Load Characteristics Modeling

• Load characteristics modeling is a technique to represent dynamic load behavior using mathematical, stochastic, and data-driven methods while incorporating internal states and external drivers.
• It spans multiple scales from appliance-level hidden semi-Markov models to aggregate power-system ZIP and composite models, supporting both forecasting and stability analyses.
• The approach enables practical applications such as demand response, privacy-preserving calibration, and adaptive control by merging physical insights with statistical learning.

Load characteristics modeling is the construction of mathematical, stochastic, or data-driven representations of how a load evolves with time, internal state, exogenous variables, and control actions. In the literature surveyed here, the modeled object ranges from individual residential appliances and dwelling-level electricity demand to aggregate responsive loads, bus-level composite $P(V,f)$ and $Q(V,f)$ behavior in power systems, authoritative DNS query load under TTL caching, and dynamic carried loads in quadruped locomotion (Chuan et al., 2015, Zhao et al., 2015, Dorji et al., 27 Aug 2025, Wang, 2016, Chang et al., 10 Jul 2025). The topic therefore spans forecasting, demand response, stability assessment, parameter identification, order reduction, privacy-preserving calibration, and control synthesis rather than prediction alone (Alizadeh et al., 2014, Ma et al., 2019, Lu et al., 8 Feb 2026).

1. Modeling scales and state representations

A recurrent feature of the field is the coexistence of several modeling scales. At the finest scale, appliance-level models represent discrete operating modes, durations, and emissions, as in hidden semi-Markov formulations for A/Cs, refrigerators, pool pumps, and EV chargers, where the generalized state is $(\tilde{x}_k,\tilde{d}_k)$ and the observation is power $y_t$ (Ji et al., 2018). At the household scale, bottom-up residential models synthesize total demand from appliance saturation, rated power, standby power, start frequencies, and hourly activity factors, so that $P_{\text{total}(t)}$ is explicitly the sum of appliance contributions (Chuan et al., 2015). At the population scale, flexible loads are represented through occupancy counts, density functions, or hybrid-state distributions, rather than by enumerating device trajectories one by one (Zhao et al., 2015, Alizadeh et al., 2014). At the transmission or distribution-bus scale, the object of interest becomes the aggregate $P$–$Q$ response of composite load mixtures containing ZIP terms, induction motors, electronic loads, and DERs (Ma et al., 2019, Lu et al., 8 Feb 2026).

The state variables used at these scales differ accordingly. Thermal loads use temperature, slip, flux, and ON/OFF mode; deferrable loads use remaining job time, deferral time, or state of charge; appliance-activity models use discrete human activity states and occupancy; data-center models add utilization, protection fractions, and cooling states; quadruped carried-load models use load position, load velocity, load mass, and load–support friction coefficient as an 8-dimensional load-characteristics vector in the robot base frame (Kundu et al., 2011, Zhao et al., 2015, Bromley-Dulfano et al., 2022, Lu et al., 8 Feb 2026, Chang et al., 10 Jul 2025). Across these cases, the modeled quantity is not simply “power” but a state-dependent response surface or dynamical law that maps internal and exogenous variables to load behavior.

A second common distinction is between intrinsic states and exogenous drivers. In power-system studies, voltage magnitude $V$, frequency $f$, and control signals $\upsilon(t)$ act as exogenous inputs to TCLs, deferrable loads, composite loads, or DER-rich large electronic loads (Zhao et al., 2015, Ma et al., 2019, Dorji et al., 27 Aug 2025, Lu et al., 8 Feb 2026). In residential studies, ambient temperature, irradiance, occupancy, and demographic attributes play the same role (Chuan et al., 2015, Bromley-Dulfano et al., 2022, Wang et al., 2021). In DNS, TTL $Q(V,f)$0 is the control variable that shapes authoritative-server query load (Wang, 2016). The concept of “load characteristics” is thus consistently relational: it denotes how load responds to operating conditions, not merely its average magnitude.

2. Canonical mathematical formalisms

A foundational bottom-up electrical formulation writes household load as a superposition of appliance trajectories,

$Q(V,f)$1

with appliance starts generated stochastically by

$Q(V,f)$2

The resulting model is neither a Markov chain nor a Poisson count process; it is a structured multiplicative start-probability model driven by Bernoulli trials and deterministic on-cycle durations, optionally with constant standby consumption (Chuan et al., 2015). This formulation is representative of bottom-up demand synthesis because it separates ownership, usage frequency, hourly activity, and temporal discretization.

For appliance-level stochastic demand, the hidden semi-Markov family replaces per-time-step starts with explicit state-duration structure. In the conditional hidden semi-Markov model, the generalized transition is factorized as

$Q(V,f)$3

and both factors are learned by multinomial logistic regression. Emissions are modeled as

$Q(V,f)$4

which makes state occupancy, duration distributions, and exogenous covariates explicit components of load characterization (Ji et al., 2018).

At the aggregate responsive-load level, stochastic hybrid systems provide a population description. With hybrid state $Q(V,f)$5, the central object is the density $Q(V,f)$6, governed by a forward PDE of the form

$Q(V,f)$7

with probability flux

$Q(V,f)$8

Aggregate power is then recovered as an integral of the density over power-consuming modes. This formalism generalizes classical TCL Fokker–Planck models to higher-order TCLs and deferrable loads such as PEV charging (Zhao et al., 2015).

A related mesoscopic formalism is the load-plasticity representation for large populations of flexible appliances. There, the feasible aggregate load is written in terms of quantized state occupancies $Q(V,f)$9 and switching flows $(\tilde{x}_k,\tilde{d}_k)$0, with aggregate load

$(\tilde{x}_k,\tilde{d}_k)$1

subject to linear conservation and feasibility constraints. This medium-grained stochastic hybrid model preserves quantized device constraints while discarding appliance identity, which makes it simultaneously scalable and anonymous (Alizadeh et al., 2014).

For homogeneous TCL populations, another canonical reduction is an aggregate transfer function from uniform setpoint shift to total power. After linearization and empirical damping augmentation, the population response is

$(\tilde{x}_k,\tilde{d}_k)$2

which is then realized in state space for LQR design. This compresses first-order thermal cycling and deadband hysteresis into a low-order input–output model suitable for ancillary-service control (Kundu et al., 2011).

3. Residential, behavioral, and AMI-driven models

Residential load characteristics modeling has developed along three main lines: appliance-based bottom-up synthesis, behavior- and population-driven simulation, and smart-meter pattern extraction. A representative bottom-up model for Singapore public housing classifies dwellings into 1–2 room, 3-room, 4-room, and 5-room flats, with average monthly energy use of approximately $(\tilde{x}_k,\tilde{d}_k)$3, $(\tilde{x}_k,\tilde{d}_k)$4, $(\tilde{x}_k,\tilde{d}_k)$5, and $(\tilde{x}_k,\tilde{d}_k)$6 kWh/month, respectively. Flat type acts as a proxy for demographic and economic status, conditioning appliance saturation, starting frequency, and profile shape. The model synthesizes a single representative day and multiplies by 30, and validates daily profiles and monthly energy against 30-minute smart-meter data from 323 NTU staff-housing units accessed through the Customer Energy Portal (Chuan et al., 2015).

A more explicit behavioral formulation begins from human activity. In the behavior- and population-driven distribution-system model, each occupant is represented by a time-inhomogeneous Markov chain over nine activity states—Away, Sleeping, Grooming, Cooking, Dishwashing, Cleaning, Laundry, Leisure, and Other—with transition probabilities estimated from the American Time Use Survey. Household occupancy is derived from activity, and appliance models for HVAC, water heaters, lighting, cold appliances, and other end uses convert activity and occupancy into minute-resolution household power. The synthetic population is then drawn from Current Population Survey data, allowing geographically differentiated load ensembles such as California and Texas cohorts of 500 households each, while holding weather and physical household parameters fixed to isolate behavioral effects (Bromley-Dulfano et al., 2022).

Seasonality can also be modeled at the level of daily pattern distributions rather than appliance physics. In the seasonal-variation framework, daily smart-meter profiles are first normalized and clustered by a two-stage K-medoids procedure into a small set of representative daily shapes. For each household and season, a probability distribution over cluster labels is then constructed, and seasonal change is quantified by relative entropy,

$(\tilde{x}_k,\tilde{d}_k)$7

Decision-tree classifiers trained on this variation label identify income level as essential across all seasonal transitions, with the number of children and the elderly also significant for specific seasonal changes (Wang et al., 2021).

AMI-driven load characterization adds a data-compression dimension. Wavelet-based load models compress hourly AMI trajectories over 31 weeks into level-3 approximation coefficients for each customer, achieving about $(\tilde{x}_k,\tilde{d}_k)$8 compression while preserving $(\tilde{x}_k,\tilde{d}_k)$9–$y_t$0 of signal energy for $y_t$1 of 323 customers. A classified wavelet load model goes further by clustering customers through 2D wavelet vertical coefficients and reconstructing them from class typical load profiles plus peak and average scalars, reaching about $y_t$2 compression (Zhong et al., 2015). The significance of this line of work is methodological: it shows that load characteristics can be modeled as multi-resolution temporal structure rather than solely as physical or econometric parameters.

At the appliance level, high-resolution submetered data support conditional hidden semi-Markov descriptions of A/Cs, refrigerators, pool pumps, and EV chargers. These models explicitly capture multi-state operation, duration distributions, and context dependence, and support short-term load forecasting and anomaly detection. The reported one-hour-ahead forecasting improvements for A/C aggregation, including a reduction of NRMSE from $y_t$3 to $y_t$4 after weighted and state-specific refinements, illustrate the importance of duration modeling for appliance cycling loads (Ji et al., 2018).

4. Composite and aggregate power-system load models

In bulk power-system studies, load characteristics modeling is dominated by composite bus-level models. The standard static representation is ZIP: $y_t$5 with coefficient sums equal to one. Dynamic extensions add induction motors, yielding ZIP+IM models, and the WECC composite load model further embeds feeder equivalents, three-phase motors, a single-phase A/C motor representation, electronic load, and DER behavior. An explicit mathematical representation of CMPLDWG and the 9-state DER_A model has been derived in differential–algebraic form, providing the basis for parameter identification, dynamic analysis, and reduction (Ma et al., 2019).

The engineering significance of model structure is evident in transfer-capability studies. When the same transient field measurements are fitted into multiple models, WECC CLM yields the lowest transfer capabilities across six cases, while a static $y_t$6 Z + $y_t$7 I + $y_t$8 P model yields the highest. ZIP+IM lies between the two. The implication is not merely numerical but conceptual: static voltage sensitivity alone does not capture the dynamic VAR demand, motor stalling, and post-fault recovery behaviors that constrain transiently secure transfer limits (Wang et al., 2020).

The same multiscale issue appears in model order. The WECC composite load model “CMPLDWG + DER_A” is a high-order nonlinear system with multi-time-scale structure. A large-signal order reduction based on singular perturbation theory separates slow and fast states, integrates fast dynamics into slow dynamics, and establishes sufficient conditions for accurate reduction. Applied to the three three-phase induction motors and DER_A, the reduced model preserves similar dynamic responses while substantially reducing runtime. For the motor example, the original 5-state machine is reduced to 3 slow states after integrating subtransient EMFs into the quasi-steady manifold; reported ODE45 runtimes drop from $y_t$9 s to $P_{\text{total}(t)}$0 s, and ODE15s runtimes from $P_{\text{total}(t)}$1 s to $P_{\text{total}(t)}$2 s (Ma et al., 2019).

A newer composite-load frontier concerns large electronic loads such as data centers, cryptocurrency mining facilities, and hydrogen electrolyzers. In the dynamic load model for data centers, total load is decomposed as

$P_{\text{total}(t)}$3

and then scaled by a retained-load fraction $P_{\text{total}(t)}$4 under protection: $P_{\text{total}(t)}$5 The IT-workload component uses a mean-reverting stochastic OU process with impulsive bursts,

$P_{\text{total}(t)}$6

cooling is represented by induction-motor dynamics, auxiliary demand by ZIP, and protection by voltage/frequency thresholds, trip delays, and ramped reconnection. System-level simulations show compound disconnection–reconnection dynamics and delayed stabilization not captured by uncalibrated or conventional load models (Lu et al., 8 Feb 2026).

5. Identification, symbolic discovery, and calibration

A major line of research concerns how to infer load characteristics from measurements while balancing fidelity, interpretability, and computational tractability. One recent approach models static composite load behavior as a function

$P_{\text{total}(t)}$7

with $P_{\text{total}(t)}$8 and $P_{\text{total}(t)}$9, and learns this map using Kolmogorov–Arnold Networks. Because KANs place trainable univariate functions on edges rather than fixed activations at nodes, they can be collapsed into explicit symbolic expressions. In a pure ZIP case, the learned equations are

$P$0

and for a WECC CLM + PV-based DG case the learned forms become

$P$1

$P$2

On test data, the KAN model achieves active-power MSE $P$3, RMSE $P$4, and MAE $P$5, compared with $P$6, $P$7, and $P$8 for an MLP, with similarly improved reactive-power errors (Dorji et al., 27 Aug 2025).

For dynamic composite loads, reinforcement-learning-based parameter identification has been proposed for ZIP+induction-motor models. The parameter vector

$P$9

is identified by minimizing the trajectory error

$Q$0

The ITQ framework couples Q-learning, imitation learning, associative memory, and transfer learning, and the paper proves convergence to the global optimal solution with probability $Q$1. The method is specifically motivated by time-varying load characteristics, where repeated re-identification under new scenarios would otherwise be computationally prohibitive (Xie et al., 2019).

A related DDQN-based methodology fits the same transient field measurements into multiple candidate structures—ZIP, ZIP+IM, and WECC CLM—to assess both model fit and operational impact. In the reported comparison, WECC CLM attains $Q$2 and $Q$3, ZIP+IM attains $Q$4 and $Q$5, and ZIP attains $Q$6 and $Q$7. The result underscores that parameter identification and model-class selection are inseparable when the end goal is dynamic-security analysis (Wang et al., 2020).

For inherently stochastic loads, the calibration target itself becomes controversial. The data-center model argues that trajectory-level alignment is ill-posed for stochastic workloads and instead introduces pattern-consistent calibration via temporal contrastive learning. Time series are split into windows, augmented, encoded into latent vectors, and summarized through first and second moments of the learned representations,

$Q$8

after which calibration minimizes $Q$9. The calibrated parameter set, rather than raw operational traces, is what gets shared with utilities, making privacy preservation part of the modeling design (Lu et al., 8 Feb 2026).

6. Validation regimes and operational applications

Validation practices vary sharply across the literature. Bottom-up residential models often validate both magnitude and temporal shape against smart-meter data. In the Singapore study, the CEP mean of $V$0 kWh/month for the NTU housing sample was compared with an EMA-based benchmark of about $V$1 kWh/month for a mixture of 1–2 and 3-room flats, and the simulated 3-room monthly value of $V$2 kWh/month was compared with the EMA figure of $V$3 kWh/month. The daily measured profile also exhibited morning and evening peaks around $V$4–$V$5 h and $V$6–$V$7 h, respectively, supporting qualitative shape validation (Chuan et al., 2015).

Aggregate responsive-load models are usually validated against Monte Carlo or digital-twin simulation rather than field AMI alone. In the stochastic-hybrid aggregate HVAC case, a population of 2000 HVACs in GridLAB-D subjected to a $V$8 setpoint increase at $V$9 hours showed about $f$0 steady-state power reduction, followed by a rebound peak when the setpoint was restored at $f$1 hours. The PDE-plus-clusters model reproduced the major oscillation cycles and rebound observed in Monte Carlo simulation, which is precisely the kind of aggregate characteristic—duty-cycle synchronization and desynchronization—that feeder-level models are meant to capture (Zhao et al., 2015).

AMI compression models validate differently again, through feeder-head power flow. With 323 customers over 5208 hours, the multi-resolution wavelet model preserved system energy within $f$2 and the classified wavelet model within $f$3, while peak-load errors remained on the order of $f$4–$f$5. This illustrates that a load-characteristics model may be acceptable for planning or energy studies even when it smooths short-duration extremes (Zhong et al., 2015).

Operational applications follow naturally from the chosen representation. Appliance- and household-level models support demand response, DSM, HEMS, and tariff experiments by exposing which end uses or activity windows drive peaks (Chuan et al., 2015, Bromley-Dulfano et al., 2022). Aggregate SHS and load-plasticity models enable ex-ante planning and real-time control of flexible portfolios under direct load scheduling or dynamic pricing (Zhao et al., 2015, Alizadeh et al., 2014). Bus-level composite and symbolic models support transient stability, RV/RMS simulation, voltage sensitivity analysis, and transfer-capability assessment (Dorji et al., 27 Aug 2025, Ma et al., 2019, Wang et al., 2020). In the data-center setting, calibrated large-electronic-load models become inputs to system-wide disturbance studies rather than to consumer-centric programs (Lu et al., 8 Feb 2026).

7. Limitations, recurrent misconceptions, and broader extensions

Several limitations recur across otherwise very different formulations. Behavioral residential models often omit weekday–weekend distinctions, holidays, and explicit occupancy state processes, instead embedding such effects in coarse activity factors or daily frequencies (Chuan et al., 2015). Population models frequently rely on homogeneity or clustering assumptions that smooth away rare but operationally important heterogeneity (Alizadeh et al., 2014, Zhao et al., 2015). Symbolic KAN models in their current form are static $f$6, $f$7 approximations, so dynamic motor stalling, protection, and delayed recovery appear only indirectly through phasor trajectories (Dorji et al., 27 Aug 2025). The DNS uniform aggregate caching model deliberately collapses resolver heterogeneity into an equivalent rate $f$8, which improves tractability but also limits fidelity when request distributions are highly skewed (Wang, 2016). Pattern-consistent calibration for large electronic loads, conversely, is motivated by the claim that exact trajectory matching can be structurally misleading for stochastic systems (Lu et al., 8 Feb 2026).

A common misconception is that static voltage dependence suffices whenever the goal is network planning. The transfer-capability study directly contradicts this: static ZIP formulations overestimate secure transfer limits relative to dynamic composite models, because they omit motor and protection dynamics that dominate post-fault recovery (Wang et al., 2020). Another misconception is that interpretability and flexibility are mutually exclusive. The KAN results show that explicit symbolic equations can be learned from disturbance data without fixing the model structure to ZIP or exponential forms in advance (Dorji et al., 27 Aug 2025). A third misconception is that load characteristics modeling is domain-specific to electric power; the DNS paper models authoritative-server load as

$f$9

in the stub-only case and

$\upsilon(t)$0

when full clients are present, making TTL the analogue of a control variable shaping aggregate load through caching (Wang, 2016). In robotics, the carried-load problem is similarly cast as load-characteristics modeling through an 8-dimensional latent state estimated from proprioceptive histories, explicitly tying load representation to downstream control (Chang et al., 10 Jul 2025).

Taken together, these works suggest a unifying view. Load characteristics modeling is best understood as a family of state-space, probability-distribution, or symbolic-equation constructions that trade off structure, identifiability, and tractability according to application. The most successful models tend to make three aspects explicit: the internal state or latent regime of the load, the exogenous variables and control actions that perturb it, and the aggregation rule that maps microscopic behavior to operationally relevant macroscopic quantities.