# The Capacity and State of Charge

The capacity, Q, of a cell is the approximate amount of charge that can be stored in the cell when it is fully charged. Thus, the capacity is the amount of electric charge that a battery can store, which is stated in units of ampere-hour (A-h). The state of charge (SOC) is defined as the percentage of the capacity (Qavaiiabie) available at any time with respect to the maximum available capacity (Q = Qmax) of a battery. The SOC of an ideal cell can be expressed in terms of the initial state of charge, the capacity, and the charging current as,

In terms of the discharging current it is,

For finite values of the SOC, the above equation is expressed as,

where g(s) is the charge-transfer efficiency which is equal to unity for low values of the SOC.

The state of charge is an extremely important parameter in determining the remaining life of a battery at any point in time. A variety of battery SOC esti­mation methods Pop et al. (2008), Han et al. (2009) have been developed, which, in general, can be classified into four categories: methods based on Coulomb counting, methods based on techniques involving computational intelligence such as artificial neural nets, fuzzy logic, and support vector machines, model-based methods, and other mixed methods. The Coulomb counting methods are, in principle, simple and easy to implement in real-time systems. In these methods, the SOC is simply calculated by integrating the measured current over time with the information of the initial SOC in a fully charged state,

Although simple in principle, the Coulomb counting methods have several disadvantages due to errors caused by factors such as a wrong initial SOC value and accumulation of estimation errors.

Moreover, the Coulomb counting methods cannot track the battery’s non-linear capacity variations due to the rate capacity and recovery effects. Model-based SOC estimation methods use a state-space model derived from the equivalent electric circuit of the battery to design a state observer such as the Kalman filter (KF) for real-time SOC estimation. In fact, the SOC is continuously and dynamically estimated from-related measurements using sophisticated recursive non-linear estimators such as the extended KF Plett (2004a, b, c) and its many variants Bhangu et al. (2005), Plett (2006). A dynamic model describing how the output of the battery behaves with time is generally used in developing the SOC estimation algorithm. The SOC equation in terms of the discharge current for complete dis­charge is generally valid only when the discharge current is constant. Hence,

SOC(r ) = 1 – ^. (7.3.5)

Hence when,

SOC(T) = 1 – ^ = 0; (7.3.6)

Q = idts. (7.3.7)

Given the state of charge and a constant discharge current, it is possible to estimate the number of hours the battery would be able to deliver the current. For example, if a fully charged cell has a capacity of 50 Ah (Amp-hours) and it is at 60 % SOC, then the cell is expected to produce a constant current of 3 A for 10 h before reaching full discharge.