The formula, as adapted from [28] , for the internal air temperature of the building after a time slot $t$ is given as by Equation (1):

$T_{int\left(t+1\right)}=T_{int\left(t\right)}-\left(Q_{loss\left(t\right)}-Q_{gain\left(t\right)}-H{P}_{SH\left(t\right)}\cdot y_{\left(t\right)}\right)\frac{\Delta t}{\Delta q}$ (1)

where:

$T_{int\left(t+1\right)}$ is the internal air temperature (˚C or K) of the building after a time slot $t$.

$T_{int\left(t\right)}$ is the internal air temperature (˚C or K) of the building in time slot $t$.

$Q_{loss\left(t\right)}$ and $Q_{gain\left(t\right)}$ are the heat loss (W) and heat gain (W) of the building in time slot $t$.

$H{P}_{SH\left(t\right)}$ is the heat output (W) of the HP in SH mode in time slot t.

$y_{\left(t\right)}$ is binary variable which determines the operational status (ON = 1 or OFF = 0) of the HP in SH mode in time slot $t$.

$\Delta t$ is the duration of the time slot in (s).

$\Delta q$ is the energy needed to change the internal air temperature of the building by 1˚C (J/˚C).

The heat loss of a building is the sum of heat loss through the fabric of the building (floors, walls, roof, windows and doors) and the heat loss due to ventilation/infiltration [29] . The heat loss, $Q_{loss\left(t\right)}$, of the building in time slot $t$ is given by Equation (2):

$Q_{loss\left(t\right)}=\left(\sum \left(AU\right)+0.3N_{ac}V\right)\times \left(T_{int\left(t\right)}-T_{ext\left(t\right)}\right)$ (2)

where:

$U$ is thermal transmittance (W/m²∙K).

$A$ is surface area through which heat transfer occurs (m²).

$N_{ac}$ is the number of air changes per hour (ac/h).

$V$ is the volume of the building (m³).

$T_{int\left(t\right)}$ is the internal air temperature (˚C or K) of the building in time slot $t$.

$T_{ext\left(t\right)}$ is external air temperature (˚C or K) in time slot $t$.

The heat gain, $Q_{gain\left(t\right)}$, of the building in time slot $t$ is given by Equation (3):

$Q_{gain\left(t\right)}=\left(Q_p\times N_p\right)+\left(A_{SW}\times SHGC\times S_{rad\left(t\right)}\right)$ (3)

where:

$Q_p$ is heat gain from one person (W).

$N_p$ is number of occupants.

$A_{SW}$ is area of window facing south (m²).

$SHGC$ is solar heat gain coefficient of window.

$S_{rad\left(t\right)}$ is solar irradiance (W/m²) in time slot $t$.

The energy needed to change the internal air temperature of the building is given by Equation (4):

$\Delta q=C_{air}\times {\rho }_{air}\times V$ (4)

where:

$C_{air}$ is specific heat capacity of air for typical room condition (J/kg∙˚C).

${\rho }_{air}$ is density of air (kg/m³).

$V$ is the volume of the building (m³).

The operational status, $y_{\left(t\right)}$, of the HP in SH mode is represented by Equation (5):

$y_{\left(t\right)}=\{\begin{array}{l}1=\text{ON},T_{int\left(t\right)}<T_{set}-T_{sg}\\ 0=\text{OFF},T_{int\left(t\right)}>T_{set}+T_{sg}\\ y_{\left(t-1\right)},T_{set}-T_{sg}\le T_{int\left(t\right)}\le T_{set}+T_{sg}\end{array}$ (5)

where:

$T_{int\left(t\right)}$ is the internal air temperature (˚C or K) of the building in time slot $t$.

$T_{set}$ is the set-point temperature of the internal air (˚C or K).

$T_{sg}$ is the swing temperature (˚C or K).

$y_{\left(t-1\right)}$ is the operational status of the HP in previous time slot.

In Equation (5), $T_{set}$ is the desired internal air temperature and therefore the thermostat set-point. If the actual internal air temperature, $T_{int\left(t\right)}$, drops below the temperature lower limit, $T_{low}$, which is the difference between $T_{set}$ and $T_{sg}$, then the HP is switched ON to raise the internal air temperature. Conversely, when the internal air temperature rises above the temperature upper limit $T_{up}$, which is the sum of $T_{set}$ and $T_{sg}$, the HP switches OFF. However, the operational status of the HP remains unchanged if the internal air temperature is between $T_{low}$ and $T_{up}$.

Ignoring losses, the heat output of the HP in SH mode is equal to the radiator output which is also equal to the condenser output. That is:

$H{P}_{SH\left(t\right)}=Q_{condenser\left(t\right)}=Q_{radiator\left(t\right)}$ (6)

where:

$H{P}_{SH\left(t\right)}$ is the heat output (W) of the HP in SH mode in time slot t.

$Q_{condenser\left(t\right)}$ is the condenser heat output (W) in time slot $t$

$Q_{radiator\left(t\right)}$ is the radiator heat output (W) in time slot $t$.

The heat flux inside the condenser of the HP can be expressed as:

$Q_{condenser\left(t\right)}=mc\left(T_{flow}-T_{return\left(t\right)}\right)$ (7)

where:

$m$ is the mass flow rate (kg/s) of water.

$c$ is the specific heat capacity (J/kg∙˚C) of water.

$T_{flow}$ is the operating temperature (˚C or K) of the working fluid reaching the condenser.

$T_{return\left(t\right)}$ is the temperature (˚C or K) of the working fluid leaving the condenser.

The heat output of the radiator can be expressed as:

$Q_{radiator\left(t\right)}=U_{rad}{A}_{rad}\left(T_{rad\left(t\right)}-T_{int\left(t\right)}\right)$ (8)

where:

$U_{rad}$ is the heat transmission coefficient (W/m²∙K) of the radiator.

$A_{rad}$ is the surface area (m²) of the radiator.

$T_{int\left(t\right)}$ is the internal air temperature (˚C or K) of the building in time slot $t$.

$T_{rad\left(t\right)}$ is the radiator temperature (˚C or K).

The radiator temperature, $T_{rad\left(t\right)}$, is the average of the temperature of the working fluid reaching the condenser ( $T_{flow}$) and the temperature of the working fluid leaving the condenser ( $T_{return\left(t\right)}$). That is:

$T_{rad\left(t\right)}=\frac{T_{flow}+T_{return\left(t\right)}}{2}$ (9)

From Equations (6) to (9) the return temperature, $T_{return\left(t\right)}$, can be expressed as:

$T_{return\left(t\right)}=\frac{T_{flow}\left(2mc-U_{rad}{A}_{rad}\right)+2U_{rad}{A}_{rad}{T}_{int\left(t\right)}}{U_{rad}{A}_{rad}+2mc}$ (10)

Based on test data, from the Heat Pump Test Centre WPZ, of 30 different models of ASHPs [22] , the expression for the Coefficient of Performance (COP) of HP can be deduced from the plot of COP against “ $T_{return}-T_{ext}$” with a coefficient of determination ( $R^2$ value) of 0.9797 by Equation (11):

$CO{P}_{\left(t\right)}=7.90471{\text{e}}^{-0.024\left(T_{return\left(t\right)}-T_{ext\left(t\right)}\right)}$ (11)

where:

$CO{P}_{\left(t\right)}$ is the coefficient of performance of the HP at time slot $t$.

$T_{ext\left(t\right)}$ is the external air temperature (˚C or K) at time slot $t$.

The COP-curve, which is here defined as the plot of COP against “ $T_{return}-T_{ext}$” derived from the test data, is shown in Figure 5 . In Figure 5 , there are 9 test points and the COP at a point is the average of COPs of 30 ASHPs at that point.

The actual electrical input, $P_{S{H}_{elect\left(t\right)}}$ (W), for the operation of the HP in SH mode is therefore given by Equation (12):

$P_{S{H}_{elect\left(t\right)}}=\frac{H{P}_{SH\left(t\right)}}{CO{P}_{\left(t\right)}}$ (12)

Two temperature regimes were used in the modelling. The set-point temperature, $T_{set}$_, of the HP between 00:00 hours and 10:30 hours is 18˚C with a swing temperature, $T_{sg}$, of 2˚C. Whereas $T_{set}$ between 11:00 hours and 23:30 hours is 20.5˚C with a $T_{sg}$ of 3˚C.

In the model formulation, single-node state is assumed since there is no occurrence of draw event large enough to trigger the transition from single-node state into two-node state. A hot water tank remains in single-node state and only changes into two-node state when a considerable volume of water is drawn in a usage event which occurs in a short interval of time [30] . In single-node state, the water in the tank is considered as a single mass of body with the heat and temperature of the water uniformly distributed. Therefore, the water in the tank is not stratified after a draw event into upper layer warm water and lower layer cold water from the inlet that replaces the drawn water. Figure 6 shows the DHW tank in single-node state as modelled in this work.

The temperature, $T_{\left(t\right)}$, of the water leaving the tank is the average temperature of the hot water inside the tank. The tank is refilled with inlet water at temperature, $T_{in}$, to replace the drawn water. The inlet water mixes with the hot water inside the tank and a new average temperature, $T_{\left(t+1\right)}$, is formed for the next water draw event. The heat (W) available inside the tank after a water draw event in time slot $t$ can be expressed in terms of heat balance equation as follows:

$Q_{\left(t+1\right)}=Q_{\left(t\right)}-Q_{use\left(t\right)}-Q_{aml\left(t\right)}+H{P}_{DHW\left(t\right)}\cdot z_{\left(t\right)}$ (13)

where:

$Q_{\left(t+1\right)}$ is the heat (W) remaining after a water draw event.

$Q_{\left(t\right)}$ is the heat (W) available before the water draw event.

$Q_{use\left(t\right)}$ is the heat (W) loss due to the water draw event.

$Q_{aml\left(t\right)}$ is the heat (W) loss to the ambience due to heat dissipation from the tank the to the environment.

$H{P}_{DHW\left(t\right)}$ is the heat output (W) of the HP in DHW mode in time slot $t$.

$z_{\left(t\right)}$ is binary variable which determines the operational status (ON = 1 or OFF = 0) of the HP in DHW mode in time slot $t$.

The heat balance equation in (13) can be written in terms of volume and change in temperature as follows:

$\begin{array}{c}\frac{Vc\left(T_{\left(t+1\right)}-T_{in}\right)}{60t}=\frac{Vc\left(T_{\left(t\right)}-T_{in}\right)}{60t}-\frac{V_{use\left(t\right)}c\left(T_{\left(t+1\right)}-T_{in}\right)}{60t}\\ -U_{ta}{A}_{ta}\left(T_{\left(t\right)}-T_{int\left(t\right)}\right)+\frac{Vc\left(T_{flow}-T_{return\left(t\right)}\right)}{60t}\cdot z_{\left(t\right)}\end{array}$ (14)

where:

$V$ is the volume (l) of the tank.

$V_{use\left(t\right)}$ is the volume (l) of the hot water used in time slot $t$.

$T_{\left(t\right)}$ is the temperature (˚C or K) of hot water inside the tank in time slot $t$.

$T_{\left(t\right)}$ is also equal to the return temperature, $T_{return\left(t\right)}$, of the working fluid.

$T_{\left(t+1\right)}$ is the temperature (˚C or K) of hot water inside the tank after the water draw event.

$T_{in}$ is the temperature (˚C or K) of the inlet cold water.

$U_{ta}$ is the heat transmission coefficient (W/m²∙K) of the tank.

$A_{ta}$ is the surface area (m²) of the tank.

$c$ is the specific heat capacity of water in kJ/kg∙˚C i.e. 4.184 kJ/kg∙˚C.

$T_{int\left(t\right)}$ is the internal air (˚C or K) of the building in time slot $t$.

$T_{flow}$ is the operating temperature (˚C or K) of the working fluid.

$t$ is the duration of the time slot in minutes.

The operational status, $z_{\left(t\right)}$, of the HP in DHW mode is represented as follows:

$z_{\left(t\right)}=\{\begin{array}{l}1=\text{ON},T_{\left(t\right)}<T_{set\left(W\right)}-T_{sg\left(W\right)}\\ 0=\text{OFF},T_{\left(t\right)}>T_{set\left(W\right)}+T_{sg\left(W\right)}\\ z_{\left(t-1\right)},T_{set\left(W\right)}-T_{sg\left(W\right)}\le T_{\left(t\right)}\le T_{set\left(W\right)}+T_{sg\left(W\right)}\end{array}$ (15)

where:

$T_{\left(t\right)}$ is the temperature (˚C or K) of hot water inside the tank time slot $t$.

$T_{set\left(W\right)}$ is the set-point temperature (˚C or K) of hot water inside the tank.

$T_{sg}$ is the swing temperature (˚C or K).

$z_{\left(t-1\right)}$ is the operational status of the HP in previous time slot.

Substituting for constant and solving for $T_{\left(t+1\right)}$ in Equation (14) yields:

$T_{\left(t+1\right)}=\frac{V{T}_{\left(t\right)}+V_{use\left(t\right)}{T}_{in}-0.0143t{U}_{ta}{A}_{ta}\left(T_{\left(t\right)}-T_{int\left(t\right)}\right)+V\left(T_{flow}-T_{\left(t\right)}\right)\cdot z_{\left(t\right)}}{V+V_{use\left(t\right)}}$ (16)

The set-point temperature, $T_{set\left(W\right)}$, of the HP for DHW is 50˚C with a swing temperature, $T_{sg\left(W\right)}$, of 5˚C. The hot water set-point temperature and the swing temperature are such that will prevent the growth of Legionella bacteria inside the tank. Legionella bacteria mostly thrives in the temperature range between 20˚C and 45˚C [31] .

The COP of the HP while working in DHW mode is as expressed in Equation (11) with $T_{return\left(t\right)}$ substituted by $T_{\left(t\right)}$. The actual electrical input, $P_{DW{H}_{elect\left(t\right)}}\left(W\right)$, for the operation of the HP in DHW mode is given by:

$P_{DW{H}_{elect\left(t\right)}}=\frac{H{P}_{DHW\left(t\right)}}{CO{P}_{\left(t\right)}}$ (17)

To ensure validity of the developed HP operational model, empirical data from credible sources were used as inputs to run the model. Decision on the number of occupants per household was based on [32] . Average daily DHW requirement of household in litres/day was estimated in line with technical guidelines from [33] and it is given by Equation (20):

$H{H}_{dail{y}_{DHW}}=25N_p+36$ (20)

where:

$H{H}_{dail{y}_{DHW}}$ is the average daily household DHW requirement in litres.

$N_p$ is the number of occupants in the household.

Normalized DHW tapping profile from [34] was used to estimate the actual DHW draw at any time of the day. Figure 9 illustrates the normalized DHW tapping profile. Data on geometric and constructional characteristics of the hot water tank came from [35] . Data about buildings parameters which consist of building type and size and U-values of building elements were from [36] and [37] respectively. Weather data were from [38] .

The model outputs, typical winter weekday and typical summer weekday average electricity demand of HPs expressed in half-hourly intervals as seen in Figure 8 , were compared with the measured daily average electricity demand of HPs, shown in Figure 10 , in the Carbon, Control and Comfort (CCC) project [14] . The comparison between the model outputs and the actual measured outputs of the CCC project showed close similarity in trends and kW values of the HPs daily average electricity demand profile. The outputs of both the model and the actual field project have morning demand spikes around 8.30 hours of 1.3 kW and 0.4 kW in both their winter and summer average electricity demand profiles respectively. This gives reasonable credence to the usefulness of the developed model. Figure 10 is the screenshot from CCC Project of average HP demand. The midnight peak observed in Figure 10 but not in Figure 8 is due to the fact that the HPs in the CCC project operate a weekly pasteurization cycle (raising the DHW temperature above 60˚C to kill Legionella bacteria) which always takes place at midnight [14] .