Power System Reactive Power Optimization Based on Fuzzy Formulation and Interior Point Filter Algorithm

doi:10.4236/epe.2013.54B134

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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

Email: [email protected]

Received March, 2013

ABSTRACT

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

1. Introduction

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 [5].

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,

12].

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-

Z. FAN ET AL.

694

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

section 5.

2. Formulation of Fuzzy Model for Reactive

Power Optimization

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:

min max

'' '

min max

min( )

..( )

()

st 





gx 0

hhxh

(1)

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;

TTTT

12 3

[,, ]xxxx

()

x is active network loss of

power system; are active balance equa-

tions( 1()gx 0

1-dimensional,

is the number of nodes in

the system); are reactive balance equa-

tions( G

2()gx 0

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-

tively(

()hx

'()hx

-dimensional , H

is the sum of 2 G

, C

and T

, C

and T

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 (

'min

h'max

dimensional S

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 [1]. The membership function

of objective function is represented as follows:

1()

()

(()) ()

0()

mmp m































(2)

where



is the ideal maximum reduction of network

loss; m

is maximum value of network loss.

The membership function of soft constraint variables

is defined as follows:

'''

min max

'' '''

max max max

'' '' '

min min min

''''

min max

1()

() ()

(()) () ()

0() ()

ivii ivi

ii ivii i

ii ii ivi

iiii

hhh

hh hhh

hhh hhh

hhorhh











 







 





















(3)

where vi



is allowable maximum offset of .

'()

On the basis of fuzzy set theor y [1], the problem (1) is

converted to maximum satisfaction degree problem [2],

namely

'min

max

min max

max

..( )

()















gx 0

hx εh

hhxh

(4)

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

Z. FAN ET AL. 695

''''

min

''''

max

min

min(,,)(lnln

ln ln

lnlnln )

..( )

()

()0 0

()

(

il iu

sl suf

pmf f

vll

vuu

sss

fSfss









 









 

 

 



xxs s

gx 0

hx εhs0s0

hx hs0s0

hx max

)

10 0

sl sl

su su

Ss s

 

 

 

hs0 s0

(5)

where ,

x''

,,,,,

luluslsu

sss ss and

are the vectors

composed of original variable s,

''T'TTT

[, ,,,,,]

luluslsufT

ssxssss.

In IPFA, the decrease of objective function

is equivalent to the satisfaction of constraints which is

denoted as:

(,, )S





min

'' ''

max

min

max

()

(,, )()

()

pmf

fSfs









 











hx εhs

xxh xεhs

hx hs





)

(6)

A set called “filter” is maintained at each iteration,

which is defined as:



111 111

((,,),(,,))

((, ,),(,,))

kkkkkkk

kkk kkk

FSS





 





xx xx

(7)

is called if the condition

111

(, ,)(,,)

kkk kkk

or SS











xx xx

xxxx (8)

is satisfied.

The iteration point is accepted only

if objective function and infinite norm of

the constraints set meet (8).

111

(, ,

kkk



(,, )S





(,, )Sxx



At iteration , the “filter” is updated according to

(9).

k

(,):(,,)

(, ,)

kkk

FFand S

 

























xx (9)

In order to solve (5), the Lagrangian function is repre-

sented as:

112 2

'T '''

min

'T '''

max

Tmin

Tmax

(ln ln

ln ln

lnlnln)

() ()

(() )

y() y(

il iu

sl suf

slsl su

sss

Ss S





 



















ygxy g x

yhxεhs

yhxh s

y(() )

fpmf

fSfs





x

(10)

where and

,,,,,,y,y

lu l uslsu

yyyyyyy

are the

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 [11].

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

Z. FAN ET AL.

696

Table 1. Comparisons of optimal results and the initial val-

ues.

Initial

values Optimal

results

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

traditional model.

fuzzy

model traditiona1

model

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

the system.

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.

5. Conclusions

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.

REFERENCES

[1] H. Yuan, G. G. Xu and J. Y. Zhou, “A Reactive

Power/voltage Optimization Based on Fuzzy Linear Pro-

gramming,” Power System Technology, Vol. 27 , No. 12,

2003, pp. 42-45.

[2] F. R. Tu and X. R. Wang, “Fuzzy Modeling for Power

System Reactive Power Optimization,” Power System

Protection and Control, Vol. 38, No. 13, 2010, pp. 46-49.

[3] Y. N. Li, L. Z. Zhang and Y. H. Yang, “Reactive Power

Optimization Under Voltage Constraints Margin,” Pro-

ceedings of the CSEE, Vol. 21, No. 9, 2001, pp. 1-4.

[4] K. Tomsovic, “A Fuzzy Linear Programming Approach

to the Reactive Power/Voltage Control Problem,” IEEE

Trans on Power Systems, 1992, Vol. 7, No. 1, pp.

287-293. doi:10.1109/59.141716

[5] M. B. Liu, Y. Cheng and S. H. Lin, “Comparative Studies

of Interior-point Linear and Nonlinear Programming Al-

gorithms for Reactive Power Optimization,” Automation

of Electric Power Systems, Vol. 26, No. 1, 2002, pp.

22-26.

[6] M. B. Liu, S. K. Tso, Y. Cheng, “An Extended Nonlinear

Primal-dual Interior-point Algorithm for Reactive-power

Optimization of Large-scale Power Systems with Discrete

Control Variables,” IEEE Trans on Power Systems, Vol.

17, No. 4, 2002, pp. 982-991.

doi:10.1109/TPWRS.2002.804922

[7] J. D. Xu, X. Q. Ding, Z. C. Qin Zhencheng, et a1., “A

Nonlinear Predictor-Corrector Interior Point Method for

Reactive Power Optimization in Power System,” Power

System Technology, Vol. 29, No. 9, 2005, pp. 36-40.

[8] K. Pan, X. S. Han and X. X. Meng, Solution Principles

Study of Nonlinear Correction Equations in Primal-dual

Interior Point Method for Reactive Power Optimization,”

Power System Technology, Vol. 30, No. 19, 2006, pp.

59-65.

[9] U. Michael, S. Ulbrich, L. N. Vicente, A Globally Con-

vergent Primal-dual Interior-point Filter Method for

Nonlinear Programming,” Mathematical Programming,

Vol. 100, No. 2, 2004, pp. 379-410.

doi:1007/s10107-003-0477-4

[10] W. Andreas and L. T. Bieglery, “Line Search Filter Meth-

ods for Nonlinear Programming: Motivation and Global

Convergence,” Yorktown Heights, USA: IBM T. J. Wat-

son Research Center, 2001.

Z. FAN ET AL.

697

[11] S. Yang, J. Y. Zhou, Q. Li, et a1., “An Interior-point Re-

active Power Optimization Based on Filter Set,” Power

System Protection and Control, 2011, Vol. 39, No. 18, pp.

14-19.

[12] Y. Y. Sun, G. Y. He and S. W. Mei, “A New Optimal

Power Flow Algorithm Based on Filter Interior Point

Method,” Advanced Technology of Electrical Engineering

and Energy, Vol. 26, No. 2, 2007, pp. 29-33.