Multiobjective optimization of heat recovery steam generator in a combined cycle power using genetic algorithm

Due to the increasing demand for electrical energy, efforts to increase the thermal efficiency of the steam and gas power plants have led to extensive reform in these cycles. One of these common reforms is employing a conventional combined gas‐steam cycle. In combined cycle power plants that are built to produce power, a significant portion of the input energy is lost. In this research to achieve the thermodynamic properties of a combined cycle power plant after modeling the cycle and determining the cycle potential independent variables, multiobjective optimization by imposing restrictions on cost functions and changing them concerning exergy efficiency has been analyzed. The results show that increasing the parameters of superheated temperature, pinch point temperature difference, pump exhaust pressure, and condenser inlet flow rate improves the system performance and increases exergy efficiency. It is shown by two‐objective optimization that when costs are increased up to 40%, exergy efficiency is increased, and when an increase is more than 40% repeated results would be obtained. Applying costs lower than 5% is not considered according to software limitations. Also, the results show that it is possible to increase exergy efficiency up to 79.7% with a 40% investment cost.


| INTRODUCTION
In most aspects of our daily life energy is a vital factor; its generation, conversion, and utilization are effective in a steady process of development.In the field of generated power, combined power plants (CPPs) are of great interest.Because of their higher thermal efficiency in comparison to separate gas or steam power plants and their less environmental impacts.In the combined cycle, the heat recovery steam generator (HRSG) boiler is the most important component.According to the increase of the expenses of the fuels and reduction of sources of fossil fuels, the optimized design of the generator of heat recovery in the combined cycle is a remarkable point.The efficiency of the combined cycle is greater compared with other cycles and their number is increasing in the world.Energy and exergy analysis has made it possible to determine suitable analysis methods of energy systems, obvious recognition of energy levels, and also unfavorable thermodynamic processes of a system.Ahmed et al. presented an HRSG design method to produce electricity using wasted energy.Their results showed the optimal design pressure in the high-pressure level for the steam generator is 100 bar and for the lowpressure level, this value is 10 bar. 1 Assad et al. used exergy and energy analysis on a power plant.On the basis of these analyses, they calculated the optimal separator temperature to obtain the maximum power of a thermodynamic analysis. 2Norouzi et al. in a conceptual design optimized an HRSG based on low entropy generation and minimum capital cost.They optimized the HRSG configuration by optimizing the diameter of the tubes and the space between them. 3Bianco et al. investigated the optimization of HRSG to minimize overall cost.They concluded by limiting the HRSG size, 20% of the costs is reduced. 4Akbari Vakilabadi et al. studied exergy analysis on a solar steam power plant.They concluded maximum exergy destruction has occurred in the solar collectors and minimum exergy destruction is in the turbines. 5In two other studies, they increased power plant efficiency with the recovery of steam power plant blow-down. 6,7Altarawneh et al. studied energy and exergy analysis of a CPP in Jordan.They have concluded that the difference between the burners and the air is the main reason for exergy loss in the boiler, which reduces the efficiency of the power plant. 8Basem et al. performed an exergy and energy analysis of a solar-bohar power plant.They concluded that the maximum energy loss is 2172.81W and the maximum exergy loss is 3650.94W. 9 Dai et al. investigated the exergy analysis for a combined cycle of ejector refrigerator and Rankine power.In this study, a combined cycle simultaneously produces power and acts as a refrigerator.They showed that the most exergy destruction is in the boiler and then in the ejector.They observed turbine inlet pressure, condenser temperature, and evaporator temperature have the greatest effect on turbine output power and combined cycle exergy efficiency. 10Li et al. investigated the thermodynamiceconomic optimization of an HRSG in a combined cycle power plant using a multiobjective genetic algorithm.They concluded that using the NSGA-II optimization method compared with other methods, power plant efficiency increases by 25.2%, exergy loss decreases by 32%, and special investment decreases by 32%. 11adpanah et al. performed a technical and economic evaluation of the use of desalination water for cooling the air entering the gas turbine.They showed that genetic algorithm leads to increasing net freshwater production and net power.Also, cooling the air entering the turbine increases the thermal efficiency from 35.5% to 36.1%. 12RSG boilers have been investigated by Ganapathy from the scientific point of view based on the temperature difference of the pinch points in different points of the boiler.Concerning the great importance of the pinch and approach points in the design of HRSG boilers, engineering problems, and also their remarkable role in correct calculations of system parameters are discussed here; one of the mentioned points here is paying attention to the design of the boiler in a nonnominal load of the power plant. 13Important states of these systems and current intervals of designing pinch and approach points have been also introduced.It was shown by Casarosa and  Franco in research that optimization of the HRSG boiler is done using steam injection and this would reduce the difference between cold and hot flows. 14A technoeconomic analysis of power plant cycles has been done by Azimian after solving governing equations and computing of complete product cost, prominence of a combined cycle to the steam power plant cycle has been shown. 15xergy analysis of a 420-MW CPP has been done; it was shown by results that using additional burners in the HRSG boiler would increase the turbine output power equal to 7.38%. 16HRSG boilers have been optimized from an exergy point of view by In and Lee 17 ; values of normalized exergies have been used to calculate the exergy destruction of any component.The energy analysis of a CPP is studied by Ahmadi et al. 18 They optimized the combined cycle using a genetic algorithm by adding a burner.The effect of each objective function and decision-making parameters on the exergy efficiency have been investigated by them.The effect of full repowering on a gradation of technical and economic properties of a steam cycle using full repowering has been analyzed by Mehrpanahi et al. 19 It was concluded by them that capacity for increasing power and efficiency of a steam cycle is determined using the full repowering method.An increase in the efficiency of the repowering cycle would also lead to better results if gas turbine cycles with higher efficiency and steam injection are used.Optimization of the HRSG boiler of a CPP has been conducted using an evaluation algorithm and based on exergy analysis by Hajabdollahi et al. 20 It is shown by results that increasing the pressure of high-and lowpressure drums would lead to an increase in pinch temperature; it would consequently cause the exergy efficiency of the HRSG boiler to be reduced.Optimization based on multiobjective (exergy efficiency and unit | 4225 cost of exergy destruction) thermodynamic modeling using a genetic algorithm is studied by Kaviri et al.They concluded increasing pinch temperature difference and superheated vapor temperature would lead to an increase and decrease in exergy destructions and cost, respectively. 21In an article, Ahmadi et al. 22 investigated different structures of HRSGs and gas turbines to rehabilitate a steam power plant unit.Their results have shown that using two heat recovery boilers and two 180-MW gas turbines for this power plant unit will give the best results.In a paper, Wang et al. 23 studied a hybrid multiple-generation system for electricity and heat generation.In this paper, they used a simple HRSG for heat recovery and SOFC-ICE-SCO 2 assembly.The results of this research have shown a thermodynamic efficiency of 65.82% for system performance.In an article, Espinosa-Cristia et al. 24 proposed the use of an ejector cooling system to reduce the inlet air temperature of the gas turbine compressor in a combined cycle with a dualpressure HRSG.Their goal is to reduce the effects of weather on the operation of the power plant.The results have shown that using this method can increase power by up to 6.26%.Strušnik and Avsec 25 proposed the use of a diesel engine in a combined cycle combination.In this research, a single-pressure HRSG is used to prepare highpressure steam.Gogoi et al. 26 evaluated the exergeoeconomic analysis of the integration of a combined cycle power plant with an organic Rankine cycle.In this article, they used an HRSG in the steam heat recovery cycle and an HRVG in the organic Rankine cycle.Their results have shown that the use of R123 fluid has the best performance and the exergy efficiency of the power plant in this case is 40.89%.In an article, Li et al. 11 evaluated a combined cycle power plant with single, double, and three-pressure HRSGs.Their main investigation is about the exergy destruction of different parts of a waste incineration power plant.Their results have shown that exhaust gases, gasifiers, and combustion chambers have the highest share of exergy destruction.Similarly, in another research, Altarawneh et al. 8 studied and evaluated a combined cycle power plant in Jordan and identified the parts with the highest exergy destruction.
It should also be mentioned that pinch and approach temperature differences have been considered in optimization.The optimized values have been extracted concerning the algorithm of optimization.One of the remarkable points in this article less discussed in other articles is the heat levels of HRSG boilers which can be a significant factor in the technical and economic analysis of power plants.According to current parameters, the optimized saturated steam temperature is obtained; the comparison between (the number of transfer unit method) and saturated temperatures is done.In this article, the optimization of the HRSG of a combined cycle power plant has been done.The pinch model has been considered to realize the results.Also, the approach temperature difference has been applied to prevent technical problems.
In this article, in addition to optimizing the structure of the HRSG, technical points have been considered, which will be the achievement of more realistic results.This article considers the pinch limit in all evaporators in the analysis and optimization procedure.Also, the approach temperature limit at the exit from the economizers is taken into account.The optimization in this article is done to provide the dependence of the objective functions and help the investor choose the right structure with the changes in the initial capital.

| COMBINED CYCLE
CPP is a combination of gas and steam turbines; in a way that gas turbine generator generates electricity while wasted thermal energy from the gas turbine (by combustion products) is used for the generation of steam required for the steam cycle; in this way, extra amount of electricity would be generated.Combining these two cycles power plant efficiency would increase.A simple cycle of a power plant without using dissipated heat commonly has efficiency between 25% and 40% 27 while efficiency in that power plant using a combined cycle would be about 50%. 28As mentioned before these power plants are constructed by combining gas and steam turbines; there are different kinds of these power plants in terms of turbines, heat recovery boilers, and recovery devices.Using gas turbines in combined cycles improves its low efficiency; consequently, it can be used to provide basic load.Nowadays gas turbines have experienced a remarkable amount of growth but the amount of exhaust gas energy has also increased.In the late 1940s, when gas turbines were introduced in the industry of electricity generation there was a transformation in power plants using fossil fuels. 16To recycle as much energy as possible from the exhaust gas, the idea of combining the Bryton and Rankine cycles has been formed.Gas turbines and steam turbines are used simultaneously to generate electricity.In this transformation high temperature of the exhaust gases in the Bryton cycle is used as a source of energy for the Rankine cycle.In Figure 1, the general structure of a gas turbine in a CPP is specified.A low amount of initial investment, low volume, reliability, high flexibility, and low emissions are some of the features of this cycle.In this structure turbine exhaust gas with high temperature (about 500°C 27 in the nominal load and volume) is directed towards the boiler; it is used instead of burner and fuel in steam units to generate heat.Generated steam rotates the steam turbine.This would increase the efficiency of the power plant, meanwhile costs of investment per kW be reduced by a significant amount.This set is used to generate basic electricity and its efficiency would increase up to 50%; it would be used only for electricity generation purposes.In cold regions turbine with high exhaust pressure is loaded instead of a condenser or cooling tower; turbine exhaust water is used to provide hot water and vapor consumed to heat industrial and urban zones; in this way, efficiency would be increased up to 80%.

| Analysis of HRSG in combined cycle
In combined cycles the exhaust hot gas from the gas turbine enters the HRSG boiler; its heat is transferred to the water inside the boiler by heat exchangers.Generated hot steam or water is used for a steam cycle; it can also be utilized in process consumptions in different units.
HRSG boilers were first used to recover generated heat in industrial units, such as refineries and industrial furnaces.From 1950 to 1960 sometimes turbine exhaust gas was used for combustion in thermal boilers.In 1952 first unit of the gas turbine was connected to the HRSG boiler in the USA, Oklahoma. 29After a while of construction of HRSG boilers, developments were seen in industries of gas turbine and capability of construction of HRSG boilers; it should also be mentioned that capacity and technology of construction of HRSG boilers was developed significantly and multipressure level boilers capable of reheating steam have been built.

| Introduction of a double-pressure cycle
In a cycle with two pressures, steam is generated in two different levels of pressure and temperature in the HRSG boiler.Depending on the kind of steam consumption in high-and low-pressure lines, this steam could be superheated or saturated.In Figure 2, a schematic of a double-pressure cycle is shown in which steam is superheated or saturated.Saturated steam is capable of doing a low amount of work but it is used for process requirements in many industrial units.In the shown figure a pump is used to transfer water from the deaerator to the economizer.The thermal level of this economizer is determined in a way that the temperature of its outgoing water would be near the saturated temperature of the drum.In some of the configurations high-pressure feed water of the economizer is provided by a low-pressure drum but there exist two pumps.
One of them sends saturated water outgoing deaerator to a low-pressure economizer and then to a low-pressure drum (evaporator).An important point in the design of double-pressure cycles is the design of an economizer for a low-pressure system.If there would be a little difference between the deaerator pressure and the drum of low-pressure line, the economizer can be omitted and the saturated water from the deaerator would be injected directly into the drum.Feed water pumps in high-and low-pressure sections can be separated and combined.Decision-making in this field is dependent on locating issues and the reliability of the boiler.Feed water would be evaporated after passing the economizer and it would be directed to the superheated section of the HRSG boiler to be converted to dry steam.

| CYCLE MODELING
The studied structure includes a double-pressure HRSG.The optimal design of the HRSG allows maximum efficiency of this part.Relations required in the modeling of a combined cycle are as follows.

| Steam turbine
Isentropic efficiency and power of steam turbine are as follows, respectively, These equations, η ise st , and W ̇st , are isentropic efficiency and production power of steam turbines, respectively.h st in , , h st out , , and h st out ise , , are the input enthalpy, real output enthalpy, and isentropic output enthalpy of steam turbines, respectively.ṁs t in , is the flow through each section of the steam turbine.

| HRSG
HRSG boiler: In combined cycles hot gas in the production of combustion is directed to the HRSG boiler after passing through the gas turbine; its heat is transferred to the water inside boiler by heat exchangers.Generated hot steam or water for electricity generation is used in a steam turbine.The trend of temperature changes of gas and vapor inside the boiler is shown in Figure 3.
This HRSG boiler is from the double-pressure kind with two steam outputs in two levels of high and low pressure.In Figure 3 boiler modeling is performed similarly to the suggested design by Kumar and Gundabattini. 30In this modeling pinch and approach points are not dependent and quantities of temperatures of exhaust gases from different sections of the HRSG boiler are dependent variables.
Equations related to pinch and approach points are defined as follows: gt out hp eva steam in hp eva pinch hp , , , , , , , gt out pre eva steam in pre eva pinch pre , , , , , , , In the high-pressure evaporator, the pinch temperature difference ( T Δ pinch hp , ) is the difference between the gas temperature at the exit from this section (T gt out hp eva , , , ) and the water saturation temperature at the exit from the evaporator (T steam in hp eva , , , ).These equations are written similarly for the high-pressure evaporator and primary preheater.
The approach temperature difference ( T Δ app ) is also the temperature difference of saturated water (T steam in eva , , ) to high-pressure economizer outlet water (T steam out eco , , ).For the three sections of the high-pressure evaporator (hp, Eva), low-pressure evaporator (lp, Eva), and primary preheater (pre, Eva), the equations are written as follows: steam out pre eco steam in pre eva app pre , , , , , , , Balance equations of gas-water between points A and B: g in HRSG p g in HRSG in HRSG out g hp sup loss st hp st out hp sup st in hp sup In the above equation, ṁg in HRSG , , , C p g in HRSG , , ,

, and T in HRSG
, are the mass flow rate, heat capacity, and temperature of the gas entering the heat recovery boiler, respectively.T out g hp , , ,sup is the temperature of the gas passing through the high-pressure superheater.E loss is F I G U R E 3 Trend of change of temperature changes of steam and gas in the HRSG.Deae, deaerator; Eco, economizer; Eva, evaporator; GT, gas turbine; HP, high pressure; HRSG, heat recovery steam generator; LP, low pressure; pre, preheater; sup, superheater.
the heat loss ratio in this section and ṁs t hp , is the flow rate of steam passing through the high-pressure section of the boiler.h st in hp , , ,sup and h st out hp , , ,sup are the inlet and outlet enthalpy of steam passing through the superheater, respectively.This equation is similarly expanded for high-pressure evaporator (hp, Eva), high-pressure economizer (hp, Eco), low-pressure superheater (lp, sup), low-pressure evaporator (lp, Eva), low-pressure economizer (lp, Eco), deaerator (Deae), and primary preheater (pre, eca) as follows: Between points B and C: g in HRSG p g in hp eva g in hp eva g out hp eva loss st hp st out hp eva st in hp eva Between points C and D: g in HRSG p g out hp eva g in hp eco g out hp eco loss st hp st out hp eco st in hp eco Between points D and E: g in HRSG p g out hp eco g in lp sup g out lp sup loss st lp st out lp sup st in lp sup Between points E and F: g in HRSG p g in lp eva g in lp eva g out lp eva loss st lp st out lp eva st in lp eva Between points F and G: g in HRSG p g in lp eco g in lp eco g out lp eco loss st lp st out lp eco st in lp eco Between points G and H: Between points H and I: g in HRSG p g in pre eco g in pre eco g out pre eco loss st lp st hp st out pre eco st in pre eco

| Pumps
Each HRSG boiler pump is composed of high-and lowpressure centrifugal pumps fed by a deaerator reservoir; these pumps are used to transfer fluid and increase its pressure.Head, capacity, and temperature of the fluid in these pumps are dependent on their performance.In all equations pressure drop for the steam passing economizer, the pressure drop of superheated, and the pressure drop in pipelines transferring steam to the steam turbine is considered equal to 3%, 3.5%, and 5%, respectively; since there is a circulation pump in the economizer for water circulation, no pressure drop has been considered for steam passing through evaporator.Effective parameters in the pump are inlet, outlet enthalpy, and isentropic outlet enthalpy.
Pump isentropic efficiency: In this regard, h in pump are the input enthalpy, real output enthalpy, and isentropic output enthalpy of the pump, respectively.

| OBJECTIVE FUNCTIONS
In this section after the modeling process, two-objective optimization has been used for power plant optimization.Two functions have been considered for optimization: first function is a set of costs (costs of buying equipment, rate of power plant costs, and investment costs); the second objective function is exergy efficiency of the HRSG boiler.The trend of the two-objective optimization is continued by restriction of costs functions and then their changes are verified in terms of exergy efficiency.Formulation of each of objective functions is as follows: inputs include temperature of turbine inlet gas, highpressure pinch temperature difference, pump pressure, condenser flow rate, and ratio of steam outgoing from high-pressure section of HRSG boiler and function outputs include costs (costs of buying equipment, rate of power plant costs, and investment costs).

| Exergy efficiency function
The exergy efficiency function is one of the functions used in the optimization of HRSG boiler.Analyzing exergy efficiency is done using the following relations: In the above relations parameters are defined as follows: + ̇+ ̇+ ̇.

HRSG D hot in HRSG hot out HRSG cold out HRSG cold in HRSG lp pump hp pump hot out HRSG
, , The above parameters are defined below:

| Electricity generation costs
The cost is a purpose parameter in economic modeling.The cost of CPPs as a function of the initial investment cost for the purchase of equipment, maintenance costs, and fuel costs are calculated as follows: The cost of the power plant for purchase of fuel (Z f ), equipment (Z c ), and maintenance (Z OM ) is as follows:

| Initial investment costs for purchasing equipment
In an analysis of an HRSG of a power plant, there is no need to buy all the equipment of a CPP.The needed equipment for this work includes an HRSG and a duct burner.The cost of purchasing each mentioned device is provided below in dollars: Cost of purchasing HRSG 31 : In the above equation, lm ∆ is a logarithmic temperature difference.Q ̇, ṁs t , and ṁg are heat transfer in each component, steam, and gas mass flow rate, respectively.The coefficients C 51 , C 52 , and C 53 are also obtained from Bejan et al. 31 Cost of purchasing duct burner 31 : .
In this equation, Q ̇DB is the amount of heat that the duct burner enters the HRSG.
The following equation calculates all of the initial investment costs of the power plant per kWh of generated electricity 31 : In the above equation, H is the annual operating period of the new plant considering the availability of 91%; it has been considered equal to 8000 h.W ̇and CRF are the net production power of the power plant and capital recovery factor, respectively.All of the investment costs at the current time are obtained as follows: n n (28)   In the above equations, r n is the annual interest rate which is equal to 5%.The presented term in parenthesis is for updating this equation.Also presented PEC term is the equipment price which is as follows: To estimate of the return factor of investment is presented by Equation (30) 31 : That i is the annual interest rate and is equal to 12%.n is the payback factor of investment and is equal to 30 years.

| Maintenance costs
In general, the maintenance costs (both fixed and constant) assume the plant operation at rated load conditions that is the function of the total cost of the initial investment 31 : Factor φ is power plant maintenance and its value depends on the type of power plant.But according to references 1.06 would be a good approximation to use.

| Cost of fuel
The annual fuel costs have been computed using the equation 31 While C f , ṁf , and LHV f are cost, mass flow rate, and lower heating value of consumed fuel, respectively; t s is plant usage nominal interest per second.Therefore, the fuel cost per kWh, owing to the net power of the power plant and annual operating hours of the plant, is given by the following equation: Heat rate of power plant is as follows: Thus, by Equations ( 33) and (34), Equation ( 35) is obtained; it expresses the fuel costs based on the heat rate of the power plant which is under investigation.
Each kWh price of generated electricity is calculated by combining Equations ( 24), ( 27), (31), and (35) and in the following way 32 : 5 | RESULTS AND DISCUSSION

| Sensitivity analysis
The goal of sensitivity analysis is to obtain optimum time paths of technoeconomic objective variables and sensitivity analysis of these time paths.To perform this task a certain optimum control model is used.Independent variables of optimization in the analysis of an HRSG boiler are as follows: Y: ratio of the division of high-pressure feed water to total feed water entering the HRSG boiler, P out,hp : exhaust vapor pressure from the high-pressure line of HRSG, T out,hp : exhaust vapor temperature from the highpressure line of HRSG, ṁf w : flow rate of feed water entering HRSG boiler, P pump,hp : pressure of the high-pressure pump, ΔT pinch : pinch temperature difference.
The minimum and maximum allowed values of these parameters are presented in Table 1.
It is shown by results obtained from optimization that if optimization policies are adopted, mentioned objective variables would remarkably have lower fluctuations.The purpose of optimization policies is to consider increasing or decreasing the initial investment cost to achieve the optimal technical structure.This method allows the investor to choose the optimal technical structure based on the initial capital.On the basis of the findings of this research when a model is solved, results are dependent on the data.It is tried in the sensitivity analysis to investigate the effects of independent variables on objective function and present solutions to increase system efficiency.To investigate pump effect and ratio of the entering feed water to the highpressure line on the exergy analysis, changes of exergy efficiency in terms of entering feed water to the highpressure line in different pressures are shown in Figure 4. Increasing pump pressure, enthalpy input to the steam turbine, generated work and exergy efficiency would also be increased.As can be seen in the steam ratio of 0.86 and the pressure of 74 bar (real circumstances of the power plant) exergy analysis is about 75.8% and if the pressure is increased up to 82 bar in the same steam ratio, efficiency would be increased up to 76.3%.
Effect of superheated vapor temperature and steam ratio of the high-pressure line on the exergy efficiency is shown in Figure 5. Increasing the turbine exhaust gas temperature, the superheated steam temperature would be increased.This enhancement of temperature would lead to a rise in steam ratio and exergy efficiency.As it is observed, an increase in superheated temperature and steam ratio would lead to the enhancement of exergy efficiency; in the steam ratio of 0.86 and temperature of 788 K (real circumstances of power plant) exergy efficiency would be about 75.9%.In a constant pressure ratio, increase in temperature of exhaust dry steam from high-pressure line of HRSG boiler would be up to 791.5 K; it would increase power plant exergy efficiency up to 77.5%.
Investigation of the effect of flow rate of condenser steam ratio on exergy efficiency is presented in Figure 6.
Increasing pump pressure, flow rate of condenser would be decreased and according to the inverse relation between steam ratio and condenser flow rate, increasing condenser flow rate, steam ratio would be decreased.Also in the steam ratio of about 0.84 and the flow rate of 56.69, exergy F I G U R 4 Changes exergy efficiency versus steam ratios in different inlet turbine pressures.HRSG, heat recovery steam generator.
F I G U R E 5 Changes in exergy efficiency versus steam ratios in different inlet turbine temperatures.HRSG, heat recovery steam generator.
F I G U R E 6 Changes in exergy efficiency versus steam ratios in different mass flow rates.HRSG, heat recovery steam generator.
efficiency would be 75.6%; in the steam ratio of 0.84 when the flow rate of condenser is increased up to 59.69, exergy efficiency would be about 77%.This flow rate increase would lead to a reduction of steam ratio.Changes in exergy efficiency in terms of pressure of the superheated pump in different flow rates are shown in Figure 6.An increase in the pressure of the superheated pump and flow rate of condenser would increase exergy efficiency; concerning this Figure 7, increasing the pressure of the superheated pump, inlet enthalpy, generated work, and exergy efficiency would be increased.A way with superheated pump pressure rising, flow rate of the condenser would be reduced.It is obvious that in the flow rate of 56.96 and pressure of 74 bar (real circumstances of power plant) exergy efficiency would be 76.2%.Effect of superheated temperature and condenser flow rate on exergy efficiency is shown in Figure 8.
Rising the turbine exhaust gas temperature would increase the superheated temperature.It should be also mentioned that this increase in temperature would increase steam ratio and exergy efficiency.Increasing condenser flow rate, superheated temperature, and steam ratio would be reduced.Increasing condenser flow rate while gas turbine exhaust gas temperature is constant, superheated steam temperature entering the turbine would be decreased.As can be seen in the temperature of 778 K and the condenser flow rate of 56.69 (real circumstances of power plant) exergy efficiency would be about 75.8%; with a flow rate of 58.69 (kg/s) in condenser superheated steam temperature and exergy efficiency are decreased.Effect of pump pressure and superheated steam temperature on exergy efficiency is shown in Figure 9. Increasing pump exit pressure and superheated steam temperature would increase exergy efficiency.The slope of the exergy efficiency curve in terms of pressure of the superheated pump implies the effect of pump pressure on exergy efficiency.In a constant temperature, the slope of the curve would be decreased if pressure is so it can be said that the effect of pressure on exergy efficiency would be reduced.The effect of pump pressure on exergy efficiency in superheated temperatures is approximately similar.In the pressure of 74 bar and temperature of 778 (real circumstances of power plant) exergy efficiency would be about 75.7%; at a temperature of 791.5 K and pressure of 74 bar, exergy efficiency would be 76.3%.
Figure 10 presents the effect of pinch temperature difference and pump pressure on the exergy efficiency.
According to this figure decreasing pinch point temperature, exergy efficiency would be increased; the reason for this increase is the fact that a decrease in pinch point temperature would increase the amount of heat absorbed from exhaust gas in the HRSG boiler, so the amount of steam generated for steam turbine would be increased; this reduction of pinch temperature would be led to an increase of heat transfer in HRSG boiler.As can be seen in a high pinch temperature difference, the pump pressure effect on exergy efficiency would be reduced; a low pinch point difference requires highpressure increase.
The effect of superheated temperature and pinch point temperature difference on exergy efficiency is shown in Figure 11.Increasing the pinch point temperature difference steam, power taken from the turbine would be reduced; the reason is that an increase of pinch point temperature difference would decrease the amount of generated steam and would lead to a decrease in power generated by the turbine.Increasing temperature of the pinch point temperature of the gas outgoing HRSG boiler would be increased; therefore some of exergy would be destructed and exergy destruction of the cycle would be increased; this increase of exergy destruction would decrease exergy efficiency in a way that effect of superheated temperature on exergy efficiency in low pinch temperature difference, can be neglected.

| Optimization
Optimization is an important and determinant activity in the design of different systems.Many optimization problems in engineering because of their complexity cannot be solvable using common methods of optimization, such as mathematical programming or any other common methods; there are different solutions to solve every problem; to compare them and choose the best answer, the objective function is defined.One of the most important optimization steps is choosing the appropriate objective function.Sometimes in optimization, multiobjectives are considered at the same time; these kinds of problems which involves several objective functions are called multiobjective problems.The purpose of optimization is determination of design variables so that the objective function is minimized or maximized.In this research genetic algorithm is used for optimization of single objective functions; then using a classic method based on multiobjective constraint optimization, optimization is extended to twoobjective optimization.Independent variables include temperature of the superheated steam, pump pressure, condenser flow rate, and pinch temperature difference.Modeling has been done in the first place; thereafter values of flow rate, pressure, temperature, and pinch are considered as independent variables.Two objective functions have been defined to perform optimization; these functions are exergy efficiency of the HRSG boiler and costs (total cost of generated electricity, rate of costs of power plant, and investment costs).
The effect of costs of buying equipment on exergy efficiency is shown in Figure 12; in this figure increase in exergy efficiency would increase the costs of buying equipment.As a result of discussions by restricting the cost functions, optimum values in different cases are investigated; considering the same weights for the two discussed functions, the nearest point to the balance point is introduced as the chosen point for the optimum state of the two-objective optimization.Exergy efficiency in the preliminary modeling (real circumstances of the power plant) towards the costs of buying equipment is about 79.1%.With a 5% decrease in costs in comparison to the real amount, exergy efficiency would be about 73%, while exergy efficiency would be about 79.7% when costs are increased up to 40%.The presented figure can be used as a measure of investment costs towards exergy function for investors.All points on the Pareto curve (presented curve) are considered as optimum points; Determiner prioritization would merely specify the preferred point in a certain design.
In Figure 13, the rate of changes of optimum points of investment and exergy efficiency in the state of using twoobjective optimization is presented.In this figure increasing the optimum point of exergy efficiency is related to the increase of rate of the investment; it means that investors can reach a system with higher values of exergy efficiency with an increasing amount of unit cost of electricity generation.Considering equal weight for each of the two discussed functions, nearest point to the balance point is considered as optimum technoeconomic point.The exergy efficiency of the HRSG boiler in the initial value of modeling (real circumstances of the power plant) is about 79.1%, while decreasing 5% of costs toward real value would decrease exergy efficiency up to 40% and increasing 40% of costs would increase HRSG boiler exergy efficiency up to 79.7%.
Presenting mathematic relations as an exact estimation of Pareto curves makes it possible to find optimum technoeconomic points among all the points; these relations are functions of changes of optimum points of exergy efficiency towards optimum points of HRSG boiler cost function in a certain range of optimization.As with every other problem of curve fitting, relations obtained from optimization curves are paid attention to as functions of the ratio of change of optimized objective functions.Regression errors of relations of these curves are estimated based on three parameters of root mean square, bias error and sum of square error.Relations between exergy efficiency function and mentioned economic functions and the quality of each of them measured using parameters of error estimation are as follows: Exergy efficiency in terms of costs of buying equipment: In this research effect of costs (costs of buying equipment, rate of power plant costs, and cost of initial investment) has been analyzed.Effects of independent variables such superheated temperature, pinch point temperature difference, pump exit pressure, and condenser flow rate on exergy efficiency have been investigated.From the above discussions and plotted curves following results can be obtained: • Increasing costs of the power plant, buying equipment, and investment would increase exergy efficiency.• Increasing costs by more than 40% of the actual amount would lead to repeated results; a decrease of costs more than 5% concerning restrictions of software cannot be investigated.• Increasing superheated temperature, pump pressure, and the steam ratio of exergy efficiency would be increased.• Increasing pressure, generated work, exergy efficiency, and enthalpy would be increased.• Decreasing pinch point temperature difference, exergy efficiency would be reduced.• In a constant steam ratio, an increase in pressure would reduce the rate of reduction of exergy efficiency.• Steam ratio effect on the efficiency of exergy in different temperatures is approximately constant.• Increasing turbine exhaust gas temperature entering the HRSG boiler, superheated temperature, and steam ratio would be increased.• Increasing condenser flow rate, steam ratio, and exergy efficiency would be reduced.• Increasing condenser flow rate, superheated temperature, and exergy efficiency would be reduced.• In a constant superheated temperature increase in pump pressure would reduce the effect of exergy efficiency.• Decreasing the pinch temperature difference, the effect of superheated temperature, and efficiency would be increased.

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exergy efficiency of HRSG, E ̇cold out HRSG , , : cold fluid exergy outgoing the HRSG, E ̇hot out HRSG , , : hot fluid exergy outgoing the HRSG, E ̇cold in HRSG , , : cold fluid exergy inning the HRSG, E ̇hot in HRSG , , : cold fluid exergy inning the HRSG, W ̇lp pump , : power of the low-pressure pump, W ̇hp pump , : power of the high-pressure pump.

F I G U R E 7
Changes in exergy efficiency in terms of superheated pump pressure in different mass flow rates.HRSG, heat recovery steam generator.F I G U R E 8 Changes in exergy efficiency in terms of superheated temperature in different mass flow rates.HRSG, heat recovery steam generator.F I G U R E 9 Effect of pressure of superheated pump on exergy efficiency in different inlet turbine temperatures.HRSG, heat recovery steam generator.

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I G U R E 10 Changes in exergy efficiency in terms of pinch temperature difference in different inlet turbine pressures.HRSG, heat recovery steam generator.F I G U R E 11 Changes in exergy in terms of pinch temperature difference in different inlet turbine temperatures.HRSG, heat recovery steam generator.

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Bias = 4.2528 10 , −= 98.70%.I G U R E 12 Relation of exergy efficiency with costs of purchasing equipment per kWh electricity.HRSG, heat recovery steam generator; PEC, purchasing equipment cost.F I G U R E 13 Relation of exergy efficiency with investment costs per kWh electricity.HRSG, heat recovery steam generator; RIC, rate of investment cost.
33A B L E 1 Minimum and maximum bounds of decision variables.33