Optimal Placement of Switching Devices in Distribution Networks Using Multi-objective Genetic Algorithm NSGAІІ

Abstract: The main reason of customer unavailability is the events occurring in distribution networks. Hence, one of the most important issues should be considered is improving reliability of distribution networks. Installation of switching devices like sectionalizers along the network, positively affects the system reliability, but, of course, it involves utility cost. The main aim of this paper is to determine optimal number and place of sectionalizers in distribution networks. By considering non-linear and non-differentiable nature of problem, using common methods to solve the problem is not possible. In this paper the optimization problem is solved by multi-objective genetic algorithm NSGAII to minimize installation cost of sectionalizers, simultaneously improving reliability indices, i.e. System Average Interruption Duration Index (SAIDI), Expected customer interruption cost (ECOST). Effectiveness of the proposed algorithm is evaluated on a 50 feeder distribution network and simulation results are assessed. Keywords: Switching devices placement, distribution networks, optimizing reliability of system, multi-objective optimization, Genetic algorithm.

1. Introduction Electric power industry restructuring has made

distribution utilities to consider reliability improvement of systems. Reliability can be improved by decreasing number of outages, decreasing outages areas and also increasing restoration speed of the system. When a fault occurs in the system, first it must be located. Then, faulted area is separated from rest of the system by applying switches and then other loads are energized. After clearing fault, switches are closed and system comes back to its normal situation [1]. The number and place of switches in the network are the most important factors for precipitating this procedure.

The selection of the number and location of the switches depends on many factors, such as reliability considerations, type of the customers connected, load variations, maintenance and installation costs. A cost- benefit analysis should be performed which includes all of the above factors.

Many articles related to optimal placement of switching devices for distribution networks are published so far. References [2] and [3] used simulated annealing and Ant Colony algorithms to find optimal placement of

switches, respectively. Reference [4] presents a cost minimization technique which includes investment, maintenance and reliability costs. Reference [5] proposes a fuzzy multi-objective approach for optimal placement of sectionalizing switches in the presence of distributed generation sources. The primary objective is reliability improvement with consideration of economic aspects. Thus, the objectives are defined as reliability improvement and minimizing cost of the sectionalizing switches. A fuzzy membership function is defined for each term in the objective function according to relevant condition and final objective function is minimized using Ant Colony Optimization algorithm. In [6] a new particles aggregation algorithm is used for optimal placement of switches. Simultaneous allocation of DGs and switches in distribution networks is done using Genetic algorithm in [7].

Most of the approaches presented above are the optimization of single objective function. Treating the problem as a multi-objective optimization problem makes the problem to be solved easier than the usual single objective optimization procedure [8]. Three objective functions are used in this paper for finding optimal switch placement. These functions are System Average Interruption Duration Index (SAIDI), System expected outage cost to customers due to supply outages (ECOST) and Switch Installation Cost (IC). Multi-objective Genetic Algorithm NSGAІІ is used in order to simultaneously minimize these functions. This algorithm is one the fastest and newest multi-objective genetic algorithms based on the non-dominated sorting [9].

In the remainder of the paper, problem modeling and mathematics expression of objective functions are described in Section 2. Thereafter, in Section 3, the proposed NSGA-II algorithm is described in details. Section 4 presents simulation results of multi-objective genetic algorithm NSGAІІ, applied to a 50 feeder distribution network. Finally, outline of the conclusions is included in Section 5.

2. Problem Modeling Practical optimization problems, especially

the engineering design optimization problems, have a

Mohammad Reza Mazidi1, Mohsen Aghazadeh2, Yousef Ahmadi Teshnizi3, Erfan Mohagheghi4

1Tehran University, 2Iran University of Science and Technology, 3Islamic Azad University of Shahrekord, 4University of Greenwich

978-1-4673-5634-3/13/$31.00 ©2013 IEEE

multi-objective nature. In some cases these multiple objectives have conflict in measuring scales and behaviors. In general, a multi-objective optimization problem can be summarized as Eq. (1):

( ) ( ) ( ) ( )( )

( )

1 2

1 2

min , ,...,

, ,...,

Tk

n

F X f x f x f x

x SX x x x

=

∈=

(1)

Where, f1(x), f2(x)…, fk(x) are objective functions. x1, x2, …, xn are optimization parameters and S ∈Rn is state space of the problem.

Classic methods of finding a set of optimal solutions in multi-objective problems, based on deciding before searching idea, merge multiple objectives by applying parameters and coefficients in the form of one objective. Another method is searching solution space of the problem to find Pareto- optimal solutions, which is used in this paper. Solution set of multi-objective optimization problem includes all non-dominant solutions, known as the Pareto-optimal set [9]. 2.1 Objective Functions

In This paper a multi-objective optimization algorithm is used to reach a maximum reliability and simultaneously minimizing costs. To calculate the reliability of distribution networks different indices are used [10]. Reliability indices are classified in two groups: Load Indices and System Indices. Using load indices, frequency and average duration of outages for specific customers are calculated. But system indices determine reliability of the whole system. Common indices to calculate reliability of distribution networks include: System Average Interruption Frequency Index (SAIFI), System Average Interruption Duration Index (SAIDI), Expected Energy Not Supplied (EENS) and Expected customer interruption cost (ECOST). In this paper two reliability indices, i.e. SAIDI and ECOST in addition to switch installation cost are selected as objective functions. Mathematical equations describing aforementioned objective functions are formulated below. 2.1.1 System Average Interruption Duration Index (SAIDI):

This indice expresses average interruption duration of all customers inside the network. Eq. (2) is used to calculate this indice:

( ) 1 11

1

n m

is is ii s

n

ii

r Nf X

N

λ= =

=

=∑∑

∑ (2)

Where isλ = average failure rate of feeder i due to contingency

on line s, isr = average outage time of contingency i occurring on

line s, iN = number of customers on feeder i,

n = number of feeders, m = number of lines. 2.1.2 Expected customer interruption cost (ECOST):

This indice is system expected outage cost to customers inside the network due to supply outages and can be calculated by Eq. (3):

( ) ( )21 1

n m

si is s si s

f X L C r λ= =

=∑∑ (3)

Where siL = cut-off load at feeder i due to contingency on line s,

sr = average outage time due to contingency on line s,

sλ = average failure rate of line s,

( )is sC r is outage cost ( $ / kW ) of customers in feeder i due to contingency on line s.

It must be noted that ( )is sC r differs for customers based on the importance of their loads. Reference [4]'s data are used in this paper to calculate ( )is sC r values. 2.1.3 Switch installation cost ( )IC :

It is obvious that by increasing the number of installed switches in the network, its reliability will increase. But distribution companies are allowed to invest up to a certain limit for switch installation. In fact this is the same cost-benefit problem for calculating reliability of distribution networks. Hence, switch installation cost in distribution network is defined in Eq. (4) as below:

( )3f X SC n= × (4) In the above equation, SC is the installation cost of

each switch and n is the number of installed switches in the network. In this paper, installation cost of each switch is considered 2000$[5].

3. Multi-objective Genetic Algorithm NSGAІІ As mentioned before, a special optimality concept is

needed to solve multi-objective optimization problem. Considering Pareto concept, optimality criterion in multi-objective optimization problem can be defined. for two 1X and 2X vectors, vector 1X dominates vector 2X if and only if two conditions are satisfied; first, 1X not to be worse than 2X in all objectives functions and second,

1X must be utterly better than 2X in at least one of objectives. The above concept is described as below in mathematics:

{ } ( ) ( )( ){ } ( ) ( )( )

1 21 2

1 2

1, 2,..., :

1, 2,..., :i i

i i

i n f X f XX X

i n f X f X

⎧ ∀ ∈ ≤⎪< ⇔ ⎨∃ ∈ <⎪⎩

(5)

Decision vector fX X∈ is called non-dominant in respect to set fA X⊆ , if and only if:

:a A X a∃ ∈ < (6) X is the optimal of Pareto if and only if it is non-

dominant in respect to fX . Hence, decision vector X is

considered an optimal one; from this point of view it’s not possible to improve any of objectives without deteriorating another objective. Such a solution is called Pareto-optimal solution. Bold points on dashed line in Fig. 1 are Pareto-optimal solutions.

Figure 1: Pareto optimal solutions

These points are indifferent to each other. The main

difference between single-objective and multi-objective problems is determined here. Multi-objective problems are not limited to just one single optimal solution, even there are a set of optimal solutions in them. None of these solutions is superior to other ones, unless the decision makers’ preferences are defined. All Pareto-optimal solution sets in a multi-objective problem and corresponding objective vectors are called Pareto-optimal set and Pareto-optimal border, respectively. One of the fastest and strongest optimization algorithms that has lower complexity than the other methods and uses the principle of non-dominance and crowding distance calculation to give Pareto-optimal points, is multi-objective genetic algorithm, NSGAІІ. In this algorithm, simultaneously elitism and scattering is taken into consideration. In this method first a population of children is generated using the population of parents with the same size of N. These two populations are merged and generate a new population of 2N members. This new population is sorted using the non-dominated sorting and finally, the new obtained population has the best N members. Each sorted population is called a front. The concept of domination of one point on another point is expressed in Eq. (7). If such a relationship is established, means that point 'a' overcomes point 'b'. fi and fj are the i-th and j-th objective functions, respectively.

(7)

Crowding distance is another criterion used in a front. To calculate this criterion, first for each objective function K, results are sorted. A value of infinite distance is allocated for the points having the maximum and minimum values of this objective function. For other points, i=2, 3…, (n-1), Eq. (8) is used. Then, crowding

distances are gathered around each point (Eq. (9)). In this equation kZ is the K-th objective function, min

kZ and maxkZ are the maximum and minimum of objective

functions.

(8)

( ) ( )kk

cd x cd x=∑ (9)

4. Simulation results

Fig. 3 shows a single-line diagram of this network which has 50 feeders and 50 lines. The length of each line, the number of consumers connected to each feeder and the average amount of each household, commercial and small industrial load in each feeder is given in Table 1. Average failure rate of lines, λ has been considered 0.08 per kilometer. Average service time per line is equal to 2 hours and average time needed to separate faulted area from the rest of the network using available switches, has been considered 0.5 hour.

[ ]( ) [ ]( )[ ]( )1, 1,

,max min

k ki k i kk i k

k k

Z x Z xcd x

Z Z+ −−

=−

Fig.2: Optimal Pareto solutions

{ } ( ) ( ){ } ( ) ( )

1,..., :

1,..., ;i i

i i

t n f a f b

i n f a f b

∀ ∈ ≥

∃ ∈ >

TABLE I: Required data associated with the test feeder

The proposed algorithm has been implemented in MATLAB software. Distribution network in [11] has been used to perform simulations. Fig. 4 shows Optimal Pareto solutions obtained from multi-objective genetic algorithm NSGAІІ.

Length axis is IC index in terms of dollar. Width axis is ECOST in terms of dollar and height axis is SAIDI in terms of outage hours for consumers for a year. These solutions are in fact a set of best solutions for optimization problem that can simultaneously minimize three objective functions. Due to space limitations, only some of non-dominant solutions are listed in Table 2.

According to needs and restrictions, distribution companies can select the appropriate solution by considering obtained values for each objective functions. But there are also some methods to select an appropriate solution among non-dominant solutions. As an example one can use Min-Max method [12]. Using the Min–Max method to select the final solution, the values of the objective functions normalized using Eq. (10), also are shown in Table 2. ( )

max k

max min

max k

max min

max k

max min

SAIDI , ECOST , IC

SAIDI -SAIDI

SAIDI -SAIDIECOST -ECOST

ECOST -ECOSTIC -IC

IC -IC

norm norm norm =

(10)

Fig. 3: A 50-bus distribution feeder

Fig. 4: Optimal Pareto solutions

Next, the result due to the application of the “Min” operator is shown in column “Max–Min”, and then using the “Max” operator to select the best multi-objective solution. According to "Min-Max" column values, the best solution is the one that has the greatest value. The best solution obtained from Min-Max method is specified in Table 2. Considering this solution, 17 switches should be installed on the lines shown in the Table 3.

Objective function values before and after the installation of switches in the network are shown in Table 4. By comparing the reliability indices it is shown that SAIDI and ECOST indices decrease about 69% and 63%, respectively.

Results show that installation of switches in obtained places increases the reliability of the network and as a result damages to consumers caused by outages fall-off.

5. Conclusions

In this paper a multi-objective optimization algorithm is used for optimal placement of switches. Objective functions used for optimization are SAIDI, ECOST and IC. These indices definitions have been noted in the paper. Multi-objective genetic algorithm, NSGAІІ, is used to optimize objective functions and simultaneously to find optimal places for switches applied in the system. Simulations have been performed on a 50 feeder distribution network. Optimal values of objective functions, i.e. a set of non-dominant solutions have been obtained. According to needs and restrictions, distribution companies can select the appropriate solution by considering obtained values for each objective functions. Also a method called Min-Max method has been introduced to find the best solution. Results show that installation of switches in obtained places increases the reliability of the network and as a result damages to consumers caused by outages fall-off.

References

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TABLE II: best solution obtained from Min-Max method

SAIDI ECOST

IC SAIDInorm

ECOSTnorm

ICnorm Min-Max

0.2062 41186 58037 0.9996 0.8438 0.3543 0.3543

0.2180 39034 41760 0.9434 0.9282 0.7235 0.7235

0.2072 37205 71378 0.9946 1 0.0516 0.0516

0.2068 37555 73262 0.9966 0.9863 0.0088 0.0088

0.2132 38070 49362 0.9663 0.9660 0.5511 0.5511

0.2164 37961 43825 0.9509 0.9707 0.6767 0.6767

0.2120 41212 41598 0.9719 0.8428 0.7227 0.7227

0.3810 59467 29583 0.1662 0.1265 0.9998 0.1265

0.2098 38309 47735 0.9823 0.9567 0.5880 0.5880

0.2487 42518 29764 0.7969 0.7915 0.9957 0.7915

0.2085 37322 71377 0.9883 0.9954 0.0516 0.0516

0.2062 41835 57739 0.9996 0.8183 0.3610 0.3610

0.2079 37902 61681 0.9912 0.9726 0.2716 0.2716

0.2068 38220 49699 0.9966 0.9602 0.5434 0.5434

0.2068 44782 41764 0.9964 0.7027 0.7234 0.7027

0.2157 38745 41785 0.9543 0.9396 0.7230 0.7230

0.2342 40571 35813 0.8663 0.8697 0.8585 0.8585

0.2107 40467 42032 0.9781 0.8720 0.7174 0.7174

TABLE IX: best solution obtained from Min-Max method

Before Installing Switches

After Installing Switches

SAIDI 0.7760 0.2342 ECOST 109380 40571

TABLE III: best solution obtained from Min-Max method

Selected Lines for Installing Switches

24-25, 5-6, 22-23, 5-21, 11-12, 2-3, 18-19, 14-15, 7-8, 28-29, 31-32, 40-41, 39-40, 36-37, 35-36, 13-34, 45-46, 43-44,

43-47, 41-42 Number of Installed

Switches 17

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