**1. Introduction**

Synchronous generators are the most important electrical machines. They produce the majority of the world's electricity. In 2017, global electricity production was 25,721 TWh [1]. Assuming that the share of solar thermal sources is negligible compared to the share of solar photovoltaic sources, it can be estimated that about 98.2% of the total global energy is produced by electric generators. After analyzing the data, it can be estimated that synchronous generators produce 93.8% of the world's electricity and induction generators 4.4% of the total production of the world's electricity. The estimate is based on data for 2017. These, and also the following data, are obtained from statistics reports of the International Energy Agency [1].

An additional important point is that electricity trading and, thus, long-distance transmission of electricity are increasing significantly. In 2017, OECD countries produced 11,051 TWh of electricity, with a trading volume of 408 TWh,

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 to 2.1% growth in overall electricity supply.

For a detailed analysis of the benefits of the advanced PSS, a mathematical model of the synchronous generator is necessary. In this work, we first present the mathematical model of the synchronous generator connected to the power system, which is convenient for analysis of the physical characteristics of the power system and is appropriate at the same time for the controller design and synthesis. We focus our work on the analysis of a system where a single synchronous generator is connected to an infinite bus. In Section 3, we attempt to estimate the amount and dynamics of the oscillations in the power systems. A thorough analysis is made and presented for the first time. The conventional PSS control system is presented in Section 4. By means of the derived mathematical model of the synchronous generator, we estimate the improvement of the power system damping due to implementation of the conventional PSS. From the analysis, it is evident that conventional PSS does not assure optimal damping in the entire operating range. Therefore, advanced control theories for PSS design and synthesis are presented in Sections 5 and 6. In Section 5, the robust control system theory is used for PSS design. The suitable direct adaptive control theory is presented in Section 6. The PSS control system developed on the basis of the presented theories and the results of the implementation of the advanced control theories for PSS design and synthesis

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

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

**2. Mathematical model of the synchronous generator connected**

model is described by sets of algebraic equations (Eqs. (1)–(10)) [5]:

*λ*AQ ðÞ¼ *t L*MQ

1 *l*d

1 *l*q

> 1 *l*F

*i*dðÞ¼ *t*

*i*qðÞ¼ *t*

*i*FðÞ¼ *t*

*λ*dð Þ*t l*d þ *λ*Fð Þ*t l*F þ *λ*Dð Þ*t l*D (1)

> *λ*qð Þ*t l*q þ *λ*<sup>Q</sup> ð Þ*t l*Q

(2)

ð Þ *λ*dðÞ�*t λ*ADð Þ*t* (3)

*<sup>λ</sup>*qðÞ�*<sup>t</sup> <sup>λ</sup>*AQ ð Þ*<sup>t</sup>* (4)

ð Þ *λ*FðÞ�*t λ*ADð Þ*t* (5)

*λ*ADðÞ¼ *t L*MD

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

are shown in Section 7.

**241**

**to the power system**

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 saving energy, it makes sense to reduce these losses.

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 field winding.

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.

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

For a detailed analysis of the benefits of the advanced PSS, a mathematical model of the synchronous generator is necessary. In this work, we first present the mathematical model of the synchronous generator connected to the power system, which is convenient for analysis of the physical characteristics of the power system and is appropriate at the same time for the controller design and synthesis. We focus our work on the analysis of a system where a single synchronous generator is connected to an infinite bus. In Section 3, we attempt to estimate the amount and dynamics of the oscillations in the power systems. A thorough analysis is made and presented for the first time. The conventional PSS control system is presented in Section 4. By means of the derived mathematical model of the synchronous generator, we estimate the improvement of the power system damping due to implementation of the conventional PSS. From the analysis, it is evident that conventional PSS does not assure optimal damping in the entire operating range. Therefore, advanced control theories for PSS design and synthesis are presented in Sections 5 and 6. In Section 5, the robust control system theory is used for PSS design. The suitable direct adaptive control theory is presented in Section 6. The PSS control system developed on the basis of the presented theories and the results of the implementation of the advanced control theories for PSS design and synthesis are shown in Section 7.
