# Sensorless Current Pulsation Compensation in a Hybrid Energy Storage
## 2 DC Link Pulsation in an Energy Storage Application
Most of the papers treat the dual active bridge converter as a voltage converter, and derive the transfer function as an output voltage to control phase function. Since the compensator requires that it control the input current of the converter, the input current required to control the phase transfer function must be derived. Figure 4: A four-wire inverter with energy storage and a pulsation compensator. For the DAB converter modulation scheme, the single phase shift (SPS) control was used [17]. This scheme can be switched to extended phase shift or, to better handle rapid changes in current, to symmetrical single phase shift [18]. In the case of the SPS, the output power can be described as follows:  
\[P_{out}=\frac{nV_{in}V_{out}\varphi(1-2|\varphi|)}{f_{s}L_{lk}} \tag{9}\]  
where \(n-\)transformer turn ratio, \(f_{s}-\)switching frequency, and \(V_{in}\), \(V_{out}-\)input and output voltages. The equivalent circuit for the reduced-order model is presented in Figure 6. At this stage, the switching cycle average currents are introduced: \(\langle i_{b1}\rangle,\ i_{b2}\rangle\). Assuming ideal efficiency and symmetry, \(P_{out}=P_{in}\), and since \(P_{in}=V_{in}\langle i_{b1}\rangle\), \(i_{b1}\) can be calculated as follows:  
\[\langle i_{b1}\rangle=\frac{nV_{out}\varphi(1-2|\varphi|)}{f_{s}L_{lk}} \tag{10}\]  
a small perturbation can be injected into the \(<\)\(i_{b1}>\):  
\[\langle i_{b1}\rangle=I_{b1}+i_{b1} \tag{11}\]  
where \(I_{b1}\) is the quiescent component and \(i_{b1}\) is the perturbation. This perturbation would be a response to the phase perturbation:  
\[\varphi=\Phi+\hat{\varphi} \tag{12}\]  
where \(\Phi\) is the phase quiescent component and \(\hat{\varphi}\) is the phase perturbation. The transfer function from phase to input current can be obtained as follows:  
\[G_{i\varphi}(s)=\frac{i_{b1}}{\hat{\varphi}}=\frac{nV_{in}(1-4\varphi)}{f_{s} L_{lk}} \tag{13}\]  
from this standpoint, the transfer function does not introduce any delays. A similar statement is presented in [19], where the transfer function (phase to output voltage) is described as:  
Figure 3:0  
where \(D-\)phase shift, \(V_{d}-\)input voltage, \(T_{s}-\)switching period, and \(Z_{o}(s)-\)transfer function of the output filter. Division by \(Z_{o}(s)\) creates a transfer function from the phase to the output current. The cited paper states that in order for this transfer function to be accurate,  
Figure 3:1 Dual active bridge topology (**a**), high-frequency voltage waves, and resulting current (**b**). Figure 3:2 Equivalent circuit of the dual active bridge converter. the control phase must change at frequencies several orders of magnitude lower than the switching frequency of the converter \(f_{s}\). The proposed solution aims to utilize the dual active bridge converter as a source of AC current to compensate for the AC pulsation in the DC link. As such, the internal converter's current control loop would require a single-phase phase locked loop (PLL) and a PI-based control loop for effective operation. A more effective approach in this application would be a PR controller, as it would remove the need for a synchronization loop in the DAB converter. In the application, the pulsation is calculated in the AC/DC converter controller and then passed via SPI protocol to the compensator (Figure 7) and used as a reference for the internal PR controller, which then controls the phase between the H1 and H2 bridges of the DAB2 converter. The transfer function of the proportional-resonant controller can be described as follows:  
Figure 3:3  
where \(K_{H}-\)resonant controller gain, \(\omega_{rH}-\)resonant span parameter, \(\omega_{0}-\)controller resonant pulsation, and \(\omega_{LPF}-\)proportional controller gain. The overall closed-loop transfer function of the compensator would be a product of the \(G_{DAB_{\text{res}}}(s)\) and the \(G_{i\varphi}(s)\), with the former acting as a linear gain in the low frequency range, depending on the converter parameters.