# MSSA-DEED: A Multi-Objective Salp Swarm Algorithm for Solving Dynamic Economic Emission Dispatch Problems

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## Abstract

**:**

## 1. Introduction

- Proposing the MSSA to find the optimal solution of the non-linear and non-convex multi-objective DEED problems.
- Solving multi-objective DEED problems considering the valve point effect, transmission loss, as well as the ramping rate.
- Applying the fuzzy decision-making approach to achieve the best compromise solutions.
- Using the 6-unit power system, 10-unit power system, and 14-unit power system as three studied cases to prove the efficacy of the MSSA.

## 2. Mathematical Model of EED Problem

#### 2.1. Objective Functions

_{2}, SOx, and NOx produced by fossil-fueled thermal generators are mathematically represented as the totality of a quadratic and an exponential function. Accordingly, the total atmospheric pollutants of ${N}_{G}$ generators through the dispatching interval is expressed as [39]:

#### 2.2. Operational Constraints

#### 2.2.1. Power Balance Constraint

#### 2.2.2. Generating Capacity Constraint

#### 2.2.3. Ramp Rate Constraint

_{i}cannot be adjusted immediately. The up and down ramp limits are represented by [40]:

## 3. Multi-Objective Optimization Algorithm

#### 3.1. Multi-Objective Optimization Problem

#### 3.2. Multi-Objective Salp Swarm Algorithm

_{i}location of the salp i, denoted by a vector of n elements x

_{i}= [ ${x}_{j}^{1}$, ${x}_{j}^{2}$, . . . , ${x}_{j}^{n}$ ], the leader’s location in the salp chain updates through this equation:

Algorithm 1 Pseudo code of MSSA |

Initialize the parameters the number of salps, Obj_no, dim, lb, ub, Archive Maxsize, $\mathit{max\_iter}$ Initialize the salp population ${x}_{i}\left(i=1,2,\dots ,n\right);$ Define the objective function (Dynamic Economic Emission Dispatch function) While $(t\le \mathit{max\_iter})$Evaluate the fitness of all salp with Ob_func; Find the ND solutions Update the repository in regard to the achieved ND agents If (repository is full);Perform the repository maintenance process to eliminate one repository neighborhood Insert the ND agent to the repository End IfSelect a source of food from repository Update ${c}_{1}$ by Equation (9);For each search agentIf (i==1)Update the location of the leading salp by Equation (8);ElseUpdate the location of the leading salp by Equation (10);End ifEnd For$t=t+1;$ End WhileReturn repository |

## 4. Simulation Results and Discussions

#### 4.1. Case 1: Six-Unit Test System

#### 4.2. Case 2: 10-Unit Test System

^{6}and 3.05994 × 10

^{5}lb, which is superior to the several compared techniques. In terms of the economy and environment, the cost and emission of the BCS of the MSSA technique was superior to the NSGA II [24], RCGA [24], MODE [25], MALO, and MOGOA. In general, the solution of the MSSA technique was superior to the comparison techniques.

_{T}, P

_{D}, and P

_{L}at each interval.

#### 4.3. Case 3: 14-Unit Test System

^{5}and 98.1415 lb, which was better than the two compared techniques. Moreover, from this figure, we can see directly that the Pareto solutions of the proposed MSSA were widely and well distributed.

_{L}and P

_{D}and that all the inequality constraints were fulfilled, which shows the efficacy of the MSSA. The output power of each generator, total power, load demand, and power loss are shown in Figure 10.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DEED | Dynamic Economic and Emission Dispatch |

FDM | Fuzzy decision-making |

MSSA | Multi-objective Salp Swarm Algorithm |

ITSA | Improved tunicate swarm algorithm |

MALO | Multi-objective ant lion optimizer |

MOGOA | Multi-objective grasshopper optimization algorithm |

EED | Economic and Emission Dispatch |

SSA | Salp Swarm Algorithm |

ODEA | The opposition-based differential evolution algorithm |

PSO | Particle swarm optimizer |

MODE | Multi-objective Differential Evolution |

CRO | Chemical Reaction Optimization |

SQP | Sequential quadratic programming |

EPFSA | Evolutionary programming-based fuzzy satisfying algorithm |

ELD | Economic Load Dispatch |

ND | Non-dominated |

BCS | Best Compromise Solutions |

VPE | Valve point effect |

NSGA-II | Non-dominated Sorting Genetic Algorithm-II |

MOP | Multi-objective optimization problem |

DED | Dynamic Economic Dispatch |

IBFA | Improved bacterial foraging algorithm |

PSOCS | Particle swarm optimization algorithm with a clone selection |

DE | Differential evolution |

MAMODE | Modified Adaptive Multi-objective Differential Evolution |

GSOMP | Group Search Optimizer with Multiple Producers |

RWS | Roulette wheel selection |

MONNDE | Multi-objective Neural Networks trained using Differential Evolution |

## Appendix A

**Table A1.**The load demands (MW) of the six-generator test system [29].

Hour | P_{D} | Hour | P_{D} | Hour | P_{D} |
---|---|---|---|---|---|

1 | 3.25 | 9 | 5.45 | 17 | 5.15 |

2 | 3.90 | 10 | 5.20 | 18 | 5.75 |

3 | 3.50 | 11 | 5.50 | 19 | 5.25 |

4 | 3.00 | 12 | 5.75 | 20 | 5.25 |

5 | 3.35 | 13 | 5.25 | 21 | 4.55 |

6 | 4.00 | 14 | 5.15 | 22 | 4.25 |

7 | 4.75 | 15 | 4.75 | 23 | 4.25 |

8 | 5.05 | 16 | 5.30 | 24 | 4.00 |

**Table A2.**Generator and emission coefficients of the six-generator test system [44].

Unit | ${\mathit{P}}_{\mathit{G}\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{P}}_{\mathit{G}\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{a}}_{\mathit{i}}(\mathbf{USD}/\mathbf{h})$ | ${\mathit{b}}_{\mathit{i}}(\mathbf{USD}/\mathbf{MWh})$ | ${\mathit{c}}_{\mathit{i}}(\mathbf{USD}/(\mathbf{MW}{)}^{2})$ | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{\beta}}_{\mathit{i}}$ | ${\mathit{\gamma}}_{\mathit{i}}$ | ${\mathit{\eta}}_{\mathit{i}}$ | ${\mathit{\delta}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|---|---|---|

P_{G1} | 150 | 5 | 10 | 200 | 100 | 4.091 | −5.543 | 6.49 | 2.00 × 10^{−4} | 2.857 |

P_{G2} | 150 | 5 | 10 | 150 | 120 | 2.543 | −6.047 | 5.638 | 5.00 × 10^{−4} | 3.333 |

P_{G3} | 150 | 5 | 20 | 180 | 40 | 4.258 | −5.094 | 4.586 | 1.00 × 10^{−6} | 8 |

P_{G4} | 150 | 5 | 10 | 100 | 60 | 5.326 | −3.55 | 3.38 | 2.00 × 10^{−3} | 2 |

P_{G5} | 150 | 5 | 20 | 180 | 40 | 4.258 | −5.094 | 4.586 | 1.00 × 10^{−6} | 8 |

P_{G6} | 150 | 5 | 10 | 150 | 100 | 6.131 | −5.555 | 5.151 | 1.00 × 10^{−5} | 6.667 |

## Appendix B

**Table A3.**The load demands (MW) of the standard 39-bus ten-unit test system [34].

Hour | P_{D} | Hour | P_{D} | Hour | P_{D} |
---|---|---|---|---|---|

1 | 1036 | 9 | 1924 | 17 | 1480 |

2 | 1110 | 10 | 2022 | 18 | 1628 |

3 | 1258 | 11 | 2106 | 19 | 1776 |

4 | 1406 | 12 | 2150 | 20 | 1972 |

5 | 1480 | 13 | 2072 | 21 | 1924 |

6 | 1628 | 14 | 1924 | 22 | 1628 |

7 | 1702 | 15 | 1776 | 23 | 1332 |

8 | 1776 | 16 | 1554 | 24 | 1184 |

**Table A4.**Generator coefficients of the standard 39-bus 10-unit test system [45].

Unit | ${\mathit{P}}_{\mathit{G}\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{P}}_{\mathit{G}\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{a}}_{\mathit{i}}(\mathbf{USD}/\mathbf{h})$ | ${\mathit{b}}_{\mathit{i}}(\mathbf{USD}/\mathbf{MWh})$ | ${\mathit{c}}_{\mathit{i}}(\mathbf{USD}/(\mathbf{MW}{)}^{2})$ | ${\mathit{d}}_{\mathit{i}}(\mathbf{USD}/\mathbf{h})$ | ${\mathit{e}}_{\mathit{i}}(\mathbf{rad}/\mathbf{MW})$ |
---|---|---|---|---|---|---|---|

P_{G1} | 150 | 470 | 786.7988 | 38.5379 | 0.1524 | 450 | 0.041 |

P_{G2} | 135 | 470 | 451.3251 | 46.1591 | 0.1058 | 600 | 0.036 |

P_{G3} | 73 | 340 | 1049.998 | 40.3965 | 0.028 | 320 | 0.028 |

P_{G4} | 60 | 300 | 1243.531 | 38.3055 | 0.0354 | 260 | 0.052 |

P_{G5} | 73 | 243 | 1658.57 | 36.3278 | 0.0211 | 280 | 0.063 |

P_{G6} | 57 | 160 | 1356.659 | 38.2704 | 0.0179 | 310 | 0.048 |

P_{G7} | 20 | 130 | 1450.705 | 36.5104 | 0.0121 | 300 | 0.086 |

P_{G8} | 47 | 120 | 1450.705 | 36.5104 | 0.0121 | 340 | 0.082 |

P_{G9} | 20 | 80 | 1455.606 | 39.5804 | 0.109 | 270 | 0.098 |

P_{G10} | 10 | 55 | 1469.403 | 40.5407 | 0.1295 | 380 | 0.094 |

**Table A5.**Emission coefficients and ramp rate of the standard 39-bus 10-unit test system [45].

Unit | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{\beta}}_{\mathit{i}}$ | ${\mathit{\gamma}}_{\mathit{i}}$ | ${\mathit{\eta}}_{\mathit{i}}$ | ${\mathit{\delta}}_{\mathit{i}}$ | $\mathit{U}{\mathit{R}}_{\mathit{i}}$ | $\mathit{D}{\mathit{R}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|

P_{1} | 103.3908 | 2.4444 | 0.0312 | 0.5035 | 0.0207 | 80 | 80 |

P_{2} | 103.3908 | 2.4444 | 0.0312 | 0.5035 | 0.0207 | 80 | 80 |

P_{3} | 300.391 | 4.0695 | 0.0509 | 0.4968 | 0.0202 | 80 | 80 |

P_{4} | 300.391 | 4.0695 | 0.0509 | 0.4968 | 0.0202 | 50 | 50 |

P_{5} | 320.0006 | 3.8132 | 0.0344 | 0.4972 | 0.02 | 50 | 50 |

P_{6} | 320.0006 | 3.8132 | 0.0344 | 0.4972 | 0.02 | 50 | 50 |

P_{7} | 330.0056 | 3.9023 | 0.0465 | 0.5163 | 0.0214 | 50 | 30 |

P_{8} | 330.0056 | 3.9023 | 0.0465 | 0.5163 | 0.0214 | 30 | 30 |

P_{9} | 350.0056 | 3.9524 | 0.0465 | 0.5475 | 0.0234 | 30 | 30 |

P_{10} | 360.0012 | 3.9864 | 0.047 | 0.5475 | 0.0234 | 30 | 30 |

**Table A6.**The B-coefficients of the standard 39-bus 10-unit test system [45].

${\mathit{B}}_{\mathit{i}\mathit{j}}$ | 0.000049 | 0.000014 | 0.000015 | 0.000015 | 0.000016 | 0.000017 | 0.000017 | 0.000018 | 0.000019 | 0.00002 |
---|---|---|---|---|---|---|---|---|---|---|

0.000014 | 0.000045 | 0.000016 | 0.000016 | 0.000017 | 0.000015 | 0.000015 | 0.000016 | 0.000018 | 0.000018 | |

0.000015 | 0.000016 | 0.000039 | 0.00001 | 0.000012 | 0.000012 | 0.000014 | 0.000014 | 0.000016 | 0.000016 | |

0.000015 | 0.000016 | 0.00001 | 0.00004 | 0.000014 | 0.00001 | 0.000011 | 0.000012 | 0.000014 | 0.000015 | |

0.000016 | 0.000017 | 0.000012 | 0.000014 | 0.000035 | 0.000011 | 0.000013 | 0.000013 | 0.000015 | 0.000016 | |

0.000017 | 0.000015 | 0.000012 | 0.00001 | 0.000011 | 0.000036 | 0.000012 | 0.000012 | 0.000014 | 0.000015 | |

0.000017 | 0.000015 | 0.000014 | 0.000011 | 0.000013 | 0.000012 | 0.000038 | 0.000016 | 0.000016 | 0.000018 | |

0.000018 | 0.000016 | 0.000014 | 0.000012 | 0.000013 | 0.000012 | 0.000016 | 0.00004 | 0.000015 | 0.000016 | |

0.000019 | 0.000018 | 0.000016 | 0.000014 | 0.000015 | 0.000014 | 0.000016 | 0.000015 | 0.000042 | 0.000019 | |

0.00002 | 0.000018 | 0.000016 | 0.000015 | 0.00016 | 0.000015 | 0.000018 | 0.000016 | 0.000019 | 0.000044 | |

${B}_{i0}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

${B}_{00}$ | 0 |

## Appendix C

**Table A7.**The load demands (MW) of the IEEE 118-bus 14-generator test system [29].

Hour | P_{D} | Hour | P_{D} | Hour | P_{D} |
---|---|---|---|---|---|

1 | 10.0 | 9 | 14.9 | 17 | 14.5 |

2 | 12.0 | 10 | 13.8 | 18 | 15.2 |

3 | 11.0 | 11 | 15.0 | 19 | 14.7 |

4 | 10.0 | 12 | 16.5 | 20 | 14.5 |

5 | 10.2 | 13 | 15.1 | 21 | 13.1 |

6 | 12.0 | 14 | 14.3 | 22 | 12.6 |

7 | 13.5 | 15 | 13.1 | 23 | 12.5 |

8 | 14.1 | 16 | 14.6 | 24 | 12.0 |

**Table A8.**Generator and emission coefficients of the IEEE 118-bus 14-generator test system [46].

Unit | ${\mathit{P}}_{\mathit{G}\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{P}}_{\mathit{G}\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{a}}_{\mathit{i}}(\mathbf{USD}/\mathbf{h})$ | ${\mathit{b}}_{\mathit{i}}(\mathbf{USD}/\mathbf{MWh})$ | ${\mathit{c}}_{\mathit{i}}(\mathbf{USD}/(\mathbf{MW}{)}^{2})$ | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\mathit{\beta}}_{\mathit{i}}$ | ${\mathit{\gamma}}_{\mathit{i}}$ |
---|---|---|---|---|---|---|---|---|

P_{G1} | 300 | 50 | 150 | 189 | 0.5 | 0.016 | −1.5 | 23.333 |

P_{G2} | 300 | 50 | 115 | 200 | 0.55 | 0.031 | −1.82 | 21.022 |

P_{G3} | 300 | 50 | 40 | 350 | 0.6 | 0.013 | −1.249 | 22.05 |

P_{G4} | 300 | 50 | 122 | 315 | 0.5 | 0.012 | −1.355 | 22.983 |

P_{G5} | 300 | 50 | 125 | 305 | 0.5 | 0.02 | −1.9 | 21.313 |

P_{G6} | 300 | 50 | 70 | 275 | 0.7 | 0.007 | 0.805 | 21.9 |

P_{G7} | 300 | 50 | 70 | 345 | 0.7 | 0.015 | −1.401 | 23.001 |

P_{G8} | 300 | 50 | 70 | 345 | 0.7 | 0.018 | −1.8 | 24.003 |

P_{G9} | 300 | 50 | 130 | 245 | 0.5 | 0.019 | −2 | 25.121 |

P_{G10} | 300 | 50 | 130 | 245 | 0.5 | 0.012 | −1.36 | 22.99 |

P_{G11} | 300 | 50 | 135 | 235 | 0.55 | 0.033 | −2.1 | 27.01 |

P_{G12} | 300 | 50 | 200 | 130 | 0.45 | 0.018 | −1.8 | 25.101 |

P_{G13} | 300 | 50 | 70 | 345 | 0.7 | 0.018 | −1.81 | 24.313 |

P_{G14} | 300 | 50 | 45 | 389 | 0.6 | 0.03 | −1.921 | 27.119 |

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**Figure 5.**Pareto Front of the first case with power loss obtained by (

**a**) MSSA; (

**b**) MALO; (

**c**) MOGOA; (

**d**) MSSA and comparison of optimal solutions.

**Figure 7.**Pareto Front of Case 2 with power loss obtained by (

**a**) MSSA; (

**b**) MALO; (

**c**) MOGOA; (

**d**) MSSA and comparison of optimum solutions.

**Figure 9.**Pareto Front of the third case with power loss obtained by (

**a**) MSSA; (

**b**) MALO; (

**c**) MOGOA; (

**d**) MSSA and comparison of optimum solutions.

Case Number | Number of Generators | Number of Buses | Number of Decision Variables | Number of Equality Constraints |
---|---|---|---|---|

Case 1 | 6 | 30 | 6 × 24 = 144 | 24 |

Case 2 | 10 | NA | 10 × 24 = 240 | 24 |

Case 3 | 14 | 118 | 14 × 24 = 336 | 24 |

Algorithm | Objectives | Total Cost (USD) | Emission (Ib) |
---|---|---|---|

MSSA | BC | 25,437.59 | 6.52773 |

BE | 26,389.76 | 5.69654 | |

BCS | 25,727.57 | 5.94564 | |

MALO | BC | 26,200.57 | 5.73927 |

BE | 26,273.83 | 5.71806 | |

BCS | 26,233.87 | 5.72757 | |

MOGOA | BC | 26,158.07 | 5.83543 |

BE | 26,637.37 | 5.73097 | |

BCS | 26,337.08 | 5.74666 | |

MAMODE | BC | 25,732 | NA |

BE | NA | 5.7283 | |

BCS | 25,912.89 | 5.97955 | |

GSOMP | BC | 25,493 | NA |

BE | NA | 5.6847 | |

BCS | 25,924.46 | 6.00415 | |

MOPSO | BC | 25,633.2 | NA |

BE | NA | 5.6863 | |

NSGA-II | BC | 25,507.4 | NA |

BE | NA | 5.6881 |

Hour | P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | P_{6} | P_{T} | P_{L} |
---|---|---|---|---|---|---|---|---|

1 | 0.3068 | 0.3926 | 0.6113 | 0.8005 | 0.6690 | 0.4950 | 3.2752 | 0.0252 |

2 | 0.3760 | 0.4722 | 0.8176 | 0.9616 | 0.7564 | 0.5511 | 3.9348 | 0.0348 |

3 | 0.3109 | 0.4573 | 0.6931 | 0.8493 | 0.6620 | 0.5560 | 3.5287 | 0.0287 |

4 | 0.2697 | 0.3719 | 0.5970 | 0.6426 | 0.6305 | 0.5080 | 3.0196 | 0.0196 |

5 | 0.3240 | 0.4369 | 0.6214 | 0.8330 | 0.6568 | 0.5060 | 3.3782 | 0.0282 |

6 | 0.3700 | 0.5111 | 0.8599 | 0.9713 | 0.7568 | 0.5666 | 4.0358 | 0.0358 |

7 | 0.4673 | 0.6018 | 0.9944 | 1.1233 | 0.9327 | 0.6861 | 4.8056 | 0.0556 |

8 | 0.5225 | 0.6361 | 1.0820 | 1.1433 | 0.9867 | 0.7439 | 5.1145 | 0.0645 |

9 | 0.5634 | 0.7228 | 1.1207 | 1.2508 | 1.0785 | 0.7930 | 5.5292 | 0.0792 |

10 | 0.5478 | 0.6738 | 1.0600 | 1.1967 | 1.0742 | 0.7190 | 5.2716 | 0.0716 |

11 | 0.6053 | 0.6953 | 1.1598 | 1.2153 | 1.1160 | 0.7898 | 5.5815 | 0.0815 |

12 | 0.6297 | 0.7469 | 1.1529 | 1.3184 | 1.1386 | 0.8572 | 5.8437 | 0.0937 |

13 | 0.5358 | 0.6647 | 1.0635 | 1.2200 | 1.0455 | 0.7936 | 5.3230 | 0.0730 |

14 | 0.5614 | 0.6853 | 1.0480 | 1.1968 | 1.0295 | 0.7012 | 5.2220 | 0.0720 |

15 | 0.5151 | 0.6013 | 0.9823 | 1.0354 | 0.9633 | 0.7117 | 4.8091 | 0.0591 |

16 | 0.5863 | 0.6697 | 1.0924 | 1.1834 | 1.0571 | 0.7879 | 5.3769 | 0.0769 |

17 | 0.5480 | 0.6413 | 1.0702 | 1.1935 | 1.0275 | 0.7392 | 5.2197 | 0.0697 |

18 | 0.6483 | 0.7825 | 1.1536 | 1.3271 | 1.0989 | 0.8356 | 5.8460 | 0.0960 |

19 | 0.5509 | 0.6653 | 1.0803 | 1.2327 | 1.0223 | 0.7717 | 5.3233 | 0.0733 |

20 | 0.5788 | 0.6293 | 1.0482 | 1.2115 | 1.1082 | 0.7498 | 5.3258 | 0.0758 |

21 | 0.4362 | 0.5833 | 0.9479 | 1.0702 | 0.9487 | 0.6131 | 4.5995 | 0.0495 |

22 | 0.3931 | 0.5030 | 0.9248 | 0.9880 | 0.8860 | 0.5946 | 4.2896 | 0.0396 |

23 | 0.3901 | 0.5804 | 0.8155 | 0.9958 | 0.9065 | 0.6071 | 4.2954 | 0.0454 |

24 | 0.3915 | 0.4850 | 0.7786 | 0.9582 | 0.8404 | 0.5858 | 4.0395 | 0.0395 |

Algorithm | Objectives | Total Cost/10^{6} (USD) | Emission/10^{5} (Ib) |
---|---|---|---|

MSSA | BC | 2.505334 | 3.083911 |

BE | 2.525936 | 3.046193 | |

BCS | 2.520778 | 3.05994 | |

MALO | BC | 2.530114 | 3.0465133 |

BE | 2.5301699 | 3.0464497 | |

BCS | 2.530128 | 3.04646 | |

MOGOA | BC | 2.5233373 | 3.178493 |

BE | 2.546446 | 3.0928948 | |

BCS | 2.5436 | 3.1126 | |

NSGA-II | BC | 2.5168 | 3.1740 |

BE | 2.6563 | 3.0412 | |

BCS | 2.5226 | 3.0994 | |

RCGA | BC | 2.5168 | 3.1740 |

BE | 2.6563 | 3.0412 | |

BCS | 2.5251 | 3.1246 | |

MODE | BC | 2.5123 | 3.0113 |

BE | 2.5436 | 2.9607 | |

BCS | 2.5224 | 3.0997 |

Hour | P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | P_{6} | P_{7} | P_{8} | P_{9} | P_{10} | P_{L} |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 150.01 | 138.07 | 166.79 | 132.28 | 153.54 | 117.74 | 57.83 | 64.50 | 54.36 | 20.46 | 19.59 |

2 | 150.09 | 142.01 | 135.31 | 167.76 | 148.41 | 134.97 | 70.44 | 74.86 | 65.90 | 42.62 | 22.38 |

3 | 151.11 | 149.39 | 186.96 | 177.14 | 160.71 | 129.73 | 96.10 | 102.74 | 79.65 | 53.16 | 28.68 |

4 | 180.36 | 173.72 | 188.94 | 189.97 | 170.87 | 159.04 | 125.90 | 118.29 | 79.96 | 55.00 | 36.05 |

5 | 152.47 | 191.60 | 185.68 | 227.73 | 219.00 | 159.49 | 129.98 | 119.07 | 79.83 | 54.92 | 39.78 |

6 | 216.24 | 195.52 | 216.39 | 264.98 | 241.20 | 157.90 | 129.89 | 119.96 | 79.93 | 54.92 | 48.92 |

7 | 212.53 | 234.47 | 271.75 | 254.13 | 238.67 | 159.30 | 130.00 | 120.00 | 79.96 | 55.00 | 53.81 |

8 | 230.46 | 237.72 | 303.67 | 277.65 | 242.83 | 159.54 | 129.50 | 119.32 | 79.66 | 54.57 | 58.92 |

9 | 283.42 | 303.83 | 325.48 | 295.22 | 242.08 | 160.00 | 129.92 | 119.88 | 79.99 | 54.99 | 70.80 |

10 | 332.92 | 342.01 | 338.70 | 299.98 | 243.00 | 160.00 | 130.00 | 120.00 | 79.99 | 54.99 | 79.58 |

11 | 376.74 | 389.20 | 339.96 | 300.00 | 243.00 | 160.00 | 129.99 | 120.00 | 80.00 | 55.00 | 87.88 |

12 | 397.87 | 416.60 | 339.99 | 300.00 | 243.00 | 160.00 | 130.00 | 120.00 | 80.00 | 55.00 | 92.46 |

13 | 356.03 | 372.61 | 339.88 | 299.98 | 242.96 | 159.99 | 130.00 | 119.99 | 79.99 | 55.00 | 84.44 |

14 | 292.50 | 296.25 | 318.86 | 299.34 | 242.96 | 159.97 | 129.99 | 119.98 | 79.99 | 54.98 | 70.82 |

15 | 230.56 | 229.74 | 292.52 | 294.79 | 242.63 | 160.00 | 129.58 | 120.00 | 80.00 | 55.00 | 58.82 |

16 | 171.31 | 174.43 | 237.51 | 245.09 | 240.74 | 157.37 | 124.65 | 118.51 | 75.72 | 52.61 | 43.94 |

17 | 157.28 | 150.24 | 203.43 | 235.55 | 238.54 | 155.80 | 127.08 | 119.30 | 77.52 | 54.81 | 39.54 |

18 | 213.11 | 203.73 | 246.62 | 245.80 | 226.31 | 158.13 | 129.67 | 119.72 | 79.40 | 54.44 | 48.94 |

19 | 232.58 | 239.14 | 295.31 | 280.12 | 242.93 | 159.98 | 129.98 | 119.92 | 79.96 | 55.00 | 58.93 |

20 | 309.10 | 311.05 | 338.90 | 300.00 | 243.00 | 159.86 | 130.00 | 120.00 | 80.00 | 55.00 | 74.90 |

21 | 286.57 | 288.13 | 334.04 | 297.93 | 243.00 | 160.00 | 130.00 | 120.00 | 80.00 | 55.00 | 70.67 |

22 | 211.19 | 217.82 | 260.45 | 248.14 | 225.32 | 146.40 | 119.67 | 116.17 | 78.09 | 53.94 | 49.19 |

23 | 151.66 | 139.84 | 197.33 | 208.20 | 202.88 | 104.38 | 129.89 | 117.26 | 71.11 | 41.56 | 32.11 |

24 | 151.96 | 137.93 | 164.64 | 182.20 | 198.39 | 109.62 | 102.80 | 90.20 | 42.34 | 29.29 | 25.38 |

Algorithm | Objectives | Total Cost//10^{5} (USD) | Emission (Ib) |
---|---|---|---|

MSSA | BC | 1.2548728 | 110.02359 |

BE | 1.3560793 | 91.69988 | |

BCS | 1.29200 | 98.1415 | |

MALO | BC | 1.3406945 | 98.15331 |

BE | 1.3638826 | 92.150718 | |

BCS | 1.34938 | 94.1988 | |

MOGOA | BC | 1.2822754 | 114.45838 |

BE | 1.3853273 | 93.275504 | |

BCS | 1.29844 | 108.6742 |

Hour | P_{1} | P_{2} | P_{3} | P_{4} | P_{5} | P_{6} | P_{7} | P_{8} | P_{9} | P_{10} | P_{11} | P_{12} | P_{13} | P_{14} | P_{L} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1.8448 | 1.0056 | 0.7031 | 0.8494 | 0.6978 | 0.7539 | 0.6777 | 0.6474 | 0.7207 | 0.8242 | 0.7970 | 1.0141 | 0.6891 | 0.6274 | 1.8522 |

2 | 1.8686 | 1.5590 | 0.7619 | 0.8887 | 0.8204 | 0.7799 | 0.8150 | 0.6967 | 0.8426 | 1.0303 | 1.0040 | 1.3621 | 0.7007 | 0.6837 | 1.8136 |

3 | 1.9159 | 1.4781 | 0.7493 | 0.8146 | 0.8097 | 0.7498 | 0.7031 | 0.6984 | 0.9535 | 0.8321 | 0.8117 | 0.8954 | 0.6964 | 0.6315 | 1.7394 |

4 | 1.5969 | 1.2879 | 0.6443 | 0.7765 | 0.7695 | 0.7960 | 0.6510 | 0.6319 | 0.6807 | 0.7502 | 0.8043 | 1.1795 | 0.7743 | 0.6046 | 1.9476 |

5 | 1.5783 | 1.2570 | 0.8429 | 0.7314 | 0.7314 | 0.7883 | 0.7075 | 0.6977 | 0.7986 | 0.8820 | 0.8005 | 1.0211 | 0.6197 | 0.6664 | 1.9229 |

6 | 1.7012 | 1.3836 | 0.8931 | 1.0650 | 0.7281 | 0.8062 | 0.8671 | 0.8092 | 1.0783 | 0.9772 | 0.9998 | 1.1138 | 0.7438 | 0.6578 | 1.8242 |

7 | 1.8590 | 1.6225 | 1.0377 | 1.0833 | 1.0213 | 1.0505 | 0.8526 | 0.7107 | 1.0057 | 1.2990 | 1.0806 | 1.1119 | 0.8145 | 0.6702 | 1.7194 |

8 | 2.2770 | 1.7767 | 0.8949 | 0.9837 | 0.9789 | 1.0381 | 0.7910 | 0.8754 | 0.9454 | 1.3403 | 1.0184 | 1.2218 | 0.8397 | 0.7222 | 1.6034 |

9 | 1.9798 | 2.1283 | 0.8953 | 1.0316 | 0.8775 | 0.9845 | 0.9566 | 0.8269 | 1.2213 | 1.3621 | 1.1799 | 1.5283 | 0.8622 | 0.7757 | 1.7100 |

10 | 2.2102 | 1.7363 | 0.9107 | 0.9018 | 0.9547 | 0.8503 | 0.8137 | 1.0557 | 0.9148 | 1.2187 | 1.0422 | 1.3154 | 0.8075 | 0.7241 | 1.6560 |

11 | 2.2647 | 1.8614 | 0.8648 | 1.1936 | 1.0429 | 0.8684 | 1.0178 | 0.9309 | 1.0312 | 1.1771 | 1.0551 | 1.5499 | 1.0144 | 0.7607 | 1.6330 |

12 | 2.2892 | 1.9223 | 1.2679 | 1.5490 | 0.9783 | 1.1875 | 0.8261 | 0.9638 | 1.2831 | 1.2737 | 1.0947 | 1.6729 | 0.9701 | 0.7903 | 1.5689 |

13 | 2.1056 | 1.9158 | 1.0940 | 1.1415 | 0.9491 | 1.2375 | 0.8647 | 0.7604 | 1.0287 | 1.1263 | 1.3653 | 1.4299 | 0.9939 | 0.7363 | 1.6489 |

14 | 1.9923 | 1.7601 | 0.9725 | 1.0530 | 1.1690 | 0.9384 | 0.8627 | 0.8816 | 0.8881 | 1.3040 | 1.1878 | 1.4874 | 0.8726 | 0.6524 | 1.7218 |

15 | 2.0599 | 1.5643 | 0.8444 | 0.9778 | 0.9095 | 0.8642 | 0.7872 | 0.7977 | 0.9421 | 1.3365 | 1.1358 | 1.0737 | 0.8018 | 0.6860 | 1.6808 |

16 | 2.1009 | 1.7972 | 1.1459 | 1.0448 | 1.1458 | 0.9029 | 0.8425 | 0.8539 | 1.0513 | 1.2255 | 1.0421 | 1.3444 | 1.0195 | 0.7369 | 1.6536 |

17 | 2.1883 | 2.0192 | 0.9134 | 1.0916 | 1.1254 | 1.0463 | 0.9255 | 0.8499 | 0.9232 | 0.9825 | 1.0277 | 1.3759 | 0.9342 | 0.7155 | 1.6186 |

18 | 2.2892 | 2.1489 | 1.0146 | 1.1794 | 0.8802 | 1.1092 | 0.8609 | 0.7866 | 1.0821 | 1.1133 | 1.3164 | 1.4556 | 0.8095 | 0.7232 | 1.5690 |

19 | 2.1990 | 1.9796 | 0.8941 | 0.9135 | 1.0583 | 1.0535 | 0.8683 | 0.8093 | 1.0409 | 1.1729 | 1.2207 | 1.6510 | 0.8146 | 0.7100 | 1.6856 |

20 | 1.9954 | 1.6575 | 0.9303 | 1.0157 | 0.9446 | 1.1513 | 1.0201 | 0.8893 | 1.1961 | 1.1178 | 1.4577 | 1.3617 | 0.7764 | 0.7405 | 1.7543 |

21 | 2.2082 | 1.6239 | 0.8441 | 0.8003 | 0.9282 | 1.0949 | 0.7419 | 0.7442 | 0.9581 | 0.9384 | 1.0039 | 1.4912 | 0.7182 | 0.7269 | 1.7224 |

22 | 1.7212 | 1.9025 | 0.8127 | 0.8846 | 1.0127 | 0.8447 | 0.7468 | 0.7491 | 0.8413 | 1.1173 | 0.9040 | 1.2863 | 0.8112 | 0.7732 | 1.8075 |

23 | 1.9478 | 1.6296 | 0.8620 | 0.9937 | 0.8662 | 0.8004 | 0.7688 | 0.6997 | 0.9438 | 1.2162 | 0.9303 | 1.2251 | 0.6933 | 0.6424 | 1.7192 |

24 | 1.7457 | 1.8476 | 0.8600 | 0.8294 | 0.8312 | 0.9356 | 0.7420 | 0.7182 | 0.8128 | 1.1049 | 0.7625 | 1.1912 | 0.7391 | 0.6600 | 1.7803 |

Case No. | MSSA | MALO | MOGOA | |
---|---|---|---|---|

Case 1 | Best | 0.807653 | 0.636744 | 0.646386 |

Worst | 0.789138 | 0.171571 | 0.185795 | |

Mean | 0.800696 | 0.39641 | 0.438754 | |

Std dev. | 0.006691 | 0.139975 | 0.14251 | |

Case 2 | Best | 0.665428 | 0.1914 | 0.617644 |

Worst | 0.08684 | 0 | 0.049874 | |

Mean | 0.3104 | 0.07108 | 0.394762 | |

Std dev. | 0.195818 | 0.080641 | 0.21163 | |

Case 3 | Best | 0.634702 | 0.704149 | 0.578087 |

Worst | 0.584107 | 0.427635 | 0.515367 | |

Mean | 0.610866 | 0.554619 | 0.542181 | |

Std dev. | 0.015048 | 0.095316 | 0.020538 |

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## Share and Cite

**MDPI and ACS Style**

Hassan, M.H.; Kamel, S.; Domínguez-García, J.L.; El-Naggar, M.F.
MSSA-DEED: A Multi-Objective Salp Swarm Algorithm for Solving Dynamic Economic Emission Dispatch Problems. *Sustainability* **2022**, *14*, 9785.
https://doi.org/10.3390/su14159785

**AMA Style**

Hassan MH, Kamel S, Domínguez-García JL, El-Naggar MF.
MSSA-DEED: A Multi-Objective Salp Swarm Algorithm for Solving Dynamic Economic Emission Dispatch Problems. *Sustainability*. 2022; 14(15):9785.
https://doi.org/10.3390/su14159785

**Chicago/Turabian Style**

Hassan, Mohamed H., Salah Kamel, José Luís Domínguez-García, and Mohamed F. El-Naggar.
2022. "MSSA-DEED: A Multi-Objective Salp Swarm Algorithm for Solving Dynamic Economic Emission Dispatch Problems" *Sustainability* 14, no. 15: 9785.
https://doi.org/10.3390/su14159785