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Robust Risk-Constrained Unit Commitment with Large-scale Wind Generation: An Adjustable Uncertainty Set Approach

Cheng Wang, Feng Liu, Jianhui Wang, Feng Qiu, Wei Wei, Shengwei Mei, Shunbo Lei

arXiv:1510.03318v5math.OC

TL;DR

Robust UC research leaves open how to quantify losses when wind realizations exceed the prescribed uncertainty set and how to choose that set’s size. This paper formulates RRUC with adjustable wind boundaries and operational risk constraints. Its formulation optimizes admissible boundaries for minimum operational risk, and its optimal value remains unchanged as K varies from 0.1 to 1.

  • Problem

    Robust UC leaves open how large out-of-set wind losses may be and how to choose the prescribed uncertainty set’s size.

    Source: §I · p. 2

  • Method

    RRUC makes wind-output boundaries first-stage decisions in a robust UC formulation with an operational-risk constraint.

    Source: C. Compact Model of RRUC · p. 5, §V · p. 9

  • Results

    The optimal value of (4) remains unchanged while K varies from 0.1 to 1.

    Source: E. Impact of penalty coefficient K · p. 8

  • Takeaways & Limitations

    RRUC’s solution provides an optimal admissible boundary for the uncertainty set in the sense of minimum operational risk.

    Source: §V · p. 9

    The two-stage modeling framework may not apply directly to power-market decision making with multiple operation stages, though it may fit decisions such as RAA or RAC.

    Source: §V · p. 9

Abstract

from arXiv · show

This paper addresses two vital issues which are barely discussed in the literature on robust unit commitment (RUC): 1) how much the potential operational loss could be if the realization of uncertainty is beyond the prescribed uncertainty set; 2) how large the prescribed uncertainty set should be when it is used for RUC decision making. In this regard, a robust risk-constrained unit commitment (RRUC) formulation is proposed to cope with large-scale volatile and uncertain wind generation. Differing from existing RUC formulations, the wind generation uncertainty set in RRUC is adjustable via choosing diverse levels of operational risk. By optimizing the uncertainty set, RRUC can allocate operational flexibility of power systems over spatial and temporal domains optimally, reducing operational cost in a risk-constrained manner. Moreover, since impact of wind generation realization out of the prescribed uncertainty set on operational risk is taken into account, RRUC outperforms RUC in the case of rare events. Three algorithms based on column and constraint generation (C&CG) are derived to solve the RRUC. As the proposed algorithms are quite general, they can also apply to other RUC models to improve their computational efficiency. Simulations on a modified IEEE 118-bus system demonstrate the effectiveness and efficiency of the proposed methodology

Decision Variables

Wind variability challenges reliable, economical day-ahead dispatch, while robust UC leaves both out-of-set operational losses and uncertainty-set sizing unresolved. RRUC makes operational risk a constraint and adjusts the uncertainty set as part of its decisions.

Decision Variables

  • Commitment variables indicate generator on/off status and startup; output variables distinguish real-time generation from day-ahead generation.The listed wind and recourse variables include normalized positive/negative deviations, wind curtailment, load shedding, wind-output bounds, and operational risk by direction.Source: Decision Variables · pp. 1–2

I. INTRODUCTION

  • Stochastic UC can miss rare, severe scenarios; robust UC guarantees feasibility within its prescribed set but may produce conservative strategies.Other approaches seek to address conservatism, including minimax regret, hybrid stochastic/interval, and multi-band uncertainty-set models.Source: §I · p. 2
  • Rare wind realizations beyond a prescribed uncertainty set may cause operational infeasibility and loss, motivating risk measures for UC decisions.The set’s appropriate size is also unresolved, while common confidence-level choices can be subjective and difficult to determine across periods.Source: §I · p. 2
  • RRUC uses expected wind-curtailment and load-shedding losses as operational risk, imposing risk as a constraint rather than an objective.Its two-stage formulation minimizes UC and forecast-wind dispatch costs in the first stage and checks feasibility against uncertainty in the second.Source: §I · pp. 2–3
  • RRUC makes uncertainty-set boundaries first-stage decisions, allocating operational flexibility across spatial and temporal domains.The risk constraint guarantees feasibility in normal scenarios within the set while limiting operational risk beyond it.Source: §I · p. 3
  • The optimal solution gives exact admissible wind-generation boundaries alongside the UC decision, without additional computation.RUC boundaries may exceed the prescribed uncertainty set, requiring a separate admissibility assessment after commitment.Source: §I · p. 3

II. MATHEMATICAL FORMULATION

The formulation defines operational loss around riskless and inadmissible wind-generation regions, then embeds a risk limit and adjustable wind uncertainty set in a two-stage robust UC model. A penalty-based reformulation selects risk-minimizing admissible boundaries, with its approximation gap controllable through K.

A. Operational Risk under Wind Generation Uncertainty

  • Wind realizations inside the admissible region are riskless, while realizations intersecting the inadmissible region may cause operational loss.The operational-risk expression accounts for under- and overestimated wind output and is approximated with piecewise linearization.Source: A. Operational Risk under Wind… · p. 3

B. RRUC Formulation

  • RRUC minimizes UC and dispatch costs at forecast wind output while requiring dispatch feasibility under an adjustable uncertainty set.The formulation is two-stage: first-stage commitment and dispatch decisions face uncertainty in the second stage.Source: B. RRUC Formulation · pp. 3–4
  • The first-stage model includes an operational-risk limit, while its objective combines UC cost and dispatch cost at forecast wind output.The risk threshold influences both the resulting strategy and solvability; operators may choose it using historical data, risk preference, or contracts.Source: B. RRUC Formulation · pp. 3–4
  • The recourse model sums load shedding and wind curtailment while enforcing generator, balance, and network constraints.Its feasibility condition prevents load shedding or wind curtailment in recourse, guaranteeing feasibility of commitment and admissibility of the selected wind boundaries.Source: B. RRUC Formulation · p. 4
  • Wind uncertainty is a decision variable described by output boundaries and uncertainty budgets over temporal and spatial domains.Unlike the cited RUC set descriptions, the proposed uncertainty set is adjustable.Source: B. RRUC Formulation · p. 4
  • Adding operational risk with penalty coefficient K makes the selected wind boundaries optimal for risk minimization under the UC strategy.The resulting model is an approximation of the original, and its optimality gap can be controlled by tuning K.Source: B. RRUC Formulation · p. 4
  • Some UC constraints have equivalent tighter forms, including minimum on/off-time constraints.The paper points to other tight UC formulations for readers seeking stronger formulations.Source: B. RRUC Formulation · p. 4

C. Compact Model of RRUC

The compact RRUC model represents the two-stage formulation with vector variables and linear constraints. Making wind-output boundaries first-stage decisions increases its computational complexity relative to RUC.

C. Compact Model of RRUC

  • The compact form expresses RRUC through a first-stage objective and constraints linking generator, wind-boundary, risk, recourse, and uncertainty variables.The paper derives the compact formulation from the detailed model.Source: C. Compact Model of RRUC · p. 5
  • The compact variables include binary generator decisions, continuous generator outputs, wind-output boundaries, operational risk, load shedding, wind curtailment, and binary wind uncertainty.The wind-boundary vector is determined in the first stage.Source: C. Compact Model of RRUC · p. 5
  • RRUC involves more variables and constraints than RUC, and choosing wind boundaries in the first stage considerably increases computational complexity.The model also uses a Hadamard product in one constraint.Source: C. Compact Model of RRUC · p. 5

III. SOLUTION METHODOLOGY

The solution method separates the compact model into a main problem and a feasibility-and-admissibility checking subproblem. It combines C&CG for the main problem with methods targeting subproblem scale and convergence speed.

III. SOLUTION METHODOLOGY

  • The solution framework separates the compact model into a Main Problem and a Feasibility & Admissibility Checking Subproblem.The latter checks feasibility and admissibility.Source: §III · p. 5
  • A C&CG-based algorithm is used to solve the Main Problem.The methodology first develops the approach for the checking subproblem, then applies C&CG to the main problem.Source: §III · p. 5
  • The methodology also derives an algorithm to reduce checking-subproblem computational scale and one to accelerate Main Problem convergence.These are presented alongside the C&CG-based Main Problem algorithm.Source: §III · p. 5

A. Solution Methodology for F&ACSP

The paper converts F&ACSP into a single-level MILP using duality and big-M linearization. Its solution efficiency depends on the scale of the auxiliary variables and constraints, which in turn relates to the sparsity of G.

A. Solution Methodology for F&ACSP

  • F&ACSP is treated as a bi-level MILP, and its inner problem is replaced by its dual to obtain a single-level bilinear program.The paper uses big-M linearization rather than outer approximation because outer approximation may fail to find a global optimum in some circumstances.Source: A. Solution Methodology for F&ACSP · p. 5
  • Big-M linearization converts the bilinear problem into a standard single-level MILP solvable by commercial solvers such as CPLEX.The linearization introduces auxiliary variables and constraints.Source: A. Solution Methodology for F&ACSP · p. 5
  • Solution efficiency for (8) is directly proportional to the scale of γ and its big-M constraints, which depends on the number of nonzero elements in G.Improving G's sparsity reduces the computational scale of (8).Source: A. Solution Methodology for F&ACSP · p. 5

B. Solution Methodology for RRUC Problem

A1 applies C&CG to RRUC by iteratively solving the master problem and F&ACSP, adding constraints based on identified worst-case scenarios. Its added master constraints differ from the original F&ACSP constraints, which are looser because recourse actions are included.

B. Solution Methodology for RRUC Problem

  • A1 initializes the scenario index and set, then solves the master problem with added constraints before proceeding to model (8).The convergence gap is represented by ε.Source: B. Solution Methodology for RRUC Problem · p. 5
  • A1 iterates between solving the master problem and model (8), terminating when the objective-value gap is below ε.Otherwise, it uses the next optimal scenario to create a variable vector and add constraints to the master problem.Source: B. Solution Methodology for RRUC Problem · p. 5
  • Each iteration adds master-problem constraints based on the identified worst-case scenario, but these differ from the original F&ACSP constraints.The original constraints can be viewed as looser because they include slack variables for recourse actions.Source: B. Solution Methodology for RRUC Problem · p. 5

C. Computational Scale Reduction

The section reduces computational scale by replacing network constraints with nodal power-balance and phase-angle constraints, then exploiting a sparser matrix representation. The resulting approach shifts some work to the inner problem while reducing the variables and constraints passed to the master problem.

C. Computational Scale Reduction

  • Replacing (3r) and (3u) with nodal power-balance and phase-angle constraints uses node admittance rather than the nodal power injection sensitivity matrix.The replacement includes line-flow limits, nodal phase-angle bounds, and a reference phase angle.Source: C. Computational Scale Reduction · p. 6
  • Formulation (13) has more continuous variables and regular constraints for η and v than (8), but substantially fewer of the remaining variables and constraints, especially big-M constraints.The comparison is listed in Table I; the passage does not provide its cell values.Source: C. Computational Scale Reduction · p. 6, Table I · p. 6
  • A2 reduces the variables and constraints generated in each master-problem iteration, while increasing the computational scale of the inner problem and decreasing F&ACSP burden.The comparison of formulations (8) and (13) is presented as a computational-scale comparison.Source: C. Computational Scale Reduction · p. 6, Table I · p. 6
  • The computational-scale reduction approach also applies to other two-stage robust optimization problems with second-stage power balance, and its effectiveness increases with wind-farm count.RUC is given as an example of an applicable model.Source: C. Computational Scale Reduction · p. 6

D. Convergence Acceleration

A3 accelerates convergence by adding a feasibility cut and F&ACSP dual information to the master problem in each iteration. The cut acts as subgradient and gradient cuts for first-stage decisions x and w.

D. Convergence Acceleration

  • A3 adds the feasibility-cut constraints when solving the master problem and terminates when successive objective values differ by less than ε.If the stopping condition is not met, it derives the next scenario and dual solution and augments the constraints.Source: D. Convergence Acceleration · p. 6
  • The feasibility cut serves as a subgradient cut for x and a gradient cut for w.Its construction uses optimal solutions of the master problem and F&ACSP from the preceding iteration.Source: D. Convergence Acceleration · p. 6
  • A3 adds a feasibility cut and the F&ACSP dual variable η to the master problem in each iteration.These are passed to the master problem in addition to the information used by A2.Source: D. Convergence Acceleration · p. 6

IV. CASE STUDIES

The case studies evaluate the proposed methods on a modified IEEE 118-bus system, using specified wind forecasts and uncertainty assumptions. The setup includes three wind farms and a piecewise-linear approximation of wind forecast-error distributions.

A. The Modified IEEE 118-bus System

  • The tested system has 54 generators and 186 transmission lines, with three wind farms connected at buses 17, 66, and 94, respectively.Fig. 2 shows forecasted values for wind farms 1–3.Source: A. The Modified IEEE 118-bus System · p. 6, Figure 2 · p. 7
  • Wind forecast errors have zero mean and farm-specific scale parameters of 20%, 15%, and 10%, with spatial and temporal confidence levels set to 95%.These settings yield uncertainty budgets of approximately 2 spatially and 8 temporally.Source: A. The Modified IEEE 118-bus System · p. 7
  • Wind forecast-error bands are derived from a Gaussian distribution, and an eight-piecewise-linear approximation is used to depict the error PDF.The experiments use the stated approximation settings and solver configuration, including a 0.1% optimality gap.Source: §IV · p. 6, A. The Modified IEEE 118-bus System · p. 7

B. Comparison with Other UC Model

Compared with other unit-commitment models, RRUC has lower operational cost and risk than RUC and the lowest combined cost and risk among the four models. Under rare events, it also yields the lowest total average operational loss; its adjustable uncertainty set corresponds to its admissibility region.

B. Comparison with Other UC Model

C. Uncertainty Set and Admissibility Region

  • RRUC’s uncertainty set varies across spatial and temporal domains, and its admissibility region is identical to that set.By contrast, RUC uses a prescribed 95% forecast-error band, while its admissibility region is larger than that band.Source: C. Uncertainty Set and Admissibility… · p. 8

D. Computational Efficiency

The algorithms improve computational efficiency, while risk level and penalty choices affect RRUC feasibility and solution behavior. The conclusion presents adjustable uncertainty boundaries as a way to manage operational risk, with a two-stage modeling scope limitation.

D. Computational Efficiency

E. Impact of penalty coefficient K

  • The optimal value of (4) remains unchanged as K varies from 0.1 to 1, supporting the proposed method for choosing K.For the described case, K is selected as 0.1 to keep the penalty term's order of magnitude no more than 10^2 and reduce the gap between problems (2) and (4).Source: E. Impact of penalty coefficient K · p. 8

F. Impact of Risk Level

  • 270$ is the minimum feasible risk level in this case; decreasing risk raises operational cost, and risk level and operational risk are not strictly linear.The paper attributes a gap between operational risk and risk level to the noncontinuity of UC variables.Source: Figure 4 · p. 8, F. Impact of Risk Level · p. 8

G. Impact of Numbers of Wind Farms

V. CONCLUSION

  • RRUC optimizes adjustable wind-generation uncertainty boundaries under an operational-risk constraint, and its solution provides an optimal admissible boundary in the sense of minimum operational risk.The formulation is a two-stage robust optimization model.Source: §V · pp. 8–9
  • Three C&CG-based iterative algorithms are proposed to solve RRUC, with simulations on the modified IEEE 118-bus system illustrating the model and algorithms.The simulations also examine the influence of risk level on RRUC.Source: §V · p. 9
  • RRUC may not apply directly to power-system decision making when practical market operations have multiple stages, though it may apply to RAA or RAC.Source: §V · p. 9
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