AVAILABLE TRANSFER CAPABILITY
ASSESSMENT IN A RESTRUCTURED
ELECTRICTY MARKET
DEPARTMENT OF ELECTRICAL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY
KURUKSHETRA
06/12/09 1

Static Modeling of Transmission Line
Bus i Bus j
Gij + j Bij
j Bsh j Bsh
Model of a simple Transmission line
06/12/09 2

Power Flow Equations
S ij = Vi ∠δ i .I ij
*
[ ]
I ij = Vi ∠δ i − V j ∠δ j Yij + Vi ∠δ i . jBsh
[ ]
S ij = Vi ∠δ i { Vi ∠ − δ i − V j ∠ − δ j ( Gij − jBij ) − jBshVi ∠ − δ i }
[ ]
= Vi 2 − ViV j ∠δ ij ( Gij − jBij ) − jBshVi 2
= (Vi
2
)
− ViV j cos δ ij − jViV j sin δ ij ( Gij − jBij ) − jBshVi 2
= {Vi
2
} { }
Gij − ViV j ( Gij cos δ ij + Bij sin δ ij ) + j − Vi 2 ( Bij + Bsh ) − ViV j ( Gij sin δ ij − Bij cos δ ij )
= Pij + jQij
The real and reactive power flow equations from bus-i to bus-
j can be written as :
Pij = Vi 2 Gij − ViV j ( Gij cos δ ij + Bij sin δ ij )
Qij = −Vi 2 ( Bij + Bsh ) − ViV j ( Gij sin δ ij − Bij cos δ ij )
06/12/09 3

Power Flow Equations
S ji = V j ∠δ j .I *
ji
[ ]
I ji = V j ∠δ j − Vi ∠δ i Yij + V j ∠δ j . jBsh
[ ]
S ji = V j ∠δ j { V j ∠ − δ j − Vi ∠ − δ i ( Gij − jBij ) − jB shV j ∠ − δ j }
[ ]
= V j2 − ViV j ∠ − δ ij ( Gij − jBij ) − jBshV j2
= (V j
2
)
− ViV j cos δ ij + jViV j sin δ ij ( Gij − jBij ) − jBshV j2
= {V j
2
} { }
Gij − ViV j ( Gij cos δ ij − Bij sin δ ij ) + j − V j2 ( Bij + Bsh ) + ViV j ( Gij sin δ ij + Bij cos δ ij )
= Pji + jQ ji
The real and reactive power flow equations from bus-j to bus-
i can be written as:
Pji = V j2 Gij − ViV j ( Gij cos δ ij − Bij sin δ ij )
Q ji = −V j2 ( Bij + Bsh ) + ViV j ( Gij sin δ ij + Bij cos δ ij )
06/12/09 4

Transmission line modeling with regulating
transformer
Bus-i Bus-j
nYij =n(Gij + jBij)
n(n-1)Yij
jBsh jBsh
(1-n)Yij
Model of simple transmission line with regulating
transformer having off nominal tap ratio ‘n’.
06/12/09 5

Power Flow Equations with off nominal tap ratio ‘n’
S ij = Vi ∠δ i .I ij
*
[
I ij = ( Vi ∠ δ i − V j ∠ δ j ) n( Gij + jBij ) + Vi ∠ δ i n( n − 1) ( Gij + jBij ) + jBsh ]
[
S ij = Vi ∠δ i {(Vi ∠ − δ i − V j ∠ − δ j ) n( Gij − jBij ) + Vi ∠ − δ i n( n − 1) ( Gij − jBij ) − jBsh ]}
= (Vi 2 − ViV j cos δ ij − jViV j sin δ ij )n( Gij − jBij ) + n 2Vi 2 Gij − nVi 2 Gij
− j ( n 2Vi 2 Bij − nVi 2 Bij + Vi 2 Bsh )
= {n 2Vi 2 Gij − nViV j ( Gij cos δ ij + Bij sin δ ij )}
+ j{− Vi 2 ( n 2 Bij + Bsh ) − nViV j ( Gij sin δ ij − Bij cos δ ij )} = Pij + jQij
The real and reactive power flow equations can be written as:
Pij = n 2Vi 2 Gij − nViV j ( Gij cos δ ij + Bij sin δ ij )
( )
Qij = −Vi 2 n 2 Bij + Bsh − nViV j ( Gij sin δ ij − Bij cos δ ij )
06/12/09 6

Transmission Loadability Curve (St. Clairs Curve)
The St. Clair curve gives the estimation of loadability of
transmission line in terms of their Surge Impedance Loading
(SIL).
It is well known that the per-unit line data normalized using
SIL and Surge Impedance is constant, i.e. independent of line
construction and voltage rating.
06/12/09 7

Loadability Curve
In the calculation of line loadability curve, the thermal rating of
conductors, line voltage drop and steady state stability are the
actual restrictive conditions.
The extensive simulations and analyses have revealed that the
thermal limitation, line-voltage-drop limitation and steady-
statestability limitation are the main controlling factors for
loadability of short, moderate and long lines respectively.
Loadability of a line is limited by :
(iii) thermal limitation (I2R losses)
(ii) voltage regulation limitation, voltage stability
limit
06/12/09 (iii) stability limitation 8

Limits on Line Loadability
06/12/09 9

Power Transfer Vs Line Length
06/12/09 10

DYNAMIC LINE RATING OF TRANSMISSION LINES
Dynamic Thermal Rating
06/12/09 11

General Dynamic Rating Considerations
The steady state thermal rating is the loading that corresponds
to the maximum allowable conductor temperature under the
assumption of thermal equilibrium.
The dynamic security rating of the line is a function of the
line admittance.
As a minimum up to a 15% or 20% increase in the thermal
rating of a transmission can be obtained with dynamic
ratings.
Even more can be expected when the transmission lines are
loaded due to wind generation since wind will be blowing
when the lines are loaded.
06/12/09 12

General Dynamic Rating Considerations
At higher wind velocities, the cooling of the transmission line
reduces the sag there by making the transmission line available
for more power transfers at the time the wind generation is
adding to the power transfers.
Dynamic rating of the line allows the transmission owner to
monitor the weather conditions, sag / tension on the line in
real time and to permit loadings that exceed the nominal
rating.
The most common practice is to calculate line nominal
ratings based on coincident high ambient temperature, full
solar radiation and effective wind speed of 0.61 m/s. Some
utilities even assume an effective wind speed of 0.91 m/s, or
higher.
06/12/09 13

IEEE and CIGRE Standards
IEEE (IEEE standard 738, 1993) and CIGRE (CIGRE, 1992,
1997, 1999) offer standard methods for the calculation of the
transmission line ampacity in the steady state and dynamic
states.
Common utility practice of rating the transmission lines is
based on the conservative assumptions of ambient temperature,
wind speed, solar radiation and maximum conductor
temperature:
Ambient temperature: 40 deg C
Wind speed: 0.61 m/s (2 ft/sec)
Solar Radiation: 1000 W/m2
Maximum conductor temperature: 80 deg C
06/12/09 14

IEEE Standard
The solution of the differential equations for the heating of a
conductor by current in the steady state and dynamic states
requires the knowledge of the following metrological data
as follows: • Ambient Temperature
• Wind Speed
• Wind Direction
• Solar radiation
The approach to increase the thermal line ratings encompasses
several active tasks:
• Develop thermal ratings standards,
• Identify sources and magnitudes of errors in as-built sags at
every temperature,
• Accurate calculation of high temperature sags,
• Probabilistic aspects of the line ratings
• Real-time rating methodologies, etc
06/12/09 15

Static and Dynamic Ratings
The static ratings of the lines are determined based on the
historical weather data in the region for different conductor types
used in the region. Generally, static line ratings are fixed for a
particular season of the year.
Dynamic line ratings are computed by online or offline methods
based on recorded data.
Methods to increase the maximum conductor temperature
involve physical modifications for the line structures to increase
ground clearance in certain spans.
The uncertainties of the sag at higher temperature can be
resolved by monitoring the sag and tension at higher current
carrying conditions.
06/12/09 16

Static and Dynamic Ratings
In the offline system, line ratings are obtained uniquely by
monitoring weather conditions along the transmission line. The
offline system may also include monitoring conductor sag by
pointing a laser beam at the lowest point of the conductor in a
span.
Transmission lines are now being equipped with fiber optic
network cables embedded in the core are used to carry useful
information regarding the sag and temperature and eliminate the
need for separate communication to supply data to the central
processing station in order to determine the dynamic rating of
the power lines.
The latest advances in computation models involve
probabilistic modeling of the conductor temperature to predict
the loss of the tensile strength and permanent elongation of the
conductor in the lifetime.
06/12/09 17

Heat Balance Equation For Calculating Allowable Conductor
Loading
M = γ.A, conductor mass, kg/m
A = conductor are, m2
Pj = joule heating, W/m
PS = solar heating, W/m
Pm = magnetic heating, W/m
Pr = radiation heat loss, W/m
PC = convection heat loss, W/m
Tav = is the average of surface temperature and the core
temperature of the conductor.
06/12/09 18

Ampacity Calculation
The ampacity is then calculated by using the formula:
Where;
Pr: Heat loss by radiation,
PC: Heat loss by convection,
PS: Heat gained by solar radiation,
Rac: AC resistance of the conductor
Effect of wind speed and direction on the Ampacity of the
conductor:
06/12/09 19

Conductor loading vs Wind speed and Direction
Line Rating (Vs.) Wind Speed And Angle Of Incidence
The above curves are for an ACSR cardinal conductor with a
mean diameter of 0.0304m. The conductor initial temperature is
assumed to be 80 deg C.
06/12/09 20

Transmission Capability under Steady State Stability
The power-carrying capability of the transmission system is
generally evaluated by line loadability curve (St. Clair curve)
which expressed in per unit of surge impedance load, SIL.
In accordance with the steady state stability criterion of
transmission Line Reference Book 345 kV and Above by EPRI,
the steady-state stability margin 30% is taken, corresponding
phase angle difference δ ＝ 44 ° across the system terminals
and margin of 5% for the voltage stability limit
06/12/09 21

Power-carrying capability of different AC power
transmission mode
06/12/09 22

Compact Transmission Lines
Operating 500 kV single-circuit compact transmission
line and double circuit compact transmission lines
06/12/09 23

Compact Transmission Lines
Comparison of conductor configuration and tower
06/12/09 24

Impact of Voltage stability limit
Loadability curves of transmission line Loadability curves of transmission line
(Uncompensated) (compensated)
06/12/09 25

TRANSIENT STABILITY TRANSFER CAPABILITY OF
DIFFERENT AC POWER
TRANSMISSION MODE
06/12/09 26

SOME TYPICAL POWER FREQUENCY PARAMETERS
OF COMPACT AND
CONVENTIONAL TRANSMISSION LINE
06/12/09 27

ACSS (Aluminum Conductor Steel Supported)
ACSS uses a fully annealed aluminum conductor around a steel
core. The steel core provides the entire conductor support. The
aluminum strands are “dead soft”, thus the conductor can be
operated in excess of 200C without loss of strength.
The maximum operating temperature of the conductor is limited
by the coating used on the steel core (conventional galvanized
coating can sustain 245C). Since the fully annealed aluminum
cannot support significant stress, the conductor has a thermal
expansion similar to that of steel.
Tension in the aluminum strands is normally low. This helps to
improve the conductor's self-damping characteristics and helps to
reduce the need for dampers.
06/12/09 28

ACSS/TW (Aluminum Conductor Steel Supported /
Trapezoidal Wire)
For ACSS/TW, the aluminum strands are trapezoidal in shape.
The wedge shaped aluminum strands enable a more compact
alignment of the aluminum wires.
Conductor designs that maintain the same circular mil cross
sectional area of aluminum as a conventional round conductor
results in a TW conductor that is 10 to 15 percent smaller in
overall diameter.
Conductor designs that maintain the same overall diameter as
a conventional round conductor result in a TW conductor that
has 20 to 25 percent more aluminum cross-sectional area
packed.
06/12/09 29

ACCR (Aluminum Conductor Composite Reinforced)
The Aluminum Conductor Composite Reinforced (ACCR) is a
new type of bare overhead conductor containing a multi-strand
core of heat-resistant aluminum composite wires, retains its
strength at high temperatures and is not adversely affected by the
environmental conditions, such as moisture and UV exposure. It’s
lightweight and reduced thermal expansion properties allow the
conductors to be installed on existing towers and requires no
additional changes.
The outer strands are composed of a temperature-resistant
material (like aluminum-zirconium alloy), which permits
operation at high temperatures (210ºC continuous, 240ºC
emergency). The Al-Zr alloy is a hard aluminum alloy with
properties and hardness similar to those of standard 1350-H19
aluminum but a microstructure designed to maintain strength after
operating at high temperatures; that is, it resists annealing.
06/12/09 30

AAAC (All Aluminum Alloy Conductors)
AAAC is a high strength aluminum alloy, concentric-lay-
stranded conductor and is similar in current carrying capacity as
the equivalent size of ACSR. AAAC is similar in construction
and appearance to ACSR.
Aluminum Alloy Conductors have a number of advantages over
the ACSR
Lower power losses than for equivalent single aluminum layer
ACSR conductors (the inductive effect of the steel core in ACSR is
eliminated).
• Simpler fittings than those required for ACSR
• Excellent corrosion resistance in environments conducive to
galvanic corrosion in ACSR.
• Strength and sags are approximately same as for equivalent 6/1
and 26/7 ACSR conductors.
06/12/09 31

Comparison of properties for different Transmission
Conductors (DRAKE SIZE)
06/12/09 32

Behavior of typical 6/1 ACSR with the behavior
of ACSR/AW under increasing load
06/12/09 33

Drake vs. Drake/AW
26/7, 795,000 cmil Drake vs. Drake/AW
54/7, 1-33,000 cmil Curlew vs. Curlew/AW
Figure 2
06/12/09 34

AVAILABLE TRANSFER CAPABILITY (ATC)
Available Transfer Capability (ATC) is a
measure of the transfer capability
remaining in the physical transmission
network for further commercial activity
over and above already committed uses.
ATC = TTC –TRM – {ETC + CBM}.
In general,
ATC = TTC – ETC.
06/12/09 35

Total Transfer Capability (TTC):
It is the amount of electric power that can be transferred
over the interconnected transmission network in a
reliable manner under a reasonable range of
uncertainties and contingencies.
It is more or less similar to the First Contingency Total
Transfer Capacity (FCTTC).
While determining TTC, system conditions, critical
contingencies, system limits, parallel path flows and
effects of non-simultaneous and simultaneous transfers
are to be considered.
06/12/09 36

Transmission Reliability Margin (TRM):
It is defined as the amount of transmission transfer
capability necessary to ensure that the
interconnected network is secure under a reasonable
range of uncertainties in the system conditions.
06/12/09 37

Capacity Benefit Margin (CBM):
It is the amount of transmission transfer capability
reserved by the load serving entities to ensure access
to generation from interconnected systems to meet
generation reliability requirements.
It also helps to reduce the installed capacity of the
plant.
06/12/09 38

Existing Transfer Commitments (ETC)
refers to the power transfer capability that must be
reserved for already committed transactions.
Considering firm and non-firm transmission contracts,
a particular load may be curtailed or recalled under
specific conditions.
ATC definition is further elaborated for such transactions
using the terms curtailability and recallability.
06/12/09 39

Curtailability
is the right of the transmission service provider to
interrupt all or part of the transmission service due to
constraints that reduce the capability of the transmission
system.
Curtailment of loads occurs only when the system
reliability is threatened or emergency conditions exist.
06/12/09 40

Recallability
is the right of a transmission service provider to interrupt all or
part of the service for any reason, including economic, in
consistence with the tariff and contract provisions and prevalent
policy.
Based on the recallability concept two commercial applications
of ATC are defined as follows:
Non-recallable Available Transfer Capability (NATC):
NATC = TTC – TRM - {Non-recallable reserved transmission
service + CBM }
Recallable Available Transfer Capability (RATC):
RATC = TTC – TRM - {Recallable reserved transmission Service
06/12/09 41
+Non-recallable reserved transmission Service + CBM }

NEED & IMPORTANCE OF ATC
• Shows the amount of maximum additional power transfer
capability between the specified interfaces.
• Ensures secure operation of the system
• Firm and non-firm reservation of transmission service can
be made on the basis of ATC.
• Gives an idea of Congestion in the specified part of the
system.
• Can be used as a tool in transmission pricing.
• The binding constraint for ATC can be used in planning
and network expansion.
06/12/09 42

Principles of ATC Determination
The basic objective of the determination of ATC is to tell the
market entities about the system limitations beforehand in terms
of additional MW of power that can be transferred from one area
to another.
Since private parties are interested only in commercial aspects, it
is the responsibility of the ISO to determine, update and post the
current value of ATC for every time interval.
Following are the governing principles for ATC determination
ATC calculation must produce commercially viable results.
ATCs produced by the calculations must give a reasonable and
dependable indication of transfer capabilities available to the
power market.
06/12/09 43

ATC calculations must recognize time-variant power flow
conditions on the entire inter-connected transmission network.
ATC calculations must recognize the dependency of ATC on
points of electric power injection, the directions of transfers
across the interconnected transmission network, and the points
of power extraction.
Regional and wide-area co-ordination is necessary to develop and
post the information that reflects the ATCs of the inter-connected
transmission network.
06/12/09 44

ATC calculations must conform to NERC (or equivalent
regulatory guidelines in the specific country), regional, sub-
regional, power pool, and individual system reliability and
operating policies, criteria or guides.
The determination of ATC must accommodate reasonable
uncertainties in system conditions and provide operating
flexibility to ensure the secure operation of the
interconnected network.
06/12/09 45

CONSTRAINTS TO ATC
Static Constraints
• Line thermal limits
• Bus voltage limits
• Reactive power generation limits
Dynamic Constraints
• Small signal stability limits
• Large signal stability limits (transient limits)
06/12/09 46

COMUPTATION OF ATC
Static Mathods
• Continuation power flow method
• DC loadflow and PTDF based methods
• Sensitivity based AC loadflow methods
• Genetic algorithm based methods
• Optimization based methods
06/12/09 47

Dynamic ATC based Method
Bifurcation approach method
Trajectory sensitivity based methods
Optimal power flow based methods incorporating
transient stability constraints, dynamic constraints
Hybrid approach (structure preserving function
and time domain method)
Adaptive wavelet neural network based approach
06/12/09 48

Enhancement of ATC Using FACTS Controllers
Types of FACTS Controllers
3. Thyristor Controlled Phase Angle Regulator
TCPAR
4. Static Var Compensator SVC
5. Thyristor Controlled Series Compensator TCSC
6. SSSC
7. UPFC
8. GUPFC
9. IPFC
06/12/09 49

Efficient Available Transfer Capability Analysis
Using Linear Methods

What is Available Transfer Capability (ATC)?
• We are familiar with the terms
– Total Transfer Capability (TTC)
– Capacity Benefit Margin (CBM)
– Transmission Reliability Margin (TRM)
– “Existing Transmission Commitments”
– Etc…
• Then ATC is defined as
– ATC = TTC – CBM – TRM – “Existing TC”
06/12/09 51

Efficient Available Transfer Capability Analysis
Using Linear Methods
weber@powerworld.com
Power Systems Software Developer
PowerWorld Corporation
Urbana, IL
http://powerworld.com

Available Transfer Capability
• In broad terms, let’s define ATC as
– The maximum amount of additional MW transfer possible between
two parts of a power system
• Additional means that existing transfers are considered part of
the “base case” and are not included in the ATC number
• Typically these two parts are control areas
– Can really be any group of power injections.
• What does Maximum mean?
– No overloads should occur in the system as the transfer is
increased
– No overloads should occur in the system during contingencies as
the transfer is increased.
06/12/09 53

Computational Problem?
• Assume we want to calculate the ATC by incrementing the
transfer, resolving the power flow, and iterating in this
manner.
– Assume 10 is a reasonable guess for number of
iterations that it will take to determine the ATC
• We must do this process under each contingency.
– Assume we have 600 contingencies.
• This means we have 10*600 power flows to solve.
• If it takes 30 seconds to solve each power flow (a
reasonable estimate), then it will take 50 hours to complete
the computation for ONE transfer direction!
06/12/09 54

Why is ATC Important?
• It’s the point where power system reliability meets
electricity market efficiency.
• ATC can have a huge impact on market outcomes and
system reliability, so the results of ATC are of great
interest to all involved.
06/12/09 55

Linear Analysis Techniques in PowerWorld
Simulator
An overview of the underlying mathematics of the
power flow
Explanation of where the linearized analysis
techniques come from

AC Power Flow Equations
• Full AC Power Flow Equations
N
∆Pk = 0 = Vk2 g kk + Vk ∑ (Vm [ g km cos( δ k − δ m ) + bkm sin ( δ k − δ m ) ] ) − PGk + PLk
m =1
m≠k
N
∆Qk = 0 = −V b + Vk ∑ (Vm [ g km sin ( δ k − δ m ) − bkm cos( δ k − δ m ) ] ) − QGk + QLk
2
k kk
m =1
• Solution requires iteration of equations
m ≠k
−1
 ∂P ∂P 
 ∆δ   ∂δ ∂V   ∆P  =  ∆P 
∆V  =  ∂Q ∂Q  ∆Q 
[ J] −1
∆Q 
       
 ∂δ ∂V 
• Note: the large matrix (J) is called the Jacobian
06/12/09 57

Full AC Derivatives
• Real Power derivative equations are
∂Pk
∂Pk
= VkVm [ − g km cos( δ k − δ m ) − bkm sin ( δ k − δ m ) ] = Vk [ g km sin ( δ k − δ m ) − bkm cos( δ k − δ m ) ]
∂δ m ∂Vm
∂Pk N
∂Pk N
= Vk ∑ [Vm [ − g km sin ( δ k − δ m ) + bkm cos( δ k − δ m ) ] ] = 2Vk g kk + ∑ [Vm [ g km cos( δ k − δ m ) + bkm sin ( δ k − δ m ) ] ]
∂δ k m =1 ∂Vk m =1
m≠k m≠k
• Reactive Power derivative equations are
∂Q k ∂Q k
= V k V m [ − g km cos( δ k − δ m ) − bkm sin ( δ k − δ m ) ] = V k [ g km sin ( δ k − δ m ) − bkm cos( δ k − δ m ) ]
∂δ m ∂V m
∂Qk N
∂Qk N
= Vk ∑ [Vm [ g km cos( δ k − δ m ) + bkm sin ( δ k − δ m ) ] ] = −2Vk bkk + ∑ [Vm [ g km sin ( δ k − δ m ) − bkm cos( δ k − δ m ) ] ]
∂δ k m =1 ∂Vk m =1
m≠k m≠k
06/12/09 58

Decoupled Power Flow Equations
δ k − δ following assumptions
• Make the m ≈ 0 Vk ≈ 1 rkm << xkm
cos( δ k − δ m ) ≈ 1 g km ≈ 0
sin ( δ k − δ m ) ≈ 0
• Derivates simplify to
∂Pk N
∂Pk
= ∑ bkm = −bkm ∂Pk
=0
∂Pk
=0
∂δ k m =1 ∂δ m ∂Vk ∂Vm
m≠k
∂Qk N
∂Qk ∂Qk ∂Qk
= −2bkk + ∑ (−bkm ) = −bkm ∂δ k
=0
∂δ m
=0
∂Vk m =1 ∂Vm
m≠ k
06/12/09 59

B’ and B’’ Matrices
∂P ∂Q
= B' = B ''
∂δ ∂V
• Define and
• Now Iterate the “decoupled” equations
[ ] ∆P
∆δ = B ' −1
∆V = [B ] ∆Q'' −1
• What are B’ and B’’? After a little thought, we can simply state that…
– B’ is the imaginary part of the Y-Bus with all the “shunt terms”
removed
– B’’ is the imaginary part of the Y-Bus with all the “shunt terms”
double counted
06/12/09 60

“DC Power Flow”
• The “DC Power Flow” equations are simply the real part
of the decoupled power flow equations
– Voltages and reactive power are ignored
– Only angles and real power are solved for by iterating
∆δ = B [ ] ' −1
∆P
06/12/09 61

Bus Voltage and Angle Sensitivities to a Transfer
• Power flow was  ∆δ  −1  ∆P 
solved by iterating ∆V  = [ J ] ∆Q 
   
• Model the transfer as a change in the injections ∆P
– Buyer: ∆T = [0 0 PF 0 PF 0] T ∑ PFBh = 1 N
B Bf Bg
– Seller: h =1
∆TS = [ 0 PFSx 0 PFSy 0 0]
N
∑ PFSz = 1
T
• Assume buyer consists of z =1
– 85% from bus 3 and 15% from bus 5, then
∆TB = [ 0 0 0.85 0 0.15 0 ]T
• Assume seller consists of
– 65% from bus 2 and 35% from bus 4, then
∆TS = [ 0 0.65 0 0.35 0 0 ]T
06/12/09 62

Bus Voltage and Angle Sensitivities to a Transfer
• Then solve for the voltage and angle sensitivities by
solving
 ∆δ S  −1  ∆TS   ∆δ B  −1  ∆TB 
∆V  = [ J ]  0  ∆V  = [ J ]  0 
 S    B  
∆δ S , ∆δ B , ∆VS , and ∆VB
• are the sensitivities of the Buyer and Seller “sending
power to the slack bus”
06/12/09 63

What about Losses?
• If we assume the total sensitivity to the transfer is the
seller minus the buyer sensitivity, then
∆δ = ∆δ S − ∆δ B ∆V = ∆VS − ∆VB
• Implicitly, this assumes that ALL the change in losses
shows up at the slack bus.
• PowerWorld Simulator assigns the change to the BUYER
instead by defining
• Then ∆Slack S Change in slack bus generation for seller sending power to slack
k= =
∆Slack B Change in slack bus generation for buyer sending power to slack
 ∆δ   ∆δ S   ∆ δ B 
 ∆V  =  ∆ V  − k  ∆ V 
   S  B
06/12/09 64

Lossless DC Voltage and Angle Sensitivities
• Use the DC Power Flow Equations
∆δ = B ∆P [ ]
' −1
• Then determine angle sensitivities
∆δ S = [B ]
' −1
∆TS ∆δ B = [B ]
' −1
∆TB
• The DC Power Flow ignores losses, thus
∆δ = ∆δ S − ∆δ B
06/12/09 65

Lossless DC Sensitivities with Phase Shifters
Included
• DC Power Flow equations [ ]
B ' ∆δ = ∆P
• Augmented to include an equation that describes the
change in flow on a phase-shifter controlled branch as
being zero.
• Thus instead of DC power flow equations we use
Line Flow Change = B δ ∆δ + B α ∆α = 0
• Otherwise process is the same.
−1
 ∆δ   B
'
0  ∆P 
 ∆α  =    
  B δ Bα   0 
06/12/09 66

Why Include Phase Shifters?
230 kV Phase Shifter D MR5 00
MDN 500
• Phase Shifters are often on lower
Canada
I NG5 00
voltage paths (230 kV or less)
R OS3 60
W AH3 60 CB K50 0
C UST ER W
S EL 500
NLY 230
with relatively small limits
SN OHO MS3
S NO HOM S4
C HIE F & 1
MON ROE
CHI EF &2
C HIE F & 1
10 00 MW
19 9 M VR
CHI EF J5
CHI EF JO 65 3 M W
EC HOL AKE 653 M W
1 05 MV R 10 5 M VR
C HI EF J3 C OUL EE 19 C OU LEE 20
C HIE F J 4
MAP LE VL
C OUL EE
SI CKL ER
M APL E &1
CO UL EE& 1
ROC KY RH
COU LEE &2
C OVI NG TN
Weak Low • They are put there in order to
C OU LEE &3
COV ING T2
COU LEE &4
115 kV Phase Shifter
RA VER
SCH ULT Z
TA COM A
O LY MPI A
Voltage Tie
HO T S PR
B ELL B& 1
BE LL BPA
VAN TAG E
67 0 M W
manage the flow on a path that
17 7 M VR
30 MW
19 MVR
SA TSO P
CE NT R G 1
P AUL
TAF T
To Canada
DWO R 2 D WOR 3
95 MW
-4 5 M VR
D WOR 1
DWO RS HAK
HA NFO RD
769 MW
26 MVR
HA TWA I
H ANF OR& 1
A SHE
LO W M ON
would otherwise commonly see
AS HE 2
L OW MON L IT GOO S
LO W G RAN
1155 MW
101 MVR L IT GOO S
LO W G RAN 769 MW
26 MV R
GAR RIS &4
7 69 MW 5 MW GA RRI S& 2
26 MV R 2 M VR
G ARR IS& 3
ALL STO N GA RRI S&1
SA CJA WEA
BPA
SAC JWA T
MC NAR Y
overloads
CO YO D1
CO YOT E
TRO UT DAL
KEE LER JOH N D AY
JOH N D AY
BI G E DDY
SL ATT
MCL OUG LN
CE LI LO
ASH E &1
PEA RL BUC KLE &2
OS TRN DER
BUC KLE Y
J OH N D &1
AS HE &2
BOA RD F
B OAR D F
540 MW 30 MW
114 MVR 15 MVR
• Without including them in the
MA RIO N
GRI ZZL &2
SAN TIA M
GR IZ ZL& 1 BUC KLE &1
AL VEY &2
sensitivity calculation, they
ROU ND BU
G RIZ ZLY
LAN E
P ON DRO SA
A LVE Y P OND RO& 1
C AP TJA &5
G RIZ ZL& 3
constantly show up as
ALV EY &1
CA PTJ A&4
GRI ZZL &4
P OND RO& 2 BUR NS
B URN S & 2
SUM MER L
BU RN S & 1
D IX ONV LE
CAP TJA &3 GR IZZ L&5
MA LI N & 2
“overloaded” when using Linear
CA PT JA& 2
GR IZZ L&6
D IXO NV& 1
CA PTJ A&1 G RIZ ZL &7
M ALI N & 1
ME RID INP
M IDP OIN T
BO RAH
ATC tools
MI DPO INT
M ALI N
C APT JAC K
ADE LAI DE AD EL TAP
OLI NDA &1
RO UND &1
115 kV Phase Shifter
I DAH O-N V
R OUN D & 2 R OUN D &4
RD MT 1M
HUM BO LDT
COY OTE CR
ROU ND MT
OLI NDA &2
V ALM Y
California
VAL MY G2
06/12/09 67

Power Transfer Distribution Factors (PTDFs)
• PTDF: measures the sensitivity of line MW flows to a MW
transfer.
– Line flows are simply a function of the voltages and
angles at its terminal buses
– Using the Chain Rule, the PTDF is simply a function of
these voltage and angle sensitivities.
• Pkm is the flow from bus k to bus m
 ∂Pkm   ∂Pkm   ∂Pkm   ∂Pkm 
PTDF = ∆Pkm =   ∆VK +   ∆Vm +   ∆δ K +   ∆δ m
 ∂Vk   ∂Vm   ∂δ k   ∂δ m 
Voltage and Angle Sensitivities that were just discussed
06/12/09 68

Pkm Derivative Calculations
• ∂PFullVAC equationsb
  ∂Pkm 

 ∂δ

km
m 

k m [
 = V g sin( δ − δ ) − km k m (
km cos δ k − δ m )] 

 ∂δ k 

[ ( ) (
 = VkVm − g km sin δ k − δ m + bkm cos δ k − δ m )]
 ∂Pkm 

 ∂Vm 
[ ( )
 = Vk g km cos δ k − δ m + bkm sin δ k − δ m ( )]  ∂Pkm 

 ∂Vk 
[ ( ) (
 = 2Vk g kk + Vm g km cos δ k − δ m + bkm sin δ k − δ m )]
• Lossless DC Approximations yield
 ∂Pkm   ∂Pkm 
  = −bkm   = bkm
 ∂δ m 
   ∂δ k 
 
 ∂Pkm   ∂Pkm 
 =0  =0
06/12/09  ∂Vk   ∂Vm  69

Line Outage Distribution Factors (LODFs)
• LODFl,k: percent of the pre-outage flow on Line K will
show up on Line L after the outage of Line K
• Linear impact of an outage is determined by modeling the
outage as a “transfer” between the terminals of the line
Change in flow on Line L
after the outage of Line K
∆Pl ,k
LODFl ,k =
Pk
Pre-outage flow on Line K
06/12/09 70

Modeling an LODF as a Transfer
Other Line l
The Rest of System
Create a transfer Switches
defined by n m
∆Pn and ∆Pm Line k ~
Pk
∆n
P ∆m
P
~
Assume Pk = ∆Pn = ∆Pm
06/12/09
Then the flow on the Switches is ZERO, thus71
Opening Line K is equivalent to the “transfer”

Modeling an LODF as a Transfer
~
• Thus, setting up a transfer of Pk MW from Bus n to Bus
m is linearly equivalent to outaging the transmission line
• ~
Let’s assume we know what P is equal to, then we can
k
calculate the values relevant to the LODF
– Calculate the relevant values by using PTDFs for a
“transfer” from Bus n to Bus m.
06/12/09 72

Calculation of LODF
• Estimate of post-outage flow on Line L
~
∆Pl , k = PTDFl * Pk
• Estimate of flow on Line K after transfer
~ ~ ~ Pk
Pk = Pk + PTDFk * Pk Pk =
• Thus we can write 1 − PTDFk
 Pk 
~ PTDFl * 
 1 − PTDF 

∆Pl ,k PTDFl * Pk  
LODFl ,k = = = k
Pk Pk Pk
• We have a simple function of PTDF values
PTDFl
LODFl ,k =
1 − PTDFk
06/12/09 73

Line Closure Distribution Factors (LCDFs)
• LCDFl,k: percent of the post-closure flow on Line K will
show up on Line L after the closure of Line K
• Linear impact of an closure is determined by modeling the
closure as a “transfer” between the terminals of the line
∆Pl , k
LCDFl , k = ~
Pk
06/12/09 74

Modeling the LCDF as a Transfer
Other Line l
The Rest of System
Net flow from rest Net flow to rest of
of the system the system
∆n
P ∆m
P
Create a “transfer” n Line k ~
m
defined by Pk
~
∆Pn and ∆Pm Assume Pk = ∆Pn = ∆Pm
Then the net flow to and from the rest of the system are
06/12/09 75
both zero, thus closing line k is equivalent the “transfer”

Modeling an LCDF as a Transfer
~
• Thus, setting up a transfer of− P MW from Bus n to Bus
k
m is linearly equivalent to outaging the transmission line
• Let’s assume we know what ~ is equal to, then we can
− Pk LODF.
calculate the values relevant to the
• Note: The negative sign is used so that the notation is
consistent with the LODF “transfer” direction.
06/12/09 76

Calculation of LCDF
• Estimate of post-closure flow on Line L
~
∆Pl ,k = PTDFl * ( − Pk )
• Thus we can write
~
∆Pl ,k − PTDFl * Pk
LCDFl ,k = ~ = ~ = − PTDFl
Pk Pk
• Thus the LCDF, is exactly equal to the PTDF for a transfer
between the terminals of the line
LCDFl ,k = − PTDFl
06/12/09 77

Modeling Linear Impact of a Contingency
1 2 ...... nc M
Contingent Lines 1 through nc Monitored Line M
• Outage Transfer Distribution Factors (OTDFs)
– The percent of a transfer that will flow on a branch M after
the contingency occurs
• Outage Flows (OMWs)
– The estimated flow on a branch M after the contingency
occurs
06/12/09 78

OTDFs and OMWs
• Single Line Outage
OTDFM ,1 = PTDFM + LODFM ,1 * PTDF1
• Multiple Line Outage
OMWM ,1 = MWM + LODFM ,1 * MW1
nC
OTDFM ,C = PTDFM + ∑ [ LODFMK * NetPTDFK ]
K =1
• What are and ?
NetPTDFK NetMWK
nC
OMWM ,C = MWM + ∑ [ LODFMK * NetMWK ]
K =1
06/12/09 79

Determining NetPTDFK
and NetMWK
• Each NetPTDFK is a function of all the other NetPTDFs
because the change in status of a line effects all other lines
(including other outages).
• Assume we know all NetPTDFs except for the first one,
NetPTDF1. Then we can write:
NetPTDF1 = PTDF1 + LODF12 NetPTDF2 + ... + LODF1nC NetPTDFnC
nC
= PTDF1 + ∑ [ LODF1K NetPTDFK ]
K =2
• In general for each Contingent Line N, write
nC
1.0 * NetPTDFN − ∑ [ LODF
K =1
NK NetPTDFK ] = PTDFN
K ≠N
06/12/09 80

Determining NetPTDFK
and NetMWK
• Thus we have a set of nc equations and nc unknowns (nc=
number of contingent lines)
Known Values
 1 − LODF12 − LODF13  − LODF1nC   NetPTDF1   PTDF1 
 − LODF 1 − LODF23  − LODF2 nC   NetPTDF2   PTDF2 
 21    
 − LODF31 − LODF32 1  − LODF3nC   NetPTDF3  =  PTDF3 
    
           
− LODFn 1 − LODFnC 2 − LODFnC 3  1   NetPTDFn   PTDFn 
 C  C   C 
• Thus
NetPTDFC = [ LODFCC ] PTDFC
−1
• Same type of derivation shows
NetMWC = [ LODFCC ] MWC
−1
06/12/09 81

Fast ATC Analysis Goal =
Avoid Power Flow Solutions
• When completely solving ATC, the number of power flow solutions
required is equal to the product of
– The number of contingencies
– The number of iterations required to determine the ATC (this is
normally smaller than the number of contingencies)
• We will look at three methods (2 are linearized)
– Single Linear Step (fully linearized)
• Perform a single power flow, then all linear (extremely fast)
– Iterated Linear Step (mostly linear, Contingencies Linear)
• Requires iterations of power flow to ramp out to the maximum
transfer level, but no power flows for contingencies.
– (IL) then Full AC
• Requires iterations of power flow and full solution of contingencies
06/12/09 82

Single Linear Step ATC
• For each line in the system determine a Transfer Limiter
Value T
 Limit M − MWM
 ; PTDFM > 0
 PTDFM

TM =  ∞ (infinite) ; PTDFM = 0

 − Limit M − MWM ; PTDFM < 0
 PTDFM

06/12/09 83

Single Linear Step ATC
• Then, for each line during each contingency determine
another Transfer Limiter Value
 Limit M − OMWM ,C
 ; OTDFM ,C > 0
 OTDFM ,C

TM ,C = ∞ (infinite) ; OTDFM ,C = 0

 − Limit M − OMWM ,C
 ; OTDFM ,C < 0
 OTDFM ,C
06/12/09 84

Important Sources of Error in Linear ATC
Numbers
• Linear estimates of OTDF and OMW are quite accurate (usually within
2 %)
• But, this can lead to big errors in ATC estimates
– Assume a line’s present flow is 47 MW and its limit is 100 MW.
– Assume OTDF = 0.5%; Assume OMW = 95 MW
– Then ATC = (100 - 95) / 0.005 = 1000 MW
– Assume 2% error in OMW (1 MW out of 50 MW change estimate)
• Actual OMW is 96 MW
– Assume 0% error in OTDF
– Actual ATC is then (100-96)/0.005 = 800 MW
• 2% error in OMW estimate results in a 25% over-estimate of the ATC
06/12/09 85

Single Linear Step ATC
• The transfer limit can then be calculated to be the
minimum value ofTM or TM ,Cfor all lines and contingencies.
• Simulator saves several values with each Transfer Limiters
• [Transfer Limit]
• Line being monitored [Limiting Element]
• Contingency T or T [Limiting Contingency]
M M ,C
• OTDF or PTDF value [%PTDF_OTDF]
• OMW or MW value [Pre-Transfer Flow Estimate]
• Limit Used (negative Limit if PTDF_OTDF < 0)
Good for • MW value initially [Initial Value]
filtering
out errors
06/12/09 86

Pros and Cons of the
Linear Step ATC
• Single Linear Step ATC is extremely fast
– Linearization is quite accurate in modeling the impact of
contingencies and transfers
• However, it only uses derivatives around the present operating point.
Thus,
– Control changes as you ramp out to the transfer limit are NOT
modeled
• Exception: We made special arrangements for Phase Shifters
– The possibility of generators participating in the transfer hitting
limits is NOT modeled
• The, Iterated Linear Step ATC takes into account these control
changes.
06/12/09 87

Iterated Linear (IL) Step ATC
• Performs the following
1.
Stepsize = ATC using Single Linear Step
2.
If [abs(stepsize) < Tolerance] then stop
3.
Ramp transfer out an additional amount of Stepsize
4.
Resolve Power Flow (slow part, but takes into account all
controls)
5. At new operating point, Stepsize = ATC using Single Linear Step
6. Go to step 2
• Reasonably fast
– On the order of 10 times slower than Single Linear Step
• Takes into account all control changes because a full AC
Power Flow is solved to ramp the transfer
06/12/09 88

Including OPF constraints in (IL) to enforce
Interface Flows
• When ramping out the transfer, Simulator can be set to
enforce a specified flow on an interface.
• This introduces a radical change in control variables that is
best modeled by completely resolving using the OPF
– The objective of the OPF is to minimize the total
controller changes (sum of generator output changes)
• Why would you do this?
– Represent a normal operating guideline that is obeyed
when transfers are changed.
06/12/09 89

Example: Bonneville Power Administration
(BPA)
Operating procedures for BPA
require them to maintain “interface”
flows into Seattle in specific ranges
1000 MW
199 MVR
653 MW
105 MVR
653 MW
105 MVR
(These are stability constraints!)
670 MW
177 MVR 30 MW
19 MVR
Seattle
95 MW
-45 MVR
769 MW
26 MVR
1155 MW
101 MVR
769 MW
26 MVR
769 MW 5 MW
26 MVR 2 MVR
Interface Grande
Flow Chief Jo
Coulee
2000 MW
6800 MW
1000 MW
199 MVR
653 MW 653 MW
105 MVR 105 MVR
A Lot of
06/12/09 Generation 90

(IL) then Full AC Method
• Performs the following
1. Run Iterated Linear Step and ramp transfer out ATC Value found
2. StepSize = 10% of the initial Linear Step Size saved during the (IL) method, or 50
MW whichever is larger.
3. Run Full Contingency Analysis on the ramped transfer state
4. If there are violations then change the sign of Stepsize
5. if [abs(stepsize) < Tolerance] then Stop
6. Ramp transfer out an additional amount of Stepsize and resolve Power Flow
7. At new operating point, Run Full Contingency Analysis
8. if [ (Stepsize > 0) and (There are Violation)] OR
[ (Stepsize < 0) and (There are NO Violations)] THEN
StepSize := -StepSize/2
9. Go to step 5
• Extremely slow.
– “Number of Contingencies” times slower than the iterated linear.
If you have 100 contingencies, then this is 100 times slower. (1
hour becomes 4 days!)
06/12/09 91

Recommendations from PowerWorld’s
Experience
• Single Linear Step
– Use for all preliminary analysis, and most analysis in general.
• Iterated Linear Step
– Only use if you know that important controls change as you ramp
out to the limit
• (IL) then Full AC
– Never use this method. It’s just too slow.
– The marginal gain in accuracy compared to (IL) (less than 2%)
doesn’t justify the time requirements
– Remember that ATC numbers probably aren’t any more than 2%
accurate anyway! (what limits did you choose, what generation
participates in the transfer, etc…)
06/12/09 92

BIFURCATION THEORY
A bifurcation occurs at any point in parameter space, at
which the qualitative structure of the system changes for a
small change in parameter vector, ‘p.’
The change may be in:
1. Number of equilibrium points
2. Number of limit cycles
3. Stability of equilibrium points or limit cycles or
4. Period of periodic solution.
06/12/09 93

SADDLE NODE BIFURCATION (SNB)
• A saddle node bifurcation is the disappearance of a
stable equilibrium as parameters change slowly. Two
equilibria – one stable and the other unstable
coalesce.
• Sensitivity w.r.t. the loading parameter of the critical
state variable is infinite – infinite slope of the
bifurcation diagram.
• System Jacobian has a zero eigen value.
06/12/09 94

HOPF BIFURCATION (HB)
• Hopf bifurcation is characterized by the onset of
oscillatory behaviour in a non-linear system. Power system
initially operating at a stable equilibrium typically starts
oscillating when parameters change slowly so that HB
occurs.
• A system which was previously in stable equilibrium
moves into stable periodic oscillations as the critical
parameter is slowly varied (Supercritical HB).
• The oscillations may be non-periodic and unstable (sub-
critical HB).
• The rate of change of the real part of the eigenvalue is non-
zero w.r.t. the loading parameter.
06/12/09 95

MODELLING OF POWER SYSTEM
COMPONENTS
The important components of a Power System are:
2. Transmission Lines and Cables
3. Transformers
4. Generator
5. Excitation System
6. Load
06/12/09 96

MODELS OF TRANSMISSION LINES,
CABLES AND TRANSFORMERS
• Nominal π-model has been considered for the
transmission lines & cables.
• The transformer with unity tap-settings are
represented by a series impedance.
• Appropriate modification in the above has been
made for off-nominal tap-settings and non-zero
phase-shifts.
06/12/09 97

2-AXIS DYNAMIC MODEL OF
A SYNCHRONOUS MACHINE

δi = ωi -ωs ..(1)
 ′[ ′ ] ′ [ ′ ]
M i ωi = TMi − Eqi − X di I di I qi − Edi − X qi I qi I di − Di ( ωi − ωs ) ..(2)
Tdoi Eqi = − Eqi − ( X di − X di ) I di + E fdi
′ ′ ′ ′ ..(3)
Tqoi Edi = − Edi + ( X qi − X qi ) I qi
′ ′ ′ ′ ..(4)
06/12/09 98

STEADY STATE PHASOR DIAGRAM
OF THE MACHINE
( )
Vi e
j π −δ i +θ i
2 ′ [ ′ ′ ′ ]
+ ( Rsi + jX di ) ( I di + jI qi ) − E di + ( X di − X qi ) I qi + jE qi = 0
′ .. .. (2.5)

( )
Pi + jQi = ( Vi ) ( I di − jI qi ) e
j π − δ i +θ i
2
− PLi ( Vi ) − jQLi ( Vi ) .. .. (6)
where,
Pi + jQi = Real and Reactive power injected into the network at bus i
n
= ∑ ViVk yik e j (θ i −θ k −α ik )
k =1
[
δ i = Angle of Vi e j( θ i ) + ( Rsi + jX qi ) I Gi ]
 PGi − jQGi 
I Gi = 
 V e - j( θ i ) 

 i 
06/12/09 100

EXCITATION SYSTEM
dE fdi
TEi = −( K Ei + S Ei ( E fdi ) ) E fdi + VRi ..(7)
dt
E fdi + K Ai (Vrefi − Vi ) ..(8)
dVRi K Ai K Fi
TAi = −VRi + K Ai R fi −
dt TFi
dR fi K Fi
TFi = − R fi + E fdi ..(9)
dt TFi
06/12/09 101

LOAD MODELLING
Two static models have been considered –
• Constant Power Load
• Voltage dependent ‘ZIP’ model
P =P +P V2 +P V
Li CLi Zli Ili
= P a + b V 2 + c V 
Li  i
 i i 
06/12/09 102

NETWORK EQUATIONS (SLFE)
0 = − Pi − jQi + ∑ ViVk Yik exp( j (θ i − θ k − α ik ) ) .... .. (10)
where,
Pi + jQi = Real and reactive power injected in the network at bus i.
Yik ∠α ik = Network Admittance between busses i & k.
Vi ∠ θ i = Voltage and phase at bus i.
06/12/09 103

JACOBIAN MATRIX FORMATION
x = f ( x, y )
 ...... (13)
0 = g( x, y ) ...... (14)
where,
x = vector of dynamic state variables
= δ ω E′ E′ E V R T t

 q d fd R f M

y = vector of algebraic variables
t
= i i V θ
q d
 

06/12/09 104

JACOBIAN MATRIX FORMATION
∆x = A∆x + B∆y
 .... (15)
0 = C∆x + D∆y .... (16)
where,
 ∂f   ∂f 
A=  , B=  ,
 ∂x  x , y  ∂y  x , y
o o o o
 ∂g   ∂g 
C=  and, D =   .
 ∂x  x , y  ∂y  x , y
o o o o
Δy = D −1 CΔ x
(
∴ Δ x = A − BD −1 C Δ x
 )
06/12/09 105

FLOWCHART
06/12/09 106

WSCC 3-MACHINE, 9-BUS SYSTEM
06/12/09 107

RESULTS
Dynamic ATC using Hopf Bifurcation Criteria - Load1 Case
(WSCC 3-machine, 9-bus System)
Transactio Low Voltage Profile
Load ATC Ploss Qloss
ns V4 V5 V6
Base Case 1.25 - 0.046 -0.922 1.026 0.996 1.013
T1 4.60 3.35 0.256 2.078 0.947 0.848 0.952
T2 3.55 2.30 0.349 1.870 0.972 0.868 0.959
T3 3.50 2.25 0.326 1.398 0.974 0.896 0.948
T4 4.30 3.05 0.285 1.579 0.970 0.866 0.966
T5 3.85 2.60 0.381 1.79 0.964 0.863 0.945
T6 4.40 3.15 0.300 1.630 0.963 0.865 0.954
T7 4.40 3.15 0.249 1.371 0.970 0.866 0.966
06/12/09 108

Static ATC using Saddle Node Bifurcation Criteria
Load1 Case
(WSCC 3-machine, 9-bus System)
Transacti Low Voltage Profile
Load ATC Ploss Qloss
ons V4 V5 V
1.02
Base Case 1.25 - 0.046 -0.922 0.996 1.013
6
0.86
T1 5.15 3.90 0.487 4.977 0.702 0.883
3
0.90
T2 4.35 3.10 0.709 5.312 0.709 0.892
3
0.86
T3 4.75 3.50 0.857 6.317 0.691 0.816
6
0.89
T4 5.25 4.00 0.570 4.575 0.707 0.904
4
0.89
T6 4.70 3.45 0.749 4.955 0.704 0.868
0
0.87
T7 5.40 4.15 0.543 4.625 0.695 0.897
9
06/12/09 109

Dynamic ATC using Hopf Bifurcation Criteria – Load2 Case
(WSCC 3-machine, 9-bus System)
Transaction Load ATC Ploss Qloss Min Voltage Profile
V4 V5
Base Case 1.25 - 0.046 -0.922 1.026 0.996 1.013
T1 2.35 1.10 0.201 0.964 0.918 0.713 0.928
T2 2.25 1.00 0.265 1.386 0.915 0.705 0.921
T3 2.25 1.00 0.245 1.075 0.918 0.721 0.921
T4 2.35 1.10 0.239 1.226 0.913 0.698 0.922
T5 2.25 1.00 0.249 1.133 0.918 0.716 0.922
T6 2.35 1.10 0.230 1.081 0.914 0.705 0.923
T7 2.35 1.10 0.221 1.068 0.916 0.706 0.925
06/12/09 110

Dynamic ATC using saddle Node Bifurcation Criteria - Load2 Case
(WSCC 3-machine, 9-bus System)
Transaction Load ATC Ploss Qloss Min Voltage Profile
V4 V5
Base Case 1.25 - 0.046 -0.922 1.026 0.996 1.013
T1 2.58 1.33 0.325 2.297 0.874 0.605 0.894
T2 2.45 1.20 0.404 2.792 0.873 0.598 0.886
T3 2.52 1.27 0.469 3.291 0.850 0.548 0.862
T4 2.55 1.30 0.372 2.600 0.870 0.588 0.888
T5 2.50 1.25 0.429 2.905 0.862 0.575 0.876
T6 2.58 1.33 0.389 2.718 0.862 0.572 0.881
T7 2.58 1.33 0.368 2.593 0.867 0.582 0.886
06/12/09 111

ATC FOR 9-BUS,3 GENERATOR SYSTEM
LOAD1 CASE
6 V limit hb limit snb limt
4
ATC (p.u.)
2
0
T1 T2 T3 T4 T5 T6 T7
Transaction (Load1 )
ATC for WSCC 9-Bus System -Load1 Case.
06/12/09 112

ATC FOR 9-BUS SYSTEM
LOAD2 CASE
V limit hb limit snb limt
1.6
1.2
ATC (pu)
0.8
0.4
0
T1 T2 T3 T4 T5 T6 T7
Transactions (Load2 at bus 5)
ATC for WSCC 9-Bus System -Load2
Case.
06/12/09 113

39-BUS, 10-GENERATOR NEW ENGLAND
SYSTEM
06/12/09 114

ATC FOR 39-BUS SYSTEM
LOAD1 CASE
25
V lim HB lim SNB lim
20
ATC (pu)
15
10
5
0
T1 T2 T3 T4 T5
Transaction (Load1 at bus 18)
ATC for 39 Bus System - Load1 Case
06/12/09 115

ATC FOR 39 BUS SYSTEM
LOAD2 CASE
16 V lim HB lim SNB lim
12
ATC (pu)
8
4
0
T1 T2 T3 T4 T5
Transaction(Load2 at bus 18)
ATC for 39 Bus System- Load2 Case
06/12/09 116

STATIC VAR COMPENSATOR (SVC)
A shunt-connected static Var generator or absorber
whose output is adjusted to exchange capacitive or
inductive current so as to maintain or control specific
parameters of the power system (typically bus voltage)

SVC OUTPUT CHARACTERISTICS
06/12/09 118

DYNAMIC MODEL OF SVC
TR B ref = − B ref + K R (V ref − Vi )
 .... (3.1)

Tb B svc = − B svc + B ref .....(3.2)
At steady state,
B svc = B ref = K R (Vref − Vi )
and Q svc = Vi 2 B svc .....(3.3)
n
Qm = V m ∑ Ymk V k sin (θ m − θ k − α mk ) − V m ⋅ B svc
2
....(3.4)
k =1
06/12/09 119

SVC PLACEMENT CRITERIA
• The load bus having the maximum participation of the
voltage state to the critical mode is selected as an optimal
bus for SVC placement.
• At the HB and SNB points, the critical eigenvalue of the
complete Jacobian was determined.
• Participation factors were computed using the respective
elements of the corresponding right and left eigenvectors.
• From these the load bus voltage state having maximum
participation to the critical mode in most of the cases was
selected.
06/12/09 120

ATC FOR 9-BUS SYSTEM, LOAD1 CASE
WITH SVC AT BUS 5
10 Vlim HBP SNB
8
ATC (pu)
6
4
2
0
T1 T2 T3 T4 T5 T6 T7
Transaction (Load1, pu)
06/12/09 121

ATC FOR 9-BUS SYSTEM, LOAD2 CASE
WITH SVC AT BUS 5
8 Vlim HBP SNB
6
ATC (pu)
4
2
0
T1 T2 T3 T4 T5 T6 T7
Transactions (Load2,pu)
06/12/09 122

Dynamic ATC using Hopf Bifurcation Criteria - Load1 Case
(39-bus, 10-Generator System with SVC at bus 18)
Transaction Load ATC Ploss Qloss Voltage Profile Critical
Base Case 5.22 - 0.471 -0.373 V32=0.945 V35=0.958 V14=0.960 V34=0.
T1 14.20 8.98 0.597 6.777 V32=0.928 V14=0.944 V16=0.945 V15=0.
T2 30.64 25.42 2.773 40.956 V19=0.786 V18=0.849 V17=0.851 V14=0.
T3 17.85 12.63 0.967 14.608 V32=0.886 V33=0.907 V31=0.908 V20=0.
T4 12.32 7.10 1.171 6.614 V32=0.935 V14=0.944 V35=0.946 V34=0.
T5 31.42 26.20 4.038 75.437 V13=0.722 V14=0.749 V38=0.776 V12=0.
T2** -- Hopf Bifurcation not observed; Results are corresponding to the SNB.
06/12/09 123

ATC FOR 39-BUS SYSTEM WITH SVC
LOAD1 CASE
30 Vlim HBP SNB
25
20
ATC (pu)
15
10
5
0
T1 Tt2 T3 T4 T5
Transation (Load1, pu)
06/12/09 124

ATC FOR 39-BUS SYSTEM WITH SVC
LOAD2 CASE.
25 Series1 Series2 Series3
20
ATC (pu)
15
10
5
0
T1 T2 T3 T4 T5
Transaction (load2 , pu)
06/12/09 125

ATC FOR 39-BUS SYSTEM WITH SVC
(LOAD1, KR=20)
25 Vlim HBP SNB
20
ATC (pu)
15
10
5
0
T1 T2 T3 T4 T5
Transaction (Load1 pu)
06/12/09 126

CONCLUSIONS
• Following are the conclusions derived from the
observations, prior to the placement of SVC in the system.
• ATC values are much higher when only the real power of
the Load is increased.
• Minimum voltage criteria is the binding constraint in most
of the cases.
• With ZIP load model, the ATC values changed in each case.
While the connected load corresp. to the occurrence of HB
and SNB increased in each case, the actual supplied values
dropped in a few as a result of voltage decline.
• For the same ATC value, losses may be very different for
different transactions.
06/12/09 127

CONCLUSIONS (Contd.)
After placement of SVC
• The ATC values increased in all the cases.
• Voltage profile in the system improved.
• Dynamic ATC corresponding to the Hopf Bifurcation
became the limiting criteria in most of the cases.
• Decrease in KR, resulted in the fall in ATC values
corresponding to the bus voltage limit and to the
static limit.
• For dynamic ATC –corresp. to Hopf bifurcation,
however, no specific trend was observed.
06/12/09 128

FUTURE RESEARCH AREAS
 Determination of closest Hopf bifurcation and
saddle node bifurcation for worst case values.
 Impact of FACTS controllers in the
enhancement of ATC along busy corridors.
 Realistic load modelling describing its nature
and dynamics.
06/12/09 129

Thank You!
06/12/09 130