DHIMAN: A novel algorithm for economic Dispatch problem based on optimization met Hod...

January 24, 2019 14:6 MPLA S0217732319500329 page 1

Modern Physics Letters AVol. 34 (2019) 1950032 (14 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0217732319500329

DHIMAN: A novel algorithm for economic Dispatch

problem based on optimization metHod usIng Monte Carlo

simulation and Astrophysics coNcepts

Gaurav Dhiman∗,§, Pritpal Singh†, Harsimran Kaur‡,¶ and Ritika Maini‡,‖

∗Computer Science and Engineering Department,

Thapar Institute of Engineering & Technology, Patiala 147004, Punjab, India†Smt. Chandaben Mohanbhai Patel Institute of Computer Applications,

CHARUSAT Campus, Changa, Anand 388421, Gujarat, India‡Department of Computer Science, Govt. Bikram College of Commerce,

Patiala 147004, Punjab, India§[email protected]; [email protected]

†[email protected]¶[email protected]‖maini [email protected]

Received 24 September 2018Revised 20 October 2018

Accepted 26 November 2018

Published 25 January 2019

This paper presents a new model using optimization approach for efficient prediction of

load in real-life environment. Monte Carlo simulation and Schrodinger equations provide

the effective number of solutions. This technique is useful in representation of relation-ships between different models. The proposed algorithm is verified and validated with

various state-of-the-art approaches for solving economic load power dispatch problem todemonstrate its efficiency. Experimental results signify that the proposed algorithm is

more precise than existing competing models.

Keywords: Astrophysics; Quantum; Schrodinger; Monte Carlo.

PACS No.: 03.70.+k

1. Introduction

The economic dispatch (ED) is the problem of minimizing the operating cost

and scheduling all the generating units while satisfying the constraints.1 The ED

problem is defined as a non-smooth optimization problem because of valve-point

§Corresponding author

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http://dx.doi.org/10.1142/S0217732319500329

mailto:[email protected]; [email protected]

mailto:[email protected]




G. Dhiman et al.

effects and multi-fuel effects. These effects may raise a problem in finding the global

optimum. Nowadays, researchers are trying to develop efficient approaches2–7 for

solving the ED problem.

The ED problem is further categorized into two categories: Classical and Non-

classical. Classical methods such as lambda iteration,8 gradient method,9 branch

and bound,10 and quadratic programming11 may increase the cost convergence

monotonically to solve ED problem. However, these methods fail to deal with the

presence of prohibited operating zones (POZ) and multiple fuel options which make

the problem discontinuous.

Various non-classical methods are able to solve these difficulties in the ED

problem. The non-classical methods are further classified into two categories as:

Evolutionary-based algorithms and Swarm-based algorithms. The evolutionary-

based algorithms are inspired by the theory of natural selection and biological evolu-

tion, such as reproduction, selection, recombination, and mutation. The well-known

evolutionary-based algorithms for ED problems are: Genetic Algorithm (GA),12

Real-Coded GA (RCGA),13 Modified Differential Evolution (MDE),14 hybrid DE

algorithm based on PSO algorithm (DEPSO),15 Improved Fast Evolutionary Pro-

gramming (IFEP),16 Improved Genetic Algorithm (IGA),17 Chaotic Differential

Evolution and Quadratic Programming (DEC-SQP),18 and hybrid differential evo-

lution with biogeography-based optimization for ED problem.19

On the other hand, Swarm-based techniques are inspired by the social behavior

of birds, animals, reptiles, and so on.20–27 The search agents or population can in-

teract with each other locally as well as globally to find the optimum solution. There

are several well-known algorithms that are based on swarm intelligence techniques

as: Particle Swarm Optimization (PSO),28 Cuckoo Search Algorithm (CSA),29 It-

eration PSO with time-varying acceleration coefficients,30 PSO for dynamic ELD

problem,31 hybrid Firefly Algorithm (FA),32 Bacterial Foraging Algorithm (BFA),33

quantum PSO,34 multi-objective CSA,35 and Geese Movement-Based Optimization

Algorithm (GMBOA).24

In this paper, quantum and astrophysics-based optimization approach is pro-

posed for solving the convex and non-convex ED problem (DHIMAN). The perfor-

mance of the proposed algorithm has been compared with BBO,36 PSO,28 CSA,29

NPSO-LRS,37 GA,28 NPSO,37 Hopfield model,38 Lambda iteration,38 and SOH-

PSO.39

The rest of this paper is structured as follows. Section 2 presents the problem

formulation of economic load power dispatch. The basic and fundamental concepts

of spotted hyena optimization algorithm is discussed in Sec. 3. In Sec. 4, the pro-

posed DHIMAN algorithm is applied to ED problems. The experimental results

and discussion are presented in Sec. 5. The conclusion and some future works are

given in Sec. 6.

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DHIMAN

2. Problem Formulation

2.1. Economic dispatch problem

The ED is considered as a nonlinear constrained optimization problem. There are

four types of ED problems which are described as follows.

2.1.1. Economic dispatch with quadratic cost function and transmission loss

(EDQCTL)

The objective function Fe with N committed generators of total cost is mathemat-

ically described as follows:

Fe =

N∑k=1

FkPk =

N∑k=1

xk + ykPk + zkP2k , (1)

where FkPk is the cost function of kth generator; xk, yk and zk are the coefficients

cost of kth generator; and Pk is the power output of kth generator. The ED problem

is the minimization problem with subject to the following constraints.

2.1.1.1. Real power balance

The mathematical formulation is as follows:N∑

k=1

Pk − PD − PL = 0 , (2)

where the transmission loss PL is calculated by using B-coefficients.

PL =

N∑k=1

N∑l=1

PkBklPl +

N∑k=1

B0kPk +B00 . (3)

2.1.1.2. Real power generation capacity

The generator power is defined within its lower limit Pmink and upper limit Pmax

k

as follows:

Pmink ≤ Pmax

k , kεN . (4)

2.1.2. Economic dispatch with quadratic cost function, prohibited operating

zones and transmission loss (EDQCPOZTL)

The objective function is to be minimized with subject to constraints using Eqs. (2)–

(4), and following POZ constraints.

2.1.2.1. Prohibited operating zones (POZ)

These zones are normally the range of a power output, which is generated by gen-

erator, where extreme vibrations occurs of the turbine shaft during operation. This

extreme vibration might cause effects on the shaft and bearings. The operating

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G. Dhiman et al.

zones of unit which is feasibly applicable is described as follows:

Pmink ≤ Pk ≤ P ll

k,1 ,

Pulk,m−1 ≤ Pk ≤ P ll

k,m , m = 2, 3, . . . , nk ,

Pulk,nk≤ Pk ≤ Pmax

k ,

(5)

where m defines the number of POZs of kth generator, Pulk,m−1 is the upper limit

and P llk,1 is the lower limit of kth generator of (m−1)th and mth POZ, respectively.

However, the total number of POZ is nk of kth generator.

2.1.3. Economic dispatch with valve point loading effect and without

transmission loss (EDVPL)

A recurring rectified sinusoid method integrates to the quadratic function to define

the effect of valve points.12 The fuel cost function including valve point loadings is

defined as follows:

Fe =

N∑k=1

FkPk =

N∑k=1

xk + ykPk + zkP2k + ‖vk × sin{wk × (Pmin

k − Pk)}‖ , (6)

where vk and wk are the cost coefficients of kth generator.

2.1.4. Economic dispatch with valve-point loading effect and multi-fuel

options (EDVPLMF )

Here, generators are supplied with multi-fuel sources where each generator rep-

resents with various quadratic functions and search the most economical fuel to

burn,40 which can be expressed as: if Pminkl ≤ Pk ≤ Pmax

kl , where j = 1, 2, . . . , Nf

and Nf describes the number of fuel types, then

FkPk =

N∑k=1

xkl + yklPk + zklP2k + |vkl × sin{wkl × (Pmin

kl − Pk)}| . (7)

2.1.5. Generation level of slack generator

Let the power loading of (k−1) generators are familiar, then the power level loading

of kth generator, i.e. the slack generator can be determined as follows:

PN = PD + PL −N−1∑k=1

Pk . (8)

The transmission loss PL including the slack generator is defined as follows:

PL =

N−1∑k=1

N−1∑l=1

PkBklPl + 2PN

(N−1∑k=1

BNkPk

)+BNNP

2N

+

N−1∑k=1

B0kPk +B0NPN +B00 . (9)

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DHIMAN

Hence, after expanding, Eq. (8) becomes

BNNP2N +

(2

N−1∑k=1

BNkPk +B0N − 1

)PN

+

(PD +

N∑k=1

N∑l=1

PkBklPl +

N−1∑k=1

B0kPk −N−1∑k=1

Pk +B00

)= 0 . (10)

Therefore, the loading of the slack generator can be found after solving Eq. (10).

3. Spotted Hyena Optimizer (SHO)

The main inspiration of SHO is social behavior and the communal relationship

of spotted hyenas.41 This algorithm mainly focuses on the attacking, encircling,

hunting, and searching behavior of the trusted group of spotted hyenas. The SHO

algorithm has successfully mimicked the behavior of spotted hyenas to obtain the

best possible optimal solution.

Encircling behavior in the SHO is captured by following the set of equations41:

Xh = |A ·Cp(x)−C(x)| , (11)

C(x+ 1) = Cp(x)−B ·Xh , (12)

where Xh represents the length spotted hyena has to cover to reach prey, and x

indicates the running iteration at given instance. The position vectors corresponding

to prey and spotted hyena are represented by Cp and C, respectively. | | and ·are the symbols used for representing absolute value and multiplication vectors,

respectively. Two coefficient vectors, viz., A and B are used in the proposed SHO.

In this study, A and B are computed as follows:

A = 2 · xd1 , (13)

B = 2h · xd2 − h , (14)

h = 5−(

Itr× 5

MaxItr

), where Itr = 0, 1, 2, . . . ,MaxItr . (15)

Here, the value of h is decremented from 5 to 0 in each iteration. Value of random

vectors xd1 and xd2 is from the range [0, 1]. One can reach different places by

changing the position vectors A and B.

The algorithm has the advantage of saving local optimal solution and asks for

position update from other agents.

Hunting behavior of spotted hyenas is mapped and potential hunting regions

are identified using following equations:

Xh = |A ·Ch −Ck| , (16)

Ck = Ch −B ·Xh , (17)

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G. Dhiman et al.

Oh = Ck + Ck+1 + · · ·+ Ck+N , (18)

where N represents the number of iterations which can be computed as follows:

N = countnos(Ch,Ch+1,Ch+2, . . . , (Ch + M)) , (19)

C(x+ 1) =Oh

N. (20)

In Eq. (19), “nos” indicate count of solutions, which are somehow similar to

optimally best solution for a search space under consideration. Oh constitutes

the cluster of optimal solutions. M indicates randomly initialized vectors within

[0.5, 1]. C(x + 1) identifies the best optimal solution and helps in the updation of

search agents.

The assurance of exploration is achieved by vector B with random values. If

the randomly assigned values > 1 or < 1 then, search agent will move away from

prey. A also helps in exploration and assigns random values which acts as weights

of prey. The exploitation of the SHO algorithm starts when |B| < 1 with random

initialization of B.

4. Proposed Quantum and Astrophysics-Based Spotted Hyena

Optimizer (DHIMAN)

In quantum computing, the entanglement of quantum particles is necessary for

quantum mechanics. This unique property of quantum has no analog in classi-

cal mechanics. Therefore, two particles can be considered as entangled, when the

energy between them has been large at any time in the past.42 However, none

of these particles are involved in the emission of other particles. Entanglement is

commonly used in quantum communication, quantum cryptography, and quantum

computing.42,43 Schrodinger introduced the concept of entanglement in quantum

mechanics.44 Based on this quantum mechanics, Singh et al.45 proposed the quan-

tum optimization algorithm (QOA). In the following, we provide the mechanism of

the QOA.45,46

For simplicity, we consider a search agent in two-dimensional search space and

solve the optimization problem using Schrodinger equation as follows:

qi = φ×Qij + (1− φ)×Qhi , 0 < φ < 1 , (21)

P (x) =1

Le−2/L , (22)

D(x) =

∫ x

−∞P (x)dx = e−2/L , (23)

where L represents the search scope of each particle. Now, obtain the position of

particles using Monte Carlo method as follows:

y = q ± L

2ln(1/v) , v ∼ U(0, 1) , (24)

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DHIMAN

where v is a random number in range lies in range of [0, 1]. Now, compute the

distance between particles as follows:

L = 2× α · |q − y| , (25)

y = q ± α · |q − y| · ln(1/v) , (26)

where α is the parameter of the algorithm.

In this paper, the concepts of quantum and path followed by planets around the

sun are discussed.47 Firstly, R numbers of spotted hyenas are randomly generated.

Thereafter, the positions of these R spotted hyenas (Ck, where k = 1, 2, . . . , R) are

updated based on the position of best optimal solution (C(x+ 1)). Hence, the new

positions of R spotted hyenas are computed as

C(x+ 1) =Oh

N+ (El(Ck)× Edistance(0, 1)) + y , (27)

El(Ck) = (C1 × C2 × · · · × CR)× π , (28)

where El(Ck) indicates the elliptical positions of spotted hyenas and Edistance is the

distance between the positions of spotted hyenas to remove the conflict between

each other.

5. Experimental Results and Discussion

The simulation and experimentation is done in Matlab R2014a version on 64 bits

Core i7 processor with 3.7 GHz and 4 GB of main memory. The proposed algorithm

is tested on four systems i.e. six, ten, 20, and 40. The results obtained by DHIMAN

as well as other approaches are described as follows.

Table 1. Power output for six generators system.

Unit power output DHIMAN BBO PSO CSA NPSO-LRS GA

P1 445.8794 447.3997 447.50 447.4768 446.9600 474.81

P2 174.0141 173.2392 173.32 173.2234 173.3944 178.64

P3 262.0356 263.3163 263.47 263.3787 262.3436 262.21

P4 133.7949 138.0006 139.06 138.9524 139.5120 134.28

P5 149.9960 165.4104 165.48 165.4120 164.7089 151.90

P6 71.4261 87.0797 87.13 87.0024 89.0162 74.18

Total poweroutput (MW) 1237.146 1274.446 1275.96 1275.446 1275.935 1276.02

Total cost ($/h) 15440.76 15443.096 15450 15443.08 15450 15459

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100

101

102

103

1.544

1.5445

1.545

1.5455

1.546

1.5465

1.547

1.5475

1.548x 10

4

Iterations

Cos

t($/

hour

)

Fig. 1. Cost convergence of six generators system.

5.1. System 1

In this system, a six generators system is examined with POZ.28 The population

size, maximum iterations and, the load demand have been selected as 100, 1000,

and 1263 MW, respectively. Table 1 reveals the comparison results for six gen-

erators system obtained by various approaches i.e. the BBO,36 PSO,28 CSA,29

NPSO-LRS,37 and GA.28 It is seen that the proposed DHIMAN algorithm obtains

minimum cost value among all the other approaches. The convergence behavior of

six generators system is also analyzed by proposed DHIMAN algorithm (see Fig. 1).

Table 2. Power output for ten generators system.

Unit power NPSO-

output DHIMAN Fuel BBO Fuel NPSO Fuel CSA Fuel LRS Fuel

P1 228.6589 2 212.96 2 220.657 2 236.4387 2 223.33 2

P2 229.0511 1 209.43 1 211.785 1 230.0000 1 212.19 1

P3 407.4309 1 332.02 3 280.402 1 417.3113 2 276.21 1

P4 129.0097 2 238.34 3 238.601 3 135.9952 1 239.41 3

P5 325.7456 1 269.25 1 277.562 1 328.6017 1 274.64 1

P6 195.0311 1 237.64 3 239.120 3 197.6450 1 239.79 3

P7 244.4100 1 280.61 1 292.139 1 257.0953 1 285.53 1

P8 224.1129 3 238.47 3 239.153 3 228.2969 3 240.63 3

P9 409.5347 3 414.85 3 426.114 3 411.4391 3 429.26 3

P10 248.4112 1 266.38 1 274.463 1 257.1768 1 278.95 1

Total power

output (MW) 2641.3961 2699.95 2699.996 2700 2699.94

Total cost

($/h) 592.4233 605.6387 624.1624 598.0243 624.1273

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DHIMAN

100

101

102

103

592

594

596

598

600

602

604

Iterations

Cos

t($/

hour

)

Fig. 2. Cost convergence of ten generators system.

5.2. System 2

In this system, the ten generators with valve point loading and multi-fuel sources40

are considered. The population size, maximum iterations, and the load demand

have been selected as 100, 1000, and 2700 MW, respectively. The transmission loss

is not considered in this system. The results obtained by proposed DHIMAN and

other competitor approaches have been described in Table 2. It has been observed

from Table 2 that the proposed DHIMAN algorithm provides the optimal cost. The

convergence analysis of ten generators system is shown in Fig. 2.

5.3. System 3

This system consists of 20 generators with quadratic cost function.38 The population

size, maximum iterations, and the load demand have been selected as 100, 1000, and

2500 MW, respectively, for this system. The results obtained by various competitive

approaches have been described in Table 3. The results show that the proposed

DHIMAN obtains minimum cost as compared with other competitor approaches.

The convergence measurement of twenty generators system is shown in Fig. 3.

5.4. System 4

This test system consists of 40 generators with valve-point loading.16 The popu-

lation size, maximum iterations and the load demand have been selected as 100,

1000, and 10500 MW, respectively, for this test system. The transmission loss is no

considered in this system. The results obtained by various methods, i.e. the CSA,29

BBO,36 NPSO-LRS,37 and SOH-PSO39 have been summarized in Table 4. It has

been observed that the proposed DHIMAN provides the minimum cost among all

methods. The convergence analysis of this system obtained by DHIMAN algorithm

is shown in Fig. 4.

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Table 3. Power output for 20 generators system.

Unit power Hopfield Lambda

output DHIMAN CSA BBO model iteration

P1 511.8541 512.8467 513.0892 512.7804 512.7805

P2 169.4410 168.8534 173.3533 169.1035 169.1033

P3 127.8750 126.8549 126.9231 126.8897 126.8898

P4 101.4411 102.8784 103.3292 102.8656 102.8657

P5 113.0047 113.6863 113.7741 113.6836 113.6386

P6 73.4498 73.5482 73.0669 73.5709 73.5710

P7 113.8400 115.4766 114.9843 115.2876 115.2878

P8 115.0021 116.4497 116.4238 116.3994 116.3994

P9 100.835 100.7505 100.6948 100.4063 100.4062

P10 106.4200 106.1438 99.99979 106.0267 106.0267

P11 148.0234 150.2221 148.977 150.2395 150.2394

P12 291.7431 292.7736 294.0207 292.7647 292.7648

P13 118.4110 118.9029 119.5754 119.1155 119.1154

P14 30.8450 30.8736 30.54786 30.8342 30.8340

P15 114.1207 115.7864 116.4546 115.8056 115.8057

P16 35.2246 36.2102 36.22787 36.2545 36.2545

P17 62.8292 66.8828 66.85943 66.8590 66.8590

P18 88.4122 87.8848 88.5470 87.9720 87.9720

P19 101.1416 100.7805 100.9802 100.8033 100.8033

P20 53.1200 54.1771 54.2725 54.3050 54.3050

Total powerouput (MW) 2577.0336 2591.9825 2592.10105 2591.967 2591.9221

Total cost ($/h) 62451.08 62456.63 62456.7926 62456.63 62456.63

100

101

102

103

6.245

6.246

6.247

6.248

6.249

6.25

6.251x 10

4

Iterations

Cos

t($/

hour

)

Fig. 3. Cost convergence of 20 generators system.

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DHIMAN

Table 4. Power output for 40 generators system.

Unit power

output DHIMAN CSA BBO NPSO-LRS SOH-PSO

P1 111.5402 112.0518 111.0465 113.9761 110.80

P2 110.4213 111.4948 111.5915 113.9986 110.80

P3 97.2253 97.5626 97.6077 97.4141 97.40

P4 177.7035 179.8000 179.7095 179.7327 179.73

P5 86.4121 88.9834 88.3060 89.6511 87.80

P6 137.6329 140.0000 139.9992 105.4044 140.00

P7 296.0278 299.9993 259.6313 259.7502 259.60

P8 285.4786 284.9506 284.7366 288.4534 284.60

P9 282.0637 284.9653 284.7801 284.6460 284.60

P10 133.1432 130.0006 130.2484 204.8120 130.00

P11 96.7479 94.0000 168.8461 168.8311 94.00

P12 93.4238 94.0000 168.8239 94.00 94.00

P13 211.7223 214.7621 214.7038 214.7663 304.52

P14 303.4222 304.5194 304.5894 394.2852 304.52

P15 392.0004 394.2799 394.2761 304.5187 394.28

P16 392.7446 394.2793 394.2409 394.2811 394.28

P17 486.9536 489.2802 489.2919 489.2807 489.28

P18 486.7413 489.2776 489.4188 489.2832 489.28

P19 510.2238 511.2797 511.2997 511.2845 511.28

P20 510.4233 511.2799 511.3073 511.3049 511.27

P21 524.4136 523.3012 523.417 523.2916 523.28

P22 522.2374 523.2928 523.2795 523.2853 523.28

P23 523.0233 523.2892 523.3793 523.2797 523.28

P24 523.7410 523.4340 523.3225 523.2994 523.28

P25 522.5212 523.2839 523.3661 523.2865 523.28

P26 523.0273 523.2810 523.4362 523.2936 523.28

P27 10.4133 10.0000 10.05316 10.0000 10.00

P28 10.0000 10.0009 10.01135 10.0001 10.00

P29 10.6541 10.0014 10.00302 10.0000 10.00

P30 87.0021 92.0666 88.47754 89.0139 97.00

P31 189.0000 190.0000 189.9983 190.0000 190.00

P32 190.8800 190.0000 189.9881 190.0000 190.00

P33 190.0000 190.0000 189.9663 190.0000 190.00

P34 167.5222 199.9998 164.8054 199.9998 185.20

P35 167.0003 199.9999 165.1267 165.1397 164.80

P36 196.0200 200.0000 165.7695 172.0275 200.00

P37 110.0200 110.0000 109.9059 110.0000 110.00

P38 110.0000 110.0000 109.9971 110.0000 110.00

P39 109.0000 110.0000 109.9695 93.0962 110.00

P40 511.4222 511.2824 511.2794 511.2996 511.28

Total powerouput (MW) 10399.9498 10499.9996 10500.00657 10499.9872 10500

Total cost ($/h) 121193.46 121425.61 121426.95 121664.43 121501.14

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100

101

102

103

1.2119

1.212

1.212

1.2121

1.2121

1.2122

1.2122

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5

Iterations

Cos

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Fig. 4. Cost convergence of 40 generators system.

6. Conclusions

This paper presents the quantum and astrophysics-based optimization algorithm

for solving both convex and non-convex ED problems of fossil fuel-fired generators.

The proposed algorithm is applied to four ED problems (i.e. 6, 10, 20, and 40 test

systems) of fossil fuel-fired generators considering transmission losses, multiple fu-

els, valve-point loading and POZs with various constraints. The results obtained by

proposed algorithm are compared with other approaches as defined in the literature.

The results show that the proposed algorithm outperforms as compared to other al-

gorithms for solving ED problem using low computational efforts. Moreover, results

reveal that DHIMAN algorithm is able to handle high-dimensional problems in the

field of power and energy. In future, this algorithm can be extended by hybridizing

it with other methods to solve real-life optimization problems.

References

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