## 2 Small-Signal Modeling of the Distributed WEDES System  
### Overview of the WEDES System and Controller Operation Principle  
Figure 1c shows the illustration of an example circuit diagram of the WEDES system, which consists of two major parts: battery modules and on-board-unit (OBU). Inside each battery module, multiple battery cells are connected in series and/or in parallel to form a battery bank, which can provide voltage/current/power to the rest of the system. The output of the battery bank is then connected to the input of a DC-DC power converter, which is used to achieve bus voltage regulation, battery module current/voltage regulation as well as SOC balancing at the same time as described later in this section. The output of the power converter is connected to an inverter stage for DC-AC power conversion. At the end of the battery module, the AC power from the inverter is applied to the transmitter (Tx) for inductive wireless power transfer (I-WPT) to the OBU. The OBU mainly consists of receiver coils (Rx) followed by an AC-DC power conversion/rectifiation stage (rectifier). The outputs of each rectifier (\(V_{\mathrm{o1}}\) through \(V_{\mathrm{oN}}\)) are connected in series to the bus/output (\(V_{\mathrm{bus}}=V_{\mathrm{o1}}+V_{\mathrm{o2}}+\ldots+V_{\mathrm{oN}}\)). Figure 1: Illustration diagrams of battery system for electric vehicle (EV) application. (**a**) The conventional battery pack and electrics drive system in EVs, (**b**) the wireless distributed and enabled battery energy storage (WEDES) battery system in EVs, and (**c**) example circuit diagram of the WEDES system [1]. To realize the functionalities of SOC balancing, bus voltage regulation, and battery module current/voltage regulation, the WEDES controller consists of three different control loops: the SOC balancing control loop (referred to be by the SOC balancing loop), battery module voltage control loop (referred to be by the module voltage loop) and bus voltage control loop (referred to be by the bus voltage loop). Figure 2 shows the diagram of the wirelessly distributed WEDES controller, where \(V_{\text{bus\_ref}}\) is the desired value of bus voltage, V\({}_{\text{MN-total}}\) is an intermediate value for voltage regulation, \(C\)_re_MX1 though \(C\)_re_MXN are the remaining capacities of battery modules for SOC calculation, \(SOC_{\text{MX1}}\) though \(SOC_{\text{MXN}}\) are the SOC values of battery modules, \(\lambda_{\text{DC1}}\) through \(\lambda_{\text{DCN}}\) are weighting factors to generate the reference values of \(V_{\text{MX1-DC-ref}}\) through \(V_{\text{MXN-DC-ref}}\) for each battery module, \(\alpha_{\text{MX1}}\) through \(\alpha_{\text{MXN}}\) are the SOC multipliers and \(\delta_{\text{MX1}}\) through \(\delta_{\text{MXN}}\) are enable/disable values. In the WEDES controller, the SOC balancing loop is used to generate multipliers \(\alpha_{\text{MX1}}\) through \(\alpha_{\text{MXN}}\) to realize SOC balancing between multiple battery modules, as given by Equation (1). \[\left\{ \begin{array}{ll}\alpha_{\text{MX1}}&=\ (SOC_{\text{MX-ref}}-SOC_{ \text{MX1}})\times G_{\text{v5 OC}}+1\\ \alpha_{\text{MX2}}&=\ (SOC_{\text{MX-ref}}-SOC_{\text{MX2}})\times G_{ \text{v5 OC}}+1\\ \alpha_{\text{MXN}}&=\ (SOC_{\text{MX-ref}}-SOC_{\text{MXN}})\times G_{ \text{v5 OC}}+1\end{array} \right., \tag{1}\]  
where \(SOC_{\text{MX-ref}}\) is the average SOC value of all battery modules, as given by Equation (2). When all battery modules are inserted and active, the sum of all \(\delta_{\text{MX1}}\) through \(\delta_{\text{MXN}}\) equals to \(N\) (\(\delta_{\text{MX1}}+\ \delta_{\text{MX2}}+\ldots+\delta_{\text{MXN}}\) = \(N\)).