P-V and Q-V Curves: Voltage Stability Margins and Reactive Power Collapse Prevention

Mathematical-Physical Foundations of Voltage Stability and Saddle-Node Bifurcation

Voltage stability in electrical power systems is defined as the ability of a power network to maintain acceptable voltages at all buses in the system under normal operating conditions and after being subjected to a disturbance. Voltage collapse is a phenomenon that is fundamentally dynamic in nature, yet its maximum power transferability limit can be rigorously analyzed using steady-state power flow equations. Voltage collapse manifests when an increase in power demand (active or reactive) drives an uncontrollable drop in voltage magnitude at load buses, caused by the inability of the transmission network and generation sources to supply the required reactive power demand.

To formalize the mathematical behavior of voltage collapse, consider the canonical two-bus Thevenin equivalent model (an infinite bus connected to a load through a line impedance). Let the voltage at the generation bus be $V_1 \angle 0^\circ$ and the voltage at the load bus be $V_2 \angle \delta$. The transmission line impedance is given by $Z_L = R + jX = |Z_L| \angle \theta$, where $R$ is the resistance, $X$ is the inductive reactance, and $\theta = \arctan(X/R)$. The complex power consumed by the load is $S_2 = P_2 + jQ_2$.

The fundamental equations governing power flow at the load bus are expressed as:

Rearranging the trigonometric terms and eliminating the phase angle $\delta$ using the Pythagorean identity $\sin^2(\theta - \delta) + \cos^2(\theta - \delta) = 1$, we obtain the implicit quadratic equation governing the load voltage magnitude $V_2$:

This is a biquadratic equation with respect to the voltage magnitude $V_2$. Defining $y = V_2^2$, the solution for the voltage roots takes the form:

The existence of physically realizable solutions in the real domain for the voltage $V_2$ requires that the discriminant $\Delta$ of the quadratic equation be strictly greater than or equal to zero ($\Delta \ge 0$):

The point at which the discriminant vanishes exactly ($\Delta = 0$) defines the critical voltage stability limit, analytically known as the Saddle-Node Bifurcation (SNB) point or the power transferability limit ("Nose Point"). At this precise point, the two voltage solutions (the stable upper branch of high voltage and low current, and the unstable lower branch of low voltage and high current) merge and collapse into a single double root.

For a purely reactive line ($R = 0$, $|Z_L| = X$), the reduced load voltage equation under a power factor $\cos\phi$ (where $Q_2 = P_2 \tan\phi$) is:

Setting the discriminant to zero under this simplified condition yields the maximum allowable active power $Pmax$ before reaching voltage collapse:

From the perspective of non-linear dynamical systems theory, the differential-algebraic system of equations (DAE) representing the power grid can be expressed as:

where $x \in \mathbb{R}^n$ is the vector of dynamic state variables (generator rotor angles, voltages behind transient reactance, AVR regulator states, tap changer states), $y \in \mathbb{R}^m$ is the vector of algebraic network variables (bus voltage magnitudes $V$ and angles $\delta$), and $\lambda \in \mathbb{R}^k$ is the system loading parameter. The reduced Jacobian of the algebraic power flow system is given by:

At the saddle-node bifurcation point, the power flow Jacobian $\mathbf{J}_R$ becomes singular. That is:

The singularity of the Jacobian implies the presence of a zero eigenvalue ($\lambda_i = 0$). The right eigenvector (u_i and left eigenvector (v_i associated with this zero eigenvalue characterize the geometry of the collapse: the right eigenvector indicates the direction of the unstable variation of bus voltage states ($\Delta V$), revealing the most critical or vulnerable buses, whereas the left eigenvector describes the sensitivity of the loading margin with respect to variations in active and reactive power injection parameters.

Methodology for Construction and Forensic Analysis of P-V and Q-V Curves

Practical analysis of voltage stability in multi-bus systems requires generating characteristic static operating curves: the P-V curve (Active Power - Voltage) and the Q-V curve (Reactive Power - Voltage). Traditional analytical resolution using the Newton-Raphson algorithm systematically fails near the collapse point due to numerical ill-conditioning and the singularity of the Jacobian ($\det(\mathbf{J}_R) \to 0$).

Continuation Power Flow (CPF) for P-V Curve Construction

To overcome numerical divergence at the nose of the P-V curve, the Continuation Power Flow (CPF) technique is used. The method redefines the system of equations by introducing a generalized loading scale parameter $\lambda \ge 0$:

where $PDi0$ and $QDi0$ are the base loads at bus $i$, and $KLi$ is the load growth direction factor. The CPF algorithm consists of a two-step iterative scheme:

1. Predictor Step: The tangent vector is calculated using the derivative of the system with respect to the arc length $s$. The augmented equation takes the form:
   $$\begin{bmatrix} \frac{\partial g}{\partial \theta} & \frac{\partial g}{\partial V} & \frac{\partial g}{\partial \lambda} \\ & \mathbf{e}_k & \end{bmatrix} \begin{bmatrix} d\theta \\ dV \\ d\lambda \end{bmatrix} = \begin{bmatrix} 0 \\ \pm 1 \end{bmatrix}$$
   where $\mathbf{e}_k$ is a row vector with a '1' in the position corresponding to the continuation variable chosen to avoid singular conditioning.
2. Corrector Step: The modified system is solved applying transverse cuts or local arc-length parameterization, allowing the algorithm to smoothly trace around the "nose point" and calculate the lower unstable branch without suffering from mathematical divergence.

The resulting P-V curve plots the bus voltage trajectory as a function of active load increase. The upper branch corresponds to the stable operating region where $\frac{dV}{dP} < 0$ in controllable magnitude and the load equivalent impedance is greater than the system's Thevenin impedance ($|Zload| > |Zth|$). The critical point $(Pcrit, Vcrit)$ defines the absolute operational limit. The lower branch represents an unstable region where an increase in required active power would cause a dramatic reduction in voltage, resulting in an actual decrease in transferred power ($\frac{dP}{dV} > 0$).

Q-V Curve Methodology and Reactive Margin Determination

Unlike P-V curves, Q-V curves evaluate system reactivity under a fixed active power condition. To construct a Q-V curve at a specific test bus (bus $k$), that bus is treated as a fictitious PV generation bus (without reactive power limits) with a scheduled voltage $V_k$. The target value of $V_k$ is systematically varied, and the power flow is solved to record the required reactive power injection $Q_k$ from the fictitious source into the grid.

The representative equation of the Q-V curve is given by the profile $Q_k = f(V_k)$. Key properties of the Q-V plot are:

• Zero Injection Point ($Q_k = 0$): Represents the actual current operating point of the system under real conditions without additional external compensation.
• Q-V Curve Minimum ($\frac{dQ_k}{dV_k} = 0$): Represents the voltage stability limit for that bus. If the curve lies entirely above the horizontal axis ($Q_k > 0$ for all $V_k$), the system is physically incapable of operating under that condition without immediate external reactive support.
• Reactive Power Margin (RPM): Defined as the distance from the current operating point ($Q_k = 0$) to the lowest point of the curve ($Qmin$):
  $$RPM = |Qmin| \quad if Qmin < 0$$
  A value of $Qmin < 0$ indicates the reactive power reserve in Mvar that the bus can support before falling into voltage collapse. If $Qmin > 0$, it represents the strict reactive power deficit that must be forcibly injected to achieve system convergence.
• Sensitivity $dQ/dV$: The slope of the curve in the upper segment describes network stiffness. A very steep slope $\frac{dQ}{dV}$ represents a "weak" node with a low Short Circuit Ratio (low SCR), highly vulnerable to load variations.

Interaction Dynamics and Collapse Mechanisms Driven by Reactive Power Deficit

Although P-V and Q-V analyses provide static boundaries, voltage collapse is a continuous evolutionary process driven by the interaction between slow and fast dynamic components of the power system. A reactive power deficit triggers a sequence of positive feedback phenomena detailed below.

Load Restoration Action and On-Load Tap Changers (OLTC)

Following a low-voltage event caused by a contingency (for example, the outage of a critical transmission line), the power consumed by loads drops temporarily if they exhibit voltage-dependent characteristics. However, on a time scale ranging from tens of seconds to minutes, two main mechanisms restore power demand, aggravating the reactive deficit:

1. OLTC (On-Load Tap Changer) Dynamics: Substation transformers equipped with automatic voltage regulators seek to maintain constant voltage on the secondary winding (distribution side). If the primary voltage drops to $V_p$, the OLTC increases the turns ratio $a = N_p / N_s$ by reducing primary turns to raise $V_s$. The secondary current increases to $I_s = Pload / V_s$, which translates into a reflected primary current:
   $$I_p = \frac{I_s}{a}$$
   By attempting to restore the secondary voltage to $1.0 p.u.$, the active and reactive power demand on the secondary side is completely restored to its pre-fault value. This drastically increases the line current in the primary transmission grid, multiplying reactive power losses across the lines by the term $I_p^2 X_L$. If the primary transmission network is near the nose point of the P-V curve, the OLTC action drives the system past the bifurcation point, provoking collapse (an effect known as "reverse tap action" or tap hunting).
2. Dynamic Load Models and Induction Motors: Industrial loads composed predominantly of induction motors demand nearly constant active power determined by the mechanical torque of the driven load ($T_m$). The electrical power developed by an induction motor is proportional to $V^2$:
   $$P_e = \frac{V^2 R_r' / s}{(R_s + R_r'/s)^2 + (X_s + X_r')^2}$$
   When the voltage decreases, the motor slip $s$ increases to maintain electromagnetic torque $T_e = T_m$. As $s$ increases, the equivalent impedance of the motor drops drastically, and the reactive power consumed by the stator and rotor surges proportionally to:
   $$Qmotor = I_m^2 (X_s + X_r')$$
   If the voltage drops below the critical stalling threshold ($Vstall \approx 0.7 - 0.8 p.u.$), the motor decelerates to a complete stop (stalling). In this stalled condition, the motor acts essentially as a short-circuit reactance with a Locked Rotor Current (LRA) 5 to 8 times nominal current and an extremely low power factor ($\cos\phi \approx 0.2 - 0.4$), consuming massive amounts of reactive power and rendering voltage recovery impossible without rapid Under-Voltage Load Shedding (UVLS).

Generator Overexcitation Limiters (OEL)

Synchronous generators are the primary and most flexible source of voltage control. They operate under Automatic Voltage Regulator (AVR) control, adjusting field current $i_f$ to maintain terminal voltage at the generation bus (PV bus). However, the overexcitation capability of the rotor field winding is thermally limited by IEEE C50.13 standards.

During a severe reactive power deficit in the network, the AVR attempts to correct the voltage drop by increasing field current $i_f$ to its maximum ceiling value. After a time delay defined by the inverse-time curve of the Overexcitation Limiter (OEL), the OEL takes control from the AVR and forcibly ramps down the field current to its maximum continuous rated limit ($i_{f,max\_cont}$).

Analytically, this instantaneously converts the generator bus in the power flow model from a PV bus (fixed voltage, unconstrained reactive power) to a PQ bus (fixed reactive power at the upper limit $Qmax$). This abrupt change in mathematical topology removes the voltage support node, drastically and instantaneously collapsing the stability margin on the P-V and Q-V curves for remaining nearby substations, accelerating the transition toward saddle-node bifurcation and final system collapse.

Comparative Table: Stability Parameters, Regulatory Limits, and Asset Impact

Parameter / Variable	Standard / Reference Norm	Critical Limit OEL / Stability	Dielectric / Operational Failure Mode	Infrastructure Consequence
P-V Load Margin ($Smargin$)	NERC TPL-001-4 / WECC Voltage Stability Criteria	$\ge 5\%$ for $N-1$ contingency; $\ge 2.5\%$ for $N-2$	Operation on the lower unstable branch of the P-V curve ($dV/dP > 0$)	Cascading distance protection tripping due to impedance encroachment (Zone 3 Load Encroachment).
Q-V Reactive Margin ($Qmargin$)	IEEE Std 1557 / CIGRE WG 38.02	$Qmargin > 0$ Mvar; Minimum $5\% - 10\%$ of bus load Mvar	$\frac{dQ}{dV} = 0$ (Bottom inflection point on Q-V curve)	Generation tripping due to undervoltage and dynamic induction motor stalling.
Rotor Excitation Limit (OEL)	IEEE C50.13 / IEEE Std 421.2	Time-current curve: $150\%$ rated current for 30s, $110\%$ continuous	Bus switching from PV to PQ due to OEL intervention	Instantaneous voltage collapse of the transmission system and loss of synchronism.
Continuous Substation Voltage	IEC 60038 / ANSI C84.1 Range B	$Vmin = 0.90 p.u.$ (Continuous); $Vcrit = 0.80 p.u.$ (Transient)	Overcurrent heating in constant power loads	Accelerated thermal degradation of insulation in cables and transformers (Arrhenius Law).
Motor Stalling Voltage	NEMA MG-1 / IEEE Std 141	$Vstall \le 0.70 - 0.75 p.u.$ sustained for $> 300 ms$	Locked-rotor trip and massive Mvar absorption ($\cos\phi < 0.3$)	Destructive overheating of inductive stators and massive tripping of low-voltage breakers.
OLTC Regulation Range	IEC 60076-10 / IEEE C57.12.00	Typical range $\pm 10\%$ to $\pm 15\%$ in steps of $1.25\%$	Reverse tap action (Tap hunting upon primary reactive exhaustion)	Premature wear of tap selector contacts and oil arcing/inflammation.

Forensic Failure Analysis and Severe Impact on Power Infrastructure

Voltage collapse is not merely an abstract power flow phenomenon; it has devastating and destructive physical repercussions on high-, medium-, and low-voltage assets. The systematic degradation of dielectric and thermal parameters during a voltage crisis is analyzed below from a forensic engineering perspective.

Power Transformers

During a prolonged low primary voltage condition caused by a reactive deficit, the current flowing through transformer windings increases considerably if transmitted power is maintained constant ($I = S / (\sqrt{3} V)$). This current increase triggers two main failure vectors:

• Thermal Stress and Cellulosic Degradation: Joule losses in the windings scale quadratically with current ($Pcu = I^2 R$). Winding Hot-Spot Temperature escalates per IEC 60076-7. Exceeding $140^\circ C$ induces accelerated depolymerization of Kraft insulating paper and potential gas bubble formation in dielectric oil (bubbling phenomenon), drastically reducing the fluid's dielectric strength and potentially causing catastrophic inter-turn internal arcing.
• On-Load Tap Changer (OLTC) Mechanical Wear: In a desperate attempt to restore secondary voltage, the motor-driven OLTC mechanism performs continuous tap changing cycles (hunting). Switching elevated currents under depressed primary voltage conditions produces longer-duration switching arcs in the arcing chamber, eroding tungsten-copper contacts and contaminating dielectric oil with free carbon particles, which lowers the chamber's dielectric breakdown voltage.

High- and Medium-Voltage Insulated Cables (XLPE)

Underground transmission and distribution cables insulated with Cross-Linked Polyethylene (XLPE) are severely compromised by voltage collapse due to sustained overcurrent:

• Thermal Overload and Mechanical Expansion: Operating under severe undervoltages ($V < 0.85 p.u.$), load current vastly exceeds continuous current-carrying capacity (ampacity) calculated per IEC 60287. Copper/aluminum conductor temperature can surpass XLPE emergency limits ($105^\circ C$) or short-circuit limits ($250^\circ C$).
• Thermo-Mechanical Degradation and Electrical Treeing: Severe thermal gradients between the conductor and outer metallic shield induce mechanical expansion cycles that deform semi-conducting screens. This structural distortion creates electric field stress risers that accelerate the growth of electrical trees through the XLPE dielectric, culminating in phase-to-ground dielectric puncture.

Switchgear and Circuit Breakers

Switchgear experiences critical operational anomalies when operating outside rated voltage envelopes:

• Overcurrent Interruption Difficulty: SF6 or vacuum circuit breakers are designed to interrupt rated currents at assigned system voltages. Although current during voltage collapse does not reach three-phase short-circuit magnitudes, the voltage-current phase angle approaches near $90^\circ$ purely inductive as the system exhausts reactive reserves. This generates a high Transient Recovery Voltage (TRV) with an extreme rate of rise ($dV/dt$) following current zero, elevating the risk of arc re-strike in the interrupter chamber.
• Control System Failures and Undervoltage Incidents: Trip coils and spring-charging motors for breaker operating mechanisms are often supplied from station service transformers or battery banks suffering coupled undervoltage. If control bus voltage drops below the operational threshold of IEC 62271-1 ($< 85\%$ rated DC/AC voltage), breakers may become inoperable precisely when the system demands emergency load shedding.

Protection Relays and Protection Schemes

Degraded voltage profiles misorient grid protection functions, triggering sympathetic tripping or failure to operate:

• Distance Relay Load Encroachment (Zone 3): Apparent impedance seen by a distance relay at a transmission bus is defined as:
  $$Zapp = \frac{Vphase}{Iline}$$
  During a voltage collapse sequence, $Vphase$ decreases while $Iline$ increases due to constant active power transfer attempts. Consequently, apparent impedance magnitude $|Zapp|$ drops drastically, and its angle $\phi = \arctan(Q/P)$ shifts toward highly inductive values. This impedance trajectory penetrates the operating characteristics (Mho or Quadrilateral) of extended distance zones (Zone 3 or Zone 2), causing false trips of healthy transmission lines. Cascading outages of these transmission lines further reduce grid capacity and accelerate catastrophic interconnected system collapse.
• Incorrect Directional Element Operation: Under extremely low voltages ($V < 0.1 p.u.$), polarising voltage magnitude in directional overcurrent relays becomes insufficient to reliably determine fault direction, leading to improper blocking or tripping in unintended directions.

Advanced Mitigation Strategies and Resilient Design

To ensure that an electrical power system operates with adequate voltage stability margins and to prevent collapse from reactive deficits, advanced engineering methodologies combining dynamic compensation equipment installation with automated protection schemes are applied.

Static and Dynamic Reactive Power Compensation Devices

Local reactive power injection is the primary method to elevate P-V and Q-V curves, shifting the saddle-node bifurcation point to higher power levels and enhancing bus stiffness. Compensation systems are categorized by response speed and V-I characteristics:

• Shunt Capacitor Banks (Fixed or Mechanically Switched Capacitors - MSC): Inject reactive power given by $Qcap = V^2 \omega C$. Their major mathematical limitation is that injected reactive power decreases quadratically with voltage drop ($Q \propto V^2$). Precisely when the system requires reactive power most (during severe voltage depression), static capacitors drastically lose effectiveness.
• Static Var Compensators (SVC): Combine Thyristor-Switched Capacitors (TSC) and Thyristor-Controlled Reactors (TCR). Their response speed is fast (1-2 cycles), but maximum reactive injection limit remains coupled to the square of nominal bus voltage.
• Static Synchronous Compensators (STATCOM): Based on Voltage Source Converter (VSC) technology with IGBT/IGCT devices. A STATCOM operates as a synthetic voltage source $Vout \angle 0^\circ$ behind a coupling reactance $X_k$. Exchanged reactive power is:
  $$QSTATCOM = \frac{Vbus (Vout - Vbus)}{X_k}$$
  Unlike an SVC, a STATCOM maintains its maximum rated output reactive current constantly even when grid voltage collapses to extremely low levels ($Iinject = Imax$, implying a linear injection relationship $Q \propto Vbus$). This provides superior support at the P-V curve "nose point".
• Synchronous Condensers: Rotating synchronous machines without mechanical load. They provide instantaneous dynamic reactive injection governed by air-gap flux physics (natural subtransient response) and high physical inertia ($H$), alongside substantially raising bus Short Circuit Ratio (SCR), making the bus more resilient to voltage variations.

Mathematical Formulation for Sizing Reactive Compensation

To restore target operational voltage $Vtarget$ at a critical load bus $k$ subjected to demand $P_L + jQ_L$, the required compensation reactive power $Qcomp$ is derived by solving for the injection variable in the Thevenin equivalent equation ($Vth, Xth$):

To determine optimal placement of these devices in large-scale networks, Modal Analysis of the Reduced Jacobian is utilized. Starting from the linearized power flow relationship with $\Delta P = 0$:

Eigenvalue and eigenvector decomposition of $\mathbf{J}_{R}$ yields:

where $\boldsymbol{\Lambda}$ is the diagonal matrix of eigenvalues $\lambda_i$, $\boldsymbol{\Xi}$ is the right eigenvector matrix, and $\boldsymbol{\Psi}$ is the left eigenvector matrix. Modal voltage variation is expressed as:

The smallest eigenvalue $\lambda_{min}$ identifies the critical voltage collapse mode. The Participation Factor of Bus $k$ ($PF_k$) in critical mode $i$ is defined as:

Buses exhibiting the highest participation factors $PF_k$ are rigorously the optimal system locations to install dynamic compensation (STATCOM/SVC), given that they exert the greatest impact on increasing the minimum eigenvalue $\lambda_{min}$ and, consequently, maximizing stability margins on P-V and Q-V curves.

Design of Under-Voltage Load Shedding (UVLS) Schemes

When dynamic reactive reserves are exhausted and the system crosses the operational stability limit, automatic Under-Voltage Load Shedding (UVLS) serves as the final line of defense against total grid collapse.

A resilient UVLS scheme must be engineered using adaptive algorithms based not only on instantaneous voltage magnitude, but also on the rate of change of voltage ($dV/dt$). Active and reactive power shed volume at step $m$ ($\Delta P_{UVLS,m}$) is calculated according to:

where $\alpha_m$ and $\beta_m$ are tuning constants adjusted via dynamic system simulations, and $\Omega_{bus}$ represents the set of buses pre-selected using participation eigenvector analysis. Active power shedding $P$ indirectly eliminates associated load reactive power ($Q = P \tan\phi$), immediately alleviating transmission voltage drop and forcing the operating point back onto the stable upper branch of the P-V curve.

Integrated Simulation and Application in Vexten Suite

Practical implementation of voltage stability analysis and reactive collapse mitigation is executed in an automated, integrated manner within the Vexten Suite ecosystem, using coordinated modules Vexten PowerFlow, Vexten Dynamic Stability Engine, and Vexten CableSizer.

Simulation and Analysis Workflow in Vexten Suite

1. Topological Model Import and Scenario Definition: The interconnected network model is imported in CIM (Common Information Model) or IEEE format. The Vexten PowerFlow engine solves base steady-state conditions and performs automated Short Circuit Ratio (SCR) evaluations across all network buses per IEC 60909 / IEEE 141, automatically highlighting weak nodes ($SCR < 2.5$).
2. Execution of Continuation Power Flow (CPF): Through the Vexten Dynamic Stability Engine module, the user configures a radial or area load growth vector ($\mathbf{K}_L$). The arc-length parameterized CPF algorithm executes active power sweeps, analytically plotting P-V curves for all network substations and identifying exact saddle-node bifurcation points ($Pmax, Vcrit$).
3. Automated Q-V Curve Generation and Reactive Reserve Diagnostics: For selected buses with low P-V margins, the software automatically executes target voltage sweep routines ($V_k$ varying from $1.15 p.u.$ to $0.50 p.u.$ in steps of $0.005 p.u.$). The algorithm plots Q-V curves and automatically extracts:
   • Reactive Power Margin ($Qmargin = |Qmin|$).
   • Nodal critical voltage ($Vcrit$).
   • Nodal sensitivity matrix $\frac{dQ}{dV}$.
4. Dynamic Compensation Sizing and Optimization: If the reactive power margin is below regulatory thresholds (e.g., NERC TPL-001-4), Vexten Suite runs an integrated multi-objective Particle Swarm Optimization (PSO) routine. The system uses modal participation factors derived from the inverted Jacobian ($\mathbf{J}_R^{-1}$) to recommend Mvar rating and optimal compensation technology (STATCOM vs. SVC vs. MSC Banks).
5. Forensic Asset Impact Calculation and Cable Derating: Using dynamic undervoltage and overcurrent trajectories obtained from collapse runs, the Vexten CableSizer module processes thermal equations per IEC 60287. The system calculates the thermal derating factor of underground cable circuits exposed to continuous undervoltage, estimating XLPE insulation life consumption using the Arrhenius degradation model.

Practical Code Example using Vexten Suite API (Python Interface)

The following script demonstrates implementation via the Vexten Suite API to automate P-V bifurcation point extraction, Q-V curve construction, and STATCOM sizing for collapse mitigation:

import vexten.suite as vxt
import numpy as np

# 1. Initialize project and load network model
app = vxt.PowerSystemEngine()
app.load_project("Industrial_Grid_Substation_500kV.vxt")

# 2. Configure Continuation Power Flow (CPF) Module
cpf = app.get_module("ContinuationPowerFlow")
cpf.set_parameter("StepSize", 0.02)
cpf.set_parameter("ParameterizedMethod", "ArcLength")
cpf.set_target_buses(["BUS_CRITICAL_138KV"])
cpf.set_load_increase_vector(direction="Uniform_Area_Growth")

# Execute CPF to plot P-V curve
pv_results = cpf.execute()
critical_power = pv_results.get_nose_point_power("BUS_CRITICAL_138KV")
critical_voltage = pv_results.get_nose_point_voltage("BUS_CRITICAL_138KV")

print(f"[P-V ANALYSIS] Nose Point Detected at P: {critical_power:.2f} MW, V: {critical_voltage:.3f} p.u.")

# 3. Configure and Execute Q-V Sweep at Critical Bus
qv = app.get_module("QVCurveAnalyzer")
qv.set_target_bus("BUS_CRITICAL_138KV")
qv.set_voltage_range(v_min=0.50, v_max=1.15, v_step=0.005)

qv_results = qv.execute()
q_margin = qv_results.get_reactive_margin()
v_at_q_min = qv_results.get_critical_voltage()

print(f"[Q-V ANALYSIS] Reactive Margin (RPM): {q_margin:.2f} Mvar at V = {v_at_q_min:.3f} p.u.")

# 4. Critical Condition Evaluation and STATCOM Sizing
MIN_REQUIRED_MARGIN_MVAR = 45.0 # Regulatory requirement

if q_margin < MIN_REQUIRED_MARGIN_MVAR:
    required_statcom_mvar = MIN_REQUIRED_MARGIN_MVAR - q_margin
    print(f"[MITIGATION] Deficit Detected! Sizing STATCOM at BUS_CRITICAL_138KV: {required_statcom_mvar:.2f} Mvar")
    
    # Insert dynamic STATCOM into Vexten model
    statcom = app.network.add_device("STATCOM", bus="BUS_CRITICAL_138KV")
    statcom.set_rating(q_rating_mvar=required_statcom_mvar)
    statcom.set_control_mode("Voltage_Control", target_v=1.00)
    
    # Re-evaluate P-V curve with integrated Vexten compensation
    pv_results_compensated = cpf.execute()
    new_p_max = pv_results_compensated.get_nose_point_power("BUS_CRITICAL_138KV")
    print(f"[VERIFICATION] New P-V Nose Point with STATCOM: {new_p_max:.2f} MW (Margin Increase: {new_p_max - critical_power:.2f} MW)")

# 5. Export cable thermal stress report per IEC 60287
cable_module = app.get_module("CableThermalSizer")
cable_module.evaluate_overcurrent_degradation(scenario="Voltage_Collapse_Transients")
cable_module.export_report("XLPE_Degradation_Report_Vexten.pdf")

Through this integrated simulation methodology in Vexten Suite, power systems engineers can predict proximity to voltage collapse with mathematical precision, quantify dynamic reactive power reserves in strict compliance with international standards (IEEE, IEC, NERC), and design highly resilient compensation systems that safeguard the physical integrity of critical high-voltage assets.