⚡ electrical engineering

VSC-HVDC setpoint adjustment for maximum grid utilisation under voltage constraints

This paper proposes an analytical, closed-form method for adjusting VSC-HVDC active power setpoints based on local measurements and wide-angle estimates to maximize grid loadability under voltage constraints without requiring global optimization.

Original authors: Gabriel Malmer, Emil Hillberg, Olof Samuelsson

Published 2026-07-16

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CC BY 4.0

Original authors: Gabriel Malmer, Emil Hillberg, Olof Samuelsson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the electrical grid as a giant, invisible highway system where electricity is the traffic. Just like cars on a road, electricity needs space to flow. If too many cars try to squeeze onto a single lane at once, you get a traffic jam. In the world of power, this is called a "thermal overload," and it's bad because wires can get too hot. But there's a second, sneakier kind of traffic jam: a "voltage collapse." Think of this not as a blocked road, but as a hill that gets so steep the cars simply run out of gas and stop moving. If the voltage (the electrical "pressure") drops too low, the whole system can shut down, causing a blackout.

To keep this grid moving, engineers use special bridges called VSC-HVDC links. These aren't just simple wires; they are smart, controllable tunnels that can decide exactly how much electricity to send and in what direction. They are like the ultimate traffic cops, able to speed up or slow down the flow instantly. However, these bridges have their own rules. They can't push too hard (a current limit) or they might break, and they can't handle too much pressure (a voltage limit) or they might get confused. The big question for scientists is: When the grid is under heavy stress and about to collapse, how should these smart bridges adjust their settings to save the day? Should they push harder, or should they back off?

This paper tackles that exact puzzle. The authors, Gabriel Malmer, Emil Hillberg, and Olof Samuelsson, developed a clever mathematical trick to figure out the perfect setting for these bridges during an emergency. They didn't just guess; they used geometry to draw a map of the bridge's limits. They found that by slightly reducing the amount of active power (the main "push" of electricity) the bridge sends, it can actually free up enough "reactive power" (the supporting "pressure") to keep the voltage stable. This allows the grid to carry more total load than if the bridge tried to push its maximum power.

The team tested this idea on a computer simulation of the Nordic Test System, a standard model used to study how power grids behave in Sweden and surrounding areas. They simulated a scenario where the demand for electricity was slowly ramping up, like a crowd of people trying to enter a stadium all at once. They tried different settings for the bridge's power output. The results showed that if the bridge kept its power at the maximum (700 MW), the grid collapsed quickly. However, if they dialed the power down to a specific sweet spot—around 550 MW—the grid could handle significantly more electricity before failing. In fact, reducing the power by 100 MW allowed the system to support an extra 150 MW of load.

The paper proposes a new rule for emergency control systems. Instead of needing a supercomputer to calculate the perfect setting for the whole country in real-time, this new method uses a simple formula. It only needs two pieces of information: the local voltage at the bridge and an estimate of the angle difference between the local grid and a distant, stable part of the network. The authors found that even if the estimate of that angle isn't perfect, the method is robust enough to work. They suggest this could be used in a System Integrity Protection Scheme (SIPS), which is like an automatic emergency brake for the grid. When the system senses it's getting shaky, this rule would instantly tell the bridge to adjust its power setpoint to the optimal level, preventing a blackout without needing complex, slow calculations.

The study confirms that this analytical approach works. In their simulations, the mathematically predicted "sweet spot" for the power setting matched exactly with the setting that produced the highest loadability in the simulation. The authors note that this works best when the grid is under stress, specifically when the voltage angle difference is between 50 and 60 degrees. They also point out that while the method is very promising, it relies on accurate measurements of voltage and angle, and future work will need to test how well it works in a real, live grid with all the messy, unpredictable variables of the real world. For now, the math holds up, suggesting that sometimes, to save the most power, you have to be willing to let go of a little bit of it.

Technical Summary: VSC-HVDC Setpoint Adjustment for Maximum Grid Utilisation Under Voltage Constraints

Problem Statement

As transmission capacity demands grow and the share of converter-interfaced resources increases, efficient grid utilization while maintaining secure operation is imperative. While Voltage-Source Converter High-Voltage Direct Current (VSC-HVDC) links offer controllable active and reactive power, existing congestion management strategies often focus primarily on thermal overload mitigation, neglecting voltage constraints. Furthermore, in emergency controls addressing voltage instability, methods often fail to explicitly balance the trade-off between active and reactive power support within the converter's capability limits. When voltage collapse approaches, resources are often depleted or not fully utilized because the optimal operating point—maximizing loadability under combined current and voltage constraints—has not been established as a general setpoint rule.

Methodology

The paper proposes an analytical method to determine the optimal active power setpoint ( $P^*_g$ ) for a VSC-HVDC station to maximize system loadability during voltage-stressed conditions.

System Modelling:
The VSC-HVDC converter station is modeled at the system bus ( $U_s$ ) using geometric representations in the $P-Q$ plane. The feasible operating region is defined as the intersection of two circular limits:

Current Limit: A circle defined by the maximum converter current ( $I_{cmax}$ ).
Voltage Limit: A circle defined by the maximum converter voltage ( $U_{cmax}$ ), representing the PWM saturation limit.

The centers and radii of these circles are functions of the system bus voltage magnitude ( $U_s$ ) and the converter's impedance parameters (phase reactor, filter, and transformer).

Derivation of the Optimal Setpoint:
Building on prior work by Johansson et al. regarding local generators, the authors demonstrate that the optimality condition for maximizing load is translation invariant. For a circular capability limit with center $(P_0, Q_0)$ , the optimal operating point satisfies:
$\frac{Q_g - Q_0}{P_g - P_0} = \frac{1}{\tan \delta}$
where $\delta$ is the voltage angle difference between the local bus and a remote, voltage-stable reference bus.

By analyzing the intersection of the current and voltage limit circles, the authors derive a closed-form, piecewise expression for the optimal active power setpoint $P^*_g(\delta)$ :

Voltage-Limited Regime ( $\delta < \delta_v$ ): The optimum is constrained by the voltage limit circle.
Intersection Regime ( $\delta_v \le \delta < \delta_i$ ): The optimum lies at the intersection of the two limits, independent of $\delta$ .
Current-Limited Regime ( $\delta \ge \delta_i$ ): The optimum is constrained by the current limit circle.

The transition angles $\delta_v$ and $\delta_i$ are calculated analytically based on the circle parameters.

Implementation Framework:
The method is designed for integration into a System Integrity Protection Scheme (SIPS) or Emergency Power Control (EPC). It requires only:

Local measurement of the system bus voltage magnitude ( $U_s$ ).
An estimate of the wide-area voltage angle difference ( $\delta$ ), obtainable via Phasor Measurement Units (PMUs) or state estimators.

The control logic updates the setpoint in real-time without requiring global optimization or a full system-wide model.

Key Contributions

Generalization of Control Laws: The paper establishes that the control law relating active and reactive power to the angle difference (originally derived for local generators) is translation invariant and applies to any circular capability limit, including VSC-HVDC current and voltage limits, accounting for phase reactors, filters, and transformers.
Closed-Form Optimal Setpoint: A closed-form expression for the optimal active power setpoint ( $P^*_g$ ) under combined current and voltage constraints is derived. This expression explicitly accounts for the trade-off between active and reactive power support.
Low-Complexity EPC Strategy: The proposed approach provides a setpoint rule suitable for fast emergency control. Unlike Optimal Power Flow (OPF) based redispatch methods, it avoids the need for system-wide models and online optimization, relying instead on local measurements and a single angle estimate.

Results and Validation

The proposed method was validated using dynamic phasor (RMS) simulations on the Nordic Test System, a standard benchmark for voltage stability. A 700 MVA VSC-HVDC link was connected to the Central area of the system.

Loadability Improvement: Simulations compared various fixed HVDC active power setpoints ( $P^{ref}_g$ ) ranging from 400 MW to 700 MW. Results showed that reducing the setpoint from the maximum (700 MW) to an intermediate value (550 MW) increased the maximum loadability ( $P_d$ ) by approximately 150 MW.
Consistency with Theory: The setpoint yielding the highest loadability in the simulations (550 MW) fell within the analytically predicted optimal interval (510–610 MW) for the observed operating conditions (angle difference $\delta \approx 50\text{--}60^\circ$ and $U_s \approx 0.95\text{--}1.0$ p.u.).
Robustness: Sensitivity analysis indicated that the method is robust to uncertainties in the angle estimate ( $\delta$ ). In the current-limited regime (typical near collapse), the derivative of the optimal setpoint with respect to $\delta$ is less than 1, partially mitigating error propagation. Furthermore, in the intersection regime, the optimal setpoint is independent of $\delta$ .
Non-Proportional Gains: The study confirmed that adjustments to the active power setpoint can yield a more-than-proportional increase in loadability, particularly when the system is operating near the current limit.

Significance and Claims

The paper claims that the derived analytical method provides a practical, low-computational-burden solution for maximizing grid loadability during voltage-stressed emergencies. By exploiting the geometric structure of converter limits, the method enables VSC-HVDC stations to dynamically adjust their active power setpoints to preserve reactive power margins, thereby delaying or preventing voltage collapse.

The authors emphasize that this approach is distinct from traditional OPF-based methods due to its reliance on local data and lack of global optimization requirements, making it suitable for real-time implementation in SIPS. The validation on the Nordic Test System confirms that the analytically predicted optimum aligns with the setpoint that yields the highest loadability in a complex, meshed system. The paper concludes that while closed-loop implementation and coordination with other remedial actions (like load shedding) warrant further investigation, the derived setpoint rule offers a robust foundation for emergency voltage support.