# Sensorless Current Pulsation Compensation in a Hybrid Energy Storage
## 2 DC Link Pulsation in an Energy Storage Application  
In order to symmetrize the grid, the inverter requires a four-wire structure (3p-4w). The fourth wire works as a neutral wire and can be connected to either a split capacitance of the DC Link or an additional active leg of the inverter (Figure 3). Additionally, independent phase current control needs to be implemented. A common practice for the manufacturers is to use three separate single-phase inverters in a Y-connection to serve as a 3p-4w inverter. The analysis of such an inverter is similar: from a control standpoint, it is viewed as a trio of single-phase, Y-connected inverters, each driven by a separate control loop, for example by a proportional-resonant controller [12]. However, in order to estimate the pulsation level, the system needs to be treated as a three-phase inverter. In that case, the instantaneous output power of the inverter can be described as the sum of its powers in each of the phases, Equation (1):  
\[p_{3f}=\sum_{k=1}^{3}\frac{1}{2}V_{k}I_{k}(\cos(\varphi_{vk}-\varphi_{ik})- \cos(2\omega t+\varphi_{vk}+\varphi_{ik})) \tag{1}\]  
Figure 2: DC link current in the case of balanced (**a**,**b**) and unbalanced (**c**,**d**) operation of the four-wire inverter. where \(V_{k}-\)magnitude of the phase voltage, \(I_{k}-\)magnitude of the phase current, \(\varphi_{vk}-\)voltage phase, \(\varphi_{ik}-\)current phase, and \(\omega=2\pi f\), with \(f\) being the grid frequency. If inverter efficiency is omitted, then this equation can be used to approximate the power drawn from the DC link. This equation can be split into time-invariant, and time-variant components. During symmetric operation of the inverter, the three time-variant components add up to zero, while during asymmetric operation they do not, which causes the \(\cos(2\omega t+\varphi_{vk}+\varphi_{ik})\) component of the equation to reappear and transfer into the DC link as an AC current pulsation. In order to extract the AC pulsation from the DC link current, the symmetrical decomposition, Clarke and Park transforms are used (Equations (2)-(7)):  
\[i_{abc}= \begin{bmatrix}i_{a}\\ i_{b}\\ i_{c}\end{bmatrix} =I^{+} \begin{bmatrix}\cos(\omega t)\\ \cos(\omega t-\frac{2\pi}{3})\\ \cos(\omega t+\frac{2\pi}{3})\end{bmatrix} +I^{-} \begin{bmatrix}\cos(\omega t )\\ \cos(\omega t-\frac{2\pi}{3})\\ \cos(\omega t+\frac{2\pi}{3})\end{bmatrix} +I^{0} \begin{bmatrix}\cos(\omega t )\\ \cos(\omega t)\\ \cos(\omega t)\\ \cos(\omega t)\end{bmatrix}  \tag{2}\]  
\[i_{a\beta}= \begin{bmatrix}i_{a}\\ i_{\beta}\end{bmatrix} =I^{+} \begin{bmatrix}\cos(\omega t+\varnothing^{+})\\ \sin(\omega t+\varnothing^{+})\end{bmatrix} +I^{-} \begin{bmatrix}\cos(-\omega t +\varnothing^{-})\\ \sin(-\omega t+\varnothing^{-})\end{bmatrix}  \tag{3}\]  
where \(I^{+}\), \(I^{-}\), and \(I^{0}\) are the magnitudes of the currents of symmetrical components; \(\mathcal{O}^{+}\) and \(\mathcal{O}^{-}\) are the phases of symmetrical components. Positive and negative sequences can be derived using two Park transforms with reference frames rotating in opposite directions:  
\[i_{dq}{}^{+}= \begin{bmatrix}i_{d}{}^{+}\\ i_{q}{}^{+}\end{bmatrix} =I^{+} \begin{bmatrix}\cos(\varnothing^{+})\\ \sin(\varnothing^{+})\end{bmatrix} +I^{-} \begin{bmatrix}\cos(2\omega t)\sin(2 \omega t)\\ -\sin(2\omega t)\cos(2\omega t)\end{bmatrix}\begin{bmatrix}\cos(\varnothing^{- })\\ \sin(\varnothing^{-})\end{bmatrix} , \tag{4}\]  
\[i_{dq}{}^{-}= \begin{bmatrix}i_{d}{}^{-}\\ i_{q}{}^{-}\end{bmatrix} =I^{-} \begin{bmatrix}\cos(\varnothing^{+})\\ \sin(\varnothing^{+})\end{bmatrix} +I^{+} \begin{bmatrix}\cos(2\omega t)(-\sin(2 \omega t))\\ \sin(2\omega t)\cos(2\omega t)\end{bmatrix}\begin{bmatrix}\cos(\varnothing^{+ })\\ \sin(\varnothing^{+})\end{bmatrix} . \tag{5}\]  
These two components can be split into time invariant and time variant components:  
\[i_{dq}{}^{+}= \begin{bmatrix}\overline{i_{d}{}^{+}}\\ \overline{i_{q}{}^{+}}\end{bmatrix} + \begin{bmatrix}\widetilde{i_{d}{}^{+}}\\ \overline{i_{q}{}^{+}}\end{bmatrix} =\overline{I^{+}