april_2020.pdf

METHODOLOGIES AND APPLICATION
Lightning attachment procedure optimization algorithm for nonlinear
non-convex short-term hydrothermal generation scheduling
Maha Mohamed1 • Abdel-Raheem Youssef1 • Salah Kamel2,4 • Mohamed Ebeed3
Published online: 17 April 2020
Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract
Short-term hydrothermal scheduling (STHS) is considered an important problem in the field of power system economics.
The solution of this problem gives the hourly output of power generation schedule of the available hydro and thermal
power units, which leads to minimization of the total fuel cost of thermal units for a given period of a time. The optimal
generation of STHS is considered as a complicated and nonlinear optimization problem with a set of equality and
inequality constraints such as the valve point loading effect of thermal units, the power transmission loss and the load
balance. This paper proposes lightning attachment procedure Optimization (LAPO) algorithm for solving the nonlinear
non-convex STHS optimization problem in order to minimize the operating fuel cost of thermal units with satisfying the
operating constraints of the system. The performance of LAPO algorithm is validated using three different test systems
considering the valve point loading effects of thermal units and the power transmission losses. The obtained results prove
the effectiveness and superiority of LAPO algorithm for solving the STHS problem compared with other well-known
optimization techniques.
Keywords Short-term hydrothermal scheduling Non-convex optimization problem Lightning attachment procedure
optimization Valve point loading effect
List of symbols
F Total fuel cost from all thermal plants
Ns Total number of thermal plants
T Total time of whole scheduling period
ai; bi; ci Power generation coefficients of thermal
plant
Pt
si Output power generation from thermal plant
di; ei Coefficients of the valve point effects of the
thermal plant
Pmin
si
Lower power generation limit of thermal
plant
Vt
hj Reservoir storage volume of hydropower
plant jth at a period of time t
It
hj External inflow to reservoir jth at a period of
time t
Qt
hj Water discharge amount of hydropower plant
j at a period of time t
S Spillage discharge rate of reservoir jth at
time interval t
Ruj Number of upstream hydropower plant
Nh Number of hydropower plant
Pt
D Power demand at a period of time t
Communicated by V. Loia.
Salah Kamel
skamel@aswu.edu.eg
Maha Mohamed
mahamohamed21@yahoo.com
Abdel-Raheem Youssef
abou_radwan@hotmail.com
Mohamed Ebeed
mohamedebeed11@gmail.com
1
Department of Electrical Engineering, Faculty of
Engineering, South Valley University, Qena, Egypt
2
Engineering, Aswan University, Aswân 81542, Egypt
3
Engineering, Sohag University, Sohâg, Egypt
4
State Key Laboratory of Power Transmission Equipment and
System Security and New Technology, Chongqing
University, Chongqing, China
123
Soft Computing (2020) 24:16225–16248
https://doi.org/10.1007/s00500-020-04936-2(0123456789().,-volV)
(0123456789().
,- volV)

Pt
hj Power generation of hydropower plant j at a
period of time t
Pt
L Power transmission loss of the system at a
period of time t
Vmin
hj
Minimum storage volume of hydro plant j
Vmax
hj Maximum storage volume of hydro plant j
Qmin
hj
Minimum water discharge of hydro plant j
QMax
hj Maximum water discharge of hydro plant j
Pmin
hj ; Pmax
hj
Minimum and maximum power generation
of hydro plant j
Pmin
si ; Pmax
si
Minimum and maximum power generation
of thermal plant i
Vbegin
hj ; Vend
hj
Initial and final reservoir storage volumes of
hydropower plant j
VT
hj Reservoir storage of hydro plant j at a period
of time from (0 to 24)
Abbreviations
STHS Short-term hydrothermal scheduling
LAPO Lightning attachment procedure
optimization
VPL Valve point loading
LP Linear programming
NLP Nonlinear programming
DP Dynamic programming
GS Gradient search
GA Genetic algorithm
EP Evolutionary programming
DE Differential evolution
PSO Particle swarm optimization
IPSO Improved particle swarm optimization
MAPSO Modified adaptive PSO
SSPSO Small population-based particle swarm
optimization
ABC Artificial bee colony
LR Lagrange relaxation
IDE Improved differential evolution
FAPSO Fuzzy adaptive particle swarm
optimization
RCGA Real-coded genetic algorithm
HIS Improved harmony search
RCGA-IMM Real-coded genetic algorithm based on
improved Mühlenbein mutation
CPSO Couple-based particle swarm optimization
TLBO Teaching learning-based optimization
ACABC Adaptive chaotic artificial bee colony
MDNLPSO Modified dynamic neighborhood learning-
based particle swarm optimization
RCGA–
AFSA
Hybrid of real-coded genetic algorithm and
artificial fish swarm algorithm
ORCCRO Oppositional real-coded chemical reaction
based optimization
DRQEA Differential real-coded quantum-inspired
evolutionary algorithm
MHDE Modified hybrid differential evolution
ACDE Adaptive chaotic differential evolution
ALO Ant lion optimization
1 Introduction
The optimal power generation of short-term hydrothermal
scheduling (STHS) has a great importance in the electric grid
systems. The main objective of STHS problem is to minimize
the total operation fuel cost of the thermal units through
determining the optimal power generation of hydro and
thermal units in each scheduling interval, while satisfying the
various equality and inequality constraints on the hydraulic
power plants and the power system network. The STHS is
considering a complicated problem, which includes the dif-
ferent equality and inequality constraints. The equality con-
straints include power balance, water storage balance, and
initial and terminal reservoir storage volumes. Also, the
inequality constraints are limitations of hydrothermal power
generation, limitations of water storage volumes and limita-
tions of water discharge rate. These constraints with the valve
point loading effect (VPLE) make the STHS problem a
nonlinear, non-convex and complicated constrained opti-
mization problem. Several optimization techniques have been
presented for solving the STHS problem. Firstly, analytical
optimization techniques have been implemented for obtaining
the optimal solution of hydrothermal scheduling problem
such as linear programming (LP) (Chang et al. 2001; Wu
et al. 2009), nonlinear programming (NLP) (Catalão et al.
2011), dynamic programming (DP) (Homem-de-Mello et al.
2011), gradient search (GS) (Wood and Wollenberg 2003),
Newton’s method (Zaghlool and Trutt 1988) and Lagrange
relaxation (LR) (Dieu and Ongsakul 2009). Linear program-
ming (LP) is applied to the problems which has linear
objective function and constraints, but the STHS problem is a
difficult and nonlinear optimization problem; therefore, this
will lead to errors in the result of the scheduling problem. The
NLP method requires large memory to reach the ideal solu-
tion of the nonlinear optimization problem and has slow
convergence. The DP is a popular method for overcoming the
difficulty of nonlinearity and non-convexity of the STHS
problem. However, the DP method suffers from the curse of
dimensionality when the size of the system increases and this
will lead to large memory storage and long computational
time. To overcome the handling constraints, the LR is more
accurate. However, the main drawback in LR is the
16226 M. Mohamed et al.
123

oscillation of solutions. The main shortage of these methods is
that they may stuck in local optima and suffer from
stagnation.
In order to overcome the drawbacks of analytical opti-
mization techniques, heuristic algorithms have been
implemented to solve the non-convex nonlinear STHS
problem such as genetic algorithm (GA) (Nazari-Heris
et al. 2017a; Haghrah et al. 2014), evolutionary program-
ming (EP) (Hota et al. 1999; Türkay et al. 2011), differ-
ential evolution (DE) (Malik et al. 2016), particle swarm
optimization (PSO) (Ramesh 2016; Mahor and Rangnekar
2012), improved particle swarm optimization (IPSO) (Hota
et al. 2009), modified adaptive PSO (MAPSO) and small
population-based particle swarm optimization (SSPSO)
(Amjady and Soleymanpour 2010), artificial bee colony
(ABC) (Liao et al. 2013; Zhou et al. 2014).
In Nazari-Heris et al. (2017a), the authors improved the GA
for finding the ideal solution of the STHS optimizationproblem
with considering the valve point loading effect of the thermal
power units and the power transmission losses. The real-coded
genetic algorithm with random transfer vectors-based mutation
(RCGA-RTVM) has been presented in Haghrah et al. (2014),
and the authorsrepresented with an innovatedmutation method
utilizing genetic algorithm (GA) to solve the nonlinear non-
convexSTHSproblem.InHotaetal.(1999;Türkayetal.2011),
the authors proposed the EP optimization algorithm to find the
optimal power generation scheduling for thermal and hydro
plants.InMaliketal.(2016),theauthorspresentedanimproved
hybrid approach based on the chaos theory in the differential
evolution (DE) algorithm for solving the STHS problem to
minimize the emission of the thermal units. The improved PSO
technique for solving the STHS problem has been presented in
Ramesh (2016), Mahor and Rangnekar (2012) and Hota et al.
(2009). The modified adaptive particle swarm optimization
(MAPSO) for determining the optimal thermal and hydro
power generation is presented in Hota et al. (x2010). To solve
the STHS problem, an adaptive chaotic artificial bee colony
(ACABC) algorithm has been considered in Liao et al. (2013).
In Zhou et al. (2014), the authors have been studied a multi-
objective artificial bee colony (MOABC) algorithm for solving
the nonlinear STHS optimization algorithm. Predator–prey-
based optimization (PPO) technique to obtain optimal genera-
tion scheduling of short-term hydrothermal system has been
offered in Narang et al. (2014). Table 1 shows the different
definitions of test systems uses for solving the STHS problems.
Moreover, literature reviews articles related to solve the STHS
optimization algorithm are summarized by Table 2.
Lightning attachment procedure optimization (LAPO) is
a new physical-based algorithm presented by Nematollahi
et al. (2017, 2019). LAPO is conceptualized from Light-
ning occurrence steps. The simulated steps of the LAPO
include trail spots, leader upward motion, section fading,
downward leader motion and the final strike point of
lightning which mimics the optimal solution.
In this paper, the authors present a new application of
lightning attachment procedure Optimization (LAPO)
technique to find the hourly optimal power generation of
thermal units and hydro power units for minimizing the
total fuel cost. The effect of valve point loading and the
power transmission loss are taken into consideration for
finding the optimal solution of the STHS optimization
problem. To evaluate the performance of proposed algo-
rithm, it is applied on three test systems including four
hydro power plants with single equivalent thermal units
and four hydro plants with three thermal units and four
hydro plants with ten thermal units.
The rest of paper is organized as follows. The formu-
lation of STHS problem is presented in Sect. 2. Section 3
presents the overview of proposed algorithm. The simula-
tion results in different studied cases are presented in
Sect. 4. Finally, the conclusion is presented in Sect. 5.
2 Problem formulation of hydrothermal
system
The STHS problem aims to minimize the total fuel cost of
thermal plants by use the hydropower as much as possible
and with negligible cost of the hydro power generation
units. The scheduling generation of hydro and thermal units
is provided during STHS process for a given period of time
for meeting the load demand and satisfying the all equality
and inequality constraints. The objective function and the
different constraints of the STHS problem are formulated
as follows.
2.1 Objective function
The objective function of total fuel cost of thermal units,
which is expressed as quadratic and a sinusoidal function
(Nazari-Heris et al. 2017b), can be represented as follows:
Table 1 Definition of test
systems studied for the solution
of STHS problem
Test system Number of hydrothermal generation units
Test system 1 One equivalent thermal unit and four cascaded hydro units
Test system 2 Four cascaded hydro power plants and three thermal plants
Test system 3 Four cascaded hydro power plants and ten thermal plants
Lightning attachment procedure optimization algorithm for nonlinear non-convex short-term… 16227
123

F ¼ min
X
T
t¼1
X
Ns
i¼1
ai þ biPt
si þ ci Pt
si
2
ð1Þ
where F is the total power generation fuel cost from all
thermal units at a time t, Ns is the total number of thermal
units, T is the total time of whole scheduling period, ai,bi,ci
are the power generation coefficients of thermal unit,Pt
si is
the output power generation from thermal unit of the ith
thermal plant at period t, respectively. The fuel cost
function of the ith thermal plant, and it is usually
Table 2 Objective functions and main contribution of researches in the area of STHS problem solution
Reference Method Year Test system Main consideration
Nazari-Heris et al. (2018) IHS 2018 Test system 1,
Test system
2
The cost of thermal units is commonly studied as a quadratic function,
valve point loading effect, transmission losses
Chang (2010) FAPSO 2010 Test system 1 The cost of thermal units is commonly studied as a quadratic function
Basu (2014b) Improved DE 2014 Test system 1,
Test system
2
Prohibited discharge zones (PDZs) of water reservoir of the hydro
units and valve point loading effect, ramp rate limits of thermal
generators, transmission losses
Mandal and Chakraborty
(2011)
SOHPSO_TVAC 2011 Test system 1 Economic emission, the cost of thermal units is commonly studied as
a quadratic function
Wu et al. (2019) CPSO 2019 Test system 1,
Test system
2
The cost of thermal units is commonly studied as a quadratic function,
valve point loading effect, prohibited discharge zones (PDZs) of
hydro units
Rasoulzadeh-Akhijahani
and Mohammadi-Ivatloo
(2015)
MDNLPSO 2015 Test system 1,
Test system
2
units and Valve point loading effect, transmission losses
Zhang et al. (2012) SPPSO 2012 Test system 1,
Test system
2
Valve point loading effect, transmission losses
Fang et al. (2014) RCGA–AFSA 2014 Test system 1,
Test system
2
Valve point loading effect, transmission losses, prohibited discharge
zones (PDZs) and ramp rate limits
Roy (2013) TLBO 2013 Test system 1,
Test system
2
units and Valve point loading effect
Roy (2014) HCRO-DE 2014 Test system 1 Valve point loading effect, emission of thermal units
Lu et al. (2010) MHDE 2010 Test system 1 Valve point loading effect, transmission losses
Kang et al. (2017) TLPSOS 2017 Test system 2 Valve point loading effect
Dubey et al. (2016) ALO 2016 Test system 2 Valve point loading effect, transmission losses
Bhattacharjee et al. (2014a) ORCCRO 2014 Test system 1,
Test system
2
Valve point loading effect
Zhang et al. (2015) MCDE 2015 Test system 2 Valve point loading effect, transmission losses
Bhattacharjee et al. (2014b) RCCRO 2014 Test system 2 Valve point loading effect, prohibited discharge zones (PDZs) of
hydro units and ramp rate limit
Gouthamkumar et al.
(2015)
DGSA 2015 Test system 2 Valve point loading effect
Swain et al. (2011) CSA 2011 Test system 2 Valve point loading effect
Lakshminarasimman and
Subramanian (2006)
MDE 2006 Test system 2 Valve point loading effect, prohibited discharge zones (PDZs) of
hydro units, emission of thermal units
Mandal et al. (2008) PSO 2008 Test system 2 Valve point loading effect
Roy et al. (2013) QTLBO 2013 Test system 2 Prohibited discharge zones (PDZs) of hydro units and valve point
loading effect
Mandal and Chakraborty
(2009)
DE 2009 Test system 2 Valve point loading effect, economic emission
Basu (2004a) EP 2004 Test system 2 Valve point loading effect
123

represented as follows with consideration of valve loading
point effect (Liao et al. 2013).
F ¼ min
X
T
t¼1
X
Ns
i¼1
ai þ biPt
si þ ci Pt
si
2
þ di sin ei Pmin
si Pt
si

n o
ð2Þ
where di and ei are the coefficients of the valve point
effects of the thermal unit i, Pmin
si is the lower power gen-
eration limit of thermal unit i.
2.2 Constraints
The objective function of STHS optimization problem is
subjected to the following equality and inequality con-
straints. The equality constraints include power balance,
water storage balance, and initial and terminal reservoir
storage volumes. Also, the inequality constraints are limi-
tations of hydrothermal power generation, limitations of
water storage volumes and limitations of water discharge
rate.
2.2.1 Water storage balance constraint
The reservoir storage of hydro plant is determined by
inflow and spillage, reservoir storage at previous period
and discharges from upstream reservoir. They must meet
the hydraulic continuity equations as follows (Wang et al.
2012).
Vt
hj ¼ Vt1
hj þ It
hj Qt
hj St
hj þ
X
Ruj
l¼1
Q
tdlj
hl þ S
tdlj
hl

jNh
tT:
ð3Þ
where Vt
hj is storage volume of hydropower plant jth at a
time t, It
hj is the external inflow rate to reservoir jth at time
t,Q
tdlj
hl is the water discharge rate from lth to jth reservoir
during the time delay dlj, dlj is the water transport delay
from lth to jth reservoir ; St
hj is the spillage discharge rate of
reservoir jth at time t, Ruj is the number of upstream
hydropower plants of jth reservoir.
2.2.2 Load demand balance constraint
Power generations of hydro and thermal power units must
meet the load demands of the hydrothermal including the
power transmission losses. Hence, load balance constraint
is expressed as follows:
X
Ns
i¼1
Pt
si þ
X
Nh
j¼1
Pt
hj Pt
L ¼ Pt
D tT ð4Þ
where Nh is the number of hydropower units, Pt
D represents
the power load demand at a period of time t,Pt
hj is the
power generation of hydropower unit j at a period of time t,
Pt
L is the transmission loss of the system at a period of time
t; Pt
hj is formulated as the following equation:
Pt
hj ¼ C1j Vt
hj
2
þC2j Qt
hj
2
þC3jVt
hjQt
hj þ C4jVt
hj þ C5jQt
hj
þ C6j jNh tT
ð5Þ
where Vt
hj,Qt
hj represent the storage volume and water
discharge amount of hydropower unit j at a period of time t,
C1j, C2j, C3j, C4j, C5j and C6j are the power generation
coefficients of hydropower unit j, respectively. The power
transmission loss Pt
L is expressed by the following
equation:
Pt
L ¼
X
NhþNs
i¼0
X
NhþNs
j¼0
Pt
iBijPt
j þ
X
NhþNs
i¼0
BoiPj
i þ Boo ð6Þ
where Bij; Boi and Boo are the power transmission loss
coefficients.
2.2.3 Reservoir storage volumes constraint
0Vmin
hj Vt
hj Vmax
hj ; jNh; tT: ð7Þ
where Vmin
hj ; Vmax
hj represent the minimum and maximum
storage volume limits of the jth hydro plant.
2.2.4 Water discharge constraint
0Qmin
hj Qt
hj Qmax
hj jNh; tT: ð8Þ
where Qmin
hj ; Qmax
hj represent the minimum and maximum
water discharge limits of the jth hydro plant.
2.2.5 Power generation constraint
Pmin
hj Pt
hj Pmax
hj jNh tT: ð9Þ
Pmin
si Pt
si Pmax
si jNs tT: ð10Þ
where Pmin
hj ; Pmax
hj are the minimum and maximum power
generation of the jth hydro plant, respectively and Pmin
si ;
Pmax
si are the minimum and maximum power generation of
the ith thermal plant, respectively.
The initial and terminal reservoir storage volumes:
Vend
hj ¼ VT
hj ð11Þ
VT
hj ¼ Vbegin
hj ð12Þ
123

where Vbegin
hj ,Vend
hj are the initial and final reservoir storage
volumes of the jth hydro plant and VT
hj is the reservoir
storage of the jth hydro plant at a period of time from (0 to
24).
2.2.6 Handling constraints
It should be highlighted here that Michalewicz and Schoe-
nauer have presented a review survey for constraints handling
methods in optimization algorithms including preserving
feasibility method, penalty functions method, feasible and
infeasible solutions method and a hybrid method. In the pre-
sent, work the dependent systems have been considered using
the penalty functions method as follows:
Fg ¼ F þ KV
X
Nh
j¼1
DVhj
2
þKP
X
Nh
i¼1
DPhj
2
ð13Þ
where KV and KP represent the penalty factors for the water
discharge limits of the hydro plant and the power genera-
tion of the hydro plant, respectively. Fg is the augmented
objective function. DVhj and DQGi are given as follows:
DVhj ¼
Vt
hj Vmax
hj

Vt
hj [ Vmax
hj
Vmin
hj Vt
hj

Vt
hjVmin
hj
0 Vmin
hj Vt
hjVmax
hj
8

:
ð14Þ
DPhj ¼
Pt
hj Pmax
hj

Pt
hj [ Pmax
hj
Pmin
hj Pt
hj

Pt
hjPmin
hj
0 Pmin
hj Pt
hjPmax
hj
8

:
ð15Þ
3 Lighting attachment procedure
optimization (LAPO)
Lightning attachment procedure optimization (LAPO) is a
novel optimization technique conceptualized from Light-
ning phenomena where huge amounts of electric charges
are cumulated in the cloud. The distribution of these
charges in the cloud is depicted in Fig. 1. Lightning is
created with increasing the amount of charges in the cloud
which lead to increase the electrical strength consequently.
Lightning strike will occur, and it may emanate at several
points.
The procedure of lightning attachment includes four
steps which are: (1) breakdown of air at surface of cloud,
(2) lightning channel downward motion, (3) upward leader
extension and (4) final strike point.
As mentioned before, huge amounts of positive and
negative charges exist in the cloud where the highest
amount of the negative charges exist in the upper portion of
the cloud and the huge positive charges will be in the lower
portion of the cloud including also small amount of posi-
tive charges as depicted in Fig. 1. With increasing the
amount of the charges, the electrical potential will also
increase. Consequently, the breakdown between the char-
ges occurs. Moreover, the negative charges at the bottom of
the cloud increase more and potential gradient between the
cloud edge and the ground rises, leading to formation of the
lightning. The lightning starts from one or more points
from the cloud. The downward leaders of the lightning
move to the earth in a gradual motion due to the collapse
caused by air contact with the cloud surface and the leaders
do not continue in one direction as depicted in Fig. 1.
3.1 Mathematical presentation of LAPO
algorithm
Step 1 Trail spots.
The trial spots represent the initial points of the down-
ward leaders which can be found as follows:
Xi
ts ¼ Xi
min þ Xi
max Xi
min

rand ð16Þ
where Xi
ts denotes the initial trial spots. Xmin is the mini-
mum value of the control variable, while Xmax is its max-
imum value. rand is a random value in the range [0,1]. The
fitness function for the initial spots is calculated as:
Fi
ts ¼ obj Xi
ts

ð17Þ
Step 2 Determination of the next jump
All initial points are averaged, and fitness values are
calculated as follows:
Xavr ¼ mean Xts
ð Þ ð18Þ
Favr ¼ obj Xavr
ð Þ ð19Þ
Downward Leader
Upward Leader
+
+
+
+
+
+
+
+ +
+
+
+ +
+ + + +
+
+
+
+ +
+
+
+
- -
+
- -
- -
- -
-
-
- -
-
- +
-
- - - -
- -
- -
- -
-
- -
-
-
-
-
-
- - -
-
- - -
-
- -
-
-- -
-
-
-
-
-
-
-
-
+
+
+ +
+
+ +
+
-
Fig. 1 Charges form in the cloud
123

Fig. 2 Solution process of
STHS problem using proposed
algorithm
123

Xavr is the averaged point, while Favr is the objective
function of the averaged point. As mentioned before, the
lightning has several tracks where the lightning is jumped
to the next high optional point. For updating the point i, a
random solution j is selected (potential point), so i = j.
Then the obtained solution is compared with the potential
solution. Hence, the next jump can be calculated as
follows:
Xi
ts new ¼ Xi
ts þ rand Xavr þ X j
PS

IF FjFavr ð20Þ
Xi
ts new ¼ Xi
ts rand Xavr þ X j
PS

IF Fj [ Favr ð21Þ
Step 3 Section fading
The branch will remain continuous if the critical value is
less than the electric field of the new test point; otherwise,
it will fade, which can be expressed as follows:
Xi
ts ¼ Xi
ts new IF Fi
ts newFi
ts ð22Þ
Xi
ts new ¼ Xi
ts otherwise ð23Þ
Test points are executed in this process, and all the
remaining points in the first stage are moving down.
Step 4 Leader upward motion
In this procedure, the points move up mimics the motion
of upward leader which is distributed exponentially along
the channel. Hence, an exponent operator can be repre-
sented as follows:
S ¼ 1
t
tmax
exp
t
tmax
ð24Þ
where t denotes the iteration number, while tmax is the
maximum number of iterations, and next jump depends on
the charge of the channel and the next point is given as
follows:
Xi
ts new ¼ Xi
ts new þ rand S Xi
best Xi
worst

ð25Þ
where Xi
best and Xi
worst are the best and the worse solutions
among the populations.
Step 5 Final strike point
The lightning operation pauses when the down leader
and the up leader gather each other and the striking point is
assigned.
The flowchart of the LAPO algorithm for obtaining the
optimal solution is shown in Fig. 2.
4 Simulation results and discussion
The effectiveness of the proposed LAPO algorithm is
validated using two hydrothermal test systems. The first
test system focuses on a multi-chain cascade of four hydro
units and one thermal power generating unit. There are two
case studies in this system. In case 1, the objective function
is smooth quadratic operation cost of thermal power gen-
eration as presented in Eq. (1). The valve point loading
effect of the thermal unit is considering in case (2) as given
1
I 2
I
3
I
4
I
1
Q 2
Q
3
Q
4
Q
Reservior 1 Reservior 2
Reservior 3
Reservior 4
Fig. 3 Scheme of the hydraulic network of the hydrothermal test
system
Table 3 Reservoir inflows of
hydropower plants for test
systems 1 and 2
Hour Reservoir Hour Reservoir Hour Reservoir
1 2 3 4 1 2 3 4 1 2 3 4
1 10 8 8.1 2.8 9 10 8 1 0 17 9 7 2 0
2 9 8 8.2 2.4 10 11 9 1 0 18 8 6 2 0
3 8 9 4 1.6 11 12 9 1 0 19 7 7 1 0
4 7 9 2 0 12 10 8 2 0 20 6 8 1 0
5 6 8 3 0 13 11 8 4 0 21 7 9 2 0
6 7 7 4 0 14 12 9 3 0 22 8 9 2 0
7 8 6 3 0 15 11 9 3 0 23 9 8 1 0
8 9 7 2 0 16 10 8 2 0 24 10 8 0 0
123

in Eq. (2). The second test system consists of four cascaded
hydro and three thermal generating units. In the second
system, two different case studies are considered. In the
first case study, the STHS problem is solved considering
the valve point loading effect without considering the
power transmission losses. In the second case study, the
STHS problem is solved considering the valve point
loading effect and the power transmission losses of the
system. The hydraulic network of these test systems is
shown in Fig. 3. The total period is 1 day that is divided
into 24 intervals. The coefficients of hydropower generat-
ing units, reservoir inflows, water discharge limits, initial
and terminal reservoir storage limits and hourly load
demands of power systems are given in Tables 3, 4, 5, 6, 7,
8 and 9. The cost coefficient of thermal and hydro gener-
ating units is adopted from Nazari-Heris et al. (2017a).
4.1 Test system 1
The first test system consists of four cascaded hydro units
and an equivalent thermal unit. In this system, the power
transmission losses are neglected for simplicity. To eval-
uate the performance of the LAPO, two different case
studies have been taken into account as follows;
Table 4 The coefficients of hydropower generation for test systems 1
and 2
Plant C1j C2j C3j C4j C5j C6j
1 - 0.0042 - 0.42 0.030 0.90 10.0 - 50
2 - 0.0040 - 0.30 0.015 1.14 9.5 - 70
3 - 0.0016 - 0.30 0.014 0.55 5.5 - 40
4 - 0.0030 - 0.31 0.027 1.44 14.0 - 90
Table 5 Hydro power generation unit characteristics
Plant Vmin
hj
Vmax
hj Vbegin
hj
Vend
hj Qmin
hj
Qmax
hj pmin
hi
pmax
hi
1 80 150 100 120 5 15 0 500
2 60 120 80 70 6 15 0 500
3 100 240 170 170 10 30 0 500
4 70 160 120 140 6 25 0 500
Table 6 The coefficients of thermal units power generation for test
system 1
Plant ai bi ci di ei pmin
si
pmax
si
1 0.002 19.2 5000 700 0.085 500 2500
Table 7 Load demands of hydrothermal system for test system 1
Hour Load Hour Load Hour Load Hour Load
1 1370 7 1650 13 2230 19 2240
2 1390 8 2000 14 2200 20 2280
3 1360 9 2240 15 2130 21 2240
4 1290 10 2320 16 2070 22 2120
5 1200 11 2230 17 2130 23 1850
6 1410 12 2310 18 2140 24 1590
Table 8 The coefficients of thermal units power generation for test
system 2
Plant ai bi ci di ei pmin
si
pmax
si
1 0.0012 2.45 0.0012 160 0.038 20 175
2 0.0010 2.32 0.0010 180 0.037 40 300
3 0.0015 2.10 0.0015 200 0.035 50 500
Table 9 Load demands of hydrothermal system for test system 2
Hour Load Hour Load Hour Load Hour Load
1 750 7 950 13 1110 19 1070
2 780 8 1010 14 1030 20 1050
3 700 9 1090 15 1010 21 910
4 650 10 1080 16 1060 22 860
5 670 11 1100 17 1050 23 850
6 800 12 1150 18 1120 24 800
123

4.1.1 Test system 1 case 1
The first case study is solved without considering the valve
point loading effect of the thermal units. The fuel cost
function of thermal unit is a quadratic function of the STHS
problem as shown in Eq. (1). The optimal hourly water
discharge and hydrothermal power generation obtained by
LAPO method for solving the STHS problem during 24 h
scheduling are reported in Table 10. It is obvious from
Table 10 that the optimal solution satisfies all the con-
straints on hydro discharges and thermal power generation.
The best results of the STHS problem proposed by LAPO
method are compared with different optimization tech-
niques in Table 11. The minimum fuel cost obtained by the
LAPO method is 871,910.67 $ which shows the capability
of the proposed method for obtaining the optimal solution
of the STHS problem with respect to other optimization
methods. The minimum cost is obtained by the proposed
method better than the recent optimization algorithm with
3483.56 $/day. The optimal hourly hydro and thermal
power generation for each hour for the first case study is
shown in Fig. 4. It is obvious that the load demand is equal
to the sum of the power generation for each hour. Figure 5
shows the convergence characteristic of the LAPO method
for this case study.
The effect of valve point loading has been taken into
account in this case to illustrate the performance of the
LAPO method. Table 12 presents the optimal variables of
water discharges and the optimal power generation of
hydro and thermal generating units obtained by LAPO
method. The best results obtained by LAPO method are
compared with recent meta-heuristic method like a real-
coded genetic algorithm based on improved Mühlenbein
mutation (RCGA-IMM) (Nazari-Heris et al. 2017a) as
illustrated in Table 13. The minimum cost found by LAPO
Hydro Power Units
Thermal
Power Unit
Electric Power
System
Load
123

Table 10 Optimal water discharge of hydro and thermal power generation for case 1 of test system 1
Hours Water discharge rates (104
m3
/s) Hydro plant power generation (MW) Thermal plant
generation (MW)
Total power
generation (MW)
Plant 1 Plant 2 Plant 3 Plant 4 Plant 1 Plant 2 Plant 3 Plant 4
1 13.248 13.377 13.574 22.250 94.981 81.167 37.384 242.897 913.567 1370
2 13.553 14.806 11.524 23.506 93.146 78.868 38.674 213.683 965.630 1390
3 14.988 14.997 11.479 24.971 90.038 74.092 38.674 199.588 957.607 1360
4 14.995 14.999 11.510 24.883 86.620 72.499 38.674 199.550 892.642 1290
5 14.852 14.999 10.544 24.990 86.640 72.499 41.245 288.581 801.027 1290
6 14.940 14.985 11.924 24.997 86.630 72.478 44.823 327.843 878.223 1410
7 14.853 15 29.934 24.998 86.640 72.500 0 327.845 1162.995 1650
8 14.925 14.998 13.595 24.999 86.632 72.498 37.357 327.849 1475.660 2000
9 14.892 14.989 11.691 24.989 86.636 72.485 38.664 327.819 1714.390 2240
10 14.724 14.994 10.663 24.999 86.643 72.491 38.465 327.848 1794.545 2320
11 14.797 14.993 29.669 24.999 86.643 72.491 0 327.848 1743.013 2230
12 14.994 14.999 29.965 24.994 86.621 72.499 0 314.280 1836.598 2310
13 14.990 14.999 11.522 24.999 86.621 72.499 38.674 327.849 1704.354 2230
14 14.988 14.998 11.474 24.978 86.622 72.497 41.700 327.789 1671.386 2200
15 14.999 14.999 12.298 24.989 86.620 72.499 44.511 327.819 1598.550 2130
16 14.953 14.999 12.307 24.991 86.628 72.499 46.881 327.825 1536.171 2070
17 14.992 14.992 12.048 24.999 86.621 72.490 49.108 327.847 1593.925 2130
18 14.990 14.976 13.147 24.981 86.621 72.466 51.001 327.795 1602.111 2140
19 14.910 14.980 12.553 24.993 86.634 72.472 52.829 327.828 1700.235 2240
20 14.954 14.999 12.194 24.996 86.628 72.499 57.180 327.839 1735.846 2280
21 14.788 14.997 12.805 24.975 86.643 72.496 60.775 327.779 1692.306 2240
22 14.489 14.887 13.477 24.961 86.612 72.339 63.265 327.703 1570.078 2120
23 14.960 14.982 13.761 24.998 86.627 72.475 64.612 327.845 1298.445 1850
24 14.999 14.846 12.812 24.972 107.01 80.704 58.974 303.488 1039.811 1590
Table 11 Comparison of the best results of the STHS problem for case 1 of test system 1
Algorithm Minimum cost
($)
Average cost
($)
Maximum cost
($)
Computation time
(s)
LAPO 871,910.67 873,820.11 878,850.11 4.08
IHS (Nazari-Heris et al. 2018) 875,394.2288 875,687.1443 876,371.0758 NA
RCGA-IMM (Nazari-Heris et al. 2017a) 875,856.41 NA NA NA
RCGA-RTVM (Haghrah et al. 2014) 877,735.9 878,597.2406 880,948.518 NA
FAPSO (Chang 2010) 914,660.00 NA NA 4.73
Improved DE (Basu 2014b) 917,250.1 NA NA NA
PSO (Chang 2010) 921,920 NA NA 10.67
SOHPSO_TVAC (Mandal and Chakraborty 2011) 922,018.24 NA NA NA
CPSO (Wu et al. 2019) 922,328.64 922,367.85 922,564.52 12.9
MDNLPSO (Rasoulzadeh-Akhijahani and Mohammadi-Ivatloo
2015)
922,336.3 922,676.2 923,404.5 35
SPPSO (Zhang et al. 2012) 922,336.31 922,668.45 927,203.63 16.3
RCGA–AFSA (Fang et al. 2014) 922,339.625 922,346.323 922,362.532 NA
TLBO (Roy 2013) 922,373.39 922,462.24 922,873.81 NA
HCRO-DE (Roy 2014) 922,444.79 922,513.62 922,936.17 NA
IPSO (Hota et al. 2009) 922,553.49 NA NA 38.46
MDE (Zhang et al. 2012) 922,556.38 923,201.13 923,813.99 53
RCGA (Fang et al. 2014) 923,966.285 924,108.731 924,232.072 NA
123

method is 881,184.5 $ which is found to be superior to all
other reported methods in Table 13. The total fuel cost can
be saved when compared to recent optimization techniques
which is 10,595.35 $/day. In addition, the proposed method
successful to maintain the load demand is equal to the total
power generation. The optimal power generation for hydro
and thermal generating units is depicted in Fig. 6. The
optimal cost convergence characteristic for this test system
is shown in Fig. 7. It is clear from these tables and fig-
ures that the best solution obtained by the LAPO method
satisfies all the constraints of the STHS problem for this
case study.
4.2 Test system 2
To evaluate the performance of the proposed LAPO
method, it is applied to another system. This system
includes four hydro and three thermal power generating
units, but this test system is more complex than the first test
system because this system includes the effect of valve
point loading and the power transmission losses.
0
500
1000
1500
2000
2500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Generation
(MW)
Time ( hour)
hydro 4
hydro 3
hydro 2
hydro 1
Thermal
Fig. 4 Hourly optimal power
generation of hydro and thermal
units for test system 1 case 1
Hydro Power Units
Thermal
Power Units
Electric Power
System
Load
123

In this case, the valve point loading effect of thermal units
is considered and the power transmission losses are
neglected. The optimal solution of the STHS problem is
given in Table 14. The water discharge and hydro power
generation of four hydro units are reported in this table. In
addition, thermal power generation of three thermal units is
provided in this table. It is obvious that the scheduling
results obtained by LAPO method satisfy all hydraulic and
electric system constraints. The minimum fuel cost of test
system 2 case 1 with recent optimization method is
40,204.32 $ which is reduced to 38,800.75 $ with the
proposed LAPO method as shown in Table 15. In the other
words, the total daily saving is 1403.57 $ compared to the
recent optimization method in Fang et al. (2014). Figure 8
shows the hourly hydro and thermal power generation of
0 500 1000 1500 2000 2500 3000
0.85
0.9
0.95
1
1.05
1.1
1.15
x 10
6
Iteration
Total
cost
($)
Fig. 5 Optimal cost of STHS problem for case 2 of test system 1
Table 12 Optimal water discharge of hydro and thermal power generation for case 2 of test system 1
m3
generation (MW)
Total power
generation (MW)
1 7.4753 12.880 10.321 19.324 72.400 80.171 38.258 235.651 943.518 1370
2 12.798 8.0651 29.437 23.685 95.012 59.213 0 218.334 1017.434 1390
3 14.061 14.997 11.571 24.990 93.993 80.183 38.673 199.596 947.517 1360
4 8.2914 13.378 13.259 23.542 73.938 73.379 37.746 198.373 906.561 1290
5 7.0003 14.732 13.828 23.904 65.434 72.103 37.049 282.771 832.640 1290
6 7.3898 14.177 28.302 24.860 67.888 71.146 0 327.449 943.513 1410
7 11.230 14.831 11.075 24.316 85.029 72.255 38.620 325.779 1128.313 1650
8 9.4620 13.681 17.170 24.089 77.900 70.132 29.029 325.025 1497.917 2000
9 12.576 12.977 10.490 24.603 86.827 68.441 38.369 326.684 1719.677 2240
10 13.639 14.938 11.775 25 86.794 72.412 39.335 327.850 1793.595 2320
11 13.167 10.335 10.600 24.668 85.575 59.442 38.432 326.881 1719.668 2230
12 11.135 14.872 10.333 24.835 81.120 72.316 44.582 318.382 1793.597 2310
13 10.405 14.676 11.610 22.920 78.992 72.016 38.671 320.643 1719.676 2230
14 10.593 14.905 11.294 24.799 80.422 72.365 38.662 325.832 1682.718 2200
15 14.804 14.145 13.874 24.543 86.643 71.085 36.983 326.496 1608.792 2130
16 14.861 14.991 27.340 24.999 86.639 72.487 0 327.849 1583.036 2070
17 12.518 14.925 11.706 24.267 84.530 72.394 38.662 325.617 1608.798 2130
18 14.844 15 11.447 24.999 86.640 72.500 38.674 327.849 1614.295 2140
19 13.142 14.834 12.917 23.920 85.541 72.260 38.072 324.445 1719.679 2240
20 14.533 13.074 16.371 18.966 86.621 68.691 31.554 299.549 1793.581 2280
21 14.473 11.346 10.251 24.597 86.608 63.381 43.667 326.666 1719.673 2240
22 14.668 8.2687 10.463 24.080 86.640 50.054 49.516 324.995 1608.796 2120
23 14.288 7.6359 11.623 24.595 86.549 46.764 39.951 326.658 1350.071 1850
24 14.184 14.706 10.063 21.786 105.92 80.468 56.178 293.020 1054.406 1590
123

the optimal solution for test system 2 case 1. The conver-
gence characteristics of STHS problem by employing the
LAPO method are shown in Fig. 9.
The valve point loading effect and the power transmission
lossesofthehydrothermalsystemareconsideredinthiscasefor
obtaining the optimal generation scheduling. The optimal
generation scheduling for four hydro and three thermal units,
($)
Average cost
($)
Maximum cost
($)
Computation
time(s)
LAPO 881,184.5 885,721.3 889,151.6 5.06
RCGA-RTVM (Haghrah et al. 2014) 917,222.73 NA NA NA
IDE (Basu 2014b) 923,016.29 923,036.28 923,152.06 547.07
2015)
923,961 925,258 926,230 119
CPSO (Wu et al. 2019) 924,042.14 925,086.38 926,213.26 18.6
MAPSO (Amjady and Soleymanpour 2010) 924,636.88 926,496 927,431 NA
DRQEA (Wang et al. 2012) 925,485.21 NA NA 7.5
MHDE (Lu et al. 2010) 925,547.31 NA NA 9
IPSO (Hota et al. 2009) 925,978.84 NA NA 31
RQEA (Wang et al. 2012) 926,068.33 NA NA 7.6
DE (Wang et al. 2012) 928,662.84 NA NA 8.7
RCGA (Fang et al. 2014) 930,565.242 930,966.356 931,427.212 NA
0
500
1000
1500
2000
2500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Generation
(MW)
Time (hour)
Hydro 4
Hydro 3
Hydro 2
Hydro 1
Thermal
0 500 1000 1500 2000 2500 3000
0.9
1
1.1
1.2
1.3
x 10
6
Iteration
Total
cost
($)
123

Table 14 Optimal water discharge of hydro and thermal power generation for case 1 of test system 2 (without losses)
Hours (h) Water discharge rates (104
m3
/s) Hydro plant power generation (MW) Thermal plant generation
(MW)
Total load (MW)
Plant 1 Plant 2 Plant 3 Plant 4 Plant 1 Plant 2 Plant 3 Plant 4 Plant 1 Plant 2 Plant 3
1 11.881 14.880 17.128 10.441 92.370 83.227 29.169 177.984 102.638 124.877 139.724 750
2 5.299 6.082 22.175 17.215 55.483 46.549 4.485 216.396 102.675 124.887 229.522 780
3 13.802 13.592 11.488 23.312 95.463 78.521 38.674 207.574 20.012 209.757 50.007 700
4 14.552 14.352 27.639 24.236 91.949 75.821 0 199.121 102.669 40.001 139.801 650
5 5.8224 14.9813 25.821 15.967 56.246 72.473 0 258.852 102.664 40.0032 139.766 670
6 14.999 13.753 10.080 24.148 87.053 70.288 38.070 325.225 99.596 40 139.778 800
7 14.061 14.363 22.972 24.989 86.437 71.486 0 327.820 20.148 124.834 319.271 950
8 13.674 8.094 29.754 24.582 86.147 48.528 0 326.620 20.004 209.427 319.263 1010
9 14.951 12.124 10.746 14.313 86.628 65.992 38.504 262.311 102.671 124.897 409.013 1090
10 12.068 14.793 22.851 16.660 83.597 72.197 0 282.772 102.626 40.000 498.795 1080
11 10.949 12.881 29.978 22.735 81.118 68.188 0 319.873 102.653 209.805 318.360 1100
12 12.959 14.548 12.827 23.469 85.279 71.807 43.305 322.807 102.587 294.706 229.505 1150
13 7.645 6.000 23.631 17.610 66.860 37.085 0 290.082 102.277 294.397 319.2940 1110
14 12.479 12.934 10.477 23.523 86.153 68.327 38.361 323.010 154.915 40.0035 319.270 1030
15 5.250 10.724 10.642 23.184 51.663 61.030 38.454 321.706 102.663 294.730 139.783 1010
16 14.999 14.999 14.427 24.872 89.021 72.499 36.103 313.277 20.000 209.811 319.274 1060
17 14.489 11.211 16.659 24.999 86.612 62.890 30.688 327.847 102.638 209.830 229.509 1050
18 14.999 14.990 10.003 20.425 86.620 72.486 38.052 308.464 174.906 209.925 229.517 1120
19 11.788 9.4846 28.025 24.983 82.931 55.652 0 327.803 164.339 209.750 229.523 1070
20 14.671 6.787 12.142 24.791 86.640 41.686 42.707 327.2481 102.671 40.000 409.035 1050
21 6.073 14.969 29.999 24.984 55.318 72.457 0 327.806 20.000 294.674 139.751 910
22 14.999 13.959 12.685 18.017 86.620 70.717 38.253 293.043 102.663 40.007 228.700 860
23 8.985 9.7949 10.115 22.697 72.635 57.085 38.099 319.711 102.658 209.802 50.007 850
24 10.875 14.999 16.860 24.925 95.751 80.948 54.837 303.378 174.997 40.096 50.000 800
Table 15 Comparison of simulation results obtained by different methods for case 1 of test system 2 (without losses)
($)
Average cost
($)
Maximum cost
($)
Computation
time(s)
LAPO 38,800.75 38,915.23 39,520 6.634
2015)
40,179 40,637 41,182 123
CPSO (Wu et al. 2019) 40,204.32 40,592.73 40,831.55 15.1
TLPSOS (Kang et al. 2017) 40,298.28 40,298.28 40,298.28 102
ALO (Dubey et al. 2016) 40,780.05 41,094.3414 40,905.8259 15.01
RCGA–AFSA (Fang et al. 2014) 40,913.82 41,362.57 41,235.72 21
ORCCRO (Bhattacharjee et al. 2014a) 40,936.65 41,127.6819 40,944.2938 10.48
MCDE (Zhang et al. 2015) 40,945.75 41,380.54 41,977.04 50.8
ACABC (Liao et al. 2013) 41,074.42 NA NA 16
RCCRO (Bhattacharjee et al. 2014b) 41,497.85 41,502.3669 41,498.2129 15.51
DGSA (Gouthamkumar et al. 2015) 41,751.15 41,989.02 41,821.49 31.99
CSA (Swain et al. 2011) 42,244.057 NA NA 109
MDE (Lakshminarasimman and Subramanian 2006) 42,611.14 NA NA 125
PSO (Mandal et al. 2008) 44,740 NA NA 232.73
123

hourly water discharge and the power transmission losses are
shown in Table 16. The LAPO algorithm is the best for solving
the STHS problem by obtaining the minimal total fuel cost with
efficiency as shown in Table 17. The minimum cost obtained
by LAPO is 39,691.86 $ which helps in daily saving the cost by
234.87 $ as compared the RCGA-IMM (Nazari-Heris et al.
2017a). The optimal results obtained by LAPO method satisfy
all constraints of STHS problem considering valve point
loading effect and the power transmission losses. The optimal
power generation for hydrothermal units is shown in Fig. 10. It
is clearly seen from Fig. 10 that the total power generation
satisfies the power load demand. Figure 11 shows convergence
characteristicsofSTHSproblemforcase 2oftestsystem2.The
power transmission loss coefficients are as follows:
Bij ¼
0:34 0:13 0:09 0:01 0:08 0:01 0:02
0:13 0:14 0:10 0:01 0:05 0:02 0:01
0:09 0:10 0:31 0:00 0:01 0:07 0:05
0:01 0:01 0:00 0:24 0:08 0:04 0:07
0:08 0:05 0:01 0:08 1:92 0:27 0:02
0:01 0:02 0:07 0:04 0:27 0:32 0:00
0:02 0:01 0:05 0:07 0:02 0:00 1:35
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
104
MW1
ð20Þ
Boj ¼ 0:75 0:06 0:7 0:03 0:27 0:77 0:01
½
106
ð21Þ
B00 ¼ 0:55 MW ð22Þ
4.3 Test system 3
Test system 3 consists of four hydro and ten thermal power
generating units. Here valve point loading effect of thermal
plants is considered, but the power transmission loss is not
considered. The data of this system have been taken from
Ref. (Mandal and Chakraborty 2008). The optimal cost
obtained by LAPO method for this system is 165,675.084
$. The hourly water discharge of hydro units and the
optimal power generation scheduling of hydro and thermal
0
200
400
600
800
1000
1200
1400
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Generation
(MW)
Time (hour)
Hydro 4
Hydro 3
Hydro 2
Hydro 1
Thermal 3
Thermal 2
Thermal 1
0 500 1000 1500 2000 2500 3000
3.5
4
4.5
5
5.5
6
6.5
7
x 10
4
Iteration
Total
cost
($)
Fig. 9 Optimal cost of STHS problem for case 1of test system 2
123

Table
16
Optimal
water
discharge
of
hydro
and
thermal
power
generation
for
case
2
of
test
system
2
Hours
(h)
Water
discharge
rates
(10
4
m
3
/s)
Hydro
plant
power
generation
(MW)
Thermal
plant
generation
(MW)
Total
generation
(MW)
PLoss
(MW)
Total
load
(MW)
Plant
1
Plant
2
Plant
3
Plant
4
Plant
1
Plant
2
Plant
3
Plant
4
Plant
1
Plant
2
Plant
3
1
13.129
7.020
24.403
23.688
94.820
56.523
0.823
244.326
102.646
117.783
139.827
756.751
6.712
750
2
10.247
13.583
12.779
24.981
85.422
82.265
38.184
210.528
20.015
209.417
139.544
785.377
5.396
780
3
9.3590
9.296
12.800
13.519
80.786
65.915
38.167
154.264
102.731
124.908
139.488
706.262
6.244
700
4
11.934
8.337
14.221
18.471
88.387
61.235
36.453
183.842
20.0136
124.916
139.567
654.416
4.391
650
5
8.414
14.248
12.611
18.478
72.841
79.228
38.304
236.000
72.4551
124.871
50.019
673.720
3.739
670
6
7.564
13.567
18.444
16.740
67.604
72.387
24.207
275.349
102.753
124.837
139.770
806.910
6.915
800
7
8.581
12.940
11.408
23.787
73.263
68.344
38.672
323.976
102.638
125.021
229.428
961.3420
11.339
950
8
8.522
14.442
12.717
18.234
73.148
71.626
38.229
294.581
20.042
209.770
318.754
1026.154
16.153
1010
9
10.196
11.4781
20.471
22.847
80.966
63.848
14.531
320.345
102.642
124.501
409.021
1115.856
25.856
1090
10
8.278
9.910
13.529
20.674
72.843
57.602
37.438
309.851
20.007
280.251
319.223
1097.219
17.208
1080
11
8.377
14.386
15.519
19.871
74.822
71.5276
35.225
305.2348
102.595
209.752
319.280
1118.439
18.441
1100
12
8.151
12.767
19.589
18.181
74.169
67.880
19.044
294.211
102.664
209.808
409.032
1176.8109
26.787
1150
13
12.171
13.126
19.444
22.634
90.860
68.824
19.742
319.444
102.617
294.680
227.853
1124.023
13.978
1110
14
10.999
10.872
17.402
22.780
87.718
61.609
28.221
320.062
20
209.808
319.024
1046.446
16.442
1030
15
9.65197
7.763
10.0424
21.133
82.660
47.614
38.037
312.312
102.654
124.797
319.292
1027.36
17.358
1010
16
12.992
10.684
19.119
15.337
92.358
60.873
21.256
271.660
20.015
209.699
408.644
1084.507
24.501
1060
17
11.357
11.429
11.548
24.719
87.078
63.678
42.777
327.034
102.576
209.815
229.524
1062.484
12.482
1050
18
12.770
9.060
10.073
24.939
88.100
53.602
38.064
327.677
102.104
209.788
319.273
1138.611
18.578
1120
19
7.4295
12.092
15.0289
18.283
66.160
65.892
34.938
294.923
99.816
124.804
408.929
1095.466
25.464
1070
20
8.690
6.594
20.970
21.302
72.205
40.596
13.377
313.184
102.393
124.920
409.040
1075.719
25.740
1050
21
9.670
8.646
17.632
23.898
75.755
52.873
27.392
310.022
102.673
209.797
139.793
918.307
8.317
910
22
13.039
10.968
18.356
23.640
85.397
61.979
24.570
323.443
20.0589
124.648
229.513
869.611
9.599
860
23
11.360
10.921
18.994
22.380
81.785
61.799
21.824
318.335
20.018
126.337
229.453
859.554
9.530
850
24
7.595
12.187
13.549
12.829
76.585
74.216
58.953
229.887
20.505
209.527
135.675
805.3480
5.336
800
123

units are shown in Table 18 and 19. It can be observed
from Tables 18 and 19 that the power load demand during
24-h scheduling time is satisfied by total power generation
of four hydro units and ten thermal unit. Table 20 shows
the results obtained by different methods for test system 3.
The convergence characteristics of the proposed method
for this system are presented in Fig. 12. The optimal hourly
hydro and thermal power generation for each hour for test
system 3 is shown in Fig. 13.
($)
Average cost
($)
Maximum cost
($)
Computation
time(s)
LAPO 39,691.86 40,150.23 40,563.5 7.5328
RCGA-RTVM (Haghrah et al. 2014) 40,486.6676 NA NA NA
Improved DE (Basu 2014b) 40,627.92 40,708.53 40,860.70 627.06
ACABC (Liao et al. 2013) 41,074.42 NA NA 16
2015)
41,183 41595 41994 192
CPSO (Wu et al. 2019) 41,215.47 41682.92 41843.55 45.5
DRQEA (Wang et al. 2012) 41,435.76 NA NA 18
MCDE (Zhang et al. 2015) 41,586.18 42,022.67 42,365.84 100.05
ACDE (Lu et al. 2010) 41,593.48 NA NA 29
MHDE (Lakshminarasimman and Subramanian 2006) 41,856.50 NA NA 31
QTLBO (Roy et al. 2013) 42,187.49 42,193.46 42,202.75 NA
DE (Wang et al. 2012) 42,801.04 NA NA 21
ALO (Dubey et al. 2016) 42,833.908 NA NA 55.63
RCGA (Fang et al. 2014) 42,886.352 43,261.912 43,032.334 NA
CABC (Liao et al. 2013) 43,362.68 NA NA 21
DE (Mandal and Chakraborty 2009) 43,500.00 NA NA 72.9570
IPSO (Swain et al. 2011) 44,321.236 NA NA NA
DE (Lakshminarasimman and Subramanian 2006) 44,526.10 NA NA NA
EP (Basu 2004a) 45,063.04 NA NA NA
123

4.4 Spillage effect
The spillage effect appears in the third hydropower only
(Kang et al. 2017), an quantity suitable of spillage from the
third hydropower plant will lead to more hydropower
production. The problem formulation of the STHS involves
the spillage effect in Eq. (3), so the spillage rate for the
hydraulic system is taken into account in short-term
hydrothermal. The STHS problem involving the spillage
effect has been solved with the proposed algorithm LAPO.
The spillage effect is taken on two test systems. Table 21
shows the effect of the spillage at different test system.
Moreover, the minimum fuel cost reduces with the pres-
ence of spillage effect compared to the spillage effect not
taking into account in short-term hydrothermal. The min-
imum cost value with considering the spillage effects
illustrates in Table 21. Table 22 illustrates the spillage
0 500 1000 1500 2000 2500 3000
3.5
4
4.5
5
5.5
6
6.5
7
x 10
4
Iteration
Total
cost
($)
Table 18 Optimal water
discharge and power generation
of hydro units for test system 3
Hours (h) Water discharge rates (104
m3
/s) Hydro plant power generation (MW)
1 11.660 12.757 29.986 9.538 91.787 79.899 0.000 169.112
2 9.634 14.997 29.970 7.152 83.435 79.521 0.000 136.154
3 6.237 14.973 29.987 7.661 62.614 74.452 0.000 136.285
4 8.207 14.908 17.102 7.967 75.858 72.369 37.725 131.459
5 5.402 14.973 29.959 15.492 55.715 72.462 0.000 219.372
6 12.716 9.468 14.811 16.966 92.170 55.575 45.658 245.181
7 7.543 14.925 10.112 10.320 70.159 72.393 50.191 204.712
8 12.436 14.931 29.965 11.903 90.047 72.402 0.000 226.682
9 12.394 14.881 18.114 20.282 88.697 72.330 41.897 304.037
10 11.741 9.678 20.047 17.257 86.681 56.551 37.258 281.721
11 10.496 14.974 14.595 21.218 83.329 72.464 56.297 293.952
12 11.588 14.935 17.550 12.999 86.208 72.408 53.622 249.307
13 12.686 14.990 29.998 22.011 87.918 72.486 0.000 312.352
14 12.561 11.202 14.130 21.883 87.359 62.854 60.839 309.742
15 10.622 9.711 17.746 15.658 82.112 56.704 57.874 268.119
16 13.021 13.738 18.970 21.129 86.624 70.256 56.324 300.998
17 14.866 14.983 18.466 20.720 86.639 72.476 58.537 308.891
18 12.503 14.953 29.974 17.282 84.501 72.434 0.000 283.482
19 14.208 10.966 21.739 19.979 86.515 61.972 47.161 299.075
20 8.983 14.975 17.290 19.772 72.618 72.465 62.561 297.025
21 14.713 11.351 12.981 22.524 86.643 63.395 64.299 306.165
22 11.263 12.472 20.979 24.904 81.501 67.044 53.678 320.156
23 14.989 14.986 15.978 24.957 86.622 72.480 64.816 316.434
24 9.920 14.827 19.095 12.504 91.104 80.674 48.342 226.653
123

Table 19 Thermal power generation for test system3
Hours (h) Thermal plant generation (MW)
Ps1 Ps2 Ps3 Ps4 Ps5 Ps6 Ps7 Ps8 Ps9 Ps10
1 139.483 274.084 94.830 119.645 274.517 139.718 104.493 35.022 98.221 127.403
2 229.530 199.397 94.750 69.812 224.884 289.329 163.735 35.056 98.322 75.790
3 319.194 199.580 59.568 119.770 174.263 139.932 104.393 35.032 97.789 176.927
4 319.379 124.750 94.864 20.443 174.549 189.384 222.951 35.015 25.052 126.202
5 139.698 200.485 20.068 119.957 374.240 139.676 104.616 72.370 25.096 126.027
6 319.179 125.239 129.979 70.467 224.782 89.853 104.411 35.003 98.939 126.839
7 229.392 349.080 20.151 69.720 324.150 89.629 223.051 35.016 33.673 176.893
8 409.009 199.506 94.769 69.823 224.599 139.757 222.721 35.018 98.330 126.840
9 229.596 273.976 94.848 119.869 274.368 189.309 163.504 35.017 25.033 177.375
10 319.535 124.856 94.779 119.726 224.701 139.675 223.043 35.082 159.098 176.961
11 409.285 274.076 94.722 69.697 124.656 139.824 222.667 35.047 97.993 125.784
12 139.905 349.450 20.829 20.062 324.149 239.822 222.566 35.041 159.538 176.733
13 319.450 424.328 20.346 119.553 75.169 190.068 222.940 35.001 98.840 129.484
14 229.625 274.406 94.644 20.077 324.286 89.632 104.279 35.141 159.973 176.875
15 319.409 199.872 129.468 119.623 174.320 289.190 104.382 35.001 97.962 76.025
16 319.087 74.606 94.969 69.758 174.496 388.855 163.456 35.268 97.713 126.542
17 229.925 199.798 95.416 119.522 224.785 239.288 104.416 35.137 98.325 176.664
18 229.412 349.129 94.713 119.799 223.997 189.656 162.815 35.004 98.128 176.926
19 139.599 274.159 94.393 69.889 224.227 289.189 163.579 35.394 159.728 125.617
20 319.043 423.640 20.107 69.896 224.187 189.186 163.504 35.040 25.004 75.774
21 139.718 349.069 94.719 69.868 224.138 139.833 163.540 35.138 97.743 75.691
22 139.730 274.382 94.802 119.598 124.951 139.741 163.169 35.109 159.850 85.853
23 229.338 274.253 95.721 69.853 124.645 89.622 104.211 35.574 159.922 126.160
24 229.359 199.509 20.052 69.692 273.385 189.624 163.499 35.096 97.763 75.564
Table 20 Comparison of
simulation results obtained by
different methods for test
system
Algorithm Minimum cost ($) Average cost ($) Maximum cost ($)
LAPO 165,675.084 167,665.23 169,564.12
SPPSO (Zhang et al. 2011) 167,710.56 168,688.92 170,879.30
IDE (Basu 2014b) 170,576.5 170,589.6 170,608.3
DE (Zhang et al. 2011) 170,964.15 NA NA
MDE (Zhang et al. 2011) 177,338.60 179,676.35 182,172.01
SPSO (Zhang et al. 2011) 189,350.63 190,560.31 191,844.28
123

effect on the optimal hydro and thermal power generation
for case 1 of test system 1.
5 Conclusion
In this paper, the optimal solution of the nonlinear non-
convex STHS problem has been solved LAPO as a recent
optimization technique. To examine the effectiveness of
the proposed LAPO algorithm, three different test systems
consisting of multi-chain cascaded of hydro power plants
and different thermal units have been used. The effect of
the valve point loading effect and power system trans-
mission losses has been considered. Moreover, the per-
formance of proposed algorithm has been compared with
various well-known optimization techniques: IHS and
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
x 10
5
Iteration
Total
Cost
($)
Fig. 12 Optimal cost of STHS problem for test system 3
0
500
1000
1500
2000
2500
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Power
Generation
(MW)
Time (Hour)
Hydro4
Hydro3
Hydro2
Hydro1
Thermal 10
Thermal 9
Thermal 8
Thermal 7
Thermal 6
Thermal 5
Thermal 4
Thermal 3
Thermal 2
Thermal 1
units for test system 3
Table 21 Spillage reduces fuel
cost
No allowed spillage allowed spillage
Test system 1 Case 1 871,910.67 867,946.554
Test system 1 Case 2 881,184.5 877,858.408
Test system 2 case 1 38,800.75 38,615.12
Test system 2 case 2 39,691.86 39,512.974
123

RCGA-IMM. However, the numerical results and simula-
tions prove the efficacy and superiority of the proposed
algorithm compared with these techniques. Using the
proposed algorithm, the minimum cost value for test sys-
tem 1 without considering the valve point loading effects is
3483.56 $/day compared to the best technique, while the
total daily saving is 234.87 $ for test system 2 with con-
sidering the valve point loading effect and transmission
power losses. Moreover, the proposed algorithm succeeded
to minimize the fuel cost with the presence of the spillage
effect compared to the spillage effect not taking into
account in short-term hydrothermal.
Compliance with ethical standards
Conflict of interest Authors declare that they have no conflict of
interest.
Ethical approval This article does not contain any studies with human
participants or animals performed by any of the authors.
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