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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
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What is a Power Flow?
A numerical model of the electricpower system in steady-state
Also known as load flow
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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
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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
The Power Flow Formulation
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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
The Power Flow Formulation
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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
The Power Flow Formulation
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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
The Power Flow Formulation
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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The Power Flow Formulation
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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The Power Flow Formulation
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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The Power Flow Formulation
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)]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
Newton Raphson Method
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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Newton Raphson Method
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 Solutions
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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Power Flow Solutions
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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.
Power Flow Solutions
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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
Non-Convergent Power Flows
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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Non-Convergent Power Flows
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
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