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Nodal Inertia Pricing for RoCoF Security with Integration of Synchronous and Virtual Resources

Bomiao Liang, Yu Yang, Licheng Wang, Lei Wan, Dipti Srinivasan, Fushuan Wen, Youbing Zhang, Beiping Hou

arXiv:2610.33590v1eess.SYcs.SY

TL;DR

As renewable generation increases, inertia shortages raise RoCoF and frequency-security risks, while existing inertia valuation does not adequately address the immediate post-disturbance period or locational differences. The paper proposes nodal RoCoF-based pricing and coordinated treatment of synchronous and virtual inertia. Case studies report that the framework identifies weak areas and maintains nodal frequency security more effectively than a COI-based method.

  • Problem

    Inertia pricing remains insufficiently developed for the immediate post-disturbance period, when security depends on online conventional and already deployed virtual inertia, and COI models miss locational differences.

    Source: §I · pp. 1–2

  • Method

    The paper replaces the COI representation with a nodal RoCoF model and coordinates synchronous and virtual inertia in a nodal pricing framework.

    Source: §I · p. 2

  • Results

    The proposed framework more accurately identifies inertia-weak nodes and maintains nodal frequency security compared with the COI-based method.

    Source: 1) High-Inertia Scenario: Proposed vs.… · p. 11, COMPARISON OF UNIT OPERATING STATUS… · p. 13, §V · p. 16

  • Takeaways & Limitations

    Nodal inertia prices reflect location-specific security value and improve consistency between market signals and frequency-security needs.

    Source: §I · p. 2, §V · p. 16

Abstract

from arXiv · show

Frequency regulation is becoming a more significant challenge in the undergoing low-carbon transition of modern power systems, which is heavily rely on system inertia provision. In the context of market-based energy allocation, reasonable valuation of the system inertia is crucial to its procurement, namely a worthy price should be determined for inertia providers. However, current research on the value of inertia remains insufficient, particularly during the initial phase immediately following a contingency, a critical period when the system primarily relies on the inherent inertia from conventional units and pre-engaged virtual inertia resources. Furthermore, it is found in existing studies that the Center of Inertia (COI) models exhibit significant deviations from actual physical conditions. For market-oriented reforms that explore individual differences, COI models also fail to capture the heterogeneous contributions of inertia at different nodes to system security. Given these circumstances, this paper proposes a nodal inertia pricing mechanism via a day-ahead unit commitment and inertia co-optimization model with nodal resolution. Following the proposed pricing scheme, the day- or hour-ahead inertia provision is monetized and promoted, providing regional guidance to units and devices with virtual inertia provision, which is further verified by simulation results of the South East Australia Power System.

I. INTRODUCTION

The paper argues that first-stage frequency security depends on inertia whose value varies by location, while existing market approaches often rely on aggregated representations. It proposes nodal pricing and coordinated scheduling of synchronous and virtual inertia, with provision and remuneration reflecting their different characteristics.

I. INTRODUCTION

  • Prior market studies advance inertia and fast frequency-support services, but many still use aggregated security representations that do not fully capture locational heterogeneity.The cited literature also describes joint clearing with energy and other services, and performance-sensitive valuation.Source: §I · pp. 1–2
  • In the first stage after a disturbance, governor response remains largely inactive, leaving online conventional units and already deployed virtual inertia to suppress initial RoCoF.Later, as governor action develops and nodal RoCoF disparities diminish, aggregated models become more adequate.Source: §I · p. 2, §II · p. 3
  • The proposed nodal RoCoF model identifies inertia-weak nodes and prices inertia according to its location-specific contribution to first-stage security.Its prices reflect state-dependent scarcity and marginal security contribution at individual nodes.Source: §I · pp. 2–3

II. OPTIMAL ENERGY AND INERTIA COMMITMENT MODEL

  • The commitment model treats synchronous inertia as determined by unit commitment and virtual inertia as dispatchable within capacity, monetizing provision alongside normal system operation.The model imposes RoCoF constraints at the instant immediately after a contingency.Source: §II · p. 3

A. Inertia Characteristics, Provision and Payment

  • At each node, total inertia can comprise conventional and virtual components; conventional inertia is usually fixed while its unit is on, whereas virtual inertia offers greater dispatch flexibility.Virtual-inertia service costs relate to device dispatch status and reserved capacity, and pricing may be based on the actual provided magnitude.Source: A. Inertia Characteristics, Provision… · p. 3
  • Traditional inertia is eligible for payment only below a specified unit load rate, while virtual inertia is typically remunerated according to cleared quantity.The paper treats nodal inertia payment as one component of provider revenue, expected to cover costs and provide reasonable long-run returns.Source: A. Inertia Characteristics, Provision… · pp. 3–4

B. Network Equation

The network formulation uses augmented DC power-flow equations to relate post-contingency injections and voltage angles. It defines generator and load nodes and simplifies the representation by treating synchronous generators and virtual synchronous generators alike as generator nodes.

B. Network Equation

  • Augmented DC power-flow equations describe post-contingency changes in nodal injections and voltage angles following the trip of a synchronous generator.Equation (8) maps generator-node active-power variations to voltage phase-angle variations immediately after the contingency.Source: B. Network Equation · p. 4, BGG BGB BBG BBB][ · p. 5

BGG BGB BBG BBB][

  • The model defines generator nodes behind generators’ internal reactance and system-side nodes, including generator terminal nodes, as load nodes.Synchronous generators and VSGs are both represented as generator nodes.Source: BGG BGB BBG BBB][ · p. 4, Figure 1 · p. 4
  • The formulation assumes unchanged network topology during the contingency and treats load power as unchanged immediately after the disturbance.It also neglects inverter saturation and assumes each VSG has sufficient capacity to match its offered virtual inertia.Source: BGG BGB BBG BBB][ · p. 4

C. Post-Contingency Boundary Conditions and Nodal RoCoF +

The model sets post-contingency boundary conditions at the initial instant and links nodal inertia to RoCoF security.

C. Post-Contingency Boundary Conditions and Nodal RoCoF +

  • At t = 0+, the tripped generator’s active-power variation equals the negative of its pre-contingency output, while synchronous or virtual-unit mechanical-power variation is zero.The maximum power imbalance occurs at this initial instant.Source: C. Post-Contingency Boundary… · p. 5
  • At non-faulted nodes with online synchronous or virtual inertia, the generator-node phase angle cannot change instantaneously; unsupported nodes are treated as load nodes with zero injection variation.This distinction sets the boundary conditions for the post-contingency network model.Source: C. Post-Contingency Boundary… · p. 5
  • The nodal RoCoF constraint couples nodal inertia levels to the initial post-contingency response, and the resulting equations form the system’s full constraint set.For online units, the stated RoCoF constraint is equivalent to the preceding form; it also holds for offline units because their power variation is zero.Source: C. Post-Contingency Boundary… · p. 5

D. Optimal Energy and Inertia Commitment Model

The model jointly schedules energy and inertia subject to network, equipment, compensation, and nodal RoCoF constraints.

D. Optimal Energy and Inertia Commitment Model

  • The objective combines synchronous generation and inertia-compensation costs, renewable generation costs, and virtual-inertia dispatch costs.Power losses and voltage constraints are omitted to focus on first-stage energy and inertia pricing.Source: D. Optimal Energy and Inertia… · pp. 5–6

(cRES ,i pRES ,i) ⏟

  • Power balance, PTDF-based line-flow mapping, and transmission-capacity limits define the model’s network constraints.The balance constraint matches total generation to load; line flows are bounded by transmission capacity.Source: (cRES ,i pRES ,i) ⏟ · p. 6
  • Synchronous generators, renewable units, and virtual synchronous generators are restricted by rated capacities or maximum output capabilities.Virtual inertia also has lower and upper limits.Source: (cRES ,i pRES ,i) ⏟ · p. 6
  • Big M linearization encodes synchronous inertia compensation conditions, while state coupling prevents shutdown units from qualifying for compensation.The compensation state variable can equal 1 only when the unit is online.Source: (cRES ,i pRES ,i) ⏟ · p. 7
  • Nodal RoCoF security constraints combine the derived network and boundary equations, using auxiliary variables and Big M constraints to linearize the inertia coupling.The resulting objective and constraints form a MILP solvable by commercial solvers such as Cplex or Gurobi.Source: (cRES ,i pRES ,i) ⏟ · p. 7

A. Linearized Dual Model and Market Equilibrium Conditions

Because integer decisions make the original pricing problem nonconvex, the paper fixes the optimal commitment before deriving prices from a continuous model.

A. Linearized Dual Model and Market Equilibrium Conditions

  • The original MILP’s dual solution cannot directly provide marginal prices with strict economic meaning because strong duality generally does not hold.The method avoids convex-hull dimensionality growth and the loss of physical meaning from continuous relaxation.Source: A. Linearized Dual Model and Market… · pp. 7–8
  • The ex-post method first solves the MILP, then fixes its integer decisions and solves the resulting continuous LP for dual prices.This preserves the physically feasible dispatch while producing a model that satisfies strong duality.Source: A. Linearized Dual Model and Market… · p. 8
  • KKT stationarity conditions for continuous decision variables yield market-clearing equilibrium conditions, and the dual problem reformulates the system’s acceptable cost boundary under nodal frequency security.The listed stationarity conditions cover generation, renewable output, virtual inertia, and other continuous variables.Source: A. Linearized Dual Model and Market… · pp. 8–9

UPP }

The dual problem’s constraints are the stationarity conditions together with non-negativity conditions on inequality-constraint dual variables.

UPP }

  • Dual feasibility requires the stationarity conditions and nonnegative dual variables for all inequality constraints.Source: UPP } · p. 9

B. Decomposition and Implications of Nodal Inertia Pricing

Nodal inertia prices translate constraint shadow prices into the marginal cost of maintaining RoCoF security. They are positive when RoCoF constraints bind and vary with each node’s contribution to mitigating contingency risks.

B. Decomposition and Implications of Nodal Inertia Pricing

  • Dual variables from the system constraints become economic price signals: conventional locational marginal prices and the proposed nodal inertia prices.The inertia price reflects the marginal cost of maintaining the RoCoF security constraint.Source: B. Decomposition and Implications of… · p. 9
  • With high renewable penetration, declining nodal electricity prices may leave energy-market revenue insufficient to cover synchronous generators’ no-load and inertia opportunity costs.This motivates the proposed inertia ancillary service pricing mechanism.Source: B. Decomposition and Implications of… · p. 9
  • Binding RoCoF constraints make nodal inertia resources valuable, with higher prices where support mitigates violations across multiple contingencies.Electrically weak areas may have higher prices than central areas where inertia is relatively sufficient.Source: B. Decomposition and Implications of… · p. 10

A. System and Simulation Settings

The simulations compare COI-based system-level clearing with nodal energy-and-inertia clearing across high- and low-inertia conditions. The reported results show that nodal constraints identify local security needs and produce location-specific inertia prices, with outcomes varying by operating condition and contingency.

A. System and Simulation Settings

  • The test system contains 40 synchronous generators, 12 RES plants, and 15 VIRs; the low-inertia scenario replaces 26 generators with renewable units, raising RES penetration to 67.6% and reducing synchronous inertia to 18.06 GWs.The high-inertia scenario has 53.46 GWs of synchronous inertia and 0% RES penetration; the N-1 generator-trip security threshold is -1 Hz/s.Source: A. System and Simulation Settings · p. 10

B. Results and Analysis

  • With high system inertia, COI-RoCoF is -0.3183 Hz/s, but maximum nodal RoCoF reaches -1.1411 Hz/s; nodal clearing starts G203 and keeps maximum nodal RoCoF within -1 Hz/s.G203 inertia rises from 2600 MWs to 3900 MWs, and total system inertia rises from 49,040 MWs to 50,340 MWs.Source: 1) High-Inertia Scenario: Proposed vs.… · p. 11, Table I · p. 12

COMPARISON OF UNIT OPERATING STATUS UNDER HIGH INERTIA SCENARIOS

COMPARISON OF UNIT OPERATING STATUS UNDER LOW INERTIA SCENARIOS

V. CONCLUSION

The proposed framework prices nodal inertia security value and coordinates synchronous and virtual inertia to maintain nodal frequency security. Case studies show more accurate identification of inertia-weak nodes than the COI-based method, with only a modest operating-cost increase.

V. CONCLUSION

  • Across high- and low-inertia scenarios, the framework more accurately identifies inertia-weak nodes and maintains nodal frequency security with only a modest increase in operating cost.It does so through coordinated dispatch of synchronous and virtual inertia resources.Source: §V · p. 16
  • The framework replaces the traditional COI model with a nodal RoCoF model to represent post-disturbance frequency dynamics more spatially and physically.It prices nodal security value and optimizes inertia allocation across nodes in the first stage following a disturbance.Source: §V · p. 16
  • Nodal inertia prices reflect locational inertia value, and their spatial distribution is linked to network topology and virtual inertia placement.The resulting signals may inform short-term operation and long-term virtual inertia siting and investment.Source: §V · p. 16
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