Abstract
This paper focuses on the distributed control problem in a networked microgrid (NMG) with heterogeneous energy storage units (HESUs) in the environment considering multiple types of time delays, which include the state, input, and communication delays. To address this problem, a state feedback control (SFC) strategy based on nested predictor is proposed to mitigate the influence of multiple types of time delays. First, a distributed control method founded upon voltage observer is developed, which can realize proportional power distribution according to the state of charge (SOC) of the HESUs, while adjusting the average voltage of the point of common coupling (PCC) bus in the NMG to its rated value. Then, considering that there exists steady-state error resulting from the initial value of the observer and impact of time delays, an SFC strategy is proposed to further improve the robustness of the NMG against time delays. Finally, the experimental results demonstrate that the proposed distributed control method is capable of fully compensating for the state, input, and communication delay. Moreover, the NMG exhibits remarkable resistance to multiple types of time delays, which has higher reliability and robustness.
THE goal of achieving net-zero energy requires an increasing proportion of net-zero distributed energy resources (DERs). Microgrid (MG) can integrate net-zero DERs, such as wind power, solar power, and other alternative energy sources without carbon emissions to achieve this goal. Considering the limited power generation capacity and specific geographic boundaries of a single MG, the most commonly used method is to interconnect multiple MGs to construct a networked microgrid (NMG). In addition, the NMG demonstrates robust resilience against extreme events. Therefore, in order to further enhance the reliability and resilience of the power system, traditional MGs are transformed into a larger and more complex NMG [
Energy storage systems (ESSs) are indispensable for the efficient utilization of DERs, as they enable flexible energy conversion and eliminate the need for simultaneous power generation, transmission, and distribution. This plays a significant role in enhancing the efficiency of the power system [
Currently, most of the relevant studies are made for distributed control strategies in MGs without time delays [
The time delay is an inherent feature of the NMG. The distributed control method of the NMG relies on information exchange between MGs and HESUs, which inevitably causes time delay. Typically, time delay includes state, input, and communication delay, all of which affect the stability of the NMG [
Based on the above analysis, the fundamental issue faced by the NMG with multiple types of time delays is to improve the system stability and mitigate the influence of the time delays on system dynamic performance. The main control goal of the NMG is to proportionally distribute power according to the SOC of the HESUs, while simultaneously regulating the voltage of the point of common coupling (PCC) bus to the designed value. To achieve this goal, a distributed voltage control (DVC) strategy based on an observer is designed in this paper. Therein, the power controller of the DVC can ensure the SOC balance of the NMG while preventing over-charging and over-discharging of each HESU. However, the voltage controller of the DVC is based on an observer, and its control performance is influenced by both the initial value of the observer and time delays, which do not explicitly address time delays in the NMG. To address this issue, a state feedback control (SFC) strategy based on nested predictor is proposed to further mitigate the influence of time delays in the NMG. By actively compensating for input delay and communication delay, the proposed SFC strategy can improve the performance, responsiveness, and stability of the NMG. It provides a proactive solution to addressing the time delays of the NMG, which can minimize the adverse effects caused by these delays. Moreover, since the NMG with multiple types of time delays belongs with the complex network system, we analyze the large signal stability of the NMG at a systemic level. Based on this analysis, we assess the convergence and robustness of the proposed SFC strategy. Compared with the existing studies [
1) Inspired by [
2) Compared with the previous studies, the proposed SFC strategy requires neither the precise information from the communication network, nor the input signals between the HESUs. This strategy can achieve consensus control of the HESUs in the NMG and meet the requirements for SOC balance. Additionally, it regulates the voltage of the PCC bus in the NMG to its rated value. Moreover, the large-signal stability of the NMG is analyzed at a systemic level.
3) The proposed SFC strategy can actively compensate for both input delay and communication delay. Furthermore, the NMG can effectively resist the impact of multiple types of time delays, which has higher reliability and robustness.
The summary of the remaining content in this paper is provided as follows. The architecture of the NMG with HESUs is presented in Section II. Section III constructs the proposed SFC strategy to mitigate the influence of multiple types of time delays, then analyzes the stability of the closed-loop NMG. Experimental results demonstrate the reliability and robustness of the proposed SFC strategy in Section IV. Finally, the overall conclusion is summarized in Section V.
In an islanded NMG, the randomness and intermittency of the DERs cannot guarantee a stable energy output. Therefore, HESUs are necessary to compensate for the unbalanced power between photovoltaic (PV) power generation and load, as well as to stabilize the bus voltage.

Fig. 1 Configuration of NMG with HESUs.
This paper considers the communication network of the NMG with N HESUs, which is modeled as a directed graph , where is the set of N HESUs in the NMG; and is the edge set of N HESUs. According to graph theory in [
The SOC in the NMG indicates the residual capacity of HESUs, which is a crucial metric for assessing the charging and discharging of HESUs. Considering the existence of different initial values of SOC among HESUs and the influence of ambient temperature on line resistance, it is necessary to maintain the SOC balance of HESUs during the charging and discharging process. The SOC balance control proposed in [
(1) |
where is the current SOC of the
The power loss of the converters is disregarded for the HESUs, and it is assumed that the charging and discharging efficiency as well as the port voltage remains constant. Then, we can further obtain:
(2) |
where is the output power of the
Note that the SOC balance of HESUs is directly controlled, which may result in charging and discharging as well as circulation among HESUs. To address this issue, define the state variable of the power allocation and then couple the output power to the SOC through this state variable. Furthermore, we incorporate SOC information into the controllers of the NMG to achieve the SOC balance of HESUs. The defined state variable can be expressed as:
(3) |
where is the actual remaining available capacity of the
(4) |
where is the lower bound of SOC in discharging state; and is the upper bound of SOC in charging state.
Remark 1: it should be noted that the state variable ensures simultaneous charging and discharging of all HESUs in the NMG, while adjusting the output power according to residual power, so that the SOC of HESUs gradually converges to the same value. Meanwhile, the unbalanced power will be allocated among HESUs based on the real-time state of SOC when the state variables of the HESUs become consistent.
For an islanded NMG, the traditional droop control method is applied to achieve power sharing among HESUs. Then, the steady-state expression can be obtained as:
(5) |
where is the voltage of the PCC bus of the
The power sharing ratio is defined as inversely proportional to . Then, it can be obtained that:
(6) |
where and are the maximum and minimum offset voltages of the PCC bus in the NMG, respectively; is the nominal voltage of the PCC bus; and is the nominal power of the
According to (5) and (6), it is observed that accurate power sharing among HESUs can be achieved if the droop coefficient is sufficiently large. However, this will result in significant voltage deviation at the steady state. Thus, the droop coefficient of the HESUs can be restricted to:
(7) |
where is the upper limit of ; is the nominal current of the
Remark 2: for an islanded NMG, due to the different capacities and line resistances of HESUs, it is a challenge for HESUs to achieve SOC balance and accurate power allocation with multiple types of time delays. Furthermore, it is necessary to avoid over-charging and over-discharging when the power allocation of HESUs is consistent with the SOC. However, the voltage deviation in the NMG and the presence of line impedance can inevitably result in imprecise power allocation among HESUs. Thus, it is imperative to redesign the control strategy for the NMG.
The proposed DVC strategy is illustrated in

Fig. 2 Structure of proposed DVC strategy.
The nominal voltage of the PCC bus can be expressed as:
(8) |
The voltage observer of the secondary control is given as:
(9) |
where and are the average voltages of the PCC bus of the
The modified droop control with the voltage adjustment term is designed as:
(10) |
where is the compensation required for bus voltage recovery provided by the distributed secondary control of the
Next, according to (8), it can be obtained that:
(11) |
In addition, the voltage decrease resulting from the implementation of droop control can be expressed as:
(12) |
Note that the power sharing ratio of the HESU is defined to be inversely proportional to . Then, we have the following conditions:
(13) |
To obtain the voltage adjustment term of the PCC bus, the error between the designed nominal voltage and the observed voltage is calculated, and the output error is then inputted into the PI controller for regulation, as shown in (9).
(14) |
where is the designed proportional coefficient for the voltage controller of the NMG; and is the designed integral coefficient for the voltage controller of the NMG.
In the communication network layer of the NMG, the power allocation variable of each HESU exchanges information to obtain the error signal , and inputs the error value into the power controller to acquire the power adjustment term.
(15) |
where is the integral coefficient of power controller.
Remark 3: it should be noted that in the case where the variables of the HESUs are not identical, the power controller will make necessary adjustments to account for the error. Each HESU needs to continuously monitor the reference voltage of the PCC bus in real time until all HESUs converge to the same value. Furthermore, by utilizing the voltage observer described in (9), HESUs can exchange information and iterate to obtain a consistent average voltage observation value. Based on this, the HESUs have the ability to promptly adapt to power variations in the NMG, thereby ensuring a proportional relationship between output power and SOC of the HESUs. Moreover, the voltage of the PCC bus in the NMG can be adjusted to the rated value.
The existence of the state delay, input delay, and asymmetric communication delay in the NMG can affect control performance and may even lead to voltage oscillation on the PCC bus and system instability. In this subsection, the aim of this paper is to propose a control strategy that enhances the stability of the NMG with multiple types of time delays and mitigates their impacts on the dynamic performance of the system. However, due to the time delays in the NMG and the influence of the initial observer value, the estimation of the proposed DVC strategy is not accurate enough for adjusting the average voltage of the PCC bus to the nominal value. According to [
The designed controller of the NMG in this paper mainly includes primary control and distributed secondary control with cyber layer. The structure of the proposed SFC strategy is shown in

Fig. 3 Structure of proposed SFC strategy.
Definition of the consensus in the NMG: considering a communication network of an NMG with N HESUs modeled by a directed graph , the proposed SFC strategy can ensure that the HESUs can achieve consensus if
(16) |
where and are the state vectors of the
It is assumed that the HESUs in the NMG broadcast information through a directed communication network, which means that the topology of the NMG contains a directed spanning tree [
Inspired by the concept of the consensus algorithm, for the NMG without common bus, it is meaningful to propose a control strategy that ensures the gradual convergence of the average voltage of PCC buses towards its designed nominal value. Furthermore, we aim to propose an SFC strategy to address the consensus issue of HESUs in the NMG.
Considering the discrete form of the NMG with N HESUs, we can obtaion:
(17) |
where is the control vector of the
Note that voltage deviations occur in the NMG due to the line impedances and device losses. Therefore, the proposed SFC strategy can further enhance the tracking accuracy. For the considered NMG in (17), the
is the state feedback function at time , which can be expressed as:
(18) |
is also the state feedback function, which can be expressed as:
(19) |
Assumption 1: the directed network of the NMG is balanced and connected, which includes a directed spanning tree constructed with the reference signals of the HESUs as the root node.
Based on [
(20) |
where for , and ; and is the eignvalue of . The eigenvalues of are expressed as , ; and is the first column of the nonsingular matrix .
It can be observed from Assumption 1 that , . In addition, the consensus of HESUs in the NMG can also be achieved without time delay. The proposed SFC strategy is defined as:
(21) |
where and are the linear operators.
Then, we impose another assumption in the NMG.
Assumption 2: matrix and matrix can ensure the asymptotic stability of the NMG with the following series of time delay.
(22) |
where represents the dynamics of the corresponding variables.
Based on [
Lemma 1: consider a high-order discrete-time NMG that exclusively incorporates the state delays:
(23) |
Assume that the
(24) |
Then, only if Assumption 2 is fulfilled, the following SFC protocol can achieve the consensus among the HESUs in the NMG. The compact form of the feedback SFC protocol is expressed as:
(25) |
Lemma 2: suppose that the NMG in (22) can achieve asymptotic stability with multiple types of time delays, if there exist the matrices and . Furthermore, there are also matrices and satisfying the LMI, and is introduced to denote the conjugate number of .
(26) |
Based on [
(27) |
To achieve consensus among the HESUs in the NMG, an SFC protocol based on a predictor is designed as:
(28) |
The detailed design steps of the predictor are then provided. Specifically, we let:
(29) |
Thus, we can predict that:
(30) |
Next, through the prediction of , we can obtain:
(31) |
where () is the predicted value of , and if , .
If , is a causal system and therefore can be implemented. Nonetheless, in the event that , continues to rely on the next state data , . It should be noted that this designed predictor cannot achieve the control objective of the NMG under Assumption 2. Since the obtained protocol depends on future states, the predictors are acausal and therefore cannot be implemented.
Hence, to address the above issue, we assume that:
(32) |
To attain the control objective, it is noted that all types of time delays in the NMG can be unified in (32). Besides, there exist integers , such that:
(33) |
Inspired by [
Step 1: predicting ) from (30), it can be obtained that:
(34) |
When is with , it can be found that is a causal system and therefore can be implemented.
Step 2: using the previous predicted value and to predict , it can be obtained that:
(35) |
We can use state variable value () and to predict for and . Then, we can obtain that:
(36) |
It can be found that is a causal system, which can therefore be implemented.
Step 3: when , we have:
(37) |
Then, the restructured proposed SFC protocol based on a predictor is expressed as:
(38) |
, can be obtained by (37). In this way, we can predict as the step size with any integer .
Remark 4: the feedback controller proposed in this paper is based on a nested predictor with SFC protocol. It should be noted that the SFC protocol is structurally independent of the proposed feedback controller.
Remark 5: it should be noted that the proposed SFC strategy can recursively obtain future state information of the NMG with a step size of r. In this subsection, the capacity difference of the HESUs, the mismatch of the corresponding line resistance, and the existence of time delays in the NMG are considered. However, these factors may hinder the guarantee of SOC balance of the HESUs and accurate power allocation. The theoretical analysis indicates that the proposed SFC strategy can achieve precise power distribution at the steady state of the NMG with multiple types of time delays.
We transform the stability of the NMG consisting of (17) and (36) into the stability analysis presented in (22). Theorem 1 and its proof are established to guarantee the consensus of HESUs in the NMG with multiple types of time delays.
Theorem 1: suppose that Assumptions 1 and 2 are satisfied. The consensus of the HESUs in the NMG in (17) can be achieved.
Proof:considering (18), (19), and (37), can be rewritten as:
(39) |
(40) |
Then, it follows (40) that:
(41) |
where denotes the Kronecker product.
The variables and are defined as:
(42) |
Then, we further define as -transformation of , and the following fact can be obtained.
(43) |
(44) |
(45) |
Based on (33), and can be represented as (46) and (47), respectively.
(46) |
(47) |
Next, according to (44), (45), and (46), we obtain the dynamics of the closed-loop NMG as:
(48) |
Based on [
(49) |
Then, for the simplified function , we have:
(50) |
Considering (48), we can obtain:
(51) |
Note that we transform the stability analysis of the closed-loop NMG consisting of (17) and (36) into the system stability analysis presented in (22). Recalling Assumption 2, we have and . Then, we define as the first column of , which means that . The proof is completed.
This section provides four case studies to validate the robustness and superiority of the proposed SFC strategy in resisting multiple types of time delays. The experimental test setup utilized in this paper is depicted in

Fig. 4 Experimental test setup.
The hardware circuit consists of power supplies, DC/DC converters with LC filters, a DC electronic load, and transmission lines that connect the DREs. We construct the model of the test system in the MATLAB environment, and set multiple types of time delays through the delay module in the Simulink. Meanwhile, the MATLAB/Simulink model and the dSPACE real-time system are connected through real-time interface (RTI) in the dSPACE software environment, and then the automatic download of real-time hardware code from Simulink models to dSPACE can be achieved through the extended real-time workshop (RTW). The configuration of the NMG closely resembles that depicted in
Symbol | Value |
---|---|
400 V | |
0.15 Ω | |
8.00 mF | |
1.00 mH | |
5.00 | |
0.05 | |
95.00 | |
1.00 | |
0.80 | |
0.20 | |
0.10 Ω | |
0.13 Ω | |
0.15 Ω | |
0.20 Ω |
The numerical example provided in this subsection serves to validate the feasibility of the proposed SFC strategy. We consider four HESUs as individual agents, whose dynamics are represented by (1) and characterized by the following parameters.
(52) |
(53) |
(54) |
(55) |
In this case, the communication delays are randomly given by and . The state delay and the input delay . The communication topology in this subsection is illustrated in
(56) |

Fig. 5 Communication topology.
We further choose the eigenvalues of as 0, 2, 2, and 2. Moreover, based on Lemma 2, we have the following state feedback gains:
(57) |
(58) |
Next, it can be obtained that:
(59) |
(60) |
(61) |
(62) |
(63) |
The state feedback differences between HESU 1 and other HESUs are presented in

Fig. 6 State feedback differences between HESU 1 and other HESUs.
These results imply that the consensus of the HESUs can be achieved by implementing the proposed SFC strategy, which further demonstrates the correctness of Theorem 1. According to [
In this subsection, we present experimental results to validate the control performance of the proposed SFC stragety. The time span for the case study is set between [0, 30]s. During the first stage, only the droop control is activated. At s, the proposed SFC strategy is implemented in the experimental case study of the NMG. Before s, the NMG is compelled to derive power from the HESUs owing to insufficient light intensity, the equilibrium between SOC and output power remains elusive at this moment. However, at s, the light intensity is enhanced by 750 W/

Fig. 7 Output power of HESU under different light intensities and local loads.
It can be observed from

Fig. 8 SOC of HESU under different light intensities and local loads.

Fig. 9 Voltage of PCC bus of HESU under different light intensities and local loads.
To assess the robustness of the proposed SFC strategy, a comparison is made between dynamics based on the proposed DVC strategy and those based on the proposed SFC strategy with multiple types of time delays. Initially, the NMG operates stably under the primary control. Then, at s, we add multiple types of time delays to the NMG. In this case, the communication delay is chosen as 0.2 s. At this moment, the proposed SFC strategy comes into effect. The voltage of PCC bus and the change of of the HESU with proposed DVC strategy are shown in Figs.

Fig. 10 Voltage of PCC bus of HESU with proposed DVC strategy.

Fig. 11 of HESU with proposed DVC strategy.
Moreover, under the influence of multiple types of time delays, the experimental results show that the NMG with proposed DVC strategy starts to oscillate without damping and becomes unstable. However, the HESUs of the NMG can still converge according to the proposed SFC strategy. The voltage of PCC bus and the change of of the HESU with proposed SFC strategy are shown in Figs.

Fig. 12 Voltage of PCC bus of HESU with proposed SFC strategy.

Fig. 13 of HESU with proposed SFC strategy.
The NMG can maintain stable operation despite the presence of multiple types of time delays, albeit with a relatively slow convergence rate. We can observe from
This subsection aims to demonstrate the superiority of the proposed SFC strategy. A case study of the NMG under step load is introduced to compare the proposed SFC strategy with the existing control methods. Specifically, the PV unit operates in the MPPT operation mode, and at s, four local loads with a resistance of 20 are connected to the PCC bus. Figures

Fig. 14 Voltage of PCC bus of HESU with existing control method.

Fig. 15 of HESU with existing control method.

Fig. 16 Voltage of PCC bus of HESU with proposed SFC strategy.

Fig. 17 of HESU with proposed SFC strategy.
The performance of different methods in the NMG is then analyzed with multiple types of time delays. Initially, the NMG operates stably under the primary control. Then, at s, multiple types of time delays are added to the NMG, the communication delay is chosen as 0.3 s with the state delay s and the input delay s. At this moment, different control methods come into effect. When the control method in [

Fig. 18 Voltage of PCC bus of HESU with control method in [

Fig. 19 of HESU with control method in [

Fig. 20 Voltage of PCC bus of HESU with proposed SFC strategy.

Fig. 21 of HESU with proposed SFC strategy.
In this paper, an SFC strategy based on nested predictor is proposed to mitigate the influence of multiple types of time delays in the NMG. The proposed SFC strategy can distribute power according to the SOC of the HESUs while adjusting the average voltage of the PCC bus to its nominal value. In contrast to the proposed DVC strategy, the proposed SFC strategy can actively compensate for state, input, and communication delay. The experimental results demonstrate that the proposed SFC strategy can effectively mitigate the impact of time delays on system performance and enhance the stability of the NMG. In the future, we will further study and explore distributed control methods considering more complex NMGs.
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