Historically, the electrical power supply of Ndjili International Airport has depended on the Congolese national grid, managed by the Société Nationaled’Électricité (SNEL) [10]-[13]. Faced with recurrent blackouts of the public utility grid most notably during the disruptions of September 2025 the airport experienced protracted outage periods that paralyzed critical operations, including runway systems, Airfield Ground Lighting (AGL), and control tower functionalities [10]-[13]. To mitigate these systemic failures, a new 15 MVA transformer was recently commissioned on-site to enhance power supply stability and attenuate reliance on the surrounding domestic distribution grid [10]-[13]. The structural and consumption data used to configure this model were derived from an on-site characterisation campaign conducted in collaboration with the Régie des Voies Aériennes and SNEL in October 2025. Measurements of rated power and load profiles for critical loads were obtained using Class A power quality analysers recording at 15-minute intervals at the main low-voltage distribution boards serving the terminal, the control tower, and the airfield lighting system. Cable parameters for 95 to 240 mm² copper conductors with XLPE insulation and transformer leakage impedances TR1 andTR2 were sourced directly from manufacturer datasheets validated during the installation of the new 15 MVA transformer. Historical fault occurrences and timelines are based on control tower operational logs and technical incident reports from September 2025.

As illustrated in Figures 1(a)-(c), the primary electrical power supply of Ndjili International Airport is delivered by two 1600 kVA transformers operating in a redundant configuration. Emergency backup power for the technical facilities is provided by two alternating 635 kVA diesel generator units. Additionally, a 350kVA generator ensures backup power for the commercial sector [10]. Voltage regulators and Uninterruptible Power Supplies (UPS) are integrated into the network architecture to enhance the reliability of the airfield ground lighting circuits, as well as the power supply lines for navigation and communication equipment. For security and safety imperatives, the comprehensive global electrical schematic of Ndjili International Airport cannot be disclosed to the public [13]. The backup architecture modelled for the numerical simulations comprises three distinct diesel generation units (G1,G2,G3), distributed according to the hub’s active and apparent power requirements. Although the preliminary schematic (Figure 1(a)) shows cabinets configured for theoretical coupling arrangements, commercial cabinet: 2×510 kVA; technical cabinet: 2 ×630 kVA, the mathematical simulation model is based on the actual physical distribution measured at the backup switchboard (Table 1), ensuring a total available nominal capacity of 1428 kW at 1785 kVA and cosφ = 0.8:

(a)

(b)

(c)

Figure 1. Functional block diagram of the Ndjili Airport electrical system: public utility grid, power transformers, MV/LV transformers, distribution lines, backup generators, critical and non-critical loads [10][11]. (a) Topology of SNEL power sources and emergency backup generator circuits [10]-[13]; (b) HQ LVMSB topology and emergency backup generator circuits [10]-[13]; (c) Topology of critical loads powered by emergency backup generators.

Generator 1 (G1): Nominal active power of P_n1 = 280 kW, apparent power of 350 kVA; primarily dedicated to partial backup for airfield lighting and the control tower.Generator 2 (G2): Nominal active power of P_n2 = 508 kW, apparent power of 635 kVA; assigned to the aeronautical technical auxiliaries busbar.Generator 3 (G3): Nominal active power of P_n3 = 640 kW, apparent power of 800 kVA; sized to handle the residual load of the backed-up commercial terminal.

The mathematical modeling of an airport electrical system aims to represent the dynamic and static behavior of the network to enable performance analysis and perturbation simulation. The primary components of the airport electrical system can be classified into three distinct subsystems [14]-[16]:

Generation (generators);Transmission and distribution (lines);Demand (consumers).

To analyze a complex electrical power network such as an airport electrical system, equivalent models or schematics of these core components, namely generators, various types of transformers, lines, and loads, are established.

In addition to the aforementioned aspects, several other concepts are addressed, specifically [14]-[16]:

Power balance equations;Probabilistic approaches to reliability;Redundant configuration frameworks.

2.3.1. Alternator Model

To model the emergency diesel generator, the unit is represented by an internal voltage source E g in series with an internal impedance Z g . The dynamic model of the Automatic Voltage Regulator (AVR), utilized to simulate the transient response during utility grid outages, can be formulated via the following equation [17]:

The AVR adjusts the generator’s excitation voltage to maintain a constant terminal voltage. The standard first-order model (modified IEEE Type 1) is expressed by the following differential equation [14]-[16]:

where:

E g : Generator internal voltage (V). T A : Regulator time constant (s), typically between 0.02 and 0.1 s. K A : Regulator gain, typically between 20 and 100. v r e s : Desired reference voltage (V). v f : Measured voltage at the generator terminals (V).

During a grid outage, the load switches to the generator. The terminal voltage V t drops instantaneously due to the internal impedance Z g . The instantaneous algebraic relationship is given by:

I L : Current drawn by the emergency load (A). Z g : Transient impedance R g + j X ′ d , where j X ′ d represents the direct-axis transient reactance.

The generator is considered the core of the airport electrical system, ensuring the production of the electrical energy demanded by the consumer. In power flow analysis, the generator is modeled as a constant voltage source V_g injecting active power P_g and reactive power Q_g (as illustrated in Figure 2).

Figure 2. Generator modeling [14].

The reactive power of the generator is adjusted to maintain a constant bus voltage. Furthermore, the generator is characterized by two reactive power operational limits, namely Q_gmax and Q_gmin. When either of these boundaries is reached, the reactive power Q remains fixed at that limiting value, and the bus voltage is no longer strictly regulated.

2.3.2. Transmission and Distribution Line Modeling

In modeling the electrical lines of an airport power system, the selection of the appropriate model depends on the cable length and the specific objective of the simulation (steady-state or transient analysis). Airport networks predominantly utilize underground cables operating at Medium Voltage (MV: 15 kV or 20 kV) and Low Voltage (LV: 400 V), which are characterized by short distances, typically under 10 km.

The standard representation for these distances is the nominal π equivalent circuit model (applicable to short and medium lines, as illustrated in Figure 3).

Figure 3. Transmission and distribution line modeling [15].

The nodal admittance matrix of a transmission line connecting bus i to busj is given by [14]-[16]:

where r i j is the series resistance of the line, r i j is the series reactance of the line, and b i j 0 represents the shunt susceptance.

Each transmission line or underground cable is defined per kilometer by four primary electrical parameters:

R: Linear resistance (Ω/km) modeling Joule losses.L: Linear inductance (H/km) yielding the inductive reactance (X = ωL).C: Linear capacitance (F/km) modeling the capacitive charging effect, which is highly pronounced in airport underground cables.G: Linear conductance (S/km) modeling dielectric losses, typically neglected in analysis (G ≈ 0).

2.3.3. Transformer Model

An electrical power transformer is mathematically represented by an asymmetrical π equivalent two-port network. The associated operational parameters are the transformation ratio a and the leakage impedance [14]-[16]. The ratios (a = R_ij/R) are integrated directly into the elements of the bus admittance matrix. Consequently, the matrix susceptance components (b = R_ij) are formulated as functions of the transformation ratios a (as illustrated in Figure 4).

Figure 4. Transformer model [14][16]-[19].

The admittance matrix of a transformer connected between bus i and bus j is expressed as:

where:

a: The transformation ratio. y i j : The admittance of branch i-j.

In specific cases, the magnetizing branch is incorporated into the model, where it is represented as a shunt inductive susceptance [14]-[16].

The MV/LV transformers are modeled by their equivalent impedance Z t = R t + j X t , which accounts for the internal voltage drop and associated power losses. The secondary voltage is calculated by [14]-[16]:

where: N p r i m and N s e c represent the primary and secondary number of turns, respectively.

2.3.4. Electrical Load Modeling

The electrical load is commonly modeled as a load impedance Z_load that consumes constant active power P_L and reactive power Q_L (as illustrated in Figure 5) [14]-[16].

Figure 5. Electrical load model [14]-[16].

The loads within the airport electrical system are classified into two categories:

Criticalloads (runway lighting, control tower, airfield ground lighting,communication systems):

Represented as constant power loads P + jQ or by functions dependent on the voltage V α and the power factor, where α describes the sensitivity of the load to voltage fluctuations [14]-[17].

The introduction of α and β coefficients near unity [0.9, 1.1] indicates that the critical loads exhibit a behavior approximating a constant current characteristic.

If ( α = β = 0 ): The load exhibits a constant power characteristic (independent of V). This applies to modern electronic equipment equipped with switch-mode power supplies (control tower servers, radar systems, telecommunication switches) that regulate and enforce their internal power consumption.If ( α = β = 1 ): The load exhibits a constant current characteristic. The specified intervals [0.9, 1.1] accurately model modern LED airfield ground lighting or legacy systems driven by Constant Current Regulators (CCRs), which maintain a stable current despite network voltage fluctuations.If ( α = 2 or β = 2 ): The load would exhibit a constant impedance characteristic (e.g., resistive heating elements).

Non-critical loads (administrative buildings, commercial areas, logisticalzones):

Modeled as constant or voltage-dependent loads, integrated with automated load shedding capabilities in the event of power deficits.

For administrative buildings and commercial zones, the load behavior is frequently modeled as Constant Power ( α = β = 0 ) or via a classical polynomial ZIP model. The defining operational feature here lies in the Under-Frequency or Under-Voltage Automatic Load Shedding (ALS) logic function. Mathematically, the non-critical load configuration is expressed as [17]-[21]:

where S ( t ) ∈ ( 0 , 1 ) is a binary state variable governed by a protective relaying function:

Under-Frequency Load Shedding (UFLS): If the emergency generator frequency drops below a predefined threshold (e.g., f< 48.5 Hz) due to mechanical overloading of the diesel engine, then S(t) = 0.Under-Voltage Load Shedding (UVLS): If the measured voltage V remains below a critical threshold (e.g., 0.85 pu) for more than x milliseconds (implying the AVR fails to provide adequate compensation), then S(t) = 0.

By synthesizing the generator model (E_g, Z_g), the nominal π-line models (Z_ligne, Y_ligne), and the two distinct load categories, the power balance equation at the airport bus is formulated as follows:

2.3.5. Load Shedding Relay Modeling (UFLS/UVLS)

The automatic load shedding of non-critical loads is triggered based on voltage and frequency threshold logic. The equation governing the control variable S(t) is formulated as [19][20]:

Recommended values for an airport emergency power system:

Frequency threshold f t h r e s h o l d = 48.5 Hz (for a 50 Hz grid) with an operating time Δ t f = 100 ms.Voltage threshold V t h r e s h o l d = 0.85 pu with a time delay Δ t v = 500 ms (allowing adequate time for AVR response).

Unlike critical loads α, β ≈ 1, the terminal administrative buildings and commercial areas integrate numerous Heating, Ventilation, and Air Conditioning (HVAC) systems, computer terminals, and commercial lighting [19][20].

The standardized values used to model these logistical zones are given by [19][20]:

α n c = 0.25 to 0.50 (mixed behavior between constant power and constant current, induced by air conditioning compressors). β n c = 1.50 to 2.50 (highly inductive and voltage-dependent behavior, characteristic of fan and pump motors).

Logic Mechanism of the Load-Shedding Relay

Automatic shedding of non-critical and semi-critical loads is controlled by a binary signal S ( t ) ∈ { 0 , 1 } , as shown in Equation (13). This process uses instant on/off switching by opening motorised circuit breakers, such as the NS1600 in Figure 1(a), after time delays of 100ms or 500ms. The physical relay does not allow for gradual reduction; it only switches fully on or off.

While the circuit breaker works with simple on/off logic S(t) = 0, in reality, the loads decrease gradually due to the dynamic damping in the ZIP load model after disconnection. The nodal voltage drops steadily as line capacitances discharge and tertiary motors, like terminal HVAC units, release back-EMF. The absorbed power then declines along a curve set by the voltage exponents until the voltage at the disconnected node reaches zero.

The mathematical formulation of the airport electrical system investigated in this paper is governed by the power flow (Load Flow) equations adapted for an n-bus power network. These equations express the active power P i and reactive power Q i injected at bus i as a function of the elements of the bus admittance matrix [Y_bus], defined as follows [19][20]:

In this formulation, the bus voltages are expressed in polar coordinates by their magnitude V and phase angle θ . The phase angle difference between two buses is denoted as θ i j = θ i − θ j . The general power flow equations for each bus i ( i = 1 , 2 , ⋯ , n ) are formulated as follows [17]-[20]:

where:

V i and V j represent the Root-Mean-Square (RMS) voltage magnitudes at buses i and j, respectively. G i j and B i j are the conductance and susceptance elements of the transmission line connecting bus i and j. θ i j = δ i − δ j denotes the phase angle difference.n represents the total number of network buses.

This formulation is widely preferred for the Newton-Raphson method because it directly aligns with the operational specifications of PV buses (where the voltage magnitude V is maintained constant by the emergency backup generator’s AVR) [18][20].

To resolve the airport electrical power outage problem and guarantee both the stability and continuity of the airport’s priority loads, the system of power flow Equation (16) for P i , Q i is coupled with the previously defined voltage-dependent load models (refer to Equations (8) - (13)). For the airport load bus (denoted as bus i), the net power injections are formulated as [18][20]:

The Newton-Raphson algorithm aims to eliminate the mismatch between the net scheduled power injections and the calculated powers derived from Equation (16).

To incorporate voltage-dependent load models into the Newton-Raphson formulation, the conventional Jacobian matrix must be modified. Static loads uniquely alter the diagonal elements of the Jacobian, as the power consumed at bus i is independent of the voltage magnitude or phase angle of any other bus j[19][20].

The iterative linear system of the Newton-Raphson method is expressed as [17][20]:

where the submatrices represent the partial derivatives of the power mismatches [16][20]:

For ≠ i , the elements of the Jacobian matrix [ H N M L ] depend exclusively on the network topology, as defined by the bus admittance matrix Y_bus [18][20].

The modified diagonal elements for bus i incorporate the load parameters α , and β (for critical loads), as well as α n c and β n c (for non-critical loads), along with the load shedding control factor S(t).

A. Derivatives with Respect to Phase Angle θ i

Since the phase angle has no influence on static voltage-dependent loads, the analytical expressions remain identical to the conventional load flow formulation [19][20]:

B. Derivatives with Respect to Voltage Magnitude ( V i ) Impact of Electrical Loads

The voltage dependence of the loads adds a direct subtractive term onto the diagonal elements of the (N) and (L) submatrices. Utilizing the standardized formulation V i ∂ ∂ V i [16]-[36]:

Static-Dynamic Digital Coupling

The digital workflow combines steady-state power flow calculation (spatial load flow) and transient time-domain simulation according to the following algorithmic sequence:

1) Initialisation: The Newton-Raphson algorithm solves the nonlinear algebraic Equation (15) for t< 5.0 s under a strict convergence tolerance of e = 10⁻⁶ pu on residuals DP and DQ. The calculated voltage moduli and initial angles determine the internal state (E_g0,v_f0) of the alternating generators.

2) Handling reactive power limits: At each Newton-Raphson iteration, if the calculated reactive power of an alternator exceeds Q_gmax or Q_gmin, the corresponding PV-type node is immediately converted to a PQ node with Q_limit. The voltage is then released and recalculated as a free system variable until convergence.

3) Switching event t= 5.0 s: The failure of the main network instantaneously modifies the topology of the nodal admittance matrix [Y_bus], isolating the source node N1. The differential equations of the alternator (3) and the frequency response (25) are then activated in closed loop.

4) Time integration: The transient dynamics are solved using the modified Euler numerical method (or 4th-order Runge-Kutta) with a fixed time step Dt= 1 ms, ensuring numerical stability against the stiffness of the AVR time constants T_A = 0.05 sand the mechanicalconstants. At each time step, the binary load shedding variable S(t) is evaluated.

Network reliability is modeled through the availability probability R(t) of the individual components [37]:

where λ represents the failure rate of the specific component (transformer, transmission line, or generator). The overall system reliability performance is obtained by combining components in series or parallel configurations [37]:

Series Configuration:

Parallel Configuration (Redundant):

This analytical framework enables quantifying the direct impact of emergency backup generators and transformer redundancy on service continuity within the airport infrastructure.

The transient temporal response of the network frequency (typically following a load disturbance incident or alternator startup) mathematically corresponds to the step response of an underdamped second-order system, characterized by decaying pseudo-periodic oscillations. This linear second-order dynamic behavior is governed by the following canonical differential equation, expressed in terms of the frequency state variable f(t) [20].

Table 1 below summarises all the numerical parameters injected into the simulation code.

Table 1. Simulation parameters.

Three primary operational scenarios were simulated in this study using the Python programming language:

1) Normal steady-state operation (stable public utility grid).

2) Public utility grid outage with automatic transfer to backup emergency sources (diesel generator sets).

3) Simultaneous failure of the public grid and a critical backup generator.

Under normal steady-state conditions, the primary utility network supplies all electrical loads. The power balance demonstrates that critical loads consume a significant portion of the total energy. Table 2 and Figure 6 summarize these steady-state findings.

Figure 6. Power curves at the nodes of the Ndjili Airport electrical network.

Table 2.Power balance across different network buses under steady-state operation (Ndjili Airport).

The primary utility network outage scenario was simulated to evaluate the dynamic performance of the emergency backup systems (as illustrated in Figure 7).

Figure 7. Transient voltage evolution during a main grid outage with transition to backup generators.

This scenario, illustrated in Figure 8, represents an extreme condition where both the main power grid and a backup unit fail simultaneously.

Figure 8. Simulation of a simultaneous main grid and backup generation failure.

Frequency analysis constitutes a fundamental indicator of the dynamic stability of a power system, particularly within critical infrastructures such as airports. Figure 9 illustrates the time evolution of the system frequency following a combined failure of the main grid and a backup source.

Figure 9. System frequency analysis following a failure.

The load shedding strategy constitutes an essential mechanism to preserve power system stability in the event of a generation deficit. Figure 10 presents the power consumption evolution across different load categories (critical, semi-critical, and non-critical) during a major failure scenario.

Based on the steady-state energy balance presented in Table 1, the transient scenario illustrated in Figure 10 was normalised using an equivalent local load of 100 kW. The initial values at t = 0 s pertain to a specific section of the backed-up technical sub-distribution panel subject to a local overload. This approach allows for the evaluation of the relay’s dynamic behaviour on a consistent relative scale and does not represent the airport’s total cumulative load shown in Table 1. The curvilinear decrease in non-critical power confirms the relationship between the relay’s binary state S(t) = 0 and the voltage response calculated by the ZIP polynomial model.

Figure 10.Load shedding profile.

Figure 11. Nodal voltage analysis.

Nodal voltage analysis allows for the assessment of power supply quality at various points in the grid, identifying areas sensitive to voltage fluctuations. Figure 11 presents the voltage evolution at different strategic nodes of the airport power system.