Modelling temperature-dependent COP of heat pumps


I was slightly incredulous to read in Residential heat pumps in the future Danish energy system that “No existing studies have been identified in which the operation of residential heat pumps is described by temperature-dependent coefficients of performance (COP)”. It is very important to model the temperature-dependence, given that it exacerbates the peak electricity demand exactly when it is very cold and heating demand is high. I checked with the authors of the Fraunhofer IWES sector-coupling study Interaktion EE-Strom, Wärme und Verkehr and they certainly do have a T-dependent COP in their model. As do we (FIAS). Do people really model with constant COP?


Certainly in almost all MILP models I’ve come across there is no temperature dependence for COP, e.g. here, here (actually a MINLP model), and here. This is also true for cooling technologies beyond the standard heat pump, as they are all adversely affected by high external temperatures.
Equally, there are no part-load efficiency curves given for many of these technologies, which is also an issue when they operate at a range of different efficiencies if not operating at full capacity.

I’ve noticed a distinct lack of libraries to describe different technology performance parameters. It’s much quicker to stick in a single value for COP than to trawl manufacturer documentation to get a graph which you then have to use image recognition on to get hold of the curve properties!


Thanks for the links and the confirmation that this is a problem!

A review of domestic heat pumps by Staffell et al. has in Figure 9 an average of manufacturer data for the temperature dependence of the COP, along with a handy quadratic regression fitting in equation (4), so no image recognition necessary!


Not a problem, I can link a number more if necessary…

That looks good, will save for when I need to include it in my modelling. Do heat pumps update their performance regularly/significantly? Just wondering how long before this particular review is no longer relevant.


That’s a good question. I’ll invite Iain Staffell to sign up so he can answer this.


Two models that use state-dependent efficiencies for heat pumps and for HVAC systems are deeco and xeona respectively (to be fair to the Danish report, xeona has not been published).

I indicated the approach taken by deeco in a posting in late-2016 to the openmod mailing list. deeco supports arbitrary (including nonlinear) efficiency dependence as the efficiency (known in the documentation as the input-output relationship or IOR) is calculated before least cost flow is attempted.

The method used in xeona is, I would hazard a guess, completely novel in an energy model. It is based on the following paper, published in 1940:

Plank, Rudolf. 1940. Zur thermodynamischen Bewertung von Kältemitteln [On the thermodynamic evaluation of refrigerants]. Zeitschrift für die gesamte Kälte-Industrie, v6 p81-84.

A copy is held in the TU-Berlin library and I have a scan if anyone is interested.

I calibrated my model using the refrigerant R-134a. Property values were obtained using RefCalc from CoolPack, a Windows GUI application from the Technical University of Denmark (which runs under Wine). Other properties calculators are also available.

The model I used is documented in the following unpublished PDF:

Morrison, Robbie. 17 December 2010. Plank refrigeration model.

The model was implemented in C++ (file: An associated control strategy for the HVAC system is also required, but that lies outside the scope of the original posting. I should add that this method is deeper than simply introducing a single characteristic curve for COP (which I suspect the other methods discussed use).

Again let me know if you would like a copy of the Plank paper, the unpublished PDF, or my source code.

I should add that Tetyana Morozyuk, chair of exergy-based methods for refrigeration systems, Institut für Energietechnik, TU-Berlin, Germany alerted me to this method.


Hi all, interesting discussion.
This is just anecdotal evidence, but last time I checked, consumer literature on heat pumps, especially in the Netherlands, tended to be focussed on ground-based heat pumps. To make matters worse, such systems were also referred to as geo-energy systems.

Checking on the site of milieu centraal it seems that currently the focus is on hybrid gas-fired boiler / air-based heat pump combinations, so I’m not sure I can back up that statement.

At any rate, for ground-based heat pumps, the temperature dependence of COP is much less relevant because ground-based heat pumps operate with much smaller external temperature changes. And it could be that the authors of studies you’ve mentioned assumed this implicitly.


The following recent publication might be of interest. It describes the use of temperature-dependant coefficients of performance:

Reference Petrović, Stefan N; Karlsson, Kenneth B (1 November 2016). “Residential heat pumps in the future Danish energy system”. Energy. 114: 787–797. doi:10.1016/ ISSN 0360-5442.


That’s the same study linked in the post that started the thread.