Fuzzy Sliding Mode Control Design and Particle Swarm Optimization Based PSS for Multimachine Power System

The aim of this study is to design nonlinear robust controllers for multimachine power systems. A technique for the optimal tuning of Power System Stabilizer (PSS) by integrating the Particle Swarm Optimization (PSO) combined with the Fuzzy Sliding Mode Control (FSMC) is proposed in this study. The fuzzy tuning schemes are employed to improve control performance and to reduce chattering in the sliding mode. The objective of this method is to enhance the stability and the dynamic response of the multimachine power system operating in different operating conditions. In order to test the effectiveness of the proposed method, the simulation results show the damping of the oscillations of the angle and angular speed with reduced overshoots and quick settling time.


INTRODUCTION
Stability of power systems is one of the most important aspects in electric system operation.The size and complexity of modern electric power systems necessitates the construction of reduced-order dynamic models, or dynamic equivalents (Amin and Hemmati, 2012).Determination of transient stability is one of the major items of power system operation and planning (Mendiratta and Jayapal, 2010).
Most controllers PSSs used in electric power systems employ the linear control theory approach based on a linear model of a fixed configuration of the power system and thus tuned at a certain operating condition.The fixed parameter PSS, called conventional PSS, is widely used in power systems, it often does not provide satisfactory results over a wide range of operating conditions (Ben-Meziane et al., 2013) Overcome these drawbacks, a lot of different techniques have been reported in the literature pertaining to coordinated design problem of the PSS (Abido, 2010).
A Power System Stabilizer based on Particle Swarm Optimization (PSO) is proposed in this study.PSO technique is used for optimal tuning of PSS parameter to reduce the convergence time.Unlike the other heuristic techniques, it has a flexible and wellbalanced mechanism to enhance the global and local exploration abilities.Also, it suffices to specify the objective function and to place finite bounds on the optimized Parameters (Shayeghi et al., 2008).
The Sliding Mode Control (SMC) is essentially a switching feedback control where the gains in each feedback path switch between two values according to some rule.The switching feedback law drives the controlled system's state trajectory in to specified surface called the sliding surface which represents the desired dynamic behavior of the controlled system (Al-Duwaish and Al-Hamouz, 2011).The SMC has been reported as one of the most effective control methodologies for nonlinear power system applications due to its disturbance rejection, strong robustness subject to system parameter variations, uncertainties and external disturbances (Al-Duwaish and Al-Hamouz, 2011).The control gain of sliding mode may be selected high value, which causes the chattering on the sliding surface, or, this gain may be chosen smaller, which cause increasing of tracking error (Ha et al., 2001).Using Mamdani fuzzy inference method to adjust the corrective gain in sliding mode control.The fuzzy logic is used to overcome the disadvantages of the sliding mode control, while the FSMC provides better damping and reduced chattering.The effectiveness of the proposed method is tested on a multimachine power system under different operating conditions.The results evaluations show that the proposed method achieves good robust performance for damping low frequency oscillations under different operating conditions.
Multimachine power system model: Under some standard assumptions, the dynamics of n interconnected generators through a transmission network can be described by classical model with flux decay dynamics.The network has been reduced to internal bus representation assuming loads to be constant impedances and considering the presence of transfer conductance.The dynamical model of the i th machine is represented by the classical third order model (Colbia-Vegaa et al., 2008): I qi and I di represent currents in d-q reference frame of the i th generator, E' qi is the transient EMF in the quadrature axis, E fi (t) is the equivalent EMF in the excitation coil, X di and X' di are direct axis reactance and direct axis transient reactance, respectively: where, Pm i is the mechanical input power assumed to be constant, D i is the damping factor; all parameters are in p.u., H i is the inertia constant, in seconds, T' di is the direct axis transient short circuit time constant, in seconds, δ i is the rotor angle, in radians, ω i is the relative speed, ω s = 2πf is the synchronous machine speed, in rad/sec and G ij , B ij is the i th row and j th column element of the nodal conductance matrix and nodal susceptance matrix respectively, which are symmetric, at the internal nodes after eliminating all physical buses in p.u.We consider E fi (t) as the input of the system (Colbia-Vegaa et al., 2008).The state representation of the i th machine of a multimachine power system can be written in the following form: [ ] ., n, represents the state vector of i th subsystem and the control applied is given by: where,

PARTICLE SWARM OPTIMIZATION BASED ON PSS DESIGN
The PSO method is a member of wide category of Swarm Intelligence methods for solving the optimization problems (Sidhartha and Ardil, 2008).It employs an appropriate model of fitness for particle, to evaluate fitness value of each particle and to record the particle that is the highest fitness value in an iterative procedure (Liang et al., 2011).The position corresponding to the best fitness is known as pbest and the overall best out of all the particles in the population is called gbest (Sidhartha and Ardil, 2008).The modified velocity and position of each particle can be calculated using the current velocity and the distance from the pbest j,g to gbest g as shown in the following formulas (Gaing, 2004): .
where, n is the number of particles in a group, m is the number of members in a particle, t is pointer of iterations (generations), v is the velocity, x is the position of each particle, c 1 , c 2 are positive constants referred to as acceleration constants and must be c 1 + c 2 ≤4, usually c 1 = c 2 = 2, r 1 , r 2 are random numbers between 0 and 1 and w is the inertia weight, which produces a balance between global and local explorations requiring less iteration on average to find a suitably optimal solution.It is determined by the following equation (Gaing, 2004): where, w max is the initial weight, w min is the final weight, iter is the current iteration number, is the For which ∆ω is the speed deviation in radians per seconds and t sim is the range of time for the simulation.The nonlinear model was conducted for the time domain simulation over the simulation period which is aimed at minimizing the fitness function so that the system response such as settling time and overshoots and the constraints are the boundaries of PSS parameters.The transfer function of the i th PSS is given by (Hasan et al., 2012): )( ) where, The block diagram of the conventional PSS is shown in Fig. 2, in which case the generator rotor speed deviation is used as the only stabilizing signal.The Conventional PSS consists of an amplifier, a washout filter and two lead-lag compensators (Hasan et al., 2012).
The PSS parameters to be tuned within their boundaries are formulated as follows (Sanjeev and Chaturvedi, 2013).Optimize J. Subject to: The membership functions for input and output variable are triangular given in Fig. 3 to 5. The 49 rules described presented in a matrix called matrix inference given in the following Table 1 (Amer et al., 2011).
The continuous component insures the motion of the system on the sliding surface whenever the system is on the surface.The equivalent control that maintains the sliding mode satisfies the condition: The relative degree is r = 3 then, the switching function can be written as (Abera and Bandyopadhyay, 2008): where, k = [k 1 , k 2 , 1] T are the coefficients of the Hurwitz Polynomial:

S e k e k e e k e k e f x g x u
If f (x) and g (x) are known, we can easily construct the sliding mode control: The fuzzy sliding mode control term is: The control input u smc to get the state δ to track δ r is then made to satisfy the Lyapunov-like function: By the following sliding condition: where, Fuzzy sliding mode control combined with PSS based on PSO: The control law used in this study is composed of three terms, the nominal control u eq , the robust term represented by the fuzzy sliding mode controller u fsmc and the regulator PSS optimized by the PSO algorithm u * pss : ( ) where, α f is the gain of sliding mode controller tuned by a fuzzy logic rule base: The combination between the two controllers, fuzzy sliding mode controller with Power System Stabilizer based on PSO algorithm, enhances the damping of the oscillations and the stability of the network.

SIMULATION OF MULTIMACHINE POWER SYSTEM
To evaluate the performance of the proposed control, we performed simulation for the three-machine nine-bus power system as in Fig. 6, with the aim to compare the performance of the proposed controller with SMC controller and the conventional PSS.Detail of the system data are given in Table 2 (Colbia-Vegaa et al., 2008).The following equilibrium point: X ir = (x i1r , x i2r , x i3r ) = [δ i ∆ω i E' qi ] for i = 1, 2, 3 of the three-machine system is considered: In this case, the control law used constituted by the nominal control, sliding mode control and power system stabilizer.The control parameters of sliding mode control used are α i = 0.03 for i = 1, 2, 3.The specified parameters of the PSS that are used in this study given Fig. 6: Three-machine nine-bus power system 2.0, 2.0 wmax, wmin 0.9, 0.4 in Table 3.The coefficients of the Hurwitz Polynomial used in this study for the multimachine power system are: The simulation results presented in Fig. 7 to 9 shows the occurrence of chattering caused by a poor choice of controller gain.
To demonstrate performance robustness of the proposed method, two performance indices: the Integral of the Time multiplied Absolute value of the Error (ITAE) and the Integral of Time weighted Squared Error (ISTE) based on the system performance characteristics are being used as: More than the values of the performance indices ITAE and ISTE are lower, the response of the system in time domain is better.Numerical results of these indices for all cases are presented in Table 4.
The parameters used in PSO algorithm are given in Table 5.Control parameters and their boundaries are given below: The aim is to compare the performance of the conventional PSS, the sliding mode control and the   simulation results demonstrated that the proposed control is capable of guaranteeing the robust performance of the multimachine power system for a wide range of operating conditions (Table 6).
In this case, the washout time of the PSS T wi = 10 and the control parameters of sliding mode control used are α i = 0.0001 for i = 1, 2, 3.
With the proposed control, the mechanical variables such as the angles rotor (δ 1 , δ 2 ) and the deviation speed (∆ω 1 , ∆ω 2 ) in the generators (G1 and

CONCLUSION
In this study, the proposed control provides an efficient solution to damp the Low frequency oscillations in the multimachine power system and eliminates the chattering in sliding mode control.The design problem of the robustly PSS parameters selection is solved by a PSO algorithm, which enhances the stability of the power system.Also, the robustness and the performance of the fuzzy sliding mode controller design has been proved and evaluated by the dynamic simulation responses of the multimachine power system.

Fig. 1 :
Fig. 1: Particle swarm optimization algorithmmaximum iteration number.The typical range of w from 0.9 at the beginning of the search to 0.4 at the end of the search, Fig.1shows the flowchart of the PSO algorithm(Rajendraprasad and Panda, 2013).In this study, the PSS design problem is formulated as an optimization problem and solved by PSO method to improve optimization synthesis and find the global optimum value of the fitness function.Selection of a desirable fitness function is very important to optimize PSS parameters.Because, different fitness functions promote different PSO based on PSS behaviors(Manisha et al., 2013).The PSS controller is optimized by minimizing the objective function (J) in order to improve the system response in terms of oscillation and settling time.Despite there are several methods to come up with the improvement of the performance of the control system (GirirajKumar et al., 2010), such as Integral of Squared Error (ISE), Integral of Time weighted Squared Error (ITSE), Integral of Absolute Error (IAE) and Integral of Time weighted Absolute value of Error (ITAE):

Fig. 2 :
Fig. 2: Block diagram of the conventional power system stabilizer lower and upper bounds of gains PSS T min ji and T max ji = The lower and upper bounds of the time constants of all controllers Fuzzy sliding mode controller: Fuzzy controller design includes the definition the following parameters: Number of partitions of input space and output membership functions, rule base, inference method, fuzzification and defuzzification (Ben-Meziane et al., 2013).In this section, the fuzzy sliding mode controller with varying control gain α f is presented.More specifically, a fuzzy inference system is used to adjusting the control gain α f in order to improve the performance of the controller.The rules in the rule base are in the following form(Amer et al.,  sets.The fuzzy variables input are defined for the rule base as, (e, ˥Ӕ ) = {NB, NM, NS, Z, PS, PN, PB} and the fuzzy output is (α f ) = {VVS, VS, S, M, B, VB, VVB}.

Fig. 3 :
Fig. 3: Membership functions of error topology of the network has been represented by the conductance nodal matrix G and by the susceptance nodal matrix B

Fig. 13 :
Fig.13: Deviation speed ∆w1 of d axis (pu) Xq : Steady state reactance of q axis (pu) Xd : Steady state reactance of d axis (pu) T'do : Time constant of excitation circuit (s) T sim : Simulation time (s) Tw : Washout filter (s) T1-T4 : Time constants of lead-lag dynamic compensator (s) K : Gain of the Stabilizer PSS : Power System Stabilizer PSO : Particle Swarm Optimization SMC : Sliding Mode Controller FSMC : Fuzzy Sliding Mode Controller

Table 2 :
Nominal parameters values

Table 6 :
Optimized PSS parameters by PSO algorithm control law proposed in this study composed by the three terms: the nominal control, the robust term FSMC and the PSS optimized by PSO algorithm.The nonlinear