Symmetric Fuzzy Logic and IBFOA Solutions for Optimal Position and Rating of Capacitors Allocated to Radial Distribution Networks

: This paper suggests an innovative approach for the ideal placement and categorization of capacitors in radial distribution networks (RDNs) by applying symmetric fuzzy and improved bacterial foraging optimization algorithm (IBFOA) solutions. The reactive power reimbursement signiﬁcantly enhances the function of the power system, and capacitor placement is an impressive technique used to reduce loss of the system. The capacitor allocation for distribution system problems involves determining the ideal location and size of the capacitor. In this work, load ﬂow is performed at ﬁrst to compute actual losses and voltages at different nodes without compensation. In the planned technique, the loss sensitivity factor (VSF) and voltage stability index (VSI) are utilized to determine the optimal location of capacitors in RDNs. Here, the IBFOA is used to determine the proper rating of the capacitor. The suggested scheme is applied on three different types of RDNs.


Introduction
Electric power is constantly transferred from source to distribution via transmission, where active and reactive power losses occur [1]. Real power drop in any radial power distribution system is significant because reactive power losses that can be controlled by legitimate management of reactive power. To avoid reactive power accidents, nearby reactive power compensation via appropriate capacitor placement is the most powerful method employed throughout the world [2]. Advantages of capacitor placement include minimization of real and reactive power losses, power factor enhancement, appropriate voltage profile maintenance, and the release of overburden on feeders and transformers [3]. Aside from these advantages, if the size and location of the capacitor are not appropriate, then the framework might be defenseless and act in anomalous ways, and voltage increments might pass cut-off points, causing unsatisfactory power factors, poor arrangements, and parallel resonance issues [4,5]. Considering this as a top priority, many researchers are working for the best reproduction process for ideal capacitor placement and measurement [6].
Optimal capacitor placement has recently been the subject of numerous research works. The majority of papers on this issue have tackled the issue with graph search algorithms, ant colony direction, particle swarm optimization (PSO), fuzzy evolutionary programming, and genetic algorithms. In addition, cost per covariance for every capacitor shifts from one size to another [7].

Related Work
Some of the recent work related to the optimal placement and sizing of capacitor is listed below. Changes in system voltage profiles, change in system power components, and expansions through cables and transformers are the surely understood advantages of optimal positioning of capacitors in force power circulation systems. The reduction of accidents because of the remuneration of the responsive parts of the influence stream is another point of interest in the field of capacitor placement. By diminishing flow through cables, the systems' heap can be expanded without over-burdening the current links or including any new links. These advantages depend fundamentally on how shunt capacitors are placed in the system.
Elsheikh et al. [18] tended to the issue of how to ideally distinguish careful areas to introduce capacitors and buses of power distribution feeders. The ideal sizes of capacitors also needed to be introduced. Their procedure utilized accident affectability components to distinguish transports requiring remuneration. Additionally, a discrete PSO was utilized to determine capacitor sizes.
In [19], an ideal capacitor situation in power distribution systems understood by the HCODEQ (hybrid CODEQ technique) obtained from hybrid combination of chaotic search, opposition-based learning, differential evolution and quantum mechanics was proposed. The ideas of quantum mechanics, chaotic search, and resistance-based learning were utilized as a part of the CODEQ technique to defeat the downside of parameter choice in the differential evolution (DE).
Zeinalzadeh et al. [20] presented a utilization of multi objective particle swarm optimization (MOPSO) system. That strategy had a point of determining the ideal location and the shunt capacitor bank size (SCBs and circulated areas (DGs)) in distribution systems while considering load vulnerability conditions. The three target elements of multi-target advancement consist in enhancing voltage steadiness, diminishing active power losses for buses, and adjusting the current in framework segments. The vulnerability of losses was displayed utilizing fuzzy data theory. This strategy utilized Pareto ideal answers to handle issues of target capacities and limitations and to determine the best arrangement among the three distinctive target capacities. Notwithstanding this, a fuzzy-based mechanism was utilized.
Sultana and Roy [21] displayed the teaching-learning-based optimization (TLBO) algorithm to deal with minimize capacitor cost and the loss of power in power systems by the ideal arrangement of capacitors. The proposed computation depended on two essential ideas of instruction: teaching stage and learning stage. In the teaching stage, learners enhanced their insight or capacity through the arrangements of the teacher. Learners expanded their insight by cooperation among themselves in the learning stage. To check achievability, the proposed technique was tested on 22-node, 69-node, 85-node, and 141-node RDNs.
Shuaib et al. [22] presented the gravitational search algorithm (GSA) for optimal capacitor placement in RDNs. They tested their method on 33-node, 69-node, 85-node, and 141-node RDNs and compared their proposed method with an interior point (IP) algorithm and simulated annealing (SA).
A direct search algorithm (DSA) was used by Raju et al. [23] to obtain ideal sizes of capacitors in RDNs for reductions in losses of the system. They considered 22-node, 69-node, and 85-node RDNs to implement their method.
Abdelaziz [24] had used a power pollination algorithm (FPA) to locate capacitors in RDNs with a target to reduce the size of capacitors at the selected locations. They had selected the candidate node using the power loss index (PLI). They applied their method on 15-, 69-, and 118-node RDNs. They then compared their method with other algorithms.

The Proposed Methodology for Optimal Placement and Sizing of Capacitor
It is essential to integrate shunt capacitors in RDNs to retain the RDN's stability and reduce its loss. However, improper placement and sizing of the capacitor causes additional problems. Thus, the identification of ideal placement and size is essential for the deduction of power loss of the system. Hence, this methodology is intended to develop a better work for the sizing and placement of capacitors in RDNs. In the proposed system, symmetric fuzzy power flow evaluation is proposed to maximize the accuracy of the method, to reduce computational effort, and to make it possible to achieve high levels of accuracy when applied to real systems. A multi-objective IBFOA is also suggested to determine the ideal or optimal rating of shunt capacitors in RDNs. Details pertaining to the IBFOA is available in [25].

Fuzzy-Based Load Flow
The voltage stability index (VSI) and the loss sensitivity factor (LSF) are the two factors used for identifying the exact location of the capacitor. Buses with greater LSF values take highest priority for placement identification. When the voltage collapse probability of a bus is high compared with that of other buses, it is chosen as the second location for capacitor placement and is determined using VSI. Here, load flow using fuzzy modeling improves accuracy.
A distribution system sample modeled is represented in Figure 1. The LSF and VSI are dependent on voltage and power losses at the particular node [25]. The degree of symmetry fuzzy ranges between 0 and 1. The voltage membership function is described as where w v is the weighting factor for the voltage membership function; v max is the maximum value of permitted voltage; v min is the minimum value of permitted voltage. The formula below depicts the membership function for active power loss where w p is the weighting factor for membership function of active power loss; L(p) is the real power loss between l and l + 1 buses; T p is the total real power loss. The membership function for reactive power loss is given by Equation (3).
where w Q is the weighting factor for reactive power loss membership function; L(q) is the reactive power loss between l and l + 1 bus; T q is the total reactive power loss. Figure 2 explains the flow chart for the symmetry fuzzy load flow. The above formulae are the membership functions of the power loss and voltage values. The fuzzy power flow is used to compute the losses, voltages, and phase angles. The inputs of the general fuzzy power flow, i.e., the injection of active and reactive powers, are taken as the constraints in the symmetric flow. In the voltage calculation, it is possible to evaluate its highest and lowest limit; when these limits are considered, the active and reactive powers are within the limit. The membership functions are calculated from the initial load flow, and the fuzzy condition is formed using these membership functions. By using this condition, the LSF and VSI values are evaluated for each node. Based on the values, the optimal location for capacitor placement is identified.

Optimal Placement and Sizing of the Capacitor
Load flow is achieved using the fuzzy method. Based on the membership functions, the voltage and loss of the system are measured. By the measured value, the condition below is checked, and the LSF and VSI values are then measured in order to locate the capacitor. The loss sensitivity factor, which is higher for a node with a lower VSI value, is used to determine the location of the capacitor. At each node, the fuzzy condition is expressed as follows: (a) LSF: This factor is used to recognize the proper position of the capacitor. The node of the radial distribution system, which has a greater chance of determine the ideal location of the capacitor, has a supreme loss sensitivity factor value.
By arranging the LSF values of all buses and nodes in descending order, we selected the node or bus that has the highest value of LSF.
(b) VSI: To check the power system's security level, many parameters are used, one of which is the VSI. Nodes that produce more voltage collapse can be identified using VSI. Capacitor placement is determined at node with a low VSI value. The VSI at every node is computed using Equation (5) [26]: where P m is the effective active power at the node m; Q m is the effective reactive power at the node m; R lm is the resistance between the node l and m; X lm is the reactance between the node l and m; the voltage at node l is V l ; the voltage at node m is V m .

Improved Bacterial Foraging Optimization Algorithm (IBFOA)
To deal with different optimization problems in the distribution system, the IBFOA was used, which is an enhanced version of the bacterial foraging optimization algorithm (BFOA). The BFOA is an effective swarm-intelligence-based stochastic interest technique designed by Kevin Passino. The thought driving the BFOA assumes a normal determination of beings with poor foraging strategies, eliminates these beings, and favors those with productive foraging strategies. The improved BFOA is implemented by improving the foraging processes of bacteria. The proposed work is improved with chemotaxis and an elimination and dispersal process. The strategy of poor foraging is either transformed into extraordinary ones or, after numerous eras, eliminated [25]. The normal developmental procedure strategy relies on their wellness criteria and are requested into • an ability to search food • self-charging (mobile behavior) The foraging mechanism of E. coli bacteria is governed by four processes: (1) chemotaxis; (2) swarming; (3) reproduction; (4) eradication and dissemination. Parameters adopted in the IBFOA algorithm are P, S, N s , N c , N re , N ed , P ed , n, C(i), and θ i .
Chemotaxis: Chemotaxis can be achieved by swimming and tumbling movements of each bacterium. Swimming denotes the movement of a bacterium that is done in a predefined way. Tumbling refers to the random movement.

Tumble =
Step length o f that bacteria × Unit length o f random direction Swarming: Swarming refers to the richest food source margining of bacteria in a concentric pattern with high bacterial density.
Reproduction: When the unhealthy bacteria will die, the healthiest one splits into two bacteria. This makes the bacterium population as a constant value. The new bacteria are placed in the position where the least healthy bacteria were present.
Elimination and Dispersal: The bacterium population may change due to unexpected changes. For example, a significant rise in temperature can kill bacteria in some places. All bacteria in a particular location are killed or replaced into a new location. The Pseudo code of IBFOA is given below (Algorithm 1). Perform elimination and dispersal up to l ≥ N ed 7: Calculate fitness for capacitor size 8: End Step 1 Parameter Initialization The speed of convergence of the algorithm contrasts with various combinations of parameters accordingly to accomplish the fastest convergence of the algorithm, which keeps running for various lengths of time for various estimations of the parameters given above and defined now. N s is the digit of iteration (31); P is the number of optimization variables (3); S is the numeral value of bacteria used for searching the total area (50); N c is the numeral value of chemotactic steps (4); N ed is the maximum numeral value of elimination and dispersal events (2); N re is the maximum numeral value of reproduction steps (4); P sd is the probability of elimination and dispersal process (0.25); n is the digit of nodes; C(i) is the step change in size in the random direction (0.05 × ones(S, 1)); θ i is the assigned location, and the lower and upper limits of the capacitor bank θ i = θ i 1 , θ i 2 , θ i 3 where i = 1, 2, . . . , S; the height of the repellent: h repellent (0.1); the width of the attractant: w attractant (0.2); the width of the repellent: w repellent (1.0); the depth of the attractant released by the cell: d attractant (0.1).

Step 2 Optimal placement
For the optimal placement of the capacitor, conditions are checked with membership functions µ P (l)&µ Q (l) and µ v (l), and the VSI and LSF values determine the best location for capacitor placement. The process is repeated for all nodes. The optimum location is computed using the formula Optimum Location : Fitness 1 = min VSI LSF . (8) Step 3 Update of the IBFOA, Elimination and Dispersal, Reproduction and Chemotaxis Processes.
After every N re reproduction steps, the elimination and dispersal is done. Likewise, after every N cd steps, a reproduction is taken into account.

Step 4 Chemotaxis
The chemotaxis is run for all bacterium. The fitness function is In chemotaxis, by using the objective function, the fitness function is computed. The objective function is based on the repellent and attractant signal. The fitness function value is equated in the F last value. For the next fitness, it is necessary to compute the movement. For that, a random vector is generated. The movement is computed by For every bacterium, the fitness function is computed and updated in F last . When the number of iterations exceeds the number of bacteria, the chemotaxis process is ended.

Step 5 Reproduction
The half number of the aggregate bacteria with the highest costs will pass on and the leftover half of the bacteria with the foremost values will be separated and set at the same location as that of their parents.
If the value of the maximum number of reproduction is less than the k value, increments of reproduction are performed.

Step 6 Elimination and Dispersal
The low F value bacterium will be selected. Every bacterium is wiped out and dispersed with likelihood, which ensures that the quantity of the microbes in the population is a consistent value. For a conclusive bacterium population, the value of F is ascertained. The ideal size of the capacitor bank is given by the bacterium, which gives the best F value.
By combining the two fitness values, we can determine the optimal size of the capacitor at the appropriate location.

Experimental Results
The proposed methodology was implemented in MATLAB, and the system had the following parameters: Operating speed: 2.90 GHz. Processor: Intel Pentium@ 2.90 GHz. RAM: 4 GB. MATLAB Version: R2014a.
With respect to static characteristics, there were load models are different types. These were used in the RDNs. In the proposed approach, the constant power model was used for analysis. The methodology for the distribution system analysis and load modeling was implemented in the working platform of MATLAB using 33-, 69-, and 141-node RDNs. In our proposed method, the cost value is taken as the load value. MATLAB can compute the power loss of any RDN and identify a capacitor bank's optimal location using the load flow. The correctness of the proposed methodology has been verified with existing work. The voltage and power values of the base case were computed from the load flow analysis. The load modeling was done by using Equation (14).
where α is the cost of the capacitor.

The 33-Node RDN
The total load of the 33-node RDN [27] with 12.66 kV and 100 MVA as base values was 3.7 + j2.3 MVA. The results (the optimal location and size, the lowest values of voltage, and the VSI along with node and real power loss for three different types of loads-light (50%), normal (100%), and peak (160%)), before and after compensations utilizing the suggested methodology, are given in Table 1. Figure 3 plots the magnitude of the voltage vs. the number of nodes before and after compensation for light load conditions (50%), normal load conditions (100%), and peak load conditions (160%). Figure 4 plots the VSI vs. the number of nodes before and after compensation for light load, normal load, and peak load.   The results attained from the proposed method after compensation has also been juxtaposed with GSA, SA, and IP [22] as shown in Table 2 for normal load (100%). The results (minimum voltage and node number, optimal location and size, and real power loss for three different loads-light (50%), normal (100%), and peak (160%)), after compensation obtained by the proposed scheme, are given in Table 3 along with the results obtained by TLBO [21], DSA [23], and FPA [24]. Table 3. Results of the proposed method with TLBO [21], the direct search algorithm (DSA) [23] and the power pollination algorithm (FPA) [24] (at loads 50%, 100%, and 160%). The magnitude of voltage vs. the number of nodes before and after compensation for each case of the above three loads are plotted in Figure 5. Figure 6 plots the VSI vs. the number of nodes before and after compensation for the above three loads.

141-Node RDN
The third example is a 141-node RDN with 12.47 kV and 100 MVA as base values. This network is considered with an initial loss of 116 kW. The system loads are 1171.5125 kW and 725.7846 kVAr, respectively. The system data is available in [22].
The result realized by the suggested method are here juxtaposed with TLBO [21] for three types of load (light, normal, and peak) and with GSA [22] for normal load, as shown in Table 4, after compensation. The proposed method, compared to those found in [21,22], provides improved results.

Conclusions
The research in this paper was motivated by power loss diminution and improved stability in the distribution system. In RDNs, enhancement in the voltage profile, the depletion of power loss, and improvement in the system stability can be performed by optimal capacitor placement, which is presented in this paper. The combined approach of symmetric fuzzy flow and IBFOA is proposed to determine the proper size and position of the capacitor in order to decrease system loss and enhance voltage stability. The best position and size of the capacitor is found out by using LSF, VSI, and the IBFOA technique. The recommended technique was exercised on 33-node, 69-node, and 141-node RDNs. Outcomes are juxtaposed with outcomes of previous related works. The proposed method provides improved results for the curtailment of loss and the enhancement of voltage.