## 2 A Voltage Ride-Through (VRT) Control Model for Energy Storage Systems  
The energy conversion process of a battery energy storage system (BESS) involves bidirectional transformation between chemical and electrical energy. The energy storage medium generates direct current (DC) through electrochemical reactions, which is subsequently converted to alternating current (AC) via power electronic conversion devices for grid integration. The system's dynamic response characteristics are primarily governed by the grid-connected converter and its control system, operating at the electromechanical transient timescale. The grid-connected converter, serving as the interface between the BESS and the power grid, typically employs a voltage source converter (VSC). Advanced control algorithms enable precise regulation of bidirectional power flow between grid dispatching commands and energy storage units. Figure 1 illustrates the equivalent circuit of the grid-connected converter in a BESS, where \(U_{dc}\): DC-link voltage; \(I_{dc}\): DC-link current; \(L\): equivalent inductance; \(R\): equivalent resistance; \(E\): converter-side AC voltage; \(U\): grid-side AC voltage. As illustrated in Figure 1, the grid-connected converter of the energy storage system can be represented in the dq0 coordinate system as follows:  
\[ \begin{bmatrix}U_{d}\\ U_{q}\end{bmatrix} =- \begin{bmatrix}R+Ls&-\omega L\\ \omega L&R+Ls\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix} + \begin{bmatrix}E_{d}\\ E_{q}\end{bmatrix}  \tag{1}\]  
\[\left\{ \begin{array}{l}E_{d}=\Big{(}\frac{K_{il}}{s}+K_{ip}\Big{)}(i_{rd}^{* }-i_{d})-\omega Li_{q}+U_{d}\\ E_{q}=\Big{(}\frac{K_{il}}{s}+K_{ip}\Big{)}(i_{rq}^{*}-i_{q})+\omega Li_{d}+U_{q} \end{array} \right. \tag{2}\]  
Figure 1: Equivalent circuit diagram of a grid-connected converter for energy storage systems. In the equations, \(E_{d}\) and \(E_{q}\): converter-side AC voltage components in the d-axis and q-axis, respectively, \(U_{d}\) and \(U_{q}\): grid-side AC voltage components in the d-axis and q-axis, respectively, and \(\omega\): grid frequency. The control strategy for grid-connected converters in energy storage systems typically employs a dual-loop coordinated control framework. The outer-loop control, with voltage signals as inputs, generates current reference values through predefined control strategies. The inner-loop control utilizes real-time current measurements as inputs, aiming to track the reference values provided by the outer loop. By leveraging decoupling control algorithms (e.g., dq-axis feedforward decoupling) and converter hardware devices, the inner loop achieves precise current tracking, thereby ensuring the stable output of grid-connected currents. Figure 2 illustrates the mathematical model of the grid-connected converter control for the energy storage system. The main variables are defined as follows: \(P_{r}^{*},Q_{r}^{*}\): active power reference and reactive power reference; \(I_{rd}^{*},I_{rq}^{*}\): d-axis and q-axis current component references; \(I_{dHVRT}^{*},I_{dHVRT}^{*},I_{dLVRT}^{*}\): reference values for d-axis and q-axis current components during voltage ride-through (VRT) events; \(I_{1d},I_{1q}\): measured d-axis and q-axis current components; \(U_{1d},U_{1q}\): d-axis and q-axis components of the AC-side voltage at the grid-connected converter; \(U_{cd},U_{cq}\): intermediate voltage variables in the control loop; and \(I_{d},I_{q}\): d-axis and q-axis components of the grid-connected converter AC. In practical engineering applications, feedforward compensation is commonly adopted to eliminate cross-coupling effects. By combining Equations (1) and (2), the inner-loop current control depicted in Figure 2 can be simplified to the following:  
\[ \begin{cases}i_{d}=\frac{1}{Ls+R}\Big{(}\frac{K_{ll}}{s}+K_{iP}\Big{)}(i_{rd}^ {*}-i_{d})\\ i_{q}=\frac{1}{Ls+R}\Big{(}\frac{K_{ll}}{s}+K_{iP}\Big{)}(i_{rq}^{*}-i_{q}) \end{cases}  \tag{3}\]  
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