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Power Flow

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Power flow: GS and NR methods
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Power Flow Solution The Power Flow Problem The Gauss Seidel Method The Newton Raphson Method The Newton Raphson Power Flo w Power - Flow Studies in Syst ems Design and Operation Regulating Transformers The Decoupled Power Flow Met hod
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Power Flow SolutionThe Power Flow ProblemThe Gauss Seidel MethodThe Newton Raphson MethodThe Newton Raphson Power FlowPower - Flow Studies in Systems Design and OperationRegulating TransformersThe Decoupled Power Flow Method

1The Power Flow Problem 3The typical element in a bus admittance matrix :

The voltage at a typical bus i of a system is given in polar coordinates by

The net current injected into the network at bus i in terms of the elements of is given by the summation:

Let and denote the net real and reactive power entering the network at the bus i , then the complex conjugate of the power injected at bus i is:

4The Power Flow Problem 1 Power flow studies are important both in power system planning and operation . Power flow analysis is very similar to circuit analysis . Except that in the circuit analysis problem is linear and power flow analysis problem is nor-linear.

2The Power Flow Problem 4

The real power mismatch at bus i denoted by

Where, -----scheduled real power being generated at bus i -----scheduled real power demand of the load at bus i

5The Power Flow Problem 5

Likewise, for reactive power at bus i we have reactive power mismatch

-----the net real power entering the network at bus i which is calculated by

6The Power Flow Problem 6Pi,sch, Qi,sch represent the known net supply and demand at bus i.Pi,calc, Qi,calc represent the calculated power at bus i. Ideally, Pi,sch should equal Pi,calc and Qi,sch should equal Qi,calc.In the general case, let us consider that all buses can have load, generation, and compensation.7For each bus are introduced two power balance equations; one for the active power and one for the reactive power.

Power balance equationsThe power balanced equation set for a N-bus system contains 2N nonlinear algebraic equations.Regulating Transformers 1How to include the regulation transformers model in power flow studies?

ideal transformer Figure 9.7 Figure 9.8-

++-

+-+-1:t+-ijij73The Decoupled Power Flow Method 1The decoupled power flow method is an approximate version of the Newton-Raphson procedure. The approximation to the Newton-Raphson procedure only affects the iteration approach. It does not reduce the accuracy of the final solution. 81In the practical power system operation:

The generators can regulate the terminal voltage magnitude, thus each bus with generator is a voltage(magnitude)-controlled bus or simply voltage-controlled bus (PV-bus)The buses without generators and with loads are called PQ buses since P and Q are known.The Power-Flow Problem 8 1. Load buses: each non-generator bus (PQ buses) 2. Voltage-controlled buses: bus in the system at which the voltage magnitude is kept constant (PV buses) 3. Slack (Swing) bus: a bus which voltage angle serves as reference for the angles of all other bus voltages, and the voltage magnitude is specified .

12The Power Flow Problem 9To understand why and are not scheduled at the slack bus, consider the total power in the system:The term in this equation is evidently the total loss in the system.The individual current in the various branches of the network can not be calculated until after the voltage magnitude and angle are known at every bus of the system. Therefore is initially unknown and it is not possible to pre-specify all the quantities.

Total real powerTotal generationTotal loadReal power loss

13The Power Flow Problem 10Likewise, the difference between the total megavars supplied by the generators at the buses and the megavars received by the loads is given by The is called the loss in the series reactance of the transmission lines.

Total reactive powerTotal reactive supplyTotal reactive demandReactive power loss14The Power Flow Problem 11

15The Gauss Seidel Method 1General description of the method: 1.This is a iterative method. 2. Assign estimated values to the unknown bus voltage. 3. Calculate a new value for each bus voltage from the real and reactive power specified. 4. A new set of values for the voltage at each bus is used to calculate another set of bus voltages for the next iteration. 5.The process is repeated until voltage changes at each bus are less than a specified minimum value.16The Gauss Seidel Method 2Gauss Seidel Method :

17The Gauss Seidel Method 3

without

18The Gauss Seidel Method 4Nodes :1. Skip the swing bus2. Use the initial value of the bus voltage to start the calculation3.

or the improved

19The Gauss Seidel Method 5Four bus system:With the slack bus designated as number 1, computations start with bus

Soloing for gives:Assume that buses and are also load buses:234~2341slack20The Gauss Seidel Method 6Iteration method:Initial voltage estimates:In general:

bus 1 is swing bus

for all iStop criteria 21The Gauss Seidel Method 7Acceleration factor:To reduce the number of iteration an acceleration factor can be used:

Voltage-controlled buses:At P-V buses, the voltage magnitude and real power generation are specified. The reactive power entering the network from the generation and voltage angle are then determined by solving the power flow problem.

22The Gauss Seidel Method 8 From a practical point view the reactive power output of the generator must be within definite limits given by the inequality

In the course of power flow solution if the calculated value of is outside either limit, the is set equal to the limit violated, the originally specified voltage magnitude at the bus is relaxed, and the bus is then treated as a P-Q bus for which a new voltage is calculated by the computer program.

23Example 9.2(1)~2431slack~AbedulOlmoPinoLine bus to busSeries ZSeries Y=Z-1Shunt YR(per unit)X(per unit)G(per unit)B(per unit)Total charging Mvar**Y/2(per unit)1-20.010080,050403.815629-19.07814410.250.051251-30.007440.037205.169561-25.8478097.750.038752-40.007440.037205.169561-25.8478097.750.038753-40.012720.063603.023705-15.11852812.750.06375100 Base MVA, 230 kVExample 9.2(2)GenerationLoadP, MWQ, MvarP, MWQ, MvarV (pu)Remarks15030.991.00 0Slack bus200170105.351.00 0Load bus300200123.941.00 0Load bus43188049.581.02 0Voltage ControlledThe Q values of load are calculated from the corresponding P values assuming a power factor of 0.85Example 9.2(3)Bus admittance matrix:Bus123418.985190 -j44.835953-3.815629 +j19.078144-5.169561 +j25.84780902-3.815629+j19.0781448.985190-j44.8359530-5.169561 +j25.847809

3-5.169561 +j25.84780908.193267-j40.863838

-3.023705+j15.118528

40-5.169561 +j25.847809-3.023705+j15.1185288.193267-j40.863838Example 9.2(4)

Acceleration factor()More generally, for bus i during iteration k the accelerated value is given by

Example 9.2(5)Using =1.6, then

and

GS (Voltage controlled bus) (1)

The reactive power injected at bus i is,which has the equivalent algorithm expression:

GS (Voltage controlled bus) (2)In the four-bus example if bus 4 is voltage controlled, then

Subtituting for ,

Since |V4| is specified, we correct the magnitude of as follows

30Example 9.3To complete the first iteration of the Gauss-Seidel procedure, find the voltage at bus 4 of Example 9.2 computed with the originally estimated voltages at buses 2 and 3 replaced by the accelerated values indicated above.

The Newton Raphson Method 1For simplicity let us consider the equation of h1 of two variables x1 and x2 equal to a constant b1 expressed as

and a second equation involving another function h2 such thatwhere b is also a constant.g1 and g2 are introduced for convenience.u is a set of parametersThe Newton Raphson Method 3The Taylors series :

34The Newton Raphson Method 4 Neglect the partial derivatives of order greater than 1, rewrite the equations as matrix:

NewJ : Jacobian Matrixcalculate

Stopping criteria

35The Newton Raphson Method 5Example 9.4 Using the Newton-Raphson method, solve for x1 and x2 of the nonlinear equationsPartial differentiation with respect to the xs yieldsFirst iteration:

36The Newton Raphson Method 6The mismatch equations

First iteration values of x1 and x2Second iteration

37The Newton Raphson Method 7Updating the jacobian and compute the new corrections

38The Newton Raphson Method 8Power system application of Ex 9.4

To load12Figure 9.3~

39The Newton Raphson Method 9

40The Newton Raphson Method 10For simplicity, we now write the mismatch equations for a four-bus system, and it will become obvious how to extend those equations to the general case

The Newton Raphson Method 11A similar equation can be written for reactive power

The Newton Raphson Method 12

Collecting all mismatch equations into vector-matrix form yield,For the four-bus system N=4The Newton Raphson Method 13Notice that in (9.45), we have used the identity,

The Newton Raphson Method 14The solution of (9.45) is found by iteration as follows:Estimate valued i(0) and |Vi |(0) for the state variablesUse the estimates to calculate:P (0) i,cal and Q (0) i,cal from Eqs. (9.38) and (9.39)mistmatches Pi(0) and Qi(0) from Eqs. (9.40) and (9.41) and the partial derivative elements of the Jacobian J.Solve (9.45) for the initial corrections i(0), |Vi |(0)/|Vi |(0).Add the solved corrections to the initial estimates to obtain

Use the new values i(1) and |Vi |(1) as starting values for iteration 2 and so on.

The Newton Raphson Method 15In the general case,

The Newton Raphson Method 16

The elements of J11 are,

The Newton Raphson Method 17

The elements of J21 are,

The Newton Raphson Method 18

The elements of J12 are,

Comparing (9.56) And (9.57)The Newton Raphson Method 19

The elements of J22 are,The Newton Raphson Method 20

Lets now bring together the results developed above in the following definitions:

The Newton Raphson Method 21The Newton Raphson Method 22Voltage-controlled (PV) buses. In the general case if there are Ng voltage-controlled buses besides the slack bus, a row and column for each PV bus is omitted from the polar form of the system jacobian matrix. Which column? Which row?

For explanation consider a four-bus system, where the bus 4 is a PV bus, the bus 1 is the slack bus and the buses 2 and three are PQ buses.

This procedure can be easily extended to the general caseThe Newton Raphson Method 23

|V4| is already especified, then there is not correction for this voltage magnitude, thus |V4| is zero.The power-balance equations for the four-bus system are given next. Observe that the sixth term of each equation disappear since |V4| is zero.The Newton Raphson Method 24

The last power-balance equation set is now written as follow. Please notice that is not |V4| anymore. This set of nonlinear equation has 5 unknowns and 6 equations, then one equation must be deleted which one?55The Newton Raphson Method 25

Finally, the resulting power-balance equations can be written in a vector-matrix form as follow: Conclusion: if the i bus is a PV bus, then the column for |Vi| and the row for |Qi| have to be deleted from the original vector-matrix representation (9.45)The Newton Raphson Method 26Table 9.1

57The Newton Raphson Method 27Example 9.1 1. 9 bus system 2. generators at buses : and 3. choose bus as slack bus.Solution: P-Q buses: and P-V buses: andUnknowns: P-Q buses:

P-V buses:Equations:

N=9, =3, so 2N- -2=13

23145567891255758The Newton-Raphson Power Flow Solution 2The matrix form:We don't include mismatches for the slack bus since and areundefined when and are not scheduled.

DDDD=42424422424424422222422244224244244222224222||||||||||||||||||||||||||||||||QQPPVQVVQVQQVQVVQVQQVPVVPVPPVPVVPVPPddddddd...................................................DDDD222242||||||||VVVVdd.........J11J12J21J22

60The Newton-Raphson Power Flow Solution 3The iteration: Estimate values and for the state variables Use the estimates to calculate: and from

and mismatches and from

Check if If yes .then stop, If no, continue for I = 1 , . n

61The Newton-Raphson Power Flow Solution 4 Solve for the initial corrections and Add the solved corrections to the initial estimates to obtain

Use the new values and as starting values for iteration 2 and continue.

62The Newton-Raphson Power Flow Solution 5

SinceThe elements in sub-matrix

63The Newton-Raphson Power Flow Solution 6The elements in submatrix

The elements in submatrix

J21J1264The Newton-Raphson Power Flow Solution 7The elements in submatrix

Voltage-controlled buses: A row and column for each voltage controlled bus is omitted from the polar formof the system Jacobian, because the voltage correction must always be zero. Equation 9.68J2265The Newton-Raphson Power Flow Solution 8

66The Newton-Raphson Power Flow Solution 9For a four bus system if bus 3 is a PV bus we have the Jacobian:

The column and row be omitted

67Power-Flow Studies in System Design and Operation 2

Table 9.3bus data for Example 9.2

69Power-Flow Studies in System Design and Operation 3

Bus informationXXLine flowGeneration Load Figure 9.4Power Flow Software70Power-Flow Studies in System Design and Operation 4Figure 9.5

To loadMW flowMvar flow13471Power-Flow Studies in System Design and Operation 5

Line losses in line 1-3

13Line charging1

3

72Regulating Transformers 2

If t is a scalar74Regulating Transformers 3Example 9.7: Solve Example 2.13 using model of Eq.(9.74) for each of the two parallel transformers and compare the solution with the approximate results.

+-+-

Figure 9.1175Regulating Transformers 4

76Regulating Transformers 5

77Regulating Transformers 6

78Regulating Transformers 7551.041.0049.039.0551.041.010)0390.00049.1(5.10105.10)*(22)*(2221)(2jIVSjIVSjjjjVjVjIbTbaTab+=-=+=-=+-=-+=-=This example shows: 1. The transformer with the higher tap setting is supplying most of the reactive power to the load. 2. The real power is dividing equally between the transformers.

79Regulating Transformers 8Figure 9.10

MW flowMvar flow

1345580The Decoupled Power Flow Method 2The principle underlying the decoupled approach is based on two observations : Change in the voltage angle at a bus primarily affects the flow of real power in the transmission lines and leaves the flow of reactive power relatively unchanged. Change in the voltage magnitude at a bus primarily affects the flow of reactive power in the transmission lines and leaves the flow of real power relatively unchanged.

82The Decoupled Power Flow Method 3

A well-designed and properly operated power transmissions system: The angular differences between typical buses of the system are usually so small that

83

The Decoupled Power Flow Method 4 The reactive power injected into any bus of the system during normal operation is much less than the reactive power which would flow if all lines from that bus were short-circuited to reference. That is The line susceptances are many times larger than the line conductances so that

i

85The Decoupled Power Flow Method 5simplifying:

86

The solution strategy :1.Calculate the initial mismatches ,2. Solve for ,3. Update the angles and use them to calculate mismatch ,4.Solve for and update the magnitudes , and 5. Repeat the iteration until all mismatches are within specified tolerances.

_969195429.unknown

VPQ

PQ busUnknownUnknownconstantconstant

PV busUnknownconstantconstant

Swing busconstantconstantUnknownUnknown

Bus typeNo. of busesQuantities specifiedNo. of available equationsNo. of (i , |Vi| state variables

Slack: i =1 1 (1 , |V1| 0 0

Voltage Controlled (i = 2, , Ng + 1) Ng Pi , |Vi| Ng Ng

Load

(i = Ng + 2, , N)N - Ng - 1 Pi , Qi2(N - Ng - 1)2(N - Ng - 1)

Totals N 2N2N - Ng - 22N - Ng - 2

Line ,

bus to

bus Series Z

R

per unitX

pre unit Series

G

pre unit

B

per unit

Shunt Y

Total

Charging

Mval

Y/2

pre unit

1-2

1-3

2-4

3-40.01008

0.00744

0.00744

0.012720.05040

0.03720

0.03720

0.063603.815629

5.169561

5.169561

3.023075

-19.078144

-25.847805

-25.847805

-15.118528

10.25

7.75

7.75

12.75

0.05152

0.03875

0.03875

0.06375

Bus

Generation

P,MW

Q,Mvar Load

P,MW

Q,MVar

V,per unit

Remarks

1234_

0

0

318

_

0

0

_50

170

200

80

30.99

105.35

123.94

49.58

Slack bus

Load bus

(inductive)

load bus

(inductive)

voltage controled

X

Bus

no.NameVolts

(p.u.)Angle

(deg.)(MW)(MVAR)(MW)(Mvar)Bus

typeTo bus

name Line flow

(MW) (Mvar)

1

2

3

4Brich

Elm

Pine

Maple1.00

0.982

0.969

1.020

0.

-0.976

-1.872

1.523

186.81

0.

0.

318.00114.50

0.

0.

181.4350.00

170.00

200.00

80.0030.99

105.35

123.94

49.58SL

PQ

PQ

PV2

3

1

4

1

4

2

3Elm

Pine

Birch

Maple

Brich

Maple

Elm

Pine38.69

98.12

-38.46

-131.54

-97.09

-102.91

133.25

104.7522.30

61.21

-31.24

-74.11

-63.57

-60.37

74.92

56.93

(Area totals) 504.81 295.93 500.00 309.86


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