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

The block diagram of the UPFC is given in Figure 1, which will be used to model this device in the matlab-simulink environment.

Applying 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 model of serial converter. It will be used to build, under Simulink, the block which represents the serial part of the system. We also want to remind that the principle applied on the serial part will be the same on the parallel part of our system.

The interphase power regulator uses a group of three-phase inductors and capacitors each installed in series between two networks or subnetworks. What distinguishes this new class of equipment from other series compensation equipment is the way the series components are connected to the networks. For example, the A-phase inductor and capacitor of the first network could be connected to the B and C phases of the network. When all the components are energized, the magnitude and phase angle (δ) of the current is set in one of the two buses to which the controller is connected. The current control thus allows the power carried by the controller to be adjusted, as well as the reactive power absorbed or generated at one of the buses. Inductors and capacitors are always considered perfect without losses. The impedances of the series components are then reduced to their imaginary part, i.e. the reactance. In the context of the controller where the series components are arranged in parallel to each other, the term susceptance is used instead of reactance for practical reasons (B = −l/X) [6]. (Figure 2)

RPI technology has given rise to a wide range of devices that can take many different forms depending on the application. Before entering fully into the description of RPI, we felt it necessary to take a look at a few examples in order to highlight the versatility of the technology as well as the need for the analytical means presented later. Table 2 presents the main characteristics of the RPI topologies.

· Number of branches

The number of branches in an RPI is one of its most fundamental aspects. In general, the single-phase. In general, the single-phase circuit of an RPI can have n branches in parallel. In practice, however, this number is kept to a minimum in order to limit the size and cost of the device. It is the angular range of the angle δ_sr across the device that defines the number of branches. [7]

· Types of interphase power regulators

Depending on the connection bar, there are two types of interphase power controllers: synchronous interphase power controllers and asynchronous interphase power controllers.

To illustrate this, and at the same time justify the construction constraints that we will use later, the configuration of RPI highlighted in our work is the one with a phase-shifting transformer (PST) in parallel with a capacitor; it is the RPI 30P15. This configuration is illustrated in Figure 3.

The model of our RPI connected to the network can be related to a quadrupole as shown in t Figure 4.

The power balance of this circuit leads to the following equations.

· Sign convention of the powers.

The apparent powers S_S, and S_r are defined in the same direction as the currents.

While posing:

V S = V S e j δ B 1 (5)

V r = V r e j δ B 2 (6)

{ δ B 1 = δ − φ 1 δ B 2 = δ − φ 2 (7)

While for the powers S_r and S_S of the shunt admittances we obtain:

S s = V S V r e j δ B 1 Z * − V S 2 Z * (8)

S r = V S V r e j δ B 2 Z * − V r 2 Z * (9)

These power equations are based on the following assumptions: The system is symmetrical, so it is always possible to represent a three-phase element by a single-phase equivalent; the frequencies of the S and R bars are essentially the same (f_s = f_r), so that it is possible to use the phasors for the equation; the series elements are linear (they do not produce harmonics).

While posing:

Z = R + J X (10)

Y _ S = G S + j B S (11)

Y _ r = G r + j B r (12)

We can rewrite as am in trigonometric form

S _ s = 1 Z [ V s V r R cos ( δ B 1 ) − X sin ( δ B 1 ) − V s 2 R ]     + 1 Z [ V s V r R sin ( δ B 1 ) + X cos ( δ B 1 ) − V s 2 R ] (13)

S _ r = 1 Z [ V s V r R cos ( δ B 2 ) − X sin ( δ B 2 ) − V r 2 R ]               + 1 Z [ V s V r R sin ( δ B 2 ) + X cos ( δ B 2 ) − V r 2 R ] (14)

S _ s = V s 2 ( G − j B 1 ) = − P + J Q s (15)

S _ r = V r 2 ( G − j B 2 ) = P + J Q r (16)

The active power P is positive when the power flow occurs from the S side to the right side, the reactive power Q_S, and Q_r are positive when the IPC generates reactive power to the buses to which it is connected Since δ_B1 = −δ_B2 the powers P, Q_s and Q_r of the series element, formulas in terms of conductance and susceptance, become in matrix form [11]:

[ − V S V r sin δ B 1 − V S V r sin δ B 2 V S 2 − V S V r cos δ B 1 V S 2 − V S V r cos δ B 2 V r 2 − V S V r cos δ B 1 V r 2 − V S V r cos δ B 2 ] [ B 1 B 2 ] = [ P Q S Q r ] (17)

B 1 = P ( 2 V r − V S cos δ − 3 V S sin δ ) − Q r V r ( 3 cos δ − sin δ ) 3 V S V r ( V S − 2 V r cos δ ) (18)

B 2 = − P ( 2 V r − V S cos δ + 3 V S sin δ ) − Q r V r ( 3 cos δ + sin δ ) 3 V S V r ( V S − 2 V r cos δ ) (19)

Equations two and three of (3.15) indicate that the reactive powers are coupled to each other by the following simple Orelation:

Q S = ( B 1 + B 2 ) ( V S 2 − V r 2 ) + Q r (20)

Figure 5 below illustrates the three-phase model of our RPI on a three-phase network.

The short-circuit administered at this level is a short-circuit between phase and neutral, we administered it at the instant t = 0.05 ms and which lasted 0.2 ms, the observed behavior is as follows: (Figure 12).

· Visualization of voltages and currents

A single-phase short circuit of 0.2 ms duration in the presence of the UPFC controller gives rise to a transient of 0.12 ms duration with variations as shown in Table 4.

We notice in this table that a single-phase short-circuit in a network in the presence of the UPFC controller is characterized by a strong increase of the current which tends to exceed the nominal intensity of the line. On the other hand, the opposite is true for the voltages, which forces us to say that the UPFC favors a good maintenance of the voltage plan profile.

The short-circuit administered at this level is a short-circuit between phase and phase, we have always administered it at the same period t = 0.05 ms and for the same 0.2 ms fault duration the observed behavior is as follows: (Figure 13).

· Visualization of voltages and currents

A two-phase short-circuit lasting 0.2 ms in the presence of the UPFC controller gives rise to a transient regime lasting 0.24 ms. Table 5 below provides information on the evolution of the network parameters in the presence of this fault.

The observation remains the same as for the single-phase short circuit.

In this part, we have made the same considerations for the time of the beginning of the fault as for its duration. The loss of one phase of the line in the presence of a controller can generate the phenomena observed below: (Figure 14).

· Visualization of voltages and currents

We notice how the appearance of a fault in the presence of the UPFC makes all the phases oscillate and some time later, the network finds a new regime of stable functioning, the oscillations generated by the disturbance disappear with an attenuated amplitude. This during in spite of the presence of the UPFC, the peaks of intensity remain a concern in this network. Table 6 below shows the evolution of the network parameters in the presence of this fault.

In this section, we are only interested in the disturbed regime of the network in the presence of the RPI controller while considering the results before, in the presence of the fault without unchanged controller. The network plus RPI controller

set is represented as follows: (Figure 15).

The short-circuit administered at this level is a short-circuit between phase and neutral, we administered it at the instant t = 0.05 ms and which lasted 0.2 ms, the observed behavior is as follows: (Figure 16).

· Visualization of voltages and currents in the presence of an RPI

A single-phase short circuit of 0.2 ms duration in the presence of the RPI controller gives rise to a transient of 0.11 ms duration. Table 7 shows the evolution of the network parameters in the presence of this fault.

We note in this table that a single-phase short-circuit in a network in the presence of the RPI controller causes current fluctuations with damped amplitudes and increasingly lower than the rated current.

The short-circuit administered at this level is a short-circuit between phase and phase, we have always administered it at the same period t = 0.05 ms and for the same 0.2 ms fault duration the observed behavior is as follows: (Figure 17).

· Visualization of voltages and current in the presence of an RPI

A two-phase short-circuit lasting 0.2 ms in the presence of the RPI controller gives rise to a transient regime lasting 0.125 ms. Table 8 provides information

on the evolution of the network parameters in the presence of this fault.

In this part, we have made the same considerations for the time of the beginning of the fault as for its duration. The loss of one phase of the line in the presence of a controller can generate the phenomena observed below: (Figure 18).

· Visualization of voltages and currents in the presence of an RPI

Once again we can appreciate the juicy contribution of the RPI controller in improving network performance as shown in Table 9.