1. INTRODUCTION
Power transfer distribution factors (PTDFs) are commonly used in
calculations required for deciding on transmission loading relief
procedures. Elements of the PTDF matrix represent the sensitivities of
branch power flows to bus power injections.
These sensitivities can be used in various power system calculations
such as those required for approval of new power transactions, congestion
management, or finding the impact of power transactions on flowgates
[2,6]. Any
errors in PTDF calculations will directly translate into operator
decisions that are incorrect and possibly harmful for the system
security.
By definition, PTDF calculations are functions of the operating point
at which they are evaluated. Those PTDFs that are evaluated at a flat
start using the well-known decoupling assumptions of [8] are referred to as DC PTDFs. Using the full
coupled model and evaluating the sensitivities at the current operating
point will yield the AC PTDFs. Any time the network topology changes, the
PTDFs will have to be updated. Moreover, changes in network parameters
such as line impedances will also affect these sensitivities and,
therefore, should be monitored and corrected whenever necessary.
The AC PTDFs, which yield more accurate sensitivities of flows to bus
injections, can be obtained each time the state estimator is executed. The
system model and current operating conditions will be supplied by the
state estimator, which processes the system measurements and the output of
the topology processor to obtain the best estimate of the system state.
Hence, the uncertainties associated with the inputs to the estimator will
also affect the quantities computed based on the state estimator's
output, such as the AC PTDFs.
In this paper the effect of the measurement and parameter
uncertainties on AC PTDF calculations is studied. In the formulation of
the commonly used weighted least squares (WLS) state estimation method,
the measurement errors are assumed to be distributed according to a Normal
distribution with zero mean. The distribution of PTDFs can then be
obtained based on the assumed distribution of the measurement errors. This
can be done not only for Normally distributed measurement errors but also
for outliers (bad data). It has been reported in the literature
[3] that although the PTDFs remain
relatively insensitive to changes in system loading, they are
significantly affected by topology and network parameter changes. In this
paper the specific case of the transmission line parameters is considered
and the sensitivity of AC PTDFs to these parameters is also investigated.
A transmission line parameter may be incorrect due to simple database
errors or due to unmonitored changes resulting from environmental or
loading conditions. Although most existing state estimators are not
configured to estimate network parameters, this capability certainly
exists, as shown in the literature [7]. It will be shown that unless this capability
is properly exploited, PTDF calculations may contain serious errors when
network topology or parameter errors exist.
The paper is organized such that the methodology is explained in
Section 2, followed by Section 3, where the simulations on the IEEE 14 bus
test system are presented. These simulations illustrate the effects of
measurement, topology, and parameter errors on AC PTDFs. Section 4
contains the conclusions and final remarks.
2. METHODOLOGY
The PTDF matrix represents the sensitivities of power flows to bus
injections, and its (i,j)-th element can be defined
as
where x0 is the operating state of the system,
PF,i is the power flow through
branch i, and PB,j is the
power injected at bus j. The partial derivatives can be evaluated
at any operating point and the PTDF matrix can be compactly written as
where Jf is a [m ×
(2 × n − k − 1)] gradient matrix
of the real power flows with respect to states, J is the power
flow Jacobian of order (2 × n − k −
1), m is the number of the transmission lines, n is the
number of the buses, and k is the number of the generation buses
(slack and PV buses). If the operating state is assumed to be the
flat-start condition and the decoupled Pθ submatrices are
substituted for Jf and J, then (1)
will yield the so-called DC PTDF matrix.
In this study the following measurement model will be assumed:
where z is the measurement vector, h(x) is
the nonlinear state estimation function, and e is the measurement
error that has a Normal distribution with zero mean and known (assumed)
variance.
In order to study the effect of the measurement error variances on
PTDFs, the following procedure is followed:
- Gaussian noise with zero mean and a prespecified standard deviation
σ is introduced to the same set of measurements generated using a base
case power flow. All measurements are assumed to have the same error
distribution. K sample measurement sets are created.
- State estimation solutions are obtained for each of the K
sampled measurement sets and the corresponding AC PTDF matrices are
calculated using these solutions.
- Steps 1 and 2 are repeated for different choices of σ.
A similar procedure is followed for studying the effect of a single
gross measurement error on the calculation of the AC PTDFs. In this case,
Step 1 is repeated. Then, before performing the state estimation, the sign
of one of the measurements is reversed in all of the K sample
measurement sets. This introduces a persistent gross error in all of the
samples, which are otherwise free of gross errors, only having Gaussian
noise in the measurements. Step 2 is then carried out to obtain K
samples of the AC PTDFs.
The final investigation is related to the topology and network
parameter errors. Topology errors occur when a switching event is
unreported and the topology processor fails to provide the correct network
connectivity information to the state estimator. This leads to a mismatch
between the actual system model and the model assumed by the state
estimator, which subsequently yields a biased solution. Another type of
error occurs when one or more of the network parameters are incorrectly
entered into the real-time database. This sort of error can be due to many
different reasons. For instance, in the case of transmission line
parameters, they are known to vary as a result of changes in the ambient
temperature, leading to line sag and overheating of the conductors. The
amount of parameter variation may be quite significant; for example,
considering the three-phase line with Pheasant-type conductors, line
resistance will change from 0.0751 Ω/mile to 0.0890
Ω/mile due to a change in temperature from 25°C to 75°C
[4]. Hence, the effects of such
parameter errors on the calculation of AC PTDFs are expected to be
significant. The following simulation study is carried out in order to
quantify this significance.
Considering the same test system, the resistance and reactance of a
line jk are assumed to be random variables with a Normal
distribution. The mean values μ of their distributions are chosen as
the base-case values, and for each parameter the corresponding σ is
chosen as 0.3μ (i.e., 30% of its mean value). The following steps are
performed to carry out the simulations:
- K samples of network parameters are generated by treating
all line parameters deterministically except for the chosen line
jk, whose parameters are allowed to vary according to a Normal
distribution. K measurement sets are created from the power flow
solutions corresponding to the K samples of network parameters.
It is assumed that there is at least one incident measurement (terminal
injections or line flows) to line jk among the measurements in
order for the transmission line parameters to be observable [5].
- K sets of AC PTDF matrices are calculated corresponding to
the state estimation solutions obtained for the measurement sets created
in Step 1.
The following section contains the results of implementing the
above-described test procedures using a small test system and its
measurement configuration.
3. SIMULATIONS
Figure 1 shows the system diagram and the
measurement configuration for the IEEE 14 bus test system, which is used
for all simulations. Bus 1 is the slack bus and buses 2 and 3 are PV
buses. Power Educational Toolbox (PET) [1] is used to verify the results obtained in a
MATLAB environment. Measurement configuration consists of 17 flow
measurements, 2 injection measurements, and 1 voltage magnitude
measurement.
IEEE 14 bus test system measurement configuration.
The three issues that were discussed in the previous section will be
investigated using simulations. The effects of measurement noise, bad
data, and network parameter errors will be illustrated separately in the
following subsections.
3.1. Effect of Measurement Noise
The sample size K is chosen as 100 and the three-step
procedure outlined in Section 2 is carried out for this case. The
measurement noise levels are varied by changing the standard deviation of
the error distributions N(0,σ2) as follows:
The results are shown only for a chosen entry of the PTDF due to the
space limitations; however, all other entries are observed to behave in a
similar manner.
The PTDF matrix entry (12, 11) is observed for each distribution,
where the 12th row of the matrix corresponds to the transmission line
6–12 and the 11th column corresponds to bus 12. The entry (12, 11)
of the PTDF matrix when calculated by using error-free measurements is
equal to −0.5802. As expected, the mean values of the PTDFs in Table 1 are all close to this value. However, the
standard deviation of the PTDF entry (12, 11) increases linearly with the
σ of measurement errors.
Effect of Measurement Error Variances on PTDF(12, 11)
3.2. Effect of Bad Data
The effect of a biased measurement or bad data on PTDFs is presented
in this subsection. The flow measurement on line 6–11 in Figure 1 is assumed to be biased. The number of
simulated samples is again chosen as 100 and measurements are generated by
introducing errors from a Normal distribution with zero mean and 0.001
standard deviation. In each measurement set, the sign of the flow
measurement 6–11 is reversed to simulate a gross error.
In this case, the entry (11, 10) of the AC PTDF is observed. Here, the
11th row corresponds to line 6–11. The PTDF matrix entry (11, 10) is
calculated as −0.5423 by using error-free measurements. Table 2 shows the results of simulations. In this
case, there is a bias in the distribution of the calculated PTDF(11, 10).
Its value changes from −0.5423 to −0.5316 as a result of the
gross error in one of the measurements. The effect of normal variation due
to measurement noise in the remaining measurements remains the same as in
Section 3.1. As the magnitude of the gross error increases, the AC PTDFs
will become more biased. This observation highlights the importance of
robust state estimators, which can properly detect, identify, and
eliminate gross errors so that other application functions will not be
affected.
Effect of Bad Data on PTDF(11, 10)
3.3. Effect of Parameter Error
In this subsection the effect of a change in transmission line
parameter on PTDFs is studied. Similar to the previous cases, 100
measurement sets are created using the power flow solutions obtained by
changing the impedance of transmission line 6–12 for each solution.
The impedance value of 6–12 is sampled from a Normal distribution
N(μ,σ2), where μ = Z0
and σ = 0.3Z0, Z0 being the
base-case value of line 6–12.
Two different sets of state estimation solutions are obtained. The
first set of solutions is computed using the modified impedance of line
6–12 for each solution. In this case it is assumed that the changes
in the parameters of the line are somehow monitored by the state estimator
and by the subsequent PTDF calculator. The second set of solutions is
obtained using the base-case value Z0 for line
6–12 impedance. This case assumes that the changes in the line
parameters go undetected and that the state estimator as well as the
subsequent PTDF calculator will use the fixed base-case value for the line
parameter even though its actual value will be varying for each solution.
Table 3 gives the calculated values for the (12,
11)-th entry of the AC PTDF matrix for these two cases.
Effect of Parameter Change on PTDF(12, 11)
Note that when calculated using error-free measurements and the
base-case value for the line parameter, the value of PTDF(12, 11) is
−0.5802. When the changes in the line parameters are known and taken
into account in state estimation and subsequent PTDF calculations, the
PTDFs also change. For instance, in the case of PTDF(12, 11), the range of
values it assumes is between −0.7976 and −0.4547.
As expected, the effects of network parameter errors on PTDFs are more
significant than the effects of measurement errors. In Table 3, it can be observed that the standard
deviation of PTDF(12, 11) is about 10% of its base-case value. Such errors
may play a critical role in approving transactions. Consider the case
where 1 MW of power transaction is planned from bus 1 to bus 12. The
system operator will decide whether to allow this transaction, according
to the PTDFs that are calculated based on the information supplied by the
topology processor. The base case PTDF(11, 12) is −0.5802. However,
assume that the parameters of line 6–12 change, subsequently
changing PTDF(11, 12) also to −0.7976. This implies that 1 MW of
power transaction from bus 1 to bus 12 will actually create a 0.7976 MW
power flow increase on line 6–12 instead of the predicted 0.5802 MW
using the base-case parameter values. In this case, if the thermal limit
of the line happens to be 0.6, allowing this transaction may lead to
thermal overloading of the line 6–12.
Unfortunately, parameter errors are not easily detected and identified
by existing bad data processing methods. This can be demonstrated by a
modified measurement configuration of the same system. Assume that the
line flow measurement at 6–12 is removed and injection measurements
at buses 6, 11, 12, and 13 are added. In this case, an error is introduced
in the parameters of line 6–12. When state estimation is performed,
the largest normalized residual analysis yields the results shown in Table 4. Discarding the measurements with largest
normalized residuals and repeating the state estimation will result in the
removal of injections at 12 and 6, as shown in Table
4. No more bad data are suspected after the third state estimation
cycle. This results in a biased estimate for the state and, worse yet, it
will allow the parameter error to go undetected. The subsequent
calculations including those involving PTDFs will all be erroneous.
Largest Normalized Residuals
One possible approach is to incorporate proper parameter error
detection and parameter estimation algorithms [7]. A suspect set of parameters can first be
identified and their values can be estimated along with the states.
6. CONCLUSIONS
In this paper the effects of measurement and network parameter errors
on AC PTDFs are discussed. It is shown that AC PTDFs are not affected
significantly by Gaussian errors in the analog measurements, whereas gross
errors in measurements may cause biased PTDFs. The effect of the bad data
on AC PTDFs strongly depends on the severity of the bias of the
measurement. It is also shown that the parameter errors have a significant
impact on AC PTDFs. The error can be detected by largest normalized
residual analysis, but it is more difficult to identify it as a parameter
error. This topic requires further investigation. The function of the
state estimator is certainly broadened in the deregulated operation of
power systems, since it is now expected to provide not only the optimal
state estimate but also the most probable network topology and network
parameters.
