Voltage fluctuations mainly occur in the form of sags and swells. Here, we formally link the time model of the signal to the sliding effective value and the normative classification criteria. We model the voltage of a phase using a slow envelope sinusoid:

$v\left(t\right)=\sqrt{2}{V}_{\text{nom}}\left(1+\delta \left(t\right)\right)\text{sin}\left(\omega t\right),$(1)

where $V_{\text{nom}}$ is the nominal RMS voltage (50 Hz in Congo, $\omega =2\pi f$) and $\delta \left(t\right)$ describes the relative (dimensionless) deviation of the amplitude from the nominal value. Thus, $\delta \left(t\right)=0$ corresponds to normal conditions; $\delta \left(t\right)<0$ indicates a decrease in amplitude (sag) and $\delta \left(t\right)>0$ indicates an increase (swell). The sliding effective value is defined by

$V_{\text{rms}}\left(t\right)=\sqrt{\frac{1}{T}{\displaystyle {\int }_t^{t+T}{v}^2\left(\tau \right)\text{d}\tau }},T=\frac{2\pi }{\omega }.$(2)

We introduce the quantity pu $u\left(t\right)=V_{\text{rms}}\left(t\right)/V_{\text{nom}}$. By setting

$S_1\left(t\right)=\frac{2}{T}{\displaystyle {\int }_t^{t+T}\delta \left(\tau \right){\sin }^2\left(\omega \tau \right)\text{d}\tau },S_2\left(t\right)=\frac{2}{T}{\displaystyle {\int }_t^{t+T}{\delta }^2\left(\tau \right){\sin }^2\left(\omega \tau \right)\text{d}\tau }$(3)

we have the identity

$u^2\left(t\right)=1+2S_1\left(t\right)+S_2\left(t\right).$(4)

If the envelope is quasi-stationary over a period ( $\delta \left(\tau \right)\approx \delta \left(t\right)$ for $\tau \in \left[t,t+T\right]$), then

$u\left(t\right)=\left|1+\delta \left(t\right)\right|\approx 1+\delta \left(t\right)\left(\text{pour}\left|\delta \right|\ll 1\right),$(5)

In the first order (minor deviations), we obtain the linear approximation

$u\left(t\right)\approx 1+S_1\left(t\right),$(6)

that is, $u\left(t\right)$ is governed by the weighted average of $\delta$ over the window, weighted by ${\text{sin}}^2$. The normative criteria are naturally expressed in pu via $u\left(t\right)=V_{\text{rms}}\left(t\right)/V_{\text{nom}}$:

$\text{sag}:u\left(t\right)\in \left[0.1,0.9\right]\text{over}\text{an}\text{interval}\text{Δ}t\in \left[0.5T,\text{a}\text{few}\text{seconds}\right],$(7)

$\text{swell}:u\left(t\right)\in \left[1.1,1.8\right]\text{on}\text{the}\text{same}\text{type}\text{of}\text{interval}.$(8)

Under the quasi-stationary assumption (5), these conditions can be rewritten (approximately) in terms of relative deviation:

$\text{sag}\iff \delta \left(t\right)\in \left[-0.9,-0.1\right],\text{swell}\iff \delta \left(t\right)\in \left[+0.1,+0.8\right].$(9)

In the first order (6), the decision depends on the weighted average $S_1\left(t\right)$, which justifies the use of a half-period moving RMS to track the evolution of $u\left(t\right)$. The entry/exit times $\left(t_{\text{in}},t_{\text{out}}\right)$ are defined as the first times at which $u\left(t\right)$ crosses the thresholds of (7) (8) in a sliding window; the duration is $\text{Δ}t=t_{\text{out}}-t_{\text{in}}$. For robustness in the presence of noise, hysteresis thresholds $u_{\downarrow }<0.9<u_{\uparrow }$ (resp. $1.1<u_{\downarrow }^{+}<u_{\uparrow }^{+}<1.8$) are introduced.

In order to avoid decision oscillations. Equation (4) shows that classification depends on a local average of the amplitude deviation; long, weakly redundant HT lines (Congolese case) favor sequences $\delta \left(t\right)$ that are consistent over several periods, making (5) applicable and detection reliable, provided that updates are made at half-period intervals and noise filtering is applied.

Traditional approaches are mainly based on: sliding RMS thresholds, spectral estimation (DFT/FFT) of the fundamental frequency, and time-scale analysis (wavelets). Their principles are outlined below, and explicit decision rules and their intrinsic limitations in the context of African/Congolese high-voltage networks (heterogeneous infrastructure, measurement noise, intermittent transmission) are derived from them [8] - [10] .

A signal is observed

$v\left[n\right]=V_{\text{nom}}\left(1+\delta \left[n\right]\right)\text{sin}\left({\omega }_{0}n{T}_s+\varphi \right)+\eta \left[n\right],$(10)

where $\left|\delta \left[n\right]\right|$ measures the amplitude deviation (drop/surge), $\eta \left[n\right]$ is additive noise (zero-mean, variance ${\sigma }^2$). Over a window of $N$ samples, the RMS estimator is

${\hat{V}}_{\text{rms}}\left[k\right]=\sqrt{\frac{1}{N}\underset{n=k-N+1}{\overset{k}{{\displaystyle \sum }}}{v}^2\left[n\right]}.$(11)

For a 50 Hz network, a relative threshold $\tau$ (e.g., $\tau =0.10$) and a minimum duration $\text{Δ}t$ (e.g., ≥ half-period) are set. The simple rule is

$\frac{\left|{\hat{V}}_{\text{rms}}\left[k\right]-V_{\text{nom}}\right|}{V_{\text{nom}}}>\tau \text{during}\ge \text{Δ}t.$(12)

If $\delta \left[n\right]\equiv \delta$ is constant over the window and $\eta \left[n\right]$ is negligible,

$\mathbb{E}\text{}\left[{\hat{V}}_{\text{rms}}\right]\approx V_{\text{nom}}\left|1+\delta \right|.$(13)

Thus, a decrease of 20% ( $\delta =-0,2$) shifts the estimate by approximately −20%. Noting that $\hat{P}=\frac{1}{N}{\displaystyle \sum v^2\left[n\right]}$, we have (approx.) $\text{Var}\left(\hat{P}\right)\approx \frac{1}{N}\text{Var}\left(v^2\right)$; under pure sine plus white noise, ${\hat{V}}_{\text{rms}}=\sqrt{\hat{P}}$ and, by the delta method

$\text{Var}\left({\hat{V}}_{\text{rms}}\right)\approx \frac{\text{Var}\left(\hat{P}\right)}{4P},P=V_{\text{nom}}^2+{\sigma }^2.$(14)

The absolute threshold $T=\tau V_{\text{nom}}/\sqrt{2}$ gives

$\text{PFA}\approx Q\left(\frac{T}{\sqrt{\text{Var}\left({\hat{V}}_{\text{rms}}\right)}}\right),$(15)

where $Q(\cdot )$ is the Gaussian tail function. Therefore, for a given noise level $\sigma$, longer windows (large $N$) are required to maintain the FAL (false alarm rate under noise), at the cost of a minimum detection delay $\ge \left(N-1\right)T_s$. This strategy suffers from a detection delay linked to the window, high sensitivity to impulse noise and brief dips, ambiguities with inrush/switching, and fixed thresholds that are not very robust to slow drifts in $V_{\text{nom}}$. Figure 1 shows the evolution of the moving average over a time window $T$, sampled at intervals of $T/2$.

The fundamental complex amplitude is estimated by the DFT on $N$ points:

${\hat{A}}_1=\frac{2}{N}\underset{n=0}{\overset{N-1}{{\displaystyle \sum }}}v\left[n\right]{\text{e}}^{-j2\pi n/N}.$(16)

Detecting a drop/surge is equivalent to testing.

$\left|{\hat{A}}_1\right|\lessgtr \left(1\mp \tau \right)V_{\text{nom}}.$(17)

If the actual frequency is $f_0+\text{Δ}f$ (slip), the modulus is attenuated by the spectral leakage via the Dirichlet kernel:

${\hat{A}}_1\approx A_1\frac{\text{sin}\left(\pi Nϵ\right)}{N\sin \left(\pi ϵ\right)}{\text{e}}^{j\phi },ϵ=\frac{\text{Δ}f}{f_s}.$(18)

So even without defects, a slight $\text{Δ}f$ (common in Central Africa, where speed regulation can fluctuate) biases $\left|{\hat{A}}_1\right|$ and causes false detections. Furthermore, an amplitude discontinuity in the window amounts to convolving the spectrum by the TF of the window (side lobes) $\Rightarrow$ overestimations or underestimations during transients. This approach requires an integer number of periods or apodized windows (leakage/bias compromise), remains sensitive to frequency fluctuations and harmonics, and induces a delay at least equal to the window.

The CWT of $v\left(t\right)$ by a mother wavelet $\psi$ is

$W_v\left(a,b\right)=\frac{1}{\sqrt{\left|a\right|}}{\displaystyle {\int }_{ℝ}v}\left(t\right){\psi }^{\text{*}}\left(\frac{t-b}{a}\right)\text{d}t.$(19)

A voltage dip of size $S$ at time $t_0$ (Heaviside step model) generates maximized details $\left|W_v\left(a,b\right)\right|$ for $b\simeq t_0$ and scales $a$ comparable to the duration of the front. With a Gaussian-derived wavelet, we obtain analytically

$W_v\left(a,b\right)\propto S\psi \left(\frac{t_0-b}{a}\right),$(20)

which forms the basis for a robust decision rule in the face of rapid changes. Performance depends heavily on the choice of $\psi$ and the scales $\left\{a\right\}$ (sensitivity to high-frequency noise), suffers from side effects that can lead to false alarms near windows, and struggles to classify simultaneous faults (e.g., dip + harmonic) without a learning layer.

AI methods for energy quality are typically divided between artificial neural networks (ANN), which are powerful but difficult to interpret, and fuzzy systems, which can be explained by linguistic rules but require expert engineering of sets. ANFIS (Adaptive Neuro-Fuzzy Inference System) combines these two paradigms: machine learning and explainable rule structures, which is particularly relevant in a heterogeneous Congolese context (noise, losses, variable latency). Let $x\in {ℝ}^d$ be the characteristic vector (e.g., $x=\left[u,\stackrel{\dot{}}{u},\text{duration},{ℰ}_{\text{wave}},\cdots \right]$). Consider $M$ Takagi-Sugeno fuzzy rules:

${ℛ}_k:\text{Si}{x}_1\text{est}{A}_{k,1}\wedge \cdots \wedge x_d\text{est}{A}_{k,d}\text{alors}{f}_k\left(x\right)=a_k^{\top }x+b_k$(21)

with Gaussian membership functions ${\mu }_{k,m}\left(x_m\right)=\text{exp}\left(-\frac{{\left(x_m-c_{k,m}\right)}^2}{2{\sigma }_{k,m}^2}\right)$. The normalized strength of rule $k$ is

${\bar{w}}_k\left(x\right)=\frac{{\displaystyle {\prod }_{m=1}^d{\mu }_{k,m}}\left(x_m\right)}{{\displaystyle {\sum }_{\ell =1}^M{\displaystyle {\prod }_{m=1}^d{\mu }_{\ell ,m}}}\left(x_m\right)}$(22)

and the ANFIS output is

$y\left(x\right)=\underset{k=1}{\overset{M}{{\displaystyle \sum }}}{\bar{w}}_k\left(x\right)\left(a_k^{\top }x+b_k\right).$(23)

Hybrid learning:

(i) conditional least squares for $\left\{a_k,b_k\right\}$, (ii) gradient descent for premises $\left\{c_{k,m},{\sigma }_{k,m}\right\}$.

Sensibilisation la QoS de transmission:

In the presence of fiber/4G/VSAT channels, we aggregate QoS $q\in \left[0,1\right]$ (normalized latency, availability, losses) and modulate the bandwidths:

${\sigma }_{k,m}\left(q\right)={\sigma }_{k,m}^{\left(0\right)}\left(1+{\beta }_{k,m}\left(1-q\right)\right),{\beta }_{k,m}\ge 0$(24)

When QoS deteriorates ( $q\downarrow$), the sets expand (increased uncertainty), and the decision becomes more cautious; conversely, $q\uparrow$ narrows the sets and reinforces the dominant rule. This adaptation reduces false alarms related to degraded transmissions, while maintaining interpretable rules.

The Congolese grid is based on 220 kV and 110 kV lines connecting hydroelectric power plants (Imboulou, Moukoukoulou, Liouesso) to consumption centers in Brazzaville and Pointe-Noire. Consisting mainly of overhead lines crossing rural areas, it remains vulnerable to weather events and voltage disturbances. The low structural redundancy increases the risk of load shedding in the event of a fault, hence the need for an intelligent monitoring system to quickly detect sags and swells. Figure S1 , in the appendix, illustrates this network and its strategic nodes.

The Congolese network faces challenges in terms of communication and power quality. Connectivity varies greatly: fiber optics in urban areas, intermittent 3G/4G in suburban areas, and VSAT in remote areas, requiring a hybrid architecture. In terms of electricity, the most frequent disturbances are voltage sags and swells, which are transient but critical phenomena for sensitive equipment, justifying the implementation of rapid and reliable detection.

Congo-Brazzaville is a representative case study due to its expanding high-voltage network, which is subject to recurring disruptions, its heterogeneous communications infrastructure (fiber optics, 4G/LTE, and VSAT), and its critical need for service continuity in the industrial and hospital sectors. Figure S1 shows the complete diagram of the Congolese electricity network, while Figure 2 illustrates a portion extracted and used for simulations. The assumption is that the results obtained on this subnetwork can be generalized to the entire system, thus providing a robust basis for evaluating the ANFIS diagnosis coupled with hybrid transmission.

The numerical data used for modeling comes from the Congolese 220 kV network and is compiled in the appendix (16). They include transformer characteristics ( Table B1 ), the distribution of active and reactive power per node, expressed in p.u. with a loss margin of 2 % ( Table B2 ) and line parameters and their transmission capacities ( Table B3 ). Regarding Table B1 , the values reported include an estimated loss margin of 2 % on the active and reactive power generated, in order to reflect realistic conditions in a high-voltage network in operation. This assumption provides a more representative distribution of loads between nodes, particularly for Djiri, where the remaining power is concentrated. However, it should be noted that the loss-free balance has also been verified, confirming the consistency of the model since the sum of the loads expressed in p.u. corresponds exactly to the injected generation.

The dataset used to train and evaluate the ANFIS model consists of 1,200 labeled sequences, including 400 normal events, 400 sags, and 400 swells, generated from the Congolese 220 kV grid model with injected disturbances. To ensure robust evaluation, the data were randomly split into 70% for training, 15% for validation, and 15% for testing. This separation prevents data leakage and allows a statistically meaningful assessment of the model’s generalization capacity.

Three-phase voltages and currents are measured via CT/VT compliant with IEC 60044-1 and acquired by an RTU at $f_s\in \left[10,20\right]$ kHz. Local preprocessing calculates the following quantities over windows of one period with 50 % overlap:

$u=\frac{V_{\text{rms}}}{V_{\text{nom}}},\left(I_0,I_1,I_2\right)\left(\text{composantes symtriques}\right),{\text{THD}}_I,\stackrel{\dot{}}{u}\triangleq \frac{\text{d}u}{\text{d}t}$(25)

A correction for bias/variance due to noise is applied, and windows are validated if $N^{\prime }/N\ge {\eta }_{\text{min}}$.

Intelligent supervision requires reliable and predictable data transport between the field and the control center. We consider three complementary technologies: fiber optics (minimal latency, limited coverage), 4G/LTE (flexible, variable latency/jitter), and VSAT (near-universal coverage, high latency). We first formalize the quality of service (QoS), then derive a hybrid routing scheme consistent with the criticality of ANFIS flows. To ensure optimal routing, each ANFIS flow $P_i$ is assigned to a communication link $R_j$ according to its quality of service (QoS) profile and criticality, as shown in Figure 3 .

For each link $j\in \left\{\text{fib},\text{lte},\text{sat}\right\}$, the random latency $T_j$, the unavailability (complement of availability) $1-D_j$, the loss rate $p_j$, and the jitter ${\sigma }_{T,j}=\sqrt{\text{Var}\left(T_j\right)}$ are observed. The latency is normalized by

${\tilde{t}}_j=\text{clip}\left(\frac{\mathbb{E}\left[T_j\right]-t_{\text{min}}}{t_{\text{max}}-t_{\text{min}}},0,1\right),$(26)

where $t_{\text{min}}$ and $t_{\text{max}}$ are engineering bounds (e.g., $t_{\text{min}}=1$ ms, $t_{\text{max}}=1$ s). The metrics are then aggregated into a scalar QoS index

$R_j=\alpha \left(1-{\tilde{t}}_j\right)+\beta D_j+\gamma \left(1-p_j\right),\alpha +\beta +\gamma =1,\alpha ,\beta ,\gamma \ge 0,$(27)

optionally complemented by a capacity safeguard $c_j$ via $R_j\leftarrow R_j\cdot 1\left\{c_j\ge d_i\right\}$ for a flow with required throughput $d_i$.

Eacss flow $f_i$ (critical, semi-critical, non-critical) is assigned a priority $P_i\in \left\{3,2,1\right\}$ and the utility is defined as

$U_{i,j}=R_j-\lambda 1\left\{c_j<d_i\right\}$(28)

where $\lambda$ is a strong penalty if the capacity is insufficient. The single-path assignment then follows

$C_i^{\star }=\text{arg}\underset{j}{\text{max}}{U}_{i,j},$(29)

with a service threshold constraint $R_{C_i^{\star }}\ge r_{\text{min}}\left(P_i\right)$ (the higher the priority, the higher the required $r_{\text{min}}$). This rule is equivalent to an admissibility filter ( $R_j\ge r_{\text{min}}$) followed by a utility-based selection. See Figure 4 (QoS/priority routing).

For critical flows, controlled duplication is allowed over a subset $\mathcal{J}\subseteq \left\{\text{fib},\text{lte},\text{sat}\right\}$ in order to reduce delay and increase the probability of delivery before a deadline $T_{\text{D}}$. If the delays $T_j$ are independent, the first-arrival law yields

$\mathbb{P}\left\{\underset{j\in \mathcal{J}}{\text{min}}{T}_j\le t\right\}=1-\underset{j\in \mathcal{J}}{{\displaystyle \prod }}\left(1-F_{T_j}\left(t\right)\right),$(30)

where $F_{T_j}$ denotes the CDF of $T_j$. In particular, for a deadline $T_{\text{D}}$,

$\underset{\text{temporal reliability}}{\underset{︸}{\mathbb{P}\left\{\text{delivered before}{T}_{\text{D}}\right\}}}=1-\underset{j\in \mathcal{J}}{{\displaystyle \prod }}\left(1-F_{T_j}\left(T_{\text{D}}\right)\right)$(31)

and the expectation satisfies $\mathbb{E}\left[{\min }_{j\in \mathcal{J}}{T}_j\right]\le {\min }_j\mathbb{E}\left[T_j\right]$. The redundant allocation is determined by

$\underset{\mathcal{J}}{\text{max}}\mathbb{P}\left\{\underset{j\in \mathcal{J}}{\text{min}}{T}_j\le T_{\text{D}}\right\}\text{s}.\text{c}.\underset{j\in \mathcal{J}}{{\displaystyle \sum }}{\text{cost}}_j\le \text{budget},\left|\mathcal{J}\right|\le J_{\text{max}}.$(32)

In practice, the $K$ best links in terms of $R_j$ are selected (fiber first, then 4G if available), while VSAT is reserved for cases of coverage failure, given its high latency.

Due to controlled redundancy (flows with $P=3$), this motivates duplication over fiber and 4G, with VSAT used only as a last resort (degraded mode). See Figure 5 .

Let $j_t$ denote the active link at time $t$. Two thresholds are defined, $r_{\downarrow }<r_{\uparrow }$. The policy is

$\text{if}{R}_{j_t}\left(t\right)<r_{\downarrow }\Rightarrow j_{t+}\in \text{arg}\underset{j}{\text{max}}{R}_j\left(t\right)\text{such}\text{that}{R}_j\left(t\right)\ge r_{\uparrow }$(33)

and $j_t$ remains unchanged as long as $R_{j_t}\left(t\right)\ge r_{\downarrow }$. The hysteresis $r_{\uparrow }-r_{\downarrow }>0$ prevents rapid oscillations under jitter. It ensures jitter absorption and alignment of measurement windows (half-cycle RMS). In other words, switching occurs when the active link falls below $r_{\downarrow }$ and recovery is triggered only when it rises above $r_{\uparrow }$. See Figure 6 .

Received packets are buffered in FIFO and reordered using timestamps (PTP over fiber, NTP over 4G/VSAT). The re-time process, based on a Kalman filter, compensates the estimated offsets $\text{Δ}t$, ensuring temporal consistency of the half-cycle RMS windows used by ANFIS.

In urban areas, fiber maximizes $R_j$ (latency $\downarrow$, losses $\downarrow$, $D\uparrow$) and becomes the carrier for critical flows. In peri-urban zones, 4G serves either as the primary link or as redundancy depending on radio load (jitter ${\sigma }_{T,\text{lte}}$), while in isolated areas VSAT ensures minimal connectivity. The optimization (32) then selects either $\left(\text{lte},\text{sat}\right)$ or $\left(\text{fib},\text{sat}\right)$ depending on availability, while maintaining the probability of delivery before $T_{\text{D}}$ via (30). This coupling of ANFIS + hybrid routing provides a stable diagnostic despite heterogeneous and intermittent channels.

The input vector $\varphi$ concatenates both electrical and network indices:

$\varphi =\left[u,\stackrel{\dot{}}{u},\text{duration},{\text{THD}}_I,\frac{\left|I_0\right|}{\left|I_1\right|+\epsilon },\frac{\left|I_2\right|}{\left|I_1\right|+\epsilon },R_{C^{\star }},{\sigma }_{T,C^{\star }},p_{C^{\star }}\right]$(34)

The target outputs are $\left\{y_{\text{det}},y_{\text{cls}},y_{\text{sev}}\right\}$. The ANFIS (Takagi Sugeno of order 1) employs with hybrid learning (LSE + gradient). The widths of the fuzzy sets are modulated by the QoS (cf. ${\sigma }_{k,m}\left(q\right)$ defined above), which makes the decision robust to link degradations.

The sizing of data flows required to feed the ANFIS model, as well as the compatibility of transmission technologies (Fiber, 4G, VSAT), are detailed in Annex 7 in Table A1 and Table A2 . This analysis includes the computation of useful throughput after local preprocessing, QoS constraints, and usage recommendations according to diagnostic scenarios.

Figure 7 presents the technical architecture of the PQ detection system as

implemented in the Simulink environment. It details the functional blocks used to simulate the behavior of the power network, encode the signals, inject faults, and transmit the data through a hybrid channel (Fiber, 4G, VSAT). QoS parameters (latency, jitter, loss) are integrated into the processing flow, influencing the quality of the aligned windows before their injection into the ANFIS model. The latter applies hybrid learning (LSE + gradient) to produce decision outputs, which are then exploited by the alarm module. This figure allows one to visualize the complete processing chain, from signal generation to event detection, while accounting for transmission and synchronization constraints. Figure S2 , placed in the appendix, illustrates the functional logic of the PQ detection system based on ANFIS. It highlights the chronological sequence of processing steps: electrical measurements, feature extraction, hybrid transmission with QoS simulation, temporal alignment, and finally neuro-fuzzy classification. The output of the model is structured into two branches: binary detection (normal/fault) and fault-type classification (SAG or SWELL), with binary state encoding (000, 111, 110). This conceptual representation complements Figure 7 by showing how the extracted features are exploited to produce a robust decision, even in the presence of network disturbances.

The input vector is defined as $\varphi \in {ℝ}^M$:

$\varphi ={\left[u,\stackrel{\dot{}}{u},\text{duration},{\text{THD}}_I,\frac{\left|I_0\right|}{\left|I_1\right|+\epsilon },\frac{\left|I_2\right|}{\left|I_1\right|+\epsilon },R_{C^{\star }},{\sigma }_{T,C^{\star }},p_{C^{\star }}\right]}^{\top }$(35)

where $u=V_{\text{rms}}/V_{\text{nom}}$, $\stackrel{\dot{}}{u}=\text{d}u/\text{d}t$, $R_{C^{\star }}\in \left[0,1\right]$ denotes the QoS of the active link, and $\left({\sigma }_T,p\right)$ represent its jitter and packet loss rate.

Robust normalization (noise/missing-packet tolerance).

For each valid window ( $N^{\prime }/N\ge {\eta }_{\text{min}}$), a robust local normalization is applied:

${\tilde{\varphi }}_m=\frac{{\varphi }_m-{\hat{\mu }}_m}{{\hat{\sigma }}_{{\varphi }_m}+ϵ},{\hat{\mu }}_m=\text{median}\left({\varphi }_m\right),{\hat{\sigma }}_{{\varphi }_m}=1.4826\text{MAD}\left({\varphi }_m\right),$(36)

(MAD: median absolute deviation). Here, ${\hat{\sigma }}_{{\varphi }_m}$ serves as an uncertainty measure to weight the learning process.

Multi-task outputs.

Three ANFIS heads are used (layers 13 shared, consequents task-specific):

${\hat{y}}_{\text{det}}\in \left[0,1\right],{\hat{y}}_{\text{cls}}\in {ℝ}^3,{\hat{y}}_{\text{sev}}\in \left[0,1\right]$(37)

with class probabilities obtained via $\text{softmax}\left({\hat{y}}_{\text{cls}}\right)$.

Let $M$ denote the number of normalized inputs ${\tilde{\varphi }}_m$ and $R$ the number of rules. The ANFIS follows the standard five-layer architecture:

${\mu }_{r,m}\left({\tilde{\varphi }}_m\right)=\text{exp}\text{}\left(-\frac{{\left({\tilde{\varphi }}_m-c_{r,m}\right)}^2}{2{\sigma }_{r,m}^2}\right),{\sigma }_{r,m}={\sigma }_{r,m}^{\left(0\right)}\left(1+{\beta }_{r,m}\left(1-q\right)\right),$(38)

where $q\in \left[0,1\right]$ is a global QoS index (higher QoS $\Rightarrow$ larger $q$).

$w_r=\underset{m=1}{\overset{M}{{\displaystyle \prod }}}{\mu }_{r,m}\left({\tilde{\varphi }}_m\right).$(39)

${\bar{w}}_r=\frac{w_r}{{\displaystyle {\sum }_{\ell =1}^R{w}_{\ell }}}.$(40)

$f_r^{\left(q\right)}\left(\tilde{\varphi }\right)=a_{0,r}^{\left(q\right)}+\underset{m=1}{\overset{M}{{\displaystyle \sum }}}{a}_{m,r}^{\left(q\right)}{\tilde{\varphi }}_m.$(41)

${\hat{y}}^{\left(q\right)}\left(\tilde{\varphi }\right)=\underset{r=1}{\overset{R}{{\displaystyle \sum }}}{\bar{w}}_r{f}_r^{\left(q\right)}\left(\tilde{\varphi }\right).$(42)

Substituting (41), we obtain:

${\hat{y}}^{\left(q\right)}\left(\tilde{\varphi }\right)=\underset{r=1}{\overset{R}{{\displaystyle \sum }}}{\bar{w}}_r\left(a_{0,r}^{\left(q\right)}+\underset{m=1}{\overset{M}{{\displaystyle \sum }}}{a}_{m,r}^{\left(q\right)}{\tilde{\varphi }}_m\right).$(43)

The proposed ANFIS employs 27 fuzzy rules derived from three membership functions per input dimension. Gaussian membership functions were initialized with centers equally spaced in the normalized input range and widths proportional to the input variance. During training, parameters were refined using the hybrid algorithm. For illustration, one interpretable rule is: IF $u$ is Small AND Duration is Long AND $\left|I_0\right|/\left|I_1\right|$ is Large THEN Fault = SAG. This template confirms that the model preserves linguistic interpretability while adapting parameters automatically. The centers $c_{r,m}$ and widths ${\sigma }_{r,m}$ are learned, but for interpretability the rules are grouped into families:

Sag: $\left(u\text{Small}\right)\wedge \left(\text{duration Medium}/\text{Long}\right)\wedge \left(\frac{\left|I_0\right|}{\left|I_1\right|}\text{Large}\text{or}\frac{\left|I_2\right|}{\left|I_1\right|}\text{Large}\right)$.

Swell: $\left(u\text{Large}\right)\wedge \left(\text{duration}\text{Medium}/\text{Long}\right)\wedge \left({\text{THD}}_I\text{Moderate}\right)$.

Normal: $\left(u\approx 1\right)\wedge \left(\left|\stackrel{\dot{}}{u}\right|\text{Small}\right)\wedge \left(\frac{\left|I_0\right|}{\left|I_1\right|},\frac{\left|I_2\right|}{\left|I_1\right|}\text{Small}\right)$.

Multi-task Loss Weighted by QoS.

Over a batch $ℬ$, the objective minimized is

$ℒ=\underset{n\in ℬ}{{\displaystyle \sum }}w\left(q^{\left(n\right)}\right)g^{\left(n\right)}({\lambda }_{\text{det}}\text{BCE}\left({\hat{y}}_{\text{det}}^{\left(n\right)},y_{\text{det}}^{\left(n\right)}\right)+{\lambda }_{\text{cls}}\text{CE}\left(\text{softmax}\left({\hat{y}}_{\text{cls}}^{\left(n\right)}\right),y_{\text{cls}}^{\left(n\right)}\right)+{\lambda }_{\text{sev}}{\Vert {\hat{y}}_{\text{sev}}^{\left(n\right)}-y_{\text{sev}}^{\left(n\right)}\Vert }_2^2),$ (44)

where $w\left(q\right)=q^{\beta }$ ( $\beta \in \left[0,2\right]$) down-weights windows with poor QoS, and $g^{\left(n\right)}\in \left\{0,1\right\}$ invalidates windows with $N^{\prime }/N<{\eta }_{\text{min}}$.

Offline (Pre-training).

Classical ANFIS alternation:

1. Consequents $\left\{a_{m,r}^{\left(q\right)}\right\}$ by weighted least squares (LSE), conditional on the premises (Layers 1-3 frozen).

2. Premises $\left\{c_{r,m},{\sigma }_{r,m}\right\}$ by gradient descent on $ℒ$ (Eq. (44)), with regularization.

Online (Lightweight Adaptation).

On real streams, the consequents are slightly adjusted by RLS with forgetting factor ${\lambda }_{\text{RLS}}\in \left[0.98,0.995\right]$:

$K_t=\frac{P_{t-1}{\psi }_t}{{\lambda }_{\text{RLS}}+{\psi }_t^{\top }{P}_{t-1}{\psi }_t},{\psi }_t={\left[\begin{array}{cccc}{\bar{w}}_1{\tilde{\varphi }}^{\top }& \cdots & {\bar{w}}_R{\tilde{\varphi }}^{\top }& 1\end{array}\right]}^{\top },$(45)

${\theta }_t={\theta }_{t-1}+K_t\left(y_t-{\psi }_t^{\top }{\theta }_{t-1}\right),P_t={\lambda }_{\text{RLS}}^{-1}\left(P_{t-1}-K_t{\psi }_t^{\top }{P}_{t-1}\right).$(46)

The premises are either frozen or updated with a step size $\eta$ proportional to $q$ in order to avoid learning under degraded QoS. The input/output functional scheme is given in Figure 9 .

The diagnostic system is based on a two-level architecture implemented in Simulink 2023. The first level performs binary fault detection (output 1 in case of anomaly, 0 under normal conditions), while the second focuses on identification and localization, discriminating between sags, swells, and interruptions. The fuzzy logic approach uses a manually designed .fis file whereas the neuro-fuzzy (ANFIS) model is obtained through supervised learning, thereby enhancing both the accuracy and the autonomy of the diagnosis. Figure 10 shows the diagnostic system architecture, including three scenarios: a healthy electricity network ( Figure 10(a) ), a power grid with voltage drop ( Figure 10(b) ), and a power grid with surge ( Figure 10(c) ).

This modular and interpretable architecture provides a solid foundation for the deployment of intelligent monitoring systems in African power networks, where reliability and ease of implementation are essential criteria. It enables rapid adaptation to local conditions while maintaining algorithmic rigor consistent with international standards.

Figure 11 illustrates the detector response during a sequence comprising a voltage sag (SAG) followed by a voltage swell (SWELL). The sliding RMS, normalized to $V_{\text{nom}}$ and computed over a one-cycle window (20 ms), is compared to the EN 50160 reference thresholds (0.9/1.1 p.u.). This configuration enables rapid and standards-compliant identification of voltage disturbances. The first two panels show nearly symmetrical three-phase voltages and currents during

most of the record, corresponding to nominal operation. Deviations appear around $t\approx 0.10\text{s}$ and $t\approx 0.20\text{s}$, where the regime briefly departs from steady state.

The third panel shows the per-phase normalized sliding RMS relative to the 0.9 and 1.1 p.u. thresholds. A drop below 0.9 p.u. indicates a SAG, while an excursion above 1.1 p.u. indicates a SWELL. In this sequence, a SAG is observed between 0.10 and 0.16 s (minimum ≈0.85 p.u.), followed by a SWELL between 0.20 and 0.24 s (maximum ≈1.20 p.u.). In both cases, the detection latency is less than one cycle (20 ms), and the chronology shows no chatter near the thresholds, which confirms a robust tuning of the window and hysteresis.

The bottom panel provides the coded chronology of states: 000 (NORMAL) predominates, interrupted by 111 (SAG) over 0.10 - 0.16 s, and by 110 (SWELL) over 0.20 - 0.24 s. The three phases cross the thresholds synchronously, corroborating a symmetrical event and a consistent per-phase classification.

Overall, these results validate the systems ability to segment, timestamp, and encode disturbances (SAG/SWELL) with field-compatible responsiveness. The joint visualization of electrical quantities and coded states provides an immediately interpretable diagnosis and supports targeted intervention, generalizable to industrial or urban contexts where service continuity and supply security are critical. Thus, the accuracy of SAG/SWELL classification by ANFIS is directly conditioned by the latency and reliability of the studied channels.

Figure 12 shows the evolution of average latency as a function of channel occupancy for the three communication technologies used in the proposed system: fiber optic, 4G/LTE, and VSAT. Fiber optic stands out with an almost constant latency, below 5 ms, independent of channel load. This ensures transmission within real-time diagnostic constraints, i.e., below one fundamental network period (20 ms at 50 Hz). 4G/LTE exhibits an initial delay of approximately 30 ms, which increases up to 60 ms under heavy load. This delay remains compatible with online diagnostics but may induce a shift of two to three cycles, thus requiring QoS management for critical applications. VSAT, by contrast, presents a very high latency (500 to 700 ms), mainly due to orbital propagation time, making it unsuitable for instantaneous detection but relevant as a backup solution for remote sites.

These results highlight that the responsiveness of the neuro-fuzzy detector (ANFIS), illustrated in Figure 11 (SAG/SWELL events), strongly depends on the transmission infrastructure. With fiber, detection is achieved in near real-time, as latency remains below the critical threshold. With 4G/LTE, detection remains reliable but may be slightly delayed, which impacts the speed of corrective actions. With VSAT, the delay largely exceeds the dynamics of voltage disturbances, restricting its use to deferred supervision or communication redundancy. Therefore, the hierarchical integration of channels fiber as the primary medium, 4G/LTE as continuity, and VSAT as redundancy emerges as a key condition for ensuring both the relevance and timeliness of ANFIS-based decisions.

The three tables separate the performances by link (Fiber, 4G, VSAT). This facilitates readability and highlights the increasing gap between methods as QoS degrades. ANFIS consistently shows an advantage in terms of F1 score (sag/swell) and detection latency. Table 1 reports the performances obtained on a fiber link, used as a quasi real-time reference.

As shown in Table 1 , ANFIS reduces the delay to 12 ms while improving the F1 score compared to baseline approaches. Table 2 illustrates the performance under 4G transmission, where latency and jitter are significant.

The results in Table 2 show that ANFIS maintains an F1 score above 0.92 despite radio variability. Table 3 presents the results under VSAT transmission, the most unfavorable case in terms of latency.

As shown in Table 3 , even with a delay of 650 ms, ANFIS maintains robust detection ( $F_1\approx 0.90$). Tables 1-3 demonstrate that the ANFIS gain becoAmes more pronounced as QoS degrades (4G, VSAT), both in terms of $F_1$ and detection delay $\text{Δ}{t}_{\text{det}}$.

Separating the ablation by link ( Table 4 and Table 5 ) isolates the effect of QoS on the decision. This directly illustrates the benefit of modulating the membership functions under radio/satellite conditions.

These two tables summarize the gap between ANFIS and the baselines. Table 6 reports the overall accuracy of the three methods on the test set. As shown, ANFIS remains superior to the baselines by 46 accuracy points.

Table 7 summarizes the median detection delays by technology. According to this table, ANFIS reduces latency by 10 to 30 ms compared to RMS and ANN, since lower values are better.