Iterative Energy Consumption Optimization (ECO)

• Iterative ECO is a recurring optimization pattern that repeatedly adjusts energy-related decisions under dynamic, multi-objective constraints.
• It employs methodologies such as dynamic programming, constrained reinforcement learning, and convex surrogate optimization to balance energy, performance, and safety.
• Applications range from eco-driving and platoon control to energy-harvesting IoT and machine-learning workflows, yielding measurable energy and cost savings.

Iterative Energy Consumption Optimization (ECO) denotes a class of optimization and control formulations in which energy- or fuel-relevant decisions are updated sequentially, recurrently, or over receding horizons rather than fixed once for an entire task. In the literature represented here, ECO appears in urban eco-driving and platooning, plug-in hybrid eco-routing, autonomous-vehicle trajectory planning, humanoid locomotion, radio access networks, energy-harvesting IoT, and energy-aware machine-learning workflows (Zhen et al., 2022, Houshmand et al., 2020, Hadjigeorgiou et al., 5 Jun 2025, Huang et al., 6 Feb 2026, Mariegaard et al., 2023, Tuncel et al., 2021, Aashish et al., 31 Dec 2025). The terminology is not standardized: some papers use “ECO” as “Energy-Constrained Optimization,” “Enabling energy-neutrality through constrained optimization,” or “Energy Consumption Optimization,” while another uses the same acronym for the unrelated “Ecological Cycle Optimizer” (Huang et al., 6 Feb 2026, Tuncel et al., 2021, Mariegaard et al., 2023, Ma et al., 28 Aug 2025).

1. Conceptual scope and nomenclature

Across the cited literature, ECO is not a single canonical algorithm but a recurring optimization pattern. The common structure is the repeated adjustment of energy-relevant variables—such as vehicle acceleration, gear selection, lane choice, route choice, battery-mode allocation, cell ON/OFF states, humanoid joint actions, or per-interval energy budgets—under dynamic, safety, timing, or resource constraints (Zhen et al., 2022, Kerbel et al., 2022, Aoki et al., 2021, Mariegaard et al., 2023, Tuncel et al., 2021). In transportation papers, the objective is usually stated directly in terms of fuel or electrical energy, or as a weighted trade-off among energy, travel time, and comfort. In robotics and constrained RL, energy is often separated from reward and imposed as an explicit inequality constraint. In ML systems, ECO can instead denote a benchmarking-and-selection loop that jointly evaluates predictive quality, energy use, and carbon emissions (Huang et al., 6 Feb 2026, Aashish et al., 31 Dec 2025).

A recurrent misconception is that ECO is synonymous with pure energy minimization. The representative formulations here are usually multi-objective or constrained: urban platoon control balances energy consumption, mobility, and passenger comfort (Zhen et al., 2022); driver-assistance RL trades fuel use against driver demand tracking, shift frequency, and power reserve (Kerbel et al., 2022); autonomous-vehicle ECO+ enforces safety and comfort while minimizing positive control input (Hadjigeorgiou et al., 5 Jun 2025); humanoid ECO maximizes locomotion reward subject to energy and symmetry budgets (Huang et al., 6 Feb 2026). Another misconception is that every paper containing the acronym “ECO” addresses physical energy consumption. The “Ecological Cycle Optimizer” is a general-purpose metaheuristic whose “energy” is an ecological analogy rather than a model of electrical, thermal, or fuel consumption (Ma et al., 28 Aug 2025).

This diversity suggests that “iterative ECO” is best understood as a cross-domain methodological label: repeated optimization of energy-relevant decisions under task-specific constraints, rather than a discipline-specific named method.

2. Recurring mathematical structures

Despite domain differences, the literature exhibits a small number of recurring mathematical templates. Weighted-sum formulations are common in eco-driving. A representative example is the distance-domain dynamic-programming objective for a heterogeneous EV platoon leader,

$$\min_{a_k} \sum_{k=1}^{N}\left[\alpha E(v_k,a_k)+\beta M(v_k)+\gamma C(a_k)\right]+\mu P_{red},$$

with mobility penalty $M(v_k)$, comfort term $C(a_k)=a_k^2$, and a red-light penalty driven by SPaT timing (Zhen et al., 2022). Explicit constrained formulations are prominent in humanoid locomotion, where ECO defines motor-energy cost

$$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$

and imposes

$J^{C_1}(\pi_\theta)\le b_1$

alongside a symmetry/reference-motion constraint $J^{C_2}(\pi_\theta)\le b_2$ (Huang et al., 6 Feb 2026). Convex surrogate formulations also appear. ECO+ minimizes the positive part of the control input,

$$\int_{t^o}^{t^d} u^+(t)\,dt,\qquad u^+(t)=\max(u(t),0),$$

and then reformulates the problem into a convex QCP and finally an LP via piecewise-affine over-approximation of resistive forces (Hadjigeorgiou et al., 5 Jun 2025). Network-routing papers instead use resource-allocation constraints, such as the battery budget

$$\sum_{(i,j)\in\mathcal{A}}\frac{d_{ij}}{\mu_{CD_{ij}}}y_{ij}x_{ij}\le E_o$$

in joint PHEV eco-routing and power-train control (Houshmand et al., 2020).

These structures support different notions of iteration. Dynamic programming and MPC repeatedly solve local trajectory problems as position or time advances. Actor-critic RL repeatedly alternates policy evaluation and policy improvement. Lagrangian constrained RL alternates primal policy updates and dual-variable updates. Network control papers repeatedly synthesize short-horizon policies from updated traffic loads or state forecasts. Energy-harvesting IoT papers compute a nominal day-start allocation and then repeatedly correct it online using rollout (Kerbel et al., 2022, Mariegaard et al., 2023, Tuncel et al., 2021).

Structure	Energy handling	Representative papers
Weighted multi-objective control	Energy as one term among mobility, comfort, or time	(Zhen et al., 2022, Wingelaar et al., 2022)
Constrained optimization / CMDP	Energy as explicit inequality constraint	(Huang et al., 6 Feb 2026)
Convex surrogate optimization	Energy approximated by PCI or similar surrogates	(Hadjigeorgiou et al., 5 Jun 2025)
Route and resource allocation	Energy budget coupled to graph decisions	(Houshmand et al., 2020, Houshmand et al., 2018)
Measurement-centered benchmarking	Energy and carbon tracked as evaluation metrics	(Aashish et al., 31 Dec 2025)

A plausible implication is that iterative ECO methods can be distinguished less by application domain than by how they encode energy: as a weighted stage cost, a hard or soft constraint, a resource budget, or a measured deployment metric.

3. Road-vehicle control and eco-driving

Road-vehicle ECO is dominated by trajectory planning under preview information. In urban platooning, a heterogeneous EV platoon on a 2500 m arterial with signalized intersections at 600 m and 2000 m is controlled through a sequential near-optimal strategy: the leader solves a distance-domain DP problem, followers attempt PID-based CACC tracking, and any follower that cannot clear the next green phase or whose average traction efficiency falls below a threshold becomes a new leader and replans (Zhen et al., 2022). In simulation with 20 vehicles, including a heavy-duty EV in position 16, adding energy to the objective reduces average platoon energy from $1812.965\ \text{Wh/veh}$ to $1473.99\ \text{Wh/veh}$, about 19%, while average travel time increases from $M(v_k)$0 to $M(v_k)$1, about 30%; the heavy-duty vehicle’s energy falls from $M(v_k)$2 to $M(v_k)$3, about 57.1% (Zhen et al., 2022).

Heavy-duty-truck eco-driving has been formulated as a receding-horizon optimal control problem over the distance domain, with fuel and trip duration in the stage cost and mode–gear combinations handled by a dedicated Branch and Bound MPC solver (Wingelaar et al., 2022). The underlying model replaces continuous drivetrain decisions with six driving modes—cruising, eco-roll, coasting, engine brake, acceleration, and downhill—and exploits warm starts, an admissible energy/time lower bound, and state aggregation. On three routes, the method reports average fuel savings of 25.8% versus a simulated human driver and 12.9% versus a PMP-based controller, with average optimization times at $M(v_k)$4 of roughly 0.52–0.74 s in Matlab (Wingelaar et al., 2022). A related line studies eco-coasting with road-grade preview. There, offline DP and online mixed-integer MPC compare fuel cut-off against engine start/stop and show that engine start/stop outperforms fuel cut-off because it removes engine drag torque, while MPC reaches fuel consumption comparable to DP without sacrificing travel time (Yan et al., 2021).

Learning-based road ECO introduces iteration through policy updates rather than Bellman recursion on explicit route grids. A deep off-policy actor-critic for driver-assistance eco-driving and transmission control treats the state as $M(v_k)$5 and the hybrid action as wheel traction torque plus gear-change command $M(v_k)$6, with reward components penalizing driver-demand mismatch, traction torque, instantaneous fuel rate, shift frequency, and loss of power reserve (Kerbel et al., 2022). Using Retrace for critic targets and MPO for constrained policy improvement, the method learns higher-gear, lower-engine-speed behavior and reports up to 12.8% MPG improvement over a baseline controller with full knowledge of the engine fuel-efficiency map (Kerbel et al., 2022).

A more recent convex formulation is ECO+, which replaces generic smoothness objectives with Positive Control Input minimization for AVs approaching an intersection (Hadjigeorgiou et al., 5 Jun 2025). The continuous-time objective $M(v_k)$7 is transformed into a convex QCP and then a linear program by piecewise-affine approximation of the resistive-force term $M(v_k)$8. In reported experiments, the LP version differs from the quadratic version by less than 0.4% on average, reduces runtime by 36.9%, and solves in about 0.09 s on average, while direct nonlinear refinement initialized from ECO+ yields only marginal gains (Hadjigeorgiou et al., 5 Jun 2025). This makes ECO+ notable as an example of surrogate-based ECO that preserves real-time tractability without abandoning physically motivated energy structure.

4. Routing, lane selection, and corridor-scale road energy optimization

Another major branch of ECO operates at the route, corridor, or lane-choice level. In PHEV transportation networks, the central issue is not only which path to choose but where to expend limited battery energy. An early formulation, Combined Routing and Power-train Control (CRPTC), treats route choice $M(v_k)$9 and charge-depleting fraction $C(a_k)=a_k^2$0 on each link as coupled variables in a MILP and reports, on an Eastern Massachusetts subnetwork, average energy savings of 2.54% versus Charge Depleting First, 5.02% versus the minimum-time route, and 6.89% versus actual routes, with an average travel-time increase of 4.79% relative to the minimum-time route (Houshmand et al., 2018). A larger-scale successor applies a related framework to Boston using HERE Maps data over more than 110,000 links, validates route choices with SUMO speed profiles and a high-fidelity VESIM model, and reports significant energy savings around 12% in the abstract and average energy-cost savings up to 19.0% versus the fastest route for $C(a_k)=a_k^2$1 in the detailed experiments, while the bi-level decomposition remains near real time at about 0.14–0.16 s (Houshmand et al., 2020).

Corridor eco-driving under signal control often uses limited look-ahead rather than full-network optimization. A multiple-signal optimization approach parameterizes the vehicle trajectory over up to two downstream signalized intersections at a time and updates the advisory speed every second using SPaT and predicted queue lengths (Yang et al., 2020). At 100% market penetration, the paper reports fuel-consumption reduction as high as 13.8% in the abstract and up to 15% in the 16-intersection grid discussion; it also shows that low market penetration in two-lane settings can increase fuel use because controlled vehicles create lane-changing opportunities for uncontrolled traffic (Yang et al., 2020). This limited-look-ahead structure is explicitly motivated by scalability and queue-prediction uncertainty: the method is modular and sequential rather than corridor-globally optimal.

Highway ECO brings the same logic to lane choice. MultiCruise evaluates left, current, and right candidate lanes through an Eco-Cruise submodule that predicts lane-conditioned driving parameters and fuel/progress/comfort cost $C(a_k)=a_k^2$2, then adds a lane-change penalty $C(a_k)=a_k^2$3 and chooses the feasible minimum-cost lane (Aoki et al., 2021). In realistic synthetic highway scenarios, MultiCruise reports up to 8.5% fuel savings, and in a simple overtaking case about 32% fuel reduction, with larger gains in moderate traffic and on longer highway segments (Aoki et al., 2021). The method is not a monolithic optimizer; it is an online hierarchy in which discrete lane actions are repeatedly reevaluated through the continuous trajectory they induce.

Taken together, these routing and corridor papers show that iterative ECO need not operate only at the actuator level. It also appears as repeated graph search, repeated lane evaluation, or sequential local optimization over successive intersections, with energy models embedded in each transition or stage cost.

5. Embodied systems, communication infrastructure, and energy-neutral devices

In humanoid locomotion, ECO has been formalized as a constrained RL framework that removes energy from the reward and treats it as an explicit inequality constraint (Huang et al., 6 Feb 2026). The energy cost is motor energy consumption computed as $C(a_k)=a_k^2$4, the symmetry/reference-motion term is treated as a second constraint with threshold $C(a_k)=a_k^2$5, and the saddle-point problem is solved by alternating PPO-style policy updates and Lagrange-multiplier updates. On the kid-sized humanoid BRUCE, the method reports Gazebo power use of $C(a_k)=a_k^2$6 at $C(a_k)=a_k^2$7, versus $C(a_k)=a_k^2$8 for PPO and $C(a_k)=a_k^2$9 for MPC; on hardware, ECO reports $$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$0 over 10-second trials, versus $$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$1 for PPO and $$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$2 for MPC, with comparable walking speeds (Huang et al., 6 Feb 2026). The paper explicitly argues that threshold-based tuning is more physically interpretable than searching over reward coefficients.

At communication-network scale, ECO-RAN treats cell activation as a stochastic control problem over ON/OFF states of capacity-layer cells and repeatedly synthesizes short-horizon policies with UPPAAL Stratego from replayed historical traffic (Mariegaard et al., 2023). The reward is the time integral of energy plus a heavy service penalty, and the practical implementation is online strategy synthesis with, for example, hourly updates over a 180-minute horizon. In the Aalborg case studies, Stratego reduces energy from 3473 to 3191 in City Syd and from 10115 to 9347 in Frydendal–Nørre Tranders, corresponding to about 8.1% and 7.6%, while maintaining zero penalty; the abstract summarizes the potential as up to 10% (Mariegaard et al., 2023). The method is iterative in both time and space: strategies are recomputed periodically and larger networks are partitioned into subareas with at most 8 cells.

For energy-harvesting IoT, ECO denotes runtime allocation of harvested energy under energy-neutrality constraints (Tuncel et al., 2021). The day is partitioned into 24 one-hour intervals, the battery evolves according to

$$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$3

and the objective is discounted utility maximization under $$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$4, $$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$5, and end-of-day target constraints. The distinctive feature is a two-level design: a constrained iterative optimization at day start computes nominal allocations, and a lightweight rollout algorithm corrects them online. Compared with rerunning the iterative solver every interval, ECO achieves about 1000× lower runtime energy overhead—$$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$6 per interval versus $$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$7—with normalized utility 0.98 versus 1.0 for the iterative reference, and up to 35% higher utility than prior techniques in energy-limited scenarios (Tuncel et al., 2021).

These examples show that embodied ECO is not tied to propulsion. It includes any setting in which energy is a scarce physical resource whose consumption must be repeatedly allocated across time, actuators, or infrastructure elements.

6. Measurement-centered ECO, decomposition methods, and limitations

A different strand of ECO treats energy as a first-class evaluation metric in computational workflows. In cybersecurity anomaly detection, an eco-aware framework evaluates Logistic Regression, Random Forest, SVM, Isolation Forest, and XGBoost on a publicly available Carbon-Aware Cybersecurity Traffic Dataset with 2,300 flow-level observations, using CodeCarbon to log training time, inference time, energy, and CO$$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$8-equivalent emissions in a controlled Colab environment (Aashish et al., 31 Dec 2025). The paper defines an Eco-Efficiency Index,

$$C_1(\mathbf{s}_{t+1}\mid \mathbf{s}_t,\mathbf{a}_t)=\sum_{j=1}^{n} |\tau_t^j \dot{q}_t^j|$$9

and reports, among other values, baseline emissions of 0.0553 g CO$J^{C_1}(\pi_\theta)\le b_1$0eq for Random Forest, 0.0400 g CO$J^{C_1}(\pi_\theta)\le b_1$1eq for SVC, 0.0025 g CO$J^{C_1}(\pi_\theta)\le b_1$2eq for XGBoost, and 0.0001 g CO$J^{C_1}(\pi_\theta)\le b_1$3eq for Logistic Regression. Its abstract states that optimized Random Forest and lightweight Logistic Regression reduce energy consumption by more than 40% compared to XGBoost while sustaining competitive detection accuracy, and that PCA reduces computational load with negligible loss in recall (Aashish et al., 31 Dec 2025). Here the iterative ECO loop is explicit as benchmark $J^{C_1}(\pi_\theta)\le b_1$4 tune $J^{C_1}(\pi_\theta)\le b_1$5 reduce dimensionality $J^{C_1}(\pi_\theta)\le b_1$6 remeasure $J^{C_1}(\pi_\theta)\le b_1$7 reselect.

Decomposition models provide another methodological layer. A task-based sensor-centric model decomposes total sensor energy into Individual, Local, Global, Environment, and Sink constituents,

$J^{C_1}(\pi_\theta)\le b_1$8

and, in simplified learned form,

$J^{C_1}(\pi_\theta)\le b_1$9

with packet-flow variables $J^{C_2}(\pi_\theta)\le b_2$0 estimated from sensed data, monitoring, routing, overhead, harvesting, and sink-related tasks (Kamyabpour et al., 2012). The paper reports about 13% average prediction error and finds that the Global constituent dominates total energy in the simulated WSN setting (Kamyabpour et al., 2012). Although not itself a complete controller, this kind of decomposition is important because it turns ECO from a black-box minimization problem into a constituent-level diagnosis-and-reallocation problem.

The limitations reported across the literature are consistent. Transportation papers often assume deterministic SPaT, no cut-ins, no overtaking, or only longitudinal control, and they often lack runtime analysis or hard collision constraints (Zhen et al., 2022, Yang et al., 2020). Constrained humanoid RL uses $J^{C_2}(\pi_\theta)\le b_2$1 as an energy proxy rather than full electrical power with thermal and battery effects, and adding more constraints degrades convergence on humanoids (Huang et al., 6 Feb 2026). ECO-RAN depends on replayed historical data and on equipment that supports hourly scheduler updates (Mariegaard et al., 2023). The cybersecurity workflow uses a small dataset in Colab, so relative differences are more informative than absolute kWh values, and repeated-run variance is not fully quantified (Aashish et al., 31 Dec 2025). These caveats do not negate the results, but they show that many ECO methods remain near-optimal, surrogate-based, or deployment-conditional rather than exact and universally transferable.

Taken together, this suggests that iterative ECO is best characterized as an engineering pattern of repeated energy-aware decision making under explicit modeling compromises. The common research challenge is not merely minimizing an energy metric, but doing so while preserving feasibility, safety, robustness, comfort, service quality, or predictive performance across domains.