# Hybrid Design of Optimal Capacitor Placement and Reconfiguration for Performance Improvement in a Radial Distribution System

Authors: Ali Nasser Hussain; Waleed Khalid Shakir Al-Jubori; Haider Fathi Kadom
Journal: Journal of Engineering
 (2019)

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## Abstract

Distribution System Reconfiguration (DSR) and Optimal Capacitor Placement (OCP) are the most alternative techniques for increasing the power system generation and covering the growth of power demands. These techniques reduce the Radial Distribution System (RDS) losses and enhance the voltage profile. Combining both techniques gives better performance than using the individual technique. In this paper, two operation modes were implemented. First, the individual mode of OCP is applied. Second, the dual mode of DSR after OCP process is applied. Multiobjective functions with considering the weighting factors are used for minimizing real losses, improving voltage profile, and increasing saving cost. The optimal selections of open switches, location, and size of capacitors in the individual and dual design for RDS are achieved using four different optimization algorithms. These algorithms are Modified Biogeography-Based Optimization (MBBO) algorithm, Cuckoo Search (CS) algorithm, Modified Imperialist Competitive (MIC) algorithm, and Modified Bacterial Foraging-Based Optimization (MBFBO) algorithm. These algorithms are applied for two standard networks (IEEE 33- and IEEE 69-bus). Comparisons among the proposed algorithms are done, and the results demonstrated that the MBBO algorithm is the most strong and fast algorithm to attain the optimum solution. In addition, comparisons with literature works are done to validate the effectiveness of proposed algorithms.


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## Body

## 1. Introduction

The Radial Distribution System (RDS) typology is the most convenient form from protection and control sides. This section of power system suffers from higher losses due to huge branch currents and voltage deviations that caused the power supplied to customers liable to lose the quality and efficiency. The pressure to improve overall efficiency has forced utilities to seek greater efficiency in distributions systems utilizing from communication and computing technology that allow remote supervisory and control [1]. Different solutions have been used such as optimal placement of Distributed Generators (DG) [2, 3], Optimal Capacitor Placement (OCP) [4], Distribution System Reconfiguration (DSR) [5], hybrid of DG and DSR techniques [6, 7], optimal use of electrical equipment, optimal loading transformers, and removal the harmonics in order to reduce the real power losses and enhance the voltage profile. Among them, DSR and OCP are relatively the lesser operating cost [8].DSR is got by altering the status of closed branches and is frequently prepared for loss reduction. The DSR brings to a combinatorial optimization technique because of several restrictions that should not be infringed through discovering an optimum solution to the DSR technique for loss minimization. This search occurred through the related radial configurations [9]. This also will be more difficult if metaheuristic algorithms are employed. However, in spite of these limitations, the DSR technique is recommended and an effective approach that required zero working cost [8].Different works were done for DSR using different optimization algorithms, networks, and objective functions. Guardado et al. proposed an encoding scheme based on paretooptimal solutions using power losses, voltage deviations, and number of operating switches as a multiobjective optimization for DSR problem when applied to real RDS [10]. Voropai and Undraal carried out the DSR with DG problem for the power system of Mongolia with single objective function of loss reduction under normal operation condition and reliability improvement after contingency condition based on ant colony algorithm [11]. Shojaeian and Ghandehari introduced sifting algorithm for DSR with an objective of real power loss and voltage deviation reductions and applied to standard 16-bus RDS with advantage of search space reduction using the method of eliminating infeasible typologies [12]. Hong et al. presented a DSR for distribution feeder containing photovoltaic arrays using an enhanced PSO algorithm with the objective of avoidance of injection of reverse power into a substation [13]. Du et al. proposed four algorithms for DSR considering DG and applied to 33-bus standard RDS. These algorithms are an Improved Quantum Artificial Fish Swarm Algorithm (IQAFSA), Quantum Artificial Fish Swarm Algorithm (QAFSA), Basic Artificial Fish Swarm Algorithm (BAFSA), and Global Edition Artificial Fish Swarm Algorithm (GAFSA) with loss reduction as an objective function. IQAFSA has proved the efficiency over the other ones [14]. Sun and Chang discovered Improved Harmony Search (IHS) algorithm for DSR with the objective of annual cost reduction and applied it to RDS in the northern Chinese city [15]. Nguyen et al. introduced Runner-Root Algorithm (RRA) to find the optimal configuration with the objective of reduction of real power losses, Load Balance Index (LBI) among branches and Load Balance (LBF) among feeders, number of switching operations, and bus voltage deviation. This approach has been applied for 33-bus and 70-bus RDSs and provided promising results [16]. Roosta et al. presented DSR with and without DG units with the objective of loss reduction, voltage improvement, and voltage stability index (VSI) increment. These objective functions are achieved simultaneously using Harmony Search Algorithm (HSA) and applied it to unbalanced 25-bus RDS for two scenarios [17]. Shuaib et al. proposed Gravitational Search (GS) algorithm to solve the DSR problem with the objective of real power loss reduction and balancing loads on feeders. This algorithm is applied for both standard 33- and 69-bus RDSs, and results show that losses are reduced and the voltage stability increased [18]. Jahani introduced a DSR technique with multiobjective functions of reliability enhancement and power loss reduction of 33-bus RDS. After DSR technique, the results prove that power losses are reduced and reliability is improved [19]. The aforementioned previous works have several drawbacks such as large DG and photovoltaic powers are required to maintain voltage and current constraints, and also other approaches did not satisfy these constraints and have slower convergence. The objective functions in these works are either real loss reduction or voltage improvement or annual cost reduction as individual objectives and not combination of above three objectives together using weighting factors.The OCP is a reactive power injection process to compensate lower reactive power in RDS that caused the voltage reduced, losses increased, and corresponding financial benefits [20]. This compensation increases the overall stability and power factor; therefore an optimum value and place of capacitors are necessary to realize these objectives, taking into account a lower overall cost [21]. This technique contains the locations and calculation of the magnitude and number of capacitors that required to be installed in the system. This makes OCP mixed-integer nonlinear technique that has a complex nature. In addition, the radial restrictions of RDS are added to the complexity for the problem [22]. Incorrect placing of capacitors causes problems such as significant system losses and voltage drops. Therefore, the proper selection is required to locate the capacitors [23].Many of studies have been done for OCP using different optimization algorithms and typologies. Voltage Stability Index (VSI) is implemented to predetermine the optimal location of capacitor while the Cuckoo Search (CS) algorithm is proposed to determine the optimal size of the capacitor on IEEE 34-bus and 69-bus [24]. Flower Pollination (FP) algorithm has been proposed for the solution of OCP problem in a RDS while the objective is to minimize the total power loss and cost of capacitor installation. The FP algorithm is applied to 33-, 34-, 69-, and 85-bus RDS, and comparisons are made with other algorithms [25]. In another study, VSI and Loss Sensitivity Index (LSI) are used to determine the locations while Bacterial Foraging Algorithm (BFA) is used to calculate the optimal size for 33-bus RDS [26]. Artificial Bee Colony (ABC) algorithm is applied to improve VSI and increase the saving cost depending on LSI and VSI to allocate capacitors and tested for two power systems, 34- and 118-bus RDSs [27]. Some research studies adopted LSI to find the candidate locations for the capacitor while the PSO algorithm was used to find the optimal capacitor size with the objective of loss reduction and voltage improvement for 10-, 34-, and 85-bus RDSs. The load flow that is used in this study is Newton–Raphson Method (NRM), and only the fixed-type capacitors are taken into consideration [21]. Finally, a hybrid algorithm of fuzzy expert system and Genetic Algorithm are used to determine location and size of shunt capacitors in distorted RDSs, considering effect of harmonic distortion. This technique is applied for 34-bus RDS and real RDS in Saudi Electricity Corporation with the objective of increasing the saving cost, and results indicate its superiority for harmonic reduction and increment of the RDS efficiency [28].The aforementioned literature works for OCP have some drawbacks such as break of bus voltage and capacitor size constraints, and also the maintenance and operation costs are excluded from total capacitor cost calculations. In addition, the proposed objective functions are loss reduction, saving cost augmentation, and voltage profile enhancement as individual objective functions and not combined by weighting factor. The load flow used in many of these studies is NRM which is not suitable for RDS, and the capacitor cost specified constant in dollar ($) per kilo Volt Ampere reactive (kVAr) neglecting standard cost per capacitor size. The locations of capacitors for these works have been specified depending on either VSI or LSI. However, the capacitor placement based on these indices has proven less than satisfactory and not always explains the appropriate location [29].The process of combining the proposed two techniques (OCP and DSR) consecutively provides further enhancements in terms of losses, cost, and voltage profile. This process begins with a OCP technique and after that the DSR technique is applied for this compensated system. A few researchers adopted dual OCP-DSR technique, and de Oliveira et al. proposed the dual technique using different levels of load and dual integer point algorithm with the objective of energy loss reduction and applied it to 16-, 33-, and 85-bus RDSs [22]. Priyadarshini et al. employed OCP-DSR technique to reduce losses and cost for 33- and 77-bus RDSs using Genetic Algorithm (GA) for OCP and loop elimination approach for DSR [30]. Mohamed et al. applied heuristic Moth Swarm Algorithm (MSA) for both techniques and tested on both radial and ring distribution systems (33- and 69-bus) with loss and cost reduction as multiobjective functions [31]. The drawbacks of these works are based on using of NRM load flow, not considering voltage profile improvement as an objective function with the other objective functions such as loss and cost reduction to perform multiobjective task. Also, the voltage constraints are not confirmed and the capacitor locations depend on LSI and VSI.In this paper, two techniques of operations have been used, individual OCP and dual OCP-DSR techniques using four different optimization algorithms to solve multiobjective functions. These functions are real power loss reduction, annual saving cost augmentation, and voltage profile enhancement while satisfying RDS constraints. These algorithms are divided into two types: first type includes MBBO [32], MIC [33, 34], and MBFBO [35–37] which are population-based evolutionary algorithms. The second type contains CS which is a metaheuristic optimization algorithm [38–41]. These algorithms are an enhanced and modified version of original algorithms to solve major drawbacks such as slowly convergence, weakly behaviour, slip in local optimum solution, and so on. These algorithms are applied to select the optimal RDS configuration, best locations, and sizes of capacitors without using LSI or other indices specified in previous studies. The two techniques and four optimization algorithms have been applied to 33- and 69-bus standard IEEE RDSs at constant and balance three-phase loads, and the results have been compared with literature works.


## 2. Materials and Methods

The implemented load flow, multiobjective functions, RDS restrictions, RDSs, and optimization algorithms used in this work are described and explained in detail in this section.


### 2.1. Load Flow Proposed

A direct approach of load flow called Direct Backward Forward Sweep Method (DBFSM) is most suitable for RDS than NRM and Gauss–Seidal Method (GSM) which is much convenient for transmission systems. This is because of lower time requirements, unbalanced load nature, lower reactance/resistance ratio (X/R), and the radial construction. The steps for DBFSM are explained in detail in [42].


### 2.2. Objective Functions

Real loss reduction, voltage profile, and annual cost saving improvement are used as multiobjective functions for both OCP and OCP-DSR techniques in this study. These objective functions are described below.


#### 2.2.1. Real Loss Reduction

The total branch losses are determined by summing the real loss for all branches based on the resistance and current of each branch as explained in the following equations:(1)F1=minPlossT,(2)PlossT=∑b=1NbrPlossbkW,(3)Plossb=Ib2∗RbkW,where Nbr, PlossT, Plossb, Ib, and Rb are number of branches, total power losses and loss of branch b, current of branch b, and resistance of branch b, respectively.


#### 2.2.2. Voltage Profile Improvement

The condition for voltage improvements is that each bus voltage should be within acceptable limits else this condition is not achieved as illustrated in the following equation:(4)F2=∑j=1NbusVvcj∗Pvj,where Nbus, Vvcj Plossb, and Pvj are number of buses, voltage violation constraint, and voltage penalty factor (equals zero if voltage limits achieved and a higher value if these limits were broken) for bus j, respectively.


#### 2.2.3. Annual Saving Cost Increasing

The saving cost gained from loss reduction is determined from the difference between annual cost of base caseCAB and annual cost after compensation CAA as demonstrated in the following equations:(5)CTcP=∑c=1ncQc∗Cpc,(6)CTc=CTcP+∑c=1ncCIc+Coc,(7)CAB=CE∗T∗PlossTB,(8)CAA=CE∗T∗PlossTA+CTc,(9)F3=Scost=CB−CA,where Scost, CE, T, PlossTB, PlossTA, CTcp, nc, CIc, COc, Qc, and CPc are annual saving cost, energy loss cost, time interval in hour (h), total real losses before the proposed techniques ($), total real losses after the proposed techniques ($), total capacitor purchase cost ($), candidate number of capacitors, capacitor installation cost ($/location), capacitor operational cost ($/year/location), size of capacitor c (kVAr), and purchase cost of the selected capacitor ($/kVAr). CTc is the total capacitor cost containing purchase, installation, and operational costs for whole injected capacitors. The values of these parameters referred to reference [24] are inserted in Table 1.Table 1
The values of cost parameters.
Parameter
Value
C
I
c ($/location)
1600
C
O
c ($/location/year)
300
E
c ($/kWh)
0.06
T (h)
8760The aforementioned multiobjective functions are merged in single objective functionFTot by using weighting factors based on the following equations:(10)FTot=W1∗F1+W2∗F2+W3∗F3,(11)W1+W2+W3=1,where W1, W2, and W3 are weighting factors for total real losses, voltage profile, and annual saving cost functions, respectively.The percentage loss reduction%reduction is calculated based on total power losses before and after application of the proposed techniques as follows:(12)%reduction=PlossTB−PlossTAPlossTB∗100%.Also, the percentage saving%saving is calculated based on annual cost before and after application of the proposed techniques as follows:(13)%saving=CB−CACB∗100%.


### 2.3. Restrictions

The proper working of RDS should achieve the practical and working restrictions as explained below.


#### 2.3.1. Practical Restrictions

These restrictions include voltage, current, capacitor size, and number restrictions that are also known as inequality constraints and qualified as follows:(1) Voltage Restrictions. Voltage value for each bus must be within its acceptable domain to sustain power quality as(14)Vjmin≤Vj≤Vjmax,jϵNbus,where the standard minimum and maximum voltages are 0.95 and 1.05, respectively [43].(2) Current Restrictions. Branch current must not exceed about the limit value from protection side and achieve the power supply continuity as(15)Ib≤Ibmax,bϵNbr,where maximum branch capacity is specified in [43].(3) Capacitor Size and Number Restrictions. Switched type capacitors are used and only one capacitor is allowed per bus. Capacitor size and corresponding purchase cost specified in this paper are mentioned in Table 2 [23], while the number of capacitors specified in this paper is 3 capacitors based on experience and earlier studies that satisfy the final objective function. Also, the total capacitor size QcT inserted in the system must not override the total reactive power load QL [31]:(16)QcT≤QL.Table 2
Standard capacitor size and costs.
Q
c (kVAr)
C
P
c ($/kVAr)
150
0.5
300
0.35
450
0.253
600
0.220
750
0.276
900
0.183
1050
0.228
1200
0.170
1350
0.207
1500
0.201
1650
0.193
1800
0.187
1950
0.211
2100
0.176
2250
0.197
2400
0.170
2550
0.189
2700
0.187
2850
0.183
3000
0.180
3150
0.195
3300
0.174
3450
0.188
3600
0.170
3750
0.183
3900
0.182
4050
0.179


#### 2.3.2. Working Restrictions

These restrictions are classified into radial structure and power balance restrictions as follows.(1) Radial Structure Restriction. This condition has been confirmed by determining the determinant of the bus incidence matrix A as given in [43].(2) Power Balance Restriction. The power supplied PSup to the system must provide the power demand PDem and total power loss Ploss which is given by [27](17)PSup=PDem+PlossT.


### 2.4. Study Systems

The validity of the proposed strategy and optimization algorithms are tested using different sizes of RDSs. In this study, two systems are used with a constant load mode and IEEE standard distribution systems 33- and 69-bus RDSs. The line and bus data for these systems are available in the reference [44].


#### 2.4.1. 33-Bus IEEE RDS

This is a standard distribution system consisting of 32 closed branches and 5 opened branches which have switches labeledS1−S32 and S33−S37, respectively, in the base case as shown in Figure 1. The base values for kilo Volt (kV) and Mega Volt Ampere (MVA) are 12.66 kV and 100 MVA, respectively, with a constant load of 3,715 kW and 2,300 kVAr. In the first proposed technique (OCP), the status of branch switches remains constant while in the second technique (OCP-DSR), the branch switches are varied.Figure 1
Single-line scheme for 33-bus RDS.


#### 2.4.2. 69-Bus IEEE RDS

This is a standard distribution system consisting of 68 closed branches and 5 opened branches which have switches labeledS1−S68 and S69−S73, respectively, in base case as shown in Figure 2. The base values for kilo Volt (kV) and Mega Volt Ampere (MVA) are 12.66 kV and 100 MVA, respectively, with a constant load of 3,802 kW and 2,695 kVAr. In the first proposed technique (OCP), the status of branch switches remains constant while in the second technique (OCP-DSR), the opened and closed branches are varied.Figure 2
Single-line scheme for 69-bus RDS.


### 2.5. Optimization Algorithms

The proposed algorithms of this work are explained in detail in this subsection, and the previous usages of these algorithms are also provided.


#### 2.5.1. Modified Biogeography-Based Optimization (MBBO) Algorithm

It is an updated version of conventional Biogeography-Based Optimization (BBO) algorithm that was discovered in 2008 by Simon, which simulates the geographical allocation of alive organisms [32]. The original BBO has a weakness in its migration and mutation stages that affect its performance. Therefore, the modified version overrides the drawbacks of conventional form through improving and adapting the BBO approach to the DSR and OCP problems to enhance solution quality at each iteration. Conventional BBO is used for optimal placement of DG [45] and DSR [46], but the MBBO is not applied for both OCP alone, DSR alone, and dual OCP-DSR techniques.The MBBO procedures are as follows:(1)
The final choice of the control parameters for the MBBO algorithm that is considered as the optimal choice in this study is inserted in Table3.(2)
Determine Maintain NumberMN depending on population Pop using the following equation:(18)MN=Maintain rate∗Pop.(3)
Determine Habitation NumberHN:(19)HN=Pop−MN.(4)
Calculate Emigration ValueEv and Immigration Value Iv consecutively for each habitat based on their maximum values EVmaxandIVmax as follows:(20)Ev=EVmax∗1Pop−1,(21)Iv=IVmax∗1−Ev.(5)
Evaluate Emigration PossibilityEP depending on Iv:(22)EP=Ev,ifr≤Iv,wherer is a random number between 0 and 1.(6)
The steady state value ofEP can be calculated depending on Emigration Possibility of each species (EPs) as follows:(23)EP∞=EPs∑S=1PopEPs.(7)
Modernise Suitability IndexSIik+1 for each habitat for ith solution utilizing from previous ith solution, jth solution and Beta variable (β):(24)SIik+1=SIik+β∗SIjk−SIik,where theβ value is specified in Table 2.(8)
Checking the selected Mutation PossibilityMP with random number (r), if it large or equals to r, then the updated SIik+1 equals(25)SIik+1=SIik+1+EN∗r,whereEN is Elite Number.(9)
Repeat procedures 2–8 until reach maximum number of iterations (itermax).(10)
End.Table 3
The typical parameters of the MBBO algorithm.
Parameter
Value
M
P
0.1
E
N
2
I
V
max
1
E
V
max
1
Maintain rate
0.2
β
0.9


#### 2.5.2. Cuckoo Search (CS) Algorithm

It is a new metaheuristic optimization method discovered by Yang and Deb in 2009, which is inspired from parasitic performance of cuckoo birds plus levy flight performance of other birds [38]. Each egg can be viewed as a solution which is randomly generated in the initialization process. CS algorithm has been used alone for DSR and OCP techniques, and the optimal result obtained via DSR technique demonstrated the effectiveness and robustness of this algorithm. Also, CS algorithm has been applied with different typologies and objective functions for only OCP technique, in spite of confirming the total capacitor size restriction but did not satisfy the voltage restrictions [24, 39, 47]. In this paper, this algorithm is modified, especially the initialization process, to give better results than the original CS algorithm.The main steps of the CS algorithm are as follows:(1)
Initialization cuckoo parameters: ranges of Host Nest number (HN) and Discover Rate Probability (DRP) are 10–50 and 0-1, respectively [39, 40]. The values of these parameters and maximum number of iterations that are used in this study are listed in Table 4.(2)
Find the fitness value for each nest value and check the constraints of optimization problem such as radial condition of RDS and other constraints. At last, select the best host nests based on the fitness value.(3)
Start Levy flight phase to get the best value for each iteration as follows:(i)
Define Allocation Factor (AF) in the range (1–3) [41], Gamma Allocation (GA) function, where AF is specified and equals 3/2 in this study.(ii)
Calculate Sigma (Six) for solution x and Sigma (Siy) for solution y as standard variances by the following equations:(26)Six=GA1+AF∗sinπ/2∗AFGA1+AF/2∗AF∗2AF−1/21/AF,(27)Siy=1.(iii)
Determine the Step Data (SDi) for nest (i) as(28)SDi=uv1/AF,whereu and v are variables of solution x and can be calculated based on random numbers rx and ry using the following equations:(29)v=ry,(30)u=Six∗rx,whererx and ry have values which range between 0 and 1.(iv)
Calculate the Increment Value (IVinew) depending on values of SDi, best solution Xi,best, optimal solution Xi,opt, and Step Size (SZ) as(31)IVinew=SZ∗SDi∗Xi,best−Xi,opt,whereSZ is specified equals to 0.01.(v)
Determine the new value (Xi,new) depending on IVinew and Xi,best values using the following equation:(32)Xi,new=roundXi,best+r∗IVinew,whereround is an integer number.(4)
Delete the worst values and build new ones keeping the best solutions. This phase also generates a new solution as in the Levy flight phase depending on factorF which explains the ability to discover the stranger eggs that is specified as follows:(33)F=1,ifr<DRP,0,else.Then, the new solution (Xi,new) can be determined by using this factor as(34)Xi,new=roundXi,best+F∗ΔXinew,whereΔXinew is similar to IVinew in the Levy flight phase and is calculated as(35)ΔXinew=r∗rd1−rd2,whererd1 and rd2 are random disturbances for nest positions in Xi,best.(5)
Sort the solutions and find the current best solutions.(6)
The best nests with high quality of eggs (best value) will carry over to the next generation.(7)
Get the best results at the end of iteration.(8)
Repeat steps 2–7 until it reaches the limits of maximum iterations (itermax).(9)
End.Table 4
Initializing parameters of the CS algorithm.
Parameter
Value
HN
30
DRP
0.25


#### 2.5.3. Modified Imperialist Competitive (MIC) Algorithm

Imperialist Competitive (IC) algorithm is an evolutionary algorithm that was discovered by Gargari and Lucas in 2007 which is inspired by colonial simulation [33]. This algorithm wishes to reach a state that only one empire existed and its corresponding colonies have the same position and cost. A modified form (MIC) is used in this article to enhance the efficiency, convergence velocity, and accuracy of the conventional IC algorithm. The MIC algorithm improves the original IC performance by preventing premature convergence to local minima based on absorption, exchange, revolution, and competition processes. This algorithm is implemented using the following steps:(1)
Initializing process: in this process, the optimal values of MIC algorithm control parameters, such as number of countries, number of imperialists, and number of colonies, were defined and inserted in Table5. Also, the cost and position of each imperialist and colony were initialized. The normalized power PN for each imperialist and number of colonies Ncn for empire are determined as follows:(36)PN=CN∑j=1ImCj,j∈Im,(37)Ncn=roundPN∗Nc,whereCN, Cj, Im, and Nc are normalize cost for each imperialist, cost of jth imperialist, number of imperialists, and total number of colonies, respectively.(2)
Absorption process: the colonies move toward their imperialist either in straight movement or diverted by angle∅. The latter is better to improve the search ability around the imperialists. The new position Posnew of the colony can be updated by vector (V) and absorption coefficient (γ) which has a value represented in Table 5. The vector V is calculated using position of imperialist (PosIM) and position of its corresponding colonies (Posc) as(38)V=PosIM−Posc.Then,Posc,new is calculated based on the previous position (Posc,old) as follows:(39)Posc,new=Posc,old+γ∗randV∗V.The movement of colony may drift from straight line by angle∅ that has the range corresponding to absorption angle coefficient (δ) value as(40)∅∈−δ:δ.(3)
Exchange process: this process occurs when position or cost of the colony is better than that for the imperialist, and then the replacement occurs to update a new imperialist.(4)
Update total cost for a new united empireTec using cost for imperialist CIM and the sharing cost of corresponding colonies as explained in the following equation:(41)Tec=CIM+Z∗AveCc,whereZ and AveCc are sharing cost and average of colony costs for the corresponding empire, respectively.(5)
Revolution process: during this process, the locations of colonies are altered randomly to enhance behavior of IC and exclude it from fall into the local minimum by using Revolution Rate Parameter (RRP) that has a range of 0.03–0.4 [34], and its optimal value is specified in Table 5. The number of revolving colonies Rc equals(42)Rc=roundRRP∗Cc.(6)
Imperialist competition process: this process aims to select the weakest colonies of weakest empires, and comparison is made between powerful empires to own these colonies. This is made depending on ownership probabilityPoe for each empire that is proportional to the normalized total cost TecN for each empire. TecN is first determined as(43)TecN=Tec−maxTec.After that,Poe equals(44)Poe=TecN∑j=1IMTecj,j∈Im.The vectorO corresponding to ownership probability for each empire generated, also a random vector R of the same size is built then a vector C contains the remaining empires after the collapse process as(45)C=O−R.(7)
Repeat above steps until it reachesitermax limit.(8)
End.Table 5
Optimal parameters of the MIC algorithm.
Parameter
Value
Country number
20
Im
10
N
c
10
Z
0.02
RRP
0.3
γ
0.5


#### 2.5.4. Modified Bacterial Foraging-Based Optimization (MBFBO) Algorithm

A Bacterial Foraging-Based Optimization (BFBO) algorithm is of evolutionary type, first discovered by Passino in 2002, which has removable property [35]. The bacterium represents the proposed solution for optimization problem. A modified style of this algorithm is performed to overcome the drawbacks of BFBO algorithms such as lower convergence speed and trapping in local minimum solution. This modification includes both chemotaxis and swarming processes, and this enables it to handle multiobjective functions [36]. This algorithm has been applied for DSR technique with the objective function of loss reduction and applied for optimal placement of DG [44, 48]. The steps of the MBFBO algorithm are as follows:(1)
Initialization of the MBFBO parameters: the values of these parameters are inserted in Table6. In addition, the loops for chemotaxis, reproduction, and elimination processes have been generated.(2)
Chemotaxis process: this process was classified into tumble and swim operations. In tumble operation, the position forith bacteria XiNew is updated based on its chemotactic step size Ciss and direction index (Di) of cth chemotaxis as follows [37]:(46)Di=r−0.5∗2,(47)XiNew=Xiold+DiDi∗DiT∗Ciss.In swim operation, swim stepsSs are predefined and each bacteria move with same Css of previous chemotaxis, and this operation continues until it reach Ss limit.(3)
Reproduction process: during this process, the health of each bacteriaCHi is determined using equation (48) and sorting it in descending order, where the ones with least health are dead while the remaining ones each split into two to keep the constant number of bacteria.(48)CHi=∑C=1NcCi,whereNc is the number of chemotaxis steps. This procedure is repeated for each of reproductive steps (Nre).(4)
Elimination process: after the reproduction step, some of the new generations move to new positions to enhance the overall performance of this algorithm depending on Removal Probability (PR).(5)
Repeat steps 2–4 until it reachesitermax.(6)
End.Table 6
Initial parameters for the MBFBO algorithm.
Parameter
Value
Bacteria number
30
N
c
2
N
re
2
P
R
0.9
S
s
1Finally, values ofPop and itermax for all applied algorithms are inserted in Table 7. These values are used to give accurate and promise results over the literature studies.Table 7
Setting of parameters for applied algorithms.
Proposed algorithms
Parameter
Pop
iter
max, (RDS)
MBBO
30
50, (33-bus)100, (69-bus)
CS
30
MIC
20
MBFBO
30


## 2.1. Load Flow Proposed

A direct approach of load flow called Direct Backward Forward Sweep Method (DBFSM) is most suitable for RDS than NRM and Gauss–Seidal Method (GSM) which is much convenient for transmission systems. This is because of lower time requirements, unbalanced load nature, lower reactance/resistance ratio (X/R), and the radial construction. The steps for DBFSM are explained in detail in [42].


## 2.2. Objective Functions

Real loss reduction, voltage profile, and annual cost saving improvement are used as multiobjective functions for both OCP and OCP-DSR techniques in this study. These objective functions are described below.


### 2.2.1. Real Loss Reduction

The total branch losses are determined by summing the real loss for all branches based on the resistance and current of each branch as explained in the following equations:(1)F1=minPlossT,(2)PlossT=∑b=1NbrPlossbkW,(3)Plossb=Ib2∗RbkW,where Nbr, PlossT, Plossb, Ib, and Rb are number of branches, total power losses and loss of branch b, current of branch b, and resistance of branch b, respectively.


### 2.2.2. Voltage Profile Improvement

The condition for voltage improvements is that each bus voltage should be within acceptable limits else this condition is not achieved as illustrated in the following equation:(4)F2=∑j=1NbusVvcj∗Pvj,where Nbus, Vvcj Plossb, and Pvj are number of buses, voltage violation constraint, and voltage penalty factor (equals zero if voltage limits achieved and a higher value if these limits were broken) for bus j, respectively.


### 2.2.3. Annual Saving Cost Increasing

The saving cost gained from loss reduction is determined from the difference between annual cost of base caseCAB and annual cost after compensation CAA as demonstrated in the following equations:(5)CTcP=∑c=1ncQc∗Cpc,(6)CTc=CTcP+∑c=1ncCIc+Coc,(7)CAB=CE∗T∗PlossTB,(8)CAA=CE∗T∗PlossTA+CTc,(9)F3=Scost=CB−CA,where Scost, CE, T, PlossTB, PlossTA, CTcp, nc, CIc, COc, Qc, and CPc are annual saving cost, energy loss cost, time interval in hour (h), total real losses before the proposed techniques ($), total real losses after the proposed techniques ($), total capacitor purchase cost ($), candidate number of capacitors, capacitor installation cost ($/location), capacitor operational cost ($/year/location), size of capacitor c (kVAr), and purchase cost of the selected capacitor ($/kVAr). CTc is the total capacitor cost containing purchase, installation, and operational costs for whole injected capacitors. The values of these parameters referred to reference [24] are inserted in Table 1.Table 1
The values of cost parameters.
Parameter
Value
C
I
c ($/location)
1600
C
O
c ($/location/year)
300
E
c ($/kWh)
0.06
T (h)
8760The aforementioned multiobjective functions are merged in single objective functionFTot by using weighting factors based on the following equations:(10)FTot=W1∗F1+W2∗F2+W3∗F3,(11)W1+W2+W3=1,where W1, W2, and W3 are weighting factors for total real losses, voltage profile, and annual saving cost functions, respectively.The percentage loss reduction%reduction is calculated based on total power losses before and after application of the proposed techniques as follows:(12)%reduction=PlossTB−PlossTAPlossTB∗100%.Also, the percentage saving%saving is calculated based on annual cost before and after application of the proposed techniques as follows:(13)%saving=CB−CACB∗100%.


## 2.2.1. Real Loss Reduction

The total branch losses are determined by summing the real loss for all branches based on the resistance and current of each branch as explained in the following equations:(1)F1=minPlossT,(2)PlossT=∑b=1NbrPlossbkW,(3)Plossb=Ib2∗RbkW,where Nbr, PlossT, Plossb, Ib, and Rb are number of branches, total power losses and loss of branch b, current of branch b, and resistance of branch b, respectively.


## 2.2.2. Voltage Profile Improvement

The condition for voltage improvements is that each bus voltage should be within acceptable limits else this condition is not achieved as illustrated in the following equation:(4)F2=∑j=1NbusVvcj∗Pvj,where Nbus, Vvcj Plossb, and Pvj are number of buses, voltage violation constraint, and voltage penalty factor (equals zero if voltage limits achieved and a higher value if these limits were broken) for bus j, respectively.


## 2.2.3. Annual Saving Cost Increasing

The saving cost gained from loss reduction is determined from the difference between annual cost of base caseCAB and annual cost after compensation CAA as demonstrated in the following equations:(5)CTcP=∑c=1ncQc∗Cpc,(6)CTc=CTcP+∑c=1ncCIc+Coc,(7)CAB=CE∗T∗PlossTB,(8)CAA=CE∗T∗PlossTA+CTc,(9)F3=Scost=CB−CA,where Scost, CE, T, PlossTB, PlossTA, CTcp, nc, CIc, COc, Qc, and CPc are annual saving cost, energy loss cost, time interval in hour (h), total real losses before the proposed techniques ($), total real losses after the proposed techniques ($), total capacitor purchase cost ($), candidate number of capacitors, capacitor installation cost ($/location), capacitor operational cost ($/year/location), size of capacitor c (kVAr), and purchase cost of the selected capacitor ($/kVAr). CTc is the total capacitor cost containing purchase, installation, and operational costs for whole injected capacitors. The values of these parameters referred to reference [24] are inserted in Table 1.Table 1
The values of cost parameters.
Parameter
Value
C
I
c ($/location)
1600
C
O
c ($/location/year)
300
E
c ($/kWh)
0.06
T (h)
8760The aforementioned multiobjective functions are merged in single objective functionFTot by using weighting factors based on the following equations:(10)FTot=W1∗F1+W2∗F2+W3∗F3,(11)W1+W2+W3=1,where W1, W2, and W3 are weighting factors for total real losses, voltage profile, and annual saving cost functions, respectively.The percentage loss reduction%reduction is calculated based on total power losses before and after application of the proposed techniques as follows:(12)%reduction=PlossTB−PlossTAPlossTB∗100%.Also, the percentage saving%saving is calculated based on annual cost before and after application of the proposed techniques as follows:(13)%saving=CB−CACB∗100%.


## 2.3. Restrictions

The proper working of RDS should achieve the practical and working restrictions as explained below.


### 2.3.1. Practical Restrictions

These restrictions include voltage, current, capacitor size, and number restrictions that are also known as inequality constraints and qualified as follows:(1) Voltage Restrictions. Voltage value for each bus must be within its acceptable domain to sustain power quality as(14)Vjmin≤Vj≤Vjmax,jϵNbus,where the standard minimum and maximum voltages are 0.95 and 1.05, respectively [43].(2) Current Restrictions. Branch current must not exceed about the limit value from protection side and achieve the power supply continuity as(15)Ib≤Ibmax,bϵNbr,where maximum branch capacity is specified in [43].(3) Capacitor Size and Number Restrictions. Switched type capacitors are used and only one capacitor is allowed per bus. Capacitor size and corresponding purchase cost specified in this paper are mentioned in Table 2 [23], while the number of capacitors specified in this paper is 3 capacitors based on experience and earlier studies that satisfy the final objective function. Also, the total capacitor size QcT inserted in the system must not override the total reactive power load QL [31]:(16)QcT≤QL.Table 2
Standard capacitor size and costs.
Q
c (kVAr)
C
P
c ($/kVAr)
150
0.5
300
0.35
450
0.253
600
0.220
750
0.276
900
0.183
1050
0.228
1200
0.170
1350
0.207
1500
0.201
1650
0.193
1800
0.187
1950
0.211
2100
0.176
2250
0.197
2400
0.170
2550
0.189
2700
0.187
2850
0.183
3000
0.180
3150
0.195
3300
0.174
3450
0.188
3600
0.170
3750
0.183
3900
0.182
4050
0.179


### 2.3.2. Working Restrictions

These restrictions are classified into radial structure and power balance restrictions as follows.(1) Radial Structure Restriction. This condition has been confirmed by determining the determinant of the bus incidence matrix A as given in [43].(2) Power Balance Restriction. The power supplied PSup to the system must provide the power demand PDem and total power loss Ploss which is given by [27](17)PSup=PDem+PlossT.


## 2.3.1. Practical Restrictions

These restrictions include voltage, current, capacitor size, and number restrictions that are also known as inequality constraints and qualified as follows:(1) Voltage Restrictions. Voltage value for each bus must be within its acceptable domain to sustain power quality as(14)Vjmin≤Vj≤Vjmax,jϵNbus,where the standard minimum and maximum voltages are 0.95 and 1.05, respectively [43].(2) Current Restrictions. Branch current must not exceed about the limit value from protection side and achieve the power supply continuity as(15)Ib≤Ibmax,bϵNbr,where maximum branch capacity is specified in [43].(3) Capacitor Size and Number Restrictions. Switched type capacitors are used and only one capacitor is allowed per bus. Capacitor size and corresponding purchase cost specified in this paper are mentioned in Table 2 [23], while the number of capacitors specified in this paper is 3 capacitors based on experience and earlier studies that satisfy the final objective function. Also, the total capacitor size QcT inserted in the system must not override the total reactive power load QL [31]:(16)QcT≤QL.Table 2
Standard capacitor size and costs.
Q
c (kVAr)
C
P
c ($/kVAr)
150
0.5
300
0.35
450
0.253
600
0.220
750
0.276
900
0.183
1050
0.228
1200
0.170
1350
0.207
1500
0.201
1650
0.193
1800
0.187
1950
0.211
2100
0.176
2250
0.197
2400
0.170
2550
0.189
2700
0.187
2850
0.183
3000
0.180
3150
0.195
3300
0.174
3450
0.188
3600
0.170
3750
0.183
3900
0.182
4050
0.179


## 2.3.2. Working Restrictions

These restrictions are classified into radial structure and power balance restrictions as follows.(1) Radial Structure Restriction. This condition has been confirmed by determining the determinant of the bus incidence matrix A as given in [43].(2) Power Balance Restriction. The power supplied PSup to the system must provide the power demand PDem and total power loss Ploss which is given by [27](17)PSup=PDem+PlossT.


## 2.4. Study Systems

The validity of the proposed strategy and optimization algorithms are tested using different sizes of RDSs. In this study, two systems are used with a constant load mode and IEEE standard distribution systems 33- and 69-bus RDSs. The line and bus data for these systems are available in the reference [44].


### 2.4.1. 33-Bus IEEE RDS

This is a standard distribution system consisting of 32 closed branches and 5 opened branches which have switches labeledS1−S32 and S33−S37, respectively, in the base case as shown in Figure 1. The base values for kilo Volt (kV) and Mega Volt Ampere (MVA) are 12.66 kV and 100 MVA, respectively, with a constant load of 3,715 kW and 2,300 kVAr. In the first proposed technique (OCP), the status of branch switches remains constant while in the second technique (OCP-DSR), the branch switches are varied.Figure 1
Single-line scheme for 33-bus RDS.


### 2.4.2. 69-Bus IEEE RDS

This is a standard distribution system consisting of 68 closed branches and 5 opened branches which have switches labeledS1−S68 and S69−S73, respectively, in base case as shown in Figure 2. The base values for kilo Volt (kV) and Mega Volt Ampere (MVA) are 12.66 kV and 100 MVA, respectively, with a constant load of 3,802 kW and 2,695 kVAr. In the first proposed technique (OCP), the status of branch switches remains constant while in the second technique (OCP-DSR), the opened and closed branches are varied.Figure 2
Single-line scheme for 69-bus RDS.


## 2.4.1. 33-Bus IEEE RDS

This is a standard distribution system consisting of 32 closed branches and 5 opened branches which have switches labeledS1−S32 and S33−S37, respectively, in the base case as shown in Figure 1. The base values for kilo Volt (kV) and Mega Volt Ampere (MVA) are 12.66 kV and 100 MVA, respectively, with a constant load of 3,715 kW and 2,300 kVAr. In the first proposed technique (OCP), the status of branch switches remains constant while in the second technique (OCP-DSR), the branch switches are varied.Figure 1
Single-line scheme for 33-bus RDS.


## 2.4.2. 69-Bus IEEE RDS

This is a standard distribution system consisting of 68 closed branches and 5 opened branches which have switches labeledS1−S68 and S69−S73, respectively, in base case as shown in Figure 2. The base values for kilo Volt (kV) and Mega Volt Ampere (MVA) are 12.66 kV and 100 MVA, respectively, with a constant load of 3,802 kW and 2,695 kVAr. In the first proposed technique (OCP), the status of branch switches remains constant while in the second technique (OCP-DSR), the opened and closed branches are varied.Figure 2
Single-line scheme for 69-bus RDS.


## 2.5. Optimization Algorithms

The proposed algorithms of this work are explained in detail in this subsection, and the previous usages of these algorithms are also provided.


### 2.5.1. Modified Biogeography-Based Optimization (MBBO) Algorithm

It is an updated version of conventional Biogeography-Based Optimization (BBO) algorithm that was discovered in 2008 by Simon, which simulates the geographical allocation of alive organisms [32]. The original BBO has a weakness in its migration and mutation stages that affect its performance. Therefore, the modified version overrides the drawbacks of conventional form through improving and adapting the BBO approach to the DSR and OCP problems to enhance solution quality at each iteration. Conventional BBO is used for optimal placement of DG [45] and DSR [46], but the MBBO is not applied for both OCP alone, DSR alone, and dual OCP-DSR techniques.The MBBO procedures are as follows:(1)
The final choice of the control parameters for the MBBO algorithm that is considered as the optimal choice in this study is inserted in Table3.(2)
Determine Maintain NumberMN depending on population Pop using the following equation:(18)MN=Maintain rate∗Pop.(3)
Determine Habitation NumberHN:(19)HN=Pop−MN.(4)
Calculate Emigration ValueEv and Immigration Value Iv consecutively for each habitat based on their maximum values EVmaxandIVmax as follows:(20)Ev=EVmax∗1Pop−1,(21)Iv=IVmax∗1−Ev.(5)
Evaluate Emigration PossibilityEP depending on Iv:(22)EP=Ev,ifr≤Iv,wherer is a random number between 0 and 1.(6)
The steady state value ofEP can be calculated depending on Emigration Possibility of each species (EPs) as follows:(23)EP∞=EPs∑S=1PopEPs.(7)
Modernise Suitability IndexSIik+1 for each habitat for ith solution utilizing from previous ith solution, jth solution and Beta variable (β):(24)SIik+1=SIik+β∗SIjk−SIik,where theβ value is specified in Table 2.(8)
Checking the selected Mutation PossibilityMP with random number (r), if it large or equals to r, then the updated SIik+1 equals(25)SIik+1=SIik+1+EN∗r,whereEN is Elite Number.(9)
Repeat procedures 2–8 until reach maximum number of iterations (itermax).(10)
End.Table 3
The typical parameters of the MBBO algorithm.
Parameter
Value
M
P
0.1
E
N
2
I
V
max
1
E
V
max
1
Maintain rate
0.2
β
0.9


### 2.5.2. Cuckoo Search (CS) Algorithm

It is a new metaheuristic optimization method discovered by Yang and Deb in 2009, which is inspired from parasitic performance of cuckoo birds plus levy flight performance of other birds [38]. Each egg can be viewed as a solution which is randomly generated in the initialization process. CS algorithm has been used alone for DSR and OCP techniques, and the optimal result obtained via DSR technique demonstrated the effectiveness and robustness of this algorithm. Also, CS algorithm has been applied with different typologies and objective functions for only OCP technique, in spite of confirming the total capacitor size restriction but did not satisfy the voltage restrictions [24, 39, 47]. In this paper, this algorithm is modified, especially the initialization process, to give better results than the original CS algorithm.The main steps of the CS algorithm are as follows:(1)
Initialization cuckoo parameters: ranges of Host Nest number (HN) and Discover Rate Probability (DRP) are 10–50 and 0-1, respectively [39, 40]. The values of these parameters and maximum number of iterations that are used in this study are listed in Table 4.(2)
Find the fitness value for each nest value and check the constraints of optimization problem such as radial condition of RDS and other constraints. At last, select the best host nests based on the fitness value.(3)
Start Levy flight phase to get the best value for each iteration as follows:(i)
Define Allocation Factor (AF) in the range (1–3) [41], Gamma Allocation (GA) function, where AF is specified and equals 3/2 in this study.(ii)
Calculate Sigma (Six) for solution x and Sigma (Siy) for solution y as standard variances by the following equations:(26)Six=GA1+AF∗sinπ/2∗AFGA1+AF/2∗AF∗2AF−1/21/AF,(27)Siy=1.(iii)
Determine the Step Data (SDi) for nest (i) as(28)SDi=uv1/AF,whereu and v are variables of solution x and can be calculated based on random numbers rx and ry using the following equations:(29)v=ry,(30)u=Six∗rx,whererx and ry have values which range between 0 and 1.(iv)
Calculate the Increment Value (IVinew) depending on values of SDi, best solution Xi,best, optimal solution Xi,opt, and Step Size (SZ) as(31)IVinew=SZ∗SDi∗Xi,best−Xi,opt,whereSZ is specified equals to 0.01.(v)
Determine the new value (Xi,new) depending on IVinew and Xi,best values using the following equation:(32)Xi,new=roundXi,best+r∗IVinew,whereround is an integer number.(4)
Delete the worst values and build new ones keeping the best solutions. This phase also generates a new solution as in the Levy flight phase depending on factorF which explains the ability to discover the stranger eggs that is specified as follows:(33)F=1,ifr<DRP,0,else.Then, the new solution (Xi,new) can be determined by using this factor as(34)Xi,new=roundXi,best+F∗ΔXinew,whereΔXinew is similar to IVinew in the Levy flight phase and is calculated as(35)ΔXinew=r∗rd1−rd2,whererd1 and rd2 are random disturbances for nest positions in Xi,best.(5)
Sort the solutions and find the current best solutions.(6)
The best nests with high quality of eggs (best value) will carry over to the next generation.(7)
Get the best results at the end of iteration.(8)
Repeat steps 2–7 until it reaches the limits of maximum iterations (itermax).(9)
End.Table 4
Initializing parameters of the CS algorithm.
Parameter
Value
HN
30
DRP
0.25


### 2.5.3. Modified Imperialist Competitive (MIC) Algorithm

Imperialist Competitive (IC) algorithm is an evolutionary algorithm that was discovered by Gargari and Lucas in 2007 which is inspired by colonial simulation [33]. This algorithm wishes to reach a state that only one empire existed and its corresponding colonies have the same position and cost. A modified form (MIC) is used in this article to enhance the efficiency, convergence velocity, and accuracy of the conventional IC algorithm. The MIC algorithm improves the original IC performance by preventing premature convergence to local minima based on absorption, exchange, revolution, and competition processes. This algorithm is implemented using the following steps:(1)
Initializing process: in this process, the optimal values of MIC algorithm control parameters, such as number of countries, number of imperialists, and number of colonies, were defined and inserted in Table5. Also, the cost and position of each imperialist and colony were initialized. The normalized power PN for each imperialist and number of colonies Ncn for empire are determined as follows:(36)PN=CN∑j=1ImCj,j∈Im,(37)Ncn=roundPN∗Nc,whereCN, Cj, Im, and Nc are normalize cost for each imperialist, cost of jth imperialist, number of imperialists, and total number of colonies, respectively.(2)
Absorption process: the colonies move toward their imperialist either in straight movement or diverted by angle∅. The latter is better to improve the search ability around the imperialists. The new position Posnew of the colony can be updated by vector (V) and absorption coefficient (γ) which has a value represented in Table 5. The vector V is calculated using position of imperialist (PosIM) and position of its corresponding colonies (Posc) as(38)V=PosIM−Posc.Then,Posc,new is calculated based on the previous position (Posc,old) as follows:(39)Posc,new=Posc,old+γ∗randV∗V.The movement of colony may drift from straight line by angle∅ that has the range corresponding to absorption angle coefficient (δ) value as(40)∅∈−δ:δ.(3)
Exchange process: this process occurs when position or cost of the colony is better than that for the imperialist, and then the replacement occurs to update a new imperialist.(4)
Update total cost for a new united empireTec using cost for imperialist CIM and the sharing cost of corresponding colonies as explained in the following equation:(41)Tec=CIM+Z∗AveCc,whereZ and AveCc are sharing cost and average of colony costs for the corresponding empire, respectively.(5)
Revolution process: during this process, the locations of colonies are altered randomly to enhance behavior of IC and exclude it from fall into the local minimum by using Revolution Rate Parameter (RRP) that has a range of 0.03–0.4 [34], and its optimal value is specified in Table 5. The number of revolving colonies Rc equals(42)Rc=roundRRP∗Cc.(6)
Imperialist competition process: this process aims to select the weakest colonies of weakest empires, and comparison is made between powerful empires to own these colonies. This is made depending on ownership probabilityPoe for each empire that is proportional to the normalized total cost TecN for each empire. TecN is first determined as(43)TecN=Tec−maxTec.After that,Poe equals(44)Poe=TecN∑j=1IMTecj,j∈Im.The vectorO corresponding to ownership probability for each empire generated, also a random vector R of the same size is built then a vector C contains the remaining empires after the collapse process as(45)C=O−R.(7)
Repeat above steps until it reachesitermax limit.(8)
End.Table 5
Optimal parameters of the MIC algorithm.
Parameter
Value
Country number
20
Im
10
N
c
10
Z
0.02
RRP
0.3
γ
0.5


### 2.5.4. Modified Bacterial Foraging-Based Optimization (MBFBO) Algorithm

A Bacterial Foraging-Based Optimization (BFBO) algorithm is of evolutionary type, first discovered by Passino in 2002, which has removable property [35]. The bacterium represents the proposed solution for optimization problem. A modified style of this algorithm is performed to overcome the drawbacks of BFBO algorithms such as lower convergence speed and trapping in local minimum solution. This modification includes both chemotaxis and swarming processes, and this enables it to handle multiobjective functions [36]. This algorithm has been applied for DSR technique with the objective function of loss reduction and applied for optimal placement of DG [44, 48]. The steps of the MBFBO algorithm are as follows:(1)
Initialization of the MBFBO parameters: the values of these parameters are inserted in Table6. In addition, the loops for chemotaxis, reproduction, and elimination processes have been generated.(2)
Chemotaxis process: this process was classified into tumble and swim operations. In tumble operation, the position forith bacteria XiNew is updated based on its chemotactic step size Ciss and direction index (Di) of cth chemotaxis as follows [37]:(46)Di=r−0.5∗2,(47)XiNew=Xiold+DiDi∗DiT∗Ciss.In swim operation, swim stepsSs are predefined and each bacteria move with same Css of previous chemotaxis, and this operation continues until it reach Ss limit.(3)
Reproduction process: during this process, the health of each bacteriaCHi is determined using equation (48) and sorting it in descending order, where the ones with least health are dead while the remaining ones each split into two to keep the constant number of bacteria.(48)CHi=∑C=1NcCi,whereNc is the number of chemotaxis steps. This procedure is repeated for each of reproductive steps (Nre).(4)
Elimination process: after the reproduction step, some of the new generations move to new positions to enhance the overall performance of this algorithm depending on Removal Probability (PR).(5)
Repeat steps 2–4 until it reachesitermax.(6)
End.Table 6
Initial parameters for the MBFBO algorithm.
Parameter
Value
Bacteria number
30
N
c
2
N
re
2
P
R
0.9
S
s
1Finally, values ofPop and itermax for all applied algorithms are inserted in Table 7. These values are used to give accurate and promise results over the literature studies.Table 7
Setting of parameters for applied algorithms.
Proposed algorithms
Parameter
Pop
iter
max, (RDS)
MBBO
30
50, (33-bus)100, (69-bus)
CS
30
MIC
20
MBFBO
30


## 2.5.1. Modified Biogeography-Based Optimization (MBBO) Algorithm

It is an updated version of conventional Biogeography-Based Optimization (BBO) algorithm that was discovered in 2008 by Simon, which simulates the geographical allocation of alive organisms [32]. The original BBO has a weakness in its migration and mutation stages that affect its performance. Therefore, the modified version overrides the drawbacks of conventional form through improving and adapting the BBO approach to the DSR and OCP problems to enhance solution quality at each iteration. Conventional BBO is used for optimal placement of DG [45] and DSR [46], but the MBBO is not applied for both OCP alone, DSR alone, and dual OCP-DSR techniques.The MBBO procedures are as follows:(1)
The final choice of the control parameters for the MBBO algorithm that is considered as the optimal choice in this study is inserted in Table3.(2)
Determine Maintain NumberMN depending on population Pop using the following equation:(18)MN=Maintain rate∗Pop.(3)
Determine Habitation NumberHN:(19)HN=Pop−MN.(4)
Calculate Emigration ValueEv and Immigration Value Iv consecutively for each habitat based on their maximum values EVmaxandIVmax as follows:(20)Ev=EVmax∗1Pop−1,(21)Iv=IVmax∗1−Ev.(5)
Evaluate Emigration PossibilityEP depending on Iv:(22)EP=Ev,ifr≤Iv,wherer is a random number between 0 and 1.(6)
The steady state value ofEP can be calculated depending on Emigration Possibility of each species (EPs) as follows:(23)EP∞=EPs∑S=1PopEPs.(7)
Modernise Suitability IndexSIik+1 for each habitat for ith solution utilizing from previous ith solution, jth solution and Beta variable (β):(24)SIik+1=SIik+β∗SIjk−SIik,where theβ value is specified in Table 2.(8)
Checking the selected Mutation PossibilityMP with random number (r), if it large or equals to r, then the updated SIik+1 equals(25)SIik+1=SIik+1+EN∗r,whereEN is Elite Number.(9)
Repeat procedures 2–8 until reach maximum number of iterations (itermax).(10)
End.Table 3
The typical parameters of the MBBO algorithm.
Parameter
Value
M
P
0.1
E
N
2
I
V
max
1
E
V
max
1
Maintain rate
0.2
β
0.9


## 2.5.2. Cuckoo Search (CS) Algorithm

It is a new metaheuristic optimization method discovered by Yang and Deb in 2009, which is inspired from parasitic performance of cuckoo birds plus levy flight performance of other birds [38]. Each egg can be viewed as a solution which is randomly generated in the initialization process. CS algorithm has been used alone for DSR and OCP techniques, and the optimal result obtained via DSR technique demonstrated the effectiveness and robustness of this algorithm. Also, CS algorithm has been applied with different typologies and objective functions for only OCP technique, in spite of confirming the total capacitor size restriction but did not satisfy the voltage restrictions [24, 39, 47]. In this paper, this algorithm is modified, especially the initialization process, to give better results than the original CS algorithm.The main steps of the CS algorithm are as follows:(1)
Initialization cuckoo parameters: ranges of Host Nest number (HN) and Discover Rate Probability (DRP) are 10–50 and 0-1, respectively [39, 40]. The values of these parameters and maximum number of iterations that are used in this study are listed in Table 4.(2)
Find the fitness value for each nest value and check the constraints of optimization problem such as radial condition of RDS and other constraints. At last, select the best host nests based on the fitness value.(3)
Start Levy flight phase to get the best value for each iteration as follows:(i)
Define Allocation Factor (AF) in the range (1–3) [41], Gamma Allocation (GA) function, where AF is specified and equals 3/2 in this study.(ii)
Calculate Sigma (Six) for solution x and Sigma (Siy) for solution y as standard variances by the following equations:(26)Six=GA1+AF∗sinπ/2∗AFGA1+AF/2∗AF∗2AF−1/21/AF,(27)Siy=1.(iii)
Determine the Step Data (SDi) for nest (i) as(28)SDi=uv1/AF,whereu and v are variables of solution x and can be calculated based on random numbers rx and ry using the following equations:(29)v=ry,(30)u=Six∗rx,whererx and ry have values which range between 0 and 1.(iv)
Calculate the Increment Value (IVinew) depending on values of SDi, best solution Xi,best, optimal solution Xi,opt, and Step Size (SZ) as(31)IVinew=SZ∗SDi∗Xi,best−Xi,opt,whereSZ is specified equals to 0.01.(v)
Determine the new value (Xi,new) depending on IVinew and Xi,best values using the following equation:(32)Xi,new=roundXi,best+r∗IVinew,whereround is an integer number.(4)
Delete the worst values and build new ones keeping the best solutions. This phase also generates a new solution as in the Levy flight phase depending on factorF which explains the ability to discover the stranger eggs that is specified as follows:(33)F=1,ifr<DRP,0,else.Then, the new solution (Xi,new) can be determined by using this factor as(34)Xi,new=roundXi,best+F∗ΔXinew,whereΔXinew is similar to IVinew in the Levy flight phase and is calculated as(35)ΔXinew=r∗rd1−rd2,whererd1 and rd2 are random disturbances for nest positions in Xi,best.(5)
Sort the solutions and find the current best solutions.(6)
The best nests with high quality of eggs (best value) will carry over to the next generation.(7)
Get the best results at the end of iteration.(8)
Repeat steps 2–7 until it reaches the limits of maximum iterations (itermax).(9)
End.Table 4
Initializing parameters of the CS algorithm.
Parameter
Value
HN
30
DRP
0.25


## 2.5.3. Modified Imperialist Competitive (MIC) Algorithm

Imperialist Competitive (IC) algorithm is an evolutionary algorithm that was discovered by Gargari and Lucas in 2007 which is inspired by colonial simulation [33]. This algorithm wishes to reach a state that only one empire existed and its corresponding colonies have the same position and cost. A modified form (MIC) is used in this article to enhance the efficiency, convergence velocity, and accuracy of the conventional IC algorithm. The MIC algorithm improves the original IC performance by preventing premature convergence to local minima based on absorption, exchange, revolution, and competition processes. This algorithm is implemented using the following steps:(1)
Initializing process: in this process, the optimal values of MIC algorithm control parameters, such as number of countries, number of imperialists, and number of colonies, were defined and inserted in Table5. Also, the cost and position of each imperialist and colony were initialized. The normalized power PN for each imperialist and number of colonies Ncn for empire are determined as follows:(36)PN=CN∑j=1ImCj,j∈Im,(37)Ncn=roundPN∗Nc,whereCN, Cj, Im, and Nc are normalize cost for each imperialist, cost of jth imperialist, number of imperialists, and total number of colonies, respectively.(2)
Absorption process: the colonies move toward their imperialist either in straight movement or diverted by angle∅. The latter is better to improve the search ability around the imperialists. The new position Posnew of the colony can be updated by vector (V) and absorption coefficient (γ) which has a value represented in Table 5. The vector V is calculated using position of imperialist (PosIM) and position of its corresponding colonies (Posc) as(38)V=PosIM−Posc.Then,Posc,new is calculated based on the previous position (Posc,old) as follows:(39)Posc,new=Posc,old+γ∗randV∗V.The movement of colony may drift from straight line by angle∅ that has the range corresponding to absorption angle coefficient (δ) value as(40)∅∈−δ:δ.(3)
Exchange process: this process occurs when position or cost of the colony is better than that for the imperialist, and then the replacement occurs to update a new imperialist.(4)
Update total cost for a new united empireTec using cost for imperialist CIM and the sharing cost of corresponding colonies as explained in the following equation:(41)Tec=CIM+Z∗AveCc,whereZ and AveCc are sharing cost and average of colony costs for the corresponding empire, respectively.(5)
Revolution process: during this process, the locations of colonies are altered randomly to enhance behavior of IC and exclude it from fall into the local minimum by using Revolution Rate Parameter (RRP) that has a range of 0.03–0.4 [34], and its optimal value is specified in Table 5. The number of revolving colonies Rc equals(42)Rc=roundRRP∗Cc.(6)
Imperialist competition process: this process aims to select the weakest colonies of weakest empires, and comparison is made between powerful empires to own these colonies. This is made depending on ownership probabilityPoe for each empire that is proportional to the normalized total cost TecN for each empire. TecN is first determined as(43)TecN=Tec−maxTec.After that,Poe equals(44)Poe=TecN∑j=1IMTecj,j∈Im.The vectorO corresponding to ownership probability for each empire generated, also a random vector R of the same size is built then a vector C contains the remaining empires after the collapse process as(45)C=O−R.(7)
Repeat above steps until it reachesitermax limit.(8)
End.Table 5
Optimal parameters of the MIC algorithm.
Parameter
Value
Country number
20
Im
10
N
c
10
Z
0.02
RRP
0.3
γ
0.5


## 2.5.4. Modified Bacterial Foraging-Based Optimization (MBFBO) Algorithm

A Bacterial Foraging-Based Optimization (BFBO) algorithm is of evolutionary type, first discovered by Passino in 2002, which has removable property [35]. The bacterium represents the proposed solution for optimization problem. A modified style of this algorithm is performed to overcome the drawbacks of BFBO algorithms such as lower convergence speed and trapping in local minimum solution. This modification includes both chemotaxis and swarming processes, and this enables it to handle multiobjective functions [36]. This algorithm has been applied for DSR technique with the objective function of loss reduction and applied for optimal placement of DG [44, 48]. The steps of the MBFBO algorithm are as follows:(1)
Initialization of the MBFBO parameters: the values of these parameters are inserted in Table6. In addition, the loops for chemotaxis, reproduction, and elimination processes have been generated.(2)
Chemotaxis process: this process was classified into tumble and swim operations. In tumble operation, the position forith bacteria XiNew is updated based on its chemotactic step size Ciss and direction index (Di) of cth chemotaxis as follows [37]:(46)Di=r−0.5∗2,(47)XiNew=Xiold+DiDi∗DiT∗Ciss.In swim operation, swim stepsSs are predefined and each bacteria move with same Css of previous chemotaxis, and this operation continues until it reach Ss limit.(3)
Reproduction process: during this process, the health of each bacteriaCHi is determined using equation (48) and sorting it in descending order, where the ones with least health are dead while the remaining ones each split into two to keep the constant number of bacteria.(48)CHi=∑C=1NcCi,whereNc is the number of chemotaxis steps. This procedure is repeated for each of reproductive steps (Nre).(4)
Elimination process: after the reproduction step, some of the new generations move to new positions to enhance the overall performance of this algorithm depending on Removal Probability (PR).(5)
Repeat steps 2–4 until it reachesitermax.(6)
End.Table 6
Initial parameters for the MBFBO algorithm.
Parameter
Value
Bacteria number
30
N
c
2
N
re
2
P
R
0.9
S
s
1Finally, values ofPop and itermax for all applied algorithms are inserted in Table 7. These values are used to give accurate and promise results over the literature studies.Table 7
Setting of parameters for applied algorithms.
Proposed algorithms
Parameter
Pop
iter
max, (RDS)
MBBO
30
50, (33-bus)100, (69-bus)
CS
30
MIC
20
MBFBO
30


## 3. Results and Discussion

The proposed four algorithms were carried out under the MATLAB environment (Release 2018b), and these algorithms were applied for two techniques (OCP and OCP-DSR) for 33- and 69-bus RDSs. The duration of the test is one year using constant power load. The results of these algorithms during this test are inserted in the following tables and figures.


### 3.1. IEEE-33 Bus RDS

The results for the application of proposed algorithms on this system for both OCP and OCP-DSR techniques are inserted in Tables8 and 9, respectively. Figures 3 and 4 show the voltage profile for 33-bus RDS with OCP and OCP-DSR, respectively, which provide that the voltage enhanced in the dual technique is better than the individual technique, and the voltage constraints are also confirmed using these techniques. The convergence graph of real losses for the four algorithms using two techniques is shown in Figures 5 and 6, respectively, which explained a significant reduction obtained using the OCP-DSR technique.Table 8
Results and comparisons of IEEE-33-bus RDS without and with individual OCP technique.
Parameters
Without OCP
Literature algorithms
Proposed algorithms
LS [49]
NA [50]
GSA [51]
FPA [52]
MFO [53]
IBPSO [54]
MBBO
CS
MIC
MBFBO
Year
—
2018
2017
2015
2016
2018
2014
2019
2019
2019
2019
Q
c (kVAr)
—
[450, 350, 900]
[550, 480, 330]
[350, 450, 800]
[250, 400, 950]
[450, 300, 900]
[900, 300, 300, 300, 300, 600, 600]
[750, 450, 600]
[300, 750, 600]
[750, 150, 750]
[900, 600, 750]
Location
—
[12, 25, 30]
[14, 30, 32]
[26, 13, 15]
[6, 9, 30]
[8, 13, 30]
[1, 3, 14, 22, 24, 30, 31]
[27, 30, 24]
[8, 30, 23]
[27, 2, 24]
[5, 27, 2]
Q
c
T (kVAr)
—
1700
1360
1600
1600
1650
3300
1800
1650
1650
2250
C
T
c ($)
—
10800
9780
6151.5
6091
6083.55
14148.7
6152.85
6144
6189
6203.7
P
loss
T (kW)
202.6771
139.23
138.72
134.5
134.47
134.0725
134.2
108.6105
123.4426
126.3468
125.755
V
min (p. u.)
0.91306
0.9291
0.9428
0.9672
0.9365
0.94
0.9389
0.95
0.95
0.95
0.95
V
max (p. u.)
1
1
1
1
1
1
1
1
1
1
1
% reduction
—
31.3
31.556
33.638
33.65
33.8502
33.7863
46.412
39.0939
37.661
37.953
C
A ($)
106527
83979
82691
76845
76768
76552
84684
63239
71025
72597
72301
S
cost ($)
—
22548
23836
29682
29759
29975
21843
43289
35502
33930
34227
%
saving
—
21.1660
22.3760
27.8630
27.9360
28.1384
20.5050
40.6366
33.3268
31.8510
32.1299
Pop
—
20
NR∗
2000
25
20
NR∗
30
30
20
30
iter
max
—
1000
NR∗
400
100
20
NR∗
50
50
50
50
NR∗: not reported.Table 9
Results and comparisons of IEEE-33-bus RDS without and with dual OCP-DSR technique.
Parameters
Without OCP-DSR
Literature
Proposed algorithm
IBPSO [54]
MBBO
CS
MIC
MBFBO
Year
—
2014
2019
2019
2019
2019
Tie-switches
[33, 34, 35, 36, 37]
[7, 10, 34, 36, 37]
[7, 11, 34, 36, 28]
[8, 5, 37, 30, 12]
[9, 25, 14, 33, 37]
[8, 37, 14, 7, 36]
Q
c (kVAr)
—
[900, 300, 300, 300, 300, 600, 600]
[750, 450, 600]
[300, 750, 600]
[750, 150, 750]
[900, 600, 750]
Location
—
[1, 3, 14, 22, 24, 30, 31]
[27, 30, 24]
[8, 30, 23]
[27, 2, 24]
[5, 27, 2]
Q
c
T (kVAr)
—
3300
1800
1650
1650
2250
C
T
c ($)
—
14148.7
6152.85
6144
6189
6203.7
P
loss
T (kW)
202.6771
95.91
83.469
95.6628
97.6377
97.3076
V
min (p. u.)
0.91306
0.9658
0.9559
0.9704
0.95
0.95
V
max (p. u.)
1
1
1
1
1
1
%
reduction
—
52.678
58.8167
52.8003
51.8259
51.9888
C
A ($)
106527
64559
50024
56424
57507
57349
S
cost ($)
—
41968
56503
50103
49020
49179
%
saving
—
39.3970
53.0410
47.0331
46.0165
46.1657
Pop
—
NR∗
30
30
20
30
iter
max
—
NR∗
50
50
50
50Figure 3
Voltage profile for 33-bus RDS using OCP technique.Figure 4
Voltage profile for 33-bus RDS using OCP-DSR technique.Figure 5
Convergence loss graph for 33-bus RDS using OCP technique.Figure 6
Convergence loss graph for 33-bus RDS using OCP-DSR technique.Also, these tables contain comparisons with literature works with the same multiobjective functions such as Locust Search (LS) [49], Novel Analytic (NA) [50], Gravitational Search Algorithm (GSA) [51], Flower Pollination Algorithm (FPA) [52], Moth Flame Optimization (MFO) [53], and Improved Binary Particle Swarm Optimization (IBPSO) [54] algorithms.From Tables8 and 9, it is concluded that the proposed algorithms achieved better performance for losses, voltage profile, and cost aspects. The results proved the superiority of proposed algorithms, and the dual OCP-DSR technique provided better performance than individual OCP technique as explained in Table 9. Also, this table contains comparison with literature work that showed the total capacitor size and number restrictions were violated in the previous study and confirmed in proposed algorithms. Also, in spite of larger number and sizes of capacitors for the literature work, the MBBO and CS algorithms achieved significant reduction in losses with more saving. This proves the superiority and effectiveness of these algorithms for proper selection of size and location of capacitors.


### 3.2. IEEE 69-Bus RDS

The capacitor size and location are optimally implemented using the four proposed algorithms for the OCP technique in this system. Table10 demonstrates the results for these algorithms with OCP technique and confirmed the better performance for the MBBO algorithm over the other proposed algorithms. Also, this table contains comparisons with LS [49], GSA [51], FPA [52], MFO [53], and Logistic Map Differential Evolution (LMDE) [55] algorithms and indicated the superiority of the proposed algorithms in terms of losses, voltages, and saving cost over these works.Table 10
Results and comparisons of IEEE 69-bus RDS without and with individual OCP technique.
Parameters
Without OCP
Literature algorithms
Proposed algorithms
LS [49]
GSA [51]
FPA [52]
MFO [53]
LMDE [55]
MBBO
CS
MIC
MBFBO
Year
—
2018
2015
2016
2018
2019
2019
2019
2019
2019
Q
c (kVAr)
—
[350, 1200]
[150, 150, 1050]
[1350]
[300, 1350]
[200, 250, 250, 600, 300]
[300, 900, 150]
[450, 600, 900]
[1350, 150, 450]
[600, 300, 300]
Location
—
[17, 61]
[26, 13, 15]
[61]
[17, 61]
[21, 59, 61, 62, 65]
[12, 60, 21]
[15, 50, 61]
[59, 69, 15]
[59, 68, 20]
Q
c
T (kVAr)
—
1550
1350
1350
1650
1600
1350
1950
1950
1200
C
T
c ($)
—
8450
6089.4
2179.45
6084.45
10012
6044.7
6110.55
6168.3
6042
P
loss
T (kW)
224.9606
146.61
145.9
150.28
145.6945
146.31
61.5959
90.8139
120.6084
101.8776
V
min (p. u.)
0.90901
0.93
0.9519
0.9333
0.9330
0.9310
0.96
0.96
0.9501
0.955
V
max (p. u.)
1
1
1
1
1
0.9999
1
1
1
1
%
reduction
—
34.8285
35.1441
33.197
35.235
34.95
72.6192
59.6312
46.3869
54.7131
C
A ($)
118239
85508
82774
81167
82661
86913
38420
53842
69560
59589
S
cost ($)
—
32731
35465
37073
35578
31326
79820
64397
48679
58650
%
saving
—
27.6820
29.9943
31.3543
30.0899
26.4937
67.5073
54.4634
41.1700
49.6029
Pop
—
20
2500
25
20
NR∗
30
30
20
30
iter
max
—
1000
750
100
20
100
100
100
100
100The second approach utilizing from DSR technique is applied to a compensated system known as dual OCP-DSR technique. The results are given in Table11 which explain a further reduction in the real losses and more improvement of voltage and saving cost if compared with using the individual OCP technique. Also, this table contains comparisons with Fuzzy Multiobjective Approach (FMOA) [56] and indicates the superiority of the proposed algorithms in terms of losses, voltages, and saving cost. The literature work broke the voltage constraints, therefore, in spite of more loss reduction but did not confirm the RDS restrictions as done with proposed algorithms. The corresponding voltage profile and convergence graph for losses are shown in Figures 7–10, respectively.Table 11
Results and comparisons of IEEE 69-bus RDS without and with dual OCP-DSR technique.
Parameters
Without OCP-DSR
Literature
Proposed algorithms
FMOA [56]
MBBO
CS
MIC
MBFBO
Year
—
2013
2019
2019
2019
2019
Tie-switches
[69, 70, 71, 72, 73]
[69, 70, 57, 64, 12]
[58, 42, 19, 60, 45]
[49, 10, 59, 45, 19]
[70, 58, 49, 69, 14]
[10, 19, 14, 60, 54]
Q
c (kVAr)
—
400
[300, 900, 150]
[450, 600, 900]
[1350, 150, 450]
[600, 300, 300]
Location
—
61
[12, 60, 21]
[15, 50, 61]
[59, 69, 15]
[59, 68, 20]
Q
c
T (kVAr)
—
400
1350
1950
1950
1200
C
T
c ($)
—
2001.2
6044.7
6110.55
6168.3
6042
P
loss
T (kW)
224.9606
95.1
54.9369
80.4276
102.846
80.6144
V
min (p. u.)
0.90901
0.9450
0.97336
0.97189
0.97447
0.98651
V
max (p. u.)
1
1
1.0016
1
1
1.0005
%
reduction
—
57.725
75.5793
64.2482
54.2827
64.1651
C
A ($)
118239
51986
34920
48383
60224
48413
S
cost ($)
—
66253
83320
69856
58015
69826
%
saving
—
56.0331
70.4674
59.0803
49.0659
59.0550
Pop
—
NR∗
30
30
20
30
iter
max
—
NR∗
100
100
100
100Figure 7
Voltage profile for 69-bus RDS using OCP technique.Figure 8
Voltage profile for 69-bus RDS using OCP-DSR technique.Figure 9
Convergence loss graph for 69-bus RDS using OCP technique.Figure 10
Convergence loss graph for 69-bus RDS using OCP-DSR technique.From Tables8–11 and Figures 3–10, it can be seen that the proposed algorithms have achieved a better performance in the dual OCP-DSR technique as compared with an individual OCP technique. The MBBO algorithm has high ability compared with the other algorithms in terms of the quality of solution and computational efficiency. A comparison with literature works has been done for individual and dual approaches that used the same objective functions and case studies. This comparison proved that the proposed algorithms achieved the voltage constraints which were broken in some literature works. In addition, the losses and cost are better in this work. Then, the approach of selecting the capacitor location using intelligent algorithms gives robust and accurate results than using conventional indices such as LSI and VSI.


## 3.1. IEEE-33 Bus RDS

The results for the application of proposed algorithms on this system for both OCP and OCP-DSR techniques are inserted in Tables8 and 9, respectively. Figures 3 and 4 show the voltage profile for 33-bus RDS with OCP and OCP-DSR, respectively, which provide that the voltage enhanced in the dual technique is better than the individual technique, and the voltage constraints are also confirmed using these techniques. The convergence graph of real losses for the four algorithms using two techniques is shown in Figures 5 and 6, respectively, which explained a significant reduction obtained using the OCP-DSR technique.Table 8
Results and comparisons of IEEE-33-bus RDS without and with individual OCP technique.
Parameters
Without OCP
Literature algorithms
Proposed algorithms
LS [49]
NA [50]
GSA [51]
FPA [52]
MFO [53]
IBPSO [54]
MBBO
CS
MIC
MBFBO
Year
—
2018
2017
2015
2016
2018
2014
2019
2019
2019
2019
Q
c (kVAr)
—
[450, 350, 900]
[550, 480, 330]
[350, 450, 800]
[250, 400, 950]
[450, 300, 900]
[900, 300, 300, 300, 300, 600, 600]
[750, 450, 600]
[300, 750, 600]
[750, 150, 750]
[900, 600, 750]
Location
—
[12, 25, 30]
[14, 30, 32]
[26, 13, 15]
[6, 9, 30]
[8, 13, 30]
[1, 3, 14, 22, 24, 30, 31]
[27, 30, 24]
[8, 30, 23]
[27, 2, 24]
[5, 27, 2]
Q
c
T (kVAr)
—
1700
1360
1600
1600
1650
3300
1800
1650
1650
2250
C
T
c ($)
—
10800
9780
6151.5
6091
6083.55
14148.7
6152.85
6144
6189
6203.7
P
loss
T (kW)
202.6771
139.23
138.72
134.5
134.47
134.0725
134.2
108.6105
123.4426
126.3468
125.755
V
min (p. u.)
0.91306
0.9291
0.9428
0.9672
0.9365
0.94
0.9389
0.95
0.95
0.95
0.95
V
max (p. u.)
1
1
1
1
1
1
1
1
1
1
1
% reduction
—
31.3
31.556
33.638
33.65
33.8502
33.7863
46.412
39.0939
37.661
37.953
C
A ($)
106527
83979
82691
76845
76768
76552
84684
63239
71025
72597
72301
S
cost ($)
—
22548
23836
29682
29759
29975
21843
43289
35502
33930
34227
%
saving
—
21.1660
22.3760
27.8630
27.9360
28.1384
20.5050
40.6366
33.3268
31.8510
32.1299
Pop
—
20
NR∗
2000
25
20
NR∗
30
30
20
30
iter
max
—
1000
NR∗
400
100
20
NR∗
50
50
50
50
NR∗: not reported.Table 9
Results and comparisons of IEEE-33-bus RDS without and with dual OCP-DSR technique.
Parameters
Without OCP-DSR
Literature
Proposed algorithm
IBPSO [54]
MBBO
CS
MIC
MBFBO
Year
—
2014
2019
2019
2019
2019
Tie-switches
[33, 34, 35, 36, 37]
[7, 10, 34, 36, 37]
[7, 11, 34, 36, 28]
[8, 5, 37, 30, 12]
[9, 25, 14, 33, 37]
[8, 37, 14, 7, 36]
Q
c (kVAr)
—
[900, 300, 300, 300, 300, 600, 600]
[750, 450, 600]
[300, 750, 600]
[750, 150, 750]
[900, 600, 750]
Location
—
[1, 3, 14, 22, 24, 30, 31]
[27, 30, 24]
[8, 30, 23]
[27, 2, 24]
[5, 27, 2]
Q
c
T (kVAr)
—
3300
1800
1650
1650
2250
C
T
c ($)
—
14148.7
6152.85
6144
6189
6203.7
P
loss
T (kW)
202.6771
95.91
83.469
95.6628
97.6377
97.3076
V
min (p. u.)
0.91306
0.9658
0.9559
0.9704
0.95
0.95
V
max (p. u.)
1
1
1
1
1
1
%
reduction
—
52.678
58.8167
52.8003
51.8259
51.9888
C
A ($)
106527
64559
50024
56424
57507
57349
S
cost ($)
—
41968
56503
50103
49020
49179
%
saving
—
39.3970
53.0410
47.0331
46.0165
46.1657
Pop
—
NR∗
30
30
20
30
iter
max
—
NR∗
50
50
50
50Figure 3
Voltage profile for 33-bus RDS using OCP technique.Figure 4
Voltage profile for 33-bus RDS using OCP-DSR technique.Figure 5
Convergence loss graph for 33-bus RDS using OCP technique.Figure 6
Convergence loss graph for 33-bus RDS using OCP-DSR technique.Also, these tables contain comparisons with literature works with the same multiobjective functions such as Locust Search (LS) [49], Novel Analytic (NA) [50], Gravitational Search Algorithm (GSA) [51], Flower Pollination Algorithm (FPA) [52], Moth Flame Optimization (MFO) [53], and Improved Binary Particle Swarm Optimization (IBPSO) [54] algorithms.From Tables8 and 9, it is concluded that the proposed algorithms achieved better performance for losses, voltage profile, and cost aspects. The results proved the superiority of proposed algorithms, and the dual OCP-DSR technique provided better performance than individual OCP technique as explained in Table 9. Also, this table contains comparison with literature work that showed the total capacitor size and number restrictions were violated in the previous study and confirmed in proposed algorithms. Also, in spite of larger number and sizes of capacitors for the literature work, the MBBO and CS algorithms achieved significant reduction in losses with more saving. This proves the superiority and effectiveness of these algorithms for proper selection of size and location of capacitors.


## 3.2. IEEE 69-Bus RDS

The capacitor size and location are optimally implemented using the four proposed algorithms for the OCP technique in this system. Table10 demonstrates the results for these algorithms with OCP technique and confirmed the better performance for the MBBO algorithm over the other proposed algorithms. Also, this table contains comparisons with LS [49], GSA [51], FPA [52], MFO [53], and Logistic Map Differential Evolution (LMDE) [55] algorithms and indicated the superiority of the proposed algorithms in terms of losses, voltages, and saving cost over these works.Table 10
Results and comparisons of IEEE 69-bus RDS without and with individual OCP technique.
Parameters
Without OCP
Literature algorithms
Proposed algorithms
LS [49]
GSA [51]
FPA [52]
MFO [53]
LMDE [55]
MBBO
CS
MIC
MBFBO
Year
—
2018
2015
2016
2018
2019
2019
2019
2019
2019
Q
c (kVAr)
—
[350, 1200]
[150, 150, 1050]
[1350]
[300, 1350]
[200, 250, 250, 600, 300]
[300, 900, 150]
[450, 600, 900]
[1350, 150, 450]
[600, 300, 300]
Location
—
[17, 61]
[26, 13, 15]
[61]
[17, 61]
[21, 59, 61, 62, 65]
[12, 60, 21]
[15, 50, 61]
[59, 69, 15]
[59, 68, 20]
Q
c
T (kVAr)
—
1550
1350
1350
1650
1600
1350
1950
1950
1200
C
T
c ($)
—
8450
6089.4
2179.45
6084.45
10012
6044.7
6110.55
6168.3
6042
P
loss
T (kW)
224.9606
146.61
145.9
150.28
145.6945
146.31
61.5959
90.8139
120.6084
101.8776
V
min (p. u.)
0.90901
0.93
0.9519
0.9333
0.9330
0.9310
0.96
0.96
0.9501
0.955
V
max (p. u.)
1
1
1
1
1
0.9999
1
1
1
1
%
reduction
—
34.8285
35.1441
33.197
35.235
34.95
72.6192
59.6312
46.3869
54.7131
C
A ($)
118239
85508
82774
81167
82661
86913
38420
53842
69560
59589
S
cost ($)
—
32731
35465
37073
35578
31326
79820
64397
48679
58650
%
saving
—
27.6820
29.9943
31.3543
30.0899
26.4937
67.5073
54.4634
41.1700
49.6029
Pop
—
20
2500
25
20
NR∗
30
30
20
30
iter
max
—
1000
750
100
20
100
100
100
100
100The second approach utilizing from DSR technique is applied to a compensated system known as dual OCP-DSR technique. The results are given in Table11 which explain a further reduction in the real losses and more improvement of voltage and saving cost if compared with using the individual OCP technique. Also, this table contains comparisons with Fuzzy Multiobjective Approach (FMOA) [56] and indicates the superiority of the proposed algorithms in terms of losses, voltages, and saving cost. The literature work broke the voltage constraints, therefore, in spite of more loss reduction but did not confirm the RDS restrictions as done with proposed algorithms. The corresponding voltage profile and convergence graph for losses are shown in Figures 7–10, respectively.Table 11
Results and comparisons of IEEE 69-bus RDS without and with dual OCP-DSR technique.
Parameters
Without OCP-DSR
Literature
Proposed algorithms
FMOA [56]
MBBO
CS
MIC
MBFBO
Year
—
2013
2019
2019
2019
2019
Tie-switches
[69, 70, 71, 72, 73]
[69, 70, 57, 64, 12]
[58, 42, 19, 60, 45]
[49, 10, 59, 45, 19]
[70, 58, 49, 69, 14]
[10, 19, 14, 60, 54]
Q
c (kVAr)
—
400
[300, 900, 150]
[450, 600, 900]
[1350, 150, 450]
[600, 300, 300]
Location
—
61
[12, 60, 21]
[15, 50, 61]
[59, 69, 15]
[59, 68, 20]
Q
c
T (kVAr)
—
400
1350
1950
1950
1200
C
T
c ($)
—
2001.2
6044.7
6110.55
6168.3
6042
P
loss
T (kW)
224.9606
95.1
54.9369
80.4276
102.846
80.6144
V
min (p. u.)
0.90901
0.9450
0.97336
0.97189
0.97447
0.98651
V
max (p. u.)
1
1
1.0016
1
1
1.0005
%
reduction
—
57.725
75.5793
64.2482
54.2827
64.1651
C
A ($)
118239
51986
34920
48383
60224
48413
S
cost ($)
—
66253
83320
69856
58015
69826
%
saving
—
56.0331
70.4674
59.0803
49.0659
59.0550
Pop
—
NR∗
30
30
20
30
iter
max
—
NR∗
100
100
100
100Figure 7
Voltage profile for 69-bus RDS using OCP technique.Figure 8
Voltage profile for 69-bus RDS using OCP-DSR technique.Figure 9
Convergence loss graph for 69-bus RDS using OCP technique.Figure 10
Convergence loss graph for 69-bus RDS using OCP-DSR technique.From Tables8–11 and Figures 3–10, it can be seen that the proposed algorithms have achieved a better performance in the dual OCP-DSR technique as compared with an individual OCP technique. The MBBO algorithm has high ability compared with the other algorithms in terms of the quality of solution and computational efficiency. A comparison with literature works has been done for individual and dual approaches that used the same objective functions and case studies. This comparison proved that the proposed algorithms achieved the voltage constraints which were broken in some literature works. In addition, the losses and cost are better in this work. Then, the approach of selecting the capacitor location using intelligent algorithms gives robust and accurate results than using conventional indices such as LSI and VSI.


## 4. Conclusions

The effective RDS has lower real losses, better voltage profile, and more saving cost that must be achieved together. DSR and OCP techniques are the most economic solutions for the aforementioned problems and perform fast convergence. In this work, two approaches that have been used to get bigger profit are individual OCP and dual OCP-DSR techniques to select the best locations and sizes of capacitors in addition to modifying the network configuration without using LSI and VSI factors. Four different optimization algorithms, MBBO, CS, MIC, and MBFBO, are used for solving the optimization problem via MATLAB 2018b software. The load flow proposed in this paper is DBFSM instead of NRM and GSM to calculate total real losses, each bus voltage, and branch currents. The two IEEE-33 and 69-bus RDSs are used to prove the efficiency of each algorithm. The results of these algorithms for two techniques and systems are compared among each other. The comparison results demonstrate the superiority of the MBBO algorithm over the other algorithms in terms of lower real losses, better voltage profile, and saving cost. This algorithm achieved percentage reduction of 46.412% and 58.8167% for 33-bus RDS while it achieved reduction of 72.6192% and 75.5793% for 69-bus RDS using individual and dual techniques, respectively. This algorithm achieved percentage saving of 40.6366% and 53.041% for 33-bus RDS while it achieved saving of 67.5073% and 70.4674% for 69-bus RDS using individual and dual techniques, respectively. Also, a comparison with literature works has been done to validate the proposed algorithms. Finally, the dual OCP-DSR technique has higher ability than the individual OCP technique in reduction of total losses, improving both voltage profile and annual saving cost of the 33-bus and 69-bus systems.



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*Source: 1696347-2019-12-31.xml*