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voltage at PCC has to be regulated to a reference voltage. The battery is supposed to supply active or reactive power to the grid to maintain the PCC voltage. The current reference ( (i_d(ref), i_q(ref)) is obtained from an outer voltage loop. The outer voltage loop ensures that the PCC voltage is maintained at its reference value. The modeling of the converter and design of the controller under different modes is discussed in the next section.

       2.5.2.1 Modelling and Control of the VSI

      The AC voltage of the PCC is expressed as

      (2.21)

      Applying Kirchhoff ’s voltage equation in the phase, we get

      (2.22)

      (2.23)

(2.24)

      To transfer the above system to synchronous reference frame, the following transformation need to be performed

      (2.25)

      Where T_abc2αβ, T_αβ2dq and θ are known as Clark transformation, Park transformation, and the phase angle of the grid which is obtained from PLL, respectively. The resulting system after transformation is given by

(2.26)

(2.27)

      Where v_sd, u_d and i_d are PCC voltage, VSC output voltage and output current on d-axis, respectively. u_q and i_q are VSC output voltage and output current on the q-axis, respectively. One can notice from equation (2.26) and (2.27) that the system is converted to an LTI system having some coupling states. A simple control law to decouple the term is given by

      (2.28)

      Where v_id and v_iq are the feedback control input. for obtaining zero steady-state error, PI controller is applied to control input

(2.29)

Where e_id = (i_d(ref) – i_d), e_iq = (i_q(ref) – i_q), K_pi,d, K_ii,d, K_pi,q and K_ii,q are parameters/gains of PI controller for d-axis current and q-axis current controller. Since the system model is LTI, it can be easily analysed and the current controller can be easily designed using linear control techniques.

      For the case when the microgrid in islanded mode, the converter has to regulate the voltage at PCC. By applying Kirchhoff’s law at the PCC voltage and applying the d-q transform on the resulting equations, one would obtain

      (2.30)

      The coupling between V_sd and V _sq is eliminated by a decoupling feed-forward compensation. One possible control law is given by

      (2.31)

      Where i_d(ref), i_q(ref) are reference for inner current loop and v_vd, and v_v are the feedback control input which is expressed as

      (2.32)

      Where e_vd = (V_{sd ( ref )} – V_sd), e_vq = (V_sq(ref) – V_sq), K_pv,d, K_iv,d, K_pv,q, and K_iv,q are parameters/gains of PI controller for d-axis current and q-axis voltage controller. The parameter of PI controller can be easily selected. For d-q transformation in islanded system, the reference angle ρ (and thus ) is decided independently.

       2.5.2.3 Typical Case Study in MATLAB-Simulink

A MATLAB-Simulink model of the three-phase inverter system shown in Figure 2.7 is developed with the specification listed in Table 2.4. The filter parameter are selected based using following equations [16, 17]

      (2.33)

      (2.34)

Table 2.4 Specifications of battery-converter system for AC microgrid.

Parameter	Value
Nominal voltage (VB)	700 V
Rated capacity	150 Ah
Initial SOC	60%
inverter side inductance (L) and ESR (rL) of inductor	1mH, 0.005Ω
Filter capacitor (Cf)	20 µF
AC load	PL=1000 W QL =100 VAR
AC microgrid voltage (Vphase(rms)) grid side inductance Lg	220 V300 µH

(2.35)

      Where P is the rated output active power of one phase for three-phase full-bridge inverter, V_g is the phase voltage, V_dc is the dc link voltage f_sw is switching frequency, K_a = 0.2,

      The control is implemented in the Simulink platform. The commands “Mode_command” along with the power to be exchanged (P_ref, Q_ref), with microgird are sent to the controller from the EMS of the microgrid. The feedback signals PCC voltage (V_sx), inverter current (i_Lx), grid voltage (V_gx) are also input to the controller.

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