The main simplifications used in this article are:

Balanced currents.Constant power factor.Negligible transverse parameters.Invariant line parameters.

The model is susceptible to failures (e.g., compensation devices, tap changes, unbalanced load).

Modelling power lines allows us to represent their expected electrical behaviour. The main modelling 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: modelling 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 (Tesla, T).

H : Magnetic field strength (Ampere/meter², Am⁻²).

D : Electric flux density (coulomb/meter², Cm⁻²).

E : Electric field density (volt/meter, Vm⁻¹).

J : Electric current density (Ampere/meter², Am⁻²).

ρ: Electric charge density (coulomb/meter³, Cm⁻³).

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 ∇ ⋅ 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 fields [26][27] are:

With:

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

J s : is the surface current density;

ρ s : is the surface charge density.

Pi modelling of power lines (see Figure 1 below) allows us to represent their expected electrical behaviour. It is based on the telegraphers’ equations [28]-[30].

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

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).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, we define the voltage at any point a distance x from the beginning of the line and at any time t, the voltage U ( x , t ) and the current I ( x , t ) . The equations are written [36]-[40]:

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

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

The calculation of the electrical parameters used for the 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].

2.4.1. Resistance Model

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

Or:

J : Current density (A/m²)

δ : Electrical conductivity (Ω⁻¹)

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 this law.

Or:

α : The coefficient of dilation

Δ t : Temperature variation ˚C

2.4.2. 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 the 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].

2.4.3. Capacity 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]:

For the calculations of losses in the lines, we assume that the current intensities are balanced in the lines; that is, we start from the assumption of equipartition of the loads in the lines. Energy lossesin HV lines are intrinsically linked to Maxwell’s equations, in particular via Poynting’s theorem, which describes the electromagnetic energy balance. The term for dissipation due to the Joule effect appears in the local energy conservation equation.

The starting point is Poynting’s theorem in its local form, derived from Maxwell’s equations (notably Maxwell-Faraday and Maxwell-Ampère) [36]-[40]:

With: Π is the Poynting vector (surface density of electromagnetic power). ϖ e m is the volumetric density of stored electromagnetic energy. P j is the volumetric density of power dissipated by Joule effect.

By combining Maxwell’s equations, we can isolate the dissipation term. For a conducting medium obeying local Ohm’s law (J= ρE), where ρ is the electrical conductivity, the volumetric power density dissipated by Joule heating is given by:

Using Ohm’s law, it can be expressed as a function of the electric field or current density:

To obtain the total power dissipated by Joule heating in a given volume V (for example, a conductor of resistance R), the volumetric power density must be integrated over this volume:

This integration leads, in the simple case of an Ohmic circuit, to the well-known macroscopic law:

For a high-voltage line of L_c = conductor length, containing n conductors of the line and R_c = linear resistance, carrying the maximum current Imax, the Joule effect losses in this line are calculated using the following equation:

With:

P_j = Joule effect losses in lines

n = Number of conductors on the line

R_c = Linear resistance [Ω/km]

I_max = Maximum current of a conductor [A]

L_c = Length of conductor [m]

These losses are obtained primarily based on the physical characteristics of the line. Given the line voltage and the current it can carry, the active power that the line will supply can perhaps be modelled as follows [41]:

With:

P = Active power supplied by the line;U_L = Line voltage [kV];I_max = intensity of a conductor [A]; cos φ = 0.9

We know that in a conductor, the power lost due to Joule heating is “ R c I c 2 ”, with “ I c ”, the current flowing in the conductor. Considering the balanced, three-phase line, the apparent power it carries will be written as follows [42]:

With:

U_L = Line voltage (kV)

I_L = Line intensity (A)

P = Active power carried by the line (W)

Q = reactive power carried by the line (VAr)

Consider the following equality:

Thus we can draw the line current I_L:

The power loss “P_P” in the line can thus be noted as:

Substituting expression (40) into relation (41) gives us:

With: R_L, I_L and U_L, respectively the line resistance, the maximum line current and the line voltage.

The relative losses in the line are thus given by the following expression:

With:

P p r : Relative power losses in %; I L : Line intensity in A; R L : Conductor resistance in Ω; U L : Voltage composed of a V-shaped line.

The assumptions for calculating the voltage drop in AC for a high-voltage AC line are that the overvoltage is primarily caused by the longitudinal parameters, resistance and reactance in series (R_L, X_L), and that the contribution of the transverse parameters is negligible. The transmission line model adopted is therefore the equivalent circuit shown in Figure 3 below [43][44]:

Figure 3.Equivalent circuit of the HV line [43]-[47].

Legend

U O ¯ =Voltage from the origin relative to neutral in kV

I L ¯ =Maximum line intensity at A

U F ¯ =Voltage at the end relative to neutral in kV

X L = Reactance inductance of the line in Ω

R L =Line resistance in Ω

Applying Kirchhoff’s law to Figure 3 above, we find the following equations:

Thus, the relative phase-to-neutral voltage drop can be denoted by the following equation:

Integrating Equation (47) into expression (48), we find the following equations:

Multiplying the right-hand side of Equation (50) by 3 , the voltage drop relative to the line voltage gives:

We know that in three-phase:

Thus, expressions (52) and (53) in relation (51) give us:

Let’s highlight R_L in Equation (54), so we find:

Therefore, Q P = tg φ in expression (55), we find:

Since losses result, among other things, in voltage drop, we can determine the models that highlight the losses related to the active power transmitted by the line and the voltage drop related to the line voltage, based on expressions (44) and (57). The ratio between P p r these two values Δ U r gives:

Hence:

From this relation (59), we can draw the following conclusions:

X L R L = tg δ The quality factor of a high-voltage (HV) line is typically between 2 and 10 for a 220 kV HV line. tg δ It is calculated based on the linear inductance (L) and resistance (R), which are low over long distances. The quality factor of an HV power line is not a standard value, but rather the result of calculations based on its physical properties (inductance, resistance, capacitance) and the line frequency.

The tg δ = 2 , losses related to the active power transmitted by the HV line as a function of the voltage drop relative to the line voltage are quantified from expression (64) below:

To provide an accurate and reproducible description of the use of data from the SNEL Inga-Kinshasa HV line in a calculation or simulation model, the following are the standard parameters generally applied in engineering studies of the Congolese grid:

2.8.1. Dataset Description

Source: SNEL (National Electricity Company) SCADA System.Measurement Points:Transmitting End: Inga substations (Inga 1 or Inga 2).Receiving End: Kimwenza substations (depending on the specific line of the corridor).Parameters Collected: Line Voltage (U), Phase Current (I), Active Power (P), Reactive Power (Q), and Frequency (f).

2.8.2. Processing Protocol (Model)

Sampling Interval: Typically 10 seconds for SCADA historical archiving, although phase measurement units (PMUs), if present, can reach 50 - 60 samples per second.Averaging Window: Raw data is often integrated over a 10 - 15 minute window for load planning studies, or stored at the second level for transient stability analyses.Operating Conditions: Steady State for power flows. Data generally excludes periods of major maintenance unless the study focuses on reliability.Preprocessing:Filtering of outliers due to telecommunication errors.Normalization (per unit conversion) based on nominal voltages (220 kV depending on the segment).

2.8.3. Assumptions of Uncertainty and Accuracy

In the absence of specific calibration certificates, the models use the following industry standards for high-voltage sensors:

a) Voltage Measurements (TP):

Accuracy class: 0.5% to 1%.Allowable uncertainty: ±1% (including transmission line errors).

b) Current Measurements (TI):

Accuracy class: 0.5% (measurement) or 5P20 (protection).Allowable uncertainty: ±1.5% at full scale.

c) Power (P and Q):

The combined uncertainty is generally estimated at ±2% to ±3%.

2.8.4. Use in the Model

The data is primarily used for:

Validation: Adjusting the line parameters (impedance, admittance) so that the software results (Matlab) correspond to the actual measurements.State Estimation: Verify the consistency of redundant measurements between Inga and Kinshasa to identify actual line losses.

The numerical values of the HV line parameters, such as R_L = 0.07 Ω/km, X_L = 0.15 Ω/km, line length 262 km, power factor cosφ = 0.9, tgφ = 0.484, and respective active and reactive power (P = 350 MW and Q = 169.4 MVAr the line input) and (P = 279 MW and Q = 84.1 MVAr the line output), are taken from SNEL/DRC.

Energy losses are a major concern for all producers, transmission, and distribution of electrical energy. To simulate these losses in high-voltage lines, we assumed that the current intensities were balanced. Considering the proposed models (see Equations (57) and (64)), the computer tools used, the characteristic data of the 220 kV line, and the power transmission readings provided by SNEL/DG/PKB, the simulation results are presented in Figures 4-20 below:

Figure 4. Simulink model of power transmission on the 220 kV Inga-Kimwenza high-voltage line.

Figure 5. Voltage of point A the Inga dispersal substation (220 kV HV line input) in Volts.

Figure 6. Current of point A the Inga dispersal substation (220 kV HV line input) in Amperes.

Figure 7. (a) Active power in Watts and (b) Reactive power at point A the Inga dispersal station (HV line inlet).

Figure 8.Voltage at point B the Kimwenza substation (220 kV HV line output) in Volts.

Figure 9. Current at point B at the Kimwenza substation (220 kV HV line output) in Amperes.

Figure 10. (a) Active power in Watts and (b) Reactive power of point B the Kimwenza substation (high-voltage line output).

Figure 11. Simulink model of the voltage drop on the 220 kV Inga-Kimwenza high-voltage line.

Figure 12. Voltage drop in % the end of the 220 kV Inga-Kimwenza high-voltage line.

Figure 13.Voltage drop in kV the end of the 220 kV Inga-Kimwenza high-voltage line.

Figure 14.Simulink model of active power losses (in % and in Watts) in the 220 kV Inga-Kimwenza high-voltage line.

Figure 15. Results (a) active power losses in % and (b) active power losses in Watts in the 220 kV HV line obtained from the Simulink model.

Figure 16. Model of the equation Δ U r = 3 R L U L I L cos φ U L 2 ( 1 + X L R L tg φ ) (57) proposed for calculating the voltage drop on the HV line.

Figure 17.Results of the equation model Δ U r = 3 R L U L I L cos φ U L 2 ( 1 + X L R L tg φ ) (57) proposed for calculating the voltage drop in % on the HV line.

Figure 18. Results of the equation model Δ U r = 3 R L U L I L cos φ U L 2 ( 1 + X L R L tg φ ) (57) proposed for calculating the voltage drop in Volts on the HV line.

Figure 19.Proposed model (Equation (64)) for evaluating active power losses (in % and in Watts) in the HV line. Implemented in Matlab.

Figure 20.Results of equation 64 (a) active power losses in % and (b) active power losses in Watts in the HV line.