Unbalanced Voltage Suppression of Bipolar DC Microgrids with Integration of DC Zero-carbon Buildings

Xuefei Zhang; Chunsheng Guo; Yiyao Zhou; Xiaolong Xu; Jianquan Liao; Niancheng Zhou; Qianggang Wang

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Unbalanced Voltage Suppression of Bipolar DC Microgrids with Integration of DC Zero-carbon Buildings PDF

- ORCID：
Xuefei Zhang ¹
✉
- ORCID：
Chunsheng Guo ²
✉
- ORCID：
Yiyao Zhou ¹
✉
- ORCID：
Xiaolong Xu ¹
✉
- ORCID：
Jianquan Liao ²
✉
- ORCID：
Niancheng Zhou ¹
✉
- ORCID：
Qianggang Wang ¹
✉

1. State Key Laboratory of Power Transmission Equipment Technology, School of Electrical Engineering, Chongqing University, Chongqing, China； 2. College of Electrical Engineering, Sichuan University, Chengdu, China

Updated：2024-12-18

DOI：10.35833/MPCE.2023.000713

OUTLINE

Abstract

Considering the majority of electrical equipment utilized in society is driven by DC, integrating a DC system can significantly enhance the efficiency and reliability of power systems by implementing the integration of diverse loads, renewable energy sources (RESs), and energy storage systems (ESSs). In this paper, the integration of multiple DC zero-carbon buildings (DC-ZCBs) is proposed to achieve the unbalanced voltage suppression of the bipolar DC microgrid (DCMG). The photovoltaic (PV) technology, loads, and DC electric springs (DC-ESs) are adopted as a unified entity to achieve the zero-carbon emission of the building. Firstly, a new configuration of PV and DC-ESs is introduced. The energy management of PV, ESS, and load are fully considered in this new configuration, which can reduce the capacity of the ESS. Subsequently, a distributed cooperative control strategy for DC-ESs based on the modulus voltage is presented, which is implemented with integration of the new configuration into the bipolar DCMG. The proposed approach addresses the issues of unbalanced voltage to improve the operating efficiency and power quality of the bipolar DCMG. The simulation is conducted in MATLAB/Simulink platform to confirm the effectiveness of the proposed approach.

Keywords

Bipolar DC microgrid (DCMG); DC zero-carbon building (DC-ZCB); electric spring (ES); modulus voltage control; unbalanced voltage suppression

I. Introduction

CURRENTLY, AC microgrid (ACMG) dominates the market for power distribution and transmission [

1], [2]. However, ACMGs suffer from several drawbacks such as the inability to connect DC loads directly, the need for additional power conversion stages leading to the increase of power losses, difficulties in frequency and phase synchronization [3]-[5], and environmental concerns [6]. Therefore, the adoption of a bipolar DC microgrid (DCMG), which utilizes the distributed renewable energy source (RES) such as photovoltaic (PV), is becoming more efficient and reliable.

Most of the bipolar DCMGs are low-carbon systems consisting of loads, RESs, and energy storage systems (ESSs). The DC building can function as load or contribute to voltage regulation in the bipolar DCMG when combined with ESSs [

7]. The DC zero-carbon building (DC-ZCB) is a component of a broader energy system aimed at achieving overall carbon neutrality [8], [9]. The availability of the affordable and effective DC/DC converters is also contributed to the prosperity of DC-ZCBs. Recent studies show that the bipolar DCMG with the integration of DC-ZCBs can significantly benefit from technological advancements in power generation, transmission, distribution, storage, and control [1]. The present configuration of bipolar DCMG with the integration of DC-ZCBs can be classified into unipolar and bipolar DCMGs based on the number of DC buses [10]. The bipolar DCMG with the integration of DC-ZCBs has the advantage of the additional neutral bus, providing the following benefits.

1) Providing more voltage levels with fewer buses, minimizing power conversion, and boosting overall efficiency [

11], [12].

2) Enhancing the utilization of DC-ZCBs in bipolar DCMG with integration of RESs, ZCBs, and electric vehicle charging facilities [

13].

3) Enhancing stability with a three-wire structure ensures that if one of the DC buses fails, the remaining buses will continue to operate normally [

14].

Although the bipolar DCMG with the integration of DC-ZCBs has advantages, the issue of the unbalanced voltage must be addressed [

8]. The unbalanced voltage in the bipolar DCMG can result in voltage and current disturbances, which further increase power losses and haste device aging [15], [16]. Hence, it is crucial to employ suitable approaches to suppress the unbalanced voltage in the bipolar DCMG.

According to the existing studies, achieving the optimization of the bipolar DCMG voltage primarily involves utilizing flexible electronic equipment or a new control topology.

One approach to addressing the issue of unbalanced voltage is to utilize flexible electronic equipment. Reference [

17] introduces a voltage balancer (VB) with the capability of a DC/DC converter. However, the VB has limitations, as the output power is always lower than the unbalanced power, making it difficult to respond to the changes in DC load. In the case of multiple VBs, the control between different VBs and rectifiers must be coordinated [18], [19], which is complicated with multiple power supplies [20], [21]. Furthermore, when the distribution line is long and has many load nodes, this coordination control may not guarantee the voltage of the constant power load (CPL) at the end of the distribution line. An alternative approach is the consensus control system based VB proposed in [22], in which the primary and secondary controls are both integrated in bipolar DCMG. However, the consensus control system requires a spare communication system to facilitate information sharing among multiple nodes, which increases the cost and complexity of the power network. Another approach is introduced in [23], which involves using a DC electric spring (DC-ES) to regulate the voltage disturbance on the load side in the bipolar DCMG. Thus, the proposed approach in [23] enhances the dynamic performance of the control system by building a transfer matrix that adopts the DC-ES decoupling control. However, in engineering applications, the recommended approaches are both costly and complex.

Another approach to address the issue of unbalanced voltage involves adopting a new control topology. Novel power sharing strategies and voltage regulation approaches are proposed in [

18], [19], [24], and [25]. However, they cannot solve communication burden problems. In [26]-[29], power sharing and energy management approaches for DCMGs are proposed. However, they have not investigated the effect of PV system characteristics on the control strategies with high PV penetration in bipolar DCMG. Therefore, these approaches are not applicable to the unbalanced voltage suppression in DCMGs with the integration of ZCBs. Additionally, the above-mentioned approaches require communication systems during the control process. It is possible to employ the modulus voltage control to reduce the unbalanced voltage brought by droop control [13], [30], and [31]. Consequently, the unbalanced voltage issue in the bipolar DCMG needs more consideration. However, those studies have not investigated the impact of coupled parameters on stability. When the distribution lines are coupled in the bipolar DCMG, the challenge of setting control parameters can be resolved through the proposed approach in this paper. The proposed approach, which compensates for the voltage drop in the neutral line and enhances the dynamic performance of the control system.

In bipolar DCMGs with the integration of ZCBs including PV, the increasing PV penetration makes it challenging to accommodate its generation. Moreover, the stochastic and fluctuating nature of PV poses higher challenges to the unbalanced voltage suppression in bipolar DCMGs. This paper addresses the impact of high PV penetration in the DCMG with the integration of PV and DC-ES. Additionally, a modulus voltage consensus control approach is proposed to precisely suppress the unbalance voltage while simplifying the complexity of the unbalanced voltage control system. A comparison with [

24] and [32] shows that the proposed approach in this paper successfully decouples the coupling between the two poles of the power supply and the bipolar control system, effectively suppressing the unbalanced parameters. The main contributions of this paper can be summarized as follows.

1) The integration of PV and DC-ES reduces the volume and weight of the ESS in DC-ES. This integration mitigates the randomness and volatility of PV, enhancing the renewable energy absorption capability of the DC-ZCB.

2) The modular transformation is introduced, which decouples bipolar DCMGs interconnected through the neutral line resistance. This simplifies control objectives and reduces the control complexity to facilitate coordinated control among multiple devices.

3) A distributed cooperative control strategy is proposed for PV-DC-ES in bipolar DCMGs for unbalanced voltage suppression, which not only enhances the electric power quality of bipolar DCMGs but also reduces system losses.

The rest of this paper is organized as follows. Section II presents the analysis of DC-ES in bilopar DCMG with integration of DC-ZCBs. Section III details the distributed cooperative control strategy. Section IV investigates the consistency in control performance and small-signal stability analysis. Section V presents the simulation results. Finally, Section VI gives the conclusion of this paper.

II. Analysis of DC-ES in Bipolar DCMG with Integration of DC-ZCBs

A. Type, Topology, and Working Mode of DC-ES

DC-ES can be widely distributed at any node of the DCMG. When there are non-critical loads (NCLs) with a broad voltage range, a smart load may be created by connecting DC-ES in series or parallel to ensure the voltage stability of critical loads (CLs). Depending on the installation position of the NCL, DC-ES may be classified into series DC-ES and parallel DC-ES. The basic structures of different types of DC-ES are shown in Fig. 1, where V_PCC is the common-node voltage; R_C is the equivalent resistance of CL; R_NC is the equivalent resistance of NCL; R_L is the line resistance; V_s is the equivalent voltage on the source side of the bipolar DCMG; and V_es is the output equivalent voltage of DC-ES. According to [

33] and [34], the working modes of series DC-ES are divided into boost discharging, boost charging, and depressing discharging.

Fig. 1 Basic structure of different types of DC-ES. (a) Series DC-ES. (b) Parallel DC-ES.

The structure of bipolar DCMG with the integration of DC-ZCBs is shown in Fig. 2. The loads in the DC-ZCBs are divided into CLs and NCLs, which make up around 46% and 54% of the total loads in the DC-ZCBs, respectively [

35]. CLs mainly include office equipment, intelligent control equipment, etc. NCLs mainly include light loads, cooling loads, and heat loads, e.g., air conditioners and water heaters. Since CLs are sensitive to voltage quality, it is important to ensure that the voltage under the DC voltage state with little fluctuation at both ends of the load. NCLs may function in a wide voltage range and have minimal voltage quality requirements.

Fig. 2 Structure of bipolar DCMG with integration of DC-ZCBs.

The composition of power supply and load in bipolar DCMG with the integration of DC-ZCBs is shown in Fig. 3. According to Fig. 3, the response capability of demand side for NCLs is not completely utilized since both CLs and NCLs are linked in parallel with the DC bus [

35]. For instance, NCLs are unable to actively reduce their own power in response to PV power fluctuations when ESS and PV power are both inadequate.

Fig. 3 Composition of power supply and load in bipolar DCMG with integration of DC-ZCBs.

To fully exploit the demand-side response capability of NCLs, both CLs and NCLs in DC-ZCB can be reconfigured. At the same time, the installation position of PV can be reconfigured. Reference [

35] proposes a novel concept of PV-embedded DC-ES by transferring a portion of PV panels from the grid to the DC link of the DC-ES. As the output power is no longer solely derived from the energy storage device, The size and volume of ESS can be significantly reduced. In order to create a smart load, the DC converter of ESS is coupled with NCLs. This structure can adjust the power of NCLs in real time according to the bus voltage. The bipolar DCMG with the integration of DC-ZCBs contains the PV power generation system and ESS, which can supply power to the loads, as illustrated in Fig. 4(a). In addition, part of the PV connected to the DC bus is embedded into DC-ES to form a PV-DC-ES structure, as illustrated in Fig. 4(b). The bipolar DCMG coupled with DC-ZCBs can drastically lower the capacity and volume of the ESS because the output power of the DC-ES is no longer completely derived from the ESS.

Fig. 4 Structure of DC-ZCBs and PV-DC-ES. (a) Structure of DC-ZCBs with power supply and load. (b) Structure of PV-DC-ES in bipolar DCMG with integration of DC-ZCBs.

DC-ZCB with the integration of PV-DC-ES in a bipolar DCMG shows a significant promise in reducing the capacity and volume requirements of the ESS. This is attributed to the fact that the output power of the DC-ES is no longer solely dependent on the ESS. Future studies on DC-ZCB with the integration of PV-DC-ES can explore several key aspects. Firstly, the integration represents a key step towards sustainable and efficient energy solutions [

34], [36], [37]. Optimizing DC-ZCB with the integration of PV-DC-ES stands as a potential avenue for enhancing system efficiency, maximizing overall energy yield, and minimizing losses during energy conversion and energy storage processes. Secondly, DC-ZCB with the integration of PV-DC-ES brings several advantages concerning DCMG integration and resilience [38]. The PV-DC-ES can contribute to grid-friendly functionalities, explore DCMG applications, and strengthen the system resilience against fluctuations and disturbances. Lastly, PV-DC-ES with the integration of smart building technologies holds the potential to augment overall functionality and energy management capabilities.

In this paper, the PV-DC-ES is implemented as the fundamental unit to analyze the coordinated control performance under bus voltage disturbances. Subsequently, a distributed cooperative control strategy based on modulus voltage for PV-DC-ES unit is presented. Figure 5 depicts the structure of PV-DC-ES in bipolar DCMG with the integration of DC-ZCBs, where +, -, and 0 are the positive, negative, and neutral lines, respectively; and R_p_i and R_n_i are the resistances of equivalent positive and negative loads at node i, respectively. Since the control goal of PV-DC-ES is to maintain the voltage of the DC link at a constant level, PV-DC-ES is equivalent to a DC power supply under steady-state conditions. Because of the differences in the capacity and operating conditions of different PV-DC-ESs, each PV-DC-ES needs to adaptively suppress the disturbance in accordance with its own capabilities when there is a disturbance in the DC bus. Due to the interdependence between the two poles of the bipolar DCMG, control actions on one pole will have an impact on the other. This paper proposes a distributed cooperative control strategy based on the idea of modulus transformation, which enables independent adjustment of the two poles. When this approach is applied, the bipolar DCMG is equivalently operated as two unipolar DCMGs.

Fig. 5 Structure of PV-DC-ES in bipolar DCMG with integration of DC-ZCBs.

Figure 6 shows the structure of two-node equivalent circuit of bipolar DCMG with integration of DC-ZCBs and DC-ES. In this circuit, V_sp_i and V_sn_i are the positive and negative output voltages of node i, respectively; and I_p, I_m, and I_n are the currents on the positive line, neutral line, and negative line, respectively. For ease of investigation, it is assumed that all the equivalent line resistances are identical. As shown in Fig. 6, the two-node equivalent circuit also incorporates a neutral resistance, implying that the bipolar DCMG with the integration of DC-ZCBs and DC-ES cannot be perceived as two disparate unipolar systems.

Fig. 6 Structure of two-node equivalent circuit of bipolar DCMG with integration of DC-ZCBs and DC-ES.

According to Thevenin’s theorem and the equivalent model in Fig. 6, it can be known that Kirchhoff’s voltage law meets:

\{\begin{array}{l} - V_{s p 1} - I_{p} R_{L} + V_{s p 2} - I_{m} R_{L} = 0 \\ - V_{s n 1} + I_{m} R_{L} + V_{s n 2} + I_{n} R_{L} = 0 \\ I_{m} = I_{n} - I_{p} \end{array}

(1)

The expression of line currents can be obtained by solving (1).

\{\begin{array}{l} I_{p} = \frac{2 V_{s p 1} + V_{s n 1} - (2 V_{s p 2} + V_{s n 2})}{3 R_{L}} \\ I_{n} = \frac{2 V_{s n 1} + V_{s p 1} - (2 V_{s n 2} + V_{s p 2})}{3 R_{L}} \\ I_{m} = \frac{V_{s p 1} - V_{s p 2} + V_{s n 2} - V_{s n 1}}{3 R_{L}} \end{array}

(2)

The positive and negative DC-ESs interact during current distribution, hindering independent control and complicating controller design. In [

31], the modulus decomposition is introduced to exclude the coupling issues between the positive and negative poles. This facilitates the design for the controller as the common-mode circuit and differential-mode circuit are decoupled from each other. The transformation equation for modulus components is given in [32], [39] as:

[\begin{matrix} X_{0} \\ X_{1} \end{matrix}] = T [\begin{matrix} X_{p} \\ X_{n} \end{matrix}] = \frac{1}{2} [\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}] [\begin{matrix} X_{p} \\ X_{n} \end{matrix}]

(3)

where X is the voltage or current in the power system; $T$ is the transformation matrix; the subscripts “0” and “1” represent the common-mode component and differential-mode component, respectively; and subscripts “p” and “n” represent the positive pole and the negative pole in the power system, respectively.

In (3), the pole voltages of node 1 and node 2 can be transformed into common-mode components V₀₁ and V₀₂, respectively. The calculation formulas are given as:

\{\begin{array}{l} V_{01} = \frac{V_{s p 1} - V_{s n 1}}{2} \\ V_{02} = \frac{V_{s p 2} - V_{s n 2}}{2} \\ V_{11} = \frac{V_{s p 1} + V_{s n 1}}{2} \\ V_{12} = \frac{V_{s p 2} + V_{s n 2}}{2} \end{array}

(4)

where $V_{11}$ and $V_{12}$ are the differential-mode components of V₀₁ and V₀₂, respectively.

Substituting (4) into (2) yields:

\{\begin{array}{l} I_{p} = \frac{V_{s p 1} - V_{s p 2} + 2 (V_{01} - V_{02})}{3 R_{L}} \\ I_{n} = \frac{V_{s n 1} - V_{s n 2} + 2 (V_{01} - V_{02})}{3 R_{L}} \\ I_{m} = \frac{2 (V_{01} - V_{02})}{3 R_{L}} \end{array}

(5)

According to (5), the line currents are correlated with the voltage difference between two nodes and the difference in common-mode voltage. When $V_{01} = V_{02}$ , the line current meets the requirements:

\{\begin{array}{l} I_{p} = \frac{V_{s p 1} - V_{s p 2}}{R_{L}} \\ I_{n} = \frac{V_{s n 1} - V_{s n 2}}{R_{L}} \\ I_{m} = 0 \end{array}

(6)

According to (4) and (6), when $V_{01} = V_{02}$ , I_p is only determined by the positive output voltage and is independent of the negative output voltage. I_n also comes to the same conclusion. In addition, the neutral current is 0, which can minimize the line loss of the bipolar DCMG.

B. Influence of DC-ES Output Voltage on Current

Figure 7 shows the changes of line currents under different V_sp1. The parameters of bipolar DCMG are detailed in Table I. As can be observed from Fig. 7(a), when V_sp1 changes, I_p changes from -100 A to 166.7 A, I_n changes from -50 A to 83.3 A, and I_m is not equal to 0, the change of which is contrary to the change of I_n. The changing trend of Fig. 7(b) is the same as that of Fig. 7(a) and the intersection point of current changes accordingly.

Fig. 7 Changes of line currents under different V_sp1. (a) $V_{s n 2} = V_{s p 2} = 395$ V. (b) $V_{s n 2} = 400$ V and $V_{s p 2} = 395$ V.

TABLE I Parameters of Bipolar DCMG

Parameter	Value
V_sn1	400 V
R_p1	10 Ω
R_n1	20 Ω
R_L	0.1 Ω
R_p2	20 Ω
R_n2	20 Ω

When the proposed strategy is adopted to guarantee that the common-mode voltages of two nodes are the same, the voltage deviation can be reduced, and the voltage drop is offset on the neutral line. The related parameters are provided below to examine the impact of the changing V_sp1 on unbalanced voltage coefficient and voltage deviation in Fig. 6, which presents the structure of bipolar DCMG. The definition of unbalanced voltage coefficient $ε_{u}$ is given as:

ε_{u} = \frac{v_{p} - v_{n}}{\frac{v_{p} + v_{n}}{2}} \times 100 %

(7)

where v_p and v_n are the positive voltage and negative voltage, respectively.

It is crucial to note that an excessively high unbalanced voltage coefficient $ε_{u}$ can lead to the increase of power losses in the system and, in severe cases, impact the normal operation of the load. As there are currently no standardized quality criteria for DC power systems, we have adopted the unbalanced voltage coefficient from AC power systems as an indicator. In accordance with ANSI C84 standards, restricting the unbalanced voltage coefficient within 3% ensures the steady-state condition of the power system [

23]. Figure 8 shows the changes of neutral line current and unbalanced voltage coefficient under different V_sp1. Specifically, when V_sp1 changes, the relationship between neutral line current and the voltage is unbalanced at Node 1. As illustrated in Fig. 2, the trend of variation in neutral line current is opposite to the trend of unbalanced voltage. When the neutral line current is equal to 0, the unbalanced voltage is also equal to 0.

Fig. 8 Changes of neutral line current and unbalanced voltage coefficient $ε_{u}$ under different V_sp1. (a) $V_{s n 2} = V_{s p 2} = 395$ V. (b) $V_{s p 2} = 400$ V and $V_{s n 2} = 395$ V.

Figure 9(a) illustrates the changes of line current with V_sp1when $R_{L} = 0.1$ $Ω$ and $V_{s p 2} = V_{s n 2} = 400$ V. $V_{s p 1}$ varies from 380 V to 420 V. In Fig. 9(b), $V_{s p 2} = 395$ V and $V_{s n 2} = 400$ V, and the rest of parameters match those in Fig. 9(a). According to Fig. 9(a), when $V_{s p 2} = V_{s n 2}$ , V_sp1 and V_sn1 change synchronously. Therefore, I_p and I_n are always equal while I_m is always equal to 0. According to Fig. 9(b), the changing trends of I_p and I_n are the same. Since V_sp2 and V_sn2 are not equal, it can be obtained that $I_{p} \neq I_{n}$ and $I_{m} \neq 0$ . Therefore, it can be inferred that when there is a voltage deviation between the positive and negative nodes, I_m is not equal to 0, which is the primary cause of rising network loss and coupling the issues between positive and negative poles.

Fig. 9 Changes of line currents under different $V_{s p 1}$ . (a) $V_{s p 2} = V_{s n 2} = 400$ V. (b) $V_{s p 2} = 395$ V and $V_{s n 2} = 400$ V.

III. Distributed Cooperative Control in Bipolar DCMG with Integration of DC-ZCBs

The proposed strategy in this paper can be explained and analyzed from two perspectives: multi-device (i.e., PV-DC-ES) coordination control and modular decoupling of the DCMG. This paper employs distributed cooperative control strategy involving common-mode and differential-mode operations within a control loop decoupled by modulus, simplifying control objectives and reducing control complexity to facilitate coordinated control among multiple devices. The proposed strategy based on modulus voltage consensus control to coordinate the DC-ESs in each node of the bipolar DCMG for unbalanced voltage suppression while simultaneously ensuring voltage consistency of all nodes.

A. Combination of Traditional Voltage Control and Modulus Voltage Control

The investigation in Section II indicates that the characteristic of voltage droop in the bipolar DCMG with the integration of ZCBs can be represented as a droop surface that is dependent on both polar and neutral currents. To incorporate this characteristic into the control system of the bipolar DCMG, traditional voltage control is designed based on (1) and (2), as shown in Fig. 10, where v_p_i and $v_{n i}$ are the voltage outputs of the positive and negative poles of the distributed generation (DG) at node i, respectively; i_p_i and i_n_i are the positive and negative output currents at node i, respectively; v₀_i and v₁_i are the the transformed common-mode and differential-mode voltages at node i, respectively; i₀_i and i₁_i are the transformed common-mode and differential-mode currents at node i, respectively; v_0ref and v_1ref are the reference values of common-mode and differential-mode voltages, respectively; R₀ and R₁ are the common-mode and differential-mode droop resistance, respectively; K_p and K_i are the proportional and integral control ratios, respectively; $d_{0 i}$ and $d_{1 i}$ are the the pulse width modulation (PWM) signals of the converter of the transformed common-mode and differential-mode at node i, respectively; and d_p_iand d_n_i are the PWM signals of the converter of the positive and negative poles of DG at node i, respectively.

Fig. 10 Combination of traditional voltage control and modulus voltage control.

Figure 10 depicts the combination of traditional voltage control and modulus voltage control. Typically, the rated bus voltage can be chosen as the reference voltage of v₁_i, whereas the reference voltage of v₀_i is set to be 0. The modulus voltage control ensures that v_p_i and $v_{n i}$ are equal when they exceed the rated bus voltage. As a result, the unbalanced power supply can be reduced, consequently lowering the neutral current and unbalanced voltage in the load.

B. Modulus Voltage Consensus Control Approach

The consensus algorithm is adopted to develop a distributed control system for common-mode and differential-mode voltages depending on the modulus voltage control. The fundamental concept in the control system is to use the consensus algorithm to ensure that the average values are constant by collecting the modulus voltages of each node as state variables.

The bipolar DCMG with the integration of DC-ZCBs contains multiple parallel DC/DC converters with CLs. The hypothetical properties in the control system are given as follows.

1) The DC/DC converter can be regarded as an optimum controllable voltage source. The response speed of the Buck converter can be rapid enough to ignore its dynamic characteristics.

2) Since the output regulation controller of the load converter is sufficiently responsive, all loads connected to the converter can be considered as CL.

3) In the bipolar DCMG with the integration of DC-ZCBs, the inductance of the lines is negligible.

A distributed cooperative control strategy based on modulus voltage is proposed, and the stability precautions are performed to eliminate CL instability, in order to produce a balanced shunt and good load voltage regulation.

Since damping has an outstanding ability to alleviate the oscillation, virtual resistance is implemented to enhance the stability and the transient function of the system. The proposed strategy of the bipolar DCMG with the integration of DC-ZCBs is illustrated in Fig. 11 and the output modulus voltage values of each inverter are:

\{\begin{array}{l} v_{0 i} = v_{0 r e f} + δ i_{0 i} + δ v_{0 i} - c_{i} i_{0 i} \\ v_{1 i} = v_{1 r e f} + δ i_{1 i} + δ v_{1 i} - c_{i} i_{1 i} \end{array}

(8)

Fig. 11 Diagram of distributed cooperative control strategy.

where $δ i_{0 i}$ and $δ i_{1 i}$ are the common-mode and differential-mode correction currents at node i, respectively; $δ v_{0 i}$ and $δ v_{1 i}$ are the common-mode and differential-mode correction voltages at node i, respectively; and $c_{i}$ is the virtual resistance at node i.

Then, the correction currents are given as:

\{\begin{array}{l} δ i_{1 j} = \frac{b_{1}}{k_{j}} \sum_{i = 1}^{N} a_{i j} (\frac{i_{1 i}}{k_{i}} - \frac{i_{1 j}}{k_{j}}) \\ δ i_{0 j} = \frac{b_{1}}{k_{j}} \sum_{i = 1}^{N} a_{i j} (\frac{i_{0 i}}{k_{i}} - \frac{i_{0 j}}{k_{j}}) \end{array}

(9)

where $k_{j}$ is the current sharing proportional coefficient at node j; $b_{1}$ is the positive gain coefficient; and $a_{i j}$ is the communication weight between nodes i and j, respectively.

When communication occurs between nodes i and j, $a_{i j} = a_{j i} > 0$ ; if no communication exists between nodes i and j, $a_{i j} =$ 0. Formula (9) can be expressed in matrix form as:

δ j = - b_{1} \int K L K i d t

(10)

where $δ j = [δ_{1}, δ_{2}, \dots, δ_{N}]^{T}$ ; $K = d i a g {1 / k_{i}}$ ; and L is the Laplace matrix of communication structure.

The voltage regulation is provided to derive the load correction voltage as:

\{\begin{array}{l} δ v_{j 1} = b_{2} \int (v_{1 r e f} - v_{1 i}) d t \\ δ v_{j 0} = b_{2} \int (v_{0 r e f} - v_{0 i}) d t \end{array}

(11)

where b₂ is the additional gain factor. To increase the reliability and reduce the communication cost, two or three DGs are involved to regulate the voltage.

According to (8), (10), and (11), the common-mode and differential-mode components of the bus voltage are presented as:

\{\begin{array}{l} v_{1} = v_{1 r e f} 1_{N} - C i_{1} + \int [- b_{1} K L K i_{1 i} + b_{2} L (v_{1 r e f} - v_{1 i})] d t \\ v_{0} = v_{0 r e f} 1_{N} - C i_{0} + \int [- b_{1} K L K i_{0 i} + b_{2} L (v_{0 r e f} - v_{0 i})] d t \end{array}

(12)

where $v_{1} = [v_{11}, v_{12}, \dots, v_{1 N}]^{T}$ ; $v_{0} = [v_{01}, v_{02}, \dots, v_{0 N}]^{T}$ ; $C =$ $d i a g {c_{i}}$ ; and $1_{N}$ is a column vector and each element is equal to 1. In the steady state, (12) can be expressed as:

\{\begin{array}{l} - b_{1} K L K i_{1} + b_{2} L (v_{1 r e f} - v_{1 i}) = 0_{N} \\ - b_{1} K L K i_{0} + b_{2} L (v_{0 r e f} - v_{0 i}) = 0_{N} \end{array}

(13)

where $0_{N}$ is an N-dimensional zero matrix.

According to the above analysis, the voltage and current meet:

\{\begin{array}{l} \frac{i_{1}}{k_{1}} = \frac{i_{2}}{k_{2}} = . . . = \frac{i_{N}}{k_{N}} \\ v = v_{r e f} \end{array}

(14)

The fact that not all DGs are required to participate in voltage recovery is evident from (14), which indicates that the communication cost can be reduced by using a number of nearby DGs to collect the load voltage.

IV. Consistency in Control Performance and Small-signal Stability Analysis

A. Consensus Control of DC Bus Average Voltage

The convergence of the DC bus average voltage and unbalanced voltage coefficient is demonstrated by using the Laplace transform and frequency domain analysis. The positive pole is studied first in the bipolar DCMG with the integration of DC-ZCBs, and the analysis procedure for the negative pole is the same. According to the analysis of the bipolar average unbalanced voltage in [

40], the Laplace matrix of the DC bus average voltage is expressed in the frequency domain as:

\{\begin{array}{l} s V_{a p} - V_{a p} (0) = s V_{p} - V_{p} (0) - L V_{a p} \\ V_{a p} = [V_{a p 1}, V_{a p 2}, \dots, V_{a p N}]^{T} \\ V_{p} = [V_{p 1}, V_{p 2}, \dots, V_{p N}]^{T} \end{array}

(15)

where $V_{a p}$ and $V_{p}$ are the Laplace transforms of $v_{a p N}$ and $v_{p N}$ , respectively.

The initial condition is satisfied with $V_{a p} (0) = V_{p} (0)$ .

V_{a p} = s {(s E + L)}^{- 1} V_{p}

(16)

whereE is the identity matrix.

According to the characteristics of L, when L is balanced, V_ap will converge to a value, i.e., the true average voltage $V_{a m p}$ .

\begin{array}{l} V_{a m p} = \underset{t \to \infty}{l i m} V_{a p} (t) = \underset{s \to 0}{l i m} s V_{a p} = \underset{s \to 0}{l i m} s {(s E + L)}^{- 1} \cdot \underset{s \to 0}{l i m} s V_{p} = \\ Q v_{m p} = <v_{m p}> 1_{N} \end{array}

(17)

where $v_{a m p} = [v_{a m p 1}, v_{a m p 2}, \dots v_{a m p N}]^{T}$ ; and Q is the mean matrix and all the elements are 1/N. The steady-state value of vector v_ap is denoted as v_amp. The mean of each element of vector v_mp is represented by $<v_{m p}>$ :

<v_{m p}> = \frac{1}{N} \sum_{i = 1}^{N} v_{m p i}

(18)

B. Consensus Control of Unbalanced Voltage Coefficient

Based on the analysis of the control approach presented in [

40], the differential-positive voltage and differential-negative voltage are expressed by the Laplace matrix, and then the equivalent arrangement in the frequency domain can be obtained as:

V_{u n} = s {(s E - L)}^{- 1} V_{u n}^{'} = H_{u n} V_{u n}^{'}

(19)

where $V_{u n}$ and $V_{u n}^{'}$ are the Laplace transforms of $v_{u n}$ and $v_{u n}^{'}$ , respectively, $v_{u n} = [v_{u n 1}, v_{u n 2,} \dots, v_{u n N}]^{T}$ , and $v_{u n}^{'} = [v_{u n 1}^{'}, v_{u n 2}^{'}, \dots, v_{u n N}^{'}]^{T}$ .

When DCMG is balanced, v_un will converge to the value $v_{m u n}$ .

\begin{array}{l} v_{m u n} = \underset{t \to \infty}{l i m} v_{u n} (t) = \underset{s \to 0}{l i m} s V_{u n} = \underset{s \to 0}{l i m} s {(s E - L)}^{- 1} \cdot \underset{s \to 0}{l i m} s V_{u n}^{'} = \\ Q v_{u n}^{'} = <v_{m u m}^{'}> 1_{n} \end{array}

(20)

where $v_{m u n} = [v_{m u n 1}, v_{m u n 2}, \dots, v_{m u n N}]^{T}$ ;and $v_{m u n}^{'}$ is the Laplace transforms of $v_{m u n}$ , $v_{m u n}^{'} = [v_{m u m 1}^{'}, v_{m u m 2}^{'},$ $\dots,$ $v_{m u m N}^{'}]^{T}$ .

C. Analysis of Steady-state Operation Point

Consensus analysis is applied to examine the steady-state operation point in the bipolar DCMG while taking into account the consensus algorithm. According to the proposed strategy, the positive correction voltage in [

40] can be expressed as:

\{\begin{array}{l} δ_{p}^{'} = G_{u} (V_{r e f p} - V_{a p}) \\ δ_{p}^{} = G_{o} (f_{1} (V_{u n}^{'}) V_{n} - V_{p}) \end{array}

(21)

where $V_{r e f p} = (v_{r e f p} / s) 1_{N}$ is the reference value of positive bus voltage; and $G_{o} = d i a g {G_{o i}}$ and $G_{u} = d i a g {G_{u i}}$ are the transfer function matrices for DC bus average voltage controller and unbalanced controller, respectively.

\{\begin{array}{l} G_{u i} = k_{p V p i} + \frac{k_{i V p i}}{s} \\ G_{o i} = k_{p u n p i} + \frac{k_{i u n p i}}{s} \end{array}

(22)

Considering the system voltage droop control at the same time, the positive reference voltage $V_{p}^{*}$ for voltage droop control of the bipolar DCMG meets:

V_{p}^{*} = V_{r e f p} + δ_{p}^{'} + δ_{p}^{} - R_{d p} I_{l o a d p}

(23)

where R_dp=diag{ $R_{d p}$ } is the droop gain matrix; I_loadp is Laplace transform of positive load current i_loadp, $i_{l o a d p} = [i_{l o a d p 1},$ $i_{l o a d p 2}, \dots, i_{l o a d p N}]^{T}$ .

The steady-state characteristics of positive average voltage is obtained in (24), with the derivation process detailed in Supplementary Material A.

V_{a m p}^{} = <V_{a m p}^{}> 1_{N} = V_{r e f p} 1_{N}

(24)

Similarly, the steady-state characteristics of negative voltage can be obtained as:

V_{a m n}^{} = <V_{a m n}^{}> 1_{N} = V_{r e f n} 1_{N}

(25)

It is proven by the derivation that the average node voltages $v_{a p i}$ and $v_{a n i}$ tend to be converged and remain uniform under the steady state in bipolar DCMG. Specifically, $v_{a p i}$ and $v_{a n i}$ will be stabilized at the positive and negative rated reference voltages $V_{r e f p}$ and $V_{r e f n}$ , respectively. The correctness of the proposed strategy is thus confirmed.

D. Analysis of Small-signal Stability

To investigate the small-signal stability in bipolar DCMGs, this paper establishes a small-signal model and analyzes the impact of the system stability under the proposed strategy through Nyquist analysis. The derivation process of the output characteristics and the transfer function from the control to the output of DC/DC converters is provided in Supplementary Material A.

The structure diagram of the primary control can be obtained, as shown in Fig. 12(a). The input of the system is the reference voltage ${\hat{v}}_{0 r e f}^{*} (s)$ (or ${\hat{v}}_{1 r e f}^{*} (s)$ ), and the output is the node voltage ${\hat{v}}_{0}^{} (s)$ (or ${\hat{v}}_{1}^{} (s)$ ). Simplifying the block diagram, taking ${\hat{v}}_{0 r e f}^{*} (s)$ (or ${\hat{v}}_{1 r e f}^{*} (s)$ ) as input and ${\hat{v}}_{0}^{} (s)$ (or ${\hat{v}}_{1}^{} (s)$ ) as output, the transfer functions of primary control are derived as:

\{\begin{array}{l} G_{0} = \frac{2 k_{p} V_{s} R (3 R + R_{L}) s + 2 k_{i} V_{s} R (3 R + R_{L})}{C L R (3 R + R_{L}) s^{3} + L R s^{2} + (3 R^{2} + R R_{L} + 6 k_{p} V_{s} r_{d} R + 2 k_{p} r_{d} R_{L} V_{s}) s + 2 k_{i} r_{d} V_{s} (3 R + R_{L})} \\ G_{1} = \frac{2 k_{p} V_{s} R (R + R_{L}) s + 2 k_{i} V_{s} R (R + R_{L})}{C L R (R + R_{L}) s^{3} + L R s^{2} + (R^{2} + R R_{L} + 2 k_{p} V_{s} r_{d} R + 2 k_{p} r_{d} R_{L} V_{s}) s + 2 k_{i} r_{d} V_{s} (R + R_{L})} \end{array}

(26)

Fig. 12 Block diagram of control system including proposed strategy. (a) Primary control approach (positive pole). (b) With integration of secondary control approach.

where G₀ and G₁ are the common-mode and differential-mode transfer functions of the primary control, respectively. Combining and simplifying (S11), (S12) in Supplementary Material A, and (26) with (9)-(11), the block diagram of the bipolar DCMG with secondary control is obtained, as shown in Fig. 12(b). The common-mode control black diagram is taken as an example in Fig. 12, and the differential-mode control block diagram is similar. By combining (9) between the average DC bus voltage and the DC bus voltage, we can simplify the system transfer function with common-mode reference voltage ${\hat{v}}_{0 r e f}^{} (s)$ as input and common-mode average voltage ${\hat{v}}_{a v e 0}^{} (s)$ and common-mode average current ${\hat{i}}_{a v e 0}^{} (s)$ as outputs. The system is then subjected to small-signal stability analysis.

Due to the modulus voltage consensus control, the control systems of the DC/DC converters interconnected through the communication system are mutually coupled. This paper verifies the stability of the proposed strategy for a bipolar DCMG with three converters connected to each other through a communication system. The three nodes are indexed by x, y, and z.

From Fig. 12(b), it can be observed that due to the modulus voltage consensus control, the control-to-output transfer function of the bipolar DCMG including the primary control and secondary control needs to take into account the communication of the three nodes. In this paper, we take node x as an example to analyze the stability of node x by considering the control influence of nodes y and z on node x, and then extend it to the stability of the entire system. The impacts of the modulus voltage consensus control of nodes y and z on node x are defined as C_x_,_y and $C_{x, z}$ , respectively.

To study the influence of the modulus voltage consensus control coefficients a_ij on the stability of bipolar DCMG, the values of a_ij are set to be the same, and Nyquist curves for various interaction terms are plotted under different $a_{i j}$ , as shown in Fig. 13. Due to symmetry, the results of $C_{x, y}$ and $C_{x, z}$ are the same, so only C_x_,_y is given. As can be observed from Fig. 13, as a_ij increases, C_x_,_x and C_x_,_y gradually increase but do not encircle the $(- 1, j 0)$ point, and the system is stable.

Fig. 13 Nyquist curves under different a_ij. (a) C_x_,_x. (b) C_x_,_y.

The following analysis focuses on the control-to-output analysis of the bipolar DCMG considering the modeling of converters, primary control, and secondary control. Based on Fig. 13(b), the control-to-output transfer function of the common-mode control loop of node x can be written as:

{\hat{v}}_{x 0} = f_{x, x} ({\hat{v}}_{x 0 r e f}) + f_{x, y} ({\hat{v}}_{y 0 r e f}) + f_{x, z} ({\hat{v}}_{z 0 r e f})

(27)

It can be observed from (27) that nodes y and z generate interaction terms for node x. To verify the impact of control parameters on stability, the effect of voltage recovery control is examined. Different coefficients b of the voltage recovery control are set, and Nyquist curves for the control-to-output of the nodes are shown in Fig. 14. From Fig. 14, as $b$ increases, the Nyquist curve gradually begins to encircle the (-1, j0) point. When $b = 1$ , the Nyquist curve passes through the (-1, j0) point. When $b = 2$ , the Nyquist curve encircles the (-1, j0) point. Therefore, to ensure the stability of bipolar DCMG, the value of b should be set less than 1.

Fig. 14 Nyquist curves under different b. (a) $f_{x, x}$ . (b) $f_{x, y}$ .

V. Simulation Results

A topology of the bipolar DCMG with the integration of DC-ZCBs is designed by using MATLAB/Simulink to verify the efficacy of the proposed strategy, as depicted in Fig. 15. The loads at the positive node are $R_{p 1} = 5$ Ω, $R_{p 2} = 10$ Ω, $R_{p 3} = 20$ Ω, and $R_{p 4} = 20$ Ω, while the loads at the negative node are $R_{n 1} = 10$ Ω, $R_{n 2} = 20$ Ω, $R_{n 3} = 20$ Ω, and $R_{n 4} = 10$ Ω. The relevant system parameters and control parameters are provided in Table II.

Fig. 15 Topology of bipolar DCMG with integration of DC-ZCBs.

TABLE II System Parameters and Control Parameters

Parameters	Value	Parameters	Value
V_ref	400 V	R_L	0.1 Ω
R_NCL	20 Ω	r_d	0.05 Ω
k_p	0.1	k_i	0.01
a_ij	10	b	0.5
C_f	200 μF	L_f	50 mH

A. Validation of Effectiveness of Proposed Strategy

Figure 16 presents the simulation results obtained from traditional voltage droop control, which is added to the control system after 3 s.

Fig. 16 Simulation results obtained from traditional voltage droop control. (a) v_p. (b) v_n. (c) ɛ_u. (d) I_m.

In Fig. 16(a) and (b), $v_{p}$ and $v_{n}$ are less than 400 V. An increase in voltage drop between the DG side and the load side can result in an increase in voltage deviation. In contrast, Fig. 16(c) shows that the traditional voltage droop control effectively suppresses the overall unbalanced voltage coefficient ɛ_u across 4 nodes. In Fig. 16(d), the neutral line currents $I_{m 1}$ - $I_{m 3}$ vary with the unbalanced voltage coefficient depicted in Fig. 16(c). Although traditional voltage droop control can suppress the unbalanced voltage coefficient to some extent, v_p and v_n still remain below the rated voltage of 400 V.

Figure 17 depicts the comparison of simulation results between traditional voltage droop control and proposed strategy, where only traditional voltage droop control is applied from 1 s to 3 s, and the proposed strategy is used from 3 s. According to the results, the proposed strategy effectively compensates for the voltage at the load side. Figure 17(a) and (b) shows that $v_{p}$ and $v_{n}$ are both increased under the proposed strategy, indicating improved voltage compensation at the load side. In addition, Fig. 17(c) and (d) demonstrates that the average common-mode voltage v_av0 is maintained at 0 V, while the average differential-mode voltage v_av1 of nodes 1-4 increases from 398 V to 400 V under the proposed strategy. These findings illustrate that the proposed strategy is capable of correcting the voltage at the load side, making it an effective strategy for the bipolar DCMG with the integration of DC-ZCBs.

Fig. 17 Comparison of simulation results between traditional voltage droop control and proposed strategy. (a) v_p. (b) v_n. (c) v_av1. (d) v_av0.

B. Performance Comparisons with Other Advanced Voltage Control Approaches

1)　Comparison with Consensus-based Distributed Control Approach [

32]

In [

32], a consensus-based distributed control approach for DGs in bipolar DCMG is presented to mitigate the unbalanced voltage. To highlight the superiority of the proposed strategy in this paper, a comparison is conducted in this part with the approach in [32]. The simulation model remains the same, as shown in Table II. The system utilizes the consensus-based distributed control approach in [32] during the initial 3 s and transitions to the proposed modulus voltage consensus control approach thereafter. The comparative results between consensus-based distributed control approach in [32] and proposed modulus voltage consensus control approach are depicted in Fig. 18.

Fig. 18 Comparative results between consensus-based distributed control in [

32] and proposed modulus control. (a) v_p. (b) v_n. (c) v_avp. (d) v_avn.

In Fig. 18(a) and (b), v_p and v_n under the consensus-based distributed control approach in [

32] deviate further from the rated voltage than those under the proposed modulus voltage consensus control approach. Figure 18(c) and (d) shows that the modulus voltage consensus control approach can ensure that the average positive voltage v_avp and average negative voltage v_avn of different nodes are consistent. Thus, the proposed modulus voltage consensus control approach can reduce the impact of the unbalanced components between DGs, resulting in a reasonably independent control of the positive and negative DGs as well as a weakening of the coupling between the controls.

Figure 19 presents $ɛ_{u}$ under conditions $a$ and $b$ , which are specifically before 3 s and after 3 s. The results demonstrate that $ɛ_{u}$ of each node is significantly smaller when the proposed modulus voltage consensus control approach is applied. Hence, the proposed modulus voltage consensus control approach is highly effective in reducing $ɛ_{u}$ .

Fig. 19 Unbalanced voltage coefficient under conditions a and b.

In a bipolar DCMG with the integration of DC-ZCBs, the magnitude of the neutral current I_m has an impact on the losses of the bipolar DCMG. When the output voltages of the DGs in the bipolar DCMG with the integration of DC-ZCBs are equalized, I_m will be further reduced. The comparison of neutral line current when load changes under conditions a and b are shown in Fig. 20. According to Fig. 20, under condition a, I_m is not equal to 0 and can even reach a maximum current of 9.5 A. In contrast, I_m is always very small under condition $b$ . The proposed strategy can further regulate the neutral current, thereby reducing the operational losses of the bipolar DCMG.

Fig. 20 Comparison of neutral line current when load changes under conditions a and b.

2)　Comparison with Adaptive Droop Control Approach of Unbalanced Voltage Based on Fuzzy Control

In this part, a comparison is made with the adaptive droop control of unbalanced voltage based on fuzzy control proposed in [

24]. The line resistance is set as 0.1

Ω

, and other parameters of the model are the same as those in Table II. The calculated unbalanced voltage coefficients are presented in Fig. 21, where

ε_{u 1}

and

ε_{u 2}

represent the simulated results of the unbalanced voltage coefficients obtained by adaptive droop control of unbalanced voltage based on fuzzy control and the proposed strategy, respectively. From Fig. 21, it can be observed that the proposed strategy effectively suppresses the unbalanced voltage coefficients at various nodes. Specifically, the unbalanced voltage coefficients at node 3 are significantly suppressed. Although there is a slight increase in the unbalanced voltage coefficients at node 2, it remain at very low magnitudes. After using the proposed strategy, the average unbalanced voltage coefficient of the nodes is reduced from 0.62% to 0.41%, a decrease of 34%.

Fig. 21 Comparison of results between adaptive droop control approach and proposed strategy.

C. Impact of Load Variations on Bipolar DCMG with Integration of DC-ZCBs Under Proposed Strategy

In Sections V-A and V-B, the proposed strategy is shown to be superior to both positive and negative voltage control approaches as well as other advanced voltage control approaches. This subsection focuses on investigating its effectiveness under load variations on the bipolar DCMG with integration of DC-ZCBs. At 3 s, a 10 $Ω$ resistance is added at the positive terminal at node 4. Figure 22 shows the effect of load variation under proposed strategy. In Fig. 22(a) and (b), it is evident that $v_{p}$ and $v_{n}$ at all nodes stay constant after load variation at 3 s. Figure 22(c) and (d) demonstrates that $v_{a v 1}$ remains at the rated voltage of 400 V, and v_av0 remains at 0 V, which is consistent with the control objectives.

Fig. 22 Effect of load variation under proposed strategy. (a) $v_{p}$ . (b) $v_{n}$ . (c) $v_{a v 1}$ . (d) $v_{a v 0}$ . (e) $i_{m}$ . (f) $ε_{u}$ .

Based on the proposed strategy, Fig. 23 presents the effect of load variation on $ɛ_{u}$ of bipolar DCMG with the integration of DC-ZCBs under proposed strategy. All the $ɛ_{u}$ of nodes 1-4 is kept at 0 before switching the load under the proposed strategy. After the additional load is added at 3 s, $ɛ_{u}$ increases slightly, but remains well below the rated value of 3. Subsequently, all the $ɛ_{u}$ of nodes 1-4 is back to 0 at 5 s.

Fig. 23 Effect of load variation on $ɛ_{u}$ of bipolar DCMG with integration of DC-ZCBs under proposed strategy.

The values of I_m for the bipolar DCMG with integration of DC-ZCBs are presented in Fig. 24. Similar to the results in Fig. 23, the neutral line currents only increase at 3 s when the additional load is switched. Therefore, it verifies the validity of the proposed strategy even under load variations.

Fig. 24 Effect of load variation on neutral line current of bipolar DCMG with integration of DC-ZCBs under proposed strategy.

VI. Conclusion

With the acceleration of energy transition, ZCBs and localized bipolar DCMGs develop rapidly. However, this growth introduces challenges to unbalanced voltage suppression caused by high penetration of RESs. Considering this scenario, this paper proposes a distributed cooperative control strategy for DC-ESs based on the modulus voltage in bipolar DCMGs with the integration of ZCBs. The topological structure of the PV-DC-ES is introduced, which improves the absorption capacity of renewable energy and reduces the capacity and volume of the ESS of DC-ES. A mathematical equivalent model for a bipolar DCMGs incorporating PV-DC-ES is established. The coupling characteristics and influencing factors within the bipolar DCMGs system are discussed. Based on this, the proposed strategy of PV-DC-ES effectively decouples the control system of the bipolar DCMG, simplify the control architecture and efficiently suppress unbalanced voltage. Through a comparative analysis with a state-of-art control approaches, the results of this paper demonstrate that the proposed strategy significantly suppresses unbalanced voltage of the system. Furthermore, unbalanced current and power losses are also significantly reduced.

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