An intelligent method to reduce losses in distribution networks using self-adaptive harmony search algorithm

Abstract

Loss reduction planning is considered one of the most significant and sensitive distribution designing system. The planning process to reduce losses along with distributed generation sources results in a non-convex, non-linear optimization problem, mixed with integer variables, which requires complicated calculations. To solve this problem, novel optimization algorithms such as harmony Search (HS) algorithm can be used for different optimization problems. This paper presents an effective intelligent optimization method to solve the network reconfiguration problem in the presence of distributed generation (DG) with an objective of minimizing real power loss and improving voltage profile in distribution system using new set of harmony search algorithms, called self-adaptive. To verify the efficiency of the algorithm the results have been analyzed in the IEEE standard system. The results show the strength, appropriate efficiency and superior performance of this algorithm in comparison to the original harmony Search algorithm and other optimization algorithms as well.

Keywords

Loss reduction reconfiguration self-adaptive harmony search algorithm distributed generation optimization

Nomenclature

P _k

Real power flowing out of bus k

Q _k

Reactive power flowing out of bus k

P _Lk+1

Real load power at bus k + 1

Q _Lk+1

Reactive load power at bus k + 1

R _k

Resistance of the line section between buses k and k + 1

X _k

Reactance of the line section between buses k and k + 1

Y _k

Shunt admittance at any bus k

V _k

Voltage at bus k

Distance from source to the DG location in km

Total length of the feeder from source to bus k in km

P _G

Real power supplied by DG

Q _G

Reactive power supplied by DG

PAR (t)

Pitch Adjusting Rate in Generation number

PAR _min

Minimum Pitch Adjusting Rate

PAR _max

Maximum Pitch Adjusting Rate

BW (t)

Distance Bandwidth in Generation number

BW _min

Minimum Distance Bandwidth

BW _max

Maximum Distance Bandwidth

Generation number

1 Introduction

According to the most reliable statistical data of the amount of loss in different stages of the power system including generation, transmission, and distribution, the distribution stage is marked by 55 percent loss, while the amount of loss in transmission and generation stages is inevitable. Analyzing distribution networks shows that effective actions can be made to reduce the losses to a great extent [3]. Losses which are more than the standard level have always been a negative point for distribution networks, and various methods have been proposed and implemented to reduce them, including system reconfiguration using maneuvering points, which is a method that has been recently accounted. System reconfiguration is an effective method to improve the network, which is achieved by transmitting the load from the heavy load feeders to light load feeders. This technique, which is used for many purposes, is very easy to implement and is almost costless. Designing distribution networks, improving voltage profiles, service and maintenance, load recovery, balancing feeders’ loads, optimization, and system loss reduction are some of these purposes, which are achieved by network reconfiguration [2, 8].

Research on how to implement this method has been vast. Using heuristic ways and its rules, some have tried to find the optimal reconfiguration, and some have proposed analysis-based methods. M.E. Baran and F. Wu have proposed branch replacement in their method for optimal reconfiguration [14]. In this method, first all open switches are closed, and then the search for finding the loop, which causes the greatest amount of loss, begins. In the selected loop, the branch which causes the greatest amount of loss is opened. Similarly, this is done to the last loop. D. Shirmohammadi, H.W. Hong and Hong have also presented an algorithm based on heuristic rules [7].

The method proposed in the present study considers simultaneously the condition of the system switches. In this method, based on the maneuvering points and the adjacent switchless on the left and right sides, certain compounds are created and then examined, and the configuration with minimum loss is achieved. Moreover, to make it more comprehensive, the method is implemented simultaneously alongside with DG placement. This often leads in a non-convex, non- linear problem, mixed with integer variables. many of the design parameters have been examined and the type of the optimization variables is identified, solving this problem is very difficult and challenging. Also, the size of the system in the case studies increase exponentially. Conventional analytical optimization techniques, present successful strategies to solve simple and ideal models, however in reality, loss reduction problem is very complicated and difficult, and cannot be solved by these methods. Therefore, presenting appropriate new methods to solve this problem seems necessary [21].

Several meta-heuristic methods have been recently proposed for solving the problem, which can have acceptable performance in solving such problems. Multi-objective phase algorithm was first used for solving the distribution network reconfiguration problem [6]. In the past few years Genetic Algorithm (GA) [10] has been used to solve the optimization problems. To achieve better results, other meta-heuristic algorithms, including refined Genetic Algorithm (RGA) [9], Harmony Search algorithm (HS) [18, 22], ant colony optimization (ACO) [4], random walks-based technique [5] and simulated annealing have also have been examined by other researchers.

Nowadays, with the development of distributed generation sources, a new prospect has emerged in electricity generation industry. In several articles, the losses have been reduced by using distributed generation sources and placing them in distribution system. One of the main challenges in the placement of the DG is to locate and to determine their capacity [1]. To address these challenges, meta-heuristic algorithms such as Genetic and Harmony Search have been employed.

This paper presents a new method for solving distribution network losses by using system reconfiguration, and with the simultaneous use of distributed generation sources, based on the evolved type of the Harmony Search algorithm. Harmony Search algorithm is inspired by the process of natural music and happens when a musician searches for the best mode of harmony [23]. To show and examine the efficiency of the proposed algorithm, the results are tested in the IEEE standard system. The findings clearly show higher efficiency of the algorithms in comparison to other existing algorithms.

Furthermore, in the second part of the paper, the mathematical formulas are presented. In the third part, Harmony Search algorithm and the evolved form are described. The fourth part of the paper presents the application of the algorithm to the problem at hand. In the fifth part, the statistical findings are presented in the IEEE standard system, and the last part concludes the paper.

2 The mathematical formulation of the problem

2.1 Load flow equations

Load flow equations in distribution systems are presented by the simplified equation in reference [20]. These equations can be easily obtained from the one-line diagram in Fig. 1.

$\begin{matrix} P_{k + 1} & = & P_{k} - P_{loss, k} - P_{Lk + 1} \\ P_{Loss, k} & = & \frac{R_{k}}{{| V_{k} |}^{2}} {P_{k}^{2} + {(Q_{k} + Y_{k} | V_{k} |^{2})}^{2}} \end{matrix}$ (1)

$\begin{matrix} Q_{k + 1} & = & Q_{k} - Q_{Loss, k} - Q_{Lk + 1} \\ Q_{Loss, k} & = & \frac{X_{k}}{| V_{k} |^{2}} {P_{k}^{2} + {(Q_{k} + Y_{k 1} | V_{k} |^{2})}^{2}} \\ - Y_{k 1} | V_{k} |^{2} - Y_{k 2} | V_{k + 1} |^{2} \end{matrix}$ (2)

$\begin{matrix} | V_{k + 1} |^{2} & = & | V_{k} |^{2} + \frac{R_{k}^{2} + X_{k}^{2}}{| V_{k} |^{2}} (P_{k}^{2} + (A)^{2}) \\ - 2 (R_{k} P_{k} + X_{k} A) \\ A & = & Q_{k} + Y_{k} | V_{k} |^{2} \end{matrix}$ (3)

Power losses in the connecting line between bus k and k + 1 is presented as follows: $P_{Loss} (k, k + 1) = R_{k} . \frac{(P_{k}^{2} + Q_{k}^{2})}{| V_{k} |^{2}}$ (4)

The total power loss in the feeder can be calculated by summing up the losses in all lines. $P_{T, Loss} = \sum_{k = 1}^{n} P_{Loss} (k, k + 1)$ (5)

2.2 Power loss equation in the reconfigured network

The problem of network reconfiguration in distribution systems is in fact to find the best network structure of a radial network that is able to minimize energy loss. However, minimal loss must meet the following constraints.

Alignment of the voltage to an appropriate level

Supplying the capacity of the feeders flow in the distribution network

Having explained so far, the power losses in the connecting line between bus k and k + 1 in the reconfigured radial is stated as follows: ${P^{'}}_{L o s s} (k, k + 1) = R_{k} \cdot \frac{(P_{k}^{' 2} + Q_{k}^{' 2})}{{|V_{k}^{'}|}^{2}}$ (6)

The total power loss in the feeder is calculated by summing up the losses in all lines. $P_{T, Loss}^{'} = \sum_{k = 1}^{n} P_{Loss}^{'} (k, k + 1)$ (7)

By using Equations (5), and (7), the amount of network loss after reconfiguration can be illustrated as Equation (8). $Δ P_{Loss}^{R} = P_{T, Loss} - P_{T, Loss}^{'}$ (8)

2.3 Power loss equations in the presence of DG

Power loss in installing distributed generation sources in the network depends on the position of the connected source; this equation is stated in (9):

$\begin{matrix} P_{DG, Loss} = \frac{R_{k}}{V_{k}^{2}} (P_{k}^{2} + Q_{k}^{2}) + \frac{R_{k}}{V_{k}^{2}} \cdot B \cdot (\frac{G}{L}) \\ B = P_{G}^{2} + Q_{G}^{2} - 2 P_{k} P_{G} - 2 Q_{k} Q_{G} \end{matrix}$ (9)

Moreover, the amount of loss in the network after installing DG in comparison to before installing it, can be calculated by Equation (10). $Δ P_{Loss}^{DG} = \frac{R_{k}}{V_{k}^{2}} \cdot B \cdot (\frac{G}{L})$ (10)

2.4 Sensitivity analysis for installing DG

Sensitivity analysis is used to calculate the sensitivity factor of the location of the buses for installing DG units in the system [11]. Estimating the position of the candidate buses helps to reduce the search area in optimization process. Notice Fig. 2.

The losses of active power in the k-th line connected between the k and k + 1 bus can be calculated by the following Equation. $P_{LineLoss} = \frac{(P_{Lk, eff}^{2} + Q_{Lk, eff}^{2}) R_{k}}{V_{k}^{2}}$ (11)

Furthermore, Loss Sensitivity Factor (LSF) is calculated by Equation (12). $\frac{\partial P_{LineLoss}}{\partial P_{Lk, eff}} = \frac{2 \times P_{Lk, eff} \times R_{k}}{V_{k}^{2}}$ (12)

Using Equation (12) and the values obtained from the load flow, LSF is calculated for all buses. The obtained values are arranged in descending order, and according to the needed number of DGs, candidate buses are selected.

3 Optimization based on harmony search algorithm

Harmony search algorithm (HS) is a new heuristic algorithm which is developed by Dr. Gim, and Kim is inspired by the process of performing natural music. This process occurs when a musician is searching for the best harmony mode. In HS algorithm, the response vector is similar to harmony in music, and the search scheme is similar to a musician’s improvisation [15, 16]. Compared with other meta-heuristic methods, HS is mathematically less complicated, and can be well adapted to a variety of engineering optimization problems. Moreover, numerical comparisons show that the evolution rate of the HS algorithm is faster than other evolutionary algorithms such as Genetic algorithm [12, 13], and [15].

This algorithm has a good performance in the solution space and in a reasonable time, however, such a performance is not enough in numerical optimization cases. Therefore, to increase accuracy and convergence rate, some improvements are done in the HS variables, in which by introducing a new strategy for adjusting algorithm parameters, it has been done dynamically. The algorithm is designated as Improved Harmony Search (IHS) [16]. Moreover, an improved algorithm is proposed in reference [15] which is developed by using the concept of collective intelligence, and is known as Global-Best Harmony Search. The results show the higher efficiency of these two algorithms in comparison to theoriginal HS.

3.1 The structure of HS algorithm

In HS algorithm, each response vector is called a harmony, and is shown by an n-dimensional vector. The initial population of harmony vector is randomly created and is saved in harmony memory (HM). Then, the new harmony candidates are recreated by all the HM responses and in the light of harmony memory consideration rule, pitch adjusting rule, and an initial assessing process. As a result, HM is updated by comparing new candidates with the worst harmony vector. HS algorithm consists of the following three basic phases:

Step 1) Initial problem value set and algorithm parameters:

HS algorithm parameters, includes harmony memory size (HMS), harmony memory consideration rate (HMCR), pitch adjusting rate (PAR), distant bandwidth (BW), and the number of iterations (NI).

Step 2) Initial value set of the memory harmony:

HM includes harmony vectors. Suppose X_i ={ x_i (1) , x_i (2) , …, x_i (n) } shows i-th harmony vector, which is selected randomly. Thus, the HM matrix will be completed by harmony vectors asfollows: $HM = [\begin{matrix} X_{1} \\ X_{2} \\ \dots \\ X_{HMS} \end{matrix}]$ (13)

Step 3) Creating a new harmony:

A new harmony vector (X_new) is created by applying three rules: Memory consideration, pitch adjustment, and random selection. First of all, a uniform random number (r1) in the range of [1 0] is created. If r1 is less than HMCR, the decision variable X_new (j) is created with regard to memory consideration; otherwise, X_new (j) is obtained by a random selection. Then, each decision variable, provided that it is obtained by memory consideration, is pitch adjusted with the PAR possibility. Pitch adjusting rule is as follows: $X_{new} (j) = X_{new} (j) \pm r \times BW$ (14)

In which r is a random number between 0 and 1.

Step 4) Updating harmony memory.

When the new harmony vector is created, the harmony memory must be updated by comparison between the new harmony vector and the worst harmony vector in HM.

Step 5) Testing stop conditions.

If the iterations are more than a predetermined rate, or are less than a specific error range, the algorithm ends; otherwise, steps 3 and 4 will be repeated.

3.2 The structure of IHS algorithm

IHS algorithm was first developed by Dr. Mahdavi where the deficiencies of HS algorithm such as the constant parameters of PAR and BW were corrected [16]. IHS algorithm is similar to HS in terms of memory consideration, pitch adjustment, and random selection, but the PAR and BW values change dynamically. $PAR (t) = {PAR}_{min} + \frac{{PAR}_{max} - {PAR}_{min}}{NI} \times t$ (15) $BW (t) = {BW}_{max} e^{(\frac{ln (\frac{{BW}_{min}}{{BW}_{max}})}{nl} \times t)}$ (16)

3.3 The structure of GHS algorithm

This algorithm was first inspired by the particle swarm algorithm, and was presented by modifying the pitch adjustment rule [15]. Unlike the HS algorithm, this algorithm creates a new vector with the best harmony vector of X_B ={ x_b (1) , x_b (2) , …, x_b (n) }. The presented pitch adjustment rule is as follows: $X_{new} (j) = X_{B} (k)$ (17)

3.4 Improving harmony search algorithm

GHS optimization algorithm uses the best harmony vector X_B (k) to create harmony vector. However, pitch adjustment rule may upset X_B (k) structures, so much so that the new harmony vectors may become worse than X_B (k). Therefore, using GHS algorithm, a new self-adaptive algorithm, called SGHS, has been developed. Unlike GHS algorithm, this algorithm has used a new scheme to create and adjust the parameters [17].

In this algorithm, the three control parameters; BW, HMCR, and PAR depend entirely on the optimization problem, and in the light of ideal evolutionary trajectory, the search process is explained.

HMCR is the possibility of selecting one of the stored values in HM. Generally, a big HMCR will result in better solutions. PAR is the pitch adjustment rate. A bigger PAR is appropriate for transferring the X_B data to future generations. In addition, small values of PAR enables the new harmony vector to choose the related values by disassembling the corresponding values in harmony memory. In this algorithm, the HMCR and PAR values are normally within the range of [0.9 1] and [1 0], and are distributed with the mean; HMCR_m and PAR_m; and with standard deviation of 0.01 and 0.08. First HMCR_m and PAR_m are stabilized on 0.9 and 0.1. During the evolution of HMCR and PAR values, the created harmony changes according to the successful values. After a certain number of generations, HMCR_m and PAR_m are recalculated by measuring the means of all the HMCR and PAR values. BW parameter is the bandwidths. A bigger BW is suitable for searching in a vast area. While a small BW is suitable for precise adjusting of the response vectors. To strike a better balance in extracting and exploring the proposed SGHS algorithm, BW reduces dynamically as the generations increase.

This algorithm does not need to precisely adjust the parameter values according to characteristics and complexities of the problems. These parameters function self-adaptively by using learning mechanisms or by the dynamic reduction of the generation counter. Therefore, SGHS algorithm can show a good performance in problems with special characteristics.

4 The adaptation steps of the proposed algorithm in solving distribution network loss reduction problems

The process of this method is designed according to maneuvering points. Each maneuvering point and its right and left arms constitute a group which is the basis for search in various system cases. The stages to implement the program are as follows:

Finding all system compounds based on maneuver branches and the two adjacent branches.

Finding valid types from the above compounds.

Loss estimation.

Configuration identification with minimum loss.

First, the existing loops are numbered, and then it is determined that which line belongs to which loop. Then, according to the number of the loops and the replaced loops, we limit the response area by the search algorithm, and finally reach the optimal answer.

In the proposed search algorithm, each answer is made of two groups. The first group includes the n number of the algorithm which is as the number of the loops of the case study network. The second group includes the m number of the algorithm which is as the number of distributed generation sources in the network. The sum of these groups shows one of the answers of the problem. $HV = [o_{1} o_{2} o_{2} \dots o_{n} | s_{1} s_{2} \dots s_{m}]$ (18)

The adaptation steps of the proposed algorithm in solving distribution network loss reduction problems are as follows:

Step 1) Creating NI, HMS parameters based on the problem.

Step 2) Initial value assignment of BW_maxBW_min‘HMCR_m’PAR_m.

Step 3) Initial value assignment and HM assessment and adjusting the t = 1 generation counter.

Step 4) Creating HMCR, PAR based on HMCR_m‘PAR_m and obtaining BW based on BW_max’BW_min.

Step 5) Finding new harmony X_new.

Step 6) If f(Xnew) <f(Xw), then HM is updated for Xnew < Xw, and stores the values of the HCMR and PAR.

Step 7) t = t + 1.

Step 8) If NI was over, it returns the best harmony vector from HM, otherwise it goes to step 4.

The flowchart of the proposed method is shown in Fig. 3. To check the value of each vector, we use the fitting function with the following equation.

$\begin{matrix} maximize f = max . (Δ P_{Loss}^{R} + Δ P_{Loss}^{DG}) \\ Subjected to V_{min} \leq | V_{k} | \leq V_{max} \\ | I_{k, k + 1} | \leq | I_{k, k + 1, max} | \\ \sum_{k = 1}^{n} P_{Gk} \leq \sum_{k = 1}^{n} (P_{GK} + P_{Loss, k}) \end{matrix}$ (19)

5 Numerical studies

Figure 4 shows the single line diagram of the 33 bus radial distribution network standard IEEE [14]. This network has 5 open switches and 32 operational switches. Operational switches are shown with numbers from 1 to 32; these switches are normally closed. Five open switches are shown with numbers 33 to 37. Line and load data are presented in reference [13, 19].

The total amount of active and reactive consumed power in the network, are respectively; 3715 kW and 2300 kVar. In this system we connect three distributed generation units to the network. The location of the units is achieved by using sensitivity analysis. The power of candidate units has been between 0 to 2 MW, where the amount of output value is determined by the harmony search algorithm.

The network has a voltage level of 12.66 kV. Details for 33-bus network are shown in Table 1.

The results obtained with the implementation of flow can be observed for the non- process. The results are shown in Table 2.

As shown in Table 2 in all modes load switches 33, 34, 35, 36, 37 are open, in this case, no action has been taken to reduce losses and in normal times 201.89 kW of losses exist. The minimum bus voltage is shown in Fig. 5.

5.1 Value assignment of the parameters used in the algorithm

As was explained in the previous section, to obtain the best solution in the best time, BWmax, BWmin, HMCRm, PARm should be value assigned in an appropriate range, although the exact value assignment of these parameters is not required because the algorithm adjusts the values precisely with the passing of the generations.

To obtain the appropriate memory harmony, different HMS values are tested by the objective function; these results are shown in Table 3. As can be seen in Table 1, by increasing the amount of harmony memory the best possible answers change. This procedure is continued to HMS 35. From then in, the best answer remains stable and only standard deviation changes, in other words, with the increased amount of HMS, the worst answers of the algorithm are approached toward the best answers. However, this requires a longer time in implementing the program. With the data in the following table, and the explanations, the optimal value for HMS is chosen as 35.

To achieve the best possible answer as well as full coverage of solution space, BW_min BW_max values are assigned in a way that the [1 0] vector is completely covered, thus, BW_max and BW_min are respectively assigned as 1 and 0.0001. In a study conducted on a network of 33 buses the initial value of initialized HMCR_m has been set on 0.9, and PARm on 0.1. Moreover, the number of generations has been 3500. In this problem, the HMCR_m and PAR_m are updated in every 100 generations.

5.2 Simulation results

In this section, simulation results using SGHS algorithm has been presented. The results of running the program are shown in Table 4. Table 5 shows the sensitivity of the obtained answers in this system with the proposed algorithm to the different HMS populations in 10 performances. As seen in Table 5, with the increasing of harmony memory size, the obtained results and the answers average improve. The variation of the values PAR_m and HMCR_m during every 100 generations is shown in Figs. 6 and 7. This graph shows the range of change of HMCR and PAR.

To show the efficiency of the algorithm, these results are compared with the results by Genetic algorithm an RGA. The results are shown in Table 6.

Optimized structure of network after the reconfiguration and the installation of DG simultaneously shown in Fig. 8.

In normal bar times maneuver keys 7, 10, 14, 27 and 32 are opened respectively. Buses 31, 32 and 33 is the best choice for the integration of distributed generation sources. The minimum bus voltage is shown in Fig. 9.

6 Conclusion

This paper has proposed a new method for power loss reduction problem in distribution network by changing the structure of the network with the inclusion of the distributed generation sources. In order to reconfigure and install DG units simultaneously in distribution system and to examine the efficiency of SGHS algorithm, the 33 bus IEEE standard at load level nominal has been used, and implementation results have been compared with some of the other methods including, genetic algorithm(GA), improved genetic algorithm(RGA), and harmony search algorithm(HSA). The results show that simultaneous network reconfiguration and DG installation method is more effective and it is fast processing method in reducing power loss and improving the voltage profile compared to other methods. The findings indicate that the proposed algorithm can produce better results than other algorithms, and can be used as an efficient tool in thisproblem.

Footnotes

Acknowledgments

We thank our colleagues from department of electricity at University of Sistan and Baluchestan who provided insight and expertise that greatly assisted the research.

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