EXHIBIT 99.324
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A LOAD ESTIMATION METHOD FOR DISTRIBUTION SYSTEMS
Carol S. Cheng Dariush Shirmohammadi
Powcom Perot Systems
Fremont, California Los Angeles, California
Abstract-- In this paper we present a load estimation method for distribution
systems. This load estimation method uses a small number of real-time
measurements to calculate the distribution systems state variables. The proposed
method uses a power flow based method to calculate a sensitivity matrix which
quantifies the relation between the loads downstream from each measurement point
and the current or power flow monitored at that measurement point. It then uses
an iterative algorithm to calculate the amount of load change, from historical
load data, needed in order to make the flows at the monitored points match the
real-time measurements. Preliminary test results show that the method is capable
of calculating the load profile efficiently and robust enough for application in
real-time systems such as Distribution Automation systems.
Keywords: distribution automation, distribution system analysis, feeder
reconfiguration, distribution load estimation, distribution power flow,
sensitivity matrix, real-time measurements.
I. INTRODUCTION
As more advanced applications such as distribution feeder reconfiguration and
capacitor control are deployed to support the operation of distribution systems,
a real-time model of distribution system operation becomes very important to
servers as the foundation for data for these applications. The real-time system
model consists of switching statuses that determine system configuration and
load at each node. Once these are known, a power flow model can be established
and solved to determine the system operating state. This paper attempts to
address the problem of calculating the real-time loads for primary distribution
feeders using "insufficient" real-time measurements. We refer to this
calculation as load estimation.
If a redundant set of real-time measurements were available for distribution
systems, the real-time system model could be obtained using the standard state
estimation method. Unfortunately, in most distribution systems the real-time
measurements are far from sufficient, not to mention redundant, and the
situation is unlikely to change in the near term due to economic constraints.
Few real-time measurements are provided by distribution SCADA systems which
monitor and control certain automatic line equipment, such as switches,
reclosers and capacitors. Because these devices are costly to install and
maintain, and since there are just too many distribution feeders to cover,
monitoring device installation can only be justified for a few crucial locations
on primary feeders.
On the other hand, given the emphasis, the real-time data obtained by monitoring
these devices are the most credible data available and more reliable than the
seasonal load data. Seasonal load data are typically obtained by converting the
energy usage (kWh) of a customer to power usage (kW, kVAR) according to a set of
conversion curves. These curves are determined by sampling the power usage of
customers in typical customer classes such as industrial, commercial,
agricultural and residential customers.
The experienced distribution system operators often use their judgment along
with the real-time measurements, as small a number as they might be, and finally
the knowledge of seasonal load data to estimate the loads and other operating
data for areas where real-time measurements are not available. This kind of
operating practice motivated us to reach the proposed load estimation
methodology. Specifically, the proposed methodology works with a small set of
real-time measurements in order to estimate loads along distribution feeders.
The methodology uses a power flow based algorithm to calculate a sensitivity
matrix which quantifies the relation between the loads downstream of each
measurement device and the current or power flow monitored at the device. It
then uses an iterative algorithm to calculate the amount of load change needed
in order to make the calculated quantities match the real-time measurements at
all points where such measurements are available. In this methodology, we still
rely on seasonal load data as the starting point of our estimation process and
also rely on the fact that the measured data is substantially more accurate than
the seasonal data.
Compared with other work in this area [4-5] which are based on adjusting
historical load data and matching measurements, the proposed solution comes
closest to the reliable practices of distribution system operators. We also make
use of the measured data within a relatively uniform mathematical framework in
which a first order approximation of the load-to-measurement relation is derived
and used. Such a framework allows a more efficient and robust algorithm to be
developed.
II. DESCRIPTION OF THE METHOD
The proposed load estimation method uses current magnitude measurements and
power flow measurements (real and reactive) all taken from line sections.
Current measurements are typical for primary feeders while power flow
measurements have only become available in more recent years. The centerpiece of
the proposed method is the definition of a sensitivity matrix which relates load
change with measurements. The derivation of the sensitivity matrix will be
presented first, followed by the description of the load estimation algorithm.
Note that our derivation is based on radial configuration of the distribution
feeders.
2.1 The Sensitivity Matrix
For a radial feeder without shunt capacitors and inductors, the current
injection at node i is
[FORMULA OMITTED]
(1)
where Pi +jQi is the constant power load at node i, and Vi is the voltage at
node i. The current in line section l is
[FORMULA OMITTED]
(2)
where D is the set of no desin the sub-tree connected to line section l. We
refer to the nodes in D as the nodes downstream of l. Subscript L denotes that
JLl is a line section quantity. In most distribution systems, the voltage
magnitude is close to nominal value and the voltage angle is small. This means
that the current magnitude on line section l, JLl, is mainly related to the
magnitude of the apparent loads at nodes downstream of l, SDl, i.e.,
[FORMULA OMITTED]
(3)
The linear approximation of this relation is
[FORMULA OMITTED]
(4)
where dJLl/dSDl represents the sensitivity of current magnitude on line section
l with respect to the load downstream from line section l. Suppose there are m
line sections in a radial feeder on which the current magnitudes are being
monitored, the effect of each downstream load on its own upstream line section
as well as on other monitored line sections can be represented using a
sensitivity matrix in the following linear equation:
[FORMULA OMITTED]
(5)
or in a compact form as
[FORMULA OMITTED]
(6)
where [AJS] is the sensitivity matrix. Its dimension is mm. Note that in
deriving the sensitivity matrix, the sum of all loads downstream of a monitored
line section is considered as one variable. Also note that in (5) and (6), the
incremental change of the downstream load (DELTA)SD is the magnitude of the
apparent load. This implies that the load power factor remains a constant.
If measurements of real and reactive power flows are also available on other
line sections, then similar approximation may be made to establish the
relationship between the real power flow and the downstream real loads, as well
as the reactive power flow and the downstream reactive loads. Specifically, for
a radial feeder without shunt elements, the power flow on line section l can be
expressed as
[FORMULA OMITTED]
(7)
[FORMULA OMITTED]
(8)
where Pi and Qi are the real and reactive loads at node i,RLk and XLk are the
resistance and reactance of line section k, and the summation is on the set of
nodes and line sections downstream from the monitored line section. It is seen
that the power flow on line section l consists of two portions: one is due to
its downstream nodal load and the other is caused by line losses. Since in most
cases, line losses are much smaller that the loads, it is reasonable to assume
that the real power flow on line section l is mainly related to its downstream
real loads, and reactive power flow online section l is mainly related to its
downstream reactive loads. It is important to note that these assumptions are
made to simplify the process of establishing the sensitivity matrix and they do
not diminish the accuracy of the final results. With the above assumptions (6)
can be expanded to include the incremental relation between the line section
power flow and the downstream load, as follows:
[FORMULA OMITTED]
(11)
Suppose there are m line sections being monitored for their current flow
magnitudes, and there are n line sections being monitored for their real and
reactive power flow, then the dimension of the sensitivity matrix is
(m+2n)x(m+2n).
The sensitivity matrix is formed column by column using the following numerical
approach. Suppose a base case power flow has been solved using the seasonal load
data. Then the following steps are used to calculate column (FORMULA OMITTED) in
the sensitivity matrix:
1) Increase the real and reactive component of every load downstream of the
j-th monitored line section by 1.0% from its seasonal load,
2) Solve a radial power flow with the increased load, and
3) Calculate the change (in percentage) of current magnitude from the base
case power flow solution for all line sections with monitored current flow
or real and reactive power flow.
In this way, m radial power flow solutions are needed to form the first m
columns in the sensitivity matrix. Then each of the next n columns,
corresponding ton real power flow measurements, are formed in a similar way
except that instep 1) above, each of the downstream real load is increased by
1.0%. Finally, for each of the last n columns, corresponding to n reactive power
measurements, each of the downstream reactive load is increased by 1.0%. In
summary, totally m+2n radial power flow solutions are needed to form the entire
sensitivity matrix. Given the fact that the number of line sections with
measurements are small, and the solution of a radial power flow is very
efficient, as demonstrated and summarized in [1-3], this way of forming the
sensitivity matrix is feasible for all existing primary distribution systems.
Data on the performance of the proposed method can be found in Section 3.3.
This approach can be extended to also cover voltage measurements by first
converting voltage measurements to pseudo power flow measurements using
estimated currents. However, the pseudo power flow measurements should be
updated on regular basis.
2.2 Load Estimation Algorithm
An iterative procedure is used to calculate the required load change. At
iteration:
1. For each line section with monitored current magnitude, calculate the
current magnitude mismatch between the power flow solution and the
measurements, and normalize the mismatch as a percentage of the current
magnitude by power flow solution:
[FORMULA OMITTED]
(10)
Where JLiM is the i-th current measurement.
2. For each line with monitored real and reactive power flow, calculate the
power flow mismatch between the power flow solution and the measurements,
and normalize the mismatch as a percentage of the power flow by power flow
solution:
[FORMULA OMITTED]
(11)
[FORMULA OMITTED]
(12)
Where PLiM and QLiM are the i-th real and reactive power flow measurement,
respectively.
If any of the mismatches calculated in steps 1 and 2 is greater than its
corresponding threshold, then perform the next step:
3. Solve Equation (9). The solution of (9) gives the required downstream load
correction, in terms of the percentage of the existing load, that would
eliminate the mismatch at each monitored line section. For example, once
[(DELTA)SD] is available, the load at each node, Sj, downstream from the
measurement point i can be updated as
[FORMULA OMITTED]
(13)
4. Solve a radial power flow with the newly adjusted load. Go to step 1 to
calculate [FORMULA OMITTED]
Steps 1-4 will be repeated until the mismatches calculated in steps 1 and 2 are
smaller than the thresholds. This means that a power flow based on the estimated
load will give the same current or power flow values on monitored line sections
as the real-time measurements.
From (13) it is seen that the loads downstream from a monitored line section is
corrected by the same percentage. This means that the resolution of the proposed
load estimation method is limited to downstream load rather than nodal load.
This is not surprising because of the measurement shortage. On the other hand,
the relative proportion between the nodal loads downstream from a monitored line
section is preserved, which is a useful piece of information embedded in the
seasonal load data.
2.3 Effects of Shunt Capacitors
In deriving the sensitivity matrix, we have assumed that there are no shunt
elements in the system. In practice, shunt capacitors are often used on
distribution feeders to improve voltage profile. If one or more shunt capacitors
are downstream of a monitored line section, then the correlation between the
downstream load and the current or power flow on the monitored line section may
be less obvious. As shown in the numerical test later, the degree of this
correlation is indicated by the value of the diagonal entries in the sensitivity
matrix. If a value in the diagonal is close to 1.0, it means that that 1% of
downstream load change has resulted in 1% of current or power flow change, thus
the correlation is strong. We have observed numerically that if the downstream
load is large, even if there are several capacitors downstream of a measurement,
the corresponding diagonal value is still close to 1.0. Other wise, the effects
of capacitors become dominant (the diagonal value is different from 1.0). A
threshold can be used to decide whether a measurement with downstream capacitors
is to be matched. For example, if the diagonal value is outside the range of
0.80 (tilde) 1.20, then the corresponding measurement will not be included in
the sensitivity matrix.
2.4 Other Considerations
If there are relatively more real-time measurements in a system, simple
comparisons may be made to check the consistency of data. In a radial system, a
current measurement downstream of another should be smaller than that
measurement. If a power flow measurement is near a current measurement, then
their values should be closely related by the nominal system voltage. If a given
set of real-time measurements is found to be inconsistent, some data can be
removed from the measurement list by a simple set of rules embedded in the
algorithm to maintain data consistency.
III. TEST RESULTS
The proposed load estimation method has been tested using several 21kV large
radial distribution systems, constructed based on our knowledge of some existing
primary distribution feeders. To make the test general, in addition to the
actual measured points and data, we created our own measurements by randomly
moving the location of the monitored line section, and randomly generating the
measured values by deviating the base case power flow solution (again randomly)
in both direction from 1% up to 70%. Table 1 lists the major features of the
three test systems. The total P and Q load are calculated from the base case
load data (historical load data). The root current is from the base case power
flow solution.
Table 1 Test Systems
<Table>
<Caption>
System 1 System 2 System 3
------------- ------------- -------------
No. of nodes 603 660 1651
------------- ------------- -------------
No. of loads 471 519 1185
------------- ------------- -------------
No of Capacitors 6 3 7
------------- ------------- -------------
Root current (Amps) 299. 569. 723.
------------- ------------- -------------
Total P load (kW) 3622. 6461. 9003.
------------- ------------- -------------
Total Q load(kVAR) 1841. 3316. 3747.
------------- ------------- -------------
</Table>
3.1 Test Results of System 1
We randomly placed 8 current measurements and 3 pairs of power flow
measurements in the system, as shown in Table 2.
Table 2 Simulated Measurements
<Table>
<Caption>
Measurement No. Base case Deviation Simulated
type power flow from base Measure.
solution case(%) Value
- ----------- ------ ---------- --------- ---------
Amps 1 71.73 23. 88.22
------ ------ ------ ------
Amps 2 257.81 25. 321.89
------ ------ ------ ------
Amps 3 0.17 -17. 0.14
------ ------ ------ ------
Amps 4 14.77 -19. 12.00
------ ------ ------ ------
Amps 5 0.09 12. 0.10
------ ------ ------ ------
Amps 6 0.82 -7. 0.77
------ ------ ------ ------
Amps 7 18.85 13. 21.24
------ ------ ------ ------
Amps 8 9.09 14. 10.40
------ ------ ------ ------
P(kW) 9 2.60 -7. 2.41
------ ------ ------ ------
P (kW) 10 23.60 26. 29.67
------ ------ ------ ------
P (kW) 11 7.81 -7. 7.23
------ ------ ------ ------
Q(kVAR) 12 1.39 -4. 1.33
------ ------ ------ ------
Q(kVAR) 13 12.36 -1. 12.26
------ ------ ------ ------
Q(kVAR) 14 4.18 -3. 4.07
------ ------ ------ ------
</Table>
Note that the second current measurement is at a line section close to the root
node, since it carries 86% of the root current (based on the base case power
flow). There are also 5 capacitors downstream of this measurement, with shunt
admittance ranging from 0.00126 to 0.00378 (1/Ohm). The sensitivity matrix is
built column by column according to the measurement order shown in Table 2.
[FORMULA OMITTED]
It is seen that the second column is dense. This is due to the fact the second
current measurement is close to the root node, and all other measurements are
located within the downstream line sections of the second measurement. Therefore
changing the downstream load for the second current measurement causes the
change of the downstream loads of all other measurement, which are reflected in
the second column of the sensitivity matrix. It is also noted that the second
diagonal entry is 0.95 while most other diagonal entries are 1.0. The slight
drop of the second diagonal value demonstrates the effects of the 5 capacitors
downstream of the second measurement.
It took 4 iterations for the load estimation algorithm to converge using the
convergence criteria(epsilon)=0.01%. The maximum current and power flow mismatch
in each iteration are listed in Table 3.
Table 3 Convergence Pattern
<Table>
<Caption>
Initial Iter. 1 Iter. 2 Iter.3 Iter. 4
------- ------- ------- ------- -------
|JL|max 24.85 1.19 0.21 0.035 0.0057
------- ------- ------- ------- -------
|PL|max 25.68 0.00030 0.00001 0.00000 0.00000
------- ------- ------- ------- -------
|QL|max 4.36 0.00006 0.00000 0.00000 0.00000
------- ------- ------- ------- -------
</Table>
At the convergence, 412 out of the total 471 loads are modified, the maximum
nodal load change is 31% of its base case value.
Recall that a power flow is solved in each iteration based on the corrected
load. Thus at the convergence, a power flow solution is available to provide
system voltages, which are system state variables.
3.2 Test Results of System 2
In this test we examined the performance of the proposed load estimation
algorithm for different number of measurements, and different simulated
measurement values. Tables 4 and 5 summarize the number of iterations and the
maximum nodal load change.
Table 4 Number of Iterations
<Table>
<Caption>
Max.deviation-> up to 30% up to+/-50% up to +/-70%
No. of measurs.
- ---------------- --------- ----------- ------------
5J, 2P, 2Q 3 3 4
--------- ----------- ------------
10J, 5P, 5Q 3 4 5
--------- ----------- ------------
20J, 10P, 10Q 5 diverge diverge
--------- ----------- ------------
</Table>
Table 5 Maximum Nodal Load (P,Q) Change (%)
<Table>
<Caption>
Max.deviation-> up to 30% up to+/-50% up to +/-70%
No. of measurs.
- ---------------- --------- ----------- ------------
5J, 2P, 2Q 28, 28 46,46 64, 64
--------- ----------- ------------
10J, 5P, 5Q 30, 33 49, 54 68, 73
--------- ----------- ------------
20J, 10P, 10Q 95,95 -- --
--------- ----------- ------------
</Table>
As can be seen, even if the real-time measurements are drastically different
from the base case power flow solution values, the proposed load estimation
method is quite effective in matching the real-time measurements; provided that
there are enough "room" for adjusting loads (i.e., there are not too many
measurements). If there are many real-time measurements, and their values are
significantly different than the base case power flow solution, it is suggested
that a set of very reliable measurements be selected and used in the proposed
algorithm.
If the measurements are less drastically different than the base case power flow
solution, for example, all the measurement values are greater than the base case
power flow solution, then more measurements can be handled with robust
performance, as shown in Tables 6 and 7.
Table 6 Number of Iterations
<Table>
<Caption>
Max.deviation-> up to +30% up to +50% up to +70%
No. of measurs.
- --------------- ---------- ---------- ----------
5J, 2P, 2Q 2 2 3
---------- ---------- ----------
10J, 5P, 5Q 3 4 5
---------- ---------- ----------
20J,10P, 10Q 4 5 6
---------- ---------- ----------
30J, 15P, 15Q 4 5 5
---------- ---------- ----------
</Table>
Table 7 Maximum Nodal Load (P,Q) Change (%)
<Table>
<Caption>
Max.deviation-> up to +30% up to +50% up to +70%
No. of measurs.
- --------------- ---------- ---------- ----------
5J, 2P, 2Q 28, 28 46, 46 64,64
---------- ---------- ----------
10J, 5P, 5Q 39, 32 49, 52 68,71
---------- ---------- ----------
20J, 10P, 10Q 42, 35 74, 58 109,84
---------- ---------- ----------
30J, 15P, 15Q 34, 34 56, 56 77,77
---------- ---------- ----------
</Table>
3.3 Test Results of System 3
System 3 is a large system with 1651 nodes and 1185 loads. We used this system
to measure the execution time of the proposed load estimation method. The test
was performed on a Pentium 166MHz PC. The execution time includes generating
simulated measurements, performing load estimation and writing a simple report.
The measured execution time for various test scenarios are summarized in Table
8.
Table 8 Execution Time (second)
for a 1651 node system on a Pentium 166 PC
<Table>
<Caption>
Max. deviation-> up to +30% up to +50% up to +70%
No. of measurs.
- ---------------- ---------- ---------- ----------
5J, 2P, 2Q 0.35 0.33 0.38
---------- ---------- ----------
10J, 5P, 5Q 0.77 0.82 0.82
---------- ---------- ----------
20J, 10P, 10Q 1.26 1.32 1.42
---------- ---------- ----------
40J, 20P, 20Q 2.47 2.78 2.80
---------- ---------- ----------
</Table>
As can be observed, the proposed load estimation algorithm is very efficient.
The execution time is not significantly affected by the magnitude of deviation
between the measurements and the base case load data. The execution time
increases nearly proportionally as the number of measurement increases. As seen
in the last row of Table 8, even for a large system with 1651 nodes and with as
many measurements as 80 points, the algorithm takes less than 3 seconds to
complete. This demonstrates that the proposed load estimation method is well
suitable for real-time operation of distribution systems.
IV. CONCLUSIONS
In this paper we have presented a novel load estimation method for primary
distribution systems. The basic idea of the proposed method, which is also the
fundamental difference compared with most state estimation method, is its
dependence on the real-time measurements. The proposed method works with small
sets of current and power flow measurements to calculate the system load. The
advantage of the method include:
(1) it is capable of providing a load profile that reinforces the real-time
measurements as well as maintain the relative proportion between nodal
loads embedded in seasonal load data,
(2) it coincides with the intuitive and practice of distribution system
operators in dealing with real-time measurement shortage, and
(3) it is computationally efficient and robust.
The proposed load estimation method has been fully tested using simulated
practical feeders and the results show that it is efficient and robust, and
suitable for real time analysis of primary distribution feeders.
V. ACKNOWLEDGMENT
We would like to thank our former colleagues in the Pacific Gas and Electric
Company for enriching our knowledge and inspiring our thinking toward this
research.
REFERENCES
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1988, pp.753-762.
[2] C. S. Cheng and D. Shirmohammadi, "A Three-phase Power Flow Method for
Real-time Distribution System Analysis", IEEE Trans. On Power Systems,
Volume10, Number 2, May 1995, pp. 671-679.
[3] G. X. Luoand A. Semlyen, "Efficient Load Flow for Large Weakly Meshed
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[4] I. Roytelman and S. M. Shahidepour, "State Estimation for Electric Power
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CAROL S. CHENG (M'92) received her BS in Electrical Engineering from Northern
Jiaotong University in Beijing in 1982,MS in Mechanical Engineering from the
University of Cincinnati in 1986, and Ph.D. in Electrical Engineering from
Georgia Institute of Technology in 1991. From 1982 to 1985, she joined the
faculty of Northern Jiaotong University as a Teaching Assistant. From 1992 to
1997, she worked for PG&E where her last position was senior system
engineer/operational analyst. Since 1998 she become an independent consultant
and founder of Powcom. Her current research interests include power system
analysis methodologies and computer application development.
DARIUSH SHIRMOHAMMADI (SM'89) received his B.Sc. in Electrical Engineering from
Sharif University of Technology in 1975 and M.A.Sc. and Ph.D. in Electrical
Engineering from the University of Toronto in 1978 and 1982 respectively.
Between 1977and1979, he worked in Hydro Quebec Institute of Research (IREQ) on
the subject of external insulation. Between 1982and 1985, Dariush worked in
Ontario Hydro on the development and application the EMTP. Dariush worked with
the Systems Engineering Group of the Pacific Gas and Electric Company (PG&E)
between 1985 and 1991 where he developed advanced methodologies and computer
models for the analysis, optimization and costing of transmission and
distribution networks. His last position with PG&E before leaving in1995 was the
Director of Energy Systems Automation where he was responsible for the
development and implementation of automation technologies in electric
distribution system operation. Dariush ran his own company, Shir Power
Engineering Consultants, Inc., between 1995 and 1996 where he provided
consulting on utility industry restructuring and on power system planning,
operations and automation to several utilities in Canada, France, Brazil, Taiwan
and the US. He joined Perot Systems in 1996 where he is a Principal Business
Specialist involved in designing, implementing, integrating, and testing market
operations and settlements protocols and computer models for the California's
Power Exchange and Independent System Operator.