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The investigation of solidification in ternary Al-Cu-Ni alloys is carried out by means of experiments and phase-field modeling. For three alloys in the Al-rich corner of the phase diagram differential thermal analysis (DTA) is performed. Then the alloys were analyzed using scanning electron microscopy with energy dispersive X-ray microanalysis. For the understanding of the general features of the alloy solidification quantitative phase-field simulations are carried out additionally to the theoretical Scheil calculation. It is found that many experimental DTA signals and the microstructure parts cannot be explained by simple Scheil calculation. We apply the multi-phase-field model previously developed for the simulation of peritectic reaction and extended it to three components and four-phases reactions. The main advantage of this model is the application of the equilibrium parameters evaluated from the refined free energies of the phases. It is shown that the simulated microstructure is comparable to the experimental one for two investigated alloys. The final phase fractions in the modeling correspond to the theoretical predictions of Scheil calculation but the time evolution of fractions is more complicated. In particular the kinetics (relation between tangential and normal growth velocities) of the peritectic-like reaction in ternary Al-Cu-Ni alloys shows differences compared to the peritectic reaction in binary Al-Ni alloys.