[Paper Review] Distributionally Robust Chance-Constrained Approximate AC-OPF with Wasserstein Metric

Chao Duan, Wanliang Fang+3 more|arXiv (Cornell University)|Jun 17, 2017

Electric Power System Optimization23 references17 citations

TL;DR

This paper proposes a distributionally robust chance-constrained approximate AC-OPF using the Wasserstein metric to handle uncertainties from variable renewable energy (VRE). By combining exact AC power flow at the nominal point with a linear approximation for uncertainty response, and constructing an ambiguity set from historical data via the Wasserstein ball, the method ensures constraint satisfaction under worst-case distributions while remaining data-driven and less conservative with more data.

ABSTRACT

Chance constrained optimal power flow (OPF) has been recognized as a promising framework to manage the risk from variable renewable energy (VRE). In presence of VRE uncertainties, this paper discusses a distributionally robust chance constrained approximate AC-OPF. The power flow model employed in the proposed OPF formulation combines an exact AC power flow model at the nominal operation point and an approximate linear power flow model to reflect the system response under uncertainties. The ambiguity set employed in the distributionally robust formulation is the Wasserstein ball centered at the empirical distribution. The proposed OPF model minimizes the expectation of the quadratic cost function w.r.t. the worst-case probability distribution and guarantees the chance constraints satisfied for any distribution in the ambiguity set. The whole method is data-driven in the sense that the ambiguity set is constructed from historical data without any presumption on the type of the probability distribution, and more data leads to smaller ambiguity set and less conservative strategy. Moreover, special problem structures of the proposed problem formulation are exploited to develop an efficient and scalable solution approach. Case studies are carried out on IEEE 14 and 118 bus systems to show the accuracy and necessity of the approximate AC model and the attractive features of the distributionally robust optimization approach compared with other methods to deal with uncertainties.

Motivation & Objective

• Address the limitations of traditional DC-OPF models in capturing AC power flow dynamics under high renewable penetration.
• Develop a chance-constrained AC-OPF that ensures security under distributional uncertainty without assuming a known probability distribution.
• Integrate historical data into a Wasserstein-based ambiguity set to construct a robust optimization framework that adapts to data availability.
• Ensure the solution remains computationally tractable and scalable by exploiting problem-specific structures in the reformulation.
• Provide a conservative yet adaptive strategy that becomes less conservative as more historical data is collected.

Proposed method

• Formulates an approximate AC-OPF by combining exact AC power flow at the nominal operating point with a linearized power flow model to capture system response under uncertainty.
• Employs a Wasserstein ball centered at the empirical distribution of historical VRE forecasting errors to define the ambiguity set of possible distributions.
• Reformulates the distributionally robust chance-constrained problem into a tractable form using duality and convex optimization techniques.
• Uses the Wasserstein metric to quantify the distance between the empirical distribution and potential true distributions, enabling worst-case distribution analysis.
• Applies a duality-based reformulation to convert the distributionally robust chance constraints into a solvable form involving support functions and $L_1$-norm constraints.
• Exploits problem structure—particularly sparsity and separability in the uncertainty set—to enhance computational scalability and efficiency.

Experimental results

Research questions

• RQ1 How can AC-OPF be made robust to uncertainties from variable renewable energy sources without assuming a known probability distribution?
• RQ2 To what extent does combining exact AC power flow with a linear approximation improve the accuracy of AC-OPF under uncertainty compared to traditional DC models?
• RQ3 How does the use of a Wasserstein ambiguity set constructed from historical data affect the conservativeness and robustness of the solution?
• RQ4 Can the proposed method maintain computational tractability and scalability on large-scale systems like the IEEE 118-bus system?
• RQ5 How does the solution quality and constraint violation probability evolve with increasing historical data volume?

Key findings

• The proposed approximate AC-OPF achieves higher accuracy than DC-based models, with a 98.73% reliability rate on the IEEE 14-bus system and 97.67% on the 118-bus system.
• The method guarantees chance constraints are satisfied for all distributions within the Wasserstein ambiguity set, ensuring robustness against distributional shifts.
• With more historical data, the ambiguity set shrinks, leading to less conservative solutions—evidenced by reduced computational time and improved operational cost.
• The time to construct the uncertainty set scales sublinearly with data size: 291.64 seconds for 1 million samples on the 118-bus system, indicating scalability.
• The solution approach is computationally efficient, with CPU times of 0.33s and 0.32s for the 14-bus system under different scenarios, and 1.72s for the 118-bus system.
• The duality-based reformulation successfully transforms the distributionally robust problem into a tractable optimization problem, enabling practical deployment on real power systems.