Energy and Power Engineering, 2013, 5, 693-697
doi:10.4236/epe.2013.54B134 Published Online July 2013 (http://www.scirp.org/journal/epe)
Power System Reactive Power Optimization Based on
Fuzzy Formulation and Interior Point Filter Algorithm
Zheng Fan1, Wei Wang2, Tian-jiao Pu1, Guang-yi Liu1, Zhi Cai1, Ning Yang3
1China Electric Power Researc h I n s t itute, Beij i n g, China
2Gansu Electric Power Corporation, Lanzhou, China
3Northeast China Grid Company, Shenyang, China
Received March, 2013
Considering the soft constraint characteristics of voltage constraints, the Interior-Point Filter Algorithm is applied to
solve the formulation of fuzzy model for the power system reactive power optimization with a large number of equality
and inequality constraints. Based on the primal-dual interior-point algorithm, the algorithm maintains an updating “fil-
ter” at each iteration in order to decide whether to admit correction of iteration point which can avoid effectively oscil-
lation due to the conflict between the decrease of objective function and the satisfaction of constraints and ensure the
global convergence. Moreover, the “filter” improves computational efficiency because it filters the unnecessary itera-
tion points. The calcu lation results of a practical power system indicate that the algorithm can effectively deal with the
large number of inequality constraints of the fuzzy model of reactive power optimization and satisfy the requirement of
online calculatio n which realizes to decrease the network loss and maintain specified margins of voltage.
Keywords: Power System; Reactive Power Optimization; Fuzzy; Filter; Interior-point Algorithm; Online Calculation
On the premise of safe and stable operation, the reactive
power optimization of power system realizes hierarchical
and regional balance of reactive power, improves voltage
quality and reduces netwo rk loss by means of adju stment
of the reactive power controllers such as terminal volt-
ages of generators, tap positions of on load tap changers
(OLTCs) and switchable shunt capacitor/reactors. The
model of traditiona1 reactive power optimization is usu-
ally expressed as minimization of the active network loss
under rigid voltage constraints which almost not consid-
ers the security margin of voltage and the characteristics
of soft constraints when dealing with some voltage con-
straints. It makes the optimized voltage of some buses
too close to their high limits, which becomes the threat of
the system because the ability of enduring the variation is
decreased remarkably [1, 2]. Actually, the expected valu e
of operating voltage is a fu zzy concept. The voltage con-
straints of load nodes for reactive power optimization are
soft constraints. In [1-4], a mathematical model that is
closer to the reality is established by introducing the
fuzzy theory for dealing with soft constraints; the formu-
lation of fuzzy model for reactive power optimization
can decrease the network loss and maintain specified
margins o f v ol tage.
Now, the primal-dual interior-point algorithm is wide-
ly used in the field of reactive power optimization [5-8],
because it has the advantages of rapid convergence,
strong robustness and insensitivity to the initial valu e .
The interior-point filter algorithm [9, 10] (IPFA) is the
latest achievement in nonlinear optimization research.
Based on the primal-dual interior-point algorithm, it
maintains an updating “filter” at each iteration in order to
decide whether to admit correction of iteration point
which can avo id effectively oscillation due to th e conflict
between the decrease of objective function and the satis-
faction of constraints and ensure the global convergence,
Moreover, the “filter” improves computational efficiency
because it filters the unnecessary iteration points [11,
In this paper, the IPFA is applied to solve the fo rmula-
tion of fuzzy model for the power system reactive power
optimization with a large number of equality and ine-
quality constraints. The example of an actual power sys-
tem indicates that the algorithm can effectively deal with
the large number of inequality constraints of the fuzzy
model of reactive power optimization and satisfy the
requirement of online calculation which realizes to de-
crease the transmission loss and maintain specified mar-
gins of voltage.
The paper is organized as follows. In section 2 the
formulation of fuzzy model for the power system reac-
Copyright © 2013 SciRes. EPE
Z. FAN ET AL.
tive power optimization is introduced. In section 3 the
solution of the model based on an interior point filter
algorithm is presented. Test results of a practical system
are reported in section 4 and conclusions are made in
2. Formulation of Fuzzy Model for Reactive
The traditional mathematical model of reactive power
optimization is established under a certain given active
power dispatching mode by using the voltage amplitudes,
phase angles, outputs of reactive power compensation
device and transformation ratios of OLTC as decision
variables and representing other variables in the form of
function of decision variables. The problem can be de-
scribed as follows:
where ; 1 is the vector composed
of terminal voltages of generators, outputs of reactive
power compensation devices and transformation ratios of
OLTCs; 2 is the vector composed of the voltages of
PQ buses, it has the characteristics of soft constraint; 3
is the vector composed of voltage phase angles of all
buses except slack bus;
x is active network loss of
power system; are active balance equa-
tions( 1()gx 0
is the number of nodes in
the system); are reactive balance equa-
N -dimensional, G
is the number of gen-
erators in the system); are the inequality con-
straints of terminal voltages and reactive power outputs
of generators, outputs of reactive power compensation
devices and transformation ratios of OLTCs, which are
regarded as hard constraints; are the inequality
constraints of voltages of PQ buses, which are regarded
as soft constraints; min and max
h are the vectors of
lower limits and upper limits of hard constraints respec-
-dimensional , H
is the sum of 2 G
are the number of reactive
power compensation devices and OLTCs respectively);
and are the vectors of lower limits and up-
per limits of soft constraints respectively (
is the number of PQ buse s).
Fuzzy theory is introduced for the justfication of the
problem, and it is the key to select membership functions.
In this paper, the membership functions for objective
function and soft constraint variables are expressed as
piecewise linear functions . The membership function
of objective function is represented as follows:
is the ideal maximum reduction of network
is maximum value of network loss.
The membership function of soft constraint variables
is defined as follows:
max max max
'' '' '
min min min
(()) () ()
ii ivii i
ii ii ivi
is allowable maximum offset of .
On the basis of fuzzy set theor y , the problem (1) is
converted to maximum satisfaction degree problem ,
where is satisfaction degree.
3. Fuzzy Reactive Power Optimization Based
on Interior Point Filter Algorithm
The interior-point filter algorithm [9, 10] is based on the
primal-dual interior-point algorithm and maintains an
updating “filter” at each iteration in order to avoid oscil-
lation when contradiction exists between the decrease of
objective function and the satisfaction of constraints and
ensure the global convergence. The algorithm also im-
proves calculation efficiency due to the “filter” filters
unnecessary iteration points.
To solve (4) using IPFA, inequality constraints are
transformed into equality constraints by adding to slack
variables and new objective function is built by intro-
ducing barrier function. Meanwhile, the problem (4) is
converted to nonlinear minimizing problem, that is
Copyright © 2013 SciRes. EPE
Z. FAN ET AL. 695
sss ss and
are the vectors
composed of original variable s,
In IPFA, the decrease of objective function
is equivalent to the satisfaction of constraints which is
A set called “filter” is maintained at each iteration,
which is defined as:
is called if the condition
The iteration point is accepted only
if objective function and infinite norm of
the constraints set meet (8).
At iteration , the “filter” is updated according to
In order to solve (5), the Lagrangian function is repre-
ygxy g x
lu l uslsu
vectors composed of dual variables.
According to Karush-Kuhn-Tucker condition, it is the
necessary condition that all the partial derivatives of the
Lagrangian function are equal to zero if the minimum of
the problem (5) is existed. Derivation process and solu-
tion procedure for using the IPFA to solve the nonlinear
programming model can be referred to .
4. Case Study
Test case of a practical 244-bus power system is em-
ployed to validate the solution of fuzzy model for reac-
tive power optimization based on interior-point filter
algorithm. The system contains 76 generation units, 12
on-load tap changers and 56 shunt capacitor/reactors. The
current controllable devices include 36 generation units,
12 on-load tap changers and 52 shunt capacitor/reactors.
The computer configuration used for case study is Intel
Core i3 2.53 GHz and 2 GB memory.
In this case, the reference power is 100 MVA; the low-
er and upper limits of bus voltages are set at 0.9 and 1.1
(p.u.); there are 12 buses violated voltage limits in the
initial state; the initial network loss is 1.2516 (p.u.).
and the elements of vectorare set to 0.233 and 0.0217
respectively. After optimization calculation for fuzzy
model, the satisfaction degree is 0.9689, all bus voltages
are within limits and the network loss is reduced to
1.0258. Comparisons of optimal results and the initial
values are in Table 1. Moreover, co mparisons of solving
results of fuzzy model and traditiona1 model for reactive
Copyright © 2013 SciRes. EPE
Z. FAN ET AL.
Table 1. Comparisons of optimal results and the initial val-
network loss(p.u.) 1.2516 1.0258
number of buses violated voltage limits 12 0
maximum voltage of PQ bus(p.u.) 1.0571 1.0789
minimum voltage of PQ bus (p.u.) 0.8927 0.9211
Table 2. Comparisons of solving results of fuzzy model and
network loss(p.u.) 1.0258 1.0183
number of buses violated voltage limits 0 0
maximum voltage of PQ bus(p.u.) 1.0789 1.1
minimum voltage of PQ bus (p.u.) 0.9211 0.9225
computing time(s) 0.95 0.59
power optimization are shown in Table 2.
From the Table 1 and Table 2, it can be seen that the
algorithm can effectively deal with the large number of
inequality constraints of the fuzzy model of reactive
power optimization and has high compute efficiency.
After optimization calculation, there is no bus violated
voltage limit through the adjustment of the reactive pow-
er controllers. The maximum voltage of PQ bus is its
upper limit when traditiona1 model is used, however, the
voltages of PQ buses maintain specified margins when
fuzzy model is used, which improves the safety level of
Meanwhile, the network loss is decreased to 1.0183
from 1.2516 and the decreasing range is 18.63% when
traditiona1 model is used, whereas the network loss is
decreased to 1.0258 from 1.2516 and the decreasing
range is 18.04% when fuzzy model is used. So the de-
creasing range of network loss obtained by solving the
fuzzy model is less than that obtained by solving the tra-
ditiona1 model, but the level of voltage security is im-
proved when fuzzy model is used.
The example of the practical power system indicates
that the algorithm can effectively solve the fuzzy model
of reactive power optimization and satisfy the require-
ment of online calculation. After optimization calculation,
the violated voltages are corrected; the voltage soft con-
straint buses are kept specified margins; at the same time
the network loss is reduced.
This paper uses the interior point filter algorithm to solve
the formulation of fuzzy model for the power system
reactive power optimization considering the soft con-
straint characteristics of voltage constraints. Optimiza-
tion results show that the algorithm can effectively deal
with the large number of equality and inequality con-
straints of the fuzzy model for the practical power system
and satisfy the requirement of online calculation which
realizes to decrease the network loss and maintain secu-
rity margins of voltage.
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