Fuzzy OPF incorporating UPFC

B. Venkatesh, M.K. George and H.B. Gooi

Abstract: A new optimal reactive power flow (ORPF) method is proposed which considers theinclusion of unified power flow controllers (UPFC). The modelling and inclusion of UPFC in thesolution of power flow equations is presented. The ORPF problem is formulated as a fuzzyoptimisation problem considering the objectives of minimising system transmission loss andobtaining the best voltage profile. The fuzzy formulation of the ORPF problem is solved using anEP algorithm. The proposed method is applied on the 6-bus and 57-bus IEEE test systems and ona 191-bus Indian electric power system. The results demonstrate the applicability of the method.

1 Introduction

Optimal reactive power flow has been well researched and aplethora of methods have been reported in the literature [1,2]. Fuzzy formulation of the ORPF has been proposed anddocumented [3, 4]. This offers several advantages for powersystems that have low voltages and lack of reactive power toalleviate the same [5, 6]. Static compensation schemes havebeen used for the last couple of decades [7, 8]. Withadvances in the field of power electronics, full form UPFCdevices are in use [9, 10]. Inclusion of the unified power flowcontroller (UPFC) in the solution of ORPF poses fewchallenges. Numerous models of UPFC, their applicationsand different steady-state optimal power flow methods havebeen extensively described in the literature [11–16].

A robust ORPF method has to account for all theelements that affect the flow of reactive power. Theseinclude continuous variables like voltage magnitude ofgenerators, reactive power output of static VAr sources andsetting of UPFC devices apart from discrete variables liketransformer taps. This problem is of constrained mixedinteger nonlinear programming nature and is fraught withhigh combinatorial complexity. Artificial intelligence-basedsearch methods are amenable for solving such problems.

2 UPFC model and incorporation into FDPF

In this Section, a simple injection model is presented. Asimple circuit model of the Mth UPFC of a power system isshown in Fig. 1. It has shunt and series connected voltagesources VSM+dSM and VUM+dUM, respectively. They areinterconnected such that the real power is absorbed in one,transferred through a DC coupling as PDCM, and given outby the other, in the loss-less case. An equivalent circuit isshown in Fig. 2. The line reactance of the circuit betweenbuses i and j is xLij. In the definition above and in thefollowing discussion line resistance is neglected. It isassumed that xSM would consume negligible vars. dSM

may be appropriately set such that real power absorbed bythe shunt source equals the power given out by the seriessource. A complex number Vi+di is interchangeablyexpressed as Vi. Defining the voltage at U as ViU ¼ Vi þVUM and xij¼ xUM+xLij, the power flowing out of bus iinto the series branch of the UPFC may be expressed asbelow:

S0ij ¼ P 0ij þ jQ0ij ¼ V i� ViU � Vj� �

jxij

� ��ð1Þ

The power flowing in the circuit from bus i accounting forUPFC through the point U (Fig. 2), retaining direction offlow is expressed as below:

Sij ¼ Pij þ jQij ¼ ViU �ViU � Vj� �

jxij

� ��ð2Þ

The power flowing from bus j through the connected circuitmay be expressed as below:

Sji ¼ Pji þ jQji ¼ Vj �Vj � ViU� �

jxij

� ��ð3Þ

As shown in Fig. 3, the total power emanating from bus i Sij

goes partly through the line and partly through the UPFC.To model this circuit within the p-model framework oftransmission lines, the power injection model is presented inFig. 4. The elements of Fig. 4 are detailed below using (2)and (3). It may be borne in mind that the shunt elements ofthe p-model of transmission lines are reflected as before in

xSM

VSM � SM

VUM ∠

∠

� UM

xUM

PDCM

Fig. 1 UPFC circuit model

UVi ∠ �i Vj ∠ �jbu

s i

VSM ∠ �SM

VUM ∠ �UM

xSM

PDCM

xUM xLij bus

j

Fig. 2 UPFC between two buses

B. Venkatesh and M.K. George are with the Faculty of Engineering andTechnology, Multimedia University, Bukit Beruang, 75450 Melaka, Malaysia

H.B. Gooi is with the School of Electrical and Electronic Engineering, Block S1,Nanyang Technological University, Nanyang Avenue, 639798, Singapore

r IEE, 2004

IEE Proceedings online no. 20040611

doi:10.1049/ip-gtd:20040611

Paper first received 1st October 2003 and in revised form 26th February 2004.Originally published online: 2nd November 2004

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 5, September 2004 625

the bus admittance matrix and are not shown in thediagram.

The complex power at bus i is as below:

Sij ¼ Pij þ jQij ¼ SLij þ SUijM ð4Þ

where

SLij ¼ Vi �Vi � Vj

jxij

� ��ð5Þ

SUijM ¼ PUijM þ jQUijM

¼ V i �VUM

jxij

� ��þVUM �

ViU � Vj

jxij

� ��ð6Þ

The complex power at bus j is as below:

Sji ¼ Pji þ jQji ¼ SLji þ SUjiM ð7Þ

where

SLji ¼ Vj �ðVj � ViÞ

jxij

� ��ð8Þ

SUjiM ¼ PUjiM þ jQUjiM ¼ �Vj �VUM

jxij

� ��ð9Þ

The traditional equations to compute the net power flowfrom a bus into the connected lines in the polar form FDPFalgorithm are altered as below to account for the UPFCdevices connected to the ith bus:

Pi ¼ Vi �XN

j¼1VjYij � cosðdi � dj � yijÞ� �

þXM2i

PUijM ð10Þ

Qi ¼ Vi �XN

j¼1VjYij � sinðdi � dj � yijÞ� �

þXM2i

QUijM ð11Þ

where N is the number of buses and Yijffyij is the (i,j)thelement of the bus admittance matrix. In the computationof the bus admittance matrix, B0 and B00 matrices, one needsto include the UPFC series reactance values as given before(1). Taking set values of the voltage phasor of the UPFCdevices, the values of the terms

PM2i PUijM and

PM2i QUijM

may be computed in all the iterations. This allows inclusionof the UPFC devices in the polar form of the FDPFalgorithm [17].

3 Optimal reactive power flow

Optimal reactive power flow changes the setting of thevariables that affect the flow of reactive power. The flow isaltered in such a way that the minimum system transmissionloss and the best voltage profile are obtained. Themathematical problem is as stated below.

3.1 Mathematical problem statementThe objectives are to minimise the system transmissionloss and obtain the best voltage profile. The problemvector is expressed as X ¼ ½V t

G QtC T tV t

UdtU � where VG is a

vector of voltage magnitudes at all the generator buses; QC

is a vector of output of shunt reactive power sources; T is avector of taps settings of all the controllable transformersand VU+dU is a vector of voltage phasors of UPFCdevices.

The problem of optimal siting minimises the systemtransmission loss whilst obtaining the best voltage profile.The mathematical problem statement is as below:

MinimiseXN

j¼1PiðX ;Y ; dÞ ð12Þ

subject to the constraints on:

FðX ;Y; dÞ ¼ 0: ðpower flow equationÞ ð13Þ

Xmin � X � Xmax ðcontrol vectorÞ ð14Þ

Ymin � Y � Ymax ðdependent vectorÞ ð15Þwhere Pi in (12) is the sum of real powers flowing from bus iinto the connected lines and N is the number of buses. d in(12) is the vector of bus phase angles. The dependent vectoris formed as follows:

Y V tLQG t� �

where VL is a vector of load bus voltage magnitudes; QG isa vector of reactive power generation at generator buses;Xmin/Xmax and Ymin/Ymax in (14)–(15) are lower/upperlimits on X and Y, respectively.

For a given solution, an index that quantifies the extentof voltage violation in the power system is defined below:

VDI ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNVB

i¼1ðVLi � VLiLIM Þ2

N

vuuutð16Þ

where NVB is the number of buses that violate theprescribed voltage limits and VLiLIM is the upper limit ofthe ith load bus voltage if there is a upper limit violation orlower limit if there is a lower limit violation.

3.2 Fuzzy EP methodThe above problem is solved using the fuzzy EP technique.The data structure used to represent the problem vector inthis method of solution is shown in Fig. 5.

VG T

�UQC VU

Fig. 5 Data structure for the EP solution string

UPFC

u

PDCM XSM

bu

s i

bus i

bus

j

VUM ∠ �UM

XUM XLij

VSM ∠ �SM

Sij SijSji

Vj ∠ δj Vi ∠ �i

′

Fig. 3 Power flow between two buses

bus

i

bus

j

Xij

SLji

SUjiM SUjiM

SLjiVi ∠ �i

Vj ∠ �j

Fig. 4 Power injection model

626 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 5, September 2004

3.2.1 Fuzzy modelling of the objectives: Asshall be seen in an EP algorithm, a set of solutions [X1, X2,X3,y,XNEP] is generated at first. NEP is the number ofsuch generated solutions. Thereafter the algorithm generatesanother set of an equal number of solutions. Then the EPalgorithm chooses NEP best solutions from amongst thetwo sets. This process continues until the optimum isreached. Thus, the evaluation of the objective function foreach of the 2*NEP solutions and their ranking is done ineach EP iteration. Keeping in mind these 2*NEP solutions[X1, X2, X3,y,X2NEP], the fuzzy objective function isdeveloped as below. The paper addresses two objectives,which are minimisation of system transmission loss andobtaining the best voltage profile. Their modelling isoutlined below.

The first objective is the minimisation of the systemtransmission loss is defined as function f(X):

f ðXÞ ¼XN

i¼1PiðX ;Y ; dÞ ð17Þ

Consider a fuzzy set that is defined as

F ¼ f½f ðXÞ; mf ðf ðXÞÞ�; f ðXÞ

2 ½set of all permissible values�g: ð18ÞUpon evaluating all the solutions [X1, X2, X3,y,X2NEP]using (18), one gets the corresponding values as [f(X1), f(X2),f(X3),y,f(X2NEP)], which forms the set of permissible valuesof (19). The membership function mfðfðXÞÞ in (19) is thendefined as below:

mf ðf ðXÞÞ ¼fmax � f ðXÞfmax � fmin

ð19Þ

where fmax and fmin are the maximum and minimum valuesamongst the set of permissible values of ‘f(X)’ in (18). Thefunction is graphically presented in Fig. 6.

The second objective is to obtain the best voltage profile.An index that quantifies the extent of voltage violation isdefined in (16). A solution X would yield the VDI valuev(X). Considering all the solutions [X1, X2, X3,y, X2NEP]and evaluating them using (16), one gets the correspondingvalues as [v(X1), v(X2), v(X3),y,v(X2NEP)]. Using thesevalues to form the set of permissible values for v(X), a fuzzyset is defined with a membership function mvðvðXÞÞ as:

V�¼ v; mvðvðXÞÞ½ �; vðXÞ 2 set of all permissible values½ �f g:

ð20ÞThe membership function in (20) is defined as

mvðvðXÞÞ ¼vmax � vðXÞvmax � vmin

� 0:2þ 0:8 ð21Þ

The terms vmax and vmin in (21) assume the maximum andminimum values amongst the set of permissible values of‘v(X)’. The function is graphically presented in Fig. 7.

The overall fuzzy objective function considering the twoobjectives is developed in this Section. It requires a methodto combine the two objectives using a fuzzy intersectionoperator. In this paper, the simple product is chosen as theintersection operator. Thus the overall objective function isdefined as below:

mxðXÞ ¼ mf ðf ðXÞÞ � mvðvðXÞÞ ð22Þ

The objective function defined above quantifies thesatisfaction with the solution string X and is used toevaluate a solution Xj from amongst 2*NEP solutions X1 toX2NEP in the evolutionary programming algorithm pre-sented in the following Section.

3.3 Fuzzy EP algorithmAn illustrative flowchart for the algorithm is presented inFig. 8. The steps are as below.

1. Randomly generate NEP combination of solutionvectors X1, X2,yXNEP.

2. Set iteration count K¼ 1.

3. Evaluate the objective function value for each of thecombinations Xj for j¼ 1 to NEP.

4. Generate NEP more solution strings XNEP+1,XNEP+2,y,X2NEP through a random process:

XNEPþj ¼ b2r � rM

rMðXmax � XminÞ þ X j ð23Þ

where

� f ,

(f (

x))

1.0

0

f , xf min f max

Fig. 6 Membership function of transmission loss minimisationobjective

� v (

v (X

))

1.0

0.8

0 v (X)

V min V max

Fig. 7 Membership function of voltage violation minimisationobjective

select the best NEP solutions from X1

to X 2NEP (22) and assign to X1 to X NEP

no

Yes

start

evaluate NEP solutions X1 to X NEP (22)

generate NEP solution X to

X2NEP (23) and evaluate them (22)

max iterations?

select the best solutions from X1 to XNEP

end

K=K

+1 NEP+1

Fig. 8 Flowchart for fuzzy EP algorithm

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 5, September 2004 627

b is a factor appropriately chosen (in this study, it hasbeen chosen to be 0.001 to 0.1 for the variables of X),

r is a random number between 0 to rM andXmin and Xmax are the minimum and maximum values of

X as defined earlier.

5. Evaluate the newly generated solution vectors XNEP+1,XNEP+2,y,X2NEP.

6. Choose the best NEP solution vectors amongst the set of2NEP solution vectors X1, X2,y,X2NEP and designate thechosen set of NEP solution vectors as X1, X2,y,XNEP.

7. Increment the iteration count, K¼K+1.

8. If Komaximum go to Step 4.

9. Choose the best from the NEP solutions X1, X2,y,XNEP.

The flowchart indicates a maximum number of iterations. Itis required to terminate the solution process. It must bepointed out that this evolved from experience for a givensystem. The number of iterations for each test system ispresented in Section 4. In a similar fashion, the value of b asdiscussed in Step 4 above is also arrived at after several trialruns. The number of solutions taken at a time is NEP. Inthis study, NEP was taken as 5. Other alternatives may alsobe attempted.

4 Results

The proposed algorithm was tested on the modified IEEE6-bus and 57-bus systems and on a 191-bus Indian system.The test results are reported below.

4.1 IEEE 6-bus systemThe data assumed for the 6-bus system is given in Appendix7. The modified system has two UPFC devices. The systemwas optimised considering several combinations of objec-tives. The bar chart in Fig. 9 gives the results with all thesecombinations. The letters L, Q and V in Fig. 9 indicateobjectives of system real power transmission loss minimisa-tion, system reactive power transmission loss minimisationand VDI minimisation, respectively. On analysis, theobjective combination of system real power transmissionloss minimisation and VDI minimisation were chosen to bethe most effective. The convergence pattern of the algorithmwith the chosen objectives considering a maximum of 1000iterations is presented in Fig. 10. The algorithm took 31seconds for execution.

4.2 IEEE 57-bus systemThe algorithm was tested on the IEEE 57-bus test system.TwoUPFC devices were included in the system. The systemtransmission loss reduced from 31.37 MW to 28.45 MW

while the VDI reduced from 0.088 to 0.046 in 1000iterations. The results of the study are summarised inFig. 11. The algorithm took 243 seconds for execution.

4.3 191-bus Indian systemThe Indian system has 20 generators, 55 transformers, 200transmission lines, 10 fixed shunt reactive power sources and12 variable reactive power sources. The system was modifiedto include two UPFC devices. The results show that optimalscheduling effectively brings down the system transmissionloss. Figure 12 shows the reduction of system transmissionloss from 81.15 MW to 76.03 MW in 1000 iterations. Thesystem inherently has large voltage violations that cannot becompletely cleared and it is reduced from 0.0066 to 0.0017.The algorithm took 422 seconds for execution.

5 Conclusions

This paper presents a new optimal reactive power flowmethod. It incorporates the UPFC device in the ORPF.The equations for the inclusion of the UPFC devices arepresented with an appropriate circuit model. A mathema-tical model of the ORPF problem is presented along with afuzzy optimisation formulation. The solution of the fuzzyORPF problem through an evolutionary programmingmethod is presented with a flowchart. The method is testedon the IEEE 6-bus and 57-bus test systems and their resultsare presented. Comparison of the algorithm using different

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

VL QLV QV QL Q L V0

0.5

1.0

1.5

2.0

2.5

3.0

Q-losses VDI P-losses

P-lo

sses

Q-lo

sses

and

VD

I

objectives chosen

Fig. 9 Comparison of performance of different combination ofobjectives for the 6-bus system

1.9

2.4

2.9

3.4

3.9

1 251 501 7510.1

0.15

0.2

loss

es in

MW

VD

I

generations

loss

VDI

Fig. 10 Convergence characteristics of the 6-bus system

loss

es in

MW

VD

I

VDI

losses

generations

32

31

30

29

281 251 501 751

0.09

0.06

0.03

Fig. 11 Evolution in the 57-bus system

loss

es in

MW

VD

I

80

78

76

generations1 251 501 751

0.0061

0.004

0.002

0losses

VDI

Fig. 12 Evolution in the 191-bus system

628 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 5, September 2004

objectives was simulated and the analysis is also presented.A test on a 191-bus Indian system is also conducted and itsresults are presented. It is concluded that the methodperforms well with the different test cases. The choice oftuning parameter like b is made from experience.

6 References

1 Mamundur, K.R.C., and Chenoweth, R.D: ‘Optimal control ofreactive power flow for improvement in voltage profile and for realpower loss minimization’, IEEE Trans. Power Appar. Syst., 1981, 100,(7), pp. 3185–3194

2 Sadasivam, G., and Abdullah Khan, M.: ‘A fast method for optimalreactive power flow solution’, Electr. Power Energy Syst., 1990, 12, (1),pp. 65–68

3 Venkatesh, B., Sadasivam, G., and Khan, M.A.: ‘A new optimalreactive power scheduling method for loss minimization and voltagestability margin maximization using successive multi-objective fuzzyLP technique’, IEEE Trans. Power Syst., 2000, 15, (2), pp. 844–851

4 Venkatesh, B., Sadasivam, G., and Abdullah Khan, M.: ‘Fuzzy logicbased successive LP method for reactive power optimization’, Int. J.Electr. Machines Power Syst., 1999, 27, (10), pp. 1141–1160

5 Abdullah Khan, M., Kuppusamy, K., Sadasivam, G., and Venkatesh,B.: ‘Study on transmission system for power evacuation of Cuddalorethermal power project’. Consultancy reports on feasibility study, May1996

6 Venkatesh, B.: ‘Optimal reactive power planning and schedulingthroughmulti-objective fuzzy LP and ANNmodels’. PhD Thesis, 2000

7 Miller, T.J.E.: ‘Reactive power control in electric power systems’,(Wiley, New York, 1982)

8 Song, Y.H., and Johns, A.T.: ‘Flexible AC transmission (FACTS)’(The Institution of Electrical Engineers, London, 1999)

9 Gyugyi, L.: ‘Control of shunt compensation with reference to newdesign concepts’, IEE Proc. C, 1981, 128, (6)

10 Gyugyi, L.: ‘Unified power-flow controller concept for flexible ACtransmission systems’, IEE Proc., 1992, 139, (4), pp. 323–331

11 Keri, A.J.F., Lombard, X., Edris, A.A., Mehraban, A.S., andElriachy, A.: ‘Unified power flow controller (UPFC): modeling andanalysis’, IEEE Trans. Power Deliv., 1999, 14, (2), pp. 648–654

12 Fuerte-Esquivel, C.R., and Acha, E.: ‘Unified power flow controller: acritical comparison of Newton–Raphson UPFC algorithms in powerflow studies’, IEE Proc. Gener. Transm. Distrib., 1997, 144, (5),pp. 437–444

13 Xiao, Y., Song, Y.H., and Sun, Y.Z.: ‘Power flow control approach topower systems with embedded FACTS devices’, IEEE Trans. PowerSyst., 2002, 17, (4), pp. 943–950

14 Liu, J.Y., Song, Y.H., and Mehta, P.A.: ‘Strategies for handlingUPFC constraints in steady-state power flow and voltage control’,IEEE Trans. Power Syst., 2000, 15, (2), pp. 556–571

15 Li, N., Xu, Y., and Chen, H.: ‘FACTS-based power flow control ininterconnected power systems’, IEEE Trans. Power Syst., 2000, 15, (1),pp. 257–262

16 Gotham, D.J., and Heydt, G.T.: ‘Power flow control and power flowstudies with FACTS devices’, IEEE Trans. Power Syst., 1998, 13, (1),pp. 859–869

17 Stott, B., and Alsac, O.: ‘ Fast de-coupled load flow’, IEEE Trans.Power Appar. Syst., 1974, 93, (3), pp. 859–869

7 Appendix

See Tables 1–7

Table 1: Generator bus data

BUS PG (MW) PG limits (MW) QG (Mvar) V Sch. (p.u.)

Max Min Max Min

1001 – 50 0 100 �100 1.0

2002 – 80 50 100 �40 1.0

Table 2: Load bus data

BUS PD (MW) QD (Mvar)

2003 55 39

1004 0 0

2005 30 54

1006 50 15

Table 3: Transformer data

From To R (p.u.) X (p.u.) Tap

1004 2003 0 0.133 1.0

1006 2005 0 0.3 1.0

Table 4: Line data

From To R (p.u.) X (p.u.)

1001 1006 0.0100 0.5180

1001 1004 0.0300 0.3700

1004 1006 0.0000 0.4070

2005 2002 0.0600 0.6400

2002 2003 0.1000 1.0500

Table 5: Fixed shunt capacitor data

Bus Mvar

1004 19.0

Table 6: Adjust var source data in Mvar

Bus Max Min Step Switched in

2005 20 0 1.0 0

Table 7: UPFC data

On Line Vu (p.u.) du (Deg.) Xu (p.u.) Xs (p.u.)

3 0.0 0.0 0.02 0.0

4 0.0 0.0 0.02 0.0

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 5, September 2004 629