an improved marine predators algorithm for short-term
Post on 04-Apr-2022
1 Views
Preview:
TRANSCRIPT
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: chenggpower@126.com).
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:
yingxiao996@163.com).
Fangjia Long is a senior engineer of State Grid Chongqing Electric Power
Company, Chongqing 400015, China (e-mail: 157990759@qq.com).
Xiaorui Hu is a professor level senior engineer of Marketing Service
Center, State Grid Chongqing Electric Power Company, Chongqing 401123,
China (e-mail: xiaorui4832@sina.com).
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:
longhongyu20@163.com).
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
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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.
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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:
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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.
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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
Algorithm Fuel cost ($)
Minimum Maximum Mean
IMPA 41739.72 42185.21 41968.94
GSA 42012.24 43699.88 42913.62
GSO 42316.39 42379.18 42339.35
AABC 42381.24 − −
CDE 42452.99 − −
MPA 42429.01 42690.91 42561.37
MHDE 42679.87 − −
SPPSO 42740.23 43622.14 44346.97
Furthermore, the results of IMPA are compared other
means, including MPA, ORCCRO [34], RCCRO [34],
QOGSO [36], SPPSO [38], MCDE [40], SOS [41] and
IPCSO [42], and listed in ascending order in TABLE XII. The
minimum, maximum and mean expenses obtained in this case
are 162582.26 ($), 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.
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
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
Algorithm Fuel cost ($)
Minimum Maximum Mean
IMPA 162582.26 162711.38 162649.06
MPA 162662.92 163278.42 163048.17
IPCSO 162714.00 162953.00 162813.00
SOS 162834.38 163147.87 162846.92
ORCCRO 163066.03 163134.54 163068.77
GSA 163498.15 163587.38 163542.51
RCCRO 164138.65 164182.35 164140.40
MCDE 165331.70 167061.60 166116.40
SPPSO 167710.56 170879.30 168688.92
QOGSO 170293.21 170349.34 170321.57
0 100 200 300 400 5001.62
1.63
1.64
1.65
1.66
1.67
1.68x 10
5
Generations
Fu
el e
xp
ense
($
)
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.
REFERENCES [1] G. Chen, M. Gao, Z. Zhang and S. Li, "Hybridization of Chaotic Grey
Wolf Optimizer and Dragonfly Algorithm for Short-Term
Hydrothermal Scheduling". IEEE Access, vol. 8, pp. 142996-143020,
2020.
[2] X. Yuan, B. Ji, Z. Chen, and Z. Chen, "A novel approach for economic
dispatch of hydrothermal system via gravitational search algorithm".
Applied Mathematics and Computation, vol. 247, pp. 535-546, 2014. [3] Y. Jin-Shyr and C. Nanming, "Short term hydrothermal coordination
using multi-pass dynamic programming". IEEE Transactions on
Power Systems, vol. 4, pp. 1050-1056, 1989. [4] S. Chang, C. Chen, I. Fong, and P. B. Luh, "Hydroelectric generation
scheduling with an effective differential dynamic programming
algorithm". IEEE Transactions on Power Systems, vol. 5, pp. 737-743,
1990. [5] R. Liang and Y. Hsu, "A hybrid artificial neural network—differential
dynamic programming approach for short-term hydro scheduling".
Electric Power Systems Research, vol. 33, pp. 77-86, 1995. [6] E. Xi, X. Guan and R. Li, "Scheduling hydrothermal power systems
with cascaded and head-dependent reservoirs". IEEE Transactions on
Power Systems, vol. 14, pp. 1127-1132, 1999. [7] J. Jian, S. Pan and L. Yang, "Solution for short-term hydrothermal
scheduling with a logarithmic size mixed-integer linear programming
formulation". Energy, vol. 171, pp. 770-784, 2019. [8] D. Sjelvgren, H. Brännlund and T. S. Dillon, "Large-scale non-linear
programming applied to operations planning". International Journal
of Electrical Power & Energy Systems, vol. 11, pp. 213-217, 1989. [9] X. Guan, P. B. Luh and L. Zhang, "Nonlinear approximation method
in Lagrangian relaxation-based algorithms for hydrothermal
scheduling". IEEE Transactions on Power Systems, vol. 10, pp.
772-778, 1995. [10] J. M. Ngundam, F. Kenfack and T. T. Tatietse, "Optimal scheduling of
large-scale hydrothermal power systems using the Lagrangian
relaxation technique". International Journal of Electrical Power and
Energy System, vol. 22, pp. 237-245, 2000.
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
[11] G. W. Chang, M. Aganagic, J. G. Waight, J. Medina, T. Burton, S.
Reeves, and M. Christoforidis, "Experiences with mixed integer linear
programming based approaches on short-term hydro scheduling".
IEEE Transactions on Power Systems, vol. 16, pp. 743-749, 2001. [12] K. K. Mandal and N. Chakraborty, "Short-term combined economic
emission scheduling of hydrothermal power systems with cascaded
reservoirs using differential evolution". Energy Conversion and
Management, vol. 50, pp. 97-104, 2009. [13] Y. Lu, J. Zhou, H. Qin, Y. Wang, and Y. Zhang, "An adaptive chaotic
differential evolution for the short-term hydrothermal generation
scheduling problem". Energy Conversion and Management, vol. 51,
pp. 1481-1490, 2010. [14] B. Yu, X. Yuan and J. Wang, "Short-term hydro-thermal scheduling
using particle swarm optimization method". Energy Conversion and
Management, vol. 48, pp. 1902-1908, 2007. [15] K. K. Mandal, M. Basu and N. Chakraborty, "Particle swarm
optimization technique based short-term hydrothermal scheduling".
Applied Soft Computing Journal, vol. 8, pp. 1392-1399, 2008. [16] P. K. Hota, A. K. Barisal and R. Chakrabarti, "An improved PSO
technique for short-term optimal hydrothermal scheduling". Electric
Power Systems Research, vol. 79, pp. 1047-1053, 2009. [17] M. Basu, "An interactive fuzzy satisfying method based on
evolutionary programming technique for multiobjective short-term
hydrothermal scheduling". Electric Power Systems Research , vol. 69,
pp. 277-285, 2004. [18] N. Sinha, R. Chakrabarti and P. K. Chattopadhyay, "Fast evolutionary
programming techniques for short-term hydrothermal scheduling".
IEEE Transactions on Power Systems, vol. 18, pp. 214-220, 2003. [19] V. S. Kumar and M. R. Mohan, "A genetic algorithm solution to the
optimal short-term hydrothermal scheduling". International Journal
of Electrical Power and Energy Systems, vol. 33, pp. 827-835, 2011. [20] K. P. Wong and Y. W. Wong, "Short-term hydrothermal scheduling.
Part I: Simulated annealing approach". IEE Proceedings: Generation,
Transmission and Distribution, vol. 141, pp. 497-501, 1994.
[21] K. B. O. Medani, S. Sayah and A. Bekrar, "Whale optimization
algorithm based optimal reactive power dispatch: A case study of the
Algerian power system," Electric Power Systems Research, vol. 163,
pp. 696-705, 2018. [22] L. Lakshminarasimman and S. Subramanian, "A modified hybrid
differential evolution for short-term scheduling of hydrothermal
power systems with cascaded reservoirs". Energy Conversion and
Management, vol. 49, pp. 2513-2521, 2008. [23] J. Zhang, S. Lin and W. Qiu, "A modified chaotic differential
evolution algorithm for short-term optimal hydrothermal scheduling".
International Journal of Electrical Power and Energy Systems, vol.
65, pp. 159-168, 2015. [24] Y. Q. Wu, Y. G. Wu and X. Liu, "Couple-based particle swarm
optimization for short-term hydrothermal scheduling". Applied Soft
Computing Journal, vol. 74, pp. 440-450, 2019. [25] S. Lu, C. Sun and Z. Lu, "An improved quantum-behaved particle
swarm optimization method for short-term combined economic
emission hydrothermal scheduling". Energy Conversion and
Management, vol. 51, pp. 561-571, 2010. [26] T. T. Nguyen and D. N. Vo, "Modified cuckoo search algorithm for
short-term hydrothermal scheduling". International Journal of
Electrical Power and Energy Systems, vol. 65, pp. 271-281, 2015. [27] J. Sasikala and M. Ramaswamy, "Optimal gamma based fixed head
hydrothermal scheduling using genetic algorithm". Expert Systems
with Applications, vol. 37, pp. 3352-3357, 2010. [28] A. Faramarzi, M. Heidarinejad, S. Mirjalili, and A. H. Gandomi,
"Marine Predators Algorithm: A nature-inspired metaheuristic".
Expert Systems with Applications, vol. 152, pp. 113377, 2020. [29] M. Abdel-Basset, R. Mohamed, M. Elhoseny, R. K. Chakrabortty, and
M. Ryan, "A Hybrid COVID-19 Detection Model Using an Improved
Marine Predators Algorithm and a Ranking-Based Diversity
Reduction Strategy". IEEE Access, vol. 8, pp. 79521-79540, 2020. [30] M. Abdel-Basset, D. El-Shahat, R. K. Chakrabortty, and M. Ryan,
"Parameter estimation of photovoltaic models using an improved
marine predators algorithm". Energy Conversion and Management,
vol. 227, pp. 113491, 2021. [31] A. H. Yakout, H. Kotb, H. M. Hasanien, and K. M. AboRas, "Optimal
Fuzzy PIDF Load Frequency Controller for Hybrid Microgrid System
Using Marine Predator Algorithm". IEEE Access, pp. 54220 - 54232,
2021. [32] N. Gouthamkumar, V. Sharma and R. Naresh, "Disruption based
gravitational search algorithm for short term hydrothermal
scheduling". Expert Systems with Applications, vol. 42, pp.
7000-7011, 2015.
[33] P. K. Roy, "Teaching learning based optimization for short-term
hydrothermal scheduling problem considering valve point effect and
prohibited discharge constraint". International Journal of Electrical
Power and Energy Systems, vol. 53, pp. 10-19, 2013. [34] K. Bhattacharjee, A. Bhattacharya and S. Halder nee Dey,
"Oppositional real coded chemical reaction based optimization to
solve short-term hydrothermal scheduling problems". International
Journal of Electrical Power and Energy Systems, vol. 63, pp. 145-157,
2014. [35] A. Rasoulzadeh-Akhijahani and B. Mohammadi-Ivatloo, "Short-term
hydrothermal generation scheduling by a modified dynamic
neighborhood learning based particle swarm optimization".
International Journal of Electrical Power and Energy Systems, vol.
67, pp. 350-367, 2015.
[36] M. Basu, "Quasi-oppositional group search optimization for
hydrothermal power system". International Journal of Electrical
Power and Energy Systems, vol. 81, pp. 324-335, 2016.
[37] P. K. Roy, "Teaching learning based optimization for short-term
hydrothermal scheduling problem considering valve point effect and
prohibited discharge constraint". International Journal of Electrical
Power and Energy Systems, vol. 53, pp. 10-19, 2013.
[38] J. Zhang, J. Wang and C. Yue, "Small population-based particle swarm
optimization for short-term hydrothermal scheduling". IEEE
Transactions on Power Systems, vol. 27, pp. 142-152, 2012.
[39] X. Liao, J. Zhou, S. Ouyang, R. Zhang, and Y. Zhang, "An adaptive
chaotic artificial bee colony algorithm for short-term hydrothermal
generation scheduling". International Journal of Electrical Power and
Energy Systems, vol. 53, pp. 34-42, 2013.
[40] J. Zhang, S. Lin and W. Qiu, "A modified chaotic differential evolution
algorithm for short-term optimal hydrothermal scheduling".
International Journal of Electrical Power and Energy Systems, vol. 65,
pp. 159-168, 2015.
[41] S. Das and A. Bhattacharya, "Symbiotic organisms search algorithm for
short-term hydrothermal scheduling". Ain Shams Engineering Journal,
vol. 9, pp. 499-516, 2018.
[42] G. Cavazzini, G. Pavesi and G. Ardizzon, "A novel two-swarm based
PSO search strategy for optimal short-term hydro-thermal generation
scheduling". Energy Conversion and Management, vol. 164, pp.
460-481, 2018.
IAENG International Journal of Applied Mathematics, 51:4, IJAM_51_4_15
Volume 51, Issue 4: December 2021
______________________________________________________________________________________
top related