LEO satellites offer reduced signal loss, fast movement, multi-beam, typically providing single coverage. This paper introduces a novel multi-beam power positioning method for low-orbit single-satellite, addressing the slow convergence and low accuracy of Doppler positioning. It establishes a power observation equation system, initializes with the nearest neighbor algorithm, and refines with the least squares method. Monte Carlo simulations indicate that with good initial values, the method converges in under 10 iterations, achieving 88.06% availability at 20° elevation with errors of 5331m (vertical) and 8798m (horizontal), and a timing error of 205μs. At 70° elevation, all users converge with errors of 1614m and 1088m, and a timing error of 31.3μs, demonstrating high power positioning availability. The statistical results show that power positioning users can obtain the positioning accuracy of kilometers and the timing accuracy of microseconds, which meets initial timing needs under strong confrontation, enhancing the medium and high orbit satellite navigation.