# Low-Order Electrochemical State Estimation for Li-Ion Batteries
###### Abstract  
Batteries are complex systems involving spatially distributed microscopic mechanisms on different time scales whose adequate interplay is essential to ensure a desired functioning. Describing these phenomena yields nonlinearly coupled partial differential equations whose numerical solution requires considerable effort and computation time, making it an infeasible solution for real-time applications. Anyway, having information about the internal electrochemical states of the battery can pave the way for many different advanced monitoring and control strategies with a big potential for improving efficiency and longevity. For such purposes, in the present paper, a combination of a low-order representation of the essential dynamics associated to the internal electrochemical mechanisms based on Dynamic Mode Decomposition for control (DMDc) is proposed to obtain an improved equivalent circuit model (ECM) representation with continuously updated parameters and combined with an extended Kalman Filter (EKF). The model-order reduction step extensively exploits the model structure, yielding a well structured low-order representation without artificial numerical correlations. The performance of the proposed method is illustrated with numerical simulations based on a well-established reference model, showing its potential usefulness in real-time applications requiring knowledge of the internal electrochemical states besides the state-of-charge. complex systems; reduced-order models; process modelling and identification; data-driven models; lithium-ion batteries; equivalent circuit model; extended kalman filter +
Footnote †: journal:  
The models based on first-principles incorporate the different chemical, electrical and physical factors into mathematical equations to predict the main internal phenomena as accurate as possible, see, e.g., [5; 6; 7; 8; 9; 10; 11; 12; 13]. These models have been used mainly to improve the understanding of the underlying mass transport and electrochemical kinetics phenomena on a quantitative-qualitative level. They have been validated in many independent experimental studies and their potential use in (i) state estimation for monitoring, (ii) optimal charging, and (iii) state feedback control design have been investigated by different groups (see, e.g., [14; 15; 16; 17; 18; 19]). While these studies on the one side outline the great advantages of having explicit information about the internal electrochemical states at hand, on the other side they involve complex and advanced analytical skills and typically still require a complete partial differential equation system to be solved online, what can lead to significant real-time limitations and impose a high computational burden that is undesired in particular for embedded solutions and becomes untractable when concerning distributed battery packs and the associated complex battery management systems [20]. The simplest of the first-principle models is the so-called equivalent circuit model (ECM), (see, e.g., [21; 22]), which can be used in fast simulation scenarios but neglects all internal electrochemical processes and in consequence has a lower prediction capability, typically showing offsets in comparison to detailed numerical models. Because of the above other approaches are being explored in order to find a balance between a detailed electrochemical representation, simplicity and real-time capability. The data-driven modeling approaches offer such a balance. Data-driven modeling treats the battery basically as a black box and through model trainings comes up with a representation of the behaviour of the battery by using algorithms that process the measured data and iteratively improve the prediction capability. As a result, minimal or zero knowledge of the battery internal parameters is needed. Neural networks and deep learning are being used to form a relation between the input information and the output measurement (see, e.g., [23]).