Design and experimental investigation of a decentralized GA-optimized neuro-fuzzy power system stabilizer

Electrical Power and Energy Systems 32 (2010) 751–759

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

Design and experimental investigation of a decentralized GA-optimized neuro-fuzzy power system stabilizer Hossam E.A. Talaat *, Adel Abdennour, Abdulaziz A. Al-Sulaiman Electrical Engineering Department, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 10 January 2008 Received in revised form 10 July 2009 Accepted 28 January 2010

Keywords: Decentralized control Neuro-fuzzy control Genetic Algorithms Power system stabilizer Multi-machine lab model

a b s t r a c t The aim of this research is the design and implementation of a decentralized power system stabilizer (PSS) capable of performing well for a wide range of variations in system parameters and/or loading conditions. The framework of the design is based on Fuzzy Logic Control (FLC). In particular, the neuro-fuzzy control rules are derived from training three classical PSSs; each is tuned using GA so as to perform optimally at one operating point. The effectiveness and robustness of the designed stabilizer, after implementing it to the laboratory model, is investigated. The results of real-time implementation prove that the proposed PSS offers a superior performance in comparison with the conventional stabilizer. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Power system stabilizers (PSSs) have been popularly used to damp out the low frequency oscillations in the system. The conventional PSS was mainly introduced as a lead–lag compensator [1]. The parameters of a conventional PSS are normally fixed at values determined based on classical control theory in the frequency domain. This class of PSSs always suffers from a poor performance for a wide range of operating conditions. To mitigate the shortcomings of conventional PSS, many control strategies applying various techniques have been proposed over the last four decades. Examples of the applied techniques are: linear quadratic regulator [2], self-tuning regulator [3,4], model reference adaptive control [5], and robust control [6]. More recently, the concepts of artificial intelligence (AI) techniques and evolutionary algorithms were applied in order to create higher degree of robustness and adaptability. Three AI techniques were widely applied: Artificial Neural Networks (ANNs) [7,8], Fuzzy Logic Control (FLC) [9–11], Genetic Algorithms (GA) [12] and Ant Colony Optimization (ACO) [13]. Merging more than one AI technique is also common in the literature [14–19]. The evaluation of the performance of any of these techniques should be carried out in view of the robustness of the PSS. By robustness we mean that the PSS has to perform well against the wide domain of variations of the system parameters, the loading * Corresponding author. Tel.: +966 1 467 3117; fax: +966 1 467 6757. E-mail address: [email protected] (H.E.A. Talaat). 0142-0615/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2010.01.011

conditions, and the disturbance size and location. The evaluation of any proposed PSS should not consider ‘‘How recent the applied technique is?” but instead ‘‘to what extent did the application use the full capability of the selected technique?” With respect to the environment of evaluation, some researches rely on computer simulation of a single-machine-infinite-bus system to test their design [3–7,9,13–14]. Others, use computer simulation of multi-machine systems [8,10–12,15,16]. However, the computer simulation environment is not adequate since it does not consider practical constraints and/or considerations. Some studies have implemented the proposed PSS on a laboratory multi-machine system [20]. This research aims at: firstly designing and implementing a decentralized PSS capable of satisfying the abovementioned requirements and, secondly, investigating its performance experimentally on an environment similar to that of a real power system. The framework of the design is based on FLC. The fuzzy control rules of the proposed PSS are derived from training three classical PSSs. Each classical PSS is tuned using GA so as to perform optimally at one operating point. The training process is carried out using Adaptive Neuro-based Fuzzy Inference (ANFIS) principles. To achieve the project objectives, four major steps were undertaken: (i) construction of the laboratory multi-machine model, (ii) modeling, simulation and model validation of the set-up, (iii) design and preliminary evaluation of the proposed stabilizer via computer simulation, and (iv) real-time implementation and testing of the designed stabilizer on the laboratory set-up.

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2. Laboratory multi-machine system 2.1. Study system The multi-machine power system considered for laboratory simulation is shown in Fig. 1. It is composed of three identical machines rated 1000 MV A each. The transmission network is composed of four 380 kV overhead transmission lines of different lengths. The PSS under development, which is assumed to apply a decentralized control concept, is installed at machine#2. Appendix A describes the main components of the laboratory system developed during this project.

1. Selection of a very simple control structure such as a first order linear compensator for each of the operating points chosen for design. Three points have been selected: light load, medium load, and heavy load. 2. Optimal tuning of the compensators’ parameters using Genetic Algorithms (GA). The simulation environment used for this purpose is the SimPowerSystem model described in Section 2.2. 3. Obtain a single neuro-fuzzy PSS that replaces the optimal compensators designed in the previous step. This PSS is trained with the control actions generated by the optimized compensators. The resulting stabilizer should capture the performance of the single compensators while offering even a better performance due to its nonlinear structure.

2.2. Dynamic model A comprehensive computer dynamic modeling of the laboratory set-up is crucial in order to create computational environment appropriate for the design, testing and evaluation of various forms of PSSs. Two main obstacles have been observed during the course of this computer modeling; the highly nonlinear characteristic of the lab system, and the high noise contents superimposed on the signals especially when the measured signal has a small value. The three-machine system constructed in the laboratory is simulated using the SimPowerSystem toolbox [21] as portrayed in Fig. 2. The synchronous machine model used in the simulation is an eighth order model: two for electromechanical and six for electromagnetic. The machines are modeled as standard per unit models in dq rotor reference frame. The details of the model are included in Appendix B.

3.2. GA tuning of the compensators GA is an attractive derivative-free optimization tool capable of attaining optimal solutions even when the search space is large. Multi-objective performance measures can also be incorporated with ease. The GA used here is similar to what is now called the classical GA and which can be found in the standard literature in the subject [22]. The PSS structure implemented with the system at hand is described by

U stab ¼

Kðs þ zÞ Dx ðs þ pÞ

ð1Þ

where Ustab is the PSS output, Dx is the angular speed deviation, and K, z and p are the parameters of the stabilizer. The objective is to tune the three parameters with the following requirements:

2.3. Model verification This study conducts the verification of the SimPowerSystem model of the lab system. The results obtained from the simulated model are compared to the measurements recorded from the lab system under different dynamic conditions. A self clearing threephase short circuit is applied to bus#2 for three cycles under three different loading conditions: light load, medium load and heavy load. The angular speed responses under these conditions are illustrated in Fig. 3. The curves of Fig. 3 reveal that the field measurements that are obtained from the lab system are very close to the responses of the computer simulation. 3. Control strategy 3.1. General The control strategy employed in this project follows the following three basic steps:

 optimal dynamic performance of rotor speed (max damping, min settling time, min overshoot);  optimal dynamic performance of load angle and;  minimal control action The fitness function used by the GA should reflect all the above requirements. One choice of such function is based on the sum of the squared error (sse), where error here means the deviation in the variable. The fitness function is given by:

fitness ¼

4

1

L24 (216 km)

L14 (216 km)

G1

G2 Load_4

L24 (144 km) PSS

ð2Þ

The first and second terms of the denominator express the deviation of the load angle and the angular speed respectively, while the third term is used to minimize the stabilizer output. The coefficients Wd, Wx and Wu are used to weigh the importance of each of the three quantities and to balance these terms to more or less the same order of magnitude. In this work these coefficients are selected as: 105, 1 and 102, respectively.

2 AVR / Exciter

1 W d sseðdÞ þ W x sseðDxÞ þ W u sseðuÞ

L23 (360 km)

3

G3 Load_3

Fig. 1. System under study.

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AR40 - Model of Lab's 3-Machine (Heavy Load) Bus 4 Bus 2

PSS_out

A

A

A

A

B

B

B

B

C

C

C

C

Machine 2

Line L24 - 216 km

Bus 1

Line L14 - 216 km Machine 1

Bus 3 PSS A

A

B

B

A

B

B

C

C

C

A

C Line L13 - 144 km

Line L23 - 360 km

Machine 3

Phasors

Scopes

powergui

STOP Unstable

Fig. 2. Dynamic simulation of the laboratory system.

The GA tuning of the stabilizer in this study is multi-objective in a sense that it inherently optimizes different types of oscillations resulting from changing the size and location of the disturbance. 3.3. Architecture of the neuro-fuzzy PSS Once the classical PSSs are optimized for the selected operating conditions, they should be ‘‘blended” in a single neuro-fuzzy PSS that not only captures their performances but also brings up the advantage of its nonlinearity in generalizing the optimal performance of each single classical PSS. By the end of this step we will end up with only one PSS capable of performing hopefully well for a wide range of operating conditions. To achieve this step, we resort to a system called Adaptive Neuro-Fuzzy Inference System (ANFIS). ANFIS is a fuzzy Sugeno model put in the framework of adaptive systems to facilitate learning and adaptation. Such framework makes the design of a neuro-fuzzy PSS more systematic and less relying on expert knowledge. To present the ANFIS architecture, let’s consider two fuzzy ifthen rules based on a first order fuzzy Sugeno model [23]:

Rule 1 : if ðx is A1 Þ and ðy is B1 Þ then ðf 1 ¼ p1 x þ q1 y þ r 1 Þ Rule 2 : if ðx is A2 Þ and ðy is B2 Þ then ðf2 ¼ p2 x þ q2 y þ r 2 Þ

ð3Þ

These two rules give

8 > < f1 ¼ p1 x þ q1 y þ r 1 w1 f1 þ w2 f2  1 f1 þ w  2 f2 ¼w f2 ¼ p2 x þ q2 y þ r 2 ) f ¼ > w1 þ w2 :

ð4Þ

A possible ANFIS architecture is to implement these two rules as shown in Fig. 4. The explanation of this architecture is given below. Layer 1: All the nodes in this layer are adaptive nodes. The output of each node is the degree of membership of the input to the fuzzy membership function (MF) represented by the node. If the bell MF is used then the degree of membership is

lAi ðxÞ ¼ 1þ

h

1 i 2 ðxc Þ ai

ib i

i ¼ 1; 2

ð5Þ

where ai, bi, ci, are the parameters for the MF. Layer 2: The nodes in this layer are fixed (not adaptive). They are labeled M to indicate that they play the role of a simple multiplier. The output of each node in this layer represents the firing strength of the rule. Layer 3: The nodes in this layer, which are also fixed, are labeled N to indicate that they perform a normalization of the firing strength from the previous layer. Layer 4: All the nodes in this layer are adaptive nodes. The output of each node in this layer is simply the product of the normalized firing strength and a first order polynomial (for first order Sugeno model):

 i ðpi x þ qi y þ r i Þ i ¼ 1; 2  i fi ¼ w w

ð6Þ

Layer 5: This layer has only one node labeled S to indicate that it performs the function of a simple summer. The ANFIS architecture is not unique. Some layers can be combined and still produce the same output. Architectures for the Mamdani fuzzy model are also available but are not adopted in this project. In this ANFIS architecture, there are two adaptive layers (layers 1 and 4). Layer 1 has three modifiable parameters (ai, bi, and ci) pertaining to the input MFs. These parameters are called premise parameters. Layer 4 has also three modifiable parameters (pi, qi, and ri) pertaining to the first order polynomial. These parameters are called consequent parameters. The task of the training or leaning algorithm for this architecture boils down to tuning all the modifiable parameters to make the ANFIS output match the training data. 3.4. Training of the neuro-fuzzy PSS The structure of the neuro-fuzzy stabilizer adopted here is shown in Fig. 5. This stabilizer has two inputs. The first input

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Angular Speed Responses to a 0.05s Short Circuit for Light Load (P=0.2) Sum Errors = 2.73

Angular Speed in p.u

0.2 Computer Sim. Experimental Sim.

0.1

0

-0.1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Angular Speed Responses to a 0.05s Short Circuit for Medium Load (P=0.4)

Angular Speed in p.u

0.2

0.1

0

-0.1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s)

Angular Speed Responses to a 0.05s Short Circuit for Heavy Load (P=0.6)

Angular Speed in p.u

0.2

0.1

0

-0.1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (s) Fig. 3. Estimated versus measured rotor speed responses.

x

A1

M

w1

N

w1 w1f1

A2

f x

y

tiation of high frequency noise typically existing in the speed signal and the second is to avoid unnecessarily long simulation time caused by small integration steps employed by the variable-step integration algorithm. The acceleration is synthesized using the following transfer function:

S HðsÞ ¼

B1 y B2

M

Layer 1

Layer 2

w2

N

w2

Layer 3

w2f2 Layer 4

Layer 5

Fig. 4. ANFIS architecture for a two-rule fuzzy system.

(input 1) is the angular speed and the second input (input 2) is the angular acceleration. The acceleration is obtained from the speed using an approximate derivative instead of a pure differentiator. This choice is made for two reasons. The first is to avoid differen-

s ð1 þ 0:01sÞ

ð7Þ

Each of the two inputs is represented by seven fuzzy membership functions, resulting in a total of 49 fuzzy rules. The system has a single output representing the stabilizing signal. To train this network, the angular speed, the angular acceleration, and the corresponding control action are collected through running the SimPowerSystem model with the GA-optimized compensators for the three operating conditions. The collected data are used to train ANFIS with the objective of automatically generating the fuzzy rules that match a corresponding output for each given pair of inputs. Since this fuzzy model is of the Sugeno type, unlike

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H.E.A. Talaat et al. / Electrical Power and Energy Systems 32 (2010) 751–759 Table 1 Average load values for considered loading conditions. Loading condition

Load at bus#3 *

Light Medium Heavy *

Load at bus#4

P (p.u. )

Q (p.u.)

P (p.u.)

Q (p.u.)

0.34 0.68 1.18

0.24 0.35 0.48

0.33 0.6 0.89

0.14 0.31 0.36

The base is 1 kVA.

5

No PSS Convl. PSS Fuzzy PSS

Fig. 5. Structure of the fuzzy stabilizer.

the Mamdani model, the output is crisp rather than fuzzy. Therefore, there will be an output value for each of the 49 fuzzy rules generated by the ANFIS structure instead of fuzzy output membership functions. However, the fuzzy decision surface is concisely represented here in a three-dimensional space to examine the degree of nonlinearity that this fuzzy stabilizer is capable of capturing. This fuzzy decision surface, shown in Fig. 6, plots the fuzzy output versus the two inputs.

Rotor Angle Deviation (rad)

4 3 2 1 0 -1 -2 0

0.5

1

1.5

Time(s) Fig. 7. Responses to a 3-cycle fault at bus#2 – light loading.

the operating conditions. A simple first order compensator in the form of Eq. (8) has been used.

4. Experimental investigations The performance of the fuzzy-based PSS developed in the previous section is to be tested in an environment similar to the real power system. The lab power system, which has been built specially for this project, is used for this purpose. The performance of the stabilizer under design is evaluated under various loading conditions and/or disturbance type. 4.1. Applied stabilizers To provide a fair base of comparison, a conventional type PSS should be designed to perform optimally over a wide range of

GPSS ðsÞ ¼

K s ðs þ ZÞ ðs þ PÞ

ð8Þ

The input to this stabilizer is the angular speed deviation of the machine under control. The extensive on-line testing of the conventional PSS, while several difficulties have been encountered, leads to the best parameters: Ks = 1, Z = 6 and P = 3. The second stabilizer to be tested is the Neuro-fuzzy (simply fuzzy) PSS, which is the subject of this research. The input to this stabilizer is globally the angular speed deviation. The computer controller produces internally the second output, i.e. the acceleration as indicated in Section 3.4. The output of any of the

Fig. 6. Fuzzy decision space.

H.E.A. Talaat et al. / Electrical Power and Energy Systems 32 (2010) 751–759

Rotor Angle Deviation (rad)

756

a- Rotor Angle Deviation 2 0 -2

No PSS Convl. PSS Fuzzy PSS

-4 0

0.5

1

1.5

1

1.5

1

1.5

Speed Deviation (pu)

b- Rotor Speed Deviation 0.2 0.1 0 -0.1 -0.2 0

0.5

Ustab (pu)

c- Stabilizing Signal 0.1 0.05 0 -0.05 -0.1 0

0.5 Time (s)

Fig. 8. Responses to a 3-cycle fault at bus#2 under medium loading.

3

5

No PSS Convl. PSS Fuzzy PSS

2 1.5 1

0.5 0 -0.5 -1 -1.5 -2 -2.5 0

0.5

1

No PSS Convl. PSS Fuzzy PSS

4 Rotor Angle Deviation (rad)

Rotor Angle Deviation (rad)

2.5

1.5

Time(s)

3 2 1 0 -1 0

0.5

1

1.5

Time(s) Fig. 9. Responses to a 3-cycle fault at bus#2 – heavy loading. Fig. 10. Responses to a 6-cycle fault at bus#2 – medium loading.

applied stabilizers is fed to a limiter of +0.15 per unit. The stabilizing signal is added to the reference of the automatic voltage regulator of machine#2. The dynamic performance of the lab power system has been compared for three control case studies: without PSS, with conventional PSS and with fuzzy PSS. To provide a fair comparison, and since the system is subject to continuous variations, the three control cases are always compared under the same experiment before any significant variations to the system variables take place. 4.2. Loading conditions Three loading conditions are applied for the experimental investigations: light, medium and heavy loading. Table 1 lists the

average values of the system loads at different loading conditions. The word ‘‘average” here refers to the nature of the operation of the three-machine lab system where it is difficult to operate the system under fixed operational variables. 4.3. Responses to a 3-cycle three-phase fault A 3-cycle three-phase short circuit was applied to the terminals of the machine under control (bus#2) for the medium loading condition, under three control cases. The rotor angle deviation, rotor speed deviation and stabilizing signal responses are given in Fig. 7. The rotor angle deviations for the light and heavy loading conditions are given in Figs. 8 and 9 respectively. The performance of the stabilizers under this fault condition is better than that of the system without stabilizers for both the maximum overshoot and

H.E.A. Talaat et al. / Electrical Power and Energy Systems 32 (2010) 751–759

the damping characteristic. For instance, the developed fuzzy stabilizer exhibited superior performance having the lowest overshoot and highest damping for the three loading conditions. It is interesting here to mention that the investigators have experienced the excellent performance of the adopted PSS from just hearing the sound resulting from the machine upon applying short circuit to its terminals. The fuzzy PSS absorbs the disturbance such that the resulting sound is the softest. The robustness of the designed fuzzy PSS has been proved through extensive testing under different operating conditions where its response is always better than that of the conventional PSS.

757

by the industry. Aside from its desirable performance, this stabilizer possesses impressive features such as robustness to changes in operating condition, capabilities in accommodating model variations and external disturbances, and simplicity of real-time implementation. The inherent nonlinearity of this stabilizer has the advantage of capturing the performance of many linear stabilizers of the type typically used in real systems. In light of the achieved results and the experience gained over the course of this project the investigators are convinced that the proposed approach enjoys a great deal of qualities making it stand as an excellent control strategy for the problem at hand.

4.4. Responses to a 6-cycle Three-Phase Fault Acknowledgement A larger disturbance is applied by increasing the duration of the short circuit to bus#2 to six cycles, under medium loading. The corresponding responses are given in Fig. 10. The neuro-fuzzy PSS is still offering better responses characterized by higher damping and lower overshoot. 5. Conclusions

The authors thank the Deanship of Scientific Research – King Saud University for the financial support of this project.

Appendix A. Laboratory three-machine power system A.1. Main components

The lab power system, which was developed using micro-synchronous machines having low inertia constants, exhibits modes of oscillations that are comparable to those exist in large multi-machine power systems. In the process of developing the proposed PSS, a comprehensive approach empowered by experimentation schemes, mathematical methods, analysis techniques, and simulation tools was employed. This stabilizer offered a superior performance, characterized by higher damping and lower overshoot, in comparison with the conventional stabilizer presently adopted

Fig. A1. A part of the laboratory multi-machine system.

The multi-machine power system considered for laboratory simulation is depicted in Fig. 1. The laboratory simulation uses scale 1:1000 for the voltage, 1:1000 for the current, thus, 1:106 for the power. Therefore, each power plant of the studied system is represented by a dc motor simulating prime-mover, coupled to a 1 kVA micro-synchronous machine simulating the generator (Fig. A1). The transmission line simulators are constructed as p-circuits. Three-phase resistive and inductive load simulators are used to simulate the system loading. A voltage control loop has been constructed, for the machine under control (Fig. A2). It incorporates a voltage transducer, an amplifier, a damping feedback block, a firing circuit and a full-wave thyristor bridge. The electronic components of the AVR are implemented on a Printed Circuit Board (PCB) using operational amplifiers. The speed/angle transducer is composed of two parts; a shaft encoder, which is coupled to the machine under control, generating 2048 pulse per revolution and a PCB for processing the pulses with two outputs: the rotor angle and the rotor speed. The starting of transducer function is controlled by a logic signal. A controlled circuit breaker is adapted to operate as a fault application unit. The fault unit can be used to apply any type of short circuits with a controlled duration at the bus connected to the unit. The data acquisition system used in this study is a 12-bit resolution Analog Input/Analog Output card having eight differential/ 16 single-ended Analog Input channels, 2 Analog Output channels, and 24 Digital Input/Output channels. The data acquisition card connections to the study system are shown in Fig. A3.

(From Computer System)

Ustab + -

Ka 1+ s Ta

Firing Circuit

-

Vref

s Kf 1+ s Tf

Fig. A2. Schematic diagram of the constructed AVR loop.

Synch. Gen.

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Fault Application Unit

Apply fault

Shaft Encoder

Data Acqu isition card

SG

Real time Windows Target Toolbox

DO DO

Start

δ

A/D A/D

AVR / Exciter

Angle/Speed Transducer

ω

Vref +

D/A

+

Ustab Fig. A3. Data acquisition card interfaced to the experimental set-up.

3 - machine Lab Syst em ( Light ) 377.0 kV

377 kV

- 5 Deg

103 MW - 233 Mvar

75 MW - 87 Mvar

G2

0 Deg

L14

Switch# 3

L24

255 MW - 63 Mvar

4

Switch# 2

2

383.3 kV

- 7 Deg

598 MW - 114 Mvar

G1

28 MW - 146 Mvar

1 326 MW 139 Mvar

L23 L13 3

380.0 kV

Switch# 1

- 7 Deg

G3 - 36 MW 330 MW 146 Mvar

3 Mvar

Fig. A4. Static simulation of the laboratory system.

Table A1 Load flow comparison for medium loading. Voltage (V)

Real power (P)

Reactive power (Q)

Point of measurement

Measured (V)

Simulated (kV)

Measured (W)

Simulated (MW)

Measured (V Ar)

Simulated (MV Ar)

G1 G2 G3 Load 3 Load 4 Line 1–4 Line 2–3 Line 2–4 Max mismatch

379.5 384.5 380.2 380.7 360.1 380.4 385 385 0.008 p.u.

379.5 384.5 380.2 380.2 363 379.5 384.5 384.5

842 430 80.5 676 614 350 145 284 0.038 p.u.

804 430 80.5 675 616 344 145.6 284

55 113 167 246 219 0 130 31.8 0.028 p.u.

83 116 158 241 220 4 134 18

A micro-computer system is used as the digital controller of the developed lab power system. A complete control of the data acquisition card is obtained using the real-time windows target toolbox of Simulink/MATLAB. A.2. Static model verification The modeling of the integrated multi-machine set-up needs to be verified for both static and dynamic operations. The ver-

ification of the static model implies conducting a load flow study. To get a system model valid for a wide range of operation, three loading conditions; light, medium and high, are considered. The proper adjustment of the loading condition is a hard task because it needs the control of DC motor field and synchronous excitation for each of the three machines. Occasionally, the readings of the measuring instruments are not stable due to the system oscillations at some operating conditions.

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H.E.A. Talaat et al. / Electrical Power and Energy Systems 32 (2010) 751–759 Table A2 Estimated parameters of the synchronous generator. Parameter

R (p.u.)

Xl (p.u.)

Xd (p.u.)

X 0d (p.u.)

X 00d (p.u.)

Xq (p.u.)

X 00q (p.u.)

T 0d (s)

T 00d (s)

T 00q (s)

Estimated value

0.07

0.08

0.68

0.28

0.15

0.62

0.15

0.019

0.009

0.009

The lab system is simulated using the ‘‘PowerWorld Simulator program” [24]. The parameters of the each component of the PowerWorld model are set to the measured parameters of the corresponding lab component. The simulated model is shown in Fig. A4. The results of the load flow study for the lab and the computer models are given in Table A1. The results reveal that there is a good matching between measured and calculated values leading to the approval of the static model of the lab system. A.3. Estimation of the synchronous machine’s parameters The static parameters approved from the load flow study are used as fixed parameters of the model whereas the dynamic parameters under estimation are expressed as variables. It is more practical to obtain the dynamic parameters of the system from on-line measurements under different conditions through applying a parameter estimation algorithm. The three machines of the system are assumed to be identical. The rotor angle and rotor speed responses of machine#2 obtained from applying three-phase fault to its terminals under the considered loading conditions are recorded. The set of parameters that yield the best fit to the recorded responses are then estimated. The estimated parameters are listed in Table A2. Appendix B. Synchronous machine dynamic model The electrical part of the machine is represented by a sixth-order state-space model and the mechanical part is represented by a second-order model. The model takes into account the dynamics of the stator, field, and damper windings. The equivalent circuit of the model is represented in the rotor reference frame (qd frame). All rotor parameters and electrical quantities are viewed from the stator. They are identified by primed variables. The subscripts used are defined as follows:    

d, q d and q axis quantity R, s rotor and stator quantity l, m leakage and magnetizing inductance f, k field and damper winding quantity The electrical model of the machine is

  d 0 0 /  xR /q /d ¼ Ld id þ Lmd ifd þ ikd dt d d 0 V q ¼ Rs iq þ /q  xR /d /q ¼ Lq iq þ Lmq ikq dt  d 0 0 0  V 0fd ¼ R0fd ifd þ /0fd /0fd ¼ L0fd ifd Lmd id þ ikd dt   d 0 0 0 V 0kd ¼ R0kd ikd þ /0kd /0kd ¼ L0kd ikd Lmd id þ ifd kt d 0 0 0 /0kq1 ¼ L0kq1 ikq1 Lmq iq / V 0kq1 ¼ R0kq1 ikq1 þ kq1 kq1 d 0 0 0 V 0kq2 ¼ R0kq2 ikq2 þ /0kq2 ¼ L0kq2 ikq2 Lmq iq / kq2 kq2 V d ¼ Rs id þ

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