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Soft computing based multi-objective optimization of steam cycle power plant using NSGA-II and ANN
Applied Soft Computing 12 (2012) 3648–3655
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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Soft computing based multi-objective optimization of steam cycle power plant using NSGA-II and ANN Farzaneh Hajabdollahi, Zahra Hajabdollahi, Hassan Hajabdollahi ∗ Department of Mechanical Engineering, Behbahan Branch, Islamic Azad University, Behbahan, Iran
a r t i c l e
i n f o
Article history: Received 30 May 2011 Received in revised form 18 May 2012 Accepted 5 June 2012 Available online 3 July 2012 Keywords: Steam turbine cycle Thermal efficiency Total cost rate NSGA-II Artificial Neural Network
a b s t r a c t In this paper a steam turbine power plant is thermo-economically modeled and optimized. For this purpose, the data for actual running power plant are used for modeling, verifying the results and optimization. Turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure are selected as fifteen design variables. Then, the fast and elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) is applied to maximize the thermal efficiency and minimize the total cost rate (sum of investment cost, fuel cost, and maintenance cost) simultaneously. The results of the optimal design are a set of multiple optimum solutions, called ‘Pareto optimal solutions’. The optimization results in some points show 3.76% increase in efficiency and 3.84% decrease in total cost rate simultaneously, when it compared with the actual data of the running power plant. Finally as a short cut to choose the system optimal design parameters a correlation between two objectives and fifteen decision variables with acceptable precision are presented using Artificial Neural Network (ANN). © 2012 Elsevier B.V. All rights reserved.
1. Introduction The optimization of power generation systems is one of the most important subjects in the energy engineering field. Owing to the high prices of energy and the decreasing fossil fuel resources, makes the optimum design of energy consumption management methods important. Recently, thermo-economic and multi-objective analysis and optimization of thermal systems have become a key solution in providing a better system in optimal energy consumption and optimal system configuration [1–5]. In addition, in the last decade the application research of soft computing such as fuzzy set, Genetic Algorithm and Artificial Neural Network has become one of the most important topics in industrial applications and decision making [6–10]. In particular, in the field of industrial power plant application for energy production including the gas turbine and combined cycle power plant, it has been mainly applied for system identification [6,7]. Rosen and Dincer [11] used thermo-economic analysis of power plants and applied it to a coal fired electricity generating power plant. Ameri et al. applied exergy and Genetic Algorithm tools for thermal modeling and optimal design of steam cycle power plant [12]. Ahmadi and Dincer applied multi objective optimization for a gas turbine and combined cycle power plant [13–15]. They considered the total cost and exergy efficiency or
∗ Corresponding author. Tel.: +98 913 2924318; fax: +98 913 2924318. E-mail address: [email protected] (H. Hajabdollahi). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.06.006
rate of exergy destruction as two objective functions. Sahin and Ali [16] performed an optimal performance analysis of a combined Carnot cycle in a cascade form, including internal irreversibility for steady-state operation. Koch et al. used the Genetic Algorithm to find the optimum design configuration and process variables in a combined cycle power plant [17]. Bracco and Siri optimized a combined cycle power plant with a single pressure HRSG in exergy point of view [18]. Valdes et al. applied Genetic Algorithm to find the optimum thermodynamic parameters of a heat recovery steam generator (HRSG) in a combined cycle power plant [19]. Aljundi performed the energy and exergy analysis of steam power plant to find the points with energy and exergy losses [20]. Ogaji and Singh performed the system identification for a two-shaft aeroderivative gas turbine to specify the fault by means of Artificial Neural Network [6]. Bertini et al. introduced a novel fitness approximation technique coupled with Fuzzy Logic and Genetic Algorithm to optimize the conflicting objectives in a combined cycle power plant [7]. In this paper after thermo-economic modeling of steam cycle power plant (SCPP), this equipment is optimized by maximizing the thermal efficiency as well as minimizing the total cost rate, simultaneously. Turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure are taken as fifteen design parameters and fast and elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) is applied to provide a set of Pareto multiple optimum solutions. After the thermal modeling
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Turbine: Nomenclature a Cin Ctotal Cfuel h cfuel ˙ m Q˙ ˙ W LHV P TTD y r
˙ T,a = W
annualized factor (–) investment cost ($) total cost rate ($/s) fuel cost ($/year) enthalpy (kJ/kg K) price of fuel ($/kg) mass flow rate (kg/s) rate of heat transfer (kW) power (kW) fuel lower heating value (kJ/kg) pressure (kPa) terminal temperature difference (◦ C) depreciation time (year) interest rate (–)
T =
˙ i hi − m
˙ e he m
˙ T,a W h − he,a = i ˙ T,s hi − he,s W
Condenser: Q˙ cond =
˙ i hi − m
3649
(5)
(6)
˙ e he m
(7)
Pump: p =
˙ p,s W (Pe − Pi ) = i ˙ he − hi Wp,a
(8)
Feed water heater:
Greek abbreviation ε total cycle thermal efficiency (–) hours of operation per year efficiency (–) ϕ maintenance factor (–)
˙ i hi = m
˙ e he m
(9)
Terminal temperature difference or TTD is used for the closed feed water heaters and defined as the temperature difference between saturated steam from extraction and feed water leaving the heater [21]. In addition the total thermal efficiency of the cycle is computed from:
Subscripts i inlet o outlet b boiler s isentropic actual a cond condenser p pump turbine T
ε=
˙ net W ˙ fuel .LHV m
(10)
˙ fuel are fuel lower heating value and fuel mass flow where LHV and m ˙ net is output net power estimated as follow: rate respectively and W ˙ net = W
˙ T,a − W
˙ p,a W
(11)
3. Genetic Algorithm for multi-objective optimization and optimization of the SCPP, the Artificial Neural Network is used to find a closed form equation for the optimum design parameters (decision variables) versus the total cost rate and efficiency. This closed form equation enables designers to find the optimum design parameters for the optimum total cost rate and efficiency. 2. Thermal modeling In order to do the thermal modeling, mass and energy balances on the system are required to determine the flow rates and energy transfer rates at the control surface. Appling the first law of thermodynamic in the steady state, one can find the formula for mass and energy balance as follow [21]: Mass balance equation:
˙i= m
˙e m
(1)
Energy balance equation: ˙ = Q˙ − W
˙ e he − m
˙ i hi m
(2)
where subscripts i and e refer to streams entering and leaving the control volume, respectively. The energy balance equations for the various parts of the steam turbine cycle as shown in Fig. 1 are as follow: Boiler: Q˙ b b = ˙ fuel .LHV m
(3)
˙ 27 (h32 − h27 ) + m ˙ 30 (h31 − h30 ) Q˙ b = m
(4)
3.1. Definition of multi-objective optimization A multi-objective optimization problem requires the simultaneous satisfaction of a number of different and often conflicting objectives. It is required to mention that no combination of decision variables can optimize all objectives, 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. Pareto optimality is the key concept to establish a hierarchy among the solutions of a multi-objective optimization problem, in order to determine whether a solution is really one of the best possible trades-off [22]. A multi-objective problem consists of optimizing (i.e., minimizing or maximizing) several objectives simultaneously, with a number of inequality or equality constraints. The problem can be formally written as follows: Find x = (xi ) ∀i = 1, 2, . . . , Nparam such as fi (x) is a minimum (respectively maximum) ∀i = 1, 2, . . . , Nobj Subject to: gj (x) = 0 ∀j = 1, 2, . . . , M,
(12)
hk (x) ≤ 0 ∀k = 1, 2, . . . , K,
(13)
where x is a vector containing the Nparam design parameters, (fi )i=1, ..., Nobj the objective functions and Nobj the number of objectives. The objective function (fi )i=1, ..., Nobj returns a vector containing the set of Nobj values associated with the elementary objectives to be optimized simultaneously. The first multi-objective GA, called vector evaluated GA (or VEGA), was proposed by Schaffer [23]. An algorithm based on non-dominated sorting was proposed
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Fig. 1. Schematic diagram of steam turbine cycle studied in this paper.
by Srinivas and Deb [24] and called non-dominated sorting geneticalgorithm (NSGA). This was later modified by Deb et al. [25] which eliminated higher computational complexity, lack of elitism and the need for specifying the sharing parameter. This algorithm is called NSGA-II which is coupled with the objective functions developed in this study for optimization.
3.4. Controlled elitism sorting To preserve diversity, the influence of elitism is controlled by choosing the number of individuals from each subpopulation, according to the geometric distribution [27]: Sq = S
3.2. Non-dominated sorting and Pareto front As defined by Deb and Goel [26], an individual X(a) is said to constrain-dominate an individual X(b) , if any of the following conditions are true: (1) X(a) and X(b) are feasible, with (a) X(a) is no worse than X(b) in all objective, and (b) X(a) is strictly better than X(b) in at least one objective. (2) X(a) is feasible while individual X(b) is not. (3) X(a) and X(b) are both infeasible, but X(a) has a smaller constraint violation. Here, the constraint violation (X) of an individual X is defined to be equal to the sum of the violated constraint function values [27]
(X) =
B
(gj (X))gj (X),
(14)
1 − c q−1 c , 1 − cw
(15)
To form a parent search population, Pt+1 (t denote the generation), of size S, where 0 < c < 1 and w is the total number of ranked non-dominated. 3.5. Crowding distance The crowding distance metric proposed by Deb and Goel [26] is utilized, where the crowding distance of an individual is the perimeter of the rectangle with its nearest neighbors at diagonally opposite corners. So, if individual X(a) and individual X(b) have same rank, each one has a larger crowding distance is better. 3.6. Crossover and mutation Uniform crossover and random uniform mutation are employed to obtain the offspring population, Qt+1 .The integer-based uniform crossover operator takes two distinct parent individuals and interchanges each corresponding binary bits with a probability, 0 < pc ≤ 1. Following crossover, the mutation operator changes each of the binary bits with a mutation probability, 0 < pm < 0.5.
j=1
3.7. Historical archive where is the Heaviside step function. A set of non-dominated individuals is used to form a Pareto-optimal fronts.
The NSGA-II algorithm has been modified to include an archive of the historically non-dominated individuals, Ht . Archive is used to update the data at each iteration.
3.3. Tournament selection 4. Artificial Neural Network Each individual competes in exactly two tournaments with randomly selected individuals, a procedure which imitates survival of the fittest in nature.
The feed-forward neural networks are the most popular architectures due to their structural flexibility, good representational
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Table 1 Purchase equipment cost functions [12]. Components
Steam turbine
Investment cost
CST =
0.7 a51 .PST
1+
a51 = 3880.5 $k W
0.05 1 − ST
3 ×
1 + 5 exp
Pe − P¯ e a3
,
−1
10.42 K
−0.7
a
˙ Boiler ) 2 p T SH/RSH CBoiler = a1 (m P = exp
Ta − 866 K−1
,
= 1 +
1 − ¯ 1 1 − 1
a4
,
T = 1 + a5 exp
Te − T¯ e a6
˙ RSH TeRSH − TiRSH Te − TiSH m + . Te TeRSH ˙ Boiler m −1 ◦ T¯ e = 593 C, P¯ e = 28 bar, ¯ 1 = 0.9, a1 = 208582 $kg s−1
Boiler
SH/RSH = 1 +
a2 = 0.8, Condenser
CCond
a3 = 150 bar,
a4 = 7,
a5 = 5,
a6 = 10.42 ◦ C
Q˙ Cond ˙ CW + 70.5.Q˙ Cond × (−0.6936.Ln(T¯ CW − Tb ) + 2.1898) = a61 . + a62 .m k. Tin −1
a61 = 280.74 $ m−2 , a2
a62 = 746 $kg
Deaerator
CDA = a1 (mwater ) ,
Pump
0.71 CPump = a71 .PPump 1+
s−1 ,
a1 = 145315 $k W 0.2 1−Pump
,
−0.7
,
k = 2200 Wm−2 K−1 a2 = 0.7 −1
a71 = 705.48 $kg
capabilities and availability of a large number of training algorithms [28]. Here the individual element inputs are I1 , I2 , . . . , IR multiplied by weights w11 , w12 , . . . , w1R and the weighted values are fed to the summing junction. The neuron has a bias b, which is summed with the weighted inputs to form the net input n. This sum, n, is the argument of the transfer function F:
s−1
6. Discussion and results 6.1. Model verification
In this study, the total cycle efficiency and total cost rate are considered as two objective functions. The efficiency is defined in Eq. (10) and the total cost rate is computed from:
To have a good verification results from the developed code, the results in this study are compared with the Montazer Ghaem steam cycle power plant that is schematically shown in Fig. 1. This power plant is located near the Tehran city, the capital of Iran. For this purpose, 3% pressure drop are considered in each side of the feed water heaters, 1% pressure drop from turbine outlet to the inlet of heater, 5% pressure drop in boiler, 0.7 for pumps isentropic efficiency, TTD = +3 degree for LP feed water heaters and TTD = −3 degree for HP feed water heaters. The results of our simulation and the actual power plant data with the above assumption along with the same input values listed in Table 5, are shown in Table 2. Results show that the difference percentage points of two mentioned modeling output results are acceptable.
Ctotal = (a.ϕ.Cin + Cfuel ) ×
6.2. Optimization
a = F(n) = F(w11 I1 + w12 I2 + · · · + w1R IR + b)
(16)
This network consists of neurons arranged in layers in which every neuron is connected to all neurons of the next layer. 5. Objective functions, design parameters and constraints
(17)
where ϕ = 1.06 is considered for the maintenance factor, Cin is the purchase cost of kth component in US dollar listed in Table 1 and a is the annual cost coefficient defined as: a=
r 1 − (1 + r)−y
(18)
where r and y are the interest rate and depreciation time respectively. Cfuel and in Eq. (17) are fuel cost in a year and a coefficient for converting the total annual cost into the cost per time unit defined as below: ˙ f × × 3600 Cfuel = cfuel × m
(19)
= ( × 3600)−1
(20)
where N is the annual number of the operating hours of the unit and cfuel is the fuel unit cost. To maximize the efficiency value and to minimize the total cost rate, fifteen design parameters including the turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure are selected. The constrains are introduced to insure that the temperature and steam quality in point 7 (T7 and x7 ) are selected higher than the 40 ◦ C and 0.95 respectively.
The steam cycle optimum design parameters are obtained for actual running power plant described in the previous section. The power station should deliver 160 MW output net power. The Efficiency and total cost rate are considered as two objective functions. Design parameters (decision variables) and the range of their variations are listed in Table 3. The number of iterations for finding the global extremum in the whole searching domain is about 2.9 × 1034 . System is optimized for depreciation time y = 20 years, interest rate r = 0.1 and 0.1 $/kg as the fuel cost. The Genetic Algorithm Optimization is performed for 700 generations, using a
Table 2 The comparison of modeling output and the corresponding results from actual running power plant. Output variables
Actual
Present paper
Difference (%)
Qb (MW) Qc (MW) WHPT (MW) WIPT (MW) WLPT (MW) WNet (MW) WP1 (kW) WP2 (kW) WP3 (kW) ε
376.93 222.21 44.167 65.405 49.158 155.74 87.182 25.64 2877.81 0.3718
372.16 218.03 46.427 64.131 48.370 156.04 98.5302 22.1437 2769.2 0.3773
1.265 1.88 5.12 6.53 1.95 0.192 13.01 13.63 3.77 1.48
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Table 3 The design parameters and their range of variation. Variables
From
To
T32 (C) P1 (MPa) P3 (MPa) P5 (MPa) P7 (MPa) P8 (MPa) P10 (MPa) P28 (MPa) P32 (MPa) P1 P2 P3 HPT IPT LPT
500 1 0.4 0.3 0.005 0.05 0.1 2 8 0.5 0.5 0.5 0.7 0.7 0.7
560 2 1 0.4 0.05 0.1 0.3 4 18 0.8 0.8 0.8 0.9 0.9 0.9
Fig. 3. The distribution of Pareto-optimal points solutions using NSGA-II.
Optimum values of two objectives for five typical points from A to E (Pareto-optimal fronts) along with the actual data from the running power plant are listed in Table 4. The optimization results in design point E show 3.76% increase in efficiency and 3.84% decrease in total cost rate when it compared with the actual data of the running power plant which is a noticeable improvement. It is clear that the design point E is dominant over the actual data of the running power plant in both efficiency and total cost rate. To provide a useful tool for the optimal design of the steam turbine power plant, the following equation for efficiency versus the total cost rate is curved for the Pareto curve (Fig. 3). Ctotal ($/s) =
4.007ε2 − 42.67ε + 18.12 ε2 − 37.78ε + 16.12
(21)
Eq. (21) is valid in the range of 0.3915 < ε < 0.4177 for efficiency. The interesting point in Eq. (21) is that considering a numerical value for the efficiency in mentioned range, provides the minimum total cost rate for that optimal point. Optimum values of fifteen design parameters for five typical points from A to E (Pareto-optimal fronts) are listed in Table 5. It results show that the all turbines isotropic efficiency should be located at their maximum range to makes the system thermoeconomically optimized. To have a better insight in some important optimum design parameters such as boiler exhaust steam pressure, HP turbine inlet temperature (TIT) and reheat pressure, the distribution of efficiency versus the mentioned parameters for optimum points in Pareto curve are shown in Figs. 4–6 respectively. It can be concluded that the efficiency increases by increase of boiler pressure, TIT and reheat pressure in the optimum selected range with the fixed net power.
Fig. 2. Results of all evaluations during 700 generations using NSGA-II. A clear approximation of the Pareto front is visible on the lower part of the figure.
search population size of M = 150 individuals, crossover probability of pc = 0.9, gene mutation probability of pm = 0.035 and controlled elitism value c = 0.55. The results of optimum efficiency and total cost rate for all points evaluated over 700 generations are depicted in Fig. 2. The Pareto-optimal curve (best rank) is clearly visible in the lower part of the figure which is separately shown in Fig. 3. The Pareto optimum results clearly reveal the conflict between two objectives, the efficiency and the total cost rate. Any change that increases the efficiency, leads to an increase in the total cost rate and vice versa. This shows the need for multi-objective optimization technique in optimal design of steam cycle power plant. It is shown in Fig. 3, which the maximum efficiency exists at design point A (0.4177), while the total cost rate is the biggest at this point. On the other hand the minimum total cost rate occurs at design point E (1.376 $/s), with a smallest efficiency value (0.3915) at that point. Design point A is the optimal situation at which, efficiency is a single objective function, while design point E is the optimum condition at which the total cost rate is a single objective function.
6.3. Selection of final optimum design Each point on the Pareto frontier has the potential of final optimum design. However, selection of a single optimum point from existing points on the Pareto front needs a process of decision-making. In fact, this process is mostly carried out based
Table 4 The optimum values of efficiency and total cost rate for the design points A–E in Pareto- optimal fronts.
Efficiency Total cost rate ($/s)
A
B
C
D
E
Actual data
0.4177 1.954
0.4137 1.749
0.4086 1.597
0.4027 1.489
0.3915 1.376
0.3773 1.431
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Table 5 The optimum values of design parameters for optimum selected points A–E in Pareto optimum front. Variables
A
B
C
D
E
Actual data
T32 (C) P1 (MPa) P3 (MPa) P5 (MPa) P7 (MPa) P8 (MPa) P10 (MPa) P28 (MPa) P32 (MPa) P1 P2 P3 HPT IPT LPT
559.9635 1.0479 0.5514 0.3530 0.0078 0.0521 0.3000 2.2177 17.7773 0.6475 0.6481 0.7982 0.8996 0.8994 0.9000
557.2884 1.0108 0.5502 0.3510 0.0078 0.0504 0.2996 2.1892 14.8543 0.6834 0.7870 0.7979 0.8994 0.8994 0.9000
550.1593 1.0055 0.5201 0.3413 0.0078 0.0509 0.2999 2.1129 12.7300 0.7975 0.5755 0.7967 0.8994 0.8997 0.9000
548.1850 1.0653 0.5188 0.3434 0.0078 0.0504 0.3000 2.1352 10.5678 0.7346 0.6235 0.7965 0.8997 0.8956 0.9000
545.7967 1.0201 0.5049 0.3301 0.0078 0.0507 0.2999 2.0886 8.0183 0.7125 0.6213 0.7795 0.8992 0.8738 0.9000
537.7778 1.4989 0.8191 0.4216 0.00847 0.0778 0.2764 3.1336 12.5110 0.7 0.7 0.7 0.82 0.86 0.8
In the LINMAP method, each objective is nondimensionalized using the following relation: Fijn =
Fij
2
(22)
m (F )2 i=1 ij
where i is the index for each point on the Pareto frontier, j is the index for each objective in the objectives space and m denotes the number of points in the Pareto front. After nondimensionalization of two objectives, the distance of each solution on the Pareto frontier from the ideal point obtained. The closest point of Pareto frontier to the equilibrium point (the design point D) might be considered as a desirable final solution with the 0.4027 thermal efficiency and 1.489 $/s total cost rate along with its optimum design parameters listed in Table 5. Fig. 4. Distribution of thermal efficiency versus boiler exhaust steam pressure for optimum points in Pareto optimum front in Fig. 3.
on engineering experiences and importance of each objective for decision makers. The process of final decision-making in Fig. 3 is usually performed with the aid of a hypothetical point named as equilibrium point, that both objectives have their optimal values independent of the other objectives [29]. It is clear that it is impossible to have both objectives at their optimum point, simultaneously. The equilibrium point is not a solution located on the Pareto frontier. In this paper, LINMAP method was used to find the final optimum solution in Pareto front [29].
Fig. 5. Distribution of thermal efficiency versus turbine inlet temperature (TIT) for optimum points in Pareto optimum front in Fig. 3.
6.4. Correlations between the objective functions and design variables The main goal of this section is to estimate a correlation between two optimum objective functions and fifteen decision variables for a specified case study. Actually the designer can find the minimum of total cost rate for any choosing efficiency in the range of 0.3915 < ε < 0.4177 using Eq. (21). Using these two optimum values (total coat rate and efficiency) and this correlation, designers can find the corresponding values of fifteen optimum design parameters. For this purpose, the Artificial Neural Network by means of a hidden layer with two neurons for two input parameters (efficiency
Fig. 6. Distribution of thermal efficiency versus reheat pressure for optimum points in Pareto optimum front in Fig. 3.
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Table 6 The values of constants in Eq. (22) for the estimated optimum design parameters in the presented case study.
T32 (C) P1 (MPa) P3 (MPa) P5 (MPa) P7 (MPa) P8 (MPa) P10 (MPa) P28 (MPa) P32 (MPa) P1 P2 P3 HPT IPT LPT
b1
b2
b3
b4
b5
w1
w2
w3
w4 × 103
−2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045
157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77
93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286
−1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558
882.270 0.808 .359 0.299 0.00775 0.0426 0.300 6.597 214.015 1.356 0.342 0.895 0.936 0.910 0.904
−65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027
−337.287 0.2226 0.154 0.0010777 0.00006784 0.00875 0.00065 −5.562 −204.982 −0.6535 0.3133 −0.1023 − 0.0388 0.01123 −0.00492
505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5
822.28 −10.991 −1.4298 −0.023943 −0.0012728 −0.14148 0.1233 9.9576 −1111.2 32.139 −17.67 0.4927 0.72803 0.15629 0.11332
Fig. 7. The schematic diagram of neural network with two input neurons and fifteen output neurons.
and total cost rate) and an output layer with fifteen neurons for fifteen output parameters (decision variables) were applied using feed forward algorithm (Fig. 7). Tangent-sigmoid transfer function (−1 + 2/(1 + e−2n )) was used at the input layer with n as the transfer function input. Furthermore
Fig. 9. Comparison of optimization results and predicted results by ANN for total cost rate.
the linear transfer function (n) was applied at the output layout. A close form equation for decision variables versus objective functions (efficiency and total cost) was derived by training the network as follows: Decision variables =
b1 1 + e(w1 ε+w2 Ctotal +b2 ) +
b3 1 + e(w3 ε+w4 Ctotal +b4 )
+ b5
(22’)
where b is the bias value and w is weighting function shown in Table 6, for the each decision variables. The corresponding errors for the estimated efficiency and the total cost rate (two objective functions) are shown in Figs. 8 and 9. Results show that with applying the above proposed correlations (Eq. (22)), the estimated total cost rate and efficiency, are accurate within −0.8% to +0.3% and −2% to 1% respectively, which are acceptable for engineering problems. 7. Conclusions
Fig. 8. Comparison of optimization results and predicted results by ANN for efficiency.
A steam cycle power plant was optimally designed using the fast and elitist Non-dominated Sorting Genetic Algorithm (NSGAII) technique. The design parameters (decision variables) were the turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure. In this presented optimization
F. Hajabdollahi et al. / Applied Soft Computing 12 (2012) 3648–3655
problem, the efficiency and total cost rate were considered as two objective functions (the efficiency was maximized and total cost rate was minimized). A set of Pareto optimal front points were extracted. The results revealed the level of conflict between the two objectives. The optimization results in design point E show 3.76% increase in efficiency and 3.84% decrease in total cost rate simultaneously, compared with the actual data of the running power plant. It was also observed that the efficiency increases by the increase of boiler pressure, TIT and reheat pressure in the optimum selected range with the fixed net power. Furthermore a correlation between the optimal values of two objective functions was proposed. Finally a correlation between two optimum objective functions and fifteen decision variables for the presented case study were proposed with acceptable accuracy using Artificial Neural Network. This correlation enables designers to find the corresponding fifteen optimum design parameters for minimum total cost rate and maximum efficiency. References [1] H. Hajabdollahi, P. Ahmadi, I. Dincer, An exergy-based multi objective optimization of a heat recovery steam generator (HRSG) in a combined cycle power plant (CCPP) using evolutionary algorithm, International Journal of Green Energy 8 (2011) 44–64. [2] S. Sanaye, H. Hajabdollahi, Multi-objective optimization of rotary regenerator using genetic algorithm, International Journal of Thermal Sciences 48 (2009) 1967–1977. [3] S. Sanaye, H. Hajabdollahi, Thermal-economic multi-objective optimization of plate fin heat exchanger using genetic algorithm, Applied Energy 87 (2010) 1893–1902. [4] S. Sanaye, H. Hajabdollahi, Multi-objective optimization of shell and tube heat exchangers, Applied Thermal Engineering 30 (2010) 1937–1945. [5] P. Ahmadi, H. Hajabdollahi, I. Dincer, Cost and entropy generation minimization of a cross flow Plate-Fin Heat Exchanger (PFHE) using multi-objective genetic algorithm, Journal of Heat Transfer – Transactions of the ASME 133 (2011) 021801. [6] S.O.T. Ogaji, R. Singh, Advanced engine diagnostics using artificial neural networks, Applied Soft Computing 3 (2003) 259–271. [7] I. Bertini, M.D. Felice, A. Pannicelli, S. Pizzuti, Soft computing based optimization of combined cycled power plant start-up operation with fitness approximation methods, Applied Soft Computing 11 (2011) 4110–4116. [8] T. Ganesan, P. Vasant, I. Elamvazuthi, Optimization of nonlinear geological structure mapping using hybrid neuro-genetic techniques, Mathematical and Computer Modelling 54 (2011) 2913–2922. ˜ D. Peidro, P. Vasant, Vendor selection problem by using [9] M. Díaz-Madronero, an interactive fuzzy multi-objective approach with modified S-curve membership functions, Computers and Mathematics with Applications 60 (2010) 1038–1048.
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Contents lists available at SciVerse ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Soft computing based multi-objective optimization of steam cycle power plant using NSGA-II and ANN Farzaneh Hajabdollahi, Zahra Hajabdollahi, Hassan Hajabdollahi ∗ Department of Mechanical Engineering, Behbahan Branch, Islamic Azad University, Behbahan, Iran
a r t i c l e
i n f o
Article history: Received 30 May 2011 Received in revised form 18 May 2012 Accepted 5 June 2012 Available online 3 July 2012 Keywords: Steam turbine cycle Thermal efficiency Total cost rate NSGA-II Artificial Neural Network
a b s t r a c t In this paper a steam turbine power plant is thermo-economically modeled and optimized. For this purpose, the data for actual running power plant are used for modeling, verifying the results and optimization. Turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure are selected as fifteen design variables. Then, the fast and elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) is applied to maximize the thermal efficiency and minimize the total cost rate (sum of investment cost, fuel cost, and maintenance cost) simultaneously. The results of the optimal design are a set of multiple optimum solutions, called ‘Pareto optimal solutions’. The optimization results in some points show 3.76% increase in efficiency and 3.84% decrease in total cost rate simultaneously, when it compared with the actual data of the running power plant. Finally as a short cut to choose the system optimal design parameters a correlation between two objectives and fifteen decision variables with acceptable precision are presented using Artificial Neural Network (ANN). © 2012 Elsevier B.V. All rights reserved.
1. Introduction The optimization of power generation systems is one of the most important subjects in the energy engineering field. Owing to the high prices of energy and the decreasing fossil fuel resources, makes the optimum design of energy consumption management methods important. Recently, thermo-economic and multi-objective analysis and optimization of thermal systems have become a key solution in providing a better system in optimal energy consumption and optimal system configuration [1–5]. In addition, in the last decade the application research of soft computing such as fuzzy set, Genetic Algorithm and Artificial Neural Network has become one of the most important topics in industrial applications and decision making [6–10]. In particular, in the field of industrial power plant application for energy production including the gas turbine and combined cycle power plant, it has been mainly applied for system identification [6,7]. Rosen and Dincer [11] used thermo-economic analysis of power plants and applied it to a coal fired electricity generating power plant. Ameri et al. applied exergy and Genetic Algorithm tools for thermal modeling and optimal design of steam cycle power plant [12]. Ahmadi and Dincer applied multi objective optimization for a gas turbine and combined cycle power plant [13–15]. They considered the total cost and exergy efficiency or
∗ Corresponding author. Tel.: +98 913 2924318; fax: +98 913 2924318. E-mail address: [email protected] (H. Hajabdollahi). 1568-4946/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.asoc.2012.06.006
rate of exergy destruction as two objective functions. Sahin and Ali [16] performed an optimal performance analysis of a combined Carnot cycle in a cascade form, including internal irreversibility for steady-state operation. Koch et al. used the Genetic Algorithm to find the optimum design configuration and process variables in a combined cycle power plant [17]. Bracco and Siri optimized a combined cycle power plant with a single pressure HRSG in exergy point of view [18]. Valdes et al. applied Genetic Algorithm to find the optimum thermodynamic parameters of a heat recovery steam generator (HRSG) in a combined cycle power plant [19]. Aljundi performed the energy and exergy analysis of steam power plant to find the points with energy and exergy losses [20]. Ogaji and Singh performed the system identification for a two-shaft aeroderivative gas turbine to specify the fault by means of Artificial Neural Network [6]. Bertini et al. introduced a novel fitness approximation technique coupled with Fuzzy Logic and Genetic Algorithm to optimize the conflicting objectives in a combined cycle power plant [7]. In this paper after thermo-economic modeling of steam cycle power plant (SCPP), this equipment is optimized by maximizing the thermal efficiency as well as minimizing the total cost rate, simultaneously. Turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure are taken as fifteen design parameters and fast and elitist Non-dominated Sorting Genetic Algorithm (NSGA-II) is applied to provide a set of Pareto multiple optimum solutions. After the thermal modeling
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Turbine: Nomenclature a Cin Ctotal Cfuel h cfuel ˙ m Q˙ ˙ W LHV P TTD y r
˙ T,a = W
annualized factor (–) investment cost ($) total cost rate ($/s) fuel cost ($/year) enthalpy (kJ/kg K) price of fuel ($/kg) mass flow rate (kg/s) rate of heat transfer (kW) power (kW) fuel lower heating value (kJ/kg) pressure (kPa) terminal temperature difference (◦ C) depreciation time (year) interest rate (–)
T =
˙ i hi − m
˙ e he m
˙ T,a W h − he,a = i ˙ T,s hi − he,s W
Condenser: Q˙ cond =
˙ i hi − m
3649
(5)
(6)
˙ e he m
(7)
Pump: p =
˙ p,s W (Pe − Pi ) = i ˙ he − hi Wp,a
(8)
Feed water heater:
Greek abbreviation ε total cycle thermal efficiency (–) hours of operation per year efficiency (–) ϕ maintenance factor (–)
˙ i hi = m
˙ e he m
(9)
Terminal temperature difference or TTD is used for the closed feed water heaters and defined as the temperature difference between saturated steam from extraction and feed water leaving the heater [21]. In addition the total thermal efficiency of the cycle is computed from:
Subscripts i inlet o outlet b boiler s isentropic actual a cond condenser p pump turbine T
ε=
˙ net W ˙ fuel .LHV m
(10)
˙ fuel are fuel lower heating value and fuel mass flow where LHV and m ˙ net is output net power estimated as follow: rate respectively and W ˙ net = W
˙ T,a − W
˙ p,a W
(11)
3. Genetic Algorithm for multi-objective optimization and optimization of the SCPP, the Artificial Neural Network is used to find a closed form equation for the optimum design parameters (decision variables) versus the total cost rate and efficiency. This closed form equation enables designers to find the optimum design parameters for the optimum total cost rate and efficiency. 2. Thermal modeling In order to do the thermal modeling, mass and energy balances on the system are required to determine the flow rates and energy transfer rates at the control surface. Appling the first law of thermodynamic in the steady state, one can find the formula for mass and energy balance as follow [21]: Mass balance equation:
˙i= m
˙e m
(1)
Energy balance equation: ˙ = Q˙ − W
˙ e he − m
˙ i hi m
(2)
where subscripts i and e refer to streams entering and leaving the control volume, respectively. The energy balance equations for the various parts of the steam turbine cycle as shown in Fig. 1 are as follow: Boiler: Q˙ b b = ˙ fuel .LHV m
(3)
˙ 27 (h32 − h27 ) + m ˙ 30 (h31 − h30 ) Q˙ b = m
(4)
3.1. Definition of multi-objective optimization A multi-objective optimization problem requires the simultaneous satisfaction of a number of different and often conflicting objectives. It is required to mention that no combination of decision variables can optimize all objectives, 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. Pareto optimality is the key concept to establish a hierarchy among the solutions of a multi-objective optimization problem, in order to determine whether a solution is really one of the best possible trades-off [22]. A multi-objective problem consists of optimizing (i.e., minimizing or maximizing) several objectives simultaneously, with a number of inequality or equality constraints. The problem can be formally written as follows: Find x = (xi ) ∀i = 1, 2, . . . , Nparam such as fi (x) is a minimum (respectively maximum) ∀i = 1, 2, . . . , Nobj Subject to: gj (x) = 0 ∀j = 1, 2, . . . , M,
(12)
hk (x) ≤ 0 ∀k = 1, 2, . . . , K,
(13)
where x is a vector containing the Nparam design parameters, (fi )i=1, ..., Nobj the objective functions and Nobj the number of objectives. The objective function (fi )i=1, ..., Nobj returns a vector containing the set of Nobj values associated with the elementary objectives to be optimized simultaneously. The first multi-objective GA, called vector evaluated GA (or VEGA), was proposed by Schaffer [23]. An algorithm based on non-dominated sorting was proposed
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Fig. 1. Schematic diagram of steam turbine cycle studied in this paper.
by Srinivas and Deb [24] and called non-dominated sorting geneticalgorithm (NSGA). This was later modified by Deb et al. [25] which eliminated higher computational complexity, lack of elitism and the need for specifying the sharing parameter. This algorithm is called NSGA-II which is coupled with the objective functions developed in this study for optimization.
3.4. Controlled elitism sorting To preserve diversity, the influence of elitism is controlled by choosing the number of individuals from each subpopulation, according to the geometric distribution [27]: Sq = S
3.2. Non-dominated sorting and Pareto front As defined by Deb and Goel [26], an individual X(a) is said to constrain-dominate an individual X(b) , if any of the following conditions are true: (1) X(a) and X(b) are feasible, with (a) X(a) is no worse than X(b) in all objective, and (b) X(a) is strictly better than X(b) in at least one objective. (2) X(a) is feasible while individual X(b) is not. (3) X(a) and X(b) are both infeasible, but X(a) has a smaller constraint violation. Here, the constraint violation (X) of an individual X is defined to be equal to the sum of the violated constraint function values [27]
(X) =
B
(gj (X))gj (X),
(14)
1 − c q−1 c , 1 − cw
(15)
To form a parent search population, Pt+1 (t denote the generation), of size S, where 0 < c < 1 and w is the total number of ranked non-dominated. 3.5. Crowding distance The crowding distance metric proposed by Deb and Goel [26] is utilized, where the crowding distance of an individual is the perimeter of the rectangle with its nearest neighbors at diagonally opposite corners. So, if individual X(a) and individual X(b) have same rank, each one has a larger crowding distance is better. 3.6. Crossover and mutation Uniform crossover and random uniform mutation are employed to obtain the offspring population, Qt+1 .The integer-based uniform crossover operator takes two distinct parent individuals and interchanges each corresponding binary bits with a probability, 0 < pc ≤ 1. Following crossover, the mutation operator changes each of the binary bits with a mutation probability, 0 < pm < 0.5.
j=1
3.7. Historical archive where is the Heaviside step function. A set of non-dominated individuals is used to form a Pareto-optimal fronts.
The NSGA-II algorithm has been modified to include an archive of the historically non-dominated individuals, Ht . Archive is used to update the data at each iteration.
3.3. Tournament selection 4. Artificial Neural Network Each individual competes in exactly two tournaments with randomly selected individuals, a procedure which imitates survival of the fittest in nature.
The feed-forward neural networks are the most popular architectures due to their structural flexibility, good representational
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Table 1 Purchase equipment cost functions [12]. Components
Steam turbine
Investment cost
CST =
0.7 a51 .PST
1+
a51 = 3880.5 $k W
0.05 1 − ST
3 ×
1 + 5 exp
Pe − P¯ e a3
,
−1
10.42 K
−0.7
a
˙ Boiler ) 2 p T SH/RSH CBoiler = a1 (m P = exp
Ta − 866 K−1
,
= 1 +
1 − ¯ 1 1 − 1
a4
,
T = 1 + a5 exp
Te − T¯ e a6
˙ RSH TeRSH − TiRSH Te − TiSH m + . Te TeRSH ˙ Boiler m −1 ◦ T¯ e = 593 C, P¯ e = 28 bar, ¯ 1 = 0.9, a1 = 208582 $kg s−1
Boiler
SH/RSH = 1 +
a2 = 0.8, Condenser
CCond
a3 = 150 bar,
a4 = 7,
a5 = 5,
a6 = 10.42 ◦ C
Q˙ Cond ˙ CW + 70.5.Q˙ Cond × (−0.6936.Ln(T¯ CW − Tb ) + 2.1898) = a61 . + a62 .m k. Tin −1
a61 = 280.74 $ m−2 , a2
a62 = 746 $kg
Deaerator
CDA = a1 (mwater ) ,
Pump
0.71 CPump = a71 .PPump 1+
s−1 ,
a1 = 145315 $k W 0.2 1−Pump
,
−0.7
,
k = 2200 Wm−2 K−1 a2 = 0.7 −1
a71 = 705.48 $kg
capabilities and availability of a large number of training algorithms [28]. Here the individual element inputs are I1 , I2 , . . . , IR multiplied by weights w11 , w12 , . . . , w1R and the weighted values are fed to the summing junction. The neuron has a bias b, which is summed with the weighted inputs to form the net input n. This sum, n, is the argument of the transfer function F:
s−1
6. Discussion and results 6.1. Model verification
In this study, the total cycle efficiency and total cost rate are considered as two objective functions. The efficiency is defined in Eq. (10) and the total cost rate is computed from:
To have a good verification results from the developed code, the results in this study are compared with the Montazer Ghaem steam cycle power plant that is schematically shown in Fig. 1. This power plant is located near the Tehran city, the capital of Iran. For this purpose, 3% pressure drop are considered in each side of the feed water heaters, 1% pressure drop from turbine outlet to the inlet of heater, 5% pressure drop in boiler, 0.7 for pumps isentropic efficiency, TTD = +3 degree for LP feed water heaters and TTD = −3 degree for HP feed water heaters. The results of our simulation and the actual power plant data with the above assumption along with the same input values listed in Table 5, are shown in Table 2. Results show that the difference percentage points of two mentioned modeling output results are acceptable.
Ctotal = (a.ϕ.Cin + Cfuel ) ×
6.2. Optimization
a = F(n) = F(w11 I1 + w12 I2 + · · · + w1R IR + b)
(16)
This network consists of neurons arranged in layers in which every neuron is connected to all neurons of the next layer. 5. Objective functions, design parameters and constraints
(17)
where ϕ = 1.06 is considered for the maintenance factor, Cin is the purchase cost of kth component in US dollar listed in Table 1 and a is the annual cost coefficient defined as: a=
r 1 − (1 + r)−y
(18)
where r and y are the interest rate and depreciation time respectively. Cfuel and in Eq. (17) are fuel cost in a year and a coefficient for converting the total annual cost into the cost per time unit defined as below: ˙ f × × 3600 Cfuel = cfuel × m
(19)
= ( × 3600)−1
(20)
where N is the annual number of the operating hours of the unit and cfuel is the fuel unit cost. To maximize the efficiency value and to minimize the total cost rate, fifteen design parameters including the turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure are selected. The constrains are introduced to insure that the temperature and steam quality in point 7 (T7 and x7 ) are selected higher than the 40 ◦ C and 0.95 respectively.
The steam cycle optimum design parameters are obtained for actual running power plant described in the previous section. The power station should deliver 160 MW output net power. The Efficiency and total cost rate are considered as two objective functions. Design parameters (decision variables) and the range of their variations are listed in Table 3. The number of iterations for finding the global extremum in the whole searching domain is about 2.9 × 1034 . System is optimized for depreciation time y = 20 years, interest rate r = 0.1 and 0.1 $/kg as the fuel cost. The Genetic Algorithm Optimization is performed for 700 generations, using a
Table 2 The comparison of modeling output and the corresponding results from actual running power plant. Output variables
Actual
Present paper
Difference (%)
Qb (MW) Qc (MW) WHPT (MW) WIPT (MW) WLPT (MW) WNet (MW) WP1 (kW) WP2 (kW) WP3 (kW) ε
376.93 222.21 44.167 65.405 49.158 155.74 87.182 25.64 2877.81 0.3718
372.16 218.03 46.427 64.131 48.370 156.04 98.5302 22.1437 2769.2 0.3773
1.265 1.88 5.12 6.53 1.95 0.192 13.01 13.63 3.77 1.48
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Table 3 The design parameters and their range of variation. Variables
From
To
T32 (C) P1 (MPa) P3 (MPa) P5 (MPa) P7 (MPa) P8 (MPa) P10 (MPa) P28 (MPa) P32 (MPa) P1 P2 P3 HPT IPT LPT
500 1 0.4 0.3 0.005 0.05 0.1 2 8 0.5 0.5 0.5 0.7 0.7 0.7
560 2 1 0.4 0.05 0.1 0.3 4 18 0.8 0.8 0.8 0.9 0.9 0.9
Fig. 3. The distribution of Pareto-optimal points solutions using NSGA-II.
Optimum values of two objectives for five typical points from A to E (Pareto-optimal fronts) along with the actual data from the running power plant are listed in Table 4. The optimization results in design point E show 3.76% increase in efficiency and 3.84% decrease in total cost rate when it compared with the actual data of the running power plant which is a noticeable improvement. It is clear that the design point E is dominant over the actual data of the running power plant in both efficiency and total cost rate. To provide a useful tool for the optimal design of the steam turbine power plant, the following equation for efficiency versus the total cost rate is curved for the Pareto curve (Fig. 3). Ctotal ($/s) =
4.007ε2 − 42.67ε + 18.12 ε2 − 37.78ε + 16.12
(21)
Eq. (21) is valid in the range of 0.3915 < ε < 0.4177 for efficiency. The interesting point in Eq. (21) is that considering a numerical value for the efficiency in mentioned range, provides the minimum total cost rate for that optimal point. Optimum values of fifteen design parameters for five typical points from A to E (Pareto-optimal fronts) are listed in Table 5. It results show that the all turbines isotropic efficiency should be located at their maximum range to makes the system thermoeconomically optimized. To have a better insight in some important optimum design parameters such as boiler exhaust steam pressure, HP turbine inlet temperature (TIT) and reheat pressure, the distribution of efficiency versus the mentioned parameters for optimum points in Pareto curve are shown in Figs. 4–6 respectively. It can be concluded that the efficiency increases by increase of boiler pressure, TIT and reheat pressure in the optimum selected range with the fixed net power.
Fig. 2. Results of all evaluations during 700 generations using NSGA-II. A clear approximation of the Pareto front is visible on the lower part of the figure.
search population size of M = 150 individuals, crossover probability of pc = 0.9, gene mutation probability of pm = 0.035 and controlled elitism value c = 0.55. The results of optimum efficiency and total cost rate for all points evaluated over 700 generations are depicted in Fig. 2. The Pareto-optimal curve (best rank) is clearly visible in the lower part of the figure which is separately shown in Fig. 3. The Pareto optimum results clearly reveal the conflict between two objectives, the efficiency and the total cost rate. Any change that increases the efficiency, leads to an increase in the total cost rate and vice versa. This shows the need for multi-objective optimization technique in optimal design of steam cycle power plant. It is shown in Fig. 3, which the maximum efficiency exists at design point A (0.4177), while the total cost rate is the biggest at this point. On the other hand the minimum total cost rate occurs at design point E (1.376 $/s), with a smallest efficiency value (0.3915) at that point. Design point A is the optimal situation at which, efficiency is a single objective function, while design point E is the optimum condition at which the total cost rate is a single objective function.
6.3. Selection of final optimum design Each point on the Pareto frontier has the potential of final optimum design. However, selection of a single optimum point from existing points on the Pareto front needs a process of decision-making. In fact, this process is mostly carried out based
Table 4 The optimum values of efficiency and total cost rate for the design points A–E in Pareto- optimal fronts.
Efficiency Total cost rate ($/s)
A
B
C
D
E
Actual data
0.4177 1.954
0.4137 1.749
0.4086 1.597
0.4027 1.489
0.3915 1.376
0.3773 1.431
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Table 5 The optimum values of design parameters for optimum selected points A–E in Pareto optimum front. Variables
A
B
C
D
E
Actual data
T32 (C) P1 (MPa) P3 (MPa) P5 (MPa) P7 (MPa) P8 (MPa) P10 (MPa) P28 (MPa) P32 (MPa) P1 P2 P3 HPT IPT LPT
559.9635 1.0479 0.5514 0.3530 0.0078 0.0521 0.3000 2.2177 17.7773 0.6475 0.6481 0.7982 0.8996 0.8994 0.9000
557.2884 1.0108 0.5502 0.3510 0.0078 0.0504 0.2996 2.1892 14.8543 0.6834 0.7870 0.7979 0.8994 0.8994 0.9000
550.1593 1.0055 0.5201 0.3413 0.0078 0.0509 0.2999 2.1129 12.7300 0.7975 0.5755 0.7967 0.8994 0.8997 0.9000
548.1850 1.0653 0.5188 0.3434 0.0078 0.0504 0.3000 2.1352 10.5678 0.7346 0.6235 0.7965 0.8997 0.8956 0.9000
545.7967 1.0201 0.5049 0.3301 0.0078 0.0507 0.2999 2.0886 8.0183 0.7125 0.6213 0.7795 0.8992 0.8738 0.9000
537.7778 1.4989 0.8191 0.4216 0.00847 0.0778 0.2764 3.1336 12.5110 0.7 0.7 0.7 0.82 0.86 0.8
In the LINMAP method, each objective is nondimensionalized using the following relation: Fijn =
Fij
2
(22)
m (F )2 i=1 ij
where i is the index for each point on the Pareto frontier, j is the index for each objective in the objectives space and m denotes the number of points in the Pareto front. After nondimensionalization of two objectives, the distance of each solution on the Pareto frontier from the ideal point obtained. The closest point of Pareto frontier to the equilibrium point (the design point D) might be considered as a desirable final solution with the 0.4027 thermal efficiency and 1.489 $/s total cost rate along with its optimum design parameters listed in Table 5. Fig. 4. Distribution of thermal efficiency versus boiler exhaust steam pressure for optimum points in Pareto optimum front in Fig. 3.
on engineering experiences and importance of each objective for decision makers. The process of final decision-making in Fig. 3 is usually performed with the aid of a hypothetical point named as equilibrium point, that both objectives have their optimal values independent of the other objectives [29]. It is clear that it is impossible to have both objectives at their optimum point, simultaneously. The equilibrium point is not a solution located on the Pareto frontier. In this paper, LINMAP method was used to find the final optimum solution in Pareto front [29].
Fig. 5. Distribution of thermal efficiency versus turbine inlet temperature (TIT) for optimum points in Pareto optimum front in Fig. 3.
6.4. Correlations between the objective functions and design variables The main goal of this section is to estimate a correlation between two optimum objective functions and fifteen decision variables for a specified case study. Actually the designer can find the minimum of total cost rate for any choosing efficiency in the range of 0.3915 < ε < 0.4177 using Eq. (21). Using these two optimum values (total coat rate and efficiency) and this correlation, designers can find the corresponding values of fifteen optimum design parameters. For this purpose, the Artificial Neural Network by means of a hidden layer with two neurons for two input parameters (efficiency
Fig. 6. Distribution of thermal efficiency versus reheat pressure for optimum points in Pareto optimum front in Fig. 3.
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Table 6 The values of constants in Eq. (22) for the estimated optimum design parameters in the presented case study.
T32 (C) P1 (MPa) P3 (MPa) P5 (MPa) P7 (MPa) P8 (MPa) P10 (MPa) P28 (MPa) P32 (MPa) P1 P2 P3 HPT IPT LPT
b1
b2
b3
b4
b5
w1
w2
w3
w4 × 103
−2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045 −2.045
157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77 157.77
93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286 93.286
−1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558 −1634.558
882.270 0.808 .359 0.299 0.00775 0.0426 0.300 6.597 214.015 1.356 0.342 0.895 0.936 0.910 0.904
−65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027 −65.027
−337.287 0.2226 0.154 0.0010777 0.00006784 0.00875 0.00065 −5.562 −204.982 −0.6535 0.3133 −0.1023 − 0.0388 0.01123 −0.00492
505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5 505.5
822.28 −10.991 −1.4298 −0.023943 −0.0012728 −0.14148 0.1233 9.9576 −1111.2 32.139 −17.67 0.4927 0.72803 0.15629 0.11332
Fig. 7. The schematic diagram of neural network with two input neurons and fifteen output neurons.
and total cost rate) and an output layer with fifteen neurons for fifteen output parameters (decision variables) were applied using feed forward algorithm (Fig. 7). Tangent-sigmoid transfer function (−1 + 2/(1 + e−2n )) was used at the input layer with n as the transfer function input. Furthermore
Fig. 9. Comparison of optimization results and predicted results by ANN for total cost rate.
the linear transfer function (n) was applied at the output layout. A close form equation for decision variables versus objective functions (efficiency and total cost) was derived by training the network as follows: Decision variables =
b1 1 + e(w1 ε+w2 Ctotal +b2 ) +
b3 1 + e(w3 ε+w4 Ctotal +b4 )
+ b5
(22’)
where b is the bias value and w is weighting function shown in Table 6, for the each decision variables. The corresponding errors for the estimated efficiency and the total cost rate (two objective functions) are shown in Figs. 8 and 9. Results show that with applying the above proposed correlations (Eq. (22)), the estimated total cost rate and efficiency, are accurate within −0.8% to +0.3% and −2% to 1% respectively, which are acceptable for engineering problems. 7. Conclusions
Fig. 8. Comparison of optimization results and predicted results by ANN for efficiency.
A steam cycle power plant was optimally designed using the fast and elitist Non-dominated Sorting Genetic Algorithm (NSGAII) technique. The design parameters (decision variables) were the turbine inlet temperature, boiler pressure, turbines extraction pressures, turbines and pumps isentropic efficiency, reheat pressure as well as condenser pressure. In this presented optimization
F. Hajabdollahi et al. / Applied Soft Computing 12 (2012) 3648–3655
problem, the efficiency and total cost rate were considered as two objective functions (the efficiency was maximized and total cost rate was minimized). A set of Pareto optimal front points were extracted. The results revealed the level of conflict between the two objectives. The optimization results in design point E show 3.76% increase in efficiency and 3.84% decrease in total cost rate simultaneously, compared with the actual data of the running power plant. It was also observed that the efficiency increases by the increase of boiler pressure, TIT and reheat pressure in the optimum selected range with the fixed net power. Furthermore a correlation between the optimal values of two objective functions was proposed. Finally a correlation between two optimum objective functions and fifteen decision variables for the presented case study were proposed with acceptable accuracy using Artificial Neural Network. This correlation enables designers to find the corresponding fifteen optimum design parameters for minimum total cost rate and maximum efficiency. References [1] H. Hajabdollahi, P. Ahmadi, I. Dincer, An exergy-based multi objective optimization of a heat recovery steam generator (HRSG) in a combined cycle power plant (CCPP) using evolutionary algorithm, International Journal of Green Energy 8 (2011) 44–64. [2] S. Sanaye, H. Hajabdollahi, Multi-objective optimization of rotary regenerator using genetic algorithm, International Journal of Thermal Sciences 48 (2009) 1967–1977. [3] S. Sanaye, H. Hajabdollahi, Thermal-economic multi-objective optimization of plate fin heat exchanger using genetic algorithm, Applied Energy 87 (2010) 1893–1902. [4] S. Sanaye, H. Hajabdollahi, Multi-objective optimization of shell and tube heat exchangers, Applied Thermal Engineering 30 (2010) 1937–1945. [5] P. Ahmadi, H. Hajabdollahi, I. Dincer, Cost and entropy generation minimization of a cross flow Plate-Fin Heat Exchanger (PFHE) using multi-objective genetic algorithm, Journal of Heat Transfer – Transactions of the ASME 133 (2011) 021801. [6] S.O.T. Ogaji, R. Singh, Advanced engine diagnostics using artificial neural networks, Applied Soft Computing 3 (2003) 259–271. [7] I. Bertini, M.D. Felice, A. Pannicelli, S. Pizzuti, Soft computing based optimization of combined cycled power plant start-up operation with fitness approximation methods, Applied Soft Computing 11 (2011) 4110–4116. [8] T. Ganesan, P. Vasant, I. Elamvazuthi, Optimization of nonlinear geological structure mapping using hybrid neuro-genetic techniques, Mathematical and Computer Modelling 54 (2011) 2913–2922. ˜ D. Peidro, P. Vasant, Vendor selection problem by using [9] M. Díaz-Madronero, an interactive fuzzy multi-objective approach with modified S-curve membership functions, Computers and Mathematics with Applications 60 (2010) 1038–1048.
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