Abstract

The integration of utility-scale photovoltaic generation into weak power systems can significantly affect power-flow distribution and voltage-stability margins, particularly when supplying reactive industrial loads. This paper investigates a 12 MW grid-connected photovoltaic plant supplying a 22 MVA industrial load under weak-grid conditions. The proposed assessment combines Newton-Raphson power flow, continuation-based P-V analysis, Jacobian minimum singular-value monitoring, and static var generator (SVG) compensation. For the adopted reduced benchmark with a short-circuit ratio of SCR = 5, the uncompensated system exhibits severe voltage degradation and limited loadability. A 6 MVAr SVG increases the minimum bus voltage from 0.617 to 0.855 p.u. and the maximum loading factor from 0.023 to 0.316. The corresponding maximum industrial active load increases from 19.13 to 24.61 MW, while active-power losses decrease from 0.924 to 0.210 MW. However, the compensated voltage remains below the 0.95 p.u. criterion, indicating that nominal reactive-power compensation alone does not ensure adequate voltage performance in a weak grid. The results highlight the need for SVG sizing based jointly on voltage regulation and static loadability constraints. Here, λ is the fractional load increase, with λ = 0 at nominal demand. The reported results require verification using the complete network dataset and actual SVG control settings.

Keywords

Photovoltaic Integration, Voltage Stability, Power Flow, Industrial Load, Weak Grid, Static Var Generator, Continuation Power Flow, Reactive-Power Compensation

Share and Cite:

1. Introduction

Large-scale photovoltaic integration modifies active-reactive power flows and voltage regulation, particularly in weak grids supplying industrial loads. Low short-circuit strength and high reactive-power demand can produce significant voltage deviations and reduce the static loadability margin [1]-[3]. This issue is relevant to the Republic of the Congo, where renewable generation and transmission-system reinforcement are being pursued. The integration of utility-scale PV therefore requires assessment of both power-flow feasibility and voltage-stability constraints. For industrial loads, active-power balance alone is insufficient. Reactive-power transfer, grid strength, and network impedance directly affect bus voltages and proximity to voltage instability [4] [5]. Moreover, increasing inverter-based generation can reduce available reactive-power reserves, making voltage stability increasingly dependent on local reactive support and network strength [6]. Recent studies confirm that renewable penetration and reactive-power availability must be jointly considered when evaluating voltage-stability margins [7]. SVG/STATCOM compensation can provide local reactive support [8] [9]; however, a rating derived solely from nominal reactive demand does not necessarily guarantee an acceptable voltage-stability margin. Under weak-grid conditions, the compensator rating must therefore be assessed against both the required voltage range and maximum transferable load [10]. This paper investigates a 12 MW grid-connected PV plant supplying a 22 MVA industrial load under weak-grid conditions. Newton-Raphson power flow, continuation-based P-V analysis, the minimum singular value of the power-flow Jacobian, and SVG compensation are jointly applied [11]-[13]. The main contributions are:

• A unified PV-grid-PCC-industrial load-SVG power-flow model;

• Critical-bus and loadability-limit identification using P-V continuation and σmin(J);

• Quantitative assessment through Vmin, network losses, λmax, and PL, max;

• Evaluation of SVG sizing under simultaneous voltage-regulation and loadability constraints.

2. Technical Background and Research Gap

2.1. PV Integration and Weak-Grid Voltage Stability

High PV penetration modifies active-reactive power exchanges and increases voltage sensitivity when grid short-circuit strength is limited [1]-[5]. For inverter-based generation, voltage stability is strongly influenced by grid impedance, reactive-power capability, and load characteristics. Static voltage stability is assessed from P-V trajectories and the maximum loading point. Near the stability boundary,

$\lambda \to {\lambda }_{\max },{\sigma }_{\min }\left(J\right)\to 0$

where J denotes the power-flow Jacobian [11]-[13]. Recent continuation-power-flow developments extend this assessment to networks containing distributed and inverter-based resources [14]. The 2025 IEEE work associated with [14], for example, develops an integrated CPF formulation accounting for DERs, load types, network imbalance and renewable fluctuations.

These indicators are subsequently adopted for the investigated PV-industrial system.

2.2. Reactive-Power Support

Reactive-power availability directly affects voltage regulation in weak networks [6] [7]. SVG/STATCOM devices provide controllable reactive support at critical buses and can modify both the voltage profile and system loadability [8]-[10].

The compensation requirement cannot therefore be reduced to nominal reactive-power balance. Under weak-grid conditions, the SVG rating must additionally satisfy

$V_{\min }\ge V_{\lim }$

while maintaining an adequate static loading margin.

The preliminary compensation rating must therefore be evaluated against both voltage performance and the static loadability boundary, rather than nominal power-factor correction alone.

2.3. Research Gap

Existing studies address weak-grid strength, renewable penetration, continuation-based voltage-stability assessment, and reactive-power compensation [1]-[15]. However, the engineering problem considered here requires these quantities to be evaluated within the same operating framework.

Accordingly, this study considers

$12\text{ MW PV}\to \text{Weak Grid/PCC}\to 22\text{ MVA Industrial Load}\to \text{SVG}$

and evaluates the compensation requirement simultaneously against bus-voltage, network-loss, and static-loadability constraints.

3. Overall System Architecture

High PV penetration affects power-flow distribution and voltage regulation, particularly in weak grids, where adequate reactive-power support becomes critical for maintaining acceptable voltage conditions.

The investigated system consists of a 12 MW grid-connected PV plant, an upstream utility grid, a point of common coupling (PCC), transformation and transmission interfaces, industrial loads, and a reactive-power compensation unit, as illustrated in Figure 1. The PCC constitutes the main electrical interface for assessing active- and reactive-power exchanges between the PV plant, utility grid, and industrial network.

The PV plant operates at a rated active power of

$P_{PV,rated} =12MW$ (1)

The industrial subsystem comprises aggregated loads connected downstream of the PCC. Since industrial loading may impose significant reactive-power requirements, the compensation device is located at the critical bus identified from the uncompensated power-flow and voltage-stability analyses rather than being predefined.

The system assessment is based on the electrical quantities

$M = \left\{V_i,P_{ij},Q_{ij},P_{loss},Q_{loss}\right\}$ (2)

where $V_i$ denotes the voltage magnitude at bus i, $P_{ij}$ and $Q_{ij}$ are the active and reactive power transfers between buses i and j, and $P_{loss}$ and $Q_{loss}$ represent the network active- and reactive-power losses. The critical bus and voltage-stability margins are subsequently determined using the assessment methodology developed in Section 5.

Figure 1. Conceptual single-line architecture with reactive compensation at the industrial bus. This equipment diagram does not replace the numerical solver bus and branch dataset.

3.1. Grid and Network Specifications

The upstream network is represented by a 220 kV transmission grid interconnected with the 12 MW PV-industrial system through the PCC. The Congolese transmission network includes 220 kV and 110 kV backbone levels, with 30 - 35 kV sub-transmission interfaces. For the investigated configuration, the upstream source is represented at 220 kV, while a 20 kV PCC is adopted for the PV and industrial subsystems.

The nominal system frequency is fixed at

$f_n = 50\text{ Hz}$ (3)

To investigate a moderately weak operating condition, the base-case short-circuit ratio is set to

$SCR = 5$ (4)

With a 12 MW PV rated power, the corresponding short-circuit capacity adopted at the PCC is

$S_{sc} = SCR P_{PV,rated} =5\times 12=60MVA$ (5)

An X/R ratio of 10 is adopted for the equivalent upstream impedance. The values SCR = 5 and X/R = 10 are treated as simulation assumptions rather than measured parameters of the Congolese transmission system. Their influence can subsequently be examined through sensitivity analysis, as shown in Table 1.

Table 1. Electrical specifications of the studied grid.

The 220/20 kV and 0.8/20 kV transformer ratios define the adopted study configuration; they are not presented as measured ratings of a specific 12 MW Congolese PV installation. Detailed equivalent impedances, per-unit conversion, network admittances, and power-flow equations are developed separately in Section 4.

Reproducibility requires a numerical bus and branch dataset in addition to the conceptual architecture. The voltage-profile figures distinguish the upstream source, PCC, and industrial bus; they do not specify the complete solver connectivity. The 220/20 kV and 0.8/20 kV transformer ratios alone do not determine leakage impedance, tap position, or transformer losses. The PCC Thévenin impedance derived in Section 5.2 is an aggregate upstream equivalent: if it already includes the grid transformer, that impedance must not be counted again as a separate series element. The PV plant is represented by fixed P and Q injections, not by a voltage-controlled PV bus merely because the generator is photovoltaic.

3.2. 12 MW Photovoltaic Plant Specifications

The photovoltaic plant is represented as a utility-scale generation unit with a rated AC active-power capacity of

$P_{PV,rated} =12MW$ (6)

The conversion chain comprises the PV array, DC-link stage, grid-connected voltage-source converter (VSC), output filter, and step-up transformer interfacing the plant with the 20 kV PCC. For system-level power-flow analysis, the internal switching dynamics are neglected and the PV plant is represented at its AC terminal by its active- and reactive-power operating setpoints.

Under nominal irradiance and temperature conditions, the available active power is bounded by

$0\le P_{PV}\le P_{PV,rated}$ (7)

The inverter apparent-power capability imposes the operating constraint

$P_{PV}{}^2 + Q_{PV}{}^2 \le S_{inv}{}^2$ (8)

which defines the reactive-power capability available for voltage support at a given active-power operating point. Consequently, the maximum inverter reactive-power magnitude is

$\left|Q_{PV}\right| \le \sqrt{S_{inv}{}^2 - P_{PV}{}^2}$ (9)

For the base operating condition, the PV plant is operated at unity power factor,

$P_{PV}=12MW,Q_{PV}=0$ (10)

thereby allowing the effect of the external reactive-power compensation system to be evaluated independently. Additional operating cases may subsequently exploit the inverter reactive-power capability to assess coordinated voltage support, as shown in Table 2.

Table 2. Main specifications of the 12 MW PV plant.

The inverter apparent-power rating S_inv is intentionally left unspecified until a documented converter rating or a clearly justified study value is selected. This prevents an arbitrary reactive-power capability from influencing the subsequent voltage-stability assessment.

3.3. Industrial Load Specifications

The industrial subsystem is represented by aggregated three-phase loads connected downstream of the 20 kV PCC. A nominal apparent-power demand of

$S_{L0} = 22MVA$ (11)

is adopted as the reference operating condition. The load is considered predominantly inductive, with a reference lagging power factor of

$p{f}_{L0} = 0.85$ (12)

Accordingly, the nominal active-power demand is

$P_{L0} = S_{L0}p{f}_{L0} = 22\times 0.85=18.70MW$ (13)

while the corresponding reactive-power demand is

$Q_{L0} = S_{L0}\surd \left(1-p{f}_{L0}{}^2\right)=22\surd \left(1-{0.85}^2\right)=11.59MVAr$ (14)

The reference industrial operating point is therefore

$S_{L0} = P_{L0} + j{Q}_{L0}=18.70+j11.59MVA$ (15)

The loading parameter λ denotes the fractional increase relative to the nominal 22 MVA demand: S_L(λ) = (1 + λ)S_L0. Thus λ = 0 is nominal, λ = −0.50 is half load, and λ = 0.50 is 150% of nominal load. Table 3 uses this same convention as Equation (56) and the P-V figures. The total load multiplier is α = 1 + λ; α and λ must not be interchanged. The listed load levels are screening cases, not evidence of feasible power-flow solutions; feasibility is checked separately for each compensation case as shown in Table 3.

Table 3. Industrial load operating conditions.

At the nominal operating point, the ratio between the rated PV generation and industrial active-power demand is

$K_{PV} = \frac{P_{PV,rated}}{P_{L0}} = \frac{12}{18.70} = 0.642$ (16)

Thus, the rated PV generation corresponds to approximately 64.2% of the nominal industrial active-power demand. Neglecting network losses, the remaining active-power requirement at rated PV production is approximately

$P_G \approx 18.70-12=6.70MW$

This operating condition maintains simultaneous participation of the PV plant and the upstream utility grid, providing a consistent reference state for the subsequent power-flow and voltage-stability assessments.

The equations used here represent the aggregated industrial demand as a balanced, constant-power PQ load: at any fixed λ, P_L and Q_L do not depend on bus voltage. This is an assumption of the static model, not an identified composition of the actual industrial process. No induction-motor dynamics, stalling, protection action, or load recovery is represented. A ZIP sensitivity study should distinguish constant-power, constant-current, and constant-impedance contributions. At depressed voltage, current and impedance components reduce demand relative to the constant-power case; their effect on the fold must be recalculated rather than assigned a universal numerical correction. Static loadability does not establish motor ride-through or dynamic voltage stability.

3.4. Reactive Power Compensation System

A 6 MVAr shunt-connected SVG is considered at the critical industrial bus, downstream of the PCC. This location is used consistently in Figure 1 and Figure 2 and Sections 4.5 and 6.3. It provides local reactive-power support in the static power-flow model; dynamic converter performance is outside the present assessment.

For the nominal industrial load,

$P_{L0}=18.70MW,Q_{L0}=11.59MVAr,p{f}_{L0}=0.85$ (17)

The preliminary reactive-power requirement is determined for a target power factor pf_t = 0.95 as

$Q_{comp} = P_{L0}\left[\tan \left({\cos }^{-1}p{f}_{L0}\right)-\tan \left({\cos }^{-1}p{f}_t\right)\right]$ (18)

Substituting the nominal operating conditions gives

$Q_{comp}=18.70\left[\tan \left({\cos }^{-1}0.85\right)-\tan \left({\cos }^{-1}0.95\right)\right]=5.44MVAr$ (19)

Accordingly, the initial SVG rating is selected as

$Q_{SVG,rated} = 6MVAr$ (20)

at the 20 kV industrial bus. The SVG operating range is defined by

$-6\le Q_{SVG}\le 6MVAr$ (21)

Positive Q_SVG denotes capacitive injection and negative Q_SVG denotes inductive absorption. The ±6 MVAr rating is a capability bound, not the actual injection. Actual nominal and critical injections must be exported from the converged solutions. The target power factor of 0.95 used for preliminary sizing is distinct from the SVG voltage reference (Table 4).

Table 4. SVG specifications.

4. Mathematical Modeling and Power-Flow Formulation

4.1. Network Admittance Formulation

For an N-bus balanced three-phase AC network, the nodal equations are obtained from Kirchhoff’s current law at each bus i:

$I_i={\sum }_{j\in {\Omega }_i} I_{ij}+I_i{}^{sh},i=1,\cdot \cdot \cdot ,N$ (22)

where Ω_i denotes the set of buses directly connected to bus i, I_ij is the current flowing from bus i to bus j, and $I_i{}^{sh}$ represents the total shunt current connected to bus i.

For a transmission branch (i, j), the series impedance is

$Z_{ij} = R_{ij} + j{X}_{ij}$ (23)

and the associated series admittance is

$y_{ij} = \frac{1}{Z_{ij}} = \frac{R_{ij} - j{X}_{ij}}{R_{ij}{}^2 + X_{ij}{}^2} = g_{ij} + j{b}_{ij}$ (24)

with

$g_{ij}=\frac{R_{ij}}{R_{ij}{}^2 + X_{ij}{}^2},b_{ij}=-\frac{X_{ij}}{R_{ij}{}^2 + X_{ij}{}^2}$ (25)

Using the conventional π-equivalent model, the total line-charging susceptance $B_{ij}{}^c$ is equally divided between the two terminal buses. The branch currents are therefore

$I_{ij} = \left(y_{ij} + j\frac{B_{ij}{}^c}{2}\right)V_i - y_{ij}{V}_j$ (26)

$I_{ji} = \left(y_{ij} + j\frac{B_{ij}{}^c}{2}\right)V_j - y_{ij}{V}_i$ (27)

Substituting the branch-current expression into Kirchhoff’s current law gives

$I_i = {\sum }_{j\in {\Omega }_i}\left[\left(y_{ij} + j\frac{B_{ij}{}^c}{2}\right)V_i - y_{ij}{V}_j\right] + y_i{}^{sh}{V}_i$ (28)

Rearranging the voltage-dependent terms yields

$I_i = \left[{\sum }_{j\in {\Omega }_i}\left(y_{ij} + j\frac{B_{ij}{}^c}{2}\right) + y_i{}^{sh}\right]V_i - {\sum }_{j\in {\Omega }_i}{y}_{ij}{V}_j$ (29)

Hence, the injected current at bus i can be expressed as

$I_i = {\sum }_{j=1}{}^N Y_{ij}{V}_j$ (30)

The off-diagonal elements of the bus-admittance matrix are given by

$Y_{ij} = -y_{ij},i\ne j$ (31)

while each diagonal element is obtained from the sum of all series and shunt admittances incident on bus i:

$Y_{ii} = y_i{}^{sh} + {\sum }_{j\in {\Omega }_i}\left(y_{ij} + j\frac{B_{ij}{}^c}{2}\right)$ (32)

The complete nodal relationship is therefore written as

$I = Y_{bus} V$ (33)

$I={\left[I_1,I_2,\cdot \cdot \cdot ,I_N\right]}^T,V={\left[V_1,V_2,\cdot \cdot \cdot ,V_N\right]}^T$ (34)

$Y_{bus} = {\left[Y_{ij}\right]}_{N\times N}$ (35)

Because the studied system includes both the 220/20 kV grid transformer and the 0.8/20 kV PV step-up transformer, off-nominal transformer ratios are explicitly incorporated into the nodal admittance formulation. For a transformer connected between buses i and j, the complex tap ratio is defined as

$a_{ij} = {\tau }_{ij}{e}^{j{\theta }_{ij}}$ (36)

where τ_ij denotes the tap magnitude and θ_ij the phase-shifting angle. The corresponding transformer series admittance is

$y_{ij}{}^T = \frac{1}{R_{ij}{}^T + j{X}_{ij}{}^T}$ (37)

and the transformer terminal-current relationship becomes

$\left[\begin{array}{c}{I}_i\\ I_j\end{array}\right] = \left[\begin{array}{cc}\frac{y_{ij}{}^T}{\left|a_{ij}\right|²}& -\frac{y_{ij}{}^T}{a_{ij}{}^{*}}\\ -\frac{y_{ij}{}^T}{a_{ij}}& y_{ij}{}^T\end{array}\right]\left[\begin{array}{c}{V}_i\\ V_j\end{array}\right]$ (38)

For transformers without phase shifting, θ_ij = 0, and the complex tap ratio reduces to

$a_{ij} = {\tau }_{ij}$ (39)

The transformer contribution can then be written compactly as

$Y^T{}_{ij} = y^T{}_{ij} \left[\begin{array}{cc}\frac{1}{{\tau }_{ij}²}& -\frac{1}{{\tau }_{ij}}\\ -\frac{1}{{\tau }_{ij}}& 1\end{array}\right]$ (40)

After assembling all transmission-line, transformer, and shunt elements, the complete bus-admittance matrix is obtained as

$Y_{bus} = Y_{line} + Y_{tr} + Y_{sh}$ (41)

and expressed in rectangular form as

$Y_{bus}=G+jB,Y_{ij}=G_{ij}+j{B}_{ij}$ (42)

The resulting $Y_{bus}$ matrix contains the complete steady-state electrical representation of the studied network and provides the algebraic basis for the nonlinear active- and reactive-power equations developed in Section 4.2.

4.2. Active and Reactive Power Equations

Using the admittance formulation established in Section 4.1, the current and complex-power injections at bus i are

$I_i={\sum }_{j=1}{}^N Y_{ij}{V}_j,S_i=P_i+j{Q}_i=V_i{I}_i{}^{*}$ (43)

Substituting the nodal current into the complex-power expression yields

$S_i=V_i\left({\sum }_{j=1}{}^N Y_{ij}{V}_j\right)*=V_i{\sum }_{j=1}{}^N{Y}_{ij}{}^{*}{V}_j{}^{*}$ (44)

With

$V_i=\left|V_i\right|e^{j{\delta }_i},Y_{ij}=G_{ij}+j{B}_{ij},{\delta }_{ij}={\delta }_i-{\delta }_j$ (45)

the complex-power injection becomes

$S_i = {\sum }_{j=1}{}^N\left|V_i\right|\left|V_j\right|\left(G_{ij} - j{B}_{ij}\right)e^{j{\delta }_{ij}}$ (46)

Using Euler’s identity and separating the real and imaginary components of (46), the nonlinear active- and reactive-power equations are obtained as

$\left\{\begin{array}{l}{P}_i={\sum }_{j=1}{}^N\left|V_i\right|\left|V_j\right|\left[G_{ij}\text{cos}{\delta }_{ij}+B_{ij}\text{sin}{\delta }_{ij}\right];\\ Q_i={\sum }_{j=1}{}^N\left|V_i\right|\left|V_j\right|\left[G_{ij}sin {\delta }_{ij}-B_{ij}cos {\delta }_{ij}\right]\end{array}\right\}$ (47)

For the power-flow solution, the specified net injections are defined according to the adopted positive-injection convention as

$\begin{array}{l}{P}_i{}^{spec} = P_{Gi} + P_{PV,i} - P_{Li}\hfill \\ Q_i{}^{spec} = Q_{Gi} + Q_{PV,i} + Q_{SVG,i} - Q_{Li}\hfill \\ \Delta P_i = P_i{}^{spec} - P_i\left(V,\delta \right)\hfill \\ \Delta Q_i = Q_i{}^{spec} - Q_i\left(V,\delta \right)\hfill \end{array}$ (48)

The steady-state operating point is obtained when the power mismatches $\Delta P_i$ . and $\Delta Q_i$ converge to zero within the prescribed numerical tolerance. The nonlinear system defined by (47) - (48) constitutes the basis of the Newton-Raphson formulation developed in Section 4.3.

4.3. Newton-Raphson Power-Flow Solution

The nonlinear system defined by (47) - (48) is solved using the Newton-Raphson method. For a network containing one slack bus, n_PV PV buses, and n_PQ PQ buses, the state vector is defined as

$x = {\left[{\delta }_{PV}{}^T {\delta }_{PQ}{}^T V\_P{Q}^T\right]}^T,\dim \left(x\right)=n_{PV}+2n_{PQ}$ (49)

where the slack-bus voltage magnitude and angle are fixed, PV-bus voltage magnitudes are specified, and PQ-bus voltage magnitudes and phase angles are unknown.

At iteration k, the active- and reactive-power mismatches are assembled as

$\Delta F^{\left(k\right)}={\left[\Delta P^{\left(k\right)}\Delta Q^{\left(k\right)}\right]}^T={\left[P^{spec}-P\left(x^{\left(k\right)}\right)Q^{spec}-Q\left(x^{\left(k\right)}\right)\right]}^T$ (50)

Linearization around the current operating point gives

$\Delta F^{\left(k\right)}=J^{\left(k\right)}\Delta x^{\left(k\right)},\Delta x^{\left(k\right)}={\left[\Delta {\delta }^{\left(k\right)T}\Delta V\_P{Q}^{\left(k\right)T}\right]}^T$ (51)

where the Newton-Raphson Jacobian is

$J=\left[HN;ML\right]=\left[\partial P/\partial \delta \partial P/\partial V;\partial Q/\partial \delta \partial Q/\partial V\right]$ (52)

For i ≠ j, the off-diagonal Jacobian elements follow directly from (47):

$\begin{array}{l}{H}_{ij} = \left|V_i\right|\left|V_j\right|\left[G_{ij} sin {\delta }_{ij} - B_{ij} cos {\delta }_{ij}\right]\hfill \\ N_{ij} = \left|V_i\right|\left[G_{ij} cos {\delta }_{ij} + B_{ij} sin {\delta }_{ij}\right]\hfill \\ M_{ij} = -\left|V_i\right|\left|V_j\right|\left[G_{ij} cos {\delta }_{ij} + B_{ij} sin {\delta }_{ij}\right]\hfill \\ L_{ij} = \left|V_i\right|\left[G_{ij} sin {\delta }_{ij} - B_{ij} cos {\delta }_{ij}\right],i\ne j\hfill \end{array}$ (53)

The corresponding diagonal elements are

$\begin{array}{l}{H}_{ii}=-Q_i-B_{ii}{\left|V_i\right|}^2;N_{ii}=P_i/\left|V_i\right|+G_{ii}\left|V_i\right|\hfill \\ M_{ii}=P_i-G_{ii}{\left|V_i\right|}^2;L_{ii}=Q_i/\left|V_i\right|-B_{ii}\left|V_i\right|\hfill \end{array}$ (54)

Rather than explicitly forming the inverse of the Jacobian, the state correction is obtained by solving the linear system and updating the state as

$J^{\left(k\right)}\Delta x^{\left(k\right)}=\Delta F^{\left(k\right)},x^{\left(k+1\right)}=x^{\left(k\right)}+\Delta x^{\left(k\right)}$ (55)

Iterations are terminated when the infinity norm of the mismatch vector satisfies

$\Vert \Delta F^{\left(k\right)}\Vert \infty \le \epsilon$

where ε is the prescribed convergence tolerance. The converged operating point provides the bus-voltage magnitudes, phase angles, and power transfers required for the subsequent voltage-stability assessment.

4.4. Voltage-Stability Formulation

Voltage stability is assessed with λ = 0 at the nominal operating point and λ > 0 for demand increases. Only industrial demand is scaled; the PV active-power injection remains 12 MW and Q_PV remains zero. The slack source balances the additional demand and network losses. At each stressed load bus,

$P_{Li}\left(\lambda \right)=P_{Li0}\left(1+\lambda \right),Q_{Li}\left(\lambda \right)=Q_{Li0}\left(1+\lambda \right)$ (56)

where P_Li0 and Q_Li0 denote the base-case load demand. The reference power factor is preserved through

$Q_{Li}\left(\lambda \right)/P_{Li}\left(\lambda \right) = Q_{Li0}/P_{Li0} = \tan \left({\cos }^{-1}p{f}_{L0}\right)$ (57)

The parameterized power-flow system is

$F\left(x,\lambda \right)={\left[\Delta P\left(x,\lambda \right)\Delta Q\left(x,\lambda \right)\right]}^T=0$ (58)

The state vector x contains the unknown voltage angles and voltage magnitudes, with the SVG control variable and equation added when voltage regulation is active. Starting from a converged high-voltage solution at λ = 0, successive solutions of (58) generate the P-V trajectories. If the nominal Newton-Raphson solution is ambiguous, a lighter-load solution must first be continued to λ = 0 to establish the selected branch.

$\left|V_i\right| = f_i\left(\lambda \right),i\in {\Omega }_{PQ}$ (59)

At a smooth fold of the equilibrium branch, the fixed-λ power-flow Jacobian becomes singular. This condition is evaluated within the current control regime; a change caused by an SVG limit must be distinguished from a smooth fold:

$\det \left[J\left(x,\lambda \right)\right]\to 0,{\sigma }_{\min }\left(J\right)\to 0$ (60)

The minimum singular value is a supporting proximity indicator, not a stand-alone stopping test with an unspecified threshold. The maximum loading parameter ${\lambda }_{\max }$ gives the incremental margin from the nominal point:

$\Delta P_{\text{margin}}={\lambda }_{\max } P_{L0},P_{L,\max }=\left(1+{\lambda }_{\max }\right)P_{L0}$ (61)

For the nominal industrial demand,

$P_{L0}=18.70MW$

Consequently, ${\lambda }_{\max }$ = 0.023 represents a 2.3% demand increase and a total multiplier α_max = 1.023; ${\lambda }_{\max }$ = 0.316 represents a 31.6% increase and α_max = 1.316. Using the reported rounded values, the corresponding active demands are 19.13 MW and 24.61 MW, with incremental margins of 0.43 MW and 5.91 MW. These are arithmetic conversions, not new simulation results.

4.5. SVG Steady-State Model

The SVG is modeled as a controllable shunt reactive-power source connected to the critical bus k, with negligible active-power exchange:

$S_{SVG}=P_{SVG} + j{Q}_{SVG},P_{SVG}\approx 0$ (62)

The reactive-power balance at the compensated bus becomes

$\Delta Q_k = Q_{Gk} + Q_{PV,k} + Q_{SVG} - Q_{Lk} - Q_k\left(V,\delta \right)$ (63)

Voltage regulation is represented by

$\left|V_k\right|=V_k{}^{ref}-K_Q Q_{SVG},K_Q>0$ (64)

subject to the adopted SVG capability:

$-6\le Q_{SVG}\le +6MVAr$ (65)

For positive droop K_Q, the unsaturated SVG solves Equation (64) together with the reactive-power balance; the bus is not an ideal fixed-voltage PV bus. K_Q has units of p.u./MVAr when Q_SVG is expressed in MVAr. With K_Q = 0, the unsaturated model becomes ideal voltage regulation. At +6 or −6 MVAr, reactive injection is fixed at the reached bound, the voltage-control equation is released, and the industrial-bus voltage is solved as a PQ variable. A saturated point is accepted only while the unconstrained control demand would remain beyond that bound; otherwise, voltage regulation is restored. No numerical voltage reference or droop is inferred from the nameplate rating.

[Enter the actual V_ref and K_Q used in the simulation, with units, and export Q_SVG at λ = 0 and at the critical point. The reported voltage of 0.855 p.u. does not by itself establish saturation at +6 MVAr. These values cannot be reconstructed uniquely from the present tables.]

5. Simulation Methodology and Study Cases

5.1. Simulation Framework

The developed power-flow and voltage-stability models are implemented for the 12 MW PV-industrial network using the parameters defined in Section 3. The base operating condition is

$P_{PV}=12MW,S_{L0}=22MVA,p{f}_{L0}=0.85$ (66)

corresponding to

$P_{L0}=18.70MW,Q_{L0}=11.59MVAr$ (67)

The power-flow solution is obtained using the Newton-Raphson formulation of Section 4.3. Convergence is imposed when

${\max }_{i}abs\left(\Delta F_i\right)\le {10}^{-6}$ (68)

A reproducible continuation calculation should use the predictor-corrector protocol specified below. The implementation and settings must be checked against the solver that produced the reported curves before those curves are accepted as validated results. The following two compensation configurations are evaluated:

$\begin{array}{l}{C}_{OFF}:Q_{SVG}=0\hfill \\ C_{ON}:-6\le Q_{SVG}\le +6MVAr\hfill \end{array}$ (69)

The technical performance vector is defined as

$\Gamma = \left[ V_{\min },P_{\text{loss}},Q_{\text{loss}},{\lambda }_{\max },\Delta P_{\text{margin}}\right]$ (70)

Figure 2. Revised assessment workflow with SVG at the industrial bus and λ = 0 at nominal load. Predictor-corrector and event settings require verification against the original implementation.

Table 5. Simulation cases.

The comparison between C1 and C3 quantifies the effect of reactive-power compensation on the maximum loading capability and voltage-stability margin (Table 5).

Continuation verification protocol. Use y = (x, λ) and a normalized tangent t satisfying [F_x F_λ]t = 0. Predict y_p = y_n + h × t_n, then correct F(x, λ) = 0 with the pseudo-arclength condition $t_{n^T}$ (y – y_p) = 0. Choose the tangent orientation continuously from the previous step. Reduce h and repeat a failed corrector; enlarge it only after well-converged steps, within documented bounds. Monitor the mismatch tolerance of 10⁻⁶ p.u. and the SVG reactive limits at every accepted point. Locate an SVG-limit crossing before continuing, fix Q_SVG at the crossed bound, release voltage regulation, rebuild the active equations and tangent, and continue with the updated PQ model. Refine the maximum λ by bracketing the sign change of the tangent component dλ/ds through the nose and reducing the step; compare the refined limit with σ_min(F_x). Solver nonconvergence alone is not a demonstrated fold. A limit-induced termination must be reported separately. The 0.95 p.u. voltage criterion is also separate from the fold: the nominal point already violates it in both reported configurations.

5.2. Network and Simulation Parameters

A common per-unit base is adopted at the PCC as

$S_B=100MVA,V_B=20kV$ (71)

The corresponding base impedance is

$Z_B=V_B{}^2/S_B={\left(20\times {10}^3\right)}^2/\left(100\times {10}^6\right)=4\Omega$ (72)

For the adopted short-circuit ratio SCR = 5, referenced to the 12 MW PV plant,

$S_{sc}=SCR·P_{PV,rated}=5\times 12=60MVA$ (73)

The equivalent grid impedance magnitude referred to the 20 kV PCC is

$\left|Z_{th}\right|=V_{PCC}{}^2/S_{sc}={\left(20\times {10}^3\right)}^2/\left(60\times {10}^6\right)=6.667\Omega$ (74)

With X_th/R_th = 10,

$R_{th}=\left|Z_{th}\right|/\surd \left(1+{10}^2\right)=0.663\Omega ,X_{th}=10R_{th}=6.634\Omega$ (75)

Table 6. Main simulation parameters.

On the 100 MVA, 20 kV base, the derived upstream equivalent is R_th ≈ 0.1658 p.u. and X_th ≈ 1.6584 p.u. Its magnitude is 1.6667 p.u. SCR and X/R constrain this aggregate equivalent but do not determine the downstream feeder, transformer, or shunt parameters (Table 6).

5.3. Performance Indices and Comparative Assessment

The uncompensated and SVG-compensated configurations are evaluated using the performance indices summarized in Table 7. Voltage regulation is assessed from the minimum bus voltage and maximum voltage deviation, while network efficiency is quantified from active and reactive losses. Voltage-stability improvement is evaluated from the critical loading factor and the corresponding loading margin.

Table 7. Performance indices.

The two compensation configurations must use identical network data, slack settings, industrial-load scaling, and PV injections. Only the industrial-bus SVG is enabled or disabled. Voltage-compliance margin and static fold margin are distinct indices; neither reported nominal solution meets V_min ≥ 0.95 p.u.

6. Results and Discussion

6.1. Base-Case Power-Flow Assessment

The base case is evaluated for P_PV = 12 MW, S_L = 22 MVA, pf_L = 0.85, and Q_SVG = 0. The corresponding industrial demand is P_L = 18.70 MW and Q_L = 11.59 MVAr. The reduced benchmark yields a minimum voltage of 0.617 p.u. at the industrial bus, indicating a severely stressed operating condition for the adopted SCR = 5, as shown in Figure 3.

Figure 3. Base-case bus-voltage profile without SVG compensation.

The active-power losses are 0.924 MW in the uncompensated base case. The industrial bus is therefore retained as the critical bus for the continuation analysis.

6.2. Uncompensated Voltage-Stability Assessment

The industrial demand is progressively increased at constant power factor. The uncompensated P-V trajectory is shown in Figure 4.

Figure 4. Industrial-bus P-V characteristic without SVG; λ = 0 denotes 22 MVA nominal demand.

The fold condition is reached at approximately

${\lambda }_{max}{}^0 \approx 0.023;P_{L,max}{}^0=18.70\left(1+0.023\right)\approx 19.13MW$

The reported incremental loading margin is approximately 0.43 MW, or 2.3% of nominal demand. The original 19.11 MW figure is inconsistent with the reported λ_max = 0.023; 19.13 MW follows from Equation (61). Final values should be calculated from the unrounded continuation export. The listed λ = 0.25 and 0.50 load cases lie beyond this reported uncompensated limit and must not be presented as feasible solutions on the same branch.

6.3. SVG Compensation Assessment

A 6 MVAr SVG is connected at the critical industrial bus while all other operating parameters remain unchanged. The compensated base-case voltage rises from 0.617 p.u. to 0.855 p.u., as shown in Figure 5.

The reported active-power losses decrease from 0.924 MW to 0.210 MW, while the nominal industrial voltage remains 0.855 p.u., below 0.95 p.u. The SVG nameplate rating does not specify its realized injection. These numerical comparisons remain conditional on verification of the network dataset and control settings.

Figure 5. Base-case voltage profiles with and without 6 MVAr SVG compensation.

6.4. Voltage-Stability Margin Enhancement

The continuation analysis is repeated with the 6 MVAr SVG in service. Figure 6 compares the uncompensated and compensated P-V trajectories.

Figure 6. Industrial-bus P-V comparison; the same nominal convention λ = 0 and industrial-bus SVG location apply to both curves.

The calculated stability boundary shifts from

${\lambda }_{\max }{}^0\approx 0.023to{\lambda }_{\max }{}^{SVG}\approx 0.316$

Using Equation (61) with the rounded loading parameters, the maximum industrial active demand increases from 19.13 MW to 24.61 MW. The reported fold margin therefore increases from approximately 0.43 MW to 5.91 MW. This static loadability improvement does not establish compliance with the nominal 0.95 p.u. voltage criterion.

6.5. Comparative Performance and Rating Implication

Table 8. Comparative performance for the reduced benchmark.

The reported comparison indicates improved voltage support and static loading margin with the initial 6 MVAr SVG rating. A final size cannot be validated until the numerical network and SVG settings are documented and the continuation results reproduced. For the reported compensated limit λ_max = 0.316, the λ = 0.25 screening point is within the indicated fold range, whereas λ = 0.50 is outside it; voltage compliance must still be checked independently (Table 8).

$Q_{SVG,rated} \in \left\{6,9,12\right\}MVAr$

The minimum rating satisfying the voltage criterion while preserving the largest practical loading margin should be retained as the final compensation size.

7. Conclusion

This study formulates a static voltage-stability assessment for a 12 MW PV plant and a nominal 22 MVA constant-power industrial load. With λ = 0 at nominal demand, the reported uncompensated and compensated limits correspond to incremental loading margins of 2.3% and 31.6%. A 6 MVAr SVG at the industrial bus is associated with reported minimum voltages of 0.617 and 0.855 p.u. without and with compensation, respectively; neither nominal point satisfies the 0.95 p.u. criterion. The associated maximum active demands are 19.13 and 24.61 MW when calculated from the reported rounded λ values. Complete network data, actual SVG settings and injections, and a continuation log are required before these numerical conclusions can be regarded as reproducible. The static constant-power model does not establish dynamic industrial-load performance.

Author Contributions

Rodolphe Gomba: Conceptualization, data curation, formal analysis, acquisition funding, survey, methodology, project administration, resources, supervision, validation, visualization and writing: original draft.

Eyandzi Amos Omboua: Software, supervision, validation, visualization, writing—revision and editing.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References