PaperlayerView exact revision on arXiv

Source-linked AI summary

Distributionally Robust Chance-Constrained Approximate AC-OPF with Wasserstein Metric

Chao Duan, Wanliang Fang, Lin Jiang, Li Yao, Jun Liu

arXiv:1706.05538v2math.OC

TL;DR

Variable renewable energy creates operational risk that conventional deterministic or DC-based uncertainty models may not adequately address. The paper develops a data-driven distributionally robust chance-constrained approximate AC-OPF using a Wasserstein ambiguity set, combining exact nominal AC power flow with an approximate linear uncertainty response. Case studies on IEEE 14- and 118-bus systems demonstrate improved precision and scalable computation, while more historical data yields less conservative solutions.

  • Problem

    VRE uncertainty creates security risks, while existing chance-constrained OPF studies using DC power flow omit voltage magnitudes and reactive power.

  • Method

    The paper combines exact nonlinear AC power flow at the nominal point with an approximate linear uncertainty-response model and uses a data-driven Wasserstein ambiguity set.

  • Results

    IEEE 14- and 118-bus case studies demonstrate improved precision of the proposed AC formulation and an efficient, scalable solution approach.

  • Takeaways & Limitations

    The approach uses historical data without assuming a distribution, and more available data produces a less conservative solution.

Abstract

from arXiv · show

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.

NOTATION

The paper uses vector, matrix, random-variable, distribution, and power-system notation to formulate its optimization model. It also frames the need for risk-aware OPF under variable renewable energy uncertainty.

  • NOTATION: Boldface lowercase letters denote real vectors, uppercase letters denote matrices, and tildes mark random vectors.Vector elements and matrix entries are indexed explicitly, while subvectors use an index set.
  • NOTATION: The notation defines buses, lines, reference, PV, and PQ bus sets used to describe network structure.nb and nl count buses and lines; R, S, and L identify reference, PV, and PQ buses.
  • NOTATION: Nominal states and network quantities include bus voltage angles and magnitudes, line MW flows, admittance matrices, and active or reactive injections.The notation distinguishes nominal quantities from generator, VRE, and load components.
  • NOTATION: Random VRE forecasting errors, probability distributions, expectations, supports, positive parts, and Hadamard products are defined for the robust formulation.These symbols support distributional ambiguity and worst-case expectation calculations.
  • CONTEXT: VRE uncertainty motivates risk-aware operation because deterministic OPF cannot adequately guide system operation under fluctuating injections.The paper identifies AGC and AVR as control mechanisms for responding to renewable and load uncertainty.

B. Linear Power FLow Response Model under Uncertainties

The model uses linear power-flow relationships to represent system responses to VRE and load uncertainties around a nominal operating point. AVR implementation further simplifies voltage responses, while the formulation focuses on VRE active-power forecasting errors.

  • The LPF model provides explicit formulas for system responses under uncertainty.These responses are derived from incremental power-flow relationships around the nominal operation point.
  • The state and injection variables are organized by reference, PV, and load buses for deriving the response equations.The reference bus fixes phase angle, while reference and PV buses maintain scheduled voltage magnitudes.
  • AVR implementation sets the voltage responses at reference and PV buses to zero under perturbations.Specifically, Δv_R = Δv_S = 0.
  • The framework can accommodate general power-injection uncertainties but uses VRE active-power forecasting errors for exposition.VRE reactive-power effects are represented through the mandated wind-farm power factor.
  • System-response expressions use constant matrices determined by network parameters and nominal line MW flows.The component-wise formulas provide the corresponding voltage, reactive-power, and line-flow responses.

C. Problem Formulation of Distributionally Robust Chance-constrained Approximate AC-OPF

The proposed formulation combines exact nominal AC power flow with an approximate linear uncertainty-response model in a distributionally robust chance-constrained OPF. Its affine-policy structure reduces the uncertainty dimensions needed for the chance constraints.

  • The objective minimizes worst-case expected generation and reserve costs over an ambiguity set built from historical data.The generation cost functions are convex quadratic functions.
  • The chance constraints enforce reserve adequacy, voltage quality, reactive-power adequacy, and transmission-line capacity across the modeled uncertainty distributions.Nominal power balance and voltage limits are enforced alongside distributionally robust operating-security constraints.
  • The formulation combines exact nonlinear AC equations at nominal operation with a linear model for deviations caused by VRE uncertainty.This preserves nominal AC-OPF behavior while maintaining a tractable uncertainty-response representation.
  • Affine AGC policies reduce the uncertainty dimension for chance constraints to one- or two-dimensional random variables.This reduction applies regardless of the number of uncertain renewable sources installed across the network.
  • The formulation addresses ambiguity arising because finite historical data cannot precisely characterize the underlying probability distribution.Historical data instead supplies probabilistic information used to construct the ambiguity set.

A. Wasserstein Metric and Ambiguity Set

The Wasserstein ambiguity set represents distributions consistent with finite historical data while shrinking toward the empirical distribution as sample size increases. The selected radius controls the robustness–conservatism trade-off.

  • Finite historical samples cannot precisely characterize the true distribution of the uncertainty variable.The empirical distribution is therefore used as an estimate rather than treated as exact.
  • The Wasserstein metric measures distance between probability distributions through joint distributions sharing the two distributions as marginals.The paper uses the l1 norm for numerical tractability in distributionally robust optimization.
  • The ambiguity set is a Wasserstein ball centered at the empirical distribution, representing distributions consistent with the available historical information.Its radius is a sample-dependent function that decreases toward zero as the number of samples grows.
  • The Wasserstein-ball radius is chosen to upper-bound the distance between the empirical and true distributions and strongly affects optimization performance.The radius depends on the confidence level and the support diameter in the cited construction.
  • The initial radius choice was found overly conservative, limiting the DRO advantage over conventional robust optimization with reasonable data volumes.The paper therefore revisits the derivation to reduce conservatism.
  • A special-case constant in the initial radius bound can be much larger in practice, creating a major source of conservatism.The paper instead estimates the relevant quantity from data.

IV. SOLUTION APPROACH

The solution approach uses strong duality to reformulate worst-case expectations and distributionally robust chance constraints. It then exploits problem structure to avoid computational growth with the historical sample size.

  • Strong duality provides the reformulation engine for worst-case expected costs and distributionally robust chance constraints.The result is tailored to the voltage-concerned chance-constrained AC-OPF setting.
  • A naive application of the reformulation would make problem size grow with the number of historical samples.This scaling is computationally undesirable for data-rich settings.
  • The method replaces worst-case expected cost with a tight upper bound and safely approximates chance constraints by robust constraints.These steps exploit the structure of the proposed OPF formulation.
  • The resulting computational burden almost remains unchanged as more historical data becomes available.The implementation also estimates the uncertainty support from the sample set, with σ_max = 10 in practice.

A. Evaluation of Worst-case Costs

The paper evaluates worst-case expected costs using a Wasserstein-based reformulation, then replaces the sample-dependent formulation with a scalable upper approximation. The approximation keeps computational size fixed as historical data grows, while its optimum approaches the original formulation as sample size increases.

  • Convex reformulation: The cost function is convex quadratic in both the decision variable and the uncertainty variable.Convexity supports the deterministic reformulation based on the Wasserstein dual representation.
  • Sample-dependent formulation: The worst-case cost formulation introduces auxiliary variables and quadratic constraints whose numbers grow with the historical sample set.This scaling can prevent full use of large datasets.
  • Scalable approximation: The authors replace the sample-dependent formulation with a close upper approximation by selecting λ from the endpoint derivatives of the cost function.The chosen λ is the smallest value satisfying the relevant conditions in the reformulation.
  • Scalable approximation: The approximation’s numbers of decision variables and constraints remain unchanged as the sample set grows.This makes the approximation more computationally favorable than the original formulation.
  • Approximation quality: The optima of the original and approximated formulations differ very little in practice, with the difference diminishing as sample size increases.The paper therefore uses the approximation to reduce computational burden while retaining a close objective value.

B. Reformulation of Distributionally Robust Chance Constraints

The paper converts distributionally robust chance constraints into tractable linear robust constraints by constructing a small data-driven uncertainty set. Standardization, hypercube selection, and nested bisection yield a scalable deterministic approximation.

  • Uncertainty-set design: The paper seeks a deterministic uncertainty set whose robust constraint implies the distributionally robust chance constraint.The set is designed to be cheap to obtain, tractable in robust constraints, and as small as possible.
  • Data standardization: Historical samples are standardized using their sample mean and covariance, producing a variable with sample mean 0, covariance I, and bounded support.The standardized samples and empirical distribution are used to construct the ambiguity-set-based procedure.
  • Uncertainty-set design: The standardized uncertainty set V is restricted to a hypercube whose side length σ is minimized to reduce conservatism.The original uncertainty set is recovered as U = ˆΣ^1/2V + ˆµ.
  • Numerical search: A worst-case probability reformulation enables a nested bisection search over two one-dimensional decision variables.The worst-case probability is non-decreasing in σ, which gives the search a unique minimum.
  • Deterministic reformulation: The hypercube is represented by its vertices, and the resulting robust constraints become linear inequalities in the decision variables.For the considered cases m = 1 and m = 2, these constraints are computationally efficient and introduce no scalability problems.
  • Resulting OPF: The complete formulation has a quadratic objective, nonlinear AC equality constraints, and linear constraints added to the conventional AC-OPF structure.The added decision variables include α and reserve-related variables.

V. NUMERICAL RESULTS

The numerical section discusses implementation issues and benchmarking before evaluating the proposed formulation on IEEE 14-bus and 118-bus systems. The two case studies emphasize model validity and distributionally robust uncertainty handling, respectively.

  • IEEE 14-bus system: The IEEE 14-bus tests emphasize the validity and necessity of the proposed approximate AC-OPF formulation.These tests focus on the approximate AC model’s role in the formulation.
  • IEEE 118-bus system: The IEEE 118-bus studies focus on distributionally robust optimization features and comparisons with other methods for handling uncertainties.The section also reports implementation issues and related benchmarking approaches.

A. Implementation and Benchmarking

The study benchmarks the proposed WDRO method against RO, MDRO, and GSP using simulated VRE forecasting errors, then evaluates approximate AC-OPF accuracy and scalability on IEEE systems.

  • Implementation: The implementation uses MATPOWER, YALMIP, and KNITRO to solve the proposed formulation within an extensible AC-OPF structure.The formulation fits MATPOWER's extensible OPF structure, with YALMIP supporting modifications and KNITRO serving as the default interior-point solver.
  • Benchmarking: RO enforces security constraints for every support realization, MDRO uses mean and covariance information, and GSP assumes a Gaussian distribution.MDRO and GSP chance constraints are reformulated as SOCP constraints and handled through iterative linear cutting planes.
  • Experimental design: The evaluation generates Laplace-distributed historical forecasting errors secretly from the methods, constructs strategies from finite data, and assesses them with larger Monte Carlo samples.The same underlying random-number generator produces historical data and out-of-sample Monte Carlo data, while its distribution parameters remain undisclosed to the methods.
  • IEEE 14-bus system: On the modified IEEE 14-bus system, the complete model guarantees reserve, voltage, reactive-power, and line-flow chance constraints within allowable ranges.Removing voltage constraints exposes bus 12 to substantial over-voltage risk, while the study compares the proposed strategy with a DC model.
  • IEEE 14-bus system: The approximate AC model has zero operation-cost error at zero forecasting error and at least one-order-of-magnitude higher voltage and reactive-power accuracy than LPF.It retains LPF-level line-flow accuracy while improving operation-cost, voltage-magnitude, and reactive-power calculations toward full-AC behavior.
  • IEEE 118-bus system: As data increases, WDRO becomes less conservative, while method conservatism follows RO>WDRO(10^2∼3)>MDRO>WDRO(10^4∼6)>GSP.At higher wind penetration, some conservative methods become infeasible; WDRO's solver time is nearly RO-like and independent of wind-farm count and historical-data volume.

VI. CONCLUSION

The paper presents a data-driven distributionally robust chance-constrained approximate AC-OPF that combines AC-model precision with numerical tractability. Historical data construct the ambiguity set, while Wasserstein robustness and specialized reformulations support scalable, less conservative operation as more data become available.

  • Monte Carlo comparisons assess the lowest reliability of all security constraints, while a separate procedure determines the uncertainty set U.
  • The proposed formulation combines full AC power flow with a linear response model to improve precision while retaining numerical tractability.
  • The approach constructs a Wasserstein-based ambiguity set from historical data without assuming a probability-distribution type.
  • The operating strategy is immunized against all distributions in the ambiguity set, and more available data yield a less conservative solution.
  • Special problem structures are exploited to improve the efficiency and scalability of the numerical solution approach.

APPENDIX

The appendix proves a reformulation result by expressing an indicator function through upper-semicontinuous functions and evaluating the resulting supremums using the problem's ambiguity-set definitions and l1 norm.

  • The proof defines l(˜ϑ) as the indicator function of the complement of V(σ), converting the violation probability into an expectation.
  • Upper semicontinuity of l(˜ϑ) allows Lemma 1 to be applied to the probability expression.
  • The proof derives inequalities involving λ, σ, and ˆϑ(k)i before evaluating the associated supremums.
Loading 1706.05538v2…