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Adaptive Robust Optimization with Dynamic Uncertainty Sets for Multi-Period Economic Dispatch under Significant Wind

Alvaro Lorca, Andy Sun

arXiv:1409.2936v2math.OC

TL;DR

High wind penetration creates significant operational uncertainty because wind output is intermittent and stochastic. The paper proposes adaptive robust multi-period economic dispatch with dynamic uncertainty sets and a rolling-horizon simulation platform. Experiments using real wind data report improved cost efficiency and system reliability over look-ahead and static-set robust dispatch models.

  • Problem

    High wind penetration makes intermittent and stochastic wind output a significant uncertainty for power-system economic dispatch, while robust ED benefits and dynamic uncertainty modeling remain insufficiently explored.

  • Method

    The paper combines a two-stage adaptive robust multi-period ED model, data-driven dynamic uncertainty sets, statistical procedures, and a rolling-horizon simulation platform.

  • Results

    Extensive real-wind-data simulations show that robust ED outperforms look-ahead ED and static-set robust ED in cost efficiency and system reliability.

  • Takeaways & Limitations

    Dynamic uncertainty sets provide a tractable way to represent temporal and spatial wind correlations while supporting robust dispatch decisions.

Abstract

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The exceptional benefits of wind power as an environmentally responsible renewable energy resource have led to an increasing penetration of wind energy in today's power systems. This trend has started to reshape the paradigms of power system operations, as dealing with uncertainty caused by the highly intermittent and uncertain wind power becomes a significant issue. Motivated by this, we present a new framework using adaptive robust optimization for the economic dispatch of power systems with high level of wind penetration. In particular, we propose an adaptive robust optimization model for multi-period economic dispatch, and introduce the concept of dynamic uncertainty sets and methods to construct such sets to model temporal and spatial correlations of uncertainty. We also develop a simulation platform which combines the proposed robust economic dispatch model with statistical prediction tools in a rolling horizon framework. We have conducted extensive computational experiments on this platform using real wind data. The results are promising and demonstrate the benefits of our approach in terms of cost and reliability over existing robust optimization models as well as recent look-ahead dispatch models.

I. INTRODUCTION

High wind penetration makes stochastic, intermittent production a significant challenge for short-term and real-time dispatch. The paper addresses gaps in robust economic dispatch by proposing adaptive decisions and dynamic, data-driven uncertainty sets.

  • Wind’s stochastic and intermittent output introduces significant uncertainty into day-ahead unit commitment and real-time economic dispatch.
  • Existing uncertainty research has focused mainly on day-ahead unit commitment, while stochastic and robust economic dispatch remain less explored.
  • The paper proposes a two-stage adaptive robust model for multi-period economic dispatch in a rolling-horizon framework.
  • The framework introduces data-driven dynamic uncertainty sets and integrates them with statistical procedures in a simulation platform using real-time data.
  • Static uncertainty sets do not explicitly capture temporal and spatial correlations or distinguish wind uncertainty from conventional demand uncertainty.

B. Dynamic uncertainty sets

Dynamic uncertainty sets model uncertainty at each stage as dependent on earlier realizations, allowing temporal dynamics and cross-resource correlations to be represented. The paper specializes this framework for wind speed and wind power using tractable linear models, time-series analysis, and power-curve approximations.

  • Dynamic uncertainty sets explicitly model correlations among uncertain resources within a period and the evolution of each resource across periods.
  • The uncertainty vector at time t depends on earlier uncertainty stages through convex dynamic constraints, with semidefinite representability providing computational tractability.
  • Linear dynamic uncertainty sets provide a tractable specialization that combines linear dynamics with time-series analysis tools.
  • Constructing dynamic uncertainty sets for wind power: Wind-speed uncertainty is constructed from seasonal patterns, autoregressive residual relationships, and error terms across a selected history length L.
  • Constructing dynamic uncertainty sets for wind power: Wind-power uncertainty is obtained from wind-speed sets through convex piecewise-linear approximations of each wind farm’s increasing power-curve region.
  • Constructing dynamic uncertainty sets for wind power: The resulting sets capture temporal wind dynamics and spatial correlations while distinguishing wind-power uncertainty from conventional demand uncertainty.

A. Mathematical formulation

The model formulates multi-period economic dispatch as a two-stage adaptive robust optimization problem, with current-period decisions implemented immediately and future dispatch adapted to uncertainty. It incorporates generation, ramping, transmission, and energy-balance constraints while allowing wind generation to be dispatchable.

  • The first stage observes current demand and available wind power, then determines the dispatch implemented at time t = 1.
  • The second stage computes worst-case dispatch cost and future dispatch decisions for periods t = 2, ..., T under uncertain demand and wind power.
  • Transmission line-flow limits and energy-balance constraints ensure network feasibility at each dispatch period.
  • Thermal and wind generation constraints enforce output limits, while ramping constraints link dispatch decisions across consecutive time periods.
  • The fully adaptive formulation allows second-stage dispatch to respond to every uncertainty realization and supports rolling-horizon implementation with periodically updated uncertainty sets.

B. Solution method

The solution method combines constraint generation for the outer robust problem with an alternating-direction heuristic for evaluating the nonconvex second-stage problem. The overall procedure iteratively adds worst-case scenarios and associated second-stage decisions.

  • The outer-level formulation has a structure suitable for constraint generation by iteratively adding extreme uncertainty realizations and associated second-stage decisions.
  • The second-stage evaluation is a bilinear max-min problem over separate polyhedral regions for uncertainty and dual variables.
  • The alternating-direction algorithm optimizes over dual variables with uncertainty fixed, then over uncertainty with dual variables fixed, solving a linear program at each step.
  • The alternating-direction method converges to a KKT point and shows empirical good solution quality and fast convergence compared with the MIP method.
  • The overall two-level algorithm repeatedly solves the separation problem, evaluates the second stage, and updates the master problem with new variables and constraints.

IV. SIMULATION PLATFORM AND EVALUATION METRICS

The simulation platform integrates robust economic dispatch with data-analysis procedures in a rolling-horizon environment. It repeatedly updates dispatch decisions and dynamic uncertainty sets as new demand and wind observations arrive.

  • The platform integrates the dispatch optimization engine with data-analysis tools that dynamically update optimization and uncertainty-model parameters.
  • At each time period, the robust economic dispatch model is solved over a window of T periods, but only the current first-stage dispatch is implemented.
  • The horizon advances by one time interval as new demand and available wind-power realizations are observed.
  • Dynamic uncertainty sets are periodically reestimated and updated using the newly observed data.
  • Production-cost evaluation uses the average and standard deviation over each 10-minute dispatch interval, including generation and penalty costs.

A. Estimating the parameters of the dynamic uncertainty set for wind speeds

The dynamic wind-speed uncertainty model separates deterministic seasonal patterns from autoregressive deviations. Its parameters are estimated using regression and multivariate time-series inference, while the framework can accommodate more sophisticated models.

  • Wind-speed vectors are represented as a deterministic seasonal pattern plus deviations that follow a multivariate autoregressive process of order L.
  • Daily and semi-daily seasonalities can be modeled using trigonometric functions over 10-minute intervals.
  • Seasonal-pattern coefficients are estimated by linear regression.
  • Autoregressive matrices and the innovation covariance matrix are estimated using statistical inference techniques for time series, with B obtained by Cholesky decomposition.
  • The dynamic-uncertainty-set framework can incorporate more sophisticated statistical models beyond the simple linear dynamic model.

V. COMPUTATIONAL EXPERIMENTS

The experiments evaluate adaptive robust economic dispatch with dynamic uncertainty sets on modified 14-bus and 118-bus systems using real wind data in a rolling-horizon simulation. Results indicate substantial cost, variability, and shortage-reliability advantages over deterministic and existing robust dispatch models.

  • Experiments compare robust ED with dynamic uncertainty sets against deterministic and existing robust dispatch models on modified 14-bus and 118-bus systems.The systems incorporate significant wind penetration.
  • The simulation uses nine 10-minute periods, a 1.5-hour look-ahead, and rolling-horizon optimization over 35 days of real wind data.Dynamic uncertainty-set parameters are updated using observations available up to each day.
  • Each robust ED solves in less than a second, while simulating 5040 periods takes about 40 minutes on a laptop.
  • Adaptive robust ED substantially reduces average production cost, cost variability, and shortage-event probability relative to the alternatives.

A. Robust ED versus look-ahead ED

Against deterministic look-ahead dispatch, robust ED with dynamic uncertainty sets improves cost and reliability by preparing for adverse wind trajectories. It does so by increasing thermal generation and moderately curtailing wind to preserve ramping capability.

  • Cost and reliability performance: At Γw = 0.5, Rob-ED reduces average cost by 7.1% and cost standard deviation by 41.2% relative to LA-ED.
  • Cost and reliability performance: At Γw = 1.0, Rob-ED reduces cost standard deviation by 82.1%, average cost by 3.75%, shortage frequency by up to 80.1%, and penalty cost by 97.3%.
  • Operational behavior: At Γw = 0.5, Rob-ED increases average thermal generation by 4.3% and decreases wind generation by 8.1% relative to LA-ED.
  • Operational behavior: When available wind drops rapidly, LA-ED incurs a penalty-cost spike from insufficient ramping capacity, whereas Rob-ED is much less affected.
  • Operational behavior: Rob-ED hedges against large interperiod wind swings by selecting detrimental future scenarios and preserving ramping capability through thermal generation and wind curtailment.

3) Comparing to look-ahead ED with reserve:

Reserve rules improve deterministic look-ahead dispatch when properly calibrated, but adaptive robust ED performs better in the reported comparisons. Dynamic uncertainty sets also dominate static alternatives across cost-reliability tradeoffs.

  • Comparing to look-ahead ED with reserve: A reserve factor of 2.5% improves LA-ED cost effectiveness and reliability, while larger reserve factors reduce reliability metrics further but increase average cost.
  • Comparing to look-ahead ED with reserve: Compared with the best tested reserve cases, Rob-ED reduces average cost by at least 7.14%, cost standard deviation by at least 37.4%, penalty cost by at least 57.2%, and penalty frequency by at least 50.3%.
  • Dynamic versus static uncertainty sets: DUS models temporal and spatial uncertainty correlations, whereas SUS1 ignores temporal correlation and SUS2 additionally ignores spatial correlation.
  • Dynamic versus static uncertainty sets: As Γw increases, cost average and standard deviation initially decrease; beyond Γw around 0.4 to 0.5, standard deviation continues decreasing while average cost rises.
  • Dynamic versus static uncertainty sets: DUS has the lowest Pareto frontier, achieving lower cost at equal reliability or lower variability at equal average cost than static uncertainty sets.
  • Dynamic versus static uncertainty sets: With added time budgets, higher-budget SUS1 curves improve their Pareto frontiers, but DUS still clearly dominates the SUS1 curves.
  • Dynamic versus static uncertainty sets: DUS dominance over static uncertainty sets with time budgets is more pronounced for SUS2 comparisons.

C. Impact of system ramping capacity

Robust ED retains a cost advantage over look-ahead ED across reduced, baseline, and increased generator ramping capacities. When demand uncertainty is added, moderate conservatism improves the cost-reliability frontier, while excessive conservatism produces inferior solutions.

  • Impact of system ramping capacity: The ramping experiments show a robust ED cost advantage across a wide range of system ramping conditions.
  • Impact of system ramping capacity: Rob-ED saves 7.1% average cost in the base case, 21.2% with 25% reduced ramping, and 3.7% with 25% increased ramping relative to LA-ED.
  • Considering both demand and wind uncertainty: Demand realizations are independently generated as nonnegative normal variables with standard deviation equal to 5% of each load’s mean, so they can fall outside the uncertainty set.
  • Considering both demand and wind uncertainty: Setting Γd = 1 shifts the cost-reliability curve downward and consistently dominates the wind-only curve, while Γd = 3 yields inferior solutions from excessive load conservatism.
  • Considering both demand and wind uncertainty: With Γd = 1 and Γw = 0.6, the best robust ED policy reduces average cost by 13.1% relative to deterministic LA-ED with Γd = Γw = 0.

E. Performance of the alternating direction method for solving the second-stage problem

The alternating direction (AD) algorithm is proposed as a fast heuristic for the bilinear second-stage problem arising in the robust ED algorithm. Experiments show reliable convergence and substantially lower runtime than the exact MIP method, with a modest optimality gap.

  • Algorithmic motivation: The inner bilinear program is solved by an alternating direction method because exact MIP approaches are time consuming and rely on uncertainty-set structures unavailable for dynamic sets.The AD method is designed for general dynamic uncertainty sets and is used within the outer master-problem iterations.
  • Computational experiment: The experiment solves 1529 inner bilinear programs from 720 Rob-ED models over a 5-day rolling horizon, comparing AD and exact MIP runtime and solution quality.Both methods are evaluated on the same instances.
  • Computational results: The AD algorithm converges on all 1529 instances with an average running time of 0.12s.The MIP method converges on only 257 instances within the reported 60-second limit, averaging 13.28s on those converged cases.
  • Computational results: For the remaining 1272 instances, representing 83.2% of all cases, MIP does not converge after 60s and its solution quality is 1.02% worse than AD on average.Running those instances for another 10 minutes produces little improvement.
  • Solution quality: AD solutions have a 3.73% average optimality gap relative to the global optimum of converged MIP solutions, supporting AD as an effective and efficient heuristic.The larger 118-bus simulations support similar overall conclusions for the proposed robust ED framework.
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