**6. Adaptive PSS**

Many examples with utilization of different adaptive techniques for realization of PSS can be found in publications. The majority of PSS realizations are based on usage of indirect adaptive control, where explicit identification of a mathematical model of a synchronous generator is needed to be carried out [16, 17]. A transparent structure of the adaptive control system with the separated identification algorithm and the control law represents an advantage of indirect adaptive PSS. There are significantly less publications available where usage of direct adaptive control for PSS is presented [18]. The methods of direct adaptive control are more difficult to be utilized for the conventional PSS structure than those for the indirect adaptive control. However, their advantage is in not requiring explicit identification of the SG, and they are, therefore, computationally less demanding. In this article, the developed robust PSS will be compared with direct adaptive PSS which was studied in detail in [2].

The theoretical foundation for the used direct adaptive PSS is represented by a theory of model reference adaptive control for almost strictly positive real plants.

The implemented direct adaptive control is considered for the controlled plant, which is described by

$$\dot{\mathbf{x}}\_{\mathbf{p}}(t) = \mathbf{A}\_{\mathbf{p}} \mathbf{x}\_{\mathbf{p}}(t) + \mathbf{B}\_{\mathbf{p}} \mathbf{u}\_{\mathbf{p}}(t) \tag{31}$$

$$\mathbf{y}\_{\mathbf{p}}(t) = \mathbf{C}\_{\mathbf{p}} \mathbf{x}\_{\mathbf{p}}(t) \tag{32}$$

where **<sup>x</sup>**pð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*<sup>n</sup>* is the controlled plant state-space vector, **<sup>u</sup>**pð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*m*is the controlled plant input vector, **<sup>y</sup>**pð Þ*<sup>t</sup>* <sup>∈</sup> <sup>ℜ</sup>*<sup>r</sup>* is the controlled plant output vector, and

**A***p*, **B***p*, and **C***<sup>p</sup>* are the matrices of the appropriate dimensions. It is assumed that:


The reference model is described by

$$
\dot{\mathbf{x}}\_{\rm m}(t) = \mathbf{A}\_{\rm m} \mathbf{x}\_{\rm m}(t) + \mathbf{B}\_{\rm m} \mathbf{u}\_{\rm m}(t) \tag{33}
$$

$$\mathbf{y}\_{\mathbf{m}}(t) = \mathbf{C}\_{\mathbf{m}} \mathbf{x}\_{\mathbf{m}}(t) \tag{34}$$

where **x**m(*t*) is the model state vector, **u**m(*t*) is the model command vector, **y**m(*t*) is the model output vector, and **A**m, **B**m, and **C**<sup>m</sup> are matrices of appropriate dimensions. The model is assumed to be stable. The dimension of the model state may be less than the dimension of the plant state.

The output tracking error is defined as

$$\mathbf{e}\_{\mathbf{y}}(t) = \mathbf{y}\_{\mathbf{m}}(t) - \mathbf{y}\_{\mathbf{p}}(t) \tag{35}$$

The control **u**p(*t*) for the plant output vector **y**p(t) to approximate "reasonably well" the output of the reference model **y**m(*t*) without explicit knowledge of **A**p, **B**p, and **C**<sup>p</sup> is generated by the adaptive algorithm:

$$\mathbf{u\_p}(t) = \mathbf{K\_e}(t)\mathbf{e\_y}(t) + \mathbf{K\_x}(t)\mathbf{x\_m}(t) + \mathbf{K\_u}(t)\mathbf{u\_m}(t) \tag{36}$$

$$\mathbf{u}\_{\mathbf{p}}(t) = \mathbf{K}(t)\mathbf{r}(t) \tag{37}$$

where

system described with (Eqs. (21), (22)), there exists matrix **D**, which ensures the

determines the equation of discontinuous sliding surfaces (Eq. (27)).

In the first stage of design of the sliding mode, we chose the desired eigenvalues of the system described with (Eq. (27)). From the desired eigenvalues, we determined matrix **D** as the solution to the (*n-m*)-th-order eigenvalue task. Matrix **D**

The second stage of the design procedure is the selection of the discontinuous control law, such that the sliding mode always arises at manifold **s**(*t*) = 0, which is equivalent to the stability of the origin in *m*-dimensional space **s**(*t*). The dynamics

<sup>¼</sup> **Ex**RFðÞþ*<sup>t</sup>* **<sup>B</sup>**RF2**u**pð Þ*<sup>t</sup>* (28)

RF2 sgn **s**ð Þ*t* (29)

**s**\_ðÞ¼ *t* **Ex**RFðÞ�*t* gj j **x**RFð Þ*t* sgn **s**ð Þ*t* (30)

An appropriate choice of the control law represents the discontinuous control

**s**\_ðÞ¼ *t* ½ � **DA**RF11 þ **A***RF*<sup>21</sup> **x**RF1ðÞþ*t* ½ � **DA**RF12 þ **A**RF22 **x**RF2ðÞþ*t* **B**RF2**u**ð Þ*t*

**<sup>u</sup>**pðÞ¼� *<sup>t</sup>* <sup>g</sup>j j **<sup>x</sup>**RFð Þ*<sup>t</sup>* **<sup>B</sup>**�<sup>1</sup>

where j j **x**RFð Þ*t* is the sum of vector **x**RFð Þ*t* component moduli and g is the

signs. It means that the sliding mode will occur on a discontinuity surface. The influence of discontinuity of the control signal is reduced by varying the amplitude

There exists such positive value of g that the functions **s**ð Þ*t* and **s**\_ð Þ*t* have different

Many examples with utilization of different adaptive techniques for realization of PSS can be found in publications. The majority of PSS realizations are based on usage of indirect adaptive control, where explicit identification of a mathematical model of a synchronous generator is needed to be carried out [16, 17]. A transparent structure of the adaptive control system with the separated identification algorithm and the control law represents an advantage of indirect adaptive PSS. There are significantly less publications available where usage of direct adaptive control for PSS is presented [18]. The methods of direct adaptive control are more difficult to be utilized for the conventional PSS structure than those for the indirect adaptive control. However, their advantage is in not requiring explicit identification of the SG, and they are, therefore, computationally less demanding. In this article, the developed robust PSS will be compared with direct adaptive PSS which was studied in detail in [2].

The theoretical foundation for the used direct adaptive PSS is represented by a theory of model reference adaptive control for almost strictly positive real plants. The implemented direct adaptive control is considered for the controlled plant,

**x**\_ <sup>p</sup>ðÞ¼ *t* **A**p**x**pðÞþ*t* **B**p**u**pð Þ*t* (31)

**y**pðÞ¼ *t* **C**p**x**pð Þ*t* (32)

desired eigenvalues of the system in (Eq. (27)).

on the **s**(*t*) space are described by the equation

The selected discontinuous control leads to

described with

*Automation and Control*

of the control signal.

**6. Adaptive PSS**

which is described by

**260**

constant.

$$\mathbf{K}(t) = [\mathbf{K}\_{\mathbf{e}}(t), \mathbf{K}\_{\mathbf{x}}(t), \mathbf{K}\_{\mathbf{u}}(t)] \tag{38}$$

$$\mathbf{r}^{\mathbf{T}}(t) = \left[\mathbf{e}\_{\mathbf{y}}^{\mathbf{T}}(t), \mathbf{x}\_{\mathbf{m}}^{\mathbf{T}}(t), \mathbf{u}\_{\mathbf{m}}^{\mathbf{T}}(t)\right]. \tag{39}$$

The adaptive gains **K**(*t*) are obtained as a combination of the "proportional" and "integral" terms

$$\mathbf{K}(t) = \mathbf{K}\_{\mathbf{P}}(t) + \mathbf{K}\_{\mathbf{l}}(t) \tag{40}$$

$$\mathbf{K}\_{\mathbf{P}}(t) = \mathbf{e}\_{\mathbf{y}}(t)\mathbf{r}^{T}(t)\mathbf{T} \tag{41}$$

$$\dot{\mathbf{K}}\_{\rm I}(t) = \mathbf{e}\_{\rm y}(t)\mathbf{r}^{T}(t)\overline{\mathbf{T}} - \sigma \mathbf{K}\_{\rm l}(t) \tag{42}$$

where σ term is introduced in order to avoid divergence of the integral gains in the presence of disturbance and **T** and **T** are positive definite and positive semidefinite adaptation coefficient matrices, respectively.

The necessary condition for asymptotic tracking when **u**m(*t*) is a step command is that the controlled plant is almost strictly positive real (ASPR) [19]. If the controlled plant is not ASPR, the augmenting of the plant with a feedforward

compensator is suggested, such that the augmented plant is ASPR. In this case, the previously described adaptive controller may be utilized.

For the non-ASPR plant described by the transfer matrix

$$\mathbf{G}\_{\mathbf{p}}(\boldsymbol{\varepsilon}) = \mathbf{C}\_{\mathbf{p}} \left(\boldsymbol{\kappa}\mathbf{I} - \mathbf{A}\_{\mathbf{p}}\right)^{-1} \mathbf{B}\_{\mathbf{p}} \tag{43}$$

the feedforward compensator is defined by the strictly proper transfer function matrix **G**ff(*s*) with the realization

$$\dot{\mathbf{s}}\_{\mathbf{p}}(t) = \mathbf{A}\_{\mathbf{s}} \mathbf{s}\_{\mathbf{p}}(t) + \mathbf{B}\_{\mathbf{s}} \mathbf{u}\_{\mathbf{p}}(t) \tag{44}$$

$$\mathbf{r}\_{\mathbf{p}}(t) = \mathbf{D}\_{\mathbf{s}} \mathbf{s}\_{\mathbf{p}}(t) \tag{45}$$

Instead of the plant output **y**p(*t*), augmented output **z**p(*t*) is to be controlled:

$$\mathbf{z\_p}(t) = \mathbf{y\_p}(t) + \mathbf{r\_p}(t) \tag{46}$$

The augmented system is defined as

$$\mathbf{G}\_{\mathbf{a}}(\boldsymbol{\kappa}) = \mathbf{G}\_{\mathbf{p}}(\boldsymbol{\kappa}) + \mathbf{G}\_{\mathbf{ff}}(\boldsymbol{\kappa}) \tag{47}$$

Feedforward compensator **G**ff(*s*) is an inverse of a (fictitious) stabilizing controller for the plant and must be selected such that the resulting relative degree of augmented plant **G**a(*s*) is indeed 1. For example, if SISO plant Gp(*s*) is stabilizable by a PD controller, one can use its inverse in a manner that is just a simple firstorder low-pass filter.

#### **7. Results**

The effectiveness of the proposed sliding mode PSS and direct adaptive PSS was tested with the simulations of the seventh-order nonlinear model of the synchronous generator in the entire operating range, numerically, as well as experimentally, in the laboratory.

#### **7.1 Robust PSS**

A block diagram of the sliding mode PSS is shown in **Figure 21**.

A sliding mode controller requires measurements of three synchronous generator's quantities: electrical power, rotor speed, and terminal voltage. Input filters are low-pass filters to eliminate the measured noise. From these measured variables, the state-space variables for the regular form model are calculated by means of state transformation. State transformation is carried out by Eqs. (24) and (25). The obtained regular form state-space variables are used in the control law described with Eqs. (26) and (29). The output of the discontinuous control law is conducted in the limiter. Hard type saturation of the PSS output was utilized, with a limited value of �35% of the value of a nominal rotor excitation voltage. The set value represents a limitation in a real excitation system.

For a synchronous generator with the data listed in Section 3, we selected desired eigenvalues *λ*1,2 ¼ �2 for the system in (Eq. (27)). The following control law parameters were calculated [2]:

$$\mathbf{D} = \begin{bmatrix} 4 & 4 \end{bmatrix} \quad \mathbf{B}\_{\text{RF2}}^{-1} = -0.06 \quad \mathbf{g} = \mathbf{350} \tag{48}$$

*7.1.1 Nominal operating point*

**Figure 22.**

**Figure 21.**

*Block diagram of the sliding mode PSS.*

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

**Figure 23.**

**263**

*[pu], with robust PSS.*

**Figures 22**–**24** show the responses of the seventh-order nonlinear model of the considered 160 MVA synchronous generator equipped with an excitation system

*Rotor angle* δ*(*t*) [pu] at nominal operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu], with robust PSS.*

*Electrical power* P*e(*t*) [pu] and rotor speed* ω*(*t*) [pu] at nominal operating point* P *= 1.0 [pu] and* Q *= 0.62*

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

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

compensator is suggested, such that the augmented plant is ASPR. In this case, the

�<sup>1</sup>

**s**\_pðÞ¼ *t* **A**s**s**pðÞþ*t* **B**s**u**pð Þ*t* (44)

**r**pðÞ¼ *t* **D**s**s**pð Þ*t* (45)

**z**pðÞ¼ *t* **y**pðÞþ*t* **r**pð Þ*t* (46)

**G**aðÞ¼ *s* **G**pðÞþ*s* **G**ffð Þ*s* (47)

the feedforward compensator is defined by the strictly proper transfer function

Instead of the plant output **y**p(*t*), augmented output **z**p(*t*) is to be controlled:

Feedforward compensator **G**ff(*s*) is an inverse of a (fictitious) stabilizing controller for the plant and must be selected such that the resulting relative degree of augmented plant **G**a(*s*) is indeed 1. For example, if SISO plant Gp(*s*) is stabilizable by a PD controller, one can use its inverse in a manner that is just a simple first-

The effectiveness of the proposed sliding mode PSS and direct adaptive PSS was tested with the simulations of the seventh-order nonlinear model of the synchronous generator in the entire operating range, numerically, as well as experimentally,

A sliding mode controller requires measurements of three synchronous generator's quantities: electrical power, rotor speed, and terminal voltage. Input filters are low-pass filters to eliminate the measured noise. From these measured variables, the state-space variables for the regular form model are calculated by means of state transformation. State transformation is carried out by Eqs. (24) and (25). The obtained regular form state-space variables are used in the control law described with Eqs. (26) and (29). The output of the discontinuous control law is conducted in the limiter. Hard type saturation of the PSS output was utilized, with a limited value of �35% of the value of a nominal rotor excitation voltage. The set value represents

For a synchronous generator with the data listed in Section 3, we selected desired eigenvalues *λ*1,2 ¼ �2 for the system in (Eq. (27)). The following control law

RF2 ¼ �0*:*06 *g* ¼ 350 (48)

**<sup>D</sup>** <sup>¼</sup> ½ � 4 4 **<sup>B</sup>**�<sup>1</sup>

A block diagram of the sliding mode PSS is shown in **Figure 21**.

**B**<sup>p</sup> (43)

**G**pðÞ¼ *s* **C**<sup>p</sup> *s***I** � **A**<sup>p</sup>

previously described adaptive controller may be utilized. For the non-ASPR plant described by the transfer matrix

matrix **G**ff(*s*) with the realization

*Automation and Control*

The augmented system is defined as

order low-pass filter.

**7. Results**

in the laboratory.

**7.1 Robust PSS**

a limitation in a real excitation system.

parameters were calculated [2]:

**262**

**Figure 21.**

*Block diagram of the sliding mode PSS.*

**Figure 22.**

*Electrical power* P*e(*t*) [pu] and rotor speed* ω*(*t*) [pu] at nominal operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu], with robust PSS.*

**Figure 23.** *Rotor angle* δ*(*t*) [pu] at nominal operating point* P *= 1.0 [pu] and* Q *= 0.62 [pu], with robust PSS.*

## *7.1.1 Nominal operating point*

**Figures 22**–**24** show the responses of the seventh-order nonlinear model of the considered 160 MVA synchronous generator equipped with an excitation system
