Load Estimation-Compensation (LEC) Methods

• Load Estimation-Compensation (LEC) is a framework that reconstructs latent or disturbed loads using historical data, statistical models, or machine learning to guide corrective actions.
• It employs diverse methodologies such as recursive least squares, ANFIS, and ARIMA, providing accurate pseudo-measurements and compensation across electric, robotic, and communication systems.
• Applications span electric distribution networks, robotic surgery, microgrids, EV charging, and transit, enhancing system performance by addressing sensing errors and operational uncertainties.

Load Estimation-Compensation (LEC) denotes, in the literature surveyed here, a family of methods in which a load, load disturbance, or load-related latent state is first estimated from incomplete or indirect observations and is then used for compensation, disturbance rejection, pseudo-measurement construction, restoration planning, or resource reallocation. The estimated quantity varies substantially across domains: Representative Load Curves (RLCs) for electric distribution networks, non-contact force and torque components in wrist-mounted force/torque sensing, residential EV charging demand, lifting load in assistive exoskeletons, power deficit in microgrids, load current disturbances in PWM buck converters, LTE cell resource occupancy, cold load pick-up after outages, and passenger load trajectories in public transit (Eyisi et al., 2021, Pirrera et al., 2 Sep 2025, Shaker et al., 26 Apr 2026, Xu et al., 19 May 2026). This suggests that LEC is best understood not as a single algorithm, but as an architectural pattern that couples estimation with a corrective mechanism.

1. Terminological scope and defining characteristics

In strict usage, the term LEC appears explicitly in the DC-DC converter literature as a “load estimation-compensation (LEC) scheme” inserted as an inner disturbance-rejection loop, where an estimator $\mathcal{E}$ reconstructs the output current disturbance and a compensator $\mathcal{F}$ injects a corrective signal into the PWM control path (Pirrera et al., 2 Sep 2025). In other literatures, the same composite idea appears under adjacent formulations: estimated RLCs are used as pseudo-measurements for real-time load estimation in distribution networks (Eyisi et al., 2021); recursive least squares is used for real-time non-contact force compensation in robotic surgery training (Shaker et al., 26 Apr 2026); EV charging profiles are estimated to support “real-time or day-ahead compensation for expected peaks” (Fang et al., 17 Nov 2025); and passenger-load estimation frameworks incorporate “estimation and error compensation at every step” through physical projection, trust allocation, and closed-loop calibration (Xu et al., 19 May 2026).

A recurring technical characteristic is that the target load is either not directly measured or is corrupted by bias, drift, aggregation, sparsity, or heterogeneous sensing errors. In the robotic surgery setting, the raw sensor reading is explicitly decomposed as

$${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$

with gravity and bias treated as non-contact components to be estimated and removed (Shaker et al., 26 Apr 2026). In the transit setting, passenger load is treated as a latent state $L_k$ updated recursively stop by stop, with physical feasibility enforced so that drift does not produce negative loads or loads above capacity (Xu et al., 19 May 2026). In LTE, cell load is inferred passively from control-channel scheduling messages rather than from operator-internal measurements (Chirkov et al., 2020).

The literature therefore separates two functions that are sometimes conflated. Load estimation reconstructs a hidden, future, or disturbance component from available evidence. Compensation uses that estimate to alter control action, improve observability, correct state drift, reduce unnecessary shedding, or support planning. A common misconception is that LEC always denotes direct actuator compensation in a feedback loop. The cited work shows a broader usage: pseudo-measurement generation, staged restoration, probabilistic demand anticipation, and dynamic trust reweighting are also compensation mechanisms in the operational sense (Eyisi et al., 2021, Xu et al., 2017, Fang et al., 17 Nov 2025, Xu et al., 19 May 2026).

2. Recurring architecture across domains

Across the surveyed papers, the workflow is structurally consistent even though the mathematics differs. First, accessible variables are measured or decoded. Second, a model-based, statistical, or learning-based estimator reconstructs the relevant load. Third, a compensatory mechanism uses the estimate to modify control or decision-making. Fourth, several systems add guards against corruption, such as contact detection, physical clipping, or validation datasets.

Domain	Estimated quantity	Compensatory use
Distribution networks	Representative Load Curves	Pseudo-measurements for real-time load estimation
Robotic surgery	Gravity, bias, installation offsets, associated torques	Real-time non-contact force/torque compensation
Buck converters	Output load current disturbance	Injected corrective signal in inner loop
EV charging	Residential EV charging profile	Compensation for expected peaks, feeder estimates
Microgrids	Global power deficit, weighted loads, utilization levels	Gradual priority-based load shedding with power compensation
Transit	Passenger load state	Fusion, macro-correction, residual-driven calibration

In electric distribution load estimation, the estimator is built from historical hourly measurements, temporal correlations such as $Y(t)$, $Y(t-24)$, and $Y(t-168)$, and model factors. ANFIS then estimates candidate RLC sections, and the one with the least MAPE is selected for pseudo-measurement use (Eyisi et al., 2021). In the converter setting, the architecture is explicitly two-loop: an outer $\mathcal{H}_{\infty}$ voltage controller and an inner LEC disturbance-rejection loop that estimates the load disturbance using only $v_o(t)$ and $i_L(t)$, then injects $\mathcal{F}$0 through low-order analog blocks (Pirrera et al., 2 Sep 2025).

Adaptive identification is another recurring pattern. The robotic surgery paper uses recursive least squares with pause-on-contact logic. At each time step,

$\mathcal{F}$1

with updates frozen when the residual exceeds a threshold so that contact forces do not contaminate non-contact parameter estimation (Shaker et al., 26 Apr 2026). The passenger-load framework closes the loop differently: unconstrained boarding and alighting predictions are projected into a feasible state, disagreement with exogenous anchors modulates trust, and the resulting load estimate is fed back as the initial condition for the next stop (Xu et al., 19 May 2026).

This suggests that LEC methods differ less by application domain than by the form of estimator used—least-squares, recursive least squares, ANFIS, ARIMA, KDE, machine learning regressors, physical projection, or control-theoretic observers—and by the kind of compensation attached to the estimate.

3. Electric power systems and grid-facing LEC

In distribution management systems, Distribution Load Estimation (DLE) is described as a key function. “Load Estimation for Electric Power Distribution Networks” proposes constructing RLCs from two years of historical hourly load measurements, partitioned into training and checking datasets, with ANFIS used to model nonlinear relationships under Takagi-Sugeno first-order rules and generalized bell-shaped membership functions. The estimated RLCs are then used as pseudo-measurements at unmetered nodes. On an 11kV radial distribution network, the reported overall MAPE for the total system weekly RLC is 2.51% with RMSE of $\mathcal{F}$2 kW, while for node 5 the daily RLC gives MAPE of 3.35% and RMSE of 363.0 kW (Eyisi et al., 2021).

For EV demand, “Data-Driven EV Charging Load Profile Estimation and Typical EV Daily Load Dataset Generation” presents two data-driven methods. The first uses weighted least-squares on feeder-level counts of EV and non-EV meters,

$\mathcal{F}$3

to infer 6.46 kW as non-EV meter average peak power, 16.99 kW as EV meter average peak power, and 10.53 kW as average EV charging power. The second isolates EV charging signatures and applies KDE to charging energy, average power, and start and end times. Both methods generate a distinct “u-shaped” daily profile with most charging overnight, and the paper explicitly identifies these profiles as useful for LEC, including disaggregated feeder or transformer load estimates and “real-time or day-ahead compensation for expected peaks” (Fang et al., 17 Nov 2025).

Cold load pick-up estimation after outages introduces a restoration-centered form of LEC. “Local Cold Load Pick-up Estimation Using Customer Energy Consumption Measurements” uses ARIMA models on local smart-meter data, with dynamic order adjustment to account for changing customer behavior and weather, to forecast foregone energy over outage windows of 1 to 12 hours. It further estimates CLPU duration as

$\mathcal{F}$4

Using actual smart meter measurements from 50 residential customers over one year, the paper reports that ARIMA-based forecast offers superior performance in both accuracy and computation speed relative to LSTM and HWES, and positions the outputs—CLPU peak power, duration, and energy—as enablers for sequencing restoration and implementing compensation schemes (Bajic et al., 2024).

Microgrid load shedding shows yet another formulation: estimation of deficit and priorities is coupled to compensation from generators or ESS. “Distributed Load Shedding for Microgrid with Compensation Support via Wireless Network” formulates active-power coordination as a distributed optimization problem and introduces DLSS, a subgradient-based algorithm that sheds gradually so that power compensation has more time to lower the deficit. The remaining load to be shed is dynamically corrected as

$\mathcal{F}$5

The paper states that smaller step size yields more, smaller shedding actions, allowing more compensation and less load loss, and that MMST can respond 2–3× faster than round-robin or deterministic protocols (Xu et al., 2017).

Taken together, these power-system examples show that compensation need not mean a direct analog control signal. It may instead mean pseudo-measurement construction, peak anticipation, staged restoration, or deficit-aware load shedding.

4. Control, disturbance rejection, and mechatronic compensation

In PWM DC-DC buck converters, LEC is formulated most explicitly as a disturbance-rejection mechanism. The power-stage model is decomposed so that the output load disturbance can be algebraically recovered from output voltage and inductor current. The resulting estimator is

$\mathcal{F}$6

with

$\mathcal{F}$7

The compensator approximates the ideal cancellation law with a proper transfer function by adding a high-frequency pole $\mathcal{F}$8, and robust stability is preserved under a sufficient small-gain condition,

$\mathcal{F}$9

The paper compares LEC against DOB and UIO/ESO structures, arguing that LEC has lower analog complexity, requires only $${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$0 and $${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$1, preserves the nominal outer-loop transfer function, and offers explicit guidance for robustness tuning via $${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$2. Experimental implementation on a prototype board reports measured voltage deviations at load steps of less than 15 mV (Pirrera et al., 2 Sep 2025).

In haptic-enabled robotic surgery training, the compensatory objective is removal of non-contact forces and torques from a wrist-mounted F/T sensor. The proposed recursive least squares method estimates gravitational force in the robot base frame, sensor force bias, center of gravity, and sensor torque bias online, without dataset collection or repeated recalibration. Force parameters are estimated first, then torque parameters are estimated using those force estimates. Contact detection monitors the residual norm and pauses RLS updates when contact is suspected. The paper reports over 95% error reduction in non-contact force compensation and more than 91% in non-contact torque compensation, with convergence in ~250 samples at 1 kHz, i.e. sub-second, and notes stable long-duration behavior with Standard Deviation <0.05 N and no significant drift (Shaker et al., 26 Apr 2026).

Upper-limb exoskeletons present a user-centric LEC problem: assistance should depend on the actual lifted load so that over-compensation and under-compensation are both avoided. “Real-Time Load Estimation for Load-lifting Exoskeletons Using Insole Pressure Sensors and Machine Learning” uses 36-channel insole pressure sensors embedded in a rigid “pancake” structure, differential pressure features that subtract baseline standing pressure, and two model families. In the channel-based method, Elastic Net, SVR, and MLP are trained on 36-dimensional differential features; the average WMAPE across three subjects is 13.46% for SVR, 14.91% for MLP, and 18.76% for Elastic Net. In the map-based method, a fully fine-tuned MobileNetV2 reaches WMAPE of 9.74% for one subject, while linear probing reaches 22.13%. Cross-subject testing degrades substantially, with WMAPE increases of 39% for SVR, 64% for MLP, and 51% for Elastic Net, indicating that personalized calibration or broader training data is needed (Wu et al., 10 Mar 2025).

These works jointly indicate that, in control and mechatronics, LEC frequently has a direct closed-loop meaning: estimate an unwanted load component or external load, then cancel it or adapt assistance accordingly.

5. Communications, sensing, and mobility applications

In LTE systems, cell load estimation is obtained by decoding DCI messages on the PDCCH and extracting unique C-RNTIs together with their resource block assignments. The frame-level information load is defined as

$${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$3

where the numerator is the number of assigned downlink resource blocks and the denominator is the total available resource blocks from the MIB. The method is implemented with MATLAB LTE Toolbox and a USRP 2920. In a real-cell example, 154 RBs out of 500 over 10 frames yield a load of 30.8\%, and the paper argues that such estimates can support dynamic LEC strategies including load balancing, resource optimization, QoS improvement, and infrastructure planning (Chirkov et al., 2020).

Passenger-load estimation in public transit generalizes LEC into a multi-agent state-estimation framework. A unified stop-event backbone indexes trip evolution by stops, with latent load state $${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$4 initialized at $${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$5. A Perception agent produces unconstrained boarding and alighting estimates; a Physical agent enforces feasibility,

$${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$6

and computes

$${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$7

The framework then computes a physics-violation residual $${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$8, dynamically allocates trust between the recursive estimate and an optional external anchor such as Wi-Fi-based load proxy, and fuses the state as

$${}^{S}\boldsymbol{f} = {}^{S}\boldsymbol{f}_{contact} + {}^{S}\boldsymbol{f}_{g} + {}^{S}\boldsymbol{f}_{0}, \qquad {}^{S}\boldsymbol{\tau} = {}^{S}\boldsymbol{\tau}_{contact} + {}^{S}\boldsymbol{\tau}_{g} + {}^{S}\boldsymbol{\tau}_{0},$$9

Optional trip-level macro-correction applies a gated additive shift under persistent drift, while closed-loop calibration reweights training samples according to physics residuals. The paper describes this as a framework that “integrates estimation and error compensation at every step,” preventing infeasible drift that would arise from open-loop accumulation of stop-level count errors (Xu et al., 19 May 2026).

These communication and mobility applications broaden the meaning of compensation. The corrective action is not necessarily physical actuation; it may instead be recursive state correction, trust modulation, or resource-management intervention driven by estimated load.

6. Validation criteria, limitations, and interpretive cautions

The validation metrics used in LEC-related work are domain-specific. Distribution-network RLC estimation emphasizes MAPE and RMSE, with

$L_k$0

Exoskeleton studies use WMAPE to weight errors by load magnitude. Robotic-surgery compensation emphasizes force and torque MAE, standard deviation, and convergence speed. CLPU forecasting compares MSE, execution time, and variability against LSTM and HWES. LTE cell-load estimation uses decoded resource-block occupancy, while passenger-load estimation focuses on feasibility, residual violations, and drift resilience (Eyisi et al., 2021, Wu et al., 10 Mar 2025, Shaker et al., 26 Apr 2026, Bajic et al., 2024, Chirkov et al., 2020, Xu et al., 19 May 2026).

Several limitations recur. The distribution-network ANFIS method requires high-quality extended historical load data, uses heuristic feature selection, excludes the first week because of input requirements, and needs validation to guard against overfitting; the paper also notes that broader deployment requires further study beyond the chosen 11kV radial network (Eyisi et al., 2021). The exoskeleton work is explicitly preliminary, with only three healthy subjects and map-based deep-learning results reported for one subject; generalization across users is weak because foot size, shape, and pressure distribution vary (Wu et al., 10 Mar 2025). The CLPU study finds that adding temperature via ARIMAX increases complexity and computational cost with minimal gains and may cause convergence issues (Bajic et al., 2024). In the surgical compensation method, inertial and Coriolis terms are neglected in low-speed, quasi-static haptic scenarios, which delimits the modeled operating regime (Shaker et al., 26 Apr 2026). In the buck-converter setting, continuous conduction mode is assumed, output capacitor ESR must be accounted for, and the choice of $L_k$1 defines a trade-off between stronger nominal disturbance rejection and greater sensitivity to parameter mismatch (Pirrera et al., 2 Sep 2025).

A second interpretive caution concerns what counts as “load.” In the surveyed literature, the term ranges from electric demand and load current to payload mass, passenger occupancy, and radio resource utilization. This suggests that LEC is fundamentally a methodological concept centered on estimation-conditioned correction, rather than a domain-specific noun with a single physical meaning.

7. Research trajectory and cross-domain significance

The surveyed works indicate three major trajectories. First, there is a movement from offline calibration and static models toward adaptive online identification. Recursive least squares in surgical sensing eliminates the need for datasets and repeated recalibration; EV estimation uses meter data to recover hidden charging demand; passenger-load estimation reallocates trust dynamically and recycles violation residuals into retraining (Shaker et al., 26 Apr 2026, Fang et al., 17 Nov 2025, Xu et al., 19 May 2026). Second, there is a growing emphasis on embedding physical feasibility directly into estimation. The passenger framework enforces mass conservation and capacity bounds; the buck-converter LEC preserves the nominal outer-loop transfer function while providing a robustness condition; microgrid DLSS explicitly coordinates generator limits and power balance (Xu et al., 19 May 2026, Pirrera et al., 2 Sep 2025, Xu et al., 2017). Third, multiple papers emphasize deployability with limited sensing: distribution-network pseudo-measurements compensate for unmetered nodes, wrist-mounted sensing avoids tip-mounted sensors, LTE cell load is estimated passively from over-the-air control traffic, and EV charging profiles are inferred from aggregated meter data rather than proprietary vehicle data (Eyisi et al., 2021, Shaker et al., 26 Apr 2026, Chirkov et al., 2020, Fang et al., 17 Nov 2025).

A plausible implication is that LEC has become a useful organizing concept wherever the operational variable of interest is only indirectly observable and yet must be acted upon in real time or near real time. In some domains the compensatory action is an injected control signal; in others it is a pseudo-measurement, a forecast-guided restoration policy, a staged shedding schedule, or a state-correction mechanism. The unifying principle is the same: estimation is not treated as an end in itself, but as an intermediate step that enables compensation under uncertainty.