**2. Solar parabolic trough systems**

Solar thermal power plants (STPP) are one of the promising options for electricity supply as demonstrated in some countries during the past decades (Singh & Kaushik, 1994). These plants with parabolic trough type of solar collectors featuring gas burners and Rankine steam cycles have been successfully demonstrated by California's Solar Electric Generating Systems (SEGS).

Trough systems use linear concentrators of parabolic shape with highly reflective surfaces, which can be turned in angular movements towards the sun position and concentrate the radiation onto a long-line receiving absorber tube. The absorbed solar energy is transferred by a working fluid, which is then piped to a conventional power conversion system. The used power conversion systems are based on two technologies:

Rankine-cycle plants are a mature technology that offers a high solar contribution. Recently, integrating the solar collector system with a gas-fired combined-cycle system has been proposed as a lower cost alternative for generating solar-powered electricity.

#### **2.1 Rankine-cycle systems**

The Rankine-cycle STPP is a steam-based power plant with solar energy as the heat source. The system is a typical Rankine cycle (see Figure 1). The hot collector heat transfer fluid transfers its heat in the heat exchanger to the water/steam. The steam drives the turbine to produce electricity. The spent steam is condensed into water in the condenser. The water is reheated in the heat exchanger and the cycle repeats. Because of the seasonal and daily variation in solar radiation, a Rankine-cycle system can only be expected to operate at full load for approximately 2400 hours annually (25% capacity factor) without the use of thermal storage. In most cases, it makes sense to add a fossil-fuel heater so that the system can operate at full load for more hours. Back-up fuels can be coal, oil, naphtha and natural gas. The number of hours that such plant can operates depends on the local conditions. In most cases, however, it makes sense to operate this type of plant to meet the daily periods of high demand for electricity (10 to 12 hours per day). However, Rankine-cycle systems suffer from relatively low efficiencies (whether solar or fossil-fuel powered). The conversion of heat to electricity has an efficiency of about 40%. If the conversion efficiency from fossil fuel to heat is included, the plant efficiency drops to approximately 35% (Status Report on Solar Thermal Power Plants, 1996).

#### **2.2 Integrated solar combined cycle systems**

Between 1984 and 1990, Luz International Limited developed, built, and sold nine parabolic trough solar power plants in the California Mojave Desert. These plants, called Solar Electric Generating Stations and referred to as SEGS I– IX, range in size from 14 MWe to 80 MWe and make up a total of 354 MWe of installed generating capacity. The details of these plants are summarized in Table 1 (www.greenpeace.org, 2005).

combined cycle system (ISCCS) are illustrated. It has been shown that the new optimization schemes are strong tools which can be used to find optimum operating condition based on

Solar thermal power plants (STPP) are one of the promising options for electricity supply as demonstrated in some countries during the past decades (Singh & Kaushik, 1994). These plants with parabolic trough type of solar collectors featuring gas burners and Rankine steam cycles have been successfully demonstrated by California's Solar Electric Generating Systems (SEGS). Trough systems use linear concentrators of parabolic shape with highly reflective surfaces, which can be turned in angular movements towards the sun position and concentrate the radiation onto a long-line receiving absorber tube. The absorbed solar energy is transferred by a working fluid, which is then piped to a conventional power conversion system. The

the main objectives of any thermal plant.

**2. Solar parabolic trough systems**

Rankine-Cycle STPP

**2.1 Rankine-cycle systems**

used power conversion systems are based on two technologies:

Integrated Solar Combined-Cycle Systems (ISCCS) and other hybrid systems.

proposed as a lower cost alternative for generating solar-powered electricity.

approximately 35% (Status Report on Solar Thermal Power Plants, 1996).

**2.2 Integrated solar combined cycle systems**

are summarized in Table 1 (www.greenpeace.org, 2005).

Rankine-cycle plants are a mature technology that offers a high solar contribution. Recently, integrating the solar collector system with a gas-fired combined-cycle system has been

The Rankine-cycle STPP is a steam-based power plant with solar energy as the heat source. The system is a typical Rankine cycle (see Figure 1). The hot collector heat transfer fluid transfers its heat in the heat exchanger to the water/steam. The steam drives the turbine to produce electricity. The spent steam is condensed into water in the condenser. The water is reheated in the heat exchanger and the cycle repeats. Because of the seasonal and daily variation in solar radiation, a Rankine-cycle system can only be expected to operate at full load for approximately 2400 hours annually (25% capacity factor) without the use of thermal storage. In most cases, it makes sense to add a fossil-fuel heater so that the system can operate at full load for more hours. Back-up fuels can be coal, oil, naphtha and natural gas. The number of hours that such plant can operates depends on the local conditions. In most cases, however, it makes sense to operate this type of plant to meet the daily periods of high demand for electricity (10 to 12 hours per day). However, Rankine-cycle systems suffer from relatively low efficiencies (whether solar or fossil-fuel powered). The conversion of heat to electricity has an efficiency of about 40%. If the conversion efficiency from fossil fuel to heat is included, the plant efficiency drops to

Between 1984 and 1990, Luz International Limited developed, built, and sold nine parabolic trough solar power plants in the California Mojave Desert. These plants, called Solar Electric Generating Stations and referred to as SEGS I– IX, range in size from 14 MWe to 80 MWe and make up a total of 354 MWe of installed generating capacity. The details of these plants

Fig. 1. Schematic diagram of a Rankine- Cycle STTP

The SEGS plants in California utilized a solar steam system to provide inlet steam for a conventional (Rankine) cycle steam turbine power plant. In addition to the standard SEGS configuration, Luz International Limited conceived a system configuration for a solar field integrated with a gas-fired combined cycle plant. This concept, known as the Integrated Solar Combined Cycle System (ISCCS), is derived from a conventional combined cycle design in which the exhaust heat from the combustion turbine generates steam in a heat recovery steam generator (HRSG) to drive a steam turbine connected to a generator, with supplemental heat input from the solar field to increase the steam to the steam turbine (Baghernejad & Yaghoubi, 2010). This approach offers a potentially more cost effective and thermodynamically efficient method to utilize solar thermal energy to produce electricity compared to the use of solar energy with a conventional boiler fired (Rankine) cycle plant. In comparison to existing Rankine cycle power plants with parabolic trough technology (SEGS), ISCCS plants offer three principal advantages: First, solar energy can be converted to electric energy at a higher efficiency. Second, the incremental costs for a larger steam turbine are less than the overall unit cost in a solar-only plant. Third, an integrated plant does not suffer from the thermal inefficiencies associated with the daily start-up and shutdown of the steam turbine. Crucial issues in the effective utilization of parabolic trough solar fields in combination with combined cycle plants are the ability to achieve a significant reduction in global emissions, the effective annual heat rate of the combined system, and the cost impact on the plant output (Baghernejad & Yaghoubi, 2011a).

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 69

The economic model takes into account the cost of the components, including amortization and maintenance, and the cost of fuel consumption. In order to define a cost function which depends on the optimization parameters of interest, component costs have to be expressed as functions of thermodynamic variables (Baghernejad & Yaghoubi, 2011a, 2011b). These relationships can be obtained by the statistical correlations between costs and the main thermodynamic parameters of the component performed on real data series. The expressions of purchase components costs and amortization factor are accepted here similar to (Schwarzenbach & Wunsch, 1989). Its format is widely used by various authors but some coefficients were adapted to quotation made by manufacturers. The new coefficients also taken into account the installation, electrical equipment, control system, piping and local

In the analysis and design of energy systems, the techniques which combine scientific disciplines with economic disciplines to achieve optimum design are growing in the energy industries (Valero, 2004). Exergoeconomic analysis as a powerful scheme is such a method that combines exergy analysis with economic studies. This method provides a technique to evaluate the cost of inefficiencies or cost of individual process streams, including intermediate and final products of any system. These costs are applicable in feasibility studies, for investment decisions, on comparing alternative techniques and operating conditions, in a cost-effective section of equipments during an installation, and an exchange or expansion of an energy system (Johansson, 2002; Verda, 2004). Also it can be utilized in optimization of thermodynamic systems, in which the task is usually focused on minimizing

Fig. 2. Typical solar collector-receiver subsystem

**4. Economic model**

assembly.

**5. Exergoeconomic principles**

the unit cost of the system product.

Table 1. Characteristics of Luz SEGS plants (www.greenpeace.org, 2005)

#### **3. Exergy analysis**

Exergy is defined as the maximum possible reversible work obtainable in bringing the state of a system to equilibrium with that of environment (Bejan et al., 1996). The physical exergy component is associated with the work obtainable in bringing a stream of matter from its initial state to a state that is in thermal and mechanical equilibrium with the environment. The chemical exergy component is associated with the work obtainable in bringing a stream of matter from the state that is in thermal and mechanical equilibrium with the environment to a state that is in the most stable configuration in equilibrium with the environment. Therefore the exergy of any state of system is:

$$
\dot{E} = \dot{E}\_{PH} + \dot{E}\_{CH} \tag{1}
$$

The physical exergy component is calculated using the following relation:

$$\dot{E}\_{\rm PH} = \dot{m} [(h - h\_0) - T\_0(s - s\_0)] \tag{2}$$

The exergy of the air and gas streams per unit mass are defined by (Moran & Sciubba, 1994):

$$\dot{e}\_i = C\_{p,Air(Gas)} [T\_i - T\_0 - T\_0 \ln(\frac{T\_i}{T\_0})] + R\_{Air(Gas)} T\_0 \ln(\frac{P\_i}{P\_0}) \tag{3}$$

Energy and exergy analyses for a solar collector-receiver subsystem, Fig. 2 have been carried in Baghernejad and Yaghoubi (2010).

Fig. 2. Typical solar collector-receiver subsystem

## **4. Economic model**

68 Modeling and Optimization of Renewable Energy Systems

**Capacity (MW)** 13.8 30 30 30 30 30 30 80 80 **Land Area (ha)** 29 67 80 80 87 66 68 162 169

**aperture area (ha)** 8.3 19 23 23 25.1 18.8 19.4 46.4 48.4

**temperature (0C)** 307 321 349 349 349 391 391 391 391 **Annual Performance (Design value)**

**Unit cost (\$/kW)** 4490 3200 3600 3730 4130 3870 3870 2890 3440

Exergy is defined as the maximum possible reversible work obtainable in bringing the state of a system to equilibrium with that of environment (Bejan et al., 1996). The physical exergy component is associated with the work obtainable in bringing a stream of matter from its initial state to a state that is in thermal and mechanical equilibrium with the environment. The chemical exergy component is associated with the work obtainable in bringing a stream of matter from the state that is in thermal and mechanical equilibrium with the environment to a state that is in the most stable configuration in equilibrium with the environment.

The exergy of the air and gas streams per unit mass are defined by (Moran & Sciubba, 1994):

,() 0 0 ( )0

Energy and exergy analyses for a solar collector-receiver subsystem, Fig. 2 have been carried

*i p Air Gas i Air Gas T P e C TT T R T*

[ ln( )] ln( ) *i i*

Table 1. Characteristics of Luz SEGS plants (www.greenpeace.org, 2005)

The physical exergy component is calculated using the following relation:

35 43 53 53 53 53 53 53 50

9.3 10.7 10.2 10.2 10.2 12.4 12.3 14 13.6

30.1 80.5 91.3 91.3 99.2 90.9 92.6 252.8 256.1

*EE E PH CH* (1)

00 0 [( ) ( )] *E mh h Ts s PH* (2)

0 0

*T P* (3)

**Solar field**

**Solar field thermal efficiency (%)**

**Solar to net electrical efficiency (%)**

**Net electricity production (GWh/yr)**

**3. Exergy analysis**

Therefore the exergy of any state of system is:

in Baghernejad and Yaghoubi (2010).

**Solar field outlet**

**Unit I II III IV V VI VII VII IX**

The economic model takes into account the cost of the components, including amortization and maintenance, and the cost of fuel consumption. In order to define a cost function which depends on the optimization parameters of interest, component costs have to be expressed as functions of thermodynamic variables (Baghernejad & Yaghoubi, 2011a, 2011b). These relationships can be obtained by the statistical correlations between costs and the main thermodynamic parameters of the component performed on real data series. The expressions of purchase components costs and amortization factor are accepted here similar to (Schwarzenbach & Wunsch, 1989). Its format is widely used by various authors but some coefficients were adapted to quotation made by manufacturers. The new coefficients also taken into account the installation, electrical equipment, control system, piping and local assembly.

#### **5. Exergoeconomic principles**

In the analysis and design of energy systems, the techniques which combine scientific disciplines with economic disciplines to achieve optimum design are growing in the energy industries (Valero, 2004). Exergoeconomic analysis as a powerful scheme is such a method that combines exergy analysis with economic studies. This method provides a technique to evaluate the cost of inefficiencies or cost of individual process streams, including intermediate and final products of any system. These costs are applicable in feasibility studies, for investment decisions, on comparing alternative techniques and operating conditions, in a cost-effective section of equipments during an installation, and an exchange or expansion of an energy system (Johansson, 2002; Verda, 2004). Also it can be utilized in optimization of thermodynamic systems, in which the task is usually focused on minimizing the unit cost of the system product.

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 71

, , *k*

*k Dk Lk Z f ZC C*

A function optimization problem may be of different types, depending on the desired goal of the optimization task. The optimization problem may have only one objective function (known as a single-objective optimization problem), or it may have multiple conflicting objective functions (known as a multi-objective optimization problem). Some problems may have only one local optimum, thereby requiring the task of finding the sole optimum of the function. Other problems may contain more than one local optima in the search space,

Mathematically, a multi-objective optimization problem (Rao, 1996) having *m* objectives and *n* decision variables requires the minimization of the components of the vector *F*(*x*)=F(*f*1(*x*),*f*2(*x*),…,*f*m(*x*)), where *F* is the evaluation function that maps the points of the decision variable space, such as *x*=x(*x*1,…,*x*n), to the points of the objective function space.

> ( ( ), ( ),..., ( )) ( ) 0, 1,2,..., ( ) 0, 1,2,...,

*hx k K x xx i n*

, 1,2,...,

*m*

1 2

*Minimize f x f x f x Subject to g x j J*

*j k*

the population converge to the entire set of optimal solutions in a single run.

() ( )

A multi-objective optimization problem requires the simultaneous satisfaction of a number of different and often conflicting objectives. These objectives are characterized by distinct measures of performance that may be (in) dependent and/or incommensurable. A global optimal solution to a multi-objective optimization problem is unlikely to exist: this means that there is no combination of decision variables values that minimizes all the components of vector *F* simultaneously. Multi-objective optimization problems generally show a possibly uncountable set of solutions, whose evaluated vectors represent the best possible trade-offs in the objective function space. The Pareto approach to multi-objective optimization (Pareto, 1896) is the key concept to establish optimal set of design variables, since the concepts of Pareto dominance and optimality are straightforward tools for determining the best trade-off solutions among conflicting objectives. An evolutionary algorithm is then chosen to carry out the search for the Pareto optimal solution because evolutionary optimization techniques already deal with a set of solutions (a population) to pursue their task, so a multi-objective Pareto-based evolutionary algorithm is able to make

*L U i i i*

*k*

thereby requiring the task of finding multiple such locally optimal solutions.

**6. Optimization problem**

*f* , ,, *k f k* / *f k cCE* (8)

*p*, ,, *k p k* / *p k cCE* (9)

*C cE D k*, ,, *f k Dk* (10)

*C cE L k*, ,, *f k Lk* (11)

(12)

(13)

The prerequisite for the exergoeconomic analysis is a proper 'fuel–product–loss' (F-P-L) definition of the system to show the real production purpose of its subsystems by attributing a well defined role, i.e. fuel, product or loss, to each physical flow entering or leaving the subsystems. The fuel represents the resources needed to generate the product and it is not necessarily restricted to being an actual fuel such as natural gas, oil, and coal. The product represents the desired result produced by the system. Both the fuel and the product are expressed in terms of exergy. The losses represent the exergy loss from the system. Once the F– P–L of a system is defined according to (Baghernejad & Yaghoubi, 2011a; Lozano & Valero, 1993), appropriate cost can be allocated to the products, fuels and losses occurring in the system. A detailed exergy analysis includes calculation of exergy destruction, exergy loss, exergetic efficiency and exergy destruction ratio in each component of the system along with the overall system. Mathematically, these are expressed as follows (Tsatsaronis & Pisa, 1994):

$$
\dot{E}\_{D,k} = \dot{E}\_{F,k} - \dot{E}\_{P,k} - \dot{E}\_{L,k} \tag{4}
$$

$$\varepsilon = \frac{\dot{E}\_P}{\dot{E}\_F} = 1 - \frac{\dot{E}\_D + \dot{E}\_L}{\dot{E}\_F} \tag{5}$$

$$\mathcal{X}\_k = \frac{(\dot{E}\_D)\_k}{(\dot{E}\_D)\_{\text{Tot}}} \tag{6}$$

Exergy costing involves cost balances formulated for each system component separately. A cost balance applied to the kth component shows that the sum of cost rates associated with all exiting exergy streams equals the sum of cost rates of all entering exergy streams plus the appropriate charges (cost rate) due to capital investment and operating and maintenance expenses. The sum of the last two terms is denoted by*Z* . Accordingly, for a kth component:

$$
\sum\_{e}^{N} \{\mathbf{c}\_{e} \dot{\mathbf{E}}\_{e}\} \ \_{k} + \mathbf{c}\_{w,k} \dot{\mathbf{V}} \dot{\mathbf{V}}\_{k} = \mathbf{c}\_{q,k} \dot{\mathbf{E}}\_{q,k} + \sum\_{i}^{N} \{\mathbf{c}\_{i} \dot{\mathbf{E}}\_{i}\} \ \_{k} + \dot{\mathbf{Z}}\_{k} \tag{7}
$$

In the above equation *k k fI Z H* , where *f* and *k I* are the annuity factor and investment cost which are calculated from those given in (Bejan et al., 1996). is maintenance factor and *H* is operation period. In general, if there are *Ne* exergy streams exiting the component being considered, we have *Ne* unknowns and only one equation, the cost balance. Therefore, we need to formulate 1 *Ne* auxiliary equations. This is accomplished with the aid of the F and P principles in the SPECO approach (Lazzaretto & Tsatsaronis, 2006).

Developing Eq. (7) for each component of a plant along with auxiliary costing equations (according to P and F rules) leads to a system of *Ne* equations. By solving this system of equations, the costs of unknown streams of the system will be obtained. These are the average unit cost of fuel ( *f* ,*k c* ), average unit cost of product ( *p*,*k c* ), cost rate of exergy destruction (*CD k*, ), cost rate of exergy loss ( *CL k*, ), and the exergoeconomic factor ( *kf* ). Mathematically, these are expressed as (Tsatsaronis & Pisa, 1994):

$$\mathcal{L}\_{f,k} = \dot{\mathbf{C}}\_{f,k} \ne \dot{\mathbf{E}}\_{f,k} \tag{8}$$

$$\mathcal{L}\_{p,k} = \dot{\mathsf{C}}\_{p,k} \;/ \dot{\mathsf{E}}\_{p,k} \tag{9}$$

$$
\dot{\mathcal{C}}\_{\mathcal{D},k} = \mathfrak{c}\_{f,k} \dot{E}\_{\mathcal{D},k} \tag{10}
$$

$$
\dot{\mathbf{C}}\_{L,k} = \mathbf{c}\_{f,k} \dot{\mathbf{E}}\_{L,k} \tag{11}
$$

$$f\_k = \frac{\dot{Z}\_k}{\dot{Z}\_k + \dot{\mathcal{C}}\_{D,k} + \dot{\mathcal{C}}\_{L,k}} \tag{12}$$

#### **6. Optimization problem**

70 Modeling and Optimization of Renewable Energy Systems

The prerequisite for the exergoeconomic analysis is a proper 'fuel–product–loss' (F-P-L) definition of the system to show the real production purpose of its subsystems by attributing a well defined role, i.e. fuel, product or loss, to each physical flow entering or leaving the subsystems. The fuel represents the resources needed to generate the product and it is not necessarily restricted to being an actual fuel such as natural gas, oil, and coal. The product represents the desired result produced by the system. Both the fuel and the product are expressed in terms of exergy. The losses represent the exergy loss from the system. Once the F– P–L of a system is defined according to (Baghernejad & Yaghoubi, 2011a; Lozano & Valero, 1993), appropriate cost can be allocated to the products, fuels and losses occurring in the system. A detailed exergy analysis includes calculation of exergy destruction, exergy loss, exergetic efficiency and exergy destruction ratio in each component of the system along with the overall system. Mathematically, these are expressed as follows (Tsatsaronis & Pisa, 1994):

> 1 *P DL F F E EE E E*

> > ( ) ( ) *D k*

Exergy costing involves cost balances formulated for each system component separately. A cost balance applied to the kth component shows that the sum of cost rates associated with all exiting exergy streams equals the sum of cost rates of all entering exergy streams plus the appropriate charges (cost rate) due to capital investment and operating and maintenance expenses. The sum of the last two terms is denoted by*Z* . Accordingly, for a kth component:

> , ,, ( ) ( ) *N N*

operation period. In general, if there are *Ne* exergy streams exiting the component being considered, we have *Ne* unknowns and only one equation, the cost balance. Therefore, we need to formulate 1 *Ne* auxiliary equations. This is accomplished with the aid of the F and P

Developing Eq. (7) for each component of a plant along with auxiliary costing equations (according to P and F rules) leads to a system of *Ne* equations. By solving this system of equations, the costs of unknown streams of the system will be obtained. These are the average unit cost of fuel ( *f* ,*k c* ), average unit cost of product ( *p*,*k c* ), cost rate of exergy destruction (*CD k*, ), cost rate of exergy loss ( *CL k*, ), and the exergoeconomic factor ( *kf* ).

*e e wk k q k q k ii k e i k k*

*cE c W c E cE Z* (7)

, where *f* and *k I* are the annuity factor and investment cost

*D Tot E E*

*k*

In the above equation *k*

*k fI Z H*

which are calculated from those given in (Bejan et al., 1996).

principles in the SPECO approach (Lazzaretto & Tsatsaronis, 2006).

Mathematically, these are expressed as (Tsatsaronis & Pisa, 1994):

*E EEE Dk Fk Pk Lk* , ,,, (4)

(5)

(6)

is maintenance factor and *H* is

A function optimization problem may be of different types, depending on the desired goal of the optimization task. The optimization problem may have only one objective function (known as a single-objective optimization problem), or it may have multiple conflicting objective functions (known as a multi-objective optimization problem). Some problems may have only one local optimum, thereby requiring the task of finding the sole optimum of the function. Other problems may contain more than one local optima in the search space, thereby requiring the task of finding multiple such locally optimal solutions.

Mathematically, a multi-objective optimization problem (Rao, 1996) having *m* objectives and *n* decision variables requires the minimization of the components of the vector *F*(*x*)=F(*f*1(*x*),*f*2(*x*),…,*f*m(*x*)), where *F* is the evaluation function that maps the points of the decision variable space, such as *x*=x(*x*1,…,*x*n), to the points of the objective function space.

$$\begin{aligned} \text{Minimize} \quad & (f\_1(\mathbf{x}), f\_2(\mathbf{x}), \dots, f\_m(\mathbf{x}))\\ \text{Subject to} \quad & g\_j(\mathbf{x}) \ge 0, \quad & j = 1, 2, \dots, J \\ & h\_k(\mathbf{x}) = 0, \quad & k = 1, 2, \dots, K \\ & \mathbf{x}\_i^{(L)} \le \mathbf{x}\_i \le \mathbf{x}\_i^{(L)}, \quad i = 1, 2, \dots, n \end{aligned} \tag{13}$$

A multi-objective optimization problem requires the simultaneous satisfaction of a number of different and often conflicting objectives. These objectives are characterized by distinct measures of performance that may be (in) dependent and/or incommensurable. A global optimal solution to a multi-objective optimization problem is unlikely to exist: this means that there is no combination of decision variables values that minimizes all the components of vector *F* simultaneously. Multi-objective optimization problems generally show a possibly uncountable set of solutions, whose evaluated vectors represent the best possible trade-offs in the objective function space. The Pareto approach to multi-objective optimization (Pareto, 1896) is the key concept to establish optimal set of design variables, since the concepts of Pareto dominance and optimality are straightforward tools for determining the best trade-off solutions among conflicting objectives. An evolutionary algorithm is then chosen to carry out the search for the Pareto optimal solution because evolutionary optimization techniques already deal with a set of solutions (a population) to pursue their task, so a multi-objective Pareto-based evolutionary algorithm is able to make the population converge to the entire set of optimal solutions in a single run.

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 73

problem. Using of this technique for optimizing district heating network-using genetic algorithm (GA) demonstrated by (Cammarata et al., 1998). The objective function for this

> *sys k Dk Lk* , , *k k*

The two objective functions of this multi-criteria optimization problem are the total exergetic efficiency of the plant (to be maximized) and the total cost rate of product (to be minimized). The mathematical formulation of objective functions is as following (Baghernejad &

*Tot*

*W*

*F E E*

*Tot k k*

Sensitivity analysis is a general concept which aims to quantify the variations of an output parameter of a system regarding to the changes imposed on some input important parameters. A comprehensive sensitivity analysis can be performed to examine the impact of the variation of important factors on electricity costs of any STPP. The most important factors which influence electricity cost of a solar thermal power plant are fuel specific cost, interest rate, plant lifetime, solar field operation period and construction period

Fig. 3 shows a schematic diagram of an integrated solar combined cycle system. This power plant contains two 125 MW gas turbines, a 150 MW steam turbine, and a 17 MW solar plant.

 Two heat recovery steam generators with two pressure lines. The high and low pressure steam conditions are: 84.8 bar and 506 0C and 9.1 bar and 231.6 0C respectively. A design stack temperature of 113 0C is selected to recover as much energy from the

The solar field considered in this site is comprised of 42 loops and for each loop, 6 collectors from type of LS-3 (Kearney, 1999) which are single axis tracking and aligned on a north– south line, thus tracking the sun from east to west. Various design parameters of these

In this system a combined cycle unit with the following equipments is used:

*C Z CC* (16)

(17)

*C Z* (18)

problem is defined as to minimize the total cost function*Csys* , which can be modeled as:

**8.2 Multi objective exergoeconomic optimization**

Yaghoubi, 2011b):

**9. Sensitivity analysis**

(Baghernejad & Yaghoubi, 2011a, 2011b).

**10.1 Integrated solar combined cycle system (ISCCS)**

Two V94.2 gas turbine units with natural gas fuel

**10. Example of application**

turbine exhaust as possible

A no reheat two pressure steam turbine

#### **7. Evolutionary algorithms for optimization**

The evolutionary algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution (Rezende et al., 2008). As a first step, parent selection is performed with each individual having the same probability of being chosen. Suppose *NP* is the size of generated population. Then *NP* numbers of parents enter the reproduction step, generating *NP* offspring through a crossover strategy in which the decision variable values of the offspring fall in a range defined by the decision variable values of the parents. Some of the offspring are also produced by adding a Gaussian random variable ( *N* ) with zero mean and a standard deviation proportional to the scaled cost value of the parent trial solution, i.e,

$$P'\_{\mathcal{g},i} = P\_{\mathcal{g},i} + N(\mathbf{0}, \sigma\_i^2) \tag{14}$$

The standard deviation *i* indicates the range over which the offspring is created around the parent trial solution and is given by

$$
\sigma\_i^2 = \varphi \frac{f(P\_i)}{f(\overline{P}\_{\text{min}})} \tag{15}
$$

Where min *f P*( )is the minimum value of the objective function among the *NP* trial solution, ( )*i f P* is the objective function value associated with the trial vector *Pi* and is a scaling factor. These offspring *Pi* , 1,2,..., *P i N* and their parents *Pi* , 1,2,..., *P i N* form a set of 2*NP* trial solutions and they contend for survival with each other within the competing pool. After competition, the 2*NP* trial solutions including the parents and the offspring are ranked in descending order of the score. The first *NP* trial solutions survive and are transcribed along with their objective functions ( )*i f P* into the survivor set as the basis of the next generation. Finally, the number of generations elapsed is compared to the established maximum number of generations. If the termination condition is met, the process stops, otherwise the surviving solutions become the starting population for the next generation (Beghi et al., 2011).

#### **8. Exergoeconomic optimization**

#### **8.1 Single objective exergoeconomic optimization**

In general, a thermal system requires two conflicting objectives: one being increase in exergetic efficiency and the other is decrease in product cost, to be satisfied simultaneously. The first objective is governed by thermodynamic requirements and the second by economic constraints. Therefore, objective function should be defined in such a way that the optimization satisfies both requirements. For a single objective optimization, the optimization problem should be formulated as a minimization or maximization problem. The exergoeconomic analysis gives a clear picture about the costs related to the exergy destruction, exergy losses, etc. Thus, the objective function in this optimization becomes a minimization problem. Using of this technique for optimizing district heating network-using genetic algorithm (GA) demonstrated by (Cammarata et al., 1998). The objective function for this problem is defined as to minimize the total cost function*Csys* , which can be modeled as:

$$
\dot{\mathbf{C}}\_{sys} = \sum\_{k} \dot{\mathbf{Z}}\_{k} + \sum\_{k} \dot{\mathbf{C}}\_{D,k} + \dot{\mathbf{C}}\_{L,k} \tag{16}
$$

#### **8.2 Multi objective exergoeconomic optimization**

The two objective functions of this multi-criteria optimization problem are the total exergetic efficiency of the plant (to be maximized) and the total cost rate of product (to be minimized). The mathematical formulation of objective functions is as following (Baghernejad & Yaghoubi, 2011b):

$$
\varepsilon\_{\text{Tot}} = \frac{\dot{E}\_{\text{VV}}}{\dot{E}\_{\text{F}}} \tag{17}
$$

$$
\dot{\mathbf{C}}\_{Tot} = \sum\_{k} \dot{\mathbf{Z}}\_{k} \tag{18}
$$

#### **9. Sensitivity analysis**

72 Modeling and Optimization of Renewable Energy Systems

The evolutionary algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution (Rezende et al., 2008). As a first step, parent selection is performed with each individual having the same probability of being chosen. Suppose *NP* is the size of generated population. Then *NP* numbers of parents enter the reproduction step, generating *NP* offspring through a crossover strategy in which the decision variable values of the offspring fall in a range defined by the decision variable values of the parents. Some of the offspring are also produced by adding a Gaussian random variable ( *N* ) with zero mean and a standard deviation proportional to the scaled cost value of the parent trial solution,

> , , (0, ) *PPN g i g i i*

> > 2

*i*

( )*i f P* is the objective function value associated with the trial vector *Pi* and

solutions become the starting population for the next generation (Beghi et al., 2011).

2

*i* indicates the range over which the offspring is created around

, 1,2,..., *P i N* and their parents *Pi* , 1,2,..., *P i N* form a set of 2*NP*

(14)

is a scaling

(15)

min ( ) ( ) *i*

*f P f P*

Where min *f P*( )is the minimum value of the objective function among the *NP* trial solution,

trial solutions and they contend for survival with each other within the competing pool. After competition, the 2*NP* trial solutions including the parents and the offspring are ranked in descending order of the score. The first *NP* trial solutions survive and are transcribed along with their objective functions ( )*i f P* into the survivor set as the basis of the next generation. Finally, the number of generations elapsed is compared to the established maximum number of generations. If the termination condition is met, the process stops, otherwise the surviving

In general, a thermal system requires two conflicting objectives: one being increase in exergetic efficiency and the other is decrease in product cost, to be satisfied simultaneously. The first objective is governed by thermodynamic requirements and the second by economic constraints. Therefore, objective function should be defined in such a way that the optimization satisfies both requirements. For a single objective optimization, the optimization problem should be formulated as a minimization or maximization problem. The exergoeconomic analysis gives a clear picture about the costs related to the exergy destruction, exergy losses, etc. Thus, the objective function in this optimization becomes a minimization

**7. Evolutionary algorithms for optimization**

the parent trial solution and is given by

**8. Exergoeconomic optimization**

**8.1 Single objective exergoeconomic optimization**

i.e,

The standard deviation

factor. These offspring *Pi*

Sensitivity analysis is a general concept which aims to quantify the variations of an output parameter of a system regarding to the changes imposed on some input important parameters. A comprehensive sensitivity analysis can be performed to examine the impact of the variation of important factors on electricity costs of any STPP. The most important factors which influence electricity cost of a solar thermal power plant are fuel specific cost, interest rate, plant lifetime, solar field operation period and construction period (Baghernejad & Yaghoubi, 2011a, 2011b).

#### **10. Example of application**

#### **10.1 Integrated solar combined cycle system (ISCCS)**

Fig. 3 shows a schematic diagram of an integrated solar combined cycle system. This power plant contains two 125 MW gas turbines, a 150 MW steam turbine, and a 17 MW solar plant.

In this system a combined cycle unit with the following equipments is used:

The solar field considered in this site is comprised of 42 loops and for each loop, 6 collectors from type of LS-3 (Kearney, 1999) which are single axis tracking and aligned on a north– south line, thus tracking the sun from east to west. Various design parameters of these

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 75

Fig. 3. Schematic diagram of a Integrated Solar Combined Cycle System

Table 2. LS-3 Collector specifications (Kearney, 1999)

Aperture Area per SCA (m2) 545 HCE Transmittance 0.96

Mirror Segments 224 Mirror Reflectivity 0.94

Aperture (m) 5.76 Length 99

HCE Diameter (m) 0.07 Concentration Ratio 82

Average Focal Distance (m) 0.94 Peak Collector Efficiency (%) 68

HCE Absorptivity 0.96 Annual Thermal Efficiency (%) 53

HCE Emittance 0.17 Optical Efficiency (%) 80

collectors are given in Table 2. In this study, numerical results are based on site design condition with ambient temperature of 19 °C and a relative humidity of 32 percent and wind speed of 3 m/s. The analysis is carried out on 21 Jun at 12:00 noon (LAT). At this hour, solar radiation intensity at the plant site is about 800 W/m2. Also, Therminol VP-1 is used as heat transfer fluid (HTF) in the solar field. The state properties and exergies calculated for the system of Fig.3 are given in Table 3. In this table, states 0, 0' and 0'' are the dead states for air, water and oil, respectively. In this model, the variables selected for the optimization are (Baghernejad & Yaghoubi, 2011a, 2011b):

The compressor pressure ratio 2 1 ( ) *r P P P* , isentropic efficiency of the compressor *AC* , temperature of the combustion products entering the turbine *T*3 , isentropic efficiency of the gas turbine *GT* in the gas cycle, isentropic efficiency of the oil pump *OILP* , and oil outlet temperature in the collectors *T*28 in the solar field, temperature and pressure of the steams leaving the heat recovery steam generator *T*23 , *P*23,17 , isentropic efficiency of the condensed extraction pump *CEP* and isentropic efficiency of the boiler feed water pump *BFP* in the steam cycle. This model is treated as the base case and the following nominal values of the decision variables are selected based on the operation program of the constructed site. 11 *Pr* , 0.85 *AC* , 3 *T K* 1404.8 , 0.875 *GT* , *OILP* 0.8 , *T K* 28 666.5 ,

*ST* , 0.8

## **10.2 Exergoeconomic analysis**

23 *T K* 779.15 , 23 *P bar* 84.8 , 17 *P bar* 9.1 , 0.85

The system used in Fig. 3 consists of 16 components and has 38 streams (32 for mass and 6 for work). Therefore 22 boundary conditions and auxiliary equations are necessary. For example in this system, for steam turbine (ST) and solar heat exchanger (SHE), cost balance and auxiliary costing equations (according to P and F rules) are formulated as follow:

Steam turbine:

$$
\dot{\mathbf{C}}\_{11} + \dot{\mathbf{C}}\_{38} = \mathbf{2} (\dot{\mathbf{C}}\_{17} + \dot{\mathbf{C}}\_{23}) + \dot{Z}\_{\text{ST}} \tag{19}
$$

*CEP* , 0.8 *BFP*

$$c\_{17} + c\_{23} = c\_{11} \quad or \quad \frac{\dot{C}\_{17} + \dot{C}\_{23}}{\dot{E}\_{17} + \dot{E}\_{23}} = \frac{\dot{C}\_{11}}{\dot{E}\_{11}} \tag{20}$$

Solar heat exchanger:

$$
\dot{\mathbf{C}}\_{25} + \dot{\mathbf{C}}\_{26} = \dot{\mathbf{C}}\_{24} + \dot{\mathbf{C}}\_{28} + \dot{\mathbf{Z}}\_{\text{SHE}} \tag{21}
$$

$$\mathbf{c}\_{26} = \mathbf{c}\_{28} \quad \text{or} \quad \frac{\dot{\mathbf{C}}\_{26}}{\dot{\mathbf{E}}\_{26}} = \frac{\dot{\mathbf{C}}\_{28}}{\dot{\mathbf{E}}\_{28}} \tag{22}$$

In the same way developing cost balance equation for other element of oil, gas and steam cycles along with auxiliary costing equations leads to a system of equations. By solving the system of 38 equations and 38 unknowns, the cost of unknown streams of the system will be obtained. More details of cost balance equations can be seen in (Baghernejad & Yaghoubi, 2011a, 2011b).

collectors are given in Table 2. In this study, numerical results are based on site design condition with ambient temperature of 19 °C and a relative humidity of 32 percent and wind speed of 3 m/s. The analysis is carried out on 21 Jun at 12:00 noon (LAT). At this hour, solar radiation intensity at the plant site is about 800 W/m2. Also, Therminol VP-1 is used as heat transfer fluid (HTF) in the solar field. The state properties and exergies calculated for the

water and oil, respectively. In this model, the variables selected for the optimization are

temperature of the combustion products entering the turbine *T*3 , isentropic efficiency of the

temperature in the collectors *T*28 in the solar field, temperature and pressure of the steams leaving the heat recovery steam generator *T*23 , *P*23,17 , isentropic efficiency of the condensed

steam cycle. This model is treated as the base case and the following nominal values of the decision variables are selected based on the operation program of the constructed

The system used in Fig. 3 consists of 16 components and has 38 streams (32 for mass and 6 for work). Therefore 22 boundary conditions and auxiliary equations are necessary. For example in this system, for steam turbine (ST) and solar heat exchanger (SHE), cost balance and auxiliary costing equations (according to P and F rules) are formulated as follow:

*E E*

In the same way developing cost balance equation for other element of oil, gas and steam cycles along with auxiliary costing equations leads to a system of equations. By solving the system of 38 equations and 38 unknowns, the cost of unknown streams of the system will be obtained. More details of cost balance equations can be seen in (Baghernejad & Yaghoubi, 2011a, 2011b).

*CEP* and isentropic efficiency of the boiler feed water pump

*GT* ,

*ST* , 0.8

17 23 11

17 23 11 *C C C*

*EE E*

26 28

26 28 *C C*

1 ( ) *r P P*

*AC* , 3 *T K* 1404.8 , 0.875

17 23 11

*c c c or*

26 28

*c c or*

23 *T K* 779.15 , 23 *P bar* 84.8 , 17 *P bar* 9.1 , 0.85

*GT* in the gas cycle, isentropic efficiency of the oil pump

and 0'' are the dead states for air,

*AC* ,

*BFP* in the

*OILP* , and oil outlet

*OILP* 0.8 , *T K* 28 666.5 ,

*P* , isentropic efficiency of the compressor

11 38 17 23 2( ) *CC CC Z ST* (19)

*CCCCZ* 25 26 24 28 *SHE* (21)

(20)

(22)

*CEP* , 0.8 *BFP*

system of Fig.3 are given in Table 3. In this table, states 0, 0'

(Baghernejad & Yaghoubi, 2011a, 2011b):

The compressor pressure ratio 2

**10.2 Exergoeconomic analysis**

gas turbine

extraction pump

Steam turbine:

Solar heat exchanger:

site. 11 *Pr* , 0.85

Fig. 3. Schematic diagram of a Integrated Solar Combined Cycle System

Table 2. LS-3 Collector specifications (Kearney, 1999)

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 77

Although the decision variables may be varied in optimization procedure, each decision variable is normally required to be within a given practical range of operation as follow

9 16 *Pr* 28 640 670 *T K* 23 80 100 *P bar* 3 1350 1500 *T K* 17 7 10 *P bar*

Also, the values of the economic parameters and fixed parameters for the optimization of

k (years) 25 ( ) *W MW out*

H (hour) for solar field 2000 H (hour) for Combined cycle 7500

Table 4. Fixed parameters for the system shown in Fig. 3 (Baghernejad & Yaghoubi, 2011a)

For a single objective optimization, with the objective function indicated in Eq. (16) in the first generation, 100 vectors 3 28 23 23 17 [, ,, , , , , , , , , ] *P P T TTPP i r AC GT OILP*

randomly generated within the operating range. For each trial, the objective function is evaluated through exergy analysis and exergoeconomic formulations after passing through the system constraints. Performance of the system with each vector is evaluated. The vector having the best system performance is stored for future comparison. The algorithm selects a group of vectors in the current generation, called parents that have better objective function values for the next generation (second generation). These parents are modified using Eq. (14) to generate the offsprings. The performance of the offsprings and the parent vectors are compared to select the best vector in the generation. The process of selecting parents and

The structure of the multi objective evolutionary algorithm (MOEA) used in the present work is illustrated in (Schwefwl, 1995). Also, the tuning of MOEA is performed according to the values indicated in Table 5 (Baghernejad & Yaghoubi, 2011b). Fig. 4 presents the Pareto optimum solutions for the integrated solar combined cycle system with the objective function indicated in Eqs. (17) and (18). As shown in this figure, while the total exergetic efficiency of the plant is increased to about 44%, the total cost rate of products increases very slightly. Increasing the total exergetic efficiency from 44% to 47% corresponds to the moderate increase in the cost rate of product. Further increasing of exergetic efficiency from 47% to the higher value leads to a drastic increasing of the total

In multi-objective optimization, a process of decision-making for selection of the final optimal solution from the available solutions is required. The process of decision-making is usually performed with the aid of a hypothetical point in Fig. 5 named as equilibrium point that both objectives have their optimal values independent to the other objective. It

then generating the offsprings repeats till the specified number of generations.

*ST* 0.75 0.9

*CC* 0.99 LHV (kJ/kg) 47966

 1.06 ri (%) 8 in (%) 10 CP (year) 3

*GT* 0.75 0.9 *CEP*

400

 *ST CEP BFP* are

*BFP* 0.75 0.9

**10.3 Result and discussion**

0.75 0.9

cost rate.

(Baghernejad & Yaghoubi, 2011a):

*AC* 0.75 0.9

23 723.15 823.15 *T K* 0.75 0.9

*OILP*

system are given in Table 4 (Baghernejad & Yaghoubi, 2011a).

Table 3. State properties and calculated exergy of the system corresponding to Fig. 3

#### **10.3 Result and discussion**

76 Modeling and Optimization of Renewable Energy Systems

0 - 292.15 1.013 292.43 - 0' - 292.15 1.013 79.82 - 0'' - 292.15 1.013 12 - 1 421.81 292.15 1.013 292.43 0 2 421.81 630.35 11.14 631.8 132.86 3 429.56 1404.80 10.58 1409 388.29 4 429.56 821.46 1.05 823.6 105.8 5 429.56 729.67 1.07 468.15 80.24 6 429.56 536.86 1.04 261.64 31.75 7 429.56 479.67 1.04 200.65 20.59 8 429.56 477.12 1.02 197.92 19.39 9 429.56 431.1 1.02 148.67 11.87 10 429.56 386.15 1.013 100.58 6 11 144.32 321.19 0.112 2304.5 28.25 12 144.32 321.19 0.112 201.15 0.80 13 144.32 321.55 25.5 204.36 1.09 14 72.16 390.02 1.8 490.52 4.09 15 9.25 390.15 9.3 491.52 0.53 16 9.25 449.9 9.3 2774.3 7.82 17 9.25 504.74 9.1 2906 8.27 18 62.91 391.71 119 506 4.39 19 62.91 488.15 118 923.8 13.17 20 53.66 488.15 118 923.8 10.22 21 53.66 578.66 92.77 2738.2 53.1 22 62.91 578.66 92.77 2738.2 68.38 23 62.91 779.15 84.8 3408.6 91.22 24 14.06 488.15 118 923.8 2.94 25 14.06 578.66 92.77 2738.2 15.28 26 218.42 571.15 11 550.34 36.54 27 218.42 573.82 16 552.63 36.83 28 218.42 666.5 26 790 73.16 29 2575 292.15 1.013 79.82 0 30 2575 320.35 1.013 197.71 13.77 31 7.75 292.15 20 292.43 401.89

Table 3. State properties and calculated exergy of the system corresponding to Fig. 3

P (bar)

h (kJ/kg)

*E* (MW)

T (K)

State *m*

(kg/sec)

Although the decision variables may be varied in optimization procedure, each decision variable is normally required to be within a given practical range of operation as follow (Baghernejad & Yaghoubi, 2011a):

9 16 *Pr* 28 640 670 *T K* 23 80 100 *P bar* 3 1350 1500 *T K* 17 7 10 *P bar* 0.75 0.9 *AC* 0.75 0.9 *BFP* 0.75 0.9 *ST* 0.75 0.9 *GT* 0.75 0.9 *CEP* 23 723.15 823.15 *T K* 0.75 0.9 *OILP*

Also, the values of the economic parameters and fixed parameters for the optimization of system are given in Table 4 (Baghernejad & Yaghoubi, 2011a).

Table 4. Fixed parameters for the system shown in Fig. 3 (Baghernejad & Yaghoubi, 2011a)

For a single objective optimization, with the objective function indicated in Eq. (16) in the first generation, 100 vectors 3 28 23 23 17 [, ,, , , , , , , , , ] *P P T TTPP i r AC GT OILP ST CEP BFP* are randomly generated within the operating range. For each trial, the objective function is evaluated through exergy analysis and exergoeconomic formulations after passing through the system constraints. Performance of the system with each vector is evaluated. The vector having the best system performance is stored for future comparison. The algorithm selects a group of vectors in the current generation, called parents that have better objective function values for the next generation (second generation). These parents are modified using Eq. (14) to generate the offsprings. The performance of the offsprings and the parent vectors are compared to select the best vector in the generation. The process of selecting parents and then generating the offsprings repeats till the specified number of generations.

The structure of the multi objective evolutionary algorithm (MOEA) used in the present work is illustrated in (Schwefwl, 1995). Also, the tuning of MOEA is performed according to the values indicated in Table 5 (Baghernejad & Yaghoubi, 2011b). Fig. 4 presents the Pareto optimum solutions for the integrated solar combined cycle system with the objective function indicated in Eqs. (17) and (18). As shown in this figure, while the total exergetic efficiency of the plant is increased to about 44%, the total cost rate of products increases very slightly. Increasing the total exergetic efficiency from 44% to 47% corresponds to the moderate increase in the cost rate of product. Further increasing of exergetic efficiency from 47% to the higher value leads to a drastic increasing of the total cost rate.

In multi-objective optimization, a process of decision-making for selection of the final optimal solution from the available solutions is required. The process of decision-making is usually performed with the aid of a hypothetical point in Fig. 5 named as equilibrium point that both objectives have their optimal values independent to the other objective. It

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 79

Fig. 5. Pareto frontier: best trade-off values for the objective functions (total exergetic

Fig. 6. Selecting procedure for optimal solution from Pareto frontier in the system shown in

The cost of the streams in the base case and optimum cases (single and multi objective optimization) are given in Table 6. Unit cost of the electricity produced by steam turbine is reduced from 29.57 cents/kWh in the base case to 27.47 and 27.63 cents/kWh in the

The related values of decision variables in both optimum cases are given in Table 7. These new parameters obtained in the optimized cases will help the designer to select components,

i.e. turbines, compressor, as close to the optimum configuration.

Fig. 3

optimum cases respectively.

efficiency and total cost rate of products) in the system shown in Fig. 3

is clear that it is impossible to have both objectives at their optimum point, simultaneously and as shown in Fig. 5, the equilibrium point is not a solution located on the Pareto frontier. The closest point of Pareto frontier to the equilibrium point might be considered as a desirable final solution. In selection of the final optimum point, it is desired to achieve the better magnitude for each objective than its initial value of the base case problem. On the other hand, the stability of the selected point when one of objective varies is highly important. Therefore a part of Pareto frontier is selected in Fig. 6 for decision-making and coincided with domain between horizontal and vertical lines. A final optimum solution with 45.19% exergetic efficiency and the total cost rate of product equal to 10920.43 \$/h as indicated in Fig. 6 is selected. It should be noted that the selection of an optimum solution depends on the preferences and criteria of each decision-maker. Therefore, each decision-maker may select a different point as optimum solution to better suits with his/her desires.

Table 5. The turning parameters in MOEA optimization program (Baghernejad & Yaghoubi, 2011b)

Fig. 4. Pareto optimal solutions for solar combined cycle system shown in Fig. 3

is clear that it is impossible to have both objectives at their optimum point, simultaneously and as shown in Fig. 5, the equilibrium point is not a solution located on the Pareto frontier. The closest point of Pareto frontier to the equilibrium point might be considered as a desirable final solution. In selection of the final optimum point, it is desired to achieve the better magnitude for each objective than its initial value of the base case problem. On the other hand, the stability of the selected point when one of objective varies is highly important. Therefore a part of Pareto frontier is selected in Fig. 6 for decision-making and coincided with domain between horizontal and vertical lines. A final optimum solution with 45.19% exergetic efficiency and the total cost rate of product equal to 10920.43 \$/h as indicated in Fig. 6 is selected. It should be noted that the selection of an optimum solution depends on the preferences and criteria of each decision-maker. Therefore, each decision-maker may select a different point as optimum solution to better

Turning parameters Value

Population size 100

Selection process Tournament

Tournament size 2

Maximum number of generations 100

Pc (probability of crossover) (%) 50

Pm (probability of mutation) (%) 1

Table 5. The turning parameters in MOEA optimization program (Baghernejad & Yaghoubi,

Fig. 4. Pareto optimal solutions for solar combined cycle system shown in Fig. 3

suits with his/her desires.

2011b)

Fig. 5. Pareto frontier: best trade-off values for the objective functions (total exergetic efficiency and total cost rate of products) in the system shown in Fig. 3

Fig. 6. Selecting procedure for optimal solution from Pareto frontier in the system shown in Fig. 3

The cost of the streams in the base case and optimum cases (single and multi objective optimization) are given in Table 6. Unit cost of the electricity produced by steam turbine is reduced from 29.57 cents/kWh in the base case to 27.47 and 27.63 cents/kWh in the optimum cases respectively.

The related values of decision variables in both optimum cases are given in Table 7. These new parameters obtained in the optimized cases will help the designer to select components, i.e. turbines, compressor, as close to the optimum configuration.

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 81

Compressor efficiency 85 85.98 82.98 Compressor pressure ratio 11 11.98 10.86

Gas turbine efficiency (%) 87.5 90 87.98 Oil outlet temperature (K) 666.5 658.17 650.28

turbine (K) 1404.8 1449.9 1437.8

turbine (bar) 84.8 99.91 96.24

steam turbine (K) 779.15 823.13 808.66

turbine (bar) 9.1 9.97 9.95

CEP efficiency (%) 80 88.84 80 BFP efficiency (%) 80 84.02 77.84 OILP efficiency (%) 80 83.44 86.16 Table 7. Comparison of the decisions variables for optimum and base cases in the system

(%) 85 89.94 89.99

The values of objective functions in the base and optimum cases including the total cost of product and the total exergy efficiency are listed in Table 8. This table indicates that the optimization process leads to 10.98% decrease in the objective function for single objective optimization and 3.2% increase in the exergetic efficiency and 3.82% decrease for the rate of product cost in multi objective optimization. Therefore, improvement for all objectives has

The comparative results of the base case and the optimum cases are presented in Table 9. According to this table, optimization process improves the total performance of the system in a way that the rate of fuel cost is decreased by 7.23 and 3.46% in optimum cases. Also exergy destructions is reduced about 12 and 5.7%, the related cost of the system inefficiencies is decreased about 14.8 and 7.32% and exergetic efficiency is increased from 43.79 to 46.8 and 45.19% in both optimum cases, although the total owning and operation cost in single objective optimization is increased about 13.3%. Moreover, it can be found from this table that the optimization increases the overall exergoeconomic factor of this system from 12.18 in the base case to 15.51 (27.34% increases) and 12.56 (3.1% increase) implying that optimization process mostly reduced the associated cost of thermodynamic inefficiencies rather to increase the capital investment and operating and maintenance cost

(Single objective)

Optimum case (Multi objective)

Decision variables Base case Optimum case

Inlet temperature of gas

Inlet pressure to HP steam

Inlet temperature to HP

Inlet pressure to LP steam

Steam turbine efficiency

been achieved using optimization process.

of the system components

shown in Fig. 3

Table 6. Cost of streams in the system shown in Fig. 3

**(cents/kWh) C (\$/h) c (cents/kWh) C (\$/h) c (cents/kWh) C (\$/h)**

**(Single objective) Optimum case**

**(Multi objective)**

**Base case Optimum case**

**State points (Fig. 2)**

**c**

Table 6. Cost of streams in the system shown in Fig. 3

Table 7. Comparison of the decisions variables for optimum and base cases in the system shown in Fig. 3

The values of objective functions in the base and optimum cases including the total cost of product and the total exergy efficiency are listed in Table 8. This table indicates that the optimization process leads to 10.98% decrease in the objective function for single objective optimization and 3.2% increase in the exergetic efficiency and 3.82% decrease for the rate of product cost in multi objective optimization. Therefore, improvement for all objectives has been achieved using optimization process.

The comparative results of the base case and the optimum cases are presented in Table 9. According to this table, optimization process improves the total performance of the system in a way that the rate of fuel cost is decreased by 7.23 and 3.46% in optimum cases. Also exergy destructions is reduced about 12 and 5.7%, the related cost of the system inefficiencies is decreased about 14.8 and 7.32% and exergetic efficiency is increased from 43.79 to 46.8 and 45.19% in both optimum cases, although the total owning and operation cost in single objective optimization is increased about 13.3%. Moreover, it can be found from this table that the optimization increases the overall exergoeconomic factor of this system from 12.18 in the base case to 15.51 (27.34% increases) and 12.56 (3.1% increase) implying that optimization process mostly reduced the associated cost of thermodynamic inefficiencies rather to increase the capital investment and operating and maintenance cost of the system components

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 83

The final optimal solution that was selected in this system belongs to the region of Pareto frontier with significant sensitivity to the costing parameters. However, the region with the lower sensitivity to the costing parameter is not reasonable for the final optimum solution. Also a small change in exergetic efficiency of the plant (exergetic efficiency from 47% to the higher value) due to the variation of operating parameters may lead to the danger of

**0.05 0.075 0.1 0.125 0.15 0.175 0.2**

Fuel cost (\$/kWh)

**1000 1500 2000 2500 28.0**

Fig. 8. Sensitivity of unit cost of electricity to the solar field operation periods

Solar field operation period (hour/year)

increasing the cost rate of product, drastically.

**15.0**

**29.0**

**30.0**

**31.0**

Unit cost of electricity produced by

steam turbine (cent/kWh)

**32.0**

**33.0**

**34.0**

**35.0**

Fig. 7. Sensitivity of unit cost of electricity with specific fuel cost

**20.0**

**25.0**

**30.0**

Unit cost of electricity produced by

steam turbine(cent/kWh)

**35.0**

**40.0**

**45.0**

**50.0**

Table 8. Comparison of the objective functions for optimum and base cases in the system shown in Fig. 3

Table 9. Comparative results of the optimum and base cases in the system shown in Fig. 3

#### **10.4 Sensitivity analysis of the ISCCS**

#### **10.4.1 Sensitivity analysis for single objective optimization**

Additional runs of the optimization algorithm were performed on the system in order to investigate the influence of the unit cost of fuel, the construction period and solar operation period on the solution. Fig. 7 shows sensitivity with respect to fuel cost which is linear. Fig. 8 illustrates variation of unit cost with increasing solar contribution which is significant (Baghernejad & Yaghoubi, 2011a).

#### **10.4.2 Sensitivity analyses for multi objective optimization**

Fig. 9 shows the sensitivity of the Pareto optimal Frontier to the variation of specific fuel cost. A comparison of the Pareto frontiers for the three optimizations shows that the economic minimum at higher unit costs of fuel is shifted upwards as expected. Similar behavior is observed for sensitivity of Pareto optimal solution to the construction period in Figures 10. In fact the exergetic objective has no sensitivity to the economic parameters such as the fuel cost and construction period (Baghernejad & Yaghoubi, 2011b).

optimization Objective optimization (\$/h) 93179.46 82942.52 -10.98

Table 8. Comparison of the objective functions for optimum and base cases in the system

Optimum case

(MW) 511.02 481.94 -5.69 497.07 -2.73

(MW) 456.39 401.31 -12.07 430.31 -5.71

(\$/h) 48593 45077 -7.23 46908 -3.46

(\$/h) 79255.75 67506.68 -14.82 73454.3 -7.32

(\$/h) 11354.02 12866.14 +13.31 10920.43 -3.82

(%) 43.79 46.80 +6.87 45.19 +3.2

(%) 12.18 15.51 +27.34 12.56 +3.1

Table 9. Comparative results of the optimum and base cases in the system shown in Fig. 3

Additional runs of the optimization algorithm were performed on the system in order to investigate the influence of the unit cost of fuel, the construction period and solar operation period on the solution. Fig. 7 shows sensitivity with respect to fuel cost which is linear. Fig. 8 illustrates variation of unit cost with increasing solar contribution which is significant

Fig. 9 shows the sensitivity of the Pareto optimal Frontier to the variation of specific fuel cost. A comparison of the Pareto frontiers for the three optimizations shows that the economic minimum at higher unit costs of fuel is shifted upwards as expected. Similar behavior is observed for sensitivity of Pareto optimal solution to the construction period in Figures 10. In fact the exergetic objective has no sensitivity to the economic parameters such

case

Single objective Multi objective

Total cost of product (\$/h) 11354.02 10920.43 -3.82 Total exergy efficiency (%) 43.79 45.19 +3.2

> Variation %

Optimum case

Optimum case

Variation (%)

Variation %

optimization Objective functions Base

case

Type of

Single objective

Multi object optimization

shown in Fig. 3

Properties Base

Fuel exergy (solar+gas)

Exergy destruction cost

Capital investment cost

Exergoeconomic factor

**10.4 Sensitivity analysis of the ISCCS**

(Baghernejad & Yaghoubi, 2011a).

**10.4.1 Sensitivity analysis for single objective optimization**

**10.4.2 Sensitivity analyses for multi objective optimization**

as the fuel cost and construction period (Baghernejad & Yaghoubi, 2011b).

Exergy efficiency

Exergy destruction

Fuel cost

The final optimal solution that was selected in this system belongs to the region of Pareto frontier with significant sensitivity to the costing parameters. However, the region with the lower sensitivity to the costing parameter is not reasonable for the final optimum solution. Also a small change in exergetic efficiency of the plant (exergetic efficiency from 47% to the higher value) due to the variation of operating parameters may lead to the danger of increasing the cost rate of product, drastically.

Fig. 7. Sensitivity of unit cost of electricity with specific fuel cost

Fig. 8. Sensitivity of unit cost of electricity to the solar field operation periods

Exergoeconomic Analysis and Optimization of Solar Thermal Power Plants 85

Fig. 10. Sensitivity of Pareto optimum solutions to the construction period

unit costs of electricity produced from combined gas and steam turbines.

base case design and discussed. The analysis of the ISCCS shows that:

The presented chapter demonstrates the basic of exergoeconomic modeling of any thermal power plant and application of the exergoeconomic concept to single and multi objective optimization of an Integrated Solar Combined Cycle System (ISCCS). The exergy-costing method is applied to a 400 MW Integrated Solar Combined Cycle System to estimate the

The application of single objective optimization process shows that exergy and

1. Objective function decreased by about 11% and overall exergoeconomic factor of

2. Unit cost of electricity produced by steam turbine reduced by about 7.1%.

3. Exergy destruction cost reduced by 14.82% and exergetic efficiency of the system

Also, it is found that multi-criteria optimization approach, which is a general form of single objective optimization, enables us to consider various and ever competitive objectives for more improvement of any thermal power plant. An example of decision-making process for selection of the final optimal solution from the Pareto frontier in the multi objective optimization is presented. This final optimum solution requires a process of decisionmaking, which depends on the preferences and criteria of each decision-maker. Each decision maker may select different points as optimum solution which better suits with their desires. The final optimum solution for a typical ISCCS is determined and compared to the

exergoeconomic analysis improved significantly for optimum operation as follows:

This is achieved, however, with 13.3% increase in the capital investment.

**11. Conclusion**

system increased by 27.34 %.

increased from about 43.79 to 46.8%.

Fig. 9. Sensitivity of Pareto optimum solutions to the specific fuel cost

Fig. 10. Sensitivity of Pareto optimum solutions to the construction period