6.2 Rectifier Model

6.2.1 Space Vector Model

The AFE rectifier is modeled as shown in Figure 6.3. The rectifier is a fully controlled bridge with power transistors, connected to the three-phase supply voltages v_s using the filter inductances L_s and resistances R_s.

Figure 6.3 AFE rectifier

Considering the circuit shown in Figure 6.3, the equations for each phase can be written as

6.1

6.2

6.3

Then, considering the space vector definition for the grid voltage

6.4

where a = e^j2π/3, and by substituting (6.1)–(6.3) into (6.4), the vector equation for the grid current dynamics can be obtained as

6.5

Note that the last term of this equation is equal to zero

6.6

The input current dynamics equation (6.5) can be simplified by considering the following definitions for the grid current vector and the voltage vector generated by the AFE:

6.7

6.8

Voltage v_afe is determined by the switching state of the converter and the DC link voltage, and can be expressed by the equation

6.9

where V_dc is the DC link voltage and S_afe is the switching state vector of the rectifier, defined as

6.10

where S₁, S₂, and S₃ are the switching states of each rectifier leg, as shown in Figure 6.3, and take the value of 0 if S_x is off, or 1 if S_x is on (x = 1, 2, 3).

The input current dynamics equation (6.5) can be rewritten in the stationary αβ frame as the following vector equation:

6.11

where i_s is the input current vector, v_s is the supply line voltage, and v_afe is the voltage generated by the converter.

6.2.2 Discrete-Time Model

The predicted current is calculated using the discrete-time equation

6.12

obtained from discretizing (6.11) for a sampling time T_s. The discretization is done by approximating the derivative as the difference over one sampling period as considered in the previous chapters and explained in Chapter 4.

Considering the input voltage and current vectors in orthogonal coordinates, the predicted instantaneous input active and reactive power can be expressed by the following equations:

6.13

6.14

where is the complex conjugate of the predicted input current vector i_s(k + 1), for a given voltage vector generated by the rectifier v_afe.

For a small sampling time, with respect to the grid fundamental frequency, it can be assumed that v_s(k + 1) ≈ v_s(k). However, if the sampling time is not small enough to consider the grid voltage as constant between two sampling intervals, the future grid voltage v_s(k + 1) can be calculated by compensating the angle of the voltage vector for one sampling time:

6.15

where Δθ = ωT_s is the angle advance of the grid voltage vector in one sampling interval and ω is the angular frequency of the grid voltage.