Optimal operation of transmission power networks by using improved stochastic fractal search algorithm

Abstract

This paper presents the application of an improved stochastic fractal search algorithm (ISFSA) for optimizing five single objectives of optimal power flow (OPF) problem and satisfying all constraints consisting of operating limits of electric components, power balance and load voltage magnitude limits. The proposed ISFSA is formed by implementing three improvements on the conventional stochastic fractal search algorithm (SFSA). The first improvement cancels one ineffective formula but keeps another one in diffusion process. The second improvement selects some worst solutions in the first update and some best solutions in the second update for producing new solutions. In the third improvement, a proposed technique is applied for carrying out the update processes. Comparisons of obtained results from three standard IEEE power systems indicate that the proposed method is superior to SFSA in terms of optimal solution quality, execution speed as well as success rate. The performance comparisons with other existing methods available in previous studies also lead to the conclusions that the proposed method can reach lower generation fuel cost, smaller total power losses, less amount of emission, better voltage profile and faster execution process. As a result, it can be recommended that the proposed ISFSA should be used for OPF problem in high-voltage power system field.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Abbreviations

\(F_{i}\) :

Fuel cost function of the ith thermal unit

\(\phi_{i} ,\phi_{j}\) :

Phase angles of voltage at the ith bus and the jth bus

\(a_{fi} ,b_{fi} ,c_{fi} ,d_{fi} ,e_{fi}\) :

Fuel cost coefficients of the ith thermal unit

\(a_{fim} ,b_{fim} ,c_{fim}\) :

Fuel cost coefficients of the fuel type m of the ith thermal unit

\(a_{ei} ,b_{ei} ,c_{ei} ,d_{ei} ,e_{ei}\) :

Emission function coefficients of the ith thermal unit

\(FF_{s,j}\) :

Fitness function of new solution s at the jth diffusion

\(FF_{s}^{new}\) :

Fitness function of the new solution s

\(FF_{s}\) :

Fitness function of the sth retained solution

\(FF_{average}\) :

Average fitness function of the whole population

\(G_{ij} ,B_{ij}\) :

Conductance and susceptance of a branch connecting the ith bus and the jth bus

\(K_{1} ,K_{2} ,K_{3} ,K_{4} ,K_{5}\) :

Penalty factors

\(N_{fs}\) :

Number of fuel sources

N VPZi :

Number of violated power zones of the ith thermal unit

\(N_{bus}\) :

Number of buses in considered system

\(N_{lb}\) :

Number of load buses

\(N_{tb}\) :

Number of transformer buses

\(N_{cb}\) :

Number of compensator buses

\(N_{tl}\) :

Number of transmission lines in the considered power system

\(N_{di}\) :

Maximum number of diffusion

\(N_{ps}\) :

Population size

\(P_{i}^{\hbox{min} } ,P_{i}^{\hbox{max} }\) :

Lower and upper limitations of real power of the ith thermal unit

\(P_{i}\) :

Real power output of the ith thermal unit

\(P_{im}^{\hbox{min} } ,P_{im}^{\hbox{max} }\) :

Lowest and the highest generations of the ith thermal unit corresponding to the mth fuel type

\(P_{loadi} ,Q_{loadi}\) :

Real and unreal power of load at the ith bus

\(P_{{i,VPZ_{j} }}^{\hbox{min} } ,P_{{i,VPZ_{j} }}^{\hbox{max} }\) :

Lower and upper bounds of the jth violated power zone of the ith thermal unit

\(Q_{sci}^{\hbox{min} } ,Q_{sci}^{\hbox{max} }\) :

Minimum and maximum reactive power output of the capacitor banks at the ith bus

\(Q_{i}^{\hbox{min} } ,Q_{i}^{\hbox{max} }\) :

Lower and upper limitations of reactive power of the ith thermal unit

\(Q_{i} ,V_{i}\) :

Currently working unreal power and voltage magnitude of the ith thermal unit

\(rand_{s,j}\) :

Random number for the solution s at the jth diffusion

\(S_{br}^{\hbox{max} }\) :

Maximum apparent power flow of the brth transmission line

\(Sol_{s,j}^{new}\) :

The sth new solution at the jth diffusion

\(T_{i}^{\hbox{min} } ,T_{i}^{\hbox{max} }\) :

Minimum and maximum setting of tap changer at the ith bus

\(V_{i}^{\hbox{min} } ,V_{i}^{\hbox{max} }\) :

Lower and upper limitations of voltage magnitude of the ith thermal unit

VPZj :

The jth violated power zone

\(V_{li}^{\hbox{min} } ,V_{li}^{\hbox{max} }\) :

Lower and upper bounds of operation voltage of the ith bus

Iter, NIt :

Current iteration and the maximum number of iterations

Pro s :

Ratio of rank of the sth solution to population size

N cv :

Number of control variables

βs, εs :

Random number within 0 and 1 for the sth solution

ε :

Random number within 0 and 1

EIL:

Emission improvement level

FC:

Fuel cost

FCIL:

Fuel cost improvement level

OPF:

Optimal power flow

TPL:

Total power losses

TPLIL:

Total power loss improvement level

VD:

Voltage deviation

VDIL:

Voltage deviation improvement level

References

  1. 1.

    Momoh JA, Adapa R, El-Hawary ME (1993) A review of selected optimal power flow literature to 1993. I. Nonlinear and quadratic programming approaches. IEEE Trans Power Syst 14(1):96–104. https://doi.org/10.1109/59.744492

    Article  Google Scholar 

  2. 2.

    Momoh JA, El-Hawary ME, Adapa R (1993) A review of selected optimal power flow literature to 1993. II. Newton, linear programming and interior point methods. IEEE Trans Power Syst 14(1):105–111. https://doi.org/10.1109/59.744495

    Article  Google Scholar 

  3. 3.

    Yuryevich J, Wong KP (1999) Evolutionary programming based optimal power flow algorithm. IEEE Trans Power Syst 14(4):1245–1250. https://doi.org/10.1109/59.801880

    Article  Google Scholar 

  4. 4.

    Ongsakul W, Tantimaporn T (2006) Optimal power flow by improved evolutionary programming. Electr Power Compon Syst 34(1):79–95. https://doi.org/10.1080/15325000691001458

    Article  Google Scholar 

  5. 5.

    Abido MA (2002) Optimal power flow using particle swarm optimization. Int J Electr Power Energy Syst 24(7):563–571. https://doi.org/10.1016/S0142-0615(01)00067-9

    Article  Google Scholar 

  6. 6.

    Attous DB, Labbi Y (2009) Particle swarm optimization based optimal power flow for units with non-smooth fuel cost functions. In: International conference on IEEE electrical and electronics engineering. ELECO 2009, pp I–377. https://doi.org/10.1109/ELECO.2009.5355329

  7. 7.

    Niknam T, Narimani MR, Aghaei J, Azizipanah-Abarghooee R (2012) Improved particle swarm optimisation for multi-objective optimal power flow considering the cost, loss, emission and voltage stability index. IET Gener Transm Distrib 6(6):515–527. https://doi.org/10.1049/iet-gtd.2011.0851

    Article  Google Scholar 

  8. 8.

    Vaisakh K, Srinivas LR, Meah K (2013) Genetic evolving ant direction particle swarm optimization algorithm for optimal power flow with non-smooth cost functions and statistical analysis. Appl Soft Comput 13(12):4579–4593. https://doi.org/10.1016/j.asoc.2013.07.002

    Article  Google Scholar 

  9. 9.

    Vo DN, Schegner P (2013) An improved particle swarm optimization for optimal power flow. In: Meta-heuristics optimization algorithms in engineering, business, economics, and finance. IGI Global, pp 1–40. https://doi.org/10.4018/978-1-4666-2086-5.ch001

  10. 10.

    Roberge V, Tarbouchi M, Okou F (2016) Optimal power flow based on parallel metaheuristics for graphics processing units. Electr Power Syst Res 140:344–353. https://doi.org/10.1016/j.epsr.2016.06.006

    Article  Google Scholar 

  11. 11.

    Younis U, Khaliq A, Saleem M (2018) Weights aggregated multi-objective particle swarm optimizer for optimal power flow considering the generation cost, emission, transmission loss and bus-voltage profile. Int J Innov Comput Inf Control 14(4):1423–1441. https://doi.org/10.24507/ijicic.14.04.1423

    Article  Google Scholar 

  12. 12.

    El-Fergany AA, Hasanien HM (2015) Single and multi-objective optimal power flow using grey wolf optimizer and differential evolution algorithms. Electr Power Compon Syst 43(13):1548–1559. https://doi.org/10.1080/15325008.2015.1041625

    Article  Google Scholar 

  13. 13.

    El Ela AA, Abido MA, Spea SR (2010) Optimal power flow using differential evolution algorithm. Electr Power Syst Res 80(7):878–885. https://doi.org/10.1016/j.epsr.2009.12.018

    Article  Google Scholar 

  14. 14.

    Thitithamrongchai C, Eua-Arporn B (2007) Self-adaptive differential evolution based optimal power flow for units with non-smooth fuel cost functions. J Electr Syst 3(2):88–99

    Google Scholar 

  15. 15.

    Sayah S, Zehar K (2008) Modified differential evolution algorithm for optimal power flow with non-smooth cost functions. Energy Convers Manag 49(11):3036–3042. https://doi.org/10.1016/j.enconman.2008.06.014

    Article  Google Scholar 

  16. 16.

    Shaheen AM, El-Sehiemy RA, Farrag SM (2016) Solving multi-objective optimal power flow problem via forced initialised differential evolution algorithm. IET Gener Transm Distrib 10(7):1634–1647. https://doi.org/10.1049/iet-gtd.2015.0892

    Article  Google Scholar 

  17. 17.

    Biswas PP, Suganthan PN, Mallipeddi R, Amaratunga GA (2018) Optimal power flow solutions using differential evolution algorithm integrated with effective constraint handling techniques. Eng Appl Artif Intell 68:81–100. https://doi.org/10.1016/j.engappai.2017.10.019

    Article  Google Scholar 

  18. 18.

    Saini A, Chaturvedi DK, Saxena AK (2006) Optimal power flow solution- a GA-fuzzy system approach. Int J Emerg Electr Power Syst 5(2):1–21. https://doi.org/10.2202/1553-779X.1091

    Article  Google Scholar 

  19. 19.

    Bakirtzis AG, Biskas PN, Zoumas CE, Petridis V (2002) Optimal power flow by enhanced genetic algorithm. IEEE Trans Power Syst 17(2):229–236. https://doi.org/10.1109/TPWRS.2002.1007886

    Article  Google Scholar 

  20. 20.

    Kumari MS, Maheswarapu S (2010) Enhanced genetic algorithm based computation technique for multi-objective optimal power flow solution. Int J Electr Power Energy Syst 32(6):736–742. https://doi.org/10.1016/j.ijepes.2010.01.010

    Article  Google Scholar 

  21. 21.

    Reddy SS, Bijwe PR, Abhyankar AR (2014) Faster evolutionary algorithm based optimal power flow using incremental variables. Int J Electr Power Energy Syst 54:198–210. https://doi.org/10.1016/j.ijepes.2013.07.019

    Article  Google Scholar 

  22. 22.

    Reddy SS, Bijwe PR (2016) Efficiency improvements in meta-heuristic algorithms to solve the optimal power flow problem. Int J Electr Power Energy Syst 82:288–302. https://doi.org/10.1016/j.ijepes.2016.03.028

    Article  Google Scholar 

  23. 23.

    Bhattacharya A, Chattopadhyay PK (2011) Application of biogeography-based optimisation to solve different optimal power flow problems. IET Gener Transm Distrib 5(1):70–80. https://doi.org/10.1049/iet-gtd.2010.0237

    Article  Google Scholar 

  24. 24.

    Kumar AR, Premalatha L (2015) Optimal power flow for a deregulated power system using adaptive real coded biogeography-based optimization. Int J Electr Power Energy Syst 73:393–399. https://doi.org/10.1016/j.ijepes.2015.05.011

    Article  Google Scholar 

  25. 25.

    Nayak MR, Nayak CK, Rout PK (2012) Application of multi-objective teaching learning based optimization algorithm to optimal power flow problem. Procedia Technol 6:255–264. https://doi.org/10.1016/j.protcy.2012.10.031

    Article  Google Scholar 

  26. 26.

    Shabanpour-Haghighi A, Seifi AR, Niknam T (2014) A modified teaching–learning based optimization for multi-objective optimal power flow problem. Energy Convers Manag 77:597–607. https://doi.org/10.1016/j.enconman.2013.09.028

    Article  Google Scholar 

  27. 27.

    Pulluri H, Naresh R, Sharma V (2018) A solution network based on stud krill herd algorithm for optimal power flow problems. Soft Comput 22(1):159–176. https://doi.org/10.1007/s00500-016-2319-3

    Article  Google Scholar 

  28. 28.

    Chen G, Lu Z, Zhang Z (2018) Improved krill herd algorithm with novel constraint handling method for solving optimal power flow problems. Energies 11(1):76. https://doi.org/10.3390/en11010076

    Article  Google Scholar 

  29. 29.

    Ladumor DP, Trivedi IN, Bhesdadiya RH, Jangir P (2017) A grey wolf optimizer algorithm for Voltage Stability Enhancement. In: Third international conference on IEEE advances in electrical, electronics, information, communication and bio-informatics (AEEICB), pp 278–282. https://doi.org/10.1109/AEEICB.2017.7972429

  30. 30.

    Abdo M, Kamel S, Ebeed M, Yu J, Jurado F (2018) Solving non-smooth optimal power flow problems using a developed Grey Wolf Optimizer. Energies 11(7):1692. https://doi.org/10.3390/en11071692

    Article  Google Scholar 

  31. 31.

    Abido MA (2002) Optimal power flow using tabu search algorithm. Electr Power Compon Syst 30(5):469–483. https://doi.org/10.1080/15325000252888425

    Article  Google Scholar 

  32. 32.

    Roa-Sepulveda CA, Pavez-Lazo BJ (2003) A solution to the optimal power flow using simulated annealing. Int J Electr Power Energy Syst 25(1):47–57. https://doi.org/10.1016/S0142-0615(02)00020-0

    Article  Google Scholar 

  33. 33.

    Niknam T, Narimani MR, Aghaei J, Tabatabaei S, Nayeripour M (2011) Modified honey bee mating optimisation to solve dynamic optimal power flow considering generator constraints. IET Gener Transm Distrib 5(10):989–1002. https://doi.org/10.1049/iet-gtd.2011.0055

    Article  Google Scholar 

  34. 34.

    Niknam T, rasoul Narimani M, Jabbari M, Malekpour AR (2011) A modified shuffle frog leaping algorithm for multi-objective optimal power flow. Energy 36(11):6420–6432. https://doi.org/10.1016/j.energy.2011.09.027

    Article  Google Scholar 

  35. 35.

    Ara AL, Kazemi A, Gahramani S, Behshad M (2012) Optimal reactive power flow using multi-objective mathematical programming. Sci Iran 19(6):1829–1836. https://doi.org/10.1016/j.scient.2012.07.010

    Article  Google Scholar 

  36. 36.

    Duman S, Güvenç U, Sönmez Y, Yörükeren N (2012) Optimal power flow using gravitational search algorithm. Energy Convers Manag 59:86–95. https://doi.org/10.1016/j.enconman.2012.02.024

    Article  Google Scholar 

  37. 37.

    Adaryani MR, Karami A (2013) Artificial bee colony algorithm for solving multi-objective optimal power flow problem. Int J Electr Power Energy Syst 53:219–230. https://doi.org/10.1016/j.ijepes.2013.04.021

    Article  Google Scholar 

  38. 38.

    Ghasemi M, Ghavidel S, Ghanbarian MM, Gharibzadeh M, Vahed AA (2014) Multi-objective optimal power flow considering the cost, emission, VD and TPL using multi-objective modified imperialist competitive algorithm. Energy 78:276–289. https://doi.org/10.1016/j.energy.2014.10.007

    Article  Google Scholar 

  39. 39.

    Le Anh TN, Vo DN, Ongsakul W, Vasant P, Ganesan T (2015) Cuckoo optimization algorithm for optimal power flow. In: Proceedings of the 18th Asia Pacific symposium on intelligent and evolutionary systems, pp 479–493. https://doi.org/10.1007/978-3-319-13359-1_37

  40. 40.

    Ghasemi M, Ghavidel S, Ghanbarian MM, Gitizadeh M (2015) Multi-objective optimal electric power planning in the power system using Gaussian bare-bones imperialist competitive algorithm. Inf Sci 294:286–304. https://doi.org/10.1016/j.ins.2014.09.051

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Daryani N, Hagh MT, Teimourzadeh S (2016) Adaptive group search optimization algorithm for multi-objective optimal power flow problem. Appl Soft Comput 38:1012–1024. https://doi.org/10.1016/j.asoc.2015.10.057

    Article  Google Scholar 

  42. 42.

    El-Hana Bouchekara HR, Abido MA, Chaib AE (2016) Optimal power flow using an improved electromagnetism-like mechanism method. Electr Power Compon Syst 44(4):434–449. https://doi.org/10.1080/15325008.2015.1115919

    Article  Google Scholar 

  43. 43.

    Bouchekara HREH, Chaib AE, Abido MA, El-Sehiemy RA (2016) Optimal power flow using an Improved Colliding Bodies Optimization algorithm. Appl Soft Comput 42:119–131. https://doi.org/10.1016/j.asoc.2016.01.041

    Article  Google Scholar 

  44. 44.

    Abaci K, Yamacli V (2016) Differential search algorithm for solving multi-objective optimal power flow problem. Int J Electr Power Energy Syst 79:1–10. https://doi.org/10.1016/j.ijepes.2015.12.021

    Article  Google Scholar 

  45. 45.

    Mohamed AAA, Mohamed YS, El-Gaafary AA, Hemeida AM (2017) Optimal power flow using moth swarm algorithm. Electr Power Syst Res 142:190–206. https://doi.org/10.1016/j.epsr.2016.09.025

    Article  Google Scholar 

  46. 46.

    Yuan Y, Wu X, Wang P, Yuan X (2018) Application of improved bat algorithm in optimal power flow problem. Appl Intell 48(8):2304–2314. https://doi.org/10.1007/s10489-017-1081-2

    Article  Google Scholar 

  47. 47.

    El-Fergany AA, Hasanien HM (2018) Tree-seed algorithm for solving optimal power flow problem in large-scale power systems incorporating validations and comparisons. Appl Soft Comput 64:307–316. https://doi.org/10.1016/j.asoc.2017.12.026

    Article  Google Scholar 

  48. 48.

    Attia AF, El Sehiemy RA, Hasanien HM (2018) Optimal power flow solution in power systems using a novel Sine-Cosine algorithm. Int J Electr Power Energy Syst 99:331–343. https://doi.org/10.1016/j.ijepes.2018.01.024

    Article  Google Scholar 

  49. 49.

    Nguyen TT, Quynh NV, Van Dai L (2018) Improved firefly algorithm: a novel method for optimal operation of thermal generating units. Complexity. https://doi.org/10.1155/2018/7267593

    Article  Google Scholar 

  50. 50.

    Ahmed F, ALMOATAZ A (2018) Single-objective optimal power flow for electric power systems based on crow search algorithm. Arch Electr Eng 67(1):123–138. https://doi.org/10.24425/118996

    Article  Google Scholar 

  51. 51.

    Nguyen TT, Vo DN, Vu Quynh N, Van Dai L (2018) Modified cuckoo search algorithm: a novel method to minimize the fuel cost. Energies 11(6):1328. https://doi.org/10.3390/en11061328

    Article  Google Scholar 

  52. 52.

    Salimi H (2015) Stochastic fractal search: a powerful meta-heuristic algorithm. Knowl Based Syst 75:1–18. https://doi.org/10.1016/j.knosys.2014.07.025

    Article  Google Scholar 

  53. 53.

    H Mosbah, M El-Hawary (2016) Power system tracking state estimation based on stochastic fractal search technique under sudden load changing conditions. In: IEEE Canadian conference on electrical and computer engineering (CCECE), pp 1–6. https://doi.org/10.1109/CCECE.2016.7726788

  54. 54.

    Khanam I, Parmar G (2017) Application of SFS algorithm in control of DC motor and comparative analysis. In: 4th IEEE Uttar Pradesh section international conference on electrical, computer and electronics (UPCON), pp 256–261. https://doi.org/10.1109/UPCON.2017.8251057

  55. 55.

    Alomoush MI, Oweis ZB (2018) Environmental-economic dispatch using stochastic fractal search algorithm. Int Trans Electr Energy Syst 28(5):e2530. https://doi.org/10.1002/etep.2530

    Article  Google Scholar 

  56. 56.

    Çelik E (2018) Incorporation of stochastic fractal search algorithm into efficient design of PID controller for an automatic voltage regulator system. Neural Comput Appl 30:1991–2002. https://doi.org/10.1007/s00521-017-3335-7

    Article  Google Scholar 

  57. 57.

    Dubey HM, Pandit M, Panigrahi BK (2018) An overview and comparative analysis of recent bio-inspired optimization techniques for wind integrated multi-objective power dispatch. Swarm Evol Comput 38:12–34. https://doi.org/10.1016/j.swevo.2017.07.012

    Article  Google Scholar 

  58. 58.

    Nguyen TP, Vo DN (2018) A novel stochastic fractal search algorithm for optimal allocation of distributed generators in radial distribution systems. Appl Soft Comput 1:1. https://doi.org/10.1016/j.asoc.2018.06.020

    Article  Google Scholar 

  59. 59.

    Awad NH, Ali MZ, Suganthan PN, Jaser E (2016) Differential evolution with stochastic fractal search algorithm for global numerical optimization. In: IEEE Congress on Evolutionary Computation (CEC), pp 3154–3161. https://doi.org/10.1109/CEC.2016.7744188

  60. 60.

    Padhy S, Panda S (2017) A hybrid stochastic fractal search and pattern search technique based cascade PI-PD controller for automatic generation control of multi-source power systems in presence of plug in electric vehicles. CAAI Trans Intell Technol 2(1):12–25. https://doi.org/10.1016/j.trit.2017.01.002

    Article  Google Scholar 

  61. 61.

    Lin J, Wang ZJ (2017) Parameter identification for fractional-order chaotic systems using a hybrid stochastic fractal search algorithm. Nonlinear Dyn 90(2):1243–1255. https://doi.org/10.1007/s11071-017-3723-7

    MathSciNet  Article  Google Scholar 

  62. 62.

    Mosbah H, El-Hawary M (2018) Power system static state estimation using modified stochastic fractal search technique. In: IEEE Canadian Conference on Electrical & Computer Engineering (CCECE), pp 1–4. https://doi.org/10.1109/CCECE.2018.8447826

  63. 63.

    Nguyen TP, Vo ND (2018) Improved stochastic fractal search algorithm with chaos for optimal determination of location, size, and quantity of distributed generators in distribution systems. Neural Comput Appl. https://doi.org/10.1007/s00521-018-3603-1

    Article  Google Scholar 

  64. 64.

    Das S, Suganthan PN (2011) Differential evolution—a survey of the state-of-the-art. IEEE Trans Evol Comput 15(1):4–31. https://doi.org/10.1109/TEVC.2010.2059031

    Article  Google Scholar 

  65. 65.

    Zou D, Li S, Wang GG, Li Z, Ouyang H (2016) An improved differential evolution algorithm for the economic load dispatch problems with or without valve-point effects. Appl Energy 181:375–390. https://doi.org/10.1016/j.apenergy.2016.08.067

    Article  Google Scholar 

  66. 66.

    Korda N, Szörényi B, Shuai L (2016) Distributed clustering of linear bandits in peer to peer networks. In: Proceedings of the 33rd international conference on machine learning, New York, USA, vol 48, pp 1301–1309

  67. 67.

    Li S, Karatzoglou A, Gentile C (2016) Collaborative filtering bandits. In: Proceedings of the 39th international ACM SIGIR conference on Research and Development in Information Retrieval, pp 539–548

  68. 68.

    Gentile C, Li S, Zappella G (2014) Online clustering of bandits. In: International conference on machine learning, Beijing, China, pp 757–765

  69. 69.

    Zimmerman RD, Murillo-Sánchez CE, Thomas RJ (2017) https://www.pserc.cornell.edu/matpower. Accessed 1 Jan 2018

  70. 70.

    Kar P, Li S, Narasimhan H, Chawla S, Sebastiani F (2016) Online optimization methods for the quantification problem. In: Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining. ACM, pp 1625–1634. https://doi.org/10.1145/2939672.2939832

  71. 71.

    Li S (2016) The art of clustering bandits. Doctoral dissertation, University of Insubria

  72. 72.

    Nguyen TT, Vo DN (2019) Improved social spider optimization algorithm for optimal reactive power dispatch problem with different objectives. Neural Comput Appl. https://doi.org/10.1007/s00521-019-04073-4

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Thang Trung Nguyen.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest with other individuals or particles.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

See Tables 20, 21, 22, 23 and 24.

Table 20 Optimal solutions obtained by the proposed method for case 1 of IEEE 30-bus power system
Table 21 Optimal solutions obtained by the proposed method for case 2 of IEEE 30-bus power system
Table 22 Optimal solutions obtained by the proposed method for case 3 of IEEE 30-bus power system
Table 23 Optimal solutions obtained by the proposed method for IEEE 57-bus power system
Table 24 Optimal solution obtained by the proposed method for IEEE 118-bus power system

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nguyen, T.T., Nguyen, T.T., Duong, M.Q. et al. Optimal operation of transmission power networks by using improved stochastic fractal search algorithm. Neural Comput & Applic 32, 9129–9164 (2020). https://doi.org/10.1007/s00521-019-04425-0

Download citation

Keywords

  • Stochastic fractal search
  • Diffusion process
  • Update process
  • Optimal power flow
  • Objective function
  • Operating limitations