 7. COST ALLOCATION IN A STEAM-TURBINE COGENERATION Step 1. Analysis of Each State. The heat balance diagram of a steam-turbine co-generation 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 T-s diagram, are illustrated in Figure 8. Step 2. Input Data. All the input data and equations for the cost allocation of the steam-turbine co-generation 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 left-upper 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. [top] Figure 7. A heat balance diagram on a steam turbine co-generation. Figure 8. Operating state on the steam-turbine co-generation system. [top] Table 7. Thermodynamic properties of the steam-turbine co-generation Table 8. Input data for the cost allocation of the steam-turbine co-generation system [top] Table 9. Results of the cost allocation on the steam-turbine co-generation  Figure 9. Cost allocation chart on the steam-turbine co-generation system. [top] 8. COST ALLOCATION  IN A COMBINED-CYCLE CO-GENERATION Step 1. Analysis of Each State. The heat balance diagram of a combined-cycle co-generation 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 T-s diagram, are illustrated in Figure 11. Step 2. Input Data. All the input data and equations for the cost allocation of the combined-cycle co-generation 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 gas-cycle and the steam-cycle. In Table 11 and Table 12, the superscript G means the gas-cycle, and the superscript S means the steam-cycle. 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 gas-turbine and the heat is outputted in the HRSG. From the input of the heat in the HRSG, the work is outputted in the steam-turbine 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 gas-cycle is equal to the analysis of the gas-turbine co-generation system, and the analysis of the steam-cycle is equal to the analysis of the steam-turbine co-generation system. Therefore, we must understand the cost allocations of the gas-turbine cogeneration and the steam-turbine co-generation. 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 gas-turbine and steam-turbine co-generation 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. [top] Figure 10. A heat balance diagram on a combined-cycle co-generation system. Figure 11. Operating state on the combined-cycle cogeneration system. [top] Table 10. Thermodynamic properties of the combined-cycle co-generation Table 11. Input data for the cost allocation of combined-cycle co-generation system [top] Table 12. Equations of the cost allocation of the combined-cycle co-generation system Table 13. Results of the cost allocation on the fuel input cost [top] Table 14. Results of the cost allocation on the combined-cycle co-generation system Figure 12. Cost allocation chart on the combined-cycle co-generation system. [top] 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 heat-to-work ratio is 1.00. That is, Table 14 is the results analyzing the heat-to-work ratio of 1.00 in Figure 10. In general, the operating mode of the co-generation system can be divided into electricity-only mode, electricity-heat mode (co-generation mode), and heat-only mode. Form the combination of these modes, the heat-to-work 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 electricity-only mode, the input costs are equal to the output cost of electricity. In the heat-only mode, the input costs are equal to the output cost of heat. In the co-generation mode, we should accomplish the common cost allocation. In the electricity-heat mode, the average heat-to-work ratio is 1.04. Therefore, it is desirable to analyze the heat balance diagram closest to the heat-to-work ratio 1.04. The ambient temperature in Figure 10 system is 15.0℃. However, the electricity-heat mode is mainly operated in the winter season, and the average temperature can be 0℃. Therefore, the heat balance diagram of the ambient temperature 0℃ is more appropriate. Figure 13. Heat-to-work ratio during a year.   Table 15. Average cost allocation during a year(8760 hour) [top] 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) [top] 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  co-generation 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. [top] 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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