## 4 Control Strategies
### Control of the Microgrid Inverter  
The main objective of the microgrid inverter is to transfer to the microgrid AC bus the excess of power produced by the DG units connected to the microgrid DC bus (or absorb the deficit). Neglecting the system losses, if the net power \(P_{INV\_in}\) injected into the microgrid DC bus (produced by DG units minus the power consumed by the loads and battery being charged) is higher (lower) than the active power \(P_{INV\_out}\) delivered to the microgrid AC bus by the inverter, the difference is absorbed (delivered) by the microgrid DC bus capacitors, and the microgrid DC bus voltage rises (falls). Therefore, the control  
Figure 8: Wind generation schemes for MPPT or adaptation to load power demand. Figure 7: Wind turbine typical efficiency characteristic, considering the fixed blades’ pitch angle. objective of the microgrid inverter can be accomplished by regulating its DC bus voltage. The effect of \(P_{INV\_in}\) and \(P_{INV\_out}\) on the microgrid DC bus voltage can be written as:  
\[P_{INV\_in}-P_{INV\_out}=\frac{d}{dt}\left(\frac{1}{2}CV_{DC}^{2}\right)=\frac{1 }{2}C\frac{d}{dt}\left(V_{DC}^{2}\right) \tag{15}\]  
According to (15), \(P_{INV\_out}\) can be used to control \(V_{DC}^{2}\) (and therefore to control \(V_{DC}\)). Due to the linear relationship between \((P_{INV\_in}-P_{INV\_out})\) and \(V_{DC}^{2}\), and since the reference voltage \(V_{DC}^{*}\) is constant in a steady-state condition, then a proportional-integral (PI) controller is adopted, taking the error \(\left(V_{DC}^{2*}-V_{DC}^{2}\right)\) as input and the negative of the inverter output power \(\left(-P_{INV\_out}^{*}\right)\) as the output control action. \(P_{INV\_in}\) is considered a disturbance to be rejected by the controller. As a second goal, the reactive power delivered by the microgrid AC side should be maintained equal to zero (\(Q_{INV\_out}^{*}=0\)) to avoid unnecessary reactive current components flowing through the inverter. The above shows that it is necessary to regulate the inverter output of active and reactive powers to achieve the objectives. According to the instantaneous power theory [12], the active and reactive powers delivered to the AC bus of the microgrid can be written, in space-vector \(\alpha\beta\) Clarke coordinates, in terms of the inverter AC output current vector \(\overset{\rightarrow}{i}\) and microgrid AC bus voltage vector \(\overset{\rightarrow}{\
i}_{BUS}\) as follows:  
\[\overset{\rightarrow}{s}_{INV}=p_{INV}+jq_{INV}=\overset{\rightarrow}{v}_{BUS }\overset{\rightarrow}{i}^{\prime}=\big{(}v_{BUSa}+jv_{BUS\beta}\big{)} \big{(}i_{a}-ji_{\beta}\big{)}. \tag{16}\]  
From (16), the inverter output current components necessary to impose the reference values of active and reactive powers are  
\[\left[ \begin{array}{c}i_{a}^{*}\\ i_{\beta}^{*}\end{array} \right]=\frac{1}{\left|\overset{\rightarrow}{v}_{BUS }\right|^{2}}\left[ \begin{array}{cc}v_{BUSa}&v_{BUS\beta}\\ v_{BUS\beta}&-v_{BUSa}\end{array} \right]\left[ \begin{array}{c}P_{INV\_ out}^{*}\\ 0\end{array} \right], \tag{17}\]  
Alternatively, using complex space vector notation:  
\[\overset{\rightarrow}{i}^{*}=P_{INV\_out}^{*}\frac{\overset{\rightarrow}{v }_{BUS}}{\left|\overset{\rightarrow}{v}_{BUS}\right|^{2}}. \tag{18}\]  
Assuming the reference active power \(P_{INV\_out}^{*}\) is constant (or slowly varying), if the voltages at the microgrid AC bus are not balanced or contain harmonic components, the currents calculated through (18) would contain the same levels of unbalance and harmonic contamination.