7. COST ALLOCATION IN A STEAMTURBINE
COGENERATION
Step 1. Analysis
of Each State. The
heat balance diagram of a steamturbine cogeneration is illustrated in Figure 7. The mass
flow rate, temperature, pressure, enthalpy, entropy, and exergy on each state
are shown in Table 6, and the results, plotted on a Ts diagram, are
illustrated in Figure 8.
Step 2. Input Data.
All the input data and equations for the cost
allocation of the steamturbine cogeneration system are given in Table 7. The
work efficiency in the
condensing mode of Figure 7 was given as 35.00%. However, this value is not
the analyzed result of Figure 7. Additionally, the work efficiency in a
condensing mode changes as often as measurements are taken. Therefore, there is
ambiguity in the methods using the work efficiency.
Step 3. Cost Allocation.
In the cost allocation, the keypoint is to find
the values of and . These values can be calculated by Table 1 and Table 8.
The values of and can be
calculated by Equations (1) and (2). The values of and can be
calculated by Equations (23) and (24).
Step 4. Analysis of the Results. The results of the cost allocation are shown in Table 9, and the charts are illustrated in Figure 9. We can confirm that the result of the heat method (1) is equal to the result of the enthalpy method (6), the result of the work method (2) is almost equal to the result of the exergetic work method (9), the result of the efficiency method (4) is almost equal to the result of the exergy method (7), and the result of the equal method (5) is almost equal to the result of the exergetic equal method (8). The results of the heat method (1) and the work method (2) are located at both endpoints, and the other results are located between them. The result of the equal method (5) is exactly located in the middle of the heat method (1) and the work method (2) results.The equalmethod presupposes that the work efficiency is equal to the
heat efficiency in the merit
distribution of Table 1. The result of the benefit distribution method (3) is
located at the leftupper side, which means that the work efficiency is higher than
the heat efficiency in the merit
distribution of Table 1. Therefore, the result of the benefit distribution
method can be considered unreasonable.
We conclude that the results of the exergy method are most reasonable within the plant boundary of Figure 1. However, the rationality in overall boundary is very different because there is the competition with other products. Therefore, the exergy method cannot be an absolutely reasonable answer. The most reasonable answer in Table 9 is that the method led to an agreement between the producer and the purchaser. In a certain case, even the heat method cannot lead to an agreement. The reason may be because a heat reservoir was not installed. We recommend the installation of a heat reservoir.
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Figure 7. A heat balance diagram on a steam turbine cogeneration.
Figure 8. Operating state on the steamturbine cogeneration
system.
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Table
7. Thermodynamic properties of the steamturbine cogeneration
Table 8. Input
data for the cost allocation of the steamturbine cogeneration system
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Table 9. Results of the cost allocation on the steamturbine cogeneration
Figure
9. Cost allocation chart on the steamturbine cogeneration system.
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8. COST ALLOCATION IN A COMBINEDCYCLE
COGENERATION
Step
1. Analysis of Each State. The
heat balance diagram of a combinedcycle cogeneration system is illustrated in
Figure 10. The mass flow rate, temperature, pressure, enthalpy, entropy, and
exergy on each state are shown in Table 10, and the results, plotted
on a Ts diagram, are illustrated in Figure 11.
Step
2. Input Data. All the input data and equations for
the cost allocation of the combinedcycle cogeneration system are given in
Table 11 and Table 12. The work efficiency in a condensing
mode of Figure 10 was given as 50.80%. This system is composed of the gascycle and
the steamcycle. In Table 11 and Table 12, the superscript G means the
gascycle, and the superscript S means the
steamcycle. To understand these symbols and equations, we must understand the
symbols and equations of Table 5 and Table 8.
Step
3. Cost Allocation. From the input of the fuel in
the combustion chamber, the work is outputted in the gasturbine and the heat
is outputted in the HRSG. From the input of the heat in the HRSG, the work is
outputted in the steamturbine and the heat is outputted in the D. H. supply.
The cost flows are equal to the energy flows. That is, the analysis of the gascycle
is equal to the analysis of the gasturbine cogeneration system, and the
analysis of the steamcycle is equal to the analysis of the steamturbine cogeneration
system. Therefore, we must understand the cost allocations of the gasturbine
cogeneration and the steamturbine cogeneration.
This system is composed of two cycles. Therefore, the cost allocation equations of Equations (27) and (28) should be used. To use Equations (27) and (28), we should calculate the cost allocation on only the fuel cost first. The results are shown in Table 13.
Step
4. Analysis of the Results. The final results of
the cost allocation are shown in Table 14, and the charts are
illustrated in Figure 12. The results of the heat method and the work
method are located at both endpoints, and the other results are located between
them. We can confirm that the result of the heat method (1) is equal to the
result of the enthalpy method (6), the result of the work method (2) is almost
equal to the result of the exergetic work method (9), the result of the
efficiency method (4) is almost equal to the result of the exergy method (7),
and the result of the equal method (5) is almost equal to the result of the
exergetic equal method (8).
In contrast to the results
of the gasturbine and steamturbine cogeneration systems, the result of the
equal method (5) is not located in the middle of the heat method (1) and the
work method (2), which means that the average of two cycles is different with
the average of one cycle.
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Figure 10.
A heat balance diagram on a combinedcycle cogeneration system.
Figure 11. Operating state on the
combinedcycle cogeneration system.
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Table 10. Thermodynamic properties of the combinedcycle cogeneration
Table 11. Input
data for the cost allocation of combinedcycle cogeneration system
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Table 12. Equations of the cost allocation of the
combinedcycle cogeneration system
Table 13.
Results of the cost allocation on the fuel input cost
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Table 14.
Results of the cost allocation on the combinedcycle cogeneration system
Figure 12. Cost allocation chart on the
combinedcycle cogeneration system.
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9. COST ALLOCATION PER A MONTH
OR A YEAR
Until now, we allocated the input
costs to the electricity and heat costs per an hour. However, the data needed for
a producer is the average data per a month or per a year. Practically, the amount
of the electricity and heat changes as in Figure 13. The system of Figure
10 produces 1159.1 GJ/h of work and 1157.9 GJ/h of heat, and the
heattowork ratio is 1.00. That is, Table 14 is the results analyzing
the heattowork ratio of 1.00 in Figure 10.
In general, the operating mode of the
cogeneration system can be divided into electricityonly mode,
electricityheat mode (cogeneration mode), and heatonly mode. Form the
combination of these modes, the heattowork ratios change as in Figure 13, and the
work cost and the heat cost also change. Therefore, we should accomplish the
cost allocation according to each mode.
The data according to each mode in Figure
13 are given in Table 15. In the electricityonly mode, the input costs
are equal to the output cost of electricity. In the heatonly mode, the input
costs are equal to the output cost of heat. In the cogeneration mode, we
should accomplish the common cost allocation.
In the electricityheat mode, the average
heattowork ratio is 1.04. Therefore, it is desirable to analyze the heat
balance diagram closest to the heattowork ratio 1.04. The ambient temperature
in Figure 10 system is 15.0¡É. However, the electricityheat mode is mainly operated in the
winter season, and the average temperature can be 0¡É. Therefore, the heat balance
diagram of the ambient temperature 0¡É is more appropriate.
Figure 13. Heattowork ratio during a
year.
Table 15. Average
cost allocation during a year(8760 hour)
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10. STRATEGY OF ELECTRICITY
SALE AND HEAT SALE
10.1. Fixed Electricity Market and Fixed Heat Market
In case there are no competitions in the electricity
market and heat market, each production cost of electricity and heat can be
estimated by the cost allocation methodology, and each sale cost can be determined
by the agreement of the producer and the purchaser.
10.2. Free Electricity Market and Fixed Heat Market
In case there are competitions in the electricity
market and there are no competitions in the heat market, the production cost of
heat can be estimated by the cost allocation methodology, and the sale unit
cost can be
determined by the agreement of the producer and the purchaser. In this case,
the minimum unit cost of electricity can be
calculated by the equation that the profit cost is zero asfollow.
(35)
(36)
where the value of changes
according to the fuel input cost , the capital cost , the amount of heat , and the amount of electricity .
The trade unit cost of electricity in a free
market changes continuously. Therefore, we should sell the electricity when the
trade unit cost is larger than the minimum unit cost as follows.
(37)
10.3. Fixed Electricity Market and Free Heat Market
In case there are no competitions in the electricity
market and there are competitions in the heat market, the production cost of
electricity can be estimated by the cost allocation methodology, and the sale
unit cost can be
determined by the agreement of the producer and the purchaser. In this case,
the minimum unit cost of heat can be
calculated by the equation that the profit cost is zero as
follow.
(38)
(39)
where the value of changes
according to the fuel input cost , the capital cost , the amount of electricity , and the amount of heat .
The trade unit cost of heat in a free
market changes continuously. Therefore, we should sell the heat when the trade
unit cost is larger than the minimum unit cost as follow.
(40)
10.4. Free Electricity Market and Free Heat Market
In case there are competitions in the electricity
market and heat market, there are only competitions, that is, there are no
agreements. Therefore, we cannot apply the cost allocation methodology to
determine the sale cost.
In this case, the equation of the
profit cost is as follows, and producers should sell the electricity and heat
when the profit is positive.
(41)
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11. GENERALITY OF THE EXERGY
METHOD
The complex energy system of Figure 15 is composed of eleven kinds of working fluids, and it produces twenty kinds of energy. All the methods except the exergy method in Table 1, can only analyze cogeneration systems. That is, only the exergy method can analyze the complex energy system of Figure 15. Therefore, the exergy method can be most reasonable.
Figure 15. Schematic diagram of a
complex energy composed of eleven types of working fluid.
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12. CONCLUSION
The cost allocation methodology can
be divided into accounting methods and engineering methods. The accounting
methods include the heat method, the work method, the benefit distribution
method, the efficiency method, and the equal method. The engineering methods
include the enthalpy method, the exergy method, the exergetic equal method, and
the exergetic work method.
The result of the enthalpy method is
exactly equal to the result of the heat method, the result of the exergy method
is almost equal to the result of the efficiency method, the result of the
exergetic equal method is almost equal to the result of the equal method, and
the result of the exergetic work method is almost equal to the result of the
work method. Therefore, we can know that the equation approximating the engineering
method is similar to the accounting method. The disadvantage of the engineering
method is that it is difficult to understand and calculate the exergy. In
contrast, the advantage of the accounting method is that it is very easy.
Therefore, we recommend the exergetic method to engineers and the accounting
method to economists.
In the case where electricity and
heat are in demand simultaneously, the most reasonable method is the exergy
method or the efficiency method. However, the greatest challenge for
cogeneration systems is that electricity and heat may not be in demand
simultaneously. In this case, we recommend the exergetic equal method or the
equal method. Under these conditions, the producer, the electricity purchaser,
and the heat purchaser will be necessarily faced with a loss of income. All the
persons concerned with the cogeneration system should collectively try to avoid
this situation.
We conclude that the results of the
exergy method are most reasonable within the plant boundary. However, the
rationality in overall boundary including the free market is very different
because there is the competition with other products. Therefore, the exergy
method cannot be absolutely reasonable answer. The most reasonable answer is
the method led to an agreement between the producer and the purchaser.
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