Introduction

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Transmission and distribution system losses are considered the main consumption in any power network. Due to the growth in the load, and environmental limits, the transmission, and distribution networks are being worked under overloaded situations, and losses in the distribution systems have become the main concern. To attain economic advantages, the fundamental conditions to get agreeable power quality and enhanced efficiency have formed a very auspicious environment for the matter of loss minimization approaches and using recent operational practices. Power loss reduction is the only alternative to enhance the efficacy of the distribution network. Therefore, it is noted that in the last few decades many researchers have concentrated on distribution network loss reduction and voltage stability. There are many helpful techniques in the literature for distribution network loss reduction [1]. However, the most often used mechanisms like (a) capacitor siting, (b) network restructure [2] (c) DG siting [3], (d) DSTATCOM siting and its mixed versions to realize upper potential interests are (e) simultaneous restructure and capacitor siting, (f) simultaneous restructure and DG siting, (g) simultaneous DG and DSTATCOM siting, and (h) simultaneous restructure, capacitor, and DG siting are presented in [1]. Conventionally, loss reduction has focused mainly on network restructure optimizing or capacitor siting for reactive power policy. Since installing capacitors are the simplest and most famous solution, they are getting steadily to be important components of the distribution network [1].

Pending last years, diverse algorithms are presented to find the proper locations and optimal sizes of shunt capacitors. Simulated Annealing (SA) [4], Tabu Search (TS) [5], Genetic Algorithm (GA) [6], Mixed Integer Nonlinear Programming Approach (MINPA) [7], Direct Search Algorithm (DSA) [8], Teaching Learning Based Optimization (TLBO) [9], Plant Growth Simulation Algorithm (PGSA) [10], Heuristic Algorithm [11], Cuckoo Search Algorithm (CSA) [12], Particle Swarm Optimization (PSO) [13], Fuzzy Genetic Algorithm (FGA) [14], Differential Evolution (DE) [15], Flower Pollination Algorithm (FPA) [16-17], Improved Harmony Search (IHS) [18], Mine Blast Algorithm (MBA) [19], and Moth Swarm Algorithm (MSA) [20] are developed to deal with the capacitor placement task. However, these techniques may drop to compass the optimum cost. To conquer these abuses, the SMA is chosen in this article to treat the process of optimum capacitor placement.

SMA is a vigorous population-based optimizer based on the oscillation mode of slime mould in nature [21-23]. It proves its effectiveness in many applications [24-25]. It is suggested here as a modern optimization approach to minify the net active power losses, the net cost and to promote the voltage profiles for an IEEE distribution network. The stations of the shunt capacitors procedure are acquired firstly by inspecting the points according to many indices. Then SMA is submitted to deduct the optimum siting and sizing of capacitors from specific nodes. The validation of the suggested algorithm in progressing the voltage profile and lowering resistive losses is given for an IEEE distribution system. The results of the SMA are compared with several algorithms to assert its superiority

The rest of the paper is constituted as follows: Section 2 shows the different indices for capacitor installing techniques. Section 3 presents the cost function and limits. Section 4 introduces Slime Mould Algorithm. Section 5 examines the results on voltage profiles and power loss. Section 6 gives the conclusion and future works to treat the distribution network optimization process.

Various Indices

In this section, three different indices with their equations are introduced.

Power Loss Index

In this article, PLI is utilized to specify the nominee points for capacitors. The region of inspection is diminished greatly and then wasted time in the optimization procedure. The demerit of this index is the pivotal computations.It is desired to execute load flow and define the attenuation in power losses by intromission reactive power at every bus except the slack one [26-27]. The PLI is given as the following equation.

Where

lr_u: The upper attenuation in actual power losses.

lr_l: The lower attenuation in actual power losses.

lr_i: The attenuation in actual power losses at the bus i.

The buses of greater PLI will have the primacy to be the nominee bus to constitute compensator apparatus.

Loss Sensitivity Factors

LSFs are appointed to specify the nominee buses to install capacitors [28]. Figure 1, shows a transmission line 'l ' linked between 'i ' and ' k ' buses.

Figure 1

The actual power loss is obtained by in this line that can be computed by

The LSFs can be given from the following equation:

The typical voltages are specified by dividing the base voltages by 0.95. If these voltages are lower than 1.01 they can be suggested as nominee nodes for capacitor installation. It is account note that the LSFs judge the series in which nodes are to be prepared to install capacitors.

Voltage Stability Index

VSI amount is close to 1 so the least VSI amounts, the mightily sensitive points to voltage collapse. Thus, VSI is utilized to elect the lowest points that have more probability of voltage collapse along all points. VSI amount is designated as the following equation [29-31]

where

V_i: The magnitude of the voltage at the bus i.

Cost Function

The developed cost function of the optimum capacitor position task is to lessen the net cost which is planned as the following equation:

Where the parameters are given as [17].

K_p: The price per KW-Hours equalizes to 0.06 $/KW-Hours,

P_Loss: The net losses after compensation,

T : The time in Hours equalizes to 8760,

D : The depreciation agent equalizes to 0.2,

CB : The number of compensated points,

K_C: The price per Kvar equalizes to 25 $/Kvar,

K_I: The price per inauguration equalizes to 1600$,

Q_C: The value of inaugurated reactive power in Kvar,

K_O: The working price equalizes to 300 $/year/position.

The overhead equation is constricted whereas accepting the following equality and inequality limits.

Equality Limits

Load flow limit

Conventional methods cannot be utilized in distribution networks due to ill conditions. The forward sweep method has been presented in [29] to treat the load flow process of distribution networks. The equality limit is shown by the following equation:

Where

P_Slack: The active power of the slack node,

Q_Slack: The reactive power of the slack node,

L : The size of the transmission line in a distribution network,

Pd_(q): The request for active power at bus q,

Qd_(q): The request for reactive power at bus q,

N: The size of total nodes.

Inequality Limits

Voltage Limits

The magnitude of the voltage at every node must be constrained by the following limit:

0.90 ≤ V ≤ 1.05

Compensation Limit

The injected reactive power at every nominee node should be lower than its efficient reactive power.

Power Factor Constraint

Power Factor (PF) should override the lower amount and less than the upper amount as given by the following limit.

Total Reactive Power Limit

Remarkably, the net injected reactive power is lower than 0.7 of the net reactive power request to extend the operating of the power system with lagging PF and prohibition the leading one.

Conventional SMA

Physarum polycephalum has been named slime mould because it is considered a fungus [21].

Approach food

SMA is illustrated by the mathematical equations of [21-22]. The equations represent the contraction mechanism as follows:

Where is a value [− a, a],

is reduced linearly from 1 to 0,

t is the t_th iteration,

is the individual location with the most concentrated scent?

is the location of slime mould (solution)?

and are the random selection of two individuals from the population,

is the weight solution,

p is computed as:

∈ 1,2,3,...,n,S(i) is to the competence of

DF is the most competence calculated over all iterations?

is computed as follows:

is computed as follows:

S(i) ranks the first half of the given solutions,

r a random number [0,1],

bF is the optimal competence existing in the current iteration?

wF is the worst competence existing in the current iteration?

smellIndex is the sequence of the sorted competence values?

Figure 2: shows the impacts of (11).

Figure 2

Wrap food

When the search area extends to an area with a rather low concentration of food, its importance will decrease and the food group will go to explore other areas. Fig. 3 shows the competence evaluation functions for slime mould [23-24].

Figure 3

The solution location is updated according to the following equation:

Where, LB is the lower limit of the search space,

UB is the upper limit of the search space?

r is a random value [0, 1].

Grabble food

is a random vector between [− a,a] and regularly accessed zero as the repetitions advance. The values lay between [−1,1] and are driven to zero in the end. Synergistic cooperation between and imitate the specific manner of slime mould. Although sticky mould is a better source of nutrition, it is preferable to distribute the organic matter to explore other areas to find the best food source rather than pouring it all into one source to find a more reliable source of food. The Pseudo-code of the SMA algorithm is shown below [25].

Pseudo-code of SMA

Initialize population_size , dim, LB, UB, z , Max _ iter ;

Initialize a set of Slime Mould random locations X (i 1,2, ,n) i = 1;

While (t ≤ Max _ iter )

Compute and sort the competence of all Slime Mould;

Update the best and the worst competence

Calculate the weight of Slime Mould

Update the best competence, the best location

For each search agent

Update p ,

Update the position of search agent

End For

t = t +1

End While

Return the best competence, best location

Results and Discussion

The prevalence of the suggested SMA is applied to an IEEE distribution system. The results of 15 radial distribution systems are offered below in detail. The suggested algorithm has been completed via Matlab [32]. Simulations were performed under the Matlab environmental (release 2013 a) and done on a Lenovo laptop with Intel core i7 CPU 2.90 GHz processor with 4 GB RAM and a 64-bit operating system.

Figure 4

15 Bus Test System

The tested case is 15 bus system as given in Figure (4). The system data are displayed in [31]. The total load for this system is 1752 KVA with PF=0.7. The losses without compensation are 60.5844 KW. Figure (5) gives the nominee buses according to their PLI. The ordered of these buses are 15, 11, 4, 7, 6, 12, 14, 3, 8, 13, 2. The nominee buses are obtained in Figure (6) according to LSF. The ordered of buses are 6, 3, 11, 4, 12, Figure (7) shows the nominee buses according to VSI values. The improvements in system voltages due to install one and two capacitors are shown in Figures (8, 9) respectively. A comparison between various indices is performed and shown in Table (1, 2) for installing one and two capacitors respectively. It is clear that, PLI gives better results than VSI and LSF for this system. Based on these results, SMA is proposed with PLI to give the better response in terms of cost and losses. The notability of the suggested SMA is demonstrated compared with other algorithms in [13,14,15,33,34]. The value of installed capacity of reactive power is 850 KVAR. The minimum voltage is increased from 0.9424 to 0.9679 p.u. The losses with compensation are decreased to 32.2499 KW due to capacitors installation as given in Table (3). The percentage reduction in losses is increased to be 46.768%. Moreover, the value of total cost due to the proposed objective algorithm is 23060.54 $ which is the smallest one. Also, the net saving with the proposed SMA is improved to 29.183 % which is the maximum one compared with other algorithms.

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Table 1

Table 2

Table 3

Conclusion

In this paper, SMA has been applied successfully to solve the tasks of optimum positions and sizing of capacitors in distribution system that have been established as an objective optimization task with competing power losses, cost of installation, operation and injected vars. The superiority of the suggested approach is clarified by using IEEE test system. Also, the results have been compared with those obtained using recent optimization techniques. Moreover, it provides a preferable performance over others in terms of voltage profiles, active power losses, net cost, and total saving. Application of the network reconfiguration and distributed generation with the most novel optimization algorithm to reduce the active losses is the future scope of this work.

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