2.1 Objective Functions

2.2 Operating Constraints

2.2.1 Power Flow Calculations

The equivalent single line diagram of a radial distribution network is shown in Figure 1. The basic backward/forward load flow program has been utilized for calculating real and reactive power flow between the buses and terminating bus voltage using Eqs. (3), (4), and (5), respectively:

Figure 1:

Equivalent Branch of an Electrical Distribution Network.

(3) Pi+1=Pi−PLi+1−Ri,i+1×(Pi2+Qi2)|Vi|2.

(4) Qi+1=Qi+Qjc−QLi+1−Xi,i+1×(Pi2+Qi2)|Vi|2.

(5) |Vi+1|2=|Vi|2−2(Ri,i+1×Pi+Xi,i+1×Qi)+(Ri,i+12+Xi,i+12)×(Pi2+Qi2)|Vi|2.

3.1 Proposed Approach

A new approach is introduced to identify candidate buses as well as the sizes of SCs in distribution networks simultaneously for capacitor installation. The load flow calculations are required to evaluate the total power loss, bus voltage, and branch current of the network, and these are associated with the fitness function. The SC locations and sizes are considered as decision variables in the optimization algorithm. Therefore, these are defined in the available search space of the optimization algorithm using Eq. (9). The optimal fitness function value totally depends upon the random selection of decision variables within the specified upper and lower limits. Out of several combinations, one combination shows the minimum function value, which would be the optimal location for capacitor placement. For obtaining the optimal solution, the optimization algorithm should be run multiple times. An optimal topology that has been used in this study can be represented as follows:

Here, Cap_Location and Cap_Size are the capacitor location and capacitor size, respectively. The capacitor locations and sizes are defined in a 1×2 column matrix.

3.2 IVM

This method has been implemented to identify optimal buses for SC placement in distribution networks. A load flow program is necessary to evaluate the real and imaginary components of current across each branch. The IVM value for the i^th bus can be evaluated using Eq. (10). Based on the values, this method identifies candidate buses for SC installation [24].

The following are the steps for implementation of the IVM approach to find candidate buses for SC placement:

• Step 1: Run the load flow program without SC installation.

• Step 2: Store the values of voltages of each bus as well as the real and imaginary components of the current of each branch.

• Step 3: Then, calculate the IVM values of all buses using Eq. (10) and sort these in a decreasing manner.

• Step 4: Evaluate the normalized voltage, norm(i)=V_i/0.95, for all buses. Those buses that have higher IVM values and least normalized voltage <1.01 are picked up as candidate buses for SC integration.

3.3 LSF

This technique has been employed to find the sensitivity of each bus of the distribution network [28]. It is helpful for mitigating the search space of the algorithm during the optimization process. The LSF values for real and reactive power support is determined using Eqs. (11) and (12), respectively:

The steps for implementation of the LSF approach are as follows:

• Step I: Calculate the LSF for all buses from base case load flow using Eq. (13):

  (13) LSF=dPi,i+1line loss/dQeff(i).

• Step II: Sort the LSF values of all buses in descending order and store them in bus position vector.

• Step III: Then, evaluate the normalized voltage for each bus, norm(i)=V_i/0.95.

• Step IV: Choose those buses with normalized voltage <1.01 as candidate buses for SC installation.

The possible standard sizes of SCs with cost in $/kVAr are available in Ref. [23], which have been utilized to determine the total annual cost of the network.

3.4 PLI

The PLI approach has been implemented to determine candidate buses for capacitor installation. The load flow is required to find loss reduction (LR) values by compensating the total reactive load across each bus, considering one bus at a time excluding the slack bus [14]. The PLI value for the i^th bus is evaluated using Eq. (14):

Those buses having higher PLI values are chosen as candidate buses for SC installation. The procedure for implementation of the PLI approach is as follows:

• Step 1: Run the load flow program of base case and evaluate real power losses.

• Step 2: Provide reactive power compensation across each bus, which is equal to the entire reactive load of the network.

• Step 3: Run the load flow and calculate the real power loss and store these values.

• Step 4: Calculate LR=(base case real power loss−real power loss of each bus after compensation) and store the value.

• Step 5: Evaluate the maximum and minimum LR. Then, compute the PLI values of each bus using Eq. (14).

• Step 6: Sort these PLI values in descending order. Those buses with voltage <0.95 pu are selected as candidate buses for capacitor installation.

3.5 IVS

The IVS is a numerical solution to compute the security level of the distribution system. The purpose of IVS is to calculate the stability of the buses and locate sensitive buses in the network [3]. Voltage collapse begins at the most sensitive bus and expand to other sensitive buses. IVS at bus i+1 can be determined using Eq. (15):

The condition for stable operation of the network is IVS_(i+1)≥0, whereas i=1, …, (N_B−1). Those buses having minimum IVS values are selected as the most sensitive buses for capacitor placement.

To determine the sensitive buses for capacitor allocation, the following steps are taken:

• Step 1: Calculate IVS values for all buses through a base case load flow program.

• Step 2: Sort these IVS values in ascending order. Those buses having the least IVS values are chosen as candidate buses for SC installation.

6.1 Conventional PSO

This algorithm was proposed by Eberhart and Kennedy in 1995 [9]. In this algorithm, the population is known as swarm and it is generated in a random manner, and the swarm consists of individuals named as particles. Each and every particle moves in the search space for finding an optimal solution. In this, the particles have two important parameters, such as position (x_i) and velocity (v_i). The position and velocity of the i^th particle in available d-dimensional search space is defined as xik=xi1k, xi2k, …, xidk and vik=vi1k, vi2k, …, vidk, respectively. In each and every iteration, the i^th particle fitness is calculated, i.e. Pbest,ik=Pbest,i1k, Pbest,i2k, …, Pbest,idk. P_best,i is the best position that has been visited through the i^th particle unto the current iteration (k). Moreover, the fitness position associated with the best particle (P^k_best-i ) is considered as particle best position (xik). The global best fitness value (G_best-i ) is the best solution among the Pbest,ik in a group of i^th particles at iteration (k). Thus, the new position of particles (xik+1) is updated, given by Eq. (19) based on the velocity (vik+1) values using Eq. (20):

6.2 IPSO

In the iteration-based PSO method, a new parameter I_best is incorporated into the velocity equation for improving the solution quality. It was developed by Lee and Chen [21]. This I_best value is the P_best value that has been chosen in random manner among all particles in the current population. Moreover, another coefficient, “c3,” called dynamic accelerating constant, has also been introduced and is evaluated using Eq. (21). Hence, the final updated velocity equation can be defined using Eq. (22):

6.3 GABC Algorithm

The GABC algorithm is one of the prevalent meta-heuristic optimization techniques; it is inspired by the social nature of honeybees for searching a food source. This was developed by Zhu and Kwong [33] in 2010. It consists of a combination of three varieties of bees, namely employed bee, onlooker bee, and scout bee, where onlooker and scout bees are considered as unemployed bees. An employed bee searches and exploits a food source position while the onlooker bees wait in the hive. Employed bees distribute information with the onlooker bees regarding a food source position. As per the information received via the employed bees, the onlooker bees find a better food source location [33]. The probability to select a particular food source through the onlooker bee is evaluated using Eq. (23). The location of each food source indicates the feasible outcomes of the defined optimization task:

where fit_i represents an objective function value of the i^th position solution and k ε {1, 2, …, dv}. The term dv belongs to the total number of decision variables. The scout bees identify a better food source position in a random manner using Eq. (24):

where γ_min,j and γ_max,j indicate the maximum and minimum values of the j^th variable at the i^th solution, and rand indicates the random number, which lies between 0 and 1. The entire population has a solution. Equation (25) represents the solution of the i^th food source:

The searching process of the GABC algorithm is classified into four simple steps, as follows: (i) initialization of parameters, (ii) employed bee phase, (iii) onlooker bee phase, and (iv) scout bee phase. In the first step, the candidate solution is determined in a random manner using Eq. (24). Both employed and onlooker bees search a new food source position using Eq. (26):

Here, ϕ_ij represents the uniform random number [−1, 1], γ_new,ij indicates the updated food source position, γ_kj is a food source that is related to the employed bees or nearer to γ_ij, c is a random number between [0, 2], and γ_j represents a global best solution of the present cycle.

6.4 Implementation of the GABC Algorithm for the OSCsP Problem

This section indicates the application and implementation of GABC algorithm to solve the optimal SC installation problem in distribution networks. The necessary steps to solve this problem are as follows:

• Step I: Initialize the power system data and the parameters of the GABC algorithm, i.e. load and line data, operating voltage and base MVA, number of employed bees, number of onlooker bees, limit, total number of iterations (MCN), etc.

• Step II: Calculate the size of SCs using Eq. (24). These are the random sizes of SCs, γ_ij. Then, convert them into standard ratings via the pseudo code below:

  if Cs1≤γij≤Cs2−Cs2−Cs12 γij=Cs1else if Cs2−Cs2−Cs12≤γij≤Cs3−Cs3−Cs22 γij=Cs2 ⋮else if Cs(n−1)−Cs(n−1)−Cs(n−2)2≤γij≤Csn−Csn−Cs(n−1)2 γij=Cs(n−1)else if γij≥Csn−Csn−Cs(n−1)2 γij=Csnend

  Here, C_s1, C_s2, …, C_s(n−1) and C_sn are the standard sizes of the capacitor.

• Step III: Run the GABC optimization algorithm including load flow program, and evaluate fitness function value (2). Initially, the iteration count is set as 1 and is repeated till the MCN is reached.

• Step IV: In this step (employed bee phase), modify the randomly generated solution in the above step using Eq. (26).

• Step V: After the employed bee phase, modify the value and repeat step III.

• Step VI: Apply a greedy search mechanism and memorize the best solution obtained from steps II and IV, and discard the worst one.

• Step VII: Onlooker bee phase. In this step, the onlooker bee selects an employed bee food source location and evaluates the probability to find a better food source using Eq. (23).

• Step VIII: After the onlooker bee phase, the values become modified. Using these modified capacitor values, execute step III. Apply the greedy mechanism and memorize the best solution obtained from steps VI and VIII, and discard the worst.

• Step IX: In the scout bee phase, if the result quality is not mended in predefined trials, the values are discarded. Then, the scout bees find a better solution in a random manner using Eq. (24).

• Step X: Memorize the best obtained solution and increment the iteration cycle: iter=iter+1. If iter<MCN, go to step IV; otherwise, stop and display the best optimal solution.

The flowchart diagram of the GABC algorithm for solving the SC installation problem is illustrated in Figure 2.

Figure 2:

Flowchart of the GABC Algorithm to Solve the OSCsP Problem.

7.1 Thirty-Four-Bus Radial Distribution System

This standard test network contains 34 buses and 33 branches, and has one main feeder and four laterals (subfeeder). The total network details, i.e. load and line data, have been taken from Ref. [5]. The total network load is 4.636+2.873 MVA, the rated voltage is 11 kV, and the base is 100 MVA.

Proposed Approach: In this, the optimal allocations and sizes of SCs are determined simultaneously through optimization algorithms for minimizing the total annual cost of the system. As a result, it improves the power loss reduction and enhances the voltage level significantly. The numerical outcomes obtained through optimization algorithms with and without consideration of load growth are tabulated in Table 3. Between the two algorithms, the GABC algorithm generates better-quality solutions.

IVM: The IVM approach is applied for determining optimal buses for SC placement. Those buses with higher IVM values and lower normalized voltage are chosen as candidate buses for SC installation. These buses are {23, 24, 25, and 26} and the optimal sizes of SCs on these respective buses are evaluated using optimization algorithms. The detailed numerical outcomes obtained through different optimization algorithms with and without considering load growth are tabulated in Table 4. Between the two algorithms, GABC provides good-quality solutions.

LSF: This methodology is applied to find the sensitivity of the buses of the distribution network. Those buses with higher LSF values and lower normalized voltage are chosen as candidate buses for SC placement. These buses are {19, 20, 21, and 22}, and the optimal sizes of SCs on these buses are evaluated through optimization algorithms. The obtained numerical outcomes via both optimization algorithms are tabulated in Table 5.

PLI: In this, the locations of SCs are determined through the PLI approach. Those buses with high PLI values and voltage <0.95 pu are chosen as candidate buses for SC installation, and these are {22, 25, 26, and 27}, and the respective size on these buses is evaluated through optimization algorithms. The numerical results obtained through various algorithms are mentioned in Table 6. From the numerical outcomes, it is noted that both optimization algorithms show the same results.

IVS: In this, those buses with the least IVS values are chosen as candidate buses for SC placement. The candidate buses for SC installation are {24, 25, 26, and 27}, and the respective sizes of SCs on these buses are computed through optimization algorithms. From Table 7, it is observed that both optimization algorithms give the same solution including load growth.

The obtained numerical outcomes through the proposed technique using the GABC algorithm are compared with the other published intelligent algorithms available in the published literature, i.e. ABC, PGSA, heuristic search, mixed-integer non-linear program (MINLP), GA, BFO, and PSO, as tabulated in Table 8. These realized numerical results indicate that the proposed technique is superior to other computation techniques. The minimum system voltage is appreciably improved after optimal compensation. In addition, Table 9 indicates the simulation results comparison between various approaches after incorporating the SCs. The percentage loss reduction is 28.14% and the percentage annual saving is 23.20% without consideration of load growth. In the presence of load growth, it becomes 27.68% and 25.36%, respectively. Among all approaches, the proposed approach shows better results. During load growth, the system draws more real and reactive power from the substation. As a result, the total network load increases, the power loss increases, and the voltage profile deteriorates. Year-wise load growth analysis and comparison without any compensation are mentioned in Table 10. In the same manner, year-wise load growth analysis and comparison after optimal compensation are mentioned in Table 11. Year-wise voltage profile comparisons without and with optimal capacitive compensation under load growth are depicted in Figure 3. The computation time of the CPU to find good-quality solution through the proposed methodology is nearly 43.02 s, including load flow. Furthermore, the simulation results of the 118-bus distribution system obtained through various approaches are indicated in Tables 12–16 after SC installation. Figure 4 indicates the performance comparison between the GABC and IPSO algorithms for the 34-bus and 118-bus distribution systems with and without considering load growth.

Figure 3:

Voltage Profile of the 34-Bus Distribution System.

(A) Before compensation considering load growth. (B) After compensation considering load growth.

Figure 4:

Convergence Characteristic Performance of GABC and IPSO.

(A) For 34 buses without load growth. (B) For 34 buses with load growth. (C) For 118 buses without load growth. (D) For 118 buses with load growth.