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Sensitivity analysis of energy demands on performance of CCHP system
Energy Conversion and Management 49 (2008) 3491–3497
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Sensitivity analysis of energy demands on performance of CCHP system C.Z. Li, Y.M. Shi *, X.H. Huang School of Mechanical and Power Engineering, Shanghai Jiaotong University, Dongchuan Road 800, Minhang District, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 1 January 2008 Accepted 5 August 2008 Available online 2 October 2008 Keywords: Energy demands Sensitivity analysis Optimization CCHP
a b s t r a c t Sensitivity analysis of energy demands is carried out in this paper to study their influence on performance of CCHP system. Energy demand is a very important and complex factor in the optimization model of CCHP system. Average, uncertainty and historical peaks are adopted to describe energy demands. The mix-integer nonlinear programming model (MINLP) which can reflect the three aspects of energy demands is established. Numerical studies are carried out based on energy demands of a hotel and a hospital. The influence of average, uncertainty and peaks of energy demands on optimal facility scheme and economic advantages of CCHP system are investigated. The optimization results show that the optimal GT’s capacity and economy of CCHP system mainly lie on the average energy demands. Sum of capacities of GB and HE is equal to historical heating demand peaks, and sum of capacities of AR and ER are equal to historical cooling demand peaks. Maximum of PG is sensitive with historical peaks of energy demands and not influenced by uncertainty of energy demands, while the corresponding influence on DH is adverse. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction CCHP (combined cooling, heat and power) is an energy supply system that produces cooling, heat and electricity simultaneously from a single source of fuel. Because this new type of energy supply system has advantages in aspects of energy saving and environment protection, the study and application of CCHP in China increase speedily for the urgent needs of energy saving and environment protection. The knowledge of the performance of CCHP system is necessary for developing and spreading this technology. Parameter sensitivity analysis is an important method to understand the characters of CCHP system, and is helpful to the facility evaluation, optimization of sizing and operation strategy of the system. Parameters sensitivity analysis of CCHP system studies the influence of the change of parameter on the optimization results of the system. There are sensitivity analyses about fuel price, electricity price, efficiency of generator etc. [1–3]. The sensitivity analysis of energy demands on performance is necessary to study since the main aim of energy supply system is to meet the energy demands, and which are key constants in the constraints of the optimization model. In the aforementioned sensitivity analysis, electrical price, fuel price, etc. are considered as known quantities. However, it is difficult to give accurate values to energy demands because they depend on uncontrollable factors such as weather. A rational description of energy demands is needed before the sensitivity * Corresponding author. E-mail address: [email protected] (Y.M. Shi). 0196-8904/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2008.08.006
analysis. Normally, a time series of historical energy demands are used to represent the energy demands of a building. Hawkes studied the influence of sampling time interval of energy demands on the optimization of micro-cogeneration system [4]. Nowadays, hourly sampling energy demands are mostly adopted in the study of CCHP system. Piacentino [5] and Cardona [6] studied the influence of variable energy demands on performance of CCHP system with data of energy demands of 8760 h. However, their model can only be used for simple facility scheme which is optimized just from a small set of layouts. There are 26280 values of three kinds of energy demands in the 8760 sampling times. The amount of so many values will lead to dimension disaster in the MILP (mixedinteger linear programming) integrated optimization model of facility scheme and operation strategy, which is the optimization model commonly used nowadays [7,8] and is adopted in this paper. It is impossible to do analysis of each datum of energy demands. Takahshi [9] uses five variables calculated from 1152 energy demand values to describe the energy demands of cogeneration system. Similar method is adopted in this paper to describe the energy demands. In the MILP model of CCHP system, the hourly sampling energy demands of a year are not inputted directly. The averages of energy demands of each of daily sampling times of each season (or month) compose the energy demands patterns of representative day. Yun uses the energy demands of representative days in the optimization model [10]. In fact, energy demands of same season and same hour of different day present uncertainty and most of them are not equal to the average. Gamou uses a normal distribution function to describe the uncertainty of energy demands of
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Nomenclature GB GT HE AR ER EC PG Q C E D q c e P m k x y z COST CC
gas boiler gas turbine heat exchanger absorption refrigerator electrical refrigerator electricity to drive the CCHP system purchased electricity heating demands cooling demands electricity demands energy demands index of heating demands index of cooling demands index of electricity demands probability density function mth season kth sampling time design variable of CCHP continuous operation variable integer operation variable annual total cost annual fixed charge
each sampling time [11]. In addition, historical peaks of energy demands are also considered in most MILP model of CCHP. So, energy demands can be described by average, uncertainty and historical peaks. Qualitative sensitivity analyses of them about a hotel and a hospital are carried out in this paper to study their respective influence on sizing and economic feasibility of CCHP system. If one facility capacity or economic index is sensitive with one of the three aspects, the aspect shall be paid more attention when dealing with the index, otherwise, attention and more accurate is unnecessary. And it is helpful to acknowledge the relation between the performance of CCHP system and energy demands. 2. CCHP system configuration The CCHP system adopted is shown in Fig. 1. The key facility of the system is the gas turbine, which is widely adopted in CCHP system for its characters of high efficiency, low noise and emission, high feasibility, stability, etc. As far as the system in Fig. 1 is concerned, gas turbine (GT) generates electricity and heat simultaneously; heat from GT is used for heating through heat exchanger (HE) or cooling through absorption refrigerator (AR), or exhausted to atmosphere by exhaust heat exchanger (DH). Gas boiler (GB) and HE meet the heat demand
Fig. 1. Schematic diagram of gas turbine cogeneration system.
Tm DT Co R i
m k
c f N n PF F PP PG d OP IP
g d
l
days of mth season duration of sampling time operation cost of sampling time capital recovery factor interest rate salvage value to initial capital cost ratio of maintenance to initial capital unit price of facility fth facility capacity of facility service life of facility unit price of natural gas natural gas flow rate function unit price of purchased electricity purchased electricity power function combination of energy demands output of facility input of facility characteristic parameter of facility on-off status of facilities characteristic parameter of facility
jointly; electrical refrigerator (ER) and AR meet the cooling demand jointly; GT and purchased electricity (PG) supply all the electricity needed in the system. 3. Model of CCHP optimization The optimization model of CCHP system is composed of design variables x and operation variables y, z. Design variables x consist of facility capacities, maximum purchased gas and maximum purchased electricity, which are determined at the planning stage. Operation variables y and z indicate the operation status, and y, z represent continuous variables and integer variables, respectively. The relation between the two types of variables is complex and an integrated optimization model is necessary to optimize all the variables simultaneously. 3.1. Description of energy demands Energy demands are described by averages, uncertainty and historical peaks. Average energy demands are described by energy demands patterns of representative days. Historical peaks of energy demands are just three constants. The energy demands are most difficult to describe when uncertainty is considered.
Fig. 2. An example of real data of the heating demands.
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Fig. 2 is the sketch of the heating demands of transitional seasons of a hotel in Shanghai. Average heating demands of each sampling time of different days constitute the heating demands pattern of representative day. The disperse points denote the real heating demands. It is shown that the average energy demand is almost located at the center of the distribution of real data of daily sampling time. The more heating demands approach the average demand, the larger the frequency of heating demands is. It is impossible to find an accurate function to describe the probability distributions above. The probability distributions are deemed to obey roughly normal ones in paper [11] and binominal ones in paper [12]. Normal distributions are adopted to describe the energy demands of a sampling time in this paper. The average energy demands are calculated firstly. It is assumed that the probability distributions of heating demands at each sampling time obey normal distribution in which 95% of the whole area is within the range of ±20% of the average energy demands [11]. Continuous energy demands within the range of ±20% of the average energy demands are discretized by setting several representative points at the same interval. It is assumed that these discretized energy demands obey the normal distribution aforementioned. Fig. 3 shows an example of discrete probability distribution assumed of energy demands. The probability of each point is corrected so that the sum of probabilities becomes one. The heating demands of other seasons and energy demands of cooling and electricity are described similarly. Further, it is assumed that heating, cooling and electricity demands are independent of each other. So, discrete energy demands and corresponding probability are used to describe energy demands of each sampling time of representative day when uncertainty of energy demands is needed to consider.
Expected value of annual cost is adopted as the objective function to be minimized from the viewpoint of long term economics. The objective function degenerates to annual cost when uncertainty of energy demands is not considered. Annual cost includes annual initial capital, maintenance cost and operation cost. The initial capital and maintenance cost are function of facility capacities. The expected value of annual cost is expressed as
X m
Tm
X
CC ¼ f½Rð1 vÞ þ iv þ kg
X
cf Nf
ð2Þ
f
R ¼ ið1 þ iÞn =fð1 þ iÞn 1g
ð3Þ
where R denotes the capital recovery factor; v denotes the ratio of salvage value to initial capital cost; i denotes the interest rate; k denotes the ratio of maintenance to initial capital cost; f denotes the fth facility, c denotes unit price of facility; N denotes capacity of facility; n denotes service life of facility and it is assumed that the service life of all the facilities is equal. COmk is defined as
Comk ¼
XXX ½PFk q
c
e
F mk ðxqce ; ymkqce ; zmkqce ; Q mkq ; C mkc ; Emke Þ þ PPk PGmk ðxqce ; ymkqce ; zmkqce ; Q mkq ; C mkc ; Emke Þ P dmk ðQ mkq ; C mkc ; Emke Þ DQ mk DC mk DEmk
ð4Þ
P P P where q , c , e denote, respectively, the sum of heating, cooling and electricity demands of corresponding sampling time; PFk and PPk denote, respectively, the unit price of natural gas and purchased electricity; Fmk and PGmk denote, respectively, the hourly natural gas flow and hourly purchased electricity; Pdmk denotes the probability density function of energy demands, which is equal to the product of probability density functions of heat, cooling and electricity demands; Qmkq, Cmkc and Emkc denote, respectively, discrete energy demands of heating, cooling and electricity. 3.3. Constraints There are two categories of constraints, one is the facility performance constraint and the other is the energy balance constraint.
3.2. Objective function
COST ¼ CC þ
CC is expressed as [13]
DT Comk
ð1Þ
k
where COST denotes the expected value of annual total cost; CC denotes annual fixed charge; Co denotes the operation cost of the kth sampling time of mth season and DT denotes duration of sampling time.
(1) Facility constraintsConstraint of fth facility at kth sampling time is expressed as follows:
OPfmk ¼ gðxÞ IPfmk þ dfmk lðxÞ 0 6 OPfmk 6 Nf dfmk
dfmk 2 f0; 1g
ð5Þ
where OPfmk and IPfmk denote, respectively, output and input of the facility. g(x) and l(x) indicate the characteristic parameters of the facility. Nf denotes the rated capacity of facilities. Binary variable dfmk denotes the on-off status of facilities. (2) Energy balance constraintsEnergy balance constraints can be written as
AY mkqce ¼ Dmkqce
ð6Þ
where A is the matrix composed of constants related to the coefficients of facilities. Ymkqce denotes the vector of operation variables of facilities. Right side of Eq. (5) expresses the energy supply of CCHP system. Dmkqce denotes the vector of energy demands.When historical peaks of energy demands are considered, their corresponding constraints are similar with other energy demands. What different is their time duration is zero.And all variables in the Eqs. (1)–(6) are not minus. Eqs. (1)–(6) constitute the mix-integer nonlinear programming model (MINLP) of CCHP system. The model is solved with the barrier method combined with the branch and bound method after linearization the nonlinear items [10]. 4. Numerical examples 4.1. Input data
Fig. 3. An example of discrete probability distributions of heating demands.
A hotel and a hospital in Shanghai are investigated as numerical example. The hospital is composed of four independent buildings.
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The areas and historical peaks of energy demands of the hotel and hospital are shown in Table 1. Fig. 4 shows the average hourly energy demands on each representative day of the hotel. Figs. 5–7 show the average hourly energy demands on each representative day of the hospital [14]. H, C and E denote energy demands of heating, cooling and electricity. S, W and SA denote, respectively, summer, winter and transitional seasons. The discrete energy demands and corresponding probability representing can be gained with the average energy demands given above and the method in 3.1 when uncertainty of energy demands is needed to considered. There is obvious difference between the energy demands of the hotel and the hospital. In the hotel, there are three types of energy demands in each season and the uncertainty of each energy demand pattern is needed to consider. In the hospital, the cooling demands only occur in summer and electricity demand patterns are nearly same in all seasons, so only uncertainty of heating demands of all seasons and cooling demands of summer is considered. The sensitivity analyses of energy demands with obvious difference are helpful to comprehend the relation between energy demands and performance of CCHP system, and ensure the universality of corresponding conclusions. The price of electricity in Shanghai is 0.964 Yuan/kWh from 6:00 to 21:00 and 0.435 Yuan/kWh from 22:00 to 5:00 next day. The price of natural gas is 1.9 Yuan/Nm3 and the combustion value is 35.2 MJ/Nm3. The performance characters and unit price of facilities are given in Table 2. The other parameters are as follows: v = 0.1, i = 0.10, n = 15 and k = 0.03.
Fig. 5. Hourly energy demand patterns of the hospital in summer.
Fig. 6. Hourly energy demand patterns of the hospital in winter.
4.2. Studied cases
Table 1 Areas and peaks of energy demands
Hotel Hospital
Area (m2)
Heating (kW)
Cooling (kW)
Electricity (kW)
78 200 83 745
4593 4373
4666 5360
2381 2086
In the sensitivity analyses, average energy demands, as primary data, are always considered. There are four models as follows according to whether the historical peaks and uncertainty of energy demands are considered or
Fig. 4. Hourly energy demand patterns of the hotel.
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5. Results and discussion 5.1. Influence on optimal capacities of facilities The facilities are classified into two categories, core facilities and assistant facilities, for convenience to analyses. Core facilities are facilities to achieve cogeneration, including gas turbine (GT), absorption refrigerator (AR) and heat exchanger (HE). Assistant facilities include gas boiler (GB), electrical refrigerator (ER), and exhaust heat exchanger (DH). And maximum purchased electricity (PG) can be seen as a special assistant facility.
Fig. 7. Hourly energy demand patterns of the hospital in transitional seasons.
Table 2 Efficiencies and unit price of the facilities Facility
Efficiency coefficient
GT
gGT gRE gGB gHE
GB HE DH AR ER
Unit price (Yuan/kW) 0.27 0.48 0.90 0.98 1.00 1.2 4.50
– COPAR COPER
6800 300 200 100 1200 970
not. Eight cases are carried out, respectively, for the hotel and the hospital by the four models. Cases 1–4 are for the hotel and cases 5–8 are for the hospital.
5.1.1. Core facilities The optimal capacities of core facilities are shown in Figs. 8 and 9. GT are the key facility in CCHP system. It is a vital step to choose the capacity of GT in the design of CCHP system. The success of CCHP system lies on the optimal capacity of GT in a great degree. In Fig. 8, GT’s capacities in different cases have only a little difference. GT’s capacity decreases slightly after uncertainty of energy demands is considered and increases slightly after historical peaks are considered. In Fig. 9 GT’s capacity keep constant in all cases. So GT’s capacity is not sensitive with uncertainty and historical peaks of energy demands. It mainly lies on average energy demands. This is because that the marginal benefit decreases with increase of the GT’s capacity due to the operation time of the capacity newly increased decreases. It is unreasonable that GT’s capacity is chosen only according to historical peaks of energy demands in practical design of CCHP system. From Figs. 8 and 9, it is known that the capacities of HE and AR are sensitive with uncertainty and historical peaks of energy demands, but capacities of HE and AR have different change trend in the two different buildings.
Model 1: minimization of the annual cost using the energy demands of representative days. Model 2: minimization of the expected value of the annual cost with consideration uncertainty of the energy demands. Model 3: minimization of the annual cost using the energy demands of representative days and the max. energy demands. This is the model adopted mostly in corresponding study [7,15]. Model 4: minimization of the expected value of the annual cost with consideration uncertainty and peaks of energy demands. This is the most complex and most rational model. The aspects of energy demands considered in models are shown in Table 3. The influence of uncertainty of energy demands can be investigated by comparing models 1 and 2 or models 3 and 4. The influence of peaks of energy demands can be investigated by comparing models 1 and 3 or models 2 and 4. If one index studied is not sensitive with uncertainty or peaks of energy demands in all models, it must lie on average energy demands. It is necessary to note that energy demands in model 1 and model 2 have their own maximums although historical peaks of energy demands are not considered. And maximum energy demands in model 2 are larger than that in model 1.
Fig. 8. Optimal capacities of core facilities of the hotel.
Table 3 Aspects of energy demands considered in models Model
Average Uncertainty Peak
1
2
3
4
U – –
U U –
U – U
U U U
Fig. 9. Optimal capacities of core facilities of the hospital.
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5.1.2. Assistant facilities Figs. 10 and 11 show the optimal capacities of assistant facilities. It is shown that GB, ER are sensitive with both of uncertainty and historical peaks of energy demands, but their capacities have different change trend in the two different buildings. Maximum of PG are sensitive with historical peaks (or maximum) of energy demands. It increases when historical peaks are considered or the maximums of energy demands increase. Uncertainty of energy demands has no influence on maximum of PG. DH’s capacity is sensitive with uncertainty of energy demands. It increases when uncertainty of energy demands is considered. This implies that uncertainty of energy depresses efficiency of primary energy utilization. Historical peaks of energy demands have no influence on DH’s capacity. It is known from Figs. 8–11 that HE, AR, GB and ER are sensitive with both of uncertainty and historical peaks of energy demands, but their capacities have different change trend in the two different buildings. Is there universal relation between these facilities’ capacities and energy demands in the two buildings? In the hypothesis of Thermal Demand Management [16], the sum of capacities of GB and HE is equal to peaks heating demand and the sum of capacities of ER and AR is equal to peaks cooling demand. These relations are often adopted in practical project designs. Are these relations correct for the facility scheme by optimization? Historical peaks of energy demands are only considered in model3 and model4. Maximum energy demands are used in model1 and model2 to compare with sum of facilities’ capacities. The corresponding data are shown in Tables 4 and 5. Sum_h and Sum_c denote, respectively, sum of capacities of GB and HE and sum of capacities of AR and ER. Peak _h and Peak_c denote, respectively, historical heating demands and historical cooling demand. Max_h and Max_c denote, respectively, the maximums of heating and cooling energy demands.
Table 4 Facilities’ Capacities and peaks of heating and cooling demands in the hotel (kW) Capacity
Sum_h Max_h/Peak_h Sum_c Max_c/Peak_c
Case 1
2
3
4
3676 3626 3725 3684
4579 4109 4175 4175
4593 4593 4670 4666
4593 4593 4670 4666
Table 5 Facilities’ capacities and peaks of heating and cooling demands in the hospital (kW) Capacity
Sum_h Max_h/Peak_h Sum_c Max_c/Peak_c
Case 5
6
7
8
3553 3385 3686 3686
3836 3836 4178 4178
4373 4373 5360 5360
4373 4373 5360 5360
The Sum_h and Sum_c must be equal to or bigger than Max_h/ Peak_h and Max_c /Peak_c, respectively, in all four models. When the formers are bigger, the flexibility of operation strategy increases and the initial capital increases too. The optimization of capacities of GB, HE, AR and ER are virtually the balance of the flexibility and the economy. In the hotel, Sum_h is equal to Peak_h. Sum_c is very close to Peak_c with difference of 0.08%, so they can be seen as equal. Sum_h and Sum_c are, respectively, larger than Max_c and Max_h. In the hospital, Sum_h and Sum_c are, respectively, equal to Peak_h and Peak_c. Sum_h of case 6 is equal to Max_h, and Sum_h of case 5 is smaller than Max_h. Sum_c of four cases are, respectively, equal to corresponding Max_c and Peak_h. The relations aforementioned are correct for the numerical examples of the hotel and hospital. If the Peak_h and Peak_c are smaller than those in the examples, for example, they are equal to Max_h and Max_c in model 1 and model 2, which implies that the variance of energy demands is small, and then the relations aforementioned may be not correct. However, generally, the variance of energy demands of civil buildings is big, so the relations aforementioned are practical. 5.2. Influence on economy evaluation
Fig. 10. Optimal capacity of assistant facilities of the hotel.
Fig. 11. Optimal capacities of assistant facilities of the hospital.
Figs. 12 and 13 show the annual cost saving rate (CSR), which is the index to evaluate the economic feasibility adopted in this paper. As a whole, economic feasibility lies mainly on the average energy demands. The uncertainty and historical peaks of energy demands have only a little influence on the CSR. The economic feasibility will be overestimated by about 1% if historical peaks of energy demands are not considered. CSR of model 3 is very close to CSR of model 4. So, in economic evaluation, uncertainty of energy
Fig. 12. Annual cost saving rate of the hotel.
C.Z. Li et al. / Energy Conversion and Management 49 (2008) 3491–3497
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The energy demands’ structure of the hotel and the hospital are different, and the parameters are normal and representative in the model, so the conclusions have universality. References
Fig. 13. Annual cost saving rate of the hospital.
demands is not essential, and approximate evaluation can be gained even only by average energy demands. 6. Conclusions Average, uncertainty and peaks are used to describe the complex energy demands used in CCHP optimization. The MINLP model with consideration of uncertainty of energy demands is established. The sensitivity analysis about energy demands is done with consideration of the three aspects of energy demands by the four models on two types of buildings. It is shown in the results that the capacity of the key facility of the CCHP system lies mainly on average energy demands. Sum of capacities of GB and HE is equal to historical peaks of heating demand, and sum of capacities of AR and ER is equal to historical peaks of cooling demand. Maximum of PG is sensitive with historical peaks of energy demands and not influenced by uncertainty of energy demands, while the corresponding influence on DH is adverse. The uncertainty of energy demands has little influence on economy evaluation of CCHP. CSR decreases slightly after historical peaks of energy demands are considered. The economy evaluation of CCHP lies mainly on average energy demands.
[1] Sundberg G. Ranking factors of an investment in cogeneration: sensitivity analysis ranking the technical and economical factors. Int. J. Energy Res. 2001;25(3):223–38. [2] Gamou S, Ito K, Yokoyama R. Parametric study on economic feasibility of microturbine cogeneration systems by an optimization approach. J. Eng. Gas Turbines Power 2005;127(3):606–14. [3] Feng Z-B, Jin H-G. Technology of gas turbine tri-generation (cooling, heating and electric power) system and its economic analysis. J. Eng. Thermal Energy Power 2005;20(4):425–9. [4] Adam Hawkes, Matthew Leach. Impacts of temporal precision in optimization modeling of micro-combined heat and power. Energy 2005;30:1759–79. [5] Piacentino A, Cardona F. On thermoeconomics of energy systems at variable load conditions: integrated optimization of plant design and operation. Energy Convers. Manage. 2007;48:2341–55. [6] Cardona E, Piacentino A. A new approach to exergoeconomic analysis and design of variable demand energy system. Energy 2006;31:490–515. [7] Yokoyama, R. Optimal configuration design of gas turbine cogeneration plants by a MILP decomposition approach. In: Proceedings of the ASME turbo expo, vol. 3; 2007. p. 691–700. [8] Si-Doek Oh, Hoo-Suk Oh, Ho-Young Kwak. Economic evaluation for adoption of cogeneration system. Appl. Energy 2007;84:266–78. [9] Kazuke Takahshi. Application of information theory for the analysis of cogeneration-system performance. Appl. Energy 1998;61:147–62. [10] Li Y, Huang XH. Optimization of facility scheme and operation strategy of CCHP system. Dyn. Eng. 2006;26(6):894–8. [11] Gamou S, Yokoyama R, Ito K. Optimal unit sizing of cogeneration systems in consideration of uncertain energy demands as continuous random variables. Energy Convers. Manage. 2002;43:1349–61. [12] Satoshi Gamou, Ryohei Yokoyama, Koichi Ito. Optimal unit sizing of cogeneration systems under the toleration for shortage of energy supply. Int. J. Energy Res. 2000;24:61–75. [13] Zhang SJ, Li YH, Ye DJ. Optimization research on facilities scheme of gas turbine cogeneration. China Power Eng. Trans. 2004;24(10):183–8. [14] Zhang BH. Study of facility scheme optimization and operation strategy of cogeneration system. Shanghai: Tongji University; 2005. [15] Zhang BH, Long WD. An optimal sizing method for cogeneration plants. Energy Build. 2006;38:189–95. [16] Cardona E, Piacentino A. Methodology for sizing a trigeneration plant in Mediterranean areas. Appl. Thermal Eng. 2003;23:1665–80.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Sensitivity analysis of energy demands on performance of CCHP system C.Z. Li, Y.M. Shi *, X.H. Huang School of Mechanical and Power Engineering, Shanghai Jiaotong University, Dongchuan Road 800, Minhang District, Shanghai 200240, China
a r t i c l e
i n f o
Article history: Received 1 January 2008 Accepted 5 August 2008 Available online 2 October 2008 Keywords: Energy demands Sensitivity analysis Optimization CCHP
a b s t r a c t Sensitivity analysis of energy demands is carried out in this paper to study their influence on performance of CCHP system. Energy demand is a very important and complex factor in the optimization model of CCHP system. Average, uncertainty and historical peaks are adopted to describe energy demands. The mix-integer nonlinear programming model (MINLP) which can reflect the three aspects of energy demands is established. Numerical studies are carried out based on energy demands of a hotel and a hospital. The influence of average, uncertainty and peaks of energy demands on optimal facility scheme and economic advantages of CCHP system are investigated. The optimization results show that the optimal GT’s capacity and economy of CCHP system mainly lie on the average energy demands. Sum of capacities of GB and HE is equal to historical heating demand peaks, and sum of capacities of AR and ER are equal to historical cooling demand peaks. Maximum of PG is sensitive with historical peaks of energy demands and not influenced by uncertainty of energy demands, while the corresponding influence on DH is adverse. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction CCHP (combined cooling, heat and power) is an energy supply system that produces cooling, heat and electricity simultaneously from a single source of fuel. Because this new type of energy supply system has advantages in aspects of energy saving and environment protection, the study and application of CCHP in China increase speedily for the urgent needs of energy saving and environment protection. The knowledge of the performance of CCHP system is necessary for developing and spreading this technology. Parameter sensitivity analysis is an important method to understand the characters of CCHP system, and is helpful to the facility evaluation, optimization of sizing and operation strategy of the system. Parameters sensitivity analysis of CCHP system studies the influence of the change of parameter on the optimization results of the system. There are sensitivity analyses about fuel price, electricity price, efficiency of generator etc. [1–3]. The sensitivity analysis of energy demands on performance is necessary to study since the main aim of energy supply system is to meet the energy demands, and which are key constants in the constraints of the optimization model. In the aforementioned sensitivity analysis, electrical price, fuel price, etc. are considered as known quantities. However, it is difficult to give accurate values to energy demands because they depend on uncontrollable factors such as weather. A rational description of energy demands is needed before the sensitivity * Corresponding author. E-mail address: [email protected] (Y.M. Shi). 0196-8904/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2008.08.006
analysis. Normally, a time series of historical energy demands are used to represent the energy demands of a building. Hawkes studied the influence of sampling time interval of energy demands on the optimization of micro-cogeneration system [4]. Nowadays, hourly sampling energy demands are mostly adopted in the study of CCHP system. Piacentino [5] and Cardona [6] studied the influence of variable energy demands on performance of CCHP system with data of energy demands of 8760 h. However, their model can only be used for simple facility scheme which is optimized just from a small set of layouts. There are 26280 values of three kinds of energy demands in the 8760 sampling times. The amount of so many values will lead to dimension disaster in the MILP (mixedinteger linear programming) integrated optimization model of facility scheme and operation strategy, which is the optimization model commonly used nowadays [7,8] and is adopted in this paper. It is impossible to do analysis of each datum of energy demands. Takahshi [9] uses five variables calculated from 1152 energy demand values to describe the energy demands of cogeneration system. Similar method is adopted in this paper to describe the energy demands. In the MILP model of CCHP system, the hourly sampling energy demands of a year are not inputted directly. The averages of energy demands of each of daily sampling times of each season (or month) compose the energy demands patterns of representative day. Yun uses the energy demands of representative days in the optimization model [10]. In fact, energy demands of same season and same hour of different day present uncertainty and most of them are not equal to the average. Gamou uses a normal distribution function to describe the uncertainty of energy demands of
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Nomenclature GB GT HE AR ER EC PG Q C E D q c e P m k x y z COST CC
gas boiler gas turbine heat exchanger absorption refrigerator electrical refrigerator electricity to drive the CCHP system purchased electricity heating demands cooling demands electricity demands energy demands index of heating demands index of cooling demands index of electricity demands probability density function mth season kth sampling time design variable of CCHP continuous operation variable integer operation variable annual total cost annual fixed charge
each sampling time [11]. In addition, historical peaks of energy demands are also considered in most MILP model of CCHP. So, energy demands can be described by average, uncertainty and historical peaks. Qualitative sensitivity analyses of them about a hotel and a hospital are carried out in this paper to study their respective influence on sizing and economic feasibility of CCHP system. If one facility capacity or economic index is sensitive with one of the three aspects, the aspect shall be paid more attention when dealing with the index, otherwise, attention and more accurate is unnecessary. And it is helpful to acknowledge the relation between the performance of CCHP system and energy demands. 2. CCHP system configuration The CCHP system adopted is shown in Fig. 1. The key facility of the system is the gas turbine, which is widely adopted in CCHP system for its characters of high efficiency, low noise and emission, high feasibility, stability, etc. As far as the system in Fig. 1 is concerned, gas turbine (GT) generates electricity and heat simultaneously; heat from GT is used for heating through heat exchanger (HE) or cooling through absorption refrigerator (AR), or exhausted to atmosphere by exhaust heat exchanger (DH). Gas boiler (GB) and HE meet the heat demand
Fig. 1. Schematic diagram of gas turbine cogeneration system.
Tm DT Co R i
m k
c f N n PF F PP PG d OP IP
g d
l
days of mth season duration of sampling time operation cost of sampling time capital recovery factor interest rate salvage value to initial capital cost ratio of maintenance to initial capital unit price of facility fth facility capacity of facility service life of facility unit price of natural gas natural gas flow rate function unit price of purchased electricity purchased electricity power function combination of energy demands output of facility input of facility characteristic parameter of facility on-off status of facilities characteristic parameter of facility
jointly; electrical refrigerator (ER) and AR meet the cooling demand jointly; GT and purchased electricity (PG) supply all the electricity needed in the system. 3. Model of CCHP optimization The optimization model of CCHP system is composed of design variables x and operation variables y, z. Design variables x consist of facility capacities, maximum purchased gas and maximum purchased electricity, which are determined at the planning stage. Operation variables y and z indicate the operation status, and y, z represent continuous variables and integer variables, respectively. The relation between the two types of variables is complex and an integrated optimization model is necessary to optimize all the variables simultaneously. 3.1. Description of energy demands Energy demands are described by averages, uncertainty and historical peaks. Average energy demands are described by energy demands patterns of representative days. Historical peaks of energy demands are just three constants. The energy demands are most difficult to describe when uncertainty is considered.
Fig. 2. An example of real data of the heating demands.
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Fig. 2 is the sketch of the heating demands of transitional seasons of a hotel in Shanghai. Average heating demands of each sampling time of different days constitute the heating demands pattern of representative day. The disperse points denote the real heating demands. It is shown that the average energy demand is almost located at the center of the distribution of real data of daily sampling time. The more heating demands approach the average demand, the larger the frequency of heating demands is. It is impossible to find an accurate function to describe the probability distributions above. The probability distributions are deemed to obey roughly normal ones in paper [11] and binominal ones in paper [12]. Normal distributions are adopted to describe the energy demands of a sampling time in this paper. The average energy demands are calculated firstly. It is assumed that the probability distributions of heating demands at each sampling time obey normal distribution in which 95% of the whole area is within the range of ±20% of the average energy demands [11]. Continuous energy demands within the range of ±20% of the average energy demands are discretized by setting several representative points at the same interval. It is assumed that these discretized energy demands obey the normal distribution aforementioned. Fig. 3 shows an example of discrete probability distribution assumed of energy demands. The probability of each point is corrected so that the sum of probabilities becomes one. The heating demands of other seasons and energy demands of cooling and electricity are described similarly. Further, it is assumed that heating, cooling and electricity demands are independent of each other. So, discrete energy demands and corresponding probability are used to describe energy demands of each sampling time of representative day when uncertainty of energy demands is needed to consider.
Expected value of annual cost is adopted as the objective function to be minimized from the viewpoint of long term economics. The objective function degenerates to annual cost when uncertainty of energy demands is not considered. Annual cost includes annual initial capital, maintenance cost and operation cost. The initial capital and maintenance cost are function of facility capacities. The expected value of annual cost is expressed as
X m
Tm
X
CC ¼ f½Rð1 vÞ þ iv þ kg
X
cf Nf
ð2Þ
f
R ¼ ið1 þ iÞn =fð1 þ iÞn 1g
ð3Þ
where R denotes the capital recovery factor; v denotes the ratio of salvage value to initial capital cost; i denotes the interest rate; k denotes the ratio of maintenance to initial capital cost; f denotes the fth facility, c denotes unit price of facility; N denotes capacity of facility; n denotes service life of facility and it is assumed that the service life of all the facilities is equal. COmk is defined as
Comk ¼
XXX ½PFk q
c
e
F mk ðxqce ; ymkqce ; zmkqce ; Q mkq ; C mkc ; Emke Þ þ PPk PGmk ðxqce ; ymkqce ; zmkqce ; Q mkq ; C mkc ; Emke Þ P dmk ðQ mkq ; C mkc ; Emke Þ DQ mk DC mk DEmk
ð4Þ
P P P where q , c , e denote, respectively, the sum of heating, cooling and electricity demands of corresponding sampling time; PFk and PPk denote, respectively, the unit price of natural gas and purchased electricity; Fmk and PGmk denote, respectively, the hourly natural gas flow and hourly purchased electricity; Pdmk denotes the probability density function of energy demands, which is equal to the product of probability density functions of heat, cooling and electricity demands; Qmkq, Cmkc and Emkc denote, respectively, discrete energy demands of heating, cooling and electricity. 3.3. Constraints There are two categories of constraints, one is the facility performance constraint and the other is the energy balance constraint.
3.2. Objective function
COST ¼ CC þ
CC is expressed as [13]
DT Comk
ð1Þ
k
where COST denotes the expected value of annual total cost; CC denotes annual fixed charge; Co denotes the operation cost of the kth sampling time of mth season and DT denotes duration of sampling time.
(1) Facility constraintsConstraint of fth facility at kth sampling time is expressed as follows:
OPfmk ¼ gðxÞ IPfmk þ dfmk lðxÞ 0 6 OPfmk 6 Nf dfmk
dfmk 2 f0; 1g
ð5Þ
where OPfmk and IPfmk denote, respectively, output and input of the facility. g(x) and l(x) indicate the characteristic parameters of the facility. Nf denotes the rated capacity of facilities. Binary variable dfmk denotes the on-off status of facilities. (2) Energy balance constraintsEnergy balance constraints can be written as
AY mkqce ¼ Dmkqce
ð6Þ
where A is the matrix composed of constants related to the coefficients of facilities. Ymkqce denotes the vector of operation variables of facilities. Right side of Eq. (5) expresses the energy supply of CCHP system. Dmkqce denotes the vector of energy demands.When historical peaks of energy demands are considered, their corresponding constraints are similar with other energy demands. What different is their time duration is zero.And all variables in the Eqs. (1)–(6) are not minus. Eqs. (1)–(6) constitute the mix-integer nonlinear programming model (MINLP) of CCHP system. The model is solved with the barrier method combined with the branch and bound method after linearization the nonlinear items [10]. 4. Numerical examples 4.1. Input data
Fig. 3. An example of discrete probability distributions of heating demands.
A hotel and a hospital in Shanghai are investigated as numerical example. The hospital is composed of four independent buildings.
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The areas and historical peaks of energy demands of the hotel and hospital are shown in Table 1. Fig. 4 shows the average hourly energy demands on each representative day of the hotel. Figs. 5–7 show the average hourly energy demands on each representative day of the hospital [14]. H, C and E denote energy demands of heating, cooling and electricity. S, W and SA denote, respectively, summer, winter and transitional seasons. The discrete energy demands and corresponding probability representing can be gained with the average energy demands given above and the method in 3.1 when uncertainty of energy demands is needed to considered. There is obvious difference between the energy demands of the hotel and the hospital. In the hotel, there are three types of energy demands in each season and the uncertainty of each energy demand pattern is needed to consider. In the hospital, the cooling demands only occur in summer and electricity demand patterns are nearly same in all seasons, so only uncertainty of heating demands of all seasons and cooling demands of summer is considered. The sensitivity analyses of energy demands with obvious difference are helpful to comprehend the relation between energy demands and performance of CCHP system, and ensure the universality of corresponding conclusions. The price of electricity in Shanghai is 0.964 Yuan/kWh from 6:00 to 21:00 and 0.435 Yuan/kWh from 22:00 to 5:00 next day. The price of natural gas is 1.9 Yuan/Nm3 and the combustion value is 35.2 MJ/Nm3. The performance characters and unit price of facilities are given in Table 2. The other parameters are as follows: v = 0.1, i = 0.10, n = 15 and k = 0.03.
Fig. 5. Hourly energy demand patterns of the hospital in summer.
Fig. 6. Hourly energy demand patterns of the hospital in winter.
4.2. Studied cases
Table 1 Areas and peaks of energy demands
Hotel Hospital
Area (m2)
Heating (kW)
Cooling (kW)
Electricity (kW)
78 200 83 745
4593 4373
4666 5360
2381 2086
In the sensitivity analyses, average energy demands, as primary data, are always considered. There are four models as follows according to whether the historical peaks and uncertainty of energy demands are considered or
Fig. 4. Hourly energy demand patterns of the hotel.
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5. Results and discussion 5.1. Influence on optimal capacities of facilities The facilities are classified into two categories, core facilities and assistant facilities, for convenience to analyses. Core facilities are facilities to achieve cogeneration, including gas turbine (GT), absorption refrigerator (AR) and heat exchanger (HE). Assistant facilities include gas boiler (GB), electrical refrigerator (ER), and exhaust heat exchanger (DH). And maximum purchased electricity (PG) can be seen as a special assistant facility.
Fig. 7. Hourly energy demand patterns of the hospital in transitional seasons.
Table 2 Efficiencies and unit price of the facilities Facility
Efficiency coefficient
GT
gGT gRE gGB gHE
GB HE DH AR ER
Unit price (Yuan/kW) 0.27 0.48 0.90 0.98 1.00 1.2 4.50
– COPAR COPER
6800 300 200 100 1200 970
not. Eight cases are carried out, respectively, for the hotel and the hospital by the four models. Cases 1–4 are for the hotel and cases 5–8 are for the hospital.
5.1.1. Core facilities The optimal capacities of core facilities are shown in Figs. 8 and 9. GT are the key facility in CCHP system. It is a vital step to choose the capacity of GT in the design of CCHP system. The success of CCHP system lies on the optimal capacity of GT in a great degree. In Fig. 8, GT’s capacities in different cases have only a little difference. GT’s capacity decreases slightly after uncertainty of energy demands is considered and increases slightly after historical peaks are considered. In Fig. 9 GT’s capacity keep constant in all cases. So GT’s capacity is not sensitive with uncertainty and historical peaks of energy demands. It mainly lies on average energy demands. This is because that the marginal benefit decreases with increase of the GT’s capacity due to the operation time of the capacity newly increased decreases. It is unreasonable that GT’s capacity is chosen only according to historical peaks of energy demands in practical design of CCHP system. From Figs. 8 and 9, it is known that the capacities of HE and AR are sensitive with uncertainty and historical peaks of energy demands, but capacities of HE and AR have different change trend in the two different buildings.
Model 1: minimization of the annual cost using the energy demands of representative days. Model 2: minimization of the expected value of the annual cost with consideration uncertainty of the energy demands. Model 3: minimization of the annual cost using the energy demands of representative days and the max. energy demands. This is the model adopted mostly in corresponding study [7,15]. Model 4: minimization of the expected value of the annual cost with consideration uncertainty and peaks of energy demands. This is the most complex and most rational model. The aspects of energy demands considered in models are shown in Table 3. The influence of uncertainty of energy demands can be investigated by comparing models 1 and 2 or models 3 and 4. The influence of peaks of energy demands can be investigated by comparing models 1 and 3 or models 2 and 4. If one index studied is not sensitive with uncertainty or peaks of energy demands in all models, it must lie on average energy demands. It is necessary to note that energy demands in model 1 and model 2 have their own maximums although historical peaks of energy demands are not considered. And maximum energy demands in model 2 are larger than that in model 1.
Fig. 8. Optimal capacities of core facilities of the hotel.
Table 3 Aspects of energy demands considered in models Model
Average Uncertainty Peak
1
2
3
4
U – –
U U –
U – U
U U U
Fig. 9. Optimal capacities of core facilities of the hospital.
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5.1.2. Assistant facilities Figs. 10 and 11 show the optimal capacities of assistant facilities. It is shown that GB, ER are sensitive with both of uncertainty and historical peaks of energy demands, but their capacities have different change trend in the two different buildings. Maximum of PG are sensitive with historical peaks (or maximum) of energy demands. It increases when historical peaks are considered or the maximums of energy demands increase. Uncertainty of energy demands has no influence on maximum of PG. DH’s capacity is sensitive with uncertainty of energy demands. It increases when uncertainty of energy demands is considered. This implies that uncertainty of energy depresses efficiency of primary energy utilization. Historical peaks of energy demands have no influence on DH’s capacity. It is known from Figs. 8–11 that HE, AR, GB and ER are sensitive with both of uncertainty and historical peaks of energy demands, but their capacities have different change trend in the two different buildings. Is there universal relation between these facilities’ capacities and energy demands in the two buildings? In the hypothesis of Thermal Demand Management [16], the sum of capacities of GB and HE is equal to peaks heating demand and the sum of capacities of ER and AR is equal to peaks cooling demand. These relations are often adopted in practical project designs. Are these relations correct for the facility scheme by optimization? Historical peaks of energy demands are only considered in model3 and model4. Maximum energy demands are used in model1 and model2 to compare with sum of facilities’ capacities. The corresponding data are shown in Tables 4 and 5. Sum_h and Sum_c denote, respectively, sum of capacities of GB and HE and sum of capacities of AR and ER. Peak _h and Peak_c denote, respectively, historical heating demands and historical cooling demand. Max_h and Max_c denote, respectively, the maximums of heating and cooling energy demands.
Table 4 Facilities’ Capacities and peaks of heating and cooling demands in the hotel (kW) Capacity
Sum_h Max_h/Peak_h Sum_c Max_c/Peak_c
Case 1
2
3
4
3676 3626 3725 3684
4579 4109 4175 4175
4593 4593 4670 4666
4593 4593 4670 4666
Table 5 Facilities’ capacities and peaks of heating and cooling demands in the hospital (kW) Capacity
Sum_h Max_h/Peak_h Sum_c Max_c/Peak_c
Case 5
6
7
8
3553 3385 3686 3686
3836 3836 4178 4178
4373 4373 5360 5360
4373 4373 5360 5360
The Sum_h and Sum_c must be equal to or bigger than Max_h/ Peak_h and Max_c /Peak_c, respectively, in all four models. When the formers are bigger, the flexibility of operation strategy increases and the initial capital increases too. The optimization of capacities of GB, HE, AR and ER are virtually the balance of the flexibility and the economy. In the hotel, Sum_h is equal to Peak_h. Sum_c is very close to Peak_c with difference of 0.08%, so they can be seen as equal. Sum_h and Sum_c are, respectively, larger than Max_c and Max_h. In the hospital, Sum_h and Sum_c are, respectively, equal to Peak_h and Peak_c. Sum_h of case 6 is equal to Max_h, and Sum_h of case 5 is smaller than Max_h. Sum_c of four cases are, respectively, equal to corresponding Max_c and Peak_h. The relations aforementioned are correct for the numerical examples of the hotel and hospital. If the Peak_h and Peak_c are smaller than those in the examples, for example, they are equal to Max_h and Max_c in model 1 and model 2, which implies that the variance of energy demands is small, and then the relations aforementioned may be not correct. However, generally, the variance of energy demands of civil buildings is big, so the relations aforementioned are practical. 5.2. Influence on economy evaluation
Fig. 10. Optimal capacity of assistant facilities of the hotel.
Fig. 11. Optimal capacities of assistant facilities of the hospital.
Figs. 12 and 13 show the annual cost saving rate (CSR), which is the index to evaluate the economic feasibility adopted in this paper. As a whole, economic feasibility lies mainly on the average energy demands. The uncertainty and historical peaks of energy demands have only a little influence on the CSR. The economic feasibility will be overestimated by about 1% if historical peaks of energy demands are not considered. CSR of model 3 is very close to CSR of model 4. So, in economic evaluation, uncertainty of energy
Fig. 12. Annual cost saving rate of the hotel.
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The energy demands’ structure of the hotel and the hospital are different, and the parameters are normal and representative in the model, so the conclusions have universality. References
Fig. 13. Annual cost saving rate of the hospital.
demands is not essential, and approximate evaluation can be gained even only by average energy demands. 6. Conclusions Average, uncertainty and peaks are used to describe the complex energy demands used in CCHP optimization. The MINLP model with consideration of uncertainty of energy demands is established. The sensitivity analysis about energy demands is done with consideration of the three aspects of energy demands by the four models on two types of buildings. It is shown in the results that the capacity of the key facility of the CCHP system lies mainly on average energy demands. Sum of capacities of GB and HE is equal to historical peaks of heating demand, and sum of capacities of AR and ER is equal to historical peaks of cooling demand. Maximum of PG is sensitive with historical peaks of energy demands and not influenced by uncertainty of energy demands, while the corresponding influence on DH is adverse. The uncertainty of energy demands has little influence on economy evaluation of CCHP. CSR decreases slightly after historical peaks of energy demands are considered. The economy evaluation of CCHP lies mainly on average energy demands.
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