Usual method to know the beam energy: calculation from the known magnetic fields.

Energy calibration by resonant depolarization (from The Energy Calibration of LEP in the 1993 Scan paper.)

Resonant depolarization, and several systematics affecting the beam energy

In an ideal e+e- storage ring the beams naturally polarize along the direction of the bending field due to the emission of synchrotron radiation. The polarization vector is defined as the ensemble average of the spin vectors of all the electrons in the bunch.

In a magnetic field, this polarization vector experiences precession.
The spin tune depends on energy:

where a_e=(g_e-2)/2.
So, if we measure ν₀ we have E.

How to measure ν₀: radial oscillating field from a coil.
If the perturbation from the radial field is in phase with the spin precession then the spin rotations about the radial direction add up coherently from turn to turn.
You have to constantly measure the polatization P.
At some point, the polarization vector is rotated away from the vertical axis (you see a drop in the polarization).
One resonance condition between the perturbing radial field and the nominal spin precession is f_field=[ν]f_rev., where f_field is the frequency of the oscillating field, f_rev. is the revolution frequency of the particles, which is precisely known, and [ν] denotes the non-integer part of the spin tune. Its integer part is known accurately enough from the setting of the bending field.

Dedicated runs are needed, in which several ν values are finely scanned.
While you do the scan, you cannot do physics. So you have to extrapolate in time (i.e. you need a model for the time depency of the beam energy; here is where tide effects, or TGV or whatever, enter).

Ambiguity: if you take [ν], you cannot determine whether the spin tune is below or above the half integer. This ambiguity is solved by increasing the beam energy with an RF-frequency change and by measuring the direction of the change in the measured spin tune.

How to measure polarization:
Photons (low energy) from a laser hit the beam at low angle. The backscattered photons (large angle) are detected and their angular distribution retains information about the polarization of the incident electron/positron.

Other methods for beam energy measurement:

Flux loop
The flux loop consists of closed electrical loops threading all the LEP dipoles and is used to measure the magnetic field of the bending magnets

Bunch length and energy spread
They are related:

So you have to measure the bunch lenght σ_Z:
The bunch length can be observed most precisely from the RMS scatter of the positions of reconstructed Z decays in the experimental detectors.