---
title: Power-Sharing Ratios
url: https://www.emergentmind.com/topics/power-sharing-ratios
type: topic
---

# Power-Sharing Ratios

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 [2509.24051] [1604.04154] [2503.02866] [2507.13272].

## 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 \(\sum_j \mu_j = 1 + L/|P_{\mathrm{out}}|\) because the allocation must also cover pack losses. It is likewise false in comparative voting-power analysis, where ratios such as \(R_i=P_i/s_i\) compare a power index with relative weights instead of partitioning a conserved flow [2503.02866] [1802.00497]. In deliberative democracy models, the normalized quantities \(s_i(t)\) and \(s_i^C(t)\) are not physical flow fractions but governance shares that bound proposal corrections and veto power [2109.01436].

## 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 \(p_j^{G*}=-\tilde Q_{e,jj}\omega^*\) yields
\[
r_j^G=\frac{1/Q_{e,jj}}{\sum_k 1/Q_{e,kk}}.
\]
For heat pumps in Mode 1, \(p_j^{P*}=a_{1,j}\omega^*\) gives
\[
r_j^{hp}=\frac{a_{1,j}}{\sum_k a_{1,k}},
\]
and for conventional heat sources,
\[
r_j^{heat}=\frac{1/Q_{h,jj}}{\sum_k 1/Q_{h,kk}}.
\]
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 \(\alpha_j=m_j/C_{o,j}\); increasing \(m_j\) reduces the per-unit contribution of heat source \(j\) because \(h_j^{G*}=-\tilde Q_{h,jj}(\omega^*/m_j)\) [2509.24051]. An earlier formulation states the same structure with separate and joint optimization problems and emphasizes that the average temperature \(\bar T\) acts as the thermal analogue of synchronous frequency, enabling optimal sharing without prior disturbance knowledge [2502.03990].

In islanded and grid-connected microgrids, the same normalization logic appears in several distinct forms. For droop-controlled inverters, proportional sharing is obtained when \(P_i^*/D_i=P_j^*/D_j\) and ratings are selected proportionally, so that \(P_{\mathrm{e},i}/\overline P_i = P_{\mathrm{e},j}/\overline P_j\); under the constructive choice \(P_i^*=\alpha D_i\), steady injections satisfy \(P_{\mathrm{e},i}/P_{\mathrm{e},j}=D_i/D_j\) [1206.5033]. For distributed generators with time-varying maximum capacities \(C_i\), proportional active-power sharing is expressed as \(P_i=\alpha C_i\) with \(\alpha=P_{\mathrm{load}}/\sum_k C_k\) whenever feasible, so \(P_i/P_j=C_i/C_j\) [2003.00565]. Reactive-power sharing admits further variants: inverter-based resources under voltage limits seek equality of utilization ratios \(\bar Q_i/S_i^{\mathrm{rated}}=\alpha_Q\) among unsaturated units [2302.09241], while in unbalanced multi-microgrids an ESS-based controller enforces
\[
\frac{Q_{PV,i}^{\phi}}{Q_{PV}^{3\phi}} \to \frac{P_{PV,i}^{\phi}}{P_{PV}^{3\phi}},
\]
which becomes \(0.25\,Q_{PV}^{3\phi}\) for a \(3\ \mathrm{kW}\) single-phase inverter paired with a \(12\ \mathrm{kW}\) three-phase inverter [2104.04202].

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
\[
i_{i,\mathrm{ref}}(t)=r_i(t)\,I_{\mathrm{load}}(t)
\]
makes the desired ratio \(r_i(t)\) 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 \(i_{L,i}\approx r_i I_{\mathrm{load}}\) when the reference sum matches the load [1604.04154].

A closely related decentralized design prescribes average current shares \(\alpha_k\) and ripple shares \(\beta_k\) through inner-loop shaping. Average power sharing is enforced with
\[
\gamma_k=\frac{\alpha_k D_n'}{D_k'},
\]
while \(120\ \mathrm{Hz}\) ripple sharing is obtained by
\[
\zeta_1^{(k)}=\frac{\beta_k \zeta_{1,n}}{\alpha_k}.
\]
This construction makes average power ratios and ripple ratios independent design targets, both realized through an equivalent single-converter \(H_\infty\) problem [1604.03573]. A further extension to DC microgrids introduces time-varying \(\gamma_k(t)\) directly into each converter’s outer current loop; if \(\sum_k \gamma_k(t)=1\) and \(i_{\mathrm{ref}}=i_{\mathrm{load}}\), then steady-state bus-current and power contributions satisfy \(i_k\approx \gamma_k i_{\mathrm{load}}\) and \(P_k\approx \gamma_k P_{\mathrm{load}}\) [1701.03065].

Large-scale BESS introduces a different constraint structure. Here the power-sharing ratio is the cell-level utilization factor
\[
\mu_j=\frac{P_{b_j}}{P_{\mathrm{out}}},
\]
and because losses are explicitly modeled,
\[
\sum_{j=1}^n \mu_j(t)=1+\frac{L(t)}{|P_{\mathrm{out}}(t)|}.
\]
The top-level controller does not optimize \(n\) free ratios directly; it parameterizes them as
\[
\mu_j(t)=
\begin{bmatrix}
\phi_{q,j}(t) & \phi_{T,j}(t) & \phi_{R,j}(t)
\end{bmatrix}
\theta,
\qquad \mathbf 1^\top \theta=1,
\]
so the high-dimensional allocation becomes a three-parameter problem balancing state of charge, temperature, and resistance [2503.02866]. 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 \(\eta\in(0,1)\) that partitions its power budget between estimation and payload transmission [1812.11413].

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 \(|\eta_{ij}^*|<\pi/2\), and the same conclusions extend to higher-order input-strictly-passive generation dynamics with convex effective costs \(\hat C_e\) and \(\hat C_h\) [2509.24051].

Converter-based sharing often uses robust optimal-control synthesis instead of direct economic dispatch. The DC-DC frameworks above formulate weighted \(H_\infty\) 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 [1604.04154] [1604.03573]. 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 \(\lambda_i\) toward the normalized reactive injections \(Q_i/S_i^{\mathrm{rated}}\), and practical stability is derived by singular perturbation plus an LMI condition on a grounded reduced model [2302.09241].

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 \(\theta\), using a virtual dynamic system and barrier functions for state and power constraints. Ensemble Kalman inversion then estimates \(\theta\) online, and a low-level PI loop converts the resulting \(\mu_j^*\) into feasible cell power references under output-power uncertainty [2503.02866]. Droop-e introduces yet another structure: a nonlinear primary frequency–power law
\[
f(p)=f_{\mathrm{set}}+f_b \alpha \left(e^{\beta p_{\mathrm{set}}}-e^{\beta p}\right),
\]
so the effective droop slope \(M_i^{\mathrm{eff}}(p_i)=f_b\alpha_i\beta_i e^{\beta_i p_i}\) becomes operating-point dependent. Incremental sharing is therefore
\[
\Delta P_i=
\frac{1/M_i^{\mathrm{eff}}(p_i)}{\sum_j 1/M_j^{\mathrm{eff}}(p_j)}\,
\Delta P_{\mathrm{total}},
\]
and a secondary integrator drives the steady state back to a chosen target sharing law [2207.03564].

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
\[
\phi_i^{(\alpha,\beta)}(E,\Pi)=
\alpha d_i(S^{-i})+
\beta\sum_{j\in N(i,S^{-i})}\frac{d_j(S^{-j})}{d_j(S^i)}
\]
combines direct inter-union degree and rescaled inter-union neighborhood influence. The normalized firm and union shares are then
\[
R_i=\frac{\phi_i^{(\alpha,\beta)}}{P^{\mathrm{tot}}},
\qquad
R_{S_\ell}=\frac{\phi_{S_\ell}^{(\alpha,\beta)}}{P^{\mathrm{tot}}},
\]
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 [2507.13272].

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 \(\alpha_{i,t}(f)\) in Stage 1 and community-level coefficients \(\beta_{i,t}\) in Stage 2, with equal, proportional, and rank-based rules implemented in both static and dynamic forms. Dynamic equal sharing uses \(\alpha_{i,t}(f)=1/|N_{\mathrm{need},f,t}|\) within a feeder, while dynamic proportional sharing uses \(\alpha_{i,t}(f)=D_{i,t}/\sum_{k:f(k)=f} D_{k,t}\) and \(\beta_{i,t}=D_{i,t}/\sum_k D_{k,t}\) [2509.12847]. In coordinated power–traffic systems, the analogous ratio is economic: a profit-sharing coefficient \(\alpha\in[0,1]\) splits the DNO’s cost reduction as \(\Pi_{\mathrm{TNO}}=\alpha \Delta\eta\) and \(\Pi_{\mathrm{DNO}}=(1-\alpha)\Delta\eta\), with \(\alpha^*=20\%\) minimizing the DNO’s total cost in the reported case [2503.11967]. For two battery-equipped renewable bundles, Pareto-optimal energy sharing is parameterized by peak sharing rates \((c_1,c_2)\), and the paper proves that every Pareto-optimal arrangement lies on the boundary where at least one bundle shares at its maximum feasible rate [2005.09222].

Institutional power-sharing adopts still different normalizations. In weighted voting games, one studies deviations between power indices and weight shares through ratios such as \(R_i=P_i/s_i\) and bounds such as
\[
\|P-w\|_1 \le \frac{c\,\Delta}{\min\{q,1-q\}}
\]
for the nucleolus and certain representation-compatible indices, together with impossibility results showing that no generally sharp approximation exists if only \(q\) or \(\Delta\) is controlled [1802.00497]. In deliberative democracy with dilutive voting power, the relevant shares are \(s_i(t)=P_i(t)/S(t)\), within-committee shares \(s_i^C(t)=P_i(t)/P_C(t)\), and the rule that a correction proposed by committee member \(i\) must satisfy \(f(d\ell_{t+1}^i,\ell_t)\le s_i^C(t)\); outside agents can veto if more than half of the total voting power outside the committee is against the correction [2109.01436].

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 \(9.77\ \mathrm{s}\) in Mode 1 versus \(11.9\ \mathrm{s}\) in Mode 2, and the aggregate electric adjustment of heat pumps stayed below \(0.035\ \mathrm{p.u.}\) [2509.24051]. In unbalanced multi-microgrids, activating the ESS multifunction controller reduced PCC voltage unbalance factor from \(4.3\%\) to \(0.03\%\) while enforcing reactive sharing proportional to the \(3\ \mathrm{kW}:12\ \mathrm{kW}\) inverter rating ratio [2104.04202]. In robust DC converter control, three-converter ratios of \(10\!:\!4\!:\!6\), \(4\!:\!8\!:\!8\), and \(6\!:\!4\!:\!10\) were tracked over successive time intervals while maintaining \(V_{\mathrm{bus}}\) around \(240\ \mathrm{V}\) [1604.04154]. In feeder-aware energy communities, the best-performing dynamic proportional, dynamic rank-based, and static rank-based schemes all achieved \(13.942\ \mathrm{MWh}\) of shared energy with \(58.702\ \mathrm{MWh}\) imported and \(2.777\ \mathrm{MWh}\) exported in the reported dataset [2509.12847]. In C-RAN, optimal pilot/data power-sharing factors increased sum-rate by \(33\%\) relative to a non-optimized baseline in the \(32\)-antenna RRU and \(128\)-antenna BBU example [1812.11413]. 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 \(2\) remained in experimental comparisons [2406.19684].

The same literature also identifies persistent limitations. Average-temperature-based CHP sharing requires measurement or estimation of the weighted average \(\bar T\), 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 [2509.24051]. 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 [1604.04154]. BESS ratio control presumes cell-level actuation through DC/DC converters and depends on identified \(u_j(q)\) and \(R_j(q)\) models [2503.02866]. In voting theory, weights can approximate power only under specific quota and maximum-weight conditions; outside those regimes, large discrepancies are provable [1802.00497].

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.

Source: https://www.emergentmind.com/topics/power-sharing-ratios