Domain

• thermal storage
• energy conversion device

Intended application (including scale and resolution)

  This model is intended for MPC-based predictions of temperature trajectories.

Modelling of spatial aspects

• lumped (single device)
• averaged (multiple devices)

This model is a trade-off between model accuracy (which means capturing the relevant effects in the thermal building behaviour) and keeping the model computationally tractable for optimization-based control via MPC. In order to do so, each story of the building is modelled as a thermal zone with a unified air temperature. Components made of concrete (i.e., walls and ceilings) are modelled with their own thermal behaviour. Thus, the thermal behaviour is aggregated on a story level and neglects thermal effects within the individual stories. For tractability, a linear RC-equivalent circuit is used, which is common in the field of MPC for buildings, leading to a lumped-parameter model consisting of thermal resistances (R) and capacitances (C).

Model dynamics

• dynamic

The model is based on first-principles for heat, diffusion, dissipation and storage. All these effects are considered in a linear fashion.

Model of computation

• time-continuous

The model consists of linear ordinary differential equations (ODEs) in state-space form. This continuous-time model is then converted to a discrete-time either via exact discretization or by a numerical integration scheme like Euler’s method or Runge-Kutta with respective step size.

Functional representation

• explicit

All algebraic equations (e.g., for the influence of the outdoor temperature) are linear and therefore directly included to the state-space model. Nonlinear influences occurring for instance in the solar irradiance are pre-processed. The resulting state-space model depends only on the pre-processed inputs/disturbances, which then enter the ODE linearly.

Input variables (name, type, unit, description)

The model has two types of inputs: The first ones are the controllable inputs collected in a vector-valued function , which are used by a controller to steer the system aiming at a certain system behaviour. These inputs are mainly thermal fluxes for radiators and the concrete core activation system. The second type of input variables are external, uncontrollable disturbances , acting on the model (mainly weather influences) The controller inputs are defined as 

where are the inputs for each floor ∈ (second floor, first floor and ground floor) with concrete-core influxes and radiator influxes both in Watt.
Additionally, the disturbance vector is defined as

with  is the outdoor temperature, is the ground temperature both in Kelvin, and are the solar irradiation acting on the roof and the walls respectively, are internal gains modelling heat influxes from occupation and electrical devices for each floor in Watt.

Output variables (name, type, unit, description)

The outputs are identical to the system state , i.e.

where ? is the identity matrix. The state variables are described below.

Parameters (name, type, unit, description)

The thermal capacitances for each component are given in , the thermal resistances are given in  The input parameters are given in .

Internal variables (name, type, unit, description)

The internal (state) variables consist of the temperatures for all elements (ambient temperatures, wall temperatures, ceiling temperatures and temperature of the thermal heat buffer). More specifically,

is a temperature vector for each floor with floors k∈  as above, where the temperature of the concrete-core, is the zone air temperature, is the wall temperature and  is the bottom temperature of the k-th floor. denotes the roof temperature. The basement temperature vector is defined as because no concrete core activation is present.

Internal constants (name, type, unit, description)

Model equations

Governing equations

The linear state-space model is described as

with output equation

The entries of the matrices are derived from first principle models. These first principle models entail:

a) Modelling of the zones/wall/ceiling temperatures based on the thermal storage equationa) Modelling of the zones/wall/ceiling temperatures based on the thermal storage equation

where is the temperature and is the thermal capacity of the zone/wall/ceiling.

b) Modelling of the  heat dissipation/convection between element i and j as

c) Enforcing energy balances as

d) Deriving input equations as

where is the  input and is a given input parameter.

Constitutive equations

Derivations of the parameters from material properties are can for example be found in the following works and references therein.
Hazyuk, I., Ghiaus, C., & Penhouet, D. (2012). Optimal temperature control of intermittently heated buildings using Model Predictive Control: Part I–Building modeling. Building and Environment, 51, 379-387.
Sturzenegger, D., Gyalistras, D., Semeraro, V., Morari, M., & Smith, R. S. (2014, June). BRCM Matlab toolbox: Model generation for model predictive building control. In American Control Conference (ACC), 2014 (pp. 1063-1069). IEEE.
Oldewurtel, F., Parisio, A., Jones, C. N., Gyalistras, D., Gwerder, M., Stauch, V., ... Morari, M. (2012). Use of model predictive control and weather forecasts for energy efficient building climate control. Energy and Buildings, 45, 15-27.

Boundary conditions

Initial conditions

Model Validation

Narrative

Generally, building model validation and identification is often hard due to very limited amount of measurements in buildings and the usually small temperature gradients. Nonetheless, this test case provides simulation results for the model in order to enable at least the possibility of a consistency check.

Test system configuration

The simulation runs the 4-floor building model for 2 weeks with constant radiator powers of 13kW at the first and second floor and a constant radiator power of 10kW at the third floor. The concrete-core activation system is not used leading to zero inputs for the respective components of ?. This yields

Inputs and parameters

Control function

Initial system state

Temporal resolution

Evolution of system state

Results

The evolution of the system states (trajectories) with respective disturbances are shown in the figures below. The inputs are not shown here, as they are constant over time.

Model harmonization

Narrative

This test case assumes constant inputs (no control function).
Furthermore, it is assumed that all thermal energy comes from the
district heating system.
Because the inputs are constant here, the peak demand is simply
the sum of all inputs, i.e., p ̅ ≡ 36kW and the fluctuation in theenergy demand v≡0 as. Hence, the test case focuses on the
comfort level.

Test system configuration

Inputs and parameters

Control function (optional)

Initial system state

Temporal resolution

Evolution of system state

Results

The average comfort level over  can be calculated as

with = 48ℎ and the desired temperature = 20°C. The
resulting comfort level for the above simulation over time is
shown below.

The comfort level for each time step individually is
shown next.

The average comfort level over the whole simulation time is . One can observe that the comfort level is highly varying. Note that high values here mean a low comfort level, because the temperature deviation from the desired temperature is high.