TP-04: Space Heating Demand | HEM Reference

Overview

The space heating demand calculation determines how much energy must be delivered to (or removed from) each thermal zone in a dwelling to maintain the specified setpoint temperature. It is the central calculation in the Home Energy Model, drawing together fabric heat loss (TP-05), ventilation (TP-06), thermal mass (TP-07), and solar gains (TP-08) into a single heat balance solved at each timestep.

The methodology follows BS EN ISO 52016-1:2017, section 6.5. A heat balance matrix equation is assembled for all thermal nodes in the zone, including one node per layer of each building element plus one node for the internal air. The matrix equation is solved simultaneously to yield node temperatures and, by comparison with the setpoint, the heating or cooling demand.

Inputs

Parameter	Symbol	Unit	Description
Zone floor area	$A_{z o n e}$	m²	Useful floor area of the thermal zone
Zone volume	$V$	m³	Total volume of the thermal zone
External air temperature	$θ_{e x t}$	°C	From climate data (TP-03)
Heating setpoint	$θ_{se t, h e a t}$	°C	Target temperature for heating (TP-17)
Cooling setpoint	$θ_{se t, coo l}$	°C	Target temperature for cooling (TP-17)
Internal gains	$Φ_{in t}$	W	Total internal heat gains in the zone
Solar gains	$Φ_{so l}$	W	Directly transmitted solar gains (TP-08)
Ventilation air change rate	$n_{a c h}$	h⁻¹	Air changes per hour (TP-06)
Average supply air temperature	$θ_{s u ppl y}$	°C	Temperature of ventilation supply air
Thermal bridge coefficient	$H_{t b}$	W/K	Total thermal bridge heat transfer (TP-05)
Convective fraction (internal gains)	$f_{in t, c}$	—	Fraction of internal gains delivered convectively
Convective fraction (solar gains)	$f_{so l, c}$	—	Fraction of solar gains delivered convectively
Convective fraction (heating/cooling)	$f_{h c, c}$	—	Fraction of heating/cooling output delivered convectively
Timestep	$Δ t$	s	Calculation timestep (typically 3600 s for hourly)

Calculation

Internal Gains

Internal gains represent heat released into the dwelling by occupants, lighting, appliances, cooking, and distribution losses. Each source provides a gain in W/m² that is multiplied by the zone floor area to give a total gain in watts.

Schedule-Based Gains

For metabolic, lighting, and other fixed-schedule sources, the gain at each timestep is read directly from a time series:

$Φ_{in t} = q_{in t} (t) \times A_{z o n e}$

Where:

$q_{in t} (t)$ is the internal gain density at timestep $t$ W/m²
$A_{z o n e}$ is the zone floor area m²

Appliance Gains

Appliances consume electricity and release a fraction of that consumption as heat. The gain is:

$Φ_{a pp} = q_{a pp} (t) \times A_{z o n e} \times f_{g ain s}$

Where:

$q_{a pp} (t)$ is the appliance energy supply density at timestep $t$ W/m²
$f_{g ain s}$ is the fraction of energy supply that becomes an internal gain —

The total energy consumed by the appliance is also recorded against the relevant energy supply for fuel accounting.

Event-Based Appliance Gains

Some appliances (dishwashers, washing machines, tumble dryers) operate as discrete events rather than continuous schedules. Each event is defined by a start time, duration, and power demand. The event power is distributed across simulation timesteps, with standby power assumed during idle periods.

When an event spans a timestep boundary, the power is apportioned to each timestep according to the fraction of the event duration falling within that timestep. Standby power is subtracted from the event power to avoid double-counting, since the baseline already includes standby consumption at every timestep.

Load Shifting

Event-based appliances may optionally participate in load shifting. A weighting time series (representing, for example, time-of-use tariff costs) is multiplied by the total demand at each candidate timestep. The event is shifted forward in time, up to a configurable maximum number of hours, to the earliest timestep where the weighted demand falls below a specified limit. If no such timestep exists within the allowed window, the event is placed at the timestep with the lowest weighted demand.

Convective and Radiative Split

All gains entering the zone are split into convective and radiative components. The convective fraction is delivered directly to the air node; the radiative fraction is distributed across the internal surfaces of all building elements in proportion to their area.

The default convective fractions from BS EN ISO 52016-1:2017, Table B.11 are:

Gain source	Convective fraction
Internal gains	$f_{in t, c} = 0.4$
Solar gains	$f_{so l, c} = 0.1$
Heating/cooling	$f_{h c, c} = 0.4$ (default)

Ventilation Heat Transfer Coefficient

The ventilation heat transfer coefficient converts air change rate to a thermal conductance:

$H_{v e} = ρ_{ai r} \times c_{p, ai r} \times \dot{V}_{v e}$

Where:

$ρ_{ai r}$ is the density of air kg/m³
$c_{p, ai r}$ is the specific heat capacity of air J/(kg·K)
$\dot{V}_{v e}$ is the volumetric airflow rate m³/s, given by $\dot{V}_{v e} = n_{a c h} \times V /3600$

Internal Thermal Capacity

The internal thermal capacity of the zone, representing air and furniture, is:

$C_{in t} = κ_{m, in t} \times A_{z o n e}$

Where $κ_{m, in t} = 10, 000$ J/(m²·K) is the areal thermal capacity of air and furniture from BS EN ISO 52016-1:2017, Table B.17.

Heat Balance Matrix Equation

The heat balance is formulated as a matrix equation:

$A \cdot X = B$

Where:

$A$ is the coefficient matrix of known thermal conductances and capacitances
$X$ is the vector of unknown node temperatures
$B$ is the vector of known heat flows and previous-timestep terms

The vector $X$ contains one temperature per node per building element, plus the zone internal air temperature. The dimension of the system is $N + 1$ , where $N$ is the total number of nodes across all building elements.

External Surface Node (BS EN ISO 52016-1:2017, Equation 41)

For the external surface node (node 0) of each building element:

$(\frac{κ _{0}}{Δ t} + h_{ce} + h_{r e} + h_{pl i, 0}) θ_{0} - h_{pl i, 0} θ_{1} = \frac{κ _{0}}{Δ t} θ_{0}^{p r e v} + (h_{ce} + h_{r e}) θ_{e x t} + α_{so l} (I_{so l, d i f} F_{s h, d i f} + I_{so l, d i r} F_{s h, d i r}) - Φ_{s k y}$

Where:

$κ_{0}$ is the thermal capacity of the external surface node J/(m²·K)
$h_{ce}$ , $h_{r e}$ are the external convective and radiative heat transfer coefficients W/(m²·K)
$h_{pl i, 0}$ is the conductance between node 0 and node 1 W/(m²·K)
$θ_{0}^{p r e v}$ is the node temperature from the previous timestep °C
$θ_{e x t}$ is the external temperature seen by the element °C
$α_{so l}$ is the solar absorption coefficient —
$I_{so l, d i r}$ , $I_{so l, d i f}$ are direct and diffuse solar irradiance W/m²
$F_{s h, d i r}$ , $F_{s h, d i f}$ are direct and diffuse shading factors —
$Φ_{s k y}$ is the net thermal radiation to sky W/m²

Inside Nodes (BS EN ISO 52016-1:2017, Equation 40)

For each inside node $i$ (nodes between external and internal surfaces):

$- h_{pl i, i - 1} θ_{i - 1} + (\frac{κ _{i}}{Δ t} + h_{pl i, i} + h_{pl i, i - 1}) θ_{i} - h_{pl i, i} θ_{i + 1} = \frac{κ _{i}}{Δ t} θ_{i}^{p r e v}$

This is a tridiagonal structure: each inside node couples only to its immediate neighbours.

Internal Surface Node (BS EN ISO 52016-1:2017, Equation 39)

For the internal surface node of each building element $e l$ :

$- h_{pl i, n - 1} θ_{n - 1} + (\frac{κ _{n}}{Δ t} + h_{c i} + h_{r i} \sum \frac{A _{e l k}}{A _{t o t}} + h_{pl i, n - 1}) θ_{n} - \sum_{k} \frac{A _{e l k}}{A _{t o t}} h_{r i} θ_{s, k} - h_{c i} θ_{ai r} = \frac{κ _{n}}{Δ t} θ_{n}^{p r e v} + \frac{( 1 - f _{in t, c} ) Φ _{in t} + ( 1 - f _{so l, c} ) Φ _{so l} + ( 1 - f _{h c, c} ) Φ _{h c}}{A _{t o t}}$

Where:

$h_{c i}$ is the internal convective heat transfer coefficient W/(m²·K), which depends on the direction of heat flow (determined by the element pitch and the temperature difference between the air node and the internal surface)
$h_{r i}$ is the internal radiative heat transfer coefficient W/(m²·K)
$A_{e l k}$ is the area of building element $k$ m²
$A_{t o t}$ is the total area of all building elements in the zone m²
$θ_{s, k}$ is the internal surface temperature of element $k$ °C
$θ_{ai r}$ is the internal air temperature °C
$Φ_{h c}$ is the heating or cooling power W

The radiative coupling between internal surfaces means each internal surface node has off-diagonal terms for every other internal surface node in the zone. This is the source of the full (non-tridiagonal) coupling in the matrix.

Zone Air Node (BS EN ISO 52016-1:2017, Equation 38)

$(\frac{C _{in t}}{Δ t} + \sum_{e l} A_{e l} h_{c i, e l} + H_{v e} + H_{t b}) θ_{ai r} - \sum_{e l} A_{e l} h_{c i, e l} θ_{s, e l} = \frac{C _{in t}}{Δ t} θ_{ai r}^{p r e v} + H_{v e} θ_{s u ppl y} + H_{t b} θ_{e x t} + f_{in t, c} Φ_{in t} + f_{so l, c} Φ_{so l} + f_{h c, c} Φ_{h c}$

Where:

$C_{in t}$ is the internal thermal capacity of the zone J/K
$H_{v e}$ is the ventilation heat transfer coefficient W/K
$H_{t b}$ is the thermal bridge heat transfer coefficient W/K
$θ_{s u ppl y}$ is the average supply air temperature °C

The zone air node equation couples the air temperature to every internal surface temperature (via convective heat transfer) and to the external temperature (via ventilation and thermal bridges).

Matrix Solution

The matrix equation $A \cdot X = B$ is solved by Gaussian elimination (or equivalent direct solver) to obtain all node temperatures and the internal air temperature simultaneously.

Optimised Solver

The full matrix has dimension $N + 1$ , where $N$ can be large for dwellings with many building elements (each contributing 2 to 5 nodes). However, the matrix is highly sparse: each building element's nodes form a tridiagonal band, and the only full coupling occurs among the internal surface nodes and the air node.

The specification permits an algebraic reduction that eliminates all intermediate nodes from the matrix. For each building element, a forward sweep from external surface to internal surface reduces the element's contribution to a single modified equation at the internal surface node.

The forward sweep computes adjusted coefficients:

$A_{i, i}^{'} = A_{i, i} - \frac{A _{i - 1, i} \times A _{i, i - 1}}{A _{i - 1, i - 1}^{'}}$

$B_{i}^{'} = B_{i} - \frac{B _{i - 1}^{'} \times A _{i, i - 1}}{A _{i - 1, i - 1}^{'}}$

After the forward sweep, the reduced matrix has dimension $M + 1$ , where $M$ is the number of building elements (one internal surface node each) plus one air node. This smaller system is solved directly.

Once the internal surface temperatures and air temperature are known, a backward sweep recovers each intermediate and external node temperature:

$θ_{i} = \frac{B _{i}^{'} - A _{i, i + 1} \times θ _{i + 1}}{A _{i, i}^{'}}$

This approach reduces the computational cost from $O (N^{3})$ to $O (M^{3} + N)$ , where $M ≪ N$ .

Operative Temperature

The operative temperature combines the air temperature and the area-weighted mean radiant temperature, following BS EN ISO 52016-1:2017, section 6.5.5.3:

$θ_{o p} = \frac{θ _{ai r} + θ ˉ _{r a d}}{2}$

Where the mean radiant temperature is:

$\overset{ˉ}{θ}_{r a d} = \frac{\sum _{e l} A _{e l} θ _{s, e l}}{A _{t o t}}$

The setpoint comparison may be based on either operative temperature or air temperature, depending on the control specification.

Heating and Cooling Demand

The demand calculation follows BS EN ISO 52016-1:2017, section 6.5.5.2, in four steps.

Step 1: Free-floating temperature. Solve the heat balance with zero heating/cooling gains ( $Φ_{h c} = 0$ ) to obtain the free-floating operative temperature $θ_{o p, f r ee}$ .

Step 2: Cooling potential from ventilation. If additional ventilation is available (e.g. opening windows), calculate the operative temperature achievable at maximum ventilation rate. If the free-floating temperature exceeds the ventilation cooling setpoint and the supply air is cooler than the internal air, interpolate the required air change rate to reach the ventilation cooling setpoint. If the resulting temperature still exceeds the active cooling setpoint, revert to the base ventilation rate and allow the mechanical cooling system to operate.

Step 3: Determine demand direction. Compare the free-floating operative temperature against the heating and cooling setpoints:

If $θ_{o p, f r ee} < θ_{se t, h e a t}$ : heating is required
If $θ_{o p, f r ee} > θ_{se t, coo l}$ : cooling is required
Otherwise: no demand; return zero

Step 4: Interpolate demand. Apply a reference heating/cooling load of $\pm 10 \times A_{z o n e}$ W and solve the heat balance again to obtain $θ_{o p, u pp er}$ . The actual demand is then found by linear interpolation:

$Φ_{h c} = Φ_{h c, u pp er} \times \frac{θ _{se t} - θ _{o p, f r ee}}{θ _{o p, u pp er} - θ _{o p, f r ee}}$

The demand in kWh for the timestep is:

$Q_{h c} = \frac{Φ _{h c}}{1000} \times Δ t_{h}$

Where $Δ t_{h}$ is the timestep in hours. A positive result is heating demand; a negative result is cooling demand.

When the setpoint basis is air temperature rather than operative temperature, the same interpolation procedure applies using $θ_{ai r}$ values in place of $θ_{o p}$ .

Temperature Update

After the heating or cooling demand has been determined and the actual heating/cooling output calculated by the plant models (TP-12, TP-14, etc.), the heat balance is solved one final time with the actual heating/cooling gains to obtain the node temperatures that carry forward to the next timestep:

$θ_{i}^{n e x t} = solve (A \cdot X = B with actual Φ_{h c})$

Node Temperature Initialisation

Before the first timestep, node temperatures must be initialised. All nodes are set to the average of the external air temperature and the initial setpoint:

$θ_{ini t} = \frac{θ _{e x t, ini t} + θ _{se t, ini t}}{2}$

A warm-up loop then iterates the heat balance using a large timestep ( $Δ t_{h} = 8760$ hours, i.e. one year) under steady-state conditions (constant external temperature, constant setpoint, zero gains) until all node temperatures converge to within a relative tolerance of $1 0^{- 8}$ . This rapid convergence strategy avoids the need to simulate multiple calendar years before the fabric temperatures stabilise.

Outputs

Quantity	Symbol	Unit	Description
Space heating demand	$Q_{h e a t}$	kWh	Heating energy required in the timestep (positive)
Space cooling demand	$Q_{coo l}$	kWh	Cooling energy required in the timestep (negative)
Operative temperature	$θ_{o p}$	°C	Combined air and mean radiant temperature
Internal air temperature	$θ_{ai r}$	°C	Zone air node temperature
Node temperatures	$θ_{i}$	°C	Temperature at each element node (carried to next timestep)
Ventilation air change rate	$n_{a c h, coo l}$	h⁻¹	Adjusted air change rate after ventilative cooling assessment

Assumptions

The convective fraction of internal gains is fixed at $f_{in t, c} = 0.4$ for all sources (BS EN ISO 52016-1:2017, Table B.11). The specification permits source-specific values, but the default is applied uniformly.
The convective fraction of solar gains is fixed at $f_{so l, c} = 0.1$ (BS EN ISO 52016-1:2017, Table B.11).
The areal thermal capacity of air and furniture is fixed at $κ_{m, in t} = 10, 000$ J/(m²·K) (BS EN ISO 52016-1:2017, Table B.17).
Radiative gains are distributed uniformly across internal surfaces in proportion to element area, not view factors.
The reference heating/cooling load for demand interpolation is $\pm 10 \times A_{z o n e}$ W. The specification terms this the "otherwise" case (no maximum power constraint from the plant).
The operative temperature is the simple average of air temperature and mean radiant temperature (equal weighting), consistent with BS EN ISO 52016-1:2017, section 6.5.5.3.
The warm-up procedure uses a single annual timestep under constant boundary conditions. Gains are set to zero during warm-up.
Internal convective heat transfer coefficients ( $h_{c i}$ ) depend on the direction of heat flow, which is evaluated from the temperature difference between the air node and the internal surface node at the previous timestep. The values are as defined in TP-05.

Cross-references

TP-01: Overview and Climate Data -- simulation timestep and climate parameters
TP-03: External Conditions -- external air temperature and element-level external temperatures
TP-05: Fabric Heat Loss -- building element node models, inter-node conductances, surface resistances, and thermal bridge coefficients
TP-06: Ventilation and Infiltration -- air change rates and supply air temperature
TP-07: Thermal Mass -- element thermal capacities distributed across nodes
TP-08: Solar Gains and Shading -- solar irradiance on element surfaces and transmitted solar gains through glazing
TP-12: Heat Pumps -- heating/cooling output that feeds back into the temperature update
TP-14: Boilers -- heating output that feeds back into the temperature update
TP-16: Heat Emitters -- convective fraction of heating system output
TP-17: Controls -- heating and cooling setpoint schedules, ventilation cooling setpoint