**2. Mathematical model of the synchronous generator connected to the power system**

The seventh-order nonlinear model of the synchronous machine connected to the infinite bus is the most detailed mathematical model of the synchronous generator connected to the large power system with constant frequency and constant voltage (=infinite bus) through the transmission line [3]. Park's matrix transformation is used to transform the origin windings' equations into a model with orthogonal axes. On this basis, the magnetic coupling of the stator, field, and damper windings is represented as a function of the position of the machine's rotor. The seventh-order model is represented in the form of a nonlinear state-space model [4]. The model's inputs are mechanical torque *T*m(*t*) and rotor excitation winding voltage *E*fd(*t*). The model's state-space variables are stator d-axis flux linkage *λ*d(*t*), stator q-axis flux linkage *λ*q(*t*), rotor excitation winding flux linkage *λ*F(*t*), rotor d-axis damper winding flux linkage *λ*D(*t*), rotor q-axis damper winding flux linkage *λ*D(*t*), mechanical rotor speed *ω*(*t*), and electric rotor angle *δ*(*t*). The seventh-order model is described by sets of algebraic equations (Eqs. (1)–(10)) [5]:

$$
\lambda\_{\rm AD}(t) = L\_{\rm MD} \left( \frac{\lambda\_{\rm d}(t)}{l\_{\rm d}} + \frac{\lambda\_{\rm F}(t)}{l\_{\rm F}} + \frac{\lambda\_{\rm D}(t)}{l\_{\rm D}} \right) \tag{1}
$$

$$
\lambda\_{\rm AQ}(t) = L\_{\rm MQ} \left( \frac{\lambda\_{\rm q}(t)}{l\_{\rm q}} + \frac{\lambda\_{\rm Q}(t)}{l\_{\rm Q}} \right) \tag{2}
$$

$$i\_{\mathbf{d}}(t) = \frac{\mathbf{1}}{l\_{\mathbf{d}}} (\lambda\_{\mathbf{d}}(t) - \lambda\_{\mathbf{AD}}(t)) \tag{3}$$

$$i\_{\mathbf{q}}(t) = \frac{\mathbf{1}}{l\_{\mathbf{q}}} \left(\lambda\_{\mathbf{q}}(t) - \lambda\_{\mathbf{AQ}}(t)\right) \tag{4}$$

$$i\_{\rm F}(t) = \frac{1}{l\_{\rm F}} (\lambda\_{\rm F}(t) - \lambda\_{\rm AD}(t)) \tag{5}$$

representing 3.7% of total production. Even more interesting is the growth rate of electricity trade. In the OECD, imports of electricity grew from 89 TWh in 1974 to 480 TWh in 2018, representing an average annual growth rate of 4.0%, compared

The facts that the majority of the world's electricity is produced by synchronous generators and that a large amount of the world's electricity is transmitted over long distances result in significant oscillations of the produced and transmitted power. Despite the relatively small oscillations—the ratio of the amplitude of the oscillations of the transmitted power relative to the mean value of the transmitted power is mainly smaller than 10%—the total global losses due to the extremely large volume of production and transmission of electricity are not negligible. In terms of

The amount of the transmitted power oscillations can be affected by optimizing the topology of the new networks, by reconfiguring of the existing networks, by selection of the better damped new synchronous generators, and by replacement of the existing synchronous generators with the better damped ones. These solutions are expensive, and their realization also depends on other social and ecological factors. Therefore, it is a much more suitable solution to use a control system to damp the power system oscillations. In power systems, control systems called power system stabilizers (PSS) are used to suppress oscillations. PSS represent the best and the most economical solution for damping of the power systems' oscillations. PSS are simple to realize—they are mainly a part of the controller of the synchronous generator's static semiconductor excitation system. PSS, based on information of the oscillations of the transmitted power, rotor speed, rotor angle, or rotor acceleration, generate an additional reference signal for the rotor current control system. This additional reference signal represents the supplementary input to the static semiconductor excitation system, which is connected to rotor

Conventional PSS design is based on a linear control theory. Conventional PSS is simple to realize, but its application shows nonoptimal damping through the entire operating range; by varying the operating point, the synchronous generator's dynamic characteristics also vary; the fact is that PSS, which was determined for the nominal operating point, does not assure optimal damping in the entire operating range. Such a PSS reduces transmission losses optimally only at the operating point for which the PSS parameters are selected. Due to the large changes in the transmitted power and the large variations in power generation of the synchronous generators, conventional PSS are not satisfactory for use in modern power systems. To improve PSS performance, major modern control theories have been tested in the past decade for the purposes of PSS design. Of all the methods, robust and adaptive control has been implemented to be the most suitable for the design of PSS. Both control methods have been used in order to assure optimal damping through the entire operating range of the synchronous generators. The use of adaptive control is possible because the loading variations and, consequently, the variations of the dynamic characteristics of the synchronous generators are, in most cases, substantially slower than the dynamics of the adaptation mechanism [2]. Reduction of losses is not the sole and basic task of PSS. Even more important is that the PSS improves the stability of the power system and allows the transfer of power from the synchronous generator to the power system or between different points in the power system as near as possible to the stability limit of the transmission. In the presented work, however, we show the results of our study, which will show the applicability of the developed robust and adaptive PSS, mainly for the

improvement of the damping of the power system oscillations.

to 2.1% growth in overall electricity supply.

*Automation and Control*

saving energy, it makes sense to reduce these losses.

field winding.

**240**

$$\dot{a}\_{\rm D}(t) = \frac{1}{l\_{\rm D}} (\dot{\lambda}\_{\rm D}(t) - \dot{\lambda}\_{\rm AD}(t)) \tag{6}$$

denote the deviations (subscript <sup>Δ</sup>) from the equilibrium state. The model is written

2 6 4

*δ*Δð Þ*t ω*Δð Þ*t E*0 <sup>q</sup>Δð Þ*t*

0 0 0 0 � � *<sup>T</sup>*mΔð Þ*<sup>t</sup>*

� � (19)

*T*mΔð Þ*t E*fdΔð Þ*t* � �

(18)

],*T*do<sup>0</sup>

0 1 *T*0 d0

*E*fdΔð Þ*t*

2*H*

*K*3*T*<sup>0</sup> d0

2 6 4

*δ*Δð Þ*t ω*Δð Þ*t E*0 <sup>q</sup>Δð Þ*t*

where *T*mΔ(*t*) represents mechanical torque deviation [pu], *P*eΔ(*t*) is electrical power deviation [pu], *ω*Δ(*t*) is rotor speed deviation [pu], *δ*Δ(*t*) is rotor angle deviation [rad], *E'*qΔ(*t*) is the voltage behind the transient reactance [pu], *E*fdΔ(*t*) is field excitation voltage deviation [pu], *V*tΔ(*t*) is the terminal voltage [pu], *H* is an inertia constant [s], *D* is a damping coefficient representing total lumped damping effects from damper windings [pu/pu], *ω*<sup>s</sup> is rated synchronous speed [rad s�<sup>1</sup>

is a d-axis transient open circuit time constant [s], and *K*<sup>1</sup> through *K*<sup>6</sup> are linearization parameters. All parameters and variables in a Heffron-Phillips model are nor-

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

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

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.

0 *ω<sup>s</sup>* 0

<sup>2</sup>*<sup>H</sup>* � *<sup>K</sup>*<sup>2</sup>

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

<sup>0</sup> � <sup>1</sup>

<sup>2</sup>*<sup>H</sup>* � *<sup>D</sup>*

<sup>¼</sup> *<sup>K</sup>*<sup>1</sup> <sup>0</sup> *<sup>K</sup>*<sup>2</sup> *K*<sup>5</sup> 0 *K*<sup>6</sup> � �

as follows:

\_ *δ*Δð Þ*t ω*\_ <sup>Δ</sup>ð Þ*t E*\_ 0 <sup>q</sup>Δð Þ*t*

*P*eΔð Þ*t V*tΔð Þ*t* � �

**3.1 Steady-state analysis**

**243**

� *<sup>K</sup>*<sup>1</sup>

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

� *<sup>K</sup>*<sup>4</sup> *T*0 d0

malized, except for electric rotor angle *δ*Δ(*t*).

$$\dot{a}\_{\mathbf{Q}}(t) = \frac{1}{I\_{\mathbf{Q}}} \left(\lambda\_{\mathbf{Q}}(t) - \lambda\_{\mathbf{AQ}}(t)\right) \tag{7}$$

$$v\_{\mathbf{d}}(t) = -\sqrt{3}V\_{\infty}\sin\left(\delta(t)\right) + R\_{\mathbf{e}}i\_{\mathbf{d}}(t) + o(t)L\_{\mathbf{e}}i\_{\mathbf{q}}(t) \tag{8}$$

$$v\_{\mathbf{q}}(t) = \sqrt{3}V\_{\infty}\cos\left(\delta(t)\right) + R\_{\mathbf{e}}i\_{\mathbf{q}}(t) + o(t)L\_{\mathbf{e}}i\_{\mathbf{d}}(t) \tag{9}$$

$$T\_{\mathbf{e}}(t) = \frac{1}{3} \left( i\_{\mathbf{q}}(t)\lambda\_{\mathbf{d}}(t) - i\_{\mathbf{d}}(t)\lambda\_{\mathbf{q}}(t) \right) \tag{10}$$

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

$$\dot{\lambda}\_{\rm d}(t) = \alpha\_{\rm s} \left( -R\_{\rm s} i\_{\rm d}(t) - \alpha(t) \lambda\_{\rm q}(t) - v\_{\rm d}(t) \right) \tag{11}$$

$$\dot{\lambda}\_{\mathbf{q}}(t) = \alpha\_{\mathbf{s}} \left( -R\_{\mathbf{s}} \dot{\mathbf{q}}\_{\mathbf{q}}(t) + \alpha(t) \lambda\_{\mathbf{d}}(t) - \nu\_{\mathbf{q}}(t) \right) \tag{12}$$

$$\dot{\lambda}\_{\rm F}(t) = a\_{\rm s}(-R\_{\rm F}i\_{\rm F}(t) + E\_{\rm fd}(t)) \tag{13}$$

$$\dot{\lambda}\_{\rm D}(t) = \alpha\_{\rm s}(-R\_{\rm D}i\_{\rm D}(t))\tag{14}$$

$$\dot{\lambda}\_{\mathcal{Q}}(t) = \alpha\_{\mathcal{s}} \left( -R\_{\mathcal{Q}} i\_{\mathcal{Q}}(t) \right) \tag{15}$$

$$
\dot{\boldsymbol{\phi}}(t) = \frac{1}{2H} (T\_{\rm m}(t) - T\_{\rm e}(t)) \tag{16}
$$

$$\dot{\delta}(t) = \alpha\_{\text{s}}(\alpha(t) - \mathbf{1}) \tag{17}$$

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 synchronous speed [rad s�<sup>1</sup> ]. All variables are normalized on the base quantities except the electric rotor angle *δ*(*t*) having unit [rad].

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 Heffron-Phillips model.

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 transient reactance *E*<sup>0</sup> <sup>q</sup>Δð Þ*t* deviations. Additional outputs are electric power *P*eΔ(*t*) and terminal stator voltage *V*tΔ(*t*) deviations. All the inputs and the state-space variables *Robust and Adaptive Control for Synchronous Generator's Operation Improvement DOI: http://dx.doi.org/10.5772/intechopen.92558*

denote the deviations (subscript <sup>Δ</sup>) from the equilibrium state. The model is written as follows:

$$
\begin{bmatrix}
\dot{\delta}\_{\Delta}(t) \\
\dot{w}\_{\Delta}(t) \\
\dot{E}\_{\text{q}\Delta}'(t)
\end{bmatrix} = \begin{bmatrix}
0 & \dot{w}\_{\circ} & 0 \\
\end{bmatrix} \begin{bmatrix}
\delta\_{\Delta}(t) \\
w\_{\Delta}(t) \\
E\_{\text{q}\Delta}'(t)
\end{bmatrix} + \begin{bmatrix}
0 & 0 \\
\frac{1}{2H} & 0 \\
0 & \frac{1}{T\_{\text{d0}}'}
\end{bmatrix} \begin{bmatrix}
T\_{\text{m}\Delta}(t) \\
E\_{\text{fd}\Delta}(t)
\end{bmatrix} \tag{18}
$$

$$
\begin{bmatrix} P\_{\text{e}\Delta}(t) \\ V\_{\text{t}\Delta}(t) \end{bmatrix} = \begin{bmatrix} K\_1 & \mathbf{0} & K\_2 \\ K\_5 & \mathbf{0} & K\_6 \end{bmatrix} \begin{bmatrix} \delta\_\Delta(t) \\ \alpha\_\Delta(t) \\ E\_{\text{q}\Delta}'(t) \end{bmatrix} + \begin{bmatrix} \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \begin{bmatrix} T\_{\text{m}\Delta}(t) \\ E\_{\text{fd}\Delta}(t) \end{bmatrix} \tag{19}
$$

where *T*mΔ(*t*) represents mechanical torque deviation [pu], *P*eΔ(*t*) is electrical power deviation [pu], *ω*Δ(*t*) is rotor speed deviation [pu], *δ*Δ(*t*) is rotor angle deviation [rad], *E'*qΔ(*t*) is the voltage behind the transient reactance [pu], *E*fdΔ(*t*) is field excitation voltage deviation [pu], *V*tΔ(*t*) is the terminal voltage [pu], *H* is an inertia constant [s], *D* is a damping coefficient representing total lumped damping effects from damper windings [pu/pu], *ω*<sup>s</sup> is rated synchronous speed [rad s�<sup>1</sup> ],*T*do<sup>0</sup> is a d-axis transient open circuit time constant [s], and *K*<sup>1</sup> through *K*<sup>6</sup> are linearization parameters. All parameters and variables in a Heffron-Phillips model are normalized, except for electric rotor angle *δ*Δ(*t*).
