Analysis of Virtual Inertia in DC Microgrid Based on Matching Control Bandwidth

Abstract

The mismatch between the DC voltage control bandwidth and the low-pass filter control bandwidth results in a non-virtual inertia phenomenon in the DC voltage of the DC microgrid. For this purpose, a transfer function model of the DC microgrid is established in this paper, and the causes of the non-virtual inertia phenomenon are explained from the perspective of control bandwidth. Secondly, a virtual inertia response criterion based on control bandwidth matching is presented in this paper. Then, the concept and solution method of the control bandwidth matching domain are also provided in this paper. This control bandwidth matching domain can not only effectively ensure the virtual inertia characteristics of the DC microgrid but also be used to evaluate the system’s virtual inertia strength under different low-pass filter control bandwidths. Experimental results show that when the ratio of the voltage control bandwidth to the low-pass filter control bandwidth is greater than 10, the DC microgrid presents virtual inertia characteristics; otherwise, it exhibits non-virtual inertia (damped oscillation) characteristics.

1. Introduction

Renewable energy sources and distributed energy storage systems are typically integrated into DC microgrids through power electronic converters [1,2,3,4,5,6]. However, power electronic converters do not possess the inertia physical characteristics of traditional synchronous machines [7,8,9]. Consequently, the lack of inertia physical characteristics in power electronic converters leads to a more prominent issue of low inertia in DC microgrids [10,11,12,13]. For this reason, how to enhance the virtual inertia of DC microgrids has become a research hotspot.

Current researchers typically enhance the virtual inertia of DC microgrids by optimizing the control structure. Due to the negative influence of DC output current on the virtual inertia response of DC bus voltage, a virtual inertia control method based on current feedforward control is presented in reference [14]. This control method is applied in the AC/DC converter of a DC microgrid. Furthermore, this feedforward control introduces the DC output current into the AC current control loop via a proportional controller. Additionally, a virtual inertia control method based on feedforward disturbance compensation of DC output current is presented in reference [15], which is applicable to DC microgrids composed of DC/DC converters. Compared to reference [14], reference [15] introduces the output current into the virtual inertia controller through a proportional controller. Actually, the compensation loop is parallel to either the current controller or the virtual inertia controller in the literature [14] and [15]. In contrast, a novel virtual inertia control structure is presented in references [16,17,18]. In this control architecture, the virtual inertia controller is connected in series between the voltage controller and the current controller. In reference [16], the novel virtual inertia control takes into account the state of charge (SOC) of each energy storage device. Therefore, the differences in SOC among various energy storage devices can be effectively reduced. A virtual inertia control strategy based on virtual capacitance is provided in reference [17], which can effectively mitigate the rate of change in the DC bus voltage. Based on reference [17], the electric power in virtual inertia control is replaced by virtual synchronous machine power in reference [18]. Additionally, virtual inertia control methods based on differential feedforward compensation of DC bus voltage are presented in references [19,20]. In reference [19], the output power of the AC/DC converter is dynamically compensated, which is obtained through a proportional-derivative control component of the DC bus voltage. Unlike in document [19], there are three differential compensation points in document [20]. One of these differential compensation points is located at the inductor current reference of the DC/DC converter. However, when significant fluctuations occur in the DC bus voltage, the differential compensation based on it may cause stability issues in the DC microgrid. For this purpose, a virtual inertia compensation control method based on capacitor current is presented in reference [21]. To enhance the virtual inertia of AC/DC microgrids, a virtual inertia control method based on output power is provided in reference [22]. To enhance the small-signal stability of AC-DC microgrids, a control method based on adaptive adjustment of the virtual inertia coefficient is presented in reference [23]. Similarly, research has also been conducted to apply model predictive control in DC microgrids to enhance their virtual inertia [24,25,26,27]. To increase the virtual inertia response time of the DC microgrid, an optimization design method for virtual capacitor based on model predictive control is provided by reference [24]. To enhance the regulation capability of the DC microgrid on the DC bus voltage, an adaptive adjustment method for current reference based on output current prediction is proposed in reference [26]. Based on references [25,26], a model predictive control suitable for AC/DC grid-connected converters is presented in reference [27]. In order to enhance the inertia support capability of the DC microgrid for the DC bus voltage, a virtual inertia control method based on nonlinear disturbance observation is proposed in reference [28]. Considering the constraint limitations of DC bus voltage variations during load power disturbances, a design method for virtual inertia parameters is provided in reference [29]. It is indicated in the literature [30] that the interaction between control loops of different time scales affects the virtual inertia of DC microgrids. In summary, research is urgently needed on how to match the voltage control bandwidth and the virtual control bandwidth to enable the DC microgrid to exhibit virtual inertia characteristics.

Reasonable virtual inertia control parameters are an important prerequisite for ensuring the strength of the virtual inertia in DC microgrids. Based on the unit step response of the DC microgrid’s transfer function model, it is found in the literature [18] that a larger virtual natural frequency or droop coefficient will cause the DC voltage to exhibit virtual inertia characteristics. Likewise, the impact of the inertia constant and damping constant on the dynamic response of DC bus voltage or AC frequency is studied in reference [22] through a simplified transfer function model. However, the Nyquist criterion and eigenvalue analysis method are applicable for studying the variations in the stability of DC microgrids under different virtual inertia control parameters [31,32]. For example, as indicated by the research in the literature [18], a larger virtual natural frequency or droop coefficient will lead to a greater stability margin in DC microgrids. Based on the zero-pole movement of the impedance model, the influence of system parameters such as the current proportional-integral coefficient and virtual resistance on the response characteristics of the DC microgrid is presented in reference [21]. Similarly, the impact of the virtual inertia coefficient and PI control parameters on the small-signal stability of AC/DC microgrids is also examined using the root locus method in reference [23]. The theoretical study in reference [23] indicates that a larger virtual inertia coefficient can cause instability issues in AC/DC microgrids. Similarly, it is found in reference [24] that the stable poles in the DC microgrid will transform into unstable poles as the virtual capacitance increases. To avoid the emergence of unstable poles in the DC microgrid, a feedback compensation control method based on an extended state observer is presented in the literature [25]. Moreover, through the frequency-domain response results of the DC microgrid’s transfer function model, it is found by reference [25] that the stability of the DC microgrid is influenced by the control bandwidth. To satisfy the specific dynamic response characteristics of the DC microgrid, the virtual inertia constant and the virtual damping constant are designed by utilizing a parameter feasible region solution method in reference [31]. However, the dynamic response characteristics of the DC bus voltage exhibit a damped oscillatory response under the designed virtual inertia parameters. In summary, in order to enhance the virtual inertia of DC microgrids, current research still lacks an intuitive and effective method for designing virtual inertia parameters.

For the issue of virtual inertia in DC microgrids mentioned above, the transfer function model of the DC microgrid is established in this paper. Based on this model, a theoretical explanation is provided for the non-virtual inertia phenomenon from the perspective of the mismatch between the DC voltage control bandwidth and the low-pass filter control bandwidth. To address the issue of virtual inertia caused by mismatched control bandwidth, a virtual inertia response criterion is provided. Furthermore, the concept and solution method for the control bandwidth matching domain based on this virtual inertia response criterion are also presented. Finally, the effectiveness of the control bandwidth matching domain is verified using the RT-Box experimental platform.

2. Non-Virtual Inertia Phenomenon of DC Microgrid

2.1. Topology of DC Microgrid

The topological structure of a DC microgrid with n parallel-connected converters is shown in Figure 1.

Here R_fn is the filter resistance of the n-th converter, L_fn is the filter inductance of the n-th converter, I_fn is the filter inductance current of the n-th converter, V_n is the output voltage of the n-th converter, C_fn is the filter capacitance of the n-th converter, R_cn is the line resistance of the n-th converter, L_cn is the line inductance of the n-th converter, I_on is the output current of the n-th converter, C_cpl is the filter capacitance of constant power load (CPL), P_cpl is the power of CPL, V_b is the DC bus voltage, I_cpl is the current of CPL, k_dn is the droop coefficient of the n-th converter, ω_f is the control bandwidth of low-pass filter, V_g is the voltage reference, V_r is the rated voltage, k_pin is the current proportional gain, k_iin is the current integral gain of the n-th converter, s is the Laplace operator, k_pvn is the voltage proportional gain of the n-th converter, k_ivn is the voltage integral gain of the n-th converter, I_rn is the current reference of the n-th converter, d_n is the duty factor of the n-th converter, U_sn is the input voltage of the n-th converter, g_1n, g_2n are the modulation signals for the two IGBTs of the n-th converter, respectively. The specific numerical values of main system parameters are listed in Table 1.

Table 1. Parameters of PVESS.

Device	Parameters	Value
Battery storage converter	Output voltage	400 V
	Input voltage	200 V
	Filter inductor/filter resistor	0.373 mH/0.005 Ω
	Switching frequency	10 kHz
	Filter capacitor	0.08 F
	Droop coefficient	0.5
	Low-pass filter bandwidth	5 rad/s
	Sampling time	0.1 ms
	RT-Box Solver time step	25 μs
Constant power load	Input filter capacitor	0.08 F

Table 1. Parameters of PVESS.

Device	Parameters	Value
Battery storage converter	Output voltage	400 V
	Input voltage	200 V
	Filter inductor/filter resistor	0.373 mH/0.005 Ω
	Switching frequency	10 kHz
	Filter capacitor	0.08 F
	Droop coefficient	0.5
	Low-pass filter bandwidth	5 rad/s
	Sampling time	0.1 ms
	RT-Box Solver time step	25 μs
Constant power load	Input filter capacitor	0.08 F

2.2. Two Classic Virtual Inertia Control Methods

The two classic virtual inertia control methods shown in Figure 2 are usually adopted in current research to enhance the virtual inertia of DC microgrids [33].

Here J_v is the virtual inertia coefficient, D_v is the virtual damping coefficient, V_r is the rated DC bus voltage, and k_d is the droop coefficient. It can be derived that the low-pass filter-based virtual inertia control and the DC-type virtual synchronous generator (DC-VSG) control are mathematically equivalent, and their parameter relationships can be established through frequency-domain analysis.

For the low-pass filter-based virtual inertia control, the transfer function describing the dynamic response of the current reference is

$$G_{\mathrm{lpf}}(s)={\displaystyle \frac{I_{\mathrm{ref}}}{I_{\mathrm{in}}}}={\displaystyle \frac{{\omega }_f}{s+{\omega }_f}}$$

(1)

where I_in is the current signal corresponding to input power, and ω_f is the low-pass filter control bandwidth.

For the DC-VSG-based virtual inertia control, the electromechanical dynamic equation considering droop characteristics is

$$J_v{\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}+D_v(V-V_r)=-k_d{V}_r(P_{\mathrm{ref}}-P)$$

(2)

In DC microgrids, voltage is linearly mapped to frequency (V∝ω), thus converting (3) into current dynamics, where P and P_ref are the actual and reference power, respectively. In DC microgrids, the power–voltage–current relationship satisfies P ≈ V_rI under small-signal conditions. Substituting this relationship and performing the Laplace transformation, the equivalent current-oriented transfer function of the DC-VSG control is obtained as

$$G_{\mathrm{VSG}}(s)={\displaystyle \frac{I(s)}{I_{\mathrm{in}}(s)}}={\displaystyle \frac{1/J_v}{s+D_v/(J_v{V}_r)}}$$

(3)

By comparing the denominators of G_lpf(s) and G_VSG(s), the pole positions are required to be identical to ensure equivalent dynamic responses. Combined with the droop coefficient definition, the complete parameter mapping relationship between the two control methods is established as

$$\left\{\begin{array}{l}{\omega }_f={\displaystyle \frac{D_v}{J_v{V}_r}}\hfill \\ k_d={\displaystyle \frac{1}{D_v}}\hfill \end{array}\right.$$

(4)

Under this mapping, the two control methods exhibit identical frequency-domain characteristics and can thus be treated as equivalent and interchangeable in controller design and parameter tuning. While J_v and D_v in the DC-VSG control possess clear physical interpretations from the perspective of synchronous generator emulation, the low-pass filter control bandwidth ω_f provides a more intuitive and engineering-friendly parameter for voltage and current loop design in DC microgrids.

2.3. Non-Virtual Inertia Experiment Results

The DC microgrid in Figure 1 is established by the RT-box HIL platform shown in Figure 3. The voltage and current controllers are discretized using the bilinear transform.

The CPL steps up from 1 kW to 2 kW at 1 s. The DC voltage experiment results are shown in Figure 4. It can be observed from the figure that the experimental waveform exhibits damped oscillation characteristics within the time range of roughly [1.0 s, 2.0 s], with an oscillation frequency of about 19.22 rad/s. In general, a waveform with exponential decay represents the virtual inertia characteristic, while that with damped oscillation corresponds to the non-virtual inertia characteristic [13]. Therefore, from what theoretical perspective can this non-virtual inertia phenomenon be explained? And how can the DC voltage be endowed with virtual inertia response characteristics? Both of these issues are worthy of in-depth investigation.

3. Theoretical Explanation of Non-Virtual Inertia Phenomenon Based on the Virtual Inertia Response Criterion

To explain the non-virtual inertia phenomenon shown in Figure 4, a transfer function model is developed in this subsection. Based on the frequency-domain response of this model, a criterion for virtual inertia response is provided. Furthermore, this criterion explains the non-virtual inertia phenomenon in Figure 4 from the perspective of the mismatch between the DC voltage control bandwidth and the low-pass filter control bandwidth.

3.1. The Transfer Function Model of DC Microgrid

When the virtual inertia control loop, voltage control loop, and current control loop in Figure 1 are not taken into account, the open-loop transfer function G_vd(s) from the duty factor d to DC voltage V can be expressed in the following form [34,35,36]:

$$G_{\mathrm{vd}}\left(s\right)={\displaystyle \frac{\left\{\left(1-d\right)-\frac{\left(R_{\mathrm{f}}+L_{\mathrm{f}}s\right)}{\left(R_{\mathrm{c}}+R_{\mathrm{load}}\right)\left(1-d\right)}\right\}V}{\left(C_{\mathrm{f}}s+G_{\mathrm{in}}\left(s\right)\right)\left(R_{\mathrm{f}}+L_{\mathrm{f}}s\right)+{\left(1-d\right)}^2}}$$

(5)

where R_load is the equivalent resistor, the expression for the function G_in(s) is presented in Equation (5). Additionally, G_vdm(s) and G_vdd(s) are designated as the numerator and denominator expressions of the transfer function G_vd(s), respectively.

$$G_{\mathrm{in}}\left(s\right)={\displaystyle \frac{\left(C_{\mathrm{cpl}}s-\frac{1}{R_{\mathrm{load}}}\right)}{\left\{L_{\mathrm{c}}{C}_{\mathrm{cpl}}{s}^2+\left(R_{\mathrm{c}}{C}_{\mathrm{cpl}}-\frac{L_{\mathrm{c}}}{R_{\mathrm{load}}}\right)s+\left(1-\frac{R_{\mathrm{c}}}{R_{\mathrm{load}}}\right)\right\}}}$$

(6)

Similarly, the transfer function G_id(s) from duty factor d to filter inductor current I_f can be obtained as follows:

$$G_{\mathrm{id}}\left(s\right)={\displaystyle \frac{\left\{V{G}_{\mathrm{vdd}}\left(s\right)-\left(1-d\right)G_{\mathrm{vdm}}\left(s\right)\right\}}{\left(R_{\mathrm{f}}+L_{\mathrm{f}}s\right)G_{\mathrm{vdd}}\left(s\right)}}$$

(7)

As can be seen from Figure 1, the expression for the current controller G_ic(s) can be written in the following form:

$$G_{\mathrm{ic}}\left(s\right)=\left(k_{\mathrm{pi}}+{\displaystyle \frac{k_{\mathrm{ii}}}{s}}\right)$$

(8)

Furthermore, the expressions for the loop gain T_ii(s) and the closed-loop transfer function G_ii(s) can be derived separately, with the details presented in Equations (9) and (10), respectively.

$$T_{\mathrm{ii}}\left(s\right)=G_{\mathrm{id}}\left(s\right)G_{\mathrm{ic}}\left(s\right)$$

(9)

$$G_{\mathrm{ii}}\left(s\right)={\displaystyle \frac{G_{\mathrm{id}}\left(s\right)G_{\mathrm{ic}}\left(s\right)}{1+G_{\mathrm{id}}\left(s\right)G_{\mathrm{ic}}\left(s\right)}}$$

(10)

For convenience, G_iim(s) and G_iid(s) are denoted as the numerator and denominator expressions of the transfer function G_ii(s), respectively. Furthermore, the open-loop transfer function of the voltage control loop G_iv(s) can be obtained as

$$G_{\mathrm{iv}}\left(s\right)=\left\{{\displaystyle \frac{G_{\mathrm{iim}}\left(s\right)\left(1-d\right)}{G_{\mathrm{iid}}\left(s\right)}}-{\displaystyle \frac{V}{\left(R_{\mathrm{c}}+R_{\mathrm{r}}+R_{\mathrm{cpl}}\right)\left(1-d\right)}}\right\}\left(Z\left(s\right)\right)$$

(11)

where the mathematical expressions for transfer functions Z (s) are shown in Equation (12).

$$Z\left(s\right)={\displaystyle \frac{R_{\mathrm{load}}+\left(R_{\mathrm{c}}+L_{\mathrm{c}}s\right)\left(C_{\mathrm{cpl}}{R}_{\mathrm{load}}s-1\right)}{\left\{\left(C_{\mathrm{cpl}}{R}_{\mathrm{load}}s-1\right)+\left\{R_{\mathrm{load}}+\left(R_{\mathrm{c}}+L_{\mathrm{c}}s\right)\left(C_{\mathrm{cpl}}{R}_{\mathrm{load}}s-1\right)\right\}{C}_{\mathrm{f}}s\right\}}}$$

(12)

As can be seen from Figure 1, the mathematical expression for the voltage controller G_vc(s) can be written in the following form:

$$G_{\mathrm{vc}}\left(s\right)=\left(k_{\mathrm{pv}}+{\displaystyle \frac{k_{\mathrm{iv}}}{s}}\right)$$

(13)

Furthermore, the mathematical expressions for the loop gain T_viv(s) and the closed-loop transfer function G_viv(s) can be derived separately. The details are presented in Equations (14) and (15), respectively.

$$T_{\mathrm{viv}}\left(s\right)=G_{\mathrm{iv}}\left(s\right)G_{\mathrm{vc}}\left(s\right)$$

(14)

$$G_{\mathrm{viv}}\left(s\right)={\displaystyle \frac{G_{\mathrm{iv}}\left(s\right)G_{\mathrm{vc}}\left(s\right)}{1+G_{\mathrm{iv}}\left(s\right)G_{\mathrm{vc}}\left(s\right)}}$$

(15)

In the virtual inertia control shown in Figure 1, the mathematical expression for the low-pass filter controller G_lpf(s) is presented in Equation (16).

$$G_{\mathrm{lpf}}\left(s\right)={\displaystyle \frac{{\omega }_{\mathrm{f}}}{s+{\omega }_{\mathrm{f}}}}$$

(16)

Similarly, the mathematical expression for the closed-loop voltage transfer function G_vivf(s) considering virtual inertia control can be derived, as shown in the following equation.

$$G_{\mathrm{vivf}}\left(s\right)={\displaystyle \frac{G_{\mathrm{vc}}\left(s\right)G_{\mathrm{ic}}\left(s\right)\left(G_{\mathrm{c}}\left(s\right)G_{\mathrm{load}}\left(s\right)+1\right)G_{\mathrm{f}}\left(s\right)G_{\mathrm{vdm}}\left(s\right)}{\left\{\begin{array}{l}{G}_{\mathrm{vc}}\left(s\right)G_{\mathrm{ic}}\left(s\right)\left\{\begin{array}{l}{G}_{\mathrm{load}}\left(s\right)k_{\mathrm{d}}{G}_{\mathrm{lpf}}\left(s\right)\\ +G_{\mathrm{c}}\left(s\right)G_{\mathrm{load}}\left(s\right)+1\end{array}\right\}{G}_{\mathrm{f}}\left(s\right)G_{\mathrm{vdm}}\left(s\right)\\ +\left\{\begin{array}{l}\left\{G_{\mathrm{f}}\left(s\right)+G_{\mathrm{ic}}\left(s\right)V\right\}{G}_{\mathrm{vdd}}\left(s\right)\\ -G_{\mathrm{ic}}\left(s\right)\left(1-d\right)G_{\mathrm{vdm}}\left(s\right)\end{array}\right\}\left(G_{\mathrm{c}}\left(s\right)G_{\mathrm{load}}\left(s\right)+1\right)\end{array}\right\}}}$$

(17)

where the mathematical expressions for transfer functions G_c(s), G_f(s), and G_load(s) are shown below, respectively:

$$\left\{\begin{array}{l}{G}_{\mathrm{c}}\left(s\right)=\left(R_{\mathrm{c}}+L_{\mathrm{c}}s\right)\\ G_{\mathrm{f}}\left(s\right)=\left(R_{\mathrm{f}}+L_{\mathrm{f}}s\right)\\ G_{\mathrm{load}}\left(s\right)=\left(C_{\mathrm{cpl}}s-1/R_{\mathrm{load}}\right)\end{array}\right.$$

(18)

3.2. Theoretical Analysis of the Non-Virtual Inertia Phenomenon

The zero-pole diagram and Bode plot of the transfer function G_vivf(s) are shown in Figure 5 and Figure 6, respectively. The oscillation frequency of the dominant conjugate poles in Figure 5 is roughly 22 rad/s (with a damping factor of roughly 0.307), which remains largely consistent with the experimental result of 19.22 rad/s presented in Figure 4.

Based on the above analysis, it can be seen that the zero-pole diagram is suitable for evaluating the damped oscillation characteristics. The impact of control bandwidth on the virtual inertia is difficult to explain by the zero-pole diagram. According to Figure 6, it can be seen that the DC voltage control bandwidth ω_v of transfer functions G_vivf(s) is approximately 26.5 rad/s. The low-pass filter control bandwidth ω_f of the transfer function G_lpf(s) is 5 rad/s.

3.3. Criterion for Virtual Inertia Response

To evaluate the matching relationship between the DC voltage control bandwidth ω_v and the low-pass filter control bandwidth ω_f, a virtual inertia response criterion is proposed in this article, as detailed in Figure 7.

η is the bandwidth coefficient, and its mathematical calculation equation is shown below

$$\eta ={\displaystyle \frac{{\omega }_{\mathrm{v}}}{{\omega }_{\mathrm{f}}}}$$

(19)

where ω_v is the DC voltage control bandwidth, and ω_f is the low-pass filter control bandwidth.

The threshold η > 10 is established based on an engineering rule of thumb for cascade control systems: the inner loop (voltage controller) should have a bandwidth at least 5 to 10 times higher than the outer loop (low-pass filter) to ensure negligible interaction. In this paper, the conservative value of 10 is adopted. To verify this choice, consider the closed-loop transfer function G_vivf(s).

(1)

Matched voltage control bandwidth:

When the bandwidth coefficient η is greater than 10, the response speed of the voltage controller G_vc(s) is much faster than that of the low-pass filter G_lpf(s). That is, the voltage controller G_vc(s) enables the DC voltage V to quickly track the voltage reference value V_g generated by the low-pass filter G_lpf(s). At this point, the impact of the voltage controller G_vc(s) on the dynamic response of DC voltage can be largely ignored. That is, the dynamic response of the system is predominantly determined by the low-pass filter G_lpf(s). Consequently, the DC voltage will primarily exhibit virtual inertia characteristics, and the virtual inertia response time T_f of the DC voltage is approximately 5/ω_f seconds.

(2)

Unmatched voltage control bandwidth:

When the bandwidth coefficient η is less than 10, the dynamic response of the system is jointly determined by the voltage controller G_vc(s) and the low-pass filter G_lpf(s). In that case, the DC voltage may exhibit non-virtual inertia phenomena, and the virtual inertia response time T_f of the DC voltage is approximately 0 s.

It can be deduced that when the bandwidth coefficient η is greater than 10, the DC microgrid will exhibit virtual inertia characteristics. That is, the ratio of DC voltage control bandwidth ω_v to low-pass filter control bandwidth ω_f should be greater than 10. According to Figure 6, it can be inferred that the ratio between the DC voltage control bandwidth ω_v and the low-pass filter control bandwidth ω_f is approximately 5.3. Since 5.3 is less than 10, the dynamic response characteristics of the DC voltage are jointly determined by the low-pass filter and the voltage controller. The frequency-domain responses of transfer functions G_lpf(s) and G_vivf(s) exhibit virtual inertia characteristics and damped oscillation characteristics, respectively. Therefore, the experimental results in Figure 4 exhibit damped oscillation characteristics (i.e., non-virtual inertia characteristics). In summary, the mismatch between the DC voltage control bandwidth ω_v and the low-pass filter control bandwidth ω_f is the fundamental reason for the non-virtual inertia phenomenon in DC microgrids.

4. Virtual Inertia Characteristics Based on Matched Control Bandwidth

4.1. PI Control Parameter Design Based on Transfer Function

Based on the established transfer function model, four sets of PI control parameters have been designed in this paper. Furthermore, the PI control parameters listed in reference [34] are denoted as the second set of PI control parameters. The details of the parameters and the Bode plot are shown in

Table 2. Performance indices of four groups with different PI parameters.

Transfer Functions	Group 1	Group 2	Group 3	Group 4
Current proportional coefficient	0.0006	0.0006	0.0029	0.0029
Current integral coefficient	0.0730	0.0730	1.8401	1.8401
Voltage proportional coefficient	0.4549	1.8183	96.9715	176.9877
Voltage integral coefficient	4.7634	76.1647	48,743.2162	166,807.0128

and Figure 8, respectively.

As can be seen from Figure 8, the DC voltage control bandwidth ω_v of PI control parameters for Groups 1 to 4 are 6.24 rad/s, 26.5 rad/s, 532 rad/s, and 1480 rad/s, respectively.

In this article, the minimum ω_{f_min} and maximum ω_{f_max} values of the low-pass filter control bandwidth are set to 1 rad/s and 10 rad/s, respectively. Consequently, the bandwidth coefficients for the PI control parameters of Group 1, Group 2, Group 3, and Group 4 are designed to be 0.62, 2.65, 53.20, and 148.00, respectively, as detailed in

Table 3. The PI control parameters for Groups 1 to 4.

Transfer Functions	Control Bandwidth ωv	Bandwidth Coefficient η	The Response Characteristics of DC Voltage
Group 1	6.24 rad/s	0.62	Damped oscillation
Group 2	26.5 rad/s	2.65	Damped oscillation
Group 3	532 rad/s	53.20	Virtual inertia
Group 4	1,480 rad/s	148.00	Virtual inertia

. Since the bandwidth coefficients η of PI control parameters for both Group 1 and Group 2 are less than 10, the DC voltage will exhibit a non-virtual inertia phenomenon. On the other hand, the bandwidth coefficients η of PI control parameters for both Group 3 and Group 4 are greater than 10. Therefore, it can be inferred that the DC voltage will demonstrate virtual inertia characteristics.

In previous studies, the relationship between the virtual inertia characteristics and control parameters of DC microgrids has not been well established. Based on the virtual inertia response criterion proposed in this paper, this relationship is defined by the bandwidth coefficient.

4.2. Control Bandwidth Matching Domain of DC Microgrid

To address the non-virtual inertia issues caused by the mismatch between the DC voltage control bandwidth ω_v and the low-pass filter control bandwidth ω_f, the concept of a control bandwidth matching domain is proposed in this paper. In addition, a method for solving the control bandwidth matching domain is proposed. The specific solution process is shown in Figure 9.

According to the construction process outlined in Figure 9, the control bandwidth matching domain is characterized by this article, as illustrated in Figure 10. In Figure 10, the closed-loop domain formed by connecting Point A, Point B, Point C, Point D, and back to Point A is represented as the control bandwidth mismatch domain. The closed-loop domain created by connecting Point E, Point I, Point J, Point F, and back to Point E is labeled as the control bandwidth matching domain. The dashed line between Point D and Point C is designated as the boundary between the control bandwidth mismatch domain and the control bandwidth matching domain.

(1)

Control bandwidth matching domain: The bandwidth coefficient η within the control bandwidth matching domain is greater than 10. In the control bandwidth matching domain, the dynamic response characteristics of the DC microgrid are almost entirely determined by the low-pass filter G_lpf(s). Under such circumstances, the DC microgrid exhibits a virtual inertia phenomenon, and the virtual inertia response time T_f is approximately 5/ω_f seconds. Simultaneously, the length of the virtual inertia response time signifies the strength of the system’s virtual inertia. Specifically, a longer virtual inertia response time indicates a stronger virtual inertia strength of the system. Additionally, under the condition of a fixed DC voltage control bandwidth ω_v, as the low-pass filter control bandwidth ω_f decreases, the virtual inertia response time increases accordingly. Consequently, the virtual inertia strength of the DC microgrid is enhanced.

For instance, in Figure 10, the DC voltage control bandwidth ω_v is set to 532 rad/s. When the low-pass filter control bandwidth ω_f is set to 5 rad/s, 2 rad/s, and 1 rad/s, the virtual inertia response times are approximately observed to be 1 s, 2.5 s, and 5 s, respectively. As the virtual inertia response time increases, the virtual inertia strength of the DC microgrid is gradually being enhanced. Within the reasonable regulation range, increasing the DC voltage control bandwidth can raise the bandwidth coefficient, effectively strengthen the system virtual inertia characteristic and further improve the system stability margin.

(2)

Control bandwidth mismatch domain: In the control bandwidth mismatch domain, the dynamic response characteristics of the DC microgrid are jointly determined by the voltage controller G_vc(s) and the low-pass filter G_lpf(s). That is, the DC microgrid will exhibit damped oscillation characteristics (or non-virtual inertia phenomenon), and the virtual inertia response time of DC voltage is approximately 0 s.

5. Experimental Verification

To verify the validity of the virtual inertia response criterion and the effectiveness of the control bandwidth matching domain, multiple sets of experimental results for DC microgrids are presented in this section.

5.1. Parameters for Group 1 and Group 2

At 1 s, the load power is varied from 1 kW to 2 kW. The detailed zero and pole values of PI control parameters for Group 1, Group 2 and Group 3 are listed in Table 4. The low-pass filter control bandwidth ω_f is fixed at 5 rad/s. Furthermore, the experimental waveforms for the parameters of Group 1, Group 2, and Group 3 are shown as the blue, black, and red curves in Figure 11, respectively. In Figure 11, the experimental waveform of the PI control parameters in Group 1 exhibits second-order damped oscillation characteristics within the time range of approximately [1.0 s, 4.5 s]. Furthermore, the experimental oscillation frequency within the time range of [1.0 s, 4.5 s] is approximately 4.62 rad/s, which is largely consistent with the theoretical oscillation frequency of 5.01 rad/s presented in Figure 12. It is concluded that the zero-pole diagram is indeed suitable for evaluating second-order damped oscillation characteristics. In addition, the validity of the control bandwidth mismatch domain is verified by the blue and black experimental waveforms. Specifically, both points (6.24 rad/s, 5 rad/s) and (26.5 rad/s, 5 rad/s) are located within the control bandwidth mismatch domain shown in Figure 10.

Compared to the PI control parameters of Group 1 and Group 2, the experimental waveform of the PI control parameters in Group 3 exhibits virtual inertia characteristics within the time range of approximately [1.0 s, 2.0 s]. The correctness of the virtual inertia response criterion is verified based on the experimental waveforms in Figure 11. The validity of the control bandwidth matching domain is verified by the red experimental waveform. Specifically, the point (532 rad/s, 5 rad/s) is located within the control bandwidth matching domain depicted in Figure 10.

Table 4. Zeros and poles of the PI control parameters from Group 1 to Group 3.

Control Parameters	Poles	Zeros
Group 1 PI Parameters	−155.00; −13.40; −4.20; −1.87 ± 4.64i;	−126.00; −13.40; −10.50; −5.00
Group 2 PI Parameters	−156.00; −13.40; −6.76 ± 20.90i; −3.48	−126.00; −41.90; −13.40; −5.00
Group 3 PI Parameters	−764.00; −386.00 ± 480.00i; −13.40; −3.44	−628.00; −503.00; −13.40; −5.00

Table 4. Zeros and poles of the PI control parameters from Group 1 to Group 3.

Control Parameters	Poles	Zeros
Group 1 PI Parameters	−155.00; −13.40; −4.20; −1.87 ± 4.64i;	−126.00; −13.40; −10.50; −5.00
Group 2 PI Parameters	−156.00; −13.40; −6.76 ± 20.90i; −3.48	−126.00; −41.90; −13.40; −5.00
Group 3 PI Parameters	−764.00; −386.00 ± 480.00i; −13.40; −3.44	−628.00; −503.00; −13.40; −5.00

5.2. The Third Set of Control Parameters

At 1 s, the load power is varied from 1 kW to 2 kW, and the experimental waveform of the DC microgrid is shown in Figure 13. From Figure 13, it can be observed that the experimental results of the PI control parameters in Group 3 all exhibit virtual inertia characteristics. Based on the experimental waveforms in Figure 13, the correctness of the virtual inertia response criterion is verified.

Based on the experimental results provided in

Table 5. Virtual inertia response time under mismatch-domain filter bandwidths for Group 3 PI parameters.

Low-Pass Filter Control Bandwidth ωf	6.24 Rad/s	26.5 Rad/s	532 Rad/s
The time range of virtual inertia characteristics	almost 0 s	almost 0 s	[25.0 s, 26.0 s]
Virtual inertia response time	almost 0 s	almost 0 s	1.0 s

and Figure 13 can be obtained. To provide more details, when the low-pass filter control bandwidth ω_f is set to 5 rad/s, 2 rad/s, and 1 rad/s, the experimental results approximately exhibit virtual inertia characteristics within the time ranges of [1.0 s, 2.0 s], [1.0 s, 3.5 s], and [1.0 s, 6.0 s] respectively. Then, the virtual inertia response times for various low-pass filter control bandwidths can all be calculated. Specifically, for low-pass filter control bandwidths of 5 rad/s, 2 rad/s, and 1 rad/s, the corresponding virtual inertia response times are 1.0 s, 2.5 s, and 5.0 s, respectively. In other words, the virtual inertia response time increases from 1.0 s to 5.0 s as the low-pass filter control bandwidth ω_f decreases from 5 rad/s to 1 rad/s. Then, the virtual inertia strength of the system will be gradually enhanced. The effectiveness of the control bandwidth matching domain is also verified through the experimental results shown in Figure 13.

5.3. The Fourth Set of Control Parameters

At 1 s, the load power is varied from 1 kW to 2 kW. The experimental waveform is detailed in Figure 14. The experimental waveforms in Figure 14 all exhibit virtual inertia characteristics. Based on this, the correctness of the virtual inertia response criterion is further verified.

Similarly,

Table 6. Virtual inertia response time for PI control parameters of Group 3.

Low-Pass Filter Control Bandwidth ωf	5 Rad/s	2 Rad/s	1 Rad/s
The time range of virtual inertia characteristics	[1.0 s, 2.0 s]	[1.0 s, 3.5 s]	[1.0 s, 6.0 s]
Virtual inertia response time	1.0 s	2.5 s	5.0 s

provides the corresponding data of Group 3, and Table 7 lists the virtual inertia response time data of Group 4 PI parameters. According to Table 5, Table 6 and Table 7, when the low-pass filter control bandwidth ωf is 5 rad/s, the virtual inertia response times for both the third and fourth groups of PI control parameters are 1.0 s. As the low-pass filter control bandwidth ω_f decreases from 5 rad/s to 2 rad/s, the virtual inertia response times for both the third and fourth groups of PI control parameters increase from 1.0 s to 2.5 s. Similarly, when the low-pass filter control bandwidth is set to 1 rad/s, the virtual inertia response times for both the third and fourth groups of PI control parameters are 5.0 s. In summary, when the DC voltage control bandwidth ω_v is increased from 532 rad/s to 1480 rad/s, the virtual inertia response time remains almost unchanged. Therefore, the correctness of the control bandwidth matching domain is also verified.

Table 7. Virtual inertia response time for PI control parameters of Group 4.

Low-Pass Filter Control Bandwidth ωf	5 Rad/s	2 Rad/s	1 Rad/s
The time range of virtual inertia characteristics	[1.0 s, 2.0 s]	[1.0 s, 3.5 s]	[1.0 s, 6.0 s]
Virtual inertia response time	1.0 s	2.5 s	5.0 s

Table 7. Virtual inertia response time for PI control parameters of Group 4.

Low-Pass Filter Control Bandwidth ωf	5 Rad/s	2 Rad/s	1 Rad/s
The time range of virtual inertia characteristics	[1.0 s, 2.0 s]	[1.0 s, 3.5 s]	[1.0 s, 6.0 s]
Virtual inertia response time	1.0 s	2.5 s	5.0 s

5.4. PI Parameter Design of Group 5 and Group 6

To further verify the validity of the DC voltage virtual inertia response criterion around the bandwidth coefficient η=10, this paper designs the sixth and seventh groups of PI control parameters. The PI parameters are given in Table 8, and relevant indices are listed in Table 9. The design process and corresponding Bode diagrams are shown in Figure 15. It can be seen that the DC voltage control bandwidth ω_vb of the transfer function G_viv (s) corresponding to Group 5 and Group 6 are 82.9 rad/s and 124 rad/s, and the calculated bandwidth coefficients η are 8.29 and 12.40 respectively.

The experimental waveforms with load power stepping from 1 kW to 2 kW at 1 s and the low-pass filter control bandwidth ω_f fixed at 5 rad/s are depicted in Figure 16. In the experimental results, the blue curve shows damped oscillation characteristics, whereas the red curve basically manifests typical virtual inertia characteristics [35].

Table 8. Current and voltage PI coefficients for Groups 5 and 6.

Transfer Functions	Group 5	Group 6
Current proportional coefficient	0.0006	0.0029
Current integral coefficient	0.0730	1.8401
Voltage proportional coefficient	5.4252	17.9832
Voltage integral coefficient	681.7447	1,614.1690

Table 8. Current and voltage PI coefficients for Groups 5 and 6.

Transfer Functions	Group 5	Group 6
Current proportional coefficient	0.0006	0.0029
Current integral coefficient	0.0730	1.8401
Voltage proportional coefficient	5.4252	17.9832
Voltage integral coefficient	681.7447	1,614.1690

Figure 16. Experimental waveforms for PI control parameters of Groups 5 to 6.

Figure 16. Experimental waveforms for PI control parameters of Groups 5 to 6.

Table 9. The PI control parameters for Groups 5 to 6.

Transfer Functions	DC Voltage Control Bandwidth ωv	Bandwidth Coefficient η	The Response Characteristics of DC Voltage
Group 5	82.9 rad/s	8.29	Damped oscillation
Group 6	124.0 rad/s	12.40	Virtual inertia

Table 9. The PI control parameters for Groups 5 to 6.

Transfer Functions	DC Voltage Control Bandwidth ωv	Bandwidth Coefficient η	The Response Characteristics of DC Voltage
Group 5	82.9 rad/s	8.29	Damped oscillation
Group 6	124.0 rad/s	12.40	Virtual inertia

5.5. Load Disturbance Performance Under Various Droop Coefficients

At 1 s, the load power is varied from 1 kW to 2 kW, and the experimental waveform of the DC microgrid under different droop coefficients is shown in Figure 17 and Figure 18, respectively. Compared with the case in Figure 13, all waveforms exhibit typical virtual inertia response characteristics, while significant differences exist in the steady-state voltage before and after load disturbance when the droop coefficient is varied. When the droop coefficient is doubled, the steady-state voltage before disturbance is 396.7 V and drops to approximately 394.1 V after the disturbance. When the droop coefficient is quadrupled, the steady-state voltage decreases from 393.4 V (pre-disturbance) to approximately 388.1 V (post-disturbance). It can be seen that both the pre- and post-disturbance steady-state voltages decrease as the droop coefficient increases.

5.6. Load Disturbance Characteristics of Multi-Parallel Converters

At 1 s, the load power is varied from 1 kW to 2 kW, and the experimental waveforms of the DC microgrid with three and four parallel converters are shown in Figure 19, respectively. Compared with the results in Figure 13, all measured waveforms exhibit virtual inertia characteristics, while obvious differences exist in the steady-state DC voltage before and after load variation. Specifically, the steady-state voltages before disturbance are approximately 398.30 V, 398.85 V and 399.14 V for two, three and four parallel units, and the corresponding post-disturbance steady-state voltages are about 397.00 V, 397.98 V and 398.49 V respectively.

6. Conclusions

For the non-virtual inertia phenomenon in DC microgrids, this paper proposes a method for enhancing virtual inertia and a method for constructing the feasible region of virtual inertia in DC microgrids based on control bandwidth matching. The main conclusions are as follows:

(1)

A virtual inertia response criterion based on matching control bandwidth is proposed in this paper. When the bandwidth coefficient η is greater than 10, the response speed of the voltage controller G_vc(s) is much faster than that of the low-pass filter G_lpf(s). That is, the dynamic response of DC voltage is predominantly determined by the low-pass filter G_lpf(s). Then, the DC voltage will primarily exhibit virtual inertia characteristics, and the virtual inertia response time T_f of the DC voltage is approximately 5/ω_f seconds. When the bandwidth coefficient η is less than 10, the dynamic response of DC voltage is jointly determined by the voltage controller G_vc(s) and the low-pass filter G_lpf(s). In that case, the DC voltage may exhibit non-virtual inertia phenomena, and the virtual inertia response time T_f of the DC voltage is approximately 0 s.

(2)

The concept and solution method of the control bandwidth matching domain are also provided in this paper. Within the control bandwidth matching domain, the bandwidth coefficient η is greater than 10. In this domain, the dynamic response characteristics of the DC microgrid are almost entirely determined by the low-pass filter G_lpf(s). Under such circumstances, the DC microgrid exhibits a virtual inertia phenomenon, and the virtual inertia response time T_f is approximately 5/ω_f seconds. For instance, in Figure 10, the DC voltage control bandwidth ω_v is set to 532 rad/s. When the low-pass filter control bandwidth ω_f is set to 5 rad/s, 2 rad/s, and 1 rad/s, the virtual inertia response times are observed to be 1 s, 2.5 s, and 5 s, respectively. As the virtual inertia response time increases, the virtual inertia strength of the DC microgrid is gradually being enhanced.

Future work will concentrate on exploring how measurement noise and sampling characteristics affect bandwidth estimation accuracy in DC microgrids. In addition, the negative effects of controller saturation and actuator constraints will be analyzed and mitigated. Relevant optimization methods will be presented to suppress disturbances, strengthen the robustness of bandwidth-matching control, and improve the dynamic performance and engineering practicability of DC microgrid control systems.