# Thermoeconomic Optimization with PSO Algorithm of Waste Heat Recovery Systems Based on Organic Rankine Cycle System for a Natural Gas Engine

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Description of the System

#### 2.2. Thermoeconomic Analysis

_{0}and s

_{0}were evaluated under reference conditions.

#### 2.3. Thermoeconomic Indicator

#### 2.4. Particle Swarm Optimization (PSO)

_{i}) in the search region. The two random numbers r

_{1}, r

_{2}in Equation (4) are generated independently in the range [0, 1].

_{1}and c

_{2}are the acceleration constants in Equation (14) and characterize the weighting of the stochastic acceleration relations that ensure that each particle is directed towards the pbest and gBest positions. The cognitive parameter (c

_{1}) signifies the confidence that the particle has in itself, and the social parameter (c

_{2}) is the confidence that the particle has in the swarm. Therefore, the adjustment of these constants changes the dispersion and integration of particles in the system [28]. Low constant values allow particles to move away from target regions before they are removed, while high values result in abrupt movement beyond target regions [29].

## 3. Results and Discussion

#### 3.1. Parametric Study on Thermoeconomic Indicator

#### 3.2. Optimization Method Selection

^{®}software (R2018b, MathWorks, Massachusetts, USA) [38]. The PSO algorithm presented the shortest computation times as shown in Table 4, with an average computational time of 14.2 min for the SORC considering all the objective functions and evaluated, which is 58.3% less than the solution with RPS, and 110.5% less than the solution obtained with the FMINCON function. This result with the PSO method was obtained using a particle number of 40, and a generation number of 30. This method emulates the behavior of flocks of birds and generates a population of candidate solutions, denoted as “particles,” and is based on the position and speed of these to look for the best global among the best local. On the other hand, the RPS algorithm, which is a variant of the PSO, also present good results but it requires more computational resources. This method does not have the tendency of best performance. However, it presents the repulsion and the best position of another randomly chosen particle, which makes it more robust than the PSO but with drawbacks in the recovery of the non-existence of solution. Finally, the longest time was presented for the FMINCON function, which is an optimization function belonging to the MathWorks mathematical software library, which satisfies limits in all iterations and is based on the Hessian of the objective function supported by the structure of the Lagrange multiplier.

#### 3.3. Thermoeconomic Optimization

- $\mathrm{FObj}1=\mathrm{LCOE}\left({\mathrm{n}}_{\mathrm{t}},\text{}{\mathrm{n}}_{\mathrm{t}}\text{}\mathrm{Ap},\text{}\mathrm{Tcond},\text{}\mathrm{rp}\right)$,
- $\mathrm{FObj}2=\mathrm{PBP}\left({\mathrm{n}}_{\mathrm{t}},\text{}{\mathrm{n}}_{\mathrm{t}}\text{}\mathrm{Ap},\text{}\mathrm{Tcond},\text{}\mathrm{rp}\right)$,
- $\mathrm{FObj}3=\mathrm{SIR}\_\mathrm{Var}\left({\mathrm{n}}_{\mathrm{t}},\text{}{\mathrm{n}}_{\mathrm{t}}\text{}\mathrm{Ap},\text{}\mathrm{Tcond},\text{}\mathrm{rp}\right)$, and the objective energetic function to maximize is
- $\mathrm{Fobj}4=\mathrm{Wneto}\left({\mathrm{n}}_{\mathrm{t}},\text{}{\mathrm{n}}_{\mathrm{t}}\text{}\mathrm{Ap},\text{}\mathrm{Tcond},\text{}\mathrm{rp}\right)$.

#### 3.3.1. SORC Optimization

#### 3.3.2. Optimization RORC

#### 3.3.3. DORC Optimization

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DORC | Double pressure Organic Rankine Cycle |

ORC | Organic Rankine Cycle |

RORC | Regenrative Organic Rankine Cycle |

SORC | Simple Organic Rankine Cycle |

WHR | Waste Heat Recovery |

Nomenclature | |

${C}_{p}$ | Specific heat at constant pressure [J/kg·K] |

$E$ | Energy [J] |

${n}_{t}$ | Turbine efficiency [%] |

${n}_{p}$ | Pump efficiency [%] |

$ex$ | Specific exergy [kJ/kg] |

$h$ | Specific enthalpy [kJ/kg] |

$\dot{m}$ | Mass flow rate [kg/s] |

$Q$ | Heat [J] |

$R$ | Universal gas constant [atm·L/mol·K] |

$T$ | Temperature [K] |

$t$ | Time [s] |

$\dot{W}$ | Power [kW] |

${X}_{i}$ | Molar gas fraction |

Greek Letters | |

$\tau $ | Hours of operation per year |

Subscripts | |

$D$ | Destroyed |

in | Input |

out | Output |

G | Gases |

VC | Control volume |

o | Reference condition |

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**Figure 2.**Graphic representation of the different system configurations for heat recovery in exhaust gases.

**Figure 4.**Effect of evaporation pressure on thermoeconomic indicators, (

**a**) leveled cost of energy (LCOE), (

**b**) specific investment cost (SIC), and (

**c**) payback period (PBP).

**Figure 5.**Effect of turbine efficiency on thermoeconomic indicators, (

**a**) LCOE, (

**b**) SIC, and (

**c**) PBP.

**Figure 6.**Effect of evaporator and condenser pinch point temperature on thermoeconomic indicators; (

**a**,

**d**) LCOE, (

**b**,

**e**) SIC, and (

**c**,

**f**) PBP.

**Figure 7.**Variation of the target function with the number of generations and particles for the simple organic Rankine cycle (SORC) system; (

**a**) minimization of LCOE, (

**b**) SIC, (

**c**) PBP, and (

**d**) power maximization.

**Figure 8.**Distribution of decision variables with the number of generations and a population of 10 for the minimization of LCOE (

**a**–

**c**); SIC (

**d**–

**f**); PBP (

**g**–

**i**) and power maximization (

**j**–

**l**) in the SORC.

**Figure 9.**Variation of the target function with the number of generations and particles for the regeneartive organic Rankine cycle (RORC) system; (

**a**) minimization of LCOE, (

**b**) SIC, (

**c**) PBP, and (

**d**) power maximization.

**Figure 10.**Distribution of decision variables with the number of generations and a population of 10 for the minimization of LCOE (

**a**–

**c**); SIC (

**d**–

**f**); PBP (

**g**–

**i**) and power maximization (

**j**–

**l**) in the RORC.

**Figure 11.**Variation of the target function with the number of generations and particles for the double-stage ORC (DORC) system; (

**a**) minimization of LCOE, (

**b**) SIC, (

**c**) PBP, and (

**d**) power maximization.

**Figure 12.**Distribution of decision variables with the number of generations and a population of 10 for the minimization of LCOE (

**a**–

**c**); SIC (

**d**–

**f**); PBP (

**g**–

**i**) and power maximization (

**j**–

**l**) in the DORC.

Economic Constant | Value | Ref |
---|---|---|

Interest rate, ${\mathrm{i}}_{\mathrm{eff}}$ | 5% | [22,23] |

Nominal Scaling Ratio, ${\mathrm{r}}_{\mathrm{n}}$ | 5% | [19,24] |

Lifetime of the project, n | 20 years | [19,25] |

Hours of operation per year, $\mathsf{\tau}$ | 7446 h | [26] |

Parameter | Value or Range |
---|---|

Number of particles (Population) | 10, 20, 30, 40 |

Number iterations (Generations) | 50 |

Weight Inertia (w) | 0.1–0.9 |

Cognitive parameter (c_{1}) | 0.5–1.5 |

Social parameter (c_{2}) | 0.5–1.5 |

Variables | SORC/RORC | DORC | |||
---|---|---|---|---|---|

Symbol | Unit | Minimum | Maximum | Minimum | Maximum |

${\mathbf{n}}_{\mathbf{t}}$ | % | 60 | 90 | 60 | 80 |

${\mathbf{n}}_{\mathbf{p}}$ | % | 55 | 85 | 60 | 85 |

$\mathbf{AP}$ | °C | 20 | 60 | 30 | 35 |

$\mathbf{Tcond}$ | °C | 55 | 88 | 11 | 16 |

${\mathbf{P}}_{\mathit{e}\mathit{v}\mathit{a}\mathit{p}}$ | MPa | 0.2 | 3.58 | 1.01 | 1.68 |

Configuration | Objective Function | PSO Time (Min) | RPS Time (Min) | FMINCON Time (Min) |
---|---|---|---|---|

SORC | LCOE min | 18.5 | 25.6 | 28.4 |

SIC min | 12.2 | 18.6 | 19.6 | |

PBP min | 15.8 | 31.3 | 21.2 | |

Power max | 10.2 | 14.4 | 14.7 | |

Average time (min) | 14.2 | 22.5 | 29.8 | |

RORC | LCOE min | 10.4 | 11.9 | 19.6 |

SIC min | 12.5 | 18.2 | 22.3 | |

PBP min | 11.7 | 9.2 | 16.3 | |

Power max | 8.1 | 4.5 | 14.7 | |

Average time (min) | 10.7 | 11.0 | 14.0 | |

DORC | LCOE min | 6.4 | 5.6 | 12.8 |

SIC min | 8.9 | 7.8 | 12.0 | |

PBP min | 10.4 | 9.8 | 13.5 | |

Power max | 19.4 | 12.8 | 29.3 | |

Average time (min) | 11.3 | 9.0 | 18.7 |

Parameter | Optimized Value | ||||
---|---|---|---|---|---|

Symbol | Unit | Solution I | Solution II | Solution III | Solution IV |

${\mathbf{n}}_{\mathbf{t}}$ | % | 90 | 90 | 90 | 90 |

${\mathbf{n}}_{\mathbf{p}}$ | % | 85 | 85 | 85 | 85 |

$\mathbf{AP}$ | °C | 31.36 | 31.24 | 31.19 | 35 |

${\mathbf{T}}_{\mathbf{pinch}-\mathbf{co}}$ | °C | 15 | 11 | 11 | 11 |

${\mathbf{P}}_{\mathbf{evap}}$ | MPa | 3.58 | 3.58 | 3.58 | 3.58 |

$\mathsf{\eta}\mathbf{t}{\mathbf{h}}_{\mathbf{engine}-\mathbf{ORC}}$ | % | 41.38 | 41.38 | 41.38 | 41.38 |

$\mathbf{\Delta}{\mathsf{\eta}}_{\mathbf{ther}}$ | % | 2.79 | 2.79 | 2.79 | 2.80 |

${\mathsf{\eta}}_{\mathbf{I},\mathbf{ORC}}$ | % | 24.61 | 24.61 | 24.61 | 24.67 |

${\mathsf{\epsilon}}_{\mathbf{hr}}$ | % | 40.77 | 40.77 | 40.77 | 40.77 |

${\mathsf{\eta}}_{\mathbf{I},\mathbf{global}}$ | % | 10.04 | 10.04 | 10.04 | 10.06 |

${\mathsf{\eta}}_{\mathbf{II},\mathbf{ORC}}$ | % | 51.82 | 51.81 | 51.81 | 51.93 |

$\mathbf{\Delta}\mathbf{BSFC}$ | % | 6.72 | 6.726 | 6.726 | 6.74 |

LCOE | $USD/kWh | 0.1222 | 0.1222 | 0.1222 | 0.1223 |

SIC | $USD/kWh | 1224.54 | 1220.64 | 1220.65 | 1221.94 |

PBP | years | 5.12 | 5.1035 | 5.1034 | 5.1096 |

${\dot{\mathbf{W}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ | kW | 126.83 | 126.82 | 126.82 | 127.09 |

Parameter | Optimized Value | ||||
---|---|---|---|---|---|

Symbol | Unit | Solution I | Solution II | Solution III | Solution IV |

${\mathbf{n}}_{\mathbf{t}}$ | % | 90 | 90 | 90 | 90 |

${\mathbf{n}}_{\mathbf{p}}$ | % | 85 | 85 | 85 | 85 |

$\mathbf{AP}$ | °C | 15 | 14.52 | 15 | 14.52 |

${\mathbf{T}}_{\mathbf{pinch}-\mathbf{co}}$ | °C | 11 | 11 | 11 | 11 |

${\mathbf{P}}_{\mathbf{evap}}$ | MPa | 3.58 | 3.58 | 3.58 | 3.58 |

$\mathsf{\eta}\mathbf{t}{\mathbf{h}}_{\mathbf{engine}-\mathbf{ORC}}$ | % | 42.41 | 42.41 | 42.41 | 42.41 |

$\mathbf{\Delta}{\mathsf{\eta}}_{\mathbf{ther}}$ | % | 3.82 | 3.82 | 3.82 | 3.82 |

${\mathsf{\eta}}_{\mathbf{I},\mathbf{ORC}}$ | % | 33.7 | 33.73 | 33.7 | 33.75 |

${\mathsf{\epsilon}}_{\mathbf{hr}}$ | % | 40.7 | 40.77 | 40.7 | 40.77 |

${\mathsf{\eta}}_{\mathbf{I},\mathbf{global}}$ | % | 13.75 | 13.76 | 13.75 | 13.75 |

${\mathsf{\eta}}_{\mathbf{II},\mathbf{ORC}}$ | % | 70.9 | 71 | 70.9 | 71 |

$\mathbf{\Delta}\mathbf{BSFC}$ | % | 8.98 | 8.99 | 8.98 | 8.99 |

LCOE | $USD/kWh | 0.1110 | 0.111 | 0.1110 | 0.111 |

SIC | $USD/kWh | 1114.07 | 1113.73 | 1114.07 | 11137 |

PBP | years | 4.68 | 4.686 | 4.6879 | 4.68 |

${\dot{\mathbf{W}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ | kW | 173.64 | 173.79 | 173.64 | 173.79 |

Parameter | Optimized Value | ||||
---|---|---|---|---|---|

Symbol | unit | Solution I | Solution II | Solution III | Solution IV |

${\mathbf{n}}_{\mathbf{t}}$ | % | 80 | 80 | 80 | 80 |

${\mathbf{n}}_{\mathbf{p}}$ | % | 85 | 85 | 85 | 85 |

$\mathbf{AP}$ | °C | 30 | 32.7 | 32 | 35 |

${\mathbf{T}}_{\mathbf{pinch}-\mathbf{co}}$ | °C | 16 | 16 | 16 | 16 |

${\mathbf{P}}_{\mathbf{evap}}$ | MPa | 1.68 | 1.68 | 1.68 | 1.68 |

$\mathsf{\eta}\mathbf{t}{\mathbf{h}}_{\mathbf{engine}-\mathbf{ORC}}$ | % | 40.78 | 40.78 | 40.78 | 40.78 |

$\mathbf{\Delta}{\mathsf{\eta}}_{\mathbf{ther}}$ | % | 2.19 | 2.19 | 2.19 | 2.19 |

${\mathsf{\eta}}_{\mathbf{I},\mathbf{ORC}}$ | % | 19.31 | 19.32 | 19.32 | 19.33 |

${\mathsf{\epsilon}}_{\mathbf{hr}}$ | % | 40.86 | 40.85 | 40.85 | 40.86 |

${\mathsf{\eta}}_{\mathbf{I},\mathbf{global}}$ | % | 7.89 | 7.89 | 7.89 | 7.9 |

${\mathsf{\eta}}_{\mathbf{II},\mathbf{ORC}}$ | % | 40.62 | 40.64 | 40.64 | 40.66 |

$\mathbf{\Delta}\mathbf{BSFC}$ | % | 5.35 | 5.35 | 5.35 | 5.35 |

LCOE | $USD/kWh | 0.1613 | 0.1613 | 0.1613 | 0.1613 |

SIC | $USD/kWh | 1610.63 | 1610.50 | 1610.51 | 1610.7 |

PBP | years | 5.67 | 5.67 | 5.67 | 5.67 |

${\dot{\mathbf{W}}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ | kW | 99.43 | 99.48 | 99.48 | 99.52 |

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## Share and Cite

**MDPI and ACS Style**

Valencia Ochoa, G.; Acevedo Peñaloza, C.; Duarte Forero, J. Thermoeconomic Optimization with PSO Algorithm of Waste Heat Recovery Systems Based on Organic Rankine Cycle System for a Natural Gas Engine. *Energies* **2019**, *12*, 4165.
https://doi.org/10.3390/en12214165

**AMA Style**

Valencia Ochoa G, Acevedo Peñaloza C, Duarte Forero J. Thermoeconomic Optimization with PSO Algorithm of Waste Heat Recovery Systems Based on Organic Rankine Cycle System for a Natural Gas Engine. *Energies*. 2019; 12(21):4165.
https://doi.org/10.3390/en12214165

**Chicago/Turabian Style**

Valencia Ochoa, Guillermo, Carlos Acevedo Peñaloza, and Jorge Duarte Forero. 2019. "Thermoeconomic Optimization with PSO Algorithm of Waste Heat Recovery Systems Based on Organic Rankine Cycle System for a Natural Gas Engine" *Energies* 12, no. 21: 4165.
https://doi.org/10.3390/en12214165