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

arXiv:2607.13889 · eess.SY, cs.SY, math.OC · Submitted 2026-07-15 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa, and with me are Dev and Taro, guest researcher.

Dev: Today's paper: "VSC-HVDC setpoint adjustment for maximum grid utilisation under voltage constraints".

Rosa: Voltage-source converter high-voltage direct current (VSC-HVDC) links offer controllable active and reactive power output, making them a promising asset for emergency voltage support.

Dev: First, who's behind it and why it matters.

Paper summary: Rosa: So, Dev, I'm really interested in this paper titled "VSC-HVDC setpoint adjustment for maximum grid utilisation under voltage constraints." The core idea seems to be about using a VSC-HVDC system to actively support the grid when voltage is stressed by finding the best way to control its active power output. What specifically does the paper claim they managed to do?

Dev: It claims they've developed an analytical method to adjust the active power setpoint of a VSC-HVDC station specifically for maximizing loadability when there are both current and voltage limits acting on it, which is pretty important for emergency support. The authors derive a closed-form expression for this optimal setpoint by looking at the geometric constraints imposed by those limits in the P-Q plane.

Taro: Maximizing loadability during voltage-stressed conditions sounds like it has big implications when we think about system resilience; it suggests a way to proactively manage power injection to keep the grid stable when things get shaky. I wonder how practical this is outside of a controlled lab setting, Rosa? Can we actually implement this kind of real-time adjustment in the messy reality of an operational power grid?

Rosa: That's exactly what I want to know, Taro; if it works outside the lab for long enough to be useful for emergency control, that would be huge. Dev, from your control engineering standpoint, how does the paper handle those constraints—the current limit and voltage limit—when you're trying to figure out this optimal setpoint?

Dev: The paper sets up the system limits geometrically as the intersection of two circles in the P-Q plane, where those circles define the feasible region based on parameters like admittances and maximum converter current or voltage magnitudes. They then use an optimality condition from previous work, which states that for any circular limit with an arbitrary center, there's a relationship between P g, Q g, and the center point (P zero Q zero) and the angle delta <ref:2607.13889#pg1,for any circular limit with>.

Taro: So it's not just about hitting a target power level; it’s about finding a specific location within the feasible region defined by those limits that gives the best loadability, based on how far you are from some reference point. That sounds like a sophisticated way to handle complex operational boundaries. What kind of relationship is that with the voltage angle difference delta mentioned in their equations?

Paper summary: Rosa: The paper defines transition angles delta i and delta v by looking at the intersection points of the current and voltage limits, which then dictates whether the optimal active power setpoint lies on the voltage limit, current limit, or right at that intersection. It sounds like a piecewise function based on those angles.

Dev: Exactly; they get this piecewise function for P* g in equation six which tells you exactly which constraint is active depending on the measured angle delta <ref:2607.13889#pg1>. This whole thing is framed within an EPC scheme called SIPS, which is triggered when we see voltage-stressed conditions like heavily loaded points without a long-term stable equilibrium.

Taro: The SIPS framework seems designed for precisely those moments when the grid starts showing signs of instability, using local voltage measurements U s and wide-area angle differences delta to feed into that setpoint calculation. When the world misbehaves, this paper offers a specific rule derived from these constraints to guide the control action.

Rosa: It’s interesting how they manage to derive this without needing a global optimization solver running constantly; it relies only on local voltage measurements and an estimate of delta from PMUs or state estimators. That makes it much more suitable for real-time emergency control than something that requires massive computational power across the whole system.

Dev: The implementation involves a five-step process: first, measurement and estimation of U s and delta; second, updating the limits based on those measurements to get delta i and delta v; third, computing the optimal setpoint using that piecewise rule; fourth, adjusting the HVDC outer-loop current reference i ref d; and finally, constraining the reactive power Q g by picking the most restrictive limit.

Taro: That reactive power coordination step sounds crucial for safety because it prioritizes the active power setpoint first and then ensures we don't violate any physical limits on reactive injection. If we can reliably use this to manage loadability during a contingency, that’s a tangible benefit for grid operators when things go south.

Rosa: Speaking of benefits, the validation using dynamic phasor simulations on the Nordic Test System showed that reducing the active power setpoint P ref g by one hundred MW could increase loadability P d by about one hundred fifty MW, which is a significant gain. The highest loadability was found for a specific setpoint of five hundred fifty MW, or approximately zero point eight p.u., when the angle delta was in the range of fifty to sixty degrees.

Paper summary: Dev: That simulation result gives us some concrete numbers to look at, Rosa; it shows a measurable improvement in loadability tied directly to this setpoint adjustment mechanism derived from their paper, "VSC-HVDC setpoint adjustment for maximum grid utilisation under voltage constraints." The sensitivity analysis also suggested that uncertainties in delta only propagate weakly to the loadability P d when it's needed, which speaks to its practical robustness.

Taro: That weakness in sensitivity is important because it means we don't need perfect, instantaneous knowledge of the wide-area angle difference for the control loop to still perform well during a voltage stress event; it suggests a good degree of operational tolerance. What are the actual limitations they flag regarding its accuracy?

Rosa: The authors point out that the accuracy of this method really depends on having good local measurements for U s and getting a reasonable estimate for delta, which is the main input. Also, they noted that the VSC-HVDC voltage limit, specifically equation (two), is much more sensitive to variations in U s than generator over-excitation limiters are, meaning we have to be careful maintaining a positive voltage difference U cmax - U s, ideally larger than (X c + X tf)I cmax <ref:2607.13889#pg0>.

Dev: That sensitivity warning is critical for me because it highlights where the control loop might struggle if our local voltage measurements are even slightly off, especially under dynamic conditions. We also saw that P* g is independent of delta when we are in the intersection regime, specifically when delta v < delta < delta i, which gives us some stability there regarding angle uncertainty.

Taro: If we consider the broader impact, this analytical approach provides a specific rule for how VSC-HVDC links can be optimally utilized during contingencies without needing a massive centralized optimization engine running constantly. This suggests that decentralized, local decision-making based on measurable system states can effectively manage voltage stability issues in high-voltage transmission.

Rosa: That's the big picture, Taro; it moves the control strategy from being purely reactive to actively optimizing power injection based on real-time constraints. I'm thinking this kind of mechanism could be deployed in areas prone to instability where traditional slower control schemes might not react fast enough.

Dev: It certainly offers a way to address the need for fast response times by providing a direct, calculated setpoint derived from the physical limits of the converter itself, which is much faster than waiting for a full system-wide optimization routine. The paper "VSC-HVDC setpoint adjustment for maximum grid utilisation under voltage constraints" gives us that specific calculation.

Paper summary: Taro: So we have a method that uses geometry to define the best operating point under combined current and voltage limits, which is then fed into an emergency control scheme SIPS triggered by real measurements of U s and delta. This suggests a pathway for more intelligent, constraint-aware power management in large VSC-HVDC networks when stability is threatened.

Rosa: I'm really excited about the idea of testing this outside the lab; if we can get this validated in a dynamic environment like the Nordic Test System, it opens up possibilities for real grid applications. We need to see how long these measurements and control loops can run reliably in practice.

Dev: The paper itself focuses on deriving that setpoint rule, but realizing its potential hinges on the loop rate and latency of the actual implementation; we'd need to ensure those timing constraints are met for effective emergency response.

Taro: That's a fair point about latency; even the best theoretical control scheme is useless if it takes too long to execute when things are happening quickly. The SIPS framework needs to be fast enough to respond meaningfully during transient voltage dips or surges, which is the core challenge in applying this type of autonomy.

Rosa: So, we’ve covered the summary of what the paper claims, how it uses geometric constraints and angle differences to derive a setpoint, and touched on the implications for real-time control. We’re going to wrap up with some broader thoughts on what this means for power system management in general.

Dev: Before we move on, I just want to reiterate that the success of applying "VSC-HVDC setpoint adjustment for maximum grid utilisation under voltage constraints" depends heavily on maintaining accurate local voltage sensing and managing the sensitivity to U s as discussed in their analysis.

Taro: And from my angle, this work provides a specific mathematical tool for autonomy researchers to feed into how we design intelligent controllers that need to react intelligently when the grid state is volatile, moving beyond simple threshold-based responses.

Rosa: It sounds like a very promising direction for applying these kinds of analytical derivations into actual power system defense mechanisms. We'll be watching how this moves from the Nordic Test System simulations to field testing in the coming years.

Conclusion: Rosa: So, we've been diving deep into this paper about finding the optimal active power setpoint for VSC-HVDC links when voltage is stressed, and now it’s time to wrap up by really thinking about what this title means and where it might take us.

Dev: I agree that understanding the core mechanics of how they derive that optimal setpoint is crucial before we jump into the big picture implications of this work.

Taro: I think focusing on how this method handles situations when the grid starts behaving unpredictably really highlights its value for autonomy researchers.

Rosa: Exactly, and looking at the authors, it gives us a sense of who's been working on these kinds of power system control issues.

Dev: The methodology they presented is quite rigorous because it uses established relationships to find that optimal operating point under those tricky current and voltage constraints.

Taro: It seems like this research could be very useful for developing autonomous control systems that need to make rapid decisions when the system faces voltage instability, which is a key area for my work.

Rosa: Thinking about the implications, this paper suggests a way to move beyond static control strategies toward dynamic optimization in emergency situations.

Dev: If we can translate this analytical method into a robust, low-latency control loop, it could significantly improve grid resilience by allowing VSC-HVDC assets to act proactively under voltage stress.

Taro: I see that potential for decentralized decision-making during contingencies, which is something we need to explore more deeply in our autonomy framework.

Rosa: It really shows how mathematical modeling can translate directly into a practical tool for maintaining system stability when things go wrong.

Lund University · RISE Research Institutes of Sweden

eess.SY, cs.SY, math.OC

Submitted: 2026-07-15

Updated: 2026-07-15

Comments: Accepted for 8th International Conference on Smart Energy Systems and Technologies (SEST26) in Ciudad Real, 6 pages, 8 figures

DOI: 10.1109/SEST67798.2026.11712298

Code: https://github.com/thematt199310/NorthEuropeanAC-DCPowerSystem-Model

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 87/100

The gist: Voltage-source converter high-voltage direct current (VSC-HVDC) links offer controllable active and reactive power output, making them a promising asset for emergency voltage support.

Key concepts

VSC-HVDC: Voltage Source Converter High-Voltage Direct Current is a system that converts AC power from the grid into DC power for long-distance transmission. It is used here because it can control both active and reactive power, making it ideal for supporting voltage stability during emergencies.
Loadability (Pd): Loadability measures how much additional active power the system can safely handle before violating its current or voltage limits. The goal of the paper is to find a setpoint that maximizes this loadability, ensuring the grid can supply more power when stressed.
SIPS Framework: System Integrity Protection Scheme is an emergency control strategy triggered when the grid experiences voltage stress, like after a major event. This framework uses measurements to calculate optimal power setpoints in real-time to maintain system stability and maximize usage.

Terminology

VSC-HVDC — 39× in this paper · explained in 2 episodes
Transformer — 10× in this paper · explained in 2 episodes
Robustness — 5× in this paper · explained in 5 episodes
Accuracy — 4× in this paper · explained in 1 episode
Core — 4× in this paper · explained in 2 episodes
Adaptation — 2× in this paper · explained in 1 episode
System Integrity Protection Scheme — 2× in this paper
Benchmark — 1× in this paper · explained in 2 episodes

Summary

Voltage-source converter high-voltage direct current (VSC-HVDC) links offer controllable active and reactive power output, making them a promising asset for emergency voltage support. The gist: A closed-form expression for the optimal active power setpoint is derived under combined current and voltage constraints to maximize loadability during voltage-stressed conditions.

Problem Formulation and Constraints

The paper addresses the challenge of maximizing loadability in VSC-HVDC stations when operating under both current and voltage constraints. The system limits are represented geometrically as circles in the P–Q plane, where the feasible region is defined by the intersection of these two circles (Eqs. 1 and 2). The VSC-HVDC current limit is described by Equation (1), while the voltage limit is given by Equation (2). These limits depend on system parameters such as admittances and maximum converter current/voltage magnitudes, denoted as Icmax and Ucmax.

Derivation of the Optimal Setpoint

The core contribution involves deriving a closed-form expression for the optimal active power setpoint, denoted as P∗g, under these combined constraints. This derivation starts by relating the optimality condition to a relationship derived from previous work (Johansson et al. [4]), which shows that for any circular limit with an arbitrary center (P0, Q0), the condition is: Qg − Q0 / Pg − P0 = 1 / tan δ (Eq. 4). By computing the intersection points of the current and voltage limits, transition angles δi and δv are defined (Eq. 5). The optimal active power setpoint P∗g(δ) is then expressed as a piecewise function of δ (Eq. 6), which dictates whether the optimum lies on the voltage limit, current limit, or at the intersection point.

Implementation within an Emergency Power Control Framework

The derived setpoint rule in Equation (6) is designed to be used within an emergency power control (EPC) scheme known as SIPS (System Integrity Protection Scheme). This framework is triggered under voltage-stressed conditions, which are defined as heavily loaded or post-contingency operating points lacking a long-term voltage stable equilibrium. The process involves five steps:

Measurement and estimation: Obtaining the local voltage magnitude Us and the wide-area voltage angle difference δ via PMU measurements or state estimators.
Parameter update: Updating the limits (Eqs. 1–2) and calculating transition angles δi and δv (Eq. 5).
Optimal setpoint computation: Identifying the operating regime based on measured δ to compute P∗g from Equation (6).
Setpoint adjustment: Sending P∗g as an input signal to the HVDC outer-loop control for the d-axis current reference i ref d.
Reactive power coordination: Constraining Qg by the most restrictive limit (Eq. 7), prioritizing the d-axis power.

Validation and Results

Validation was performed using dynamic phasor (RMS) simulations on the Nordic Test System in DIgSILENT PowerFactory [9]. Simulations showed that reducing the active power setpoint P ref g by 100 MW (e.g., 700 MW → 600 MW) could increase the loadability Pd by about 150 MW. The highest loadability was found for P ref g = 550 MW ≈ 0.8 p.u., which is consistent with the analytically predicted optimal setpoint P∗g calculated under specific voltage conditions (δ in the interval 50–60°). A sensitivity analysis confirmed that uncertainties in δ propagate only weakly to the loadability Pd in regimes where it is required, supporting practical feasibility.

Practical Considerations and Limitations

The method's accuracy depends on local measurements of Us and an estimate of δ. Sensitivity analysis showed that P∗g is "independent of δ in the intersection regime (δv < δ < δi), which offers robustness against angle uncertainty. However, the paper notes that the VSC-HVDC voltage limit (2) is much more sensitive to variations in Us" compared to generator over-excitation limiters. To avoid excessive reactive power reduction, it is important to maintain a positive voltage difference Ucmax − Us, ideally larger than (Xc + Xtf)Icmax. Furthermore, the paper concludes that Pd can be increased by voltage setpoint adaptation of Us, with a maximum effect observed at Us = 1.05 p.u. and highlights that P∗g is not dependent on E or X.

Summary and Conclusion

The paper successfully presents an analytical method for determining the optimal active power setpoint P∗g under combined current and voltage constraints, providing a low-complexity, real-time control mechanism for SIPS targeting long-term voltage instability. The derived expression "

Improvements for AI systems

Based on the provided scientific paper, here are specific ways an AI system (specifically one designed for power grid operation, control, or emergency response) can be improved:

The AI system should be equipped with a module capable of performing a real-time calculation of the optimal active power setpoint, denoted as the closed-form expression in Equation (6).
This module must ingest only two primary inputs: local voltage magnitude measurements at the system bus and an estimate of the wide-area voltage angle difference (derived from PMU/WAMS data).
The AI should be designed to dynamically identify which operating regime (voltage-limited, intersection, or current-limited) is active based on the relationship between the measured angle difference and pre-calculated transition angles, and subsequently apply the correct piecewise function of Equation (6) to determine the optimal active power setpoint, ensuring it maximizes loadability.
The AI should be capable of operating within an Emergency Power Control (EPC) framework, where this calculated setpoint is fed directly as an input signal to the HVDC outer-loop control for the d-axis current reference, enabling fast corrective action without requiring computationally expensive global optimization (like OPF).
The system must incorporate a reactive power coordination layer that constrains the reactive power output to be within the most restrictive capability limit (Equation 7), prioritizing active power support while ensuring voltage stability margins are maintained.
The AI should include a sensitivity analysis routine that evaluates how uncertainties in the voltage angle difference estimate propagate through Equation (6) and Equation (9–10), allowing it to adjust its setpoint calculation based on the assessed uncertainty levels, thereby increasing robustness against measurement errors.
The system can be enhanced by incorporating a mechanism to adapt the voltage magnitude input, treating the local voltage measurement as a tunable parameter that allows for voltage setpoint adaptation, which has been shown to maximize loadability up to a specific point (Us = 1.05 p.u.).

This improved AI system can perform the following:

Perform high-speed, localized emergency response by calculating the mathematically optimal active power injection required from a VSC-HVDC station during voltage instability events.
Maximize the total loadability of a stressed grid by providing precise, real-time setpoint adjustments that are analytically derived and computationally lightweight (suitable for fast control loops).
Act as an intelligent controller within an EPC scheme, bridging the gap between local measurements and HVDC power references to proactively prevent voltage collapse.
Provide enhanced operational transparency by quantifying the benefit of its actions (loadability increase) based on system state and allowing for a degree of uncertainty management in its critical calculations.

Abstract

Voltage-source converter high-voltage direct current (VSC-HVDC) links offer controllable active and reactive power output, making them a promising asset for emergency voltage support. This paper presents an analytical method for adjusting the active power setpoint of a VSC-HVDC station to maximise loadability during voltage-stressed conditions. By exploiting the geometric structure of converter capability limits, a closed-form expression for the optimal setpoint is derived under combined current and voltage constraints. The method requires local voltage measurements and an estimate of a wide-area voltage angle difference, making it suitable for real-time emergency control without global optimisation. Validation on the Nordic Test System confirms that the analytically predicted optimum is consistent with the setpoint yielding the highest loadability, and that adjustments of active power setpoint can yield a more-than-proportional increase in loadability. The results further indicate robustness to angle estimate uncertainty.