Chaos driven instability control using interval type-2 fuzzy logic controller for better performance

Abstract

Engineering systems are governed by set of differential equations and work under various uncertainties of parameters. Type–1 fuzzy logic controller are widely used in engineering systems for expert control of system using logic and mathematics together. This paper proposes the first-ever use of Interval Type–2 fuzzy logic controller design to control chaos and associated instability in a nonlinear dynamical power system. Interval Type–2 fuzzy designs have an edge over type-1 fuzzy sets and interval type–2 fuzzy logic controller is well suited for the uncertainty present in weak and chaos sensitive systems. Uncertainty in parameters affects differential equation very badly for such systems. Uncertainty in engineering applications needs an extra layer of handling system control mathematically and logically. Comparison of Type–1 and Type–2 fuzzy logic controller based on time domain waveform, phase plane trajectory and integral square error, clearly proves the efficacy of Type–2 fuzzy logic controller in controlling dynamic behavior of a nonlinear dynamical system as demonstrated through results discussed in the paper. This paper discusses control of chaos driven voltage instability issue in the case of nonlinear dynamical power system, as an application.

Keywords

Type-2 fuzzy logic controller interval type-2 fuzzy logic controller fuzzy logic set nonlinear dynamical system

1 Introduction

Systems in general are broadly classified as linear, nonlinear, robust, predictive, stochastic, weak, uncertain and sensitive from the perspective of response to input excitation. Out of these systems, predictive systems are governed by differential equations which show predictive trend in solution, while firm prediction can not be made for response of weak systems. Such weak systems are prone to chaotic solutions owing to specific range of parameters. One such example is behavior of nonlinear dynamical power system. Voltage in electrical systems is prone to chaotic response for some specific range of reactive power and mechanical power input.

Type–2 fuzzy handle imprecision and uncertainty in better way. Uncertainty in parameter values of system could be due to various reasons such as nature and environmental conditions. Measurement sensors also introduces errors in output signal.

Uncertainties are classified as process uncertainty, measurement uncertainty, model uncertainty, estimate uncertainty and implementation uncertainty.

Type–2 Fuzzy sets were introduced in 1975 by L. Zadeh. There is no mention of uncertainty in membership function of linguistic variables in Type–1 fuzzy systems. Type–2 FIS is an extension of Type–1 fuzzy set theory with inclusion of footprint of uncertainty. Mendel and Karnik et al. [16 , 27] further developed the theory of T2 FLSs and came up with new concepts in fuzzy membership functions to solve issues of uncertainty.

Certainty of membership function in Type–1 fuzzy sets became issue of debate. The word “fuzzy” has the connotation of being uncertain. Thus Type–2 fuzzy sets introduced by Zadeh in 1975 were characterized by fuzzy nature of membership functions. A type–2 fuzzy set is characterized by a fuzzy membership function, the membership value for each element of type–2 is a fuzzy set spread in range [0,1]. The membership functions of type–2 fuzzy sets are three dimensional (3D) and include a footprint of uncertainty (FOU) (which is shaded in gray in Fig. 1).

Fig.1

Difference between type-1 and type-2 fuzzy logic membership functions.

Uncertainty is better handled by the use of extra degree of freedom related to third dimension of membership functions. Therefore, a Type–2 FLS has an edge over Type–1 FLS [12 , 18].

Higher order nonlinear dynamical power system models suffer from poor voltage stability due to insufficient reactive power support. As system works close towards maximum loadability limit, it suffers voltage instability or onset of limit cycle and is prone to chaos. Very sensitive controllers which can differentiate between different situations as per expert rules can help in handling the situation in better way. Fuzzy logic controllers are suitable for such applications. But crisp membership functions as required in type 1 system are not well suited for handling voltage instability. Voltage is affected by reactive power Q, whose value is uncertain due to uncertainty in customer demand. Not only this, reserved resources of reactive power may push sensitive system into instability due to their dynamics. Nonlinear dynamics behavior of power system will push system near to voltage collapse as indicated by Proximity to Voltage Collapse index. This index should be high enough to indicate safe distance of operating voltage from point of voltage collapse.

Benchmark model of power system helps in analyzing the situation of onset of various types of bifurcations and route to chaos leading to instability. We suggest use of interval type 2 fuzzy logic controllers to handle voltage instability issue in an uncertain environment wisely. Previous attempts are based on type–1 fuzzy logic based system which helps in controlling voltage collapse but membership functions used for voltage was crisp and no uncertainty in voltage. But practical situations clearly has margin of uncertainty in voltage. Hence extra fuzziness in voltage membership function force us to design Interval Type–2 fuzzy logic controller.

Application area of type-2 fuzzy logic controller related to electrical power system are

IT2FLC based power system stabilizer (PSS) for power system models [25].

An interval type-2 fuzzy logic controller for FACTS devices such as Thyristorized Switched Capacitor (TCSC) to damp oscillations in power system [26].

Speed control of DC motors using Interval Type-2 fuzzy logic controller design [13].

Interval type 2 fuzzy logic control system for multi-machine power systems [1].

Section 2 of the paper discusses basics of type-2 fuzzy set, its architecture along with three dimensional membership function with footprint of uncertainty. Section 3 gives brief idea about different chaos control schemes. Section 4 introduces power system model along with governing state equations. Section 5 is regarding results obtained and discussion which advocates use of type–2 design followed by conclusion based on simulation results obtained.

Objective of the paper is to shift focus from conventional type–1 fuzzy logic designs for the systems to type–2 FLC designs. Type–2 FLC gives better performance, where uncertainty affects the system dynamics badly (like chaotic systems). Type–2 fuzzy logic controllers (FLC) are better suited for such systems.

2 Interval type–2 fuzzy logic set

Difference in structure of Type–2 FIS is three dimensional membership functions and presence of Type Reducer before defuzzification as shown in Figs. 1 and 2.

Fig.2

Internal architecture of interval type–2 fuzzy logic design.

Figure 1 shows that primary membership function and secondary membership function as well in third dimension. Membership function of type–1 is crisp and membership function value for a given system parameter x’ is unique. Membership function of type–2 has footprint of uncertainty and for a given system parameter x’, there exist upper and lower limit of primary membership value, while in another dimension it has its own membership (secondary) value spread in interval 0 to 1. General type–2 fuzzy membership function represents triangular shape in third dimension and it is usually highly complex in computation. To simplify it if secondary membership function is uniform rectangle with value 1, then it is simple to compute and hence popularly used in design and named as interval type 2 fuzzy set.

Internal architecture of type–2 differs slightly from general architecture of fuzzy logic controller as shown in Fig. 2. It consist of four basic blocks: fuzzifier, rule base, fuzzy inference engine, and output processor. Output processor contains a type reducer to derive a type-1 set from the type-2 set.

Practical applications rarely use general type 2 fuzzy logic sets due to massive computational complexity. Hence Interval Type 2 Fuzzy Logic Sets (IT2FLS) [27] came in picture. IT2FLS design is characterized by secondary membership functions that only take uniform value of 1 over their domain. Such modification reduces the computational burden of performing inference compared with previous general type 2 fuzzy logic sets.

IT2FLS have drawn more attention from practical applications related to academia and industry. IEEE Computational Intelligence Magazine, February 2007, contains several papers reviewing theory and application of T2FLS and IT2FLS [16 –28].

3 Chaos control schemes for systems

Following are various methods to control chaos in different systems.

Feed-forward method of control [5]

Linearization method (OGY method) [2]

Delayed feedback [34]

Linear and nonlinear control [3 , 12]

Frequency domain methods [6]

Artificial Neural Network based [31]

Fuzzy Logic Based [9 , 29]

ANFIS based controls [10, 11]

These control schemes vary in performance and way in which they are implemented. Modern control approach is Fuzzy based, Neural Network, ANFIS or hybrid technique. Randomness and aperiodic nature of chaos makes the mathematical control laws difficult and formal design of control with analytical means is complicated.

Numerical techniques such as Windowed Lyapunov Exponents are preferred for sample based computation of exponential tendency of system parameter going out of control. Fuzzy based controls are rarely of type–2. Various engineering systems are affected by crisp input assumption. Better controller design is type–2 based. Further sections of the paper will prove the efficacy of type 2 FLC over type–1 FLC.

Instability can be controlled using different types of controllers. Conventional tuned PID controller is first most preferred choice for control system designers owing to its simplicity of control law and reliable use in several engineering system with well defined mathematics. For a big system tuned PID controller focusses only on correction of error signal fed at its input and tunes itself so as to minimize the error between reference signal and actual output signal. For small systems this works quite well, while for large complex nonlinear dynamical system new intelligent controllers are designed for achieving multi prong objective apart from usual control of parameter. For example, in case of power system voltage control is possible using conventional PID controller but with violations slightly above prescribed limits of tolerable voltage magnitude. Further subsections here will throw light on merits and demerits of control scheme chosen based on simulation carried on nonlinear dynamical power system.

4 Nonlinear dynamical electrical power system model for instability control

Chaos, bifurcation and chaos driven instability studies can very well be demonstrated on benchmark model of power system [7 , 33]. Any power system model can be converted to generic equivalent model represented by three node model for investigating bifurcation, chaos and instability due to different initial conditions or parameter variation. Three node model of the power system in Fig. 3 is having the generator bus at node 1, infinite bus bar at node 3 and load bus at node 2. Bus 1 connects to the equivalent of several generators represented by a single machine with prime mover as a turbine with the governor. This model has an induction motor represented by dynamic load demand P and Q, in parallel with capacitor at bus 2. Bus 3 is an infinite bus-bar.

Fig.3

Generic power system model.

Strategies and algorithmic steps in this model are applicable to other practical large scale power system as well. Understanding of dynamics and control of the power system model, will help control system designers to formulate strategies for other large scale power systems.

This model is generic in nature since it has generator bus, load bus and connection to infinite grid represented by infinite bus bar. The reactive loading of the model causes bifurcation, chaos and instabilities due to mathematical nature of nonlinear dynamical power system. Jacobian matrix, eigen values, floquet multipliers and Lyapunov Characteristic exponents are analytical approaches which confirm the onset of bifurcation, chaos and instabilities. In Fig. 3, δ is rotor angle, ω _m is angular speed, δ _L load Angle, V _L is load voltage and Q is reactive power demand at load bus. The conventional model of generator at bus 1, can be represented by two state variables rotor angle δ and angular speed ω _m . Bus 3 load involves two more additional parameters, i.e. voltage at load bus V_L and electrical angle at bus 2 i.e. δ_L. Thus, there are four state variables for the fourth order model of power system i.e. $[x] = [δ ω_{m} δ_{L} V_{L}]$ (1)

An associated set of state equation for 4-D model of the power system are $\dot{δ} = ω_{m}$ (2) $\begin{matrix} {\dot{ω}}_{m} & = & K_{1} \sin (δ_{L} - δ + θ_{1}) V_{L} - K_{2} ω_{m} + \frac{P_{m}}{J} \end{matrix}$ (3) $\begin{matrix} {\dot{δ}}_{L} & = & K_{3} V_{L}^{2} - K_{4} \cos (δ_{L} - δ + θ_{1}) V_{L} - K_{5} V_{L} \\ - K_{6} \cos (δ_{L} - θ_{2}) V_{L} + K_{7} Q + K_{8} \end{matrix}$ (4) $\begin{matrix} {\dot{V}}_{L} & = & - K_{9} V_{L}^{2} + K_{10} \cos (δ_{L} - δ + θ_{3}) V_{L} \\ + K_{11} V_{L} + K_{12} \cos (δ_{L} - θ_{4}) V_{L} \\ - K_{13} Q - K_{14} \end{matrix}$ (5) $Q_{add} = Q_{SVC} = B_{SVC} ★ (△ V_{L})^{2}$ (6)

Where K₁, K2,... K₁₄ are constants with values mentioned in Appendix, B _SVC is static VAR compensator (SVC) susceptance and Q _SVC is reactive power injected or withdrawn at load bus due to controller action. Four state equations have four Lyapunov Characteristic Exponents (LCE) for system. These can be computed by using MATHEMATICA code [30]. Table 1 entries are the values of LCE for two different sets of initial condition. The nature of sign of LCE found is of type (+,0,-,-) which confirms that chaos due to the presence of one positive LCE. The specific range of 10.89 ≤ Q ≤ 10.894 and 11.377 ≤ Q ≤ 11.38 is reported to have one positive sign of LCE and thus confirm the existence of chaos [33].

Table 1

Lyapunov Characteristic Exponents for system under different initial conditions

Q	Initial Conditions	LE 1	LE 2	LE 3	LE 4	Nature of LCE
11.377	[0.3, 0, 0.2, 0.97]	0.24299	-0.02493	-4.6399	-87.2635	(+,0,-,-)
11.377	[0.35, 0, 0.2, 0.97]	0.13006	0.052379	-5.6881	-88.8286	(+,0,-,-)

4.1 Control and simulation strategy of system

Schematic diagram is as shown in Fig. 4. Four state variables are used to represent three bus power system model. Integrator blocks are used to integrate the derivative of state variable, thus state variable behavior with respect to time is available. Voltage is chosen as electrical quantity to be used for state feedback. Voltage signal is compared with reference voltage magnitude of 1.0 per unit. Difference in actual voltage and reference voltage is fed as input to different controllers. Output of controller is used to perturb the system to come out of bifurcation or chaos since sensitive nonlinear dynamical system have multiple states of equilibrium near to unstable chaotic point of operation of system. The variable amount of reactive power decided by controller will modify the reactive power loading to new modified values of reactive power Q _mod . Thus controller acts as a watchdog for monitoring deviation of voltage from reference value. Four situations were analyzed.

System without Controller.

System with PID controller.

System with Type–1 fuzzy logic controller.

System with Type–2 fuzzy logic controller.

Fig.4

Schematic diagram of system in presence of Interval Type–2 Fuzzy Logic Controller.

Matlab-Simulink software platform is used to solve the state equations of power system in presence of different controllers. Each state equation is modeled in Simulink platform provided in MATLAB along with integrator blocks and functional blocks of PID, fuzzy logic type–1 controller and interval type–2 fuzzy logic controller. Figures 4 and 5. shows different blocks and their interconnections to realize the solution of four state equations. First order derivative of all state variables are integrated to get state variables. The linear combination of state variables results in derivative of state variable itself. Hence a closed loop realization of four state equation is possible. Most important parameter in state equation is term of reactive power Q, if we perturb this parameter with the help of controller then voltage can be controlled effectively. Figure 5 is model of system as implemented in simulation, without controller. Figure 6 represents model with Interval Type–2 FLC for correcting voltage error and adjusting reactive power to new value by modification in Q by an amount dQ. Thus system is fed with Q_mod for controlling voltage instability.

Fig.5

Simulink Model of System without Controller.

Fig.6

Simulink Model of System with interval type 2 fuzzy logic controller (IT2FLC).

4.2 Without controller

System without controller follows its natural dynamics which shows onset of bifurcation, chaos and instability for specific values of parameters and initial conditions [33]. Reactive power demand at load bus 2 along with initial conditions mentioned in Table 1 causes chaos in load voltage profile as discussed in further sections.

4.3 Conventional PID controller for voltage instability control

Proportional-Integral-Derivative controller design requires setting of three controller gains named K _p , K _I and K _D . Analytical techniques such as Zeigler Nicol method exist for selection of proper controller gains but rarely used for large systems due to auto tuning feature present in modern software tools such as MATLAB. Software itself checks different controller gains and arrives at best suitable combination of above mentioned controller gains. Merit of such controller is that, their focus is reduction in error due to difference in reference signal and actual signal. Controller in Figs. 4 and 6. is replaced by PID controller for analyzing the effect of conventional PID controller.

4.4 Interval type II fuzzy system for voltage instability control

Actual voltage from power system is fed as state variable feedback signal and compared with reference voltage magnitude of 1 per unit. Error in voltage signal is passed through different types of controllers and dynamics of voltage variation are analyzed for any presence of bifurcation, chaos or instability. As in Figs. 4 and 6, interval Type–2 FLC is placed in front of voltage error signal. Membership function of voltage error is having uncertainty in the form of upper and lower limit. Proper rules are framed to inject reactive power as an output of fuzzy logic controller. Membership functions are not crisp as in case of Type–1 FLC. Equations (4) and (5) change to Equations (7) and (8)

\begin{matrix} {\dot{δ}}_{L} & = & K_{3} V_{L}^{2} - K_{4} \cos (δ_{L} - δ + θ_{1}) V_{L} \\ - K_{5} V_{L} - K_{6} \cos (δ_{L} - θ_{2}) V_{L} \\ + K_{7} (Q \pm dQ) + K_{8} \end{matrix}

(7)

\begin{matrix} {\dot{V}}_{L} & = & - K_{9} V_{L}^{2} + K_{10} \cos (δ_{L} - δ + θ_{3}) V_{L} \\ + K_{11} V_{L} + K_{12} \cos (δ_{L} - θ_{4}) V_{L} \\ - K_{13} (Q \pm dQ) - K_{14} \end{matrix}

(8)

5 Results and discussion

Voltage output without controller fluctuates chaotically and leads to instability at around 7.61 second as shown in Fig. 7. This is due to nature of state equation, specific parameter values and specific initial conditions. The fluctuation in voltage are reduced and instability also does not occur when PID controller acts and modify the reactive power. Type–1 fuzzy logic controller design further reduces the voltage fluctuations and no evidence of instability. Type–2 FLC design results in nearly flat voltage profile with least fluctuations in voltage and no instability within 7.61 seconds as shown in Fig. 7. Figure 8 is magnified version of Fig. 7, where one can compare response of Type–1 FLC and Type–2 FLC. From Fig. 8, it is clear that type–2 FLC shows less fluctuation than Type–1 design due to proper care of uncertainty in voltage error signal. Phase plane trajectories reveal dynamical interaction of state variables very well. It means trend or progressive growth of a particular state variable is easily predictable a priory since system parameters progress on a peculiar trajectory as tine increases. State variables of a system can not suddenly leave the trajectory for small disturbances.

Fig.7

Voltage behavior as a function of time.

Fig.8

Comparison of Type–1 FLC and Type–2 FLC effect on voltage profile.

Tuning of an individual controller requires computation of minimum integral error computations. Even for comparison between different types of controllers same performance parameters are used and a better controller is designed for system. Difference between reference signal and actual signal is termed as error. Controller efforts are to minimize this error. If area under error curve is minimum, it means actual output signal is following input command with less control efforts.

The integral of the absolute magnitude of error (IAE) criterion is defined as $IAE = \int_{0}^{\infty} ∣ e (t) ∣ dt$ (9)

The integral of the square of error (ISE) criterion is defined as $ISE = \int_{0}^{\infty} {∣ e (t)}^{2} ∣ dt$ (10)

The integral of the product of time and absolute magnitude of error (ITAE) criterion is defined as $ITAE = \int_{0}^{\infty} t ∣ e (t) ∣ dt$ (11)

Sensitivity of ITSE (integral time-square error) index is less and computationally not so easy as compared to ITAE (integral time-square error). While ITAE is best suited due to small overshoot feature, compared to IAE (integral of the absolute error) and ISE (integral square error) performance indices [15, 32].

5.1 Integral errors for evaluation of controllers

Table 2 shows performance index of controller based on integral error discussed in previous section. It is very clear error performance of system with type–2 fuzzy logic controller is much better than type–1, PID controller and without controller situation. This is due to better handling of uncertainty in error for most of the engineering systems. Since errors in system are uncertain due to different initial conditions and parameter values, better performance is ensured by system design with type–2 fuzzy logic controller.

Table 2
Error performance of different controllers

IAE ISE ITAE

Without Controller 0.3799 0.1305 1.865

With PID 0.251 0.008302 0.9496

With Fuzzy Type 1 0.2405 0.007715 0.9096

With Fuzzy Type 2 0.1224 0.002733 0.4386

	IAE	ISE	ITAE
Without Controller	0.3799	0.1305	1.865
With PID	0.251	0.008302	0.9496
With Fuzzy Type 1	0.2405	0.007715	0.9096
With Fuzzy Type 2	0.1224	0.002733	0.4386

5.2 Comparison of phase plane trajectory for different controllers

Dynamics of state trajectory can be controlled using state of art controller such as type-2 fuzzy logic controller. Voltage is one of the important parameter for electrical power system and it is chosen as state variable related to control system approach of state variable. Progressive growth of state variables can be judged from state variable trajectory. These trajectories may be inward or outward as time progresses. Controller job is to destroy them according to need of system if found going outward beyond a permissible limit. Figure 9 shows voltage trajectory for differentsituations i.e. without controller, with PID controller, with type–1 FLC and type–2 FLC.

Fig.9

Phase plane trajectory without controller and in presence of different controllers.

Figure 9 shows, for system without controller, voltage trajectory is a large spiral progressing outwards towards instability and this happens within few seconds. Figure 10 is magnified view of Fig. 9 and it is easy to compare type–1 and type–2 fuzzy design.

Fig.10

Comparison of phase plane trajectory with Type–1 FLC and Type–2 FLC.

System with PID controller voltage trajectory is a smaller spiral and it does not progresses outwards, which means voltage fluctuations are reduced and it is not leading to instability within same time frame.

System with fuzzy logic controller type 1, voltage trajectory is compressed spiral and it also does not progresses outward, which means voltage fluctuations are further reduced and instability does not occur.

System with fuzzy logic controller type 2 is straight line with small slope which means voltage does not change much and voltage profile with respect to time is flat.

6 Conclusion

System complexity in the form of chaos appearing at any point of time may lead to instability due to large disturbance of energy. Uncertainty in parameter values causing such chaos driven instability is a matter of concern. Type–2 fuzzy logic controller is key to control chaos in an uncertain environment of parameters and initial condition. Dynamics and uncertainties of system can very well be controlled with the help of type–2 fuzzy logic controllers. In paper, power system was chosen as a fit case for understanding severity of instability affecting a large customer base and proposed use of type–2 fuzzy logic controller. Power system is one of the largest nonlinear dynamical system.

After careful observation of obtained results in the form of time domain waveform, phase plane trajectory and integral square error, it can be easily concluded that systems with interval type–2 fuzzy controllers are the best choice to avoid chaos driven instabilities since it avoids instability as well as results in flat voltage profile in uncertain error condition of system. Careful design and framing of rules is essential for getting better response than type–1 fuzzy logic controller. Comparative performance demonstrated in paper advocates the use of intelligent and more realistic interval type-2 fuzzy logic controller. Control error is an uncertain parameter hence its membership function can not be taken as crisp value as in case of type 1 fuzzy logic controller. Type–2 fuzzy set theory is very well justified for such engineering systems.

Results and discussions for the case study and improvement in control scheme using interval type–2 fuzzy inference system is demonstrated in paper. The suggested controller will help in healthy and secure operation of power system under the fluctuating scenario of nonlinear dynamical system.

Footnotes

Appendix

Parameters used in simulation model of power system [15] are as mentioned in Table 3. Initial conditions at time t = 0, are assumed to be δ (0) =0.3 rad, ω_m (0) =0 rad/sec, δ_L (0) =0.2 rad and V_L (0) =0.97 pu.

References

Abbadi

, Nezli

and Boukhetala

, A nonlinear voltage controller based on interval type 2 fuzzy logic control system for multimachine power systems, International Journal of Electrical Power and Energy Systems45(1) (2013), 456–467.

Andrievskii

B.R.

and Fradkov

A.L.

, Control of chaos: Methods and applications. I. Methods, Avtomatika | Telemekhanika5 (2003), 3–45.

Aguirre

L.A.

and Torres

L.A.B.

, Control of nonlinear dynamics: Where do models fit in?International Journal of Bifurcation and Chaos10(03) (2000), 667–681.

Ahmad

W.M.

and Harb

A.M.

, On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos, Solitons & Fractals18(4) (2003), 693–701.

Chacon

, Maintenance and suppression of chaos by weak harmonic perturbations: A unified view, Physical Review Letters86(9) (2001), 1737–1740.

Chen

, Basso

, Genesio

, Giovanardi

and Tesi

, A Frequency domain methods for chaos control, Controlling Chaos and Bifurcations in Engg System (1999), 179–204.

Lal

D.K.

and Swarup

, Modeling and simulation of chaotic phenomena in electrical power systems, Applied Soft Computing11(1) (2011), 103–110.

Deepyaman

, Ayan

, Mithun

, Amit

and Ramdoss

, Tuning PID and PI^λ D^μ controllers using the integral time absolute error criteria, 4th International Conference on Information and Automation for Sustainability ICIAFS, 2008, pp. 457–462.

Fradkov

A.L.

and Evans

R.J.

, Control of chaos: Methods and applications in engineering, Annual Reviews in Control29(1) (2005), 33–56.

10.

Ginarsa

I.M.

, Soeprijanto

and Purnomo

M.H.

, Controlling chaos and voltage collapse using an ANFIS-based composite controller-static var compensator in power systems, International Journal of Electrical Power & Energy Systems46 (2013), 79–88.

11.

Ginarsa

I.M.

, Muljono

A.B.

and Nrartha

I.M.A.

, Controlling chaos and voltage collapse using layered recurrent network-based PID-SVC in power systems, Telkomnika11(3) (2013), 451–462.

12.

Hagras

, A hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots, IEEE Trans Fuzzy Syst12(4) (2004), 524–539.

13.

Hassani

and Zarei

, Interval type-2 fuzzy logic controller design for the speed control of DC motors, Systems Science and Control Engineering3(1) (2015), 266–273.

14.

Dobson

, Chiang

H.-D.

, Thorp

J.S.

and Fekih-Ahmed

, A model of voltage collapse in electric power systems, in Proceedings of the 27th IEEE Conference on Decision and Control, 1988, pp. 2104–2109.

15.

Nagrath

I.J.

and Gopal

, Control Systems Engineering, Fifth Edition, ISBN: 81-224-2008-7, New Age International Publishers, 2007, pp. 193–268, 297–343, 425–451.

16.

Mendel

, Type-2 fuzzy sets and systems: An overview, IEEE Computat Intell Mag2(1) (2007), 20–29.

17.

Mendel

and John

, Type-2 fuzzy sets made simple, IEEE Trans Fuzzy Syst10(2) (2002), 117–127.

18.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning—I, Information Sciences8(3) (1975), 199–249.

19.

, et al., Anticontrol of chaos for discrete TS fuzzy system, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications49(2) (2002), 249–253.

20.

Nangrani

S.P.

, Sunil

and Bhat, Instability, chaos and bifurcation control in nonlinear dynamical system behavior using perturb-boost fuzzy logic controller, Journal of Intelligent & Fuzzy Systems32(4) (2017), 3017–3029.

21.

Nangrani

S.P.

and Bhat

S.S.

, Numerical study of optimized fractional-order controller for chaos control of nonlinear dynamical power system, International Transactions on Electrical Energy Systems27(8) (2017), e2336.

22.

Nangrani

S.P.

and Bhat

S.S.

, Fractional order controller for controlling power system dynamic behavior, Asian Journal of Control20(1) (2018), 403–414.

23.

Karnik

, Mendel

and Liang

, Type-2 fuzzy logic systems, IEEE Trans Fuzzy Syst7(6) (1999), 643–658.

24.

Karnik

N.N.

and Mendel

J.M.

, Centroid of a type-2 fuzzy set, Inf Sci132(1-4) (2001), 195–220.

25.

Panda

M.K.

, Pillai

G.N.

and Kumar

, Interval type-2 fuzzy logic controller as a power system stabilizer, IEEE Conference on Advances in Power Conversion and Energy Technologies (APCET), 2012, pp. 1–6.

26.

Panda

M.K.

, Pillai

and Kumar

, An interval type-2 fuzzy logic controller for TCSC to improve the damping of power system oscillations, Frontiers in Energy7(3) (2013), 307.

27.

Liang

and Mendel

, Interval type-2 fuzzy logic systems: Theory and design, IEEE Trans Fuzzy Syst8(5) (2000), 535–550.

28.

John

and Coupland

, Type-2 fuzzy logic: A historical view, IEEE Computat Intell Mag2(1) (2007), 57–62.

29.

Salarieh

and Shahrokhi

, Indirect adaptive control of discrete chaotic systems, Chaos, Solitons & Fractals34(4) (2007), 1188–1201.

30.

Sandri

, Numerical calculation of Lyapunov exponents, Mathematica Journal6(3) (1996), 78–84.

31.

Song

, Chen

and Yuan

, New chaotic PSO-based neural network predictive control for nonlinear process, IEEE Transactions on Neural Networks18(2) (2007), 595–601.

32.

Schultz

W.C.

and Rideout

V.C.

, Control system performance measures: Past, present and future, IRE Trans on Automatic Control1 (1961), 22–35.

33.

, Jia

, Li

and Su

, Power system instability and chaos, Electric Power Systems Research65(3) (2003), 187–195.

34.

Zhu

and Tian

Y.P.

, Necessary and sufficient conditions for stabilizability of discrete-time systems via delayed feedback control, Physics Letters A343(1) (2005), 95–107.

Chaos driven instability control using interval type-2 fuzzy logic controller for better performance

Abstract

Keywords

1 Introduction

4 Nonlinear dynamical electrical power system model for instability control

4.3 Conventional PID controller for voltage instability control

4.4 Interval type II fuzzy system for voltage instability control

Table 2 Error performance of different controllers IAE ISE ITAE Without Controller 0.3799 0.1305 1.865 With PID 0.251 0.008302 0.9496 With Fuzzy Type 1 0.2405 0.007715 0.9096 With Fuzzy Type 2 0.1224 0.002733 0.4386

Footnotes

Appendix

References

Table 2
Error performance of different controllers

IAE ISE ITAE

Without Controller 0.3799 0.1305 1.865

With PID 0.251 0.008302 0.9496

With Fuzzy Type 1 0.2405 0.007715 0.9096

With Fuzzy Type 2 0.1224 0.002733 0.4386