Application of Monte Carlo-based predictive stochastic model for energy efficiency retrofit in building clusters

In reality, building energy consumption data is characterized by a large amount of information and complexity. Similar to an organism, objects in a building, especially those related to people, are always complex and variable. In addition, there are many parameters related to building energy consumption, for example, in order to conduct a detailed energy simulation, engineers need to enter 2000~4000 parameters. To summarize, it is time-consuming and laborious to collect all the information of a building in a complete manner.

The hierarchical characterization of the parameters is an effective means to reduce the difficulty of collection. Grading data is also a common methodology used in building energy audits, such as the ASHRAE Energy Audit Standard, which categorizes energy audits into three levels. The first level (Level 1) is the most basic review record, requiring a simple survey of the building to analyze low-cost energy efficiency measures in the building and to identify what additional analysis is needed in the future. Level 2 is an intermediate level of review that requires a more detailed study of the building, to itemize energy consumption, analyze the energy savings potential and cost of all energy efficiency measures, and identify elements that need to be analyzed further. The third level (Level 3) is the most demanding audit, requiring a focused analysis of energy efficiency measures on buildings with large investments, more detailed on-site research, and a more accurate analysis process.

High-quality data are essential for subsequent high-quality data analysis. There are a variety of factors that can affect the quality of the data collected, notably:

Lack of complete historical information on design and operation. For historic buildings, design information may have been discarded. For past operational information, building owners may lack a tradition of collecting, organizing and managing energy consumption information. Limited knowledge and practice of data collectors. Different results may be obtained when conducting data collection due to different conceptual understanding of the collectors.

High-quality data is the basis for data-driven building energy analysis. Poor communication between the building design team, energy analysis engineers, and the owner’s team may lead to degradation of data quality when performing building energy analysis.

A Markov chain is a mathematical system that undergoes transitions from one state to another according to certain probability rules. In Eq. Let x₀, x₁, x₂, …, x_n be a sequence of random variables taking values in state space {S1,S2,…,Sn}$$\left\{ {{S_1},{S_2}, \ldots ,{S_n}} \right\}$$ and the sequence be a Markov chain or Markov process. Pointing out that the sequence of rocks conforms to the characteristics of a Markov chain, i.e., the state at a given moment in a state space is only related to the previous moment T₀ and not to the state before T₀, the conditional probability of the variable is expressed as the following equation p(xi+1|x1,x2,…,xi−1,xi)=p(xi+1|xi)$$p\left( {{x_{i + 1}}|{x_1},{x_2}, \ldots ,{x_{i - 1}},{x_i}} \right) = p\left( {{x_{i + 1}}|{x_i}} \right)$$. where x_i+1 denotes the pixel point to be predicted and x_i is the current pixel point, both belonging to a Markov chain X, and the states of x_i are in the set of states S.

A transfer matrix is the set of probabilities of transitioning from one state to another, represented as a matrix. A Markov chain has a finite state space, then the transfer probabilities of all states can be arranged in a matrix in a single-step evolution to obtain the transfer matrix: 1Pn,n+1=(Pin,in+1)=[ p0→0 p0→1 ⋯ p1→0 p1→1 ⋯ ⋯ ⋯ ⋯]$${P_{n,n + 1}} = \left( {{P_{{i_n},{i_{n + 1}}}}} \right) = \left[ {\begin{array}{*{20}{c}} {{p_{0 \to 0}}}&{ {p_{0 \to 1}}}& \cdots \\ {{p_{1 \to 0}}}&{ {p_{1 \to 1}}}& \cdots \\ \cdots & \cdots & \cdots \end{array}} \right]$$

Row i_n of the matrix represents the probability of X_n+1 taking all possible states at time Xn=Sin$${X_n} = {S_{{i_n}}}$$ (discrete distribution), while a single element p_i,j of the matrix with a value of p(j ∣ i) represents the probability of changing from state i to state j, where each element is non-negative and the sum of the elements in each row is 1, i.e. p_i,j ≥ 0, ∑ipi,j=1$$\sum\limits_i {{p_{i,j}}} = 1$$.

Some non-Markovian processes can be constructed by expanding the notion of “present” and “future” states, which is called a second-order Markov process. By analogy, higher order Markov processes can also be constructed. For a rock image, the set of states is the set of all possible values for each pixel. In this paper, the rock is divided into two parts, skeleton and pore, and the CT scan image of the rock is binarized according to these two parts, where the state value of 0 is the skeleton, and 1 is the pore, so there are only 2 elements 0 and 1, i.e., S = {0, 1}, in the state set of the rock, and then there are only four state transfer probabilities in the rock image transfer matrix, i.e., 0 → 0, 0 → 1. 1 → 0, and 1 → 1 as shown in Eq. (2): 2P=[ p0→0 p0→1 p1→0 p1→1]$$P = \left[ {\begin{array}{*{20}{l}} {{p_{0 \to 0}}}&{ {p_{0 \to 1}}} \\ {{p_{1 \to 0}}}&{ {p_{1 \to 1}}} \end{array}} \right]$$

Let X⁽ⁱ⁾ need to take s discrete state values and satisfy a stochastic process called Markov chain: 3p(x(i)|x(i−1),…,x1)=T(x(i)|x(i−1))$$p\left( {{x^{(i)}}|{x^{(i - 1)}}, \ldots ,{x^1}} \right) = T\left( {{x^{(i)}}|{x^{(i - 1)}}} \right)$$

If there is no change in i and T, then equation (3) is called a chi-squared Markov chain, and the chain change in x is related to the constant transfer matrix. The following Markov chain contains 3 state values, assuming that the transfer matrix is: 4T=[ 0.25 0.5 0.25 0 0.13 0.87 0.2 0.05 0.75]$$T = \left[ {\begin{array}{*{20}{c}} {0.25}&{ 0.5}&{ 0.25} \\ 0&{ 0.13}&{ 0.87} \\ {0.2}&{ 0.05}&{ 0.75} \end{array}} \right]$$

The irreducible case of randomized transfer matrices ensures that the chain gets the simplest independent matrices. The irreducible case prevents Markov from falling into a loop. p(x) needs to be satisfied: 5p(x(i))T(x(i−1)|x(i))=p(x(i−1))T(x(i)|x(i−1))$$p\left( {{x^{(i)}}} \right)T\left( {{x^{(i - 1)}}|{x^{(i)}}} \right) = p\left( {{x^{(i - 1)}}} \right)T\left( {{x^{(i)}}|{x^{(i - 1)}}} \right)$$

Summing the above equations yields: 6p(x(i))=∑x(t−1)p(x(i−1))T(x(i)|x(i−1))$$p\left( {{x^{(i)}}} \right) = \sum\limits_{x(t - 1)} p \left( {{x^{(i - 1)}}} \right)T\left( {{x^{(i)}}|{x^{(i - 1)}}} \right)$$

Monte Carlo method is a numerical simulation method that takes probabilistic phenomena as the object of study, and is a computational method for inferring unknown characteristic quantities based on the sampling method of obtaining statistical values. Monte Carlo methods are usually more efficient than traditional numerical methods and are also known as multi-probability models. The basic idea is: in order to solve the problem, a probabilistic model or a stochastic process is established, so that its parameters or numerical characteristics are equal to the solution of the problem, and then the model or the stochastic process is computed through observation or sampling tests, which results in these parameters or numerical characteristics and finally gives an approximate result of the solution of the problem. The precision of the solution is expressed as the standard error of the estimate.

When faced with severe uncertainty when making predictions or estimates, some methods replace uncertain variables with a single mean, but Monte Carlo method simulations instead use multiple values and then average the results. The probability of changing the outcome cannot be determined due to random variables interfering. Monte Carlo simulation therefore focuses on constantly repeating random samples. Monte Carlo simulation takes variables with uncertainty and assigns a random value to them, then runs the model and provides results. This process is repeated again and again until the requirements are met, in other words the Monte Carlo method is about constant experimentation. Probability distributions are statistical functions that represent a set of values distributed between limits and are often used to predict a possible uncertain variable that may consist of discrete or continuous values. In the MCMC method, the probability distribution for Monte Carlo simulation is the transfer probability matrix of the Markov chain. For rock images, the value of the requested point of the image can be regarded as a random variable, and the value of the point is requested by sampling a large number of random samples of the point and estimating the value of the point by the value of these samples.

The traditional Monte Carlo method is no longer applicable when the random variables are multivariate or the probability density is in non-standard form. Monte Carlo methods require a large number of samples, whereas Markov chains can sample a large number of samples because a random walk is equivalent to a sample, and the combination of the two can be used to solve complex problems. The MCMC method can be used to sample from any probability distribution. Mostly, we use it to sample from tricky posterior distributions for inference. A two-dimensional image is essentially a matrix, where the height and width of the image are the rows and columns of the matrix, respectively [36]. In general, a binarized image is used to represent a core that includes only pore and skeleton information, i.e., rock pores and rock skeleton are represented by 1 and 0, respectively, so that each pixel point of this image matrix has only two state values, 1 and 0. The structural features of the image can be characterized by a probability distribution function. Let the number of pixels of a two-dimensional image be n, x=(x1,x2,…,xn)$$x = \left( {{x_1},{x_2}, \ldots ,{x_n}} \right)$$ represents the value at that pixel, and x_i can only be 0 (skeleton) or 1 (pore space) in a binarized core image. Based on the original image, reconstructing an image of a 2D porous medium with similar properties using the MCMC method theoretically requires obtaining a complete probability distribution function of x, i.e., p(x).

In this paper, the prior distribution is selected using a stratified prior, which ensures that the established model can avoid over-reliance on the prior distribution to a certain extent and enhance the robustness of the model estimation. The overall population is divided into m layer according to a certain characteristic, and the overall value of each layer is set as N_k. A sample of capacity n is randomly selected from the target population s, in which the sample value of each layer is n_k, and the probability of entry is θ_k. Considering the probability of entry θ_k as an unknown parameter, the unknown parameter θ_k is given in the form of a density function, i.e., θk~π1(θk|λ)$${\theta_k}\sim {\pi_1}\left( {{\theta_k}|\lambda } \right)$$ is used as the first layer of the a priori distribution, in which λ is the hyperparameter, and Λ is the range of the values. Secondly a second layer prior (super-prior) π₂(λ) is given for the hyperparameter λ. The multilayer prior for the unknown parameter θ_k is expressed as: 8π(θk)=∫Λπ1(θk|λ)π2(λ)dλ$$\pi \left( {{\theta_k}} \right) = \int_\Lambda {{\pi_1}} \left( {{\theta_k}|\lambda } \right){\pi_2}(\lambda )d\lambda$$

The following Bayesian inference procedure is carried out based on π(θk)$$\pi \left( {{\theta_k}} \right)$$.

Assume that the prior distribution given to the parameter to be estimated θ_k is the gamma distribution of parameters α, β, i.e., θ_k ~ Gamma(α, β), k = 1, 2, …, m. Then the expression for the prior distribution of θ_k is: 9p(θk|α,β)=βαθkα−1e−θkΓ(α),θk>0$$p\left( {{\theta_k}|\alpha ,\beta } \right) = \frac{{{\beta^\alpha }\theta_k^{\alpha - 1}{e^{ - {\theta_k}}}}}{{\Gamma (\alpha )}},{\theta_k} > 0$$

where α and β are hyperparameters, but both are unknown. The second level a priori assumes respectively that hyperparameter α obeys an exponential distribution, i.e., α ~ Exponential(λ), and hyperparameter β obeys a Gamma distribution, i.e., β ~ Gamma(a, λ), as follows: 10p(α)=λeλa,α>0,λ>0Scale Parameters p(β)=λaβa−1e−λβΓ(a),β>0,a>0Shape parameters λ>0Shape parameters$$\begin{array}{l} p(\alpha ) = \lambda {e^{\lambda a}},\alpha > 0,\lambda > 0\ {\text{Scale Parameters}} \\ p(\beta ) = \frac{{{\lambda^a}{\beta^{a - 1}}{e^{ - \lambda \beta }}}}{{\Gamma (a)}},\beta > 0,a > 0\ {\text{Shape parameters}} \\ \lambda > 0\ {\text{Shape parameters}} \\ \end{array}$$

In Bayesian inference, the prior distributions of the parameters to be estimated are the basis for modeling, while the posterior distributions are the basis for statistical inference. Based on the prior distribution function of parameter θ_k and hyperparameters α, β, the joint posterior distribution function of parameter θ_k and hyperparameters α, β can be obtained: 11p(α,β,θk|data)=p(α)p(β)p(θk|α,β)p(nk|θk)$$p\left( {\alpha ,\beta ,{\theta_k}|data} \right) = p(\alpha )p(\beta )p\left( {{\theta_k}|\alpha ,\beta } \right)p\left( {{n_k}|{\theta_k}} \right)$$

When the hyperparameters α, β are fixed, the provincial entry probabilities θ_k are independent of each other and the joint posterior distribution has covariance, i.e., the posterior distribution function of the entry probability θ_k is Gamma(nk+α,Nk+β)$$Gamma\left( {{n_k} + \alpha ,{N_k} + \beta } \right)$$. Therefore, the joint posterior distribution function of the entry probability θ_k is: 12p(θk|α,β,data)=∏i=1m(Nk+β)nk+αθknk+α−1e−(Nk+β)θkΓ(nk+α)$$p\left( {{\theta_k}|\alpha ,\beta ,{\text{ }}data} \right) = \prod\limits_{i = 1}^m {\frac{{{{\left( {{N_k} + \beta } \right)}^{{n_k} + \alpha }}\theta_k^{{n_k} + \alpha - 1}{e^{ - \left( {{N_k} + \beta } \right){\theta_k}}}}}{{\Gamma \left( {{n_k} + \alpha } \right)}}}$$

Suppose that a random sequence θⁿ of parameter θ is obtained by the MCMC algorithm, and when n is large enough, θⁿ⁺¹, θⁿ⁺², θⁿ⁺³⋯ is considered to be a sample from the posterior distribution. This sample discards the first θ¹, θ², …, θⁿ⁻¹ cells of the sequence to eliminate the effect of initial values on the results. The most commonly used MCMC methods are the Metropolis-Hastings algorithm (H-T algorithm) and Gibbs sampling. M − H The algorithm is relatively simple and effective, but it is too demanding for density function selection, and its effectiveness relies on the premise that the selected density function should be close to the true posterior distribution. In contrast, Gibbs sampling has better simulation performance and specifies the conditional distribution at the time of sampling for each element of the parameter vector.

Suppose that the stochastic processes {Xk,k∈T}$$\left\{ {{X_k},k \in T} \right\}$$, T = {0, 1, 2, …}, corresponding to X_k take the value {x0,x1,x2,…}$$\left\{ {{x_0},{x_1},{x_2}, \ldots } \right\}$$ if for any k ∈ T the conditional probabilities are satisfied: 13p{Xk=xk|X0=x0,X1=x1,…,XT−1=xT−1} =p{Xk=xk|XT−1=xT−1}$$\begin{array}{l} p\left\{ {{X_k} = {x_k}|{X_0} = {x_0},{X_1} = {x_1}, \ldots ,{X_{T - 1}} = {x_{T - 1}}} \right\} \\ = p\left\{ {{X_k} = {x_k}|{X_{T - 1}} = {x_{T - 1}}} \right\} \\ \end{array}$$

Then {Xk,k∈T}$$\left\{ {{X_k},k \in T} \right\}$$ is called a Markov chain.

Thus it can be obtained: 14p{X0=x0,X1=x1,…,XT=xT} =p{X0=x0}∏i=1Tp{Xi=xi|Xi−1=xi−1}$$\begin{array}{l} p\left\{ {{X_0} = {x_0},{X_1} = {x_1}, \ldots ,{X_T} = {x_T}} \right\} \\ = p\left\{ {{X_0} = {x_0}} \right\}\prod\limits_{i = 1}^T p \left\{ {{X_i} = {x_i}|{X_{i - 1}} = {x_{i - 1}}} \right\} \\ \end{array}$$

Gibbs sampling also draws samples in a Markov chain, when the number of simulations is sufficiently high, the sampling distribution eventually converges to the joint distribution, which is suitable for occasions where the target distribution is multidimensional, as exemplified by the sampling of the a posteriori distribution f(θ ∣ x), where θ=(θ1,θ2,…,θk)$$\theta = \left( {{\theta_1},{\theta_2}, \ldots ,{\theta_k}} \right)$$. Gibbs sampling of parameter θ=(θ1,θ2,…,θq)$$\theta = \left( {{\theta_1},{\theta_2}, \ldots ,{\theta_q}} \right)$$ is shown in Fig. 2.

Figure 2.

Gibbs sampling

Gibbs sampling is performed based on the posterior distribution p(θk|α,β,data)$$p\left( {{\theta_k}|\alpha ,\beta ,data} \right)$$ of parameter θ_k to simulate the generation of a martesian chain θk(0),θk(1),……$$\theta_k^{(0)},\theta_k^{(1)}, \ldots \ldots$$ about θ_k. Gibbs sampling is a type of MCMC algorithm that involves computing the marginal, conditional, and joint distributions of a random variable θ₁, θ₂, …, θ_n. Next, the complete conditional distribution is sampled such that the sequentially drawn samples form a smooth martensian chain. Assume that the joint density probability function of θ₁, θ₁, …, θ_n depends only on the conditional density function f(θkθ−k),θ−k=(θk,…,θk−1,θk+1,θn)T$$f\left( {{\theta_k}{\theta_{ - k}}} \right),{\theta_{ - k}} = {\left( {{\theta_k}, \ldots ,{\theta_{k - 1}},{\theta_{k + 1}},{\theta_n}} \right)^T}$$ given θ_−k under θ_k. Gibbs sampling entails generating samples from the conditional density function f(θkθ−k)$$f\left( {{\theta_k}{\theta_{ - k}}} \right)$$. First, a set of random numbers θ1(0),θ2(0),…,θn(0)$$\theta_1^{(0)},\theta_2^{(0)}, \ldots ,\theta_n^{(0)}$$ is chosen as initial values, which in turn can be generated θ1(1)$$\theta_1^{(1)}$$ from the conditional density function f(θ1|θ−1(0))$$f\left( {{\theta_1}|\theta_{ - 1}^{(0)}} \right)$$. By analogy, θ2(1),θ3(1),…,θn(1)$$\theta_2^{(1)},\theta_3^{(1)}, \ldots ,\theta_n^{(1)}$$ is generated sequentially from f(θ2θ1(1),θ3(0),…,θn(0)),…,f(θnθ−n(1))$$f\left( {{\theta_2}\theta_1^{(1)},\theta_3^{(0)}, \ldots ,\theta_n^{(0)}} \right), \ldots ,f\left( {{\theta_n}\theta_{ - n}^{(1)}} \right)$$, respectively. Thus, the first set of iterated values θ1(1),θ2(1),…,θn(1)$$\theta_1^{(1)},\theta_2^{(1)}, \ldots ,\theta_n^{(1)}$$ is obtained, and after t iterations of this kind, the distribution of (θ1(t),θ2(t),…,θn(t))$$\left( {\theta_1^{(t)},\theta_2^{(t)}, \ldots ,\theta_n^{(t)}} \right)$$ converges to the distribution of (θ1,θ2,…,θn)$$\left( {{\theta_1},{\theta_2}, \ldots ,{\theta_n}} \right)$$ under certain conditions.

The prior distributions based on parameter θ_k and hyperparameters α, β are represented as directed relationship graphs by Bayesian graphical modeling method, called Doodle model at WinBUGS, consisting of graphs such as nodes, arrows, and flat panels, where each parameter, the overall value, and the sample observations are regarded as model nodes, connected by arrows. Where α is represented by alpha, β by beta, θ_k by theta[k]$$theta\left[ k \right]$$, the scale parameter λ by lambda[k]$$lambda\left[ k \right]$$, and the overall values N_k and sample observations n_k of each stratum are represented by NN[k] and nn[k]. Parameter estimates are generated sequentially from the conditional distributions that match each parameter by setting initial values for each parameter. The Doodle model is shown in Figure 3.

Figure 3.

Doodle model

Step 1: Build and test the Doodle model

A Doodle model is built based on the distribution of each parameter α, β, θ_k, and λ and ensures that each node, arrow, and plate passes the model test.

Step 2: Data Loading, Model Integration and Initial Value Generation and Assignment

The overall and sample values N_k and n_k of each stratum of the variable are defined as arrays and loaded into the model and integrated, and the initial value of the parameter (θ1(0),θ2(0),…,θm(0))$$\left( {\theta_1^{(0)},\theta_2^{(0)}, \ldots ,\theta_m^{(0)}} \right)$$ is set when the integrity and consistency of the data have been tested, but the value does not need to be necessarily approximated by the actual expected value of the parameter.

Step 3: Model Iteration and Posterior Sample Generation

Gibbs sampling values are generated sequentially from the full conditional distribution of each parameter. After t simulated sampling, a Mahalanobis chain is generated θk(0),θk(1),⋯,θk(t0),⋯$$\theta_k^{\left( 0 \right)},\theta_k^{\left( 1 \right)}, \cdots ,\theta_k^{\left( {{t_0}} \right)}, \cdots$$. Whenever the number of simulated sampling is sufficiently high, more accurate estimation results can be obtained. Meanwhile, in order to ensure the accuracy of the estimation, the pre-sampling values of the results of the first t₀ initial iterations are considered to be excluded, and the sampling values of the last n = t − t₀ times are used to calculate the marginal a posteriori densities of each parameter, and the mean value of the a posteriori estimation is chosen to be used as the estimation value of the target parameter θ_k in this paper. The estimated value θk^$$\widehat {{\theta_k}}$$ of the target variable θ_k of each layer is obtained, and the estimated weight of each layer is obtained according to dk=1/θk^$${d_k} = 1/\widehat {{\theta_k}}$$ and the next step of calibration estimation is carried out.

Step 4: Convergence discrimination of MCMC algorithm

Convergence discrimination refers to, when estimating each parameter of the model using Gibbs sampling algorithm, judging whether it is converged or not based on the trajectory graph of the parameter in the process of finite iterations, i.e., when the image obviously tends to be stable as the number of iterations increases, it can be determined that the Markov chain of the parameter is converged and the posterior estimation based on the parameter is valid.

We assumed that the cost of a traditional home is 5000 yuan and the incremental cost is 100, 200 and 300 yuan for simulation. The empirical study concludes that the service life of a home in the 1960s is 25.5 years, in the 1970s is 35.7 years, and in the 1980s is 40.4 years, and with the development of building technology and the ecological concept of building design, the service life of an eco-home should be more than 40 years, and in this paper we use a conservative estimate of 40 years.

The operating costs of a home consist primarily of energy costs. Energy costs are the costs incurred as a result of the energy consumed in heating, ventilating, cooling and lighting the home. In this paper, these two indicators are measured by the amount of water and electricity consumed per unit area (m²) of the residence per year. In this paper, domestic water consumption, residential electricity consumption and residential housing area in a city from 2000 to 2023 are selected as the basis of simulation data. From the results of the data, it can be seen that electricity costs have shown a rising trend in recent years, while water costs are relatively stable and decreasing year by year. Based on the analysis of the actual data, the variance of electricity consumption costs is 22% of the expected variance, which is due to the fact that the cost of electricity varies greatly over time, and the longer the period of time, the more factors that may affect the residential electricity consumption. In contrast, the cost of water consumption fluctuates less and the variance is fixed at 0.7. Based on the Housing Security and Housing Administration (HSA) statistics, it can be concluded that the average annual maintenance cost of a home in 2023 is 3.75yuan per square meter. Since the maintenance cost tends to increase over time and the volatility increases. Therefore, based on the actual data, the expectation of the annual maintenance cost Mt per unit area of the house is satisfied. The standard deviation is 21% of the expectation.

The most important feature of eco-house is energy saving, and the energy saving effect is generally responded by the energy saving rate. According to China’s ecological building design standards, the energy saving rate of some existing ecological buildings in China reaches more than 65%, and the power saving rate of general ecological houses is more than 50%, and the water saving rate is about 20%. Therefore, it is assumed that the power saving rate is evenly distributed between 50% and 60%, and the water saving rate is evenly distributed between 15% and 25%.

In this paper, we simulate the total cost of a 100m² eco-house and a conventional house of the same size in a city for 40 years and the energy-saving payback period of the eco-house. First, the end-of-period total costs of the eco-house with a construction cost of 5100 yuan/m² per unit area and the traditional house with 5000 yuan/m² are simulated. The full life cycle cost of the conventional house is shown in Figure 4. It can be seen that the total cost of the ecological residence has a 95% confidence interval of 540802 to 547936 yuan, which is significantly smaller than that of the traditional residence of 585904 to 605372 yuan. This indicates that eco-homes are economically feasible from a full life cycle perspective. However, in the current construction market, due to differences in design, construction, and choice of materials, the construction cost per unit area of many eco-homes is higher than that of traditional homes by 200 to 300 yuan. We simulate the total end-of-period costs in the case of eco-homes with a construction cost of 5,200 yuan/square meter and 5,300 yuan/square meter, respectively. The full life cycle cost of the eco-homes (5100 yuan/square meter) (5200 yuan/square meter) (5300 yuan/square meter) is shown in Fig. 5, Fig. 6, and Fig. 7. Comparison of the full life cycle cost of ecological and traditional houses is shown in Table 1. From the data in the table, it can be concluded that even if the construction cost of eco-households is at 5,300 yuan/m², the total cost of traditional houses (595,582 yuan) is significantly higher than that of eco-houses (564,251 yuan). The eco-house can save almost 50% of the cost in the process of using it.

Figure 4.

Traditional residential life cycle cost

Figure 5.

Ecological residential full life cycle cost

Figure 6.

Ecological residential full life cycle cost

Figure 7.

Ecological residential full life cycle cost

The next simulation is the energy-saving payback period of the eco-house. In this paper, the total end-of-period costs of eco-homes are simulated for each year from year 1 to year 40 under the conditions of construction costs of 5100 yuan/m², 5200 yuan/m², and 5300 yuan/m², respectively, and the energy-saving payback period is calculated in comparison with that of a conventional house in the same period. The energy-saving payback periods of eco-homes with different construction costs are shown in Table 2. It can be seen that the average energy-saving payback period lengthens and increases in volatility as the unit price of construction cost rises. When the construction cost is 5100 yuan/square meter, it takes at most 10 years to recover the energy-saving investment. And when the construction cost is 5,300 yuan/square meter, it takes about 25 years. It can be seen that the lower the initial construction cost control, the shorter the energy-saving payback period, and the more obvious the advantages of eco-housing are reflected. However, it is difficult to reduce the incremental cost of eco-house construction to less than 100 yuan/square meter due to the influence of technology, design, materials, construction and environment. Based on this paper, the construction unit can consider controlling the incremental cost between 100 and 200 yuan/square meter, at which time the energy-saving payback period is 9 to 13 years, which is still economically beneficial relative to the 40-year life cycle. The payback period will be even shorter if the social and environmental benefits that cannot be calculated are taken into account.

To verify the accuracy of the stochastic model’s energy efficiency retrofit predictions and the consistency of the database energy efficiency retrofit potential data. Within the study area, this paper evaluates the energy consumption data of 15 building clusters that have completed energy efficiency retrofits, with a floor area accounting for 24% of the buildings in the entire region, and demonstrates the accuracy of the model’s prediction outputs of energy efficiency retrofits on a large regional scale by observing the distribution of the building energy efficiency rates obtained from these retrofitted building samples, following the adoption of a comprehensive package of energy efficiency measures. Since the overall operating conditions and environments of these retrofit building samples do not change much in the study area, the monthly energy billing method is used to compare the results with the predictive model output. The building information and total energy consumption information in the sample pool of 15 building clusters against the energy prediction model is shown in Table 3. The total energy consumption of the observed 15 building sample pre-retrofit buildings is 510 TJ, which is 20% of the total regional building energy consumption. A total of 93 energy saving measures were taken, reducing energy consumption by 93 TJ, and the overall observed sample building energy saving rate was 17.2%.

The results of the energy consumption comparison of the 15 buildings before and after the energy efficiency retrofit are shown in Figure 9.

Figure 9.

Energy consumption comparison results

A graph comparing the LSBESR database with the actual energy savings rates for the 15 building retrofit sample is shown in Figure 10. Compared to the measure energy savings in the database, the energy savings rate data for the two groups are very close. When comparing the arithmetic mean of the two sets of data, the deviation between them is 15%. Taking building number 5 as an example, four energy-saving technical measures were implemented to retrofit the building. For the energy-saving retrofit prediction model, the corresponding prediction factors in the lighting density decreased by 5.5W/m², the winter heating efficiency index increased from 0.85 to 3, and the summer cooling source COP index increased by 22%.

Figure 10.

Comparison of LSBESR database and the actual energy saving rate

The monthly electricity consumption data (in 10,000 k Wh) of a building is shown in Table 4. The month-by-month energy consumption before and after the energy-saving retrofit of the case is shown in Fig. 11, with (a)~(b) showing the energy consumption of the building before the retrofit and the energy consumption of the building after the retrofit, respectively. According to the energy consumption observation comparison data, the energy consumption of a building before and after the retrofit decreased by 22.1%, especially when the energy-consuming electric boiler was replaced with a high-efficiency air-cooled heat pump, which increased the heating efficiency by 200% in winter under the same heating load.

Figure 11.

Energy consumption and energy consumption

Through the simulation model simulation of this paper, the comparison of the energy consumption before and after the renovation is obtained, and the comparison between the simulated value and the actual results shows Figure 12 (Figure a is the distribution of the total annual energy consumption before the renovation of the renovated building complex, Figure b is the distribution of the total energy consumption frequency of the renovated building complex before the transformation, Figure c is the annual total energy consumption distribution of the renovated building complex, and Figure D is the distribution of the total annual energy consumption frequency after the renovation of the renovated building complex). The following results are obtained:

Figure 12.

Comparison of simulation results with actual observations

The average value of the energy consumption simulated by the prediction model before the retrofit is 526TJ, while the observed value of the actual energy consumption is 511TJ, with a deviation of 2.9%, and the actual value of the energy consumption falls within the simulation value of the interval μ ± δ.

After adjusting the retrofit energy parameters, the average value of total annual energy consumption simulated by the prediction model is 405TJ, while the observed value of actual energy consumption is 412TJ, with a deviation of -1.7%, and the actual value of energy consumption falls within the interval of simulation value μ ± δ.

The average energy saving before and after the retrofit is 121 TJ, and the energy saving rate is 23%, while the actual observed energy saving is 99 TJ, and the observed energy saving rate is 19.4%, with a deviation of 22.22%, and the deviation of the energy saving rate is larger than the deviation of the total energy consumption.