Control of Heat Pumps with CO2 Emission Intensity Forecasts

Abstract

An optimized heat pump control for building heating was developed for minimizing CO2 emissions from related electrical power generation. The control is using weather and CO2 emission forecasts as inputs to a Model Predictive Control (MPC)—a multivariate control algorithm using a dynamic process model, constraints and a cost function to be minimized. In a simulation study, the control was applied using weather and power grid conditions during a full-year period in 2017–2018 for the power bidding zone DK2 (East, Denmark). Two scenarios were studied; one with a family house and one with an office building. The buildings were dimensioned based on standards and building codes/regulations. The main results are measured as the CO2 emission savings relative to a classical thermostatic control. Note that this only measures the gain achieved using the MPC control, that is, the energy flexibility, not the absolute savings. The results show that around 16% of savings could have been achieved during the period in well-insulated new buildings with floor heating. Further, a sensitivity analysis was carried out to evaluate the effect of various building properties, for example, level of insulation and thermal capacity. Danish building codes from 1977 and forward were used as benchmarks for insulation levels. It was shown that both insulation and thermal mass influence the achievable flexibility savings, especially for floor heating. Buildings that comply with building codes later than 1979 could provide flexibility emission savings of around 10%, while buildings that comply with earlier codes provided savings in the range of 0–5% depending on the heating system and thermal mass.

Full text

energies
Article
Control of Heat Pumps with CO2Emission
Intensity Forecasts
Kenneth Leerbeck 1,* , Peder Bacher 1, Rune Grønborg Junker 1, Anna Tveit 1and
Olivier Corradi 2, Henrik Madsen 1,3 and Razgar Ebrahimy 1
1
Department of Applied Mathematics and Computer Science, Technical University of Denmark, 2800 Lyngby,
Denmark; pbac@dtu.dk (P.B); rung@dtu.dk (R.G.J); annatveit@hotmail.com (A.T); hmad@dtu.dk (H.M);
raze@dtu.dk (R.E.)
2Tomorrow (Tmrow IVS), Njalsgade, 2300 Copenhagen, Denmark; olivier.corradi@tmrow.com
3Faculty of Architecture and Design, Norwegian University of Science and Technology,
NO-7491 Trondheim, Norway
*Correspondence: kenle@dtu.dk; Tel.: +45-61427386
Received: 29 April 2020; Accepted: 26 May 2020; Published: 3 June 2020


Abstract:
An optimized heat pump control for building heating was developed for minimizing
CO
2
emissions from related electrical power generation. The control is using weather and CO
2
emission forecasts as inputs to a Model Predictive Control (MPC)—a multivariate control algorithm
using a dynamic process model, constraints and a cost function to be minimized. In a simulation
study, the control was applied using weather and power grid conditions during a full-year period
in 2017–2018 for the power bidding zone DK2 (East, Denmark). Two scenarios were studied; one with
a family house and one with an office building. The buildings were dimensioned based on standards
and building codes/regulations. The main results are measured as the CO
2
emission savings relative
to a classical thermostatic control. Note that this only measures the gain achieved using the MPC
control, that is, the energy flexibility, not the absolute savings. The results show that around
16% of savings could have been achieved during the period in well-insulated new buildings with
floor heating. Further, a sensitivity analysis was carried out to evaluate the effect of various building
properties, for example, level of insulation and thermal capacity. Danish building codes from 1977
and forward were used as benchmarks for insulation levels. It was shown that both insulation
and thermal mass influence the achievable flexibility savings, especially for floor heating. Buildings
that comply with building codes later than 1979 could provide flexibility emission savings of around
10%, while buildings that comply with earlier codes provided savings in the range of 0–5% depending
on the heating system and thermal mass.
Keywords:
heat pumps; model predictive control (MPC); buildings; dynamic systems; CO
2
-emissions;
electrical grid power
1. Introduction
Energy flexibility on the electricity market is a high focus area in modern energy policies scoping
in on storage (e.g., batteries, fuel cells, hydro reservoirs, thermal) and flexible demand (e.g., heat
pumps, electric cars) [
1
]. The aim is to decrease CO
2
emissions by meeting the fluctuating proportion
of renewable sources (e.g., solar, wind) vs nonrenewable sources (e.g., coal, gas, nuclear). Ideally, in the
future, electricity users (the demand) will respond to the renewable power generation levels in an
attempt to minimize emissions—in a 100% renewable scenario storage and flexibility is a must
for operating the power system [2].
Therefore, methods for identifying the flexibility potential in various applications are developed.
In Reference [
3
], the energy flexibility potential in buildings is identified by the so-called Flexibility
Energies 2020,13, 2851; doi:10.3390/en13112851 www.mdpi.com/journal/energies

Energies 2020,13, 2851 2 of 19
Index, which is the energy cost, from a penalty-aware control, relative to a penalty-ignorant control.
The penalty could be for example, a CO
2
or price signal. The present paper investigates the energy
flexibility potential in buildings with a focus on heat pumps.
Heat pumps have different sizes and applications, from small single building heat pumps
to large heat pumps for district heating. The scope of the present study is limited to investigate
the increasing potential in single building heat pumps—which has been almost four-fold from 2011 to
2019 while the number of oil-fired boilers have decreased by roughly one third in the same period [
4
].
Many oil-fired boilers are replaced with heat pumps—due to both economic and environmental
benefits and political pressure (bans of oil-fired boilers in certain districts for new buildings [
5
].)
The control of the heating, however, are often simple thermostatic controls. This often results in heating
when electricity demand is high (e.g., afternoon and evening peaks), leading to increased system stress,
resulting in increased fossil fuel consumption. It is, therefore, an opportunity to shift the demand away
from peak hours using the heat-storing potential of the buildings.
In a power system, the generator which is responding to small changes in demand (e.g., start-up
of a heat pump) is called the marginal generator. A reasonable estimate of the marginal generator
is achieved by using price signals, see Figure 1—the merit order illustrated with a supply/demand
curve; the x-axis has the accumulated supply generators and the y-axis is the corresponding price.
A small increase in demand (dashed blue line) illustrates the marginal generator—in this case, a coal
fired Combined Heat and Power (CHP) plant.
10 20 30 40
0 20 40 60 80 100
Volume [GWh]
Price (EUR/MWh)
Renewables
Nuclear
CHP/Coal
Gas
Demand
Market price
Small increase in demand Marginal
Generators
Figure 1.
The figure is the merit order illustrated with a supply and demand curve. The x-axis
is the accumulated generators in the power system and the y-axis is their corresponding costs.
The highest generator in the merit order is the one crossing with the demand curve—a coal Combined
Heat and Power (CHP) plant in this example. The average emissions are a weighted average from
all activated generators. The
marginal generator
is the generator that will be activated by moving
the demand line slightly to the right (dashed blue line). Data source: Nord Pool AS.
Due to both grid stability, economic and environmental benefits, day-ahead spot price-based
control strategies have been proposed in recent papers [
6
–
8
], using occupancy mode detection
and rule-based price control and Model Predictive Control (
MPC
) (a multivariate predictive control
algorithm using a dynamic process model, constraints and a cost function to be minimized).
In Reference [
6
],
MPC
is used with varying electricity prices to minimize the cost of operating a heat

Energies 2020,13, 2851 3 of 19
pump connected to a storage unit and a floor heating system. The control only heats at night, where
the prices are low, and it is assumed that the heat pump and storage are large enough to accumulate
enough heat for the whole day. Cost savings of between 25% to 30% are obtained.
MPC
is a well-known
concept in building automation control literature [
9
–
13
], and proven to be promising with respect
to minimizing costs, but a broad practical implementation still has various challenges discussed
in Reference [14].
In Reference [
10
], the importance of occupancy information is highlighted and evaluated on
a daily basis. However, a higher resolution is needed to incorporate variations throughout the day
(e.g., when people are at work). In the study in Reference [
7
] occupancy modes are used together
with price signals to control a heat pump. The occupancy modes were developed in The Olympic
Peninsula project [
15
] and describe work, night and home mode, each with a corresponding set point
and price sensitivity. The study showed a significant level of load shifting, leveling out the normal
peaks in the daily demand curve. A self-learning controller was applied and adapts easily to changing
consumer habits.
There is a problem with spot prices though, known as the merit order emission dilemma,
as illustrated in Reference [
16
] for the German-Austrian power market: The price for coal is low
but the emissions are high. A price-based control, therefore, only leads to a decrease in emissions
if there is a surplus of renewable energy (more renewable energy than needed)—otherwise coal
is favored, and it is therefore encouraged to use CO2emission signals instead.
For CO
2
emissions, two distinct measures are used: average and marginal emission
intensities, both with the units
kgCO2-eq
MWh 
. Average emissions correspond to the overall, for
example, region-wide, electricity production including net imports. The marginal reflects the emissions
of the marginal generator. The concepts are compared in Reference [
17
] and the importance of
distinguishing between the two is highlighted due to their very opposing patterns. It is emphasized
that the marginal emission is the most optimal signal to use for control.
In Reference [
8
], the average CO
2
emission intensity and price signals are used in heat pump
control of residential buildings in Norway (known for low emissions due to large amount of
hydropower) with Predictive Rule-Based Control (uses predefined thresholds to give information
about when the emissions are low). it is concluded that with price-based control, the overall CO
2
emissions have actually increased (evaluated using the average emissions). It is argued to result from
the load being shifted to the night time, where cheap carbon-intensive electricity is imported from the
continental European power grid. This is either a great example of the merit order dilemma or a result
that may have been different if marginal emissions had been used.
A recent study [
18
] investigated marginal emissions and uses estimates provided by Tomorrow (www.
tmrow.com) to develop a 24-hour forecast using a machine learning approach on historical data. The CO
2
estimates are calculated with the empirical method developed in Reference [
19
] using historical data
from European bidding zones. The chain of imports (the so-called flow tracing, originally introduced
in Reference [
20
,
21
]) is followed to assess the impact of a specific generator or load on the power
system. This is a large scale solution using data from the majority of bidding zones around the world.
In the present study,
MPC
is used for control for heating a building. This allows using knowledge
of future indoor climate states, CO
2
emissions and weather conditions to schedule heat pump
operation. It is a linear approach, which has its limits and requires simplifications, investigated
and discussed in Reference [
22
]. The simplifications include neglecting the effect on the efficiency
from factors, for example, frequency variations in the compressor (a main component in a heat pump)
and temperature variations. The paper concludes that neglecting these factors can lead to significant
errors. The frequency variations are, however, are not used in this study. From Reference [
22
],
the frequency is noticed to be the least important factor and is specifically justified when using varying
electricity prices, because the heat pump mostly operates at nominal speed to maximize the heat output
when prices are low. The impact from the outdoor temperature is accounted for—this is important
because it means the efficiency is lower during the night, where also the emissions are low.

Energies 2020,13, 2851 4 of 19
In order to model the heat dynamics in the building, a lumped dynamic process model is applied [
23
].
A tricky part is to determine the values of the parameters appropriately, for example, insulation level and heat
capacity: if the right type of measured data is available, the parameters can be estimated [
24
], or they can
be calculated according to physics. In the present study physics are used and a sensitivity analysis
is carried out to map the impact of parameters on the CO
2
savings potential. Such a sensitivity analysis
is lacking in the literature. In some papers transparency is lost, since the impact of the parameters is
not elucidated, thus increasing the risk of biased results. This paper addresses both of these issues
by using historic danish building codes from 1977 and later to describe the insulation thickness
as a parameter along with the heat pump size and thermal capacity of floor in two hypothetical
buildings: a family house and an office building. Further, the impact of using forecasts is assessed
by comparing the savings achieved with known future weather (perfect forecasts) vs. real forecasts.
It is noted that the emission saving potential using an
MPC
, that is, flexible demand, is measured
as CO
2
emission savings relative to classical thermostatic control, that is, non-flexible demand.
Hence, the results express only the potential of energy flexibility, not the absolute emission savings.
In Section 2, the weather data and marginal CO
2
emissions are presented. The dynamic process
model is presented in Section 3as an RC-diagram together with the
MPC
which is written up as a linear
programming formulation. The efficiency of the heat pump is modelled as a temperature-dependent
variable but neglects the compressor frequency. In Section 4the building codes, temperature settings
and model parameters are discussed. The results are presented in Section 5as graphs showing the CO
2
emission reductions vs selected parameters—for example, heat pump size, concrete and building
regulation year.
2. Data
Data provided by Tomorrow (www.tmrow.com) is used in the study. It comprises the marginal
CO
2
emission data and weather forecasts (temperature and solar irradiation) to model the building
thermodynamics and heat pump planning—see Figure 2. The emissions show close to none seasonality
except the winter, where the intensity peaks. Both the temperature and solar radiation are highly
seasonal with a hot climate and high solar radiation in the summer—reversed in the winter.
Only the CO
2
emissions are provided in both real-time values and forecasts. The real-time weather
conditions are, however, also essential for the model to describe the actual building thermodynamics.
The solar irradiation forecast is plotted in Figure 3(left plot) for the 21st of June. Every sixth hour a new
forecast is provided, at 2 a.m., 8 a.m., 2 p.m. and 8 p.m. The gaps between the forecasts are significant
and illustrate the inaccuracy of the prediction for long horizons.
For modelling purposes, the real-time weather conditions are modelled from the forecasts.
Assuming horizons for
h=
1 are the most accurate forecasts, a kernel smoothing process using
splines and a weight for short horizon favouritism is applied—see Appendix Afor a description
of the approach. The result for solar radiation is the smoothing curve seen in Figure 3, right plot.
The temperature forecasts did not show any significant gaps, suggesting all horizons, 1–6, are more
correct models of the real-time condition than the solar radiation.

Energies 2020,13, 2851 5 of 19
CO2Intensity
[kgCO2−eq/kWh]
100 250 400
Temperature [C]
−5 5 15
Irradiation
[W/m2]
Apr Jun Aug Oct Dec Feb
0 300 700
2017 2018
Figure 2.
Marginal CO
2
emissions (real-time), Temperature and solar irradiation (forecasts) plotted
for the evaluated period.
Figure 3.
Solar irradiation 6-hour horizon forecasts (left) for the 21st of June 2017 (updated at 2 a.m., 8
a.m., 2 p.m. and 8 p.m.). Model of the real-time irradiation derived from a kernel smoothing approach
on the updated forecasts (right).
3. Model
The model is a state-space model, derived from thermodynamic state equations describing the heat dynamics
in the building as a lumped dynamic process model [
25
]. The model parameters are defined based on the building
composition and structure. However, if the correct measurement data is available the model parameters could
be estimated as in Reference [
26
], thus it would be easy to use the applied control setup in existing buildings,
without the need for information about the building composition.

Energies 2020,13, 2851 6 of 19
3.1. Assumptions
It is assumed that the building is just one big room with a flat roof. Thereto the following assumptions
have been made; one uniform air temperature; no ventilation; no influence from the humidity of the air;
no influence from the wind. The heat pump is assumed to be static because its dynamics are much faster
than those of the building. Heat pumps require power to move heat from a cold space to a hot space using
a refrigerant (it extracts heat from a cold space through an evaporator and delivers that heat to the hot space
through a condenser). The heat pump efficiency is described using the COP-factor (Coefficient Of Performance).
The maximum efficiency is modelled as the Carnot Efficiency [27] by
COPCarnot =1−Tcold
Thot −1
, (1)
where
Thot
represents the condensation temperature, which is the temperature of the water flowing in the heating
system (fluctuating in reality, but simplified to a constant
=
40
◦
C ).
Tcold
is the ambient evaporation temperature.
However, the COP-factor is smaller in reality and therefore it is multiplied by another efficiency
η
, which can
be assumed to be between 50% and 70% [
28
]. To be conservative, it has been set to 50% in the calculations.
Therefore, the COP factor is expressed as
fcop(Thot,Tcold) = η·COPCarnot . (2)
3.2. State Space Equations
The state-space model is defined by dividing the building into three sections; ’Floor’, ’Interior’,
and ’Inner envelope’. Inner refers to the part of the walls and roof that is on the inside of the insulation.
The goal is to determine the temperature dynamics in the three sections. Thermodynamics can very well
be explained in the same way as electric circuits, which will provide a nice analytic approach. The model is shown
as the commonly used RC-diagram [24], in Figure 4.
Cf
Tf
(1−Ψh)Φh
(1−Ψs)gAΦs
Rf,i
Ci
Ti
ΨhΦh
ΨsgAΦs
Ri,e Re,a
Ce
Te
−
+Ta
Floor Interior Envelope Ambient
Figure 4. RC -network diagram of the model (floor heating).
dTf
t
dt =1
Cf Ti
t−Tf
t
Rf,i + (1−Ψs)gAΦs
t+ (1−Ψh)Φh
t!(3)
dTi
t
dt =1
Ci Tf
t−Ti
t
Rf,i +Tw
t−Ti
t
Ri,e +ΨsgAΦs
t+Φh
tΨh!(4)
dTw
t
dt =1
Ce Ti
t−Tw
t
Ri,e +Ta−Tw
t
Re,a !(5)
Refer to the nomenclature in Section 1for variable definitions.
Ψh
is a logic variable integer (0,1) defining
the heat system (
Ψ=
1 for radiators and
Ψ=
0 for floor heating). The heat delivered to the respective zones from
the heat pump is
Φh
t=fcop(40, Ta
t)·Pe
t, (6)

Energies 2020,13, 2851 7 of 19
where
Pe
t
is the electrical power used by the heat pump. The solar irradiation
Ps
t
is going through the windows
and heating both the floor and the room. Ψsdescribes the fraction that heats the room.
3.3. MPC
The model is transformed from continuous into discrete-time, see Reference [
24
]. The discrete-time linear
state space model is written as
xt=Axt−1+But−1+Edt−1(7)
y=Cxt−1+et−1, (8)
where
x
is the state vector (building temperatures) and
u
is the controllable input vector describing is the electrical
power to the heat pump, since the heat output from the heat pump,
Φh
, is a function of both the power input
signal and the COP factor.
d
is the disturbances, which in this case is the outdoor temperature and solar irradiation.
ytis thus the controllable variable Ti
tplus some error et. Hence
xt=


Ti
t
Tf
t
Tw
t


ut=hPe
tidt="Ta
t
Gt#. (9)
The matrix
A
states the dynamic behavior of the system, whereas matrix
B
specifies how the controllable
input signals enter the system, and
E
specifies the uncontrollable input signals. Furthermore,
C
is a constant
matrix that specifies the controllable state(s), in this case,
C=h100i
. For a deeper explanation, please refer
to References [6] and [24].
The MPC then becomes a linear programming problem formulated as
arg min
us,vk
N
∑
k=1
λt+kut+k+pkvt+k
subject to Xs+1=AX s+Bus+EDs
Tmin ≤CXs+1+vs+1
Tmax ≥CXs+1−vs+1
0≤us≤Pmax
vs≥0
∀s∈ {t,t+1, t+2, ..., t+N}
, (10)
where
λt+k
is the penalty at time
t+k
, which in this case is the marginal CO
2
emission intensity.
N
is the prediction horizon. At each sampling time, the linear program is solved to obtain the heating schedule
[ut+1
,
ut+2
, ...,
ut+N]
. P
max
is the maximum power input signal the heat pump can receive. As it may not always
be possible to meet the temperature demand, a slack variable
vk
is introduced and connected to the violation
penalty pv. This value is set relatively high to avoid temperature violations.
This linear program is solved using lpsolve interfaced with the R-package lpSolve, [29].
4. Inputs for the Model
In this section, the reference buildings and input data are presented.
The Danish building codes specify the building law requirements and contain detailed requirements for all
construction work. This study evaluates the building codes from 1977 to 2018, denoted BC
year
. The minimum
requirements for outer walls, roof, windows and doors are listed by year in Table 1.
Two different types of buildings are considered; a family house and an office building, they differ
in size and minimum temperature time settings during a nightly setback. The night time setpoint is 18
◦
C
for (11 p.m.:5 a.m.) and (6 p.m.:7 a.m.) respectively for the family house and office building. During the day,
the set point is 20
◦
C . The specific model parameters for each building are listed in Table 2for a default case
complying with BC2015:2018.
For the sake of simplicity, the buildings are squares with one story. Typical building part constructions
are used, illustrated in Appendix B. The thickness of the concrete layers will be varied in the analysis.
The construction material properties are listed in Table A1 in Appendix C, where also the building dimensions
and model parameters are defined.

Energies 2020,13, 2851 8 of 19
The windows are defined as equal sizes on each side of the house pointing in north, east, south and west
respectively with a window-to-wall ratio of 0.11 (the proportion of the wall that is windows). The R-package solaR
is used for solar radiation inclination angle calculations. Ψsis set to 0.1 as in Reference [6].
Dimensioning of the heat pump is based on the heat loss from the buildings, specified in Appendix C.
Table 1.
Building code requirements to insulation properties by year. Windows requirements are
speficied with both
U
-values, glazing (
g
) and the estimated net heat transfer through the window into
the room, Eref (follows; Eref =194.4 g−90.36 U).
BC Walls Roof Doors Windows
Year U hW
m2KiUhW
m2KiUhW
m2KiUhW
m2KiEre f hkWh
m2ig[−]
1977 1 0.45 3.6 3.6 −174.314 ** 0.777 **
1979–1985 0.4 0.2 2 2.9 −117.378 ** 0.744 **
1995–1998 0.4 0.2 2 2.3 −69.934 ** 0.709 **
2008 0.4 0.2 2 2 −46.909 ** 0.688 **
2010 0.3 0.2 2 1.8 ** -33 0.673 **
2015–2018 0.14 *0.1 *2 1.6 ** −17 0.654 **
*U
-values from ROCKWOOL A/S [
30
] needed to comply with the building envelope
requirements. For the considered buildings the given U-values in BC
1977:2010
are strict
enough to comply with the respective envelope requirements.
**
Estimated values: From 2010, the windows insulation properties are described by
Eref
.
An exponential relationship between
U
and the glazing (
g
) is found from key numbers
of different window types to be;
U=
0.0205
e6.6545g
[
31
]. This essentially allows
U
and
g
to be calculated from Eref.
Table 2.
Building dimensions and model parameters for both buildings based on BC
2018
—calculations
are as specified in Appendix C.
Family House Office Building
Afm2156 1250
Awm2107 302
Adoors m24 13
Awindows m214 39
Re,a K
kW 10.398 2.379
Ri,e K
kW 1.190 0.269
Rf,i K
kW 1.442 0.180
CekWh
K7.508 39.527
CfkWh
K3.198 25.623
CikWh
K0.876 6.944
* The window and door area is determined from a window-to-wall ratio of 0.11 and a door-to-wall ratio of 0.04.
Forecasts
24 h horizon CO
2
emission forecasts presented in the related paper, [
18
], are used. The real-time values are
presented in Section 2along with estimated real-time values of the temperature and solar radiation. The weather
forecasts have horizons of 24 h too.
To evaluate the MPC and the impact of using forecasts, different extreme cases are defined:
•CaseIdeal
: This takes the exact value of a future CO
2
emission intensity as prediction hence, a perfect forecast.
This provides an upper limit of CO2savings.
•CaseReal
: This takes the CO
2
emission forecast developed in Reference [
18
] and represents the performance
of the MPC with real forecasts.
•CaseTrivial
: This makes no use of forecasts and will thus result in a non predictive controller that simply
controls the heat pump keeping the temperature at the lower limit if possible.

Energies 2020,13, 2851 9 of 19
5. Results and Discussion
In this section, various conditions and parameters are evaluated, for example, effect of horizon length, heat
pump size, insulation and concrete thicknesses. The radiator and floor heating system is compared throughout
the analysis along with the family house versus the office building. The criteria to be optimized is the CO
2
emission savings. The total emissions are calculated from:
Λ(Case) = ∑n−N
t=1uCase,tλt
, where
n
is the number of
data points presented in Section 2(
n=
8688) and the ’Case’ denotes one of the three cases defined in Section 4.
The savings are thus calculated as the flexibility index [32] by
Savings(Case) = Λ(CaseTrivial)−Λ(Case)
Λ(CaseTrivial)(11)
As previously noted, this measure indicates the relative savings from utilizing the energy flexibility, hence not
the absolute savings. If not otherwise stated the results are calculated with a building complying with BC
2018
,
see Table 1.
The parameters that will be investigated are:
•Heating system and varying set points:
Both radiator and floor heating are considered and the use of
varying set points (lower temperature during the night).
•Horizon of forecasts:
To get an idea about how long horizons actually are needed to get a well
performing MPC.
•Size of heat pumps:
Essential for comparing the buildings and to know whether the potential is reached.
Also economically, this is important because as the price increases with larger heat pumps. This will
become a compromise between price and CO
2
emission. The default values for the family house and office
are the minimum sizes required to meet the heat demand on the coldest day (
−
12
◦
C ); 3 and 13 kW
heat
respectively Appendix (see Cfor calculations). Requirements are thus a 1 and 4.3 kW input signal respectively
according to Equation (1).
•Insulation and concrete thicknesses:
These will be adjusted to see the impact of levels of insulation and heat
capacity. The default thicknesses and material properties are shown in Appendix C.
[°C]
18 20 22
[kW]
0 1
CaseTrivial
Floor heating
[gCO2−eq/kWh]
0.1 0.3
Room [°C]
Floor [°C]
Wall/roof [°C]
Minimum [°C]
Maximum [°C]
Power [kW]
CO2 [gCO2−eq/kWh]
CaseReal
Floor heating
[°C]
18 20 22
[kW]
0 1
Mar 4 Mar 5 Mar 6 Mar 7
Radiator heating
[gCO2−eq/kWh]
0.1 0.3
Mar 4 Mar 5 Mar 6 Mar 7
Radiator heating
Room [°C]
Floor [°C]
Wall/roof [°C]
Minimum [°C]
Maximum [°C]
Power [kW]
CO2 [kg CO2−eq/kWh]
Figure 5.
Varying temperature constraints–night [11pm:5am]; 18
◦
C . Note the CO
2
emission intensity
and heating does not follow the Y-axis range, but rather the specified range in the colour legend.
An example of the differences between the cases, and the radiator and floor heating, is illustrated in Figure 5.
The resulting electrical power and temperature for Case
Trivial
and Case
Real
on a four day period for the family
house is shown. The result of Case
Trivial
is slightly different for the two heating systems. With radiators, it needs
to heat more continuously than the floor heating throughout the day. This is because the radiators transfer the heat
directly to the internal air, and not through the large heat capacity in the floor, resulting in a much faster response .
In both cases, the heating is switched off during the night time to reach the lower set point. However, the floor
heating violates the temperature restrictions more during the morning while heating the house, which is due to its
slow response. Case
Real
seeks to only switch on the heat pump during low emission periods. The radiator system

Energies 2020,13, 2851 10 of 19
does this well, but it is clearly limited by the maximum indoor temperature limit and the power input decays
immediately to avoid temperature violation. In the floor heating system, the heat pump can operate at full load
for a longer time using the floor as storage. An interesting point is that using day and night profiles, Case
Real
has
no benefits of letting the temperature drop during the night because of: i) the temperature response is too slow
and ii) the emissions are usually lowest during the night, so this is the best time to use the heat pump. Contrarily,
the indoor temperature in the radiator system occasionally drops during the night if there is no significant drop
in CO2emissions.
As expected, throughout almost a year (
n=
8688 h) the floor heating system provides slightly more flexibility
and reaches savings of 11% against 9% using radiators for the whole year for CaseReal.
hours
4 8 12 16 20 24
Savings [%]
−5 0 5 10 15 20
Family house
hours
4 8 12 16 20 24
Real
Ideal
Floor:
Radiator:
Office building
Figure 6.
Savings with respect to the forecast horizon. Shown for both an ideal (perfect) and the
Real forecasts.
Time
4:00 8:00 12:00 16:00 20:00
0.16 0.185 0.21
[gCO2−eq/kWh]
Power [kW]
0 0.2 0.4
CaseTrivial [kW]
CaseReal [kW]
CO2 [gCO2−eq/kWh]
Figure 7.
Average heat pump load in a family house with floor heating (200 mm floor concrete)
versus the hour of the day for both CaseTriviall and CaseReal. Hourly average marginal CO2emissions
are shown in green.
The control horizon needs to be sufficiently long for the MPC to provide flexibility to the system (Figure 6).
Note, the savings from floor heating become negative when using low control horizons. The nightly setback
causes that; Case
Trivial
switches on at six AM every morning and the emission peak is happening already at four
AM (Figure 7). For example, when using a two-hour control horizon, the heat pump will be forced to switch on at
four AM instead and thus increase the emissions.
Interesting to note is the changing behaviour of the curves around the eight hours horizon: Case
Ide al
in the family house with radiators has no savings up until this point. Like any other energy storage, a loss
is introduced, in this case, by an increase in temperature resulting in a higher heat loss. Therefore, the MPC
will only store heat if the CO
2
variations are large enough for the resulting emissions to break even with

Energies 2020,13, 2851 11 of 19
the increased losses. These variations become sufficiently large around eight hours. This is less of a problem
for the floor heating system, as the loss is much lower. This behaviour is less pronounced in the office building
with radiators because the volume to surface area increases with larger buildings, hence the heat capacity increases
relative to the surface area.
The loss in savings due to forecast errors, can be found from the difference between Ideal and Real.
The radiator system is close to reaching its full potential, where the floor heating still can improve maximum
about 5% savings from better forecasts.
Power input [kW]
1 2 3 4
Savings [%]
10 15 20
Family house
Power input [kW]
4.5 7.5 10.5 13.5
Real
Ideal
Floor:
Radiator:
Office building
Figure 8.
CO
2
emission reduction as a function of the size of the heat pump in kW shown
for the real forecasts.
Finally, it is noted, that there is still an increase in savings at the 24 h horizon, for all scenarios, indicating
that even longer horizons will lead to further increase in savings.
Increasing the heat pump size can lead to slightly higher savings (Figure 8), as more heat can be produced
around the small-time slots with low emission. This is especially true when using floor heating, due to the high
heat capacity in the floor. For the radiator systems, there is no significant gain for either the family house nor
the office building. For the family house with floor heating, there is a relatively large gain when considering
Case
Ide al
, while Case
Real
only reaches a slight improvement. There are significantly higher savings to achieve from
Case
Real
in the office building; because of the larger floor to wall area ratio (more heat capacity relative to the area
the heat can escape through), a larger heat pump can accumulate more heat and thereby increase the flexibility.
The impact from insulation is evaluated with respect to the development of building codes from 1977
and forward, which has been an increase in insulation, window and door requirements (see Table 1for building
code specs and corresponding physical values). Another important aspect is the concrete thickness in the floor
because it increases the heat capacity and thus the heat storage capabilities. Generally, BC
1977
houses have very
minimal because of the low insulation restrictions—significantly improved in BC
1979
(Figure 9and Table 3).
However, the office building using floor heating can provide savings of around 9% with 200 mm of floor concrete
and following BC
1977
restrictions. This is due to the floor to wall ratio—the larger it is, the more the concrete
thickness in the floor can contribute, and the less the insulation in the walls contributes.
For radiator heating, the concrete thickness in the floor is not very important for the savings. Still, adding
50–80 mm can increase the savings by around 2.5–4% (Table 3), but any thicker layer will not increase the savings
at all (i.e., the red colored area in the graphs for radiator heating can hardly be seen— Figure 9). The insulation
thickness in the office building using radiators seems saturated after BC
1979
. This may seem counter-intuitive,
but in reality, with high thermal resistance, the model becomes more rigid and therefore, less flexibility is allowed
to occur. Of course, the absolute CO2emission savings will increase with more insulation.
Floor heating in a thick concrete floor provides the most flexibility in both buildings (around 16–17% in both
with 200 mm concrete and BC
2015:2018
). This setup exploits the full potential of the large heat storage in the floor,
but lacks to exploit the nightly decrease in set point, because of the slow temperature response. With only 10 mm
of concrete, the savings are almost identical for radiator and floor heating (about 6% with BC
2015:2018
in all cases).
Note that a concrete thickness of 10 mm is not common practise.

Energies 2020,13, 2851 12 of 19
Floor heating
Savings [%]
Family house
5 10 15
Radiator heating
10
200
Concrete [mm]
Building Code
1977 1995−1998 2010
1979−1985 2008 2015−2018
Office building
5 10 15
1977 1995−1998 2010
1979−1985 2008 2015−2018
Figure 9.
Savings with respect to the building code year and concrete thickness in the floor. Refer to Table 1.
Table 3.
Min. savings are derived from a concrete thickness of 10 mm. Max. savings are derived from
concrete thicknesses of 200 mm by default.
Floor Radiator
BC1977 BC2015:2018 BC1977 BC2015:2018
Family house
Min. savings 0.5% 9.4% 1.8% 9.9%
Max. savings 2.7% 17.4% 2.8% 12.4%
Office building
Min. savings 1.4% 7.8% 2.4% 8.1%
Max. savings 9.2% 16.0% 4.1% 12.3%
* Optimal concrete thickness of 50 mm. ** Optimal concrete thickness of 80 mm.
This study only measured the relative savings, but the absolute savings will, of course, also increase with
more insulation. Therefore, if a house is not well insulated, the first attempt to decrease CO
2
emissions should
be to reduce the heat demand by adding insulation before trying to optimize the control. As a result of new
regulations in BC
1979
, buildings in Denmark built before 1979 have been increasingly re-insulated to decrease
the heat demand and costs. In Reference [
33
], it is found that the actual heat demand is on average lower
than the theoretical calculations based on registered building data. This could imply that buildings indeed
are re-insulated without further registration.
Most buildings in Denmark are built before 1979 [
4
], and this group is therefore the best representative
for the potential buildings—floor heating is rarely the main heating system in this group. It would thus
follow the early end of the radiator curves in Figure 9with re-insulated buildings likely to comply with BC
1979
.
Therefore, between 4% and 7% of savings can be expected. However, new buildings complying with BC
2015−2018
envelope requirements and floor heating installed may reach savings of nearly 17%. Often, buildings combine
radiators and floor heating and rely not only on one or the other, thus in reality have the advantages from
both systems can be utilized depending on the heated rooms. Recall from Figure 5, the radiators are good
at quick responses and therefore allows the temperature to decrease during the night contrarily to the floor
heating. However, it has little capability of storing heat for more extended periods. The potential for this is open
to further studies.
The average daily load for a BC
2015−2018
family house has shifted significantly (Figure 7). The trivial control
follows to a certain degree the emissions throughout a day. The natural need for heating is therefore very
inconvenient for the emissions and illustrates why energy flexibility is important. The MPC smooths the load
during the day, decreasing it at otherwise peak hours and shifts most of the load to operate at midnight despite
the lower temperature set point. This illustrates the importance of predictive control—if all houses follow the same
schedule, there will be a need for much more additional storage capacity.

Energies 2020,13, 2851 13 of 19
In general, this study can be relevant to buildings and scenarios with similar characteristic and climate
conditions. Especially in the Nordic countries where the heating system comprises a large chunk of the electricity
consumption. In warmer climates the study could be converted into air conditioning rather than heating. The wind
and solar power production share was 36% in the examined region for the one year period in 2017 and 2018.
The results will change depending on the conditions for example, with higher levels of fluctuating renewable
generation in the future [
34
]. In a 100% renewable scenario it is of course not meaningful to use the CO
2
emissions
as minimizing objective, however utilizing energy flexibility will be vital [35].
Model Simplification
The results are based on calculations using a simplified building model of a simplified building with only
a single room. If more rooms are considered, it would add more heat capacity from walls dividing the rooms.
This could increase the savings. Furthermore, a storage tank could have been added, which also could increase
the savings as it would boost energy flexibility.
Disturbances, other than solar radiation and ambient temperatures, have been neglected in this study, for
example, human activity. It is left as a point for future studies to include and assess the impact of building usage
in the models and analysis.
6. Conclusions
The potential of achieving CO
2
savings using the energy flexibility of buildings has been analysed in a range
of relevant scenarios including, both a family house and an office building. The simulated buildings were heated
with heat pumps and the characteristics were varied according to the historical development of Danish building
codes. The CO
2
level of the electricity generation for a year-long period in the Danish area DK2 was used as input,
together with weather data from the area.
The results show considerable CO
2
savings for both radiator and floor heating systems. Forecast horizons
should be 24 h or longer to obtain the full savings potential. Over-dimensioning the heat pump to increase
flexibility turns out only to yield significant savings with floor heating, especially in the office building due to its
larger storage area relative to the wall area.
Complying with BC
2015:2018
requirements, savings can reach around 17% using floor heating with 200 mm
of floor concrete and 12% using radiators (Table 3). Generally, buildings must comply with building codes later
than 1977 to achieve any considerable savings due to the low requirements in BC1977.
Predictive control is vital to eliminate the peak hours, especially happening in the morning after the nightly
setback, which is common practise. It is able to shift the demand to periods, where the coal power production
is low, typically out of cooking peaks and during night time.
Author Contributions:
conceptualization, R.G.J. and P.B.; methodology, K.L. and A.T.; software, R.G.J., K.L.
and R.E.; validation, K.L.; formal analysis, K.L.; investigation, K.L.; resources, R.G.J., P.B. and H.M.; data curation,
K.L. and O. Corradi; writing—original draft preparation, K.L.; writing—review and editing, R.G.J., P.B., O.C., R.E.
and H.M.; visualization, K.L.; supervision, R.G.J. and P.B.; project administration, P.B., H.M.; funding acquisition,
P.B., H.M. All authors have read and agreed to the published version of the manuscript.
Funding:
This research was supported through the project “Smart Cities Accelerator 2016–2020” funded by the EU
program Interreg Öresund-Kattegat-Skagerrak, the European Regional Development Fond and the CITIES project
(DSF1305-00027B).
Acknowledgments:
We are thankful for Tomorrow (www.tmrow.com) who has provided the data used in this
study (including emission calculations for the bidding zone DK2).
Conflicts of Interest:
The authors declare no conflict of interest. The funders had no role in the design of the study;
in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish
the results.
Abbreviations
The following abbreviations are used in this manuscript:
MDPI Multidisciplinary Digital Publishing Institute
CHP Combined Heat and Power
MPC Model Predictive Control

Energies 2020,13, 2851 14 of 19
Appendix A. Kernel Smoothing
The 1 to 6 h horizon solar radiation forecasts are smoothed in order to remove large shifts in value, thus
providing a better representation of real conditions.
Assuming horizons for
h=
1 are the most accurate forecasts, a kernel smoothing process using a weight for
short horizon favouritism is applied. The kernel weight, w1, is defined as the Epanechnikov kernel;
w1=3
4(1−u2), (A1)
where
u=|xi−x|
b
.
x
is a vector
[
1, 2, ..,
n]
, where
n
is the number of data points and
b
is the bandwidth. The short
horizon favoritism weight is defined as
w2=e−a
h−1, (A2)
where
h∈[
1, 2, ..., 6
]
represents the hours in advance to the observation the forecast was received. Together,
w1
and w2define the final weight function; w=w1·w2. This is applied into a linear regression model
y=Xβ+e, (A3)
for e∼N(0, σ2I),
where
X
is the input matrix (explanatory variables; hour, day and month),
y
is the output vector (response
variable: Solar irradiation) and
β
is a vector of regression coefficients to be found.
e
represents the normally
distributed errors in the model.
The least square regression is performed to minimize
S(β) = ||y−Xβ||2
and obtain the weighted least-squares solution
β= (XTwX)−1XTwy. (A4)
where
W
is the weight vector
w
. Using
a=
1.5 and
b=
7, the results are illustrated in Figure A1 (left plot). This is
not sufficient on its own, as it does not manage to capture the curvature and midday peak. Therefore, the hour is
converted into base splines (local polynomials between specified points called knots [36]);
bs(hourt) = hbs0(hourt)bs1(hourt). . . bsd f (hourt)i>. (A5)
where
hourt
is the hour corresponding to the time step
t
.
d f
is the degrees of freedom (essentially the number of splines;
the higher the better it will fit the actual values). Using d f =7, the final result is illustrated in Figure A1 (right plot).
●
●●●●
●
●
●
●
●
●
●●
●●
●
●
●
●
●●●●●
Irradiation [W/m2]
0:00 8:00 16:00 23:00
●
●●●●●
●
●
●
●
●
●●
●●
●
●
●
●
●●●●●
0:00 8:00 16:00 23:00
Figure A1.
Solar irradiation 6 h horizon forecasts for the 21st of June 2017 (updated at 2 a.m., 8 a.m.,
14 p.m. and 20 p.m.) and estimates of the real time irradiation (solid line) w/o splines (left). Estimates of
the real-time irradiation using splines and the kernel smoothing approach (right)

Energies 2020,13, 2851 15 of 19
Appendix B. Building Parts
The buildings consist of three parts; the walls, the floor and the ceiling. The material composition in each
part is illustrated in Figure A2. Thereto, the model parameters (heat capacities and thermal resistances) are defined
in Figure A3. The wall and roof are divided into the inner and outer part where only the inner temperature
is modelled. The inner part is everything between the room and the insulation.
Figure A2. Typical constructions of walls (top), floor (middle) [37] and roof (bottom) [38].

Energies 2020,13, 2851 16 of 19
Figure A3.
Definition of model parameters (heat capacities and thermal resistances) in the roof (left),
floor (middle) and wall (right).
Appendix C. Parameter Estimation and Heat Pump Dimensions
The model parameters are defined illustratively in Figure A3. The materials used and corresponding properties
are listed in Table A1.
Table A1.
Thickness (
ζ
), density (
ρ
), heat capacity (
C
), thermal conductivity (
k
) and thermal resistance
(
R
) of all the building parts. The thermal resistance is;
R=ζ
k
. Thicknesses are examples for a
BR18 building.
Wall
Roof
Floor
Index Material ζ[m] ρ[kg
m3]C[J
kg·K]k[W
m·K]R[m2·K
W]
Surface, outer w,s,o - - - - - 0.06
Outer w,o Bricks 0.15 1920 790 0.9 0.167
Insulation w,insul Rockwool 0.12 240 710 0.042 2.693
Inner w,i
Concrete,
light weight 0.10 1600 840 0.79 0.127
Surface, inner w,s,i - - - - - 0.12
Surface, outer r,s,o Waterproof layer 0.01 0 0 0 0.06
Insulation r,insul Rockwool 0.25 144 1000 0.058 4.304
Air space r,air Air 0.05 1.225 1000 0.026 0.400
Concrete r,con
Concrete,
light weight 0.05 1600 840 0.79 0.063
Ceiling r,cei Plaster light 0.01 1680 840 0.81 0.012
Inner surface r,s,i - - - - - 0.16
Inner surface f,s,i - - - - - 0.11
Cover f,cov Plywood 0.01 545 1210 0.12 0.083
Concrete f,con
Concrete,
light medium 0.05 1600 840 0.79 0.063
Insulation f,insul Rockwool 0.30 240 710 0.042 7.143
Based on that, the parameters are calculated from Equation (A6) to (A15).

Energies 2020,13, 2851 17 of 19
The thermal resistances, Re,a,Re,i and Rf,i ;
Re,a =1
Uwindows ·Awindows +Udoors ·Adoors +Uw,a ·Aw+Ur,a ·Ar(A6)
Uw,a =1
Rw,s,o +Rw,o +Rw,insul +Rw,i
2
r(A7)
Ur,a =1
Rr,con
2+Rr,insul +Rr,s,o
(A8)
Re,i =1
Uw,i ·Aw+Ur,i ·Ar(A9)
Uw,i =1
Rw,i
2+Rw,s,i
(A10)
Ur,i =1
Rr,con
2+Rair +Rcei +Rr,s,i
(A11)
Rf,i =1
Uf,i ·Af(A12)
Uf,i =1
Rf,con
2+Rcov +Rf,s,i
(A13)
As for the heat capacities;
Ce
is the sum of the heat capacity [kWh] in all the material on the inside of the
insulation in the building envelope (concrete, air gap and ceiling).
Cf
is the sum of the heat capacity [kWh] in the
floor concrete and tiles.
Ci
is estimated from a key number of 20
kJ
K·m2
[
39
], accounting for everything inside the
room for example, air and furniture.
Ce=1
3.6E6(ζw,iρw,iCw,i ·Aw(A14)
+ (ζr,airρr,airCr,air +ζr,conρr,conCr,con +ζr,cei ρr,ceiCr,cei)·Ar)
Cf=Af
3.6E6(ζf,covρf,covCf,cov +ζf,con ρf,conCf,con)(A15)
Heat Pump Dimensions
The minimum heat pump sizes are estimated from the heat loss on the coldest day. This is defined in Danish
Standard DS-418 as an outdoor temperature of −12 °C with an indoor set point of 20 °C, thus dT =32 °C.
Qloss = (Uwalls ·Awalls +Uwindows ·Awindows (A16)
+Udoors ·Adoors +Uroof ·Aroof)·dT (A17)
Note, the floor is neglected because the model assumes no heat loss through the floor. The heat losses
are 2.9 kW and 13.4 kW for the family house and office building (danish building code of 2018) respectively.
The maximum power input to the heat pump Pmax is then fcop (40,−12)
Qloss , hence 1 kW and 4.5 kW.
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