Stabilization Method Considering Disturbance Mitigation for DC Microgrids with Constant Power Loads

Abstract

In this paper, the stability of direct current (DC) microgrids with a constant power load (CPL) and a non-ideal source is investigated. The CPL’s negative impedance will destabilize the system, and disturbances in the non-ideal source will degrade the load voltage quality. In this study, we aim to: (1) overcome the instability of the CPL; (2) mitigate disturbances from the non-ideal source; (3) prevent the discontinuous harmonic current of high-frequency switching regulator from interfering with the source. Then, a stabilization method based on active damping which can achieve the above three objectives simultaneously is proposed. To obtain the stability conditions, the small-signal model of the system near the high-voltage equilibrium is established. Then, stability conditions are derived by eigenvalue analysis, and the domain of attraction near equilibrium is also obtained using the quadratic Lyapunov function. For the second objective, the key is to choose the optimal parameters to achieve disturbance attenuation. For the third objective, the active damper can separate the source from the switching regulator, which can prevent the discontinuous harmonic current. Moreover, the proposed method can be extended to multiple cases, and simulation results verify the effectiveness of the proposed method.

1. Introduction

Microgrids, which primarily rely on renewable energy such as photovoltaic (PV) and wind power, have been identified as an effective complement to traditional power systems [1,2,3]. Usually, they can be worked in two modes: grid-connected mode, and islanded mode. Usually, the main control objectives of a DC microgrid include maintaining stability [4,5,6,7,8,9], minimizing cost [10,11,12], sharing power and regulating voltage [13]. For these objectives, there are mainly three control methods: decentralized control [10,12,14,15,16,17], distributed control [18,19,20,21,22,23,24,25,26] and centralized control [11,27].

In general, there are two kinds of microgrids: direct current (DC) microgrids and alternating current (AC) microgrids. At present, domestic research on microgrids has primarily focused on AC microgrids [12,13,14,15,16,17]. However, DC microgrids have the following advantages: high transmission efficiency, high reliability, no frequency synchronization issues, easier integration of renewable energy, and ease of stabilization, therefore, DC microgrids are increasingly being used in applications such as aircraft, spacecraft, and electric vehicles [4,24].

In DC microgrids, the load is typically connected to the DC bus through a DC/DC or DC/AC converter. When the response of the load-end converter is rapid, the load exhibits a negative impedance, which is equivalent to a constant power load (CPL) [4]. To ensure stable and reliable operation of a DC microgrid, voltage stability is essential. To address this issue, numerous studies have been performed to evaluate modeling, stability analysis, and control strategies [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49]. These studies can be categorized as two groups based on the number of distributed generation units: 1) one DG unit and one CPL; 2) n DG units and m CPLs.

The first category focuses only on stability analysis and stabilization methods for a single converter with one CPL [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. Several linear stabilization methods have been used to stabilize such systems. Because voltage oscillations can be mitigated by increasing damping and reducing negative impedance, methods based on the addition of resistive loads, filters, and energy storage have been proposed [28]. Moreover, several linear stabilization methods have also been proposed, such as passivity-based control strategy [29,30], active damping method [31,32,33], and virtual impedance method [34,35]. To fulfill small-signal stability conditions, some impedance-based stability criteria have been proposed [34,36]. To analyze the large-signal stability of a system, several Lyapunov function-based methods have been proposed, including the Lure Lyapunov function [37], Brayton-Moser’s mixed potential [38,39,40], and the Popov criterion [41]. To ensure that the system is globally asymptotically stable, novel nonlinear techniques based on feedback linearization have been introduced, which transform an unstable nonlinear system into a stable linear system [42,43]. Meanwhile, nonlinear stabilization methods based on sliding modes have been investigated [5,44]. Comparative speaking, the linear stabilization methods such as PBC, virtual RLC damper are simple, easy and effective, but the equilibrium is not global asymptotically stable (GAS); the nonlinear controller such as feedback linearization and sliding mode control can achieve GAS, but it is more complicated and may need communication in some occasions [45].

The second group extends the stability analysis to a system with n converters [45,46,47,48,49]. The structures of DC microgrids with multiple DG units can be divided into two types: single bus and multiple bus. For a DC microgrid with a single-bus structure, the loads throughout the entire bus are equivalent to a common load when the resistance of the DC bus can be neglected. The topology of this type of DC microgrid is equivalent to the star type. Likewise, the topology of a DC microgrid with multiple DG units is meshed. The stability of a DC microgrid with star topology under droop control was analyzed in [45,46,47]. To simplify this problem, a reduced-order linearized model was derived, and transient processes were ignored [46]. Another stability condition has been derived by assuming that all DG units have the same ratio between inductance and resistance [47]. These studies show that the system is stable if the droop coefficient is larger than the equivalent negative impedance of CPL. To study the exact behavior of a CPL in a parallel multi-converter system, a high-dimensional model is estabilished to analyze system stability. Using results for quadratic eigenvalue problems, a more accurate stability condition has been obtained [45]. The stability of a star-type DC microgrid with distributed control has been analyzed in [48], and analytical stability conditions were established as a function of the system parameters by using the inertia theorem. The stability of a meshed DC microgrid with multiple buses has been analyzed in [49]. Based on Tarski’s fixed-point theorem, the existence conditions for a system equilibrium were obtained. Moreover, stability conditions ensuring that the system allows a robust stable equilibrium were obtained [49].

In the above studies, the DG units are modeled as ideal controlled voltage sources (CVSs) or CVSs with DC/DC converters. However, in a DC microgrid, DG units are primarily renewable energy sources with bounded uncertain disturbances. To our best knowledge, the stability and disturbances attenuation of the DC microgrid with a CPL and non-ideal sources has hardly been considered. Moreover, the discontinuous harmonic current produced by the high-frequency switching of the DC/DC converter will damage the source. This issue should also be considered. In this paper, an active damping method is proposes to overcome the CPL’s instability, mitigate disturbances in the DG and prevent the discontinuous harmonic current simultaneously. The main contributions of this paper can be summarized as follows:

First, a stabilization method based on active damping, which can overcomes the CPL’s instability and mitigates disturbances in non-ideal sources, is proposed. Moreover, the active damper can also protect the source input from the influence of a discontinuous harmonic current produced by high-frequency switching.

Second, the stability of the system under the proposed stabilization control strategy is analyzed, and a small-signal stability condition is obtained based on the Routh-Hurwitz criterion. This paper shows that the stability criterion derived in [40] is incorrect from two aspects of theoretical analysis and simulation. Moreover, an approach based on the quadratic Lyapunov function is developed to analyze the large-signal stability and to estimate the domain of attraction (DOA) at equilibrium.

Finally, this paper extends the proposed stabilization method to a system with multiple paralleled DC/DC converters. Further, it demonstrates that the dynamic model of a system with multiple identical converters can be reduced to a model with a single converter. The stability of the system with multiple non-identical converters is also investigated in simulation, and the simulation results shows that the proposed method can also be applicable.

This paper is organized as follows: Section 2 introduces the topologies and the proposed stabilization technique for a DC microgrid. Section 3 provides a stability analysis of a DC microgrid under the proposed control. Section 4 presents a generalization of the proposed stabilization method. Section 5 provides case studies and simulations, and the conclusions are drawn in Section 6.

2. Problem Formulation and Stabilization Method

2.1. Problem Formulation

A DC microgrid, whose simplified model is presented in Figure 1, is composed of a load, a DC/DC converter, and an energy input. In most previous studies, the energy input is assumed as a constant voltage source. In general, the load is connected to the DC bus through a converter to regulate the voltage. When the response of the load converter is rapid, the load can be treated as an ideal CPL. An equivalent average model is presented in Figure 2. Here, a large-signal model of the CPL is given as a controlled current source whose output current is equal to the constant power divided by the voltage. The energy input and the switch are equivalent to a CVS, which is the control input of the system. According to [45], the state equations of the system in figure are given by:

$$\{\begin{array}{c}L\frac{d{x}_1}{dt}=u-x_2\\ C\frac{d{x}_2}{dt}=x_1-\frac{P}{x_2}\end{array}$$

(1)

where L, C, x₁, x₂, P, and u are the inductance, capacitance, inductance current, output voltage, CPL, and control input, respectively.

2.2. Proposed Stabilization Method

The mechanism by which the CPL leads to voltage instability has been explained in [28], and numerous stabilization methods based on the system in (1) have been proposed [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. However, the following two important issues have not been considered in these studies. First, when the energy input comes from a renewable energy source, disturbances in the input will cause output voltage fluctuations, which will reduce the quality of the power supply. Second, unlike a boost circuit, which has an LC filter between the source and the converter, the source of a Buck circuit can easily experience interference from an insulated gate bipolar transistor (IGBT) with a high switching frequency. The high-frequency switching of an IGBT will produce a discontinuous harmonic current, which degrades the power source. To address these issues, this paper proposes an active damping method, which is presented in Figure 3. Here, R, L₁, and C₁ are the resistance, inductance, and capacitance of the active damping circuit, respectively, and L₂ and C₂ are the inductance and capacitance of the filter circuit of the Buck converter, respectively. Term $\delta \left(t\right)$ represents the uncertain disturbance in the energy input, which is time-varying and bounded.

The resistance of the active damper will increase the damping of the system, which can overcome the negative impedance of the CPL. Moreover, the active damper can act as a filter to suppress high-frequency disturbances in the non-ideal input for the Buck converter. The active damper can also isolate the input sources from the IGBT, thereby reducing the high-frequency current harmonics due to the pulse-width modulation (PWM) control operation.

An equivalent average model of the system in Figure 3 is presented in Figure 4. The model can be decomposed into two sub-circuits. Here, x₁ and x₃ are the currents through L₁ and L₃, respectively, and x₂ and x₄ are the voltages across C₁ and C₂, respectively. The constant d corresponds to the duty cycle, where $0\le d\le 1$.

According to Kirchhoff’s voltage law, the dynamics of the system in Figure 4 can be described by the following differential equations:

$$\{\begin{array}{l}{L}_1\frac{d{x}_1}{dt}=E+\delta \left(t\right)-R{x}_1-x_2\\ C_1\frac{d{x}_2}{dt}=x_1-d{x}_3\\ L_2\frac{d{x}_3}{dt}=d{x}_2-x_4\\ C_2\frac{d{x}_4}{dt}=x_3-\frac{P}{x_4}\end{array}$$

(2)

Let us assume that the maximal value of the disturbance is far less than E. Then, the operation points of the system are determined by:

$$\{\begin{array}{l}1-\frac{R}{E}{x}_1-\frac{1}{E}{x}_2=\frac{\delta \left(t\right)}{E}\approx 0\\ x_1-d{x}_3=0\\ d{x}_2-x_4=0\\ x_3-\frac{P}{x_4}=0\end{array}$$

(3)

Solving (3), the operation points $\left(x_1,x_2,x_3,x_4\right)$ of the system are obtained as:

$$\{\begin{array}{l}{e}_1=\left(\frac{E-\sqrt{E^2-4RP}}{2R},\frac{E+\sqrt{E^2-4RP}}{2},\frac{E-\sqrt{E^2-4RP}}{2Rd},\frac{d\left(E+\sqrt{E^2-4RP}\right)}{2}\right)\\ e_2=\left(\frac{E+\sqrt{E^2-4RP}}{2R},\frac{E-\sqrt{E^2-4RP}}{2},\frac{E+\sqrt{E^2-4RP}}{2Rd},\frac{d\left(E-\sqrt{E^2-4RP}\right)}{2}\right)\end{array}$$

(4)

Clearly, the system has two operation points. Operation point e₁ has a higher voltage and lower current than e₂, which results in a lower line loss for e₁. Hence, this paper investigates the stability of the system near e₁. We linearize (2) about e₁, and the system Jacobian matrix is given by:

$$J=\left[\begin{array}{cccc}-\frac{R}{L_1}& -\frac{1}{L_1}& & \\ \frac{1}{C_1}& 0& -\frac{d}{C_1}& \\ & \frac{d}{L_2}& 0& -\frac{1}{L_2}\\ & & \frac{1}{C_2}& \frac{P}{C_2{v}^2}\end{array}\right]$$

(5)

where $v=\frac{d\left(E+\sqrt{E^2-4RP}\right)}{2}$.

Compared with Equation (1), the system in (2) is a higher-order system in the presence of an active damper, which will increase the difficulty of stability analysis. Next, we analyze the stability of operation point e₁ and determine the stability conditions.

3. Stability Analysis

3.1. Recent Related Results

The mixed potential theory developed by Brayton and Moser is generally used to construct Lyapunov-type functions to determine stability criteria for nonlinear electrical networks. In [40], the stability criteria for an electrical system with multistage LC filters (presented in Figure 5) were obtained based on the Brayton–Moser mixed potential theory. In Figure 5, $i_i$ and $u_i$ denote the inductor current and capacitor voltage of the i-th LC filter, respectively.

According to the Equation (8) in Reference [40], the state equation of the DC microgrid with N-stage LC filters is given by:

$$\{\begin{array}{l}{L}_1\frac{d{i}_1}{dt}=E-R{i}_1-u_1\\ C_1\frac{d{u}_1}{dt}=i_1-i_2\\ L_2\frac{d{i}_2}{dt}=u_1-u_2\\ C_2\frac{d{u}_2}{dt}=i_2-i_3\\ \hspace{1em}\hspace{1em}⋮\\ L_N\frac{d{i}_N}{dt}=u_{N-1}-u_N\\ C_N\frac{d{u}_N}{dt}=i_N-\frac{P}{u_N}\end{array}$$

(6)

The main results in [40] can be briefly derived as follows: firstly, according to Equation (9) in [40], the mixed potential function is equivalent to the following expression:

$$P\left(i,v\right)=-\frac{1}{2}R{i}_1^2+{\displaystyle {\int }_0^{u_N}\frac{P}{u_N}}d{u}_N+i^{T}Mv-i^T\gamma$$

(7)

where $i={\left[\begin{array}{cccc}{i}_1& i_2& \cdots & i_N\end{array}\right]}^T,u={\left[\begin{array}{cccc}{u}_1& u_2& \cdots & u_N\end{array}\right]}^T,\gamma ={\left[\begin{array}{cccc}E& 0& \cdots & 0\end{array}\right]}^T$, and M is given by:

$$M=\left[\begin{array}{ccccc}1& & & & \\ -1& 1& & & \\ & -1& 1& & \\ & & \ddots & \ddots & \\ & & & -1& 1\end{array}\right]$$

(8)

Then, according to the Equation (14) in [24], the system in (6) is stable if the following holds:

$$\frac{L_{\alpha }}{C_{\alpha }}\frac{P}{v^2}<R<\frac{E^2}{4P}$$

(9)

where $L_{\alpha }=\max \left\{L_i\right\}$ and $C_{\alpha }=\min \left\{C_i\right\}$.

Clearly, if we take d = 1 and N = 2, then the system presented in Figure 3 is completely equivalent to the system in Figure 5. Therefore, if the stability analysis method proposed in [40] is correct, then the stability criterion in (9) is also applicable to the system described in (2). Unfortunately, the stability analysis method is not applicable to this situation, and the stability criterion in (9) is inaccurate. There is a fatal mistake in the stability analysis presented in [40]. The generalized integral term ${\displaystyle {\int }_0^{u_N}\frac{P}{u_N}}d{u}_N$ does not converge. Therefore, Brayton–Moser’s mixed potential function may not be applicable for these systems. In the next section, we will analyze the stability of the system via eigenvalue analysis and the quadratic Lyapunov function.

3.2. Small-Signal Stability Conditions

According to the Hartman−Grobman theorem, the behavior of a dynamical system near a hyperbolic equilibrium point is qualitatively equivalent to the behavior of its linearization near this point. Hence, the system is stable if and only if J is a Hurwitz matrix.

The characteristic polynomial of J is determined by:

$$|\lambda I-J|=\left|\begin{array}{cccc}\lambda +\frac{R}{L_1}& \frac{1}{L_1}& & \\ -\frac{1}{C_1}& \lambda & \frac{d}{C_1}& \\ & -\frac{d}{L_2}& \lambda & \frac{1}{L_2}\\ & & -\frac{1}{C_2}& \lambda -\frac{P}{C_2{v}^2}\end{array}\right|={\lambda }^4+a_1{\lambda }^3+a_2{\lambda }^2+a_3\lambda +a_4=0$$

(10)

where $a_1,a_2,a_3$, and $a_4$ are given by:

$$\begin{array}{l}{a}_1=\frac{R}{L_1}-\frac{P}{C_2{v}^2},\hspace{1em}{a}_2=\frac{1}{L_1{C}_1}+\frac{d^2}{L_2{C}_1}+\frac{1}{L_2{C}_2}-\frac{R}{L_1}\frac{P}{C_2{v}^2}\\ a_3=\frac{R}{L_1}\left(\frac{d^2}{L_2{C}_1}+\frac{1}{L_2{C}_2}\right)-\frac{d^{2}P}{L_2{C}_2{C}_1{v}^2}-\frac{P}{L_1{C}_1{C}_2{v}^2}\\ a_4=\frac{1}{L_1{L}_2{C}_1{C}_2}\left(1-\frac{RP{d}^2}{v^2}\right)\end{array}$$

(11)

According to the Routh-Hurwitz criterion, the system is stable if and only if:

$$\{\begin{array}{l}R>\max \left\{\frac{L_1}{C_2}\frac{P}{v^2},\frac{L_1{d}^2+L_2}{C_2{d}^2+C_1}\frac{P}{v^2}\right\}\\ R<\min \left\{\frac{L_1{C}_2{v}^2}{P}\left(\frac{1}{L_1{C}_1}+\frac{d^2}{L_2{C}_1}+\frac{1}{L_2{C}_2}\right),\frac{v^2}{P{d}^2},\frac{E^2}{4P}\right\}\\ \left(\frac{R}{L_1}-\frac{P}{C_2{v}^2}\right)\left(\frac{1}{L_1{C}_1}+\frac{d^2}{L_2{C}_1}+\frac{1}{L_2{C}_2}-\frac{R}{L_1}\frac{P}{C_2{v}^2}\right)>\frac{R}{L_1}\left(\frac{d^2}{L_2{C}_1}+\frac{1}{L_2{C}_2}\right)-\frac{d^{2}P}{L_2{C}_2{C}_1{v}^2}-\frac{P}{L_1{C}_1{C}_2{v}^2}\\ a_1{a}_2{a}_3>a_3^2+a_1^2{a}_4\end{array}$$

(12)

Clearly, the system is stable if (and only if) the conditions in (12) hold. In addition, the form of the stability condition is complex, demonstrating that the results in [40] are incorrect. Thus, a specific system can be stabilized by choosing the appropriate R, L₁, and C₁ values according to Equation (12).

3.3. Large-Signal Stability Analysis and DOA Estimation

If the system satisfies (12), the system is small-signal stable. However, the disturbance $\delta \left(t\right)$ persists and continues to affect the system. Hence, it is important to select the appropriate R, L₁, and C₁ value to ensure that the influence of the disturbance is within an acceptable range. Clearly, it is essential to estimate the DOA of the operation point.

Let us define the variables $z_1,z_2,z_3$, and $z_4$ as:

$$\{\begin{array}{l}{z}_1=x_1+\frac{dP}{v}\\ z_2=x_2+\frac{v}{d}\\ z_3=x_3+\frac{P}{v}\\ z_4=x_4+v\end{array}$$

(13)

Substituting (13) into (2) and neglecting the disturbance term, the following can be obtained:

$$\frac{dz}{dt}=Jz+f\left(z\right),\hspace{1em}f\left(z\right)=\left[\begin{array}{c}0\\ 0\\ 0\\ -\frac{P}{C_2{v}^2\left(v+z_4\right)}{z}_4^2\end{array}\right]$$

(14)

If the system satisfies (12), then J is a Hurwitz matrix. According to the Lyapunov stability theorem, there is a real-symmetric positive definite Q such that:

$$QJ+J^{T}Q=-I$$

(15)

where I is the identity matrix. Then, the quadratic Lyapunov function can be constructed as:

$$V\left(z\right)=z^{T}Qz$$

(16)

where $z={\left[\begin{array}{cccc}{z}_1& z_2& z_3& z_4\end{array}\right]}^T$. We define $D\left(r\right)=\left\{z|z^{T}Qz<r\right\}$, where r is an undetermined positive scalar. If $\dot{V}\left(z\right)\le 0$ always holds for any $r\in D\left(r\right)$, then $D\left(r\right)$ is a DOA for the operation point. We consider:

$$\dot{V}\left(z\right)={\dot{z}}^{T}Qz+z^{T}Q\dot{z}=z^T\left(J^{T}Q+QJ\right)z^T+2f{\left(z\right)}^{T}Qz=-z^{T}z+f{\left(z\right)}^{T}Qz$$

(17)

Obviously, $\underset{z^{T}z\to 0}{\lim }\frac{f{\left(z\right)}^{T}Qz}{z^{T}z}=0$. Therefore, $\dot{V}\left(z\right)$ must be negative definite in $D\left(r\right)$ as long as r is sufficiently small. A less conservative estimate can be obtained by solving the following optimization problem:

$$\begin{array}{l}\min r\\ s.t. \{\begin{array}{l}{z}^{T}Qz-r=0\\ -z^{T}z+2f{\left(z\right)}^{T}Qz=0\\ z^{T}z>0\end{array}\end{array}$$

(18)

Then, by solving the optimization problem in (18), the DOA of the operation can be obtained.

3.4. Transfer Function of the Disturbance

To investigate the effect of disturbance on the output voltage, the transfer function of the disturbance should be analyzed. Neglecting the high-order term and adding the disturbance term to (14), the following can be obtained:

$$\{\begin{array}{l}\frac{dz}{dt}=Jz+b\delta \left(t\right)\\ y=b^{T}z\end{array}, b={\left[\begin{array}{cccc}0& 0& 0& 1\end{array}\right]}^T$$

(19)

By a Laplace transform, the transfer function can be obtained as:

$$G\left(s\right)=b^T{\left(sI-J\right)}^{-1}b$$

(20)

If the system satisfies Equation (12), J is a Hurwitz matrix. Then, we can select the appropriate parameters to improve the steady state and transient performance of G(s).

4. Generalizability of the Proposed Stabilization Method

4.1. Stability Analysis of the DC Microgrid with Multiple Identical Converters

In this section, we extend the proposed stabilization method to a DC microgrid with multiple parallel-connected converters. First, the DC microgrid in Figure 6 is considered. Here, all of the converters are assumed identical.

Clearly, all of the capacitors of the Buck converter can be equivalent to a common capacitance C_eq as follows:

$$C_{eq}=N{C}_2$$

(21)

Let us define $x_i,y_i,z_i$ as the current of inductor L₁, the voltage of capacitor C₁, and the current of inductor L₂ for the i-th converter, respectively. We define $u_C$ as the voltage of the capacitance C_eq. According to Kirchhoff’s voltage law, the dynamics of the i-th converter can be described by the following differential equations:

$$\{\begin{array}{l}{L}_1\frac{d{x}_i}{dt}=E+{\delta }_i\left(t\right)-R{x}_i-y_i\\ C_1\frac{d{y}_i}{dt}=x_i-d{z}_i\\ L_2\frac{d{z}_i}{dt}=d{y}_i-u_C\\ C_{eq}\frac{d{u}_C}{dt}={\displaystyle \sum _{i=1}^N{z}_i}-\frac{P}{u_C}\end{array}$$

(22)

According to (22), the following can be easily derived:

$$\{\begin{array}{l}\frac{d}{dt}\left({\displaystyle \sum _{i=1}^N{x}_i}\right)={\displaystyle \sum _{i=1}^N\frac{E+{\delta }_i\left(t\right)}{L_1}}-\frac{R}{L_1}\left({\displaystyle \sum _{i=1}^N{x}_i}\right)-\frac{1}{L_1}\left({\displaystyle \sum _{i=1}^N{y}_i}\right)\\ \frac{d}{dt}\left({\displaystyle \sum _{i=1}^N{y}_i}\right)=\frac{1}{C_1}{\displaystyle \sum _{i=1}^N{x}_i}-\frac{d}{C_1}{\displaystyle \sum _{i=1}^N{z}_i}\\ \frac{d}{dt}\left({\displaystyle \sum _{i=1}^N{z}_i}\right)=\frac{d}{L_2}{\displaystyle \sum _{i=1}^N{y}_i}-\frac{N}{L_2}{u}_C\\ C_{eq}\frac{d{u}_C}{dt}={\displaystyle \sum _{i=1}^N{z}_i}-\frac{P}{u_C}\end{array}$$

(23)

We define:

$$\overline{x}=\frac{1}{N}{\displaystyle \sum _{i=1}^N{x}_i},\overline{y}=\frac{1}{N}{\displaystyle \sum _{i=1}^N{y}_i},\overline{z}=\frac{1}{N}{\displaystyle \sum _{i=1}^N{z}_i}$$

(24)

Substituting (24) into (23), the following can be obtained:

$$\{\begin{array}{l}{L}_1\frac{d}{dt}\overline{x}=E-R\overline{x}-\overline{y}+\frac{1}{N}{\displaystyle \sum _{i=1}^N{\delta }_i\left(t\right)}\\ C_1\frac{d}{dt}\overline{y}=\overline{x}-d\overline{z}\\ L_2\frac{d}{dt}\overline{z}=d\overline{y}-u_C\\ C_2\frac{d{u}_C}{dt}=\overline{z}-\frac{P}{N{u}_C}\end{array}$$

(25)

Thus, a reduced-order model of the DC microgrid with multiple parallel-connected converters is obtained in (25). Obviously, the system described in (25) is equivalent to the system in (2). Hence, the small-signal stability conditions can be obtained by replacing P with P/N in (12), and the large-signal stability and DOA estimation can be assessed via the same method presented in Section 3.3.

4.2. Stability Analysis of the DC Microgrid with Multiple Non-identical Converters

In this section, we consider the DC microgrid with multiple non-identical converters. Denotes L_1,i, C_1,i, L_2,i, and C_2,i as the active damper’s inductance, capacitance, Buck converter’s inductance and capacitance of the i-th DG, respectively. Similarly, all of the capacitors of the Buck converter can be equivalent to a common capacitance C_eq as follows:

$$C_{eq}={\displaystyle \sum _{i=1}^N{C}_{2,i}}$$

(26)

According to Kirchhoff’s voltage law, the dynamics of the i-th converter can be described by the following differential equations:

$$\{\begin{array}{l}{L}_{1,i}\frac{d{x}_i}{dt}=E_i+{\delta }_i\left(t\right)-r_i{x}_i-y_i\\ C_{1,i}\frac{d{y}_i}{dt}=x_i-d_i{z}_i\\ L_{2,i}\frac{d{z}_i}{dt}=d_i{y}_i-u_C\\ C_{eq}\frac{d{u}_C}{dt}={\displaystyle \sum _{i=1}^N{z}_i}-\frac{P}{u_C}\end{array}$$

(27)

where x_i and z_i are the currents through L_1,i and L_2,i, respectively, and z₂ and u_C are the voltages across C_1,i and C_eq, respectively. E_i, r_i, d_i and ${\delta }_i\left(t\right)$ are the input voltage, resistance, duty cycle and uncertain disturbance of the i-th DG. Write (27) into compact form, the whole dynamics of the system can be obtained:

$$\{\begin{array}{l}{\mathit{L}}_{\mathbf{1}}\frac{dx}{dt}=E+\mathit{\delta }\left(t\right)-\mathit{R}x-y\\ {\mathit{C}}_{\mathbf{1}}\frac{dy}{dt}=x-\mathit{D}z\\ {\mathit{L}}_{\mathbf{2}}\frac{dz}{dt}=\mathit{D}y-{\mathit{u}}_C\\ C_{eq}\frac{d{u}_C}{dt}={\displaystyle \sum _{i=1}^N{z}_i}-\frac{P}{u_C}\end{array}$$

(28)

where ${\mathit{L}}_{\mathbf{1}}=diag\left\{L_{1,i}\right\},{\mathit{C}}_{\mathbf{1}}=diag\left\{C_{1,i}\right\},{\mathit{L}}_{\mathbf{2}}=diag\left\{L_{2,i}\right\},E={\left[\begin{array}{cccc}{E}_1& E_2& \cdots & E_N\end{array}\right]}^T,\mathit{\delta }\left(t\right)={\left[\begin{array}{cccc}{\delta }_1& {\delta }_2& \cdots & {\delta }_N\end{array}\right]}^T$, $x={\left[\begin{array}{cccc}{x}_1& x_2& \cdots & x_N\end{array}\right]}^T,y={\left[\begin{array}{cccc}{y}_1& y_2& \cdots & y_N\end{array}\right]}^T,{\mathit{u}}_C={\left[\begin{array}{cccc}{u}_C& u_C& \cdots & u_C\end{array}\right]}^T$ and $z={\left[\begin{array}{cccc}{z}_1& z_2& \cdots & z_N\end{array}\right]}^T$. Obviously, the operation point of the system is determined by:

$$\{\begin{array}{l}E-\mathit{R}x-y=\mathbf{0}\\ x-\mathit{D}z=\mathbf{0}\\ \mathit{D}y-{\mathit{u}}_{\mathit{C}}=\mathbf{0}\\ {\displaystyle \sum _{i=1}^N{z}_i}-\frac{P}{u_C}=0\end{array}$$

(29)

where 0 = ${\left[\begin{array}{cccc}0& 0& \cdots & 0\end{array}\right]}^T$. By solving (29), the load voltage of the operation point is given by:

$$v=\frac{{\displaystyle \sum _{i=1}^N\frac{E_i}{d_i{r}_i}}+\sqrt{{\left({\displaystyle \sum _{i=1}^N\frac{E_i}{d_i{r}_i}}\right)}^2-4P{\displaystyle \sum _{i=1}^N\frac{1}{d_i^2{r}_i}}}}{2{\displaystyle \sum _{i=1}^N\frac{1}{d_i^2{r}_i}}}$$

(30)

Linearizing (28) around the operation point, the equivalent linearized model of the system is given by:

$$\{\begin{array}{l}{\mathit{L}}_{\mathbf{1}}\frac{d\Delta x}{dt}=\mathit{\delta }\left(t\right)-\mathit{R}\Delta x-\Delta y\\ {\mathit{C}}_{\mathbf{1}}\frac{d\Delta y}{dt}=\Delta x-\mathit{D}\Delta z\\ {\mathit{L}}_{\mathbf{2}}\frac{d\Delta z}{dt}=\mathit{D}\Delta y-\Delta {\mathit{u}}_{\mathit{C}}\\ C_{eq}\frac{d\Delta u_C}{dt}={\displaystyle \sum _{i=1}^N\Delta z_i}+\frac{P}{v^2}\Delta u_C\end{array}$$

(31)

where $\Delta x,\Delta y,\Delta z$ and $\Delta u_C$ are the small-signal variables. According to (31), the Jacobian matrix of the system is given by:

$${\mathit{J}}_1=\left[\begin{array}{cccc}-{\mathit{L}}_1^{-1}\mathit{R}& -{\mathit{L}}_{\mathbf{1}}^{-1}& & \\ {\mathit{C}}_{\mathbf{1}}^{-1}& \mathit{O}& -{\mathit{C}}_{\mathbf{1}}^{-1}\mathit{D}& \\ & {\mathit{L}}_{\mathbf{2}}^{-1}\mathit{D}& \mathit{O}& -{\mathit{L}}_{\mathbf{2}}^{-1}{\mathbf{1}}_N\\ & & C_{eq}^{-1}{\mathbf{1}}_N^T& \frac{P}{C_{eq}{v}^2}\end{array}\right]$$

(32)

where ${\mathbf{1}}_N={\left[\begin{array}{cccc}1& 1& \cdots & 1\end{array}\right]}^T$.

Then, the system is stable if and only if J₁ is Hurwitz. However, the characteristic polynomial of J₁ is a high-order polynomial which analytic stability conditions are very complicated. Hence, this paper will investigate the stability in simulations. If there exist L₁, C₁ and R such that J₁ is Hurwitz, the large-signal stability and DOA estimation can also be assessed via the same method presented in Section 3.3.

5. Simulation

To verify the effectiveness of the proposed stabilization method, simulations were carried out in MATLAB/Simulink. Let us consider the DC microgrid presented in Figure 3, where L₂ = 5 mH, C₂ = 5 mF, P = 500 W, d = 0.5, and E = 120 V. Term $\delta \left(t\right)$ is a bounded high-frequency disturbance such that $\left|\delta \left(t\right)\right|\le 5$, and R, L₁, and C₁ are the parameters that needs to be designed.

5.1. Small-Signal Stability Conditions

Let us take R = 1 Ω and C₁ = 5 mF. According to (4), the operation point of the system can be obtained as (4.32, 115.68, 8.64, 57.84). According to the stability analysis, the system is stable if and only if (12) holds. Substituting the parameters into (12), the following can be obtained:

$$5\times {10}^{-4}<L_1<7.25\times {10}^{-3}$$

The root locus of J is presented in Figure 7 as L₁ increases from $5\times {10}^{-4}$ to $7.25\times {10}^{-3}$ H.

The stability condition derived in [40] shows that the system is stable if (9) holds. Substituting the parameters into (12), the following can be obtained:

$$R>\frac{L_{\alpha }}{C_{\alpha }}\frac{P}{v^2}\Rightarrow L_1<33.45\times {10}^{-3}$$

However, the root locus presented in Figure 7 shows that the system is stable if and only if $5\times {10}^{-4}<L_1<7.25\times {10}^{-3}$ H. Thus, the stability criterion proposed in [40] is incorrect.

5.2. Transfer Function of the Disturbance

According to (20), the transfer function of the disturbance is given by $G\left(s\right)=b^T{\left(sI-J\right)}^{-1}b$. Figure 8 shows a Bode diagram of G(s) when L₁ = 2, 3, 4, 5, and 6 mH. When the frequency of the disturbance is within $\left[{10}^2,{10}^4\right]$ rad/s, the magnitude of the transfer function ranges from −20 dB to −80 dB. Therefore, the system has a strong ability to mitigate disturbances.

5.3. DOA of the Operation Point

Let us take L₁ = 5 mH. Then, the eigenvalue of the system Jacobian matrix is given by 77.34 ± 153.68i, 7.71 ± 227.98i. According to the Lyapunov stability theorem, there is a real-symmetric positive definite matrix Q that satisfies (15). By solving (15), Q is given by:

$$Q=\left[\begin{array}{llll}0.0219& 0.0194& -0.0077& -0.0319\\ 0.0194& 0.0452& 0.0339& -0.0348\\ -0.0077& 0.0339& 0.0994& 0.0144\\ -0.0319& -0.0348& 0.0144& 0.0799\end{array}\right]$$

Then, $D\left(r\right)=\left\{z|z^{T}Qz<r\right\}$ is an estimation of the DOA, where r is the solution of the optimization problem described in (18). Clearly, the optimization problem in (18) can also be described as follows:

$$\begin{array}{l}\min z^{T}Qz\\ s.t. -z^{T}z+2f{\left(z\right)}^{T}Qz=0;z^{T}z>0;\\ f\left(z\right)=\left[\begin{array}{cccc}0& 0& 0& \frac{149.5}{57.84+z_4}{z}_4^2\end{array}\right]\end{array}$$

By using the MATLAB Optimization Toolbox, the optimal solution of (18) is obtained as r = 2.2336. Therefore, the DOA of the operation point is estimated as $D=\left\{z|z^{T}Qz<2.2336\right\}$.

5.4. Compared with Virtual Resistance Method

Virtual resistance is a typical and effective stabilization method in DC microgrid [35]. A DC microgrid under virtual resistance stabilization method is presented in Figure 9b.

In Figure 9, all the parameters of the two microgrids in Figure 9 are the same except for the control strategy. i₁ and i₂ are the output currents of the energy supply sources. Next, we will compare the above two stabilization methods from the following three aspects: stability; the disturbance in output voltage; high-frequency harmonic components in output current of energy supply sources.

5.5. Simulation Results

Let us take R = 1 Ω, L₁ = 5 mH, and C₁ = 5 mF. According to the theoretical analysis in Section 5.1 and Section 5.2, the system in Figure 9a is stable and has a strong ability to mitigate disturbances. The disturbance is assumed as $\delta \left(t\right)=5\cos 200\pi t+\kappa \left(t\right)$, where $\kappa \left(t\right)$ is a random noise that satisfies $\left|\kappa \left(t\right)\right|<5$. The frequency of the PWM signal is 10 kHz. The simulation results are presented in Figure 10.

Figure 10(a1) shows that the system is stable, verifying that the proposed stabilization method can overcome the instability of the CPL. Compared with Figure 10(b1), the disturbance of the output voltage in Figure 10(a1) is smaller, which shows that the proposed method can mitigate fluctuations from a non-ideal input and has a better performance in disturbance mitigation than the traditional virtual resistance method. Figure 10(b2) shows that output current of the energy source contains large high-frequency discontinues harmonics from switching, which will damage the energy supply source. Figure 10(b1) shows that the output current of the energy input source does not contain high-frequency harmonics from switching, demonstrating that the proposed method can mitigation the discontinuous harmonic current from the PWM.

In summary, simulation results shows that the proposed active damping method can overcome the CPL’s instability, mitigate disturbance and protect the high-frequency discontinues harmonic current from interfering with the source simultaneously.

5.6. The DC Microgrid with Multiple Identical Converters

To verify the extendibility of the proposed method, we consider the DC microgrid with three converters presented in Figure 11. Take L_2,1 = L_2,2 = L_2,3 = 5 mH, C_2,1 = C_2,2 = C_2,3 = 5 mF, P = 1500 W, d₁ = d₂ = d₃ = 0.5, and E₁ = E₂ = E₃ = 120 V. ${\delta }_1\left(t\right)=5\cos 200\pi t+\kappa \left(t\right),{\delta }_2\left(t\right)=10\cos 400\pi t+\kappa \left(t\right)$, and ${\delta }_3\left(t\right)=10\cos 200\pi t+\kappa \left(t\right)$, where $\kappa \left(t\right)$ is a random noise that satisfies $\left|\kappa \left(t\right)\right|<5$.

Let us take r₁ = r₂ = r₃ = 1 Ω, L_1,1 = L_1,2 = L_1,3 = 5 mH, and C_1,1 = C_1,2 = C_1,3 = 5 mF. Then, the converters in Figure 11 are identical. According to the theoretical analysis in Section 4, the system is stable.

The simulation results presented in Figure 12 also show that the system is stable, and the disturbance of the output voltage is small. Therefore, the proposed method and the stability condition in (12) can be applied to a DC microgrid with multiple identical converters.

5.7. The DC Microgrid with Multiple Identical Converters

Take L_2,1 = 1 mH, L_2,2 = 2 mH, L_2,3 = 5 mH, C_2,1 = 2 mF, C_2,2 = 4 mF, C_2,3 = 8 mF, P = 500 W, d₁ = 0.6, d₂ = 0.5, d₃ = 0.5, E₁ = 100 V, and E₂ = E₃ = 120 V. ${\delta }_1\left(t\right)=5\cos 200\pi t+\kappa \left(t\right),{\delta }_2\left(t\right)=10\cos 400\pi t+\kappa \left(t\right)$, and ${\delta }_3\left(t\right)=10\cos 200\pi t+\kappa \left(t\right)$, where $\kappa \left(t\right)$ is a random noise that satisfies $\left|\kappa \left(t\right)\right|<10$. Then, all the converters in Figure 11 are non-identical.

Take L_1,1 = 5 mH, L_1,2 = 2.5 mH, L_1,3 = 1 mH, r₁ = r₂ = r₃ = 2 Ω, C_1,1 = C_1,2 = C_1,3 = k, where k is undetermined parameter. If J₁ is Hurwitz, the system will be stable. The root locus of J₁ is presented in Figure 13 as k increases from $5\times {10}^{-4}$ to $5.1\times {10}^{-3}$ F.

Figure 13 shows that J₁ is Hurwitz when 0.5 mF < k < 5.1 mF. Take C_1,1 = C_1,2 = C_1,3 = 5 mH, then, the system is stable.

The simulation results presented in Figure 14 also show that the system is stable, and the disturbance of the output voltage is small. Therefore, the proposed method can be applicable to the DC microgrid with multiple converters.

6. Conclusions

The stability and disturbance mitigation of a DC microgrid with a CPL are investigated in this paper. Stabilization based on active damping is proposed, and the necessary and sufficient stability conditions are established. In addition, this paper demonstrates that the stability criterion derived in [40] is incorrect. Moreover, this paper develops an approach for analyzing the large-signal stability of a system based on the Lyapunov function. This paper transforms the problem of DOA estimation into an optimization problem, which can be solved by numerical techniques. This paper demonstrates that the proposed method and stability conditions can be applied to a DC microgrid with multiple converters. Finally, simulation results verify the effectiveness of the proposed method.