1

Online Thevenin Equivalent Parameter Estimation using
Nonlinear and Linear Recursive Least Square Algorithm
Md. Umar Hashmi
Department of Energy Science & Engineering
IIT Bombay
Bombay, Maharashtra
umar.hashmi123@gmail.com

Rahul Choudhary
Department of Systems and Control
IIT Bombay
Bombay, Maharashtra
rahulchoudhary.2002@gmail.com

Abstract—This paper proposes method for detection, estimation of
Thevenin equivalent parameters to describe power system behavior.
Thevenin equivalent estimation is a challenge due to variation in
system states caused by power flow in the network. Thevenin
equivalent calculation based on changes in system with multiple
sources integrated with grid, isolated distributed generator system is
analysed and nonlinear least square fit estimation technique for
algorithm is adopted. Linear least square fit is used with a linearized
model. Performance evaluation of proposed method is carried out
through mathematical model, nonlinear and linear least square fit
based
algorithm
technique
and
simulation
through
MATLAB/SIMULINK package. Accurate grid and source side
impedance estimation technique is applicable for distributed
generation sources interfaced with grid to improve dynamic response,
stability, reliability when subjected to faults or any other
disturbances in network. Algorithm can accurately estimate Thevenin
equivalent of multiple sources connected in parallel simultaneously
with voltage and current phasor measurements at point of common
coupling. Mathematical analysis and simulation results validate the
effectiveness of proposed method.
Index Terms—Adaptive controller, Impedance, Least square fit
estimation, Point of common coupling, Thevenin equivalent.

I. INTRODUCTION

W

ith integration of large number of distributed generation
sources (DG) into distribution network has raised concerns
related to stability, power quality and reliability of the
supply [1-3]. Distributed power generation sources depending
upon their mode of operation affects line impedance of distributed
electricity network. Performance of grid connected converter
depends upon line impedance [4]. Islanding operation due to
faults or grid disturbances due to voltage or frequency sag if not
detected properly leads to poor power quality at point of common
coupling (PCC).Online measurement due its numerous advantages
is preferred to fulfill anti islanding requirement. The distortion in
network caused due to electrical loads and distributed generation
sources strongly depends upon grid impedance and amount of
apparent power connected [5].To optimize the operation and
performance of distributed generation sources in microgrid
network, time and frequency dependent grid parameter estimation/
calculation is key [5].
In literature several methods are proposed for impedance
estimation. Estimation based on control loop variations which
include new values of inductance and resistance in control loop to
improve performance of system is proposed in [6]. Grid
impedance estimation based on use of extra devices is proposed in
[7]. Injection of harmonics signal into the grid and use of
mathematical models to obtain new impedance parameters are

Jayesh G. Priolkar
Department of Electrical Engineering
Goa College of Engineering
Farmagudi, Ponda Goa
jayeshpriolkar@gmail.com

addressed in [8-10].Grid impedance using controlled excitation
based on frequency characteristics of inductance -capacitance–
inductance (LCL) filter resonance is reported in [3]. The various
methods used to determine grid impedance characteristics offline
by means of frequency response analysis is reported in [11-13].
The methods of equivalent impedance estimation can be
broadly classified as active and passive type [14]. Passive
methods utilize the existing disturbance present in power
networks, for example detection of low order harmonic frequency
impedance. In active methods in addition to regular operation,
forced disturbance is injected into the grid or distributed
generation network for parameter estimation. Regression analysis
is used to extract parameters from measured data to define
physical characteristics of system. Thevenin equivalent estimation
based on concept of regression is also reported in literature.
Recursive least square estimation technique based on varying
system states to determine Thevenin equivalent is proposed in
[15].
Detection and estimation of Thevenin equivalent impedance for
different application is reported in literature. For example, power
system fault detection, load matching for maximum power
transfer, state of charge estimation for battery bank [16],
simultaneous estimation of Thevenin equivalent of multiple
sources, for load management by load shedding in power system
network based on detection of undervoltage [17], voltage stability
margin adjustment and analysis for prevention of voltage collapse
by a real time voltage instability identification algorithm based on
local phasor measurements [18].
The main aim of our work is to develop the algorithm for
online Thevenin equivalent parameter estimation. It is active
method which is available to estimate Thevenin equivalent
voltage and current. To evaluate the performance of proposed
method, mathematical model, nonlinear least square fit based
algorithm
technique
and
simulation
through
MATLAB/SIMULINK package are done.
Paper is organized as follows, Introduction of paper is given in
section I, section II discusses in brief applications of online
Thevenin equivalent estimation algorithm, section III presents
parameter estimation for isolated power system, and section IV
discusses the parameter estimation for grid connected system.
Simulation results are discussed in section V. Section VI
concludes the paper.

II. APPLICATIONS OF THEVENIN EQUIVALENT ESTIMATION
Online Thevenin equivalent estimation algorithm finds
application in adaptive control of electrical processes for
performance improvement and preemptive corrective action can be
taken for safe system operation. Some of the applications have
been discussed in this section.

2
i) Power system fault detection: - The Thevennin equivalent of
power system will indicate the change of estimatedd parameters due
to fault in power system. The relationships betweeen the real fault
distance and the varying Thevenin equivalennt impedance is
presented in paper [19]. The proposed Thevvenin equivalent
estimation algorithm in our work uses balancced symmetrical
components for correct estimation. Balanced ground faults,
balanced short circuits and balanced impedancee change can be
identified using the presented algorithm.
ii) Load matching for maximizing power transsfer: - Maximum
power is transferred to a load resistor (RL) when the value of the
load resistor is selected to match the value oof the Thevenin
resistance (Rth) of the power source. Figure 1 dem
monstrates this for
a value of Rth of 10 ohms.
Load Power Response to Changes in Load Resistance forr Rth = 10 ohms
1000

Power (Watts)

800
600

iii) State of charge estimation of battery or battery bank: A
6] between the open circuit
linear relationship is presented in [16
voltage (VOC) and SOC
)
%) =
The values of a and b are found experrimentally [16]. Open circuit
voltage is the Thevenin equivalent voltage of a battery under no
load condition.
ustment and analysis for
iv) Voltage stability margin adju
prevention of voltage collapse: Voltage instability is a major
concern for power systems operatiion. A Real-Time voltage
instability identification algorithm Based on Local Phasor
Measurements is proposed in [18] which recognize instabilities in
voltage. The power transferred to the bus reaches its voltage
stability limit when that Thevenin impedance has the same
magnitude as the load impedance at the bus [18].
The apparent power supplied is S, Y = 1/ZL, than

400

=

200
0

0

10

20
30
Load Resistance(Ohms)

40

50

Figure 1: Maximum Power Transfer at Zload=
=Zsource
Maximum Power Transfer with Complex Impedance

)
))^

(2)

The condition for maximum load ap
pparent power is dS/dY =0,
hence the critical point of voltage insstability is Zth = ZL. Hence
maximum loading point can be accurrately monitored online by
calculating dS/dY. The value of dS/d
dY close to zero indicates
proximity to voltage collapse point [20]].
The proposed algorithm assumes no simplification in system for
estimating Thevenin equivalent parameters.

Power Transfer (Watts)

1000

III. PARAMETER ESTIMATION FOR ISOLATED
M
SYSTEM

800

600

400

200

0
5
20
15

0

Description of the Proposed Algorith
hm:
Consider a voltage source feeding a lo
oad which has an impedance
of itself, line and filter impedance befo
ore it’s actually connected to
the load. The Thevenin equivalent of the
t circuit is shown adjacent
to it in the figure below.

10
Impedance (Ohms)

5
-5

Resiistance (Ohms)

0

Figure 2: Maximum Power Transfer with complexx impedance
2

P = I R, PL = [Vth/ (Rth+ RL)]2(RL)
(1)
One example is load matching is for wirelesss power transfer
circuitry design using inductive coils. Four coill wireless power
transfer simulation diagram along with the matcching network is
shown in Figure 3. Parameter values of the matchhing network are
dependent on the Thevenin equivalent impedancee of the four coil
system.
Matching network shown in figure 3 is one of thhe many possible
architectures of a matching network.

Figure 3: Matching network design for 4 coil wireless pow
wer transfer circuit

Figure 4: Isolated system with itss Thevenin equivalent

Using Kirchhoff’s Voltage Law we can write the voltage loop
equation as:
VT
cos θ j sin θ) = VPCC cos
j sin ) IPCC cos φ
j sin φ) × R T
jXT
)
(3)
We can separate the real and imaaginary component of the
equation as,
VT
cos θ = VPCC cos
IPCC cos φ × R T
sin φ ×
XT
)
(4)
VT
sin θ = VPCC sin
IPCC co
os φ × XT
RT
× sin φ)
(5)

3
The algorithm will be able to estimate the Theevenin equivalent
voltage and current. The point of common coupliing is the point at
which the load is connected. The known and unkn
known parameters
are shown below in table 1:
Table 1: State of Parameters
Known Parameters
Voltage Magnitude at PCC (VPCC)

Start

Unknown P
Parameters
Thevenin Voltage (VThevenin) ( 1)

Voltage Angle at PCC ( )

Thevenin Voltagee Angle (θ) (

Current Magnitude at PCC (IPCC)

Thevenin Resistancee (RThevenin) ( 3)

Current Angles at PCC (φ)

Thevenin Inductancee (XThevenin)(

VPCC cos = VT
XT
)

cos θ

IPCC cos φ × R T

VPCC sin = VT
RThevenin×sinφ)

sin θ

IPCC cos φ × XT

Equation (6) and (7) are non-linear in nature.
cos )
=
sin )
Where, y = VPCC cos , = VPCC sin , a = IPCC cos φ,
b = IPCC sin φ are known parameters and = VT
,
= RT
, = XT
are unknown param
meters.
=
Where

, , )
=

Estimation of Thevenin equivalent uses local measurements of at
least two different voltage and curreent vectors (magnitude and
phase) pairs measured at different time with different values
nin equivalent parameters.
associated with same reference Theven

Initialize Simulation
Variables

2)

For i = 1: number of iterations
s

4)

1. Define Load
2. Run Simulation
p
3. Get Voltage and Current (both phase
& magnitude ) at PCC

sin φ ×
(6)

Yes

(7)

No

Initial Guess for unknown
parameters X

Use
U
previous value of
unknown
u
parameter X
No

(8)
= ,

1. Minimize the error function E 2 for X
[E2 = f(Vpcc,Ipcc, load, Vsourse
e)]
2. Get X using least square meth
hod
Store
unknown
parameter X

Display X

(9)
and

If i = 1

If i = iteration limit

=

Yes
end

= min ∑

)

(10)

It is intended to solve these equations to obtaain the values X
which satisfy the system of equations describedd in equation 10.
Initial guess values are random numbers. Equattion 10 has to be
minimized for optimal values of unknown param
meters. Ideally β
has to be zero but since the algorithm recursiveely optimizes the
unknown parameters, there is a residual error w
which has to be
maintained as low as possible for accurate ressults. Residual is
defined by equation (11).
=

(11)

Nonlinear Recursive algorithm needs two Phasoor measurements
of voltage and current at the point of commoon coupling. The
figure below shows the circuit diagram for two different loading
scenarios. Phasor measurements are done at steaddy state.

Figure 7: Flow chart for the proposed algorithm

Parameters Estimation using linear regression
Non-linearity of the Equation (8) can be
b taken care by eliminating
the trigonometric variables.
=
=

(12)
1
0

0
1

×

(13)

Where, y = VPCC cos , = VPCC sin , a = IPCC cos φ, b =
IPCCsinφ are known parameters and 1=VThevenincosθ,
= VT
sin θ,
= RT
, = XT
are unknown
parameters.
a
Additional variable can be calculated as
(14)
VT
=
)
= tan

(15)

For n different loads, Equation (13) caan be written in following
structures (augmentation of matrix),
Figure 5: Circuit description of two data set coollection

=

(16)

Where dimension of the matrices are as,
a
For given initial estimate of

Figure 6: Phasor diagram of at least two different measurem
ments is required for
operation of estimation algorithm.

=
Estimation error
=
=
=
=

×

=

×

×

= ,
(17)
(18)
(19)

4
Error between actual and estimate will be minimuum for
)

=0

=0
)

=
Vector

(20)
(21)
(22)

provides good estimate of unknown paraameters.

IV. PARAMETER ESTIMATION FO
OR GRID
CONNECTED SYSTEM
Consider DG source connected to grid as shown in figure 8.
The total impedance of network includes line,, filter and load
impedance.
Thevenin Estimation Results:
Positive sequence voltage and current angle annd magnitude are
measured. It is assumed that the input voltagee magnitude and
angle and the Thevenin impedance as seen from
m the load side is
constant while the algorithm is in operation. A
As the estimation
gives an output which is within ±1% error wiith just one load
change; and hence it does not take more than a few seconds for
accurate prediction.
The positive sequence extraction is done too eliminate the
unbalanced components in voltage and currentt. The impact of
negative sequence is in terms of losses or heeating caused by
unbalances, for this case it’s not that relevant.
The results of the estimation with multiple numbbers of iterations
are tabulated below table.
Table 2: Estimation of Parameters for Isolatedd system
No. of Phasor
Generator
Voltage
Resistaance
Inductance
Measurements
Voltage
angle
Actual
70.7107
1
0.377
2 Phasor
70.642
1.98E-05
0.984442
0.3747
Error %
0.09706
1.557333
0.5993
3 Phasor
70.642
2.98E-05
0.9843399
0.374741
Error %
0.09715
1.56001
0.5992
5 Phasor
70.6254
1.11E-05
0.978555
0.373026
Error %
0.12057
2.144227
1.05397
10 Phasor
70.576
2.76E-05
0.96009
0.3691
Error %
0.1905
3.911
2.0955
100 Phasor
70.5762
1.34E-06
0.961
0.3691
Error %
0.19015
3.899449
2.09503

The estimated values tabulated in Table 2 shows accurate
prediction with just two Phasor measurem
ments of known
parameters at point of common coupling.

•

Current from individual so
ources (Igen and Igrid in the
figure above) is needed
Equivalent Thevenin voltage (maagnitude and angle) and
impedance can be estimated. The resullts of estimation are listed in
Table 4 and 6. Generator Side Estim
mation known and unknown
parameters are listed in table 3. A hypothetical scenario is
simulated in which voltage levels off two sources connected in
parallel are drastically different. The
T
generator side voltage
magnitude is set at 70.7 V and the grid is assumed at 49.5 V. The
algorithm accurately estimates the volttage level difference.
Table 3: State of Parameters on Generator sid
de
Known Parameters
Unknown Parameters
Voltage Magnitude at PCC
Generator Voltage
Voltage Angle at PCC
Generator Voltage Angle
Current Magnitude of generator
Generator resistance
Current Angles
Generator Inductance

Parameter Estimated with their errors percentages
p
for generator
side parameter estimation is tabulated in Table 4.
Table 4: Estimation of Parameters with error on Generator side
No. of Phasor
Generator
Voltage
Resistance
Measurements
Voltage
angle
Actual
70.7107
1
2 Phasor
70.70932
-6.285
0.999933
Error %
0.00195
0.0067
3 Phasor
70.70948
-6.279
0.999942
Error %
0.00172
0.00577
5 Phasor
70.71005
-6.290
0.999971
Error %
0.00092
0.00289

0.377
0.376977
0.0061
0.376974
0.007004
0.376978
0.0059

Grid Side Estimation known and unkn
nown parameters are listed in
table 5.
Table 5: State of Parameters on grid side
Known Parameters
Voltage Magnitude at PCC
Voltage Angle at PCC
Current Magnitude from grid
Current Angles

Unknown Parameters
Grid Voltage
Grid Voltage Angle
Grid Resistance
Grid Inductance

Parameter Estimated with their errors:
Table 6: Estimation of parameters with errors on grid side
No. of Phasor Grid Voltage
Voltagee
Grid
Measurements
angle
Resistance
Actual
49.4975
0.5
2 Phasor
49.4973
-0.0019
9
0.5
Error %
0.0004
0
3 Phasor
49.49726
-6.29
0.500017
Error %
0.000483
-0.00334
5 Phasor
49.49754
-6.29
0.499
Error %
-8.26E-05
0.00112

Grid
Inductance
0.0377
0.0377
0
0.037699
0.00285
0.03769
0.00157

'n' number of Power Sources co
onnected in Parallel

Superposition theorem implementation of proposeed algorithm:

140
Known Parameters for Parame
eter Estimation
Unknown Parameters Estimat ed

120
Number of Parameters

The algorithm can be used for multiple soources parameter
estimation simultaneously. The information nneeded from the
system is that we should know the currents cominng out of each of
the sources and the common voltage vector at ppoint of common
coupling.

Inductance

100
80
60
40
20
0

Figure 8: Generator and grid connected in parallel tto feed a load

For parameter estimation for a system with multipple sources:
• Voltage magnitude and angle at PCC is needed.

0

5

10
15
20
0
25
Number of Sources in
n Parallel

30

35

Figure 9: Known & Unknown parameters forr ‘n’ sources connected in parallel

5

Generator
Voltage
70.711
15.395
78.228
47.984
32.140
70.602
0.154
70.651
0.084
70.599
0.158
70.583
0.181
70.636
0.105

Voltage
angle
39.479
-30.161
-0.151
-0.112
-0.143
-0.151
-0.110

Resistance
1.000
-8.829
982.878
-1.342
234.184
1.005
-0.483
1.006
-0.557
0.999
0.091
0.996
0.374
0.998
0.219

Inductance
0.377
-3.351
988.831
-6.005
1692.739
0.390
-3.496
0.392
-4.019
0.374
0.665
0.367
2.730
0.371
1.603

Linear recursive least square method derived for thevenin
equivalent estimation requires at least three phasor measurements
for accurate prediction of unknown parameters. The result for
linear recursive least square estimation is listed in table 7. The
results with 1st and 2ndphasor measurements are not reliable,
however the results converge with 3 phasor measurements and
own wards.
Compared to linear recursive fit, nonlinear recursive fit is better as
the error percentages are much lower as per results tabulated in
table 4, 6 and 7.
Nonlinear recursive method uses exact equation, therefore
computationally exhaustive, for estimation and hence the overall
accuracy of estimation is higher. Furthermore nonlinear recursion
estimation converges with 2 phasor measurement data but linear
method takes at least 3 phasor measurements for convergence.
The results generated using proposed algorithms has an upper
bound of 5000 as maximum function evaluations and maximum
number of iterations as 8000.

Power System Grid Impedance Change Detection
Power system undergoes various kinds of faults which lead to
overall impedance change of the system. Simulations are
conducted in MATLAB Simulink for isolated system to verify the
capability of proposed algorithm to detect impedance change in

Volts

70.711

70.7105

70.71

70.7095

0

2

4

6

8

10

Time(s)

Figure10: Voltage magnitude estimation

The voltage magnitude from source side is unchanged and only
line impedance is changed. Therefore the Thevenin voltage
magnitude and phase estimated value remains fairly stable.
-4

-9

Voltage Angle

x 10

-9.2

-9.4

-9.6

-9.8

-10

0

2

4

6

8

10

Figure 11: Voltage Phase estimation
Resitance
2.2
2
1.8
Ohms

No. of Phasor
Measurements
Actual
1 Phasor
Error %
2 Phasor
Error %
3 Phasor
Error %
4Phasor
Error %
5 Phasor
Error %
6 Phasor
Error %
10 Phasor
Error %

RMS Voltage Magnitude
70.7115

Angle

Table 7: Parameter Estimation with linear regression

power system. Single source simulations with impedance change
from 1+0.377j to 2+0.755jat 5 second in a 9.6 second simulation
is conducted. The algorithm is capable to detect the impedance
change very accurately. As the source voltage and phase remains
unchanged, the estimation remains fairly stable at its initial
values.

1.6
1.4
1.2
1
0.8

0

2

4

6

8

10

Time(s)

Figure12: Power source Resistance estimation
Reactance
0.8

0.7

Ohms

The results of estimation indicate accurate prediction of both
generator and grid side parameters simultaneously. This algorithm
can be extrapolated to ‘n’ number of parallel sources feeding a
load. The algorithm uses the shared current from the source and
voltage at point of common coupling for estimating Thevenin
equivalent parameters. Figure 9 shows the number of estimated
parameters and number of known parameters used for estimating
the unknown values using the proposed algorithm. For very large
number of sources connected in parallel, the algorithm proposed
estimates twice the number of unknown parameters. This aspect
of the proposed algorithm can help in controller design and tuning
for multiple sources i.e. inverters and/or UPS to be connected in
parallel. Virtual impedance design requires equivalent Thevenin
impedance seen at from load side for proper compensation of
parameter mismatches (Line and/or filter parameters).
Simultaneous estimation of multiple sources Thevenin’s
equivalent will help in proper selection of virtual impedance for
minimizing circulating current flowing among power sources.

0.6

0.5

0.4

0

1

2

3

4

5
Time(s)

6

7

Figure 13: Power System reactance

8

9

10

6
As the resistance and reactance values are doubled at 5sec, the
estimated values changes level around 5 sec. The error % for the
estimated quantities is well below 0.5% under steady state. The
positive sequence voltage and current angle and magnitude are
measured. It is assumed that the input voltage magnitude and
angle and the Thevenin impedance as seen from the load side is
constant while the algorithm is in operation. The positive
sequence extraction is done to eliminate the unbalanced
components in voltage and current. Proposed algorithm can be
used for detection of balanced power system faults using the
proposed parameter estimation algorithm using linearized and
nonlinear recursive least square algorithms evaluated through
simulation results shown from Figure 10 to 13.

VI. CONCLUSION
Online Thevenin equivalent parameter estimation using
nonlinear and linear recursive least square algorithm are proposed
and evaluated in this paper. Simultaneous parameter estimation of
‘n’ parallel source with known parameters at point of common
coupling i.e. current and voltage phasor are evaluated and
analysed. The proposed algorithms require at least two phasor
measurements for accurate prediction. The robustness of the
algorithms has also been tested and verified. Fault in power
system leads to impedance change. The Thevenin equivalent
estimation proposed in the paper can help to determine power
system behavior and adopt the corrective actions to improve
response of system network and stability. Simulation results
validating the performance of the proposed algorithms have also
been verified for power system impedance change detection.
From the analysis carried out it is found that nonlinear recursion is
comparatively better than linear recursion in terms of error
percentage of the estimation. However linear method is
computationally less exhaustive, more viable for online controller
implementation.

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