SOC Estimation

For the SOC-based analysis in 2.1.4, an accurate SOC estimate is required. As discussed within [7, 58, 77], an SOC estimate is also critical for several aspects of battery pack operation. It identifies how much charge is remaining (which can be extrapolated to other metrics such as time remaining or driving range). It may also be part of the vehicle’s energy management strategy, for example HEVs typically maintain the battery pack SOC inside a narrow window. SOC cannot be measured directly. If the cell is fully relaxed, then the terminal voltage is equal to the OCV, and so SOC can be directly obtained from a lookup table using data as

in section 3.3.1. However, even this can be subject to error if the OCV-SOC data is incorrect or simplified owing to interpolation or averaging the charge and discharge OCV-SOC curves. SOC can be calculated using (17), also known as coulomb counting, but there are several sources of error in this equation as summarised in Table 2Error! Reference source not found..

Table 2: Sources of error from coulomb counting

Source Errors

Initial SOC s SOC is an integrator and so accumulates charge relative to a starting point. If the initial SOC is incorrect, the future values will also be incorrect.

Current icell Errors in current can arise from incorrect calibration and quantisation. Additionally, discrete sampling means that the sample taken is assumed constant for the following time-step, when in practice it may differ.

Cell capacity Qcap The cell capacities well generally be an estimate and as such subject to some uncertainty. Incorrect estimates will incorrectly scale conversion from charge to SOC.

The above methods can be combined, by coulomb counting until the cells come to (or close to) a relaxed state, at which time the cell voltage is used to look-up SOC. However, this is not a viable solution unless the cells reach a near-relaxed state frequently, which is not guaranteed with an EV, particularly a BEV which is designed to cover a wide DOD. These methods can more tightly integrated using an estimation algorithm. There have been many methods proposed for cell SOC estimation, ranging from data-driven methods such as neural networks [78] and support vector machines [79] to model-based methods such as the Kalman Filter (KF) [75, 80, 81]. They all have the same premise, which is to continually use voltage measured to improve (correct) the SOC estimate, even when the cell is not relaxed. The KF was chosen because data-driven methods require a large amount of training data, which is time-consuming to generate and must be carefully chosen such that it represents the final application, whereas the KF integrates well with standard Catapult characterisation tests mentioned in section 3.4. Additionally, data-driven methods are more complex to develop and more

computationally expensive to execute. The literature suggests there is little difference in accuracy between the KF and data-driven methods.

2.1.5.1 Kalman Filter Framework

The KF is a commonly used algorithm for estimating states in physical systems and has been proven over time to be efficient and robust for a wide variety of applications [82–84]. The general data fusion method for a KF is outlined below. The ^ accent denotes that the variable is an estimate not a measurement, the - superscript indicates that the estimate is prior to correction, and the + superscript indicates the estimate is post correction. The k subscript denotes the relative point in time. The input vector u, state vector x, and process noise vector w are passed into the state update function to obtain the uncorrected state estimate at the next step using (9)Error! Reference source not found.. This is stored and used at the next time step. The uncorrected state vector, input vector, and output noise vector v are passed through the output function (10) to obtain an estimate of the measured output. The Kalman gain matrix Lk provides a systematic way of correcting the state estimates using the measurement data (11)Error! Reference source not found..

𝒙Œ_𝒌Ž𝟏• _{= 𝑓(𝒙Œ} ‘ Ž_{, 𝒖} ‘, 𝒘Œ‘) (9) 𝒚Œ_‘• _{= 𝑔(𝒙Œ} ‘ •_{, 𝒖} ‘, 𝒗Œ‘) (10) 𝒙Œ_‘Ž _{= 𝒙Œ} ‘ •_{+ 𝐿} ‘(𝒚‘− 𝒚Œ‘•) (11)

The specific calculation of L depends on the type of filter used. It is calculated according to two matrices: the state process noise matrix and output noise matrix. The process noise matrix indicates the accuracy of the model: small matrix values imply confidence in the model, meaning that the states will not be corrected much for a given measurement error, whereas larger values acknowledge that there are a number of model errors or unmodelled effects in Error! Reference source not found. [82]. Similarly, the output noise matrix indicates noise on the output measurements. In theory, these matrices are the covariance of w and v

respectively, but in practice often become tuning factors whose values are adjusted until the desired performance characteristics are reached. One of the difficulties of the KF is that the v and w values are generally unknown and can vary with time, and often may not be Gaussian white noise as the filter assumes [82]. Implementation of a nonlinear KF for real-time SOC estimation is detailed in section 4.3.