An arXiv preprint describes a way for an operator to choose fixed power-factor settings for distributed energy resources (DERs) before active-power injections are known. In the paper’s linear model, a closed-form “cancellation” setting reaches the best possible worst-case result for the total size of departures from nominal voltage, but only when the calculated setting is feasible and the theorem’s other conditions hold.
A game played before the injections are known
Power factor is represented here by a reactive-to-active power ratio and a reactive-power direction. The operator chooses one constant setting for each DER first; the active-power owner then chooses an injection within the device’s limits. The objective is the largest aggregate absolute voltage deviation among those possible choices.
Once the settings are fixed, the worst active-power choice at each node is an endpoint of its feasible interval: either zero or the largest allowed injection, depending on whether increasing that injection raises or lowers the aggregate voltage deviation.
The formula has a boundary
To solve the operator’s side of the game, the authors replace discrete power-factor magnitude and direction choices with continuous reactive-to-active coefficients and divide the problem into regions according to the signs of the voltage residuals. In the nominal-orthant region—the region matching the nominal sign pattern—the cancellation vector is calculated from the model’s coefficients. If every relevant coefficient is nonzero and that vector is feasible, the theorem says it is globally minimax and reaches the model’s initial nominal-offset value.
That guarantee does not apply automatically outside those conditions. In one reported direct minimax test at a DER rating of 0.05 p.u. and a minimum power-factor setting of 0.7, the lower bound was reached for Hawaii 37, IEEE 118 and Illinois 200. RTS-GMLC had a 1.14 ratio to the lower bound and no feasible operator setting in that condition, so no minimax value was available.
Numerical checks matched the formula
The main numerical table covered four network cases: Hawaii 37, RTS-GMLC, IEEE 118 and Illinois 200. Their reported participating-node counts were 27, 40, 64 and 162, respectively. The analytical and numerical cancellation settings agreed at solver precision; runtimes ranged from 0.002 to 0.05 seconds, while reported relative errors ranged from 2.89 × 10^-14 to 3.15 × 10^-6.
The authors also report the same cancellation computation on network instances ranging from 14 to 2,000 buses. In that broader check, relative errors stayed between 10^-14 and 10^-7, and every instance was solved in under one second. The check is evidence about implementation and scaling, not a separate demonstration of global minimax optimality outside the theorem’s conditions.
The nonlinear test narrows the claim
The optimization uses a first-order linear voltage approximation, with voltage sensitivities taken from nonlinear AC power-flow equations at initial operating points. In selected tests at a uniform DER rating of 0.01 p.u. per participating node, maximum voltage-magnitude differences between the linear and nonlinear results ranged from 5.4 × 10^-6 to 2.1 × 10^-3 p.u., while relative differences in aggregate deviation ranged from 0.01% to 6.1%. For Illinois 200 at 0.05 p.u., the maximum voltage-magnitude difference reached 0.057 p.u.
Restrictions change the answer
The modeled objective stayed at its lower-bound value while unconstrained cancellation remained feasible. It rose when a minimum power-factor floor clipped the cancellation setting, with a sharp increase as the floor approached 1.
DER ratings produced a network-specific pattern. The fixed cancellation setting reached the lower bound for Hawaii at every reported rating and for Illinois 200 by 0.1 p.u. At 0.05 p.u., RTS-GMLC and IEEE 118 remained above the bound, at ratios of 1.14 and 1.22.
A centralized curtailment benchmark gave a different voltage-deviation profile, but it optimized active-power injection rather than the robust voltage-deviation objective. The comparison was therefore not like-for-like, and the ordering differed between Hawaii and RTS-GMLC.
The result is a conditional mathematical guarantee tied to the linear approximation, the stated feasibility conditions and the reported network tests—not a demonstrated operating outcome. The work is an arXiv preprint dated 20 Aug 2026.
Paper data and sources
Original title: Zero-Sum Power Factor Games
Authors: Cameron Khanpour, Samuel Talkington, Mathieu Dahan, Daniel K. Molzahn
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text