For the energy system model, a wind turbine of type Enercon E-160 EP3 E2 was selected which provides the electrical energy for operating the HP in addition to the grid connection. The electrical output is calculated hourly using the system-specific performance curve and the wind speeds from [17] extrapolated to 120 m hub height with Eq. (1), which specifies the electrical power as a function of wind speed at hub height. (1)vHH,i(hHH)=vMH,i⋅ln⁡(hHHz0)ln⁡(hMHz0)where vHH(hHH) is the wind speed at hub height, vMH is the wind speed at measure height, h is the height, i indexes the time step, and z0 takes the environmental conditions into account. Following [4], the surroundings around the city of Göttingen can be assumed to be z0 = 1 m.

The wind turbine supplies the HP with electrical energy via a direct line, thereby avoiding grid fees. This results in electricity procurement costs from the wind turbine of 0.0735 EUR/kWh, which corresponds to the “[…] maximum value for tenders for onshore wind energy for the bidding dates in 2025 […]” [18] set by the Federal Network Agency. Although the wind turbine is grid-connected, only the direct electricity supply to the HP is considered within the system boundaries of this study. Electricity that is not consumed by the HP can be fed into the grid; however, the associated revenues are not included in the economic evaluation, as electricity market participation is outside the scope of this analysis. The grid electricity procurement costs, amounting to 0.166 EUR/kWh, are close to the industrial electricity price for 2024 according to BDEW [19].

According to [20,21], the waste water at the outlet of a waste water treatment plant is on average 5 to 10 Kelvin warmer than the outside air over the course of a year. The difference is greatest during the winter months from October to April. This is also seen at the outlet of the waste water treatment plant in the city of Göttingen in Fig. 2. There, the waste water flow at the outlet of the treatment plant is given with daily temperature and daily volume flow for the year 2019.

For the simulation, the daily values are distributed evenly across the hours of the day, with the heat extractable from the waste water being calculated using Eq. (2). (2)Q˙WW,i=m˙WW,i⋅cp⋅ΔTWW,iwhere Q˙WW,i is the amount of waste heat, and m˙WW,i is the mass flow rate of the waste water at time step i. cp is the thermal heat capacity of water with 4.185 kJ/kgK, and ΔT is the temperature drop due to heat extraction.

It is assumed that the heat exchanger is capable of transferring the required heat load. The reduction in heat transfer performance due to biofilm formation is accounted for by appropriate over-sizing of the heat transfer area. This corresponds to common engineering practice, where fouling resistances are considered during the design stage using fouling factors or safety margins [22].

In practical operation, biofilm growth and removal processes (e.g., due to shear stress) typically reach a dynamic equilibrium, resulting in an asymptotic fouling resistance and a quasi-steady heat transfer performance [22] (p. 11–15). This approach is also consistent with industrial design practices as described in heat exchanger design guidelines [23].

The HP is modelled on the basis of a standard thermodynamic cycle. The working medium evaporates in the heat exchanger on the heat source side, absorbing thermal energy in the process shown in Fig. 3 points ① to ②. A compressor in ② uses electrical energy to increase the temperature and pressure level of the working medium. Finally, the working medium releases its heat for heating purposes via a second heat exchanger, also referred to as the condenser on the heat sink side, between ② and ③. After passing through an expansion valve in ④, the temperature and pressure of the working medium drop back to their initial state. After condensation, the working medium begins the cycle anew. Overall, the HP raises the temperature of the treated waste water flow and feeds the thermal energy thus generated into the supply line of the district heating network. Ammonia (NH3) is currently the most commonly used natural refrigerant for flow temperatures of around 90∘C on the heat sink side [24].

Various key figures are used for technical and economic evaluation: the Coefficient of Performance (COP), the Annual Performance Factor (APF)¹ 1

Alternative terms used in the literature include Seasonal COP (SCOP) or Seasonal Performance Factor (SPF).

Following [26], the operation of a HP can be described in a practical yet idealized manner by the Lorenz cycle. The corresponding COPLorenz,i expresses the amount of heat (Qi) that can be supplied per unit of work (Wi) at time step i and is defined by Eq. (3). Further discussions on the use of the Carnot or Lorenz cycle as a reference for HP performance can be found in [4,27]. (3)COPLorenz,i=QiWi

According to [26], the COPLorenz can also be expressed by the average temperatures of the low-temperature heat source and the high-temperature heat sink. The average temperatures (T¯) on the waste water side (WW) and the DH side are calculated for each time step i using Eqs. (4) and (5), respectively. Based on these average temperatures, the COPLorenz can be determined using Eq. (6) [28] (p. 349). (4)T¯WW,i=TWW,in,i−TWW,out,iln⁡(TWW,in,iTWW,out,i) (5)T¯DH,i=TDH,in,i−TDH,out,iln⁡(TDH,in,iTDH,out,i) (6)COPLorenz,i=T¯DH,iT¯DH,i−T¯WW,i

Losses in the system components and peripherals are simplified and taken into account using the degree of quality (η) in order to indicate the actual efficiency (COPi) of the overall system in accordance with Eq. (7). Based on the literature in Table 1, an average factor of 0.55 is reasonable.(7)COPi=COPLorenz,i⋅η

With Eq. (8) the energy balance of the HP can be expressed. Depending on the consumed electric power (Pi) and the used waste heat (Q˙WW,i), the heat delivered by the HP (Q˙HP,i) can be calculated for time step i. (8)Q˙HP,i=Pi+Q˙WW,i

The APF shows the annual (a) efficiency of a HP and is the quotient of the annual thermal energy (Qa) supplied divided by the sum of the annual electrical energy (Pa) consumed [32] (p. 338). (9)APF=QHPaPa

The FLH is calculated from the annual amount of thermal energy generated (QHPa) divided by the nominal thermal output of the HP (Q˙N), as shown in Eq. (10). (10)FLH=QHPaQ˙N

The investment costs are derived from the costs of tapping the heat source, the HP itself, and other auxiliary equipment. The literature provides figures of different existing HP systems, given in Table 2. From these key metrics a regression function can be derived as shown in Fig. 4.

Using the result from Fig. 4, the specific investment costs (cI,HP∗) can be calculated as a function of the nominal heat capacity of the HP (Q˙N) given in Eq. (11). Thus, the investment costs (CI,HP∗) are calculated by Eq. (12). (11)cI,HP∗(Q˙N)=1.3331⋅Q˙N−0.322 (12)CI,HP∗(Q˙N)=cI,HP∗(Q˙N)⋅Q˙N

Additionally, the costs of Operation and Maintanance (O&M), excluding energy costs, must be determined. Literature assumes the O&M costs to be between 1.5% and 3% of the investment costs (see Table 3). In this paper, 3% is used.

Given an expected lifetime (n) of 20 years and a discount rate (z) of 5%, the investment costs are increased by the annuity factor that is z⋅(1+z)n(1+z)n−1. As the O&M are due yearly, they are assumed to be interest-free. Finally, the total annual costs of the system (CI) can be determined by Eq. (13). (13)CI=CI,HP∗(Q˙N)⋅z⋅(1+z)n(1+z)n−1+CI,HP∗(Q˙N)⋅0.03

Following Eq. (14), the quotient of total costs (∑inCI,i) divided by the amount of energy generated during lifetime (∑inQHP,i) is the LCoH. The LCoH is the most important key figure when dimensioning the system and should be as low as possible. (14)LCoH=∑inCI,i∑inQHP,i

The heat pump’s DSs is defined as the ratio of the electricity supplied to the HP by the wind turbine to the total electrical energy used by the HP.

The Agora Energiewende study “The roll-out of large-scale heat pumps in Germany” [11] compares different heat sources for large HPs. They conclude that the investment costs for HPs that use the heat at the outlet of a waste water treatment plant are the lowest [11] (p. 75). Furthermore, plants are designed for a high annual operating time in order to achieve low heat production costs. Up to 7500 FLH are achieved, for instance in [43]. In 2023, HPs with a total rated thermal output of 60 MW were installed in Germany and 600 MW were in planning or under construction [34].

The hourly heat load curve for the DH network from 2016 is adjusted to 2019 using the specific heating load curve. The geometric mean of the outside temperatures over the preceding four days, calculated in Eq. (15), is used to generate a smoothed outside temperature that takes into account the thermal inertia of buildings. Using the hourly DH demand and the smoothed hourly outside temperature, three specific heating load curves with the highest possible coefficient of determination are derived, see Eq. (16). With these three specific heating load curves, the DH demand is approximated for 2019 as a function of the outside temperature.(15)Tg,i=∑i=0i−23Ti24+∑i−24i−47Ti24⋅0.5+∑i−48i−71Ti24⋅0.25+∑i−72i−95Ti24⋅0.1251+0.5+0.25+0.125where Tg,i is the outside temperature according to geometric mean and Ti is the measured temperature at time step i.(16)Q˙DH,i={−155.89kW/∘C⋅Ti+5487.1kWfor Ti<15.17∘C−1070.7kW/∘C⋅Ti+19371kWfor 15.17∘C≤Ti≤24.59∘C−1700kWfor 24.59∘C<Tiwhere Q˙DH,i is the heat load of the DH system, and Ti is the outside temperature at time step i.

Data from the existing biomass plant of the Stadtwerke Göttingen show quarterly supply and return temperatures of approximately 90∘C and 60∘C, respectively, for 2019 and 2020. The emission factor for DH generation in 2025 is specified as 0.164 kg/kWh according AGFW FW 309-6 [44].

Fig. 5 shows that the amount of waste heat from the waste water increases linearly with the extraction capacity to just under 26 MW at a maximum temperature spread of ΔT 10 Kelvin. At a ΔT of 6 K, the annual heat quantity stagnates at an extraction capacity of 17 MW, at ΔT 5 K at 15 MW, and at ΔT 4 K at 11 MW.

The temperature spread (ΔT) has a significant impact on the amount of recoverable waste heat. For a variable extraction capacity of up to 26 MW, the annual recovered energy is 200 GWh/a at ΔT = 10 K, 120 GWh/a at 6 K, 100 GWh/a at 5 K, and 80 GWh/a at 4 K.

This indicates that, with few exceptions, the waste water temperature remains above 10∘C throughout the year, since higher temperature spreads always allow more heat to be extracted at the maximum controllable capacity. Conversely, smaller temperature differences limit the total recoverable heat over the year.

The extraction capacity also influences the number of FLHs in an almost linear fashion. In the worst case, the FLHs begin to decrease from 8760 to 3000 h in the case of a ΔT 4 K and an extraction capacity of 7 to 26 MW. Furthermore, it can be observed that the temperature spread (ΔT) strongly affects the number of full-load hours (FLHs), shifting their decline toward lower extraction capacities. For example, with ΔT = 10 K, FLHs of 8760 can be achieved up to an extraction capacity of 17 MW. For ΔT = 6 K, this applies only up to 10 MW, for 5 K up to 8 MW, and for 4 K up to 7 MW. As a result, the limit of 5000 FLHs is first exceeded in scenario ΔT 4 K at a withdrawal capacity of 16 MW, followed by ΔT 5 K at 20 MW and ΔT 6 K at 24 MW. A ΔT of 10 K remains above 5000 FLH even at maximum withdrawal capacity.

The maximum heat quantity of 80 GWh/a is provided by a HP with a maximum PL range of 100% and a nominal thermal output between 21 and 26 MW (see Fig. 6). At a PL range of 75%, this is 70 GWh/a at a power rating of 17 MW, at 50% it is 62 GWh/a at 15 MW, and at 25% it is 52 GWh/a at 12 MW. In general, the lower the PL range, the greater the drop in heat output after the peak has been reached.

Up to a power rating of 2 MW, 8760 full utilization hours can be achieved. At maximum power rating, these are reduced to around 3000 h at 100% PL range and to just over 100 h at 25% PL range. FLHs decrease more significantly with increasing power rating and lower PL ranges.

According to Fig. 5, up to 7 MW of heat output can theoretically be extracted from the waste water stream, with the heat quantity and FLHs remaining the same across all scenarios. According to Fig. 6, however, the annual heat quantity increases sharply up to a nominal output of 3 MW. The increase slows down above 3 MW. At 3 MW, the ratio of heat quantity to nominal output is optimal with maximum annual heat quantity. Looking at Fig. 6, the FLHs are greatest up to a nominal thermal power of 2 MW. Therefore, an optimal LCoH can be assumed for a nominal output of up to 2 MW.

This is confirmed by the overview of the LCoH in Table 4. In accordance with the maximum achievable FLH in Fig. 6, all scenarios up to a nominal output of 2 MW have a favorable LCoH. These remain constant for increasing temperature spreads or PL ranges. The greater the nominal output, the greater the influence of the PL range on the LCoH. In general, the greater the PL range, the greater the nominal output of the HP can be. The temperature spread has a negligible influence on the LCoH. In summary, it should be noted that the HP is designed for the most advantageous LCoH at 2 MW.

The sensible choice of a rated power of 2 MW is also confirmed by the course of the APF. Fig. 7 shows the thermal energy and the APF over the nominal thermal power of the HP. The APF drops down strongly from nearly 4 to approximately 3.5 until a rated power of around 2 MW is reached for every scenario. For higher nominal thermal power, it levels out at around 3.

As the temperature spread has no impact on the LCoH, Fig. 8 is important to determine ΔT as it gives the degree of self-sufficiency. Fig. 8 plots the electrical energy generated by the wind turbine against the temperature reduction of the waste water. Analysis of the overlap between the maximum hourly waste heat available and the hourly wind energy shows that the electrical energy from wind power that can be used by a HP increases from 12 to 13.2 GWh/a at a temperature spread of ΔT 5 K. This corresponds to an increase of 1.2 GWh. From the ΔT 5 K scenario to the ΔT 6 K scenario, the increase is reduced to 0.6 GWh and falls to 0.1 GWh from the ΔT 6 K to the ΔT 10 K scenario. The degree of self-sufficiency is greater than or equal to 0.6 for the ΔT = 4 or 5 Kelvin scenarios.

Fig. 9 shows DH generation in 2019, totaling 80 GWh, broken down by the types of heat generators used and the scenarios analysed in this study. In addition, the amount of heat generated by the HP is shown, which cannot be fed into the district heating network due to a lack of simultaneity between generation and demand.

As an example, the scenario with a temperature spread of 4 Kelvin shows that 76% of the potential HP energy could be used, while 24% of the DH was provided by other heat generators. A further 23% of the total amount of district heat demand could also have been covered by the HP if demand and generation had been simultaneous.

It also shows that with a temperature spread of 5 Kelvin, the ratio of usable HP heat to total HP heat generated is highest at 1.59.

Taking everything into account, the system would be optimally designed according to the operating parameters in Table 5. This results in the costs and emission reductions shown in Table 6.

The HP has a power rating of 2 MW, and a PL capacity of 90% ensures a high degree of flexibility. The amount of heat provided over the course of the year 2019 is 17.5 GWh/a. The electrical energy required to operate the system comes from two sources: On the one hand, around 1.18 GWh/a is provided by the power grid, and on the other hand, 3.60 GWh/a is sourced directly from the connected wind turbine. This results in a total electricity consumption of 4.78 GWh/a. With a total of 8741 FLH, the HP is used intensively. The degree of electricity self-sufficiency is particularly noteworthy: with a self-sufficiency rate of around 75%, the majority of the electricity demand can be covered by renewable, locally generated wind energy. This results in low electricity costs and low CO2 emissions.

The economic parameters in Table 6 show that the annual investment costs for the HP amount to 171,000 EUR/a, plus 43,000 EUR/a for O&M. The electricity costs are divided proportionally between grid electricity, at approximately 200,707 EUR/a, and electricity from the wind turbine, at around 264,721 EUR/a. The total annual electricity costs therefore amount to approximately 465,311 EUR/a. The total annual system costs for the HP are therefore around 679,200 EUR/a. These total costs result in a LCoH of 38.85 EUR/MWh. The system achieves annual CO2 savings of around 2867 t. The specific costs of the avoided emissions are 0.237 EUR/kgCO2.