Global optimization algorithms applied to solve a multi-variable inverse artificial neural network to improve the performance of an absorption heat transformer with energy recycling

Applied Soft Computing Journal 85 (2019) 105801

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Global optimization algorithms applied to solve a multi-variable inverse artificial neural network to improve the performance of an absorption heat transformer with energy recycling ∗

∗

J.E. Solís-Pérez a , J.F. Gómez-Aguilar b , , J.A. Hernández c , , R.F. Escobar-Jiménez a , E. Viera-Martin a , R.A. Conde-Gutiérrez c , U. Cruz-Jacobo d a

Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico CONACyT-Tecnológico Nacional de México/CENIDET. Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico c Centro de Investigación en Ingeniería y Ciencias Aplicadas (CIICAp-IICBA), Universidad Autónoma del Estado de Morelos (UAEM), Av. Universidad 1001 Col. Chamilpa, 62209, Cuernavaca, Morelos, Mexico d Posgrado en Ingeniería y Ciencias Aplicadas, Centro de Investigación en Ingeniería y Ciencias Aplicadas (CIICAp), Universidad Autónoma del Estado de Morelos (UAEM), Av. Universidad 1001 Col. Chamilpa, 62209, Cuernavaca, Morelos, Mexico b

article

info

Article history: Received 31 March 2019 Received in revised form 2 August 2019 Accepted 22 September 2019 Available online 25 September 2019 Keywords: Optimization Absorption heat transformer Genetic algorithms Cuckoo search Particle Swarm Optimization Simulated Annealing

a b s t r a c t In this research, global optimization algorithms were applied to solve the inverse artificial neural network (ANNi) for obtaining the best inputs values of an absorption heat transformer with energy recycling (AHTER) and improving its performance. The ANNi was obtained by inverting an artificial neural network (ANN) which architecture was 16 input variables, 3 neurons in the hidden layer and 1 output variable. The ANNi’s aim was optimizing 1, 2, 3, and up to 4 manipulated input variables, as well as calculating the other 12 input variables not manipulated in the system (AHTER) considering a coefficient of performance (COP) desired. The Cuckoo Search (CS), Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Simulated Annealing (SA) algorithms were used to find the optimal inputs. The results showed that the four algorithms used (ANNi-CS, ANNi-PSO, ANNi-GA, and ANNiSA) satisfactorily optimize of 1 up to 16 inputs of the ANNi. However, the algorithms of ANNi-CS and ANNi-SA were slightly faster with acceptable accuracy. Additionally, they were carried out two analyses using different COPs values. These analyses showed that both algorithms optimize the AHTER’s inputs for different COP, as well as R > 0.988 were obtained with the COP experimental data against COP obtained data by both ANNi models. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In general, optimization is highlighted as the estimation of a value or a set of them to obtain the minimum or maximum potential in the performance of a process, searching within an objective function the optimal point that complies with certain established restrictions so that the solution is valid [1–4]. Under this criterion, the optimization contributes to improving the operation of processes, reducing production costs and maximizing efficiency [5]. In the other hand, the inverse artificial neural network (ANNi) is a method that has been distinguished by the capability to optimize processes based on experimental data and computational methods of artificial intelligence. The development of the ANNi methodology has evolved to the extent of ∗ Corresponding authors. E-mail addresses: [email protected] (J.F. Gómez-Aguilar), [email protected] (J.A. Hernández). https://doi.org/10.1016/j.asoc.2019.105801 1568-4946/© 2019 Elsevier B.V. All rights reserved.

optimizing multiple input variables to improve the performance of a process [6]. However, the methodology has focused on using only some optimization algorithms to solve the objective function, for example: Nelder–Mead simplex method has been used to solve problems with 1 input variable; genetic algorithms and ant colony have also been used to solve problems with 1, 2 and up to 3 input variables [7–10]. The latest above mentioned algorithms come from an optimization method known as meta-heuristic algorithms, which are suitable for global search due to its ability to explore and find regions in the search space [11]. The metaheuristic algorithms are iterative and search optimal values in the shortest possible time. They are ideal for solving multivariable functions raised through the ANNi model. Among the various meta-heuristic algorithms used to solve optimization problems are Genetic algorithm (GA), Particle Swarm Optimization (PSO), Cuckoo Search (CS) and Simulated Annealing (SA). Regarding the optimization algorithms GA, PSO, CS, and SA, the following references have found in the literature, Cachón and Vazquez [12] proposed a genetic algorithm for improving

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J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

the path search of possible solutions over the space of domain of the problem. Carneiro et al. [13] used a genetic algorithm to create the possible solutions to solve a conflict determined, as a result of the approach, it is possible to generate a large number of solutions. Manoranjan et al. [14] proposed Genetic Algorithms in the production industry, in order to maximize profits, minimizing carbon emissions to the atmosphere, through a model that takes into account carbon emission regulation policies. He et al. [15] presented an improved genetic algorithm to determine the optimal subset of channels, in the area of diagnostic imaging by electroencephalogram (EEG). Li et al. [16] proposed a genetic algorithm to obtain an approximation to the optimal solution of the expansion of the point-to-point connection tree. Dasgupta et al. [17] presented a Genetic Algorithm with the purpose of predicting the performance index of the membrane, evaluating the retained dye and the diffusion flow in an aqueous solution. Marini and Walczak [18] presented the potential of particle swarm optimization by solving various problems in chemometrics. Soesanti and Syahputra [19] applied a particle swarm optimization in the production of batik for minimizing the cost of production due to the time and materials used, obtaining the maximum benefit. Nguyen and Truong [20] proposed a cuckoo search algorithm to solve the distribution network reconfiguration problem; proving to be an efficient method of power loss minimization and voltage profile improvement. Naik and Panda [21] presented an adaptive cuckoo search algorithm for intrinsic discriminant analysis-based face recognition. Vincent and Lin [22] proposed a simulated annealing algorithm for solving the open locationrouting problem. It was capable of minimize the total cost of the vehicle operation and the traveling cost. Isakov et al. [23] performed several applications of simulated annealing algorithm to optimize Ising spin glasses on sparse graphs in computers. Marotta and Avallone [24] proposed the simulated annealing algorithm for increasing the cost-benefit of virtual computing to reach for the best combination of energy consumption and image quality. Recently, Monshizadeh et al. [25]; Allyson and Gosselin [26]; and Hussain et al. [27] have employed PSO and GA algorithms for the minimization of total power losses, thermoelectric generator designs and emission dispatch of an independent power plant respectively. In these works, we can find that PSO has fast convergence and lower number of computational formulations in comparison with GA. However, in the works of Sahu et al. [28]; Singla and Shukla [29], they report that CS has better performance compared to PSO. In addition, CS works better for higher dimensions and it converges faster than the others. Kerr and Mullen [30] tested and compared, SA and GA in thermal conductance of harmonic lattices by considering their vast sets of possible hyperparameters. They concluded that GA can find high conductance candidates in a vast pool. By considering the applications, the advantages and the disadvantages of these algorithms we choose them to develop an ANNi methodology. On the other hand, continuing the research proposed by Conde et al. [6], the ANNi methodology was applied to optimize the coefficient performance of an absorption heat transformer (AHT) with energy recycling. In this device, it is interesting to analyze the interaction of various input variables to get efficient operation. The modeling of these devices has been carried out theoretically by means mass and energy balances [31,32]. However, different operating problems or heat losses prevent the process to reach the ideal theoretical operating conditions. To optimize an AHT, by determining multiple optimal input variables, Jain and Sachdeva [33] proposed a multi-objective optimization algorithm to analyze an absorption heat transformer through energy, exergy, and economy variables. The results showed that the multi-objective design was better than the single objective

design. Yari et al. [34] developed a three-objective optimization algorithm for a new type of absorption heat transformer, considering: cost, exergy, and mass flow variables. The results showed an elevation of the gross maximum temperature of 18– 27◦ C respect to other systems. The results of these works were of great interest because they provide information that favors the analysis of AHT. The above-mentioned investigations applied AHT’s theoretical model to develop the multi-objective optimization algorithms. It may imply that the results of the methods, experimentally could not be satisfactory. In this research, it is proposed to maximize the COP in an absorption heat transformer with energy recycling (AHTER), by the application of meta-heuristic optimization algorithms (CS, PSO, GA, and SA) to solve ANNi methodology. In this case, an ANN model reported by Conde-Gutiérrez et al. [6] and Hernández et al. [35] was used and inverted to optimize 16 input variables from a COP desired. Consequently, the optimal multivariable operating conditions to estimate were the following: the temperature in the generator (TGE ), the temperature in the evaporator (TEV ), the temperature in the condenser (TCO ), and temperature in the absorber (TAB ), these temperatures were the only ones manipulated during the operation of the device. Also, the remaining 12 input variables were optimized using the ANNi and by the four optimization algorithms. The optimization algorithms chosen to solve the multivariable function proposed in the present research were promising since they have been tested and validated in other complex processes. In summary, the main contributions of this research were: 1. To propose a methodology that improves the COP of an AHTER. This methodology considers the ANN inverse to obtain a multivariable function that optimize the four manipulated input temperatures simultaneously, as well as, the other twelve not-manipulated input variables of the system. 2. The performance of four different optimization algorithms (CS, PSO, GA, and SA) was analyzed, allowing to solve the multivariable function of the ANNi model and proposing the best algorithm. 3. The COP of an AHTER was maximized using the multivariable function considering constraints which test the operation limits of the device (AHTER) in a sustainable way and use the best optimization algorithm. 2. Description of the geometries used in the components of the AHTER In the researches of Conde et al. [6] and Hernández et al. [35], the authors made a description of the AHTER and its operation. Based on the thesis of Morales [36], this section is dedicated to describes the overall operation of the AHTER and to give the geometries of the components. Also, the section purpose was to give the reader aware of the complexity to operate the AHTER, in turn, identify the temperatures that can be optimized. To carry out the operation of the AHTER, the cycle uses a working solution (LiBr–H2 O) composed of an absorbent fluid (LiBr) and a refrigerant (H2 O). As shown Fig. 1, the cycle begins in the generator, where the objective of the component consisted in separating the working solution, in order to send a concentrated working fluid to the absorber. The generator was designed by a serie of horizontally placed tubes, within which provided heating water was supplied through an external source. The tubes were flooded by the working solution, to achieve separation by heat transfer. The refrigerant fluid leaves the generator in vapor phase towards the main condenser which was designed by two concentric tubes in

J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

3

Fig. 1. Schematic diagram of the AHTER considering the geometries of the components.

the form of a coil with 5 turns. The function of this geometry consisted of passing the vapor through the inner tube and cooling water through the outer tube, coming from an external cooling source. Once condensed, the fluid was sent to the evaporator, this component has the same geometry as the condenser, but designed with 4 turns. The condensed fluid was passed through the inner tube and heating water passes through the external tube, coming from the same external source that supplies heating water to the generator. Subsequently, the exothermic reaction was carried out in the absorber, this occurs when the vapor generated from the evaporator and the concentrated solution coming from the generator come into simultaneous contact. This component was designed in the same way as the generator, through a serie of tubes placed horizontally, but with the difference of having a drop distributor on top. The concentrated solution passes through the distributor, the vapor enters for the lower part and the impure water to purify passes inside the tubes. Once the exothermic reaction was generated, the working fluid returns to the generator in a diluted manner to restart the cycle. The heat generated in the absorber was transferred directly to the impure water, which later passes through a separator, whose purpose was to separate the fluid that was about to saturation and the vapor. The fluid at saturation point was sent back to the absorber to make it evaporate. On the other hand, the generated vapor was sent to a first auxiliary condenser, where it performs a fundamental role in recycling the energy in the external source that supplies the generator and evaporator, as shown in Fig. 1. Finally, once the heat exchange was carried out, the vapor passes through a second auxiliary condenser, to obtain purified water at an ambient temperature, using the same external source of cooling of the main condenser. In Fig. 1, it is important to highlight the manipulated temperatures that will be optimized by the algorithm: temperature in the generator (TGE ), temperature in the evaporator (TEV ), temperature in the condenser (TCO ) and temperature in the absorber (TAB ). These input variables were the only ones that can be manipulated

during the operation of the device. In the case of the generator and evaporator, the temperature can be manipulated by an external heating system through electrical resistances. In the case of the condenser, the temperature can be manipulated through a cooling tower. In the case of the absorber, the temperature can be manipulated by increasing or decreasing the flow of impure water. 3. Proposed optimization problem Once the experimental tests have been carried out, the analysis of the operating coefficient of the device continues. To perform this calculation, it was necessary to know the heat load of each component of the device, through the following Eqs:

˙ 6 h6 + m ˙ 5 h5 − m ˙ 4 h4 , Q˙ GE = m

(1)

˙ 7 (h7 − h8 ) , Q˙ CO = m ˙ ˙ 9 (h10 − h9 ) , QEV = m

(2)

˙ 2 h2 + m ˙ 1 h1 − m ˙ 3 h3 . Q˙ AB = m

(4)

(3)

In Eqs. (1)–(4), the mass flows and the experimental temperatures of inlet and outlet of the components were necessary to calculate the enthalpies. Once knowing the heat loads, it was possible to calculate the value of the COPs using the Eqs. (1)–(4), represented as follows: COP =

˙ 2 h2 + m ˙ 1 h1 − m ˙ 3 h3 m ˙ 6 h6 + m ˙ 5 h5 − m ˙ 4 h4 + m ˙ 9 (h10 − h9 ) m

.

(5)

From Eq. (5), the experimental calculation of the COP was performed. With these values, the development of the ANNi methodology was proposed. This optimization methodology begins with the training of an artificial neural network (ANN) model to simulate the experimental COP. Within the input variables to train the ANN model, the temperatures that can be manipulated during the operation of the AHTER should be considered. Subsequently, the ANN model was inverted, generating a multivariable

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J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

Fig. 2. Schematic diagram of the optimized temperatures by the ANNi model and 4 optimization algorithms.

function. From this point, the optimization problem was proposed by minimizing the difference between the COP desired (COPD ) and the COP reached (COPN (x)) i.e.: minimize

{COPD − COPN (x)} ,

x ∈ R4 ,

(6)

Table 1 Simulation parameter for each algorithm. Algorithm

Parameters

Value

GA (4.1)

Generations (ngen ) Population (npop ) Mutation Crossover Selection

150 100 Uniform Uniform Random

PSO (4.2)

Steps (ns ) Particles (np ) Cognitive parameter (c1 ) Social parameter (c2 ) Inertia weight (w )

150 100 1.49 1.49 0.5 +

CS (4.3)

Generations (ng ) Nests (n) Discovery rate (pa ) Step size (α )

150 100 0.25 0.01

SA 4.4

Initial temp (t0 ) Iterations (imax ) Reduction rate (α )

0.1 150 0.99

subject to the constraints x1 ∈ [86.3710, 90.9220], x2 ∈ [29.0990, 31.1870], x3 ∈ [81.5670, 86.6390], x4 ∈ [91.2080, 95.8880], hence x1 := TOUT .GE −AB , x2 := TOUT .CO , x3 := TOUT .EV −AB , x4 := TOUT .AB−GE . and COPN (x) = b2(j) −

S ∑

Wo(j)

+

S ∑

⎝ j=1

[37]

4. Methodology to solve the multivariable function

j=1

⎛

rand 2

2Wo

⎞

( (∑ ((j) )) ⎠ ) K 1 + exp −2 · k=1 Wi(j,k) · x k + b1(j) (7)

where S is the number of neurons in the hidden layer (S = 3), W , b are weights and biases and K is the number of neurons in the input layer (K = 16). In addition, once the optimization problem was solved, it is possible to have the following benefits on the operation of the device: 1. Energy saving in the heat supply in the generator and evaporator. 2. Optimization of multiple variables of an experimental test in real time. 3. Control the input variables in the components of an AHTER.

Fig. 2 shows schematically the input variables used to train the ANN model. Subsequently, the ANNi model was proposed to optimize the input temperatures of the main components. This section describes the meta-heuristic optimization algorithms used to solve the multivariable function, as well as the parameters (see Table 1) and the pseudocodes developed in each of these. 4.1. Genetic Algorithm (GA) Genetic Algorithms are a set of computational algorithms inspired by the evolution of species. These algorithms encode the potential solution for a specific problem in a data structure individually, this possible solution resembles a chromosome to which a recombination was applied by the genetic operators, conserving genetic information as a possible solution in the individuals of the next generation [38]. The pseudocode of GA used to solve the ANNi model (ANNi-GA) is presented in Algorithm 1.

J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

on similarity provides a subtle form of crossover. The algorithm of CS applied in to ANNi model (ANNi-CS) is summarized in Algorithm 3.

Algorithm 1 Genetic Algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

Initialize population npop = 100 for g = 1 to ngen do for i = 1 to npop do { } g Evaluate fitness of individual i: COPD − COPN (xi ) end for Save best individual to population g + 1 for i = 2 to npop do Select 2 individuals Crossover: create 2 new individuals Mutate the new individuals Move new individuals to population g + 1 end for end for

Algorithm 3 Cuckoo Search via Lévy’s Flights 1: 2: 3:

Initialization of hots n = 100 nests while t < ng do Choose a cuckoo egg using Lévy’s flights xik+1 = xik + α ⊕ Lévy(s, λ),

α > 0,

L(λ) =

4:

4.2. Particle Swarm Optimization (PSO) 5:

Particle Swarm Optimization is a population-based swarm intelligence algorithm proposed by Kennedy and Eberhart [39] in 1995. This algorithm simulates the social behavior of organisms such as bird flocking and fish schooling using physical movements of the individuals in the swarm. Individuals evolve by cooperation and competition among themselves to find a solution to optimization problem. Each particle adjusts its flight in accordance to its experience flight and experience flight of the population. The pseudo code of PSO employed in ANNi model (ANNi-PSO) is shown in Algorithm 2. Algorithm 2 Particle Swarm Optimization 2: 3: 4: 5: 6: 7: 8:

Initialize the inertia weight w = 0.5 + rand 2 for k = 1 to np do Initialize particles’ positions (xk ) and velocities (vk ) g Initialize global best pk and local best pik end for while t < ns do for k = 1 to np do Update particle’s velocity by

vki +1 = w · vki + c1 · R1 · (pik − xik ) + c2 · R2 · (pgk − xik ), 9:

10: 11: 12: 13: 14: 15: 16: 17: 18: 19:

(8)

Update particle’s position by xik+1 = xik + vki +1 ,

(9)

if COPD − COPN (xik ) < COPD − COPN (pik ) then pik = xik { } { } g if COPD − COPN (pik ) < COPD − COPN (pk ) then g pk = pik end if end if end for t ←t +1 end while Return pg

{

}

{

(10)

where

λΓ (λ) sin (πλ/2) 1 , π s1+λ

s ≫ s0 > 0

{

1:

5

}

4.3. Cuckoo Search Algorithm (CS) Cuckoo Search is a nature-inspired metaheuristic algorithm developed by Xin-She Yang [40] in 2009. It was based on the parasitism of some cuckoo species by Lévy flights rather than random walks. The principal feature of this algorithm is that the similarity between eggs produces better solutions, that is to say, mutation realized by Lévy flights and the solutions based

6: 7: 8: 9: 10: 11:

(11)

}

and evaluate your fitness, Fi = COPD − COPN (xik+1 ) randomly and calculate your fitness, Fj = { Choose an egg } COPD − COPN (xirnd ) if Fi > Fj then replace the j-th egg for the i-th egg A fraction (pa = 0.25) of the worst nests are demolished and replaced by new Good nests are preserved (better solutions) end if t ←t +1 end while

4.4. Simulated Annealing (SA) Simulated annealing not only is a probabilistic algorithm of local search but also is an heuristic for approximate global optimization in the space of search. It was introduced in 1982 by Krkpatrick et al. [41] to allow the approximation to the global optimum of an objective function. Here, the objective function is static and can be maximized or minimized easily using this algorithm. The formulation of SA allowing in specific circumstances, movement that worsen the solution, which depends of a probability calculated. Moreover, it considers several neighboring state of the current state, and probabilistically decides between moving the system to neighboring state or staying in the current state. This step is repeated until the system reaches a state that solves the problem, or until a given computation budget has been finished. The pseudocode [42] for the ANNi model (ANNi-SA) is summarized in Algorithm 4. 5. Results It is important to mention that there is an experimental database of 4702 data that represents the coefficient of performance of an AHTER which will be able to validate the methodology implemented applying 4 optimization algorithm (CS, PSO, GA and SA). All the methodologies presented (ANNi-CS, ANNi-PSO, ANNi-GA and ANNi-SA) were evaluated considering the average of the groups of COPs and their standard deviations. Figs. 3 and 4 show 1, 2, 3 and up to 4 physically manipulated input variables versus its temperatures to different coefficient of operation (COP = 0.18 and 0.23). In this case: temperature in the generator (TGE was to the input variable (1), temperature in the condenser (TCO was to the input variable (2), temperature in the evaporator (TEV was to the input variable (3), and temperature in the absorber (TAB was to the input variable (4). Furthermore, from Tables 2 to 5, the 4 applied algorithms to 4 input variables at COP = 0.31 were presented. MAPE was used to determine the accuracy of the methodology. As it can see in these figures (Figs. 3 and 4) and tables (Tables 2 to 5), the 4 optimization algorithms

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Fig. 3. 4 manipulated input variables versus its temperatures: 4 proposed methodologies (ANNi-CS, ANNi-PSO, ANNi-GA and ANNi-SA), and temperatures average with its standard deviations experimental to a COP = 0.18.

Algorithm 4 Simulated Annealing 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

Sc ← CreateInitialSolution(ProblemSize) Sb ← Sc tmpc ← t0 for i = 1 to imax do Si ← CreateNeighborSolution(Sc ) if (COPD − COPN (Si )(7)) <= (COPD − COPN (Sc )(7)) then Sc ← Si if (COPD − COPN (Si )(7)) <= (COPD − COPN (Sb )(7)) then Sb ← Si end if else ( ) (COPD −COPN (Sc )(7))−(COPD −COPN (Si )(7)) if exp > rand ∈ tmp

[0, 1) then 13: 14: 15: 16: 17: 18:

c

Sc ← Si end if end if tmpc ← α ∗ tmpc ) end for return Sb

can optimize from 1, 2, 3 and up to 4 in highly accordance with MAPE (see Table 6). With respect to the non-manipulated variables in the AHTER, an analysis was carried out to observe the behavior of the 4 optimization algorithms evaluated (to COP = 0.35) considering

Table 2 One manipulated variable ∗TOUT .GE −AB to COP = 0.31. Algorithm

Experimental

∗Estimated

MAPE

CS PSO GA SA

87.901 87.901 87.901 87.901

88.609 88.609 88.609 88.611

0.80492 0.80492 0.80492 0.80742

simultaneously the 4 manipulated inputs variables. Table 7 shows the 4 manipulated variables and 3 non-manipulated variables to COP = 0.35 and 4 proposed methodologies. Table 8 presents the 4 manipulated variables and 6 non-manipulated variables with 4 methodologies. Table 9 describes 4 manipulated and 9 non-manipulated variables and Table 10 illustrates 4 manipulated variables and 12 non-manipulated variables to 4 methodologies. The above Tables 7 to 10 show that the 4 applied methodologies can obtain satisfactorily from 1 to 16 input variables of the ANNi. Fig. 5 shows how each one of the errors of the optimization algorithms behaves respect to the solutions number and the time. All the calculus were carried out with a computational characteristic of: AMD A12-9720P, 8 GB Archlinux + KDE. The times found were between 0.018 to 1.2 s. The SA was the fastest algorithm meanwhile GA was the slowest algorithm. CS and PSO were similar in time around 0.4 s. These optimization algorithms confirm that the time was very short and it is possible to consider the 4 algorithms to be used in the AHTER since the time is sufficiently adequate to operate the system online. Regarding the obtained error, it was between 8 × 10−4 to 0.05. The minor

J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

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Fig. 4. 4 manipulated input variables versus its temperatures: 4 proposed methodologies (ANNi-CS, ANNi-PSO, ANNi-GA and ANNi-SA), and temperatures average with its standard deviations experimental to a COP = 0.23.

Table 3 Two manipulated variables ∗TOUT .GE −AB and ⋆TOUT .CO to COP = 0.31. Algorithm

Experimental

∗Estimated

MAPE

Experimental

⋆Estimated

MAPE

CS PSO GA SA

87.901 87.901 87.901 87.901

88.713 88.646 88.694 88.801

0.92301 0.84724 0.90173 1.0231

31.066 31.066 31.066 31.066

31.118 30.914 31.06 31.383

0.16764 0.48904 0.017456 1.0213

Table 4 Three manipulated variables ∗TOUT .GE −AB , ⋆TOUT .CO and •TEV −AB to COP = 0.31. Algorithm

Experimental

∗ Estimated

MAPE

Experimental

⋆ Estimated

MAPE

CS PSO GA SA

87.901 87.901 87.901 87.901

88.874 88.962 88.412 88.761

1.1071 1.207 0.5808 0.97841

31.066 31.066 31.066 31.066

31.391 31.588 30.968 30.799

1.0467 1.6818 0.31328 0.85986

Algorithm

Experimental

• Estimated

MAPE

CS PSO GA SA

84.156 84.156 84.156 84.156

84.192 84.147 84.808 84.03

0.042656 0.010697 0.77439 0.15032

error was obtained with the GA algorithm while applying the SA

optimization algorithms were satisfactory due to the errors were

algorithm the obtained error was 0.05. Similarly all the applied

acceptable.

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Fig. 5. Solutions number versus time (s) and error evaluated to the 4 optimization algorithms. Table 5 Four manipulated variables ∗TOUT .GE −AB , ⋆TOUT .CO , •TEV −AB and ⋄TAB−GE to COP = 0.31. Algorithm

Experimental

∗ Estimated

MAPE

Experimental

⋆ Estimated

MAPE

CS PSO GA SA

87.901 87.901 87.901 87.901

88.393 88.865 88.127 88.727

0.55987 1.0966 0.25669 0.93905

31.066 31.066 31.066 31.066

31.343 31.341 30.7 30.734

0.89351 0.8874 1.1792 1.0691

Algorithm

Experimental

• Estimated

MAPE

Experimental

⋄ Estimated

MAPE

CS PSO GA SA

84.156 84.156 84.156 84.156

83.981 84.943 84.734 83.259

0.20854 0.93504 0.68646 1.066

94.503 94.503 94.503 94.503

94.029 94.777 94.29 94.155

0.501 0.29062 0.22503 0.36756

Table 6 MAPE prediction levels proposed by [43].

Table 7 7 optimized variables to COP = 0.35.

MAPE (%)

Level of prediction

Variable

EXP

ANNi-CS

ANNi-PSO

ANNi-GA

ANNi-SA

< 10

Highly accurate Good Reasonable Imprecise

Tout .AB−GE Tin.AB−GE Tout .GE −CO Tin.CO Tout .CO Tin.EV Tout .EV −AB

94.1497 ± 2.5743 86.4523 ± 1.2048 88.7000 ± 2.8299 48.0307 ± 2.5981 30.7675 ± 0.7568 34.2686 ± 6.7535 83.9207 ± 2.0206

96.7136 85.2479 88.6117 46.5565 30.5919 30.6186 83.5015

95.7189 86.5099 87.8067 47.5320 30.8672 41.0222 82.6741

95.4534 85.8309 86.4610 45.4326 30.8762 34.3723 84.7136

94.1977 87.2627 85.9086 46.2482 31.3759 29.5883 82.6086

10–20 20–50 > 50

Fig. 6 shows how each one of the errors of the optimization algorithms behaves respect to the iterations number and the time. Similarly, time was minor to SA with error of 0.05. While, GA had a time of 1.2 s with error minor of 8 × 10−4 . CS and PSO were also similar in time (minor to 0.4 s) and error (minor to 0.04). Consequently, iterations and solutions number versus time and error to the 4 optimization algorithms were accuracy to determine the coefficient of performance in the AHTER. However,

minor time and error acceptable were the CS and SA. Therefore ANNi-CS and ANNi-SA were considered to be compared. Fig. 7 represents different COPs (0.21–0.34) versus 4 manipulated temperatures optimized freely comparing the ANNi-CS and ANNi-SA: to TGE the difference between both methodologies was around 0.7 ◦ C; to TEV was the difference of approximately 2.7 ◦ C; TAB was of 0.2 ◦ C and to TCO was of 1 ◦ C. Consequently, the TEV

J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

Fig. 6. Iterations number versus time (s) and error evaluated to the 4 optimization algorithms.

Fig. 7. 4 manipulated variables optimized freely considering 2 methodologies (ANNi-CS and ANNi-SA).

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J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

Fig. 8. 2 manipulated variables (TGE and TEV ) optimized freely and 2 manipulated variables restricted (TCO and TAB ) considering as limit smaller constant temperatures in both ANNi-CS and ANNi-SA and to COPs from 0.16 to 0.3.

was the biggest difference between both algorithms while the TAB was the smallest difference. In this case, the 2 proposed methodologies show that each COP the values obtained were slightly different. This is normal since both algorithms are different. Fig. 8 shows the theoretical COPs versus 4 optimized input variables simultaneously. However, in this case, considering as restriction 2 low constant temperatures (TCO ) and (TAB ) in order to minimize energy consumption and the other 2 temperatures (TEV and TGE ) were optimized freely. It is important to remark that the applied methodology (ANNi-CS and ANNi-SA) were evaluated to determine the COPs from 0.16 to 0.40. Both optimization algorithms follow the same behavior, obtaining the 4 easily manipulated temperatures. In order to maximize the COP applying the ANNi-CS and ANNi-SA and considering to optimize the 4 manipulated input temperatures, Fig. 9 describes the behavior of the COP from 0.36 to up 0.40 versus 2 easy manipulated temperatures (TEV and TGE ) optimized freely and the 2 remaining temperatures (TCO and TAB ) were limit low constant values to minimize energy consumption. In this Fig. 9, both manipulated temperatures were very similar applying ANNi-CS and ANNi-SA. This confirms that both methodologies proposed were accuracy to maximize the COPs in AHTER. Finally, considering that the 4 applied methodologies (ANNiCS, ANNi-SA, ANNi-PSO and ANNi-GA) were evaluated, it is possible to mention that the 4 methodologies were accuracy to determine the COP in the AHTER. In addition, to validate ANNi-CS and ANNi-SA with experimental data, Fig. 10 shows COP experimental against COP simulated for both ANNi models, regression coefficients were greater than 0.988.

Table 8 10 optimized variables to COP = 0.35. Variable

EXP

ANNi-CS

ANNi-PSO

ANNi-GA

ANNi-SA

Tin.GE −AB Tout .AB−GE Tin.AB−GE Tout .GE −CO Tin.CO Tout .Co Tin.EV ToutEV −AB Xout .AB−CO Xout .GE −CO

88.7000 ± 2.8299 94.1497 ± 2.5743 86.4523 ± 1.2048 88.7000 ± 2.8299 48.0308 ± 2.5981 30.7675 ± 0.7568 34.2686 ± 6.7535 83.9207 ± 2.0206 52.1900 ± 1.4968 54.3937 ± 1.1609

87.6364 92.4650 86.8489 87.3689 50.6288 30.8533 29.2668 84.6099 51.5470 55.1376

87.1522 96.2898 86.5008 87.6287 49.3376 30.8277 29.2618 84.7318 52.0922 54.3373

88.2862 93.9552 87.0249 85.8701 45.4326 30.4609 33.7796 85.3772 53.6869 55.3857

87.2021 94.1190 85.7122 87.5822 45.9490 30.1796 28.1027 82.4427 53.3808 55.3069

6. Conclusions Four different optimization algorithms (CS, PSO, GA and SA) were applied to solve the multi-variable inverse artificial neural network (ANNi) with the purpose of improve the coefficient of performance of an AHTER system. In this case, the 4 proposed methodologies (ANNi-CS, ANNi-PSO, ANNi-GA and ANNiSA) were satisfactory to determine from 1 to up 16 input variables of the system. However, only four manipulated input variables were obtained to achieve the COPs desired. All proposed methodologies have a computational cost ≤ 1.2 s to obtain from 1 to up 4 simultaneously input variables to an specific number of iterations and solutions. This time is sufficient to determine: 4 optimized manipulated temperatures, maximize

J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

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Table 9 13 optimized variables to COP = 0.35. Variable

EXP

ANNi-CS

ANNi-PSO

ANNi-GA

ANNi-SA

Tin.GE −AB Tout .AB−GE Tin.AB−GE Tout .GE −CO Tin.CO Tout .Co Tin.EV ToutEV −AB Xout .AB−CO Xin.GE −CO Xout .GE −CO PAB PGE

88.7000 ± 2.8299 94.1497 ± 2.5743 86.4523 ± 1.2048 88.7000 ± 2.8299 48.0308 ± 2.5981 30.7675 ± 0.7568 34.2686 ± 6.7535 83.9207 ± 2.0206 52.1900 ± 1.4968 52.1900 ± 1.4968 54.3937 ± 1.1609 9.3000 ± 1.0000 20.4000 ± 0.5100

89.2336 95.1788 86.0110 87.6034 49.3377 30.6903 36.6468 85.1946 51.7162 52.0056 54.5006 8.9844 20.3374

88.2419 92.1061 87.1090 87.6705 49.3704 30.5752 36.3887 84.9098 51.3612 53.4371 55.0719 8.9966 20.8792

89.0109 94.1424 86.5649 86.1222 45.4326 30.7953 31.4893 84.8129 52.0650 51.8823 54.5131 8.9948 20.3976

87.8852 96.3607 87.5595 87.6908 50.2459 30.3756 40.9660 83.9740 51.1270 53.0717 53.6313 9.1574 20.6462

Table 10 16 optimized variables to COP = 0.35. Variable

EXP

ANNi-CS

ANNi-PSO

ANNi-GA

ANNi-SA

Tin.GE −AB Tin.EV −AB Tout .AB−GE Tin.AB−GE Tout .GE −CO ToutGE −AB Tin.CO Tout .Co Tin.EV ToutEV −AB Xin.AB−CO Xout .AB−CO Xin.GE −CO Xout .GE −CO PAB PGE

88.7000 ± 2.8299 83.9207 ± 2.0206 94.1497 ± 2.5743 86.4523 ± 1.2048 88.7000 ± 2.8299 81.7151 ± 0.6139 48.0308 ± 2.5981 30.7675 ± 0.7568 34.2686 ± 6.7535 83.9207 ± 2.0206 53.3695 ± 1.3761 52.1900 ± 1.4968 52.1900 ± 1.4968 54.3937 ± 1.1609 9.3000 ± 1.0000 20.4000 ± 0.5100

88.1956 82.0237 95.3701 85.3859 87.8030 81.6297 46.9894 30.2831 40.1433 83.3333 52.6237 53.0625 52.8798 54.5326 8.3000 20.2780

89.2540 81.9322 93.2138 85.9202 87.8588 81.5523 48.8900 30.9932 29.0144 83.9985 52.7220 51.0489 53.3311 54.7526 8.9584 20.7004

88.0325 82.0981 94.6189 86.5114 88.5531 81.5294 45.4326 30.2541 35.5131 83.6868 54.2459 51.8537 52.9576 54.7834 9.7836 20.2571

89.0342 82.3161 95.8409 86.7298 89.6439 81.8084 45.8543 30.3542 36.8613 83.5564 52.9020 53.5080 53.1603 53.5976 8.5126 20.4203

the COPs and improve the AHTER. Also, the errors obtained from the 4 methodologies were low (≤ 0.05). It is important to confirm that all methodologies implemented can maximize the COPs in AHTER online. With these methodologies it is possible to have the following benefits on the device operation: saving energy in the heat supply in the generator and evaporator; optimize multiple variables of an experimental test in real-time in order to control the input variables in the components of an AHTER. Consequently, the above mentioned methodologies allows to improve the operation of process, reducing production costs and maximizing efficiency online of the

Fig. 9. Maximize a COP’s from 0.36 to up 0.4 comparing both methodologies (ANNi-CS and ANNi-SA) with 2 manipulated variables (TGE and TEV ) optimized freely and 2 manipulated variables restricted (TCO and TAB ) considering as limit low constant temperatures.

AHTER system. In addition, these methodologies can be used in other different systems. Declaration of competing interest No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105801. Acknowledgments The authors thanks CONACYT for the financial support received in the scholarships of the graduated students. Jesús Emmanuel Solís-Pérez, Ulises Cruz-Jacobo and Edumis Viera-Martin acknowledges the support provided by CONACyT through the assignment doctoral and Master’s fellowships, respectively. José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: Cátedras CONACyT para jóvenes investigadores 2014.

Fig. 10. COP experimental (COPexp ) against COP simulated (COPANNi ) values for both ANNi models.

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J.E. Solís-Pérez, J.F. Gómez-Aguilar, J.A. Hernández et al. / Applied Soft Computing Journal 85 (2019) 105801

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