Abstract
The unbalanced state of charge (SOC) of distributed energy storage systems (DESSs) in autonomous DC microgrid causes energy storage units (ESUs) to terminate operation due to overcharge or overdischarge, which severely affects the power quality. In this paper, a fuzzy droop control for SOC balance and stability analysis of DC microgrid with DESSs is proposed to achieve SOC balance in ESUs while maintaining a stable DC bus voltage. First, the charge and discharge modes of ESUs are determined based on the power supply requirements of the DC microgrid. One-dimensional fuzzy logic is then applied to establish the relationship between SOC and the droop coefficient Rd in the aforementioned two modes. In addition, when integrated with voltage-current double closed-loop control, SOC balance in different ESUs is realized. To improve the balance speed and precision, an exponential acceleration factor is added to the input variable of the fuzzy controller. Finally, based on the average model of converter, the system-level stability of microgrid is analyzed. MATLAB/Simulink simulation results verify the effectiveness and rationality of the proposed method.
DISTRIBUTED generation technology has attracted considerable attention from researchers because of its advantages of environmental friendliness and high energy efficiency. However, when distributed energy resources are directly connected to the existing power system, the probability of power imbalance is increased. Therefore, the concept of a microgrid has been proposed [
As the output of distributed energy resources such as photovoltaic and wind power is intermittent and random, it is difficult to balance the supply and demand power of the microgrid. In other words, the bus voltage is unstable, which severely affects the output power quality. Therefore, distributed energy storage systems (DESSs) with certain capacities should be configured to absorb or release electrical energy to solve these problems [
The control strategies for rational load power distribution among ESUs include centralized control, decentralized control, and distributed control. Centralized control is performed through a central controller to collect the information related to the integration and processing of each unit following the issuance of control instructions. In [
Many studies have been conducted on the stability of microgrid that contains a DESS, and various stability analysis methods have been used. The equivalent models and stabilities of DESS are discussed in [
A DESS in a DC microgrid can ensure that the energy supply of the system will not fluctuate significantly under load switching, circuit failure, or environmental change. However, it is necessary to consider a reasonable distribution of load power among all ESUs to ensure a consistent SOC and to avoid early withdrawal of some ESUs due to overcharge and overdischarge. These problems are described in [
1) The relationship between the SOC deviation of each ESU and the corresponding droop coefficient is established through one-dimensional fuzzy logic. The droop coefficient can be automatically adjusted according to the changes in the external environment, which greatly simplifies the complexity of the system control structure.
2) When the SOC of the ESUs tends to be consistent, an exponential acceleration factor is added, which improves the balance speed and precision to a certain extent.
3) Based on the average model theory, the frequency domain stability of a DC microgrid with DESSs is studied, and the obtained results can provide relevant guidance for system parameter design.
DESS should be configured with a specific capacity to provide energy support and increase system redundancy, thereby ensuring safe and stable operation of the DC microgrid [

Fig. 1 System structure of a DC microgrid with a DESS.
The system is mainly composed of renewable energy sources (RESs), DESS, loads, various converters, and their control circuits. The photovoltaic panel and wind turbine transmit power to the DC bus through unidirectional DC/DC and AC/DC converters, respectively. Each ESU in the DESS absorbs or releases power to the DC bus via a bidirectional DC/DC converter. Loads are the main power-consumption components in a DC microgrid, which can directly or indirectly access the DC bus through the converter. When RES produces more power than the load consumption, DESS absorbs the excess power in the form of a charge. When RES produces less power than the load consumption, DESS generates a power deficit in the form of a discharge.
When the DESS operates, the load power of each ESU should be balanced, i.e., should have SOC balance, to avoid the premature exit of some ESUs due to overcharge or overdischarge, which may affect the service life of DESS and the stable operation of DC microgrid [
Droop control has the advantages of high reliability and strong flexibility while offering plug-and-play capabilities and easy implementation. It can be used in the control process of ESU energy storage converters [

Fig. 2 Control structure of a DC microgrid with a DESS.
In
DESS in DC microgrid often adopts the I-U droop control, which can be expressed as:
(1) |
where and are the reference output voltage and virtual resistance, i.e., droop coefficient, of the energy storage converter, respectively.
The operation of two ESUs in parallel is used as an example to illustrate the basic principle of I-U droop control.

Fig. 3 Corresponding Thevenin equivalent circuit of two ESUs in parallel.
In
(2) |
Since and the reference voltage of the energy storage converter is usually determined by the DC bus voltage reference value, , the relationship between the output currents of two ESUs can be obtained by:
(3) |
When and are assumed, the droop curve considering the effects of line impedance can be generated, as shown in

Fig. 4 Droop curve considering effects of line impedance.
In
When the line impedance is significantly less than the virtual impedance, its effect on the line impedance can be ignored.
(4) |
Thus, the relationship between output power P1 and P2 of the two ESUs can be obtained as:
(5) |
In addition, the voltage drop can be expressed by the virtual resistance as:
(6) |
The voltage drop is proportional to the virtual resistance, and the value range of the virtual resistance can be obtained by:
(7) |
where is the maximum allowable voltage deviation; and is the full-load current of the energy storage converter.
The defects in the I-U droop control lead to its inability to meet the DESS control requirements, which are mainly reflected in the uneven load power distribution and difficult SOC balance in ESUs. If the relationship between the battery SOC and droop coefficient is established using fuzzy logic, will automatically adjust according to the battery SOC. This dynamic adjustment to the output power of each ESU results in a consistent SOC.
Common methods for estimating battery SOC include the ampere-hour integral method, open-circuit voltage method, internal resistance method, extended Kalman filter method, and neural network method [
The calculation for estimating the battery SOC using the ampere-hour integral method is:
(8) |
where and are the current and initial SOCs of the battery, respectively; is the rated capacity of the battery; and is the output current of the battery, which is positive when discharged and negative when charged.
We can take the derivative from both ends of (8) to obtain:
(9) |
It is difficult to quantify the relationship between the battery SOC and Rd. However, it can be effectively expressed through a fuzzy logic relationship.

Fig. 5 SOC fuzzy droop control scheme.
In

Fig. 6 Basic structure of a fuzzy controller.
In
To ensure that the SOC of each ESU achieves regular consistency, fa and fb should be selected as the variables associated with the SOC and Rd:
(10) |
where is the average SOC of all ESUs; and superscript n denotes a constant.
The SOCs of the ESUs during DESS operation are not different. Therefore, the numerical domain of the input variable is . To ensure the effect of drop control, the numerical domain of fb is determined using (7), and the range of fb is .
Seven fuzzy subsets {NB, NM, NS, ZO, PS, PM, PB} are selected to cover the fuzzy domain of fa, and all fuzzy subsets are selected as triangular membership functions. NB, NM, NS, ZO, PS, PM, and PB represent negative large, negative medium, negative small, zero, positive small, positive medium, and positive large, respectively. The membership function expression for fa is given by (11). Similarly, seven fuzzy subsets {NB, NM, NS, ZO, PS, PM, PB} are selected to cover the fuzzy domain of fb, and all fuzzy subsets are selected as triangular membership functions. The membership function expression for fb is given by (12). The distributions of the membership functions of fa and fb are shown in Fig. SA1 of Supplementary Material A.
(11) |
(12) |
where i and j are the numbers of fuzzy subsets of and , respectively; and are the membership degrees of and , respectively; and and are the distances between the center points of adjacent membership functions in input and output membership functions, respectively. Fuzzification is the process of identifying the relative fuzzy subsets of and their membership functions.
The purpose of establishing fuzzy rules is to select an appropriate according to the different inputs . For an ESU with a high SOC (where the corresponding input fa is larger), a smaller should be allocated during discharge to increase the discharge power, and a larger should be allocated during charge to reduce the charge power. For an ESU with a low SOC (where the corresponding input is small), a larger should be allocated during discharge to reduce the discharge power, and a smaller should be allocated during charge to increase the charge power. The SOC balance in ESUs can be realized by meeting these requirements.
The change from NB to PB in the fuzzy subset corresponds to a change in the language variables from small to large. Therefore, linguistic fuzzy rules can be obtained. For example, when ESU is discharged, is PB (large) and is NB (small), and so on. The fuzzy rules are obtained by sorting the obtained linguistic fuzzy rules, as shown in
ESU charge () | ESU discharge () | ||
---|---|---|---|
Fa | Fb | Fa | Fb |
NB | NB | NB | PB |
NM | NM | NM | PM |
NS | NS | NS | PS |
ZO | ZO | ZO | ZO |
PS | PS | PS | NS |
PM | PM | PM | NM |
PB | PB | PB | NB |
The process of reasoning in a signed syllogism is as follows: ① major premise: ; ② minor premise: ; and ③ conclusion: .
The
(13) |
(14) |
where is the membership degree.
The fuzzy set in the conclusion is determined by using the minor premise , as shown in (15), and the corresponding fuzzy membership function is given by (16).
(15) |
(16) |
where represents the compositional operation; and m is the number of active rules.
The output obtained by the approximate inference is a fuzzy set , which must be converted into an exact numerical quantity fb to control the controlled object, i.e., defuzzification. When the area-center method is used for defuzzification, fb is expressed as:
(17) |
Following this process, a one-dimensional fuzzy relationship between fa and fb is obtained. The relationship between fa and fb during the charge and discharge of the ESU is shown in Fig. SA2 of Supplementary Material A.
Under the gradual SOC balance process, the SOC difference among ESUs becomes increasingly smaller. The fuzzy input fa of each ESU tends to be the same. Therefore, the fuzzy output fb, i.e., Rdi, also tends to be the same. According to (4) and (5), the output current and power of each ESU change slowly, which are not conducive to load power distribution and greatly reduce the speed and accuracy of the SOC balance.
To solve this problems, an acceleration factor is added to the late stage of the SOC balance. The value of the fuzzy input fa is dynamically changed mainly by multiplying it with other functions to improve the resolution between the SOC and Rd of each ESU. The designed acceleration factor K is expressed as:
(18) |
where m1 and m2 are the constants.

Fig. 7 Relationships among different variables. (a) K, m2, and fa. (b) K, m1, and fa.
The change process of K from small to large not only can avoid the excessive output current of some ESUs caused by the large difference in the droop coefficient in the early stage of equalization, but also can ensure the resolution of the droop coefficient in the late stage of equalization to improve the SOC balance speed and accuracy. The modified fuzzy input can be expressed as:
(19) |
The converters in the DESS are bidirectional DC/DC converters, and their specific circuit structures are shown in
(20) |
where is the average duty cycle of the upper bridge arm switch tube ; and is the equivalent voltage of the ESU.
The calculation for the inductance voltage and capacitance current in circuit theory is:
(21) |
where is the terminal voltage of .
(22) |
A structural block diagram of the bidirectional DC/DC converter derived from (22) is presented as

Fig. 8 Structural block diagram of a bidirectional DC/DC converter.
The control of the bidirectional DC/DC converter has a double-loop control structure, in which the outer and inner loops adopt droop and voltage-current double closed-loop PI controls, respectively. A control block diagram of the bidirectional DC/DC converter is given in

Fig. 9 Control block diagram of a bidirectional DC/DC converter.
In
(23) |
where and are the proportional and integral coefficients of the voltage controller, respectively; and and are the proportional and integral coefficients of the current controller, respectively.
(24) |
where is the equivalent output impedance during discharge; is the equivalent output impedance during charge; and is the equivalent transfer function in the simplification process, which is expressed as:
(25) |
In addition, the closed-loop transfer function of the
(26) |
The impedance characteristics and the ESU stability can be analyzed using (24) and (26), respectively.
Impedance ratio analysis is commonly used to study the stability of a microgrid. The steps are as follows. First, the microgrid is divided into multiple submodules, each of which can be regarded as a two-terminal network. Then, each two-terminal network undergoes an equivalent treatment based on the Thevenin theorem. The two-terminal network is represented by the series form of the voltage source and resistance. Finally, the impedance ratio of the system is calculated using the circuit equivalent principle, and the stability of the microgrid is determined based on the Nyquist curve of the impedance ratio. The DC microgrid with a DESS as shown in

Fig. 10 Thevenin equivalent circuit of a DC microgrid with a DESS.
The microgrid is often equivalent to the power supply and load subsystems. This equivalence enables system stability to be analyzed under the disturbance of each frequency band. When switch S is turned off, each ESU is in the charge mode, which is regarded as a load and integrated into the load subsystem. When switch S is turned on, each ESU is in the discharge mode, which is regarded as a micropower supply and is integrated into the power supply subsystem. This process is illustrated in Figs.

Fig. 11 Equivalent circuit of a microgrid during ESU charge or discharge. (a) ESU charge. (b) ESU discharge.

Fig. 12 Unified circuit model of a microgrid.
In
(27) |
(28) |
In
(29) |
Droop control methods based on the SOC exponential and SOC power functions are proposed in [
(30) |
(31) |
where is the droop coefficient of the energy storage converter at full-load current; and is a constant. The ESU is charged when and is discharged when .
These two methods are compared with the proposed method. With two ESUs used as examples to conduct the simulation research, the specific parameter settings are listed in
Parameter | Value |
---|---|
Reference value of DC bus voltage | 400 V |
Bus voltage fluctuation range | ±5% |
ESU1: L1, C1, kup,1(kui,1), kip,1(kii,1), SOC1 | 0.3 mH, 300 μF, 2(20), 3(3), 60% |
ESU2: L2, C2, kup,2(kui,2), kip,2(kii,2), SOC2 | 0.3 mH, 300 μF, 2(20), 3(3), 58% |
m1, m2, n | 50, 20, 3 |
Load | 8 kW |
When the output power of RES exceeds the load demand, the DESS is in a stable charge mode. This situation can be simulated by closing switch S, as shown in

Fig. 13 DESS state during steady charge. (a) SOC. (b) Output power.
The DESS state during steady discharge under different droop control methods is shown in Fig. SA3 of Supplementary Material A.
When the power of the supply and demand sides of the DC microgrid is not equal, the charge and discharge modes of the DESS change. The supply- and demand-side power refers to the output power of the RES and power consumed by the loads, respectively. In this paper, the simulation duration is 60 s, and the state of switch S is changed from closed to open at 30 s to simulate a power change situation.

Fig. 14 DESS state when charge and discharge modes change. (a) SOC. (b) Output power.
As
The acceleration factor is introduced to improve the resolution of the SOC differences among the ESUs. This enables a fast equalization speed to be maintained in the later stages. The duration of the simulation is 40 s, and the state of switch S is changed from closed to open at 20 s. Accordingly, the SOC fuzzy drop control method is simulated with and without the acceleration factor.

Fig. 15 DESS state with and without an acceleration factor. (a) SOC. (b) Output power.
As
The design method for the acceleration factor is not unique and mainly consists of an exponential function, logarithmic function, power function, and other basic functions. The exponential and logarithmic functions are suitable for describing the change laws of fast and slow growth rates, respectively, and the power function resides between the two and is thus suitable for describing the change law of the general growth rate. Therefore, several types of the acceleration factors are designed as:
(32) |
Different types of acceleration factors have different effects on the SOC balance speed and power distribution accuracy. Under the condition in which the charge and discharge modes of the DESS change, i.e., where the charge mode changes to the discharge mode at 30 s, the SOC fuzzy drop control under different types of acceleration factors is simulated. The DESS state under different acceleration factors is shown in

Fig. 16 DESS state with different acceleration factors. (a) SOC. (b) Output power.
As
As
To obtain an accurate evaluation of how well the fuzzy controller performs with the added exponential acceleration factor and to assess the performances of the other controllers, the performance index J is introduced, as shown in (33), where J reflects the SOC balance accuracy of multiple ESUs at a specific time during the DESS balance process. A smaller J indicates better controller performance, and faster SOC balance can then be achieved.
(33) |
where and are the current and initial SOCs of the
Situation | Method | J | ||||
---|---|---|---|---|---|---|
t=10 s | t=20 s | t=30 s | t=40 s | t=50 s | ||
1. Stable charge | M1 | 0.895 | 0.800 | 0.715 | 0.635 | 0.565 |
M2 | 0.820 | 0.640 | 0.460 | 0.285 | 0.120 | |
M3 | 0.295 | 0.100 | 0.025 | 0.005 | 0 | |
2. Stable discharge | M1 | 0.875 | 0.745 | 0.620 | 0.500 | 0.390 |
M2 | 0.720 | 0.450 | 0.200 | 0.015 | 0 | |
M3 | 0.250 | 0.055 | 0.010 | 0 | 0 | |
3. Change of charge and discharge modes | M1 | 0.895 | 0.800 | 0.715 | 0.630 | 0.540 |
M2 | 0.820 | 0.640 | 0.460 | 0.205 | 0.015 | |
M3 | 0.295 | 0.100 | 0.025 | 0.020 | 0 | |
4. Effects of acceleration factor | M3-K0 | 1.000 | 0.995 | 0.990 | 0.880 | 0.755 |
M3-K1 | 1.580 | 1.585 | 1.580 | 1.390 | 1.195 | |
M3-K2 | 0.970 | 0.945 | 0.915 | 0.805 | 0.690 | |
M3-K3 | 0.295 | 0.100 | 0.025 | 0.020 | 0 |
As
The DESS is set to contain three ESUs with initial SOCs for the three ESUs of 60%, 58%, and 56%, respectively. The DESS changes from the charge mode to discharge mode at 30 s.

Fig. 17 DESS state with three ESUs. (a) SOC of M1. (b) SOC of M2. (c) SOC of M3. (d) Output power of M1. (e) Output power of M2. (f) Output power of M3.
As

Fig. 18 Effects of droop coefficient on closed-loop output impedance of ESU.
As

Fig. 19 Effects of droop coefficient on ESU stability.
As
Because the bandwidth of the voltage PI controller is much smaller than that of the current PI controller, the effects of the voltage PI controller parameters on ESU stability are more obvious. The corresponding Bode diagram is given as Fig. SA5 in Supplementary Material A.
The DC microgrid with DESS in this paper has two working modes. ① Mode 1: when the output power of the RES is greater than the load power, the DESS is in the charge mode. ② Mode 2: when the output power of the RES is less than the load power, the DESS is in the discharge mode. If the DC microgrid is stable under both working modes, the overall system is stable.
A DC microgrid with two ESUs is next taken as an example. The equivalent output impedances of the power and load sides of the microgrid in Modes 1 and 2 are given by (34) and (35), respectively.
(34) |
(35) |
Equations (

Fig. 20 Nyquist curves of system impedance ratio of microgrid. (a) Mode 1. (b) Mode 2.
As
In addition, the smaller amplitude and phase angle margins in Modes 1 and 2 are at the lower limits, ensuring the stable operation of the microgrid ( dB, at 0.0372 rad/s). Thus, the conditions of the system input and output impedances for ensuring the stable operation of the microgrid can be obtained when and the phase angle of ZLi is not subject to any constraints. When the aforementioned conclusions are applied, it is necessary to obtain the amplitudes and phase angles of the input and output impedances of the system , as shown in (27) and (28). Then, a calculation must be performed to determine whether the condition is satisfied. If the condition is satisfied, the designed microgrid is stable; otherwise, the parameters of the microgrid must be redesigned.
Autonomous DC microgrids usually contain multiple ESUs for maintaining bus voltage stability and power balance. Therefore, reasonable power distribution and fast SOC balance in ESUs are critical. An SOC fuzzy droop control with an acceleration factor is proposed, and the system stability is investigated based on the average model of the converter.
The following conclusions are obtained. ① The SOC fuzzy droop control has a faster balance speed and accuracy as compared with the SOC exponential and SOC power function droop control. ② The design of the acceleration factor can resolve the SOC between ESUs in the entire equalization process, particularly in the later stages, and this can in turn improve the balance speed. ③ The transfer function of the ESU in the droop control mode is derived, and this paper shows that the ESU continues to operate in a stable manner when the droop coefficient changes. ④ A stability criterion for the DC microgrid with DESS is given, which can provide a reference for parameter design.
In future research, we will address the following challenges. ① The main variables affecting the variation in droop coefficient will be analyzed, and a multidimensional fuzzy logic relationship will be constructed. ② The effects of different fuzzy membership functions and their distributions on the SOC balance performance will be studied to obtain a better fuzzy controller. ③ The form of the acceleration factor is not unique, where a combined acceleration factor is developed to achieve autonomous control of the balance speed. ④ Various modeling theories are used to analyze the stability of the microgrid, and a reasonable application range of the corresponding criteria is provided.
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