A typical topology and control structure of the three-phase voltage-controlled MG inverter is shown in
Figure 1. The application control layer realizes the droop control or VSG control, generates the amplitude reference,
, and frequency reference,
, according to different control targets. The voltage control layer then tracks the instantaneous reference voltage rapidly in the SRF and adjusts the output impedance. The PWM algorithm is used to generate the driving signals of the switching devices.
The direct current (DC) side is usually equipped with an energy storage system (ESS) or bus capacitors, and the voltage stabilization control algorithm is adopted to maintain a constant bus voltage, ; is the voltage of the inverter side; and and are the output voltage and current of the inverter. The LC filter includes the filter inductor, , equivalent series resistance, , and the filter capacitor, ; and represent the current of the filter inductor and capacitor. and are line impedances; is the output of the voltage loop and is the output of the current loop; local loads are paralleled on the point of common coupling (PCC).
2.2. Structure of LADRC-Based Voltage Loop
The voltage controller scheme based on LADRC is shown in
Figure 2. A dual-loop control (DLC) structure is used. The inner current loop is utilized to control the inductor current with the PI-based regulator, and the outer voltage loop is used to control the capacitor voltage with the LADRC-based regulator.
To simplify the analysis, the delay caused by sampling and calculation is ignored. The PWM inverter gain
, so
. To improve the response speed, the current loop uses a proportional controller with a gain of
. The current loop regulator is:
where
and
are the output of the current loop and are compared to the PWM carrier to generate the switching signal;
and
are the output of the voltage loop and the reference of the current loop.
The closed loop transfer function of the current loop is:
Combined with Equations (1) and (3), the voltage loop is designed based on LADRC, as shown in
Figure 2. Ignoring
, the controlled object of the outer voltage loop is obtained:
where,
and
are the defined disturbance of
dq axis respectively,
.
From
Figure 2 and Equation (5), the controlled object of the outer voltage loop is a second-order system, so two second-order LADRC controllers with third-order LESO need to be designed in the
dq axis. The coupling between the
dq axis can be regarded as a part of the total disturbance. Decoupling can be performed through feedforward compensation, so the controllers of the
dq axis can be designed as separate channels with the same structure as LADRC. Simply take the
d axis as an example.
An LADRC comprises LESO and LSEF. Wherein, the total disturbance is expanded into a new state. Then LESO is adopted to estimate the total disturbance and other state information. Finally, LSEF integrates all the state information obtained from LESO and reconstructs the system state equation.
From Equation (5), the state variables and output are defined as:
Then, the following state-space equation can be derived:
where
is the system input and
is the system control gain (CG). Considering the uncertainty of
, its estimated value
is used, which is generally
;
is often called the compensation factor (CF);
denotes the total disturbance, which includes external disturbance, internal dynamics, and the estimation error between CG and CF [
29].
By extending
as a separate state,
, of the system, and assuming that
is differentiable with
, then the system model can be represented in state space:
The third-order LESO of the system can be designed as:
where
,
, and
are the state variables of LESO.
,
, and
are the observer gain.
By choosing the appropriate observer gain, the state variables of the system can be tracked quickly by the LESO in Equation (9), which is , , .
Through taking the Laplace transformation of Equation (9), we have:
To achieve good disturbance rejection, the estimation variable is added into the control input; LSEF can be designed as:
where
and
are gains of LSEF,
is the intermediate variable, and
is the reference input.
If the LESO is reliable and the estimation error of CG is negligible, then
,
,
, and
. According to Equations (7) and (11), the closed-loop transfer function of the system can be deduced:
Combining Equations (10) and (12), the characteristic Equation of LESO and the closed-loop transfer function are:
According to the parameterization technique proposed [
18], the eigenvalues of the LESO and the closed-loop transfer function can be located at
and
. Then,
is the bandwidth of the LESO and
is the bandwidth of the controller in LADRC. The tuning details are given by:
With the above configuration, the dynamic performance of the system is dependent on only two parameters, which are the controller bandwidth, , and the observer bandwidth, . With appropriate bandwidth parameters, the system can track the reference input without overshoot.
The design process of the outer voltage loop controller based on LADRC in the
d-axis is completed, including Equations (9), (11), and (14). The simplified structure of the original LADRC-based controller is shown in
Figure 3. The controller in the
q axis can be designed regarding the
d axis and will not be described in detail.
2.3. Influence of Observer Bandwidth and Compensation Factor
After designing the LADRC-based voltage loop completely, the next step is tuning the parameters to satisfy the requirements of stability and dynamic performance in the system; is chosen by the requirements of dynamic performance and is generally not an adjustable parameter.
Affected by discrete period and measurement noise, there exists a trade-off between observation accuracy and noise rejection ability when choosing . Furthermore, CG in the actual system is hard to obtain. It can be seen from Equation (7) that the selection of and the perturbation of some parameters, such as inductance and capacitance, will also lead to deviation from the theoretical value, resulting in . Therefore, it is essential and important to study the influence of CF and on system stability, anti-disturbance, and noise rejection ability.
According to Equations (10) and (14), and considering the system model (7) and the measurement noise
, the control structure of LADRC can be simplified, as shown in
Figure 4. The control input of the system is:
where
,
, and
are the Laplace transformation of
r,
y, and
n, and
,
.
Let
, which represents the ratio of CF to CG. The output of the system is:
where
.
From Equation (16), the system output consists of a tracking term, a disturbance term, and a measurement noise term; , , and are related to the system’s stability, dynamic performance, anti-disturbance, and noise rejection ability. In order to study the observer bandwidth constraints and the influence of CF, the amplitude-frequency characteristics (called Bode diagram) and root locus diagram are adopted respectively in this subsection.
The Bode diagram of the transfer function
with different
has been studied in [
19,
20,
21,
22,
25,
26], so they are not mentioned in this paper. The Bode diagrams of the transfer function
and
are given in
Figure 5, where
and
. It can be seen that increasing
can reduce the low-frequency gain of
, which can enhance the system’s anti-disturbance ability. On the contrary, the high-frequency gain of
is increased, so the system becomes more sensitive to the measurement noise. Thus, there is a trade-off between anti-disturbance and noise rejection ability when choosing
.
In Equation (16),
does not affect the closed-loop zero position; only affecting the pole position, so a root locus diagram is adopted to analyze the dynamic performance and robustness of
. To obtain better observation accuracy,
is generally chosen between 2
to 10
[
30]. Let
rad/s; the root locus with different
is shown in
Figure 6. The yellow area represents the unstable regions caused by the right-half-plane poles. The solid arrow represents the pole change when
increases from 0 to 1, while the dashed arrow represents the situation when
increases from 1 to infinity.
When , the poles are located at points A and C on the real axis, corresponding to and , and the system has no overshoot and a short settling time;
When
changes from 0 to infinite, if
is smaller than 1.02 (point D), or greater than 5.24 (point E), the system will be unstable. The range of
that makes the system stable is listed in
Table 1. As
increases, the stable range of
is expanded.
When increases from 0 to 1, poles enter the stable regions and approach point A and C. Poles , , and are closer to the imaginary axis, so are marked as dominant poles. With increasing, and gradually move away from the imaginary axis and approach the real axis, so the response time and overshoot decreases and the damping increases.
When increases from 1 to 1.02 (point B), , , and are on the real axis and there is no overshoot in response. When they continue to increase, poles approach the imaginary axis, and the damping becomes smaller. This shows that the response becomes slower, and there is both overshoot and damped oscillation, then poles cross the negative half axis, and the system becomes unstable.