Optimal Power Flow Incorporating FACTS Devices using Gravitational Search Algorithm

Yusuf SONMEZ Department of Electrical Technology,

Gazi Vocational Collage, Gazi University, Ankara, TURKEY

ysonmez@gazi.edu.tr

Ugur GUVENC Department of Electrical and Electronic Engineering,

Technology Faculty, Duzce University Duzce, TURKEY

ugurguvenc@duzce.edu.tr

Serhat DUMAN Department of Electrical Education,

Technical Education Faculty, Duzce University Duzce, TURKEY

serhatduman@duzce.edu.tr

Nuran YORUKEREN Department of Electrical Engineering,

Engineering Faculty, Kocaeli University Kocaeli, TURKEY

nurcan@kocaeli.edu.tr

Abstract—This paper aims to solve the optimal power problem (OPF) incorporating flexible AC transmission systems (FACTS) devices using Gravitational Search Algorithm (GSA) that minimizes the fuel cost function in the problem. In the optimization problem, Thyristor controlled series compensation (TCSC) and thyristor controlled phase shifter (TCPS) FACTS devices are considered to find their optimum location in transmission lines. In order to evaluate the effectiveness of proposed algorithm, it has been tested on modified IEEE 30 bus system and compared with particle swarm optimization (PSO) and hybrid tabu search and simulated annealing (TS/SA) approach which are used in solving the same problem and reported before in the literature. Results show that GSA produces better results than others and has fast computing time for solving OPF problem with FACTS.

Keywords-optimal power flow; flexible AC transmission systems; gravitational search algorithm; optimization

I. INTRODUCTION Optimal Power Flow (OPF) is generally designated to a

non-linear optimization problem. It is aimed in this optimization problem that control variables for a power network are set to their optimum values minimizing chosen objective functions such as fuel cost or network loses while meeting some equality and inequality constraints. Therefore, OPF problem solution has great attention last three decades in power markets and many researchers have studied on this topic using different methods like conventional mathematical models or evolutionary algorithms [1-3].

Flexible AC transmission system (FACTS) is a power electronics based system composed of static devices such as SVC (static VAR compensation), TCSC (thyristor controlled series compensation) and TCPS (thyristor controlled phase shifter) etc. used for the AC transmission of electrical energy. These FACTS devices can increase transmission line capability, improve the system stability and security of

transmission system [4]. Also, in the classical OPF problem, since there is an increasing in load demands, it causes an overloading of transmission lines and losing optimality due to forecasted errors [5]. Therefore, in recent years, researchers have considered using FACTS incorporated with OPF in order to improve the static performance of power systems. There are several papers in dealing with OPF incorporating FACTS. However, classical OPF problem is already a non-differential, non-linear and non-convex problem. Moreover, inclusion of FACTS devices into the OPF, it makes the problem more complex. In order to overcome this negativity, researchers have intensified their efforts on artificial intelligence and evolutionary methods for determining the parameters of FACTS. There are some studies in the literature using these methods such as GA [7,8], PSO [8, 9] non-linear interior point [10], fuzzy logic [5] and some hybrid methods created by combining a few of them [11, 12, 4]. But there is a main disadvantage that these methods doesn’t exactly converge the local optimum and have huge computational time.

In this paper, a new method is proposed to solve OPF with FACTS problem using Gravitational Search Algorithm. In the OPF problem, TCSC and TCPS devices are considered to minimize the fuel cost function while meeting the constraints, namely; equality constraints, in equality constraints, generation constraints, transformer constraints, security constraints and FACTS devices constraints. In order to show the effectiveness of proposed algorithm, it is tested on IEEE 30 bus system with two different cases and compared the PSO reported in [9] for test case1 and TS/SA (tabu search/simulated annealing) reported in [4] for the other test case. Results show that GSA produce better solution than others in solving such a problem.

978-1-4673-1448-0/12/$31.00 ©2012 IEEE

II. OPF INCORPORATING FACTS DEVICES

A. Modeling of TCSC and TCPS In this paper, static TCSC and TCPS FACTS devices are

used to minimize the cost function based on [4,9]. In the TCSC model, a series capacitor bank can be seen as a controllable reactor by thyristor. Equivalent circuit of TCSC is illustrated in Fig. 1.

-jXC Rij+jXij

i j

Vi Vj

Figure 1. Equivalent circuit of TCSC

Cij XXX −=mod (1)

( ) ( )( )ijijijijjiijiij bgVVgVP δδ sincos2 +−= (2)

( ) ( )( )ijijijijjiijiij bgVVbVQ δδ cossin2 −−−= (3)

( ) ( )( )ijijijijjiijjji bgVVgVP δδ sincos2 −−= (4)

( ) ( )( )ijijijijjiijjji bgVVbVQ δδ cossin2 ++−= (5)

where Vi and Vj are voltage magnitude of i-th and j-th buses, δij is the voltage angle difference between i-th and j-th bus and, gij and bij are described as follows.

( )2mod

2 XRRg

ij

ijij += (6)

( )2mod

2mod

XRXb

ijij += (7)

In the TCPS model, there is a phase shifting transformer having a control parameter which is voltage shift angle φ. Equivalent circuit of TCPS is illustrated in Fig. 2 [4].

Rij+jXij

i j

Vi Vjϕ∠1:1

Figure 2. Equivalent circuit of TCPS

The power flow equations [4] are described as follows.

( ) ( )( )ϕδϕδ +++−= ijijijijjiiji

ij bgKVV

KgV

P sincos2

2

(8)

( ) ( )( )ϕδϕδ +−+−−

= ijijijijjiiji

ij bgKVV

KbV

Q cossin2

2

(9)

( ) ( )( )ϕδϕδ +−+−= ijijijijji

ijjji bgKVV

gVP sincos2 (10)

( ) ( )( )ϕδϕδ ++++−= ijijijijji

ijjji bgKVV

bVQ cossin2 (11)

where )cos(ϕ=K . The TCPS effect on a power system is generally represented by the injected power model shown in Fig 3 [4].

Figure 3. Injected model of TCPS

( ) ( )( )ijijijijjiijiis bgmVVgVmP δδ cossin22 −−−= (12)

( ) ( )( )ijijijijjiijiis bgmVVbVmQ δδ sincos22 ++= (13)

( ) ( )( )ijijijijjijs bgmVVP δδ cossin +−= (14)

( ) ( )( )ijijijijjijs bgmVVQ δδ sincos −−= (15)

where )tan(ϕ=m

B. OPF Formulation In this paper, The OPF with FACTS devices problem aims

to minimize the fuel cost function. The mathematical model of the problem [4] is described as follows.

( )∑=

++=NG

iGiiGiii PcPbaF

1

2min i=1, 2, …,NG (16)

where ai, bi and ci are fuel cost parameters and PGi are active power generations at i-th bus and NG is the number of total generator.

subject to:

Equality constraints:

( ) ( )

( ) ( )( )∑

∑∑

=

==

=−−

+−

bus

TCPSbus

N

jijCijCijji

N

iii

N

iDiGi

XXYVV

PtcPP

1

11

0cos δθ

ϕbusNi ∈∀ (17)

( ) ( )

( ) ( )( )∑

∑ ∑

=

= =

=−+

+−

bus

bus TCPS

N

iijCijCijji

N

i

N

iiiDiGi

XXYVV

QtcQQ

1

1 1

0sin δθ

ϕbusNi ∈∀ (18)

where QGi is reactive power generations at i-th bus, PDi and QDi are active and reactive power demand at i-th bus, Ptci and Qtci are injected active and reactive power, φi is phase shift angle of i-th TCPS, Yij(XC) and θij(XC) are magnitude and angle of ij-th elements in Ybus matrix, Nbus is the set of bus indices, NTCPS is number of TCPSs.

Inequality constraints:

max,min, GiGiGi PPP ≤≤ i=1, 2, …, NG (19)

max,min, GiGiGi QQQ ≤≤ i=1, 2, …, NG (20)

max,min, iii VVV ≤≤ i=1, 2, …, Nbus (21)

max,ii SlSl ≤ i=1, 2, …, Ntl (22)

max,0

ii CC XX ≤≤ i=1, 2, …, NTCSC (23)

max,0 ii ϕϕ ≤≤ i=1, 2, …, NTCPS (24)

where Ntl is number of transmission lines, NTCSC is number of TCSCs.

III. IMPLEMENTAION OF GRAVITATIONAL SEARCH ALGORITHM

GSA is a brand new heuristic optimization algorithm based on law of gravity and law of motion [14,15]. GSA is different from other optimization methods like PSO and GA and a useful method for solving non-linear functions. It is reported in some studies [14-26] that GSA improves the solution quality for solving non-linear optimization problems when compared other methods. In GSA, a set of agents are described as objects and their masses are introduced by using law of gravity and law of motion in order to find the local optima in solution. Here, it is expressed that step by step how the GSA works based on [14].

GSA searches the best fitness value for a given function. The searching process consists of six steps. In the first step, assume that there is a system with N agents; the position (masses) of the i-th agents is described as follows.

),...,,..,( 1 ni

diii xxxX = i=1, 2, …, N (25)

where xid is the position of the i-th masses in the d-th

dimension and n is the total number of agents. In the second step, the fitness evolution for all agents at each cycle is performed. It is done via calculating best and worst fitness values at each cycle. For a minimization problem it is described as follows.

( ) ( )tfittbest imin= Ni ∈∀ (26)

( ) ( )tfittworst imax= Ni ∈∀ (27)

where fiti(t) is the fitness value of the i-th agent at t time. In the third step, the gravitational constant G(t) at time t is calculated using a function G of the initial value G0 and time t described as follows.

( ) ( )tGGtG ,0= (28)

( ) Tt

eGtGα−

= 0 (29)

In the fourth step, mass of each agents Mi(t) are updated as follows.

NiMMMM iiipiai ,...,2,1, ==== (30)

( ) ( ) ( )( ) ( )tworsttbest

tworsttfittm i

i −−

= (31)

( ) ( )( )∑

=

= N

kk

ii

tm

tmtM

1

(32)

In the fifth step, in order to calculate the acceleration of an agent, the force acting on i-th mass Fi

d(t) is computed first based on the law of gravity as follows.

( ) ( ) ( ) ( )( ) ( )

( ) ( )( )txtxtXtX

tMtMtGrandtF d

idj

ji

ijN

ijkbestjj

di −

+= ∑

≠= ε2

, ,

. (33)

where kbest is the set of first k agents with the best fitness value and biggest mass, randj is a random number in the interval of [0,1], Mi(t) and Mj(t) are the gravitational masses of the i-th agent and the j-th agent,

2)(),( tXtX ji is the

Euclidian distance between i-th and j-th agents, ε is a small constant. Then acceleration of an agent ai

d(t) is calculated as follows.

( ) ( )( )tMtF

tai

did

i = (34)

In the last step of the searching process, the next velocity and the position of the agents are updated as follows.

( ) ( ) ( )tatvrandtv di

dii

di +×=+1 (35)

( ) ( ) ( )11 ++=+ tvtxtx di

di

di (36)

The searching process of GSA expressed above repeats until the stopping criterion is reached. Thus, the optimum solution is obtained for a specified function when the searching process stops. The flow diagram of the GSA is illustrated in Fig. 4 [14].

IV. SIMULATION RESULTS The proposed meta-heuristic approach has been applied to

solve the OPF problem incorporating flexible AC transmission system. In order to verify the effectiveness of the proposed GSA approach which is tested on standard IEEE 30-bus test system shown in Fig. 2 for different operating scenarios and the test system data is given in [27]. In this simulation study, G0 is set to 100 and α is set to 10, and T is the total number of iterations. Maximum iteration numbers are 200 for operating scenarios.

Case 1 is the OPF with TCSC and TCPS at line 3-4, which is simulated by the GSA, the results obtained from the proposed approach are compared to GA, SA, TS and TS/SA,

respectively. The results of this comparison and the average and maximum results of the GSA technique are given in Table I and Table II. In case study, the reactance limits of the TCSC and phase shifting angle of the TCPS are considered to be 0-0.02 in p.u and 0-0.1 radian, respectively [4]. Fig. 5 shows the convergence of the best total cost result obtained from the GSA approach.

Figure 4. Flow diagram of the GSA

Figure 5. Convergence of GSA for case 1

TABLE I. THE RESULTS OF GSA APPROACH FOR CASE 1

Method Total cost ($/h) Min. Average Max.

GSA 803.31234015 803.31234017 803.31234022

The minimum fuel cost obtained from the proposed approach is 803.31234015 $/h. GSA is approximately less by 0.09885%, compared to previously reported results 804.1072 $/h.

In Case 2, the OPF is considered with two TCSCs at branches 4 and 24 together with two TCPSs at branches 4 and 8. Minimum and maximum limits for the control variables, TCSCs and TCPSs are taken from [9]. The results obtained

from the GSA are compared to the other method in the literature.

TABLE II. BEST CONTROL VARIABLES SETTINGS FOR CASE 1

Control variables settings GA [4] SA [4] TS [4] P1 (MW) 192.5105 192.5105 192.5105 P2 (MW) 48.3951 48.3951 48.3951 P5 (MW) 19.5506 19.5506 19.5506 P8 (MW) 11.6204 11.6204 11.6204 P11 (MW) 10.0000 10.0000 10.0000 P13 (MW) 12.0000 12.0000 12.0000

TCSC line 3-4 0.0200 0.0200 0.0200 TCPS line 3-4 0.0141 0.0141 0.0141 Total PG (MW) 294.0766 294.0766 294.0766

Ploss (MW) 10.6766 10.6766 10.6766 Cost($/h) 804.1072 804.1072 804.1072

CPU time (min) 10:24 6:40 6:24 Control variables settings TS/SA [4] GSA

P1 (MW) 192.5105 177.18623041 P2 (MW) 48.3951 48.89254087 P5 (MW) 19.5506 21.49421605 P8 (MW) 11.6204 21.56424278 P11 (MW) 10.0000 12.07831745 P13 (MW) 12.0000 12.00000000

TCSC line 3-4 0.0200 0.02000000 TCPS line 3-4 0.0141 0.01228762 Total PG (MW) 294.0766 293.2155476

Ploss (MW) 10.6766 9.81554756 Cost($/h) 804.1072 803.31234015

CPU time (min) 4:43 1:05

TABLE III. BEST CONTROL VARIABLES SETTINGS FOR CASE 2

Control variables settings (p.u.) PSO [9] GSA

P1 1.7472 1.74782623 P2 0.4851 0.46409423 P5 0.2380 0.21501375 P8 0.2106 0.22838850 P11 0.1210 0.13032508 P13 0.1200 0.13132134 V1 1.0811 1.08513531 V2 1.0618 1.04658378 V5 1.0314 1.03541025 V8 1.0391 1.07661053 V11 1.0617 0.95395895 V13 1.0715 0.96257641 T11 1.0040 1.08481924 T12 0.9918 1.09820187 T15 0.9955 1.04395236 T36 0.9686 1.06035021

TCSC4 0.1687 0.13302452 TCSC24 0.2903 0.23014370 TCPS4 -0.0272 0.02345993 TCPS8 -0.0336 0.03369910

Cost($/h) 800.931 799.056077 Voltage deviations 1.070 0.87277107

The results of this comparison and the average and maximum results of the GSA technique are given in Table III and Table IV. The convergence characteristic curve of the best total fuel cost result obtained from the GSA is shown in Fig.6. From the results in Table III, it can be seen that the best fuel cost result obtained from the proposed method is 799.056077 $/h, which is less in comparison to reported result the literature.

Figure 6. Convergence of GSA for case 2

TABLE IV. THE RESULTS OF GSA APPROACH FOR CASE 2

Method Total cost ($/h) Min. Average Max.

GSA 799.056077 799.615973 800.129561

V. CONCLUSION In this paper a new method is proposed to solve the optimal

power flow problem with FACTS devices using Gravitational Search Algorithm. In the optimization problem fuel cost function is minimized by fixing location of TCSC and TCPS FACTS devices. Experimental studies show that GSA has fast computing time and finds better results than compared others in solving such a problem. In our future work, GSA will be implemented on more case studies with different FACTS devices and compared other popular optimization algorithms.

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