Design and Analysis of Decentralized Virtual Impedance Based Controller for Enhancing Power Sharing and Stability in an Islanded Microgrid

Abstract

This paper investigates the role of a virtual impedance controller in improving the small-signal stability and power-sharing for an inverter-based islanded microgrid. The stability of microgrids is badly affected due to undamped low-frequency oscillations, which are often caused due to improper power-sharing among micro sources. The droop control approach has provided a viable solution to improve the power-sharing and hence the system’s stability. However, the droop gains suffer an inherent trade-off between power-sharing and stability, with a change in system dynamics like step load change. An increase in the droop gains improves power-sharing but causes low-frequency oscillating modes at the load terminals. It also deteriorates the micro sources voltage stability at the output terminals. This paper overcomes the trade-off faced by the droop controller with an approach of virtual impedance modeling along with voltage and current controller loops. A novel approach has been adapted by considering the virtual impedance as one state variable and investigating system stability by plotting the eigenvalue spectrum as a function of the droop coefficient. Participation factor-based sensitivity analysis for the proposed system has been carried out to measure the relative participation of corresponding state variables in respective oscillating modes. Time-domain simulation validates the role of virtual impedance in improving the dam** performance and enhancing the participation of DG’s during the transient mode. It also provides an insight into the role of a virtual impedance controller in improving reactive power sharing and voltage stability.

Appendices

Appendix

Power Controller Modelling

$$\begin{aligned} A_{P}=\left[ \begin{array}{ccc} 0 &{} -m_{p} &{} 0 \\ 0 &{} -w_{c} &{} 0 \\ 0 &{} 0 &{} -w_{c} \end{array}\right] \end{aligned}$$

(A.1)

$$\begin{aligned} B_{P}=\left[ \begin{array}{cccccc} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} w_{c} I_{o d} &{} w_{c} I_{o q} &{} w_{c} V_{o d} &{} w_{c} V_{o q} \\ 0 &{} 0 &{} w_{c} I_{o q} &{} -w_{c} I_{o d} &{} -w_{c} V_{o q} &{} w_{c} V_{o d} \end{array}\right] \end{aligned}$$

(A.2)

$$\begin{aligned} B_{P c o m}=\left[ \begin{array}{c} -1 \\ 0 \\ 0 \end{array}\right] \end{aligned}$$

(A.3)

$$\begin{aligned} C_{P w}=\left[ \begin{array}{lll} 0&-m_{p}&0 \end{array}\right] \end{aligned}$$

(A.4)

$$\begin{aligned} C_{P v}=\left[ \begin{array}{ccc} 0 &{} 0 &{} -n_{q} \\ 0 &{} 0 &{} 0 \end{array}\right] \end{aligned}$$

(A.5)

Virtual Impedance Modelling

$$\begin{aligned} A_{Z}=\left[ \begin{array}{ll} 0 &{} 0 \\ 0 &{} 0 \end{array}\right] \end{aligned}$$

(B.1)

$$\begin{aligned} B_{Z 1}=\left[ \begin{array}{ll} 1 &{} 0 \\ 0 &{} 1 \end{array}\right] \end{aligned}$$

(B.2)

$$\begin{aligned} B_{Z 2}=\left[ \begin{array}{llllll} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right] \end{aligned}$$

(B.3)

$$\begin{aligned} C_{Z}=\left[ \begin{array}{ll} 0 &{} 0 \\ 0 &{} 0 \end{array}\right] \end{aligned}$$

(B.4)

$$\begin{aligned} D_{Z 1}=\left[ \begin{array}{cc} -I_{o d} Z_{V} &{} 0 \\ 0 &{} -I_{o q} Z_{V} \end{array}\right] \end{aligned}$$

(B.5)

$$\begin{aligned} D_{Z 2}=\left[ \begin{array}{cccccc} 0 &{} 0 &{} 0 &{} 0 &{} -Z_{V} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -Z_{V} \end{array}\right] \end{aligned}$$

(B.6)

Current Controller Modelling

$$\begin{aligned} A_{c}=\left[ \begin{array}{ll} 0 &{} 0 \\ 0 &{} 0 \end{array}\right] \end{aligned}$$

(C.1)

$$\begin{aligned} B_{C 1}=\left[ \begin{array}{ll} 1 &{} 0 \\ 0 &{} 1 \end{array}\right] \end{aligned}$$

(C.2)

$$\begin{aligned} B_{C 2}=\left[ \begin{array}{cccccc} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right] \end{aligned}$$

(C.3)

$$\begin{aligned} C_{c}=\left[ \begin{array}{cc} K_{i c} &{} 0 \\ 0 &{} K_{i c} \end{array}\right] \end{aligned}$$

(C.4)

$$\begin{aligned} D_{C 1}=\left[ \begin{array}{cc} K_{p c} &{} 0 \\ 0 &{} K_{p c} \end{array}\right] \end{aligned}$$

(C.5)

$$\begin{aligned} D_{C 2}=\left[ \begin{array}{llllll} -K_{p c} &{} -w_{n} L_{f} &{} 0 &{} 0 &{} 0 &{} 0 \\ w_{n} L_{f} &{} -K_{p c} &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right] \end{aligned}$$

(C.6)

Filter Modelling

$$\begin{aligned} A_{L C L}=\left[ \begin{array}{cccccc} -\frac{r_{f}}{L_{f}} &{} w_{0} &{} -\frac{1}{L_{f}} &{} 0 &{} 0 &{} 0 \\ -w_{0} &{} -\frac{r_{f}}{L_{f}} &{} 0 &{} -\frac{1}{L_{f}} &{} 0 &{} 0 \\ \frac{1}{C_{f}} &{} 0 &{} 0 &{} w_{0} &{} -\frac{1}{C_{f}} &{} 0 \\ 0 &{} \frac{1}{C_{f}} &{} -w_{0} &{} 0 &{} 0 &{} -\frac{1}{C_{f}} \\ 0 &{} 0 &{} \frac{1}{L_{c}} &{} 0 &{} -\frac{r_{c}}{L_{c}} &{} w_{0} \\ 0 &{} 0 &{} 0 &{} \frac{1}{L_{c}} &{} -w_{0} &{} -\frac{r_{c}}{L_{c}} \end{array}\right] \end{aligned}$$

(D.1)

$$\begin{aligned} B_{L C L 1}=\left[ \begin{array}{cc} \frac{1}{L_{f}} &{} 0 \\ 0 &{} \frac{1}{L_{f}} \\ 0 &{} 0 \\ 0 &{} 0 \\ 0 &{} 0 \\ 0 &{} 0 \end{array}\right] \end{aligned}$$

(D.2)

$$\begin{aligned} B_{L C L 2}=\left[ \begin{array}{cc} 0 &{} 0 \\ 0 &{} 0 \\ 0 &{} 0 \\ 0 &{} 0 \\ -\frac{1}{L_{c}} &{} 0 \\ 0 &{} -\frac{1}{L_{c}} \end{array}\right] B_{L C L 3}=\left[ \begin{array}{c} I_{l q} \\ -I_{l d} \\ V_{o q} \\ -V_{o d} \\ I_{o q} \\ -I_{o d} \end{array}\right] \end{aligned}$$

(D.3)

System Parameters

See Table 7.

Table 7 System parameters

Inverter Modelling

$$\begin{aligned} B_{I N V i}=\left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ B_{L C L 2} T_{s}^{-1} \end{array}\right] B_{i w c o m}=\left[ \begin{array}{c} B_{P w c o m} \\ 0 \\ 0 \\ 0 \end{array}\right] \end{aligned}$$

(F.1)

$$\begin{aligned} \begin{aligned} C_{I N V w_{i}}&=\left[ \begin{array}{lll} C_{P w}&0&0 \end{array}\right] \text{ for } i=1 \\ C_{I N V c i}&=\left[ \begin{array}{lllll} T_{c}&0&0 \end{array}\right]&0 \quad 0&\left. \left[ \begin{array}{lll} 0&0&T_{s} \end{array}\right] \right] \\&=\left[ \begin{array}{llll} 0&0&0&0 \end{array}\right] \text{ for } i \ne 1 \end{aligned} \end{aligned}$$

(F.2)