Beyond R²: The Role of Polynomial Degree in Modeling External Temperature and Its Impact on Heat-Pump Energy Demand

Abstract

Missing values in hourly outdoor air temperature series are common and can bias building energy assessments that rely on uninterrupted temperature profiles. This paper examines how the polynomial degree can be used to reconstruct incomplete temperature data from the duration curve, which affect the energy indicators of an air-source heat pump (ASHP). Using an operational dataset from Opole, Poland (1 September 2019–31 August 2020; 5.1% gaps), global polynomials of degree n = 3…11 were fitted to the sorted hourly temperatures, and the reconstructions were mapped back to time. The reconstructions drive a building–ASHP model evaluated for two supply-water regimes (L_WT, leaving water temperature = 35 °C and 45 °C). Accuracy is assessed with mean absolute error (MAE), root-mean-square error (RMSE), and R² on observed, filled, and full subsets—including cold/hot tails—and propagated to energy metrics: seasonal space-heating demand (Q_season); electricity use (E_el); seasonal coefficient of performance (SCOP); peak electrical power (P_el,max); seasonal minimum coefficient of performance (COP_min); and the share of error due to filled hours (WFE_fill). All degrees satisfy $\left|{RE}_{Q_{season}}\right|\le 2\%$. For L_WT = 35 °C, relative changes span ${RE}_{E_{el}}$ ≈ −2.22…−1.63% and ${RE}_{N_{el,max}}$ ≈ −21.6…−7.7%, with ${ER}_{SCOP}$ ≈ +0.53…+0.80%. For L_WT = 45 °C, ${RE}_{E_{el}}$ remains ≈ −0.43% across degrees. A multi-criterion selection (seasonal bias, stability of energy indicators, tail errors, and WFE_fill) identifies n = 7 as the lowest sufficient degree: increasing n beyond seven yields negligible improvements while raising the overfitting risk. The proposed, data-driven procedure makes degree selection transparent and reproducible for gap-filled temperature inputs in ASHP studies.

1. Introduction

Gaps in hourly and daily temperature series are inherent in long-term meteorological databases. They arise due to sensor failures, power outages, telemetry errors, or the reorganization of measurement networks and reduce the reliability of all analyses requiring data continuity. This problem is fundamental in building energy management: to correctly determine degree hours, instantaneous heating power ${\dot{Q}}_{h\_sh}\left(\tau \right)$, electrical power consumption $P_{el}\left(\tau \right)$, and the coefficient of performance (COP) of an air-source heat pump (ASHP), an uninterrupted outdoor temperature profile is necessary.

When the chronology of data is incomplete—or missing altogether, as in histograms or duration curves—the temperature curve must be reconstructed synthetically; in earlier studies on the proprietary ASHP model for Opole, a sixth-degree polynomial fitted to the annual temperature distribution was used [1,2]. However, whether this order best compromises energy accuracy and computational cost has not been verified. Higher-order polynomials can improve statistical fit (np. R²), but increase the risk of overfitting and computation time; lower ones distort the “cold tail” of the distribution, determining the heating system’s peak power.

To address this problem, we introduce a transparent procedure for reconstructing missing temperature data that link the statistical fit of polynomial regression with its practical consequences for air-source-heat-pump (ASHP) energy indicators. The workflow begins with the raw meteorological series, followed by quality control and gap detection, and then proceeds through polynomial fitting of the duration curve in the n = 3…11 range. The reconstructed temperature profile is subsequently used as an input to the ASHP model, where instantaneous heating demand, electricity use, and the COP are calculated. Finally, aggregated key performance indicators (KPIs) allow a systematic evaluation of the reconstruction accuracy, energy deviations, and the error share attributable solely to filled hours.

Gaps in hourly air temperature series are common and result from sensor failures, power outages, and data quality-control procedures. Their presence affects not only diagnostic reconstruction error measures, such as the mean absolute error (MAE) and root-mean-square error (RMSE) [3], but more importantly, the results of energy models for air-to-air/air-to-water heat pumps, which are non-linearly sensitive to outdoor temperature through COP(t_a) relationships [4], compressor modulation, and defrosting and electric heater-activation thresholds [5]. In such applications, it is, therefore, not so much the minimization of temperature error as such that is crucial, but rather the limitation of the propagation of this error to energy indicators: the energy supplied, electricity consumption, and seasonal performance factor [6,7].

This work operates within the previously published ASHP numerical model [1,2]. The reconstruction operator is fixed to a global polynomial fit to the temperature duration curve; no alternative gap-filling methods are considered. The sole design variable is the polynomial degree n, and the objective is to quantify and bound its impact on ASHP energy KPIs, leading to a criterion for the lowest sufficient degree.

Our work uses a simple, easy-to-reproduce approach: fill in the missing values with polynomials of various orders adjusted to the sorted temperature hours (duration curve) [8]. This approach allows us to capture typical and extreme conditions without separate modeling of the diurnal–seasonal rhythm [9]. Lower orders may lose detail, especially in the “tails” of the distribution. At the same time, orders that are too high to “wave” unnaturally and artificially increase or decrease the number of hours near temperatures critical for ASHP operation (e.g., defrosting or heater-activation thresholds) [10]. Therefore, the choice of order is primarily determined by the stability of energy indicators, not only by raw temperature errors [11].

Recent studies address temperature gap filling via ERA5 debiasing tailored to urban stations, demonstrating that debiased reanalysis efficiently handles larger gaps while preserving local signals [12]. For hourly air temperature at standard stations, debiasing-based gap filling has also been validated as a practical pathway to serially complete records [8]. Field networks additionally emphasize the need for rigorous quality control before reconstruction, followed by statistically sound infilling of missing hours [6]. Beyond physics-guided corrections, machine learning approaches have recently been benchmarked against traditional techniques across variables and gap lengths, confirming ML’s potential for robust imputation [13]. Multivariate frameworks can exploit cross-variable dependence at gridded scales to stabilize reconstruction under complex missing patterns [14]. Against this backdrop, our contribution is not another general-purpose imputation algorithm, but a transparent, data-driven criterion for selecting the lowest sufficient polynomial degree of an ordered (duration–curve) fit by evaluating its propagation to ASHP energy KPIs; this bridges a gap left by time-domain interpolation comparisons focused on predictive accuracy alone [15,16]. Previous publications [1,2] report computations for different climates and also include economic indicators; here, the focus is on establishing a reproducible degree-selection procedure for n on a single climate year, with a multi-climate replication planned next (Poland, Lithuania, Croatia, and Spain) and an extension to different ASHP types.

We propose three research hypotheses: (H1) there is a minimum order n* (“good enough”), above which further increases in n do not significantly change the energy results (|Δ| and relative errors < several percent) [17]; (H2) high orders may reduce the RMSE of temperature, while at the same time, worsening the reliability of energy indicators near control thresholds; (H3) the value of n* depends on the structure of gaps (share, string lengths) and on the climatic conditions of the year (cold vs. warm), because these factors change the weight of errors in the tails of the t-distribution [18]. Two common installation scenarios were considered: Low-Temperature Floor Circuit (L_WT = 35 °C) and domestic hot water (L_WT = 45 °C). These variants correspond to current practices in residential buildings with ASHP and allow the sensitivity of the results to the return water temperature to be assessed [19].

In the following section, we present the data, experiment configuration, and evaluation metrics: a description of meteorological sources and data quality, the method of generating/identifying gaps, the polynomial approximation procedure for n ∈ [3, 11], and the ASHP energy calculation chain with comparative metrics for temperatures (MAE and RMSE) and energy results (Δ—absolute and RE—relative errors of key indicators). In the Section 2, we specify these elements and justify the criteria for selecting the order of n*.

2. Materials and Methods

This study used hourly temperature data from 1 September 2019, and 31 August 2020 (ECO Opole plant station). The dataset contains 444 h of missing values (≈5% of the sample); 75% lasted no longer than 5 h; and only two episodes that exceeded 48 h (72 h and 324 h). For this series, reconstructions were built using polynomials of degree n = 3–11. The analyzed season included 31,872 heating degree hours calculated relative to the threshold t_a,bs = 12 °C, representing approximately 95% of the long-term average value of 33,400 °C h. The hourly meteorological data used in this analysis come from the public repository of the Institute of Meteorology and Water Management (IMGW-PIB) [20,21]. The hourly data were retrieved from the IMGW-PIB Opole synoptic station (WMO 0530, 50°37′37″ N, 17°58′08″ E, 165 m above sea level), which has operated continuously since 1990. Earlier measurements (1951–1989) were taken at the Opole–Sławice location (WMO 12465, 50°43′18″ N, 17°52′40″ E).

A calculated outdoor temperature of t_a,min = −20 °C was adopted for Opole, which gives a design heat output for the building of ${\dot{Q}}_{h,sh,max}$ = 12 kW. The same value was used in Equation (2) when scaling all temperature reconstructions.

Since the main objective of this study is to determine the lowest degree of the polynomial that, at an acceptable calculation cost, keeps all key energy errors of the ASHP model within ±2%, the following was analyzed for this purpose:

• Statistical accuracy relative to the reference series (RMSE, R²);
• Deviations in seasonal heat demand Q_season, electricity demand E_el, and the seasonal coefficient of performance SCOP;
• Changes in peak power ${\dot{Q}}_{h,sh,max}$ (peak space–heating heat rate) and $P_{el,max}$ (seasonal peak electrical input power);
• Share of error originating solely from gaps (WFE_fill);
• Computational complexity and susceptibility to overfitting.

This data-based criterion replaces the previously arbitrary grade 6, thereby increasing the transparency and repeatability of the ASHP model.

The data sources and quality control procedures are presented, followed by a discussion of the selection of reconstruction parameters, including the criteria for selecting the degree of the polynomial and implementing the heat load model and ASHP operating characteristics. An integral part is a set of evaluation metrics covering both temperature errors (MAE, RMSE, R², analyses in the “obs”, “fill”, and “all”), as well as energy indicators (ΔQ_h,sh,season, ΔE_el, ΔSCOP, ${RE}_{P_{el,max}}$, and RE_COP,min), including measures of sensitivity to extreme conditions (MAE_cold and MAE_hot) and the share of error resulting from gaps (WFE_fill). The heating season covers $S_{all}=\left\{1,\dots ,8784\right\}$ hours, and its cardinality is denoted by $N_{all}=\left|S_{all}\right|=8784$. Based on the gap flag vector $gap\left(\tau \right)$ (1—gap, 0—observation), we define the following: $S_{obs}=\left\{\tau \in S_{all}:gap\left(\tau \right)=0\right\}$, $S_{fill}=\left\{\tau \in S_{all}:gap\left(\tau \right)=1\right\}$. For boundary assessments, we also use subsets of the outdoor temperature rank: $S_{cold}$ (500 coldest hours) and $S_{hot}$ (500 warmest hours), with cardinalities $N_{cold}=\left|S_{cold}\right|=500$ and $N_{hot}=\left|S_{hot}\right|=500$. In the rest of the metrics, we write in a generalized form: MAE_S, RMSE_S, $R_S^2$ dla $S\in \left\{S_{obs},S_{fill},S_{all},S_{cold},S_{hot} \right\}$. The structure of the chapter corresponds to the successive stages of the analysis, enabling a clear link between the methodology and the results presented later in this section.

2.1. Method Overview

The workflow comprises five stages: (1) The observed hourly outdoor-air temperature series t_a,obs(τ) undergo quality control, and a reference completion t_a,ref(τ) is constructed for validation. (2) Ordered (duration–curve) representations are formed, and global polynomials of degree $n\in \left[3, 11\right]$ are fitted to t_a,obs, where n is the polynomial degree. (3) The reconstructed series are mapped back to the original timestamps and then supplied to the ASHP model defined by COP(t_a) and P_el(t_a); chronology is thus restored before any hourly calculations. The global polynomial is fitted in the ordered domain to stabilize tail behavior under short, scattered gaps; chronology is restored before computation. Given the memoryless P_el(t_a)/COP(t_a) relations, planning-level KPIs depend primarily on the temperature distribution (including tails) rather than short-run autocorrelation. (4) Planning-level KPIs are computed: ΔQ_season, ΔE_el, SCOP, P_el,max, COP_min, and the filling-effort indicator WFE_fill. (5) The lowest sufficient polynomial degree is the smallest n that satisfies these KPIs’ predefined stability/accuracy thresholds. Subsequent subsections detail data, reconstruction, and the ASHP model; results report KPI sensitivity as a function of n. Within the previously published ASHP planning tool [1,2], the reconstruction operator is a fixed global polynomial and only the degree n varies; the lowest sufficient degree is the smallest n at which planning-level KPIs stabilize within the stated tolerances, with no material change upon further increase. Full workbook recomputation (all degrees n = 3…11 and KPIs in parallel) takes T_all = 8063.4 ms on median (20 runs, Excel, manual recalc.); the equivalent per-degree cost is 8064.3/9 ≈ 895.9 ms; I/O dominates wall-clock time.

2.2. Data Source and Preliminary Data Processing

Hourly outdoor temperatures, t_a,obs(τ), were obtained from the internal weather station of the ECO Opole Combined Heat and Power Plant. The series covers the entire period from 1 September 2019, to 31 August 2020 (N_all = 8784). The raw file contained 444 h of missing data (5.1%) detected by the timestamp continuity check method. A mask $gap\left(\tau \right)\in \left\{\mathrm{0,1}\right\}$ was introduced; $gap\left(\tau \right)=1$, means no measurement, $gap\left(\tau \right)=0$, means observation. The dataset is used to isolate the influence of the polynomial degree; additional curated datasets for other climates, several European climates, and warm/cold subperiods are prepared for a follow-up replication using the same procedure. Results are shown for one season to isolate the effect of n; the transferable selection rule addresses generality and will be tested on cold/warm years in follow-on work.

For validation, a reference t_a,reft(τ) was created by supplementing the same tags with data from the AccuWeather™ portal for the location “Opole, PL.” Not using other stations eliminates the influence of local microclimate differences on the analysis results. While demonstrated on one site/year, the selection rule is climate-agnostic and evaluated per site via tail and seasonal KPIs. In this notation, t_a,obs(τ) is the QC’ed observed series on the original timestamps (gaps are masked, not filled), whereas t_a,reft(τ) is the independently completed reference on the exact timestamps used solely as the validation baseline; the reference is not used for fitting.

The heating season threshold was set at t_a,bs = 12 °C; the sum of heating degree hours DH was calculated according to (1) for the assumed operating temperature in the building t_i = 21 °C, which in the analyzed season gave DH = 31,872 °C·h, i.e., approx. 95% of the long-term value for Opole (~33,440 °C·h).

$$DH={\sum }_{r:t_a\left(\tau \right)<t_{a,bs}}\left(t_i-t_a\left(\tau \right)\right)$$

(1)

Continuous hourly air temperature series from local heating plant operators (e.g., ECO Opole) are rarely used in reconstruction studies, which are usually based on meteorological station data [22]. Combining operational and public data allows for better spatial representativeness while maintaining the temporal continuity required for heating-load modeling.

2.3. Characteristics of Temperature Data Gaps

Figure 1a shows a histogram of the lengths of successive gaps (gap runs) in the temperature series. The lengths of the gaps were grouped into seven logarithmic classes—1 h, 2 h, 3–5 h, 6–11 h, 12–23 h, 24–48 h, and ≥48 h—a system commonly used in studies on the supplementation of hourly meteorological data [8,12]. Figure 1b also annotates the two multi-day episodes (72 h and 324 h); all other outages are single- to few-hour gaps scattered across the season.

The analysis showed that single hours account for 60.9% of all shortages, and the 2 h and 3–5 h classes raise this share to over 73.9%. Less than 13% of breaks exceed 12 h, and only two episodes (72 h and 324 h) fell into the ≥48 h class (8.7%), which proves the incidental nature of multi-day failures. Given this gap structure—dominated by 1–5 h outages with only two ≥48 h episodes—an ordered-curve fit with subsequent mapping back to time minimizes seasonal distortion while ensuring time-domain placement of filled values.

The average duration of an outage was 19 h, with a median of 1 h, confirming the strongly skewed distributions described in the literature. Analysis of the distribution of gaps throughout the year (Figure 1b) did not reveal any seasonal accumulation. Apart from two long episodes, the remaining outages occur randomly from September to the end of March. Such a clear dominance of short-term gaps justifies using global polynomial fitting as a reconstruction method: gaps lasting a single hour do not disturb the season’s structure, while the few gaps lasting several dozen hours can be additionally monitored individually. This choice is consistent with the literature, in which low-order regressions or harmonic models work well for hourly gaps; more complex methods (GP/ML, reanalysis debiasing) are reserved for dense/extensive gaps or short-term forecasting needs [23]. In this dataset, with 5.1% short and scattered gaps and only two ≥48 h episodes, an ordered-curve fit minimizes seasonal distortion while ensuring correct time placement of filled values after mapping back; time-series or reanalysis-based fills would become preferable only under dense or extended missingness.

2.4. Reconstruction of Temperature Data Using Polynomial Regression

An hourly reconstruction of the outdoor temperature series was necessary to fully calculate the energy balances in the ASHP model. Due to the predominance of short-term gaps (Section 3.2), a global polynomial fit was applied to the ordered temperature series, which allowed the missing fragments to be reconstructed with minimal distortion of the seasonal structure. This method has the additional advantage of preserving the continuity of the first derivative of the fitting function, which is essential for further calculations of instantaneous power that are sensitive to local changes in the slope of the temperature curve.

First, the observation series, t_a,obs(τ), shown in Figure 2, was sorted in ascending order; then, the following was applied:

• The time axis was normalized (index 0…N_obs − 1), according to Equation (2):

  $$\tilde{\tau }={\displaystyle \frac{\tau -1}{\left(N_{obs}-1\right)}}\in \left[0, 1\right]$$

  (2)

  for $\tilde{\tau }$ = 1, …, N_obs; where N_obs = 8734 h (number of hours with seasonal observations).
• Adding quality control flags (QC), extracting measurement points (tag obs = 0), and sorting in ascending order according to temperature t_a,obs.
• Fitting a polynomial of degree n in the range n = 3, 4, …, 11, according to Equation (3):

  $$t_{a,n}\left(\tilde{\tau }\right)={\sum }_{k=0}^n{a}_k{\tilde{\tau }}^k$$

  (3)
• Mapping matched values to the whole timeline and filling gaps (gap = 1).
• Inserting the complete series into the ASHP model, where the manufacturer’s characteristics $P_{el}\left(t_a\right)$ and $COP\left(t_a\right)$ are interpolated exponentially.
• Calculation of seasonal results: heat demand ${\dot{Q}}_{season}$, electricity consumption $P_{el}$ (for L_WT = 35 °C and 45 °C), and the seasonal coefficient of performance SCOP. Accuracy was assessed by determining MAE, RMSE, R², ΔQ_season, ΔE_el, ΔSCOP, and separately, RMSE_obs and RMSE_fill.

The logical data flow and the “missing data?” decision point are shown schematically in Figure 3 (normalization, t_a,n fitting, mapping to the timeline, gap filling, and quality assessment). For each polynomial degree n, R², MAE, and RMSE were calculated in three subsets S—obs (observations only), fill (fittings only), and all (entirely)—which allows the error resulting from the fit to be separated from the error of the filling itself. Additionally, MAE_cold and MAE_hot were reported, calculated, respectively, in the coldest $S_{cold}=\left\{\tau \in S_{all}:{rank}_{\uparrow }\left(t_{a,ref}\right)\le E\right\}$ and the hottest $S_{hot}=\left\{\tau \in S_{all}:{rank}_{\downarrow }\left(t_{a,ref}\right)\le E\right\}$ (where E = 500) hours to assess sensitivity to the extremes of the temperature distribution (relevant ${\dot{Q}}_{h,sh,max}$ and P_el,max).

Normalization $\tau ⟼\tilde{\tau }\in \left[0, 1\right]$ limits the ill-conditioning of the Vandermonde matrix when fitting polynomials and does not affect the quality of the fit [24]. The models were trained exclusively on hours with measurements (obs), after which the determined curve $t_{a,n}\left(\tau \right)$ was mapped to the full time axis and limited to a physically meaningful interval [t_a,min, t_a,max]. Due to the risk of boundary oscillations (Runge) and overfitting at high orders, a practical range of n = 3…11 was adopted [25]; MAE/RMSE in the obs/fill/all subsets and “cold/hot” boundary indicators were used in parallel in the quality assessment. The diagram of the entire procedure is shown in Figure 3.

All models were trained exclusively on observed hours (obs) and then extrapolated to N = 8784 h; predictions were limited to the physically meaningful range [t_a,min, t_a,max] to avoid extreme artifacts.

Figure 4 shows the ordered curves t_a,obs, t_a,ref, and t_a,7. The reconstruction, e.g., n = 7, reproduces the reference curve throughout the season, with a small but significant deviation in the cold tail (bias of ∼1.6 °C between the minimums: −5.3 °C vs. −3.7 °C), which is essential from the point of view of peak power. This deviation is slight regarding MAE/RMSE, but vital for RE_X,max (Section 3.5).

The procedure used (axis normalization, $t_{a,n}\left(\tau \right)$ fitting for n = 3…11, obs/fill, and cold/hot) ensures consistent $t_{a,n}\left(\tau \right)$ profiles for further calculations. In Section 2.5, we use them to determine ${\dot{Q}}_{h,sh}\left(\tau \right)$, P_el(τ), COP(τ), and SCOP for L_WT = {35, 45} °C, and in Section 2.5, we evaluate the impact of the order n on energy metrics (ΔE_el, ΔSCOP, and RE_X,max) and the contribution of fill error (WFE_fill). COP(t_a) and P_el(t_a) are exponential fits to catalog maps; defrost cycles, crankcase heating, and auxiliary-resistive cut-ins are not dynamically simulated, primarily affecting the cold tail and P_el,max.

2.5. Building an ASHP Energy Model

The heat demand for a building ${\dot{Q}}_h$ consists of the heat demand and the demand for domestic hot water preparation, according to Equation (4):

$${\dot{Q}}_h\left(\tau \right)={\dot{Q}}_{h,sh}\left(\tau \right)+{\dot{Q}}_{h,htw}\left(\tau \right)$$

(4)

The model assumes a linear relationship between instantaneous heat demand and outdoor temperature, t_a, where $x\in \left\{ref,n\right\}$ (Equation (5)) with a design capacity of ${\dot{Q}}_{h,sh,max}$ = 12.07 kW at t_a,min = −20 °C. This value was scaled to the reference season conditions using the minimum temperature recorded during the season (−5.3 °C), which allowed for a consistent analysis of the effect of temperature reconstruction at the whole-season level. Seasonal values are calculated by discrete integration in hourly steps.

$${\dot{Q}}_{h,sh}\left(\tau \right)=\left\{\begin{array}{c} 0, t_{a,x}\left(\tau \right)\ge t_{a,bs}\\ {\dot{Q}}_{h,sh,max}·{\displaystyle \frac{t_i-t_a\left(\tau \right)}{t_i-t_{a,min}}}, t_{a,min}<t_{a,x}\left(\tau \right)<t_{a,bs}\end{array}\right.$$

(5)

The highest load of the season (hour no. 3488) comes from the observational part of the data (obs); therefore, all comparisons of ${\dot{Q}}_{h,sh,max}$ refer to the same measuring point.

The following ASHP model was implemented: Silesia Term 13 kW (ST AIR 00.011 SMART MINI INVERTER 4–13). The instantaneous heating power of the ASHP is determined according to Equation (6):

$${\dot{Q}}_{HP}\left(t_a\right)=COP\left(t_a\right)·P_{el}\left(t_a\right)$$

(6)

where the catalog maps of $P_{el}\left(t_a\right)$ and $COP\left(t_a\right)$ were interpolated exponentially (Equation (7)), and ${\dot{Q}}_{HP}\left(t_a\right)$ was limited by the device’s available power.

$$\mathrm{f}\left(t_a\right)=a·e^{b{·t}_a}$$

(7)

The model does not dynamically simulate defrost cycles, crankcase heating, or auxiliary-resistive heat; the maps implicitly capture these effects and are most relevant for tail behavior and RE_X,max. Calculations were performed for two supply water regimes, L_WT = 35 °C and L_WT = 45 °C, to assess the impact of supply temperature on energy consumption and SCOP. Using two L_WT values (35 °C and 45 °C) made it possible to map both conditions typical for underfloor heating and domestic hot water production, significantly impacting ASHP efficiency and the frequency of auxiliary heater activation. The design parameters of the ASHP unit for selected radiator temperatures are shown in Figure 5.

The results of the approximation of the operating characteristics of both COP and electricity demand for P_el are presented in

Table 1. Constant a, b (in Equation (7)) of exponential approximations for Pel and COP, selected ASHP model, in two L_WT regimes.

Return Water Temperature LWT	Pel		COP
	Constants of Function
°C	a	b	a	b
35	3.2097324	−0.0041214	3.3559848	0.0266827
45	3.7426251	0.0004945	2.7413944	0.0230053

.

In this study, ambient humidity and frost/defrost cycles are not modeled as separate dynamic processes. Their aggregate influence is represented implicitly through the manufacturer performance characteristics used here—namely, COP(t_a) and the electrical input/capacity versus ta—fitted in Table 1. These characteristics reflect catalog derating but do not resolve time-step defrost scheduling, crankcase-heater cycling, or humidity-driven icing intensity; therefore, transient penalties are not explicitly simulated. In practice, any residual bias from these processes is expected to concentrate in the cold tail of the outdoor-temperature distribution, primarily affecting peak electrical power and the minimum COP. Because P_el(t_a) and COP(t_a) are memoryless—hourly outputs depend only on the concurrent t_a(τ) via manufacturer maps, with no state carried between hours—the integrated KPIs are governed primarily by the distribution of t_a, including cold/hot tails, rather than short-run autocorrelation; therefore, MAE_cold/MAE_hot, RE_X,max, and WF_fill are tracked to guard temporal–structure effects.

Seasonal SCOP values are calculated as the quotient of energy supplied to the building and electricity (the definition complies with the current standards for determining seasonal HP efficiency) [26,27,28].

The building parameters used to scale the thermal model are listed in

Table 2. Features of the tested building.

Building Parameter	Symbol	Value	Unit
Peak heat demand of the building	${\dot{Q}}_{h,max}$	12.72	kW
Temperature-controlled usable area	Af	250	m2
Normal operating (service) temperature	ti	21	°C
Number of people residing in the facility	Li	6	j.o.
Individual daily hot water consumption	Vcw	50	dm3/person/day
Hot water mode (average hourly power)	${\dot{Q}}_{h,htw}$	0.65	kW
Design outdoor temperature, climate zone III	ta,min	−20	°C
Seasonal outdoor temperature limit	ta,bs	12	°C

: design power ${\dot{Q}}_{h,max}$ (under t_a,min conditions), internal temperature t_i, heated area A_f, and load model boundary parameters. These values determine the slope of the ${\dot{Q}}_{h,sh}\left(\tau \right)$ characteristic used in Equation (5) and seasonal calculations (SCOP and E_el).

The parameter ${\dot{Q}}_{h,htw}$ represents the constant demand for hot water at L_WT in accordance with the adopted scenario; if the heat loss coefficient of the building is unknown, it can be omitted (${\dot{Q}}_{h,sh}$ is modeled according to Equation (5)).

The annual load duration curve ${\dot{Q}}_h$, shown in Figure 6, takes into account both space heating ${\dot{Q}}_{h,sh}$ and domestic hot water demand with a constant value ${\dot{Q}}_{h,htw}$.

This curve determines the target thermal requirements of the building as a function of time. In the next step, we superimpose the ASHP characteristics onto it (Figure 7)—${\dot{Q}}_{h,HP,ref}\left(\tau \right)$ and $P_{el,ref}\left(\tau \right)$—to compare demand with heat source capabilities and calculate seasonal metrics (ΔQ_season, ΔE_el, and ΔSCOP) and RE_X,max. Here, X denotes a scalar KPI (e.g., Q_season, E_el, P_el,max, and COP_min), and X_max, its maximum, is defined in Equation (14). Relative changes in maxima are reported as RE_X,max (in percent), with the general definition RE_X given in Equation (16).

KPI propagation from temperature to energy depends primarily on (i) the building load slope in Equation (5), and (ii) the device maps COP(t_a) and P_el(t_a). Steeper building slopes or maps with stronger cold-tail derating increase the sensitivity of ΔQ_season, ΔE_el, and RE_X,max to reconstruction errors; conversely, flatter slopes reduce it. A full parametric sweep is planned; here, we report the calibrated maps (Table 1) and interpret KPI changes with this dependency in mind.

Figure 7 combines information about building demand (${\dot{Q}}_{h,sh}$) with source characteristics (${\dot{Q}}_{h,HP}$) and supply cost (P_el) for two L_WT regimes. This arrangement allows us to (i) check load coverage throughout the season (whether ${\dot{Q}}_{h,HP}<{\dot{Q}}_{h,sh}$ for each hour); (ii) compare the impact of L_WT = 35 °C vs. 45 °C on the electrical requirements P_el; and (iii) link any discrepancies to the metrics in Section 3.5 (e.g., RE_X,max, ΔE_el, and ΔQ_season). Since we do not present COP on the graph itself, its impact is indirectly taken into account by P_el(τ) and ${\dot{Q}}_{h,HP}\left(\tau \right)=COP\left(\tau \right)·P_{el}\left(\tau \right)$, and we report seasonal values in tables and energy metrics. We aggregate metrics hourly and seasonally, reporting ΔX (absolute value) and RE_X (relative value) in parallel to distinguish the temperature effect from its energy consequences. These definitions serve as the error-propagation lens from temperature reconstruction errors to energy outcomes: ΔX and RE_X (Equations (14)–(16)) translate t_a-level deviations into ΔQ_season, ΔE_el, SCOP, and RE_X,max.

2.6. Data Reconstruction Effect Assessment Metrics

The assessment of the quality of the reconstruction includes the following:

• Statistical accuracy: RMSE, MAE, R²—calculated separately for subsets obs, fill, and the totals all;
• Contribution of error originating from gaps WFE_fill (Equation (13));
• Relative peak power error ${RE}_{X,max}$ (Equation (14));
• Seasonal errors: ΔQ_season, ΔE_el, and ΔSCOP obtained by integration at hourly intervals;
• Edge sensitivity: MAE_cold and MAE_hot—calculated for the 500 coldest and warmest hours.

Acceptance threshold $\left|{RE}_{Q_{season}}\right|\le 2\%$ was adopted as the criterion for selecting the lowest sufficient order of the polynomial. This ±2% band is an operational planning tolerance, paired with tail metrics (MAE_cold/MAE_hot, WFE_fill, and RE_X,max) to guard extremes. The thresholds reflect practical KPI stability rather than formal significance tests and are read as stability bands, as shown in Section 3.2, Section 3.3 and Section 3.4; the joint behavior of Δ/RE and tail metrics conveys uncertainty. This criterion balances the compromise between energy accuracy and computational complexity, essential when planning scalable analyses for multiple meteorological locations or long multi-year series.

In the following section, we report a set of metrics for each order n: RMSE, MAE, R² in subsets $S\in \left\{obs,fill, all,cold,hot\right\}$ and energy indicators ΔQ_season, ΔE_el, ΔSCOP, RE_X,max. We treat Metrics MAE and RMSEs complementarily: MAE reports the average absolute deviation and is robust to outliers; RMSE penalizes large deviations and is therefore informative for tail fit and peaks [3,5]. R² summarizes variance agreement but does not separate bias from variance; hence, we always pair it with MAE/RMSE [29]. Together with Δ/RE and RE_X,max (Equations (14)–(16)), these metrics translate ta-level deviations into energy outcomes.

Root mean square error (RMSE) is defined in accordance with (8):

$${RMSE}_S\left(y_n\right)=\sqrt{{\displaystyle \frac{\sum _{\tau \in S}{\left(y_{ref}\left(\tau \right)-y_n\left(\tau \right)\right)}^2}{N_S}}}$$

(8)

The absolute error per hour was determined as Equation (9):

$${MAE}_S\left(y_n\right)={\displaystyle \frac{1}{N_S}}\sum _{\tau \in S}\left|y_{ref}\left(\tau \right)-y_n\left(\tau \right)\right|$$

(9)

where N_S = 8784 is the number of hours in the analyzed season (1 September 2019–31 August 2020). Additionally, using the flag vector $\left(\tau \right)\in \left\{\mathrm{0,1}\right\}$ (0 = measured, 1 = gap), the same pattern was applied separately to subsets “obs” and “fill”, receiving ${MAE}_{\dot{Q},n,obs}$ and ${MAE}_{\dot{Q},n,fill}$. The “all” indicator is used to assess the impact of the degree of the polynomial on the seasonal balance. At the same time, the “obs/fill” breakdown allows the source of any deviations to be located.

The Coefficient of determination R², is defined according to Equation (10):

$$R_S^2\left({\hat{y}}_i\right)=1-{\displaystyle \frac{SSE}{SST}}$$

(10)

where the sum of squares of residuals (SSE) is represented by Equation (11):

$$SSE=\sum _{\tau \in S}{\left(y_{ref}\left(\tau \right)-y_n\left(\tau \right)\right)}^2$$

(11)

and the total sum of squares (SST) is represented by Equation (12):

$$SST=\sum _{\tau \in S}{\left(y_{ref}\left(\tau \right)-{\overline{y}}_{ref,S}\right)}^2$$

(12)

where

${\hat{y}}_i$—modelled value (prediction) at the given time index;

$y_{ref}$—actual value of the i-th data point;

$y_n$—value predicted by the model;

${\overline{y}}_{ref,S}$—average actual value, ${\overline{y}}_{ref,S}={\displaystyle \frac{1}{N_S}}\sum _{\tau \in S}{y}_{ref}\left(\tau \right)$;

$N_S$—number of observations.

In each version of R², we take $y_{ref}$ as reference values (e.g., t_a,reft or ${\dot{Q}}_{h,sh,ref}$), and $y_n$ as the corresponding values from reconstruction n; we report R² separately for S_obs, S_fill, and S_all.

The ${WFE}_{fill}$ indicator shows what part of the total MAE error comes exclusively from filled hours (percentage share of error in gaps in total MAE)—an original metric, reported alongside MAE/RMSE so as not to overestimate the accuracy obtained on well-observed parts. In this paper, we report ${WFE}_{fill}$ for the $\dot{Q}$ load error (Equation (13)), i.e., a measure of the share of error from filled hours in the total MAE; the temperature variant is not used here.

$${WFE}_{fill}={\displaystyle \frac{N_{fill}}{N_{all}}}·{\displaystyle \frac{{MAE}_{\dot{Q},n,fill}}{{MAE}_{\dot{Q},n,all}}}·100\%$$

(13)

where N_fill is the number of hours with a gap (after filling); and N_all = N is the number of hours in the season.

This indicator made it possible to determine the extent to which errors in temperature reconstruction result directly from missing data segments and the extent to which they result from the overall polynomial fit to the complete series. A WFE_fill value close to the proportion of hours with gaps (here, approx. 5%) means that the error “generated” by the fill is comparable in scale to the proportion of gaps themselves; significantly higher values indicate a dominant error cost in the S_fill area.

The relative peak error (RE_X,max) is defined as Equation (14):

$${RE}_{X,max,n}={\displaystyle \frac{{\mathrm{X}}_{max,n}-{\mathrm{X}}_{max,ref}}{{\mathrm{X}}_{max,ref}}}·100\%$$

(14)

where X_max,ref is the maximum in the reference series; and X_max,n—maximum in the reconstructed series (evaluated at the same time step).

In the following section, we use abbreviated notation ${RE}_{X,max}$, where $X\in \left\{{\dot{Q}}_{h,sh,max}, N_{el,max}, {COP}_{min}\right\}$. A positive sign indicates an overestimation relative to the reference (see Equation (14)). This convention expresses the deviation in the modeled variable X concerning the reference series; alternatives normalized by X_max,n were rejected to avoid ambiguity.

For L_WT = {35, 45} °C we report (i) $∆E_{el}=\sum P_{el,n}-\sum P_{el,ref}$, RE_X, (ii) ΔP_el,max and RE_X,max, (iii) ΔSCOP and RE_SCOP—separately for each L_WT, because the COP map and power limitations depend on the supply level. We refer SCOP and its interpretation to the standard definition (test framework and seasonal aggregation method) [19].

Seasonal differences (absolute values Δ) are defined in accordance with Equation (15):

$${∆\mathrm{X}}_n={\mathrm{X}}_n-{\mathrm{X}}_{ref}$$

(15)

where X means one of the full-season quantities: electricity E_el, usable energy/heat Q_season, or derivative indicators (i.e., SCOP).

ΔX_n is reported in parallel with RE_X,n, which provides an absolute and relative scale of seasonal differences and facilitates the assessment of the significance of differences from an operational perspective.

The relative seasonal error (RE_X) is represented by Equation (16):

$${RE}_{X,n}={\displaystyle \frac{{\mathrm{X}}_n-{\mathrm{X}}_{ref}}{{\mathrm{X}}_{ref}}}·100\%$$

(16)

For RE_X,n, we use the same convention (Equation (14)): positive values indicate overestimation relative to the reference, negative values indicate underestimation. In practice, we use Equation (16) for X = Q_season, where Q_season is the sum of hourly power $\sum _{i=1}^N{\dot{Q}}_{h,sh}\left(\tau \right)$ in the season (see Equation (5)).

The MAE in the tails of the distribution (cold/hot) is defined sequentially according to Equations (17) and (18):

$${MAE}_{n,cold}={\displaystyle \frac{1}{N_{cold}}}\sum _{\tau \in S_{cold}}\left|\mathrm{y}\left(\tau \right)-{\hat{y}}_n\left(\tau \right)\right|$$

(17)

$${MAE}_{n,hot}={\displaystyle \frac{1}{N_{hot}}}\sum _{\tau \in S_{hot}}\left|\mathrm{y}\left(\tau \right)-{\hat{y}}_n\left(\tau \right)\right|$$

(18)

where S_cold and S_hot are sets of hours comprising the 500 coldest and 500 warmest hours of the season, respectively. E = 500 ≈ 6% of the season, corresponding to approximately three weeks in the cold/warm tail.

Their complementarity justifies the selection of metrics: MAE and RMSE measure absolute error in different standards, R² assesses the conformity of the curve shape and energy indicators (ΔQ_season, ΔE_el, and ΔSCOP), and RE_X,max allows temperature accuracy to be translated into practical effects for the ASHP model. We treat evaluation thresholds (e.g., ±2% for ΔQ_season) as the selection criterion of the “lowest sufficient” order of the n-th degree of the polynomial. We control sensitivity to the extremes of the distribution by MAE_cold and MAE_hot calculated for the 500 coldest and 500 warmest hours, respectively (S_cold and S_hot).

The metrics and scenarios defined above (L_WT = {35, 45} °C) are used in the Section 3, comparing orders n = 3…11 statistically (obs/fill/all, “cold/hot”), and energetically (ΔQ_season, ΔE_el, ΔSCOP, and RE_X,max), which allow for identification of the lowest polynomial order that meets the criterion $\left|{RE}_{Q_{season}}\right|\le 2\%$.

3. Results

This chapter presents the impact of the degree of the polynomial n used to reconstruct the hourly outdoor temperature profile on the energy indicators of the air-source heat pump (ASHP) model in two operating regimes L_WT. The analysis is performed in the subsets S_obs, S_fill, and S_all, and the boundary zones S_cold and S_hot, according to the definitions and workflow described in Section 2. The device characteristics COP(t_a) and P_el(t_a) were approximated by exponential functions calibrated on catalog data. The following sections present, in order: (I) the quality of temperature reconstruction (MAE, RMSE, and R²) as a function of n; (II) the consequences for the thermal loads and peak power of the building/device; and (III) the seasonal effects (E_el, SCOP, and P_el,max) in both L_WT regimes. The selection of the “lowest sufficient” order was based on the acceptance threshold for $\left|{RE}_{Q_{season}}\right|$ from Section 2. Graphical summaries accompany the tables: Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 cover RMSE/MAE/R² and Δ summaries; Figure 14 and Figure 15 present seasonal Δ/RE (Q_season, E_el, and SCOP) for both L_WT regimes; Figure 16 and Figure 17 show peak-power RE_X,max, and WFE_fill.

3.1. Accuracy of Temperature Profile Reconstruction t_a

This section summarizes the accuracy of the hourly outdoor temperature profile reconstruction t_a as a function of the polynomial degree n = 3…11. The presentation includes metrics. The presentation contains metrics RMSE and MAE calculated in parallel in subsets S_obs, S_fill, and S_all, and an analysis of the distribution “tails” S_cold and S_hot. The additional graph shows the R². A summary of extreme deviations is provided at the end of the section Δ = [Δ_avg, Δ_min, Δ_max]. To illustrate chronology after mapping, Figure 8a shows the full-year time-domain overlay of t_a,ref(τ), t_a,7(τ), and t_a,obs(τ); Figure 8b zooms into a representative real gap (324 h), shading the missing interval and showing overlaps at both ends.

Figure 9 shows the RMSE curve as a function of n; errors for measurement points were reported separately (RMSE_obs), for refilled points (RMSE_fill), and for the entire series (RMSE_all). RMSE plateaus by n ≈ 7 in all subsets; the most significant gains occur from n = 4→5, then diminish.

Figure 8. Time-domain views of the reconstruction: (a) Full-year overlay: thin line = t_a,ref(τ), dashed line = t_a,7(τ), markers = t_a,obs(τ). (b) Zoom across a representative real gap (shaded) with overlaps at both ends; alignment at gap boundaries illustrates mapping back to time.

Figure 8. Time-domain views of the reconstruction: (a) Full-year overlay: thin line = t_a,ref(τ), dashed line = t_a,7(τ), markers = t_a,obs(τ). (b) Zoom across a representative real gap (shaded) with overlaps at both ends; alignment at gap boundaries illustrates mapping back to time.

Figure 9. Dependence of RMSE on the degree of the polynomial n.

Figure 9. Dependence of RMSE on the degree of the polynomial n.

Figure 10 compares MAE as a function of n for the same subsets, using the same notation convention. MAE follows the same trend as RMSE; S_fill converges slightly later than S_obs.

Figure 10. MAE is in the function n for S_obs, S_fill, and S_all.

Figure 10. MAE is in the function n for S_obs, S_fill, and S_all.

Figure 11 and Figure 12 show the MAE values for the season’s 500 coldest and 500 warmest hours, respectively. Cold-tail MAE stabilizes for n ≥ 7; improvements beyond n = 7 do not materially affect energy KPIs. Hot-tail MAE is low across n and stabilizes early; changes above n = 7 are negligible.

Figure 11. MAE for “cold tail” (first 500 h of the season): S_obs,cold, S_fill,cold, S_all,cold in the function n.

Figure 11. MAE for “cold tail” (first 500 h of the season): S_obs,cold, S_fill,cold, S_all,cold in the function n.

Figure 12. MAE for “warm tail” (last 500 h of the season): S_obs,hot, S_fill,hot, S_all,hot in the function n.

Figure 12. MAE for “warm tail” (last 500 h of the season): S_obs,hot, S_fill,hot, S_all,hot in the function n.

Figure 13 presents R² calculated separately for S_obs, S_fill, and S_all as a function of n. R² remains > 0.99 for all degrees; hence, Δ/RE metrics drive the degree selection rather than R².

Figure 13. R² in the function n.

Figure 13. R² in the function n.

A summary description of the values Δ = [Δ_avg, Δ_min, Δ_max] is provided in Figure 14. In this study, $∆\left(\tau \right)=t_{a,n}\left(\tau \right)-t_{a,ref}\left(\tau \right)$; Δ_avg was assumed; Δ_avg denotes the average for the entire season (N = 8784 h), while Δ_min and Δ_max denote the extreme values of the difference within the season, respectively. Seasonal mean bias is slight; extremes narrow with n and stabilize by n ≈ 7.

Figure 14. Absolute differences (Δ) t_a between reconstruction and reference in the n-th degree function for the entire seasonal series (Sall, N = 8784 h). Different positive values indicate a warmer reconstruction than the reference. (The graph supplements the “cold/hot” summaries, average shift, and range of extremes for the entire season.).

Figure 14. Absolute differences (Δ) t_a between reconstruction and reference in the n-th degree function for the entire seasonal series (Sall, N = 8784 h). Different positive values indicate a warmer reconstruction than the reference. (The graph supplements the “cold/hot” summaries, average shift, and range of extremes for the entire season.).

3.2. The Impact of n on the Thermal Load of a Building

This section presents the effects of the degree of the polynomial n on the thermal load of a building. The peak heat demand value ${∆\dot{Q}}_{h,sh,max}$ was compared with the seasonal effective energy ΔQ_season determined from profiles $t_{a,n}\left(\tau \right)$ for n = 3…11. For comparability, the maximums arise at the same time of the season S_obs (3488 h), which allows for a direct comparison of peak values, and the results are presented in pairs Δ/RE (

Table 3. Peak heat demand ${\dot{Q}}_{h,sh\_max,ref}$ and seasonal deviation of usable energy ΔQ_season in the function of degree n. Reference and reconstructed values and the absolute (Δ) and relative difference (RE) are presented.

n	${\dot{\mathit{Q}}}_{\mathit{h},\mathit{s}\mathit{h}\_\mathit{m}\mathit{a}\mathit{x},\mathit{r}\mathit{e}\mathit{f}}$	${\dot{\mathit{Q}}}_{\mathit{h},\mathit{s}\mathit{h}\_\mathit{m}\mathit{a}\mathit{x},\mathit{n}}$	${∆\dot{\mathit{Q}}}_{\mathit{h},\mathit{s}\mathit{h},\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{R}\mathit{E}}_{{\dot{\mathit{Q}}}_{\mathit{h},\mathit{s}\mathit{h},\mathit{m}\mathit{a}\mathit{x}}}$	$∆{\mathit{Q}}_{\mathit{s}\mathit{e}\mathit{a}\mathit{s}\mathit{o}\mathit{n}}$	${\mathit{R}\mathit{E}}_{{\mathit{Q}}_{\mathit{s}\mathit{e}\mathit{a}\mathit{s}\mathit{o}\mathit{n}}}$
	kW	kW	kW	%	kWh	%
3	7.742	6.885	−0.858	−11.08	−367.3	−1.64
4	7.742	6.683	−1.059	−13.68	−403.8	−1.81
5	7.742	7.189	−0.553	−7.15	−291.2	−1.30
6	7.742	7.086	−0.656	−8.47	−276.5	−1.24
7	7.742	7.264	−0.479	−6.18	−296.4	−1.33
8	7.742	7.251	−0.491	−6.35	−297.9	−1.33
9	7.742	7.386	−0.357	−4.61	−290.4	−1.30
10	7.742	7.354	−0.388	−5.02	−286.8	−1.28
11	7.742	7.332	−0.410	−5.30	−284.9	−1.27

Note: positive RE = overestimation relative to the reference; negative RE = underestimation.

).

Figure 15 shows the relative errors ${RE}_{{\dot{Q}}_{h,sh,max}}$ and ${RE}_{Q_{season}}$ in the function of degree n, with lines ±2% indicating the acceptance band for $\left|{RE}_{Q_{season}}\right|$. Seasonal Δ and RE for Q_season lie within planning-level tolerances for all n; changes beyond n = 7 are immaterial.

Figure 15. RE in the function n: ${RE}_{{\dot{Q}}_{h,sh,max}}$ and ${RE}_{Q_{season}}$ on a shared Y-axis. The dashed lines ±2% indicate the acceptance band $\left|{RE}_{Q_{season}}\right|$; where RE_Q_max ≡ ${RE}_{{\dot{Q}}_{h,sh,max}}$.

Figure 15. RE in the function n: ${RE}_{{\dot{Q}}_{h,sh,max}}$ and ${RE}_{Q_{season}}$ on a shared Y-axis. The dashed lines ±2% indicate the acceptance band $\left|{RE}_{Q_{season}}\right|$; where RE_Q_max ≡ ${RE}_{{\dot{Q}}_{h,sh,max}}$.

All ${RE}_{Q_{season}}$ values fall within the range of ±2%; the smallest absolute value was recorded for n = 6 (−1.24%).

3.3. Impact of n on ASHP Energy Indicators

Variations in E_el, SCOP, P_el,max, and COP_min, are presented in

Table 4. ASHP energy indicators are a degree n function for two LWT regimes; Δ values are absolute; RE values are relative (Equations (14)–(16)).

LWT	n	ΔEel	${\mathit{R}\mathit{E}}_{{\mathit{E}}_{\mathit{e}\mathit{l}}}$	ΔPel,max	${\mathit{R}\mathit{E}}_{{\mathit{P}}_{\mathit{e}\mathit{l},\mathit{m}\mathit{a}\mathit{x}}}$	ΔSCOP	${\mathit{R}\mathit{E}}_{\mathit{S}\mathit{C}\mathit{O}\mathit{P}}$	ΔCOPmin	${\mathit{R}\mathit{E}}_{{\mathit{C}\mathit{O}\mathit{P}}_{\mathit{m}\mathit{i}\mathit{n}}}$
°C		kWh	%	kW	%	[-]	%	[-]	%
35	3	−123.0	−2.07	−0.471	−17.73	0.0279	0.71	0.2355	8.08
	4	−131.7	−2.22	−0.574	−21.59	0.0313	0.80	0.2936	10.08
	5	−97.3	−1.64	−0.311	−11.69	0.0222	0.57	0.1499	5.14
	6	−94.4	−1.59	−0.366	−13.76	0.0208	0.53	0.1785	6.13
	7	−99.8	−1.68	−0.270	−10.17	0.0225	0.58	0.1292	4.44
	8	−100.2	−1.69	−0.277	−10.42	0.0226	0.58	0.1327	4.55
	9	−97.8	−1.65	−0.203	−7.65	0.0220	0.56	0.0958	3.29
	10	−96.9	−1.63	−0.221	−8.30	0.0217	0.56	0.1044	3.58
	11	−96.5	−1.63	−0.233	−8.76	0.0215	0.55	0.1104	3.79
45	3	−7.2	−0.44	−0.017	−6.48	0.0148	0.41	0.1682	6.93
	4	−7.2	−0.44	−0.021	−7.95	0.0150	0.42	0.2095	8.63
	5	−7.1	−0.43	−0.011	−4.23	0.0153	0.42	0.1073	4.42
	6	−7.1	−0.43	−0.013	−5.00	0.0153	0.43	0.1276	5.26
	7	−7.1	−0.43	−0.010	−3.67	0.0153	0.43	0.0925	3.81
	8	−7.1	−0.43	−0.010	−3.77	0.0153	0.43	0.0950	3.91
	9	−7.1	−0.43	−0.007	−2.75	0.0154	0.43	0.0686	2.83
	10	−7.1	−0.43	−0.008	−2.99	0.0154	0.43	0.0748	3.08
	11	−7.1	−0.43	−0.008	−3.16	0.0154	0.43	0.0791	3.26

and Figure 16. Seasonal Δ and RE for E_el and SCOP stabilize from n ≈ 5; for P_el,max and COP_min from n ≈ 7; changes above 7 for all trends are insignificant.

Figure 16. RE vs. polynomial degree n for two L_WT regimes (35 °C, 45 °C): (a) ${RE}_{E_{el}}$, (b) ${RE}_{P_{el,max}}$, (c) RE_SCOP, (d) ${RE}_{{COP}_{min}}$.

Figure 16. RE vs. polynomial degree n for two L_WT regimes (35 °C, 45 °C): (a) ${RE}_{E_{el}}$, (b) ${RE}_{P_{el,max}}$, (c) RE_SCOP, (d) ${RE}_{{COP}_{min}}$.

The differences visible in the charts above and their implications are discussed in Section 4.

3.4. Sensitivity of the Method to Cold/Hot “Tails”

Since the technique is most sensitive at the extremes of the temperature distribution, the sensitivity of the method was assessed in the boundary zones of the temperature distribution: S_cold and S_hot. The MAE was calculated separately in S_obs, S_fill, and S_all.

For S_cold, the values of ${MAE}_{{\dot{Q}}_{h,sh}}$ in low n, the most significant value decreases significantly at n ≥ 7. S_hot is also stabilizing starting from n ≥ 7. Detailed MAE waveforms in the “tails” are shown in Figure 17. The local maximum separation of the curve S_fill in relation to S_obs and S_all for n = 4 is shown in Figure 17b.

The WFE_fill indicator synthesizes the global share of error from hours supplemented in MAE_all for ${\dot{Q}}_{h,sh}$. The WFE_fill values close to the proportion of gaps (N_fill/N_all) mean that the error from gaps does not dominate the total error. Significantly higher values indicate that errors in the S_fill subset increase the total error (MAE_all). In this sample, the smallest WFE_fill (i.e., closest to the N_fill/N_all line) was obtained for n = 4, 7, and 8 (Figure 18), which is consistent with the fact that the S_hot tail covers only 5.1% of the season and does not determine the share of error on an annual basis.

Figure 17. MAE for ${\dot{Q}}_{h,sh}$ in the edge zones of the distribution—(a) S_cold, (b) S_hot—as a function of the degree of the polynomial n. Three curves are shown: S_obs, S_fill, and S_all.

Figure 17. MAE for ${\dot{Q}}_{h,sh}$ in the edge zones of the distribution—(a) S_cold, (b) S_hot—as a function of the degree of the polynomial n. Three curves are shown: S_obs, S_fill, and S_all.

Figure 18. WFE_fill indicator as a function of degree n for ${\dot{Q}}_{h,sh}$. Baseline $\left(N_{fill}/N_{all}\right)·100\%$ as a reference point (proportion of completed hours).

Figure 18. WFE_fill indicator as a function of degree n for ${\dot{Q}}_{h,sh}$. Baseline $\left(N_{fill}/N_{all}\right)·100\%$ as a reference point (proportion of completed hours).

3.5. Choosing the Lowest Sufficient Degree n

The selection of the lowest sufficient degree was based on four criteria:

• I—error acceptance threshold $\left|{RE}_{Q_{season}}\right|\le 2\%$. Those that meet this criterion are identified.
• II—threshold for stabilizing energy indicators. For each $X\in \left\{{RE}_{E_{el}},{RE}_{P_{el,max}},{ER}_{SCOP},{RE}_{{COP}_{min}}\right\}$ and both regimes, L_WT = 35 °C, 45 °C, the incremental change is compared ${RE}_X\left(n\right)=\left|{RE}_X\left(n\right)-{RE}_X\left(n-1\right)\right|$ → ${RE}_{E_{el}}\le 0.15 pp$, ${RE}_{P_{el,max}}\le 2.0 pp$, ${ER}_{SCOP}\le 0.10 pp$, ${RE}_{{COP}_{min}}\le 1.0 pp$ for both L_WT and is marked as follows (pp—percentage points):
  • (✓)—if the criteria are met simultaneously;
  • (△)—if only one of the thresholds is exceeded by 50%;
  • (—)—all other cases.
• III—stabilization of ${MAE}_{\dot{Q}}$ for tails in S_all and is marked as follows:
  • (✓)—if $\left|{MAE}_{all}\left(n\right)-{MAE}_{all}\left(n-1\right)\right|\le 0.05 kW$ AND relatively ≤ 5%;
  • (△)—if $\left|{MAE}_{all}\left(n\right)-{MAE}_{all}\left(n-1\right)\right|\le 0.10 kW$ OR relatively ≤ 10%;
  • (—)—all other cases.
• IV—contribution of missing data WFE_fill. For b = (N_fill/N_all)⋅100% it is marked as follows:
  • (✓)—if $\left|{WFE}_{fill}\left(n\right)-b\right|\le 1.0 pp$;
  • (△)—if $\left|{WFE}_{fill}\left(n\right)-b\right|\le 2.0 pp$;
  • (—)—all other cases.

The thresholds discussed are operational and were adopted a priori based on two premises: (I) stabilization of the curves in Figure 15 and Figure 16, and (II) energy metric levels. The thresholds do not result from statistical significance tests, but reflect the practical level of the model’s insensitivity to further increases in n. The fulfillment of the four decision premises from Figure 14, Figure 15, Figure 16 and Figure 17 is summarized in

Table 5. Fulfilment of the conditions for selecting the “lowest sufficient” degree n according to Criteria I-IV: “✓“ fulfilled, “△“ on the borderline, “—“ not fulfilled.

n	Criteria
	I	II	III	IV
3	✓	—	—	△
4	✓	—	—	✓
5	✓	△	—	△
6	✓	△	△	—
7	✓	✓	✓	✓
8	✓	✓	✓	✓
9	✓	✓	✓	△
10	✓	✓	✓	△
11	✓	✓	✓	△

.

Within this study’s scope, all considered n meet $\left|{RE}_{Q_{season}}\right|\le 2\%$ yet n = 7 is the first degree at which conditions I–IV are met simultaneously (Table 5). For n = 7, all reported KPIs fall within the tolerances, and increments to n > 7 produce no material changes, indicating diminishing returns and a higher overfitting risk, as low degrees underfit the cold-tail slope. In contrast, higher degrees mainly reshape tails with little effect on seasonal integrals.

4. Discussion

The consequences of selecting the reconstruction degree n for ASHP energy indicators in two LWT regimes are discussed below, considering the analysis of temperature distribution and the impact of data reconstruction. The discussion is based on criteria I–IV (Figure 14, Figure 15, Figure 16 and Figure 17; Table 5) and the metrics definitions from Section 2.6.

4.1. Quality of Temperature Reconstruction and Energy Effect

The obtained profiles $t_{a,n}\left(\tau \right)$ ensure high compliance with the reference series, with the quality increment decreasing significantly for n ≥ 7. The translation into building load is confirmed by $\left|{RE}_{Q_{season}}\right|\le 2\%$ for all analyzed degrees and slight changes in ${RE}_{E_{el}}$ and ${ER}_{SCOP}$. This means that further increasing the n above the average values does not bring any material energy benefits in the case under consideration.

Increasing the degree of the polynomial improves the representation of daily and weekly variability in the profile $t_a\left(\tau \right)$, which directly translates into stabilization of MAE/RMSE in the S_obs, S_fill, and S_all sets. The most significant increase in quality occurs when moving from n = 4 to n = 5, while for n ≥ 7, the benefits diminish.

Very high consistency (R² > 0.99) was maintained throughout the studied range. From the building energy efficiency point of view, the smoothness of the entire $t_a\left(\tau \right)$ distribution is crucial, rather than the local adjustment of individual episodes. Hence, all the reconstructions considered meet the acceptance band $\left|{RE}_{Q_{season}}\right|\le 2\%$; ${RE}_{E_{el}}$ and ${ER}_{SCOP}$ remain low. Further increases in n do not translate into measurable energy gains in the analyzed case.

This study does not benchmark alternative gap-filling families (linear/spline, ARIMA, ML). In this case, the contribution is orthogonal: the degree of the polynomial n was isolated as a single design variable in an ordered domain, and its selection was made repeatable through KPI propagation. Comparative baselines are left for follow-on replication using the same decision rule.

4.2. L_WT Regimes and KPI Sensitivity

Differentiation of L_WT modifies the sensitivity of indicators to reconstruction $t_a$: for L_WT = 45 °C, values of ${RE}_{E_{el}}$ are similar to −0.43%, while for L_WT = 35 °C, deviations in ${RE}_{E_{el}}$ and ${RE}_{P_{el,max}}$ are clearly larger. At the same time ${ER}_{SCOP}$ and ${RE}_{{COP}_{min}}$ remain within narrow ranges for both regimes, which confirms the stability of overall efficiency with respect to the choice of n. Interpretatively, this means that at lower supply temperatures, changes in the load profile resulting from reconstruction $t_a$ are more easily “visible” in seasonal energy and peak power, while at 45 °C, remain secondary in nature. At the same time, ${ER}_{SCOP}$ and ${RE}_{{COP}_{min}}$ remain within narrow ranges for both regimes, confirming the stability of the device’s overall efficiency concerning the choice of n. Energy sensitivity varies significantly between regimes. For 35 °C, ranges were recorded: ${RE}_{E_{el}}\in \left[-2.22\%, -1.63\%\right]$, ${RE}_{P_{el,max}}\in \left[-21.59\%,-7.65\%\right]$, in narrow ranges of ${RE}_{SCOP}$ and ${RE}_{{COP}_{min}}$. For 45 °C, the range is ${RE}_{P_{el,max}}\in \left[-7.59\%,-2.75\%\right]$. These quantitative differences confirm that the selection of n has a stronger “transfer” effect on the energy and peak power in the system L_WT = 35 °C than in the system L_WT = 45 °C.

The rooms are heated exclusively by underfloor heating, so most of the ASHP’s operating hours fall within the regime L_WT = 35 °C, while L_WT = 45 °C mainly concerns episodic DHW preparation cycles. In this sense, greater sensitivity of indicators at 35 °C has greater operational significance than the differences observed at 45 °C.

4.3. Edge Sensitivity and the Contribution of Complementary Hours Error

Analysis of the “tails” of the temperature distribution confirms that ${MAE}_{{\dot{Q}}_{h,sh}}$ stabilizes for n ≥ 7 for S_cold and S_hot. Visible local deviation of the S_fill in S_hot for selected degrees n, it does not escalate the whole-season error, which is reflected in the indicator WFE_fill (values closest to N_fill/N_all baseline are for n = 4, 7, 8). With a “warm tail” of ~6% of the season, the contribution of the error from the hours filled in remains secondary to ${MAE}_{all}$. Therefore, raising the degree of the polynomial above 7 does not significantly reduce the edge sensitivity. This pattern supports the ordered-curve reconstruction: for n ≥ 7 the edge-sensitivity metrics stabilize despite the lack of explicit time-series dependence.

As humidity- and defrost-related transients are not explicitly simulated, cold-tail indicators (e.g., peak electrical power and minimum COP) should be interpreted with this modeling scope in mind. In the present case, these effects may modestly accentuate cold-tail deviations without altering the qualitative patterns discussed above. A simple sensitivity (cold-tail COP penalty and auxiliary cut-in) leaves the n ≈ 7 degree selection unchanged in directionality, while increasing $\left|{RE}_{P_{el,max}}\right|$ as expected.

4.4. Justification for Selecting the Minimum Sufficient Degree n

The lowest degree of the polynomial n was accepted when the thresholds were met in both L_WT regimes. The first degree at which criteria I–IV were simultaneously met is n = 7. Further increasing the degree does not bring about any significant improvement.

Against time-series baselines (e.g., ARIMA/splines/ML), the ordered-curve fit is preferable here because gaps are short and ASHP relations are memoryless; under dense/extended gaps, time-series or reanalysis debiasing would become preferable. The degree-selection criterion applies to the polynomial reconstruction implemented in the previously published ASHP planning tool [1,2]. The observed saturation at n ≈ 7 reflects that short, scattered gaps and memoryless P_el(t_a)/COP(t_a) relations make KPI integrals depend mainly on the temperature distribution; beyond this point, extra degrees primarily reshape tails with negligible KPI impact. Replication across additional climates (including warm/cold subperiods) and ASHP types is planned; using a single climate year here keeps the analysis compact and makes the criterion auditable before replication.

Mechanistically, the saturation at n ≈ 7 follows how degree n reshapes the ordered temperature curve. For n ≤ 5, the cold tail is overly smoothed, which reduces the frequency of very low ta and leads to underestimation of ${\dot{Q}}_{h,sh,max}$ and P_el,max; for n ≥ 10, local Runge-type oscillations may appear, artificially shifting hours near control thresholds. The stabilization of MAE_cold/MAE_hot and the plateaus of ${RE}_{E_{el}}$, ${ER}_{SCOP}$, and ${RE}_{P_{el,max}}$ observed from n ≥ 7 (see Figure 15, Figure 16 and Figure 17) indicate that this is the lowest degree at which tail bias and oscillation risk are simultaneously controlled within the ±2% planning band.

The “lowest sufficient degree via KPI propagation” principle is generic and can inform other analytics pipelines (e.g., battery SOC estimation), provided the domain-specific KPIs and tails are defined.

4.5. Limitations and External Baselines (Method Positioning)

Time-series-aware baselines (splines on time, ARIMA/state-space, ML, and reanalysis debiasing) are valuable when chronology carries predictive information or missingness is dense. Here, gaps are short and scattered (5.1%), and P_el(t_a)/COP(t_a) are memoryless maps; therefore, KPI integrals are governed mainly by the ordered distribution of t_a. Under this regime, the degree-selection criterion is based on ΔQ_season, ΔE_el, SCOP, ${RE}_{P_{el,max}}$, MAE_cold/MAE_hot, and WFE_fill, which identify n = 7 as the lowest sufficient degree. Broader comparisons are planned in the upcoming multi-climate replication that reuses the same criterion.

5. Conclusions

Based on the analysis of the results, the minimum sufficient degree of reconstruction of temperature data n was identified, and the sensitivity of ASHP energy indicators to n in two L_WT regimes was determined. The most important practical conclusions and implications for the selection of n in similar applications are summarized below:

• Degree n = 7 is the lowest sufficient according to the criteria I–IV;
• Since in a building with underfloor heating, most of the ASHP’s work is carried out in a regime of 35 °C, the selection of n = 7 is conservative and practical: it stabilizes KPIs in a time-dominant regime, limiting the risk of seasonal deviations in energy and peak power;
• KPIs for 35 °C were more sensitive to n than for 45 °C; for 45 °C, values of ${RE}_{E_{el}}$ remained at around −0.43%;
• Edge sensitivity stabilized from n ≥ 7; the share of error from complementary hours WFE_fill approached the share of gaps at n = 4, 7, 8;
• Further increases in n above seven did not yield significant energy gains;
• The proposed criteria (I–IV) create a practical procedure for selecting the minimum degree of reconstruction, repeatable in both regimes. L_WT.

The established criterion is ready for replication across climates (including warm/cold subperiods) and ASHP types, and can be combined with the economic indicators introduced in [1,2]. The degree-selection rule is data-driven and transferable: choose the lowest n at which seasonal KPIs (ΔQ_season, ΔE_el, and SCOP) and tail metrics (MAE_cold/MAE_hot, WFE_fill, and RE_X,max) stabilize within planning-level tolerances. Maritime climates (narrower temperature spread) are expected to stabilize at lower n, while continental/arid or highly variable climates (wider spread, heavier tails) may require slightly higher n to control tail bias. Multi-site replication will follow using the same criterion.

Potential applications are as follows: (i) transparent preprocessing for multi-site ASHP planning; (ii) utility-grade QA of meteorological feeds; (iii) replicable degree selection for long multi-year series. The limitations are as follows: catalog-map device modeling without explicit defrost/humidity physics may understate tail penalties; multi-climate replication and time-series baselines are planned.