This paper presents a mathematical (small-signal) model of an electronically interfaced distributed generator (DG) by considering the effect of voltage and frequency variations of the prime source. Dynamic equations are found by linearization about an operating point. In this study, the dynamic of DC part of the interface is included in the model. The stability analysis shows with proper selection of system parameters; the system is stable during steady-state and dynamic situations, and oscillatory modes are well damped. The proposed model is useful to study stability analysis of a standalone DG or a Microgrid.
Distributed generation (DG) systems have been expected to be an important electric power supply system for next generation. DGs are able to be installed near the loads, so they can increase the power quality and reliability of electricity delivered to sensitive loads. Some of the DG technologies require a power electronics interface in order to convert the energy into the grid compatible AC power. These interface devices make the sources more flexible in their operation and control compared to the conventional electrical machines. However, due to their negligible physical inertia, they also make the system potentially susceptible to oscillation resulting from network disturbances [
The coordinated operation and control of DGs together with loads and storage devices are central to the concept of microgrid [
Reference [
In most articles, DC part of the DG (DC filter’s voltage and current) is not considered in dynamic studies, or it is assumed to be a constant value [
The objective of this paper is to find a comprehensive dynamic model of DG, including the prime source, power electronically interfaces, output filter, and controller. The proposed model represents all components of the DG in a
The dynamic stability of the DG is investigated by the small signal and step response analysis. In this paper, the DG is connected only to the static load, but it can be extended into the network or typical microgrid for more applications.
A simplified single-line diagram of the studied DG is illustrated in Figure
Single line equivalent of DG.
As shown in Figure
In this study, the sin-triangle modulation strategy is used to prepare applied voltage in the converter. The next parts of the control system are the outer voltage and inner current control loops, which are designed to reject high-frequency disturbances and provide sufficient damping for the output filter by using the proportional integral (PI) compensators.
The dynamic model of AC side of DG in three-phase
The average voltage and current of capacitor and inductance of dc filter will change with time during transient. For the rectifier under operation without phase delay and with commutating inductance, we have [
We will see that the
Figure
Inverter power controller.
Common reference frame.
The power measurement outputs are given by (
All equations of DG have been written in its reference frame. In power system analysis, it is necessary to relate all components’ reference frames to a common frame. This common reference frame would be represented as constant voltages in the synchronous reference frame. In microgrid studies, DG with the biggest nominal power is considered as a common reference frame.
Figure
By liberalization of nonlinear differential equations around the operating steady-state point, the small-signal model of the system has been created [
For inverter equations, first output variables must be transferred to the common reference frame and then be linearized. By using (
The output vector of the system is defined as a linear combination of control and state variables in the form of
Finally, the transfer function is defined by
In power systems small-signal stability analysis is aimed to determine the properties of operation parameter variations that are independent from disturbance intensity. Eigenvalue analysis is used to show the information of different stability modes for power system small-signal stability problems. A system is stable when all of its modes are stable. Furthermore, it is required that all oscillations are well and quickly damped. This technique is declared in [
In this paper, the participation factors and eigenvalues of the state matrix are computed and analyzed. Eigenvalues with negative real parts show that the system is stable. the state variable with the highest normalized participation factor is the best choice for feedback signal if it may be a measurable physical variable.
To study small-signal stability of the model, a 10 kVA DG connected to a 9 kW static load is considered as the case study. All DG information is taken from [
The rectifier’s input inductance
The DC filter is involved in a small series inductance,
In order to realize the sensitivity and dynamic behaviour, eigenvalues trajectory plots are drawn as a function of some system parameters. Figure
Loci of eigenvalues according to kp changing.
Figure
Loci of eigenvalues according to kv changing.
Trajectory plots show instability in modes of 1, 2, 3, and 4 if the
With regard to
Loci of eigenvalues according to wf changing.
Results are used to adjust the gains of the power controller. With optimal values of controller gains and appropriate selection of the DG parameters, all oscillatory modes are well damped, and the proposed model is stable.
By using QR-decomposition method in the MATLAB software environment, the eigenvectors and eigenvalues of the state matrix have been computed in operating point. The DG system along with its controller is described by six eigenvalues:
The normalized participation factors of the dynamic model are represented in Table
Participation factors.
|
|
|
|
|
| |
---|---|---|---|---|---|---|
|
0.2428 | 0.2428 | 0.2001 | 0.2001 | 0.0781 | 0.0 |
|
0.0497 | 0.0497 | 0.3646 | 0.3646 | 0.0829 | 0.0 |
|
0.4458 | 0.4458 | 0.0528 | 0.0528 | 0.0 | 0.0 |
|
0.2522 | 0.2522 | 0.2025 | 0.2025 | 0.0463 | 0.0 |
|
0.0094 | 0.0094 | 0.1782 | 0.1782 | 0.7923 | 0.0005 |
|
0.0001 | 0.0001 | 0.0019 | 0.0019 | 0.0004 | 0.9995 |
Figures
Step response of current to the source voltage.
Step response of current to the source angular velocity.
The mathematical model of the inverter-based DG has been proposed in this paper. The proposed model has included power and control circuits of the DG. The dynamics of the DC filter and the effects of the prime source have been also considered in the DG model. By using eigenvalue analysis, the stability of proposed model has been studied in detail. This study presented that all oscillatory modes are well damped with suitable selection of system parameters. Since the proposed model is stable, it can be used in the dynamic and steady-state study of a microgrid.
Linear equations of the DG.
By substituting (
By rearranging (
Linear form of (
In the same way of the active power control, from (