AN INTEGRATED COMPUTATIONAL APPROACH FOR SOLVING DYNAMIC ECONOMIC DISPATCH PROBLEMS

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AN INTEGRATED COMPUTATIONAL APPROACH FOR SOLVING DYNAMIC ECONOMIC DISPATCH PROBLEMS

Ji-Pyng Chiou*

Department of Electrical Engineering Ming Chi University of Technology New Taipei City, Taiwan 243, R.O.C.

Key Words: CODEQ, differential evolution, sequential quadratic programming,

dynamic economic dispatch.

ABSTRACT

An integrated computational approach proposed to solve the dynamic economic dispatch (DED) systems is presented in this paper. This computa- tional approach was organized into two separate parts. First, one point was searched by CODEQ method. In addition, this point as an initial condition used in the sequential quadratic programming (SQP) method finds the solution. The CODEQ method is a new population-based, parameter-free, meta-heuristic algorithm integrating concepts from Chaotic searches, opposition-based learn- ing, differential evolution (DE) and quantum mechanics. The use of the CODEQ method can overcome the parameters selection drawback of DE method. The SQP method was used to speed up convergence due to its fast convergence property. One benchmark function was used to compare the performance of the CODEQ and DE methods. The computation results showed that the performance of the CODEQ method outperformed the DE method. One 10-unit DED system was solved by the proposed computation approach.

I. INTRODUCTION

Differential evolution (DE) as developed by Stron et al. [1] is one of the best evolutionary algorithms (EAs), and has proven to be a promising candidate to solve real valued optimization problems [2]. This method also turned out to be one of the best genetic algorithms for solving the real-valued test function suite of the first International Con- test on Evolutionary Computation, which was held in Na- goya in 1996. DE is a stochastic search and optimization method. The fittest of an offspring competes one-to-one with that of the corresponding parent, which is different from the other EAs. This one-to-one competition gives rise to a faster convergence rate. However, this faster convergence also leads to a higher probability of obtaining a local optimum because the diversity of the population descends faster during the solution process. To overcome this drawback, the parameters selection is very important for the DE

algorithm. That is a drawback for all the evolutionary algorithms. To increase the search ability and alleviate the parameters selection problem in the DE method, many variant methods [3-13] are proposed. Among of the variant methods, Omran and Salman [7] proposed a CODEQ method to overcome the parameter selection problem and increase the convergence property of DE method. Five constrained benchmark systems from the literature are investigated to prove the performance of CODEQ. The experiments con- ducted show the CODEQ provides excellent with the added advantage of no parameter tuning [7].

The aim of the dynamic economic dispatch (DED) problem is to schedule the generator outputs with predi- cated load demand and the various systemic and operating constraints over a certain period of time economically.

Thus, the objective of the DED is to minimize the cost of energy subject to the various constraints. Different from the conventional economic dispatch problem, the DED problem

* Corresponding author: Ji-Pyng Chiou, e-mail: jipyng@mail.mcut.edu.tw

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takes into consideration the limits on the generator ramping rate to maintain the life of generation equipment [14-31].

It is a dynamic optimization problem that is difficult to solve because of its non-convex property. To speed the solution search of the DED problems, the DED problem can divide the entire dispatch period into a number of small time intervals and a static economic dispatch has been em- ployed to solve the problem in each interval. The hybrid DE-SQP and hybrid PSO-SQP methods are proposed by Elaiw et al. [32] for solving the DED problem with valve- point effects. Cai et al. [33] proposed a hybrid CPSO and sequential quadratic programming (SQP) method to solve the DED problem considering the valve-point effects.

Victoire and Jeyakumar [34] proposed a hybrid solution me- thodology integrating particle swarm optimization (PSO) algorithm with the sequential quadratic programming (SQP) method for the DED problem. However, the drawback of the parameter selection problem is still exists in these hybrid methods. And, the DED problems are only considering the valve-point effects.

In this study, an integrated computational approach that combining CODEQ algorithm [7] and SQP method is proposed to solve the dynamic economic dispatch problem. The concepts of chaotic search, opposition-based learning, and quantum mechanics are used in the CODEQ method to overcome the drawback of selection of the crossover factor and scaling factor used in the original DE method. So, only two parameters, the population size and maximum iteration number, are needed in the CODEQ method. One point searched by the CODEQ method can be used in the SQP method as an initial condition. Then, the optimal solution can be found fast based on the convergence property of the SQP method. One benchmark function is used to compare the performance of the CODEQ and DE methods. And, one 10-unit DED system from the literature is solved by the proposed computational approach.

II. PROBLEM FORMULATION

The dynamic economic dispatch problem considering the various systemic and operating constraints can be mathe- matically described as follows:

1 1

2 1 1

min

it

T n

it it

P t i

T n

i i it i it i i itmin it

P t i

F P

c b P a P e sin f P P º ª »

¬ ¼

¦¦

(1) subject to the following constraints:

1. Power Balance Constraint

1 n

it Dt Lt

i

P P P

¦

⁽²⁾

2 1 n

Lt ii it

i

P

¦

B P ⁽³⁾

2. Ramping Rate Limits

itmin it itmax

P dP dP (4)

( 1)

( )

itmin imin i t i

P max P , P DR (5)

( 1)

( )

itmax imax i t i

P min P ,P UR (6)

3. System Spinning Reserve Constraints

1 n

it Rt

i

S tS

¦

⁽⁷⁾

^ `

it itmax it imax

S min P P ,S i Ȍ ȥ (8)

Sit 0 (9) i ȥ

4. Generation Limits of Units

Units with prohibited operating zones:

1 l

itmin it i,

P dP dP or (10)

1 2

u l

i, j it i, j i

P dP dP , j !, , n or (11)

i u

i,n it itmax

P dP dP , i (12) ȥ

Units without prohibited operating zones:

itmin it itmax

P dP dP (13)

(3)

In this study, the treatment of constraints is performed with the penalty method. The penalty method is among the most popular techniques used to handle constraints, is easy to implement, and is considered efficient. The penalty method is usually a close degree to the nearest solution in a reasonable region, and it can allow an objective function effort to arrive at the optimum solution.

III. CODEQ METHOD

The main idea of the CODEQ method is using the concepts of chaotic search, opposition-based learning, and quantum mechanics into the original DE method to overcome the parameters selection drawback. The CODEQ algorithm is briefly described in the following.

1. Step 1: Initialization

Input system data and generate the initial population.

The initial population is chosen randomly and would at- tempt to cover the entire parameter space uniformly. The uniform probability distribution for all random variables as following is assumed

0 ( ), 1

i min Vi max min i , , Np

Z Z Z Z ! (14)

Where

V

i (0, 1] is a random number. Zmax and Zmin

are the maximum bounds and minimum bounds, respec- tively. The initial process can produce Np individuals of

0

Zi randomly.

2. Step 2: Mutation Operation

The essential ingredient in the mutation operation of the DE algorithm is the difference vector. Each individual pair in a population at the G-th generation defines a differ- ence vector Djk as

G G

jk j k

D Z Z (15)

The mutation process at the G-th generation begins by randomly selecting either two or four population individuals

G

Zj, Z^G_k , Z^G_l and Z^G_m for any j, k, l and m. These four individuals are then combined to form a difference vector Djklm as

( ^G ^G) ( ^G ^G)

jklm jk lm j k l m

D D D Z Z Z Z (16)

A mutant vector is then generated based on the present individual in the mutation process by

ˆ^G_i1 ^G_p F _jklm, i 1, , N_p

Z Z D ! (17)

Where scaling factor, F, is a constant. And, j, k, l and m are randomly selected.

However, the scaling factor value is depends on the problem. Different from the original DE algorithm, the concept of the quantum mechanics [7, 36-37] is used to generate the noisy replica from the parent individual in CODEQ algorithm as follows:

1

1 2

ˆ^G_i ^G_i ( ^G_i ^G_i ) ln( )1 , i 1, , N , i_p 1 i2 i u

z z

Z Z Z Z !

(18)

Where u (0, 1] is a random number.

3. Step 3: Estimation and Selection

The parent is replaced by its offspring, if the fitness of the offspring is better than that of its parent. Contrarily, the parent is retained in the next generation if the fitness of the offspring is worst than that of its parent. Two forms are represented as follows

^ `

1 ( ) (ˆ 1)

G G G

i arg min f i , f i

Z Z Z (19)

^ `

1 ( )

G G

b arg min f i

Z Z (20)

Where argmin means the argument of the minimum.

4. Step 4: Exclude Operation If Necessary

To increase the convergence of the CODEQ algorithm, the exclude operation is considered. First, a new individual is created as follows:

1 1

1 1 1 1

2

0.5 (2 1),

G

min max worst

G

w G G G G

best i1 i

Ȗ , if į

c otherwise

d

®°°¯ Z Z Z

Z Z Z Z

(21)

Where

J

^and

G

are randomly generated numbers uniformly distributed in the range of (0, 1). Z^G_worst¹ and Z^G_best¹ are the worst and best individual in the (G 1)th generation. c^G¹ is the chaotic variable defined as follow:

1 / 0,

(1 )/(1 ) , 1

G G

G

G G

c p, if c p

c c p , if c p

®°°¯ ª¬ (22)

Where c⁰ and p are initialized randomly within the interval (0,1).

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Start

Read System data

Generate the initial population

Apply mutation operation to find the offspings

Choose the individuals of the next generation

by estimation and selection operations

Reaching to terminate conditions?

End Execute exclude

operation

No

Yes

Fig. 1 Main calculation procedures of the CODEQ method

The worst individual in the G-th generation is replaced by the generated individual, if the fitness of the generated individual is better than that of worst individual in the G-th generation.

5. Step 5

Repeat step 2 to step 4 until the maximum iteration quantity or the desired fitness is accomplished. The computational process of the CODEQ is as shown in Fig. 1.

IV. SEQUENTIAL QUADRATIC PROGRAMMING METHOD

Sequential quadratic programming (SQP) is a popular method for solving nonlinear programming problems [38-42]. Many hybrid methods [43-46] are also proposed to solve the various problems because the property of fast convergence. The nonlinear programming problems can be expressed as follows:

0 1 0 1

j h

k g

min max

min J( ) s.t.

h ( ) , j , , n g ( )d , k , , n

d d

z z

z z z z z

!

(23)

Where hj(z) is the jth equality constraint, nh is the number of equality constraints, gk(z) is the kth inequality constraint,ng is the number of inequality constraints, zmin

andzmax are the decision variables of the lower and upper

bounds and can be expressed in the semi-formulas of inequality constraints.

The Lagrange function [47] for this approximate problem is defined as:

1 1

( ) ( ) ( ) ( )

h ng

n

L j j k k

j k

J ^z J ^z

¦

Į h ^z

¦

Ĳ g ^{z (24)} Where

D

^j^and

W

^k are the Lagrange multipliers of the

jth equality and kth inequality constraints, respectively.

And, we can redefine the objective function as follows:

^ `

2 2

1

2 2

1

1 2 1 2

h

g n

a k k k k

k n

k k k k

k

J J Ȧ h ș ș

ȕ g ȣ ȣ

ª º

¬ ¼

¦

z z z

z

(25)

Where ¢gk² is defined as max{gk, 0}. From the definition of Equation (25), the Equation (25) can be written as Equation (26).

2 2

1 1

2 2

1 1

1 1 2 2 0

h g

h g n n

L k k k k k k

k k

a n n

k

L k k k k

k k k

J Ȧ h ȕ g , if g ȣ

J J Ȧ h Ĳ , if g ȣ

ȕ

t

°° ®°

°¯

¦ ¦

(26)

The effect of changes in the multipliers

T

k and v_k upon the shape of contours of the augmented Lagrange function J_a for each individual is not obvious but can be seen in an expression for the Hessian matrix of J_a. Assume that the objective function J(z) and the constraints, h_j(z) and g_k(z), are twice continuously differentiable. Using the definition of the bracket operator, the gradient of J_a for each individual is as Equation (27).

1 1

h ng

n

a k k k k k k k k

k k

J J Ȧ h ș h ȕ g ȣ g

^z

¦

ª¬ º¼

¦

(27)

An additional differentiation produces the Hessian matrix of J_a

^ `

2

2 2 2

1

2 2 1

h

g a

n

k k k k k

k n

k k k k k

k

J

J Ȧ h ș h h

ȕ g Ȟ g g

ª¬ º¼

¦

z

(28)

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Table 1 Computational results for one hundred runs of example 1

Mutation strategy 1 2 3 4 5 6 CODEQ Largest 38.8503 38.8486 38.8503 38.8503 38.8503 38.8503 38.8503 Smallest 33.8467 35.8129 31.1505 38.7328 38.6328 33.8505 38.7247 Average 38.1874 38.1573 36.6145 38.8105 38.8140 37.7809 38.8132 STD 0.9780 0.5552 2.0062 0.0551 0.0565 1.1941 0.0416 Count 10 2 6 65 70 14 55

Table 2 Standard deviation for various population size

Population size 1 2 3 4 5 6 CODEQ 10 1.7443 1.1668 3.1256 0.4549 0.9143 1.7201 0.1346

5 2.6685 2.0677 3.5187 1.2180 1.2985 2.9524 0.4363

It can be shown that the radius of curvature of a par- ticular contour of Ja varies as ²Ja. If both the equality and inequality constraints are linear, then the Hessian matrix of Ja becomes

2 2

1 1

h ng

n

a k k k k

k k

J J Ȧ h ȕ g

¦

ª¬ º¼

¦

ª¬ º¼ ⁽²⁹⁾ As a result, the Hessian matrix is independent of the

multipliers,

T

^k^{and v}^k^.

V. APPLICATION OF THE PROPOSED METHOD

One benchmark functions and one modified 10-unit dynamic economic dispatch (DED) system are used to in- vestigate the performance of the proposed method.

1. Example 1

Let us consider the maximization problem is described by:

1 2

1 2 21.5 1 4 1 2 20 2

max J z , zz ,z z sin ʌz z sin ʌz (30) Where -3 z1 12.1 and 4.1 z1 5.8. This problem has been solved by the simple genetic algorithm using the population size of 20. The best result in generation 396 had the best objective function value of 38.827553 [48].

The convergence property of the CODEQ algorithm and the original DE method are compared via this example firstly. The population size Np = 20, scaling factor F = 0.1, factorCR = 0.5, and maximum iterations of 300, are used

in the DE method for solving this example. Six strategies of mutation operation are respectively used to solve this example. The solution of this example is repeatedly solved one hundred times. The largest and smallest values among the best solutions of the one hundred runs are respectively expressed in Table 1. The average for the best solutions of the one hundred runs and the standard deviation with respect to the average are also shown in this table. A smaller standard deviation implies that almost all the best solutions are close to the average best solution. From the Table 1, the standard deviation for the CODEQ algorithm is smaller than the other mutation strategy. That implies the convergence property of the CODEQ algorithm is better than the original DE method. Five parameters including the population size, mutation operation, crossover factor, scaling factor, and the maximum iteration number must be set in the original DE method. But, only two parameters including the population size and the maximum iteration number need to set in the CODEQ algorithm. The best solutions for these one hundred runs are compared to the best objective function value obtained by the simple genetic algorithm. The number of times that these best solutions were greater than 38.827553 is shown in Table 1. From the Table 1, the number of the successful runs that the best solutions were greater than 38.827553 is 10, 2, 6, 65, 70, and 14 for six different strategies of mutation operation.

The number of the successful runs that the best solutions were greater than 38.827553 is 55 in the CODEQ algorithm.

Only fourth and fifth mutation strategy is better than the CODEQ algorithm. Table 2 lists the standard deviation values when the population size is reassigned to 10 and 5 to solve this example one hundred times, respectively.

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Table 3 Input data of the 10-unit system

Unit 1 2 3 4 5 6 7 8 9 10

ai 0.00043 0.00063 0.00039 0.00070 0.00079 0.00056 0.00211 0.0048 0.10908 0.00951 bi 21.60 21.05 20.81 23.90 21.62 17.87 16.51 23.23 19.58 22.54 ci 958.20 1313.6 604.97 471.60 480.29 601.75 502.70 639.40 455.60 692.40 ei 450 600 320 260 280 310 300 340 270 380 fi 0.041 0.036 0.028 0.052 0.063 0.048 0.086 0.082 0.098 0.094

Pi min 150 135 73 60 73 57 20 47 20 55

Pi max 470 460 340 300 243 160 130 120 80 55

Si max 50 0 30 30 0 0 50 50 0 0

Bii 0.000042 0.000069 0.000093 0.000093 0.000085 0.000053 0.000092 0.000065 0.000042 0.000070 URi 80.00 80.00 80.00 50.00 50.00 50.00 30.00 30.00 30.00 30.00 DRi 80.00 80.00 80.00 50.00 50.00 50.00 30.00 30.00 30.00 30.00

Table 4 Prohibited zones of units

Unit Zone 1 (MW) Zone 2 (MW) Zone 3 (MW) 2 [185, 225] [305, 335] [420, 450]

5 [90, 100] [153, 168] [195, 210]

6 [77, 85] [122, 132] [143, 153]

9 [30, 40] [55, 65]

Table 5 Load demand for 24 hours

Hour Load (MW) Hour Load (MW) Hour Load (MW)

1 1036 9 1924 17 1480

2 1110 10 2072 18 1628

3 1258 11 2146 19 1776

4 1406 12 2179 20 2072

5 1480 13 2072 21 1924

6 1628 14 1924 22 1628

7 1702 15 1776 23 1332

8 1776 16 1554 24 1184

From the above discussion, the convergence property of the CODEQ algorithm is better than the original DE method.

2. Example 2

The modified 10-unit DED system [35] is used to il- lustrate the performance of the proposed algorithm. In this system, four units have prohibited operating zones, and the remaining six units contribute the required spinning reserve to the system. The valve-point effects are also considered in this example. Input data of the 10-unit system are listed in Table 3. Table 4 lists the prohibited

zones of units 2, 5, 6, and 9. These zones result in four disjoint feasible sub-regions for each of units 2, 5, and 6, and three for unit 9. Hence those zones result in a non-convex decision space which consists of 192 convex sub-spaces for every dispatch interval of the example system. The load demand of the system is divided into 24 dispatch intervals, shown in Table 5. The spinning reserve is required to be greater than 5% of the load demand at every dispatch interval. The fuel cost obtained by the proposed method is 1,051,964.4$/day that is better than that of [35].

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VI. CONCLUSIONS

The performance of the CODEQ algorithm and DE method is compared with a benchmark function. And, the hybrid method of the CODEQ algorithm and SQP method is used to solve the 10-unit dynamic economic dispatch system in this paper. The concepts of chaotic search, opposition- based learning, and quantum mechanics are used in the CODEQ method to overcome the drawback of selection of the crossover factor, scaling factor, and mutation operator used in the original differential evolution (DE) method.

The SQP method is a local search method which has a fast convergence property. From example 1, the convergence property of the CODEQ algorithm is better than the original DE method. From example 2, the proposed hybrid method is properly used to solve dynamic economic dispatch problems.

NOTATION

i index of dispatchable units t index of time intervals

Fit fuel cost function of unit i at the t-th time interval Pit power generation of unit i at the t-th time interval n the number of generating units

T the number of intervals in the entire dispatch period

ai,bi,ci fuel cost coefficients of unit i

ei,fi constants from the valve-point effects of unit i

Pitmin minimum generation limit of unit i at the t-th

time interval

PDt load demand at the t-th time interval PLt power loss at the t-th time interval Bii power loss coefficient

Pitmax maximum generation limit of unit i at the t-th

time interval

Pi min minimum generation limit of unit i

Pi max maximum generation limit of unit i

Pi(t-1) power generation of unit i at the (t 1)-th time

interval

URi the up ramp limit of the i-th unit DRi the down ramp limit of the i-th unit

Sit spinning reserve of unit i at the t-th time interval SRt system spinning reserve requirement at the t-th

time interval

Si max maximum spinning reserve of unit i

\

set of all dispatchable units with prohibited zones

< set of all dispatchable units

,1 l

Pi lower bound of the first prohibited zone of unit i

, l

Pi j lower bound of the j-th prohibited zone of unit i

, 1 u

Pi j upper bound of the (j 1)-th prohibited zone of uniti

,_i u

Pi n upper bound of the ni-th prohibited zone of unit i ni number of prohibited zone in unit i

ACKNOWLEDGEMENTS

Financial research support from the National Science Council of the R.O.C. under grant NSC 99-2221-E-131-037 is greatly appreciated.

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Manuscript Received: Feb. 16, 2016 First Revision Received: Feb. 16, 2016 Second Revision Received: Mar. 29, 2016 and Accepted: Apr. 13, 2016