Internatio nal Journal on Electrical Engineering and Informatics - Volume 15, Number 1, March 2023



Security-Constrained Temperature-Dependent Optimal Power Flow Using
Hybrid Pseudo-Gradient based Particle Swarm Optimization and
Differential Evolution

Minh-Trung Dao
1.2
, Khoa Hoang Truong
3.4,*
, Duy-Phuong N. Do
2
, Bao-Huy Truong
5
, Khai
Phuc Nguyen
1.4
, and Dieu Ngoc Vo
1.4

1
Department of Power Systems, Ho Chi Minh City University of Technology (HCMUT), 268
Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam
2
Department of Electrical Engineering, College of Engineering Technology, Can Tho
University, Can Tho City, Vietnam
3
Department of Power Delivery, Ho Chi Minh City University of Technology (HCMUT), 268
Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam
4
Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc City , Ho Chi
Minh City, Vietnam
5
Institute of Engineering and Technology, Thu Dau Mot University, Binh Duong Province, Viet Nam
*Corresponding author: Email: [email protected]
Abstract: Conventional optimal power flow studies neglect the effect of temperature on
resistance for simple calculation. However, the branch resistance changes with the change of
temperature. Thus, the optimal power flow (OPF) should consider the temperature effect for
accurate calculation. Moreover, contingency cases should be considered to ensure system
security. Accordingly, the security- constrained temperature- dependent optimal power flow (SC -
TDOPF) emerges as a critical and practical issue in power systems. To deal with the SC-TDOPF
problem, this study suggests a hybrid method, namely pseudo- gradient based particle swarm
optimization and differential evolution method (PGPSO-DE). The suggested PGPSO-DE
method is applied to the standard IEEE 30 bus system under normal condition as well as
contingency condition. The findings have shown that the PGPSO-DE method provides better
solution quality than other studied optimization methods. Consequently, the PGPSO-DE method
proves its effectiveness in solving the complex SC-TDOPF problem.

Keywords: Power flow analysis; security-constrained; temperature- dependent power flow
(TDPF); differential evolution; particle swarm optimization; optimal power flow; pseudo
gradient; hybrid method.

1. Introduction
Power flow is always performed to calculate the fitness function when solving the important
optimization problems in power systems such as economic dispatch, unit commitment, optimal
power flow, optimal reactive power dispatch, and hydrothermal scheduling. Power flow analysis
is also used to perform the contingency analysis and transient stability study. Therefore, the
accuracy of power flow study is a very essential concern. In conventional power flow calculation,
the temperature effect is ignored and the resistance of the elements of power systems is treated
as a constant. However, the resistance is a function of temperature. As the temperature rises, the
resistance of a metallic conductor rises as well. Therefore, there is always an error related to
temperature in conventional power flow and branch loss calculation. To obtain more accurate
results, power flow calculation should consider the temperature effect. This study investigates
the optimal power flow (OPF) problem considering the temperature effect. A fully coupled
temperature- dependent power flow (TDPF) algorithm in [1] is used for power flow calculation
in this study.
The OPF problem is one of the most important tools in power system operation and planning.
Its solution offers the optimal settings for generators, transformers, and shunt capacitors which
Received: October 8
th
, 2022. Accepted: March 23
rd
, 2023
DOI: 10.15676/ijeei.2023.15.1.6
82

minimize the considered objective function (e.g., total fuel cost) while satisfying various system
operating limits [2] . The OPF problem has a long history of development since it was first
introduced in 1962 [3] . This problem has been initially solved by traditional methods based on
mathematical programming such as Newton-based techniques [4], linear programming [5] , non-
linear programming [6] , quadratic programming [7] , and interior point methods [8] . In general,
these methods have short computational time and effectively solve the simple OPF problem with
convex and continuous objective functions. However, the practical OPF is a large- scale, non-
linear and non- convex optimization problem. This is a challenge for solution methods, especially
for the traditional methods. They may suffer difficulty in finding a global solution or cannot
successfully solve the complex OPF problem. To overcome these barriers, advanced
optimization methods (i.e., meta-heuristic methods) have been developed based on biological
simulation to cope with the complex OPF problem. The meta-heuristic methods have the
advantage of obtaining near-optimum solutions for any type of optimization problem. Therefore,
these methods have been widely implemented to various engineering fields. For the OPF
problem, the meta-heuristic methods have also been successfully solved this problem such as
genetic algorithm (GA) [9], particle swarm optimization (PSO) [10], differential evolution (DE)
[11], artificial bee colony (ABC) algorithm [12] , biogeography- based optimization (BBO) [13] ,
grey wolf optimizer (GWO) [14] , moth swarm algorithm (MSA) [15] , krill herd algorithm
(KHA) [16], and moth- flame optimization (MFO) [17], etc. These methods have shown their
effectiveness in dealing with this problem. However, they can consume long computational time
and still suffer sub- optimal solutions when facing complicated and large-scale problems. Besides
single methods, hybrid methods based on the combination of different algorithms have been also
developed to efficiently solve the complex OPF problem. Some hybrid methods are mentioned
such as a hybrid of shuffle frog leaping algorithm (SFLA) and simulated annealing (SA)
algorithm (SFLA-SA) [18], a hybrid method of modified imperialist competitive algorithm
(MICA) and teaching learning algorithm (TLA) (MICA-TLA) [19], and a hybrid of particle
swarm optimization (PSO) and gravitational search algorithm (GSA) (PSOGSA) [20]. In
general, hybrid methods have the advantage of achieving high- quality solutions for complicated
optimization problems. However, their disadvantage lies in the setting of many control
parameters from the combined methods. Improper setting of control parameters can lead the
algorithm not converge to the global solution.
The primary OPF problem was formulated under normal operating conditions of the power
system. However, contingency cases (e.g., outage of a transmission line) may occur. In such a
situation, the solution from the OPF problem can violate the system operating constraints.
Therefore, system security should be considered for the OPF problem to ensure the reliability
and economic operation of power systems. In this context, the conventional OPF problem
becomes a security-constraint OPF (SC-OPF) problem. By adding the security constraints, the
solution of the SC-OPF problem is not only feasible for the normal case (N - 0) but also for the
contingency case (N - 1). However, the problem becomes very complex and a great challenge
for solution methods. Recently, many solution methods have been suggested for dealing with the
SC-OPF problem. The study [21] proposed a modified bacteria foraging optimization algorithm
(MBFA) to optimally operate the wind-thermal generation system with minimum cost reduction
of the system loss while satisfying a voltage secure operation. Although a detailed cost model
was introduced, the valve loading effects was not included in the cost model. In [22] , a
contingency partitioning approach was proposed for the preventive-corrective SC-OPF problem.
The authors used a DC network model for calculation and the valve point loading effects was
not considered in this study. A fuzzy based harmony search algorithm (FHSA) was suggested in
[23] to determine the best solution for the SC-OPF problem. The objective function in this study
was a quadratic fuel cost function, without considering the valve point loading effects of thermal
generating units. In [24] , an adaptive flower pollination algorithm (APFPA) was successfully
solved the SCOPF problem with the different objective functions of minimizing the fuel cost,
power losses and voltage deviation. A cross-entropy (CE) method in [25] was introduced to
assess the SCOPF solutions. The corresponding SC-OPF stochastic problem was first defined, Minh-Trung Dao, et al. 83

and the CE was then applied to solve the formulated problem. The results obtained from the CE
method for the IEEE 57 bus and IEEE 3000 bus systems showed that this method offered betters
solutions with fewer evaluations than other compared algorithms. Moreover, the SC-OPF
problem has been also solved by a hybrid method such as a hybrid canonical differential
evolutionary particle swarm optimization (hC-DEEPSO) [26]. This hybrid method was tested on
the IEEE standard systems including 57, 118, and 300 buses, showing its effectiveness from the
comparison of obtained results with other evolutionary methods. In general, the SC-OPF
problem is a highly nonlinear and non- convex optimization problem, posing a great challenge
for finding a globally optimal solution.
In literature, regarding the OPF problem with temperature effect, there is a few studies have
investigated this problem. In [27], the authors proposed a gbest-guided artificial bee colony
(GABC) algorithm to solve the OPF problem as well as the temperature-dependent OPF
(TDOPF) problem. Also, the TDOPF problem was solved by the chaotic whale optimization
algorithm (CWOA) [28]. Both GABC and CWOA were tested on the IEEE 30 bus system, the
2383 bus winter peak Polish system, and the 2736 bus summer peak Polish system. However,
these two studies did not consider the valve point loading effects in the objective function as well
as the security constraint in the problem formulation. In this study, the OPF problem is
investigated with the temperature effect. In addition, the valve point loading effects and the
security constraint are also considered for the OPF problem. The new OPF problem is called the
security constraint temperature-dependent OPF (SC-TDOPF) problem. To solve the SC-TDOPF
problem, this study proposes a hybrid pseudo- gradient particle swarm optimization and
differential evolution method (PGPSO-DE) [29]. The proposed PGPSO-DE method utilizes the
search ability of PGPSO and DE to find the near-optimum solution. The PGPSO is used to
explore the global search while the DE method is used to exploit the local search. As a result,
the PGPSO-DE method has a key advantage of balance between exploration and exploitation.
PGPSO-DE can deal with optimization problems having many control variables and complicated
constraints such as the SC-TDOPF problem. The proposed method has been tested on the IEEE
30-bus system and their obtained results have been compared with other optimization methods.

2. Mathematical problem formulation of TDPF
The resistance of a metallic conductor increases as temperature increases. Thus, the resistance
is proportional to the temperature and expressed as follows [1]:

F
Ref
Ref F
TT
RR
TT
+
= ×
+
(1)
where R is the conductor resistance; R
Ref is the conductor resistance at the reference temperature;
T is the conductor temperature; T
Ref is the reference temperature; and T F is the temperature
constant depending on the conductor metal.
A. Thermal model of elements in power systems
The thermal characteristic of the devices in power systems is modelled by a generalized
thermal resistance model as shown in Figure 1.


Figure 1. Thermal resistance model for branch element in power system.
Security-Constrained Temperature-Dependent Optimal Power Flow Using 84

In a thermal resistance model, the device temperature rise of the device is linearly
proportional to the loss of the device. The ratio between the steady- state temperature rise and the
loss of that device is the thermal resistance.

Rise RatedRise
Loss RatedLoss
TT
R
PP
θ
= =
(2)
where, R
θ
is the thermal resistance; T Rise is the device temperature rise above ambient; P Loss is
the power loss within the device; T
RatedRise is the rated (or reference) device temperature rise; and
P
RatedLoss is the corresponding rated (or reference) loss.
The temperature of the device (T) equals the ambient temperature (T
Amp) plus the device
temperature rise above ambient (T
Rise). Rearranging Eq. (2), the temperature of the device is
expressed as follows:

Loss
Amp RatedRise
RatedLoss
P
TT T
P

= +

(3)
If P
Loss is suitably expressed as a function of voltage and temperature state, Eq. (3) can be directly
used in power flow calculation.
Besides the thermal model of branch element, the other thermal models of power system
elements such as overhead lines, underground cables, and transformers are given in [1] .

B. Equations of TDPF problem
In the TDPF problem, it is assumed that the system must operate in both thermal and steady-
state cases. In addition, there are three main modifications to the conventional power flow as
follows [1]:

B.1. State Vector:
Besides the conventional state variables like V and δ, an additional state variable T for the
temperature is considered for each temperature- dependent branch. Accordingly, the state vector
has the form:

V
x
T
δ


=



(4)
in which, all state variables are expressed in per-unit.

B.2. Mismatch Equations:
Conventional power flow employs two mismatch equations including real and reactive power
mismatch equations. The TDPF problem requires two those convention mismatch equations and
an additional mismatch equation of temperature difference. Three mismatch equations of the
TDPF are described as follows:

,,
( ) (, , )
i Gen i Load i i
P P P P VT δ∆= − −
(5)

,,
( ) (, , )
i Gen i Load i i
Q Q Q Q VT δ∆= − −
(6)

0 (, , )
ij ij
H H VTδ∆=−
(7)
B.3. Jacobian Matrix
The Jacobian matrix must be restructured due to the addition of the state variable T . Partial
derivatives of real power, reactive power and temperature difference equation are taken with respect to each variable in the state vector (δ, V, and T). The restructured Jacobian matrix is
expressed as follows: Minh-Trung Dao, et al. 85

(, , )
PPP
VT
QQQ
J VT
VT
HHH
VT
δ
δ
δ
δ
∂∂∂

∂∂∂

∂∂∂
=
∂∂∂

∂∂∂

∂∂∂
(8)

B.4. Overall procedure of Fully Coupled TDPF
There are four types of TDPF as described in [1] . This paper employs fully coupled TDPF
(FC-TDPF) for calculation. If the Jacobian matrix in Eq. (8) is used, the overall procedure of
FC-TDPF is described as follows:

Step 1: Initialize all state variables ( ie., δ, V, and T ).
Step 2: Update all branch resistances according to the most recent temperature. Calculate the
admittance matrix Y
bus.
Step 3: Calculate the Jacobian matrix as in Eq. (8).
Step 4: Calculate the mismatch (ie., ΔP, ΔQ, and ΔH ) using Eqs. (5-7)
Step 5: Update δ , V, and T as follows:

1
11
1
( , , ).
vv v
v v v vv v
vv v
P
V V J VT Q
TT Hδδ
δ
+
+−
+
   ∆
   
=−∆
   
   
∆
   
(9)
Step 6: If ΔP , ΔQ, ΔH < ε (ε is the specified tolerance), stop the loop. Otherwise, go to Step 2.
3. Problem formulation
The SC-TDOPF is a very complicated optimization problem. This problem has many control
variables and complex constraints that need to be handled. The goal of this problem is to
determine an optimal set of control variables so as the total cost of thermal generating units is
minimized while satisfying various constraints of the system in both normal and contingency
cases. In general, the SC-TDOPF problem is mathematically formulated as follows:

Minimize ( , )FU X (10)
subject to the following constraints for the normal case:
(, ) 0hU X= (11)
(, ) 0gU X≤ (12)
and the following constraints for the contingency cases:
( , )0
SS
hU X =
(13)
( , )0
SS
gU X ≤
(14)
where F(.) represents the fuel cost function of generators, U and X are the set of state and control
variables, respectively; h(.) and g(.) are the set of the equality and inequality constraints,
respectively; and S is the set of outage lines.

A. Objective function
The objective function is to minimize the total fuel cost of all thermal generating units.

1
Min = Min ( )
g
N
i gi
i
F FP
=
∑
(15)
where N
g is the total number of thermal units; and P gi is the power output of thermal unit i
In Eq. (15), a quadratic function is used to expressed the fuel cost function of a thermal generating unit i as follows:

2
()
i gi i i gi i gi
F P a bP cP=++ (16) Security-Constrained Temperature-Dependent Optimal Power Flow Using 86

The solution of the SC-TDOPF is more accurate and practical when considering the valve
point effects (VPEs) of thermal generating units. As a result, the fuel cost function is represented
as a quadratic function adding a sinusoidal function:

2
, min
( ) sin( ( ))
i gi i i gi i gi i i gi gi
F P a bP cP e f P P=+++ − (17)
where a
i, bi, ci, ei and f i are fuel cost coefficients N g is the total number of generators; and
P
gi,min is the minimum power output of generator i .
B. System Constraints
For both normal and contingency cases, the objective function of the SC-TDOPF problem is
subject to the following constraints:

i. Equality constraints:

( )
1
( ) cos( ) ( ) sin( )
b
N
gi di i j ij i j ij i j
j
P P V V GT BT δδ δδ
=
−= × − + × −∑
i = 1, 2, …, Nb (18)

( )
1
( ) sin( ) ( ) cos( )
b
N
gi di i j ij i j ij i j
j
Q Q V V GT BT δδ δδ
=
−= × −− × −∑
i = 1, 2, …, Nb (19)

( )( )
22
,
() ( ) 2 () cos( ) 0
ij Amp ij ij i j ij i j i j
T T R gT V V gT VV
θ
δδ− + ×+− × − =
i = 1, 2, …, Nb (20)
where Q
gi is the reactive power outputs of generator i ; V i and δi are the magnitude and angle of
voltage at bus i , respectively; V
j and δj are the magnitude and angle of voltage at bus j ,
respectively; P
di and Q di are the active and reactive power demands at load bus i , respectively;
G
ij and B ij are the real and imaginary components of elements in the admittance matrix; g ij is the
conductance of line ij; and N
b is the number of buses in the system.

ii. Power generation output limits:
i = 1, 2, …, N
g (21)
i = 1, 2, …, N
g (22)
where P
gi,min and P gi,max denote the limits of active power outputs while Q gi,min and Q gi,max denote
the limits of reactive power outputs of generator i .

iii. Bus voltage limits:
i = 1, 2, …, N
g (23)
i = 1, 2, …, N
d (24)
where V
gi and V li are the voltage magnitude at generation bus i and load bus i , respectively; V gi,max
and V
gi,min represent the limits of voltage magnitudes at generation bus i ; V li,max and V li,min present
the limits of voltages at load bus i ; and N
d is the number of load buses.

iv. Shunt VAR compensator limits:
i = 1, 2, …, N
c (25)
where Q
ci is the shunt VAR compensation at bus i ; Q ci,max and Q ci,min denote the capacity limits
of shunt VAR compensator; and N
c is the number shunt VAR compensator.
Transformer tap settings limits:
k = 1, 2, …, N
t (26)
where; T
k,min and T k,max present the limits of tap settings of transformer k ; T k is the value of tap
setting of transformer k ; and N
t is the number of transformer tap settings.



max,min, gigigiPPP≤≤
max,min, gigigiQQQ≤≤
max,min, gigigi
VVV≤≤
max,min, lilili
VVV≤≤
, min , maxci ci ci
Q QQ≤≤
max,min, kkkTTT≤≤Minh-Trung Dao, et al. 87

v. Transmission line limits
l = 1, 2, …, N
l (27)
where S
l and S l,max are the apparent power flow and the rating of transmisson line l , respectively;
and N
l is the number of transmission lines.
For the security constraint, the values of the severity index (SI) are calculated to rank the severe
cases of line outage as follows:
l = 1, 2, …, N
l (28)

4. The hybrid PGPSO and DE method
A. Pseudo-Gradient Particle Swarm Optimization Method
PSO is a well-known meta-heuristic optimization method. This method simulates the
behaviors of birds or fishes in finding their food [30] . The PSO method is widely implemented
to various optimization problems due to its simple structure and it is applicable to large-scale
problems. PSO initializes randomly a population (swarm) containing individuals (particles).
Each particle has a position vector X
i and a velocity vector V i, indicating that each particle has a
specific velocity for moving from its position to another.

Mathematically, the position of each particle d is expressed as follows:

( 1) () () ()
13 2 4
( ) ()
nn n n
id id d id id
V V c rand Pbest X c rand Gbest Xω
+
=× +× × − +× × −
(29)
and the corresponding velocity of this particle is updated by:

( 1) () ()n nn
id id id
X XV
+
= +
(30)
where c
1 is the coefficient of the individual cognitive component and c 2 is the coefficient of the
social cognitive component;
ω is the inertia weight parameter; Pbest d is the best position of
particle d at iteration n , and Gbest is the best position in the population.
To enhance the search ability of PSO, Clerc and Kennedy introduced a constriction factor.
[31]. The velocity for particles is modified as follows:

( 1) () () ()
13 2 4
( ( ) ( ))
nn n n
id id d id id
V V c rand Pbest X c rand Gbest Xχω
+
= × +× × − +× × −
(31)
where
χ is the constriction factor and is determined as follows:

12
2
2
; ,4
24
cc
χ ϕϕ
ϕϕ ϕ= =+>
−− −
(32)
Besides the modification of the particle's velocity, the particle’s position is updated by using the concept of pseudo-gradient [32] . In non- differentiable problems, the pseudo- gradient is used to
determine whether the current particle's direction in the search space is good or not. Suppose that a particle moves from a point x
k to another x l, the pseudo-gradient g p(x) is determined by the
following rules [33]:
i. If f(x
k) ≥ f(x l): The particle's direction is good, and the particle should keep moving in that
direction. As a result, at point l , the pseudo- gradient is nonzero, (i.e., g
p(xl) ≠0).
ii. If f(x
k) < f(x l): The particle's direction is not good, and the particle should move in a different
direction. As a result, at point l , the pseudo- gradient is zero, (i.e., g
p(xl) = 0).
The rules above are used to update the new position of each particle as follows:

( ) ( 1) ( 1) ( ) 1
( 1)
( 1) ( ) 1
( ) | | if 0
otherwise
n nn n
n id p id id p id
id nn
id id
X g X V g (X )
X
XV
++ +
+
−+
+× ≠
=
+

(33)
This study uses the pseudo- gradient guided PSO (PGPSO) method for the suggested hybrid
approach.



max,llSS≤
2
1 max,
∑
=








=
l
N
l l
l
S
S
SISecurity-Constrained Temperature-Dependent Optimal Power Flow Using 88

B. Differential Evolution Method
Differential evolution (DE) is also a well- known meta-heuristic optimization method used
for solving complex optimization problems. DE also initializes with a random population. A new
population is created via three main stages as follows [34]:
• Mutation stage: In this stage, a new individual is generated based on other random individuals
as in Eq (34). Thus, the search space of the problem is effectively explored.
'() () () ()
1 23
()
nn n n
id r d r d r d
X X FX X= +× −
(34)
where,
)('n
id
X
is the newly created individuals; r1 , r2, and r3 are random indexes of the population;
and F is the mutation factor selected in [0,1].
• Crossover stage: To increase the diversity, this stage mixes the mutant vector
)('n
id
X
and the
current solution
()n
id
X
to create a trial individual
''( )n
id
X
:


 =≤
=
otherw
iseX
D dCR randX
X
n
id
rand
n
idn
id )(
5
)('
)(''
or if
(35)
where CR is the crossover rate in [0,1]; and D
rand is a random index of the population.
• Selection stage: The fitness values are computed for the current and trial individuals. The
individual with a better fitness value is chosen for the next generation. In this way, a new generation has better individuals than the previous one.
C. The Hybrid PGPSO and DE Method
In general, PGPSO and DE have their advantages and disadvantages when dealing with
different optimization problems. PGPSO is capable of finding the near-optimal solution in a short
time for a considered problem, however, it is not guaranteed to provide high solution quality for
complex problems. On the contrary, DE can easily find a high- quality solution for small- scale
problems but may not be able to solve large-scale ones. In other words, it can be said that PGPSO
has the advantage of good exploration while DE has the advantage of good exploitation.
Therefore, this study proposes a combination of PGPSO and DE to form a hybrid method. The
hybrid PGPSO-DE method utilizes the advantages of PGPSO and DE to become an effective
method to solve complicated optimization problems. The following are the main steps of the
proposed methods:
• Initialization: Like other meta-heuristic methods, PGPSO-DE randomly initializes a
population of Np individuals in their boundaries.
• Creating the first new generation: Based on the initialized generation, the mechanism of
PGPSO is used to create the first new generation. The fitness values are calculated for newly
generated individuals. The individual with the best fitness value is selected for the next
generation.
• Creating the second new generation: In this step, the DE method is used to create the second
new generation. The new individuals are also evaluated via the fitness values to choose the
best one for the next iteration.

5. Implementation of PGPSO-DE to the SC-TDOPF problem
This section describes the implementation of PGPSO-DE to the SC-TDOPF problem for the
objective of minimizing the total fuel cost.

A. Initialization of Population
In the SC-TDOPF problem, there are two types of variables including control and state
variables. In the population of PGPSO-DE, each individual includes the control variables
described by a vector below:

2 3 1 2 1 2 12
[ , ,..., , , ,..., , , ,..., , , ,..., ]
g g ct
id g g gN g g gN c c N N
X P P P V V V Q Q Q TT T= (36) Minh-Trung Dao, et al. 89

where P
g1 is the real power output of the generator at the slack bus; i = 1, 2, …, N with N = 2N g
+ N
c + Nd -1 and d = 1, 2, …, N p with N p is the number of individual in the population.
Also, the state variables are represented as in Eq. (37).

1 1 2 12 1 2
[ , , ,..., , , ,..., , , ,..., ]
gd l
g g g gN l l lN l l lN
U PQ Q Q VV V S S S= (37)
The position and velocity of each individual are initialized as in Eq. (38) and Eq. (39),
respectively.

( 0) min max min
1
()
id id id id
X X rand X X=+×−
(38)

( 0) min max min
2
()
id id id id
V V rand V V=+ ×−
(39)
where
max
id
X
and
min
id
X
are the maximum and minimum values for individual d, respectively;
max
id
V
and
min
id
V
denote velocity limits for individual d , respectively, which are calculated by:

max max min
()
id id id
V RX X=×− (40)

maxmin
idid
VV−=
(41)
where R is the scale factor for the velocity.

B. Fitness function
Initial population and next created populations are used to solve the power flow problem to
evaluate the quality of individuals. The fitness value is calculated for each individual via a fitness
function in the normal case as follows:

( 0 ) lim 2 lim 2
0 11 0
11
lim 2 2
0 0 ,max
11
() ( ) ( )
() ( )
gg
dl
NN
d i gi p g g q gi gi
ii
NN
v li li s l l
il
FT F P K P P K Q Q
K VV K SS
= =
= =
= +×− +× −
+× − + × −∑∑
∑∑
(42)
where K
p0, Kq0, Kv0, and K s0 are the penalty factors for the normal case.

For the contingency case, the fitness function is calculated by:

( 0 ) ( 0 ) lim 2 lim 2 2
_ ,max
1 11
( ) () ( )
g dl
N NN
s ss
d outage d q gi gi v li li s l l
i il
FT FT K Q Q K V V K S S
= = =
= +× − +× − +× −∑∑∑
(43)
where, K
q, Kv, and K s are the penalty factors for the outage case,
s
gi
Q
is the reactive power output
of generator i in the contingency case;
s
li
V
is the voltage at load bus i in the contingency case;
s
l
S
is the apparent power flow in transmission line l in the contingency case.
In Eq. (42- 43), the limitations of state variables are handled by:






<
>
=
other
wise
if
if
minmin
maxmax
lim
X
XXX
XXX
X
(44)
Where X
lim
denotes
lim
1g
P
,
lim
gi
Q
,
lim
li
V
, and
lim
li
S
while X denotes P g1, Qgi, Vli, and S li.
C. Overall procedure
The overall procedure for the implementation of the proposed PGPSO-DE method to solve
the SC-TDOPF problem are described by the following steps:
Step 1: Set PGPSO-DE parameters: N
p, Itermax, c1 and c 2, R, F, and CR .
Perform the contingency analysis to obtain the SI values corresponding to each branch.
Select the outage lines corresponding to the highest SI values.
Step 2: Initialize a random population as described in Section 5.1.
Step 3: Solve the power flow and calculate the fitness value for each individual in the initial
population using Eq. (42). The position of the individual having the best f itness value is Security-Constrained Temperature-Dependent Optimal Power Flow Using 90

set to Gbest. The initial population and the corresponding fitness function are set to
Pbest
d and FT d
best, respectively. Set k = 1, k is the iteration counter.
Step 4: In this step, the mechanism of the PGPSO method is used to create the first new
population. The new velocity of individuals is first computed by using Eq. (31). Then the
new position of individuals is updated by using Eq. (33). A repair action as in Eq (44) is
applied if the new created velocity and position of individuals violate their limits.
Step 5: Run the power flow using the newly generated population and calculate the fitness
function (42) for the normal case, and fitness function (43) for the outage case.
Step 6: The second new population is created in this step, based on the first population created
by the PGPSO mechanism, by using the mutation stage of DE as in Eq. (34). A repair
action as in Eq (44) is applied if the new position violates its limits.
Step 7: The crossover stage of DE is applied to create new individuals

from the second new
created population using Eq. (35).
Step 8: Run the power flow using the new individuals generated from the crossover stage and
calculate the fitness function (42) for the normal case, and fitness function (43) for the
outage case.
Step 9: The best individuals are selected via the selection stage of DE for the next generation.
The fitness values of individuals from the first generated populations are compared to
those from the second generated populations. The new individual is selected by:



 ≤
=
otherwiseX
FT FTX
X
k
id
k
d
k
d
k
idknew
id )(
)()('')(''
)(
if
(45)
Update the new individual
()new k
d
X
and the corresponding fitness value
()new d
d
FT

accordingly. Step 10: The best population is updated in this step. The update of the best position of each
individual is described as follows:



 ≤
=
otherw
isePbest
FT FTX
Pbest
d
best
d
knew
d
knew
id
d
)()(
if
(46)
Update the corresponding better fitness function
best
d
FT
. Update Pbest d to Gbest.
Step 11: If the current iteration k is lower than the maximum iteration Iter
max, increase k and
return step 4. Otherwise, stop.

6. Numerial results
The suggested PGPSO-DE has been tested on the IEEE 30- bus system for both the normal
case and selected outage cases. The fuel cost function is considered with a quadratic function and
a function with VPEs. The test system has six thermal generating units, four tap changing
transformers, and 41 transmission lines. Besides, there are nine shunt power compensators at
buses 10, 12, 15, 17, 20, 21, 23, 24 and 29. The data of the test system and the transmission line
limits are from [6]. Table A.1 gives the generator data with the quadratic fuel cost function while
Table A.2 gives the fuel cost coefficients for valve point loading effects. Table A.3 represents the
limits for the bus voltage and transformer tap settings. Shunt power compensators have a lower
limit of 0 MVar and an upper limit of 5 MVar. The power flow problem in this study is solved by
the TDPF toolbox [35].

Table 1. Control parameters of PGPSO-DE for the TDOPF with normal and contingency cases
Parameters Itermax Np c1 c2 R F CR
Case 1 150 10 2.05 2.05 0.15 0.7 0.5
Case 2 200 50 2.05 2.05 0.15 0.7 0.5
Case 3 250 10 2.05 2.05 0.15 0.7 0.5
Case 4 300 20 2.05 2.05 0.15 0.7 0.5
Minh-Trung Dao, et al. 91

For implementing the proposed PGPSO-DE method to the TDOPF problem, its control
parameters are selected for different cases of normal and contingency as shown in Table 1 as
follows:
Case 1: Normal case with the quadratic objective function.
Case 2: Normal case with the objective function taking into account VPEs.
Case 3: Selected outage case with the quadratic objective function.
Case 4: Selected outage case with the objective function taking into account VPEs.
The code of the suggested PGPSO-DE method was written in Matlab software. To find the best
solution, each case was run in 50 separate trials.

A. Normal case
The proposed PGPSO-DE approach is implemented to deal with the normal TDOPF problem.
The objective function is taken into account the quadratic function and VPEs.

A.1. Quadratic objective function
To verify the effectiveness of the proposed PGPSO-DE method, it is firstly tested on the IEEE
30-bus system to solve the conventional OPF problem with the quadratic objective function. For
this case, the number of individuals N
p of PGPSO-DE is set to 50 for fair appraisal since most
compared algorithms also set this parameter to 50. The fuel costs obtained from PGPSO-DE
including minimum, average and maximum values are compared to those from other optimization
algorithms as shown in Table 2. It can be seen that the minimum fuel cost of PGPSO-DE is better
than other compared methods and close to GWO [14] and IKHA [16]. In addition, the average
fuel cost of PGPSO-DE is the best value among compared methods and the standard deviation is
rather small. Hence, the proposed method provides a very high and robustness solution quality in
this case. It is further confirmed as seen from Figure 2 which shows the algorithm’s robustness
for 50 independent runs. Figure 3 shows the convergence characteristic of PGPSO-DE for the
IEEE 30 bus test system with the conventional OPF problem. The minimum fuel cost achieved
by the PGPSO-DE method is 800.4141 ($/h).

Table 2. Result comparison for the IEEE 30 bus test system with the conventional OPF problem
Algorithms
Minimum
Fuel cost
($/h)
Average
Fuel cost
($/h)
Maximum
Fuel cost
($/h)
Standard
deviation
Parameters
Setting
Np Itermax
ABC [12] 800.6600 800.8715 801.8674 - 50 200
ARCBBO [13] 800.5159 800.6412 800.9262 - 50 200
GWO [14] 801.413 801.655 801.958 0.1129 - 300
MSA [15] 800.5099 - - - 50 100
IKHA [16] 800.4143 - - - 30 500
IMFO [17] 800.3848
*
- - - 50 500
GABC [27] 800.4401 800.6390 800.7959 1.58142 50 100
CWOA [28] 800.1998
*
- - - 50 100
PGPSO-DE 800.4141 800.5708 802.8181 0.3978 50 150
*
Violated solution
To investigate the effect of temperature, the TDOPF problem is performed with each value
of temperature rise (T
RatedRise). The base temperature (T Base) is selected as 100
o
C. The ambient
temperature (T
Amp) and the reference temperature (T Ref) are selected as 25
o
C. The temperature
constant (T
F) is 228.1
o
C because all conductors are considered hard-drawn aluminium [1] . For
a fair comparison with GABC [27] , the upper voltage limit of load bus, in this case, is set to 1.06
p.u. as the same in [27] . Table 3 presents the minimum fuel cost and the power loss obtained by
the proposed PGPSO-DE method corresponding to each value of T
RatedRise for the IEEE 30 bus
test system. It can be observed that PGPSO-DE offers a better solution than GABC [27] and Security-Constrained Temperature-Dependent Optimal Power Flow Using 92

CWOA [28]. For 0
o
C temperature rise, the fuel cost and the real power loss obtained by PGPSO-
DE are 799.8667 ($/h) and 8.8670 MW, respectively. These values become 802.9909 ($/h) and
9.4663 MW, respectively, for 100
o
C temperature rise. For the 30
o
C temperature rise, the
increase in fuel cost and power loss are 0.14% and 2.61%, respectively. Figure 4 depicts the
effect of temperature rise on fuel cost and power loss. When the temperature rise ( T
RatedRise)
increases, the fuel cost and power loss also increase. In general, the fuel cost increase by 0.04%
approximately for every 10
o
C temperature rise. Figure 5 depicts the convergence characteristic
of PGPSO-DE for the IEEE 30 bus system with the quadratic objective function corresponding
to T
RatedRise = 30
o
C. It can be observed that the objective function converges smoothly to the near
optimal solution.


Figure 2. Fuel cost of fifty independent runs for the conventional OPF problem
for the IEEE 30 bus system


Table 3. Fuel cost and power loss obtained for the TDOPF problem
with the quadratic objective function
Fuel cost ($/h) Power loss (MW)
TRatedRise GABC [27] CWOA [28] PGPSO-DE GABC [27] CWOA [28] PGPSO-DE
0 800.0627 800.0227 799.8667 8.912071 8.7731 8.8670
10 800.4531 800.4010 800.2735 9.029315 8.9258 8.9716
20 800.8292 800.7836 800.6067 9.142254 9.0924 9.0335
30 801.1922 801.1336 801.012 9.251205 9.1346 9.1047
40 801.5429 801.4729 801.3203 9.356453 9.2749 9.1788
50 801.8822 801.8016 801.6144 9.458257 9.3417 9.1934
60 802.2109 802.1204 801.941 9.556852 9.4663 9.2650
70 802.5297 802.4294 802.1624 9.652453 9.5133 9.3602
80 802.8392 802.7296 802.4328 9.745256 9.6674 9.3678
90 803.14 803.0297 802.7393 9.83544 9.7513 9.4046
100 803.4327 803.3224 802.9909 9.923173 9.8017 9.4663 Minh-Trung Dao, et al. 93

Figure 3. Convergence characteristic of PGPSO-DE for the IEEE 30 bus system
with the conventional OPF problem.


Figure 4. Temperature rise effect on fuel cost and power loss for the TDOPPF problem with
the quadratic objective function.


Figure 5. Convergence characteristic of PGPSO-DE for the TDOPF problem with the quadratic
objective function corresponding to T
RatedRise = 30
o
C.

A.2. Objective function considering VPEs.
The valve point loading effects is taken into account in the objective function of the TDOPF
problem in this case. Table 4 shows the obtained results of fuel cost and power loss for the rise in
temperature for the TDOPF problem with VPEs. For 0
o
C temperature rise, the fuel cost obtained
by the PGPSO-DE method is 919.825 ($/h) and the real power loss is 9.8902 MW. For 30
o
C
temperature rise, the increase in fuel cost is 0.22% and 4.12 % for power loss. The obtained fuel
cost becomes 925.5945 ($/h) and the real power loss is 11.3242 MW when the temperature rise Security-Constrained Temperature-Dependent Optimal Power Flow Using 94

T
RatedRise is 100
o
C. Figure 6 illustrates the effect of temperature rise on fuel cost and power loss
for the IEEE 30 bus system with valve point loading effects. The general trend is for increasing
fuel cost and power loss as the temperature rises. Figure 7 shows the convergence characteristic
of PGPSO-DE for the IEEE 30 bus system with the valve point loading effects for T
RatedRise = 30
o
C. It can be observed that the objective function converges to the near-optimal solution after 150
iterations.

Table 4. Fuel cost and power loss obtained for the TDOPF problem with VPEs
TRatedRise Fuel cost ($/h) Power loss (MW)
0 919.8525 9.8902
10 920.7833 10.0651
20 920.8674 10.1003
30 921.8648 10.3154
40 922.9310 10.4745
50 922.9798 10.6180
60 923.5945 10.7661
70 923.6734 10.8103
80 924.4393 11.0610
90 924.5678 11.1348
100 925.0736 11.3242


Figure 6. Temperature rise effect on fuel cost and power loss
for the TDOPF problem with VPEs.


Figure 7. Convergence characteristic of PGPSO-DE for the IEEE 30 bus system
with VPEs corresponding to T
RatedRise = 30
o
C
Minh-Trung Dao, et al. 95

B. Outage cases
To consider the outage of the transmission line, contingency analysis is carried out before
solving the SC-TDOPF problem. The SI value is determined for each N-1 outage line.

Table 5. Contingency analysis of the IEEE 30 bus system
Outage line Overload line
Line flow
(MVA)
Line flow limit
(MVA)
Overload rate
(%)
Severity index
1-2
2 307.0136 130 236.1643
16.3035
4 281.3522 130 216.4248
7 178.4014 90 198.2238
10 46.5144 32 145.3575
1-3
1 274.0264 130 210.7895
9.4474
3 86.1203 65 132.4928
6 92.7203 65 142.6466
10 35.2567 32 110.1773
3-4
1 271.0750 130 208.5192
9.2390
3 84.8816 65 130.5871
6 91.7672 65 141.1803
10 34.9449 32 109.2027
2-5
1 165.4421 130 127.2632
8.5614
3 74.6652 65 114.8695
6 102.9619 65 158.4030
7 123.6755 90 137.4172
10 35.4150 32 110.6719
4-6
1 200.5759 130 154.2892
5.7600 6 98.5645 65 151.6377
15 67.5536 65 103.9286
The IEEE 30-bus system has 41 transmission lines, thus, there are 41 obtained SI values. The
severe cases with the highest SI values will be selected as outage cases for the SC-TDOPF
problem. In this study, five transmission lines 1- 2, 1-3, 3-4, 2-5 and 4-6 are determined as severe
cases since their SI values are higher than the other transmission lines. The contingency analysis of these five outage cases is given in Table 5, where each of the severe cases is considered in one
outage case.The obtained results for both cases with a quadratic objective function and a function
of VPEs are reported in the following subsections:

B.1. Quadratic objective function
For the SC-TDOPF problem with the quadratic function, the fuel cost and real power loss
obtained by the proposed PGPSO -DE method corresponding to each value of T
RatedRise are
presented in Table 6 for the outage lines 1- 2, 1-3, and 3- 4, and in Table 7 for the outage lines 2- 5
and 4- 6. For 0
o
C temperature rise, the proposed PGPSO-DE method provides the minimum fuel
costs for outage lines 1- 2, 1-3, 3-4, 2-5, and 4- 6 are 823.7949 ($/h), 820.4042 ($/h), 819.5505
($/h), 806.5713 ($/h), and 801.4946 ($/h), respectively. These values, in order, become 828.0905
($/h), 823.3478 ($/h), 822.6376 ($/h), 808.7184 ($/h), and 804.1846 ($/h) for 100
o
C temperature
rise. For the real power loss, the proposed PGPSO-DE method provides the values of 6.1357 MW,
6.3038 MW, 6.6004 MW, 7.4138 MW, and 8.3652 MW for 0
o
C temperature rise, and 6.3978
MW, 6.6481 MW, 7.2708 MW, 7.9547 MW, and 9.1687 for 100
o
C temperature rise
corresponding to the outage lines 1- 2, 1-3, 3-4, 2-5, and 4- 6. For 30
o
C temperature rise, the
increase in fuel cost and real power loss are 0.17% and 0.46 % for outage line 1- 2, 0.11 % and
1.81% for outage line 1-3, 0.13 % and 2.49% for outage line 1-3, 0.1 % and 2.28% for outage line
2-5, and 0.12 % and 3.58% for outage line 4- 6, respectively. Figure 8 shows the effect of
temperature rise on fuel cost and power loss for the SC-TDOPF problem with the quadratic
objective function. It can be seen that the fuel cost and power loss increase following the increase
of T
RatedRise. Figure 9 depicts the convergence characteristic of PGPSO-DE for outage lines 1-2, 1-Security-Constrained Temperature-Dependent Optimal Power Flow Using 96

3, 3-4, 2-5, and 4-6 for T
RatedRise = 30
o
C. The convergence characteristics yielded by the proposed
method for those cases are found to be stable. The optimal solution obtained by the proposed
method is given in Appendix.

Table 6. Fuel cost and power loss obtained for the SC-TDOPF problem with quadratic
objective function for outage lines 1- 2, 1-3, and 3-4.
Outage
case
Line 1-2 Line 1-3 Line 3-4
TRatedRise
Fuel cost
($/h)
Power
loss
(MW)
Fuel cost
($/h)
Power
loss
(MW)
Fuel cost
($/h)
Power
loss
(MW)
0 823.7949 6.1357 820.4042 6.3038 819.5505 6.6004
10 824.4885 6.2004 821.0448 6.4217 819.8601 6.6526
20 824.9029 6.2339 821.1104 6.3891 820.2708 6.7493
30 825.2031 6.1641 821.3032 6.4199 820.6233 6.7688
40 825.6662 6.2837 821.8879 6.4684 820.8518 6.7809
50 826.0028 6.2699 822.1185 6.5183 821.1988 7.0280
60 826.5077 6.3237 822.3815 6.5540 821.5800 6.9197
70 826.9197 6.3724 822.4503 6.4644 821.9318 6.9503
80 827.3582 6.2803 822.5881 6.5298 822.0921 6.9387
90 827.6874 6.2975 822.9092 6.6259 822.5010 7.0695
100 828.0905 6.3978 823.3478 6.6481 822.6376 7.2708


Line 1-2

Line 1-3

Line 3-4


Line 2-5

Line 4-6
Figure 8. Temperature rise effect on fuel cost and power loss for the SC-TDOPPF problem
with the quadratic objective function.
Minh-Trung Dao, et al. 97

Figure 9. Convergence characteristic of PGPSO-DE for the SC-TDOPF problem with the
quadratic objective function corresponding to T
RatedRise = 30
o
C.

Table 7. Fuel cost and power loss obtained for the SC-TDOPF problem with quadratic
objective function for outage lines 2- 5 and 4-6.
Outage case Line 2-5 Line 4-6
TRatedRise
Fuel cost ($/h) Power loss
(MW)
Fuel cost ($/h) Power loss
(MW)
0 806.5713 7.4138 801.4946 8.3652
10 806.7956 7.5068 801.8090 8.5623
20 807.0974 7.5545 802.0934 8.6289
30 807.3633 7.5871 802.4289 8.6759
40 807.5542 7.6766 802.6870 8.7421
50 807.8132 7.7412 802.9846 8.8857
60 808.0300 7.8139 803.2857 8.9063
70 808.1800 7.8076 803.5020 8.9634
80 808.3591 7.9252 803.6682 9.0475
90 808.6439 7.9897 803.9590 9.1169
100 808.7184 7.9547 804.1846 9.1687

B.2. Objective function considering VPEs.
In this case of the SC-TDOPF problem, the objective function comprises VPEs, which makes
the SC-TDOPD becomes a non- convex optimization problem. Table 8 tabulates the fuel cost and
power loss corresponding to each value of T
RatedRise for the outage lines 1- 2, 1-3, and 3-4. Similarly,
Table 9 presents the obtained results for the outage lines 2- 5 and 4- 6. The fuel costs obtained by
PGPSO-DE for 0
o
C temperature rise are 1034.8118 ($/h), 1030.2897 ($/h), 1025.6910 ($/h),
962.5737 ($/h), 952.1968 ($/h) for outage lines 1- 2, 1-3, 3-4, 2-5, and 4- 6, respectively. These
values are 1035.8552 ($/h), 1034.2135 ($/h), 1030.9578 ($/h), 961.7055 ($/h), and 954.3138 ($/h)
for 100
o
C temperature rise. For 30
o
C temperature rise, the increases in fuel cost are 0.05%, 0.16%,
0.18%, 0.03%, and 0.09% for outage lines 1- 2, 1-3, 3-4, 2-5, and 4- 6, respectively.
Regarding the real power loss, for 0
o
C temperature rise, PGPSO-DE achieves the values of 5.3209
MW, 5.9945 MW, 6.3729 MW, 8.0992 MW, and 8.3652 MW for the outage lines 1- 2, 1-3, 3-4,
2-5, and 4- 6. For 100
o
C temperature rise, the values of power loss become 5.5706 MW, 6.3244
MW, for 6.8565 MW, 8.7871 MW, and 7.8437 corresponding to the outage lines 1- 2, 1-3, 3-4,
2-5, and 4-6. For 30
o
C temperature rise, the increases in power loss are 1.63%, 2.47%, 2.90%,
2.14%, and 2.82% for corresponding outage lines.
Figure 10 shows the effect of temperature rise on fuel cost and power loss for the SC-TDOPF
problem with valve point loading effects for outage lines 1- 2, 1-3, 3-4, 2-5, and 4-6. It can be seen
that the fuel cost and power loss increase with the increase of T
RatedRise. Figure 11 illustrates the
convergence characteristic of PGPSO-DE for outage cases for T
RatedRise = 30
o
C. It can be observed Security-Constrained Temperature-Dependent Optimal Power Flow Using 98

that the proposed method converges smoothly to the near-optimal solution for outage lines 1- 2,
1-3, 3-4, 2-5, and 4-6. The optimal solution obtained by PGSO-DE for this case is given in


Line 1-2

Line 1-3

Line 3-4

Line 2-5

Line 4-6
Figure 10. Temperature rise effect on fuel cost and power loss
for the SC-TDOPPF problem with VPEs.

Appendix.
Table 8. Fuel cost and power loss obtained for the SC-TDOPF problem with VPEs
for outage lines 1- 2, 1-3, and 3-4.
Outage
case
Line 1-2 Line 1-3 Line 3-4
TRatedRise
Fuel cost
($/h)
Power
loss
(MW)
Fuel cost
($/h)
Power
loss
(MW)
Fuel cost
($/h)
Power
loss
(MW)
0 1034.8118 5.3209 1030.2897 5.9945 1025.6910 6.3729
10 1034.9668 5.3466 1030.8810 6.0989 1026.3124 6.4584
20 1035.0830 5.3813 1031.1819 6.0967 1026.8841 6.5023
30 1035.2897 5.4093 1031.9249 6.1463 1027.5598 6.5637
40 1035.3090 5.4237 1032.1796 6.2157 1027.8454 6.5829
50 1035.3977 5.4380 1032.3729 6.1726 1028.4009 6.6460
60 1035.5541 5.5031 1032.9630 6.2767 1029.0285 6.7297
70 1035.6378 5.5070 1033.2300 6.2698 1029.4535 6.7469
80 1035.6697 5.5015 1033.7299 6.3547 1030.0198 6.8218
90 1035.8174 5.5395 1033.8801 6.3397 1030.5906 6.7837
100 1035.8552 5.5706 1034.2135 6.3244 1030.9578 6.8565
Minh-Trung Dao, et al. 99

Figure 11. Convergence characteristic of PGPSO-DE for the SC-TDOPF problem with the
VPEs corresponding to T
RatedRise = 30
o
C.

Table 9. Fuel cost and power loss obtained for the SC-TDOPF problem with VPEs for outage
lines 2-5 and 4-6.
Outage case Line 2-5 Line 4-6
TRatedRise
Fuel cost
($/h)
Power loss
(MW)
Fuel cost
($/h)
Power loss
(MW)
0 961.4261 8.0992 952.1968 7.2400
10 961.4625 8.1205 952.6466 7.3755
20 961.7055 8.1547 952.7685 7.4288
30 961.7517 8.2767 953.0724 7.4499
40 961.8854 8.3563 953.2266 7.5389
50 961.9792 8.3842 953.3388 7.5391
60 962.0529 8.6210 953.5756 7.6115
70 962.4567 8.6214 953.7074 7.6258
80 962.5144 8.7538 953.9941 7.7289
90 962.5737 8.7589 954.1395 7.7605
100 962.7802 8.7871 954.3138 7.8437

7. Conclusion This study has investigated the OPF problem considering the temperature effect. The
objective function has been examined with the quadratic cost function and a function comprising of VPEs. In addition, the security constraint has also been considered for the OPF problem.
Considering the temperature effect increases the accuracy of the OPF problem. The SC-TDOPF
is a non- linear, non- convex, and large-scale problem which is a real challenge for solution
methods. In this paper, the proposed PGPSO-DE method has been successfully dealt with the
considered SC-TDOPF problem. The proposed method has been tested on the IEEE 30-bus
system with normal and outage cases for objective functions of quadratic and valve point effects.
The obtained results have shown that the proposed method is effective in dealing with the SC-
TDOPF problem with quadratic and VPEs of fuel cost function. As a result, the suggested
PGPSO-DE method could be a favourable method for dealing with the large-scale and difficult
optimization problems in power systems. The large-scale test systems and other complex
objective functions would be examined in future works for the TDOPF problem.

8. Acknowledgments
This research is funded by Vietnam National University HoChiMinh City (VNU-HCM)
under grant number C2021-20-14.
Security-Constrained Temperature-Dependent Optimal Power Flow Using 100

9. Appendix
Table A.1. Cost coefficients of thermal units with with a quadratic
cost function in the IEEE-bus system
Unit
Pi,max
(MW)
Pi,min
(MW)
Qi,max
(MVAr)
Qi,min
(MVAr)
Cost coefficients
ai ($/h) bi ($/MWh) ci ($/MW
2
h)
1 200 50 200 -20 0 2.00 0.00375
2 80 20 100 -20 0 1.75 0.01750
5 50 15 80 -15 0 1.00 0.06250
8 35 10 60 -15 0 3.25 0.00834
11 30 10 50 -10 0 3.00 0.02500
13 40 12 60 -15 0 3.00 0.02500

Table A.2. Cost coefficients of of thermal units with VPEs in the IEEE 30- bus system
Unit
P
i,max
(MW)
P i,min
(MW)
Cost coefficients
ai ($/h) bi ($/MWh) ci ($/MW
2
h) ei ($/h) fi (1/MW)
1 200 50 150 2.00 0.00160 50 0.063
2 80 20 25 2.50 0.01000 40 0.098
5 50 15 0 1.00 0.06250 0 0
8 35 10 0 3.25 0.00834 0 0
11 30 10 0 3.00 0.02500 0 0
13 40 12 0 3.00 0.02500 0 0

Table A.3. Limits transformer tap setting and bus voltage for the IEEE 30- bus system
Lower limit (p.u.) Upper limit (p.u.)
Transformer tap setting (Tk) 0.90 1.10
Slack bus voltage (Vg1) 0.90 1.10
Generator bus voltage (Vgi) 0.90 1.10
Load bus voltage (Vli) 0.95 1.05

Table A.4. Optimal solutions by HPSO-DE for TDOPD problem (Normal and Outage cases)
with quadratic fuel cost function corresponding to T
RatedRise = 30
o
C
Optimal
solution
Normal
case
Outage
line 1-2
Outage
line 1-3
Outage
line 3-4
Outage
line 2-5
Outage
line 4-6
PG1 (MW) 176.3517 122.8073 127.3085 129.4705 157.0478 169.3141
PG2 (MW) 48.8281 62.7836 61.2513 60.7798 42.4111 47.1001
PG5 (MW) 21.7061 25.6834 24.4011 25.3942 24.9709 21.6438
PG8 (MW) 21.5351 35.0000 35.0000 34.8064 34.5102 28.6375
PG11 (MW) 12.0837 22.0504 21.1753 20.3922 17.1942 13.3805
PG13 (MW) 12.0000 21.2393 20.6837 19.3256 14.8529 12.0000
VG1 (p.u.) 1.0945 1.0740 1.0802 1.0494 1.0770 1.0860
VG2 (p.u.) 1.0746 1.0556 1.0581 1.0292 1.0587 1.0669
VG5 (p.u.) 1.0429 1.0287 1.0282 1.0048 1.0244 1.0389
VG8 (p.u.) 1.0469 1.0390 1.0368 1.0128 1.0334 1.0394
VG11 (p.u.) 1.0465 1.0830 1.1000 1.0819 1.0630 1.0813
VG13 (p.u.) 1.0561 1.0656 1.0480 1.0533 1.0735 1.0376
QC10 (MVAr) 5.0000 0.0000 1.9495 0.0146 0.1535 3.7475
QC12 (MVAr) 2.1018 0.0000 2.5254 1.7606 4.1345 0.8124
QC15 (MVAr) 3.9448 3.5929 2.3804 3.0843 0.1873 1.9432
QC17 (MVAr) 5.0000 4.6862 4.5985 5.0000 1.9768 5.0000
QC20 (MVAr) 3.2950 3.8692 4.5061 3.8157 4.1076 0.0019
QC21 (MVAr) 4.9877 2.6067 0.4104 4.9866 4.2745 5.0000 Minh-Trung Dao, et al. 101

Optimal
solution
Normal
case
Outage
line 1-2
Outage
line 1-3
Outage
line 3-4
Outage
line 2-5
Outage
line 4-6
QC23 (MVAr) 5.0000 4.0042 4.2849 1.4906 4.2358 4.1888
QC24 (MVAr) 5.0000 1.7054 5.0000 5.0000 5.0000 5.0000
QC29 (MVAr) 5.0000 2.6949 1.5807 0.2004 1.4451 4.1379
T11 (p.u.) 1.0574 1.0014 1.0811 1.0119 1.0436 1.0610
T12 (p.u.) 0.9000 0.9916 0.9227 0.9361 0.9031 0.9384
T15 (p.u.) 0.9713 0.9958 0.9683 1.0093 1.0099 0.9450
T36 (p.u.) 0.9875 0.9740 0.9724 0.9621 0.9778 0.9949

Table A.5. Optimal solutions by HPSO-DE for TDOPD problem (Normal and Outage cases)
with VPEs corresponding to T
RatedRise = 30
o
C
Optimal
solution
Normal
case
Outage
line 1-2
Outage
line 1-3
Outage
line 3-4
Outage
line 2-5
Outage
line 4-6
PG1 (MW) 149.7332 99.8696 127.5941 129.8608 149.7352 149.7283
PG2 (MW) 52.0594 80.0000 52.0571 52.0565 51.2808 52.0546
PG5 (MW) 22.3130 26.8057 27.6889 26.7472 27.1427 24.3538
PG8 (MW) 33.2914 35.0000 35.0000 35.0000 35.0000 31.7920
PG11 (MW) 16.6501 24.4057 25.4596 23.6783 16.6122 16.4309
PG13 (MW) 17.0357 22.7283 21.7466 22.6209 12.0004 16.4903
VG1 (p.u.) 1.0694 1.0706 1.0809 1.0492 1.0848 1.0727
VG2 (p.u.) 1.0492 1.0596 1.0580 1.0262 1.0577 1.0576
VG5 (p.u.) 1.0137 1.0307 1.0278 1.0020 0.9906 1.0327
VG8 (p.u.) 1.0337 1.0408 1.0285 1.0091 1.0261 1.0429
VG11 (p.u.) 1.0827 1.0964 1.0767 1.0759 0.9919 1.0935
VG13 (p.u.) 1.0781 1.0428 1.0641 1.0677 1.1000 1.0469
QC10 (MVAr) 1.4073 2.7162 5.0000 2.9103 5.0000 2.6873
QC12 (MVAr) 0.4978 3.2551 0.0469 2.3769 1.1882 3.2422
QC15 (MVAr) 2.2904 4.9133 4.3504 3.1394 2.0455 3.6043
QC17 (MVAr) 2.6629 4.5545 4.2242 2.6968 5.0000 4.0109
QC20 (MVAr) 1.0981 2.2348 4.9931 3.2844 5.0000 4.9691
QC21 (MVAr) 4.7504 4.6913 0.0358 3.6180 3.9394 3.0023
QC23 (MVAr) 2.8429 5.0000 0.6435 0.0036 2.4749 4.9121
QC24 (MVAr) 3.8930 2.7292 5.0000 5.0000 2.7409 4.0099
QC29 (MVAr) 3.4787 2.7533 2.5568 3.0209 2.3062 2.5498
T11 (p.u.) 1.0318 1.0837 0.9922 0.9493 1.0928 1.0135
T12 (p.u.) 0.9000 0.9114 0.9887 1.0368 1.0277 1.0067
T15 (p.u.) 1.0812 0.9761 0.9989 0.9857 1.0035 0.9791
T36 (p.u.) 0.9690 0.9747 0.9684 0.9673 1.0316 0.9812

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Minh-Trung Dao received a B.Sc. degree in Electrical Engineering from Can Tho
University, Vietnam, in 2005 and M.Eng. degree in Electrical Engineering from
University of Technology - Viet Nam National University Ho Chi Minh City in 2012.
Since 2005, he has been with the Department of Electrical Engineering, Can Tho
University in Can Tho city, Viet Nam, where he is currently a lecturer. His research
interests include power system optimization, power system operation and control,
power system analysis, and renewable energy systems. Now, he is also a PhD student
at University of Technology - Viet Nam National University Ho Chi Minh City (from
8/2021).



Khoa Hoang Truong received his B.Eng. degree in electrical engineering from Ho
Chi Minh City University of Technology, VNU- HCM, Vietnam, in 2012. He also
received his MSc degree (Applied Sciences) and Ph.D. degree in Electrical and
Electronics Engineering from Universiti Teknologi PETRONAS, Malaysia, in 2016
and 2020, respectively. He is currently a Lecturer at Department of Power Delivery,
Faculty of Electrical and Electronics Engineering, Ho Chi Minh City University of
Technology, VNU-HCM, Vietnam. His research interests are power system operation
and control, distributed generation, microgrids, and artificial intelligence-based
algorithms.
Security-Constrained Temperature-Dependent Optimal Power Flow Using 104

Duy-Phuong N. Do was born in Can Tho city, VietNam in 1982. He received B.S,
M.Sc and Ph.D degree in electrical engineering from Can Tho University, Ho Chi Minh
University of Technology, Vietnam and Gyeongsang National University, Korea in
2005, 2008 and 2017 respectively. His research interests include wind speed forecast,
voltage stability evaluation of power system, and power loss minimization in
distribution network.




Bao-Huy Truong received the B.Eng. degree in Electrical and Electronics Engineering
from Ho Chi Minh City University of Technology (HCMUT), VNU- HCM, Vietnam,
in 2017 and the M.S. degree in Electrical and Electronics Engineering from Universiti
Teknologi PETRONAS (UTP), Malaysia, in 2020. He is currently a Research Officer
with Institute of Engineering and Technology, Thu Dau Mot University, Binh Duong
Province, Vietnam. His research interests are renewable energy power generation,
artificial intelligence-based algorithms and their application in optimization problems.



Khai Phuc Nguyen received his B.Eng. and M.Eng. degrees from Ho Chi Minh City
University of Technology, Vietnam, in 2010 and 2012, respectively, and his Ph.D. in
electrical engineering from Shibaura Institute of Technology, Tokyo, Japan, in 2017.
His research interests are Artificial Intelligence (AI) in power system optimization,
operation and control, power system analysis, and automation in power systems.





Dieu Ngoc Vo received his B.Eng. and M.Eng. degrees in electrical engineering from
Ho Chi Minh City University of Technology, Ho Chi Minh city, Vietnam, in 1995 and
2000, respectively and his D.Eng. degree in energy from Asian Institute of Technology
(AIT), Pathumthani, Thailand in 2007. He is Research Associate at Energy Field of
Study, AIT and Head of Department of Power Systems, Faculty of Electrical and
Electronic Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh
city, Vietnam. His research interests are applications of AI in power system
optimization, power system operation and control, power system analysis, and power
systems under deregulation and restructuring.


Minh-Trung Dao, et al. 105