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System Strength-Constrained Scheduling with Switchable Grid-Forming and Grid-Following Generation Resources

Jiaxin Wang, Fei Teng, Huanhai Xin, Jing Dai, Tomislav Capuder, Ning Zhang

arXiv:2609.29412v1eess.SY

TL;DR

IBR scheduling must account for system strength because IBR operating behavior is flexible and coupled to strength, complicating optimization. The paper reformulates the strength constraint as an exact MISDP and solves it with Rayleigh cuts; compared with direct MISDP solution, the cut method reaches a 0.17% MIP gap in 32.96 seconds.

  • Problem

    IBR operating flexibility is coupled with system strength, making it difficult to capture these decisions together in scheduling.

    Source: §I · p. 2

  • Method

    The paper derives an equivalent LMI reformulation of the system-strength constraint and combines it with operational constraints in an MISDP solved using a Rayleigh Cut method.

    Source: §I · p. 3, §VI · p. 13

  • Results

    A 0.17% MIP gap is reached within 32.96 seconds with the Rayleigh-cut method, versus an 85.78% gap after 2 hours for direct MISDP solution.

    Source: E. Comparison with Direct MISDP Solution · p. 12

  • Takeaways & Limitations

    The Rayleigh Cut method progressively tightens the feasible domain and is compatible with standard MILP solvers.

    Source: E. Comparison with Direct MISDP Solution · p. 12, §VI · p. 13

Abstract

from arXiv · show

Inverter-based resources (IBRs) are increasingly dominating modern power systems, posing significant challenges to cost-effectively maintain system strength for stability. At the same time, the operating behaviors of IBRs are software-defined, including both their steady-state power outputs and control modes, e.g. grid-forming (GFM) and grid-following (GFL). Such flexibility has not been fully explored to efficiently operate future power systems. This paper develops a novel framework that simultaneously optimizes IBR operating behaviors and ensures adequate system strength. A comprehensive solution is provided to integrate system strength constraints into scheduling models, despite their inherent strong non-convexities. We derive a rigorous linear-matrix-inequality (LMI) reformulation of the system strength constraint, effectively addressing non-explicit formulations and dimension variation issues caused by GFM/GFL mode switching of IBRs. Then, we equivalently convert the original non-convex implicit system strength-constrained scheduling problem into an explicit mixed-integer semi-definite programming (MISDP) problem by incorporating the reformulated system strength constraint along with other operational constraints. We further provide a Rayleigh Cut method, which is compatible with standard mixed-integer linear programming (MILP) solvers, to solve this system strength-constrained scheduling problem. Case studies on a modified IEEE 118-bus system and a practical Jiangsu power system demonstrate the performance of the proposed methods.

A. Abbreviations

The paper frames system-strength-constrained scheduling as a way to coordinate IBR operating choices with adequate system strength. Its notation distinguishes resource sets, operating modes, and network matrices used to express that coupling.

A. Abbreviations

  • The notation defines gOSCR, GFM, GFL, IBR, MIP, MILP, MISDP, PV, LMI, SDP, SG, and SOC as abbreviations used in the paper.The definitions include generalized operational short-circuit ratio, grid-forming and grid-following, and mixed-integer semi-definite programming.Source: A. Abbreviations · p. 1

B. Sets

  • The sets distinguish storage buses E, renewable buses R, synchronous-generator buses G, remaining buses L, all buses N, IBR buses I, other buses J, and time horizons T.At time t, I_t denotes GFL buses and J_t denotes non-GFL buses.Source: B. Sets · p. 1

C. Variables

  • The variables include the gOSCR threshold γ0, mode indicator x_i,t, network matrices B_pf and B_sys, Schur complements B and ˆB, and GFL-power matrices P and ˆP.The definitions specify that x_i,t is 1 for GFM and 0 for GFL, and ˆP extends GFL power entries across IBR buses with zeros at non-GFL buses.Source: C. Variables · p. 1

I. INTRODUCTION

  • System-strength-constrained scheduling is motivated by the sensitivity of system strength to IBR operating choices and the limits of post-scheduling checks and manual adjustments.IBR power outputs and GFM/GFL modes are software-defined, while their coupling with system strength complicates optimization.Source: §I · p. 2
  • Earlier scheduling approaches either use approximations without strict feasibility guarantees or simplify IBR operations, leaving mode switching and related flexibility incompletely modeled.The paper positions its framework as addressing this gap in system-strength-constrained scheduling.Source: §I · p. 3
  • The framework combines a scheduling model of IBR behavior with an exact MISDP reformulation of system-strength constraints and a Rayleigh Cut method compatible with standard MILP solvers.The reformulation is stated to avoid approximation error and the method guarantees optimality within a prescribed tolerance.Source: §I · p. 3

II. PROBLEM STATEMENT AND THE PROPOSED SOLUTION

The problem minimizes operating cost while scheduling generation, storage, and IBR modes subject to network, generation-side, and system-strength constraints. The gOSCR condition couples GFL power injections and the network representation to operating decisions.

A. Notational Conventions

  • The notation partitions buses into storage, renewable, synchronous-generator, and remaining sets, then distinguishes GFL IBR buses from all other buses over time.The system considered has inverter-interfaced storage and renewable resources alongside synchronous generators.Source: A. Notational Conventions · pp. 3–4

B. System Strength-Constrained Scheduling Problem

  • The section introduces the problem structure before presenting the complete formulation after the theoretical results.The stated purpose is to provide a conceptual overview.Source: B. System Strength-Constrained… · p. 4
  • The schedule minimizes total operating cost using continuous and binary decisions for IBR modes, synchronous-generator commitment, and mutually exclusive storage charging or discharging.Constraints cover network power flow and transmission limits, generation-side operations, and system strength.Source: B. System Strength-Constrained… · p. 4
  • The system-strength metric is gOSCR, based on the positive spectrum of P^-1B, where P contains GFL power and B is the Schur-complement reduction of B_sys to GFL buses.Both matrix dimensions vary with GFM/GFL mode decisions.Source: B. System Strength-Constrained… · p. 4
  • B_sys incorporates internal admittances of online synchronous generators and GFM IBRs, while GFL renewable and storage injections determine entries of P.The storage injections can represent charging or discharging; synchronous-generator status and IBR modes affect B.Source: B. System Strength-Constrained… · p. 4

C. Challenges and the Proposed Solution

The paper identifies mode-dependent IBR operating constraints, a non-convex and dimension-varying system-strength constraint, and solver compatibility as key scheduling challenges. It formulates GFM/GFL switching and power limits, then outlines solutions for integrating these decisions with system strength.

C. Challenges and the Proposed Solution

  • System-strength scheduling must handle a non-convex, implicit constraint whose matrix dimensions vary with GFM/GFL choices, while remaining compatible with mainstream solvers.The proposed solutions include an equivalent linearization, a fixed-dimension reformulation, and a Rayleigh Cut method.Source: C. Challenges and the Proposed Solution · p. 5

III. FORMULATION OF THE OPERATING CONSTRAINTS FOR IBRS WITH GFM/GFL MODE SWITCHING

  • IBR mode switching requires distinct operational constraints, including GFM capacity headroom and power limits that vary with the selected mode.The formulation introduces binary mode variables and corresponding power-output constraints.Source: §III · p. 5, A. Switching Between GFM and GFL Modes · p. 5

A. Switching Between GFM and GFL Modes

  • For storage, the model distinguishes charging from discharging and constrains power by its limit while accounting for stored energy and energy balance.GFM operation also requires reserved capacity.Source: A. Switching Between GFM and GFL Modes · p. 6
  • The system-strength decision matrices depend on mode choices, and the paper notes that their size-change issue is addressed later.The power matrix P includes scheduled power only for IBRs operating in GFL mode.Source: A. Switching Between GFM and GFL Modes · p. 6

B. Equivalent Linearization of the Operational Constraints

The operational constraints contain bilinear terms and a matrix inverse linked to switching decisions. The paper introduces auxiliary variables and equivalent linearizations to handle these nonlinearities without approximation error or additional binary variables.

B. Equivalent Linearization of the Operational Constraints

  • The operational constraints contain bilinear terms and matrix inverses arising from mode-switching and on/off decisions.The subsection introduces equivalent linearization techniques to address the bilinear terms.Source: B. Equivalent Linearization of the… · p. 6
  • An auxiliary continuous variable linearizes the storage term involving mode and charging/discharging status, while another handles the mode-times-power term.The storage status product is represented using an auxiliary variable with linear constraints.Source: B. Equivalent Linearization of the… · p. 6
  • An auxiliary matrix represents the inverse result, and auxiliary continuous variables linearize its products with mode variables.The bound M is chosen at least as large as the maximum absolute matrix entry.Source: B. Equivalent Linearization of the… · p. 6
  • The stated linearizations are exact, add no binary variables beyond the original ones, and use an M value that can be estimated in practical applications.These properties are reported in Remark 4.Source: B. Equivalent Linearization of the… · p. 6

IV. INTEGRATING THE SYSTEM STRENGTH CONSTRAINT IN THE SCHEDULING PROBLEM

The paper reformulates the system-strength requirement as a fixed-size LMI to address its implicit, non-convex form and mode-dependent dimensions. It incorporates that reformulation into an MISDP scheduling model and proposes a Rayleigh Cut solution method.

IV. INTEGRATING THE SYSTEM STRENGTH CONSTRAINT IN THE SCHEDULING PROBLEM

  • The reformulated constraint is incorporated with operational constraints to convert the original scheduling problem into an MISDP, and a Rayleigh Cut method is provided to solve it.The authors describe the resulting formulation as tractable and without additional approximation error.Source: §IV · pp. 6–7, A. Exact Reformulation of the System… · pp. 7–8

A. Exact Reformulation of the System Strength Constraint

  • The original system-strength constraint is non-convex and implicit, and its decision-matrix dimensions vary with the GFM/GFL-dependent index set.These varying dimensions complicate its use in optimization.Source: A. Exact Reformulation of the System… · p. 7
  • The system-strength constraint is equivalently reformulated as an exact, explicit LMI of fixed size, independent of mode-switching decisions.The matrices can be constructed without an inverse operation, addressing the original size-change and non-convexity issues.Source: A. Exact Reformulation of the System… · p. 7

B. The Complete Scheduling Model

The scheduling model minimizes operating costs while coordinating generator commitment, renewable dispatch, storage operation, and system-strength requirements. The reformulated model is an exact MISDP, without approximation errors.

B. The Complete Scheduling Model

  • The complete model minimizes synchronous-generator, renewable-energy, and storage operating costs subject to operational and system-strength constraints.The cost terms include generation, commitment, curtailment, charging/discharging, and additional costs for GFM operation.Source: B. The Complete Scheduling Model · p. 8
  • The model uses DC power-flow equations for network constraints and includes classic unit-commitment constraints for synchronous generators.Renewable, storage, and system-strength constraints are also incorporated.Source: B. The Complete Scheduling Model · p. 8
  • The original non-convex, non-explicit scheduling problem is converted into an MISDP without approximation errors.The reformulated system-strength constraint is incorporated into the complete scheduling model.Source: B. The Complete Scheduling Model · p. 8

C. Rayleigh Cut Solution Method

The Rayleigh Cut method solves the system-strength-constrained scheduling problem by iteratively adding linear cuts within an MILP branch-and-bound framework. It guarantees feasibility and optimality for the original MISDP within the accepted MIP-gap tolerance.

C. Rayleigh Cut Solution Method

  • The method solves the MISDP iteratively using an MILP branch-and-bound framework, which can be provided by commercial solvers such as Gurobi.It replaces the system-strength constraint with a subset of Rayleigh Cuts during the iterative solve.Source: C. Rayleigh Cut Solution Method · p. 8, Figure 2 · p. 9
  • When an incumbent violates the system-strength constraint, the method computes a minimum eigenvector and adds corresponding Rayleigh Cut constraints.The process repeats until a strength-feasible solution is found within the accepted MIP-gap tolerance.Source: C. Rayleigh Cut Solution Method · p. 8
  • The resulting solution is guaranteed feasible and optimal for the original MISDP within the accepted MIP-gap tolerance.The cuts are linear inequalities equivalent to the LMI constraint and remove infeasible solutions.Source: C. Rayleigh Cut Solution Method · pp. 8–9

A. Basic Information

The modified IEEE 118-bus case evaluates the proposed scheduling and solution methods. The results show that Rayleigh Cuts solve the model quickly, while coordinated IBR mode and storage dispatch maintain system strength as renewable generation varies.

A. Basic Information

  • The modified IEEE 118-bus case has 118 buses, 186 lines, and 50 sources, with a 24-hour horizon and required gOSCR of 2.0.The sources include wind, PV, storage, and thermal generation; GFM power headroom is set to 10% of rated capacity and storage SoC reserve to 8%.Source: A. Basic Information · p. 9

B. Model Performance

  • 0.17% MIP gap was reached in 32.96 seconds, compared with a direct MISDP solve that still had an 85.78% gap after 2 hours.The proposed method used 39 iterations and added 249 Rayleigh Cut constraints.Source: Figure 5 · p. 10, B. Model Performance · p. 10
  • All 1,000 randomly generated instances, with matrix dimensions from 5 to 1,000, showed matching feasibility outcomes for the original and reformulated constraints.All points fell in the first or third quadrant of the gOSCR-threshold difference versus minimum-eigenvalue plot.Source: Figure 6 · p. 10, B. Model Performance · p. 10

C. Interactions Between IBR Operating Behaviors and System Strength Requirement

  • Post-scheduling gOSCR stays at or above the required 2.0 across all horizons, with strength relatively low during periods 11–18 and high-renewable periods.The model responds by scheduling some renewables and storage in GFM mode, increasing storage charging, and reducing discharging.Source: Figure 7 · p. 10, C. Interactions Between IBR Operating… · p. 10
  • Storage charging during high-renewable periods supports system strength and absorbs surplus generation, while discharging during low-renewable periods helps balance power.The paper describes power-balance and system-strength constraints as synergistic in shaping this scheduling pattern.Source: Figure 8 · p. 11, C. Interactions Between IBR Operating… · p. 11
  • Storage and renewable resources are more likely to operate in GFM mode during high-renewable, low-gOSCR periods; raising the strength threshold by one increased operating cost by 207,541 CNY.Seven renewable generators were scheduled in GFM mode, with reserved power reducing peak renewable output to some extent.Source: Figure 9 · p. 11, C. Interactions Between IBR Operating… · p. 11, Figure 10 · p. 11
  • Increasing required power headroom from 10% to 15% reduced renewable GFM periods from 9 to 3 and increased storage GFM periods from 12 to 24.Raising storage SoC reserve from 8% to 13% changed renewable GFM periods from 9 to 10 and storage GFM periods from 12 to 7; the impacts are difficult to predict precisely in advance.Source: C. Interactions Between IBR Operating… · pp. 11–12

D. Effects of Unit Commitment on IBR Mode Switching

Unit commitment and IBR mode switching coordinate to maintain system strength, and the Rayleigh-cut method substantially outperforms direct MISDP solution on the reported case. The Jiangsu study also maintains the required strength at every time horizon, while the conclusions describe an exact LMI reformulation and a systematic scheduling framework.

D. Effects of Unit Commitment on IBR Mode Switching

  • With reduced startup costs, two 300-MW thermal generators shut down during high-renewable periods, while five renewable sources operate in GFM mode in each hour from 12–17.In the baseline, GFM renewable-source counts in hours 12–17 are 1, 1, 1, 1, 3, and 2; the modified case varies which five sources operate in GFM mode.Source: D. Effects of Unit Commitment on IBR… · p. 12, Figure 11 · p. 12
  • Different combinations of thermal commitment and IBR GFM/GFL modes can meet system-strength requirements, and the framework selects the lowest-cost option for the specified inputs.The source identifies costs, load forecasts, and other model inputs as determinants of the selected solution.Source: D. Effects of Unit Commitment on IBR… · p. 12

E. Comparison with Direct MISDP Solution

  • Within 32.96 seconds, Rayleigh cuts reach a 0.17% MIP gap, versus direct MISDP's 85.78% after 2 hours and first feasibility after 45.7 minutes.The direct MISDP result remains far from optimal after two hours.Source: E. Comparison with Direct MISDP Solution · p. 12
  • Direct MISDP reaches a maximum gOSCR of 8.2 against the 2.0 threshold, with over 27 renewable sources in GFM mode and four unnecessary thermal commitment transitions.The reported over-conservatism adds GFM opportunity costs and thermal startup costs.Source: E. Comparison with Direct MISDP Solution · p. 12, Figure 12 · p. 13

F. Application to a Real-World System

  • The Jiangsu test system has 194 buses and 238 transmission lines, and the proposed method solves its 24-hour scheduling problem in 207.83 seconds.The system is derived from a practical provincial grid, with geographically sensitive data removed and renewable penetration increased.Source: F. Application to a Real-World System · p. 13, Figure 13 · p. 13
  • Jiangsu system-strength levels remain above the required threshold at every time horizon, and raising that threshold by one unit increases total operating cost by 1,041,309 CNY.The paper attributes the increase to additional opportunity costs from scheduling more IBRs in GFM mode.Source: F. Application to a Real-World System · p. 13, Figure 14 · p. 13

VI. CONCLUSIONS

  • The conclusions present an exact LMI reformulation that handles GFM/GFL mode switching and dimension variation, integrated into a MISDP scheduling model solved by a Rayleigh-cut method.The method iteratively adds cuts within a MILP branch-and-bound framework and is described as optimality-guaranteed within the MIP gap.Source: §VI · pp. 13–14

APPENDIX

The appendix gives a computable bound for setting the big-M parameter and illustrates how the system-strength constraint becomes a fixed-dimension LMI. The illustrative reformulation retains IBR mode and synchronous-generator commitment effects without approximation.

A. Parameter Setting for the Big M in (16)

B. Illustrative Example

  • In the three-bus example, the original system-strength constraint is equivalently transformed into the fixed-dimension LMI ˆB−γ0 ˆP ⪰0, retaining mode-switching and generator-commitment effects exactly.The example has one synchronous generator and two IBRs, with all line reactances set to 0.05 p.u.Source: B. Illustrative Example · p. 14, Figure 15 · p. 14

C. Derivative of the gOSCR to GFL Power Injections

The derivation differentiates the gOSCR with respect to GFL power injections, using the corresponding eigenvalue relation.

C. Derivative of the gOSCR to GFL Power Injections

  • The derivation starts from the gOSCR eigenvalue relation and differentiates with respect to p_i to examine sensitivity to GFL power injections.It uses ξᵀB = γξᵀP in the subsequent derivation.Source: C. Derivative of the gOSCR to GFL… · p. 14
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