A New Optimal Power Flow Model Considering the Active Power
Constraints of Transmission Interfaces
XIUQIONG HU
School of Information and Electric Engineering,
Panzhihua University,
Panzhihua, 617000,
CHINA
Abstract: - An optimal power flow model as well as algorithm is provided for interconnected power systems
considering active power constraints of transmission interfaces. To ensure the optimum operation of an
interconnected power system, the optimal goal of the proposed model is minimizing the loss of active power.
For satisfying transaction power constraints, the active power constraints of transmission interfaces are taken
into account as inequality constraints. Considering the multi-area characteristic of interconnected power
systems, a decomposition-coordination optimal model is proposed on the base of the model previously
established. And the decomposition-coordination interior point method is used to solve the decomposition-
coordination optimal model. Simulations of two test systems illustrate that the proposed model as well as the
algorithm can improve computational efficiency, which can provide operation schedule decisions for
interconnected power systems.
Key-Words: - Interconnected Power System, Optimal Power Flow, Active Power Constraints of Transmission
Interfaces, Decomposition-coordination Optimal Model, Decomposition-coordination Interior Point Method,
computational efficiency
Received: April 6, 2022. Revised: February 8, 2023. Accepted: March 10, 2023. Published: April 11, 2023.
1 Introduction
Under the power market environment, the
interconnection of power grids becomes the
developing direction for the power system. In the
interconnected power system, though each area has
its independent system operation, it still needs to be
controlled coordinately to guarantee optimal
operation of the system. These situations provide an
opportunity for the development of optimal power
flow. At present, optimal power flow is mainly used
for pricing, [1], [2], [3], congestion management,
[4], [5], [6], [7], available transfer capability, [8],
[9], etc. For interconnected power systems,
however, to achieve a wider range of optimal
assignment of resources, and greater competition in
the market, the active power of transmission
interfaces must also be adjusted to the specified
value to meet the transaction power constraints.
However, this is barely considered in the present
optimal power flow model.
The calculation methods of optimal power flow
include the centralized algorithm, [10], [11], [12],
and the distributed algorithm, [1], [4], [5], [13],
[14], [15]. In the interconnected power system, the
centralized algorithm has difficulty in real-time data
collection, and disadvantages of heterogeneous data
resources, large data traffic, and large data storage.
Whereas the distributed algorithm can complete
calculations independently, according to the local
data and target within each area. So, the distributed
algorithm has the advantages of small data traffic
and small data storage, and it can guarantee the
global simulation precision and speed requirements
while avoiding the leak of internal important data.
Therefore, the distributed algorithm becomes an
important calculation tool to solve the integration
simulation for the interconnected power system.
For the distributed algorithms, the auxiliary
problem principle (APP) algorithm is widely used,
[13], [14]. However, the research of APP is not yet
mature. In [15], decomposition-coordination interior
point method was compared with the APP
algorithm, and found that this method was better
than the APP algorithm in computing time, the
accuracy of the objective function, and iteration
number.
Based on existing research, an optimal power
flow model, as well as the algorithm is proposed
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2023.22.3
Xiuqiong Hu
E-ISSN: 2224-266X
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considering the active power constraints of
transmission interfaces. This model uses minimizing
the loss of active power as an optimization goal.
And the inequality constraints include active power
constraints of transmission interfaces. Then,
considering the disadvantages of the centralized
algorithm and APP, the decomposition-coordination
model is established, and calculated by the
decomposition-coordination interior point method.
2 The Proposed Optimal Power Flow
Model
To take into account the optimal economic
operation for the power system, the proposed
optimal power flow model uses minimizing the loss
of active power as an optimization goal. The
optimized variables include the reactive power
generated by the generator, the active power
generated by the generator, the reactive power
output of shunt reactive power compensation
equipment, the ratio of transformer fitted with an
on-load tap changer (OLTC) as well as bus voltage.
All constraints include the power flow equality
constraints, the active power constraints of
transmission interfaces, and the inequality
constraints of optimized variables.
The proposed optimal power flow model can be
formulated by Equations (1)-(12).
min
BB
GD
ii
i N i N
PP
(1)
s.t
LT
G D L T B
0
ii
i i ij ij
ij S ij S
P P P P i N
(2)
LT
G C R D L T B
0
ii
i i i i ij ij
ij S ij S
Q Q Q Q Q Q i N
(3)
T
0,
i m m i
e f e f i m N
(4)
T
0 , ,
i t m
e k e i m t N
(5)
link,
, cut, cut
0
n
ij n n
ij S
P P n N
(6)
2 2 2 2 2
min max Bi i i i i
V V e f V i N
(7)
min max Tt t t
k k k t N
(8)
G min G G max Gi i i
Q Q Q i N
(9)
C min C C max Ci i i
Q Q Q i N
(10)
R min R R max Ri i i
Q Q Q i N
(11)
G min G G max G
i i i
P P P i N
(12)
In the above equations, NB, NG, NT, NC, NR, and Ncut
respectively represent the number of buses,
generators, and transformers fitted with OLTC,
shunt capacitor, shunt reactor, and transmission
interface. Slink,n, SLi, and STi respectively represent
the set of tie lines included in the nth transmission
interface, line branch connected with ith bus, and
OLTC connected with ith bus. PG, QG, PD, QD, QC,
QR, k, V, e, and f respectively represent the active
power generated by the generator, the reactive
power generated by the generator, the active load,
the reactive load, the reactive power generated by
shunt capacitor, the reactive power generated by
shunt reactor, the ratio of transformer fitted with
OLTC, the amplitude of bus voltage, the real parts
and imaginary parts of bus voltage. PLij, QLij, PTij,
and QTij respectively represent the active power and
reactive power of the line branch, and the active
power and reactive power of the branch with a
transformer fitted with OLTC. Pij,n represents the
active power of the nth transmission interface. Pcut,n
represents the set value of transaction power for the
nth transmission interface.
Furthermore, Equations (2) and (3) represent
power flow equality constraints; Equations (4) and
(5) represent the voltage conversion relation of
branches with transformer fitted with OLTC;
Equation (6) represents the active power constraints
of transmission interfaces; Equations (7)-(12)
respectively represent the inequality constraints for
each optimized variable.
The present interconnected power system has
characteristics of hierarchical and partition
management, and each dispatching center is only
responsible for the maintenance and management of
its data. If the above model uses the traditional
centralized optimization algorithm, it is bound to
encounter the splicing problem of basic data.
Especially, when the scale of interconnected power
is increasing, the above model could meet the
convergence problem. The decomposition-
coordination algorithm based on multi-area can well
solve these problems.
3 The Decomposition-coordination
Model and Algorithm
3.1 The Decomposition-coordination Model
Using a two-area interconnected power system, for
example, xI1, and xI2 respectively represent the inner
variables of each area without boundary buses, and
xB represents the variable of boundary buses. And
for area 1, the variable of boundary buses can be
denoted as
1 1 1 1
T
B1 = ( , , , )
i i i i
e f P Qx
; for area 2, the
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Xiuqiong Hu
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variable of boundary buses can be denoted
as
T
B2 = ( , , , )
2 2 2 2
i i i i
e f P Qx
. The specific
decomposition-coordination model can be expressed
by Equations (13) - (18).
min
1 I1 B1 2 I2 B2
( , ) ( , )ffx x x x
(13)
s.t.
1 I1 B1
( , ) 0g x x
(14)
2 I2 B2
( , ) 0 g x x
(15)
1 1 I1 B1 1
( , )h h x x h
(16)
2 2 I2 B2 2
( , )h h x x h
(17)
B1 B2 0 x x
(18)
In the above equations, Equation (13) represents the
sum of the objective function of each area which can
be expressed by Equation (1); Equation (14) and
Equation (15) respectively represent the equality
constraints for area1 and area2; Equation (16) and
Equation (17) respectively represent the inequality
constraints for area1 and area2; Equation (18),
which is the boundary coordination equation,
represents the coupling constraints for the two areas,
and it can be further stated as Equation (19).
A
0
mm
m
mN
Ax
(19)
In Equation (19), NA represents the number of areas,
Am represents the coupling relationship matrix
between area m and the other areas, and xm
represents the variables which include the inner
variables and boundary variables.
3.2 The Solving Steps based on
Decomposition-coordination Interior Point
Method
Decomposition-coordination interior-point method,
which was proposed in [13], is used to solve
Equations (13)-(19). The solving steps are shown
below.
1) Iteration number donated by K for the
decomposition-coordination interior point method
should be set as 0 and its maximum value is given.
The tolerance error for complementary gap and KT
condition are set as 10-6. The initial values of
variables and the Lagrange multiplier are
determined.
2) The complementary gap and residue of the
optimal model for each area are calculated. And if
they are less than the set tolerance error, the solution
will stop, and the optimal value is achieved.
Otherwise, it will turn to step 3).
3) The K is set as K + 1. If K is greater than the
maximum number of iterations for the
decomposition-coordination interior point method, it
shows the solution is not converged, and the
solution will stop. Otherwise, it will turn to step 4).
4) According to the following steps, the original
variables and the dual variables for each area will be
updated:
According to the Jacobin matrix and Hessian
matrix of each area, the correction equation Mm
and residue Bm with reduced order are solved.
Combined with the coupling relationship
matrix Am, the matrix
[ , ]
m
m m q N
0EA
, in which
Nm and q respectively represent the number of
equality constraints and coupling constraints, is
achieved.
According to the following
equation
A A A
A A A
1 T 1 T
1 1 1 d
1 T 1 T
1 1 1 d
( ... )
...
N N N
N N N
E M E E M E y
E M B E M B D
, Δyd,
which is the Lagrange multiplier of coupling
constraints, is calculated.
After that, according to the following
equation
TT
d
[ , ]
m m m m m
M x y B E y
, the
increment of the optimal variable for each area
can be achieved. In this equation, ym represents
the Lagrange multiplier of equality constraints.
According to the iterative step and the
increment of the optimal variable for each area,
the optimal variables can be updated.
5) Return to step 2).
Detailed steps of the decomposition-coordination
interior-point method can be referred to [15].
4 Simulations for Test Systems
4.1Introduction of Test Systems
The correctness and effectiveness of the proposed
optimal model and method are demonstrated using
the simulations for the following two test systems
which are shown in Fig.1. In Fig.1 (a), the IEEE 30-
bus system is divided into two areas, and the
information of the transmission interfaces between
the two areas is shown in Table 1. In Fig.1 (b), the
IEEE 118×2-bus system consists of two IEEE 118-
bus systems which are represented using two cycles,
and each IEEE118-bus system has the same
topology structure and parameters. For the
IEEE118×2-bus system, the bus number of the area2
is the original number adding 118. The information
on the transmission interfaces between the two areas
is shown in Table 1 too.
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area 1 area 2
tie line
bus
4,6,25
tie line
bus
9,10,12,
27
transmission
interface
(a)IEEE30-bus
interconnected power system
area 1 area 2
tie line
bus
30,68,81
tie line
bus
148,186,
199
transmission
interface
(b)IEEE118×2-bus
interconnected power system
Fig. 1: Information on the two test systems
Table 1. Transmission interface information on
the test system
Test
system
Tie line of the
transmission
interface
The set value of active
power of transmission
interface /p.u.
IEEE30
6-9
6-10
4-12
25-27
0.5
IEEE118×2
30-148
68-186
81-199
1.0
To guarantee the set value of active power for
each transmission interface, each area should
have generators that can regulate the active
power output. These generators are called
frequency regulation units. The number of
frequency regulation units is shown in Table 2.
And the limit of the active power output for
these generators is shown in Table 2 too.
Table 2. The number and active power outputs
limit of the slack generator in each test system
Test
system
The frequency
regulation units
in each area
The limit of the active
power output of frequency
regulation units /p.u.
IEEE30
1,1
area1
area2
(0, 3)
(0, 0.5)
IEEE118×2
1,1
area1
area2
(0, 8)
(0, 8)
4.2 Analysis of Simulations
Firstly, the power flow operation result is
obtained by using the Newton-Raphson method.
And the initial active power of each
transmission interface can be achieved, which is
shown in Table 3.
Table 3. Initial active power of each interface
Test system
Initial value of active power of each
transmission interface/p.u.
IEEE30
0.5272
IEEE118×2
0.0177
From Table 3, it can be found that the initial
active power has a large difference from the set
value for the two test systems. It will need a lot
of work if the traditional power flow calculation
is used to regulate the active power of each
transmission interface. What’s more, this
method can’t guarantee the optimal operation of
the system and the operational feasibility of
each variable.
According to the proposed method in Section 2
and Section 3, the model of optimal power flow
considering the active power constraints of
transmission interfaces is established, and the
decomposition-coordination interior point method is
adopted to solve the model. Meanwhile, the
centralized algorithm, which is the prediction-
correction primal dual interior point method, is used
to solve the model too. The tolerance error for the
complementary gap is 10-6 for the two methods.
For the two methods, solving the correction
equation with reduced order is a critical aspect
affecting the computation time of the algorithm. The
higher dimension of the coefficient matrix in the
correction equation, the more time it takes to solve
the modified equation. The dimension of the
coefficient matrix in the correction equation with
reduced order for the two methods is shown in
Table 4. When the scale of the system is increasing,
the dimension of the coefficient matrix in the
correction equation increases sharply for the
centralized method. This increases the time of
solving the correction equation and the data storage
of the computer. In the decomposition-coordination
interior point method, because the system is divided,
the dimension of the coefficient matrix in the
correction equation of each area is greatly reduced
compared with the centralized algorithm, which
greatly reduces the time of solving the correction
equation in each area and reduces the data storage of
the computer.
Table 4. Coefficient matrix dimensions of the
corrected equations of the two algorithms
Test system
Centralized
method
Decomposition-
coordination interior
point method
Dimension of the
coefficient matrix
in reduced order
modified equation
Dimension of the
coefficient matrix in
reduced order modified
equation in each area
IEEE30
134
81, 90
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IEEE118×2
1224
657, 645
The optimization results of the decomposition-
coordination interior point method and
centralized method are shown in Table 5, in
which the active power loss of test systems
calculated through the decomposition-
coordination interior point method is consistent
with the result obtained by the centralized
method.
In Table 5, the number of iterations of the
decomposition-coordination interior point
method is more than that of the centralized
method. This is due to the need for continuous
interaction and coordination of the boundary
variables of each area.
Table 5. Simulation results of the two algorithms
Test
system
Centralized method
Decomposition-coordination interior point method
Number
of
iteration
Active
power
loss/p.u.
Computing
time/s
Number
of
iteration
Active
power
loss /p.u.
Computing
time /s
The active power output of
frequency regulation
unit/p.u.
IEEE
30
9
0.1225
7.559
16
0.1225
7.661
area1
area2
2.1324
0.4242
IEEE
118×2
12
2.1316
12.802
18
2.1316
13.073
area1
area2
3.861
5.8906
In Table 5, the computing time of the
decomposition-coordination interior point
method is slightly more than that of the
centralized computing method, because the
simulation analysis in this paper is completed in
serial computing mode. If it can be carried out
in the parallel mode, the decomposition-
coordination interior point method should have
less computing time than the centralized
computing method. The reason is as follows: on
the one hand, in the calculation process of the
decomposition-coordination interior point
method, the order of the coefficient matrix in
the correction equation for each area is greatly
reduced and the amount of calculation is
reduced too, which can also be seen from Table
4; On the other hand, the optimization
calculation for each area can be carried out
simultaneously in parallel computing mode,
which can further reduce the computing time of
the decomposition-coordination interior point
method.
The active power output of frequency
regulation units is also shown in Table 5.
Compared to Table 2, it can be found that the
final active power output of these generators is
within their operation constraints while
ensuring that the active power flowing in each
transmission interface is consistent with the
transaction power constraint.
Therefore, it can be seen from Table 5 that
satisfactory results can be obtained using the
proposed model and method with a few
iterations. And it does not take a lot of time to
repeatedly adjust the control variables in the
system. Furthermore, the optimization results
are consistent with that of the centralized
algorithm, which shows that the proposed
optimal power flow model and algorithm are
correct and effective.
5 Conclusion
This paper presents an optimal power flow
model and algorithm considering active power
constraints of transmission interfaces. The
proposed model considers equality constraints
of power flow, active power constraints of
transmission interfaces, and operational
constraints of each variable. Compared with the
traditional power flow calculation method, it
can save time to adjust the system to satisfy
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various operation constraints, and it can
guarantee optimal operation of the power
system. Considering the multi-area
characteristic of interconnected power systems,
a decomposition-coordination model is
established and calculated through the
decomposition-coordination interior point
method, which further improves the
computational efficiency of the proposed
optimal model. Furthermore, the method can
also achieve calculation accuracy equivalent to
the centralized method. Therefore, the proposed
model, as well as the algorithm has a certain
reference value for the operation schedule
decision of the interconnected power system.
The following research work for this paper
includes two aspects:
1) The data of the actual interconnected power
system should be used for modeling and
simulation to prove the practicability of the
proposed model as well as its algorithm.
2) The feasibility of the proposed model as well
as the algorithm should be verified for
interconnected power systems containing
renewable energy.
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The author contributed in the present research at all
stages from the formulation of the problem to the
final findings and solution.
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a Scientific Article or Scientific Article Itself
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Conflicts of Interest
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is relevant to the content of this article.
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