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Regional Frequency Constrained Dispatch Method Considering Spatial-joint Stochastic Disturbances and Contingencies

Nian Liu, Yubing Chen, Kai Jiang, Jiahao Liu, Cheng Wang, Tianshu Bi

arXiv:2609.05087v1eess.SY

TL;DR

The paper addresses frequency-security dispatch under declining inertia, renewable-output uncertainty, regional frequency heterogeneity, and spatially correlated disturbances. It combines regional frequency modeling, Vine-Copula uncertainty characterization, PLB-PINN approximations, and CVaR-based constraints, improving regional frequency security on the IEEE 118-bus system.

  • Problem

    Renewable growth creates insufficient regulation and frequency-support capability, while spatial heterogeneity and renewable-output uncertainty challenge frequency-security-constrained dispatch.

  • Method

    The method combines a multi-regional frequency-response model, Vine-Copula disturbance characterization, PLB-PINN frequency-term approximation, and CVaR-based stochastic frequency constraints.

  • Results

    32.76% and 2.06%: regional frequency nadir and RoCoF respectively improved versus the traditional COI-based method, with a 1.31% increase in total dispatch cost.

  • Takeaways & Limitations

    The proposed dispatch method keeps regional frequency indicators within safe thresholds while balancing frequency security and dispatch economy.

Abstract

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The increasing penetration of renewable energy challenges frequency stability due to high variability and declining inertia. Traditional frequency security constrained dispatch methods fail to capture regional frequency heterogeneity and spatially correlated stochastic disturbances, resulting in inaccurate frequency security enforcement. To address this, a regional frequency constrained dispatch method is proposed, considering the spatially joint stochastic disturbances and contingencies. Firstly, a multi-regional frequency response model is constructed, incorporating the Vine-Copula based characterization of regional stochastic disturbances and regional frequency support. Then, a progressive latent-bottleneck physics-informed neural network is applied to characterize differential frequency nadir terms and regional frequency support integral terms via an encoder-decoder architecture, which enables optimization compatible expressions of system frequency dynamics. Finally, a day ahead dispatch model is developed in which regional stochastic frequency constraints are embedded using a CVaR-based formulation. A case study on the IEEE 118-bus system shows that the proposed method outperforms the unified center of inertia embedded approach in mitigating regional frequency violations, with the regional frequency nadir and RoCoF improved by 32.76% and 2.06%, respectively.

NOMENCLATURE

The nomenclature defines indices, variables, parameters, functions, and security thresholds used in the regional dispatch and frequency-response formulations.

  • Indices: Indices identify dispatch and transient time slots, units and regions, vine-tree edges, and neural-network training epochs.
  • Parameters: Parameters cover inertia, damping, response time constants, droop coefficients, costs, operating limits, reserve capacity, and energy-storage bounds.
  • Variables: Variables describe generation, renewable output, energy-storage power, forecast-error disturbances, contingencies, regional transmission, voltage states, and frequency states.
  • Functions and security quantities: Functions and security quantities include density and distribution functions, copulas, neural-network terms, RoCoF and nadir thresholds, and their maximum exceeding risk levels.

Functions and operators

The paper defines the probability, copula, activation, and step-function operators used in its modeling framework.

  • The notation includes PDF, CDF, KDE, two-dimensional normal CDF, pair-copula density, Gaussian copula distribution, and ReLU operators.

I. INTRODUCTION

The introduction motivates regional frequency-constrained dispatch by highlighting spatially heterogeneous support and correlated renewable disturbances, then presents a framework combining stochastic modeling, regional dynamics, neural-network representation, and CVaR constraints.

  • Motivation: Renewable growth and power-electronic resources reduce regulation and frequency-support capability, making inertia and reserve capacity relevant to economic dispatch.
  • Motivation: Frequency-support resources are unevenly distributed, producing regional differences in inertia, primary response, and frequency resilience.
  • Motivation: Renewable forecast errors can create disturbances comparable to major contingencies and therefore matter in frequency-security-constrained dispatch.
  • Contributions: The proposed method models joint regional disturbances using Vine-Copula decomposition of renewable forecast-error biases and contingency effects.
  • Contributions: A PLB-PINN represents regional frequency dynamics through differential and integral frequency-response terms in a physics-informed encoder-decoder model.
  • Contributions: The dispatch framework co-optimizes unit commitment and regional reserves under heterogeneous inertia, inter-regional support, and CVaR-based tail-risk constraints.

III. REGIONAL FREQUENCY RESPONSE MODEL UNDER JOINT DISTRIBUTION OF STOCHASTIC DISTURBANCES

The regional response model combines contingency and renewable forecast-error disturbances, models their spatial dependence with Vine Copula, and links them to regional frequency deviations, primary response, and inter-regional exchange.

  • Stochastic disturbance model: Equivalent regional disturbances combine contingency-induced active-power changes with stochastic renewable forecast-error biases.
  • Regional frequency response: The equivalent disturbance initiates regional frequency deviations, primary frequency response, and inter-regional power exchange in the response model.
  • Stochastic disturbance model: Kernel density estimation constructs marginal disturbance distributions from random samples.
  • Spatial dependence: A Vine Copula represents high-dimensional spatial dependence through Z-1 hierarchical trees, edges between regions, and conditioning sets.
  • Spatial dependence: Conditional Gaussian pair-copula densities are combined into a joint copula density and then into the multi-regional disturbance PDF using marginal densities.

B. Regional System Frequency Response Model

The regional frequency response model represents inter-regional power exchanges, regional equivalent inertia, and generator frequency-support processes under disturbances. It formulates regional frequency-security indicators, including RoCoF and frequency nadir, from these coupled dynamics.

  • The model formulates a multi-regional frequency response process using regional coupled swing equations.
  • Regional transmission power: Inter-regional phase-angle differences produce oscillatory power exchanges that modify regional power balances and disturb other regions.
  • Generator frequency response: Inertia control models fast frequency support from synchronous generators, wind turbines, and energy storage systems through rotor-dynamic response representations.
  • Regional equivalent inertia: Each region uses an equivalent COI, while separate COIs across regions capture spatially heterogeneous inertial characteristics.
  • Generator frequency response: Droop control represents primary frequency regulation from synchronous generators, wind turbines, and energy storage systems.
  • Frequency-security indicators: The model defines RoCoF as regional frequency's post-disturbance time derivative and obtains frequency nadir by searching the minimum along the response curve.

IV. PLB-PINN FOR FREQUENCY SECURITY CONSTRAINTS

The paper addresses the difficulty of embedding nonlinear frequency-security dynamics in dispatch optimization by using a physics-informed encoder–decoder with a frequency-nadir latent bottleneck. The encoder links dispatch variables to security indicators, while the decoder reconstructs physical responses for consistency.

  • Nonlinear frequency-nadir dynamics and integral inter-regional support relationships make direct embedding of frequency-security constraints difficult.
  • The PLB-PINN treats frequency nadir as a causal latent variable linking dispatch decisions to frequency security.
  • Its encoder maps high-dimensional dispatch variables to a nadir-centered latent representation, while its decoder reconstructs primary support and inter-regional power exchange.
  • The decoder enforces consistency with frequency dynamics while the encoder provides the frequency-security representation used for optimization.

A. PLB-PINN Formulation

The PLB-PINN uses reserve-capacity inputs, a frequency-nadir encoder output, and a decoder for associated physical responses. Its training combines encoder, decoder, and physics-informed losses with adaptive and progressive weighting.

  • The encoder takes primary frequency reserve capacities from synchronous generators, wind turbines, and energy storage systems as input features.
  • The encoder outputs regional frequency nadir values as the latent frequency-security representation.
  • The decoder uses predicted frequency nadir as a latent bottleneck to map into primary frequency-response power and related physical response variables.
  • The physics-informed regularization term is formed by substituting the encoder-predicted dynamic trajectory into the regional frequency-response model.
  • The overall loss combines encoder loss, decoder loss, and physics-informed regularization, with adaptive weighting balancing reconstruction and physical constraints.
  • A ramp-up schedule progressively activates physics-informed regularization, while early training suppresses physics-term influence using a scaling factor.

B. Frequency Security Constraints Linearization

The method linearizes the PLB-PINN and embeds stochastic frequency-security constraints into day-ahead dispatch. ReLU activations enable exact Big-M reformulation, while CVaR represents violation risk tractably under uncertainty.

  • Network linearization: ReLU activations replace the nonlinear network functions and permit exact Big-M linearization within a mixed-integer linear dispatch model.
  • Network linearization: Only encoder outputs representing frequency-security indicators are embedded in dispatch optimization, so linearization is performed exclusively on the encoder.
  • Network linearization: The decoder remains auxiliary because its outputs reconstruct frequency trajectories without affecting dispatch decisions.
  • Dispatch formulation: The day-ahead dispatch objective minimizes total cost subject to the dispatching model's constraints.
  • Stochastic security constraints: CVaR provides a probabilistic and optimization-tractable representation of stochastic frequency-security risk.

L RoCoF RoCoF

The dispatch formulation combines regional stochastic frequency-security constraints with DC power-flow, generator, renewable, storage, and commitment constraints. CVaR constraints are convexified and linearized for integration into the dispatch problem.

  • Stochastic frequency constraints: CVaR represents regional frequency-security losses using a confidence-level threshold and the Rockafellar–Uryasev convex formulation.The formulation introduces slack variables for the convex CVaR representation.
  • Stochastic frequency constraints: The stochastic frequency-security constraints are incorporated into the dispatch model after deriving regional nadir and RoCoF expressions.The formulation explicitly introduces the stochastic frequency-security constraint set before the operational constraints.
  • Network and operating constraints: The regional dispatch model maintains power balance using a DC optimal power-flow representation with each region modeled as an equivalent node.Regional transmission power constraints are included alongside the balance equations.
  • Unit constraints: Generator constraints cover allowable output, ramping, start-up, shut-down, and minimum operating-duration conditions.These constraints regulate both continuous generation levels and commitment transitions.
  • Storage constraints: Energy-storage constraints limit charging, discharging, reserve, and energy states across time.The storage formulation links consecutive energy states while enforcing output and capacity limits.
  • Renewable constraints: Renewable outputs are bounded by their minimum and maximum operating limits.The renewable-output constraints are included alongside conventional-generator and storage operating restrictions.

VI. CASE STUDY

The case study validates the regional dispatch method on a modified IEEE 118-bus system divided into three heterogeneous regions. Results show region-specific reserve allocation, critical frequency periods, and dynamic support through local resources and inter-regional tie-lines.

  • System and disturbance setting: The method is tested on a modified IEEE 118-bus system divided into three geographic and electrically separated regions.Region 1 is thermal-dominated, Region 2 has higher wind penetration, and Region 3 is a high-demand load center relying on support from other regions.
  • System and disturbance setting: The disturbance model superposes a 10% regional-maximum-load contingency in Region 3 with 15-minute renewable forecast errors.Region 3 is selected because its largest load produces the greatest absolute contingency deficit and frequency impact.
  • Day-ahead dispatch: 80.33% to 88.80% of installed renewable capacity is generated during time slots 40 to 52, while ESSs discharge during peaks and charge during high-PV periods.Thermal generation approaches upper limits during the evening peak when wind and photovoltaic output decline.
  • Frequency-support schedule: Reserve allocation reflects load, renewable forecast-error uncertainty, and equivalent frequency disturbance rather than load demand alone.Average scheduled reserve is 2063.98 MW during time slots 30–45 versus 1424.78 MW during time slots 75–85, and 2019.90 MW during time slots 35–40 versus 1800.62 MW during time slots 55–60.
  • Frequency-support schedule: Regional reserve composition differs substantially: Region 1 relies mainly on thermal units, Region 2 on wind reserves, and Region 3 predominantly on ESS reserves.Region 1 allocates 563.79 MW of thermal reserve, while Region 2 averages 200.71 MW of wind reserve and Region 3 averages 103.06 MW of ESS reserve.
  • Frequency response: Frequency security deteriorates most during time slots 36–52, when Region 3 reaches a −0.32 Hz nadir margin and −0.33 Hz/s RoCoF with about 240.09 MW reserve.The response is jointly supported by local resources and neighboring regions through inter-regional tie-lines.

C. Analysis of regional COI representation

The regional COI representation is evaluated against unified-COI and dispersed-disturbance dispatch models, with additional sensitivity analyses for contingency location, magnitude, and transmission capacity. The results connect regional modeling with frequency performance, reserve allocation, and operating cost.

  • Regional versus unified COI: The proposed regional-COI dispatch method is compared with a unified-COI frequency-model dispatch during the critical time slots 36–52.The comparison includes day-ahead schedules and regional operating costs.
  • Regional versus unified COI: The proposed method costs $5.43 million versus $5.36 million for the unified-COI method, increasing total operating cost by 1.31%.Generation cost rises 4.40%, while reserve cost falls 4.41% under regional frequency-security constraints.
  • Spatial-joint versus dispersed disturbances: The dispersed-disturbance method improves minimum nadir by 0.001 Hz but reserves 1240.88 MWh more on average over the scheduling horizon.The additional reserve reflects conservative allocation caused by failing to represent spatially joint renewable disturbances accurately.
  • Spatial-joint versus dispersed disturbances: Joint-distribution modeling reduces unnecessary unit commitment and lowers generation and reserve costs compared with dispersed-distribution scheduling.The reported reductions are 148.36 in generation cost and 34.14 in reserve cost, while unit-commitment cost increases by 4.06 thousand dollars.
  • Contingency sensitivity: Increasing contingency magnitude worsens nadir values across regions, with Region 3 producing the strongest system-wide effect and local RoCoF deterioration.Region 2 is more affected by renewable forecast-error disturbances because of its high renewable penetration.
  • Transmission-capacity sensitivity: Reducing inter-regional line capacity raises reserve requirements: average reserve increases from 1801.05 MW at the basic case to 2243.29 MW at 80% capacity.At 120% capacity, average reserve decreases to 1683.94 MW, a 6.50% reduction relative to the basic case.

F. Analysis of PLB-PINN computing performance

PLB-PINN provides a more accurate and computationally efficient alternative to Bernstein linearization for embedding regional frequency constraints. The broader dispatch results also show improved regional security and reduced costs under the proposed framework.

  • Prediction accuracy: PLB-PINN reduces MAE by 92.94%, 91.94%, and 82.26% in Regions 1-3, respectively, compared with the Bernstein method.The method achieves lower MAE values in all three regions.
  • Optimization model size: 78.17% fewer additional constraints are required by PLB-PINN than by the Bernstein method.PLB-PINN introduces 83,520 additional constraints, compared with 382,656 for Bernstein.
  • Computational efficiency: 82.23% lower computational time is achieved, decreasing from 5021.98s with Bernstein to 892.58s with PLB-PINN.The smaller optimization model accompanies the shorter computation time.
  • Method comparison: The Bernstein method has lower fitting accuracy because its uniform COI formulation weakens regional dynamics and omits explicit inter-regional transmission-line power-exchange integrals.These limitations contribute to poorer representation of the studied multi-region coupled frequency-response system.
  • Dispatch outcomes: 32.76% relative improvement in vulnerable-region frequency nadir is obtained while RoCoF improves by 2.06% against the traditional COI-based method.All regional frequency indicators remain within safe thresholds, with only a 1.31% increase in total dispatch cost.
  • Stochastic disturbance modeling: 2.14% lower total dispatch cost is achieved through Vine-Copula-based spatial-joint stochastic disturbance modeling while maintaining frequency security.The modeling supports more precise reserve allocation than dispersed distribution methods.
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