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Whenever I look up IEEE 34-bus test systems, the following images are what is presented. I often find the first image in papers related to optimal capacitor studies, the second image I find in any other kind of studies. Is there a difference between both images? In optimal capacitor studies, when they cite the source of the 1st image, the paper detailing the IEEE 34 by W.H. Kersting shows the 2nd image.
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To answer your question directly, I believe that these are two completely different distribution system models, related only by the fact that they both happen to have 34 buses. The second is the one that should be correctly known as the "IEEE 34-node test feeder".

It looks to me like the first image is based on this paper:

Chis, M.; Salama, M.M.A.; Jayaram, S.: 'Capacitor placement in distribution systems using heuristic search strategies', IEE Proceedings - Generation, Transmission and Distribution, 1997, 144, (3), p. 225-230, DOI: 10.1049/ip-gtd:19970945 IET Digital Library, https://digital-library.theiet.org/content/journals/10.1049/ip-gtd_19970945

Chis, et al do not cite any source for this 34-bus radial distribution system. They also do not call it the IEEE 34-bus distribution system. They introduce it as "a 34-bus three-phase radial feeder with lateral branches" (emphasis added). It seems to me that subsequent publications may have incorrectly labeled this as the IEEE 34-bus distribution system.

The second image is from the 34-bus test case prepared by the IEEE and available publicly from https://cmte.ieee.org/pes-testfeeders/resources/. I believe the original publication for this is this following:

Kersting, William H. "Radial distribution test feeders." IEEE Transactions on Power Systems 6.3 (1991): 975-985.

Interestingly, Dugan and Kersting cite the source of the test feeder(s) like this:

[1] IEEE PES Distribution Systems Analysis Subcommittee Radial Test Feeders, http://ewh.ieee.org/soc/pes/dsacom/testfeeders.html.

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