mathematical modeling and simulation analysis of a dc

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11242-11251

© Research India Publications. http://www.ripublication.com

11242

Mathematical Modeling and Simulation Analysis of A DC Motor Based

Wind Turbine Emulation System

Shefali Jagwani*, L. Venkatesha

Department of Electrical and Electronics Engineering, BMSCE, Bengaluru, Karnataka 560019, India. * Corresponding Author

Abstract

The cost of implementing an actual wind turbine is quite high,

therefore a wind turbine is emulated to obtain the

characteristics of a real wind turbine. The concept of emulation

involves the torque control of a DC motor according to the

reference torque generated from the turbine. In this paper, a

separately excited DC motor is controlled using a first quadrant

DC Chopper to obtain the wind turbine characteristics. First, a

MATLAB program is written to obtain the theoretical power

and torque characteristics of the turbine. Then, the turbine is

simulated using mathematical modeling of DC motor and

chopper to obtain the desired theoretical torques. The model is

tested for two different wind turbines and is validated using the

simulation results obtained. Also, both the turbines are

simulated with a three-phase rectifier and results are presented

which can be used for laboratory purpose. Hence, this paper is

intended to propose an accurate model which saves the cost of

actual implementation of turbine and provides a study platform

for researchers in this domain.

Keywords: Wind Energy Systems, DC Chopper, PI controller,

Wind Turbine Emulator

INTRODUCTION

In recent scenario, where there is a limited availability of

conventional sources, the renewable energy sources are

significant members of a power system. Wind power is one of

the commonly used renewable energy resource which has

various advantages like availability, cost, clean and green

energy, etc. [1]. A lot of research activities are being carried out

in this area, but it is difficult to implement an actual wind

turbine for research in laboratories. Therefore, a wind turbine

emulator is required which can emulate the actual behavior of

a turbine. Several authors have published their work in this

field. Researchers have used squirrel cage induction motors,

permanent magnet synchronous motors and separately excited

DC motors etc. to emulate the wind turbine. To emulate the

wind characteristics, the input wind speed profile is an

important aspect. Studies consider the input wind as a

stochastic or a deterministic profile. Authors have modelled the

wind by decomposing it in an average speed component and

high frequency fluctuations. Some studies have considered it as

a sum of several harmonics or variable scalar function versus

time.

Nomenclature

𝑃𝑚 mechanical output power (W)

𝜌 air density (kg / m3)

𝑣 input velocity of the wind (m/s)

𝐴 area swept by the rotor blades (m2)

𝐶𝑝 Coefficient of performance

𝜆 Tip Speed Ratio

𝑅 radius of the blade (m)

𝛽 pitch angle of the blade (degree)

𝜔𝑡 rotational speed of the turbine (radians per second)

ω rated turbine speed

G gear ratio

S scaling factor

𝑇𝑡 turbine torque

Authors have preferred inverter controlled induction machine

as a wind simulator over DC machine [2-3]. The paper [4] has

presented an induction motor driving a dc generator for

emulation. The effect due to wind shear and tower shadow are

studied in the paper [5]. In paper [6], flux and torque control of

induction motor is incorporated. Some of the papers have used

permanent magnet synchronous motor [7-8] in preference to

DC motor. Authors have used boost converters [8], rectifiers

[9-11] or other converters for the control of motor. Papers [12-

13] use forth quadrant chopper to obtain the wind simulator and

paper [14] implements the same using FPGA.

In paper [15], dc motor is used to drive self-excited induction

generator. Authors in Refs. [16], discuss the harmonics in the

torque generated due to torsional oscillation. A. Mahdy and

others [17] describes the implementation using analog

electronic circuits to generate a specific reference wind speed.

In [18], a three-mass system including a turbine, a gear box and

a generator connected with two elastic shafts is constructed. An

armature and field controlled motor is used in paper [19]. The

experimental verification for DC motor coupled to a double fed

induction generator for a 300W prototype is performed in [20].

A small wind conversion system is studied using a dc motor-

generator set which can be connected to a micro-grid [21].

Authors in [22] present the simulation, considering, the pitch

control and additional wind effects and shaft dynamics.

mailto:[email protected]



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In this paper, a separately exited DC motor is opted to emulate

a wind turbine. The obvious reasons for using a DC motor are

easy to understand concepts and less complicated speed and

torque control. As discussed above, the previous works done

use converters like a rectifier or a four-quadrant chopper. Some

of the authors have used mathematical modeling of turbine, but

the converter is generally not modelled using equations. In this

study, a mathematical model of a first quadrant DC chopper is

developed and applied to mathematical model of wind turbine.

PI based control is implemented to control the IGBT switches

of chopper. PI controller is also simulated using mathematical

equations. As the complete model uses mathematical

modelling for all the subcomponents, i.e. turbine, chopper, PI

controller and dc motor, the accuracy of the model can be

considered high compared to the ones available in literature.

Also, the model is simulated with a rectifier instead of a

chopper and results are presented for both the ac-dc and dc-dc

converters.

Fig.1 shows the block diagram for basic concept of wind

emulation system. To implement this work, the wind turbine is

first modelled using mathematical equations. Then, the

theoretical power and torque characteristics are obtained using

these equations in a MATLAB program. This theoretical torque

acts as the reference torque for closed loop control of DC motor

using mathematical model of first quadrant DC chopper. Thus,

an easy to implement wind emulator is mathematically

simulated using MATLAB and results are obtained for

deterministic wind speed profiles. Two different wind turbines

are simulated so that the results obtained can be considered

closer to the actual turbine. The torque obtained from dc motor

can be fed to the generator and the electricity can be supplied

to the load.

Figure 1. Basic concept of wind turbine emulation

WIND TURBINE MODEL

The mechanical output power obtained from the wind

generators is given by [10]:

𝑃𝑚 =1

2𝜌𝐴𝑣3𝐶𝑝(𝜆, 𝛽) (1)

where, 𝜌 is the air density (kg / m3), 𝑣 is the wind speed (m/s),

A is the area swept by the rotor blades (m2) and 𝐶𝑝(𝜆, 𝛽)is the

performance coefficient.

The parameter 𝐶𝑝 defines efficiency of power extraction. It is a

nonlinear function of Tip Speed Ratio (𝜆) and pitch angle of

the blade (𝛽). 𝜆 is the ratio of rotational speed of turbine (𝜔𝑡) to

linear speed of blade tips and is given by:

𝜆 = 𝜔𝑡 ∗ 𝑅/𝑣 (2)

where, R is the radius of the blade. In this paper, a fixed pitch

turbine with varying speed is taken into consideration, hence

𝛽 is kept fixed at 0°. Various versions of equation for obtaining

𝐶𝑝 have been defined by authors in previously published

papers. In this paper, the following two equations (3), (4) are

used for 𝐶𝑝 [23], [10]:

Turbine 1:

𝐶𝑝 = 0.00234 + 0.03227𝜆 − 0.076354𝜆2 + 0.061857𝜆3 −

0.015155𝜆4 + 0.001514𝜆5 − 0.000055𝜆6 (3)

Turbine 2:

𝐶𝑝 = 0.5176(116(1/ 𝜆𝑖 − 0.4 𝛽 − 5)𝑒(−21 ∗ 1/ 𝜆𝑖) +

0.0068 𝜆 (4)

Where, 1/ 𝜆𝑖 = (1/( 𝜆 + 0.08𝛽)) − (0.035/( 𝛽3 + 1)) (5)

Fig. 2a and Fig. 2b show the simulated plot of power coefficient

𝐶𝑝 with Tip Speed Ratio for turbine 1 and turbine2 respectively.

It can be observed that the maximum 𝐶𝑝 (0.3921) for turbine1

occurs at 𝜆 = 5.321 and for the turbine2, the maximum 𝐶𝑝 (0.48)

occurs at 𝜆 = 8.148.



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Figure 2a. Power Coefficient curve for turbine1

Figure 2b. Power Coefficient curve for turbine2.

Fig.3a and Fig.3b show the power characteristics for both the

turbines for different wind speeds from 1m/s to 14 m/s and

rated speed is considered as 12m/s. These are the theoretical

values of power and torque which are obtained from a

MATLAB program using equations mentioned in this paper.

Figure 3a. Power Characteristics for turbine1.



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Figure 3b. Power Characteristics for turbine2.

The torque of the turbine is stated as:

𝑇𝑡=𝑃𝑚/ 𝜔𝑡 (6)

Fig. 4a and Fig. 4b show the torque characteristics for both the turbines.

Figure 4a. Torque Characteristics for turbine1.



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Figure 4b. Torque Characteristics for turbine2

The other parameters of both the turbines which are used in this

paper are given in Table 1.

Here, G is the gear ratio and S is the scaling factor and can be

defined as:

𝐺 = 𝜔/𝜔𝑡 (where ω is the rated turbine speed) (7)

𝑆 = 𝑟𝑎𝑡𝑒𝑑 𝑚𝑎𝑐ℎ𝑖𝑛𝑒 𝑝𝑜𝑤𝑒𝑟 / 𝑟𝑎𝑡𝑒𝑑 𝑡𝑢𝑟𝑏𝑖𝑛𝑒 𝑝𝑜𝑤𝑒𝑟 (8)

Table 1. Wind turbine parameters

Parameter Turbine 1 Turbine 2

No. of blades 3 3

𝛽 0 0

𝜌 1.08 1

R 2.5 1.5

G 7.4 0.5

S 1 1/3

Emulation of Wind Turbine Characteristics using DC

motor

A separately excited DC motor, whose parameters are given in

appendix, is controlled to follow the reference torque generated

by wind turbine at a given wind speed and rotor speed. The

emulation of the actual turbine means the machine torque must

be followed according to the generated torque and speed. To

obtain the closed loop torque of DC motor, the PI control is

used. The first quadrant DC chopper in continuous current

mode is used to control the motor. Also, the model is simulated

with a three-phase rectifier instead of DC chopper and results

are shown for both the turbines. The control strategy is shown

in Fig.5. In Fig.6, the simulation model with DC Chopper is

presented. Fig. 7 shows a subsystem which includes the

mathematical modelling of a first quadrant DC chopper. Fig.8

shows the input wind speed profile applied to the system.

Figure 5. Control Strategy for Emulation of wind turbine using chopper



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Figure 6. Simulink model of wind turbine emulation using DC chopper

Figure 7. Mathematical Model of Chopper Subsystem

Figure 8. Input wind speed

SIMULATION RESULTS

The proposed model is simulated using MATLAB/SIMULINK

to obtain the desired torque and rotor speed. Fig 9,10, 11 show

the results obtained for turbine1. The torque is obtained by

simulating the turbines with rectifier and chopper. Fig. 9 shows

the generated torque of DC machine with chopper. Fig. 10

shows the torque obtained for same turbine with rectifier

controlled motor. It can be observed that the torque

characteristics obtained with simulation vary according to the

input wind speed and closely follows the theoretical torque

characteristics of the turbine 1 (Fig. 4). Fig. 11 shows the rotor

speed for turbine1 which also follows the speed obtained in

Fig.4.



11248

Figure 9. Generated Torque of DC machine with chopper for turbine1

Figure 10. Generated Torque of DC machine with rectifier for turbine1

Figure 11. Rotor speed of DC machine for turbine1

Fig.12, Fig.13 and Fig. 14 present the simulated results for

turbine 2. The generated torque of DC motor is shown with

chopper and rectifier respectively. Also, the rotor speed is

shown. It can be observed that the torque of the dc motor at 12

m/s is 8 Nm and it follows the theoretical turbine torque (shown

in Fig. 5). In Fig. 14, rotor speed at 12m/s is 50 radians per

second and it matches with the rotor speed shown in Fig. 5. This

can be verified for all the speeds. As the torque and speed are

following the reference turbine values, the results obtained are

closer to actual system. The simulated values are little lesser

than the theoretical values due to the frictional losses in DC

motor. The results are also verified for other input speed

profiles.



11249

Figure 12. Generated Torque of DC machine with chopper for turbine2

Figure 13. Generated Torque of DC machine with rectifier for turbine2

Figure 14. Rotor speed of DC machine for turbine2

CONCLUSIONS

The proposed model illustrated in this paper accurately

emulates the turbine characteristics for all the wind speeds. The

theoretical reference torque obtained from turbine is accurately

followed by the generated torque from the DC machine. The

mathematical model is tested for two turbines and results are

presented. Also, the model is tested for other input wind

profiles. The accuracy of the model is high as the complete

mathematical model is used for all the components including

turbine, first quadrant chopper, PI controller and motor. Also,

the results are shown for rectifier controlled DC motor. The

results presented in this paper can be used to study wind energy

systems and can save a huge cost of implementing actual wind

turbine.

ACKNOWLEDGMENT

The authors are grateful to Technical Education Quality

Improvement Program(TEQIP-III) for providing necessary

facilities for this research work.



11250

Appendix

DC Motor Parameters

Rated power 1 KW

Rated speed 1450 rpm

Rated armature voltage 120 V

Armature resistance 0.2Ω

Armature inductance 73mH

Rated field voltage 240 V

Rated field current 0.88A

Mutual inductance 1.068H

Inertia 0.014kgm2

Friction coefficient 0.01Nms

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http://dx.doi.org/10.1007/978-3-642-03454-1_13

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mathematical modeling and simulation analysis of a dc

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