Power line modeling allows us to represent their expected electrical behaviour. The main modeling method is based on a system of two partial differential equations that describe the evolution of voltage and current on a power line as a function of distance and time. Two methods, with completely distinct foundations, are used:

Field theory, developed from Maxwell’s equations [23], is rigorous but not always easy to apply and allows us to establish the second theory (circuit theory).Circuit theory: modeling propagation along the line by constructing an equivalent circuit (inductors, capacitors, resistors), the analysis of which is simple.

In this article, we will limit ourselves to field theory. Indeed, field theory, developed from Maxwell’s equations [27], is rigorous but not always easy to apply and allows us to establish the second theory (circuit theory). At any point in space, which is not located on a surface separating two media, that is, in a linear, homogeneous, and isotropic (LHI) medium, Maxwell’s general equations specify that [28]:

The basic variables of these equations are:

B : Magnetic induction (T);

H : Magnetic field strength (A/m);

D : Electric flux density (C/m²);

E : Electric field density (V/m);

J : Electric current density (A/m²);

ρ : Electric charge density (C/m³).

With:

and:

μ : Magnetic permeability in Tm²/A;

ε : Electrical permeability in Cm⁻³/V;

σ : Electrical conductivity.

In this form, called local or differential, Maxwell’s equations express relationships between spatial variations of certain fields and temporal variations of other fields [26][27].

In a vacuum:

Then Maxwell’s equations become [26][27]:

The differential operator nabla ¨ ∇ ¨ is used to express the curl operation ∇   × = rot and the divergence operation ∇   ⋅ = div . Maxwell’s equations can also be expressed in “global form” as follows [26][27]:

After integrating Equations (13) and (14), using the divergence theorem, the two Equations (4) and (6) then become [26][27]:

The boundary conditions for the fields are [26][27]:

or:

n : is the normal to the separation surface, going from medium 2 to medium 1;

j S : is the surface current density in (A/m²);

ρ S : is the surface charge density in (C/m²).

Modeling of High-Voltage AC Power Lines

The π modeling of power lines (see Figure 1 below) allows us to represent their expected electrical behaviour. It is based on telegraphers’ equations [28]-[30]. This model provides accurate results within a certain range of assumptions corresponding to its use; these are black boxes (four-pole models) for which the notion of representing the physical reality of the phenomena disappears, and only the data-results relationship matters. Models used for the design and operation of networks fall into this category. Their limited use leads to simpler models requiring less computing power. Consequently, they are more easily integrated into large simulation suites that allow us to discover a wide range of phenomena [31].

A portion of a power line can be represented by the four-terminal network in Figure 2 below, where [31]-[35]:

• The linear resistance (per unit length) R of the conductor is represented by a series resistance (expressed in ohms per unit length).

The linear inductance L is represented by an inductance (Henry per unit length).

Figure 1. Simplified model of a high-voltage power line [28]-[30].

Figure 2. Two-port network model of a high-voltage power line [36]-[40].

The linear capacitance C between the two conductors is represented by a shunt capacitor C (Farad per unit length).The linear conductance G of the dielectric medium separating the two conductors is represented by a shunt resistance (Siemens per unit length). The resistance in this model has a value of 1/G Ohms.

In this model, the voltage at any point a distance x from the beginning of the line is defined, and at any timet, the voltage U ( x , t ) and the current I ( x , t ) are defined. The equations can be written [36]-[40]:

From the formulation above, we can derive two partial differential equations, each involving only one variable [36]-[40]:

High-Voltage Line Parameters

The calculation of the electrical parameters used for modelling is based on Maxwell’s equations. The model with a single Pi section is only valid for low frequencies and short power lines; otherwise, several Pi sections must be connected in series [34]-[37].

1)Resistance model

Starting from the local Ohm’s law in local form [40]:

or:

j : Current density (A/m²);

δ : Electrical conductivity (Ω⁻¹);

E : Electric field strength (V/m).

The resistance of conductors depends on temperature and frequency; it is defined by [39]-[40]:

The resistivity of a material increases with temperature according to the following law.

or:

α : The coefficient of thermal expansion;

Δ t : The temperature change in ˚C;

ρ o : is the resistivity of the conductor at 20˚C (Ωm).

2) Inductance Model

A conductor carrying a varying current “I” generates a magnetic flux “φ”. The variation of this flux is the cause of the appearance of induced voltage. To account for these effects, the linear inductance of a conductor alone is defined as follows [36]-[40]:

With: h(m): height of the line relative to the ground. D(mm): diameter of the conductor. n: Number of conductors in the bundle [36]-[40].

3) Capacitance Model

The electric field established between the line and ground is the cause of the appearance of capacitive current (leakage current). To account for these effects, capacitance is defined as follows [36]-[40]:

where: C is the line capacity in Farads; Q is the electrical charge carried by the line in Coulombs; and v is the line voltage in kV.

In electrical engineering, each electrical circuit can be represented as a four-terminal network, for which the constants A, B, C, and D, which link the input and output parameters, must be mathematically determined (see Figure 3) [36].

Figure 3. Power line represented by a two-port network [41]-[44].

From Figure 3 associated with the theory of two-port networks, the general equations are [45][46]:

The constants A, B, C, and D depend on the electrical parameters of the line and are generally complex numbers. These equations in matrix form can be written as follows [45]:

The validity of Equations (29), (30) is based on the possibility of representing the transport line by a linear, passive and bilateral network, which imposes [46]:

On the other hand, we can write [45][46]:

Starting from Equations (29)-(33), we derive the values of the constants A, B, C, and D from the line considered [45][46].

Hence:

The π model stipulates that the capacitance of the line is divided into two parts, C/2 each, which are connected to the two ends of the line, while the resistance and inductive reactance (series impedance of the line) of the line are totally concentrated in the center of the line as shown in the electrical diagram below Figure 4 [47]-[49].

Figure 4. The π model of a long-distance AC high-voltage line [47]-[49].

From the electrical model in Figure 4, by applying Kirchhoff’s current law at current node “A”, the current I flowing through branch “AC” is therefore of the form [49]:

The source voltage is modeled by the following relationship [47]-[49]:

It can be shown from Figure 4 that [48]:

The current source is given by the following Equation (38):

or:

In matrix notation, Equation (39) can be written as follows [48][49]:

Also, the constants A, B, C, and D of the network of an electrical line in the π model are deduced from the following expression (41) [49]:

Let the uniformly distributed constants of line [50] be denoted by R: resistance in (Ω); L: inductance in (H); G: conductance in (Ω⁻¹), and C: capacitance of one phase in (μF). We define Z(impedance) and Y (admittance) by the system of Equations (42) following [51].

The operating equations of the lines with uniformly distributed constants characterising V₁ andI₁ at the input of the AC HV line, established from the solution of Equations (21) and (22), are then given by the expressions (43), (44) [51]:

If we consider that:

And, as we know that, cosh 2 a − sinh 2 a = 1 , Equations (43) and (44) satisfy the passive two-port network equation.

For the transmission of electrical energy over very long distances, the Joule effect must be minimised, and consequently, the ohmic resistance (R) and conductance (G) are extremely low and negligible compared to (jLω and jCω). Furthermore, all high-voltage lines are constructed with a surface potential gradient of the conductors well below the critical gradient in order to reduce losses due to the corona effect. Lateral losses in the insulators are also negligible. In short, we can write [45].

Taking into account these relationships, the quantity (a) defined by Equation (45), we therefore find:

Furthermore, the approximate models of expressions (27-28) are respectively [39][40]:

If we neglect the internal flux of the cables.

In these expressions, (a') is the fictitious equivalent radius of a beam of (n) conductors and (a') is given by:

or:

r: is the outer radius of the cable used in mm,S: is the equivalent phase spacing in mm, and D: is the conductor diameter in mm.

This phase spacing can be determined by [39][40][45]:

Multiplying terms (49-50) gives the value of 3.10⁵, which is close to the numerical value of the speed of light and, given the units used, is indeed (c) the speed of light.

Under these conditions (48) is transformed as follows:

where (λ) is the wavelength of the sinusoidal alternating current, which, if f= 50 hertz, is 300,000 ÷ 50 = 6000 km. Therefore, we finally have:

According to [45], complex hyperbolic equations possess the mathematical property of equivalence with ordinary trigonometric functions given by:

Taking into account expressions (43-44) and Equations (55)-(57), the simplified form of equations (43-44) is:

For a high-voltage AC line operating at no load, I₂ = 0 and Equation (58) becomes:

Equation (60) shows that, for ( l ) equal to 1000 km, the voltage at the end of the AC HV line is twice that of the input:

When alternating current power transmission lines exceed a certain length, as shown in Equation (60), they become sensitive to distance and frequency, generating electromagnetic waves. These waves are the basis of capacitive and inductive effects that can cause high no-load overvoltages of almost 200%.

For ( l ) reaching a quarter-wavelength, i.e., 1500 km, the overvoltage is infinite. For ( l ) exceeding a quarter-wavelength, the overvoltages along the line take on excessive values when the line is unloaded [45]-[57].

Assume that the series capacitor C₁ is inserted at the end of the line and that the resulting situation is that defined by the following Figure 5 [45]:

Figure 5. Insertion of a serial capacitor C₁ at the end of a line segment considered in (n) segments. [45]-[58].

The vector equations relating to Figure 5 are:

By eliminating V ′ we have:

or:

Equations (65) and (66) obviously satisfy expressions (29)-(30). Let us adopt the following simplified expressions as equations of the lines:

For very long lines, the capacitive current is significant. It causes overvoltages (see Equation (60)), which can be excessive, and it is necessary to cancel this capacitive current completely or partially. An insertion (or several suitably distributed along the line) of a shunt reactance can serve this purpose. Suppose we introduce a shunt reactance X at the incoming end of the line in Figure 5, where the capacitor C₁ is inserted. The resulting situation will be that of Figure 6 below, allowing us to deduce the vector equation below:

The presence of the shunt reactance in Figure 6 allows the HTAC line to be artificially transformed into a DC connection, with A being negligible. This results in compensation of the capacitive current I₂, provided the line is not compensated at 100%. Consequently, at no load, I ′ 2 = 0 , and the overvoltage at the end of the line is therefore reduced (see Equation (73)).

Figure 6. Insertion of a shunt reactance X at the end of a line (or line segments).

Therefore, the capacitive current of the line, equal to C ⋅ V 2 , is reduced and could even become inductively reactive, which is not desirable. For a long line, the introduction of a shunt reactance produces an overly abrupt concentrated effect, and it is more appropriate to distribute the insertions of shunt reactances and series capacitors evenly along the line. Consider a line of length (L) divided into (n) equal segments. Let (d) be the length of a segment. We have [29]:

In a practical implementation, the intersection points could be determined in a mandatory manner:

Presence of an electrical power supply station. The intersection of the reactances and capacitors should preferably be made at such a location because it is under surveillance, and the connection would also be less expensive.Geographical (topographical) impossibility of easily accessing the theoretical intersection point. For example, a river crossing at that location or the presence of a mountain that is difficult to access.Favourable location for accessing such a point due to the presence of a road, railway, or navigable river.

If this is the case, Equation (74) will be replaced by the following expression (75):

To illustrate our ideas, we will consider the simple case of dividing the line into (n) equal segments. For segment (1)-(2) we have:

And,

And the insertion node of the shunt reactance X, we have:

Following the same reasoning, for the segment (2)-(3), we have:

and,

Substituting V 2 and I ′ 2 into Equations (76) and (77) with their expressions, we obtain the following results after grouping:

and similarly, I 1 is equal to:

The next node at the end of the second segment is another reactance X, and we proceed in the same way as before. Finally, the various A, B, C, and D coefficients can be obtained from the algorithms shown in Table 1 below, clearly illustrating the general law governing the formation of the A, B, C, and D coefficients of the two-port network.

Table 1. Coefficients of the various equations as a function of the number of segments.

Obtaining the recurrence model for a high-voltage AC line segmented into n segments using the finite difference method relies on discretising the telegrapher’s equations and applying an iterative four-pole formalism. The continuous transmission line is governed by the telegrapher’s equations in the steady state (frequencyf, angular frequencyω), as a function of the position xalong the line:

where Z = R + j ω L is the series impedance per unit length, and Y = G + j ω C is the parallel admittance per unit length.

For a segment of length Δ x = L / n , we approximate the derivatives at the discretisation points k and k+1 (where k varies from 0 to n − 1):

Voltage: V k + 1 − V k Δ x ≈ − Z ⋅ I k (or a more precise centered approximation)

Current: I k + 1 − I k Δ x ≈ − Y ⋅ V k + 1 (or another approximation)

Modeling a segment (ElementaryQuadripole)

The use of the chain of transfer matrices (ABCD or transmission matrices), where the total matrix T t a t o l = T 1 ⋅ T 2 ⋯ T n , links the input parameters ( V 1 , I 1 ) to the output parameters ( V n + 1 , I n + 1 ) via a recursive relation between each segment where, each segment i (length Δx) is represented by a transmission matrix T i , the input parameters ( V i , I i ) and output parameters ( V i + 1 , I i + 1 ) are linked by the following relation (85):

Recurrence relation for n segments

The complete line is a cascade of n transmission matrices T i . If the segments are identical ( T i = T for all i), the total matrix is:

The overall input/output relationship becomes:

The recurrence model allows us to calculate the state of the row at any point by iteratively applying the matrix Tbased on the initial or final conditions. We note that the matrix T n equals:

Substituting expression (88) into Equation (87), we find:

From model (89), we can deduce the recurrence model for an HT AC line segmented into n segments.

The coefficients A n − 1 and C n − 1 define the no-load overvoltage at the end of the line, while B n − 1 and D n − 1 define the capacitive current.

Line voltage: length U = 400 kV;Line length l = 1200 km;Line inductance L = 0.292 H/km;Line capacitance C = 3.58 × 10⁻⁶ F/km;Characteristic line impedance Z₀ = 286 Ω.

1) Parameters (A,B,CandD) of the Two-PortNetwork

The parameters of (A, B, C and D) of the four-terminal network modelling the AC high-voltage line in our study for the 3 scenarios are presented in Tables 2-4 below:

Table 2. Coefficients of the various equations as a function of the number of segments(Scenario 1: Case of n = 4 segments).

Table 3. Coefficients of the various equations as a function of the number of segments(Scenario 2: Case of n = 6 segments).

Table 4. Coefficients of the various equations as a function of the number of segments(Scenario 3: Case of n = 8 segments).

2) Voltage and Capacitive CurrentBehaviourat the End of the EHV Line

By implementing models (76)-(82) in the Matlab environment, we obtain the analytical results presented in the following Figures 7-9:

Figure 7. Variation of current as a function of voltage (Scenario 1: Case of n = 4 segments).

Figure 8. Voltage variation as a function of current for a line arranged in six sections (Scenario 2: Case of n = 6 sections).

Figure 9. Voltage variation as a function of current for a line divided into eight sections (Scenario 1: Case of n = 8 sections).

1) Parameters (A,B,C, andD) of the Two-Port Network

The parameters (A, B, C, and D) of the two-port network modeling the AC high-voltage transmission line in our study for the 3 scenarios are presented in Figures 10-12 below.

2) Voltage and Capacitive CurrentBehaviourat the Output of the EHV Line

By clicking the “display voltage-current curves” interface shown in the GUI interface of Figures 10-12, we obtain the analytical results presented in the following Figures 13-15 and Table 5.