SoC-Based Dynamic Droop Control for Battery Energy Storage Systems in DC Microgrids Feeding CPLs

In this paper, a dynamic state-of-charge (SoC)-based droop control and bus voltage stabilization strategy has been proposed for battery energy storage systems (BESSs) feeding constant power loads (CPLs). Firstly, a SoC-based dynamic droop control is proposed to realize proportional distribution of power. Secondly, a dynamic average consensus (DAC) algorithm is adopted to ensure that all SoC converge to the same value simultaneously. Thirdly, a nonlinear disturbance observer (NDO) and a backstepping controller are integrated to stabilize the DC bus voltage. In this way, the rapid voltage recovery and power distribution are obtained when droop coefficient is time-varying. Simulation results verify the feasibility and effectiveness of the proposed control strategy.


Introduction
In recent years, DC microgrids have been widely applied to power generations. The distributed DC generation becomes a critical research direction with the utilization of renewable energy sources (RESs), such as solar PV, wind turbine, water, and tidal. A typical DC microgrid is illustrated in Fig. 1. In general, RESs, BESSs, and loads are typically connected to the common DC bus through AC/DC or DC/DC converters. The intermittence of RESs and uncertainties of the loads cause certain destabilizing effects for DC microgrids. Given these conditions, the battery energy storage systems (BESSs) are widely deployed in DC microgrids to improve system stability and reliability [1]. Although BESSs can maintain the bus voltage at a desirable value and buffer the RES power, it should also consider the power distribution between BESSs to improve the batteries' lifetime and efficiency. There are many research results been proposed to solve the power distribution issue in the literature. See for example, [2][3][4][5]. In [2][3], the sources and loads power management problem was dealt with by tuning the fuzzy logic parameters. Meanwhile, droop control is widely used for power  [4]. However, the traditional droop control coefficient is a constant, it cannot adapt to state-of-charge (SoC) variations of BESSs. Therefore, the BESSs with low SoC may run out of their energy, which will cause the system to become unstable.
To solve problem of BESS over-discharging, the SoC of battery has been considered in many energy storage strategies. The SoC balance issue is addressed in [6] by proper energy sharing through adjusting charging and discharging rates. An SoC-based adaptive droop control is reported to ensure SoC balance with an adjustable convergence speed. The SoC of each battery needs to be estimated and controlled. Reference [7] proposed mode switching strategies to adjust the SoC of the energy storage devices. This method can flexibly adjust the control strategy according to SoC real-time value, while it has strict requirements for seamless mode transition. In [8], with the intention of solving the problem of unbalanced output power of multiple energy storage batteries, dynamic droop control strategy based on the SoC of battery has been used.
The BESSs need to keep power distribution and SoC balance to stabilize DC bus voltage. Meanwhile, the load variations will affect system stability. For example, constant power load (CPL) variations may cause instability of DC microgrids. Multiple types of research have been conducted for stabilizing DC bus voltage with CPLs. In [9], a decentralized composite controller (DCC) is proposed to realize global system large signal stability of the DC microgrids. In [10], backstepping control is also used for large signal stability. Considering that the nonlinear disturbance observer (NDO) estimates parameters faster and more accurate than traditional ones, this paper uses NDO and backstepping controller to regulate the DC bus voltage. The contributions of this paper can be summarized as follows.
(1) To regulate the power distribution of BESSs, the local SoC information can be shared through the dynamic average consensus (DAC) algorithm, which ensures the dynamic allocation of the preserved capacity of all batteries.
(2) By employing a compound stabilizer to regulate the DC bus voltage, the system large signal stability is enhanced with fast voltage convergence rate. The voltage deviation caused by the droop control can be compensated by a consensus algorithm.
The remaining of this paper is organized as follows: Section II describes the method of dynamic droop control and BESSs configuration. Section III presents a distributed control strategy with CPL. In Section IV, case studies are verified with Simulink. Finally, the conclusions are drawn in Section V.

THE PROPOSED DYNAMIC DROOP CONTROL
The architecture of distributed control for voltage regulation in a DC microgrid with multiple BESSs is shown in Fig. 2. The energy storage devices consist of batteries and bidirectional DC/DC converters, which mainly use boost converters. Batteries are connected to the common bus by DC/DC converters respectively. They will supply power to load and maintain the DC bus voltage. In this section, a dynamic power management strategy is proposed to effectively distribute power between BESSs.

SoC-Based Droop Control
This section proposes a dynamic power distribution method. By employing basic droop control, the voltage reference of ith converter is expressed as where is the nominal voltage, and are output voltage and load power of the converter for the ith battery, and m is the corresponding droop coefficient. The subscript i (i =1, 2, 3,4) is the variable associated with the ith BESS.
Let SoC t be the SoC of the ith BESS at time t and 0 be the initial SoC value. Also, let and be the battery capacity and output current, respectively. is estimated by using the basic Coulomb counting method as 0 . (2) To achieve SoC balancing among BESSs, higher SoC supply higher power while lower SoC batteries provide less power in discharging mode. Inspired by [11], the droop coefficient which is based on ith SoC is expressed as where is a positive constant, n represents the equalization velocity adjustment coefficient, is the average value of .

Dynamic SoC Consensus
The dynamic average consensus algorithm has been used to balance all SoC of BESSs. In addition, in order to obtain ith local SoC information for dynamic droop cofficient, the dynamic average consensus algorithm is designed as ∑ ， where i, j=1, 2, 3, 4. is the ith local measurement data, is an internal state, and 0 is a constant.
The dynamic consensus algorithm shown in (6) is adopted, and each local communication agent can exchange information without a central controller. In order to obtain average consensus, the error vector is obtained as , where i =1, 2, …, n. As time goes to infinity, the goes to zero. The (6) can be achieved in [13].

System Model
The simplified DC/DC boost converters diagram is demonstrated in Fig. 2. The DC/DC boost converter circuits dynamic model is expressed as where is the battery output voltage, represents the power of CPL, and d is the duty ratio. The control objective is , by setting to make track . In order to observe immeasurable quantities, through feedback linearization of DC/DC converters, the system model in (8) has been transformed into the standard form as where , are state variables, ℎ , ℎ are uncertain items.
where is total load power. According to (9), ℎ can get as

Nonlinear Disturbance Observer design
In (10), it can be observed that reference value is an uncertain value, which varies with load power and may affect the stability of the system. Due to this cause, the NDO is employed to estimate uncertain items ℎ , ℎ and to achieve accurate tracking and fast dynamic response. Due to the ℎ , ℎ are uncertain items, the uncertain item ℎ can be estimated based on NDO of the following form [10] ℎ , ℎ where is an auxiliary state of the NDO, 0 is the NDO gain. Similarly, the uncertain item ℎ is estimated by ℎ , ℎ where is an auxiliary state of the NDO, 0 is the NDO gain. According to (12) and (13), the estimation error of ℎ , ℎ and their derivative are expressed as ℎ ℎ ℎ , Considering accuracy and stability of the observer, NDO-based tracking control is designed for uncertain systems (9), the uncertainties ℎ , ℎ are related to and . Thus, the rate of disturbances change must be bounded. The following assumption is required: Assumption 1: The uncertain state ℎ , ℎ for the system (9) satisfy the following two conditions:

Backstepping Control Design
In this part, a nonlinear backstepping control is designed to obtain an optimized tracking performance of the main DC-DC bus voltage. Based on the standard form (9)  (26) According to (25) and (26), the derivative of is expressed as ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ .

Voltage Consensus Algorithm
The DC bus voltage cannot back to the voltage reference due to the virtual impedance. To improve the power distribution quality, the voltage of each agent should be consistent. From Eqn. (3), the droop coefficient can be achieved. By adopting consensus algorithm to get the error between and , voltage state can be expressed as where i, j=1, 2, 3, 4. is an internal state, and 0 is a constant. Define tracking error . .
(30) In order to achieve global stability, the voltage compensation input is given by . According to global information, the DC bus voltage can track its reference value. Besides, local SoC information is adopted to regulate power distribution.

Simulation
In this section, simulations are conducted to verify the feasibility and effectiveness of the proposed control method with three cases. In this paper, the working range of BESS sets as [10%, 90%]. The system parameters are listed in Table I.

Case 2: Performance of SoC-based droop control with unbalanced SoC
It is assumed that the BESSs are working under different SoC values, the initial SoC values for 4 BESSs are respectively set as 90%, 88%, 86%, 84%. The CPL variations are the same with case 1. Fig. 5 shows that BESSs have different power distribution ratio. 1st Battery which has the highest SoC provides the most power under this case. The DC bus voltage will soon return to the steady state under the condition of CPL changes.
According to the dynamic consensus algorithm, the results are shown as Fig. 6. BESSs with unbalanced SoC will converge to the same value around 170s. All BESSs provide power in proportion to the preserved SoC of BESSs.

Case 3: Removal of BESS
The third case shows results with removal of BESS. Considering that if the SoC of BESS is less than 10%, this BESS should unplug to avoid over-discharging.
In this case, the initial SoC values for 4 BESSs are respectively set as 20%, 15%, 14%, 11%. As Fig.  7 shows, when the lowest BESS's SoC less than 10%, that BESS will stop supplying energy for load. At the same time, the remaining BESSs need to bear the demand power and dynamically distribute it in proportion.

Conclusion
A SoC-based dynamic droop control approach has been presented for power distribution and batteries' healthy in DC Microgrids with CPL. The proposed control scheme can share local SoC information with other BESSs by a dynamic average consensus algorithm. By adopting the NDO and backstepping controller, the system stability is improved. The stability and rapidity of the system are verified using Matlab / Simulink models, inferring that the proposed control algorithm is able to maintain good voltage regulation and appropriate power distribution.