Shadow Pricing of Static Voltage Stability Services within Unit Commitment for Inverter-Dominated Power Systems

Jul 16, 2026

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Modern power systems increasingly rely on inverter-based resources (IBR) operating in grid-following mode, which does not directly regulate terminal voltage and can weaken static voltage stability at corresponding buses. The issue is intensified under constant power-factor behavior without voltage-adaptive reactive support, especially when weather-driven IBR are deployed in remote areas with low short-circuit ratio (SCR). A mechanism is proposed that assigns shadow prices to a second-order-cone static voltage stability constraint within unit commitment, and pricing schemes are compared on an IEEE 30-bus case.

LanguageEnglish

CategoryResearch & Report

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Pages11

Reading Time28 min

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Introduction

Shadow Pricing of Static Voltage Stability Services within Unit Commitment for Inverter-Dominated Power Systems Peng Wang, Graduate Student Member, IEEE, Luis Badesa arXiv :2607 .05209v1 [ ee ss . SY] 6 Jul 2026 Abstract—Modern power systems are increasingly dominated by Inverter-Based Resources (IBR), most of which work in Gridfollowing (GFL) mode. This implies that they do not directly control their terminal voltage, so the static voltage stability at these buses may be compromised, especially under constant-powerfactor operation that lacks voltage-adaptive reactive support. In addition, weather-driven IBR are often installed in electrically remote areas with low Short-Circuit Ratio (SCR), further exacerbating voltage issues. To address this challenge, grid-forming control can be utilized to enhance low-SCR buses, while GFLIBR could be explicitly required to provide voltage support through grid codes. As an alternative, a market mechanism could be devised that incentivizes relevant generators to proactively adjust their operating points as a service to maintain voltage stability, while the theoretical framework for such a market has not been developed. To fill this gap, this work adopts a second-order cone-based static voltage stability constraint for GFL-IBR buses within a unit commitment problem, and proposesa mechanism to assign shadow prices to this ancillary service. To determine appropriate price values under non-convex conditions, different pricing schemes are assessed. Using a modified IEEE 30-bus system, we demonstrate that both the dispatchable and restricted pricing methods can yield revenue-adequate service prices, though the former may deliver less efficient price signals and the latter may require well-defined uplift payments. This implies that, given differentiated pricing mechanisms and price signals, operators need to select a suitable pricing method in accordance with actual system conditions and market rules. Index Terms—Ancillary services, inverter-based resources, non-convexity, shadow prices, static voltage stability. NOMENCLATURE Indices and Sets gc , Gc Index, Set of conventional SGs gf , Gf Index, Set of GFL-IBR gv , Gv Index, Set of VSGsi Index of buses m, M Index, Set for products of units’ operating states Φ(·) Index of generator buses Constants and Parameters cmgc cnglc cstgc , cshgc PDi , QDi Pm{ in} , Pm{ ax} Qm{ in} , Qm{ ax} Sm{agxc,gf ,gv }αgf , αgv Marginal generation cost of SG ( ~~ C~~/MWh) No-load cost of SG ( ~~ C~~/h) Startup/shutdown costs of SGs ( ~~ C~~) Active/reactive demand at bus i (MW, Mvar) Generators’ active power output limits (MW) Generators’ reactive power output limits (Mvar) Rated apparent power of generators (MVA) Capacity factor of GFL-IBR and VSGs Primal and Dual Variables Cstgc , Csghc Costs incurred by startup/shutdown of SG ( ~~ C~~) ˆPgf , ˆQgf Equivalent active/reactive power injected by GFL-IBR (MW, Mvar) P{·} , Q{·} Active/reactive power output of generators (MW, Mvar) ugc Binary variable, commitment of SG Γgf Variable associated with SCR ηm Product of operating states of each two (V)SGs λ 1,gf ,λ2,gf ,µgf Dual variables, associated with voltage stability constraints λE Dual variable, energy price ( ~~ C~~/MWh) λgc ,commit Dual variable, commitment price for SG ( ~~ C~~) I. INTRODUCTION VOLTAGE stability analysis conventionally focuses on load buses, under the assumption that voltages at synchronous-type controlled buses are properly regulated [1] . As a result, voltage stability is largely affected by system loading conditions. However, as Grid-Following (GFL) InverterBased Resources (IBR) become increasingly prevalent in lowcarbon power systems, voltage stability at GFL-IBR buses must also be considered to ensure secure system operation. This consideration arises from the fact that, unlike GridForming (GFM) generators, which behave as voltage sources and establish system voltages, GFL-IBR interfa

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What do the experiments on the IEEE 30-bus system show about pricing schemes?

Both dispatchable and restricted pricing methods can produce revenue-adequate service prices. The dispatchable method may send less efficient price signals, while the restricted method may need well-defined uplift payments.

Jul 16, 2026

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