Dynamic virtual resistance-based droop control for seamless transition operation of multi-parallel microgrid inverters

Abstract

Inverter is required to operate at both grid-connected and grid-forming mode for microgrid. When an unplanned microgrid disconnecting to grid circumstance happens, the transition will cause severe current shock to system with multi-parallel microgrid inverters. This paper adopts a dynamic virtual resistance-based droop control strategy and small signal analysis method to reduce the current shock, improve the system stability and safety. The large dynamic virtual resistance is added to the control loop when the mode transition event occurs so the shock current is limited. Comparison between the capacitor current feedback for seamless mode transition method and the proposed method is made to prove the validation, and parameters’ impact on the system is also analyzed. Results, which are conducted on MATLAB/Simulation and Hardware-in-the-Loop (HIL) experimental platform verify the feasibility of the proposed method in this paper, effectively reduce the current shock by more than 60%.

Appendices

Appendix A Abbreviation and variable list

PLL	Phase lock loop
HIL	Hard ware in the loop
PCC	Point of common coupling
I cc	Circling current among inverters
R v	Dynamic virtual resistance
R s	Initial value of virtual resistance
R 0	Eventual value of virtual resistance
τ	Time constant of proposed method
t 0	Initial time for mode transition
u * c,dq	Input reference of inner voltage and current loop
ũ * c,dq	Output voltage reference from droop loop
i 2,dq	Output current of the inverter
L v	Virtual inductance
Guref(s)	Transfer function from voltage reference to output voltage in voltage-oriented scheme
Zo(s)	Output impedance in voltage-oriented scheme
Yo(s)	Output conduction in current-oriented scheme
Giref(s)	Transfer function from current reference to output current in current-oriented scheme
L 1	Machine side inductor in LCL filter
L 2	Grid side inductor in LCL filter
C 1	Capacitor in LCL filter
i 1,dq	Current on L1
i 2,dq	Current on L2
u c,dq	Voltage on C1
u s,dq	Output voltage behind the IGBT bridge
u pcc,dq	Voltage on PCC
x filt	State space variable for the LCL filter
A f , B f1 , B f2	Parameter matrices for state space model of LCL filter
k pin	Proportional gain for the current loop from inner loop
k po	Proportional gain for the voltage loop from inner loop
k io	Integral gain for the voltage loop from inner loop
i * 1,dq	Current reference for current loop from inner loop
x s	State space variable for inner loop controller
As, Bs1, Bs2, Cs, Ds1, Ds2	Parameter matrices for inner loop controller
T hpf	Time constant of high pass filter in virtual impedance method
x v	State space variable for virtual impedance loop
Av, Bv1, Bv2, Cv, Dv1, Dv2	Parameter matrices for virtual impedance loop
n Q	Reactive power droop coefficient
n P	Active power droop coefficient
Q o	Reactive power output
P o	Active power output
T lfp	Time constant of low pass filter from droop loop
x dq	State space variable for droop loop
A pq , B pq , C pq	Parameter matrices for droop loop
[Ps, Qs]	Power reference array for grid-connected mode

Appendix B Transfer function in Sect. 3

For voltage-oriented mode controller

$$ \begin{gathered} G_{{\text{u}}}^{{{\text{ref}}}} \left( s \right) = \frac{{c_{1} s + c_{0} }}{{n_{4} s^{4} + n_{3} s^{3} + n_{2} s^{2} + n_{1} s + n_{0} }} \hfill \\ Z_{{\text{o}}} \left( s \right) = \frac{{m_{3} s^{3} + m_{2} s^{2} + m_{1} s + m_{0} }}{{n_{4} s^{4} + n_{3} s^{3} + n_{2} s^{2} + n_{1} s + n_{0} }} \hfill \\ \qquad + \left( {L_{2} + L_{{\text{v}}} } \right)s + r_{2} + r_{{\text{v}}} \hfill \\ \end{gathered} $$

where,

$$ \begin{gathered} c_{1} = k_{{{\text{pin}}}} k_{{{\text{po}}}} \hfill \\ c_{0} = k_{{{\text{pin}}}} k_{{{\text{io}}}} \hfill \\ m_{3} = L_{1} T_{{\text{s}}} \hfill \\ m_{2} = L_{1} + T_{{\text{s}}} k_{{{\text{pin}}}} + T_{{\text{s}}} r_{1} \hfill \\ m_{1} = r_{1} \hfill \\ m_{0} = 0 \hfill \\ n_{4} = C_{1} L_{1} T_{{\text{s}}} \hfill \\ n_{3} = C_{1} L_{1} + C_{1} T_{{\text{s}}} k_{{{\text{pin}}}} + C_{1} T_{{\text{s}}} r_{1} \hfill \\ n_{2} = C_{1} k_{{{\text{pin}}}} + C_{1} r_{1} \hfill \\ n_{1} = k_{{{\text{pin}}}} k_{{{\text{po}}}} \hfill \\ n_{0} = k_{{{\text{pin}}}} k_{{{\text{io}}}} \hfill \\ \end{gathered} $$

And T_s is the control period time of inner PWM. For current-oriented mode controller

$$ \begin{gathered} G_{I}^{{{\text{ref}}}} \left( s \right) = \frac{{a_{1} s + a_{0} }}{{q_{5} s^{5} + q_{4} s^{4} + q_{3} s^{3} + q_{2} s^{2} + q_{1} s + q_{0} }} \hfill \\ Y_{{\text{o}}} \left( s \right) = \frac{{p_{4} s^{4} + p_{3} s^{3} + p_{2} s^{2} + p_{1} s + p_{0} }}{{q_{5} s^{5} + q_{4} s^{4} + q_{3} s^{3} + q_{2} s^{2} + q_{1} s + q_{0} }} \hfill \\ \end{gathered} $$

where,

$$ \begin{aligned} a_{1} = & R_{{\text{v}}} k_{{\text{p}}} \\ a_{0} = & R_{{\text{v}}} k_{{\text{i}}} \\ p_{4} = & C_{1} L_{1} R_{{\text{v}}} T_{{\text{s}}} \\ p_{3} = & L_{1} T_{{\text{s}}} + C_{1} L_{1} R_{{\text{v}}} + C_{1} R_{{\text{v}}} T_{{\text{s}}} r_{1} \\ p_{2} = & L_{1} + R_{{\text{v}}} T_{{\text{s}}} + T_{{\text{s}}} r_{1} + C_{1} R_{{\text{v}}} r_{1} \\ p_{1} = & r_{1} \\ p_{0} = & 0 \\ \end{aligned} $$

$$ \begin{aligned} q_{5} = & C_{1} L_{1} L_{2} R_{{\text{v}}} T_{{\text{s}}} \\ q_{4} = & L_{1} L_{2} T_{{\text{s}}} + C_{1} L_{1} L_{2} R_{{\text{v}}} + C_{1} L_{1} R_{{\text{v}}} T_{{\text{s}}} r_{2} + C_{1} L_{2} R_{{\text{v}}} T_{{\text{s}}} r_{1} \\ q_{3} = & L_{1} L_{2} + L_{1} R_{{\text{v}}} T_{{\text{s}}} + L_{2} R_{{\text{v}}} T_{{\text{s}}} + L_{1} T_{{\text{s}}} r_{2} + L_{2} T_{{\text{s}}} r_{1} + \\ & C_{1} L_{1} R_{{\text{v}}} r_{2} + C_{1} L_{2} R_{{\text{v}}} r_{1} + C_{1} R_{{\text{v}}} T_{{\text{s}}} r_{1} r_{2} \\ q_{2} = & L_{1} r_{2} + L_{2} r_{1} + L_{1} R_{{\text{v}}} + L_{2} R_{{\text{v}}} + \\ & R_{{\text{v}}} T_{{\text{s}}} r_{1} + R_{{\text{v}}} T_{{\text{s}}} r_{2} + T_{{\text{s}}} r_{1} r_{2} + C_{1} R_{{\text{v}}} r_{1} r_{2} \\ q_{1} = & R_{{\text{v}}} k_{{\text{p}}} + R_{{\text{v}}} r_{1} + R_{{\text{v}}} r_{2} + r_{1} r_{2} \\ q_{0} = & R_{{\text{v}}} k_{{\text{i}}} \\ \end{aligned} $$

Appendix C Parameter description in Sect. 4.1

Define the following matrix for simple.

$$ J_{2} = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right),\;K_{2} = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right) $$

From Eq. (7)

$$ A_{{\text{f}}} = \left( {\begin{array}{*{20}c} { - \frac{{r_{1} }}{{L_{1} }}J_{2} - \omega K_{2} } & { - \frac{1}{{L_{1} }}J_{2} } & {0_{2 \times 2} } \\ {\frac{1}{{C_{1} }}J_{2} } & { - \omega K_{2} } & { - \frac{1}{{C_{1} }}J_{2} } \\ {0_{2 \times 2} } & {\frac{1}{{L_{1} }}J_{2} } & { - \frac{{r_{2} }}{{L_{2} }}J_{2} - \omega K_{2} } \\ \end{array} } \right) $$

$$ B_{{{\text{f}}1}} = \left( {\begin{array}{*{20}c} {\frac{1}{{L_{1} }}J_{2} } \\ {0_{2 \times 2} } \\ {0_{2 \times 2} } \\ \end{array} } \right),\;B_{{{\text{f2}}}} = \left( {\begin{array}{*{20}c} {0_{2 \times 2} } \\ {0_{2 \times 2} } \\ { - \frac{1}{{L_{2} }}J_{2} } \\ \end{array} } \right) $$

From Eq. (8)

$$ \begin{gathered} D_{{{\text{aux}}1}} = k_{{{\text{pin}}}} J_{2} ,\;D_{{{\text{aux2}}}} = \left( {\begin{array}{*{20}c} { - k_{{{\text{pin}}}} J_{2} + \omega L_{1} K_{2} } & {J_{2} } & {0_{2 \times 2} } \\ \end{array} } \right) \hfill \\ A_{\sigma } = 0_{2 \times 2} ,\;B_{\sigma 1} = k_{{{\text{io}}}} J_{2} ,\;B_{\sigma 2} = \left( {\begin{array}{*{20}c} {0_{2 \times 2} } & {0_{2 \times 2} } & {0_{2 \times 2} } \\ \end{array} } \right) \hfill \\ C_{\sigma } = J_{2} ,\;D_{\sigma 1} = k_{{{\text{po}}}} J_{2} ,\;D_{\sigma 2} = \left( {\begin{array}{*{20}c} {0_{2 \times 2} } & {\omega C_{1} K_{2} } & {J_{2} } \\ \end{array} } \right) \hfill \\ D_{01} = J_{2} ,\;D_{02} = \left( {\begin{array}{*{20}c} {0_{2 \times 2} } & { - J_{2} } & {0_{2 \times 2} } \\ \end{array} } \right) \hfill \\ \end{gathered} $$

From Eq. (9)

$$ \begin{gathered} \omega_{{{\text{hpf}}}} = \frac{1}{{T_{{{\text{hpf}}}} }} \hfill \\ A_{{\text{v}}} = - \omega_{{{\text{hpf}}}} J_{2} ,\;B_{{{\text{v}}1}} = 0_{2 \times 2} ,\;B_{{{\text{v2}}}} = \left( {\begin{array}{*{20}c} {0_{2 \times 2} } & {0_{2 \times 2} } & {J_{2} } \\ \end{array} } \right) \hfill \\ C_{{\text{v}}} = - L_{{\text{v}}} \omega_{{{\text{hpf}}}}^{2} J_{2} ,\;D_{{{\text{v}}1}} = J_{2} , \hfill \\ D_{{{\text{v}}2}} = \left( {\begin{array}{*{20}c} {0_{2 \times 2} } & {0_{2 \times 2} } & {L_{{\text{v}}} \omega_{{{\text{hpf}}}} J_{2} - R_{{\text{v}}} J_{2} - \omega L_{{\text{v}}} K_{2} } \\ \end{array} } \right) \hfill \\ \end{gathered} $$

From Eq. (10)

$$ \begin{gathered} \omega_{{{\text{lpf}}}} = \frac{1}{{T_{{{\text{lpf}}}} }} \hfill \\ A_{{{\text{pq}}}} = - \omega_{{{\text{lpf}}}} J_{2} , \hfill \\ B_{{{\text{pq}}}} = 1.5\left( {\begin{array}{*{20}c} {0_{2 \times 2} } & {I_{d} J_{2} + I_{q} K_{2} } & {\left( {\begin{array}{*{20}c} {U_{d} } & {U_{q} } \\ {U_{q} } & { - U_{d} } \\ \end{array} } \right)} \\ \end{array} } \right) \hfill \\ C_{{{\text{pq}}}} = \left( {\begin{array}{*{20}c} 0 & { - n_{{\text{Q}}} } \\ 0 & 0 \\ { - n_{{\text{P}}} } & 0 \\ \end{array} } \right) \hfill \\ \end{gathered} $$

A_inv is gotten by state space expansion for the sake of simplicity.

$$ \begin{gathered} A_{2} = A_{{\text{f}}} + B_{{{\text{f}}1}} D_{{{\text{aux}}2}} \hfill \\ B_{21} = B_{{{\text{f}}1}} D_{{{\text{aux1}}}} ,\;B_{22} = B_{{{\text{f2}}}} \hfill \\ A_{3} = \left( {\begin{array}{*{20}c} {A_{2} + B_{21} \left( {D_{\sigma 1} D_{02} + D_{\sigma 2} } \right)} & {B_{21} C_{\sigma } } \\ {B_{\sigma 1} D_{02} + B_{\sigma 2} } & {A_{\sigma } } \\ \end{array} } \right), \hfill \\ B_{31} = \left( {\begin{array}{*{20}c} {B_{21} D_{\sigma 1} D_{01} } \\ {B_{\sigma 1} D_{01} } \\ \end{array} } \right),\;B_{32} = \left( {\begin{array}{*{20}c} {B_{22} } \\ {0_{2 \times 2} } \\ \end{array} } \right), \hfill \\ D_{3} = \left( {\begin{array}{*{20}c} {D_{{{\text{v1}}}} } & {0_{2 \times 2} } \\ \end{array} } \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} A_{4} = \left( {\begin{array}{*{20}c} {A_{3} + B_{31} D_{3} } & {B_{31} C_{{\text{v}}} } \\ {B_{{{\text{v1}}}} } & {A_{{\text{v}}} } \\ \end{array} } \right) \hfill \\ B_{41} = \left( {\begin{array}{*{20}c} {B_{31} D_{{{\text{v2}}}} } \\ {B_{{{\text{v2}}}} } \\ \end{array} } \right),B_{42} = \left( {\begin{array}{*{20}c} {B_{32} } \\ {0_{2 \times 2} } \\ \end{array} } \right), \hfill \\ B_{5} = \left( {\begin{array}{*{20}c} {B_{{{\text{pq}}}} } & {0_{2 \times 2} } & {0_{2 \times 2} } \\ \end{array} } \right), \hfill \\ A_{5} = \left( {\begin{array}{*{20}c} {A_{4} } & {B_{41} C_{{{\text{pq}}}} } \\ {B_{5} } & {A_{{{\text{pq}}}} } \\ \end{array} } \right), \hfill \\ A_{{{\text{inv}}}} = A_{5} \hfill \\ \end{gathered} $$