1. Introduction
The conventional power system presents strong inertia to support primary frequency stability. Hence, inertia is required to limit frequency variations during a disturbance due to a change in system conditions [
1]. The concept of virtual inertia was first presented in 2007. The authors used an electronic power interface to emulate the behavior of a synchronous generator (SG), known as a virtual synchronous machine (VSM) and designated as “VISMA” by the authors [
2]. Virtual inertia is the ability to replicate the inertial response of SGs in microgrid systems.
Other methods, such as VSM or VISMA [
2,
3], also called synchronverters [
4], use a control algorithm based on the mathematical model of the machine to replicate the dynamic properties of the SG in the voltage source converter (VSC) [
2,
3,
5]. The SG model is represented by equations to describe its electrical and mechanical behavior. To simplify the analysis, several assumptions are considered, such as the steady state of voltages and currents, the neglect of power losses, etc.
The swing equation describes the effect of the balance between the mechanical torque and the electromagnetic torque, establishing the relationship between the accelerating torque and the product of the moment of inertia of the rotor by its angular acceleration [
6]. It is often desirable to include the damping or stopping force in the swing equation (Equation (
1)), which acts on the system only when it is moving; this component is known as the damping torque (
), which is included in Equation (
2) [
7].
Equation (
2) can be written based on the parameters of a VSC, such as power and the angular frequency, so it can be expressed as:
where
is the nominal angular velocity (frequency) of the grid,
is the damping coefficient,
H is the virtual inertia constant,
is the virtual mechanical torque of the VSM,
is the electrical torque of the VSM,
is the angular acceleration of the VSM,
is the reference setpoint mechanical power, and
is the measured active power output from the grid.
Microgrids are made up of groups of interconnected loads and distributed generators (DGs) with defined limits [
8]. VSG-type systems are increasingly added to provide virtual inertia to microgrids. Using this topology, DGs emulate small synchronous machines connected by short lines. Hence, the connection lines exhibit the characteristics of resistance and inductance components [
9,
10]. Thus, the assumption that the line impedance is predominantly inductive is not always true in microgrid systems [
11,
12]. Therefore, VSG controllers in low-voltage microgrids with complex line impedances present challenges, such as poor power dynamic performance and voltage and frequency deviations due to the lack of the resistance component consideration in the electrical parameters of the connecting lines [
10,
12,
13].
Due to these features, other approaches have been addressed in the literature, such as the adaptive control structures of virtual synchronous generators (AVSGs) [
14], whose objective is based on online calculations for the adjustment of controller parameters to surf in strong and weak grid conditions.
The proposed approach differs from the AVSG because it does not involve modifying the VSG control parameters. Instead, we calculate the active and reactive power by utilizing the line parameters. As a result, the power coupling can be significantly reduced with the proposed method.
In order to improve the power decoupling caused by the characteristics of a complex line impedance connection, this document introduces the RVSG controller for grid-connected microgrids, keeping the same principle of virtual inertia imposition for frequency restoration. In addition, the proposed RVSG compensates the voltage and angle to fit the new reference frame that takes into account the impedance characteristics of the line. Considering the above issues, this paper initially presents a novel mathematical approach that uses the power line characteristics in a microgrid to rotate the power control reference frame and proposes a new control method called the “Rotated Virtual Synchronous Generator” (RVSG).
The advantages of the proposed control method are summarized as follows:
The use of this proposed mathematical approach allows microgrids that lack inertia and operate with complex line impedance to emulate virtual inertia.
The RVSG adjusts the voltage and angle of the VSC to the new reference frame and, consequently, improves the stability and performance of the microgrid in which it is implemented.
The method minimizes the coupling of active and reactive power, improving the dynamic behavior, such as lower steady-state error, fewer oscillations, shorter settling time, and minimum overshoot.
Active and reactive power control with the RVSG improves power sharing, power quality, stability, and performance, mainly in microgrids with complex line impedance.
The rest of this paper is organized as follows:
Section 2 provides a brief description of the reference frame for power sharing by introducing the proposed orthogonal rotational transformation matrix. In
Section 3, the proposed RVSG utilizing the rotated reference frame approach is presented.
Section 4 presents a rotated state-space model of a voltage source converter. In
Section 5, the stability study of the RVSG approach is developed, along with a comparison with the traditional VSG approach, and then
Section 6 presents the simulations and experimental results obtained with each VSG method. Finally, conclusions and highlights of the RVSG are presented in
Section 7.
3. Integration of Rotation into VSG Control
This section presents a rotated virtual synchronous generator (RVSG) control method, which uses the new rotated reference frame approach described in the previous section. The RVSG control method allows the emulation of the same behavior of the virtual inertia in a VSC, improving the VSG method presented by Driesen and Visscher [
17]. This control scheme is similar to the VSG method, which is located in the active power control loop [
18]. The proposed control approach allows for fulfilling the premise of decoupling active and reactive power, achieving better performance.
Figure 2 depicts how the RVSG is implemented.
The new rotational reference frame (
8) and the behavior of a traditional swing equation, (
3) are executed in the proposed RVSG. The rotated power droop control of
G, which is responsible for controlling the voltage amplitude (
13), is similar to proportional reactive power droop control. As a result, the power control law of the RVSG is as follows:
where
is the damping coefficient,
H is the virtual inertia constant,
is the angular acceleration of the RVSG,
is the RVSG controller output frequency, which is integrated to obtain the angle
, and
v is the RVSG controller output voltage.
is the slope of the rotated droop controller of
G.
and
are the frequency and voltage references, respectively.
For practical purposes, the line impedances
and
are used in the transformation matrix of the new frame of reference (
8), where
and
contain the sum of the output impedance of the LCL filter and the grid impedance end to the AC bus connection point. Therefore, the line impedance values (
R) and (
X) equate to
and
.
To interface with the voltage and frequency coming from outside the RVSG, the voltage and reference angle required for the inverter operation must be converted to the new reference frame. For this purpose, Equations (
14) and (
15) are used to solve the nominal voltage
and delta angle
. Using the concept that when the VSC is synchronized, the phase angle between the inverter voltages and the Point of Common Coupling (PCC) must be close to zero, it is assumed that
and
.
These equations are implemented in block
, as depicted in
Figure 2, where (
14) and (
15) can be expressed as follows:
where
is the grid voltage magnitude, which is measured once at the PCC, before connecting the inverter, and
is the inverter reference output voltage.
4. Model of a Voltage Source Converter with RVSG Control
Following the same approach presented in [
19,
20], the small-signal state-space model for the RVSG controller was achieved. Considering the external power control loop the output filter, and the impedance grid used as mentioned previously, where the external power control loop setpoints for the amplitude of the voltage and frequency on the inverter output are determined according to the RVSG control law using the Equations (
10), (
12), and (
13), as explained in the previous section. Due to their quick reaction times in comparison to the rest of the system, the inner control loops (voltage and current) are not taken into account in this modeling, so a simplified state-space model is presented. Therefore, having a simplified model offers the advantage of focusing on the behavior of the eigenvalues corresponding to the power dynamics and thus identifying the factors that affect the stability of the system. A linear system with inputs
u, outputs
y, and state variables
x has the following generic structure in the state space:
where
A is the state matrix,
B is the input matrix,
C the output matrix, and
D the direct transmission matrix.
By taking into account a complex line impedance, the small-signal state-space power controller model for the RVSG may be described as follows:
where the matrices in (
17) and (
18) are explained in (
19).
,
, and
are made up of several sub-matrices, which can be found by means of Equations (
12) and (
13).
where
,
,
,
,
,
It is crucial to emphasize that this approach’s state-space model is based on a grid-connected system, unlike [
19,
20]. The component matrix
from matrix
, which incorporates the rotation reference frame into the state-space model, is another distinction. The
values from the
input matrix are shown in (
18).
The linearized state-space model may be described as follows using the same approach as the control power and the linearized small-signal model of the output (LCL) filter [
19].
where
,
, and
are the state, input, and output matrices of the state-space model of the output LC filter and the coupling line impedance, respectively. It is important to mention that matrix
is an identity matrix of length six used to see the state values, and
is equal to the zero matrices. The matrices of (
22) are defined in (
24). It can be seen that matrix
is composed of different sub-matrices, which are defined in (
25).
where
The whole model of an inverter may be generated by combining the output LC filter, coupling grid impedance, and the small-signal state-space model of the power controller. By combining all the state-space models presented in (
17), (
18), (
22), and (
23), it can be described as follows:
where
and
are the state and input variables, and
and
are the state and input matrices of the state-space of the model of the inverter. The output of the model variables is given by
, where
is the identity matrix and
is a zero matrix. As a result, a whole inverter’s state variables and related matrices are shown as:
5. Small-Signal Stability
In this part, the eigenvalue concept is used to examine the stability of the VSG controllers and the proposed RVSG controller. By changing the line impedance and VSG settings, the eigenvalue behavior of the two controllers is compared. To make the comparison fair, the inner loop parameters, initial conditions, and filter settings are fixed and the same for both scenarios. Each case’s state matrix A is used to determine the system’s eigenvalues using a MATLAB program. In the RVSG case, the matrix used is
given by Equation (
27). In the VSG case, the matrix used is
without internally implementing the proposed rotation. The parameters utilized to conduct the analysis are provided in
Table 1. The modeling uses the grid impedance values and VSG controller parameters shown in
Table 1, while the inverter hardware parameters are shown in
Table 2.
5.1. Analysis of the Behavior of the Eigenvalues When Changing the Line Impedance Parameters
Figure 3 illustrates how the eigenvalues for the RVSG and VSG systems respond to variations in the inductance of the line parameter for a case where
,
s,
Ns/m, and
. In the figure, it can be observed that the increase in the line inductance value
= (
) makes the system unstable by moving the dominant eigenvalues of the VSG to the right side of the origin. These correspond to the
cluster represented in the close-up plot made at the poles near the origin. Conversely, it is noted that in the RVSG situation of the
cluster, the same dominating eigenvalues also move to the right side, but in this instance, the poles stay on the left side of the origin and the system continues to be stable.
In the VSG model, the cluster of real eigenvalues shifts to the left side as the line inductance increases, causing those poles to have less influence on the system. Compared to the case of the RVSG, the same eigenvalues corresponding to cluster also shift to the left side, but these start closer to the origin, causing those poles to have more influence on the system. Therefore, the transient responses with the cluster have small amplitudes and pass away quickly.
In order to see how the eigenvalues behaved when the line resistance parameter was the only one changed, a new study was conducted using the same methodology as the prior analysis. This study shows that the displacement of the entire locus diagram (
Figure 3) to the left is caused by an increase in the line resistance value. As a result, every system eigenvalue is moved into a new location that offers quicker dynamics. As the
and
clusters are dominant and enhance the stability of the system when rotation is taken into account, this effect provides greater advantages for the RVSG system than for the VSG system.
5.2. Analysis of the Behavior of the Eigenvalues When Changing the VSG Parameters
In this instance,
Figure 4 illustrates how the eigenvalues respond to variations in the reactive power coefficient
for the case where
,
H,
Ns/m, and
s. Changes in the damping coefficient
and in the virtual inertia constant
H have minimal impact on the system’s stability and cause the poles to shift relatively little. Variations in these parameters are thus not taken into account. Additionally, it can be seen in
Figure 4 that the system becomes unstable for high values of
because the eigenvalues of the
cluster from the VSG move to the right side of the origin. Conversely, in the RVSG scenario, the cluster
eigenvalues greatly enhance the system performance in comparison to the VSG, keeping the system stable in the face of high values of
. This indicates that the RVSG permits dealing with quick reactive power responses, in contrast to the VSG. Moreover, the
and
eigenvalue clusters behave similarly to the preceding example when the slope of reactive power rises.
6. Simulation and Experimental Results
As illustrated in
Figure 2, a scenario using a VSI is suggested in order to assess the effectiveness of the proposed RVSG controller. The mathematical state-space model that was previously described was used via simulation and was compared behaviorally to the experimental testbed in order to validate the controller. The area marked as Software in
Figure 2 corresponds to the RVSG and VSG control algorithms, which were implemented with Matlab Simulink® and compiled using ConfigurationDesk®, downloaded to the dSPACE SCALEXIO platform. The testbed of the microgrid laboratory of the University of Puerto Rico at Matyaguez was used. This testbed contains a Danfoss VLT-302 inverter with LCL filters, a sensor box, and other components. The testbed and the most commonly used control blocks are described in more detail in [
18,
21,
22]. The inverter was synchronized and coupled to the grid at the Point of Common Coupling (PCC) at the beginning of the experiment (before the active power step). The settings in
Table 1 and
Table 2 were used for both systems to obtain the findings shown in
Figure 5 and
Figure 6. To monitor the experimental power output, a Low-Pass Filter (LPF) with a bandwidth of 18.85 rad/s was employed.
The active and reactive power responses output from the mathematical model and from the experiment are shown in
Figure 5 and
Figure 6, respectively, where both RVSG and VSG controllers use the same power step.
The behavior of the experimental system’s output power and the mathematical model using the RVSG controller is shown in the upper part of
Figure 5 and
Figure 6. The behavior of the output power of the mathematical model and the experimental system using the VSG controller can be seen in the lower part of the same figures (
Figure 5 and
Figure 6).
Figure 5 and
Figure 6 were developed to show that the mathematical model follows the experimental behavior of the power, so the eigenvalues of the previous section agree with the dynamics of the experimental power responses.
Figure 7 shows a comparison between the experimental responses of the active and reactive power for the RVSG and VSG methods; it also shows the changes in the power that are due to power steps and the events that occur due to the power coupling.
The upper part of
Figure 7 shows that after 0.5 s, when the active power step occurs, the proposed RVSG reaches the steady state six times faster than the VSG controller, and the overshoot decreases by 30% when comparing the transient responses of the two controllers. Moreover, the active power disturbance in the RVSG controller is 60% less than that in the VSG controller when a reactive power step occurs after 10 seconds. As a result, the RVSG controller’s active and reactive power are less coupled than those of the VSG controller.
The lower part of
Figure 7 shows the reactive power response to a 0.5 p.u. step in the reactive power reference, where the steady-state inaccuracy of the reactive power both before and after applying the step in the VSG controller is observed. In contrast, the RVSG controller has a steady-state error of no more than 4%, but the VSG has a steady-state error of more than 20%.
As seen in
Figure 4, systems that require quick power responses, particularly for reactive power, are susceptible to entering unstable zones. When VSG controllers are used, the system is slower and performs poorly, which is caused by the utilization of small slope values in the reactive power droop controller. The proposed RVSG control enables the use of greater slope values in reactive power VSG parameters, and thus, a quick reactive power response is expected.
Figure 7 shows that the reactive power in the RVSG controller reaches a steady state two times faster than the VSG controller. A summary of the comparison of the dynamic characteristics of the power responses between RVSG and VSG is shown in
Table 3.
7. Conclusions
This paper presents an RVSG controller that improves the transient response, performance, and precision in power-sharing strategies in microgrids with complex line impedances. It is important to note that the power quality is enhanced by employing a rotational reference frame technique since the coupling of power is reduced.
The rotated reference frame is incorporated into the state-space model provided in this paper.
In order to improve stability under stronger reactive power VSG parameters, this innovative approach enables power-sharing controllers such as the RVSG to be used in microgrids without ignoring the line impedance resistance parameter.
Compared with VSG controllers, the experimental results show that the proposed RVSG has positive effects on the stability and dynamic behavior in power sharing.
A lower steady-state deviation error is observed in the active and reactive power performance of the proposed RVSG for microgrids with complex line impedances.