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

By Nasim Ullah, Anwar Ali, Haider Ali, Asier Ibeas and Jorge Herrera

Submitted: May 23rd 2018Reviewed: June 18th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.79615

Downloaded: 195

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, 10, 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, 15, 16, 17]. In the above cited work, the authors proposed several control techniques such as adaptive backstepping, fuzzy logic; Lyapunov-based 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, 21, 22, 23, 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 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.

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.

Figure 1.

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

2.1. Definitions of fractional derivatives and integrals

Basic definition of the fractional operator can be denoted by a general fundamental operator aDtαas a generalization of the differential and integral operators, and it is defined as [28].

aDtαDα=dαdtα,Rα>01,Rα=0atα,Rα<0E1

Here αrepresents the order of fractional operator and Rαrepresents 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].

aDtαft=dαdtαft=1Γmαdmdtmatfτtταm+1E2

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

aDtαft=Iαft=1Γαatfτtτ1αE3

Here mis the first integer larger than α, such that m1<α<m, with tais the interval of integration and Γis the Euler’s Gamma function. The Caputo fractional derivative expression of a continuous function is expressed as

aDtαft=1Γnαatfnτtταn+1n1  α < ndndtnftα=nE4

The GL definition is given as:

aGLDtαft=limh01hαj=0tα/h1jαjftjhE5

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

αj=Γα+1Γj+1Γαj+1E6

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

0Dtαx=Ax,x0=x0E7

Here α is differential order, x  Rnand A  Rn×n. The system is asymptotically stable if argeigA > απ/2. If the condition is satisfied then the system converges toward zero like tα. The system stability is guaranteed if argeigA  απ/2and the critical Eigen values that satisfy argeigA=απ/2have geometric multiplicity. Moreover, the stable region of fractional system with 0 < λ < 1is larger than that of 1 < λ < 2, λ=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.

Figure 2.

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.

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 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].

BSVC=BTCR+BCE8

In Eq. (8), BTCRrepresents the susceptance of the thyristor-controlled reactor and BCis the susceptance of the fixed capacitor. The susceptance of the thyristor-controlled reactor depends on the degree of firing angle of the thyristor and it can be expressed as

BTCR=BL2π2α+sin2απE9

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

BSVCf=BTCRf+BCfE10

Here BSVCfrepresents the fractional susceptance of the SVC configuration, BCfand BTCRfrepresent the fractional susceptance of the inductive and capacitive elements. From Eq. (9), the fractional inductance of the inductor is expressed as BLf[31].

BLf=12πfLfE11

The voltage dynamics across a fractional inductor is written as

DγILVL=1LfLf=VLDγILE12

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

BLf=12πfVLDγILE13

By multiplying the operator Dγon both hand sides of Eq. (13) yields

DγBLf=Dγ12πfVLDγILE14

Using Eq. (14), and by equating DγBLf=DγBL, Eq. (9) in fractional order sense is written as

BTCRf=DγBL2π2α+sin2απE15

The voltage and the current dynamics across the fractional capacitor is written as Vc=1CfDγIc. Here, Vcrepresents the voltage, Icis the current across the capacitor, Cfis the fractional capacitance, and Dγrepresents the fractional integrator. Now, the current–voltage dynamics across the fractional capacitor are expressed as [31].

Cf=IcDγVcE16

The fractional capacitive susceptance is written as

BCf=2πfCfBCf=2πfIcDγVcE17

Simplifying Eq. (17) yields

DγBCf=2πfDγIcDγVcE18

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

BSVCf=DγBL2π2α+sin2α+πDγBCfπE19

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

δ̇=wẇ=1MPmV1V2sinδXDwV̇2=1τV22X+V22DγBL2π2α+sin2α+πDγBCfπ+V1V2cosδXkPdDγBL=1τVrefV2E20

In Eq. (20), δrepresents the power angle of the generator, wis the angular speed, Pmis the input mechanical power to the generator, Dis the damping component, Mis the moment of inertia, Pdrepresents the load demand over time, V1is the sending end voltage, and V2is the voltage over infinite bus.

3. Controller formulation and stability proof

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, the control objective is the regulation of the bus voltage V2regardless of the variation in the generation side and load side. Let e=V2rV2, ė=V̇2rV̇2and the control signal is BSVCf=DγBSVC, then the fractional order sliding manifold is defined as

S=C1Dα/2e+C2ėE21

In the above expressions erepresents the error signal, V2ris the reference signal and V̇2represents the first derivative of the command signal. Differentiating Eq. (21) with respect to Dαyields

DαS=C1Dα/2e+C2DαėE22

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

DαS=C1Dα/2e+C2DαV̇2r+1τV22X1τV22DγBSVCV1V2cosδτX+1τkPdE23

The control law is derived as

DγBSVC=τV22V22τX+V1V2cosδτXkτPdV̇2rC1C2Dα/2eKsDαsgnSE24

The following inequality holds [30].

j=1Γ1+αΓ1j+αΓ1+jDjSDαjSτSE25

Here τis a positive constant. To prove the stability of the closed loop system, the Lyapunov function is chosen as V=12S2. Applying operator Dαto the Lyapunov function yields

DαV=SDαS+j=1Γ1+αΓ1j+αΓ1+jDjSDαjSE26

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

DαV  SDαS+τSE27

Using Eqs. (23) and (27), DαVis calculated as

DαV SC1Dα/2e+C2DαV̇2r+1τV22X1τV22DγBSVCV1V2cosδτX+1τkPd+τSE28

By combining Eqs. (24) and (28), one obtains

DαVKsS+τSE29

The first term of Eq. (29) is negative, so if the discontinuous gain KS>τ, then it is shown that the fractional derivative of the Lyapunov function is always less than zero, that is, DαV0, which means that reaching condition of the sliding surface is satisfied and S=0. The following lemmas are defined that will be used in the convergence proof.

Lemma 1. If integral of the fractional derivative aDtαof a function f(t) exists then according to [30].

aDtαaDtαft=ftJ=1KaDtαJfta=ttaαJΓαJ+1E30

Here K1  α < Kand Γrepresents the standard gamma function.

Lemma 2. The fractional integral operator aDtαwith α > 0is bounded such that the expression in Eq. (31) is valid [29].

aDtαfPKfP;1P;1KE31

From Eq. (29) it is proved that the sliding surface is zero, that is, S=0. With this condition Eq. (31) can be simplified as

Dα/2e=C2C1ėD1+α/2e=C1C2eE32

Multiplying Eq. (32) by D1α/2and by further simplification, one obtains

e=C2C1D1α/2eDα/2Dα/2e=C2C1D1α/2eE33

Using Lemma 1, Eq. (33) can be expressed as

etrDtα/21et=trttrα/21Γα/2=C2C1D1α/2eE34

At time t=tr, the term under the fractional integral of Eq. (34) is equal to zero, that is, trDtα/21et=trttrα/21Γα/2=0, then the remaining expression of Eq. (34) can be written as

e=C2C1D1α/2eE35

Eq. (35) can be expressed as

D2D2e=C2C1D1α/2eE36

Using Lemma 1, Eq. (36) is expanded as

ettrDt21ett=trttr212etr=C2C1D1α/2etE37

Application of Lemma 2 to the right hand side of Eq. (37) yields

C2C1D1α/2etC2C1KetE38

Combination of Eqs. (37) and (38) yields

ettrDt1ett=trttr2etrC2C1KetE39

If St=ts=0and et=0, then the necessary condition of convergence is trts<. Using this concept Eq. (39) can be expressed as

trDt1ett=trtstr2etrėtt=trtrts2etrE40

From Eq. (40), one obtains

trts2etrėtt=trE41

From Eq. (41), it is proved that the error will converge near to the equilibrium points in finite time.

4. MATLAB/Simulink response optimization tool box

MATLAB/Simulink offers built-in response optimization tool box, which is extensively used for the optimal parameters selection of the control system [38]. To start the GUI, go to analysis tab of the simulink and click response optimization tool box. Define the parameters to be optimized in the MATLAB workspace. Then, use the response optimization tool box for importing the workspace defined parameters as design variables. The minimum and maximum search space of the design variables are adjusted. In the second step, the input reference signal is defined and imported using the toolbox. Finally, the output signal is imported from the simulink model that will follow the reference imported signal such that the error between the two signals is minimum. The error is minimized in an iterative manner and in this case, the error signal is the cost function. While minimizing the cost function, the parameters of the control system are adjusted online. The toolbox is associated with different optimization methods such as pattern search, simplex search, and gradient descent methods [39, 40].

5. Optimization of the control system for fractional order SVC-based system

The derived fractional order control system is shown in Eq. (24). For best control performance, it is vital to choose the optimal control parameters. This work compares the performance of the proposed controller (Eq. 24) to the conventional PID control system. The parameters of the proposed control scheme are optimized using Latin Hypercube method and Simulink response optimization toolbox. The order of the fractional operator is chosen constant as α=0.8.

To optimize the parameters, the cost function chosen is the integral of square of the error (ISE) between the reference command and the feedback signal. The initial values of the controller parameters are chosen as C1=0.58, C2=0.78, and Ks=0.48. The convergence of the parameters to the optimal values is shown in Figure 3. The optimal parameters after convergence of the optimization algorithm are: C1=0.35, C2=0.86and Ks=0.625. The optimized step response of the closed system is shown in Figure 4.

Figure 3.

Optimized parameters of the proposed scheme.

Figure 4.

Step response optimization.

The step response and optimized parameters of the PID controllers are shown in Figures 5 and 6. Figure 5 shows the response of the system with optimized PID under nominal conditions. The optimal parameters are given in the table of Figure 6.

Figure 5.

PID step response optimization.

Figure 6.

Parameters of the optimized PID controller.

6. Results and discussions

For numerical simulations, the nominal parameters of the power system including the SVC configuration are given as following [18]: M=1, X=0.5, V1=1, τ=8, k=0.25, D=0.1, and Pm=Pd. The Simulink simulation setup is shown in Figure 7 and the profile of the load demand is shown in Figure 8.

Figure 7.

MATLAB/SIMULINK simulation setup.

Figure 8.

Load demand profile.

6.1. Step response comparison

The control performance of the SVC-based power system under the step response is compared in Figure 9. The comparison is done for the error of the bus voltage stabilization response under the proposed fractional order control system and optimized PID. From the simulation results, it is concluded that the proposed control system is more robust against the load demand variation profile as compared to the optimized PID control system.

Figure 9.

Step response error (a) proposed scheme (b) optimized PID.

6.2. Tracking response comparison

In this section, the bus voltage tracking performance of the SVC-based power network is compared under the action of the proposed control and the optimized PID. The comparative results are shown in Figure 10. From the results provided, it is clear that the system under the proposed control scheme perfectly tracks the reference command while the system under the PID controller exhibits tracking errors of considerable magnitudes.

Figure 10.

Tracking response (a) proposed scheme (b) optimized PID.

6.3. Performance comparison under inductive variable loading condition

The system’s bus voltage stabilization under heavy inductive loads is compared under the action of the proposed control and the optimized PID. The simulation results are shown in Figure 10. The inductive load is varied gradually from time t = 60 sec and increased step wise till t = 100 sec. The proposed controller is robust against the dynamics introduced due to the heavy inductive loads. From Figure 11, it is clear that the response of the power system with PID-based SVC system suffers from voltage sage condition. The proposed controller-based SVC system is efficient in handling the situation and thus, the bus voltage error is very small with no voltage dips.

Figure 11.

Step response under heavy inductive loads (a) proposed scheme (b) optimized PID.

The control signals comparison is given in Figure 12. The proposed control system offers low frequency oscillations, which shows the robustness of the proposed method as compared to the PID method. Usually, in classical sliding mode control method, the robust term excites high frequency oscillations. Due to the high frequency oscillations, the classical sliding mode control is not feasible for practical implementations. The proposed control method is noninteger in nature and from the control law given in Eq. (24), there is an integral term around the robust term, that is, signum function. So, it excites low frequency oscillations and thus it is very feasible for practical implementation over the microprocessor.

Figure 12.

Control signal comparison.

7. Conclusions

In this chapter, a fractional order SVC-based power system is discussed. Based on the fractional sliding manifold, a fractional order sliding control system is proposed. The robustness of the proposed control system is tested under the variable load demand and heavy inductive loading. The performance of the proposed control system is compared with a classical PID control system. From the results and discussion section, it is clear that the proposed control system is more robust as compared to the classical PID control under nonlinear loading profile. Moreover, the control chattering phenomenon is significantly reduced. The proposed control system is feasible for practical implementations.

Acknowledgments

The authors are thankful to the University of Technology Nowshera, Pakistan for providing the financial support for the publication of this chapter.

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Nasim Ullah, Anwar Ali, Haider Ali, Asier Ibeas and Jorge Herrera (November 5th 2018). Static Var Compensator with Fractional Order Dynamics for Enhanced Stability and Control [Online First], IntechOpen, DOI: 10.5772/intechopen.79615. Available from:

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