Zero-Sum Power Factor Games

We formulate the operator’s decision as a robust minimax design. The operator first selects one constant power factor setting for each DER. The maximizing active power player, termed the owner, then selects an injection vector within the device limits. The operator minimizes the largest aggregate absolute voltage deviation that the owner can produce. In the robust interpretation, the inner maximization identifies the worst feasible realization, and no strategic or malicious actor is required. In the cybersecurity interpretation, the same formulation is a zero-sum Stackelberg game in which a stealthy attacker observes the installed power factor settings and then selects compromised active power references. The reactive power parameter channel retains integrity, and the injections remain consistent with the commanded power factor setting to avoid immediate detection. Both interpretations lead to the same minimax problem with the operator moving first.

This paper makes the following three contributions.

1. 1.

   We formulate constant power factor control as a robust minimax design while enforcing each DER’s apparent power limit. Because the power factor determines both the reactive response and the feasible active power set, the model captures capacity effects omitted when active and reactive power are treated independently. The formulation applies to robust voltage regulation and related hosting capacity and curtailment assessments.

2. 2.

   We derive the solution structure by representing the power factor decisions with continuous reactive-to-active power ratios (Lemma 2) and exactly decomposing the payoff by the sign pattern of the voltage residual (Proposition 1). At fixed operator settings, each worst-case active power injection is either zero or its largest feasible value, depending on whether increasing that injection raises or lowers the aggregate voltage deviation (Proposition 2). When no bus voltage crosses its nominal value, the optimal minimax ratio cancels the corresponding injection’s contribution and yields a closed form strategy (Theorem 1).

3. 3.

   We characterize how DER ratings and restrictions on the admissible power factor range affect regulation performance. Numerical tests check the cancellation computation, solve the complete minimax problem directly at a representative DER rating, and compare the linear voltage predictions with nonlinear AC power flow. The analytical power factors also resemble empirical recommendations in [33, 28], providing a game theoretic characterization.

Section II develops the network model, voltage approximation, and robust power factor design. Section III decomposes the lifted payoff by voltage orthant, characterizes the optimal minimax strategy in closed form when the voltage residual remains in the nominal orthant, and interprets the resulting power factors. Section IV verifies the results on realistic network models. Section V concludes the paper.

We first define the network operating point, voltage approximation, constant power factor model, and regulation target.

The operator first installs one constant power factor setting for each DER. The active power realization then occurs within the device limits, and the reactive power is determined by the installed setting. The operator chooses the settings to minimize the largest aggregate voltage deviation from nominal that can occur afterward.

The action set $\mathcal{A}$ in (9) includes a discrete choice for the direction of reactive power. The coefficient $\kappa_{i}$ in (4) combines this direction with the power factor magnitude. Under Assumption 1, its range is the box

Define $\boldsymbol{\tilde{S}}(\boldsymbol{\kappa})\triangleq\boldsymbol{R}+\boldsymbol{X}\operatorname{diag}(\boldsymbol{\kappa})$. For a fixed $\boldsymbol{\kappa}$, the feasible active power set is

The corresponding operator action set is

For $\boldsymbol{\kappa}\in\mathcal{K}$ and $\boldsymbol{p}\in\mathcal{P}(\boldsymbol{\kappa})$, the lifted objective is

The scalar $\kappa_{i}$ is the installed coefficient in $q_{i}=\kappa_{i}p_{i}$. When $p_{i}>0$, it also equals $q_{i}/p_{i}$. This interpretation does not require a realized power factor at zero output.

Thus, the partly discrete description $(\boldsymbol{\alpha},\boldsymbol{\xi})$ can be replaced by $\boldsymbol{\kappa}$ without changing the model.

Define the exact inner worst-case value

Lemma 2 makes (12) equivalent to

For $\boldsymbol{s}\in\{-1,0,1\}^{n}$ and $\boldsymbol{\kappa}\in\mathcal{K}$, define

where $\operatorname{sign}$ is applied componentwise with $\operatorname{sign}(0)=0$. These regions form an exact disjoint partition of $\mathcal{P}(\boldsymbol{\kappa})$.

The sign pattern $\boldsymbol{s}^{\bullet}\triangleq\operatorname{sign}(\boldsymbol{m}^{\bullet})$ is the sign of the voltage residual at zero DER output. Since $\boldsymbol{m}^{\bullet}=\boldsymbol{v}^{\bullet}-\boldsymbol{v}^{0}$, its entries show whether initial voltages are below, above, or at nominal. For fixed $\boldsymbol{\kappa}$, $\mathcal{R}_{\boldsymbol{s}^{\bullet}}(\boldsymbol{\kappa})$ contains injections that leave every nonnominal voltage strictly on its initial side. A voltage reaching nominal has a zero entry in its exact sign pattern.

The closure in $\mathbb{R}^{n}$ allows voltages to reach nominal without crossing to the opposite side. For each $j$ with $s_{j}^{\bullet}\neq 0$, the condition is equivalent to

If $s_{j}^{\bullet}=0$, the corresponding row of $\boldsymbol{\tilde{S}}(\boldsymbol{\kappa})$ must be zero. Smaller ratings, weaker voltage sensitivity toward nominal, and greater initial distance favor membership; voltage limit headroom separately determines whether $\boldsymbol{\kappa}\in\mathcal{K}$. For uniform apparent power ratings $\bar{s}_{i}=\bar{s}$ and every $m_{j}^{\bullet}\neq 0$, the threshold for fixed $\boldsymbol{\kappa}$ is

where $\boldsymbol{R}$ and $\boldsymbol{X}$ are evaluated at unit ratings, and a zero denominator imposes no restriction. For $\boldsymbol{\kappa}\in\mathcal{K}^{\bullet}$, the $\ell_{1}$ objective is affine over the entire feasible set, so the inner maximization is separable. Define

For a fixed power factor setting, $\sigma_{i}+\omega_{i}\kappa_{i}$ is the contribution of one unit of active power at node $i$ to the affine expression for the aggregate voltage deviation. It accounts for the reactive power that accompanies the active power. A positive coefficient makes the largest feasible injection worst for the operator, while a negative coefficient makes zero injection worst.

We call a vector a cancellation setting if it satisfies $\sigma_{i}+\omega_{i}\kappa_{i}=0$ for every $i\in\mathcal{N}$. Such a vector cancels every injection coefficient in (25). The componentwise cancellation conditions alone do not guarantee feasibility. The resulting vector must satisfy the range and robust voltage constraints in (15) and the nominal orthant condition in Definition 2.

If $\omega_{i}=0$ and $\sigma_{i}\leq 0$, the $i$th summand in (25) is zero for every feasible $\kappa_{i}$, so this coordinate can be chosen to help satisfy the coupled constraints. If the coefficients at all other coordinates are canceled and the resulting vector belongs to $\mathcal{K}^{\bullet}$, the same minimax value $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$ follows. If $\omega_{i}=0$ and $\sigma_{i}>0$, the summand is $\sigma_{i}/\sqrt{1+\kappa_{i}^{2}}$ and cannot be reduced to zero by cancellation. In this case, the operator must minimize the exact value (18) over $\mathcal{K}$.

If the cancellation vector does not belong to $\mathcal{K}^{\bullet}$, the theorem does not necessarily solve the exact minimax problem. It still cancels the nominal affine branch, but another voltage sign pattern may determine $I(\boldsymbol{\kappa})$. In that case, the exact objective (18) must be minimized over $\mathcal{K}$.

The operator cannot control the active power injections, but it controls the reactive power that accompanies each unit of active power. The setting $\kappa_{i}^{*}=-\sigma_{i}/\omega_{i}$ cancels the contribution of the $i$th injection to the nominal affine branch of the aggregate voltage deviation. When $\boldsymbol{\kappa}^{*}\in\mathcal{K}^{\bullet}$, no feasible injection raises the aggregate deviation above the fixed initial offset $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$. This is the sense in which the minimax power factors neutralize worst-case active power injections.

The minimax power factors (26) depend explicitly on the topology-dependent quantities $\boldsymbol{\sigma}$ and $\boldsymbol{\omega}$. Moreover, (26) is identical to the recommended power factor settings prescribed by [33, 28]. Thus, these empirical rules can be interpreted as robust reactive power responses to worst-case active power perturbations.

Table I compares the analytical cancellation setting $\boldsymbol{\kappa}^{*}$ from Theorem 1 with a numerical implementation check that does not use the ratio $-\sigma_{i}/\omega_{i}$. We compute

where $\operatorname{range}(\boldsymbol{k})$ is the lifted box in (13). This is the program used to produce Table I. Its Hessian is $2\operatorname{diag}(\omega_{1}^{2},\ldots,\omega_{n}^{2})$, so it is a convex quadratic least squares problem with box constraints. The program checks the numerical model of the cancellation equations and the inverse map to the power factor variables.

A direct numerical solution of (19) must account for every feasible voltage sign pattern. The operator setting also changes both the voltage response and the owner’s active power bounds. Across all four networks, $\boldsymbol{\kappa}^{\mathrm{num}}$ and its power factor representation agree with the analytical setting at solver precision. This agreement verifies the cancellation computation but is not independent evidence of global minimax optimality. When $\boldsymbol{\kappa}^{*}\in\mathcal{K}^{\bullet}$, Theorem 1 proves global optimality. Section IV-D separately evaluates the fixed cancellation setting and solves the complete minimax problem directly.

We also apply the injections that saturate the apparent power constraints at $\boldsymbol{\kappa}^{*}$ to the nonlinear AC power flow equations and compare the resulting voltages with the linear approximation. When $\boldsymbol{\kappa}^{*}\in\mathcal{K}^{\bullet}$, Theorem 1 identifies these injections as representative worst case realizations. Otherwise, they are boundary test points without that guarantee. We measure agreement using the maximum absolute voltage magnitude difference across participating nodes and the relative difference between the two aggregate deviations. At a uniform DER rating $\bar{s}=0.01$ p.u. per participating node, the first metric ranges from $5.4\times 10^{-6}$ to $2.1\times 10^{-3}$ p.u. across the four networks. The second ranges from $0.01\%$ to $6.1\%$. Both metrics come from evaluating the injections at the boundary of the feasible set with the linear and nonlinear models. The discrepancies increase with the uniform apparent power rating $\bar{s}$. On the Illinois 200-bus case, the maximum voltage magnitude difference reaches $0.057$ p.u. at $\bar{s}=0.05$ p.u. Thus, the linear model is accurate for small deviations from the initial operating point, while the analytical result may be less applicable to large injections under the nonlinear AC power flow model. The cancellation result fixes the aggregate deviation at its initial offset under its assumptions, but it does not require every voltage to remain at its initial value.

We next examine how the nominal orthant expression varies with the minimum operator power factor $\underline{\alpha}$ that bounds the lifted action set. Let $\widehat{\boldsymbol{\kappa}}$ be the cancellation setting projected onto the lifted box allowed by $\underline{\alpha}$. Define the excess above the fixed initial offset as

Fig. 2 plots $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}+V^{\bullet}(\widehat{\boldsymbol{\kappa}})$ for $\underline{\alpha}\in[0.7,0.99]$. When the unconstrained cancellation setting $\boldsymbol{\kappa}^{*}$ is realizable within the floor, every summand vanishes and the value equals $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$. As $\underline{\alpha}$ tightens and clips $\boldsymbol{\kappa}^{*}$, the plotted expression increases, quantifying the regulation capability forgone by restricting reactive power. The floor at which the value departs from $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$ is network dependent and is governed by the smallest cancellation power factor. The value rises sharply as $\underline{\alpha}\to 1$, where the operator is driven to unity power factor and retains no reactive authority.

As an operational reference, we formulate an approximate optimal power flow model that minimizes active power curtailment. Maximizing the total normalized active power is equivalent to minimizing curtailment from the available upper bounds in $\mathcal{P}$. The centralized planner selects the active power and reactive to active power ratios jointly. In particular, we solve

The centralized benchmark (28) gives one planner complete knowledge of the active and reactive power decisions. Its objective differs from the voltage deviation objective in (12). It provides a conventional curtailment reference for comparing the voltage behavior that results from centralized coordination with the behavior of the robust power factor setting.

We compare the centralized curtailment solution with the cancellation setting in Fig. 3 using the Hawaii and RTS-GMLC test networks. We show a representative realization that saturates each apparent power constraint at the cancellation setting. This is a worst case realization when the setting lies in $\mathcal{K}^{\bullet}$. On Hawaii, the centralized planner injects slightly more total active power than the cancellation realization ($\boldsymbol{1}^{\top}\boldsymbol{p}=26.7$ versus $25.0$ p.u.), but incurs substantially larger voltage deviations. The mean distance to nominal is $0.089$ p.u. under the centralized benchmark and $0.027$ p.u. at the cancellation setting. Indeed, the realized $\ell_{1}$ deviation, $27\times 0.027\approx 0.72$ p.u., matches the robust minimax value $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}=0.71$ p.u. predicted by Theorem 1. On RTS-GMLC, the ordering reverses. The mean distance is $0.235$ p.u. at the cancellation setting and $0.031$ p.u. under the centralized benchmark. At full normalized rating, $\mathcal{K}$ is empty under the voltage limits supplied with this case. Therefore Theorem 1 is not globally optimal on this instance.

On the nominal orthant subset $\mathcal{K}^{\bullet}$ of Definition 2, where the owner’s entire feasible set $\mathcal{P}(\boldsymbol{\kappa})$ remains in the nominal orthant, Theorem 1 shows that the cancellation setting $\boldsymbol{\kappa}^{*}$ attains the global lower bound $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$ on the worst case objective. The operator cannot reduce its worst case deviation below this value because zero injection is always feasible. No owner realization can raise the deviation above it at $\boldsymbol{\kappa}^{*}$. Thus, the exact value in (18) satisfies $I(\boldsymbol{\kappa}^{*})=\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$.

We first fix the operator action at $\boldsymbol{\kappa}^{*}$ and compute the inner maximization (18) exactly as a mixed integer program. This is the inner maximization of the game. It evaluates $I(\boldsymbol{\kappa}^{*})$ without solving the outer minimization over $\mathcal{K}$. We begin with a uniform device rating $\bar{s}=1$ p.u. on the system base at every participating node. The ratio $I(\boldsymbol{\kappa}^{*})/\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$ is $1.00$ for Hawaii, $2.47$ for Illinois 200, $13.05$ for RTS-GMLC, and $15.24$ for IEEE 118. A ratio above one means that an injection near the device rating can push the voltage residual across the nominal profile into another orthant. The value $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$ is then only the nominal orthant prediction at the cancellation setting, while its exact inner value is strictly larger. The choice $\bar{s}=1$ p.u. is a deliberate stress test in which every participating node can inject apparent power equal to the system base. The robust model does not require this rating. This fixed action calculation alone does not solve the outer minimization or establish that $\boldsymbol{\kappa}^{*}$ belongs to $\mathcal{K}$.

The quantity $\bar{s}_{i}$, introduced in Section II-A, is the physical apparent power rating at node $i$ expressed on the system base. In practice, these ratings are far below the system base. On a 100 MVA base, $\bar{s}_{i}=0.05$ to $0.1$ p.u. corresponds to 5 to 10 MVA of devices at every participating node, or hundreds of MVA of aggregate DER capacity on these systems. Normalizing each device output by $\bar{s}_{i}$ is equivalent to scaling the corresponding columns of the voltage sensitivity matrices $\boldsymbol{R}$ and $\boldsymbol{X}$ by $\bar{s}_{i}$. This scaling leaves $(\boldsymbol{\kappa}^{*},\boldsymbol{\alpha}^{*},\boldsymbol{\xi}^{*})$ and the nominal value $\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$ unchanged because $\bar{s}_{i}$ cancels in the ratio $\boldsymbol{\sigma}/\boldsymbol{\omega}$. Table II reports $I(\boldsymbol{\kappa}^{*})/\left|\left|\boldsymbol{m}^{\bullet}\right|\right|_{1}$ as a uniform rating $\bar{s}$ is reduced. The fixed cancellation value equals the lower bound for Hawaii at all reported ratings and for Illinois 200 by $\bar{s}=0.1$. RTS-GMLC and IEEE 118 approach the lower bound with ratios $1.14$ and $1.22$ at $\bar{s}=0.05$. For RTS-GMLC, (23) gives $\bar{s}\leq 0.0151$ p.u. at $\boldsymbol{\kappa}^{*}$, but $\mathcal{K}^{\bullet}$ remains empty at every positive rating because three initial voltages equal their upper limits and the corresponding robust constraints are incompatible. The power factors are invariant to within $10^{-14}$ under changes in $\bar{s}$. These values describe the performance of the cancellation setting.

We next solve the complete outer problem (19) by constraint generation over voltage sign patterns. The master problem enforces the full operator set $\mathcal{K}$ and provides a global lower bound. The exact mixed integer inner problem finds a worst case sign pattern and provides an upper bound. Gurobi solves both the global nonconvex master and the inner problem. Agreement of the bounds certifies the minimax value.

For a concise comparison, define

Table III reports the representative rating $\bar{s}=0.05$ p.u. with $\underline{\alpha}=0.7$. This power factor floor is also the lower endpoint used in Fig. 2. The cancellation value $C$ is a fixed action diagnostic even when $\boldsymbol{\kappa}^{*}$ is inadmissible. The direct value $V$ is reported only when $\mathcal{K}$ is nonempty.

The global lower and upper bounds agree at solver precision for the three nonempty cases. On Hawaii and Illinois 200, the cancellation setting belongs to $\mathcal{K}^{\bullet}$, and the direct solution confirms the value established by Theorem 1. On IEEE 118, the cancellation setting is not an admissible operator action at this power factor floor. The theorem therefore does not apply to that setting. A different admissible action nevertheless attains $V=L$, which shows that the nominal offset can remain achievable outside the cancellation regime. For RTS-GMLC, no power factor setting keeps the full reference box (8) within the voltage limits supplied with the case. Thus, $\mathcal{K}$ is empty, the assumed nonempty operator set does not hold, and this model instance has no minimax value. Its reported cancellation ratio is not the value of a feasible policy.

Finally, the cancellation setting can be computed well beyond these representative cases. Across networks ranging from $14$ to $2000$ buses, the relative error between (26) and the numerical implementation check remains between $10^{-7}$ and $10^{-14}$, with each instance solved in under one second.