Allocation of Capacitors and Voltage Regulators in Unbalanced Distribution Systems: A Multi-objective Problem in Probabilistic Frameworks

Guido Carpinelli; Christian Noce; Angela Russo; Pietro Varilone

doi:10.1515/ijeeps-2014-0106

Article

Allocation of Capacitors and Voltage Regulators in Unbalanced Distribution Systems: A Multi-objective Problem in Probabilistic Frameworks

Guido Carpinelli , Christian Noce , Angela Russo and Pietro Varilone

Published/Copyright: November 12, 2014

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal International Journal of Emerging Electric Power Systems Volume 15 Issue 6

Abstract

Capacitors and series voltage regulators are used extensively in distribution systems to reduce power losses and improve the voltage profile along the feeders. This paper deals with the problem of contemporaneously choosing optimal locations and sizes for both capacitors and series voltage regulators in three-phase, unbalanced distribution systems. This is a mixed, non-linear, constrained, multi-objective optimization problem that usually is solved in deterministic scenarios. However, distribution systems are stochastic in nature, which can lead to inaccurate deterministic solutions. To take into account the unavoidable uncertainties that affect the input data related to the problem, in this paper, we have formulated and solved the multi-objective optimization problem in probabilistic scenarios. To address the multi-objective optimization problem, algorithms were used in which all the objective functions were combined to form a single function. These algorithms allow us to transform the original multi-objective optimization problem into an equivalent, single-objective, optimization problem, an approach that appeared to be particularly suitable since computational time was an important issue. To further reduce the computational efforts, a linearized form of the equality constraints of the optimization model was used, and a micro-genetic algorithm-based procedure was applied in the solution method.

Keywords: optimal allocation of capacitors and voltage regulators; unbalanced distribution systems; probabilistic formulation; multi-objective formulation

Appendix A – Micro-genetic algorithm

The µGA evolves from a population of five individuals and uses selection and crossover. In particular, the selection is based on the roulette-wheel method. In this algorithm, mutations are not implemented, since diversity is guaranteed by periodically refreshing the population, i.e. the complete replacement of the population (except for the best individual) [26, 29]. After the application of genetic operators, the amount of diversity in the population is measured by counting the total number of genes that are unlike the genes possessed by the best individual. If the diversity of the population is less than a selected threshold, four individuals are deleted and regenerated randomly; the next step is performed with this new population, which contains four new random individuals and the best individual. Of course, if the diversity of the population exceeds the selected threshold, no actions are executed.

Appendix B – The linearization method

The linearization method is based on the linearization of the equality constraints around a region of expected values. With reference to the equality constraints (9), the three-phase load flow equations are linearized around a region of expected values. In the following, for the sake of conciseness, the procedure is described only for a generic load level. Here, we let the vector μ(Y) represent the expected values of Y. If a deterministic, three-phase load flow is calculated using μ(Y) as input data, the solution of eq. (9) in the field of the deterministic model will be given by the vector X₀, such that:

(18)f(X0)=μ(Y)

Linearizing eq. (9) around point X₀ and keeping in mind eq. (18), the results become

(19)X≅Xo+AΔY=X′o+AY,

where

A=|∂f∂XX=X0−1,ΔY=Y−μ(Y),X′o=Xo−Aμ(Y).

Equation (19) expresses each random element of the state vector X as a linear combination of the random elements of the input vector Y. It follows that the magnitude and argument of the phase voltages can be approximated by jointly normally distributed variables for which statistical characterization can be conducted in terms of the mean values and covariance matrices. (Note that the input load powers are normal random variables.) In particular, for the mean values, it can be assumed that:

(20)μ(X)≅Xo

while the covariance matrix of X is given by:

(21)CovX=A˙covYAT

A similar procedure can be used for the equality constraints (10). Then, the dependent variables also can be approximated by jointly normally distributed variables for which statistical characterization can be conducted in terms of mean values and covariance matrices.

References

1. CarpinelliG, NoceC, ProtoD, RussoA, VariloneP.Probabilistic approach for optimal capacitor allocation in three phase unbalanced distribution systems. In Proceedings of 10th PMAPS, 2008, Puerto Rico (USA).Search in Google Scholar

2. CarpinelliG, NoceC, RussoA, VariloneP.A Probabilistic approach for multiobjective optimal allocation of voltage regulators and capacitor placement in three-phase unbalanced distribution systems. part i: theoretical aspects. In Proceedings of 12th PMAPS, 2012, Istanbul (Turkey).Search in Google Scholar

3. MarlerRT, AroraJS.Survey of multi-objective optimization methods for engineering. Struct Multidisc Optim2004;26:369–95.10.1007/s00158-003-0368-6Search in Google Scholar

4. CaramiaP, CarpinelliG, VariloneP, VerdeP.Probabilistic three-phase load flow. Int J Elect Power Energy Syst1999;21:55–69.10.1016/S0142-0615(98)00030-1Search in Google Scholar

5. CarpinelliG, ProtoD, NoceC, RussoA, VariloneP.Optimal allocation of capacitors in unbalanced multi-converter distribution systems: a comparison of some fast techniques based on genetic algorithms. Electr Power Syst Res2010;80:642–50.10.1016/j.epsr.2009.10.029Search in Google Scholar

6. CarpinelliG, NoceC, ProtoD, RussoA, VariloneP.Single-objective probabilistic optimal allocation of capacitors in unbalanced distribution systems. Electr Power Syst Res2012;87:47–57.10.1016/j.epsr.2012.01.008Search in Google Scholar

7. ChiangH-D, WangJ-C, DarlingG.Optimal capacitor placement, replacement: and control in large-scale unbalanced distribution systems: system modeling and a new formulation. IEEE Trans Power Syst1995;10:356–62.10.1109/59.373956Search in Google Scholar

8. WangJ-C, ChiangH-D, Nan MiuK, DarlingG.Capacitor placement and real time control in large scale unbalanced distribution systems: numerical studies. IEEE Trans Power Deliv1997;12.10.1109/61.584420Search in Google Scholar

9. Nan MiuK, Chiang H-D, Darling G.Capacitor placement replacement and control in large-scale distribution systems by a ga-based two-stage algorithm. IEEE Trans Power Syst1997;1210.1109/59.630457Search in Google Scholar

10. KimK-H, RheeS-B, KimS-N, YouS-K.Application of ESGA hybrid approach for voltage profile improvement by capacitor placement. IEEE Trans Power Deliv2003;18:1516–22.10.1109/TPWRD.2003.817814Search in Google Scholar

11. GhoseT, GoswamiSK.Effects of unbalances and harmonics on optimal capacitor placement in distribution system. Electr Power Syst Res2003;68:167–73.10.1016/S0378-7796(03)00149-4Search in Google Scholar

12. EajalAA, El-HawaryME.Optimal capacitor placement and sizing in unbalanced distribution systems with harmonics consideration using particle swarm optimization. IEEE Trans Power Deliv2010;25:1734–41.10.1109/TPWRD.2009.2035425Search in Google Scholar

13. SeguraS, RomeroR, RiderM.Efficient heuristic algorithm used for optimal capacitor placement in distribution systems. Int J Electr Power Energy Syst2010;33:71–8.10.1016/j.ijepes.2009.06.024Search in Google Scholar

14. CarpinelliG, NoceC, RussoA, VariloneP.Trade-off methods for capacitor placement in unbalanced distribution systems. In Proceedings of Int Conf on Future Power Systems, 2005, Amsterdam.10.1109/FPS.2005.204307Search in Google Scholar

15. CarpinelliG, NoceC, ProtoD, RussoA, VariloneP.Multiobjective optimal allocation of capacitors in distribution systems: a new heuristic technique based on reduced search space regions and genetic algorithms. In Proceedings of CIRED, 2009, Prague.10.1049/cp.2009.0819Search in Google Scholar

16. AntunesCH, PiresDF, BarricoC, GomesÁ, MartinsAG. A multi-objective evolutionary algorithm for reactive power compensation in distribution networks. Appl Energy2009;86:977–984.10.1016/j.apenergy.2008.09.008Search in Google Scholar

17. CarpinelliG, NoceC, ProtoD, RussoA, VariloneP. AProbabilistic approach for multiobjective optimal allocation of capacitors in distribution systems based on genetic algorithms. In Proceedings of 11th PMAPS, 2010, Singapore.10.1109/PMAPS.2010.5528422Search in Google Scholar

18. CarpinelliG, NoceC, VariloneP. AProbabilistic approach for voltage regulators and capacitor placement in three phase unbalanced distribution systems. In Proceedings of CIRED, 2011, Frankfurt (Germany).Search in Google Scholar

19. MadrugaEP, Neves CanhaL.Allocation and integrated configuration of capacitor banks and voltage regulators considering multi-objective variables in smart grid distribution system. In Proceedings of 9th IEEE/IAS International Conference on Industry Applications, 2010.10.1109/INDUSCON.2010.5740055Search in Google Scholar

20. HsiaoYT, ChienCY.Optimisation of capacitor allocation using an interactive trade off method. IEE Proc Gen Transm Distrib2001;148:371–374.10.1049/ip-gtd:20010358Search in Google Scholar

21. ScottB, AlsacO.Fast Decoupled Load Flow. IEEE Trans Power Apparatus Syst1974;93: 859–69.10.1109/TPAS.1974.293985Search in Google Scholar

22. Chang S-K, Brandwaijn V.Adjusted solution in fast decoupled load flow. IEEE Trans Power Syst1988;3: 726–733.10.1109/59.192928Search in Google Scholar

23. AburA, SinghH, LiuH, and KlingensmithWN.Three phase power flow for distribution system with dispersed generation. In Proceedings of 14th PSCC, 2002, Sevilla (Spain).Search in Google Scholar

24. AthanTW, PapalambrosPY.A note on weighted criteria methods for compromise solutions in multi-objective optimization. Eng Optim1996;27:155–76.10.1080/03052159608941404Search in Google Scholar

25. JiaJ, FischerGW, DyerJS.Attribute weighting methods and decision quality in the presence of response error: a simulation study. J Behav Decis Making1998;11:85–160.10.1002/(SICI)1099-0771(199806)11:2<85::AID-BDM282>3.0.CO;2-KSearch in Google Scholar

26. DelfantiM, GranelliGP, MaranninoP, MontagnaM.Optimal capacitor placement using deterministic and genetic algorithms. IEEE Trans Power Syst2000;15:1041–6.10.1109/59.871731Search in Google Scholar

27. Abou-GhazalaA.Optimal capacitor placement in distribution systems feeding nonlinear loads. In Proceedings of IEEE PowerTech Conf, 2003, Bologna (Italy).Search in Google Scholar

28. HaghifamM-R, MalikOP.Genetic algorithm-based approach for fixed and switchable capacitors placement in distribution systems with uncertainty and time varying loads. IEE Proc Gen Transm Distrib2007;1:244–52.10.1049/iet-gtd:20045267Search in Google Scholar

29. De SouzaA, do NascimentoH, FerreiraHA. Microgenetic algorithm and fuzzy logic applied to the optimal placement of capacitor banks in distribution networks. IEEE Trans Power Syst2004;19:942–7.10.1109/TPWRS.2004.825901Search in Google Scholar

30. KerstingWH. Radial distribution test feeders. In Proceedings of IEEE Power Engineering Society Winter Meeting, 2001, 908–12. http://ewh.ieee.org/soc/pes/dsacom/testfeeders/index.html. Accessed January 2014.Search in Google Scholar

Published Online: 2014-11-12

Published in Print: 2014-12-1

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/ijeeps-2014-0106

Keywords for this article

optimal allocation of capacitors and voltage regulators; unbalanced distribution systems; probabilistic formulation; multi-objective formulation