The Future Circular Collider (FCC) is a proposed large-scale collider programme at CERN, based on a new tunnel of about 91km circumference. Its first stage, the electron-positron collider FCC-ee, is the preferred option for the next flagship collider at CERN. Designed as a Higgs, electroweak and top factory running at several centre-of-mass energies from the Z pole to above the top-quark pair threshold, it would greatly improve the precision of the electroweak observables.

Due to the unprecedented luminosity, a statistical precision of about 4keV is foreseen for the Z mass, and 100keV for the W mass. The aim is therefore to bring the systematic uncertainties down to the same level as the statistical ones. A leading contribution to this systematic budget is the knowledge of the average beam energy, from which the true centre-of-mass energy must be carefully inferred. Meeting the goal set at the Z energy calls for the average beam energy to be known to a relative precision of the order of 10^-8.

Such a demanding target is met by exploiting the spin of the stored beams. In a perfectly aligned flat machine without solenoids, the closed orbit spin tune, meaning the number of spin precessions per turn on the closed orbit, stays strictly proportional to the average beam energy. Although machine imperfections introduce slight deviations from this exact proportionality, the spin tune nonetheless remains the most direct and precise observable for measuring the beam energy to very high accuracy.

This thesis follows the key steps toward the target. It begins with a systematic study of how machine imperfections affect the polarization, assessing whether the equilibrium polarization in realistic lattice models, including imperfections and corrections, remains sufficient for the energy calibration experiments. The results show that as long as a closed orbit solution can be established, the equilibrium polarization generally remains high in the presence of various imperfections, even when the correction procedure addresses only the orbit. In parallel, the systematic uncertainties produced by imperfections, which degrade the precision of the measured energy, are evaluated. As a rough estimate of this systematic error, the closed orbit spin tune shift stays at the keV or even sub-keV level across most imperfect lattices, an indication that reaching the precision target is promising. The application of Harmonic Spin Matching (HSM) is then explored as a way to raise the polarization further beyond what conventional lattice correction achieves. Three candidate HSM schemes for the FCC-ee are examined, and their performance is assessed through simulation. Methods for determining the beam energy by measuring the spin tune are simulated to verify their applicability to the FCC-ee. The standard approach, Resonant Depolarization (RDP), is evaluated using the FCC-ee lattice. Furthermore, Near Resonance Polarization Modulation (NRPM), a novel technique proposed and developed in this thesis, is introduced as a complementary method to RDP for the FCC-ee. NRPM offers distinct advantages in both the precision achievable in simulation and certain operational aspects, and it has the potential to be generalized to other electron or positron storage rings. To compare their performance and characteristics, RDP and NRPM are also applied to the SLS 2 and ELSA lattices. These studies lay the groundwork for a high precision beam energy measurement at the FCC-ee.