## 3 Off-board Charger Control
### VSM Control
#### 3.1.1 Droop Control Based on Charging Mode  
Primary frequency/voltage regulation is a fundamental feature of the VSM control. In previous related studies [15; 16], the active power \(P\) and reactive power \(Q\) of the rectifier are leveraged to achieve the droop control, and the \(P\)-\(\omega\) and \(Q\)-\(U\) curves is shown in Figure 3a. The above-mentioned commonly-used droop control only incarnates the primary regulation characteristics of VSM, but it cannot bring about the proposed fast charging function. Thus, in order to accomplish the above goal, this paper presents the droop control based on charging mode, whose \(P_{\text{bat}}\)-\(\omega\) and \(Q\)-\(U\) curves is expressed in Figure 3b, in which \(Q\)-\(U\) curve remains invariable while the charging power \(P_{\text{bat}}\) substitutes the active power of the rectifier \(P\). Figure 1: The main circuit topology and fast charger control strategy. Figure 2: Applied control structure of virtual synchronous motor (VSM). \(P\) and \(P_{\text{bat}}\) represent the active power of the rectifier and charging power of the power battery, respectively. \(\omega\) is the angular frequency. \(Q\) and \(U\) are the reactive power and the AC voltage amplitude of the rectifier. \(D_{\text{P}}\) and \(D_{\text{q}}\) are the frequency and voltage droop coefficients. Reference values for any given signal x are indicated by a superscript asterisk as x*. According to the droop curves based on the charging mode in Figure 3b, the droop relations of the \(P_{\text{bat}}\)-\(\omega\) and \(Q\)-\(U\) are given by  
\[D_{\text{P}}=\frac{\omega^{*}-\omega}{P_{\text{bat}}{}^{*}-P_{\text{bat}}} \qquad(D_{\text{P}}>0) \tag{1}\]  
\[D_{\text{q}}=\frac{U^{*}-U}{Q^{*}-Q}\qquad(D_{\text{q}}>0) \tag{2}\]  
Equations (1) and (2) are transformed to  
\[\omega=\omega^{*}-D_{\text{P}}({P_{\text{bat}}}^{*}-P_{\text{bat}})\ \ \ (D_{\text{P}}>0) \tag{3}\]  
\[U=U^{*}-D_{\text{q}}(Q^{*}-Q)\ \ \ (D_{\text{q}}>0) \tag{4}\]  
To integrate with the constant-current charging, we alter the setting form of \(P_{\text{bat}}{}^{*}\) in Equation (3), as shown in Equation (5). \[{P_{\text{bat}}}^{*}=U_{\text{bat}}\times{I_{\text{bat}}}^{*} \tag{5}\]  
Battery Model is adopted in the system. \(U_{\text{bat}}\) is the feedback value of the real-time charging voltage of Battery, which changes with variation of state of charge (SOC) and charging current during the charging process. \(I_{\text{bat}}{}^{*}\) is the charging current reference value. The product of \(U_{\text{bat}}\) and \(I_{\text{bat}}{}^{*}\) constitutes the charging power reference \(P_{\text{bat}}{}^{*}\). Since the unit of the battery capacity is Ah, which is the product of Charging current (A) and Time (h), i.e., Capacity (Ah) = Charging current (A) \(\times\) Time (h), Charging current (A) is equal to the quotient of Capacity (Ah) and Time (h), i.e., Charging current (A) = Capacity (Ah)/Time (h). The fast charging time is usually less than an hour, i.e., Time (h) < 1 h, so if the constant-current fast charging intends to be fulfilled, we should assign that the set value of the charging current \(I_{\text{bat}}{}^{*}\) is greater than the value of the power battery capacity, i.e., \(I_{\text{bat}}^{*}\ >\ |\text{Capacity (Ah)}|\). Figure 3: Comparison chart of the droop curves (**a**) Traditional droop curves; (**b**) Droop curves based on charging mode. Thus, in accordance with the above analysis, the active droop relation combined with constant-current fast charging is obtained in Equation (6). \[\omega=\omega^{*}-D_{\rm p}({I_{\rm bat}}^{*}\times U_{\rm bat}-P_{\rm bat})(D_{ \rm p}>0) \tag{6}\]  
The actual angular frequency is acquired through a PLL measurement in engineering control, so \(\omega\) is expressed by \(\omega_{\rm g}\). Integrating the relationships of Equation (4), Equation (6) and \(\omega_{\rm g}\), the new primary regulation features of the proposed VSM are shown by Equation (7). \[\left\{