A method to predict cooling load of large commercial buildings based on weather forecast and internal occupancy

Abstract

Considering the fact that customers of large commercial buildings have the characteristics of the higher density and randomness, this paper presented an air-conditioning cooling load prediction method based on weather forecast and internal occupancy density. The multiple linear feedback regression model was applied to predict, with precision, the air conditioning cooling load. Case analysis showed that the largest mean relative error of hourly and the daily predicting cooling load maximum were 18.1% and 5.14%, respectively.

Keywords

commercial building / load prediction / multiple linear regression

Introduction

With the problem of high energy consumption in large public buildings becoming prominent, energy efficiency diagnosis and retrofit of air-conditioning system is being much more cared, and is conducted in more and more existing public buildings [ 1]. Commercial buildings, which are the main type of large-scale public buildings, have the characteristics of high power density per unit area and yearly large power consumption, and the air conditioning energy consumption covers the overwhelming proportion of energy consumption in commercial buildings. The annual electrical consumption by commercial buildings increased from 45% to 60% of the total electrical energy use in Hong Kong from 1990 to 2000, which accounted for the most sector-wide electricity consumption in the past 20 years. Approximately 30% to 50% of energy consumption in commercial buildings is consumed by air conditioning systems [ 2]. The prediction of the air conditioning load in buildings is significant in order to optimize control strategies and estimate the quantity of energy storage in energy storage systems with the aim of energy saving.

Quite a few researchers have proved that weather conditions have a significant effect on the air conditioning load [ 3]. The traditional air conditioning load prediction method incorporated weather conditions into the prediction model without considering indoor disturbance, affecting the prediction accuracy. To improve the prediction accuracy and avoid complicating the prediction model, it is necessary to select indoor disturbance which has a significant influence on load changes. This paper developed a comprehensive prediction model based on the indoor and outdoor disturbance.

Analysis of impact factor of air conditioning load

In general, air conditioning load can be divided into two categories, the one mainly caused by architectural features, such as the building orientation, the height, shape coefficient, ratio of window to wall and the envelope structure, and one mainly caused by indoor disturbance and outdoor disturbance, such as the occupancy density, lighting density, heating equipment power density, outdoor temperature and humidity, the sun radiation and the fresh air volume. Obviously, building envelope is a constant parameter. Lighting and equipment power maintain constant in business, which will not be considered as the independent variables in the prediction model, in view of the fact that they have less effect on the air conditioning load change. Since the majority of commercial buildings operate with constant fresh air volume, the fresh air volume could be considered as the function of indoor occupancy density and outdoor meteorological parameters. Based on such a consideration, this paper only considers indoor occupancy density and outdoor meteorological parameters in the air conditioning load prediction model.

Basic principle of prediction methods

There are many load prediction methods, such as the multiple linear regression, the time series analysis method, the load factor method, and the artificial neutral network [ 4]. The modeling process of time series analysis is more complex and need higher theoretical knowledge. The factors in the load factor method are hard to obtain and the prediction made is not accurate. Artificial neural network cannot avoid less learning generated in the training process. The regression prediction method is a common prediction method because there is a nonlinear relationship between the most factors and prediction values of the air conditioning load [ 5]. Zhao et al. [ 6] has proposed multivariate linear regression model with feedback to predict air conditioning load. The results indicate that the multivariate linear regression model with feedback has a better prediction accuracy than the traditional nonlinear model.

Prediction model of multivariate linear regression with feedback

In air conditioning load prediction, the air conditioning load is defined as y, the effect factors are x_i (i = 1, 2, …, m). The relationship can be approximately expressed as

(1) y_i={\beta }_{\mathrm{0}}+{\beta }_{\mathrm{1}}{x}_{\mathrm{1}i}+{\beta }_{\mathrm{2}}{x}_{\mathrm{2}i}+\mathrm{...}+{\beta }_j{x}_{ji}+\mathrm{...}+{\beta }_m{x}_{mi}+{\epsilon }_i,

where i represents the serial number of the data in the sample, i=1,2,...,n; β_m represents the regression coefficient, m=1,2,...,n; and ε_i is random error, i=1,2,...,n.

The mathematical expectation of random error is zero, namely E(ε_i )=0 (i=1,2,...,n). Defining \stackrel{⌢}{y}=E(y) , then the traditional multivariate linear regression equation can be derived, namely Eq. (2).

(2) \stackrel{⌢}{y}={\stackrel{⌢}{\beta }}_{\mathrm{0}}+{\stackrel{⌢}{\beta }}_i{x}_{\mathrm{1}}+{\stackrel{⌢}{\beta }}_{\mathrm{2}}{x}_{\mathrm{2}}+\mathrm{...}+{\stackrel{⌢}{\beta }}_m{x}_m.

A residual error {\xi }_i is defined as

(3) {\xi }_i=y_{i-1}-{\stackrel{⌢}{y}}_{i-1},

where y_i−1 is the actual values, {\stackrel{⌢}{y}}_{i-1} is the prediction value obtained by using the multiple regression method. Equation (2) can be converted into Eq. (4) by incorporating {\xi }_i into the model, namely the multivariate linear regression model with feedback.

(4) \stackrel{⌢}{y}={\stackrel{⌢}{\beta }}_{\mathrm{0}}+{\stackrel{⌢}{\beta }}_i{x}_{\mathrm{1}}+{\stackrel{⌢}{\beta }}_{\mathrm{2}}{x}_{\mathrm{2}}+\mathrm{...}+{\stackrel{⌢}{\beta }}_m{x}_m+{\xi }_i.

Establishment procedure of air conditioning load prediction model

Prediction of outdoor dry bulb temperature

According to Ref. [ 7], the prediction of the hourly dry bulb temperature in next day can be calculated using the simple ASHRAE coefficient method, which is given by Eq. (5).

(5) T_i=T_{\mathrm{h}}-{\alpha }_i\times (T_{\mathrm{h}}-T_{\mathrm{l}}),

where T_i is the outdoor temperature prediction value of time i, °C; T_h is the highest weather forecasting temperature, °C; T_l is the minimum weather forecasting temperature, °C; and α_i is the temperature coefficient of prediction on time i.

Prediction of relative humidity

The prediction of relative humidity can be calculated using the index weighted moving average method presented by Kawashima et al. [ 8] in 1995 as

(6) \phi \text{'}=\phi {\text{'}}_{i-\mathrm{24}}+\lambda ({\phi }_{i-\mathrm{24}}-\phi {\text{'}}_{i-\mathrm{24}}),

where \phi \text{'} is the predicted relative humidity on time i, %; φ_i−24 is the measured relative humidity on time i of the day before, %; λ is the smoothing factor, generally taking 0.3; \phi {\text{'}}_{i-\mathrm{24}} is the predicted relative humidity on time i of the day before, %.

Prediction of solar radiation

The solar radiation is predicted by a method based on the weather forecast presented by Kawashima et al. [ 8] in 1996. First, the daily solar total radiation is figured out through the experimental data using the multiple linear equation expressed as

(7) Q_{\mathrm{s}}={\alpha }_{\mathrm{0}}+{\alpha }_{\mathrm{1}}\times T_{\mathrm{h}}+{\alpha }_{\mathrm{2}}\times (T_{\mathrm{h}}-T_{\mathrm{l}})+{\alpha }_{\mathrm{3}}\times I,

where, I is weather conditions divided into four kinds of situations like hot, sunny, cloudy and rainy respectively, corresponding to the value of 1, 2, 3 and 4 respectively. α₀, α₁, α₂, and α₃ is the temperature coefficient of prediction on time 0, 1, 2, and 3.

Then, the forecast hourly solar radiation of the day can be obtained by multiplying the hourly coefficient with the daily solar total radiation.

Prediction of indoor occupancy density

Lei et al. [ 9] has conducted many surveys on occupancy density in commercial buildings. The results indicate that there is no obvious linear relationship between the density and the area in commercial buildings, nonetheless, the types of commercial buildings, urban economic level and operating period have major impacts on occupancy density. Guo et al. [ 10] has pointed out that occupancy density is also related to commercial intensive index, and the distance index of the store away from the downtown of the city. Based on 59 groups of data collected and the conclusion made by Guo et al., this paper has established a commercial buildings occupancy density prediction model based on the method of multiple linear regression, as shown in Eq. (8), in which some symbols are explained in Table 1.

(8) D=\mathrm{0.277}-\mathrm{0.106}{{\displaystyle B}}_{\mathrm{1}}-\mathrm{0.131}{{\displaystyle B}}_{\mathrm{2}}-\mathrm{0.069}{{\displaystyle C}}_{\mathrm{1}}-\mathrm{0.065}{{\displaystyle C}}_{\mathrm{2}}-\mathrm{0.136}{{\displaystyle T}}_{\mathrm{1}}-\mathrm{0.069}{{\displaystyle T}}_{\mathrm{2}}+\mathrm{0.007}{{\displaystyle D}}_i+\mathrm{0.008}S.

Establishment of load predicting model

First, the whole parameter prediction equation can be obtained by using multivariate linear fitting with air conditioning load historical data. To simplify the air conditioning load prediction model, the stepwise regression method has been used to eliminate the non-significant factors. The regression coefficients and the significant of all parameters have been calculated. The factors have been eliminated if the Sig value (Sig represents the significant level in statistics.) of the parameters is greater than 0.1.Eventually, appropriate outdoor meteorological parameters and occupancy density have been used as the model input parameters until the Sig value decreases to less than 0.1. Figure 1 shows the prediction flowchart.

Evaluation standard of predicted results

The correlation coefficient R is used for evaluating the goodness of fit of the regression model ranging between –1 and+ 1, the more significant effect of fitting with larger R. According to engineering experience, the prediction is effective when the correlation coefficient of the model reaches above 0.75. The prediction has a fairly high accuracy when the correlation coefficient of the model reaches above 0.85 [ 11].

To verify the accuracy of the multiple linear regression models, evaluation indicators have been introduced.

The hourly mean relative error is defined as the absolute of the real value minus the predicted value divided by the real value, then multiplied by 100%. The smaller the mean relative error is, the more precisely the model can predict.

(9) \mathrm{MRE}={\displaystyle \frac{{\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^n\left|{\stackrel{⌢}{y}}_i-y_{\mathrm{ave}}\right|}}{y_{\mathrm{ave}}}}\times \mathrm{100}\%,

where MRE is the hourly mean relative error, y_{\mathrm{ave}}={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^n{y}_i} is the measured load average in predicted period, \stackrel{⌢}{y} is the predicted load value on time i, and n is the number of the total time during the predicted period.

The root mean square error is also employed to assess the performance of the prediction model.

(10) \mathrm{MRSE}=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^n{(y_i-y_{\mathrm{ave}})}^{\mathrm{2}}}}{n}}},

where MRSE is the hourly root mean square error.

Engineering application of established prediction model

To explore the applicability of the prediction method proposed in this paper, the air conditioning history monitoring data of a commercial building A have been used to establish the load prediction model, and the monitoring data have been used to analyze the accuracy of prediction results.

The commercial building A is a department store completed in 1998 with a construction area of 50000 m² in Heping District, Tianjin. The information of building envelope is listed in Table 2. The heat transfer coefficients of external wall, roof, and external window are 1.88, 0.58, and 4.7 W/(m²•K) by calculation. The air conditioning period is from April 14 to October 26 in 2012. The design cooling load is 5596.6 kW in summer.

According to the operation monitoring records of direct-fired absorption chillers, including supply and return water temperature and chilled water flow, the monitoring data can be used to calculate the hourly air conditioning load.

Based on the description in Section 2, the historical data in May have been dedicated to multivariate linear fitting of building A, and some of the training data are tabulated in Table 3. The meteorological parameters and the occupancy density of current moment and 1 to 3 h in advance are temporarily taken as the basic input parameters of the model. Moreover, the whole parameter prediction model of air conditioning load is established. The regression coefficients and significant of parameters are presented in Table 4. As can be seen from Table 4, some parameters, whose Sig value is greater than 0.1, become the non-significant factor, which are eliminated by using the stepwise regression method. The simplified model is obtained with non-significant factors. Eventually, the model has been taken with the outdoor dry bulb temperature t on current time, relative humidity RH on current time and RH₁ of 1 h in advance, the sun radiation R on current time and R₁, R₂ of 1 to 2 h in advance, and indoor occupancy density D on current time. Note that the hysteresis effect of outdoor dry bulb temperature and occupancy density on the air conditioning load is not obvious. Therefore, the prediction model can ignore the hysteresis effect. The simplified model is expressed in Eq. (11), while the coefficients in Eq. (11) are illustrated in Table 5. Since the multiple correlation coefficient is 0.964, higher than the limit of the effective prediction 0.75, this model can be applied to air conditioning load prediction.

(11) Q=-\mathrm{3027.488}+\mathrm{137.899}t+\mathrm{13.121}RH+\mathrm{10.522}{{\displaystyle RH}}_{\mathrm{1}}-\mathrm{0.525}R-\mathrm{0.295}{{\displaystyle R}}_{\mathrm{1}}-\mathrm{0.289}{{\displaystyle R}}_{\mathrm{2}}+\mathrm{2065.993}D.

The predicted data in June have been calculated by using the traditional multivariate linear regression model, and the error ε is made linear or nonlinear based on the predicted data. Finally, the multivariate linear regression prediction with feedback have been obtained by using multivariate linear regression based the prediction error. The load changing trend of two prediction methods on June 26 to June 28 is illustrated in Fig. 2. Obviously, the prediction values are obtained by using multiple linear regression model with feedback tally with the actual values; nevertheless, most prediction values are obtained by using the traditional multiple linear regression model deviate from the actual values. The hourly mean relative error of two prediction methods is depicted in Fig. 3. The mean relative error, the root mean square error, and the largest mean relative error of traditional multivariate linear regression model is 11.3%, 197, 38.4%, respectively; nevertheless, the mean relative error, the root mean square error, and the largest mean relative error of the multivariate linear regression model with feedback is 0.6, 137, 18.1%, respectively. This implies that the multivariate linear regression model with feedback have a better prediction accuracy than the traditional multivariate linear regression model.

In addition, the daily air conditioning load in June could be obtained by accumulating the hourly load prediction data. As can be seen from Fig. 4, the prediction made by the multiple linear regression prediction model with feedback is more accurate than that made by the traditional multivariate linear regression prediction model. The calculation indicates that the daily maximum relative error of the multivariate linear regression prediction model with feedback reaches 5.14%.

Conclusions

Verified by examples, in view of commercial buildings with large customer flow density and time-variation, the load prediction method based on the meteorological forecast and indoor occupancy density has a good prediction accuracy. The multivariate linear regression model with feedback has a better prediction accuracy than the traditional multivariate linear regression model. The mean relative error, the root mean square error, and the largest mean relative error of the multivariate linear regression model with feedback is 0.6, 137, and 18.1%, respectively.

For the daily air conditioning load, it is obtained that the daily maximum relative error of the multivariate linear regression prediction model with feedback is only 5.14%.

The multiple linear regression method with feedback not only can be used in air conditioning load prediction, but also applied to other areas. For specific applicable cases, the number of independent variables is just not certain. One or more independent variables should be selected according to specific circumstances. This method is simple, less time consuming, changeable in number of independent variables, and more accurate, etc.

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