**ABSTRACT:**** **

The analysis of the small-signal stability of conventional power systems is well established, but for inverter based microgrids there is a need to establish how circuit and control features give rise to particular oscillatory modes and which of these have poor damping. This paper develops the modeling and analysis of autonomous operation of inverter-based microgrids.

Each sub-module is modeled in state-space form and all are combined together on a common reference frame. The model captures the detail of the control loops of the inverter but not the switching action. Some inverter modes are found at relatively high frequency and so a full dynamic model of the network (rather than an algebraic impedance model) is used. The complete model is linearized around an operating point and the resulting system matrix is used to derive the eigenvalues.

The eigenvalues (termed “modes”) indicate the frequency and damping of oscillatory components in the transient response. A sensitivity analysis is also presented which helps identifying the origin of each of the modes and identify possible feedback signals for design of controllers to improve the system stability.

With experience it is possible to simplify the model (reduce the order) if particular modes are not of interest as is the case with synchronous machine models. Experimental results from a microgrid of three 10-kW inverters are used to verify the results obtained from the model.

**KEYWORDS:**

- Inverter
- Inverter model
- Microgrid
- Power control
- Small-signal stability

**SOFTWARE:** MATLAB/SIMULINK

**BLOCK DIAGRAM:**

** **

Fig. 1. Typical structure of inverter-based microgrid.

** ****EXPECTED SIMULATION RESULTS:**

** **

** **Fig. 2. Active power (filtered) response of micro-sources with 3.8 kW of step

change in load power at bus 1.

Fig. 3. Reactive power exchange between the micro sources with 3.8 kW of

step change in load power at bus 1 (Initial values: Q1 =0, Q2 = 200, Q3 =

+200; Final values: Q1 = +600, Q2 = 300, Q3 = 200).

Fig. 4. Active power (filtered) response of micro-sources with 16.8 kW and

12 kVAR RL load step change at bus 1.

Fig. 5. Reactive power (filtered) response of micro-sources with 16.8 kW and

12 kVAR RL load step change at bus 1.

Fig. 6. Output voltage (d-axis) response with 27 kW of step change in load

power at bus 1.

Fig. 7. Inductor current (d-axis) response with 27 kW of step change in load

power at bus 1.

** ****CONCLUSION:**

** **In this paper, a small-signal state-space model of a microgrid is presented. The model includes inverter low frequency dynamics dynamics, high frequency dynamics, network dynamics, and load dynamics. All the sub-modules are individually modeled and are then combined on a common reference frame to obtain the complete model of the microgrid.

The model was analyzed in terms of the system eigenvalues and their sensitivity to different states. With the help of this analysis the relation between different modes and system parameters was established. It was observed that the dominant low-frequency modes are highly sensitive to the network configuration and the parameters of the power sharing controller of the micro sources. The high frequency modes are largely sensitive to the inverter inner loop controllers, network dynamics, and load dynamics.

Results obtained from the model were verified experimentally on a prototype microgrid. It was observed that the model successfully predicts the complete microgrid dynamics both in the low and high frequency range.

Small signal modeling has had a long history of use in conventional power systems. The inverter models (and the inclusion of network dynamics) illustrated in this paper allow microgrids to be designed to achieve the stability margin required of reliable power systems.

** ****REFERENCES:**

[1] R. H. Lasseter, “Microgrids,” in *Proc. Power Eng. Soc.Winter Meeting*, Jan. 2002, vol. 1, pp. 305–308.

[2] A. Arulapalam, M. Barnes, A. Engler, A. Goodwin, and N. Jenkins, “Control of power electronic interfaces in distributed generation microgrids,” *Int. J. Electron.*, vol. 91, no. 9, pp. 503–523, Sep. 2004.

[3] R. Lassetter, “Integration of Distributed Energy Resources: The CERTS Microgrid Concept,” CERT Rep., Apr. 2002.

[4] M. S. Illindala, P. Piagi, H. Zhang, G. Venkataramanan, and R. H. Lasseter, “Hardware Development of a Laboratory-Scale Microgrid Phase 2: Operation and Control of a Two-Inverter Microgrid,” Nat. Renewable Energy Rep., Mar. 2004.

[5] Y. Li, D. M. Vilathgamuwa, and P. C. Loh, “Design, analysis and realtime testing of a controller for multibus microgrid system,” *IEEE Trans. Power Electron.*, vol. 19, no. 5, pp. 1195–1204, Sep. 2004.