We simulate the grand-canonical partition funtion for intersecting and nonintersecting one-dimensional interfaces by a Monte Carlo technique. The canonical partition function for nonintersecting interfaces is also simulated. We show that the total length of the interface grows longer as the chemical potential increases, which is one of the free two parameters in the system, while another parameter, rigidity, remains fixed. We find that there are no drastic changes in the total length, the internal energy and their fluctuations when the two parameters vary gradually. In the canonical MC simulations for nonintersecting interfaces, we find that there are two distinct phases, ordered and disordered, that there is no phase transition between them, like the case of the intersecting interfaces, and also that the disordered phase is characterized by the fractal dimension H=4/3. Moreover, we find, in the grand-canonical MC for nonintersecting and intersecting interfaces, that there are no phase transitions between the ordered and disordered phases.