**3.2 Dynamic analysis**

losses by active power generation. The results are presented in two different operating points. In the first one, the electric angle between generator internal voltage and infinite bus voltage was 10°, and, in the second one, the angle amounts to 40°. The results of the rotor speed oscillations' variations are presented in the amplitude range from 0 to 20% of the synchronous speed by frequencies 1 and 2 [Hz]. From the obtained results, it is evident that there is no influence of the oscillations on

**Table 5** shows the impact of the rotor speed oscillations on the losses by reactive power generation. The results are presented at two different operating points. In the first one, the generator's internal voltage was 10% bigger than the infinite bus voltage, and, in the second one, the difference amounts to 50%. The results of rotor speed oscillations' variations are presented in the amplitude range from 0 to 20% of the synchronous speed by frequencies 1 and 2 [Hz]. From the results, it is obvious

*V***1e** *V2***<sup>e</sup>** *φ***<sup>1</sup>** *φ***<sup>2</sup>** *A***ao** *A***fo** *F***ao** *F***fo** *P***IB** *P***<sup>L</sup>** *p***<sup>L</sup>** 1.0 1.0 10 0 0.0 0.0 0 0 0.40 1.1<sup>10</sup><sup>3</sup> 0.3 1.0 1.0 10 0 0.1 0.1 1 1 0.40 1.5<sup>10</sup><sup>3</sup> 0.4 1.0 1.0 10 0 0.1 0.1 2 2 0.40 1.5<sup>10</sup><sup>3</sup> 0.4 1.0 1.0 10 0 0.2 0.2 1 1 0.40 2.6<sup>10</sup><sup>3</sup> 0.6 1.0 1.0 10 0 0.2 0.2 2 2 0.40 2.5<sup>10</sup><sup>3</sup> 0.6 1.0 1.0 40 0 0.0 0.0 1 1 1.47 1.8<sup>10</sup><sup>2</sup> 1.2 1.0 1.0 40 0 0.1 0.1 1 1 1.47 1.8<sup>10</sup><sup>2</sup> 1.2 1.0 1.0 40 0 0.1 0.1 2 2 1.46 1.9<sup>10</sup><sup>2</sup> 1.3 1.0 1.0 40 0 0.2 0.2 1 1 1.46 1.9<sup>10</sup><sup>2</sup> 1.3 1.0 1.0 40 0 0.2 0.2 2 2 1.46 1.9<sup>10</sup><sup>2</sup> 1.3

*Mean values of three-phase transmitted power to the infinite bus (*P*IB) and three-phase power losses in the transmission line* P*<sup>L</sup> as a function of the angle between the generator's internal voltage and infinite bus voltage (*φ*1-*φ*2) by different amplitudes (*A*ao,* A*fo) and frequencies (*F*ao,* F*fo) of amplitude and frequency oscillations*

*V***1e** *V2***<sup>e</sup>** *φ***<sup>1</sup>** *φ***<sup>2</sup>** *A***ao** *A***fo** *F***ao** *F***fo** *P***IB** *P***<sup>L</sup>** *p***<sup>L</sup>** 1.1 1.0 0 0 0.0 0.0 0 0 3.7<sup>10</sup><sup>3</sup> 3.7<sup>10</sup><sup>4</sup> <sup>10</sup> 1.1 1.0 0 0 0.1 0.1 1 1 3.6<sup>10</sup><sup>3</sup> 7.9<sup>10</sup><sup>4</sup> <sup>22</sup> 1.1 1.0 0 0 0.1 0.1 2 2 3.6<sup>10</sup><sup>3</sup> 7.7<sup>10</sup><sup>4</sup> <sup>21</sup> 1.1 1.0 0 0 0.2 0.2 1 1 3.3<sup>10</sup><sup>3</sup> 2.0<sup>10</sup><sup>3</sup> <sup>61</sup> 1.1 1.0 0 0 0.2 0.2 2 2 3.3<sup>10</sup><sup>3</sup> 2.0<sup>10</sup><sup>3</sup> <sup>61</sup> 1.5 1.0 0 0 0.0 0.0 0 0 1.9<sup>10</sup><sup>2</sup> 9.4<sup>10</sup><sup>3</sup> <sup>50</sup> 1.5 1.0 0 0 0.1 0.1 1 1 1.9<sup>10</sup><sup>2</sup> 1.0<sup>10</sup><sup>2</sup> <sup>54</sup> 1.5 1.0 0 0 0.1 0.1 2 2 1.9<sup>10</sup><sup>2</sup> 1.0<sup>10</sup><sup>2</sup> <sup>54</sup> 1.5 1.0 0 0 0.2 0.2 1 1 1.8<sup>10</sup><sup>2</sup> 1.2<sup>10</sup><sup>2</sup> <sup>66</sup> 1.5 1.0 0 0 0.2 0.2 2 2 1.8<sup>10</sup><sup>2</sup> 1.2<sup>10</sup><sup>2</sup> <sup>66</sup>

*Mean values of three-phase transmitted power to the infinite bus (*P*IB) and three-phase power losses in the transmission line* P*<sup>L</sup> as a function of the generator's internal voltage (*V*1e) by different amplitudes (*A*ao,* A*fo) and frequencies (*F*ao,* F*fo) of amplitude and frequency oscillations in the generator's internal voltage by reactive*

*in the generator's internal voltage by active power transmission.*

power losses in the transmission line.

*Automation and Control*

**Table 4.**

**Table 5.**

**250**

*power transmission.*

The Heffron-Phillips model described in Section 2 is used for the analysis of the dynamic characteristics of the synchronous generator. The impact of the inputs' variations on the oscillations is studied on the synchronous generator with the following data [4]:

New parameters in **Table 6** are as follows: *S*<sup>N</sup> is nominal power [MVA], *V*<sup>N</sup> is nominal voltage [kV], cos *φ*<sup>N</sup> is a nominal power factor, and *x*'<sup>d</sup> is unsaturated d-axis transient reactance [pu].

From the data in **Table 6**, the equilibrium state for the Heffron-Phillips model is calculated by means of a phasor diagram. Phasor equations permit the solution of the initial conditions that exist prior to the application of the inputs' variations. The linearization coefficients of the Heffron-Phillips model are calculated for the synchronous generator with data in **Table 6** and for the calculated equilibrium state's data. The linearization coefficients for nominal operating point (*P*<sup>N</sup> = 1 [pu], cos *φ*<sup>N</sup> = 0.85), and eigenvalues of the Heffron-Phillips model (λ1, λ2, and λ3) are presented in **Table 7**.

The linearized Heffron-Phillips model of a synchronous generator has three eigenvalues. The damping ratio and frequency of rotor angle oscillation are seen directly from the dominant conjugate complex eigenvalues. Therefore, it is very transparent to investigate the dependence of the synchronous generator's oscillation dynamics from the operating point by means of eigenvalue analysis. By changing the operating point, the Heffron-Phillips model's eigenvalues also change.

Transient response of the synchronous generator with data in **Table 6** and nominal operating point with data in **Table 7** are shown in **Figures 11**–**13**. Step changes are simulated in both generator's inputs. **Figure 11** shows the simulated


#### **Table 6.**

*Data of the synchronous generator used for dynamic analysis [4].*


**Table 7.**

*Linearization parameters and eigenvalues of the Heffron-Phillips model in the nominal operating point.*

### *Automation and Control*

trajectory of the turbine mechanical torque and rotor excitation voltage. Step changes are selected in order to maximize the excitation of oscillations.

The responses of the generated electrical power and the stator voltage are shown in **Figure 12**. The oscillations are seen clearly from the response of the generated electrical power.

**Figure 13** shows the response of the rotor speed and rotor angle on the inputs' trajectories shown in **Figure 11**.

*Synchronous generator inputs' trajectories: Mechanical torque* T*m(*t*) [pu] and rotor excitation voltage* E*fd(*t*) [pu], nominal operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu].*

To evaluate the influence of the operating point on the synchronous generator's

*Linearization parameters and eigenvalues of the Heffron-Phillips model at operating point* P *= 1.0 [pu] and*

dynamics, the Heffron-Phillips model was calculated, and simulations were performed at different operating points for different types of generators with different nominal values. From the obtained results, it was obvious that the dynamics

*P* = 1.0 [pu] *Q* = 0.1 [pu] cos *φ* = 0.995 *K*<sup>1</sup> = 1.2506 *K*<sup>2</sup> = 1.5867 *K*<sup>3</sup> = 0.3072 *K*<sup>4</sup> = 2.2164 *K*<sup>5</sup> = 0.0730 *K*<sup>6</sup> = 0.3693 λ<sup>1</sup> = 0.4493 + 9.9715i λ<sup>2</sup> = 0.4493 9.9715i λ<sup>3</sup> = 0.0750

*Synchronous generator outputs' trajectories: Rotor speed* ω*(*t*) [pu] and rotor angle* δ*(*t*) [degrees], nominal*

*Robust and Adaptive Control for Synchronous Generator's Operation Improvement*

In this text, results for the synchronous generator with data in **Table 6** are

• The operating point with high active power and low reactive power: *P* = 1.0 [pu] and *Q* = 0.1 [pu]. This point represents the strongly damped operating point and, therefore, the less problematic case for testing of the PSS.

• The operating point with low active power and high reactive power *P* = 0.1 [pu] and *Q* = 1.0 [pu]. This point represents the weakly damped real operating

**Table 8** shows the linearization data and Heffron-Phillips model eigenvalues for the operating point where active power is generated with very high power factor

The results of the simulation of the synchronous generator at operating point

point and, therefore, the worst case for testing of the PSS.

of the synchronous generator vary significantly.

*operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu].*

*DOI: http://dx.doi.org/10.5772/intechopen.92558*

cos *φ.*

**253**

**Table 8.**

**Figure 13.**

Q *= 0.1 [pu].*

presented for the two most distinctive operating points:

*P* = 1 [pu] and *Q* = 0.1 [pu] are shown in **Figures 14**–**16**.

#### **Figure 12.**

*Synchronous generator outputs' trajectories: Generated electrical power* P*e(*t*) [pu] and stator terminal voltage* V*t(*t*) [pu], nominal operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu].*

*Robust and Adaptive Control for Synchronous Generator's Operation Improvement DOI: http://dx.doi.org/10.5772/intechopen.92558*

#### **Figure 13.**

trajectory of the turbine mechanical torque and rotor excitation voltage. Step changes are selected in order to maximize the excitation of oscillations.

electrical power.

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**Figure 11.**

**Figure 12.**

**252**

trajectories shown in **Figure 11**.

The responses of the generated electrical power and the stator voltage are shown in **Figure 12**. The oscillations are seen clearly from the response of the generated

**Figure 13** shows the response of the rotor speed and rotor angle on the inputs'

*Synchronous generator inputs' trajectories: Mechanical torque* T*m(*t*) [pu] and rotor excitation voltage* E*fd(*t*)*

*Synchronous generator outputs' trajectories: Generated electrical power* P*e(*t*) [pu] and stator terminal voltage*

*[pu], nominal operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu].*

V*t(*t*) [pu], nominal operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu].*

*Synchronous generator outputs' trajectories: Rotor speed* ω*(*t*) [pu] and rotor angle* δ*(*t*) [degrees], nominal operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu].*


#### **Table 8.**

*Linearization parameters and eigenvalues of the Heffron-Phillips model at operating point* P *= 1.0 [pu] and* Q *= 0.1 [pu].*

To evaluate the influence of the operating point on the synchronous generator's dynamics, the Heffron-Phillips model was calculated, and simulations were performed at different operating points for different types of generators with different nominal values. From the obtained results, it was obvious that the dynamics of the synchronous generator vary significantly.

In this text, results for the synchronous generator with data in **Table 6** are presented for the two most distinctive operating points:


**Table 8** shows the linearization data and Heffron-Phillips model eigenvalues for the operating point where active power is generated with very high power factor cos *φ.*

The results of the simulation of the synchronous generator at operating point *P* = 1 [pu] and *Q* = 0.1 [pu] are shown in **Figures 14**–**16**.

**Table 9** shows the linearization data and Heffron-Phillips model eigenvalues for other interesting operating points. In this case, almost only reactive power is generated—Power factor cos *φ* is very small.

The results of the simulation of the synchronous generator in operating point *P* = 0.1 [pu] and *Q* = 1.0 [pu] are shown in **Figures 17**–**19**.

*P* = 0.1 [pu] *Q* = 1.0 [pu] cos *φ* = 0.099 *K*<sup>1</sup> = 1.2614 *K*<sup>2</sup> = 0.1631 *K*<sup>3</sup> = 0.3072 *K*<sup>4</sup> = 0.1219 *K*<sup>5</sup> = 0.0185 *K*<sup>6</sup> = 0.6207 λ<sup>1</sup> = 0.2123 + 10.0141i λ<sup>2</sup> = 0.2123 10.0141i λ<sup>3</sup> = 0.5490

*Synchronous generator outputs' trajectories: Rotor speed* ω*(*t*) [pu] and rotor angle* δ*(*t*) [degrees], operating*

*Robust and Adaptive Control for Synchronous Generator's Operation Improvement*

*DOI: http://dx.doi.org/10.5772/intechopen.92558*

*Linearization parameters and eigenvalues of the Heffron-Phillips model at operating point* P *= 0.1 [pu] and*

*Synchronous generator inputs' trajectories: Mechanical torque* T*m(*t*) [pu] and rotor excitation voltage* E*fd(*t*)*

**Table 9.**

**Figure 16.**

*point* P *= 1.0 [pu] and* Q *= 0.1 [pu].*

Q *= 1.0 [pu].*

**Figure 17.**

**255**

*[pu], operating point* P *= 0.1 [pu] and* Q *= 1.0 [pu].*

**Figure 14.** *Synchronous generator inputs' trajectories: Mechanical torque* T*m(*t*) [pu] and rotor excitation voltage* E*fd(*t*) [pu], operating point* P *= 1.0 [pu] and* Q *= 0.1 [pu].*

#### **Figure 15.**

*Synchronous generator outputs' trajectories: Generated electrical power* P*e(*t*) [pu] and stator terminal voltage* V*t(*t*) [pu], operating point* P *= 1.0 [pu] and* Q *= 0.1 [pu].*

*Robust and Adaptive Control for Synchronous Generator's Operation Improvement DOI: http://dx.doi.org/10.5772/intechopen.92558*

#### **Figure 16.**

**Table 9** shows the linearization data and Heffron-Phillips model eigenvalues for other interesting operating points. In this case, almost only reactive power is gen-

The results of the simulation of the synchronous generator in operating point

*Synchronous generator inputs' trajectories: Mechanical torque* T*m(*t*) [pu] and rotor excitation voltage* E*fd(*t*)*

*Synchronous generator outputs' trajectories: Generated electrical power* P*e(*t*) [pu] and stator terminal voltage*

erated—Power factor cos *φ* is very small.

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*[pu], operating point* P *= 1.0 [pu] and* Q *= 0.1 [pu].*

V*t(*t*) [pu], operating point* P *= 1.0 [pu] and* Q *= 0.1 [pu].*

**Figure 14.**

**Figure 15.**

**254**

*P* = 0.1 [pu] and *Q* = 1.0 [pu] are shown in **Figures 17**–**19**.

*Synchronous generator outputs' trajectories: Rotor speed* ω*(*t*) [pu] and rotor angle* δ*(*t*) [degrees], operating point* P *= 1.0 [pu] and* Q *= 0.1 [pu].*


#### **Table 9.**

*Linearization parameters and eigenvalues of the Heffron-Phillips model at operating point* P *= 0.1 [pu] and* Q *= 1.0 [pu].*

#### **Figure 17.**

*Synchronous generator inputs' trajectories: Mechanical torque* T*m(*t*) [pu] and rotor excitation voltage* E*fd(*t*) [pu], operating point* P *= 0.1 [pu] and* Q *= 1.0 [pu].*

**4. Conventional control system for synchronous generator's rotor**

*Robust and Adaptive Control for Synchronous Generator's Operation Improvement*

Two principal control systems affect a synchronous generator directly: a governor control system and an excitation control system. The governor control system controls the mechanical power from a steam or water turbine by opening or closing valves regulating the steam or water flow. The response of the governor control system is too slow to damp the synchronous generator's oscillations, which are mainly in the frequency range 0.5–2.5 Hz. Damping the oscillations is possible only with the excitation control system. The excitation control system (also called an automatic voltage regulator) changes the rotor field voltage (and current) in such a way that the generator's output voltage is the same (or close enough) to the reference voltage. In modern power plants, the thyristor or transistor rectifiers are used mainly to generate the required voltage for rotor winding. The electrical power flow from the excitation system is much smaller than the mechanical power flow. This, and the fact that semiconductor components are used in the excitation system instead of mechanical ones, is the reason that the excitation system is significantly faster than the governor system. Therefore, an exciter is used for the damping of the

A conventional linear PSS approach is based on utilization of the static excitation

*Block diagram of the Heffron-Phillips model of synchronous generator equipped with excitation system, voltage*

system. Through this system, the PSS changes the field excitation voltage of a synchronous generator. An additional component of an electrical torque is generated as a consequence. This torque must be in phase with the rotor speed and thus increases damping of the synchronous generator [9]. **Figure 20** presents a block diagram of the Heffron-Phillips model of synchronous generator equipped with an excitation system, voltage controller, and power system stabilizer [10]. The generator's output voltage is compared with a reference voltage, and the calculated error is driven to the rectifier with an integrated voltage controller. The rectifier with the voltage controller is presented with the first-order model. The PSS input represents one or more signals in which oscillations are visible. The PSS generates an additional

**excitation equipment**

*DOI: http://dx.doi.org/10.5772/intechopen.92558*

oscillations.

**Figure 20.**

**257**

*controller, and power system stabilizer.*

signal, which is added to the voltage error.

*Synchronous generator outputs' trajectories: Generated electrical power* P*e(*t*) [pu] and stator terminal voltage* V*t(*t*) [pu], operating point* P *= 0.1 [pu] and* Q *= 1.0 [pu].*

**Figure 19.**

*Synchronous generator outputs' trajectories: Rotor speed* ω*(*t*) [pu] and rotor angle* δ*(*t*) [degrees], operating point* P *= 0.1 [pu] and* Q *= 1.0 [pu].*

From the analysis of the effect of different loadings on the synchronous generator dynamic characteristics, it can be concluded that the variations in the machine dynamics are considerable in the entire operating range, and, therefore, a control system is necessary for damping of the oscillations. From the comparison of the responses across different operating points, it is obvious that the present system is nonlinear and that the conventional linear control theory does not provide adequate damping throughout the entire operating area. Therefore, the implementation of a robust or adaptive control theory is meaningful.
