Topic A1. Building energy demand and energy performance of buildings, systems, and components
Application of Lorenz Curve and Gini Index in the Analysis of Load Feature in HVAC Systems Xin Zhou1, Da Yan1, and Yi Jiang1 1
Department of Building Science, School of Architecture, Tsinghua University, Beijing, China *
Corresponding email:
[email protected]
Keywords: Load distribution, Lorenz curve, Gini index, HVAC system, Performance SUMMARY Load feature includes how load requirements distribute among different zones and how load requirements change along the time. These characteristics influence the energy consumption and efficiency of the HVAC system significantly. Traditional method can hardly describe the load feature in a clear and straightforward way. This paper examines the load feature and its influence using Lorenz curve and Gini index. Lorenz curve can describe the load distribution among zones or times graphically, and Gini index can identify the unevenness degree in a quantitative way. Different Gini indexes, which correspond to different load features, are compared to reveal the influence of load features on the supply, distribution and adjustment process in HVAC systems. Higher Gini index reveals that the load requirements distribute unevenly among zones and along times, which usually leads to the oversupply, the loss of distribution power, the entransy dissipation and a lower heating/cooling source efficiency. INTRODUCTION Centralized HVAC (Heating, ventilation and Air Conditioning) systems have been widely applied in buildings, and are approved and supported by certain policies (Zhang et al. 2009). In some opinions, centralized HVAC systems can consume less energy with better service, so the development of future indoor environment control in the residential buildings should take the centralized HVAC systems into consideration (Aste et al. 2013). One of the main advantages of centralized HVAC systems is that it can satisfy the cooling requirements for multi-zones at the same time (Chow et al. 2004(b)). This kind of cooling supply mode has the advantage of applying refrigeration equipment with large capacity and high efficiency. Compared with split AC systems, the installed power of equipment can be reduced (Chow et al. 2004(c); Shimoda et al. 2008; Soederman 2007; Jordi et al. 2013). Moreover, as to renewable energy, like underground water or seawater, influenced by the type of cooling source, centralized cooling systems are simpler and less expensive, and from the view of energy use and urban landscape, centralized HVAC systems are effective and should be promoted (Rezaie and Rosen 2012; Chow et al. 2004(a)). However, in some other opinions, decentralized HVAC systems are more advantageous. With decentralized HVAC systems, users are more flexible to control the AC terminals according to their requirements. Under this kind of control method, the heating/cooling consumption supplied by the HVAC system would be reduced effectively (Li and Jiang 2009). Meanwhile,
there is no distribution system in decentralized HVAC systems, which means that the total energy consumption would not include the consumption of fans or pumps. As Fig.1 and Fig.2 shows, many researches (Long et al. 2003; Chen et al. 2008; Building Energy Research Center in Tsinghua University 2013) have been conducted that examined the energy consumption in commercial and residential buildings in different districts in China. From the comparison, the energy consumption of centralized HVAC systems seems larger than that of decentralized systems in general. The largest discrepancies between the energy consumption of centralized and decentralized systems can reach 10 times. AC electricity consumption/ （kWh/m2）
25 20 15 10 5 0 Wuhan Guanzhou Shanghai Hangzhou Hunan Split AC system
Xi'an
Beijing
Suzhou
Beijing
Wuhan
Nanjing
Nanjing
Henan
Beijing
Centralized AC system
AC electricity consumption (kWh/m2)
Figure 1. Energy consumption comparison in residential buildings 70 60 50 40 30 20 10 0 Beijing Beijing Beijing Shanghai Shanghai Beijing Beijing Beijing Beijing Shanghai Shanghai Shanghai Shanghai VRF
centralized AC system
Figure 2. Energy consumption comparison in commercial buildings Some researchers have realized that the building thermal load has an important effect on the performance of centralized HVAC systems (Chow et al. 2004 (b); Nagota et al. 2008; Shimoda et al. 2008). However, it is still not clear how to define the load feature and how to connect the load feature with its influence in a quantitative way. In this paper, the uneven distributed load feature of HVAC systems is identified and analyzed firstly, and a new methodology is applied to describe the characteristics in a graphic and quantitative way. Meanwhile, the influence of the uneven degree of load feature is discussed to figure out the main reason for the differences of energy consumption between the two types of HVAC systems. METHODS Centralized HVAC systems serve several zones together at the same time. How these loads distribute among these zones and how load changes along the time would influence the operation of the system greatly. Namely, the characteristics of heating/cooling requirements reflect on both time and space dimensions. Meanwhile, the extreme requirement of thermal load would also influence the consumption and operation of HVAC systems greatly. Therefore, the
main load characteristics in HVAC systems should be analyzed from three aspects, as Fig. 3 shows.
Load feature
Load distribution at space dimension
Load distribution at time dimension
Extreme requirement of load
Figure 3. Load features Lorenz curve and Gini coefficient (Corrado 1921) are applied to evaluate the uneven feature of thermal load from time and space dimensions in a quantitative way. Lorenz curve and Gini coefficient are commonly used in economics to represent the inequality of the wealth distribution. The concept is also useful in describing inequality among the size of individuals in ecology (Christian and Weiner 2000) and in studies of biodiversity, where the cumulative proportion of species is plotted against the cumulative proportion of individuals (Lieven et al. 2009). In this paper, it is used to describe the inequality of load distribution among zones and load change among times. Quantitative description of load distribution When Lorenz curve is used to describe the load distribution among different zones, as Fig. 4 shows, the x-axis represents the cumulative share of the number of zones in the order of lowest to highest load requirement, and the y-axis is the cumulative share of load requirements. The load requirement is expressed in a dimensionless way, which equals the load needed at this moment divides the rated load. When the Lorenz curve deviates more from the 45o line, the load distribution has significant differences among zones. And if the Lorenz curve overlaps the 45o line, it means that all the zones have the same load requirement. The Lorenz curve is a graphical representation, and the Gini coefficient is a quantitative index to represent the distribution. The Gini coefficient can be thought of as the ratio of the area that lies between the line of equality and the Lorenz curve over the total area under the line of equality, namely the ratio A/(A+B) in Fig. 4, where A and B are the indicated areas. So the larger the Gini coefficient, the more uneven the load distribution is. Gini index can be calculated by the equation:
Gini 1
1 n2 y
n
(2n 2i 1) y i 1
i
(1)
Where, n is the number of zones, y is the average value of the dimensionless load requirements, yi is the dimensionless load requirement of each zone. Taking the heating loads in a typical hour as an example, 100 zones are studied and three different climate zones are analyzed in this case. All the zones have the same size (10mx5mx3m). It is assumed that there is only one external wall (10mx3m), and all other walls are heat insulation. Other settings are shown in Table 1.
The Lorenz curves and Gini indexes of the three kinds of load distribution in this hour are shown in Fig. 5. It can be detected that as the outdoor temperature increases (from location A to C), the variation of heating load in different zones become significant, which reveals that the influence of indoor heat gain become evident gradually. The results prove the effectiveness of the application of Lorenz curve and Gini index to express the load distribution in a typical moment.
Cumulative share of load requirements
120%
100%
80%
60%
A
40%
B 20%
0% 0%
20% 40% 60% 80% 100% Cumulative share of zones from lowest to highest load requirements
Figure 4. Application of Lorenz curve in load distribution Table 1. Settings of the heating load calculation Location Heat transfer coefficient of external walls W/m2/K Outdoor dry bulb temperature Correction coefficient of solar radiation Indoor heat gain
A
B
C
0.4
0.5
1
-26 -9 5 -0.15 -0.25 -0.35 Uniform distributed among [0 400]W
Quantitative description of load change The Lorenz curve and the Gini coefficient can also be applied to reflect the load change among different moments. The x-axis represents the cumulative share of the number of time in the order of lowest to highest load requirement, and the y-axis is the cumulative share of load requirements. The load requirement here equals to the load requirement of the system divides the rated load that the heating/cooling source on operation can supply at the corresponding time. The calculation of Gini index is similar to that for load distribution. For example, two kinds of AC use modes are studied in a residential community in Beijing, China. As Table 2 shows, under AC use mode A, the AC would be used part time and the setting temperature is relatively high, while under AC use mode B, the AC would be always on operation, and the setting temperature is lower. Table 2. Two kinds of AC use modes Use-mode
Opening T of AC Living room Bedroom
Setting T of AC Living room Bedroom
Use time of AC Living room Bedroom
A
29℃
28℃
Only at night
29℃
27℃
Only at night
-
B
-
24℃
24℃
All day long
All day long
With the two kinds of AC use modes, the variation of cooling load in the community along time has large differences, as Fig. 5 shows. Under AC use mode A, for the low AC usage and the frequent adjustment of indoor environment by residents, the cooling requirement changes a lot in a day, which can be detected from the Lorenz curve (Fig. 5 (a)) and results in a larger Gini index. Meanwhile, with AC use mode B, the residents seldom adjust the AC equipment, and the variation of cooling load in a day is relatively small. These phenomena can also be described graphically in Fig. 5 (b), and denoted quantitatively by a smaller Gini index. 1 Cumulative share of load requirements
Cumulative share of load requirements
1
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0
0
0
0.2
0.4
0.6
0.8
Cumulative share of time from lowest to highest load requirements
(a) AC use mode A Gini = 0.8
1
0
0.2
0.4
0.6
0.8
1
Cumulative share of time from lowest to highest load requirements
(b) AC use mode B Gini = 0.5
Figure 5. The application of Lorenz curve and Gini index in cooling load change RESULTS The influence of the load feature on the HVAC system includes three parts: 1) oversupply; 2) distribution loss; 3) the efficiency of cooling/heating source. Load features’ influence on the oversupply In the system level, although the load requirements among zones are different, the supplied air temperature is unified, and in CAV system, even the supply air volume is constant. The unified supply and diversified requirements would inevitably lead to oversupply. The relationship between the unevenness of load requirement and the amount of oversupply is shown in Fig. 6. The oversupply ratio is defined as the supplied load divided by the required load, and the Gini index is used to identify the unevenness of load requirements. It can be detected that as the diversity of load requirements increases, the amount of oversupply also increases. And the oversupply ratio in CAV systems is larger than that in VAV systems.
(b) VAV system
(a) CAV system 3.5 Ratio vs. Index fit 1
Ratio vs. Index fit 1
1.5
Oversupply ratio
Oversupply ratio
3
2.5
2
1.5
1.4
1.3
1.2
1.1
1
0.1
0.2
0.3
0.4
0.5
0.6
0.1
0.15
0.2
0.25
Gini Index
0.3
0.35
0.4
0.45
0.5
0.55
0.6
Gini Index
Figure 6. The relationship between load requirement and oversupply Load features’ influence on the distribution process A typical FCU (Fan Coil Unit) system with 200 FCUs is studied. Load features with different Gini indexes are studied, and the DeST programmer (Yan et al., 2008; Zhang et al., 2008; Zhu et al. 2013; Zhou et al, 2014) is applied to calculate the loss of power. Define the percentage of loss as the ratio of power loss caused by uneven distributed load feature to the total power loss. Then the relationship between Gini index and the percentage of loss can be analyzed, as shown in Fig. 7. 0.7 Percentage of Loss
0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4 0.6 Gini Index
0.8
1
Figure 7. The relationship between Gini index and the percentage of loss The result in Fig. 7 proves that as the increase of Gini index, which reflects the uneven degree of load distribution becomes higher, the percentage of loss increases as well. For the rest part of the loss of power is almost constant, it reveals that a more uneven distributed load feature would result in a larger loss of power, and Gini index can be used as a quantitative index to calculate the loss of power. Load features’ influence on the adjustment process To deal with the unevenness of load change among different times, the heating/cooling source needs to make several adjustments, such as the changes in the number and collocation of equipment on operation and so on. However, when there are great variations of load requirements along the time, it is inevitable that during some times, the heating/cooling source would operate under low load ratio, which would lead to a lower equipment efficiency and higher energy consumption.
7 6
COP
5 4 3 2 1 0 0
0.2
0.4 0.6 TGini Index
0.8
1
Figure 8. COP of chillers under different load changes Different load distributions among moments are studies, and Gini index is applied to describe the uneven degree. When the Gini index is small, which reflects that the load change is flat during a day, it is easy for the chillers to keep a high load ratio, which results in a relatively good performance. Meanwhile, when the Gini index is high, the load changes greatly during a day. In order to satisfy the peak load requirement, the chillers have to operate under low load ratios most of the time, which reflects on the low COP of the chillers. A typical chiller model (Zhang et al., 2008) is applied, and the climate in Beijing is studied. In this case, the influence of load change on the performance of chillers is shown in Fig. 8. It can be detected that as the Gini index increases, the COP of chillers decreases, which reflects that the performance of the chillers deviates from the design situation. CONCLUSIONS Load feature influences the energy consumption and efficiency of the HVAC system significantly. This paper examines the load feature and its influence using Lorenz curve and Gini index. The main conclusions can be summarized as follows: 1) Load feature includes how load requirements distribute among different zones and how load requirements change along the time. Lorenz curve and Gini index are suitable for identifying the uneven degree of load features in a quantitative way; 2) How load requirements distribute among zones has a significant influence on the performance of the water/air distribution process in HVAC systems. Based on Lorenz curve and Gini index, it is clear and straightforward to illustrate the load feature and judge its influence on the oversupply, distribution loss and entrasy dissipation; 3) The change of load requirements along time plays a key role on the efficiency of the heating/cooling source. Lorenz curve and Gini index can be used to evaluate the effect of the load feature on the performance of the heating/cooling source, and a lower Gini index corresponds to a higher COP in general. ACKNOWLEDGEMENT Supported by Beijing Natural Science Foundation (8142022). REFERENCES Aste N., Adhikari R.S., and Manfren M. 2013. Cost optimal analysis of heat pump technology adoption in residential reference buildings. Renewable Energy, 60, 615-624
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