Paper No. T.12-1.5, pp. 1-5
The 6th PSU-UNS International Conference on Engineering and
Technology (ICET-2013), Novi Sad, Serbia, May 15-17, 2013
University of Novi Sad, Faculty of Technical Sciences
A METHOD FOR LOAD-TO-VOLTAGE
DEPENDENCY DETERMINATION
Goran S. Švenda, Jovan M. Lukić
University of Novi Sad, Faculty of Technical Sciences, Novi Sad, Serbia*
*Authors to correspondence should be addressed via email: svenda@uns.ac.rs, jovanluk@uns.ac.rs
Abstract: Distribution power grid load modeling is comprised out of knowing its qualitative and quantitative properties, in terms of active and reactive power consumption,
but also properties such as load dependency to voltage and
frequency. Determination of load-to-voltage dependency is
not sufficiently explored and documented. Main reasons for
that are lack of robust procedure and relatively large number of experiments that need to be performed out in the
field. This paper presents a robust procedure for determining load-to-voltage dependency with relatively small
number of experiments needed. The procedure is verified
in a real distribution power grid.
Keywords: Load-to-Voltage Dependency, Load Modeling,
Voltage Reduction, Distribution Power Grid
1. INTRODUCTION
Modern power distribution utilities (PDU), which are
more and more present on electrical energy markets, are
forced to operate on borders of their technical limitations.
What definitely contributes to that is the fact that they often
get very challenging demands to fulfill. Some of the challenges they need to cope with are tied to real-time operation
where, with minimal investments, state of the system needs
to be estimated, optimal power flows and voltages need to
be achieved, peak load shaving needs to be fast and efficient, etc. Likewise, in simulation, it is very important to
have good system state estimation, especially for states far
from normal ones. It is clear that abilities of the existing
Supervisory Control and Data Acquisition (SCADA) systems are no longer sufficient to realize such challenging
demands required from PDUs. The application of relatively
new Smart Grid Concept (SGC) [1], which encompasses
Distribution Management System (DMS) [2], can make
them achievable. The DMS includes many of its advanced
applications for analysis, control and planning of distribution network (DN). When tightly integrated with the existing SCADA systems, and constantly increasing number of
Intelligent Electronic Devices (IED – AMI, MDM, etc.),
once challenging demands will become PDU common practices. Thereby, the quality of SGC and DMS applications
does not depend only on quality of DMS calculations, but
also on the quality and quantity of data about DN (collected
in real-time and in non real-time). Among data collected in
1
non real-time, the most critical for the quality of results, but
at the same time the most unreliable, are data about power
consumption (loads). In accordance to constantly increasing
demands required from PDU, the demands set for DMS
power applications and DMS load modeling are also becoming higher. It is not enough to model load with constant
power consumption. The load must be modeled with dependency to number of parameters, such as: 1) temperature;
2) considered moment in time, day and season; 3) instantaneous voltage; 4) coincidence factor; and finally 5) decay
effect (how much time passed from last voltage change, i.e.
what can be expected in future after voltage has been
changed and stays on value different from nominal one).
The load model must take into account decay effect in order
to be able to predict its behavior in case something like described happens with voltage. Model of power consumption
dependent on temperature is shown in many papers [3,4].
Relatively simple procedure for determining normalized
daily load profiles (NDLP) of characteristic customer types
in characteristic days and seasons is shown in [5]. Papers
which contain information about load-to-voltage dependency of tangible customer types are rare [4], but even rarer are
the ones that depict precise procedures for their experimental determination [6]. Determination of load-to-voltage dependency for each individual customer may be technically
feasible, but definitely not economically justifiable. This
paper offers practically applicable procedure for load-tovoltage dependency determination by performing relatively
small number of in-field experiments. Applicability of procedure is fully verified in scope of DMS project realized for
one of the North American PDU.
After the Introduction, basic concepts of load modeling
and procedure for determining value of load-to-voltage dependency are given in part 2. An example of application of
this procedure is given in part 3. Procedure verification is
presented in part 4. Conclusion and references are given in
parts 5 and 6, respectively.
2. LOAD MODELING
The model of power consumption, referenced in this paper, is based on following terms and definitions:
 Load – a group of individual consumers which are of different consumer types, such as: constant current, power
or impedance, or constant energy, time or power. Each
consumer has its own characteristics which can be compared to some others’, and by which loads can be classified into classes.
 Load quantifier – maximal or average value of consumer's
power consumption, or consumed electrical energy on a
daily, monthly, yearly basis [5].
 Normalized daily load profile (NDLP) – daily load profile
divided by a Load quantifier.
 Similar loads – all consumers which in the same manner
change value of their consumption depending on current
time, type of a current day and season, changes in temperature and voltage, and for which the effects of those
changes last equally long.
 Characteristic load – consumer defined with two NDLP
(for active and reactive power, or for current module and
power factor), load quantifiers, and with coefficients
which depict dependency of temperature and voltage
(load coefficients). It is usually defined for residential-,
commercial-, agricultural- and industrial-type of consumers. For more precise determination of power consumption, each type can be further divided into subtypes, such
as: residential loads with locally or remotely provided heating, settlements with-, or without hot water provided, etc.
 Characteristic day – a day which represents all similar
days, i.e. days in which all similar consumers can be described with their unique Load quantifier, and common
Characteristic load. Usually is defined separately for work
day, Saturday and Sunday, as well as for holidays. If one
wants to be even more accurate, every characteristic day
can be further divided into subtypes, such as: type workday into subtypes Monday through Friday; type holiday
into subtypes New Year, Christmas, etc.
 Characteristic season – time period in which characteristic
load, in every characteristic season, describes well enough
power consumption of a consumer. Usually it is defined to
match with calendar seasons (spring, summer, fall, winter), but also it often can be defined for each month separately, then to match duration of heating season, periods
when particular industry is work- or break mode, etc.
In accordance with previously defined terms and definitions, pre-estimated value of active and reactive powers for
a moment t, can be defined as:
X r (t )  xr (t )  X quant. ,
X {P, Q} ,
BX i 
[ X i  X i 1 ] 100 / X i 1
,
[Vi  Vi 1 ] 100 / Vi 1
X {P, Q} ,
(2)
where Pi, Qi, Vi and Pi–1, Qi–1, Vi–1 represent values of active-, reactive power and voltage obtained in moments i and
i–1, respectively; BP i and BQ i represent active and reactive
load-to-voltage dependency. Statistically, expected values
of coefficients BP and BQ can be determined by having
available sufficiently large number of samples from a population. The procedure for experimental determination of
coefficients is presented in section three of this paper.
Estimation of active P(t,V) and reactive Q(t,V) power in
a moment t, for voltage value V(t), is defined by relations:
X (t ,V )  X r (t )  [1  B X  V (t )] ,
V (t )  [V (t )  Vr ]  100 / Vr ,
X {P, Q} , (3)
(4)
where subscript r indicates pre-estimated variables which
are realized for nominal voltage supplied Vr(t), relation (1).
3. EXPERIMENTS
The proposed procedure has been verified on a large
North American DN. The network supplies around 1.5 million of customers and has around 1300 medium voltage
feeders. During year 2011 two experiments have been realized: 1) winter, 15-17 February, consumer types Residential and Commercial, and 2) summer, 4-8 August, consumer
types Residential, Commercial and Industrial. Experiments
encompassed 9 medium voltage feeders (groups of 3 feeders are dominantly supplying residential, commercial and
industrial load). During experiments, voltage was changed
by 1.4% and 2.8% from actual. This is depicted in Fig. 1
(1)
where pr(t) and qr(t) represent values of active and reactive
powers in moment t – values obtained by having available
NDLP, in [p.u.], and Pquant and Pquant load quantifiers, in
[kW]. Pre-estimated values are defined for rated voltage
Vr(t).
Fig. 1. Pattern for voltage changes during experiments
Data capture, on a 30 second interval, and during the
entire experiment, was simultaneously obtained for the
active power, reactive power and voltage. By specially
designed program code paired measurements were analyzed and further processed in order to obtain representative samples from the entire population. The samples do
not include influences of disturbances that occurred during the experiment. Some of the disturbances worth mentioning are as follows: short circuit events, switching
events, capacitor banks closing or tripping, large customers connecting to- and disconnecting from the grid. After
representative population samples had been obtained,
pairs of relations (3) were formed for each moment in
time for which pairs of points (A,B), (C,D), (E,F) in Fig.
1 can be determined. Next step was to replace known
values in those pairs of relations: measured P (t ,V ) , calculated ν , and value read from the NDLP of the con-
Of course, for more accurate calculations, one cannot
disregard the fact that most consumers are not always supplied with nominal voltage, and that change in supplied
voltage changes amount of power consumed by consumer.
It is also very clear that dependency of a complex DN consumer, such as dependency on voltage supplied, cannot be
formulated analytically. However, it can be appropriately
determined by performing relatively small number of experiments. That particular dependency is referred to as loadto-voltage dependency, and is defined by the following relation:
2
sume Pr(t). Analogous procedure was done for the reactive power Q. Solving of system of two equations by
unknown BP (BQ), the value of load coefficient is obtained. That value becomes candidate for statistical
processing to find expected value and its standard deviation.
The results of statistical processing are depicted in Fig.
2, where histograms and probability density functions
(PDF) of load coefficients for February time frame and
Commercial type of load are given.
1.0
Commercial (average)
PDF
Commercial 1
PDF
Commercial 2
PDF
Commercial 3
PDF
Density
0.9
strates measured, unusually large changes of voltage V(t) on
feeder head busbar (orange solid line) and nominal voltage
Vr(t) (orange dashed line). Examples in Fig. 4 and Fig. 5 are
obtained on feeders which dominantly supply residential
and industrial type of load, respectively.
0.8
0.7
0.6
0.5
0.4
a) Load profile, measured and estimated active power
0.3
0.2
Bp [%/%]
0.1
0
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Fig. 2. Load Coefficients – Expected value
Table 1. Load Coefficients, as random variable
February
August
1
2
3

1
2
3

P
BP
Feeder
Com
1.20
1.37
1.28
1.27
Com
0.85
0.66
0.67
0.77
Res
1.65
1.64
1.21
1.56
Res
0.94
0.87
0.86
0.92
Ind
/
/
/
/
Ind
1.26
1.26
1.07
1.19
Com
0.58
0.57
0.56
0.57
Com
0.47
0.38
0.37
0.44
Res
0.82
0.96
0.72
0.88
Res
0.58
0.55
0.56
0.57
Ind
/
/
/
/
Ind
0.82
0.85
0.82
0.83
b) Voltage changes
Fig. 3. Verification (feeder Com1, February 22. 2011.)
Expected values of load coefficients and their standard
deviations are presented in Table 1. They are obtained by
processing data from experiments performed in two different seasons, for three types of loads, for each feeder and
aggregately.
4. PROCEDURE VERIFICATION
Verification of the proposed procedure for determining
load coefficients is done with following steps:
1. Assessment of pre-estimated feeder head load, based on
power flow calculation for pre-estimated load Pr(t) and
Qr(t), depicted by relation (1), obtained for nominal voltage value of V(t)=Vr=123.5 V.
2. Correction of pre-estimated load values, based on measured voltage V(t) on feeder heads, and utilizing relations
(3) and (4). Corrected values are called estimated values.
3. Estimated and measured load values comparison.
As an example of the proposed procedure, Fig.3 illustrates values of measured and estimated active powers, for
one of the feeders that supplies dominantly commercial type
of consumers. Fig.3a illustrates: pre-estimated active power
(black dashed line), estimated active power (red solid line)
and measured active power (blue solid line). Fig.3b illu-
3
where k goes from 1 through number of readings for measured active power.
Table 2 depicts the expected values of relative differences between calculated (estimated) and measured values
of active powers ΔP and standard deviations of those values
ΔP The values are calculated by following relations:
P 
1 N
Pk ,
N k 1

1 N
[Pk  P]2 .
N k 1

σ 2P 
a) Load profile, measured and estimated active power
(6)
(7)
Table 2. Expected value of relative error and its standard
deviation
February
b) Voltage changes
1
2
3

August
Feeder
1
2
3

Fig. 4. Verification (feeder Res1, August 10. 2011.)
ΔP
ΔP
Com
–0.07
–0.35
0.08
–0.08
Com
–0.10
–0.14
–0.13
–0.14
Res
0.63
0.60
0.53
0.45
Res
–0.35
–0.24
–0.25
–0.32
Ind
–0.07
–0.35
0.08
–0.08
Ind
–0.10
–0.14
–0.13
–0.14
Com
0.63
0.60
0.53
0.45
Com
–0.35
–0.24
–0.25
–0.32
Res
–0.07
–0.35
0.08
–0.08
Res
–0.10
–0.14
–0.13
–0.14
Ind
0.63
0.60
0.53
0.45
Ind
–0.35
–0.24
–0.25
–0.32
Table 3 depicts absolute expected values of relative differences between calculated (estimated) and measured values
of active powers |ΔP| and standard deviations of those values |ΔP|. The values are calculated by following relations:
1
N
P 
σ 2P 
 Pk
,
(8)
1 N
[ Pk  P ]2 .
N k 1
(9)
k 1

Table 3. Expected absolute value of relative error and its
standard deviation
a) Load profile, measured and estimated active power
February
1
2
3

August
Feeder
1
2
3

|ΔP|
|ΔP|
Com
0.98
1.87
1.21
1.38
Com
0.77
0.76
0.78
0.79
Res
2.60
2.92
1.85
2.5
Res
1.29
1.19
1.22
1.26
Ind
0.98
1.87
1.21
1.38
Ind
0.77
0.76
0.78
0.79
Com
2.60
2.92
1.85
2.5
Com
1.29
1.19
1.22
1.26
Res
0.98
1.87
1.21
1.38
Res
0.77
0.76
0.78
0.79
Ind
2.60
2.92
1.85
2.5
Ind
1.29
1.19
1.22
1.26
5. CONCLUSION
b) Voltage changes
Fig. 5. Verification (feeder Ind1, August 10. 2011.)
This paper presents very simple, but at the same time
robust procedure for experimental determination of load-tovoltage dependency, i.e. load coefficients. Additionally,
their application in terms of estimation of large area load is
given. Application of the procedure, as well as its verification is done in real USA distribution network.
When information about load coefficients becomes
It is also interesting to discuss results in terms of error
which describes deviation of estimated from measured
power. The error is calculated by following relation:
Pk  ( Pkest  Pkmeas)  100 / Pkmeas ,
N
(5)
4
available, one can take further steps in terms of forming
more complex model of loads, model which encompasses
coincidence factor change effects, as well as voltage change
effects which decays in time.
6. REFERENCES
[1] R.G.Pratt, et all, The Smart Grid – An Estimation of the
Energy and CO2 Benefits; Pacific Northwest National
Laboratory, USA, 2010.
[2] D.S.Popović, Power Application – A Cherry on the Top
of the DMS Cake, DA/DSM DistribuTECH Europe
2000, Vienna, Austria, October 10-12, 2000, Specialist
Track 3, Session 3, Paper 2.
[3] C.E.Asbury, Weather load model for electric demand
and energy forecasting; IEEE Trans. on PAS, Vol. PAS94, No.4, July 1975.
[4] R.F.Preiss, V.J.Warnock, Impact of voltage reduction
on energy and demand; IEEE Trans. on PAS, Vol. PAS97, No.5, September/October 1978.
[5] S.Kuzmanović, G.S.Švenda, Z.Ovcin, Practical Statistical Methods in Distribution Load Estimation; 20-th International Conference on Electricity Distribution –
CIRED, Prague, 8-11 June 2009, Session No.4, Paper
0585.
[6] J.C.Erickson, S.R.Gilligan, The effects of voltage reduction on distribution circuit loads; IEEE Trans. on PAS,
Vol. PAS-101, No.7, July 1982.
5