PaperlayerView exact revision on arXiv

Source-linked AI summary

A Robust Approach to Chance Constrained Optimal Power Flow with Renewable Generation

Miles Lubin, Yury Dvorkin, Scott Backhaus

arXiv:1504.06011v3math.OC

TL;DR

Intermittent renewable generation challenges least-cost, risk-averse power-system operations, while conventional chance-constrained formulations often rely on uncertain distribution parameters. The paper develops a distributionally robust chance-constrained OPF with a cutting-plane solution method and reports cost and technical benefits in a BPA case study.

  • Problem

    Intermittent renewable generation complicates least-cost, risk-averse management, while distribution parameters are often estimated and not known exactly.

  • Method

    The paper develops a distributionally robust chance-constrained OPF and solves its potentially infinite convex constraint set with a cutting-plane algorithm.

  • Results

    Compared with deterministic and chance-constrained OPF, RCC OPF can deliver cost savings, fewer transmission-limit and generator-ramping violations, and smaller Area Control Error values.

  • Takeaways & Limitations

    RCC OPF provides a scalable approach for accounting for uncertainty in renewable-generation distribution parameters in large-scale power-system operations.

  • Takeaways & Limitations

    Results may differ across systems because the BPA network benefits from flexible, low-cost hydro generation.

Abstract

from arXiv · show

Optimal Power Flow (OPF) dispatches controllable generation at minimum cost subject to operational constraints on generation and transmission assets. The uncertainty and variability of intermittent renewable generation is challenging current deterministic OPF approaches. Recent formulations of OPF use chance constraints to limit the risk from renewable generation uncertainty, however, these new approaches typically assume the probability distributions which characterize the uncertainty and variability are known exactly. We formulate a Robust Chance Constrained (RCC) OPF that accounts for uncertainty in the parameters of these probability distributions by allowing them to be within an uncertainty set. The RCC OPF is solved using a cutting-plane algorithm that scales to large power systems. We demonstrate the RRC OPF on a modified model of the Bonneville Power Administration network, which includes 2209 buses and 176 controllable generators. Deterministic, chance constrained (CC), and RCC OPF formulations are compared using several metrics including cost of generation, area control error, ramping of controllable generators, and occurrence of transmission line overloads as well as the respective computational performance.

I. INTRODUCTION

The paper motivates probabilistic OPF formulations for integrating intermittent renewable generation and proposes RCC OPF to address uncertainty in wind-error distribution parameters. It evaluates deterministic, CC, and RCC OPF on a modified BPA system.

  • A. Motivation: Intermittent wind and solar generation expose drawbacks in deterministic operating procedures and challenge least-cost, risk-averse management of grid assets.Transmission and controllable-generation investments can add flexibility but are costly and constrained by regulation and policy.
  • Contribution: RCC OPF generalizes CC OPF by allowing uncertainty in the mean and variance of wind forecast errors and compares both against deterministic OPF on a modified BPA system.The implementation uses a cutting-plane approach and JuMPChance to study large-scale interconnections.
  • A. Motivation: Deterministic OPF uses central wind forecasts without endogenously accounting for wind variability and uncertainty, while heuristic reserves may be cost-inefficient.These policies were designed for relatively small load deviations and are not expected to manage large net-load fluctuations cost effectively.
  • Related work: Chance-constrained and stochastic formulations require accurate statistical models, whereas interval and robust methods can become overly conservative or lack a systematic uncertainty-budget choice.Scenario-based stochastic programming can also lead to impractically large computing times.
  • Organization: The paper reviews deterministic OPF, CC OPF, robustified chance constraints, and the proposed RCC OPF before comparing their performance on the BPA test system.The comparison covers cost, technical performance, and computational behavior.

B. CC OPF Formulation

The CC OPF extends deterministic dispatch by modeling generator and line-limit violations probabilistically under Gaussian wind deviations. Its convex chance constraints can be represented as SOC constraints and solved with specialized outer approximation.

  • Deterministic OPF: Deterministic OPF minimizes generation cost for forecast wind and demand subject to generator and transmission operating constraints.Wind deviations are incorporated through proportional generator response using participation factors.
  • CC OPF: CC OPF introduces probabilistic constraints for generator outputs, generator ramping, and transmission flows while optimizing participation factors.The ramping constraints bound the probability that real-time response exceeds continuous ramp-up or ramp-down limits.
  • Line-flow constraints: Adjusted phase angles and line flows are expressed explicitly as functions of wind deviations and decision variables after fixing a reference bus.This reformulation enables chance constraints on implicitly varying line flows.
  • Gaussian chance constraints: Wind deviations are assumed independent and normally distributed with zero mean and known variance, making the chance constraints convex and computationally tractable.The resulting expected quadratic production-cost formulation is representable using second-order cone programming.
  • Computational treatment: Each chance constraint is treated independently; enforcing multiple linear constraints jointly with high probability remains an open convexity question even under normality.Large-scale instances motivate sequential outer approximation rather than relying solely on off-the-shelf SOC solvers.

C. Cutting-plane algorithm to solve distrbutionally robust chance constraints

The distributionally robust chance constraints form potentially infinite convex constraint families over uncertainty sets. The paper solves them by iteratively finding violated worst-case distributions and adding linearizations.

  • Robust chance constraints: Distributionally robust chance constraints require high-probability feasibility for every distribution whose parameters lie in an uncertainty set.This addresses the fact that Gaussian mean and covariance parameters are often estimated rather than known exactly.
  • Cutting-plane algorithm: For partitioned uncertainty sets, the robust formulation is convex but lacks a known compact deterministic reformulation, motivating a cutting-plane algorithm.The algorithm is designed to handle large-scale instances.
  • Cutting-plane algorithm: At each iteration, the method solves a relaxation, optimizes over uncertainty sets to test feasibility, and adds a linearization when a violating mean-covariance pair is found.The process repeats until robust feasibility holds within numerical tolerances.
  • Algorithmic enhancements: The implementation applies cutting planes to the mean term and adds covariance linearizations only when the original robust constraint is violated.This avoids unnecessary iterations, although polynomial convergence is not guaranteed in general.
  • Special structure: Generator-output and ramping constraints exploit scalar Gaussian structure to reduce to a single linear constraint, but this simplification does not apply to line-capacity constraints.The special structure can avoid applying cutting planes to constraints (13)-(16).

D. Formulation for RCC OPF

The RCC OPF models wind forecast-error means and variances within structured uncertainty sets and robustifies the CC OPF constraints. Uncertainty budgets control conservatism while preserving low computational overhead for the selected sets.

  • Uncertainty model: RCC OPF assumes independent wind-site fluctuations and represents mean and variance uncertainty with intervals for each wind farm.Aggregate uncertainty is represented using polyhedral sets constructed for the mean and variance.
  • RCC OPF formulation: The RCC OPF replaces generator and transmission chance constraints with distributionally robust counterparts.The robustification requires constraints to hold across the allowed distribution-parameter uncertainty.
  • Uncertainty budgets: Uncertainty budgets Γµ and Γσ adjust conservatism: Γ = 0 recovers standard CC OPF, while Γ = 1 is the most conservative limit described.The budgets can be interpreted as bounding the proportion of wind farms taking worst-case distributions.
  • Computational implications: For the chosen uncertainty sets, robust feasibility checks reduce to a sorting operation, adding little computational expense relative to CC OPF.The formulation leaves the nominal variance term in the objective unchanged.
  • Modeling convention: The robustification is applied separately to each constraint, so the worst-case distribution may differ across constraints.The paper sets Γ = Γµ = Γσ when investigating results as a function of Γ.
  • Robust-optimization interpretation: Under Gaussian uncertainty, CC OPF corresponds to robust linear constraints with ellipsoidal uncertainty, while RCC OPF adds uncertainty sets around their nominal values.This provides a robust-optimization interpretation of the two formulations.

III. CASE STUDY

The study evaluates RCC OPF against CC and deterministic OPF in vertically integrated operations on a modified BPA system. It uses Γ to examine robust conservatism and operational outcomes.

  • RCC OPF, CC OPF, and deterministic OPF are implemented for a modified Bonneville Power Administration system.
  • The vertically integrated grid-operations setting enables evaluation of probabilistic methods in OPF.
  • Γ is varied to study robust conservatism's effects on generation cost and Area Control Error statistics.

A. Test System and Data

The case study uses a modified BPA network and winter operational data, with assumptions filling unavailable information and data-driven uncertainty ranges for wind power.

  • The modified BPA system contains 2209 buses, 2866 transmission lines, 176 controllable generators, and 24 wind farms.Controllable generation capacity is 40.6 GW, while wind capacity is 4.6 GW.
  • The study covers December 2012 through March 2013, using hour-ahead forecasts and actual five-minute load realizations.The seasonal window is chosen so wind, load, and hydro factors are relatively stationary.
  • Wind forecast-error ranges are estimated from historical data by fitting a generalized normal distribution and converting wind-speed ranges into wind-power units.The conversion uses a typical wind-turbine power curve.
  • Gas and coal outputs and participation factors are decision variables, whereas the nuclear unit is treated as must-run at 95% of nameplate capacity.The nuclear unit's real-time balancing participation factor is set to zero.
  • Hydro outputs are fixed exogenously in all OPF formulations and disaggregated from historical system-level BPA data.Hydro dispatch depends on water-flow considerations rather than power-system operations.

B. Evaluation Procedure

The evaluation procedure separates hour-ahead scheduling from real-time dispatch and applies the same framework across the tested OPF formulations.

  • The evaluation has two steps that emulate hour-ahead scheduling and real-time dispatch, respectively.
  • For each hour, RCC OPF, CC OPF, and deterministic OPF are solved using hourly wind and load forecasts.The CC formulation is RCC OPF with Γ=0.

1) Step 1:

The second-stage simulation applies actual five-minute wind and demand realizations, computes generator responses and power flows, and aggregates operational statistics across the study period.

  • Step 2:: Actual five-minute wind and demand realizations are used at twelve intervals within each operating hour.
  • Step 2:: Controllable-generator outputs are computed from the actual realizations during each five-minute interval.
  • Step 2:: DC power flow calculations use the resulting generator outputs to obtain transmission-line flows.
  • Step 2:: Actual hourly operating cost is calculated by summing generation costs over all five-minute intervals.The emulation allocates costs according to energy delivered without a markup for regulation.
  • Step 2:: The procedure is repeated from December 2012 through March 2013, and results are analyzed for ACE, generator ramping, and line-flow statistics.

C. Cost Performance

The RCC OPF’s cost depends on the uncertainty budget Γ and risk parameters ϵi and ϵmn. In the case study, actual generation cost is minimized at Γ*=0.6, while the cost response to risk parameters reflects a trade-off between generator and network risk.

  • Parameter selection: The RCC OPF uses Γ to budget uncertainty in wind forecast probability distributions, while ϵi and ϵmn limit generator and network constraint-violation probabilities.Γ is treated as a short-term policy parameter, whereas the risk parameters are fixed for the analysis.
  • Parameter selection: Γ*=0.6 produces the lowest actual generation cost Ca in the case study.The study evaluates Γ from 0.03 to 1.0 and compares against deterministic OPF.
  • Risk-parameter sensitivity: For ϵmn=0.01, decreasing ϵi spreads dispatch more uniformly across generators, reducing individual ramping duty but potentially increasing operating cost through var(Ω).The cost increase is suppressed in this case study by its high percentage of flexible hydro generators, a result not expected to transfer to all systems.
  • Risk-parameter sensitivity: For ϵmn=0.0001, congestion avoidance plays a larger role, making cost less sensitive to generator cost and ϵi; intermediate ϵmn values show a strong trade-off in Ca.At the smallest network-risk parameter, cost remains elevated and relatively flat as ϵi approaches 1/48.
  • Evaluation procedure: The cost analysis computes realized generation cost Ca over December 2012–March 2013 from dispatches and participation factors produced by the two-step procedure.Table I reports changes relative to Γ=0, while Table II reports RCC OPF cost sensitivity to ϵi and ϵmn.

D. Technical Analysis

Technical performance is evaluated using ACE distributions and averages while varying the RCC uncertainty budget Γ. The RCC OPF improves ACE control relative to CC and deterministic OPF and reduces generator ramp-rate violations.

  • 1) ACE Performance:: ACE is evaluated for each five-minute interval, using its cumulative distribution and average over the study period to compare technical performance.Figure 2a presents the CDF, while Figure 2b presents the average ACEt(τ).
  • 1) ACE Performance:: The RCC OPF outperforms CC and deterministic OPF in controlling ACE by progressively accounting for uncertainty in wind-distribution parameters.CC already outperforms deterministic OPF by accounting for wind fluctuations.
  • 1) ACE Performance:: Γ≈0.6 marks the point where ACE statistics first significantly deteriorate as conservatism is relaxed and RCC OPF cost reaches its minimum.Above Γ=0.6, additional ACE-control performance is small relative to the additional cost.
  • 1) ACE Performance:: For Γ<0.6, ACE events become more frequent across sizes and the distribution develops a longer tail of large ACEt(τ) values.The CDF differences among Γ=1.0, 0.8, and 0.6 are not very significant.
  • 2) Generator Ramping:: The RCC OPF reduces generator ramp-rate violations relative to both CC OPF and deterministic OPF, potentially avoiding generator wear-and-tear effects.This result complements the ACE comparison with a generator-ramping reliability metric.

2) Ramping Performance:

Increasing RCC robustness reduces both the number of generators affected by ramp-rate violations and the violations per generator, while Γ has little effect on the frequency of power-line overloads.

  • Ramping Performance:: 11 generators have ramp-rate violations under CC OPF, compared with 4 at Γ=0.2 and 2 at Γ=1.0.The reduction is observed even at the relatively small robustness level Γ=0.2.
  • Ramping Performance:: Increasing RCC robustness greatly reduces the number of ramp-rate violations per affected generator.The greatest count at Γ=0.2 occurs for generator 159, representing approximately 0.004 of the study intervals.
  • Ramping Performance:: The deterministic OPF produces ramp-rate violations on 39 generators, including 24 with violations in more than one 5-minute interval.These violations are not shown in Figure 3.
  • Ramping Performance:: Figure 3 compares histograms of ramp-rate violations per generator for CC OPF and RCC OPF at Γ=0.2, 0.6, and 1.0.All results use ǫ* mn = 0.0025 and ǫ* i = 1/6.
  • Transmission Overload Performance:: Γ does not significantly affect overload frequency for the most frequently overloaded lines shown in Figure 4.The figure compares CC and RCC OPF overloads occurring during more than one 5-minute interval; line 1813 has the greatest observed violation fraction at Γ=0.

E. Computational Performance

The RCC OPF scales to a 2209-bus BPA case and can improve cost and technical performance relative to deterministic and CC OPF. Results remain system-dependent, motivating extensions to other generation mixes, richer constraints, and longer planning horizons.

  • Computational performance: The RCC OPF was solved on a modified BPA system with 2209 buses, demonstrating scalability to a large power-system case.The case study indicates solution within seconds and potential compatibility with short-term planning tools.
  • Scope and assumptions: The reported cost and technical results may differ on other systems because the BPA case benefits from flexible, low-cost hydro generation.The authors expect greater value in systems dominated by less-flexible fossil generation.
  • Computational performance: Compared with deterministic and CC OPF, RCC OPF can reduce generation cost while improving transmission-limit, generator-ramping, and ACE performance.The reported comparison covers both economic and technical metrics rather than computational time alone.
  • Future work: Future work includes extending RCC OPF to reactive-power effects, time-extended ramping, day-ahead UC tools, and market-based regulation procurement.The day-ahead extensions may require further algorithmic development for tractable solution.
Loading 1504.06011v3…