{i}$ for $i \in {1, 2, \dots, N}, α > 0, β > 0$ and $κ > 0$ are design parameters. In addition, the nominal frequency controller can be constructed in (7):,

{\overset{\cdot}{ω}}_{i}^{*} = - κ (\sum_{j = 1}^{N} a_{ij} (ω_{i}^{*} - ω_{j}^{*}) + b_{ii} (ω_{i}^{*} - {\hat{P}}_{a, i} - ω^{r})) .

(7)

Due to the fact that the nominal frequency is not a variable that needs to be measured by a BESS, the use of this controller can reduce the installation of measurement equipment and measurement errors. Figure 1 shows the secondary frequency control structure with SoC balance and power-sharing estimators designed in detail.

Figure 1.

The proposed distributed hierarchical control for a battery energy storage system (BESS).

To illustrate the stability analysis of (6) and (7), the following useful lemmas are given.

Lemma 2. George and Freeman (2019) consider the system dynamics as

{\overset{\cdot}{x}}_{i} = {\overset{\cdot}{u}}_{i} - α (x_{i} - u_{i}) - β \sum_{j = 1}^{N} L_{ij} x_{j} - v_{i},

{\overset{\cdot}{v}}_{i} = α β \sum_{j = 1}^{N} L_{ij} x_{j} .

Let

y_{i} = x_{i} - \frac{1}{N} \sum_{j = 1}^{N} u_{j}, i \in {1, \dots, N},

w = v - \bar{v}, \bar{v} = Π_{N} (\overset{\cdot}{u} + α u),

where $u = [u_{1}, u_{2}, \dots, u_{N}]$ , $Π_{N} = I_{N} - (1 / N) 1_{N} 1_{N}^{T}$ . Then, the system can be rewritten as follows:

[\begin{matrix} \overset{\cdot}{y} \\ \overset{\cdot}{w} \end{matrix}] = A [\begin{matrix} y \\ w \end{matrix}],

where

A = [\begin{matrix} - α I_{N} - β L & - I_{N} \\ α β L & 0 \end{matrix}] .

If $α$ , $β > 0$ , and the communication topology satisfies Assumption 1, then $y_{i} (t) \to - (α^{- 1} / N) \sum_{j = 1}^{N} w_{j} (0)$ , $w_{i} (t) \to (1 / N)$ $\sum_{j = 1}^{N} w_{j} (0)$ , as $t \to \infty$ , $\forall i \in {1, \dots, N}$ .

Lemma 3. Kia et al. (2014) based on Lemma 2, there exist a positive constant $γ_{1}$ , if $α > 0$ and $β > 0$ , $y_{i}$ satisfies

lim_{t \to \infty} | y_{i} | \leq \frac{γ_{1}}{β λ_{2}} .

Lemma 4. Nguyen and Khazaei (2021) consider a dynamic system

{\overset{\cdot}{x}}_{i} = - κ (\sum_{j = 1}^{N} a_{ij} (x_{i} - x_{j}) + b_{ii} (x_{i} - x_{0})) .

There exist a positive constant $γ_{2}$ , if $κ > 0$ , $x_{i}$ satisfies

lim_{t \to \infty} | x_{i} - x_{0} | \leq \frac{γ_{2} Φ}{κ},

where $Φ = max_{t < \infty} | {\overset{\cdot}{x}}_{0} |$ .

Based on the above analysis, we have the following result about the distributed control schemes (6) and (7). And the relevant proof will be given next.

Theorem 1. Consider a multi-BESS system contains $n$ BESS equipped with the dynamics (5a)- to (5e). Let the communication topology satisfies Assumptions 1. Then, if $α$ , $κ$ , $β > 0$ , Problem 11 will be solved with the distributed control algorithm (6) and (7) in asymptotic manner. And we have $lim_{t \to \infty} | {\hat{E}}_{a, i} - E_{a} | \leq γ / β λ_{2}$ , where $γ = ∥ E ∥_{\infty} < + \infty$ .

Proof. Define estimation error and frequency modulation error as follows:

Δ P_{i} = {\hat{P}}_{a, i} - P_{a},

Δ E_{i} = {\hat{E}}_{a, i} - E_{a},

Δ ω_{i} = ω_{i} - ω^{r} .

Denote some stack vectors $Δ P = [Δ P_{1}, Δ P_{2}, \dots, Δ P_{n}]^{T}$ , ${\hat{P}}_{a} = [{\hat{P}}_{a, 1}, {\hat{P}}_{a, 2}, \dots, {\hat{P}}_{a, n}]^{T}$ , ${\hat{E}}_{a} = [{\hat{E}}_{a, 1}, {\hat{E}}_{a, 2}, \dots, {\hat{E}}_{a, n}]^{T}$ , $Δ ω = [Δ ω_{1}, Δ ω_{2}, \dots, Δ ω_{n}]^{T}$ , and $ω = [ω_{1}, ω_{2}, \dots, ω_{n}]^{T}$ .

Next, the convergence of the average load power estimator will be explained as follows. Construct the following candidate Lyapunov function:

V_{P} = \frac{1}{2} \sum_{i = 1}^{n} Δ P_{i}^{2} .

Derivative of $V_{P}$ with respect to time is formed as

\begin{matrix} {\overset{\cdot}{V}}_{P} = \sum_{i = 1}^{n} Δ P_{i} Δ {\overset{\cdot}{P}}_{i} = - κ \sum_{i = 1}^{n} Δ P_{i} \sum_{j = 1}^{n} a_{ij} ({\hat{P}}_{a, i} - {\hat{P}}_{a, j}) \\ = - κ \sum_{i = 1}^{n} Δ P_{i} \sum_{j = 1}^{n} a_{ij} (Δ P_{i} - Δ P_{j}) \\ = - \frac{κ}{2} \sum_{i, j = 1}^{n} a_{ij} (Δ P_{i} - Δ P_{j})^{2} < 0 . \end{matrix}

So, $V_{P}$ converges to 0 asymptotically and $lim_{t \to \infty} Δ P_{i} = 0$ . That is, $lim_{t \to \infty} {\hat{P}}_{i, a} = P_{a}, \forall i \in {1, 2, \dots, N}$ .

Combined with (6a) and (6b), it can be concluded that

{\overset{\cdot}{\hat{E}}}_{a, i} = {\overset{\cdot}{E}}_{i} - α q_{i} - β \sum_{j \in N_{i}} a_{ij} ({\hat{E}}_{a, i} - {\hat{E}}_{a, j}) - v_{i} .

With the help of the mapping in Lemma 2 and (6c), the following state-space model can be obtained:

[\begin{matrix} \overset{\cdot}{y} \\ \overset{\cdot}{w} \end{matrix}] = A [\begin{matrix} y \\ w \end{matrix}],

where $y = {\hat{E}}_{a} - E_{a}$ , $w = v - \bar{v}$ , $\bar{v} = Π_{N} (\overset{\cdot}{E} + α E)$ ,

A = [\begin{matrix} - α I_{N} - β L & - I_{N} \\ α β L & 0 \end{matrix}] .

Then, it can be inferred in terms of Lemma 2 that $y_{i} (t) \to - (α^{- 1} / N) \sum_{j = 1}^{N} w_{j} (0)$ , $w_{i} (t) \to (1 / N) \sum_{j = 1}^{N} w_{j} (0)$ , as $t \to \infty$ , $\forall i \in {1, \dots, N}$ . According to Lemma 3, $lim_{t \to \infty} | y_{i} | \leq γ / β λ_{2}$ , that isi,e., $lim_{t \to \infty} | {\hat{E}}_{a, i} - E_{a} | \leq γ / β λ_{2}$ .

Lastly, according to Lemma 4 and (7), we have

lim_{t \to \infty} | ω_{i}^{*} - {\hat{P}}_{a, i} - ω^{r} | = lim_{t \to \infty} | ω_{i} - ω^{r} | \leq \frac{γ_{2} Φ}{κ},

where $Φ = max_{t < \infty} | {\overset{\cdot}{\hat{P}}}_{a, i} |$ . From a different perspective, combined with (5e), (6c), and (7), the following derivation is obtained:

\begin{matrix} Δ \overset{\cdot}{ω} = \overset{\cdot}{ω} = {\overset{\cdot}{ω}}^{*} - {\overset{\cdot}{\hat{P}}}_{a} = - κ L ω^{*} - κ B (ω^{*} - {\hat{P}}_{a} - ω^{r}) + κ L {\hat{P}}_{a} \\ = - κ H (ω - ω^{r}) = - κ H Δ ω . \end{matrix}

Then, $Δ ω$ can be obtained by integrating $Δ \overset{\cdot}{ω}$ with respect to time $Δ ω (t) = e^{- κ H t} (ω (0) - ω^{r})$ , where $ω (0)$ is the synthesis vector of the initial value of each BESS’s frequency. It can be deduced that the eigenvalues of $H$ are non-zero with a negative real part. Accordingly, $lim_{t \to \infty} Δ ω (t) = 0$ , that is $lim_{t \to \infty} ω_{i} = ω^{r}, \forall i \in {1, 2, \dots, N}$ . Then, Problem 1 has been solved in asymptotic manner.

So far, we have completed the proof of Theorem 1.

Remark 2. The dynamic and steady-state performance of ${\hat{E}}_{a, i}$ are influenced by different parameters, respectively. Dynamic tracking performance is closely positively correlated with $α$ , that is, as $α$ increases, the tracking rate becomes faster. The consensus steady-state performance, according to Lemma 3, can be improved by selecting a larger $β$ under a given communication graph. The consensus dynamic performance depends on $α$ and $β$ .

Remark 3. This solution is completely distributed, as each agent utilizes information from itself and its neighbors. In addition, the derivative of SoC, that is, proportional power, is not required in real-time. But this does not mean that there is no need to measure power. Occasionally, when switching between constant loads, the estimated power-sharing state needs to be reset to the power at the time of load switching. So, the complete dynamic of the system is a hybrid integrator with load-driven jumps, that is,

{\begin{matrix} {\hat{P}}_{a, i} (t^{+}) = P_{i}, if D changs at t, \\ {\overset{\cdot}{\hat{P}}}_{a, i} = - κ \sum_{j \in N_{i}} a_{ij} ({\hat{P}}_{a, i} - {\hat{P}}_{a, j}), otherwise . \end{matrix}

The estimator below can serve as an alternative to this solution,

{\overset{\cdot}{\hat{P}}}_{a, i} = - κ (\sum_{j \in N_{i}} a_{ij} ({\hat{P}}_{a, i} - {\hat{P}}_{a, j}) + b_{ii} ({\hat{P}}_{a, i} - P_{a})) .

At this point, simply update $P_{a}$ at the time of load switching.

Distributed finite-time control scheme

In this section, we present the distributed finite-time scheme to solve Problem 1. In order to illustrate the subsequent design and related proof, we give the following lemmas.

Lemma 5. Nguyen and Khazaei (2021) consider if $y_{1}, y_{2}, \dots,$ $y_{n} \geq 0$ and $0 < r < p$ , then

{(\sum_{i = 1}^{n} y_{i}^{p})}^{1 / p} \leq {(\sum_{i = 1}^{n} y_{i}^{r})}^{1 / r} .

Lemma 6. Zhang et al. (2012) denote $x$ an $n$ -dimensional real column vector. Then we have $x^{T} L x \geq λ_{2} x^{T} x$ , $x^{T} H x \geq λ_{m} x^{T} x$ , and $∥ x ∥_{1} \geq ∥ x ∥_{2}$ .

Lemma 7. Forti et al. (2006) consider the following system dynamic as $\overset{\cdot}{x} = u$ . If the Lyapunov function $V$ with respect to the state $x$ satisfies $\overset{\cdot}{V} \leq - K V^{α}$ , where $K > 0$ and $0 < α < 1$ . Then $V$ will converges to 0 in finite time $T = V^{1 - α} (0) / K (1 - α)$ . At the same time, the state $x$ reaches stable.

A finite-time estimator of the SoC balance of each battery can be designed as (8a) and (8b):

{\hat{E}}_{a, i} = \sum_{j = 1}^{n} a_{ij} (q_{i} - q_{j}) + E_{i},

(8a)

{\overset{\cdot}{q}}_{i} = - α sign (\sum_{j = 1}^{n} a_{ij} ({\hat{E}}_{a, i} - {\hat{E}}_{a, j})),

(8b)

where ${\hat{E}}_{a, i}$ is the estimation of the average of SoC of battery $i$ , $q_{i}$ is a internal state, $α > 0$ is a designed parameter, $sign (•)$ represents the symbolic function. In addition, the power-sharing estimator and the nominal frequency controller are designed as (9a) and (9b),

{\overset{\cdot}{\hat{P}}}_{a, i} = - β_{1} \sum_{j = 1}^{n} a_{ij} sign ({\hat{P}}_{a, i} - {\hat{P}}_{a, j}) | {\hat{P}}_{a, i} - {\hat{P}}_{a, j} |^{η},

(9a)

{\overset{\cdot}{ω}}_{i}^{*} = - β_{2} sign (\sum_{j = 1}^{n} a_{ij} (ω_{i}^{*} - ω_{j}^{*}) + b_{ii} (ω_{i}^{*} - {\hat{P}}_{a, i} - ω^{r})) .

(9b)

With these above lemmas in hand, the following result can be easily obtained based on the designed estimators.

Theorem 2. Consider a power network with $n$ BESSs subjecting to the dynamics (5). Let the communication topology satisfy Assumption 1. Then, if $α > \sqrt{n} P_{Σ} / 3600 λ_{2}$ , $β_{1} > 0$ , $β_{2} > ϕ$ and $0 \leq η < 1$ , where $ϕ = ∥ \overset{\cdot}{\hat{P}} (0) ∥_{\infty}$ , Problem 1 will be solved with the distributed control algorithm (8) and (9) in finite time. That is, active power sharing and SoC balance states can be estimated at $t = T_{P}$ and $t = T_{E}$ , respectively. Besides, frequency restores to $ω^{r}$ at $t = T_{ω}$ , where the nominal frequency of each BESS converge to $(ω^{r} + P_{a})$ at $t = T_{ω}$ .

Proof. Firstly, the convergence of the average load power estimator is explained under the conditions of $0 \leq η < 1$ and $β > 0$ . Select the following Lyapunov function:

V_{P} = Δ P^{T} Δ P .

Find its derivative as

\begin{matrix} {\overset{\cdot}{V}}_{P} = 2 \sum_{i = 1}^{N} Δ P_{i} Δ {\overset{\cdot}{P}}_{i} \\ = 2 \sum_{i = 1}^{N} Δ P_{i} (- β_{1} \sum_{j = 1}^{n} a_{ij} sign ({\hat{P}}_{a, i} - {\hat{P}}_{a, j}) | {\hat{P}}_{a, i} - {\hat{P}}_{a, j} |^{η}) \\ = - β_{1} \sum_{i, j = 1}^{N} (Δ P_{i} - Δ P_{j}) sign (Δ P_{i} - Δ P_{j}) | Δ P_{i} - Δ P_{j} |^{η} \\ = - β_{1} \sum_{i, j = 1}^{N} a_{ij} | Δ P_{i} - Δ P_{j} |^{η + 1} . \end{matrix}

According to Lemmas 5 and Lemma 6, $\overset{\cdot}{V}$ follows the following derivation:

\begin{matrix} {\overset{\cdot}{V}}_{P} = - β_{1} \sum_{i, j = 1}^{N} a_{ij} | Δ P_{i} - Δ P_{j} |^{η + 1} \\ \leq - β_{1} {(\sum_{i, j = 1}^{n} a_{ij}^{2 / (η + 1)} | Δ P_{i} - Δ P_{j} |^{2})}^{1 / 2} \\ = - β_{1} [2 Δ P^{T} L Δ P]^{(η + 1) / 2} \\ \leq - β_{1} (2 λ_{2})^{(η + 1) / 2} [V_{P}]^{(η + 1) / 2} . \end{matrix}

By Lemma 7, $lim_{t \to T_{P}} V_{P} = 0$ , where

T_{P} = \frac{∥ Δ P (0) ∥^{1 - η}}{(1 - η) β_{1} {(2 λ_{2})}^{(η + 1) / 2}} = \frac{{(∥ Δ P (0) ∥ \sqrt{2 λ_{2}})}^{1 - η}}{2 λ_{2} β_{1} (1 - η)} .

(10)

That is $lim_{t \to T_{P}} | {\hat{P}}_{a, i} - P_{a} | = 0$ and ${\hat{P}}_{a, i} = P_{a}$ when $t \geq T_{P}$ for $\forall i \in {1, 2, \dots, N}$ .

Rewrite (9a) in a matrix form as

{\overset{\cdot}{ω}}^{*} = - β_{2} sign (H (ω^{*} - {\hat{P}}_{a} - ω^{r})) .

Define a new variable $δ ω = H (ω^{*} - {\hat{P}}_{a} - ω^{r})$ . Select a candidate Lyapunov function as

V_{ω} = \frac{1}{2} δ ω^{T} H^{- 1} δ ω,

whose derivative is, according to Lemma 6

\begin{matrix} {\overset{\cdot}{V}}_{ω} = δ ω^{T} H^{- 1} H ({\overset{\cdot}{ω}}^{*} - {\overset{\cdot}{\hat{P}}}_{a}) \leq δ ω^{T} {\overset{\cdot}{ω}}^{*} - δ ω^{T} {\overset{\cdot}{\hat{P}}}_{a} \\ = - β_{2} δ ω^{T} sign (δ ω) - δ ω^{T} {\overset{\cdot}{\hat{P}}}_{a} \\ = - β_{2} \sum_{i = 1}^{n} | δ ω_{i} | - δ ω^{T} {\overset{\cdot}{\hat{P}}}_{a} \leq - β_{2} ∥ δ ω ∥_{1} + ϕ ∥ δ ω ∥_{1} \\ \leq - (β_{2} - ϕ) (2 λ_{M})^{1 / 2} V_{ω}^{1 / 2}, \end{matrix}

(11)

which implies, according to Lemma 7, if $β_{2} > ϕ$ , the nominal frequency $ω_{i}^{*}$ converges to $({\hat{P}}_{a, i} + ω^{r})$ in finite time

T_{ω} = \frac{\sqrt{2 V_{ω} (0)}}{(β_{2} - ϕ) \sqrt{λ_{M}}}

(12)

for $\forall i \in {1, 2, \dots, N}$ . Thus, under the action of droop controller, $ω_{i} = ω^{r}$ when $t \geq T_{ω}$ .

Next, the finite-time convergence of the SoC balance estimator will be proved.

By constructing a matrix form of (8a) and (8b), one can get

\begin{matrix} {\overset{\cdot}{\hat{E}}}_{a} = L u^{q} + \overset{\cdot}{E}, \\ \overset{\cdot}{q} = u^{q}, \\ u^{q} = - α sign (μ), \\ μ = L {\hat{E}}_{a} . \end{matrix}

As for the estimation error $Δ E$ of SoC, the following two equations hold true,

\begin{matrix} Δ \overset{\cdot}{E} = {\overset{\cdot}{\hat{E}}}_{a} - \frac{1}{N} 1_{N} 1_{N}^{T} \overset{\cdot}{E} = L u^{q} + (I_{N} - \frac{1}{N} 1_{N} 1_{N}^{T}) \overset{\cdot}{E}, \\ L Δ E = L ({\hat{E}}_{a} - E_{a}) = L {\hat{E}}_{a} . \end{matrix}

Denote $L (L^{+}) = I_{N} - (1 / N) 1_{N} 1_{N}^{T}$ , where $L^{+}$ is the generalized inverse of $L$ . Then, we arrive at

(L^{+}) μ = (L^{+}) L {\hat{E}}_{a} = {\hat{E}}_{a} .

Select the Lyapunov function with respect to $Δ E$ as

V = \frac{1}{2} Δ E^{T} Δ E .

Because $V$ is a well-defined continuous function with a continuous first derivative function, the first derivative function can be derived as

\begin{matrix} \overset{\cdot}{V} = Δ E^{T} Δ \overset{\cdot}{E} = Δ E^{T} ({\overset{\cdot}{\hat{E}}}_{a} - {\overset{\cdot}{E}}_{a}) = Δ E^{T} (L u^{q} + L (L^{+}) \overset{\cdot}{E}) \\ = μ^{T} u^{q} + μ^{T} (L^{+}) \overset{\cdot}{E} . \end{matrix}

The second term on the right side of the above equation satisfies the following inequality:

μ^{T} (L^{+}) \overset{\cdot}{E} \leq σ_{max} (L^{+}) \sqrt{n} ∥ μ ∥_{1} ∥ \overset{\cdot}{E} ∥_{\infty},

where $σ_{max} (L^{+})$ denotes the maximum singular value and $σ_{max} (L^{+}) = 1 / λ_{2}$ . Also,

μ^{T} u^{q} = - α \sum_{i = 1}^{n} | μ_{i} | = - α ∥ μ ∥_{1} .

As a result, $\overset{\cdot}{V}$ satisfies the following derivation:

\begin{matrix} \overset{\cdot}{V} \leq \frac{1}{λ_{2}} \sqrt{n} ∥ μ ∥_{1} ∥ \overset{\cdot}{E} ∥_{\infty} - α ∥ μ ∥_{1} = (\frac{1}{λ_{2}} \sqrt{n} {| \overset{\cdot}{E} |}_{max} - α) \\ ∥ μ ∥_{1} \leq (\frac{1}{λ_{2}} \sqrt{n} \frac{1}{3600} {| P |}_{max} - α) ∥ μ ∥_{1} . \end{matrix}

In general, $P_{i} \leq P_{Σ}$ . If $α > \sqrt{n} P_{Σ} / 3600 λ_{2}$ , according to Lemma 6, we arrive at

\begin{matrix} \overset{\cdot}{V} \leq (\frac{\sqrt{n} P_{Σ}}{3600 λ_{2}} - α) ∥ μ ∥_{1} \leq (\frac{\sqrt{n} P_{Σ}}{3600 λ_{2}} - α) \\ ∥ μ ∥_{2} = (\frac{\sqrt{n} P_{Σ}}{3600 λ_{2}} - α) \sqrt{Δ E^{T} L^{2} Δ E} \\ \leq (\frac{\sqrt{n} P_{Σ}}{3600 λ_{2}} - α) λ_{2} V^{1 / 2} . \end{matrix}

(13)

According to Lemma 7, SoC balance error converges to 0 within a finite-time denoted by $T_{E}$ and

T_{E} = \frac{2 ∥ Δ E (0) ∥}{α λ_{2} - (\sqrt{n} P_{Σ} / 3600)} .

(14)

So far, we have completed the proof of Theorem 2.

Remark 4. From (10), it can be seen that the convergence rate of $Δ P$ can be accelerated by selecting a larger parameter $β_{1}$ (or $β_{2}$ ) and a communication topology with a larger minimum non-zero eigenvalue, respectively. The role of parameter $η$ is not yet clear. This is because given different system initial values, communication topology, and parameters $β_{1}$ , $β_{2}$ , the influence of parameter $η$ on convergence time is different. Therefore, this requires further analysis based on different situations and inevitably requires the use of trial and error methods.

Remark 5. From (12), it can be seen that the convergence time of frequency decreases with the increase of the maximum eigenvalues and parameters of the communication topology, respectively. However, $β_{2}$ no longer affects the convergence time by selecting $β_{2} \geq ϕ + 1$ . That is, (11) can be further deduced as ${\overset{\cdot}{V}}_{ω} \leq - (2 λ_{M})^{1 / 2} V_{ω}^{1 / 2}$ . Then, according to Lemma 7, $T_{ω} = \sqrt{2 V_{ω} (0)} / \sqrt{λ_{M}}$ . Similarly, (27) shows that the convergence time decreases with the increase of $α$ and $λ_{2}$ . When $α λ_{2} \geq (\sqrt{n} P_{Σ} / 3600) + 1$ (or $α \geq (\sqrt{n} P_{Σ} / 3600 λ_{2}) + 1$ ) is selected, (??) can be further deduced as $\overset{\cdot}{V} \leq - V^{1 / 2}$ (or $\overset{\cdot}{V} \leq - λ_{2} V^{1 / 2}$ ), which leads to $T_{E} = 2 ∥ Δ E (0) ∥$ (or $T_{E} = (2 ∥ Δ E (0) ∥ / λ_{2})$ ). This implies that by selecting $α λ_{2} \geq (\sqrt{n} P_{Σ} / 3600) + 1$ or $α \geq (\sqrt{n} P_{Σ} / 3600 λ_{2}) + 1$ , the convergence time $T_{E}$ is not affected by $α$ and $λ_{2}$ or only by $λ_{2}$ .

Some simulation cases

In this section,several cases are designed to verify the two proposed schemes. The modified IEEE-57 bus is selected as the load distribution exhibited in Figure 2(a), where seven BESSs are set on buses 1, 2, 3, 6, 8, and 9. The communication topologies are illustrated in Figure 2(b). $Matpower$ , which is used to calculate power flow in the simulation, is a $Matlab$ toolkit. $MatDyn$ is selected as the Simulation toolkit based on $Matpower$ . Here are six case studies designed to verify and test the performance of the two designed solutions.

Figure 2.

The modified IEEE-57 bus system and its communication topology: (a) the modified IEEE-57 bus system; (b) the communication topology graph.

These two schemes, under a fixed graph, are tested in Cases 1 and Case 4, respectively.

To test the effectiveness of two schemes under time-varying communication topologies, a switching sequence with an interval of 20 minutes, i.e.that is, (a) → (b) → (c) → (d) in Figure 2(b), is executed in Cases 2 and 5.

In Cases 3 and 6, the influence of parameters on the results is studied trial and error.

Case 1. The test on the asymptotic algorithm

In order to verify the effectiveness of the strategy proposed in an asymptotic manner, some operations are implemented during the simulation. The whole system works in discharging mode. The secondary controllers based on droop control work at $t = 10$ . When $t = 40$ , there is a total load decrease of $1.26 p . u$ ., which is evenly distributed at each load node. Besides, $0.06 p . u$ . increase at each load node occurs at $t = 75$ .

Figure 3 is the simulation result. Figure 3 reveals that the estimation of SoC balance, and active power sharing are reached, and the frequency restores to the reference. In addition, the SoC balance can be well maintained regardless of how the load changes, and the active power of each BESS output stably.

Figure 3.

The simulation result of the asymptotic algorithm in Case 1.

To illustrate the progressiveness of the scheme we designed, two existing distributed schemes are compared, one of which is fully driven Khazaei and Nguyen (2019) and the other is under driven Nguyen and Khazaei (2021). The simulation results are shown in Figure 4. Comparing Figure 3, it can be seen that previous schemes have performed poorly in addressing the secondary control and SoC balance of BESSs with capacity degradation. Especially in the underactuated scheme, it makes it difficult to achieve precise consensus on the SoC, proportional power, and frequency of each BESS, respectively.In contrast, our designed solution improves the traditional droop control approach of distributing power according to rated capacity, so that power is shared according to the current SoC level. This enables all the SoCs of BESS to converge to 0 simultaneously, while ensuring accurate frequency restoration.

Figure 4.

Simulation results under previous schemes: (a) simulation results under an underactuated scheme in Nguyen and Khazaei (2021); (b) simulation results under a full drive scheme in Khazaei and Nguyen (2019).

Case 2. The test on the asymptotic algorithm with time-varying communication topologies

In this case, the performance of the asymptotic scheme in dealing with switching topologies is studied. After the controller is activated at $t = 10$ , a total load decrease of $1.26 p . u$ . and an increase of $2.52 p . u$ . occur at $t = 35$ and $t = 55$ , respectively. According to the previously predetermined topology switching frequency and order, the relevant simulation results are shown in Figure 5.

Figure 5.

The simulation result of the asymptotic algorithm with timevarying communication topologies in Case 2.

From this result,it can be seen that each variable can still converge under time-varying topologies, but with slightly different rates. This is related to the current communication topology. This indicates the robustness of the designed scheme to switching topologies. In addition, as long as connectivity is maintained, even if some communication links fail, this algorithm remains effective, ensuring the stability and security of BESS.

Case 3. Influence of parameters on the asymptotic algorithm

Different parameters value groups of $α$ , $β$ , and $κ$ in the distributed asymptotic controller are set in this case to grasp the influence of each parameter on the results and verify the correctness of the discussion in Remark 2. The simulation results of the real-time SoC balance estimation, power sharing state estimation, and frequency restoration are shown in Figure 6(a) and 6(b).

Figure 6.

The influence of parameters of asymptotic algorithms on simulation results: (a) effect of different $α$ and $β$ on ${\hat{E}}_{a, i}$ under the asymptotic algorithm in Case 3; (b) effect of different $κ$ on $ω_{i}$ and ${\hat{P}}_{a, i}$ under the asymptotic algorithm in Case 3.

From the simulation results in Figure 6(a), different values of $α$ have no obvious effect on the results. However, the real-time SoC balance estimation is mainly affected by $β$ . Specifically, the slow speed of SoC balance estimation becomes more serious with a smaller $β$ . For power-sharing estimation and frequency restoration, selecting a larger $κ$ can accelerate the convergence rate. All of these indicate the correctness of the discussion in Remark 2.

Case 4. The test on the finite-time control algorithm

To test theffect of the proposed finite-time control scheme, the same events and load changes as Case 1 are implemented in this case. Figure 7 are the simulation results in the discharging mode.

Figure 7.

The simulation result of the finite-time algorithm in Case 4.

Compared with Case 1, although the estimation of the real-time SoC balance and power sharing can be well estimated as described in Figure 7. Besides, frequency can be restored to the reference value. SoCs can ultimately reach 0 simultaneously and BESSs have stable power output. At this time, the high-frequency chatting behavior is not obvious.

Case 5. The test on the finite-time algorithm with time-varying communication topologies

In this case, the performance of the finite-time scheme in dealing with switching topologies is studied. After the controller is activated at $t = 10$ , a total load decrease of $1.26 p . u$ . and an increase of $2.52 p . u$ . occur at $t = 35$ and $t = 55$ , respectively. According to the previously predetermined topology switching frequency and order, the relevant simulation results are shown in Figure 8.

Figure 8.

The simulation result of the finite-time algorithm with time-varying communication topologies in Case 5.

From this result, it can be seen that each variable can still converge under time-varying topologies, and with no significantly changed rate. This indicates, similar to the conclusion in Case 3, that this algorithm exhibits robustness to time-varying topologies and solves the problems proposed in this article while ensuring the connectivity of the communication topology.

Case 6. Influence of parameters on the finite-time algorithm

To explore the impact relevant parameters $α$ , $β$ , and $η$ on the system behavior and validate the discussion in Remarks 4 and 5, simulations are conducted using different values of each parameter based on Case 1. Corresponding to Figure 2(b), $λ_{2} = 2$ and $λ_{m} = 0.2412$ . Besides, $∥ Δ P (0) ∥ = 1.0671$ , $P_{Σ} = 2.575$ , and $ϕ = 3.057 β_{1}$ . Hence, through a simple verification of (10) with $0 \leq η < 1$ , $T_{P}$ is negatively correlated with $η$ . In addition, if selecting $α \geq 1.0557$ , according to Remark 5, $T_{E}$ remains unchanged. And according to Remark 5, $T_{ω}$ remains unchanged by selecting $β_{2} \geq 3.057 β_{1} + 1$ . The simulation results are shown in Figures 9(a) and 10.

Figure 9.

The influence of $β_{1}$ and $α$ of the finite-time algorithm on simulation results: (a) effect of different $β_{1}$ and $η$ on ${\hat{P}}_{a, i}$ under the finite-time algorithm in Case 6; (b) effect of different $α$ on ${\hat{E}}_{a, i}$ under the finite-time algorithm in Case 6.

Figure 10.

The influence of $β_{2}$ of the finite-time algorithm on simulation results.

It can be deduced from Figure 9(a) that with a larger $β_{1}$ , consensus on ${\hat{P}}_{a, i}$ is accelerated. However, the conclusion about $η$ is exactly the opposite. From the simulation results of ${\hat{E}}_{a, i}$ , it can be seen that the increase of $α$ accelerates the consensus rate, which is, however, by selecting $α > 1.0557$ , no longer changes. On the contrary, the shaking phenomenon becomes more pronounced. For $β_{2}$ , from Figure 10, a larger value can indeed accelerate frequency recovery, but after exceeding a certain value, there is no significant change in the convergence rate. Also, it is necessary to select the proper $α$ in order to obtain high-quality system performance and ideal consensus speed. The above phenomena all verify the correctness of the discussion in Remarks 4 and 5.

Conclusions

In this article, we propose a novel droop controller with a dynamic droop gain for a BESS to allocate power, and design a distributed asymptotic and finite-time secondary controller for it. These efforts are made to address the issue of secondary control and SoC balance considering battery cell capacity degradation. As a result, our designed scheme promotes the SoCs of all BESSs to converge to 0 simultaneously and ensures frequency restoration to the reference value through reasonable power allocation. This scheme ensures all BESSs to work in the discharging mode before depletion, thus preventing some BESSs working modes from being changed before the entire network is depleted of energy. We do not advocate for consistent SoCs for all BESSs, as this may cause some batteries to enter the next cycle ahead of schedule, leading to modeling errors. Two distributed schemes are designed namely asymptotic and finite-time schemes. The simulation results show that the power is shared according to the SoC ratio, and the frequency can be restored to the reference value, while the SoCs of all BESSs simultaneously drop to 0. In addition, based on this work, new explorations can be explored in the future, such as intermittent communication and event-triggering schemes.

Footnotes

Appendix I

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant no. 62103203), the Natural Science Foundation of Tianjin (grant no. 22JCQNJC01440), and the General Terminal IC Interdisciplinary Science Center of Nankai University.

ORCID iDs

Yalin Zhang

Zhongxin Liu

Fuyong Wang

Zengqiang Chen

References

Calero

Cañizares

Bhattacharya

, et al. (2022) A review of modeling and applications of energy storage systems in power grids. Proceedings of the IEEE 111: 1–26.

Chen

Liu

Chen

, et al. (2022) Multi-agent reinforcement learning for decentralized resilient secondary control of energy storage systems against dos attacks. IEEE Transactions on Smart Grid 13: 1739–1750.

Ding

Han

Ning

, et al. (2020) Distributed resilient finite-time secondary control for heterogeneous battery energy storage systems under denial-of-service attacks. IEEE Transactions on Industrial Informatics 16: 4909–4919.

Forero-Quintero

Villafafila-Robles

Barja-Martinez

, et al. (2022) Profitability analysis on demand-side flexibility: A review. Renewable & Sustainable Energy Reviews 169: 112906.

Forti

Grazzini

Nistri

, et al. (2006) Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-lipschitz activations. Physica D: Nonlinear Phenomena 214: 88–99.

George

Freeman

(2019) Robust dynamic average consensus algorithms. IEEE Transactions on Automatic Control 64: 4615–4622.

Gonzalez-Garrido

Gaztanaga

Saez-de Ibarra

, et al. (2020) Electricity and reserve market bidding strategy including sizing evaluation and a novel renewable complementarity-based centralized control for storage lifetime enhancement. Applied Energy 262: 114591.

Hossain Lipu

Ansari

Miah

, et al. (2022) A review of controllers and optimizations based scheduling operation for battery energy storage system towards decarbonization in microgrid: Challenges and future directions. Journal of Cleaner Production 360: 132188.

Kang

Jung

Lee

, et al. (2022) How to better share energy towards a carbon-neutral city? A review on application strategies of battery energy storage system in city. Renewable & Sustainable Energy Reviews 157: 112113.

10.

Kebede

Kalogiannis

Van Mierlo

, et al. (2022) A comprehensive review of stationary energy storage devices for large scale renewable energy sources grid integration. Renewable & Sustainable Energy Reviews 159: 112213.

11.

Khazaei

Nguyen

(2019) Multi-agent consensus design for heterogeneous energy storage devices with droop control in smart grids. IEEE Transactions on Smart Grid 10: 1395–1404.

12.

Kia

Cortes

Martínez

(2014) Dynamic average consensus under limited control authority and privacy requirements. International Journal of Robust and Nonlinear Control 25: 1941–1966.

13.

Lin

Zamora

(2022) Controls of hybrid energy storage systems in microgrids: Critical review, case study and future trends. Journal of Energy Storage 47: 103884.

14.

Liu

, et al. (2023) A distributed control strategy for state-of-charge balance of energy storage without continuous communication in AC microgrids. IEEE Transactions on Sustainable Energy 14: 206–216.

15.

Meng

Lin

Shamash

(2021) Distributed cooperative control of battery energy storage systems in DC microgrids. IEEE/CAA Journal of Automatica Sinica 8: 606–616.

16.

Meng

Lin

Wan

, et al. (2022) State-of-charge balancing for battery energy storage systems in DC microgrids by distributed adaptive power distribution. IEEE Control Systems Letters 6: 512–517.

17.

Mesbahi

Egerstedt

(2010) Graph theoretic methods in multiagent networks. In: Princeton Series in Applied Mathematics. Princeton, NJ: Princeton University Press. pp. 14–33 DOI:10.1515/9781400835355.

18.

Nguyen

Khazaei

(2021) Unified distributed control of battery storage with various primary control in power systems. IEEE Transactions on Sustainable Energy 12: 2332–2341.

19.

Pei

Zhang

Deng

, et al. (2022) Review of operational control strategy for DC microgrids with electric-hydrogen hybrid storage systems. CSEE Journal of Power and Energy Systems 8: 329–346.

20.

Qays

Buswig

Hossain

, et al. (2022) Recent progress and future trends on the state of charge estimation methods to improve battery-storage efficiency: A review. CSEE Journal of Power and Energy Systems 8: 105–114.

21.

Raeispour

Atrianfar

Baghaee

, et al. (2020) Resilient distributed control of BESSs and voltage source converter-based microgrids considering switching topologies and non-uniform time-varying delays. IET Generation, Transmission & Distribution 14: 5060–5071.

22.

Rouholamini

Wang

Nehrir

, et al. (2022) A review of modeling, management, and applications of grid-connected Li-ion battery storage systems. IEEE Transactions on Smart Grid 13: 4505–4524.

23.

Chai

Tian

(2022) Distributed multirate control of battery energy storage systems for power allocation. IEEE Transactions on Industrial Informatics 18: 8745–8754.

24.

Xing

Mishra

Tian

, et al. (2019) Distributed state-of-charge balance control with event-triggered signal transmissions for multiple energy storage systems in smart grid. IEEE Transactions on Systems, Man, and Cybernetics: Systems 49: 1601–1611.

25.

Guerrero

Lashab

, et al. (2022) A review of DC shipboard microgrids-part iI: Power architectures, energy storage, and power converters. IEEE Transactions on Power Electronics 37: 5155–5172.

26.

Yang

, et al. (2021) Adaptive cooperative terminal sliding mode control for distributed energy storage systems. IEEE Transactions on Circuits and Systems I-Regular Papers 68: 434–443.

27.

Zhou

, et al. (2021) Frequency synchronization and power optimization for microgrids with battery energy storage systems. IEEE Transactions on Control Systems Technology 29: 2247–2254.

28.

Zeng

Zhang

Liu

, et al. (2022) An improved distributed secondary control strategy for battery storage system in DC shipboard microgrid. IEEE Transactions on Industry Applications 58: 4062–4075.

29.

Zhang

Lewis

(2012) Lyapunov, adaptive, and optimal design techniques for cooperative systems on directed communication graphs. IEEE Transactions on Industrial Electronics 59: 3026–3041.

30.

Zhang

Song

Fei

(2023) A distributed step-by-step finite-time consensus design for heterogeneous battery energy storage devices with droop control. CSEE Journal of Power and Energy Systems 9: 1893–1904.