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MODIFIED ALGORITHM OF LOAD FLOW SIMULATION FOR LOSSMINIMIZATION IN POWER SYSTEMS
D. Lukman T.R. BlackburnSchool of Electrical and Telecommunication Engineering
University of New South Wales, KensingtonSydney, NSW-2052, Australia
e-mail : [email protected] / [email protected] paper focuses on the use of load flow to minimize the losses in an electrical power system. A Matlabload flow analysis program has been developed using a modified Newton Raphson algorithm based on aY-bus admittance matrix to determine the voltage level of the system bus. Losses are then calculatedusing B-loss coefficient formula and verified with the traditional I2R or differential power methods.Voltage control using either switched capacitor bank or load tap changer of transformer shall be done toimprove the voltage level whilst minimizing losses. A network of 5-bus system is used as the test case.Modeling of capacitor bank and tap changer are carried out and implemented in the program. Severalcase studies with different values of capacitances and tap settings of transformers are conducted todetermine the minimum losses.
1. INTRODUCTIONLoad flow deals with the flow of electrical powerfrom one or more sources to loads consuming energythrough available paths as commonly shown in a oneline diagram. Electric energy flow in a networkdivides among branches according to their respectiveimpedances until a voltage balance is reached inaccordance to Kirchoff’s Laws. The flow will shiftanytime the circuit configuration is changed ormodified, generation is shifted or load requirementschange. Information about these changes areimportant for industrial plants and electric utilityoperators to ensure efficient operation, minimizelosses, maintain reliability of service and coordinateprotective relaying for unexpected and emergencyconditions.
The losses in electrical network distribution as well asreal and reactive power flows for all equipmentconnecting the buses can be computed by means ofload flow simulation. The quantification andminimization of losses is important because it willdetermine the economic operation of the powersystem[5]. If we know how the overall losses occur,we can take steps to minimize them. Active powerlosses can be determined by various methods. It cansimply be computed as I2R. The power loss in a linecan also be calculated by taking the algebraic sum ofthe total power flows in either direction and the totalloss would be the sum of all the line losses[3].
Two methods to reduce the losses on the systemnetwork, which will be discussed in this paperinclude:(1) the change of transformer tap settings
(2) addition of different values of capacitor banks tocontrol reactive power distribution
System changes can then be simulated using a Newton-Raphson load flow computer program developed. Theresults of such changes are described.
The other method used in this work to calculate losses isusing B-losses coefficients, which express the transmissionlosses as a function of the outputs of all thegeneration/power plants.[2] Hypothetically, B-lossescoefficients can bias the operation of transformer tapchangers and/or capacitive reactive poweradjustment/FACTS devices inside the traditional Newton-Raphson load flow algorithm. Instead of having certaintarget voltages, we allow voltages to vary within a 5%tolerance of 1 per unit rating in order to obtain minimumlosses whilst improving the voltage level.
2. NEWTON-RAPHSON LOAD FLOWAND MODIFIED ALGORITHM
A power flow or load flow program computes the voltagemagnitude and angle at each bus in a power system underbalanced three phase steady state conditions. Once theyare calculated, real and reactive power flows for allequipment interconnecting the buses, as well as equipmentlosses are also computed.
There are two ways to represent the bus voltage equationsto solve the Newton-Raphson load flow problem. Mostreferences use rectangular coordinates of bus voltages.[2,3,4]. We prefer to use polar coordinates of bus voltageas used in [1] as it will be implemented in the Matlabsimulation program developed.
Consider first the non-linear equation y = f(x)
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( ) ( )( )
( )
( )
( )
=
=
=
=
=
=
xQ
QxP
xP
xQxP
xf
Q
QP
P
QP
y
V
VVx
N
N
N
N
N
N
.
.
.
.
;
.
.
.
.
;
.
.
.
.
2
2
2
2
2
2
δ
δ
δ
Eq. [2.1]
where V (voltage), P (real power) and Q (reactivepower) terms are given in per unit and δ (phaseangle) terms are in radians. The swing bus variablesδ1 and V1 are omitted from Equation [2.1] becausethey are already known. This equation shows that thereal and reactive powers at every bus except the slackbus can be expressed as a function of voltagemagnitude and phase angles.
The outputs of Newton-Raphson load flow algorithmwhich give the voltage levels at each bus, power flowin the line connecting two buses in either directionand line losses can be controlled by applying themodified algorithm to the original Newton-Raphsonalgorithm by means of the followings:
1. Changing transformer tap changers2. Additional of switched capacitor bank3. Application of B-losses formulaThese three methods will be outlined on the followingsections and implemented in the program developed.
3. APPLICATION OF TAPCHANGERS OF TRANSFORMER
Tap changing can control the reactive var flow sooptimum bus voltages can be determined and reduceline losses. A method of controlling the voltages in anetwork makes the use of transformers, the turns ratioof which may be changed. A schematic diagram ofan off-load tap changer is shown in Figure 1 (a)which requires disconnection of the transformer whenthe tap setting is to be changed. Many transformersnow have on-load tap changers as can be shown inFigure 1 (b).
Figure 1. (a) Off-load Tap changing transformer. (b)On-load tap-changing transformer with S1 and S2transfer switches, T centre-tapped reactor [4]
The presence of a tap changer allows manual or automaticchange of the turn ratio, and hence of the output voltage.Because of the impedance of the lines, the voltage at thereceiving end is slightly lower than the voltage at thesending end for most loads. In order to get a constant andrated voltage at the secondary of a ‘normally’ step-downtransformer automatically, an on load tap changer withadditional S1 and S2 transfer switches and R centre-tappedreactor is mounted at the primary side of it as shown inFigure 1 (b).
Assume that an automatic load tap changing transformer(OLTC) is connected to a particular bus to keep loadvoltage constant. It is possible to run the load flowprogram employing one tap setting and without mentioningthe magnitude of load voltage. If the voltage magnitudedetermined by the load flow program run exceeds the givenlimits, a new tap setting is then selected for the next run.In general, when the automatic tap-changing feature isemployed to represent a manual tap-changing transformer,the output of the load flow program will specify the tapsetting that gives the required bus voltage. The change oftap setting or turn ratio will change the system impedancematrix. Therefore, after each tap ratio adjustment, the Ybusadmittance matrix has to be adjusted.
Another means of taking into account the LTC transformeris to represent it by its impedance, or admittance,connected in series with an ideal autotransformer, as shownin Figure 2 (a). A model of a load tap changer needs to bedeveloped. An equivalent Π circuit, as shown in Figure 2(b) [12], can be developed in load flow studies. Thepresence of the tap changing transformer causes necessary
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modifications to the Newton-Raphson power flowtechnique. The elements of the equivalent Π circuit,can then be treated in the same manner as lineelements.
(b)
Figure 2. LTC transformer representations:(a) equivalent circuit; (b) equivalent ΠΠΠΠ circuit [12]
The following parameters of the equivalent Π circuit(Figure 2 (b)) in terms of admittances and off-nominal turns ratio T can be derived:[3]
Ty
A ij= ; ijyT
B
−= 11 ; ijy
TTC
−= 111
T = per unit turns ratio (i.e tap setting is +1.25% thenT = 1.0125) [10].
B and C can either be an inductor or capacitor. If wewant to increase the voltage of the transformer output,normally by taking positive tap, B is chosen asinductor and C is a capacitor and vice versa. Thepresence of a tap changing transformer changes theelements of both diagonal and off-diagonal of busadmittance matrix where the transformer is connectedbetween two buses. The Newton Raphson load flowsimulation is then rerun to obtain the required output.
4. APPLICATION OF SWITCHEDCAPACITOR BANKS
Capacitors are used in the transmission/distributionline to increase line loadability (maximum power
transfer) and to adjust the system voltage.[1,2] Shuntcapacitors are used to deliver reactive power and increasethe voltage magnitudes during heavy load conditions.Figure 3 shows the effect of adding a shunt capacitor bankto a power system bus. The system is represented by itsThevenin Equivalent at the node, where the capacitor willbe applied by closing the switch. With the switch open, thenode voltage Vt is equal to the Thevenin voltage Eth.
Figure 3. Effect of adding a shunt capacitor to a powersystem bus
From the power flow standpoint, the addition of a shuntcapacitor bank to a load bus corresponds to the addition ofa negative reactive load. The power flow programcomputes the increase in bus voltage magnitude along withthe small change in phase angle.
The additional capacitor is modeled with the susceptanceB. Given a required reactive power injection of Q, thesusceptance B can be calculated from Q = V2B. V is theinitial voltage of the bus where the shunt capacitor needs tobe installed.
The addition of capacitor bank changes the bus admittancematrix similar to the change of tap setting of transfomer.However, it will only affect the element of the diagonaladmittance matrix of the bus where the capacitor is added.
5. B-LOSSES CALCULATIONThe B matrix loss formula was originally introduced in theearly 1950s as a practical method for loss and incrementalloss calculations[7]. In this method, the results of powerflow is used to account for power transmission losses in thepower system. It is important in terms of the economicdispatch problem[11] to express the system losses in termsof active power generations only. This is commonlyreferred to as the loss formula or B-coefficient method.
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The simplest form of loss equation is George’sformula[2], which is:
∑ ∑= =
=k
m
k
nnmnmL PBPP
1 1
Eq. [5.1]
where PL is the power lossesPm, Pn is the power generation from allgenerator sources
The coefficients Bmn are commonly referred as theloss coefficients with the units of reciprocalWatt/MWatt. The B coefficients are not truly constantbut vary with unit loadings. A more general formula(Kron’s loss formula) is given by:
∑∑∑= ==
++=k
m
k
nnmnm
k
mmmLL PBPPBKP
1 1100 Eq. [5.2]
A linear term ΣBm0Pm and a constant KL0 have beenadded to the original quadratic equation. This showsthat losses depend on the active power generations Ponly. Bmn is called the loss coefficient and is given bya general expression:
( )( )( )∑
−=
kkknkm
nmnm
nmmn RNN
pfpfVVB σσcos
Eq. [5.3]
where σm, σn are phase angles of currents Im,InVm, Vn are voltages at bus m and nNkm, Nkn are current distribution factorspfm, pfn are power factors
For a simple system consisting of two generatingplants and one load as shown in Figure 4, losses PL interms of power output of the plants and B-lossescoefficient can be derived as[5]:PL = P1
2B11 + 2P1P2B12 + P22B22 Eq. [5.4]
where
( )21
21
11pfVRR
B ca +=
( )( )212112 pfpfVV
RB c=
( )22
22
22pfVRR
B cb +=
Figure 4. A simple radial system of twogenerators and one load bus
Knowing that the real power losses are a function ofgenerations and B-losses coefficient, varying thegenerations to fulfill the power demand will change thelosses accordingly. If B-losses are reduced, the losses canbe minimized. Since B-losses coefficients are functions ofresistances of every line, voltage magnitudes and powerfactors at each generation, phase angle of generatorcurrents and current distribution factors from eachgeneration, while resistances are physical properties ofelectrical equipment, which tend to be constant, improvingthe voltage at certain points will minimize B-lossescoefficients. Voltage control using either variable tapchanging transformers or capacitors as explained before arenecessary to improve the voltage levels and minimizelosses. These can be implemented in the load flowsimulation developed as explained in Section 6.
6. MATLAB SIMULATION ANDSENSITIVITY ANALYSIS
Matlab was chosen as the simulation tool for this researchbecause of the ease of manipulation of matrix structuresand inputs. It has in-built routines such as inverse function,abs function, and so on, graphing facilities to plotconvergence of load flow.
A single line diagram of a five bus system shown in Figure5 [1] was chosen to be tested. This system is selectedbecause it represents a typical meshed network where aload bus is supplied from alternative sources. It has typicalline ratings. The diagram has two generators of 400 MVAand 800 MVA power ratings, two step down transformersof 400 MVA and 800 MVA ratings, three long distancetransmission lines and one remote load bus. Overall, thissystem consists of five buses and five branches. Bus 1 isassigned as the slack/swing bus. Bus 3 is the voltagecontrolled PV bus while Bus 2,4 and 5 are load buses. Thebase of apparent power is 400 MVA. Vbase = 15 kV atbuses 1,3 and 345 kV at buses 2,4,5.
Figure 5. One line diagram of 5 bus power system[1]
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Input data for the simulation is shown Table 1.
Table 1. Bus input data, line input data andtransformer input data of Figure 5.[1]
A Newton-Raphson load flow algorithm has beenimplemented in Matlab and the five bus powersystem was simulated. The load flow programconverges after 5 iterations.
Plot of convergence of Newton-Raphson in terms ofpower mismatches errors versus number of iterationsare shown in Figure 6. The error drops rapidly fromiteration 1 to iteration 2 and then reduces at slowerrate and finally reaches zero sluggishly after fiveiterations. The tolerable power mismatch error wasassigned to be 5e-5.
Figure 6. Convergence of Newton Raphson
Voltages at bus 1-5 were calculated and the results areshown in Table 2. Notice that the voltage magnitude at bus2 of 0.8338 per unit is under-voltage. Voltage level at bus2 needs to be improved by applying voltage control.V = magV = angleV = 1.0000 1.0000 0 0.7708 - j0.3178 0.8338 -22.4063 1.0499 - j0.0109 1.0500 -0.5973 1.0181 - j0.0504 1.0193 -2.8340 0.9712 - j0.0773 0.9743 -4.5479Table 2. Bus Voltage Outputs in pu & degreesReal power losses were calculated using differentialpowers method. Losses turn out to be 0.0871 per unit or3.81%. This value has been verified by calculating lossesfrom I2R. Calculation using B-loss formula gives lossesequal to 0.0823 pu or error = 5.5%.
6.1 Applying voltage control by changing tap setting ofTransformer between bus 1 and 5
Using Matlab, parameters of the equivalent Π circuit of thetap changing transformer between bus 1 to 5 are calculatedto be A = 0.9208 – j12.2766, B = 0.0115 – j0.1535 and C =-0.01137 + j0.1516. This shows that B is an inductor andC is a capacitor.
The tap setting is increased in step of 1.25 % and the loadflow is rerun to obtain an acceptable voltage level within ±5 % of unity which gives the minimum losses. Two waysof calculating real power losses using traditional I2R andB-loss formula are compared. The minimum losses of0.0704 pu using I2R as shown in Figure 7 occurs using tapsetting = 15% with voltage of bus 2 of 0.941 pu. Minimumlosses of 0.0676 pu using B-losses as shown in Figure 7occur at tap = 10% with voltage of bus 2 of 0.9098 pu.These two voltage levels are still not acceptable. Minimumlosses with acceptable voltage magnitude of 0.9553 pu isobtained with tap = 17.5% where the losses are slightlydifferent using both methods.
Figure 7. Graph of Real Power Losses based on I2R andB-losses vs Tap Setting of Transformers.
Real Power Losses using B-losses and I2R vs Tap Setting
0.06
0.065
0.07
0.075
0.08
0.085
0.09
1.0125 1.025 1.0375 1.05 1.0625 1.075 1.0875 1.1 1.1125 1.125 1.1375 1.15 1.1625 1.175 1.1875 1.2
Tap Setting of Transformer (1.25 - 20 %)
Rea
l Pow
er L
osse
s (p
u)
I2RB-losses
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The two curves are slightly different. They intersectat tap = 15% where the losses are equal to 0.0704 pu.
6.2 Applying voltage control by adding shuntcapacitor at bus 2
Now we want to improve the voltage level at bus 2whilst minimizing the total real power losses. Thecapacitor has been modeled with susceptance B. Thefirst run of load flow program gave a voltage at bus 2of 0.8338 magnitude. The sensitivity analysis iscarried out by adding 5 steps of 0.2 pu reactive powerinjection from 0.2 to 1 pu. Susceptance can then becalculated from B = Q/V2.
Minimum losses of 0.0583 pu calculated from I2R asshown in Figure 8 is attained by adding 0.6 pu or 300Mvar with acceptable bus 2 voltage of 1.0595 pu.Using B-loss formula, minimum losses 0.0733 asshown in Figure 8 is achieved by adding 0.4 pu ofreactive power with acceptable voltage of bus 2 of0.9791 pu.
Figure 8. Graph of Real Power Losses using I2Rand B-losses vs Capacitor Bank Reactive Power
The two curves have similar shapes although thelosses calculated using I2R are smaller than the lossescalculated using B-loss coefficients.
It has been shown that the real power losses varyparabolically with either tap setting of transformer orcapacitor reactive power. Hence, there is a pointwhere minimum real power losses occur. This is anoptimum point for the operation of an electricalpower system as long as the voltage is withinallowable range.
7. CONCLUSION
A Matlab load flow simulation program has beendeveloped using a modified Newton-Raphson algorithm tocalculate and control the voltage, determine real andreactive power flows and compute real power losses.
Voltage control using tap changers can be implemented inthe load flow analysis by using a Π equivalent circuit.Optimum tap setting can be determined by load flowsimulation, which gives minimum real power losses andacceptable voltage level. Optimum voltage control bymeans of switched capacitor bank can also improve thevoltage level at a bus to result in minimum power losses.The shunt capacitor is added at the bus where the busvoltage is under voltage before load flow simulation isrerun. Optimum value of capacitance is obtained fromsimulation, which gives minimum losses at acceptablevoltage level.
Results of losses using I2R are slightly different than theresults calculated using B-losses formula. At this stage itcan be said that the calculation of losses using I2R is moreaccurate because it has been verified with differentialpower method which gives the same results. Although it isnot so accurate, calculation of losses based on B-lossescoefficients is useful because it allows optimizationconfiguration to achieve minimum losses.
Future research will explore the use of B-lossescoefficients to determine the tap settings of the transformeror the capacitance of capacitor banks to satisfy the requiredvoltage level whilst minimizing losses.
8. REFERENCES1. Glover, J.D. and Sarma, M. 1994, Power System
Analysis and Design, 2nd ed., PWS PublishingCompany, Boston
2. Stevenson, W.D. 1975, Elements of Power SystemAnalysis, 3rd ed., McGraw-Hill Kogakusha, Ltd.,Tokyo.
3. Stagg, G.W. and El-Abiad, A.H. 1968, ComputerMethods in Power System Analysis, McGraw-HillBook Company, New York.
4. Weedy, B.M. and Cory, B.J., 1998, Electric PowerSystems, 4th ed., John Wiley & Sons, West Sussex.
5. Lukman, D., Blackburn, T.R and Walshe, K, LossMinimization in Industrial Power System Operation,Proceedings of the Australasian Universities PowerEngineering Conference (AUPEC’94), Brisbane,Australia, 24-27 September 2000.
Real Power Losses using B-losses and I2R vs Capacitor Bank
0.05
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.2 0.4 0.6 0.8 1
Capacitor Bank Reactive Power (pu)
Rea
l Pow
er L
osse
s (p
u)
B-lossesI2R
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6. Penny, J. and Lindfield, G., 1995, Numericalmethods using MATLAB, Ellis Horwood Limited,Hertfordshire
7. Wood, A.J. and Wollenberg, B.F., 1996, PowerGeneration, Operation and Control, 2nd ed., JohnWiley and Sons, Inc., New York.
8. Del Toro, V., 1992, Electric Power Systems,Prentice Hall, Inc., New Jersey.
9. ANSI/IEEE Standard 399-1980, IEEErecommended practice for industrial andcommercial power systems analysis, 1980, PowerSystem Technologies Committee of the IEEEIndustry Applications Society.
10. Parker, A.M., The Modeling of Power SystemComponents, 1997 Residential School inElectrical Power Engineering, UNSW, Australia,26 Jan – 14 Feb 1997.
11. Jabr, R., Coonick, A.H and Cory, B.J., A Study ofthe Homogeneous Algorithm for DynamicEconomic Dispatch with Network Constraintsand Transmission Losses, IEEE Transactions onPower Systems, Vol. 15, No. 2, May 2000, pp605-611
12. Gonen, T. 1988, Modern Power System Analysis,John Wiley and Sons, Inc., New York
APPENDIX
Power flow solutions by Newton-Raphson are basedon the non-linear power flow solutions. It can beshown that the power flow equations of y = f(x) canbe written as:
( ) ( )∑=
−−===N
nknnknknkkkk VYVxPPy
1
cos θδδ Eq.[A.1]
( ) ( )∑=
+ −−===N
nknnknknkkkNk VYVxQQy
1sin θδδ Eq.[A.2]
where k = 2, 3, …, NYkn is the element of the bus admittancematrix between buses k and n
Hence, there are two non-linear simultaneousequations for each node. The real and reactivepowers depend on the product of the sum of thevoltages connected between two buses and theadmittance between the buses. The bus admittancematrix can be first formed from the impedancesconnected to a bus or between two buses.
Changes in P and Q are related to changes in V and δby Equations [A.1] and [A.2], e.g.
NN
PPPP δδ
δδ
δδ
∆∂∂
++∆∂∂
+∆∂∂
=∆ 23
3
22
2
22 ... Eq.[A.3]
Similar equations hold in terms of ∆P and ∆V, and ∆Q interms of ∆δ and ∆V.
Hence, the Newton-Raphson method requires that a set oflinear equations be formed expressing the relationshipbetween the changes in real and reactive powers and thecomponents of bus voltages and phase angles. TheJacobian matrix can be partitioned into four blocks J1, J2,J3 and J4.
∆∆
=
∆∆
VJJJJ
QP δ
43
21 Eq. [A.4]
The partial derivatives in each block can be derived fromEquations [A.1] and [A.2]. The unknown quantities inEquation [A.4] are the elements of the column matrix ofthe changes in the phase angle and voltage of each bus.Convergence criteria are often based on ∆y(i) or powermismatches rather than ∆x(i) or phase angle and voltagemagnitude mismatches.
Once the voltage at each bus is computed, line flows can becalculated. The current at bus k in the line connecting kand n is given by:
( )2'kn
kknnkknyVyVVi +−= Eq. [A.5]
where ykn = line admittance, y’kn = total line chargingadmittance and (Vk×y’kn)/2 = current contribution at bus kdue to line charging. The real and reactive power flowfrom k to n is found to be:
( )2'*** kn
kkknnkkknkknknyVVyVVViVjQP +−==− Eq. [A.6]
The power loss in line k-n is the algebraic sum of thepower flows in either direction.
