⚡ electrical engineering

Zero-Sum Power Factor Games

This paper formulates and solves a robust minimax problem for an operator to optimally select constant power factor settings for distributed energy resources, thereby minimizing worst-case voltage deviations caused by uncertain active power injections in electric power networks.

Original authors: Cameron Khanpour, Samuel Talkington, Mathieu Dahan, Daniel K. Molzahn

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cameron Khanpour, Samuel Talkington, Mathieu Dahan, Daniel K. Molzahn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Electricity flows through a vast, invisible web of wires, and for the grid to work safely, the electrical pressure at every point—known as voltage—must stay within a very narrow range. If the pressure gets too high, it can damage equipment; if it drops too low, lights flicker and motors stall. In the past, this pressure was managed by large, centralized power plants that could be told exactly how much to push. Today, however, the grid is changing. It is increasingly filled with smaller, scattered devices like rooftop solar panels, home batteries, and electric vehicles. These devices are wonderful for clean energy, but they introduce a new kind of uncertainty. Their output depends on the weather, the time of day, and the decisions of their owners, making the electrical pressure harder to predict and control.

To keep the grid stable, operators can adjust how these devices handle a specific type of electrical flow called reactive power. Think of this as a balancing act: while the main power does the work of running appliances, reactive power acts like a stabilizer that keeps the voltage steady. A key rule in the industry, known as IEEE Standard 1547, allows operators to set a fixed relationship between the main power a device produces and this stabilizing reactive power. This setting is called the power factor. The challenge arises because the operator must set this relationship before knowing exactly how much main power the device will actually produce. If the setting is wrong, the stabilizing effect might be too weak or too strong, leading to voltage swings that could disrupt the neighborhood.

In this study, researchers approached this timing problem as a strategic game. They imagined two players: an operator who sets the power factor rules, and a "worst-case" scenario where the actual power output behaves in the most difficult way possible for the operator. The goal was to find a setting that would keep the voltage as stable as possible, even if the power output turned out to be the most troublesome version allowed by the device's physical limits. The researchers used a simplified, straight-line model of how voltage changes to solve this puzzle. They discovered that there is a specific, perfect setting for the power factor that effectively cancels out the worst possible voltage swings. When this setting is used, the voltage deviation never grows larger than it was before the devices even started producing power. It is as if the operator finds a position where the device's own reactive power perfectly neutralizes the disturbance caused by its active power, leaving the grid exactly where it started.

The team tested this idea on several realistic computer models of electrical grids, ranging from small local networks to large regional systems. They found that for many of these networks, the perfect setting could be calculated directly using a simple formula, without needing to run complex, time-consuming simulations. This formula relies on the specific layout of the wires and the sensitivity of the voltage to power changes at each location. The results showed that when the devices are small relative to the size of the grid, this perfect setting works beautifully, keeping the voltage stable no matter how the power output fluctuates. However, if the devices are very large, the simple formula might not be enough, and the operator would need to solve a more difficult problem to find the best setting.

The researchers also looked at what happens if the rules for the power factor are restricted. In the real world, devices cannot always provide the full range of reactive power needed; they are often limited to a certain minimum power factor. The study showed that as these limits become tighter, the ability to stabilize the voltage decreases. The researchers quantified exactly how much regulation capacity is lost when the operator is forced to choose a less flexible setting. Interestingly, the perfect settings they calculated mathematically closely matched recommendations that engineers had already arrived at through experience and observation. This suggests that the intuitive rules used by practitioners are actually a form of robust defense against the worst-case uncertainties of the modern grid.

While the study relied on mathematical models and computer simulations rather than physical experiments on a live grid, the results offer a clear path forward. The researchers demonstrated that by treating the problem as a game against uncertainty, it is possible to find a strategy that guarantees stability. They also noted that their findings depend on the assumption that the devices follow the rules set by the operator. If a device were to ignore the power factor setting and change its reactive power independently, the strategy would no longer hold. The work provides a solid theoretical foundation for how grid operators can set their controls today to ensure that the transition to a cleaner, more distributed energy future remains safe and reliable.

Technical Summary: Zero-Sum Power Factor Games

Problem Formulation
The paper addresses the challenge of voltage regulation in electric power networks with distributed energy resources (DERs) where active power injections are uncertain or potentially compromised. The core problem arises when a network operator must configure reactive power control parameters (specifically constant power factor settings) before observing the realized active power injections. This timing creates a vulnerability: if the operator sets a fixed power factor, an adversary (or worst-case uncertainty) could select active power injections that maximize voltage deviations, subject to the physical limits of the DERs.

The authors model this interaction as a robust minimax problem, interpreted either as a worst-case uncertainty scenario or as a zero-sum Stackelberg game. In this game:

The Operator (Leader): Selects a constant power factor magnitude ( $\alpha$ ) and reactive power direction ( $\xi$ ) for each DER. This setting couples the device's reactive power response to its active power output ( $q = K(\alpha, \xi)p$ ).
The Owner/Attacker (Follower): Observes the installed settings and selects an active power injection vector ( $p$ ) within the device's apparent power limits ( $p^2 + q^2 \leq 1$ ).
Objective: The operator aims to minimize the maximum aggregate absolute voltage deviation ( $\ell_1$ norm) from the nominal profile that the follower can induce.

Methodology
The authors utilize a linear voltage approximation model ( $v = v^0 + Rp + Xq$ ) centered at an initial operating point. The methodology proceeds through three main analytical steps:

Continuous Reformulation: The discrete choice of power factor direction and magnitude is transformed into a continuous coefficient vector $\kappa$ (where $q_i = \kappa_i p_i$ ). This allows the problem to be treated as a continuous optimization over a convex set, simplifying the analysis of the coupling between active and reactive power limits.
Orthant Decomposition: The authors decompose the inner maximization problem (the worst-case active power selection) based on the sign pattern of the voltage residual ( $m^\bullet - \tilde{S}(\kappa)p$ ). They identify a specific subset of settings, termed the "nominal orthant" ( $K^\bullet$ ), where no feasible active power injection causes a bus voltage to cross its nominal value (i.e., the sign of the deviation remains constant).
Closed-Form Solution (Cancellation): Within the nominal orthant, the objective function becomes affine. The authors derive a "cancellation setting" ( $\kappa^*$ ) where the coefficient of each active power injection in the aggregate deviation equation is zero. This is achieved by setting $\kappa^*_i = -\sigma_i / \omega_i$ , where $\sigma$ and $\omega$ are vectors derived from the network sensitivity matrices and the initial voltage deviations. When this setting is feasible, it neutralizes the impact of any active power injection on the aggregate voltage deviation.

Key Contributions

Robust Minimax Formulation: The paper formulates constant power factor control as a robust minimax design that explicitly enforces DER apparent power limits. This captures the physical tradeoff where reserving capacity for reactive power reduces the maximum feasible active power injection, a coupling often omitted in independent optimization approaches.
Analytical Solution Structure: The authors derive the exact structure of the solution. They prove that when the optimal setting keeps all voltages within the nominal orthant, the minimax strategy is a closed-form "cancellation" setting. This strategy effectively cancels the contribution of every active power injection to the aggregate voltage deviation, yielding a worst-case deviation equal to the initial offset ( $||m^\bullet||_1$ ).
Characterization of Ratings and Constraints: The paper quantifies how DER ratings and restrictions on the admissible power factor range affect regulation performance. It demonstrates that restricting the power factor range (e.g., forcing a higher minimum power factor) reduces the operator's ability to cancel injection effects, thereby increasing the worst-case voltage deviation.

Results
Numerical tests were conducted on four test networks (Hawaii 37-bus, Illinois 200-bus, RTS-GMLC 73-bus, and IEEE 118-bus).

Validation of Cancellation: The analytical cancellation settings matched numerical least-squares solutions to within solver precision ( $10^{-10}$ to $10^{-14}$ ).
Linear vs. Nonlinear: When the cancellation setting was feasible, the linear model predictions closely matched nonlinear AC power flow results for small DER ratings (e.g., 0.01 p.u.). Discrepancies increased with larger ratings, confirming the linear model's accuracy is limited to small deviations from the operating point.
Feasibility and Optimality: For networks with realistic DER ratings (e.g., 0.05 p.u.), the cancellation setting was often feasible and globally optimal, achieving the theoretical lower bound of the initial voltage offset. However, for networks with tighter voltage constraints or larger ratings (e.g., IEEE 118 at 0.05 p.u.), the cancellation setting sometimes fell outside the feasible set, requiring a direct numerical minimax solution which still achieved the lower bound in some cases.
Comparison to Benchmarks: The robust power factor setting was compared to a centralized optimal power flow benchmark. While the centralized approach could sometimes inject more total active power, it often incurred significantly larger voltage deviations compared to the robust minimax setting.

Significance and Claims
The paper claims that the derived analytical power factor settings provide a game-theoretic justification for empirical rules found in prior literature (specifically [10], [11]). The authors note that their derived "cancellation" settings are identical to those empirically recommended, offering a rigorous robust minimax interpretation for these practices.

The significance lies in providing a tractable, closed-form strategy for operators to configure DERs in a way that is robust against worst-case active power variations, without requiring complex iterative optimization in real-time. The authors emphasize that this approach is particularly effective when device ratings are small enough that voltage deviations do not cross nominal values, a condition often met in practical distribution systems with moderate DER penetration. The work also highlights the value of maintaining the integrity of the reactive power control channel, as a compromised channel would prevent the operator from enforcing the necessary coupling between active and reactive power.

Technical Summary: Zero-Sum Power Factor Games

More like this