# Analysis of Synchronous Generators' Local Mode Eigenvalues in Modern Power Systems
## 4 Local Mode Analysis of Synchronous Generators with AVR
### Synchronous Generators' Voltage Control Systems
#### 4.3.1 Hydro-Type Synchronous Generators  
The same hydro-type synchronous generators were analyzed as in Section 3.2.1. All synchronous generators were equipped with the AVR system with the AVR and exciter, modeled with Equation (5). Exciters with the time constant \(T_{\text{AVR}}\) = 0.05 s were used in all cases. The gains \(k_{\text{AVR}}\) were calculated according to the theoretically, numerically, and empirically confirmed equation \(k_{\text{AVR}}\) = \(T^{\prime}{}_{d0}/(10\cdot T_{\text{AVR}})\). The coefficients of the H-P models for operating points in the entire operating range were calculated for all synchronous generators. Active power ranged from 0.0 pu to 1.2 pu, and reactive power ranged from 0.0 pu to 1.2 pu at lagging power factor. The local mode eigenvalues were calculated from (8). Figures 11-14 show the eigenvalue loci as a function of the operating point for synchronous  
Figure 10: Influence of exciter and AVR gain \(k_{\text{AVR}}\) on the local mode eigenvalue of the hydro-generator with nominal power 158 MVA for nominal operating point S = 158 MVA, \(\cos\varphi\) = 0.9.  
generators with nominal powers 9 MVA, 12 MVA, 158 MVA, and 615 MVA. The individual curves correspond to the constant reactive power (values from 0.0 pu to 1.2 pu in steps of 0.1 pu). Active power ranged from 0.0 pu to 1.2 pu. Figure 11: Eigenvalue loci of the local mode of 9-MVA hydro-type synchronous generator with AVR. Figure 12: Eigenvalue loci of the local mode of 12-MVA hydro-type synchronous generator with AVR. Figure 14: Eigenvalue loci of the local mode of 615-MVA hydro-type synchronous generator with AVR. Figure 13: Eigenvalue loci of the local mode of 158-MVA hydro-type synchronous generator with AVR. eigenvalues on the operating point have similar forms for hydro-generators of different powers. While in hydro-generators without AVR systems, the increase in active power in all cases led to a decrease in the real part of the eigenvalue of the local mode, and thus to an increase in damping of oscillations, in hydro-generators with AVR systems, an increase in active power leads to an increase in the local mode eigenvalue's real part, and thus to a reduction in damping or even instability. The root locus diagram for open-loop transfer function \(G_{\rm o}(s)\) clearly shows the influence of the AVR on the stability of a closed-loop transfer function \(G_{\rm c}(s)\):  
\[G_{\rm o}(s)=\frac{V_{\rm t\Delta}(s)}{E(s)}=\frac{b_{\rm o2}s^{2}+b_{\rm o1 }s+b_{\rm o0}}{a_{\rm o4}s^{4}+a_{\rm o3}s^{3}+a_{\rm o2}s^{2}+a_{\rm o1}s+a_{ \rm o0}} \tag{9}\]  
\[G_{\rm c}(s)=\frac{V_{\rm t\Delta}(s)}{V_{\rm t\Delta,ref}(s)}=\frac{b_{\rm c2 }s^{2}+b_{\rm c1}s+b_{\rm c0}}{a_{\rm c4}s^{4}+a_{\rm c3}s^{3}+a_{\rm c2}s^{2}+ a_{\rm c1}s+a_{\rm c0}} \tag{10}\]  
where \(E(s)\) represents the input in the AVR, \(V_{\rm t\Delta}(s)\) is the terminal stator voltage deviation, \(V_{\rm t\Delta,ref}(s)\) is the reference terminal voltage deviation, and \(a_{\rm i}\) and \(b_{\rm i}\) are transfer functions' coefficients. Figure 15 shows the root locus diagrams of local mode eigenvalue for a hydro synchronous generator with nominal power 158 MVA at three operating points: (i) \(P=0.0\) pu, \(Q=0.0\) pu; (ii) \(P=0.0\) pu, \(Q=0.0\) pu; and (iii) \(P=0.0\) pu, \(Q=0.0\) pu. All curves start in the local mode poles of \(G_{\rm o}(s)\) and end in the corresponding zeros of \(G_{\rm o}(s)\). With \"square \" are denoted closed-loop poles at gain \(k_{\rm AVR}=10.4\). The findings obtained from the numerical analysis of all synchronous generators can be summarized in four conclusions:  
* An increase in active power results in a decrease in local mode damping and can lead to instability. Figure 15: Root locus diagrams of local mode eigenvalue for a hydro-type synchronous generator with nominal power 158 MVA for operating points: (i) \(P=0.5\) pu, \(Q=0.0\) pu, (ii) \(P=1.0\) pu, \(Q=0.0\) pu and (iii) \(P=1.2\) pu, \(Q=0.0\) pu.