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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004

Design of a Nonlinear Variable-Gain Fuzzy Controller for FACTS Devices P. K. Dash, Stella Morris, and S. Mishra

Abstract—The paper presents the design of a nonlinear variable-gain fuzzy controller for a flexible ac transmission systems (FACTS) device like the unified power flow controller (UPFC) to enhance the transient stability performance of power systems. The fuzzy controller uses a numerical consequent rule base of the Takagi–Sugeno type, which can be either linear- or nonlinear-producing control-gain variation over a very wide range. This type of fuzzy control is expected to be more robust and effective in damping electromechanical oscillations of the power systems in comparison with the conventional PI regulators used for UPFC control. Computer simulation results validate the superior performance of this controller. Index Terms—Flexible ac transmission systems (FACTS), fuzzy control, local and inter-area modes, oscillation damping, series-connected voltage source, Takagi–Sugeno (TS) fuzzy control, Takagi–Sugeno (TS) rules, unified power flow controller (UPFC), voltage-source converters.

I. INTRODUCTION

W

ITH the increasing electric power demand, power systems can reach stressed conditions, resulting in undesirable voltage and frequency conditions. Flexible ac transmission systems (FACTS) devices are one of the recent propositions to alleviate such situations by controlling the power flow along transmission lines and improving power oscillations damping. For many years, power system stabilizers (PSSs) have been one of the most common controls used to damp out oscillations and to offset the negative damping of the automatic voltage regulators. The major role of PSSs is to introduce a modulating signal acting through the excitation system to add to rotor oscillation damping. However, during some operating conditions, this device may not produce enough damping, particularly to inter-area modes [1] and, hence, other effective alternatives are needed in addition to PSSs. Recently, FACTS technology is emerging as an interesting approach to help in alleviating several power system operating difficulties, such as inter-area oscillations and controlling voltages at critical buses. Amongst the available FACTS devices for transient stability enhancement, the unified power flow controller (UPFC) is the most versatile one [2]–[5]. The UPFC is a solid-state controller based on high-power electronics to control active and reactive power flows in a transmission line. The UPFC comprises a series voltage-source converter and a shunt voltage-source converter, each of which can be modeled as a controllable voltage source. This is realized by connecting a voltage-source converter through a transformer in series with the transmission line and

Manuscript received September 4, 2002. Manuscript received in final form April 24, 2003. Recommended by Associate Editor D. A. Schoenwald. P. K. Dash was with Multimedia University, Cyberjaya, Malaysia. He is now with Silicon Institute of Technology, Bhubaneswar, India. S. Morris is with Multimedia University, Cyberjaya, Malaysia. S. Mishra is with the Department of Electrical Engineering, University College of Engineering, Burla, India. Digital Object Identifier 10.1109/TCST.2004.824332

another one in shunt at the same point of connection through a similar transformer. The shunt branch of the UPFC provides the necessary voltage support to the connected bus and exchanges real power from the bus with the series-connected voltage source. The power balance between the series and the shunt converters is a prerequisite to maintain a constant voltage across the dc capacitor connected between the two converters. As the series branch of the UPFC injects a voltage of variable magnitude and phase angle, it can exchange real power with the transmission line and, hence, improves the power flow capability of the line and its transient and small-signal stability limits. The shunt branch, however, can independently exchange reactive power with the system. Several control strategies for controlling the magnitude and phase angle of the series-voltage source and the shunt-reactive current magnitude have been reported recently [3]–[7]. The injected voltage can be split into two components, which are in phase (real voltage) and the quadrature (reactive voltage) with the line current. Controlling the quadrature component of the series voltage can effectively control the real power through the line. In a similar way, by varying the component of the voltage in phase with the line current, which can be measured locally, one can control the reactive power flow through the line. The real and reactive power references are obtained from the steady-state power flow requirements. The PI regulators used for control of FACTS devices [6] suffer from the inadequacies of providing suitable control and transient stability enhancement over a wide range of power system operating conditions. A radial-basis-function neural network control scheme has also been suggested for the UPFC to damp the electromechanical oscillations of the power system [7]. Linearized power system models with UPFCs have been developed in [8] and [9] to generate control signals to damp out inter-area oscillations. Since these controllers are derived from a small-signal model at a given operating point, they are not globally optimal. Besides the nonlinear nature of the power system operation necessitates the development of a nonlinear controller. The fuzzy-logic [10] approach, on the other hand, provides a model-free approach for UPFC 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. The fuzzy-logic-based approach, which is used in the design of a FACTS controller, uses linguistic rules for both antecedent and consequent parts. This controller is not able to provide a wide variation of the control gains as may be required for the operation of the UPFC as an impedance compensator, phase-angle controller, or a voltage regulator. Instead, a Takagi–Sugeno (TS)-type fuzzy controller, which provides a wide variation of the control gains and could use either a linear consequent rule base or a nonlinear one based on the power

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(a)

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

or voltage error, and its integral or derivative. A recent version of this type of controller [11] is found to be very efficient for a wide variety of nonlinear control problems. Both the series and shunt power flow controllers of the UPFC using TS fuzzy control provide excellent damping and dynamic performance improvement in case of power systems subjected to a variety of transient disturbances. Several case studies are presented in this paper. The approach is, however, computationally intensive, particularly with a multi-neuron architecture.

(b) Fig. 2. (a) UPFC equivalent circuit with controlled voltage sources. (b) UPFC equivalent circuit with controlled current sources.

Since to

for the shunt converter operation, (1) simplifies

II. SYSTEM MODEL

(2)

A. Modeling the UPFC Fig. 1 shows the schematic diagram of a single-machine infinite-bus power system operating with a UPFC between the buses s and r. The UPFC consists of an excitation transformer (ET), a boosting transformer (BT), a dc link capacitor, and two three-phase GTO-based voltage source inverters. Assuming that the voltage induced across BT and ET are and , respectively, the UPFC is represented as two current sources (one positive and the other negative) connected across the buses s and r, as shown in Fig. 2(b). Fig. 2(a) shows the equivalent circuit with controlled voltage sources where series and shunt reactances of UPFC converter transformers, respectively; series and shunt susceptances of UPFC converter transformers, respectively; series voltage magnitude ratio angle of with respect to shunt voltage magnitude ratio angle of with respect to

and

as (3) However, if some real power loss is accounted for (4) the real component of the shunt converter current. This is also true as the shunt and series converters are independently controllable and the exact power balance between the two is never achieved. To take care of the mismatch in real power . as the between the converters, the injection modes use additional term in (4). With the above representation of UPFC between bus s and bus r, the equivalent circuit of the single-machine infinite-bus power system is shown in Fig. 3. In Fig. 3, the controllable loads at the buses s and r are ob, and as tained from

and

The real and reactive powers injected at the buses s and r are obtained as

(5) Further, in (1), the parameters and of the series voltage control circuit are controllable within limits and

(6)

B. UPFC Control (1)

1) Power Flow Control Mode: The UPFC has the unique capability of independently controlling both real and reactive

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dc-link voltage, using a simple PI controller, the current obtained as

is

(13)

C. Synchronous Machine Model

Fig. 3. Reduced equivalent circuit of one machine-infinite-bus system with UPFC.

power flow on a transmission line at a specified point. The local control scheme assumes that both the series and shunt inverter generate controllable voltage sources and the dc bus voltage remains substantially constant. The series inverter power flow into two control is achieved by splitting the series voltage components and , with in phase with the transmisin quadrature with it. is obtained sion line current and from

Each synchronous generator in a single-machine infinite-bus system or a multimachine power system is modeled as a thirdorder model equipped with a simple automatic voltage regulator (AVR) for excitation control. A PSS is also used for controlling the local modal oscillations, as mentioned in the Section I. No speed governor is used for highlighting the effectiveness of UPFC control. No damper winding is modeled, as we are investigating the performance of UPFC controllers. The dynamics of each synchronous machine is differential operator

(14) and

(7) and (8)

The control

in (14) is obtained from the PSS control loop as (15)

where

(9)

, and are the direct- and quadrature-axis components of current and voltages, respectively. and Further, controllable parameters and are related to as (10) (11) and for UPFC The phase and quadrature components series-voltage control are usually obtained from PI regulators using reactive and real power deviations from reference reactive and real power , respectively. power Another control mode of UPFC can be realized by controlusing the shunt-inverter-reactive curling the bus voltage rent component. The dynamics of the dc-link-voltage neglecting losses can be represented as

where all the gains and time constants are given in the Appendix. The algebraic equations for both single and multimachine power systems are straightforward after incorporating two conand for the UPFC. For the multimachine trollable loads case, only the generator buses are retained finally for transient stability study. The next section describes the TS fuzzy controller for generating a wide variation of nonlinear gains for conand quadrature components of trolling the phase the UPFC. III. DESIGN OF TS FUZZY CONTROLLER As shown in the literature, regardless of the type, fuzzy controllers are just conventional nonlinear controllers and can produce satisfactory results when constructed properly. Compared with the existing types, a recent version of a nonlinear variable-gain proportional and derivative type (PD) controller using the TS control rule scheme has been presented. 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. For controlling the magnitude and phase angle of the series-converter voltage source, the following procedure is adopted: the active or reactive power deviations are fuzzified using two input fuzzy sets named (positive) and (negative), and the membership functions are

(12) where is the in-phase component (with respect to ) of the current drawn by the shunt converter. Further, to control the

(16)


431

Fig. 4. Membership function. Fig. 5.

where

denotes error at the th sampling instant given by or

and

Conventional PI control scheme.

Therefore, the proportional and integral gains at any instant depends on error and its integration. If the maximum values of and , respectively, then error and its integration are

For the negative set (17) The membership functions for the error and integration of error are shown in Fig. 4. The TS fuzzy controller uses four simplified rules as If

is positive and

is positive, then

A. Design of TS Fuzzy Parameters As our interest is to design a variable-gain PI controller through a TS fuzzy scheme, the conventional PI controller is designed at first. The and are then obtained from the PI controller by the equations

If

is positive and

is negative, then

If

is negative and

is positive, then

If

is negative and

is negative then

(21)

In the above rule base and represent the output of the TS fuzzy controller. Using Zadeh’s rules for the AND operation and the general defuzzifier, we get (18) However, for given as

This TS fuzzy controller is a highly nonlinear variable-gain produce a controller, and the coefficients wide variation of the controller gains. If the fuzzy controller and , it is termed as a uses all the three coefficients and only, nonlinear rules (NLR) controller, and if it uses it is termed as a linear rule (LR) controller.

, we get the centroid defuzzifer with

where

(19) and (20)

Fig. 5 shows the conventional PI control scheme. The gains of the PI controller are optimized by taking the performance index. The ITAE of the ITAE system at a particular operating condition is calculated for different values of and parameters. The tuned proportional and integral parameters are those for which ITAE is minimum. and are tuned by taking as the error, and the error tunes and . The parameters and are chosen to be equal to the optimized proportional and integral gains of the conventional PI controller. To have a variation of twice the nominal value of the PI controller gains for real power conand are chosen as 1, and . For troller, and , the values of and these values of are and . It is to be noted that the and is based upon the physchoice of the values of ical behavior of the power system. The real power error becomes suddenly positive when a fault occurs and negative when cleared in a power network. Due to this nature, the integration of error remains positive for some time after the fault. As the gains should be larger immediately after the inception of the and dominate. Due to this reason, the fault, the rules gains associated with these rules are chosen as 1. Using a similar argument, the parameters for reactive power controller are

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Fig. 6. TS fuzzy controller with error and integral of error.

Fig. 8. Transient performance for a three-phase fault near the infinite bus (P 1.2 p.u., Q 0.3 p.u.).

=

=

considerably damped using the nonlinear consequent part in the rule base of the TS fuzzy controller TS (NLR) in comparison with the conventional PI regulator as well as TS (LR) controller.

Fig. 7. Transient performance for a three-phase fault near the infinite bus (P 0.8 p.u., Q 0.2 p.u.).

=

=

and as the error in this case becomes negative immediately after the fault. The value of is decided by taking the ITAE criterion as previously noted. The TS fuzzy controller is presented in Fig. 6. IV. SIMULATION RESULTS A. Single-Machine Infinite-Bus The single-machine infinite-bus system shown in Fig. 1 is taken for simulation studies. Large disturbances are created by initiating a three-phase fault near the infinite bus on the transmission line. The conventional PI controller is tuned at a low 0.8 p.u. and p.u. The controller power level with gains are given in the Appendix. The following fault studies are performed for evaluating the performance of the new controller. Case 1) The loading conditions of the generator are at a power level 0.8 p.u., 0.2 p.u. and a three-phase fault of 0.1-s duration is simulated near the infinite bus. In this case, the response with active and reactive power flow control is shown in Fig. 7, from which it can be seen that the TS fuzzy (NLR) as well as TS fuzzy (LR) perform much better compared with the conventional PI controller. Case 2) To prove the effectiveness of the new controller for higher loading condition, the power level is 1.2 p.u., 0.3 p.u. The same increased to three-phase fault occurs near the infinite bus at the high power level. Fig. 8 shows the transient response of the power system with UPFC under TS (NLR), TS (LR) fuzzy control and the conventional PI control. From the response, it is observed that the first undershoot and the second overshoot are

B. Multimachine Power System To validate the performance of the proposed controller, the four-machine two-area power system of Fig. 9 is subjected to different transient disturbances. This system has been specially designed for fundamental studies of inter-area oscillations. The multimachine data is given in [12] and [13]. As the TS (NLR) gives the best damping to speed deviation in the single-machine infinite-bus power system case, its performance in damping the inter-area mode (difference in speed of Gen.2 and Gen.3 and difference in speed of Gen.1 and Gen.3) and local mode (difference in speed of Gen.1 and Gen.2 and difference in speed of Gen.3 and Gen.4) is examined in the multimachine power system. Taking generator G3 as the reference and the predisturbance operating condition in per units as 0.4444 0.2056 0.6667 0.2611 1.5767 0.2186 0.3333 and 0.2244, the response of the network to different disturbances is presented to establish the superiority of the TS (NLR) controller over the conventional PI controller tuned at this operating condition by an ITAE criterion as before. A modulating signal is generated either from the or phase angle difference power flowing between buses 7 and 8. This modulating signal is added to the constant power flow control of the series element. The value of can be obtained from (22) and if and hence

is small and (23)

By measuring the bus voltage and the real power flowing toward bus 8, the value of can be determined. The overall control block diagram is shown in Fig. 10. The parameters of TS (NLR) are decided with the same principle as that of the single-machine infinite-bus case.


433

Fig. 9. Multimachine power system with UPFC.

Fig. 10. Damping controller added to the active power control loop of the series element.

Fig. 11.

Fig. 12.

Local mode of oscillation.

Fig. 13.

Inter-area mode of oscillation.

Fig. 14.



The controller parameters and systems data are provided in the Appendix. The coefficient of the modulating signal is opby an integral time error criterion as timized at before. The modulating signal is also used for the PI control loop. The following case studies are undertaken for evaluating the performance of the proposed controller. Case 1) A three-phase fault of 100-ms duration is simulated at the middle of the line connecting bus 7 and bus 8. Figs. 11 and 12 and Figs. 13 and 14 present the local and inter-area mode of oscillations, respectively. These figures show the comparison of the TS (NLR) controller over its conventional counterpart in UPFC control. The performance of the TS (NLR) is quite prominent in comparison with the PI controller, even if gains are optimized at this operating point with the previously mentioned fault. The overshoots and settling time are significantly improved with TS fuzzy control. The variation of voltage across the dc capacitor is shown in Fig. 15, and it is seen that the overshoots and the settling

time are well controlled by the proposed controller. The dc capacitor voltage is a very important factor

434

Fig. 15.

Fig. 16.

Fig. 17.


DC voltage variation. Fig. 18.


Fig. 19.


Fig. 20.


Fig. 21.




for successful operation of the series and shunt converters of UPFC. Figs. 16 and 17 show the perforas the supmance with phase-angle difference plementary modulating control signal. It is seen that the transient response of the multimachine power as a system is practically similar in case of supplementary modulating control signal instead of . a phase-angle difference Case 2) A three-phase fault of 100-ms duration is simulated at the middle of one of the transmission lines connecting bus 9 and bus 10 with the same operating condition as Case 1. The inter-area and local modes of oscillations are shown in Figs. 18–21. The variation of UPFC bus voltage is shown in Fig. 22. The


Fig. 22.

Fig. 23.

Fig. 24.

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Fig. 25.


Fig. 26.


Fig. 27.


UPFC bus voltage variation.



performance of TS (NLR) is found to be significant in damping both the inter-area and local mode of oscillations. Further, as the fault location changes, the PI controller fails to give satisfactory performance. Case 3) To validate the effectiveness of TS (NLR) in the multimachine environment, the operating conditions (in per units) of the power network are changed to 0.5556 0.2056 0.5556 0.2611 1.3739 0.1502 0.5556 0.2244. The three-phase fault of Case 2 is initiated for 100-ms duration. Figs. 23–26

presents the inter-area and local mode of oscillations for the transient disturbance. The settling time is almost the same as that of Case 2. From these responses, it is well established that the TS (NLR) controller is highly effective in its performance. Case 4) The operating conditions (in per units) of the 0.5556 power network are changed to 0.2056 0.5556 0.2611 1.3739 0.1502 1.5556 0.2244. The three-phase fault of Case 3 is initiated for 100-ms duration. In this case, nearly 900 mW of extra power is getting transported from area 1 to area 2, and this produces large deviation in relative angular speed. Figs. 27–30 show the inter-area and

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Fig. 28.


Fig. 29.


Fig. 30.


local mode of oscillations. From these responses, it is observed that the proposed controller performs significantly in damping inter-area oscillations. Case 5) In order to test the effectiveness of the proposed controller, the same fault of Case 3 is simulated with a fault clearing time of 250 ms. Figs. 31–34 shows the inter-area and local mode of oscillations. In addition, the three-phase fault of Case 4 is simulated with a fault clearing time of 250 ms, and the response is shown in Figs. 35–38. From the response shown in Figs. 31–38, it can be seen that the instability of PI controller is overcome by the proposed controller even at high-power loading condition.

Fig. 31.


Fig. 32.


Fig. 33.


Fig. 34.



Fig. 35.


Fig. 38.

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UPFC with the proposed TS fuzzy-control scheme is evaluated vis-à-vis the conventional PI control to validate its superior performance in respect of transient stability enhancement both in a single-machine and a multimachine power system. This controller is found to be very effective to fault location and provides significant transient stability performance improvements over a wide range of operating conditions. Both inter-area and local modals of power system oscillations are damped much faster using this new controller compared with the conventional PI controller. In addition, the fault clearing time is improved considerably with this new controller. Fig. 36.

APPENDIX


A. Single-Machine Infinite-Bus Data

UPFC data in per units 31.113 Kv

MVA

5500 F

1) Controller Data: Tuned PI controller both

and

controllers

TS fuzzy controller: Fig. 37.


controller controller

V. CONCLUSION In this paper, a nonlinear variable-gain controller for the UPFC has been proposed. The design of controller parameters is dealt with in great detail for improving the stability performance of a power system using an efficient version of the TS fuzzy-control scheme. The new fuzzy-logic-based control scheme produces a wide variation of the control gains, depending on the operating condition of the power system and, hence, a superior performance in comparison with the linear PI controllers used in the UPFC. In the TS fuzzy-control scheme, the rule consequent could be either a linear or a nonlinear function of input variable, and, hence, a superb nonlinear variable-gain controller can be realized. The performance of the

B. Multimachine Data 1) Generator Data: See first table at the top of the next page. 2) UPFC Data: 31.113 kV

5500 F

3) Time Constants of PSS Transfer Function: 4) Controllers Data: Damping controller data: See second table at the top of the next page.

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Conventional PI controller both

and

controllers

TS fuzzy controller

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