# Analysis of Heuristic Optimization Technique Solutions for Combined Heat-Power Economic Load Dispatch

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Problem Formulation of CHPED

_{k}denote the number of power only units, CHP units, and heat only units, respectively. Similarly, i, j and k show the number of power-only units, CHP units, and heat-only units.

^{th}generating units, and a

_{i}, b

_{i}, and c

_{i}are the cost coefficients of power-only units.

#### 3.1. Problem Formulation with Valve-Point Effects

#### 3.2. Constraints

#### 3.2.1. Power Balance

^{th}generating units, and ${P}_{d}$ and ${P}_{Loss}$ are the demand of power and power loss in the transmission line, respectively. Power loss in the transmission line is given as

#### 3.2.2. Heat Balance

_{d}is the head demand.

#### 3.2.3. Generation Limit Due to Power-Only Units

#### 3.2.4. Capacity Limits of Power and Heat Due to Combined Heat-Power Units Only

## 4. Heuristic Optimization Techniques Analysis

## 5. Comparative Results and Analysis

**Test case 1**

**Test case 2**

**Test Case 3**

**Test Case 4**

**Test Case 5**

**Test Case 6**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AI | Artificial immune |

BCO | Bee colony optimization |

BLPSO | Biogeography-based learning particle swarm optimization |

CHPED | Combined heat-power economic dispatch |

${\mathrm{C}}_{\mathrm{i}}$ | Total generation cost |

${\mathrm{C}}_{\mathrm{j}}$ | Generation cost with CHP units |

${\mathrm{C}}_{\mathrm{k}}$ | Generation cost using heat-only units |

CSA-BA-ABC | Artificial bee colony |

C-PSO | Co-evolutionary particle swarm optimization |

CSO | Civilized swarm optimization |

COA | Cuckoo optimization algorithm |

Np | Number of power-only units |

Nc | CHP units |

Ck | Heat-only units |

ECSA | Elitist cuckoo search algorithm |

GWO | Grey wolf optimization |

GSO | Group search optimization |

GAMS | General algebraic modeling system |

HBOA | Heap-based optimization algorithm |

HTSA | Heat transfer search algorithm |

HBJSA | Hybrid heap-based and jellyfish search algorithm |

IGA-NCM | Improved genetic algorithm |

IDE | Improved differential evolution |

IMPA | improved marine predators algorithm |

MPSO | Modified particle swarm optimization |

MGSO | Modified group search optimizer |

OQNLP | OptQuest/NLP |

PPS | Powell’s pattern search |

SPSO | Selective particle swarm optimization |

TVAC-PSO | Time varying acceleration coefficient particle swarm optimization |

WOA | Whale optimization algorithm |

SFS | Stochastic fractal search algorithm |

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Ref. No. | Optimization Techniques | Constraints | Taken Case Study for Optimization | Advantages and Disadvantages |
---|---|---|---|---|

[1] 1994 | Quadratic programming | Generation limits | 15 traditional power units, 9 boilers, and 15 co-generation units | Fast response and does not depend on the size of the data |

[2] 1996 | Lagrangian relaxation | Power balance and generation limits | 7-unit system | Best suitable for small generating unit system optimization |

[3] 2013 | Benders decomposition | Inequality constraint | 4- and 5-unit systems | Performing well for a small data set |

[4] 2009 | SPSO | Equality and inequality | 4 units | Best performing for small test data |

[5] 2011 | Bee colony | Generation limits | 4 units | Fast and effective |

[6] 2012 | Artificial immune system | Power balance and generation limits | 4 units | Gives an optimum solution and takes less CPU time, but does not test the big test data set. |

[7] 2013 | Firefly algorithm | Power balance and generation limits | 4 units | Simple and effective |

[8] 2011 | Mesh adaptive direct search | Power balance and generation limits | Single- as well as multi-heat area and power area systems | Conceptually, it is very straightforward, easily implementable, and computationally effective. |

[9] 2015 | TVAC-PSO | Valve point, generation limit, power balance, and heat balance | 4- and 84-unit system | Effective for CHPED issues that are non-convex and non-linear |

[10] 2015 | Crisscross optimization | Valve point, transmission losses, and prohibited operating zones | 4,7, 24, and 48units | Effective for large test data also |

[11] 2016 | Exchange market | Valve-point loss along with power balance and generation limits | 4,5, 7, 24, and 48units | Powerful and robust algorithm |

[12] 2017 | WOA | Valve-point effect, generation limits | 24, 84, and 96 units | Easily handles large test data and gives a global solution |

[13] 2019 | IGA-NCM | Power balance | 4-, 5-, 7-, 24- and 48-unit system | It can handle small and large data and give optimal solutions easily. |

[14] 2019 | Advanced modified PSO | Valve-point effect, power balance, and generation limits | 4- and 7-unit system | The suggested technique can locate the ideal solution and avoid local minima. |

[15] 2020 | Hybrid NSGA II-MOPSO | Power balance and generation limits | 4- and 7-unit system | It can handle single- as well as multi-objective problems. |

[17] 2021 | HBOA | Transmission losses and the valve point | 4, 24, 84, and 96 generating units | Compared to other optimization techniques, the feasibility, capability, and efficiency are better for large-scale systems. |

[18] 2021 | HBJSA | Power balance and generation limits | 24-, 48-, 84- and 96-unit systems | The method used by HBJSA to calculate the lowest minimum, average, and maximum generation costs is very stable and efficient. |

[19] 2015 | Opposition-based group search | Valve-point loading and prohibited operating zones | 4-, 7-, 24-, and 28-unit systems | Best situated for small and large data sets to solve nonlinear problems |

[20] 2016 | Gravitational search algorithm(GSA) | Valve-point effect, power balance, and generation limits | 5-, 7-, 24- and 48-unit systems | Ability to solve large data sets of CHPED problems, good convergence characteristics, and efficiency in computation |

[21] 2020 | BLPSO | Power and heat limitations and prohibited operating zones. | 5, 7, 24, and 48 units | This approach prevents premature convergence and increases the precision of the solution. |

[22] 2106 | Cuckoo search algorithm (CSA) | Valve point, power losses, and power balance | 4 and 5 units | Controls parameters in such a way that they evaluate the high-quality solution and take less computational time. |

[23] 2017 | CPSO | Prohibited operating zones, valve point, and transmission losses | 4, 7, and 24 units | Enhances the quality of the answer while requiring fewer function evaluations. |

[24] 2017 | MGSO | power balance and valve point | 5-, 24-, 48-, 72-, and 96-unit test system | The suggested approach provides a better solution and outperforms existing methods computationally. |

[25] 2017 | Hybrid TVAC-GSA-PSO | Power balance and generation limits | 24 units, 48 units, | This technology is robust in evaluating the minimum generation cost with less expensive solutions. |

[26] 2018 | CSA-BA-ABC | Power and heat balance and prohibited operation zones | 5- and 7-unit test system | Delivering a high-quality solution with more economic benefits and no convergence issues |

[27] 2020 | SFS | Power balance and generation limits | 5- and 7-unit test system | It is possible to avoid local minima and require less computing time. |

[28] 2020 | Kho-Kho optimization (KKO) | Power balance and prohibited operation zones | 5- and 7-unit test system | This method imitates the special technique the chasing squad used to touch the runners team. |

[29] 2020 | OQNLP | Valve-point loading effect and power balance | 48-unit system | This technique provides an effective tool for dealing with optimization problems. |

[30] 2022 | Improved marine predators optimization algorithm | Power balance and generation limits | 5, 48, 84, 96 units | Convergence characteristics of IMPOA are stable, and computation is also fast. |

[31] 2023 | Comprehensive learning wavelet-mutated slime mold algorithm | Valve loading, prohibited operating zones, and generation limits | 24-, 48-, 84- and 96-unit system | The suggested technique solves the local search issue of population concentration. |

[32] 2020 | Direct Optimization algorithm | Power balance and generation limits | 4-unit system | Good convergence characteristics are suitable for small test data sets. |

[33] 2022 | C-PSO | Power balance and generation limits | 7 units (period of 24 h) | Performs well, and results show it is effective compared to other optimization techniques used for the same test data set. |

[34] 2020 | Heat transfer search (HTS) | Transmission loss, valve point, and prohibited operating zones | 7, 24, and 48 units | Stable operation and less computation time |

[35] 2022 | Differential evolution | Power generation limits, heat limits, prohibited operating zone | 11, 33 and 165 units | This method can hasten the removal of constraint violations and the decrease in the value of the goal function for each solution. |

[36] 2019 | TVAC-PSO | Prohibited operating zones, spinning reserve, valve point, power loss, and ramp rate | 5, 7, and 48 units | It can handle the various constraints and gives a global solution for the considered case. |

[37] 2022 | ETLBO with IPSO | Valve effects, prohibited operating zones, and power transmission loss | 4, 24, and 48 units | Handles constraints easily |

[38] 2020 | Multi-verse optimization algorithm | Valve point, transmission losses, and ramp limits | 4-, 7-, 10-, and 40-unit system | The exceedingly challenging combined economic emission dispatch is solved by the suggested method. |

[39] 2016 | Group search optimization | Prohibited operating zones and valve-point loading | 4, 7, and 24 units | Effective for multiobjective nonlinear problem solutions |

[41] 2013 | SALCSSA | Ramp rate | 10, 30, 150 units (for 24 h) | Gives an optimum solution with a good convergence speed |

[42] 2017 | Cuckoo optimization algorithm | Valve-point effects | 7, 24, 48 | Handles the loading effect and gives optimum results. |

[43] 2016 | IDE | Valve-point effects | 13, 38 units | Easily handles the equality constraints |

[44] 2014 | Teaching–learning-based optimization | Valve-point loading | 7, 24, and 48 units | For multiobjective problems, this approach effectively enhances the overall performance of the solutions. |

[45] 2016 | Grey wolf optimization | Ramp rate, valve point, and spinning reserve | 4, 7, 11, and 24 units | The recommended method works more consistently and with higher-quality solutions. |

[46] 2013 | Fuzzy logic | Ramp-rate limits | 7 units | This technique has the potential to solve a larger, multi-objective problem. |

[47] 2005 | IGA-MU | Change fuels and valve point | 4-, 7-unit system | This approach has a straightforward idea that makes it easier to use and more successful. |

[48] 2020 | Cuckoo optimization | Power generation and heat limits | 4 units | Enhances the exploration on the search space |

[49] 2015 | Coded genetic algorithm | Valve point and transmission losses | 4, 5, 7, and 24 units | Effective for small and large test data |

[50] 2019 | MPHS | 24 and 84 units | Handles large data easily | |

[51] 2013 | MPSO | Valve point and prohibited operating zones | 24 and 48 units | Toimprove the efficiency and simulation solution, Gaussian random variables were used. |

**Table 2.**Results obtained for the four generating units with two co-generation units and one heat unit for the load demand of 200 MW and 115 MWth.

Results of Generating Units | FA [7] | MADS–DACE [8] | TVAC- PSO [9] | CSO [10] | EMA [11] | IGA-NCM [13] | SFS [27] | ETLBOI PSO [37] | GWO [45] |
---|---|---|---|---|---|---|---|---|---|

P1 (MW) | 0.0014 | 0 | 0 | 0 | 0 | 0 | 0 | 0.8473 | 0 |

P2 (MW) | 159.99 | 160 | 160 | 160 | 160 | 160 | 160 | 159.338 | 160 |

P3 (MW) | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 39.8150 | 40 |

H2 (MWth) | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 | 40 |

H3 (MWth) | 75 | 75 | 75 | 75 | 75 | 75 | 75 | 75 | 75 |

H4 (MWth) | 0.0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.18 | 0 |

Total cost ($) | 9257.1 | 9257.07 | 9257.07 | 9257.07 | 9257.07 | 9257.07 | 9257.07 | 9178.9934 | 9257.07 |

CPU time (s) | 1.25 | 3.27 | 1.78 | 1.18 | 0.9846 | 1.44 | 3.78 | 1.59 | 2.17 |

**Table 3.**Comparative performances of the various optimization techniques for the 7-unit system for the load demand of 600 MW and heat demand of 150 MWth.

Optimum Results of Generating Units | AIS [6] | TVAC- PSO [9] | CSO [10] | EMA [11] | IGA-NCM [13] | HTS [34] | GSO [39] | GWO [45] | RCGA-I [49] |
---|---|---|---|---|---|---|---|---|---|

P1 (MW) | 50.1325 | 47.3383 | 45.2 | 52.684 | 45.155 | 44.2825 | 45.6188 | 52.8074 | 45.6614 |

P2 (MW) | 95.5552 | 98.5398 | 98.539 | 98.5398 | 98.5398 | 100.110 | 98.5401 | 98.5398 | 98.5398 |

P3 (MW) | 110.751 | 112.673 | 112.67 | 112.673 | 112.673 | 112.621 | 112.672 | 112.6735 | 112.6735 |

P4 (MW) | 208.768 | 209.815 | 209.81 | 209.815 | 209.815 | 209.700 | 209.815 | 209.8158 | 209.8158 |

P5 (MW) | 98.8 | 92.3718 | 94.183 | 93.8341 | 94.5549 | 94.0105 | 94.1027 | 93.8115 | 93.9960 |

P6 (MW) | 42 | 40 | 40 | 40 | 40 | 40.0235 | 40.0001 | 40 | 40 |

H5 (MWth) | 19.4242 | 37.8467 | 27.178 | 29.242 | 29.2388 | 28.262 | 27.6600 | 29.3704 | 28.2842 |

H6 (MWth) | 77.0777 | 74.9999 | 75 | 75 | 75 | 74.7432 | 74.9987 | 75 | 75 |

H7 (MWth) | 53.498 | 37.1532 | 47.82 | 45.75 | 45.7612 | 46.9948 | 47.3413 | 29.3704 | 46.7158 |

Total cost ($) | 10,355 | 10,100.3 | 10,094.12 | 10,111.07 | 10,107.90 | 10,094.7 | 10,094.26 | 10,111.24 | 10,094.05 |

CPU time (s) | 5.2956 | 3.48 | 3.09 | 2.06 | 3.47 | 2.01 | 2.4203 | 5.2618 | 3.15 |

**Table 4.**Competitive results for a 24-unit system for the power and heat demands of 2350 MW and 1250 MWth.

Output | CSO [10] | EMA [11] | WOA [12] | IGA-NCM [13] | HBOA [17] | HBJSA [18] | HTS [34] | ETLBOI- PSO [37] | TLBO [44] |
---|---|---|---|---|---|---|---|---|---|

P1 | 448.7 | 628.31 | 628.3185 | 628.318 | 538.5587 | 448.818 | 539.5724 | 458.4 | 628.324 |

P2 | 225.2 | 299.18 | 299.1993 | 299.198 | 300.2175 | 299.2188 | 298.9487 | 291.93 | 298.7686 |

P3 | 299.2 | 299.16 | 299.1993 | 29.1665 | 301.08255 | 300.7211 | 297.9085 | 228.1 | 298.9086 |

P4 | 109.86 | 109.86 | 109.8665 | 109.867 | 159.777 | 60.10963 | 110.082 | 93.74 | 110.1919 |

P5 | 109.86 | 109.86 | 109.8665 | 109.866 | 63.2173 | 159.7451 | 110.2645 | 180 | 110.0846 |

P6 | 159.73 | 109.865 | 109.8665 | 60 | 60.6889 | 159.7769 | 110.2381 | 124.06 | 110.1379 |

P7 | 159.73 | 60 | 109.8665 | 109.86 | 160.20652 | 159.7718 | 110.2745 | 115.92 | 110.1045 |

P8 | 159.73 | 109.86 | 60.00003 | 109.823 | 111.5383 | 60 | 110.2452 | 116.68 | 110.2444 |

P9 | 109.86 | 109.856 | 109.8665 | 109.852 | 11.25395 | 159.751 | 110.1592 | 180 | 110.1992 |

P10 | 40 | 40 | 40.00003 | 40.0001 | 40 | 77.41183 | 77.3992 | 65.38 | 77.4989 |

P11 | 77.399 | 77.019 | 76.9485 | 77.0316 | 40.00025 | 40.00109 | 77.8364 | 40 | 77.7367 |

P12 | 92.399 | 55 | 55.00003 | 55.0098 | 55.657936 | 55.00862 | 55.0023 | 79.44 | 55.1036 |

P13 | 55 | 55 | 55.00003 | 55 | 55.284 | 55.6611 | 55.0109 | 89.23 | 55.1107 |

P14 | 87.554 | 81 | 81.00003 | 81.0035 | 87.944 | 85.84419 | 81.0524 | 81 | 81.0624 |

P15 | 40 | 40 | 40.00165 | 40.0003 | 41.2662 | 42.75199 | 40.0015 | 40 | 40.3515 |

P16 | 90.609 | 81 | 81.00003 | 81.0003 | 84.034 | 95.88869 | 81.003 | 81.1 | 81.262 |

P17 | 40 | 40 | 40.00003 | 40.0001 | 43.1437 | 44.46837 | 40.0009 | 40 | 40.0119 |

P18 | 10 | 10 | 10.00003 | 10.0002 | 11.0824 | 10.04622 | 10.0002 | 10 | 10.0011 |

P19 | 35 | 35 | 35.00003 | 35.0003 | 35.044 | 35.00512 | 35.0001 | 35.012 | 35.0012 |

H14 | 108.47 | 104.82 | 104.8 | 104.801 | 108.697 | 107.4915 | 105.2219 | 104.76 | 105.211 |

H15 | 75 | 75 | 75.0014 | 75.0001 | 76.0921 | 77.37645 | 76.5205 | 75 | 76.5306 |

H16 | 110.19 | 104.82 | 104.8 | 104.799 | 106.47627 | 113.1557 | 105.5142 | 104.74 | 105.511 |

H17 | 75 | 75 | 75 | 74.9988 | 77.7146 | 78.85075 | 75.4833 | 74.99 | 75.4706 |

H18 | 40 | 40 | 40 | 39.9993 | 40.4643 | 40.02 | 39.9999 | 40 | 39.9999 |

H19 | 20 | 20 | 20 | 20.0001 | 20.0204 | 20.00127 | 18.3944 | 18.25 | 18.4014 |

H20 | 461.32 | 470.39 | 470.3986 | 470.409 | 460.53781 | 453.1093 | 468.9043 | 473 | 468.902 |

H21 | 59.999 | 60 | 59.99998 | 60 | 60 | 60 | 59.9994 | 60 | 59.9995 |

H22 | 59.999 | 60 | 59.99998 | 60 | 60 | 59.99883 | 59.9999 | 59.96 | 59.9995 |

H23 | 119.99 | 120 | 119.9999 | 120 | 119.99644 | 119.9964 | 119.9854 | 119.35 | 119.9856 |

H24 | 120 | 120 | 119.9999 | 119.991 | 120 | 119.9995 | 119.9768 | 119.99 | 119.986 |

Total cost ($) | 57,907.1 | 57,825.5 | 57,830.52 | 57,826.09 | 57,994.51 | 57,968.54 | 57,842.99 | 57,758.66 | 57,843.52 |

CPU (s) | 24.98 | 1.167 | 2.71 | 1.72 | 3.62 | 4.04 | 5.47 | 2.63 | 5.4106 |

**Table 5.**Costs obtained by different heuristic optimization techniques for the 48-unit system (power and heat demands of 4700 MW and 2500 MWth).

Methods | Min. Cost ($) | Methods | Min. Cost ($) |
---|---|---|---|

CSO [10] | 115,967.7205 | OQNLP [29] | 116,993.2 |

EMA [11] | 115,611.84 | IMPAO [30] | 116,640.6 |

IGA_NCM [13] | 115,685.2 | CLWSMA [31] | 116,389.588 |

HBJSA [18] | 116,140.34 | TVAC-PSO [36] | 115,610.465 |

OGSO [19] | 116,678.2 | ETLBOIPSO [37] | 115,126.32 |

MGSO [24] | 115,606.5482 | COA [42] | 116,789.91535 |

TVAC-GSA-PSO [25] | 116,393.4034 | TLBO [44] | 116,739.3640 |

KKO [28] | 115,422 | OTLBO [44] | 116,579.2390 |

MPSO [51] | 116,919 |

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## Share and Cite

**MDPI and ACS Style**

Singh, N.; Chakrabarti, T.; Chakrabarti, P.; Panchenko, V.; Budnikov, D.; Yudaev, I.; Bolshev, V.
Analysis of Heuristic Optimization Technique Solutions for Combined Heat-Power Economic Load Dispatch. *Appl. Sci.* **2023**, *13*, 10380.
https://doi.org/10.3390/app131810380

**AMA Style**

Singh N, Chakrabarti T, Chakrabarti P, Panchenko V, Budnikov D, Yudaev I, Bolshev V.
Analysis of Heuristic Optimization Technique Solutions for Combined Heat-Power Economic Load Dispatch. *Applied Sciences*. 2023; 13(18):10380.
https://doi.org/10.3390/app131810380

**Chicago/Turabian Style**

Singh, Nagendra, Tulika Chakrabarti, Prasun Chakrabarti, Vladimir Panchenko, Dmitry Budnikov, Igor Yudaev, and Vadim Bolshev.
2023. "Analysis of Heuristic Optimization Technique Solutions for Combined Heat-Power Economic Load Dispatch" *Applied Sciences* 13, no. 18: 10380.
https://doi.org/10.3390/app131810380