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Chance-Constrained AC Optimal Power Flow: Reformulations and Efficient Algorithms

Line Roald, Göran Andersson

arXiv:1706.03241v2math.OC

TL;DR

Renewable forecast uncertainty makes deterministic OPF insufficient for secure operation, while chance-constrained AC OPF remains difficult to solve. The paper combines full AC equations at the forecasted point with a local uncertainty linearization, then develops analytical and iterative solution methods. Across the reported case studies, the iterative approach scales to large systems and the analytical reformulation efficiently enforces chance constraints relative to sample-based alternatives.

  • Problem

    Renewable forecast uncertainty requires stochastic OPF, but chance-constrained AC OPF is difficult to solve while accurate AC equations remain important.

  • Method

    The paper retains full nonlinear AC equations at the forecasted operating point, linearizes uncertainty effects around it, and solves the reformulated problem using one-shot or iterative algorithms.

  • Results

    The iterative method solves the Polish test case with 941 uncertain loads in 30s on a desktop computer, while the analytical reformulation has lower solution times than alternatives and outperforms Monte Carlo enforcement.

  • Takeaways & Limitations

    Separating uncertainty assessment from deterministic AC OPF enables scalable implementations and accommodates more general chance-constraint reformulations.

Abstract

from arXiv · show

Higher levels of renewable electricity generation increase uncertainty in power system operation. To ensure secure system operation, new tools that account for this uncertainty are required. In this paper, we formulate a chance-constrained AC optimal power flow problem, which guarantees that generation, power flows and voltages remain within their bounds with a pre-defined probability. We then propose an accurate, yet tractable analytical reformulation of the chance constraints. The reformulation maintains the full, non-linear AC power flow equations for the forecasted operating point, and models the impact of uncertainty through a linearization around this point. We discuss different solution algorithms, including one-shot optimization with and without recourse, and an iterative algorithm which enables scalable implementations. We further discuss how more general chance constraint reformulations can be incorporated within the iterative solution algorithm. In a case study based on four different IEEE systems, we compare the performance of the solution algorithms, and demonstrate scalability of the iterative scheme. We further show that the analytical reformulation accurately and efficiently enforces chance constraints in both in- and out-of-sample tests, and that the analytical approach outperforms two alternative, sample based chance constraint reformulations.

I. INTRODUCTION

The paper develops chance-constrained AC OPF methods to address renewable forecast uncertainty while retaining accurate AC modeling and computational scalability. It compares analytical and sample-based reformulations and evaluates one-shot and iterative algorithms.

  • Motivation: Renewable forecast uncertainty motivates treating OPF as a stochastic problem rather than a deterministic one.OPF supports market clearing and security assessment while enforcing transmission, voltage, and generation limits.
  • Problem: AC chance constraints are harder to solve than DC approximations but can enforce voltage, reactive-power, apparent-power, and current limits probabilistically.The paper targets accurate AC equations because many distribution and transmission applications require them.
  • Method: The analytical reformulation retains full nonlinear AC equations at the forecasted operating point and linearizes uncertainty effects around that point.This partial linearization enables analytical chance-constraint reformulation without the full linearization or relaxation used in alternative approaches.
  • Results: The iterative approach finds solutions similar to one-shot local optima and solves the Polish test case with 941 uncertain loads in 32s.The study compares one-shot and iterative solution times and benchmarks analytical performance against sample-based alternatives.
  • Algorithms: The iterative algorithm separates uncertainty assessment from AC OPF solution, enabling scalable implementations and incorporation of more general reformulations.The paper also introduces Monte Carlo and scenario-based sample reformulations within the iterative framework.

2) Generation and Voltage Control:

The operating model represents how generators respond to uncertain injections through active-power participation factors and local reactive-power voltage control. These responses determine deviations in generation, voltage, and line currents.

  • Generation and Voltage Control:: Active-power deviations are distributed among generators according to participation factors α, with reference-bus generation also balancing changes in losses.The loss change δp is treated as a secondary effect and assigned to the reference-bus generator.
  • Generation and Voltage Control:: Reactive power is locally adjusted at PV and θV buses, while generators at PQ buses keep reactive output constant.The corresponding reactive adjustments are denoted δq_i.
  • Generation and Voltage Control:: Voltage magnitude remains fixed at the reference and PV buses but varies at PQ buses.The paper notes that centralized automatic voltage regulation could also be incorporated.
  • Generation and Voltage Control:: Uncertain power injections change transmission current magnitudes, which are modeled through deviations δi_ij from the forecasted currents.Transmission constraints are stated as current constraints for a thermally constrained system.

III. CHANCE-CONSTRAINED AC OPTIMAL POWER FLOW

The AC CC-OPF minimizes generation cost subject to AC power balance and probabilistic limits on generation, voltage, and transmission currents. Its chance constraints bound violation probabilities but do not specify outcomes after violations occur.

  • III. CHANCE-CONSTRAINED AC OPTIMAL POWER FLOW: The full AC CC-OPF minimizes quadratic, linear, and constant generation costs subject to power-flow and operating constraints.The objective uses cost coefficients c_2, c_1, and c_0.
  • III. CHANCE-CONSTRAINED AC OPTIMAL POWER FLOW: AC power balance equations must hold for all uncertainty realizations in the uncertainty set D.They depend on uncertain voltage angles, voltage magnitudes, and nodal injections.
  • III. CHANCE-CONSTRAINED AC OPTIMAL POWER FLOW: Nodal active and reactive injections combine generator outputs, demand, forecasted uncertainty sources, and active or reactive fluctuations.Reactive fluctuations enter through the power-ratio parameter γ.
  • III. CHANCE-CONSTRAINED AC OPTIMAL POWER FLOW: Generation, voltage, and current limits are formulated as chance constraints with separate acceptable violation probabilities.The probabilities are denoted ϵ_P, ϵ_Q, ϵ_V, and ϵ_I.
  • III. CHANCE-CONSTRAINED AC OPTIMAL POWER FLOW: Chance constraints limit the probability of violation but do not describe what happens when a violation occurs.The paper distinguishes constraint types to interpret the consequences of violations.

1) Interpretation of the Chance Constraints:

The paper interprets chance constraints as probabilistic safeguards for hard and soft operating limits under renewable forecast uncertainty, using AC-feasible operating points and local sensitivity-based approximations.

  • Constraint interpretation: Hard generation constraints probabilistically preserve AGC capacity by limiting the chance that generators reach active or reactive power limits.Such violations can leave the system imbalanced and require manual intervention.
  • Constraint interpretation: Soft voltage and current constraints limit the probability of undervoltage, overvoltage, or transmission overload requiring tolerance or corrective redispatch.The chance constraint represents the probability that additional control action may be needed.
  • Modeling assumptions: The formulation treats AGC participation factors α and power ratio γ as controllable variables, although practical systems may pre-specify them.The paper mostly assumes these parameters are fixed.
  • Analytical approximation: The forecasted operating point is obtained from the full AC power flow equations, preserving an accurate AC-feasible representation at ω = 0.The operating point is represented by x = (θ, v, p, q).
  • Analytical approximation: A first-order Taylor expansion models uncertainty around the forecasted operating point and is expected to be accurate when forecast errors are small.The method concerns small forecast errors rather than necessarily small renewable generation.
  • Analytical approximation: Sensitivity factors ΓP, ΓQ, ΓV, and ΓI describe how generator outputs, voltages, and currents respond to uncertainty and depend on x, α, and γ.Their dependence on the operating point is nonlinear, while their dependence on α and γ is linear.
  • Analytical approximation: Under multivariate normal uncertainty, linear dependence on ω enables analytical chance-constraint reformulation despite nonlinear decision-variable dependence.The current constraint is approximated using the line-current sensitivity row ΓI(ij,·).

A. Reformulated Chance-Constrained Problem

The reformulated AC CC-OPF applies uncertainty margins to the relevant generation, voltage, current, and power constraints while retaining compact analytical expressions.

  • Constraint reformulation: Current uncertainty margins λI are defined separately from voltage margins λV and active- or reactive-power margins λP and λQ.The passages identify distinct margin definitions for currents, voltages, active power, and reactive power.
  • Constraint reformulation: The reformulation includes specialized conditions for generator and bus types, including zero reactive-power margin for PQ generators.The displayed constraints include conditions indexed by GPV, GPQ, and GθV.
  • Constraint reformulation: The standard deviation of total active-power imbalance is computed as σΩ = ∥1TΣ1/2∥2.The vector 11,m is a row vector of m ones.

V. SOLUTION ALGORITHMS

The paper compares direct one-shot optimization with an iterative scheme that repeatedly solves deterministic AC OPF subproblems and updates uncertainty margins.

  • A. One-Shot Optimization: One-shot optimization jointly optimizes scheduled generation, reserves, and voltage-control responses through a continuous non-convex formulation.The sensitivity factors depend on the forecasted operating point, participation factors, and power ratios.
  • A. One-Shot Optimization: Because the one-shot problem is non-convex, the solver is not guaranteed to find a global optimum.Direct solution may also be computationally prohibitive.
  • B. Iterative Solution Algorithm: The iterative algorithm alternates between deterministic AC OPF with fixed margins and recalculation of margins from the resulting operating point.It can use existing AC OPF software rather than solving the full reformulated problem directly.
  • B. Iterative Solution Algorithm: Each iteration solves the deterministic AC OPF, evaluates uncertainty margins, and checks their maximum change against stopping criteria.The process repeats until the criteria are satisfied.
  • B. Iterative Solution Algorithm: The iterative scheme offers scalability but does not optimize α or γ because those parameters do not enter its deterministic AC OPF subproblem.They must therefore be specified before running the algorithm.
  • B. Iterative Solution Algorithm: The iterative algorithm is not guaranteed to converge or to reach a locally optimal solution.In the reported simulations, it converged within a few iterations under stated margin tolerances and found solutions similar to one-shot optimization.

VI. ALTERNATIVE CHANCE CONSTRAINT FORMULATIONS

The iterative framework can incorporate Monte Carlo and scenario-based margin calculations, extending beyond the analytical reformulation while exposing differences in accuracy, conservatism, and assumptions.

  • VI. ALTERNATIVE CHANCE CONSTRAINT FORMULATIONS: The iterative method separates AC OPF solution from uncertainty-margin calculation, allowing alternative margin estimators without sacrificing tractability and scalability.This separation enables computationally heavier methods to be used inside iterations.
  • 1) Uncertainty Margins from Monte Carlo Simulation: Monte Carlo margins estimate empirical constraint quantiles by sampling uncertainty and solving the resulting nonlinear AC power flows.The simulation must be rerun at every iteration because margins depend on the AC OPF solution.
  • 2) Uncertainty Margins for Joint Chance Constraints: Joint chance constraints require all constraints to hold simultaneously with probability 1 − ǫJ.Scenario-based reformulations use a sample count depending on decision-variable dimension and acceptable joint violation probability.
  • 2) Uncertainty Margins for Joint Chance Constraints: The confidence level 1 − β for satisfying the chance constraint is typically chosen to be very small.
  • 2) Uncertainty Margins for Joint Chance Constraints: Figure 1 compares analytical, Monte Carlo, and scenario margins using colored lines relative to the forecasted voltage and an empirical voltage histogram.The analytical approach is green, Monte Carlo purple, and scenario approach blue.
  • 2) Uncertainty Margins for Joint Chance Constraints: Scenario and robust sample-set representations rely on convexity, so applying them to a convex relaxation does not guarantee the actual non-convex AC problem.
  • 2) Uncertainty Margins for Joint Chance Constraints: The paper instead solves full non-convex AC power flow for each sampled scenario while assuming solutions remain near a locally convex optimum.Margins are defined from the highest and lowest observed magnitudes relative to the forecasted value.
  • 2) Uncertainty Margins for Joint Chance Constraints: The analytical margins are symmetric, whereas Monte Carlo margins follow empirical quantiles and scenario margins are much larger than both alternatives.For the example voltage constraint, symmetry produces a larger upper and smaller lower analytical margin than Monte Carlo.

3) Comparison of Uncertainty Margins:

The study compares deterministic, one-shot, and iterative AC CC-OPF formulations using several uncertainty-margin definitions and assesses their cost, computational time, and constraint satisfaction.

  • Relaxed power-flow equations may produce solutions that are not physical AC power-flow solutions when the relaxation is not tight.The paper notes that AC feasibility should be checked for solutions obtained from a relaxation.
  • Scenario-based margins are not directly comparable with analytical and Monte Carlo margins because they enforce a joint rather than separate violation probability.The scenario approach uses a joint violation probability ǫJ, whereas the other approaches limit each constraint's violation probability ǫ.
  • The case study evaluates operating cost, computational time, chance-constraint satisfaction, and expected violation size across in-sample and out-of-sample tests.
  • The compared formulations include deterministic AC OPF, one-shot AC CC-OPF with fixed or optimized α, γ, and iterative AC CC-OPF variants.The iterative variants use analytical, Monte Carlo, or scenario-based uncertainty margins.
  • The analytical, Monte Carlo, and scenario-based iterative variants differ in how their uncertainty margins are defined.Analytical margins use closed-form expressions, while the other variants use empirical margins from simulations or limiting scenarios.
  • The uncertainty model treats forecast uncertainty as fluctuations in net load, combining fluctuations in load and renewable generation.The test cases use standard deviations specified as percentages of forecasted load, with uncertainty levels selected to produce congested but feasible systems.

2) Test systems:

The experiments use four increasingly large IEEE-based systems with uncertain loads, then compare solution approaches by generation cost and solution time.

  • Test systems: The four test systems are IEEE RTS96, IEEE 118 Bus, IEEE 300 Bus, and the Polish 2383 Bus Winter Peak case.The systems contain 17, 99, 131, and 941 uncertain loads, respectively.
  • Test systems: The uncertainty assumptions vary across systems, including standard deviations of 5% or 10% and either zero or within-zone correlation.The IEEE 118 Bus system uses correlation coefficient ρ = 0.3 within each zone and zero correlation between zones.
  • Test systems: The Polish test case contains 941 uncertain loads representing 67% of total system demand.
  • Experimental settings: Chance constraints use base violation probabilities ǫP = ǫQ = ǫV = ǫI = 0.01.The iterative algorithms also use specified tolerances for active power, voltage, and current uncertainty margins.
  • Comparison of solution algorithms: The iterative approach and one-shot optimization with fixed α, γ produce similar solutions, while optimizing α, γ lowers cost but substantially increases computational complexity and solution time.The comparison is made relative to the deterministic AC OPF and uses generation cost and solution time.

B. Scalability of the Iterative Approach

The iterative approach scales by repeatedly solving deterministic AC OPF problems, while accuracy tests examine linearization and distribution assumptions through in-sample and out-of-sample simulations.

  • Scalability: The iterative implementation uses the standard Matpower 5.1 runopf function with the default MIPS solver across all four test systems.The study reports solution times, iteration counts, and generation costs across successive iterations.
  • Scalability: The iterative problems converge within 4–5 iterations and finish within half a minute, including the Polish case with 941 uncertain loads and 2383 buses.Most of the cost change occurs between the first and second iterations, followed by minor adjustments until convergence.
  • Accuracy: The analytical reformulation has two accuracy limitations: linearization of uncertainty effects and the assumption of normally distributed uncertainty.
  • In-sample accuracy: In-sample tests show that empirical violation probabilities are closer to acceptable values for smaller standard deviations and larger acceptable violation probabilities.For large standard deviations and small ǫ, empirical violation probabilities can exceed ǫ; for larger ǫ, the solutions can be conservative.
  • In-sample accuracy: The maximum empirical violation probability remains within ±0.01 of the prescribed acceptable ǫ in the in-sample tests.
  • In-sample accuracy: The AC CC-OPF also limits the joint violation probability to a relatively small percentage despite enforcing separate chance constraints.
  • Out-of-sample accuracy: Out-of-sample tests using 8492 historical Austrian Power Grid samples produce only a slightly higher maximum violation probability than in-sample tests, remaining within ±0.01 of acceptable ǫ.The observed joint violation probability is also slightly higher but remains in the same range.
  • Violation size: The Monte Carlo figure reports maximum expected violation size across generator active power, generator reactive power, voltage magnitude, and current magnitude constraints.It uses 2000 multivariate normal samples over different acceptable violation probabilities ǫ.

D. Controlling Violation Size Through Chance Constraints

The paper examines how acceptable violation probability affects violation size and compares analytical, Monte Carlo, and scenario-based uncertainty margins. Lower acceptable violation probabilities generally reduce violations, while the analytical approach avoids the scenario method’s conservatism.

  • Violation-size assessment: 2000 Monte Carlo samples estimate expected constraint-violation sizes across acceptable probabilities 0.01–0.15.Violation size is the amount by which a limit is exceeded, with nonviolations assigned zero.
  • Violation-size assessment: Maximum expected violations generally increase as acceptable violation probability increases, but the relationship is non-monotonic.Reducing acceptable violation probability generally leads to smaller violations, but does not determine violation size one-to-one.
  • Uncertainty-margin comparison: The analytical and Monte Carlo approaches produce relatively similar solutions, while Monte Carlo has slightly lower cost but higher violation probabilities.Monte Carlo can violate the acceptable probability more substantially despite using full AC power flow equations and no explicit distributional assumption.
  • Uncertainty-margin comparison: The scenario approach has higher cost but significantly lower violation probabilities, with actual joint violation probability ǫJ,emp = 0.007 versus prescribed ǫJ = 0.1.Its conservative solution may be far from cost optimal despite guaranteeing chance-constraint feasibility.

B. Compatibility with Existing Tools

The iterative AC CC-OPF separates deterministic AC OPF solution from uncertainty assessment, enabling scalable implementations with existing solvers and accommodating more general reformulations. The paper demonstrates practical performance on large test cases while identifying data and convergence limitations.

  • Compatibility with existing tools: The iterative algorithm can use existing deterministic AC OPF or industrial AC SCOPF solvers without major changes to the solution approach.This supports practical implementation of the proposed method.
  • Practical boundaries: Practical deployment requires high-quality probabilistic forecasts capturing geographical correlations among uncertainty sources.The iterative algorithm also needs further extensions with guarantees for convergence and solution quality.
  • Iterative solution method: The iterative algorithm alternates deterministic AC OPF optimization with computation of uncertainty margins for the resulting operating point.This separation decouples uncertainty assessment from AC OPF solution.
  • General reformulations: The uncertainty-assessment step can incorporate Monte Carlo, scenario, or other more general reformulations without sacrificing computational tractability.The sample-based alternatives require samples rather than distributional assumptions, but the scenario approach has very high cost.
  • Comparison of reformulations: The analytical reformulation had lower solution times and outperformed Monte Carlo for chance-constraint enforcement, while the scenario approach gave more rigorous guarantees at very high cost.The comparison highlights sensitivity of sampling-based solutions to sample choice.
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