## 2 Ultra-Low-Frequency Oscillation Suppression Based on Transient Energy Function
### Calculation of System Transient Energy  
In a system with \(n\) units, there are theoretically \(n-1\) oscillation modes, and each unit may participate in multiple oscillation modes at the same time. Now, the left eigenvector \(\psi\) and the right eigenvector \(\phi\) of the system are introduced. According to the definition of the left eigenvector of the system, \(\psi_{\lambda_{k}}\) can be used to measure the influence weight of each unit on each unit of oscillation mode \(\lambda_{k}\). \[ \begin{array}{l}\psi=[\psi_{1}^{\text{T}}\cdots\psi_{\lambda_{k}}^{\text{T} }\cdots\psi_{n}^{\text{T}}]\\ \phi=[\phi_{1}\cdots\phi_{\lambda_{k}}\cdots\phi_{n}]\end{array} , \tag{7}\]  
The rotor motion equation is linearized near the stable operating point,  
\[\left\{ \begin{array}{l}T_{\text{j}}\Delta\dot{\omega}_{j}=-\Delta P\\ \Delta P_{i}=\Delta P_{\text{mi}}-\Delta P_{\text{ei}}-D_{i}\Delta\omega_{i} \end{array} \right., \tag{8}\]  
where \(\Delta P_{i}\) is the acceleration power deviation. For oscillation mode \(\lambda_{k}\), the acceleration power deviation of unit \(i\) can be expressed in a generalized phasor form, namely:  
\[\Delta\overrightarrow{P}_{i,\lambda_{k}}=-T_{\text{j}i}(\sigma_{k}+\text{j} \chi_{k})\Delta\overrightarrow{\omega}_{i,\lambda_{k}}^{\rightarrow}. \tag{9}\]According to literature [20], under oscillation mode, the ratio of acceleration power deviation of any two units a and b is as follows:  
\[\frac{\Delta\overset{\rightarrow}{P}_{\mathrm{a},\lambda_{k}}}{\Delta\overset{ \rightarrow}{P}_{\mathrm{b},\lambda_{k}}}=\frac{T_{\mathrm{ja}}\Delta\overset{ \rightarrow}{\omega}_{\mathrm{a},\lambda_{k}}}{T_{\mathrm{b}}\Delta\overset{ \rightarrow}{\omega}_{\mathrm{b},\lambda_{k}}}. \tag{10}\]  
According to the free motion analysis theory of dynamic systems with zero input, for a certain mode, the ratio of corresponding terms of a and b units in the right characteristic phasor is equal to the ratio of corresponding terms of state variables, that is:  
\[\frac{\phi_{\mathrm{a},\lambda_{k}}}{\phi_{\mathrm{b},\lambda_{k}}}=\frac{ \Delta\overset{\rightarrow}{\omega}_{\mathrm{a},\lambda_{k}}}{\Delta\overset{ \rightarrow}{\omega}_{\mathrm{b},\lambda_{k}}}=\frac{\Delta\overset{ \rightarrow}{P}_{\mathrm{a},\lambda_{k}}/T_{\mathrm{ja}}}{\Delta\overset{ \rightarrow}{P}_{\mathrm{b},\lambda_{k}}/T_{\mathrm{jb}}}. \tag{11}\]  
In the actual system, the left characteristic phasor and the right characteristic phasor have the following approximate relationship [21]. \[\psi_{\lambda_{k}}\approx[diag(T_{\mathrm{j}}/\omega_{s})]\phi_{ \lambda_{k}}\] \[diag(T_{\mathrm{j}}/\omega_{s})=\frac{1}{\omega_{s}} \begin{bmatrix} T_{\mathrm{j}}&0&\cdots&0\\ 0&T_{\mathrm{j}}&\cdots&0\\ \vdots&\vdots&&\vdots\\ 0&0&\cdots&T_{\mathrm{jn}}\end{bmatrix} , \tag{12}\]  
where \(\omega_{s}\) is the steady-state value of the generator speed. Then the ratio of corresponding terms of units a and b in the left characteristic phasor is:  
\[\frac{\phi_{\mathrm{a},\lambda_{k}}}{\phi_{\mathrm{b},\lambda_{k}}}=\frac{ \Delta\overset{\rightarrow}{\omega}_{\mathrm{a},\lambda_{k}}}{\Delta \overset{\rightarrow}{\omega}_{\mathrm{b},\lambda_{k}}}=\frac{\Delta \overset{\rightarrow}{P}_{\mathrm{a},\lambda_{k}}/T_{\mathrm{ja}}}{\Delta \overset{\rightarrow}{P}_{\mathrm{b},\lambda_{k}}/T_{\mathrm{jb}}}. \tag{13}\]  
Based on the law of conservation of energy, this paper defines the integration of power to time as an energy calculation formula, and it obtains the acceleration power \(p_{i}\) of each unit by time domain simulation. The maximum accelerated power deviation of the unit under the leading oscillation mode is selected as the reference value, and the ratio of the accelerated power deviation amplitude of the unit \(i\) to the reference value is obtained.