Case 1348453228

TRANSIENT STABILITY ANALYSIS OF POWER SYSTEMS

WITH ENERGY STORAGE

by

CHI YUAN WENG

Submitted in the partial fulfillment of the requirements For the degree of Master of Science

Thesis Advisor: Dr. Kenneth A. Loparo

Department of Electrical Engineering & Computer Science

CASE WESTERN RESERVE UNIVERSITY

January 2013

CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

_____CHI YUAN WENG____

candidate for the ____Master of Science___ degree*

Committee Chair: (signed)_Kenneth Loparo______________________________________________ Dissertation Advisor Professor, Department of Electrical Engineering & Computer Science Committee: (signed) _Vira Chankong_______________________________________________ Committee: (signed) _Marc Buchner________________________________________________ Committee: (signed) _________________________________________________

(date) __________09/19/2012____________ *We also certify that written approval has been obtained for any proprietary material

contained therein.

iii

Table of Contents

Table of Contents ......................................................................................................... iii

List of Tables ................................................................................................................. vi

List of Figures ............................................................................................................ viii

Acknowledgement ......................................................................................................... x

Abstract ......................................................................................................................... xi

Chapter 1 Introduction

1.1 Motivation and Literature Survey ..................................................................... 1

1.2 Outline of the Dissertation .............................................................................. 3

Chapter 2 Power System Stability

2.1 Definition of Stability and Classification ........................................................ 4

2.2 Swing Equations ............................................................................................ 5

2.3 Load Flow ...................................................................................................... 6

2.4 Multi-machine transient stability ................................................................... 7

2.5 The method to increasing stability ................................................................... 8

iv

Chapter 3 Power System Modeling

3.1 Synchronous Generator with and without load ............................................. 9

3.2 Transient Stability and three-phase fault ..................................................... 12

3.3 Conclusions ................................................................................................. 21

Chapter 4 Power System Modeling with Energy Storage

4.1 Transient Stability and Energy Storage .......................................................... 22

4.2 Three-Phase Fault on SMIB with ES ........................................................... 22

4.3 Simulation and Results ................................................................................ 24

4.4 Conclusions .................................................................................................. 43

Chapter 5 Conclusions and Future Work

5.1 Summary ........................................................................................................ 44

5.2 Future Development ..................................................................................... 44

v

List of Tables

3.1 Initial conditions for SMIB without load impedance .......................................... 10

3.2 Initial conditions for SMIB with load impedance................................................ 11

3.3 Initial conditions for SMIB with load impedance .............................................. 11

3.4 Observing how CCT (sec) changes for SMIB with different load impedance .. 19

3.5 Observing how load power changes during prefault, fault, and postfault states

with different load impedances .............................................................................. 19

4.1 Changes in CCT (sec) with and without ES when load impedance changes from

0.42 to 0.55 (per unit) ............................................................................................ 19


0.64 to 0.94 (per unit) ............................................................................................ 26


1.16 to 2.39 (per unit) ............................................................................................ 27

4.4 Observing how CCT (sec) changes with and without ES when load impedance is

4.84 (per unit)......................................................................................................... 28

4.5 Load fault power changes with ES when load impedance varies from 0.42 to

0.55 (per unit)......................................................................................................... 29


vi

0.94 (per unit)......................................................................................................... 30


2.39 (per unit)......................................................................................................... 31

4.8 Load fault power changes with ES when load impedance is 4.84 (per unit) ....... 32

4.9 Load fault energy changes with ES when load impedance varies from 0.42 to

0.55 (per unit)......................................................................................................... 33


0.92 (per unit) …………………………………………………………………………………………………. 34


2.39 (per unit)……………………………………………………………………………………………………35

4.12 Load fault energy changes with ES when load impedance is 4.84 (per unit)…. 36

4.13 Energy absorbed during a fault when load impedance varies from 0.42 to 0.55

(per unit) ………………………………………………………………………………………………………….37

4.14 Energy absorbed during a fault when load

impedance varies from 0.64 to 0.92 (per unit) ....................................................... 38

4.15 Energy storage absorbed during a fault when load

impedance varies from 1.16 to 2.39 (per unit) …………………………………………………..39

4.16 Energy storage absorbed during a fault when load

impedance is 4.84 (per unit) ……………………………………………………………………………..40

vii

List of Figures

1.1 Classification of Power System Stability [4] ........................................................... 3

3.1 SMIB without load impedance ................................................................................ 9

3.2 SMIB with constant load impedance ..................................................................... 10

3.3 Thevenin circuit for SMIB with constant load impedance .................................... 12

3.4 Fault occurs in the middle of one parallel SMIB ................................................... 13

3.5 Simplified diagram of fault circuit ......................................................................... 14

3.6 Diagram of postfault circuit ................................................................................... 15

3.7 Rotor angle versus time as impedance (per unit) varies (0.42, 0.48) ..................... 16



3.10 Rotor angle versus time as impedance (per unit) varies (1.16, 1.57) ................... 17

3.11 Rotor angle versus time as impedance (per unit) varies (2.39, 4.84) ................... 18

3.12 Rotor angle versus time for unstable case............................................................ 18

3.13 Diagram of three-phase fault for SMIB without load impedance ........................ 21

3.14 Diagram of postfault state of SMIB without load impedance ............................. 21

4.1 Diagram of fault state for SMIB with ES .............................................................. 25

4.2 Rotor angle versus time with ES real (reactive) power equal to 0.03 and 0.04 (per

viii

unit) ........................................................................................................................ 42


unit) ........................................................................................................................ 42


unit) ........................................................................................................................ 43


unit) ........................................................................................................................ 43

4.6 Rotor angle versus time with ES real (reactive) power equal to 0.2 and 0.3

(per unit)................................................................................................................. 44

ix

ACKNOWLEDGEMENTS

I can finish my research; Thanks for my advisor, Professor Kenneth A. Loparo,

providing me numerous discussions, exact direction, and careful correction of this

thesis, making this thesis more perfect.

I appreciate Professsor Vira Chankong and Professor Marc Buchner for my

advisory committee and correcting my thesis.

I also need to appreciate my professors in Tatung University. Dr. Tsung Chun

Kung gives me some control theorem. Dr. Wen Cheng Ju shares his American

studying life to me and let me adapt the environment soon.

During the period of my studying in Case Western Reserve University, I also

thanks to my friends who give me a lot of favors, especially Adirak Kanchanahruthai,

Ye Lei Li, Feng Ming Li, and Feng Din who share precious experiences and help.

Finally, I would appreciate my parents support and encouragement, and my brothers

share his experiences for my thesis. Owing to them, I can focus on my thesis and

finish it.

x

Transient Stability Analysis of Power System with Energy Storage

Abstract

by

CHI YUAN WENG

Power systems can effectively damp power system oscillations through

appropriate management of real or reactive power. This thesis addresses some

problems in power system stability with and without energy storage.

A power system model with energy storage is used to analyze the influence of

three-phase faults on the transient stability of the systems using simulation to

determine the Critical Clearing Time (CCT) using the following approach:

(1) Prefault Period: Solve the power flow equation to obtain initial values

xi

(2) Fault Period: With and without energy storage, use SMIB (single machine infinite

bus) power system model with constant impedance load to determine how CCT

(critical clearing time), real and reactive power change during transients. Dynamic

and algebraic power flow equations (DAE) are solved simultaneously.

(3) Postfault Period: Solve DAEs to determine system response.

Simulation results show how energy storage affects CCT and real and reactive

power supplied to the load during disturbances such as faults and changes in load.

1

Chapter 1 Introduction 1.1 Motivation and Literature Survey

Due to exploiting large amounts of traditional energy sources, like natural gas and

petroleum, there is increased interest in developing more efficient ways to generate

electricity, and renewable energy generation is a good alternative.

From a power system operating perspective, operational reliability and stability

are key performance objectives. Power system stability [1], the ability of the system

to recovery to a new operating equilibrium after a disturbance, is important for secure

system operation [2][3]. Power system stability studies can be divided into categories

steady-state stability (or dynamic stability) and transient stability. Steady-state

stability refers to small disturbances, like small variations of power or rotor angle,

over long time periods. Transient stability addresses the impact of large disturbances

such as symmetrical three- phase short circuit transmission line faults, on the ability

of the system to converge to a stable equilibrium after the fault is cleared from the

2

system. As shown in Figure 1.1, power system stability [2] can be classified as (1)

Voltage stability, (2) Rotor angle stability, and (3) Frequency stability. We can see

from Figure 1.1, (1) and (2) can be subdivided into small-signal and transient stability

under occurrence of any disturbances. Therefore, it is possible that one form of

instability may cause the other.

The purpose of a power system is to generate and deliver electricity in a secure

and economic manner to consumers. So, the method of controlling and operating the

power system is important, especially dynamic state estimation (DSE), short-term

load forecasting, and yearly peak load forecasting.

State estimation involves estimating unobservable state variables from measured

system data, and can be divided into static state estimation (SSE) and dynamic state

estimation (DSE). DSE is an important state estimation function in energy

management to provide the information required for control and to estimate how the

load may change in the next time period. The Extended Kalman (EKF) [5, 6, 7] is

often used in DSE applications.

Short-term load forecasting estimates how the load demand will change within

one hour to one week in the power system. The accuracy of short-term load

forecasting has a direct impact on the generation cost. Therefore, how to increase the

efficiency of forecasting is also an important issue. The method of short-term

3

forecasting can be divided into the following: (1) Stochastic Time Series [8, 9, 10], (2)

Exponential Smoothing [11], (3) Linear Regression [12], (4) Expert Systems [13, 14],,

and (5) Artificial Neural Networks [15, 16, 17, 18, 19, 20].

Yearly peak load forecasting refers to predicting electricity demand periods of

five to 10 years. There are several methods for calculating yearly peak load forecasts,

such as the Holt-Winter Method [21], the Logistic Method [21, 22], and the Gompertz

Method [21].

A topic of considerable interest is how energy storage can be integrated into

existing and future power systems. There have four major energy storage system (ESS)

technologies: Superconducting Magnetic Energy Storage (SMES), Flywheel Energy

Storage (FES), Super Capacitors, and Battery Energy Storage Systems (BESS) [23].

These ESS are used in combination with distributed renewable generation resources

such as wind and solar to address problems related to the intermittency of these

generation resources [24, 25, 26].

Southern California Edison (SCE) has successfully to suppress power system

oscillations using Energy Storage Power System Stabilizer (ESPSS) installed on a

10MW 40MWh BESS at its Chino substation [37]. BESS are also used with wind

farms [38], to make the wind energy resource more dispatchable. In [39], a

STATCOM integrated with BESS is used to improve power quality and stability

4

margins. As reported in [38][39], the performance of traditional FACTS is compared

to BESS/FACTS (STATCOM, UPFC, SSSC), showing that BESS/FACTS enhance

voltage and power flow control. This thesis investigates the role of energy storage

during power system transients.

1.2 Outline of the Thesis

The rest of the thesis is organized as follows.

(a) Chapter 2: definition of power system stability and swing equation

(b) Chapter 3: SMIB without ES during transient

(c) Chapter 4: SMIB with ES during transient

(d) Chapter 5: Conclusion and summary

5

Figure 1.1: Classification of Power system stability [4]

6

Chapter 2 Power System Stability

Power system stability refers to the ability of three-phase synchronous generators

to remain synchronized during transients such as sudden change in load of network

topology. System stability is determined by the dynamics of the rotor angles and

voltages. Section 2.1 provides definitions of power system stability. Section 2.2

provides the swing equations. Section 2.3 discusses the Power Flow equations.

Section 2.4 discusses multi-machine power system stability. Section 2.5 discusses

methods for improving power system stability.

2.1 Definitions of Stability

Stability refers to the ability of the system to return to a suitable operating point

after the occurrence of a disturbance. Power system stability can be divided into two

categories [27]:

a. Transient Stability: When a major disturbance, such as a three-phase short circuit

7

to ground fault, occurs the frequency of the synchronous generators temporarily

deviates from the synchronous speed, and the power angle are also changing. The

system is said to be transiently stabile is if each synchronous generator returns to

suitable set of power angles at the synchronous frequency. Transient stability

analysis generally requires the full nonlinear model of the system.

b. Steady-State Stability: This type of stability refers to the ability of the system to

continue to meet demand under small signal disturbances, such as continuously

changing load. Steady-state, or small signal, stability can be determined from a

linearized model of the power system in the neighborhood of an operating point.

2.2 Swing Equations

The Swing Equations defining the dynamics of the synchronous generators

connected to the power system. The trajectories of the swing equations are called

swing curves, and by observing the swing curves for all the synchronous generators,

we can determine the stability of the system.

Consider a single synchronous generator with synchronous speed ω𝑠𝑚 ,

electromagnetic torque T𝑒, and mechanical torque T𝑚 means mechanical torque. In

steady-state,

T𝑚=T𝑒 (2.1)

8

When a disturbance occurs, the torque deviates from steady-state, causing an

accelerating (T𝑚>T𝑒) or decelerating (T𝑚<T𝑒) torque:

T𝑎 (accelerating torque) = T𝑚 - T𝑒 (2.2)

Assume J is the combined inertia of generator and prime mover, neglecting friction

and damping torque we have:

J 𝜃𝑚′′＝ T𝑎 = T𝑚 - T𝑒 (2.3)

where θ is the angular displacement of the rotor relative to the stator, the suffix m

means generator. The rotor speed relative to synchronous speed, is given by:

𝜃𝑚 = 𝜔𝑠𝑚t + 𝛿𝑚 (2.4)

From equation (2.4), we obtain the angular speed of the rotor:

𝜔𝑚 = 𝜃𝑚′ = ω𝑠𝑚 + 𝛿𝑚′ (2.5)

9

Where

𝜃𝑚′′ = 𝛿𝑚′′ (2.6)

Substituting (2.6) into (2.3), we obtain:

J 𝛿𝑚′′＝ T𝑎 = T𝑚 - T𝑒 (2.7)

Multiply eq. (2.7) by 𝜔𝑚 :

𝜔𝑚 J 𝛿𝑚′′= 𝜔𝑚 T𝑚 - 𝜔𝑚 T𝑒 = P𝑚 - P𝑒 (2.8)

J 𝜔𝑚 is called the constant of inertia, referenced by “M” and associated with W𝑘

(kinetic energy):

W𝑘= 0.5 J𝜔𝑚2 = 0.5M𝜔𝑚 (2.9)

or M = (2W𝑘/ω𝑠𝑚 ) (2.10)

10

For small changes ωm, it is reasonable to assume that M is constant, so

M= (2Wk)/ (ω𝑠𝑚) (2.11)

Then we obtain the standard from of the swing equation:

M 𝛿𝑚′′ = P𝑚 - P𝑒

(2.12)

2.3 Load Flow

Generally, a power system can be divided into subsystems that include

generation, transmission, and distribution.

Load flow analysis refers to solving for the real and reactive power flows in the

system, including the complex voltages (magnitude and angle) in each line [28].

Generally speaking, load flow analysis requires identifying slack buses,

voltage-controlled buses, and load buses. Then based on these designations, we

construct each line flow equation. Gauss-Siedel, Newton-Raphson, or Fast-Decoupled

load flow method are used to obtain a solution [27].

11

Because transmission system load has high balance in load flow problem, then we

always assume the system operate in three-phase balance condition, called

three-phase balanced. So it can be simplified into single-phase load flow problem.

Then, we explain three different bus styles categorized by physical property.

(1) Slack bus: Also called the infinite or reference bus. When solving the power flow

equation, the magnitude and phase of the slack bus voltage is set to 1.0∠0 (p.u.)

and the injected real and reactive powers are unknown.

(2) Voltage-Controlled bus: Also called a machine or P-V bus. The magnitude of

voltage and real power are fixed, but phase of the voltage and reactive power are

unknown.

(3) Load bus: Also called P-Q bus. Real and reactive powers are known, but the

magnitude and phase of the voltage are unknown.

Stability is a necessary condition for power system security. The first step to

improving system security is to ensure the system is stable for both small signal and

large signal disturbances.

2.4 Multi-machine transient stability

For transient stability, estimating the critical clearing time (CCT) is important.

When a system fault occurs, the fault should be cleared before the CCT, or the system

12

can become unstable. In a multi-machine generator system during a transient, each

generator can oscillate, and the complexity of calculating the system trajectory during

a transient increases with the number of generators [28].

To simplify the analysis of a multi-machine power system for transient stability

studies we have the following assumptions:

(a) During the transient, the machine power to each generator is constant.

(b) Damping power is neglected.

(c) Each generator is modeled as a fixed transient reactance in series with a fixed

internal voltage.

(d) The rotor angle of each generator is equal to the angle of each internal voltage.

(e) Each load is modeled as a constant reactance, equal to its prefault value.

Assumptions (a) to (e) are referred to as the classical stability model. Transient

stability analysis has the following steps:

(1) Before the system occur fault, solve the load flow equations to determine the

initial value.

(2) Given the network model before fault, determine the model during fault and for

the postfault situation.

(3) Solve the swing equations and determine if the system is stable or unstable.

13

2.5 The method to increasing stability

Improving power system stability includes the following [28]:

(1) Increasing transmission capacity during prefault conditions.

(2) Rapid fault clearance, improves transient stability margins.

(3) Rapid circuit breaker re-closure, increases transmission system capacity in the

post fault state and improve transient stability.

(4) Increased mechanical inertia of generators, decreases angular acceleration, slows

down rotor angle oscillations, and thereby increases CCT.

14

CHAPTER 3 Power System Models

In this Chapter, we develop simplified dynamic models of a single-machine

infinite bus power system, and investigate how CCT changes for different

configurations for a three-phase fault.

3.1 Synchronous Generator (SG) with and without load

Figure 3.1 is the model of synchronous generator connected to an infinite bus

(SMIB) without load. Bus 1 connects to the generator, bus 2 connects to buses 1 and 2,

and bus 3 is the slack bus. Initial conditions and parameters are given in table 3.1

[35].

15

Figure 3.1 SMIB without load impedance

H 𝑃𝑚=𝑃1 𝑥𝑑′ 𝑥1 𝑥2 𝑄1

5.0 1.0 0.2 0.1 0.2 0.8

𝑉1 𝜃1 𝑉2 𝜃2 𝐸′ 𝛿

1.0 17.458 0.990 11.659 1.0499 28.4389

Table 3.1 initial conditions and parameters for SMIB without load impedance

16

Figure 3.2 SMIB with constant impedance load

Figure 3.2 is the model of a SMIB with impedance load [29]. The generator is

modeled by the classical model E’∠δ. Bus 1 connects to the generator, bus 2

connects to the constant impedance load, and bus 3 is the slack bus. The DAE model

for the system can be written as follows:

�̇�=𝜔0∆ω= 0 (3.1)

2H∆ω ̇ = 𝑃𝑚 - 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) = 0 (3.2)

0 = - 𝑉12

𝑥𝑑′+ 𝑉1𝐸′

𝑥𝑑′cos(𝜃1 − 𝛿) -

𝑉12

𝑥1 +

𝑉1𝑉2𝑥1

cos(𝜃1 − 𝜃2) (3.3)

0 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) - 𝑉1𝑉2𝑥1

sin(𝜃1 − 𝜃2) (3.4)

0= -Re(𝑉2∠𝜃2(𝑉2∠𝜃2𝑍𝐿

)∗) - 𝑉1𝑉2𝑥1

sin(𝜃2 − 𝜃1)- 𝑉3𝑉2𝑥2

sin(𝜃2 − 𝜃3) (3.5)

0 = - I𝑚(𝑉2∠𝜃2(𝑉2∠𝜃2𝑍𝐿

)∗) - 𝑉12

𝑥1 +

𝑉1𝑉2𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2

cos(𝜃2 − 𝜃3)

(3.6)

17

𝑃𝑚 = 𝑃𝑒 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) (3.7)

𝑄𝑚 = - 𝑉12


𝑥𝑑′cos(𝜃1 − 𝛿) (3.8)

Bus 2 is the constant impedance load, bus 3 is the slack bus 𝑉3=1∠0, and base

power is 100MVA. Equations (3.1) and (3.2) are the swing equations. Equations (3.3)

to (3.6) are the load flow equations. During steady-state, �̇� 𝑎𝑎𝑎 ∆ω ̇ are zero. 2H (p.u.)

is the constant mechanical inertia of the generator, and 𝑍𝐿 (p.u.). δ is the electrical

angle of the rotor, and ∆ω=ω-1 is angular velocity with respect to infinite bus. Using

Matlab™, we can solve the DAE model [36]. The initial conditions for the power

system simulations are listed in Table 3.2 and Table 3.3.

H 𝑃𝑚=𝑃1 𝑥𝑑′ 𝑥1 𝑥2 𝑄1

5.0 1.0 0.2 0.1 0.2 0.8

Table 3.2 Initial conditions for Figure 3.2

𝐸′ δ 𝑉1 𝜃1 𝑉2 𝜃2 𝑍𝐿(R,X)

1.1801 15.9712 1.0 6.213 0.924 0.00 0.42,0.42

1.1647 17.2792 1.0 7.392 0.932 1.23 0.48,0.48

1.1500 18.5694 1.0 8.554 0.939 2.441 0.55,0.55

1.1356 19.8449 1.0 9.702 0.946 3.635 0.64,0.64

1.1220 21.1046 1.0 10.837 0.953 4.814 0.75,0.75

1.1088 22.3516 1.0 11.96 0.96 5.98 0.92,0.92

1.0962 23.5873 1.0 13.075 0.966 7.134 1.16,1.16

1.0839 24.8129 1.0 14.18 0.972 8.278 1.57,1.57

18

1.0723 26.0281 1.0 15.278 0.,978 9.412 2.39,2.39

1.0609 27.2369 1.0 16.371 0.984 10.539 4.84,4.84

Table 3.3 Initial conditions for Figure 3.2 (unit of 𝐸′, 𝑉1, 𝑉2,𝑍𝐿:𝑝. 𝑢, unit of δ,

𝜃1,𝜃2: degree)

An alternate approach is to determine 𝐸′, δ and then substitute these values into

equations (3.5) to (3.8) to obtain (𝑉1, 𝑉2,𝜃1 ,𝜃2). In Figure 3.3, we will simplify

Figure 3.2 by determining the Thevenin equivalent circuit for the load and slack bus.

Figure 3.3 Thevenin equivalent circuit incorporated into Figure 3.2

𝑍𝑡ℎ = 𝑍𝐿 ∥ 𝑋2

𝑉𝑡ℎ = (1∠0)( 𝑍𝐿𝑍𝐿+𝑋2

)

𝑃𝑒 = 𝑃𝑚 = 𝐸′ 𝑉𝑡ℎ 𝑋𝑑′+𝑋1+𝑍𝑡ℎ

sin(𝛿)

19

𝑄𝑒= 𝑄𝑚 = 𝐸′ 𝑉𝑡ℎ 𝑋𝑑′+𝑋1+𝑍𝑡ℎ

cos(𝛿) - 𝑉𝑡ℎ2

𝑋𝑑′+𝑋1+𝑍𝑡ℎ

Solving these four equations, we obtain 𝐸′, δ.

3.2 Transient Stability for a three-phase fault

The objective of transient stability analysis is to observe the dynamic behavior of

power system from prefault to postfault. The CCT (critical clearing time) is of interest

because it is the maximum time that the fault can be present on the system before

instability. If a fault occurs on the system and the clearing time exceeds the CCT, the

rotor angle exit the domain of attraction of the postfault equilibrium state, and the

system will be unstable. Therefore increasing the CCT, improve the stability margin

of the system.

In Figure 3.2, the generator delivers 1.0 p.u. power to the infinite bus. When a

three-phase fault occurs, assume that the magnitude of 𝐸′ is constant. In Figure 3.2,

the impedance 𝑥1 is replaced by one parallel branch of line 2𝑥1, and a three-phase

fault occurs at location F, causing the rotor angle to accelerate and the voltage to

collapse. During the fault, the circuit breaker will open on the impacted line, and it

will automatically try to re-close and will remain closed if the fault is cleared.

Transient stability analysis includes the following steps:

I. Steady-state operation during prefault.

20

II. The fault occurs at time 𝑡𝑓 starts.

III. The line impacted by the fault is isolated by the circuit breaker at time 𝑡𝑐𝑐𝑒𝑎𝑐.

Thus, CCT is 𝑡𝑐𝑐𝑒𝑎𝑐- 𝑡𝑓.

IV. The system us restored at 𝑡 = 𝑡𝑐𝑒, and the rotor angle stabilizes at 𝑡 = 𝑡𝑝𝑝𝑠𝑡,

indicating postafault system operation.

Figures 3.4, Figure 3.5, and Figure 3.6 show the diagram of the fault location, model

of the faulted system, and the postfault condition.

Figure 3.4 Fault occurs in one parallel branch of line 2𝑥1

We use steady-state initial conditions to calculate the DAE during fault. The

values from faulted condition are then used as the initial values for the postfault

system.

During fault, the DAE equations are as follows:

21

�̇�=𝜔0∆ω (3.9)

2H∆ω ̇ = 𝑃𝑚 - 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1)-𝐾𝑑 ∆ω (3.10)

0 = - 𝑉12



𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃1 − 𝜃2)- I𝑚(𝑉1∠𝜃1(𝑉1∠𝜃1𝑗𝑥1

)∗)

(3.11)

0 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) - 𝑉1𝑉22𝑥1

sin(𝜃1 − 𝜃2) - 𝑅𝑒(𝑉1∠𝜃1(𝑉1∠𝜃1𝑗𝑥1

)∗) (3.12)

0= -Re(𝑉2∠𝜃2(𝑉2∠𝜃2𝑗𝑥1∥𝑍𝐿

)∗) - 𝑉1𝑉22𝑥1


sin(𝜃2 − 𝜃3) (3.13)

0 = - I𝑚(𝑉2∠𝜃2(𝑉2∠𝜃2𝑗𝑥1∥𝑍𝐿

)∗) - 𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2


(3.14)

𝑃𝑚 = 1.0≠𝑃𝑒 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) (3.15)

From Equations (3.9) to (3.15), V, δ, ω will change with time. The algebraic variables

are (δ, θ), the state variables are (𝑉1 , 𝑉2, 𝜃1 ,𝜃2 ), and the input variables are

(𝐸′,𝑃𝑚). 2𝐻 is the mechanical inertia constant, and the other variables are the same as

pre-fault variables.

22

Figure 3.5 Simplified diagram during faulted operation

Figure 3.6 Post-fault circuit

Postfault DAE:

�̇�=𝜔0∆ω (3.16)

23

2H∆ω ̇ = 𝑃𝑚 - 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) (3.17)

0 = - 𝑉12



𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃1 − 𝜃2) (3.18)

0 = 𝐸′𝑉1𝑥𝑎′

sin(𝛿 − 𝜃1) - 𝑉1𝑉22𝑥1

sin(𝜃1 − 𝜃2) (3.19)


)∗) - 𝑉1𝑉22𝑥1


sin(𝜃2 − 𝜃3) (3.20)

0 = - I𝑚(𝑉2∠𝜃2(𝑉2∠𝜃2𝑍𝐿

)∗) - 𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2


(3.21)

The pre-fault, fault, and post-fault equations are solved to obtain the rotor angles

and the different CCTs for different load conditions.

Figures 3.7 to Figure 3.11 show the rotor angles versus time for different load

impedances.

In Figure 3.12 the actual clearing time is greater than the CCT, and the rotor angle

accelerates and is unstable. Table 3.4 shows different CCTs for different loads.

24

Figure 3.7 Impedance (p.u.) = (0.42, 0.48), rotor angle v.s. time, CCT, stable

post-fault system.

25


post-fault system.

26


post-fault system

27


post-fault system.

28

Figure 3.11 Impedance (p.u.) = (2.39, 4.84), rotor angle v.s. time, CCT, stable state

after fault.

29

Figure 3.12 Example of the unstable state with load impedance 4.84 (p.u.), and

clearing time =0.25 >0.2429 (CCT)

Z (R=X) (p.u.) 0.42 0.48 0.55 0.64 0.75 0.92 1.16 1.57 2.39 4.84

CCT (sec)

0.2518

0.2508

0.2498

0.2487

0.2478

0.2468

0.2458

0.2448

0.2439

0.2429

Table 3.4 shows how different load impedances affect CCTs for the system shown in

Figure 3.2. In this model, the value of resistance is equal to the impedance.

30

Constant impedance load Z (R=X),(p.u).

Load prefault real and reactive power (P=Q),(p.u.)

Load fault real and reactive power (P=Q) (p.u.)

0.42 1.0 0.0291

0.48 0.9 0.0259

0.55 0.8 0.0226

0.64 0.7 0.0195

0.75 0.6 0.0165

0.92 0.5 0.0135

1.16 0.4 0.0107

1.57 0.3 0.0079

2.39 0.2 0.0052

4.84 0.1 0.0026

Table 3.5 the table shows how load power changes during prefault, fault in different

load impedance condition

Therefore, from table 3.4, we observe that as the load impedance increases, the CCT

decreases. Table 3.5 shows that during the fault, the load power decreases. If we can

increase the CCT for different load conditions, this will enhance the stability of the

system. Consequently, in the next chapter, we add Energy Storage (ES) to bus 1 to

determine its effect CCT, to observe how load power changes, and investigate the role

of ES during faulted system operation.

We can also simulate the three-phase fault illustrated in Figure 3.1 by solving the

following equations.

During pre-fault, the algebraic equations are as follows:

31

0 = - 𝑉12



𝑉12

𝑥1 +

𝑉1𝑉2𝑥1

cos(𝜃1 − 𝜃2) (3.22)

0 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) - 𝑉1𝑉2𝑥1

sin(𝜃1 − 𝜃2) (3.23)

0= - 𝑉1𝑉2𝑥1


sin(𝜃2 − 𝜃3) (3.24)

0 = - 𝑉12

𝑥1 +

𝑉1𝑉2𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2


(3.25)

𝑃𝑚 = 𝑃𝑒 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) (3.26)

𝑄𝑚 = - 𝑉12


𝑥𝑑′cos(𝜃1 − 𝛿) (3.27)

During the fault, the DAE is as follows:

0 = - 𝑉12



𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1


)∗)

(3.28)

0 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) - 𝑉1𝑉22𝑥1


)∗) (3.29)

0= -Re(𝑉2∠𝜃2(𝑉2∠𝜃2𝑗𝑥1

)∗) - 𝑉1𝑉22𝑥1


sin(𝜃2 − 𝜃3) (3.30)

0 = - I𝑚(𝑉2∠𝜃2(𝑉2∠𝜃2𝑗𝑥1

)∗) - 𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2


(3.31)

During post-fault, the DAE is as follows:

�̇�=𝜔0∆ω (3.32)

2H∆ω ̇ = 𝑃𝑚 - 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) (3.33)

32

0 = - 𝑉12



𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃1 − 𝜃2) (3.34)

0 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) - 𝑉1𝑉22𝑥1

sin(𝜃1 − 𝜃2) (3.35)

0= - 𝑉1𝑉22𝑥1


sin(𝜃2 − 𝜃3) (3.36)

0 = - 𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2


(3.37)

Solving equations (3.22) to (3.37), gives a CCT = 0.16 seconds.

Figure 3.13 and 3.14 show the system during the fault and post-fault.

Figure 3.13 Diagram of three-phase fault for the system shown in Figure 3.1

33

Figure 3.14 Diagram for post-fault operation of the system shown in Figure 3.1 3.3 Conclusions

In this chapter, we have developed a model of a SMIB system with and without

load. We have derived the DAE models for the SMIB system during pre-fault, fault,

and post-fault condition, observing how CCT changes for different load impedances.

When the load impedance increases, CCT decreases. Graphs of rotor speed verses

time are used to determine if the system is stable or unstable. In the next chapter we

investigate the role of ES on enhancing system stability by increasing CCT.

34

Chapter 4

Power system model with ES

In this chapter, we investigate the role of Energy Storage (ES) on the transient

stability of a single-machine infinite bus power system by observing how the CCT

changes with load, and the capacity of the ES system.

4.1 Transient stability and energy storage

Integration of distributed renewable energy resources (DRERs) into the power

system is a challenge. Several large-scale integration projects have been demonstrated

in Europe, e.g. in Denmark and Greece, where the operation of wind power resources

has been assisted by wind forecasting [30]. Investigating the impact of intermittent

DRERs on the transient stability of power systems is an important problem. This

problem can be exacerbated by increases in energy demand. Energy storage

technologies have the potential to improve power stability, as demonstrated in [31].

Battery Energy Storage is the most common technology and includes the

interconnection of batteries, along with control and power conditioning systems

35

(C-PCS). Battery Energy Storage Systems (BESS) can be used to provide frequency

regulation [32] and changes in real power to enhance the system [31].

StatCom devices and BESS can be combined to improve reactive and real power

separately [33]. R. Kuiava [34] also combined Statcom/BESS, and a Supplementary

Damping Controller (SDC) into reactive power control scheme to improve

transmission power quality and the damping of oscillations. Nikkhajoei and Abedini

[35] have provided that energy storage cannot only be a subsidiary source to alleviate

power fluctuations but also to control load changes using a PM synchronous

generator.

Flexible AC Transmission System (FACTS) devices are also useful in dealing

with transient power stability and reduce the cost of power delivery. FACTS devices

can supply real or reactive power to the grid, improving efficiency of power

transmission [31]. It includes devices such as Series Compensators (SC), Static Var

Compensators (SVC), Mechanically Switched Capacitors (MSC/MSCDN), Static

Synchronous Compensators (STATCOM). Among of these devices, the STATCOM is

frequently used in power system because it can supply reactive power compensation

by modulating its voltage, improving transient stability.

4-2 Three-phase fault on SMIB (single-machine infinite bus) with ES

In this section, we investigate the system given in Figure 3.2, assuming a

36

three-phase fault at F, and energy storage that can absorb (or deliver) constant power

during fault, other parameters have the same values in given in Chapter 3. Next we

provide the pre-fault, fault, and post-fault DAEs with ES.

Pre-fault DAE:

�̇�=𝜔0∆ω (4.1)

2H∆ω ̇ = 𝑃𝑚 - 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) (4.2)

0 = - 𝑉12



𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃1 − 𝜃2) (4.3)

0 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) - 𝑉1𝑉22𝑥1

sin(𝜃1 − 𝜃2) (4.4)


)∗) - 𝑉1𝑉22𝑥1


sin(𝜃2 − 𝜃3) (4.5)

0 = - I𝑚(𝑉2∠𝜃2(𝑉2∠𝜃2𝑍𝐿

)∗) - 𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2


(4.6)

Fault DAE:

�̇�=𝜔0∆ω (4.7)

2H∆ω ̇ = 𝑃𝑚 - 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1)-𝐾𝑑 ∆ω (4.8)

0 = - 𝑉12

𝑥𝑎′

+ 𝑉1𝐸′

𝑥𝑎′

cos(𝜃1 − 𝛿) - 𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1


)∗)-𝑄𝐸𝐸

(4.9)

37

0 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) - 𝑉1𝑉22𝑥1


)∗) -𝑃𝐸𝐸 (4.10)

0= -Re(𝑉2∠𝜃2(𝑉2∠𝜃2𝑗𝑥1∥𝑍𝐿

)∗) - 𝑉1𝑉22𝑥1


sin(𝜃2 − 𝜃3) (4.11)

0 = - I𝑚(𝑉2∠𝜃2(𝑉2∠𝜃2𝑗𝑥1∥𝑍𝐿

)∗) - 𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2


(4.12)

𝑃𝑚 = 1.0≠𝑃𝑒 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) (4.13)

Post-fault DAE:

�̇�=𝜔0∆ω (4.14)

2H∆ω ̇ = 𝑃𝑚 - 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) (4.15)

0 = - 𝑉12



𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃1 − 𝜃2) (4.16)

0 = 𝐸′𝑉1𝑥𝑑′

sin(𝛿 − 𝜃1) - 𝑉1𝑉22𝑥1

sin(𝜃1 − 𝜃2) (4.17)


)∗) - 𝑉1𝑉22𝑥1


sin(𝜃2 − 𝜃3) (4.18)

0 = - I𝑚(𝑉2∠𝜃2(𝑉2∠𝜃2𝑍𝐿

)∗) - 𝑉12

2𝑥1 +

𝑉1𝑉22𝑥1

cos(𝜃2 − 𝜃1) - 𝑉22

𝑥2 +

𝑉3𝑉2𝑥2


(4.19)

The diagram of pre-fault and post-fault condition without ES are the same as the same

as that with ES. The only change of diagram is during the fault. Therefore, Figure 4.1

shows the diagram of the circuit.

38

Figure 4.1 Fault period: SMIB with ES

In this study, the ES system can absorb energy during the fault, and want to

investigate how CCT changes with and without ES.

4.3 Simulation Results

Tables 4.1, 4.2, 4.3 and 4.4 show CCT, the percentage increase in CCT with and

without ES for different (constant impedance) load conditions.

39

ES LOAD POWER

0.42

0.48

0.55

0.00

0.2518 (CCT) (0.00%)

0.2508(CCT) (0.00%)

0.2498(CCT) (0.00%)

0.03

0.2576(CCT) (2.30%)

0.2566(CCT) (2.31%)

0.2555(CCT) (2.28%)

0.04

0.2596(CCT) (3.10%)

0.2585(CCT) (3.07%)

0.2575(CCT) (3.08%)

0.05

0.2616(CCT) (3.89%)

0.2605(CCT) (3.87%)

0.2595(CCT) (3.88%)

0.06

0.2637(CCT) (4.73%)

0.2626(CCT) (4.70%)

0.2615(CCT) (4.68%)

0.07

0.2657(CCT) (5.52%)

0.2646(CCT) (5.50%)

0.2635(CCT) (5.48%)

0.08

0.2678(CCT) (6.35%)

0.2667(CCT) (6.34%)

0.2656(CCT) (6.33%)

0.09

0.2699(CCT) (7.19%)

0.2688(CCT) (7.18%)

0.2677(CCT) (7.17%)

0.10

0.2721(CCT) (8.06%)

0.2709(CCT) (8.01%)

0.2698(CCT) (8.01%)

0.20

0.2955(CCT) (17.36%)

0.2940(CCT) (17.22%)

0.2927(CCT) (17.17%)

0.30

0.3222(CCT) (27.96%)

0.3203(CCT) (27.71%)

0.3184(CCT) (27.46%)

Table 4.1 CCT for load impedance from 0.42 to 0.55 with without ES power (unit of

ES POWER and Load: p.u.).

40

ES LOAD POWER

0.64

0.75

0.92

0.00

0.2487 (CCT) (0.00%)

0.2478(CCT) (0.00%)

0.2468(CCT) (0.00%)

0.03

0.2544(CCT) (2.29%)

0.2534(CCT) (2.26%)

0.2524(CCT) (2.27%)

0.04

0.2564(CCT) (3.10%)

0.2553(CCT) (3.03%)

0.2543(CCT) (3.08%)

0.05

0.2583(CCT) (3.86%)

0.2605(CCT) (5.13%)

0.2562(CCT) (3.81%)

0.06

0.2603(CCT) (4.66%)

0.2626(CCT) (5.97%)

0.2582(CCT) (4.62%)

0.07

0.2624(CCT) (5.51%)

0.2646(CCT) (5.50%)

0.2602(CCT) (5.43%)

0.08

0.2644(CCT) (6.31%)

0.2667(CCT) (6.78%)

0.2622(CCT) (6.24%)

0.09

0.2665(CCT) (7.16%)

0.2688(CCT) (7.63%)

0.2642(CCT) (7.05%)

0.10

0.2686(CCT) (8.00%)

0.2709(CCT) (9.30%)

0.2663(CCT) (7.90%)

0.20

0.2912(CCT) (17.09%)

0.2940(CCT) (18.64%)

0.2884(CCT) (17.17%)

0.30

0.3165(CCT) (27.26%)

0.3203(CCT) (29.26%)

0.3128(CCT) (26.74%)

Table 4.2 represents load impedance from 0.64 to 0.94 with and without ES power,

how CCT changes (unit of ES POWER and LOAD: p.u.).

41

ES LOAD POWER

1.16

1.57

2.39

0.00

0.2458(CCT) (0.00%)

0.2448(CCT) (0.00%)

0.2439(CCT) (0.00%)

0.03

0.2514(CCT) (2.28%)

0.2503(CCT) (2.25%)

0.2494(CCT) (2.26%)

0.04

0.2533(CCT) (3.05%)

0.2522(CCT) (3.02%)

0.2513(CCT) (3.03%)

0.05

0.2552(CCT) (3.82%)

0.2541(CCT) (3.80%)

0.2532(CCT) (3.81%)

0.06

0.2572(CCT) (4.64%)

0.2561(CCT) (4.62%)

0.2551(CCT) (4.59%)

0.07

0.2591(CCT) (5.41%)

0.2580(CCT) (5.39%)

0.2570(CCT) (5.37%)

0.08

0.2611(CCT) (6.22%)

0.2600(CCT) (6.21%)

0.2590(CCT) (6.19%)

0.09

0.2631(CCT) (7.04%)

0.2620(CCT) (7.03%)

0.2610(CCT) (7.01%)

0.10

0.2652(CCT) (7.89%)

0.2641(CCT) (7.88%)

0.2630(CCT) (7.83%)

0.20

0.2871(CCT) (16.80%)

0.2857(CCT) (16.71%)

0.2844(CCT) (17.17%)

0.30

0.3165(CCT) (28.76%)

0.3203(CCT) (30.84%)

0.3075(CCT) (26.74%)

Table 4.3 CCT for load impedance from 1.16 to 2.39 with and without ES power (unit

of ES POWER and LOAD: p.u.).

42

ES LOAD POWER

4.84

0.00

0.2429(CCT) (0.00%)

0.03

0.2484(CCT) (2.26%)

0.04

0.2503(CCT) (3.05%)

0.05

0.2522(CCT) (3.83%)

0.06

0.2541(CCT) (4.61%)

0.07

0.2560(CCT) (5.39%)

0.08

0.2579(CCT) (6.18%)

0.09

0.2599(CCT) (7.00%)

0.10

0.2619(CCT) (7.82%)

0.20

0.2831(CCT) (16.55%)

0.30

0.3056(CCT) (25.81%)

Table 4.4 CCT for load impedance equal to 4.84 with and without ES (unit of ES

POWER and LOAD: p.u. ).

Therefore, CCT progressively increase from 2.26% to 27.46% with increases in ES

power from 0.03 to 0.3 for different constant impedance loads. Tables 4.5 to 4.8 show

43

how the power to the load changes during the fault with different amounts of ES

power.

ES LOAD POWER

0.42

0.48

0.55

0.00

0.0299 (P=Q) (fault power)

0.0261(P=Q) (fault power)


0.03




0.04




0.05




0.06




0.07




0.08




0.09




0.10




0.20




0.30




Table 4.5 Power to the load during the fault with ES when load impedance is from

0.42 to 0.55 (unit of ES POWER and LOAD: p.u.).

44

ES LOAD POWER

0.64

0.75

0.92

0.00




0.03




0.04




0.05




0.06




0.07




0.08




0.09




0.10




0.20




0.30




Table 4.6 Power to the load during the fault with ES when load impedance is from

0.64 to 0.92 (unit of ES POWER and LOAD: p.u.).

45

ES LOAD POWER L

1.16

1.57

2.39

0.00




0.03




0.04




0.05




0.06




0.07




0.08




0.09




0.10




0.20




0.30




Table 4.7 Power to the load during the fault with ES power when load impedance is

from 1.16 to 2.39 (unit of ES POWER and LOAD: p.u.).

46

ES LOAD POWER

4.84

0.00


0.03


0.04


0.05


0.06


0.07


0.08


0.09


0.10


0.20


0.30


Table 4.8 Power to the load during the fault with ES when load impedance 4.84 (unit

of ES POWER and LOAD: p.u.).

Therefore, power to the load during the fault increases from 0.0026 to 0.036 when ES

47

power increases from 0.03 to 0.3 for different constant impedance loads.

Tables 4.9 to 4.12 show the energy to the load during the fault with ES power.

ES LOAD POWER

0.42

0.48

0.55

0.00

7.5 KJ (fault energy)



0.03




0.04




0.05




0.06




0.07


7.3 KJ (fault power)


0.08




0.09




0.10




0.20




0.30




Table 4.9 Energy delivered to the load during the fault with ES power when load

impedance is from 0.42 to 0.55 (unit of ES POWER and LOAD: p.u.).

48

ES LOAD POWER

0.64

0.75

0.92

0.00




0.03




0.04




0.05




0.06




0.07




0.08




0.09




0.10




0.20




0.30






49

ES LOAD POWER

1.16

1.57

2.39

0.00




0.03




0.04




0.05




0.06




0.07




0.08




0.09




0.10




0.20




0.30






50

ES LOAD POWER

4.84

0.00


0.03


0.04


0.05


0.06


0.07


0.08


0.09


0.10


0.20


0.30



impedance is 4.84 (unit of ES POWER and LOAD: p.u.).

51

Thus, energy delivered to the load progressively increases from 0.6KJ to 11.6KJ

when ES power increases from 0.03 to 0.3 for different constant impedance loads.

Next, we determine how energy is absorbed by the ES system during fault for

different constant impedance load. Tables 4.13 to 4.16 show the results.

52

ES LOAD POWER

0.42

0.48

0.55

0.00




0.03




0.04




0.05




0.06




0.07




0.08




0.09




0.10




0.20




0.30




Table 4.13 Energy absorbed by the ES power system during the fault when load


53

ES LOAD POWER

0.64

0.75

0.92

0.00




0.03




0.04




0.05




0.06




0.07




0.08




0.09




0.10




0.20




0.30






54

.

ES LOAD POWER

1.16

1.57

2.39

0.00




0.03




0.04




0.05




0.06




0.07




0.08




0.09




0.10




0.20




0.30






55

ES LOAD POWER

4.84

0.00


0.03


0.04


0.05


0.06


0.07


0.08


0.09


0.10


0.20


0.30



impedance is 4.84 (unit of ES POWER and LOAD: p.u.).

56

The energy stored progressively increases from 7.5KJ to 96.7KJ with increases in ES

power from 0.03 to 0.3 for different constant impedance load. Figures 4.2 to 4.6 show

rotor angle time trajectories for different amounts of ES power with load impedance

equal to 0.42.

Figure 4.2 Rotor angle versus time with ES power equal to 0.03 and 0.04 (p.u.).

57



58


59


4.4 Conclusions

In this chapter, we have shown that for a SMIB power system including ES to

absorb power during a fault can improve transient stability of synchronous generators.

Simulation results show that an ES system that can absorb constant power 0.3 can

increase CCT by approximately 27% compared with no ES. These results suggest that

the design and operation of ES systems should include important issues such as the

time response of the ES system during both charging and discharging, as well energy

management issues that address the ability of the ES system to store energy during

60

faults as well as deliver energy during periods of limited supply.

Chapter 5

Conclusions and Future Work

5.1 Summary

In this thesis, we have used a DAE model of SMIB power system to study the role of

ES systems in improving transient stability. We assume a constant impedance load

and that the ES system can absorb constant power during a fault. We focus on the

following problems:

(1) Determining CCT for different fault scenarios

(2) Determining the power to the load during a fault

(3) Determining the energy absorbed by ES system during a fault

(4) Determining rotor angle trajectories during pre-fault, fault, and post-fault

conditions.

5.2 Future Developments

The following list can be implemented for future developments:

(a) Enlarge to multi-machine power systems with ES, including different generator

61

types such as by DFIG, SG, or SG-DFIG.

(b) Use STATCOM/Battery models for ES.

(c) Analyze the model of large-scale power systems with ES using DAE models

(d) Consider advanced control design methods such as IDA-PBC, for managing the

ES system

62

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