*5.1. Conventional VSC*

The conventional VSC is also referred to as an average model, which is represented in Figure 2. This model consists of one controlled voltage source on the AC side, and another controlled current source on the direct current (DC) side. The state space model of the conventional VSC converter is derived by the connection to the grid through a power transformer; therefore, its control can be performed by the impedance *Zf* among two buses *vg* and *vc*, defined by *Lf* and *Rf* . The equivalent circuit of the grid is represented by a grid impedance *Zs* and a voltage source *Vs*.

**Figure 2.** Average VSC model.

According to Figure 2 and considering the Kirchhoff's voltage law in the *abc* reference frame, the voltage drop along the impedance *Zs* is:

$$
\sigma\_{\mathcal{S}}^{\rm abc} - \sigma\_{\mathcal{c}}^{\rm abc} = L\_f \frac{d i\_f^{\rm abc}}{dt} + R\_f i\_f^{\rm abc}.\tag{16}
$$

Applying Park's Transformation *dq*0 and considering *ω<sup>g</sup>* as the angular frequency of the rotating system; Equation (16) can be rewritten dividing the real and imaginary parts, giving the next expressions as a result [28]:

$$L\_f \frac{d\dot{\mathbf{r}}\_f^d}{dt} = \omega\_\mathcal{g} L\_f \dot{\mathbf{r}}\_f^d - R\_f \dot{\mathbf{r}}\_f^d + \mathbf{v}\_\mathcal{g}^d - \mathbf{v}\_\mathcal{c}^d \tag{17}$$

$$L\_f \frac{d i\_f^q}{dt} = \omega\_\ $ L\_f \mathbf{i}\_f^d - R\_f \mathbf{i}\_f^q + \mathbf{v}\_\$ ^q\_\S - \mathbf{v}\_\circ^q. \tag{18}$$

The *d*-axis is aligned to the *dq*0-rotating frame, which provokes the *q*-component equals zero in steady-state [29]. In this context, to synchronize the *dq*0-rotating frame a Phase-Locked Loop (PPL) must be used. Therefore, the real and reactive powers in the *dq*0-rotating frame can be expressed as follows:

$$P\_{\mathcal{S}} = \upsilon\_{\mathcal{S}}^d \mathfrak{i}\_f^d \tag{19}$$

$$Q\_{\mathcal{S}} = -v\_{\mathcal{S}}^d i\_f^q. \tag{20}$$

A VSC is capable of controlling independently active power *P* (or DC voltage) and reactive power *Q* (or AC voltage) [24]. Basically, the control consists of two control loops: the outer-control loop and the inner-control loop. Both are derived from the mathematical equations developed previously. The outer-control loop calculates the reference signals for *i d*∗ *<sup>f</sup>* and *i q*∗ *<sup>f</sup>* , which are employed as inputs to the inner-control loop. The inner-control loop gives the reference voltage [28] to *Vc* (controlled voltage source). Depending on the target of control, the VSC can work either as a rectifier or inverter. Considering a lossless converter, the power balance equation in the *dq*0 reference frame is [30]:

$$P\_{dc} = P\_{ac} = \upsilon\_{dc} i\_{dc} = \upsilon\_{\mathcal{g}}^d i\_f^d + \upsilon\_{\mathcal{g}}^q i\_f^q. \tag{21}$$

Expression (21) can also be rewritten as:

$$\dot{a}\_{d\varepsilon} = \frac{v\_{\mathcal{S}}^d \dot{\mathbf{r}}\_f^d + v\_{\mathcal{S}}^q \dot{\mathbf{r}}\_f^q}{v\_{\alpha d}}.\tag{22}$$

In this case, Equation (22) provides the input signal to the controlled current source on the DC side of the VSC.
