## 3 Conventional and Proposed Control Schemes of the Battery-SC HESS
Figure 3: Schematic diagram of the conventional control scheme of the co-located battery-supercapacitors (SCs) hybrid energy storage systems (HESSs). current bursts for assisting with dynamic voltage regulation, its droop factor should be much smaller (approximately 2% to 10%) than that of the battery converter. On the other hand, it is desirable that this current burst lasts just long enough to mitigate the DC bus voltage variations so as not to drain/supply much energy from/to the SC unit. This can be accomplished with the appropriate choice of the time constant (\(\tau_{F}\)) of the HPF. In order to select these parameters, a dynamic model of the system, including the small-signal model of the SC interface (class C DC-DC converter) as shown in Figure 5, is considered. Assuming that the solar converter operates in the MPPT mode, it is modeled by a virtual constant current source. The battery converter operates with droop control and can be represented by a Thevenin model with a voltage source equal to the threshold voltage and a resistor equal to the droop factor (\(R_{dB}\)) of the battery converter [17]. DC bus voltage variations shall occur mostly due to load variations. Figure 5 shows the small-signal model of the system, based on [6; 18]. It includes the inner boost inductor current (\(I_{L}\)) control loop with a PI-type controller and the duty-cycle to inductor current transfer function (\(Gdi(s)\)) of the converter. The outer (droop) voltage control loop is based on a droop factor (\(R_{dSCI}\)), that concerns the DC bus voltage (\(V_{dc}\)) and the reference injected current (\(I_{dc}\)). The slow SC voltage control loop is neglected in this analysis. A first-order high-pass filter (HPF) with time constant \(\tau_{F}\) is used in this work, where  
\[\tau_{F}=\frac{1}{2\pi f_{c}}. \tag{3}\]  
The power converter is assumed to be ideal and a gain factor (1-\(D\)) is employed to refer quantities from the DC bus to the SC side of the power converter, and vice-versa. A time constant for the DC nanogrid side (\(\tau_{S}\)) is defined from the droop factor of the battery converter (\(R_{dB}\)), which is much lower than the equivalent load resistance (\(R_{L}\)), and the output capacitance of the SC interface (\(C_{DC3}\)). Thus,  
\[\tau_{S}=C_{DC3}R_{dB}, \tag{4}\]  
Since the outer voltage (droop) loop has a much lower bandwidth than the inner current loop, the latter, which presents a zero error in steady-state due to the PI controller, can be simplified to a unity gain. Then, one can assume that there will be no variations in \(V_{NL\_SC}\) and that load and PV current variations can be combined in a single element (\(I_{\Delta}\)). Finally, the block diagram describing the dynamics of the system can be further simplified as shown in Figure 6.  
Figure 5: Small-signal model for the analysis of the proposed SC control scheme. The system's transfer function (\(G_{T}\)) is given by:  
\[G_{T}(s)=\frac{V_{dc}(s)}{I_{\Delta}(s)}=\frac{G(s)}{1+kG(s)H(s)}, \tag{5}\]  
where  
\[G(s)\;H(s)=\frac{s\tau_{F}R_{dB}}{(1+s\tau_{F})(1+s\tau_{S})}, \tag{6}\]  
and  
\[k=\frac{1}{R_{dSC1}}. \tag{7}\]  
The characteristic equation of the system is:  
Figure 4:0  
It can be represented in terms of the damping ratio (\(\zeta\)) and the natural frequency (\(\omega_{n}\)) as:  
Figure 4:1  
where  
Figure 4:2  
and  
Figure 4:3  
The impact of the value of the droop factor (\(R_{dSC1}\)) and time constant (\(\tau_{F}\)) of the HPF on the dynamic response of the system can be observed by means of the root locus method. The latter determines the location of one of the open loop roots while the former affects the position of the closed loop roots. Figure 7 shows the root locus of the system for \(R_{dB}=0.289\;\Omega\) and \(C_{DC3}=1500\;\mathrm{\SIUnitSymbolMicro\SIUnitSymbolMicro}\). For the analysis, the cut-off frequency of the HPF is taken as 50 Hz. Since transfer function \(G\), (6), has two negative real poles and one zero at the origin, the system is stable for any gain \(k\).