a fuzzy variable structure controller for statcom

A fuzzy variable structure controller for STATCOM

Stella Morris , , P. K. Dash and K. P. Basu

Multimedia University, Selangor, Malaysia

Received 24 January 2002;

revised 26 September 2002;

accepted 9 October 2002. ;

Available online 14 January 2003.

Abstract

Two new variable structure fuzzy control algorithms are presented in this paper for

controlling the reactive component of the STATCOM current in a power system. The

control signal is obtained from a combination of generator speed deviation and

STATCOM bus voltage deviation fed to the variable structure fuzzy controller. The

parameters of these fuzzy controllers can be varied widely by a suitable choice of

membership functions and parameters in the rule base. Simulation results for typical

single-machine and multimachine power systems subject to a wide range of operating

condition changes confirm the efficiency of the new controllers.

Author Keywords: Multimachine; Power systems; Generator; STATCOM; Fuzzy

controller

Article Outline

1. Introduction

2. System model

3. Derivation of variable structure fuzzy controller

4. Takagi sugeno fuzzy controller

5. Simulation results

5.1. A. single-machine infinite-bus power system with STATCOM

5.1.1. Case 1

5.1.2. Case 2

5.1.3. Case 3

5.1.4. Case 4

5.2. B. Multimachine power system operating with STATCOM

5.2.1. Case 1

5.2.2. Case 2

6. Conclusion

Appendix A. Parameters of the studied system (in per unit unless indicated specially)

References

1. Introduction

In recent years, various types of FACTS devices (UPFC, STATCOM, TCSC, SVC, etc.)

have been studied for their use in the existing power systems with a view to improve the

flexibility, controllability and to enhance system stability. Reactive power compensation

is an important issue in electrical power systems [1, 2, 3, 4 and 5] and STATCOM plays

an important role in controlling the reactive power flow to the power network and hence

the system voltage fluctuations and angle stability. One of the most important

advantage of the STATCOM is its behavior during the voltage collapse at the bus where

it is located as it supplies almost a constant reactive power without being affected by

voltage variation across it.

Normally the STATCOM comprises a voltage source shunt converter connected through

a transformer and filter across a load bus where the voltage is to be regulated. The

shunt converter is usually modeled as a controllable voltage source generated by the

inverting action of the converter with a DC voltage applied through a charged capacitor.

The converter controls the current injected to the power system and as the energy

exchanges by the STATCOM is limited by the capacitor stored energy, only reactive

power can be exchanged in steady state.

In essence, the desired reactive power exchange is achieved by a reactive current

component Is to maintain DC capacitor voltage constant. However, certain amount of

real power flows into the STATCOM and the real component of the STATCOM current

provides the losses in the resistive elements of the converter and maintains the

capacitor voltage. The STATCOM, therefore, is modeled as a controllable active and

reactive current source; the active part It is obtained from the DC link voltage error

through a PI controller and the reactive part through a time delay unit.

The control of active and reactive components of STATCOM current are normally

achieved through a PI controller. However, these controllers suffer from inadequacies of

providing a robust control and transient stability enhancement over a wide range of

power system operating conditions. Hence other state feedback control methods have

been considered in reference [4], which provide superior performance in comparison to

the PI controller. Both the pole placement technique and linear quadratic regulator are

based on a small signal linearized model of the power system including STATCOM and

hence are suboptimal in nature. Thus, there is the need of a nonlinear controller to

stabilize the inherently nonlinear power system under dynamic operating conditions.

The fuzzy logic approach [7 and 8] on the other hand, provides a model free approach

for STATCOM control and can be effective over the entire range of power system

operation. Furthermore, the fuzzy logic approach allows the knowledge from

experiences to be incorporated to the control scheme as a set of linguistic rules and

membership functions. Also multimachine simulation studies under fault conditions have

been included in this paper to highlight the effectiveness of this new controller.

In this paper, a new fuzzy variable structure control algorithm is used, in which there is a

fuzzy controller derived from a sliding surface and a supervisory control which is

designed to satisfy the sliding conditions. This new controller is able to compensate the

effects of uncertainties, disturbances and unmodeled system dynamics. This is verified

by performing simulation studies on a single-machine infinite-bus power system

subjected to wide range of operating conditions like faults and mechanical torque

changes, etc.

2. System model

In order to illustrate the performance of the power system with a current controlled

STATCOM detailed models of exciter, turbine control loops are omitted and the

transient emf in the quadrature axis, e′q, and mechanical power input, Pm, remain

constants. The single-machine infinite-bus power system shown in Fig. 1 comprises a

synchronous generator connected to the infinite-bus through a double-circuit

transmission line. The STATCOM is located at a bus between the generator and the

infinite-bus.

Full-size image (3K)

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

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An equivalent circuit is shown in Fig. 2, where both active and reactive components of

the STATCOM current are shown.


Fig. 2. Equivalent circuit with real and reactive components of the STATCOM currents.

View Within Article

The differential and algebraic equations of the power system are given by

(1)

ω=ω0+pδ

where, p=d/dt (differential operator)

(2)

(3)

(4)

(5)

where, It and Is are active and reactive components of the STATCOM current,

respectively and Vdc is the DC capacitor voltage.

In the above equations ks, and k1 are gains and ts and t1 are time constants of the

STATCOM current control loop; Rdc and C are resistance and capacitance of the DC

link capacitor; Rs is the shunt converter transformer resistance. The electrical power

output Pe of the generator is given by:

(6)

and STATCOM bus voltage Vm and angle θm are

(7)

where

(8)

(9)

The symbols used above have the usual meaning as in the single-machine infinite-bus

power system and

(10)

3. Derivation of variable structure fuzzy controller

Recently, combinations of fuzzy control and variable structure controller (VSC)

approaches have achieved superior performance [9, 10, 11, 12, 13, 14 and 15]. Briefly,

for instance, Hwang and Lin [13] developed a non-adaptive fuzzy controller, and Wu

and Liu [14] used the switching manifold as a reference, where sliding modes are used

to determine the optimal values of parameters in fuzzy control rules. Othani and

Yoshimura [15] also presented a fuzzy control law using the concept of sliding mode,

where fuzzy rules are tuned by learning.

For designing a linear sliding mode current controller for STATCOM, a linearized model

of the system equations is developed in the form

(11)

where a1, a2, a3, b depend on power system operating condition and machine and line

parameters.

A time varying sliding surface is defined to track the speed error (∆ω=ω−ω0) of the

generator. However, if the speed is not available at the STATCOM bus due to its

location away from the generator, the bus angle θm is used. In either case, the sliding

surface σ is defined as

(12)

where error, e=δ−δ0, or e=ω−ω0, or e=θm−θm ref, λ1>0.

Another definition of σ also could be

(13)

The parameters λ1 and λ2 are obtained from the principles of sliding mode control

design using pole placement technique as follows:Eq. (13) is rewritten as

(14)

For the existence of sliding mode,

(15)

Thus with σ=0, ∆Is is eliminated from Eq. (13) and substituted in Eq. (11) to yield

(16)

which is of the form

(17)

The characteristic equation of the A matrix is designed to have two eigenvalues p1 and

p2, which are found as

(18)

By suitably choosing the values of p1 and p2, the constants λ1 and λ2 are found as

(19)

where d1, d2, and d3 depend on power system operating condition and machine and line

parameters.

After evaluating λ1 and λ2, is rewritten from Eq. (13) as

(20)

The reachability condition for the sliding mode control is used to obtain u; since

(21)

The control u is obtained as

(22)

Due to sgn(σ) term in the control law there is the possibility of undesirable chattering

and hence fuzzy control is used.

In designing fuzzy sliding mode control scheme, the fuzzy controller employs two inputs:

the sliding signal σ and the rate of change of the sliding signal . This fuzzy controller

has only one control output u. The value of is not obtained easily due to system

uncertainties and hence an approximation is used as

(23)

where h is a small positive constant (h=0.01).

Another alternative is to use a first order filter given in Fig. 3 to obtain .

(24)

replaces in the calculations(Tf is a small positive constant, Tf=0.1).

Full-size image (<1K)

Fig. 3. Representation of first order filter.

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Based on the chosen membership functions, the fuzzy control rules are written as

R1 : If σ is P and is P then u is NB.

R2 : If σ is P and is N then u is NS.

R3 : If σ is N and is P then u is PS.

R4 : If σ is N and is N then u is PB.

In the above rule base K is a scaling factor to be suitably chosen to limit σ and

values to lie within −L and +L. the membership functions for positive and negative sets

are given by

(25)

where P and N stand for positive and negative sets; PB, NB, PS, NS stand for positive

big, negative big, positive small, and negative small fuzzy sets, respectively. The value

of K is 1/50.

Similar positive and negative sets are used for the fuzzification of the input . Fig.

4(a) shows the membership functions of σ and . For the control output u, the

membership functions are shown in Fig. 4(b).


Fig. 4. (a,b) Membership function.

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The maximum values of the output sets PB, PS, NB, NS are u1, u2, u3, and u4,

respectivelywhere

(26)

From the fuzzy rule base it can be understood for rule-1, the system states are above

the sliding surface and are moving away from the sliding surface. Hence the control

action needs to be negative and big enough to turn the system states downwards. In a

similar way rules 2, 3, and 4 can be interpreted in terms of the sliding mode control

strategy.Using Zadeh's rules for antecedent part of the rule base (µA and µB=µA×µB) and

general defuzzification formula, the output u is obtained as

(27)

where is the consequent vector, in which uj is the output of the jth rule,

µj is the membership value of the jth rule, and with

(28)

In the above formulation, if γ=1, we get the centroid defuzzifier. The fuzzy controller, if

properly tuned will result in minimizing the error terms e and in Eq. (13) when u is

replaced by uf. However, the uncertainty in the bus voltage variation Vm or due to

unmodeled dynamics of E′q might introduce an error in the STATCOM current reference

Is ref and hence will not become zero to guarantee the sliding of the states of the

power system to the origin. Hence the total current control of the STATCOM will

comprise of a fuzzy control uf and a supervisory control term us in the form as

u=uf+us

The supervisory control is chosen by following the power rate reaching law and

boundary layer approach as

(29)

Due to the power rate reaching law, this control ensures the higher reaching speed

when states are far away from the switching manifold and reduces the rate when the

states are near the manifold. The result is a fast reaching and low chattering reaching

mode [9]. The boundary layer is automatically constructed from the σ values between

sets Big and Small.

The above expression is motivated by boundary layer approach in reducing chattering.

The value of δi is chosen using a fuzzy rule base as follows:

If |σ| is Big, δi is Big

If |σ| is Small, δi is Small

The membership function of the fuzzy sets Big and Small are

(30)

µBig(|σ|)=1−e−|aσ|

µSmall(|σ|)=e−|aσ|

Fig. 5 shows the membership grades for the sets Big and Small. The constants a, β,

and Ks are chosen as a=2, β=0.2, Ks=0.05.


Fig. 5. Membership grades for the sets Big and Small.

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The control block diagram for both reactive and active current controller are shown in

Fig. 6(a) and (b), respectively.


Fig. 6. (a) Control block diagram for reactive current controller. (b) Control block

diagram for active current controller.

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A simple condition for stability can be established for this control law by using the sliding

mode relation

Thus

(31)

Knowing the bounds of , the maximum value of uf can be estimated. Further since uf

is obtained from , the fuzzy control will tend to make both σ→0, and and

hence stability will be achieved. This is verified from actual nonlinear simulations of a

power system.

4. Takagi sugeno fuzzy controller

In comparison to the conventional Mamdani type fuzzy controller presented in Section

3, a Takagi Sugeno (TS) fuzzy control scheme [6 and 7] is presented in this section to

provide a wide range of variation of the nonlinear gain of the current controller for

STATCOM. The simplified TS rules are shown to parameterize the characteristics of the

gain variation and consequently an infinitely large number of gain variation

characteristics can be produced. The inputs to the TS fuzzy controller are the sliding

surface σ and its derivative . The membership values µP(σ) and µN(σ) or µP( ) and

µN( ) are obtained using similar membership functions given in Eq. (25). The following

fuzzy rule base is used:

In the above rule base, the computation of the strength of the antecedent part of the rule

with an AND operator is performed by multiplication of membership values µ(σ)×µ( ),

respectively for each rule.

The effective control u is obtained using a centroid defuzzifier

(32)

which when simplified yields

(33)

and

(34)

The value of λ, K, K1, K2, K3, and K4 are chosen as λ=0.2, K=1/50, K1=1, K2=0, K3=0,

and K4=0.5, respectively for this study.

5. Simulation results

5.1. A. single-machine infinite-bus power system with STATCOM

The single machine infinite-bus system shown in Fig. 1 is considered for simulation

studies. The various transient disturbances due to faults and mechanical power

changes are created to study the performance of the variable structure fuzzy controller

(VSFC), VSFC with supervisory control, and VSC with Takagi Sugeno (TS) fuzzy

control for STATCOM placed at bus no.3 of the power system. The power system data

is given in the Appendix A. The conventional PI controller is used for comparison. The

control output from the PI controller is obtained as

(35)

where the PI gains are optimized using ITAE criterion as Kp=0.1, Ki=0.1, and Kw=10.The

values of K, L, λ1, β, and h are chosen as 1/50, 50, 0.2, 0.2, and 0.01, respectively.

5.1.1. Case 1

The operating power level of the generating system is P=0.8, Q=0.6 and a 3-phase fault

of 0.1 s duration is simulated on bus no.3. In this condition, the voltage across the

STATCOM bus suddenly goes to zero and hence there is a significant rise in swing

angle δ from nearly 80° to 135°. The transient response of the PI controller, VSFC with

a supervisory control, and VSC with TS fuzzy control are shown in Fig. 7. From the

figure it is observed that the VSFC with a supervisory control and VSC with TS fuzzy

produce significant improvement in damping of the electromechanical oscillations of the

generator in comparison to the conventional PI controller. In comparison to the

conventional Mamdani type fuzzy controller with a supervisory control, the TS fuzzy

controller provides significant damping and reduces the overshoot in nearly 1.5 s. The

terminal voltage of the generator and the DC voltage across the STATCOM capacitor

are also damped fast with these two fuzzy controllers.


Fig. 7. Transient response with P=0.8, Q=0.6.

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The power level of the generating system is then increased to P=1.0 and Q=0.4 and the

reactance is decreased from 0.7 to 0.5 pu. A 3-phase fault of 0.1 s. duration is

simulated at bus no.2 (near the infinite-bus). The performance of PI controller, VSC, and

VSFC with supervisory control are shown in Fig. 8. Here, since the power level is high

the load angle swings to nearly 140° and to maintain the stability the reactance of the

line was decreased. From Fig. 8 it can be observed that the VSFC performs well in

reducing the system oscillations. The conventional PI controller, on the other hand,

shows large oscillations and takes more time to damp out the system oscillations. Thus

an uncertain reactance change condition is handled very well by the fuzzy controller.


Fig. 8. STATCOM performance with P=1.0, Q=0.4

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Another loading condition which produces severity in transient oscillations by changing

the operating power levels to P=0.8, Q=−0.2 and reducing the line reactance from 0.7 to

0.6 is shown in Fig. 9. A 3-phase fault of 0.1 s duration is simulated at bus no.2 (near

the infinite bus). The VSFC with supervisory control performs very well in producing

significant damping to the electromechanical oscillations of the system. The

performance of the PI regulator is very oscillatory.


Fig. 9. Transient performance with negative reactive power (P=0.8, Q=-0.2).

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5.1.2. Case 2

A 3-phase fault of 0.1 s duration is simulated on line no.2 at bus no.3 of the power

system. The operating power level is taken P=0.8, Q=0.2. The performance of the PI

controller, VSC, and VSC with TS fuzzy control are shown in Fig. 10. The performance

of both the VSC and the VSC with TS fuzzy control are found to provide significant

damping to the system oscillations. However, the VSC exhibits larger overshoot in the

first swing and takes a little more time than TS fuzzy control scheme for the STATCOM.


Fig. 10. STATCOM performance with P=0.8, Q=0.2.

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Now the transmission line with reactance x21(x21=0.3) is removed from the power

system for a duration of 1 s and its transient response is depicted in Fig. 11. It is seen

from the response that the effect of uncertainty is very well controlled by the proposed

controller.



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5.1.3. Case 3

The mechanical torque input to the generator is increased by 30% for 0.5 s and then

brought to normal and after 4 s, the torque input is reduced by 30% for 0.5 s. The

operating power level is taken as P=0.8, Q=0.6. The transient performance of the VSFC

with supervisory control, and VSC with TS fuzzy control shown in Fig. 12 are found to

be quite satisfactory in reducing transient excursions in the states of the power system

line load angle, terminal voltage and DC voltage for the STATCOM. The performance of

the PI controller is very oscillatory.



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5.1.4. Case 4

To take care of the unmodeled dynamics of the generator voltage E′q behind the

transient reactance, the value of E′q is increased by 50% and the system oscillations are

depicted in Fig. 13. The proposed controller is practically immune to the increase in the

value of E′q and damping is achieved very fast.



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5.2. B. Multimachine power system operating with STATCOM

To verify the performance of the proposed variable structure fuzzy supervisory

controller, a two-area four-generator power system of Fig. 14 is considered. This system

has been specially designed for fundamental studies of inter-area oscillations due to

power flow from one area to the other area. Each generator of the multimachine system

is equipped with an ordinary voltage regulator and a power system stabilizer (PSS) for

damping local mode oscillations during transient conditions due to faults. The PSS

alone cannot provide sufficient damping to the inter-area oscillations and hence the

STATCOM is used at the bus 8 to provide extra damping during system oscillations. In

this study a third order synchronous machine model is used and the system equations

are given in the Appendix A. The system data is given in Ref. [16]. The STATCOM is

modeled as before providing real and reactive current injections at the STATCOM bus.

The sliding surface σ used in this case takes the form

(36)


Fig. 14. Multi-machine power system with STATCOM.

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Here θ78 (θ7−θ8=phase angle difference between the bus 7 and bus 8) is obtained in the

following way.

The active and reactive power flowing in the line 7–8 towards bus 8 are given by:

(37)

(38)

x78=reactance in the line 7–8.V7, and V8 are bus 7 and bus 8 voltages, respectively. θ78

is now calculated as

(39)

Measurement of P78, Q78 and V8 yields the value of θ78. In Eq. (36) ∆ signifies the small

changes in the values of the variables. The constants c1 and c2 are chosen as, c1=2.3,

c2=1.3, for best stability performance of the multimachine power system.

5.2.1. Case 1

Taking generator-3 as reference and pre-disturbance loading conditions in p.u. as

P1=0.44, Q1=0.20, P2=0.66, Q2=0.26, P3=1.57, Q3=0.21, P4=0.33, Q4=0.22, a 3-phase

fault is created on the middle of one of the transmission line connecting bus-9 and bus-

10 and cleared after 0.1 s. The local mode and inter-area mode of oscillations are

presented in Fig. 15. From the figure it is clearly found that the system oscillation are

damped much faster using fuzzy supervisory control VSFC in comparison to

conventional PI control. Fig. 16 and Fig. 17 show the variation of voltage across the DC

capacitor and the variation of STATCOM bus voltage, V8, respectively. It is seen from

the figures that the overshoots and the settling time are well controlled by the proposed

controller.


Fig. 15. Local and inter-area mode of oscillations.

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Fig. 16. DC voltage variation.

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Fig. 17. STATCOM bus voltage variation.

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5.2.2. Case 2

The power loadings of the generators are then changed to P1=0.55, Q1=0.20, P2=0.55,

Q2=0.26, P3=1.37, Q3=0.15, P4=0.55, Q4=0.22. A three-phase fault of 100 ms duration is

simulated at the middle of one of the transmission line connecting Bus-7 and Bus-8. Fig.

18 shows the local mode and inter-area mode of oscillation. Both the modes of

oscillations are damped very well and overshoots reduced fast with VSFC with

supervisory control.


Fig. 18. Local and inter-area mode of oscillations.

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6. Conclusion

The paper presents new variations of the fuzzy control scheme for the control of

STATCOM current. The parameters of fuzzy reactive current controller are adapted by

using a sliding surface similar to the sliding mode control instead of the error and its

derivative. Further, the parameters of the sliding surface can also be varied by using a

pole-placement technique. These new fuzzy controllers for STATCOM provide a wide

range of gain variations for controlling the electromechanical oscillations of a single-

machine infinite-bus and multimachine power systems. The new controllers are found to

provide significant improvement in damping electromechanical oscillations of both the

single and multimachine power systems over a wide range of operating conditions.

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Appendix A. Parameters of the studied system (in per unit unless indicated specially)

Single-machine infinite-bus data

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Multi-machine data. Generator data

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Power system stabilizer

upss=Kδ∆ω

Kδ=0.24, tq=0.4, t3=0.03, t2=0.01

The value of p1 and p2 for best stability performance of the single-machine infinite-bus

power system are chosen as

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Corresponding author

a fuzzy variable structure controller for statcom

Documents

pole placement technique

small positive constant

fuzzy logic approach

bus power system

power system

fuzzy controller

fuzzy control

simulation results