# Maximizing Total Profit of Thermal Generation Units in Competitive Electric Market by Using a Proposed Particle Swarm Optimization

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## Abstract

**:**

## 1. Introduction

- (1)
- Propose the effective NCHM for handling constraints.
- (2)
- Propose the high performance PPSO.
- (3)
- Consider valve effects on thermal generation units for the considered problem.

- (1)
- PPSO method has few control parameters, population and the number of iterations. Therefore, the setting of the two parameters is simple.
- (2)
- The process of evaluating solution quality is easily and simply performed by calculating fitness function.

- (1)
- Reach very high success rate with 100%: Implemented methods using NCHM always reaches all successful runs but the same implemented methods without using NCHM must suffer much lower than 100% for success rate.
- (2)
- Converge to high quality solutions: NCHM supports implemented methods to find global optimum solutions with fast speed and reach high stability.
- (3)
- PPSO method always reaches better results than other PSO, SSA, MDE and previous methods.
- (4)
- PPSO method is faster than approximately all other methods for study cases.

- (1)
- The most appropriate values for the population and the number of iterations are not easy to select. In fact, higher values can result in better results but simulation time is still increased correspondingly. If high values are set, all methods have the same best solution and the evaluation is not exactly performed. In this case, real performance of PPSO method cannot be shown.
- (2)
- The procedure of applying PPSO method is a long iterative algorithm. Therefore, the implementation procedure must be careful and verification procedure must be serious.

## 2. Problem Formulation

#### 2.1. Objective Function

_{RP}) is higher than the price of delivered power (Price

_{DP}). The total revenue (TR) and total cost (TC) are calculated by [19]:

_{n}is on/off operation status of the nth thermal generation unit. The parameter has two values only, 1 for on operation status and 0 for off operation status. In addition, FSU

_{n}is start-up fuel cost of the nth thermal generation unit. The determination of the status and the start-up fuel cost are really important for finding optimal generation and optimal reserve as unit commitment problem is considered. However, in the study, we consider pure economic load dispatch problem and the assumption is that all thermal generation units are working. Therefore, the start-up fuel cost and on/off operation status can be neglected, and the result does not influence the task of determining optimal generation and optimal reserve.

_{DP}and Price

_{RP}are not fixed values and they are different for different time periods in a day [18,19,44]. In fact, this issue was demonstrated [18] and different values for these prices were then applied [19]. In this paper, we consider only one period for pure economic load dispatch problem. Therefore, we only considered one fixed value for Price

_{DP}and one fixed value for Price

_{RP}for each study case.

#### 2.2. The Consisdered Constraints

_{loss}is the total power losses in all transmission lines; TP

_{G}is total power generation of all power sources [46] that can supply electricity to loads. TP

_{G}can be expressed in detail as follows:

_{i}is the power generation of the ith hydropower plant; PP

_{j}is the jth power generation of photovoltaic system; PW

_{k}is the power generation of the kth wind turbine; and PG

_{n}is the power generation of the nth thermal generation unit; I is the number of hydropower plants; J is the number of photovoltaic systems; and K is the number of wind turbines

## 3. Applied PSO Methods

#### 3.1. CF-PSO and IW-PSO

#### 3.2. TVIW-PSO and TVAC-PSO

_{1}and c

_{2}, and the new velocity of TVAC-PSO is determined by:

_{1initial}and c

_{1End}are initial and final cognitive acceleration factors, respectively. c

_{2initial}and c

_{2End}are initial and final social acceleration factors, respectively. The factors are predetermined and fixed during the application of TVAC-PSO for a typical optimization problem.

#### 3.3. PG-PSO

#### 3.4. The Proposed PSO Method

## 4. Implementation of PPSO Method for the Considered Problem

#### 4.1. The New Constriaint Handling Method for Reseve Power

_{n}-LB

_{n}) but it is totally different if Equation (17) is considered as another main constraint. In fact, Equation (17) indicates that the reserve must not be higher than (UB

_{n}-PG

_{n}) while PG

_{n}can be higher than LB

_{n}. So, we suggest that upper bound of reserve should be determined first and then real reserve can be constrained by the upper bound. Namely, the two formulas below should be used.

#### 4.2. Main Steps of the Proposed Method for the Implementation

#### 4.2.1. Selection of Control Variables and Population Initialization

_{p}) as the following expression:

_{p}is not infinite and seriously constrained by the upper bound Po

^{UB}and lower bound Po

^{LB}, which are, respectively, determined by:

_{p}(where p = 1, …, No

_{p}) in the population is randomly produced by:

#### 4.2.2. Calculation of Dependent Variables

#### 4.2.3. Correction for Produced Control Variables

#### 4.2.4. Handling Violation of Power Demand and Reserve Demand

#### 4.2.5. Handling Violation of the First Thermal Generating Unit

#### 4.2.6. Fitness Function

_{1}, K

_{2}, K

_{3}and K

_{4}are penalty factors and determined by experiment.

#### 4.3. Establishing Limits of Velocity and Producing Initial Velocity

#### 4.4. Termination Criterion for Iterative Algorithm

_{max}), it is a successful run. On the contrary, if fitness function is higher than total profit (corresponding to penalty terms in (41)–(44) are higher than zero), there is at least one violated constraint and this is an unsuccessful run.

#### 4.5. The Entire Search Process of PPSO for the Considered Problem

_{p}and G

_{max}for the proposed method.

_{1}and RG

_{1}by using (38) and (39), respectively.

_{n}(n = 1, …, N) by using (33).

_{Gbest}.

_{Gbest}and current iteration G to 1.

^{LB}if $V{e}_{p}^{new}$ < Ve

^{LB}and $V{e}_{p}^{new}$ is set to Ve

^{UB}if $V{e}_{p}^{new}$ > Ve

^{UB}.

_{n}(n = 2, …, N) by using (40).

_{1}and RG

_{1}by using (38) and (39), respectively.

_{n}(n = 1, …, N) by using (33).

_{best,p}and $P{o}_{p}^{new}$(p = 1, …, No

_{p}) to keep better one, and set kept one to Po

_{best,p}.

_{best,p}, determine the best solution with the lowest and set to Po

_{Gbest}.

_{max}, stop search process. Otherwise, set G = G + 1 and back to Step 10.

## 5. Numerical Results

- Test system 1: Three units with convex fuel cost function shown in Equation (1)
- Test system 2: Ten units with convex fuel cost function shown in Equation (1)
- Test system 3: Twenty units with nonconvex fuel cost shown in Equation (2)
- Case 1: Total revenue and total fuel cost are obtained by using Equations (5) and (6)
- Case 2: Total revenue and total fuel cost are obtained by using Equations (7) and (8)

- (1)
- No
_{p}= 5 and G_{max}= 5 for test system 1 - (2)
- No
_{p}= 20 and G_{max}= 100 for test system 2 - (3)
- No
_{p}= 30 and G_{max}= 500 for test systems 3

#### 5.1. The Impact of the Proposed NCHM on Results

- (1)
- Methods using NCHM can reach the highest SR with 100% but SR of the methods without using NCHM is much lower, only from 83.3% to 92.5%.
- (2)
- NCHM can support methods to find the global optimum solutions, high search stability and low possibility to low quality solutions.

#### 5.2. Comparison for Test System 1

#### 5.3. Comparison for Test System 2

#### 5.4. Comparison for Test System 3

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

IW-PG-PSO | Inertia weight factor and pseudo gradient -based particle swarm optimization |

CF-PG-PSO | Constriction factor and Pseudo gradient-based particle swarm optimization |

TVIW-PSO | Time varying inertia weight factor-based particle swarm optimization |

LF-HLN-EF | Lagrange function-based Hopfield neuron network method with Error function |

LF-HLN-THF | Lagrange function-based Hopfield neuron network method with hyperbolic tangent function |

LF-HLN-GdF | Lagrange function-based Hopfield neuron network method with Gudermanian function |

LF-HLN-GF | Lagrange function-based Hopfield neuron network method with Gompertz function |

LF-HLN-LF | Lagrange function-based Hopfield neuron network method with Logistic function |

## Nomenclature

${\mathsf{\alpha}}_{n},\text{}{\mathsf{\beta}}_{n},\text{}{\mathsf{\chi}}_{n},\text{}{\mathsf{\delta}}_{n},\text{}{\mathsf{\epsilon}}_{n}$ | Known coefficients of fuel cost function of the nth unit |

c_{1}, c_{2} | Acceleration constants |

D | Forecasted power demand |

ΔD_{p} | Penalty term for the violation of power demand corresponding to the pth solution |

ΔRD_{p} | Penalty term for the violation of reserve demand corresponding to the pth solution |

ΔPG_{1,p} | Penalty term for the violation of power generation of the first thermal generation unit corresponding to the pth solution |

ΔRG_{1,p} | Penalty term for the violation of reserve of the first thermal generation unit corresponding to the pth solution |

ε_{1}, ε_{2}, ε_{3} | Random numbers generated in range of [0,1] |

F_{p} | Fitness function of old position Po_{p} |

${F}_{p}^{new}$ | Fitness function of new position |

F_{n} | Fuel cost function of the nth thermal generation unit as producing power only |

${F}_{n}^{\prime}$ | Fuel cost function of the nth thermal generation unit as producing power and reserve |

G | Current iteration |

G_{max} | Maximum iteration |

LB_{n} | Lower bound of generation of the nth thermal generating unit |

N | Number of thermal generating units |

n | Unit index |

No_{p} | Population size |

$P{o}_{p}^{new},\text{}V{e}_{p}^{new}$ | New position and new velocity of the pth particle |

$P{o}_{p}^{\mathrm{Pr}e}$ | The previous position of old position |

PG_{n} | Power generation of the nth thermal generating unit |

PG_{1} | Power generation of the first thermal generation unit |

Po_{best,p} | The so-far best position of the pth particle |

Po_{Gbest} | The so-far best position of all particles |

RD | Forecasted reserve power demand |

RG_{n} | Reserve generation of the nth thermal generating unit |

RG_{1} | Reserve of the first thermal generation unit |

TC | Total cost |

TP | Total profit |

TR | Total revenue |

UB_{n} | Upper bound of generation of the nth thermal generating unit |

Ve^{LB}, Ve^{UB} | Lower bound and upper bound of velocity |

Ve_{p}, Po_{p} | Old velocity and position of the pth particle |

ω | Inertia weigh factor |

ω_{min}, ω_{max} | Minimum and maximum value of inertia weigh factor |

## Appendix A

n | ${\mathsf{\chi}}_{\mathit{n}}$ | ${\mathsf{\beta}}_{\mathit{n}}$ | ${\mathsf{\alpha}}_{\mathit{n}}$ | $\mathit{L}{\mathit{B}}_{\mathit{n}}^{}$ (MW) | $\mathit{U}{\mathit{B}}_{\mathit{n}}^{}$ (MW) |
---|---|---|---|---|---|

1 | 0.002 | 10 | 500 | 100 | 600 |

2 | 0.0025 | 8 | 300 | 100 | 400 |

3 | 0.005 | 6 | 100 | 50 | 200 |

n | ${\mathsf{\chi}}_{\mathit{n}}$ | ${\mathsf{\beta}}_{\mathit{n}}$ | ${\mathsf{\alpha}}_{\mathit{n}}$ | $\mathit{L}{\mathit{B}}_{\mathit{n}}^{}$ (MW) | $\mathit{U}{\mathit{B}}_{\mathit{n}}^{}$ (MW) |
---|---|---|---|---|---|

1 | 0.0004800 | 16.19 | 1000 | 150 | 455 |

2 | 0.0003100 | 17.26 | 970 | 150 | 455 |

3 | 0.00200 | 16.60 | 700 | 20 | 130 |

4 | 0.0021100 | 16.50 | 680 | 20 | 130 |

5 | 0.0039800 | 19.70 | 450 | 25 | 162 |

6 | 0.0071200 | 22.26 | 370 | 20 | 80 |

7 | 0.0007900 | 27.74 | 480 | 25 | 85 |

8 | 0.0041300 | 25.92 | 660 | 10 | 55 |

9 | 0.0022200 | 27.27 | 665 | 10 | 55 |

10 | 0.0017300 | 27.79 | 670 | 10 | 55 |

n | ${\mathsf{\chi}}_{\mathit{n}}$ | ${\mathsf{\beta}}_{\mathit{n}}$ | ${\mathsf{\alpha}}_{\mathit{n}}$ | ${\mathsf{\delta}}_{\mathit{n}}$ | ${\mathsf{\epsilon}}_{\mathit{n}}$ | $\mathit{L}{\mathit{B}}_{\mathit{n}}^{}$ (MW) | $\mathit{U}{\mathit{B}}_{\mathit{n}}^{}$ (MW) |
---|---|---|---|---|---|---|---|

1 | 1000 | 18.19 | 0.00068 | 100 | 0.0840 | 150 | 600 |

2 | 970 | 19.26 | 0.00071 | 100 | 0.0840 | 50 | 200 |

3 | 600 | 19.8 | 0.00650 | 150 | 0.0630 | 50 | 200 |

4 | 700 | 19.1 | 0.00500 | 120 | 0.0770 | 50 | 200 |

5 | 420 | 18.1 | 0.00738 | 100 | 0.0840 | 50 | 160 |

6 | 360 | 19.26 | 0.00612 | 0 | 0 | 20 | 100 |

7 | 490 | 17.14 | 0.00790 | 0 | 0 | 25 | 125 |

8 | 660 | 18.92 | 0.00813 | 0 | 0 | 50 | 150 |

9 | 765 | 18.27 | 0.00522 | 0 | 0 | 50 | 200 |

10 | 770 | 18.92 | 0.00573 | 0 | 0 | 30 | 150 |

11 | 800 | 16.69 | 0.00480 | 0 | 0 | 100 | 300 |

12 | 970 | 16.76 | 0.00310 | 0 | 0 | 150 | 500 |

13 | 900 | 17.36 | 0.00850 | 0 | 0 | 40 | 160 |

14 | 700 | 18.7 | 0.00511 | 0 | 0 | 20 | 130 |

15 | 450 | 18.7 | 0.00398 | 0 | 0 | 25 | 185 |

16 | 370 | 14.26 | 0.07120 | 0 | 0 | 20 | 80 |

17 | 480 | 19.14 | 0.00890 | 0 | 0 | 30 | 85 |

18 | 680 | 18.92 | 0.00713 | 0 | 0 | 30 | 120 |

19 | 700 | 18.47 | 0.00622 | 0 | 0 | 40 | 120 |

20 | 850 | 19.79 | 0.00773 | 0 | 0 | 30 | 100 |

Parameters | System 1 | System 2 | System 3 | |||
---|---|---|---|---|---|---|

D (MW) | 1100 | 1100 | 1500 | 1500 | 2500 | 2500 |

RD (MW) | 100 | 100 | 150 | 150 | 300 | 300 |

Price_{DP} ($/MWh) | 11.3 | 11.3 | 31.65 | 31.65 | 31.6 | 30 |

Price_{RP} ($/MWh) | 33.9 | 0.0452 | 158.25 | 0.3165 | 158.25 | 0.12 |

r | 0.005 | 0.005 | 0.05 | 0.005 | 0.05 | 0.005 |

n | Case 1 | Case 2 | ||
---|---|---|---|---|

PG_{n} (MW) | RG_{n} (MW) | PG_{n} (MW) | RG_{n} (MW) | |

1 | 324.5042 | 100 | 324.5076 | 100 |

2 | 400 | 0 | 400 | 0 |

3 | 200 | 0 | 200 | 0 |

n | Case 1 | Case 2 | ||
---|---|---|---|---|

PG_{n} (MW) | RG_{n} (MW) | PG_{n} (MW) | RG_{n} (MW) | |

1 | 455 | 0 | 455 | 0 |

2 | 455 | 0 | 455 | 0 |

3 | 130 | 0 | 130 | 0 |

4 | 130 | 0 | 130 | 0 |

5 | 162 | 0 | 162 | 0 |

6 | 80 | 0 | 80 | 0 |

7 | 25 | 60 | 25 | 60 |

8 | 42.9997 | 12.0003 | 43 | 12 |

9 | 10 | 45 | 10 | 45 |

10 | 10 | 32.9997 | 10 | 33 |

n | Case 1 | Case 2 | ||
---|---|---|---|---|

PG_{n} (MW) | RG_{n} (MW) | PG_{n} (MW) | RG_{n} (MW) | |

1 | 599.5146 | 0 | 600 | 0 |

2 | 50.1367 | 148.7997 | 199.2433 | 0 |

3 | 50.0686 | 3.7282 | 50 | 0.0056 |

4 | 50 | 0 | 50.6444 | 0 |

5 | 92.4212 | 0 | 90.7484 | 13.0514 |

6 | 27.7061 | 3.4081 | 20.0254 | 5.194 |

7 | 123.4713 | 0 | 125 | 0 |

8 | 51.7433 | 29.7412 | 50.5536 | 0 |

9 | 140.986 | 0 | 107.5 | 0 |

10 | 30 | 0.249 | 50.4308 | 0 |

12 | 278.2058 | 0 | 300 | 0 |

13 | 463.971 | 0 | 401.3243 | 0 |

14 | 139.3174 | 0 | 110.9913 | 0 |

15 | 20 | 36.7794 | 67.3715 | 0.0325 |

16 | 185 | 0 | 57.1231 | 0 |

17 | 40.4855 | 0 | 35.2791 | 0.0142 |

18 | 35.6628 | 18.5147 | 30 | 0.2627 |

19 | 30 | 0 | 44.9579 | 0.7649 |

20 | 54.6993 | 0.0504 | 78.7799 | 0.8343 |

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**Figure 1.**The flowchart of solving the considered problem by using proposed particle swarm optimization (PPSO) method.

**Figure 2.**Maximum total profit (MTP) and average total profit (ATP) obtained by particle swarm optimization (PSO) methods with and without using NCHM for case 1 of system 1.

**Figure 8.**Convergence characteristic of implemented methods corresponding to the best run for case 1 of system 1.

**Figure 9.**Convergence characteristic of implemented methods corresponding to the best run for case 2 of system 1.

**Figure 12.**Convergence characteristic of implemented methods corresponding to the best run for case 1 of system 2.

**Figure 13.**Convergence characteristic of implemented methods corresponding to the best run for case 2 of system 2.

**Figure 18.**Higher MTP and ATP obtained by PPSO as comparing to other methods for case 1 of system 3.

**Figure 19.**Higher MTP and ATP obtained by PPSO as comparing to other methods for case 2 of system 3.

**Figure 22.**Convergence characteristic of implemented methods corresponding to the best run for case 1 of system 3.

**Figure 23.**Convergence characteristic of implemented methods corresponding to the best run for case 2 of system 3.

**Table 1.**Better maximum total profit (MTP) and average total profit (ATP) in $/h and % by using NCHM for test system 1.

Method | Case 1 | Case 2 | ||||||
---|---|---|---|---|---|---|---|---|

Higher MTP | Higher ATP | Higher MTP | Higher ATP | |||||

In $/h | In % | In $/h | In % | In $/h | In % | In $/h | In % | |

PSO | 1.45 | 0.13 | 130.40 | 15.85 | 1.23 | 0.11 | 236.62 | 30.86 |

IW-PSO | 0.23 | 0.02 | 146.86 | 18.49 | 0.60 | 0.05 | 193.05 | 22.92 |

CF-PSO | 0.33 | 0.03 | 114.68 | 13.76 | 0.65 | 0.06 | 151.21 | 17.39 |

PG-PSO | 0.10 | 0.01 | 125.65 | 14.67 | 1.45 | 0.13 | 194.86 | 23.19 |

IW-PG-PSO | 0.23 | 0.02 | 76.28 | 8.38 | 0.40 | 0.04 | 178.47 | 20.32 |

CF-PG-PSO | 0.75 | 0.07 | 88.61 | 9.66 | 0.54 | 0.05 | 142.25 | 15.82 |

TVIW-PSO | 0.13 | 0.01 | 168.25 | 21.31 | 1.02 | 0.09 | 314.01 | 42.90 |

TVAC-PSO | 0.37 | 0.03 | 179.72 | 22.30 | 0.90 | 0.08 | 261.25 | 33.78 |

PPSO | 0.02 | 0.00 | 69.06 | 7.35 | 0.45 | 0.04 | 199.24 | 23.04 |

Method | Case 1 | Case 2 | ||||||
---|---|---|---|---|---|---|---|---|

Higher MTP | Higher ATP | Higher MTP | Higher ATP | |||||

In $/h | In % | In $/h | In % | In $/h | In % | In $/h | In % | |

PSO | 391.03 | 2.76 | 3580.52 | 34.21 | 196.80 | 1.46 | 4146.69 | 45.99 |

IW-PSO | 787.98 | 5.72 | 3300.27 | 31.08 | 115.63 | 0.86 | 4766.99 | 52.09 |

CF-PSO | 938.57 | 6.89 | 4026.77 | 40.35 | 114.78 | 0.85 | 4932.34 | 54.35 |

PG-PSO | 53.79 | 0.37 | 2694.35 | 23.64 | 75.03 | 0.55 | 2667.70 | 23.35 |

IW-PG-PSO | 275.97 | 1.93 | 2893.48 | 25.89 | 160.95 | 1.19 | 3027.44 | 27.41 |

CF-PG-PSO | 178.00 | 1.24 | 3533.46 | 33.79 | 79.73 | 0.59 | 3137.16 | 28.90 |

TVIW-PSO | 331.59 | 2.33 | 4117.14 | 41.75 | 124.70 | 0.92 | 4838.27 | 52.94 |

TVAC-PSO | 414.13 | 2.93 | 3291.83 | 30.42 | 114.91 | 0.85 | 3335.88 | 30.96 |

PPSO | 39.15 | 0.27 | 1741.05 | 13.98 | 11.13 | 0.08 | 1343.79 | 10.46 |

Method | Case 1 | Case 2 | ||||||
---|---|---|---|---|---|---|---|---|

Higher MTP | Higher ATP | Higher MTP | Higher ATP | |||||

In $/h | In % | In $/h | In % | In $/h | In % | In $/h | In % | |

PSO | 20.24 | 0.10 | 2263.72 | 12.56 | 281.76 | 1.95 | 2277.23 | 19.11 |

IW-PSO | 28.26 | 0.14 | 2173.71 | 12.07 | 146.23 | 1.01 | 2376.21 | 20.24 |

CF-PSO | 425.54 | 2.09 | 2155.86 | 11.89 | 174.51 | 1.21 | 1852.81 | 14.95 |

PG-PSO | 241.38 | 1.17 | 1302.57 | 6.84 | 146.42 | 1.01 | 919.01 | 6.87 |

IW-PG-PSO | 296.20 | 1.44 | 1070.75 | 5.62 | 286.23 | 1.97 | 612.94 | 4.47 |

CF-PG-PSO | 30.34 | 0.15 | 834.64 | 4.29 | 124.28 | 0.85 | 850.62 | 6.40 |

TVIW-PSO | 357.07 | 1.75 | 2101.92 | 11.59 | 73.69 | 0.51 | 628.74 | 4.68 |

TVAC-PSO | 315.84 | 1.54 | 2078.20 | 11.34 | 209.16 | 1.44 | 1286.83 | 9.85 |

PPSO | 83.30 | 0.40 | 611.88 | 3.07 | 83.32 | 0.57 | 670.91 | 4.81 |

Method | MTP ($/h) | ATP ($/h) | STD | G_{max} | No_{p} | Cpu Time (s) |
---|---|---|---|---|---|---|

LF-HLN-EF [31] | 1102.45 | 1102.45 | - | - | - | 0.017 |

LF-HLN-THF [31] | 1102.45 | 1102.45 | - | - | - | 0.02 |

LF-HLN-GdF [31] | 1102.45 | 1102.45 | - | - | - | 0.06 |

LF-HLN-GF [31] | 1102.45 | 1102.449 | - | - | - | 0.062 |

LF-HLN-LF [31] | 1102.45 | 1102.45 | - | - | - | 0.069 |

PSO [31] | 1102.45 | 938.8674 | - | 500 | 5 | 0.383 |

CSA [31] | 1102.45 | 1099.229 | - | 500 | 5 | 0.765 |

DE [31] | 1102.45 | 635.3542 | - | 500 | 5 | 0.808 |

ELF-HNM [30] | 1102.45 | - | - | - | - | 0.16 |

SSA | 1102.45 | 935.3537 | 193.9 | 5 | 5 | 0.0055 |

MDE | 1102.45 | 1001.462 | 108.5 | 5 | 5 | 0.0235 |

PSO | 1102.024 | 953.201 | 186.4 | 5 | 5 | 0.0027 |

IW-PSO | 1102.45 | 941.067 | 183.6 | 5 | 5 | 0.0027 |

CF-PSO | 1102.45 | 948.201 | 176.9 | 5 | 5 | 0.0023 |

PG-PSO | 1102.444 | 981.901 | 190.7 | 5 | 5 | 0.0052 |

IW-PG-PSO | 1102.367 | 986.454 | 101.6 | 5 | 5 | 0.0051 |

CF-PG-PSO | 1102.442 | 1006.033 | 195.4 | 5 | 5 | 0.0054 |

TVIW-PSO | 1102.449 | 957.905 | 196.7 | 5 | 5 | 0.0028 |

TVAC-PSO | 1102.45 | 985.487 | 185.1 | 5 | 5 | 0.0026 |

PPSO | 1102.451 | 1008.994 | 96.4 | 5 | 5 | 0.0051 |

Method | MTP ($/h) | ATP ($/h) | STD | G_{max} | No_{p} | Cpu Time (s) |
---|---|---|---|---|---|---|

LF-HLN-EF [31] | 1095.648 | 1095.648 | - | - | - | 0.07 |

LF-HLN-THF [31] | 1095.647 | 1095.647 | - | - | - | 0.1 |

LF-HLN-GdF [31] | 1095.61 | 1095.61 | - | - | - | 0.18 |

LF-HLN-GF [31] | 1095.589 | 1095.589 | - | - | - | 0.185 |

LF-HLN-LF [31] | 1095.59 | 1095.59 | - | - | - | 0.32 |

PSO [31] | 1095.648 | 943.7049 | - | 500 | 5 | 0.77 |

CSA [31] | 1095.648 | 1088.329 | - | 500 | 5 | 0.82 |

DE [31] | 1095.648 | 745.1618 | - | 500 | 5 | 0.95 |

ELF-HNM [30] | 1095.648 | - | - | - | - | 0.16 |

SSA | 1094.993 | 950.3221 | 190.3 | 5 | 5 | 0.0043 |

MDE | 1095.412 | 872.2816 | 212.5 | 5 | 5 | 0.0238 |

PSO | 1095.624 | 1003.32 | 185.1 | 5 | 5 | 0.0022 |

IW-PSO | 1095.648 | 1035.424 | 115 | 5 | 5 | 0.0053 |

CF-PSO | 1095.648 | 1020.58 | 170.1 | 5 | 5 | 0.0029 |

PG-PSO | 1095.648 | 1035.26 | 141.9 | 5 | 5 | 0.0045 |

IW-PG-PSO | 1095.648 | 1056.866 | 123.1 | 5 | 5 | 0.0043 |

CF-PG-PSO | 1095.647 | 1041.161 | 142.2 | 5 | 5 | 0.0058 |

TVIW-PSO | 1095.648 | 1045.962 | 110.1 | 5 | 5 | 0.0025 |

TVAC-PSO | 1095.648 | 1034.671 | 120.3 | 5 | 5 | 0.0025 |

PPSO | 1095.648 | 1063.955 | 97.3 | 5 | 5 | 0.0049 |

Method | MTP ($/h) | ATP ($/h) | STD | G_{max} | No_{p} | Cpu Time (s) |
---|---|---|---|---|---|---|

LF-HLN-EF [31] | 14,564.73 | 14,564.73 | - | 194 | - | 0.08 |

LF-HLN-THF [31] | 14,564.73 | 14,564.73 | - | 225.6 | - | 0.1 |

LF-HLN-GdF [31] | 14,564.72 | 14,564.72 | - | 256.81 | - | 0.11 |

LF-HLN-GF [31] | 14,564.71 | 14,564.71 | - | 195 | - | 0.08 |

LF-HLN-LF [31] | 14,564.71 | 14,564.71 | - | 279.57 | - | 0.22 |

PSO [31] | 14,182.19 | 9771.186 | - | 500 | 5 | 1.5 |

CSA [31] | 14,564.05 | 14,101.86 | - | 500 | 5 | 1.7 |

DE [31] | 14,053.03 | 8416.163 | - | 500 | 5 | 1.9 |

ELF-HNM [30] | 14,564.73 | - | 5000 | - | 0.18 | |

SSA | 14,370.95 | 14,128.56 | 237.1 | 100 | 20 | 0.1537 |

MDE | 14,527.64 | 14,041.82 | 240.4 | 100 | 20 | 0.8143 |

PSO | 14,563.76 | 14,046.23 | 411.4 | 100 | 20 | 0.0224 |

IW-PSO | 14,563.73 | 13,918.55 | 417.3 | 100 | 20 | 0.0119 |

CF-PSO | 14,563.77 | 14,007.25 | 381.5 | 100 | 20 | 0.0124 |

PG-PSO | 14,563.74 | 14,091.15 | 416.1 | 100 | 20 | 0.019 |

IW-PG-PSO | 14,563.74 | 14,071.23 | 398.1 | 100 | 20 | 0.0189 |

CF-PG-PSO | 14,563.76 | 13,992.12 | 617.6 | 100 | 20 | 0.0207 |

TVIW-PSO | 14,563.77 | 13,977.68 | 299.4 | 100 | 20 | 0.0133 |

TVAC-PSO | 14,563.41 | 14,111.44 | 357.5 | 100 | 20 | 0.0122 |

PPSO | 14,564.74 | 14,193.08 | 236.9 | 100 | 20 | 0.0148 |

Method | MTP ($/h) | ATP ($/h) | STD | G_{max} | No_{p} | Cpu Time (s) |
---|---|---|---|---|---|---|

LF-HLN-EF [31] | 13,635.11 | 13,635.11 | - | 187 | - | 0.08 |

LF-HLN-THF [31] | 13,635.11 | 13,635.11 | - | 227.56 | - | 0.1 |

LF-HLN-GdF [31] | 13,635.11 | 13,635.11 | - | 270.48 | - | 0.12 |

LF-HLN-GF [31] | 13,635.11 | 13,635.11 | - | 195 | - | 0.09 |

LF-HLN-LF [31] | 13,635.11 | 13,635.11 | - | 278.86 | - | 0.22 |

PSO [31] | 13,158.07 | 9824.841 | - | 500 | 5 | 1.6 |

CSA [31] | 13,635.11 | 13,448.05 | - | 500 | 5 | 1.7 |

DE [31] | 13,093.19 | 8346.24 | - | 500 | 5 | 2 |

ELF-HNM [30] | 13,635.11 | - | - | 5000 | - | 0.18 |

SSA | 13,597.06 | 13,454.77 | 109.8 | 100 | 20 | 0.1535 |

MDE | 13,626.02 | 13,353.64 | 515.4 | 100 | 20 | 0.84575 |

PSO | 13,634.83 | 13,163.26 | 528 | 100 | 20 | 0.0138 |

IW-PSO | 13,603.95 | 13,138.09 | 605.7 | 100 | 20 | 0.0172 |

CF-PSO | 13,635.02 | 13,303.35 | 359.9 | 100 | 20 | 0.0119 |

PG-PSO | 13,635.00 | 13,319.49 | 351.4 | 100 | 20 | 0.021 |

IW-PG-PSO | 13,635.04 | 13,306.97 | 386.6 | 100 | 20 | 0.0251 |

CF-PG-PSO | 13,635.02 | 13,321.75 | 348.8 | 100 | 20 | 0.0292 |

TVIW-PSO | 13,618.66 | 13,086.99 | 591.6 | 100 | 20 | 0.0141 |

TVAC-PSO | 13,634.92 | 13,326.12 | 315.9 | 100 | 20 | 0.0146 |

PPSO | 13,635.12 | 13,525.28 | 105.1 | 100 | 20 | 0.0206 |

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**MDPI and ACS Style**

Kien, L.C.; Duong, T.L.; Phan, V.-D.; Nguyen, T.T.
Maximizing Total Profit of Thermal Generation Units in Competitive Electric Market by Using a Proposed Particle Swarm Optimization. *Sustainability* **2020**, *12*, 1265.
https://doi.org/10.3390/su12031265

**AMA Style**

Kien LC, Duong TL, Phan V-D, Nguyen TT.
Maximizing Total Profit of Thermal Generation Units in Competitive Electric Market by Using a Proposed Particle Swarm Optimization. *Sustainability*. 2020; 12(3):1265.
https://doi.org/10.3390/su12031265

**Chicago/Turabian Style**

Kien, Le Chi, Thanh Long Duong, Van-Duc Phan, and Thang Trung Nguyen.
2020. "Maximizing Total Profit of Thermal Generation Units in Competitive Electric Market by Using a Proposed Particle Swarm Optimization" *Sustainability* 12, no. 3: 1265.
https://doi.org/10.3390/su12031265