Distributed Fixed Time Control for DC Microgrid with Input Delay

Feng, Yi Wei; Liu, Jia Li

doi:https://doi.org/10.1155/2023/8837110

International Transactions on Electrical Energy Systems

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 8837110 | https://doi.org/10.1155/2023/8837110

Distributed Fixed Time Control for DC Microgrid with Input Delay

Yi Wei Feng¹and Jia Li Liu¹

Academic Editor: Michele De Santis

Received08 Dec 2022

Revised09 Jan 2023

Accepted07 Feb 2023

Published25 Feb 2023

Abstract

This paper develops a distributed fixed-time quadratic coordinated control strategy for isolated DC microgrids with time delays. The proposed control scheme based on the droop control strategy is intended to realize bus voltage recovery and current balance distribution under time delays. We first model the isolated DC microgrids as a single integrator system with input delays by the Artstein’s transform. The current balance and voltage recovery are achieved by using fixed time control, which does not depend on the initial state of the system. Based on the Lyapunov functional method, the stability analysis criterion of the system is derived. In order to verify the effectiveness of the proposed control method, the DC microgrid system is simulated in Matlab/Simulink.

1. Introduction

In recent years, researchers around the world have paid more attention to microgrids. DC microgrids are also attracting architectures integrating various renewable energy sources, distributed power sources, hybrid energy storage systems, and loads [1]. Generally, DC microgrid can be divided into grid-connected and isolated island operation modes [2]. In the grid-connected mode, the DC microgrid is connected to the public grid. In the application of isolated island operation mode of the DC microgrid, it operates as an independent unit and is not integrated with the public power grid. Most emerging applications, such as electric helicopters, microwave ovens, military warships, nuclear submarines, and transportation information centers, will use DC island microgrids. Therefore, the control and stability of DC microgrids become particularly important.

Droop control is a common simple control strategy in DC microgrids. In order to realize local current sharing of distributed DC generators, we can select appropriate droop coefficient. Due to the line resistance, the droop control alone can introduce an important problem, that is, the bus voltage deviation [3]. Due to the existence of voltage deviation, it is necessary to compensate the voltage to realize the accurate parallel connection of each converter. A secondary control strategy is proposed and applied to the microgrid [4]. In the proposed secondary control structure, the centralized secondary control structure has high requirements on network connectivity and is not suitable for application in the DC microgrid when the system scale is expanded due to its disadvantages such as heavy communication burden and poor scalability [5]. In order to overcome the disadvantages of centralized control, a distributed secondary control strategy is proposed and applied to DC microgrid. In [6], a distributed secondary voltage control strategy based on the information of DC bus voltage and its adjacent units is proposed. A mixed dynamic controller skeleton frame is established in [7], and the controller is divided into continuous-time part and discrete-time part. The controller eliminates the voltage error of the DC bus and realizes the voltage accuracy of the DC microgrid. Distributed secondary control structures generally demand few and scattered communication networks between converters, and converters only exchange information with neighboring converters [8]. Therefore, in the distributed secondary control mode of a DC microgrid, the multiagent control method is used to define each converter as an agent [6]. In [9], the feedback linearization method is adopted, which can convert the quadratic control problem into a consistency problem, which is convenient for designing a distributed quadratic controller. The above distributed quadratic control method can only achieve asymptotic or exponential convergence. In practice, some sensitive loads need to operate at rated voltage [10]. Due to the uncertainty and intermittence of distributed new energy sources such as light and wind, it is necessary to achieve rapid convergence and finite-time convergence of secondary control. In recent years, the finite-time distributed quadratic control strategy has attracted the attention of researchers. In [11], finite-time frequency and voltage recovery and active power distribution are realized simultaneously. In [12], a distributed finite-time secondary controller is proposed to realize voltage regulation and current sharing in a DC microgrid. However, finite time control is closely related to the initial conditions of the system, and obtaining the initial conditions is also a difficulty. To solve this problem, the concept of fixed-time convergence was proposed in the literature [13] and studied in the consistency algorithm. The fixed-time control design has nothing to do with the initial state of the system but only with the design parameters, which makes it possible to estimate the set time in advance. However, whether the finite-time control strategy is adopted or the fixed-time control strategy is adopted, it will inevitably introduce a time delay to the low-level output control signal. The increase in delay will affect the transient time of the system and even have a close relationship with the stability of the system. Therefore, the existence of delays must be considered when designing the control system. However, the control method discussed above neglects the delay. For the system with time delay, Artstein’s transformation can transform a system with a time delay into a system without time delay. It can ensure that the converted system is reversible and, if appropriate control methods are used, make the converted system stable. Then, the control method can also keep the original system stable. At the same time, the delay in communicating by letter may be looked upon as an input delay in control [14].

In conclusion, the distributed control strategy has a significant effect on the current sharing and voltage recovery control of multiple DC microgrids. Unfortunately, the fixed time control has not been further studied in the above literature, and the influence of the controller on the input delay has not been studied. Therefore, we further propose an improved fixed-time control strategy that takes into account the existence of the input time delay of the controller. The contribution of this paper mainly lies in the following aspects:(1)A fixed-time secondary controller for DC microgrid is designed using symbolic functions. This design can accelerate the convergence of the system and achieve current sharing and voltage regulation within a fixed, stable time.(2)The control strategy considers the input delay of the controller. The problem of voltage regulation and current sharing is transformed into a linear system with nonuniform and unbounded input delay by using the feedback linearization method. Because it is difficult to find a suitable Lyapunov function, Artstein’s transformation is used to deal with the delay, which brings great convenience to the design of the control strategy.(3)The proposed control strategy ensures the stability of fixed time using the Lyapunov method.

The rest of this paper is organized as follows: Section 2 transforms the system with delay into a delay-freefirst-order integral system through linear feedback and Artstein’s transformation. In Section 3, the fixed time control of the input delay is given. The Section 4 carries on the system stability analysis. Section 5 simulations and validates the performance of the proposed results. Conclusions are given in Section 6.

2. Problem Description

The interconnection of multiple DC microgrids makes the DC microgrid model more complex. The traditional droop control has some limitations on the voltage and current control of multiple DC microgrids. The control target can be further realized through secondary control. The existence of communication delay in the secondary control has a great impact on the stability of the system, and also brings great difficulties to the control. However, the delay in the communication link can be regarded as the input delay in consensus control. Therefore, in this section, based on droop control, the problem of voltage and current is transformed into a linear system with input delay by feedback linearization, so that the problem of voltage and current becomes a consistency problem, and the system is transformed into a delay-free system by Artstein’s transformation to solve the control complexity caused by time delay.

The DC microgrid model is simplified and the common bus resistance is set to zero. As shown in Figure 1, the DC microgrid considered in this paper is composed of N distributed generation units (DGS), each of which has a buck DC/DC converter [7]. The average switching model obtained by using Thevenin equivalent circuit is as follows [15]:where d_i is the duty cycle of the switch, L_i and C_i are the inductance and electric capacity of the wave filter in the i-th DG unit, _si and i_Li are the DC source voltage and the current of the filter inductance, and _i and i_i are the output voltage and out current for the i-th DG, respectively.

Rewrite equation (1) to obtain the following equation:where V = [₁, ₂, …, _i]^T, I = [i₁, i₂, …, i_i]^T, d = [d₁, d₂, …, d_i]^T, = diag{_si}, C_f = diag{C_i}, L_f = diag{L_i}, i [1, 2, …, N].

In the conventional droop control, the reference voltage value _i needs to be designed for each DC/DC converter, and the droop function is as follows:where is the expected value of the common bus voltage, k_i is the droop gain of the i-th DG, and i_i is the output current of the i-th converter. R_i is the feeder resistance of the i-th converter connected to the common DC bus. We use external voltage and internal current PI controllers to control the converter. Then, the bus voltage _BUS is

From (4), it is obtained that

From (5) we further have

The value of k_i is much larger than that of R_i, i.e., .

However, when current sharing is realized, it can be seen from (4) that, with the increase of k, the error between the output voltage of each DG unit and the reference voltage becomes larger and larger. In order to make the error zero, secondary control is required to realize voltage recovery and current sharing, namely,and

Because the dynamic characteristics of DG in DC microgrid are nonlinear and the dynamic characteristics of each unit are not completely the same, the direct relationship between output and input can be obtained through time differentiation by feedback linearization method [11], which can be converted into a consistency problem to achieve the purpose of secondary control design. To apply feedback linearization, differential (3) gives an auxiliary control input for nonuniform delay voltage regulation (9) and current balancing (10).where and are the control inputs of the secondary voltage recovery and current sharing controller with time delay , respectively.

Rewrite the control input with nonuniform delay aswhere

R _vi is the virtual sag coefficient of the i-th DG unit.

Remark 1. The feedback linearization method transforms the voltage and current control problem into a consistency problem. At the same time, based on the information of DG unit and its communication neighbors, a distributed secondary voltage controller is designed by using nonlinear functions.
In order to realize voltage regulation and current balance, we designed controllers and , and introduced Artstein’s transformation to transform the system with time-delay control into ordinary differential control equation (16). In this paper, we use Artstein’s transform to reduce the output voltage and current of each DG cell to the first-order integrator without delay. Thus, we can obtain the first-order integrator of the output voltage as follows:In the same way as above, the first-order integrator of the output current is

Remark 2. The controllers designed using (14) and (16) can stabilize the converted system equations (13) and (15), respectively, and the original system equations (9) and (10) can also be stabilized under the same controller (14) or (16). Therefore, a distributed timing controller for voltage regulation and current sharing will be discussed below based on equations (13) and (15), respectively.

3. Distributed Fixed Time Secondary Control

The communication topology between DGs can be represented by the undirected graph = (, E). The vertex of the graph = {1, 2, …, n} represents DG, and the edge indicates the communication connection. The undirected edge between vertex i and j is expressed as (i, j), and the neighbor set of vertex I is expressed as N_i = {j | (i, j) E}. The adjacency matrix A = [a_ij] R^n×n indicates the communication weights. If the , then a_ij = a_ji> 0; Otherwise, a_ij = a_ji = 0. The Laplacian matrix of graph theory is defined as L = [l_ij] R^n×n, where and l_ij = −a_ij, for . The Laplacian matrix L is the same as the admittance matrix. If there’s one between any two different vertices, then the undirected graph G is connected. Define the pin matrix B = diag {b₁, b₂ ,…, b_N}, where b_i ⩾ 0.

The block diagram of the proposed distributed secondary fix control with time delay for the i-th DG is shown in Figure 2. The control strategy is divided into two parts: voltage regulation part and current balance part.

3.1. Voltage Regulation

The control input recovers each output voltage within a fixed time and input delay to eliminate the error of droop control on the DC microgrid bus voltage. It is assumed that at a fixed time T, all DGS satisfy the control objective of equation (7).

The purpose of secondary voltage control is to design the reference value of primary voltage control, so that the output voltage amplitude _i of each DG can be restored to the rated value . In order to realize voltage fixed time recovery control, based on DGs own information and its neighbor’s information. The auxiliary controller is designed as follows:where and are the control gains. and are scalars verifying .

3.2. Current Balancing

In addition, the control input balances the current of each DG within a fixed time and with the input delay. Set fixed time T, and all DGs meet the control target of equation (8).

The fixed time auxiliary control of current balance is defined as follows:where c₁ > 0 and c₂ > 0 are the current balancing control gains.

4. System Stability Analysis

Subsequent analysis requires the following lemma.

Lemma 1 (see [17]). Consider the following systems:where . If the nonlinear function is discontinuous, then the solution of system (21) is defined in the sense of Filippov. Suppose the origin is an equilibrium point of the system (21). If there is a radial unbounded function: Rⁿ ⟶ R⁺∪{0} which makes satisfy (1) (x) = 0 ⇔ x = 0;(2)Existence constant α; β; p; q; k > 0: pk < 1; qk > 1, any solution x (t) of the system equation (21) satisfies the inequality D⁺ (x (t)) ⩽ −(αV^p(x(t))−βVq(x(t)))^k, then the system equation (21) is globally fixed time stable and the set time estimate satisfies

Lemma 2 (see [18]). Let ξ₁, ξ₂, …, ξ_n ≥ 0. Then, we have

Lemma 3 (see [18]). For an undirected connected graph, the Laplace matrix L is semidefinite. 0 is a simple eigenvalue of L and 1_n is its eigenvector. The second smallest eigenvalue of L, which is denoted by λ₂, is larger than zero, and therefore, if , then

Theorem 1. If the undirected communication topology between DGS is connected, and at least one DG can obtain the reference voltage information, under the action of auxiliary controller equations (18) and (20), use the voltage control reference value obtained by the formula (11), the system can realize voltage recovery and proportional sharing of current within a fixed limited time.

Proof. First, the error term is defined as follows: .Take the derivative with respect to e_vi, and we getLet us go to the Lyapunov function given as follows:where , .
Differentiating (27), we can getFor the voltage regulation part, can be further written as follows:Define and forAccording to Lemma 2, it follows thatwhere L^µ and L^ν represent the corresponding Laplace matrices of the adjacent matrices and , respectively. B^µ and B^ν represent the pin matrix with and as diagonal elements, respectively.
According to Lemma 3, equations (30), (31) can be as follows:ThusSimilarly, can be obtained from formula equation (28), as follows:As with the voltage regulation part analysis, can be obtained as follows:It can be obtained by combining the expressions equations (28), (33), and (35)Furthermore, η, ρ, p, and q can be expressed separately as follows:By Lemma 2 again, we get furtherwhere , .
According to Lemma 1, V converges to 0 in fixed time, and the estimated setting time is as follows:When e_vi settles down to the origin, at which e_vi = 0, and become zero after T₁ + τ, Therefore, within the upper bound of fixed time, precise fixed time consensus can be obtained as follows:which completes the proof.

Remark 3. In this section, the stability of the designed quadratic controller is discussed based on Lyapunov stability theory. At the same time, the controller ensures that the bus voltage of the DC microgrid can be restored to its rated value and is not affected by its initial state.

5. Simulation Studies

In Matlab/Simulink environment, Figure 3 shows the structure diagram of the DC microgrid during simulation. A DC microgrid model composed of four parallel DG units, including the voltage source, a Buck DC/DC converter, and two common loads, is established. The model is to verify the effectiveness of the proposed secondary control scheme. In the droop control of the first stage, two standard PI controllers are used for the voltage control loop and the current control loop, respectively. The communication topology is shown in Figure 4.

It can be seen from the figure that only DG1 can directly obtain the reference current information, that is, the pin gain b₁ = 1, while the other modules can only obtain the reference current information indirectly through DG1. The simulation parameters and secondary controller are shown in Table 1.

5.1. Experimental Results with Communication Delay

Considering the communication delay in the generation of the secondary control signal, it is set to 10 ms, and the output current ratio of the DGs is set to 1 : 1: 1 : 1. As can be seen from Figure 5, when the secondary control delay is 10 ms, the voltage auxiliary controller can adjust the bus voltage of DGs to the rated value of 48 V. It can be seen from Figure 6 that the output current of DGs can be shared at 1 : 1 : 1 : 1. Therefore, voltage and current auxiliary controllers are effective.

5.2. The Current Is Shared Proportionally

On the basis of 5.1, set the current ratio to 1 : 2 : 3 : 6. As can be seen in Figure 7, the voltage of each DGs can still be restored to its rated value of 48 V. It can be seen from Figure 8 that the output current of each DGs can also be shared in a ratio of 1 : 2 : 3 : 6.

5.3. The Load Change

In the case of 5.1, only one resistive load R_load1 = 5 Ω is connected to the bus between 0 and 0.3 s. In order to test the robustness of the load conversion system, a R_load2 = 5 Ω resistor is connected at 0.3 s, and the load changes from 5 to 10 Ω. As can be seen from Figures 9 and 10, even if the load changes, the DC bus voltage is still well maintained at the rated value, and the output current of each DGS is reasonably shared according to the design.

5.4. Power Production System Changes

In this case, we test the ability of the control scheme to change the power production system. The setting parameters are the same as those in 5.1 case. At 0.3 s, converter ^#3 was disconnected from the power grid, and DG3 stopped working. The results of the case study are shown in Figures 11 and 12, which, respectively, represent the output voltage and output current of each unit. It can be clearly seen that when converter ^#3 is disconnected from the power grid, the other three converters can quickly share the current. Since the DG3 unit stops working, the current is also rapidly zero, and the voltage can still be kept at the reference value of the DC bus. It shows that the control scheme still has good control ability despite the changes of the power production system.

5.5. Comparison with the Existing Schemes

This case study compares the control scheme proposed in this paper with the finite time control scheme proposed in [16] and the progressive consistency control scheme in [15]. It is assumed that the DC system will go through the following three stages: Stage 1. (0–0.4 s): Complete secondary control. Stage 2. (0.4 s–0.7 s): The other resistive load R_load2 = 5 Ω is connected to the power grid at t = 0.4 s. Stage 3. (0.7 s–1 s): At 0.7 s, the power production system changes, and converter ^#3 is disconnected from the power grid.

The results of the case study are shown in Figures 13–18. In the first stage, the control scheme proposed in this paper is obviously faster than the control scheme 0.1 s-0.2 s in [15, 16] in the convergence of voltage and current. There is a small fluctuation in the voltage in [15], and the current fails to achieve 1 : 1:1 : 1 sharing. In the second stage, the three control schemes have good control over load changes, but the voltage in the [15] control scheme is not smooth enough and has small fluctuations. In the third stage, the voltage control is basically consistent. In contrast to the current control, the control scheme proposed in this paper has less fluctuation, while the current control fails in [15]. In contrast, the control scheme proposed by us can realize the voltage regulation and current sharing of each unit faster and more accurately.

6. Conclusions

This paper studies the secondary control of a DC microgrid. Based on the information of each DGS and its communication neighbors, a distributed secondary fixed time control is designed to achieve voltage recovery and current proportional distribution. However, the existence of a delay in secondary control cannot be ignored. On the basis of droop control, the time-delay system is transformed into a time-free system by feedback linearization and Artstein’s transformation, which is convenient for secondary control of voltage and current. Theoretical analysis and numerical simulation verify the effectiveness of the proposed control scheme.

The fixed-time quadratic control problem in the case of directed communication topology will be further considered in the future.

Data Availability

The data used to support this study can be obtained from the corresponding authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was funded in part by The Science and Technology Foundation of Gansu Province, China (grant no. 22CX3JA002), in part by the National Natural Science Foundation of China (grant no. 62141304).

References

A. Tah and D. Das, “An enhanced droop control method for accurate load sharing and voltage improvement of isolated and interconnected DC microgrids,” IEEE Transactions on Sustainable Energy, vol. 7, no. 3, pp. 1194–1204, 2016.
View at: Publisher Site | Google Scholar
T. Dragicevic, X. Lu, J. C. Vasquez, J. M. Guerrero, A. I. U. S. Argonne, and A. N. L. National Lab, “DC microgrids-Part II: a review of power architectures, applications, and standardization issues,” IEEE Transactions on Power Electronics, vol. 31, no. 5, pp. 3528–3549, 2016.
View at: Publisher Site | Google Scholar
J. W. Simpson-Porco, F. Dörfler, and F. Bullo, “Synchronization and power sharing for droop-controlled inverters in islanded microgrids,” Automatica, vol. 49, no. 9, pp. 2603–2611, 2013.
View at: Publisher Site | Google Scholar
J. M. Guerrero, J. C. Vasquez, J. Matas, L. G. de Vicuna, and M. Castilla, “Hierarchical control of droop-controlled AC and DC microgrids-A general approach toward standardization,” IEEE Transactions on Industrial Electronics, vol. 58, no. 1, pp. 158–172, 1982.
View at: Publisher Site | Google Scholar
M. Yazdanian and A. Mehrizi-Sani, “Distributed control techniques in microgrids,” IEEE Transactions on Smart Grid, vol. 5, no. 6, pp. 2901–2909, 2014.
View at: Publisher Site | Google Scholar
F. Guo, Q. Xu, C. Wen, L. Wang, and P. Wang, “Distributed secondary control for power allocation and voltage restoration in islanded DC microgrids,” IEEE Transactions on Sustainable Energy, vol. 9, no. 4, pp. 1857–1869, 2018.
View at: Publisher Site | Google Scholar
X. K. Liu, H. He, Y. W. Wang, Q. Xu, and F. Guo, “Distributed hybrid secondary control for a DC microgrid via discrete-time interaction,” IEEE Transactions on Energy Conversion, vol. 33, no. 4, pp. 1865–1875, 2018.
View at: Publisher Site | Google Scholar
D. Yue, H. Zhang, and C. Dou, “Mas-based distributed cooperative control for dc microgrid through switching topology communication network with time-varying delays,” in Cooperative Optimal Control of Hybrid Energy Systems, pp. 251–268, Springer, Singapore, 2021.
View at: Publisher Site | Google Scholar
A. Bidram, A. Davoudi, F. L. Lewis, and J. M. Guerrero, “Distributed cooperative secondary control of microgrids using feedback linearization,” IEEE Transactions on Power Systems, vol. 28, no. 3, pp. 3462–3470, 2013.
View at: Publisher Site | Google Scholar
Q. F. Yuan, Y. W. Wang, X. K. Liu, and Y. Lei, “Distributed fixed-time secondary control for DC microgrid via dynamic average consensus,” IEEE Transactions on Sustainable Energy, vol. 12, no. 4, pp. 2008–2018, 2021.
View at: Publisher Site | Google Scholar
S. Zuo, A. Davoudi, Y. Song, and F. L. Lewis, “Distributed finite-time voltage and frequency restoration in islanded AC microgrids,” IEEE Transactions on Industrial Electronics, vol. 63, no. 10, pp. 5988–5997, 1982.
View at: Publisher Site | Google Scholar
S. Sahoo and S. Mishra, “A distributed finite-time secondary average voltage regulation and current sharing controller for DC microgrids,” IEEE Transactions on Smart Grid, vol. 10, no. 1, pp. 282–292, 2019.
View at: Publisher Site | Google Scholar
A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2106–2110, 2012.
View at: Publisher Site | Google Scholar
Z. Zuo, C. Wang, and Z. Ding, “Robust consensus control of uncertain multi‐agent systems with input delay: a model reduction method,” International Journal of Robust and Nonlinear Control, vol. 27, no. 11, pp. 1874–1894, 2017.
View at: Publisher Site | Google Scholar
M. Dong, L. Li, Y. Nie, D. Song, and J. Yang, “Stability analysis of a novel distributed secondary control considering communication delay in DC microgrids,” IEEE Transactions on Smart Grid, vol. 10, no. 6, pp. 6690–6700, 2019.
View at: Publisher Site | Google Scholar
J. Ni, L. Liu, C. Liu, and J. Liu, “Fixed-timeleader-following consensus for second-order multiagent systems with input delay,” IEEE Transactions on Industrial Electronics, vol. 64, no. 11, pp. 8635–8646, 1982.
View at: Publisher Site | Google Scholar
S. Chen, Z. Zou, Z. Zhang, and L. Zhao, “Fixed-time scaled consensus of multi-agent systems with input delay,” Journal of the Franklin Institute, vol. 480, no. 4, 2022.
View at: Publisher Site | Google Scholar
L. Wang and F. Xiao, “Finite-time consensus problems for networks of dynamic agents,” IEEE Transactions on Automatic Control, vol. 55, no. 4, pp. 950–955, 2010.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2023 Yi Wei Feng and Jia Li Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies