The equivalent circuit of an inverter in microgrid is shown in Figure 1. In Figure 1, *L* and *R* are the line parameters; *V*, *E* and *I* denote inverter side voltage, load side voltage and the line current respectively; *ϕ*_{V} and *ϕ*_{E} represent the phase angle of *V* and *E*; *P* and *Q* are the average real and reactive powers.

Figure 1 Equivalent circuit of an inverter in microgrid

The following relations can be obtained for highly inductive transmission line

$$\begin{array}{}{\displaystyle V=L\frac{dI}{dt}+E}\end{array}$$(1)

$$\begin{array}{}{\displaystyle S=P+jQ={I}^{\ast}E}\end{array}$$(2)

*Ψ*_{V} and *Ψ*_{E} are the inverter side flux and load side flux which are obtained by the time integration of *V* and *E*. They satisfy

$$\begin{array}{}{\displaystyle {\psi}_{V}=\underset{-\mathrm{\infty}}{\overset{t}{\int}}Vdt}\end{array}$$(3)

$$\begin{array}{}{\displaystyle {\psi}_{E}=\underset{-\mathrm{\infty}}{\overset{t}{\int}}Edt}\end{array}$$(4)

Combining (1) (3) (4), the following formula can be obtained
$$\begin{array}{}{\displaystyle I=\frac{{\psi}_{V}-{\psi}_{E}}{L}}\end{array}$$(5)

By substituting (4) (5) into (2), powers supplied to the load can be obtained

$$\begin{array}{}{\displaystyle P=\frac{w\left|{\psi}_{V}\right|\left|{\psi}_{E}\right|}{L}\mathrm{sin}\delta}\end{array}$$(6)

$$\begin{array}{}{\displaystyle Q=\frac{w\left|{\psi}_{E}\right|}{L}(\left|{\psi}_{V}\right|\mathrm{cos}\delta -\left|{\psi}_{E}\right|)}\end{array}$$(7)

Considering the value of *δ* is typically small, active and reactive powers can be controlled by *δ* and |*Ψ*_{V}|, respectively. Thus the virtual flux droop formulas can be obtained

$$\begin{array}{}{\displaystyle \delta ={\delta}_{n}-m(P-{P}_{n})}\end{array}$$(8)

$$\begin{array}{}{\displaystyle \left|{\psi}_{V}\right|={\left|{\psi}_{V}\right|}_{n}-n(Q-{Q}_{n})}\end{array}$$(9)

where *δ*_{n} and |*Ψ*_{V}|_{n} are the nominal flux angle difference and amplitude of inverter flux; *P*_{n} and *Q*_{n} are the rated active and reactive powers; *m* and *n* are the droop coefficients of the *P* - *δ* and *Q* — |*Ψ*_{V}|.

Figure 2 is the overall control structure of flux droop method, which consists of two parts: flux droop controller and direct flux controller. As shown in Figure 2, |*Ψ*_{V}|_{ref}and *δ*_{ref} from flux droop controller are compared with their actual values to get deviation signals. Then, deviation signals combined with the current position of inverter flux are used to select the proper voltage vectors according to a switching table during each period [9]. The control system is simple and has excellent control performance.

Figure 2 The overall control block diagram of flux droop control

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