## 2 Split DC Link Voltages in Zero-Sequence-Restricted Three-Phase Three-Level AC-DC Converter  
A common dual-stage three-phase power conversion system, consisting of a three-level AC-DC (or DC/AC) converter, a DC link, and a DC-DC converter, is depicted in Figure 1. System power flow is bidirectional in general, i.e., energy may be transferred from the AC to the DC side or in the reverse direction [36]. Consequently, the 'Load' in Figure 1 may be regenerative or serve as a source in case the power is displaced from DC to AC side. The DC link contains three terminals (X, O, Y) at the side of the AC-DC converter and can be either two-level or three-level at the DC/DC converter side, as shown. Underbalanced sinusoidal conditions (which are considered in this paper), steady-state AC-side electrical parameters may be expressed as  
\[\overset{\rightarrow}{v}_{RST}(t)= \begin{pmatrix}v_{RN}(t)\\ v_{SN}(t)\\ v_{TN}(t)\end{pmatrix} =V_{M} \begin{pmatrix}\sin(\omega t)\\ \sin(\omega t-\theta)\\ \sin(\omega t+\theta)\end{pmatrix} \] (1a) and \[\overset{\rightarrow}{i}_{RST}(t,\varphi)= \begin{pmatrix}i_{R}(t,\varphi)\\ i_{S}(t,\varphi)\\ i_{T}(t,\varphi)\end{pmatrix} =I_{M} \begin{pmatrix}\sin(\omega t-\varphi)\\ \sin(\omega t-\theta-\varphi)\\ \sin(\omega t+\theta-\varphi)\end{pmatrix}  \tag{1b}\]  
with \(V_{M}\)[V], \(I_{M}\)[A], respectively, symbolizing phase voltage and current magnitudes, \(\omega\) [rad/s] standing for base angular frequency, \(\varphi\) [rad] denoting the load angle (i.e., the phase difference between voltage and current vectors), and \(\theta=2\pi/3\) [rad]. Considering (1), instantaneous AC-side phase power vector is given by  
\[\overset{\rightarrow}{p}_{RST}(t,\varphi)= \begin{pmatrix}p_{R}(t,\varphi)\\ p_{S}(t,\varphi)\\ p_{T}(t,\varphi)\end{pmatrix} = \begin{pmatrix}v_{RN}(t)i_{R}(t,\varphi)\\ v_{SN}(t)i_{S}(t,\varphi)\\ v_{TN}(t)i_{T}(t,\varphi)\end{pmatrix} =0.5V_{M}I_{M} \begin{pmatrix}\cos\varphi -\cos 2\big{(}\omega t-\frac{\varphi}{2}\big{)}\\ \cos\varphi-\cos 2\big{(}\omega t-\frac{\varphi}{2}-\theta\big{)}\\ \cos\varphi-\cos 2\big{(}\omega t-\frac{\varphi}{2}+\theta\big{)}\end{pmatrix} . \tag{2}\]  
As a result, total instantaneous AC-side power is given by  
\[p_{RST}(t,\varphi)=p_{R}(t,\varphi)+p_{S}(t,\varphi)+p_{T}(t,\varphi)=S\cos \varphi\triangleq P_{RST} \tag{3}\]  
with \(S=3/2\)\(V_{M}I_{M}\) denoting the total apparent power and is low-frequency ripple free. Considering lossless conversion (for brevity only, without loss of generality), total instantaneous AC-side power \(P_{RST}\) is equal to the absolute value of the load power \(P_{L}=V_{WZ}I_{L}\) (cf. Figure 1) in steady state due to energy balance. As a result, there is no instantaneous power flow (besides switching frequency-related components) into the DC link in Figure 1.  
General three-phase three-level AC-DC conversion topology is depicted in Figure 2[35]. Here, the DC link consists of two split capacitors (namely, \(C_{\text{DC1}}\) and \(C_{\text{DC2}}\)) with corresponding instantaneous voltages vector given by  
\[\overset{\rightarrow}{v}_{DC}(t,\varphi)= \begin{pmatrix}v_{DC1}(t,\varphi) \\ v_{DC2}(t,\varphi)\end{pmatrix} . \tag{4}\]  
According the previous paragraph outcomes, total instantaneous DC link voltage \(v_{DC}\) is low-frequency-ripple-free since there it does not cope with instantaneous power flow. Yet, this outcome does not imply that each of the split DC link voltages \(v_{DC1}\), \(v_{DC2}\) is low-frequency-ripple-free (only their sum is) [36]. According to the voltages Equation (1a),  
Figure 1: Dual stage power conversion system under consideration. the controller (cf. Figure 1) generates shifted sinusoidal pulse-width modulation (PWM) signals which may be approximated by  
\[\overset{\rightarrow}{m}_{RST}(t)= \begin{pmatrix}m_{R}(t)\\ m_{S}(t)\\ m_{T}(t)\end{pmatrix} \approx M(t) \begin{pmatrix}\sin(\omega t)\\ \sin(\omega t-\theta)\\ \sin(\omega t+\theta)\end{pmatrix}