A novel power system reconfiguration for a distribution system with minimum load balancing index using bacterial foraging optimization algorithm

Abstract

In this paper, the objective of minimum load balancing index (LBI) for the 16-bus distribution system is achieved using bacterial foraging optimization algorithm (BFOA). The feeder reconfiguration problem is formulated as a non-linear optimization problem and the optimal solution is obtained using BFOA. With the proposed reconfiguration method, the radial structure of the distribution system is retained and the burden on the optimization technique is reduced. Test results are presented for the 16-bus sample network, the proposed reconfiguration method has effectively decreased the LBI, and the BFOA technique is efficient in searching for the optimal solution.

Keywords

bacterial foraging optimization algorithm (BFOA) / distribution system / network reconfiguration / load balancing index (LBI) / radial network

Introduction

Fault location and isolation of a faulty section in a distribution network in real time is made possible with the help of monitoring and control functions in an automated distribution system. But, service restoration to the non-faulted-out-of-service area in real time poses a real challenge. Now-a-days, fast restoration strategies are essential to reduce the inconvenience to the user during such interruptions. There has been a considerable interest in the recent past for service restoration through feeder reconfiguration.

Distribution network reconfiguration for load balancing is illustrated in this paper. The problem is formulated as a non-linear optimization problem where LBI is minimized subject to security and operational constraints. Optimization problems in the steady state analysis of power systems aim at minimizing or maximizing an objective function. Traditional methods like Gauss-Siedel method, Newton Raphson method, and Lambda iteration method are used to solve linear, continuous, and differential objective functions. To solve non-linear objective functions, evolutionary algorithms (EAs) came into existence. The EAs are random, stochastic, and robust algorithms used for optimization of nonlinear problems. One of these EAs is bacterial foraging optimization algorithm (BFOA).

This paper discusses the problem of calculating LBI [1,2] in distribution feeders for feeder reconfiguration through BFOA approach. The 16-bus sample network [3,4] has been adopted for solving LBI which is divided into subsystems of radial feeders equipped by numbers of sectionalising switches and tie switches. The power system can be operated more reliably by changing the configuration of the network. A number of algorithms including mathematical programming and artificial intelligent methods have been proposed to reconfigure distribution feeders with the objective of calculating LBI of a 3-feeder network [5] while avoiding transformer and feeder overloads and inadequate voltages. LBI values for the various configurations are obtained and finally the network having least LBI value is selected for optimal solution.

Using BFOA the LBI are reduced substantially. In this study, feeder switching status and consequently, network configuration, are adjusted to keep LBI at a minimum. BFOA has been widely accepted as a global optimization algorithm of current interest for distributed optimization and control.

To achieve non-linear objective functions, EAs came into existence. In artificial intelligence, an evolutionary algorithm (EA) is a subset of evolutionary computation, a generic population based metaheuristic optimization algorithm. An EA uses some mechanisms inspired by biological evolution: reproduction, mutation, recombination, and selection. Candidate solution to the optimization problem play the role of individuals in a population, and the fitness function determines the environment within which the solutions live. Evolution of the population then takes place after the repeated application of the above operators. Artificial evolution (AE) describes a process involving individual EAs; EAs are individual components that participate in an AE. Evolutionary computation uses iterative progress, such as growth or development in a population. This population is then selected in a guided random search using parallel processing to achieve the desired end. Such processes are often inspired by biological mechanisms of evolution. As evolution can produce highly optimized processes and networks, it has many applications in computer science. Artificial life started with the work of Nils Aall Barricelli in the 1960s, and was extended by Alex Fraser, who published a series of papers on simulation of artificial selection. Artificial evolution became a widely recognized optimization method as a result of the work of Ingo Rechenberg in the 1960s and early 1970s, which used evolution strategies to solve complex engineering problems. Genetic algorithms in particular became popular through the writing of John Holland. As academic interest grew, dramatic increases in the power of computers allowed practical applications, including the automatic evolution of computer programs. EAs are now used to solve multi-dimensional problems more efficiently than software produced by human designers, and to optimize the design of systems. EAs are random, stochastic, and robust algorithms used for optimization of nonlinear problems.

Problem formulation

This paper is aimed at finding a way to keep load balancing through feeder reconfiguration so that stability and reliability of the distribution network could be enhanced. The system is reconfigured in such a way that all the lines should carry an optimum value of current. The objective of this optimization problem can be expressed by the load balancing index (LBI) as

(1)\mathrm{L}\mathrm{B}\mathrm{I}={\displaystyle \sum _{i=\mathrm{1}}^N{L}_i}{\left({\displaystyle \frac{\left|I_i\right|}{I_i^{\mathrm{R}}}}\right)}^{\mathrm{2}},

where N is the total number of branches in the system, I_i is the complex current flow in each branch, I_i^{\mathrm{R}} is the current rating of bus i, and L_i is the length of each branch, which is subject to:

1) No feeder section can be left out of service;

2) Radial network structure must be retained always;

3) Bus voltage magnitude can’t exceed upper and lower limits.

In this study feeder switches status, and consequently, network configuration, are adjusted to keep load balancing in the area. I_i is the current flowing through sectionalizing switches in the simplified model.

A distribution system network carries electricity from theβtransmission systemβand delivers it to consumers. Distribution network is divided into subsystems of radial feeders equipped by a number of sectionalizing switches and tie switches.

Network reconfiguration is the process of changing the topology of distribution systems by altering the open/closed status of switches.

Distribution networks are built as interconnected meshed networks, while in operation they are arranged into a radial tree structure. This means that distribution systems are divided into subsystems of radial feeders, which contain a number of normally-closed switches (sectionalizing-switches) and a number of normally open switches (tie-switches). Many recent researches on network reconfiguration have focused on the minimum LBI configuration problem.

Of the several operational schemes in electrical distribution systems, distribution feeder configuration can minimize LBI, real power loss, improve voltage profile, and relieve overloads in the network. The power system can be operated more reliably by changing the configuration of the network by closing and opening the sectionalizing and tie switches. The load on the feeders of a distribution system is generally a combination of industrial, commercial, residential and lighting loads. Substation transformers and feeders undergo peak loading at different times of the day, and therefore, the distribution system becomes heavily loaded at certain times of the day and lightly loaded at other times. This is detrimental to the operating conditions of the network and leads to unbalancing of loads, high real losses and poor voltage profile. Therefore one of the feasible measures taken to reduce the system LBI and improve voltage profile is to reconfigure the distribution network.

Steps of bacterial foraging algorithm

Bio-mimicry of foraging activities of bacteria such as E.coli etc is creating this BFOA algorithm. If it is in a neutral medium (one not too rich in nutrients), the E.coli alternate between running (move straight) and tumbling (change direction). If swimming up a nutrient gradient, swim longer and seek increasingly favorable environments; if swimming down a nutrient gradient, search and avoid unfavorable environments. It is subdivided into chemotaxis (swimming and tumbling), reproduction and elimination and dispersal. The parameters initialized for the run are number of chemotactic steps (N_c), number of reproduction steps (N_re), number of elimination and dispersal steps (N_ed), dispersal probability (P_ed), and number of bacteria (N) and swim length (N_s).

There are three steps in bacterial foraging algorithm after the swimming or tumbling searching strategies, which are chemotaxis, reproduction, and elimination dispersal.

Chemotaxis

This process simulates the movement of an E.coli cell through swimming and tumbling via flagella. Biologically an E.coli bacterium can move in two different ways. It can swim for a period of time in the same direction or it may tumble, and alternate between these two modes of operation for the entire lifetime. Suppose θⁱ(j, k, l) represents i-th bacterium at j-th chemotactic, k-th reproductive and l-th elimination-dispersal step. C(i) is the size of the step taken in the random direction specified by the tumble (run length unit). Then in computational chemotaxis [7] the movement of the bacterium may be represented by

(2){\theta }^i(j+\mathrm{1},k,l)={\theta }^i(j,k,l)+C(i){\displaystyle \frac{\mathit{\Delta }(i)}{\sqrt{{\mathit{\Delta }}^{\mathrm{T}}(i)\mathit{\Delta }(i)}}},

where \mathit{\Delta } indicates a vector in the random direction whose elements lies in [-1, 1].

An interesting group behavior has been observed for several motile species of bacteria including E.coli and S. typhimurium, where intricate and stable spatio-temporal patterns (swarms) are formed in semisolid nutrient medium. A group of E.coli cells [6-8] arrange themselves in a traveling ring by moving up the nutrient gradient when placed amidst a semisolid matrix with a single nutrient chemo-effecter. The cells, when stimulated by a high level of succinate, release an attractant aspertate, which helps them to aggregate into groups and thus move as concentric patterns of swarms with high bacterial density. The cell-to-cell signaling in E. coli swarm may be represented by

(3)\begin{array}{l}{J}_{\mathrm{c}\mathrm{c}}(\theta ,P(j,k,l))={\displaystyle \sum _{i=\mathrm{1}}^S{J}_{\mathrm{c}\mathrm{c}}(\theta ,{\theta }^i(j,k,l))}=\left[-d_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\tan \mathrm{t}}\exp \left(-w_{\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}{\displaystyle \sum _{m=\mathrm{1}}^p({\theta }_m-{\theta }^i)}\right)\right]\hfill \\ \hfill \\ \hfill \end{array}+\left[-d_{\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}\exp \left(-w_{\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{t}}{\displaystyle \sum _{m=\mathrm{1}}^p({\theta }_m-{\theta }^i)}\right)\right],

d_attractant, w_attractant, d_repellant, w_repellant are the coefficients.

Reproduction

The least healthy bacteria eventually die while each of the healthier bacteria (those yielding lower value of the objective function) asexually split into two bacteria, which are then placed in the same location. This keeps the swarm size constant.

Elimination and dispersal

Gradual or sudden changes in the local environment where a bacterium population lives may occur due to various reasons, e.g., a significant local rise of temperature may kill a group of bacteria that are currently in a region with a high concentration of nutrient gradients. Events can take place in such a fashion that all the bacteria in a region are killed or a group is dispersed into a new location. To simulate this phenomenon in BFOA some bacteria are liquidated at random with a very small probability while the new replacements are randomly initialized over the search space.

Implementation of bacterial foraging algorithm in distribution system

After the load flow solution by Newton Raphson method, the pre and post fault currents are found. LBI value is calculated using the load flow values with different length values in Step 1 (Initialization process). Now by elimination of BFOA values of LBI that is greater than their other values i.e. S/2 values of LBI will be eliminated by this loop in Step 2 (elimination and dispersal). In Step 3, S/2 values of LBI having least values will be reproduced, keeping the net strength of LBI bacteria constant (reproduction). Finally, a lower LBI is found and the lower LBI will balance the load.

The steps of the bacterial foraging algorithm are as follows:

Step 1　Initialization of the following parameters:

P: dimension of the search space-Number of parameters used in the objective function and every dimension signifies a parameter (P dimension of search space = 3):

\begin{array}{l}{P}_{\mathrm{1}}:L_i;\\ P_{\mathrm{2}}:I_i;\\ P_{\mathrm{3}}:I_i^R.\end{array}

S stands for the number of bacteria in the population (20); N_c, the number of chemotactic steps (10); N_s, the length of a swim when it is on a gradient (4); N_re, the number of reproduction steps (4); N_ed, the number of elimination/dispersal events (2); P_ed, the probability that each bacterium will be eliminated / dispersed (0.75); C(i, j)|_j=1, the initial run-length unit; C(N_c), the run-length unit at the end of the chemotactic steps (j = N_c); and θⁱ, the initial random location of each bacterium.

After the load flow solution by Newton Raphson method, the pre and post fault currents have been calculated and substituted in the objective function equation. LBI value is calculated using the load flow values with different length values.

Step 2　Now by elimination of BFOA values of LBI that are greater than their other values, i.e., S/2 values of LBI will be eliminated by this loop,

Elimination/dispersal loop, l = l + 1.

Step 3　 S/2 values of LBI having least values will be reproduced, keeping the net strength of LBI bacteria constant.

Reproduction loop, k = k + 1.

Step 4　Chemotaxis loop, j = j + 1.

For i = 1, 2, …, S, execute the chemotactic step for each bacterium as follows:

Evaluate the objective function used in Eq. (1), which is equaled to J(i, j, k, l).

Let J_last = J(i, j, k, l) so that a lower LBI could be found. A lower LBI will balance the load and minimize losses in the system.

Tumble　Generate a random vector BoldItalic(i) and BoldItalic_m(i), m =1, 2, ..., p is a random number in the range of [-1, 1].

Compute BoldItalic(i):

(4)\varphi (i)={\displaystyle \frac{\mathit{\Delta }(i)}{\sqrt{{\mathit{\Delta }}^{\mathrm{T}}(i)\mathit{\Delta }(i)}}}.

Move using

(5){\mathit{\theta }}^i(j+\mathrm{1},k,l)={\mathit{\theta }}^i(j,k,l)+C(i){\displaystyle \frac{\mathit{\Delta }(i)}{{\mathit{\Delta }}^T(i)\mathit{\Delta }(i)}}.

Compute J(i, j + 1, k, l) and J_cc(θ, P(j + 1, k, l)), then use these two parameters to find the new J(i, j + 1, k, l).

Swim　Let m=0 (counter for swim length).

While m<N_s (no climbing down too long).

Let m = m + 1.

If J(i, j + 1, k, l )<J_last Let J_last = J(i, j + 1, k, l ) then take another step in the same direction and compute the new J(i, j + 1, k, l ). Hence the bacteria take one step and fetch a new LBI.

Go to the next bacterium (i =i + 1 if i = S).

Update the run-length unit using

(6)C(i,j+\mathrm{1})={\displaystyle \frac{C(i,j)-C(N_{\mathrm{c}})}{N_{\mathrm{c}}+C(N_{\mathrm{c}})}}(N_{\mathrm{c}}-j).

Compute the best (lower) cost obtained (J_best ( j) ).

Compute the difference in cost achieved in the current chemotactic step (Diff( j)).

(7)Diff(j)=J_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}(j)-J_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}(j-\mathrm{1}).

If j<N_c/n (e.g. n=2).

If |Diff (j)-Diff (j-h)|<Ɛ, h=1, 2, ..., h_m, h_m<N_c/n. j = N_c (i.e., end chemotactic operations).

Step 5

If j<N_c go to Step 4 (j = j + 1).

Step 6　Reproduction:

For the given k and l, evaluate the health of each bacterium i as follows:

(8)J{(i)}_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{h}}={\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{c}}+\mathrm{1}}J(i,j,k,l)}.

The health of the bacterium i measures how many nutrient it got over its lifetime. Sort bacteria according to their health J{(i)}_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{h}} in ascending order. Hence all the bacteria are arranged in such a way that the best bacteria have the least LBI which increases in the coming bacteria.

The bacteria with the highest J{(i)}_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{h}} values, computed by S_r=S/2 die while the other S_r with the lowest values split and take the same location of their parents.

Step 7

If k<N_re, go to Step 3 (k = k + 1).

Step 8　Elimination/dispersal:

With probability P_ed, randomly eliminate and dispersal each Bacterium i, keeping the size of the population constant. So the dispersed bacterium gives brand new values to the parameters of the objective function (LBI).

Step 9　If l<N_ed, go to Step 2 (l = l + 1), otherwise end. These are the steps in BFOA. Bacterial foraging algorithm is robust, stochastic and is one of the most efficient EA.

16-bus distribution system

Table 1 gives the system data of a 16 bus distribution system illustrated in Fig. 1.

The problem of calculating LBI in distribution feeders for feeder reconfiguration by using BFOA approach has been considered here as a main objective. The 16-bus sample network is adopted for solving LBI and it is divided into subsystems of radial feeders equipped by numbers of sectionalising switches and tie switches. The power system can be operated more reliably by changing the configuration of the network. A number of algorithms including mathematical programming and artificial intelligent methods have been proposed to reconfigure distribution feeders with the objective of calculating LBI of a 3-feeder network while avoiding transformer and feeder overloads and inadequate voltages. LBI values for the various configurations are obtained and finally the network having least LBI value is selected for optimal solution.

Due to their inherent characteristics, BFO algorithms are well suited for combinatorial optimization problem [9]. The results listed in Tables 2-4 for load balancing demonstrate the effectiveness of BFO algorithm in solving combinatorial optimization problems. The effectiveness of this method can be further demonstrated by applying this method to larger systems. The base value of the 16-bus test system is 23 kV and 100 MVA.

Test results and comparison

For an assumed fault at bus 11, Table 2 presents the test results obtained by using BFOA and the results are compared with those from Refs. [1] and [3].

From Table 2, the reconfigured network with the least LBI value is considered as optimal solution and its switch statuses are revealed in Table 3.

The results obtained from Refs. [1] and [3] based on feeder reconfiguration are used for comparison. Table 4 shows the LBI values and the locations for the switches which are opened and closed. The objective of minimizing LBI, for 16-bus distribution system is solved using BFOA. The feeder reconfiguration problem is formulated as a non-linear optimization problem and the optimal solution is obtained using BFOA. With the proposed reconfiguration method, the radial structure of the distribution system is retained and the burden on the optimization technique is also reduced. Test results are presented for 16-bus sample network, the proposed reconfiguration method has effectively decreased the LBI. The BFOA technique is efficient in searching for the optimal solution. Load flow values are obtained using N-R method. Then the various LBI values are obtained for 16-bus distribution system with different configurations by substituting the load flow values in LBI equation. Finding the minimum LBI with feasible configuration helps to keep the load at a balanced state.

Conclusion

A BFOF for restoration and reconfiguration is adopted to keep the load in balance. A novel model is used to simplify the distribution network. The problem has been formulated as a non-linear optimization problem with an objective function of minimizing LBI subject to security constraints. Test results have shown that, using BFOA method, the feeder reconfiguration problem can be solved efficiently and the LBI is decreased effectively by reconfiguration. The fast and effective convergence of this approach proves that it is a highly suitable technique to use in service restoration procedures of distribution automation system.

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