Small-signal stability analysis of DFIG based wind power system using teaching learning based optimization

Electrical Power and Energy Systems 78 (2016) 672–689

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Small-signal stability analysis of DFIG based wind power system using teaching learning based optimization Shamik Chatterjee, Abishek Naithani, V. Mukherjee ⇑ Department of Electrical Engineering, Indian School of Mines, Dhanbad, Jharkhand, India

a r t i c l e

i n f o

Article history: Received 18 May 2015 Received in revised form 17 November 2015 Accepted 24 November 2015

Keywords: Doubly fed induction generator Eigenvalues Low voltage ride through Teaching learning based optimization (TLBO) Wind turbine generator

a b s t r a c t The present paper formulates the state space modelling of doubly fed induction generator (DFIG) based wind turbine system for the purpose of small-signal stability analysis. The objective of this study is to discuss the various modes of operation of the DFIG system under different operating conditions such as three phase fault and voltage sags with reference to variable wind speed and grid connection. In the present work, teaching learning based optimization (TLBO) algorithm optimized proportional–integral (PI) controllers are utilized to control the dynamic performance of the modelled DFIG system. For the comparative analysis, TLBO based simulated results are compared to those yielded by particle swarm optimization (PSO) method for the same DFIG model. The simulation results show that the proposed TLBO based PI controller effectively works in minimizing the damping phenomena, oscillation in rotor currents and fluctuation in electromagnetic torque for the studied DFIG model. It is also observed that TLBO is offering better results than the PSO for the dynamic performance analysis of the studied model. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction In recent years, wind energy has witnessed a large surge in research and development. The drawback of wind energy is that electrical energy is obtained only when the wind blows. Even though modern wind turbines regulate power well and level off at their rated capacity, the amount of power produced by them varies throughout the day. Many installations have established that utility systems are able to accommodate the change in wind generation just as they modify their output to follow dynamic demand. Specialists predict that wind power can constitute up to 30% of present energy demands before reliability of the system would be an issue. Generation of kinetic energy is done by utilizing the atmospheric air’s energy. Wind energy had been used from centuries to perform many different functions such as grain grinding, sailing and for irrigation purposes. The main function of wind power system is conversion of kinetic energy present in the wind into various sources of power. In ancient times, milling and irrigation were also done by wind power systems. During twentieth century, wind power started to generate electricity. Similarly, wind mills were used in several countries to pump water from the ground. ⇑ Corresponding author. Tel.: +91 0326 2235644; fax: +91 0326 2296563. E-mail addresses: [email protected] (S. Chatterjee), abhinaithani@gmail. com (A. Naithani), [email protected] (V. Mukherjee). http://dx.doi.org/10.1016/j.ijepes.2015.11.113 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

Wind turbines can be used as single unit as well as in groups (also known as wind farms). Wind turbines which are smaller in size are also called as aero generators. These can be used for charging largesized batteries. Five countries in the world has greater than 80% of the installed global wind energy capacity, among them India is at the 5th position [1]. The output power can be improved by 2–6% for a variable speed turbine as compared to a fixed speed turbine [2] whereas it may go up to 39% according to [3]. It is revealed that the energy generation gain of the variable-speed turbine as compared to the fixed-speed turbine may fluctuate by 3–28% according to the condition of site and design consideration [4]. The energy capture is enhanced by 20% in case of doubly fed induction generator (DFIG) when compared to variable speed turbine using a cage bar induction machine and by nearly 60% from fixed speed system. As the assumptions used while performing the study of DFIG varies vastly from one person to another, therefore, the results may also vary accordingly. The controlling of DFIG is far more tedious than controlling any other machine. The rotor current in the DFIG is controlled by power converters. It is controlled by using vector control techniques. Till date, various vector control techniques has been suggested for the controlling of DFIG. The stator flux orientation can be used to control the rotor currents according to the system parameters [5,6]. According to [7,8], the eigenvalues of the DFIG are poorly damped having a corresponding natural frequency close to the line frequency. In addition to this, the DFIG system is not

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

673

Nomenclature A A1 Asys B C C dc c1
c6 C P ðk; bÞ D Dtran Fb fr fs H idr;ref iqr;ref Ids

state or system matrix turbine swept area, m2 complete system matrix input matrix output matrix DC link capacitance, mF power co-efficients performance co-efficient feed forward matrix transmission line distance, km base frequency, Hz rotor frequency, Hz supply frequency, Hz total inertia of generator, kg m2 reference rotor current in direct axis reference, A reference rotor current in quadrature axis reference, A three phase stator current in direct axis reference frame, A Iqs three phase stator current in quadrature axis reference frame, A K opt constant co-efficient K i1 ; K i2 integral controller gains K P1
K P3 proportional controller gains Lm magnetizing inductance, H Llr leakage inductance of rotor winding, H leakage inductance of stator winding, H Lls Lr self inductance of rotor, H Ls self inductance of stator, H n synchronous speed, rpm p pair of poles Pm mechanical output power, W Pr rotor power, W stator power, W Ps Rblade blade radius, m Re transmission line resistance, Ohm Rr rotor resistance of machine per phase, Ohm stator resistance of machine per phase, Ohm Rs RT total resistance, Ohm s slip of the machine, p.u. Sb base MV A, MV A T transpose of matrix

stable for various operating conditions. The poorly damped poles of the DFIG affect the dynamics of rotor current from the back electromotive force. The response of wind turbines to grid disturbances is a crucial issue, particularly, since the rated power of windturbine installations will increase slowly. Therefore, it is vital for utilities to be capable to study the results of several voltage sags and also the resultant turbine response. It is desirable to have a simple model that is able to model the dynamics of concern. In [9–11], a third-order model has been planned that neglects the stator-flux dynamics of the DFIG. This model provides an accurate mean value [9]. However, a disadvantage is that some of the crucial dynamics of the DFIG system are also neglected. So, as to preserve the dynamic behaviour of the DFIG system, a rather totally different modelling approach should be followed. As discussed previously in the literature, a dominating feature of the DFIG system is that the natural frequency of the flux dynamics is near the line frequency. Since the dynamics of the DFIG are influenced by two poorly damped eigenvalues (poles), it might be natural to condense the model of the DFIG to the flux dynamics described by a secondorder model. This is one typical way to scale back model of DFIG in typical control system stability analysis [8]. The chance of its usage

T em Tm T sp u V dc

v dgrid v dr v_ dr v ds v qgrid v qr v_ qr v qs v s;ref vw

wb wr ws x1
x3 Xe X ls X lr Xm X rr X ss XT X TR y b k

q

wdr wds wqr wqs x_ 1
x_ 3

electromagnetic torque, N m mechanical torque developed, N m set point torque, N m input vector of the model DC link voltage, V grid voltage in direct axis reference frame, V three phase rotor voltage in direct axis reference frame, V direct axis rotor voltage signal from PI controller, V three phase supply voltage in direct axis reference frame, V grid voltage in quadrature axis reference frame, V three phase rotor voltage in quadrature axis reference frame, V quadrature axis rotor voltage signal from PI controller, V three phase supply voltage in quadrature axis reference frame, V stator reference voltage, V wind speed, m/s base angular frequency, radians per minute rotational speed of generator, revolutions per minute synchronous angular frequency, radians per minute state vectors transmission line reactance, Ohm stator leakage reactance, Ohm rotor leakage reactance, Ohm magnetization reactance, Ohm rotor reactance, Ohm stator reactance, Ohm total reactance, Ohm transformer reactance, Ohm output vector of the model blade pitch angle, deg tip speed ratio, p.u. air density, kg/m3 rotor flux in direct axis reference frame, Wb stator flux in direct axis reference frame, Wb rotor flux in quadrature axis frame, Wb stator flux in quadrature axis reference frame, Wb first derivative of x1
x3

as a simulation model is yet to be shown, so as to preserve the behaviour of the oscillatory response. It is clear that secondorder simulation model is the easiest one to use. At present, the DFIG wind turbine is disconnected from the grid when huge voltage sag appears in the system. When wind turbine is disconnected from the system, it needs few seconds before it can be reconnected with the system. This means that the wind turbine needs to have extra protection to avoid these voltage dips. Today’s DFIG system encompasses a crowbar within the rotor circuit that at large grid disturbances must short circuit the rotor so as to shield the converter. This highlights that the turbine should be separated out from the grid, if large voltage sag occurs. According to the works reported earlier, there are different ways to change the DFIG system so that it can withstand the voltage sags. In [12], thyristors are placed in anti-parallel topology within the stator so as to achieve fast (less than 10 ms) discontinuation of the stator and, thus, enabling it to re-magnetize the turbine generator as well as rejoin the stator from the grid as quickly as possible. The technique, proposed in [13], uses an ‘‘active” crowbar which disintegrates the short circuit current to a minimum value. All of these systems have completely different dynamic

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performance. Also, the selection of the system depends on the efficiency and the price of different voltage sag ride through capability. Therefore, when DFIG system is modified for different voltage sag ride through, it is essential to keep in mind the constraints of cost as well as efficiency. Any analysis, relating to voltage sag ride-through capabilities with their immediate effect on DFIG system efficiency, is really tough to search out within the existing literature. The improved awareness of individuals towards renewable energy support from governmental establishments and speedy advancement within power industries, which form the core of wind power generation systems, are the foremost contributing factors for its expansion. As a result, the share of wind power generation with respect to total installed power capacity is increasing worldwide. Particularly, grid-connected wind generation system plays an important role in the development [14]. As the use of wind turbine is increasing, the wind turbines are required to remain connected during the fault and contribute to the stability of the system. The variable speed system, which utilizes wind energy, is the main type of grid-connected wind generation system [15]. Different PI controllers installed in the studied power system model of the present work requires optimal tuning for proper functioning of the model. Recently, computational intelligence based algorithms have been applied to different fields of power engineering applications. Rao et al. [16] have introduced a novel optimization technique and named it as teaching learning based optimization (TLBO). It is based on the teaching–learning process in a classroom. TLBO has been found to be very efficient in solving various engineering optimization problems with very fast convergence rate and less computational time. In view of the above, the objectives of this paper are: (a) formulation of the state space model of the DFIG, which is connected to the grid, for its small-signal and transient stability assessment, (b) optimization of the gain of the PI controllers, using TLBO and particle swarm optimization (PSO), for steady state stability performance improvement, (c) assessment of the impact of the optimized controller gains on the nature of modes of oscillations with varying wind conditions and with different strength of transmission network and (d) investigation of the fault ride through capabilities of the DFIG with new optimized controller gains both under three phase fault as well as under voltage dip conditions. The rest of the paper is organized as follows. Section ‘Mathema tical modelling of DFIG’ describes the modelling concepts of wind turbine generating system associated with DFIG along with its control strategies adopted. Section ‘Interfacing of DFIG with grid’ describes the interfacing of DFIG with the grid. Section ‘Design of objective function’ defines the objective function of the present work. An overview of the proposed TLBO algorithm is presented in ‘TLBO: an overview’ Section. Section ‘Results and discussions’ presents and discusses the results obtained. Finally, the conclusions are drawn in Section ‘Conclusion’.

Mathematical modelling of DFIG

maximum energy from the wind even at low wind speed by optimizing the speed of the turbine while minimizing the mechanical stress on the turbine during stormy wind. The maximum power that can be produced by the DFIG is directly proportional to the speed of the wind. Power electronics converter can generate as well as absorb reactive power and, hence, additional reactive power compensator is not required which is the advantage of the DFIG. The rotor-side controller (RSC) controls the electromagnetic torque by controlling the injected rotor voltage. The electromagnetic torque should follow the reference speed provided by the DFIG system. RSC also provides the control of reactive power as well as the voltage and power factor of the system. It helps in obtaining maximum power in case of variable speed operation of DFIG. The stator is connected to the grid through grid side converter which is used to control the dc-link voltage and to exchange the reactive power with the electrical grid. Induction generator model The assumptions made during modelling of DFIG are mentioned below. (a) Synchronous reference frame is used to derive the equations. (b) The d-axis is lagging q-axis by 90°. The three phase windings of the induction machine are placed in the stator. Rotating magnetic field is provided by the three phase windings placed in the stator which rotates at synchronous speed. The equations for stator and rotor voltage in d–q synchronous reference frame are presented in (1)–(4), in order, while flux equations are described in (5)–(11) [17,18].

V ds ¼ Rs Ids
ws wqs þ

dwds dt

ð1Þ

V qs ¼ Rs Iqs þ ws wds þ

dwqs dt

ð2Þ

V dr ¼ Rr Idr
sws wqr þ

dwdr dt

ð3Þ

V qr ¼ Rr Iqr þ sws wdr þ

dwqr dt

ð4Þ

wds ¼ Ls Ids þ Lm Idr

ð5Þ

wqs ¼ Ls Iqs þ Lm Iqr

ð6Þ

wdr ¼ Lr Idr þ Lm Ids

ð7Þ

wqr ¼ Lr Iqr þ Lm Iqs

ð8Þ

where

Ls ¼ L1s þ Lm

ð9Þ

Lr ¼ L1r þ Lm

ð10Þ

sws ¼ ws
wr

ð11Þ

The notations used in (1)–(11) are the same as in [17,18]. The schematic diagram of a DFIG connected with the grid through transmission line and transformer is shown in Fig. 1. DFIG consists of a wound rotor induction generator and an AC-DC-AC power electronics converter. The stator winding is directly connected to the electrical grid while the variable frequency is fed to the rotor through the power electronic converters. DFIG extracts

Wind turbine model for DFIG For complete DFIG modelling, it is required to combine the equations that describe electrical voltage and current components of the machine with swing equation that provides rotor speed as

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DFIG Transformer

PWM converter Wind turbine

Grid

PWM converter

C

PWM

PWM

Rotor side controller

Grid side controller

Fig. 1. Schematic diagram of DFIG integrated with grid.

state variable. In power system studies, drive trains are modelled as a series of rigid disks connected via mass less shafts. Maximum power point tracking The aim of the DFIG wind turbine is to extract maximum power from the wind. The mechanical power extracted from the wind and tip speed ratio ðkÞ is given by (12) and (13) [19–21]

1 Pm ¼ C p ðk; bÞ  qA1 m3w 2 k¼

ð12Þ

Rblade  wr

ð13Þ

vw

where Pm denotes the mechanical output power (W) and it depends upon the performance coefficient, air density (q), turbine swept area (A) and wind speed ðv w Þ. In (12), the term 12 qAm3w denotes the kinetic energy contained in the wind at a particular speed and the term C p ðk; bÞ denotes the performance coefficient which depends on tip speed ratio (k) and blade pitch angle (b). It decides how much of the kinetic energy of the wind can be captured by the wind turbine system. In (13), Rblade denotes the blade radius. A nonlinear model describes C p ðk; bÞ as given in (14)

C p ðk; bÞ ¼ c1 ðc2
c3 b
c4 b
c5 Þe


c6

ð14Þ

where

1 1 0:035
¼ ki k þ 0:08b b3 þ 1

ð15Þ

Operating principle of DFIG According to the principle of AC induction generator, the supply frequency is related to rotor frequency and may be stated by (16) [19].

np  fr 60

Pr ¼ T m  wr

ð17Þ

Ps ¼ T em  ws

ð18Þ

Mechanical equation for a loss less generator is given by (19).

J

dwr ¼ T m þ T em dt

ð16Þ

ð19Þ

In steady-state, mechanical and electromagnetic torques are equal for a loss less generator and is stated by (20)

T m ¼ T em

A typical wind turbine characteristic and turbine power characteristic curve is shown in Fig. 2(a) which corresponds to the maximum energy to be captured from the wind. When generator speed is less than the lower limit or higher than the rated value, the reference speed is set to the minimal value or rated value, respectively.

fs ¼

The AC-excited wind power generation scheme is shown in Fig. 1. The system consists of a turbine, a DFIG, AC–DC–AC converters, transformer and the grid. If the speed of rotor is less than the speed of stator then the system operates at sub-synchronous state. In that case, AC excitation to the rotor of DFIG is given by inverter and grid power is supplied by the stator. If the speed of the rotor is greater than the speed of the stator then the system runs at a super-synchronous state and flow of power through the converter is reversed. Now, grid power is supplied by both the stator as well as the rotor. If the speed of the rotor equals the synchronous speed of the stator rotating magnetic field then the system runs at synchronous state. During this condition, DFIG runs as a synchronous motor, rotor frequency becomes equal to zero and the converter gives DC excitation to the rotor. With the change in the speed of the DFIG, the stator frequency can always be maintained equal to the grid frequency by changing the rotor frequency. The mechanical power ðP r Þ as well as the stator electrical power output ðPs Þ can be calculated by using (17) and (18) [20], in order.

ð20Þ

where

Pm ¼ Ps þ Pr

ð21Þ

Eq. (22) may be obtained by using (19) and (20) as follows:

Pr ¼ P m
Ps ¼ T m  wr
T em  ws ¼
sP s

ð22Þ

where s ¼ ðxs
xr Þ=xs denotes the slip of the generator. Normally, the value of slip is much lower than 1.0 and, consequently, Pr is only a fraction of P s . Since T m is positive for power generation and since xs is also positive and constant for a constant frequency grid voltage, the sign of Pr depends on the sign of slip. P r is positive for negative slip and vice versa. For super-synchronous speed of operation, Pr is transmitted to DC bus capacitor and tends to raise the DC voltage.

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Fig. 2. DFIG control strategies: (a) power speed characteristics, (b) turbine power characteristics, (c) torque control scheme and (d) voltage control scheme.

Control strategies for DFIG The wind turbine system consists of two power electronics converters which are connected to the grid side and rotor side, respectively. Designing of both of these two converters are required for effective control of active and reactive power [21,22].

characteristic so that power output can be the maximum. The rotor current is further splitted into two orthogonal elements, d-axis and q-axis. The torque of the DFIG is controlled by q-axis current and the power factor or terminal voltage is controlled by d-axis current.

Torque control scheme RSC control scheme The main function of RSC is to control the output power of wind turbine as well as the voltage of the grid. The power of the wind turbine is controlled in such a way that it follows the tracking

Reference torque ðT sp Þ is obtained by maximum power characteristics as given in Fig. 2(b). Reference current ðiqr;ref Þ can be calculated with the help of T sp . The reference current is then compared with iqr and the obtained value is fed into a PI controller which

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changes it to v 0qr . The rotor voltage ðv qr Þ required to generate the pulse for the RSC is obtained by addition of the obtained value of PI controller and the compensation term. This compensation term is essential for the minimization of cross-coupling between the speed and the voltage control loops. The complete model of torque control scheme is given in Fig. 2(c). All the variables shown in this model are in p.u. The torque control scheme is used to change the electromagnetic torque of the turbine generator in accordance with the variation in wind speed and bring the system to the optimal operating reference point. Neglecting stator resistance and stator transients, the d-axis and q-axis stator voltages can be obtained as in (23) and (24), in order, by using (1), (2) and (5), (6).

v ds ¼
ws ð
X ss iqs þ X m iqr Þ

ð23Þ

v qs ¼ ws ð
X ss ids þ X m idr Þ

ð24Þ

Using (23) and (24), q-axis and d-axis stator currents are obtained as in (25) and (26), respectively.

iqs ¼

1 X v ds þ m iqr ws X ss X ss

ids ¼


1 X v qs þ m idr ws X ss X ss

ð25Þ

ð26Þ

Eq. (25) can be described by (27) after taking d-axis component of stator voltage v ds ¼ 0, because of the stator flux oriented reference frame.

iqs ¼

Xm iqr X ss

ð27Þ

(a) There should be compensation of reactive power consumed by the DFIG. (b) If the terminal voltage becomes too low or too high in comparison with the reference value, then idr;ref ought to be adjusted suitably. Reference current ðidr;ref Þ can be calculated with the help of v s;ref as given in (33).

idr;ref ¼ x3 þ

vs

ð33Þ

X m ws

The reference current ðidr;ref Þ is then compared with idr and the obtained value is fed into the PI controller which changes it to v 0dr as given in (34).

v 0dr ¼ x2 þ K p2 ðidr;ref
idr Þ

ð34Þ

The rotor voltage ðv dr Þ, required to generate the pulse for the RSC, is obtained by subtraction of the obtained value of PI controller and the compensation term. From Fig. 2(d), Eq. (35) follows.

x3 ¼ K p3 ðv s;ref
v s Þ

ð35Þ

This compensation term is essential for minimization of crosscoupling between speed and voltage control loops. Interfacing of DFIG with grid The system used for the analysis of DFIG is presented in Fig. 1 where DFIG is integrated with the grid through the transmission line. The voltage and frequency of the grid are considered as constant. For steady state stability analysis of the system shown in Fig. 1, the linearized equation of the model may be given by (36) and (37)

The value of electromagnetic torque may be derived by employing (28).

x_ ¼ Ax þ Bu

ð36Þ

T e ¼ X m ðidr iqs
iqr ids Þ

y ¼ Cx þ Du

ð37Þ

ð28Þ

Using (26) and (27) in (28), the electromagnetic magnetic torque of DFIG may be derived as in (29).



 Xm 1 X T e ¼ X m idr iqr
iqr
v qs þ m idr ws X ss X ss X ss

ð29Þ

Using reference torque set point ðT sp Þ, the reference current ðiqr;ref Þ is given by (30).

iqr;ref ¼

ws X ss T sp X m v qs

ð30Þ

The final equation for q-axis rotor voltage ðv qr Þ may be derived after removing the transient term from (4) as presented in (31).

v qr ¼ Rr iqr þ sws

! X 2m Xm idr
X rr
v qs X ss ws X ss

ð31Þ

From Fig. 2(c), Eq. (32) follows:

v 0qr ¼ x1 þ K p1 ðiqr;ref
iqr Þ

where T x_ ¼ ½ids iqs idr iqr wr 

ð38Þ

Input and output vectors of the model are given by (39) and (40), respectively.

u ¼ ½v ds

v qs v dr v qr T

y ¼ ½idr iqr 

ð39Þ

T

ð40Þ

For studying the integration of DFIG with the transmission network, d-axis and q-axis stator voltage equations are given in (41) and (42), respectively

v ds ¼ v dgrid
X T iqs þ RT ids

ð41Þ

v qs ¼ v qgrid þ X T ids þ RT iqs

ð42Þ

where

ð32Þ

Voltage control scheme The complete block diagram of the DFIG terminal voltage controller is shown in Fig. 2(d). All the variables shown in Fig. 2(d) are in p.u. The RSC is used to control the voltage or the power factor at the terminal of the DFIG. As the reactive power delivered to the grid is increased or decreased, the terminal voltage is also increased or decreased. The following mentioned two rules are required to be addressed.

X T ¼ X TR þ X e

ð43Þ

RT ¼ Re þ Rs

ð44Þ

In (41) and (42), v dgrid and v qgrid are grid voltages, respectively. State variables for control loops, used in Fig. 2(c) and (d), are given by (45)–(47).

x_ 1 ¼
K i1 iqr þ ðK i1 X ss =ws X m Þ

T sp

vs

x_ 2 ¼
K i2 x_ 3
K i2 idr þ ðK i2 =ws X m Þv s

ð45Þ ð46Þ

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S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689 Table 1 System parameters. Parameters

Type of grid

VAsc X=R Ze Re Xe Rt Xt

Scenario 1: Strong grid

Scenario 2: Weak grid

20 MV A 10 0.05 0.0050 0.0498 0.0099 0.0998

8 MV A 10 0.125 0.0124 0.1244 0.01733 0.1744

x_ 3 ¼
K p3 v s þ K p3 v sref

ð47Þ

The state variables and control inputs, as used in DFIG after its integration with the transmission network, are given in (48) and (49), respectively. Fig. 3. D-shaped sector in the negative half of s-plane [23].

T x_ ¼ ½ids iqs idr iqr wr x2 x1 x3 

Initialize the population, design variables and termination criterion Evaluate the initial population Calculate the mean of each design variable Select the best solution Calculate the difference mean and modify the solutions based on best solution

Keep the previous solution

No

Is new solution better than existing?

Yes

Accept

Select the solutions randomly and modify them by comparing with each other

Keep the previous solution

No

Is new solution better than existing?

Yes

Is termination criterion fulfilled?

Yes Final value of solution Fig. 4. Flowchart of TLBO algorithm.

No

Accept

ð48Þ

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S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689 Table 2 Initial condition for test system. Parameters

Scenario 1: 20 MV A (Strong grid)

wr0 ids0 iqs0 idr0 iqr0

v ds0 v qs0 v dr0 v qr0 v ds: inf v qs;inf Dv ds Dv qs Te

v s0

Scenario 2: 8 MV A (Weak grid)

Case a

Case b

Case c

Case a

Case b

Case c

0.8
0.024 0.35 0.229 0.367
0.035 0.999
0.02 0.206 0 1
0.0351 1.001 0.3516 0.9996

1.1
0.055 0.661 0.197 0.693
0.066 0.998 0.02
0.098 0 1
0.0664 1.0010 0.6653 1.0001

1.29
0.097 0.977 0.154 1.024
0.098 0.995 0.085
0.286 0 1
0.0984 0.9999 0.9873 0.9998

0.8
0.035 0.343 0.217 0.367
0.06 0.998
0.025 0.206 0 1
0.0604 0.9998 0.3449 0.9998

1.1
0.084 0.649 0.166 0.693
0.114 0.993 0.025
0.097 0 1
0.1146 0.9965 0.6559 0.9995

1.29
0.151 0.959 0.096 1.024
0.1169 0.998 0.105
0.28 0 1
0.1698 0.9902 0.9751 0.9929

The eigenvalues of Asys are used to formulate the objective function. Thus, the objective of the present work is to formulate an objective function which will, in turn, help to tune the gains of the PI controllers installed in the studied power system model. The objective function of the present work is designed in the next section.

Table 3 Original and optimized controller parameters. Parameters

Original values [21]

PSO based values [Studied]

TLBO based values [Proposed]

K p1 K i1 K p2 K i2 K p3

0.05 10 0.05 10 7

0.1579 3.6933 0.1000 14.0907 20

0.3844 12.8539 0.1277 13.6576 19.4144

u ¼ ½idr iqr

v s v s;ref

T

T sp 

Design of objective function

ð49Þ

The complete system matrix ðAsys Þ of DFIG, when connected with the grid, is given in (50) for steady state stability analysis. The non-zero elements of system matrix ðAsys Þ are given in Appendix B.

2

Asys

A11

6 A21 6 6 6 A31 6 6A 6 41 ¼6 6 A51 6 6A 6 61 6 4 A71 A81

A18

3

A12

A13

A14

A15

A16

A17

A22

A23

A24

A25

A26

A27

A32

A33

A34

A35

A36

A37

A42

A43

A44

A45

A46

A47

A52

A53

A54

A55

A56

A57

A62

A63

A64

A65

A66

A67

A72

A73

A74

A75

A76

A77

A28 7 7 7 A38 7 7 A48 7 7 7 A58 7 7 A68 7 7 7 A78 5

A82

A83

A84

A85

A86

A87

A88

The optimization of PI controller gains is done to achieve high level of stability as well as damping of oscillation. To optimize the studied system model, an objective function defined in (51), is formulated to minimize both the undershoot as well as the overshoot and to attain quicker settling time of the oscillations of the transient response [23].

OF ¼ 10  OF 1 þ 10  OF 2 þ 0:01  OF 3 þ OF 4

ð50Þ

ð51Þ

In (51), the components involved in the right hand side are described below. P OF 1 ¼ i ðr0
ri Þ2 , if ri > r0 , where r0 ¼
2:0 and ri is the real part of the ith eigenvalue. The value of r0 determines the relative stability. P OF 2 ¼ i ðn0
ni Þ2 , if (imaginary part of the ith eigenvalue) > 0.0, ni < n0 and ni is the damping ratio of the ith eigenvalue.

Table 4 Results of calculated original small-signal stability analysis of DFIG at different wind speed. Cases

Mode number

Strong connection (VAsc = 20 MV A)

Weak connection (VAsc = 8 MV A)

Eigenvalue

Frequency of oscillations (Hz)

Damping ratio

Eigenvalue

Frequency of oscillations (Hz)

Damping ratio

Case (a): wr = 0.8 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


80.03 ± 509.2i
2.47 ± 379.24i
2.83 ± 21.97i
0.09
0.66

81 60.3579 3.4966 0 0

0.1553 0.0066 0.1280 1 1


87.60 ± 590.52i
3.28 ± 379.17i
1.79 ± 19.10i
1.14
0.09

93.9842 60.3468 3.0399 0 0

0.1467 0.0087 0.0932 1 1

Case (b): wr = 1.1 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


80.17 ± 508.33i
2.46 ± 379.26i
2.51 ± 22i
0.13
0.64

80.9032 60.3611 3.5014 0 0

0.1558 0.0066 0.1133 1 1


87.66 ± 589.01i

3.28 ± 379.19i
1.21 ± 19.10i
1.10
0.14

93.7439 60.350 3.0399 0 0

0.1472 0.0087 0.0921 1 1

Case (c): wr = 1.29 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


80.23 ± 507.63i
2.45 ± 379.27i
2.25 ± 21.99i
0.17
0.63

80.7918 60.3627 3.5 0 0

0.1561 0.0066 0.1019 1 1


87.78 ± 587.69i
3.28 ± 379.21i
0.89 ± 19.09i
1.08
0.17

93.5338 60.3531 3.083 0 0

0.1477 0.0087 0.0464 1 1

680

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

Table 5 Results of PSO based [Studied] small-signal stability analysis of DFIG at different wind speed. Cases

Mode number

Strong connection (VAsc = 20 MV A)

Weak connection (VAsc = 8 MV A)

Eigenvalue

Frequency of oscillations (Hz)

Damping ratio

Eigenvalue

Frequency of oscillations (Hz)

Damping ratio

Case (a): wr = 0.8 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


198.2190 ± 510.0210i
3.0662 ± 381.4815i
11.9605 ± 29.3500i
0.1014
1.8728

81.1724 60.7147 4.6712 0 0

0.3623 0.0079 0.3774 1 1


180.7391 ± 502.1646i
3.2854 ± 381.2482i
6.0442 ± 20.0383i
0.1012
1.8530

79.9220 60.6775 3.1892 0 0

0.3387 0.0086 0.2888 1 1

Case (b): wr = 1.1 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


181.0458 ± 502.4425i
3.2220 ± 381.3058i
6.0642 ± 21.2978i
0.1551
1.8142

79.9662 60.6867 3.3897 0 0

0.3390 0.0085 0.2738 1 1


132.4015 ± 510.4452i
3.0221 ± 380.4687i
5.4561 ± 25.3888i
0.1012
1.8596

81.2399 60.5535 3.7224 0 0

0.2511 0.0079 0.2101 1 1

Case (c): wr = 1.29 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


162.7021 ± 504.8247i
3.1146 ± 381.0802i
6.0000 ± 23.8941i
0.2077
1.7846

80.3453 60.6508 3.8029 0 0

0.3068 0.0081 0.2435 1 1


257.4066 ± 574.6309i
6.0000 ± 383.5299i
9.8162 ± 26.8680i
3.0337
0.2069

91.4553 61.0407 4.2762 0 0

0.4088 0.0101 0.3431 1 1

Table 6 Results of TLBO based [Proposed] small-signal stability analysis of DFIG at different wind speed. Cases

Mode number

Strong connection (VAsc = 20 MV A)

Weak connection (VAsc = 8 MV A)

Eigenvalue

Frequency of oscillations (Hz)

Damping ratio

Eigenvalue

Frequency of oscillations (Hz)

Damping ratio

Case (a): wr = 0.8 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


313.5459 ± 525.6455i
6.0053 ± 385.5469i
9.5108 ± 22.2264i
2.8720
0.4183

83.6591 61.3617 3.5374 0 0

0.5123 0.0155 0.3934 1 1


505.0621 ± 398.3863i
6.0290 ± 387.1397i
14.1898 ± 9.3774i
0.1639
0.2155

63.4373 61.6464 1.49 0 0

0.7851 0.0156 0.8343 1 1

Case (b): wr = 1.1 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


321.2293 ± 526.7301i
6.0000 ± 385.6305i
8.3111 ± 18.5650i
2.7762
0.4141

83.8317 61.3750 2.9547 0 0

0.5207 0.0155 0.4086 1 1


430.8688 ± 445.5332i
6.0046 ± 387.0451i
11.8227 ± 10.2798i
0.3557
0.1353

70.94477 61.6313 1.6369 0 0

0.6952 0.0155 0.7546 1 1

Case (c): wr = 1.29 p.u.

k1 ; k2 k3 ; k4 k5 ; k6 k7 k8


173.5807 ± 488.3757i
3.3356 ± 381.3223i
5.8310 ± 16.70i
0.2239
0.0546

78.0042 60.72 2.66 0 0

0.3349 0.0087 0.7438 1 1


385.3523 ± 460.5606i
6.0022 ± 386.9679i
11.9294 ± 8.4870i
0.4574
0.1466

73.337 61.6190 1.35 0 0

0.6417 0.0155 0.8148 1 1

Generally, the minimum value of damping ratio is assumed as 0.3. Minimization of this objective function will minimize the maximum overshoot. P OF 3 ¼ i (imaginary part of ith eigenvalue)2, if ri P
2:0. Prevention of high value of imaginary part of ith eigenvalue to the right of vertical line r0 ¼
2:0 is required. The damping will further increase on zeroing the value of OF 3 . OF 4 is an arbitrary constant which is of very high fixed value, around 106 , which will show some ri values of greater than 0.0. This indicates that for particular model parameters, unstable oscillations will occur. In (51), to transmit more weights to OF 1 and OF 2 and to reduce the weight value of OF 3 , the weighting factors are embroiled and, hence, during optimization, making them bilaterally competitive with the other component (like OF 4 Þ. The main aim of choosing the objective function, defined in (51), is focussed on selection of all the closed loop poles of the system to lie within a D-shape sector shown in Fig. 3.

TLBO: an overview TLBO algorithm is inspired by teaching–learning process in the classroom. It is developed by Rao et al. [16] in 2011. A brief overview of this algorithm is provided in the next three sub-sections. TLBO: features TLBO algorithm is inspired by teaching–learning process, as teaching–learning process is the most powerful instrument of education to bring about the desired changes in the students. Similarly, TLBO algorithm is augmented by the effect of influence of a teacher on the output of student (learner). The results of student, in terms of marks or grade obtained, are considered to be the output of TLBO algorithm. Teacher is assumed to be highly learned person who teaches learners so that they can improve their marks or grades. Thus, if the students achieve good grades in different subjects, their learning will get enhanced. Moreover, learners also

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

681

learn by interacting with each other which, in turn, helps them to achieve better result. TLBO is a population based algorithm. In this algorithm, a group of learners are considered as the population while subjects or courses offered to the learners are considered as different design variables. The output produced by the learners is analogous to the fitness function of the optimization problem. TLBO algorithm consists of two parts; one is ‘teacher phase’ while the other is ‘learner phase’. Teacher phase In this phase, the teacher imparts knowledge to the learners and tries to increase the mean learning value of the class according to the quality of teaching imparted. Let, there are ‘m’ numbers of design variables offered to ‘n’ number of learners. Let T i be the result of the teacher and M i be the mean result at any instant i. The value of T i will try to match the level of M i to its own level and the new mean is defined as M new . The difference between the existing and the new mean is given by (52)

difference meani ¼ r i ðM new
T F M i Þ

ð52Þ

where r i is a random number in the range varying from 0 to 1 and T F is the teaching factor that decides the value of mean to be changed whose value can either be 1 or 2. T F is defined by (53).

T F ¼ round½1 þ randð0; 1Þ  ð2
1Þ

ð53Þ

The modification of existing solution owing to this difference is done according to the expression given by (54)

X new;i ¼ X old;i þ difference mean

ð54Þ

Learner phase This phase consists of the mutual interaction between the learners which, in turn, tends to increase the knowledge of the learner. Each learner interacts randomly with other learners which facilitates knowledge sharing. A learner learns something new if the other learner has more knowledge than him or her. The learning phenomenon of this phase is expressed below.

The flowchart for TLBO algorithm is presented in Fig. 4. TLBO in power system application Recently, TLBO algorithm has been used in different areas of electrical field of applications like multi area economic load dispatch (ELD) problem [24], optimal power flow problem [25], short term hydro thermal scheduling problem [26], automatic load frequency control problem of multi source power system [27], etc. TLBO has been applied in ELD problem, that incorporates in non conventional energy sources like wind power [28]. A modified form of TLBO algorithm such as modified TLBO has been proposed to determine the optimal placement and size of the distributed generation units in [29] and solution of complex highdimensional benchmark problems may be found in [30]. The common parameters like population size ðN p Þ and maximum iteration cycles are chosen as 50 and 100, respectively, for both PSO and TLBO. The chosen values of PSO parameters are c1 ¼ c2 ¼ 2:05; wmax ¼ 0:9; wmin ¼ 0:4 [31]. Results and discussions Test scenario The effects of DFIG on the behaviour of power system when operated under different wind speed conditions is assessed

Fig. 5. Comparison of damping performances of all three cases pertaining to Scenarios 1 and 2.

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

2

5

T (p.u.)

0

0

e

Vabcstator (p.u.)

682

-2

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

1

1.1

(p.u.)

0

1.5

1.6

1.4

1.5

1.6

1.4

1.5

1.6

qs

i

i

1

1.1

1.2

1.3

1.4

1.5

-10

1.6

1

1.1

1.2

1.3

Time (sec) 10

(p.u.)

5 0

0

i

i

qr

dr

-5

1

1.1

1.2

1.3

1.4

1.5

Time (sec) 5000 0

V -5000

1

1.1

1.2

1.3

1.4

1.5

-10

1.6

1.6

fitness function

(p.u.)

1.4

0

Time (sec)

(Volt)

1.3

10

ds

(p.u.)

5

-5

dc

1.2

Time (sec)

Time (sec)

1

1.1

1.2

1.3

Time (sec) 3800 3750 3700

0

20

40

60

80

100

No.of iterations

Time (sec) TLBO PSO

2

5

T (p.u.)

0

0

e

Vabcstator (p.u.)

Fig. 6. Comparative PSO and TLBO based transient responses of Case a for Scenario 1 considering three phase fault.

-2

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

1

1.1

Time (sec)

1.5

1.6

1.4

1.5

1.6

1.4

1.5

1.6

(p.u.) qs

0

i

i

ds

(p.u.)

1.4

5

0 -5

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

1

1.1

1.2

1.3

Time (sec)

Time (sec) 5

(p.u.)

5

0

0

i

i

qr

dr

(p.u.)

1.3

Time (sec)

5

-5 1.1

1.2

1.3

1.4

1.5

5000

V

dc

0 -5000

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

Time (sec)

1.6

fitness function

1

(Volt)

1.2

1

1.1

1.2

1.3

Time (sec) 3900 3800 3700

0

20

40

60

80

100

No.of iterations

Time (sec)

TLBO PSO Fig. 7. Comparative PSO and TLBO based transient responses of Case b for Scenario 1 considering three phase fault.

through Fig. 1. The strength of the transmission network also affects the dynamic performance of the power system. Therefore, the effect of strong and weak transmission network are studied with short circuit level of 20 MV A and 8 MV A, respectively.

MATLAB/SIMULINK 2013 is used to implement the model of the present work. The model parameters used for the simulation are given in Appendix A. The two different scenarios taken for the analysis are:

683

2

5

T (p.u.)

0

0

e

Vabcstator (p.u.)

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

-2

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

1

1.1

1.3

1.4

1.5

1.6

1.4

1.5

1.6

1.4

1.5

1.6

10

(p.u.)

(p.u.)

5

0

i

i

ds

qs

0 -5

1

1.1

1.2

1.3

1.4

1.5

-10

1.6

1

1.1

Time (sec)

1.3

10

(p.u.)

(p.u.)

1.2

Time (sec)

5

0

i

i

qr

dr

0

1

1.1

1.2

1.3

1.4

1.5

Time (sec)

5000

V

dc

0 -5000

1

1.1

1.2

1.3

1.4

1.5

-10

1.6

1.6

fitness function

-5

(Volt)

1.2

Time (sec)

Time (sec)

1

1.1

1.2

1.3

Time (sec)

3900 3800 3700

0

20

40

60

80

100

No.of iterations

Time (sec) TLBO PSO

2

2

T (p.u.)

0

0

e

Vabcstator (p.u.)

Fig. 8. Comparative PSO and TLBO based transient responses of Case c for Scenario 1 considering three phase fault.

-2

1

1.1

1.2

1.3

1.4

1.5

-2

1.6

1

1.1

Time (sec)

1.6

1.4

1.5

1.6

1.4

1.5

1.6

(p.u.) qs

i

i

ds

(p.u.)

1.5

0

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

1

1.1

1.2

1.3

Time (sec)

Time (sec) 5

(p.u.)

5

0

i

i

qr

0

dr

-5

1

1.1

1.2

1.3

1.4

1.5

Time (sec) 5000

V

0 -5000

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

1.6

Time (sec)

fitness function

(p.u.)

1.4

5

0 -5

(Volt)

1.3

Time (sec)

5

dc

1.2

1

1.1

1.2

1.3

Time (sec) 750 700 650

0

20

40

60

80

100

No.of iterations TLBO PSO

Fig. 9. Comparative PSO and TLBO based transient responses of Case a for Scenario 2 considering three phase fault.

(a) Scenario 1: Short circuit level of 20 MV A (corresponds to strong grid). (b) Scenario 2: Short circuit level of 8 MV A (corresponds to weak grid). The parameters used to represent strong and weak grid are presented in Table 1 while the considered initial conditions for the chosen two scenarios are included in Table 2. The behaviour of

the DFIG is, additionally, analysed for the subsequent rotor speeds taking into account both the above two scenarios and, thus, the following mentioned three cases are considered. (a) Case a: Rotor speed ðxr Þ ¼ 0:8 p.u. (b) Case b: Rotor speed ðxr Þ ¼ 1:1 p.u. (c) Case c: Rotor speed ðxr Þ ¼ 1:29 p.u.

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

5

2

T (p.u.)

0

0

e

Vabcstator (p.u.)

684

-2

1

1.2

1.4

1.6

1.8

-5

2

1

1.2

i

1.8

2

1.8

2

1.8

2

80

100

(p.u.)

0

qs

0

i -5

1

1.2

1.4

1.6

1.8

-5

2

1

1.2

Time (sec)

1.4

1.6

Time (sec) 5

(p.u.)

5

0

qr

0

i

i

dr

(p.u.)

1.6

5

ds

(p.u.)

5

1

1.2

1.4

1.6

1.8

5000

V

dc

0 -5000

1

1.2

1.4

1.6

-5

2

Time (sec)

1.8

2

fitness function

-5

(Volt)

1.4

Time (sec)

Time (sec)

1

1.2

1.4

1.6

Time (sec)

4000 2000 0

0

20

40

60

No.of iterations

Time (sec) TLBO PSO

2

5

T (p.u.)

0

0

e

Vabcstator (p.u.)

Fig. 10. Comparative PSO and TLBO based transient responses of Case b for Scenario 2 considering three phase fault.

-2

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

1

1.1

1.4

1.5

1.6

1.4

1.5

1.6

1.4

1.5

1.6

(p.u.) qs

0

i

i

ds

(p.u.)

0 -5

-5 1

1.1

1.2

1.3

1.4

1.5

1

1.6

1.1

1.2

1.3

Time (sec)

Time (sec) 5

(p.u.)

5

0

qr

0

i

i

dr

-5

1

1.1

1.2

1.3

1.4

1.5

Time (sec) 5000

V

dc

0 -5000

1

1.1

1.2

1.3

1.4

1.5

-5

1.6

1.6

Time (sec)

fitness function

(p.u.)

1.3

5

5

(Volt)

1.2

Time (sec)

Time (sec)

1

1.1

1.2

1.3

Time (sec) 4000 2000 0

0

TLBO PSO

20

40

60

80

100

No.of iterations

Fig. 11. Comparative PSO and TLBO based transient responses of Case c for Scenario 2 considering three phase fault.

Small-signal stability analysis Small-signal analysis is, commonly, used in the power system study for finding the low frequency oscillations. The system equations are linearized around the operating point and, hence, modes of oscillations of system response can be obtained from the eigen-

values of the system state matrix. This analysis of the eigen properties of the system state matrix gives information about the stability of the test system. Table 3 includes the original controller parameters used for the purpose of analysis. The same table also includes PSO and TLBO optimized controller gains. Table 4 presents the results of eigenvalue analysis including frequency of oscillation

685

Vabcstator (p.u.)

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

2

T e (p.u.)

1 0 -1 0.4

0.6

0.8

1

1.2

1.4

0 -2 0.4

1.6

0.6

0.8

1

Time (sec)

0

0.6

0.8

1

1.2

1.4

0 0.4

1.6

0.6

0.8

1

1.2

1.4

1.6

1.2

1.4

1.6

Time (sec) 5

i qr (p.u.)

2 0 -2 0.4

0.6

0.8

1

1.2

1.4

1400 1200 1000 0.4

0.6

0.8

1

1.2

1.4

1.6

0 -5 0.4

1.6

Time (sec)

fitness function

i dr (p.u.)

1.6

2

Time (sec)

V dc (Volt)

1.4

4

i qs (p.u.)

i ds (p.u.)

2

-2 0.4

1.2

Time (sec)

0.6

0.8

1

Time (sec) 4000 3800 3600 0

10

20

30

40

Time (sec)

50

60

70

80

90

100

No.of iterations TLBO PSO

1

2

T e (p.u.)

Vabcstator (p.u.)

Fig. 12. Comparative PSO and TLBO based transient responses of Case a for Scenario 1 considering voltage sag.

0

-1 0.4

0.6

0.8

1

1.2

1.4

1.6

0 -2 0.4

0.6

0.8

0

0.6

0.8

1

1.2

1.4

0 0.4

1.6

0.6

0.8

i qr (p.u.)

i dr (p.u.)

1

1.2

1.4

1.6

1.2

1.4

1.6

5

0

0.6

0.8

1

1.2

1.4

0 -5 0.4

1.6

0.6

0.8

1400 1200

0.6

0.8

1

1.2

1.4

1.6

fitness function

Time (sec) V dc (Volt)

1.6

Time (sec)

2

1000 0.4

1.4

2

Time (sec)

-2 0.4

1.2

4

i qs (p.u.)

i ds (p.u.)

2

-2 0.4

1

Time (sec)

Time (sec)

1

Time (sec) 4000 3800 3600

Time (sec)

0

10

20

30

40

50

60

70

80

90

100

No.of iterations TLBO PSO

Fig. 13. Comparative PSO and TLBO based transient responses of Case b for Scenario 1 considering voltage sag.

and damping ratio of the considered two scenarios (viz. strong grid and weak grid) when original controller gains presented in Table 3 are used. Tables 5 and 6 are pertaining to the obtained results of small-signal stability analysis of the studied power system model focussing on eigenvalues, frequency of oscillation (Hz) and damping ratio as yield by PSO and the proposed TLBO algorithm, respectively. The results obtained for the system reveal that system is stable for all the scenarios. The eigenvalues like k1 and k2 have the highest real part magnitude and, therefore, the frequency of

oscillation is found to be the highest. On the other hand, while adopting the proposed TLBO algorithm, k3 and k4 have the lowest damping ratios among all the modes. Fig. 5 shows the comparison of the damping ratios of all the three cases for scenarios 1, 2 using original PSO and TLBO optimized controller gains presented in Table 3. The results presented in Fig. 5 clearly illustrate that the proposed TLBO optimized controller gains yield better damping as compared the original controller gains and PSO optimized controller gains.

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

Te (p.u.)

1 0 -1 0.4

0.6

0.8

1

1.2

1.4

1.6

1 0 -1 0.4

0.5

0.6

0.7

Time (sec) 1 0

-1 0.4

0.6

0.8

1

1.2

1.4

1.6

i qr (p.u.) 1

1.2

1.4

1.6

Vdc (Volt)

Time (sec) 1400 1300 1200 1100 0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

fitness function

i dr (p.u.)

0

0.8

1

1.1

1.2

1.3

1.4

1.1

1.2

1.3

1.4

1.1

1.2

1.3

1.4

70

80

90

100

1 0 -1 0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec)

1

0.6

0.9

2

Time (sec)

-1 0.4

0.8

Time (sec) i qs (p.u.)

i ds (p.u.)

Vabcstator (p.u.)

686

2 1 0 -1 0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec) 3800 3700 3600

0

10

20

30

40

50

60

No.of iterations

Time (sec)

TLBO PSO

2

Te (p.u.)

1 0 -1 0.4

0.6

0.8

1

1.2

1.4

1.6

0 -2 0.4

0.6

0.8

Time (sec) 2

0.6

0.8

1

1.2

1.4

1.6

i qr (p.u.)

0 0.6

0.8

1

1.2

1.4

0 0.4

0.6

0.8

1

1.2

1.4

1.6

Time (sec)

1.2

1.4

1.6

1500

2 1 0 0.4

1.6

fitness function

i dr (p.u.)

1.6

Time (sec)

2

Vdc (Volt)

1.4

1

Time (sec)

-2 0.4

1.2

2

0 -2 0.4

1

Time (sec) i qs (p.u.)

i ds (p.u.)

Vabcstator (p.u.)

Fig. 14. Comparative PSO and TLBO based transient responses of Case c for Scenario 1 considering voltage sag.

0.6

0.8

1

Time (sec)

800 600

1000 0.4

0.6

0.8

1

1.2

1.4

1.6

400

0

Time (sec)

10

20

30

40

50

60

70

80

90

100

No.of iterations TLBO PSO

Fig. 15. Comparative PSO and TLBO based transient responses of Case a for Scenario 2 considering voltage sag.

Transient response of DFIG under short circuit Since the usage of wind power in present electrical power system is increasing, the behaviours of DFIG wind turbine under different faults, voltage dips and various other disturbances are becoming very much pertinent and vital ones also, particularly, for those having power electronic converters. The decrease in the grid voltage, due to voltage dips executed at the connection point of the DFIG ends up in high current in rotor. This high rotor current will harm the RSC and should cause a huge rise in the dc-link voltage. Because of such huge dc-link voltage, the rotor current and the

oscillations in torque arising due to fault in the grid are quite harmful for the DFIG-based wind power system. For the protection of grid, either the DFIG or the RSC has to be separated out from the system by using the crowbar resistance. There is a rapid change in frequency due to the sudden loss of wind power caused due to the fault in the electrical grid. Due to this, as the RSC is removed from the system, DFIG will start to act as squirrel cage induction generator. As a result, DFIG will start consuming a lot of additional reactive power and may also lead to voltage instability. Therefore, it is desired to have wind turbines connected to the grid actively contribute to the system stability throughout

687

2

2

T (p.u.)

0

0

e

Vabcstator (p.u.)

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

-2 0.4

0.6

0.8

1

1.2

1.4

1.6

-2 0.4

0.6

0.8

(p.u.)

(p.u.)

2

1.2

1.4

1.6

1.2

1.4

1.6

1.2

1.4

1.6

2 1

i

i

ds

qs

0 -2 0.4

0.6

0.8

1

1.2

1.4

0 0.4

1.6

0.6

0.8

Time (sec) (p.u.)

(p.u.)

1

Time (sec)

1

2 1

i

i

dr

qr

0

0.6

0.8

1

1.2

1.4

Time (sec)

1500

V

dc

1000 500 0.4

0.6

0.8

1

1.2

1.4

0 0.4

1.6

1.6

fitness function

-1 0.4

(Volt)

1

Time (sec)

Time (sec)

0.6

0.8

1

Time (sec) 4000 2000 0

0

20

40

60

80

100

No.of iterations

Time (sec) TLBO PSO

1

2

T (p.u.)

0

0

e

Vabcstator (p.u.)

Fig. 16. Comparative PSO and TLBO based transient responses of Case b for Scenario 2 considering voltage sag.

-1 0.4

0.6

0.8

1

1.2

1.4

-2 0.4

1.6

0.6

0.8

(p.u.) 0.6

0.8

1

1.2

1.4

1.2

1.4

1.6

1.2

1.4

1.6

2 0 0.4

1.6

0.6

0.8

1

Time (sec) 2

(p.u.)

2 0

1

i

i

qr

dr

(p.u.)

1.6

4

Time (sec)

0.6

0.8

1

1.2

1.4

Time (sec) 1500

1000 0.4

0.6

0.8

1

1.2

1.4

0 0.4

1.6

1.6

fitness function

-2 0.4

(Volt)

1.4

i

i -2 0.4

dc

1.2

qs

0

ds

(p.u.)

2

V

1

Time (sec)

Time (sec)

0.6

0.8

1

Time (sec) 4000 2000 0

0

20

Time (sec)

40

60

80

100

No.of iterations TLBO PSO

Fig. 17. Comparative PSO and TLBO based transient responses of Case c for Scenario 2 considering voltage sag.

the disturbances and faults. Low voltage ride through (LVRT) ability is defined as the ability of wind power system to remain connected to the grid throughout the faults and voltage dips [32]. At present scenario, to ensure the safety of power system network, many countries have already introduced and enforced their grid codes for LVRT capability while integrating wind power generation system into the grid.

Three phase fault A three phase fault is applied at the terminal of the DFIG to study the transient performance of the DFIG connected to the electrical grid. A three phase fault is applied at 1.0 s for a duration of 40 ms after which the normal operation is resumed. The steady state responses of stator voltage ðV abcstator Þ, electromagnetic torque

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S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

ðT e Þ, stator currents ðids ; iqs Þ, rotor currents ðidr ; iqr Þ and dc-link voltage ðV dc Þ are shown in Figs. 6–11. These figures, basically, illustrate a comparative study of the transient responses among PSO and TLBO optimized controller gains. It may be clearly observed from the results obtained that the dc-link voltage, electromagnetic torque and currents attain their normal values after the fault is cleared. The dc-link voltage, currents and electromagnetic torque were operating at their initial settled value before the occurrence of the three phase fault. As the three phase fault is applied suddenly, it causes transient responses of dc link voltage, currents and electromagnetic torque. The optimized values of the controllers used, effectively, limit the peak values in these transients and suppress them very quickly to their normal position. Finally, the test system comes back to its original operating points as soon as the fault is removed. While examining these figures it is revealed that the proposed TLBO optimized controller gains yield better transient response as compared to PSO optimized ones. It is further noted that promising convergence profile of fitness function is achieved while adopting the proposed TLBO algorithm for all the considered cases of both the scenarios viz. strong as well as weak grid condition.

Voltage sag The LVRT capability stated in grid codes, additionally, needs the wind turbine generators to be able to operate at reduced voltage for some hundreds of ms to several seconds. The voltage sag is also performed in the DFIG system from 0.5 s to 1.21 s, taking into account both the scenarios by decreasing the terminal voltage of DFIG from its original value to 30% of its original value, as shown in Figs. 12–17. Finally, it may be established from the results presented in these figures that the steady state responses exhibit superior performance with the proposed TLBO optimized controller gains for the voltage sag condition. Promising convergence profile of fitness function values may be observed for the proposed TLBO approach as compared to PSO counterpart.

Appendix B. Elements of system matrix Asys The non-zero elements of Asys , as defined in (50), are presented in (B.1)–(B.40).

A11 ¼ A12 ¼

A13 ¼ A14 ¼ A15 ¼

A16 ¼

A18 ¼ A21 ¼

A22 ¼

n

wb X rr X ss


X 2m


RT X rr þ ðK p2 =v s0 ws ÞðRT v ds0 þ X T v qs0 Þ

ðX rr X ss
s0 X 2m Þws þ X T X rr X rr X ss
X 2m o  þ K p2 =v s0 ws ÞðRT v qs0
X T v ds0 Þ n

wb X 2m

n

wb X rr X ss


X 2m n

wb X rr X ss


X 2m

o
X m ðRr þ K p2 Þ ð
X m X rr þ s0 X m X rr Þws ðX m iqs0
X rr iqr0 ÞX m

o

o

ðB:4Þ ðB:5Þ

wb X m

ðB:6Þ

X rr X ss
X 2m wb X m K p2

ðB:7Þ

X rr X ss
X 2m n

wb

ð
X rr X ss þ s0 X 2m Þws
X T X rr X rr X ss
X 2m o  þ K p1 K opt X ss ws w2r0 =v 3s0 ÞðRT v ds0 þ X T v qs0 Þ wb X rr X ss
X 2m

ðB:8Þ

n o
RT X rr þ ðK p1 K opt X ss ws w2r0 =v 3s0 ÞðRT v qs0
X T v ds0 Þ ðB:9Þ

Conclusion

A24 ¼ A25 ¼

ðB:10Þ n

wb X rr X ss


X 2m n

wb X rr X ss


X 2m

o
X m ðRr þ K p1 Þ

ðB:11Þ

ð
X m ids0 þ X rr idr0 ÞX m

o

A27 ¼ A16 A31 ¼

ðB:12Þ ðB:13Þ

n

wb X rr X ss


X 2m

o
RT Xm þ ðK p2 X ss =X m v s0 ws ÞðRT v ds0 þ X T v qs0 Þ ðB:14Þ

A32 ¼

A33 ¼ Appendix A. Parameter of DFIG (in p.u. otherwise specified) H ¼ 4:32; K opt ¼ 056; Sb ¼ 1:67 MV A, F b = 60 Hz, ws ¼ 1; wb ¼ 2  pi  F b ; Rs ¼ 0:023; X ls ¼ 0:18; Rr ¼ 0:016; X lr ¼ 0:16; X m ¼ 2:9; v w = 13 m/s, pitch-rate = 10 deg/s, pitch-max = 27 deg, V grid = 120 kV, V dc = 1150 V, C dc = 10 mF, F ¼ 0:01; np ¼ 3, Dtran = 30 km.

ðB:2Þ ðB:3Þ

A23 ¼
A14

Taking consideration of the restraints due to the limited nonrenewable energy sources and with the ever increasing demand of electricity, it has become an absolute necessity to integrate the wind power systems in the grid, for which a complete and detailed study of wind power generation models as well as the grid codes are required, in which simulation studies under various expected operating conditions are highly needed to prevent any harmful impact of the power system network to the grid where it is connected. In this paper, a dynamic model of DFIG and the controllers used with the reduced order illustration is presented which is helpful for the steady state as well as the transient stability analysis of the power system. It is, finally, observed that optimization of controller gains of the DFIG model by using TLBO is much better than using PSO technique for the studied power system model under different scenarios considered.

ðB:1Þ

n

wb

X rr X ss


o

A34 ¼ A35 ¼

n

wb

ðX m X ss
s0 X m X ss Þws þ X T X rr X rr X ss
 o þ K p2 X ss =X m v s0 ws ÞðRT v qs0
X T v ds0 Þ X 2m

n

wb X rr X ss
X 2m

n

wb X rr X ss


X 2m n

wb X rr X ss


X 2m

o
X ss ðRr þ K p2 Þ ð
X 2m þ s0 X ss X rr Þws ðX m iqs0
X rr iqr0 ÞX ss

ðB:15Þ ðB:16Þ

o o

ðB:17Þ ðB:18Þ

S. Chatterjee et al. / Electrical Power and Energy Systems 78 (2016) 672–689

A36 ¼

A38 ¼ A41 ¼

wb X ss

ðB:19Þ

X rr X ss
X 2m wb X ss K p2

ðB:20Þ

X rr X ss
X 2m wb X rr X ss


X 2m

n ð
X m X ss þ s0 X m X ss Þws
X T X rr

o þ ðK p1 X ss K opt X ss ws w2r0 =v 3s0 X m ÞðRT v ds0 þ X T v qs0 Þ A42 ¼

wb X rr X ss
X 2m

ðB:21Þ

n o
RT X m þ ðK p1 X ss K opt X ss ws w2r0 =v 3s0 X m ÞðRT v qs0
X T v ds0 Þ

ðB:22Þ A43 ¼ A44 ¼ A45 ¼

wb X rr X ss


X 2m

wb X rr X ss


X 2m

wb X rr X ss


X 2m

n o ðX 2m
s0 X ss X rr Þws

ðB:23Þ

n o
X ss ðRr þ K p1 Þ

ðB:24Þ

n o ð
X m ids0 þ X rr idr0 ÞX ss þ 2ðK p1 X ss =X m Þ

ðB:25Þ

A47 ¼ A36

ðB:26Þ

A51 ¼

X m iqr0 2H

ðB:27Þ

A52 ¼


X m idr0 2H

ðB:28Þ

A53 ¼


X m iqs0 2H

ðB:29Þ

A54 ¼

X m ids0 2H

ðB:30Þ

A61 ¼ ðK i2 =v s0 ws X m ÞðRT v ds0 þ X T v qso Þ

ðB:31Þ

A62 ¼ ðK i2 =v s0 ws X m ÞðRT v qs0
X T v ds0 Þ

ðB:32Þ

A63 ¼
K i2

ðB:33Þ

A68 ¼ K i2

ðB:34Þ

A71 ¼ ðK i1 K opt X ss ws w2r0 =v 3s0 X m ÞðRT v ds0 þ X T v qs0 Þ

ðB:35Þ

A72 ¼ ðK i1 K opt X ss ws w2r0 =v 3s0 X m ÞðRT v qs0
X T v ds0 Þ

ðB:36Þ

A74 ¼
K i1

ðB:37Þ

A75 ¼ 2K i1 K opt X ss ws wr0 =X m v s0

ðB:38Þ

A81 ¼ ð
K p3 =v s0 ÞðRT v ds0 þ X T v qs0 Þ

ðB:39Þ

A82 ¼ ðK p3 =v s0 ÞðRT v qs0
X T v ds0 Þ

ðB:40Þ

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