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Contract No. DE-AC36-08GO28308

DWT-Based Aggregated Load 

Modeling and Evaluation for 

Quasi-Static Time-Series 

Simulation on Distribution 

Feeders 

Preprint 

Xiangqi Zhu and Barry Mather 

National Renewable Energy Laboratory 

Presented at the IEEE Power and Energy Society General Meeting

Portland, Oregon

August 5–9, 2018

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Conference Paper

NREL/CP-5D00-70975

August 2018
NOTICE

This work was authored by the National Renewable Energy Laboratory, operated by Alliance for Sustainable 

Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding 

provided by the U.S. Department of Energy’s Office of Energy Efficiency and Renewable Energy (EERE), Solar 

Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE 

or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, 

acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish 

or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.

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DWT-Based Aggregated Load Modeling and 

Evaluation for Quasi-Static Time-Series Simulation 

on Distribution Feeders

Xiangqi Zhu and Barry Mather

Power Systems Engineering Center

National Renewable Energy Laboratory 

Golden, CO, USA

xiangqi.zhu@nrel.gov, barry.mather@nrel.gov

Abstract—This paper presents an extension of a previously 

reported discrete wavelet transform-based load modeling 

methodology that targets modeling time-series load profiles to 

enable more effective and realistic quasi-static time-series (QSTS)

simulations of distribution feeders. The time-series load model is 

composed of two major parts: 1) the low-resolution field-measured 

load data, which usually have resolutions of 30 or 15 min; and 2) 

the high-resolution load variability model data extracted from the

established variability database, which can have resolutions up to 1 

s. A load aggregation methodology is developed to aggregate the 

load profiles so that they can be effectively modeled at different 

transformer ratings. Validation, evaluation, and analysis of the 

developed load modeling approach has been performed on the 

IEEE 123-node test feeder and an actual utility feeder from 

California. The analysis completed on the two feeders has 

demonstrated the effectiveness and revealed the value of the 

developed model for distribution feeder QSTS simulations at high 

temporal resolutions. 

Keywords—Load modeling; quasi-static time-series; QSTS;

high-resolution load models; distribution system simulation

I. INTRODUCTION

Load modeling has always played a critical role in distribution 

system simulation, but in recent yearsload models that can enable 

quasi-static time-series (QSTS) simulations are in high demand. 

This is because utilities and grid operators desire effective load 

models, at high temporal resolutions, to develop and test new

control algorithms and applications that are used to inform the 

challenges brought by the increasing integration of rooftop 

photovoltaic (PV) systems and other distributed energy resources

(DERs) onto the electric power grid [1]–[-2]. Traditional load 

modeling and load modeling aggregation methods are discussed 

in [3], but such methods typically characterize loads based on 

assumed load classes and dominant behaviors. This work focuses 

on an empirical, data-driven method that leverages utility data.

Previously, when performing distribution system simulations 

or conducting new control algorithm testing, all the sub-node 

profiles were typically populated with a scaled version of the 

substation load because only substation load profiles were 

available; however, actual profiles of individual distribution 

transformer loads are usually “spikier” and contain more 

variability. Nodes with higher aggregations of load would have 

smoother profiles. Therefore, representing all the sub-node load 

profiles with only one substation level profile is not able to 

capture the variability and diversity of the load profiles along a 

feeder. Now, with the advent of advanced metering infrastructure 

(AMI), data with increased spatial resolution are often available,

but the data are still of relatively low temporal resolution (e.g., 30 

min), which is not sufficient to capture the variability of the 

feeder loads when performing the feeder simulation or 

conducting algorithm testing at high temporal resolutions. Such 

input data, when used in a study, would not be able to accurately 

reflect the parameter results, including tap changes, voltage 

ramping, line loss, etc. [4]. In addition, even if the high-resolution 

data could be obtained, the original user load data could not be 

used to develop algorithms and test the feeder because the pattern 

of the original load data would reveal the living/energy usage

pattern of the customers, thus causing privacy concerns.

To tackle the challenge of obtaining appropriate load models 

for conducting QSTS simulations, we developed a load modeling 

approach to build load profiles that could not only represent the 

variability of the realistic sub-node load but also effectively 

protect customer privacy. The developed load model consists of 

two major parts: 1) one is the low-resolution field-measured load 

data (i.e., AMI-type data), and 2) the other is the high-resolution 

load variability model data extracted from the established 

variability database, which has a resolution up to 1 s. A load 

aggregation methodology has also been developed to aggregate 

the load profiles so that load profiles at different transformer 

ratings could be modeled. The developed load model has been 

tested on the standardized IEEE 123-node test feeder and an 

actual utility feeder from California. The testing results on these 

two feeders demonstrated the effectiveness of the proposed load 

model.

This paper is organized as follows: Section II introduces the 

load modeling methodology. Section III presents the feeder 

testing, including the test on the IEEE 123-node test feeder and 

an actual utility feeder. Section IV concludes the paper and 

discusses future work.

II. LOAD MODELING METHODOLOGY

This section introduces the load modeling approach, the load 

database used to build the variability library, and the 

methodology of establishing the library. In addition, this section 
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presents the load aggregation approach that is used to develop 

load models for different load levels. 

The database we used in this paper is for residential load 

profiles because the data obtained from a utility partner primarily 

consisted of residential transformer data. A database of 

commercial or industrial loads could be established if such data 

were available. The modeling methodology we developed here 

could be applied to the commercial and industrial loads as well. 

A. Modeling Approach

A discrete wavelet transform (DWT)-based variability 

extraction methodology has been developed to establish a 

variability library using previously measured high-resolution 

transformer data [5]. As shown in Fig. 1, base low-resolution load 

profiles of hourly or 30 min-resolution are used along with highresolution data extracted from a developed variability library. In 

this way, load variability characteristics for all loads can be

modeled while protecting the privacy of customers. 

Fig. 1. Load modeling approach

B. Load Database 

Currently, 1-s resolution data for 3 years of a transformer with 

approximately 10–12 houses connected are used to establish the 

database on which the variability library is built. The data have 

been divided into four seasons, and the variability library is built 

for each season. A sample synthetic profile of a day in fall is 

shown in Fig. 2. The field-measured data in this database also 

contain the rooftop PV data (i.e., generation), therefore the profile 

shown in Fig. 2 has some negative data points. Because the 

measured area usually has a clear sky and solar is present only

during daytime, it is reasonable to assume that the variability in 

this database is mostly caused by load changes instead of the solar 

intermittency for short time frames.

Fig. 2. One-day sample from fall in realistic load database

C. Variability Library 

The variability library is built for each season separately. 

Multi-resolution DWTs are used to decompose the load data into 

detailed coefficients and approximate coefficients at different 

levels [6]–[7]. An example of three-level decompostion is shown 

in Fig. 3; input data s will be decomposed into detailed and 

approximate coefficients. The values of a1, a2, and a3 represent 

approximate coefficients, which are obtained by low-pass 

filtering and contain the low-frequency signal. The values of d1, 

d2, and d3 represent detailed coeffcients, which are obtained by 

high-pass filtering and contain the high-frequency signal. As 

shown in (1), the summation of all the detailed coeffcicents and 

the approximate coefficient from the last level of decompostion 

will reconstruct the origial signal. 

To extract the load variability characteristics in the resolution 

range from 1-s to 30-min, 12-level DWTs have been performed 

for the daily load profiles in the database. The summation of the

first 12 detailed coefficients define the load variability model, and 

the 12th approximate coefficient will be kept asthe baseload. Each 

load profile in the database will have one corresponding load 

variabilty model and a baseload term.

Fig. 3. Wavelet decompostion illustration

 ss = dd1 + dd2 + dd3 + aa3 (1)

After obtaining the variability models and baseloads, the next 

step is to construct the variability library. Using the fall season as

an example, for all the 273 1-s field-measured daily load profiles 

in fall, the variability of each load profile will be quantified by 

calculating the ramp profile. As shown in (2), the ramp is 

calcuated as the difference between the load magnitude at the next 

time step (t1) and the current time step (t0). In this way, the 273 

ramp profiles for fall are obtained: 

RR RRRR = PP(tt1) − PP(tt0) (2)

Then the correlation between one ramp profile and the other 

272 load profiles is calcuated. When calculating the correlation 

coefficent of two ramp profiles A and B, each profile will be 

divided into N bins, and the number of data points in each bin are

recorded as AAii and BBii. As shown in (3), the correlation coefficient 

of the two ramp profiles is calculated using the Pearson 

correlation coefficient [8], in which μμAA and σσAArepresent the mean 

and standard deviation of A, respectively; and μμBB and 

σσBBrepresents the mean and standard deviation of B, respectively. 

 ρρ(AA, BB) = 1

NN−1 ∑ �

AAii−μμAA

σσAA

� �

BBii−μμBB

σσBB

� NN

ii=1 (3)

 If the correlation coefficient of ramp profiles A and B is 

greater than 0.99, then these two profiles are recorded as 

correlated. The ramp profiles are ranked by the correlation record. 

If one profile has the highest number of correlated profiles, then 

this profile is ranked top of the list. The ramp profiles that are 

correlated with at least half the total number of profiles are ranked 

on the list. 

The load variability models associated with the ramp profiles 

on the list are recognized as variability classes and are included 

in the variability library for the fall season. The varaibility library 

of the other three seasons are established in the same way. 
3

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D. Load Aggregation Approach

Different sub-nodes should have different load profiles. If we 

have the peak load for each node, then a load aggregation 

approach could be used to model the load profiles at various load 

levels [9]. 

As shown in the flowchart in Fig. 4, first, the maximum 

noncoincident load is calculated by multiplying the peak load 

with the diversity factor (DF). Second, the number of loads N is 

calculated by dividing the maximum noncoincident load by the 

average peak load. The number N will be rounded to the closest 

integer. Third, if N is greater than 1, N load profiles will be 

randomly selected from the realistic load database. Then the 

selected profiles will be summed to get the total load profile. The 

difference between the peak total load and the given peak load 

will be compared with the error tolerance. The third step is

repeated until the difference is smaller than the tolerance. After 

the tolerance is satisfied, the corresponding load model is

developed by adding the baseloads of the selected realistic loads 

and the variability models extracted from the variability library. 

Note that if N is smaller than 1, the load profile with the peak load 

closest to the node peak is directly determined from the load 

database. 

DF is calculated as shown in (4), where PP1~PPnn represents the 

profiles in the load database, and n is the total number of profiles:

DDDD = mmmmmm mmmmmm nnoo oooo oo oooo oo

mmmmmm mmmmmm dd dddd dd (4)

= max(PP1)+max(PP2)+⋯ max(PPnn)

max(PP1+PP2+⋯ PPnn)

 

Fig. 4. Load aggreagation flowchart

In this work, because we have only the node peak load 

information of the feeder, we use the above practical and 

simplified load aggregation approach to model the load at 

different levels. We are also developing an advanced load 

aggregation methodology that could capture salient points of the 

load profile so that the key characteristics, such as PV impact,

could be modeled. We will present this methodology in a followup paper. 

III. FEEDER TESTING

 The proposed load modeling approach has been tested on two 

feeders: 1) the standardized IEEE 123-node test feeder [10], and 

2) a realistic utility feeder from California. Realistic load data

and the modeled load data built from the variability database and 

baseload pool were used to perform tests on the feeders. A series 

of evaluation criteria have been selected to evaluate the 

effectiveness of the modeled load profiles on feeder testing. 

A. Evaluation Criteria 

As shown in Table I, four metrics have been used to evaluate 

the performance of the developed load models on feeder testing. 

Maximum and minimum voltage are compared between the 

testing results of the original load and modeled load to check if

the voltage range remains the same if performing the testing using

the modeled load. The number of regulator operations and the 

voltage ramp distribution are also compared to check if the 

modeled load effectively reflects the variability characteristics of

the original load. The error rate for the comparison of voltage 

regulator operations and the voltage ramp is calculated in (5) and 

(6), respectively. 

TABLE I. PERFORMANCE EVALUATION METRICS

1 Voltage ramp 

distribution 3 Minimum voltage

2 Maximum 

voltage 4 Number of regulator tap change

EEEEEEEEEE RR RR = RR RR RR MM (oo )−RR RR RR MM (MM )

RR RR RR MM (oo ) (5)

VV VV VV RR RRRR = VV(tt1) − VV(tt0) (6)

B. Testing on IEEE 123-Node Test Feeder

The topology of the IEEE 123-node test feeder is shown in 

Fig. 5. A total of 91 nodes have load. Four voltage regulators are 

installed along the feeder.

Fig. 5. IEEE 123-node test feeder 

Load profiles for n days were randomly selected from the 

orignal load database to perform the testing. Then the 

corresponding n approximate coefficients at Level 12 from the 

baseload database were extracted as the low-resolution baseloads. 

As shown in Fig. 6(a), the n load variability models, which are 

ranked on the top of the list from the variability library, were

selected and added onto the top of the baseloads to construct the 

high-resolution load profiles. The value of n can be a random 

integer that is greater than the number of load nodes. Fig. 6(b) 

shows a sample load profile with constituent components from

the descibed approach. The blue line is the load variability model 

extracted from the library, and the black line represents the 

baseload from the baseload pool. Adding the above two 

components gets the modeled load profile, which is the red line. 

Then the load aggregation approach is presented in Section II. 

D is used to obtain the 91 load profiles at the 91 load nodes of the 

IEEE 123-node test feeder according to the node peak load 

information. 
4

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 (a) (b)

Fig. 6. Load modeling for IEEE 123-node test feeder 

Evaluations have been completed for various temporal 

resolutions, including 1 s, 1 min, 5 min, 10 min, 15 min, and 30 

min. The data with lower resolutions are obtained by averaging 

the data with higher resolutions. Fig. 7 shows one sample of the 

original load data and modeled load data, which has a resolution 

of 1 min.

Comparisons between the results of the original data testing 

and the modeled data testing were analyzed via the 

aforementioned metrics. The number of regulator operation 

comparisons is shown in Table II. The minimum and maximum 

voltage comparison is shown in Table III. Table IV shows the 

correlation coefficients of the voltage ramp profiles. The 

coeffcients were also calcuated using the Pearson correlation

coefficient, as shown in (3). Fig. 8 shows the comparison of the 

histograms for the voltage ramp profiles. 

Fig. 7. Load sample of 1-minute resolution

From Table II, we can see that the modeled load has captured 

most of the impact on voltage regulators caused by load 

variability. For the test cases of resolution ranging from 5 min to 

30 min, the number of regulator operations for the modeled data 

testing case is almost the same as that of the original data testing 

case. Also, it is observered that the higher resolution data has 

captured more regulator operationsthan the lower resolution data. 

This reveals the importance of modeling using high-resolution 

data for feeder testing, particularly for voltage regulation 

equipment impacts. The error rates for 1-s and 1-min data testing 

are more than 20%, which might be caused by the size of the load 

variability library. Because only the fall data were used for this 

evaluation, there might not be enough unique variability models 

to capture the system-level diversity characteristics. This could 

be improved in future work. 

Table III shows that the minimum and maximum voltages 

match well for the original case and modeled case. The high

values for the correlation coefficient shown in Table IV 

demonstrate that the voltage ramp distribution of the two cases 

for different resolutions align well. This agreement is also shown 

by the the histograms in Fig. 8, which cover the histograms for 

the original and modeled cases with resolutions from 1 s to 10 

min. 

TABLE II. COMPARISON OF NUMBER OF REGULATOR OPERATIONS

Voltage Regulator Operations

Resolution Original Load Modeled Load Error Rate

1 second 126 91 27.78%

1 minute 114 87 23.68%

5 minutes 59 60 1.69%

10 minutes 51 52 1.95%

15 minutes 53 50 5.66%

30 minutes 49 48 2.04%

TABLE III. COMPARISON OF MAXIMUM AND MINIMUM VOLTAGE

Original Load Modeled Load

Resolution

Maximum 

Voltage 

(p.u.)

Minimum 

Voltage 

(p.u.)

Maximum 

Voltage 

(p.u.)

Minimum 

Voltage 

(p.u.)

1 second 1.0518 0.9773 1.0516 0.9768

1 minute 1.0509 0.9798 1.0513 0.9787

5 minutes 1.0520 0.9777 1.0524 0.9797

10 minutes 1.0504 0.9758 1.0531 0.9764

15 minutes 1.0507 0.9809 1.0532 0.9742

30 minutes 1.0523 0.9844 1.0521 0.9797

TABLE IV. CORRELATION COEFFICIENT OF VOLTAGE RAMP DISTRIBUTION

Resolution Correlation Coefficient

1 second 0.9882

1 minute 0.9999

5 minutes 0.9985

10 minutes 0.9926

15 minutes 0.9950

30 minutes 0.9897

(a) 1-second (b) 1-minute

(c) 5-minute (d) 10-minute

Fig. 8. Histogram of voltage ramp for various resolutions

C. Testing on Realistic Feeder

 A realsitic utility feeder from California has also been used to 

test the modeled load. There are 619 load nodes in total and 3 

regulators installed on this feeder. The load profies for this feeder 

are constructed in a way that is similar to the IEEE 123-node test

feeder.

The tests have also been done for various resolutions,

including 1 s, 1 min, 5 min, 10 min, 15 min, and 30 min. Metric 

comparisons between the results of the original data testing and 

the modeled data testing are summarized in Table V, Table VI, 

and Table VII, repesectively. Fig. 8 shows the comparison of the 

histograms for the voltage ramp profiles.

Table V shows that the modeled load profiles have 

successfully captured most of the influence on voltage regulators 

caused by the the load variability. The error rates and the absolute 
5

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difference between the number of regulator operations for the 

original and modeled cases are both small. 

Table VI shows that the minimum and maximum voltages 

match well for the original case and modeled case. Table VII

shows that the voltage ramp distribution of the two cases with 

different resolutions all have very strong correlation. The 

histograms in Fig. 9 also demonstrate that the voltalge ramp 

distributions of the orignal and modeled cases align very well. 

These results demonstrate that the modeled loads could 

effectively represent the load variability influence on the feeder 

and could capture most of the charactersictics brought by the 

original loads. 

TABLE V. COMPARISON OF NUMBER OF REGULATOR OPERATIONS

Voltage Regulator Operations

Resolution Original Load Modeled Load Error Rate

1 second 130 123 5.38%

1 minute 124 111 10.48%

5 minutes 103 98 4.85%

10 minutes 101 102 0.99%

15 minutes 95 98 3.16%

30 minutes 77 75 2.6%

TABLE VI. COMPARISON OF MAXIMUM AND MINIMUM VOLTAGE

Original Load Modeled Load

Resolution

Maximum 

Voltage 

(p.u.)

Minimum 

Voltage 

(p.u.)

Maximum 

Voltage 

(p.u.)

Minimum 

Voltage 

(p.u.)

1 second 1.0544 0.9533 1.0519 0.9539

1 minute 1.0600 0.9748 1.0530 0.9637

5 minutes 1.0547 0.9713 1.0517 0.9691

10 minutes 1.0534 0.9721 1.0514 0.9768

15 minutes 1.0528 0.9751 1.0510 0.9776

30 minutes 1.0525 0.9511 1.0508 0.9501

TABLE VII. CORRELATION COEFFICIENT OF VOLTAGE RAMP DISTRIBUTION

Resolution Correlation Coefficient

1 second 0.9882

1 minute 0.9984

5 minutes 0.9985

10 minutes 0.9968

15 minutes 0.9700

30 minutes 0.9813

 (a) 1-second (b) 1-minute

(c) 5-minute (d) 10-minute

Fig. 9. Histogram of voltage ramp for various resolutions

IV. CONCLUSION AND FUTURE WORK

This paper presents a DWT-based load modeling 

methodology to model the time-series loads to enable effectual

QSTS simulations of distribution feeders. The load model is 

constructed from two major parts: the low-resolution fieldmeasured load data, which usually have resolutions of 15 or 30 

min; and the high-resolution load variability model extracted 

from a previously developed variability database, which could 

have resolutions up to 1 sec. A load aggregation methodology is 

also developed to aggregate the load profiles so that load profiles 

at different transformer ratings can be modeled effectively. The 

developed load modeling approach has been tested on the IEEE 

123-node test feeder and an actual utility feeder from California. 

The analysis completed on the two feeders demonstrated the 

value of the developed model for distribution feeder QSTS

simulations. Future work will focus on refining the load 

variability library and developing models with more effective

customer diversity characteristics. 

ACKNOWLEDGMENT 

This work was authored by Alliance for Sustainable Energy, 

LLC, the manager and operator of the National Renewable 

Energy Laboratory for the U.S. Department of Energy (DOE) 

under Contract No. DE-AC36-08GO28308. Funding provided 

by U.S. Department of Energy Office of Energy Efficiency and 

Renewable Energy Solar Energy Technologies. The views 

expressed in the article do not necessarily represent the views of 

the DOE or the U.S. Government. The U.S. Government retains, 

and the publisher, by accepting the article for publication, 

acknowledges that the U.S. Government retains a nonexclusive, 

paid-up, irrevocable, worldwide license to publish or reproduce 

the published form of this work, or allow others to do so, for U.S. 

Government purposes.

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