An intelligent approach for optimal capacitor placement problem as a reliability reinforcement strategy

Abstract

This paper proposes a new method based Cuckoo Search Algorithm (CSA) to investigate the optimal capacitor placement and sizing from the reliability point of view. In this regard, System Average Interruption Frequency Index (SAIFI) and System Average Interruption Duration Index (SAIDI) as the “customer-orientated indices” and Average Energy Not Supplied (AENS) as the “load and energy-orientated index” are utilized as the reliability objective functions. Meanwhile, to reach a compromise between the original objective functions and reliability ones, the total system cost including the MW cost of power losses as well as the capacitor investment cost is also considered as an objective function. In order to handle the proposed nonlinear constraint multi-objective optimization problem suitably, a new self adaptive modification method based on Cuckoo Search Algorithm (CSA) is proposed. The feasibility and effectiveness of the proposed method is investigated through a standard test system.

Keywords

Reliability enhancement capacitor placement and sizing self adaptive modification method Cuckoo Search Algorithm (CSA)

Abbreviations

CSA:

Cuckoo Search Algorithm

SAIFI:

System Average Interruption Frequency Index

SAIDI:

System Average Interruption Duration Index

AENS:

Average Energy Not Supplied

SAMCSA:

Self Adaptive Modified CSA

GA:

Genetic Algorithm

PSO:

Particle Swarm Optimization

Nomenclature

Symbol

Description

Decision variable

N _Cus

Total number of customers

U _i

Annual outage time of the ith component

N _b

The number of the buses

Q_{i}^{c}

Amount of reactive power compensated in ith bus

P_T, loss

Total power losses of the network

P_{ij, max}^{Line}

Maximum power flow between the buses i and j

Real integer value

V _max

Maximum value of bus voltage

λ _i

Failure rate of the ith line

N _ueq

Number of inequality constraints of the MOP

h_i (X)

Equality constraints of the MOP

μ_fi (X)

Membership function for ith objective function

Number of the solutions

⊕

Entrywise product

P _a

Portion of the nest abandoned

Gbest

Best global solution found to date

n _d

Number of variables

r _i

Random term producer for ith solution

Log

Mathematic logarithm operator

Prb _θ

probability of θth sub-modification

n _{Mod
_θ}

number of cuckoos which have chosen θth sub-modification method

λ _i

Average failure rate of the ith component

f_i (X)

The value of the ith objective function

La _i

Average load connected to ith load point in kW

K_{i}^{c}

Annual capacitor installation cost

K _p

Equivalent annual cost per unit of power loss

T _i

A constant corresponding to ith bus equal to 0/1

| P_{ij}^{Line} |

Absolute power flow between buses iand j

V _min

Minimum value of bus voltage

Failure Reduction coefficient

N _eq

Number of equality constraints of the MOP

g_i (X)

Inequality constraints of the MOP

μ _ref,I

reference membership function

N _fun

Number of objective functions

x _i,j

The jth element of the ith solution

α > 0 is the step size

Iteration number

T _F

A random integer equal to 0 or 1

Dis _i,j

Distance between X _i and X _j in Cartesian

N _t

Total number of iterations

Rep _θ

Accumulator of θth sub-modification method

Learning factor

1 Introduction

In recent years, a wide range of researches have been implemented to see different aspects of the optimal capacitor placement problem. In a technical classification, there are four main approaches to handle optimal capacitor placement problem [1]: analytical techniques [2], heuristics [3, 4], numerical programming based methods [5] and artificial techniques [6]. Nevertheless, most of the studies have been paid to the traditional objective functions such as power losses, voltage deviation, cost, etc. In [7], a new method based on fuzzy dynamic programming is proposed to minimize the power losses, voltage deviation and harmonic distortion in the system. In [8], a fuzzy-based approach is proposed with the purpose of total cost minimization including the cost of power losses, energy losses and shunt capacitors. In [9], a hybrid method based on the fuzzy theory and simulated annealing algorithm is proposed to find the optimal operating point by minimizing the active and reactive power losses and the voltage deviation as well. Similar work based on Ant Colony Search algorithm is proposed to reduce the total power losses in [10]. A reconfiguration based approach is proposed in [11] to solve the capacitor placement and reconfiguration problems from the active power loss minimization point of view. Similar work based on BF-PSO in unbalanced distribution system is examined in [12] for loss reduction. In [13], capacitor placement problem is formulated as a nonlinear mix integer optimization problem when the discrete variables are modeled in the form of sigmoid function. In [14], a discrete version of Particle Swarm Optimization method is mixed with the radial distribution power flow to search for the minimal cost of the power system in the harmonic situation. In [15], a hybrid method using fuzzy set theory and Differential Evolution (DE) and Multi Agent Particle Swarm Optimization (MAPSO) is utilized to solve the capacitor placement problem by minimizing the power losses, voltage deviation and total cost, simultaneously.

As it can be seen, all the above researches have paid their attention around similar targets as traditional objective functions such as power loss minimization, voltage profile enhancement and cost minimization. In [16], the authors propose a new method to consider the capacitor placement problem for improving the reliability of the distribution systems. In their work, shunt capacitors are supposed as redundant lines and so by compensating the reactive loads can increase the load-carrying capability of the feeders. However, the work [16] has just assessed the AENS index as an energy-orientated reliability criterion. In fact, neglecting some significant criteria such as SAIFI and SAIDI as the customer-orientated indices can be a big barrier in deep assessment of the capacitor placement problem from reliability enhancement point of view. In [17], the authors introduce a state-space method to calculate the reliability indices for compensated and uncompensated systems including two objective functions. The first objective function is the cost of reliability and investment cost while the second is the cost of power losses. Here again neglecting the customer-orientated indices can limit the research.

According to the above discussion, the main purpose of this paper is to investigate the role of the optimal capacitor placement and sizing to improve the reliability of the distribution systems. Meanwhile, the total system cost as an attractive and significant issue to the utility is also considered as an objective function. Therefore, the objective functions to be investigated are SAIFI, SAIDI, AENS and total cost including both the power losses cost and the investment cost. Also, a new self adaptive modification technique based on CSA is proposed to explore the total search space globally. In a typical multi-objective optimization problem, there can be a number of non-inferior points. In this regard, an interactive fuzzy satisfying method is proposed to find the most satisfying solution among the non-inferior solutions such that the decision maker can apply his/her preferences to the degree of satisfying each objective function. The feasibility and satisfying performance of the proposed method is tested on the 33-bus IEEE distribution test system.

2 Problem formulation

In this section, the investigated objective functions and the relevant constraints are discussed.

2.1 Objective functions

- System Average Interruption Frequency Index (SAIFI):

The SAIFI criterion evaluates the average number of interruptions that a customer may experience as follows: $f_{1} (X) = SAIFI = \frac{\sum_{i = 1}^{N_{Cus}} λ_{i} N_{i}}{\sum_{i = 1}^{N_{Cus}} N_{i}}$ (1)

- System Average Interruption Duration Index (SAIDI):

The SAIDI criterion as a popular reliability index (especially by utilities) shows the average outage duration time that a customer may experience in the system as follows: $f_{2} (X) = SAIDI = \frac{\sum_{i = 1}^{N_{Cus}} U_{i} N_{i}}{\sum_{i = 1}^{N_{Cus}} N_{i}}$ (2)

- Average Energy Not Supplied (AENS):

The AENS shows the average electrical energy which is not supplied for each customer as follows: $\begin{matrix} ENS = \sum_{i = 1}^{N_{Cus}} {La}_{i} U_{i} \\ f_{3} (X) = AENS = \frac{ENS}{\sum_{i = 1}^{N_{Cus}} N_{i}} \end{matrix}$ (3)

- Total Annual Cost:

This objective function evaluates the total system cost including both of the active power losses and the capacitor investment cost as follows [18]: $\begin{matrix} f_{4} (X) = Cost (X) = K_{p} \times P_{T, loss} + \sum_{i = 1}^{N_{b}} K_{i}^{c} \times Q_{i}^{c}; \\ i = 1, \dots, N_{b} \end{matrix}$ (4)

According to the literature, the variable K _p is supposed to 168 $ [18]. In the above equations, the variable X as the control vector is consisted of the status of each bus to be/not to be a candidate for capacitor placement and the capacitor sizes as follows: $\begin{matrix} X = [T, Q]_{1 \times (2 N_{b})} \\ T = [T_{1}, T_{2}, \dots, T_{N_{b}}] \\ Q = [Q_{1}^{c}, Q_{2}^{c}, \dots, Q_{N_{b}}^{c}] \end{matrix}$ (5)

2.2 Constrains

- Capacitor sizes:

Commercially, the sizes of the capacitor are discrete values as multiples of the smallest available capacitor size. From the economical view, a capacitor with a larger size costs lower unit price in comparison to the smaller capacitors. Therefore, the available capacitors are multiples of the smallest capacitor ( $Q_{0}^{c}$ ) as follows: $Q_{Max}^{c} = L \times Q_{0}^{c}$ (6)

- Distribution Feeder Power Capacity:

The amount of power flow in the lines should not exceed the line maximum capacity as follows: $| P_{ij}^{Line} | < P_{ij, max}^{Line}$ (7)

- Bus voltage limitation:

The voltage magnitude for each bus is limited in the pre-determined values as follows: $V_{min} \leq V_{i} \leq V_{max}$ (8)

3 Reliability enhancement via capacitor placement

Optimal capacitor placement problem has been investigated in recent years widely. However, most of the attentions have been paid to some traditional targets such as power loss minimization, total cost reduction, voltage profile enhancement, load balance improvement, etc. The high tendency to the traditional targets caused to forget that capacitor placement problem can also be a suitable and significant strategy to reinforce the reliability of the distribution system. In fact, when the optimal capacitor placement can be implemented with the aim of optimizing traditional objective functions, considering the reliability targets can increase the efficiency of the planning and so reducing the total costs.

Shead and underline cables. In the case of underline cables, a maximum operating temperature is defined which should not be exceeded otherwise the insulation destruction rate is increased. This event would increase the underline cable’s failure rate. As the electrical power flow is increased in the line, the resistive losses is also increased proportionally to the square of the line current magnitude. The life expectancy of the insulation material decreases exponentially as the operating temperature increases [19]. The high temperature can also increase the feeder failure rate in another way. At high temperature, the moisture absorption takes place more rapidly. When moisture invades extruded dielectrics such as cross-linked polyethylene or ethylene–propylene rubber, the voltage withstand capability of the cable is reduced and the probability of dielectric breakdown increases, and consequently, the failure rate of the cable is increased [16]. Similar matter can happen for the overhead lines. High currents can cause the overhead lines to sag which will reduce the ground clearance and increase the probability of occurring an electric break [20].

According to the above explanations, any strategy which can reduce the feeder current can reduce the line failure rate and improve the system reliability. Technically, the line current is composed of two parts of active and reactive components. In this regard, the capacitor placement problem can reduce the reactive current component and so the current magnitude. Reducing the current magnitude would decrease the resistive losses and finally can reduce the line failure rate. In addition, the optimal capacitor placement problem by compensating the reactive power component of the system would increase the loadability of the network and therefore can be considered as a redundant line. Consequently, the main idea to consider the capacitor placement problem is to be used as a failure rate reduction strategy. Before applying the capacitor placement to the system, any line i has an uncompensated failure rate value as λ ^init. The best failure rate which can be reached by fully compensating the reactive power component of the feeder is called λ ^best. Now, a compensation coefficient can be defined as follows [23]: $χ = \frac{I_{R - new}^{i}}{I_{R - old}^{i}}$ (9)

Therefore, the new failure rate of each line can be evaluated proportionally as follows [23]: $λ_{i}^{new} = χ (λ_{i}^{init} - λ_{i}^{best}) + λ_{i}^{best}$ (10)

According to the above equation, any planning strategy which can reduce the current magnitude of the feeders can engage to enhance the reliability of the distribution systems, effectively.

4 Solution methodology

4.1 Multi-objective optimization problem

Mathematically, the multi-objective optimization problem is defined as follows [23]: $\begin{matrix} min F = [f_{1} (X), f_{2} (X), \dots, f_{N_{fun}} (X)]^{T} \\ s . t . \\ g_{i} (X) < 0 i = 1, 2, \dots, N_{ueq} \\ h_{i} (X) = 0 i = 1, 2, \dots, N_{eq} \end{matrix}$ (11)

In a general multi-objective problem, there are several non-inferior points. In order to choose the most suitable solution among the set of non-inferior points, an interactive fuzzy satisfying method is utilized in this paper. The proposed method will let the operator to apply his/her preferences to satisfy each objective function individually. $F (X) = min_{x \in Ω} {max_{i = 1, \dots, N_{fun}} | μ_{ref, i} - μ_{f, i} (X) |}$ (12)

After evaluation of the membership function value of each objective function, the decision maker is asked to determine the achievement degree of the objective function called reference membership function (μ _ref,i). The role of the reference membership function is to determine the significance of each objective function which is in the range of (0,1). By the use of the above equation, the multi-objective optimization problem is converted to a single objective optimization problem.

4.2 Self Adaptive Modified CSA (SAMCSA)

4.2.1 Original CSA

CSA is a population based optimization algorithm which was first introduced by Yang et al. in 2009 [21]. It mimics the behavior of some cuckoos to put their eggs in the nests of other species. CSA performs mainly based on three ideas [21]: 1) Each cuckoo lays one egg at a time which leaves it in a host nest randomly; 2) The best nests with high quality of eggs will carry over to the next generations & 3) The number of host nests is fixed and a host can discover an alien egg with the probability of P _a∈ [0,1]. If the host bird discovers the alien egg, it will through it away or fly to a new nest. Each egg in the nest indicates an existing solution and the cuckoo egg indicates a new promising solution. During the optimization, CSA employs the Lévy flight to update the position of each cuckoo. A Lévy flight is a random walk in which the step-lengths are distributed according to a heavy-tailed probability distribution [22]. It is shown that Lévy flight movement can be a suitable tool in the optimization applications. Initially, n hosts are generated randomly as follows: $X_{i} = [x_{i, 1}, x_{i, 2}, \dots, x_{i, n_{d}}]; i = 1, 2, 3, n$ (13)

The position of each cuckoo is updated by the use of Lévy flight as follows: $X_{i}^{t + 1} = X_{i}^{t} + α \oplus Le ’ vy (ω)$ (14) ${Le}^{'} vy (ω) \sim u = t^{- ω}; (1 < ω \leq 3)$ (15)

The fitness value (objective function) of each new cuckoo should be evaluated now. Each cuckoo would choose a nest randomly to put her egg in it. The best nest is chosen as the optimal solution and a portion of the worst nests (P _a) are left to choose new ones.

4.2.2 Self adaptive modification method based on CSA

In this part, a new self adaptive modification method is proposed which will improve the total ability of the CSA in the optimization process to escape from the promising local optima as well as the premature convergence. The proposed self adaptive modification method consists of three sub-modification techniques which each of them can be chosen to be utilized during the optimization process.

Sub-modification Method 1: $X_{i}^{new} = X_{i}^{old} + T_{F} (Gbest - X_{i}^{old})$ (16)

Sub-modification Method 2: $X_{i}^{new} = (1 - {Dis}_{i, j}) X_{i}^{old} + {Dis}_{i, j} \times Gbest + r_{i}$ (17) ${Dis}_{i, j} = \sqrt{\sum_{h = 1}^{n_{d}} (x_{i, h} - x_{j, h})^{2}}$ (18)

Sub-modification Method 3: $α^{t + 1} = (1 / 2 N_{t})^{1 / N_{t}} α^{t}$ (19)

Initially, the probabilities of all sub-modification methods are supposed equal (Prb _θ = 0.33 & θ = 1, 2, 3). Also, an accumulator (Rep _θ) is supposed for each method which is initially zero. In each iteration, the population is sorted according to the objective function value (therefore X ₁ is the best and X _n is the worst) and a weighting factor is designated to each solution as follows: ${Wgt}_{j} = \frac{Log (n - j + 1)}{\sum_{i = 1}^{n} Log (i)}; j = 1, \dots, n$ (20)

Now, the accumulator of each sub-modification method is updated as follows: ${Rep}_{θ} = {Rep}_{θ} + \frac{{Wgt}_{l}}{n_{{Mod}_{θ}}} l = 1, \dots, n_{{Mod}_{θ}}$ (21)

Here n _{Mod
_θ} is the number of cuckoos which have chosen θth sub-modification method. Now, the probability variable of each method is updated as follows: ${Prb}_{θ} = (1 - Δ) \times {Prb}_{θ} + Δ \times \frac{{Rep}_{θ}}{Iter}, (θ = 1, 2, 3)$ (22)

Finally, the Prb _θ is normalized as follows: ${Prb}_{θ} = {Prb}_{θ} / (\sum_{θ = 1}^{3} {Prb}_{θ})$ (23)

In each iteration, the ith cuckoo chooses the suitable sub-modification method by the use of Roulette Wheel Mechanism (RWM). The process of selection by RWM is shown in Fig. 1.

5 Application of SAMCSA to the capacitor placement

The proposed SAMCSA can be applied to the capacitor placement and sizing as follows:

Step 1: Input the initial data.

Step 2: Generate the initial population.

Step 3: Run power flow to evaluate the objective functions and their membership functions. Now, the proposed interactive fuzzy method is used to convert the multi-objective problem to single objective one.

Step 4: Select Gbest from the population.

Step 5: By the use of Lévy flights a cuckoo is chosen randomly. Also, the jth nest is chosen randomly so that the cuckoo will lay her egg (new solution) in it.

Step 6: If the new solution is better than jth nest, replace it. Steps 5 to 6 should be repeated until the population is updated.

Step 7: A specific portion of the worst nests (P _a) are put aside and new ones are reconstructed.

Step 8: Apply the self adaptive modification method as described in the last section.

Step 9: update the best solution.

Step 10: Check the termination criterion. If it is satisfied finish the algorithm; otherwise return to step 5.

6 Simulation results

In this part, the 34-bus IEEE test system is utilized to see the performance of the proposed method [24]. The single line diagram of the test system is shown in Fig. 2. The test system has 1 main feeder and 4 sub-feeders to supply the total active and reactive loads of 4636.5 (kW) and 2873.5 (kVar) respectively. The nominal voltage of the system is 11 (kV).

6.1 Assumptions

As mentioned before, this paper aims to investigate the capacitor placement problem from the reliability enhancement point of view. In this regard, the following assumptions are made to evaluate the reliability of the system. It is supposed that the failure rate of the line with the greatest impedance is 0.4 (f/yr) and the line with the least impedance has the failure rate of 0.1 (f/yr). Therefore, the failure rate of all the other lines can be calculated proportionally. It is worth to note that the failure rate is proportional to the length of the line and more impedance belongs to longer lines [16]. A main breaker is supposed to be in the main feeder. Also, it is supposed that there is a sectionalizer at the beginning of each branch. Since capacitor installation will just affect the lines’ failure rates, it is supposed that all other equipments such as transformers, busbars, disconnects, breakers are completely reliable. The repair time and the switching time are supposed to be 6 hours and 0.5 hours, respectively. It is assumed that if the reactive component of the line current is compensated fully, its failure rate would be reduced to 0.85 of its initial value. The number of customers served by each busbar is shown in Table 1. Also, the available capacitor sizes and their costs are shown in Table 2. The equivalent annual cost per unit of power loss is considered 168 $/(kW-yr) [25]. The initial size of the SAMCSA population is 15 with the maximum iteration number of 100. The reason for such a termination criterion is that it was seen that there is no any progress after about 100 iterations.

6.2 Analysis

Since the proposed SAMCSA is utilized in this paper for the first time to solve the optimal capacitor placement problem, at the beginning, single objective optimization analysis is performed. In this regard, at first, the cost objective function is optimized, individually. The initial power loss and cost of the system are 221.67 (kW) and 37,241.54 ($/yr), respectively. In Table 3, the results of single cost optimization are shown. For better comparison, the results of some other well-known methods in the area are also shown in the table.

As it can be seen, the proposed SAMCSA has reduced the total cost of the system more than the other methods. Using the proposed method has caused 26.52% cost reduction which means 9,880 ($/yr). As it can be seen, the process of the cost minimization has reduced the active power losses of the system by 28.18 (kW) to the suitable value of 159.203 (kW). Therefore, the superiority of the proposed method over the other methods can be deduced from Table 3 easily. The simulation results of optimizing SAIFI, SAIDI & AENS objective functions are shown in Table 4. Since in this paper for the first time, reliability objective functions are evaluated for the test system, Genetic Algorithm (GA), PSO algorithm and the original CSA as popular optimization tools are used for comparison. Also, for better comparison, the values of all objective functions in the single optimization of each one is shown in Table 4. As it can be seen, the performance of the proposed SAMCSA in all cases of single optimization of SAIFI, SAIDI and AENS is better than the other algorithms. In Table 4, the forth column belongs to the cost objective function. From the reliability enhancement point of view, capacitor placement problem has improved the system reliability, effectively. The initial value of the SAIFI, SAIDI and AENS before capacitor placement are 2.300938 (failurer/Customer.yr), 16.563724 (hr/Customer.yr), 43.574078 (kWh/Customer.yr). However, the optimal capacitor placement (single objective cases) problem has reduced the SAIFI, SAIDI and AENS to 1.816565 (failurer/Customer.yr), 13.550773 (hr/Customer.yr) and to 37.088633 (kWh/Customer.yr), respectively. It should be noted that this reduction in reliability indices is so significant. In fact, when the main focus of capacitor placement problem in the system is just to satisfy some original objective functions such as power loss reduction, cost reduction and voltage profile enhancement, it is seen that it can improve the reliability of the system, indirectly. In other words, by considering the reliability targets in the capacitor placement problem, the efficiency of the planning can be improved.

According to the forth column of Table 4, the cost values of capacitor placement for each of the SAIFI, SAIDI and AENS objective function optimizations are shown. As it can be seen, almost all cases have resulted to the high cost values for capacitor placement problem. This event roots in the high number of capacitors which are required to compensate the reactive current component the network feeders (the basic idea of reliability enhancement). In this regard, in Table 5, the relevant capacitors evaluated by single objective optimization of each objective function are shown.

As it can be seen, in the case of cost minimization, the number of capacitors is low while it is high for reliability indices. This event shows a relative relationship between the capacitor investment costs (number of capacitors), the reliability indices and the total system cost. In order to better understanding of the problem, Fig. 3 is shown here. As it can be seen from Fig. 3, the capacitor investment cost and the customer cost (reliability associated indices) have diverse relationship with each other such that reducing one would result to incremental cost in the other. Therefore, the main idea should be to find the optimal operating point which can give a suitable balance between the reliability cost and the capacitor placement cost. This point is shown in Fig. 3 by a small circle.

At this point, the necessity of multi-objective optimization becomes evident. Therefore, by the use of the proposed interactive fuzzy satisfying method, the simulation results of multi-objective optimization are shown in Table 6. In order to have better comparison, the initial values of the objective functions are also shown in Table 6. As it can be seen, the proposed method has reduced the total cost of the system greatly when the reliability indices of the system are also improved suitably. It is worth to note that this amount of reduction in the reliability indices is valuable for both of the customers and system. Note it that the amount of the reliability enhancement achieved is an indirect result of optimal capacitor placement which is a wise strategy under the philosophy that it is better to consider a few significant reliability indices than neglecting all.

In order to better understanding the significance of the problem, Fig. 4 is utilized. Figure 4 shows Composite Customer Damage Function (CCDF) [28] evaluating the cost of customer interruption per time duration. Here, we limit our discussion on the case of SAIDI.

As it can be seen from Table 6, after capacitor placement, the value of SAIDI is reduced about 80 minutes (16.563724-15.243778 = 1.319946 (hr) ≈ 79.19676 (min)). According to Fig. 4, if we suppose it as 1 dollar off per customer (which it is of course more!), multiplying it with the total number of customers which are served by the network, it becomes a valuable amount. Consequently, it is completely economical to consider the reliability indices in the planning. The relevant optimal places and sizes of the capacitors are shown in Table 7. As it can be seen, the number of capacitors used is less than those of the single objective optimization for reliability indices.

In order to see the effect of capacitor placement on the voltage profile, Fig. 5 shows the voltage magnitude of the system before and after compensation. As it can be seen, the voltage profile is also improved, sufficiently.

7 Conclusion

This paper assessed the role of the optimal capacitor placement problem to reinforce the reliability of the distribution systems. In this regard, a suitable method based on failure reduction technique was utilized to see the positive effect of the capacitor placement on some of the most significant reliability indices such as SAIFI, SAIDI and AENS. Meanwhile, since the total MW cost is an attractive issue to the utilities, the total system cost including the active power losses and the capacitor investment cost was also considered as an objective function. The problem was then formulated in a multi-objective framework based on the interactive fuzzy satisfying method. Therefore, the investigated problem is a type of complex, discreet, nonlinear, multi-objective optimization problem which requires a powerful too to escape from local optima as well as premature convergence. In this regard, a novel self adaptive modification method based on CSA was proposed. The simulation results showed that the utilization of the optimal capacitor placement with the target of reliability enhancement can be a powerful tool to reduce both of the customer and network reliability costs. In addition, it was seen that a suitable balance between the traditional objective functions with the reliability ones is required to improve the efficiency of the system notably. From the optimization point of view, the proposed SAMCSA has much better performance and ability than some of the most popular evolutionary algorithms such as PSO, GA and original CSA.

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Evaluation of interrupted energy assessment rates in distribution systems

IEEE Trans Power Deliv 6 4 1876 1882