Heat load factors
The heat load factor (à) is a dimensionless ratio that represents the residential heat demand (L Total) compared to the maximum possible heat output (P Max) over a specified time period (∆) This metric is essential for understanding energy efficiency in residential heating systems.
P Max ã∆ (1) where the total residential heat demand,L Total , is the sum of the individual space heating and hot water components,L Space Heat andL Hot Water , respectively:
L Total =L Space Heat +L Hot Water
The decentralized nature of heating leads to a lack of readily available consumption data, limiting further research opportunities The theory of heating degree-days (HDD) is commonly used in literature, such as in studies by Berger and Worlitschek (2018) and Christenson et al (2006), as a proxy for daily heat demand variations This study posits a direct proportionality between total residential space heat demand (L Total) and heating degree-days (HDD Space Heat).
L Total =L Space Heat +L Hot Water (2)
=αãHDD Space Heat +L Hot Water αis a constant of proportionality in units of energy per heating degree-day Inspired by Kozarcanin et al.
(2019), the accumulated heating degree-days, HDD SpaceHeat ∆,x , for a single grid location, x, over a period of time,
The expression (T 0−T x (t))+ is defined as a positive value or zero, as noted by Thom (1954) Specifically, when T 0 exceeds T x (t), the result contributes to the heating degree-days Conversely, if T 0 is less than or equal to T x (t), the value is set to zero.
The base temperature, T0, refers to the external temperature at which a building is considered to require heating, typically set at 17 °C for simplicity However, research indicates that this value may differ based on regional factors and specific studies (Kozarcanin et al., 2019) The variable Tx(t) represents the gridded time series of ambient air temperature.
The maximum heat output, P Max, is directly related to the total heat consumption, which includes the maximum output for space heating, P Space Heat, and hot water, P Hot Water.
P Max =P Space Heat +P Hot Water (4)
The maximum output of hot water (P Hot Water) is equal to the normalized consumption of hot water (L Hot Water) adjusted to a specific threshold (∆) Similarly, P Space Heat is defined in relation to L Space Heat The design temperature (T x, design) for the system is determined as the 0.05% quantile of the gridded daily ambient temperature (T x(t)).
To ensure optimal operation above the design temperature, a small quantile is utilized, achieving this in 95.95% of cases Specifically, a 0.05% quantile indicates an expected operational time of about 5 hours annually Utilizing a 100% quantile could lead to an overestimation of the technology's capacity, resulting in unnecessary increases in capital investments.
Finally, replacingL Total andP Max in Eq 1 by the expressions given in Eq 2 and 4 leads to: à x αãHDD Space Heat ∆,x +L Hot Water ³ α Ă T 0 − T x, design  ã 1day
This can be simplified by removing the∆in the denominator and dividing both the numerator and denomina- tor byαas: à x HDD Space Heat ∆,x + L
Hot Water α ¡T 0 −T x, design ¢ ã1day+ L Hot Water α
Due to the lack of available measurements for hot water consumption that align with the needs of this study, L Hot Water is utilized as a proxy, represented in heating degree-days This approach aims to provide the most accurate approximation for analyzing hot water usage.
In this study, we analyze measured data on hot water and space heating consumption in Stockholm, as detailed by Levihn (2018) Our primary objective is to determine the ratio of hot water usage to space heating in the city, which will enable us to make informed estimates regarding overall energy consumption patterns.
L Hot Water α Based on the measured consumption data, the ratio of hot water to space heat is 26%, i.e.:
L Hot Water Stockholm =0.26ãL Space Heat Stockholm
L Hot Water Stockholm =0.26ãαãHDD Space Heat ∆,Stockholm
L Hot Water Stockholm α =0.26ãHDD Space Heat ∆,Stockholm
L Hot Water Stockholm α can be estimated using Equation 3, taking into account the duration of time If data from other regions is available, it can similarly be applied to estimate the heating degree-day proxy for hot water consumption We also assume that hot water consumption remains constant over time and space, meaning that L Hot Water α is uniform across all grid locations, x, and is set to the value specific to Stockholm.
Techno-economic standpoint of heat generation
The hourly accumulated cost,X x,θ TOT , for a technology,θ, and grid location, x, depends linearly on the heat load factor,à x , as:
The capital expense, denoted as X CAP θ, is directly proportional to the installed capacity, κ θ, according to Equation 6 The costs associated with equipment, installation, and maintenance per MW are represented by x κ θ, x I θ, and x FM θ, respectively To account for the capital cost, it is annuitized over the technology's lifespan using a discount rate of 4%.
The marginal expense, X OP x,θ , is proportional to the installed capacity,κ θ as well as the ratio between the fuel price, x Fuel θ , and efficiency, effx,θ :
The efficiency, denoted as η θ, remains constant for technologies other than heat pumps, represented by eff x,θ For heat pumps, the efficiency is indicated by COPx,θ(t), where θ specifies the type of heat pump and t denotes time Detailed information on prices and technology characteristics can be found in Table 1 of the main paper.
Coefficient of performance (COP)
Heat pumps operate with a Coefficient Of Performance (COP), which measures the ratio of heat output to electricity input and is significantly influenced by temperature variations Long-term average COP values are not reliable due to this temperature dependency An empirical formula relates COP to the temperature difference between the heat source and heat sink (∆T = T sink - T source), specifically for air and ground-based heat pumps, as derived from Staffell et al (2012) and updated with new data from NTB Buchs (2019) The COP for air source heat pumps varies depending on whether defrosting is necessary, which occurs when outdoor temperatures drop below 5°C, resulting in a 4% decrease in COP For both air and ground source heat pumps, T source represents the gridded air and soil temperatures, with ground temperature calculated as the average air temperature over a 20-year period, reflecting conditions at approximately 50 meters below ground level T sink is set at 30°C for air-to-air heat pumps and 55°C for large area hot water heating, as noted by Staffell et al (2012).
COPAir driven, x(t)(0.0012∆ T 2 − 0.1702∆ T + 7.855 if T air≤5°C 0.0019∆ T 2 − 0.2258∆ T + 9.073 if T air>5°C
During winter, the Coefficient of Performance (COP) for air-to-air and air-to-water heat pumps decreases due to high heating demands, while in summer, the COP significantly increases with lower heating needs The annual average COP for these heat pumps, denoted as COPASHP, is adjusted based on heating degree-days (HDD) for space heating However, the hot water component remains unaffected by this adjustment, as its consumption is assumed to be constant year-round In contrast, ground-to-water heat pumps do not require similar adjustments, as the temperatures at a depth of 50 meters remain consistent throughout the seasons.
HDD Space Heat x (t)ãCOPAir driven, x(t) (8)
Climate model temperature data
The combinations of regional and global climate models are shown in Tab 1.
Table 1: Overview of the CMIP5 climate models implemented into this study along with the available climate projections, RCP26, RCP45 and RCP85.
The key projections RCP2.6, RCP4.5, and RCP8.5 illustrate different climate futures, as depicted in Fig 1 RCP2.6 aligns with the Paris Agreement's goal to limit global temperature rise to well below 2°C above pre-industrial levels by the end of the 21st century, achievable through reduced CO2 emissions and concentrations In contrast, RCP8.5 forecasts a significant rise in CO2 emissions, potentially leading to a European average temperature increase of up to 5°C RCP4.5 represents a middle-ground scenario that incorporates strict climate policies, including economic penalties for CO2 emissions.
Recent studies highlight the challenges faced by Global Climate Models (GCMs) in accurately simulating near-surface air temperatures Cattiaux et al (2013) found that 33 CMIP5 GCMs exhibit negatively biased winter temperatures in Northern Europe when compared to ground observations from ECAD (Van Den Besselaar et al., 2015), while positively biased summer temperatures are noted in Eastern and Central Europe The ensemble mean bias for winter months is approximately −1°C ± 9°C, and for summer months, it is around 0.5°C ± 6°C Similar biases are observed in Northern Eurasia, with the largest discrepancies occurring during winter and summer (Miao et al., 2014) Although small improvements have been made since the CMIP3 models (Meehl et al., 2007), addressing these biases remains crucial To tackle these issues, a bias adjustment method adapted from Kozarcanin et al (2019) has been implemented to refine temperature profiles.
The IPCC climate projections reveal critical metrics regarding future climate scenarios Panel a) illustrates the anticipated CO2 emissions, measured in gigatons per year (GtCO2/yr), while panel b) indicates the resulting atmospheric CO2 concentration in parts per million (ppm) Additionally, panel c) displays the projected temperature increases for Europe, derived from the HIRHAM5-ICHEC-EC-EARTH model, with the right axis reflecting the temperature rise compared to the average European temperature from 1950 to 1970.
Extended discussions on the original pricing scheme
In this section, we further discuss the unperturbed pricing scheme and the single technology dominance across Europe.
The screening curves for various technologies, as illustrated in Fig 2, reveal that heat pump coefficients of performance vary with ambient temperatures, leading to distinct screening curves (refer to Fig 4-6 in Section 2.2) For instance, air-to-water heat pumps exhibit performance coefficients ranging from 2.0 to 3.0, resulting in defined upper and lower screening curves that outline the cost region (Fig 2) Similar trends are observed for soil-to-water and air-to-air heat pumps, while biomass, oil, and gas boilers maintain uniform efficiencies, resulting in a single screening curve The black shaded region in Fig 2 indicates the range of heat load factors across Europe, with a detailed spatial distribution presented in Fig 3 Notably, gas boilers emerge as the only cost-optimal technology within these heat load factors, prompting the implementation of a balanced pricing scheme to promote a more diverse distribution of technologies, which is essential for assessing the potential impacts of climate change on heat generation.
The annual accumulated heating costs, expressed in 1000 Euro/kW, are illustrated through screening curves that vary with the heat load factor The data utilized for technology prices and properties is sourced from Table 1 of the main article For air-to-water heat pumps (ASHP), the upper and lower screening curves correspond to coefficients of performance of 2.0 and 3.0, while ground-to-water heat pumps (GSHP) exhibit coefficients of 2.5 and 4.5 Additionally, the hybrid system combining air-to-air heat pumps and electricity-driven boilers shows combined efficiencies ranging from 3.0 to 5.0 The black-shaded region on the graph indicates the heat load factor range in Europe, confined between 0.25 and 0.50.
The heat load factors across Europe, as illustrated in Fig 3, reveal significant increases in the British Isles and Scandinavia during the historical period, primarily due to the cold oceanic climate Additionally, the Iberian Peninsula experiences high heat load factors, attributed to its warm Mediterranean climate This may appear contradictory, yet it highlights the rising temperatures affecting these regions.
To naturally decrease the need for space heating, it is essential to recognize that consistent hot water usage constitutes a substantial portion of overall heat demand, thereby elevating the heat load factors.
A thorough analysis of Fig 3 indicates that identifying a clear trend in heat load factors relative to climate change is challenging The variations in heat load factors stem from the interplay between heating degree-days and design temperature changes Notably, the Balkan countries exhibit similar heat load factors, whereas other European regions are more sensitive to shifts in ambient temperatures Under the RCP2.6 climate projection, slight temperature increases by the century's end correspond to minor adjustments in heat load factors In contrast, the RCP4.5 scenario shows a moderate reduction in heat load factors, particularly affecting the British Isles and Scandinavia The RCP8.5 projection, with its extreme temperature rise, results in substantial decreases in heat load factors in certain areas, such as the British Isles, while regions like the Iberian Peninsula and Eastern Europe remain largely unaffected.
Extended discussions on the heat pump coefficients of performance
Figures 4 to 6 illustrate the spatial distribution of performance coefficients for three types of heat pumps studied Notably, air-to-water heat pumps exhibit the lowest coefficients in Scandinavia, while the Mediterranean region shows the highest values, ranging from 2.0 to 2.7 The modest temperature rise projected for the end of the century under RCP2.6 has minimal impact on performance coefficients In contrast, the RCP4.5 and RCP8.5 scenarios indicate a substantial increase in coefficients, with RCP8.5 showing gains of up to 2.5 in Scandinavia and 3.0 in southern Iberia These enhancements are expected to significantly boost the adoption of heat pumps across Europe by the century's end, a trend also reflected in the performance of ground-to-water and air-to-air heat pumps.
When comparing the coefficients of performance (COP) of various heat pumps, ground-to-water heat pumps demonstrate significantly higher values than air-to-water heat pumps This advantage is primarily due to the stable ground temperatures that ensure a high COP throughout the year, regardless of seasonal changes Additionally, air-to-air heat pumps achieve the highest COP owing to their low sink temperature.
The spatial distributions of heat load factors are analyzed, focusing on the historical period from 1970 to 1990 The climate scenarios RCP2.6, RCP4.5, and RCP8.5 are examined for the climatic period of 2080 to 2100 These findings are derived from the ICHEC-EC-EARTH HIRHAM5 climate model.
The spatial distribution of performance coefficients for air-to-water heat pumps operating at a sink temperature of 55 °C is illustrated in Figure 4 These coefficients are adjusted based on the annual space heating demand, as outlined in Equation 8 The analysis covers a historical period from 1970 onward.
1990 RCP2.6, RCP4.5 and RCP8.5 spans a climatic period from 2080-2100 The figures are based on the ICHEC-EC-EARTH HIRHAM5 climate model.
The spatial distributions of performance coefficients for ground-to-water heat pumps operating at a sink temperature of 55 °C are illustrated, focusing on the historical period from 1970 to 1990 The analysis incorporates climate scenarios RCP2.6, RCP4.5, and RCP8.5, which project conditions for the years 2080 to 2100, utilizing data from the ICHEC-EC-EARTH HIRHAM5 climate model.
The spatial distribution of performance coefficients for air-to-air heat pumps operating at a sink temperature of 30 °C is illustrated in Figure 6 These coefficients are weighted based on the annual space heating demand, as outlined in Equation 8 The analysis covers a historical period from 1970 onwards.
1990 RCP2.6, RCP4.5 and RCP8.5 spans a climatic period from 2080-2100 The figures are based on the ICHEC-EC-EARTH HIRHAM5 climate model.