Many studies have attempted to optimize integrated Solid Oxide Fuel Cell -Gas Turbine (SOFC -GT), although different and somehow conflicting results are reported employing various algorithms. In this study , Multi -Objective Optimization (MOO) is employed to approach the optimal design of SOFC -GT considering all prevailing factors . The emphasis is placed on the evaluation of the Parti cle Swarm Optimization (PSO) and Genetic Algorithm (GA) performance as two effective approaches for solving the multi -objective and non-linear optimization problems. Multi - objective optimization is carried out on two vital objectives; the electrical efficiency and the overall output power of the system . The considerable achievements are the set of optimal points that aim to identify t he system optimal performance which provide s a practical basis for the decision -makers to choose the appropriate target functions . For the studied conditions, the two algorithms nearly exhibit similar performance , while the PSO is faster and more efficient in terms of computational effort . The PSO appears to achieve its ultimate parameter values in fewer generations compared to the GA algorithm under the examined circumstances . It is found that the maximum power of 410 kW is accomplished employing the GA op timization method with an efficiency of 64%, while PSO method yields the maximum power of 419.19 kW at the efficiency of 58.9%. The results stress that PSO offers more satisfactory convergence and fidelity of the solution for the SOFC -GT MOO problems. Keywords : Modeling ; Solid oxide fuel cell; Gas turbine; Multi -Objective Optimization ; Heat recovery steam generation 1. INTRODUCTION Renewable energy generation methods have been intensively studied in recent years. However , solid oxide fuel cells (SOFC) have been extensively interrogated and developed since they work at high temperatures . This provides the opportunity of hybrid concepts employment such as the integrated gas turbine (GT) and SOFC leading to a considerable improvement of the overall system 2 efficiency [1]. Combining internal combustion engine s or SOFC with GT, the gaseous fuel can be converted to heat and power. However, the combination of SOFC and GT yield s better performance [2-5]. This hybrid energy conversion system will provide superior utilization of natural resources that considerably decreases harmful impacts on the environment as well as making higher profit [6-8]. Typically, the optimization of the integrated SOFC/GT deals with various objectives which sometimes may be conflicting like capital investment , operating cost, output power, efficiency, energy consumption, and environmental considerations . In such circumstances , multi -objective optimization (MOO) turns out a promising approach for finding various Pareto optimal solutions, to comprehend the quantitative tradeoffs between the objectives and also to acquire the optimal values of decision variables [6, 9, 10] . However, various arrangements of SOFC -GT combined systems have been studied with respect to several design parameters [11-15]. The SOFC -GT hybrid systems are usual ly fueled by natural gas, converted by means of an internal reformer. This will dramatically reduce the system capital expense due to eliminating any external fuel processing [16]. Nevertheless, some researche rs have interrogated the thermodynamic modeling of SOFC/GT combined system [16-20], while others focused on the optimization of these systems, suggesting various design configurations . A synthesis/design optimization of a n integrated GT -SOFC power plant has been performed by Calinse et al. [14], where a classical single -level method created on the GA basis has been developed . A thermo -economic optimization that employs a Lagrange Multiplier Approach (LMA) has been examined by Cheddie [15]. In this work , an indirectly coupled integrated SOFC -GT power plant was considered where sizing and operating condition s were employed as the decision variables to obtain the minimum lifecycle cost of the system. Autissier et al. [21] have performed the optimization for the design of hybrid SOFC -micro GT considering two objectives; boosting the energy efficiency and deceasing the cost. The optimization results exhibit designs with costs ranging from 2400 $/kW for a 44% energy efficienc y to 6700 $/kW for a 70% energy efficiency. Sabramanyan et al. [22] have performed multi -objective optimization for a conceptual layout of a SOFC -proton exchange membrane fuel cell (PEMFC) hybrid power plant. Optimizing the design of SOFC system involves consideration of the cost and performance of a SOFC -PEMFC based on a novel algorithm for minimization of single objective optimization problems (M INSOOP). 3 Table 1 lists a review of the literature regarding the optimization methods and strategies for SOFC/GT hybrid systems. It offers the decision variables, parameters that have been taken into consideration, objectives , the optimization method and the outstanding results for hybrid applications [23]. Although different optimization methods have been applied to SOFC/GT systems, none has compared the performance of these optimization methods. This work presents two optimization algorithms based on the intelligent approaches for an integrated SOFC -GT system to optimize the efficiency of the system , i.e. simultaneo usly maximiz ing the power output and the overall efficiency. At first, the integrated system was modeled in MATLAB to assess the performance characteristics of the system and thereby the system components have been evaluated to examine the efficiency of the system. Since the efficiency of the SOFC -GT hybrid system is defined as the ratio of the generated powe r of the system to the fuel input, it is a function of several variables. Two different optimization algorithms were employed to accomplish the optimal operating condition s of the entire system. As the system is extremely nonlinear, due to the combination of two extremely nonlinear systems, artificial intelligence should be employed to optimize the overall system performance. GA and PSO were used as two most common and effective approaches among various existing intelligent algorithms . Both algorithms are population -based heuristic search methods that proceed toward s the optimal solution by updating generations . Also, they are assumed to obtain a solution to the given objective function s by emplo ying different methodologies while interactivel y comparing their implementation and the results. The concurrent implementation of GA and PSO for the optimization of the combined SOFC -GT system provides an optimization framework that can be easily extended to the optimization of other complicated energy systems [24]. As discussed earlier, t here are a variety of studies that have aimed to optimize SOFC/GT cost versus power or cost versus efficiency, while only a few of them examine these two objective functions by a MOO approach for the optimization purposes. Since not all prevailing factors have been comprehensibly studied , it is still necessary to apply MOO on the SOFC/GT system. Although some researches examined the output power and cost or the system efﬁciency and cost as the objective functions, there is no reported study to focus on the optimization of efficiency and output power, as two more impactf ul and conflicting objectives while considering fuel utilization, 4 pressure ratio, steam to carbon ratio, and fuel flow rate as the decision variables. Also, to the best of our knowledge, PSO and GA have been rarely compared directly in the previous studies , although they are applied individually for the other renewable energy systems like SOFC/GT CHP system. To establish the optimization framework , first the thermodynamic model of SOFC -GT hybrid system is described. Then, the multi-objective optimization is discussed and implemented to the proposed system. The MOO is applied based on the simultaneous determination of the optimal performance and operation conditions through employing two different optimization methods , i.e. the GA and the PSO. The considerabl e accomplishments of the recommended MOO approaches are set of optimal points (Pareto frontier) that target to determine the SOFC/GT optimal performance and operation conditions. Undoubtedly, Pareto frontiers offer critical information for the decision -making process , as the most remarkable and beneficial achievement of this work. Briefly, the main purposes of this research can be explained as: • Developing an efficient energy system based on SOFC/GT for CHP applications • Inter rogating the effects of steam to carbon ratio, compressor pressure ratio, fuel utilization factor on the electrical efficiency and the output power • Applying MOO to examine the optimal operation conditions of the designed system using both PSO and GA for determin ing the best solutions • Systematically e xploring and comparing the Pareto frontiers with regard to two heuristics algorithms to investigate the optimal range of the influential parameters • Providing a practical basis for decision -makers to choose am ong the target functions in accordance with the optimal performance requirements Eventually, the results of optimization methods are presented and compared. The results include the changes in power output and efficiency of the hybrid system over the iterat ions employing GA and PSO. Also , the Pareto diagram for both GA and PSO approach is presented. In summary, the innovations of this paper are: • Implementation of two new objective functions (efficiency and output power) to maximize the power generation and energy efficiency of the system 5 • Optimization of the system including SOFC, gas and steam cycle by two optimization algorithms, i.e. PSO and GA • Comparison of the optimization results from the employed algorithms to sele ct the best approach as the most effective tool in this type of cycle s • Sensitive analysis of the optimization results 2. MATERIAL AND METHODS 2.1 System description The schematic block diagram of the suggested SOFC -GT hybrid system is depicted in Fig . 1. The system layout is divided into two main subsystems: SOFC and GT. The system includes other auxiliary equipment such as a heat recovery steam generator (HRSG), fuel compressor, inverter, and heat exchangers. Compressed fuel is reformed into a rich hydrogen gas in an internal reforming unit, then the gas is transported into the anode compartment of the SOFC stack. In the SOFC, reformed fuel will be converted to electricity directly and DC power will be generate d by an electrochemical reaction occurring within the stack, and subsequently, the generated power will be turned to AC power by employing an inverter. A portion of this electricity is utilized to provide the power source for the fuel compressor. The compressed air is used as SOFC cathode gas, after being preheated in the heat exchanger. Unreacted gases of the anode side and preheated compressed air of the cathode participate in the combustion reaction. Combus tion products will be expand ed in the gas turbine for both generating additional electrical power and producing mechanical work for running the compressor. Gas turbine exhaust flow will be recovered to preheat the compressed air as well as delivering the neces sary heat in the HRSG to produc e steam used by the internal reformer. 6 1211 13 910 7 8 Heat Exchanger 1 GE SOFC Hot gasesCompressor Turbine CCFilter GE 5Flue gas Fuel compressor Methane1 246 1214Heat Exchanger 2 315 Internal reformer Anode CathodeAir Anode gas E Air Fuel Water Air fuel mixture ElectricityPumpWater16 17 HRSGEvaporator Economizer 14PInv E 17 14 17P 13 Fig. 1. The Schematic of SOFC -GT hybrid system. 2.2. Model assumptions Principal assumptions for developing the pr oposed SOFC -GT hybrid system are listed as: • Ideal gas assumption is considered for a ll gases. • Steady state condition is considered for the operation of all system components. • The working fluid in all parts of the cycle is assumed to work as an ideal gas • Temperature, pressure, and gas compositions are uniformly distributed in each component. • 5% pressure drop is assumed in the combustion chamber and the fuel cell stack. • Pressure drop is presumed to be 4% and 5% for the heat exchanger s and the HRSG , respectively. • The inverter efficiency is assumed 0.98 , the turbine mechanical efficiency is considered 0.8, the generator efficienc y is presumed 0.97 and the efficiency of both air compressor and fuel compressor is assumed 0.78 . 7 • Methane is selected as t he supplied fuel to the system. • The dead conditions are assumed P 0=101.3 kPa and T 0= 298 K These assumptions are employed throughout the analysis to derive the corresponding e quations of each system component [25, 26] . 2.3. System modeling The model is structure d according to the zero-dimensional method in which the mass and energy balance equations are solved simultaneously. 2.3.1. Solid oxide fuel cell In order to optimize the SOFC -GT integrated system parameters, it is vital to develop a SOFC model that is capable of predict ing the performance based on its operating condition. The proposed SOFC is built on a tubular design which has been designed and demonstrated by Siemens - Westinghouse [27]. An electrochemical model that evaluates the power and efficiency of SOFC is developed . The geometric and material properties are selected from the literature [28]. The gas flow leaving the fuel compressor will enter the internal pre-reformer, then reforming and shifting reactions occur , i.e. (CH4 + H2O ↔ 3H2 +CO) and (CO+ H2O ↔ H2+CH4), respectively . The hydrogen produced from such reactions participates in the cell electrochemical reaction i.e. (H2+1 2 O2→H2O); as a result, a portion of methane will be turned into hydrogen. These reactions are considered to proceed to chemical equilibrium. The chemical and electrochemical reactions that take place at the anode and the cathode of the fuel cell may be obtained from the following equilibrium equations [29]: xr→(CH4 + H2O ↔ 3H2 +CO) (1) yr→(CO+ H2O ↔ H2+CH4) (2) zr→(H2+1 2 O2→H2O) (3) where xr, yr and zr are the internal reformer conversion rates. The fuel utilization factor, Uf, and the air utilization ratio, Uo, can be defined as follows [29]: Uf=(Fuel)Consumed (Fuel)Supplied=zr 3xr+yr (4) Uo=(Air)Consumed (Air)Supplied=zr2⁄ ṁO2.9 (5) 8 Employing the mass balance equations for each gas component , the molar flow rate will be achieved as follow s [29]: ṁCH4.5=xr (6) ṁH2O.5=rsc∙ṁCH4.5 (7) ṁH2.6=3xr+yr−zr (8) ṁCO.6=xr−yr (9) ṁCO2.6=yr (10) ṁH2O.6=ṁH2O.5−xr−yr+zr=(1⋅5xr−yr+zr) (11) ṁO2.9=zr 2Uo (12) ṁO2.10=ṁO2.9−zr 2 (13) ṁN2.9=79 21×ṁO2.9 (14) ṁN2.10=ṁN2.9 (15) ṁ9=ṁO2.9+ ṁN2.9 (16) ṁ10=ṁO2.10+ ṁN2.10 (17) ṁ5=ṁH2O.5+ ṁCH4.5 (18) ṁ6=ṁH2.6+ṁH2O.6+ ṁCO.6+ṁCO2.6 (19) The constants xr and yr can be obtained employing equilibrium constant as well as the above equations , while shifting equilibrium constant can be written as [29]: lnK𝑠=−∆G0 R̅T= ln [yr×(3xr+yr−zr) (xr−yr)×(1⋅5xr−yr+zr)] (20) Maximum cell voltage is determined from equation (21). Therefore, Nernst voltage can be formulated as follows [25]: Vnernst=−∆G0 2F−R̅T 2F ln [pH20 pH2√po2] (21) where R̅ is the universal gas constant, ∆G0 indicates Gibbs free energy at standard conditions for the electrochemical reaction, F refers to Faraday’s constant ( 96,485 C mol−1), cell temperature is introduced by T and p indicates the partial pressure of respective species. 9 The operating voltage drops w hen the electric current is drawn from the cell, mainly due to some loss such as ohmic , activation and concentration polarization . They are individually categorized as electrochemical reaction activation, Vact; ohmic polarization, Vohm; and concentration depletion, VCon. Operating cell voltage , VCell, can be calculated as [25]: Vcell=Vnernst−(Vact+Vohm+Vcon) (22) Activation polarization is calculated by the Butler -Volmer equation , express ed as [25]: j=j0[exp(βneFVact RT)−exp(−(1−β)(neFVact RT))] (23) where j indicates the current density, β is the transfer coefficient, and j0 refers to exchange current density. Ohmic polarization result ing from the electron and ion transmission processes is determined by the following equation s based on Ohm's l aw [28]: VOhm=i∑rk k (24) rk=∑ρkδk A k (25) ρk=∑akexp(bkT⁄) k (26) where rk is the ohmic resistance, δk refers to the current flow length and ρk indicates the material resistivity. The coefficients of a and b are the characteristics of constituent materials The concentration loss which is a result of the transport of gases to the electrodes will be calculated as follows [28]: Vcon=RT 2F Ln (1+pH2.j pH2.jlim)−RT 2FLn (1−j jl) (27) where jl is the limiting current density and j refers to the current density that will be estimated by the following electrochemical reaction [28]: I=j⋅Ac (28) j=ne∙F∙zr Ac (29) 10 where ne is the number of electrons that are formed per hydrogen mole by electro -chemical reactions , Ac is the cell activation area , and I is the current . More detail s about calculation procedures of voltage , polarizations and current density may be found in [25, 28] . The electrical power generated from the SOFC may be calculated as [25]: ẆSOFC=N ⋅I⋅VCell (30) where N indicates number of cells . The net electrical power of the SOFC can be given as [25]: ẆSOFC,net=ẆSOFC⋅Ƞinv−ẆFC (31) Where, Ƞinv is the inverter efficiency which is assumed to be 0.98, and ẆFC is the electrical power used by the fuel compressor . Mass balance of SOFC considering input and output flows may be written as follow s [30]: ∑ṁi=∑ṁe (32) ṁ9+ṁ4=ṁ10+ṁ6=ṁ9+(ṁ4Uf)+ṁ4(1−Uf) (33) where ṁ4 is the fuel mass flow rate injected to the SOFC and ṁ9 is the air mass flow rate injected to SOFC. Alternatively, applying the first law of thermodynamic equation ( 33) can be rewritten as [30]: ṁ9h9+(ṁ4Uf)LHV+ṁ4(1−Uf)h4−ẆFc−ṁ10h10−ṁ6h6=0 (34) where LHV stands for the lower heating value of methane that is about 50 ,000 (kJ kg⁄). It is noteworthy that all reactions are considered to be adiabatic . The energy balance for the proposed hybrid system can be written as [30]: ṁ1h1+ṁ3h3+ṁ4UfLHV+Q̇CC−ṁ15h15−Q̇Loss−ẆFc−ẆP−ẆGT=0 (35) where Q̇CC is the generat ed heat in combustion chamber , Q̇Loss is the heat losses of the hybrid cycle, ẆFc and ẆP are power used by the fuel compressor and pump , respectively , and ẆGT is the output power generated by the gas turbine . In order t o solve the equations of the SOFC stack, first the cell current is calculated at a given current density. The quantity of hydrogen and oxygen necessary for the electrochemical reaction can be calculated at this identiﬁed current amount. Later, the fuel utilization as well as the air utilization can be calculated by equations (4,5), followed by determining the inlet molar ﬂow rates 11 of anode and cathode employing the mass balance equations for each gas component equations (6 - 20). Then , the cell voltage may be found by Nernst equation (21) , also the polarization equations at given operating cell temperature and pressure can be calculated by solving the set of equations (21-29). Finally, the heat and electrical power and efficiency will be determined. More details about the calculation process may be f ound in [31]. 2.3.2. Air compressor The gas turbine model is comprised of the combined gas turbine and compressor with a n interconnecting shaft. The irreversibility effect of the compressor is usually defined by isentropic efficiency. Accordingly , the outlet characteristics (T8,h̅8) of the air compressor can be determined from isentropic efficiency, ηAC and ẆC, the used power in the air compressor and as follows [25]: ηAC=WS Wa=h̅8s−h̅7 h̅8−h̅7 (36) T8S T7=(P8 P7)γ−1γ⁄ (37) ẆAC=ṁ7(h̅8−h̅7) (38) where T is temperature , P is pressure, h is specific enthalpy, ṁ is mass flow rate (kgs⁄), ηAC is the air compressor efficiency and h̅8s is the specific enthalpy of air. 2.3.3. Gas turbine Then, the gas turbine outlet may be identified by considering the turbine isentropic efficiency, ηtis and expansion ratio, Prt equations [25]: ηtis=Wa Ws−h̅11−h̅12 h̅11−h̅12s (39) Prt=P11 P12 (40) where h̅12s is the specific enthalpy of fuel gas . Further ẆT, the generated power by the turbine, can be determined by the following equations [25]: ẆT=∑(ṁ11h̅11)in i−∑(ṁ12h̅12)out i (41) Also, t he net power output of the gas turbine, ẆGT can be written as: 12 Ẇ GT=(ẆT−WAĊ)μm∙μg (42) where μm refers to the mechanical efficiency and μg indicates generator efficiencies which are assumed 0.80 and 0.98, respectively. [25]. 2.3.4. Combustion chamber The exhausted fuel from the anode side of the SOFC and the off -air from the cathode side are introduced to a combustor in order to make the unreacted gases to be burnt completely for the heat recovery [30]. The c ombustion equati on of the combustion chamber is as follows [30]: CH4+XO20O2+XN20N2+XCO20CO2+XH2O0H2O →(f+XCO20)CO2+(2f+XH2O0)H2O+(XO20−2f)O2+XN20N2 (43) where, f is the molar ratio of fuel to air. Also, the energy balance of the combustor is expressed as [30]: ∑miin6 i=1(∆Hf0+∫CpdTTin 298)=∑miout4 i=1(∆Hf0+∫CpdTTout 298) (44) It should be noted that t he adiabatic combustion temperature may be calculated by equation (44). 2.3.5. Heat recovery steam generator (HRSG) The HRSG is used to recover energy from turbine exhaust gas as well as to generat e some auxiliary steam to be utilized in an internal reformer where water is pump ed to the HRSG. The steam flow rate generated by the HRSG, ṁs, may be determined by considering the following equations simultaneously [32]: ṁg=ṁa+ṁf (45) ṁs(h3−h8)=ṁg(h13−h14) (46) ṁs(h3−h17P)=ṁg(h13−h14P) (47) P0=P13(1−∆PHRSG) (48) where s represents to steam and g implies gas. 13 2.3.6. Heat exchangers The outlet temperature of the air flow that leaves the air preheaters , i.e. T13 and T4, may be calculated by solving the air preheater effectiveness, εAPH, as follows [30]: εAPH1=T13−T8 T12−T8 εAPH2=T4−T2 T14−T2 (49) (50) The energy conservation equation for the air preheater may be rearranged as [30]: ṁ8(h13−h8)=ṁ12(h12−h13)εAPH1 (51) ṁ2(h4−h2)=ṁ14(h14−h4)εAPH2 (52) 2.3.7. Fuel compressor The isentropic efficiency and work of the f uel compressor can be determined as follows [26]: ηFC=WS Wa=h̅2s−h̅1 h̅2−h̅1 (53) ẆFC=ṁf(h̅2−h̅1) (54) Where 𝛈𝐅𝐂 refers to the mechanical efficiency of fuel compressor , assumed to be 0.78. 2.3.8. Water pump The isentropic efficiency and work of the w ater pump can be calculated as [26]: ηwp=h̅17s−h̅16 h̅17−h̅16 (55) ẆWP=ṁW(h̅17−h̅16) (56) 2.3.9. Power and efficiency The total energy efficiency and net power output of the proposed hybrid system including the SOFC and GT hybrid cycle is determined as following [30]: Ẇ net=(ẆSOFC+ẆGT−ẆFC−ẆWP) (57) Q̇ tot=(ṁf⋅Uf⋅LHV) (58) 14 ηHS=Ẇ net Q̇ tot (59) 2.4 Simulation and validation The proposed model developed in the preceding sections is employed to simulate and assess the performance of the SOFC -GT hybrid system. The nominal model inlet parameters of the studied SOFC -GT CHP system are summarized in Table 2 while more details may be found in [25, 28] . In 15 Table 3, in order t o indicate the model validation and accuracy, some simulation results of the proposed model (SOFC -GT) are compared to the existing studies [28]. It is observed that the presented results reasonabl y agree with the counterparts in literature. 3. OPTIMIZATION METHODS To obtain the optimum solution for both maximum efficiency and output power of the SOFC - GT system, a mathematical method is required. Generally, two categories of optimization methods are discussed in literature [33]; the first category is the classical optimization methods like the linear programming (LP) ; while the next type is the heuristic method. Since finding the optimum solution for the integrated SOFC -GT system is defined under nonlinear categories and determining the optimal values with conventional methods is slightly complicated, the heuristic method has been implemented in this work . Both the GA and the PSO are employed in this optimization problem. Since both optimization approaches are supposed to determine the optimal solutions to a given objective function while employing different procedures, it is essential and helpful to compare their performance. 3.1. Multi -objective optimization Recently, multi -objective optimization problems have been extensively studi ed, and various viable solutions have been found in the literature [2, 3, 6, 9, 22, 34] . Multi -objective problem appears virtually in different fields and has been widely applied in a variety of contexts such as, designing aircra ft control systems, designing structures , and vehicles , as well as dealing with pollution control and management problems [22]. A multi -objective optimization approach is capable of finding the solution to a problem with numerous objectives simultaneously where such objectives are regularly in conflict with each other , such as the seek for maximum efficiency or maximum power generation and minimum cost at the same time [34]. Although there is no unique correct solution to MOO problems , instead there is a set of solutions whereas one is called the Pareto set. Therefore, decision making is required to select the best answers from the Pareto solutions. The MOO problem may be demonstrated as [34]: Minxf(x)=[ f1(x),f2(x),f(x),…, fn(x)]T (60) Subject to: 16 hj(x)=0 j=1,2,…,n (61) gi(x)≤0 i=1,2,…,m (62) where, fn(x) refers to the 𝑛𝑡ℎ objective function, h(x) and g(x) are the equality and inequality constraints, respectively, and x indicates the vector of decision variables. 3.2 The MINSOOP algorithm Although , there is a vast pool of analytical method s to solve the multi -objective problem s, they generally fall into two main categories : the preference -based and the generating methods. Preference -based methods like goal programming are conventional optimization techniques that takes into account the decision maker's p