To carry out our work, we made the following considerations: the generator and the switches of our converter will be assumed ideal, the line will be balanced, the voltage drops across the line represented by the reactance X, the inductance of the line is represented by the inductance L. the characteristics of the line are given in the following Table 1.

Subsequently, we will proceed with the modeling of the serial converter while knowing that for the parallel converter the principle will be the same.

The application of Kirchhoff's laws [4]. To the meshes of the circuit in Figure 1 gives us the mathematical equations governing our system as follows:

{ V s a − V c a − V r a = r i s a + L d i s a d t V s b − V c b − V r b = r i s b + L d i s b d t V s c − V c c − V r c = r i s c + L p d i s c d t (1)

In order to minimize the computing time, we have switched from the three-phase system to the two-phase system, i.e. from three-phase reference frames with coordinates a, b and c, to two-phase reference frames with coordinates d and q. The transformation matrix is the following (Park’s matrix) [5]:

K = 2 3 [ cos w t cos ( w t − 2 π 3 ) cos ( w t + 2 π 3 ) sin w t sin ( w t − 2 π 3 ) sin ( w t + 2 π 3 ) 1 2 1 2 1 2 ] (2)

So the system becomes:

{ V p s d − V c d − V r d = r i p s d + L d i s d d t − w L i q V s q − V c q − V r q = r i s q + L d i s q d t + w L i d (3)

Using the matrix representation on the system we have:

d d t ⋅ [ i s d i s q ] = [ − r L w − w − r L ] ⋅ [ i s d i s q ] + 1 L ⋅ [ V s d − V c d − V r d V s q − V c q − V r q ] (4)

The equation above corresponds to our serial converter model. It will be used to build, under Simulink, the block that represents the serial part of the system. Therefore, this equation is represented by the following block diagram (r = R): (Figure 2)

The DC-Link connects the serial converter and the shunt converter of the UPFC. To model the DC-Link, we will use the single-phase model of the following converter (Figure 3).

We will first define the input/output relationships of our converters. We are going to assimilate them to a quadripole whose modulation function w(t) connects their inputs and outputs in the following way:

➢ For the shunt converter

The equation verified by the quadrupole is as follows:

V e = ω e ( t ) V i n     and     I e = 1 ω e ( t ) I i n (5)

Due to the independence of the current I_e from the voltage V_in and since the current I_in does not depend on the voltage V_e either, the matrix representation of the input system is therefore:

[ I e V e ] = [ 1 ω e ( t ) 0 0 1 ω e ( t ) ] ⋅ [ I i n V i n ] (6)

In this quadrupole, the instantaneous power is preserved. The following relationship is therefore verified:

V e I e = V i n I i n .

➢ For the serial converter

In the same way as for the shunt, one can write:

[ I s V s ] = [ 1 ω s ( t ) 0 0 1 ω s ( t ) ] ⋅ [ I o u t V o u t ] (7)

The equations verified by the DC-Link shown in the figure above are the following:

{ d U D C d t = 1 C I C I C = I i n − I R − I o u t I R = V i n R (8)

Because of the equality of tensions U D C , V i n and V o u t ( U D C = V i n = V o u t ), we obtain, from the previous equations, the following expression of U D C :

d ( U D C ) 2 d t = − 2 R C ( U D C ) 2 + 2 C P S h − 2 C P S e (9)

this equation will be used in MATLAB/SIMULINK to form a block having the following form: (Figure 4).

Active and reactive power

P s = 3 2 ( V s d i s d + V s q i s q ) (10)

Q s = 3 2 ( V s q i s d + V s d i s q ) (11)

To drive our converter correctly, we need to remove this interdependence of the d and q axes. So we are going to create a decoupling block that will reproduce the coupling signal that we want to eliminate and then we are going to introduce this signal with the opposite sign at the input of our system. Our decoupling algorithm will therefore have the following form:

{ X 1 = 1 L s ( V d − V s d ) X 2 = 1 L s ( V q − V s q ) (14)

With

{ X 1 = ( K p − K i S ) ⋅ ( i s d * − i s d ) − w i s q X 2 = ( K p − K i S ) ⋅ ( i s q * − i s q ) − w i s d (15)

The association of the decoupling block and the system model will have the form of the following Figure 6 and Figure 7. We will only be left with our PI with a serial model of the system without coupling [9].

The classical proportional-integral regulator used in figure (9) ensures

maximum control of the error when its gains are maximal. It is also the most widely used regulator on an industrial scale due to its simplicity of implementation and its relatively low cost compared to advanced models [10]:

K P = K P max , K i = K i max

By introducing the following simple function linking the controller parameters

K i K p = R L (16)

We obtain a first-order transfer function with a time constant of

T = 1 K p (17)

F 11 ( P ) = F 22 ( P ) K P P + K P (18)

Given the above equations and a suitable choice of the value of T(T Td) with Td = 0.1 ms (critical time of fault elimination). The gains K_P and K_I can be obtained as follows

K P = 1 T     et     K I = R L ⋅ 1 T (19)

The purpose of the control is to generate the commands to open and close the switches so that the voltages created by the inverter are as close as possible to the reference voltage. By multiplying the number of intermediate levels, the amplitude of each rising or falling edge of the output voltage is reduced: the amplitude of the harmonic lines is therefore lower. In the specific case of PWM operation, the use of multilevel converters combined with a judicious control of the power components also makes it possible to suppress certain families of harmonic lines [11] (Figures 8-10).