An Enhanced McCormick Envelopes to Represent Kron's Loss Formula

Document Type : Original Article


Department of Electrical and Computer Engineering, Hakim Sabzevari University, Sabzevar, Iran


Recently, some researchers have employed the McCormick envelopes method to convexify some NP-hard optimization problems with bilinear terms. However, few publications concentrate on its variants to derive a more tight convex relaxation for practical applications. This paper proposes a new viewpoint on Kron’s loss formula, also known as the B-matrix formula, as an equation having bilinear terms. Relying on the perspective, we transform the loss equation to some linear constraints using an enhanced McCormick relaxation. In the technique, the domain of bilinear variables is divided into some smaller parts to improve the relaxation tightness. Some case studies with different nonconvex terms are considered to verify the effectiveness of the enhanced envelopes for capturing Kron’s loss formula. The findings from the numerical simulations suggest that the proposed approach can represent Kron’s loss equation precisely. Moreover, the method performs more effectively than the other methods available in the literature as it usually converges to more optimal solutions.

Graphical Abstract

An Enhanced McCormick Envelopes to Represent Kron's Loss Formula


  1. Valinejad, J., Oladi, Z., Barforoushi, T. and Parvania, M., "Stochastic unit commitment in the presence of demand response program under uncertainties", International Journal of Engineering, Transactions B: Applications, Vol. 30, No. 8, (2017), 1134-1143. doi: 10.5829/ije.2017.30.08b.04.
  2. Lotfi, M.M., "Short-term price-based unit commitment of hydrothermal gencos: A pre-emptive goal programming approach", International Journal of Engineering, Transactions C: Aspects, Vol. 26, No. 9, (2013), 1017-1030. doi: 10.5829/idosi.ije.2013.26.09c.09.
  3. Shakouri G, H., Amin Naseri, M. and Neshat, N., "A game theoretic approach for sustainable power systems planning in transition", International Journal of Engineering, Transactions C: Aspects, Vol. 30, No. 3, (2017), 393-402. doi: 10.5829/idosi.ije.2017.30.03c.09.
  4. Sharifzadeh, H., "Sharp formulations of nonconvex piecewise linear functions to solve the economic dispatch problem with valve-point effects", International Journal of Electrical Power & Energy Systems, Vol. 127, (2021), 106603. doi: 10.1016/j.ijepes.2020.106603.
  5. Bhattiprolu, P.A. and Conejo, A.J., "Multi-period ac/dc transmission expansion planning including shunt compensation", IEEE Transactions on Power Systems, Vol. 37, No. 3, (2021), 2164-2176. doi: 10.1109/TPWRS.2021.3118704.
  6. Saleh, S.A. and Chowdhury, M.R., "Survivability analysis of impacts of load-side activities on power systems", IEEE Transactions on Industry Applications, Vol. 58, No. 2, (2022), 1869-1878. doi: 10.1109/TIA.2022.3146102.
  7. Su, W., Yu, S.S., Li, H., Iu, H.H.-C. and Fernando, T., "An mpc-based dual-solver optimization method for dc microgrids with simultaneous consideration of operation cost and power loss", IEEE Transactions on Power Systems, Vol. 36, No. 2, (2020), 936-947. doi: 10.1109/TPWRS.2020.3011038.
  8. Sharifzadeh, H. and Amjady, N., "A review of metaheuristic algorithms in optimization", Journal of Modeling in Engineering, Vol. 12, No. 38, (2014), 27-43. doi: 10.22075/JME.2017.1677.
  9. Zou, D. and Gong, D., "Differential evolution based on migrating variables for the combined heat and power dynamic economic dispatch", Energy, Vol. 238, (2022), 121664.
  10. Niknam, T., Azizipanah-Abarghooee, R. and Aghaei, J., "A new modified teaching-learning algorithm for reserve constrained dynamic economic dispatch", IEEE Transactions on Power Systems, Vol. 28, No. 2, (2012), 749-763. doi: 10.1109/TPWRS.2012.2208273.
  11. Ellahi, M., Abbas, G., Satrya, G.B., Usman, M.R. and Gu, J., "A modified hybrid particle swarm optimization with bat algorithm parameter inspired acceleration coefficients for solving eco-friendly and economic dispatch problems", IEEE Access, Vol. 9, No., (2021), 82169-82187. doi: 10.1109/ACCESS.2021.3085819.
  12. Braik, M.S., Awadallah, M.A., Al-Betar, M.A., Hammouri, A.I. and Zitar, R.A., "A non-convex economic load dispatch problem using chameleon swarm algorithm with roulette wheel and levy flight methods", Applied Intelligence, (2023), 1-40.
  13. Sutar, M. and Jadhav, H., "An economic/emission dispatch based on a new multi-objective artificial bee colony optimization algorithm and nsga-ii", Evolutionary Intelligence, (2022), 1-36.
  14. Ali, M.H., El-Rifaie, A.M., Youssef, A.A., Tulsky, V.N. and Tolba, M.A., "Techno-economic strategy for the load dispatch and power flow in power grids using peafowl optimization algorithm", Energies, Vol. 16, No. 2, (2023), 846.
  15. Goudarzi, A., Fahad, S., Ni, J., Ghayoor, F., Siano, P. and Haes Alhelou, H., "A sequential hybridization of etlbo and ipso for solving reserve‐constrained combined heat, power and economic dispatch problem", IET Generation, Transmission & Distribution, Vol. 16, No. 10, (2022), 1930-1949.
  16. Iqbal, M.N., Bhatti, A.R., Butt, A.D., Sheikh, Y.A., Paracha, K.N. and Ashique, R.H., "Solution of economic dispatch problem using hybrid multi-verse optimizer", Electric Power Systems Research, Vol. 208, (2022), 107912. doi.
  17. Beirami, A., Vahidinasab, V., Shafie-khah, M. and Catalão, J.P., "Multiobjective ray optimization algorithm as a solution strategy for solving non-convex problems: A power generation scheduling case study", International Journal of Electrical Power & Energy Systems, Vol. 119, (2020), 105967.
  18. Gaing, Z.-L., "Particle swarm optimization to solving the economic dispatch considering the generator constraints", IEEE Transactions on Power Systems, Vol. 18, No. 3, (2003), 1187-1195. doi: 10.1109/TPWRS.2003.814889.
  19. Selvakumar, A.I. and Thanushkodi, K., "A new particle swarm optimization solution to nonconvex economic dispatch problems", IEEE Transactions on Power Systems, Vol. 22, No. 1, (2007), 42-51. doi: 10.1109/TPWRS.2006.889132.
  20. Elsayed, W.T., Hegazy, Y.G., El-bages, M.S. and Bendary, F.M., "Improved random drift particle swarm optimization with self-adaptive mechanism for solving the power economic dispatch problem", IEEE Transactions on Industrial Informatics, Vol. 13, No. 3, (2017), 1017-1026. doi: 10.1109/TII.2017.2695122.
  21. Ciornei, I. and Kyriakides, E., "A ga-api solution for the economic dispatch of generation in power system operation", IEEE Transactions on Power Systems, Vol. 27, No. 1, (2011), 233-242. doi: 10.1109/TPWRS.2011.2168833.
  22. Reddy, A.S. and Vaisakh, K., "Shuffled differential evolution for large scale economic dispatch", Electric Power Systems Research, Vol. 96, (2013), 237-245.
  23. Roy, P.K. and Bhui, S., "Multi-objective quasi-oppositional teaching learning based optimization for economic emission load dispatch problem", International Journal of Electrical Power & Energy Systems, Vol. 53, (2013), 937-948.
  24. Bhattacharjee, K., Bhattacharya, A. and nee Dey, S.H., "Oppositional real coded chemical reaction optimization for different economic dispatch problems", International Journal of Electrical Power & Energy Systems, Vol. 55, (2014), 378-391. doi.
  25. Mandal, B., Roy, P.K. and Mandal, S., "Economic load dispatch using krill herd algorithm", International Journal of Electrical Power & Energy Systems, Vol. 57, (2014), 1-10. doi.
  26. Sharifzadeh, H., "Solving economic load dispatch by a new hybrid optimization method", International Journal of Industrial Electronics Control and Optimization, Vol. 3, No. 4, (2020), 469-474.
  27. Floudas, C.A., "Deterministic global optimization: Theory, methods and applications, Springer Science & Business Media, Vol. 37,  (2013).
  28. Bergamini, M.L., Aguirre, P. and Grossmann, I., "Logic-based outer approximation for globally optimal synthesis of process networks", Computers & Chemical Engineering, Vol. 29, No. 9, (2005), 1914-1933.
  29. Castro, P.M., "Tightening piecewise mccormick relaxations for bilinear problems", Computers & Chemical Engineering, Vol. 72, (2015), 300-311.