###### Abstract
However, the ampere-hour integral method is the open loop control algorithm, and its performance is greatly affected by the measurement noises, environmental conditions, as well as the variation of initial SOC caused by self-discharging [8; 9]. Hence, some periodic calibration methods are recommended by [PERSON] et al. [2]. The model-based method is a close-loop algorithm and has more accurate prediction results. Based on the battery model and algorithms, the model-based method is realized. The common battery models include the electrochemical model (EM) [10] and equivalent circuit model (ECM) [11]. ECM may achieve a better balance between circuit structural complexity and prediction accuracy [2]. The popular algorithms are Kalman filter series [2; 12], particle filters [13], sliding mode observer (SMO) [14], and so on. For the model-based method, an accurate battery model, depending on circuit structures and parameters, is critically important. Based on the previous research works, we know that circuit structures are sensitive to battery systems [15], and circuit parameters change with battery charge capacity and environment temperature [16]. Hence, EM and ECM of lithium-ion batteries cannot be applied directly to primary lithium batteries. However, the specific circuit models of primary lithium batteries do not yet exist, so the applicability of the model-based method in engineering needs to be further explored. Without understanding the internal mechanism, the data-driven methods take the battery as a black box. Relying on a great deal of experimental data and artificial intelligence algorithms, it can accurately estimate the battery capacity. The typical algorithms consist of support vector regression (SVR) [17], relevance vector machine (RVM) [18], and Gaussian process regression (GPR) [19]. Based on the pulse load test, [PERSON] et al. used the back propagation neural network to build a primary lithium battery capacity estimator. Meanwhile, the estimator has been fully validated by experiments under various load conditions of lithium thionyl chloride batteries with different storage times [20]. Nonetheless, this method has its limitations in practice, i.e., heavy computing burden. Moreover, tremendous battery data must be accumulated to offline the train estimation model, and the model accuracy depends entirely on training data. Hence, whether the data-driven method can be applied under different practical operating conditions still needs to be further validated. From the above analysis, there are relatively few specific capacity estimation models of primary lithium batteries, and the existing methods, ignoring model complexity and acquisition difficulty of corresponding data, are still in the theoretical research stage. Owing to the underwater working environment, the spatial size and weight of underwater vehicles are constrained strictly. It means that measurement sensors cannot be equipped optionally and data processing capability of embedded chip is limited. Hence, it is extremely urgent to develop simple, accurate, fast, and practical capacity estimation methods of the primary lithium batteries. After comprehensive consideration of the advantages and disadvantages of the five methods, the characteristics of primary lithium batteries, and the working environment of underwater vehicles, the ampere-hour integral method is the most practical for the detection of the charge capacity of the primary lithium battery in engineering. Hence, a revised ampere-hour integral method is proposed in this research to estimate capacity of primary lithium batteries in underwater vehicles, which can overcome the effect of temperature and age of the common ampere-hour integral algorithm. The detailed arrangement of the manuscript is as follows: the method is presented in Section 2. Storage and discharge experiments are introduced in Section 3.