On Using Linear Approximation and Distribution Factors
In the accelerated environments of today’s electric energy markets,
fast analyses of power flows are a must. Emerging realtime and
dayahead markets require that analysis of infrastructure capacity be
performed in a compressed timeframe. Whereas the electric demand of
consumers and industry may retain its wellknown cyclical nature,
varying by time of day, by season and by local weather and social
patterns, the supply side of the equation has drastically changed.
Competition has engendered even the traditional suppliers of energy to
be more flexible and anticipatory to pricing and demand signals,
affecting operating and bidding strategy in timeframes that range from
the next operating hour to when the next new generation facility can be
interconnected. In addition, newer energy sources such as wind and solar
power introduce new dependencies which vary hourtohour to the supply
mix.
Power flows are the tools by which system analysts determine the
impact of coincident electric demand and supply bids on the
infrastructure of the AC interconnected transmission grid. The power
flow tool used most prevalently, known as the AC power flow, requires
specialized knowledge to use, is subject to convergence issues and
modeling constraints and requires continuous monitoring for data
integrity. Databases used for transmission analysis model, for example,
the whole Western or Eastern U.S. Interconnection, and parts of Canada
and Mexico, maintained through cooperative interchange among many
operating entities.
In compressed timeframes, approximate but fast and consistent power
flow solutions tend to be preferred over accurate but unwieldy ones.
Hence, we find the increasing popularity of linear approximations to the
power flow. The reason for widespread use may have to do with the fact
that linear methods are much easier to apply, are not subject to
convergence issues, and require less data.
Assumptions
A linear approximation, sometimes also referred to as a DC power
flow, applies assumptions to the AC power flow equations that simplify
the calculation and makes it noniterative, and thus not subject to
convergence issues that plague the AC power flow. Some of the key
assumptions include:
 The resistance component of transmission lines is much smaller
than the reactive component.  Reactive power flows are negligible thus accurate representation
of generator reactive capability, reactive demand and power
conditioning devices or static var devices is not required.  Voltage magnitudes are fixed at either nominal or some assumed
initial vector.  Line losses are negligible, constant or can be approximated with
a heuristic method that does not require solving the AC power flow.
Various methods for handling losses lead to various forms of DC
power flows as evidenced by a variety of commercial software.
Distribution Factors
The DC power flow maintains energy balance by ensuring that the total
MW supplied equals the total MW consumed. Any changes to the initial
energy balance condition may be approximated by extending the
linearization paradigm through the use of distribution factors.
Distribution factors, or dfax, are a measure of the impact of
injections and network changes on the grid applied over the initial
energy balance condition or base case power flow. They are a function of
network topology (noting that the assumption that resistance is a small
component relative to the reactance of transmission lines is still
applicable here) alone. There are a number of forms of dfax, including:
 Power injection dfax – measure the change in MW flow on
transmission branches as a function of an injection of unit MW at a
certain node. (See Figure 1)
(Note: To maintain energy balance, a positive MW injection is
offset by a negative injection, or absorption, at another location.
The balancing location is specified either explicitly as a pair to
the injection, or explicitly via a “swing bus” defined in the power
flow data.)
Figure 1: Illustration of Power Injection DFAX.
 Line outage dfax – measure the change in flow on branches as a
function of the outage of a transmission branch. (see Figure 2)
Figure 2: Illustration of Line Outage DFAX.

Power transfer dfax – measure the total change in MW
flow on a transmission branches for a set of injections and
absorptions at various buses. 
Line outage transfer dfax – are a form of power
transfer dfax with specific line outages or contingencies. Hence,
this form of dfax apply only to the postcontingency state.
Since dfax are linear, when dfax are superposed, the
resulting impact is the algebraic sum of the individual impacts. Hence,
line outage transfer dfax may be determined from the PIdfax and LOdfax
of Figures 1 and 2.
The attractiveness of dfax for realtime and transaction
analysis applications may be attributed to its conceptual simplicity and
uncomplicated solutions. Software that rely primarily on dfax methods
are widely used, not only at control centers but in planning
environments.
Exactly how approximate?
Dfax are approximations. Errors may vary from small to
significant. Some factors that impact the magnitude of the approximation
error from dfax are:

Voltage profile of the starting power flow model.
Approximation errors increase the larger the deviations from nominal
voltage. 
Voltage level of transmission lines. Error is
greater for lower voltage lines than higher voltage lines. 
Utilization level of the grid. Error increases with
a more severely loaded grid such as may be experienced during the
peak load hour of the year. 
Large injections tend to result in larger errors. In
combination with the previous bullet, the higher the ratio of the
amount of injection with respect to the loading level or demand, the
higher the error. 
The presence of underground or submarine cables or
series compensated lines also contribute to approximation error.
Sometimes, there is a mitigation effect to the errors in
that errors that overestimate may compensate for errors that under
estimate, resulting in less net error than expected. This is more a
coincidental than an intrinsic characteristic of the method.
In some applications, dfax that are calculated without
consideration of losses may be supplemented with loss factors, or lfax,
which estimate the change in transmission losses as a function of power
injections or line outages. Loss factors are themselves an approximation
of the losses that would be determined if a full AC power flow solution
were applied.
Case Studies
Both case studies used for illustrative purposes are
based on reallife cases in the Eastern Interconnection (North America).
For our first case study, we consider the system shown
in Figure 3, wherein a new power plant A is proposed for
interconnection. Substation A is an existing facility that serves as
termination for four 161 kV transmission lines.
Figure 3: First case study.
The limiting contingency for power injections from Plant
A is the outage of Line 4. Before any new generation, this contingency
results in a thermal load on line 3 of 85% of rating. What is the
maximum MW injection from Plant A that will not overload line 3?
Using dfax, the predicted limit is 95 MW. Checking with
AC power flow solutions, there are three possible answers, as follows:

The voltage at Substation A is 102% of nominal
before the addition of Plant A. If this voltage is maintained, the
limit based on an AC power flow solution is 75 MW.If Plant A is able to provide the reactive power to raise the
voltage at Substation A to 105%, the limit on injected power is 100
MW. 
If Plant A is a reactive power sink such as an
induction generator and reduces the voltage at Substation A to 100%
nominal, the maximum power at Plant A is 60 MW. 
The answer predicted by dfax is close to no. 2 above
and appears to assume that Plant A supplies significant reactive
power to raise the voltage at Substation A.
For our second case study, we consider a 500 kV
interface that is the path for transactions of power from west to east.
The limiting constraint is the thermal rating of one of the EHV lines on
loss of the other line as shown in Figure 4. For this contingency, the
starting case shows an AC power flow loading on the remaining EHV line
of 89% of its thermal rating.
Figure 4: Second case study.
A power transfer dfax calculation on the starting case
predicts that an incremental transfer of 1869 MW can be supported before
the EHV line hits its thermal limit.
Using AC power flows to gradually increment the west to
east power transfer shows a resulting transfer limit of 1500 MW. The dfax and AC power flow results are plotted in Figure 5. As in the first
case study, the dfax solution assumes voltages are either steady or
slightly rising as transfers are increased. In reality, increasing
reactive losses on the EHV line results in dropping voltages along the
transfer path, as shown in Figure 6. The dfax method in this case thus
overestimates the actual transfer limit.
Figure 5: Power Transfer Limits determined by dfax and AC power flow.
Figure 6: Underlying reactive power losses as transfers increase.
Conclusions
Proponents of linear models have justified their use by
the fact that market uncertainty already embeds an inherent
approximation in the power flow calculations that is not overcome by
slower but more accurate power flow solutions. However, there is the
risk of the error from uncertainty adding on to the error from linear
approximation moving analytical results further from the reality that it
purports to represent.
However, the ease of use and relative simplicity of
linear models make it attractive for realtime and short timeframe
assessments of power transactions, market dispatch and loading relief.
Where reactive losses and voltages are expected to be a factor,
additional margins need to be applied to ensure that the linear methods
do not give an overly optimistic estimate.
In general, some caution is needed when applying linear
models. Where the application does not require fast turnaround or
involves future conditions that are not day ahead but months or years
ahead, using more accurate methods remains the safe and prudent
approach.