Electric Power Systems Research 77 (2007) 276–283
Application of particle swarm optimization technique for optimal location of FACTS devices considering cost of installation and system loadability M. Saravanan ∗ , S. Mary Raja Slochanal, P. Venkatesh, J. Prince Stephen Abraham Electrical and Electronics Engineering Department, Thiagarajar College of Engineering, Madurai 625015, India Received 13 April 2005; received in revised form 5 November 2005; accepted 8 March 2006 Available online 18 April 2006
Abstract This paper presents the application of particle swarm optimization (PSO) technique to find the optimal location of flexible AC transmission system (FACTS) devices with minimum cost of installation of FACTS devices and to improve system loadability (SL). While finding the optimal location, thermal limit for the lines and voltage limit for the buses are taken as constraints. Three types of FACTS devices, thyristor controlled series compensator (TCSC), static VAR compensator (SVC) and unified power flow controller (UPFC) are considered. The optimizations are performed on the parameters namely the location of FACTS devices, their setting, their type, and installation cost of FACTS devices. Two cases namely, single-type devices (same type of FACTS devices) and multi-type devices (combination of TCSC, SVC and UPFC) are considered. Simulations are performed on IEEE 6, 30 and 118 bus systems and Tamil Nadu Electricity Board (TNEB) 69 bus system, a practical system in India for optimal location of FACTS devices. The results obtained are quite encouraging and will be useful in electrical restructuring. © 2006 Elsevier B.V. All rights reserved. Keywords: FACTS; TCSC; SVC; UPFC; Particle swarm optimization; Maximum system loadability
1. Introduction The electric supply industry is undergoing a profound transformation worldwide. Market forces, scare natural resources and an ever increasing demand for electricity are some of the drivers responsible for such an unprecedented change. Particularly in the case of transmission systems, it requires non-discriminatory open access to transmission resources. Therefore, sufficient transmission capacity for supporting transmission services is of great demand to transmission network’s requirement. Flexible AC transmission system (FACTS) can provide benefits in increasing system transmission capacity and power flow control flexibility and rapidity [1–3]. FACTS devices are solid-state converters that have the capability of control of various electrical parameters in transmission circuits. FACTS devices include thyristor controlled series compensator (TCSC), static VAR compensator (SVC), unified power flow controller (UPFC), static compensator (STATCOM), etc. Modeling of FACTS devices for power flow studies and the inte-
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gration of those devices into power flow studies were reported [4,5]. TCSC is connected in series with the line conductor to compensate for the inductive reactance of the line. The SVC can be made to generate or absorb reactive power by means of thyristor controlled elements [6,7]. The UPFC is capable of providing active and reactive power control and voltage magnitude control and it regulates all the three variables simultaneously or any combination of them, provided no operating limits are violated [8]. Population based, cooperative and competitive stochastic search algorithms are very popular in the recent years in the research arena of computational intelligence. Some well established search algorithms such as GA [9] and Evolutionary Programming (EP) [10,11] are successfully implemented to solve complex problems efficiently and effectively. The PSO algorithm was introduced by Kennedy and Eberhart [12,13] and further modifications in PSO algorithm were carried out [14]. PSO is applied for solving various optimization problems in electrical engineering [15–18]. Optimal location of different types of FACTS devices in the power system has been attempted using different techniques such as Genetic Algorithm (GA), hybrid tabu approach and simulated annealing (SA). The best location for a set of phase
M. Saravanan et al. / Electric Power Systems Research 77 (2007) 276–283
shifters was found by genetic algorithm to reduce the flows in heavily loaded lines resulting in an increased loadability of the network and reduced cost of production [19]. The best optimal location of FACTS devices in order to reduce the production cost along with the device’s cost using real power flow performance index was reported [20]. A hybrid tabu search and simulated annealing was proposed to minimize the generator fuel cost in optimal power flow control with multi-type FACTS devices [21]. The best location of UPFC to minimize the generation cost function and the investment cost on the UPFC device was found using steady state injection model of UPFC, continuation power flow technique and OPF technique [22]. Power flow algorithm with the presence of TCSC and UPFC has been formulated and solved [23]. A hybrid GA approach to solve optimal power flow in a power system incorporating FACTS devices has been reported [24]. In this paper, applying PSO technique, the optimal location of FACTS devices to achieve minimum cost of installation of FACTS devices and to improve system loadability (SL), while satisfying the power system constraints, for single- and multitype FACTS devices were found. The variables for the optimization for each device are its location in the network, its setting and the installation cost, in the case of single-type devices. In the case of multi-type devices, the type of device used is taken as additional variable for optimization. TCSC has been modeled as a variable reactance inserted in the line and SVC is modeled as a reactive source added at both ends of the line. UPFC is modeled as combination of a SVC at a bus and a TCSC in the line connected to the same bus. Computer simulations were done for IEEE 6, 30, 118 bus systems and Tamil Nadu Electricity Board (TNEB) 69 bus test system. In both single- and multitype FACTS devices, it is observed that SL cannot be increased beyond a limit after placing certain number of FACTS devices and the maximum value of SL that can be achieved without violating the constraints is known as maximum system loadability (MSL). The MSL, minimum number of FACTS devices required to attain the MSL and the optimal installation cost of FACTS devices are obtained for single- and multi-type FACTS devices.
installation of various FACTS devices are given by (2).
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CUPFC = 0.0003S 2 − 0.2691S + 188.22 CTCSC = 0.0015S 2 − 0.7130S + 153.75 CSVC =
0.0003S 2
(2)
− 0.3051S + 127.38
where S is the operating range of the FACTS devices in MVAR S = |Q2 | − |Q1 |
(3)
where Q2 is the reactive power flow in the line after installing FACTS device in MVAR and Q1 is the reactive power flow in the line before installing FACTS device in MVAR. The cost is optimized with the following constraints. 2.2. Line flow and bus voltage constraints
J=
OVLLINE ×
LINE
VSBUS
(4)
BUS
J is the factor indicating violation of line flow limits and bus voltage limits, where OVL denotes line overload factor for a line and VS denotes voltage stability index for a bus ⎧ max 1; if Ppq ≤ Ppq ⎪ ⎨ (5) OVL = Ppq max ⎪ ⎩ exp λ 1 − max ; if Ppq > Ppq Ppq
VS =
1;
if 0.9 ≤ Vb ≤ 1.1
exp(μ|1 − Vb |);
otherwise
(6)
max the where Ppq is the real power flow between buses p and q, Ppq thermal limit for the line between buses p and q, Vb the voltage at bus b, and λ and μ are the small positive constants both equal to 0.1
2.3. FACTS device’s constraints (i)
−0.8XL ≤ XTCSC ≤ 0.2XL p.u.
(7)
(ii)
−100 MVAR ≤ QSVC ≤ 100 MVAR
(8)
2. Problem formulation
(iii)
(7) and (8) for UPFC
2.1. Objective
where XTCSC is the reactance added to the line by placing TCSC, XL the reactance of the line where TCSC is located and QSVC is the reactive power injected at the bus by placing SVC.
Optimal placement of FACTS devices to minimize the cost of installation of FACTS devices has been mathematically formulated and is given by (1). Minimize IC = C × S × 1000
2.4. Power flow constraints g(V, θ) = 0
(1)
where IC is the optimal installation cost of FACTS devices in US$ and C is the cost of installation of FACTS devices in US$/KVAR. The cost of installation of UPFC, TCSC and SVC are taken from Siemens database and reported in [25,26]. The costs of
where
⎧ ⎪ Pt (V, θ) − Ptnet ⎪ ⎪ for each PQ bus t ⎪ ⎨ Q (V, θ) − Qnet t t g(V, θ) = ⎪ Pm (V, θ) − P net for each PV bus m, ⎪ m ⎪ ⎪ ⎩ not including reference
(9)
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Pt is the calculated real power for PQ bus, Pm the calculated real power for PV bus, Qt the calculated reactive power for PQ bus, Ptnet the specified real power for PQ bus, Qnet t the specified reactive power for PQ bus, V the voltage magnitude at different buses and θ is the voltage phase angle at different buses. 3. Overview of PSO and its implementation for optimal location of FACTS devices PSO is developed through simulation of bird flocking in two-dimensional space. The position of each agent is represented in x–y plane with position (Sx , Sy ), Vx (velocity along x-axis) and Vy (velocity along y-axis). Modification of the agent position is realized by the position and velocity information. Bird flocking optimizes a certain objective function. Each agent knows its best value so far, called ‘Pbest ’, which contains the information on position and velocities. This information is the analogy of personal experience of each agent. Moreover, each agent knows the best value so far in the group, ‘Gbest ’ among all Pbest . This information is the analogy of knowledge, how the other neighboring agents have performed. Each agent tries to modify its position by considering current positions (Sx , Sy ), current velocities (Vx , Vy ), the individual intelligence (Pbest ) and the group intelligence (Gbest ). The following equations are utilized, in computing the position and velocities, in the x–y plane: Vik+1 = W × Vik + C1 × rand1 × (Pbesti − Sik ) + C2 × rand2 × (Gbest − Sik ) Sik+1 = Sik + Vik+1
(11) (12)
where Vik+1 is the velocity of ith individual at (k + 1)th iteration, Vik the velocity of ith individual at kth iteration, W the inertia weight, C1 and C2 the positive constants having values (0, 2.5) [13,14], rand1 and rand2 the random numbers selected between 0 and 1, Pbesti the best position of the ith individual, Gbest the best position among the individuals (group best) and Sik is the position of ith individual at kth iteration. The velocity of each particle is modified according to (11) and the minimum and maximum velocity of each variable in each particle is set within the limits of Vmin and Vmax , respectively. The position is modified according to (12). The inertia weight factor ‘W’ is modified using (13) to enable quick convergence [13,14]. W = Wmax −
Wmax − Wmin × iter itermax
(13)
where Wmax is the initial value of inertia weight equal to 0.9, Wmin the final value of inertia weight equal to 0.4, iter the current iteration number and itermax is the maximum iteration number. The implementation of PSO algorithm for optimal location of FACTS devices is given below. 3.1. Initialization Initially the type of FACTS device and number of FACTS devices to be used are declared. SL is set to 101% (i.e. load
factor = 1.01) which means that the load bus real power is increased by 1% from the base case value. The initial population of particles (Sk ) is generated randomly such that the variables of each particle are in normalized form (i.e. between 0 and 1). The variables of each particle in the population correspond to the FACTS devices setting and their location, when single-type FACTS devices are used. If ‘N’ number of FACTS devices (either TCSC or SVC) is to be installed then each particle has ‘2 × N’ variables (N-FACTS device settings and N-locations). When ‘N’ number of UPFCs are to be installed, each particle has ‘3 × N’ variables (N-TCSC settings, N-SVC settings and N-locations), since the UPFC is modeled as combination of TCSC and SVC in this work. When multi-type FACTS devices are used, each particle has one more additional variable for each FACTS device, indicating the type. Here for indicating the type of FACTS device, the value “1” is used for TCSC, value “2” is used for SVC and value “3” is used for UPFC. 3.2. Calculation of fitness function The constrained optimization problem of optimal location of FACTS devices is converted into unconstrained optimization problem using penalty factor (PF) as given in (14). This becomes the fitness function in PSO technique. Fitness function = IC + PF × ||J − 1||
(14)
It consists of two terms. The first term corresponds to installation cost of FACTS devices given by (1). The second term corresponds to constraint violation and it is multiplied by penalty factor. To calculate the fitness function given by (14) for each particle, the normalized value of each variable (Xnorm ) in the particle are first denormalized to actual value (Xactual ) according to (15). Xactual = Xmin + (Xmax − Xmin ) × Xnorm
(15)
where Xmin is the minimum value of the variable and Xmax is the maximum value of the variable. Since the variables such as location (line number) and type of FACTS device are integer, their denormalized value is rounded to nearest integer to get the actual value. When more than one FACTS devices are to be installed, after generating the initial population or new population, it is verified that only one device is placed in a line. If two FACTS devices are placed in the same line, one of the FACTS devices is removed from that line and it is placed in some other line where FACTS device is not present. For each particle, the line data is updated according to its FACTS device’s (TCSC) setting and location and the bus data is updated according to its FACTS device’s (SVC) setting and location and the current SL. Load flow is performed using Newton–Raphson method and line flows and voltage at buses are obtained. Using these values, the value of J for each particle is found out using (4) and the fitness function of each particle is calculated using (14). The particle that gives minimum value for the fitness function in the population, is considered as Gbest particle.
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3.3. Generation of new population The new velocity is calculated using Eq. (11) and the new position of each particle is found using (12). The procedures said in 3.2 and 3.3 are repeated until maximum number of iterations is reached. 3.4. Finding MSL After the maximum number of are iterations are reached, the value of J for the Gbest particle is checked. If it is equal to 1 then
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using that Gbest particle, the current value of SL can be met out without violating line flow and bus voltage limit constraints and the Gbest particle is saved along with its cost of installation and SL. SL is then increased by 1% and again the PSO algorithm is run. If the value of J for the Gbest particle is not equal to 1 then the Gbest particle is unable to meet out the current SL and the Gbest particle with J = 1, obtained in the previous run is considered as the best optimal settings and the SL corresponding to that Gbest particle is considered as the MSL. The step by step procedure to find optimal installation cost of FACTS devices and the MSL is shown in the flowchart (Fig. 1).
Fig. 1. Flowchart showing PSO algorithm implementation for optimal location of FACTS devices.
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Table 1 Line flows before and after installing single- and multi-type FACTS devices, optimal setting and optimal cost of installation of FACTS devices (IC) and MSL in IEEE 6 bus system Qpqa (MVAR)
Device setting (p.u. for TCSC, MVAR for SVC)
IC (×106 US$)
39.7681 59.9769 39.9259 28.9459 39.9239
−21.1384 19.3805 10.8897 48.8949 13.2763
−0.009306 −0.029088 0.021361 −0.002585 −0.073963
0.368 (MSL = 115%)
−15.4187 20.1201 46.0541
33.5555 54.331 29.967
−17.4429 −3.8646 2.3951
−79.50954 96.571101 53.015082
9.73 (MSL = 110%)
28.6897 43.5849 2.9303 33.0909 26.2489
−15.4187 20.1201 −12.2687 46.0541 12.3995
39.7034 59.5295 −10.0356 36.5896 61.5946
−17.9088 1.4559 −8.5234 42.9759 36.6098
2 4
28.6897 43.5849
−15.4187 20.1201
39.5855 58.6958
−21.0394 10.0134
0.448685 28.936442
3 5 6
2.9303 15.5145 43.7732
−12.2687 15.3532 60.7242
4 28.9139 48.1295
−14.4129 24.107 59.0285
0.043657
Case
Type of device used
From line
To line
Ppqb (MW)
Qpqb (MVAR)
Single type
TCSC
1 1 1 2 2
2 4 5 4 6
28.6897 43.5849 35.6009 33.0909 26.2489
−15.4187 20.1201 11.2547 46.0541 12.3995
SVC
1 1 2
2 4 4
28.6897 43.5849 33.0909
UPFC
1 1 2 2 2
2 4 3 4 6
SVC
1 1
TCSC UPFC SVC
2 2 3
Multi-type
Ppqa (MW)
32.3 (MSL = 122%)
9.42 (MSL = 116%)
5.993867
4. Results and discussions The solutions for optimal location of FACTS devices to minimize the cost of installation of FACTS devices and to find the MSL for IEEE 6, 30, 118 and TNEB utility systems were obtained and discussed below. The simulation studies were carried out on Pentium-IV, 2.4 GHz system in MATLAB 6.5 environment. 4.1. IEEE 6 bus system The bus data and line data of the six bus sample system are taken from [27] and it contains 11 lines. The location, settings of FACTS devices and optimal installation cost are obtained using the PSO technique for single- and multi-type devices and it is given in Table 1. Ppqb and Qpqb are the real and reactive power flow in the line p–q before placing FACTS device, respectively. Ppqa and Qpqa are the real and reactive power flow in the line p–q after placing FACTS device, respectively. The effect of number of FACTS devices on SL and the installation cost are also observed and are shown in Figs. 2 and 3, respectively. In the case of TCSC, it is observed that placing TCSC in lines (1–2, 1–4, 1–5, 2–4 and 2–6) gives MSL of 115% and the cost of installation is US$ 0.368 × 106 . This is indicated as point ‘A’ in Figs. 2 and 3. Among the five lines, it is observed that the improvement in power flow is maximum in the line 1–4 which is represented in bold case in Table 1 and the corresponding XTCSC setting is −0.029088 p.u. Similarly for the other cases, bold case in Table 1 represents the line in which maximum improvement in power flow occurs after placing FACTS device.
Fig. 2. System loadability curve for TCSC, SVC, UPFC and multi-type device in IEEE 6 bus system.
In the case of SVC, the MSL obtained is 110% and the minimum installation cost for SVC is US$ 9.73 × 106 and this is represented as point ‘B’ in Figs. 2 and 3. The SVC has to be placed in lines (1–2, 1–4 and 2–4). Placing SVC with QSVC setting of 96.571101 MVAR in line (1–4) gives large improvement in power flow among the lines, where SVC is placed. In the case of UPFC, to achieve MSL of 122%, UPFC have to be placed in lines (1–2, 1–4, 2–3, 2–4 and 2–6) and the installation cost is US$ 32.3 × 106 and it is shown as point ‘C’ in Figs. 2 and 3.
Fig. 3. Installation cost (US$) curve for placing various FACTS devices in IEEE 6 bus system.
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In the case of single-type of devices, UPFC shows best performance with MSL of 122%. Next to UPFC, TCSC gives MSL of 115%. SVC gives lowest MSL. In the case of multi-type devices, the values tabulated in Table 1 are the best combination with minimum cost, minimum number of devices required for attaining MSL and their type. In this case, MSL of 116% is reached and placing UPFC in line 2–5, it is observed that there is a large improvement in power flow from the base case. The installation cost for placing SVC in three lines, TCSC in one line, and UPFC in one line is US$ 9.42 × 106 and it is shown as point ‘D’ in Figs. 2 and 3. Placing UPFC in line 2–5 improves the power flow by about 100% of base case value compared to other lines where FACTS devices are placed. In all the cases, it is observed that FACTS devices improve the line flows of the lines even to their thermal limit. It is concluded that for six bus system, TCSC is cost wise cheaper with better improvement in SL, but UPFC gives largest MSL. In all the cases (single- and multi-type), one of the device is placed in lines 1–2 and 1–4 and hence it is inferred that placing any type of FACTS device in these lines will be beneficial in increasing SL. The optimal location of FACTS devices for IEEE 6 bus system is also carried out by PSO algorithm reported in [14], by linearly decreasing the acceleration coefficient C1 from 2.5 to 0.5 and linearly increasing the acceleration coefficient C2 from 0.5 to 2.5 as the iteration proceeds, which gives good results for most of the benchmark functions [14]. The comparison of the results for IEEE 6 bus system obtained by PSO [13] in which C1 and C2 are kept constant at 2 and PSO [14] is shown in Table 2. From the results, it is observed that the cost obtained by PSO [13] is less than the cost obtained by PSO [14]. But the SL is improved while using PSO [14]. Hence, PSO algorithm [13] is used for the simulation studies of remaining systems. 4.2. IEEE 30 bus system The bus data and line data of 30 bus system are taken from [11] and it contains 41 lines. Table 3 shows the MSL, optimal cost of installation and minimum number of devices needed for 30 bus system, obtained by PSO technique. Both in single- and multitype of FACTS devices, after placing 8 numbers of devices, the SL is saturated and it does not increase further. Using single type of device, UPFC improves the SL to 139% and TCSC gives MSL of 138%. SVC improves the SL to 128%. Comparing the cost and SL, TCSC is the best option.
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Table 3 Optimal installation cost (IC), MSL and minimum number of FACTS devices (N) needed in IEEE 30 bus system Type of device used
MSL (%)
N
IC (×106 US$)
TCSC SVC UPFC Multi-type
138 128 139 138
8 8 8 8
3.57 0.52 276.7 12.61
Table 4 Optimal installation cost (IC), MSL and minimum number of FACTS devices (N) needed in IEEE 118 bus system Type of device used
TCSC SVC UPFC Multi-type
Results obtained in this work
Results reported in [9]
MSL (%)
N
IC (×106 US$)
MSL (%)
N
IC (×106 US$)
135 118 140 136
32 32 32 32
15.1 3.26 197 21.1
135 119 – 140
30 40 – 30
– – – –
4.3. IEEE 118 bus system The bus data and line data values are taken from [11]. Simulations are carried out for optimal location of single- and multitype FACTS devices. In all the cases, after using 32 devices, the SL saturates. Table 4 shows the MSL in using single- and multi-type devices and the optimal cost of installation. Fig. 4 shows the variation of system loadability with respect to number of FACTS devices used. The MSL obtained for TCSC, SVC, UPFC and multi-type of devices are indicated as points ‘A’, ‘B’, ‘C’ and ‘D’, respectively, in Fig. 4. After placing 32 devices, there is no considerable improvement in the SL. In this system, UPFC gives largest MSL but cost-wise it is too costlier. Multitype device gives next higher MSL followed by TCSC. TCSC is cost wise cheaper. SVC gives least MSL. The results obtained in IEEE 118 bus system are compared with the results reported in [9] and it is given in Table 4. The MSL and the minimum number of FACTS devices obtained in this work are nearly equal to the results reported in [9] for TCSC and multi-type of device. But in the case of SVC, the minimum number of SVC required for achieving nearly same value of MSL, obtained in this work is less, when compared with that of the result reported in [9]. Also the cost of installation of FACTS
Table 2 MSL and the optimal installation cost (IC) of FACTS devices obtained by PSO [13] and PSO [14] algorithm in IEEE 6 bus system Type of device used TCSC SVC UPFC Multi-type
PSO [13]
PSO [14] (×106
MSL (%)
IC
115 110 122 116
0.368 9.73 32.3 9.42
US$)
MSL (%)
IC (×106 US$)
160 150 160 165
0.3616 9.796 33.184 10.745
Fig. 4. System loadability curve for TCSC, SVC, UPFC and multi-type devices in IEEE 118 bus system.
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Table 5 Optimal installation cost (IC), MSL and minimum number of FACTS devices (N) needed in TNEB utility system Type of device used
MSL (%)
N
IC (×106 US$)
TCSC SVC UPFC Multi-type
131 116 124 130
33 33 8 35
21.1 32.2 35.4 55.0
Fig. 7. Functional evaluation graph in IEEE 30 bus system for TCSC, when system loadability is 138%.
Fig. 5. System loadability curve for TCSC, SVC, UPFC and multi-type devices in TNEB system.
devices is not considered while finding MSL and for UPFC, the results are not reported in [9], which are considered in this work. 4.4. TNEB 69 bus system The bus data and line data are taken from [28] and it has 99 lines. The MSL, minimum number of FACTS devices needed to attain MSL and installation costs are shown in Table 5. Fig. 5 shows the results obtained for TNEB 69 bus system. From the figure, it is observed that TCSC gives highest MSL when com-
pared with all the cases and the installation cost is also lowest. In the case of UPFC, minimum numbers of devices are required to attain MSL of 124% when compared with all other cases. MSL obtained using SVC is lowest, the value being 116%. In case of multi-type of device, the MSL obtained was 130% but the cost of installation is highest. The MSL obtained for TCSC, SVC, UPFC and multi-type of devices are indicated as points ‘A’, ‘B’, ‘C’ and ‘D’, respectively, in Fig. 5. Fig. 6 shows the single line diagram of TNEB 69 bus system. 4.5. PSO parameters The population size (Np ) for IEEE 6, 30 and 118 and TNEB system are taken as 20, 30, 50 and 30, respectively, and maximum
Fig. 6. Single line diagram of TNEB 69 bus system.
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number of iterations (Ni ) for the above systems are taken as 50, 75, 100 and 100, respectively. The graphs showing the functional evaluation for different values of population size and maximum number of iterations for 30 bus system using only TCSCs for MSL of 138% are shown in Fig. 7. 5. Conclusion In this work, the optimal location of FACTS devices are found to minimize the cost of installation of FACTS devices and to improve system loadability, for single- and multi-type FACTS devices using PSO technique. Simulations were performed on IEEE 6, 30 and 118 bus systems and TNEB practical system. Optimizations were performed on the parameters namely location of the FACTS devices, their settings in the line, and the cost of installation of FACTS devices for single-type FACTS devices. In the case of multi-type FACTS devices, the type of device to be placed is also considered as a variable in the optimization. In both single- and multi-type devices, it is observed that system loadability cannot be improved further after placing certain number of FACTS devices. In IEEE test systems, UPFC gives maximum system loadability but the cost of installation is high when compared with all other cases and TCSC requires minimum cost of installation with better improvement in system loadability. SVC gives lowest cost of installation in IEEE 30 and 118 bus systems but with minimum improvement in system loadability. In TNEB system, TCSC gives maximum system loadability and cost of installation is also minimum when compared with all other devices. Acknowledgements The authors express their sincere thanks to the Management of Thiagarajar College of Engineering and All India Council for Technical Education for providing necessary facilities and project grant to carry out this research work. References [1] N.G. Hingorani, L. Gyugyi, Understanding FACTS Concepts and Technology of Flexible AC Transmission Systems, IEEE Press, 2000, ISBN 0-7803-3455-8. [2] R.M. Mathur, R.K. Varma, Thyristor Based FACTS Controllers for Electrical Transmission Systems, John Wiley & Sons Inc., 2002. [3] Y.H. Song, X.F. Wang, Operation of Market Oriented Power System, Springer-Verlag Ltd., 2003, ISBN 1-85233-670-6. [4] J.G. Douglas, G.T. Heydt, Power flow control and power flow studies for systems with FACTS devices, IEEE Trans. Power Syst. 13 (1) (1998) 60–65. [5] D. Povh, Modeling of FACTS in power system studies, in: IEEE Power Engineering Society Winter Meeting, vol. 2, January 2000, pp. 1435–1439. [6] A. Kazemi, B. Badrzadeh, Modeling and simulation of SVC and TCSC to study their limits on maximum loadability point, Electr. Power Energy Syst. 26 (2004) 619–626. [7] S.N. Singh, Role of FACTS devices in competitive power market, in: Proceeding of Short Term Course on Electric Power System Operation and Management in Restructured Environment, 2003, pp. a71–a80.
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