Modeling, Analysis and Testing of Autonomous Operation of an Inverter-Based Microgrid


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.

  1. Inverter
  2. Inverter model
  3. Microgrid
  4. Power control
  5. Small-signal stability



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


 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.


 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.


[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.

Distributed Cooperative Control and Stability Analysis of Multiple DC Electric Springs in a DC Microgrid


Recently, dc electric springs (dc-ES s) have been proposed to understand voltage regulation and power quality improvement in dc micro grids. This paper establishes a distributed unified control framework for multiple dc- ES s in a dc micro grid and presents the small-signal stability  separation of the system. The primary level implements a droop control to coordinate the operations of multiple dc-ES s. The secondary control is based on a unity algorithm to regulate the dc-bus voltage reference, incorporating  the state-of-charge (SOC) balance among dc-ES s.


With the design, the cooperative control can achieve average dc-bus voltage consensus and maintain SOC balance among different dc-ES s using only neighbor-to-neighbor  information. Furthermore, a small-signal model of a four dc-ES s system with the primary and secondary controllers is developed. The eigenvalue analysis is presented to show   the effect of the communication weight on system stability. Finally, the effectiveness of the proposed control scheme and the small-signal model is verified in an is landed dc micro grid under different scenarios through simulation and  experimental studies.

  1. Consensus
  2. Dc micro grid
  3. Distributed control
  4. Electric springs (ES)
  5. Small-signal stability



 Fig.1.Distributed network with multiple dc -ES s



Fig. 2.SE Z. Controller comparison. (a) Node bus voltage, (b) dc-ES s output

power, (c) SOC, and (d) state variables xi .

Fig. 3. Proposed controller with different a i j . (a) and (d) Average  bus voltages with a i j = 0.5 and a i j = 10. (b) and (e) State variables with a i j = 0.5 and a i j = 10. (c) and (f) Bus voltages with a i j = 0.5  and a i j = 10.

Fig. 4. dc-ES 4 failure at 5 s. (a) Node bus voltage, (b) output power,  and (c) SOC.

Fig. 5. Proposed controller with communication delay τ . (a) Node bus  average voltage, (b) SOC, and (c) state variables xi .


Fig. 6. Proposed controller with five dc-ES s. (a) Node bus voltage, (b) output power, and (c) SOC.


A hierarchical two-level voltage control scheme was proposed for dc-ES s in a micro grid using the consensus algorithm to estimate the average dc-bus voltage and promote SOC balance among different dc-ES s. The small-signal model of four dc-E S s system incorporating the controllers was developed for eigenvalues analysis to investigate the stability of the system. The consensus of the observed average voltages and the defined state variables has been proven.


Results show that the control can improve the voltage control accuracy of dc-E S s and realize power sharing in proportion to the SOC. The resilience of the system against the link failure has been improved and the system can still maintain operations as long as the remaining communication graph has a spanning tree. Simulation and experimental results also verify that the correctness and effectiveness of the proposed model and controller strategy.


[1] X. Lu, K. Sun, J. M. Guerrero, J. C. Vasquez, and L. Huang, “State-of charge balance using adaptive droop control for distributed energy storage systems in DC micro grid applications,” IEEE Trans. Ind. Electron., vol. 61, no. 6, pp. 2804–2815, Jun. 2014.

[2] Q. Sh a f i e e, T. Drag ice v i c, J. C. Vasquez, and J. M. Guerrero, “Hierarchical control for multiple DC-micro grids clusters,” IEEE Trans. Energy Con v e r s., vol. 29, no. 4, pp. 922–933, Dec. 2014.

[3] W. Ya o, M. Chen, J. Mat as, J. M. Guerrero, and Z. M. Q i an, “Design and analysis of the droop control method for parallel invert er s  considering the impact of the complex impedance on the power sharing,” IEEE Trans. Ind. Electron., vol. 58, no. 2, pp. 576–588, Feb. 2011.

[4] V. Na s i r i an, S. M o a y e d i, A. D a v o u d i and F. Lewis, “Distributed cooperative control of DC micro grids,” IEEE Trans. Power Electron., vol. 30, no. 4, pp. 6725–6741, Dec. 2014.

[5] J. M. Guerrero, J. C. Vasquez, J. Mat as, L. G. d e Vi cu˜n a, and M. Cast i l la, “Hierarchical control of droop-controlled AC and DC micro grids—A general  approach toward standardization,” IEEE Trans. Ind. Electron., vol. 58, no. 1, pp. 158–172, Jan. 2011.