Power-Sharing Ratios

• Power-sharing ratios are normalized descriptors that define how physical or institutional resources are allocated among diverse agents.
• They are derived from optimization and control techniques, ensuring steady-state performance in microgrids, battery systems, and networks.
• Designs adapt to system dynamics using methods like droop control, Bayesian inference, and KKT optimization to balance contributions effectively.

Power-sharing ratios are normalized descriptors of how a total burden, resource, or influence is apportioned among multiple agents. In physical energy systems, they commonly represent steady-state or time-varying shares of active power, reactive power, current, or heat, for example $r_i := P_i/\sum_k P_k$ or $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$. In coupled electro-thermal systems they may be induced by inverse-cost coefficients, droop gains, or temperature–frequency couplings; in battery systems they can be loss-aware utilization factors; and in cooperative or institutional settings they can denote normalized bargaining or voting power derived from centrality measures, Shapley values, or accumulated voting units (Qin et al., 28 Sep 2025, Baranwal et al., 2016, Farakhor et al., 4 Mar 2025, Aleandri et al., 17 Jul 2025).

1. Core definitions and normalization conventions

The term “power-sharing ratio” is not tied to a single normalization. What remains invariant across domains is the role of the ratio: it maps heterogeneous agents into a comparable allocation rule. In engineering applications, the numerator is usually a physical power or current contribution; in network and institutional applications, it is often an index of influence or entitlement.

Setting	Ratio form	Normalization property
Combined heat and power networks	$r_i := P_i/\sum_k P_k$	Shares follow inverse-cost or droop-like coefficients
Parallel DC converters	$$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$	$\sum_i r_i(t)=1$
Large-scale BESS	$\mu_j=P_{b_j}/P_{\mathrm{out}}$	$\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$
Sharing networks with a priori unions	$R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$	Firm and union shares are normalized power indices
Deliberative voting power	$s_i(t)=P_i(t)/S(t)$, $s_i^C(t)=P_i(t)/P_C(t)$	Shares determine committee influence and correction size

A recurrent misconception is that power-sharing ratios must always sum to one. That is false in loss-aware battery dispatch, where the cell-level utilization factors satisfy $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$0 because the allocation must also cover pack losses. It is likewise false in comparative voting-power analysis, where ratios such as $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$1 compare a power index with relative weights instead of partitioning a conserved flow (Farakhor et al., 4 Mar 2025, Kurz, 2018). In deliberative democracy models, the normalized quantities $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$2 and $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$3 are not physical flow fractions but governance shares that bound proposal corrections and veto power (Karoukis, 2021).

2. Steady-state laws in power, heat, and microgrid systems

In combined power and heating networks with heat pumps, steady-state power-sharing ratios are induced by decentralized droop-like controllers that solve explicit optimization problems. For electric generators, the equilibrium law $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$4 yields

$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$5

For heat pumps in Mode 1, $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$6 gives

$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$7

and for conventional heat sources,

$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$8

Mode 1 separates optimal sharing in the electric and thermal subsystems, whereas Mode 2 solves a joint electric–heat allocation in which the cross-sector split is shaped by $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$9; increasing $r_i := P_i/\sum_k P_k$0 reduces the per-unit contribution of heat source $r_i := P_i/\sum_k P_k$1 because $r_i := P_i/\sum_k P_k$2 (Qin et al., 28 Sep 2025). An earlier formulation states the same structure with separate and joint optimization problems and emphasizes that the average temperature $r_i := P_i/\sum_k P_k$3 acts as the thermal analogue of synchronous frequency, enabling optimal sharing without prior disturbance knowledge (Qin et al., 6 Feb 2025).

In islanded and grid-connected microgrids, the same normalization logic appears in several distinct forms. For droop-controlled inverters, proportional sharing is obtained when $r_i := P_i/\sum_k P_k$4 and ratings are selected proportionally, so that $r_i := P_i/\sum_k P_k$5; under the constructive choice $r_i := P_i/\sum_k P_k$6, steady injections satisfy $r_i := P_i/\sum_k P_k$7 (Simpson-Porco et al., 2012). For distributed generators with time-varying maximum capacities $r_i := P_i/\sum_k P_k$8, proportional active-power sharing is expressed as $r_i := P_i/\sum_k P_k$9 with $$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$0 whenever feasible, so $$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$1 (Aalipour et al., 2020). Reactive-power sharing admits further variants: inverter-based resources under voltage limits seek equality of utilization ratios $$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$2 among unsaturated units (Abdolmaleki et al., 2023), while in unbalanced multi-microgrids an ESS-based controller enforces

$$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$3

which becomes $$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$4 for a $$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$5 single-phase inverter paired with a $$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$6 three-phase inverter (Akdogan et al., 2021).

These formulations show that a steady-state ratio need not be tied to identical devices. It can instead encode inverse marginal costs, nameplate capacities, utilization equalization, or a prescribed dispatch proportional to ratings. This suggests that “power-sharing ratio” is best understood as a model-dependent equilibrium notion rather than as a single universal formula.

3. Time-varying and constrained implementations

A major development in converter control is the treatment of desired ratios as external references instead of fixed internal droop slopes. In robust DC-DC converter architectures, the centralized law

$$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$7

makes the desired ratio $$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$8 a time-varying signal supplied to each current loop. The same architecture also admits decentralized operation by replacing direct load-current measurement with a scheduled reference plus a voltage-error compensation term. Under the single-converter equivalence theorem, the multi-converter plant inherits voltage-regulation performance from a nominal single-converter design, and low-frequency sharing satisfies $$r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)=I_i(t)/I_{\mathrm{load}}(t)$$9 when the reference sum matches the load (Baranwal et al., 2016).

A closely related decentralized design prescribes average current shares $\sum_i r_i(t)=1$0 and ripple shares $\sum_i r_i(t)=1$1 through inner-loop shaping. Average power sharing is enforced with

$\sum_i r_i(t)=1$2

while $\sum_i r_i(t)=1$3 ripple sharing is obtained by

$\sum_i r_i(t)=1$4

This construction makes average power ratios and ripple ratios independent design targets, both realized through an equivalent single-converter $\sum_i r_i(t)=1$5 problem (Baranwal et al., 2016). A further extension to DC microgrids introduces time-varying $\sum_i r_i(t)=1$6 directly into each converter’s outer current loop; if $\sum_i r_i(t)=1$7 and $\sum_i r_i(t)=1$8, then steady-state bus-current and power contributions satisfy $\sum_i r_i(t)=1$9 and $\mu_j=P_{b_j}/P_{\mathrm{out}}$0 (Baranwal et al., 2017).

Large-scale BESS introduces a different constraint structure. Here the power-sharing ratio is the cell-level utilization factor

$\mu_j=P_{b_j}/P_{\mathrm{out}}$1

and because losses are explicitly modeled,

$\mu_j=P_{b_j}/P_{\mathrm{out}}$2

The top-level controller does not optimize $\mu_j=P_{b_j}/P_{\mathrm{out}}$3 free ratios directly; it parameterizes them as

$\mu_j=P_{b_j}/P_{\mathrm{out}}$4

so the high-dimensional allocation becomes a three-parameter problem balancing state of charge, temperature, and resistance (Farakhor et al., 4 Mar 2025). In massive-MIMO C-RAN, an analogous power-sharing concept appears in the split between pilot training and data transmission, where each user device and each remote radio unit is assigned a factor $\mu_j=P_{b_j}/P_{\mathrm{out}}$5 that partitions its power budget between estimation and payload transmission (Zhang et al., 2018).

Across these implementations, the central shift is from fixed ratio laws to ratio trajectories subject to bandwidth, saturation, and estimation limits. This suggests that power-sharing ratios increasingly function as control references embedded in layered architectures rather than as static operating points.

4. Optimization, inference, and stability structure

In many formulations, power-sharing ratios are not heuristic. They are the KKT solution of explicit optimization problems. In combined heat-and-power networks, Mode 1 steady states solve separate convex quadratic programs for the electric and heating subsystems, while Mode 2 solves a joint optimization over generators, heat sources, and frequency-dependent loads. Stability is established locally around equilibria satisfying $\mu_j=P_{b_j}/P_{\mathrm{out}}$6, and the same conclusions extend to higher-order input-strictly-passive generation dynamics with convex effective costs $\mu_j=P_{b_j}/P_{\mathrm{out}}$7 and $\mu_j=P_{b_j}/P_{\mathrm{out}}$8 (Qin et al., 28 Sep 2025).

Converter-based sharing often uses robust optimal-control synthesis instead of direct economic dispatch. The DC-DC frameworks above formulate weighted $\mu_j=P_{b_j}/P_{\mathrm{out}}$9 problems, with performance channels for bus-voltage tracking, current-tracking error, control effort, and high-frequency attenuation. The resulting controllers guarantee internal stability and explicit sharing-error bounds, while preserving a single-converter interpretation of the networked plant (Baranwal et al., 2016, Baranwal et al., 2016). Reactive-power sharing under voltage limits combines a local nonlinear integral controller with a distributed primal–dual optimizer. There the ratio objective is encoded as consensus of $\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$0 toward the normalized reactive injections $\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$1, and practical stability is derived by singular perturbation plus an LMI condition on a grounded reduced model (Abdolmaleki et al., 2023).

Battery dispatch pushes the optimization layer further by replacing direct NMPC solution with Bayesian inference. The constrained nonlinear predictive problem is rewritten as a MAP estimation problem for the parameter vector $\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$2, using a virtual dynamic system and barrier functions for state and power constraints. Ensemble Kalman inversion then estimates $\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$3 online, and a low-level PI loop converts the resulting $\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$4 into feasible cell power references under output-power uncertainty (Farakhor et al., 4 Mar 2025). Droop-e introduces yet another structure: a nonlinear primary frequency–power law

$\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$5

so the effective droop slope $\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$6 becomes operating-point dependent. Incremental sharing is therefore

$\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$7

and a secondary integrator drives the steady state back to a chosen target sharing law (Kenyon et al., 2022).

A plausible implication is that ratio design has evolved from static coefficient tuning toward formally optimized, state-dependent, and provably stable allocation mechanisms.

5. Networked, market, and institutional interpretations

Power-sharing ratios also appear where the “power” being shared is economic, relational, or institutional rather than electrical. In sharing networks with a priori unions, the firm-level power index

$\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$8

combines direct inter-union degree and rescaled inter-union neighborhood influence. The normalized firm and union shares are then

$\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|$9

and the index is characterized as the Shapley value of a transferable-utility game. The paper further shows structural stability of rankings under restricted spillovers and reports that the core is often empty for a broad class of network structures (Aleandri et al., 17 Jul 2025).

In feeder-aware renewable energy communities, sharing coefficients determine how surplus energy is allocated first within a feeder and then across the community. The framework uses feeder-level coefficients $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$0 in Stage 1 and community-level coefficients $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$1 in Stage 2, with equal, proportional, and rank-based rules implemented in both static and dynamic forms. Dynamic equal sharing uses $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$2 within a feeder, while dynamic proportional sharing uses $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$3 and $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$4 (Shooshtari et al., 16 Sep 2025). In coordinated power–traffic systems, the analogous ratio is economic: a profit-sharing coefficient $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$5 splits the DNO’s cost reduction as $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$6 and $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$7, with $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$8 minimizing the DNO’s total cost in the reported case (Sima et al., 15 Mar 2025). For two battery-equipped renewable bundles, Pareto-optimal energy sharing is parameterized by peak sharing rates $R_i=\phi_i^{(\alpha,\beta)}/P^{\mathrm{tot}}$9, and the paper proves that every Pareto-optimal arrangement lies on the boundary where at least one bundle shares at its maximum feasible rate (Deulkar et al., 2020).

Institutional power-sharing adopts still different normalizations. In weighted voting games, one studies deviations between power indices and weight shares through ratios such as $s_i(t)=P_i(t)/S(t)$0 and bounds such as

$s_i(t)=P_i(t)/S(t)$1

for the nucleolus and certain representation-compatible indices, together with impossibility results showing that no generally sharp approximation exists if only $s_i(t)=P_i(t)/S(t)$2 or $s_i(t)=P_i(t)/S(t)$3 is controlled (Kurz, 2018). In deliberative democracy with dilutive voting power, the relevant shares are $s_i(t)=P_i(t)/S(t)$4, within-committee shares $s_i(t)=P_i(t)/S(t)$5, and the rule that a correction proposed by committee member $s_i(t)=P_i(t)/S(t)$6 must satisfy $s_i(t)=P_i(t)/S(t)$7; outside agents can veto if more than half of the total voting power outside the committee is against the correction (Karoukis, 2021).

These examples show that “power-sharing ratio” generalizes well beyond physical flow allocation. It can denote a normalized claim on surplus, influence, or rights, provided the normalization is defined by the governing institutional or cooperative rule.

6. Empirical behavior, trade-offs, and common cautions

Reported case studies show that ratio design materially changes transient behavior, efficiency, and distributional outcomes. In an IEEE-39 bus case with four heating areas, both heat-pump participation modes stabilized frequency; bus 30 reached steady state in about $s_i(t)=P_i(t)/S(t)$8 in Mode 1 versus $s_i(t)=P_i(t)/S(t)$9 in Mode 2, and the aggregate electric adjustment of heat pumps stayed below $s_i^C(t)=P_i(t)/P_C(t)$0 (Qin et al., 28 Sep 2025). In unbalanced multi-microgrids, activating the ESS multifunction controller reduced PCC voltage unbalance factor from $s_i^C(t)=P_i(t)/P_C(t)$1 to $s_i^C(t)=P_i(t)/P_C(t)$2 while enforcing reactive sharing proportional to the $s_i^C(t)=P_i(t)/P_C(t)$3 inverter rating ratio (Akdogan et al., 2021). In robust DC converter control, three-converter ratios of $s_i^C(t)=P_i(t)/P_C(t)$4, $s_i^C(t)=P_i(t)/P_C(t)$5, and $s_i^C(t)=P_i(t)/P_C(t)$6 were tracked over successive time intervals while maintaining $s_i^C(t)=P_i(t)/P_C(t)$7 around $s_i^C(t)=P_i(t)/P_C(t)$8 (Baranwal et al., 2016). In feeder-aware energy communities, the best-performing dynamic proportional, dynamic rank-based, and static rank-based schemes all achieved $s_i^C(t)=P_i(t)/P_C(t)$9 of shared energy with $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$00 imported and $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$01 exported in the reported dataset (Shooshtari et al., 16 Sep 2025). In C-RAN, optimal pilot/data power-sharing factors increased sum-rate by $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$02 relative to a non-optimized baseline in the $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$03-antenna RRU and $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$04-antenna BBU example (Zhang et al., 2018). In double-null plasma simulations and TCV validation, the divertor outer-target asymmetry scaling tracked trends across magnetic imbalance and turbulence level, although deviations of up to a factor of about $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$05 remained in experimental comparisons (Lim et al., 2024).

The same literature also identifies persistent limitations. Average-temperature-based CHP sharing requires measurement or estimation of the weighted average $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$06, which implies aggregation of temperatures weighted by volumes and hence some communication or supervisory infrastructure; capacity constraints and saturation are not part of the core CHP proofs and alter the realized ratios when active (Qin et al., 28 Sep 2025). Time-varying converter ratios are most accurate in the low-frequency band where the current-loop sensitivity is small; very fast ratio changes beyond current-loop bandwidth incur tracking error, and decentralized operation depends on the quality of local load estimates (Baranwal et al., 2016). BESS ratio control presumes cell-level actuation through DC/DC converters and depends on identified $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$07 and $r_i(t)=P_i(t)/P_{\mathrm{tot}}(t)$08 models (Farakhor et al., 4 Mar 2025). In voting theory, weights can approximate power only under specific quota and maximum-weight conditions; outside those regimes, large discrepancies are provable (Kurz, 2018).

A common misunderstanding is that a desirable ratio can be specified independently of system dynamics. The surveyed work repeatedly shows the opposite: the same nominal ratio can produce different realizations depending on passivity, voltage saturation, load uncertainty, headroom, communication structure, or institutional feasibility. This suggests that a power-sharing ratio is not merely a target fraction but part of a closed allocation mechanism whose meaning is inseparable from the dynamics, constraints, and normalization that generate it.