The ATLAS and CMS collaborations announced the discovery of the Higgs boson in July 2012, completing the particle content of the Standard Model. Although the Standard Model is a great triumph, it is not considered to be the complete theory of particle physics. Among the newly-developed theories, Supersymmetry (SUSY) is one of the most promising ones. SUSY predicts the existence of supersymmetric partner particles and it is one of the best-motivated extensions of the space-time symmetry of particle interactions. There are supersymmetric partner particles associated with each SM particles in which the spin differs by 1/2. This dissertation focuses on a search for electroweak production of supersymmetric particles with compressed mass spectra in the final states with exactly two low-momentum leptons and missing transverse momentum. The proton-proton collision data is recorded by the ATLAS detector at the Large Hadron Collider in 2015 and 2016, corresponding to 36.1~{\ifb} of integrated luminosity at s=13~{\TeV}. Events with same-flavor and opposite electric charge lepton pairs are selected. The data are found to be consistent with the Standard Model prediction. Results are interpreted using the non-universal Higgs mass model with two extra parameters (NUHM2) with small mass differences between the masses of produced supersymmetric particles. Upper limits of the cross-section at 95% confidence level are set for the NUHM2 model as a function of the universal gaugino mass m1/2.