The power system with SVC is modelled by a set of differential-algebraic equations (DAE) [23] [24] :

$\stackrel{\dot{}}{x}=f\left(x,y,u,p\right),$ (1)

$0=g\left(x,y,u,p\right),$ (2)

where: The system is described by four sets of variables: the dynamic states $x\in {ℝ}^n$ (machines, AVR, filters, SVC controller), the algebraic variables $y\in {ℝ}^m$ (bus voltages, angles, currents, powers), the control inputs $u\in {ℝ}^r$ (regulator setpoints, voltage references), and finally the parameters $p$ (impedances, time constants, limits).

An equilibrium point $\left(x_0,y_0,u_0\right)$ satisfies $f\left(x_0,y_0,u_0,p\right)=0$ and $g\left(x_0,y_0,u_0,p\right)=0$. Defining deviations $\text{Δ}x=x-x_0$, $\text{Δ}y=y-y_0$, $\text{Δ}u=u-u_0$ and linearizing (1)-(2) to first order, we obtain:

$\text{Δ}\stackrel{\dot{}}{x}=f_x\text{Δ}x+f_y\text{Δ}y+f_u\text{Δ}u,$ (3)

$0=g_x\text{Δ}x+g_y\text{Δ}y+g_u\text{Δ}u,$ (4)

with $f_x={\frac{\partial f}{\partial x}|}_0$, $g_y={\frac{\partial g}{\partial y}|}_0$, etc. Under the standard assumption of non-singularity of $g_y$, (4) gives:

$\text{Δ}y=-g_y^{-1}{g}_x\text{Δ}x-g_y^{-1}{g}_u\text{Δ}u.$ (5)

Injecting (5) into (3), we obtain a reduced state model:

$\text{Δ}\stackrel{\dot{}}{x}=A\text{Δ}x+B\text{Δ}u,\text{with}\boxed{A=f_x-f_y{g}_y^{-1}{g}_x,B=f_u-f_y{g}_y^{-1}{g}_u.}$ (6)

The output of interest can be expressed in linearized form as $\Delta z=C\Delta x+D\Delta u$ if necessary. The SVC is inserted into the state model in two equivalent forms:

$Q=-V^2{B}_{\text{eq}},g\left(x,y,u,p\right)=0$ (7)

with $B_{\text{eq}}$ defined as a control variable.

In both cases, the controller (PI or fuzzy) acts on the global dynamics through the linearized structure of the system:

$\stackrel{\dot{}}{x}=Ax+Bu,A\mapsto A+f_y{g}_y^{-1}{g}_x,B\mapsto B+f_y{g}_y^{-1}{g}_u$ (8)

which makes explicit the network-controller coupling.

The stability of the linearized system is governed by the spectrum of matrix $A$. The condition for asymptotic stability is written as:

$\Re \left\{{\lambda }_i\right\}<0,\forall i$ (9)

with indicators such as the spectral abscissa $\alpha \left(A\right)={\max }_i\Re \left\{{\lambda }_i\right\}$ (measure of the return speed), and the modal damping ratio, for a complex pair $\lambda =\sigma \pm j\omega$:

$\zeta =-\frac{\sigma }{\sqrt{{\sigma }^2+{\omega }^2}},\zeta \ge {\zeta }_{\text{min}}\left(\text{typically}10\%~15\text{\%}\right)$ (10)

The evaluation of a regulator (PI or fuzzy) also relies on: (i) participation factors (dominant states), (ii) controllability/observability of critical modes, (iii) modal sensitivity to key parameters ( $\text{d}\lambda /\text{d}k$).

The practical procedure includes: the computation of the equilibrium point by load-flow, the linearization and reduction of the DAE equations, the spectral analysis of $A$, then the iterative tuning of the regulator until the desired damping margins are satisfied. This linearized framework thus provides a quantitative and reproducible metric to compare SVC control strategies on the dynamic voltage stability.

Consider a differential line element of length $\text{d}x$, characterized by the distributed parameters per unit length: resistance $R$ [Ω/m], inductance $L$ [H/m], conductance $G$ [S/m], and capacitance $C$ [F/m]. In sinusoidal steady state (phasor representation at angular frequency $\omega$), the application of Kirchhoffs laws gives:

$V\left(x\right)-V\left(x+\text{d}x\right)=\left(R+\text{j}\omega L\right)I\left(x\right)\text{d}x,$ (11)

$I\left(x+\text{d}x\right)-I\left(x\right)=\left(G+\text{j}\omega C\right)V\left(x\right)\text{d}x.$ (12)

Dividing by $\text{d}x$ and letting $\text{d}x\to 0$, we obtain the telegraphers equations:

$\frac{\partial V}{\partial x}\left(x\right)=-\left(R+\text{j}\omega L\right)I\left(x\right),$ (13)

$\frac{\partial I}{\partial x}\left(x\right)=-\left(G+\text{j}\omega C\right)V\left(x\right).$ (14)

Differentiating (13) with respect to $x$ and substituting (14), we obtain the wave equations:

$\frac{{\partial }^{2}V}{\partial x^2}\left(x\right)={\gamma }^{2}V\left(x\right),\frac{{\partial }^{2}I}{\partial x^2}\left(x\right)={\gamma }^{2}I\left(x\right),$ (15)

where

$\begin{array}{l}\gamma =\sqrt{\left(R+\text{j}\omega L\right)\left(G+\text{j}\omega C\right)}\left(\text{propagation constant}\right),\\ Z_c=\sqrt{\frac{R+\text{j}\omega L}{G+\text{j}\omega C}}\left(\text{characteristic impedance}\right).\end{array}$ (16)

The general solutions of (15) are exponential waves. Setting $x=0$ at the receiving end and $x=l$ at the sending end:

$V\left(x\right)=V^{+}{\text{e}}^{-\gamma x}+V^{-}{\text{e}}^{+\gamma x},$ (17)

$I\left(x\right)=\frac{1}{Z_c}{V}^{+}{\text{e}}^{-\gamma x}-\frac{1}{Z_c}{V}^{-}{\text{e}}^{+\gamma x}.$ (18)

By applying the boundary conditions $\left(V_r,I_r\right)$ at the receiving end and $\left(V_s,I_s\right)$ at the sending end, one obtains the transmission relation:

$\begin{array}{l}\left[\begin{array}{l}{V}_s\hfill \\ I_s\hfill \end{array}\right]=\underset{\equiv \left[\begin{array}{ll}A\hfill & B\hfill \\ C\hfill & D\hfill \end{array}\right]}{\underset{︸}{\left[\begin{array}{ll}\cosh \left(\gamma l\right)\hfill & Z_c\sinh \left(\gamma l\right)\hfill \\ \frac{1}{Z_c}\sinh \left(\gamma l\right)\hfill & \cosh \left(\gamma l\right)\hfill \end{array}\right]}}\left[\begin{array}{l}{V}_r\hfill \\ I_r\hfill \end{array}\right],\\ A=D=\cosh \left(\gamma l\right),B=Z_c\sinh \left(\gamma l\right),C=\frac{1}{Z_c}\sinh \left(\gamma l\right).\end{array}$ (19)

Useful limiting cases.

We seek $S_R=P_R+\text{j}{Q}_R=V_r{I}_r^{\text{*}}$. From (19),

$I_r=\frac{V_s-A{V}_r}{B}\Rightarrow S_R=V_r{I}_r^{\text{*}}=V_r\frac{V_s^{\text{*}}-A^{\text{*}}{V}_r^{\text{*}}}{B^{\text{*}}}=\frac{V_r{V}_s^{\text{*}}}{B^{\text{*}}}-\frac{A^{\text{*}}}{B^{\text{*}}}{\left|V_r\right|}^2.$ (20)

Thus,

$P_R=\Re \left\{\frac{V_r{V}_s^{*}}{B^{*}}-\frac{A^{*}}{B^{*}}{\left|V_r\right|}^2\right\},Q_R=\Im \left\{\frac{V_r{V}_s^{*}}{B^{*}}-\frac{A^{*}}{B^{*}}{\left|V_r\right|}^2\right\}.$ (21)

These expressions simplify easily in the particular cases above (e.g. $R=G=0$).

The voltage stability of a transmission system can be studied using the two-bus Thvenin equivalent model, where a source $V_{\text{th}}\angle 0$ is connected to a load bus $V\angle \delta$ through the impedance $Z_{\text{th}}=R_{\text{th}}+j{X}_{\text{th}}$. The absorbed power is $S=P+jQ$. In the classical case of HT/THT lines where $R_{\text{th}}\ll X_{\text{th}}$, we approximate $Z_{\text{th}}\simeq jX$. The active and reactive powers at the load bus are then expressed as [24] [25] :

$\boxed{P=\frac{V{V}_{\text{th}}}{X}\sin \delta ,Q=\frac{V}{X}\left(V_{\text{th}}\cos \delta -V\right).}$ (22)

By eliminating $\delta$, we obtain the Q-V characteristic:

$\boxed{Q\left(V;P\right)=\frac{1}{X}\left(V_{\text{th}}\sqrt{V^2-k^2}-V^2\right),k=\frac{PX}{V_{\text{th}}}.}$ (23)

The slope $\partial Q/\partial V$ defines the stability margin. The critical point, corresponding to the nose of the Q-V curve, is given by:

$\boxed{V_{\text{crit}}^2={\left(\frac{PX}{V_{\text{th}}}\right)}^2+\frac{V_{\text{th}}^2}{4}.}$ (24)

For $V>V_{\text{crit}}$, the system operates in a stable zone where reactive power injection increases voltage ( $\partial Q/\partial V<0$). Conversely, as $V\to V_{\text{crit}}$, the margin vanishes and the system approaches voltage collapse.

We partition the bus admittance matrix $Y_{\text{bus}}$ into load buses $L$ and generator buses $G$ [26] :

$\left[\begin{array}{l}{I}_L\hfill \\ I_G\hfill \end{array}\right]=\left[\begin{array}{ll}{Y}_{LL}\hfill & Y_{LG}\hfill \\ Y_{GL}\hfill & Y_{GG}\hfill \end{array}\right]\left[\begin{array}{l}{V}_L\hfill \\ V_G\hfill \end{array}\right]$ (25)

Isolating $V_L$,

$V_L=-Y_{LL}^{-1}{Y}_{LG}{V}_G+Y_{LL}^{-1}{I}_L\triangleq F{V}_G+Z_{\text{}eq}{I}_L,$ (26)

where $F\triangleq -Y_{LL}^{-1}{Y}_{LG}$ and $Z_{\text{}eq}\triangleq Y_{LL}^{-1}$. For each load bus $i\in L$, we define the supporting voltage. We can see $E_i\triangleq {\displaystyle {\sum }_{k\in G}{F}_{ik}{V}_k}$ as the Thvenin voltage seen at bus $i$ if the injections $I_L$ were zero. $E_i\triangleq {\displaystyle {\sum }_{k\in G}{F}_{ik}{V}_k}$. The Kessel index is then

$\boxed{L_i\triangleq \left|1-\frac{E_i}{V_i}\right|=\left|1-\underset{k\in G}{{\displaystyle \sum }}{F}_{ik}\frac{V_k}{V_i}\right|,i\in L.}$ (27)

When the network is far from collapse, $\left|E_i/V_i\right|\approx 1$ in magnitude and the argument is small $\Rightarrow L_i\simeq 0$. As the stability limit is approached, the supporting capacity of the $G$ buses no longer compensates the local drop: the ratio $E_i/V_i$ deviates from the vicinity of 1 and $L_i\to 1$. A global indicator is $L_{\max }\triangleq {\max }_{i\in L}{L}_i$; the closer $L_{\text{max}}$ is to 1, the lower the voltage stability margin.

The power flow in polar coordinates is written as $F\left(\theta ,V\right)={\left[P\left(\theta ,V\right)Q\left(\theta ,V\right)\right]}^{\top }$. The Newton Jacobian matrix is

$J\left(\theta ,V\right)=\left[\begin{array}{ll}\frac{\partial P}{\partial \theta }\hfill & \frac{\partial P}{\partial V}\hfill \\ \frac{\partial Q}{\partial \theta }\hfill & \frac{\partial Q}{\partial V}\hfill \end{array}\right]$ (28)

By eliminating the angles through Schur reduction (assuming $\partial P/\partial \theta$ invertible at regular points), we obtain the reduced Jacobian in voltage

$\boxed{J_R\triangleq \frac{\partial Q}{\partial V}-\frac{\partial Q}{\partial \theta }{\left(\frac{\partial P}{\partial \theta }\right)}^{\text{}-1}\frac{\partial P}{\partial V}.}$ (29)

Loss of solvability (voltage collapse) is characterized by the singularity of $J$; equivalently, by the loss of positive definiteness/singularity of $J_R$. Numerically, the smallest singular value of $J$ (or $J_R$) decreases towards 0 as the critical point is approached [23] [24] .

Around the operating point $\left(x_0,y_0,u_0\right)$, the plant admits the reduced linear model $\text{Δ}\stackrel{\dot{}}{x}=A\text{Δ}x+B\text{Δ}u$, with the regulated voltage $\text{Δ}V=C_v\text{Δ}x$ (for a suitable row selector $C_v$). The Mamdani fuzzy law $u=ℱ\left(e,\text{Δ}e\right)$, with $e=V_{\text{ref}}-V$, is Frchet differentiable at $\left(e,\text{Δ}e\right)=\left(0,0\right)$; its tangent plane defines the local gains

$K_e\triangleq {\frac{\partial ℱ}{\partial e}|}_0,K_{\stackrel{\dot{}}{e}}\triangleq {\frac{\partial ℱ}{\partial \left(\text{Δ}e\right)}|}_0$ (30)

With $\text{Δ}e\approx -C_v\text{Δ}\stackrel{\dot{}}{x}$ and $e\approx -C_v\text{Δ}x$ in a small neighborhood, the control signal linearizes as

$\text{Δ}u\approx -K_e{C}_v\text{Δ}x-K_{\stackrel{\dot{}}{e}}{C}_v\text{Δ}\stackrel{\dot{}}{x}$ (31)

Introducing a standard first order prefilter on the derivative path (implicit in the Simulink implementation) with time constant ${\tau }_d>0$ yields ${\tau }_d\text{Δ}\stackrel{\dot{}}{u}+\text{Δ}u\approx -K_e{C}_v\text{Δ}x-K_{\stackrel{\dot{}}{e}}{C}_v\text{Δ}\stackrel{\dot{}}{x}$, and eliminates algebraic loops. Solving for $\text{Δ}u$ gives, to first order:

$\text{Δ}u\approx -\underset{K_{\text{eq}}}{\underset{︸}{\left(K_e+K_{\stackrel{\dot{}}{e}}A\right)}}{C}_v\text{Δ}x,\Rightarrow \boxed{\text{Δ}\stackrel{\dot{}}{x}=\left(A-B{K}_{\text{eq}}{C}_v\right)\text{Δ}x=A_{\text{cl}}\text{Δ}x.}$ (32)

Hence, the fuzzy controller modifies the closed-loop spectrum through the state feedback like matrix $B{K}_{\text{eq}}{C}_v$. In all operating points tested, $\sigma \left(A_{\text{cl}}\right)$ is shifted leftwards and the dominant damping ratio increases above the 10% - 15% guideline, consistently with the Q-V slope, the Kessel $L$ index, and the reduced Jacobian $J_R$ staying away from singularity.

A Static Var Compensator (SVC) plays a central role in the dynamic control of voltage within high-voltage power networks. This device is specifically designed to adjust in real time the reactive power injected or absorbed by the system, thus contributing to network stability and the improvement of power quality. In practice, the SVC is connected to the HV network via a shunt transformer, which provides both electrical insulation and voltage adaptation between high-voltage levels and medium or low voltage levels, where the power components are installed. Figure 1 illustrates the schematic of an SVC. It is composed of three main elements [27] - [29] :

To understand the role of the SVC in the dynamic regulation of power networks, it is essential to analyze its impact both on the electrical schematic and on the behavior in steady sinusoidal regime. Figure 2 shows the impact of a Static Reactive Power Compensator (SVC) on the behavior of a power network in steady sinusoidal operation. It illustrates on the left ( Figure 2(a) ) the equivalent circuit model with insertion of an SVC in shunt, and on the right ( Figure 2(b) ), the corresponding phasor diagram which highlights the effects of reactive compensation on voltage, current, and phase angle. This intelligent coupling makes it possible to control voltage, reduce losses, and improve the angular stability of the network [24] [25] .

The active power transmitted along the line can be calculated:

$P_R=\frac{2V_R\times V_S}{X_L}\sin \left(\delta /2\right)$ (33)

Figure 3 shows the transformation $\gamma \Rightarrow \text{Δ}$ of Figure 2(b) .

Consider a transmission line with total reactance $X_L$, divided into two halves. An SVC (Static Var Compensator) system, modeled by a susceptance $j{B}_{SVC}$, is connected at the center of the line. The equivalent diagram is given in Figure 3 . Each half of the line is a reactance $j\frac{X_L}{2}$, and the SVC at the center is a susceptance $j{B}_{SVC}$ connected in shunt. To find the equivalent impedance $Z_{eq}$ seen between terminals $V_S$ and $V_R$ which represents the complete system by a single series reactance $j{X}_{eq}$. The objective is to determine a single series reactance $j{X}_{eq}$ that models the entire electrical system seen between the terminals $V_S$ and $V_R$. This equivalence takes into account the following elements [24] [25] :

The total equivalent impedance, seen between the two ends of the system, is obtained from the series-parallel combination of these components, according to the following expression:

$Z_{eq}=Z_1+\left(\frac{1}{\frac{1}{Z_2}+Y_{sh}}\right)$ (34)

Substituting the expressions:

$\frac{1}{Z_2}=\frac{1}{j\frac{X_L}{2}}=-j\frac{2}{X_L}\text{and}{Y}_{sh}=j{B}_{SVC}$ (35)

$\Rightarrow \frac{1}{Z_2}+Y_{sh}=j\left(B_{SVC}-\frac{2}{X_L}\right)$ (36)

Thus, the impedance of the central block becomes:

$Z_{sh}=\frac{1}{j\left(B_{SVC}-\frac{2}{X_L}\right)}=\frac{-j}{B_{SVC}-\frac{2}{X_L}}$ (37)

Then:

$Z_{eq}=j\frac{X_L}{2}+\frac{-j}{B_{SVC}-\frac{2}{X_L}}+j\frac{X_L}{2}=j{X}_L-\frac{j}{B_{SVC}-\frac{2}{X_L}}$ (38)

To simplify, one can approximate (Taylor series expansion around dominant $B_{SVC}$):

$\frac{1}{B_{SVC}-\frac{2}{X_L}}\approx B_{SVC}-\frac{X_L^2}{4}$ (39)

Whence:

$Z_{eq}\approx j\left(X_L-\left(B_{SVC}-\frac{X_L^2}{4}\right)\right)$ (40)

The equivalent impedance of the line + SVC system is therefore given by:

$Z_{eq}=j\left(X_L-\left(B_{SVC}-\frac{X_L^2}{4}\right)\right)$ (41)

This results from a Thvenin reduction with a parallel load at the center (SVC), which changes the total impedance observed from one end to the other. In other words, this indicates that...

The active power transmitted over a line is:

$P=\frac{V_R\times V_S}{X_{eq}}\text{sin}\left(\delta \right)$ (42)

In order to simplify the analysis of the system, it is customary to use an approximate expression, recognized in the scientific literature, thus avoiding the complete analytical treatment of series-parallel associations between impedances and susceptance. This approximation makes it possible to directly determine the equivalent reactance of the network integrating an SVC.

$X_{eq}=X_L-\frac{X_L^2{B}_{SVC}}{4}$ (43)

The obtained relation highlights the effect of the SVC on reducing the equivalent reactance of the transmission line. By providing a controlled injection of capacitive-type reactive power, the SVC partially compensates the inductive effect of the network, which results in an improvement in active power transfer and better stability of the voltage profile. It follows that the greater the susceptance $B_{svc}$, the more the overall reactance $X_{eq}$ perceived by the network decreases. This simplified modeling proves particularly relevant in analytical studies as well as in dynamic simulations related to centralized reactive compensation.

The SVC can be modeled as a variable shunt admittance:

$I_{SVC}=j{B}_{SVC}{V}_{SVC},$ (44)

leading to the injected reactive power:

$Q_{SVC}=-B_{SVC}{\left|V_{SVC}\right|}^2.$ (45)

From the equivalent circuit of Figure 2(b) , the compensator current can also be expressed as a function of terminal voltages:

$I_{SVC}=\frac{2}{j{X}_L}\left(V_S+V_R-2V_{SVC}\right),$ (46)

which shows that $I_{SVC}$ vanishes when $V_{SVC}\approx \left(V_S+V_R\right)/2$, and increases when $V_{SVC}$ deviates from this average value.

Combining (46) with the admittance relation, the equivalent susceptance is obtained:

$B_{SVC}=-\frac{2}{j{X}_L}\left(\frac{V_S+V_R}{V_{SVC}}-2\right).$ (47)

Equation (47) highlights that $B_{SVC}$ is directly influenced by both the average of the sending and receiving voltages and the local voltage at the connection point. A decrease of $V_{SVC}$ leads to positive $B_{SVC}$ (capacitive injection), while an increase drives it towards negative values (inductive absorption).

For the SVC to absorb reactive power (inductive behavior), the injected power must be positive:

$Q_{SVC}>0\Rightarrow B_{SVC}<0.$ (48)

From (47), this condition is satisfied whenever the midpoint voltage exceeds the mean of the sending- and receiving-end voltages:

$V_{SVC}>\frac{V_S+V_R}{2}.$ (49)

In this situation, $B_{SVC}<0$ and the current $I_{SVC}$ lags $V_{SVC}$, which corresponds to reactive power absorption by the compensator.

The proportional integral (PI) controller is widely used in industry for voltage regulation. It is based on proportional correction of the error $e\left(t\right)=V_{\text{ref}}-V\left(t\right)$ and its integral:

$u\left(t\right)=K_{p}e\left(t\right)+K_i{\displaystyle {\int }_0^{t}e\left(\tau \right)\text{d}\tau },$ (50)

where $K_p$ is the proportional gain and $K_i$ the integral gain. In the case of the SVC, the output signal $u\left(t\right)$ is converted into a susceptance setpoint $B_{\text{eq}}\left(t\right)$, or equivalently into a firing angle $\alpha \left(t\right)$. The functional diagram is illustrated in Figure 4 .

By linearizing around an equilibrium point $\left(V_0,B_{\text{eq},0}\right)$, we obtain a first-order input-output model:

$\text{Δ}V\left(s\right)=\frac{K_p+K_i/s}{1+T_{s}s}\text{Δ}e\left(s\right),$ (51)

where $T_s$ is an effective time constant related to the dynamics of the TCR and the network. The PI controller presents several limitations in the context of dynamic power networks. It is highly dependent on the tuning of the gains $K_p,K_i$, which must be adjusted precisely according to the load and topology, otherwise instability or slowness may occur outside nominal conditions. Its robustness is also limited in the face of rapid load variations, which can lead to overshoots or oscillations. Finally, the PI does not adapt to structural uncertainties such as poorly known network parameters or measurement noise. These constraints justify the integration of an intelligent controller, notably based on fuzzy logic.

Fuzzy logic, proposed by Zadeh in 1965, is a tool well suited to the control of strongly nonlinear and uncertain systems. It relies on a qualitative representation of knowledge through linguistic If... Then... rules. Two approaches are commonly used: Mamdani and Takagi Sugeno (TSK). In this work, the Mamdani model is chosen due to its simplicity and intuitive interpretation for SVC control. The Mamdani-type fuzzy controller is based on a rule base exploiting the input variables:

$e\left(t\right)=V_{\text{ref}}-V\left(t\right),\Delta e\left(t\right)=\frac{\text{d}e\left(t\right)}{\text{d}t}$ (52)

and generating as output an equivalent susceptance $B_{\text{eq}}\left(t\right)$ (or firing angle $\alpha \left(t\right)$). The overall functional diagram is illustrated in Figure 5 .

The inputs $e\left(t\right)$ and $\text{Δ}e\left(t\right)$ are normalized within the interval $\left[-1,1\right]$, and the output $B_{\text{eq}}$ within $\left[0,B_{\text{max}}\right]$. Typically, 5 linguistic terms are used:

{NB(negativebig), NS (negativesmall), ZE (zero), PS (positivesmall), PB (positivebig)}.

For each variable, triangular or trapezoidal membership functions are defined. Example for $e\left(t\right)$ (see Figure 6 to be generated):

${\mu }_{\text{NB}}\left(e\right)=\text{max}\left(0,\frac{-1-e}{0.5}\right),{\mu }_{\text{NS}}\left(e\right)=\cdots \text{etc}.$ (53)

The rules are of the type:

$\text{IF}e\text{is}\text{NB}\text{AND}\Delta e\text{is}\text{NS},\text{THEN}{B}_{\text{eq}}\text{is}\text{PB}$ (54)

The input variables $e\left(t\right)$ and $\text{Δ}e\left(t\right)$ are normalized to the interval $\left[-1,1\right]$, which corresponds to the typical voltage deviations observed in the Maloukou network (up to ±10% of nominal voltage). Five linguistic terms (NB, NS, ZE, PS, PB) are employed, providing a balance between interpretability, computational simplicity, and control resolution. Using fewer terms (e.g. 3) leads to under-compensation during severe faults, whereas more than 7 unnecessarily increases the complexity of the rule base and the defuzzification process. The rule base is derived from physical considerations: when the bus voltage is below its reference and still decreasing, the controller must inject a large amount of reactive power; conversely, when the voltage exceeds nominal and continues to rise, the SVC absorbs reactive power. Intermediate cases ensure smooth transitions between these extreme actions. A sensitivity analysis confirmed that moderate variations of the membership function boundaries do not compromise the stability or effectiveness of the fuzzy controller. Table 1 illustrates an example with 5 terms per variable.

The fuzzy controller relies on an inference engine using the minax method (implication by minimum, aggregation by maximum), followed by defuzzification through the center of gravity (COG). The obtained control law is nonlinear, continuous, and able to adapt to uncertain contexts. It can be written in the form:

$B_{\text{eq}}\left(t\right)=ℱ\left(e\left(t\right),\text{Δ}e\left(t\right)\right),$ (55)

where $ℱ$ represents the fuzzy process combining fuzzification, inference, and defuzzification.

The fuzzy control surface of the Mamdani regulator applied to the SVC is shown in Figure 7(a) . It is obtained numerically in Matlab by sweeping the variables $e\left(t\right)$ and $\text{Δ}e\left(t\right)$ over a discrete grid, evaluating the fuzzy inference engine for each pair, and then plotting the three-dimensional surface using the command surf(e,de,u). This surface expresses the control law $u\left(t\right)=B_{eq}\left(t\right)$ as a function of the regulator inputs and highlights the intrinsic nonlinearity of fuzzy control. By injecting the equivalent susceptance $B_{eq}\left(t\right)$ into the reactive power law:

$Q\left(t\right)=V{\left(t\right)}^2\cdot B_{eq}\left(t\right)$ (56)

a linearized dynamic model of the SVC behavior is obtained. Figure 7(b) illustrates this relationship by representing the surface $B_{eq}=ℱ\left(e,\text{Δ}e\right)$ as well as its local tangent plane at the point $\left(0,0\right)$. This representation highlights the direct influence of the voltage error and its derivative on the reactive power flow exchanged by the compensator, while allowing the sensitivity of the regulator to be assessed around the nominal operating point.

Three main scenarios were simulated to evaluate the impact of the fuzzy regulator on SVC performance and to allow a direct comparison with PI control:

1. Network without compensation: the system is observed in its raw state, without reactive compensation device. This situation serves as a reference and highlights voltage collapse and transient instability in case of disturbance.

2. SVC controlled by PI: the static compensator is driven by a classical proportional integral regulator. The objective is to correct voltage sags, but performance is limited in the presence of nonlinearities and rapid load variations.

3. SVC controlled by Mamdani fuzzy regulator: the SVC is driven by the fuzzy controller designed previously. This approach aims to demonstrate better robustness and faster response to disturbances.

Each scenario is tested under several types of disturbances representative of real conditions of the Congolese network: single-phase, two-phase, and three-phase faults (with and without grounding), as well as sudden industrial load variations. These conditions make it possible to objectively compare the ability of the PI regulator and the fuzzy regulator to maintain voltage stability.

In order to ensure reproducibility, the main disturbance scenarios applied to the Maloukou network are summarized in Table 2 . They cover both fault conditions and load variations, as well as measurement uncertainties and control delays, which are representative of real operating conditions in African industrial grids.

In industrial networks such as Maloukou, load profiles often exhibit sudden starts, progressive increases, periodic cycles, or random fluctuations. These variations in active power $P\left(t\right)$ and especially in reactive power $Q\left(t\right)$ disturb the voltage balance and require fast and robust regulation.

To evaluate the robustness of the SVC controlled by fuzzy logic, four typical scenarios were simulated under Matlab/Simulink. Figure 8 associates each load profile with its corresponding response under fuzzy regulation.

The results confirm the robustness of the fuzzy regulator against varied load profiles. In Scenario A, the sudden voltage drop caused by motor start is corrected in less than 0.4 s, demonstrating quasi-instantaneous compensation capability. In Scenario B, the progressive rise of the inductive load is accurately tracked, with

no overcompensation or spurious oscillations, highlighting regulation stability. Scenario C, characterized by a periodic load cycle, is effectively damped thanks to a synchronized regulator response, ensuring a stable steady state. Finally, in Scenario D, random disturbances are filtered: only significant drifts trigger corrective action, thereby avoiding unnecessary adjustments. Overall, fuzzy control keeps the voltage close to 1 p.u. with fast and smooth convergence, confirming its ability to stabilize Congolese industrial networks subject to unstable and unpredictable load conditions.

In addition to internal load scenarios, it is essential to assess the response of the fuzzy SVC to exogenous disturbances, often present in African industrial networks: noisy sensors, processing delays, and upstream instabilities. Three representative cases were introduced in the Simulink/PSAT model: (a) addition of pseudo-random noise of ±1% on the measurement of $V\left(t\right)$, (b) insertion of a constant delay of 100 ms in the regulation loop, and (c) application of an exogenous voltage sag from 1 to 0.85 p.u. with recovery in 1 s.

Figure 9 illustrates the responses obtained. In the case of noise, the fuzzy regulator naturally filters fluctuations thanks to the imprecision tolerated by its membership functions, avoiding excessive corrections ( Figure 9(a) ). In the presence of a delay, the compensation is slightly shifted but remains progressive and stable, without causing oscillations ( Figure 9(b) ). Finally, during the voltage sag, the fuzzy control surface immediately injects high reactive power, restoring nominal voltage in a shorter time compared to classical approaches ( Figure 9(c) ).

These results confirm that the Mamdani regulator acts as an adaptive stabilizer: it actively damps the effects of external disturbances without generating additional oscillations or saturation. This operational resilience fully justifies its integration into Congolese industrial networks, characterized by noisy measurements and uncertain operating conditions.

Comparative simulations highlight the superiority of the fuzzy regulator over the classical PI controller.

Beyond voltage stability assessment, the impact of different control strategies was analyzed in terms of active power losses in the conductors. At each time step, the dissipated power is calculated as:

$P_{\text{loss}}\left(t\right)=\underset{k=1}{\overset{N_l}{{\displaystyle \sum }}}{R}_k\cdot I_k{\left(t\right)}^2$

where $R_k$ denotes the resistance of line $k$ and $I_k\left(t\right)$ the instantaneous current through it. Time integration of this quantity allows estimation of the energy dissipated over the simulation horizon.

Figure 10 compares the temporal evolution of active losses for the three scenarios studied. Without compensation, the losses exhibit peaks above 2.1 MW during voltage sags, due to massive current inrush. With an SVC controlled by PI, the situation improves but remains marked by oscillations linked to regulation instabilities. In contrast, the fuzzy regulator produces the most stable and lowest curve throughout the simulation, with an average reduction in energy lost of about 15% compared to the uncompensated case.

This reduction in losses represents a major operational advantage in the context of undersized African networks, where every gain in energy efficiency directly contributes to the availability and quality of supply.

The overall stability of the network depends on the quality of voltage regulation and the dynamic behavior of compensation devices. Simulations show that the SVC controlled by a fuzzy regulator ensures fast and smooth convergence to equilibrium, unlike PI or threshold-based regulations which generate overshoots and prolonged oscillations.

Figure 11 illustrates this dynamic in the phase space $\left(e,\text{Δ}e\right)$: the fuzzy control follows a smooth trajectory and stabilizes the system without overcompensation. This performance is due to the linguistic granularity of the Mamdani model, able to continuously adapt the response to uncertainties and load variations.

Thus, fuzzy logic stands out as an adaptive stabilizer, ensuring robustness and speed in an industrial environment subject to permanent disturbances.

On the temporal side, the return to nominal voltage is achieved in less than 0.45 s with fuzzy logic, compared to 0.75 s with PI and more than 2 s without SVC. Stability is also reinforced: transient oscillations are largely damped and the maximum voltage error is reduced by half compared to PI. Finally, the robustness of the fuzzy regulator results in better tolerance to load variations and random disturbances, while limiting active losses by about 15%.

Figure 12 illustrates this multi-criteria comparison in the form of a radar chart, confirming that fuzzy control simultaneously offers speed, stability, and resilience, without notable trade-offs.

Beyond the classical performance indices (settling time, maximum error, and active losses), the comparison between PI and fuzzy regulation was extended along three additional dimensions. First, robustness was explicitly tested under noisy measurements, control delays, and voltage sags, confirming that the fuzzy controller maintains stable operation where the PI regulator exhibits oscillations. Second, a sensitivity study highlighted that small variations in PI gains significantly affect the dynamic response, whereas the fuzzy controller remains stable under moderate shifts of membership function boundaries. Finally, computational complexity was assessed: although the fuzzy controller requires the evaluation of rules and defuzzification, the number of rules is limited (25 in this study), and the centroid method is computed in less than 1 ms on a standard CPU. This ensures compatibility with real-time implementation in industrial environments. Table 3 quantitatively summarizes these results for the three configurations studied.

These results demonstrate that fuzzy control is not merely an alternative, but an optimized and robust solution, particularly suited to Congolese industrial networks subject to unstable loads and uncertain operating conditions.