Optimal placement and sizing of multiple APLCs using a modified discrete PSO

Electrical Power and Energy Systems 43 (2012) 630–639

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Optimal placement and sizing of multiple APLCs using a modified discrete PSO Iman Ziari, Alireza Jalilian ⇑ Department of Electrical Engineering, Centre of Excellence for Power System Automation and Operation, Iran University of Science and Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 5 November 2011 Received in revised form 9 April 2012 Accepted 9 June 2012 Available online 7 July 2012 Keywords: APLC Distribution system Harmonics Optimization PSO

a b s t r a c t In this paper, a new method is proposed to optimize the placement and size of multiple Active Power Line Conditioners (APLCs). The objective is to minimize the investment cost, while the voltage total harmonic distortion and the individual harmonic distortion as constraints are maintained within the standard level. Since the APLC size is a discrete value, the objective function has a number of local minima. To deal appropriately with this, a Modified Discrete Particle Swarm Optimization (MDPSO) has been proposed. In this optimization method, DPSO has been developed by Genetic Algorithm (GA) operators in order to increase the diversity of the optimizing variables and decrease the risk of trapping in local minima. To evaluate the proposed method, two cases are studied. In case 1, the robustness and accuracy of the MDPSO are compared with other methods, DPSO, GA, Simulated Annealing (SA), and Discrete Nonlinear Programming (DNLP). The IEEE 14-bus distribution system is modified and used in this case. In case 2, a comparison is performed between the results obtained when the APLC cost function is assumed to be proportional with its rating, as done in almost all previously published papers, and when a more realistic APLC cost is considered. For this case, the IEEE 18-bus distribution system is employed as the test system. It is illustrated that the proposed method is more accurate and robust than other methods. This is also highlighted that the lowest cost planning is achieved when the realistic APLC cost function is used. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction By developing the power electronic technology, the use of nonlinear loads, such as rectifiers, converters, adjustable speed drives, arc furnaces, and computer power supplies, is growing rapidly in distribution networks. These loads inject the harmonic currents into the system and distort the voltage waveforms. This distortion causes various unwanted effects for the linear loads connected at the same point of common coupling. Overheating of motors, transformers, cables, mal-operation of protection devices and resonance with the capacitors are among these effects [1]. To alleviate these consequences, mainly passive filters and APLCs are used (sometimes capacitors [2,3]). Passive filters create a low impedance path to divert the harmonic currents from the feeder to the filter path. APLCs inject equal-but-opposite currents to the PCC to purify the load side feeder currents. Although using passive filters is a low cost solution, they have fixed compensation, large size and may cause resonance [4–6]. The APLCs do not have these demerits, but their cost is a critical issue. In this paper, the aim is to find the optimal placement and rating of APLCs in distribution networks. One solution is to provide each nonlinear load with an APLC. This results in a large investment cost for installation of the required APLCs. On the other hand, ⇑ Corresponding author. Tel.: +98 21 7352 5616; fax: +98 21 7352 5600. E-mail addresses: [email protected] (I. Ziari), [email protected] (A. Jalilian). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.06.055

using only one APLC is not enough for keeping the network in a standard condition, particularly when the network is highly polluted by nonlinear loads. For this purpose, an optimization procedure is required to find the optimal placement and rating of APLCs in distribution networks. This procedure will result in the lowest possible investment cost for APLC installation; while, the technical constraints are satisfied within the standard limits. This necessitates solving the Optimal Allocation and Sizing of APLCs (OASA) problem in distribution networks. OASA problem is a discrete problem which needs a discrete optimization method. Almost all available methods in this area employ discrete optimization methods and there are only a few papers which solve the problem in continuous [7,8]. Assumption of continuousness is not realistic when APLCs are commercially discrete. The optimization methods can be classified into two main groups, the analytical and heuristic based methods. Although the analytical methods can find the minimum solution in a shorter time, they need an appropriate initial solution which is not easy to find. Particularly in discrete problems, their final solution highly depends on the initial values of optimizing variables. Since the allocation and sizing problem is severely non-smooth, using analytical methods [7–13] is not recommended. In [7,8], Discrete Nonlinear Programming (DNLP) and in [9–13], mixed integer programming are used for solving the OASA problem. The heuristic methods are based on the random values so that even if only one of these random values is located at a point close to the global

I. Ziari, A. Jalilian / Electrical Power and Energy Systems 43 (2012) 630–639

minimum, the results can be acceptable. Differential evolution method is used in [14,15] to solve the OASA problem. In [16,17], Genetic Algorithm (GA) and chance constrained programming are employed for planning the APLCs respectively. Among the heuristic methods, the Particle Swarm Optimization (PSO) is becoming more popular in literatures [18,19]. The discrete version of this optimization method [19,20] is used for solving the discrete problems such as the OASA. In [20], DPSO is employed for planning APLCs. It is shown that this method is more accurate than GA and DNLP when it is applied to the realistic OASA problem; however, the local minima problem is still a suffering issue. In order to further mitigate the local minimum problem, the DPSO is developed by GA operators in this paper to increase the diversity of the optimizing variables. The realistic investment cost of an APLC is separated into two different parts, constant cost and the incremental cost [17]. The constant cost, called fixed installation cost, is constant and is not related to the APLC rating. The incremental cost, e.g. the purchase cost, is proportional with the APLC rating. Although this assumption influences the results, almost all papers [7–16,20] ignore the fixed installation cost. This leads the optimization method to result in use of a number of APLCs so higher investment cost. In this paper, a DPSO developed by GA is proposed to find the optimal placement and size of APLCs in a distribution network. The objective function is the investment cost of APLCs and the constraints are voltage THD and the individual voltage harmonic distortion which should be maintained less than 5% and 3% respectively. The Motor Load Losses (MLL), as one of the effects of voltage harmonics on the induction motors, is reduced during the optimization procedure. This paper is organized as follows. The OASA problem formulation is presented in Section 2. Section 3 explains the employed optimization method and its implementation. The simulation results are illustrated and discussed in Section 4 and conclusions are given in Section 5.

OF ¼

NB X

where CC and CI are the constant and the incremental cost of APLCs, SAPLC m is the rating of an APLC located at bus m, NB is the number of buses, and DP is the constraint penalty factor. The APLC rating is proportional with its current.

IAPLC m ¼

NH X jIAPLCm j2

ð3Þ

h¼2

where IAPLC m is the current amplitude of an APLC installed at bus m, and NH is the number of harmonic orders. As constraints, THD along with the individual harmonic distortion at buses should be maintained less than 5% and 3%.

THDm  0:05 V hbusm V 1busm

 0:03

ð4Þ

ð5Þ

where V hbusm is the hth harmonic voltage at bus m and THDm is the THD at bus m. No limitation is applied to the APLC size at a bus; however, it can be lumped in the constraints. The Death Penalty method is used in this paper to include the constraints [21]. In this method, the constraints are included in the objective function with a penalty factor, called DP, in (2). If all constraints are satisfied, DP will be zero. Otherwise, DP is set as a large number and is added to the objective function to exclude the relevant solution from the search space [22]. During the optimization procedure, the average THD and the average MLL are reduced significantly. These factors are calculated as follows:

PNB ATHD ¼

m¼1 THDm

NB

ð6Þ

PNB

In this section, the APLC model along with the objective function and constraints are formulated. 2.1. APLC model

AMLL ¼

m¼1 MLLm

NB

ð7Þ

where MLLm is the MLL at bus m. 3. Implementation of PSO for OASA Problem

The APLC, in this paper, is modeled in a set of current sources which inject different order of harmonics to the PCC, as assumed in all available papers in this area [7–17,20]. To assume amplitude and phase angle for an APLC current, the phasor model is used as shown by h;i

ð2Þ

m¼l

2. Problem formulation

h;r IhF;m ¼ IF;m þ jIF;m

C c þ C I SAPLC m þ DP

631

ð1Þ

where IhF;m is the hth harmonic current for the APLC located at bus m, h;i Ih;r F;m and IF;m are the real and imaginary parts of the hth harmonic current for an APLC located at bus m respectively. Indices r and i present the real and imaginary parts. 2.2. Objective function and constraints The main objective of the OASA problem is to minimize the total investment cost of APLCs while the harmonic constraints are satisfied. The investment cost of an APLC is composed of the constant cost and the incremental cost. The constant cost, called fixed installation cost, is constant and is not related to the APLC rating, e.g. the required cost for securing and purchasing land. The incremental cost, e.g. the purchase cost, is proportional with the APLC rating. Given this point, the objective function is formulated as follows:

3.1. Overview of PSO PSO is a population-based and self adaptive technique introduced originally by Kennedy and Eberhart [22]. This stochasticbased algorithm handles a population of individuals in parallel to search the area of a multi-dimensional space where the optimal solution is located. The individuals are called particles and the population is called a swarm. Each particle in a swarm moves towards the optimal point with an adaptive velocity. Mathematically, the particle i in an n-dimensional vector is represented as Xi = (xi,1, xi,2, . . ., xi,n). The velocity of this particle is also an n-dimensional vector as Vi = (vi,1, vi,2, . . ., vi,n). Alternatively, the best position related to the lowest value of the objective function for each particle is represented as Pbesti = (pbesti,1, pbesti,2, . . ., pbesti,n) and the best position among all particles or best pbest is denoted as Gbesti = (gbesti,1, gbesti,2, . . ., gbesti,n). During the procedure, the velocity and position of particles are updated [19]. The discrete version of PSO, called DPSO, is an optimization method which can be applied to the discrete problems where discrete variables should be considered for particles. In this situation, the optimal solution can be achieved by rounding off the real particle value to the nearest discrete value, as done in this paper. In [19], it is mentioned that the performance of DPSO is not influenced in this rounding off process compared with other methods.

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3.2. Applying PSO to OASA problem The APLC current at each bus for each harmonic is regarded as the position of a particle during the optimization process. The proposed PSO-based algorithm as an APLC location and size optimizer is illustrated in Fig. 1. The coding is written in Matlab programming language and executed by using a desktop computer with a Core 2 Duo CPU E8600 at 3.33 GHz and 3.46 GB of RAM. The description and comments of the steps are presented as follows. Step 1 (Input System Data and Initialization) In this step, the distribution network configuration and data are input. The maximum allowed THD and individual harmonic distortion are also specified. The DPSO and GA parameters are initialized. The optimizing variables are the real and imaginary parts of the APLC current at each bus for each harmonic order. In this paper, the IEEE 14-bus and 18-bus distribution system are used as the test systems. All buses in 14-bus systems and 16 buses in 18-bus

Input Data Initialize Iteration Number = 1

Calculate OF Equation (2)

Calculate Pbest Equation (10 )

system are assumed as candidate for installation of APLCs and eight harmonic orders are included. Therefore, the number of optimizing variables is 2  14  8, 224, and 2  16  8, 256, in which ‘2’ is because of the real and imaginary parts of the APLC current, ’14 and 16’ are the number of candidate buses, and ‘8’ is the number of harmonic orders. Step 2 (Calculate the Objective Function) After determination of the location and size of APLCs from the previous step, the currents injecting to buses are found. The impedance matrix is calculated by the test system data. The bus voltage for all harmonic orders is then calculated by multiplying the injecting currents and the impedance matrix as:

2

V hbus1

3

2

7 6 6 6 h 7 6 h 6 V bus 7 6 Y bus 2;1 2 7 6 6 7 6 6 6  7 6  7¼6 6 7 6 6 6  7 6  7 6 6 7 6 6 6  7 6  5 4 4 Y hbusNB;1 V hbusNB

Y hbus1;2



Y hbus2;2















Y hbusNB;2



Y hbus1;NB

7 7 Y hbus2;NB 7 7 7 7  7 7 7  7 7 7  5 h Y busNB;NB

Ihbus ¼ IhAPLC  IhNLD

Calculate Gbest Equation (11 )

Update Velocity Equation (12 ) and (13 ) Increase Iteration Number Update Position Equation (14 )

Mutation

Last Iteration Number?

Yes Print Results Fig. 1. Algorithm of proposed PSO-based approach.

3

7 6 6 h 7 6 Ibus 7 2 7 6 7 6 6  7 7 6 7 6 6  7 7 6 7 6 6  7 5 4 IhbusNB

ð8Þ

ð9Þ

IhAPLC

( Do Not Change Half of Population

Ihbus1

where and are the injecting current vectors related to the nonlinear loads and APLCs respectively which have been specified in the previous step. After calculation of the bus voltages by (8), THD and MLL are calculated. Subsequently, the objective function is computed using (2). The constraints are also determined by (4) and (5). Similar to this procedure is implemented for all particles and their corresponding objective function and constraints are found. Step 3 (Calculate pbest) The objective function value associated with a particle is compared with its corresponding value in previous iteration and the position with lower objective function is recorded as pbest for that particle for the current iteration. kþ1 pbest j

Crossover

31 2

This equation can be rewritten as V hbus ¼ ðY hbus ÞI  Ihbus in which V hbus and Ihbus are the bus voltage and the injecting current vectors for hth harmonic order, respectively. Y hbus is the admittance matrix for hth harmonic order. As an input from the previous step, Y hbus is multiplied by the injecting current, (8), to calculate V hbus as the Step 2’s output. In this procedure, the current injecting to buses, Ihbus , is already obtained using the following equation:

IhNLD

No

Y hbus1;1

¼

k

pbestj

if OF jkþ1 P OF kj

X kþ1 j

if OF jkþ1  OF kj

ð10Þ

where k is the number of iterations, OFj is the objective function value related to particle j, and X kþ1 is the APLC currents vector associj ated with particle j for iteration k + 1. Step 4 (Calculate gbest) In this step, the objective functions associated with particles in the current iteration are compared with those in the previous iteration and the lower ones are labeled as gbest.

( gbest

kþ1

¼

Gbest

k

kþ1 Pbestj

if OF kþ1  OF k if OF kþ1  OF k

ð11Þ

where OFk is the minimum objective function value among particles in iteration k. Step 5 (Update Velocities and Particles) The position of particles for the next iteration can be calculated using the current pbest and gbest as follows:

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I. Ziari, A. Jalilian / Electrical Power and Energy Systems 43 (2012) 630–639 k

k

V kþ1 ¼ xV kj þ c1 randðpbestj  X kj Þ þ c2 randðgbest  X kj Þ j

ð12Þ

where V kj is the velocity of particle j at iteration k, x is the inertia weight factor, and c1 and c2 are the acceleration coefficients. The inertia weight factor, x, is defined as shown in (13). This factor is to adjust the effect of velocity calculated in the previous iteration on the velocity value in the current iteration.

x ¼ xmax 

xmax  xmin Iter max

 Iter

ð13Þ

where xmax and xmin are the initial and final inertia weight factors, Iter is the current iteration number, and Itermax is the maximum iteration number. Regarding the obtained velocity of each particle, (12), the particles are updated for the next iteration using from following equation:

X kþ1 ¼ X kj þ V kþ1 j j

ð14Þ

After this step, half of the population continues DPSO procedure and other half goes through the genetic algorithm operators. The first half continues their route at Step 7. While, the second half go through Step 6. Step 6. (Apply GA Operators) In this step, the crossover and mutation operators are applied to half of the individuals. This is done to increase the diversity of the optimizing variables to improve the local minimum problem. Step 7. (Check Convergence Criteria) If Iter = Itermax or if the output does not change for a specific number of iterations, the program is terminated and the results are printed, else the program continues from Step 2. 4. Simulation results The IEEE 14-bus and 18-bus distribution systems are modified and used to evaluate the proposed technique. These test systems are commonly used as the test system in many papers. Their corresponding data are available in [7–10,15–17,20,23–25]. These systems are composed of several linear and nonlinear loads. The 5th, 7th, 11th, 13th, 17th, 19th, 23rd and 25th harmonic orders are considered. The nonlinear loads are modeled as identical harmonic current sources. Fig. 2 shows the harmonic contents of the employed harmonic current sources (the nonlinear loads).

In this paper, two cases are examined. In case 1, the accuracy and robustness of the proposed MDPSO are compared with DPSO, GA, Simulated Annealing (SA), and DNLP. The robustness with respect to the changes of the PSO parameters is used to compare the proposed MDPSO and DPSO. Furthermore, a robustness study with respect to the changes of initial values is done to compare the MDPSO robustness with DPSO, GA, and SA. During these procedures, the accuracy of these methods is also evaluated. In case 2, the effect of including the fixed cost of APLCs on the final solution is investigated. In Sections 1 of case 2, the APLC fixed cost is assumed to be zero and the APLC investment cost is proportional with its rating, as done in almost all papers [7–16,20]. In Section 2 of this case, a realistic function is used to model the APLC cost. A comparison is finally implemented to show the influence of the APLC fixed cost on the results. Based on [17], the fixed cost of an APLC is assumed $90,000 and incremental cost of an APLC is assumed to be $720,000 per 1 pu. In this paper, the number of population members and iterations are normally selected 100 and 3000 for PSO-based methods. To make the problem more realistic, the APLC current rating is assumed as integer multiples of 0.05 pu. The APLC rating is rounded to the nearest bigger available APLC. For instance, if the optimal rating of an APLC is calculated 0.17, a 0.20 pu APLC is selected. For this purpose, the real and imaginary parts of the APLC currents are modified as follows: h;r Ih;r APLC m ¼ I APLC m  K C m

ð15Þ

h;i Ih;i APLC m ¼ I APLC m  K C m

ð16Þ

K Cm ¼

  I m BR  int APLC BR

where operator ‘‘int’’ shows rounding the variable to the nearest integer. K C m is a correction factor to correct the rating of the APLC located at bus m as integer multiples of BR, which is the base rating of APLCs. As mentioned, BR is assumed to be 0.05 pu in this paper. 4.1. Case 1 In this case, the IEEE 14-bus distribution system is modified and used to evaluate the robustness and accuracy of the proposed MDPSO. For this purpose, MDPSO robustness, with respect to the changes of PSO parameters is studied and compared with DPSO. The robustness with respect to the changes of the initial values is

0.14

x 10

6

M DPSO DPSO

1.8

Objective Function ($)

0.12 0.1

Current (PU)

ð17Þ

IAPLCm

0.08 0.06 0.04

1.7 1.6 1.5 1.4

0.02 1.3 0

5

7

11

13

17

19

23

Harmonic Order Fig. 2. Harmonic contents of used nonlinear loads.

25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Acceleration Coefficient (C1) Fig. 3. OF versus acceleration coefficient c1.

0.9

1

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I. Ziari, A. Jalilian / Electrical Power and Energy Systems 43 (2012) 630–639

1.8

x 10

6

1.75 1.7 1.65 1.6 1.55 1.5 1.45 1.4 1.35

x 10

6

M DPSO DPSO

1.65

Objective Function ($)

Objective Function ($)

1.7

M DPSO DPSO

1.6 1.55 1.5 1.45 1.4 1.35 1.3

1.3 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.2

0.3

Acceleration Coefficient (C2)

0.4

0.5

0.6

0.7

0.8

0.9

0.99

Final Inertia Weight Factor

Fig. 4. OF versus acceleration coefficient c2.

Fig. 6. OF versus final inertia weight factor.

also done to compare the proposed MDPSO with DPSO, GA, and SA. It is assumed that 10 nonlinear loads are located at buses 2, 3, 4, 5, 6, 7, 9, 12, 13, and 14. Fig. 3 illustrates the trend of objective function versus an acceleration coefficient, c1 in (12), for both DPSO and the proposed MDPSO. During the computations, the rest of parameters are kept constant. Additionally, identical initial values are considered for both DPSO and MDPSO to have a fair comparison. As observed in Fig. 3, the changes of objective function versus c1 for MDPSO is lower than DPSO. To assess the robustness of the methods, the ‘Relative Standard Deviation’ index (%RSD is defined as the standard deviation divided by the average) is used. This index illustrates how much the points are close to the average value. The %RSD of the objective function values resulted from a range of c1 for MDPSO is 4.4% which is less than half of 9.2% for DPSO. This demonstrates that selecting different values for this parameter can influence 4.4% on the final result in the proposed method. In other words, the proposed method is more robust, with respect to the changes of this parameter, compared with DPSO. The ‘average’ index is also used to show the accuracy of the proposed method. The higher this index is, the more accurate a method will be. Given the objective function values for a range of c1, the average value of the objective function for MDPSO is $1,364,400 which is about 10% lower than $1,513,800 for DPSO. This highlights the higher accuracy of the proposed MDPSO in terms of c1.

Similar to this procedure is performed for the second acceleration coefficient, c2. The trend of objective function for a range of this PSO parameter is shown in Fig. 4. %RSD of the objective function values for different values of c2 is 4.6% and 9.1% for the proposed MDPSO and DPSO, respectively. In terms of accuracy, the average of the objective function values is $1,350,000 and $1,441,800 for MDPSO and DPSO, respectively. These underline that the proposed MDPSO is more robust and accurate compared with DPSO in terms of c2. Figs. 5 and 6 depict the trend of objective function for a range of the initial and final inertia weight factors. The average and %RSD of the objective function values for a range of initial inertia weight factors are $1,333,800 and 2.98% for MDPSO and $1,492,120 and 6.57% for DPSO. In terms of final inertia weight factors, the average and %RSD of the objective function values for a range of final inertia weight factors are $1,315,800 and 3.43% for MDPSO and $1,492,200 and 8.03% for DPSO. Similar to acceleration coefficients, it is observed that the proposed MDPSO is more robust and accurate with respect to the inertia weight factors. The above results highlight that both MDPSO and DPSO are almost insensitive to a wide range of PSO parameters. This verifies the selection DPSO-based methods for solving OASA problem. In particular, the proposed MDPSO is more robust and accurate for this purpose. As mentioned in Step 6 of the proposed MDPSO (also

1.65

x 10

6

x 10 M DPSO DPSO

1.5 1.55

Objective Function ($)

Objective Function ($)

1.6

6

1.55

1.5 1.45 1.4 1.35 1.3 0.1

1.45 1.4 1.35 1.3 1.25

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Initial Inertia Weight Factor Fig. 5. OF versus initial inertia weight factor.

0.9

0.99

0

0.1

0.2

0.3

0.4

0.5

0.6

Ratio Fig. 7. OF for different ratios.

0.7

0.8

0.9

1

635

I. Ziari, A. Jalilian / Electrical Power and Energy Systems 43 (2012) 630–639

in Fig. 1), GA operators are applied to half of the population members. Fig. 7 shows why this ratio, 0.5, is selected. In this figure, the objective function versus a range of ratios is depicted. During this computation, all other parameters are kept constant. As observed, the minimum objective function is found when this ratio is equal to 0.5. In this figure, when this ratio is equal to zero, this means the optimization method is DPSO. After assessment of the proposed method robustness and accuracy with respect to the changes of parameters, these factors with respect to the changes of initial values (while the PSO parameters are kept constant) are investigated. For this purpose, MDPSO, DPSO, SA, and GA are run 25 times with randomly different initial values while the optimization method parameters are constant (Fig. 8). In this procedure, GA could not improve the results from some initial points so its corresponding results are not lumped in Fig. 8. It should be noted that the optimization methods are adjusted so that their computation time becomes identical. As observed, SA suffers from local minima problem so that it could not minimize the objective function significantly. As a criterion to compare the robustness of MDPSO and DPSO for different initial values, %RSDs are found to be 4.46% and 5.71% for MDPSO and DPSO, respectively. Similarly, their average is calculated to be $1,318,320 and $1,398,240 which are much less than $2,340,000 and $2,285,280 for GA and SA. These two points illustrate the higher accuracy and robustness of the PSO-based methods in general and the proposed MDPSO in particular compared with other methods for different initial values. Table 1 shows a comparison among the employed heuristic optimization methods. As observed in Table 1, $79,920, $966,960, and $1,021,680 cost benefits are gained by employing MDPSO instead of DPSO, SA, and GA, respectively. Table 1 highlights the priority of PSO-based methods over other methods for solving the OASA problem. In addition to the heuristic methods, DNLP is applied to solve the OASA problem in this case study. It should be noted that if a discrete size is assumed for APLCs in DNLP, the program traps in the initial values. To solve this problem, a continuous size is assumed for the APLCs. The DNLP related results are based on the continuous APLC size. After deriving the final results, the size of APLCs is rounded up to the closest available APLC rating. Table 2 demonstrates a detailed comparison among the employed optimization methods along with the ‘Zero THD’ case (all nonlinear loads are provided with an APLC) and ‘No APLC’ case (No APLC is installed). In the employed heuristic methods, the median solution among 25 runs is selected as the average value and the corresponding results are given.

x 10

6

M DPSO DPSO SA

Objective Function ($)

2.4 2.2 2 1.8 1.6 1.4

1.2 1

3

5

7

9

11

13

15

17

Run Number Fig. 8. OF for different initial values.

19

21

23

25

Table 1 Comparison of heuristic optimization methods.

GA SA DPSO MDPSO

Worst ($)

Best ($)

Average ($)

%RSD

2,340,000 2,340,000 1,512,000 1,440,000

2,340,000 2,232,000 1,296,000 1,206,000

2,340,000 2,285,000 1,398,000 1,318,000

0 2.43 5.71 4.46

Table 2 Comparison of optimization methods. Average THD (%) DNLP GA SA DPSO MDPSO Zero THD NO APLC

Average MLL (%)

Total rating

Total cost ($)

1.92 1.92 4.41 4.56 4.48 0

0.55 0.55 1.34 1.54 1.46 0

2.00 2.00 1.90 1.35 1.45 2.50

2,340,000 2,340000 2,268,000 1,422,000 1,314,000 2,700,000

14.59

4.17

0

Penalty

Table 2 shows that the total cost decreases from $2,700,000 to $1,314,000 by optimizing the APLCs. This underlines the importance of optimal allocation and sizing of APLCs. That DNLP, GA, and SA could not minimize the total cost significantly shows that the objective function has several local minima. After applying DNLP in the continuous mode, the result is 10 APLCs with the rating of 0.167 pu. However after rounding up to the closest available APLC rating, the optimal rating of APLCs has been raised to 0.20 pu. As observed in Table 2, the total cost related to the MDPSO is lower than DPSO while the total APLC rating is higher. This is because 3 APLCs with the rating of 0.50, 0.50, and 0.45 pu are found as the optimal solution for MDPSO while 5 APLCs with the rating of 0.35, 0.30, 0.25, 0.15, and 0.30 pu are the final solution for DPSO. This higher number of APLCs has applied higher installation cost; however, the total rating is lower.

4.2. Case 2 The IEEE 18-bus distribution system is modified and used in this case. This case is studied in two sections. In Section 1, it is assumed that the cost of an APLC is composed of only the incremental cost, which is proportional with the rating of APLC. A realistic cost function is assumed for an APLC in section 2. This cost function is composed of a fixed cost and an incremental cost. It is assumed that 11 nonlinear loads are located at buses 3, 4, 5, 6, 7, 8, 21, 23, 24, 25, and 26. Table 3 illustrates the results when no APLC is located in the network. As shown in Table 3, the average THD at all buses is 12.3% which represents an unallowable harmonic distortion level regarding to the IEEE standard (the standard limit is 5%). The maximum THD is related to bus 16, 17.11%. If no optimization method is employed, the APLCs can be simply located at the nonlinear load buses with the same size of the corresponding nonlinear load. Given the nonlinear load current which is 0.2303 pu, 11 APLCs with the rating of 0.25 pu should be placed at nonlinear load buses which results in an investment cost equal to $2,970,000. The optimization methods are employed to reduce this amount. A Modified DPSO (MDPSO) is proposed in this paper. To evaluate the proposed method, the MDPSO based results are compared with those obtained by DPSO, GA, SA, and DNLP. It should be noted that the number of population members and iterations in DPSO, GA, and SA are adjusted to have an identical computation time. It is observed that GA could find no feasible solution,

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Table 3 Voltage conditions at different buses in no APLC state. Bus number

THD (%)

MLL (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ATHD Max

11.23 10.60 10.78 10.18 9.34 10.36 12.32 12.18 9.51 11.64 13.13 12.55 14.36 15.85 15.67 17.11 12.30 17.11

3.27 3.09 3.15 2.98 2.74 3.04 3.61 3.58 2.78 3.93 3.83 3.66 4.19 4.61 4.58 5.00 3.59 5.00

Voltage distortion for each harmonic order (%) 5

7

11

13

17

19

23

25

3.15 2.98 3.05 2.89 2.67 2.97 3.53 3.51 2.68 3.26 3.69 3.53 4.05 4.42 4.43 4.86

3.55 3.35 3.43 3.25 2.99 3.33 3.97 3.93 3.01 3.68 4.15 3.97 4.56 4.99 4.99 5.47

4.62 4.36 4.45 4.20 3.86 4.30 5.11 5.06 3.91 4.79 5.40 5.17 5.92 6.51 6.47 7.08

5.06 4.78 4.86 4.59 4.21 4.68 5.57 5.50 4.28 5.24 5.91 5.65 6.47 7.13 7.07 7.72

4.96 4.68 4.74 4.47 4.10 4.54 5.39 5.32 4.20 5.14 5.79 5.53 6.32 7.00 6.89 7.52

4.68 4.41 4.47 4.21 3.85 4.26 5.06 4.99 3.96 4.85 5.46 5.22 5.96 6.61 6.49 7.07

2.72 2.57 2.59 2.44 2.22 2.46 2.91 2.86 2.30 2.82 3.17 3.03 3.46 3.85 3.76 4.09

1.69 1.59 1.61 1.51 1.37 1.52 1.80 1.77 1.43 1.75 1.97 1.89 2.15 2.40 2.33 2.53

Table 4 Comparison of APLCs rating current in zero fixed cost state. Bus number

Zero THD

SA

DNLP

DPSO

Proposed method1

Proposed method2

1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 Total rating ATHD AMLL OF (M$)

0 0 0.25 0.25 0.25 0.25 0.25 0.25 0 0 0.25 0 0.25 0.25 0.25 0.25 2.75 0 0 1.980

0 0 0.20 0.20 0.20 0.15 0.15 0.20 0 0 0.20 0 0.20 0.20 0.20 0.20 2.10 4.44% 1.47% 1.512

0 0 0.10 0.10 0.15 0.15 0.20 0.20 0 0 0.15 0 0.10 0.20 0.25 0.20 1.80 4.57% 1.63% 2.286

0 0 0 0.15 0.10 0.15 0.15 0.15 0 0 0.20 0 0.15 0.15 0.20 0.15 1.55 4.28% 1.38% 1.296

0 0 0.20 0 0 0.15 0.15 0.20 0 0 0.05 0 0.15 0.20 0.20 0.20 1.50 4.56% 1.41% 1.080

0 0 0.20 0 0 0.10 0.15 0.20 0 0 0.05 0 0.15 0.20 0.20 0.20 1.45 4.57% 1.43% 1.044

1.6

x 10

6

1.6

Objective Function ($)

1.5

Objective Function ($)

6

1.55

1.55

1.45 1.4 1.35 1.3 1.25 1.2 1.15

1.5 1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.1

1.1 1.05

x 10

1.05 50

100

150

200

250

Iteration Fig. 9. Convergence characteristic of MDPSO (300 iterations).

300

200

400

600

800

1000

1200

1400

Iteration Fig. 10. Convergence characteristic of MDPSO (2000 iterations).

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I. Ziari, A. Jalilian / Electrical Power and Energy Systems 43 (2012) 630–639 Table 5 THDs and MLLs in different buses in zero fixed cost state. Bus number

THD (%)

MLL (%)

1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 ATHD Max

4.35 4.30 4.67 4.97 4.81 4.98 4.98 4.39 3.86 4.38 4.80 4.59 4.53 4.50 4.30 4.67 4.57 4.98

1.37 1.35 1.47 1.54 1.48 1.54 1.56 1.37 1.22 1.38 1.51 1.45 1.45 1.45 1.36 1.45 1.44 1.56

unless one of the population members (APLC currents) is initialized equal to the nonlinear loads current. In this condition, this optimization method converges on this initial point. 4.2.1. Zero fixed cost In this section, it is assumed that the APLC cost is proportional with its rating and fixed cost is zero. This leads the program to converge in many APLCs. The APLC currents obtained by different optimization methods are shown in Table 4. Table 4 shows that the optimal number of APLCs is close to the number of nonlinear loads in all methods. As mentioned before, the APLC current in ‘Zero THD’, which is not based on an optimization method, is 0.25 pu because it is the nearest available APLC rating to the nonlinear load current, 0.2303 pu. Given this, the total rating of APLCs is calculated 11  0.25 = 2.75 pu. This value is considerably reduced when an optimization method is employed. As observed in Table 4, the total APLC rating is 2.10 pu when SA is employed. This value is decreased to 1.80, 1.55, 1.50, and 1.45 pu by using DNLP, DPSO, and the proposed algorithms as the optimization methods. The proposed algorithm shows that 9 APLCs with the rating of 0.2, 0.1, 0.15, 0.2, 0.05, 0.15, 0.2, 0.2, and 0.2 pu should be installed at buses 3, 6, 7, 8, 21, 23, 24, 25, and 26, respectively. Similar to case 1, if APLCs with discrete size is used in DNLP, the program traps in the initial values. Therefore, a continuous size is

assumed for the APLCs and it is finally rounded up to the closest available APLC rating. To evaluate the convergence of the proposed MDPSO, two versions of this method are lumped in the tables. ‘Proposed Method1’ is the MDPSO with the iteration number set on 300 and ‘Proposed Method2’ in which the iteration number is set on 2000. About 3% decrease is observed when the iteration number is increased from 300 to more than 400 (Figs. 9 and 10). However, the computation time is a point which should be considered. As mentioned, THD as a constraint is used in the optimization procedure which should be maintained within the standard level, 5%. This index along with MLL is shown in Table 5. A key point in all optimization methods is that the maximum THD is obtained close to 5% in all methods. This is to satisfy the THD criteria while lowest amount of APLC currents is found (Table 4). For instance, the maximum THD obtained in the proposed method is 4.98%. As expected, the higher THDs are related to the buses where the nonlinear loads are located. Figs. 9 and 10 demonstrate that the objective function decreases significantly and converges close its final solution after 300 iterations. This quick convergence is a main characteristic of the PSO-based optimization methods. As mentioned, a non-realistic cost function is assumed in this section so that the APLC investment cost, used in this section, is composed of only the incremental cost. The realistic cost function is studied in the second section. 4.2.2. Non-zero fixed cost Table 6 shows a comparison among the APLC currents obtained by different optimization methods when the realistic cost function is used to model the APLCs investment cost. The realistic APLC investment cost is composed of a constant cost and an incremental cost. The constant cost is a fixed cost such as the cost required for securing and purchasing land and the incremental cost is the cost which is proportional with the APLC rating (e.g. purchase cost). Therefore, in practice, the investment cost for two APLCs with the rating of S1/2 is more than the cost of one APLC with the rating of S1 because the installation of two APLCs requires two fixed costs. As a result, it is expected that the program converges in a lower number of APLCs. As shown in Table 6, the results obtained by the proposed method are more accurate than other optimization methods. Although the optimal rating of APLCs by the ‘Proposed Method1’, 1.50 pu, is more than it by DPSO, 1.40 pu, the total cost by the proposed method1, $1,530,000, is less than DPSO, $1,548,000. Since, the

Table 6 Comparison of results in non-zero fixed cost state. Bus number

SA

DNLP

DPSO

Proposed method1

Proposed method2

1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 Total rating ATHD AMLL OF (M$)

0 0 0.20 0.20 0.20 0.15 0.20 0.20 0 0 0.20 0 0.20 0.15 0.20 0.20 2.10 4.15% 1.33% 2.502

0 0 0.10 0.15 0.10 0.20 0.15 0.15 0 0 0.15 0 0.10 0.20 0.30 0.15 1.75 4.43% 1.61% 2.250

0 0 0.20 0 0 0 0.30 0.15 0 0 0 0 0.20 0.20 0 0.35 1.40 4.59% 1.57% 1.548

0 0 0.25 0 0 0 0.50 0 0 0 0 0 0 0.25 0.30 0.20 1.50 4.37% 1.48% 1.530

0 0 0.20 0 0 0 0.45 0 0 0 0 0 0 0.35 0 0.40 1.40 4.47% 1.56% 1.368

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F

8 7

Substation 12.5 kV / 138 kV 1 50

F 6

5

4

3

Swing Bus 51

2

9

20

21

22

F 26 23

25

24

F

Fig. 11. The 18-bus test system configuration after APLC planning.

optimal number of APLCs by the proposed method1 is 5 as it is 6 by DPSO which applies another fixed cost, $90,000. Compared with zero fixed cost state, in which the optimal number of APLCs is 9 for the proposed methods, the optimal number of APLCs is found to be 5 and 4 in ’Proposed Method1’ and ’Proposed Method2’. It should be noted that the total APLC currents in this state is close to those in zero fixed cost state as expected. As observed in this table, 4 APLCs with the rating of 0.2, 0.45, 0.35, and 0.4 pu are required to be installed at buses 3, 7, 24, and 26, respectively. Fig. 11 shows the configuration of APLCs resulted from the proposed method. The red elements in this figure illustrate the location of APLCs. Table 7 illustrates THD and MLL at all buses when the proposed optimization method is applied. Similar to the preceding state, the maximum THD is maintained close to 5% to employ lower total APLCs rating. The convergence characteristic of the MDPSO based optimization methods are shown in Figs. 12 and 13. A 12% decrease is seen when the iteration number increases from 300 (Fig. 12) to more

than 600 (Fig. 13). However, this 12% difference applies at least twice more computation time. Minimizing the objective function value to about half of the initial value in less than 300 iterations demonstrates the quick convergence of the proposed algorithm. In this paper, APLCs are planned for the worst case when the nonlinear loads inject their maximum harmonic currents. To show the reliability of APLCs, it is assumed that the nonlinear loads inject half of the current they inject in the worst harmonic polluted case. As an illustration, as shown in Table 6, the optimal solution is to allocate 4 APLCs with the rating of 0.20, 0.45, 0.35, and 0.40 at buses 3, 7, 24, and 26, respectively to handle the worst harmonic polluted case. The maximum and average THDs in this case are 4.96% and 4.47%. Now if the nonlinear loads inject half of the worst case current, the optimal solution is that the APLC located at bus 3 injects 0 pu, and the APLCs located at buses 7, 24, and 26 inject 0.10, 0.10, and 0.15 pu. The maximum and average THDs in this case are 4.69% and 4.32%. As observed, the APLCs found for the worst harmonic polluted case are reliable even if the nonlinear loads inject lower harmonic currents.

Table 7 THDs and MLLs in different buses in non-zero fixed cost state. THD (%)

MLL (%)

1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 ATHD Max

4.36 4.27 4.62 4.83 4.56 4.71 4.28 4.96 3.83 4.44 4.94 4.73 4.64 3.86 4.30 4.18 4.47 4.96

1.52 1.48 1.59 1.62 1.50 1.57 1.42 1.66 1.33 1.55 1.72 1.65 1.68 1.35 1.65 1.63 1.56 1.72

x 10

6

2.4

Objective Function ($)

Bus number

2.6

2.2 2 1.8 1.6 1.4

50

100

150

200

250

300

Iteration Fig. 12. Convergence characteristic of MDPSO (300 iterations).

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I. Ziari, A. Jalilian / Electrical Power and Energy Systems 43 (2012) 630–639 Table 8 Comparison of APLCs investment cost in cases 1 and 2. Bus number

APLCs investment cost (M$)

Case 1 Case 2

2.6

x 10

Zero THD

SA

DNLP

DPSO

Proposed method1

Proposed method2

2.970 2.970

2.502 2.502

2.286 2.250

2.016 1.548

1.890 1.530

1.854 1.368

method get benefit from higher accuracy and robustness compared with other methods. The benefit of the assuming a realistic cost function for an APLC for solving OASA problem is demonstrated in case 2.

6

Objective Function ($)

2.4

References

2.2 2 1.8 1.6 1.4 1.2

200

400

600

800

1000

1200

1400

Iteration Fig. 13. Convergence characteristic of MDPSO (2000 iterations).

4.2.3. Comparison of zero and non-zero fixed cost states Based on Ref. [17], CC and CI are equal to k$90 and k$720 respectively. Using these values, the realistic investment cost of APLCs in zero and non-zero fixed cost states are calculated as shown in Table 8. As revealed in Table 8, considering the fixed cost leads the problem to lower investment cost. The total investment cost decreases from $1,890,000 to $1,530,000 in the proposed method1 and from $1,854,000 to $1,368,000 in the proposed method2, when the fixed cost is included. This illustrates the importance of including this factor in the planning procedure. This table also shows a cost benefit about 54% over the ‘Zero THD’ case. This highlights the importance of planning the APLCs. Compared with other optimization methods, cost benefits $1,134,000, $882,000, and $180,000 are gained by using the proposed MDPSO instead of SA, DNLP, and DPSO, respectively. This underlines the priority of the proposed algorithm over other methods for solving the OASA problem.

5. Conclusions To allocate and sizing of multiple APLCs in distribution systems, a modified DPSO is proposed in this paper. In this optimization method, DPSO is developed by Genetic Algorithm operators to increase the diversity of the optimizing variables. This decreases the risk of trapping in local minima in DPSO. The objective function is the APLCs investment cost which should be minimized during the optimization procedure while the voltage total harmonic distortion and individual harmonic distortion as constraints should be maintained less than 5% and 3% respectively. To evaluate the proposed algorithm, two cases are studied. The IEEE 14-bus and 18-bus distribution systems are employed as the test systems in cases 1 and 2, respectively. In case 1, the robustness and accuracy of the proposed MDPSO are evaluated and compared with DPSO, Genetic Algorithm, Simulated Annealing, and Discrete Nonlinear Programming. The results illustrate that the proposed

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