Efficient power-saving schemes are essential in extending the battery life of the portable devices in wireless communication systems. Accordingly, this study proposes an adaptive power-saving protocol for asynchronous wireless ad hoc networks in which the sleep---awake patterns of the wireless devices are determined using a grid quorum-based approach. In the proposed protocol, the grid size of each mobile node is adjusted dynamically in accordance with changes in the mobile node's active ratio so as to minimize the total energy consumption within the system. Previous studies on grid power-saving systems have shown that if the mobile nodes arbitrarily choose one row and one column of quorum intervals, in which they must be awake, then the quorum intervals of any two mobile nodes necessarily have intersections. However, this fact does not hold for systems in which the grid size varies dynamically. To resolve this problem, this study proposes a fixed-column selection scheme, in which each node randomly chooses one row of quorum intervals in each quorum cycle, but chooses the same column in every cycle. It is shown that under this scheme, the quorum intervals of two asynchronous nodes with adaptive grid sizes always have intersections. The performance of the proposed protocol is compared with that of three existing protocols, namely the quorum-based power-saving protocol (QPS) (with fixed grid size), the dominating-awake-interval protocol (DAI), and the non-sleep (NS). It is shown that under both light and heavy traffic load conditions, the proposed protocol outperforms the QPS, DAI and NS protocols in terms of higher sleep ratio, lower energy consumption, and better efficiency of awakeness. Besides, as for neighbor discovery time and transmission latency, the proposed protocol also outperforms or has about the same performance as the QPS, DAI and NS protocols. In other words, the proposed protocol has the advantage of achieving higher sleep ratio, lower energy consumption, and better efficiency of awakeness without incurring much overhead in neighbor discovery time and transmission latency. Consequently, the proposed protocol provides an effective and practical power saving solution for asynchronous wireless ad hoc networks.