**2. Mathematical model of TCR-FC type SVC by inclusion of fractional inductor and capacitor**

**Figure 1** shows the block diagram of the fractional order SVC-based power system. Before going into the details of the mathematical models of fractional order SVC, this section presents the definitions of the fractional calculus.

#### **2.1 Definitions of fractional derivatives and integrals**

Basic definition of the fractional operator can be denoted by a general fundamental operator *aD<sup>α</sup> <sup>t</sup>* as a generalization of the differential and integral operators, and it is defined as [28].

$$\,\_{a}D\_{t}^{a} \cong D^{a} = \begin{cases} \frac{d^{a}}{dt^{a}} & \,, \, R(a) > 0\\ 1 & \,, \, R(a) = 0\\ \int\_{a}^{t} (d\tau)^{-a} \,, \, R(a) < 0 \end{cases} \tag{1}$$

Here *α* represents the order of fractional operator and *R*ð Þ *α* represents set of real numbers. The following three definitions used for the general fractional operator are the Riemann–Liouville (RL) definition, the Caputo definition, and the Grunwald–Letnikov (GL) definition. The αth order Riemann–Liouville fractional derivative of function is given by [28].

**Figure 1.** *(a) Configuration of fractional SVC (TCR-FC), (b) single machine infinite bus configuration.*

a transmission network is subject to the variations due to the stochastic nature of variable inductive load demands. The bus voltages are much dependent on the reactive power demands [3]. The utilization of the SVC as shunt compensation for voltage management and related concepts are discussed by the authors of reference [4]. SVC has a simple structure, which provides controlled reactive power compensation over the transmission network. Other shunt compensator such as static synchronous compensator (STATCOM), which is based on the power electronics converter concept, is an advanced version of the FACTS controllers [5]. The power electronics-based shunt compensator can provide dynamic stability of the power network over a wider range as compared to the conventional SVC-based system. However, the closed loop control structure in case of the power electronics-based FACTS controller is more complex as compared to the SVC. In [6], the authors proposed a coordinated control strategy for the dynamic stability of the SVC-based power network. In order to enhance the transient stability, damping of power oscillations, and economic operation of the power network, several researchers have proposed the utilization of the unified power flow controller (UPFC) that simultaneously provide the series and shunt compensation over the transmission network [7, 8]; however, the structure of the closed loop control system is very complex for the UPFC. Moreover, the solution is very costly as compared to the conventional SVC-based power system. Different applications of the UPFC and STACOM controllers in the power networks have been discussed in detail by the authors of Ref. [9–11]. Apart from the applications of the FACTS controllers, another important issue is to choose the optimal location for the installment of these controllers [12]. A detailed review on the optimal placement of the FACTS devices is presented in [13]. Apart from the benefits of the FACTS controllers, feedback system plays vital role in achieving the control objectives. Several integer order robust control methods have been proposed for the SVC-based power system control problem. The detailed discussion of the power flow stability using closed loop FACTS controllers are discussed by the authors of [14–17]. In the above cited work, the authors proposed several control techniques such as adaptive backstepping, fuzzy logic; Lyapunovbased nonlinear controller and the H∞ Control. In [18], the authors have proposed robust variable structure control system for the SVC-based power system. An important issue in the design of any control system is how to select the optimal parameters. Different approaches are used in the literature such as PSO-based parameters optimization [19] and genetic algorithm-based parameters selection in [20–24]. In [25, 26], two different variants of the robust controllers have been proposed for VSC-based HVDC system. Fractional calculus is finding numerous applications in the area of the modeling and control of the dynamic systems. Fractional order systems have some additional advantages over integer order systems such as high degree of freedom in the parameters selection, robustness to noise, offer less chattering in the control signal, and wide stability margins [27]. In [28, 29], the authors proved that the fractional order systems are stable even when the systems roots lie in the right half of the complex plane. The formulation of fractional order controllers based on fractional order models offer additional advantages such as reduced computational costs and more robustness against uncertainties [30, 31]. Several authors proposed fractional order model-based controllers such as robotic manipulators [32], thermal modeling of buildings [33], aircraft [34], and pneumatic actuators [35]. Based on the above literature survey, it is concluded that the wider stability region concepts of the fractional order systems can be applied in several fields of interests. Particularly, the fractional order dynamics can be introduced to the SVC-based power system dynamic systems for enhanced stability margins. Fractional order filters consisting of inductive and capacitive elements have been practically realized and discussed in the literature.

*Control Theory in Engineering*

**268**

*Control Theory in Engineering*

$$\_aD\_t^af(t) = \frac{d^a}{dt^a}f(t) = \frac{1}{\Gamma(m-a)}\frac{d^m}{dt^m}\Big|\_a^t \frac{f(\tau)}{(t-\tau)^{a-m+1}}d\tau\tag{2}$$

Riemann–Liouville formula of the αth-order fractional integration can be written by

$$\, \_a D\_t^{-a} f(t) = I ^af(t) = \frac{1}{\Gamma(a)} \int\_a^t \frac{f(\tau)}{\left(t - \tau\right)^{1-a}} d\tau \tag{3}$$

susceptance (**B**). The block diagram of the system and single machine infinite bus

In Eq. (8), *BTCR* represents the susceptance of the thyristor-controlled reactor and *BC* is the susceptance of the fixed capacitor. The susceptance of the thyristorcontrolled reactor depends on the degree of firing angle of the thyristor and it can

*BTCR* <sup>¼</sup> *BL*ð Þ <sup>2</sup>*<sup>π</sup>* � <sup>2</sup>*<sup>α</sup>* <sup>þ</sup> sin 2ð Þ *<sup>α</sup>*

In Eq. (9), *BL* represents the susceptance of the inductor. If the capacitor and

Here *BSVCf* represents the fractional susceptance of the SVC configuration, *BCf* and *BTCRf* represent the fractional susceptance of the inductive and capacitive elements. From Eq. (9), the fractional inductance of the inductor is expressed as

*BLf* <sup>¼</sup> <sup>1</sup>

The voltage dynamics across a fractional inductor is written as

0

BB@

*Dγ IL VL*

2*πf L <sup>f</sup>*

¼ 1 *L f* 1

CCA

*Lf* <sup>¼</sup> *VL Dγ IL*

Combining Eqs. (12) and (11) yields the fractional order susceptance of the

*BSVC* ¼ *BTCR* þ *BC* (8)

*BSVCf* ¼ *BTCRf* þ *BCf* (10)

*<sup>π</sup>* (9)

(11)

(12)

For the integer type of SVC-based power system, the expression of the

*Stability region of (a) integer order system (b) fractional order system with order between 1 and 2*

*Static Var Compensator with Fractional Order Dynamics for Enhanced Stability and Control*

configuration is shown in **Figure 1a** and **b**.

*(c) fractional order systems with order between 0 and 1.*

*DOI: http://dx.doi.org/10.5772/intechopen.79615*

inductor are noninteger, then Eq. (8) is expressed as

susceptance is expressed as [18].

be expressed as

**Figure 2.**

*BLf* [31].

inductor as

**271**

Here *m* is the first integer larger than *α*, such that *m* � 1 *<α* < *m*, with *t* � *a* is the interval of integration and Γ is the Euler's Gamma function. The Caputo fractional derivative expression of a continuous function is expressed as

$$\,\_aD\_{L}^{a}f(t) = \begin{cases} \frac{1}{\Gamma(n-a)} \int\_a^t \frac{f''(\tau)}{\left(t-\tau\right)^{a-n+1}}d\tau & (n-1 \le a < n) \\\\ \frac{d^n}{dt^n}f(t) & (a=n) \end{cases} \tag{4}$$

The GL definition is given as:

$$\sideset{}{^{GL}}{}{\mathop{D}}^{a}\_{t}f}(t) = \lim\_{h \to 0} \frac{1}{h^{a}} \sum\_{j=0}^{\left[ (t-a)/h \right]} (-\mathbf{1})^{j} \binom{a}{j} f(t-jh) \tag{5}$$

Here, *h* represents the time step that is increasing with time and [.] is the integer part,

$$
\binom{a}{j} = \frac{\Gamma(a+1)}{\Gamma((j+1)\Gamma(a-j+1))}\tag{6}
$$

*Theorem 1: The following equation shows an autonomous system* [29]:

$$\_0D\_t^a \mathfrak{x} = A\mathfrak{x}, \ \mathfrak{x}(\mathbf{0}) = \mathfrak{x}\_0 \tag{7}$$

*Here α is differential order, x* ∈ *R<sup>n</sup> and A* ∈ *R<sup>n</sup>*�*n. The system is asymptotically stable if* j j argð Þ *eig A*ð Þ > *απ=*2*. If the condition is satisfied then the system converges toward zero like t*�*<sup>α</sup>. The system stability is guaranteed if* j j argð Þ *eig A*ð Þ <sup>≥</sup> *απ=*<sup>2</sup> *and the critical Eigen values that satisfy* j j argð Þ *eig A*ð Þ ¼ *απ=*2 *have geometric multiplicity. Moreover, the stable region of fractional system with* 0 < *λ* < 1 *is larger than that of* 1 < *λ* < 2, *λ* ¼ 1.

From **Figure 2**, it is clear that the fractional order systems with the fractional orders range of 0–1 have wider stability margins as compared to the integer order systems.

#### **2.2 Mathematical model of fractional SVC-based power system**

The goal of this work is to include the fractional components in the SVC mathematical model [18]. In order to modify the conventional design of TCR-FC type SVC, the fractional order inductor and capacitor are used to analyze and improve the system performance. Controllable part of the SVC is the susceptance, so the mathematical model derived in this section shall be represented in terms of the

*Static Var Compensator with Fractional Order Dynamics for Enhanced Stability and Control DOI: http://dx.doi.org/10.5772/intechopen.79615*

**Figure 2.**

*aD<sup>α</sup>*

*Control Theory in Engineering*

*aD<sup>α</sup> <sup>t</sup> f t*ðÞ¼

The GL definition is given as:

*GL <sup>a</sup> D<sup>α</sup>*

written by

part,

systems.

**270**

*<sup>t</sup> f t*ðÞ¼ *<sup>d</sup><sup>α</sup>*

*aD*�*<sup>α</sup>*

8 >>>><

>>>>:

*<sup>t</sup> f t*ðÞ¼ *I*

derivative expression of a continuous function is expressed as

1 Γð Þ *n* � *α*

*dn*

*<sup>t</sup> f t*ðÞ¼ lim *h*!0

> *α j* � �

> > 0*D<sup>α</sup>*

**2.2 Mathematical model of fractional SVC-based power system**

ð*t*

*a*

1 *hα*

*Theorem 1: The following equation shows an autonomous system* [29]:

*dt<sup>α</sup> f t*ðÞ¼ <sup>1</sup>

Γð Þ *m* � *α*

Riemann–Liouville formula of the αth-order fractional integration can be

*<sup>α</sup>f t*ðÞ¼ <sup>1</sup>

*f n* ð Þ*τ*

½ � ð Þ *t* X�*<sup>α</sup> <sup>=</sup><sup>h</sup> j*¼0

Here, *h* represents the time step that is increasing with time and [.] is the integer

<sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>* <sup>þ</sup> <sup>1</sup>

*Here α is differential order, x* ∈ *R<sup>n</sup> and A* ∈ *R<sup>n</sup>*�*n. The system is asymptotically stable if* j j argð Þ *eig A*ð Þ > *απ=*2*. If the condition is satisfied then the system converges toward zero like t*�*<sup>α</sup>. The system stability is guaranteed if* j j argð Þ *eig A*ð Þ <sup>≥</sup> *απ=*<sup>2</sup> *and the critical Eigen values that satisfy* j j argð Þ *eig A*ð Þ ¼ *απ=*2 *have geometric multiplicity. Moreover, the stable region of fractional system with* 0 < *λ* < 1 *is larger than that of* 1 < *λ* < 2, *λ* ¼ 1. From **Figure 2**, it is clear that the fractional order systems with the fractional orders range of 0–1 have wider stability margins as compared to the integer order

The goal of this work is to include the fractional components in the SVC mathematical model [18]. In order to modify the conventional design of TCR-FC type SVC, the fractional order inductor and capacitor are used to analyze and improve the system performance. Controllable part of the SVC is the susceptance, so the mathematical model derived in this section shall be represented in terms of the

Γð Þ *α*

Here *m* is the first integer larger than *α*, such that *m* � 1 *<α* < *m*, with *t* � *a* is the interval of integration and Γ is the Euler's Gamma function. The Caputo fractional

*dtn f t*ð Þ ð Þ *<sup>α</sup>* <sup>¼</sup> *<sup>n</sup>*

ð Þ �<sup>1</sup> *<sup>j</sup> <sup>α</sup> j* � �

ð*t*

*a*

*dm dtm* ð*t*

*a*

*f*ð Þ*τ*

ð Þ *<sup>t</sup>* � *<sup>τ</sup> <sup>α</sup>*�*n*þ<sup>1</sup> *<sup>d</sup><sup>τ</sup>* ð Þ *<sup>n</sup>* � <sup>1</sup> <sup>≤</sup> *<sup>α</sup>* < *<sup>n</sup>*

<sup>Γ</sup>ð Þ *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>* � *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> (6)

*<sup>t</sup> x* ¼ *Ax*, *x*ð Þ¼ 0 *x*<sup>0</sup> (7)

*f*ð Þ*τ*

ð Þ *<sup>t</sup>* � *<sup>τ</sup> <sup>α</sup>*�*m*þ<sup>1</sup> *<sup>d</sup><sup>τ</sup>* (2)

ð Þ *<sup>t</sup>* � *<sup>τ</sup>* <sup>1</sup>�*<sup>α</sup> <sup>d</sup><sup>τ</sup>* (3)

*f t*ð Þ � *jh* (5)

(4)

*Stability region of (a) integer order system (b) fractional order system with order between 1 and 2 (c) fractional order systems with order between 0 and 1.*

susceptance (**B**). The block diagram of the system and single machine infinite bus configuration is shown in **Figure 1a** and **b**.

For the integer type of SVC-based power system, the expression of the susceptance is expressed as [18].

$$B\_{\rm SVC} = B\_{\rm TCR} + B\_{\rm C} \tag{8}$$

In Eq. (8), *BTCR* represents the susceptance of the thyristor-controlled reactor and *BC* is the susceptance of the fixed capacitor. The susceptance of the thyristorcontrolled reactor depends on the degree of firing angle of the thyristor and it can be expressed as

$$B\_{\rm TCR} = \frac{B\_L(2\pi - 2a + \sin(2a))}{\pi} \tag{9}$$

In Eq. (9), *BL* represents the susceptance of the inductor. If the capacitor and inductor are noninteger, then Eq. (8) is expressed as

$$B\_{SV\mathbb{Q}'} = B\_{T\mathbb{C}\mathbb{R}\mathbb{f}} + B\_{\mathbb{Q}'} \tag{10}$$

Here *BSVCf* represents the fractional susceptance of the SVC configuration, *BCf* and *BTCRf* represent the fractional susceptance of the inductive and capacitive elements. From Eq. (9), the fractional inductance of the inductor is expressed as *BLf* [31].

$$B\_{Lf} = \frac{1}{2\pi f \, L\_f} \tag{11}$$

The voltage dynamics across a fractional inductor is written as

$$\begin{pmatrix} \frac{D^\circ I\_L}{V\_L} = \frac{1}{L\_f} \\\\ L\_f = \frac{V\_L}{D^\circ I\_L} \end{pmatrix} \tag{12}$$

Combining Eqs. (12) and (11) yields the fractional order susceptance of the inductor as

$$B\_{Lf} = \frac{1}{2\pi f \frac{V\_L}{D^\gamma I\_L}}\tag{13}$$

In Eq. (20), *δ* represents the power angle of the generator, *w* is the angular speed, *Pm* is the input mechanical power to the generator, *D* is the damping component, *M* is the moment of inertia, *Pd* represents the load demand over time,

*Static Var Compensator with Fractional Order Dynamics for Enhanced Stability and Control*

The control objective is to formulate a robust control system for the fractional order SVC-based power system that must behave as insensitive to the disturbances and uncertainties, thus ensure a stabilized voltage over the transmission lines. Thus,

*BSVC*, then the fractional order sliding manifold is

*e* þ *C*2*e*\_ (21)

*<sup>e</sup>* <sup>þ</sup> *<sup>C</sup>*2*Dαe*\_ (22)

*τX* þ

1 *τ kPd*

*<sup>e</sup>* � *KsD*�*<sup>α</sup>*sgnð Þ*<sup>S</sup>*

≤*τ*∣*S*∣ (25)

� � � � � (24)

(26)

*BSVC* � *<sup>V</sup>*1*V*2cos*<sup>δ</sup>*

*C*2

� � � � � *D*�*α=*<sup>2</sup>

. Applying operator *D<sup>α</sup>* to the Lyapunov

*Dj SD<sup>α</sup>*�*<sup>j</sup> S*

*<sup>D</sup><sup>α</sup><sup>V</sup>* <sup>≤</sup> *SD<sup>α</sup><sup>S</sup>* <sup>þ</sup> *<sup>τ</sup>*∣*S*<sup>∣</sup> (27)

� � (23)

*Pd* � *<sup>V</sup>*\_ <sup>2</sup>*<sup>r</sup>* � *<sup>C</sup>*<sup>1</sup>

*Dj SD<sup>α</sup>*�*<sup>j</sup> S*

Γð Þ 1 þ *α* Γð Þ 1 � *j* þ *α* Γð Þ 1 þ *j*

Here *τ* is a positive constant. To prove the stability of the closed loop system, the

� �

the control objective is the regulation of the bus voltage *V*<sup>2</sup> regardless of the variation in the generation side and load side. Let *<sup>e</sup>* <sup>¼</sup> *<sup>V</sup>*2*<sup>r</sup>* � *<sup>V</sup>*2, *<sup>e</sup>*\_ <sup>¼</sup> *<sup>V</sup>*\_ <sup>2</sup>*<sup>r</sup>* � *<sup>V</sup>*\_ <sup>2</sup> and

*<sup>S</sup>* <sup>¼</sup> *<sup>C</sup>*1*D*�*α=*<sup>2</sup>

and *V*\_ <sup>2</sup> represents the first derivative of the command signal. Differentiating

*<sup>D</sup>α<sup>S</sup>* <sup>¼</sup> *<sup>C</sup>*1*Dα=*<sup>2</sup>

1 *τ V*<sup>2</sup> 2 *<sup>X</sup>* � <sup>1</sup> *τ V*<sup>2</sup> 2 *Dγ*

*V*1*V*2cos*δ <sup>τ</sup><sup>X</sup>* � *<sup>k</sup>*

Γð Þ 1 þ *α* Γð Þ 1 � *j* þ *α* Γð Þ 1 þ *j*

*j*¼1

� � � � � 2 *S*2

*τ*

In the above expressions *e* represents the error signal, *V*2*<sup>r</sup>* is the reference signal

*V*<sup>1</sup> is the sending end voltage, and *V*<sup>2</sup> is the voltage over infinite bus.

**3. Controller formulation and stability proof**

*DOI: http://dx.doi.org/10.5772/intechopen.79615*

By combining Eqs. (20) and (22), one obtains

*<sup>e</sup>* <sup>þ</sup> *<sup>C</sup>*2*D<sup>α</sup> <sup>V</sup>*\_ <sup>2</sup>*<sup>r</sup>* <sup>þ</sup>

2 *τX* þ

the control signal is *BSVCf* <sup>¼</sup> *<sup>D</sup><sup>γ</sup>*

Eq. (21) with respect to *D<sup>α</sup>* yields

The control law is derived as

<sup>2</sup> � *<sup>V</sup>*<sup>2</sup>

The following inequality holds [30].

� � � � �

Lyapunov function is chosen as *<sup>V</sup>* <sup>¼</sup> <sup>1</sup>

X∞ *j*¼1

*<sup>D</sup><sup>α</sup><sup>V</sup>* <sup>¼</sup> *SD<sup>α</sup><sup>S</sup>* <sup>þ</sup> <sup>X</sup><sup>∞</sup>

Using Eq. (25), we can simplify Eq. (26) as

*V*<sup>2</sup>

defined as

*<sup>D</sup>α<sup>S</sup>* <sup>¼</sup> *<sup>C</sup>*1*Dα=*<sup>2</sup>

*BSVC* ¼ � *<sup>τ</sup>*

*Dγ*

function yields

**273**

By multiplying the operator *D<sup>γ</sup>* on both hand sides of Eq. (13) yields

$$D^\gamma B\_{L\circ} = D^\gamma \begin{bmatrix} \mathbf{1} \\ \frac{\mathbf{1}}{2\pi \mathbf{f}' \frac{V\_L}{D^\gamma I\_L}} \end{bmatrix} \tag{14}$$

Using Eq. (14), and by equating *D<sup>γ</sup> BLf* <sup>¼</sup> *<sup>D</sup><sup>γ</sup> BL*, Eq. (9) in fractional order sense is written as

$$B\_{\rm TCR} = \frac{D^{\prime}B\_{L}(2\pi - 2a + \sin(2a))}{\pi} \tag{15}$$

The voltage and the current dynamics across the fractional capacitor is written as *Vc* <sup>¼</sup> <sup>1</sup> *C <sup>f</sup> D*�*<sup>γ</sup> Ic*. Here, *Vc* represents the voltage, *Ic* is the current across the capacitor, *Cf* is the fractional capacitance, and *D*�*<sup>γ</sup>* represents the fractional integrator. Now, the current–voltage dynamics across the fractional capacitor are expressed as [31].

$$\mathbf{C}\_{f} = \frac{I\_{c}}{D^{\mathbf{r}} \mathbf{V}\_{c}} \tag{16}$$

The fractional capacitive susceptance is written as

$$\begin{pmatrix} B\_{\text{Cf}} = 2\mathfrak{af}\,\,\mathbf{C}\_f \\\\ B\_{\text{Cf}} = 2\mathfrak{af}\,\frac{I\_c}{D^\circ V\_c} \end{pmatrix} \tag{17}$$

Simplifying Eq. (17) yields

$$D^{\gamma}B\_{\widetilde{\mathcal{G}}} = 2\pi \mathfrak{f}' D^{\gamma} \left(\frac{I\_c}{D^{\gamma}V\_c}\right) \tag{18}$$

By combining Eqs. (10), (15) and (18), one obtains

$$B\_{\rm SVQ\overline{f}} = \frac{D^{\rm r}B\_{L}(2\pi - 2a + \sin(2a)) + \pi D^{\rm r}B\_{\rm C\overline{f}}}{\pi} \tag{19}$$

The modified state space representation of the power system with the fractional order SVC-based system configurations written as

$$\begin{aligned} \dot{\delta} &= w\\ \dot{w} &= \frac{1}{M} \left( P\_m - \frac{V\_1 V\_2 \sin \delta}{X} - Dw \right) \\ \dot{V}\_2 &= \frac{1}{\tau} \left( \frac{-V\_2}{X} + V\_2^2 \left( \frac{D^\gamma B\_L (2\pi - 2a + \sin(2a)) + \pi D^\gamma B\_{\zeta^\gamma}}{\pi} \right) + \frac{V\_1 V\_2 \cos \delta}{X} - kP\_d \right) \\ D'B\_L &= \frac{1}{\tau} \left( V\_{\eta f} - V\_2 \right) \end{aligned} \tag{20}$$

*Static Var Compensator with Fractional Order Dynamics for Enhanced Stability and Control DOI: http://dx.doi.org/10.5772/intechopen.79615*

In Eq. (20), *δ* represents the power angle of the generator, *w* is the angular speed, *Pm* is the input mechanical power to the generator, *D* is the damping component, *M* is the moment of inertia, *Pd* represents the load demand over time, *V*<sup>1</sup> is the sending end voltage, and *V*<sup>2</sup> is the voltage over infinite bus.
