Lower Hybrid waves have shown to have a high current drive (CD) efficiency suitable for current profile modification [1, 2]. These results were obtained at rather low density. Recent results on C-Mod [3], FTU [4] and JET [5, 6] indicate that CD efficiency could decrease at high density. On JET, the analysis of the contribution of fast electrons to the electron cyclotron emission (ECE) spectrum suggests that the efficiency is reduced for pulses performed with the JET steady state scenario characterized by high density at the pedestal and rather high line-averaged density. On Tore Supra, the plasma is generally leaning on the bottom toroidal pumped limiter (TPL) but a few experiments were carried out with a small outboard limiter (LPA). All the pulses were performed with the same plasma current (1MA) and toroidal field (3.8T). LH power launched from the PAM launcher [7] was 2MW (unless specified) and the main parallel index of the wave N//=1.8 or 1.9. Fast electron population is diagnosed from the multi-channel HXR diagnostic [8] and the ECE diagnostic [9].