IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO.

2, MAY 2000 541

New Approach with a Hopfield Modeling Framework

to Economic Dispatch
Ching-Tzong Su, Senior Member, IEEE, and Chien-Tung Lin

Abstract—This paper presents a new Hopfield model based ap- to overcome the above two drawbacks. Which applies a linear
proach for the economic dispatch problem of power systems. To input-output model for neurons. But the above method can not
solve the economic dispatch problem using the Hopfield model, an take transmission line losses into account.
energy function composing power mismatch, total fuel cost and the
transmission line losses is defined. The weighting factors associated In the proposed approach, transmission losses are consid-
with the terms of the energy function can be either appropriately ered. An energy function composed of power mismatch, total
selected or directly estimated in the proposed model. Which, how- fuel cost and transmission line losses is defined. Each term of
ever, are determined by trial and error in the conventional Hopfield the energy function is multiplied by a weighting factor which
method. To minimize the value of the energy function, the compu- represents the relative importance of that term. Selecting of the
tational procedures including a series of adjusting the weighting
factor associated with the transmission line losses and updating weighting factors closely affects whether the energy function
the unit generations and power losses are carried out. Because the can converge to a minimum, which is decisive in obtaining
weighting factors are governed by some relationships developed, optimal solutions. In the conventional Hopfield method, the
adjustment of the weighting factor is much simpler and more effec- weighting factors are found by trial and error. Furthermore,
tive in steadily achieving solutions than adjustment of the -multi- because available weighting factors usually lie within small
plier in the lambda-iteration method for economic dispatch prob-
lems. Computational results reveal that this approach can find ac- ranges for economic dispatch problems, locating these factors
curate solutions more simply and fast compared with the conven- is not an easy task. In the proposed approach, the corresponding
tional lambda-iteration method. weighting factors are either appropriately selected or directly
Index Terms—Economic dispatch, Hopfield model, energy func- estimated according to the specified power mismatch.
tion, linear input, output model, sigmoidal function. The computational procedures include a series of adjustments
of the weighting factor associated with the transmission line
losses. After each adjustment of the weighting factor, unit power
I. INTRODUCTION
generations and incremental losses are re-evaluated. The newly

T HE economic dispatch problem is one of the important op-

eration. Traditionally, to solve the problem, a Lagrangian
augmented function is first formulated [1]–[4], and the optimal
obtained power generations and incremental losses are applied
to compute power mismatch and re-adjust power generations if
necessary. Similar adjustments are repeated until optimal gener-
conditions are obtained by partial derivation of this function. ations are attained. The optimal power generations are the ones
In the traditional method, calculation of the penalty factor as which result in the minimum energy function value. Because
well as the incremental loss is always the key points in the the weighting factors are governed by some relationships devel-
solution algorithm. In the past, the incremental loss (and thus oped, adjustment of the weighting factor is much simpler and
the penalty factor) was determined by the B-coefficient method more effective than the adjustment of the -multiplier in the
[1], [2], which states that the transmission losses can be ex- lambda-iteration method for economic dispatch problems. Two
pressed in quadratic forms of the generation powers, and the examples are employed to demonstrate the application of the
problem is solved by employing the -iteration method [5] or proposed approach. Computational results reveal that the pro-
Newton-Raphson method. posed method is relatively simple and fast compared with the
Recently, the economic dispatch problem has been solved conventional lambda-iteration method.
by using the Hopfield neural network approach [6], [7]. In this
way, the objective function of the economic dispatch problem is II. THE ECONOMIC DISPATCH MODEL
transformed into a Hopfield energy function, and numerical it-
erations are applied to minimize the energy function. However, The economic dispatch problem can be mathematically de-
due to the use of a sigmoidal neuron model, two drawbacks arise scribed as follows
in the solving procedures. The first is the selection of appro-
(1)
priate weighting factors for the energy function. The second is
the requirement of large computational burden to obtain an op-
timal solution. In [7], an analytic Hopfield method is proposed where
: index of dispatchable units
Manuscript received November 5, 1998; revised May 6, 1999. : input-output cost function of unit
C.-T. Su is with the Institute of Electrical Engineering, National Chung Cheng : generated power of unit
University, Chiayi 621, Taiwan.
C.-T. Lin is with Taiwan Power Company. : cost coefficients of unit
Publisher Item Identifier S 0885-8950(00)03784-6. : set of all dispatchable units
0885–8950/00$10.00 © 2000 IEEE

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542 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000

subject to the following constraints: The time derivative of the energy function can be proven to
a) Power balance constraint be negative. Therefore, in the computational process the model
state always moves in such a way that the energy function grad-
(2) ually reduces and converges to a minimum.

where IV. MAPPING OF THE ECONOMIC DISPATCH MODEL ONTO THE

: total load demand HOPFIELD MODEL
: transmission losses To solve the ED problem using the Hopfield method, an en-
b) Generation limits of each unit ergy function is defined as follows
Each unit has its maximum and minimum generation limits
i.e.,

(3) (9)
where
: minimum generation limit of unit This energy function is composed of power mismatch, total fuel
: maximum generation limit of unit cost and transmission losses. The positive weighting factors ,
The transmission losses are traditionally represented by and introduce the relative importance for their respective
associated terms.
By changing the generation output of unit from to ,
(4) the transmission losses change from to , which may be
expressed as
where
: total number of units (10)
: transmission loss coefficient
The well known solution method [5] to this problem using the where is the linearized power loss change, and is the
coordination equation is incremental loss of unit at power generation of .
To obtain the connective conductances and the external in-
(5) puts, substituting (10) into (9) yields

where is the penalty factor of unit given by

(6)

and is the incremental loss of unit . The penalty

factors can be computed from losses formula (4).

III. THE HOPFIELD MODEL

The Hopfield model is a mutual coupling neural network and
nonhierarchical structure. The dynamic characteristic of each
neuron can be described by the following differential equation
[8]
(11)
(7)

where where
: input of neuron
: interconnection conductance from the output of , is a constant.
neuron to the input of neuron By comparing (8) and (11) and letting the output of neuron
: selfconnection conductance of neuron represent the power generation of unit , the connective conduc-
: external input to neuron tances and external input of neuron for the Hopfield network
: output of neuron may be given as
The energy function of the continuous Hopfield model can be
defined as

(8)
(12)

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SU AND LIN: NEW APPROACH WITH A HOPFIELD MODELING FRAMEWORK TO ECONOMIC DISPATCH 543

TABLE I
COMPUTATIONAL RESULTS FOR EXAMPLE 1

Fig. 1. The linear input-output neuron model.

V. THE LINEAR NEURON MODEL

TABLE II
Instead of applying the conventional sigmoidal neuron INPUT DATA OF EXAMPLE 2
model, we will employ a linear input-output model for the
neuron. Fig. 1 shows the linear input-output function for the
neuron

(13)

where both and are constants, and

and

(14)

(15)

VI. THE SOLUTION FORMULATIONS AND COMPUTATIONAL

PROCEDURES
From (7), the dynamic equation of a neuron may be given as
By differentiating (13) and substituting it into (17), the dy-
(16) namic equation becomes

(19)
Substituting (12) into (16), the dynamic equation becomes
Solving (19), the generation of unit is given as

(20)

(17) and

Where is the power mismatch, which is expressed as (21)

(18) Here and are the values of as and

, respectively. represents the optimal generation

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544 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 15, NO. 2, MAY 2000

TABLE III
LOSS COEFFICIENTS FOR EXAMPLE 2 (THE ENTRY OF ROW i AND COLUMN j REPRESENTS THE COEFFICIENTS B IN 10 p.u.)

of unit , which is the solution for the problem. Also note that (21), (22) and (26) constitute a new approach for economic dis-
is not related to the parameters of and , and patch problems. The main computational procedures for the pro-
is a meaningless variable which does not represent time. posed approach are summarized below:
Substituting (21) into (18) yields Step 1 Input data of load demand , loss coefficients ,
weighting factors and , unit fuel cost parameters
and neuron model parameters and
.
Step 2 Initialize power generation of each unit; set
and initialize counter .
Step 3 Determine by applying (4); calculate increment
(22) loss; determine by applying (22); determine
From (21) and recalling that , we have (i.e. ) for each unit by applying (21).
Step 4 Applying (26), let .
(23)
Step 5 Is power mismatch toler-
The equal-incremental-cost criterion states ance ? If yes, go to step 6; otherwise, go to step 3.
Step 6 Is tolerance for all units ? If
yes, go to step 7; otherwise, let , then go to
(24) step 3.
substituting (23) into (24) gives Step 7 Stop and output the results.

VII. APPLICATION EXAMPLES

(25) To illustrate the application and demonstrate the effectiveness
of the proposed method, two example systems are employed.
Remembering that , we have from (25) the
Example 1: The example system has three generating units
following relationship
to supply a total load demand of 210 MW. Input data including
(26) fuel cost functions and the loss expression are given in [5]. The
weighting factors and should be appropriately selected to
It is worth noting that (25) is a necessary condition assuring sat- achieve a small power mismatch. The amount of power mis-
isfaction of the equal-incremental-cost criterion. Together, (4), match is related to the ratio of to . The higher the ratio is,

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SU AND LIN: NEW APPROACH WITH A HOPFIELD MODELING FRAMEWORK TO ECONOMIC DISPATCH 545

TABLE IV VIII. CONCLUSIONS

COMPUTATIONAL RESULTS FOR EXAMPLE 2
A new approach to economic dispatch problems using
the Hopfield model is proposed. The approach employs a
linear model to describe the input-output relationship for the
neuron. The proposed method essentially obeys the equal-in-
cremental-cost criterion followed by conventional economic
dispatch methods. However, the proposed method does not re-
quire to compute incremental fuel costs in the solving process.
The Hopfield neural network is a mutual coupling network
and nonhierarchical structure. Its connective conductances and
external input can be determined by employing the system
data. Thus, the proposed model, unlike other neural networks,
requires no training. Furthermore, because the method has a
Hopfield modeling framework, hardware implementation for
the proposed approach is promising because of the advantage
of the real time response. Still, there are a lot of efforts to be
devoted.
The B-coefficients loss formula employed in the paper is not
an accurate representation. However, there is no great difficulty
in making some changes in the formulations developed so that
the proposed approach can employ different loss representa-
tions instead of the B-coefficients formula. Two example sys-
tems have been employed to illustrate the application of the pro-
posed method. Computational results reveal that the proposed
method is superior to the conventional lambda-iteration method
in computational requirement.

the smaller the power mismatch becomes. By appropriately se- REFERENCES

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eration method and the proposed method are given in Table IV. Ching-Tzong Su received his B.S., M.S. and Ph.D. degrees all in Electrical En-
Computational results from example 2 reconfirm that the gineering from National Cheng Kung University, Tainan, Taiwan. He is a Pro-
fessor of the Department of Electrical Engineering at National Chung Cheng
proposed method is more efficient than the lambda-iteration University, Chiayi, Taiwan. His research interests are in the areas of power
method in computational requirement. system planning, reliability engineering, and optimal control. Dr. Su is a Se-
Basically, the proposed method is computationally faster than nior Member of IEEE Power Engineering Society and Reliability Society, Vice-
Chairman of Bulk Power Network Committee, EMSPA. Taiwan.
the lambda-iteration method. Because the solution procedures
employed by the former is more efficient than that employed Chien-Tung Lin was born in Taiwan on March 1, 1969. He graduated from
by the latter. However, the difference of CPU time required be- the Department of Electrical Engineering, National Yuen-Lin Institute of Tech-
tween the two methods varies, because the computational re- nology, Yuen-Lin, Taiwan. He received his M.Sc. degree in Electrical Engi-
neering from National Chung Cheng University, Chiayi, Taiwan. Since 1991
quirement of the lambda-iteration method is related to its initial he has been with Taiwan Power Company. His research interests include secu-
value of guessed. rity control and application of artificial intelligence to power systems.

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