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Multistage Robust Unit Commitment with Dynamic Uncertainty Sets and Energy Storage

Alvaro Lorca, Xu Andy Sun

arXiv:1604.04890v2math.OC

TL;DR

Wind and solar variability makes reliable, cost-effective unit commitment difficult, while existing two-stage formulations assume knowledge of future uncertainty and often underrepresent temporal and spatial correlations. The paper proposes a multistage adaptive robust UC model with storage, dynamic uncertainty sets, affine dispatch policies, and an efficient solution framework. Computational experiments show scalable solution of a large Polish system and benefits in cost and reliability over traditional robust and deterministic alternatives.

  • Problem

    Reliable and cost-effective UC under abundant wind and solar requires decisions that respect sequential uncertainty revelation while representing temporal and spatial correlations and storage operation.

  • Method

    The paper combines a multistage robust UC model with energy storage, dynamic uncertainty sets, a simplified affine dispatch policy, and constraint-generation and duality-based solution methods.

  • Results

    The proposed robust UC model with policy-guided look-ahead economic dispatch improves operational cost and system reliability over deterministic UC with reserve and look-ahead dispatch, while solving large-scale high-dimensional instances within a suitable day-ahead time budget.

  • Takeaways & Limitations

    The framework provides an approach for operating large-scale power systems with many wind and solar farms and storage devices under high-dimensional uncertainty.

  • Takeaways & Limitations

    The uncertainty model assumes the innovation vector is i.i.d. multivariate normal with zero mean and covariance matrix Σ, and future work must incorporate security constraints.

Abstract

from arXiv · show

The deep penetration of wind and solar power is a critical component of the future power grid. However, the intermittency and stochasticity of these renewable resources bring significant challenges to the reliable and economic operation of power systems. Motivated by these challenges, we present a multistage adaptive robust optimization model for the unit commitment (UC) problem, which models the sequential nature of the dispatch process and utilizes a new type of dynamic uncertainty sets to capture the temporal and spatial correlations of wind and solar power. The model also considers the operation of energy storage devices. We propose a simplified and effective affine policy for dispatch decisions, and develop an efficient algorithmic framework using a combination of constraint generation and duality based reformulation with various improvements. Extensive computational experiments show that the proposed method can efficiently solve multistage robust UC problems on the Polish 2736-bus system under high dimensional uncertainty of 60 wind farms and 30 solar farms. The computational results also suggest that the proposed model leads to significant benefits in both costs and reliability over robust models with traditional uncertainty sets as well as deterministic models with reserve rules.

1 Introduction

The paper develops a multistage robust UC framework for renewable-rich power systems that respects sequential uncertainty revelation, models temporal and spatial renewable correlations, and incorporates storage. It combines affine dispatch policies with an efficient computational framework and evaluates the approach against robust and deterministic alternatives.

  • Motivation: Two-stage robust UC models unrealistically assume dispatch decisions know all future uncertainty, whereas multistage models enforce dependence only on uncertainty revealed up to each decision period.This non-anticipativity is especially relevant for systems with limited ramping capacity and significant renewable variations.
  • Model: The paper proposes a multistage robust UC model with wind and solar availability uncertainty, energy storage, and a simple affine policy for adaptive dispatch decisions.Wind and solar generation are dispatchable, while their availability is the uncertain component.
  • Dispatch: A policy-guided look-ahead economic dispatch model uses the affine dispatch policy from robust UC to improve the robustness of real-time operation.The dispatch process accompanies the day-ahead robust UC model.
  • Uncertainty sets: Dynamic uncertainty sets capture joint temporal and spatial correlations across multiple wind and solar farms while directly modeling renewable power.The approach includes a dimensionality-reduction enhancement and avoids relying on explanatory factors such as wind speed or solar irradiance.
  • Solution method: The solution framework combines constraint generation, duality-based reformulation, outer approximation, one-tree Benders, and constraint screening.It can solve the Polish 2736-bus system with high-dimensional uncertainty within a couple of hours on a modest personal computer.
  • Computational evaluation: Extensive simulations compare the proposed models and algorithms with other robust and deterministic approaches, including deterministic UC with reserve and look-ahead economic dispatch.The conclusion reports improvements in operational cost and system reliability over those alternatives.

2 Multistage Robust Unit Commitment Model

The paper formulates a multistage robust UC model in which commitment decisions precede uncertainty, while dispatch and storage decisions adapt only to renewable power revealed over time. Affine policies make this formulation tractable and support a policy-guided look-ahead economic dispatch process.

  • Fully-Adaptive Model: The model minimizes commitment costs plus worst-case dispatch cost while jointly representing generator, renewable, storage, transmission, and timing decisions.The dispatch is a policy function, and the objective can use piecewise-linear dispatch costs without changing the problem structure.
  • Multistage Adaptation: Dispatch at time t depends only on renewable power realized through time t, enforcing non-anticipativity unlike two-stage robust UC.Two-stage models allow dispatch at time t to depend on uncertainty from all time periods, including future realizations.
  • Affine Dispatch Policy: Affine policies approximate adaptive recourse by making dispatch decisions affine functions of uncertain renewable power deviations.A full affine policy can depend on uncertainty at all buses and prior time periods, but is computationally difficult for large systems.
  • Affine Dispatch Policy: The simplified affine policy makes generator and storage dispatch depend on total available renewable power, while renewable dispatch depends on local available power.This aggregation reduces computational difficulty while retaining an affine representation of dispatch decisions.
  • Policy-Guided Look-Ahead ED: The policy-guided look-ahead ED uses the UC-derived affine policy in robust ramping constraints and enforces multi-period storage levels, with T′ = 3 look-ahead periods.It combines realized renewable availability with forecasts for future periods and guides real-time dispatch using the robust UC policy.

3 Dynamic Uncertainty Set for Wind and Solar Power

The proposed dynamic uncertainty set models available wind and solar power with seasonal components, temporal dynamics, and spatial correlations. Its parameters can be estimated statistically, while dimension reduction controls the computational burden of high-dimensional uncertainty.

  • Dynamic Uncertainty Set: The dynamic uncertainty set captures temporal and spatial correlations in wind and solar power through matrices governing dependence across time and renewable units.The set also includes seasonal components and a budget-over-time parameter ρ.
  • Dynamic Uncertainty Set: A static uncertainty set emerges when temporal coupling, spatial coupling, and the budget-over-time restriction are removed through specific parameter choices.With B as the identity, Al = 0, and ρ = 1, the set becomes separable over time and ignores temporal and spatial correlations.
  • Parameter Interpretation: In the basic box-set case, Γ determines the allowed renewable-power variation in standard-deviation units at each unit and time period.This interpretation applies when the norm is ℓ∞ and git represents the standard deviation of available power.
  • Modeling Choices: The paper directly models renewable-power uncertainty for both wind and solar, rather than transforming wind-speed uncertainty through a power curve.The stated motivation is improved computational efficiency while modeling correlations for both renewable types.
  • Estimation and Dimension Reduction: Parameters are estimated using regression, time-series inference, and Cholesky decomposition, while principal component analysis reduces uncertainty dimension.The retained dimension Nv can range from 1 to the number of renewable units; dimensions near that upper bound produce high-dimensional sets.

4 Solution Method

The solution framework converts the affine multistage robust UC into a tractable deterministic formulation by combining duality-based reformulation with constraint generation. It exploits model structure and specialized reformulations to improve efficiency.

  • Problem Reformulation: The affine multistage robust UC is a semi-infinite mixed-integer problem with finitely many variables but infinitely many robust constraints.A deterministic counterpart is therefore required for computation.
  • Combined Framework: The proposed method combines duality-based reformulation and dynamic generation of violated scenarios, while exploiting special structures in the robust UC model.These two reformulation approaches are integrated rather than used independently.
  • Algorithmic Improvements: Specialized procedures address generation-limit and energy-balance constraints, inter-temporal constraints, and further efficiency enhancements.The solution framework includes an outer approximation method for inter-temporal constraints.

4.1 Constraint Generation Framework

The constraint-generation framework iteratively solves a master robust UC problem, identifies violated robust constraints through worst-case scenarios, and adds the associated scenarios until all constraints are satisfied.

  • Master Problem: The master problem represents robust constraints using current sets of uncertainty-set extreme points.The variable z represents worst-case dispatch cost, K is the number of robust constraints, and Pk stores points for constraint k.
  • Scenario Separation: After solving the master problem, the algorithm checks each robust constraint for violation under the current solution.The check is performed across all K robust constraints.
  • Scenario Addition: When a violation is found, the associated worst-case scenario is added to that constraint’s extreme-point set and the master problem is solved again.This iterative process expands the deterministic master problem only where needed.
  • Termination: The algorithm terminates when every robust constraint satisfies its feasibility condition.The stopping test requires all robust constraints to pass simultaneously.
  • Structure Exploitation: The practical algorithm uses constraint generation for transmission-line flow limits while reformulating other robust constraints more efficiently.This structure-aware design extends the basic constraint-generation framework.

4.2 Reformulation of generation limit and balance constraints

The simplified affine policy enables direct reformulation of robust generation, storage, renewable-output, and energy-balance constraints without fully dualizing the uncertainty set.

  • Deterministic counterparts of robust generation-limit and energy-balance constraints can be derived explicitly without dualizing the uncertainty set.
  • The simplified affine policy directly identifies worst-case scenarios for robust generation-limit constraints.
  • Under affine policy (2), robust generation-limit constraints are equivalent to the system given in Proposition 1.
  • The same type of reformulation applies to storage input/output limits and renewable-unit output limits.
  • Under affine policy (2), robust energy-balance constraints are equivalent to a system of equations for every time period.

4.3 Outer approximation for inter-temporal constraints

The method uses outer approximations and duality to reformulate inter-temporal robust constraints efficiently, addressing dynamic uncertainty sets and energy-storage coupling across time.

  • Inter-temporal cost, ramping, and storage-capacity constraints couple dispatch decisions across consecutive time periods or up to T periods.
  • Direct dualization introduces many variables and constraints, while direct constraint generation may converge slowly for these coupled constraints.
  • The formulation replaces the original projected uncertainty set with an outer approximation involving total renewable-power variables.
  • The outer approximation preserves robust feasibility and uses fewer additional variables than the original uncertainty-set representation.
  • Duality reformulates the resulting robust constraint using nonnegative auxiliary vectors.
  • Unlike static uncertainty sets, dynamic sets are not time-separable, making outer approximation and duality critical for handling storage and dynamic uncertainty efficiently.

4.4 Further Algorithmic Enhancements

The constraint-generation framework is accelerated through one-tree Benders, fast upper-bound screening, and selective checking of potentially active robust constraints.

  • Constraint generation is enhanced with techniques designed to reduce the computational burden of repeatedly checking robust constraints.
  • One-tree Benders adds generated constraints during a single branch-and-bound tree instead of solving many mixed-integer programs from scratch.
  • Each separation problem is a linear program over the dynamic uncertainty set for each robust constraint.
  • An upper bound screens constraints without solving the separation linear program when ub_k(W) ≤ b_k(w,z).
  • Interval-type outer-approximation sets allow rapid upper-bound computation, especially for transmission constraints whose coefficient signs can be checked directly.
  • The algorithm temporarily checks only robust constraints violated or nearly violated in the previous iteration, returning to all constraints after master convergence.

5 Computational Experiments

Experiments evaluate the proposed solution method and compare dynamic and static multistage robust UC against deterministic UC on a large Polish power-system instance. The results show computational tractability, small outer-approximation quality loss, and improved cost, reliability, storage utilization, and renewable utilization for the dynamic robust approach.

  • Experimental setup: The test system has 289 generators, 60 wind farms, 30 solar farms, 10 storage units, 2,011 demand nodes, and 100 transmission lines.It has 28,880 MW generation capacity, 10,689 MW wind capacity, 6,299 MW solar capacity, and 600 MW storage output capacity.
  • Experimental setup: The 24-hour experiments estimate dynamic uncertainty-set parameters from 30 days of NREL data, using L = 1, Nv = 25, and ρ = 0.1.These settings produce a polyhedral uncertainty set and balance uncertainty representation with computational tractability.
  • Solution-method performance: All solution-method enhancements improve efficiency by integrating robust reformulations, outer approximation, one-tree Benders, and constraint screening.Constraint screening reduces separation problems by quickly identifying robust constraints that are not violated.
  • Solution-method performance: Outer approximation incurs small worst-case-cost quality loss, especially for small Γ, compared with runs exceeding the six-hour limit without it.The comparison supports outer approximation as an effective computational technique for these large-scale models.
  • Robust UC versus deterministic UC: Compared with DetUC at Γ = 4, RobUC-Dynamic at Γ = 1 reduces average cost by 7.62%, cost std by 91.64%, and CVaR by 35.49%, while eliminating penalty.RobUC-Static at Γ = 3 achieves corresponding reductions of 7.04%, 90.77%, and 34.87%, and also eliminates penalty.
  • Dynamic versus static robust UC: RobUC-Dynamic at Γ = 1 outperforms RobUC-Static at Γ = 3, with 0.62% lower average cost, 9.57% lower cost std, and 0.96% lower CVaR.It also uses storage more, utilizes more renewable power, and reduces renewable curtailment relative to the static model.

6 Conclusion

The paper presents an effective framework for large-scale multistage robust UC with dynamic uncertainty sets, storage, and policy-guided dispatch. Its approach improves operational cost, reliability, storage utilization, and renewable curtailment relative to existing approaches.

  • 6 Conclusion: The proposed algorithm solves large-scale multistage robust UC models with high-dimensional uncertainty within a day-ahead-operation time budget.The model includes significant wind and solar power and storage units.
  • 6 Conclusion: The proposed robust UC model with novel ED dominates deterministic UC with reserve and look-ahead ED in operational cost and system reliability.The comparison is reported as an outcome of extensive computational experiments.
  • 6 Conclusion: Dynamic uncertainty sets capture temporal and spatial correlations of wind and solar power, while the new ED method increases storage utilization and reduces renewable curtailment.These features further improve the performance and operational use of renewable resources and storage.
  • 6 Conclusion: The proposed framework combines multistage robust UC, dynamic uncertainty sets, policy-guided look-ahead ED, and an improved solution methodology.It targets large-scale systems with many wind and solar farms and storage devices.
  • 6 Conclusion: The method significantly improves over existing deterministic and multistage robust UC models and solution methods.The conclusion characterizes the approach as novel and effective for operating large-scale power systems.
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