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A Primal-Dual Formulation for Pricing Static Voltage Stability Services within a Unit Commitment Model

Peng Wang, Luis Badesa

arXiv:2609.11436v1eess.SYcs.SY

TL;DR

GFL-IBR connections to weak-grid buses raise static voltage-stability concerns, while existing pricing methods may not ensure cost recovery for participating units. The paper proposes a primal-dual pricing framework with explicit profitability constraints. Its case-study conclusion is that the framework enables units to remain profitable without uplift payments.

  • Problem

    GFL-IBR integration into weak-grid regions raises voltage-instability risks, and existing pricing methods do not guarantee adequate revenues for participating units.

    Source: §1 · pp. 2–3

  • Method

    A primal-dual pricing formulation uses explicit non-negative-profit constraints and duality-gap minimization to recover binary commitment decisions.

    Source: §1 · p. 3

  • Results

    The P-D framework internalizes fixed costs into transparent market prices, enabling generating units to remain profitable without uplift payments.

    Source: §5 · p. 11

  • Takeaways & Limitations

    (V)SGs can be remunerated for improving SCR at weak buses, while GFL-IBR receive payments for reactive-power support that sustains voltage stability.

    Source: §5 · p. 11

    Future work would incorporate frequency reserves and inertia and examine the effects of strategic bidding by self-interested market agents.

    Source: §5 · p. 11

Abstract

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In modern power systems with high penetration of Inverter-Based Resources (IBR), most converters operate in Grid-Following (GFL) mode. Some buses exhibit inherently low Short-Circuit Ratios (SCRs), a property majorly shaped by network topology. The integration of GFL-IBR onto such weak buses thus demands attention to static voltage stability. To address this issue, market mechanisms have been proposed to incentivize generators to provide voltage stability services, such as commitment of synchronous generators for reducing the equivalent impedance at low SCR buses and adaptive reactive power support from GFL-IBR. To compute shadow prices for these services, previously proposed methods such as the `restricted' and `dispatchable' approaches may fail to guarantee operating cost recovery for voltage-stability service providers. As the resulting prices are determined purely from a social surplus maximization objective, the profitability of units is entirely overlooked. This suggests that new pricing methodologies are needed to satisfy cost-recovery requirements. Therefore, this paper proposes a pricing method based on a primal-dual formulation. Case studies demonstrate that the proposed method can consistently produce revenue-adequate shadow prices, enabling all participating units to recover their costs without supplementary uplift payments.

Nomenclature

The acknowledgments identify support from Spanish and European funding programs and from the China Scholarship Council.

Funding

  • The work received support from MICIU/AEI and ERDF/EU under grant PID2023-150401OA-C22.Source: Constants and Parameters · p. 2
  • The Madrid Government supported the work under its 2023–2026 Multiannual Agreement with Universidad Politécnica de Madrid.The support was under “Line A - Emerging PIs” (grant 24-DWGG5L-33-SMHGZ1).Source: Constants and Parameters · p. 2
  • Peng Wang’s work was also supported by the China Scholarship Council under grant 202408500065.Source: Constants and Parameters · p. 2

1. Introduction

The paper frames static voltage stability at GFL-IBR buses as an operational concern in weak-grid regions and proposes a primal-dual pricing method intended to ensure participating units recover costs without uplift payments.

1. Introduction

  • In weak-grid regions, GFL-IBR connections can lower local SCR and raise voltage-instability risks because these converters lack inherent terminal-voltage regulation.Source: §1 · p. 2
  • Earlier support strategies include installing synchronous condensers or switching IBR from GFL to GFM, but the latter requires coordinated transitions to avoid disrupting converter control loops.Source: §1 · pp. 2–3
  • An operating-point approach uses online SGs and VSGs to reduce equivalent impedance while GFL-IBR provide reactive power when required.Source: §1 · p. 3
  • Dispatchable pricing can partially commit thermal units and yield negative net profits, while restricted pricing may still require uplift when energy and ancillary-service revenues are inadequate.Source: §1 · p. 3
  • Unit-specific uplift payments are not public in real time, and their cost allocation remains unclear.Source: §1 · p. 3
  • The proposed P-D method uses profitability constraints and duality-gap minimization to preserve binary commitment decisions while ensuring non-negative profits and eliminating uplift payments.Source: §1 · p. 3

2. State of the Art in Pricing Static Voltage Stability Services

This section presents a static voltage-stability constraint for GFL-IBR buses, derives associated shadow prices, and reviews restricted and dispatchable pricing alongside their limitations.

2.1. Representation of Static Voltage Stability Constraints

  • The GFL-IBR bus constraint links power injections with SCR; support can come from increased reactive power at fixed SCR or reduced equivalent impedance through (V)SG operation.Because SCR remains nonlinear in the formulation, an offline training approach approximates it linearly.Source: §2.1 · p. 4

2.2. Shadow Prices for Static Voltage Stability Services

  • SOC reformulation and KKT stationarity provide the dual variables used to derive voltage-stability service prices; active power is not remunerated as a service.The paper states that active power serves energy demand and consumes available voltage-stability margin.Source: §2.2 · p. 4

2.3. Existing Methods for Computing Shadow Prices

  • The section reviews existing shadow-pricing methods and their limitations in addressing non-convexity.Source: §2.3 · p. 4

2.3.1. Restricted Pricing Method

  • Restricted pricing fixes the optimal commitment schedule in a second relaxed solve and adds a commitment price that covers only part of committed SG uplift.Uplift allocation lacks a well-defined mechanism, and IBR without commitment variables are ineligible for this compensation.Source: §2.3.1 · p. 4

2.3.2. Dispatchable Pricing Method

  • Dispatchable pricing relaxes binary variables, but bilinear terms make the problem non-convex and relaxed commitments can understate SG voltage-stability contributions, requiring make-whole payments.Source: §2.3.2 · pp. 4–5
  • The proposed P-D method aims to preserve binary variables and reward generators adequately without uplift payments.Table 1 provides a comparative overview of the pricing methods’ main features.Source: §2.3.2 · p. 5, Table 1 · p. 5

3. Primal-Dual Formulation for Pricing Static Voltage Stability Services

The framework combines a voltage-stability-constrained unit commitment model with primal-dual pricing and explicit non-negative-profit constraints. It accounts for IBR investment costs and linearizes bilinear pricing terms before solving.

3.1. Mathematical Background of Primal-Dual Formulation

  • The primal-dual formulation minimizes the duality gap while satisfying primal and dual constraints, with strong duality guaranteed when ε = 0.Additional constraints can compromise strong duality, but the formulation remains feasible and its solution stays close to the original problem's optimum.Source: §3.1 · p. 5

3.2. Primal of Static Voltage Stability Constrained UC

  • The primal MISOCP minimizes system operating costs subject to voltage stability at GFL-IBR buses and generator operating constraints.It includes startup costs, active and reactive power bounds, apparent-power capacities, power balances, and binary commitment requirements.Source: §3.2 · pp. 5–6

3.3. Dual of Relaxed Static Voltage Stability Constrained UC

  • The dual is derived after relaxing binary variables, with dual constraints corresponding to primal variables and SOC constraints.Auxiliary dual terms account for equations used to linearize products of commitment variables.Source: §3.3 · pp. 6–7

3.4. Primal-Dual Formulation for Pricing Static Voltage Stability Services

  • The pricing formulation adds non-negative-profit constraints so units can recover costs through energy and ancillary-service revenues.These constraints are intended to provide sufficient incentives for generators under nonconvex operating conditions.Source: §3.4 · p. 7
  • The method relaxes binary variables, derives the dual, adds profit constraints, and merges primal and dual formulations.The resulting nonlinear terms are linearized before solving, producing an MISOCP.Source: §3.4 · p. 7
  • Including LCOE captures IBR capital investment costs that short-run operating-cost pricing omits.VSG and GFL-IBR investment costs are amortized daily, independent of wind generation.Source: §3.4 · p. 7

4. Case Studies

Case studies evaluate the pricing methods on a modified IEEE 30-bus system. The proposed method prevents losses for the units that lose money under the alternatives, while service volumes and prices vary with system conditions.

4.1. Test System Setting

  • The modified IEEE 30-bus test system places wind units at buses 1, 23, and 24, with 420 MW of aggregated wind capacity.The bus-1 unit uses VSG control; the other two use GFL-IBR modes, and synchronous generators are placed at buses 2, 3, 4, 5, 27, and 30.Source: §4.1 · pp. 7–8
  • The voltage-stability constraint approximation has overall MAPE below the 5% threshold defined in.Offline training for the test system takes 3.99 seconds.Source: §4.1 · p. 8, Table 3 · p. 8

4.2.1. Volumes of Voltage Stability Services

  • Synchronous-unit SCR support at bus 24 varies with commitment, while VSG support is nearly complementary to that of other generators.At bus 23, lower SCR is associated with greater reactive support: average output is 54.13 Mvar/h versus 44.86 Mvar/h at bus 24.Source: §4.2.1 · p. 8

4.2.2. Prices of Voltage Stability Services

  • Dispatchable and P-D reactive-support prices are generally zero from 06:00 to 19:00, while restricted-method ancillary-service prices are zero throughout this scenario.Off-peak prices diverge between dispatchable and P-D pricing when the P-D profit constraints bind; setting LCOEs to zero makes their energy and ancillary-service prices identical.Source: §4.2.2 · pp. 8–9

4.2.3. Profitability of Generating Units

  • All units except the VSG at bus 1 and thermal unit at bus 5 earn non-negative total profits; the P-D method prevents both from incurring losses.Under restricted pricing, the VSG lacks commitment payments and does not recover its LCOE; under dispatchable pricing, both units receive inadequate incentives.Source: §4.2.3 · p. 9, Table 4 · p. 9

4.3. Sensitivity of Market Clearing to Electricity Demand

As demand rises, ancillary-service prices under the primal-dual formulation fall as energy revenues and synchronous-generator commitment increase; its prices ensure non-negative profits for all units.

4.3.1. Prices vs. Electricity Demand

  • P-D produces higher reactive-support prices than the dispatchable method to ensure non-negative profits for all units; restricted commitment prices can be zero when they cover synchronous-generator costs.Source: §4.3.1 · p. 9
  • As load rises, ancillary-service prices decline because wind growth raises energy revenues while more committed synchronous generators raise system SCR and reduce additional support needs.Source: §4.3.1 · p. 9
  • Above 60% of base load, restricted commitment payments move inversely with energy prices; below it, both rise, and dispatchable prices still require uplift.Source: §4.3.1 · p. 9
  • By integrating cost-recovery obligations into market prices, P-D provides transparent price signals independent of commitment remuneration.Source: §4.3.1 · p. 9

4.3.2. Net Profits vs. Electricity Demand

  • At light load, thermal deficits peak at -137.55 €, while 𝑔𝑣−𝑏1, 𝑔𝑓−𝑏23, and 𝑔𝑓−𝑏24 lose money; the latter two profit only at high loads.Source: §4.3.2 · p. 10
  • P-D price signals captured by wind units guarantee non-negative profits in all cases, providing incentives for wind assets to participate in market operations.Source: §4.3.2 · p. 10

4.4. Sensitivity of Market Clearing to Reactive Power Support

The sensitivity analysis examines how reactive-power capacity affects prices and generator profits. The proposed pricing method protects unit profitability while adjusting the balance between energy and ancillary-service prices.

4.4.1. Prices vs. Reactive Power Support

  • As reactive-power capacity increases, P-D and dispatchable voltage-stability prices decline; prices remain higher at bus 23 than bus 24 because its weaker SCR increases reliance on reactive support.Increased reactive support can compensate for weakened SCR and facilitate greater wind integration, making both SCR enhancement and reactive support less scarce.Source: §4.4.1 · p. 10
  • At some periods, P-D ancillary-service prices at bus 24 are below dispatchable prices because the operator adjusts energy prices to satisfy unit profits.Source: §4.4.1 · p. 10
  • Under restricted pricing, energy prices rise with reactive capacity while commitment prices fall; P-D and dispatchable energy prices are less sensitive because fixed costs are recovered through market prices and thermal scarcity drives energy prices.Under restricted pricing, increased energy revenues recover more thermal operating costs, reducing the need for commitment payments.Source: §4.4.1 · p. 10

4.4.2. Net Profits vs. Reactive Power Support

  • Unit gc-b5 earns [−8.10, 28.96] €/day only under dispatchable pricing, while the other two mechanisms yield positive earnings.The result exposes dispatchable pricing’s inability to guarantee full cost recovery; the proposed framework provides sound incentives without uplift payments.Source: §4.4.2 · p. 10

4.5. Computational Performance

  • Dispatchable and restricted clearing are faster than P-D, but P-D adds only a modest solving-time increase versus an energy-only market.P-D coordinates energy and ancillary-service provision while optimizing prices to ensure non-negative unit profits; Table 5 compares computational times.Source: §4.5 · p. 10, Table 5 · p. 10

5. Conclusion and Future Work

The paper proposes primal-dual pricing to address cost-recovery limits in voltage-stability service markets. It reports that participating units can remain profitable without uplift payments and identifies additional services and strategic bidding for future study.

5. Conclusion and Future Work

  • The primal-dual scheme enforces non-negative unit profits by optimizing shadow prices, remunerating (V)SGs for SCR enhancement and GFL-IBR for reactive-power support.Source: §5 · p. 11
  • Sensitivity analyses find that primal-dual prices internalize fixed costs and keep all generating units profitable without uplift payments, whereas dispatchable prices can leave units with negative profits and restricted uplift allocation excludes IBR.Source: §5 · p. 11
  • Future work would add frequency reserves and inertia to pricing and examine how strategic bidding affects prices in practice.Source: §5 · p. 11

A. Offline Training for Approximating Nonlinear Terms in Voltage Stability Constraints

The appendix describes an offline procedure for approximating nonlinear impedance terms used in voltage-stability constraints, fitting candidate parameters against exact network-based ratios.

A. Offline Training for Approximating Nonlinear Terms in Voltage Stability Constraints

B. Dual Formulation of McCormick Envelopes

This appendix derives dual terms for relaxed commitment variables using auxiliary constraints, illustrated with two generators and indexed pairwise combinations.

B. Dual Formulation of McCormick Envelopes

C. Binary Expansion for Nonlinear Terms in Non-negative Profit Constraints

Binary expansion linearizes energy-price products in non-negative-profit constraints, and analogous bounded linearizations handle reactive-power and commitment products.

C. Binary Expansion for Nonlinear Terms in Non-negative Profit Constraints

  • Binary expansion decomposes synchronous-generator dispatch into auxiliary binary variables, rewrites its energy-price product, and linearizes the resulting bilinear term.The discretization uses N segments; larger N approximates the continuous dispatch variable with higher accuracy.Source: C. Binary Expansion for Nonlinear… · pp. 11–12
  • Energy prices are constrained non-negative, while M is chosen large enough that constraint (18e) is inactive at the optimum.M can be set to twice the maximum marginal cost across synchronous generators.Source: C. Binary Expansion for Nonlinear… · p. 12
  • Analogous bounded linearizations apply to reactive-power and commitment products, with the multiplier bound set to twice the largest combined no-load and startup cost.This bound is intended to provide sufficient compensation for thermal units scheduled to support voltage stability.Source: C. Binary Expansion for Nonlinear… · p. 12
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