The DC microgrid passes through only one level of the DC-DC voltage converter between each distributed power source and the load, which effectively reduces the power transmission loss and has a low system cost and high reliability [1,2]. Multiple DC-DC converters are internally connected according to a certain topology to form a small power supply network [3,4]. Voltage amplitude stability in a DC microgrid is an important measure of voltage quality, and it is also key to the reliable operation of a DC microgrid system [5,6]. Therefore, the output voltage synchronization of the converter is an important prerequisite for the stable operation of the DC microgrid system. At the same time, the controller design and energy management of the DC microgrid is also the key factors to maintain the stable operation of the microgrid, and the appropriate energy storage configuration and efficient energy management system can realize the dynamic balance between the distributed energy supply and the diversified load demand of the microgrid. A new energy management algorithm and a multi-objective nonlinear controller in the microgrid environment of the gym was proposed in Reference [7] to achieve a dynamic balance between the intermittent power generation of distributed generators and the randomness of load changes.

In DC microgrid, many studies focus on the output energy distribution of distributed generators and the stable control of bus voltage. Reference [8] proposes an improved hierarchical control method for DC microgrid, which realizes the balanced distribution of load power and the stability control of bus voltage through the coordination between controllers. In Reference [9], a distributed controller based on virtual voltage difference was designed, and voltage stabilization and power distribution could be achieved simultaneously through the global sharing of virtual voltage difference. In Reference [10], a droop control method was designed based on the reference voltage deviation on the basis of considering the energy storage unit, which realized the dynamic balance between the load and the power supply and the unbiased control of the bus voltage. Reference [11] proposes an adaptive virtual inertial control method for DC microgrid bus voltage, which solves the problem of small interference oscillation of bus voltage and bus voltage offset under large interference. Reference [12] proposes an adaptive droop control strategy based on the state of charge of the battery, which realizes the smooth switching of the DC microgrid system in different working modes and can maintain the stability of the bus voltage. In the context of islanded DC microgrids, a distributed cooperative control strategy considering the voltage regulation and economic dispatching objectives was proposed in Reference [13], which realized bus voltage stability and optimal load distribution under the condition of considering energy storage constraints. The above literature proposes many stability control methods for DC microgrid bus voltage, but the influence of the complex topology of DC microgrid compared with traditional power grid on the synchronization characteristics of DC bus voltage is not fully considered.

Currently, the state-space method and the coupled synchronization method based on state equations or the Kuramoto model is mainly used for the synchronization characterization of microgrid systems. Bellini et al. proposed a method to synchronize the output voltage of microgrid converters by cascading coupled power lines under the microgrid islanding operation mode [14]. References [15,16] take the output voltage of the generator set as a reference for the output voltage of the microgrid converter to realize the tracking and synchronization of the microgrid bus voltage. Reference [17] explores how the interaction between DC microgrid converters affects their voltage synchronization characteristics. The mechanism by which the dynamic interaction between DC microgrid converters affects the voltage stability of the bus is reported. However, the interconnected DC microgrid system, which utilizes multiple DC-DC converters in a specific topology, exhibits an intricate nonlinear behavior [18]. This complexity leads to challenges in synchronizing the output voltage oscillation of converters and poses new obstacles for ensuring the stability and control of the bus voltage within the DC microgrid system [19,20]. Reference [21] used microgrid converters as network nodes and studied the bifurcation phenomenon of converters and the coupling synchronization method of the bus voltage of a photovoltaic microgrid system. In Reference [22], authors investigated the effect of microgrid topology on network synchronization stability based on Kuramoto’s first-order model and provided a formula for the critical coupling strength after adding new connections and new node units to the microgrid. Reference [23] investigates the effect of adding new connections on voltage synchronization in power networks, and the results show that increasing the number of nonuniformly connected edges is positively correlated with the voltage synchronization performance. Moreover, a microgrid, as a new type of electric energy network, has a topology with small-world characteristics in complex networks, and the synchronization problem among its nodes has attracted extensive attention from many scholars [24−26]. Reference [27] investigated the enhancement of voltage synchronization characteristics by changing the switching state of circuit breakers to change the topology of the entire power network. Literature [28] shows that shortcuts in small-world networks enhance the synchronization characteristics of power networks. Reference [29] investigated the impact of power network structure on the vulnerability of transmission networks from the perspective of network hierarchy characteristics and their relationships. The above literature mainly focuses on the effect of the power grid structure on voltage synchronization, whereas, the strong nonlinear characteristics of microgrid converters have not been considered fully in the analysis process.

In the above literature, References [14–17] have studied the bus voltage oscillation of microgrid, but not the asynchronous bus voltages caused by the topological connection relationship of microgrids. In order to solve the problem that the synchronization method of bus voltage in References [21–23] has certain limitations and is not suitable for DC microgrid, this paper proposes a principal stability function analysis method suitable for the study of bus voltage synchronization characteristics of DC microgrids. References [27–29] only study the synchronization characteristics of bus voltages from the perspective of complex networks, but they do not provide appropriate control methods for the phenomenon of failure to achieve synchronization. In view of the shortcomings of the above literature, this paper proposes an improved adaptive synchronous control method for DC microgrid bus voltage control for the first time, and the bus voltage realizes synchronous control and oscillation suppression.

In this paper, we focus on a DC microgrid with hierarchical control and study the influence of the structural parameters in the network on the synchronization characteristics of converter output voltages based on the master stabilization function method and a small-world network model, to better control the DC microgrid bus voltage oscillation caused by converter bifurcation behavior. With respect to the voltage desynchronization phenomenon caused by the capacity limitation of the contact line, a new improved adaptive synchronization control method was developed. This control method can quickly realize voltage synchronization between converter nodes in a DC microgrid small-world network.

The innovation of this paper is mainly in the following three aspects:

1. A new modeling method of DC microgrid based on complex network is proposed to adapt to the actual topology of DC microgrid.

2. A new principal stability function method suitable for DC microgrid bus voltage synchronization is applied to study the synchronization characteristics of DC bus voltage synchronization.

3. A new and improved adaptive controller is designed for synchronous control and oscillation suppression of DC bus voltage.

The remainder of this paper is organized as follows: Section 2 builds a small-world network model based on a hierarchically controlled DC microgrid; Section 3 analyzes the synchronization characteristics of the converter output voltages based on the master stability function method; Section 4 investigates the improved adaptive synchronization control for DC microgrid small-world networks; Section 5 verifies the theoretical analysis via simulation; and Section 6 concludes the paper.

The synchronized manifold of the DC microgrid converter network model (12) is a linear subspace in RN×N: A={x=(x1,…,xN):∀xi,xj∈Rn,(i,j∈N)}, and network (12) is said to be fully synchronized, or synchronized for short if the solution x of converter network (12) converges to set A. That is, when time t→∞, for all nodes of the network (12), under any initial condition, when t→∞, ‖xi(t)−xj(t)‖→0, (i,j∈N). This leads to ‖xi(t)−x¯(t)‖→0,(i∈N) when t→∞.

In this paper, voltage synchronization analysis of DC microgrid small-world networks is based on the master stability function method proposed by Sivaganesh [32]. The network satisfies the following conditions: ① all DG nodes have the same dynamics equations; ② the coupling functions between each converter node are the same; ③ the synchronization pop is constant and can be linearized near the steady-state solution. From this, the network model (12) can be variationally differentiated such that ξi(t)=xi(t)−s(t), and the variational equations of (12) are obtained. (13)ξ˙i=Df(s)ξi−k∑j=1NlijDH(s)ξj,i∈N

In Eq. (13), the Jacobian matrices of functions f(x) and H(x) at their solution s are Df(s) and DH(s), respectively. And letting ζ=[ζ1,ζ2,…,ζN], Eq. (13) can be rewritten: (14)ζ˙=ζDf(s)−ζkDH(s)LT

Obviously, the matrix L can be diagonalized by noting LT=PΛP−1, Λ=diag(λ1,λ2,…,λN), where λi(i∈N) is the eigenvalue of L, and then letting η=[η1,η2,…,ηN]=ζP, we have: (15)η˙i=[Df(s)−kλiDH(s)]ηi,i=2,3,…,N

Considering that the L matrix of the converter network is a symmetric matrix that has characteristics and that only one value is 0, the rest are positive real numbers, i.e., 0=λ1<λ2≤…≤λN; then, its main stabilization equation is: (16)y˙=[Df(s)−βDH(s)]y

Then, the region in Eq. (16) with the maximum Lyapunov exponent Lmax=Lmax(β), where Lmax(β) is negative, is the synchronization region, denoted as SR={β|Lmax(β)<0}. Therefore, if kλi∈SR in Eq. (15), then network (12) reaches synchronization.

For the case where the synchronization region is unbounded, SR=(β1,+∞), i.e., when β1<kλ2≤kλ3≤…≤kλN, Lmax<0, i.e., kλ2>β1. That is, the synchronization manifold is stable when the coupling strength k is greater than a certain value. Therefore, in this case, λ2 of the Laplacian matrix can be used as an indicator to measure the network synchronization capability; if λ2 is larger, the synchronization time of the network is shorter, and the synchronization ability is stronger.

For the actual network constituted by the converter, the coupling strength is limited by objective factors such as the line capacity, which cannot be increased indefinitely; at this time, it is necessary to use control methods to synchronize the voltage amplitude. Under the linear coupling function, the adaptive controller can synchronize the output voltage of each node, but the control effect is poor. In this paper, by improving the adaptive controller, the voltage of the whole DC microgrid with a better synchronization effect than adaptive control has been achieved.

Consider a small-world network with linear dissipative coupling consisting of N identical converter nodes with controlled network equations: (17)x˙i=f(xi,t)+∑j=1NaijHxj+ui(x1,…,xN)

The synchronization error of the voltage of the network can be written as: (18)e˙i=f(xi,t)−f(s,t)+∑j=1NaijHxj+ui(x1,…,xN)

Adaptive controllers are added to each node in the converter network in the following form: (19)uik=−dikeik,d˙ik=hik‖eik‖2,1≤k≤N

An improved adaptive controller is applied to each node in the converter network in the following form: (20)uik=−dikeik,d˙ik=−exp⁡δ(e˙ik−eik),1≤k≤N

The linearization of the error in the neighborhood of the steady-state solution is obtained: (21){e˙ik=Df(s,t)ei+∑j=1NaijHej−diei,1≤i≤Nd˙i=−exp⁡δ(e˙ik−eik),1≤i≤N

Suppose there is a positive constant α such that ‖Df(s,t)‖2≤α, and if there exists a natural number l satisfying λl+1<−αγ, then the network is synchronized under the improved adaptive controller, where γ=‖H‖2>0.

The proof procedure is as follows. For the above system, the Lyapunov function is constructed as follows: (22)V=12∑i=1NeiTei+12∑i=1L(di−d)2hi

Then: V˙=12∑i=1N(e˙iTei+eie˙iT)+∑i=1L(di−d)d˙ihi=∑i=1Ne˙iT(Df(s,t))Sei+∑i=1N∑j=1NaijeiTHej−∑i=1ldeiTei=∑i=1Ne˙iT(Df(s,t))Sei+∑i=1N∑j=1j≠iNaijeiTHej+∑i=1NaiieiTHSei−∑i=1ldeiTei≤∑i=1NαeiTei+∑i=1N∑j=1j≠iNγaij‖ei‖⋅‖ej‖+∑i=1Naiiλmin(HS)eiTei−∑i=1ldeiTei=eT(αIN+γA˘−D)e

To make αIN−l+γAl+1<0, choose an appropriate positive number D such that αIN+γAl+1−D<0. Suppose that there exists a natural number l that satisfies the condition αIN+γλl+1<0 such that αIN−l+γAl+1<0, which leads to αIN+γA˘−D<0. Thus, when t→∞, ei(t)→0. The proof is complete.

To verify the correctness and feasibility of the analysis and control methods proposed in this paper, simulation verification is carried out below. Under the small-world network structure of a DC microgrid with hierarchical control, the bifurcation behavior of converters and the effects of different reconnection edge probabilities, coupling strengths, and network sizes on the voltage synchronization characteristics of DC microgrids under the small-world network structure are analyzed. The system parameters selected are shown in Table 1.

A bifurcation diagram with DGi DC input voltage E as the bifurcation parameter is shown in Fig. 3. Under certain conditions of other parameters, as E decreases, the output voltage Uo near E=52V undergoes inverse doubling cycle bifurcation and enters two-cycle operation; Uo near E=40V undergoes boundary collision bifurcation; and the inverse doubling cycle bifurcation continues to occur near E=34V, which causes the converter to enter a chaotic working state. When the converter enters chaotic motion due to bifurcation, the output voltage of the converter becomes unstable, which makes it difficult to synchronize the output voltage of each converter.

A DC microgrid small-world network model is established to simulate and verify its voltage amplitude synchronization characteristics. The Ode45 method is adopted to solve the differential equation system of the converter nodes, with an integration step of h = 1 × 10⁻⁶ s and network parameters of N = 10 and P = 0.2. Γ=diag(0,1) indicates that only the coupling relationship between the voltage variables is considered, and the coupling strength is k. The simulation duration is 0.2 s. The output voltage timing diagrams under different coupling strengths are shown in Figs. 4–6.

As can be seen from Figs. 4 to 6, in a 10-node microgrid network, when k = 0, the voltage between the converters cannot be synchronized at all, when k = 1, the synchronization deviation between the voltages of each node decreases to ±4 V, and when k = 2, the voltage synchronization deviation of each node decreases to ±2.5 V. From the data analysis, it can be seen that with the increase of the coupling strength between each node, the synchronization deviation of the bus voltage of each node of the microgrid has a significant decreasing trend.

To observe the synchronization characteristics of each converter node voltage more intuitively, this paper compares the synchronization performance of the DC microgrid converter voltage amplitude using variance, which is defined as follows [33]:(23)σ=1N∑i=1N(Uoi−U¯on)2,(i∈N)where Uoi denotes the converter output voltage of node i, U¯on denotes the average value of the output voltage amplitude of node N in the converter network, and σ=0 when the converter output voltages are synchronized.

As can be seen from Fig. 7, when k = 5, the maximum value of voltage synchronization variance reaches the maximum value of σ = 1.65 at t = 0.023 s, and decreases with time, when t = 0.05 s, σ = 0.78, when k = 10, at t = 0.022 s, the maximum value of voltage synchronization variance decreases to σ = 1.65, when t = 0.05 s, σ = 0.35, when k = 50, at t = 0.018 s, the maximum value of voltage synchronization variance decreases to σ = 0.02, when t = 0.05 s, σ = 0.002. From the above data analysis, it can be seen that with the increase of network coupling strength k, the peak value of the variance of each converter gradually decreases, and the larger k is, the shorter the output voltage synchronization time of the converter network node.

The Laplacian matrix of the DC microgrid small-world network structure can be obtained from the definition of Eq. (4) combined with the converter network structure. The variations in the eigenvalues of the Laplacian matrix of the small-world network with the number of nodes at different P values are calculated via MATLAB, and the results are shown in Fig. 8. The synchronization ability of the DC microgrid small-world network decreases as the network size increases. For the same number of converter nodes, λ2 increases with increasing P, indicating that the shortcuts in the small-world network have an enhanced effect on the synchronization ability of the DC microgrid network, which is stronger than that of the traditional radial and ring networks.

To better study the synchronization of the DC microgrid small-world network model system, the synchronization time change rule with the network size is studied by a numerical calculation method with node N as the change parameter. The synchronization time of the output voltage of each node of the converter of the DC microgrid small-world network model under different reconnection edge probabilities is shown in Fig. 9. In the 10-node microgrid network, when P = 0.1 increases to 0.3, the voltage synchronization time of each node is about 0.012 s due to the small scale of the network. When the network size increases to 40 nodes, from P = 0.1 to P = 0.2, the voltage synchronization time decreases from 0.16 to 0.06 s, and from 0.06 to 0.04 s when the reconnect probability from P = 0.2 to P = 0.3. When the network size increases to 100 nodes, from P = 0.1 to P = 0.2, the voltage synchronization time decreases from 0.37 to 0.28 s, and the voltage synchronization decreases from 0.28 to 0.18 s when the reconnect probability from P = 0.2 to P = 0.3. Through the comparative analysis of the data, it can be seen that with the increase of the number in nodes, the output voltage synchronization time increases, but with the increase of the reconnection probability, the output voltage synchronization time is shorter under the same network scale.

The control performance of the DC microgrid’s second layer control of the converter using improved adaptive control and adaptive control is analyzed in the following figures by comparison. The simulation of the second layer control using adaptive control is shown in Fig. 10; the simulation of the second layer control using improved adaptive control is shown in Fig. 11. In both control modes, to facilitate the comparison of their control performance, the coupling strength is k = 1, the rest of the circuit parameters are the same as the aforementioned parameters, the control is applied at t = 0.06 s, and a unit step perturbation is applied to the node voltage at t = 0.1 s to test the immunity performance of the controller.

Compared with Figs. 10 and 11, it can be observed that the voltage deviation of each node is ±0.65 V and gradually decreases when the traditional adaptive controller is applied at t = 0.06 s, the bus voltage fluctuation is ±0.48 V when the input voltage is disturbed at t = 0.09 s, and the synchronization error of the bus voltage is still 0.12 V at t = 0.2 s, and the voltage deviation of each node is ±0.46 V and gradually decreases rapidly when the improved adaptive controller designed in this paper is applied at t = 0.06 s. When the input voltage is disturbed at t = 0.09 s, the bus voltage fluctuation is only ±0.17 V, and the synchronization error of the bus voltage is only ±0.05 V when t = 0.15 s. Through comparison, it is found that the output voltage of the DC microgrid converter network can be synchronized under the two control modes, but the improved adaptive control has a better control effect. When the input voltage of one of the nodes fluctuates, the fluctuation of the converter output voltage is smaller under the improved adaptive control, and it can also quickly restore the smoothness.

In the following, the perturbation resistance analysis of the above model is performed. The primary layer control of the microgrid is droop control, where W=diag(0,1), the perturbation function is Gi=3sin⁡(wt), the perturbation is applied at t = 0.5 s, and the simulation is shown in Fig. 12. Second layer control with adaptive control returns the voltage to smoothness after a longer period, and the elimination of the output voltage difference generated by droop control is slower. The maximum voltage fluctuation at the time of controller addition is 4.4 V, which is eliminated after 0.07 s, but there is still a voltage deviation of 0.9 V, and the deviation is still not eliminated at 0.8 s. In addition, in Fig. 13, the secondary control with improved adaptive control returns the voltage to smoothness faster and eliminates the voltage difference of the droop control very well. With the application of the improved adaptive controller, the maximum fluctuation of the bus voltage is only ±0.6 V, and the deviation of the bus voltage has been eliminated at 0.55 s.

In this paper, a small-world network model of a DC microgrid is established based on hierarchical control. The effects of the coupling strength and node connection relationship on the voltage synchronization characteristics and voltage synchronization control method are investigated. Based on References [14–17], this paper considers the complex topology of DC microgrid, establishes an adaptable small-world network model, and proposes a principal stability function method suitable for the analysis of the voltage synchronization characteristics of DC microgrid bus based on References [21–23]. Based on References [27–29], an improved adaptive voltage synchronous control method is proposed, and the following conclusions are drawn through simulation comparison and analysis with the traditional adaptive control method:

(1) With increasing coupling strength and connection edges between converter nodes, the voltage synchronization performance of the DC microgrid small-world network is enhanced, and the bus voltages tend to be synchronized. In the microgrid network model, when the coupling strength k = 50, the amplitude of the voltage synchronization error of each node is controlled within 0.5%. When the reconnection probability P = 0.3, for a typical 100-node microgrid network, the voltage synchronization of each node is reached in 0.18 s.

(2) Hierarchical control of DC microgrid system voltage secondary control using improved adaptive control can eliminate the voltage difference of droop control faster than adaptive control and has better anti-interference performance. The voltage fluctuation of the improved adaptive controller designed in this paper is only ±0.6 V under disturbance, which is 3.8 V lower than that of the traditional adaptive control, and the unbiased control of the bus voltage is realized within 0.05 s, which is 0.12 s faster than the traditional adaptive controller.