Instruction


FACTA UNIVERSITATIS 
Series: Electronics and Energetics Vol. 29, N

o
 2, June 2016, pp. 159 - 175 

DOI: 10.2298/FUEE1602159A 

CHARACTERIZATION OF NONLINEAR LOADS 

 IN POWER DISTRIBUTION GRID

 

Miona Andrejević Stošović
1
, Marko Dimitrijević

1
, Slobodan Bojanić

2
, 

Octavio Nieto-Taladriz
2
, Vančo Litovski

1
 

1
University of Niš, Faculty of Electronic Engineering, Niš, Serbia 
2
Escuela Técnica Superior de Ingenieros de Telecomunicación,  

Universidad Politecnica de Madrid, Madrid, Spain  

Abstract. Electronic devices are complex circuits, consisting of analog, switching, and 

digital subsystems that require direct current (DC) for polarization. Since they are 

connected to the mains delivering alternating current (AC), however, AC-to-DC 

converters are to be introduced between the mains and the electronics to be fed. A 

converter is an electric circuit containing several subsystems, the most important being 

the switch-mode power supply, drawing power from the mains in pulses hence it is highly 

nonlinear. That happens, in reduced amplitude, even when the electronics to be fed is 

switched off. The process of AC-to-DC conversion is not restricted to feeding electronic 

equipment only. It is more and more frequently encountered in modern smart-grid 

facilities giving rise to the importance of the studies referred hereafter. The converter can 

be studied (theoretically or by measurements) as two-port network with reactive and 

nonlinear port-impedances. Characterization is performed after determining the port 

electrical quantities which are voltages and currents. Based on these data power and 

power quality parameters – power factor and total harmonic distortion- may be extracted. 

When nonlinear loads are present, one should introduce new ways of thinking into the 

considerations due to the existence of harmonics and related power components. In that 

way the power factor can be generalized to total or true power factor where the apparent 

power, involved in its calculations, includes all harmonic components. After introducing a 

wide range of definitions used in contemporary literature, here we describe our 

measurement set-up both as hardware and a software solution. The results reported 

unequivocally confirm the importance of the subject of characterization of small 

nonlinear loads to the grid having in mind their number which is rising without saturation 

seen in the near and even far future. 

Key words: smart grid, nonlinear loads, load characterization, power factor, harmonic 

distortions 

 

                                                           
Received September 29, 2015 

Corresponding author: Miona Andrejević Stošović 
University of Niš, Faculty of Electronic Engineering, Aleksandra Medvedeva 14, 18000 Niš, Serbia 

(e-mail: miona.andrejevic@elfak.ni.ac.rs) 



160 M.ANDREJEVIĆ-STOŠOVIĆ, M. DIMITRIJEVIĆ, S. BOJANIĆ, O. NIETO-TALADRIZ, V. LITOVSKI 

1. INTRODUCTION 

With the advent of modern diversified sources of electrical energy, the issue of power 

quality becomes both more ambiguous and more complicated. We will address here first 

the new aspects that are coming in fore thanks to the new ways of producing electrical 

energy, which are becoming more and more popular, and thanks to the emergence of a 

new paradigm known as smart-grid which involves mutual interaction of power electrical 

systems and electronic systems for its proper functionality [1]. 

Nowadays we are witnessing changes in the demand and energy use which in fact means 

“new” load characteristics, and trends changing the nature of the aggregate utility 

consumption. All of that is mostly due to the electronic devices that became ubiquitous. It is 

presumed that the overall household consumption for electronic appliances will rise with a 

rate of 6% per year so reaching 29% of the total household consumption in the year 2030. In 

the same time the household consumption is expected to reach 40% of the overall electricity 

demand. The immense rise of the office consumption due to the enormous number of 

computers in use is also to be added. That stands for educational, administrative, health, 

transport, and other public services, too. One may get the picture if one multiplies the average 

consumption of a desk-top (about 120 W) with the average number of hours per day when the 

computers is on (about 7), and the number of computers (billion(s)?).  

Electronic loads are strongly related to the power quality thanks to the implementation 

of AC/DC converters that in general draw current from the grid in bursts. The current 

voltage relationship of these loads, looking from the grid side, is nonlinear, hence nonlinear 

loads. 

In fact, while keeping the voltage waveform almost sinusoidal, they impregnate pulses 

into the current so chopping it into seemingly arbitrary waveform and, consequently, 

producing harmonic distortions. Having all this in mind the means for characterization of 

the load from the nonlinearity point of view becomes one of the inevitable tools of quality 

evaluation of smart grid. 

The problem is further complicated when different power generation technologies and 

resources are combined leading. New subsystem in the power production, transport, and 

consumption emerge named micro-grids and the overall system is supposed to become a 

smart-grid. For example, due to the rise of the number of different kind of electricity 

sources even the frequency of the grid voltage may be considered as “unknown” asking 

for algorithms and software to be implemented in real time to extract the frequency value 

[2] and, based on that, to compute the amplitudes of the harmonics [3, 4, 5]. 

Due to the nonlinearities, measurement of power factor and distortion, however, usually 

requires dedicated equipment. For example, use of a classical ammeter will return incorrect 

results when attempting to measure the AC current drawn by a non-linear load and then 

calculate the power factor. A true RMS multi-meter must be used to measure the actual RMS 

currents and voltages and apparent power. To measure the real power or reactive power, a 

wattmeter designed to properly work with non-sinusoidal currents must be also used. 

Contemporary methods and algorithms for spectrum analysis are presented in this paper. 

The basic definitions of parameters describing nonlinear loads are introduced. Alternative 

definitions for reactive power and their calculation methods are elaborated, also. 

In our previous research we were first developing a tool for efficient measurements 

that would allow for proper and complete characterization of the nonlinear loads [6, 7]. 

Namely we found that the tools for characterization of modern loads available on the 

market, most frequently, lack at least one of the following properties: low price, ability of 

implementation of complex data processing algorithms (versatility), ability to store and 



 Characterization of Nonlinear Loads in Power Distribution Grid 161 

statistically analyze the measured data, and ability to communicate with its environment 

no matter how distant it is. All these were achieved by the system reported in [6, 7] and 

the measurement results demonstrated here were obtained by these tools. 

Next, we implemented these tools for characterization of small loads. The results 

obtained, as reported in [8] and [9] for example, were, in some cases, surprisingly different 

from what expected. That stands for the power components which are not the active power 

and for the abundance of harmonics. In [10] and [11] we demonstrated that based on the 

main's current, by proper data processing, despite the complex signal transformation 

between the mains and the components of a computer via the power supply chain, one may 

deduce the activities within the computer. Even more, one may recognize a software 

running within the computer. Such information is distributed via the grid. 

Here we will for the first time summarize the theoretical background of all computations 

necessary to be performed for complete characterization of small loads. Then, we will 

demonstrate our new results in the implementation of the theory and the measurement tools 

on a set of nonlinear loads. 

The definitions used in modern characterization of the main's current, voltage, and 

power which are implemented by our system will be listed in the second section so 

enabling the main attention to be devoted to the set of measured results and their analysis, 

which will be given next. 

The paper will be organized as follows. First a short description of the measurement 

experiment will be given. To preserve conciseness, for this purpose, we will mainly refer 

to our previous work.  

2. PARAMETER DEFINITIONS 

Although power quality is a relatively ambiguous concept, limited mostly to conversations 

among utility engineers and physicists, as electronic appliances take over the home, it may 

become a residential issue as well. 

2.1. Linear loads with sinusoidal stimuli 

A sinusoidal voltage source 

 
RMS

( ) 2 sin(ω )v t V t  (1) 

supplying a linear load, will produce a sinusoidal current of 

 
RMS

( ) 2 sin(ω φ)i t I t   (2) 

where VRMS is the RMS value of the voltage, IRMS is the RMS value of the current, ω is 

the angular frequency, φ is the phase angle and t is the time. The instantaneous power is 

 ( ) ( ) ( )p t v t i t   (3) 

and it can be represented as 

 
RMS RMS

( ) 2 sin ω sin(ω ) .
p q

p t V I t t p p      (4) 

Using trigonometric transformations, we can write: 

 
RMS RMS

cosφ (1 cos(2ω )) (1 cos(2ω ))
p

p V I t P t         (5) 

and 



162 M.ANDREJEVIĆ-STOŠOVIĆ, M. DIMITRIJEVIĆ, S. BOJANIĆ, O. NIETO-TALADRIZ, V. LITOVSKI 

 
RMS RMS

sin φ sin(2ω ) sin(2ω )
q

p V I t Q t         (6) 

where 

 
RMS RMS

RMS RMS

cos φ,

sin φ

P V I

Q V I

  

  
 (7) 

represent real (P) and reactive (Q) power. 

It can be easily shown that the real power presents the average of the instantaneous 

power over a cycle: 

 
0

0

t +T

t

1
( ) ( )P v t i t dt

T
    (8) 

where t0 is arbitrary time (constant) after equilibrium, and T is the period (20ms in 

European and 1/60s in American system, respectively).  

The reactive power Q is the amplitude of the oscillating instantaneous power pq. The 

apparent power is the product of the root mean square value of current times the root mean 

square value of voltage: 

 
RMS RMS

S V I   (9) 

or: 

 
2 2

.S P Q   (10) 

Power factor is simply defined as the ratio of real power to apparent power [12, 3]: 

 / .TPF P S  (11) 

For pure sinusoidal case, using (7), (10) and (11) we can calculate: 

 cosφ.TPF   (12) 

2.2. Nonlinear loads 

When there is a nonlinear load in the system, it operates in non-sinusoidal condition 

and use of well known parameters such as power factor, defined as cosine of phase 

difference, does not describe system properly. In that case, traditional power system 

quantities such as effective value, power (active, reactive, apparent), and power factor need 

to be numerically calculated from sampled voltage and current sequences by performing 

DFT, FFT or Goertzel algorithm [3]. 

The RMS value of some periodic physical entity X (voltage or current) is calculated 

according to the well-known formula [13, 14]: 

 
0

0

t +T

2

RMS

t

1
( ( ))

T
X x t d t   (13) 

where x(t) represents time evolution, T is the period and t0 is arbitrary time. For any 

periodic physical entity x(t), we can give Fourier representation: 

 0
1

( ) ( cos( ω ) sin( ω ))
k k

k

x t a a k t b k t




      (14) 



 Characterization of Nonlinear Loads in Power Distribution Grid 163 

or 

 0
1

( ) cos( ω )
k

k

k
x t c c k t 





     (15) 

where
0 0

c a represents DC component, 
2 2

k k k
c a b  magnitude of k

th
 harmonic, 

k = arctan(bk/ak) phase of k
th

 harmonic and  = 2/T, angular frequency. 

Fourier coefficients ak, bk are: 

 

T / 2 T / 2

0

T / 2 T / 2

1 2 2 π
( ) ,     ( ) cos

T T T
k

k t
a x t dt a x t dt

 

 

 
    

 
   (16) 

and 

 

T / 2

T / 2

2 2 π
( ) sin .

T
k

k t
b x t dt

T





 
   

 
  (17) 

The RMS value of k
th

 harmonic is 

 
k, RMS

/ 2.
k

X c  (18) 

We can calculate total RMS value 

 
2 2 2

RMS , RMS 1, RMS H, RMS

1

  
M

k

k

X X X X


    (19) 

where M is the highest order harmonic taken into calculation. Index “1” denotes first or 

fundamental harmonic, and index “H” denotes contributions of higher harmonics. 

Equations (13) – (19) need to be rewritten for voltage and current. Practically, we 

operate with sampled values and integrals (16) and (17) are transformed into finite sums.  

For a single-phase system where k is the harmonic number, k phase difference between 
voltage and current of k

th
 harmonic and M is the highest harmonic, the total active power is 

given by: 

 
,RMS ,RMS 1 H

1

cos φ .
M

k k k

k

P I V P P


      (20) 

The first addend in the sum (20), denoted with P1, is fundamental active power. The 

rest of the sum, denoted with PH, is harmonic active power [13]. 

In the literature, there exists a number of definitions of reactive power for non-

sinusoidal conditions that serve to characterize nonlinear loads and measure the degree of 

loads’ non-linearity [14]. As more general term, non-active power N, was introduced. 

Each definition has some advantages over others. But, although there is tendency to 

generalize, there is no generally accepted definition. 

The most common definition of reactive power is Budeanu’s definition [15], given by 

following expression for single phase circuit: 

 
B ,RMS ,RMS

1

sin φ .k k k
k

Q I V




    (21) 



164 M.ANDREJEVIĆ-STOŠOVIĆ, M. DIMITRIJEVIĆ, S. BOJANIĆ, O. NIETO-TALADRIZ, V. LITOVSKI 

Budeanu proposed that apparent power consists of two orthogonal components, active 

power (20) and non-active power, which is divided into reactive power (21) and distortion 

power: 

 
2 2 2

B .D U P Q    (22) 

It should be noted that the actual contribution of harmonic frequencies to active and 

reactive power is small (usually less than 3% of the total active or reactive power). The 

major contribution of higher harmonics to the power comes as distortion power. 

The apparent power, for non-sinusoidal conditions conventionally denoted as U, can 

be written: 

 
2 2
1

2 2
H

2 2 2 2 2

1,RMS 1,RMS 1,RMS H,RMS

2 2 2 2

1,RMS H,RMS ,RMS H,RMS

V

I

S D

H

D S

U I V I V

V I V I

    

   
 (23) 

where S1 represents fundamental apparent power, DV voltage distortion power, DI current 

distortion power and SH harmonic apparent power. S1 and SH are 

 
2 2 2 2 2

1 1 1 H H H H
,  S P Q S P Q D      (24) 

where DH represents harmonic distortion power. The total apparent power, denoted with 

U, is 

 
2 2 2

RMS RMS
.U P Q D I V      (25) 

We can also define non-active power N, defined with equation 

 
2 2

N Q D   (26) 

and phasor power S, defined in the same way as apparent power for sinusoidal conditions 

(10). It is obvious that for sinusoidal conditions, apparent power and phasor power are 

equal, and (25) reduces to (10). 

The total harmonic distortions, THD, are calculated from the following formula [12, 13]: 

 
H, RMS 2

, RMS

2

2 2

RMS  1, RMS

2

 1, RM1, RMS 1, RMS S

1 M

I j

j

I
THD I

I

I I

II 



   (27) 

and 

 
H, 2

, RMS

2 2

RMS 1, RMS

2

1, RM, S21, 1

1 MRMS
V k

kRMS RMS

VV
TH

V

V
D

V V
V




    (28) 

where Ij, Vk j, k=1, 2, …, M stands for the harmonic of the current or voltage. It can be 

shown that: 

 

1, RMS H, RMS 1

H, RMS 1, RMS 1

1

  

  

.

I I

V V

H I V

D V I S THD

D V I S THD

S S THD THD

   

   

  

 (29) 



 Characterization of Nonlinear Loads in Power Distribution Grid 165 

Fundamental power factor or displacement power factor is given by the following 

formula: 

 
1

1 1

1

cos .
P

PF
S

   (30) 

Total power factor TPF [12, 13], defined by equation (12), taking into calculation (11) 

and (23), is 

 1 H
2 2 2 2

1 HI V

P PP
TPF

U S D D U


 

  
 (31) 

and substituting (29) and (30): 

 

 

H
1

1

22 2

1 cos φ

.

1
I V I V

P

P
TPF

THD THD THD THD

 
 

 


   

 (32) 

Total power factor can be represented as product of distortion power factor DPF and 

displacement power factor PF1, i.e. cos1: 

 
1

cosφTPF DPF   (33) 

Therefore, distortion power factor is [12, 13] 

 

 

H

1

22 2

1

.

1
I V I V

P

P
DPF

THD THD THD THD





   

 (34) 

In real circuits, PH << P1 and voltage is almost sinusoidal (THDV < 5%), leading to 

simpler equation for TPF [12, 13]: 

 1
2

cos φ
.

1
I

TPF
THD




 (35) 

2.3. Other definitions of reactive power  

Budeanu’s definition 

The most common definition of reactive power is Budeanu’s definition [16], given by 

following expression for single phase circuit, as mentioned earlier in the text: 

 B ,RMS ,RMS
1

sink k k
k

Q I V 




    (36) 

Budeanu proposed that apparent power consists of two orthogonal components, active 

power and non-active power, which is divided into reactive power (36) and distortion power: 

 2 2 2B B .D U P Q    (37) 

IEEE Std 1459-2010 proposes reactive power to be calculated as: 

 
,RMS ,RMS

2 2 2
IEEE

1

sin
k k k

k

Q I V 




    (38) 



166 M.ANDREJEVIĆ-STOŠOVIĆ, M. DIMITRIJEVIĆ, S. BOJANIĆ, O. NIETO-TALADRIZ, V. LITOVSKI 

Equation (38)eliminates the situation where the value of the total reactive power Q is 

less than the value of the fundamental component. 

Kimbark’s definition 

Similar to Budeanu’s definition, Kimbark [17] proposed that apparent power consists 

of two orthogonal components, non-active and active power, defined as average power. 

The non-active power is separated into two components, reactive and distortion power. 

The first is calculated by equation 

 k 1,RMS 1,RMS 1sinQ I V     (39) 

It depends only on fundamental harmonic. The distortion power is defined as non-active 

power of higher harmonics: 

 2 2 2
k k .D U P Q    (40) 

Sharon’s definition 

This definition [18], introduces two quantities: reactive apparent power, Sq, and 

complementary apparent power Sc, defined as: 

 
2 2

q RMS ,RMS

1

sink k
k

S V I 




    (41) 

and 

 
2 2 2

c qS U P S    (42) 

where S is apparent power (9) and P active power(8). 

Fryze’s definition 

Fryze’s definition [19] assumes instantaneous current separation into two components 

named active and reactive currents. Active current is calculated as 

 a 2
RMS

( ) ( )
P

i t v t
V

  (43) 

and reactive current as: 

 r a( ) ( ) ( ).i t i t i t   (44) 

Active and reactive powers are  

 
RMS a

f RMS r

P V I

Q V I

 

 
 (45) 

where Ia and Ir represent RMS values of instantaneous active and reactive currents. 

Kusters and Moore’s power definitions 

Kusters-Moore definition [20] presents two different reactive power parameters, inductive 

reactive power: 



 Characterization of Nonlinear Loads in Power Distribution Grid 167 

 

,RMS ,RMS

1
L RMS

2
,RMS

2
1

1
sink k k

k

k

k

V I
k

Q V
V

k










  

 





 (46) 

and capacitive reactive power: 

 

,RMS ,RMS

1
C RMS

2 2
,RMS

1

sin

.

k k k

k

k

k

k V I

Q V

k V










  

 







 (47) 

There are other power decompositions, not considered in this paper: Shepard-Zakikhani 

[21], Depenbrock [22] and Czarnecki decomposition [23, 24]. More comprehensive 

comparison of reactive power definitions, obtained by means of simulation, can be found in 

[25].  

3. MEASUREMENT SYSTEM 

In order to establish a comprehensive picture about the properties of a given load one 

needs to perform complete analysis of the current and voltage waveforms at its terminals. 

In that way the basic and the higher harmonics of both the current and the voltage may be 

found. More frequently, however, indicators related to the power are sought in order to 

quantitatively characterize the load. Namely, a linear resistive load will have voltage and 

current in-phase and will consume only real power. Any other load will deviate from this 

characterization and one wants to know the extent of deviation expressed by as much 

indicators as necessary to get a complete picture. All these were implemented in our 

measuring system which will be shortly described in the next. 

The solution, as described in full details in [6, 7], is based on a real time system for 

nonlinear load analysis. The system is based on virtual instrumentation paradigm, keeping 

main advantage of legacy instruments – determinism in measurement. 

The system consists of three subsystems: acquisition subsystem, real time application for 

parameter calculations, and virtual instrument for additional analysis and data manipulation 

(Fig. 1).  

TCP/IPPCIFPGA

RTOS
GPOS

N

Ni9225L1

 

L3
L2

Ni9227
 

Fig. 1 The system architecture 

The acquisition subsystem, Fig. 2, is implemented using field programming gate array 

(PXI chassis equipped with PXI-7813R FPGA card with Virtex II FPGA) in control of 

data acquisition [26]. Acquisition is performed using NI 9225[27] and NI 9227 [28] c-

series acquisition modules connected to PXI-7813R FPGA card [26]. A/D resolution is 



168 M.ANDREJEVIĆ-STOŠOVIĆ, M. DIMITRIJEVIĆ, S. BOJANIĆ, O. NIETO-TALADRIZ, V. LITOVSKI 

24-bit, with 50 kSa/s sampling rate and dynamic range ±300 V for voltages and ±5 A for 

currents. The FPGA provides timing, triggering control, and channel synchronization 

maintaining high-speed, hardware reliability, and strict determinism. 

The FPGA code is implemented in a LabVIEW development environment. The 

function of the FPGA circuit is acquisition control. 

A

A

A

A

V

V

V

DUT

NI 9225 NI 9227

L1

L2

L3

N

 

Fig. 2 Connection diagram of acquisition subsystem 

The software component is implemented in two stages, executing on real-time 

operating system (PharLap RTOS, [29, 30]) and general purpose operating system 

(GPOS). Described system enables calculation of a number of parameters in real-time 

that characterize nonlinear loads, which is impossible using classical instruments. The 

measured quantities are calculated from the current and voltage waveforms according to 

IEEE 1459-2000 and IEEE 1459-2010 standards [12, 13]. 

Real time application (Fig. 3) calculates power and power quality parameters 

deterministically and saves calculated values on local storage. The application is executed 

on real time operating system. 

 

Fig. 3 Part of real-time application in G code, alternative reactive power calculations 



 Characterization of Nonlinear Loads in Power Distribution Grid 169 

Virtual instrument, implemented in National Instruments LabVIEW [30, 31] environment, 

is used for additional analysis and data manipulation represents user interface of described 

system. It runs on general purpose operating system, physically apart from the rest of the 

system. Communication is achieved by TCP/IP. Parameters and values obtained by 

means of acquisition and calculations are presented numerically and graphically (Fig. 4). 

 

Fig. 4 Virtual instrument provides measurements of various parameters 

4. MEASUREMENT RESULTS 

We have performed measurements on various small loads. The parameters obtained 

may be used for decision making of various kinds, such as verification of compliance to 

some standards or categorization within quality frames. As small loads here we consider 

various devices: CFL and LED lamps, power supply devices and battery chargers in case of 

personal communication and computing devices. These devices are ubiquitous and in 

everyday use, thus their cumulative effect on power distribution grid is not negligible [32], 

[33]. Various parameters that characterize nonlinearity, efficiency and quality are measured 

and calculated. 

Table 1 shows measured results obtained on small loads such as various compact 

fluorescent lamps (CFL, 7 W – 20W), incandescent lamps (100W and 60W), two low-power 

1 W indoor LED (light emitting diode) lamps, prototype of street 34 W LED lamp and CRT 

computer monitor for reference. 

Compact fluorescent lamp is good example of nonlinear load [34]. It brings reduction in 

total energy consumption (about 20%, comparing to incandescent lamp of equivalent 

luminosity), but with harmonic currents and increased harmonic loss on distribution 

transformer. Measurements show that CFL lamps have good correction of displacement 

power factor, but significant distortion leading to low total power factor (Table 1). CFLs are 

equipped by power supply units which conduct current only during a very small part of 

fundamental period, so the current drawn from the grid has the shape of a short impulse. 



170 M.ANDREJEVIĆ-STOŠOVIĆ, M. DIMITRIJEVIĆ, S. BOJANIĆ, O. NIETO-TALADRIZ, V. LITOVSKI 

Table 1 CFL and LED lamps 

Type 
N

o
m

in
a
l 

p
o

w
e
r 

(W
) 

F
re

q
u
e
n

c
y

 

(H
z
) 

V
R

M
S
(V

) 

I R
M

S
 (
m

A
) 

A
c
ti

v
e
 

p
o

w
e
r 

(W
) 

I 
D

C
 (

m
A

) 

V
o

lt
a
g

e
 

T
H

D
(%

) 

C
u

rr
e
n

t 

T
H

D
(%

) 

C
u

rr
e
n

t 

C
R

E
S

T
 

V
o

lt
a
g

e
 

C
R

E
S

T
 

D
P

F
 (

%
) 

c
o

s(
φ

) 

T
P

F
 (

%
) 

Incandescent 100 50.03 230.20 421.66 97.02 0.62 3.11 3.05 1.52 1.47 99.95 1.00 99.95 

CFL bulb 20 50.03 231.49 134.87 18.64 0.24 2.58 112.17 3.38 1.41 66.55 0.90 59.70 

CFL tube 20 49.95 231.20 145.89 19.66 0.25 2.84 114.01 4.33 1.44 65.94 0.88 58.28 

CFL bulb 15 49.99 231.47 92.16 12.60 0.13 2.82 115.52 3.52 1.41 65.45 0.90 59.08 

Incandescent 60 49.97 231.15 257.88 59.58 0.42 2.87 2.84 1.57 1.41 99.96 1.00 99.96 

CFL spot 7 49.97 232.48 50.86 7.23 0.19 2.81 104.24 3.24 1.40 69.23 0.88 61.20 

CFL bulb 7 50.06 230.95 52.46 7.21 0.28 2.83 112.26 3.42 1.40 66.51 0.90 59.54 

CFL bulb 9 50.01 233.20 60.54 8.25 0.11 2.87 116.93 3.60 1.39 64.99 0.90 58.44 

CFL tube 11 50.01 233.17 84.34 11.66 0.16 2.79 112.27 3.37 1.45 66.51 0.89 59.28 

CFL tube 18 50.01 221.32 135.56 18.40 0.38 2.82 107.35 4.52 1.45 68.16 0.90 61.32 

CFL tube 11 50.01 221.14 115.00 14.06 0.16 3.01 119.30 4.06 1.46 64.24 0.86 55.41 

CFL helix 11 50.00 221.83 76.73 10.23 0.25 2.96 109.26 4.90 1.47 67.51 0.89 60.09 

CFL bulb 9 49.99 232.52 70.06 9.70 0.19 2.84 110.87 3.52 1.43 66.98 0.89 59.53 

CFL helix 18 50.01 221.46 138.68 19.01 0.35 2.89 105.56 3.94 1.43 68.77 0.90 61.71 

CFL helix 20 50.03 231.19 156.43 21.02 0.20 2.79 111.36 3.91 1.44 66.82 0.87 58.13 

CFL tube 15 50.01 221.00 105.09 13.96 0.29 3.16 112.13 4.46 1.40 66.56 0.90 60.11 

LED white 1 50.00 217.24 14.96 0.35 0.09 2.36 21.14 1.72 1.38 97.84 0.11 10.79 

LED cold w. 1 49.94 217.33 14.95 0.35 0.08 2.36 21.14 1.72 1.38 97.84 0.11 10.79 

LED street 34 49.99 216.63 246.12 32.87 0.05 2.53 102.98 3.28 1.38 69.66 0.89 61.66 

CRT   50.03 232.63 475.86 107.46 1.60 2.93 13.24 1.65 1.49 99.14 0.98 97.69 

Characterization of nonlinear loads can be accomplished by analyzing reactive and 

distortion power. Table 2 shows reactive power and distortion power values, calculated 

using alternative definitions, for compact fluorescent lamps, two incandescent lamps and 

indoor LED lamps. Following values are displayed: active power (P), apparent power (S), 

non-active power (N), Budeanu’s reactive power (QB), Budeanu’s distortion power (DB), 

Fryze’s reactive power (Qf), IEEE Std 1459-2010 proposed definition for reactive power 

(QIEEE), Shanon’s  apparent power (Sq), Kimbark’s reactive power (Qk), Kusters-Moore’s 

capacitive (QC) and inductive (QL) reactive power. 

Comparison of Budeanu’s reactive and distortion power suggests that all examined CFL 

and LED lamps are non-linear loads (DB>QB). Reactive power calculated from Fryze’s 

definition (45) is equal to non-active power, 2 2N S P  . Kimbark’s equation (39) for 

reactive power, which takes only fundamental harmonic into account, gives approximately 

±3% deviance from Budeanu’s formula (QB). It suggests that the actual contribution of 

harmonic frequencies to reactive power is small – less than 3% of the total reactive power. 

IEEE proposed definition always provides value of the total reactive power greater 

than the value of the fundamental component. 



 Characterization of Nonlinear Loads in Power Distribution Grid 171 

Table 2 CFL and LED lamps 

No. Type 

P
o

w
e
r 

P
 (

W
) 

U
(V

A
) 

N
 (

V
A

R
) 

Q
B
(V

A
R

) 

D
B
(V

A
R

) 

Q
f 
(V

A
R

) 

Q
IE

E
E
 (

V
A

R
) 

S
q
 (

V
A

R
) 

Q
k
 (

V
A

R
) 

Q
C
 (

V
A

R
) 

Q
L
 (

V
A

R
) 

1 CFL Rod   11.56 17.84 13.58 -6.16 12.10 13.58 6.16 10.24 -6.16 -4.43 -6.11 

2 CFL bulb E27 20 17.14 27.72 21.78 -8.43 20.08 21.78 8.43 14.48 -8.43 -6.46 -8.37 

3 CFL tube E27 20 16.77 28.46 23.00 -8.44 21.39 23.00 8.45 14.55 -8.45 -6.07 -8.39 

4 CFL bulb E27 15 11.59 18.91 14.94 -5.31 13.97 14.94 5.32 9.22 -5.32 -4.00 -5.28 

5 Inc E27 100 86.77 86.78 0.80 -0.50 0.63 0.80 0.50 0.56 -0.50 -0.36 -0.49 

6 CFL spot E14 7 5.87 9.32 7.25 -2.83 6.67 7.25 2.81 4.23 -2.81 -2.17 -2.80 

7 CFL bulb E27 7 6.16 9.86 7.71 -2.64 7.24 7.71 2.65 4.83 -2.65 -2.03 -2.63 

8 CFL bulb E14 9 6.46 10.78 8.63 -2.72 8.19 8.63 2.72 5.45 -2.72 -2.08 -2.70 

9 CFL tube E14 11 9.89 16.11 12.72 -4.71 11.82 12.72 4.69 7.89 -4.69 -3.61 -4.66 

10 CFL tube E27 18 17.10 28.86 23.24 -8.73 21.54 23.24 8.75 13.27 -8.75 -6.64 -8.68 

11 CFL tube E27 11 10.63 17.67 14.12 -5.83 12.85 14.12 5.83 8.85 -5.83 -4.41 -5.79 

12 CFL helix E27 11 9.58 16.27 13.16 -4.93 12.20 13.16 4.95 8.75 -4.95 -3.68 -4.90 

13 Inc E14 60 55.06 55.06 0.61 -0.37 0.49 0.61 0.37 0.37 -0.37 -0.27 -0.37 

14 CFL helix E27 18 17.21 28.87 23.18 -8.82 21.43 23.18 8.83 15.55 -8.82 -6.77 -8.76 

15 CFL helix E27 20 18.41 30.68 24.54 -9.95 22.43 24.54 9.93 16.14 -9.93 -7.56 -9.86 

16 CFL tube E27 15 12.66 21.97 17.95 -6.32 16.80 17.95 6.33 11.63 -6.33 -4.80 -6.28 

17 Spot E27 15 16.92 34.24 29.77 -3.88 29.52 29.77 4.14 20.01 -4.13 -1.98 -4.06 

18 Spot E27 10 13.23 26.33 22.76 -2.97 22.56 22.76 3.17 15.45 -3.17 -1.51 -3.12 

19 Bulb W E27 8 10.00 19.53 16.77 -2.81 16.54 16.77 2.94 11.52 -2.93 -1.74 -2.89 

20 Bulb W E27 6 8.51 9.45 4.11 0.08 4.11 4.11 0.07 3.29 0.07 0.08 0.07 

21 Bulb E27 6 8.69 9.58 4.04 0.09 4.04 4.04 0.08 3.28 0.08 0.08 0.08 

22 Bulb E27 3 4.07 7.70 6.54 -0.84 6.48 6.54 0.90 4.35 -0.90 -0.45 -0.88 

23 RGB E27 3 1.92 3.17 2.52 0.01 2.52 2.52 0.01 1.39 0.00 0.05 0.00 

24 Spot E14 3 4.00 8.05 6.99 -0.98 6.92 6.99 1.04 4.86 -1.04 -0.52 -1.02 

 

Further, personal devices such as tablet computer, mobile phone, laptop computer and 

cordless telephone containing rechargeable batteries are analyzed regarding operating 

conditions. Measured results are presented in Table 3. Working conditions are standby 

(device turned off and battery not charging), working and charging (device turned on and 

battery charging) and charging only (device turned off and battery charging). A standalone 

battery charger is also tested. 

Following values are measured and shown in the table: voltage RMS (V), current RMS 

(I), frequency (f), cosine of 1st harmonic phase difference (cosφ1), TPF – total power factor 

(%), DPF – distortion power factor (%), THDV – voltage total harmonic distortion 

(%),THDI – current total harmonic distortion (%), active power (P), Budeanu’s reactive 

power (QB), apparent power (U), distortion power (D), non-active power (N), phasor power 

(S), first harmonic active power (P1) and higher harmonics active power (PH). 



172 M.ANDREJEVIĆ-STOŠOVIĆ, M. DIMITRIJEVIĆ, S. BOJANIĆ, O. NIETO-TALADRIZ, V. LITOVSKI 

In the next we will pay some attention to the very results depicted in Table 3. Let's 

first have a glimpse at the distortions of the current (THDI). As can be seen even in the 

best cases the THDI is larger than 20%. There is a case, a mobile phone battery charger 

while charging, where the THDI is 154.51% which means the harmonics exceed by a 

large margin the fundamental. Note that this is not an isolated case. One may observe 

several THDIs of similar value. To summarize, THDI is exposing the nonlinear character 

of all small loads, some of which are extremely nonlinear producing harmonics larger 

than the fundamental one. 

Table 3 Personal devices in different working conditions 

N
o

. 

Device description 

V
 (

V
) 

I 
(m

A
) 

f 
(H

z
) 

1 Charger 230V 1.7A - 2XAAA NiCd battery charging. 850mAh 236.06 9.89 50.02 

2 Tablet computer turned on. Li-Polimer 8220 mAh battery charging 235.70 80.92 49.98 

3 Tablet computer turned off. Li-Polimer 8220 mAh battery charging 236.59 61.65 49.99 

4 Tablet computer turned off. charger 230V/2A connected. not charging 236.51 1.70 50.00 

5 Mobile phone charger connected. not charging 230V/0.2A 236.62 1.33 9.99 

6 Mobile phone turned on. Li-Ion 1230 mAh battery charging 235.65 53.72 49.98 

7 Mobile phone turned off. Li-Ion 1230 mAh battery charging 236.09 48.05 50.01 

8 Laptop comp. (type 1) turned on. charger 230V. 1.7A connected, not charging 233.49 22.99 50.01 

9 Laptop comp. (type 1) turned on. Li-ION 2200mAh battery charging 232.81 231.39 50.00 

10 Laptop comp. (type 1) turned off. Li-ION 2200mAh battery charging 233.52 106.52 49.99 

11 Laptop comp. (type 2) turned on. Charger 230V 1.5A connected, not charging 233.07 15.71 49.99 

12 Laptop computer (type 2) turned on. Li-ION 4400mAh battery charging 232.05 436.60 49.97 

13 Cordless telephone base charger 230V/40mA disconnected 232.77 21.05 49.97 

14 Cordless telephone base. 2XAAA. NiCd. 550mAh battery not charging 233.68 21.71 50.00 

15 Cordless telephone base. 2XAAA. NiCd. 550mAh battery charging 233.55 25.60 49.99 

N
o

. 

T
P

F
 (

%
) 

D
P

F
 (

%
) 

T
H

D
V
 (

%
) 

T
H

D
I 
(%

) 

P
 (

W
) 

Q
B
 (

V
A

R
) 

U
 (

V
A

) 

D
 (

V
A

R
) 

N
 (

V
A

R
) 

S
 (

V
A

R
) 

P
1
 (

W
) 

P
H
(W

) 

1 32.93 70.81 1.70 94.47 0.77 1.77 2.33 1.62 2.20 1.68 0.78 -0.02 

2 57.36 58.15 1.73 137.76 10.94 -1.74 19.07 15.53 15.62 11.08 11.08 -0.14 

3 55.12 55.54 1.70 146.23 8.04 -0.93 14.59 12.13 12.17 8.09 8.17 -0.12 

4 21.43 79.20 1.67 114.80 0.09 0.18 0.40 0.35 0.39 0.20 0.05 0.00 

5 12.64 101.35 1.69 59.01 0.04 0.17 0.31 0.26 0.31 0.18 0.02 0.00 

6 52.73 53.66 1.71 154.51 6.67 -1.18 12.66 10.69 10.76 6.78 6.73 -0.05 

7 51.18 51.98 1.77 161.72 5.81 -0.96 11.34 9.70 9.75 5.88 5.87 -0.06 

8 7.00 95.18 1.78 29.07 0.38 1.38 5.37 1.61 5.36 5.12 0.38 -0.01 

9 53.67 54.76 2.00 147.11 28.91 -6.10 53.87 45.04 45.45 29.55 29.65 -0.71 

10 47.51 50.62 1.92 164.35 11.82 -4.64 24.87 21.39 21.89 12.70 12.18 -0.28 

11 12.69 99.22 1.94 40.82 0.46 1.46 3.66 1.42 3.63 3.37 0.43 0.00 

12 96.74 97.30 1.83 20.90 98.01 -10.67 101.31 23.32 25.65 98.59 97.86 0.02 

13 23.50 90.76 1.80 43.70 1.15 4.33 4.90 1.97 4.76 4.48 1.16 -0.01 

14 47.31 92.64 1.78 36.64 2.40 4.09 5.07 1.81 4.47 4.74 2.43 -0.01 

15 70.29 92.99 1.82 37.24 4.20 3.66 5.98 2.16 4.25 5.57 4.23 -0.02 



 Characterization of Nonlinear Loads in Power Distribution Grid 173 

The next very important and also interesting set of data is related to the power factor. In 

early days it was known as cos
 
of the load while only linear loads were considered 

supposedly having reactive component introducing phase shift between the voltage and the 

current. The total power factor (TPF) encompasses the whole event including the distortions 

of both the voltage and the current and their mutual phase shift. As can be seen from Table 

1, there is only one case where the TPF is approaching unity which is supposed to be its 

ideal value. In many of the cases the value of TPF is smaller than 50% meaning that the 

active power is smaller than a half of the total power drawn from the main which, as we 

could see from the previous paragraph, is mainly due to the distortions. In general, since 

most of the chargers are considered of small power (look to the column P1 in Table 3), no 

power factor correction is built in so that significant losses are allowed. That, to repeat 

once more, would not be a problem if the number of such devices, being attached to the 

mains all the time, is not in the range of billion(s).
 

The next column, the distortion power factor (DPF), represents the percentage of power 

taken by the harmonics. As we can see, except for a small number of cases where the 

harmonics are approximately on the level of half of the total power, in most cases they are 

taking as large power as the fundamental. Note, the harmonics are unwanted not only 

because of efficiency problems. In fact, in the long term, the presence of harmonics on the 

grid can cause:  

 Increased electrical consumption 
 Added wear and tear on motors and other equipment 
 Greater maintenance costs 
 Upstream and downstream power-quality problems, 
 Utility penalties for causing problems on the power grid 
 Overheating in transformers, and similar. 
Similar conclusion may be drawn in by comparison of the Distortion (D) and the 

power of the first (fundamental) harmonic (P1). There are only three cases where the second is 

larger than the former. 

To summarize the data from Table 3 one may say that an electronic load to the grid 

which in fact represents a power supply of a telecommunication or IT device, represents a 

small but highly nonlinear load. In many cases the TPF of such a load is in favor of everything 

but not the active power to be delivered to the device. 

 

5. CONCLUSION 

Due to the changes in the nature of the electrical loads to the grid new aspects of the 

characterization of the loads to the electrical grid are emerging. These are related mainly 

to the nonlinearities of modern electronic loads and to the subsystems used for conversion 

from DC to AC and vice versa that is becoming unavoidable in modern production and 

distribution systems. 

To qualify and quantify the properties of the modern power electrical systems new 

tools are to be developed being able to cope with the new properties of the signals arising 

at the grid-to-load and grid to power-producing-facility interface. That stands for both 

theoretical algorithms for computation and for the very measurement equipment. 

In these proceedings we represent our results in development and implementation of a 

measurement system for small loads that are becoming ubiquitous and consequently of big 

concern for the quality of the delivered electrical energy. We also present the measurement 



174 M.ANDREJEVIĆ-STOŠOVIĆ, M. DIMITRIJEVIĆ, S. BOJANIĆ, O. NIETO-TALADRIZ, V. LITOVSKI 

results for a broad set of electronic loads revealing many secrets hidden behind the 

prejudice that these loads are small and unimportant. 

Our hardware and software solutions may be characterized as advanced, accurate and 

versatile while at the same time of low price making them very attractive for practical use 

being it in laboratory or in field conditions. 

Acknowledgement: This research was partly funded by The Ministry of Education and Science of 

Republic of Serbia under contract No TR32004. 

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