loss minimization-_euro.journ.of sc.res

7/28/2019 Loss Minimization-_Euro.journ.of Sc.res.

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LOSS MINIMIZATION IN OPTIMAL POWER FLOWby

AHIAKWO C.O. and IGWEH C.O.Department of Electrical and Computer Engineering

Rivers State University of Science and TechnologyNkpolu-Oroworukwo, Port Harcourt.

ABSTRACTLoss minimization in optimal power flow (OPF) is the minimization of the

losses on the line due to current heating. This is done by selecting andaugmenting the already power solution from the cost minimization done before

on the system.

This paper has a set of algorithm for the analysis of congested powersystems, which will provide for the most efficient operational cost with

minimum line losses. The algorithms are simulated in the computer usingMATLAB and convergence is achievable at ranges.

This paper therefore is geared towards developing suitable OPF

algorithms to solve this congestion management problem.

Keyword:- Optimal Power flow (OPF), control variables, congestionmanagement, line losses

INTRODUCTION

Optimal Power Flow (OPF) is a generic term that describes a broad class ofproblems. It determines optimal control variables and system quantities for efficientpower system planning and operation.[1] When we talk of power system

optimization, it has to always relate to optimal power flow, which is simply amathematical model for the calculation of power flow in a power system,

taking into account the line losses, lagging or leading power factor effect,regular disturbance due to switching etc.

For over one hundred years, the electric power industry in nearly

every country worldwide operated as a regulated industry. But in ourown country, Nigeria the difference is the case, since the government is

the sole operator of the power system in operation.

Many methods have been used in the computation of the operationand planning of a power system, some of which are dealt with in thiswork. There is a full analysis of a case study using the power loss

minimization equation(2).The need for a non-stop power supply, made it very necessary to have

a very effective computational method which will allow both the additionof new loads/ generator input and proper running of any number of sub-stations without exceeding the transmission limits.

THEORETICAL ANALYSIS

3.1 RESEARCH GOAL
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The need for the control of the ever-growing demand of powervia congestion management could be analyzed in this work by

formulating the following problems which are the usual controlvariables in the following problems which are the usual control

variables in loss minimization.(a) Generator bus voltage magnitudes(b)

Trans former tap ratios(c) Switch able shunt capacitors and inductors

(d) Phase-shifter anglesIn the formulation of loss minimization, generator voltages

and transformer tap ratios are used as control variables.During the optimization, transformer Tap ratios are treated as

continuous variable, and are later adjusted to the nearestphysical tap position and reiterated holding the taps at the

adjusted values which is justified based on the small stepsizeusually found in transformers (3).

The following assumptions are made in the formulation ofthe loss minimization objective.

1. Loss minimization is done after a cost minimization,holding the active power generations excluding the slackbus generation at their optimal values.

2. Generator bus voltages and transformer tap ratios are usedas control variables, shunt reactance s and phase shifter

angels (where available) are held at nominal values.3. During the optimization the transformer tap ratios are

treated as continuous variables, after which they are

adjusted to the nearest physical tap position andreiterated.

4. Restrictions made on the real and imaginary components

of the complex voltage across the lines control the currentflow approximately.

5. Contingency constraints are neglected.MATHEMATICAL MODEL FOR LOSS MINIMIZATION

The objective function to be minimized is given by the sum ofall line losses.

(1)Where Pik ,the individual line losses is given in terms of voltagesand phase angles as

Pik=gk (v i2+vj2-2v ivj cos (i- j)) k=1,.., Nl (2.)Which can be transformed to equivalent rectangular form as

Pik=gk (ei2+fi2+ej2+fj2)-2(eiej+fifj))(3.)K=1,..,Nl

Hence, simplified to be,Pik=gk ((ei-ej)2+(fi-fj)2) K=1,.N (4)

Pik

Nl

PlK=1S=

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Therefore the objective function can be written asgk ((e i-ej)2+(fi-fj)2),, ..(5)

The equality constraints are given byPi-Pgi + Pdi = 0 i = 1,...,Nb ..(6)

i genQiQgi + Qdi = 0 i = 1,...,Nb ..(7)

i

gen-synchWhere Pi =

Nb

j 1=S (GijeiBijfi) + fi

Nb

j 1=S (Gijfj +Bijei) .(8)

Qi + FiNb

j 1=S (GijeiBijej) ei

Nb

j 1=S (Gijfj +Bijei)

i = 1,...,Nb..(9)

andkp1kp2 = 0 kp1 = 1,,Npi

kp2 =- 1,.Npikp1 = kp2 ..(10)

The inequality constraints are

vimin Vi vimax i = 1,...,Nb ..(11)

Pgimin Pgi Pgimax i slack bus .(12)

Qgimin Qgi Qgimax i = 1,.,Ngq ..(13)

iminiimax i =1,,Ni ..(14)

-kei . Iimax eiej kei . Iimax l = 1,,N1 ...-(15)i,j defined by l

-kfi . Iimax fi fj kfi . Iimax l = 1,,N1 .(16)i,j defined by 1.

It should be noted that equation (2.24) is not linear in therectangular formulation, hence an approximate linear form is

Keimin . Vimax ei keimax Vimax i = 1,,Nb ..(17)

Kfimin . Vimin fi kfimax . Vimax i = 1,,Nb (18)V in exact form given as

Vi2 = ei2 + fi2 i = 1,,Nb (19)

The transformer tap ratio controls the optimization via the admittancematrix. The relationship is as follows

Gli = )~(2 Li

m

Li

LiNL

gg

gLii

++S

te

i = 1,,Nb.. (20)

Nl

PlK=1S=
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RESULTS

Using data from the Nigeria Power System.

The following results are enclosed:

==========================================================================

System Summary==========================================================================

P (MW) Q (MVAr)------------- -----------------

Buses 20 Total Gen Capacity 335.0 -95.0 to 405.9Generators 6 On-line Capacity 335.0 -95.0 to 405.9Committed Gens 6 Generation (actual) 159.7 -850.5Loads 14 Load 143.3 82.6

Fixed 14 Fixed 143.3 82.6Dispatchable 0 Dispatchable 0.0 of 0.0 0.0

Shunts 1 Shunt (inj) 0.0 0.2Branches 22 Losses (I^2 * Z) 2072.71 -8728.84Transformers 0 Branch Charging (inj) - 855.9Inter-ties 0 Total Inter-tie Flow 0.0 0.0Areas 1

Minimum Maximum------------------------- --------------------------------

Voltage Magnitude 0.114 p.u. at bus 15 16.725 p.u. @ bus 17Voltage Angle -171.28 deg at bus 13 167.85 deg @ bus 10P Losses (I^2*R) - 19097.22 MW @ line 12-17Q Losses (I^2*X) - 7895.73 MVAr @ line 12-17

================================================================================| Bus Data |================================================================================Bus Voltage Generation Load# Mag(pu) Ang(deg) P (MW) Q (MVAr) P (MW) Q (MVAr)

----- ------- -------- -------- -------- -------- --------1 1.000 0.000 -5.98 -64.10 - -2 1.000 -34.772 60.97 104.03 21.70 12.703 1.000 17.987 21.59 -274.49 2.40 1.20

4 1.000 150.945 26.91 -484.30 7.60 1.605 1.000 -146.875 19.20 -64.07 - -6 1.000 -170.527 37.00 -67.54 - -7 1.013 -27.107 - - 22.80 10.908 2.212 -148.473 - - 30.00 30.009 0.627 -33.653 - - - -

10 0.136 167.850 - - 5.80 2.0011 0.998 -0.192 - - - -12 8.175 119.762 - - 11.20 7.5013 0.143 -171.276 - - - -
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14 0.985 152.678 - - 6.20 1.6015 0.114 41.845 - - 8.20 2.5016 8.177 131.637 - - 3.50 1.8017 16.725 54.964 - - 9.00 5.8018 0.858 -30.515 - - 3.20 0.9019 1.447 -67.428 - - 9.50 3.4020 2.227 -51.645 - - 2.20 0.70

-------- -------- -------- --------Total: 159.69 -850.46 143.30 82.60

================================================================================| Branch Data |================================================================================Brnch From To From Bus Injection To Bus Injection Loss (I^2 * Z)

# Bus Bus P (MW) Q (MVAr) P (MW) Q (MVAr) P (MW) Q (MVAr)----- ----- ----- -------- -------- -------- -------- -------- --------

1 11 1 0.00 -0.00 0.00 -1.10 0.002 -0.00

2 4 1 -35.61 -52.22 -5.98 -63.00 -41.593 -114.343 5 13 8.34 -61.28 2.36 8.03 10.701 -50.194 6 13 15.82 -65.53 -2.14 9.27 13.678 -55.405 13 10 -0.94 -0.89 1.12 0.39 0.179 -0.416 13 18 6.32 -9.23 -5.52 -67.90 0.800 -75.817 18 8 -158.57 -239.91 350.84 -660.56 192.274 -887.348 5 6 -25.39 -13.42 27.78 -2.01 2.389 -11.139 5 9 67.23 18.44 28.04 -34.38 95.271 -14.0210 9 4 72.99 -89.36 127.36 -133.58 200.350 -219.8511 4 12 -710.39 -457.78 6900.78 105.70 6190.387 -270.2812 12 16 3202.46 5718.49 -1946.22 -6368.94 1256.243 -537.4513 12 17 4959.47 -10063.57 14137.75 17480.74 19097.222 7895.73

14 2 7 23.37 2.19 -22.80 -10.90 0.573 -3.1815 5 2 -99.79 -7.62 -43.95 89.14 -143.736 82.9516 4 14 6.34 1.09 -6.20 -1.60 0.136 -0.1717 4 19 -187.09 156.59 -68.87 342.85 -255.958 508.8118 9 19 -42.91 -83.84 176.80 31.58 133.891 12.5619 9 15 268.82 -12.11 -14.28 -46.77 254.541 -58.3920 15 19 9.29 -33.86 444.94 -32.99 454.231 -64.3121 19 20 54.70 -168.95 -150.89 219.82 -96.183 57.3722 17 3 -27114.70 -14710.06 1822.01 -275.69-25292.688 -14923.97

-------- --------Total: 2072.709 -8728.84

>> | System Summary |

================================================================================

How many? How much? P (MW) Q (MVAr)--------------------- ------------------- ------------- -----------------Buses 20 Total Gen Capacity 335.0 -95.0 to 405.9Generators 6 On-line Capacity 335.0 -95.0 to 405.9Committed Gens 6 Generation (actual) 159.7 -850.5Loads 14 Load 143.3 82.6

Fixed 14 Fixed 143.3 82.6
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Dispatchable 0 Dispatchable 0.0 of 0.0 0.0Shunts 1 Shunt (inj) 0.0 0.2Branches 22 Losses (I^

INTERIOR POINT RESULTS

[ e1, e2, e3, e4, e5, e6, e7, e8, e9, e10, e11, e12, e13, e14, e15, e16, e17, e18, e19,e20]

1.200, 0.9000,0.9000,0.9000,0.9000,0.9000,0.9000,0.9000,0.9000,0.9000,0.90000.9000 , 0.9000,0.900, 0.900 , 0.9000, 0.9000, 0.9000, 0.9000, 0.9000

[ f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12, f13, f14, f15, f16, f17, f18, f19, f20]

0 , 0.1380, 0.0567,-0.1512,-0.0796,0.2382,-0.2270,-0.0255,0.1682,0.0066,0.01070.0876, -0.1800,-0.2157,0.2165,0.1004,0.1621,-0.245, 0.1809,-0.2011

slack variable-0.000, 4.9349,-0.0000,-0.0000,0.0000, -0.000, 0.0000,0.0000,-0.0000,0.0000,-0.000,

0,0.0000,0.0000,-0.0000,0.0000,0.0000,-0.0000,0.0000,0.0000,-0.0000,-0.0000,0.0000, 0.0000,0,-

0.0000, 0.0000,-0.0000,-0.0000,-0.0000,-0.0000,0.0000, 0,-0.0000,0.0000,-0.0000,-0.000, -0.0000,0.0000,-0.0000, 0.0000

Total Power loss=0.7100

CONCLUSION


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This paper has identified a class of decompositions and hasselected a super-hybrid that is well suited to problems. The algorithm

have the following attractive features.

(i) Nonlinear objective functions and constraints can beaccommodation directly, without tricks or intricate manoevres.

(ii) It is convenient to chooses these variables to be the same as the

ones calculated in a load flow program.(iii) The algorithm will force convergence from profoundly infeasiblestarting points.

(iv) The algorithm has been proved to be fast in tests on largeproblems.

(v) To calculate the sensitivity of the optimal solution to parametervariations, the algorithm has all the required information.

REFERENCE

1. Momah. J.A., Guo S. X., Ogbuobiri C.E. and Adapa R. (1993) The

Quadratic Interior point method for solving power systemproblems IEEE Trans. On power system, Vol. 9.

2. British Britannica Encyclopedia (1980)

3. Burechett R. C., Happ H.H. and Wirgau K.A. (1982) Large ScaleOptimal Power Flow, IEEE Transactions on Power Apparatus and

System, Vol. PAS 101, (No, 10) 3722-3732.

4. Dommel H.W. and Tinney W.F 91968) Optimal Power Flowsolutions IEEE Transactions on Power Apparatus and Systems,

Vol. PAS 87, 1866-1876.

5.Momoh J.A. (2001) Electric Power System Applications ofOptimization. Marcel Deker Inc. New York 339-399.

6. Chukwu, Nwodo, and Ahiakwo (2005) Modelling of Phase ShiftingTransformer for Load Flow Studies. AMSE Journals .

7. P.E.GILL,w.Murray,M.A Saunders,andM.H.Wright, Quadratic

Programmming-Based Methods for Large scale Nonlinearly constrained

Optimization,Technical Report SOL 81-1,Systems Optimization

Laboratory,Stanford University,January 1981.

8. http://www.pserc.cornell.edu/matpower/matpower.html

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