Simplifications in the Optimization of Heat Pumps and Their Comparison for Effects on the Accuracy of the Results ^†

Abstract

This work presents a model that calculates temperature-dependent heat pump performances as a circular heat pump process as a reference model. The model is then systematically simplified by making assumptions or applying functional approximations to key variables. These simplifications include linearization of the substance database calculations and modeling of the compressor efficiency as a function or constant. The effects of these simplifications on the accuracy of results are quantified and compared with other modeling approaches from the literature suitable for linear and bilinear optimization issues. Initial comparisons show that the root mean square error of the model achieves better results than comparable methods. While the root mean square error of the COP in linearized models in the compared literature ranges from 0.433 to 1.233, it can be improved to a maximum of 0.335 using the approach presented.

1. Introduction

Heat pumps are vital for the transition to sustainable energy systems. They offer high efficiency in heating applications. However, the modeling and optimization of heat pump systems is based on simplifying assumptions that can significantly impact the accuracy of the simulation and design outcomes. A majority of optimization tools for energy systems are based on linear programming, which involves the linearization and simplification of nonlinear relations [1,2]. It is crucial to understand the impact of these assumptions to enhance optimization methods and guarantee reliable performance predictions. This work is an extended version of our paper published in the 20th Conference on Sustainable Development of Energy, Water and Environment Systems [3].

Modeling is usually not the definitive goal but serves as a subsequent optimization issue. Typical operational optimizations have a given demand and calculate the electrical compressor power via the coefficient of performance (COP), the use of which must be minimized. The COP or electrical power can be calculated using various approaches. A distinction can be made between whether the COP is specified exogenously or determined endogenously [2]. For potential subsequent optimizations, it is of course important whether the ambient and supply temperatures are defined exogenously or endogenously, but this has no influence on the investigation of model accuracy carried out in this work.

One common exogenous method of modeling the COP as an independent constant is used in [4,5,6], among others. It was also shown in [7] that energy planning tools often rely on a constant description of the COP.

Another approach frequently used for optimization issues is to describe the COP as a dependent variable of the source or ambient conditions, i.e., T_S, sometimes under the assumption or condition that the supply temperature is constant [8,9,10]. While a constant flow temperature T_F may correspond to practice in some cases, potential optimization issues in these cases are restricted by one variable. If the ambient and flow temperatures are not decision variables, the COP can also be regarded as an exogenous constant here.

An approach for modeling the COP according to the authors of [11] uses two operating states of a heat pump according to EN14511 and creates a linear function through these two states. The presumably selected states A-7/W35 and A7/W35 result in temperature differences of 42 K and 28 K with corresponding COP values, from which the COP can be modeled as a dependent linear function of the differences [11,12]. This approach is similar to the developed method in terms of the fact that operating states depending on temperatures or temperature differences were also selected for parameterization. However, in contrast to our own model, in which the cycle was modeled as a dependent variable including compressor efficiency, among other factors, the COP was simplified and directly represented as a dependent variable of the temperatures. In addition, with only two operating states, the database is significantly smaller. In general, linear fits of the COP depending on temperature are a commonly found method [10,13,14].

In addition to modeling the COP, its combination with the electrical compressor power is also decisive for the choice of the optimization approach. In the common formula, the heat pump capacity is calculated by multiplying COP by electrical capacity. However, in order for the optimization problem to remain linear in this formula, COP must be specified exogenously. An alternative common formula is the generic linear equation, in which COP and electrical capacity are simply added together and offset with pre-factors. An excerpt of possible approaches can be seen in

Table 1. Overview of possible modeling approaches.

Modeling Approaches	Constraint	Used in
Linear	QD = COP · Pel	[4,5,6]
	QD = COP(TS,TF) · Pel	[8,9,10]
	QD = a · COP(TS,TF)linear + b · Pel + c	[10]
Bilinear/quadratic	QD = COP(TS,TF)linear · Pel	[10]
	QD = COPCarnot(TS,TF)linear · Pel	[15]
	QD = COP(Δh{ TS,TF }) linear · Pel	Own appr.
Nonlinear	QD = COP(TS,TF)nonlinear · Pel	[16,17]

Optimization variables marked in red.

. If the COP is determined endogenously in the optimization model by temperature dependencies, the same model becomes bilinear, since two optimization variables are multiplied together.

There is a wealth of literature devoted to modeling heat pump processes in as much detail as possible. These are often complex semi-physical models described using highly nonlinear formulas, which lack implementation in practical applications [18]. On the other hand, a great deal of research is being conducted into the optimization of energy technology processes, including the integration of heat pumps. Numerous assumptions and simplifications are often applied in order to make the model underlying the optimization solvable for common (linear) solvers. In most cases, the models are described using a simple input–efficiency–output relationship (as shown in Table 1). Other influencing factors, such as temperatures, which can be used to better represent these relationships, are integrated when necessary, but their influences are not usually examined. It can also be assumed that, if sufficient accuracy is achieved, no further search for more precise approaches will be conducted, even if this could further improve the optimization itself. The systematic review by the authors of [2] emphasizes that future research into COP modeling should focus more on in-field performance that goes beyond generic temperature dependence.

The present work aims to address precisely this research gap. To this end, an initially nonlinear approach in the form of the heat pump cycle is modeled nonlinearly. This approach is parameterized with an underlying database to form a reference model and then transformed into a question suitable for optimization. Simplifications are then carried out systematically, from which further models are derived. The models are first briefly compared with each other using various model quantities. However, the main focus is on comparing the models with optimization models commonly found in the literature with regard to the relevant variables, COP and electrical power. The work presented here is intended to contribute to a better assessment and comparison of the effects of the various assumptions and simplifications used in upstream modeling during optimization. This will help to select the right approach depending on the problem, the optimization approach, and the desired accuracy of the results.

2. Methods

Firstly, an extensive literature search was carried out to gain an overview of the approaches that can be used to model heat pump processes. In this context, modeling approaches for heat pumps are limited to work aimed at optimizing their operation. An overview of this was already provided in the previous chapter. Specific requirements, such as temperature variability of the environment and demand, have been formulated for the models, which are based on the comparison of bilinear optimization with linear and nonlinear optimization for various issues relating to energy systems. In order to be able to apply different modeling variants on the basis of the model, it was decided to describe the heat pump cycle as a circular process and to simplify it if necessary. While linear and bilinear optimization models often rely on simplified relationships as described in Table 1, more complex nonlinear approaches can be based on the circular process. By systematically linearizing this process until the model is completely linear or bilinear, it is possible to effectively evaluate the effects of individual assumptions.

Figure 1 shows the heat pump cycle with the corresponding defined states before and after the technical components in the pressure–enthalpy diagram using the example of refrigerant R410A. State 1 describes the state of saturated vapor. State 1_sup is after the refrigerant is superheated and before it enters the compressor. State 2 is after the compressor and before entering the condenser and is now at the higher pressure level. State 2_is a theoretical point that would result from isentropic compression. It is included because it can be used later to determine the isentropic compressor efficiency. States 3 (with complete condensation) and 3_sub (subcooled liquid) are located in or after the condenser. State 4 describes the state after the expansion valve. The state variables are temperature, pressure, enthalpy, and entropy. As the figure shows, the relationships between these state variables are not linear.

Detailed measurement datasets for five ground source heat pumps serve as the data basis, including heat output and evaporator/compressor power at varying source and sink temperatures [19]. Key optimization variables include source and sink temperatures. Dependent variables, such as compressor efficiency, pressure levels, enthalpies at all states, and mass flow rate, are treated as either parameters or as linear or nonlinear functions of the source and sink temperatures. The model is then tested across all temperature ranges, comparing the modeled COP to real-world data to evaluate deviations caused by the assumptions.

Based on the datasets, a cycle process can be calculated as a reference model. Since both input variables (electrical power of the compressor and heat power of the source) and output variables (heat output of the heat pump) are known here, these results are used to validate the subsequently derived models. The reference model is then adapted so that system variables and key values (temperatures, pressures, enthalpies, compressor efficiency, and COP) can be determined on the basis of a heat demand and specified temperatures. Systematic simplifications and assumptions are made in the cycle process, and six models are defined and compared.

To illustrate the explanatory power in regressions used to describe the linearized relations, the coefficient of determination R² in comparison to the reference model is always specified for the functions. The root mean square error (RMSE) is then used for comparison in order to compare the models with each other and other models later in the paper and to focus on the absolute errors. The RMSE function (1) calculates the root mean square error between actual values y_i and model predictions ŷ_i, whereby larger deviations are weighted more heavily. The square root at the end produces an error value in the same unit as the original data. The smaller the RMSE, the better the model agrees with the real data.

$$RMSE=\sqrt{MSE}=\sqrt{{{\displaystyle \sum }}_{i=1}^n{\displaystyle \frac{{\left(y_i-{\widehat{y}}_i\right)}^2}{n}}}$$

(1)

The refrigerant-dependent calculations are carried out using the CoolProp substance database (Version 6.7.0) [20]. The model was programmed in Python 3.13.1.

3. Heat Pump Modeling

Detailed manufacturer data from five different geothermal heat pumps in a series is available for the real reference variables of the model [19]. Demand heat output, electrical compressor power, and heat power of the source and the resulting COP in various operating states were provided. The operating states differ in source temperature T_S = [−5, −2, 0, 2, 5, 7, 10, 15] °C and demand temperature T_F = [30, 35, 40, 45, 50, 55, 62] °C. Data were available for this temperature range, covering an operating range in which heat pumps are commonly used in buildings. The data are available for every combination of temperatures (with the exception of −5 to 62 °C), resulting in 55 operating states per heat pump. The heat output supplied ranges from 46 to 209 kW. The COP varies between 2.25 and 7.93. The number of operating states and the range of data appear to provide a good database. This database was chosen because the data are freely accessible and provide a basis that significantly exceeds the energy consumption labeling requirements under Regulation (EU) No. 813/2013. The heat pumps tested in accordance with this regulation and EN 14511 have four operating points with a COP for air-to-water heat pumps and two operating points for water-to-water heat pumps. The methodology presented can also be applied here, but this does not appear to be useful due to the small number of data points and the resulting insufficient representation of temperature dependencies. Formulas (2)–(9) presented later are exclusively dependent on the refrigerant and are, therefore, readily transferable. Only Formulas (10)–(13) are heat pump-specific and are parameterized by input and output power. This makes the methodology transferable, even if the RMSE results have to be checked individually.

In addition to the known data, assumptions must be made for the temperature differences in the condenser and evaporator, the superheating and subcooling temperatures, and the efficiency of the condenser and evaporator. For this purpose, typical temperature differences within the usual ranges for water–water heat pumps have been assumed [21,22,23,24]. An overview of the modeling parameters can be seen in

Table 2. Modeling parameters for the cycle modeling.

Symbol	Description	Value
R	Refrigerant	R410A
TF	Flow temperature (demand side)	[30, 62] °C
TS	Source temperature (supply side)	[−5, 15] °C
QD	Heat pump output	[46.46, 209.38] kW
Pel 1	Electrical compressor power	[10.85, 51.77] kW
QS 1	Power of the heat source	[27.3, 181.37] kW
ΔTcon	Temperature difference in the condenser	2 K
ΔTev	Temperature difference in the evaporator	2 K
Tsub	Subcooling temperature	2 K
Tsup	Superheating temperature	5 K
ηcon	Condenser efficiency	1
ηev	Evaporator efficiency	1

¹ Only parameters in the reference model and variables in the derived models.

. Technical losses in the components are neglected here. The pressure losses in the cycle depend, among other things, on the choice of refrigerant and the ambient temperature, and can reduce the COP by up to 5% [25]. These losses are already implicitly accounted for in the data but are not explicitly modeled here. Since the input and output data are measured values, it seems sufficient to offset the corresponding losses in the cycle process instead of assigning them to individual components or process steps. The influence on the unmeasured modeling results is addressed in Section 5. In the available datasets, the sum of compressor power and heat power of the source corresponded to the heat pump power, which is why these efficiencies are not modeled within the cycle and are assumed to be 1. The isentropic compressor efficiency represents a difference to this and must, of course, be modeled.

Further modeling is based on being able to solve potential tasks for optimization with variable supply or flow temperatures T_S/T_F for the given parameter heat demand Q_D. Therefore, all functions should be limited to dependencies on these two variables.

3.1. Reference Model and Model Derivation

Firstly, the temperatures T₃ and T_3sub can be calculated using the demand temperature T_F, temperature difference in the condenser ΔT_con, and subcooling temperature T_sub.

$$T_3=T_F+\Delta T_{con},T_{3_{sub}}=T_3-T_{sub}$$

(2)

The now known temperature T₃ provides the upper pressure level, which is assumed to be the same at states 3, 3_sub, and 2. The pressure level is determined using the CoolProp substance data model for the refrigerant R410A [20]. These data models correspond to nonlinear functions, which are characterized by f_NL. f_NL does not indicate a specific function but only shows that an arbitrary nonlinear calculation is performed for a state variable.

$${\mathrm{p}}_3={\mathrm{f}}_{\mathrm{NL}}\left({\mathrm{T}}_3\right),{\mathrm{p}}_{3_{\mathrm{sub}}}={\mathrm{p}}_2={\mathrm{p}}_3$$

(3)

The enthalpies h₃, h_3sub, and h₄ result from the nonlinear functions of temperature and pressure. Since state 3 is located exactly on the boiling line, the enthalpy can be determined here without the associated pressure or other state variables. A calculation from temperature and pressure is just as possible. In state 3_sub, both state variables are necessary. The enthalpy h₄ is calculated because the change in state in the expansion valve is described as an isenthalpic pressure reduction in the cycle.

$${\mathrm{h}}_3={\mathrm{f}}_{\mathrm{NL}}\left({\mathrm{T}}_3\right),{\mathrm{h}}_{3_{\mathrm{sub}}}={\mathrm{f}}_{\mathrm{NL}}\left({\mathrm{T}}_{3_{\mathrm{sub}}},{\mathrm{p}}_{3_{\mathrm{sub}}}\right),{\mathrm{h}}_4={\mathrm{h}}_{3_{\mathrm{sub}}}$$

(4)

The calculation for the temperature T₁ and T_1sup is similar to before. The temperature T₄ corresponds to temperature T₁, as the changes in state between these two states take place entirely in the wet vapor region.

$${\mathrm{T}}_1={\mathrm{T}}_{\mathrm{S}}-\Delta {\mathrm{T}}_{\mathrm{ev}},{\mathrm{T}}_{1_{\sup }}={\mathrm{T}}_1+{\mathrm{T}}_{\sup },{\mathrm{T}}_4={\mathrm{T}}_1$$

(5)

The lower pressure level for p₁, p_1sup, and p₄ results from temperature T₁.

$${\mathrm{p}}_1={\mathrm{f}}_{\mathrm{NL}}\left({\mathrm{T}}_1\right),{\mathrm{p}}_{1_{\sup }}={\mathrm{p}}_4={\mathrm{p}}_1$$

(6)

As before, the enthalpies h₁ and h_1sup can also be determined by nonlinear functions of temperature or temperature and pressure.

$${\mathrm{h}}_1={\mathrm{f}}_{\mathrm{NL}}\left({\mathrm{T}}_1\right),{\mathrm{h}}_{1_{\sup }}={\mathrm{f}}_{\mathrm{NL}}\left({\mathrm{T}}_{1_{\sup }},{\mathrm{p}}_{1_{\sup }}\right)$$

(7)

The state variables temperature, pressure, and enthalpy are determined for all states, with the exception of state 2. The entropies for all states can be calculated using two of these variables, in this case, pressure and enthalpy. The index i here indicates any state from 1 to 4, as the calculation is always identical.

$${\mathrm{s}}_{\mathrm{i}}={\mathrm{f}}_{\mathrm{N}\mathrm{L}}({\mathrm{p}}_{\mathrm{i}},{\mathrm{h}}_{\mathrm{i}})$$

(8)

Using the upper pressure level and the entropy s_1sup, the theoretical isentropic enthalpy s_2is can be calculated. This is necessary in order to be able to calculate the isentropic compressor efficiency later.

$$h_{2_{is}}=f_{NL}(p_2,s_{1_{sup}})$$

(9)

Up to this point, the entire refrigerant cycle can be described on the basis of two temperatures and the assumption of certain temperature differences. The state variables that are not on the dew or boiling line can only be determined unambiguously with two other state variables. However, one of these is always a known temperature, and the other can be determined in advance using only the temperatures.

There are two ways to determine the missing state variables in state 2. The first is to determine the state changes caused by the compressor starting from state 1_sup. The second option is to calculate back from state 3 or 3_sub via the change in state in the condenser. In both cases, an efficiency must be assumed or known. This is feasible in the reference model, as the heat supplied via the evaporator and the enthalpy change are known. Since this method offsets the modeling of losses within the cycle, but the isentropic compressor efficiency is part of the research question, the second approach is chosen. Before h₂ can be determined, however, the refrigerant mass flow must be calculated. This is obtained by dividing the heat flow from the environment by the enthalpy change in the evaporator, while taking into account the evaporator efficiency η_ev.

$$\dot{m_R}={\displaystyle \frac{{\dot{Q}}_S·{\eta }_{ev}}{h_{1_{sup}}-h_4}}$$

(10)

The enthalpy upstream of the condenser inlet h₂ can then be determined using the mass flow and the heat flow on the demand side.

$$h_2={\displaystyle \frac{{\dot{Q}}_D·{\eta }_{con}}{\dot{m_R}}}+h_{3_{sub}}$$

(11)

The enthalpy and the upper pressure level can be used to determine the missing state variables in state 2.

$$T_2=f_{NL}\left(p_2,h_2\right), s_2=f_{NL}(p_2,h_2)$$

(12)

The isentropic compressor efficiency can be calculated by comparing the real and theoretical isentropic enthalpy change in the compressor.

$${\eta }_{comp}={\displaystyle \frac{h_{2_{is}}-h_{1_{sup}}}{h_2-h_{1_{sup}}}}$$

(13)

Six models are derived from the reference model:

• Model 1: Nonlinear functions, exact values from process calculation for compressor efficiency.
• Model 2: Linearized functions, exact values from process calculation for compressor efficiency.
• Model 3: Nonlinear functions, function calculation for compressor efficiency.
• Model 4: Linearized functions, function calculation for compressor efficiency.
• Model 5: Nonlinear functions, constant compressor efficiency.
• Model 6: Linearized functions, constant compressor efficiency.

These models differ from the reference model in that they only assume the heat output of the heat pump Q_D as known, but not the resulting input variables Q_S and P_el. The models shown differ with regard to the calculation of the system variables (nonlinear and linearized) and the determination of the compressor efficiency (reference model values, determination with a function or a constant value). An overview can be seen in

Table 3. Differences in the models of the heat pump circuit process.

Var.	Reference Model	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6
QD	P	P	P	P	P	P	P
QS	P	NL	L	NL	L	NL	L
Pel	P	NL	L	NL	L	NL	L
ηcon	P	P	P	P	P	P	P
ηev	P	P	P	P	P	P	P
ηcomp	NL	P	P	NL/L	NL/L	C	C
mR	NL	NL	L	NL	L	NL	L
T1–4	NL	NL	L	NL	L	NL	L
h1–4	NL	NL	L	NL	L	NL	L
p1–4	NL	NL	L	NL	L	NL	L
s1–4	NL	NL	L	NL	L	NL	L

P = Predefined, NL = Nonlinear, L = Linearized, C = Constant.

. Models 1 and 2 use the real efficiency of the reference model. Models 3 and 4 are based on a function for the isentropic compressor efficiency, which is presented later and can be linear or nonlinear. Models 5 and 6 assume a constant compressor efficiency. The models with odd numbers (1, 3, and 5) describe the other system variables nonlinearly, while the models with even numbers (2, 4, and 6) are based on linearized functions.

3.2. Nonlinear Modeling of State Variables

Models 1, 3, and 5 calculate the state variables, if necessary, using nonlinear functions via the CoolProp library [20]. The difference between these models is in the calculations of the compressor efficiency. The calculations in Formulas (2)–(9) are accordingly the same in these models as the calculation of the reference model. However, without knowing the heat power of the source, no refrigerant mass flow and, therefore, no enthalpy for state 2 can be derived in this way. The calculation is, therefore, based on the change in state in the compressor. For this purpose, a compressor efficiency is calculated or assumed using various methods, which is presented in Section 3.3. This allows the enthalpy h₂ to be determined.

$$h_2=h_{1_{sup}}+{\displaystyle \frac{h_{2_{is}}-h_{1_{sup}}}{{\eta }_{comp}}}$$

(14)

The temperature T₂ and entropy s₂ can then be determined as in Equation (12). The mass flow now results from the enthalpy changes in the condenser and the heat output Q_D.

$$\dot{m_R}={\displaystyle \frac{{\dot{Q}}_D}{\left(h_2-h_{3_{sub}}\right)·{\eta }_{con}}}$$

(15)

The mass flow can be used to calculate the heat power of the source Q_S and the compressor power P_el. It should be noted that the overall technical compressor efficiency is not included in this equation. The isentropic compressor efficiency is calculated by the real entropy h_2, and the focus of this work is on the cyclic process and the comparison of the state variables. In practice, a technical efficiency would be added, and an overall efficiency would be formed.

$${\dot{Q}}_S={\displaystyle \frac{\left(h_{1_{sup}}-h_4\right)·\dot{m_R}}{{\eta }_{ev}}} , P_{el}=\left(h_2-h_{1_{sup}}\right)·\dot{m_R}$$

(16)

3.3. Linearized Modeling of State Variables

The sequence of calculations and regressions and how they build on each other is shown in Figure 2. For models 2, 4, and 6, the temperatures T₃ and T_3sub are calculated as in Equation (2). Then, regressions for functions (3), (4), (6)–(9) have been carried out for the specified temperature ranges of T_S (−5 to 15 °C) and T_F (30 to 62 °C) in steps of 1 °C. This means that one data point corresponds to one integer temperature and the associated function value in the value range. The functions can also be applied to temperatures outside this range, but they have not been parameterized for these values, and it is then a matter of extrapolation or a wider regression. In order to avoid distorting the regressions by conditions that lie outside the valid operating range of the heat pumps, this range was kept as narrow as possible. For the following linear regressions, the coefficient of determination R² is listed next to all equations and shows the correlation with the nonlinear functions based on substance data models. Starting from the basic linear equation f(x) = ax + b, the following Equations (17)–(27) in

Table 4. Linearized system of equations for the state variables.

f(x)	a	x	b	R2
p3	68,553.771	T3	−302,487.743	0.9942	(17)
h3	2071.823	T3	183,600.007	0.9962	(18)
Δhsub	−47.035	T3	−33.532	0.8584	(19)
h3sub	$h_3+{∆h}_{sub}·T_{sub}$				(20)
p1	27,436.360	T1	806,690.021	0.9960	(21)
h1	280.005	T1	421,291.198	0.9950	(22)
Δhsup	9.310	T1	1120.363	0.9964	(23)
h1sup	$h_1+{∆h}_{sup}·T_{sup}$				(24)
s1	−1.927	T1	1810.671	≈1	(25)
Δssup	0.019	T1	4.083	0.9923	(26)
s1sup	$s_1+{∆s}_{sup}·T_{sup}$				(27)

result.

The upper pressure level and enthalpy h₃ can be calculated from the temperature T₃. The enthalpy change from state 3 to 3_sub, resulting from the subcooling, must then be calculated. Equation (19) has been set up for this purpose, which, starting from a temperature level of T₃, indicates how large the enthalpy change is per temperature change. The result of this equation multiplied by the subcooling temperature T_sub is added to the enthalpy h₃ to calculate the enthalpy h_3sub (20). Δh_sub, Δh_sup, and Δs_sup describe the rate of change in enthalpy or entropy per change in temperature during subcooling or superheating at a given temperature level.

The temperatures T₁, T_1sup, and T₄ can be determined as in (5). The lower pressure level and enthalpy h₁ are calculated analogously to Equations (17) and (18). The same procedure is used for the enthalpy difference for superheating in state 1_sup. Not all entropies are relevant for the further calculations, which is why only the entropies relevant for the calculation of the compressor efficiency s₁ and s_1sup are determined. These are calculated in the same way as the enthalpy in Equations (23) and (24). The enthalpy h_2is the result of an initial level in state 1_sup and an isentropic compression. It was, therefore, decided to model it as a function of the two variables p₁ and p₃ rather than a single variable. This equation, with the associated coefficient of determination, can be seen in (28). By substituting p₁ and p₃ with Equations (17) and (21), h_2is can also be described as a function of the input variables.

$$h_{2_{is}}=-0.023·p_1+0.009·p_3+\mathrm{449,916.826} R^2=0.9915$$

(28)

The enthalpy h₂, mass flow, heat power of the source, and the compressor power are calculated according to Formulas (14)–(16).

3.4. Approaches for Compressor Efficiency

The previous modeling and calculation of the state variables are, with the exception of the assumption of efficiency, independent of the actual heat pump used. The influence of the efficiency of the evaporator and condenser is estimated to be rather low and was, therefore, assumed to be constant. In [26], the effects of heat exchanger effectiveness have been investigated, with the result that, with an effectiveness range of 0.84 to 0.96, the COP changes by only a maximum of 0.07. This assumption is not fully transferable to the heat pumps in question, but it does provide an indication of the order of magnitude of the deviations. For this reason, and due to the fact that no data are available on the temperature dependence of the individual components, and modeling this would go beyond the scope of this work, constant efficiencies are assumed here. This assumption is not feasible for the compressor efficiency.

3.4.1. Calculated Compressor Efficiency

In order to analyze the influence of the nonlinear and linearized calculation of the state variables and data of the model without further assumptions and inaccuracies, the previously calculated, correct compressor efficiency can be used. The isentropic compressor efficiency determined from the reference model was, therefore, used for models 1 and 2. These values range from 0.497 to 0.791.

3.4.2. Function for the Compressor Efficiency

The compressor efficiency in a heat pump cycle process is influenced by several factors. These include, for example, the compression ratio, intake pressure and temperature, refrigerant overheating, and the mechanical behavior of the compressor, such as speed control and partial load behavior [27]. No specific key figures are known about the mechanical behavior and technical modeling of individual components, so this is not part of this work. It is, therefore, examined whether the compressor efficiency can be modeled as a dependent variable of the input variables. The other state variables, pressure, enthalpy, and entropy, were also analyzed with regard to their predictive power of η_comp, but they did not show a stronger correlation, which is why they are not listed separately. It is assumed that this is due to the fact that these variables were also determined on the basis of the temperatures. However, this is not mathematically proven in the context of this work.

Table 5. Comparison of different regressions for heat pump compressor efficiencies.

Var.	Polynomial Degree	R2
		HP 1	HP 2	HP 3	HP 4	HP 5
TS	1	0.117	0.004	0.016	<0.001	0.041
	2	0.168	0.101	0.134	0.173	0.206
TF	1	0.124	0.103	0.085	0.052	0.071
	2	0.140	0.133	0.110	0.075	0.085
TF-TS	1	0.017	0.062	0.036	0.039	0.116
	2	0.337	0.376	0.373	0.405	0.506
TF/TS	1	0.007	0.047	0.024	0.029	0.105
	2	0.385	0.416	0.424	0.464	0.560

presents several options for modeling compressor efficiency of the five heat pumps (HP1 to HP5) as a dependent variable using regression. T_S and T_F are included in the regression either individually as variables or as a subtraction or division of these variables. Regressions were performed with first and second-degree polynomials, and the coefficient of determination was analyzed.

All first-degree regressions have a coefficient of determination that has hardly any explanatory power. The second-degree polynomial with T_F/T_S as the independent variable has the best value, which is why this calculation is used for the compressor efficiency in the following, even if the R² values between 0.385 and 0.560 only offer limited predictive power. Figure 3 shows the isentropic compressor efficiency of all heat pumps in dependency of the temperature ratio, which has the highest coefficient of determination.

The trend in the data points also clearly shows why first-degree polynomials have such low explanatory power. Both a bell-shaped curve and a large scatter of points can be seen, which makes regression challenging. The extent to which this inaccuracy also affects the accuracy of the modeling results is shown in Section 4.

3.4.3. Constant Compressor Efficiency

A common approach is to assume a constant compressor efficiency. For this reason, a constant value per heat pump is assumed in models 5 and 6. This efficiency results from the mean value of all compressor efficiencies in the operating states. Each heat pump has its own value. The values are very close to each other, ranging from 0.637 to 0.658. The RMSE is between 0.036 and 0.045.

4. Results

The RMSE of selected variables can be seen in

Table 6. RMSE comparison of the models.

Var.	Unit	Value Range		RMSE of the Models
		Min.	Max.	1	2	3	4	5	6
ηcomp	-	0.50	0.79	0.000	0.000	0.040	0.040	0.041	0.041
mR	kg/s	0.22	1.03	0.010	0.010	0.013	0.013	0.013	0.013
p1	bar	6.34	11.84	0.000	0.121	0.000	0.121	0.000	0.121
p2	bar	19.82	41.85	0.073	0.615	0.073	0.615	0.073	0.615
Pel	kW	10.85	51.77	0.304	0.337	1.237	1.331	1.268	1.353
QS	kW	27.30	181.37	1.535	1.570	1.937	2.016	1.972	2.045
COP	-	2.25	7.93	0.054	0.068	0.266	0.313	0.289	0.335
s1uh	J/kgK	1794.42	1832.30	0.000	0.142	0.000	0.142	0.000	0.142
T2	°C	51.47	121.96	0.181	3.322	2.769	4.233	2.637	4.004
h1sup	kJ/kg	421.23	427.18	0.000	0.121	0.000	0.121	0.000	0.121
h2	kJ/kg	452.98	510.10	0.120	0.415	3.455	3.736	3.301	3.581
h2is	kJ/kg	440.91	472.99	0.077	0.259	0.077	0.259	0.077	0.259
h3sub	kJ/kg	248.32	312.24	0.024	1.084	0.024	1.084	0.024	1.084

. As the RMSE is always in the original unit, the value ranges of the variables have been added. It should be noted that the RMSE only compares the accuracy of the models with each other, i.e., in relation to the reference model. Deviations between the reference model and reality are not part of this work due to a lack of further measurement data. The comparison between models 1 and 2 shows how only the linearizations affect the model accuracy in the case of known compressor efficiency. It can be seen that the lower pressure level can be better linearized than the upper one. Only minor deviations can be observed for mass flow, compressor, and source power. Larger differences can be seen in the enthalpies and temperature at the compressor outlet. In general, however, the RMSE values appear to be at a low and, therefore, good level.

To analyze the effects of the simplifications of the isentropic compressor efficiency, models 1, 3, and 5 or 2, 4, and 6 can be compared with each other. There is little or no effect on mass flow, pressures, entropy, and enthalpies, with the exception of h₂. The relative changes in P_el, Q_S, COP, and h₂ are more visible, but the absolute figures still seem reasonably accurate. The deviation of T₂ is the most significant.

However, the key figures that are further used in optimization issues, usually the COP or the electrical power, are probably more relevant. Here, the RMSE shows values, which can correspond to a deviation of up to 15% for COP and 13% for P_el in the worst case.

The reported RMSE values should be interpreted in the context of optimization-oriented modeling rather than as absolute engineering acceptance criteria. In energy system optimization, model accuracy is application-dependent and must be assessed with respect to the optimization objective, time horizon, and the role of the heat pump within the overall system. Accordingly, simplified modeling approaches commonly used in the literature, such as constant or weakly temperature-dependent COP formulations, can be fully adequate for certain planning or system-level applications, while more accurate representations may be required for detailed operational optimization. The purpose of this work is, therefore, not to define universal error thresholds, but to systematically quantify how different modeling simplifications affect accuracy relative to a consistent reference and to commonly applied optimization models. This enables practitioners to select an appropriate modeling approach based on the required accuracy class and computational tractability for their specific application.

In order to be able to compare these figures in particular, it is not only necessary to compare the models with each other, but also to compare them with other approaches, which is completed below. The methods presented in the Introduction chapter are used for comparison and have been applied to our own data as additional comparison models. The three compared approaches are a constant COP (for an overview, referred to as approach E1) used in [4,5,6], a COP dependent on T_S (E2) as in [8,9,10], and the COP modeled depending on two operating points (E3) as in [11,12].

This comparison shows that all the approaches considered achieve worse RMSE values than the models presented, as shown in

Table 7. RMSE comparison to other models.

Var.	Unit	Value Range		RMSE of the Models
		Min.	Max.	6	E1	E2	E3
Pel	kW	10.85	51.77	1.353	7.382	6.506	8.030
QS	kW	27.30	181.37	2.045	7.382	6.506	8.030
COP	-	2.25	7.93	0.335	1.233	1.078	0.433

. The two approaches with constant COP (E1) and COP dependent on the source temperature T_S (E2) show consistently poor RMSE values. Surprisingly, the model E3 with the temperature-dependent COP shows a poor comparative value for compressor and source performance. This result can probably be attributed to the fact that, in the range of medium temperature differences, where many operating states are located, good COP values are calculated, while in the limit ranges with very high temperature differences and thus high outputs, the COP is significantly underestimated, and, therefore, too high electrical outputs are determined. The operating states A-7/W35 and A7/W35 selected in E3 result in temperature differences of 42 K and 28 K. Of the 55 operating states, seven are below the smallest temperature difference of 28 K, and 25 are above the highest temperature difference of 42 K. In these areas, modeling is a matter of extrapolation, and even small deviations in the COP can lead to large deviations in the result.

5. Discussion

Firstly, the limitations and restrictions should be mentioned here, which are prerequisites for why the methodology presented can generate good key figures. All assumptions are only valid for the specified refrigerant R410A, and the functions of other refrigerants must be parameterized and described separately. However, since these databases are freely and easily accessible, the effort required after the initial modeling is very low.

In addition, the partial load behavior of the heat pump is not taken into account, as the available key figures were defined for operating states at full load. The heat pump-specific calculations are also not transferable to other heat pumps without testing and adaptation. Various water heat pumps with different outputs but from the same series were analyzed here. However, the methodology presented here can also be adapted without any effort or individual adjustments. All that is required is the appropriate database, which is likely to be the biggest hurdle and disadvantage. Heat pump manufacturers often only provide a minimum amount of operating data. A database with 55 operating points in different temperature combinations is rather the exception and makes it difficult to apply the method in many cases. The approach described can also be applied with fewer available data points. Most of the model equations are unaffected by this regardless.

As shown in Section 1, the approaches E1, E2, and E3 presented are suitable for modeling linear or bilinear optimization problems, depending on whether T_S and T_F are exogenous or endogenous. Model 6 (constant compressor efficiency and a linearized equation system), shown in Table 7, is the most simplified model. Nevertheless, it can be seen that modeling as a cycle consistently calculates better RMSE values. Model E3 achieves approximate RMSE values for the COP but cannot replicate these when modeling electrical and source power. This suggests that the own model is more accurate in representing temperature-dependent power. The model’s own inadequacies can essentially be attributed to poor modeling of the isentropic compressor efficiency.

While the flow and source temperature, as well as the input and output power, are known as measured variables, some assumptions and simplifications (temperature differences and efficiencies for the condenser and evaporator, superheating and subcooling temperatures) have also been made. The method described is well suited for typical optimizations in which only input and output variables are relevant. However, the other variables, as presented in Table 6, have only been validated using the reference model, which is also based on the assumptions mentioned above. These variables have been included in order to systematically identify inaccuracies in the method itself and highlight the qualitative differences in the models. A comparison of the reference model with real measurement data, except for power and COP, is not part of this work. Without measurements of the corresponding variables or comparison with established thermophysical simulations, it is not possible to draw conclusions about the accuracy of these variables in comparison with reality. When modeling and optimizing heat pump systems, it is, therefore, always important to ask in advance which variables should be modeled and what level of model accuracy is sufficient for the given problem.

6. Conclusions and Future Work

The COP in particular is often the focus of optimization issues, the results of which are heavily dependent on the selected modeling approach and its precision. With the methodology presented, the COP can be modeled depending on the application:

• Linearized correlations and T_S or T_F exogenous: Linear optimization.
• Linearized correlations, T_S or T_F endogenous: Linear optimization or bilinear optimization; depending on whether one or both variables are endogenous, the requirement is that the compressor efficiency is modeled as constant or as a function of the temperature difference.
• Nonlinear correlations: Nonlinear optimization.

The model presented is capable of working with all of these optimization approaches. The comparisons made also show that modeling as a linearized cycle process achieves better RMSE values than common models in the literature. The method presented provides a good opportunity to gain an impression of how linearizations and different modeling approaches for compressor efficiency affect model accuracy in comparison to the reference model. A modeling approach has also been developed that helps to close the research gap presented in [2] in the more realistic modeling of temperature-dependent COP. This enables a better estimation of the discrepancies between exact nonlinear models and simplified linear optimization approaches, while at the same time, presenting a method for gradually approaching a model that is applicable to the specific optimization problem from the nonlinear cycle process.

Future work will, therefore, build on this model to address various optimization issues for heat pumps in operation with variable temperatures in order to investigate these further with regard to model accuracy, calculation time, and model complexity, as presented in [28], especially looking at the potential of bilinear optimization. It is also conceivable to supplement and analyze other approaches, such as piecewise constant variables, Willan lines, or Carnot efficiency, in combination with a quality factor. Furthermore, parameters and simplifications that were previously simplified but not investigated, e.g., condenser and evaporator efficiency or isobaric and isenthalpic changes in state, can be modeled as functions, and the model can thus be further developed.