It is essential to bias a thin-film lithium-niobate Mach-Zehnder electro-optic (EO) modulator at the desired operation condition to ensure optimal performance of the modulator. While thermo-optic (TO) control can solve the problem of bias drift, it consumes significant electric power. In this paper, we propose a technique to largely reduce bias power consumption by combining passive bias and TO bias. In our design, waveguide sections with different widths are introduced in the two arms of the MZ modulator to produce a desired phase difference of π/2 rad (the desired operation condition), and local heating with electrode heaters placed on the waveguides is employed to provide compensation for any phase drift caused by fabrication errors and other effects. As the TO control only serves to compensate for small errors, the electric power required is low and the response is fast. To demonstrate our technique experimentally, we fabricate several modulators of the same design on the same chip. Our experimental modulators can operate up to ∼40 GHz with a half-wave voltage of ∼2.0 V over a wide optical bandwidth, and the performances are insensitive to ambient temperature variations. The TO bias powers required range from 1 mW to 15 mW, and the thermal rise and fall times are 47 µs and 14 µs, respectively. Our technique can facilitate the development of practical high-speed EO modulators on the lithium-niobate-on-insulator platform.