Carbon-Aware Scheduling of Heat Pumps in Apartment Buildings: A Digital-Twin Methodology for Multiobjective Energy, Carbon, and Comfort Control

1. Introduction

Electrified space heating can reduce direct fossil-fuel combustion in buildings, but the climate effect of electrification depends partly on when electricity is consumed. Heat pumps introduce operational flexibility because the thermal mass of the building, emitters, and any associated storage can shift part of the heating load across time. In apartment buildings, however, this flexibility is constrained by many-zone thermal interactions, partial state observability, shared electrical or hydraulic limits, and uneven occupancy. A carbon-aware scheduler in this setting cannot be reduced to simple load shifting or a single setback rule. It must evaluate the relation among electricity use, the carbon intensity of that electricity, and the risk of discomfort in occupied apartments.

Adjacent building research supports this framing. A recent industrial retrofit study argued that optimization focused on energy alone misses trade-offs among energy consumption, carbon emissions, and thermal comfort, and it built a tri-objective framework combining simulation, surrogate models, evolutionary optimization, and multi-criteria decision analysis. That work concerned retrofit design rather than operational control, but its transferable lesson is methodological: low-carbon building decisions often require explicit representation of energy, carbon, and comfort together. The same study also identified future need to include active systems and dynamic occupant behavior, which points toward operational scheduling. [1]

This article develops a conceptual methodology for carbon-aware scheduling of heat pumps in apartment buildings. It does not claim measured carbon reductions or completed field validation. Its contribution is a technically specified control architecture that combines occupant-centered inference, uncertainty-aware prediction, and evidence-constrained digital-twin integration into a supervisory scheduling logic suitable for future testing where sensing is incomplete and direct personal instrumentation may be undesirable.

2. Research question and evidence scope

The research question is: how should a digital-twin control framework schedule apartment-building heat pumps when the aim is to reduce operational carbon emissions without unacceptable comfort degradation and when occupancy and thermal state are only partially observed? The question is narrower than general smart-building optimization. It concerns supervisory scheduling over short receding horizons, winter heating as the primary use case, and apartment buildings rather than factories, schools, or urban infrastructure.

The evidence base is a targeted narrative synthesis of adjacent method families, not a systematic review and not direct proof of heat-pump scheduling effectiveness. Multiobjective building optimization provides formal treatment of energy, carbon, and comfort trade-offs and shows how surrogate models can reduce repeated simulation cost. Occupant-centered environmental assessment from fixed sensing and timetables demonstrates a way to infer exposure without wearables, which is transferable to occupancy-weighted comfort scheduling. Uncertainty-aware predictive ensembles show why point forecasts alone are insufficient under operational risk and why calibrated intervals can support more reliable decisions. Digital-twin studies contribute two narrower lessons relevant here: one shows how a five-dimensional twin model can support dynamic scheduling in a cyber-physical system, and another shows how explainable assessment can be organized around structured risk prototypes, scene state, observable evidence, threshold bases, and explicit open evidence gaps. [1] [2] [3] [4] [5]

None of these sources directly evaluates apartment-building heat-pump scheduling. This manuscript therefore uses them as bounded methodological precedents, limited to points supported by the cited passages: tri-objective energy-carbon-comfort framing, timetable-based occupant inference from fixed sensing, calibrated prediction intervals, and evidence-constrained twin representation. Whether these ingredients improve apartment heat-pump control remains an empirical question. [1] [2] [3] [5]

3. Problem definition, variables, and assumptions

Consider an apartment building with either a centralized heat-pump plant serving multiple apartments or coordinated apartment-level units connected to a common electrical infrastructure. Time is discretized into intervals t = 1, ..., T of duration Δt. Let z index thermal zones, including apartments and relevant common spaces. The supervisory scheduler issues an action vector u_t at each interval. The framework distinguishes three actuation modes: mode A, plant-level continuous targets such as supply-water temperature; mode B, plant-level discrete equipment decisions such as compressor stage or unit enable state; and mode C, zone-allocation decisions such as apartment heating enable signals or valve-priority weights. A deployment may optimize one mode only or combine them.

This distinction resolves an ambiguity often hidden in conceptual control papers. Define u_t = (u_t^A, u_t^B, u_t^C), where absent components are fixed by the building configuration. In a centralized hydronic system, a realistic implementation might optimize u_t^A and u_t^B while holding u_t^C as a downstream rule. In apartment-level split systems with feeder constraints, the supervisory layer may optimize u_t^B and u_t^C while omitting u_t^A. Comparators must match the available actuation mode; a fixed-setpoint baseline for mode A is not a valid comparator for a building where only mode C exists.

Let P_t(u_t) denote electrical power, CI_t the grid-carbon-intensity forecast, T_z,t the operative-temperature estimate, O_z,t the inferred occupancy state, and q_z,t the confidence assigned to that inference. For a prediction horizon H beginning at interval t, the scheduler evaluates candidate action sequences U_t = (u_t, ..., u_{t+H-1}). To avoid double-counting energy in the scalar objective, the default score contains two normalized terms rather than three: J(U_t) = Σ_{τ=t}^{t+H-1}[λ_C Ĉ_τ + λ_D Ḋ_τ], where Ĉ_τ = CI_τE_τ/C_ref, Ḋ_τ = D_τ/D_ref, and E_τ = P_τ(u_τ)Δt. Here C_ref and D_ref are fixed positive scaling constants declared before testing, preferably the corresponding baseline totals over a preregistered reference period. Discomfort is D_τ = Σ_z w_z,τ{max[0, T_min,z,τ - T_z,τ] + max[0, T_z,τ - T_max,z,τ]}. The default weights are λ_D = 0.70 and λ_C = 0.30, reflecting the rule that protected comfort dominates carbon after normalization. Energy use remains visible through separate reporting and may be added only in a preregistered sensitivity analysis defined on the same horizon sum: J_E(U_t) = J(U_t) + λ_E Σ_{τ=t}^{t+H-1} Ē_τ, where Ē_τ = E_τ/E_ref and E_ref is a fixed positive scaling constant declared in advance. In that sensitivity analysis, λ_E must be reported explicitly and interpreted as a cost-of-energy regularizer rather than as an independent climate objective.

Other formulations remain admissible, but they should be treated as sensitivity analyses rather than an undefined menu. A secondary analysis may use Pareto search followed by transparent ranking, or a lexicographic rule with hard comfort protection and soft carbon minimization. The default operational rule in this manuscript is the normalized weighted score above, because it preserves a direct interpretation: one term represents operational carbon and the other discomfort. Energy consumption is still an essential reported outcome and constraint-related quantity, but not a separate default objective because operational carbon already scales with electricity use. The broader lesson from tri-objective retrofit optimization is retained in bounded form: energy, carbon, and comfort should be reported separately even when the supervisory controller optimizes a scalar rule for implementation. [1]

The framework assumes partial sensing rather than full thermal-state observability. An adjacent indoor-monitoring study noted that low-cost deployments commonly measure air temperature and relative humidity, whereas air velocity, metabolic rate, and clothing insulation are often unavailable; it therefore used plausible nominal assumptions for comfort-related estimation. The same principle is useful here, with a stricter reporting rule: comfort should be labeled as inferred comfort unless all required variables are directly measured. When air speed is low and mean radiant temperature is close to air temperature, operative temperature may be approximated by air temperature; if those assumptions are doubtful, the scheduler should widen prediction intervals and apply more conservative comfort protection. [2]

Further assumptions should remain explicit. First, the present article focuses on winter heating, although the definitions can be mirrored for cooling. Second, domestic hot water is excluded unless coupled thermal storage is part of the optimized asset set. Third, apartment-level personal identification is not required; the method is compatible with aggregate or pseudonymized occupancy inference. Fourth, carbon-aware scheduling is most meaningful where carbon intensity varies within or across days. If the electricity mix is nearly constant or near-zero carbon, the optimization problem may reduce in practice to energy-comfort scheduling.

4. Occupant-centered inference from fixed sensing and timetables

Apartment-building heating schedules frequently rely on generic occupied periods, but household routines differ by age, work pattern, caregiving responsibility, and seasonal behavior. A transferable alternative comes from occupant-centered indoor air-quality monitoring that infers occupant location from timetables and combines that information with fixed room sensors instead of requiring wearables. The core transferable idea is not the air-quality metric itself. It is the shift from room-centric monitoring to occupant-centered inference using existing infrastructure. [2]

In apartment buildings, the proposed inference stack has three levels. Level 1 is deterministic schedule input: declared household routines, vacancy notices, known caretaker rounds, booking schedules for shared rooms, and building service periods. Level 2 is probabilistic correction from fixed signals, which may include indoor temperature and humidity trajectories, domestic-hot-water draw signatures where permitted, apartment-door events, corridor motion, elevator calls, or non-intrusive electrical patterns. Level 3 is confidence assignment. Every inferred occupancy state is accompanied by a confidence score q_z,t in [0,1], and decisions should revert toward conservative fallback protection as q_z,t falls.

Operationally, let O_z,t ∈ {0,1} denote inferred occupancy and let Ō_z,t ∈ [0,1] denote a fallback prior occupancy probability for zone z at time t. Define the raw comfort weight as w'_z,t = α_z[(1-β)+βq_z,t]O_z,t + γ_z(1-q_z,t)Ō_z,t, with β ∈ [0,1], α_z > 0, and 0 ≤ γ_z ≤ α_z. The default setting is β = 0.5, which gives equal influence to active-occupancy confidence and residual protection under uncertainty. To prevent excessive penalty magnitudes across buildings, use the clipped normalized weight w_z,t = min(w_max, w'_z,t)/w_max with default w_max = max_z α_z. Zone-priority coefficients α_z are fixed ex ante by an operator-resident governance rule; a simple default is α_z = 1 for standard apartments and α_z = 1.5 for predefined vulnerability classes. If building-specific justification is lacking, set γ_z = α_z so that uncertainty does not lower minimum protection below the zone-priority level.

This occupant-centered formulation differs from room-centric control in two respects. First, it emphasizes occupant exposure over nominal room status, paralleling the adjacent study's argument that acceptable environmental conditions at room level may still fail to represent individual exposure over time. Second, it avoids treating all apartments as equally critical at all hours. Carbon-aware preheating is justified only when inferred future occupancy, model confidence, and estimated thermal inertia together indicate that later comfort can still be protected. [2]

5. Multiobjective scheduling logic for heat pumps

The control architecture should separate fast local regulation from slower supervisory scheduling. Local controllers preserve equipment safety, anti-short-cycling logic, defrost handling, and actuator stability. The supervisory layer updates every 15 to 60 minutes and determines plant or zone targets consistent with the available actuation mode. This separation is necessary because carbon-aware scheduling should not compromise equipment protection or low-level control stability.

A practical receding-horizon sequence is as follows. Step 1: assemble the current twin state from sensors, meters, weather forecast, carbon-intensity forecast, occupancy inference, and evidence-quality indicators. Step 2: generate candidate action sequences U_t from the admissible action set or constrained continuous ranges. Step 3: predict zone temperatures, electricity use, and shared plant or feeder loads over horizon H, together with calibrated lower and upper prediction bounds. Step 4: reject any candidate violating hard constraints, including equipment limits, minimum on/off durations, feeder limits, hydraulic limits, freeze protection, and mandatory comfort floors for protected zones. Step 5: rank feasible candidates by the default score J(U_t) or a preregistered sensitivity formulation. Step 6: apply only the first action and repeat at the next interval.

Define the feasible-action set at time t as F_t = {U_t ∈ A_t^H : all hard constraints hold for every τ in [t, t+H-1] after uncertainty propagation and protection-state resolution}. Hard constraints should be explicit. For shared electrical limits, Σ_{i∈HP} P_i,τ(U_t) + P_other,τ ≤ P_feeder,max,τ. For hydronic systems, require ṁ_min ≤ ṁ_τ(U_t) ≤ ṁ_max and Δp_τ(U_t) ≤ Δp_max, or another plant-specific hydraulic feasibility map declared in advance. For cycling protection, if unit i turns on at τ, then it must remain on for at least L_i^on intervals; if it turns off, it must remain off for at least L_i^off intervals. Protection status should be resolved by a single rule rather than by overlapping exceptions. First compute S_z,τ = 1 if z belongs to a predefined vulnerability class; otherwise S_z,τ = 0. Next compute B_z,τ = 1 if O_z,τ = 1 and q_z,τ ≥ q_min, or if Ō_z,τ ≥ p_protect; otherwise B_z,τ = 0. A zone is protected at interval τ if G_z,τ = max(S_z,τ, B_z,τ) = 1. Distinguish two lower bounds declared before testing: T_stat,z,τ, the non-negotiable statutory or habitability minimum when such a rule exists, and T_comf,z,τ, the preferred occupied comfort minimum used in discomfort scoring. Let T_pref,z,τ = max(T_stat,z,τ, T_comf,z,τ). Define the non-negotiable protection floor by rule as follows: if S_z,τ = 1, set T_floor,z,τ = T_pref,z,τ - Δ_vuln; else if B_z,τ = 1, set T_floor,z,τ = T_pref,z,τ - Δ_std; else set T_floor,z,τ = max(T_stat,z,τ, T_pref,z,τ - Δ_unprot). The default is Δ_vuln = 0 °C, Δ_std = 2 °C, and Δ_unprot = 4 °C. Thus vulnerability status has priority when signals conflict, high-confidence or high-prior occupancy triggers standard protection for other zones, and unprotected zones still retain at least the statutory or habitability minimum where one exists. Feasibility requires T̂_z,τ^L(U_t) ≥ T_floor,z,τ for every zone with G_z,τ = 1 and every horizon step. Operationally, the sequence is fixed: generate joint occupancy and thermal scenarios first, resolve G_z,τ and T_floor,z,τ within each scenario or under the more protective documented approximation, then test candidate feasibility against the resulting lower temperature bounds.

An implementable conservative deferment rule can then be stated exactly. Let CI_τ^pt be the carbon-intensity point forecast for interval τ and let [CI_τ^L, CI_τ^U] be its calibrated interval. Because the scheduler chooses the current action only, the screening comparison should be interval-to-interval. Define the future reference level on the same uncertainty basis as the current interval: R_CI,t^L = median{CI_τ^L : τ = t+1, ..., t+H-1}. Let δ_CI be a preregistered deferment threshold. Carbon deferment is permitted only if CI_t^L ≥ R_CI,t^L + δ_CI; otherwise the controller skips deferment and ranks feasible candidates by the default score J(U_t). If deferment is permitted and there exists at least one candidate in F_t satisfying T̂_z,τ^L(U_t) ≥ T_floor,z,τ for all protected zones and all τ, define that subset as F_t^0 and choose U_t* = argmin_{U_t ∈ F_t^0} Σ_{τ=t}^{t+H-1} E_τ(U_t). If multiple candidates are within a predefined energy tolerance, break ties by lower expected carbon Σ_{τ=t}^{t+H-1} CI_τ^ptE_τ(U_t); if a tie remains, choose the smaller predicted compressor switching count. If F_t^0 is empty, choose U_t* = argmin_{U_t ∈ F_t} Σ_{τ=t}^{t+H-1} Σ_{z∈Z_prot,τ} max[0, T_floor,z,τ - T̂_z,τ^L(U_t)]. Here zero violations means zero predicted lower-bound floor violations over the horizon for all protected zones. The rule is conservative because it requires a clearly high current carbon signal relative to later intervals on a matched interval basis and lower-bound thermal feasibility before deferment is allowed.

This rule is narrower than full model-predictive control and is proposed as the default baseline scheduler. A more general optimizer may replace the deferment ranking with J(U_t), Pareto dominance, or another preregistered criterion, but it should preserve the same protected-zone feasibility logic and uncertainty-aware prediction bounds so that comparisons are meaningful.

For later empirical comparison, two baselines are appropriate. The primary comparator is conventional fixed scheduling: occupied-period setpoints with nighttime setback and no carbon signal. The secondary comparator is tariff-aware but carbon-blind optimization, because price and carbon intensity may correlate without being equivalent. The carbon-aware scheduler should be considered beneficial only if it improves carbon outcomes relative to these baselines without materially worsening protected-zone comfort or causing excessive switching.

6. Digital-twin architecture and evidence-constrained data integration

A carbon-aware scheduler requires more than a prediction model. It requires a structured representation of what the building measures, what it infers, and what remains uncertain. Digital-twin work on urban drainage is relevant at the architectural level because it combines a five-dimensional twin model with dynamic scheduling and real-time data exchange. The drainage physics do not transfer to buildings, but the methodological lesson does: supervisory scheduling should be attached to an explicit cyber-physical state representation rather than implemented as an isolated optimizer. [4]

For apartment heat pumps, the proposed twin has five layers. The asset layer stores apartments, common spaces, heat pumps, emitters, storage tanks if present, meters, and sensors. The state layer stores temperatures, humidity, power, inferred occupancy, forecast quantities, and associated timestamps. The behavior layer stores thermal response models, equipment performance maps, and exogenous forecast interfaces such as weather and carbon intensity. The rule layer stores comfort policies, feeder and equipment constraints, privacy restrictions, and fallback logic. The evidence layer stores data provenance, confidence scores, missingness flags, sensor-health indicators, and conflict records.

Evidence-constrained integration is necessary because apartment data are often sparse, delayed, or contradictory. A recent residential digital-twin safety framework organized assessment around a computable Risk Prototype and a hierarchical SceneState structure that binds geometric measurements, semantic labels, image observations, and regulatory knowledge. It also retained unresolved cases as open evidence gaps requiring follow-up. The transferable lesson for heating control is that automated judgments should record whether they rest on direct measurement, model-based estimation with adequate support, or fallback assumptions under evidence gaps. Scheduling may continue under evidence gaps, but the logic should tighten safety margins or request human review rather than present unsupported certainty. [5]

This architecture supports auditability as well as control. If underheating is reported, the operator can inspect whether the relevant decision depended on low-confidence occupancy inference, a failed sensor, a wide thermal interval, or an optimistic carbon forecast. Repeated evidence-gap patterns also inform investment decisions about additional sensing. In that sense, the twin is not only a computational host for scheduling but also a governance mechanism for operational accountability. [5]

7. Uncertainty-aware model construction and interpretation

Scheduling under uncertainty should not rely only on point predictions. The relevant control question is whether a candidate action keeps protected zones within acceptable bounds despite forecast error. Work on search-guided regression ensembles in construction estimation is instructive because it combined dynamic learner selection, calibrated prediction intervals, and post hoc interpretability rather than treating uncertainty as secondary. The application domain differs, but the methodological message is relevant: operational forecasts should be accurate, uncertainty-aware, and inspectable. [3]

The proposed predictive module contains three model classes trained per building or apartment cluster. Class 1 predicts zone thermal response T_z,t+1:t+H under candidate actions and exogenous inputs. Class 2 predicts electrical demand P_t:t+H and any feeder or plant peak indicators. Class 3 predicts occupancy correction or occupancy-confidence updates from fixed sensing. Each class may be implemented as an ensemble of complementary learners, such as regularized linear models, tree-based regressors, kernel methods, and neural networks. Dynamic learner selection is a suitable option because apartment buildings change across seasons, retrofit states, and occupancy patterns, but simpler fixed ensembles remain acceptable if interval calibration is maintained.

Each predictive output should include both a point forecast and calibrated lower and upper bounds. Interval calibration is especially important for the lower temperature bounds used in protected-zone feasibility checks, because under-covered lower bounds create false assurance. A consistent uncertainty-propagation rule is therefore required. The default rule is scenario propagation with quantile extraction: for each candidate sequence U_t, generate K joint forecast scenarios by sampling from the calibrated residual distributions of the thermal, occupancy-confidence, demand, and carbon-intensity models while preserving any declared cross-correlations through a shared random seed or copula-like dependence map. For each zone and horizon step, compute T̂_z,τ^L(U_t) as the α-quantile of the resulting scenario distribution of temperature, with α preregistered and fixed across evaluation; a default α = 0.10 yields a conservative lower bound. If occupancy uncertainty affects internal gains or protection status, use the same sampled occupancy scenario consistently within that simulation path rather than mixing independent marginal draws. When joint scenario generation is not feasible, the scheduler should use a documented worst-case approximation, such as combining the thermal lower bound with the more protective occupancy state, and label the result as conservative approximation rather than calibrated joint propagation. Before any test, the operator should preregister one parameter-selection rule and one recalibration trigger. The default selection rule is a two-stage blocked historical procedure: split the available predeployment record into an earlier training segment and a later tuning segment without interleaving; fit models on the training segment; then choose all free constants from fixed candidate grids by minimizing, on the tuning segment, the lexicographic criterion of (1) zero statutory-floor violations, if any floor is defined, otherwise minimum protected-zone lower-bound violations; (2) minimum occupancy-weighted discomfort severity; and (3) minimum operational carbon. The chosen values are then frozen for back-testing and shadow mode. The default candidate grids are λ_D ∈ {0.60, 0.70, 0.80}, λ_C = 1 - λ_D, λ_E ∈ {0, 0.05, 0.10}, q_min ∈ {0.50, 0.70}, p_protect ∈ {0.30, 0.50}, Δ_std ∈ {1, 2} °C, Δ_unprot ∈ {3, 4} °C with Δ_vuln fixed at 0 °C, and δ_CI equal to the 50th, 60th, or 70th percentile of positive pairwise differences CI_t^L - R_CI,t^L observed in the tuning segment. If no tuning data exist, retain the manuscript defaults unchanged and report that no data-driven tuning was performed. The default recalibration trigger is observed lower-bound coverage below its nominal target on a rolling holdout window declared in advance. [3]

Interpretability also serves operations. Feature-attribution methods analogous to SHAP can help engineers check whether predictions respond plausibly to weather, recent indoor conditions, occupancy indicators, or supply settings. These attributions are explanatory aids rather than causal proof. If important features fluctuate implausibly across retraining windows, the scheduler should widen comfort buffers or revert to simpler fallback control until model stability is restored. [3]

8. Prospective evaluation protocol

Because this article is conceptual, a prospective evaluation protocol replaces an empirical results section. The protocol distinguishes outputs observable in simulation or shadow mode from those requiring field confirmation. It also distinguishes model-based comfort metrics from resident-reported experience. This separation keeps the proposal falsifiable without pretending that inferred comfort is identical to lived comfort.

Phase 1 is historical back-testing within a digital twin. Archived meter, weather, and indoor-sensor data are used to reconstruct plausible states and replay candidate policies. Observable outputs include estimated operational carbon emissions, electricity use, predicted occupied discomfort hours, comfort-floor violation time, predicted feeder peaks, switching counts, and carbon per delivered thermal unit when delivered heat can be estimated. In this manuscript, occupied discomfort hour is one interval Δt counted whenever O_z,t^eval = 1 and T̂_z,t falls outside [T_min,z,t, T_max,z,t], summed across zones with optional occupancy weights; comfort-violation severity is Σ_zΣ_t O_z,t^eval{max[0, T_min,z,t - T̂_z,t] + max[0, T̂_z,t - T_max,z,t]}Δt. Delivered thermal energy Q_del is defined operationally by estimator class. For centralized hydronic mode A or B with heat metering, use measured thermal output Q_del = Σ_τ ρc_p ṁ_τ(T_sup,τ - T_ret,τ)Δt. If direct heat metering is absent but whole-plant electrical input and COP estimates are available, use inferred Q_del = Σ_τ COP̂_τP_τΔt and label the result as model-based. For apartment-level unit coordination in mode C, use unit-level metered heat if available; otherwise estimate Q_del per unit from runtime, compressor state, and a preregistered performance map, then aggregate across units. Report the estimator class separately from the outcome. For heterogeneous portfolios with mixed actuation modes, preregister one aggregation rule: compare buildings on a within-building basis against a mode-matched baseline, compute percentage change in carbon per delivered thermal unit only within the same estimator class, and aggregate portfolio summaries by inverse-variance weighting or, if variance estimates are unavailable, by delivered-heat weighting. Do not pool raw Q_del ratios across different estimator classes without stratification.

Phase 2 is supervisory shadow mode. The scheduler processes live data and produces recommended actions while incumbent building control remains authoritative. This phase can directly evaluate decision latency, forecast calibration, action stability, and disagreement frequency with existing control. It can also quantify the fraction of recommended decisions made under low-confidence evidence and the rate at which the proposed scheduler would have triggered conservative fallback logic. Comfort metrics in this phase remain model-based unless independent resident feedback or portable verification measurements are collected. If resident-reported experience is collected, discrepancies should be resolved by a preregistered adjudication rule: protected-zone floor violations and resident reports of unacceptable cold take precedence over model-based comfort success, whereas the absence of complaints does not erase recorded floor violations.

Success criteria must be declared before testing, but their thresholds are context-specific and therefore remain hypothetical here. A pilot can be judged provisionally acceptable if (i) the median apartment-level occupied-discomfort-hour difference relative to baseline does not exceed a predeclared non-inferiority margin m_DH, (ii) the 90th-percentile apartment-level difference does not exceed a predeclared upper-tail margin m_90, and (iii) operational carbon per evaluation period, and carbon per delivered thermal unit when Q_del is estimable within a common estimator class, are directionally lower than baseline. The values of m_DH and m_90 should be fixed ex ante by the operator and resident-governance process using one explicit rule: select the smallest margins that exceed the 95th percentile of absolute baseline week-to-week variation in the corresponding apartment-level metrics on a preregistered historical window; if such historical variation cannot be estimated, do not claim non-inferiority and treat the pilot as calibration only.

Validation should also test subgroup fairness. Apartment orientation, top-floor exposure, vulnerable occupancy, and irregular schedules may affect who bears the burden of carbon-aware shifting. A building-level carbon benefit obtained through recurrent discomfort in a small set of apartments would be unacceptable even if the average outcome looked favorable. Apartment-level reporting is therefore required alongside building averages.

9. Failure cases, conflicting conditions, and limits of generalization

Several failure cases should be anticipated. Occupancy inference may be wrong, especially where routines are irregular, privacy rules restrict data sources, or visitors are frequent. Simplified thermal models may fail during defrost events, very cold weather, open-window airing, or unobserved use of portable electric heaters. Older apartment buildings may also exhibit strong inter-apartment thermal coupling that obscures the effect of supervisory actions on any one zone.

Conflicting operational conditions are likely. The carbon-minimizing action may increase electrical peak demand, may diverge from tariff incentives, or may cause localized overheating during preheating. A low-carbon period may coincide with hours of low occupancy, which encourages load shifting, yet some apartments may have insufficient thermal storage capacity to carry that shift without later discomfort. The tri-objective framing is useful precisely because it does not assume alignment among these goals. Adjacent retrofit optimization emphasized the existence of energy-carbon-comfort trade-offs and also noted the need to incorporate active systems and dynamic occupant behavior. That caution is even more relevant in operational scheduling. [1]

Limits of transfer from the evidence base should be stated plainly. Timetable-based occupant inference transfers imperfectly from schools or workplaces to residential settings because household routines are less institutionalized. Digital-twin scheduling ideas from drainage systems transfer at the architectural level, not through their physical equations or dispatch priorities. Uncertainty-aware ensemble methods from construction estimation transfer as a strategy for interval-aware modeling, not as validated thermal predictors. Evidence-constrained twin reasoning from residential safety assessment supports explicit handling of evidence gaps, but not the thermal decisions themselves. This article should therefore be read as a bounded methodological proposal and prospective test protocol for apartment heat-pump control, not as validation of the framework. [2] [4] [3] [5]

10. Conclusion

This article proposes a conceptual methodology for carbon-aware scheduling of heat pumps in apartment buildings under partial observability. The claim is bounded: apartment heat-pump scheduling can be formulated as a multiobjective energy-carbon-comfort problem and organized through a digital-twin architecture that integrates occupant-centered inference, evidence-constrained data management, and uncertainty-aware forecasting. The manuscript specifies variables, a normalized objective, comfort-weight calibration rules, shared constraints, protected-zone feasibility logic, a conservative deferment rule, comparator policies, and a staged evaluation design. [1] [2] [3] [4] [5]

What remains untested is equally important. The methodology has not been validated in apartment-building heat-pump operation, benchmarked against existing thermostatic or model-predictive controls, or shown robust across tenancy types, climates, and electricity-carbon regimes. The accuracy of occupancy inference, calibration of thermal prediction intervals, fairness of apartment-level burden distribution, and acceptability of comfort-risk trade-offs all require empirical study. The next step is staged implementation with falsifiable tests of which elements materially improve carbon outcomes while preserving resident comfort and operational trust.

References

1. Duan, Zhongcheng; Li, Binhao; Zi, Yilun; Yao, Gang (2025). Building retrofit multiobjective optimization using neural networks and genetic algorithm three for energy carbon and comfort. Scientific Reports. 15(1), 38076. doi:10.1038/s41598-025-21871-0.
2. Ruffa, Filippo; Iadarola, Grazia; De Capua, Alberto; De Capua, Claudio (2026). A Method for Optimized Monitoring of Indoor Air Quality in Public Buildings. Sensors. 26(14), 4559. doi:10.3390/s26144559.
3. Chen, Lifei; Lim, Zhi Min; Lim, Wei Hong; Tiang, Sew Sun; Sharma, Abhishek; Syamsunur, Deprizon; Alharbi, Amal H.; Eid, Marwa E.; et al. (2026). Search-guided regression ensembles for accurate, interpretable, and uncertainty-aware construction cost estimation. Scientific Reports. 16(1), 21456. doi:10.1038/s41598-026-51706-5.
4. Zhao, Yuanjing; Yang, Min (2026). Modelling and simulation of smart city drainage system based on digital twin five-dimensional models. PLOS One. 21(7), e0352787. doi:10.1371/journal.pone.0352787.
5. Jing, Zijian; Zhu, Liyi; Chen, Tianyi; Hovestadt, Ludger; Li, Li (2026). Vision-Based Digital Twin and AI Agent Framework for Low-Cost, Explainable Indoor Building Inspection and Safety Assessment. Sensors. 26(15), 4992. doi:10.3390/s26154992.