THERMAL BUILDING
Thermal building- Title
Thermal Building Component Model
Thermal building - Model overview
Model Overview
Author / organization: Alexander Engelmann / KIT
Domain:
- Thermal storage
- Energy conversion device
Intended application: MPC-based predictions of temperature trajectories
Modelling of spatial aspects:
- Lumped (single device)
- Averaged (multiple devices)
Model dynamics: Dynamic
Model of computation: Time-continuous
Functional representation: Explicit
Thermal building- input and output
Input and Output
Input variables :
- controllable inputs collected in a vector-valued function
- Uncontrollable disturbances acting on the model
Output variables: Identical to the system state
Thermal building - related documents
Thermal building - description
Short Description
State-space model of the thermal behaviour of KIT Flexoffice. 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). The model is based on first-principles for heat, diffusion, dissipation and storage. All these effects are considered in a linear fashion
Present use / development status
prototype
Media Gallery
Thermal building
Model Details
Domain |
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Intended application (including scale and resolution) | This model is intended for MPC-based predictions of temperature trajectories.
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Modelling of spatial aspects |
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 |
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 |
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 |
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. 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.
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Internal constants (name, type, unit, description) | -
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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. | |||
Boundary conditions | - | ||
Initial conditions | In the MPC-scheme, the initial condition is usually given by measurements at the current sampling point. For forward simulation, we set all temperatures to a value of 21°C here. |
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 for all t.A sampling time of 1ℎ is used and an explicit Runge-Kutta 4/5 method to solve the ordinary differential equation.
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Inputs and parameters | Time-varying external weather influences summarized in the disturbance vector . A time series for ? is given as appendix to this document (KIT_Thermal-building_data.zip). The derivation of the system matrices is based on the principles mentioned above. The matrices are given in the SmILES data format. | ||
Control function | |||
Initial system state | All temperatures are initialized with 21°?. | ||
Temporal resolution | The sampling time is 1ℎ. | ||
Evolution of system state | Due to the constant input, the zone temperatures are determined by the disturbances consisting of the outdoor temperature, the solar irradiation and occupancy only. The constant inputs for the heat flux of the radiators are chosen in a way that the zone temperatures stay in a temperature band 19±2°C. All temperatures should start at 21°C and the basement zone temperature should drop and reach 12°C after 12 days approximately, because no heating system is present in the basement. | ||
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). | ||
Test system configuration | Same as for model validation. | ||
Inputs and parameters | Same as for model validation. | ||
Control function (optional) | None. | ||
Initial system state | Same as for model validation. | ||
Temporal resolution | Same as for model validation. | ||
Evolution of system state | Same as for model validation. | ||
Results | The average comfort level over can be calculated as with = 48ℎ and the desired temperature = 20°C. The The comfort level for each time step individually is 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. |