We report chemical abundances obtained by Sloan Digital Sky Survey (SDSS)-III/Apache Point Observatory Galactic Evolution Experiment for giant stars in five globular clusters located within 2.2 kpc of the Galactic Centre. We detect the presence of multiple stellar populations in four of those clusters (NGC 6553, NGC 6528, Terzan 5 and Palomar 6) and find strong evidence for their presence in NGC 6522. All clusters with a large enough sample present a significant spread in the abundances of N, C, Na and Al, with the usual correlations and anticorrelations between various abundances seen in other globular clusters. Our results provide important quantitative constraints on theoretical models for self-enrichment of globular clusters, by testing their predictions for the dependence of yields of elements such as Na, N, C and Al on metallicity. They also confirm that, under the assumption that field N-rich stars originate from globular cluster destruction, they can be used as tracers of their parental systems in the high-metallicity regime.