Differential evolutionary algorithm for TCSC-based

Simulation Modelling Practice and Theory 17 (2009) 1618–1634

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Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat

Differential evolutionary algorithm for TCSC-based controller design Sidhartha Panda * Department of Electrical and Electronics Engineering, National Institute of Science and Technology, Berhampur, Orissa 761008, India

a r t i c l e

i n f o

Article history: Received 18 December 2008 Received in revised form 30 May 2009 Accepted 4 July 2009 Available online 9 July 2009 Keywords: MATLAB/SIMULINK Modelling and simulation Differential evolution Thyristor controlled series compensator (TCSC) Power system stability Multi-machine system Power system stabilizer

a b s t r a c t In this paper, a systematic procedure for modelling, simulation and optimal tuning the parameters of a thyristor controlled series compensator (TCSC) controller, for the power system stability enhancement is presented. The design problem of the proposed controller is formulated as an optimization problem and differential evolution (DE) is employed to search for optimal controller parameters. A detailed analysis on the selection of objective function and controller structure on the effectiveness of the TCSC controller is carried out and simulation results are presented. The dynamic performance TCSC controller under various loading and disturbance conditions are analyzed and compared. Finally, the proposed design approach is extended to a multi-machine power system for simultaneous design of multiple and multi-type controllers. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Power system stability problem has been and continues to receive a great deal of attention over the years. Recently, with the deregulation of the electricity market, power systems are expected to perform functions well beyond its original design capacity with new loading and power flow conditions. The increase in utilization of existing power systems may get the system operating closer to stability boundaries making it subject to the risk of collapse of power systems. Synchronous generator excitation control is one of the most important, effective and economic methods to enhance the stability of power systems and to damp low-frequency electromechanical oscillations during disturbance conditions. Power system stabilizers (PSSs) are now routinely employed in the industry in conjunction with generator excitation systems to enhance the system damping and extend power transfer limits, thus ensuring secure and stable operation of the power system. However, the excitation control is restrained by excitation current limit and the time constant of excitation windings and a power system may not maintain the synchronism when a large fault occurs in the power system with generator excitation control only [1]. Recent development of power electronics introduces the use of flexible ac transmission systems (FACTS) controllers in power systems [2]. Subsequently, within the FACTS initiative, it has been demonstrated that variable series compensation is highly effective in both controlling power flow in the lines and in improving stability [3–5]. Thyristor controlled series compensator (TCSC) is one of the important members of FACTS family that is increasingly applied with long transmission lines by the utilities in modern power systems. It can have various roles in the operation and control of power systems, such as scheduling power flow; decreasing unsymmetrical components; reducing net loss; providing voltage support; limiting short-circuit currents; mitigating subsynchronous resonance; damping the power oscillation and enhancing transient stability [6–8].

* Tel.: +91 9438251162. E-mail addresses: [email protected], [email protected] 1569-190X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2009.07.002

S. Panda / Simulation Modelling Practice and Theory 17 (2009) 1618–1634

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In stability studies, the overall accuracy is primarily decided by how correctly the system is modeled. Over the years, several TCSC models have been developed and used by the researchers depending on the applications [9–11]. The Phillips–Heffron model is a well-known model for synchronous generators [12,13]. Traditionally, for the small signal stability studies of a single-machine infinite-bus (SMIB) power system, the linear model of Phillips–Heffron has been used for years, providing reliable results [14]. However, linear models cannot properly capture complex dynamics of the system, especially during major disturbances. The behavior underlying the performance of a synchronous machine during major disturbances is represented by a set of non-linear differential equations. In addition to these, other equations describing different constraints introduced by the loads and/or network, the excitation system, mechanical control system, and installed FACTS controller etc., are included. Thus the complete mathematical description of a power system becomes exceedingly difficult. Linearized models give satisfactory results under small disturbance conditions. This presents difficulties for designing the FACTS controllers in that, the controllers designed to provide desired performance at small signal condition do not guarantee acceptable performance in the event of major disturbances. In this paper TCSC controller is designed for large disturbance condition. To optimize the TCSC-based controller parameters under the most severe three-phase fault conditions, MATLAB/SIMULINK model of the power system with TCSC controller has been used. In recent years, one of the most promising research field has been ‘‘Heuristics from Nature”, an area utilizing analogies with nature or social systems. These techniques are finding popularity within research community as design tools and problem solvers because of their versatility and ability to optimize in complex multi-modal search spaces applied to non-differentiable objective functions. Differential evolution (DE) is a branch of evolutionary algorithms developed by Rainer Stron and Kenneth Price in 1995 for optimization problems [15]. It is a population-based direct search algorithm for global optimization capable of handling non-differentiable, non-linear and multi-modal objective functions, with few, easily chosen, control parameters. It has demonstrated its usefulness and robustness in a variety of applications such as, Neural network learning, Filter design and the optimization of aerodynamics shapes. DE differs from other evolutionary algorithms (EA) in the mutation and recombination phases. DE uses weighted differences between solution vectors to change the population whereas in other stochastic techniques such as genetic algorithm (GA) and expert systems (ES), perturbation occurs in accordance with a random quantity. DE employs a greedy selection process with inherent elitist features. Also it has a minimum number of EA control parameters, which can be tuned effectively [16,17]. In this paper, a comprehensive assessment of the effects of TCSC-based damping controller has been carried out. Two types of TCSC-based controller structure namely a lead-lag (LL) and a proportional–integral-derivative (PID) structure are considered. The design problem of the proposed controllers is transformed into an optimization problem. The design objective is to improve the stability of a single-machine infinite-bus (SMIB) power system, subjected to severe disturbances. Further, for the optimization purpose, two objective functions namely integral square error (ISE) and integral of time-multiplied absolute value of the error (ITAE) are considered. GA-based optimal tuning algorithm is used to optimally tune the parameters of these controllers for minimizations of ISE and ITAE. The proposed controllers have been applied, tested and compared on a weakly connected power system. The dynamic performances of both the LL and PID structured TCSC controller are analyzed at different loading conditions and under various disturbance condition. Further, the proposed approach is extended to a 3-generator system for simultaneous design of TCSC and power system stabilizers. 2. Modelling the power system under study The SMIB power system with TCSC shown in Fig. 1 is considered in this study. The generator has a local load of admittance Y = G + jB and a double circuit transmission line of total impedance Z = R + jX. In the figure VT and VB are the generator terminal and infinite-bus voltage, respectively, and XT is the reactance of the transformer. 2.1. Modelling the TCSC dynamics It is well-known that the reactance adjusting of TCSC is a complex dynamic process. Effective design and accurate evaluation of the TCSC control strategy depend on the simulation accuracy of this process. Basically a TCSC consists of three components: capacitor banks, bypass inductor and bidirectional thyristors. The firing angles of the thyristors are controlled to adjust the TCSC reactance in accordance with a system control algorithm, normally in response to some system parameter Tr. Line-1

VT

VB XT

Generator

Z = R + jX

Tr. Line-2

TCSC Infinite-bus

Y = G + jB

Fig. 1. Single-machine infinite-bus power system with TCSC.

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variations [18]. According to the variation of the thyristor firing angle (a) or conduction angle (r), this process can be modeled as a fast switch between corresponding reactance offered to the power system. Assuming that the total current passing through the TCSC is sinusoidal; the equivalent reactance at the fundamental frequency can be represented as a variable reactance XTCSC. There exists a steady-state relationship between a and the reactance XTCSC. This relationship can be described by the following equation [7]:

X TCSC ðaÞ ¼ X C

X 2C ðr þ sin rÞ 4X 2C cos2 ðr=2Þ ½k tanðkr=2Þ tanðr=2Þ þ ðX C X L Þ p ðX C X L Þ ðk2 1Þ p

ð1Þ

where XC is the nominal reactance of the fixed capacitor, XL is the inductive reactance pffiffiffiffiffiffiffiffiffiffiffiffiffiffi of inductor connected in parallel with fixed capacitor, r = 2(p a) is the conduction angle of TCSC controller, k ¼ X C =X P is the compensation ratio. Since the relationship between a and the equivalent fundamental frequency reactance offered by TCSC, XTCSC(a) is a unique-valued function, the TCSC is modeled here as a variable capacitive reactance within the operating region defined by the limits imposed by a. Thus XTCSCmin 6 XTCSC 6 XTCSCmax, with XTCSCmax = XTCSC(amin) and XTCSCmin = XTCSC(180°) = XC. In this study, the controller is assumed to operate only in the capacitive region, i.e., amin > ar, where ar corresponds to the resonant point, as the inductive region associated with 90° < a < ar induces high harmonics that cannot be properly modeled in stability studies [18]. 2.2. Modelling the power system with TCSC The generator is represented by the third-order model comprising of the electromechanical swing equation and the generator internal voltage equation. The state equations may be written as [19]:

x_ ¼ ½Pm P e Dðx 1Þ=M d_ ¼ xb ðx 1Þ

ð3Þ

V T ¼ vd þ jvq

ð4Þ

I ¼ id þ jiq

ð5Þ

ð2Þ

where Pm and Pe are the input and output powers of the generator, respectively; M and D are the inertia constant and damping coefficient, respectively; xb is the synchronous speed; VT is the terminal voltage; I is the current, d and x are the rotor angle and speed, respectively. The d-axis and q-axis components of armature current, I can be calculated as:

id iq

¼

V B R2 E0q 2 Yq Z e X 2 Yd

X1

R1

sin d

ð6Þ

cos d

where

Y d ¼ C 1 X 1 C 2 R2 Þ=Z 2e ;

C 2 ¼ RB þ XG;

Y q ¼ C 1 R1 þ C 2 X 2 Þ=Z 2e ;

Z 2e ¼ R1 R2 þ X 1 X 2 ;

X 1 ¼ X Eff þ C 1 X q ;

X 2 ¼ X Eff þ C 1 X 0d ;

C 1 ¼ 1 þ RG XB

R1 ¼ R C 2 X 0d ;

R2 ¼ R C 2 X q

X Eff ¼ X XTCSCðaÞ

The power Pe, the internal voltage E_ 0q and the terminal voltage VT can be expressed as:

Pe ¼ E_ 0q iq þ ðX q X 0d Þid iq E_ 0q ¼ EfdE0q ½X d X 0 id =T 0do d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 V T ¼ ðX q iq Þ þ ðE0q X d id Þ2

ð7Þ ð8Þ ð9Þ

In the above equations, Efd is the field voltage; T 0do is the open circuit field time constant; Xd and X 0d are the d-axis reactance and the d-axis transient reactance of the generator, respectively. The simplified IEEE Type-ST1 excitation system is considered in this work. It can be described as:

E_ fd ¼ ½K A ðV ref V T Þ Efd =T A

ð10Þ

where KA and TA are the gain and time constant of the excitation system; Vref is the reference voltage. 2.3. Modelling the TCSC-based controller structure Despite significant strides in the development of advanced control schemes over the past two decades, the conventional lead-lag (LL) structure controller as well as the classical proportional–integral-derivative (PID) controller and its variants,


σ0 +

+

Δσ

Max.

σ0 + Δσ

∑

X TCSC (α)

1 1 + sTTCSC Min. 1 + sT1T 1 + sT2T

1 + sT3T 1 + sT4T

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Two stage lead-lag Block

Output

sTWT 1 + sTWT

KT

Washout Block

Gain Block

Δω Input

Fig. 2. Lead-lag structure of TCSC-based controller.

Proportional KP

Δω Input

Max.

+ 1 / sK i + Integral KD

du dt

∑ +

Δσ

∑

+ +

σ0 + Δσ

1 1 + sTTCSC

X TCSC (α)

Output

Min.

σ0

Derivative Fig. 3. PID structure of TCSC-based controller.

remain the controllers of choice in many industrial applications. These controller structures remain an engineer’s preferred choice because of their structural simplicity, reliability, and the favorable ratio between performance and cost. Beyond these benefits, these controllers also offer simplified dynamic modelling, lower user-skill requirements, and minimal development effort, which are issues of substantial importance to engineering practice [20,21]. In view of the above, both lead-lag and PID structures are considered for analysis in the present study as a TCSC-based controller. The LL and PID structures of TCSC-based damping controller, to modulate the reactance offered by the TCSC, XTCSC(a) are shown in Figs. 2 and 3, respectively. The input signal of the proposed controllers is the speed deviation (Dx), and the output signal is the reactance offered by the TCSC, XTCSC(a). The lead-lag structured controller consists of a gain block with gain KT, a signal washout block and two-stage phase compensation blocks. The signal washout block serves as a high-pass filter, with the time constant TWT, high enough to allow signals associated with oscillations in input signal to pass unchanged. From the viewpoint of the washout function, the value of TWT is not critical and may be in the range of 1–20 s [1]. The phase compensation block (time constants T1T, T2T, T3T and T4T) provides the appropriate phase-lead characteristics to compensate for the phase lag between input and the output signals. The proportional, integral and derivative parameters of the PID controller are KP, Ki and KD, respectively. In Figs. 2 and 3, r0 represents the initial conduction angle as desired by the power flow control loop. The steady state power flow loop acts quite slowly in practice and hence, in the present study, r0 is assumed to be constant during large disturbance transient period. The desired value of line reactance is obtained according to the change in the conduction angle Dr. This signal is put through a first order lag representing the natural response of the controller and the delay introduced by the internal control which yields the reactance offered by the TCSC, XTCSC(a). The effective reactance is given by:

X Eff ¼ X X TCSC ðaÞ

ð11Þ

where XTCSC(a) is the reactance of TCSC at firing angle a. The value of firing angle a is changed according to the change in out put of TCSC controller Dr, as a = p r/2 and r = r0 + Dr, r0 being initial value of conduction angle. 3. Proposed approach 3.1. Problem formulation In case of LL controller, the washout time constants TWT and the time constants T2T and T4T are usually prespecified [14,20,22]. In the present study, TWT = 10 s and T2T = T4T = 0.1 s are used. The controller gain KT and the time constants T1T and T3T are to be determined. In case of PID controller, the parameters KP, Ki and KD are to determined. During steady state conditions Dr and r0 are constant. During dynamic conditions, conduction angle (r) and hence XTCSC(a) is modulated to improve power system stability. The desired value of compensation is obtained through the change in the conduction angle (Dr), according to the variation in Dx. The effective conduction angle r during dynamic conditions is given by:

1622


r ¼ r 0 þ Dr

ð12Þ

3.2. Objective function In this study, two different objective functions are considered for optimization of LL and PID controller parameters. First is integral squared error (ISE) and second is integral of time-multiplied absolute value of the error (ITAE), defined as follows:

ISE ¼

Z

t1

e2 ðtÞdt

ð13Þ

0

ITAE ¼

Z

t1

tjeðtÞjdt

ð14Þ

0

where ‘e’ is the error signal and t1 is the time range of simulation. In ISE, only error is considered and therefore no importance is given to time. But for the power system stability problems, it is required that settling time should be less and also oscillations should die out soon. To this end, in ITAE, while performing integration, time is multiplied with error so that oscillations die out sooner. In the present study, speed deviation Dx following a disturbance is taken as the error signal. For objective function calculation, the time-domain simulation of the non-linear power system model is carried out for the simulation period. It is aimed to minimize this objective function in order to improve the system stability. The problem constraints are the TCSC controller parameter bounds. Therefore, the design problem can be formulated as the following optimization problem:

Minimize J

ð15Þ

Subject to

K min K T K max T T

ð16Þ

max T min 1T T 1T T 1T

ð17Þ

T min 3T

T max 3T

T 3T

ð18Þ

Where J is the objective function (ISE or ITAE); lower bounds of time constants (i = 1, 3).

K max , T

K min T

is the upper and lower bounds of gain;

T max iT ,

T min iT

is the upper and

3.3. MATLAB/SIMULINK based model of system under study In order to tune TCSC controller parameters, the MATLAB/SIMULINK model of the example power system is developed using Eqs. (2)–(12) as shown in Fig. 4. The SIMULINK model for calculation of id and iq is shown in Fig. 5. The relevant

Damping

D Power angle

Mechanical power input

Electrical power output

W0

Pmech

[Pe]

Clock Pe_Fault

1 s

1/M 1/M

Electrical Power before and during fault

Rotor speed

Tw.s

T1T.s+1

T3T.s+1

Tw.s+1

T2T.s+1

T4T.s+1

TCSC controller gain Washout

Two stage Lag/Lead

Terminal voltage 1/Tdo' [vt]

KA

Reference voltage

Efd

TA.s+1

Vref

2*pi*f

1 s Integrator

1 s

-K-

[id]

Xtcsc

id

Eq'

Iq

[iq]

iq id & iq calculation

Integrator

-K-

Change over Switch for Pe

-K-

Initial speed

Delta Id

Sig Xtcsc

1 Ttcsc.s+1

Sigma Initial conduction angle Eq'

Sigma to Xtcsc calculation

TCSC delay

Xtcsc M AX/M IN

Pe = iq*Eq' +(Xq-Xd')*id*iq

Electrical power output

f(u)

[Pe]

[id] Xd-Xd' [iq] Xd-Xd' .

Mux Xq-Xd'

Vt =sqrt ((Xq*iq)^2 +((Eq'-Xd'*id))^2)

Xq-Xd'

f(u)

(Xd-Xd')id

Terminal voltage

[id]

Fig. 4. SIMULINK model of SMIB with TCSC controller.

[vt]


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Fig. 5. SIMULINK model for calculation of id and iq.

parameters and initial condition calculations are given in Appendix A and B, respectively. The objective function is evaluated for each individual by simulating the example power system under the severe disturbance. The most severe situation where a three-phase short-circuit fault occurs at the generator busbar terminal is considered in the present study for objective function calculation.

4. Overview of differential evolution (DE) algorithm Differential evolution (DE) algorithm is a stochastic, population-based optimization algorithm introduced by Storn and Price [15]. DE works with two populations; old generation and new generation of the same population. The size of the population is adjusted by the parameter NP. The population consists of real valued vectors with dimension D that equals the number of design parameters/control variables. The population is randomly initialized within the initial parameter bounds. The optimization process is conducted by means of three main operations: mutation, crossover and selection. In each generation, individuals of the current population become target vectors. For each target vector, the mutation operation produces a mutant vector, by adding the weighted difference between two randomly chosen vectors to a third vector. The crossover operation generates a new vector, called trial vector, by mixing the parameters of the mutant vector with those of the target vector. If the trial vector obtains a better fitness value than the target vector, then the trial vector replaces the target vector in the next generation. The evolutionary operators are described below [15–17]. 4.1. Initialization For each parameter j with lower bound X Lj and upper bound X Uj , initial parameter values are usually randomly selected uniformly in the interval [X Lj ; X Uj ]. 4.2. Mutation For a given parameter vector Xi,G, three vectors (Xr1,G Xr2,G Xr3,G) are randomly selected such that the indices i, r1, r2 and r3 are distinct. A donor vector Vi,G+1 is created by adding the weighted difference between the two vectors to the third vector as:

V i;Gþ1 ¼ X r1;G þ F ðX r2;G X r3;G Þ

ð19Þ

where F is a constant from (0, 2). 4.3. Crossover Three parents are selected for crossover and the child is a perturbation of one of them. The trial vector Ui,G+1 is developed from the elements of the target vector (Xi,G) and the elements of the donor vector (Xi,G).Elements of the donor vector enter the trial vector with probability CR as:

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U j;i;Gþ1 ¼

V j;i;Gþ1 X j;i;Gþ1

if if

randj;i CR or j ¼ Irand randj;i > CR or j–Irand

ð20Þ

With randj,i U (0,1), Irand is a random integer from (1, 2, . . . , D), where D is the solution’s dimension i.e. number of control variables. Irand ensures that Vi,G+1 – Xi,G

X r1,G Difference Vector

F . ( X r 2 ,G − X r 3, G )

Vi,G+1 = X r1,G + F. ( X r2,G − X r3,G )

X r 2 ,G X r 2 ,G − X r 3 ,G

X r 3,G Fig. 6. Vector addition and subtraction in DE to generate a new candidate solution.

Start

Specify the DE parameters

Initialize the population Gen.=1 Evalute the population

Create offsprings and evalute their fitness

Is fitness of offspring better than fitness of parents ? Yes Replace the parents by offsprings in the new population

No

Discard the offspring in new population

Yes Size of new population < Old population ? Gen. = Gen+1

No No

Gen. > Max. Gen ? Yes Stop

Fig. 7. Flow chart of proposed DE optimization approach.


1625

4.4. Selection The target vector Xi,G is compared with the trial vector Vi,G+1 and the one with the better fitness value is admitted to the next generation. The selection operation in DE can be represented by the following equation:

X i;Gþ1 ¼

U i;Gþ1

iff ðU i;Gþ1 Þ < f ðX i;G Þ

X i;G

otherwise:

ð21Þ

where i e [1, NP]. Fig. 6 shows the vector addition and subtraction necessary to generate a new candidate solution. 5. Results and discussions 5.1. Application of differential evolution Implementation of DE requires the determination of six fundamental issues: DE step size function, crossover probability, the number of population, initialization, termination and evaluation function. Generally DE step size (F) varies in the interval (0, 2). A good initial guess to F is in the interval (0.5, 1). Crossover probability (CR) constants are generally chosen from the interval (0.5, 1). If the parameter is co-related, then high value of CR work better, the reverse is true for no correlation [23,24]. In the present study, a population size of NP = 20, generation number G = 200, step size F = 0.8 and crossover probability of CR = 0.8 have been used. Optimization is terminated by the prespecified number of generations for DE. One more important factor that affects the optimal solution more or less is the range for unknowns. For the very first execution of the program, a wider solution space can be given and after getting the solution one can shorten the solution space nearer to the values obtained in the previous iteration. The flow chart of the DE algorithm employed in the present study is given in Fig. 7. 5.2. Lead-lag structured TCSC controller The lead-lag structure shown in Fig. 2 is first considered as the TCSC-based controller. The controller parameters are optimized considering both the objective functions ISE and ITAE. Optimization process is repeated 20 times and the controller parameters are chosen corresponding to the best fitness function obtained in the 20 runs. The bounds of unknown parameters of gain and time constants used in the present study and the optimized TCSC controller parameters obtained in for ISE and ITAE are shown in Table 1. Figs. 8 and 9 show the convergence rate of ISE and ITAE, respectively, with the number of generations. It may be noted that the higher converged values of ITAE is due to multiplication of time factor as indicated in Eqs. (13) and (14), and it does not in any ways ascertain a poor response. The actual response must be observed through the simulation results presented in the next section. In order to verify the effectiveness of the proposed TCSC controller optimized using ISE and ITAE, simulation studies are carried out. A 3-phase, self-clearing fault of 100 ms duration is applied at the generator terminal busbar at t = 1 s. The original system is restored upon the fault clearance. The system power angle response for the above contingency is shown in Fig. 10. In Fig. 10 the response without the controller is shown with the dotted line with the legend NC; and the responses with TCSC controller optimized using ISE and ITAE are shown with dashed line and solid line with legends ISE and ITAE, respectively. It is clear from Fig. 10 that, without controller even though the system is stable, power system oscillations are poorly damped. It is also clear that, proposed DE optimized TCSC controller significantly suppresses the first swing in the power angle and provides good damping characteristics to low-frequency oscillations by stabilizing the system much faster. Further, it is also obvious from Fig. 10 that, the performance of TCSC controller is better when the objective function used is ITAE as compared with ISE. Figs. 11 and 12 show the responses of electrical power output of generator and the reactance offered by the TCSC response for the above contingency. It can be observed from these results that, when ISE is used as objective function oscillations remain for little longer time. In ITAE, however, we get improved damping with lower settling time. At this stage, studies are carried out to answer the pertinent question, ‘‘Whether the proposed TCSC controller designed under large disturbance condition work satisfactorily under the small perturbation?” In order to answer this important question, the dynamic performance of the system with proposed TCSC controller is examined under small disturbance conditions.

Table 1 Optimized lead-lag controller parameters for objective functions ISE and ITAE. Parameters

Minimum range Maximum range ISE ITAE

Gain

Time constants

KT

T1T

T3T

20 100 77.2308 68.2881

0.01 0.5 0.1566 0.2736

0.01 0.5 0.0519 0.0118

1626


x 10

-3

5.55 5.5

convergence of ISE

5.45 5.4 5.35 5.3 5.25 5.2 5.15 0

20

40

60

80

100

120

140

160

180

200

180

200

Generations Fig. 8. Convergence of ISE for lead-lag structured TCSC controller.

2

Convergence of ITAE

1.9

1.8

1.7

1.6

1.5 0

20

40

60

80

100

120

140

160

Generation Fig. 9. Convergence of ITAE for lead-lag structured TCSC controller.

Fig. 10. Power angle response without and with control (ISE and ITAE) under 100 ms 3-phase fault disturbance with lead-lag structured TCSC controller.


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Fig. 11. Electrical power response without and with control (ISE and ITAE) under 100 ms 3-phase fault disturbance with lead-lag structured TCSC controller.

Fig. 12. Variation of reactance offered by TCSC without and with control (ISE and ITAE) under 100 ms 3-phase fault disturbance with lead-lag structured TCSC controller.

Fig. 13. Power angle response without and with control (ISE and ITAE) for 1 pu step increase in mechanical power input with lead-lag structured TCSC controller.

1628


The input mechanical power is increased by a step of 0.1 pu at t = 1 s. Fig. 13 shows the power angle response for the above contingency. Fig. 13 illustrates the advantage of using ITAE over ISE as the objective function. The first swing in power angle is also slightly reduced with ITAE compared to ISE. 5.3. PID structured TCSC controller As we have seen in section 5.2 that ITAE is better objective function than ISE; therefore the parameters of the PID controller (shown in Fig. 3) are optimized using ITAE as objective function. The optimized parameters (obtained in 20 runs) are shown in Table 2. Fig. 14 shows the convergence rate of ITAE with the number of generations for a PID structured TCSC controller. Comparison of converged values of ITAE for lead-lag and PID (shown in Figs. 9 and 14) shows that, in case of leadlag structured TCSC controller, ITAE converges to a lower value than that of PID structured TCSC controller. So lead-lad structure should give a better response compared to the PID structure. The performance of PID and lead-lag structured TCSC controller optimized using ITAE as objective function, are compared by applying a severe disturbance. A 3-phase fault is applied at the generator terminal busbar at t = 1 s. and removed after 100 ms. The original system is restored upon the fault clearance. Figs. 15 and 16 show the system response for the above

Table 2 Optimized PID controller parameters using ITAE as objective function. Parameters/objective function

KP

Ki

KD

ITAE

71.5052

0.3295

0.1362

2.35

ITAE convergence

2.3

2.25

2.2

2.15

2.1

0

20

40

60

80

100

120

140

160

180

200

Generations Fig. 14. Convergence of ITAE for PID structured TCSC controller.

Fig. 15. Power angle response with PID and lead-lag structured TCSC controller under 100 ms 3-phase fault disturbance using ITAE objective function.


1629

Fig. 16. XTCSC variation with PID and lead-lag structured TCSC controller.

Fig. 17. Power angle response with PID and lead-lag structure TCSC controller for 1 pu step decrease in mechanical power input using ITAE as objective function.

contingency with PID and lead-lag structured TCSC controller. In these figures the response with PID structured TCSC controller are shown in dotted line with legend ‘PID’ and the response with lead-lad structured TCSC controller are shown in solid line with legend ‘Lead-lag’. It is clear from Figs. 15 and 16 that the performance of lead-lag structured TCSC controller is better then a PID structured TCSC controller. The performance of PID and lead-lag structured TCSC controller optimized using objective function ITAE are compared under small disturbance. The input mechanical power is decreased by a step of 0.1 pu at t = 1 s. Fig. 17 shows the power angle response for the above contingency. It is clear form the simulation results that, a lead-lag structured TCSC controller with ITAE as objective function provides best results compared to all other alternatives. 5.4. Extension to multi-machine system with multi-type controllers The proposed approach is extended to a multi-machine power system with multiple controllers. The widely used Western Systems Coordinating Council (WSCC) 3-machine, 9-bus system shown in Fig. 18 is considered [25]. The simplified IEEE type-ST1A static excitation system has been considered for all three generators. The system data are given in Appendix C. Further a TCSC is assumed to be installed in line from bus 7 to bus 8. Also, to deal with the simultaneous design of multiple controllers, power system stabilizers (PSS) are assumed for machines 2 and 3. The generators are represented by a flux-decay model. The differential-algebraic equations for the m machine, n bus system with IEEE Type-I exciters are [25]:

ddi ¼ xi xs dt 2Hi dxi ¼ T mi T ei Di ðxi xs Þ xs dt 0 dEqi T 0doi ¼ E0qi ðX di X 0di ÞIdi þ Efdi dt

ð22Þ ð23Þ ð24Þ

1630


Gen2 18.0 KV 1.025 pu

Z=0.0119+j0.1008

Y=0+j0.0745

Y=0+j0.1045

5

Tap=16.5/230 Z=j0.0576

6

Z=0.039+j0.17

100MW 35 MVAR

Z=0.01+j0.085 Y=0+j0.088

125MW 50 MVAR

Y=0+j0.179

Y=0+j0.153

Tap=18/230 Z=j0.0625

Tap=13.8/230 Z=j0.0568

3

85 MW

Z=0.017+j0.092 Y=0+j0.079

4 1

Slack Bus

Gen3 13.8 KV 1.025 pu

9

Z=0.0085+j0.072

Z=0.032+j0.161

2

163 MW

8

7

90MW 30 MVAR

Gen1 16.5 KV 1.04 pu

Fig. 18. WSCC 3-machine, 9-bus system.

In this study, the loads are assumed to be constant impedance and converted to admittances as:

Li ¼ y

ðPLi jQ Li Þ

ð25Þ

V 2i

where i = 1, . . . , m. Li , since loads are assumed as injected quantities. There is a negative sign for y The network equations for the new augmented network can be written as:

"

IA 0

#

!"

YAYB

¼

YCYD

EA

#

VB

ð26Þ

Since there is no current injection at the n network buses, theses buses can be eliminated resulting in:

IA ¼ ðY A Y B Y 1 EA ¼ Y int EA D YCÞ

ð27Þ

where the elements of the IA and EA are:

Ii ¼ ðIdi þ jIqi Þejðdi p=2Þ

ð28Þ

Ei ¼ Ei \di

ð29Þ

The elements of the Y int are:

Y ij ¼ Gij þ jBij

ð30Þ

Since the network buses are eliminated, the internal nodes are renumbered as 1, . . . , m for ease of notation. So the current equation becomes:

Ii ¼

m X

Y ij Ej

ð31Þ

j¼1

The real power output of the internal node i can be written as:

Pei ¼ Re½Ei Ii ¼

m X

Ei Ej ðGij cos dij þ Bij sin dij Þ

ð32Þ

j¼1

The 3-machine, 9-bus power system shown in Fig. 18, is modeled in the MATLAB/SIMULINK environment using Eqs. (22)–(32) as shown in Fig. 19. In Fig. 19 only one machine is shown and the other two machines have been modeled in the similar manner.

1631


-KD1 Pm1

-K-

Pm1

1/s Del1

1/s w1

Pa1 [Pe1] Pe1 ' [e1]

E1

[d1]

del1

. [e2]

E2

., [d2]

del2

del

Vt1

-K-

, [e3]

E3

.,. [d3]

del3 E1_

,., [XTCSC] ..,,

|u|

Clock1

V tEfd

Terminal Voltage

IEEE-1 Excitor

-K-

Switch1

dE/dt

XTCSC

After fault clearence

[e1]

E1

" [d1]

del1

..,,. [e2]

E2

_ [d2]

del2

[e3]

E3

-[d3]

del3

''

-K-

id I

I1

1/s

I1

E1 [e1] Pe1 Re(u)

[Pe1]

Complex Power

Pe

u Clock2

Math Function

E1_

Switch5

During fault Fig. 19. MATLAB/SIMULINK model for machine 1 in a 3-machine power system.

Input

K PS Gain block

sTWP 1 + sTWP Washout block

1 + sT1P 1 + sT2 P

1 + sT3 P 1 + sT4 P

Two-stage lead-lag block

VSmax

VS Output

VSmin

Fig. 20. Structure of the power system stabilizer.

Table 3 DE Optimized PSS and TCSC-based controller parameters for multi-machine system. Parameters

TCSC

PSS 2

PSS 3

K T1 T2 T3 T4

0.3881 0.0417 0.1 0.198 0.1

8.0306 0.4157 0.1 0.1093 0.1

14.337 0.203 0.1 0.4396 0.1

1632


The lead structure of the TCSC-based controller shown in Fig. 2 is considered in the present study. The structure of the PSS is shown in Fig. 20. A 3-phase fault is applied at bus 7 and cleared by permanent tripping of the line from bus 5 to bus 7. The accelerating power of the nearest machine i.e. machine 2 is selected as the input signal to the TCSC-based controller. Accelerating powers of the individual generators are chosen as the input signals input signals to the PSSs. A washout time constant of 2 s and all the denominator time constants of 0.1 s are considered for simultaneous designing of PSSs and TCSC controller. The objective function J is expressed as:

J¼

Z

t1

ðjx2 x1 jÞ t dt þ ðjx3 x1 jÞ t dt

ð33Þ

0

where x1, x2 and x3 are the rotor speed of machine 1, 2 and 3, respectively, and t1 is the time range of simulation.The parameters of the PSSs and the TCSC-based controller are simultaneously tuned using DE technique as explained in Section 5.1. The parameters are optimized for the most severe conditions. The obtained parameters of TCSC-based controller and PSS are shown in Table 3. To assess the effectiveness of the proposed controller, simulation studies are carried out. A 3-phase fault of 100 ms duration is applied at bus 7. The fault is cleared by permanent tripping of the line between bus 5 to bus 7. The variations of the relative speed deviation of machines 2 and 3 with respect to machine 1 for the above contingency are shown in Figs. 21 and 22, respectively. In these figures, the response without controllers is shown with dotted line and the response with proposed PSS and TCSC controller is shown with solid line. It is clear from these Figs. that, the system is unstable without control under this severe disturbance. It is also clear from the figures that with proposed controllers stability of the system is maintained and low-frequency oscillations are quickly damped out.

Fig. 21. Response of (x2 x1) for a 100 ms 3-phase fault at bus 7 cleared by line tripping.

Fig. 22. Response of (x3 x1) for a 100 ms 3-phase fault at bus 7 cleared by line tripping.

1633


6. Conclusion This paper presents a systematic procedure for modelling, simulation and optimal tuning of TCSC controller for enhancing rotor angle stability of a power system. A MATLAB/SIMULINK model is presented for a single-machine infinite-bus power system installed with TCSC controller. Different controller structures, namely a lead-lag (LL) and a proportional– integral-derivative (PID) and objective functions namely integral square error (ISE) and integral of time-multiplied absolute value of the error (ITAE) are considered. The design problem is transferred into an optimization problem and differential evolution algorithm is employed to search for the optimal TCSC controller parameters. The performance of the proposed controllers under various disturbances are compared and analyzed. Simulation results show that ITAE is a better objective function than ISE for optimization problems concerning TCSC controller design. Further, it is observed that leadlag structured TCSC controller where the controller parameters are optimized using ITAE as objective function, gives the best system response compared to all other alternatives. Finally, the proposed modelling and design approach has been extended to a 3-generator power system and power system stabilizers and the TCSC-based controller are simultaneously designed.

Appendix A System data for single-machine infinite-bus power system: All data are in pu unless specified otherwise. Generator:

M ¼ 8 s; f ¼ 60;

D ¼ 4:4; V T ¼ 1:0;

X d ¼ 1:0;

X q ¼ 0:8;

X 0d ¼ 0:3;

T 0do ¼ 5:044;

Pe ¼ 0:9 pu; Q ¼ 0:1513 pu; d0 ¼ 51:7960

Exciter (simplified IEEE type-ST1):

K A ¼ 10;

T A ¼ 0:01 s:

Transmission line and transformer:

X ¼ 0:6;

X T ¼ 0:1;

Ra ¼ 0;

G þ jB ¼ 0 þ j0:

TCSC controller:

T TCSC ¼ 15 ms; X TCSC0 ¼ 0:3369;

a0 ¼ 1580 ; X C ¼ 0:5X; X P ¼ 0:25X C ; k ¼ 2; X TCSC max ¼ 0:8X; X TCSC min ¼ 0

Appendix B Calculation of initial conditions

k¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi X C =X L

X TCSC ðaÞ ¼ X C

X 2C ðr þ sin rÞ 4X 2C cos2 ðr=2Þ ½k tanðkr=2Þ tanðr=2Þ þ ðX C X L Þ p ðX C X L Þ ðk2 1Þ p

X Eff ¼ X T þ X TL X TCSC ðaÞ

P jQ Ia ¼ ; V t id ¼

0vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 !2 u u V 2t C Bt 2 V d ¼ Pe V t @ Pe þ Q e þ A; Xq

ðPe iq V q Þ ; Vd

E0q ¼ Vq þ X 0d id ;

Vq ¼

V d ¼ V d þ X Eff iq ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2t V 2d ;

iq ¼

V q ¼ V q X Eff id ;

Vd ; Xq Eq ¼ V T þ jX q Ia ;

Efd ¼ Eq þ ðX d X 0d Þid

1634


Appendix C System data for 3-machine bus power system: All data are in pu unless specified otherwise. Please see Ref. [25] for details.

H1 ¼ 23:64;

H2 ¼ 6:4;

H3 ¼ 3:01;

D1 =M1 ¼ 0:1;

X d2 ¼ 0:8958;

X d3 ¼ 1:3125;

X 0d1 ¼ 0:0608;

X q2 ¼ 0:8645;

X q3 ¼ 1:2578;

T 0do1 ¼ 8:96;

D2 =M 2 ¼ 0:2;

X 0d2 ¼ 0:1198; T 0do2 ¼ 6:0;

D3 =M 3 ¼ 0:3;

X 0d3 ¼ 0:1813;

X d1 ¼ 0:146;

X q1 ¼ 0:0969;

T 0do3 ¼ 5:89

Exciter (simplified exciter):

K A1 ¼ K A2 ¼ K A3 ¼ 20;

T A1 ¼ T A2 ¼ T A3 ¼ 0:2 s

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