**3. Analysis of the impact of the oscillations on the power system quality**

The oscillations in the power system are due to the physical properties of the synchronous generator that operates parallel to the network. These properties are reflected in the dynamical mathematical model of the synchronous generator and appear as poorly damped dominant eigenvalues. Therefore, any changes in the synchronous generators' inputs (rotor field voltage and mechanical torque), in the network loads (changes in bus voltages) and disturbances, cause oscillations with relatively high amplitude and low damping. Oscillations in the power system are visible in several physical quantities of the system: in the synchronous generators' rotor speed, rotor angle, stator voltage, stator current, and produced power and in the power system's voltages, currents, frequency, and transmitted powers. These oscillations reduce the quality of the electricity and increase the stability risk of the power system.

It is very difficult to estimate the impact of oscillations on actual losses in a power system. In a real power system operation, it is problematic to evaluate how much of the losses is due to the rotor angle oscillations and how much of the losses are due to other factors. Therefore, in the first subsection, the influence of the amplitude and frequency of the oscillations on the amount of the losses in the transmission line and on the constancy of the transmitted power is discussed in more detail. The thoroughly steady-state analysis was made for this purpose. The dynamic analysis is presented in the second subsection. Dynamic analysis shows the vulnerability of the synchronous generator on the different input changes in different operation points.

#### **3.1 Steady-state analysis**

An analysis of the impact of the oscillations on losses and on the constancy of the transmitted power is made numerically. In the case of constant rotor speed, the

*i*DðÞ¼ *t*

*i*<sup>Q</sup> ðÞ¼ *t*

1

3

3

*T*eðÞ¼ *t*

\_

except the electric rotor angle *δ*(*t*) having unit [rad].

\_

\_

*ω*\_ ðÞ¼ *t*

\_

1

where *i*d(*t*) and *i*q(*t*) are stator d- and q-axis currents [pu]; *i*F(*t*) is field current [pu]; *i*D(*t*) and *i*Q(*t*) are damping d- and q-axis currents [pu]; *v*d(*t*) and *v*q(*t*) are stator terminal d- and q-axis voltages [pu]; *λ*AD(*t*) and *λ*AQ(*t*) are d- and q-axis mutual flux linkages [pu]; *R*<sup>e</sup> and *L*<sup>e</sup> are transmission line resistance and reactance [pu]; *V*<sup>∞</sup> is infinite bus voltage [pu]; *T*e(*t*) is electromagnetic torque [pu]; *L*MD, *L*MQ, *L*AD, and *L*AQ are mutual inductances [pu]; *l*d, *l*q, *l*F, *l*D, and *l*<sup>Q</sup> are leakage inductances [pu]; *R*S, *R*F, *R*D, and *R*<sup>Q</sup> are stator, field, d-axis damping, and q-axis damping winding resistances [pu]; *H* is an inertia constant [s]; and *ω*<sup>s</sup> is electric

The seventh-order model is the superior one; although, on the other hand, it is too complicated to gain insight into the physical characteristics of the controlled plant [5]. It is also not suitable for the design and synthesis of control systems, since many control methods require linear mathematical models for the development of the control system. Many simplified models are derived from this seventh-order nonlinear model [6]. For a synchronous generator analysis and for the design of the PSS control system, a simplified linearized third-order model is still the most popular. It was presented for the first time in 1952 [7] and is, therefore, also called the

The Heffron-Phillips model is obtained from the seventh-order nonlinear model by means of linearization for an every steady-state operating point (i.e., an equilibrium point). The Heffron-Phillips model describes the synchronous generator's dynamics in the proximity of the selected equilibrium point. The Heffron-Phillips model has two inputs and three state-space variables. The inputs are mechanical torque *T*mΔ(*t*) and rotor excitation winding voltage *E*fdΔ(*t*) deviations; the statespace variables are rotor angle *δ*Δ(*t*), rotor speed *ω*Δ(*t*), and voltage behind tran-

terminal stator voltage *V*tΔ(*t*) deviations. All the inputs and the state-space variables

<sup>q</sup>Δð Þ*t* deviations. Additional outputs are electric power *P*eΔ(*t*) and

*<sup>v</sup>*dðÞ¼� *<sup>t</sup>* ffiffiffi

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*<sup>v</sup>*qðÞ¼ *<sup>t</sup>* ffiffiffi

and differential equations (Eqs. (11)–(17)):

\_

\_

synchronous speed [rad s�<sup>1</sup>

Heffron-Phillips model.

sient reactance *E*<sup>0</sup>

**242**

1 *l*D

1 *l*Q ð Þ *λ*DðÞ�*t λ*ADð Þ*t* (6)

*<sup>λ</sup>*<sup>Q</sup> ðÞ�*<sup>t</sup> <sup>λ</sup>*AQ ð Þ*<sup>t</sup>* � � (7)

<sup>p</sup> *<sup>V</sup>*<sup>∞</sup> sin ð Þþ *<sup>δ</sup>*ð Þ*<sup>t</sup> <sup>R</sup>*e*i*dðÞþ*<sup>t</sup> <sup>ω</sup>*ð Þ*<sup>t</sup> <sup>L</sup>*e*i*qð Þ*<sup>t</sup>* (8)

<sup>p</sup> *<sup>V</sup>*<sup>∞</sup> cosð Þþ *<sup>δ</sup>*ð Þ*<sup>t</sup> <sup>R</sup>*e*i*qð Þþ*<sup>t</sup> <sup>ω</sup>*ð Þ*<sup>t</sup> <sup>L</sup>*e*i*dð Þ*<sup>t</sup>* (9)

*<sup>λ</sup>*dðÞ¼ *<sup>t</sup> <sup>ω</sup>*<sup>s</sup> �*R*s*i*dðÞ�*<sup>t</sup> <sup>ω</sup>*ð Þ*<sup>t</sup> <sup>λ</sup>*qðÞ�*<sup>t</sup> <sup>v</sup>*dð Þ*<sup>t</sup>* � � (11)

*<sup>λ</sup>*qðÞ¼ *<sup>t</sup> <sup>ω</sup>*<sup>s</sup> �*R*s*i*qðÞþ*<sup>t</sup> <sup>ω</sup>*ð Þ*<sup>t</sup> <sup>λ</sup>*dðÞ�*<sup>t</sup> <sup>v</sup>*qð Þ*<sup>t</sup>* � � (12)

*λ*FðÞ¼ *t ω*sð Þ �*R*F*i*Fð Þþ*t E*fdð Þ*t* (13)

*λ*DðÞ¼ *t ω*sð Þ �*R*D*i*Dð Þ*t* (14)

*<sup>λ</sup>*<sup>Q</sup> ðÞ¼ *<sup>t</sup> <sup>ω</sup>*<sup>s</sup> �*R*<sup>Q</sup> *<sup>i</sup>*<sup>Q</sup> ð Þ*<sup>t</sup>* � � (15)

<sup>2</sup>*<sup>H</sup>* ð Þ *<sup>T</sup>*mðÞ�*<sup>t</sup> <sup>T</sup>*eð Þ*<sup>t</sup>* (16)

*δ*ðÞ¼ *t ω*sð Þ *ω*ðÞ�*t* 1 (17)

]. All variables are normalized on the base quantities

<sup>3</sup> *<sup>i</sup>*qð Þ*<sup>t</sup> <sup>λ</sup>*dð Þ�*<sup>t</sup> <sup>i</sup>*dð Þ*<sup>t</sup> <sup>λ</sup>*qð Þ*<sup>t</sup>* � � (10)

induced voltage in the stator winding is sinusoidal, with constant amplitude and frequency. The swinging of the rotor speed results in induced voltage with variable frequency and amplitude. From the solution of the swing equation, it is evident that the swinging of the rotor is sinusoidal [8]. Therefore, in a steady-state analysis, we suppose that the rotor's speed oscillates about the constant synchronous speed with sinusoidal oscillations. This results in the stator's induced voltage, which varies sinusoidally in amplitude and frequency. The amplitude increases when the frequency increases, and vice versa. For such input signal, there is no transparent analytical solution for the evaluation of the losses and constancy of the transmitted power. Therefore, the numerical solution that is based on an equivalent circuit of the synchronous generator connected by transmission line to the infinite bus is used for the analysis. The equivalent circuit is shown in **Figure 1**.

in the transmission line [pu]; *P*La(*t*), *P*Lb(*t*), and *P*Lc(*t*) represent instantaneous power losses for different phases [pu]; and *P*<sup>L</sup> represents the mean value of the instantaneous three-phase power losses in the transmission line [pu]. *p*<sup>L</sup> indicates the relative value of the mean value of the three-phase power losses (*P*L) compared to the mean value of the three-phase power flow to the infinite bus (*P*IB) in [%]. Stator voltage amplitude and frequency oscillations resulting from rotor speed swing are described with the amplitude and frequency of both oscillations. *A*ao and *F*ao denote amplitude and frequency of amplitude oscillations. *A*fo and *F*fo denote

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

For the presented results, the impedances of the synchronous generator and

**Figures 2**–**5** show the time responses of the electrical quantities if there are no oscillations in rotor speed. The parameters of the generator's internal voltage and

**Figure 2** shows the instantaneous values of the synchronous generator's internal

**Figures 6**–**10** show the time responses of the electrical quantities if oscillations occur in the rotor speed. The parameters of the generator's internal voltage and

*R*<sup>a</sup> = 0.0011 [pu] *X*<sup>s</sup> = 0.9 [pu] *R*<sup>e</sup> = 0.02 [pu] *X*<sup>e</sup> = 0.4 [pu]

*Synchronous generator's internal voltages for all three phases:* v*1a(*t*),* v*1b(*t*), and* v*1c(*t*).*

voltages for all three phases. Currents in the transmission lines are shown in **Figure 3**. Power flow to the infinite bus for all three phases separately and the sum of the phases' power flow are shown in **Figure 4**. As expected, the total three-phase power flow is constant. **Figure 5** shows power losses in the transmission line for all

three phases separately and the three-phase power losses. Again, the total

the amplitude and frequency of frequency oscillations.

transmission line are shown in **Table 1**.

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

infinite bus voltage are shown in **Table 2**.

three-phase power losses are constant.

**Table 1.**

**Figure 2.**

**245**

infinite bus voltage are shown in **Table 3**.

*The synchronous generator's and transmission line's impedances.*

where *V*1e denotes the effective value of the generator's internal voltage [pu], *V*2e is the effective value of the infinite bus voltage [pu], *φ*<sup>1</sup> is the generator's internal voltage angle [rad], *φ*<sup>2</sup> is the infinite bus voltage angle [rad], *R*<sup>s</sup> is stator (armature) resistance [pu], *X*<sup>s</sup> is synchronous reactance [pu], *R*<sup>e</sup> is transmission line resistance [pu], and *X*<sup>e</sup> is transmission line reactance [pu]. The equivalent circuit presented in **Figure 1** is a balanced symmetrical three-phase system. The impedance in any one phase is equal to that in either of the other two phases. Three voltages on the generator side are displaced 120° electrical degrees in time as a result of the phases being displaced 120° in space. Also, the three voltages on the infinite bus side are displaced 120° electrical degrees in time, so that the resulting phase currents are equal in amplitude and displaced in phase from each other by 120°. *v*1a(*t*), *v*1b(*t*), and *v*1c(*t*) denote instantaneous values of the synchronous generator's internal phase voltages; *v*IBa(*t*), *v*IBb(*t*), and *v*IBc(*t*) denote instantaneous values of the infinite bus phase voltages; and *i*a(*t*), *i*b(*t*), and *i*c(*t*) denote instantaneous values of the transmission line phase currents. Unless otherwise specified, *P*IB(*t*) represents the instantaneous three-phase power flow to the infinite bus [pu]; *P*IBa(*t*), *P*IBb(*t*), and *P*IBc(*t*) represent instantaneous power flow to the infinite bus for different phases [pu]; and *P*IB represents the mean value of the instantaneous three-phase power flow to the infinite bus [pu]. *P*L(*t*) denotes instantaneous three-phase power losses

#### **Figure 1.**

*Equivalent circuit of the synchronous generator connected by transmission line to the infinite bus used for the steady-state analysis.*

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

in the transmission line [pu]; *P*La(*t*), *P*Lb(*t*), and *P*Lc(*t*) represent instantaneous power losses for different phases [pu]; and *P*<sup>L</sup> represents the mean value of the instantaneous three-phase power losses in the transmission line [pu]. *p*<sup>L</sup> indicates the relative value of the mean value of the three-phase power losses (*P*L) compared to the mean value of the three-phase power flow to the infinite bus (*P*IB) in [%]. Stator voltage amplitude and frequency oscillations resulting from rotor speed swing are described with the amplitude and frequency of both oscillations. *A*ao and *F*ao denote amplitude and frequency of amplitude oscillations. *A*fo and *F*fo denote the amplitude and frequency of frequency oscillations.

For the presented results, the impedances of the synchronous generator and transmission line are shown in **Table 1**.

**Figures 2**–**5** show the time responses of the electrical quantities if there are no oscillations in rotor speed. The parameters of the generator's internal voltage and infinite bus voltage are shown in **Table 2**.

**Figure 2** shows the instantaneous values of the synchronous generator's internal voltages for all three phases. Currents in the transmission lines are shown in **Figure 3**. Power flow to the infinite bus for all three phases separately and the sum of the phases' power flow are shown in **Figure 4**. As expected, the total three-phase power flow is constant. **Figure 5** shows power losses in the transmission line for all three phases separately and the three-phase power losses. Again, the total three-phase power losses are constant.

**Figures 6**–**10** show the time responses of the electrical quantities if oscillations occur in the rotor speed. The parameters of the generator's internal voltage and infinite bus voltage are shown in **Table 3**.


#### **Table 1.**

induced voltage in the stator winding is sinusoidal, with constant amplitude and frequency. The swinging of the rotor speed results in induced voltage with variable frequency and amplitude. From the solution of the swing equation, it is evident that the swinging of the rotor is sinusoidal [8]. Therefore, in a steady-state analysis, we suppose that the rotor's speed oscillates about the constant synchronous speed with sinusoidal oscillations. This results in the stator's induced voltage, which varies sinusoidally in amplitude and frequency. The amplitude increases when the frequency increases, and vice versa. For such input signal, there is no transparent analytical solution for the evaluation of the losses and constancy of the transmitted power. Therefore, the numerical solution that is based on an equivalent circuit of the synchronous generator connected by transmission line to the infinite bus is used

where *V*1e denotes the effective value of the generator's internal voltage [pu], *V*2e is the effective value of the infinite bus voltage [pu], *φ*<sup>1</sup> is the generator's internal voltage angle [rad], *φ*<sup>2</sup> is the infinite bus voltage angle [rad], *R*<sup>s</sup> is stator (armature) resistance [pu], *X*<sup>s</sup> is synchronous reactance [pu], *R*<sup>e</sup> is transmission line resistance [pu], and *X*<sup>e</sup> is transmission line reactance [pu]. The equivalent circuit presented in **Figure 1** is a balanced symmetrical three-phase system. The impedance in any one phase is equal to that in either of the other two phases. Three voltages on the generator side are displaced 120° electrical degrees in time as a result of the phases being displaced 120° in space. Also, the three voltages on the infinite bus side are displaced 120° electrical degrees in time, so that the resulting phase currents are equal in amplitude and displaced in phase from each other by 120°. *v*1a(*t*), *v*1b(*t*), and *v*1c(*t*) denote instantaneous values of the synchronous generator's internal phase voltages; *v*IBa(*t*), *v*IBb(*t*), and *v*IBc(*t*) denote instantaneous values of the infinite bus phase voltages; and *i*a(*t*), *i*b(*t*), and *i*c(*t*) denote instantaneous values of the transmission line phase currents. Unless otherwise specified, *P*IB(*t*) represents the instantaneous three-phase power flow to the infinite bus [pu]; *P*IBa(*t*), *P*IBb(*t*), and *P*IBc(*t*) represent instantaneous power flow to the infinite bus for different phases [pu]; and *P*IB represents the mean value of the instantaneous three-phase power flow to the infinite bus [pu]. *P*L(*t*) denotes instantaneous three-phase power losses

*Equivalent circuit of the synchronous generator connected by transmission line to the infinite bus used for the*

for the analysis. The equivalent circuit is shown in **Figure 1**.

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**Figure 1.**

**244**

*steady-state analysis.*

*The synchronous generator's and transmission line's impedances.*

**Figure 5.**

**Table 2.**

**Figure 6.**

**247**

*Synchronous generator's internal voltages for phase a:* v*1a(*t*).*

*Instantaneous three-phase power losses in the transmission line* P*L(*t*) (constant) and instantaneous power losses*

*V*1e = 1.0 [pu] *φ*<sup>1</sup> = 40 [°] *V*2e = 1.0 [pu] *φ*<sup>1</sup> = 0 [°] *A*ao = 0.0 [pu] *F*ao = 0 [Hz] *A*fo = 0.0 [pu] *F*fo = 0 [Hz]

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

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

*The parameters of the internal voltage and infinite bus voltage, without oscillations.*

*in the transmission line for different phases* P*IBa(*t*),* P*IBb(*t*), and* P*IBc(*t*).*

**Figure 3.** *Transmission line currents for all three phases:* i*a(*t*),* i*b(*t*), and* i*c(*t*).*

#### **Figure 4.**

*Instantaneous three-phase power flow to the infinite bus* P*IB(*t*) (constant), and instantaneous power flow to the infinite bus for different phases* P*IBa(*t*),* P*IBb(*t*), and* P*IBc(*t*).*

For better insight, **Figure 6** shows the instantaneous values of the synchronous generator's internal voltage for phase a. The oscillations are visible in amplitude and in frequency.

**Figure 7** shows the instantaneous values of the synchronous generator's internal voltages for all three phases. Currents in the transmission lines are shown in **Figure 8**. Power flow to the infinite bus for all three phases separately and the instantaneous value of the three-phase power flow are shown in **Figure 9**. In this case, due to oscillations in rotor speed, and, consequently, oscillations in the internal voltages, the total three-phase power flow is not constant. **Figure 10** shows power losses in the transmission line for all three phases separately and the threephase power losses. Again, total three-phase power losses are not constant.

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

#### **Figure 5.**

*Instantaneous three-phase power losses in the transmission line* P*L(*t*) (constant) and instantaneous power losses in the transmission line for different phases* P*IBa(*t*),* P*IBb(*t*), and* P*IBc(*t*).*


#### **Table 2.**

*The parameters of the internal voltage and infinite bus voltage, without oscillations.*

**Figure 6.** *Synchronous generator's internal voltages for phase a:* v*1a(*t*).*

For better insight, **Figure 6** shows the instantaneous values of the synchronous generator's internal voltage for phase a. The oscillations are visible in amplitude and

*Instantaneous three-phase power flow to the infinite bus* P*IB(*t*) (constant), and instantaneous power flow to the*

**Figure 7** shows the instantaneous values of the synchronous generator's internal

voltages for all three phases. Currents in the transmission lines are shown in **Figure 8**. Power flow to the infinite bus for all three phases separately and the instantaneous value of the three-phase power flow are shown in **Figure 9**. In this case, due to oscillations in rotor speed, and, consequently, oscillations in the internal voltages, the total three-phase power flow is not constant. **Figure 10** shows power losses in the transmission line for all three phases separately and the threephase power losses. Again, total three-phase power losses are not constant.

*infinite bus for different phases* P*IBa(*t*),* P*IBb(*t*), and* P*IBc(*t*).*

*Transmission line currents for all three phases:* i*a(*t*),* i*b(*t*), and* i*c(*t*).*

in frequency.

**Figure 4.**

**246**

**Figure 3.**

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**Figure 7.** *Synchronous generator's internal voltages for all three phases:* v*1a(*t*),* v*1b(*t*), and* v*1c(*t*).*

**Figure 8.** *Transmission line currents for all three phases:* i*a(*t*),* i*b(*t*), and* i*c(*t*).*

From **Figures 2**–**10**, it is seen clearly that the rotor speed oscillations cause the oscillation in the transmitted power. The sum of the transmitted powers of the individual phases is no longer constant (**Figure 9**), as is the case for the balanced three-phase symmetric systems without oscillations (**Figure 4**). These oscillations reduce the transmission capability and quality. To ensure the power transmission with minimal power oscillations, it is necessary to reduce the rotor speed oscillations.

**Figure 10.**

**Figure 9.**

*different phases* P*IBa(*t*),* P*IBb(*t*), and* P*IBc(*t*).*

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

**Table 3.**

**249**

*Three-phase power losses in the transmission line* P*L(*t*) (violet) and instantaneous power losses in the*

*Three-phase power flow to the infinite bus* P*IB(*t*) (violet) and instantaneous power flow to the infinite bus for*

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

*The parameters of the internal voltage and infinite bus voltage, with oscillations in internal voltage.*

*V*1e = 1.0 [pu] *φ*<sup>1</sup> = 40 [°] *V*2e = 1.0 [pu] *φ*<sup>1</sup> = 0 [°] *A*ao = 0.1 [pu] *F*ao = 2 [Hz] *A*fo = 0.1 [pu] *F*fo = 2 [Hz]

*transmission line for different phases* P*IBa(*t*),* P*IBb(*t*), and* P*IBc(*t*).*

A thorough numerical analysis was performed to estimate the influence of the rotor speed oscillations on the power system losses. Some results are presented in **Tables 4** and **5**. **Table 4** shows the impact of the rotor speed oscillations on the

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

#### **Figure 9.**

*Three-phase power flow to the infinite bus* P*IB(*t*) (violet) and instantaneous power flow to the infinite bus for different phases* P*IBa(*t*),* P*IBb(*t*), and* P*IBc(*t*).*

#### **Figure 10.**

From **Figures 2**–**10**, it is seen clearly that the rotor speed oscillations cause the oscillation in the transmitted power. The sum of the transmitted powers of the individual phases is no longer constant (**Figure 9**), as is the case for the balanced three-phase symmetric systems without oscillations (**Figure 4**). These oscillations reduce the transmission capability and quality. To ensure the power transmission with minimal power oscillations, it is necessary to reduce the rotor speed oscilla-

*Transmission line currents for all three phases:* i*a(*t*),* i*b(*t*), and* i*c(*t*).*

*Synchronous generator's internal voltages for all three phases:* v*1a(*t*),* v*1b(*t*), and* v*1c(*t*).*

A thorough numerical analysis was performed to estimate the influence of the rotor speed oscillations on the power system losses. Some results are presented in **Tables 4** and **5**. **Table 4** shows the impact of the rotor speed oscillations on the

tions.

**248**

**Figure 8.**

**Figure 7.**

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*Three-phase power losses in the transmission line* P*L(*t*) (violet) and instantaneous power losses in the transmission line for different phases* P*IBa(*t*),* P*IBb(*t*), and* P*IBc(*t*).*


#### **Table 3.**

*The parameters of the internal voltage and infinite bus voltage, with oscillations in internal voltage.*
