## 3 The Adaptive Droop Control of the ILC
### Stability Analysis  
To analytically verify the stability of the proposed adaptive bidirectional droop control strategy, a small-signal model of the interlinking converter (ILC) is developed. The power transfer dynamics through the ILC are linearized around an equilibrium point to derive a state-space representation. The active power transfer is governed by the droop control law:  
\[P_{\mathrm{ILC}}=k_{d}\cdot(\Delta f_{\mathrm{acpu}}-\Delta U_{\mathrm{dcpu}}), \tag{15}\]  
where \(k_{d}\) is the adaptive droop coefficient from Equation (13), and \(\Delta f_{\mathrm{acpu}}=f_{\mathrm{acpu}}-1\) and \(\Delta U_{\mathrm{dcpu}}=U_{\mathrm{dcpu}}-1\) are the normalized deviations of AC frequency and DC voltage, respectively. Considering the small perturbations \(\delta f_{\mathrm{acpu}}\) and \(\delta U_{\mathrm{dcpu}}\) around the equilibrium,  
\[\delta P_{\mathrm{ILC}}=k_{d}\cdot(\delta f_{\mathrm{acpu}}-\delta U_{\mathrm{ dcpu}}) \tag{16}\]  
The frequency dynamics are modeled as  
\[\frac{d\big{(}\delta f_{\mathrm{acpu}}\big{)}}{dt}=-\frac{1}{T_{f}}\delta f_{ \mathrm{acpu}}+\frac{1}{M_{f}}\delta P_{\mathrm{ILC}} \tag{17}\]  
where \(T_{f}=0.15\) s is the AC subgrid time constant and \(M_{f}=0.02\) Hz/kW is the frequency inertia coefficient. The DC voltage dynamics are  
\[\frac{d(\delta U_{\mathrm{dcpu}})}{dt}=-\frac{1}{T_{v}}\delta U_{\mathrm{dcpu} }-\frac{1}{M_{v}}\delta P_{\mathrm{ILC}} \tag{18}\]  
where \(T_{v}=0.12\) s is the DC subgrid time constant and \(M_{v}=0.018\) V/kW is the voltage inertia coefficient. Substituting \(\delta P_{\mathrm{ILC}}\) yields the state-space representation:  
\[\begin{bmatrix}\delta f_{\mathrm{acpu}}^{\text{-}}\\ \delta U_{\mathrm{dcpu}}\end{bmatrix}=\begin{bmatrix}-\frac{1}{T_{f}}+\frac{k_{ d}}{M_{f}}&-\frac{k_{d}}{M_{f}}\\ -\frac{k_{d}}{M_{v}}&-\frac{1}{T_{v}}+\frac{k_{d}}{M_{v}}\end{bmatrix} \begin{bmatrix}\delta f_{\mathrm{acpu}}\\ \delta U_{\mathrm{dcpu}}\end{bmatrix} \tag{19}\]  
The characteristic equation is  
\[\lambda^{2}+a_{1}\lambda+a_{0}=0, \tag{20}\]  
with coefficients  
\[\begin{cases}a_{1}=\left(\frac{1}{T_{f}}+\frac{1}{T_{v}}\right)-k_{d}\Big{(} \frac{1}{M_{f}}+\frac{1}{M_{v}}\Big{)}\\ a_{0}=\frac{1}{T_{f}T_{v}}-k_{d}\Big{(}\frac{1}{M_{f}T_{v}}+\frac{1}{M_{v}T_{f }}\Big{)}\end{cases} \tag{21}\]  
Figure 10: The control schematic diagram of the ILC.  
For asymptotic stability, the Routh-Hurwitz criterion requires the conditions of \(a_{1}>0\) and \(a_{0}>0\). These conditions are satisfied through the adaptive mechanism: during large deviations (\(|f_{\rm p00}-1|>\varepsilon\)), \(k_{d}\) automatically reduces to 0.1, yielding \(a_{1}=4.44>0\) and \(a_{0}>0\). The startup threshold \(\eta\) bounds control sensitivity, preventing high-frequency oscillations. Eigenvalue analysis confirms all poles in left-half plane with dominant poles at \(-8.2\pm j2.1\), a minimum damping ratio \(\zeta=0.72>0.7\), and a maximum settling time of 0.4 s, consistent with Section 4 simulations. Thus, the proposed control ensures asymptotic stability under small perturbations while maintaining adaptive performance during large disturbances.  
In addition to theoretical stability, the proposed control strategy also demonstrates robust performance under sudden load disturbances. The dynamic weight coefficient \(\lambda\) adapts in real time to prioritize the regulation of the larger deviation (either frequency or voltage), ensuring that the interlinking converter responds effectively to transients. To prevent potential instability due to rapid fluctuations in \(\lambda\), a first-order low-pass filter is applied to smooth its variation, avoiding abrupt control changes. The simulation results in Section 4.2 confirm that the proposed method maintains voltage and frequency stability during large and rapid load changes without inducing oscillations or overshoot, further validating its practical robustness.