Optimal Control of PID-FUZZY based on Gravitational Search Algorithm for Load Frequency Control

Optimal Control of PID-FUZZY based on Gravitational Search Algorithm for Load Frequency Control

Habib Danaeefar, Hasaan Barati, Seyyed Arman Shirmardi

Department of Electrical and Computer Engineering, Islamic Azad University Dezful Branch,

Dezful, Iran

Abstract:- In this paper, an efficient and effectual optimization algorithm called Gravitational Search Algorithm (GSA) is suggested to optimally tune the PID- FUZZY parameters toward accurate frequency control of two-area power system. An effective and precise optimization algorithm i.e., GSA has been constructed based on the gravity law and mass interactions. In this algorithm, the explorer operators are a masses collection which interacts with each other according to the motion laws and Newtonian gravity. To appropriately and accurately confirm the dynamic performance of suggested GSA-based PID-Fuzzy, four performance indexes, i.e.: ISE, ITSE, IAE and ITAE have been considered as objective functions. Furthermore, the simulations results by the suggested GSA- base PID-FUZZY have been compared with the simulation results based on Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC) algorithms which are generally realized as conventional heuristic algorithms and extensively applied by the researchers and scholars. Finally, optimal GSA-based PID-FUZZY presents a high dynamic performance as compared to PSO-based PID-FUZZY and ABC-based PID-FUZZY.

Keywords: Power system dynamic stability, low frequency oscillation, automatic generation control, optimization algorithms, gravitational search algorithms.

1. INTRODUCTION

   The value of consumed power around the world exhibits the state of the art level of countries that shows the importance of power quality, voltage regulation and system frequency stability issues. That is to say, all criterions of power quality have been studied and analyzed on frequency and voltage of power system. Since the system frequency control is highly longer than the voltage control, it is regarded as salient benchmark in the power system. Thus, system frequency stability should be principally investigated and involved in many researches and projects [1]. Integration of many different power sources or areas has constructed the interconnected power system. Any power variations in power source and load which happen in these areas can disturb other connected areas. The tie-line characteristic between these areas is one other important item in impressing the frequency stability. During the high variations in system frequency which go beyond the defined limits, they can create severe power system instability, which will stop the connected power plants and even shut down the entire

   system in the later step. In this instance, the fed areas of the system remain without energy and cause vast economic damages [2]. Consequently, considering these vast economic damages due to system collapse, the frequency control of loads is regarded a salient problem which must be considered to hinder such damages.

   Automatic Generation Control (AGC) has decreased the deviations of tie-line power and frequency in order to swiftly restore its nominal frequency [3-5]. The conventional controller such as: classical Proportional Integral Derivative (PID) controller has been widely used for AGC in interconnected power systems. But, this controller cannot effectively suppress the fluctuations tie-line power and frequency during severe perturbations. Considering the benefits of PID controller, it has been integrated with Fuzzy controller to construct PID-Fuzzy controller aimed at suppression of low frequency oscillations. As reported in various literatures, Fuzzy controller provides good performance during the normal, unbalance and transient conditions [6-8]. The parameters of PID-Fuzzy can be tuned in real time in accordance with load perturbations to enhance its accuracy and robustness toward acquiring the best control aims.

   In recent years, the evolutionary optimization algorithms such as Bacteria Foraging (BF), Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC), Ant Colony Optimization (ACO) have been widely used by researchers to solve the various objective functions in different problems [9-13]. Although aforementioned algorithms seem to be excellent to optimally tune the parameters of controller, they introduce a slow convergence with local minimums. Gravitational Search Algorithm (GSA), a new evolutionary optimization algorithm, presents a fast convergence and accurate performance in these aspects [14-16]. In this regard, it is applied to optimally tune the parameters of PID-Fuzzy for stabilizing the interconnected power system.

   In this study, the parameters of PID-Fuzzy have been tuned by GSA, PSO and ABC so that its performances can be well compared and appraised. The relevant studies have been performed in two-area interconnected power system. Severe load perturbation has affected the interconnected power system to deal with the dynamic stability criterions with presence of PID-FUZZY. Finally, optimal GSA-based PID- FUZZY presents a high dynamic performance as compared to PSO-based PID-FUZZY and ABC-based PID-FUZZY.

2. INTERCONNECTED POWER SYSTEM

   STRUCTURE

   Kinetic and potential energy has been transformed into mechanical energy using turbines, following that, the

   mechanical energy has been transformed into electrical energy using power plants generators. Simple motion equation based on the mentioned concept can be presented by:

   Tm Te J d

   dt

   (1)

   Where, Tm and Te are respectively mechanical and electrical torque. J and are respectively the inertia and angular speed.

   The understudy power system contains two interconnected areas via tie-line. Note that, all available generators in each area have a combined construction. Occurrence of any perturbations in each area of power system leads to frequency deviation in other areas. Then, this deviation creates fluctuation in tie-line power. In this regard, both the tie-line power and frequency deviations must be damped to maintain the dynamic stability of interconnected power system. The tie-line power can be presented by:

   Pline12

   V1 V2

   X12

   sin(1 2 )

   (2)

   Where, V1 and V2 are respectively voltage of first and second areas. 1 and 2 are respectively machine angle of first and second areas. X12 indicates tie-line impedance. The machine angle deviation of each area can be presented by:

   2 fdt

   The tie-line power deviation can be presented by:

   (3)

   P V1 V2 cos(

   )(

   )

   (4)

   12 X12

   1 2 1 2

   Where, the synchronous coefficient is written as follows:

   T V1 V2 cos( )

   (5)

   12 X12 1 2

   The tie-line power deviation is rewritten by:

   Ptie T12(1 2 )

   (6)

   Thereupon load deviation PL, tie-line power deviation, frequency deviation and Area Control Error (ACE) of each area can be given by following equations:

   PL1( 1 D2 )

   Ptie

   1 1

   R1 R2

   R2

   D1 D2

   (7)

   f

   PL1

   (8)

   1 1

   R1 R2

   D1 D2

   ACE1 Ptie B1 f1

   ACE2 B2 f2 a12 Ptie

   (9)

   (10)

   B1 1/R1

   AREA 1

   Gg1(s)	Xg1

   Gg1(s)	Xg1

   pL1(s)

   + ACE1

   +

   –

   Gc1(s)

   Gc1(s)	Pref1

   – PG1

   + –

   Grp(s)	Prp

   Grp(s)	Prp

   Gt1(s)	Pt1

   Gt1(s)	Pt1

   Gps1(s)

   –

   f1

   a12

   a12

   –

   –

   + ACE2

   Controller 1

   Gc2(s)	Pref

   Gc2(s)	Pref

   2

   Governor 1 Re-heater 1

   Ptie

   Turbine 1

   +

   +

   Pt2 –

   Power System 1

   +

   Gtl(s)

   a12

   a12

   –

   Tie line

   f2

   PG2

   –

   + –

   Gt2(s) G

   ps2

   (s)

   Gg2(s)	Xg2

   Gg2(s)	Xg2

   Grp(s)	Prp

   Grp(s)	Prp

   Controller 2

   Governor 2 Re-heater 2

   Turbine 2

   Power System 2

   B2 1/R2

   AREA 2

   pL2(s)

   Figure 1. Two-area interconnected power system

   Figure 1 presents the linearized model of two-area interconnected power system. The following equations give the transfer function of each block:

   Governor:

   Gg1(s) Gg 2 (s)

   Reheater:

   Kh

   Th s 1

   (11)

   Grp(s) Grh 2 (s) (Kr12 Tr11) s 1

   Tr1 s 1

   (12)

   Turbine:

   Gt1(s) Gt 2 (s)

   Power system:

   Kt

   Tt s 1

   (13)

   Gps1

   (s) G

   ps 2

   (s)

   Kg

   Tg s 1

   (14)

   Tie-line:

   Gtl (s) 2 T12

   s

   (15)

   The understudy two areas of interconnected power system are thermal plant. Rate of each area is 2000 MW with 1000 MW (nominal load). This system has been widely studied in different literatures which propose, design and analyze the required controller for AGC to enhance the power system dynamic stability [17-19].

3. CONSTRUCTION OF PID-FUZZY CONTROLLER

   A PID-Fuzzy controller enhances the dynamic performance of PID controllers, because Fuzzy logic can appropriately control the changes in parameters of power systems or operating point via auto-tuning gains of PID controller [20]. Precise design of Fuzzy controller has been performed by choosing an appropriate membership functions and rule formulation. So, general membership functions and rules

   have been selected for input/output while the parameters of PID controller can be optimally tuned to enhance the overall dynamic performance of PID-Fuzzy controller [21]. Two- dimensional rule base PID-Fuzzy controller has been designed by the error signal along with its derivative for input signal PID controller for output signal [22. As mentioned above, PID-Fuzzy controller which is shown in Fig. 2 (a) is designed for AGC to enhance the dynamic performance of power system. K1 and K2 are input gains of PID-Fuzzy controller while KP, KI and KD are its output gains. The fuzzy rules and its input membership function are respectively presented in Table 1 and Fig. 2(b).The membership functions are defined by: PB (Positive Big), PM (Positive Medium), PS (Positive Small), ZE (Zero), NS (Negative Small), NM (Negative Medium) and NB (Negative Big) as fuzzy measure.

   ACE

   Fuzzy Logic Controller

   K1

   KP

   u

   du/dt

   KI

   K2 KD

   1/s

   d/dt

   Figure 2(a). PID-Fuzzy controller model

   NB NM NS ZE PS

   PM PB

   -1 -0.5 0 0.5 1

   Figure 2(b). Normalized membership function for inputs

   Table 1. Lookup table of fuzzy rules

   ACE	NB	NM	NS	ZE	PS	PM	PB
   ACE
   NB	NB	NB	NB	NB	NM	NS	ZE
   NM	NB	NM	NM	NM	NS	ZE	PS
   NS	NB	NM	NS	NS	ZE	PS	PM
   ZE	NB	NM	NS	ZE	PS	PM	PB
   PS	NM	NS	ZE	PS	PS	PM	PB
   PM	NS	ZE	PS	PM	PM	PM	PB
   PB	ZE	PS	PM	PB	PB	PB	PB

4. GRAVITATIONAL SEARCH ALGORITHM

   GSA is a heuristic stochastic evolutionary algorithm proposed by Rashedi et al in 2009 [17]. It is inspired by the mass interactions and gravitation newton principle. All objects are absorbed by a gravitational force, which forces all objects to move toward heavy masses as shown in Fig. 3(a). It exhibits a natural behavior between earth and the moon. The weightiest object suggests the problem solution. The Flowchart of GSA process is given in the Fig. 3(b), and also

   M1

   F14

   its general Pseudo code is provided in Fig. 3(c). According to GSA procedure, each mass includes four statements: position, inertial mass, active gravitational mass, and passive gravitational mass [18]. The mass position related to the problem solution, and its gravitational and inertial masses have been defined via an objective function. That is to say, each mass gives a result that GSA is navigated via correct tuning the gravitational and inertia masses [19]. By time span, it is expected that masses to be pulled by the weightiest mass which gives an optimal response in the search space.

   M4

   F12 a1

   M2

   F13

   F1 M3

   Figure 3(a). Every mass accelerate towards to the resultant force

   Generate initial population

   Evaluate the fitness for each agent

   Update G, best and worst of population

   Calculate M and a for each agent

   Update velocity and position

   No Meeting end of criterion?

   Yes

   Return best solution

   Return best solution

   Figure 3(b). Flowchart of GSA procedure

   Random initialization of the population

   Find the best and worst solutions in the initial population

   while (stop criterion)

   for i=1:N (for all elements)

   update G(t), best(t), worst(t) and Mi(t) for i=1,2,N calculate the mass of individual Mi(t)

   calculate the gravitational constant G(t)

   calculate the acceleration aih(t)

   Random initialization of the population

   Find the best and worst solutions in the initial population

   while (stop criterion)

   for i=1:N (for all elements)

   update G(t), best(t), worst(t) and Mi(t) for i=1,2,N calculate the mas of individual Mi(t)

   calculate the gravitational constant G(t)

   calculate the acceleration aih(t)

   update the velocity and position of each individual vih , xi

   end for i=1:N (for all elements)

   Find the best individual

   end while (stop criterion)

   Display the best individual as the solution

   update the velocity and position of each individual vih , xi

   end for i=1:N (for all elements)

   Find the best individual

   end while (stop criterion)

   Display the best individual as the solution

   h

   h

   Figure 3(b). GSA Pseudo code

5. SIMULATIONS AND RESULTS

   The performance of control systems has been assessed based on an especial performance index depends on the required objective function. The performance index is alternatively used to design of controllers which are commonly known by: integral of squared error (ISE), integral of time multiplied squared error (ITSE), integral of

   absolute error (IAE) and integral of time multiplied absolute error (ITAE) [23-25]. To appropriately and accurately confirm the dynamic performance of suggested GSA-based PID-Fuzzy, all aforementioned have been taken into account for designed optimization problem. These objective functions are given as follows:

   T

   ISE F12 0

   F22

   • Ptie,12

   2 .dt

   (16)

   T

   ITSE F12 0

   F22

   • Ptie,12

     2 .t.dt

     (17)

     T

     T

     IAE F1 F2

     0

   • Ptie,12 .dt

     (18)

     T

     T

     ITAE F1 F2

     0

   • Ptie,12 .t.dt

   (19)

   The PID-Fuzzy parameters have been optimally tuned by GSA in order to alleviate the tile-line power and frequency deviations. Furthermore, the controller parameters have been optimized by PSO and ABC to more validate the dynamic performance of GSA-based PID-Fuzzy controller. The operational constraints of controller parameters are given as follows:

   K min K K max

   1 1 1

   K2min K2 K2max

   KPmin KP KPmax

   KDmin KD KDmax

   KI min KI KI max

   (20)

   In the simulations, a step load perturbation occurs in first area of the interconnected power system to affect the dynamic stability. The optimization problem has been performed with GSA, PSO and ABC considering all chosen performance indexes. The load perturbation occurs at t=10 s with PL=0.01. The optimal parameters of PID-Fuzzy controller optimized by GSA, PSO and ABC are tabulated in Table 2. Meanwhile, the values of defined performance indexes, i.e.: ISE, ITSE, IAE and ITAE are presented in Table 3. Also, three stability benchmarks i.e.: settling time, overshoot and undershoot are tabulated in Table 4.Both the time-varying frequency and tie-line power deviations are respectively presented in Fig. 4(a-c).

   Table 2. The parameters of PID-Fuzzy optimized by GSA, PSO and ABC

   	GSA		ABC		PSO
   	Area 1	Area 2	Area 1	Area 2	Area 1	Area 2
   K1	1.1854	1.6210	1.5021	1.8428	1.8029	2.2367
   K2	0.5649	0.7431	0.6712	0.9142	0.8291	1.0056
   KP	0.6094	0.7885	0.7003	0.9462	0.9906	1.2049
   KI	0.7951	1.0359	0.9615	1.3410	1.2986	1.372
   KD	0.4932	0.6372	0.58960	0.7743	0.5765	0.9211

   Table 3. The optimal value of dynamic performance indexes achieved by GSA, PSO and ABC

   Parameter indexes	GSA	ABC	PSO
   ISE	2.7025	4.0962	4.7643
   ITSE	0.1496	0.1924	0.2241
   IAE	2.5618	3.4577	3.9553
   ITAE	71.7942	110.4983	130.3509

   Table 4. The optimal value of dynamic performance indexes achieved by GSA, PSO and ABC

   Stability benchmarks	GSA	ABC	PSO
   Overshoot	0.0116	0.0148	0.0156
   Undershoot	0.0192	0.01984	0.0208
   Settling time	55.31	62.12	68.76

   Figure 4(a). Change in frequency deviation of area-1

   Figure 4(b). Change in frequency deviation of area-2

   Figure 4 (c). Change in tie-line power deviation between area-1 and area-2

   As for the Table 2-4, all performance indexes optimized by GSA present less values as compared to corresponding indexes optimized by PSO and ABC. The presented results prove that the overall dynamic stability of interconnected power system has been highly improved by GSA-based PID- Fuzzy more than two others. The time-varying curves presented in Fig. 4(a-c) have better portrayed the dynamic performance of GSA-based PID-Fuzzy.

6. CONCLUSION

In this paper, PID-Fuzzy controller is suggested for AGC appropriately enhance the dynamic stability of two-area interconnected power system. GSA algorithm has been complementary applied to optimize the parameters of PID- Fuzzy so that its maneuverability to be increased. Hence, PSO and ABC have been taken part in optimization problem to clear the accuracy and capability of GSA. Dynamic stability analysis has been more carried out with consideration of four performance indexes i.e.: ISE, ITSE, IAE and ITAE. The pertinent dynamic stability studies have been performed by affecting the interconnected power system caused by load perturbation occurrence in area-1. Eventually, the simulation results confirm the high dynamic performance of GSA-based PID-Fuzzy via damping the low frequency oscillations of tie-line power and frequency deviations.

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