Gridfeeding inverter
The control objective of gridfeeding (GFD) [11] inverter is to track the specified power references. Figure 1 illustrates the control block diagram of the most common current controlled GFD inverter. For dispatchable microsources, such as microturbine and fuelcells, the inverter power references can be set directly according to practical requirements. For nondispatchable microsources, such as photovoltaic cells, the active power reference is usually decided by the voltage controller of the inverter’s DC bus. In addition, this type of sources can also export reactive power without affecting maximum power point tracking.
The GFD inverter’s power referencetrackingis realized by adjusting the output currents. The control system calculates the output current references based on the relationships among the inverter’s output power, output current and the voltage at the point of connection (PC). The threephase voltages at the PC are represented by vector v, and the inverter’s output currents are represented by vector i as
$$ v={\left[{v}_a\kern0.5em {v}_b\kern0.5em {v}_c\right]}^T $$
$$ i={\left[{i}_a\kern0.5em {i}_b\kern0.5em {i}_c\right]}^T $$
Neglecting the power consumed on the filter inductor, the output power of GFD inverter is calculated according to instantaneous power theory [7] as:
$$ \left\{{}_{Q\kern0.5em =\kern0.5em \leftv\times i\right}^{P\kern0.5em =\kern0.5em v.i}\right. $$
(1)
If the current controller in Fig. 1 is properly designed, the output currents of the GFDinverter will follow their references. Thus the current reference vector, i
_{ref}, can be obtained by solving the following equation:
$$ \left\{{}_{Q_{ref}=\kern0.5em \leftv\times {i}_{ref}\right}^{P_{ref}\kern0.5em =\kern0.5em v\cdot {i}_{ref}}\right. $$
(2)
The output currents of the GFD inverter are the same as the currents flownthrough the filter inductor. In natural reference frame, there exists the following relation:
$$ sLi={u}_{inv\kern0.5em }\kern0.5em u $$
(3)
The voltages at the PC, u, are measured using voltage transducers, the output voltages of the inverter, u
_{inv}, can then be adjusted based on u (see the voltage feedforward in Fig. 1) to control the voltage drop on the filter inductor. This implies that the filter inductor’s currents can be controlled indirectly. If the potential at the middle point of the inverter’s DC bus equal zero and ignore the delay of PWM process, the inverter is equivalent to a proportional element with gain k
_{PWM}. Thus, the closedloop transfer function of the inverter’s current control can be written as:
$$ {G}_c(s)=\frac{k_{pwm}{T}_c(s)}{sL+{k}_{pwm}{T}_c(s)} $$
(4)
T
_{c}(s) needs be designed in a way to ensure G
_{c}(s) have sufficient bandwidth. Meanwhile, the gain and phase shift of G
_{c}(s) around fundamental frequency should be close to 0 dB and 0 degree respectively. Therefore, the output current of the GFD inverter can track their references quickly and accurately.
For threephase balanced operation cases, the control system of the GFD inverter is usually designed in dq reference frame, where the voltages and currents are DC signals. In this case, using PI controller can realize the output current tracking without steadystate error. In dq reference frame, the Park transformation will result in coupling between the d and q axis inductor currentcomponents, as shown in Eq. (5). Therefore, the control system must comprise dq decoupling modules. The detailed control block diagram in dq reference frame is illustrated in Fig. 2.
$$ sL{\boldsymbol{I}}_{dq}={\boldsymbol{U}}_{inv,dq}{\boldsymbol{U}}_{dq}\left[\begin{array}{l}0\kern0.75em \omega L\\ {}\omega L\kern0.75em 0\end{array}\right]{\boldsymbol{I}}_{dq} $$
(5)
For unbalanced operation cases, the GFD inverters need simultaneously controlthe positive and negative sequence currents [8, 9]. Under such condition, using PR controller [10] in αβ reference frame might be a better choice as a single PR controller can regulate both the positive and negative sequence currents, and the control effect is similar to that of using two PI controllers in double positive/negative dq reference frames.
Gridforming inverter
The control objective of the gridforming (GFM) [11] inverters is to maintain stable voltage and frequency in a microgrid. GFM inverters are characterized by their low output impedance, and therefore they need a highly accurate synchronization system to operate in parallel with other GFM inverters [11]. GFM inverters usually equips with energy storage on their DC sides, therefore they can respond to the change of load in a short time. The control block diagram of a GFM inverter is shown in Fig. 3, including an inner inductor current loop, which is identical to that of the GFD inverter, and an outer capacitor voltage loop. GFM inverters achieve their control objective by regulating the filter capacitor’s voltage, u. In natural reference frame, there exists the following relation:
where iandi
_{o}are the inductor and grid currents, respectively.
According to the above analysis, the GFM inverters can also precisely control their inductor current by a properly designed inner current loop. The impact of the grid current on capacitor voltage is removed by current feedforward and thus, u is fully controlled by adjusting i.
As shown in Eq. (7), in thedq reference frame, the d and q axis component of the filter capacitor voltage are also coupled. Similarly, it is necessary to introduce dq decoupling modules in the voltage control loop as illustrated in Fig. 4.
$$ sC{U}_{dq}={I}_{dq}{I}_{o,dq}\left[\begin{array}{l}0\kern0.75em \omega C\\ {}\omega C\kern0.75em 0\end{array}\right]{U}_{dq} $$
(7)
Gridsupporting inverter
There exists an approximate linear droop relation between the Pω and QU of traditional synchronous generators. By emulating this output characteristics, gridsupporting (GS) [11] inverters, aimed at sharing load proportional to their power capacities, can deploy two different droop control structures, namely “PQdroop” and “ωUdroop”. The PQdroop GS inverter adjusts its output power as a function of the variation of the microgrid’s voltage and frequency. In this case, the inverter behaves like a power source and its control system is designed based on that of the GFD inverter, as shown in Fig. 5(a). On the contrary, the voltage and frequency at the PC of the ωUdroop GS inverter are adjusted according to the variations of its output power. The ωUdroop GS inverter behaves as a controlled voltage source and its control system is based on that of the GFM inverter, as shown in Fig. 5(b).
In Fig. 5, ω
_{0} and U
_{0} represent the noload frequency and noload voltage, k
_{P} and k
_{Q} represent the active and reactive power droop coefficients, respectively. In steady state, the frequency of the microgrid is a global quantity, and the voltages at different points of the microgrid are almost identical. If “ω
_{0}”and “U
_{0}”of each inverter are identical, then both the PQdroop GS inverter and the ωUdroop GS inverter can share load variations as follows:
$$ {k}_{p1}\varDelta {P}_1={k}_{p2}\varDelta {P}_2=\dots ={k}_{pn}\varDelta {P}_n $$
(8)
$$ {k}_{Q1}\varDelta {Q}_1={k}_{Q2}\varDelta {Q}_2=\dots ={k}_{Qn}\varDelta {Q}_n $$
(9)
where k
_{Pi} and △P
_{i} (i = 1,2,…,n) represent the active power droop coefficient and output active power variation of the ith GS inverter, respectively. k
_{Qi} and △Q
_{i} represent the reactive power droop coefficient and output reactive power variation of the ith GS inverter, respectively. Although both types of GS inverters shown in Fig. 5 have a good loadsharing performance, the PQdroop GS inverter cannot operate by itself. In contrast, the ωUdroop GS inverter is controlled as a voltage source, and thus can work independently regardless of the microgrid operation mode. The ωUdroop GS inverter has acquired extensive attentions for its excellent features though some problems still exist, including:

the line impedance of a lowvoltage microgrid has a large resistive component, thus Pω and QU droop control is no longer suitable.

the voltages at the PCs of each inverter are not completely equal, thus the GS inverters cannot share reactive power precisely.
Many researchers have proposed various improved methods to deal with the above problems and some typical schemes will be presented in the following sections.

A.
Decoupling transformation method
As depicted in Fig. 6, the voltage at the PC of theωUdroop GS inverter is denoted by U∠δ, and the voltage at the microgrid bus is denoted by E∠0. Z_{L} is the line impedance between the inverter’s filter capacitor and the microgrid bus with an impedance angle of θ.
Due to the small power angle δ, it is assumed that:
$$ sin\delta =\delta, cos\delta =1 $$
(10)
Thus, the output power of the GS inverter can be expressed as:
$$ \left\{{}_{Q=\frac{EU \sin \theta {E}^2 \sin \theta EU\delta \cos \theta }{Z_L}}^{P=\frac{EU \cos \theta {E}^2 \cos \theta +EU\delta \sin \theta }{Z_L}}\right. $$
(11)
If both the resistive and reactive components of the line impedance cannot be ignored, the output active and reactive power of the inverter will be dependent on both δ and U. In this case, the Pω(δ) and QU decoupling relation will no longer valid. To solve this problem, the virtual power P’, Q’ and the transformer matrix T
_{PQ} are introduced in [12, 13]:
$$ \left[{}_{Q^{\hbox{'}}}^{P^{\hbox{'}}}\right]={T}_{PQ}\left[{}_Q^P\right]=\left[\begin{array}{l} \sin \theta \kern0.75em  \cos \theta \\ {} \cos \theta \kern0.75em \sin \theta \end{array}\right]\left[{}_Q^P\right] $$
(12)
According to Eq. (11) and (12), it can be derived that:
$$ \left\{{}_{Q^{\hbox{'}}\frac{EU{E}^2}{Z}}^{P^{\hbox{'}}=\frac{EU}{Z}\delta}\right. $$
(13)
The ωUdroop control based on the virtual power is given as:
$$ \left\{{}_{U={U}_0{k}_QQ\hbox{'}}^{\omega ={\omega}_0{k}_PP\hbox{'}}\right. $$
(14)
Similarly, transforming ω(δ) and U with the matrix T
_{ωU} [14] gives the virtual frequency (phase angle) and voltage as:
$$ \left[{}_{U\hbox{'}}^{\omega \hbox{'}\left(\delta \hbox{'}\right)}\right]={T}_{\omega E}\left[{}_U^{\omega \left(\delta \right)}\right]=\left[\begin{array}{l} \sin \theta \kern0.75em \cos \theta \\ {} \cos \theta \kern0.75em \sin \theta \end{array}\right]\left[{}_U^{\omega \left(\delta \right)}\right] $$
(15)
According to Eq. (11) and (15), it can be derived that:
$$ \left\{{}_{Q=\frac{EUU\hbox{'}+\left(U{U}^2{E}^2\right)E\kern0.5em \sin \theta }{Z}}^{P=\frac{EU\delta \hbox{'}+\left(U{U}^2{E}^2\right)E\kern0.5em \cos \theta }{Z}}\right. $$
(16)
Since E and U are constant when the droop control process reaches steadystate, the output active and reactive power of the GS inverter will be regulated by the virtual frequency and voltage respectively. Thus, a novel ωUdroop can be established:
$$ \left\{{}_{U\hbox{'}=U{\hbox{'}}_0{k}_QQ}^{\omega \hbox{'}=\omega {\hbox{'}}_0{k}_PP}\right. $$
(17)
In Eq. (17), ω
_{0}’ and U
_{0}’ is the corresponding virtual noload frequency and voltage. The droop control block diagram of the GS inverters applying two types of decoupling transformation method is shown in Fig. 7.
It is worth noting that the first decoupling method is designed to share the virtual power rather than the real power. So there exists a complicated relation between the variations of each inverter’s output power and their droop coefficients when the load in the microgrid changed. The second decoupling method avoids this problem considering that all inverters have the same ω′ and U′, i.e. the R/X of each line in the microgrid must be identical. In addition, the variables directly controlled by Eq. (17) are ω′ and U′, and thus, it is necessary to carefully select the droop coefficients [14] to ensure that the real frequency and voltage are located in reasonable ranges.

B.
Virtual impedance method
The coupling between the output active and reactive power of the conventional ωUdroop control can be mitigated by introducing virtual impedance [15], as illustrated in Fig. 8. The voltage at the inverter’s PC is expressed as:
$$ U={G}_u(s)\kern0.5em {U}_{ref}{G}_u(s)\kern0.5em {Z}_V{I}_o $$
(18)
where G
_{u}(s) is the voltage closedloop transfer function of the ωUdroop GS inverter, and Z
_{V} is virtual impedance.
The total impedance between the equivalent voltage source of the inverter and the microgrid bus can be written as:
$$ Z={G}_u(s)\kern0.5em {Z}_V+{Z}_L $$
(19)
where Z
_{L} is the line impedance.
If the magnitude of the virtual impedance is much larger than the line impedance, the total impedance will be largely decided by the virtual impedance. However, the large total impedance may cause the microgrid voltage to reduce substantially. In [16], a novel method was proposed to solve this problem by introducing a negative resistive component into the virtual impedance. As the virtual negative resistor counteracts the line resistor, the total impedance can be designed to be mainly inductive and of small magnitude. According to Eq. (11), if the total impedance is mainly inductive [17], the GS inverter should adopt Pω and QU droop control. However, if the total impedance is mainly resistive [18], PU and Qω droop control should be applied.

C.
Reactive power sharing method based on communication
To improve the reactive power sharing accuracy, a common method is to revise the GS inverters’ droop control parameters, including noload voltage and droop coefficient. The following analysis takes the inductive line (cosθ ≈ 0,sinθ ≈ 1) as examples. According to Eq. (11), the relation between the output reactive power and the voltage of the GS inverter’s PC is shown as:
$$ U=E+\frac{Z_L}{E}Q $$
(20)
In the QU plane, the intersection of the operational curve described by Eq. (20) and the reactive power droop curve is the GS inverter’s stable operating point [19].
As illustrated in Fig. 9, there are two inverters, namely 1# and 2#, with the same droop coefficient. The total impedance between these two inverters’ equivalent voltage sources and the microgrid bus are Z
_{1} and Z
_{2}, respectively. If Z
_{1} is not equal to Z
_{2}, the inverters’ operating points will be different. Increasing inverters’ droop coefficient leads to new operating points. The voltage of the microgrid bus moves from E to E’, and the inverters’ output power changes move from Q
_{1} and Q
_{2} to Q
_{1}’ and Q
_{2}’, respectively It can be seen that the reactive power sharing accuracy is improved with the increase of the inverter’s droop coefficient. Decreasing the GS inverter’s noload voltage can also increase reactive power sharing accuracy, as shown in Fig. 10. To adjust each inverter’s droop curve parameters in a coordinated manner [19, 20], it is necessary to employ a centralized control system.
Different with the method of adjusting droop parameter, reference [21] proposed an improved control structure by introducing integral module, as shown in Fig. 11.
In Fig. 11, U
_{0} is the inverter noload voltage; E is the voltage of microgrid bus; k
_{Q} is the reactive power droop coefficient; K
_{u} is the integral gain. The transfer function of the inverter’s output reactive power can be written as:
$$ Q(s)=\frac{K_u\left[{U}_oE(s)\right]\kern0.5em E(s)s{E}^2(s)}{s{Z}_L+{K}_u{k}_QE\kern0.5em (s)} $$
(21)
and its steadystate value can be calculated as:
$$ \begin{array}{c}\hfill \underset{t\to \infty }{ \lim }Q(t)=\underset{s\to 0}{ \lim}\kern0.5em sQ(s)\hfill \\ {}\hfill =\underset{s\to 0}{ \lim}\frac{\left({U}_0E\right)\kern0.5em {K}_uEs{E}^2}{s{Z}_L+{K}_u{k}_QE}=\frac{U_0E}{k_Q}\hfill \end{array} $$
(22)
In this method, the output reactive power of each GS inverter is independent to the line impedance Z
_{L}. By delivering the voltage information of the microgrid bus to each GS inverter, accurate reactive power sharing can be realized. This method doesn’t require a central controller to participate, avoiding the usage of complicated algorithms. Besides, the additional parameter, K
_{u}, can be used to adjust the dynamic response of reactive power control.