an improved marine predators algorithm for short-term

Abstract—In this paper, an improved marine predators

algorithm (IMPA) is proposed to solve the short-term

hydrothermal scheduling (STHS) problem. The marine

predators algorithm (MPA) owns low diversity of the initial

population and is easy to fall into local optima in the

optimization process. Facing these challenges, three

improvements are presented. Tent map is applied to initialize

the population, which makes the population more uniformly

distributed. An average fitness preferential strategy is adopted

to improve the quality of population, which provides more

possibility for MPA to find better solutions. By segmenting the

probability factor in fish aggregating devices (FADs) effect on

the optimization process, the premature convergence of MPA is

improved. Moreover, a selective repair strategy and an

economic priority strategy are proposed to handle dynamic

water balance of reservoirs and the power balance, respectively.

Three hydrothermal test cases are employed to verify the

feasibility and effectiveness of the proposed method, and the

results show that IMPA can obtain solutions of high quality.

Compared with other methods, IMPA can get better results,

which reflects its strong competitiveness in tackling the STHS

problem.

Index Terms— hydrothermal scheduling, Tent map, average

fitness preferential strategy, selective repair strategy, economic

priority strategy.

I. INTRODUCTION

he short-term hydrothermal scheduling (STHS) is a

significant aspect in current power system operation, and

it is widely concerned because of its considerable economic

benefits. It is aimed at achieving the minimum fuel cost over a

planned schedule period, while meeting the various

Manuscript received June 2, 2021; revised October 12, 2021. This work

was supported by the National Natural Science Foundation of China under

Grant 51207064.

Gonggui Chen is a professor of Key Laboratory of Industrial Internet of

Things and Networked Control, Ministry of Education, Chongqing

University of Posts and Telecommunications, Chongqing 400065, China;

Chongqing Key Laboratory of Complex Systems and Bionic Control,

Chongqing University of Posts and Telecommunications, Chongqing

400065, China (e-mail: [email protected]).

Ying Xiao is a master degree candidate of Chongqing Key Laboratory of

Complex Systems and Bionic Control, Chongqing University of Posts and

Telecommunications, Chongqing 400065, China (e-mail:

[email protected]).

Fangjia Long is a senior engineer of State Grid Chongqing Electric Power

Company, Chongqing 400015, China (e-mail: [email protected]).

Xiaorui Hu is a professor level senior engineer of Marketing Service

Center, State Grid Chongqing Electric Power Company, Chongqing 401123,

China (e-mail: [email protected]).

Hongyu Long is a professor level senior engineer of Chongqing Key

Laboratory of Complex Systems and Bionic Control, Chongqing University

of Posts and Telecommunications, Chongqing 400065, China

(corresponding author to provide phone: +8613996108500; e-mail:

[email protected]).

constraints of hydro power and thermal power generation [1].

Since hydroelectric power nearly has no expense, the cost of

STHS is only related to the fuel cost of thermal plants.

Cascade reservoirs are used in the STHS system. Therefore,

the downstream reservoir is always affected by the upstream

reservoir. At the same time, the difficulty of STHS problem is

increased. In addition, the valve point effect of thermal plants

and the transmission loss generated by the power system

further improve the complexity, nonlinearity and

non-convexity of STHS problem [2].

In the past decades, a number of mathematical

programming approaches based on operational research have

been used for solving STHS problem, such as dynamic

programming (DP) [3-5], linear programming (LP) [6, 7],

nonlinear programming (NLP) [8, 9], lagrangian relaxation

(LR) method [10], mixed integer programming (MIP) [11].

The DP method is an effective and common method for

solving nonlinear and non-convex STHS problem. However,

when dealing with large-scale problems, the dimension

disaster is prone to occur. The LP method is merely

applicable to solving the extreme problem with linear

objective function and linear constraints, so approximate

linearization is used to deal with the nonlinear problems. This

will inevitably lead to errors in calculation results. On the

contrary to the LP method, the NLP method can directly

search the optimal solution of nonlinear issues. But this

method requires the objective function to be continuous and

differentiable. Although the LR method has a good effect in

large-scale systems, it may be influenced by duality gap

oscillation in the convergence process, contributing to the

divergence for some issues with non-convexity of incremental

heat rate curves of thermal generators. The MIP method is not

suitable for large-scale systems because its computational

complexity will increase exponentially with the increase of

variables.

In recent years, heuristic intelligent optimization algorithm

has been widely applied to cope with various optimization

issues, owing to it does not need continuous and differentiable

objective function and constraints, and has better robustness

with classical optimization algorithms. Among differential

evolution (DE) [12, 13], particle swarm optimization (PSO)

[14-16], evolutionary programming (EP) [17, 18], genetic

algorithm (GA) [19] and simulated annealing (SA) [20, 21]

have been successfully applied to solve STHS problem.

Nonetheless, these algorithms still have some shortcomings.

In terms of DE, it may be trapped in local optima, especially

when handling large-scale optimization problems. PSO is not

good at local search and is easy to fall into local optima. In

case of EP, it has a slow convergence speed when dealing with

multimodal optimization problems. In regard to GA, it has a

strong global search capability, but it exists premature

convergence. As far as SA is concerned, its convergence

An Improved Marine Predators Algorithm for

Short-term Hydrothermal Scheduling

Gonggui Chen, Ying Xiao, Fangjia Long, Xiaorui Hu and Hongyu Long*

T

IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15

Volume 51, Issue 4: December 2021

______________________________________________________________________________________

speed is slow. Thereafter, researchers have proposed a variety

of improved algorithms successfully applied to STHS

problem, for instance, modified hybrid differential evolution

(MHDE) [22], modified chaotic differential evolution

(MCDE) [23], couple-based particle swarm optimization

(CPSO) [24], improved quantum-behaved particle swarm

optimization (IQPSO) [25], modified cuckoo search

algorithm (MCSA) [26] and optimal gamma based genetic

algorithm (OGB-GA) [27]. Compared with the original

algorithm, they have more obvious advantages in dealing with

STHS problem after experimental verification.

In 2020, Faramarzi et al. proposed marine predators

algorithm (MPA), which is a nature-inspired algorithm based

on the search trajectories and strategies of predators in the

ocean when they hunt [28]. MPA is well in the global search,

can find the approximate global optimal solution, and has a

fast convergence speed [28]. Moreover, MPA has proved to

be applied successfully to diverse research areas [29-31]. This

paper comes up with an improved marine predators algorithm

(IMPA) to solve the STHS problem. For the sake of

improving the diversity of the initial population of MPA, the

population initialization approach by random number

generator is abandoned, and Tent map is adopted in

initializing the population. The Average Fitness Preferential

(AFP) Strategy is presented to obtain high quality solutions.

Furthermore, in order to decrease the influence of local

optima on MPA as much as possible, the probability factor of

fish aggregating devices (FADs) effect on the optimization

process is segmented. The experimental simulation proves

that IMPA is effective and feasible in solving STHS problem.

The remaining of this article is arranged as follows. The

STHS problem formulation is introduced in section II. In

section III, we describe outline of MPA and its improvement

measures. Section IV shows the application of IMAP in the

STHS problem. In section V, the optimization results of the

STHS problem using IMPA are presented. Eventually, the

section VI provides the conclusions of this paper.

II. PROBLEM FORMATION

The STHS problem can be modelled as a nonlinear

optimization problem containing a multimodal objective

function and a series of equations, inequalities and dynamic

constraints.

A. Objective function

The STHS aims at minimizing the fuel cost for thermal

plants over a scheduling period T by using hydro resources as

much as possible. The objective function of the STHS is as

follows.

1 1

Minimize SNT

Si it

t i

fF P

(1)

where T is the number of scheduling intervals, NS is the

number of thermal plants, PS

it represents the generating

capacity of thermal plant i at time interval t and fi (PS

it )

represents the fuel cost of thermal plant i. In general, the fuel

cost of thermal plant i is expressed as:

2

S Si it is it is it is

SP b Pf a P c (2)

where ais, bis and cis represent the fuel cost coefficients of

thermal plant i and they are all constant coefficients.

Considering the valve point effect, the fuel cost of thermal

plant i modifies by adding a sinusoidal component and it is

generally expressed as:

2

,min sin

S Si it is

S

S S

it is it is

is is i it

P a P b

Pe

P

d P

f c

(3)

where dis and eis represent the fuel cost coefficients of thermal

plant i affected by the valve point effect and ,minS

iP refers to

the minimum generating capacity of thermal plant i.

B. Constraints

1) Power balance

The total power generation of power plants in each period

should meet the power demand and the power transmission

loss, expressed as:

1 1

1,2, ,hSN N

S h

it jt d ltt

i j

P P P P t T

(4)

where Nh is the number of hydro plants, Ph

jt represents the

generating capacity of hydro plant j at time interval t, Pdt

represents the system power demand at time interval t and Plt

represents the power transmission loss at time interval t.

The generating capacity of hydro plant can be formulated

as a quadratic function of the drainage rate and the volume of

reservoir as follows.

2 2

1 2 3

4 5 6

h h h hjt j jt j jt j jt jt

h hj jt j jt j

hP c V c Q c V Q

c V c Q c

(5)

Where hjtV is the volume of reservoir of hydro plant j at time

interval t, hjtQ is the drainage rate of hydro plant j at time

interval t. Furthermore, cj1, cj2, cj3, cj4, cj5 and cj6 are all about

constant coefficients for hydro plants.

The power transmission loss can be formulated as follows.

lt 0 00

1 1 1

N N N

S S Sit ij it i it

i j i

P P B P B P B

(6)

where Bij, B0i and B00 are the transmission loss coefficient of

the corresponding power system and N is the number of

thermal plant and hydro plant combined.

2) Generation capacity limitation

,min ,max 1,2, , , 1,2, ,S SSi it i sP P P i N t T (7)

where PS

i,min and PS

i,max are the minimum and maximum

generating capacity of thermal plant i, respectively.

j,min j,max 1,2, , , 1,2, ,t

hj

h hhP P P i N t T (8)

where Ph

j,min and Ph

j,max are the minimum and maximum

generating capacity of thermal plant i, respectively.

3) Reservoir capacity limits

,min ,max 1,2, , , 1,2, ,h h hj jt j hjV V V N t T (9)

where ,minhjV and

,maxhjV are the minimum and maximum

reservoir storage capacity of hydro plant j, respectively.

4) Drainage rate limits

,min ,max 1,2, , , 1,2, ,h h hj jt j hjQ Q Q N t T (10)

where ,minhjQ and ,max

hjQ are the minimum and maximum

drainage rate of hydro plant j, respectively.

5) Dynamic water balance

(t 1) (t ) (t )

1

ju

k k

R

h h h h h h hjt j jt jt jt j j

k

V V I Q S Q S

(11)

where hjtI and h

jtS represent the inflow and spillage of hydro

plant j at time interval t, respectively, Rju is a set of reservoirs



______________________________________________________________________________________

located directly upstream of reservoir j and τk represent the

flow delay time of upstream reservoir k to its directly

connected downstream reservoir j.

6) Initial and terminal reservoir storage capacity

h,begin h,end

0 , 1,2, ,h hjj j jT hV V V NjV (12)

where h,begin

jV and h,end

jV are the initial and terminal reservoir

storage capacity of hydro plant j, respectively.

III. OUTLINE OF MPA AND ITS IMPROVEMENT MEASURES

A. Marine Predators Algorithm

MPA is an algorithm that imitates the predation process of

marine organisms, which the trajectory of prey is related to

Brownian motion and Lévy flight [28]. The nucleus of this

algorithm is to utilize the velocity ratio of prey and predator to

promote the optimization process of the whole algorithm. The

major steps of MPA are as follows.

1) Initialization

The prey matrix is constructed by Eq. (13), and the matrix

is abbreviated as Pr. Afterwards, the top predator with the

optimal fitness value forms the elite matrix with the same

dimension as the prey matrix, which is abbreviated as E.

min max mia nX RX X X (13)

where Ra represent uniform random vector i between 0 and 1,

maxX and minX represent the upper and lower boundaries

containing variables of the optimization issue.

2) Optimization scenarios

According to the diverse velocity ratios of predator and

prey, the optimization process of MPA is divided into three

phases, the specific situation is as follows.

a) High velocity ratio

When prey is moving faster than predator, a strategy of

prey moving in Brownian motion and predator remaining still

should be adopted. The mathematical model of this

exploration phase is expressed as:

1

3maxWhile Iter Iter

( )i b i b iS R E R Pr (14)

i iiPr Pr P R S (15)

where bR is a random numerical vector based on normal

distribution representing Brownian motion, the symbol

shows the entry-wise multiplication, P is a constant (0.5 is

usually recommended) and R is a random vector generated

uniformly between [0,1]. Iter represents the current iteration,

Itermax refers to the maximum number of iterations in the

entire iteration process.

b) Unit velocity ratio

When predator and prey are moving at uniform speed, prey

is in charge of exploitation and predator takes charge of

exploration. Thus, predator employs Brownian motion while

prey employs Lévy flight. Therefore, the population will be

divided into two parts for optimization.

1 2

3 3max maxWhile Iter Iter Iter

The first half of the population is formulated as follows.

i ililS E PR rR (16)

i i iPr Pr SP R (17)

where lR is a random numerical vector based on Levy

distribution representing Lévy flight.

The other half of the population is formulated as follows.

i b b i iS R R E Pr (18)

i i iPr P CFE S (19)

max

(2 )

max

( )

1Iter

IterIter

CFIter

(20)

where CF is regarded as an adaptive parameter controlling the

step size of the predator motion.

c) Low velocity ratio

When prey moves slower than predator, the best strategy of

both prey and predator moving in Lévy flight is adopted. The

mathematical model of this exploitation phase is expressed as:

2

3maxWhile Iter Iter

l li i iS R R E Pr (21)

i i iPr P CFE S (22)

3) Fish Aggregating Devices effects

There is a chance of vortices or a gathering Fish

Aggregating Devices (FADs) effects in the ocean. Generally

speaking, predators spend more than four-fifths of their time

foraging near FADs, and the rest of their time foraging at

distant prey gathering places. FADs are regarded as local

optima which is mathematically represented as:

1 2

if

[ (1- )] -

if

i

i

min m

i

i

r

ax n

r

mPr

PrP

CF X R X X U

r FADs

r FADs rr Pr

r F Ds

r

A

P

(23)

where r is a random vector created uniformly in the range of

[0,1], FADs = 0.2 embodys the probability of FADs effect on

the optimization process. U represents a binary vector which

each array includes only 0 and 1, and if the array is less than

0.2, then this array is set to 0; otherwise it is set to 1. In

addition, both r1 and r2 denote any number in the population

range.

Algorithm 1 The Marine Predators Algorithm (MPA)

1. Initialize the prey population, i = 1,…,n

2. Iter = 1

3. while (Iter < Itermax)

4. Calculate the fitness values of prey and establishment the elite matrix

5. if (Iter < Itermax/3)

6. Update the prey by Eq. (15)

7. else if (Itermax/3 < Iter < 2* Itermax/3)

8. if (i < n/2)

9. Utilize Eq. (17) to Update the prey

10. else

11. Use Eq. (19) to Update the prey

12. else if (Iter > 2* Itermax/3)

13. Update the prey based on Eq. (22)

14. end if

15. Accomplish the memory saving and update the elite matrix

16. Execute the FADs effect by Eq. (23)

17. Iter++

18. end while

MPA uses the memory saving to simulate a good memory

of marine predators in order to remember where they

successfully hunted. After updating the current solution, the

elite matrix is updated by comparing the fitness values of each

current solution and each old solution. The pseudocode of



______________________________________________________________________________________

MPA is listed in Algorithm 1.

B. Improved Marine Predators Algorithm

Although MPA is superior to PSO, GA and GSA in the

global search capability [28], it still has some defects, such as:

1) it has a low diversity of the initial population and 2) there is

a means to jump out of local optimum in MPA, but it is still

easy to fall into local optima. Next, three improvements of

MPA will be introduced in detail.

1) Chaotic initialization

Since the best individual in each generation leads the

iterative calculation of the next generation population, the

first generation population obtained by initialization has a

considerable influence on the optimal solution finally

searched. If one particle happens to be near the optimal

solution in the course of initialization, then it will inevitably

accelerate convergence speed of the population. In addition,

the chaotic sequence generated by Tent map can traverse all

the states in the population without repetition and has a high

diversity. In consequence, Tent map is selected in this paper

to initialize the population. The mathematical expression of

Tent map is as follows:

1

0 1/ 222(1 ) 1/ 2 1

k kk

k k

m mm

m m

(24)

Tent map can be expressed in the following form after

Benoulli shift transformation.

1 2 mod1k km m (25)

Therefore, the specific steps of the chaotic sequence m

generated by Tent map as follows:

Step 1: Get an initial value m0 at random (m0 should be

avoided falling into small period points, such as 4 periods (0.2,

0.4, 0.8, 0.6)), and denote as Z(1) = m0. Let i = j = 1.

Step 2: The sequence m is generated by updating Eq. (25)

and i = i+1.

Step 3: If the sequence m falls into a fixed point or a small

cycle within 5 cycles (such as m(i) = {0, 0.25, 0.5, 0.75} or

m(i) = m(i-k), k = {0, 1, 2, 3, 4}), then go to step 4; if the

maximum number of iterations is reached, then move on to

step 5; otherwise, go to step 2.

Step 4: Change the initial iteration value of the sequence m,

m(i) = Z(j+1) = Z(j) + ε and j = j+1, then go to step 2.

Step 5: Terminate the program and save the produced

sequence m.

2) Average Fitness Preferential Strategy

In the process of searching optimal solutions, because of

the uncertainty of particle mass, the search time may be

prolonged or even the optimal solution may be missed. Hence,

we recommend an average fitness preferential (AFP) strategy

to strengthen the reliability of the algorithm and the quality of

the population solution.

The AFP is a strategy to select a high quality population

solution by comparing the fitness value with the average

fitness value of the population. Additionally, the average

fitness value can be calculated by Eq. (26).

1

1= 1, ,f

i

n

iAn

i nf

(26)

where Af refers to the average fitness value of the population,

n represents the total number of prey in the population and fi is

regarded as the fitness value of prey i.

By comparing the fitness values with the average fitness

values, the fitness values of prey are divided into two

categories: low-average solutions and high-average solutions.

Low-average solutions correspond to the fitness value lower

than the average fitness value, while high-average solutions

correspond to the fitness value higher than the average fitness

value.

For low-average solutions, they are preserved. In other

words, the prey which fitness value is lower than the average

value does not transform its position.

For high-average solutions, we update the prey position by

the position of the top predator and the position of any prey in

the first half of the population, and it is as follows:

+

1, , , [1, / 2]2

ir

i

TP PrPr n r randi n (27)

where TP is the position of the top predator and r represents a

random index selected from the first half of the prey

population.

AFP improves the average fitness value of each generation

and provides more possibility to find a better solution.

Yes

Update the prey based

on Eq. (22)

Chaotic initialization, Iter=0

Iter=Iter+1

Calculate the fitness values of prey

and establishment the elite matrix

Apply AFP

Update the Elite matrix

Segment FADs with Eq. (28)

Iter<2*Itermax/3

Update the prey by Eq.

(15)

Yes Itermax/3 < Iter <

2* Itermax/3

Use Eq. (17) or Eq. (19)

to Update the prey

No

Yes

No

Accomplish memory saving and update the Elite matrix

Execute the FADs effect by Eq. (23)

Iter≤ Itermax

Return the top predator

No

Fig. 1. The program block diagram of IMPA

3) Modification of FADs effect

As far as the FADs effect is concerned, this process is to

avoid local optima of MPA. In the entire optimization process

of MPA, the probability of avoiding FADs (can also be said to

be local optima) is always set as 20%. Nevertheless, as the

amount of iterations increases, the probability of falling into

local optima will increase. Consequently, the probability

factor of avoiding local optima in the MPA is segmented, and

the segmenting method is as follows:

0.1

= 0.15

/ 3

/ 3 2* / 3

2*0.5 / 3

d

d

m

d t

ax

max max

max

Iter Iter

Iter Iter Iter

Iter It

R

FAD

R er

s R

(28)

where Rd is a random constant between [0,1].

After these improvements, the program block diagram of

IMPA is shown in Fig. 1.



______________________________________________________________________________________

C. Test for IMPA

In order to test the feasibility and rationality of the IMPA,

we used 13 well-known benchmark functions to test and

compare the performance of the IMPA and MPA [28]. The 13

benchmark functions are divided into unimodal functions

(F1-F6) and multimodal functions (F7-F13), unimodal functions

are designed to experiment the exploitation performance

while multimodal functions are used to test the exploration

ability of an algorithm.

The mathematical formulas and properties of these

benchmark functions are shown in TABLE I. The

two-dimensional views and convergence curves of four

benchmark functions of IMPA are given in Fig. 2. At the same

time, the test results of IMPA and MPA are listed in TABLE

II. From the test results, IMPA is better than MPA in all 13

benchmark test functions. It obviously verifies that the

improvement measures for MPA in this paper are effective

and feasible.

TABLE I

THE MATHEMATICAL FORMULAS AND PROPERTIES OF BENCHMARK FUNCTIONS

Function Dim Domain Fmin

21( )

m

iix xF 50 [-100,100] 0

2 1 1( ) |

mm

i ii ix x xF

50 [-10,10] 0

2

3 1( ) ( )

m i

ji jF x x

50 [-100,100] 0

4( ) max ,1i iF x x i n ∣ 50 [-100,100] 0

1 2 2 25 11( ) 100(x x ) (x 1)

m

i i iixF

50 [-30,30] 0

26 1( ) (x 0.5)

m

iiF x

50 [-100,100] 0

47 1( ) 0,1

m

iiF x ix random

50 [-1.28,1.28] 0

8 1( ) sin

m

i iiF x x x

50 [-500,500] -418.9829×5

29 1( ) 10cos(2 x ) 10

m

i iif x x

50 [-5.12,5.12] 0

210 1 1

1 1( ) 20exp( 0.2 ) exp( cos(2 x )) 20

m m

i ii iF x x e

m m

50 [-32,32] 0

211 1 1

1( ) cos( ) 1

4000

mm iii i

Fx

x xi

50 [-600,600] 0

1 2 212 1 11

2

1

( ) 10sin (y 1) 1 10sin ( y )

(y 1) (x ,10,100,4)

m

i ii

m

m ii

F x y

u

m

50 [-50,50] 0

2 2 213 1 1

2 2

1

( ) 0.1 sin (3 x ) (x 1) 1 sin (3 x 1)

(x 1) 1 sin (2 x ) (x ,5,100,4)

m

i ii

m

m m ii

F x

u

50 [-50,50] 0

Fig. 2. The two-dimensional views and convergence curves of some benchmark functions of IMPA

-200

0

200

-200

0

2000

0.5

1

1.5

2

x 1011

x1

F5 Topology

x2

F5

( x 1

, x

2 )

0 100 200 300 400 50010

0

102

104

106

108

1010

Objective space

Iteration

Bes

t sc

ore

ob

tain

ed s

o f

ar

-500

0

500

-500

0

500-1000

-500

0

500

1000

x1

F8 Topology

x2

F8

( x 1

, x

2 )

0 100 200 300 400 500-10

5

-104

-103

Objective space

Iteration

Bes

t sc

ore

ob

tain

ed s

o f

ar

-10

0

10

-10

0

100

50

100

150

x1

F12 Topology

x2

F1

2(

x 1 ,

x2 )

0 100 200 300 400 50010

-5

100

105

1010

Objective space

Iteration

Bes

t sc

ore

ob

tain

ed s

o f

ar

-100

0

100

-100

0

1000

5000

10000

15000

x1

F2 Topology

x2

F2

( x 1

, x

2 )

0 100 200 300 400 50010

-20

10-10

100

1010

1020

Objective space

Iteration

Bes

t sc

ore

ob

tain

ed s

o f

ar



______________________________________________________________________________________

TABLE II

THE RESULTS FOR BENCHMARK FUNCTIONS

Function IMPA MPA

Average Standard Average Standard

F1(x) 3.66E-22 2.08E-22 2.04E-21 2.69E-21

F2(x) 2.69E-13 3.79E-13 1.85E-12 1.89E-12

F3(x) 4.57E-03 5.58E-03 2.95E-02 3.06E-02

F4(x) 1.77E-08 1.70E-09 2.54E-08 1.18E-08

F5(x) 4.57E+01 3.02E-01 4.63E+01 4.69E-01

F6(x) 2.12E-01 7.31E-02 4.86E-01 3.16E-01

F7(x) 1.19E-03 2.27E-04 1.36E-03 9.22E-04

F8(x) -1.40E+04 1.96E+02 -1.35E+04 7.41E+02

F9(x) 0.00E+00 0.00E+00 0.00E+00 0.00E+00

F10(x) 8.19E-12 1.28E-12 1.23E-11 6.23E-12

F11(x) 0.00E+00 0.00E+00 0.00E+00 0.00E+00

F12(x) 3.51E-03 9.77E-04 1.25E-02 5.14E-03

F13(x) 2.95E-01 9.17E-02 5.91E-01 2.32E-01

IV. APPLICATION OF IMAP FOR THE STHS PROBLEM

A. Structure of solution and initialization

As far as the STHS problem is concerned, a set of decision

variables involving the drainage rate of each hydro plant and

the generating capacity of each thermal plant within each time

interval constitute the structure of an individual. The

individual Xg (g = 1, … , N) is defined as follows:

In the initialization process, a series of individuals is

formed by Tent map. Initialization can be realized by the

following formula.

11 211 21 1

1

1 1

12 22 22 22 2

1 2 1 2

h S

h S

h S

h h h S S S

N N

h h h S S S

N N

h h h S S S

t t N t

g

tt N t

Q Q Q P P P

Q Q Q P P P

Q Q Q P P P

X

(29)

,min ,max ,min

,min ,max ,min(

(

)

)

S S S

it

h h h h

jt j j j j

S

i i i i

Q Q m Q Q

P P m P P

(30)

where mj and mi are respectively the jth and the ith number in

the chaotic sequence m generated by Tent map.

B. Constraint handling

In the course of using IMPA to address the STHS problem,

certain constraints may not be satisfied for individuals after

initialization and renewed solutions. Therefore, we will

introduce some improvements to the constraints in the

following sections.

1) Generation capacity limitation

,min ,min

,min ,max

,max ,max

Si it i

S S Sit it i it i

Si

S S

S

it i

S

S S

P P PP P P P P

P P P

(31)

2) Drainage rate limits

,min ,min

,min ,max

,max ,max

h h hj jt j

h h h h hjt jt j jt j

h h hj jt j

Q Q Q

Q Q Q Q Q

Q Q Q

(32)

3) Reservoir capacity limits

,min ,min

,min ,max

,max ,max

h h hj jt j

h h h h hjt jt j jt j

h h hj jt j

V V V

V V V V V

V V V

(33)

4) Water dynamic balance

This paper adopts a selective repair strategy to handle the

water dynamic balance of reservoirs, and its procedures are as

follows:

Step 1: Set hydro plant j = 1.

Step 2: The water spillage is assumed to be zero in Eq. (11).

At this point, we can obtain the difference between the total

water discharge and the amount of available water of hydro

plant j, which can be expressed as:

h,begin h,end

(t )

1 1

1 1

ju

k

RTh

j j j j

t k

T Th h

jt jt

t t

Q V V Q

I Q

(34)

Step 3: Select a time interval b randomly in the scheduling

interval. Let count = 1.

Step 4: In order to make the current reservoir storage

capacity more accurately meet the initial and terminal

reservoir storage capacity limits, the drainage rate of hydro

plant j at time interval b is calculated as follows:

h,begin h,end

(t )

1 1

1 1

ju

k

RTh h

jb j j j

t k

T Th h

jt jt

t tk b

Q V V Q

I Q

(35)

If the drainage rate does not violate the constraint in Eq. (10)

after calculation, then go to the step 7; otherwise, go to step 5.

Step 5: If ∆Qjb > δ (δ is a given minimal constant), then go

to the next step; otherwise, go to step 7.

Step 6: Let Qh

jt = Qh

jt+∆Qj / T.

Step 7: Update the drainage rate by Eq. (32).

Step 8: Choose another new interval b and let count = count

+ 1.

Step 9: If count ≤ T, then move on to step 4.

Step 10: Let j = j + 1. If j < Nh, then go to step 2.

Step 11: Terminate the execution of the program.

5) Power system balance

After the above approach adjustment, the dynamic

reservoir volume has satisfied its constraint. At this time, we

need to deal with the problem of power balance in power

system. In this paper, the valve point effect is considered and

in order to attain the power balance constraint more

accurately and effectively in Eq. (4), we present an economic

priority strategy based on the valve point effect to deal with

the difficulty.

We put forward the index βit to judge the economy of

thermal power plant can be calculated as follows:

1

1

i i

S S

it i

i

t

S S

it it

t

P P

P

f f

P

(36)

where PS+1

it represents the generating capacity of thermal plant

i at the next position after PS

it in the generating capacity

sequence (All values in the sequence satisfy Eq. (4)) of the

thermal power plant formed according to the valve point

effect. βit represents the fuel cost of thermal plant i for every

additional 1 MW of power generation.

The smaller the value of the index βit is, the lower the fuel

cost of the thermal plant is. Therefore, in the entire scheduling

period, all thermal plants are arranged in descending order by

using index βit and form a priority sequence table. The

economic priority strategy for repairing the power system

balance is as follows:



______________________________________________________________________________________

Step 1: Based on the valve point effect, the generating

capacity sequence including PS

it satisfying Eq. (4) is generated.

In the entire scheduling period, Eq. (36) was calculated, and

all thermal plants are arranged in descending order by using

index βit to form the priority sequence table.

Step 2: Set current time interval t = 1.

Step 3: Calculate the violations of power system balance

∆Pt by the following equation.

1 1

hSN NS h

t it jt dt

i j

ltP P P P P

(37)

Step 4: If ∆Pt = 0, then go to step 13; if ∆Pt > 0, then move

on to step 9; otherwise, go to the next step.

Step 5: Set d = 1.

Step 6: Select the thermal plant z with the smallest index βit

from the priority sequence table. Afterwards, let the

generating capacity of thermal plant z at time interval t to be

PS

zt = PS

z,max, and remove thermal plant z from the priority

sequence table.

Step 7: Use Eq. (37) to calculate the new violations. If ∆Pt

= 0, then go to step 13; if ∆Pt > 0, make S S

zt zt tP P P ; if ∆Pt

< 0, set d = d + 1, and if d < NS, go back to step 6.

Step 8: Set d = d + 1. If d < NS, go back to step 6; otherwise,

move on to step 13.

Step 9: Set p = 1.

Step 10: Select the thermal plant z with the biggest index βit

from the priority sequence table. Afterwards, let the

generating capacity of thermal plant z at time interval t to be

,min

S S

zt zP P , and remove thermal plant z from the priority

sequence table.

Step 11: Use Eq. (37) to calculate the new violations. If ∆Pt

= 0, then go to step 13; if ∆Pt < 0, make S S

zt zt tP P P ;

otherwise, go to the next step.

Step 12: Set p = p + 1. If p < NS, go back to step 10;

otherwise, move on to step 13.

Step 13: Set t = t + 1. If t < T, then go back to step 3;

otherwise, go to the next step.

Step 14: Accomplish the execution of the program.

V. NUMERICAL SIMULATION RESULTS

The proposed IMPA has been implemented utilizing the

MATLAB platform on PC (Core i5, 3.3GHz and 8GB). In

order to verify the effectiveness of IMPA in dealing with the

STHS problem, we conduct experiments on two typical test

systems. One test system contains three thermal plants and

four cascaded hydro plants, the other test system contains ten

thermal plants and four cascaded hydro plants. The whole

scheduling period is 1 day, and the selected time interval of 1

hour divides a day into 24 time intervals averagely.

A. Parameters setting

Owing to the randomness of heuristic algorithms, the

experimental results are different each time. Thus, we conduct

20 simulation experiments and select the minimum, maximum

and average values to verify whether IMPA can effectively

solve the STSH problem in this paper. Meanwhile, the

population size N is set as 50 and the maximum number of

iterations Itermax is set as 500 in this paper.

In addition, in order to gain the optimal solutions, the

specific parameters of IMPA obtained after a large number of

tests are shown in TABLE III.

TABLE III

PARAMETERS OF IMPA FOR DIFFERENT HYDROTHERMAL TEST SYSTEMS

Parameters P FADs

Iter <

Itermax/3

Itermax/3 < Iter <

2* Itermax/3

Iter > 2* Itermax/3

Case 1 in system

I

0.1 0.1 0.15 0.5

Case 2 in system

I

0.5 0.1 0.12 0.5

System II 0.5 0.1 0.15 0.5

TABLE IV

OPTIMAL DISCHARGE OF EACH HYDRO PLANT FOR CASE 1 IN SYSTEM I

Hour Hydro discharge (104 m3)

1 2 3 4

1 11.66 6.05 29.79 8.29

2 8.69 11.23 29.99 10.55

3 6.05 7.20 27.63 7.20

4 5.98 11.05 16.08 14.65

5 5.21 7.07 29.62 12.97

6 8.88 10.19 19.27 8.60

7 11.06 6.70 13.65 12.55

8 7.37 6.59 13.76 11.43

9 6.48 9.47 14.94 17.87

10 7.97 7.34 16.88 16.88

11 7.51 9.27 12.86 17.18

12 8.03 7.41 15.84 16.44

13 12.06 7.61 15.33 18.09

14 7.43 6.41 18.99 12.88

15 10.61 7.75 17.66 17.37

16 6.42 9.91 21.10 16.42

17 7.30 8.07 19.11 14.42

18 11.74 9.01 13.01 17.69

19 7.25 7.84 13.04 14.88

20 10.71 10.16 10.67 18.94

21 5.50 8.06 11.99 17.46

22 7.74 9.34 13.17 19.86

23 7.69 10.25 12.01 17.43

24 5.64 8.02 11.97 15.98

B. Test system I

The test system I includes three thermal plants and four

cascaded hydro plants. In this test system, we consider two

cases: 1) only the valve point effect and 2) both the valve

point effect and the transmission loss. Besides, the data of two

test cases are all from the reference [22].

1) Case 1: Only the valve point effect is considered

The power generation of each power plant is denoted in

Fig. 3 and Fig. 4 signifies the storage capacity of each

reservoir. The optimal discharge of each hydro plant is shown

in TABLE IV. Moreover, the optimal power generation of

each hydro plant and each thermal plant is presented in

TABLE V. As can be seen from these tables, the optimal

solutions satisfy all constraints about thermal plants and

cascaded hydro plants in this case.

Meanwhile, the results of IMPA are compared other means,

including MPA, MHDE [22], MCDE [23], DGSA [32],

TLBO [33], ORCCRO [34] ，DNLPSO [35] and GSA, and

listed in ascending order in TABLE VI. From this table, we

can find that the minimum, maximum and mean expenses of

IMPA are 40695.05 ($), 40986.15 ($) and 40853.27 ($)

respectively, and these expenses of IMPA are all lower than

those of other means. For example, compared with DNLPSO,

IMPA saves 535.95($), 1380.85($) and 929.73($) in the

minimum, maximum and mean expenses. Therefore, IMPA is

superior to other means in dealing with the STHS problem.

Under the same parameters, the convergence trajectory of

the minimum expense of IMPA, MPA and GSA is shown in

Fig. 5. It can be seen from this picture, IMPA compared with

MPA and GSA has obvious advantages in terms of



______________________________________________________________________________________

convergence speed and searching for the optimal solutions

when applied in the STHS problem.

2) Case 2: Both the valve point effect and the transmission

loss are considered

The power generation of each power plant and the storage

capacity of each reservoir are signified in Fig. 6 and Fig. 7.

The optimal discharge and optimal power generation of each

hydro plant is shown in TABLE VII. In addition, the optimal

power generation of each thermal plant is represented in

TABLE VIII. As can be seen from these tables, the optimal

solutions has no violation in this case.

Meanwhile, the results of IMPA are compared other means,

including MPA, MHDE [22], GSO [36], TLBO [37], SPPSO

[38], AABC [39], GSA and CDE [40], and listed in ascending

order in Table IX. The minimum, maximum and mean

expenses obtained in this case are 41739.72 ($), 42185.21 ($)

and 41968.94 ($) respectively, which all demonstrate that

IMPA can better deal with the STHS problem.

Under the same parameters, the convergence trajectory of

the minimum expense of IMPA, MPA and GSA is represented

in Fig. 8. It can be seen from this picture, in contrast to MPA

and GSA, IMPA has the better performance in convergence

speed and the results of convergence when managed the

STHS problem.

Fig. 3. Power generation of each power plant for case 1 in system I

Fig. 4. Storage capacity of each reservoir for case 1 in system



______________________________________________________________________________________

TABLE V

OPTIMAL POWER GENERATION OF EACH HYDRO PLANT AND EACH THERMAL PLANT FOR CASE 1 IN SYSTEM I

Hour Hydro generation (MW) Thermal generation (MW) Total Demand

1 2 3 4 1 2 3

1 91.80 50.49 0.00 155.92 102.22 209.82 139.76 750.00 750.00

2 78.83 77.06 0.00 172.71 101.83 209.82 139.76 780.00 780.00

3 61.42 57.43 0.00 128.91 102.67 209.82 139.76 700.00 700.00

4 61.07 76.17 38.92 184.37 20.00 40.00 229.47 650.00 650.00

5 54.81 55.91 0.00 191.96 102.65 124.91 139.76 670.00 670.00

6 80.49 70.97 22.43 169.02 102.66 124.91 229.52 800.00 800.00

7 89.37 51.06 41.10 226.54 102.59 209.82 229.52 950.00 950.00

8 70.74 50.56 42.59 219.20 102.65 294.72 229.52 1010.00 1010.00

9 65.28 66.34 43.61 287.93 102.61 294.72 229.52 1090.00 1090.00

10 76.47 55.50 38.31 282.80 102.67 294.72 229.52 1080.00 1080.00

11 74.41 66.29 45.78 281.75 102.67 209.82 319.28 1100.00 1100.00

12 78.32 56.18 42.91 273.37 175.00 294.71 229.52 1150.00 1150.00

13 97.85 57.63 45.26 282.37 102.65 294.72 229.52 1110.00 1110.00

14 74.78 51.51 34.76 242.03 102.67 294.72 229.52 1030.00 1030.00

15 93.65 60.88 41.51 276.81 102.67 294.72 139.76 1010.00 1010.00

16 67.52 71.34 25.12 269.10 102.67 294.72 229.52 1060.00 1060.00

17 74.57 60.92 34.52 253.07 102.67 294.72 229.52 1050.00 1050.00

18 98.68 64.15 49.71 280.75 102.47 294.72 229.52 1120.00 1120.00

19 73.74 57.10 51.25 261.00 102.67 294.72 229.52 1070.00 1070.00

20 93.19 67.63 52.96 294.24 102.64 209.82 229.52 1050.00 1050.00

21 59.16 57.52 55.45 285.62 102.66 209.82 139.76 910.00 910.00

22 76.81 64.17 57.28 294.41 102.67 124.91 139.76 860.00 860.00

23 76.64 66.84 57.94 274.15 20.00 124.90 229.52 850.00 850.00

24 60.82 55.50 58.60 257.74 102.67 124.91 139.76 800.00 800.00

0 100 200 300 400 5004

4.1

4.2

4.3

4.4

4.5x 10

4

Generations

Fuel

expen

se (

$)

IMPA

MPA

GSAMPA

IMPAGSA

Fig. 5. Convergence trajectory for case 1 in system I

TABLE VI

COMPARISON OF PERFORMANCES OF VARIOUS ALGORITHMS FOR CASE 1 IN

SYSTEM I

Algorithm Fuel cost ($)

Minimum ($) Maximum Mean

IMPA 40695.05 40986.15 40853.27

ORCCRO 40936.65 41127.68 40944.29

GSA 40937.17 42582.52 41890.35

MCDE 40945.75 41977.04 41380.54

DNLPSO 41231.00 42367.00 41783.00

MPA 41501.33 41849.33 41697.92

DGSA 41751.15 41989.02 41821.49

MHDE 41856.50 − −

TLBO 42385.88. 42441.36 42407.23

Fig. 6. Power generation of each power plant for case 2 in system



______________________________________________________________________________________

Fig. 7. Storage capacity of each reservoir for case 2 in system

TABLE VII

OPTIMAL DISCHARGE AND OPTIMAL POWER GENERATION OF EACH HYDRO PLANT FOR CASE 2 IN SYSTEM I

Hour Hydro discharge (104 m3) Hydro generation (MW)

1 2 3 4 1 2 3 4

1 9.21 8.80 29.97 8.37 82.38 66.04 0.00 156.86

2 5.47 10.12 29.89 6.00 57.46 71.22 0.00 123.54

3 8.09 6.96 29.14 6.33 76.89 55.00 0.00 123.33

4 8.00 6.98 15.84 16.09 76.03 56.33 37.78 199.40

5 5.55 6.13 19.47 9.91 58.03 51.68 25.75 169.91

6 8.91 6.32 20.68 11.15 81.02 53.38 18.89 202.04

7 7.08 8.92 10.44 13.26 69.71 67.66 43.97 238.51

8 5.78 10.20 19.02 12.71 60.38 71.93 28.37 235.90

9 11.75 9.57 18.76 16.10 94.40 67.86 27.49 269.92

10 7.55 9.29 14.27 17.35 74.13 66.29 40.97 282.87

11 5.01 6.27 11.32 14.05 54.75 49.95 46.69 251.69

12 14.19 6.45 11.89 13.28 102.50 52.14 49.09 249.33

13 10.58 7.78 15.11 19.03 92.11 60.81 47.66 296.36

14 7.86 6.62 19.61 13.14 77.73 54.79 35.42 248.73

15 11.10 11.12 18.65 15.82 95.46 77.82 39.86 269.37

16 5.20 6.69 13.11 18.47 56.85 54.79 52.71 282.38

17 9.20 9.76 29.95 18.53 86.99 70.81 0.00 279.09

18 5.11 8.70 19.89 17.03 56.12 63.59 31.85 271.54

19 10.24 8.64 10.00 14.21 92.56 62.21 49.76 252.38

20 10.96 14.35 14.31 19.92 94.69 78.47 50.84 287.61

21 5.95 7.11 10.37 19.98 63.11 50.19 53.08 298.80

22 8.84 8.35 15.26 19.86 83.93 58.01 54.39 298.12

23 7.68 9.40 13.76 19.91 76.57 62.46 57.65 287.66

24 5.69 7.47 10.00 17.61 61.28 52.29 56.06 269.79

0 100 200 300 400 5004.1

4.2

4.3

4.4

4.5

4.6

4.7x 10

4

Generations

Fu

el e

xp

ense

($

)

IMPA

MPA

GSA

GSA

MPA

IMPA

Fig. 8. Convergence trajectory for case 2 in system I

C. Test system II

The test system II includes ten thermal plants and four

cascaded hydro plants. In this test system, we test the situation

that includes the valve point effect. Moreover, the data of this

test system arise from the reference [42].

The power generation of each power plant and the storage

capacity of each reservoir are shown in Fig. 9 and Fig. 10,

respectively. The optimal power generation of each thermal

plant is represented in TABLE X. Additionally, the optimal

discharge and optimal power generation of each hydro plant is

shown in TABLE XI. The optimal solutions contain no

violation in this test indicated from these tables.



______________________________________________________________________________________

Fig. 9. Power generation of each power plant in system I

TABLE VIII

OPTIMAL POWER GENERATION OF EACH THERMAL PLANT FOR CASE 2 IN SYSTEM I

Hour Thermal generation (MW) Loss Total Demand

1 2 3 (MW) (MW) (MW)

1 102.70 209.82 139.76 7.56 757.56 750.00

2 102.42 294.73 139.77 9.14 789.14 780.00

3 102.67 209.82 139.76 7.47 707.47 700.00

4 20.20 124.90 139.76 4.40 654.40 650.00

5 20.00 209.81 139.75 4.94 674.94 670.00

6 102.69 209.82 139.76 7.61 807.61 800.00

7 102.54 209.85 229.56 11.79 961.79 950.00

8 102.68 294.68 229.48 13.42 1023.42 1010.00

9 120.35 209.83 319.29 19.14 1109.14 1090.00

10 105.25 294.73 229.53 13.77 1093.77 1080.00

11 102.43 294.75 319.31 19.58 1119.58 1100.00

12 102.63 294.75 319.31 19.74 1169.74 1150.00

13 102.67 294.69 229.48 13.79 1123.79 1110.00

14 102.64 294.69 229.48 13.47 1043.47 1030.00

15 102.69 294.73 139.77 9.69 1019.69 1010.00

16 102.64 294.69 229.48 13.53 1073.53 1060.00

17 102.83 294.69 229.48 13.89 1063.89 1050.00

18 102.61 294.75 319.31 19.77 1139.77 1120.00

19 102.44 294.69 229.48 13.52 1083.52 1070.00

20 25.61 209.83 319.29 16.35 1066.35 1050.00

21 103.38 209.82 139.76 8.14 918.14 910.00

22 20.45 124.91 229.52 9.34 869.34 860.00

23 20.44 124.91 229.52 9.22 859.22 850.00

24 102.67 124.91 139.76 6.77 806.77 800.00

TABLE IX

COMPARISON OF PERFORMANCES OF VARIOUS ALGORITHMS FOR CASE 2 IN

SYSTEM I


Minimum Maximum Mean

IMPA 41739.72 42185.21 41968.94

GSA 42012.24 43699.88 42913.62

GSO 42316.39 42379.18 42339.35

AABC 42381.24 − −

CDE 42452.99 − −

MPA 42429.01 42690.91 42561.37

MHDE 42679.87 − −

SPPSO 42740.23 43622.14 44346.97

Furthermore, the results of IMPA are compared other

means, including MPA, ORCCRO [34], RCCRO [34],

QOGSO [36], SPPSO [38], MCDE [40], SOS [41] and

IPCSO [42], and listed in ascending order in TABLE XII. The

minimum, maximum and mean expenses obtained in this case

are 162582.26 ($), 162711.38 ($) and 162649.06 ($)

respectively, which all show that IMPA can better manage the

STHS problem.

The convergence trajectory of the minimum expense of

IMPA and MPA is represented in Fig. 11 when the three

algorithms have the same parameters. It can be seen from this

picture, in contrast to MPA and GSA, IMPA has a better

performance in dealing with the STHS problem. The picture

illustrates that IMPA is superior to MPA in terms of

convergence speed and finding the optimal solution.



______________________________________________________________________________________

Fig. 10. Storage capacity of each reservoir in system II

TABLE X

OPTIMAL POWER GENERATION OF EACH THERMAL PLANT IN SYSTEM II

Hour Thermal generation (MW) Total thermal generation

1 2 3 4 5 6 7 8 9 10

1 319.28 274.40 94.80 119.73 174.60 139.73 45.00 35.00 25.00 177.03 1404.58

2 319.28 199.60 94.80 119.73 124.73 139.73 104.28 35.00 160.00 177.09 1474.24

3 139.76 274.40 94.80 119.73 124.73 139.73 104.28 35.00 160.00 177.03 1369.46

4 319.28 199.60 94.80 119.73 124.73 139.73 45.00 35.00 98.06 177.13 1353.07

5 319.28 199.60 20.00 119.73 124.73 139.73 104.28 35.00 160.00 126.36 1348.71

6 319.28 199.60 94.80 119.73 124.73 139.73 104.28 35.00 98.06 177.03 1412.24

7 229.52 349.20 94.80 119.73 174.60 139.73 163.55 35.00 98.06 177.00 1581.20

8 319.28 274.40 94.81 119.73 224.47 189.60 45.00 35.00 160.00 180.00 1642.29

9 229.52 349.20 94.80 119.73 224.47 189.60 163.55 35.00 98.06 176.92 1680.85

10 319.28 274.40 94.80 119.73 174.60 139.73 163.55 35.00 160.00 177.03 1658.12

11 319.28 274.40 94.80 119.73 224.47 139.73 163.55 35.00 160.00 177.01 1707.97

12 319.28 274.40 94.80 119.73 274.33 189.60 104.28 35.00 98.06 176.90 1686.38

13 319.28 274.40 94.80 119.73 224.47 189.60 163.55 35.00 98.06 176.98 1695.87

14 319.28 274.40 94.80 119.73 174.60 139.73 104.28 35.00 160.00 179.93 1601.75

15 319.28 199.60 94.80 119.73 174.60 139.73 163.55 35.00 160.00 176.99 1583.28

16 319.28 274.40 94.80 119.73 124.73 139.73 163.55 35.00 160.00 179.81 1611.04

17 319.28 274.40 94.80 119.73 124.73 139.73 163.55 35.00 160.00 176.93 1608.16

18 319.28 274.40 94.80 119.73 274.33 189.60 104.28 35.00 98.06 176.98 1686.46

19 319.28 274.40 94.80 119.73 174.60 139.73 104.28 35.00 160.00 176.97 1598.79

20 319.28 349.20 94.80 119.73 224.47 139.73 45.00 35.00 98.06 176.89 1602.16

21 319.28 199.60 94.80 119.73 174.60 139.73 104.28 35.00 98.06 179.68 1464.76

22 319.28 199.60 20.00 119.73 174.60 139.73 104.28 35.00 98.06 179.66 1389.94

23 319.28 199.60 94.80 119.73 124.73 139.73 45.00 35.00 98.06 177.06 1353.00

24 319.28 199.60 94.80 119.73 124.73 139.73 45.00 35.00 98.06 176.80 1352.73

Abbreviations

AABC adaptive artificial bee colony algorithm

AFP average fitness preferential strategy

CDE adaptive chaotic differential evolution

CPSO couple-based particle swarm optimization

DE differential evolution

DGSA disruption based gravitational search algorithm

DNLPSO modified dynamic neighborhood learning based

particle swarm optimization

DP dynamic programming

EP evolutionary programming

GA genetic algorithm

GSA gravitational search algorithm

GSO Quasi-oppositional group search optimization

IMPA improved marine predators algorithm

IPCSO novel two-swarm based PSO search strategy

IQPSO improved quantum-behaved particle

swarm optimization

LP linear programming

LR lagrangian relaxation

MCDE modified chaotic differential evolution

algorithm

MCSA modified cuckoo search algorithm

MHDE modified hybrid differential evolution

MIP mixed-integer programming

MPA marine predators algorithm

NLP nonlinear programming

OGB-GA optimal gamma based genetic algorithm

ORCCRO oppositional real coded chemical reaction based

optimization

PSO particle swarm optimization

QOGSO quasi-oppositional group search

optimization

RCCRO real coded chemical reaction based

optimization

SA simulated annealing

SOS symbiotic organisms search algorithm

SPPSO small population-based particle swarm

optimization

STSH short-term hydrothermal scheduling

TLBO teaching learning based optimization



______________________________________________________________________________________

TABLE XI

OPTIMAL POWER GENERATION OF EACH THERMAL PLANT IN SYSTEM II

Hour Hydro discharge (104 m3) Hydro generation (MW) Total Demand

1 2 3 4 1 2 3 4 (MW) (MW)

1 6.40 10.65 25.75 13.31 64.87 73.74 4.32 202.50 750.00 750.00

2 5.62 7.60 29.72 13.14 59.18 58.20 0.00 188.39 780.00 780.00

3 9.14 8.43 23.93 14.35 83.53 63.40 1.59 182.02 700.00 700.00

4 7.60 7.66 29.58 14.46 73.95 59.74 0.00 163.24 650.00 650.00

5 6.50 9.10 20.41 13.20 65.88 67.22 17.59 170.60 670.00 670.00

6 11.08 8.78 13.82 13.18 90.72 64.33 41.74 190.97 800.00 800.00

7 8.82 6.02 15.46 13.00 79.93 47.17 39.61 202.10 950.00 950.00

8 6.28 8.27 20.23 13.03 63.69 60.67 23.22 220.12 1010.00 1010.00

9 11.43 8.50 19.61 13.28 92.16 61.63 25.78 229.58 1090.00 1090.00

10 9.22 10.41 14.80 13.03 83.10 70.11 40.68 227.98 1080.00 1080.00

11 5.91 6.59 16.94 14.50 61.95 50.64 37.02 242.41 1100.00 1100.00

12 14.17 9.86 11.49 14.51 100.13 67.99 47.34 248.17 1150.00 1150.00

13 5.14 8.72 15.46 14.26 55.57 61.75 45.74 251.07 1110.00 1110.00

14 11.92 7.44 19.51 13.03 97.31 55.43 34.59 240.92 1030.00 1030.00

15 8.52 7.02 20.78 15.30 81.70 54.06 27.45 263.51 1010.00 1010.00

16 9.53 9.23 10.06 13.37 87.68 66.03 51.18 244.07 1060.00 1060.00

17 9.33 6.42 17.94 15.25 86.53 49.68 44.16 261.47 1050.00 1050.00

18 5.13 8.09 15.03 15.39 56.05 58.81 52.13 266.56 1120.00 1120.00

19 6.17 10.67 13.18 16.76 65.14 69.03 55.42 281.62 1070.00 1070.00

20 7.32 8.46 11.13 14.80 74.05 58.06 55.00 260.73 1050.00 1050.00

21 7.68 6.58 10.54 14.98 76.51 48.29 55.41 265.04 910.00 910.00

22 6.78 6.63 17.54 20.38 70.03 50.20 51.35 298.47 860.00 860.00

23 8.46 13.22 11.01 18.30 82.04 76.98 57.52 280.46 850.00 850.00

24 6.84 7.67 12.56 16.80 70.88 53.44 58.91 264.03 800.00 800.00

TABLE XII

COMPARISON OF PERFORMANCES OF VARIOUS ALGORITHMS IN SYSTEM II


Minimum Maximum Mean

IMPA 162582.26 162711.38 162649.06

MPA 162662.92 163278.42 163048.17

IPCSO 162714.00 162953.00 162813.00

SOS 162834.38 163147.87 162846.92

ORCCRO 163066.03 163134.54 163068.77

GSA 163498.15 163587.38 163542.51

RCCRO 164138.65 164182.35 164140.40

MCDE 165331.70 167061.60 166116.40

SPPSO 167710.56 170879.30 168688.92

QOGSO 170293.21 170349.34 170321.57

0 100 200 300 400 5001.62

1.63

1.64

1.65

1.66

1.67

1.68x 10

5

Generations

Fu

el e

xp

ense

($

)

IMPA

MPA

GSA

GSA

MPA

IMPA

Fig. 11. Convergence trajectory in system II

VI. CONCLUSION

The STSH problem is a nonlinear and non-convex

optimization problem with high complexity. In this paper, an

improved marine predators algorithm (IMPA) is presented

and successfully applied to solve the STSH problem of

finding the optimal scheduling solutions. The proposed

method has been tested on three test cases. Compared with a

lot of literature, the minimum, maximum and mean expenses

of IMPA are all superior to other literature in three test cases.

For instance, in case 1 of test system I, IMPA saves 250.7($),

990.89($) and 527.27($) respectively when compared with

MCDE in the minimum, maximum and mean expenses. This

indicates that IMPA has a strong competitiveness in solving

the nonlinear and non-convex STSH problem with complex

constraints. At the same time, it can be seen from a large

number of charts in this paper that optimal results of IMPA do

not violate any constraints. This shows that a series of

proposed constraint processing strategies is effective.

Therefore, the proposed IMPA provides an effective and

feasible method for solving the STSH problem.

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an improved marine predators algorithm for short-term

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