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The influence of economic parameters on the optimal values of the design variables of a combined cycle plant
Energy 35 (2010) 911–919
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The influence of economic parameters on the optimal values of the design variables of a combined cycle plant Janusz Kotowicz*, qukasz Bartela Institute of Power Engineering and Turbomachinery, Silesian University of Technology, Konarskiego 18, 44-100 Gliwice, Poland
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 October 2008 Received in revised form 3 July 2009 Accepted 9 July 2009 Available online 12 August 2009
This paper presents the analysis of the influence of fuel price variation on the optimal values of the design variables of the steam part of a combined cycle plant. The investigated system was a power plant with a triple-pressure heat recovery steam generator and extraction-condensation steam turbine. Fourteen design variables for the steam part were identified. The variables that were optimised were the pressure levels of the working medium in the steam part of the system, and characteristic differences of temperatures in the heat recovery steam generator. Thanks to the development of an optimising programme, based on the genetic algorithms theory, it was possible to find an optimal solution. The indices of economic efficiency, in the form of the break-even price of electricity, were chosen as the objective function in the optimisations. The results of economic optimisations were compared with the results of the optimisation, where the electric efficiency was the objective function. This paper includes an analysis of the sensitivity of the economic objective function to failures in the adherence of the optimal values of decision variables. This analysis allowed the selection of variables such that a failure results in the highest increase of the break-even price of electricity. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Combined cycle plant Fuel price Optimisation Genetic algorithms
1. Introduction Combined cycle plants, which utilise gas turbines and heat recovery steam generators (HRSG), are one of the most effective technologies for generating fossil fuel based electricity. With the development of gas turbines, the energy efficiency of these systems has increased, exceeding more than 60% [1]. The steam part is also an important component and its optimisation may considerably improve the energy efficiency and economic performance of the plant [2]. There are many possible configurations of combined cycles including, among others, systems with a single, double, and threepressure boiler. The latter two can also be used together with a reheater. The solutions used in each configuration may differ from each other, both by the distribution of the heated surfaces in the boiler (in series, parallel or mixed), and by the means of preheating the condensate and feed water [2–4]. The most complicated configuration of combined cycles utilise three-pressure HRSGs; the power rating of such a system with only one gas turbine is 400 MW. Aside from their high efficiency, combined cycles are characterised by their low emission of noxious
* Corresponding author. Tel.: þ48 32 237 17 45; fax: þ48 32 26 80. E-mail address: [email protected] (J. Kotowicz). 0360-5442/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2009.07.014
substances, short construction times, high disposability, and low capital costs. In Poland, the incorporation of combined cycle plants has been hampered by the high price of gas fuel. The popularity of these installations is expected to grow, however, as a result of the development of technology to gasify hard coal. The large fluctuation in gas fuel price (almost doubled since 2003) poses the question: how much do fuel prices influence the optimal values of the design variables of energy systems? 2. Design variables of the steam part The electric efficiency in a combined cycle plant is defined by the relation
helCC ¼
Nel
¼ mf ,LHV
NelGT þ NelST Non mf ,LHV
(1)
This relation can be transformed as follows
helCC ¼ helGT ð1 þ a,helST Þ
(2)
In this equation, helST can be interpreted as the electric efficiency in the steam part of the system ðhelST ¼ ðNelST Non Þ=Q4a Þ, a is equal to Q4a =NelGT , and helST denotes the efficiency of the gas turbine ðhelGT ¼ NelGT =ðmf ,LHVÞÞ.
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Nomenclature
a AP b-e CND D el f G,g CC limit GT,ST hex HE HRSG is m n on opt PUMP PP R s sat Y ¼ H,I,L
A area, m2 Am amortisation charges, PLN (1 PLN ¼ 0.292 V) C price, PLN F interest, PLN h specific enthalpy, kJ/kg J total investment costs, PLN k unit investment costs, PLN K,Kobr,KPR costs, changes of the working capital, costs of production, PLN L salvage value, PLN LHV lower heating value, MJ/kg m mass flow rate, kg/s N power, MW, or last year of operation NPV net present value, PLN p pressure, Pa income tax, PLN Pd Q heat flux, MW r discount rate, % t years of work T temperature, K x steam quality, or decision variable D increment h efficiency, % s annual time of operation, h
air or combustion gas approach point break-even quantity condenser deaerator electrical fuel generator combined cycle limited values installation of the gas turbine, steam turbine heat exchanger hot end of the superheater heat recovery steam generator isentropic mechanical nominal conditions own needed optimal values pump pinch point steam reheater water or steam saturation parameters steam pressure levels (H – high, I – intermediate, L – low)
Indices 1,2,3,I,II,III characteristic points in the system (Fig. 1 and Fig. 2)
Generally, the steam and gas parts in the combined cycle are not autonomous. Assuming, however, that we have chosen a gas turbine with operating characteristics determined by its thermodynamic parameters, we get the values: helGT, NelGT, Q4a. In this case, the efficiency optimisation of the combined cycle reduces to the optimisation of the steam part, with the requirement
helST ; NelST ¼ max
(3)
Since the system uses a gas turbine, we know the thermodynamic parameters at point 4a (i.e., the parameters of the flue gas leaving the gas turbine), which means that by using Eqs. (4) and (5), and also using Eqs. (6)–(10), we find that
NelST ¼ f ðDTHE ; DTPP ; DTAP ; p3s Þ
(11)
Proceeding similarly for the case of a multi-pressure system (Y), we can write
A combined cycle with a single-pressure boiler is shown in Fig. 1. The power rating of the turbine of this system is equal to
NelST ¼ f DTHE;Y ; DTPP;Y ; DTAP;Y ; p3s;Y
NelST ¼ m3s ,ðh3s h4s Þ,hm ,hg
where: Y ¼ H; I; L. In the case of systems with interstage reheating, the right-hand side of Eq. (12) must be complemented by quantities which allow for the determination of the thermodynamic parameters of the reheated steam ðp3s;R ; T3s;R ¼ T4a DTHE;R Þ, which results in
(4)
The balance of the evaporator and superheater indicates that
m3s ,ðh3s h2:1s Þ ¼ m4a ,ðh4a hIIa Þ,hhex
(5)
By means of the characteristic temperature differences, we can determine the temperature of live steam, combustion gas at the pinch point, and water at the outlet of the economiser, by using relations
T3s ¼ T4a DTHE
(6)
TIIa ¼ Tsat ðp3s Þ þ DTPP
(7)
T2:1s ¼ Tsat ðp3s Þ DTAP
(8)
This permits us to determine the enthalpy at points 3s and 4s by
h3s ¼ f ðp3s ; T3s Þ h4s ¼ f ðp3s ; T3s ; p4s ; hisST Þ
(9) (10)
NelST ¼ f DTHE;Y ; DTPP;Y ; DTAP;Y ; p3s;Y ; DTHE;R ; p3s;R
(12)
(13)
Thus, in the case of a classical three-pressure system with reheating, there are 14 variables (Eq. (13)); in a double-pressure system with reheating, there are 10; and there are only 8 in a double-pressure system without reheating.
3. Characteristics of the investigated structure The system presented in Fig. 2 was subjected to multivariable optimisation. The fundamental installations in the combined cycle are the gas turbine, heat recovery steam generator, steam turbine, condenser, deaerator, condensate extraction pump and feed water pump.
J. Kotowicz, Ł. Bartela / Energy 35 (2010) 911–919
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pressure levels of the working medium was chosen. Such a system allows for minimal exergy losses during the heat exchange between the combustion gas and the working medium, which contributes to an increase in the efficiency of the production of electricity of the whole plant [5,6]. In this case, the pressure of the working media is conditioned by three parallelly positioned pumps. The distribution of the respective surfaces, at all levels of pressure, is shown in Fig. 2. The boiler is equipped with a steam reheater, which also acts as the second part of the superheater for the intermediate-pressure steam. The last heated surface in the boiler is the deaeration economiser, which is used to preheat the condensate passing into the deaerator. The characteristic quantities of the boiler, such as the pinch point, approach point, and difference of the temperatures at the hot ends of the superheaters, are decision variables in the optimisation problem. These quantities decide the distribution of thermal loads in the respective heat exchangers (economisers, evaporators, and superheaters), determining the surface of these exchangers. Besides these quantities, the values of pressure for each pressure level are also decision variables [7,8]. Fundamental parameters of the operation of other elements of the system are characterised in Table 1. Fuel gas that is passed into the gas turbine is characterised by the following composition: CH4 ¼ 97.33%, N2 ¼ 0.86%, C2H6 ¼ 0.81%, C3H8 ¼ 0.46%, CO2 ¼ 0.28%, C4H10 ¼ 0.26%, which has a lower heating value of 48 820 kJ/kg. 4. Optimisation algorithms
Fig. 1. Combined cycle with a single-pressure boiler and temperature distribution of the media in the boiler.
In terms of the analysis in this paper, the most essential element of the system is the HRSG. The structure of HRSG with three
The fundamental design variables of the steam part of the system presented in Fig. 2 were optimised. The design variables of the steam part of combined cycles were reviewed in Chapter 2. In the case of the investigated installation, it was assumed that the pressure of the reheated steam equals the intermediate-pressure of the live steam. An additional variable was the underheating of water at the outlet of the dearation economiser. It should also be noted that the high-pressure economiser was divided into two
Fig. 2. Diagram of the investigated system.
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Table 1 Characteristic parameters of installations belonging to the analysed system. Characteristic parameters
Unit
Gas turbine Westinghouse 501G (1995 GTW) Power Efficiency of producing electric energy Combustor exit temperature Expander exit temperature Pressure ratio Ratio of the cooling air to compressed air Compressor isentropic efficiency Combustor energy efficiency Turbine isentropic efficiency Generator efficiency Relative pressure loss in the air filter Relative pressure loss in the combustor
% % % % % % %
Steam turbine Isentropic efficiency part h Isentropic efficiency part i Isentropic efficiency part l Pressure of extraction steam Pressure of outlet steam
% % % kPa kPa
Generator Efficiency
%
Condenser Cooling water temperature rise Deaerator Operation pressure Pumps Isentropic efficiency
MW % C C
Table 2 Decision variables and their ranges during optimisation.
Value 228.9 38.24 1429.8 597.5 19.2 16 88.54 99.90 93.05 98.50 0.6 3 90 90 86 200 5 98
min
max
50 5 5 15 15 000
200 20 20 20 19 000
5 5 10 2000
15 15 30 7000
p3s,L, kPa
15 50 300
30 100 1000
DTHE;R , K
30
40
DTAP;D , K
5
35
DTAP1;H , K DTAP2;H , K DTPP;H , K DTHE;H , K p3s,H, kPa
DTAP;I , K DTPP;I , K DTHE;I , K p3s,I, kPa
DTPP;L , K DTHE;L , K
S ¼ Cel
Zs
ðNel Non Þds
C
10
kPa
200
%
85
Using Eqs. (14)–(16) under the condition NPV ¼ 0 we get
Pt ¼ N ½JþðKPR þPd þKobr ÞAm FLt be Cel
¼
t¼0
ð1þrÞt
Zs Pt ¼ N t ¼0
sections, resulting in five characteristic quantities for high-pressure level. Generally, keeping in mind the analogy with Sector 2, there are 14 distinct characteristic variables that influence the operation of the steam turbine in the analysed system. In the optimisation of the system, these quantities served as decision variables. Thus, for this purpose, the following decision variables have been assumed: Pressures of live steam at each pressure level – (p3s) Underheating of water at the outlets of all economisers (including the deaeration economiser approach point) – (DTAP) Minimum pinch point temperature differences at each pressure level – (DTPP) Differences in temperature between the media at the so-called hot ends of all superheaters of steam (including the reheater) – (DTHE) The ranges of the respective decision variables are presented in Table 2. The objective function in the calculations was the break-even b-e - in this case the set of selected decision price of electricity Cel variables had to ensure the minimisation of this quantity [9]. The break-even price of electricity is the price that makes the net present value equal zero (NPV ¼ 0). NPV is one of the fundamental and most widely applied economic coefficients for the assessment of economical effectiveness [10,11].
NPV ¼
tX ¼N
CFt
t¼0
ð1 þ rÞt
(16)
o
(14)
where CFt is cash flow
CFt ¼ ½S J ðKPR þ Pd þ Kobr Þ þ Am þ F þ Lt
(15)
An essential component of equation (15) is the profit from sales S, expressed as
o
(17)
ðNel Non Þds ð1þrÞt
The fundamental problem in the analysis of the economical effectiveness in the investment of energy systems is the determination of capital costs (J). In preliminary studies, in which basic decisions are made, statistical data gathered during the course of realising similar projects are generally used. The price of installations is determined by approximating the statistical data versus characteristic quantities. The following equations were used in determining the cost of the respective appliances [12]: 0:7 KGT ¼ 2:453,3832,Ne1GT
KHRSG ¼ 2:453,35; 000,
(18)
X
khexi ,Ahexi
0:6
(19)
i
1:543 þ 823:7,NelST KST ¼ 2:453, 3197280,A0:261 ST
KG ¼ 2:453,3082,
NelGT
hgGT
þ
NelST
(20)
!0:58 (21)
hgST
KCND ¼ 2:453,2754,A1:01 CND 0:8 , 1þ KPUMP ¼ 2:453,1293:44,NPUMP
(22) 0:3
0:46
1 hisPUMP
(23)
The total investment costs may be expressed as the product of the cost of machines and installations K, and coefficient B, which takes into account the costs of construction
J ¼ K,B ¼ B,
X
Ki
i
In the calculations, the value of B was assumed to be 3.5.
(24)
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The relations in Eq. (24) and Eqs. (19)–(23) connect the value of the capital costs J with the decision variables. For optimisation, a custom-made algorithm was applied, which makes use of the popular theory of genetic algorithms [13,14]. This type of optimisation is often used in engineering research and also in the optimisation of energy systems [15,16]. The thermodynamic model of the system was constructed in the GateCycle software [17]. The block diagram of the optimising algorithm is presented in Fig. 3. Additionally, for comparison, the technical optimisation was carried out. In this case, the decision variables were chosen to maximise the electric efficiency
helCC ¼
Nel
(25)
mf ,LHV
5. Assumptions and results of optimisations Aside from determining the capital costs, many assumptions must be made when analysing an investment economically. The most important of them are: 20% of the investment is self-financed and 80% is obtained from commercial credits. The actual interest of the credit amounts to 6%, the repay time is 10 years. The anticipated time of constructing the power plant is 2 years, with a repay of 40% in the first year and 60% in the second. The time of operation – 20 years.
915
The annual time of operation – 7500 h. The discount rate amounts to 5.69%. The constant of repairs were determined to be 0.5% of the capital costs for the first four years of operation, 1% for the subsequent four years, 1.5% for the next four years, 2% for further years, and 2.5% for the last four years of exploitation. The cost of fuel was calculated in compliance with the price list of PGNiG S.A. for 2007, amounting to about 25 PLN/GJ (in relation to the lower heating value of 35.5 MJ/m3); for the analysis, 25 PLN/GJ was assumed. The assumed depreciation is 6%. The excise tax is 20 PLN/MWh. The rate of income tax is 19%. The salvage value of the designed system is assumed to be 20%$J. The change of the working capital has been neglected in the calculations. Initially, two independent optimisations were analysed; the first, where the objective function was the minimised break-even price of electricity (17), and the second, where the electric efficiency was maximised (25). The results of the optimisation of the investigated system for both objective functions, with fundamental assumptions, are shown in Table 3. The maximisation of electric efficiency as an objective function of optimisation has merely a thermodynamic aspect. In such a case, the results of optimisation are reduced to the choice of boundary values (‘‘*’’ in Tables 3 and 4) in the range of the considered values of the decision variables except for approach point in the first section of the high-pressure economiser and intermediate pressure. If the break-even price of electricity is minimised, the optimisation programme indicates values of the decision variables which deviate most from the results of the preceding optimisation. The two optimisations yielded the same results only for the lowpressure part, the reheater of steam and the deaeration economiser. The susceptibility of changes of the fundamental economic assumptions to the break-even price of electricity was then analysed. It is the investors, as well as the recipients of the products and those who are affected by the results of the investment, who decide its feasibility. A representative quantity may be for them the economic objective function (17). Therefore, in the analysis of the susceptibility, a model with variables optimised in compliance with the economic objective function was applied. The susceptibility of
Table 3 Results of the optimisation for the economical and thermodynamic objective function. b-e Þ optðCel
opt(helCC )
114.8 7.2 10.3 20.0* 16 055
107.8 5.0* 5.0* 15.0* 19 000*
11.6 10.0 29.4 3112
5.0* 5.0* 10.0* 3976
p3s,L, kPa
15.0* 100.0* 300*
15.0* 100.0* 300*
DTHE;R , K
30.0*
30.0*
DTAP;D , K
5.0*
5.0*
DTAP1;H , K DTAP2;H , K DTPP;H , K DTHE;H , K p3s,H, kPa
DTAP;I , K DTPP;I , K DTHE;I , K p3s,I, kPa
DTPP;L , K DTHE;L , K
Fig. 3. Block diagram of the optimisation algorithm.
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Table 4 Results of optimisation with the thermodynamic (first column – results) and economical objective functions concerning various prices of fuels (column 2O6).
DTAP1;H , K DTAP2;H , K DTPP;H , K DTHE;H , K p3s,H, kPa
DTAP;I , K DTPP;I , K DTHE;I , K p3s,I, kPa
DTPP;L , K DTHE;L , K p3s,L,, kPa
1
2
3
4
5
6
Opt ðhelCC Þ Cf ¼ 25.0 PLN/GJ
b-e Þ C ¼ 20.0 Opt ðCel f PLN/GJ
b-e Þ C ¼ 22.5 Opt ðCel f PLN/GJ
b-e Þ C ¼ 25.0 Opt ðCel f PLN/GJ
b-e Þ C ¼ 27.5 Opt ðCel f PLN/GJ
b-e Þ C ¼ 30.0 Opt ðCel f PLN/GJ
107.8 5.0* 5.0* 15.0* 19 000*
114.2 7.7 12.2 20.0* 15 000*
113.0 7.7 10.7 20.0* 15 344
114.8 7.2 10.3 20.0* 16 055
116.9 6.4 9.9 20.0* 17 018
119.2 5.7 9.5 20.0* 17 984
5.0* 5.0* 10.0* 3976
14.2 12.5 30.0* 2878
13.5 11.3 29.9 2975
11.6 10.0 29.4 3112
9.7 9.0 28.7 3243
8.5 8.1 27.7 3381
15.0* 100.0* 300*
15.0* 100.0* 300*
15.0* 100.0* 300*
15.0* 100.0* 300*
15.0* 100.0* 300*
15.0* 100.0* 300*
DTHE;R , K
30.0*
30.0*
30.0*
30.0*
30.0*
30.0*
DTAP;D , K
5.0*
5.0*
5.0*
5.0*
5.0*
5.0*
be , PLN/MWh Cel
205.47 57.978 33.233 85.741 0.9189 99.80 119.392 587.495 1686.58
172.21 57.511 32.446 85.692 0.9346 99.92 116.242 503.516 1458.69
187.85 57.571 32.547 85.691 0.9330 99.93 116.630 507.476 1468.52
203.48 57.620 32.630 85.702 0.9309 99.89 116.978 511.155 1477.67
219.10 57.680 32.731 85.710 0.9291 99.89 117.406 515.996 1489.82
234.71 57.733 32.820 85.718 0.9273 99.86 117.795 520.767 1501.91
helCC, % helST, % hHRSG, % x5s t5a, C NelST, MW J, mlnPLN ki, PLN/kW
the objective function was tested with respect to changes of two parameters, namely, The price of supplied gas, Capital cost. Both these quantities varied within the range of 20% in relation to nominal values. The price of fuel was calculated in relation to the nominal price
Cf ¼ b, Cf
n
Cf ¼ 25 PLN=GJ n
The total investment was calculated in relation to the nominal total investment
The quality of the steam leaving the steam turbine x5s, The temperature of flue gases leaving the power plant T5a, The power of steam turbine NelST, Total capital costs J, Unit investment costs ki (ki ¼ J/Nel).
6. Sensitivity analysis of the economical objective function to changes of the optimal values of variables The presented optimisation, depending on the respective assumptions, has allowed for six sets of optimal values of the selected design variables of the steam part. We consider the question, how essential is it to keep the optimal values of decision variables for the value of the objective function? In order to answer this question, the sensitivity to changes of the respective decision variables to the attained value of the economic
J ¼ b,K,B ¼ b,Jn Jn ¼ 511:155 mln PLN The results of the susceptibility analysis are shown in Fig. 4. The results of the analysis of susceptibility indicate that among the analysed quantities, the largest influence on the value of the economical function is exerted by changes of the costs of fuel. Due to this, further investigations dealt with the identification of the effect of changes of fuel price on the values of the obtained decision variables. For this purpose, another four optimisations were carried out for various prices of gas, viz. 20, 22.5, 27.5, 30 PLN/GJ. All the remaining assumptions did not change. The results of calculations, compared with previously presented results, are shown in Table 4. Aside from the investigated objective functions and determined values of the decision variables, several quantities are presented, which allow for interpretation of the obtained results: The efficiency of the steam cycle helST, The efficiency of the HRSG hHRSG,
Fig. 4. Break-even price of electricity as a function of the relative change of fuel price and capital costs.
J. Kotowicz, Ł. Bartela / Energy 35 (2010) 911–919
objective function was analysed based on the results of an economical optimisation, which had been carried out for a fuel price of 25 PLN/GJ. In the course of these investigations, each decision variable was successively changed within the range of relative values ðDx=xopt ¼ x xopt =xopt Þ from 0.5 to 0.5. The other quantities remained unchanged from those attained during the course of the optimisation. The respective quantities could not exceed the ranges defined in Table 2. The results of calculations are shown in Figs. 5–7, which include 14 curves. The ranges of decision variables, for which these curves were not determined, resulted from the revealed boundaries of the considered ranges, or the technical restrictions, which exclude such a situation from the set of possible solutions [18]. Thus, for instance, within some ranges of decision variables, we can observe a sudden growth of the values of the objective function: approach point in the first section of the highpressure economiser ðDTAP1;H Þ, high pressure (p3s,H), and intermediate pressure (p3s,I), caused by the establishment of parameters of the heat-exchanging media, which require a considerable expansion of the heat-exchanging surface in the heat recovery steam generator. This results in a sudden increase in capital investments. As an example of restrictions, let’s take a given maximum value of the high-pressure medium (p3s,H). In Fig. 5, we see that the value of the break-even price of electricity, together with the growth of Dx=xopt up to w0.17, approaches infinity. This means that there is some limiting value of high pressure. In order to avoid such a boundary case, the following condition must be satisfied:
TVa > T2:1s;H
(26)
Knowing that
T2:1s;H ¼ Tsat p3s;H DTAP1;H
(27)
we get
TVa > Tsat p3s;H DTAP1;H
(28)
Tsat p3s;H
(29)
917
Fig. 6. The effect of changes of the respective decision variables of the intermediatepressure part on the change of the break-even price of electricity.
TVa ¼ Tsat p3s;I þ DTPP;I
(30)
we may rewrite it as
Tsat p3s;H
(31)
Thus, as a result of Eq. (31) we can, for instance, determine the limiting value of high pressure
h i plimit 3s;H ¼ psat Tsat p3s;I þ DTPP;I þ DTAP1;H
(32)
As proved, for the high pressure, determined by Eq. (32), an equalisation of the temperature of flue gases at the inlet to the economiser of the high-pressure part, and the temperature of the water at the outlet of the economiser occurs ðT2:1s;H ¼ TVa Þ. The approximation of the high pressure to the limit pressure will also have economic consequences, because the surface area of the exchanger approaches infinity.
lim
A p3s;H ¼ N
(33)
Applying the relation
p3s;H /plimit 3s;H
Fig. 5. The effect of changes of the respective decision variables of the high-pressure part on the change of the break-even price of electricity.
Fig. 7. The effect of changes of the respective decision variables of the low-pressure part, characteristic quantity of the reheater, and quantity of the deaeration economiser on the change of the break-even price of electricity.
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Fig. 8. The effect of changing the pressure in the high- and intermediate-pressure parts on the electric efficiency, unit investment costs, and break-even price of electricity.
In Fig. 5, a similar limitation is observed in the case of the function DTAP1;H (the approach point for the first section of the high-pressure economiser), as well as in Fig. 6, for the function p3s,I (the pressure in the intermediate-pressure part). In both of these cases, we have to deal with the same kind of limitations as in the analysed case; transforming, therefore, Eq. (31) we get
h i plimit 3s;I ¼ psat Tsat p3s;H DTAP1;H DTPP;I
limit DTAP1;H ¼ Tsat p3s;H Tsat p3s;I DTPP;I
(34) (35)
In the case of the optimal variables, the values of the analysed limit limit quantities amount to plimit 3s;H ¼ 18; 829 kPa, p3s;I ¼ 2447 kPa, limit ¼ 101:7 K. and DTAP1;H Based on the presented diagrams, it may be concluded that from among the optimised quantities, the most essential influence on the break-even price of electricity is effected by a change of all pressure levels (p3s,H,p3s,Iand p3s,L). The change of the pinch point in the low-pressure part ðDTPP;L Þ is also essential.
Fig. 9. The effect of changing the pressure in the high- and low-pressure parts on the electric efficiency, unit investment costs, and break-even price of electricity.
Fig. 10. The effect of changing the pressure in the intermediate- and low-pressure parts on the electric efficiency, unit investment costs, and break-even price of electricity.
The sensitivity, in the case when three pressure levels were changed, was analysed in more detail. Figs. 8–10 illustrate the effect of a simultaneous change of two pressures on the value of the economic objective function, and additionally on the electric efficiency as well as the unit investment costs. All the other quantities, from among the fourteen decision variables, have been kept on the optimal level, determined during the course of optimisation. Within certain analysed pressures (close to the darkened area of technical restrictions), a strong increase of unit capital costs can be observed, noticeable by isolines concentration of these costs. The last one is caused by the previously described increase of heat exchange surface in the heat recovery steam generator. The position of the point indicating the optimal value of the objective function confirms an obvious conclusion, that this optimisation leads to a certain compromise between high efficiency and minimal investment costs.
7. Discussion of results and conclusion 1. Genetic algorithms, constituting one of the stochastic methods of optimisation, permit determination of the optimal values of many design variables of energy installations. In the case of the analysed combined cycle, the derived algorithm permitted optimisation of 14 decision variables. The optimisation was carried out particularly for the economic objective function b-e Þ. The main objective of the research was the analysis of ðCel the influence of fuel price on the optimal values of design variables of the steam part of the system. Additionally, for comparison, the thermodynamic quantity ðhelCC Þ was assumed as an objective function. 2. Thermodynamic and economic optimisation accomplished for a fuel price of 25 PLN/GJ results in values of objective functions differing from each other by 1.99 PLN/MWh and 0.358%. 3. The aforementioned differences result from different obtained values of decision variables. For both optimisations of decision variables: a) the high and intermediate-pressure part differ from each other (inclusive 9 variables), b) the low-pressure part, the reheater, and the deaeration economiser are the same (inclusive 5 variables). 4. In this paper, the susceptibility of the total investment and price of fuel on the value of the economic objective function
J. Kotowicz, Ł. Bartela / Energy 35 (2010) 911–919
5.
6.
7.
8.
was analysed. Calculations were carried out for the basic assumptions. It was observed that an increase in the price of b-e that is 7.54 times greater than fuel causes an increase in Cel the proportional increase of total investment. Considering point 4, the influence of the price of fuel on the optimum values of the decision variables was investigated. The results show that an increase in the price of fuel causes: a) an increase of the optimum values of steam pressures in the high and intermediate-pressure part, b) a decrease of the optimum values of the characteristic differences of temperatures: DTPP and DTAP (an exception is DTAP1;H – in the first section of a high-pressure economiser), c) an increase of the optimum values of the steam temperatures in the intermediate-pressure part. Generally, all variables mentioned in points a, b, and c aim at optimal values determined during thermodynamic optimisation. Changes of the values of the variables mentioned in points a, b, and c give: I. a small increase of the steam turbine power (along with an increase of the electric efficiency), II. a small increase of the total investment cost (and also a lower increase of the unit investment cost). In the fuel price range of from 20 to 30 PLN/GJ, the increase in the steam turbine power is 1.55 MW (which results in an increase in the electric efficiency of 0.222%), the increase in the total investment costs is 17.25 mlnPLN (about 7.03 mlnV). Simultaneously, aside from the price of fuel, the comparison between the determined parameters for both optimisations mentioned in conclusion 3 are correct. The increase of power is the reason for the growth of electricity production, and revenue connected with its sale. The additional revenue, however, is compensated for by the increase of total investment costs. As a consequence, the values of the quantity b-e shown in columns 2–6 in Table 4 are almost the same as Cel those of Fig. 4. As a result of the sensitivity analysis, it is possible to select those decision variables whose change considerably influences the chosen objective function. The results of this analysis ought, however, not to be generalised. The characteristics of the respective curves, and the relations between them, depend on the range of values for which the analysis has been carried out. In the process of selecting the decision variable values, the optimisation algorithm yielded solutions outside of the set of possible solutions. These solutions resulted from the fact that
919
the condition of physical conformity of the simulated phenomena was not satisfied. The example described in the paper concerns heat exchange realised in an exchanger. Acknowledgements The investigations presented in this paper have been carried out within the frame of the research project No. 3 T10B 068 30, sponsored by the Ministry of Education and Science. References [1] Franco A, Casarosa C. Thermoeconomic evaluation of the feasibility of highly efficient combined cycle power plants. Energy 2004;29(12–15):1963–82. [2] Chmielniak T. Energy technologies. Warszawa: WNT; 2008 [in Polish]. [3] Szargut J, Ziebik ˛ A. Fundamentals of thermal engineering. Warszawa: PWN; 1998 [in Polish]. [4] Kotowicz J, Bartela q. Influence of selected criteria on characteristics of a gassteam heat and power plants. Rynek Energii 2007;5(72):33–9 [in Polish]. [5] Franco A, Giannini N. A general method for the optimum design of heat recovery steam generators. Energy 2006;31(15):3342–61. [6] Casarosa C, Donatini F, Franco A. Thermoeconomic optimization of heat recovery steam generators operating parameters for combined plants. Energy 2004;29(3):389–414. [7] Valdes M, Duran D, Rovira A. Thermoeconomic optimization of combined cycle gas turbine power plants using genetic algorithms. Applied Thermal Engineering 2003;23(17):2169–82. [8] Toffolo A, Lazzareto A. Evolutionary algorithms for multi-objective energetic and economic optimization in thermal system design. Energy 2002;27(6):549–67. [9] Kotowicz J. The analysis of chosen criteria for the choice of the configuration of a gas-steam cycle. Rynek Energii 2006;5(66):40–7 [in Polish]. [10] Liszka M, Ziebik ˛ A. Economic optimization of the combined cycle integrated with multi-product gasification system. Energy Conversion and Management 2009;50(2):309–18. [11] Kotowicz J, Bartela q. Investigations concerning the influence of selected parameters on the thermodynamic and economic characteristics of a gassteam heat and power plant. In: Proceedings of the Conference ECOS 2007, Padova, Italy, vol. II; 25–28 June 2007. p. 957–65. [12] Attala L, Facchini B, Ferrara G. Thermoeconomic optimization method as design tool in gas-steam combined plant realization. Energy Conversion and Management 2001;42(18):2163–72. [13] Michalewicz Z. Genetic algorithms þ data structures ¼ evolution programs. Warszawa: WNT; 1996 [in Polish]. [14] Goldberg DE. Genetic algorithms in search, optimization, and machine learning. Warszawa: WNT; 2003 [in Polish]. [15] Dimopoulos GG, Frangopoulos CA. Optimization of energy systems based on evolutionary and social metaphors. Energy 2008;33(2):171–9. [16] Fredriksson Mo¨ller B, Assadi M, Potts I. CO2-free power generation in combined cycles – integration of post-combustion separation of carbon dioxide in the steam cycle. Energy 2006;31(10–11):1520–32. [17] GateCycleTM. GE Enter Software, LLC, 1490 Drew Avenue, Suite 180, Davis, California 95616, U.S.A. [18] Remiorz L, Kotowicz J. Restrictions of modelling process in gas-steam combined cycle. Rynek Energii 2008;1(74):42–7 [in Polish].
Contents lists available at ScienceDirect
Energy journal homepage: www.elsevier.com/locate/energy
The influence of economic parameters on the optimal values of the design variables of a combined cycle plant Janusz Kotowicz*, qukasz Bartela Institute of Power Engineering and Turbomachinery, Silesian University of Technology, Konarskiego 18, 44-100 Gliwice, Poland
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 October 2008 Received in revised form 3 July 2009 Accepted 9 July 2009 Available online 12 August 2009
This paper presents the analysis of the influence of fuel price variation on the optimal values of the design variables of the steam part of a combined cycle plant. The investigated system was a power plant with a triple-pressure heat recovery steam generator and extraction-condensation steam turbine. Fourteen design variables for the steam part were identified. The variables that were optimised were the pressure levels of the working medium in the steam part of the system, and characteristic differences of temperatures in the heat recovery steam generator. Thanks to the development of an optimising programme, based on the genetic algorithms theory, it was possible to find an optimal solution. The indices of economic efficiency, in the form of the break-even price of electricity, were chosen as the objective function in the optimisations. The results of economic optimisations were compared with the results of the optimisation, where the electric efficiency was the objective function. This paper includes an analysis of the sensitivity of the economic objective function to failures in the adherence of the optimal values of decision variables. This analysis allowed the selection of variables such that a failure results in the highest increase of the break-even price of electricity. Ó 2009 Elsevier Ltd. All rights reserved.
Keywords: Combined cycle plant Fuel price Optimisation Genetic algorithms
1. Introduction Combined cycle plants, which utilise gas turbines and heat recovery steam generators (HRSG), are one of the most effective technologies for generating fossil fuel based electricity. With the development of gas turbines, the energy efficiency of these systems has increased, exceeding more than 60% [1]. The steam part is also an important component and its optimisation may considerably improve the energy efficiency and economic performance of the plant [2]. There are many possible configurations of combined cycles including, among others, systems with a single, double, and threepressure boiler. The latter two can also be used together with a reheater. The solutions used in each configuration may differ from each other, both by the distribution of the heated surfaces in the boiler (in series, parallel or mixed), and by the means of preheating the condensate and feed water [2–4]. The most complicated configuration of combined cycles utilise three-pressure HRSGs; the power rating of such a system with only one gas turbine is 400 MW. Aside from their high efficiency, combined cycles are characterised by their low emission of noxious
* Corresponding author. Tel.: þ48 32 237 17 45; fax: þ48 32 26 80. E-mail address: [email protected] (J. Kotowicz). 0360-5442/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2009.07.014
substances, short construction times, high disposability, and low capital costs. In Poland, the incorporation of combined cycle plants has been hampered by the high price of gas fuel. The popularity of these installations is expected to grow, however, as a result of the development of technology to gasify hard coal. The large fluctuation in gas fuel price (almost doubled since 2003) poses the question: how much do fuel prices influence the optimal values of the design variables of energy systems? 2. Design variables of the steam part The electric efficiency in a combined cycle plant is defined by the relation
helCC ¼
Nel
¼ mf ,LHV
NelGT þ NelST Non mf ,LHV
(1)
This relation can be transformed as follows
helCC ¼ helGT ð1 þ a,helST Þ
(2)
In this equation, helST can be interpreted as the electric efficiency in the steam part of the system ðhelST ¼ ðNelST Non Þ=Q4a Þ, a is equal to Q4a =NelGT , and helST denotes the efficiency of the gas turbine ðhelGT ¼ NelGT =ðmf ,LHVÞÞ.
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J. Kotowicz, Ł. Bartela / Energy 35 (2010) 911–919
Nomenclature
a AP b-e CND D el f G,g CC limit GT,ST hex HE HRSG is m n on opt PUMP PP R s sat Y ¼ H,I,L
A area, m2 Am amortisation charges, PLN (1 PLN ¼ 0.292 V) C price, PLN F interest, PLN h specific enthalpy, kJ/kg J total investment costs, PLN k unit investment costs, PLN K,Kobr,KPR costs, changes of the working capital, costs of production, PLN L salvage value, PLN LHV lower heating value, MJ/kg m mass flow rate, kg/s N power, MW, or last year of operation NPV net present value, PLN p pressure, Pa income tax, PLN Pd Q heat flux, MW r discount rate, % t years of work T temperature, K x steam quality, or decision variable D increment h efficiency, % s annual time of operation, h
air or combustion gas approach point break-even quantity condenser deaerator electrical fuel generator combined cycle limited values installation of the gas turbine, steam turbine heat exchanger hot end of the superheater heat recovery steam generator isentropic mechanical nominal conditions own needed optimal values pump pinch point steam reheater water or steam saturation parameters steam pressure levels (H – high, I – intermediate, L – low)
Indices 1,2,3,I,II,III characteristic points in the system (Fig. 1 and Fig. 2)
Generally, the steam and gas parts in the combined cycle are not autonomous. Assuming, however, that we have chosen a gas turbine with operating characteristics determined by its thermodynamic parameters, we get the values: helGT, NelGT, Q4a. In this case, the efficiency optimisation of the combined cycle reduces to the optimisation of the steam part, with the requirement
helST ; NelST ¼ max
(3)
Since the system uses a gas turbine, we know the thermodynamic parameters at point 4a (i.e., the parameters of the flue gas leaving the gas turbine), which means that by using Eqs. (4) and (5), and also using Eqs. (6)–(10), we find that
NelST ¼ f ðDTHE ; DTPP ; DTAP ; p3s Þ
(11)
Proceeding similarly for the case of a multi-pressure system (Y), we can write
A combined cycle with a single-pressure boiler is shown in Fig. 1. The power rating of the turbine of this system is equal to
NelST ¼ f DTHE;Y ; DTPP;Y ; DTAP;Y ; p3s;Y
NelST ¼ m3s ,ðh3s h4s Þ,hm ,hg
where: Y ¼ H; I; L. In the case of systems with interstage reheating, the right-hand side of Eq. (12) must be complemented by quantities which allow for the determination of the thermodynamic parameters of the reheated steam ðp3s;R ; T3s;R ¼ T4a DTHE;R Þ, which results in
(4)
The balance of the evaporator and superheater indicates that
m3s ,ðh3s h2:1s Þ ¼ m4a ,ðh4a hIIa Þ,hhex
(5)
By means of the characteristic temperature differences, we can determine the temperature of live steam, combustion gas at the pinch point, and water at the outlet of the economiser, by using relations
T3s ¼ T4a DTHE
(6)
TIIa ¼ Tsat ðp3s Þ þ DTPP
(7)
T2:1s ¼ Tsat ðp3s Þ DTAP
(8)
This permits us to determine the enthalpy at points 3s and 4s by
h3s ¼ f ðp3s ; T3s Þ h4s ¼ f ðp3s ; T3s ; p4s ; hisST Þ
(9) (10)
NelST ¼ f DTHE;Y ; DTPP;Y ; DTAP;Y ; p3s;Y ; DTHE;R ; p3s;R
(12)
(13)
Thus, in the case of a classical three-pressure system with reheating, there are 14 variables (Eq. (13)); in a double-pressure system with reheating, there are 10; and there are only 8 in a double-pressure system without reheating.
3. Characteristics of the investigated structure The system presented in Fig. 2 was subjected to multivariable optimisation. The fundamental installations in the combined cycle are the gas turbine, heat recovery steam generator, steam turbine, condenser, deaerator, condensate extraction pump and feed water pump.
J. Kotowicz, Ł. Bartela / Energy 35 (2010) 911–919
913
pressure levels of the working medium was chosen. Such a system allows for minimal exergy losses during the heat exchange between the combustion gas and the working medium, which contributes to an increase in the efficiency of the production of electricity of the whole plant [5,6]. In this case, the pressure of the working media is conditioned by three parallelly positioned pumps. The distribution of the respective surfaces, at all levels of pressure, is shown in Fig. 2. The boiler is equipped with a steam reheater, which also acts as the second part of the superheater for the intermediate-pressure steam. The last heated surface in the boiler is the deaeration economiser, which is used to preheat the condensate passing into the deaerator. The characteristic quantities of the boiler, such as the pinch point, approach point, and difference of the temperatures at the hot ends of the superheaters, are decision variables in the optimisation problem. These quantities decide the distribution of thermal loads in the respective heat exchangers (economisers, evaporators, and superheaters), determining the surface of these exchangers. Besides these quantities, the values of pressure for each pressure level are also decision variables [7,8]. Fundamental parameters of the operation of other elements of the system are characterised in Table 1. Fuel gas that is passed into the gas turbine is characterised by the following composition: CH4 ¼ 97.33%, N2 ¼ 0.86%, C2H6 ¼ 0.81%, C3H8 ¼ 0.46%, CO2 ¼ 0.28%, C4H10 ¼ 0.26%, which has a lower heating value of 48 820 kJ/kg. 4. Optimisation algorithms
Fig. 1. Combined cycle with a single-pressure boiler and temperature distribution of the media in the boiler.
In terms of the analysis in this paper, the most essential element of the system is the HRSG. The structure of HRSG with three
The fundamental design variables of the steam part of the system presented in Fig. 2 were optimised. The design variables of the steam part of combined cycles were reviewed in Chapter 2. In the case of the investigated installation, it was assumed that the pressure of the reheated steam equals the intermediate-pressure of the live steam. An additional variable was the underheating of water at the outlet of the dearation economiser. It should also be noted that the high-pressure economiser was divided into two
Fig. 2. Diagram of the investigated system.
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Table 1 Characteristic parameters of installations belonging to the analysed system. Characteristic parameters
Unit
Gas turbine Westinghouse 501G (1995 GTW) Power Efficiency of producing electric energy Combustor exit temperature Expander exit temperature Pressure ratio Ratio of the cooling air to compressed air Compressor isentropic efficiency Combustor energy efficiency Turbine isentropic efficiency Generator efficiency Relative pressure loss in the air filter Relative pressure loss in the combustor
% % % % % % %
Steam turbine Isentropic efficiency part h Isentropic efficiency part i Isentropic efficiency part l Pressure of extraction steam Pressure of outlet steam
% % % kPa kPa
Generator Efficiency
%
Condenser Cooling water temperature rise Deaerator Operation pressure Pumps Isentropic efficiency
MW % C C
Table 2 Decision variables and their ranges during optimisation.
Value 228.9 38.24 1429.8 597.5 19.2 16 88.54 99.90 93.05 98.50 0.6 3 90 90 86 200 5 98
min
max
50 5 5 15 15 000
200 20 20 20 19 000
5 5 10 2000
15 15 30 7000
p3s,L, kPa
15 50 300
30 100 1000
DTHE;R , K
30
40
DTAP;D , K
5
35
DTAP1;H , K DTAP2;H , K DTPP;H , K DTHE;H , K p3s,H, kPa
DTAP;I , K DTPP;I , K DTHE;I , K p3s,I, kPa
DTPP;L , K DTHE;L , K
S ¼ Cel
Zs
ðNel Non Þds
C
10
kPa
200
%
85
Using Eqs. (14)–(16) under the condition NPV ¼ 0 we get
Pt ¼ N ½JþðKPR þPd þKobr ÞAm FLt be Cel
¼
t¼0
ð1þrÞt
Zs Pt ¼ N t ¼0
sections, resulting in five characteristic quantities for high-pressure level. Generally, keeping in mind the analogy with Sector 2, there are 14 distinct characteristic variables that influence the operation of the steam turbine in the analysed system. In the optimisation of the system, these quantities served as decision variables. Thus, for this purpose, the following decision variables have been assumed: Pressures of live steam at each pressure level – (p3s) Underheating of water at the outlets of all economisers (including the deaeration economiser approach point) – (DTAP) Minimum pinch point temperature differences at each pressure level – (DTPP) Differences in temperature between the media at the so-called hot ends of all superheaters of steam (including the reheater) – (DTHE) The ranges of the respective decision variables are presented in Table 2. The objective function in the calculations was the break-even b-e - in this case the set of selected decision price of electricity Cel variables had to ensure the minimisation of this quantity [9]. The break-even price of electricity is the price that makes the net present value equal zero (NPV ¼ 0). NPV is one of the fundamental and most widely applied economic coefficients for the assessment of economical effectiveness [10,11].
NPV ¼
tX ¼N
CFt
t¼0
ð1 þ rÞt
(16)
o
(14)
where CFt is cash flow
CFt ¼ ½S J ðKPR þ Pd þ Kobr Þ þ Am þ F þ Lt
(15)
An essential component of equation (15) is the profit from sales S, expressed as
o
(17)
ðNel Non Þds ð1þrÞt
The fundamental problem in the analysis of the economical effectiveness in the investment of energy systems is the determination of capital costs (J). In preliminary studies, in which basic decisions are made, statistical data gathered during the course of realising similar projects are generally used. The price of installations is determined by approximating the statistical data versus characteristic quantities. The following equations were used in determining the cost of the respective appliances [12]: 0:7 KGT ¼ 2:453,3832,Ne1GT
KHRSG ¼ 2:453,35; 000,
(18)
X
khexi ,Ahexi
0:6
(19)
i
1:543 þ 823:7,NelST KST ¼ 2:453, 3197280,A0:261 ST
KG ¼ 2:453,3082,
NelGT
hgGT
þ
NelST
(20)
!0:58 (21)
hgST
KCND ¼ 2:453,2754,A1:01 CND 0:8 , 1þ KPUMP ¼ 2:453,1293:44,NPUMP
(22) 0:3
0:46
1 hisPUMP
(23)
The total investment costs may be expressed as the product of the cost of machines and installations K, and coefficient B, which takes into account the costs of construction
J ¼ K,B ¼ B,
X
Ki
i
In the calculations, the value of B was assumed to be 3.5.
(24)
J. Kotowicz, Ł. Bartela / Energy 35 (2010) 911–919
The relations in Eq. (24) and Eqs. (19)–(23) connect the value of the capital costs J with the decision variables. For optimisation, a custom-made algorithm was applied, which makes use of the popular theory of genetic algorithms [13,14]. This type of optimisation is often used in engineering research and also in the optimisation of energy systems [15,16]. The thermodynamic model of the system was constructed in the GateCycle software [17]. The block diagram of the optimising algorithm is presented in Fig. 3. Additionally, for comparison, the technical optimisation was carried out. In this case, the decision variables were chosen to maximise the electric efficiency
helCC ¼
Nel
(25)
mf ,LHV
5. Assumptions and results of optimisations Aside from determining the capital costs, many assumptions must be made when analysing an investment economically. The most important of them are: 20% of the investment is self-financed and 80% is obtained from commercial credits. The actual interest of the credit amounts to 6%, the repay time is 10 years. The anticipated time of constructing the power plant is 2 years, with a repay of 40% in the first year and 60% in the second. The time of operation – 20 years.
915
The annual time of operation – 7500 h. The discount rate amounts to 5.69%. The constant of repairs were determined to be 0.5% of the capital costs for the first four years of operation, 1% for the subsequent four years, 1.5% for the next four years, 2% for further years, and 2.5% for the last four years of exploitation. The cost of fuel was calculated in compliance with the price list of PGNiG S.A. for 2007, amounting to about 25 PLN/GJ (in relation to the lower heating value of 35.5 MJ/m3); for the analysis, 25 PLN/GJ was assumed. The assumed depreciation is 6%. The excise tax is 20 PLN/MWh. The rate of income tax is 19%. The salvage value of the designed system is assumed to be 20%$J. The change of the working capital has been neglected in the calculations. Initially, two independent optimisations were analysed; the first, where the objective function was the minimised break-even price of electricity (17), and the second, where the electric efficiency was maximised (25). The results of the optimisation of the investigated system for both objective functions, with fundamental assumptions, are shown in Table 3. The maximisation of electric efficiency as an objective function of optimisation has merely a thermodynamic aspect. In such a case, the results of optimisation are reduced to the choice of boundary values (‘‘*’’ in Tables 3 and 4) in the range of the considered values of the decision variables except for approach point in the first section of the high-pressure economiser and intermediate pressure. If the break-even price of electricity is minimised, the optimisation programme indicates values of the decision variables which deviate most from the results of the preceding optimisation. The two optimisations yielded the same results only for the lowpressure part, the reheater of steam and the deaeration economiser. The susceptibility of changes of the fundamental economic assumptions to the break-even price of electricity was then analysed. It is the investors, as well as the recipients of the products and those who are affected by the results of the investment, who decide its feasibility. A representative quantity may be for them the economic objective function (17). Therefore, in the analysis of the susceptibility, a model with variables optimised in compliance with the economic objective function was applied. The susceptibility of
Table 3 Results of the optimisation for the economical and thermodynamic objective function. b-e Þ optðCel
opt(helCC )
114.8 7.2 10.3 20.0* 16 055
107.8 5.0* 5.0* 15.0* 19 000*
11.6 10.0 29.4 3112
5.0* 5.0* 10.0* 3976
p3s,L, kPa
15.0* 100.0* 300*
15.0* 100.0* 300*
DTHE;R , K
30.0*
30.0*
DTAP;D , K
5.0*
5.0*
DTAP1;H , K DTAP2;H , K DTPP;H , K DTHE;H , K p3s,H, kPa
DTAP;I , K DTPP;I , K DTHE;I , K p3s,I, kPa
DTPP;L , K DTHE;L , K
Fig. 3. Block diagram of the optimisation algorithm.
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Table 4 Results of optimisation with the thermodynamic (first column – results) and economical objective functions concerning various prices of fuels (column 2O6).
DTAP1;H , K DTAP2;H , K DTPP;H , K DTHE;H , K p3s,H, kPa
DTAP;I , K DTPP;I , K DTHE;I , K p3s,I, kPa
DTPP;L , K DTHE;L , K p3s,L,, kPa
1
2
3
4
5
6
Opt ðhelCC Þ Cf ¼ 25.0 PLN/GJ
b-e Þ C ¼ 20.0 Opt ðCel f PLN/GJ
b-e Þ C ¼ 22.5 Opt ðCel f PLN/GJ
b-e Þ C ¼ 25.0 Opt ðCel f PLN/GJ
b-e Þ C ¼ 27.5 Opt ðCel f PLN/GJ
b-e Þ C ¼ 30.0 Opt ðCel f PLN/GJ
107.8 5.0* 5.0* 15.0* 19 000*
114.2 7.7 12.2 20.0* 15 000*
113.0 7.7 10.7 20.0* 15 344
114.8 7.2 10.3 20.0* 16 055
116.9 6.4 9.9 20.0* 17 018
119.2 5.7 9.5 20.0* 17 984
5.0* 5.0* 10.0* 3976
14.2 12.5 30.0* 2878
13.5 11.3 29.9 2975
11.6 10.0 29.4 3112
9.7 9.0 28.7 3243
8.5 8.1 27.7 3381
15.0* 100.0* 300*
15.0* 100.0* 300*
15.0* 100.0* 300*
15.0* 100.0* 300*
15.0* 100.0* 300*
15.0* 100.0* 300*
DTHE;R , K
30.0*
30.0*
30.0*
30.0*
30.0*
30.0*
DTAP;D , K
5.0*
5.0*
5.0*
5.0*
5.0*
5.0*
be , PLN/MWh Cel
205.47 57.978 33.233 85.741 0.9189 99.80 119.392 587.495 1686.58
172.21 57.511 32.446 85.692 0.9346 99.92 116.242 503.516 1458.69
187.85 57.571 32.547 85.691 0.9330 99.93 116.630 507.476 1468.52
203.48 57.620 32.630 85.702 0.9309 99.89 116.978 511.155 1477.67
219.10 57.680 32.731 85.710 0.9291 99.89 117.406 515.996 1489.82
234.71 57.733 32.820 85.718 0.9273 99.86 117.795 520.767 1501.91
helCC, % helST, % hHRSG, % x5s t5a, C NelST, MW J, mlnPLN ki, PLN/kW
the objective function was tested with respect to changes of two parameters, namely, The price of supplied gas, Capital cost. Both these quantities varied within the range of 20% in relation to nominal values. The price of fuel was calculated in relation to the nominal price
Cf ¼ b, Cf
n
Cf ¼ 25 PLN=GJ n
The total investment was calculated in relation to the nominal total investment
The quality of the steam leaving the steam turbine x5s, The temperature of flue gases leaving the power plant T5a, The power of steam turbine NelST, Total capital costs J, Unit investment costs ki (ki ¼ J/Nel).
6. Sensitivity analysis of the economical objective function to changes of the optimal values of variables The presented optimisation, depending on the respective assumptions, has allowed for six sets of optimal values of the selected design variables of the steam part. We consider the question, how essential is it to keep the optimal values of decision variables for the value of the objective function? In order to answer this question, the sensitivity to changes of the respective decision variables to the attained value of the economic
J ¼ b,K,B ¼ b,Jn Jn ¼ 511:155 mln PLN The results of the susceptibility analysis are shown in Fig. 4. The results of the analysis of susceptibility indicate that among the analysed quantities, the largest influence on the value of the economical function is exerted by changes of the costs of fuel. Due to this, further investigations dealt with the identification of the effect of changes of fuel price on the values of the obtained decision variables. For this purpose, another four optimisations were carried out for various prices of gas, viz. 20, 22.5, 27.5, 30 PLN/GJ. All the remaining assumptions did not change. The results of calculations, compared with previously presented results, are shown in Table 4. Aside from the investigated objective functions and determined values of the decision variables, several quantities are presented, which allow for interpretation of the obtained results: The efficiency of the steam cycle helST, The efficiency of the HRSG hHRSG,
Fig. 4. Break-even price of electricity as a function of the relative change of fuel price and capital costs.
J. Kotowicz, Ł. Bartela / Energy 35 (2010) 911–919
objective function was analysed based on the results of an economical optimisation, which had been carried out for a fuel price of 25 PLN/GJ. In the course of these investigations, each decision variable was successively changed within the range of relative values ðDx=xopt ¼ x xopt =xopt Þ from 0.5 to 0.5. The other quantities remained unchanged from those attained during the course of the optimisation. The respective quantities could not exceed the ranges defined in Table 2. The results of calculations are shown in Figs. 5–7, which include 14 curves. The ranges of decision variables, for which these curves were not determined, resulted from the revealed boundaries of the considered ranges, or the technical restrictions, which exclude such a situation from the set of possible solutions [18]. Thus, for instance, within some ranges of decision variables, we can observe a sudden growth of the values of the objective function: approach point in the first section of the highpressure economiser ðDTAP1;H Þ, high pressure (p3s,H), and intermediate pressure (p3s,I), caused by the establishment of parameters of the heat-exchanging media, which require a considerable expansion of the heat-exchanging surface in the heat recovery steam generator. This results in a sudden increase in capital investments. As an example of restrictions, let’s take a given maximum value of the high-pressure medium (p3s,H). In Fig. 5, we see that the value of the break-even price of electricity, together with the growth of Dx=xopt up to w0.17, approaches infinity. This means that there is some limiting value of high pressure. In order to avoid such a boundary case, the following condition must be satisfied:
TVa > T2:1s;H
(26)
Knowing that
T2:1s;H ¼ Tsat p3s;H DTAP1;H
(27)
we get
TVa > Tsat p3s;H DTAP1;H
(28)
Tsat p3s;H
(29)
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Fig. 6. The effect of changes of the respective decision variables of the intermediatepressure part on the change of the break-even price of electricity.
TVa ¼ Tsat p3s;I þ DTPP;I
(30)
we may rewrite it as
Tsat p3s;H
(31)
Thus, as a result of Eq. (31) we can, for instance, determine the limiting value of high pressure
h i plimit 3s;H ¼ psat Tsat p3s;I þ DTPP;I þ DTAP1;H
(32)
As proved, for the high pressure, determined by Eq. (32), an equalisation of the temperature of flue gases at the inlet to the economiser of the high-pressure part, and the temperature of the water at the outlet of the economiser occurs ðT2:1s;H ¼ TVa Þ. The approximation of the high pressure to the limit pressure will also have economic consequences, because the surface area of the exchanger approaches infinity.
lim
A p3s;H ¼ N
(33)
Applying the relation
p3s;H /plimit 3s;H
Fig. 5. The effect of changes of the respective decision variables of the high-pressure part on the change of the break-even price of electricity.
Fig. 7. The effect of changes of the respective decision variables of the low-pressure part, characteristic quantity of the reheater, and quantity of the deaeration economiser on the change of the break-even price of electricity.
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Fig. 8. The effect of changing the pressure in the high- and intermediate-pressure parts on the electric efficiency, unit investment costs, and break-even price of electricity.
In Fig. 5, a similar limitation is observed in the case of the function DTAP1;H (the approach point for the first section of the high-pressure economiser), as well as in Fig. 6, for the function p3s,I (the pressure in the intermediate-pressure part). In both of these cases, we have to deal with the same kind of limitations as in the analysed case; transforming, therefore, Eq. (31) we get
h i plimit 3s;I ¼ psat Tsat p3s;H DTAP1;H DTPP;I
limit DTAP1;H ¼ Tsat p3s;H Tsat p3s;I DTPP;I
(34) (35)
In the case of the optimal variables, the values of the analysed limit limit quantities amount to plimit 3s;H ¼ 18; 829 kPa, p3s;I ¼ 2447 kPa, limit ¼ 101:7 K. and DTAP1;H Based on the presented diagrams, it may be concluded that from among the optimised quantities, the most essential influence on the break-even price of electricity is effected by a change of all pressure levels (p3s,H,p3s,Iand p3s,L). The change of the pinch point in the low-pressure part ðDTPP;L Þ is also essential.
Fig. 9. The effect of changing the pressure in the high- and low-pressure parts on the electric efficiency, unit investment costs, and break-even price of electricity.
Fig. 10. The effect of changing the pressure in the intermediate- and low-pressure parts on the electric efficiency, unit investment costs, and break-even price of electricity.
The sensitivity, in the case when three pressure levels were changed, was analysed in more detail. Figs. 8–10 illustrate the effect of a simultaneous change of two pressures on the value of the economic objective function, and additionally on the electric efficiency as well as the unit investment costs. All the other quantities, from among the fourteen decision variables, have been kept on the optimal level, determined during the course of optimisation. Within certain analysed pressures (close to the darkened area of technical restrictions), a strong increase of unit capital costs can be observed, noticeable by isolines concentration of these costs. The last one is caused by the previously described increase of heat exchange surface in the heat recovery steam generator. The position of the point indicating the optimal value of the objective function confirms an obvious conclusion, that this optimisation leads to a certain compromise between high efficiency and minimal investment costs.
7. Discussion of results and conclusion 1. Genetic algorithms, constituting one of the stochastic methods of optimisation, permit determination of the optimal values of many design variables of energy installations. In the case of the analysed combined cycle, the derived algorithm permitted optimisation of 14 decision variables. The optimisation was carried out particularly for the economic objective function b-e Þ. The main objective of the research was the analysis of ðCel the influence of fuel price on the optimal values of design variables of the steam part of the system. Additionally, for comparison, the thermodynamic quantity ðhelCC Þ was assumed as an objective function. 2. Thermodynamic and economic optimisation accomplished for a fuel price of 25 PLN/GJ results in values of objective functions differing from each other by 1.99 PLN/MWh and 0.358%. 3. The aforementioned differences result from different obtained values of decision variables. For both optimisations of decision variables: a) the high and intermediate-pressure part differ from each other (inclusive 9 variables), b) the low-pressure part, the reheater, and the deaeration economiser are the same (inclusive 5 variables). 4. In this paper, the susceptibility of the total investment and price of fuel on the value of the economic objective function
J. Kotowicz, Ł. Bartela / Energy 35 (2010) 911–919
5.
6.
7.
8.
was analysed. Calculations were carried out for the basic assumptions. It was observed that an increase in the price of b-e that is 7.54 times greater than fuel causes an increase in Cel the proportional increase of total investment. Considering point 4, the influence of the price of fuel on the optimum values of the decision variables was investigated. The results show that an increase in the price of fuel causes: a) an increase of the optimum values of steam pressures in the high and intermediate-pressure part, b) a decrease of the optimum values of the characteristic differences of temperatures: DTPP and DTAP (an exception is DTAP1;H – in the first section of a high-pressure economiser), c) an increase of the optimum values of the steam temperatures in the intermediate-pressure part. Generally, all variables mentioned in points a, b, and c aim at optimal values determined during thermodynamic optimisation. Changes of the values of the variables mentioned in points a, b, and c give: I. a small increase of the steam turbine power (along with an increase of the electric efficiency), II. a small increase of the total investment cost (and also a lower increase of the unit investment cost). In the fuel price range of from 20 to 30 PLN/GJ, the increase in the steam turbine power is 1.55 MW (which results in an increase in the electric efficiency of 0.222%), the increase in the total investment costs is 17.25 mlnPLN (about 7.03 mlnV). Simultaneously, aside from the price of fuel, the comparison between the determined parameters for both optimisations mentioned in conclusion 3 are correct. The increase of power is the reason for the growth of electricity production, and revenue connected with its sale. The additional revenue, however, is compensated for by the increase of total investment costs. As a consequence, the values of the quantity b-e shown in columns 2–6 in Table 4 are almost the same as Cel those of Fig. 4. As a result of the sensitivity analysis, it is possible to select those decision variables whose change considerably influences the chosen objective function. The results of this analysis ought, however, not to be generalised. The characteristics of the respective curves, and the relations between them, depend on the range of values for which the analysis has been carried out. In the process of selecting the decision variable values, the optimisation algorithm yielded solutions outside of the set of possible solutions. These solutions resulted from the fact that
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the condition of physical conformity of the simulated phenomena was not satisfied. The example described in the paper concerns heat exchange realised in an exchanger. Acknowledgements The investigations presented in this paper have been carried out within the frame of the research project No. 3 T10B 068 30, sponsored by the Ministry of Education and Science. References [1] Franco A, Casarosa C. Thermoeconomic evaluation of the feasibility of highly efficient combined cycle power plants. Energy 2004;29(12–15):1963–82. [2] Chmielniak T. Energy technologies. Warszawa: WNT; 2008 [in Polish]. [3] Szargut J, Ziebik ˛ A. Fundamentals of thermal engineering. Warszawa: PWN; 1998 [in Polish]. [4] Kotowicz J, Bartela q. Influence of selected criteria on characteristics of a gassteam heat and power plants. Rynek Energii 2007;5(72):33–9 [in Polish]. [5] Franco A, Giannini N. A general method for the optimum design of heat recovery steam generators. Energy 2006;31(15):3342–61. [6] Casarosa C, Donatini F, Franco A. Thermoeconomic optimization of heat recovery steam generators operating parameters for combined plants. Energy 2004;29(3):389–414. [7] Valdes M, Duran D, Rovira A. Thermoeconomic optimization of combined cycle gas turbine power plants using genetic algorithms. Applied Thermal Engineering 2003;23(17):2169–82. [8] Toffolo A, Lazzareto A. Evolutionary algorithms for multi-objective energetic and economic optimization in thermal system design. Energy 2002;27(6):549–67. [9] Kotowicz J. The analysis of chosen criteria for the choice of the configuration of a gas-steam cycle. Rynek Energii 2006;5(66):40–7 [in Polish]. [10] Liszka M, Ziebik ˛ A. Economic optimization of the combined cycle integrated with multi-product gasification system. Energy Conversion and Management 2009;50(2):309–18. [11] Kotowicz J, Bartela q. Investigations concerning the influence of selected parameters on the thermodynamic and economic characteristics of a gassteam heat and power plant. In: Proceedings of the Conference ECOS 2007, Padova, Italy, vol. II; 25–28 June 2007. p. 957–65. [12] Attala L, Facchini B, Ferrara G. Thermoeconomic optimization method as design tool in gas-steam combined plant realization. Energy Conversion and Management 2001;42(18):2163–72. [13] Michalewicz Z. Genetic algorithms þ data structures ¼ evolution programs. Warszawa: WNT; 1996 [in Polish]. [14] Goldberg DE. Genetic algorithms in search, optimization, and machine learning. Warszawa: WNT; 2003 [in Polish]. [15] Dimopoulos GG, Frangopoulos CA. Optimization of energy systems based on evolutionary and social metaphors. Energy 2008;33(2):171–9. [16] Fredriksson Mo¨ller B, Assadi M, Potts I. CO2-free power generation in combined cycles – integration of post-combustion separation of carbon dioxide in the steam cycle. Energy 2006;31(10–11):1520–32. [17] GateCycleTM. GE Enter Software, LLC, 1490 Drew Avenue, Suite 180, Davis, California 95616, U.S.A. [18] Remiorz L, Kotowicz J. Restrictions of modelling process in gas-steam combined cycle. Rynek Energii 2008;1(74):42–7 [in Polish].