SOPF-Based Adaptive Droop Control for Hybrid AC--HVDC Grids Under Offshore Wind Uncertainty

Technical Summary: SOPF-Based Adaptive Droop Control for Hybrid AC–HVDC Grids Under Offshore Wind Uncertainty

Problem Statement
The integration of massive offshore wind capacity into hybrid AC–HVDC grids necessitates robust DC voltage regulation. While decentralized DC voltage droop control is the predominant architecture for Multi-Terminal HVDC (MTDC) systems, conventional implementations rely on fixed-gain coefficients calibrated at a nominal operating point. These fixed coefficients struggle under the severe stochastic volatility of offshore wind, often driving the system away from nominal conditions on a sub-hourly basis. This mismatch results in excessive voltage deviations, sub-optimal dispatch, and uneven converter stress during large fluctuations or contingencies. Existing adaptive schemes typically recalibrate gains reactively based on real-time deviations or heuristic rules, lacking a direct link to system-level economic dispatch and offering no explicit statistical voltage-security guarantees.

Methodology
The paper proposes a novel framework that bridges system-level stochastic dispatch with converter-level control using a Stochastic Optimal Power Flow (SOPF) approach. The methodology consists of three core components:

1. Zone-Wise Beta Parameterization of Wind Uncertainty:
   Recognizing that offshore wind forecast errors exhibit heteroscedasticity (variance depends on the operating regime), the authors reject single fixed-distribution assumptions. Instead, they partition the normalized wind power forecast into three zones: Low (cut-in), Mid (ramp), and High (near-rated). Within each zone, forecast errors are modeled using a Beta distribution, with shape parameters ($\alpha, \beta$) fitted to historical forecast-error statistics. This ensures the uncertainty model is physically consistent with the bounded nature of wind power and captures the distinct statistical characteristics of different operating regimes.

2. PCE-Based Chance-Constrained SOPF:
   The dispatch problem is formulated as a chance-constrained Optimal Power Flow (OPF) in the current-voltage (IV) space of the hybrid AC–HVDC grid. Uncertainty is propagated through the nonlinear power-flow equations using Polynomial Chaos Expansion (PCE) with a Jacobi polynomial basis (degree $d=2$). The stochastic states are represented analytically, allowing chance constraints (e.g., DC voltage limits with 95% confidence) to be reformulated as tractable second-order cone constraints. The optimization minimizes expected generation cost while satisfying these probabilistic security constraints.

3. Sensitivity-Based Adaptive Droop Extraction:
   A key innovation is the derivation of the adaptive droop gain directly from the first-order PCE coefficients obtained from the SOPF solution. Rather than performing explicit Jacobian formation and inversion, the method utilizes a sensitivity interpretation where the first-order coefficients represent the partial derivatives of state variables with respect to uncertainty sources. The optimal droop gain ($k_{opt}$) is calculated as a ratio of the aggregated magnitudes of the active power and DC voltage sensitivity coefficients across all wind sources. This gain is then normalized to an engineering scale and applied locally at the converter.

Key Contributions
The paper identifies three primary contributions:

• Unified Framework: It establishes a coupling between system-level stochastic dispatch and converter-level droop control, ensuring the droop coefficient inherits the probabilistic voltage-security guarantees encoded in the SOPF chance constraints.
• Data-Driven Uncertainty Modeling: It introduces a zone-wise Beta parameterization fitted from historical data, providing a heteroscedastic uncertainty input that outperforms the fixed-distribution assumptions common in existing PCE-SOPF studies.
• Jacobian-Free Sensitivity Extraction: It derives a sensitivity-based droop rule directly from PCE coefficients, bypassing the need for explicit Jacobian inversion and yielding an interpretable gain that varies continuously with the stochastic operating point.

Results
The proposed framework was validated on a 4-terminal AC–HVDC test system with two offshore wind farms. The study compared the adaptive scheme against fixed-coefficient droop settings ($k=20$ p.u. and $k=15$ p.u.) and a no-droop baseline.

• Adaptive Characteristics: The extracted adaptive gain ($k_{opt}$) demonstrated a clear dependence on the operating point, varying continuously as wind forecasts traversed different zones. The gain was observed to increase at low wind outputs and decrease toward rated regions, reflecting changing system sensitivities.
• Performance Under Disturbance: In terms of active-power tracking error ($\Delta P$) relative to the ideal post-realization dispatch, the adaptive scheme consistently outperformed fixed-coefficient and no-droop strategies.
  • The adaptive method achieved the lowest tracking error in 67%–70% of cases across Low, Mid, and High disturbance bands.
  • In the High disturbance band, the adaptive scheme reduced the mean tracking error by 13.9% compared to the no-droop baseline (38.76 MW vs. 45.01 MW).
  • While the improvement over the fixed $k=20$ setting was modest in mean error (as $k=20$ happened to align with typical Mid-zone gains), the adaptive scheme maintained a significantly higher "winner rate," demonstrating a reliable per-case advantage rather than a coincidental average benefit.

Significance and Claims
The paper claims to fundamentally bridge the operational gap between probabilistic system-wide dispatch and local converter regulation. By moving away from rigid fixed-gain assumptions, the proposed mechanism ensures that local converter responses are not heuristically tuned but are rooted in economic efficiency and probabilistic security. The authors assert that this approach effectively minimizes active-power tracking errors during extreme wind disturbances, providing a principled basis for selecting droop coefficients that reflect both system economics and the statistical nature of offshore wind uncertainty. Future work is noted to focus on extending the framework to include $N-1$ security contingencies and hardware-in-the-loop validation.