Power System Stabilizer Design Based on a Particle Swarm Optimization Multiobjective Function Implemented Under Graphical Interface

Power system stability considered a necessary condition for normal functioning of an electrical network. The role of regulation and control systems is to ensure that stability by determining the essential elements that influence it. This paper proposes a Particle Swarm Optimization (PSO) based multiobjective function to tuning optimal parameters of Power System Stabilizer (PSS); this later is used as auxiliary to generator excitation system in order to damp electro mechanicals oscillations of the rotor and consequently improve Power system stability. The computer simulation results obtained by developed graphical user interface (GUI) have proved the efficiency of PSS optimized by a Particle Swarm Optimization, in comparison with a conventional PSS, showing stable system responses almost insensitive to large parameter variations.Our present study was performed using a GUI realized under MATLAB in our work.


I. INTRODUCTION
Power system stabilizers (PSS) have been used for many years to add damping to electromechanical oscillations. The use of fast acting high gain Automatic Voltage Regulator (AVR) and the evolution of large interconnected power systems with transfer of bulk power across weak transmission links have further aggravated the problem of low-frequency oscillations. The continuous change in the operating condition and network parameters result in corresponding changes in the system dynamics [1,2]. This constantly changing nature of power systems makes the design of damping Received: XX XX, XXXX; received in revised form: xxxx xx, xxxx; accepted: xxxx xx, xxxx; available online: xxxx xx, xxxx. controllers a very difficult task. Power system stabilizers (PSS) were developed to extend stability limits by modulating the generator excitation to provide additional damping to the oscillations of synchronous machine rotors. Recent developments in the field of robust control provide methods for designing fixed parameter controllers for systems subject to model uncertainties.
Conventional PSS based on simple design principles such as PI control and eigenvalue assignment techniques have been widely used in power systems. Such PSS ensure optimal performance only at their nominal operating point and do not guarantee good performance over the entire operating range of the power system. This is due to external disturbances such as changes in loading conditions and fluctuations in the mechanical power. In practical power systems networks, a priori information on these external disturbances is always in the form of certain frequency bands in which their energy is concentrated.
PSO appeared as a promising evolutionary technique for handling the optimization problems. PSO has been popular in academia and the industry mainly because of its intuitiveness, ease of implementation, and the ability to effectively solve highly nonlinear, mixed integer optimization problems that are typical of complex engineering systems [3,4].
As mentioned above, we have developed a global optimization method based on PSO and a multiobjective function using relative and absolute stability parameters that are obtained from the system eigenvalue analysis.

A. Power System Description
The SMIB system used in our study is shown in Fig. 1 including Synchronous Generator, AVR and PSS, and Infinity Bus.

B. The Modeling of Powerful Synchronous Generators
This paper is based on the Park modeling of powerful synchronous generators. The PSG model is defined by Equations (1) to (6) [1,2]: The first two equations are obtained from the second order swing equation as with

C. PSS Model
The conventional lead-lag structure is chosen in this study as a Conventional PSS (CPSS). The structure of the CPSS controller model is shown in Fig. 2.
In this paper the PSS signal used, is given by Refs. [1,2].

A. PSO Theory
PSO is one of the methods among the smart methods for solving the optimization problems that was first introduced as an optimization method by Kennedy and Eberhart [5] and it is inspired by the bird's intelligence. In PSO algorithm, each particle has a value that is called fitness and it is calculated by the fitness function. This fitness is measured by the amount of the closeness to the target. Basically, the beginning of the PSO is in a way that a group of particles is randomly created and in each level, each particle is optimized by the use of two optimum values.
The first value is called the best personal experience or the pbest. The other best result which is used is the best position that is gained by a group of particles and it is called the gbest. The equation of the velocity update [6] is given as: The role of the weight parameter in converging the algorithm is so important because it is used for affecting the velocity at the present moment by the velocity of the previous moment. The equation of the position update is given as: The steps PSO are shown in Fig. 3.

B. PSO Numerical Application
We consider the simple case of function with two variables x 1 and x 2 belongin to the natural number set. We intend to minimize: Subject to

IV. APPLICATION OF THE PARTICLE SWARM OPTIMIZATION TO PSS A. Multiobjective Function Choice
The choice of objectives functions generally based on the needs of our controlled system. The purpose of the PSS is to ensure satisfactory oscillations damping and to ensure the overall system stability to different operation points. To meet this goal, we use a function, composed of two multiobjective functions. To understand the concept of this multiobjective function we consider two examples: Example 1: We considered system even imaginary part ω s1 = ω s2 and deferring real part σ: • System 11 : P 11,2 = −6 ± 6 • System 12 : P 12,2 = −1 ± 6 The poles systems on the imaginary axis and step responses match each system shown in Fig. 6. On learning this result we can see that the decrease real part σ improved dynamic performance and system stability.

Example 2:
We considered two systems with real part σ s1 = σ s2 and deferring imaginary part: The poles systems on the imaginary axis and step responses match each system shown in Fig. 7. The increase in damping coefficient ζ improves system stability.
In view of these results, we proposed an objective function which is composed of two functions. This function aims to maximize stability margin by increasing the damping factors while minimizing the real parts of the eigenvalues of the system, and we the must maximize the set of two objective functions.

B. Steps of Multiobjective Function Calculation
The multiobjective function calculating steps are: 1) Formulate the linear system in an open loop (without PSS). 2) Locate the PSS and its parameters initialized by the PSO through an initial population. 3) Calculate the closed loop system eigenvalues and take only the dominant modes.
The transfer function of the entire closed loop system (Fig. 8) F (s) becomes: The eigenvalues of the closed loop system are the poles of the transfer function F (s) with GPSS(s) = 1 1 + T f p 5) Gather both objective functions in a multiobjective function F as follows: 6) Returning this multiobjective function value the to the PSO program to restart a new generation.

A. Implementation of PSS-PSO under the Proposed GUI/ Matlab
To analyzed and visualized the different dynamic behaviors, we have created and developed a GUI under MATLAB, see Fig. 9. This GUI allows us to optimize the controller parameters by PSO, to perform control system from PSS controller, to view the system reg- ulation results and simulation, to calculate the system dynamic parameters, to test the system stability and robustness, and to study the different operating regime (under-excited, nominal and over-excited regime).

B. Optimizations Results
We present an example for optimization and tuning the parameters of the PSS-PSO using our proposed GUI with these parameters: the number of individuals = 10 and the number of population = 10.  The optimized parameters are: K 1 = 3.0961, K 2 = 4.2697, T 1 = 0.0230, T 2 = 0.0323, σ = −3.9885, ζ = 0.322, and multiobj = 4.3108

PSO Initialization
The obtained optimization results show that PSO optimization technique is well adapted to the multiobjective function (see Fig. 10): • Increase damping coefficient ζ. • Decrease real part of pole σ. • Increase multiobjective function. Table I shows the PSS parameters (K 1 , K 2 , T 1 , and T 2 ) that were optimized by particle swarm optimization under different operating regime (under-excited, nominal and over-excited regime) with different synchronous power generators of type: TBB-200, TBB-500, BBC-720, TBB-1000 (the parameters are shown in Appendix) [7].

C. Simulation Results
For stability study of SMIB system, we have performed perturbations by abrupt variations of turbine torque ∆T m of 15% at t = 1 second. The following results were obtained by studying the SMIB for the following cases: Opened Loop and Closed Loop System with PSS and PSS-PSO.
We have simulated three operations: the underexcited, the rated and the over-excited. Figures 11  and 12 show simulation results with: a:'s' variable speed, b:'P e' electromagnetic power system c:'delta' the internal angle, d:'U g' terminal voltage.
From the simulation results, it can be observed that the use of PSS optimized by PSO improves considerably the dynamic performances and granted the stability of the SMIB system studied even in critical situations (especially the under-excited regime).   VI. CONCLUSIONS In this article, we have optimized the PSS parameters by Particle Swarm Optimization; the optimized PSS are used for powerful synchronous generators exciter voltage control in order to improve static and dynamic performances of the power system. This technique (PSO) allows us to obtain a considerable improvement in dynamic performances and robustness stability of the SMIB studied. All results are obtained by using our created GUI/MATLAB.