Two-Tier Energy Method in Hierarchical Systems

• Two-Tier Energy Method is a structural decomposition technique that partitions complex energy-related problems into two interdependent tiers, each with specialized roles.
• It is applied in wireless networks for distributed power control, in quantum systems for kinetic-energy partitioning, and in market mechanisms for layered control.
• The method enhances computational efficiency and system fairness while clarifying the trade-offs between architectural design and resultant outcomes.

“Two-Tier Energy Method” is not a single standardized term across the research literature. In the cited corpus, it denotes a family of energy-related formulations in which a problem is decomposed into two tiers, two layers, or two steps: a distributed power-control method for uplink heterogeneous wireless networks (Lu et al., 2017), a kinetic-energy partition scheme for quantum systems with competing potentials (Mineo et al., 2012), an architectural-versus-outcome distinction in EnergyNet (Birgersson et al., 9 Sep 2025), layered energy-sharing and transactive-control markets (Su et al., 2024, Cheng et al., 2019), and several other hierarchical optimization and prediction frameworks. The common feature is structural decomposition; the meaning of “tier” depends entirely on domain.

1. Terminological scope

In the cited literature, the expression refers to distinct technical constructions rather than a universal algorithm. In wireless systems, the “two-tier” attribute often names a macrocell tier overlaid with a small-cell or picocell tier, with the energy method then operating over that hierarchy (Lu et al., 2017, Davaslioglu et al., 2016). In quantum mechanics, the analogous phrase is the “split kinetic energy / kinetic energy partition (KEP), or ‘two-tier energy,’ method,” where the Hamiltonian is reorganized into two subsystem Hamiltonians by partitioning the kinetic energy rather than the potential energy (Mineo et al., 2012). In energy-system architecture, Tier-1 may denote “components, interfaces, operating model,” whereas Tier-2 denotes “expected outcomes contingent on adoption” (Birgersson et al., 9 Sep 2025).

Domain	Representative paper	Meaning of the two tiers
Uplink heterogeneous wireless networks	(Lu et al., 2017)	Subcarrier grouping and EGT-based distributed power control
Quantum systems with competing potentials	(Mineo et al., 2012)	Two subsystem Hamiltonians formed by kinetic-energy partition
EnergyNet architecture	(Birgersson et al., 9 Sep 2025)	Tier-1 architecture versus Tier-2 contingent outcomes
Prosumers’ energy sharing	(Su et al., 2024)	Lower-level LAMs and upper-level WAM
Interconnected MES coordination	(Cheng et al., 2019)	Lower-level autonomous scheduling and upper-level coordination

This diversity suggests that “two-tier” is principally a structural descriptor. A plausible implication is that the term is best understood as denoting a decomposition principle rather than a specific mathematical object.

2. Wireless-network energy efficiency

In uplink heterogeneous wireless networks, the phrase is used most directly for a distributed energy-efficient power-control algorithm in an OFDMA two-tier network composed of one macrocell tier with a massive-MIMO base station and a small-cell tier (Lu et al., 2017). The macrocell and all small cells share the same set of $N$ orthogonal subcarriers, both the MBS and SBSs use maximum-ratio combining, and each user selects transmit power from a finite discrete set $\mathcal{L}=\{1,2,\ldots,L\}$. The per-user energy efficiency is defined as

$\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$

and the network objective is the sum of individual user EEs over users and subcarriers rather than the ratio of total throughput to total power. The algorithm is explicitly two-step: users on the same subcarrier are assigned to the same group, producing $N$ independent subproblems, and then each group is optimized through an evolutionary-game-theoretic power-control game. The replicator dynamics

$$\dot{x}_{i,a}=x_{i,a}\left(\pi_{i,a}-\bar{\pi}_i\right)$$

favor strategies with above-average payoff and are presented as fairness-improving. The brute-force complexity $\mathcal{O}\!\left(L^{(K+1)N}\right)$ is reduced to $\mathcal{O}\!\left((K+1)LN\right)$, and the reported simulations show fast convergence and “remarkable improvements in terms of fairness” (Lu et al., 2017).

A related two-tier energy-efficiency framework appears in multi-cell multi-carrier macro/pico networks (Davaslioglu et al., 2016). There, the network-wide energy-efficiency maximization problem is reformulated via fractional programming into a bi-criterion objective that balances rate maximization and power minimization, then decomposed by dual decomposition with interference pricing. The resulting closed-form updates take a multi-level or modified water-filling form, with the pricing terms inserted directly into the water-filling threshold. In the reported simulations, power control improves energy efficiency by a factor of $2.68$ in the two-tier case, and pricing adds about $39\%$ more energy-efficiency gain (Davaslioglu et al., 2016).

A third wireless interpretation is analytical rather than algorithmic. In two-tier femtocell networks with partially open channels, a Markov-chain framework is used to derive blocking probabilities, spectrum efficiency, and an energy-efficiency ratio $\eta_{EE}=C_{total}/E(PW_{FBS})$ from the stationary distribution (Ge et al., 2014). The abstract states that the number of open channels has an adverse impact on both spectrum and energy efficiency, while the detailed technical summary characterizes increasing $\mathcal{L}=\{1,2,\ldots,L\}$0 as increasing spectrum efficiency but decreasing energy efficiency (Ge et al., 2014). This discrepancy underscores that “two-tier energy method” does not guarantee a single performance narrative even within one application class.

3. Quantum kinetic-energy partition

In quantum mechanics, the two-tier formulation is explicitly the KEP method for systems with two competing potentials $\mathcal{L}=\{1,2,\ldots,L\}$1 and $\mathcal{L}=\{1,2,\ldots,L\}$2 (Mineo et al., 2012). Instead of splitting the potential into “unperturbed” and “perturbed” parts, the kinetic energy

$\mathcal{L}=\{1,2,\ldots,L\}$3

is partitioned into two equal pieces by introducing an effective mass $\mathcal{L}=\{1,2,\ldots,L\}$4, so that

$\mathcal{L}=\{1,2,\ldots,L\}$5

The full wavefunction is expanded simultaneously in the basis sets of both subsystem Hamiltonians,

$\mathcal{L}=\{1,2,\ldots,L\}$6

and the coupling between the two subsystem spaces is rewritten in terms of the potential difference $\mathcal{L}=\{1,2,\ldots,L\}$7. The method becomes exact as the basis sets are enlarged; in the special case $\mathcal{L}=\{1,2,\ldots,L\}$8, the coupling terms vanish and the KEP energies satisfy $\mathcal{L}=\{1,2,\ldots,L\}$9, which is exact (Mineo et al., 2012).

The worked examples define the method’s scope. For double delta-function potentials, a two-state approximation already yields energies and wavefunctions close to exact results. For a charged harmonic oscillator in a strong magnetic field, basis sizes $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$0 to $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$1 produce rapid convergence, with low-lying-state errors typically below about $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$2 and often much smaller. For $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$3, the method uses subsystem Hamiltonians built around atomic-like centers and a distance-dependent mass-splitting parameter $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$4; the resulting ground-state potential-energy curves agree well with accurate benchmark results and outperform conventional LCAO-MO in the reported comparison (Mineo et al., 2012).

A related but distinct two-step treatment of energies appears in multistate perturbation theory, where a first Rayleigh–Schrödinger effective-Hamiltonian step resolves quasi-degeneracy and a second state-specific Brillouin–Wigner step refines individual transition energies (Delafosse et al., 2023). This is not the same nomenclature as KEP, but it exhibits the same hierarchical logic: difficult multistate structure is handled first, and state-specific correction is deferred to a second tier.

4. Layered energy architectures, markets, and control

In EnergyNet, the two-tier method is explicitly an argumentative and architectural separation between what the system is and what it is expected to enable (Birgersson et al., 9 Sep 2025). Tier-1 comprises “components, interfaces, operating model”: the Energy Router with galvanic separation and a DC backplane, ELAN and EWAN network domains, the open Energy Protocol, the Energy Router Operating System, the EP Server, and the Energy Network Management System. Tier-2 comprises “expected outcomes contingent on adoption,” including local-first autonomy with global interoperability, near-real-time operation with local buffering, removal of EV-charging bottlenecks, freed grid capacity for data centers and industrial electrification, and a trend toward low, predictable, fixed-cost clean energy. The paper is explicit that these outcomes are contingent rather than automatic (Birgersson et al., 9 Sep 2025).

A market-theoretic two-layer energy-sharing scheme appears for massive prosumers (Su et al., 2024). The lower layer consists of local-area markets within communities, and the upper layer is a wide-area market that clears the uncleared surplus and shortage of those communities. In each LAM, the sharing price is

$\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$5

while the WAM clears the residual $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$6 subject to power balance and congestion constraints. The full problem is an MPEC, but the proposed hierarchically distributed bidding algorithm is reported on the IEEE 123-bus system with $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$7 prosumers; the runtime is $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$8 s versus $\mathrm{EE}_k^i=\frac{r_{ki}}{P_{\text{sum},ki}},$9 s for solving the full MPEC, and the wide-area sharing outcome is characterized as near-social-optimal in large-scale systems (Su et al., 2024).

An analogous upper–lower decomposition governs interconnected multi-energy systems (Cheng et al., 2019). At the lower level, each MES solves a rolling-horizon cost-minimization problem with a convexification technique for storage complementarity constraints; the paper proves exactness of the relaxation under stated conditions. At the upper level, a coordinator clears a transactive price so that aggregate bids satisfy transformer limits and system balance, and the overall procedure is implemented as a two-stage transactive-control framework. In the reported comparison, 2S-TC yields a total cost of $N$0k yuan versus $N$1k yuan for SG-RTC, while reducing hourly-stage iterations from as many as $N$2 to a maximum of $N$3 (Cheng et al., 2019).

A closely related bi-level allocation mechanism is proposed for DSO markets with aggregators and home-level agents (Faqiry et al., 2017). The DSO-level auction performs projected-gradient updates over aggregator allocations subject to transformer, line-flow, voltage, and budget-balance constraints, while each aggregator clears a local proportional allocation auction without requiring private utility functions or generation capacities. The resulting mechanism is characterized as efficient and weakly budget balanced (Faqiry et al., 2017).

5. Hierarchical engineering optimization and prediction

In heterogeneous two-tier wireless sensor networks, the two-tier energy method refers to a deployment problem over sensors, access points, and fusion centers, with objective

$N$4

(Karimi-Bidhendi et al., 2020). The optimal cell partition is a heterogeneous generalized Voronoi diagram, and the necessary conditions imply that each optimal AP lies on the line segment between the centroid of its assigned cell and its connected FC. The Heterogeneous Two-Tier Lloyd algorithm alternates among AP-to-FC assignment, generalized Voronoi partition update, FC update, and AP update, while the Limited-HTTL extension projects updates into feasible communication regions under bounded range. The reported experiments show HTTL and Limited-HTTL outperform MER, AC, DC, PSO, RNDWSN, and IRNP on the tested instances (Karimi-Bidhendi et al., 2020).

In solar forecasting, a two-tier method separates long-horizon prediction from real-time correction (Wang et al., 2015). The global tier produces a 24-hour forecast using weighted k-NN or a three-layer feedforward neural network trained only on historical power data, while the local tier computes residuals from real-time measurements, extracts low-frequency components with discrete Fourier series, and updates the remaining forecast. On the UCLA Microgrid with $N$5 kW of solar generation capacity, the weighted-k-NN-based two-tier method yields about $N$6 improvement over weighted k-NN alone, and the NN-based two-tier method yields about $N$7 improvement over NN alone (Wang et al., 2015).

Another engineering use of a two-grid energy hierarchy appears in thermal radiative transfer (Anistratov et al., 2024). The method couples a fine photon-energy grid for the multigroup RTE and LONWF equations with a coarse one-group grey grid for GLOQD and the MEB equation, with information transferred through nonlinear projections and spectrum averages. The system is discretized by fully implicit Euler, and the numerical results for the Fleck–Cummings Marshak wave with $N$8 photon groups show convergent multilevel iteration behavior and a time-step-dependent optimal number of inner V-cycles (Anistratov et al., 2024).

A two-case formulation also governs multi-unit energy-efficiency optimization for systems with similar versus different device efficiencies (Yao, 2024). For similar efficiencies, Yao’s Theorem 1 states that the optimal load distribution keeps the operating efficiency of each operating device equal. Yao’s Theorem 2 states that the optimal switching point for the number of operating units is at the point of equal efficiency or at the maximum output point of the device (Yao, 2024). This is another instance in which the tiers correspond to a higher-level configuration decision and a lower-level load-allocation rule.

6. Recurrent structures, exactness claims, and misconceptions

The surveyed formulations show that “two-tier” does not identify a single ontology. In some cases it denotes a physical hierarchy, as in macrocell–small-cell wireless networks or mother–regular HAPS-SMBS architectures (Lu et al., 2017, Khennoufa et al., 18 Apr 2026). In others it denotes an algorithmic split, such as subcarrier grouping followed by EGT power control, or a global-tier predictor followed by a local-tier residual corrector (Lu et al., 2017, Wang et al., 2015). In still others it denotes a market hierarchy, an architecture-versus-outcome distinction, or a two-basis Hamiltonian decomposition (Su et al., 2024, Birgersson et al., 9 Sep 2025, Mineo et al., 2012).

The associated optimality guarantees also differ sharply. The KEP method becomes exact as the basis sets are enlarged and is immediately exact when $N$9 (Mineo et al., 2012). The wireless power-control algorithm for massive-MIMO two-tier networks is described instead as converging to a “sub-optimal but fair and efficient equilibrium” (Lu et al., 2017). The two-layer prosumer market is near-social-optimal in large-scale systems rather than exactly socially optimal at finite scale (Su et al., 2024). In the MES framework, exactness attaches specifically to the storage-relaxation step under stated conditions (Cheng et al., 2019). This suggests that “two-tier” should not be conflated with either exact decomposition or guaranteed global optimality.

A common misconception is that the term must refer to two physical energy infrastructures. The cited literature shows otherwise. The KEP method operates in Hilbert space, not in layered infrastructure (Mineo et al., 2012). The solar prediction method operates over forecasting and correction stages (Wang et al., 2015). EnergyNet uses the tiers to separate a buildable architecture from downstream, adoption-contingent outcomes (Birgersson et al., 9 Sep 2025). A plausible synthesis is that the unifying function of the two-tier pattern is computational or conceptual manageability: it isolates subproblems, clarifies responsibilities, or separates what can be solved or specified directly from what must emerge indirectly.

Another plausible implication is that the method family is best evaluated by the property each decomposition is designed to protect. In wireless systems, the protected properties are low complexity, distributed implementation, and fairness (Lu et al., 2017). In DSO and MES frameworks, they are privacy, autonomy, and grid feasibility (Faqiry et al., 2017, Cheng et al., 2019). In KEP, the protected property is robustness under competing potentials where conventional perturbation theory becomes asymptotic (Mineo et al., 2012). In EnergyNet, the protected property is conceptual discipline: architecture is not conflated with expected societal outcome (Birgersson et al., 9 Sep 2025).