# Multiple Fuel Machines Power Economic Dispatch Using Stud Differential Evolution

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Objective Function

#### 2.2. Constraints

#### 2.2.1. Equality Constraint

#### 2.2.2. In-Equality Constraint

#### 2.3. Fuel Cost Equations

#### 2.3.1. Power Economic Dispatch considering Valve Point Loading Effects Only

#### 2.3.2. Power Economic Dispatch Considering Multiple Fuel Options Only

#### 2.3.3. Power Economic Dispatch Considering Multiple Fuel Options and Valve Point Loading Effects Together

## 3. Differential Evolution (DE)

**Step 1:**Randomly initialize the initial population ${P}_{i}^{G}$ (target vectors) of $NP$ size and of $D$ dimensions, in a feasible range.

**Step 2:**Calculate the fitness value for all generated target vectors.

**Step 3:**Generate the mutant vector ${V}_{i}^{G}$ by perturbing a randomly selected vector ${P}_{k}^{G}$ with the difference of two other randomly selected vectors ${P}_{l}^{G}$ and ${P}_{m}^{G}$ according to Rand/1/bin mutation strategy.

**Step 4:**Generate the trial vectors $({U}_{i}^{G})$ through crossover by randomly recombining the parameters of target vectors (${P}_{i}^{G}$) and mutant vectors (${V}_{i}^{G}$).

**Step 5:**Calculate the fitness value for each trial vector generated in step 4.

**Step 6:**Perform 1-1 comparison between target vectors and trial vectors and select the vectors with improved fitness value for new offspring.

**Step 7:**Check whether desired fitness value is attained or maximum number of generations is achieved, if yes then stop this optimization process, otherwise go back to step 3.

## 4. Stud Differential Evolution (SDE)

Algorithm 1: Stud Differential Evolution (SDE) |

BeginRandomly initialize the population P (target vectors) of NP size and of D dimensions, in a feasible range Set the generation counter G = 1 Allot suitable values to all other control parameters i.e., crossover rate CR, mutation probability F etc. Calculate the fitness for all generated population vectors. While G < Maximum Generation doImplement regular DE from conventional mutational and crossover all the way to selection. for I = 1: NP doPerform Mutation and generate mutant vector ${V}_{i}^{G}$ Perform the SC operator in Algorithm 2end for iSort all the vectors and find the current best vector G = G + 1; end whileDisplay the best solution. End. |

Algorithm 2: Stud Crossover (SC) Operator |

BeginPerform the Selection Select the Stud/Best vector ${P}_{best}^{G}$ for mating Perform the Crossover Generate trial vector ${U}_{i}^{G}$, taking stud ${P}_{best}^{G}$ as first parent and mutant vector ${V}_{i}^{G}$as a second parent Calculate fitness of trial vector ($f({U}_{i}^{G})$) If ($f({U}_{i}^{G})>f({P}_{i}^{G})$) doAccept the generated trial vector ${U}_{i}^{G}$ for next generationelse elseAccept the generated target vector ${P}_{i}^{G}$ for next generation end ifEnd. |

## 5. Simulation Results

#### 5.1. System 1: 10 Machine Multiple Fuel Convex PED (without Valve Point Loading Effects)

#### 5.2. System 2: 10 Machine Multiple Fuel Non-Convex PED (with Valve Point Loading Effects)

## 6. Conclusions

- SDE is a potential solution methodology for the PED problem, as it addresses the convex and non-convex PED equally.
- Results obtained from SDE are better in comparison with the current research available, which indicates the promise of the approach.
- SDE can easily be further modified and hybridized with other optimization techniques because it has fewer control parameters.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathit{N}$ | Total no. of units |

${\mathit{P}}_{{\mathit{G}}_{\mathit{i}}}$ | Power from ith unit |

${\mathit{F}}_{\mathit{i}}({\mathit{P}}_{{\mathit{G}}_{\mathit{i}}})$ | Fuel cost associated with ith unit |

${\mathit{P}}_{\mathit{d}}$ | Total power demand |

${\mathit{P}}_{{\mathit{G}}_{\mathit{i}}}^{\mathit{m}\mathit{i}\mathit{n}}$ | Minimum power generation from ith unit |

${\mathit{P}}_{{\mathit{G}}_{\mathit{i}}}^{\mathit{m}\mathit{a}\mathit{x}}$ | Maximum power generation from ith unit |

${\mathit{P}}_{\mathit{i}\mathit{k}}^{\mathit{m}\mathit{i}\mathit{n}}$ | Minimum power generation from ith unit consuming kth fuel |

${\mathit{P}}_{\mathit{i}\mathit{k}}^{\mathit{m}\mathit{a}\mathit{x}}$ | Maximum power generation from ith unit consuming kth fuel |

${\mathit{F}}_{\mathit{T}}$ | Total cost of power generation |

${\mathit{a}}_{\mathit{i}\mathit{k}}$, ${\mathit{b}}_{\mathit{i}\mathit{k}}\mathit{a}\mathit{n}\mathit{d}{\mathit{c}}_{\mathit{i}\mathit{k}}$ | Cost coefficients of the ith generating unit consuming kth fuel |

Optimization Techniques | |

AIS | Artificial Immune System |

APSO | Adaptive particle swarm optimization |

ASA | Adaptive Simulated Annealing |

BSA | Back-tracking Search Algorithm |

CBPSO_RVM | Combined particle swarm optimization with real-valued mutation |

CGA_MU | Conventional Genetic Algorithm with Multiplier Updating |

C-GRASP–DE | Continuous Greedy Randomized Adaptive Search Procedure with Differential Evolution |

CPSO | Combinatorial particle swarm optimization |

CSA-Cauchy | Cuckoo Search Algorithm with Cauchy distribution |

CSA-Gauss | Cuckoo Search Algorithm with Gaussian distribution |

DEPSO | Differential Evolution with Particle Swarm Optimization |

DSPSO_TSA | Distributed Sobol Particle Swarm Optimization and Tabu Search Algorithm |

EALHN | Enhanced Augmented Lagrange Hopfield Network |

GA_BGC | Genetic Algorithm with best of Gaussian and Cauchy mutations |

GA_C | Genetic Algorithm GA with Cauchy mutation |

GA_G | Genetic Algorithm with Gaussian mutation |

GA_MGC | Genetic Algorithm with mean of Gaussian and Cauchy mutations |

GHS | Global-best Harmony Search |

HLN | Hopfield Lagrange Network |

HNN | Hopfiled Neural Network |

IDE | Improved Differential Evolution |

IEP | Improved Evolutionary Programming |

IGA_MU | Improved Genetic Algorithm with Multiplier Updating |

IODPSO_G | improved orthogonal design particle swarm optimization with global star structure |

IODPSO_L | improved orthogonal design particle swarm optimization with local ring structure |

LI | Lamda-iteration |

MHNN | Modified Hopfield Neural Network |

MPSO | Modified Particle Swarm Optimization |

MSFLA | Modified Shuffled Frog Leaping Algorithm |

NPSO | New particle swarm optimization |

QPSO | Quantum-behaved particle swarm optimization |

RCGA | Real-coded Genetic Algorithm |

SADE_ALM | Self-adaptive Differential Evolution method with Augmented Lagrange Multiplier |

SDE | Stud Differential Evolution |

SFLA-GHS | shuffled frog leaping algorithm with global-best harmony search algorithm |

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**Figure 5.**Convergence-characteristics of SBE algorithm for system 1 (without valve point loading effects), P

_{d}= 2400 MW, 2500 MW, 2600 MW.

**Figure 6.**The convergence characteristics of the proposed for system 1 (without valve point loading effects), P

_{d}= 2700 MW.

**Figure 7.**Convergence-characteristics of SBE algorithm for system 1 (without valve point loading effects), P

_{d}= 2400 MW, 2500 MW, 2600 MW.

**Figure 8.**The convergence characteristics of the proposed for system 2 (with valve point loading effects), P

_{d}= 2700 MW.

**Figure 9.**Cost-distribution around 30 run for system 2 (with valve point loading effects), P

_{d}= 2700 MW.

Unit No. | Fuel Types | Methods | ||||
---|---|---|---|---|---|---|

MSFLA | MHNN | SaDE | IEP | SDE | ||

P1 | 2 | 226.57 | 224.50 | 218.94 | 219.54 | 218.249988 |

P2 | 1 | 215.35 | 215.00 | 212.72 | 211.44 | 211.662614 |

P3 | 1 | 291.35 | 291.80 | 282.63 | 279.68 | 280.722785 |

P4 | 3 | 242.24 | 242.20 | 239.77 | 240.32 | 239.631553 |

P5 | 1 | 293.02 | 293.30 | 277.46 | 276.53 | 278.497228 |

P6 | 3 | 242.24 | 242.20 | 240.18 | 239.87 | 239.631562 |

P7 | 1 | 302.57 | 303.10 | 287.29 | 289.00 | 288.584580 |

P8 | 3 | 242.24 | 242.20 | 239.91 | 241.31 | 239.631491 |

P9 | 3 | 355.50 | 355.70 | 426.09 | 425.14 | 428.521600 |

P10 | 1 | 288.91 | 289.50 | 275.01 | 277.17 | 274.866600 |

Power Generated | 2700.00 | 2699.70 | 2700.00 | 2700.00 | 2700.00 | |

Total Cost | 626.25 | 626.12 | 623.92 | 623.85 | 623.809154 |

Unit No. | Fuel Used | Methods | |||
---|---|---|---|---|---|

HLN | LI | SaDE | SDE | ||

P1 | 2 | 209.7882 | 209.788 | 218.23 | 216.544182 |

P2 | 1 | 207.9078 | 207.9078 | 211.71 | 210.905752 |

P3 | 1 | 269.9145 | 269.9146 | 276.77 | 278.544078 |

P4 | 3 | 236.9782 | 236.9782 | 239.37 | 239.096668 |

P5 | 1 | 263.7247 | 263.7247 | 275.65 | 275.519445 |

P6 | 3 | 236.9782 | 236.9782 | 240.18 | 239.096668 |

P7 | 1 | 274.359 | 274.3591 | 285.99 | 285.717009 |

P8 | 3 | 236.9782 | 236.9782 | 238.16 | 239.096669 |

P9 | 1 | 402.7945 | 402.7945 | 341.90 | 343.493387 |

P10 | 1 | 260.5768 | 260.5767 | 272.04 | 271.986142 |

Power Generated | 2600.00 | 2600.00 | 2600.00 | 2600.00 | |

Total Cost | 574.74 | 574.74 | 574.54 | 574.380823 |

Unit No. | Fuel Used | Methods | |||
---|---|---|---|---|---|

MPSO | EALHN | AIS | SDE | ||

P1 | 2 | 206.5 | 206.5188 | 205.88 | 206.519016 |

P2 | 1 | 206.5 | 206.4573 | 206.33 | 206.457317 |

P3 | 1 | 265.7 | 265.7392 | 266.48 | 265.739085 |

P4 | 3 | 236.0 | 235.9531 | 235.79 | 235.953146 |

P5 | 1 | 258.0 | 258.0178 | 256.87 | 258.017644 |

P6 | 3 | 236.0 | 235.9531 | 236.65 | 235.953163 |

P7 | 1 | 268.9 | 268.8636 | 269.2 | 268.863542 |

P8 | 3 | 235.9 | 235.9531 | 235.51 | 235.953149 |

P9 | 1 | 331.5 | 331.4876 | 332.23 | 331.487723 |

P10 | 1 | 255.1 | 255.0564 | 255.02 | 255.056214 |

Power Generated | 2500.00 | 2500.00 | 2500.00 | 2500.00 | |

Total Cost | 526.239 | 526.239 | 526.240 | 526.238760 |

Unit No. | Fuel Used | Methods | ||||
---|---|---|---|---|---|---|

MHNN | AIS | EALHN | MPSO | SDE | ||

P1 | 1 | 192.7 | 189.683 | 189.7397 | 189.7 | 189.740527 |

P2 | 1 | 203.8 | 202.40 | 202.3427 | 202.3 | 202.342694 |

P3 | 1 | 259.1 | 253.814 | 253.8954 | 253.9 | 253.895318 |

P4 | 3 | 195.1 | 233.019 | 233.0456 | 233.0 | 233.045560 |

P5 | 1 | 248.7 | 241.94 | 241.8299 | 241.8 | 241.829619 |

P6 | 3 | 234.2 | 233.063 | 233.0456 | 233.0 | 233.045548 |

P7 | 1 | 260.3 | 253.374 | 253.2752 | 253.3 | 253.275055 |

P8 | 3 | 234.5 | 232.851 | 233.0456 | 233.0 | 233.045563 |

P9 | 1 | 324.7 | 320.452 | 320.3831 | 320.4 | 320.383139 |

P10 | 1 | 246.8 | 239.404 | 239.3973 | 339.4 | 239.396978 |

Power Generated | 2399.8 | 2400.00 | 2399.80 | 2400 | 2400 | |

Total Cost | 487.87 | 481.723 | 481.72300 | 481.723 | 481.722624 |

**Table 5.**Comprehensive comparison of total fuel cost and computation time for system 1 (without valve point loading effects), P

_{d}= 2400 MW, 2500 MW, 2600 MW and 2700 MW.

(i) | (ii) | ||||||

2400 MW | 2500 MW | ||||||

Methods | Total Power | Min. Cost | CT | Methods | Total Power | Min. Cost | CT |

HNN [41] | 2399.80 | 481.8700 | ~60 | IEP [42] | 2500.00 | 526.4000 | NR |

SaDE [32] | 2400.00 | 481.8628 | NR | SaDE [32] | 2500.00 | 526.3232 | NR |

IEP [42] | 2400.00 | 481.7790 | NR | ELANN [24] | 2500.00 | 526.2700 | 12.25 |

ELANN [24] | 2400.00 | 481.7400 | 11.53 | DE [48] | 2500.00 | 526.2390 | NR |

EALHN [45] | 2400.00 | 481.7230 | 0.008 | EALHN [45] | 2500.00 | 526.2390 | 0.006 |

MPSO [44] | 2400.00 | 481.7230 | NR | LI [43] | 2500.00 | 526.2390 | 2.508 |

RCGA [47] | 2400.00 | 481.7230 | 49.92 | RCGA [47] | 2500.00 | 526.2390 | 49.92 |

DE [48] | 2400.00 | 481.7230 | NR | MPSO [44] | 2500.00 | 526.2390 | NR |

LI [43] | 2399.99 | 481.7217 | 7.84 | HNN [41] | 2499.80 | 526.1300 | ~60 |

SDE | 2400.00 | 481.7226 | 2.50 | SDE | 2500.00 | 526.2387 | 2.43 |

(iii) | (iv) | ||||||

2600 MW | 2700 MW | ||||||

Methods | Total Power | Min. Cost | CT | Methods | Total Power | Min. Cost | CT |

LI [43] | 2600.00 | 574.7412 | 6.871 | HNN [41] | 2599.80 | 626.1200 | ~60 |

HLN [43] | 2600.00 | 574.7413 | 0.152 | SaDE [32] | 2700.00 | 623.9225 | NR |

SaDE [32] | 2600.00 | 574.5380 | NR | ELANN [24] | 2700.00 | 623.8800 | 21.36 |

IEP [42] | 2600.00 | 574.4730 | NR | IEP[42] | 2700.00 | 623.8510 | NR |

ELANN [24] | 2600.00 | 574.4100 | ~9.99 | RCGA [47] | 2700.00 | 623.8092 | 44.56 |

RCGA [47] | 2600.00 | 574.3960 | 33.57 | DE [48] | 2700.00 | 623.8090 | NR |

DE [48] | 2600.00 | 574.3810 | NR | LI [43] | 2699.99 | 623.8089 | 6.221 |

EALHN [45] | 2600.00 | 574.3810 | 0.005 | MPSO [44] | 2700.00 | 623.8090 | NR |

MPSO [44] | 2600.00 | 574.3810 | NR | CGA-MU [28] | 2700.00 | 623.8095 | 19.42 |

HNN [41] | 2599.80 | 574.2600 | ~60 | IGA-MU [28] | 2700.00 | 623.8093 | 5.27 |

SDE | 2600.00 | 574.3808 | 2.04 | SDE | 2700.00 | 623.8092 | 2.2 |

Unit No. | Fuel Used | Methods | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

IGA_MU | MSFLA | PSO | DE | RGA | NPSO-LRS | BSA | CSA-Cauchy | BAT | SDE | ||

P1 | 2 | 219.13 | 215.50 | 219.9962 | 218.2499 | 220.9376 | 223.33 | 218.58 | 218.1322 | 217.3232 | 218.593998 |

P2 | 1 | 211.16 | 210.72 | 212.7648 | 211.6626 | 212.6096 | 212.19 | 211.22 | 211.4116 | 209.9266 | 211.464175 |

P3 | 1 | 280.66 | 284.71 | 283.7391 | 280.7228 | 283.5811 | 276.21 | 279.56 | 281.6867 | 284.5552 | 280.657064 |

P4 | 3 | 238.48 | 239.77 | 240.5205 | 239.6315 | 240.0089 | 239.41 | 239.50 | 238.7456 | 237.2677 | 239.639428 |

P5 | 1 | 276.42 | 286.45 | 282.3127 | 278.4972 | 282.8920 | 274.64 | 279.97 | 279.8622 | 279.9804 | 279.934520 |

P6 | 3 | 240.47 | 240.18 | 240.5387 | 239.6315 | 240.4739 | 239.79 | 241.12 | 240.3328 | 240.1984 | 239.639428 |

P7 | 1 | 287.74 | 278.87 | 293.0846 | 288.5845 | 292.9792 | 285.53 | 289.80 | 287.7978 | 290.0943 | 287.727493 |

P8 | 3 | 240.76 | 242.06 | 240.2886 | 239.6315 | 240.1989 | 240.63 | 240.58 | 238.3435 | 238.3427 | 239.639428 |

P9 | 3 | 429.34 | 425.32 | 406.9797 | 428.5216 | 406.9988 | 429.26 | 426.89 | 427.8687 | 425.717 | 426.835856 |

P10 | 1 | 275.85 | 276.43 | 279.7752 | 274.8667 | 279.3199 | 278.65 | 272.80 | 275.8188 | 276.5845 | 275.868609 |

Power Generated | 2700.00 | 2700.00 | 2700.00 | 2700.00 | 2700.00 | 2700.00 | 2700.00 | 2700.0 | 2700.00 | 2700.00 | |

Total Cost | 624.52 | 624.12 | 624.5074 | 624.5146 | 624.5081 | 624.13 | 623.90 | 623.8566 | 623.8425 | 623.826575 |

Unit No. | Fuel Used | Methods | |||||||
---|---|---|---|---|---|---|---|---|---|

PSO | RGA | DE | MSFLA | GHS | BAT | SaDE | SDE | ||

P1 | 2 | - | - | - | 218.59 | 209.35 | 218.1376 | 219.99 | 216.539998 |

P2 | 1 | - | - | - | 203.05 | 207.99 | 212.1547 | 212.76 | 210.721482 |

P3 | 1 | - | - | - | 271.58 | 269.63 | 279.6484 | 283.74 | 278.640638 |

P4 | 3 | - | - | - | 236.41 | 236.95 | 239.552 | 240.52 | 238.698832 |

P5 | 1 | - | - | - | 276.43 | 265.48 | 271.4263 | 282.31 | 276.157152 |

P6 | 3 | - | - | - | 241.92 | 235.88 | 237.2423 | 240.53 | 238.967574 |

P7 | 1 | - | - | - | 287.73 | 273.51 | 287.7358 | 293.08 | 285.356480 |

P8 | 3 | - | - | - | 240.85 | 237.76 | 236.4615 | 240.29 | 238.564461 |

P9 | 1 | - | - | - | 344.20 | 403.33 | 339.8086 | 406.98 | 343.645968 |

P10 | 1 | - | - | - | 279.23 | 260.11 | 277.8228 | 279.78 | 272.707417 |

Power Generated | 2600.00 | 2600.00 | 2600.00 | 2600.00 | 2700.00 | 2600.00 | 2600.00 | 2600.00 | |

Total Cost | 575.161 | 575.161 | 575.175 | 574.89 | 574.79 | 574.5609 | 574.54 | 574.387064 |

Generation Schedule for P_{d} = 2500 MW and Non-Convex Cost | ||||||

Unit No. | SDE | |||||

P1 | 206.269999 | |||||

P2 | 206.512887 | |||||

P3 | 266.542078 | |||||

P4 | 236.414526 | |||||

P5 | 258.350235 | |||||

P6 | 236.280155 | |||||

P7 | 268.759386 | |||||

P8 | 235.608300 | |||||

P9 | 331.467106 | |||||

P10 | 253.795328 | |||||

Power Generated | 2500.00 | |||||

Total Cost | 526.245078 | |||||

Comparison of Results for P_{d} = 2500 MW and Non-Convex Cost | ||||||

Method Used | DE | RGA | PSO | ASA | SDE | |

Total Cost | 527.03600 | 527.0189 | 527.01850 | 526.32310 | 526.245533 |

Generation Schedule for P_{d} = 2400 MW and Non-Convex Cost | ||||||

Unit No. | SDE | |||||

P1 | 188.517831 | |||||

P2 | 202.551856 | |||||

P3 | 253.435305 | |||||

P4 | 232.786510 | |||||

P5 | 240.439406 | |||||

P6 | 233.189623 | |||||

P7 | 254.533306 | |||||

P8 | 233.055252 | |||||

P9 | 320.395414 | |||||

P10 | 241.095497 | |||||

Power Generated | 2400.00 | |||||

Total Cost | 481.734808 | |||||

Comparison of Results for P_{d} = 2400 MW and Non-Convex Cost | ||||||

Methods Used | ACO | DE | PSO | RGA | ASA | SDE |

Total Cost | 482.5267 | 482.5275 | 482.5088 | 482.5114 | 481.86290 | 481.734808 |

Power Demand (MW) | Crossover Rate (CR) | ||
---|---|---|---|

0.5 | 0.6 | 0.7 | |

2400 | 481.747921 | 481.734808 | 481.764849 |

2500 | 526.253282 | 526.245533 | 526.277145 |

2600 | 574.402910 | 574.387064 | 574.464175 |

2700 | 623.832350 | 623.826575 | 623.843516 |

**Table 11.**Comprehensive comparison of simulation results, standard deviation and computation time for system 2 (with valve point loading effects), P

_{d}= 2700 MW.

Methods | Min. Cost | Ave. Cost | Max. Cost | St. Deviation | CT (s) |
---|---|---|---|---|---|

CGA-MU [28] | 624.7193 | 627.6087 | 633.8652 | NR | 25.65 |

IGA-MU [28] | 624.5178 | 625.8692 | 630.8705 | NR | 7.14 |

DE (a) [49] | 624.5146 | 624.5246 | 624.5458 | 0.0077 | 2.8236 |

RGA (a) [49] | 624.5081 | 624.5079 | 624.5088 | 2.9476 × 10^{−5} | 4.1340 |

PSO (a) [49] | 624.5074 | 624.5074 | 624.5074 | 1.9691 × 10^{−13} | 3.3852 |

GA [7] | 624.5050 | 624.7419 | 624.8169 | 0.1005 | 18.3 |

PSO_GM [29] | 624.3100 | 625.09 | 624.67 | 0.16 | NR |

TSA [7] | 624.3078 | 635.0623 | 624.8285 | 1.1593 | 9.71 |

PSO_LRS [27] | 624.2297 | 625.7887 | 628.3214 | NR | 0.93 |

CPSO [29] | 624.1700 | 624.78 | 624.55 | 0.13 | NR |

NPSO [27] | 624.1624 | 625.218 | 627.4237 | NR | 0.41 |

NPSO_LRS [27] | 624.1273 | 624.9985 | 626.9981 | NR | 1.08 |

MSFLA [40] | 624.11569 | 624.8958 | 628.3428 | NR | NR |

APSO [54] | 624.0145 | 624.8185 | 624.8185 | NR | 0.52 |

PSO (b) [30] | 624.0120 | 624.2055 | 624.4376 | 0.0889 | 0.308 |

CBPSO_RVM [29] | 623.9600 | 624.29 | 624.08 | 0.06 | NR |

DE (b) [30] | 623.9280 | 624.0068 | 624.0653 | 0.0271 | 0.625 |

BSA [51] | 623.9016 | 623.9757 | 624.0838 | NR | NR |

ACO [55] | 623.9000 | 624.3500 | 624.7800 | NR | 8.35 |

GA_G [56] | 623.8900 | 625.21 | 635.30 | NR | NR |

GA_MGC [56] | 623.8900 | 624.72 | 626.94 | NR | NR |

GA_C [56] | 623.8800 | 624.53 | 626.95 | NR | NR |

GA_BGC [56] | 623.8800 | 624.14 | 626.51 | NR | NR |

QPSO [30] | 623.8766 | 623.9639 | 624.4163 | 0.0688 | 0.315 |

DE_ALM [57] | 623.8716 | 626.1298 | 642.7812 | NR | 12.375 |

CSA [58] | 623.8684 | 623.9495 | 626.3666 | 0.2438 | 1.587 |

CSA_Cauchy [52] | 623.8566 | 624.1160 | 626.3440 | 0.7395 | 2.1 |

CSA_Gauss [52] | 623.8564 | 624.3618 | 626.3474 | 0.9826 | 2.2 |

GHS [40] | 623.84914 | 624.1341 | 625.3157 | NR | NR |

CQPSO [30] | 623.8476 | 623.8652 | 623.8885 | 0.0151 | 0.318 |

SFLA-GHS [40] | 623.84065 | 623.9521 | 624.7804 | NR | NR |

DSPSO_TSA [7] | 623.8375 | 623.8625 | 623.9001 | 0.0106 | 3.44 |

SQPSO [30] | 623.8319 | 623.8440 | 623.8605 | 0.0107 | 0.324 |

IODPSO_G [59] | 623.83 | 623.84 | 623.83 | 0.01 | NR |

IODPSO_L [59] | 623.83 | 623.83 | 623.83 | 0.00 | NR |

SADE_ALM [57] | 623.8278 | 624.7864 | 634.8313 | NR | 17.032 |

SDE | 623.826575 | 623.833894 | 623.8412 | 3.62 × 10^{−3} | ~10 |

Units | P_{d} = 2700 MW | P_{d} = 2600 MW | P_{d} = 2500 MW | P_{d} = 2400 MW | ||||
---|---|---|---|---|---|---|---|---|

Convex | Nonconvex | Convex | Nonconvex | Convex | Nonconvex | Convex | Nonconvex | |

1 | 218.249988 | 218.593998 | 216.544182 | 216.539998 | 206.519016 | 206.269999 | 189.740527 | 188.517831 |

2 | 211.662614 | 211.464175 | 210.905752 | 210.721482 | 206.457317 | 206.512887 | 202.342694 | 202.551856 |

3 | 280.722785 | 280.657064 | 278.544078 | 278.640638 | 265.739085 | 266.542078 | 253.895318 | 253.435305 |

4 | 239.631553 | 239.639428 | 239.096668 | 238.698832 | 235.953146 | 236.414526 | 233.045560 | 232.786510 |

5 | 278.497228 | 279.934520 | 275.519445 | 276.157152 | 258.017644 | 258.350235 | 241.829619 | 240.439406 |

6 | 239.631562 | 239.639428 | 239.096668 | 238.967574 | 235.953163 | 236.280155 | 233.045548 | 233.189623 |

7 | 288.584580 | 287.727493 | 285.717009 | 285.356480 | 268.863542 | 268.759386 | 253.275055 | 254.533306 |

8 | 239.631491 | 239.639428 | 239.096669 | 238.564461 | 235.953149 | 235.608300 | 233.045563 | 233.055252 |

9 | 428.521600 | 426.835856 | 343.493387 | 343.645968 | 331.487723 | 331.467106 | 320.383139 | 320.395414 |

10 | 274.866600 | 275.868609 | 271.986142 | 272.707417 | 255.056214 | 253.795328 | 239.396978 | 241.095497 |

TP (MW) | 2700.00 | 2700.00 | 2600.00 | 2600.00 | 2500.00 | 2500.00 | 2400.00 | 2400.00 |

TC ($/h) | 623.809154 | 623.826575 | 574.380823 | 574.387064 | 526.238760 | 526.245533 | 481.722624 | 481.734808 |

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Naila; Haroon, S.S.; Hassan, S.; Amin, S.; Sajjad, I.A.; Waqar, A.; Aamir, M.; Yaqoob, M.; Alam, I. Multiple Fuel Machines Power Economic Dispatch Using Stud Differential Evolution. *Energies* **2018**, *11*, 1393.
https://doi.org/10.3390/en11061393

**AMA Style**

Naila, Haroon SS, Hassan S, Amin S, Sajjad IA, Waqar A, Aamir M, Yaqoob M, Alam I. Multiple Fuel Machines Power Economic Dispatch Using Stud Differential Evolution. *Energies*. 2018; 11(6):1393.
https://doi.org/10.3390/en11061393

**Chicago/Turabian Style**

Naila, Shaikh Saaqib Haroon, Shahzad Hassan, Salman Amin, Intisar Ali Sajjad, Asad Waqar, Muhammad Aamir, Muneeb Yaqoob, and Imtiaz Alam. 2018. "Multiple Fuel Machines Power Economic Dispatch Using Stud Differential Evolution" *Energies* 11, no. 6: 1393.
https://doi.org/10.3390/en11061393