## 2 Ultra-Low-Frequency Oscillation Suppression Based on Transient Energy Function  
Mechanism of Hybrid DC Modulation to Suppress Ultra-Low-Frequency Oscillation Based on Transient Energy Function  
There are \(n\) units in the whole system. Each unit uses the second-order classical model and takes damping into account, which is represented by the center of inertia (COI). COI is defined as follows [19]:\[\left\{ \begin{array}{l}T_{\rm JT}=\sum\limits_{i=1}^{n}T_{\rm ji}\\ \delta_{\rm COI}=\frac{1}{I_{\rm JT}}\sum\limits_{i=1}^{n}T_{\rm ji}\delta_{i} \\ \omega_{\rm COI}=\frac{1}{I_{\rm JT}}\sum\limits_{i=1}^{n}T_{\rm ji}\omega_{i} \\ P_{\rm COI}=\sum\limits_{i=1}^{n}(P_{\rm mi}-P_{\rm ei})\end{array} \right., \tag{1}\]  
where \(T_{\rm ji}\), \(\delta_{i}\), \(\omega_{i}\), \(P_{\rm mi}\), and \(P_{\rm ei}\) are the inertia time constant, power angle, angular velocity, mechanical power, and electromagnetic power of the rotor of unit \(i\) in synchronous coordinates, respectively. \(T_{JT}\) is the total inertia time constant of rotors of all units. \(P_{\rm COI}\) is the acceleration power of COI. \(\delta_{\rm COI}\) and \(\omega_{\rm COI}\) are the equivalent rotor angle and equivalent rotor angular velocity of COI, respectively. When DC modulation is not considered, the motion equation of each unit in COI coordinates is:  
\[\left\{ \begin{array}{l}d\theta_{i}/dt=\widetilde{\omega}_{i}\\ T_{\rm ji}\frac{d\widetilde{\omega}_{i}}{dt}=P_{\rm mi}-P_{\rm ei}-\frac{T_{ ji}}{T_{JT}}P_{\rm COI}-D_{i}(\widetilde{\omega}_{i}+\omega_{\rm COI})+\\ \frac{T_{\rm ji}}{T_{JT}}\sum\limits_{j=1}^{n}D_{j}(\widetilde{\omega}_{j}+ \omega_{\rm COI})\end{array} \right. \tag{2}\]  
where \(\theta_{i}=\delta_{i}-\delta_{\rm COI}\) and \(\widetilde{\omega}_{i}=\omega_{i}-\omega_{\rm COI}\) are the rotor angle and rotor angular velocity of unit \(i\) in COI coordinates, respectively. The oscillating kinetic energy \(V_{Ki}\), oscillating potential energy \(V_{Pi}\) and damping energy \(V_{Di}\) of unit i are defined as follows:  
\[\left\{ \begin{array}{l}V_{i}(t)=V_{Ki}(t)+V_{Pi}(t)+V_{Di}(t)\\ V_{Ki}(t)=T_{ji}\widetilde{\omega}_{i}^{2}(t)/2\\ V_{Pi}(t)=\int_{\theta_{i}^{i}}^{\theta_{i}(t)}P_{mi}-P_{ei}-\frac{T_{ji}}{T_ {JT}}p_{COI}\mathrm{d}\theta_{i}\\ V_{Di}(t)=\int_{\theta_{i}^{i}}^{\theta_{i}(t)}-D_{i}\omega_{i}+\frac{T_{ji}}{ T_{JT}}\sum\limits_{j=1}^{n}D_{j}(\widetilde{\omega}_{j}+\omega_{\rm COI}) \mathrm{d}\theta_{i}\end{array} \right., \tag{3}\]  
where \(\theta_{i}(t)\) and \(\theta_{i}^{\rm s}\) are rotor angle and equilibrium rotor angle under COI coordinates of the system at time \(t\) respectively. After the system is disturbed, its transient energy is:  
\[V(T)=\sum\limits_{i=1}^{n}V_{i}(t)=\sum\limits_{i=1}^{n}\left[V_{\rm Ki}(t)+V _{\rm Pi}(t)+V_{\rm Di}(t)\right]. \tag{4}\]  
Now assume that there is only one dominant oscillation mode \(\lambda_{k}=\sigma_{k}+j\chi_{k}\) in the system, After the system is disturbed, according to the difference of \(\sigma_{k}\), the system may appear in the first oscillation period [\(t_{s}\),\(t_{s+k}\)] (where \(\widetilde{\omega}(t_{s})=\widetilde{\omega}(t_{s+k})=0\), \(t_{s+k}=t_{s}+2\pi/\chi_{k}\)) in the following three oscillation cases:  
1. \(\sigma_{k}<0\); because of the damping energy, the transient energy of the system decreases with time \(\Delta V_{t_{k}}=V(t_{s+k})-V(t_{s})<0\) (\(\Delta V_{t_{k}}\) is the change in the system energy during the first oscillation period), and the system is stable in the first oscillation period. 2. \(\sigma_{k}=0\); because the damping energy of the system \(V_{D}\approx 0\), in the process of oscillation and oscillation, the kinetic energy and potential energy components increase and decrease, and the transient energy of the system basically remains unchanged, and the system is stable. And \(V(t)\approx\) C (C is constant), so \(\Delta V_{t_{k}}=V(t_{s+k})-V(t_{s})=0\). \(\sigma_{k}>0\); because of the damping energy \(V_{D}\), the transient energy of the system increases with time, and the system becomes unstable.