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Energy consumption reduction efforts in the residential buildings sector represent socio-economical, technological and environmental preoccupations which justify advanced scientific research. These lead to use inverse models to describe thermal behavior and to evaluate the energy consumption of buildings. Their principal goal is to provide supporting evidence of enhanced energy performances and predictions. More specifically, research questions are related to building thermal modeling which is the most appropriate in a smart grid context. In this context, the models are reviewed according to three categories. The first category is based on physical and basic principle modeling (white-box). The second offers a much simpler structure which is the statistical models (black-box). The black-box is used for prediction of energy consumption and heating/ cooling demands. Finally, the third category is a hybrid method (grey-box), which uses both physical and statistical modeling techniques. In this paper, we propose a detailed review and simulation of the main thermal building models. Our comparison and simulation results demonstrate that the grey-box is the most effective model for management of buildings energy consumption.

Energy policies in the Nordic countries, such as Canada, stipulate that 50% of the energy consumed by 2025 should come from renewable and CO_{2}-free energy sources [_{2} emissions and consequently reduce the speed of global warming. Moreover, this transition is a contributing factor to protect Nordic economies from the consequences of sharply rising prices of fossil fuels which can be attributed to an increasing demand and depletion of these non-renewable resources [

Buildings energy consumption is imputable mainly to heating/cooling, lighting, and electrical appliances. In Nordic countries, heat pumps, water based heating systems, water heater and electric baseboard heaters are of the main loads for heating [

The study of buildings and grid interactions in the new smart grid scenario is then necessary. This study starts with a thermal and electric modeling of buildings. Its goal is to provide a review of the subject of energy effi-

ciency and buildings modeling in the context of smart grid. In Section 1, the state of the art is presented. Section 2 gives a review of the most important approaches used in thermal modeling of buildings. Section 3 provides a condensed comparison of the white, black and grey-boxes modeling approaches. Section 4 is devoted to summarizing the importance of the thermal modeling in the implementation of prediction techniques of energy consumption taking into account weather predictions to increase energy efﬁciency of buildings. Section 5 gives an example of simulation of black and grey-boxes approaches. Finally, Section 6 presents concluding remarks and identifies a number of research and policy needs.

In recent years, a variety of works in the area of thermal modeling of buildings has been done and can be found in the literature [

As such, many models have been made to improve the energy efficiency in buildings. Some aim for residential buildings, some for commercial buildings, and some for both [

Generally, the proposed models are characterized by two thermal behaviors. The first one is a static behavior, while the second one is dynamic. The static behavior is applied to simplify the thermal model and to overcome the limitations of computing resources. The dynamic behavior is interested in understanding the phenomenon of thermal exchange for simulation purposes.

The static behavior approach is used for steady state conditions of buildings, when all the internal and external inputs are controllable. The dynamic behavior approach is related to the transition of internal and external inputs and outputs of the building system [

In fact, considerable attention has been placed on the black, white and grey models in dynamic approach which include the thermal networks, modal analysis, differential equations, autoregressive moving average model (ARMA), Fourier series [

In practice, residential buildings are more variable in occupancy and internal heat gains due to appliances and all other electrical driven equipment. Appliances driven loads and gains are divided into two classes. The first one corresponds to the responsive loads which indicate that the customer is able to modify the behavior of the appliance due to a price signal. This class includes lights, plug loads, clothes washers, clothes dryers, dishwashers, cooking ranges, and microwaves. The second class is the unresponsive loads appliances, which indicate that the customer is typically not able to modify the behavior without investment in additional technologies. This class includes refrigerator and freezer loads [

Considering thermal phenomena, heat transmission, heat storage, fluid flow and heat flux represent the fundamental thermal properties of building elements. These phenomena are highly time-sensitive [

In the case of physical model with parameters of physical significance, we speak of a white-box (physical and basic principle modeling) approach, which require a significant amount of building knowledge [

・ Static and dynamic models;

・ Linear, nonlinear models;

・ Differentiable, continuous, non-continuous models.

In static models the output of system does not depend on the time. While in dynamic models, the output is time varying due to the dynamic heat balance time evolution. These dynamic white-box models are typically represented by differential equations. However, their mathematical representation also depends on the relation

Type | Example | Application |
---|---|---|

Static linear equations | Transmission through component | |

^{2});^{2}), | ||

^{2}∙˚C) and A: heat transfer area of the surface (m^{2}). | ||

Static nonlinear equations | Building simulation radiation exchange (walls and ceilings) | |

Dynamic linear equation Ordinary differential equation | Passive/active Energy storage | |

Dynamic linear equation Partial linear | Dynamic heat flux | |

^{−}^{1}∙˚C) |

between the parameters. These relations can be ordinary, partial, linear and non-linear differential equation, we propose to sum up these mathematical representations in

The complexity of white-box modeling depends mainly on the chosen precision`s levels of the known phenomena associated with the building system to be modeled [

If the model cannot be calibrated, this suggests that the model structure has to be questioned. In the literature, it is suggested that a too high number of parameters can be a significant source of error [

Black-box models are empirical models (statistical) without physically significant parameters. That is, contrary to the white-box, when little is known about the inner workings of the building system. This means that black-box models are derived the inputs-outputs thermal behavior. This case is black-box approach [

In the black-box, the parameters are generally adjusted automatically [

In thermal modeling of buildings, it is reasonable to combine the relative strengths of black-box coming from the statistical with the white-box strengths based on physical interpretation [

Grey-box models are therefore mixed or transitional forms of white-box and black-box models. There are in the literature several definitions; the following are the most frequently encountered:

・ Definition 1 (type of parameters)

Grey-box parameters are both empirical and have a physical significance [

・ Definition 2 (determination of the parameters)

Grey-box models are characterized by the fact that all their parameters or a part of them are determined on the basis of measured data of real system [

Definition 2 does not define any kind of model, but rather the manner of determining the parameters of a model. It should be mentioned that in the literature, often the term Hybrid model emerges. Independently of the terms or definition used, grey or hybrid models, this approach cannot inform on the particular composition of various phenomena parameters from the white or black models. That is, we just know that there is a mixture of both models without knowing which one dominates in the combination of white-box and black-box models [

Type of model | Model structure | Parameter estimation | Example |
---|---|---|---|

Static | |||

Linearly | Linear function | Linear regression (Least Squares method) | Energy signature of weekly values |

Nonlinear | Polynomials Any nonlinear function | Linear regression (Least Squares Method) Iterative process, Levenberg Marquardt | Pump curve |

Dynamic | |||

Linearly | Transfer functions models (ARMA, ARMAX, etc.) | Linear regression (least square method), an iterative procedure | Heat flow through a plane wall |

Nonlinear | Neural Networks (sigmoid, wavelet, radial basis networks) | Damped Gauss-Newton backpropagation | Arbitrary non-linear systems |

Polynomials (Wiener /Hammerstein model, Volterra model) | Linear regression (least squares method) | Linear system with static nonlinearities at the input or output (control element with saturation behavior) |

White-Box | Grey-Box | Black-Box | |
---|---|---|---|

Internal structure of the model | + | 0 | − |

Number of parameters Source of error | − | 0 | + |

Formulation of the model | + − | + 0 | + |

Processing speed | + 0 | + 0 | + |

Required training data | + | 0 − | − |

Calibration effort | + − | + − | + |

Extrapolation | + | 0 | − |

Suitable for optimization | + | 0 | − |

Parameters physical meaning | + | + − | − |

Formulation system equations | + − | + 0 | + |

(+) Advantages (−) Drawbacks (0) Not available |

As illustrated in

Grey-box modeling can be useful to understand the parameters which can lead to significant errors in predicting units’ consumption [

The next section focuses on the link between the grey-box models and the model predictive control of building systems. We will briefly describe the need of a structural model based on semi-physical laws and the basic ideas lying behind the prediction method in building to improve performance buildings and reduce the energy consumption of heating and cooling systems [

The key to the sustainability of the energy efficiency of buildings [

There are several completely different approaches to system identification (see e.g. [

We addressed the question: how to describe the response of the indoor temperature in the presence of the different internal phenomena such as the heat transmission, heat storage, and the external phenomena of consumption able to improve the energy performance of a building, in order solar gains, wind speed and outdoor temperature? [

Generally, the need of a complete and detailed heat balance equation should incorporate the effects of different sources: the weather (temperature, humidity, solar radiation and wind speed) [

As it was discussed above, it is easier in this case to describe these phenomena by grey-box modeling approach [

Describing the transit thermal behavior by grey-box and capture the evolution of the indoor temperature of building is presented in section V. Thus, it is convenient to introduce the concept of thermal capacity of building denoted by

With

And

where:

Heat balance equation solved of the (RC) model for the air temperature node

And the heat balance for the mass temperature node

Depending to the application, to the complexity of modeling and to the set of phenomena of residential building to described. The (RC) model parameters can be used to represent all homes in the population for simplicity. Or if there is a need, multiple RC models with different thermal parameters can be used. In addition to changing input parameters while simulating a population of homes, the (RC) model can also be changed to accurately represent a given building stock [

Hence, the prediction models are needed to enable optimal control of buildings phenomena [

In the literature grey-box thermal models using MPC [

On the other hand, in the case when multiple inputs multiple outputs (MIMO) systems are considered [

They have the ability to handle large amount of data. This was demonstrated for instance in the identification of a thermodynamic model of a small residential building that was equipped with tens of wireless sensors collecting temperatures, humidity and solar radiation [

In order to compare the approaches, we have assumed that the building under study can be estimated as having a single zone. The equivalent (RC) electric circuit of the global thermal zone can be represented as is shown in

The physical model used to reproduce the behavior of the indoor temperature is based on the fundamental laws of thermodynamics, heat transfer, and thermo-physical variables. For this example, the thermal mass, the thermal resistance, and the building dimensions are presented in

Physical approaches are mostly applicable to contexts in which building design data are available. However, they are handier in scopes where interpretation of physical phenomena is desired [

Building dimensions | (Pi) | (m) |
---|---|---|

Height | 8 | 2.44 |

Length | 38 | 11.58 |

Width | 29 | 8.84 |

Thermal building parameters | ||

295585.31 | 0.0180 |

We can develop the grey box model from the combination of physical and statistical models. For the black-box model which is based on a function deduced only, is implemented from samples of training data describing the behavior of the modeled building.

Accordingly, in grey-box model to capture the evolution of indoor temperature of modeled building, we have supposed that, the envelope of the building is modeled by the set of

Here, the modeled building consists of the known and unknown components. Furthermore, the corresponding model parameters such as, the thermal capacity

We describe in the

temperature

difference between the variation of heat flow from electric heater and heat loss, multiplied by the inverse of

And the heat losses equivalent to the difference between the indoor and the outdoor temperature, divided by the equivalent thermal resistance of the building.

1) Finding the parameters of RC model: RLS

The (RC) model parameters

Now we precede the discretization of the variation of the indoor temperature associated with Equation (7):

The resulting differential equation can be approximated by the Equation (9)

where,

Following in the RLS method, the error considered the total error from the measurement and estimated indoor temperature is calculated using the formula of Equation (11):

where

where

To adjust the weights of the adaptive linear combiner in the purpose adaptive algorithm RLS a general expression for mean square error as a function of the weight values has been given by Equation (11).

The input of RLS is given by the vector,

Which include the previous indoor temperature of moment

where

The adaptation of the matrix

where

Finally, the filter output is calculated using the filter weights of previous iteration and the current input vector, we obtain the estimated indoor temperature of building as Equation (19):

In contrast, the fact that we do not require any physical information, the black-box models compared to the grey-box methods, stay totally based on measures or empirical parameters. However, one among of different black-box tools employed to estimate and describe the indoor temperature of modeled building. We propose to use RPROP stand for “Resilient backpropagation” for artificial neural networks. The Rprop was used to perform a local adaptation of the weight-updates according to the behavior of the total squared error. To get the behavior of indoor temperature of building, we have based on [

1) Description of Rprop

Rprop is used to eliminate the harmful influence of the size of the partial derivative on the weight step. As a consequence, only the sign of the derivative is considered to indicate the direction of the weight update [

where

Every time the partial derivative of the corresponding weight

In order to select a “best” model based on the test comparison by means of two suitable estimators of the behavior of the indoor temperature of modeled building. First, when grey-box is tested and RLS method is employed for the estimation of parameters. Secondly, when black-box is tested and the artificial neural networks are used for estimation of indoor temperature.

The results of simulations illustrated in

In on the other side in the

comparison with the measured indoor temperature. Although a wide variety of parameter values has been used, that means that artificial neural network was not able to find the correct value of the indoor temperature. Moreover, it is the presence of the nonlinearity of the power of electric heater which complicates the behavior of indoor temperature. Whereas, with this a simple neural networks structure, we cannot remain the indoor temperature from its true behavior. On the one side, the algorithm converges considerably slowly compared to the grey-box model. But only when varying the initial parameters of the Rprop, the convergence algorithm acceleration could be changed.

In this paper, we have proposed a review of black, white and grey-boxes modeling approaches of building systems and the predictions methods enabling to improve energy efficiency of buildings. These approaches have been introduced and compared into three categories. Each of them was associated to specific paradigms and field. First approach, relying on physical models “white-box”, it is mainly correspond to a gradual rise of the level of details of building models. It is based on the laws of physics to describe the set of phenomena of residential building and to permit high fidelity modeling of the building system. Second approach, is based on observations of “black-box”, which relies on statistical and measurements treatments to describe the set of the phenomena. Those can provide little insight into the dynamics dictating the system behavior of building. It is, however, quite difficult to perform a qualitative and comparative assessment of the various techniques devised in this field. Since again their performances will depend on the training data used as input. The great power of these models is the fact that they do not need to have much knowledge on the building geometry or the detailed physical phenomena to deduce the indoor temperature. Compared to physical approaches, black-box requires less information about the building and may appear as easier to deploy. Finally, the very natural question arises, when combining between the two models to obtain a simplified model structure (grey-box) suitable for the physics principle and of the observations of phenomena of building system. Grey-box approaches appear as a very promising field for the near future. They can be appreciated in situations where a building physical model is available. Grey-box models could be of great help if there is difficult to rebuild detailed physical model.

This work was supported by Laboratoire des technologies de l’énergie (LTE) d’Hydro-Québec, Natural Science and Engineering Research Council of Canada and Fondation UQTR.