## 1 Introduction  
Today, the lithium ion battery (LiB) is used widely in the practical applications because of its advantages, such as high energy density, no memory and no scheduled cycling, less self-discharge, long battery life and low maintenance. LiBP can provide extremely high currents and can discharge very rapidly when short-circuited [1, 2]. For an application with high power, for example, electric vehicles or hybrid electric vehicles, the lithium ion battery pack (LiBP) is formed by connecting some cells in parallel and series in order to get the desired power. State of charge (SoC) is the important parameter of the cell, it is used to determine the electrical energy remaining of the cell in the battery management system (BMS). However, when it comes to the LiBP, the SoC of the pack is not the proper definition in the practice [3]. For the LiBP with many cells connected in series, the definitions average SoC of the pack and SoC of cells are used. The average SoC of the pack shows the amount of electrical energy of the pack, and the SoC of cells indicates the electrical energy in the cells. Average SoC of the pack and SoC of cells are two very important parameters for the BMS and cell balance control in the practice. In recent years, an assortment of techniques has been proposed in order to achieve accurate battery SoC estimation. The most common methods are: the terminal voltage (open loop) method, the internal resistance method and the Coulomb counting method [4; 5]; the methods are based on the equivalence circuit model of cell [6; 7], the adaptive model considering the hysteristic, the noise, the varying of cell resistance dependent on the aging process and the operating temperature [8]. The methods use the Kalman filter [9], extended Kalman filter (EKF) [10] and particle point (PF) [11; 12]. Additionally, there are SoC estimation methods based on learning algorithms such as neural networks [13; 14], fuzzy logic and the nonlinear fractional model [15; 16]. For the LiBP, the SoC estimation problem is still a hard problem [17; 18]. Not only does this problem consider the complexity of the LiBP's model, the cell SoC, the average SoC for the cells connected in series, the complexity of the estimation algorithm and the computing rate, but the influence of parameters in the cell model and measurement noise are considered too [3]. Each consideration could bring forward some different methods to solve the SoC estimation for LiBP with different accuracies. In order to estimate the SoC of the cell, we need to measure exactly the operating temperature, voltage and current through a cell. In the practice, the voltage and current measurements always are affected by Gausses noise with a zero mean. The current bias is added on current measurements which are caused by the amplifier. In this paper we present the SoC estimation for the LiBP connected by \(N_{s}\) cell modules in series; each cell module is connected by \(N_{p}\) cells in parallel (see Figure 1). In this research, we use the second dynamic model of cell considering the voltage measurement noise of cell module, noise and current bias of the current though LiBP. The SPKF, known as one of the state estimation methods, has some advantages over the EKF by application ranges, and this method estimates more accurately the states for the second-order system or higher; meanwhile, computational complexity is the same as EKF. SPKF is easily implemented, as there is no requirement of the Jacobi matrix calculation [19; 20; 21; 22]. For the EKF, the system's nonlinear dynamic is linearized at the sampling time, the input distributions are mapped via the linearized dynamic function to calculate the output distributions, and the output distributions are less accurate if there is high nonlinear of the system dynamic. For the SPKF, the whole of the input distributions are mapped thought the nonlinear dynamic functions to get the output distribution.