Research on Flexible Virtual Inertia Control Method Based on the Small Signal Model of DC Microgrid

Abstract

Renewable energy is usually connected to the DC micro-grid by a large number of power electronic devices, which have the advantages of a fast system response, but the disadvantage to reduce the inertia of the system, which makes the stability of the system worse. It is necessary to increase the inertia of DC micro-grid so that it can recover and stabilize well when it receives a disturbance. In this paper, a small-signal model of DC micro-grid with constant power load (CPL) is established, and a flexible virtual inertial (FVI) control method based on DC bus voltage real-time variation is proposed, by controlling the DC/DC converter of the energy storage system, the problem of system oscillation caused by introducing voltage differential link to the system is solved. Compared with the droop control method, the FVI control method can increase the inertia of DC micro-grid system, reduce the influence of small disturbances, and improve the stability of the system. Finally, the validity of the FVI control method based on small signal model is verified in dSPACE.

1. Introduction

Renewable energy has the advantages of a flexible site selection, environmental friendliness and high energy efficiency, and is widely used around the world [1,2,3]. However, distributed energy sources such as photovoltaic (PV) and wind energy are easily affected by factors such as season, terrain, time, etc., and are volatile and random. If connected to the traditional power grid directly, it will have a great impact on the power grid, resulting in large fluctuations and even could run the power grid out of control [4,5,6]. Distributed energy sources are generally connected to the micro-grid by power electronic equipment, such as a DC/DC converter, which makes the system have significant power electronic characteristics. That is, the introduction of lots of electronic devices will cause the system to respond quickly but with little inertia [7,8]. This will weaken the stability and anti-interference ability of the system. Compared with an AC microgrid, a DC microgrid is easier to connect to distributed energy and easier to control [9,10,11]. Therefore, the research of improving the inertia of DC microgrid is very important.

When the load increases or decreases, the distributed energy source fluctuates, or even the constant power load (CPL) is introduced into the system, the bus voltage of the DC microgrid can easily lose its stability, thus threatening the stability of the system [12,13]. Therefore, in order to increase the DC micro-grid stability and improve the robustness of the system, the research method of virtual inertia control has become one of the hot spots in the field of DC microgrid stability research [14,15]. At present, the control methods used to enhance the DC micro-grid inertia can be roughly divided into three categories: additional inertia control links [16], droop control [17], and similar virtual synchronous generators (AVSG) [18].

Reference [19] introduced the energy storage link in the DC micro-grid into the virtual inertia coefficient, and verified the change of the Bode diagram of the system by changing the value of the virtual inertia coefficient, thereby determining the system stability. However, the virtual inertia coefficient is constant and cannot be automatically adjusted according to system changes. Therefore, this method has defects in the control of system stability and cannot adapt to fluctuations of different amplitudes of the system. In reference [20], an improved series of virtual impedance control method is proposed for the control link of the grid-connected converter. It is emphasized that the system with the large virtual inertia coefficient is easy to oscillate at high frequency. According to the principle of impedance matching, reducing the output impedance of the power supply side and enhancing the damping characteristics can effectively improve the system stability. However, the adjustment of inertia size cannot meet the requirements of the system, so the paper also has the problem that it cannot adapt to the different amplitude fluctuations of the system.

Reference [21] uses a droop control method to simplify a DC micro-grid into an equivalent model including multiple DC-DC converters, CPL, and resistance loads. A small-signal model of a DC micro-grid is established. On the basis of this model, an active damping method based on low-pass filtering is given. By adjusting the output impedance of the DC-DC converter, the dominant characteristic root is moved to the left of the s-domain plane. Reference [22] combines the two methods of the droop control with the virtual inertia control, analyzes the stability of the small signal model, and adjusts its control parameters by analyzing the root locus to verify the feasibility of the method. However, in droop control, a virtual impedance is connected to the system in series, thus increasing the equivalent output impedance. Therefore, when the output impedance is greater than the line impedance, the droop control mode has certain limitations.

The virtual synchronous generator (VSG) technology introduces damping and virtual inertia into the control link, so the converters have similar electrical characteristics as the synchronous generator, such as damping and rotational inertia characteristics. VSG control is widely used in AC microgrids. Based on the traditional control method for the inverters, the characteristics of damping and inertia of synchronous generators are simulated, which can better realize the frequency regulation of active power and the voltage regulation of reactive power. It is an effective and feasible control strategy; this technology can provide the inertial for the AC microgrid.

Since there is a one-to-one correspondence of physical quantities between the AC system and the DC system, the principle of adjusting the frequency and voltage of a synchronous generator can be simulated, and by analogy reasoning the inertia can be provided for the DC microgrid. Literature [23] analogously reasoned the virtual synchronous generator control method in traditional AC system to a DC microgrid and proposed a two-way grid-connected converter (BGC) virtual inertia control strategy suitable for the DC microgrid. This control strategy improves the inertia characteristic of the DC microgrid and suppresses the fluctuation of DC bus voltage. For a hybrid microgrid connected to the wind turbines, it is not only necessary to research the influence of load fluctuations on the system, but also to customize a stable control strategy for the system according to different wind speeds [24]. Since the virtual inertia generated by the grid-connected converter can adjust the AC frequency more directly and accurately, the droop characteristics of the energy storage system, the specific control algorithm of the energy storage unit and the power given module are designed according to the frequency response characteristics of the AC measurement. This enables the system to actively provide the virtual inertial support at multi-interference conditions, which can effectively cope with various wind speed changes and sudden changes in the AC load, further suppress the frequency changes of the AC side. Reference [25] proposed a flexible virtual inertial control based on the DC bus voltage change rate and voltage deviation for grid-connected converters, studied the impact of different control parameter values on system stability. It is verified that the proposed method can effectively reduce the transient problem and voltage deviation caused by the bus voltage change. Reference [26] proposes a virtual inertial control method considered virtual capacitor voltage to realize bidirectional DC/DC converter droop control, which can enhance the inertia of the system in the case of low power disturbance, and improves the DC bus voltage stability. However, the disadvantage of this control method is that the system will lose stability under high power disturbance.

To sum up, the current virtual inertial control is mainly concentrated in the AC field, and there are few studies on the DC microgrid. At the same time, most of the virtual inertia introduced in the DC microgrid is a constant value and cannot be adjusted with the disturbance of the system. Here, a control method of flexible virtual inertia (FVI) based on bus voltage differential link is proposed. A virtual capacitor is introduced in the outer voltage loop and a comparator is connected in series, it solves the problem of the DC bus voltage oscillation caused by voltage differential circuit control link in the above-mentioned literature, and has a very high research significance.

The paper is organized as follows: In Section 2, a small-signal model of DC microgrid with constant power load is established. In Section 3, a droop control method and a FVI control method are proposed. In Section 4, the simplified model of multiple constant power loads is established, and the real-time working situation by taking the additional random disturbance as an example is simulated and the stability of the system is studied by small-signal analysis. Finally, the effectiveness of the proposed FVI control method is verified on the dSPACE hardware-in-the-loop simulation platform.

2. Small-Signal Modeling

The DC microgrid mainly consists of distributed renewable energy, controllers, rectifiers, inverters, energy storage system, DC loads, and AC loads, and the topology of a DC microgrid is shown in Figure 1. When considering the control method of DC microgrid, because the reactive power need not be considered, it is generally necessary to control the bus voltage of DC microgrid to ensure the stability of the system.

Generally, in a DC microgrid, all the units are connected to the DC bus through a DC/DC converter. Topological structure diagram of DC/DC converter is shown in Figure 2. In this paper, it is defined that the current is positive when a unit outputs power to the DC bus, and negative when the unit absorbs power from the DC bus.

Among them, u_s is the equivalent DC power supply, L is the equivalent line inductance, R_L is the equivalent line impedance, i_L is the current flowing through the equivalent circuit, R_C is the line equivalent parallel resistance, C is the line equivalent parallel capacitance, R_i is the line resistance, i_o is the load current, R_o is the effective load (it can represent both resistive loads, and can also represent CPL). There are many types of loads in DC microgrids, which can be broadly divided into resistance loads and CPLs [27].

2.1. Resistive Load

The most basic type is resistive load, which is easy to model. When considering the DC/DC converter between the DC microgrid and the energy storage unit (it can also be other distributed renewable energy sources), list the small signal equations of the DC/DC converter:

$$\{\begin{array}{l}L\frac{\mathrm{d}{i}_L}{\mathrm{d}t}=u_{\mathrm{s}}-i_L{R}_L-\left(1-d\right)\left[\left(i_L-i_{\mathrm{o}}\right)R_L+u_{\mathrm{c}}\right]\\ C\frac{\mathrm{d}{u}_{\mathrm{C}}}{\mathrm{d}t}=\left(1-d\right)i_L-i_{\mathrm{o}}\\ u_{\mathrm{o}}=\left[\left(1-d\right)i_L-i_{\mathrm{o}}\right]R_L+u_{\mathrm{c}}-i_{\mathrm{o}}{R}_{\mathrm{i}}\end{array}$$

(1)

where d represents the duty cycle of the converter. The system state equation is:

$$\{\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

(2)

where x and y is the state variable. $x=\left[i_L,u_c\right]$, $u=\left[u_s,i_o\right]$, y = u_o:

And $A=\left[\begin{array}{cc}-\frac{R_L+\left(1-d\right)R_{\mathrm{C}}}{L}& \frac{1-d}{L}\\ \frac{1-d}{C}& 0\end{array}\right]$, $B=\left[\begin{array}{cc}\frac{1}{L}& \frac{\left(1-d\right)R_{\mathrm{C}}}{L}\\ 0& -\frac{1}{C}\end{array}\right]$, $C=\left[\begin{array}{cc}\left(1-d\right)R_{\mathrm{C}}& 1\end{array}\right]$, $D=\left[\begin{array}{cc}0& -\left(R_{\mathrm{i}}+R_{\mathrm{C}}\right)\end{array}\right]$.

The transfer function of duty cycle and capacitor voltage is derived as:

$$G_{vd}=\frac{u_c(s)}{D(s)}=\frac{V_s-\left[\frac{2R_L}{1-D}+R_C-\frac{sL}{1-D}\right]I_o}{s^{2}LC+sC[(1-D)R_c+R_L]+{(1-D)}^2}$$

(3)

The parameters value of the DC microgrid are brought into the physical model, as shown in

Table 1. DC Microgrid Parameters.

Parameter	Value	Parameter	Value
RL	2 × 10−2 Ω	Ri	0.1 Ω
L	3 × 10−3 H	RC	0.2 Ω
C	9.4 × 10−4 F	Us	100 V

.

For the DC-DC converter using the traditional double closed-loop control method, its Bode diagram is obtained as shown in Figure 3. At the amplitude crossover frequency ù_c, the phase margin of the system is about ã = 29.6°; at the phase angle crossover frequency ù_g, the system amplitude margin is about g_m = 4.34 dB due to the influence of resonance, the system is stable, but there are resonance peaks in the amplitude-frequency characteristic curve, so that reduces the stability margin of the system.

For the traditional double closed-loop control method of a parallel bidirectional DC-DC converter, its open-loop transfer function is:

$$G_{\mathrm{k}}=k_1{G}_{\mathrm{PI}\_\mathrm{U}}\left(s\right)G_{\mathrm{PI}\_\mathrm{I}}\left(s\right)e^{-s{T}_{\mathrm{d}}}{G}_{\mathrm{m}}\left(s\right)G_{\mathit{vd}}\left(s\right)$$

(4)

Among them, G_{PI_U}(s) is the voltage outer loop transfer function, G_{PI_I}(s) is the current inner loop transfer function, G_m(s) is the transfer function corresponding to the transmission delay of the PWM pulse width modulator, and T_d is the delay time of PWM:

$$G_{\mathrm{m}}(s)=e^{-0.5s{T}_{\mathrm{d}}}$$

(5)

2.2. CPL

Usually, some resistance loads are directly connected to the DC bus, but most of the AC loads and new power electronic loads should be connected through the corresponding DC-AC and DC-DC converters. Since the load converter usually adopts closed-loop controls to adjust the load terminal voltage, this kind of load will exhibit the dynamic characteristics of CPL [28]. The external characteristics of CPL can be expressed as

$$i_{\mathrm{cpl}}=\frac{P_{\mathrm{cpl}}}{u_{\mathrm{dc}}}$$

(6)

where P_cpl = P_cpli − P_wt − P_v. P_cpli means the user’s load that can be regarded as a CPL. P_wt means the power provided by the wind turbine. P_v means the power provided by the PV.

A DC microgrid with CPLs can be indicated by the circuit topology diagram of Figure 4. In this paper, the distributed renewable energy can be photovoltaic, wind turbines, and so on.

Write the small signal equation for the above DC microgrid with CPLs, which is quite different from the microgrid with resistive load:

$$\{\begin{array}{c}{L}_{\mathrm{i}}\frac{\mathrm{d}{i}_{Li}}{\mathrm{d}t}=V_{si}{d}_i-v_{ci}\\ C_{\mathrm{i}}\frac{\mathrm{d}{v}_{ci}}{\mathrm{d}t}=i_{Li}-i_{oi}\\ L_{ei}\frac{\mathrm{d}{i}_{oi}}{\mathrm{d}t}=v_{ci}-i_{oi}{R}_{ei}-v_{\mathrm{bus}}\\ C_{\mathrm{e}}\frac{\mathrm{d}{v}_{\mathrm{bus}}}{\mathrm{d}t}={\displaystyle \sum _{i=1}^2{i}_{oi}}-\frac{P}{v_{\mathrm{bus}}}-\frac{v_{\mathrm{bus}}}{R}\end{array}$$

(7)

There are usually multiple constant power loads in the DC microgrid system, but the existing literature does not consider the parallel operation of multiple constant power loads in modeling. In this paper, the external characteristics of line impedance are fully considered when multiple CPLs are connected in parallel to DC microgrid. To simplify modeling, multiple CPLs are simplified to a single CPL, as shown in Figure 5:

In the Figure 5, L_eq, R_eq, C_eq and P_eq are the line inductance, resistance, capacitance, and the total power value of the load after multiple CPLs are equivalent. When building a small signal model for a CPL, considering its external characteristics shown as Equation (6), and its Taylor expansion is:

$$i=\frac{2P_{\mathrm{cpl}}}{U_L}-\frac{P_{\mathrm{cpl}}}{U_L^2}{u}_L$$

(8)

According to Figure 5, the mathematical equations of each parameter can be obtained:

$$\{\begin{array}{l}{R}_{\mathrm{o}}=-\frac{U_{\mathrm{o}}^2}{P_{\mathrm{cpl}}}\\ i_{\mathrm{cpl}}=\frac{2P_{\mathrm{cpl}}}{U_L}\end{array}$$

(9)

The small signal model suitable for DC microgrid with CPL is:

$$\{\begin{array}{c}L\frac{\mathrm{d}{i}_L}{\mathrm{d}t}=u_{\mathrm{s}}-R_L{i}_L-\left(1-d\right)u_{\mathrm{c}}\\ C\frac{\mathrm{d}{v}_{\mathrm{c}}}{\mathrm{d}t}=\left(1-d\right)i_L-i_{\mathrm{o}}\\ L_{\mathrm{eq}}\frac{\mathrm{d}{i}_{\mathrm{io}}}{\mathrm{d}t}=u_{\mathrm{c}}-i_o{R}_{e\mathrm{q}}-u_{\mathrm{o}}\\ C\frac{\mathrm{d}{u}_{\mathrm{o}}}{\mathrm{d}t}=i_{\mathrm{o}}-\frac{P_{\mathrm{cpl}}}{u_{\mathrm{o}}}-\frac{u_{\mathrm{o}}}{R_{\mathrm{o}}}\end{array}$$

(10)

Then the specific state space equation is:

$$\left[\begin{array}{c}{\dot{I}}_L\\ {\dot{U}}_{\mathrm{C}}\\ {\dot{I}}_{\mathrm{O}}\\ {\dot{U}}_{\mathrm{O}}\end{array}\right]=\left[\begin{array}{cccc}-\frac{R_L}{L}& -\frac{1-D}{L}& 0& 0\\ \frac{1-D}{C}& 0& -\frac{1}{C}& 0\\ 0& \frac{1}{L_{\mathrm{eq}}}& -\frac{R_{\mathrm{eq}}}{L_{\mathrm{eq}}}& -\frac{1}{L_{\mathrm{eq}}}\\ 0& 0& \frac{1}{C}& -\frac{P_{\mathrm{cpl}}}{U_{\mathrm{o}}^2}-\frac{1}{R_{\mathrm{o}}}\end{array}\right]\left[\begin{array}{c}{I}_L\\ U_{\mathrm{C}}\\ I_{\mathrm{O}}\\ U_{\mathrm{O}}\end{array}\right]+\left[\begin{array}{cccc}\frac{U_{\mathrm{c}}}{L}& -\frac{I_L}{L}& 0& 0\end{array}\right]D$$

(11)

Same as Equation (2):

$$A=\left[\begin{array}{cccc}-\frac{R_L}{L}& -\frac{1-D}{L}& 0& 0\\ \frac{1-D}{C}& 0& -\frac{1}{C}& 0\\ 0& \frac{1}{L_{\mathrm{eq}}}& -\frac{R_{\mathrm{eq}}}{L_{\mathrm{eq}}}& -\frac{1}{L_{\mathrm{eq}}}\\ 0& 0& \frac{1}{C}& -\frac{P_{\mathrm{cpl}}}{U_{\mathrm{o}}^2}-\frac{1}{R_{\mathrm{o}}}\end{array}\right]$$

(12)

It is the characteristic matrix of the system. When all the eigenvalues in the matrix A are located in the left half plane of the s domain, it indicates that this DC microgrid runs near the steady-state operating point, and is stable with small disturbances.

The transfer function of the established microgrid system is:

$$\begin{array}{c}\frac{U_{\mathrm{o}}\left(s\right)}{D\left(s\right)}=\frac{U_c+\left(L+R_L\right)I_L}{\left(L+R_L\right)\left[Cs\left(R_{\mathrm{eq}}+s{L}_{\mathrm{eq}}\right)Q+Q\right]+D^{\prime }\left[Q\left(R_{\mathrm{eq}}+s{L}_{\mathrm{eq}}\right)+1\right]}\\ Q=\left(Cs+\frac{P_{\mathrm{cpl}}}{U_{\mathrm{o}}{}^2}+\frac{1}{R_{\mathrm{o}}}+1\right)\end{array}$$

(13)

By calculation, the eigenvalues are all located in the left half plane of the s domain.

3. Control Method

3.1. Droop Control

Usually droop control method is the main method to realize power distribution among various renewable energy sources in microgrid. It reflects the relationship between power and voltage, current and voltage of the system, and is easy to implement. Most DC/DC converters adopt double closed-loop control strategy of voltage and current, so the working process of droop control of DC microgrid is: add droop curve control as the control outer loop in the voltage and current double closed-loop control of the converter to obtain DC Voltage reference value of the converter output side, and then feed it back to the current inner loop, complete the double closed-loop control of voltage and current [29].

Droop control is often the first step in distributed energy communication and exchange in microgrids. The DC bus voltage fluctuation directly reflects the power transmission information of the system. Especially in off-grid operation, the DC bus voltage value is not fixed, and is fluctuating in a certain range. Therefore, in the DC microgrid system, the control strategy based on the voltage signal of DC bus is adopted. Therefore, the droop control method, which reflects the relationship between current and voltage, is a simple and effective method [30]. The block diagram of control system is shown in Figure 6, where k is the droop coefficient:

In the traditional droop control method, it is usually adding a virtual impedance in series with the system, which means that the inherent output impedance increases. Therefore, the droop control method is only suitable for the case of low line impedance.

A Bode plot comparing the droop control with the traditional double closed-loop control can be seen in Figure 7. From the comparison result of the two lines, it can be seen that the phase angle margin of line 1 at the amplitude crossover frequency ù_c of the system is about ã = 31°, and at the phase angle crossover frequency ù_g, the amplitude margin of the system due to resonance is about 74.8 dB, and the system is stable at this time. The resonance has a peak, which improves the phase angle edge and amplitude of the algorithm; the line 2 phase angle edge of the system at the amplitude crossover frequency ù_c is about ã = 111°; at the phase angle crossover frequency ù_g, the resonance amplitude is about 38.8 dB. At this point the system is stable. There is no peak at this time, indicating that the traditional droop control method can effectively improve the system stability.

Change the droop coefficient size and then perform the output voltage waveform detection. From Figure 8, it indicates that as the droop coefficient increases, the bus voltage of the system drops, but the system still maintains stable operation.

Enlarging the response phase and the disturbed part of the bus voltage can be shown in Figure 8. It can be seen that the droop control method reduces the DC bus voltage fluctuation. The size of the droop coefficient will affect the DC bus voltage. The DC bus voltage will decrease as the droop coefficient increases. This is because the method is similar to connecting a resistor in series in the system, using the principle of impedance matching to keep the bus voltage stable. However, this droop control method is susceptible to the influence of the system line impedance. When the system line impedance is large, the use of the droop control method will have limitations. Therefore, the controlling idea of FVI is proposed.

3.2. FVI Control Method

There are still some defects in using the droop control method to balance current under the condition of large differences in cable impedance. Therefore, the FVI control method is used for comparative analysis. The DC microgrid has low inertia and is susceptible to CPL increase and decrease and micro-source fluctuations, which in turn leads to bus voltage instability. In the AC system, the control equation of the active power-frequency controlled by VSG is:

$$P_{\mathrm{ref}}-P_{\mathrm{m}}-K_{\mathrm{d}}\left(\omega -{\omega }_{\mathrm{n}}\right)=J\omega \frac{\mathrm{d}\omega }{\mathrm{d}t}\approx J{\omega }_{\mathrm{n}}\frac{\mathrm{d}\omega }{\mathrm{d}t}$$

(14)

where ω is angular frequency of the system; ω_n is the rated angular frequency of the public bus; t is the time, and $\frac{\mathrm{d}\omega }{\mathrm{d}t}$ is the change rate of the angular frequency; P_ref is the set value of active power; P_m is the active power measurement value by distributed energy sources; K_d is the system damping coefficient; J is the system virtual moment of inertia.

The reasoning method by analogy is shown in

Table 2. Relationship Comparison Table.

AC	DC
Frequency ω	DC bus voltage udc
Active power Pm	DC current idc
Dynamic moment of inertia J	DC side parallel capacitor C

.

Applying the FVI control strategy, the virtual inertia equation in the DC microgrid is obtained as:

$$\begin{array}{l}{i}_{\mathrm{dc}}^{\ast }-i_{\mathrm{dc}}-k_{\mathrm{G}}\left(v_{\mathrm{dc}}-U_{\mathrm{dc}\_\mathrm{G}}^{\ast }\right)=C_{\mathrm{v}}{v}_{\mathrm{dc}}\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}\approx C_{\mathrm{v}}{U}_{\mathrm{dc}\_\mathrm{G}}^{\ast }\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}\\ \begin{array}{cc}{C}_{\mathrm{v}}=\{\begin{array}{l}{C}_{\mathrm{v}0}\\ C_{\mathrm{v}0}+k_3{\left(\left|\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}\right|\right)}^{k_4}\end{array}& \begin{array}{c}\left|\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}\right|<M\\ \left|\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}\right|\ge M\end{array}\end{array}\end{array}$$

(15)

where i_dc is the current flowing from the micro source to the DC grid; i_dc^* is the reference value of i_dc; k_G is the droop coefficient of the G-VSC droop control curve; u_dc is the bus voltage of the DC microgrid; U_{dc_G}^* is the bus voltage reference value of the DC microgrid; C_v0 is the virtual capacitance value in steady state; $\frac{\mathrm{d}{u}_{\mathrm{dc}}}{\mathrm{d}t}$ is the change rate of DC bus voltage; M is the set critical value of the DC bus voltage change rate; k₃, k₄ are related parameters [23].

Usually, the introduction of a voltage differential link in the control link will bring disturbance to the system bus voltage. To solve this problem, by setting the time margin and adding a comparison link in this paper, the above disturbance is reduced. This paper selects DC microgrid including wind power, PV, diesel generator, electric vehicle charging station, and other DC loads as the research object, establishes its small-signal model, applies analogy reasoning method to obtain the virtual inertia equation of the system, and applies it to the small-signal model of the DC microgrid. The sensitivity analysis and Bode diagram analysis are carried out to verify the affecting law of the parameters of the VFI control link on the inertia of the system, and then obtain the control scheme of the bus voltage stabilization of the DC microgrid.

Bring the parameters into the entire DC microgrid system to obtain the small-signal model of the DC microgrid with FVI, the control block diagram of FVI is shown in Figure 9. A virtual capacitor is introduced in the outer voltage loop and a comparator is connected in series.

Figure 10 shows the Bode diagram of the system. Compared with Figure 7, it can be obtained that the DC microgrid with FVI has a phase angle margin of approximately γ = 103° at the amplitude crossover frequency ω_c; at the phase angle crossover frequency ω_g, the amplitude margin of the system g_m due to resonance is approximately 57.5 dB, and the system is stable.

4. Simulation and Analysis

The physical model of the DC microgrid shown in Figure 4 is established in Matlab. Take an off-grid DC microgrid as an example in this paper: the distributed energy source adopts photovoltaic arrays. In order to facilitate the analysis, the distributed energy source and energy storage links in the system are simplified as DC power supply, and the output load is CPLs.

The control methods adopt the droop control and FVI control, respectively, and the characteristic curve of output voltage is shown in Figure 11. The blue line shows the droop control, and the red line shows the FVI control. It can be seen that the system has been connected to the constant power load. At 0.5 s, the constant power load is subject to a small disturbance, increasing from 1000 kW to 1200 kW. Comparing the two control methods, the FVI control method has fast response speed, small overshoot, and short recovery time in the disturbance response stage, as shown in the enlarged view of curve part 1. In addition, after the system is stable, the output voltage characteristic curve of the FVI control mode fluctuates less, as shown in the enlarged view of curve part 2. Apparently for CPLs, the droop control method is not a good choice. The droop control method is only applicable when the line impedance is less than the load. Droop control does not convey current and voltage well when the line impedance is large.

Adding photovoltaic modules to the system, considering the influence of solar radiation changes on the system, the light irradiance is reduced from 1000 W/m² to 800 W/m² at 1 s, the system still adopts the droop control method and the FVI control method. It can be seen from the comparison that the application of the FVI control method makes voltage jump of the system smaller in the disturbance response stage, as shown in the part 3 of the curve. Similarly, after the bus voltage returns to stability, the voltage ripple is smaller and the power quality is better.

Compare and analyze the output characteristic curve of the PV module system with or without it, as shown in Figure 12. The set of FVI control method in this article will change with the fluctuation of the voltage, so the PI parameters of the control link can remain unchanged, because the FVI will provide a large stability margin for the PI parameters, which can be used when the PI parameters are fixed. Through the adjustment of the flexible virtual inertia, the system is restored to stability. It can be seen from the comparison chart of the output characteristic curves with and without photovoltaic modules that the response of the system will fluctuate due to the addition of PV, which also confirms the impact of distributed energy access on the bus voltage.

Adjust the virtual capacitance parameters k₃ and k₄ in formula 16 to verify the influence of inertia on the DC bus voltage stability.

Set k₃ unchanged and change the size of K₄. The characteristics of the output voltage of the system are as shown in the Figure 13a. When k₄ is less than 1, with the increase of k₄, the inertia increases moderately, and the system is stable; When k₄ is greater than 1, with the increase of k₄, the inertia increases greatly, the system falls into oscillation and loses stability. Similarly, set k₄ to be constant and less than 1, change the size of k₃, the characteristics of the output voltage of the system are as shown in Figure 13b. It can be seen that with the change of k₃, the change of system inertia is small, indicating that the system stability is less affected by the value of k₃. Therefore, k₄ should be less than 1, and its value range plays a decisive role in the system stability.

The above DC microgrid physical model is discretized into a mathematical model, the M function is used to represent the module in Matlab, dSPACE I/O is added to the system, and the offline model in Matlab is converted into a real-time model of dSPACE. Use the “RTW build” command to generate and download real-time code to real-time hardware. The experimental software adopts a console-integrated experimental environment. During the experiment, the parameters of each module are adjusted to verify the influence of these parameters in the FVI formula on the bus voltage stability when the system is disturbed.

At this time, the load is connected to the white noise link in the SIMULINK environment, so that the power fluctuates irregularly between 650–1400 W. The light irradiation conditions in the PV modules were changed at 1 s. The simulation in this section mainly simulates the small disturbance state of the CPL under actual working conditions. Figure 14 is the power fluctuation diagram of the system, in which Figure 14a is the load random fluctuation power diagram within 2 s, Figure 14b is the dynamic characteristic of the DC bus voltage, Figure 14c is the diagram of battery output power change within 2 s, Figure 14d is the diagram of the output power change of PV modules within 2 s.

It is obvious that both methods can maintain the system stability under the condition of random load fluctuations. As shown in Figure 14b, the DC bus voltage fluctuation of the droop control and the FVI control is 2.45% and 0.8%, respectively. As can be seen from the above figure, FVI control method can effectively suppress voltage fluctuation and improve the voltage quality of DC bus. From Figure 14c,d, it shows that the power from the PV module is greater than the demand of the system loads before 1 s, so the energy storage system is in a charged state and the output power is negative. Since the PV modules work under the MPPT control, the fluctuation of the bus voltage can be well compensated, so that the fluctuation of the battery is small. However, after 1 s the light irradiance is greatly reduced and the output power of the system cannot meet the demand of the loads, and the battery is in the state of power generation. At this time, the battery needs to balance the power fluctuation suppressed by the bus voltage. Since the bus voltage fluctuation of the FVI control method is small, the battery absorbs a lot of power, resulting in a large fluctuation of the power curve.

5. Conclusions

The stability control method of DC microgrid based on small-signal modeling is studied in this paper. A small-signal model of a DC microgrid including CPLs is established. In order to suppress the influence of small disturbances on bus voltage, provide proper inertia for the system, the FVI control method is introduced into the DC microgrid. By introducing a virtual capacitor in the outer voltage loop, adding a comparator connected in series, and setting the time margin, the disturbance to the system bus voltage caused by the voltage difference link is reduced. It is verified that the system can remain stable when subjected to small signal disturbances (such as load increase and decrease, micro-source fluctuations, etc.). In addition, by comparing the two control methods of droop control and FVI, it is verified that FVI can provide a more stable inertia for the DC microgrid with CPL, and improve the system stability. The hardware test was completed on the dSPACE platform to obtain that the FVI method can suppress the DC bus voltage fluctuation and improve the output voltage quality when the noise link is added. Since the control method considered in this paper is oriented to DC microgrid and does not consider the access of AC load, future research will consider grid connection.