Power Flow Solution Techniques

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PTERRA PHILIPPINES, INC.

March 1-3, 2016 1

Power Flow Formulation

Presented by

Pterra Philippines, Inc.

All rights reserved

Outline

What is a Power Flow?Power Flow FormulationPower Flow Solution MethodsNon-convergence of the Power FlowPower Flow Adjustments

2

What is a Power Flow?

A numerical model of the electricpower system in steady-state

Also known as load flow

3

Numerical Model

Pgen, Pload – real power generation and loadQgen, Qload – reactive power generation andloadVgen, Vbus1, Vbus2, Vload – voltage atvarious locationsTap1, Tap2 – tap step for transformersI – current on transmission lines

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Characteristics of the Power Flow

Based on one phase of a three-phase electricsystem

Assumes phases are balancedMore accurately, the positive sequencecomponent of a 3-phase systemHence, the use of SINGLE LINE DIAGRAMS

Uses a simplified notation known as the PER-UNIT system

Electric parameters are specified with respect toa standard BASE

Uses complex numbers to represent electricparameters

Real and imaginary numbers

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Sample Single-Line Diagram

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The Per-Unit System

Expression of system quantities as fractionsof a defined base unit quantity

For example, 100 MW on a 100 MVA base is 1.0per unit or 1 p.u.

Simplifies power flow modelingSimilar apparatus (generators, transformers,lines) will have similar per-unit impedancesexpressed on their own rating, regardless of their absolute size.Per-unit quantities are the same on either side of a transformer, independent of voltage levelBy normalizing quantities to a common base,power flow calculations are simplified

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8

1 2

P2 = 1.2 pu

V 1 = 1 pu V 2

P3 = -1.5 pu

V 3

30.1 pu 0.2 pu

0.25 pu

P1

+ -- +

Network impedances,generator power and loadare indicated in per-unit.

Mutual admittances areY12=- 4puY13=- 10 puY23= -5 pu

The Per-Unit System


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The Power Flow Formulation

Given: Pgen, Pload, Qload, VgenFind: Qgen, Vbus1, Vbus2, Vload, I

Also Tap1, Tap2, electrical lossesUsing:

numerical methods,positive sequence network,per-unit quantities andcomplex variables

9 10

Ohm’s Law is the relationship betweenvoltages and currents. For a network, itcan be expressed in matrix notation as:

[I] = [Y bus ] [V]Where

[I] is an array of node current injections[V] is an array of node voltages[Ybus ] is the node admittance matrix


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n = 3 = total number of nodesVi : voltage (to ground) at node iI1 & I3 : current injections at nodes 1 & 3

Yii : sum of all admittances connecting to node i Yij : - (sum of all admittances between nodes i & j)

21 3

I 1 I 3

y cy a

y b y d


12

3

2

1

3332

232221

1211

3

1

V

V

V

Y Y 0

Y Y Y

0 Y Y

I

0

I

Y11 = ya + yb

Y12 = Y21 = -yb

Y22 = yb + yd + yc

Y23 = Y32 = -yd

Y33 = yd



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The current injection is expressed as a function

of the real and reactive power as follows:

Where:Pi = P injected into the network at bus iQi = Q injected into the network at bus i|Vi| = Magnitude of voltage at bus i

i = Angle of bus i voltage*=conjugate of a complex quantity

VQ jP

VQ jP

Ii

ii*i

iii

iδ

The Power Flow Formulation The Power Flow Formulation

The resulting equation for each bus or node i is:

Complex injected power S i is a non-linear function of the complex bus voltages V.

Where V is

Direct solution is not possible; need to usenumerical iterative methods

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)( *1

*ij

N

j jiiii Y V V S jQP

Vi iδ


At each node or bus, there are four quantities: P, Q, Vand

Solution requires specification of two out of four quantitiesat each bus

Types of busBuses with generators

PV bus – P and V are input parameters, Q and are solved for PQ bus - P and Q are input parameters, V and are solved for Swing Bus - V and are input parameters, P and Q are solvedfor

Buses with no generatorsPQ bus - P and Q are input parameters, V and are solved for

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Codes for types of power flow busused in commercial software

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Type of Bus PSS/E Code PSLF Code

PV or generator bus 2 2

PQ or load bus 1 1

Swing or slackbus 3 0


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Switching from PV to PQ busInitially, the bus with a generator may bespecified as a PV bus

Regulates voltage V as long as the generator’sreactive limits have not been reached.

If a generator reaches its maximum or minimum reactive limit during the power flowsolution, the bus switches to a PQ bus

Generator reactive power is held at the reactivelimit, and the bus voltage is allowed to vary.

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18

)]cossin((:

)]sincos([:

:

)(:

1

1

*

1

*

N

jijik ijij jii

N

jijik ijij jii

ijijij

ij

N

j jiiii

BGV V Qand

BGV V Pthen

jBGY where

Y V V S jQPGiven

δδ

δδ

Outline

What is a Power Flow?Power Flow Formulation

Power Flow Solution MethodsNon-convergence of the Power FlowPower Flow Adjustments

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Power Flow Solution Methods

Methods considered most effectiveand most useful for practical power system problems are:

Gauss-Seidel - method of successivedisplacementNamed after the German mathematicians CarlFriedrich Gauss and Philipp Ludwig vonSeidel

Newton-Raphson - method for findingsuccessively better approximations to thezeroes (or roots) of a real-valued function

Named after Isaac Newton and JosephRaphson

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Power Flow Solution Methods

Characteristics of numerical methods for solving the power flowNeed an initial estimate of the unknown

parametersTypically, this is voltageIf all voltages are initialized at 1.0 p.u., this isknown as a FLAT START

Solves to a level of error known as amismatch tolerance

If the mismatch tolerance is met, the power flow isconvergent

May fail to find a solution due to modeling or numerical errors

The power flow solution is divergent

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Gauss-Seidel Method

A technique for solving the equations of the

linear system of equations one at a time insequence, and uses previously computedresults as soon as they are available.Algorithm:

Assume starting values for all unknownparameters

Typically, V=1.0 pu, = 0, Q=0Solve for bus 1 unknowns using t he assumedvalues for the other busesSolve for next bus unknowns using updatedvalues from previous busSolve all other buses in sequenceWhen convergence criteria are met, stop, case isconverged

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The voltages at each bus are:

Where V is a complex number comprising of /V i / and δ i

k1*ii V

V

1V

ik

N

ik k

ii

ii

Y Q jP

Y

Gauss-Seidel Method Gauss-Seidel Method

Convergence criteria specify theacceptable accuracy for the power flowsolution

Basis for stopping an iterative methodMax | ΔV | is below a specifiedconvergence tolerance, ξ

Where ΔV is the difference in voltage at abus between the present calculated valueand the previous value

∆V = V ne w - V ol d

Typically (0.01 pu > ξ > 0.0001 pu)

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Gauss-Seidel Method

Modifications to the method: Acceleration/deceleration factors - toeither accelerate or decelerate the rate of convergence to a solution by updatingvoltages as follows

V ne w = V ol d + a*(V new -V ol d ) 0.7 ≤ a ≤ 1.5

Commercial software may allow for separate values of a for the real andimaginary components of V

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Gauss-Seidel Method

Characteristics:Is able to converge even if the initialestimate of voltage is poor

Acceleration factor must be tuned tomatch system for optimum performanceWill not identify modeling errors

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Gauss-Seidel Method

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START

InitialEstimate of

the unknownparameters

Solve the newvalues using theprevious values

Compute themismatchtolerance

Less thanspecified

tolerance?

End

YES

NO

Newton-Raphson Method

A technique for solving nonlinear equations using successive linear

approximations

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Newton Raphson Method

Starting from the power flow equations, getthe partial derivatives of P and Q to V and δ For ∆P with respect to ∆V for a 3-bus system:

Repeat this expansion for ∆Q with respect to∆V, ∆P with respect to ∆δ, ∆Q with respect to∆δ

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3

3

12

2

11

1

11 V

V

P V

V

P V

V

PP

3

3

22

2

21

1

22

V V

P V

V

P V

V

PP

3

3

33

3

31

1

33 V

V

P V

V

P V

V

PP


In matrix notation:

[J] is called the Jacobian matrix(matrix of 1 st derivatives)

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δ

V

Q

P]J[

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Newton Raphson Method Newton Raphson Method

Convergence criteria:Max ΔP and ΔQ is below a specifiedconvergence tolerance, ξTypically (0.1 pu > ξ > 0.001 pu)

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Modification/ImprovementsNon-divergent power flow

Applies a reduction to the Jacobian multiplier whenever the solution appears to exit feasiblespace

Decoupled NewtonSimplifies computational requirement byassuming that the real and reactive componentsof the power flow equations are independent of each other

Fixed Slope NewtonSimplifies computational requirement by notupdating the Jacobian for each iteration

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Outline

What is a Power Flow?Power Flow FormulationPower Flow Solution MethodsNon-convergence of the Power FlowPower Flow Adjustments

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Power Flow Solutions

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Power Flow solution results:Convergence

All nodes have met the mismatch tolerance

DivergenceSingularity of the JacobianSteady-state voltage collapse

Non-convergenceIs it physical (singularity) or numerical?

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METHOD ADVANTAGES DISADVANTAGES

Gauss-Seidel o Tolerates well data errors

and poor voltage andreactive power conditions.o If failing to converge, it gives

an idea of problems.o Has low computer memory

requirements

o Slow convergence rate.

o Exhibits problems when thesystem is stressed due tohigh levels of active power transfer.

o Acceleration factors for better performance.

o Number of iterationsincreases with system size.

NewtonRaphson

o Converges quickly for well-conditioned systems(quadratic convergence rate).

o Computation time increaseslinearly with the system size.

o Small mismatches can beachieved.

o Intolerant of data errors.o Difficult to converge for

cases with poor voltageestimates.

o Does not indicate cause for failing to converge.


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METHOD ADVANTAGES DISADVANTAGES

Gauss-Seidel Use when:o Data is suspecto Poor voltage estimateo N-R fails to convergeo Network has reactive

power problems

Do not use when:o Network has very low

(or negative)impedance branches

o Variation of themethod works for negative branches

Newton-Raphson

Use when:o Large network size

with high solutionaccuracy.

o Network containsnegative reactancebranches

Do not use when:o Network contains

branches with low X/Rratios

o Reactive power problems

Power Flow Solutions Non-Convergent Power Flows

Power Flowsolutionresults:

ConvergenceDivergenceNon-convergence

Is it physical(singularity)or numerical?


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Non-Convergent Power Flows

What can you do if the case does notconverge?

Check convergence monitor. See whichbus has the biggest mismatch.If “terminated” check the mismatch. If it isrelatively small, change the maximumnumber of iterations and solve again.If “blown up”, identify the bus with thelargest mismatch. Also, what is the totalmismatch?Check if there is a modeling error in thevicinity of the bus


General options to address non-convergence

Try changing the swing bus Adjust local controls

transformer taps, phase shifters, switched shuntsremote bus control models

Equivalence remote portions of modelChange solution parameters

Reduce acceleration factor Increase mismatch toleranceIncrease iteration limit

Open reactive limits

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Other methods for addressing non-convergence

Non-divergent power flow

Applies a reduction to the Jacobianmultiplier whenever the solution appears toexit feasible space

Interior-Point Newton methoduses [H] - the Hessian matrix, a 2 ndapproximation of the nodal equations toimprove the convergence

Optimal Power Flow Allows parameters to change in order to finda feasible solution

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Power Flow Adjustments

Outside of the power flow formulation,adjustments may be added to thecomputer model to handle varioustypes of power system equipment.Examples:

Transformer tap adjustmentsPhase-shifter adjustmentsHVDC converter transformer tapadjustmentsSwitching of capacitor banks and reactors


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Power Flow Adjustments

Typically adjustments are added in betweeniterations and are triggered by setpointsembedded in software

The settings may not be accessible to usersDifferences in these settings lead variouscommercial to different solutions for the samepower flow case

Poorly defined adjustments can lead thepower flow solution to oscillate or diverge

Power Flow Solution Techniques

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