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Stochastic Unit Commitment in Low-Inertia Grids

Matthieu Paturet, Uros Markovic, Stefanos Delikaraoglou, Evangelos Vrettos, Petros Aristidou, Gabriela Hug

arXiv:1904.03030v1math.OC

TL;DR

The paper addresses how low rotational inertia and renewable uncertainty affect frequency-secure unit commitment. It derives frequency constraints from a uniform response model and embeds them in a two-stage stochastic UC, using bounds to linearize the nonlinear nadir constraint. In simulations, frequency constraints changed scheduling and increased costs by requiring additional synchronous generators for inertia.

  • Problem

    Low-inertia, renewable-rich systems require UC decisions that account for RoCoF, frequency nadir, and steady-state frequency deviation, not inertia alone.

  • Method

    The paper derives analytic frequency metrics from a uniform multi-machine model and incorporates them into a two-stage stochastic UC covering wind and equipment uncertainties.

  • Results

    5% increase in total expected system costs and 185% increase in start-up costs followed the addition of frequency constraints in the studied system.

  • Takeaways & Limitations

    Frequency-secure UC can require additional synchronous machines to provide inertia, substantially changing generator scheduling and operating costs.

Abstract

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In this paper, the Unit Commitment (UC) problem in a power network with low levels of rotational inertia is studied. Frequency-related constraints, namely the limitation on Rate-of-Change-of-Frequency (RoCoF), frequency nadir and steady-state frequency error, are derived from a uniform system frequency response model and included into a stochastic UC that accounts for wind power and equipment contingency uncertainties using a scenario-tree approach. In contrast to the linear RoCoF and steady-state frequency error constraints, the nadir constraint is highly nonlinear. To preserve the mixed-integer linear formulation of the stochastic UC model, we propose a computationally efficient approach that allows to recast the nadir constraint by introducing appropriate bounds on relevant decision variables of the UC model. For medium-sized networks, this method is shown to be computationally more efficient than a piece-wise linearization method adapted from the literature. Simulation results for a modified IEEE RTS-96 system revealed that the inclusion of inertia-related constraints significantly influences the UC decisions and increases total costs, as more synchronous machines are forced to be online to provide inertial response.

NOMENCLATURE

The nomenclature defines sets for network components, scenarios, and generation units, alongside variables and parameters used in stochastic unit commitment and frequency modeling.

  • Sets: Scenarios ξ combine generation outages and wind-power uncertainty.ξ = {c, ω}, with c denoting generation outages and ω wind-power uncertainty.
  • Sets: The model distinguishes conventional units, converter-based units, nodes, transmission lines, and units located at each bus.These sets support network representation and generator-specific scheduling.
  • Frequency variables: M^ξ_t and R^ξ_t denote global system inertia and global system droop factor in scenario ξ.F^ξ_t represents the global fraction of synchronous-generator turbine power, while ΔP^ξ_t denotes outage size.
  • Operational and economic quantities: The notation includes wind spillage, load shedding, reserve deployment, line susceptance and capacity, generator limits, ramp limits, and cost parameters.These quantities support operational, network, uncertainty, and objective components of the UC model.
  • Decision variables: Commitment variables u_it, start-up variables y_it, and shut-down variables z_it represent conventional-unit scheduling decisions.The notation also includes day-ahead conventional and wind dispatch variables.

I. INTRODUCTION

The introduction motivates frequency-aware UC for low-inertia, renewable-rich systems and presents a model combining detailed frequency dynamics, multiple uncertainty sources, and improved nadir handling.

  • Motivation: Higher renewable penetration reduces the frequency-stabilizing role of synchronous-generator rotational inertia.Stored kinetic energy slows frequency dynamics and reduces RoCoF after generation-demand imbalances.
  • Motivation: Low-inertia systems require minimum inertia requirements, which can alter day-ahead UC by keeping synchronous generators online solely for inertia.This creates a direct interaction between frequency security and generator scheduling.
  • Research gap: Prior UC studies derived RoCoF constraints but omitted frequency-deviation metrics, motivating inclusion of frequency nadir and quasi steady-state deviation.The introduction identifies this as a limitation of swing-equation approaches based only on center-of-inertia dynamics.
  • Contributions: The study models synchronous generators together with droop-controlled and VSM converters in a realistic low-inertia system.It derives frequency metrics as functions of inertia, damping, aggregate droop gain, and related system variables.
  • Contributions: The paper proposes bounds on nadir-related decision variables and a more comprehensive event-probability and scenario-tree structure for two-stage stochastic UC.The bounds support treatment of the nonlinear nadir constraint within the UC framework.

B. Analytic Derivation of Frequency Metrics

A uniform frequency-response model reduces the multi-machine dynamics to aggregate system parameters and yields analytic expressions for RoCoF, nadir, and steady-state deviation.

  • Model reduction: The transfer function combines traditional generators, droop converters, and VSM converters in a general-order frequency-dynamics model.The model represents the respective generator and converter control contributions.
  • Model reduction: Assuming T ≫ T_j ≈ 0 simplifies the dynamics because synchronous-machine time constants greatly exceed converter time constants.The approximation assumes similar synchronous-machine time constants and much faster converter dynamics.
  • Frequency metrics: The frequency metrics depend directly on weighted system and synchronous-generator averages M, D, F_g, and R_g.These averages summarize inertia, damping, turbine-power fraction, and droop-related behavior.
  • Frequency metrics: A stepwise electrical-power disturbance yields time-domain frequency deviation and analytic expressions for nadir, RoCoF, and steady-state deviation.The disturbance is represented as ΔP_e(s) = −ΔP/s.
  • Frequency metrics: RoCoF scales as M^-1 and steady-state deviation scales as (D + R_g)^-1, whereas nadir depends nonlinearly on M, D, R_g, and F_g.These aggregate parameters can therefore be regulated through UC decisions.

III. FORMULATION OF FREQUENCY CONSTRAINTS

The derived frequency expressions are imposed as UC constraints using prescribed thresholds, with linear RoCoF and steady-state limits and a separately approximated nonlinear nadir constraint.

  • Constraint formulation: The frequency expressions are converted into SI units and bounded by prescribed ENTSO-E thresholds within the stochastic UC.The constraints cover frequency nadir, RoCoF, and steady-state deviation.
  • Thresholds: The base frequency is 50 Hz, and the UFLS trigger is Δf_lim = 0.4 Hz.These values define the nadir-related frequency boundary.
  • Thresholds: The maximum permissible RoCoF is ḟ_lim = 0.5 Hz/s, while the maximum steady-state deviation is Δf_ss,lim = 0.2 Hz.These limits define the linear frequency constraints.
  • Constraint formulation: Linear RoCoF and steady-state constraints contrast with the nonlinear nadir constraint.The distinction determines how each frequency requirement is represented computationally.
  • Constraint formulation: A linear approximation of the nadir constraint preserves the stochastic UC formulation as a MILP with a measurable optimality gap.This avoids the computational burden of a MINLP formulation.

A. Piece-wise Linearization of Nadir Expression

The reviewed PWL approach approximates the nonlinear frequency-nadir function with linear segments so the nadir constraint can be integrated into a MILP UC model. It fixes damping and omits one degree of freedom when illustrating the approximation at fixed inertia.

  • PWL formulation: PWL approximates the frequency-nadir function to integrate its constraint into a linearized UC formulation.The nadir depends on Rg, Fg, M, and D; the reviewed approach assumes constant D because damping and droop gains are narrowly prescribed.
  • PWL formulation: The approximation minimizes deviations between selected PWL segments and the nadir function across evaluation points.An inner maximum selects the segment closest to the curve at each evaluation point.
  • MILP integration: The resulting optimal segments are converted into MILP inequalities together with the nadir threshold constraint.The approximation is illustrated for a 20-generator test system after loss of the largest unit.
  • Illustration: Fig. 2 fixes M = 9 and therefore presents a two-variable surface rather than the full nadir function.The original surface and its PWL approximation segments are compared in the figure.

B. Extracting Bounds on Relevant Variables

The proposed bound-extraction method replaces direct nadir linearization by identifying combinations of Rg, Fg, M, and D that satisfy the nadir threshold. It is faster and less error-prone for the studied 20-generator system, although its enumeration grows exponentially with system size.

  • Bound extraction: The method confines Rg, Fg, M, and D to ranges that guarantee satisfaction of the nadir threshold.Unlike the PWL approach, damping D can remain variable rather than being fixed.
  • Bound extraction: Feasible dispatch combinations are extracted from points below the shaded threshold plane in the nadir scatter plot.These values are then used to substitute the nadir constraint in the UC formulation.
  • Computational comparison: The PWL method is more time-intensive, especially when higher approximation precision is required.Table I compares the computational time needed to obtain equivalent linear nadir equations for one power-outage value.
  • Computational limitation: For every additional Δ|I| generators, the bound-extraction computation time increases by a factor of 2^Δ|I|.This growth results from enumerating 2^|I|−1 generator combinations in a system with |I| generators.
  • Computational comparison: The bound-extraction method is significantly faster and introduces less error on the investigated 20-generator test system.The paper therefore uses this method for the subsequent analysis.
  • Uncertainty modeling: The stochastic UC combines equipment-failure and wind-production uncertainty into scenarios formed from contingency and wind-realization pairs.Wind scenarios capture spatio-temporal forecast-error dependence, while credible contingencies concern synchronous-generator outages; transmission assets are assumed fully reliable.

V. STOCHASTIC UNIT COMMITMENT

The paper formulates a two-stage stochastic UC model that adds frequency constraints to day-ahead scheduling and real-time operation under uncertainty. Its formulation minimizes expected energy, balancing, start-up, shut-down, and load-shedding costs while enforcing network, reserve, ramping, and frequency requirements.

  • Model structure: The proposed stochastic UC is a two-stage optimization model with additional frequency-related constraints.The model incorporates uncertainty realizations through scenario-indexed operating decisions.
  • Decision variables: The formulation retains binary commitment, start-up, and shut-down variables alongside continuous dispatch, reserve, network, and frequency variables.The variable domains and optimization-variable set are specified explicitly in constraints (11y) and Φ.
  • Objective: The objective minimizes total expected system cost across day-ahead energy and real-time balancing operations.Costs include fuel, start-up, shut-down, reserve deployment, and involuntary load shedding.
  • Day-ahead operation: Day-ahead scheduling enforces nodal power balance, transmission limits, and minimum online and offline times for conventional units.The minimum-duration constraints depend on binary commitment variables.
  • Real-time operation: Real-time power balance is enforced for every uncertainty realization, with generator availability determining reserve provision.Generation capacity and ramping constraints account for reserve activation, while reserve offers are capacity-limited.
  • Frequency constraints: Frequency constraints model inertial response and system frequency variables for each scenario and time period.A generator contributes inertial response only when committed, available, and not experiencing an outage.

A. System Description

The study evaluates a modified IEEE RTS-96 system with stochastic UC under wind and contingency scenarios, comparing schedules with and without frequency constraints. Frequency constraints increase synchronous commitment and system inertia, raising operating costs while leaving synchronous production only slightly changed.

  • The test system has 48 buses, 2 areas, 20 synchronous generators, and 16 wind farms.
  • Six wind farms provide virtual inertia: four through VSM control and two through equivalent droop regulation.
  • The stochastic UC considers failures of four synchronous generators and 10 wind-power scenarios, producing 50 combined scenarios.
  • Frequency-constrained UC commits significantly more generators, while total synchronous production changes only slightly because additional units operate at technical minimum for inertia.
  • Frequency constraints produce a step increase in aggregate inertia at hour 67, with carryover effects on later schedules after the commitment changes.
  • Adding frequency constraints increases total expected system costs by 5% and start-up costs by 185%, with the cost impact dependent on the system and contingencies.

VII. CONCLUSION

The paper incorporates frequency constraints into stochastic unit commitment and proposes an efficient reformulation of the nonlinear nadir constraint. Results show that these constraints alter synchronous-generator dispatch and increase expected costs.

  • VII. CONCLUSION: Frequency nadir, RoCoF, and quasi steady-state deviation constraints are incorporated into a stochastic UC model for systems with high wind penetration.The model accounts for wind production and generation-outage uncertainties while minimizing expected system costs.
  • VII. CONCLUSION: The proposed bounds-based reformulation of the frequency nadir constraint is computationally superior to the adapted piecewise-linearization approach.This reformulation preserves the mixed-integer linear programming formulation of the stochastic UC problem.
  • VII. CONCLUSION: Frequency constraints significantly affect synchronous-generator dispatch and expected system costs.Critical generation-loss events require additional synchronous machines to provide sufficient inertia and damping for frequency containment.
  • VII. CONCLUSION: Additional inertia provision increases UC costs, particularly through start-up and reserve scheduling.The resulting remuneration of units committed solely for frequency regulation is identified as a future challenge.

APPENDIX A

The appendix illustrates how the piecewise-linearization method approximates a two-dimensional function. It selects nearby segments at evaluation points and minimizes the resulting shaded approximation area.

  • APPENDIX A: Fig. 10 illustrates the PWL optimization method applied to a two-dimensional function.The figure uses four evaluation points and approximates the function with four segments.
  • APPENDIX A: The evaluation points are set at -7.5, -2.5, 2.5, and 7.5.Each point is used to construct the piecewise approximation.
  • APPENDIX A: At each evaluation point, the model identifies the closest segment and minimizes the overall shaded area.The technique is subsequently extended to a function of three variables.
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