An Improved Distribution System Reconfiguration ...

An Improved Distribution System Reconfiguration Using Hybrid GA with PSO Dawit Fekadu Teshome

Kuo Lung Lian

Department of Electrical Engineering National Taiwan University of Science and Technology Taipei, Taiwan Email: [email protected]

Department of Electrical Engineering National Taiwan University of Science and Technology Taipei, Taiwan Email: [email protected]

Abstract—This paper presents an efficient and accurate way of solving radial distribution system reconfiguration (DSR), which plays an important role in distribution automation for realizing smart grids. It deploys different heuristic optimization approaches to resolve the desired optimum configuration and to efficiently reconfigure the connectivity of the distribution networks. The objective is to minimize the system power loss while the voltage in each bus is limited to some allowable range, and the topology of the system is kept radial. In this paper, a hybrid algorithm consisting of particle swarm optimization (PSO) and genetic algorithm (GA) is proposed. The first part of the hybrid approach is based on a modified PSO where the initial swarm of particles fit the radiality constraint and it introduces external randomness to velocities and locations with certain probabilities when particles are in equilibrium or close to equilibrium state. The second part is a modified GA which receives its initial population from best solutions of the modified PSO and uses adaptive mutation for introducing population diversity. In addition, the particle’s location in PSO and each chromosome in GA are repaired in such a way that the radiality constraint is always satisfied. The validity and the effectiveness of the proposed method has been tested using the standard IEEE 33-bus distribution network. The results show that the proposed method is robust and delivers a minimal average power loss of independent runs with reduced computational time. Index Terms—particle swarm optimization; genetic algorithm; power loss; network reconfiguration; distribution networks

I. I NTRODUCTION The main objective of the distribution network reconfiguration problem is to find a radial operating structure that minimizes the overall power loss while satisfying the operating constraints. The opening and closing of a number of tie switches (normally-open) and a number of sectionalizing (normally-closed) switches determine the change in the topology of the system. Altering the position of these switches results in different values of power loss in the network. Hence, the main objective is to find the best combination of switches that gives the lowest possible power loss of the system. However, this objective is subjected to several constraints which include: 1. The bus voltages have to be kept in the allowable range. 2. The current flow should not violate the ratings of the lines. 3. To ensure the power balance of the system, the total number of normally closed tie switches has to remain the same.

In earlier times, a number of scholars are tempted to solve the DSR problems. However, each algorithm is accompanied with its own merit and demerit. Merlin et al. [1] implemented the discrete branch and bound method. However, for a complex power system that involves a large number of combination of switches, the method takes high computational time to determine the minimal loss operational configuration. Civanlar et al. [2] employed a heuristic approach to reduce system losses. An estimated amount of loss change resulting from transferring a group of loads from one feeder to the other has been computed using a simplified equation. Moreover, a heuristic method based on branch exchange to solve DSR problem has been introduced by Baran and Wu [3]. Approximated power flows have been also developed to improve the search for optimal radial configuration. Generally, the discrete optimization algorithm requires high computational time to find the best solution due to the combinatorial nature of DSR problem. The heuristic methods are problem dependent techniques. They usually are adapted to the problem at hand. However, since they employed a greedy algorithm, they usually get trapped in a local optimum. To over come this difficulty, many researches have came up with concept of solving DSR with metaheuristic methods such as GA and PSO. Unlike heuristics, meta-heuristics are problem independent techniques. As such, they do not take advantage of any specificity of the problem and do not employ a greedy approach. In course of the search, a temporary deterioration of the solution may be acceptable, which allows to explore more thoroughly the solution space and thus to get a better solution. However, it is important that even though a meta-heuristic is a problem independent method, it is necessary to do some fine tuning of its intrinsic parameters in order to adapt the technique to the specific problem to be solved. Koichi et al. used GA to solve DSR problem [4]. Arc (branch) and switch numbers have been used to express the string structure of the chromosome. For large and complex networks, it is not efficient to represent every arc in the string, since its length is very long, taking high computation time to reconfigure the network. Mehdi et al. investigated the ability of PSO together with graph theory for network reconfiguration to reduce the power loss and to enhance the voltage profile of distribution system [5]. However, when the network becomes large, it converges

c 978-1-4799-7993-6/15/$31.00 ⃝2015 IEEE

prematurely without finding a local optimum. J. Zhu proposed a method to study DSR based on a refined GA (RGA). Some improvements are made on chromosome coding, fitness function and mutation pattern [6]. Nevertheless, considering the complexity of GA and the computational effort that it costs, further investigations compared with the other meta-heuristic approaches such as PSO and Simulated Annealing (SA), are needed to validate the effectiveness of the stated improvement. Esther et al. explored two methods [7]. The first method uses mixed-integer linear programming (MILP) to solve the DSR problem. The second technique retains the nonlinear objective function, dealing with the reconfiguration problem by means of a GA. However, MILP requires linearization of the objective function and the constraints, which may lead to less accurate results. Also the second method is not guaranteed since the conventional GA is not always converge to a global optimum. Trapping in the local minimum may lead to premature convergence. Moreover, other methods such as Simulated Annealing (SA), Tabu Search (TS), improved Tabu Search (ITS), Harmony Search Algorithm (HSA), Fuzzy Systems (FS) and Artificial Neural Networks (ANN) have been applied for solving DSR and power system optimization problems [8]-[14]. SA is inefficient if a problem is composed of many local minima. TS uses only one solution (individual based) and can easily miss some promising areas of the search space. FS requires more tuning while ANN imposes greater computational burden for the training algorithm.

This substantially reduces the search space and avoids the generation of infeasible topologies. The rest of the paper is organized as follows. The problem formulation of DSR is presented in Section II. Section III describes the proposed algorithm. Numerical examples and results are shown in Section IV, and Section V summarizes the main findings of this paper.

V𝑀 𝐼𝑁 ≤ V𝑖 ≤ V𝑀 𝐴𝑋 , 𝑖𝜖𝑁

(2)

In this paper, a hybrid method consisting of PSO and GA is proposed. The approach modifies both the conventional PSO and GA part of the algorithm and takes the advantage of these two non-identical algorithms which mostly likely will explore different areas of the search space. The main contributions of this paper are as follows:

I𝑚 ≤ I𝑚,𝑀 𝐴𝑋

(3)

∙

∙

∙

It modifies the initial swarm of PSO particles in such a way that the radiality constraint is not violated. During iterations, it also introduces external randomness to the velocities and locations of particles with certain probability for maintaining diversity. Moreover, adaptive mutation proposed by J. Zhu [6] is applied to the GA part of the algorithm. Population diversity is controlled smoothly by fine tuning the mutation rate in the course of iterations. It implements a bouncing strategy for handling constraints such as lower and upper boundary of particle’s location and velocity. When a particle exceeds the extreme boundaries, it directs back to the feasible region and is assigned a random location and velocity. All updated particles in PSO and generated chromosomes in GA are subjected to satisfy the radiality constraint. The location of particles and the chromosomes are repaired in such a way that only one switch is opened in each loop created by the closing of each tie switch. Each dimension of particle’s location or each chromosome of an individual represents a single variable (tie switch) and its value is approximated to the nearest value of the switch number that belongs to the corresponding loop.

II. DSR P ROBLEM F ORMULATION A. Mathematical modeling The objective function of the network reconfiguration is to minimize the total power losses as well as improving voltage profile. The optimal objective is represented as: min

𝑀 ∑

(S𝑚 I2𝑚 R𝑚 )

(1)

𝑚=1

where I𝑚 , R𝑚 and 𝑀 are the current that flows in branch 𝑚, the resistance of branch 𝑚, and the set of branches existing in the network, respectively. S𝑚 represents the status of branch 𝑚, and will be 0 if the branch is open or 1 if the branch is closed. The objective is subjected to the following constraints. The voltage magnitude at each bus must be maintained within its limits. The current on each branch has to lie within its capacity rating. These constraints are formulated as shown in (2) and (3).

where V𝑀 𝐼𝑁 , V𝑀 𝐴𝑋 , I𝑚,𝑀 𝐴𝑋 and 𝑁 are the lower and upper limits of the voltage magnitudes at bus 𝑖, the maximum current carrying capacity limit of branch 𝑚, and the number of buses in the network, respectively. The voltage and current constraints are handled as penalty functions and added to the objective (fitness) function in PSO or GA. In any DSR optimization algorithm, to calculate the fitness of each possible combination, a power flow has to be run in order to determine branch currents for any existing feasible solution, resulting in radial structure of the system topology. The next section explains the load flow analysis. B. Radial Power Flow The ratio of resistance-to-reactance of a distribution network line compared with a transmission system line is relatively large. Thus, conventional Newton-type load flow algorithms are not appropriate for solving distribution power flow. In this paper, a network topology based algorithm has been applied to determine the network parameters such as branch currents and bus voltages. The method has been known for its robustness and fast convergence rate for radial distribution systems [15]. The algorithm starts by building the Bus-Injection-to-BranchCurrent (BIBC) matrix and Branch-Current-to-Bus-Voltage (BCBV) matrix. Then it computes the Distribution Load Flow (DLF) matrix which is the multiplication of BCBV and BIBC. Afterwards, iterations commence to calculate current and voltage at each bus until the difference between

the magnitudes of bus currents in two consecutive iterations (△I) are less than the given tolerance value, 𝜖. The bus current is calculated as the ratio between consumed power (P + jQ) and voltage. In each iteration, the difference voltage (△V) is determined by multiplying the DLF matrix with bus current matrix. Then, the bus voltages are updated as the sum of no load voltages (Vno load ) and △V.

in (7) the particle is bounced back to the search space if it moves outside the feasible range .

III. P ROPOSED A LGORITHM

4) Diversity Preservation: The conventional PSO might be easily trapped if there are multiple local optima in the search space. It may occur that all particles move to the same local optima and the velocities are all decayed to zero. Thus, diversity should be properly maintained. In this proposed algorithm, a dissipative approach has been introduced. When particles are in equilibrium (i.e same 𝑝𝐵𝑒𝑠𝑡) or close-toequilibrium state which is measured through variance, external randomness is applied to the velocities and locations with certain probabilities, 𝑝𝑣 and 𝑝𝑙 , respectively. The same equations that have been used for the bouncing strategy are also implemented for maintaining diversity. For any 𝑟, uniformly generated random number between 0 and 1, if 𝑟 is less than 𝑝𝑣 , then according to (6) the velocity is updated as any random value but less than the maximum velocity. Similarly, for any 𝜌, uniformly generated random number between 0 and 1, if 𝜌 is less than 𝑝𝑙 , then according to (7) the location is updated as any random value but bounded to the lower (𝑥𝑚𝑖𝑛 ) and upper limit (𝑥𝑚𝑎𝑥 ) of the searching space. Note that diversity preservation adopts a probabilistic decision; however the bouncing strategy takes a deterministic decision if the constraint for velocity or location is violated. 5) Repairing Particle’s Location for Radiality: In DSR problem, the closing of each tie switch creates a loop and another switch has to be opened in order to restore the radial configuration. Since each dimension of PSO particle represents a tie switch, the locations are repaired in such a way that only one switch is opened in each loop created by the closing of each tie switch. Once the particle passes through the updating process and is checked for bouncing and diversity, then the location is approximated to the nearest value of the switch number that belongs to the corresponding loop. In such a way, all the particles satisfy the radiality constraint, and the use of penalty function in the case of infeasible topology which requires fine tuning of the penalty factors is avoided here. The repairing strategy significantly reduces the search space and speeds up the convergence rate. The same method is applied for the chromosomes of GA to handle the radiality restriction of DSR problem.

A. Modified PSO The flow diagram of the first stage of the proposed algorithm which is a modified PSO is given in Fig. 1. The main program starts by initializing locations in feasible region and the velocities are set to small random values to prevent particles from leaving the search space during the first iterations. The movements of the particles are guided by their own best known location in the search space as well as the entire swarm’s best known location. When improved locations are being obtained these will then come to guide the movements of the swarm. The velocities and locations of the particles are iteratively updated until the stopping criterion is met. The following subsections explain each part of the algorithm in detail. 1) Topology Representation and Initialization: In the modified PSO, each branch of the distribution network is represented by a decimal number. If the system has 𝑘 number of tie switches, then each particle is associated with a 𝑘dimensional vector. While initializing, the radial characteristics of the network shall be kept and all loads have to be in service. In view of this, the initial swarm is generated in such a way that each particle should not violate the network constraints and the structure of the system topology. 2) Updating Velocity and Location: The evaluation of each particle’s fitness is based on the value of the objective function which is the total system power loss as stated in (1). If the fitness is better than the particle’s best experience, the location vector for the particle is saved as 𝑝𝐵𝑒𝑠𝑡. If the fitness is better than the best in the entire swarm, the location vector for that particle is saved as 𝑔𝐵𝑒𝑠𝑡. Then, the particle’s velocity and location is updated according to (4) and (5). x𝑝𝐵𝑒𝑠𝑡 − x ˜(𝑖)) v ˜(𝑖 + 1) = 𝑤˜ v(𝑖) + 𝑟1 𝑐1 (˜ + 𝑟2 𝑐2 (˜ x𝑔𝐵𝑒𝑠𝑡 − x ˜(𝑖)) x ˜(𝑖 + 1) = x ˜(𝑖) + v ˜(𝑖 + 1)

(4) (5)

where v ˜(𝑖) and x ˜(𝑖) are the velocity and location vectors at iteration 𝑖. 𝑤 is an inertia weight to control influence of the previous velocity (usually the value is selected between 0 and 1). 𝑐1 and 𝑐2 are positive acceleration constants chosen by the user while 𝑟1 and 𝑟2 are random variables generated uniformly between 0 and 1. 3) Bouncing Strategy: When a particle attains a velocity beyond the value set by the user, according to (6) it is dampened by an upper limit (𝑣 𝑚𝑎𝑥 ), and multiplied by a random element. This helps to reduce fast velocity saturation that might occur in subsequent iterations. In similar fashion,

v ˜(𝑖) = 𝑟𝑎𝑛𝑑 ⋅ 𝑣 𝑚𝑎𝑥

(6)

x ˜(𝑖) = 𝑥𝑚𝑖𝑛 + 𝑟𝑜𝑢𝑛𝑑(𝑟𝑎𝑛𝑑 ⋅ (𝑥𝑚𝑎𝑥 − 𝑥𝑚𝑖𝑛 ))

(7)

B. Modified GA The flow chart of the second stage of the proposed algorithm has been shown in Fig. 2. The initial population is determined from the n-best solutions of PSO where all the retrieved configurations are feasible in terms of the network constraints and system topology. The following subsections explain the algorithm in detail.

START

STAGE 2

Represent the system topology with a number associated with each branch and define a particle with k-dimensions: p Generate the initial feasible swarm of particles satisfying radiality constraint: S = {p1, p2 , .., pn}

Represent the system topology with a number associated with each branch and generate a chromosome with n-bits length: p Swap the initial population POP from n-best solutions of PSO then code using binary representation: POP= {p1, p2, …, pn}

k-dim k dim kp1:

X1,1

X1,2

X1,3

.

k-bits k bits k.

.

.X1,k-2

X1,k-1

X1,k

p1:

0

pn:

1

……………………………………………………………… pn:

Xn,1

Xn,2

Xn,3

.

.

.

Xn,k-2

Xn,k-1

Xn,k

. . . 1 1 0 0 1 ……………………………………………………………… 1

0

1

.

.

.

1

0

Iteration: i=1

Number of generations: i=1 Compute fitness for each particle using radial power flow

Decode each chromosome in POP Compute PBest for each particle in S and GBest for S

Repair chromosomes to fit radiality constraint i>MAX_ITER NO

YES

STAGE 2

Evaluate fitness using radial power flow

NO Update velocity and location for each particle in S

i>MAX_GEN Apply bouncing strategy in extreme boundary conditions

YES

STOP

NO Encode, Select and Crossover chromosomes from POP

NO

i=i+1

Is PBest the same or nearly the same for all particles in S ?

NO

YES Assign random velocity and location for some particles with certain probability

mu_rate(i)=mu_rate(i-1)

Is Best_fitness t_fi f tness t the same betwe between two consecutive generations?

i=i+1

YES mu_rate(i)=mu_rate(i-1)-∆mu_rate

Repair particle’s location to fit radiality constraint

Apply mutation to chromosomes from POP

Fig. 1. Flow chart of the first stage of proposed algorithm (Modified PSO) Fig. 2. Flowchart of the second stage of proposed algorithm (Modified GA)

1) Chromosome Encoding and Decoding: Each of the switches is associated with a decimal number that identifies its location in the network topology. A binary representation of these switches are encoded in the gene of a chromosome. An individual (a combination of switches) in the population is composed of chromosomes representing each of the decision variables to be searched for the minimum power loss. For instance, if there are 32 branches in the network which is equivalent to the number of switches, a decision variable with a length of five bits is required to encode each switch number (25 = 32). For every generation prior to the evaluation of the individual, it has to be decoded, repaired for radiality and a power flow is run to obtain the individual’s fitness. 2) Selection and Crossover: Using roulette wheel selection method, potentially useful configurations (solutions) are chosen for crossover. Crossover and mutation rates can affect the convergence of GA, which is analogous to adaptive inertia weight manipulation in PSO. However, controlling premature convergence using mutation and crossover in GA is more

effective than altering the level of inertia weight in PSO. 3) Adaptive Mutation: Mutation operators are mostly used to provide exploration, and crossover operators are widely used to lead population to converge on the good solutions found so far (exploitation). Consequently, while crossover tries to converge to a specific point in landscape, mutation does its best to avoid convergence and explore more areas. Too high mutation rate increases the probability of searching more areas in search space. However, it may prevent population to converge to any optimum solution. On the other hand, too small mutation rate may result in premature convergence. In this paper, an adaptive mutation is put into implementation. When the best solutions in two consecutive generations are similar, the mutation rate is set to decrease dynamically from higher value to a lower value; however during the searching process if the best fitness among generations is kept decreasing, the mutation rate is not altered.

Substation

0.21

s1 1 s2 s3

s22 3

s4

22 s23

4 s5 18

23 5

s19

s24

s6 19

24

s25 6

s20

s7 20

s33

s21

s26

8

s27

s8 21

26

s9 9

s28

10

s29

0.16 0.15

s37

0.13 0

28

s11 s12

0.17

10

20

30 40 No. Iterations

27

s10 s35

0.18

0.14

25 7

Powerloss (MW)

0.19 2

50

60

70

Fig. 4. Convergence characteristics of proposed method

29 11

s30 s34

30

12

1

s31

s13

Initial System After Reconfiguration

31 13

s32

0.98

32

s14 14 s15 15 s16 16 s17 17

s36

Bus Voltage (p.u)

s18

Best Worst Average

0.2

0.96

0.94

0.92

Fig. 3. A 33-bus Test system 0.9 0

IV. N UMERICAL R ESULTS The proposed methodology is tested on a standard IEEE 33-bus distribution power system as shown in Fig. 3. The normally open tie switches are s33, s34, s35, s36, and s37, represented by dotted lines, and the normally closed switches, s1 to s32, are represented by solid lines. The system data can be found in [6]. The total number of tie switches for the test system is five which exactly forms five meshes if each of them is assumed to be closed. In modified PSO, each particle deals with five decision variables since five out of thirty-seven switches are the ones that are searched to be opened for the minimum loss configuration. In the case of modified GA, each variable (switch) is represented with six bits which in turn results in a sixty-four bit long chromosome. In the proposed method, PSO is first run for a predefined number of iterations, and the best solutions (𝑔𝐵𝑒𝑠𝑡 in each iteration) found so far are decoded into chromosomes and passed as initial population to GA. Then, GA is set to run for the rest of the iterations. Although it is assumed that increasing the number of iteration may lead to a better solution, it should be determined on the cost of computational resources. The algorithm has been tested several times to properly tune the number of particles/population size, the number of iterations and the inertia weight for the best performance of the optimization process. To demonstrate the robustness of the proposed approach, the system is solved repeatedly for 200 independent runs, and results are compared with GA [4], RGA [6], ITS [12] and HSA [14] where the performance of these algorithms is reported in [14]. The simulations for the proposed method were performed on a PC with Intel Core Duo processor (3GHz) with 3GB of RAM. The convergence characteristics of the proposed hybrid

4

8

12

16 Bus No.

20

24

28

32

Fig. 5. Voltage magnitude in each bus after reconfiguration

method for the best and worst cases of the 200 independent runs as well as the average of these two cases are shown in Fig. 4. For the first 25 iterations, the modified PSO reduces the power loss from 0.2027 MW (initial value) to 0.1463 MW. Then, GA further reduces the loss and captures the global optimum (0.1395 MW) at the 44𝑡ℎ iteration. The simulation results obtained using different algorithms and the proposed method for 200 independent runs are shown in Table I. In terms of average loss reduction, the result found using the proposed method is 70.7%, 64.24%, 58.38% and 23.26% higher than that of GA, RGA, ITS and HSA, respectively. The standard deviation (STD) for the optimal solutions of the independent runs is also 89.66%, 88.72%, 87.60% and 86.73% lower than that of GA, RGA, ITS and HSA, respectively. Furthermore, the proposed approach results in lowest computational time compared with the algorithms mentioned above. The bus voltage magnitude profiles of the original and reconfigured system are shown in Fig. 5. The voltage profile is considerably improved after reconfiguration compared to the initial system. Referring to Table I, the minimum voltage magnitude after reconfiguration using the proposed method is 2.71%, 0.95%, 0.68%, 1.82% and 0.39% higher than that of the initial system, GA, RGA, ITS and HSA, respectively. V. C ONCLUSIONS In this paper, a hybrid method comprising GA and PSO is proposed for solving the DSR problem more efficiently and accurately. Different strategies are applied to maintain

TABLE I L OSS REDUCTION AND T IE SWITCH CONFIGURATION Configuration Algorithm

Initial GA [4]

RGA [6]

ITS [12]

HSA [14]

Proposed method (Hybrid GA-PSO)

Power Loss (MW) Best Worst Average STD 0.2027 0.1416 0.2027 0.1662 0.0145 0.1395 0.1984 0.1649 0.0133 0.1393 0.1963 0.1635 0.0121 0.1381 0.1951 0.1523 0.0113 0.1395 0.1463 0.1406 0.0015

Tie Switches

Average Loss Reduction (%)

Loss Reduction (%) (Best value)

Minimum Voltage (p.u) (Best value)

CPU Time (s) (Average time)

𝑠33, 𝑠34, 𝑠35, 𝑠36, 𝑠37 𝑠7, 𝑠9, 𝑠14, 𝑠32, 𝑠37

-

30.15

0.9131 0.9290

-

18.01 𝑠7, 𝑠9, 𝑠14, 𝑠32, 𝑠37

19.1 31.20

0.9315

18.65 𝑠7, 𝑠9, 𝑠14, 𝑠36, 𝑠37

13.8 31.29

0.9210

19.34 𝑠7, 𝑠10, 𝑠14, 𝑠36, 𝑠37

8.1 31.89

0.9342

24.85 𝑠7, 𝑠9, 𝑠14, 𝑠32, 𝑠37

population diversity in the proposed approach. It also implements a repairing strategy for satisfying a radiality constraint for each PSO particle or GA chromosome, which greatly reduces the solution searching space. The proposed method results in a fast convergence rate and is able to find the global optimum without premature convergence. Compared with different algorithms, the proposed hybrid method is superior in terms of less computational time, larger average loss reduction and minimum standard deviation for the optimal solutions of several independent runs. VI. ACKNOWLEDGMENT The authors would like to sincerely thank the Ministry of Science and Technology (MOST), Taiwan for financially supporting this work (Contract Number: MOST 103-2221-E011-099-). R EFERENCES [1] A. Merlin and H. Back, “Search for a minimal-loss operating spanning tree configuration in an urban power distribution system,” in Proc. 5th Power System Computation Conference, Sept 1975. [2] S. Civanlar, J. J. Grainger, H. Yin, and S. S. H. Lee, “Distribution feeder reconfiguration for loss reduction,” Power Delivery, IEEE Transactions on, vol. 3, no. 3, pp. 1217–1223, Jul 1988. [3] M. Baran and F. Wu, “Network reconfiguration in distribution systems for loss reduction and load balancing,” Power Delivery, IEEE Transactions on, vol. 4, no. 2, pp. 1401–1407, Apr 1989. [4] K. Nara, A. Shiose, M. Kitagawa, and T. Ishihara, “Implementation of genetic algorithm for distribution systems loss minimum reconfiguration,” Power Systems, IEEE Transactions on, vol. 7, no. 3, pp. 1044–1051, Aug 1992. [5] M. Assadian, M. M. Farsangi, and H. Nezamabadi-pour, “Optimal reconfiguration of distribution system by pso and ga using graph theory,” in Proceedings of the 6th Conference on Applications of Electrical Engineering, ser. AEE’07, 2007, pp. 83–88.

7.2 31.20

30.63

0.9378 5.7

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