**Abstract**

This chapter presents a new theoretical approach for a novel static Var compensator (SVC) system using fractional order calculus. The thyristor-controlled reactor (TCR) and fixed capacitor are assumed to be noninteger. A state space model is derived for the fractional SVC and a novel fractional order sliding surface is proposed, based on which a fractional order controller is derived for bus voltage stabilization with variable loading. Keeping in view the enhanced stability margins of the system, the parameters of the control system are optimized using Simulink response optimization toolbox. The stability and the convergence proof of the control system is verified using fractional order Lyapunov theorem. The effectiveness of the proposed control scheme is verified using numerical simulations.

**Keywords:** static Var compensator, transmission line reactance, sliding mode controller, voltage stability, fractional inductor, fractional capacitor

## **1. Introduction**

In recent times, the demand and the importance of a robust transmission network is increasing at an extraordinary rate. With the advancement of the society, it is vital to ensure the enough power availability to both the common consumer and the industry. The conventional transmission system has several disadvantages that include the lack of its operation on full load capacity, performance degradation under heavy inductive loads, and voltage sagging problems. Moreover, the conventional transmission system is not efficient in resolving and isolating the fault over the lines because most of the power systems are controlled through mechanical means. A more preferable alternative to the fixed mechanical control of the transmission network is the flexible AC transmission system (FACTS). All FACTS controllers are operated in a closed loop fashion and a detailed comparison of several control schemes is presented in the work given in [1].

The shunt compensators are preliminarily used for the reactive power compensation over the transmission network system. The reactive power is supplied or absorbed using the electronic drive-based shunt flexible ac transmission system (FACTS) controllers including the static Var compensator (SVC) and STATCOM [2, 3]. Over a transmission network bus voltage stability is a prime issue and it needs to be addressed for the overall stability of the power network. The bus voltage over 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.

Fractional order tunable resonators and filters have been practically realized and the details are given in [36]. Electronically tunable all pass filter has been proposed and

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

This chapter proposes static Var compensator (SVC) using fractional order inductive and capacitive elements. The idea is exploited theoretically and the dynamic equation of the fractional order susceptance is derived. Finally, a novel fractional order sliding surface is proposed and a feedback controller is derived for the bus voltage stabilization problem of power transmission network. The parameters of the proposed control scheme are tuned using Simulink response optimization

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

**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

Basic definition of the fractional operator can be denoted by a general funda-

*dα*

8 >>>>><

> Ð*t a*

>>>>>:

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

*<sup>t</sup>* as a generalization of the differential and integral operators,

(1)

*dt<sup>α</sup>* , *<sup>R</sup>*ð Þ *<sup>α</sup>* <sup>&</sup>gt;<sup>0</sup> 1 , *R*ð Þ¼ *α* 0

ð Þ *<sup>d</sup><sup>τ</sup>* �*<sup>α</sup>* , *<sup>R</sup>*ð Þ *<sup>α</sup>* <sup>&</sup>lt;<sup>0</sup>

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

SVC, this section presents the definitions of the fractional calculus.

**2.1 Definitions of fractional derivatives and integrals**

*aD<sup>α</sup>*

derivative of function is given by [28].

*<sup>t</sup>* ffi *<sup>D</sup><sup>α</sup>* <sup>¼</sup>

discussed in [37].

**inductor and capacitor**

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

mental operator *aD<sup>α</sup>*

**Figure 1.**

**269**

and it is defined as [28].

tool box.

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

Fractional order tunable resonators and filters have been practically realized and the details are given in [36]. Electronically tunable all pass filter has been proposed and discussed in [37].

This chapter proposes static Var compensator (SVC) using fractional order inductive and capacitive elements. The idea is exploited theoretically and the dynamic equation of the fractional order susceptance is derived. Finally, a novel fractional order sliding surface is proposed and a feedback controller is derived for the bus voltage stabilization problem of power transmission network. The parameters of the proposed control scheme are tuned using Simulink response optimization tool box.
