[IEEE 2014 IEEE 8th International Power Engineering and Optimization Conference (PEOCO) - Langkawi,...

Optimal Capacitor Placement and Sizing in an Unbalanced Three-Phase Distribution System Using

Particle Swarm Optimization Technique

1Muhd Azri Abdul Razak Mohammad Murtadha Othman Muhammad Bukhari Shahidan

Junaidah Ariffin Ismail Musirin

Mohamad Fadhil Mohd Kamal Faculty of Electrical Engineering

Universiti Teknologi MARA Shah Alam, Malaysia

1email : [email protected]

2Zilaila Zakaria Ainor Yahya

Mat Nasir Kari Mohd Fazli Osman

Electrical Engineering Branch Selangor Public Work Department

Shah Alam, Malaysia 2email : [email protected]

Abstract- Installing shunt capacitors in a radial distribution system can provide a lot of benefit to utility and users. There are many ways to install capacitors as long as the sizes do not exceed the original inductive reactive power of a system. But it is very difficult to determine the best locations and size of capacitors. In addition, the unbalance nature of distribution system make the placement becomes more complicated. Optimal placement of capacitor is a highly nonlinear optimization problem which requires discrete control variables. This paper proposed an artificial intelligence approach to obtain optimal total line loss reduction and total cost of capacitor while improving the voltage profile along the feeders. It was done by integrating the circuitry distribution model in SIMULINK with particle swarm optimization (PSO) technique constructed in MATLAB. The optimization process was done in two stage where the first stage is to determine the early optimal locations while second stage is to determine optimal sizes. A modified IEEE 13-bus three phase unbalanced radial distribution system is used to validate effectiveness of the proposed technique in solving the problem.

I. INTRODUCTION In epochal years, the infrastructure of a distribution systems is extensively expanding which facing the pressure to maintain their operational efficiency and power quality in the current scenario of deregulated power system. When dealing with the distribution system containing with several feeders and different loads, deciding on the best locations and sizes of capacitors are becoming a complicated optimization problem. The electric power supply delivers from the sources to the end users may cause a substantial energy loss. The energy losses have a direct impact on the net profits of the distribution systems. Capacitors are commonly installed in distribution systems operated as a reactive power compensation have a tendency to improve efficiency of distribution system. These capacitors are also used for a voltage profile improvement and maximize transmitted power flowing through the cables and transformers. The extent of these advantages are depending on controlling the operation of capacitors in a system and also strategically placed in any possible loading conditions. The

optimal location and sizing of capacitors are an important issue because its placement involves a capital cost. The placement of capacitor banks on distribution feeders involves determination of size, type ( xed or switched) and location of capacitors at the speci c load levels. The main objective in capacitor placement problem is to minimize the total cost embodied with the amount of energy losses of the whole system and the investment cost of installed capacitors while the voltage magnitude at each node is within the limits. In addition, numerous advantages gained from the optimal location and sizing of capacitors are such as reduction in line losses will improve the power quality as well as the capacitors are reliable, dependable, sturdy, simple and rugged work-horses type of component. Many different optimization techniques have been proposed in the previous studies, since the optimal capacitor placement is a complicated combinatorial optimization. Das et al [1] presents an algorithm based on the heuristic rules and fuzzy multi objective approach for optimizing the network configuration of a balanced distribution feeder. K.Iba [2] proposed a method using Genetic Algorithm for the capacitor placement problem. However, balanced tests cases were carried out in this study and capacitor placement was not considered. Both authors utilize the positive sequence network model and associated power flows in formulating the problem where the results will not apply to the system that containing feeders with missing phases and evenly loaded feeders. Grainger et al [3] formulated the capacitor placement and voltage regulator problem and proposed a decoupled solution for the distribution system. El-kib et al [4] extended the solution to unbalanced three phase feeders. This solution formulate the problem as a non-linear programming problem by fixing the capacitor location and sizes as continuous variables. Dian et al [5] discusses about the concept, system and challenging to implement the particle swarm optimization (PSO). Based on the variant of PSO, the advantage of using PSO is that velocity clamping will limit the magnitude of particles's velocity so

2014 IEEE 8th International Power Engineering and Optimization Conference (PEOCO2014), Langkawi, The Jewel of Kedah,Malaysia. 24-25 March 2014

978-1-4799-2422-6/14/$31.00 ©2014 IEEE 624

that it can control the movement of the particle and improve the convergence rate. This paper presents a new approach of determining the best location and sizing for the capacitors using the PSO. The PSO is performed based on the objective function which includes the cost of the capacitor banks and the power losses. Robustness of the proposed method in solving the problem is verified through a case study of modified IEEE 13-bus three phase unbalanced distribution implemented in a MATLAB and SIMULINK software.

II. PROBLEM FORMULATION

A. Optimal Capacitor Placement The main purpose of installing the capacitors in an electrical distribution system is to reduce the total power loss and also improve the voltage level in a system. The formulation of total power losses utilized in this study as the constraint parameters for the optimization solution is given in equation (1).

=

=l

m

N

mlossTloss PP

1 (1)

where, m and Nl is the feeder number and total number of feeder, respectively. In the market, the size of the capacitors are given in fixed size. In this study, a complete size of capacitors are designed based on the combination of several capacitors with the smallest size of Qc

0 .

TABLE 1 : AVAILABLE DISCRETE CAPACITOR SIZES

Capacitor sizes for IEEE 13-bus system (kVar) 50 100 150 200 250 300 350 400 450 500

550 600 650 700 750 800 850 900 950 1000 Capacitor installation cost is chosen proportional to the size of the capacitor. The size of the capacitors to be installed at the selected destination is limited to the maximum size of reactive power load [6]. QLQ cc

0max ×= (2) where Qc

0 is the smallest capacitor given in Table 1, L is the multiple factor of the smallest size of capacitor to be installed. The most optimal placement and the size of the installed capacitors are referred to the cost of the capacitors and total power loss as expressed in equation (3)

=

+=l

m

b N

mlosss

N

j

cj

cj PBKQKfMin

1

. (3)

Where, Ks is the cost coefficient for power losses ($/kW) and j is the number of selected buses required for the capacitor installation. The objective function is bounded by a number of

constraints which are the allowable minimum and maximum voltage limit and limitation of capacitor size specified at each bus. Thus, the constraints considered in this study can be described as follows.

maxmin VVV i ≤≤ (4)

NbjQQQ jjj ,....,3,2,1,max,min, =≤≤ (5)

TsysLc QQ ≤max (6)

For this case study, the value for

sK is $0.168/kW and Kcj is

3.0 ($/kVar) [7]. The coefficient B is added in equation (13) so that the cost of total power losses becomes dominant. If the coefficient is taken out, there is high possibility that no capacitor will be placed at all since cost is the determinant factor in optimization. In this study, 1000 was set for the value of coefficient. The minimum and maximum voltage magnitude limits are set as 0.90 p.u. and 1.10 p.u., respectively. The limit of Qj,max and Qj,min are referred as the permissible amount of inherent minimum and maximum inductive reactive power of the load. Thus, the total size of capacitor Qc

max installed in the system will not exceed the

original total inductive reactive power of the system, QLTsys

.

But the minimum constraint of inductive reactive power for this study is set to zero to provide wide selection of capacitor sizing. The PSO technique developed for this case study will execute equations (3), (4), (5), and (6) at every computational iteration. The global best solution will not be update unless the objective function, f is improved or reduced and this condition is described in equation (7).

>+≤++

=+)()1()()()1()1(

)1(kfkfifkfkfkfifkf

kf (7)

where, k is the number of computational iteration. B. Particle Swarm Optimization Technique Particle Swarm Optimization (PSO) is a population based optimization algorithm introduced by Kennedy and Eberhart in 1995 [8]. In the current years, there are numerous studies using PSO to solve optimization problems especially related to non-linear condition [6,9]. In a PSO technique, the population is called as swarm while the individuals is called as particles. Each particle move in multi dimensional , d search space area based on their velocity. Their velocity is based on the memory of their own best positions and global best positions. In term of mathematical formulation, positions and velocities of a particle are represented by d

nddd

i pxxxx ,..., 21= and

dn

dddi p

vvvv ,..., 31= , respectively. The total number of

particles is denoted by np. Each particle also hold a value of


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their best solution so far which can be expressed by

pnpbestpbestpbestpbest ,...,, 21= . On the other hand, there

are only one global, gbest solution in a population. In this study, fitness equation was designed so that higher value of fitness will indicate a better solution. The process to obtain the optimal solution was done iteratively until stopping criteria is achieved. In this study, the stopping criteria is attained when optimization process reaches to the maximum number of iteration. To control the trajectories of the particles, particles velocity was bound in a range of -1 and +1. In addition, the positions for each particle are also kept between 0 and 1. The condition of controlling the velocity and position of a particle is shown below

<+>+

≤+≤+=+

minmin

maxmax

maxmin

)1()1(

)1()1()1(

di

di

di

di

di

di

di

di

di

di

di

vkvifvvkvifv

vkvvifkvkv (8)

≤+≤+<+=+>+=+

=+maxmin

minmin

maxmax

)1()1()1(0)1(,)1(0)1(,

)1(d

id

id

id

i

di

di

di

di

di

di

di

di

di

xkxxifkxxkxifkvxxkxifkvx

kx (9)

Some studies have found out that the original PSO has higher possibility to give premature convergence which result inaccurate optimal solution [5] . So in this study, some testing have been made on constriction factor and inertia weight coefficient to avoid premature convergence. Since optimal capacitor placement and sizing is a highly non-linear optimization problem with discrete control variables, constriction factor give a slightly better solution rather than by using the inertia weight. The equation of updating velocity and position used in this study is shown in equations (10) and (12). The constriction factor is calculated by using equation (13) with some restriction that shown below the equation.

))]()((

))()(()([)1(

22

11

kxkgbestrc

kxkpbestrckvkvd

i

dii

di

di

−+

−+=+ χ (10)

)4(22

−−−=

φφφχ K (11)

where, ]1,0[∈K

]1,0[, 21 ∈rr

2211 rcrc +=φ 4≥φ

)1()()1( ++=+ kxkxkx d

id

id

i (12)

III. METHODOLOGY

A. Construction of distribution system in SIMULINK

Most of previous studies used mathematical modelling to run load flow of electrical distribution system [6&10]. In this study, circuitry-based commercial software have been chosen to develop IEEE 13-bus three-phase unbalanced radial distribution system. The model was designed by taking into account several important electrical components such as the three phase load, distribution line, buses, incoming source and measurement blocks. The load flow simulation is performed which will provide the measurement in a time domain response at a steady state condition. To indicate the effect of new capacitor placements in this study, the original capacitors in the system were taken out. B. Optimal capacitor placement and sizing using PSO The procedure of PSO algorithm implemented in this study to obtain the optimal value of the objective function is discussed as follows. a) Perform a three-phase unbalanced load flow solution for

the original system (without the capacitor placement) to obtain the total power loss and other required data.

b) Generate a matrix consisting of '1' and '0' to represent the existences of phase current indicating whether the phase is available or not-available, respectively. Row of the matrix will indicate the bus number, while the column will indicate phases of each bus. Analogous dimension of the matrix will be used to indicate the capacitor sizes and locations.

c) Perform the initialization of PSO algorithm for its positions, velocity, pbesti and gbest where capacitor locations is randomly chosen. In this process, velocity of the particles is set to zero and is at static condition.

d) Generate randomly a set of capacitor locations at some of the available phases in a distribution system. Thus, another matrix consisting of '1' and '0' with similar dimension as the matrix of available phases was produced. This is to indicate the preselected locations of capacitor placement. Install capacitor of 0.5

max_QL

j at the selected location and

perform load flow to obtain total power loss. Repeat this process until maximum evaluation (hmax) and save the best set of location with restpect to minimum total power loss.

e) Update the velocity and position of the particles using equations ( 10) and (12), respectively. In this process, the trajectories of the particles were controlled by using equation (8), and equation (9) is used to ovoid the particles from fly away from search space border.

f) Use equation (13) to determine a set of decimal capacitor sizes at the selected locations. Then, the discrete capacitor sizes are replaced with the closest size available in the specification data. Once again the chosen discrete capacitor size will be checked with max_j

LQ to ensure the size does not violate the sizing constraints

min_min_max_)( QxQQQ L

jd

iLj

Lj

cj +−= (13)


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g) Apply the set of capacitor sizes of each particle into the SIMULINK model and perform the load flow solution to obtain a new value of total power loss and other required information of voltage magnitude.

h) Determine the fitness which requires the objective function, constraints and information obtained in step g), in order to evaluate quality of the particles solution at k-th iteration. If the particle violates one of the constraints, the fitness value is set to zero. In other word, the particle will not be considered as pbesti or gbest for that k-th iteration.

i) Update pbesti and gbest based on the fitness value using equation (7).

j) Return from steps d) to i) until maximum iteration, kmax is reached. The best solution is recorded and analyzed.

The procedure is executed for several times without changing any parameter of the optimization algorithm in order to test the reliability of convergence in its objective function. The procedure of PSO used for capacitor placement and sizing is illustrated in a flowchart shown in Fig.1.

1=k

1+= kk

maxkk =

1=h

max_QL

j

maxhh=1+= hh

Fig. 1. Optimal capacitor placement and sizing using PSO technique.

IV. RESULT AND DISCUSSION

The algorithm of PSO technique and a case study of IEEE 13-bus three-phase unbalanced distribution model were developed in MATLAB and SIMULINK software, respectively. The PSO is executed for 10 times with 100 iterations of optimization process is specified for each time. In this study, the simulation was done by assuming that the system is operating for six hours per day, 22 days per month and 12 month per year.

Fig. 2. IEEE 13-Bus Three Phase Unbalanced Distribution System

In Fig. 2, the IEEE 13-bus three-phase unbalanced distribution system is embodied with the total real and reactive power of 3676.50kW and 2560.90kVar, respectively. The system is said to be unbalanced since there are several buses connected with only single or two phase load. The sampling time of simulation is set as 50μs and 60 Hz is set for the frequency. The system is operating at the nominal voltage of 4.16kV accept at bus 634 where the voltage is step down to 480V. Five particles were used on the optimization process of PSO technique. The value of 2.05 is allocated for 1c and 2c in equation (10). Table 2 elucidates the result obtained from the PSO optimization process which consist of total energy loss, percentage of energy loss reduction, cost of energy loss, total size of capacitor installed, cost of capacitor and total cost.


627

TABLE 2 : RESULTS FOR EACH EXECUTION

Run

Total Line

Power loss

(kW)

Cost of

energy loss

($/year)

Percentage of

Reduction (%)

Total Capacitor

Size (kVar)

Capacitor Cost ($)

Total cost ($)

Base Case 226.71 60330.25 0.00 0 0.00 60330.25

1 195.76 52094.08 13.65 1000 3000.00 55094.08 2 189.25 50361.49 16.52 1050 3150.00 53511.49 3 188.16 50072.12 17.00 1150 3450.00 53522.12 4 187.22 49820.74 17.42 1200 3600.00 53420.74 5 187.63 49931.87 17.24 1000 3000.00 52931.87 6 197.75 52623.56 12.77 1100 3300.00 55923.56 7 189.08 50316.59 16.60 1350 4050.00 54366.59 8 189.77 50499.17 16.29 1000 3000.00 53499.17 9 186.44 49614.34 17.76 1450 4350.00 53964.34 10 189.49 50424.88 16.42 1250 3750.00 54174.88

By referring to Table 2, each run of PSO technique give a difference in terms of its optimization result. This is due different location of capacitor placement generated at each run. The overall comparisons of execution have shown that there is a small variation or fluctuation on the percentage of total loss reduction and total cost. Hence, the pattern indicates reliability and consistency of the proposed algorithm in solving the optimization problem without a significant difference in its randomized solution. Table 3 and table 4 envisage the minimum and maximum voltage magnitudes, and locations and sizing of capacitors generated for each run of optimization process, respectively.

Fig. 1. Graph of Total Power Loss against Computational Iteration.

TABLE 3 : MAXIMUM AND MINIMUM VOLTAGE FOR EACH CASE

Run Min. Voltage (p.u.)

Max. Voltage (p.u.)

Base Case 0.8853 1.0674

1 0.9019 1.0736 2 0.9020 1.0370 3 0.9223 1.0423 4 0.9126 1.0488 5 0.9072 1.0379 6 0.9254 1.0496 7 0..9045 1.0540 8 0.9000 1.0531 9 0.9000 1.0768

10 0.9000 1.0531 Based on Tables 2, 3 and 4, the 5th execution gives the best solution with respect to the best total cost considered as its objective function. Albeit the other cases of execution did improve the system condition in terms of voltage magnitude and total losses. However the 5th execution yielding to the least amount of total cost that is $52931.87 per year including one time investment of 1000kVar capacitor with the cost of $3000.00. Fig. 3 and 4 show the convergence characteristic of total line loss reduction and total cost based on the case of 5th execution. Fig. 4, indicates that the obtained optimal total cost is about 12.26% reduction per year from the total cost at base case system condition. The return on investment can be acquired about after five month of utilization.

Fig. 2. Graph of Total Cost againt Computational Iteration.


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TABLE 4 : LOCATIONS AND SIZES OF CAPACITOR FOR EACH CASES

Case Phases Buses Total

size (kVar) 650 632 633 634 645 646 671 692 675 684 611 652 680

1 A 100 50 B 100 100 200 1000 C 450

2 A 100 100 B 100 100 50 1050 C 150 450

3 A 200 150 B 100 100 100 1150 C 400 100

4 A 250 100 B 350 1200 C 250 100 150

5 A 250 50 B 100 100 50 1000 C 450

6 A 300 100 50 B 100 50 1100 C 300 150 50

7 A 250 50 B 200 100 100 1350 C 400 100 100 50

8 A 100 100 B 100 300 1000 C 200 100 100

9 A 250 100 B 550 100 100 50 1450 C 150 150

10 A 200 200 B 100 100 200 1250 C 350 100

V. CONCLUSION This paper has presented a two stage optimization process of PSO technique in obtaining the optimal locations and sizes of capacitors. Determination of capacitor placement was done in the first stage and followed by the determination of capacitors sizes was done in the second stage. This technique can be considered as solving the combinatorial optimization problem where two controlled variable which are locations and size are handled simultaneous. By optimally place the capacitor with proper size, the total line losses was reduced while keeping the operational constraints within the limits.

REFERENCES [1] D. Das and H. S. Nagi, “Novel method for solving radial distribution

networks.” [2] K. Iba, “Reactive power optimization by genetic algorithm,” IEEE

Trans. Power Syst., vol. 9, no. 2, pp. 685–692, May 1994. [3] K. N. Clinard and L. J. Gale, “great simplifications.,” no. 9, pp. 2714–

2722, 1984.

[4] C. Power and L. Company, “UNBALANCED THREE-PHASE FEEDERS INVOLVING LATERALS Esj,” no. 11, pp. 3298–3305, 1985.

[5] D. P. Rini, “Particle Swarm Optimization�: Technique , System and Challenges,” vol. 14, no. 1, pp. 19–27, 2011.

[6] A. A. Eajal, S. Member, and L. Fellow, “Unbalanced Distribution Systems With Harmonics Consideration Using Particle Swarm Optimization,” vol. 25, no. 3, pp. 1734–1741, 2010.

[7] S. A. Taher and R. Bagherpour, “A new approach for optimal capacitor placement and sizing in unbalanced distorted distribution systems using hybrid honey bee colony algorithm,” Int. J. Electr. Power Energy Syst., vol. 49, pp. 430–448, Jul. 2013.

[8] R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” MHS’95. Proc. Sixth Int. Symp. Micro Mach. Hum. Sci., pp. 39–43.

[9] S. G. Saranya, E. Muthukumaran, S. M. Kannan, and S. Kalyani, “Optimal capacitor placement in radial distribution feeders using fuzzy-Differential Evolution,” 2011 Natl. Conf. Innov. Emerg. Technol., pp. 85–90, Feb. 2011.

[10] T. Thanjavur, “Harmony Search Approach for Optimal Capacitor Placement and Sizing in Unbalanced Distribution,” 2012.


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