Applications Of Hopfield Neural Networks To Distribution Feeder Reconfiguration

Applications of Hopfield Neural Networks to Distribution Feeder Reconfiguration Demck Bouchard and Aziz Chikhani Department of Electrical and Computer Engineering Royal Military College of Canada

V.L John

M.M.A.Salama

Department of Electrical Engineering Queen's University Kinmton. Ontario. Canada K 7 c 3N6

Department of Electrical and Computer Engineering University of Waterloo waterloo: Ontario. Canada N2L 3G1

Abstract

the fact that the losses to be minimized are $R losses, which are nonlinear.

Dktribution feeder reconflguration k an optimization problem for loss minimization, and, in this paper; we investigate the use of a Hopfield neural network for dktribution feeder reconfigumtion. A network model is devebped andpresente4 and then the method applied to a dktribution system used by Wagner et a1 [ l ] conskting of threefeeders, thirteen normallyclosed sectionallihg swWzes, three normally open tie switches and thirteen load points. Simulation results using thk distribution system modelled a neural network are presented Keywords: dktribution automation, feeder reconfiguration, loss reduction, Hopfield neural network Introduction Distribution automation feeder reconfiguration involves the opening and closing of distribution feeder switches to minimize system power losses, while at the same time providing for improved load balancing amongst a system's feeders and satisfying the required voltage profile using voltage regulators. Switching operations are constrained by the capacities of the system transformers and feeders, voltage profiles and by the need to keep phases as closely balanced as possible. In an emergency, feeder reconfiguration allows sections without power to be switched to another feeder to restore power, providing increased reliability to customers. Even a relatively small distribution network will allow for many possible feeder configurations, and an exhaustive search of all possible configurations to determine the optimum configuration is impossible in real-time applications. Each feeder in a distribution system usually has a mix of residential, commercial and industrial cust?mers with varying needs depending on the season of the year and the time of day, and, thus, lengthy computation times are not possible in a real-time implementation. The optimization problem is compounded by

Feeder reconfiguration is an active area of research, and several algorithms have recently been proposed to solve the feeder reconfiguration problem (for example, [1-6]). Much of the research explores heuristic methods that employ rules-ofthumb to reduce the number of possible configurations to be searched. None of the algorithms presented to date guarantee that the solution found is the optimum one, and in all cases the prohibitive computational requirements limit applications to smaller systems. Recently, researchers have investigated the use of the Hopfield neural networks to solve optimization problems, such as the Travelling Sales Person problem [71. Hopfield nets are attractive for optimization problems because the solution is determined collectively in a very short time by a large number of neurons. Hopfield neural networks are beginning to find application in electric circuits [8-101 and communications [ll]. Hopfield Neural Networks Hopfield and Tank [7] have suggested that neural networks are well suited to the solution of combinatorial optimization problems. They have illustrated the basic idea by solving the travelling salesperson (TSP) problem, which requires that a route be found for a salesperson to visit n cities, visiting each city only once, and minimizing the distance to be travelled. 'rhus, the solution is to find the best order among n cities to be visited. If we introduce a square matrix containing n x n neurondbinary elements, the solution can be represented by entering a "1" into the row/column to indicate when the ith city is to be visited. For example, table 1 below represents a five-city tour. Here, the solution is to visit city 1 f i t , city 4 second, then city 5, then city 3 and finally, city 2. Note in table 1 that there is only one "1" in each row, only one "1" in each column, and that the total number of ones is equal to the number of cities.

0-7803-1217-1/93/$3.00 01993 IEEE 31 1

Position on tour

Table 1 - Possible solution to the. fivecity Travelling Salesperson Problem (TSP)

These constraints can be incorporated into an energy function, as follows [12,131: .

n

n

n

i#j

is to reconfigure the network. Figure 1 shows a three feder distribution system consisting of 3 three feeders serving 13 loads. There are 16 sectionalizing switches, with 3 of them open. The data for this system (line impedances,voltages at the busbars and MVAs at each busbar) is shown in table 2. This system was studied in [l] by Wagner et al. This distribution system could be reconfigured, for example, by closing the switch in feeder section 15, and opening the switch in feeder section 19.

FEEDER 1

FEEDER 2

FEEDER 3

1 116

122 23

The solution to the problem is to minimize this energy function. We should examine what this function represents. The first term will be zero once a solution is reached if and only if there is only a single value of 1 in each row. The second term will be zero if and only if there is a single value of 1 in each column. The third term is a summation of all output values, and there should only be n of these that have a value of 1 - all others should be zero. The final term computes a value proportional to the distance between cities, and should be a minimum when a solution is reached. Note that there are four constraints. of which three are strong and one is weak.

13 26

@-

0

1

-

LOADCENTRE FEEDER SECTION, SWITCH CLOSED

n FEEDER SECTION, SWITCH OPEN

Distribution Systems

Problem Statement

In a radial distribution system, sectionalizing switches have two purposes - the first is to isolate faults, and the other

A typical distribution network has a radial structure, with each load supplied by only one feeder. The following aiteria

312

must be met to solve the distribution network problem: a. each load is connected to only one feeder; b. the n u m b of switches closed equals the number of loads connected; and, c. the total line losses, SR,are minimized. Solution to the system shown in figure 1

To represent the system shown in figure 1 as it is connected in figure 1 as a Hopfield neural network, we could represent it using the matrix shown in table 3. Here, A "1" in a matrix element indicates a load is connected to a particular feeder. A "blank" indicates the load is not connected to that feeder.

The final term computes a value that is proportional to the losses in the feeder configuration. This will have a nonzero value, but one that should be a minimum when a solution is reached. How can we turn this energy function into a neural network? Let us define a connection weight matrix in terms of inhibitions between processing elements. We will make use of a four-index scheme instead of the double index notation that has been used up to now to describe the weight matrix. Our notation will be TXLv Let us now reconsider the energy function. The first term will be zero if and only if there is one element in each column of the output matrix. Consider modelling this as

In [ll, Wagner et al showed that the optimal configuration for this network was to move load 11 to feeder 1, and to move load 10 to feeder 3. After this reconfiguration, the matrix would be as shown in table 4. The energy function to be minimized is as follows:

E =

B "f "f " -2 i-1 CXC C Vxivyi - 1 Y-1 Y#X

+

[.5 5 X=1 i l l

vxi-n

J

"f " +, D Cdxyvxi 2

c

-B

6 , (1-6g),

Using this model, 6,, will be zero everywhere except where X = Y. The quantity in parentheses will be one except when i = j . This is to ensure that all other units in a column are inhibited except the unit itself. The second term involves a sum of all outputs, and can be said to be a global inhibition term, and can be modelled as

-C. The third term can be modeled as follows

x-1 Y-1

-D dxy 6xy

In the above expression, n is the number of loads and nf is the number of feeders. The loss associated with each load is represented by dxu. If the constants, B, C,D are all greater than zero,then E will be nonnegative.

, term ensures that inhibitory connections are made The % only between loads. The term -D dxy ensures that loads causing the biggest line losses will receive the largest inhibitory signal.

We should examine what our energy function represents. Note that the t e m containing the constant A in the original energy function for the TSP problem has disappeared, as the constraint that there be only one "1" in each row does not exist. The first term of the modified energy function, vxj yyj, will be zero if and only if each column of the processing elements matrix contains a single value of 1. This term should be zero when a solution is reached.

Combining these terms gives

TX,V =

- B 6g (1-6m) - C - D d m 6,

For small increments of time, the time evolution of the above network can be approximated by

Exxi

The second term, YM is a summation of all n x nf output values. There should only be n of these that have a value of 1 all others should be zero. Hence, this term should also be zero when a solution is reached.

-

where N is the number of processing elements, t is the system

313

time constant, and Ii is the input.

where the line impedance between branch i and branch i+l is rj + xQ the load at branch i+l is SLi = PLi + QLi, and thus Pi+r, Qi+l and Vi+l are the values at branch i + I , having determmed those values for branches U . . . i. The power loss on a line between two branches, i and i+l , is then given by pi = ri (P: + Q;)

p.u.

where the assumption is made that Vi' is approximately equal to 1 p.u. ?his, then, can be used as the distance parameter, dm, above.

where A is called the gain parameter.

Simulation results We have attempted to represent the three-feeder network shown in figure 1 as a neural network. The results have not been encouraging.

2.

= 2 (1 + tmh(hXi))

Substituting Txi previously defined, and defining Ixi = Cn', where C is as previously defined and n' is a constant,

Y#X

After several attempts and using various values of B, C, D, z, A, and n', the network did not provide a valid solution to the optimization problem. Using the hyperbolic tangent as a transfer function typically resulted in all switches being open, or all switches being closed, depending upon the value of A. We tried using other transfer functions. We had limited success using the following transfer function: 1 vi = 1 + e-pi

This expression was evaluated if& Y#X

To complete the solution, the constants B, C,D,z, A, and n ' must all be specified.

was greater than some threshold, t, and set to zero otherwise. Using this transfer function, we were able to get a mixture of switches being closed and opened. However, in this case, we usually found that some loads were connected to two feeders, while some loads were not ~ o ~ e c t to e dany feeder.

Conclusions Power Flow Equations

In [14]. Baran and Wu provide a set of power flow equations for radial distribution networks. By assuming that the losses on the lines between branches are much smaller than branch power terms, they develop the following set of branch equations: n

Pi+l = Pi

- pLi+l

=

k-i+2 n

314

'Lk

We have modelled a distribution network as a Hopfield neural network, and solved it in a fashion similar to the Travelling Salesperson Problem. However, the network did not converge to a valid solution for the network of figure 1. Future research will explore why the network did not converge.

Bus to

Section

Section

bus

Resistance @.U.)

Reactance (p.u.,

1-4

0.075

0.1

2.0

1.5

4-5

0.08

0.11

3.0

1.5

1.1

0.9881-0.561

4-6

0.09

0.18

2.0

0.8

1.2

0.986/-0.697

6-7

0.04

0.04

1.5

1.2

0.9851-0.704

2-8

0.11

0.11

4.0

2.7

0.9791-0.763

8-9

0.08

0.11

5.0

3.0

8-10

0.11

0.11

1.o

0.9

9-11

0.1 1

0.11

0.6

0.1

0.6

0.9711-1.525

0.08

0.11

4.5

2.0

3.7

0.969/-1.836

9-12

-1

~~

3-13

o.ll

13-14

0.09

~~

I

0.12

EndBus Load (Mw)

I

reoI 1.0

I

EndBus Load

EndBus Capacitor

(WAR)

(WAR)

o.9

0.7

End Bus Voltage (p.u.1

0,9911-0.370

1.2

0.9711-1.451 0.9771-0.770

I I

1.8

I I

0.9941-0.332 0.995/-0.459

~~

0.08

0.11

1.o

0.9

0.04

0.04

2.1

1.o

5-11

0.04

0.04

10-14

0.04

0.04

7-16

0.09

0.12

13-15

0.992/-0.257

~~

15-16

0.9

0.991/-0.596

~~~

Table 2 - System data for the system shown in figure 1.

Table 3 - Distribution system of figure 1 illustrated as a Hoptield neural network.

-

Table 4 Optimum configuration for the distribution system of figure 1.

315

References

[l] T.P. Wagner, A.Y. Chikhani, R. Hackam, "Feeder Reconfiguration for Loss Reduction: An Application of Distribution Automation," IEEE Transactions on Power Delivery, Vol. 6, No. 4, October 1991, pp. 1922-1933.

[8] D.W. Tank, JJ. Hopfield. "Simple 'Neural' Optimization Networks: An A/D Converter, Signal Decision Circuit, and a Linear Programming Circuit," IEEE Transactions on Circuits andsystem, Vol. 33, No. 5, May 1986, pp. 533-541.

[2] K. Aoki, H. Kuwabara, T. Satoh, M. Kanezashi, "An Efficient Algorithm for Load Balancing of Transformers and Feeders by Switch Operation in Large Scale Distribution Systems," IEEE Transactions on Power Delivery, Vol. 3, No. 4, October 1988, pp. 1865-1872.

[9] M.P. Kennedy, L.O. Chua, "Neural Networks for Nonlinear Programming," IEEE Transactionson Circuits and Systems, Vol. 35, No. 5, May 1988, pp. 554-562.

[3] S . Civanlar, J J . Grainger, H. Yin, S.S.H. Lee, "Distribution Feeder Reconfiguration for Loss Reduction," IEEE Transactions on Power Delivery, Vol. 3, No. 3, July 1988, pp. 1217-1223. [4]

S.K. Goswami, S.K. Basu, "A New Algorithm for the

Reconfiguration of Distribution Feeders for Loss Minimization," IEEEIPB Summer Meeting, San Diego, California, July 1991. [5] D. Shirmohammadi, H.W. Hong, "Reconfiguration of Electric Distribution Networks for Resistive Line Losses Reduction," IEEE Transactions on Power Delivery, Vol. 4, NO. 2, April 1989, p ~ 1492-1498. . [61 T. Taylor, D.Lubkeman, "Implementation of Heuristic Search Strategies for Distribution Feeder Reconfiguration," IEEE Transactions on Power Delivery, Vol. 5, No. 1. January 1990, pp. 239-246. [7] JJ. Hopfield, D.W. Tank, "Computing with Neural Circuits: A Model," Science, Vol. 233, 8 August 1986, pp. 625-633.

316

[lo] J. Kita, 0. Sekine, Y. Nishikawa, "A Design Principle of the Analog Neural Network for Combinatorial Optimization: Findings from the Study on the Placement Problem", Proceedings of the 1991 International Joint Conference on Neural Network;, Seattle, Washington,July 1991, p. I1 A-889. [ l l ] T.X. Brown, "Neural Networks for Switching," IEEE Communicationr Magazine, November 1989, pp. 72-81. (121 B. Muller, J. Reinhardt, Neural Networks Introduction, Springer-Verlag, 1990. [13]

-

An

J.A. Freeman, D.M. Skapura, Neural Networks

-

Algorithms, Applications, and Programming Techniques,

Addison-Wesley Publishing Company, 1991. [14] M.E. Baran and F.F. Wu, "Network Reconfiguration in Distribution Systems for Loss Reduction and Load Balancing," IEEE Transactions on Power Delivery, Vol. 4, No. 2, April 1989, pp. 1401-1407.