1. Introduction
With the development of PV, energy storage sources, and other distributed energy sources, the DC microgrid has gradually become the focus of attention in recent years because it has high power supply reliability and its transmission lines have low losses; the DC microgrid has made great progress in the fields of power generation, transmission, and distribution [
1,
2].
A distributed energy source needs a DC converter to connect with the DC bus, and the DAB DC converter stands out among many DC converter topologies because it has low switching loss, low voltage stress, and high power density [
3,
4,
5]. DAB connects the distributed energy source and the DC bus as an interface, and it realizes the bidirectional flow of power between the distributed energy source and the DC bus [
6]. In addition to being widely used in DC microgrids, instances in the literature [
7,
8,
9,
10,
11] have used DAB DC converters as battery chargers to connect battery energy storage systems with a 270 V electric aircraft power distribution bus, and proposed a predicted peak-current-based fast response control in DAB control to improve the response speed of the system.
A DAB DC converter has three control modes: single-phase shift control (SPS), dual-phase shift control (DPS), and triple-phase shift control (TPS) [
12]. SPS control is simple, but DAB has high backflow power and inductor current stress under SPS control, which leads to a converter with low efficiency. The TPS control has three phase shift ratios, so it is the most flexible control, and DAB has low backflow power and inductor current stress under TPS, but the model of TPS is so complex that we seldom put it to practical use. DPS control has less backflow power and inductor current stress compared with SPS, and the model of DPS is simpler than TPS; we can also optimize the backflow power and inductor current stress of the converter by algorithm under DPS [
13]. Literature [
14] proposes a new DAB modeling method, it uses the two-time scale discrete-type models, which segregates the dynamics of the DAB converter into fast and slow state variables, which eases the analysis of the DAB converter.
Electronic converters connect with the DC microgrid, which make the DC microgrid lack inertia and damping, so fluctuations in system power will affect the stability of the DC bus voltage [
15], thus threatening the safety and stability of the DC microgrid [
16,
17,
18]. In order to improve the stability of the DC bus, the literature [
19] introduces a virtual capacitor at the DC side, which improves the stability of the DC bus by using the voltage of a capacitor that cannot suddenly change, but virtual capacitance cannot provide additional damping for the system. In the AC microgrid, virtual synchronous machine (VSG) control provides the AC grid additional inertia and damping, which AC frequency will not drastically change. Compared with VSG control in the AC grid, the literature [
20,
21,
22] proposes the virtual DC motor control strategy (VDCM), where the mechanical rotation equation and armature equation of the DC motor are added to the control of the converter, which attributes the converter with external characteristics like a motor so it improves the stability of the DC bus; in [
23,
24], the authors use VDCM control to improve the voltage of a PV power generation system, but the inertia and damping coefficient of the system are constants in VDCM, and they cannot be adjusted according to the different states of the DC bus voltage, which leads to the system lacking good dynamic performance; in [
25], the paper designs a variable voltage drop control loop, combined with VDCM control, which improves the sensitivity of voltage control, but this control cannot reduce the fluctuation value of the DC bus when disturbed. On this basis, the literature [
26,
27] analyzes the influences of inertia and damping on system stability in VDCM by building small signal models. Studies [
28,
29,
30] propose two-parameter adaptive control of VDCM to realize the flexible change in inertia and damping, which improves the dynamic performance of the DC bus, but the optimization effect of the two-parameter adaptive VDCM is limited, and the influence of other parameters of VDCM control on DC bus stability is not analyzed in these papers.
Based on the above literature, in order to further improve the stability of the DC bus and dynamic performance of the DC system, in this paper, we take a DAB DC converter controlled by DSP as the research object and propose a three-parameter adaptive VDCM control, which realizes the flexible change in inertial, damping, and armature resistance; at the same time, we optimize the backflow power and inductor current stress of the converter to improve the efficiency of the converter.
The rest of the paper is organized as follows. In
Section 2, the topology of DAB under DSP control is analyzed. In
Section 3, the VDCM control is introduced into the converter, and a small signal model of DAB is built to analyze the influence of different armature resistance on the stability and dynamic performance of the DC bus, and the three-parameter adaptive VDCM control is proposed in this section. In
Section 4, the setting and adjustment principles of armature resistance are analyzed. In
Section 5, the backflow power and inductor current stress of the converter are optimized by constructing Lagrangian functions. Finally, the advantages of three-parameter adaptive VDCM control in improving the stability of DC bus are verified through simulation and experiments.
3. VDCM Control
The DAB converter controlled by VDCM control is shown in
Figure 3; the power side is supplied by the energy storage device.
are connected with the energy storage; they are the output voltage and the output current of the energy storage, and
is the output current of the DAB DC converter,
denote the DC bus voltage and DC bus current,
is also the output voltage of the converter, and the mechanical rotation equation of the DC motor is shown in Formulas (1) and (2).
where
J denotes the inertia constant,
H denotes the damping coefficient of the DC motor,
and
are the actual rotational angular velocity and rated rotational angular velocity of the DC motor, respectively;
are the mechanical torque and electromagnetic torque of the DC motor,
are the input mechanical power and the output electromagnetic power of the DC motor, and the actual value of the DC bus voltage corresponds to the actual value of
in the DC motor, and the rated value of the DC bus voltage corresponds to
in the DC motor.
The armature equation of the DC motor can be expressed as:
where
E is the armature electromotive force of the DC motor, it can be written as
, where
means flux per pole,
denotes the torque coefficient,
denotes the total equivalent resistance of the armature circuit, and
denotes the armature current. The electromagnetic power
of the DC motor can be written as
.
The control block diagram of the VDCM control strategy is shown in
Figure 4, on the basis of the voltage loop and the current loop, adding the VDCM control loop. In the voltage control part, the DC bus voltage feedback value
and the DC bus voltage reference value
are compared, using the voltage proportional integral (PI) controller to adjust
and obtain mechanical power deviation, and then add mechanical power reference value
to obtain
. In the VDCM control part, according to Formulas (1) and (3), which makes the converter operate as the DC motor to mimic its inertia characteristic, and then obtains the armature current
;
is also the reference value
of the output current of the converter. In the current control part, the output current feedback value
and the output current reference value
are compared using the current proportional integral (PI) controller to obtain PWM to control the operation of the switch tubes.
When , the converter works under SPS, and the SPS control only has one phase-shift ratio between . In DPS, determines the magnitude and direction of the transmitted power, in DPS and has the same effects with in SPS; therefore, DPS has a similar small signal model to SPS. We build a small signal model of SPS instead of DPS to simplify operation, and and are completely equivalent in the subsequent research.
The average model of the DAB DC converter is shown in
Figure 5, and the transmitted power of the DAB under SPS can be expressed as (4) [
31]; it can be seen that
of the DAB converter is not only affected by the the input
and
, but it is also restricted by the n, D, and
. Combined with
Figure 3, the instantaneous value of
and
of the converter can be represented as (5) and (6):
The small signal decomposition is carried out for each parameter in (6), and ignored for the change in
to obtain Equation (7).
and d denote steady state quantities of
and D,
and
denote the small signal AC components of
and D.
Ignoring the DC components and the second-order AC components in Equation (7), according to the Laplace transform, we can transform (7) into (8):
The transfer function of the VDCM control can be derived from
Figure 4:
In order to analyze the stability of the whole system, which adopts VDCM control, combined with
Figure 4, the small signal model of DAB is established in
Figure 6, where
stands for the change in each parameter.
The following formula can be obtained from
Figure 6, where
are proportion and integral coefficient of the voltage loop,
are proportion and integral coefficient of the current loop.
Adding the VDCM control to the converter, it is equivalent to introduce a first-order inertia link into the control of the converter [
32], the amplitude margin and phase margin of the system are improved, thus improving the stability of the system; it is also equivalent to add a virtual capacitor in the DC bus side, and the voltage at both ends of the capacitor cannot be suddenly changed; thus, the VDCM control can suppress the fluctuation of the DC bus voltage.
According to
Figure 6 and Equation (10), the open-loop transfer function from
to
can be expressed as
:
The literature [
33] has analyzed the influence of different values of
J and
H on system stability, so this paper only analyzes the influence of different values of armature resistance
on system stability.
We change Equation (11) into Equation (12), and the value of
affects both the second-order part a and the inertial part b of
from Equation (12).
Assuming that
and
are zero points of
, which
, the change in the value of
will make the zero points of
change. The relation between the value of
and the zero points of
is shown in (14).
With the decreases in
, the change trend of
is shown in
Figure 7; the change in zero points is two negative zero points, two equal zero points, and no zero points; the poles points change in the second-order part a in
is two negative real pole points, two equal negative real pole points, and a pair of conjugate pole points.
will increase as
decreases, and the pole points of the second-order a in
gradually move away from the real axis, which makes the system decay speed and response speed faster. The step response of
under different
is shown in
Figure 8.
We change Equation (11) into Equation (15),
K is the open-loop gain of
.
The open-loop gain
K of
can be expressed as:
It can be seen from (16) that K is irrelevant with J; it is directly proportional with the value of and inversely proportional with the value of H, and the time constant of the inertia link of Equation (15) is also directly proportional to . Comparing the position relationship between and J in Equation (15), they have the same influence on the inertia of the system, and the increase in can be equivalent to the increase in J.
Based on the small signal model of the DAB DC converter, the open-loop Bode diagram of
is drawn to analyze the influence of
on the stability of the control system. The open-loop Bode diagram with different
is shown in
Figure 9, the phase angle margin and the open-loop gain
K will increase as
increases within a certain range.
The transfer function from
to
in
Figure 6 can be expressed as:
By drawing the unit step response curve of
, we can simulate the change process of the DC bus voltage when disturbed by the load. The unit step response of
for different values of
is shown in
Figure 10; it can be seen that when the converter is disturbed by the load, the output voltage has a larger deviation when the
increases.
4. Three Parameter VDCM Adaptive Control
Taking the increase in output power of the converter as an example, the DC bus voltage
is divided into several intervals, as shown in
Figure 11; when the
deviates from the
, we need a smaller
to reduce the voltage deviation and speed up the response of the system. When the output voltage returns to the
, we need a larger
to provide additional inertia for the system. The literature [
34] analyzed how to choose
and H in different voltage intervals.
Table 1 shows the selection of three parameters, where
, and
are inertia, damping, and the armature resistance coefficient when the system is stable.
The adaptive regulation of three parameters can be realized by (18), (19), and (20), where c denotes the DC bus voltage change threshold, when the DC bus change exceeds the threshold, and the converter enters the VDCM adaptive adjustment mode; is the difference between the reference value and the actual value , and x is the rate of change in the DC bus voltage; A, B, and F are the regulating coefficients of J, H, and .
From 0 to
, the system remains stable, the DC bus voltage is
,
, and the armature resistance is
. From
to
, the load power suddenly increases, and at this time
,
,
is reduced to enhance the system response speed, and reduce the fluctuation of
. From
to
, the DC bus voltage begins to recover to the
,
,
is increased to increase the inertia of the system. From
to
,
, we still use the smaller
to reduce the voltage deviation of the DC bus. From
to
,
, we use the larger
to increase the inertia of the system, which makes the change in the DC bus voltage more gradual.
J and
H have the same adjustment principles as
[
35].
The design principles of
,
A,
, and
B have been detailed in the literature [
35], so this paper mainly analyzes the selection principles of
and
. The closed-loop transfer function
G of
is as follows:
Drawing the zero-pole distribution diagram of
G under
and
,
G has four poles, and we divide the poles into three groups,
and
, as shown in
Figure 12.
is a pair of conjugate poles, and the real part of a pole is much larger than the other two groups, so it has the least influences on the system. Compared with
, the real part of
changes more with
changes. As the dominant pole,
has the greatest influence on the stability of the converter. After analyzing the influence of different poles on the system, we make
and
, the adjustment range of
is 0.5~2, such that
and
.
7. Simulation and Analysis
MATLAB/Simulink simulation platform is used to verify the effectiveness of the control strategy. The secondary power supply
is replaced by resistance
R, and the voltage of
R is the DC bus voltage
; related parameters are shown in
Table 2.
The initial voltage
is maintained at 15 V before 0.2 s, at 0.2 s, the load
R is reduced from 5 Ω to 2 Ω, and the load power increases from 45 W to 112.5 W; the DC bus voltage is shown in
Figure 14.
Under different control strategies, the fluctuation peak and recovery time of output voltage
are shown in
Table 3.
From
Figure 14 and
Table 3, the DC bus voltage has the smallest fluctuation peak and the shortest time to recover to steady-state value
under the three-parameter adaptive VDCM.
The instantaneous power and inductor current of the converter are shown in
Figure 15.
The peak of backflow power is shown in
Table 4, and the backflow power of the converter is obviously reduced after backflow power optimization.
The inductor current stress is shown in
Table 5, and the inductor current stress of the converter is obviously reduced after inductance current stress optimization.
One thing to note is that adding VDCM control will not affect the backflow power and the inductor current stress of the converter. When adding the backflow power or inductance current stress optimization to the upper control of the converter, and are fixed, the bottom control will not change the size of , and different control strategies only affect the peak value of DC bus voltage fluctuation and the recovery time.