Energy-saving technology of BIM green buildings using fractional differential equation

Based on the first-order one-variable gray model [GM(1,1)] model, this paper proposes the fractional main model idea, and the optimal order of the model is obtained by the GM–backpropagation (GM-BP) neural network. Finally, the model is applied for the buildings’ energy-saving prediction. By adopting some improvement measures, the energy-saving potential can be estimated by using some of the model parameters to predict the energy consumption results, then comparing with the base energy value given by the country's authorities.

At present, the research on improvement of buildings’ energy-saving prediction model and calculation methods mainly include the following.

BIM is a data-based tool for engineering design and construction management. The relevant data based on the construction project is integrated through the parameters of the model. During building life-cycle management involving project planning, operation, maintenance, sharing, and delivery, engineers and technicians should have a correct understanding of information on various buildings and efficient response design teams. Building construction units, including architectural pairs, are involved in architectural collaboration work, thereby increasing production efficiency, optimizing costs, and shortening duration, thus play important roles.

In accordance with a priority design of current office buildings, the office buildings’ energy-saving design has been studied by Li, with an in-depth study of the case in architectural design technology, with a comparison of various cases. The application of BIM technology guides the energy-saving design of office buildings and related practical methods. Finally, with the author's related design practice, an example of the design of the office building using BIM technology is provided, demonstrating how to achieve the goal of buildings’ energy efficiency in architectural design, as well as reduce energy consumption during construction and operation. Green building energy contributes to the design goals of social sustainable development [4].

Yong et al. reviewed the most advanced technology that can be semiautomated or automatically created, analyzing the geometry and thermal models for buildings’ energy modeling and transformation assessment purposes. They outline the main algorithm used by these methods to indicate time and space – hot clock, convert these points cloud into the Semantic BIM in GBXML format, so that it is also a stark contrast to them. The paper also proposes the potential preparations and methods for measuring the actual thermal resistance of the building assembly and maps them to GBXML-based representations. It also introduces the latest works of the IT Drive building automation system (BAS) for energy-saving purposes. Finally, the technical gap is determined and actions needed in future research are discussed [5].

In this paper, the GM changes the first-order differential in the model to the fractional differential and proposes a fractional-based GM. Compared to the traditional GM, the recognition of fractional order is increased, thus increasing the difficulty of parameter recognition. Therefore, the author establishes a fractional gray model based on the model of the fractional system and calculates the error of the fractional GM.

There are currently three definitions of fractional order, using Caputo calculus, and the α differential of the function x(t) can be expressed as follows: (1) Dax(t)=dαx(t)dtα=1Γ(r−α)∫0tx(r)(τ)(t−τ)(α+1−r)dτ r−1<α<r. {D^a}x\left( t \right) = {{{d^\alpha }x\left( t \right)} \over {d{t^\alpha }}} = {1 \over {\Gamma \left( {r - \alpha } \right)}}\int_0^t {{{x^{\left( r \right)}}\left( \tau \right)} \over {{{\left( {t - \tau } \right)}^{\left( {\alpha + 1 - r} \right)}}}}d\tau \quad r - 1 < \alpha < r. In the expression, r – positive integer; α – score; and Γ() – gamma function. The function Γ() is intended for the following calculation: (2) Γ(β)=∫0∞τβ−1e−τdτ. \Gamma \left( \beta \right) = \int_0^\infty {\tau ^{\beta - 1}}{e^{ - \tau }}d\tau . Its Laplace transformation can be represented as follows: (3) L[Dαx(t)]=sαY(s)−∑i=0r−1sα−1−ix(i)(0). L\left[ {{D^\alpha }x\left( t \right)} \right] = {s^\alpha }Y\left( s \right) - \sum\limits_{i = 0}^{r - 1} {s^{\alpha - 1 - i}}{x^{\left( i \right)}}\left( 0 \right). In this paper, zero initial conditions are considered, when the Laplace transforms to the following: (4) L[Dαx(t)]=sαY(s). L\left[ {{D^\alpha }x\left( t \right)} \right] = {s^\alpha }Y\left( s \right). For the calculation of s^α, usually using the integer-order integral, we use the phase front filter to simulate the output results of s^α. Assuming that the fit frequency band is (ω_b, ω_h), the transfer function of the following filter can be constructed as follows: (5) sα≈Gαs∏i=1N1+s/ωie1+s/ωi. {s^\alpha } \approx {{{G^\alpha }} \over s}\prod\limits_{i = 1}^N {{1 + s/\omega _i^e} \over {1 + s/{\omega _i}}}. In this expression, 2N+1 is the selected filter order. Gα=ωb1−α, ωi=δωie, ωi+1e=ηωi, α=1−logδlogδη. {G^\alpha } = \omega _b^{1 - \alpha },\quad {\omega _i} = \delta \omega _i^e,\quad \omega _{i + 1}^e = \eta {\omega _i},\quad \alpha = 1 - {{\log \delta } \over {\log \delta \eta }}. Equation (5) may also be expressed in the form of an equation of state: (6) MIz˙I=AIzI+BIu(t) w=CIzI(t) {M_{I\dot zI}} = {A_{IzI}} + {B_I}u\left( t \right)\quad w = {C_{IzI}}\left( t \right) In this expression, z_I (t) – N+1 is the dimensional state vector. MI=[10…0−δ1⋮⋮⋮⋮00..−δ1], BI=[Gα0⋮0];AI=[00…0ω1−ω1⋮⋮⋱⋱00..ωNω−N], zI(t)=[z1(t)z2(t)⋮zN+1(t)];CI=[0…01]. \matrix{ {{M_I} = \left[ {\matrix{ 1 & 0 & \ldots & 0 \cr { - \delta } & 1 & {} & \vdots \cr {} & {} & {} & {} \cr \vdots & \vdots & \vdots & 0 \cr 0 & {..} & { - \delta } & 1 \cr } } \right],\quad {B_I} = \left[ {\matrix{ {{G^\alpha }} \cr 0 \cr \vdots \cr 0 \cr } } \right];} \hfill \cr {{A_I} = \left[ {\matrix{ 0 & 0 & \ldots & 0 \cr {{\omega _1}} & { - {\omega _1}} & {} & \vdots \cr \vdots & \ddots & \ddots & 0 \cr 0 & {..} & {{\omega _N}} & {{\omega _{ - N}}} \cr {} & {} & {} & {} \cr } } \right],\quad {z_I}\left( t \right) = \left[ {\matrix{ {{z_1}\left( t \right)} \cr {{z_2}\left( t \right)} \cr \vdots \cr {{z_{N + 1}}\left( t \right)} \cr } } \right];} \hfill \cr {{C_I} = \left[ {0 \ldots 01} \right].} \hfill \cr } Equation (6) can be converted to the following: (7) z˙I(t)=AzI(t)+Bu(t) w=CzI(t). \matrix{ {{{\dot z}_I}\left( t \right) = A{z_I}\left( t \right) + Bu\left( t \right)} \hfill \cr {\;\;\;\;w = C{z_I}\left( t \right).} \hfill \cr } Here, A=MI−1AI A = M_I^{ - 1}{A_I} , B=MI−1BI B = M_I^{ - 1}{B_I} , C = C_I.

Set the original time series as x⁽⁰⁾ = {x⁽⁰⁾ (1), x⁽⁰⁾ (2), ..., x⁽⁰⁾ (n)}. Its primary cumulative generation sequence is x = [x(1)x(2)...x (n)], where x(k)=∑i=1kx(0)(i) x\left( k \right) = \sum\limits_{i = 1}^k {x^{\left( 0 \right)}}\left( i \right) .

The gray model can be represented as (8) dx(t)dt+b1y(t)=b2. {{dx\left( t \right)} \over {dt}} + {b_1}y\left( t \right) = {b_2}. The pending constants b₁ and b₂ in Eq. (8) are determined as follows: (9) [b1b2]T=[E1TE1]−1E1TE2. {\left[ {{b_1}_{}{b_2}} \right]^T} = {\left[ {E_1^T{E_1}} \right]^{ - 1}}E_1^T{E_2}. In Eq. (9), E1=[−0.5(x(1)+x(2))1−0.5(x(2)+x(3))1⋮⋮−0.5(x(n−1)+xn)1] {E_1} = \left[ {\matrix{ { - 0.5\left( {x\left( 1 \right) + x\left( 2 \right)} \right)} & 1 \cr { - 0.5\left( {x\left( 2 \right) + x\left( 3 \right)} \right)} & 1 \cr \vdots & \vdots \cr { - 0.5\left( {x\left( {n - 1} \right) + xn} \right)} & 1 \cr } } \right] , E2=[x(0)(2)x(0)(3)⋮x(0)(n)] {E_2} = \left[ {\matrix{ {{x^{\left( 0 \right)}}\left( 2 \right)} \cr {{x^{\left( 0 \right)}}\left( 3 \right)} \cr \vdots \cr {{x^{\left( 0 \right)}}\left( n \right)} \cr } } \right] .

It can be seen from Eq. (8) that the gray model is an input-free ordinary differential equation, which is changed here in two aspects: one is to increase the input term u; and the second is the variable ordinary differential equation for the fractional differential equation, namely: (10) dαx(t)dtα+b1x(t)=b2u(t). {{{d^\alpha }x\left( t \right)} \over {d{t^\alpha }}} + {b_1}x\left( t \right) = {b_2}u\left( t \right). To maintain the characteristics of the original gray model, here, the input item u may use the unit step function, namely: (11) u(t)={1t≥00t<0. u\left( t \right) = \left\{ {\matrix{ {{1_{}}t \ge 0} \cr {{0_{}}t < 0.} \cr } } \right. Using Laplace transform on both sides of Eq. (10), the transfer function of the system is (12) G(s)=Y(s)U(s)=b2sα+b1. G\left( s \right) = {{Y\left( s \right)} \over {U\left( s \right)}} = {{{b_2}} \over {{s^\alpha } + {b_1}}}. If the frequency domain of z(t) is defined as (13) Z(s)=1sα+b1U(s), Z\left( s \right) = {1 \over {{s^\alpha } + {b_1}}}U\left( s \right), the output variable x(t) can be expressed as follows: (14) Dαz(t)=−b1z(t)+u(t) x(t)=b2z(t). \matrix{ {{D^\alpha }z\left( t \right) = - {b_1}z\left( t \right) + u\left( t \right)} \hfill \cr {\,\;\;\;\;x\left( t \right) = {b_2}z\left( t \right).} \hfill \cr } The above formula can be written equivalent to the following: (15) z˙I(t)=Gα[−b1zN+1(t)+u(t)] x(t)=b2zN+1(t) \matrix{ {{{\dot z}_I}\left( t \right) = {G^\alpha }\left[ { - {b_1}{z_{N + 1}}\left( t \right) + u\left( t \right)} \right]} \hfill \cr \;\,{x\left( t \right) = {b_2}{z_{N + 1}}\left( t \right)} \hfill \cr } That is to say, (16) z˙I(t)=A˜zI(t)+B˜u(t) x=C˜zI(t) A˜=A−b1BC B˜=B C˜=b2C \matrix{ {{{\dot z}_I}\left( t \right) = \tilde A{z_I}\left( t \right) + \tilde Bu\left( t \right)} \hfill \cr {\,\;\;\;\;\;x = \tilde C{z_I}\left( t \right)} \hfill \cr {\;\;\;\,\;\tilde A = A - {b_1}BC} \hfill \cr {\,\;\;\;\;\tilde B = B} \hfill \cr {\;\;\;\;\,\tilde C = {b_2}C} \hfill \cr }

The Gray system theory is the study of “little data, uncertainty” theory. The GM(1,1) model, established based on this theory, is currently the most widely used gray model, which helps to solve the current situation of energy consumption data of ‘small samples and poor information’ [6, 7]. The main modeling steps are as follows:

Set the original sequence as x⁽⁰⁾ = {x⁽⁰⁾ (1), x⁽⁰⁾ (2),...,x⁽⁰⁾ (n)} and build the original sequence with Eq. (17): (17) x(1)(k)=∑j=1kx(0)(j),(j,k=1,2,...,n). {x^{\left( 1 \right)}}\left( k \right) = \sum\limits_{j = 1}^k {x^{\left( 0 \right)}}\left( j \right),\left( {j,k = 1,2,...,n} \right).

The energy consumption of buildings is seriously affected over time and has the characteristics of cyclical changes. It is difficult to achieve a good prediction effect using the GM(1,1) model or BP neural network model. This paper combines both to obtain the GM-BP neural network prediction model, which uses the GM(1,1) model (weakened by the ‘small sample’ and ‘poverty information’ data) but also fully exerts the maturity of the BP neural network algorithm [8, 9].

In addition, the combined model does not require many factors that affect energy consumption and simply provides accurate and reliable energy consumption history data to conduct public buildings’ energy consumption prediction research. At the same time, in addition to the support of the MATLAB neural network toolbox, this paper provides a more convenient and fast process of establishing a GM-BP neural network prediction model. The main steps in the modeling of the GM-BP combination prediction model are as follows:

Gray: once the original energy consumption data are accumulated, the accumulation sequence weakens the randomness of the raw data and highlights the overall development trend.

This paper takes a community in Taiyuan, Shanxi Province, as the experimental object, and the number of residences in 2018 was about 297,300. The residential stock is divided into six residential types: detached houses; semidetached houses that are single families; terraced houses that share partition walls with neighbors; villas; and second floor apartments, often with a separate entrance. Also, there are two different configuration apartments, hospitality apartments and innhouse apartments.

After repeated testing, it was understood that when M = 6, n = 6 (the sample matrix is of six rows and seven columns), i.e., 6 months ago, the GM-BP combination model predicted the effect of 6 months after training.

The GM (1, 1) model and the GM-BP combination model (established by the MATLAB tool) predict the monthly power consumption (energy consumption) of the building in the second half of 2018, and the forecasted results are shown in Figure 1.

Fig. 1

In order to evaluate the effectiveness of various models, this paper evaluates the prediction accuracy and stability of three models using three performance indicators of maximum relative error absolute value E_max (see Eq. (23)), mean relative error ε_ave (Eq. (24)), and root mean square error [RMSE] (RMSE (see Eq. (25)). (23) Emax=max|Y˜(i)−Y(i)Y(i)|; {E_{\max }} = \max \left| {{{\tilde Y\left( i \right) - Y\left( i \right)} \over {Y\left( i \right)}}} \right|; (24) Eave=1n∑i=1nY˜(i)−Y(i)Y(i); {E_{ave}} = {1 \over n}\sum\limits_{i = 1}^n {{\tilde Y\left( i \right) - Y\left( i \right)} \over {Y\left( i \right)}}; (25) RMSE=1n∑i=1n[Y˜(i)−Y(i)]2. RMSE = \sqrt {{1 \over n}\sum\limits_{i = 1}^n {{\left[ {\tilde Y\left( i \right) - Y\left( i \right)} \right]}^2}} . In the equations, n is the total number of samples; Y is the predicted value for the ith sequence; and Y (i) is the actual value of the ith sequence.

Table 1 presents the calculation results of each model evaluation index. According to the table analysis, the GM-BP combination model is optimal in terms of error control and is the ideal choice [10, 11]. Although the GM(1,1) model is simple and easy, the prediction accuracy is greatly reduced and the strain capacity is poor; the BP model involves short training time, the prediction accuracy is lacking, and the model stability is insufficient. The GM-BP combination model has excellent results for various types of building predictions.

The 200 prediction experiments on natural gas and power consumption for various residential types and time periods were recorded and compared with the actual measurements. Figure 2 depicts a scatterplot of the predicted and measured values of the average residential natural gas consumption on a specific area, and Figure 3 depicts the scatterplot of the predicted and measured power consumption. Among them, the transverse coordinates represent the actual measured value, the longitudinal coordinates represent the value predicted by this model, and the dotted line represents the dividing line of ±20% error between the predicted value and the measured value. It is seen that 18% of the forecasted values showed deviation from the measured values exceeding ±20% for natural gas consumption and going down to 8% [12]. The larger differences in the gas consumption prediction are attributed to the accuracy of the original data set and the gas measurement procedures, as well as changes in building features. For natural gas, the average absolute percentage error in the model prediction is 13% and 9% for electric energy, which shows that the prediction accuracy of the prediction model has reached about 90% and thus this parameter has a high prediction accuracy. These results provide a preliminary assessment of the distribution of error spaces in statistical analysis, by performing further analysis, obtaining predictions for larger regions, and detecting possible spatial patterns of energy consumption throughout the city.

Fig. 2

Fig. 3

Table 2 shows the energy-saving potential of space heating and domestic hot water in different building years and various types of residences after implementing renovation measures. It can be seen that the larger the building's age, the greater is the building's energy savings. For example, homes built before 2016 had the highest energy-saving potential, with an average percentage of 56%, while homes built after 2018 had only 3%. For building types, apartments – inns (26%), villas (29%), and townhouses (20%) – have greater energy savings potential as these are the major parts of the current residential stock. Independent housing and nonindependent housing contribute only 3%–12% to the total energy conservation potential, respectively, indicating that implementing improvement measures in single-family houses does not bring greater energy conservation efficiency to the whole city.

The traditional gray model is an input-free ordinary differential equation, which has two aspects: increasing the input term and the variable ordinary differential equation is fractional differential equation. Combined with the engineering examples of the fractional gray model and the hyperbolic model, the error between the fractional gray theory prediction results and the project measurement results shows that the prediction models are more accurate and better fit the measured results.