With the continuous development of science, technology, and industrial technology, the global energy crisis is becoming more and more serious, and the shortage of traditional fuels and the environmental pollution problems caused by them are also increasing. At the same time, the continuous improvement and gradual maturity of renewable energy technologies such as solar power and wind power provide a new way to solve the energy crisis (Lin et al., 2018; Hoang and Lee, 2020; Huang et al., 2022). Therefore, many countries are focusing on distributed power generation using renewable energy as the energy source. Compared with the traditional large-scale centralized generation and distribution modes, distributed generation technology has the advantages of real-time control, low investment, flexible installation location, a short construction period, high energy utilization, and low environmental pollution (Morstyn et al., 2018). However, for DC microgrids operating on isolated islands, distributed power sources, due to their randomness, volatility, and uncontrollability, can greatly reduce power quality, thereby affecting the stable operation of the microgrid (Yang et al., 2021; YNan et al., 2021). Therefore, it is necessary to configure distributed energy storage systems (DESSs) to suppress energy fluctuations (Hosseinipour and Hojabri, 2018).

In DESSs, each DESU is often connected to the bus bar in parallel. Since the initial SOC or capacity of each DESU may not be consistent, the SOC of each DESU may be inconsistent during charging and discharging, resulting in some DESUs exiting the system early due to overcharging or over discharging, leading to asymmetric operation of the microgrid (Bi et al., 2019; Hoang and Lee, 2021). Therefore, it is necessary to coordinate and control the DESU’s output current and SOC, ensuring that each DESU’s output current is accurately distributed according to the capacity ratio and the SOC is balanced (Yang et al., 2022).

Droop control is widely used in energy distribution in DC microgrids for its advantages such as high reliability and plug-and-play. Zhou et al. (2020) introduced an optimal control method for multi-battery energy storage systems in islanded DC microgrids, leveraging the PI consensus algorithm to enhance robustness against time delays. However, this and other studies (Li et al., 2022) failed to address the crucial issue of SOC balancing in their control loops, potentially causing overcharging or over-discharging of energy storage units. To address SOC balancing, research has explored linking the droop coefficient and SOC. Lu et al. (2014) and Lu et al. (2015) constructed a link by a power function to dynamically adjust the droop coefficient to achieve fast SOC equalization. However, the SOC equalization accuracy is low, and the bus voltage drop introduced by the droop coefficient has not been effectively compensated. To eliminate the bus voltage drop, Li et al. (2014) and Shahbazi et al. (2016) not only modified the virtual impedance but also dynamically adjusted the output voltage reference of the DESU based on changes in SOC to achieve SOC equalization and bus voltage stabilization. Zhi et al. (2020) designed the droop coefficient to be proportional to SOC during charging and inversely proportional to SOC during discharging, thus achieving both SOC equalization and power equalization and improving the stability of the DC bus voltage by enhancing the inertia of the DC microgrid. Xu et al. (2020) achieved SOC equalization by setting the droop coefficient to be inversely and positively proportional to the nth power of SOC during charging and discharging, respectively. In addition, the given reference voltage is changed by the multi-intelligent body technique, which compensates for the bus voltage variation caused by the droop coefficient. However, the aforementioned studies did not consider the effect of line impedance on SOC equalization and power distribution.

To simultaneously stabilize the bus voltage and eliminate the influence of line impedance on SOC equalization and current distribution, an autonomous voltage regulation scheme was proposed by Yang et al. (2018), which compensates for the voltage drop caused by line impedance by setting virtual resistors to achieve stable voltage. However, the value of the virtual resistance is equal to the line impedance, so the line impedance information must be known, and the line impedance is affected by factors such as environmental changes, which leads to unequal line impedance and makes impedance information difficult to obtain accurately, further reducing the effectiveness of the control method. Liu et al. (2018) measured line resistance information by injecting pulse perturbations and detecting changes in the output voltage and current, which are used to compensate for droop coefficients and restore DC bus voltage. However, in practical operation, it is difficult to eliminate interference caused by various environmental factors, so the accuracy of this method is not high enough. To completely eliminate the influence of line impedance on SOC equalization and accurate current distribution, Chen et al. (2018) proposed a two-level layered voltage control scheme, which uses a consistency algorithm to estimate the average DC bus voltage, eliminates the influence of line impedance, and achieves SOC equalization and voltage stabilization. Shafiee et al. (2014) proposed a distributed hierarchical control framework, which adjusts the droop coefficient in real-time based on changes in SOC in the primary control layer. A voltage regulator based on the consistency algorithm is introduced in the secondary control layer to eliminate the bus deviation, and again, the power equalization distribution is achieved by a power controller. Hoang and Lee (2019) proposed a virtual power rating method based on SOC, which achieved accurate current distribution. However, the aforementioned literature did not consider the situation where the capacity of DESUs is different.

To achieve SOC equalization and accurate current distribution among different capacity DESUs simultaneously, Lin and Wang Zhi Xin (2020), Zeng et al. (2022a), and Mi et al. (2023) constructed the connection between the droop coefficient and SOC by setting the convergence function and dynamically adjusting the droop coefficient to quickly achieve SOC equalization between DESUs of different capacities. Lin and Wang Zhi Xin (2020) and Zeng et al. (2022a) designed a virtual voltage drop equalizer and a voltage compensator, which, respectively, eliminate the impact of line impedance on output current accuracy and compensate for bus voltage drop. Mi et al. (2023), on the other hand, achieved the aforementioned objectives by introducing virtual power rating compensation and voltage compensation. However, the aforementioned methods all use two PI controllers to form a secondary control to achieve energy equalization, and the parallel connection of both PI controllers affects the same variables, which makes the design of control parameters more difficult. As a result, recent research has focused on developing more simplified and efficient control strategies to address these limitations. Zhang et al. (2022) proposed a distributed cooperative control strategy that employs a single controller, simplifying system complexity. Chen et al. (2020) demonstrated that each supercapacitor or battery utilizes only neighboring information exchange rather than global communication, enjoying equal priority in participating in the voltage or SOC regulation. (Wu et al., 2018) introduced an SOC balancing strategy in AC microgrids that does not require communication or a central controller, thus enhancing system scalability. Jiang et al. (2023) proposed a double-ascent algorithm to optimize current distribution coefficients online, reducing current losses. Both Zeng et al. (2022b) and Lin et al. (2022) introduced exponential functions, where the ratio of each energy storage unit’s SOC to the average SOC of all units is used as the input to the droop coefficient. Morstyn et al. (2016) achieved SOC balancing by comparing the SOC of a unit with its neighboring units. However, the SOC balancing speed of these control strategies remains relatively slow.

The expansion strategies of DC microgrids can be classified as centralized, decentralized, and distributed. However, centralized control suffers from a single point of failure, requires a high bandwidth communication network, and is difficult to expand (Nasirian et al., 2015; Hoang and Lee, 2020), while decentralized control leads to problems such as poor voltage regulation and inaccurate load distribution. Distributed control, on the other hand, is more suitable for microgrids with its high reliability, simple communication network, and easier scalability (Anand et al., 2013). Among them, distributed control based on a consistency algorithm becomes the most attractive control scheme, which eliminates the need for a central controller and obtains global average information by simply exchanging information with neighboring nodes.

Based on the aforementioned analysis, a DESS composed of multiple DESUs mainly faces the following challenges in SOC equalization and accurate current distribution:

1. Most of the current studies on accurate output current distribution and bus voltage recovery use two or more PI controllers for regulation. However, since the controlled quantities are linked to each other, changing any of the control parameters arbitrarily will have an impact on the overall effect, increasing the difficulty of parameter design.

To solve the aforementioned problems, a multi-storage islanded DC microgrid energy balancing strategy based on the hierarchical cooperative control is proposed in this paper, and the advantages of the proposed control strategy are as follows:

1. In the primary control layer, this paper introduces a multi-storage islanded DC microgrid energy balancing strategy grounded in hierarchical cooperative control, aimed at addressing the SOC equalization issue in DESS composed of DESUs with varying capacities. This strategy innovatively leverages the properties of logarithmic functions to devise a novel adaptive droop coefficient adjustment scheme. Within this adaptive framework, a faster SOC equalization rate is achieved by harnessing the characteristics of the logarithmic function, along with the integration of a purposefully designed acceleration factor and adjustment coefficients.

The rest of this paper is organized as follows: Section 2 of this paper introduces the structure of a DC microgrid containing multiple DESUs and analyzes the shortcomings of traditional droop control. In Section 3, a hierarchical cooperative control-based energy balance strategy is introduced in detail, which achieves SOC equalization and precise current distribution through the cooperation of primary and secondary layers and the communication layer. In Section 4, a detailed exploration of the parameter selection for the adjustment scheme of the droop coefficients in the primary control layer is presented. In Section 5, the stability of the proposed control strategy is analyzed using the example of a DC microgrid consisting of two ESUs of equal capacity. Then, the effectiveness and feasibility of the proposed strategy are confirmed in Section 6 through experimental verification under three operating conditions. Finally, Section 7 summarizes the whole paper and draws conclusions.

The overall block diagram of the proposed hierarchical control strategy is shown in Figure 3, where DESU _i denotes the ith DESU; I _Li and V _i are the inductor current and virtual voltage drop of the ith DESU, respectively; and SOC _avg is the average value of the SOC of DESS obtained by the consistency algorithm.

To enhance the SOC equalization effect and ensure the stability of the ESS, it is necessary to make rational selections for the parameters in Eq. 8, primarily focusing on the choice of the initial droop coefficient R _oi , the equalization acceleration factor n, and the adjustment coefficient m. For an ESS consisting of ith ESUs, the relationship between SOC _avg and SOC _i satisfies the following condition: S O C 1 + S O C 2 + ⋯ + S O C i = i * S O C avg S O C avg S O C i = S O C 1 + S O C 2 + ⋯ + S O C i − 1 i * S O C i + 1 i > 1 i . (16)

Therefore, the ESS consisting of four ESUs is analyzed by taking the variation range of SOC _avg/SOC _i from 1/4 to 5 with SOC _avg/SOC _i as the independent variable.

For the equilibrium acceleration factor n, fixing the initial value of the droop coefficient R _oi = 4 and adjustment factor m = 14, the curve of variation in the droop factor for increasing n from 100 to 600 with each increase of 100 is obtained as shown in Figure 5A.

In order to achieve fast SOC equalization, it is required that the SOC descent rate of the ESU with larger SOC _i is fast, and the SOC descent rate of the ESU with smaller SOCi is slow. For the ESU with larger SOC _i , its output current should be larger to obtain a larger descent rate. In addition, a large output current means a small droop coefficient of the ESU, so from Figure 5A, it can be seen that for the ESU with larger SOC _i , where 1/4<SOC _avg/SOC _i <1, it is best to take a larger value of n in order to keep it in the state of lower R _i for a long period of time. In addition, for the storage unit with smaller SOC _i , where 1<SOC _avg/SOC _i , the droop coefficient should be kept. R _i is always in the high droop coefficient state, and a larger n is needed to obtain a larger droop coefficient.

In summary, the larger the value of n, the better the droop coefficient; however, from Figure 5A, it can be found that when n take a value greater than 400, its droop coefficient change range increase is particularly small, so choosing n = 400 can be better.

Because the initial value of the droop coefficient R _oi and the adjustment coefficient m both directly affect the upper and lower limits of the range of variation in the droop coefficient, so n = 400 can be fixed, and the graph of the variation in the droop coefficient of the seven combinations of R _oi and m can be obtained, as shown in Figure 5B.

As can be seen from Figure 5B, no matter what value of R _oi is taken, when m is taken as 10, for the case of larger SOC _i , a negative droop coefficient occurs, and therefore, the value of m should be larger. In addition, when fixing R _oi = 2 and m takes 20, for the case of larger SOC, the droop coefficient is the largest among the drawn curves, which does not meet the requirement of the small droop coefficient, and the fluctuation range of its droop coefficient is also very limited, which cannot satisfy the requirement of a fast SOC equilibrium, so m takes 14 as the most appropriate. When fixing m = 14, a large initial droop coefficient is required in order for the droop coefficient to have a sufficient range of variation, where n = 1 does not fulfill the requirement. However, when R _oi takes a large value, the droop coefficient R _i produces a large jump in the SOC near equilibrium (near SOC _avg/SOC _i = 1); when R _oi = 4, R _i will repeatedly jump across in 1.3, 4, and 6.3 in large steps, and since the output current size will be dynamically adjusted in the secondary control layer according to the size of the droop coefficient of the ESU, it will inevitably result in large current fluctuations before and after the equalization of the SOC, thus affecting the stabilization of the bus voltage. In summary, R _oi taking the range of 2–3 is the most appropriate.

In this section, to analyze the stability of the control strategy proposed in this paper, a DC microgrid consisting of two equally capacity ESUs is taken as an example.

The equivalent model for stability analysis is shown in Figure 6, where the SOC is equivalent to an inertial link (Li et al., 2017) and its transfer function can be expressed as G e = 1 s + 1 . (17)

The output current is connected to a low-pass filter with a transfer function of G p = ω c s + ω c , (18)

where ω _c is the cutoff frequency of the low-pass filter.

The DC voltage closed-loop transfer function can be simplified as (Gu et al., 2014) G a = 1 τ s + 1 ≈ 1 , (19)

where τ is the time constant under the action of the switch. Since the bandwidth of the droop control loop is designed to be much smaller than the switching frequency, the value of τ is very small, and its delay effect can be ignored.

As can be seen from Figure 6 and Eqs 17–19, the relationships are as follows: V dc 1 = V r e f − G e G p R v 1 i dc 1 + 1 s V r e f − ξ a v g λ 1 G a . (20)

The output current of the first converter is i dc 1 = α V dc 1 − β V dc 2 . (21)

Among them, α = R line j + R Load R line i R line j + R line i R Load + R line j R Load β = R L R line i R line j + R line i R Load + R line j R Load . (22)

Without considering the communication delay among ESUs, the output voltage of the first converter can be obtained by combining Eqs 20–22. V dc 1 = V r e f 1 + 1 s + V dc 2 G e G p R v 2 β − λ 2 2 s λ 1 G a 1 + G e G p R v 1 α + 1 2 s G a . (23)

By applying small-signal analysis to linearize the variables in Eq. 23, the small-signal model expression can be derived as follows: V ^ dc 1 V ^ r e f V ^ dc 2 = 0 = V r e f 1 + 1 s G a 1 + G e G p R v 1 α + 1 2 s G a . (24)

Figure 7 illustrates the root locus of the parameters R _v and ω _c. As can be observed from Figures 7A, B, the droop coefficient R _v and the cutoff frequency ω _c play a dominant role in system stability. It is advisable not to set the droop coefficient too high; instead, it should be selected based on the stability analysis results, along with a balance between the current sharing accuracy and bus voltage deviation. The selection of the cutoff frequency ω _c should take into account the trade-off between overshoot and adjustment time, and its value is primarily influenced by the sampling frequency and system hardware parameters.

To verify the effectiveness and feasibility of the proposed control strategy, a hardware-in-the-loop experimental platform based on RT-LAB is built in this paper, as shown in Figure 8. The DC microgrid system, shown in Figure 2, is constructed using the upper mechanism, which mainly consists of four DESUs and one load resistor. The aforementioned model is added to the real-time simulator OP5600, the proposed control strategy is deployed in the DSP controller, and the experimental waveforms are transferred to the oscilloscope through the I/O port. In order to further prove the advantages of the proposed control strategy in terms of speed, stability, and the reliable operation under different operating conditions, three different operating conditions are constructed and analyzed in this paper.

The capacity ratio of the four DESUs in the DESS is 2:2:3:3; the initial SOC values are 0.90, 0.87, 0.85, and 0.83 in order; the initial droop coefficients are 2, 2, 4/3, and 4/3, respectively; and the line impedances of the transmission lines of each DESU are 0.40Ω, 0.50Ω, 0.60Ω, and 0.70Ω, respectively, with bus voltage reference values V _ref = 400 V and load resistance R _load = 20Ω.

For DESS composed of DESUs with different capacities, this paper proposes a multi-storage islanded DC microgrid energy balancing strategy based on the hierarchical cooperative control and draws the following conclusions:

1. In the primary control layer, an adaptive droop coefficient adjustment scheme is proposed so that the DESU with large SOC has a smaller droop coefficient, while the DESU with small SOC has a larger droop coefficient. The SOC of the former will drop at a faster rate, while the latter is slower, finally achieving fast SOC equalization.